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Автор: Рыбин В.С.
Теги: подземное строительство земляные работы фундаменты строительство тоннелей строительство строительное проектирование
ISBN: 5-274-00502-0
Год: 1990
Текст
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npHMeHHIOT nplI 3Ha1JHTeJIbHbIX MOMeHTHbIX 3HaKonepeMeH-
Hblx HarpY3Kax HJIH Kor,Ua no YCJIOBHHM pa3BHToro noD.-
3eMHoro X03HHcTBa Tpe6yeTcH YMeHbIllHTb JI.JIIIHY n9JI.OIIIBbl.
<1>YHJI.aMeHTbI C TaBpoBoH nOAollIBoH HaXO,UHT nplIMeHeHHe
AJIH MOMeHTHWX HarpY30K O,UHoro HanpaBJIeHIIH H rrplI Tex
8
}Ke YCJIOBHHX, liTO II cpYH,lI,aMeHTbl C CHMMeTpHQHOH ,lI,BYT3B-
pOBOM nOJI.OllIBOM, KOrJI.a paCCT05lHHe OT IJ,eHTpa THLKeCTH
nO.nOIllBbI .nO OJI.HOrO H3 ee KpaeB ,nOJI>KHO 6bITb MHHHMaJlh-
HbIM. CPYHJI.aMeHTbI C r -06pa3HOH IIOJI.OlliBOH npHMeHHIOT B
CJIyqa5lX 3HatIHTeJIbHbIX MOMeHTHbIX HarpY30K, ,neHCTBYIO-
I.IlHX BO B3aHMHO nepneH,UHKYJIHpHbIX nJIOCKOCTHX, HJI"H KOr-
lla YCJIOBHH, OrOBOpeHHble JI.JI5I <PYH,UaMeHTOB C TaBpOBOH
nOJI.OllIBOM, JI.OJI}KHbI 6bITb COOJIIOJI.eHbI OTHOCHTeJIbHO JI.BYX
CMe:>KHbIX KpaeB nOJI.OIIlBbI.
3a:MKHYTble JIeHTOtIHble npHMoyrOJIbHOrO OQepTaHHSI B
OJIaHe <PYH,UaMeHTbI npHMeHHIOT no,U OT,UeJIbHO paCnOJIO-
LKeHHbIMH COOpYMeHHHMH H no,U aHaJIOrH1JHbIMH npHCTpOH-
RaMH K 3JI.aHHHM, OTpe3aHHbIMH OT nOCJIeJI.HHX JI.ecpOpMan H -
OHHbIMH lllBaMH. KpeCTOBble <PYH,lI,aMeHTbI" npHMeHHIOT BMe-
CTO npHMoyrOJIbHbIX, KOr,Ua B nOCJIeJI.HHX npe'uYCMaTpHBaIOT
npHJIHBW ,nJIH OnHpaHHH OJIH3KO paCnOJIO>KeHHMX CTOeK H
pyrHx YCTPOHCTB pa3JIH1JHOrO Ha3Ha1JeHHH. KOJIbIJ,eBWe H
Kpyr JIbIe cpYHJI.aMeHTbI HCnOJIb3YlOT no,n pa3JIH1JHbIe Kpyr-
JIble B nJIaHe cOOpY>KeHHH, npH 3TOM nepBhIe H3 HHX peKO-
MeH,nYIOTCH npH CpaBHHTeJIbHO MeHblliHX HarpY3KaX. MHq-
royrOJIbHbIe <PYHJI.aMeHTbI 1JaCTO npHMeHHIOT BMeCTO Kpyr-
JIbIX, TaK KaK OHH npOIII.e B npOH3BOJI.CTBe pa60T H MeHee
TpYJI.OeMKH npli H3rOTOBJIeHHH OnaJIy6KH. npHBeJI.eHHble
THnbI <PYH,UaMeHTOB ycnemHo TaK)Ke HCnOJIb3YlOTCH no,n
pa3JIH1JHOe 060pY,UOBaHHe H JI.pyrHe TeXHOJIorH1JeCKHe YCT-
pOHCTBa.
PaCqeT <PYH,UaMeHTOB C nO,UOlliBaMH; OTJIH1JaIOInHMHCH
OT paCCMOTpeHHbIX, B 1JaCTHOCTH, C JI.BYTaBpOBbIMH H TaB-
pOBbIMH nOJI.OlliBaMH C IIOJIKaMH, CMeIII.eHHbIMH OTHOCHTeJIb-
HO OCM pe6pa, B TOM 1JHCJIe. IllBeJIJIepOOOpa3HbIe, r -o6pas-
HhIe Ii JI.pyrHe MO:>KHO BbInOJIHHTb 3THMH :>Ke MeTo,UaMH.
Bee paC1JeTHbIe cpOPMYJIbI (CM. rJI. 2-5) nOJIY1JeHbI H3
YCJIOBHH paBHoBecHH YCHJIHH, JI.eHcTBYIOIII.HX Ha ypoBHe
nOJI.o.WBbI <PYHJI.aMeHTa, H: peaKlLHH OCHOBaHHH. npH 3TOM
,lI.JIR I,Bcex <PYH,UaMeHToB, pa60TalOIII.HX C OrpaHH1JeHHbIMH
KpaeBbIMH ,UaBJIeHHHMH Ha OCHOBaHHe npH ,neiicTBHH MO-
MeHTa B O,UHOl\1 HanpaBJIeHHH, HCnOJIb30BaHbI H3BeCTHbIe
<POPMYJIbI BHelLeHTpeHHoro C:>KaTH5I.
L{JIH CJIY1JaH Pmax= I,2R H Pmin=O:
N Ne x 1 2R .
Pmax== - + - = , ,
F W y
Pmln -:- lV _ Nex = o.
F W y
(1.1)
( 1.2)
9
,LLJIH CJIyqaH Pmax= I,2R II pmin =O,25pmax: Pmax BbIqIIC-
JIHeTCH no <popMY.lIe (1.1);
Prnln = - N;: = 0,25prnax = 0,25 · 1 ,2R = 0,3R. (1.3)
,LL.lIH C.lIyqaH Pmax= I,5R II Pmin=O;
__ N I Ne x + Ney - I 5R .
Pmax - T - -, ,
F W y W x
. _ N _ Ne x _Ne y -0
Prnm - F W y W x - ·
,LL.lIH C.lIyqaH Pmax= I,5R H Pmin=O,25pmax: Pmax BbI-
QHCJIHeTCH no cpopMY.lIe (1.4);
( 1.4 )
( 1.5)
_ N Ne x Ne y -- 0 25 _
Pmln------- , Pmax-
F W y W x
= 0,25 · I, 5R === O,375R. (1.6)
no cpopMY.lIaM (1.1).. .(1.6) C yqeTOM npIIBeJI.eHHOH pa-
Hee Ta6.lIHl{bI nO.lIyqeHbI paCqeTHbIe <POPMY.lIbI J(.lIH onpeJ(e-
.lIeHHH pa3MepoB nO)1,OIllBbI H ,lI.aBJIeHHH B ee xapaKTepHbIx
TOqKaX, rr03BO.lIHIO HX nocTpoHTb 3nlOpy J(aB.lIeHHH Ha oc-
HOBaHHe.
,LL.lIH <pYH,lI.aMeHTOB- Bcex THnOB, pa60TalOI..UHx C OTpbI-
BOM nOJI.OlliBbI, paCqeTHbIe <POPMY.lIbI TaKLKe nOJIyqeHbI 1I3
YC.lIOBHH paBHoBecHH. nepBOe YC.lIOBlIe npe'uycMa1:pHBaeT
paBeHcTBo 06'beMa 3nlOpbI ,UaBJIeHIIH Ha OCHOBaHHe paBHo-
,lI.eHcTBYIO eH BepTHKaJIbHOH OCH N. BTopoe YC.lIOBHe yqH-
TbIBaeT paCnOJIOLKeHlIe l{eHTpa THLKeCTII 06'beMa 3nlOpbl
JI.aBJIeHHH Ha OCHOBaHHe Ha .lIMHHH ,UeHCTBHH paBHOJI.eHCT-
BYIOIII.eH CHJIbI. TaKIIM 06pa30M npOeKl{IIH YKa3aHHoro l{eH-
Tpa THLKeCTII Ha n.lIOCKOCTb nO,lI.olliBbI <PYH,lI,aMeHTa ,lI.o.r(}KHa
cOBna,UaTb C TOllKOM nplI.lIO}KeHIIH paBHO,lI.eMCTBYIOIII.eH CII-
JIbI. CJIe,UOBaTe.lIbHO, YCJIOBHbIM CTaTlIqeCKHH MOMeHT nlO-
pw AaBJIeHHH Ha OCHOBaHHe II MOMeHT BepTIIKaJIbHOH paB-
HOJ(eHCTBYIOIII.eH CHJIbI OTHOCHTe.lIbHO .lII06oH OCII AO.lI)KHbI
6bITb paBHbI. CTaTHqeCKIIH MOMeHT Ha3BaH YCJIOBHbIM, TaK
KaK BMeCTO cpaKTHqeCKOrO TI.lIeqa nplIHHTa ero npoeKll.IIH Ha
n.lIOCKOCTb nOAoIllBbI <PYHAaMeHTa (CM. rJI. 4 H 5).
XOTH paCqeTHbIe <POPMYJIbI npH OrpaHHqeHHbIX KpaeBbIX
JI.aBJIeHHHX Ha OCHOBaHHe nOJIyqeHbI 1I3 YCJIOBHH pmax= 1,2R
AJIH CJIYlIaH ,UeHcTBIIH MOMeHTa B OAHOM HanpaBJIeHlI1I Ii
Pmax= 1,5R AJIH CJIYlIaH AeMcTBHH MOMeHTOB B J(BYX B3a-
IIMHO nepneH,lI.HKY.lIHpHbIX HanpaBJIeHHHX, TeM He MeHee
10
3TMMH cpOpMYJIaMH MO)KHO nOJIb30BaTbCH npH JII06bIX 38-
.naHHbIX 3HaqeHHHX Pmax. ,LLJIH 3TOrO CJIe,l(yeT nO.lI.CTaBJIHTb
B cpOPMYJIbI B nepBOM CJIY1Jae R=Pmax/I,2 Ii BO BTOpOM
R = Pmax/ 1,5. T aKaH Heo6xo.lI.HMOCTb HHor.lI.a B03HHKaeT,
KOr.lI.a K03cpcpH HeHTbI 1,2 H 1,5 npe)J.CTaBJIHIOTCH He.lI.OCTa-
TOqHO 060CHOBaHHbIMH. HanpHMep .lI.JIH MHoroyrOJIbHbIX
YH.lI.aMeHTOB npH .lI.eHCTBHH MOMeHTOB B )J.BYX B3aHMHO
nepneH)I.HKYJIHpHbIX HanpaBJIeHHHX rrpHHHTo Pmax= I,5R,
qTO cpopMaJIbHO COOTBeTcTByeT YKa3aHHIO n. 2.49 CHHn
2.02.01-83. O,aHaKo yqHTbIBaH, qTO ,l(JIH TaKHX CPYH.lI.aMeH-
TOB BeplllHHa C Pmax pacnOJIO}KeHa Ha O,LI.HOH JIOMaHOH C
.l(BYM5I .lI.pyrHMH CMe}KHbIMH BeplllHHaMH, r,l(e ,lI,aBJIeHHH Ha
OCHOBaHHe He3HaqHTeJIbHO OTJIHqaIOTCH OT Pmax, npaBOMOq-
HOCTb K09 cpHIl.HeHTa 1,5 CTaHOBHTCH COMHHTeJIbHOH. npe,n-
CTaBJ)51eTCH Il.eJIeC006pa3HbIM B TaKHX CJIyqa5lX CMHrqaTb
npHBe.lI.eHHOe YKa3aHHe CHHna H npHHHMaTb Pmax= I,2R
HJIM, B KpaHHeM cJIyqae, Pmax (1,2+ 1,5)R/2= 1,35R.
TaKHM o6pa30M npH cooTBeTcTBYIO eM 06ocHoBaHIili
BeJIHqHHa Pmax MO}KeT BapbHpoBaTbcH no cpaBHeHHIO C pe-
KOMeH,ayeMoH 6e3 H3MeHeHHH paCqeTH X CPOPMYA.
rnABA 2. YH AMEHTbI, PA60TAIOmHE
C OrPAHHIJEHHblMH KPAEBblMH AABnEHHSlMH
HA OCHOBAHHE UPH AEACTBHH MOMEHTA
B O,nHOM HAnPABnEHHH
2.1. 06 He paCqeTHble 4JOPMYJlbl
L{JIH <PYH.lI.aMeHTOB, y KOTOpbIX reOMeTpHqeCKHe xapaK-
TeplicTHKH nO.lI.OlliBW 3aBHCHT OT .lI.BYX ee pa3MepoB (CM.
npHBe,aeHHYIO paHee Ta6JIH Y), nOJIyqeHbI o6 He paCqeT-
HbIe cpOPMYJIbI, BbIBO,l( KOTOpbIX npHBo)J.HM HH}Ke. I1CKJIIO-
qeHMe COCTaBJIHIOT TaBpoBbIe CPYH.lI.aMeHTbI, ,l(JI5I KOTOpbIX
paCqeTHbIe cpOPMYJIbI npHBe.lI.eHbI- B n. 6 3TOH rJIaBbI.
,UJI5I cJIyqa51 Pmax= 1,2R H Pmin=O nOCJIe IIO.lI.CTaHOBKH
B OPMYJIbI (1.1) Ii (1.2) 3HaqeHHH F 11 W y 113 Ta6JIHUbI
nOJIyqHM:
P N + Nex = 1,2R;
max === k1AB k j A 2 B
( 2.1 )
N
Pmlo === k1AB
Ne x
k4 A2B
== o.
( 2.2)
11
CYMMa H pa3HOCTb KpaeBbIX ,AaBJIeHIIH
2N I (k 1 AB) == I,2R;
2Ne x /(k i A'jB) == 1,2R.
H3 BbIpa}KeHHH (2.3) H (2.4):
2N/(k 1 AB} === 2Ne x /(k",A2B), oTKy.u.a
(2.1) H (2.2):
( 2.3 )
(2.4 )
ext A == k41 k 1 ,
A=k 1 e x lk 4 .
113 BbIpa:>KeHHH (2.3):
B == 2N / (I,2klRA) == 5N / (3k 1 RA).
AHaJIOrHtIHO ,lI.JIH CJIY1JaH Pmax= 1,2R H
=O,25Pmax=O,3R, 3HatIeHIIe Pmax cooTBeTcTByeT
JIe (2.1):
( 2.5 )
(2.6 )
Pmln =
<l>OPMY-
Pmin =--== N I klAB = Nexl k4A2B == O,3R;
Pmax + Pmin = 2N I (k1AB) == l,..5R;
Pmax - Pmln -== 2Ne x / (k4 A2B ) =-== O,9R
HJIH ION e xl (3k4A 2B) === 1 ,5R, (2.9)
oTKy.na 2N/(k 1 AB)=10Ne x /(3k,fi2.B) HJIM e x /A=O,6k 4 /k l H
A == 5k 1 e X /(3k 4 ). (2.10)
113 BbIpa}KeHH5I (2.8)
B == 4 N i (3k 1 R A ) . ( 2 . 11 )
,aJIH <PYHJI.aMeHTOB, ,lI.JI5I KOTOpbIX reOMeTpH1JeCKHe xa-
paKTepHcTHKH no.nOlliBbI 3aBHCHT OT o.nHoro ee pa3Mepa A,
RH HJIH a (CM. Ta6JIH Y), TaK}Ke nOJIyqeHbI 06llI.He paC1JeTHble
4>OPMYJIbI. 3TH <PYHJI.aMeHTbI HMelOT oco6eHHOCTb, COCTOH.
IIJ.YIO B TOM, qTO ,lI.JI5I onpe,lI.eJIeHHH OJI.Horo HeH3BeCTHoro
pa3Mepa nO,lI.OIllBbI pacnOJIaraeM ,lJ,BYMH ypaBHeHHHMH paB-
HOBeCHH. CJIe.nOBaTeJIbHO, 3a.n.aqa HBJIHeTCH Heonpe,lJ,eJIeH-
HOM H O,lI.HOBpeMeHHoe y,aoBJIeTBOpeHHe paBeHcTBa KpaeBbIx
,naBJIeHHM Pmax II Pmln HopMHpyeMbIM CTaHOBHTCH HeB03-
MO:>KHbIM. n03TOMY paCqeTHbIe CPOPMYJIbI npe.nYCMaTpHBa-
lOT y.nOBJIeTBop.eHHe OrpaHHqeHHOrO 3HaqeHHH TOJIbKO O.lI..
Horo KpaeBoro .naBJleHHH, a BTopoe - paCCMaTpHBaeTCH KaK
BTopoe HeH3BeCTHoe. OpaKTHqeCKH B KaqeCTBe HeH3BeCTHo-
ro KpaeBoro .naBJleHHH y.no6Hee npHHHMaTb Pmln. ECJIH npH
12
(2.7)
(2.8)
9TOM Prnax a}l{eTCH 60JIbllIe HOpMaTHBHO AOnYCTHMOrO, TO
YBeJIHQHBalO1\ COOTBeTCTBeHHO pa3Mep cpYH,naMeHTa.
npHBe,neM bIBO,n paCqeTHbIX cpOPMYJI ,nJIH T8KHX cpYH-
A8MeHTOB npH HHTeJIbHO K CPYH,n8MeHTaM C KOJIbueBOH
nO,nOWBOH. ,QJI5I CJIYT.laH Pmln=O:
N + Hex
Pmax = k R 2 k R 3 ;
1 Ii 4 H
N Ne x O .
Pmln =:= - =,
k 1 RH 2 k4Ru 3
113 Bblpa>KeHHH (2.13):
ex == k4RH / k I ; Ru === k 1 e x / k4.
113 BbIpa>KeHHH (2.12)
N ( I + ex )
Pmax = R 2 k k R
H. -:-:.
nO,nCT8BHB B nOCJIe,nHee YP8BHeHHe 3HaqeHHe ex liS
nOCJIe npOCTeHIIlHX npeo6pa30BaHHH nOJIYQHM
Pmax = 2N / (k 1 RH2).
,aJIB CJIyqaH Pmln=O,25Pmax HMeeM
_ N _ Nex _ 0 25 ( N + Nex )
Pmln - - , ·
k 1 RH2 k 4 RH 3 k 1 RH2 k..RH 3
no aHaJIOfHH co CJIyqaeM Pmln=O:
_ O,6k 4 R H . R -- 5k 1 e x . _ 1,6N
ex - k 1 ' H - 3k4, t Pmax - k 1 RH 2 ·
(2. 12)
(2.13)
(2.14)
(2.14) ,
( 2.15 )
( 2. 16)
(2.17)
X!J1H paccMaTpHBaeMbIx YH,n8MeHToB OCTaJIbHbIX THnOB
COXp8H5IIOTCH npHBe,neHHbIe CPOPMYJIbI C 3aMeHOH B HHX Ra
Ha a (CM. Ta6JIHI.(Y Ha c. 4).
nOJIyqeHHble <pO.PMYJIbI ,nJIH CPYH,naMeHToB, pa60TalOll.\Hx
C 3a.n.aHHbIMH OrpaHHqeHHbIMH KpaeBbIMH ,naBJIeHHHMH H8
o HoBaHHe npH ,neAcTBHH MOMeHTa B O HOM HanpaBJIeHHH,
npe,nycMaTpHBalOT onpeJI.eJIeHHe MHHHM8JIbHbIX pa3MepoB
no,nOlliBbI, npH KOTOpbIX C06JIlO,n8IOTCH HopMHpyeMbIe YCJIO-
BHH. B HeKoTopbIX CJIYT.laHX pa3MepbI nO OlIIBbI H HX COOT-
HOIlIeHHH nOJIYT.laIOTCH HenpHeMJIeMbIMH IIO KOHCTPYKTHB-
HbIM co06paJKeHH5IM. B TaKHX CJIYlJaHX, eCJIH n03BOJIHIOT
YCJIOBHH pa3Mell.\eHHH <PYH,naMeHTa, CJIe,nyeT YBeJIHqHTb
,nJIHHY nOJI.OllIBbI, npH 3TOM Pmin HeCKOJIbKO B03pacTeT no
cp-aBHeHHIO C 3a,lJ.aHHbIM. lllHPHHY IIO OllIBbI KoppeKTHpy-
13
lOT TaK, lIT06bI BeJIHT.lHHa Pmax OI<a3aJIaCb paBHqH IIJIH 6JI1I3-
KOH K 3a,lJ,aHHOM. .
I
B npIIBe eHHWX ,lJ,aJIee nplIMepax pacqera onpe,lJ,eJIeHW
MHHHMaJIbHO ,lJ,OnYCTIIMbIe pa3MepbI nO,lJ,OIlJ'BbI 1I3 YCJIOBlfH
O)lHOBpeMeHHoro paBeHCTBa Pmax II Pmln. 3TII nplIMepbI IIJI-
JIIOCTPIIPYlOT nOCJIe,lI,OBaTeJIbHOCTb paCqeTa.
2.2. np.HMoyroJlbHble 4JYH,lI.aMeHTbl
,UJIH np5lMoyrOJIbHbIX <PYH,lJ,aMeHTOB kt= 1, k z =k 3 = 1/12
H k4=k5= 1/6. OpH Pmax= 1,2R If Pmin=O cpOPMYJlbI (2.5)
H (2.6). nplIHIIMalOT COOTBeTCTBeHHO BH,l( (plfC. 2.1, a) :
A==6e x ; e x ==A/6; B===5N/(3RA). (2.18)
AHaJlOrHqHO nplI Pmax= 1,2R If Pmln==O,25pmax=0,3R
H3 cpOPMYJI (2.10) II (2.11) HMeeM (pIlC. 2.1,6) :
A = 10e x ; ex == A /10; B == 4N / (3RA). (2.19)
0) y
0,5A 0,5A
.
c::s x c:::i x
CQ co CQ,
&.c) C.(') J
r::::::; c:::)
'"
A Qc
A Q:::
C"-l )( ..
-.: c:s >(
(:I
f II
..
r::
....
14-
Pac. 2.1
npHMep ,. P,aHo: YCHJIHH, .n.eMcTBYlOmHe Ha ypoBHe nO;J.OlliBbI N =
1000 KH, M x =600 KH.M; R=200 Kna, pmax= 1,2R= 1,2.200=240 Kna.
Tpe6yeTcH onpe.n.eJIHTb pa3MepbI no.n.OllIBbl H .n.aBJIeHHH Ha OCHOBaHHe.
PaC'leT: ex=M x IN=600j1000=0,6 M no ct>opMYJIaM (2.18):
A=6e x ===6.0,6==3,6M; B==5NJ(3RA)==
= 5. 1000 -; (3 . 200 . 3,6) = 2,31 Ej M.
14
.aJIH KOHTpO.nH HaHJI.eM Pmax H Pmtn:
Pmax = 1000/3,6 · 2,315 + 6 · 600/3,6 2 · 2,315 ==
=--== 120 + 120 == 240 Kna; Pmlo == 120 - 120 == O.
npHMep 2. p'aHo: ,QJIH YCJIOBHH, npHBe)J.eHHbIX B npHMepe 1, 3a HC-
KJIIOtleHHeM M x = 360 KH. M, onpe.n.eJIHTb pa3MepbI nOJI.OWBbI Ii JI.aBne-
HliH Ha OCHOBaHHe npH Pmax= 1,2R=240 Kna Ii Pmin=0,25Pmax=
=0,25.240=60 Kna.
PaCtteT: ex=360/1000=0,36 M. Do <j>opMYJIaM (2.19):
A == 10e x === 10 · 0.36 == 3,6 M;
B == 4. 1000 /3 . 200 · 3,6 === 1,85 M.
.QJlJI KOHTpOJUI onpeJI.eJIHM pmax H pmin:
Pmax = 1000 / (3,6 . 1,83) + 6 · 360 / (3,62 . 1,85) ===
=== 150 + 90 == 240 240 KIla;
Pmln == 15J - 90 = 60 60 Kna.
3.3. fl)YH,l(aMeHTbI C CHMMeTpHqHOH
u v
BYTaBpOBOH nO OllIBOH
,l:LJIH 3TMX <PYH,QaMeHTOB pa3MepbI no,nOWBbI onpeJI.eJIHIOT
no cpopMYJIaM (2.5) H (2.6) npM Pmax= 1,2R M Pmin=O
(pHC. 2.2, a) H no <popMYJIaM (2.10) M (2.11) npM Pmax=
= 1,2R M Pmin=0,25Pmax=O,3R (pRC. 2.2, 6). DpH 3TOM:
k 1 === 1 - a. (1 - );
k 2 === 1 / 12 [ 1 - a H (1 - )];
kg === 1 / 12 ( 1 - ex + (l 3);
k4 === 1 / 6 [1 - ex:i (' 1 -- )] == 2k 2 ;
k == If 6 (1 - ex + a 3) === 2k s '
r,ne a == a I A, = b / B.
3HaqeHMH K03ct><PMl.(HeHTOB kt, k4 H k5 npMBe,lLeHbI B
Ta6JI. 2.1.
,QaBJIeHMe Ha OCHOBaHHe B npoMeLKYTOqHblX xapaKTep-
HbIX TOqKaX nO,QOIlIBbI onpe,neJIHIOT no <popMYJIaM (CM.
pRC. 2.2):
(2.20)
(2.21 )
(2.22)
(2.23)
(2.24)
N MXCl
Pt = ktAB + k4A2B ;
N MxrJ.
P2 === klAB - k-tA2B ·
( 2.25 )
1.5
fi
v :x \
6)
,
"v
a I
A L
"
x
:::-
x
.....
...
D,5a J1 fJ.5a '7
I (J.5A a a
" "< iJ,5A 1J
A I:J A
C<
C\.J c::5 C'\I
'" .....:-
II
II 'I
t:: JI
"
f2
I:::L
Pf
PHC. 2.2
npHMep 3. l1.aHo: YCHJIHH, JleHcTBYlOllIJle H3 ypoBHe nOjlOllIBbI CPYH-
,D.3MeHT3 N= 1000 KH, M x =750 KH.M, R=250 Kna, Pmax= 1,2R=
= 1,2.250 = 300 Kila, proln = O. Tpe6yeTcH onpeJleJIHTb p33MepbI n01l.0-
llJBbI H Jl3BJIeHHH Ha OCHOBaHHe.
PaCtleT: ex=M x jN=750/1000=0,75 M. 38)J.3eMcH a=0,5 11 =0,4.
no T36JI. 2.1 npH a=O,5 H =0,4 HaX01l.HM k 1 =O,7 H k4=O,154. no
<}JopMYJIaM (2.5) H (2.6):
A = k1e X 1 k4 = 0,7 · 0,75/0,154 = 3,4 M;
B' = 5N-1 (3ktRA) = 5 · 1000/3 · 0,7 250 · 3,4 = 2,8 M;
a=aA=0.5. 3,4= 1,7 M; b= B=OJ4. 2,8=1,12 M.
JI.JlB KOHTpOJIB OnpeJ1eJ1HM Pm." H pmln:
Pmax=1000jO.7. 3,4. 2,8+75010,154. 3,4 2 . 2,8=
= 150 + 150 = 300 300 Kna.
Pmin = 150 - 150 = O.
16
TaCiJlHu.a 2.1
- f3=-0.3 13=0,35 (3=0,4
a k4 ks k. k4 ks k. k l
k. k.
0,164' 0.ll81 0,805\ 0,164' 0,119 I
0,3 0,79 0,82 0, 164 0,12
0,35 0,755 0, 162 0, 11 ° 0, 773 0, 162. 0, 11 ° 0,79 0, 162 0,11
0,4 0,72 0, 159 0, 102 0,74 0,160; 0,103 0,76 0, 160 0,10
o 45 0,685 0, 156 0,094 0,708 0, 1571 0,095 0,73 0, 158 0,09
,
0,5 0,65 0, 152 0,086 0,675 (\, 1531 0, 087 ° , 7 0, 154 0,08
0,55 0,615 0, 1 47 0,078 0,643 0,149 0,079. 0,67 0,15 0,081
0,6 0,58 0,141 0,069 0,61 0,143 0,071 0,64 0,145 0,07
.0 65 0,545 0,135 0,063 0,5781 0,137 0,063 0,61 0, 139 0,06
,
0,7 0,51 0,127 0,053, 0-, 545J 0, 130 0,055 0,58 0, 132 0,05
I I
2
4
7
9
3
5
8
(3=0.45 (3 = 0,5 P=-0.55
Q
k. k. k l k. k4 k6 k. k. k.
0,3 0,835 0, 164 0,121 0,85 0, 164 0, 123 0,865 0,165 0,125
0,35 0,808 ° , 163 0,114 0,825 0, 163 0,116 0,842 0, 164' 0,118
0,4 0,78 0.161 0.106 0,8 0,161 0, 1 08 0,82 0, 162 0,111
0,-45 0,753 0, 158 0,099 0,7"75 0, 159 0,101 0,798 0, 160 0,104
0,5 0, 725 0, 155 0,091 0,75 0, 136 0,094 0,775 0, 157 0,097
0,55 0,698 0,151 0,083 0,725 0, 152 0,087 0,752 ° , 154 0,09
0,6 0,67 0, 147 0,076 0,7 0, 149 0,079 0,73 0,151 0,08
0,65 0,643 0, 142 0,068 0,675 0,144 0,072 0,708 0,146 0,07
0,7 0,615 0,135 0,061 0,65 0, I 38 0,065 0,685 0,141 0,069
3
6
P=O,6
p = 0.65
(3=0,7
CI I
k. I Ie. k& k. k. k6 .Ill k4 k l
0,128\ 0,895 0,16510,130\ 0,91 :
0'3 0,88 0, 165 0, 165 o , 134
,
0,35 0,86 0, 164 0,121 0,878 0,164' 0,124 0,895 0, 165 0, 128
0,4 0,84 0, 162 0,114 0,86 0, I 63 0,118 0,88 0, 164 0, 123
0,45 0,82 0,161 0, I 08 0,842 0,161 0, 112 0,865 0,162 0, 117
0,5 0,8 0, 158 0,101 0,825 0, 159 0, 1 06 0,85 0, 160 0, t 12
0,55 0,78 0,156 0,095 0,808 0,157 0, 1 00 0,835 0, I 58 0, 107
0,6 0,76 0, I 52 0,088 0,79 0, 154 0,094 0,82 0,156 0, 101
0,65 0,74 0,148 0,082 0,773 0, 151 ° , 088 0,803 0, 153 0,096
0,7 ° 72 0, 144 0,075 (',755 0, 147 0, 082 0,79 .0,150 0,09
,
Do Q>opMYJ1aM (2.25): Pl=150+150.0,5=225 Kna; P2=150-
- 0,5.150=75 Kna.
DpHMep 4. ,[{JIH YCJIOBHH, npHBe)J.eHHbIx B npHMepe 3, 3a HCKJIIOQe.
HHeM M x =500 KI-I. M, onpe.u.eJIUTb pa3Mepbl no.u.OlllBbl H .naBJIeHHH H8
OCHOBaHHe npH Pmax , 300 ilia. Ii Pmln=75 Kna.
17
PaC'leT: e x =500/1000=0,5 M. 113 npHMepa 3 k l =0,7, k,=0,154. no
.ct>opMYJIaM (2.10) H (2.11) HMeeM:
A= 5.0,7.0,5 =3,78M, B= 4.1000 =2,02M.
3 . 0 154 3 · 0 7 . 250 . 3 78
, "
a = 0,5 · 3, 8 == 1,89 M, b = 0,4 . 2,02 = 0,8t M.
,QJIH KOHTpOJIH onpe)J.eJ1HM Pmax H pmln:
Pmax = tOOO /0,7 · 3,78. 2,02 + 500 /0,154 · 3,78 2 · 2,02 =
= 187,1 + t12,3=299,4 300 KOa;
Pm In = 187,1 - 112,3 === 74,8 75 Kna.
no q>opMYJIaM (2.25) PI = 187,1 + 112,3.0,5=243,3 l<Oa; P?= 187, I -
- 112,3.0,5= 130,9 KOa.
2.4. cl»YHAaMeHTbI C 3aMKHYToH JleHTOlfHoH
u
npHMoyrOJlbHOrO OqepTaHHH B nJlaHe nOAOWBOH
,[(JIH 3TRX CPYH,lI.aMeHTOB pa3MepbI nO,lI.OllIBbI onpeJJ.eJIH"
lOT no cpopMYJIaM (2.5) H (2.6) B CJIyqae Pmax= 1,2R If
Pmin=O (pRC. 2.3, a) If no cpopMYJlaM (2.10) H (2.11) 8
CJIyqae Pmax= 1,2R H Pmin=0,25Pmax=0,3R (pHC. 2.3, 6).
npR 3TOM:
k 1 :=!: 1 - C1 ;
k 2 = 1 / 12 (1 - C13 );
k3 = 1 / 12 (1 - C1 3) ;
k4 = 1/6 (1 - C13 ) = 2k 2 ;
k5 = 1/6 (1 - C1 3) = 2k 3 ;
f,lI.e a = a/A; == biB.
( 2.26 )
( 2.27)
(2.28 )
( 2.29 )
( 2 .30 )
(2.31 )
3HaqeHHH K03cpcpH HeHToB k1, k4 H ks npRBe.neHbI B
Ta6JI. 2.2.
,I:LaBJIeHHH Ha OCHOBaHHe B npOMe}KYTOqHbIX xapaKTep-
HbIX TOlIKaX nO,lI.OllIBbI orrpe.neJI5IIOT no CPOPMY JIaM (CM.
pRC. 2.3):
N MXCl . N MXCl 2
Pi = k1AB + k l A2B ' P2 = k.AB k,A2B (.32)
npHMep 5.. /l,aHo: YCHJIHH, JleHcTBYIOJ..U.He Ha ypoBHe nO.ll.OIlIBbI
$YH.lI.aMeHTa N=500 KH, M x =250 KH.M, R=300 KOa, Pmax=1,2R==
18
a)
--
" I
-.,t
6
) Aj
0,5A n i.J
a IA
I Wa "
:,.
I
I ...
... x
..
.. ---
t(}
'C')
"' ci. -
1 «:::::)
)0(
t::s ! ! .
i Q:::
lC') ! C"I
C'...I
y
51.
"'
...
X
::
'C) t(o)
c::::5'
l
II
)(
:
11
.t:::)
C::::f b
II
t::
. .
,. I
- t
.1
P1
PRC. 2.3
= 1,2.300 = 360 Kna, Pmin = O. TpeoyeTcH onpe)J.eJIHTb pa3MepbI noJI.O-
WBhl H )J.aBJIeHHH Ha OCHOBaHHe.
PaCtteT: e x =250j500=0,5 M. 3a)J.aeTcH a=O,7 H =0,6. no
TaOJI. 2.2 npn a=0,7 If =O,6 HaXO)J.HM k l =0,58 H k(,.=0,132. no 4>oP-
MYJIaM (2.5) H (2.6) HMeeM:
A= (0,E8. 0,-5)/0,132=2,19 M; B=5. 500/(3. o,saX
X 300 . 2,19) == 2,19 M;
a=O,7.2,19=1,53 M, b=0,6.2,19==1,31 M.
.l1JUI KOHTpOJIH onpe)J.eJIHM Pm ax H pmfn:
Pmax=500/(0,fi8. 2,19. 2,19) +250/(0,132. 2,19 2 . 2,19) =
== 179,7 + 179,7 ==359,4 360 Kna;
Ta6J1Hua 2.2
= 0,3 13=0,35 P-O.4
u
k l k. ks k l k. ks k l k. k.
0,3 0,91 0, 165 0, 166 0,895 0, 165 0, 165 0,88 0, 165 0,164
o 35 0,895 0, ] 65 0, 165 0,878 0, 164 0, ] 64 0,86 0, 164 0,163
,
0,4 0,88 0, 164 0, 1 65 0,86 0, 163 0, 164 0,84 0, 162 0, 162
0,45 0;865 0, 162 0, 165 0,843 0,161 0, 164 0,82 0,161 0, 162
0,5 0,85 0, 160 0, ] 64 0,825 0, 159 0, 163 0,80 0, 158 0, 161
0,55 0,835 0, 158 0, 164 0,808 0, 157 0, 163 0,78 0, 156 0, 161
0,6 0,82 0,156 0, 164 0,79 0, 154 0, 162 0,76 0, 152 0, 160
0,65 0,805 0, 153 0, 164 0,773 0,151 0, 162 0,74 0, 148 0, 160
0,7 0,79 0, 150 0, 164 0,755 0, 147 0, 162 0,72 0, 144 0, I 59
19
npOaOA eHue Ta6A. 2.2
11-0,45 P-0.5 P-O.&6
a
k. k.. k& k. k.. ks k. k. Ie,
0,3 0,865 0, 165 0,162 0,85 0, 164 0, 1601 O. 835 0, 164 1 0,158
0,35 0,843 0, 164 0,161 0,825 0, 163 0, 159 0,808 0, 163 0,157
-0 4 0,82 0, 162 0,161 0,8 0,161 0, 158 0,78 0,161 0, 156
,
0,45 0, 798 0, 160 0, 160 0,775 0, 159 0, 157 0,753 0, 158 0, 154
.0 5 0,775 0, 157 0, 159 0,75 0,156 0, 156 0,725 0, 155 0,152
,
.0 , 55 0,753 0, 154 0,158 0,725 0, 153 0, 155 0,698 0,151 O,15i
.0 6 0,73 0, 150 0, 158 0,7 0, 149 0, 154 0,67 0,147 0,15
,
0,65 0,708 0,146 0, 157 0,675 0, 144 0,153 0,643 0, 142 0,149
0,7 0,685 0,141 0,156 0,65 0, 138 0, 152 0,615 0, 135 0, 147
P-0,6 13=0,65 P-O,7
a I
k J k. le s k. k4 ks k. k. ks
'0,3 0,82 0,164 0,156 0,805 0, 164 0, 153 0,79 0, 164 0, 150
.0,35 0,79 0, 162 0, 154 0,773 0,162 0,151 0,755 0, 162 0, 141
-0 4 0,76 0, 160 0, 152 0,74 0, 160 0, 148 o 72 o I 159 0, 144
, ,
,0 45 0,73 0, 158 0, 151 0,708 0, 157 0, 146 0,685 0, 156 0,141
,
0,5 0,7 0, 154 0, 149 0,675 0,153 0, 144 0,65 0, 152 0, 138
.0 55 o 67 0,15 0, 147 0,643 0, 149 0, 142 0.615 0, 147 0, 135
, ,
.() 6 0,64 o , 145 0, 145 0,61 0, 143 o , 139 0,58 0, 142 0, 137
,
0,65 0,61 0, t 39 o , 143 0,578 o I 137 0,545 0, 135 0,135 0, 132
.0 7 0,58 0, 132 0,141 0,545 0, 120 0, 133 0,51 0, 127 0,120
,
Prolo = 179,7 - 179,7 = 0.
no 4>opMYJIaM (2.32):
Pt= 179,7+ 179,7 · 0,7=305,5 KOa; Ps= 179,7-
-179,7. 0,7=53,9 Kna.
npHMep 6. ,nJIH YCJIOBHH, npHBe.u.eHHbIx B npHMepe 5, onpe.u.eJIHTb
pa3MepbI no.u.OWBbI npH pmax= 1,2R H Pmin=0,25 Pmax=O,3R.=0,3.300=
= 90 Kna, a TaK>Ke .n.aBJIeHHH Ha OCHOBaHHe.
PaCtteT: H3 npHMepa 5 ex=0,5 M; k 1 =0,58 H k",= 132. no fl>OPMY-
...naM (2.1 0) H (2.11) HMeeM:
A = 5 . 0 58. 0,5/(3.0,132) =3,65 M;
B = 4 · 500/(3 · 0,58 · 300 · 3,65) := 1,05 M;
a = 0,7 · 3,65 ==- 2,56 M; b = 0,6 · 1,05 = 0,63 M.
,nJIH KOHTpOJlH onpe.u.eJlHM Pmax H pmln:
.Pms" = 500jO,58 · 3,65 . 1,05 + 250/0,132 · 3,65 2 · 1,05 =
= 224,9 + 135 === 359,9 360 KOa;
'20
!J
11) ex
I
.
c:::5
:::
"«:Q
1\
tQc:t::)
.' ....'")
"/. I --- c:::)
-/ --t
45A(t-ol.) a=d..A a5A (I-ol) _
0,5A "i
A
P2. Pf
If)
y
ex
x
I .
...
...'
(::j
E: I
t'-.,
c:::)
" t:: I
JL
+-
A
'
..
"
.30(
c::s
J
P2
P,
PRC. 2.4
Pm In = 224,9 - 135 90 KOa.
no opMynaM (2.32):
Pt = 224,9 + 135 · 0,7 = 319,4 Kna; P2 == 224,9
- 135 · 0,7 == 130,4 KOa.
2.5. YHAaMeHTbI C KpecToBoH nO,ll.owBOH
,LlJISI 3THX <PYHJJ:aMeHToB pa3MepbI nOJJ:OIlIBbI onpeJJ:eJIHIOT
no <popMYJIaM (2.5) H (2.6) npH Pmax= 1,2R II Pmln=O
(pRC. 2.4, a) II no <popMYJIaM (2.10) H (2.11) npH Pmax=
= 1,2R H Pmln=0,25pmax=0,3R (pHC. 2.4,6). npH 3TOM:
k 1 == + ex ( 1 - ) ; ( 2.33 )
k 2 = 1/12 [ + ex 3 (1 - ) ] ; ( 2.3 4 )
ka = l' 12 [(1 + 3 (1 - ex) ]; (2.35)
k" = 1/6 [ + (13 ( 1 - )] == 2k 2 ; (2.36 )
k5=1/6[a+ 3(I-a)]=2k3' (2.37)
r.ne a=aIA; =bIB.
21
3HaqeHHH
Ta6JI. 2.3.
,[{aSJIeHHH
HbIX TOqKaX
pHC. 2.4) :
PI = N + Mxa ; P2-= N
klAB k4 A2 B klAB
K03cpcp H lI.HeHTOB kl, kt" H ks npHBeJJ.eHbI B
Ha OCHOBaHHe B npOMe}KYTOqHbIX xapaKTep-
nOJJ.OIIIBbI onpeJJ.eJIHIOT IIO cpopMYJIaM (CM.
Mxa
k4A 2 8
(2.38 )
npHMep 7. p'a1to: YCHJIHH, .n.eHcTBYIOIIJ.He Ha ypoBHe no.nOWBbI «I>YH-
.n.aMeHTa N= 1000 KH, M x =430 KH.M, R=250 Kna, pmax= 1,2R= 1,2X
X 250 = 300 Kna, pmfn = 0. Tpe6yeTcH onpe.n.eJIHTb pa3MepbI nO)J.OlIIBbl
H .n.aBJIeHHH Da OCHOBaHHe.
PaCCf,eT: e x =430/1000=O,43 M. 3a)J.aeMcH a=0,5 H =O,45. no
Ta6JI. 2.3 npH a=O,5 H 13=0,45 HaXO)J.HM k 1 =O,725 H k =O,0865. no
4>opMYJIaM (2.5) H (2.6) HMeeM:
A= O,725.0,43 =3,6M; B= 5.1000 ==255M
° 0865 3 . ° 72.5 . 250 . 3 6 '
, "
a = 0,5. 3,6 = 1,8 M; b = 0,45. 2,55 = 1,]5 u.
.nJIH KOHTpOJIH onpeJleJIHM pmax H pmln:
Pmax = 1000 + 430 _
° 725 . 3 6 . 2 55 ° 0865 · 3 6 2 · 2 5.5
, " , , ,
= 150,2 + 150,4 = 300,6 300 Kna;
Prnlo == 150,2 - 150,4 = - 0,2 o.
Ta6J1HQa 2.3
==0,3 = 0.35 a ==0,4
a I
k. k. k'a k. k. k& k. k. k'r,
I
I 0,053 1 0,053
0,3 0,51 0,545 0,061 0,055 ° 58 0,069 0,05
,
0,35 0,545 0,055 0,061 0,578 0,063 0,063 0,61 0,071 ° 06-
,
0,4 ° .58 0,058 0,069 0,61 0,065 0,071 0,64 0,073 0,07
,
0,45 0,615 0,061 0,078 0,643 0,068 0,0.79 0,67 0,076 0,081
0,5 ° 65 0,065 0,086 0,675 0,072 0,087 ° 7 0,079 0,08
, ,
0,55 0,685 0,069 0,094 0,708 0,076 0,095 0,73 0,083 0,09
0,6 0,72 0,075 0, 102 0,74 0,082 0, 103 0,76 0,088 0,104
0,65 0,755 0,082 0,110 0,772 0,088 0,110 0,79 0,094 0,112
0,7 0,79 0,09 0,118 0,805 0,096 0,119 ° ,82 I 0, 1 ° I 0,120
8
a-
3
9
7
22
n pOaOA eH,Ue Ta6A. 2.3
p-0,45
P=O,5
=0.55
I I
a. I ks
k l k4 ks k l k4 k l ke k.
0,3 0,615 0,078 0,061 0,65 0,086 0,065 0,685 0,094 0,070
0,35 0,643 0,079 0,068 0,675 0,087 0,072 0,708 0,095 0,076
0,4 o 67 0,081 0,076 0,7 0,089 0,079 0,73 0,097 0,083
,
{),45 0,698 0,083 0,083 0,725 0,091 0,087 0,753 0,099 0,090
° 5 0,725 0,087 0,091 0,75 0,094 0,094 0,775 0, 101 0,091
,
055 0,753 0,090 0,099 0, 77'5 0,097 0,101 0,798 0, 104 0, 104
,
0,6 0,78 0,095 0,106 0,8 0,101 0, 1 08 0,82 0, 108 0,111
0,65 0,808 0, 100 0,114 0,825 0,106 0,1'16 0,843 0,112 0,118
0,7 0,835 0, 107 0, 121 0,85 0,112 0, 123 0,865 0,117 0,125
;."' "'
P=0,6 P =0,65
k l k4 ks k l k4 I ks
I
i
IJ=0,7
a.
k l
k.
ks
0,3 0,72 0, 102 0,075 0,765 0,110 0,082 0,79 0,118 0,09
0,35 0,74 0, 103 0,082 0,773 0,111 0,088 0,805 0,119 0,096
0,4 ° 76 0, 104 0,088 0,79 0,112 0,094 0,82 0, 120 0,101
,
0,45 0,78 0, 106 0,095 0,808 0,114 0,100 0,835 0, 121 0, 107
0,5 0,8 0, 108 0,101 0,825 0,116 0, 106 0,85 0, 123 0,112
0,55 0,82 0,111 0, 1 08 0,843 0,118 0,112 0,865 0, 125 0,117
0,6 0,84 0,114 0,114 0,86 0,121 0,118 0,88 0, 128 0,123
0,65 0,86 0,118 0,121 0,878 0, 124 0, 124 0,895 0, 130 0, 128
0,7 0,88 0, 123 0, 128 0,895 0, 128 0, 130 0,91 0, 134 0, 134
no 4>opMYJIaM (2.38):
PI === 150,2 + 150,4 . 0,5 == 225,4 Kna; p! === 150,2-
-150,4 · 0,5== 75 Kna.
npHMep 8. LlJIH YCJIOBRH, npHBe.n.eHHblx B npHMepe 7, onpeJI.eJIHTb
p83MepbI no.u.OIlIBbI H .u.aBJIeHHH Ha OCHOBaHHe npH Pmax = 1 ,2R = 300 Kna,
H Pmfn=0,25 pmax=0,25.300=75 Kna. 113 - npHMepa 1 ex=0,43 M,
k 1 =0,725, k =0,0865. no <popMYJIaM (2.10) H (2.11) HMeeM:
A = 5 . 0,725 . 0,43 === 6 aiM. B = 4 . 1000 == 1 ,22 M.
3 .0,0865 ' , 3. 0,725. 250 . 6,01 '
a = aA == 0,5 . 6,01 == 3 M; b == B ===0,45 . 1 ,22 ===O 55 M.
LlJIR KOHTpOJIfI onpeJI.eJIHM pmax R pmln:
1000 430
Pmax === ° 725 .6 01 . 1 22 + ° 0865.6 01 2 · 1 22 -
, , ,
= 188, I + 112,8 = 309,9 300 KOa;
23
Pmln = 188,1 - 112,2 === 75,3 75 Kna.
Do q>opMYJJaM (2.38) Pl= 188,1 + 112,8.0,5=244,5 Kna; P2=
= 188,1 - 112,8.0,5= 131,7 KTIa.
2.6. $YH,D.aMeHTbI C TaBpoBoH H r-06pasHoH nO,ll.OmBOH
CHaqaJIa paccMoTpHM <PYH,naMeHTbI C TaBpoBoH nOAOW-
BOH. PellleHHe )(JIH TaKHX <PYH,naMeHToB HeMHoro YCJI01KHH-
eTCH, TaK KaK HeCHMMeTpHIIHOCTb no,nOlllBbI OTHOCHTeJIbHO
OCH y npHBOAHT K CMeI.l{eHHIO ee ll.eHTpa THLKeCTII B- HanpaB-
JIeHHH nOJIKH. 003TOMY npH paCqeTe TaKHX CPYH,naMeHTOB
CJIe,nyeT YQHThIB3Tb ,nB3 B03MO}l{HbIX BapHaHTa HanpaBJIe-
HH5I .n.eiiCTBHH MOMeHTa: nepBbIH, Kor,na MOMeHT AeiicTByeT
B HanpaBJIeHHH nOJIKH nO,lI,OllIBbI, H BTOpOH - B HanpaBJIe-
HHH pe6pa. Ka}K,nbIii H3 .9THX BapHaHTOB BKJIlOqaeT B ce6H
ABa CJIYIIaH, 3aBHCHInHe OT 3a,naHHoro Pm1n.
BbIBeAeM paCIIeTHbIe CPOPMYJIbI ,lI,JIH nepBoro BapHaHTa
npH Pmin=O (pRC. 2.5, 6). 3HaqeHH5I Pmax H Pmin onpe,ne-
JIHeM no <popMynaM:
Pmax ==
N + Ne x = 1 2R;
klAB k4A2B '
N
P mln = klAB
Nex
k 4 ' A2B
=0.
( 2 .39 )
Pa3HOCTb ,naBJIeHHH Pmax II Pm in
Ne x ( + ) == 1,2R.
A2B k4 k/
YMHOLKHB BhIpa>KeHHe (2.1) Ha Ilk',. H (2.39) Ha Ilk",
n&JIYtIHM
(2.40 )
N + Nex
k1k/ An k4k./ A2B
1 ,2R .
k/
(2.41 )
N Ne x
=0.
kll4AB k4k4' A 2 B
CYMMa BbIpa>KeHHH (2.41) II (2.42):
N ( + .-!- ) = I t 2R
klAB k 4 ' k4 k 4 ' ·
Pa3,neJIHB (2.43) Ha (2.40), nOJIYQHM
AI (k1e x ) = Ilk4,"
OTKY,lLa A=k 1 e X /k 4 '.
( 2.42)
( 2.43)
( 2.44 )
24
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. ,4 t. __
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N.2 eA e J./
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PUC. 2.5
113 (2.43) Nk/ (Ilk' + Ilk,,) = 1,2R k tAB, OTKYAa
B= IV(1 +k/lk,) . (2.45)
I ,'2klRA
AHa OrHqHO no yqeHM paCqeTH e OPMYRW ARB ApyrHx
CJIyqaeB,. KOTOpbIe npHBOJI.HM 6e3 BbIBOJI.OB. nepBbIH BapH-
aHT np" Pmin = O,25Pmax (CM. pHC. 2,5, 2) :
A === k1ex ( 2... + ..!- ) . (2.46 )
3 k 4 ' k 4 '
B == N k, + k/ (2.47)
0, 3k 1 RA 4k + k4,'
BTOPOH BapHaHT npH Pmln=O (CM. pHC. 2.5, a):
A=k 1 e x lk4,; (2.48)
25
B = N (1 + k4 Ik 4') (2.49)
1 .2k 1 RA
BTOpOH BapHaHT npH Pmln=O,25Pmax (pHC. 2.5,.8):
A = k1e X ( -' ) . (2.50)
3 k I k ,.,
L .1:
B== N kl+k/ (2.51)
o .3k 1 RA k-J, + 4k 4 '
K03<P<PHl.\HeHTbI, BXO,lUIll.\He B paCqeTHble <POPMYJIbI, paB-
HbI:
kl==l-a(l- ); (2.52)
k 2 = (I - + 1X3 + (l - (l )[!1-x - 0,5 ( 1 - a)]2 +
+a (1-p.x-O.5 )2; (2.53)
ka == [1 - a (1 - 3) ] 112; ( 2.54 )
k4 == k"f P.x; (2.55)
k 4' k 2 / ( 1 - I-t x ) ; ( 2.56 )
k5 = 2ka; (2.57)
ka' = k5/ ; (2.58)
I-tx = 0,.5 [(1 - a)2 + a (2 - a)] (2.59)
k 1
r.ne a == a/ A, == b / B .
3HaqeHHH nepeqHCJIeHHbIX K03<P<PHl.\HeHTOB cBeAeHbI B
Ta6JI. 2.4.
,QaBJIeHHH Ha OCHOBaHHe B npOMe>KYT01JHblX xapaKTepHhlX TOllKaX
no.n.OIllBbl onpe,lI.eJIHIOT no <popMYJIaM:
,lI.JIH nepBoro BapHaHTa npH pmln = 0 (CM. pHC. 2.5. 6)
PI = Pmaxa..,
npH Pmln=O,25pmax (CM. pHC. 2.5. z)
Pt = O,25pmax (1 + 3a);
,lI.JIH BToporo BapHaHTa npH Pmln=O (CM. pRC. 2.5, a)
Pt -=== Pmax (1 - Cl ) ,
npH Pmln = O.25Pmax (CM. pHC. 2.5, B)
PI = O,25Pmax (4 - 3a),
26
(2.60 )
(2.61)
( 2.62)
(2.63)
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29
Ha plIC. 2.5 COBMeIII.eHbI 3nlOpbI JI.aBJIeHHH JI.JI5I JI.BYX Ba.
plIaHToB - CJIyqalI Pmin=O II Pmin=O,25pmax.
npHMep 9. P'aHo: YCHJIIUI, .QeHcTBYIOllI.He Ha ypoBHe nO)J.OlllBbI <!JYH-
.naMeHTa N=lOOO KH, NI x =600 KH'M, R=200 Kna. MOMeHT .neMcT-
ByeT B HanpaBJIeHHH nOJIKH no.n.OIlIBbI, T. e. no nepBoMY BapHaHTY. Tpe-
6yeTcH onpe.n.eJIHTb pa3Mepbl no.n.omB H .n.aBJIeHHH Ha OCHOBaHHe npH
pmax= 1,2R= 1,2.200=240 I<na H Pmin=O.
Pacq,eT: ex=600/1000=O,6 M. 3a.n.aeMcH a= =0,5. no Ta6JI. 2.4
npH'a= =0,5 Haxo.n.HM k t =0,75, k 4 =0,1375, k'4=O,0982 H J,tx=0,4167.
no <popMYJIaM (2.44) H (2.45): .
A = k 1 e X 0,15 . 0,6 == 4,58 M; B -= N(I+k/ /k. l ) ===
k 4 ' 0,0986 I ,?k 1 RA
=. 1000 (1 + 0,0982/0,1375) == 2,08 M; a == 0,5 . 4,58 = 2 29 M.
I ,2 · 0, 75 . 200 . 4, 58 '
b == O,S · 2,08 == 1,04 M.
no $opMYJIe (2.60) pt=pmax a=240.0,5=120 Kna. ,l{JIH KOHTpOJIH
onpe.n.eJIHM Pmax H Pmtn:
1000 600
Pmax = 0,75. 4,58 .2,08 + 0,1375. 4,582. 2,08 -
== 140+ 100===240 I<Oa;
== 140 - 600 140 140 °
Pmln 0,0982 . 4,.58 2 .2,08 == - =.
npHMep 10. .D.JIH YC,,'10BHH, npHBe.n.eHHbIX B npHMepe 9, 3a HCKJIIO-
qeHHeM M = 360 n:H. M, onpe.n.eJIHTb pa3MepbI no.n.OIlIBbI H .n.aBJIeHHH Ha
OCHOBaHHe npH Pmfn=0,25 Pmax=0,25.240=60 KTIa. 113 npHMepa 9:
kt=0,75, k4=0,1375, k'4=0,0982 H a= =0,5.
PaCtleT: e x =360/1000=0,36 M.- no $OPMYJl8M (2.46) H (2.47):
A = k1ex ( + ) = 0,7"). 0,36 ( 4 . + 1 _ ) ===
3 . k 1 kt . 3 0 , 0932 0, r 37.) i
=== 4,32 M;
B = N k l + kt' _ 1000 X
o , 3k 1 R A 41t; + k I' 0, 3 . 0, 75 . 200 . 4, 32
X O,137 +O,Oq8 =-= 1,87 M.
4 . 0, 1 37rj + 0, 0982
a == 0,5 · 4,32 = 2,16 M; b === O,. . 1,87 === 0,935 M.
J:{JIH KOHTpOJIH onpe.n.eJIHM Pmax H pmln:
1000 360
Pmax = 0,75. 4,32 . I,R7 + 0, 1375 . 4,32 2 · I ,87 -
= 165 -t- 75 == 240 Kna;
30
360
Pmln= 165 - === 165 - 105 = 60 Kna.
0,0982 · 4,32 2 · I, 87
no 4>opMy.:le (2.61) Pl=O,25pmax (1+3a)=0,25.240 (1+3.0,5)==
= 150 Kna.
npHMep 11.l1.aHo: YCHJIHH, .n.eHcTBYIOllI.He Ha ypoBHe no.n.OWBbI <l>YH-
JJ,aMeHTa N= 1200 KH, M x =600 KH.M, R=300 Kna, pmax= 1,2.300-
=360 Kna, Pmin=O. MOMeHT .n.eHcTByeT B HanpaBJIeHHH pe6pa, T. e. no.
BTOPOMY BapHaHTY. Tpe6yeTcH onpe.n.eJIHTb pa3MepbI no.n.owBbI H .n.aB-
neHHH Ha OCHOBaHHe.
PaCtf,eT: ex= 600/1200=0,5 M. 3a,naeTcH a= =0,6. no Ta6JI. 2.4
npH a= =0,6 HaXOD-HM k 1 =0,76; k =O,1456 H k' ,=0,1127. no <l>oP-
MYJlaM (2.48) H (2.49):
A === k 1 e X /k 4 == 0,76 · 0,5/0,1456 === 2,61 M;
B === N (1 + k4 / k/ ) == 1200 (1 + 0, 1456 /0, 1 127) == 3 85 M;
1,2k 1 RA 1,2 .0,76 . 300 .2,61 '
a=O,6. 2,61 === 1,57 M; b===0,6. 3,85==2,31 M
,U.JIH KOHTpOJIH onpe.n.eJIHM pmax H pmtn:
1200 ... + 600
Pmax == -
0, 76 . 2,61 . 3, 8a 0, 1127 . 2, 61 2 · 3, 85
= 157,1 + 203 360 KOa.
Pmln= 157,1- 600 == 157,1-157 1 =0.
o 1456 . 2 6 I 2 . 3 8.5. ,
, , ,
no cpopMyJIe (2.62) Pl=pmax (I - a) =360(1 - 0,6) = 144 Kna.
npHMep 12. ,aJIH' YCJIOBRH, npHBe.neHHblX B npHMepe 11, onpe.ne-
JlHTb pa3MepbI no.n.OIlIBbl H )J.aBJIeHHH Ha OCHOB8HHe npH Pmtn = 0,25 pmax ==
=0,25.360=90 Kna. 113 npHMepa 11 k 1 =O,76, k =0,1456, k'4=0,1127,
e4\:=0,5 M H a= p=O,6. no 4>OPMYJI8M (2.50) H (2.51):
A=_ klex ( ...!..+ ) == O,76,'0,5 ( 4_ + 1 ) =
3 k4 k4, 3 0, 1406 0 , 1127
=4,6 M; B=== N k4+ k 4' _ 1200 X
° ,3k 1 RA k4'+ 4k/ 0,3. 0,76. 300 · 4,6
X 0,1456 + 0,1127 == 1 65 M
0, 1456 + 4 . 0, 1127 '
a=O,6.4,6=2,76 M; b===0,6.1,65=O,99 M.
,llnH KOHTpOJlH onpe.neJIHM pmax H pmln:
1200 600
Pmax === 0 76 . 4 6 . 1 65 + 0 1197 . 4 6 2 . 1 65 -
, " ,-, ,
=== 208 + 152,5 360 Kna;
31
Pmln == 208 - 600 == 208 -- 118 == 90 Kna.
0, I 456 · 4, 6 2 . I, 65
no 4>opMYJIe 2.63 PI =0,25pmax. (4 - 3a) =0,25.360(4 - 3.0,6) =
198 Kna.
PaCq,eT (jJYHAaMeHTOB C f-06pa3HOH nOAOlllBOH (CM.
pHC. B Ta6JIHIl.e) npOH3BOJU1TCSI aHa.HOfIfqHO paCqeTY <PYH-
J(aMeHTOB C TaBpOBOH nOLLOIliBOH C HCnOJIb30BaHHeM Tex }l{e
$OPMYJI H Ta6JI. 2.4 co CJIeLLYIOIIJ.HMH oco6eHHocTSIMH. Pa-
Hee npHBe,neHHbIe cpOPMY JIbI (2.52), (2.53), (2.55)', (2.56),
(2.59) coxpaHSIIOTcSI 6e3 H3MeHeHHH. <I>OPMYJIbI (2.54),
(2.57) H (2.58) npHHHMalOT COOTBeTCTBeHHO BH,lJ.:
ks = (1 - )3 + a : + (l - ) [!Joy - 0,5 (1 - ))2 +
12
+ a (1 - fLy - O,5 )2;
ka == k3 1 f1y; fla' == k3 1 (1 - fLy),
rJ(e y == 0,5 [(1 - )2 + C( (2 - )] I kt;
C(===aIA, ==bIB.
3HaqeHHSI kt, k 4 , k' 4, lx no Ta6JI. 2.4 onpe,neJlSIIOT B 3aBH-
CHMOCTH OT a H , a 3HaQeHHSI k 5 , k'5 H J.,ty HaXOJI.HT ITO TOM
me Ta6J1Hu.e C 3aMeHoH a, , k 5 , k'5 H ly COOTBeTCTBeHHO
Ha , a, k 4 , k'q If J.,tx. HanpHMep, npH a=O,5 II =O,4 ITO
Ta6JI. 2.4 HaXO,lUIM k 1 =O,7, k 4 =O,128, k'4=0,082 H f.1x=
=0,3929. 3HaqeHHH k 5 , k'5 If fly HaXOJI.HM npH a=O,4 H
13=0,5; k 5 "=O,1369, k'5=O,1012 H fJ,y=O,425.
npHMepoB paCIIeTa 3JI.eCb He npHBOJ(HM, TaK KaK OHR He
OTJIHqalOTCH OT npHMepo paCqeTa TaBpOBbIX CPYHJ(aMeHTOB.
2.7. cl>YHAaMeHTbI C KOJlbQeBOH H KpyrJloH nOAowBoii
,lLJIH TaKHx CPYHJI.aMeHTOB pa3MepbI no,nOIliBbI onpe,neJISI-
lOT no cpopMYJIe (2.14) J(JISI cJIyqaSl Pmin=O (pRC. 2.6, a)
H no 4>opMYJIe (2.17) AJISI CJIYlIaSl Pmin=O,25Pmax (pRC.
2.6, 6) npH .9TOM:
k 1 == 3,1416 (1 - C(2);
k 2 == k4 == 0,785 (1 - ( 4 ),
fJ(e C( == RbI R H .
3HaQeHRSI K03cpqrHIl.HeHTOB kt, k2=k4
Ta6JI. 2.5.
32
(2.64)
(2.65)
npHBeJI.eHbI B
a) R., R., 6) R 1 R f
R 2 R 2 'R 2 R 2
- -
Iii
I I
I
x
x
R f R 1
Rz R 2
PRC. 2.8
Ta6JJHQa 2.5
a
k. I 0.65
1 O. 0.3 0.35 0.4 0,45 0.5 0.55 0,6 0,7
k l=k,
2, 191 2,011 1,814
-
0,714 0,684 0,645
i
1,602
0,591
let
3,142 2,8592,757 2,639 2,5052,356
0,785 0,1830,7740,765 0,7530,736
i
aBneHHH Ha OCHOBaHHe B npOMe}KYTOqH X XapaKTep.
HbIX TOqKaX no.nOlliBbI BhlqHCJISIeM no 4>opMYJIaM:
,nJIH CJIyqaH Pmln=O (CM. pHC. 2.6, a) :
Pt . O,5Pmax(1-a.); P2=O.5Pmax(1 + ex); (2.66)
AJIH cnyqaSl Pmln=O,25Pmax (CM. pRC. 2.6, 6) :
Pt = O,5Pmax (1 - ex) + O,5pmln (1 + a);
P2=O,5Pmax(1+a.)+O,5Pmln(1 -ex) (2.67)
33
,1J.JISI KpyrJIblX YH.naMeHTOB npHHHMalOT a=6.
npHMep t 3. J],aHo: YCHJIHH, .neHcTBYIOllI.He Ha ypoBHe nOD-OllJBhI 4>YH-
,naMeHTa N=2600 KH, M x =1950 KH.M, R=320 l<Da, Pmin=O. Tpet>y-
eTCH onpe.n.eJIHTb pa3MepbI KOJIbu.eBOH nO)lOlliBbI H )laBJ1eHHH H3 OCHO-
BaHHe.
Pac eT: e x = 1950/2600=0,75 M. 3a.n.aeMcH a=0,5. Do Ta6J1. 2.5 npH
a=0,5 Haxo.n.HM k1=2,3563 H k =0,7363. no tPopMYJIaM (2.14) H (2.15)
HMeeM:
RH===kleX/k =2,3563. 0,75/0,7363=2,4 M;
Pmax === 2N 1 (k 1 R H 2) == 2 · 2600/ (2,3563 · 2,4 2 ) = 383,2 <
< 1 ,2R = 1,2 · 320 == 384 Kna;
Rb == aR H == 0,5 . 2,4 == 1,2 M.
.lI.JIH KOHTpOJ1$1 onpe.n.eJIHM Pmax H pmln:
Pmax=26001 (2,3563. 2,4 2 ) + 1950/ (0,7363 · 2,43)==
= 191,6 + 191,6==383,2_Kna;
Prnin = 191,6 - 191,6 == 0.
lI.aBJ1eHHH H npOMe>KYTOQHbIX xapaKTepHhlX TOtlK3X no,nOllJBbI BY-
1)IICJ1HeM no 4>opMYJIaM (2.66):
Pl == 0,5Pmax (1 - ':l) == 0,5 . 383,2 (I - 0,5) = 95,8 Kna;
P2= 0,5pmax( 1 + 0:) == 0,5 .383,2 (1 + 0,5) = 287,4 Kna.
npHMep 14. J],aHo: YCHJIHH, .n.eHcTBYIOIUHe Ha ypoBHe nO,lJ.OWBhI 4»ya-
.n.aMeHTa N=4000 KH, M x =3000 KH.M, R=320 Kna. Tpe6yeTcJI on-
pe.n.eJlHTb pa3Mepbl KOJIbu.eBou no,nOWBbI H .naBJIeHHH Ha OCHOBaHHe
npH pmin = 0,25pmax.
PaCtteT: e x =3000/4000=O,75 M. 3a.n.aeMcH a=0,7. no Ta6JI. 2.8
npH a=0,7 Haxo.n.HM k1=l,602 H k =0,5968. no tPopMYJ1aM (2.17):
R == 5k 1 e x == 5 . 1,602 . 0,75 = 3,36 M.
H 3k 4 3 . 0,5968 '
Prnax === 1,6N _ 1, . 4000 = 353,8 < 1 ,2R ===
k l RH 2 1,602 · 3,36 2
=== 1,2 · 320 = 384 Kna;
Rb = aRM =:= 0,7 · 3,36 === 2,35 M.
,QJIH KOHTpOJIB onpe.n.eJIHM Pmax H pmln:
Pmax=4000/(1,602. 3,36 2 ) +3000/(0,5968. 3,363) ==
== 221,1 + 132,5=== 353,6 353,B Kna.
34
Pmln === 221,1 - 132,5 == 88,6 0,25pmax == 0,25 . 353,8=
= 88,.5 Kna.
,UaBJIeHHH B npOMe>KYTOqHbIX xapaKTepHblX TOIIK8X I10AOUIBbl Haxo-
aHM no «popMYJIaM (2.67):
PI ==--: O,5pmax (I-a) +O,5Pmln ( 1 + a) 0,5 . 353,8 (1 - 0,7) +
+ 0,5 · 88,6 (1 + 0,7) === 53,1 + 75,3 == 128,4 Kna;
P2=== 0,5pmax( 1 +a) + 0,5pmln (I-a) 0,5 .353,8 (1 +0,7)+
+ Q,5 .88,5 (1 - 0,7) == 3JO,7 + 13,3 == 314 Kna.
2.8. 4>YH,lI.aMeHTbI C wecTHyroJlbHoH nOAowBOH
,QJ151 TaKHX <PYHAaMeHToB pa3Mep CTOpOHbI nO,QOllIBbI
onpe.neJIHIOT no cpopMYJIaM (2.14) ,l(JI5I CJIyqaH Pmln=O H
no q,opMYJIaM (2.17) ,l(JIH CJIYtIaH Pmln=O,25Pmax. npH
3TOM B03MO>KHbl .u.Ba BapHaHTa HanpaBJIeHHH .u.eHcTBluI MO-
MeHTa. nepBbIH BapHaHT, Kor.u.a MOMeHT .u.eHcTByeT B nJIO -
v
KOCTH, nepeceKalOIUeH .u.Be npOTHBOnOJIO>KHbIe BepllIHHbI -
no.u.OllIBbI (pHC. 2.7, a, 6), BTOpOH - Kor.u.a MOMeHT .u.eHcT-
v
ByeT B nJIOCKOCTH, nepeCeKalOIl{eH Cepe,l(IIHbI ,l(BYX npoTH-
BOnOJIO>KHhIX CTOpOH no,nOIliBbI (pIlC. 2.7, B, 2). 3HaqeHHH
HCnOJIb3yeMblx K03cp<}>HnHeHTOB kt=2,598; k 2 =k 3 =k,.=
=0,541 H k s =0,625.
,l{aBJIeHHH Ha OCHOBaHHe B npOMe:>KYTOtIHbIX xapaKTep-
HbIX TOtIKaX no,nOlliBbI (pHC. 2.7) onpe,neJIHIOT no cJ.>OPMY-
naM:
JI.JIH nepBoro BapHaHTa npH Pmin=O
PI === O,25Pmax; P2 -== O,75pmax; (2.68)
npM Pmln=O,25Pmax
Pt === O,4375pmax; P2 === O,8125pmax; (2.69)
J(JUI BToporo BapHaHTa npH Pmln=O
Pt === O,5pmax; (2.70)
np" Prnln === O,25pmax
Pi ==0,625pmax. (2.71)
..
npHMep 15. P'aHo: YCHnHSI, AeiicTByJOIltHe Ha ypoBHe nOAOWBhI 4»YH-
J(8MeHTa N==5200 KH, M x =3900 KH.M, R=270 Kna. Tpe6yeTcH onpe-
.n:eJIHTb )J..nHHY CTOpOHbI nO.D,OIDBbl 4>YH.n.aMeHTa H .n.aBJIeHHH Ha OCHO-
BaHne npH Jl.eHCTBHH MOMeHTa no nepBoMY aapHaHTY H pmln = O.
35
f::::)"
tJ
a)
Ie')
C\I
"
t:::
) -....
o t:
y
e 1
...
...
8)
t:::)
I
/I.,
x ::
c:::5'
C;j
"':>
..
..
" .
.
e f N 1 ....
--'0&11111111111 i
Pf e 2 . t i
t::
2) Jt:
#2
I
!77
!{
! 0.866 a
\I
\..P4
PIIC. 2.7
Pae'leT: ex = Mx/N =3900/5200=0,75 M. no cl>opMYJIaM (2.14) .
(2.15) C 3aMeHoH RH Ha a HMeeM:
a = k 1 e X / k4 = 2,598 · 0,75/0,541 = 3,6 M;
pmax= 2N / (k 1 a 2 ) = 2 · 5200/ (2,598 · 3,6 2 ) = 308,9 < 1,2R=
= 1,2 · 270 = 324 Kna.
.nJIH KOHTpOJIH onpeAeJftIM pm." B pmlo:
_ N /( k 2 ) Mx _ 5200 +
pmax - 1 a + k 4 a 3 - 2,598 . 3,62
+ 3900 = 154 4 + 154 5 = 308,9 Kna;
o 541 · 3 6 3 ' ,
, ,
Pmln = 154,4 - 154,5 O.
no cl>opMYJIaM (2.68) PI == O,25pmax == 0,25.308,9 = 77,2 KJIa; Pi-
O,75.308,9=a231,7 Klla.
DpHMep 16. ,nJIH YCJIOBHH npHMepa 15, 3a HCKJIlOqell'HeM M..-
=2600 KH.M H R= 180 Kna, onpe.neJlHTb pa3Mep cropOHY DOAOIDBbI
YHAaMeHTa H .naBJIeHHH Ha OCHOBaHHe npH pm In = O,25Pm.x.
36
PaCtleT: ex = 2600/5200 = 0,5. no 4>op naM (2.17) HMeeM:
a = 5k 1 e X == 5. 2,598 . 0,5 = 4 M; Pmax == 1,6N ===
3k 4 3 . 0 541 k 1 a 2
= 1,6 · 5200 = 200,2 Kna < 1 ,2 · 18) === 216 Kna.
2,598 . 4 2
.1J.JIH KOHTpOJIH onpejJ.eJIHM pmax H pmln:
Pmax == 5'2001 (2,598 · 4 2 ) + 2,600/ (0,541 · 43) ==
== 125,1 +75,1 ==200,2<216 Kna;
Pmln == 125, t - 75,1 == 50 0,25pmax == 0,25 · 200,2 Kna.
,naBJIeHHH B npOMe>KYTOqHbIX xapaKTepHbIX TOtIKaX no.n.OlllBbI BY-
qHCJIHIOT no <PopMY 1JaM (2.69): PI =0,4375.200,2= 87,6. Kfla, P2=
=0,8125.200,2= 162,7 Kna.
npHMep 17. ,nJIH YCJIOBHM npHMepa 15 npH .neHcTBHH MOMeHTa no
BTOPOMY BapHaHTY H R=350 Kna onpe,neJIHTb JI.JIHHY CTOpOHbI no,no-
llIBbI H jJ.aBJIeHHH Ha OCHOBaHHe.
Pacq,eT: no <popMYJIaM (2.14) H (2.15) C saMeHOH RH Ha a H k.
Ha k5 HMeeM:
a == k 1 e xl k5 === 2,598 · 0,75/0,625 === 3,12 M;
Pmax === 2N / (k 1 a 2 ) === 2 · 5200/(2,598 · 3,122) =
== 411,2 < 1 ,2R === 1,2 · 350 =-= 420 K.na.
,nJIH KOHTpOJIH onpe,neJIHM Pmax H pmln:
_ 5200 + 3QOO _ 205 6 +
pmax - 2 508 . 3 122 0 625 . 3 "12 3 - ,
,.J, , ,
+ 205,5 41t,2 Kna.
Pmln == 205,6 - 205,5 0. no QJopMYJIe (2.70)
P1 == 0,5pmax --= 0,5 . 411,2 = 205,6 Kna.
npHMep 18. /l.aHo: YCHJIHH, ,neiiCTBYIOJ.IJ.He Ha ypOBHe no,nOlllBbI4>Y8-
.n.3MeHTa N = 6000 KH, M x =3000 KH. M, R=260 Kna, Pmln=0 25pm.x,
Pmax= 1,2R= 1,2.260=312 Kna.
. Tpe6yeTcH onpe.n.eJIHTb .n.JIHHY CTOpOHbI no,nOWBbI cpYH.naMeHT8 R
.n.aBJIeHHH Ha OCHOBaHHe npH .n.eMcTBHH MOMeHTa no BTOPOMY B3pHaHTY.
PaCtteT: ex = 3000/6000 = 0,5 M. TIo 4>opMYJIaM (2.17) C 3aMeHoA
RH Ha a H k", Ha k5 HMeeM
Ek1e X 5 . 2,598 . 0,5 a 46
a===-=== ==, M,
3k 5 3 . 0,625
P _1,6N _ 1,6.6000 - a n S - 7 n <312 n
max - -- - \..,;, K aKa.
k 1 a 2 2, 598 . 3, 46 2
37
.D.JIH KOHTpOJlH onpeJJ.eJlHM Pmax H pmln:
6000 3000
pmax = +
2 598 · 3 46 2 0 625 · 3 46 3
, . , .
= 192,9 + 115,9 = 308,8 Kna;
Pmln == 192,9 - 115,9 === 77 Kna O,25pmax =0,25 · 308,8 =
== 77,2 Kna.
no <J>opMyne (2.71) p1=O.625pmax=O,62S.308,8= 193 Kna.
2.9. YHAaMeHTbI C BocbMHyroJlbHoH nOAowBoH
PaClIeT TaKHx <PYHAaMeHToB aHaJIOrHlIeH paCqeTY CPYH-
AaMeHTOB C mecTHyrOJIbHOH no,nollIBoH H OTJIHqaeTCSI TOJIb-
KO 3HalleHHHMH K03<P<t>HII.HeHTOB k i , k 2 , k 3 , kl" H ks, a TaK>Ke
AaBJIeHHHMH Ha OCHOBaHHe B rrpoMe*YTOqH X xapaKTep-
HbIX TOllKax no,nOllIBbI. ,QJIH 3THX c}>YHAaMeHToB TaK>Ke B03-
MO>KHbI ,nBa BapHaHTa HanpaBJIeHHH ,neHcTBHH MOMeHTa.
BTOpOH BapHaHT, KorAa nJIOCKOCTb AeHcTBHH MOMeHTa rre-
peceKaeT cepe.nHHbI ABYX npoTHBOnOJIO>KHbIX CTOpOH IIOAO-
UIBbI (pHC. 2.8, a, 6). nepBbIH BapHaHT, Kor,na nJIOCKOCTb
.neiicTBHH MOMeHTa rrepeceKaeT ,nBe IIpOTHBOnOJIO>KHbIe Bep-
WOHbI no,nowBbI (pHC. 2.8, 8, 2).
3HalleHHH HCIIoRb3yeMbIx K03<P<I>HUHeHTOB k i =4,8259,
k 2 =k 3 =1,8595, k,,=1,4232 H k 5 =1,5405. ,QaBJIeHHH Ha
OCHOBaHHe B npOMe:>KYTot.J:HblX xapaKTepHbIx TOllKaX rrOAo-
WBbI (pRC. 2.8) paBHbI.
nepBbIH BapHaHT - npH Pmin=O:
Pt = 0, 1454p1113x; P2 === 0,5 pmax; p =:=: O,8546pmax. (2.72)
npH Pm In == O,25pmax;
Pt ==0,3591pmax; P2 ==O,625pmax; P3 =O.8935pmax. (2.73)
BTOpOH BapHaHT - IIPH Pmln=O:
Pt ' O,2929pmax; P2 == O,707pmax; (2.74)
npH pmin == O,25pmax;
PI == O,4697pmax; P2 === O,7803pmax.
(2.75)
npHMep 19. l1.aHO: YCHJIHR, .n.eHCTBYIOIl1.He Ha ypOBHe nOJI.OWBbI 4»YH-
AaMeHTa N=5200 KH; M x =3900 KH.M, R=280 Kna, Pmin=O. MOMeHT
AeHcTByeT 110 nepnOMY BapHaHTY. Tpe6yeTcH onpe.n.e.miTb .n.JIH"y CTO.
pOHbI nOJI.OWBLI 11 lI.aBJ1eHHH Ha OCHOBaHHe.
38
#2.
x
a)
C"I
'>II )(
Cj
P2
:V 2 . N 1
t:j
....... :I
..
CC:)
«:)
c:::::r
I
t:1 .
1J,38a D.9266a 0,9266a 0,J8u.
6) 111 T t
f -..tl I liiiii l 'U I '
'C) . P., . 0::
cl! e 2 N 2 R a::
II 3
2) -s
.x
'>II )(
tJ
ti
P3
PRC. 2.8
PaC'leT: e x =3900j5200=O,75 M. no 4>opMYJIaM (2.14) H (2.15) c
3aMeHOH RH Ha a H:Mee t:
a == kle xl k4 == 4,8259 . 0,75/1,4232 == 2,54 M;
Pmax=2NI(k t a 2 ) ==2. 5200/(4,8259. 2,54 2 ) ==
=== 334 < 1 ,2R == 1 ,2 . 280 == 336 Kna.
.JJ.JlH KOHTpOJIH onpeJJ.eJIHM Pmax H Pmfn:
pmax === 5200/ (4;8259 . 2,54 2 ) + 39001 (1,4232 · 2,543) =
-= 167 + 167,2 334 Kna, pmln= 167 -167,2 0.
no 4>OPMYJI3M (2.72):
PI == 0, 1454Pmax == 0,-1454 . 334 === 48,6 Kna;
P2 == O,5pmax == 0,5 · 334 == 167 Kna, P3 == 0,8546pmax =
39
= 0,8546 . 334 == 285,4 KOa.
npHMep 20. ,LLJIH YCJIOBHH npHMepa 19, 3a HCKJIlOqeHHeM M¥=
== 2080 KH. M H R = 300 Kna, onpe.l1.eJIHTb AJIHHY CTOpOHbl nO.l1.0WBbI H
A8BJIeHHH Ha OCHOBaHHe npH Pmln=O,25pmax.
Pac 'leT: ex=2080/5200 = 0,4 M. no <l>opMYJIaM (2.17) C 3aMeHOH
R. Ha a HMeeM:
_ 5k L t x __ 5 . 4, 25J · 0,4 _ 2 26 .
a-- ---- - M
3k.! 3 .. I , 4232 "
I,GN _ 1,6.5200 ==337,5Kna < 1,2R==
pmax == k 1 a 2 4,8259 . 2J26
== 1,2 . 300 == 360 Kna.
,QJIH KOHTpOJIH onpe.u.eJII1M pmax H pmln:
5200 2080
pmax == + -
4 . 8259 . 2 26 2 1,4232 .2,26 3
== 211 +.126,6 337,5 Kna,
pmln = 211 - 126,6 == 84,4 Krla == 0,25 · 337,6 === 84,4 Kna.
no <popMYJIaM (2.73): Pl=O,3591.337,5= 121,2 KDa; p2=O,625X
X337,5=210,9 Klla; P3=O,8935.337,5=300,2 KDa.
npHMep 21. l1.aHo: YCHJIHH, .neHcTBYIOIIJ.He Ha ypoBHe nOjJ.OlllBbI <PYH-
A8ueHT8 N=4000 KH, M x =3000 KH.M, R=250 Kna. MOMeHT ,lJ.eHCT-
BYeT no BTOPOMY BapHaHTY. Tpe6yeTcH onpe.neJIHTb ,lJ.JIHHY CTOpOHbI
nOJ(OIllBbI H .u.8BJIeHHH Ha OCHOBaHHe npH pmln = 0.
PaCtteT: e =3000/4000=O,75 M. ITo cpopMYJIaM (2.14) H (2.15) C
saMeHOH RH Ha a H k, Ha k5 HMeeM:
a = k 1 e x l k5 == 4,8259 · 0, 75 /1,5405 == 2,35 M;
Pmax=2Nf(k t a 2 ) =2. 4000/(4,8259. 2,35 2 ) =
= 300,2 I ,2R == 1,2 · 250 == 300 KOa.
)l.JIH KOHTpOJlH OIIpe.n JIHM pmax H pmfn:
4000 3000
pmax = 4,8259.2,35 2 + 1,5405.2,35 3 -
150,1 + l50,1 == 3)0,2 Kna;
pmln == 150,1 - 150,1 == O.
no <popMYJIaM (2.74): pl=O,2929.300,2=87,9 Kna; p2=O,7071X
X300,2=212,3 Kna.
npHMep 22. p'aHo: YCHJIHH, .n.e cTBYIOw.He Ha ypoBHe no.u.OlllBhI <PYH-
.naMeHTa N=6000 KH, M x =3QOO KH.M, R=250 Kna, .Pmln=O,25P ax.
40
OMeHT eHcTByeT no BTOPOMY BapHaHTY. Tpe6yeTcH onpe.n.eJIHTb AnR-
HY CTOpOHbI no.n.OWBbl H .naBJleHHSI Ha OCHOBaHHe.
faC'leT: ex=3000/6000=O,5 M. no <j)OPMYJlBM (2.17) C 38MeHoA R.
nB a °H k, H3 k5 HMeeM:
a === 5kte xl (3k 5 ) == 5 . 4,8259 · 0,5 1(3 . 1,5405) = 2,61 M.
pmax === 1 ,6N I (k10,2) === 1,6 . 6000/ (4,8259 · 2,61 2 -) =
= 292 < t ,2R == 1,2 . 250 == 300 Kna.
.lI.J1S1 KOHTpOJISI onpe.neJIHM pmax H pmln:
6000 3000
pmax === +
4 ,8259 . 2,6 1 2 ] ,5405 . 2,61 3
= 182,5 + 109,5 == 292 Kna;
pmln === 182,5 - 109,5 == 73 == O,25pmax === 0,25 · 292 =
= 73 Kna.
no cpopMYJIaM (2.75): pl=O,4697Pmax=O,4697.292=137,2 'Kna;
P2 = Ot 780 3pmax = 0,7803.292 = 227 ,8 Kn .,
2. t O. cJ>YHAaMeHTbl C HenOJlHblM KOHTaKTOM
nO)l.OmBbi H OCHOBaHHH
npH 60JIbllIHX 3KCneHTpHCHTeT3X npOJI.OJIbHOH CHJIhI CO-
6JIIOJI.eHHe 33,lI.3HHbIX orpaHH eHHbix Kp3eBblX ,n3BJleHHH' Ha
OCHOB3HHe npJlBOJI.HT K 3H3qH eJIbHbIM p33Mep3M <l>YH;na...
v
MeHTOB B Hanp3BJIeHHH eHCTBHH MOMeHTa H HenOJIHQMY
HCnOJIb30B3HHlO paCqeTHOrO conpOTHBJIeHHH rpYHT O HO-
BaHHH.
. npeJI.JI3r3eM3H KOHCTPYKIJ.HH cpYHJI.3MeHT3 9TJIH aetCH
OT 06bIqHO npHMeHHeMbIX cpYHJI.3MeHTOB e¥, qTO. Ha. ee .r o-
.nOlliBe npe,lI.YCM3TpHBaIOT BbIrOJI.HO p3CnOJIO}KeHHbI TY-
nbI, 06eCneQHBalOllI.He q3CTHQHbIH KOHT3KT no,nOlliBbI C OC-
HOBaHHeM. KaK 6YJI.eT nOK33aHq JI.3JIe , TaKOH KOHCTPYK-
I.I.HH cpYHJI.3MeHT3 YCTP3HHIOTCH OTMeqeHHbIe He,nOCTarKH,
CBOHCTBeHHbIe CPYHJI.3MeHT3M co CnJIOllIHbiM 'onHp3HHeM 'rib...
JI.OllIBbI H, CJIeJI.OB3TeJIbHO, ,lI.OCTHraeTCH 33MeTH3S1 3KOHOMHH
MaTepH3JIOB. .
MO>KHO peKOMeH,lI.OB3Tb JI.B3 THn3 T3KJ{X. <PYH.,n3MeUTQB:
THn I C JI.BYMH BbIcTyn3MH (pRC. 2..9,.a) H. THn II C, Tp MH
BbIc ynaMH (pHC. 2.10, a). npe.lI.nOqTeHHe CJIeJI.yeT oT.ziaB Tb
<PyiI.n.aMeHT3M THn3 I. .nHllib B CJIyq3HX 3;H3 .TeJlhH9H p c-
qeTHOH JI.JIJlHbI 4>YHJI.3MeHT3, K9 r ,na. pelli lOm C _HOBSlT H
yCJiOBHH ero npOqHOCTH 60JIee 3KbHOMH HbIMH Mo'rYT .o.K:a-
33TbC'H <PYHJI.aMeHTbI THn3 II. <l>YHJI.3MeHT COOTBeTcTByiOfue-
41
Q.
-II:::
feU
; W) )(
tf
r:
t:i"
. ....,
N
- C::f
) II c::
'.....
rf
'"<
Cj
Q
Pac. 2.9
0)
I
)(
(:J
c{
c::
)(
(:J
c:t
PRC. 2.10
ro THna BbloHpalOT no pe3YJIbTaTaM TeXHHKO-3KOHOMHIIeCKO-
fO cpaBHeHHH.
TaK KaK qaCTH nO,lI.OlIlBbI Me}K,lI.Y BbIcTynaMH He YIIaCT-
BylOT B lIepe.aaqe ,lI.aBJIeHHH Ha OCHOBaHHe, TO B ueJIHX ,lI.O-
nOJIHHTenbHOH 3KOHOMHH MaTepHaJIOB YH,lI.aMeHT THna I
npHHHT .lI.BYTaBpoBoro ollepTaHHH B nJIaHe (CM. pHC. 2.9),
a <t>YH.lI.aMeHT TIfna I I - ollepTaHHH, IIp IfBe,lI.eHHorO H3
pnc. 2.10. npH 3TOM MHHHM3JIbHYlO IIIHPHHY pe6pa b p Ha-
3HaqalOT H3 YCJIOBIIH pa3MeIlleHHH Ha HeM nO,lI.KOJIOHHHKa,
a ero BbICOTY h p onpe.neJIHIOT paCqeTOM Ha npOIlHOCTb.
113 M3BeCTHOH ct>OPMY JIbI BHeueHTpeHHoro CLKaTHH CJIe-
AyeT, liTO Pmin O y.nOBJIeTBOpHeTCH lIP" YCJIOBHH
N/F K > Nexj'X l yK , (2.76)
r,Ae F K Ii W!lK - nJIOII.{3,Ab KOHTaKTa no.n.OWBbI C OCHOBaHHeM H ee MO-
MeHT conpOTHBJIeHHH OTHOCHTenbHO OCR y.
,QnH C06nlO,neHHH ycnoBHH (2.76) Heo6xo,lI.HMO COOTBeT-
CTBeHHO nOAo6paTb IIJIOrna,lI.b FK' T. e. pa3Mep a, tlTo6bI HC-
KnlO IITb B03HHKHOBeHHe pacTHHYTOH 30HbI B nnoc:KocTH
KOHTaKTa nO.ll.OUIBbI C OCHOBaHHeM, nOCKOJIhKY C YMeHhwe-
HHeM F H no cpaBHeHHIO co Bcefl IIJIOlU8JI.blO nO,l(OWBbI F B
42
3HaQHTeJIbHOH Mepe MeHbIlIe, qeM W YR no cpaBHeHHIO C W".
OnYCKaH npOMe:>KYTOqHbIe BbIKJIa,lI.KH, npHBe.n eM 3HaqeHHH
HCnOJIb3yeMbIx reOMeTpHqeCKHX xapaKTepHCTHK no,nOWB
060HX THnOB <PYH,lI.aMeHTOB.
,aJ1H <pYHJ18MeHTa THna I:
F K = k 1 A8; (2.77)
W yK =k 4 A2B, (2.78)
f,lle k 1 =2a, k4==a[-a 2 +3(1-a)2]J3, a=aJA. (2.79)
,LLJISI <}>YH,n8MeHT8 THna II «POPMYJIbl (2.77) H (2.78) co-
xpaHHIOTcSI, npH 3TOM
k 1 = 3a; k4 == a [a 2 + 2 (1 - Cl )2] /2. (2.80)
P a3MepbI IIO,ll.OllIBbI onpe.neJ1HIOT no <POPMY JIaM (2.5) H
(2.6) ,lI.JIH CJIYQaH Pmln=O, Pmax=-1,2R H no <}>OPMYJI8M
(2.10) Ii (2.11) ,lI.JIH cJIyqaSl Pmln=O,25Pmax=O,3R, Pmax=.
= 1 ,2R.
TaKHf\f o6pa30M, 3a,naBaHCb 3HaqeHHeM a, no YK33aH-
HbIM <p0pMYJIaM BbIqliCJIHIOT pa3MepbI IIO,nOIliBbI A H B. .
,QJIH ynpOw.eHHSI paCqeTOB npHBO,lI.HM 3HaqeHHH k i H k
B 3aIaHCHMOCTH OT a, CBe.neHHbIe B Ta6JI. 2.6 ,lI.JISI <PYH.na..
MeHT,OB THna I II B Ta6JI. 2.7 .lI.JISI <PYH,lI.3MeHTOB THna II..
Ta6J1Hna 2.6.
I, I
il I
a k. k4 :1 ex. I k. k4 Ct k. k.
II
I
0,15 0,3 0, 1095 0,22 0,44 0, 1374 0,29 0,58 0, 1543
0,16 0,32 0,1143 0,23 0,46 0, 1404 0,3 0,6 0, 156
0,17 0,34 0, 1188 0,24 0,48 0,1432 0,31 0,62 0, 1515
0,18 0,36 0, 123 0,25 0,5 0, 1458 0,32 0,64 0, 1589
0,19 0,38 0, 1267 0,26 0,52 0, 1482 0,33 0,66 0,1601
0,2 0,4 0, 1307 0,27 0,54 o , 1504 0,333 0,667 0, 1605
0,21 0,42 0, 1342 0,28 0,56 0, 1525
CI k. I k4 a k. k4 a k. k4
I
TafiJlHna 2.7
.0 1 0,3 0,0815 0, 14 0,42 0, ] 049 0, ]8 0,54 0, 1239
,
0, 11 o 33 0,0878 0,15 0,4.5 0, ] 101 0, ]9 0,57 0, ] 281
,
0,12 0,36 0,0938 0,16 0,48 0, J J 49 0,2 0,6 0, 132
0, 13 0,39 o , 0 }95 0,17 0,51 O,JI!:6 - - -
43
,lLaBJIeHHSI Ha OCHOBaHHe B npOMeJKYTOIIHblX TOIIKaX no-
,I10llIBbI (Ha rpaHSIX BblcTynOB) orrpe,neJIHIOT no <popMYJIaM:
,nJISI <PYH,naMeHTa THna I npH Pmln=O (CM. pHC. 2.9,.6):.
P1 === a.pmax; P2 == (1 - ex) pmax;
(2.81 )
npH Pmin=0,25Pmax (CM. pHC. 2.9,8):
Pt == O,25pmax (1 + 3cx); P2:::::= O,25pmax (4 - 3 ). (2.82)
,nJISI <PYH,naMeHTa THrra II npH Pmin=O (CM. pHC. 2.10,6).:
Pt === apmax; P't == 0.,5 (1 -- ex) pmax;. P3 == 0,5 (1 + a )Pmax;
P4 === (1 - a.) pmax; (2.83)
npH Pmin=O,25Pmax (CM. pHC. 2.10,8):'
Pt == O,25pmax (1 + 3a); P2 === O,25pmax (2,5 -- 1,5(1);
P3 == O,25pmax (2,5 + a); P4 == O,25pmax (4 -3ct). (2.84)
3'aMeTHM, liTO npOCTOTa paCqeTa n03BOJISleT 3JIeMeHTap-
HhlMH BblqHCJIeHHSlMM H C MMHHMaJIbHOH 3aTpaTOH BpeMeHH
BapbHpOBaTb pa3MepaMH rrO,nOlliBbI <PYH,naMeHTa H BbI6H-
paTb HaH60JIee npMeMJIeMbIe.
TaK ¥aK paCqeTOM npeAYCMOTpeHO o,nHOBpeMeHHoe pa-
seHCTBO KpaeBbIX ,naBJIeHHH Pmin H Pmax peKOMeH,nyeMbIM,
TQ B HeKOTopbIX CJIyqaHX nOJIyqeHHble pa3MepbI nO,lI.OllIBbI
MorYT OKa3aTbCSI. He COBceM npMeMJIeMbIMH no KOHCTPYK-
THBHbIM C006pa)KeHMSIM. OpH 3TOM CJIe,lI.yeT HeMHoro CKOp-
peKTHpOB2Tb nOJIyqeHHbIe pa3MepbI, qTo6bI OHM C He6oJIb-
W .M H36bITOIIHbIM 3anaCOM Ha,ne)KHOCTH, BCJIe,lI.CTBMe YBe-
JIHqeHMH .Pmln H YMeHbllIeHHSI Pmax, OTBeqaJIH Heo6xo,nHMbIM
TP.e6oa:aHHSIM. He3aBHCMMO OT 3Toro ,nonYCKaeTCSI nOJIYlleH-
HbIe pa3Mep'bI rro,nOlliBbI, B TOM qHCJIe pa3Mep a, oKpyrJIHTb
Ha :=1::5:°/0.
PaccMoTpeHHbIe. THnbI <PYH,naMeHToB MorYT BblnOJIHSlTb-
CH C60PH'bIMH H MOHOJIMTHbIMH. B 060HX CJIyqaSlX YCTyrrbI
ne,lI.OIIlBbI 6eToHMpYlOTCH COBMeCTHO C <!>YH,naMeHToM. BbI-
COTY BhicTynoB hb peKOMeH,lI.yeTcH npHHHMaTb 100-150 MM,
HO He MeHee BeJIHqHHbl O)KH,lI.aeMOH oca,l(KH <PYH,lI.aMeHTa,
YBeJIHqeHHOH Ha 50 MM. EJIS! Toro IITo6bI npH 06paTHOH 33-
CblnKe KOTJIOBaHa rpYHT He 3anOJIHHJI npOCTpaHCTBO M }I{,ny
B-bIcTyna H, ero CJIe,nyeT orpa,nHTb KHpnH'IOM HJIH .npyrHM
orpa>K,n :HHeM.
npHMep 23. P'IlHO: YCHJIHB, .neHcTBYJOll\He H8 ypoBHe nOAOWBH YH.
JJ.aMeHTa N=lOOO KH, M x =1200 KH-M, R=300 KOa 2 . TpeoyeTcB on-
44
peJ1eJIHTb pa3Mep nO.nOlllBbI ct>YH.naMeHTa THna I H .naBJIeHHH ua oc-
BOB8HHe npH Pmln=-P H pmax= 1,2R= 1,2.300=360 Kna.
PaC'leT: e x = 12OQ/I000= 1,2 M. 3a.naeMcH a=0,2. no Ta()JI. 2.6
Upll 8=0,2 H8XO.nHM 1==O,4 H k,=0,13. no $opMYJIaM (2.5) H (2.6):
\
A=0,4. 1,2/0,1-3=3,67 M; B=5. 1000/(3. 0,4X
X 300. 3,67) =3,78 M.
l1JIH K9HTpOJIH onpe.neJIHM pmax H pmln:
_ 1000 + 1200 _ 180 1 +
Pmax - 0 4 . 3 67 .3 78 0 1307.3 67 2 . 3 78 - ,
, , , , , ,
+ 180,3 === 360,4 360 Kna;
pmln == 188,1 - 180,3 = - 0,2 Kna 0.
,llaBJleHHe Ha OCHOBaHHe B npOMe>KYTOqHhIX TOtIKaX n0)10WBhI BbI-
QHCJISIIOT no 4>opMynaM (2.81):
Pt == 0,2 · 360 = 72 Kna; P2 == (1 - 0,2) 360 = 288 KrIa.
npHMep 24. ,QJIH YCJIOBHH npHMepa 23 onpe.neJIHTb pasMephi n0)10-
IDBbI II .naBJIeHHH Ha OCHOBaHlie npH Pmln = O,25pmax = 0,25 . 360 = 90 KIIa.
no 4>opMYJIaM (2.10) H (2.11):
A===5.0,4.1,2/(3-0,13)-===6,12 M;
B = 4 - 1000/ (3. 0,4 .6,12 · 300) === 1,82 M.
,l1JIH KOHTpOJUI onpe.u.eJlHM pmax H pmln:
1000 1200
prnax === 0 4 . 6 12. 1 82 + 0 1307 . 6 12 2 .1 82 -
, , ,
=224,7 + 134,7 =359,'4 360 Kna;
pmln === 224,7 - 134,7 == 90 Kna,
,QaBJIeHHH Ha OCHOBaHHe B npOMe1KYTOQHIiX TOllK8X UOAOWBH BY-
'IHCJIHIOT no 4>opMYJlaM (2.82):
Pt == 0,25 · 360 (1 +3 · 0,2) == 144 KfIa;
P2 = 0,25 - 360 (4 - 3 · 0,2) .:- 306 Kna.
npHMep 25. ,UJIH YCJIOBHH npHMepa 23 onpe.neJIHTb pasMepbl no-
,nOWBbI H .naBJleHHSI H8 OCHOBaHHe YHJ1aMeHTa THna II. 3aJ1aeMCH u==
==0,12. no Ta6JI. 2.7 npH a=0,12 HaXOJ1HM kl 0,36 H k 2 ==O,094. no
cpo-pMy aM (2.5).H (2.6):
il=O,36. 1,2/0,094=4,61 M; B=5- 1000/ (3. 0,36 X
X 4,61 · 300) == 3,35 M.
.l1nH KOHTpOJUI onpe.n'eJlHM pm.x R pmln:
45
1000 1200
pmax = 0,36.4,61 .3,35 + 0 0938 · 4,612.3 35 -
= 179,9 + 179,7 == 359,6 360 Kna;
pmln = 179,9 - 179,7 = 0,2 Kna O.
.lI.aBJIeHHH Ha OCHOB8HHe B npOMe)KYTOqHbIX TOqKaX nOAOWBhI BbI-
1JHCJIHIOT no opMYJIaM (2.83):
Pt = 0,2 · 360 === 72 Kna; P2 = 0,5 (1 - 0,12) 360 ===
== 158,4 Kna; Ps =0,5 (1 + 0,12) 360 == 201,6 Kna;
p, == (1 - 0.1 ) 360 == 316,8 KOa...
npHMep 26. ,D.JIH YCJIQBHH npHMepa 25 onpe.neJIHTb pa3Mepbl no.no-
IlIBbI H .naBJIeHHH Ha OCHOBaHHe npH pmfn = 0,25Pmax = 90 KOa. no op-
MYJI3M (2.10) H (2.11):
A === 5 · 0,36 . 1,2/ (3 . 0,0938) === 7,68 M;
B == 4. . 1000 / (3 . 0,60 · 30 . 7,68) === 1,61 M
.lI.JISI KOHTpOJIH onpe.neJIHM pmax H pmln:
1000 1200
P rnax == 0 36 . 7 68 . 1 61 + 0 0938. 7 68 2 . I 61" -
, , , , , ,
=== 224,7 + 134,7 ==359.,4 360 KOa;
pmln == 224,7 - 134,7 == 90 Kna.
.lI.3BJIeHHH Ha OCHOBaHHe B npOMe1KYT01JHbIX TOIlKaX no.nOWBbI BbI-
IjHCJIHIOT no <popMYJIaM (2.84):
Pt =0,25 . 360 (1 + 3 . 0,12) == 122,4 I<na; P2 == 0,25 X
X 360 (2,5 - 1,5 · 0,12) == 208,8 Kna; Pa == 0,25 X
X 360 (2,5 + 0,12) == 235,8 Kna; P4 == 0,25 . 360 (4-
- 3 . 0, 12) == 327,6 I{ n a .
OpH npHMoyrOJIbHOM CPYH)J.aMeHTe co CnJIOllIHbIM onHp3HHeM H3
OCHOBaHHe Tpe6yeMbIe pa3MepbI nO)J.OWBbI npH Pmfn = 0, A= 7,2 M H
npH Pmin=0,25pmax A= 12 M. npeBbIWeHHe COCT3BJIHeT no cpaBHeHHIO
c <PYH)J.3MeHToM THna I - 96,1, C THnOM I I - 56,2 0/ 0 .
fJlABA 3. YHAAMEHTbI, PA60TAIOUJ.HE
C OfPAHHIJEHHblMH KPAEBblMH AABnEHH»MH
HA OCHOBAHHE, nPH AERCTDHH. MOMEHTOB B ABYX
B3AltMHO nEPnEHJJ.HKynjlPHbIX HAnPABnEHHjlX
3.1. PaCqeTHbie cjJOPMYJlbI
KaK H ,lI.JIH <PYHJI.aMeHTOB, pa60TalOIll.HX C OrpaHHlleH-
HbIMH KpaeBbIMH .lI.aBJ1eHHHMH Ha OCHOBaHHe, npH AeHCTBHH
46
MOMeHTCl B O.ll.HOM HanpaBJIeHHH (CM. rJI. 2) JIH Bcex pac-
cMaTpHBaeMbIx q.YH,naMeHToB, JI.JIH KOTOpbIX reOMeTpHqe-
CKHe xapaKTepHcTRKH 3aBHCHT OT ,nBYX ee pa3MepOB A H B
(CM. T86JIHII.Y), nOJIyqeHbI 06Ill.He paCqeTHbIe 4>opMynbI,
BbIBO,nbI KOTOpbIX np'HBe,neHbI HH>Ke. I1CKJII01J:eHHe COCTaB-
JIHIOT <PYH,naMeHTbI C TaBpoBoH nOJI.OllIBOH, ,l(JIH KOTOpblX
paC1J:eTHbIe 4>OPMYJIbI H peKOMeH.lI.alI.HH no paCqeTY ,naHbI
,naJIee B n. 3.6.
,QJIH CJIyqaH Pmln'= 0 H Pmax = I,5R nOCJIe no,nCTaHOBKH
3Ha1J:eHHH F, W y If W X H3 Ta6JIHII.bI B 4>oPMYJIbI (1.4)
H (1.5):
pmax == N + Ne x + Ne y == 1,5R; (3.1 )
k l AB k!A 2 8 k5 A 8 2
Nex _ Ne y == o.
k4A 2 8 k5 A B2
CYMMa II pa3HOCTb KpaeBblx .naBJIeHHH (3.1) H (3.2)
paBHbI:
N
pmln == klA8
(3.2)
pmax === 2N / (kt AB ) = I ,5R; ( .3)
2N ( +i ) =1,5R. (3.4)
A8 k,A kalJ
113 4>OPMYJlhI (3.4)
eX/(k4A) + ey/(kaB) = 1,5RABI(2N). (3.5)
113 <POPMYJlbI (3.3)
1,5k 1 RAB==2N. (3.6)
nO,ncTaBIIB (3.6) B (3.5), nOJlYQHM
ex I (k 4 A) + e y / (ka B ) = 1 I kt. (3.7)
113 BbIpa>KeHHH (3.6)
B=2N/(I, 5k tRA). (3.8)
nO.ncTaBHB BbIpa>KeHHe (3.8) B (3.7) t nOJIYQHM
exl(k,A) + t,5ktRAey/(2kaN) = Ilkt. (3.9)
nOcJIe npHBe,lteHHH (3.9) K o6meMY SHaMeHaTeJlIO H He-
CJIO>KHblX npeo6pasoBaHHH HMeeM:
A2 _ 2ksN A+ 2ksNe x = 0
1 ,5k 1 2 Re y 1,5k 1 k.Re y ,
47
oTKy.n.a
A = 8 1 :1: VS 1 2 - 8 2 ,
(3.10)
rAe
8 - krfi · 8 - 2 Nex , _ 8 2exk1
1- 1 5k 2 ' 2- I k - 1
, 1 Re y ,5 lk 4 Ro y k4,
,QJIH CJIyqaH Pmax= 1,5R H Pmln, 0,25Pmax nOCJIe nOA-
CTaHOBKH 3HaqeHHH F, W y If W X H3 Ta6JIH bI B 4>OPMYJIhi
(1.4) H (1.6) nOJIyqHM:
._ N + Nex + Ney _ I 5 R .
Pmax - k k A2 k -, ,
:lAB' 4 B 5AB2
( 3. 11 )
mln = N _ Ne x _ Ne y = O,375R.
p k1AB k 2B k:AB2
TIo aHaJIOfHH co CJIyqaeM Pmln=O:
B === 2N / (1 ,875k t AR); P max 1 ,6N / (ktAB);
A = 8 3 + V8 3 2 04'
( 3.12)
(3.13 )
.( 3. 14 )
r.n.e
03 = O,6k 5 N / (1 , 75k12Rey); 04 = 2k 5 Ne x / (1 ,875k 1 k 4 Re y ) =
== 8 3 1 Oe Xk1 / (3k 4 ) .
H3 <POPMYJI (3.10) H (3.14) CJIe,nyeT, qTO pa3Mep A
nO,lI.OIIIBbI, a CJIe,lI.OBaTeJIbHO, H 3aBIfCHIII.HH OT Hero pa3-
Mep B HMelOT peaJIbHbIe 3HaqeHHH npH YCJIOBIfH, KOf.n.a npH
Pmin = 0
( ! k,;,)N ) 2 '-.. 2k5N ex
. ,.. , OTl<y.n.a
\ 1 ,;:,k 1 2 ]?e y 1,5k 1 k 4 Re y
R < k4k5N / (3k,3e X e y ).
AHaJIOfHqHO npH Pmin=0,25pmax
(3.15 )
R < 2,4k,k 5 N / (25k 1 3 e xey). (3.16)
Heco6JIIO,neHHe YCJIOBHH (3.15) HJIH (3.16) 03HallaeT,
u
liTO npH 3a,naHHbIX 3Kc eHTpHCHTeTax npo,nOJIbHOH CHJIbI ex
Hey, a TaK>Ke npH paCqeTHOM ,naBJIeHHH Ha OCHOBaHHe R
npHHHT e KpaeB e ,naBJIeHHH Pmax H Pm o,nHOgpeMeHHO
He y.n.OBJIeTBOpHIOTCH. B TaKHX CJIyqaHX CJIe,lI.yeT npHHHMaTb
npH Pmin=O:
R ---:- k(k 5 N / (3k 1 3 e x e y);
(3.17 )
48
A =kf>N/(1.5k t 2 Re y ) =81' (3.18)
a ,l1J1H cJlyqaH Pmln = 0,25Pmax:
R == 2,4k4k5N/(25k.3exey): (3.19)
A == O,6k a N f (1 ,875k 1 2 Re y ) = 03. (3.20)
npM 3TOM B CPOPMYJIbI (3.8), (3.13), (3.18) H (3.20) nOA-
CT3BJIHIOT 3H3IIeHMH R, nOJIyqeHHbIe COOTBeTCTBeHHO R3
(3.17) M (3.19). 113 CPOPMYJI (3.8), (3.10), (3.13) H (3.14)
CJIe yeT T3KLKe, qTO npM COOJIIO,neHMH YCJIOBMH (3.15) H
(3.16) p33MepbI no,nOIIIBbl A II B MMelOT no ,nBa 3HalleHHH,
H3 KOTOpblX npMHHMalOT npHeMJIeMbIe no KOHCTPYKTHBHhlM
coOOpa>KeHHHM.
,QJIH <PYH.n3MeHToB, reOMeTpllqeCKMe xapaKTepHcTHKH
KOTOpbIX 33BHCHT TOJIbKO OT o,nHoro paSMepa no,nOIUBhI (CM.
Ta6JIHII.Y), TaKLKe rrOJIyqeHbI 06InHe <POPMYJIbI ,nJlH onpe,ne-
JIeHIIH p33MepOB rrO OIIIBbI. HCKJIlOqeHHe COCTaBJlHIOT KOJlh.
lI.eBbIe M Kpyr JIbIe tPYHJI.aMeHTbI, JIH KOTOpbIX JIIOOaH ,nHa..
MeTpaJIbH3H OCb HBJIHeTCH OCblO CMMMeTpHH no,nOlliBhI. no-
3TOMY rrpM .neHcTBMH Ha T3KMe 4>YH.naMeHTbI MOMeHTOB M
II A1y B BYX B3aHMHO rreprreHJI.HKYJIHpHhlX HanpaBJleHRHX
OHM paCCIIHTbIBalOTCH no peKOMeH 3u.HHM, npHBe,neHHhlM B
n. 2.7, Ha p3BHO,llet'IcTBYIOlll.YIO 3TMX MOMeHTOB, paBHYIO
(3.21 ) :
M = VMx2 + My2.
(3.21 )
3.2. n pHMoyroJlbHble 4>YHAaMeHTbI
JIH rrpHMoyrOJIb bIX CPYH.naMeHToB nOJIyqeHbl OOIIJ.He
ct>OPMYJIbI C yqeTOM, lITO k 1 = 1 H k4=k5= 1/6. TaK AnSI
CJIyqaH Pmax l,5R .Pmin=O (pIlC. 3.1, a) :
01 == Nf(9Re y ); 8 2 == 12Ne x f(9Re y ) == 8t l2e x; (3.22)
B == 2N/(l,5RA), (3.23)
a YCJIOBMe (3.15) npHHMM3eT BH,n
R<N/(108e x e y ). (3.24)
,LLJIH CJIyqaH pmax 1,5R H Pmln=O,25Pmax (pRC. 3.1,6):
()3 === Nf(18,75Re y ); 8 4 = 20Ne x /(18,75Re y ) = 03 20e x; (3.25)
.B=2N/(18,75RA), (3.26)
49
Pac. 3. t
a YCJIOBlIe (3.16) rrpHHHMaeT BHA
R < N / (375 Xey). (3.27)
aBneHRH Ha OCHOBaHRe B ABYX ApyrHx yrnax no o-
WBhI nfl 060HX cJlyqaeB paBHbI (CM. pRC. 3.1) :
Pl=Nj(AB) +6M x l(A2B) -6M y f(AB2); (3.28)
P2 = N / (AB) -- 6Mxl (A2B) +6My/ (AB2). (3.29)
RpHMep 1. p'aHo: YCHJIHH, .n.eHcTBYlOlUHe Ha ypoBHe no.n.OWBbl «!>YH-
J(aMeHTa N=500 KH, M x = 100 KHM, M y =45 KHM, R=250 Kna. Tpe-
6yeTca onpeAeJIHTb pa3Mep nO OlliBbI H aBJIeHHH Ha OCHOBaHHe npH
P..x 1,5R= 1,5.250=375 Kna H Pmln=O.
Pactler: e x = 100/500=0,2 M; e y =45/500=0,09 M. nposepaeM yc-
JlGlllle (3.24):
500/ (108 .0.2 · 0,09 ) = 257,2> R = 250 Kna.
it
YCJIOBHe y.n.OBJ1eTBopHeTCH. no «t>OPMYJI8M (3.10) t (3.22) H (3.23}
HMeeM:
0 1 = N / (9Re y ) === 500/ (9. 250.,. 0,09) = 2,47; 8 2 7=: 128tez=
== 12 · 2,47 · 0,2 = 5,928.
A === 2,47 + V .,472 - 5,928 === 2,47 + 0,4; At = 2,87 M;
A 2 = 2,07 M;
8t === 2 . 500/ (1,5 ..250 · 2,87) === 0,93 M;
B 2 =-== 2 · 500 / ( 1 ,5 · 250 · 2,07) == 1,29 M.
I]pHHHMaeM A=2,07 M, B= 1,29 M.
ITo opMYJIe (3.3) pmax=2N I(K1AB) =2.500/(1.2,07.1,29) =-
= 374,5 Kna. ,llJIH KOHTpOJIH onpe.n.eJIHeM pmax H pmln:
500 6 . 100 6 . 45
Pmax=== + +
2 07 . 1 29 2 07 2 . 1 29 2 07 · 1 29 2
, ,
=== 187,5+ tC8,8 +78,7=375 374,5 1,5R=
=== 1,5 · 250 === 375 Kna;
pmln = 187,5 - lC8,8 - 78,7 === O.
no cjJopMYJ1aM (3.28) Ii (3.29) .n.aBJ1eHHH Ha OCHOBaHHe B ABYX
JlpyrHx yr JIax no.n.OWBbI:
Pt === 187,5 + 108,8 - 78,7 === 217,6 Kna;
P2=== 187,5 - 108,8 + 78,7 = 157,4 KOa.
npHMep 2. p'aHo: YCHJIHH, .n.eHCTBYIOI.IJ.He Ha ypOBHe nOJI.OWBbI 4»YH-
J1a leHTa N=1500 I{H, M x =225 I{H.M, M y =150 KH.M, R==250 Kna.
Tpe6yeTcH onpe.n.eJ1liTb pa3MepbI no.n.OlIIBbI H .n.aBJIeHHH H8 OCHosaHHe
npH Pmax= 1,5R= 1,5.250=375 I{na H Pmln=0,25Pmax=0,25.375==
=93,8 Kna.
Pacf/,er: e x =225/1500=0,15 M, e y = 150/1500=0,1 M.
npoBepHeM YCJ10BHe (4.33): 1500/(375.0,15.0,1) =268,2>250 Kna.
YCJ10BHe y.n.oBJ1eTBopHeTcH. no cl>opMYJIaM (3.13), (3.25) H (3.26):
03 = 1500/ (18,75 .2 50 · 0,1) - 3,2; 1°4 =3,2 . 20 · 0,15 =9,6,
A === 3,2 + }-' 3,2 2 - 9,6 = 3,2 + 0,8; At === 4 u;
.A 2 = 2,4 M;
B 1 == 2 · 1500 / (1,875 · 250 . 4) = 1,6 M;
B 2 = 2. 1500/(1,875.250.2,4) = 2,67 M.
I1pHHHMaeM A=2,4 M; B=2,67 M; no 4>opMYJIe (3.13)
Pmax == 1,6 · 1500/ (2,4 · 267) == 374,5 Kna.
51
AnSI KOHTpOJUI onpeJJ.eJIHeM Pmax H pmln:
1500 6,225 6 · 150
Pmax= + +
2 ,4 · 2,67 2 ,4 2 · 2,67 .2 ,4 . 267 2
== 243,2 + 87,8 + 52,6 = 375 Kna, Pmto = 234,2,-
-87,8 - 52,6 === 93,8 === 0,25 · 375 = 93,8 Kna.
no <popMynaM (3.28) H (3.29) .n.aaneHHH Ha OCHosaHHe s JJ.BYX
.l.pyrHx yr JI8X nO,AOWBbI:
Pt === 234,2 + 87,8 - 52,6 == 322 Kna; P2 = 234,2 - 87,8 +
+ 52,6 == 199 Kfla.
npHMep 3. l1.aHo: YCHJIHH, .n.eHcTBYIOIIJ.He Ha ypoBHe no.n.OWBbl «pya-
".MeHTa N=1000 KH, M x =250 KH.M, M y =150 KH.M, R=300 Kna.
Tpe6yeTcSI onpe.n.eJIHTb pa3MepbI no.n.OWBbI H JJ.aBJIeHHH Ha OCHOBaHHe
npH Pmax= 1,5R= 1,5.300=450 Kna H Pmln=O. .
Pac 'leT: e x =250/1000=0,25 M; e y = 150/1000=0,15 M.
npoBepHeM YCJIOBHe (3.24): 1000/(108.0,25.0,15) =246,9<R=
==300 Kna. YCJIOBHe (3.24) He YJJ.OBJIeTBOpHeTCH. B STOM CJIyqae no
fl>opMYJIe (3.22) A=lh = 1000/(9.246,9.0,15) =3 M; no <popMYJIe (3.8)
B=2.1000/(1,5.246,9.3) = 1,8 M.
AJIH KOHTponH onpe.neJIHeM Pmax H pmln:
= 1000 + 6. 250 + 6. 150 = 185,2 +
pm ax 3 . I 8 3 2 . 1 8 3 . 1 8 2
, , ,
+ 92,6 + 92,6 == 370,4 == 1,5 · 246,9 = 370,4 Kna;
Pml == 185,2 - 92,6 - 92,6 == o.
no opMYJIaM (3.28) H (3.29): PI = 185,2+92,6 - 92,6= 185,2 Kna;
P2 = 185,2 - 92,6 + 92,6 = 185,2 Kna.
3.3. .YHAaMeHTbI C cHMMeTpHqHOA ABYTaBpoBoA
.,
nOAOWBOH
,l{JIH TaKHX <PYH,lI.aMeHTOB pa3MepbI nO,lI.OIllBbI onpeAe-
JlDIOT no QJop"MYJIaM (3.8) H (3.10) C yqeTOM YCJIOBHH (3.15)
.nnH CJIyqaH Pmax= 1,5R H Pmln=O (pHC. 3.2, a) H no clJop-
MYJI3M (3.13) II (3.14) ,lI.JIR CJIYQ3H Pmln=O,25pmax=
==O,375R (pHC. 3.2, 6). Bxo,nHIUHe B 4JOPMYJIbI K09CPcl»HUH-
eHTbI kit k4 H k5 npHHHMalOT no TaOJI. 3.1 B 3aBHCHMOCTH
OT 3HaqeHHH a=a/A H =bIB.
,[LaBJIeHHH Ha OCHOB8HHe B npOMe>KYTOqHhlX xapaKTep-
HhIX TOqKaX no,nOWBhI ,nJIH OOOHX 3HaqeHHH Pmln=O H
Pmta=O,25Pmax=O,375R onpe,neJlHIOT no 4JopMynaM:
52
x
0)
-....,'.,
r
x
'\t)
Q.:. ,
.
PRC. 8.2
pt=l+.m-n; p2=l-m+n; P3=l+ma+n;
P4=1+ma+n ; P5=l+ma-n ; P6=l+ma-n; (3.30)
p, = 1 - ma + n; Ps = 1 - ma + n ;
P. = 1 - ma - n ; P10 == l - ma - n,
rAe a aqeH'UI I, m H n Te >Ke, qTO H paHee npuHBThIe.
DpHMep 4. J1.aHo, YCHJIHH, .neHcTBYIOllI.He Ha ypoBHe no.n.OWBhI 4>YHc
AaMeHTa N=1800 KH, M x =540 KH.M, M y =450 KH.M, R=300 Kna.
Tpe6yeTeB onpeJJ.eJIHTb pa3MepbI no.n.OllIBbI H .n.aBJIeHHB Ha OCHOBaHHe
DPH '.ax= 1,5R= 1,5.300=450 Kna H Pmln=O.
53
PaCtteT: ex=540/1800=O,3 M; ey=450/1800=O,25 M. 3aJJ.aeMCfl3H3-
QeHlU1MH a=O,6 H =O,4. TIo Ta6011. 2.1 np" a=O,6 H p=O,4 Haxo.n.HM
k t =0,64; k 4 =O,145 H k s =O,073. TIpoBepHeM YCJ10BHe (3.15): O,145X
xO,073.1800/(3.0,64 3 .O,30.0,25) =323,6>300 Kna. YCJIOBHe y.u.oBJIe-
TBOpfleTCfl:
01 == 0,073 · 1800/ ( 1,5 . 0,64 2 · 300 · 0,25) = 2,86;
B 2 2,86 (2 · 0,3 · 0,64/0,145) = 7,57.
no cj>opMynaM ( 3.3), (3.8) H (3.10):
A == 2,86 + 1/2,86 2 -7,57 == 2,86 + 0,77; A= 3,t3M;
,
A 2 = 2,09 M;
B 1 =2. 1800/(1,5. 0,64. 300. 3,(3) =3,44 M;
B 2 == 2 . 1800/ (1,5..0,64 · 300 · 2,09) == 5,98 M.
npHHHMaeM - A=3,63 M H B=3,44 M; a=O,6.3,63=2,18 M, b=O,4X
X3,44= 1,38 M;
prnax = 2 · 1800/ (0,64 · 3,63 · 3,44) = 450,5 Kfla 450 Kfla.
-aJIH KOHTpOJUI onpe.n.eJIfleM pmax H pmin:
1800 540
pmax == 0 64 . 3 63 . 3 · 44 + 0 1451. 3 63 2 . 3 44 +
" ",
450
+ == 225,2 +82,1 + 143,3 ==
o 0731 . 3 63. 3 44 2
, , ,
=450,6 450 Kfla; Prnln=225,2-82,1-143,3 0.
no 4JopMYJIaM (3.30) Haxo.n.HM .n.aBJIeHIUI Ha OCHOBaHHe B npoMe-
)KYTOt:lHbIX xapaKTepHbIx TOLJKaX no.n.OWBbi. 113 Bblt:lHCJIeHHH Pmax cne-
.llyeT:
I == 225,2 Kfla; m == 82, I KIla; n == 143,3 Kfla, Tor.na:
Pt == 225,2 +82,1 - 143,3 == 164 Kna; P2 == 225,2 - 82,1 +
+ 143,3==286,4 Kna; P3== 225,2+82,1 .0,6 +.1.43,3 ==
==417,8 Kfla; P4 === 225,2 + 82,1 .0,6 + 143,3. 0,4 = 33°1,8 Kfla;
p5==225,2+821 .0,6-143,3.0,4===217,2 I<na; p6=225,2+
+82,1 · 0,6- 143,3. 0,4 === 131,2 Kfla; P7=225,2 - 82,1 X
XO,6+143,3== 319,2 Kfla; p8=225,2-82,1.0,6+J
+ 143,6. 0,4 =-= 233,2 I<na; P9=== 225,2 - 82,1.0,6 - 143,3X
X 0,4 == 118,6 Kna; Pl0 === 225,2 - 82,1 . 0,6 -
- 143,3 = 32,6 Kfla.
54
npHMep 5. D.ano: YCHJIIUI, lI.eHcTBYlOlIUre Ha ypoBHe nOlI.OlIIBbI <!>YH-
.naMeHTa 1V=2000 KH, M x =400 KH.M, M y =300 KH.M, R=200 Kna.
TpeoyeTcH onpe.ll.eJIHTb pa3MepbI no.n.OIIIBbI H .n.aBJIeHHH Ha OCHOB8HHe
n-PH Pmax= 1,5R= 1,5.200=300 Kna H Pmlu=0,25pmax=0,25.300=75 Kna.
PaC'leT: ex=400/2000=0,2 M, e y =300/2000=0,15 M. 3a.n.aeMcR
3HaqeHHHMII a= =0,5. no Ta6JIHue npH a= 13=0,5 Haxo.n.HM k 1 =0,75,
k,=8 156, k 5 =0,094.
npOBepHeM YCJIOBHe (3.16): 2,4.0,156.0,094.2000/(25.0,75 3 .O,2X
XO,15) =203>200 KOa. OHO y.u.OBJIeTBopHeTCH.
no <)JopMYJIaM (3.14) H (3.13):
03 === 0,6 .0,094 · 2000/ (1,875 · 0,75 2 · 200 · 0,15) == 3,56;
&4 =2 · 0,094 · 2000 · 0,2/ (1 ,875 . 0,75 · 0 ,156 · 200 · 0,15) =
= 11,38; A == 3,56 + V3,56 2 - 11,38 == 3,56 + 1,14;
A t ==4,7M; A 2 =2,42 M; B t ==2.2000/(1,875.0,75X
X 4,7 . 2CO) == 3,03 M; B 2 === 2 · 2000/ (1,875 · 0,75 X
X 2,42 · 2CO) ==- 5,88 M.
npHHHMaeM A=4,7 M; B=3,03 M; a=O,5.4,7=2,35 M; b=O,5.3,03=
==1,515 M; Pmax=I.6.2000/(0,75.4,7.3,03) =299,8 Kna 300 KOa.
,[{nH KOHTpOJIH onpe.n.eJIHeM pmax H pmln:
2000 400
pmax = 0 75 . 4 7 · 3 03 + 0 156 . 4 7 2 . 3 03 +
, " , , ,
300 2 .
+ = 187,3 +38, + 74,1=
o ,094 . 4, 7 . 3,03 2
== 299,6 Kna 300 KOa;
pmln == 187,3 - 38,2 - 7,41 == 75 KOa.
no <popMYJIaM (3.30) HaXO.ll.HM AaBJIeHHH B npOMe>KYTOqHblX xa-
paKTepHbIx TOqKaX no.n.OIIIBbI.
113 BblQHCneHHH pmax Cne.lI.yeT: 1= 187,3 Kna; m=38,2 Kna; n=
==74,1 KOa, TOr.lI.a:
P1 === 187,3 + 38,2 - 74,1 = 151,4 Kna; P2= 187,3 - 38,2 +
+ 74,1 = 223,2 Kf1a; P3 === 187,3 + 38,2 · 0,5 + 74,1 =
= 280,5 Kna; P4 = 187,3 + 38,2 · 0,5 + 74,1 · 0,5 =
= 243,5 Kna; P5 == 187,3 + 38,2 . 0,5 - 74,1 · 0,5 =
= 169,3 Kna; P6 = 187,3 + 38,2 · 0,5 - 74,1 = 132,3 Kna;
P1= 187,3 - 3d,2. 0,5+74,1 = 242,3 Kna; Ps= 187,3-
- 38,2 · 0,5 + 74,1 .0,5 = 205,3 Kna; P9 = 187,3-
-,38,2 · 0,5 -74,1 · 0,5 == 131,1 Kna; Ptt = 187,3-
- 58,2 · 0,5 - 74,1 = 94,1 KOa.
55
Q.. 'q
,.
a)
)(
t::I
J
ce-
J 0,5A .I
I-
/ a
J
A f
I
I
17
I
I
y
PmL.n
o,25J
5)
_J "'! 'j l,lJ r.
"'I""'!-" J II
--1 1"" Ul , 1\' j II ,)" )II(
J ... "iJ tt
). lIV J "E
J.
c:t' lJ
J
j T / I,
. .
I ), , V t II l VII
C:ll' rl Q.... "'>
.: V II ," ..a7 - _ x
N I .J l
cu":n / , I - J
J ..1 J
'I t A , 'If! -
I / J 0... co
I<) I i I i 1
n: j ' If
, I
i ,1 , I
/ /
V I / iJ
=j J / a ./ -...-
a5A r / --,.. "
.,
"!r AI ./
I' I "
PHC. 3.3
3.4. YHAaMeHTbI C 3aMKHYToH JleHTOQHOH
npBMoyroJlbHoro OQepTaHHB B nJlaHe nOAomBoH
,QJlSI TaKHX $YH,naMeHToB pa3Mepbl no,nOWBIi oape,neJIJl-
lOT no <l>opMynaM (3.10) H (3.8) c yqeTOM ycnoBHH (3.15)
56
AJISI CJlYllaSl Pmax= 1,5R H Pmln=O (pHC. 3.3, a) H no <pop-
MYJlaM (3.14) H (3.13) ,nJISI CJIYllaSl Pmln=O,25Pmax=
=O,375R (pHC. 3.3, 6). Bxo,nSlIl.\He B paCqeTHble $OPMYJIbI
K03$<pH HeHTbI k i , k" H k5 npHHHMalOT no Ta6JI. 2.2 B 3aBH-
CHMOCTH OT 3HaqeHHH a=a/A H =b/B. ,LLaBJIeHHSI Ha
OCHOBaHHe B npOMe:>KYTOllHbIX xapaKTepHbIx TOllKaX no,no-
WBhI AJISI 060HX CJIYllaeB paBHbI (CM. pHC. 3.3) :
pt=l+m-n; P2===l-m+n; P3==l+m(l+n ; } (3.31)
P4=l-ma+n ; P5==l-ma-n ; P6==l+ma.. -n ;
rAe 3HaqeHHH I, m, n Te >Ke, qTO H paHee npHHHTbIe.
npHMep 6. p'aH,o: YCHJIHH, .n.eHcTBYIOlll.He Ha ypoBHe no.n.OIlIBbI 4>YH-
A8MeHTa N=2000 KH, M x = 1000 KH.M, M y =800 KH.M, R=3000 Kfla.
Tpe«5yeTcH onpe.n.eJIHTb pa3MepbI no.n.OlIIBbI H .n.aBJIeHHH Ha OCHOBaHHe
DPH Pmax= 1,5R= 1,5.300=450 Kna H Pmln=O.
PaC'l.eT: e x = 1000/2000=0,5 M; e y =800/2000=0,4 M. 3a.n.aeMcH
SH8QeHHBMH a=0,7 H (3=0,6. no Ta6JI. 2.2 npH a=0,7 H =O,6 Haxo-
AHM k 1 ==0,58; k 4 =O,132 H k s =0,142. npoBepHeM YCJIOBHe (3.l5):
0,132.0,142.2000 =320,3>R == 300 Kna.
3 . 0 58 3 . 0 5 . 0 4
, , ,
YCJlOBHe (3.15) y.n.OBJIeTBopHeTCB. n03TOMY
0t == 0,142 .2000 == 4,67;
I 5. 0 58 2 . 300 · 0 4
" ,
02 == 2 . 0, 142 · 2000.. 0, 5 == 20,47.
I 5. 0 58 . 0 132 · 300 . 0 4
, " ,
no cpopMYJIaM (3.3), ( 3.8) H (3.l0):
A == 4,67 + 11 4,67 2 - 20,47 == 4,67 + 1,16.
2 . 2000
A t =5,83M; A 2 ==3,61 M, Bt= -=2,63M
I 5 . 0 58 . 300 · 5 83
" ,
B 2 == 2 . 2000 = 4,25 M.
I ,5 . 0, 58 . 300. 3,61
npHHHMaeM A=3,61; B=4,25; a=O,7 .3,61 =2,53 M; b=0,6.4,25=
-=2,55 M; Pmax=2.2000/(0,58:3,61.4,25)=449,5 Kna 450 Kna.
,l{JIB KOHTpOJIH onpe.n.eJIHM pmax H pmln:
2000 1000
Pmax == 0 58 . 3 61 . 4 25 + 0 132 . 3 61 9 . 4 25 +
, , , , , ,
800
+ =225,1 +-137,9+87,3==450,3
o 142 . 3 61 . 4 25 2 I
, , ,
450 Kna, Pmln ===225,4 - 137,9 - 87,3 == O.
57
Do <popMYJIaM (3.31) onpe.neJIHM .naBJIeHHH Ha OCHOBaJUle B npo-
Me:mYT01JHbIX xa paKTepHbIx T01JKaX no.n01UBbI:
1 == 225,1 I<na; m == 137,9 Kna; n == 87,3 Kna;
Pt = 225,1 + 137,9 - 87,3 =-:= 275,6 Kna; P2 = 225,1 -
-137,9 +87,3= 174,7 Kna; P3=225,1 + 137,9. 0,7-
-87,3.0,6 ="= 374 Kna; P4 == 225,1 - 137,9. 0 7 +87,3 X
X 0,6== 181 KOa; P5 == 225,1 -137,9 .0,7 -87,3 · 0,6 =
=76,2Kna; P6==225,1+137,9. 0,7-87,3. 0,6==269,2 KOa.
npHMep 7. p'aHo: YClIJ1HH, lleHcTBYIOllJ.He Ha ypo">He OOJlOlllBbI tYH-
.naMeHTa lV=3100 KH, Mx=M y = 1240 KH.M, R= 150 I{Da. Tpe(5yeTcJI
OnpeAeJIHTb pa3Mepw nOJlOlllBH H aBJIeHHH Ha OCHOBaHHe np" pmax=
= 1,5R= 1,5.150=225 KDa II Pmin=0,25pmax 0,25.225=56,3 Kna.
PaC'LeT: ex=e y = 1240/3100=0,4 M. 3aJlaeTCH a=0,7 H (3=0,6. no
Ta6JI. 2.2 npn a= 0,7 II (3=0,6 HaXO.llHM k I =0,58; kt.=O,122; k 5 =O,142.
npOBepHeM YCJIOBHe (3.16):
') 4 . (: l - q -) . ( ) 1 4 :) . 31 00
- , -', IJ-. , & 178 > 1 h: o n
== u KIa.
25 . 0 58 3 . 0 4 . 0 4
, , ,
Y CJIOBHe (3.16) Y.llOBJIeTBOpHeTcH. TorJla
o == 0,6 . 0,142 .'3100 = 6,96.
3 1 875. 0 58 2 . 150 . 0 4 '
" ,
" 2 .0,142 . .3]00 . 0,4 == 40 62.
0 4 =: 1, 875 . 0, 58 . 0, 132 . 0,4 . 150 '
no <popMYJIaM (3.13) H (3.14):
A == 6,96 + V6,96 2 - 40,62 == 6,96 + 2,81; At == 9,76 M;
A 2 == 4, 16 M; B t === 2 . 3100 / ( 1 ,875 · 0,58 · 150 · 9,76) ==
== 3,89 M;
B 2 = 2 · 3100 / ( I ,875 · 0,58 · 150 · 4, 16) = 9, 13 M.
DpHHHMaeM A=9,76 M; B=3,89 M; a=O,7.976=6,83 M; b=f),6X
x3,89=2,33 M;
pmax = 1,6 · 3100/ (0,58 .9 · 76.3,89) == 225,2 225 Kna.
.D.JIH KOHTpOollSi onpe eJIHM pmax H pmin:
3100 , + 1240 +
pmax ==
0,58 . 9,76 · 3,89 0,132 . 9,76 2 . 3,89
+ 1240 == 140,8 +25,3 + 59,3 == 225,4 KOa;
0, 142 . 9, 76 . 3,89 2
Pmln == 140,8 - 25,3 - 59,3 == 56,2 z 56,3 Kna;
1 == 140,8 KOa; m == 25,3 KOa; n == 59,3 KOa.
58
no 4>opMYJlaM (3.31):
Pt = 140,8 + 25,3 - 59,3 == 106,8 Kna; P2 === 140,8 - 25,3 +
+ 59,3 == 174,8 Kna; P3 == 140,8 + 25,3 . 0,7 +59,3 · 0,6 ==
= 194 Kna; P4== 140,8-25,3 · 0,7 + 59,3 . 0,6== 158,7 Kna;
P5=== 140,8 - 25,3 . 0,7 -59,3 · 0,7 - 59,3 · 0,6 == 87 Kna;
P6 == 140,8 + 25,3 -.0,7 - 59,3 · 0,6 == 122,9 Kna.
3.5. .YH,lI.aMeHTbI C KpecToBoH nO,ll.OWBOH
B 3aBHCHMOCTH OT COOTHOllleHHfI MOMeHTOB Mx H My,
a TaK}Ke OT pa3MepoB no,nOlliBbI Pmax MO>KeT 6bITb B TOq-
Kax 1 HJIH 2. npH 3TOM Pm In 6y,neT COOTBeTCTBeHHO B KOCO-
CHMMeTpHqHO pacnOJIO}KeHHbIX TOqKaX (pHC. 3.4).. KpOMe
Toro, npH ,neHCTBHH Mx ,nJIH Bcex npOMe}KYTOqHbIX TOqeK
nOAOlliBbI B paCqeTHbIe CPOPMYJIbI BMeCTO k4 no,nCTaBJIHIOT
k4/a. AHaJIOrHqHO npH ,neHCTBHH My - BMeCTO k5 nO CTaB-
JIHIOT k5/ n. B pe3YJIbTaTe OCHOBHbIe paCqeTHbIe CPOPMYJIbI,
npHBe.lleHHbIe B n. 3.1, npHHHMalOT CJIe YlOrnHH BH.lI.: npH
Pmax= 1,5R B TOqKe 1 H Pmln=O (CM. pHC. 3.4, a)
0- 1 === k5N / (1 ,5k12Rey ); 02 = 0l2e x k 1 / k . (3.32)
Pa3MepbI nO OIllBbI A H B, a TaK}Ke Pmax onpe,neJIHIOT
COOTBeTCTBeHHO no $opMYJIaM (3.10), (3.8) H (3.3), a KPH-
TepHH (3.15) npHHHMaeT BHA
R == 0,6k 3 N I (1 ,875k13eXey ).
(3.33)
3HaqeHHH K03<P<PHUHeHToB kt, kl" H k5 npHHHMalOT no
Ta6JI. 3.3.
npH pmax= 1,5 B TOqKe 1 H Pmin=0,25Pmax (pHC. 3.4,6):
03 == 0,6k a N / (1 ,875k12Rey ); 04 == 03 (10e x k t /3k 4 ). (3.34)
Pa3MepbI no,nOlI1B A H B, a TaKiKe pmax onpe,neJIHIOT COOT-
B TCTBeHHO no cpopMYJIaM (3.13) H (3.14), a KpHTepHH
(3.16) npHHHMaeT BHA
R < 2,4k4ka/ (25k13eXey ).
( 3.35 )
,QaBJIeHHH Ha OCHOBaHHe B npOMe>KYTOqHbIX xapaKTep-
HbIX TOQKaX nOAOlliBbI He3aBHCHMO OT 3HaqeHHH Pmin onpe-
AeJIHIOT no cpopMYJIaM (CM. pllC. 3.4; a) :
.59
a)
c.... ...,
Q..C\4
)(,
ti
77
/ I
/
i
)C
5)
C\I ")(
. t:I
J
c:l
1
Pt = 1 + ma-n ; P2 == l +mo: + n; P3= l- mo: +n
P4==l-mo:+n ; P5==l-m+n ; P6=l-ma-n :
P7=l-ma-n; ps=l+ma,-:-n; P9=l+ma-n :(3.36)
t
PtO == l + m - n ,
r.ne
l=N/(ktAB), m==Mx/(k4A2B), n=My/{k5A ).
npH Pmax=1,5R B TOtIKe 2 H Pmln=O (CM. pHC. 3.4,B)
60
cr
I)
-:-
)(
-.
;
x
2) !I
11
......
R I ,r,
.
x
PRC. 3.4
eJ
8 1 =kaN/(1,5kt2Rey); O 2 =8. (2exkta/k4). (3.37)
Pa3MepbI no,nOllIBhI A H B, a TaK>Ke Pmax HaXO,l(HT COOT-
BeTCTBeHHO no 4>opMynaM (3.10), (3.8) H (3.3), a KpHTe-
PHH (3.15) npHHIIMaeT BHJ(
R,k4kaNj(3k.:Jexey':J.). (3.38)
npH.Pmax=I,5R B TOqKe 2 H Pmin=O,25Pmax (CM. pHC.
3.4, 2) :
03 == O,6kf}N / (25, t3exe)'a); 04 =--== 03 e x k t a j (O,3k 4 ). (3.39)
Pa3MepbI nO,LlOlliBbI A If B, 3 T3K>Ke Pmax onpe,l\eJIHIOT
COOTBeTCTBeHHO no cpopMYJIaM (3.14) II (3.13)t a KpHTe-
PHH (3.16) rreHHHMaeT BHA
R < 2, 4k 4 k a N / (25k 1 3 e xeya). (3.40)
,[{aBJIeHHH Ha OCHOB3HHe B rrpOMe>KYTOtlHbIX xapaKTep-
u
HblX TOqKaX IIO)J.OIlIBbI He3aBHCHMO OT 3HaqeHHH Pmtn Bbl-
qHCJIHIOT no cpopMYJIaM (CM. pRC. 3.4, 8, 2) :
pt=l-'ma+n; P2==l-ma+n ; P3=l-m+n ;
P4=1-m-n ;/ P5=l-ma-n ; P6=1+ma-n; (3.41)
P7 = 1 + ma - n ; Ps === 1 +:m - n ;
P9 = 1 + m + n ; P10:= 1 - ma + n ,
r.n.e
1 =="N / (ktAB); m == M xl (k4A2B); n == My/ (kaAB2).
YCT3HOBHTb KpHTepHH, n03BOJIHIOInHH 3ap3Hee oIIpe,ne-
JlHTb, B KaKOH H3 TOlleK 1 HJIH 2 HaXO,LlHTCH Pmax, He rrpe)1-
CTaBJIHeTCH B03MO>KHbIM, TaK KaK OH 3aBHCHT OT HeH3BeCT-
HbIX HCKOMbIX pa3MepOB no,nOIliBbI H COOTHomeHHSI Mx H
My. n03TOMY CHaqaJI3 rrpOH3Bo,nHT p3CQeT B npe,nnoJlo}Ke-
v
HHH, 'ITO Pmax H3XO,nHTCSI B O,nHOM H3 YK333HHbIX TOlleK, If
eCJIH npHHSlToe npe,nnOJIO>KeHHe He no,nTBep,nHTCSI, TO pac-
'leT nOBTopSiIOT ,nJISI BTOpOH TOqKH. npe,nB3pHTeJIbHO MO}l(-
HO npH6JIH>KeHHO C'IHTaTb Pmax B TO'IKe 1 rrpH ex>e y H
a<p.
npHMep 8. p'ano: YCHJlHH, .n.eHcTBYIOJ..U.He Ha ypoBHe nOJ{OWBbI ct»YH-
.naMeHTa N=2000 KH, M x =600 KH.M; M y =200 KH.M, R=200 Kna.
Tpe6yeTcSI onpe.n.eJlHTb pa3MepbI no.n.OIlIBbI H J{aBJIeHHH Ha OCHOBaHHe
UpH pmax= 1,5R= 1,5.200=300 Kna H I Pm'n=0,25Pmax=0,25.300=
c::75 Kna.
PaCtteT: e x = 600/2000=0,3 M; e y =200/2000=O,l M. 3a.l(aeTCH
«=0,4, p=0,6. no Ta6.1J.3.3 npli a=0,4 H p=0,6 HaXOD.HM k l =0,76,
62
k =0,le4 H k s =0,088. TaK KaK ex>e y H a< , nOJIaraeM pmax II Teq-
Ke 1. npoBepHeM YCJIOBHe (3.35):
2,4 . 0,104 .0,088 . 2000 == 223,8 > R = 200 Kna.
25 - 0 76 3 . ° 3 . ° I - ° 6
, , , ,
YCJIOBHe (3.35) YAOBJIeTBOp5leTcH.
no opMYJlaM (3.34), (3.14) H (3.13) HMeeM:
03 == 0,6 .0,088 . 2000 =8,15;
I 875 . ° 76 2 . 200 . 0 I . 0 6
" "
o = 8, 15 . 10 · 0,3 . 0, 7 6 = 59 39.
4, 3 . 0 1 04 ' ,
,
A = 8,15 + VB, 15 2 -.59,39 == 8,15 + 2,64; At = 10,79 M;
A 2 == 5,51 M;
B - 2.2000 _ 13 .
1- - , M,
I ,875 · 0,76 · 200 · 10,79
B 2 __ 2 . 2000 == 2,55 M.
- 1,875. 0, 76 . 200 . 5,5 I
npHHHMaeM A=5,51 M; B=2,55 M; a=0,4.5,51 =2,2 M; b=O,6x
X 2,55= 1,53 M; Pmax= 1,6-2000/(0,76.5,51.2,55) =299,7 300 Kna.
,l{JlH KOHTpOJlH BblqHCJIHM Pmax H Pmln:
2000 + 600 -L ,
prnax =
0,76 - 5,51 · 2,55 0,104 . 5,51 2 . 2,55
+ 200.0,6 =187,3+743 + 379=
0,088 . 5,51 . 2,55 2 "
= 299,5 300 Kna;
prnln == 187,3 - 74,3 - 37,9 == 75, 1 75 Kna.
OnpeAenHeM aBJleHHH Ha OCHOBaHHe B ApyrHx xapaKTepHNX TOq-
Kax no,nOlliBbI. Tor.n.a 83 BbIQHCJIeHHH Pmax 8MeeM:
1= 187,3 KfIa; m=74,3 Kfla; n=37,9jO,6=63,2 Kna;
ma=74,3-0,4=29,7 KfIa; n =37,9 Kna;
P1= 187,3+ 29,7 +37,9=254,9 KfIa; P2= 187,3+29,7 +
+63,2=280,2 KfIa; P3= 187,3 - 29,7 +63,2=220,8 Kna;
p!j.= 187,3 - 29,7 +37,9= 195,5 KfIa; P5= 187,3 - 74,3+
+37,9= 150,9 KfIa; P6= 187,3 - 29,7 - 37,9= 119,7 Kna;
P7 " 187,3 - 29,7 - 63,2=94,4 KfIa; ps= 187,3+29,7-
-63,2=153,8 KfIa; P9=187,3+29,7-37,9=179,1 Kna
P10=187,3+74,3-37,9=223,7 Kna.
63
npHMep 9. p'ano: YCHJlHH, .n.eHcTBYIOllJ.He Ha ypoBHe no.n.OWBbI 4>YH-
A8MeHTa N=2100 KH, M x =M y =630 KH.M; R=240 Kna. Tpe6yeTcJI
oupe.neJIHTb pa3Mep . no.n.OWB H aBJIeHHH Ha OCHOBaHHe npH pmax=
-=-1,5R= 1,5.240=360 Kna Ii pmln=O B npe.nnOJIO>KeHHH, qTO pmax B TOq-
Ke 1.
PaC'leT: ex=e y =630/2100=O,3 M. 3a.naeMcH a=O,4, =0,7. no
T8()JI. 2.7 npH a=0,4 H =0.7 Haxo.nHM k 1 =O.82; k,=0,12; ks=0,101.
fipoBepSleM YCJIOBHe (3.33):
0,12 .0,101 . 2100 = 244,1 > 240 Kna.
3 . 0 82 3 . 0 3 . 0 3 . 0 7
, , , ,
Y CJIOBHe (3.33) y.noBJIeTBOpHeTcH. no 4>opMYJIaM (3.32) :
0 1 = 0, 101 . 2100 = 4, 173;
1 5. 0 82 2 . 240 . 0 3 . 0 7
" "
8 =4173 2 . 0 ,3.0,82 =17122.
2' 0 12 '
,
A = 4,173 + 11 4,173 2 -17,122 = 4,173 + 0,54;
At = 4,71 M; A 2 = 3,63 M;
B 1 = 2.2100 =3,02 M;
1 ,5 . 0,82 · 240 . 4,71
B 2 = 2 · 2100 = 3,92 M.
1 , 5 . 0, 82 · 240 . 3, 63
npHHHMaeM A=4,71 M; B=3,02 M; a:::;:O,4.4,71 = 1,88 M; b=O,7X
g3,02=2,ll M.
.nJISI KOHTpOJISI BbIqHCJIHM Pmax H pmfn:
2100 + . 630 +
prnax =
0, 82 . 4, 71 . 3, 02 0, 1 20 . 4, 712 · 3, 02
630 . 0,7
+ = 180 + 78,4 + 101,6 == 360 Kna;
0, 101 . 4, 71 · 3, 02 2
pmln = 180 - 78,4 - 101,6 = O.
Onpe.n.eJIHeM .naBJIeHHH H8 OCHOBaHHe B .npyrHx xapaKTepHHX TOq-
:Kax no.nOWBhI. 113 BbIqHCJIeHHH Pmax H Pmln HMeeM: 1== 180 Kna; m=
c:78,4 Kna; n= 101,6/0,7= 145,2 Kna; am=0,4.78,4=31,4 Kna; n=
-0,7 -145,2= 101,6 Klla.
no fj>opMynaM (3.36):
P1= 180+31,4+ 101,6=313 KIla; P2= 180+31,4+ 145,2=
=356,6 Kna; Pa=180-31,4+145,2=293,8 Kna; p,=
=180-31,4+101,6=250,2 KIla; P5=180-78,4+
+ 101,6=203,2 RITa; P6=.l80 - 31,4 - 101,6=47 Kna;
4
P7= 180 - 31,4 - 145,2=3,4 Kna; pa= 180+31,4-
- 145,2=66,2 Kna; P9= 180+31,4 - 101,6= 109,8 KIla;
P10= 180+ 78,4 - 101,6= 156,8 Kna.
npe.LlnOJI01KeHHe 0 TOM, qTO Pmax HaxO,lJ.HTCH B TOt:lKe 1, nO,lJ.TBep,nHJIOcb.
ECJIH .lI.JIH YCJIOBHH npHMepa 9 3aMeHHTb MeCT8MH MOMeHTbI Mx
H My, TO pmax OKa:>KeTCH B TOqKe 2, npH 3TOM p83MepbI nO,lJ.OWBbI A H
B TaK}Ke nOMeHHIOTCH MeCTaMH. B pe3YJIbTaTe nOJIYQHM A=3,02 M;
8=4,71 M (CM. npHMep 9).
B cJIyqae eCJIH rrpe.nnOJIaraeMaH TOqKa C Pm ax He no,a-
TBep,lJ.HTCSI H rrpH aTOM HeYBH3Ka OKa}KeTCH He3HaqHTeJIb-
HOM, J(onYCKaeTCH He YTOqHHTb paCqeT ,aJIH ,apyroH TOqKH.
He3HaqHTenbHOH HeYBH3KOH cne,ayeT CqMTaTb TaKYro, npH
KOTOpOH C06J1IO,lI.aIOTCH CJIe,ayIOInHe YCJIOBMH: Pmax<P
1,05Pmax; Pmin> p I,05pmln,
Pmin=O,25 max H Pmin>P - 10 Kna npH" Pmin=O.
lJacTHbfH CJIyqaH I)JYH,l(aMeHTa C KpecToBoii nO,l(owBoii.
B cJIYlIae Kor,aa Ha c}>YH.naMeH'f C KpecToBoH nOJI.OIl1BOH
,neifcTByroT ,l{Ba O,l{HHaKOBbIX MOMeHTa Mx=M y , pa3MepbI
nO,QOIlIBbI npHHHMaIOT A=B, a=b HJIH a= . DpH 3TOM
paCqeT $YH,l{aMeHTa Be,U.YT Ha paBHo,neMcTBYIO YIO MOMeH-
TOB Mx M My, paBHYIO M ' X = 1,4142M x , ,neMcTBYIO YIO B
nJIOCKOCTH OCM x' (CM. pHC. 3.4, a). ,l!JIH TaKHX $YH,naMeH-
TOB pa3MepbI nO,lI.OlIIBbI onpe.n.eJIHIOT no cpopMYJIe (3.14)
J]}IH CJIyqaH Plnin=O H no cpopMYJIe (3.17) ,U.JIH CJIyqaH
Pn1in=O,25Pmax, 3aMeHMB B 3THX cpopMYJIaX RH Ha A. MaK-
CHManbH e M MHHHManbH e KpaeBue ,naBneHHH Ha OCHO-
BaHRe B03HHKaIDT o HoBpeMeHHo B BYX TOqKaX CMMMeT-
PMQHO paCIIOJIOLKeHHbIx OTHOCnTeJIbHO OCM x' (CM. pRC.
3.4, a).
f1 pHBO.lI.HM 6e3 npOMe)KYTOQRbIX BbIKJIa,nOK reOMeTpMqe-
CKHe xapaKTepHCTHKH IIO.n.OIlIBbI:
F=k1A2; k 1 =a(2 - a);
lx'=/y'=k 2 A'; k2=a(1+a 2 -a 3 )/12;
(J. I + (12 - (13
W I W I == k 4 A3. k , ==-:
x Y , 4, 2426 1 + (1
rl1e a=aIA.
3HaQeHHH K03cpcpHlI.HeHTOB k i , k 2 H k, npMBe,l{eHbI B Ta6-
JIHlI.e. ,[taBJIeHHe Ha OCHOBaHHe E .n.pyrHx xapaKTepHbIX TOq-
Kax nO,lJ.olllBbI onpe.neJ151IOT no <POPMY JIa M (CM. pHC. 3.4, a)
npH Pmin=O:
(3.42}
(3.43)
(3.44 )
65
a ° 5 1 -a 5
Pt = Pmax ' P2 == , ilmax , P3 === 0,. Pmax,
l+a I+a
1 + 3a 1
P4 == 0,.5pmax I ' P5 == Pmax ;
+a I+a
IIPH Pmln = O,25pmax
2 - I + 4a 2,5 - 0, Sa
P1 == 0, tJpmax , P2 == 0,25pmax
I+a l+a
P3 == 0,625pmax,
025 2,5+5,5a 025 4+a
P 4 === , pmax , P5 = , , pmax ·
l+a I+a
(3.45 )
, 1
} (3.46)
I
J
Ta6J1HQa
II I I
I
k. k 2 k. I k l k 2 k. k l k 2
ex II a. ! a. k.
I
0,3 0,51 0,0270,038 0,450,6980,0420,081 0,6 0,84 0,0570,101
0,35 0,5780,0320,066 0,5 0,75 0,047 0,088 0,650,8780,0620,107
° , 4 0,64 0,037 0, 074 0,55 0, 798 0,052 0,095 ° , 7 0, 91 0,067 0, 111
npHMep 9a. aH,o: YCHJIHH, .n.eHcTBYIOll.{He Ha ypoBHe nO,llOlliBbI <PYH-
.llaMeHTa N=2000 KH, M'x= 1000 KH.M, R=300 Kna, Pmax= 1,2R=
= 1,2.300 = 360 Kna, pmin = O. Tpe6yeTcH onpe,lleJIHTb pa3MepbI no.n.o-
llIBbI H .n._aBJIeHHH Ha OCHOBaHHe.
Pac'ier: e'x= 1000/2000=0,5 M. 3a.n.aeMcH a=0,4. no Ta6JIHue npH
a=0,4 HaXO,llHM k 1 =0,64 Ii k 4 =0,074. no 4>opMYJIe (2.14):
A=(O,64.0,5)/O,074=4,34 M; a=O,4.4,34=1,74 M.
J1.JIH KOHTpOJIH onpe,lleJIHM pmax H pmin:
2000 + 1000 == 165.8 _L 165)8 =--==
Pmax == ° 64 . 4 34 2 ° 074 . 4 34 3 I
, , , ,
-= 331,6 < 360 I<na; pmtn == 165,8 - 165,8 = 0.
LI.aB.neHHH B .n.PyrHx xapaKTepHbIX TOqKaX no.n.OllIBbI. TIo <POPMY-
.naM (3.45):
o 4 1 - 0,4
PI = 331,6 ' === 94,7 Kna; P2 -;- 0,5 · 331,6 ==
I +0,4 1+0,4
== 71,1 Kna;
P3 == 0,5 . 331,6 == 165,8 Kna;
66
P4 === 0,5 · 331,6 1 + 3 . 0,4 == 260,5 Kna;
1 + 0,4
1
P5 331,6 == 236,9 Kna.
1 + 0,4
npHMep 96. p'aHo: YCHJIHH, .n.eHcTBYIOIIJ.He Ha ypoBHe no.n.OillBbl 4>YH-
.n.aMeHTa N=2500 KH, M'x=625 KH.M; R 400 Kna; pmax= 1,2R=
=400.1,2=480 Kna; Pmln=0,25pmax. Tpe6yeTcH onpe.n.enHTb pa3MepbI
no.n.OIllBbI Ii .n.aB.neHHH Ha OCHOBaHlie.
PaC'leT: e'x=62,5/2500=0,25 M. 3a,llaeMCH a=0,4. no Ta6n. 3.1
kl=O,64; k 2 =0,074. no cl>opMyne (2.17):
A = 5.0,64 0 .0,25 = 3,61 M; a = 0,4 · 3,61 = 1,44 M.
3.0,74-
.IJ.JIH KOHTpOJIH Bbl1JHCJIHM pmax H pmln:
Pm ax == 2500 + 625 === 299,8 +
° 64 . 3 61 2 0 074 . 3 61 3
" , ,
+ 180 === 479,8 480 I<na;
Pmln=299,8 - 180= 119,8 479,8 .0,25= 119,9 Kna.
.IJ.aBJIeHliH B ,llPyrHx xapaKTepHbIX T01JKaX no.n.OillBbI onpe.ll.eJIHIOT
o 4>opMYJIaM (3.46):
1+4.0,4
PI === 0,25 · 479,8 = 222,8 Kna,
I + 0,4
P2 = 0,25 · 479,8 · 2,5. - 0,5 .0,4 = 197,1 Kna;
1 + 0,4
P3 == 0,625 · 479,8 = 299,9 Kna;
02 - 4798 2,5+6,5.Q,4 4027 n .
P4 == , . , == , K a,
1 +0,4
P == 0,25 · 479,8 4 + 0,4 == 377 Kna.
;) I + 0,4
3.6. YHAaMeHTbI C TaBpoBoH H r -06pa3HoH nOAowBoii
CHaqaJIa paccMoTpHM <PYH,lI.aMeHTbI C TaBpoBoH nOAOW-
80U. PaCqeT TaKHX <PYH,lI.aMeHTOB HeMHoro YCJIO:>KHHeTCH B
CBH3H C HecHMMeTpHtlHOCTblO rrO,lI.OIlIBbI, KOTopaH rrpHBo,lI.HT
K ,lI.BYM 3HaqeHHHM MOMeHTOB corrpOTHBJIeHHH ee rrJIO aJI.H
OTHOCHTeJIbHO Kbop,lI.HHaTH x oceM x H y. KaK 6Y,lI.eT rro-
Ka3aHO ,lI.aJIee, orrpe,lI.eJIeHHe pa3MepoB rrO,lI.OIlIBbI TaBpoBbIX
67
YH aMeHTOB CBH3aHO C Heo6xo HMOCTbro perneHHH Ky6H-
qeCKOrO ypaBHeHHH.
B 3aBHCHMOCTH OT HanpaBJIeHHH eHCTBHH MOMeHTa Mx
H 3a,l1.aHHorO KpaeBoro aBJIeHHH Pmtn B03MO}KHbI ,l1.Ba Ba-
pHaHTa paCqeTa, Ka}K}1,bIH H3 KOTOpbIX CO,l1.ep}KHT no Ba
CJIyqaH. PaCCMOTpHM Ka}K bIH H3 HHX.
BapHaHT I. MOMeHT Mx .n.eHcTByeT B HanpaBJIeHHH nOJI-
KH nO OrnBbl.
CIlYtLau 1 Pmin=O H cllYtLau 2 Pmin=O,25Pmax. BbIBe-
,lJ,e!\1 06Ill.He CPOPMYJIbI ,l1.JIH 060HX YKa3aHHbIx CJIyqaeB, ,l1.JIH
4ero o603Ha4HM C= I,5R H D=O ,l1.JIH CJIyqaH 1 H D=
==O,25Pmax=0,375R JIH CJIyqaH 2.
3anHIlIeM 3HaqeHHH Pmax H Pmin:
_ N + Nex + Ney - c. ( 3 47 )
Pmax - k 1 AB k,A2B k5AB2 - , ·
_ N _ Ne x _ Ne y === D ( 3 48 )
Pmln - k1AB k/.4 2 B k 5 ' AB2 . ·
H3 Q>OPMYJIbI (3.47)
+!!£. +2 == -.£ AB. (3.49)
k I k4 A k5B N
113 CPOPMYJIbI (3.48)
_ ex e y = E- AR. (3.50)
k 1 k./ A k 5 ' B .N
,LLe"JIHM Bblpa}KeHHe (3.49) Ha k'", H (3.50) Ha k",:
1 + ex + e y = AR. (3.51)
k1k/ k4 k / A kfjk/ B N k/ '
1
klk4
ex
k..k/ A
e y ===.!2.. AR.
kfj' k4 B N k4
( 3.52)
CYMMHPYH (3.5 I) C (3.52), nOJIYtIHM
( + ) +e ( 1 _ 1 ) l.= AB ( £+ E... ) . (3.53)
k 1 k/ k.t y k5 k / k 5 'kd, B N k/ k.
CYMMHPYH npoH3Be eHHH (3.47) Ha Ilk'5 H (3.48) Ha
II k 5 , nOJIyqHM
-.!.... ( ...!.. + ) + e ( 1 _ I ) = A B ( S- + E-' ) . (3.54)
k i k 5 ' k5 x k4 k 5' k/ k5 A N k 5 ' k5
113 BbIpaLKeHHH (3.53) H (3.54):
68
e ( 1 - I ) I __ A B ( C + D ) 1 ( 1 + I ) (3
y - - -- - - - -- - -. .55)
k5k/ k 5 'k.j. B N k/ k4 k 1 k/ k,'
e ( 1 _ 1 . ) == AB ( + ) _ ( -J... ) .(3.56)
x k,k 5 ' k/ k5 A N k 5 ' k;) k 1 ,k;/ k5
YMHO>KHM Q>OPMYJIY (3.55) H3 (3.56):
( 1 I ) ( I I \ I (AB)2 ( C
e xl! y krfi/ - k 5 ' k/ k i k 5 ' - k/ kJ AB = N2 k/ +
+ )( k + )-[+( k:' +i)( k + )+ I (k:' +
+ J ( k ' + )] . + k: 2 ( k:' + i-) (k:' + J . (3.57)
YMHO>KHM cpOPMYJIY (3.57) H3 AB:
( + ) ( + E.. ) (AB)2 _ [( J... + _ __ ) ( + E- ) +
k,/ k, k./ k5 N2 k 1 ,k/ k4 k 5 ' k5
( I 1 \ ( CD )] (AB) 2 + I ( I 1 ) ( I
+ k/ + k5 ) k/ + k, N k 1 2 . . k/ + k, k 5 -; +
+ ) AB+eXey ( I _ I ) 2=0. (3.5S)
k5 kJl,./ k 5 'k 4
nO.ll.CT3BHB C -- l,5R, D=O,375R II 0603H31IHB:
4] 4 1 1 I )
k7 + k = CPt' k ----; -1-- - k === CP2' k + k7 = CP3;
'4 5 5 4 4
+ _ . I 1 _ . (O,375R)2 _.
k k ' - Cf'4, k-k' - k-'k .- Cf'5' N2 Cf'1 Cf'2 - 'lJJt' (3.59)
5 5 OJ 4 ;} 4,
O,375R ( ) . I 1J . 2 "1
N Cf'2Cf'3+Cf'tCf'4 .= 2' k 1 2 Cf'3Cf'4 =', 8' e xeyCf' " = '(4'
nOJIYlIHM .lI.JISI B3PH3HT3 I npH Pmax= 1,5R II Pmln=0,375R
(pHC. 3.5, 6) :
11'1 (AB)3 -1jJ2(AB)2.+'I' 3 AB +'I' =O. (3.60)
H3 CPOPMYJIbI (3.54)
A=
e .xC{>5
1 0 ,375
- en. - Rm.)AB
k "t' "A N "t' -
1
(3.61 )
,LLJISI B3PH3HT3 I npH pnlax= 1,5R H Pmln=O D=O, Tor-
.Aa (pHC. 3.5, a)
69
)(
tj
t' n
Q: --
{J
j' ;d'
d\jJ""\;
a)
x
a
A
6)
'ln =a25p max
7
Puc. 3.5
CPt =4fkt/; cp:.! =4/k 5 '.
(3.62)
Bce OCTaJIbHbIe <POPMYJIbI coxpaHHIOTcH, KaK H .n.JIH CJIYlI3H
Pmln=O,375R.
,l:LJIH B3PH3HT3 II, Kor.n.a MOl\1eHT Mx .n.efIcTByeT B Ha-
npaBJIeHHH pe6pa IIO.n.OIlIBbI (pHC. 3.6), COXp3HHIOTCH Te >Ke
Q>OPMYJIbI, liTO H ,lI,JIH BapH3HTa I, C 3aMeHOH B HHX MeCT3-
MH K03<p<PHlI.HeHTOB k 6e3 lIITpHXOB H3 cooTBeTcTBYIOIIJ.He
K03<p<pHUHeHTbI k C TeMH me HH,lI.eKC3MH co llITpHX3MH H
70
a)
c:t'
.Q..C\I
x
t:S
Q..E
L__1 a
_ _ ..-- / a
P",m - 0.25"1---f-_
5)
r::f:
PUC. 3.6
Hao6opOT. CJIe,lI,OBaTeJIbHO, JJ.JIH BapHaHTa I I npH Pmln=
=O,25Pmax (CM. pHC. 3.6, 6) : ..
cpt =4/k,/ + I/k 4 ; <P2=4/k 5 ' + I/k 5 ;
qJ5= II (k5' k 4 ) - 1/ (k 5 k 4 '). (3.63)
Bce OCTaJIbHbIe cpOPMYJIbI. (3.53) coxpaHHIOTcH 6e3 H3-
MeHeHHH.
,[lJIH BapHaHTa II npH Pmln=O (CM. pHC. 3.6, a):
fP1= 4J k 4'; cp2=4/k 5 '; cp3=1/(k 5 'k 4 ) -1/ (k 5 k 4 '). (3.64)
Bce OCTaJIbHbIe cpOPMYJIbI (3.59) TaK>Ke coxpaHHIOTcH 6e3
H3MeHeHHH.
71
T3KHM 06P330M, .lI.JIH Onpe,lleJIeHHH p33MepOB nO,llOlliBbI
CHaq3JI3 H3 yp3BHeHH5I (3.60) MeTO,lI.OM nOCJIe.lI.OB3TeJIbHbIX
npH6JIH}KeHHH BbIQHCJI5IIOT npOH3Be.lI.eHHe AB. 33TeM no
<}>opMYJIe (3.61) HaXO,lI.HT p33Mep A H .n.3JIee p33Mep B K3K
Q3CTHOe OT ,lleJIeHHH AB H3 A.
AH3JIOrHQHO nOCJIe BbIQHCJIeHHH npOH3Be,lI.eHHH AB MO)K-
HO onpe,lleJIHTb p33Mep B no CJIeLLYIOlll.ei1 cpopMYJIe, nOJIY-
QeHHOH H3 (3.53):
B == e ),CJ>5
0,375 1
N Rcpt AB - k) CPt
(3.65)
a 33TeM p33Mep
A =A.BIB. (3.66)
B K3IJeCTBe nepBoro npH6JIH)KeHHH 3HaIJeHH5I A H B onpe-
,lI.eJI5IIOT no <POPMYJI3M (3.10) H (3.8) ,llJIH CJIYQ35I Pmin=O
H COOTBeTCTBeHHO no <POPMYJI3M (3.14) H (3.13) ,llJIH CJIY-
q35I Pmln=O,25Pmax. BXO,lI.HmHe B p3CQeTHbIe CPOPMYJIbI
K03<P<P H II.HeHTbI kit k(u k41, k 5 , k5 1 H J1x npHHHM310T no
T36JI. 2.4. ,a3BJIeHHH B .lI.pyrHx x3p3KTepHbIx TOQK3X no,lI.O-
IIlBbI BbIQHCJIHIOT no <POPMYJI3M:
.lI.JIH B3PH3HT3 I (CM. pHC. 3.5):
pt=l+m-n; P2=l-m'A-n; P3=l-m'A-n', } (3e67)
P4=1- m' - n'; P5=1- m'A+n'; P6=l- m'A+n; I
r e l=N/(k 1 AB); m=Mx/(k4A2B); n=M y f(k:AB2)-;
m'=M x /(k 4 'AB2); n'=M y /(k 5 'AB2); 'A=(l-a- x)/ x;
.lI.JIH B3PH3HT3 II (CM. pHC. 3.6):
p1=I+m'A+n'; P2=1+m'A+n; P3=1- m+n; }
p4=1+m')., - n; P5=1+m"A - n; p6=1+m' _ n'. (3.68)
n pH a + f.1x> 1 K03<P<PliII.HeHT 'A - OTpHII.3TeJIbHbIH Ii ero
CJIeJ1.yeT nO.lI.CT3BJIHTb B <POp MYJIbI (3.67) Ii (3.68) co CBO-
HM 3H3KOM.
H3 pemeHHH MHOrOIJHCJIeHHbIX npHMepoB YCT3HOBJIeHO,
IITO npHHHToe, K3K YK333HO BbIIIle, nepBoe npH6JIH}KeHHOe
3H3IJeHHe npOH3Be,lleHHH AB BCerJ1.3 y.n.oBJIeTBOp5IeT yp3BHe-
HHIO (3.60) H, CJIe,lI.OB3TeJIbHO, ero MO)KHO CIJHT3Tb OKOHIJ3-
TeJIbHbIM. 06'bHCHHeTcH 3TO, nO-BH.lI.HMOMY, TeM, IJTO npH
33.n.3HHbIX yCHJIHHX, .n.eHcTBYIOll.{HX H3 <PYH.n.3MeHT, H npH
BbI6p3HHbIX rr3p3MeTp3X a H nJIOll.{3,lI.b rrO,lI.OIIlBbI HBJISI-
72
eTCH rrOCTOHHHOH, H3MeHHIOTCH TOJIbKO pa3MepbI A H B,
C TeM QT06bI COOTBeTCTBYIOIII.He MOMeHTbI corrpOTHBJIeHHH
rrJIOIII.a,L:LH rrO,lI,OlllBbI Y,L:LOBJIeTBOpHJIH OCHOBHbIM ypaBHeHHHM
paBHoBecHH (3.47) H (3.48). ,LLoKa3aTb 3TO rrpe,lI,rroJIo1Ke-
HHe B 06Il1.eM BH,Zle He npe,lI,CTaBJI5IeTC51 B03MO}KHbIM H3-3a
CJIO)KHOt'I 3aBHCHMOCTH l\!HO)l{eCTBa BeJIHqHH, yqaCTBYIOIII.HX
B YKa3aHHbIX ypaBHeHHHX. TI03TOMY ,lI,JIH COKpaIII.eHHH TPY-
AoeMKOCTH paCqeTa peKOMeH,lI,yeTcH, nOJIyqHB 3HaqeHHe
npOH3Be,lI,eHH5I AB, He rrpOBepHTb ero COOTBeTCTBHe ypaB-
HeHHIO (3.60), a onpeJI.eJIHTb pa3Mep A no cpopMYJIe (3.61)
M 3aTeM pa3Mep B H rrpoBepMTb, y.n.oBJIeTBOpflIOT JIH OHM
ypaBHeHHHM (3.47) M (3.48). JIMllib B cJIyqae Hey.n.oBJIeT-
BOpeHH5I YKa3aHHwM ypaBHeHMHM 3HaqeHMH A M B CJIeAyeT
YTOqH5ITb C rrOMOIII.blO ypaBHeHHH (3.60).
B npMBe.n.eHHbIx ,lI,aJIee rrpHMepax ,lI.JIH Har JIH,lI,HOCTM ,lI,3 I-I
paCqeT B nOJIHOM 06'beMe, BKJIlOqaH npoBepKY COOTBeTCTBH5i
npoH3Be,lI,eHHH AB ypaBHeHHIO (3.62).
<l>YH)J.aMeHTbl c r -o6pa3HoH nO)J.owBoH (CM. TaOJIHUY).
KaK M ,lI.JIfl cJIyqa5I jI.eHcTBH5I MOMeHTa B O,lI.HOM HarrpaBJIe-
HHM, paCqeT TaKHX <pYH.lI.al\feHTOB rrpH ,lI,eHCTBMH fvlOMeHTOB
B .n.BYX B3aHMHO neprrelIJI.HKY HPHbIX HarrpaBJIeHMHX aHaJIO-
rHqeH paCqeTY CPYH,lI,aMeHTOB C TaBpoBoH nO,L:LOIliBOH C yqe-
TOM oco6eHHOCTeii, H3JIO>KeHHbIX B IT. 2.5. TaK KaK pac-
C:MaTpMBaeMbIe Q>YH.n.aMeHTbI HecHMMeTpMqHbI OTHOCMTeJIbHO
oceM x H y, TO rrOMHMO .n.BYX BapHaHTOB HanpaBJIeHMH .n.eH-
CTBHH MOMeHTa Mx B03M01KHbI TaK1Ke ,lI.Ba BapHaHTa Ha-
npaBJIeHMH ,lI.eHCTBHH MOMeHTa My: Ia - MOMeHT My ,lI.eii-
cTByeT B HarrpaBJIeHHH rrOJIKM rro.z(oIliBbI; IIa - MOMeHT My
.n.eiicTByeT B HanpaBJIeHHM pe6pa rrO,lI.OIllBbI. n pH 3TOM CJIe-
.n.yeT MMeTb B BH,lI.y, qTO JI.JIH CJIyqa5I ,lI,eHCTBHH 1M!! Ha3Ba-
HMH «rrOJIKa» M «pe6po» rro.n.OlliBW 3aMeHeHW MeCT3MM rro
cpaBHeHMIO co CJIyqaeM ,lI.eHCTBMH Mx.
TaKHl\1 oopa30M, npM BapHaHTe Ia coxpaHHIOTCH Bce
CPOPMY JIbI M TaOJI. 2.4 ,lI.JIH TaBpoBoii rrO,lI.OIlIBbI C yqeTOl\1
BbIllleoroBopeHHbIx ocooeHHocTeH. npM BapMaHTe IIa ,lI,eH-
CTBHH MOMeHTa My CJIe.n.yeT ,lI,onOJIHMTeJIbHO BO Bcex <pop-
MYJIaX 3aMeHHTb MeCTaMH K03cpcpHUHeHTbI k5 H k'5.
OpMMepoB paCqeTa He npMBO,lI.MM, TaK KaK OHH He OT-
JIHqalOTCH OT paCqeTa <pYH.n.aMeHTOB C TaBpOBOH rrOJI.OllIBOH.
npHMep to. p'aHo: YCHJIHH, .u.eHCTBYIOllI.He.Ha ypoBHe nO,QOlIIBbI CPYH-
.u.aMeHTa N= 1500 KH, M x =450 KH.M, M y =300 KH'M, R=300 Kna.
TpeoyeTcH onpe.u.eJIHTb pa3MepbI nO)l.OIIIBbI H ,llaBJIeHHH Ha OCHOBaHHe
npH .n.eHcTBHH Mx no BapHaHTY I H npH pmax = 1 ,5R = 1,5.300 = 450 Kna
H pmln=O.
73
PaCtieT: ex=450/1500=O,3 M; e y =300/1500=0,2 M. 3a.ll.aeMCH a=
=0,6 H (3=0,4. no Ta6JI. 2.4 npH a=0,6 H (3=0,4 Haxo.n.HM k 1 =0,64;
1?4=0,129; k'4=0,082; k5=O,073; k'5=O,183; f,tx=0,388.
no <popMYJIaM (3.10) H (3.8) onpe.ll.e.nHeM B nepBoM npH6JIH>KeUHH
pa3MepbI no,nOWBbI A H B, JJ.JIB qero npe.n.BapHTeJIbHO npoBepHeM yc-
JIOBHe (3.15):
0,129 · 0,073 . 1500 === 299,8 300 Kna.
3 . 0 64 2 . 0 3 . 0 2
, , ,
YCJlOBHe (3.15) Y.lI.OBJIeTBOpSleTCH. TaK KaK )J.aB.neHHe no YCJIOBHIO
(3.15) paBHo R, TO BTopoe CJlaraeMOe B «popMYJIe (3.10) paBuo UYJIIO H:
A === 0 == 0,073 . 1500 === 2 97.
1 1 5 . 0 64 2 . 300 . 0 2 "
" ,
B == 2 . 1500 == 5,5.1 M;
1 5 . 0 64 . 300 . 2 97
" ,
AB == 2,97 · 3,51 == 10,41 M 2 ;
(AB)2 == 10,41 2 === 108,16 M 4 ; (AB)3.== 10,41 3 == 1124,86 M 6 .
no <popMYJIaM (3.59) Ii (3.62):
QJt=4/0,082=49,02; CP2= /0,183=21,88; CP3= 1/0,129+
+ 1/0,082=20; CP4= 1/0,073+ 1/0,183= 19,15; Q)5=
= 1/ (0,073. 0,082) + 1/ (0,183. 0,129) = 125,25;
'\PI = (0,375. 300) 249,02.21,86/1500 2 = 6,03; 1P2=
=0,375-300(21,88-20+49,02.19,15)/(0,64-1500) = 161,29;
1P3 = 20. 19,15/0,64 2 = 935,06; '\P4 = 0,3. 0,2 - 125,25 2 =941,25.
nO.ll.CTaBJIHeM nOJIyqeHHbIe 3HaqeHHH B ypaBHeHHe (3.60): 6,03 X
X 1124,86 - ]61,29.108,16+935,06.10,41 +941,25=0
6783 -17455+9725+941,25= +4 0.
TaKHM 0.6pa30M npliHHToe B nepBoM npH6JIH>KeHHM AB = 10,41 M Z
Y.lI.OBJIeTBopHeT ypaBHeHHIO (3.60). no <popMYJIe (3.61):
44 == 0,3 · 125,25 = 2,93 M;
19,15/0,04 - 0,375 . 300 . 21,88 . 10,41 /1500
B= 10,41/2,93=3,55 M; a=0,6.2,93= 1,76 M; b=
=0,4-3,55= 1,42 M.
.lI.JIH KOHTpOJIH onpeJJ.eJIHeM pmax H pmln:
_ 1500 4- 450
pmax - 0,64. 2,93 . 3,55 ' 0,129. 2,93 2 . 3,55 +
+ 300 == 225,3 + 114,5 + 111,1 ==
o 073 . 2 93 . 3 55 2
, , '.
== 450,9 450 Kna;
74
450 300
pm In == 220,3- +
o 082 · 2 93 2 . 3 55 0 183. 2 93 . 3 55 2
, , , , , ,
= 225,3-181-44,4 0.
OnpeAenHeM .n.aBneHHe Ha OCHOBaHHe B .n.pyrHx xapaKTepHblX TOq-
Kax nOAOlliBbl. 113 BblqHCJIeHHH pmax H pmfn, I = 225,3 KITa, m = 114,5 Kna;
m' = 18 KITa; n= 111,1 KITa; n' =44,4 KITa; A= (1 - 0,6 - 388) {O,388=
= 0,0323.
no 4>opMYJIaM (3.67):
pt=225,3+ 114,5 - 111,1=228,7 Kna; P2=225,3-
-114,5.0,0323-111,1=117,9 Kna; P3=225,3-114,5X
xO,0323-44,4===177,2 Kf1a; P4===225,3-181+44,4=
=88,7 KDa; p5=225,3 - 114,5.0,0323+44,4=266 Kna;
P6 -=-- 225,3 - 114,5.0,0323+ 111,1 =332,6 Kna.
npHMep 11. J1.alio: YCHJIHH, .QeHCTBYlOllI.He Ha ypOBHe nOAOlIIBbf 4>YH-
aMeHTa 1\'=2500 KH, M x =500 KH.M, M y =250 KH.M, R=300 Kna.
TpeoyeTcH onpe.Qe.nHTb pa3MepbI no.n.OlIIBbI H .n.aBJIeHHH Ha OCHOBaHHe
npH .n.eHcTBHH Mx no BapHaHTY I H pmax= 1,5R= 1,5.300=450 Kna;
Pmin = 0,25pmax = 0,25.450= 112,5 KITa.
PaC'ieT: ex=500/2500=O,2 M; ey=250/2500=0,1 M. 3a.n.aeMcH a=
= (3=0,5. no Ta6JI. 2.4 npH a= =O,5 HaXO,lI.HM: k 1 =0,75; k,=0,138;
k', = 0,098; ks = 0,094; k' 5 = 0, 188, f.tx = 0,417.
ITo 4>opMYJIaM (3.14) H (3.13) onpe.QeJIHM B nepBoM npHOJIH}I{eHHH
pa3MepbI nO.QOllIBb{ A H B, .n..nH qero npe.QBa pHTeJIbHO npoBepHM yc.no-
BHe (3.16):
2,4 . 0,138 .0,094 . 2500 == 366,8 > 300 Kna.
25 . 0 75 3 . 0 2 . 0 1
. , , ,
YCJIOBHe (3.16) y.n.OBJIeTBopHeTCH:
o --- 0 ,6 . 0,094 . 2500 4 47
3 "-- I 875 . 0 75 2 . 300 . 0 I == , ;
" ,
(3 ==447 ]0.0,2.0,75 =1625.
4 , 3 . 0 138 "
,
A =4,47 + 1'4,47 2 - 16,25=4,47 + 1,9; At=6,37 M,
A z =2,57 M;
B --- 2 . 2500 == 1,86 M;
1 --- 1,875. 0,75 .6,37 .300
B 2 . 2500 = 4 61 M . AB -- 6 . -
2 == I 875. 0 75 . 2 57 . 300 ' , "-,37 1 ,86 -
, , ,
== 11,85; (AB)2 == 11,85 2 == 149,423;
(AB)3 == 11,85 3 == 1664-,013.
75
no cpopMYJIaM (3.59):
Q)1=4/0,098+ 1/0,138=48,006; CP2=4jO,188+ 1/0,094=
=31,983; CP3= 1/0,138+ 1/0,098= 17,456; CP4= 1/0,094+
+ 1/0,188= 15,992; <p'a= 1/(0,094-0,098) - 1/(0,188X
XO,138) =69,792;
1!Jl = (0,375 - 300) 248,006.31,983/2500 2 = 3, 1 09; "'2=
=[0,375. 300/ (0,75 - 2500)]/ (31,983.17,456+48,006 - 15 X
X992) =79 589; 1V3= (1/0,75 2 ) 17,456.15,992=496,277;
"'4=0,2 - 0, 1- 69,79 '= 97,419.
TIo.n.CTaBJIHeM nOJIyqeHHbIe 3HaqeHHH B ypaBHeHHe (3.60):
3,109. 1664,013 - 79,589. 140,423 + 496,277 - 11,85 +
+97,419=5173,416 - 11176,126+5880,882+97,419=
=- 25,109 0.
TaKHM o6pa30M npHHHToe B nepBOM npH6JIH>KeHHH AB'= 11,85 M 2
Y J1.0BJIeTBOpSleT ypaBHeHHIO (3.60). no <popMYJIe (3.61):
A == 0,1 . 69,792 = 3,27 M,
15,992/0,75 - (0,375/2500) 300 . 31,983 . 11,85
B = 11,85/3,27 = 3,62 M, a = 0,5 - 3,27 = 1,64 M;
b == 0,5 · 3,62 -=- 1,81 M.
1l.'IH KOHTpO,,'IH BbIqHCJIHeM pmax H pmln:
2500 + 500 +
pm ax ==
0,75 . 3,27 · 3,62 0,138 . 3,27 2 · 3,62
+ 250 ==2al ,6+95,9+66,2=443,7 < 450 Kna
o ,094 . 3, 27 . 3, 62 2
-2816 500 + 250 _
Pmto - ,- 0 098 .3 27 2 . 3 62 0 188 .3 27 · 3 62 1
, , , , , ,
== 281,6 - 132,5 - 33, t == 116 112,5 Kna.
PaCXO>KJ1.eHHe 3HaqeHHH H3-3a oKpyrJIeHHH npH BbIqHCJIeHHSlX He
npeBbIlliaeT 4 % .
Onpe.n.eJIHeM .n.aBJIeHHH Ha OCHOBaHHe B .n.pyrHx xapaKTepH x TOq-
Kax no.n.OlIIBbI. 113 BbIQHCJIeHHH pmax H pmln:
l=281,6 Kna; m=95,9 Kna; m' = 132,5 Kna; n=66,2 Kna3
n'=33,1 Kna; A= (1- 0,5 - 0,417)/0,417=0,2.
no 4>opMYJIaM(3.67):
Pl=281,6+95,9 - 66,2=311,3 Kna; P2=281,6 -95,9 X
76
XO,2-66,2=196,2 Kna; P3=281,6-19,2-33,1=
= 229,3 Kna; p,=281,6 - 132,5+33,1 = 182,2 Kna; P5=
===281,6 - 19,2+33,1 =295,5 Kna; P6=281,6 - 19,2+
+66,2=328,6 Kna.
npHMep 12. l1.aHo: YCHJIHH, .n.eHcTBYIOUJ.He Ha ypoBHe no.n.OlIIBbI YH-
.n.aMeHTa N=2000 KH, M x =1200 KH.M, M y =300 KH.M, R=200 Kna.
Tpe6yeTcH onpe.n.eJIHTb pa3MepbI no.n.OlIIBbI H .n.aBJIeHHH Ha OCHOBaHHe
npH AeHcTBHH Mx no BapHaHTY II H pmax= 1,5R= 1,5.200=300 Kna,
Pmin=O.
PaCtf.eT: e x = 1200/2000=0,6 M; e y =300j2000=O,15 M. 3a.n.aeMcH
a=0,5 H =0,7. no Ta6JI. 2.4 npH a=O,5 H =O,7 Haxo.n.HM k 1 =0,85;
k,=0,152; k',=O,127; k 5 =0,112; k's=0,160; f,.tx=O,456. no opMYJIaM
(3.10) H (3.8) onpe.n.eJIHeM B nepBOM npH6JIH:>KeHHH pa3MepbI Uo.n.o-
WBbl A H B, .n.JIH qero npe.n.BapHTeJIbHO npOBepHeM YCJIOBHe (3.15):
0,127 · 0,160 . 2000 = 245,7 > R == 200 KOa.
3 . 0 85 3 . 0 6 · 0 15
, , ,
YcnOBHe (3.15) Y.llOBJIeTBOpHeTcn:
0 1 === 0, 160 · 2000 =-.: 9 837.
, ,
I 5. 0 85 2 . 200 · 0 15
" ,
02 =--= 9,837 2 .0,6 .0,85 = 78,755;
0, 127
A -== 9,837 + V 9,837 2 -78,755 = 9,837 + 4,244;
At = 14,081 M; A 2 == 5,593 M;
B == 2 · 2000 = 1, 114 M.
1 1,5 . 0,85 . 200 . 14,081 t
B = 2 · 2000 == 2,804 M.
2 I ,5 · 0, 85 . 200 . 5,593
npHHHMaeM: A =5,593 M; B=2,804 M; AB=5,593.2,804= 15,683 M 2 ;
(AB)2= 15,683 2 =246,956 M 4 ; (AB) 3= 15,683 3 =3873,011 M 6 . no <pop-
MYJIaM (3.59) H (3.64): qJl =4/0,152=26,316; q>2=4/0,112=35,746; qJ3=
= 1/0,152+ 1/0,127 = 14,428; qJ4 = 1/0, r12+ 1/0,160= 15,191; qJs =
= 1/(0,160.0,152) - 1/(0,112.0,127) = -29,001; )1=[(0,375.200)2/
/2000 2 ]26,316.35,746= 1,323; '\1.'2=[0,375.200/ (0,85.2000)] (35,746 X
X 14,428+26,316.15,191) =40,39; )3= 1/0,85 2 .14,428.15,191 =303,357;
'\P4 = 0,6 . 0,15 (-29,001) 2 = 75,695.
nO.n.CTaBJIHeM nOJIyqeHHbIe BeJIHqHHbI B ypaBHeHHe (3.60): 1,323 X
X3873,011 40,39.246,956+303,357.15,683+75,695 = 5123,994-
- 9974,578+4757,548+75,695= -17,341 O.
TaKHM 0(5pa30M npHHHToe B nepBoM npH6JIH:>KeHHH AB = 15,683 M 2
Y.LlOBJIeTBOpSleT ypaBHeHHIO (3.60).
no <popMYJIe (3.62):
77
A = 0,6 (-29,001) . = 5 52 M.
15, 101 /0, 85 - 0,375 . 200 · 35, 746 . 15,683/2000 ' ,
B= 15,683/5,52 === 2,84 M; a = 0,5 · 5,52 === 2,76 M;
b = 0,7 · 2,84 === 1,99 M.
,llJIH KOHTpOJIH onpe.n.eJIHeM pmax H pmln:
2000 + 1200 +
pmax ===
° ,85 · 5,52 . 2, 84 0, 127 . 5,52 2 . 2, 84
+ 300 =150,1+108,8+42,1=301 300Kna;
0, 160 . 5, 52 . 2, 84 2
-1001- 1
Prnln - , 0 152 . 5 52 2 . 2 84 0 112 . 5 52 . 2 84 2
, , , , , ,
=== 150,1-91,2- 60,2 == -1,3 Kna O.
,L{aBJIeHH.H Ha OCHOBaHHe B .npyrHx xapaKTepHbIX TOqKaX no.n.OlIIBbi. 113
BbIqHCJIeHHH Pmax H Pmin 1= 150,1 Klla; m= 108,8 KTIa; m'=91,2 Klla;
n=42,1 Kna; n'=60,2 KDa; 'A=(1-O,5-0,456)jO,456=O,098. llo
<popMYJIaM (3.68): Pl=150,l+0,098.108,8+60,2=221 Klla; p2=150,1+
+0,098.108,8+42,1 =202,9 Kna; P3= 150,1- 108,8+42,1 =83,8 Kna;
P4= 150,1 +0,098.108,8 - 42,1 = 118,7 Klla; P5= 150,1 +0,098.108,8-
- 60,2= 100,6 Klla; P6= 150,1 +91,2 - 60,2= 181,1 Kna.
DpHMep 13. afto: YCHJIHH, ,lI.eH<;TBYIO He H8 ypoBHe nO)J.OIlIBbI YH-
.n.aMeHTa N=2500 KH, M x =500 KH.M, M y =250 KH.M, R=300 Kna.
Tpe6yeTcH onpe.n.eJIHTb pa3MepbI no.n.OlIIBbI H .n.aBJIeHHH Ha OCHOBaHHe
npH .n.eHcTBHH Mx no BapHaHTY II, pmax= 1,5R= 1,5.300=450 !{na H
Pmin=O,25pmax=0,25.450= 112,5 Kna.
PaCtf.eT: e x =5'00/2500=O,2 M; ey=250/2500=0,1 M. 3a.n.aeMcH a=
c:: =O,5. no T8()JI. 2.4 npH a= ?=0,5 H8XO,lI.HM k 1 =O,75; k 4 =O,138;
k'4=O,098; k s =0,094; k's=O,188; J.l,x=O,417. no 4>opMYJIaM (3.14) H
(3.13) onpe.n.eJIHeM B nepBoM npH6JIH:>KeHHH pa3MepbI no.n.OnIBbI A H B,
.n.JIH qero npe.n.BapHTeJIbHO npoBepHeM YCJIOBHe (3.16):
2,4 · 0,138 .0,094 . 2500 = 366,8 > R ===300 Kna.
25 . 0 75 3 . 0 2 . ° I
, , ,
YCJIOBHe (3.16) y.n.oBJIeTBopHeTcH:
B - 0, 6 . 0, 188 · 2500 = 8 ,89;
3 - I 875. 0 75 2 . 300 · 0 I
" ,
0===889 10.0,2.0,75 =45264.
4 ' 3 . 0 098 "
,
r
A=8,89 + / 8,89 2 -45,264=8,89 + 5,81; A 1 =14,7 M;
A 2 =3,08 M;
B J === 2 . 2500 = 0,81 M;
I 875 ..0 75. 14 7. 300
, , ,
78
2 - 2500
8 2 == = 3,85 M.
1,875 . 0,75 · 3,08 . 300
DpHHHMaeM: A=3,08 M; B=3,85 M; AB=3,08.3,85= 11,86 M; (AB) 2=
= 11,86 2 = 140,66 M ; (AB) 3 = 11,86 3 = 1668,228 M 6 .
no <popMYJIaM (3.59) H (3.63):
fPt=4/0,138+ 1/0,098=39,274; Q)2=4/0,094 + 1/0,188=
=47,974; fP.3= 1/0,098+ 1/0,138= 17,456; fP4= 1/0,188+
+ 1/0,094= 15,992; Q)5= 1/(0,188-0,138) -
- 1/ (0,094. 0,098) == - 69,8;
1!Jl = [(0,375. 300)2/2500 2 ] 39,274. 47,974 =3,815; 1!J2=
= [0,375 - 300/ (0,75. 2500) ] (47,974. 17,456+ 39,274 X
X 15,992) =87,933; 1l'3= (1/0,75 2 ) 17,456-15,992=496,277;
f
1j)4=0,2. 0, 1 (-69,8)2=97,441.
nO.n.CTaB.nHeM nOJIyqeHHbIe 3HaqeHHH B ypaBHeHHe (3.60): 3,815 X
X 1668,228 - 87,933.140,66+496,277.11,86+97,441 = 6364,29-
- 12368,655 + 5885,845 + 97,441 = -21 ,269 0.
TaKHM 06pa30M ypaBHeHHe (3.60) y.n.oBJIeTBOpHeTcH npH AB= 11,86.
no <popMYJIe (3.61):
A= 0,2(-69,8) =3,26 M.
15,992/0,75 - (0,375/2500) 300 . 47,974 - 11,86 '
B=11,86/3,26=3,64 M; a=0,5-3,26=1,63 M; b=0,5X
X3,64= 1,82 M.
.LI.J1H KOHTpOJIH BbIqHCJIHeM pmax H pmin:
2500 + 500 +
pmax =
o , 75 . 3, 26 . 3, 64 0 ,098 . 3, 26 2 . 3, 64
+ 250 =280,9+132,2+32=445,1 450Knal
0, 188 · 3, 26 - 3, 64 2
_ 280 9 _ 500 + 250 _
pmln - , 0,138 . 3,26 2 - 3,64 0,094 . 3,26 . 3,64 2
=== 230,9-99,1-64 = 117,8 112,,5 Kna_
PaCXO>KJJ.eHHe 3HaqeHHH H3-3a oKpyrJIeHHH npH BbIqHCJIeHHHX He
npeBbIlliaeT 3 0/0.
Onpe.n.e.nHeM .n.3BJIeHHH Ha OCHOBaHHe B JJ.pyrHx xapaKTepHbIx TOq-
K3X nOJ1.01IIBbi. 113 BbIqHCJIeHHH Pmax H pmin:
1=280,9 Kna; m=132,2 Kna; m'=99,1 Kna; n=32 Klla; n'=64 Klla;
'A = (I - 0,5 - 0,417) /0,417 = 0,2. no <popMYJIaM (4.68): .
pI=280,9+0,2.132,2+64=200,9 Kna; p2=280,9+0,2.132,2+32=
=339,3 Kna; p3=280,9 - 132,2+32= 180,7 Klla; p =280t9+0,2.132,2-
- 32=275,3 Kna; ps=280,9+0,2.132,2 - 64=318,5 Kna; p6=280,9+
+99,1 - 64=316 Klla.
79
3.7. YHAaMeHTbl C wecTHyroJlbHoH nOAowBoH
B OTnHqHe OT cnyqaH eHcTBHH MOMeHTa B O HOM Ha-
npaBneHHH nH TaKHX YH aMeHToB HeT Heo6xo HMOCTM
paccMaTpHBaTb Ba BapMaHTa paCqeTa 3aBMCHInHX OT Ha-
I1paBneHHH eHcTBMH MOMeHTOB Mx M My. ,[LocTaTOqHO yc-
OBHO nOBepHYTb YH aMeHT Ha 90° BOKpyr eHTpa nOAO-
lllBbI H 3aMeHMTb Mx Ha My H Ha060poT, KaK OAHH 113 Ba-
pHaHTOB CTaHOBHTCH PyrHM. 003TOMY paCCMOTpHM TOnbKO
BapHaHT, Kor a MOMeHT Mx eHcTByeT B nnOCKOCTH, rrepe-
ceKalOII1.eM Be npOTHBorronOLKHbIe BeplllHHbI nO OlllBbI, rrpM
3TOM My 6y eT eHcTBoBaTb BO B3aHMHO neprreHAHKY HP-
HOM nnOCKOCTH, nepeCeKalOI.U.eM cepe MHbI BYX npOTHBorro-
OLKHbIX CTOpOH nO OIlIBbl. KpoMe Toro, B 3aBHCHMOCTH OT
COOTHOllIeHHH ex H e y KpaeBoe aBJIeHHe Pmax MOLKeT B03-
HHKHYTb B O HOH Ji3 BepIIIHH rrO OIlIBbI 1 H H 2 (pHC. 3.7
H pHC. 3.8), npH 3TOM Pmin OKa:>KeTCH COOTBeTCTBeHHO B O,lI.-
HOH H3 npoTHBOnOnO:>KHbIX BepIlIHH.
BbIBe,lI.eM KpHTepHH, n03BOJIHIOInHH 3apaHee onpe,lI.enHTb,
B KaROH H3 YKa3aHHbIX ,lI.BYX BepllIHH B03HHKH T Pmax. 3a-
I1HIIIeM .n.Ba 3HaqeHHH Pmax, nepBoe H3 KOTOpbIX npe,lI.noJIa-
raeT Pmax B BepllIHHe 1, a BTopoe B BepllIHHe 2:
Nt (k 1 a 2 )' + Nex/ (k 4 a 3 ) =pmax;
N/(k 1 a 2 ) + N e x/(2k"a 3 ) + Neyl(k 5 a 3 ) =Pmax
r)J.e kl =2,598; k 2 =ks=k 4 =0,541; k5 =0,625.
DpHpaBHHB 3TH 3HaqeHHH Pmax, nocne COOTBeTCTBYIO-
HX npe06pa30BaHHH nonyqHM: 0,5e x /k 4 =e y /k 5 HnH
0,5e x /0,541 =e y /O,625, oTKy a e x = 1,7312e y .
CJIe OBaTenbHO, npH
ex> 1,7312ey (3.71)
Pmax B BeplIIHHe 1 H rrpH Hec06nlO eHHH YCJIOBHH (3.71) -
B BepllIHHe 2.
npH Pmax B BeplllHHe 1 aBJIeHHe B 3TOH TOqKe OT .n.eM-
CTBHH My paBHo HYJIIO, n03TOMY paCqeT npOH3BO,lI.HTCH no
cpopMynaM (2.15) H (2.16), KaK JIH CJIyqaH .n.eHcTBIIH
TonbKO MOMeHTa Mx B O.n.HOM HanpaBJIeHHH. HCKJIlOlJeHHe
COCTaBnHIOT .n.aBJIeHHH B .n.pyrHx BepllIHHax no.n.OIlIBbI, KO-
TopbIe onpe,lI.enHIOT no cpopMynaM (CM. pHC. 3.7, a II 3.8, a):
P1=l+O,924m+ 1,8484n; P2=l- O,9242m+ 1,8484n;
P3=l- 0,9242m - 1,8484n;
P4=l+O,9242m - 1,8484n, (3.72)
( 3.69)
(3.70)
80
a)
0)
y
ex
J
t.511. \
!!OJ;
Pac. 3.7
a)
QJ
y
JC
.
.
c:j
«::::$.
P.nzi... o..Z5pf!!.
"
o.5a n " "
PHC. 3.8
rAe
l=O,3849N/a 2 ; m=M x /a 3 , n=M y la 3 .
PaccMoTpHM cJIyqaii, Kor,11.a Pmax B03HHKaeT
He 2 (CM. pHC. 3.7,6 H 3.8, 6), npH Pmin=O:
_ N + Nex + Ney
pmax - k 2 O k 3 _ Ii 3'
la L 4 a 5 a
---L_ Ne x _Ne y -0
pmln - k 2 O k 3 k - ·
la L ,a 5a
113 CPOPMYJIbI (3.74)
2-_ _ e y = 0
- J
k l '2k 4 a k 5 a
B BepllIH-
(3.73)
(3.74 )
81
QTKY,Aa
a=[ex/ (2k 4 ) +ey/ks]kt
", nO,ACTaBHB 3HaqeHHH k 1 , k4 H k 5 , nOJIYQHM:
a=2,4011ex - 4,1568e y . (3.75)
CYMMa ,11.aBJIeHHH Pmax H Pmln:
Pmax=2N/ (k 1 a 2 ) =2N / (2,598a 2 ') =O,7698N/a 2 . (3.76)
AHaJIOrHQHO npH Pmin=0,25Pmax:
a=4,0018 e x+6,928e y ; (3.77)
Pmax= 1,6N/(k 1 a 2 ) = 1,6N/(2,598a 2 ) =0,6159N/a 2 , (3.78)
,QaBJIeHHH Ha OCHOBaHHe B ,11.pyrIlx BeplllHHax nO,11.0111BbI
(CM. pIlC. 3.7,6 II 3.8,6):
Pt=1+O,9242m+ 1,6n; P2=1- 1,8484m; } (3.79)
P3=1+0,9242m - 1,6n; P4= l+ 1,8484m.
npHMep 14. J1.aHo: YCHJIHH, .n.eHcTBYIOllI.He Ha ypoBHe no,nOWBbl cl>YH-
rl(aMeHTa N= 1100 KH, M x =220 KH'M; M y =330 KH.M, R=200 Kna.
'Tpe6yeTcH onpe.n.eJIHTb .lI.JIHHY CTOpOHbI no.n.OlIIBbI H .n.aBJIeHHH Ha oc-
HOBaHHe npH pmfn = O.
PaCtleT: e x =220jll00=0,2 M; e y =330/1100=0,3 M. TaK KaK e y =
=O,3>e x =O,2, TO no YCJIOBHIO (3.71) pmax 6y.n.eT B BeplIIHHe 2. no
<popMYJIaM (3.75) H (3.76): a=2,4011.0,2+4,1568.0,3= 1,73 M; Pmax=
=2.1100/(2,598.1,73 2 ) =282,9 Klla < 1,5R= 1,5.200=300 Klla.
,QJIH KOHTpOJIH onpe.n.eJIHeM pmax H pmln:
_____ 1100 220 1L 330
pmax ----- 2 598 . 1 73 2 + 2 . 0 541 . 1 73 3 0 625. 1 73 3
" " , ,
= 141,4 + 39,2 + 102 = 282,6 282,9 KDa;
Pmin= 141,4 - 39,2 - 102=0,2 Kna O.
JJ.aBJIeHHH B .n.pyrHx BepIIIHHax no.n.OllIBbI:
[==0,3840.1100/1,73 2 "= 141,4 Kna; m=M x a 3 =220/1,73 3 =
=42,5 Kfla; n=330/1,73 3 =63,8 Kfla; 0,9242m=0,9242X
X42,5=39,2 Kfla; 1,6n= 1,6.63,8= 102 Kfla; 1,8484m=
= 1,8484.42,5=78,6 Kna.
no opMy.naM (3.79):
Pl= 141,4 - 39,2+ 102=204,2 Kfla; P2= 141,4 -78,6=
=62,8 Kfla; P3= 141,4+39,2 - 102=78,6 Kna; P4=
== 141,4+78,6=220 Kna.
82
npHMep 15. p'afto: YCHJIHH, )J.eHcTBYIOllI.He Ha ypoBHe n0.n.oWBbI YH-
.n.aMeHTa N=5400 KH, M x =1350 KH.M, M y =p1836 KH.M, R=200 Kna.
Tpe6yeTcH onpe.n.eJIHTb .n.JIHHY CTOpOH no.n.OWB H aBneHHH Ha OCHO-
BaHHe npH pmlo = O,25pmax.
PaC'ieT: e x = 1350/5400=0,25 M; e y = 1836/5400=0,34 M. npOB pH-
eM YCJIOBHe (3.71): l,7312.0,34=0,588>e x =O,25. CJIe.n.oBaTeJIbHO, Pma
6Y.lI.eT B BepllIHHe 2. no <popMYJIaM (3.77) H (3.78);
a=4,0018. 0,25+6,928. 0,34=3,36 M;
Pmax= 1,6. 5400/ (2,598. 3,36 2 ) ==294,6< 1 ,5R = 1,5. 200 =
= 300 Kna.
,QnH KOHTpOJIH onpe.n.eJIHeM pmax H pmlo:
5400 1350 1836
pmax == + + -
2 598 . 3 36 2 2 . 0 54 I . 3 36 3 0 625 . 3 36 3
, , , ,
== 184,1 +32,9 +77,4==294,4 294,6 Kna;
pmin= 184,1 - 32,9 - 77,4=73,8=0,25pmax=0,25X
X294,6=73,8 Kna.
,LLaBJIeHHH B .lI.pyrHx BeplIlHHax no.n.OllIBbI: l= 184,1; m= 1350/3,36 3 =
=35,6 KOa; n= 1836/3,36 3 =48,4 Kna; O,9242m=0,9242.35,6=32,9 Kna;
l,6n= 1,6.48,4=77,4 KTIa; 1,8484m= 1,8484.35,6=65,8 Kna.
no opMYJIaM (3.79): Pl= 184,1 - 32,9-r 77,4=228,6 Kna; P2=
=184,1-65,8=118,3 Kna; p3=184,1+32,9-77,4=139,6 Kna; P4=
= 184,1 +65,8=249,9 KTIa.
3.8. YHAaMeHTbI C BocbMHyrOJIbHOH nOAowBoH
,l:LJlH TaKHX cpYH,lJ.aMeHTOB B03MOLKHbI ,lJ.Ba BapHaHTa pac-
qeTa, 3aBHCHmHe OT HanpaBJIeHHH ,lJ.eHCTBHH MOMeHTOB Mx
H My, npH 3TOM KaLK,lJ.bIH H3 BapHaHToB CO,lJ.epLKHT no ,lI.Ba
CJlyqaH, yqHTbIBalOIII.He 3HaqeHHH Pmln. BapHaHT I npe,lly-
cMaTpHBaeT ,lJ.eHCTBHe Mx H My B ,lJ.BYX B3aHMHO neprreH,lJ.H-
KYJlHpHbIX nJlOCKOCTHX, Ka>KAaH H3 KOTOpbIX nepeceKaeT
,lJ.Be npoTHBOnOJIO:>KHbIe BepIIlHHbI nO,lI.OIlIBbI (pHC. 3.9).
B 3aBHCHMOCTH OT COOTHOllIeHHH ex H ey KpaeBoe ,lJ.aBJIeHHe
Pmax MOLKeT B03HHKHYTb B O,lJ.HOH H3 BepllIHH 1, 2 HJIJI 3,
npH 3TOM Pmin OKa:>KeTCH cooTBeTcTBeHHo B O,lJ.HOH JI3 npo-
THBOnOJIO}KHbIX BepIlIHH. BbIBe,lJ.eM KpHTepJIH, n03BOJIHIO-
III.HH 3apaHee onpe,lJ.eJIHTb, B KaKOH JI3 YKa3aHHbIX BepIIlHH
B03HHKaeT Pmax. 3anHIIleM TPH 3HaqeHJIH pmax, nepBoe JI3
KOTOpbIX npe.n.nOJIaraeT Pmax B BepllIHHe 1, BTopoe - B Bep-
llIHHe 2 H TpeTbe - B BeplIlHHe 3:
N Ne x
k ---; + - k = pmax;
1 a 4 a 3
83
0)
0)
I
:
aJ8a
t::::f
« .
J PlTJin= 2.5Pmax
, aJ8a . . aJ8at
PRC. 3.9
!:!-. + O,9266Ne x + O,9266Ne y
k I a 2 k 2 a 3 k"2 a3
. + Ne,_
k 2 k - - pmax,
la 4 a3
r.n.e ki =4,8259; k 2 = 1,8595; k6.= 1,4253; k5= 1,5405.
CpaBHHBaH nepBoe H TpeTbe 3HaqeHHH Pmax, npHXO.lI.HM
K BbIBO.n.y 0 TOM, qTO 60JIbIIIee .lI.aBJIeHHe Pmax B03HHKaeT
B BepllIHHe 1 npH ex>e y H B BepllIHHe 3 npH ex<e y . OpH
Pmax B BepIIIHHe 1, T. e. ex>ey, ,lI.aBJIeHHe B 9TOH BepllIHHe
OT ,lI.eHCTBHH My paBHO HYJIIO. n09TOMY paCqeT npoH3BO,LIJIT
no cpopMYJIaM (2.15) H (2.19) KaK ,lI.JIH CJIyqaH JleHCTBHH
TOJIbKO Mx B O,lI.HOM HarrpaBJIeHHH. HCKJIlOqeHHe COCTaBJIH-
lOT ,naBJIeHHH B ,npyrHx BepllIHHax no.n.OllIBbI, KOTopbIe on-
pe,neJISnOT no cpopMYJIaM:
P1=l+O,4983m+O,4983n; P2=l+O,7016n; P3=l-
- 0,4983m+0,4983n; P4=l- O,4983m - O,4983n; P5=
== l- O,7016n; P6=l+O,4983; m -O,4983n, (3.80)
=-:: pm2X;
r.n.e
l=O,2072N/a 2 , m=M x la 3 , n=M y la 3 .
DpB Pmax B BepllIHHe 3, T. e. Kor,na ex<ey, ,naBJIeHHe
B 9TOH BepllIHHe OT .n.eHcTBHH Mx paBHo HYJIIO, n09TOMY
paCqeT npoH3Bo,nHT no TeM :>Ke cpopMYJIaM, qTO H ,nJIH npe-
,nbI,lI.ymero CJIyqaH C 3aMeHOH Mx Ha My. OpH 9TOM .lI.aBJIe-
84
HHe B .n.pyrHx BepllIHHaX nO.lI.OIlIBbI onpe.lI.eJIHIOT no CPOPMY-
JIaM:
Pl=l+0,7016n; P2=1- 0,4983m+O,4983n; P3=l-
- O,7016m; P4=l- 0,7016n; P5=l -t- O,4983m - 0,4983n;
P6=l+0,7016m. (3.81)
TaKHM 06pa30M OCTaeTC paCCMOTpeTb TOJIbKO CJIyqaH,
KOf,LI,a Pmax B03HHI<aeT B BeplllHHe 2. CpaBHHBaH nepBoe H
BTopoe npHBe.n.eHHbIe BbIllie 3HaqeHH51 Pmax. nOJIyqHM
Nex < O,9266Nex + O,9266Ne y
I,4253a 3 1,8595a 3 1,8595a 3
OTI(y.n.a I1PH ex < e y
e x <2 J 4511ey. (3.82)
AHaJIOfHqHO, cpaBHHBaH BTopoe H TpeTbe 3HaqeHHH
Pmax, I10JIyqHM I1PH ey<ex
ey<2,4511ex. (3.83)
TaKHM o6pa30M Pmax B03HHKaeT B TOqKe 2 I1PH co6JIIO-
tleHHH YCJIOBHH (3.82) H (3.83). r
YCJIOBHSI paBHoBecHH npH Pmln=O:
_ N + O,9266Ne x + O,9266Ne y . ( 3 84 )
pmax - 4, 8259a 2 I ,8595a 3 I , 8595a 3 ' .
__ N _ O,9266Ne x _ O,9266Ne y 0 ( 3 85 )
pmln - 4,8259a 2 . I,8595a 3 1,8595a 3 ==, .'
oTKy.n.a
a=2,4047 (ex+ey).
CYMMa .n.aBJIeHHH Pm ax H Pmin:
Pmax = 2N --- O,4144 .
4, 8259a 2 a 2
.A.HaJIOrlflIHO npH Pmin=0,25Pmax (CM.
(3.86)
(3.87)
a=4,0084 (ex+e )'>;
pmax= 0,33 1 5Nla 2 .
pHC. 3.9, 6) :
(3.88)
(3.89)
aB eHHH Ha OCHOBaHHe B .n.PYfHX TOQKaX no.n.OlliB
(pnc. 3.9):
Pl=l+O,7016n; P2=1- 0,4983m+O,4983n; P3=1- }
- O,7016m; P4=1- O,7016n; P5=1+O,4983m - O,4983n;
P6=l+O,7016m. (3.90)
85
i!}t'
c:::;:)
.
f '
.. {[5a"" o.5a 0:70710.
PUC. 3.10
BapHaHT 11"- MOMeHTbI Mx H My )],eHCTBYIOT B .nBYX
B3aHMHO rrepneH)],HKYJIHpHbIX nJIOCKOCTHX, Ka}K,lI.aH H3 KO-
TOpbIX nepeceKaeT cepe)],HHbI ,lI.BYX npOTHBOrrOJIO}KHbIX CTO-
pOH nO,lI,OIlIBbI (pIlC. 3.10). no aHaJIOrHH C BapHaHToM 1
Pmax 6Y,lI.eT B BepllIHHe 1 npH ex>e y H B BepllIHHe 2 npH
ex<e y .
PaCCMoTpHM CJIyqaH, KOf,lI.a Pmax 6Y,lI.eT B BepllIHHe 1,
T. e. KOr,lI.a ex>e y npH Pmin=O (pHC. 3.10, a). 113 BbIpa>Ke-
HHH ,lI.JIH Pmax H Pmin no aHaJIOrHH C BapHaHTOM I:
a=3, 1 327e x + 1,2976ey;
pmax=O,4144N la 2 .
,QaBJIeHHH B .LI.pyrHx BeplllHHax nO.LI.OIIlBbI:
P1=1+O,2689m+O,6491n; P2=1- O,2689m+O,6491n; Ps=
=1- O,6491m+O,2689n; p&:=l- O,2689m - O,6491n;
P5=1+0,2689m - O,6491n; P6=1+O,6491m-
- O,2689n. (3.93)
(3.91 )
(3.92)
rJI.e
l=O,2072Nla 2 ; m=Mx/a 3 , n=M y fa 3 .
,QJIH Pmin=O,25Pmax, Kor,lI.a Pmax B BepIIIHHe 1, T. e.
ex>e y , nOJIYlIIlM:
a=5,221gex+ 2, 1631ey;
pmax = O,3315N la 2 .
(3.94 )
(3.95)
86
,aaBJIeHHH B JI.pyrHx BepllIHHaX nOJI.OllIBbI onpeJI.eJIHlOT
no <popMYJIaM (3.93).
npH Pmax, HaXOJI.H eMCH B BepllIHHe 2 (pHC. 3.10,6), T. e.,
Kor,lI.a ex<e y , Bce <POPMYJIbI JI.JIH CJIyqaH ex>e y coxpaHHIOT-
.CR, HO C 3aMeHOH MeCTaMH Mx H My.
npHMep 16. l1.ano: YCHJ1HH, .lI.eHcTBYlOllI.He H3 ypoBHe nO.ll.OIlIBbI CPYH-
.JJ.aMeHTa, N=1700 KH; M x =680 KH.M; M y =540 KH.M; R=240 Kna.
Tpe6yeTcH onpe.ll.eJIHTb .lI.JIHHY CTOpOHbI nO.ll.OIlIBbI H JI.aBJIeHHH Ha oc-
HOB3HHe npH .lI.eHCTBHH MOMeHTOB no BapH3HTY I H pmln = o.
PaCtf,eT: e x =680/1700=0,4 M, ey=340/1700=O,2 M. TaK KaK e x =
=0,4>e y , TO npOBepHeM YCJIOBHe (3.81): 2,4511e y =2,4511.0,2=O,49>
>e x = 0,4 M. CJIe.ll.OB3TeJIbHO, pmax B03HHKaeT B BepIlIHHe 2.
no cpopMYJIaM (3.86) H (3.87):
a=2,4047(0,4+0,2) = 1,44 M;
Pmax=0,4144.1700/1,44 2 =339,7 Kna< 1,5.240=360 Kna.
JJ.JIH KOHJ'pOJIH onpe.ll.eJIHM pmax H pmln:
1700 680 . 0 9266 340 . 0, 9266
pmax = 4 8259 . 144 2 + 1 8595: 1 44 3 + 1 8595 · 1 44 3 -
, ""
= 169,9 + 113,5 +56,7 = 340,1 339,7 < 1,5R =
= 1,5.240=360 KOa;
Pmin=169,9-113,5-56,7=-0,3 Kna O.
JJ.aBJIeHHH B JI.PyrHx BepIlIHHax n0.ll.OIlIBbI:
1= 169,9; m=680/1,44 3 =227,7 Kna; n=340/1,44 3 =
= 113,8 Kna; 0,7016n=0,7016.113,8=79,8 Klla;
O,4983m=0,4983.227,7= 113,5 Kna; 0,4983n=0,4983X
X 113,8=56,7 Kna; 0,7016.227= 159,3 Kna.
no cpopMYJIaM (3.90):
P1=169,9+79,8=249,7 Kna; P2=169,9-113,5+56,7=
= 113,1 KDa; P3= 169,9 - 159,3= 10,6 Kna; P,,= 169,9-
- 79,8==90,1 KOa; P5= 169,9 - 113,5 - 56,7=226,7 KDa;
P6= 169,9+ 159,3=329,2 KDa.
npHMep 17. l1.aHo: YCHJIHH, .lI.eHCTBYIOIJ.J.He Ha ypOBHe nOJI.OIlIBbI CPYH-
.n.3MeHTa, N=4500 KH; Mx=675 KH.M; M y =1350 KH.M; R=320 Kna.
Tpe6yeTcH onpe.ll.eJII1Tb .lI.JIHHY CTOpOHbI nO.ll.OlllBbI H .lI.aBJIeHHH Ha OC-
IIOBaHHe npH JI.eHCTBHH MOMeHTOB no BapHaHTY I H pmln = O,25pmax.
PaCtteT: e x =675/4500=0,15 M; ey= 1350/4500=0,3 M. TaK KaK
e y =0,3>e x =0,15, TO npOBepHeM YCJIOBHe (3.83): 2,451Ie x =2,4511 X
X 0, 15 = 0,37> e y = 0,3 M. CJIe.D.OBaTeJ1bHO, pmax B03HHKaeT B BeplllHHe 2.
110 <popMYJIaM (3.88) H (3.89):
87
a=4,0084(O,3+0,15) = 1,8 M;
Pmax=0,3315.4500fl,8 2 =460,4 Kna< 1,5R= 1,5.320=
=480 Klla_
AJIH KOHTpOJIH onpe.n.e.m!:\'I Pmax l pmin:
4500 675 . 0, 9266 1350 . 0,9266
pmax = 4 8259 . I 8 2 + 1 8595. 1 8 3 + I 8595. 1 8 3 =
, , ., ,
-= 287,8 + 57,7 + 115,4 =:=: 460,9 460 KfIa;
pmin=287,8 - 57,7 - 115,4= 114,7 O,25Pmax=O,25X
X460,4= 115,1 Kna.
AaBJIeHHH B )J.pyrHx BepIlIHHax nO)J.OIlIBbI:
1=287,8 Kna; m= 675/1,8 3 === 115,7 KfIa;
n == 1350 / 1 ,8 3 === 231 ,4 K n a;
p1=287,8+0,7016.231,4=450,2 Kna; P2=287,8-
- O,4983X 115,7 +0,4973-231,4=345,4 Kna; P3=287,8-
- 0,7016.115,7=206,6 Kna; p =287,8 - 0,7016.231,4=
= 125,4; P5 . 287,8+0,4983- 115,7 - 0,4983-231,4=
=230,2 KfIa; P6==287,8+0,7016.115,7=369 Klla.
npHMep 18. p'aHo: YCHJIHH, .ll.ei1cTBYIOIJ.J.He Ha ypoBHe no.n.OIlIBbI <PYH-
.n.aMeHTa, N= 1700 KH; M x =680 KH.M; M y =340 KH.M;' R=220 Kna.
Tpe6yeTcH onpe.n.e,,"IHTb )J.JIHHY CTOpOHbI nO.ll.OIlIBbI H )J.aBJIeHHH Ha oc-
HOBaHHe npn )J.ei1cTBHH MOMeHTOB no BapHaHTY II H pmfn=O.
PaCtleT: ex=680{1700=0,4 M; ey=340/1700=0,2 M. TaK KaK ex=
=0,4>e y =O,2 M, TO pmax GY)J.eT B BepIlIHHe 1. no cpopMYJIaM (3.91) H
(3.92) :
a=3,1327 .0,4+ 1,2976.0,2= 1,51 M;
pmax =0,4144.1700/1,51 2 =308,9< 1,5R= 1,5.220=330 Kna.
AJIH KOHTpOJIH onpe.n.eJIHM Pmax II pmln:
1700 680 340 · 0,5
pmax= + +
4 8259 - 1 51 9 I ,5405 . 1,5 I 3 1 , 8595 . 1,51 3
= 154,5 + 128,2 + 26,5 === 309,2 3C8,9 KfIa;
pmln= 154,5 - 128,2 - 26,5 O_
.lI.aSJIeHHH B OCTaJIbHbIX BeplllHHax nOJlOlliBbI:
l=O,2072.1700/1,51 2 = 154,5 Kna; m=680/1,51 3 =
=== 197,5 Kna; n=340jl,51 3 =98,8 Kna; P1= 154,5+
+0,2689.197,5+0,6491-98,8=271,7 KOa; P2= 154,5-
- 0,2689-197,5+0,6491-98,8= 165,5 Kna; P3= 154,5-
88
-0,6491-197,5+0,2689X98,8=52,8 Kna; p = 154,5-
- 0,2689.197,5 - 0,6491 098,8=37,3 KTIa; P5= 154,5+
+0,2689-197,5 - 0,6491-98,8= 143,5 Kna; P6= 154,5+
+ 0,6491. 197,5 - 0,2689 - 98,8= 256,2 Klla_
npHMep 19. p'aHo: YCHJIHH, )J.eMcTBYIOIll.He Ha ypoBHe nO)J.OlllBbI <l>YH-
:n.aMeHTa, N = 6000 KH, Mx = 1200 KH. M, My = 2400 KH. M,. R = 220 Kna.
Tpe6yeTcH onpe.n.eJIliTb )J.JIHHY CTOpOHbI nO)J.OlllBbI H )J.aBJIeHHH Ha oc-
HOBaHHe npH )J.eMcTBHH MOMeHTOB no BapHaHTY I I H pmin O,25pmax.
Pac'teT: e x = 1200/6000=0,2 M; e y =2400/6000=O,4 M. TaK KaK
ex=0,2<e y =0,4 M, TO B paCqeTHbIX <l>opMYJIaX MeHHeM MeCTaMH Mx H
My, npH 3TOM pmax B03HHKaeT B BeplllHHe 2. no <l>opMYJIaM (3.94) H
(3.95) : .
a 5,2219.0,4+2,1631.0,2=2,52 M;
Pmax = 0,3315.6000/2,52 2 = 313,2 Kna <'1 ,5R = 1,5.220 = 330 KOa.
,[{JIH KOHTpOJIH onpe.n.eJIHM pmax H pmin:
6000 2400 1200 . 0,5
pmax = 4,8259. 2,52 11 + 1,5405. 2,52 3 + 1 8589 · 2 52 3 -
=== 195,8 + 97,4 + 20,2 313,2 KOa;
pmln= 195,8 - 97,4 - 20,2=78,2 O,25pmax=O,25X
X 313,2=78,3 Kfla.
,[{aBJIeHHH B OCTaJIbHbIX BeplllHHax no.n.OlllBbI BbIqHCJIHIOT no <l>op"
MYJIaM (3.93). npH 3TOM )J.JIH coxpaHeHHH HH)J.eKCOB Y Pi (pHC. 3.10, 6)
CMeIllaeM HX Ha o.n.HY e.n.HHHu.y:
1= 195,8 Kna; m=2400/2,52 3 = 150 Kna; n= 1200/
/2,52 8 =75 Kfla.
no <l>opMYJIaM (3.92):
P1= 19,58+0,2689.15+0,6491.7,5=284,8 Kfla; P2=
= 195,8 - 0,2689-150+0,6491.75=204,2 KOa; P3=
=195,8-0;6491XI50+0,2689-75=118,6 Kfla; P4=
== 195,8-0,2689-150-0,6491-75=106,2 Kna; P5=
==195,8+0,2689.150-0,6491.75=187,4 Kfla; P6=
== 195,8+0,6491.150 - 0,2689-75=273 Klla_
r JlABA 4. cl>YH,LlAMEHTbl, PA60TAIOUJ.HE C OTPbIBOM
nO,LlOlliBbI nPH AEflCTBHH MOMEHTA
B O,nHOM HAnPABJlEHHH
4.1. n peAUOCblJlKH, npHHHTble UpH BblBOAe
paClleT"HbIX 4JOPMYJI
LLJIH BceX <PYH,lI,aMeHToB paccMaTpHBaeMblx THrrOB He-
3aBHCHMO OT cpOpMbI 01IepTaHHH nOJI.OlliBbI rrpHMeHeHo e,lI,H-
89
Hoe pellleHHe. BC5I nJlOlll.aJI.b nOJI.OlllBbI qJleHHJIaCb Ha OT-
lleJlbHbIe yqaCTKH nOCT05IHHOH lllHpHHbI JlH60 lllHpHHOii, H3-
MeHSlIOIII-eHCSI no OJI.HOMY II TOMY :>Ke 3aKOHY. B npeJI.eJlax
Ka:>KJI.OrO TaKoro yqaCTKa npOBOJI.IIJlIICb JI.Be BepTHKaJlbHbIe
nJIOCKOCTII, napaJlJIeJlbHbIe OCII Y II YJI.aJIeHHbIe OllHa OT
JI.PyroM Ha 3J1eMeHTapHOM paCCTOHHlI1I dx.
n pOH3BeJI.eHHeM dx Ha IIIHPHHY paCCMaTpHBaeMoro yqa...
CTKa H Ha nepeMeHHoe JI.aBJleHHe Ha OCHOBaHlIe px, Bblpa-
:>KeHHOrO KaK CPYHKUIISI Pmaxt onpeJI.eJIHJICH 3JIeMeHTa pHbIH
06"beM 3nlOpbI JI.aBJleHHH Ha OCHOBaHHe. npH rrepeMeHHOH
1lIIIpHHe nOJI.OllIBbI Ha paCCMaTpHBaeMOM yqaCTKe, HO H3Me-
H5IIOIIJ,ei1:cH no OJI.HOMY 3aKOHY, OHa TaK)Ke BbIpa:>KaJlaCb KaK
CPYHKUHSI COOTBeTCTBYIOIUHX pa3MepOB nO.llOlllBbI. HanpH-
Mep .llJlSI Kpyr JIbIX II KOJlbueBbIX nOJI.OIllB - B 3aBHCHMOCTII
OT paJI.HYCOB RH II RBt JI.JIH MHoroyrOJIbHbIX nO,lJ.OIIIB - OT
JI.JIMHbI CTOpOHbI a. 06'beM 3nlOpbI .llaBJIeHHH Ha OCHOBaHHe
B npe.lleJIax Ka:>K,lJ.oro yqaCTKa orrpeJI.eJISIJICH KaK HHTerpan
3JIeMeHTapHbIX 06'beMOB t a BeCb 06'beM V 3nlOpbI JI.aBJIeHHH
Ha OCHOBaHHe JlH paCCMaTpHBaeMoro cJlyqaSI nOJlOllieHHH
HeHTpaJIbHOH OCII KaK CYMMa 06'beMOB Bcex yqaCTKOB Ha
,QJlHHe C:>KaTOH 30HbI nOJI.oIllBbI.
H3 YCJlOBHH paBeHCTBa 06'beMa V npOJI.OJlbHOH CHJle N
nOJlyqeHO OJI.HO ypaBHeHHe paBHOBeCHH. AHaJlOrHqHO nOJlY-
qeH YCJlOBHbIM CTaTHCTHqeCKHH MOMeHT S 06'beMa V OTHO-
CHTeJlbHO OCH y. YCJIOBHbIM OH Ha3BaH nOTOMY, qTO npH ero
onpeJI.eJleHHH nplIHHMaJIOCb npOH3BeJI.eHlIe 06'beMa He Ha
cpaKTHlJeCKOe nJleqo - paCCTOHHHe OT ueHTpa TH:>KeCTH no-
CJIeJI.HerO )1.0 OCII Y (no HaKJIOHY), a Ha npoeKUHIO 3Toro
paCCTOSIHHH Ha nJlOCKOCTb nO,lI.OllIBbI.
113 YCJlOBHH paBeHCTBa S MOMeHTY npOJI.OJIbHOM CHJlbI N
OTHOCIITeJlbHO TOM :>Ke OCli y nOJlyqeHO BTopoe ypaBHeHHe
paBHoBeCHSI. COBMeCTHbIM peIlleHHeM 3THX ,QBYX ypaBHeHlIH
onpe,QeJleHW MaKCHMaJIbHOe .llaBJIeHMe Ha OCHOBaHHe Pmax
II ,lI.JlHHa C:>KaTOR- 30HbI nO,lJ.OlllBbI VA. Ha OCHOBe nOJlyqeH-
HWX rpoM03,lI.KHX CPOPMYJI .llJIH"npaKTHlJeCKHX paCqeTOB co-
CTaBJleHbI Ta6JIIIl1.bI, n03BOJlHIOIII.He BeCbMa npOCTO onpelle-
JIHTb HCKOMbIe BeJIlItI1IHbI 1 .
Tal{ KaK oTpbIBaeMaH qaCTb nO,lJ.OIllBbI OrpaHHqeHa yc-
JIOBIISIMH FOTP O,25F H FOTp O,5F, TO npM BbIBO.lle pac-
qeTHbIX CPOPMYJI CYMMlIpOBaHHe 06'beMOB 3nlOpbI )1.aBJIeHIIH
1 T86JIHllbI cO)J.ep}KaT p83rp8HHqHTeJIbHbIe JIMHHH, BbIllie KOTOpbIX
Ta6JIHqHbIe K09ct>ct> HUHeHTbI COOTBeTCTBYIOT YCJIOBIUO F OTp O,25F, HH-
>Ke - YCJIOBHIO F OT"p O,5F.
90
Ha OCHOBaHHe H COOTBeTCTBYID HX CTaTHqeCKHX MOMeHTOB
npOH3BO,lI.HJIOCb ,llJISI qaCTH CLKaTOH 30HbI nO,lI.OllIBbI He Me-
Hee O,5F.
OpH nOJIb30B3HHH T36JIHIJ.3MH 6e3 33MeTHoro yrnep6a
J(JISI TOQHOCTH p3CqeTa ,llOnYCKaeTCSI oKpyrJISlTb OTHOCHTeJIb-
HbIe 3KCIJ.eHTpHCHTeTbI I:IpO)J.OJIbHOH CHJIbI ex/A, extRa H ex/a
JI.O 6JIHLK3HllIHX T36JIHQHbIX 3H3qeHI1H H no HHM npH Heo6-
XOJI.HMOCTH orrpe.n.eJISlTb HCKOMbIe K03<pcpHl1.HeHTbI JIHHeHHOH
HHTeprrOJISlIJ.HeH.
4.2. npHMoyroJlbHble clJYHAaMeHTbI
06'beM 3rrlOpbI .n.aBJIeHHSI H3 OCHOB3HHe Hero YCJIOBHbIH
CT3THQeCKHH MOMeHT OTHOCHTeJIbHO OCH y paBHbI (pHC. 4.1) :
('A
a5 A IJ
A
T
1
y
PUC. 4.1
V=O,5pmaxVAB; (4.1)
S=O,5PmaxVAB. t/3yA= Pmaxv2A2B. (4.2)
6
113 YCJIOBHH p3BHOBeCHSI HMeeM:
N= V=O,5Pmaxl'AB; (4.3)
o 5A - e == == Pmax1 2 .4 2 B - IA (4.4)
, x V 6 .O,5PmaxIAB - 3'
OTKy.n3
V A =3(O,5A - ex)
H H3 (4.3) C YQeTOM (4.5)
( 4.5)
91
pmax=2N/{V AB ) .
( 4.6)
YCTaHoBH 1 KpHTepHH, Orp3HHqHBalOIlI.He M3KCHM3JIbHO
,l{OnYCTH 1 bIe IIJIOIlI.3)IJI OTpbIBa nO,lI.OllIBbI <l>YH,lI.aMeHT3.
IlpH
FQTP 0,25F, V»O,75. (4.7)
FOTp O,5F, O,75 V O,5. (4.8)
npHMep 1. jJ,aHO: pa3MepbI no.n.OlllBbI A=2,1 M, B= 1,5 M, YCHJUIH,
)J.eHcTBYlOlllHe H8 ypOBHe no.n.OlllBbI YH.lI.aMeHTa, N = 450 KH, Mx =
= 180 KH. M. Tpe6yeTcH onpe.ll.eJUITb .lI.JIHHY C>KaTOH 30HbI nO.ll.OlllBbI "
M8KCHMaJIbHOe .n.aBJIeHHe Ha OCHOB8HHe.
PaCtleT: ex= 180/450=0,4 M. T8K K8K 6e x =6.0,4=2,4>A=2.1 l\f,
TO HMeeT MeCTO OTpbIB no.aOWBbI (CM. cl>OPMYJIY 2.22). no opMYJIaM
(4.5) H (4.6):
VA=3(0,5.2,1 - 0,4) = 1,95 M; pmax=2.450/(I,95.1,5) =
=307,7 KIla; v= 1,95/2,1=0,929>0,75,
CJIe)J.OBaTeJlbHO, F OTp < 0,25F.
npHMep 2. LlJIH Tex >Ke YCJIOBlIH, \ITO H B npHMepe I, HO Mx =
= 240 KH. M, onpe)J.eJIHTb .lI.JIHHY C>KaTOH 30HbI nO.ll.OIlIBbI H MaKCHM8JIb-
Hoe )J.aBJIeHHe H8 OCHOBaHHe.
PaCfleT: ex=240j450=O,53. no cl>OPMYJI8M (4.5) H (4.6):
IlpH
VA=3(O,5-2,1-- 0,53) = 1,56 M; Pmax=2.450!1,56.1,5=
==330 Kna; y= 1,56/2,1 ==0,743_
CJIe,lJ.oBaTeJIbHO, 0,75>v=O,743>0,5 Ii 0,25F<F oTp <
<0,5F.
4.3. <J>YH aMeHTbl C cHMMeTpH1JHOH ABYTaBpoBoH
nOAowBoH
B 33BHCHMOCTH OT pa3MepoB IIO,lJ.OIlIBbI <PYH,lJ.3MeHTa II
BeJIHqHH ,LI.eHcTBYlOIlI.HX YCHJIHH B03MO}l{HbI ,lI.Ba cJIyqaSl
OqepTaHHSI cmaToH 30HbI nO,lI.OllIBbI. npHHIIMaeM a=ajA,
=b/B H v=CjA.
CJIyq3H 1 IIPH l v 0,5(I+a) (pIlc.4.2). 06neM
aIIlOpbI ,lI.3BJIeHHH H3 OCHOBaHHe Hero YCJIOBHbIH CTaTHqe-
CKHH MOMeHT OTHOCHTeJIbHO OCH y paBHbI:
1 A O,6A (1 +a)
V= f BPxdx - .f (I - )Bpxdx;
o O,5A(1-a)
-t A O. 5A (1 + a) ( 4.9 )
s= f Pxxdx- 1 (I - )Bpxxdx.
o O,5A (I-a)
92
113 pHC. 4.2:
px=pmax{VA - x)/(VA). (4.10)
PellIHB HHTerpaJlbI (4.9) .c yqeTOM (4.10), nOCJIe HeKO-
TOpbIX npeo6pa30BaHHH (onYI1l.eHHbIX )J.JISI KpaTKOCTH) no-
JIyqHM:
V = ABpm ;x;
S=l1 A2B pmax,
(4.11 )
( 4.12),
k
a 'v
"''2 X 2
t" r
. ,
: I
i , 'dx 1 .X:,
1_ 1
I
I
) . .- .
:-i
t:Q
a5A
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A
PHC. 4.2
r.lle
1:=0,5 [V - a(1- f})(2- 7 )} (4.13)
"1/= 2 -a(l- )(O,5_ 3-;; '} (4.14)
113 ypaBHeHHH paBHOBeCHSI HMeeM:
V=N=CABpmax; (4.15)
S=N( 0,5 - )A = 1'JA2BPmax. (4.16}
OTKY,1J.a nOCJIe npOCTeHlliHX npeo6pa30BaHHH nonYQHM:
Pmax=N/(CAB); (4.17)"
ex/A =0,5 - rilC. (4.18)
93
113 pliC. 4.2:
F oTP =AB(I-y); (4.19)
F=AB[l- a(l- )]. (4.20)
CJIe.noBaTeJIbHO, )J.JIH C06JIIO.!I.eHHH YCJIOBHH F OTp O,25F
Heo6xo,llHMO, qTo6w
1> '\' > 0,25[3 + a (1 - )].
(4.21 )
x
::
{f,6A "
A
.r
X t
JI
II
Puc. 4.3
CJIyqaM 2 npli V O,5 (1 + a) (pHC. 4.3). npli 9TOM <pop-
MYJIbI (4.17) H (4.18) coxpaHHIOTcH 6e3 H3MeHeHHH, a <pop-
MYJIbI (4.13), (4.14) Ii (4.21) no aHaJIOfHH co CJIyqaeM 1
npHHHMalOT COOTBeTCTBeHHO BH)J.:
C=O,5fr + (1- ) [1-(;(- O.25( -a)2 ]}; (4.22)
= +O,125(1- ) [ (I-C1)2- (l-a)3 ] ; (4.23)
6 31
J
I V O,25[3 - (I - (3) (2+a)],/(3. (4.24)
Ha OCHOBe <POPMYJI (4.13), (4.14), (4.22) Ii (4.23) BbI-
QHCJIeHbI K03<P<PHIJ.J1eHTbI II OTHOCHTeJIbHbIe 3Kcu.eHTpHcH-
TeTbI ex/A B 3aBHCHMOCTH OT pa3JIHQHbIX 3HaQeHHM a,
H v. nOJIYQeHHbIe pe3YJIbTaTbI HCnOJIb30BaHbI )J.JIH COCTaB-
JIeHHH Ta6JIHIJ.bI 4.1, n03BOJIHIOI..II-eH BeCbMa npocTo onpe,lI.e-
JIHTb pa3MepbI cLKaToH 30HbI nO)J.OIIIBbI <PYH.naMeHTa C=VA
H MaKCHMaJIbHOe )J.aBJIeHlIe Ha OCHOBaHlIe Pmax=N/ (r;,AB)
B "3aBHCHMOCTH OT a, (3 Hex/A. Ta6JIIIu.a COCTaBJIeHa C yqe-
TOM YCJIOBHH (4.21) H (4.24).
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PI - pmax , P'l. - pmax . ·
l' l'
,[{JIH CJIYQ3H 2 HMeeM P2=O.
npHMep 3. P.aHo: pa3MepbI nO)J.OWBbI <PYH.LI.aMeHTa A= 8 M, B =
=6 M, a=5,6 M; b=2,4 M; YCHJIIHI, .QeHcTBYlOlllHe Ha ypoBHe no.n.OWBbI,
N=2450 KH, M x =552,7 KH.M. Tpe6yeTcH onpe.QeJUITb .n.JIHHY C>KaTOH
30HbI no.n.OlllBbI H .n.aBJ1eHHH Ha OCHOBaHHe.
PaC'leT: e x /A=552,7/2450=0,282; a=5,6/8=O,7; =2,4/6=O,4.
no Ta6JI. 4.1 npH a = 0,7, = 0,4 Hex/ A= 0,282 H8XO.n.HM V = 0,86 H
t=0,254. F OTP <O,25F, TaK KaK 3HaqeHHH '\' H pacnOJIO>KeHbI B Ta6JIHue
BbIllie pa3rpaHHqHTeJIbHOH JIHHHH. LI.JIHHa C>KaTOH 30HbI no.n.OlllBbI V A =
=C=O,86.8=6,88 M>0,5A{1+a)=O,5.8{1+0,7)=6,8 M, (cJIyqaii 1).
no cpopMYJIaM (4.17) H (4.25):
2450 .
Prr.ax = == 201 Kna;
o "254 . 8 . 6
,
_ _ 0,85 - 0,5 (I - 0,1) 201 -- 16 1""" 9 n .
PI - - .. ) K a
o 86 "
,
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P2 0 86 -::-.= , K a.
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u
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(pHC. 4.4).
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o 0,5.4(1-a)
.4 0,5-4 (1+a) _
S= J Bpxxdx - f Bpxxdx.
o 0,5.4 (I-a)
( 4.27)
113 pHC. 4.4 HMeeM
px=pmax(VA - x) /VA .
( 4.28)
102
A
A
0,5A
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a
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0.5 a J1
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11
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x
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PRC. 4.5
PUC. 4.4
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nOCJIe HeKOTopbIX npeo6pa30BaHHH (onYIII.eHHbIX )1.:'151 KpaT-
HOCTH) nOJIYQHM:
v= :ABpmax;
S = 'r)A2.B pmax,
( 4 .29l
( 4.30)'
(4.31 }
r)J.e
=0,5 [,V - a( 2 - + )];
. 1'2 ( h 3 + (12, )
"1/ 7 =c (; -?a 0,,) - 12 1 .
113 ypaBHeHHH p BHOBeCH5I HMeeM:
V=N=:ABpmax;
S=N (O,5A - ex) =flA2Bpmax,
oTKy a nacne npocTeHlliHx npeo6pa30BaHHH
Pmax=Nf:AB;
ex/A =0,5 - ,,/:.
(4.32)
( 4.33)'-
(4.34 )'
113 pHC. 5.4 JlMee!\1:
FOTP=AB (1 - V);
F=AB(I- a(3).
nOJIY1.JHM:
( 4 .35 )
( 4.36)"
( 4.37)'
( 4.38)'
CJle,l1.0BaTeJlbHO, JI.JIH C06JIIO,lI.eHHH YCJIOBHH F oTP 0,25F
Heo6xo,lJ.HMO, lJTo6bI
103
1 V O,25(3+ap). (4.39)
.nH c.nyqaH 2 "PH V O,5(1 +a) (pHC. 4.5) <POp MYJIbl
(4.35) H (4.36) coxpaHHIOTCH 6e3 H3MeHeHHH, a CPOPMYJIbl.
(4.31), (4.32) H (4.38) no aHaJIorHH co CJIyqaeM 1 npHHH-
MalOT COOTBeTCTBeHHO BH,lJ.:
_ 0 5 { ( 1 , r 0 ,25 (I -- ex) 2 ] } .
- , l' - ) + ll-a---. 31 ' (4.40)
'1J = 2 (I - ) + 0, 125 [(l _ a ) _ (1 Ia )3 ]; (4.41)
1 >/ l' > 0,2) [3 - (2 + ex)] . (4.42)
1-
Ha OCHOBe CPOPMYJI (4.31), (4.32), (4.40) H (4.41) Bbl-
QHCJIeHbI K03<P<P H II.HeHTbI II OTHOCHTeJIbHble 3KCU.eHTPHCH-
TeTbi ex/A B 3aBHCHMOCTH OT pa3JIHQHbIX 3HaQeHHH a,
H v. nOJIYQeHHble pe3YJIbT3Tbl HCnOJIb30BaHbI ,lJ.JIH COCTaB-
JIc.HHH Ta6.n. 4.2, n03Bo.nHIOll(eti BeCbM3 npocTo onpe,lJ.e.nHTb
AJIHHY C}I{aTOH 30HbI nO,lJ.OIIIBbI <PYH,lJ.aMeHTa C=VA II MaK-
CHMaJIbHOe ,lJ.aBJIeHHe Ha OCHOBaHHe Pmax==N/ ( B) B 3a-
BHCHMOCTH OT a, Hex/A. Ta6JIHu.a COCTaBJIeHa C yqeTOM
YCJIOBHH (4.39) H (4.42).
,QaBJIeHHH Ha OCHOBaHHe B ,lJ.pyrHx npoMe}l{YTOqHblX TOq-
Kax nOJJ.OIIIBbI onpe,lJ.eJIHIOT no CPOPMY JIaM:
1'-O,, (] -ex) j-O,F\(I + ex)
0 1 .= Pmax ; P2 = Pmax ·
l' l'
,QJIH CJIyqaH 2 HMeeM P2.=O.
npHMep 4. 1I.UIlO: pa3MepbI no.n.OlIIBbI <PYH,lJ.aMeHTa A= 8 M, B-==
=6 M" a=5,6 M; b=3,6 M; YCHJIHH, .n.eHCTBYIOllI.He H8 ypOBHe no.n.o-
WBhI, rv =2450 KH, M x =550 KH. M. Tpe6yeTcH onpe.n.eJIHTb .n.JIMHY C>Ka-
TOM 30Hbl no.n.olliBbI H ,lJ.aBJIeHHH Ha OCHOBaHHe.
PaCtteT: e x /A=550/(2450.8) =0,282; a=5,6/8=O,7; =3,6/6=0,6.
no Ta6JI. 4.2 npH a=O,7; =O,6 H e x /A=0,282 Haxo.n.HM =O,254 H
y=0,86. F oTP<O,25F, TaK K8K 3HaqeHHH H l' pacnOJIO>KeHbI B Ta6JIH-
D-e BbIllie pa3rpaHHqHTeJIbHOH JIMH"H.
i{JIHHa C>KaTOH 30HbI no.n.OlllBbI C=1'A=O,86.8=6,6'8 M>O,5A(1 +
+a)=O,5.8(1+0,7)=6,8M (CJI)'qaH 1).nocpopMYJI8M (4.35)" (4.43):
Pmax == 24.30/(0,2-54 .8,6) = 201 KDa;
P 1 = 201 0 · 86 - O. fi (I - O. 7 ) === 165 K n a;
0,86
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P2 -=---=: 201 == ,.. K a.
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CTBYIOlI.{HX YCHJIHH B03MomHbI ,nBa CJIyqaH 01lepTaHHH c}I{a-
TOB 30Hb! no,nOWBbI. npHHHMaeM a=a/A, P=b/B H-
V=C/A. CJIyqaH 1 npH O,5 V O,5(1+a) (pHc.4.6).
x
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y
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f A
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06'beM 3TIIOpbI .n.3BJIeHHH Ha OCHOB3HHe Ii ero YCJIOBHblH
cT3TH ecKliH MOMeHT OTHOCHTeJIbHO OCM y:
O,5A (I-IX) jA
V= J pBpxdx+ J BPxdx;
o O,fA(1-a)
O,!)A (t - a) jA
S = f Bpxxdx+ f Bpxxdx.
o 0, 5A (1 - a)
( 4.44 )
( 4.45)
113 pIlC. -4.6 CJIe,lI.yeT
Px=Pmax(V A - x)/(VA). (4.46)
PellIHB HHTerpaJIbI (4.44) H (4.45) C y eToM (4.46),
nOCJIe HeKOTopbIX npeo6pa30BaHHH (onymeHHbIX ,l.JISI KpaT-
KOCTH) nOJIY1fHM:
112
V=CABpmax;
S =1}A2Bpmax,
rJle t = 0,125 \.1 - ex) (1 -?) C a -4) + 0,51;
== 0, 125 (1 - Cl) 2 (1 - ) ( ] - ex - 1 ) + J:.
3"( 6
113 YCJIOBHft: paBHOBeCH}I:
V=N=CABpmax;
S =N (O,5A - ex) =flA2Bpmax,
OTKy,n3
pmax=N/ AB;
ex/A =0,5 - f1/C.
( 4.47}
{ 4.48}
( 4.49)
(4.50)
(4.51):
( 4.52J
( 4.53J
( 4 .54
113 pRC. 4.6 npn
F OTp =O,25F, 'V >O,25[2+a+ (I- a)]. (4.55)
,L{JIH CJIyqaH 2 npH 1 1' 0,5(I+a) (pRc.4.7) coxpa-
HRIOTCH <pop MYJIbI (4.53) H (4.54) 6e3 H3MeHeHHH, a <l>op-
MYJIbI (4.49), (4.50) II (4.55) no aH3JIOrHH co CJIyqaeM 1
npHHHMalOT COOTBeTCTBeHHO BHJI.:
;:=a{l- ) ( l- ) +0,5 V;
2"(
==0 12.5(1_ ) [ 2a(G"(-ex2-- 3) + 4 "(2 ] .
"'l, . 3, 3(1- ) ,
'V , __ 1 _ 0, 2.5 [ex + (] - ex) ].
,
(4.56J
( .57)
( 4 '58)
DpHBeJI.eHHbIe cpOPMYJIbI HCnOJIb30B3HbI JI.JI}I BhlIIHc.ne-
HHH K03<p<pHI.lHeHTOB H OTHOCHTeJIbHbIX 3Kco.eHTpHcHTeToB
ex/A B 3aBRCHMOCTH OT p33JIRIIHbIX 3HalleHHH a, p H v. ,aJISI
npaKTHllecKHx paclleToB COCT3BJIeHa Ta6JI. 1 npHJI. 1, n03-
BOJIHIO aH BeCbMa npocTo onpeJI.eJIHTb JI.JIHHY C1K8TOH 30HY
nOLlOIliBbI C=VA H MaKCHMaJIbHOe JI.aBJIeHHe Ha OCHOBaHHe
Pmax=N/( B) B 33BHCHMOCTH OT a, p Hex/A. Tao.nHu.a
COCT3BJIeHa C yqeTOM YCJIOBHH (4.55) H (4.58).
ilaBJIeHHH 'Ha OCHOBaHHe B JI.pyrHx npOMe>KYT01fHbIX
T01JK3X nOJI.OllIBbI onpeJI.eJIHIOT no <popMYJIaM:
P. =--= Pmax ,- OJ> (I - It) ; P2==pmax "(-0,5 (1 + «) . (4.59)
"( "(
113
c>
.
aXJ I i X
r -==t
t.%fl t/./, . %
'/'A/ 7// r >
.x 1// ///:'.r/ :I/{ }!/fJ /_
". // - .. - f" / "/ "', / 'IIi.......
;-:,:, -%: !/ . / tt -P0J' J.
, ,.: ; ai / ?J: : a .
:, /:'y r-F --
;/}1 J k1 2
I 1 ;1 I I '-
I ; ! dX 11 X 1 $)
I ,- .Il'
Ii! 1--.( --:
i dX",11 IX?
I --r -t-
to i . -L. fA;
D,5A (i-of.) I a = _l :7,5 1': )
I t7. iq.__.----J__ -;--I
I: iA!
ex N Y
.P1
t:Q
I.(")
c:$
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jI
-
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.llJIH CJIyqaH 1; P2 = O.
npHMep 5. p'aHo.: pa3MepbI nOJl.OlllBbI YH)J.aMeHTa A=4 M, B=3 M,
a= 1,6 M, b = 1,8; YCH.nHH, )J.eHcTBYIOUJ.He Ha ypoBHe nO)J.OlllBbI N =
= 1000 KH, M x =720 KH.M. Tpe6yeTcH onpeJl.eJlHTb ,aJlHHY C)KaTOH 30-
HbI nO.lI,OIllBbI Ii .u.aB.neHHH Ha OCHOBaHHe.
PaCtteT: ex/A=720/1000.4=O,18; a= 1,6/4'=0,4; = 1,8/3=0,6. no
Ta6 TJ. 1 npHJI. 1 npn a=0,4, =0,6 H e x /A=0,18 HaXO.ll.HM =0,321 H
,\,=0,85. CJIyqaH 2 OqepTaHHH C>KaTOH 30HbI nOJl.OWBbI, TaK KaK 1>'V=
=O,85>0,5(I+a) =0,5(1+0,4) =0,7. TaR KaK K034> Hl{HeHTbI t 11 'V
pacnOJIO:>KeHbI B Ta6J1Hu.e BbIllie pa3rpaHHqHTeJIbHOH JlHHHH, TO FOTP<-
<O,25F.
,[{JIHHa C>KaTOH 30HbI no.u.OWBbI: '\' A = 0,85.4 = 3,4 M;
pmax= 1000/(0,321.4.3) =259,6 Klla.
no opMYJIaM (4.59):
= 259 6 0,8.) - 0,5 (1 - 0, 4 ) === 169 I{ n a ·
Pl, 0 85 '
,
=== 259 6 0,85 - 0,5 (I + 0,4) 45 8 n
P2, 0 85 ===, I{ a.
,
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BapHaHT3, 3aBHCHII.J.He OT HarrpaBJIeHHH ,lI.eHCTBHH MOMeHTa
M x , Ka>K,nbIH H3 KOTOpbIX cO,QepLKHT no ,QBa CJIyqaH, or;rpe-
,lteJIHIOlll.HX 01IepTaHHe CLKaTOH: 30HbI rrO,QOIIIBbI. OnycKaH
npOMe}l{YTOIIHbIe BbIKJIa,l(KH, nOJIYIIeHbI CJIe,QYIOInHe BhIpa-
)KeHHH JJ.JI5I HCnOJIb3yeMbIx reOMeTpH1JeCKHX napaMeTpoB no-
,ltOJllBbI:
ITJIOm3,l(b rrO,fLOIllBbI
F=AB(l+a - a),
( 4.60 J
paCCT05lHHe OT KpaH nO,fLOIllBbI ,fL0 ee l.(eHTpa TH}I{eCTH
Xc = J.txA, ( 4.61 J
rAe
o ,5 [1 - a 2 (1 - )]
a===ajA; =b/B; P,x= ·
1 + a - ex
115
BapHaHT I, c.Ilyttat1 1. MOMeHT Mx ,neiicTByeT B Ha-
npaBJleHI!H pe6pa, HeHTpaJIbHaH JIHHHH nepeceKaeT nOJIKY
1 ,, a (pHC. 4.8). 06'beM 3nlOpbI ,l(aBJIeHHH Ha OCHOBa-
HHe Ii efo YCJIOBH H CTaTHQeCKHH MOMeHT OTHOCWTeJIbHO
OCR Y:
aA rA
V= J bpxdx+ J Bpx dx ;
o aA
( 4.62)
a.A r A
S= J bpxxdx+ J Bpxxdx.
(9 «A
( 4".63)
n OCJIe pemeHHH IfHTerp aJIOB (4.62) H (4.63) If COOTBeT-
CTBYIOllI-HX npeo6pa30BaHHH C yqeTOM Toro, QTO
px=Pma:l:[(V A - x)f{yA)],
V= ABpmax,
S = l1A 2Bpmax ,
( 4.64 )
( 4.65)
( 4.66)
BOJIYllRM
rAe
C=O,5y - a(l - ) (1- );
( 2'
'I1=y2 - a 2 (1 - ) 3 --f)-
113 YCJIOBHH paBHOBeCHH HMeeM:
V =N =CABpmax;
N[(l - J.l,x)A - ex]=NS/V,
( 4.67)
(4.68)
( 4.69)
( 4.70)
OTKYJI.a
pmax=NI( ,AB);
exlA= 1 - f.tx -111 c.
nJI01I.{aAb OTpbIBaeMOH qaCTH nO,l(OIliBbI:
F OTP = (1 - V)AB;
(4.71 )
( 4.72)
np. F OTP =O,25F
npH F OTP =O,5F
V=O,25[3+a(1- )];
( 4.73)
(4.74)
,\,=0,5[1 +a(1 - P)]. (4.75)
AUftJlOPHQHO nOJIyqeHbI cJleJ(YIO He <t>OPMYJlhI J(JIH py-
rHx BOIM01KHbIx cJlyqaeB g1{epTal'lftH cJKaToA 30HbI nOAo-
WBhI.
116
A
t } a -" _ - ---a---j
'- n ' It
! L \ I.
I ;;/ . > /// I
i \ /j T ' , //;.: '
x : . " »> > .(,. ; J :
,.,....1 : . '/"',/,'k '//
"""'-J I .... . . /.
1 - \ : I : ---!' - -
! . e----J.
LLJ,-.. h
I Jt'! A : j ( 1-.,,1.1. ) A l
+----fj "1
r .ftl!Htt pI
I , .
I Y'A :
iJ of
PRC. 4.9
BapHaHT I, cAytlaU 2. HeHTpaJIbHaH JIHHHH nepeceKaeT
pe6po v<a (pHC. 4.9) :
f: =0,5 V;
11 = 1 /613v 2 ;
ex/A = 1 - J.Lx - (l1l C .);
( 4.76l
( 4.77)
( 4.78)
npH F OTp =O,25F
,\,=0,75[1- a(l- )]/ ,
,\,=0,5[1- a(l- )]/ .
( 4.79)
( 4 .80 )
npH F OTp - 0,5F
BapHaHT II, cJlYtlau 1. MOMeHT Mx eHcTByeT B Ha-
npaBneHHH nOJIKH, HeHTpaJIbHaH JIHHHH nepeceKaeT pe6po
l-a y 1 (pHc.4.10):
= 21 +(I-a)(I- )(I- I IX) (4.81)
12 + (I - IX)2 (I - ) [3 - 2 (I ; IX) 1
== 6 ' (4.82)
ex/A=f1x - (f)/C);
( 4.83)
DPH F OTp =O,25F
i= -O,25[1-Cl(1- )]/ ,
(4.84 )
111
A
A
r
I
a
x
C()
e
!I pA (f-,p)A
PHC 4.10
PHC. 4.11
npH F OTP =0,5F
3 0 _ [ '" ' I ' ] 'Q
""= - . i-a.' - ) j .:J
& " \ , .
( 4 . 85 )
BapHaHT II, cAyttaU 2. HeHTpaJIbHaH JIHHHH nepeceKaeT
nOJIKY V I-a (pHC. 4.11):
C=O 51\,.
, r ,
TI.=1/ 6r ;
ex/A =Jlx - (Tl/C);
( 4.86)
( 4 .87)
( 4.88)
npH F O 'fP=O,25F
1'=0,75[1- a(l- )],
( 4.89)
npH F OTP =O,5F
V=0,5[1 - a(l - )].
( 4.90)
fIpHBe,lI.eHHble CPOPMYJIbI HCnOJIb30BaHbI ,lJ.JIH BbltIHCJIe..
HHH K03cpcpHI.lHeHTOB , TI H f.1x, a TaK}I{e OTHOCHTeJIbHblX
3KClI.eHTpHCHTeToB exlA B 3aBHCHMOCTH OT pa3JIHtIHbIX 3Ha..
qeHHH a, H V ,lI.JIH Bcex lIeTbIpex paccMoTpeHHblx cJIyqaes
01IepT3HHH C>KaTOH 30HbI nO,lJ.OIlIBbI.
,[{JIH npaKTHlIeCKHX paC1.IeTOB COCTaBJIeHbI T36JI. 2, co-
oTBeTcTBYIO aH BapHaHTY I, H Ta6JI. 3 (npHJI. 2), COOT BeT-
118
CTBYIOLUaH BapHaHTY II, n03BOJIHIO He BeCbMa npocTo Ha-
XO,lI.HTb ,lJ.JIHHY C}I{aTOH 30HbI nO,lJ.OIlIBbI VA H MaKCHMaJIbHOe
,lI.aBJIeHHe Ha OCHOBaHHe Pmax=N/ ( AB). ,I:LJIH ,lI.pyrHx npo-
Me}l{YTOQHblX TOqeK nO,lI.OlllBbI ,lI.aBJIeHHH Ha OCHOBaHHe on-
pe,lI.e TI5IIOT no cpopMYJIaM:
BapnaHT I, CJIyqaM 1 p1=Pmax[(V - a)/v];
CJIyqaM 2 P1 = 0;
BapnaHT II, CJIyqaH 1 P1=pmax[V - (1- a)]/v;
CJIyqaM 2 P1=0.
3HaQeHHH f.1x npHHHMalOT no Ta6JI. 2.4.
(4.91 )
npHMep 6. JI.aHo: pa3Mepbl nOJI.OlIIBbI Q>YHJI.aMeHTa A= 3 M, B =
=2,4 M, a = 1,5 M, b= 1,2 M. YCHJIHH, )J.eHcTBYIOIll.He Ha ypoBHe no.lI.O-
WBbl <PYHD.8MeHTa, N =800 KH, M x =450 KH. M. BapHaHT II HanpaB-
JIeHHH .n.eflcTBHH MOMeHTa. Tpe6yeTcH onpeJI.eJIHTb .lI.JIHHY C1KaTOH 30.Hbl
no.n.OWBbI II JI.aBJIeHHH Ha OCHOBaHHe.
PacfleT: e x /A=456j800.3=0,19; a "- l,5/3=O,5; = 1,2/2,4=0,5. no
Ta6JI. 3 npHJ1. 2 3HaqeHHHM e x /A=O,19, a=0,5 H =0,5 COOTBeTCTSY-
lOT 1'=0,735 II =0,349. .lI.JIHHa C1KaTOH 30HbI nOJI.OlIIBbI "A=O,795.3=
=2,21 M; Pmax=800j(0,349.3.2,4)=318 KTIa. TaK KaK 3HaqeHHH V H
t paCn0J10LKeHbI B Ta6JIHu.e BbIllie pa3rpaHHqHTeJIbHOH JIHHHH, TO F OTP<
<0,25F.
4.7. YHAaMeHTbI C KOJlb",eBOH H KpyrJloH
u
nOAOWBOH
PaccMoTpHM CHaqaJla YH,lJ.aMeHTbI C KpyrJIOH nO,lI.Olll-
BOM. Bce 0603HaQeHH5I o6 enpHHHTbIe (pHC. 4.12).
I13 pnc. 4.12 HMeeM
px=pmax(VRt- X)/(V R 1).
I13 ypaBHeHHH OKPY}I{H OCTH {x - R1)2+ y2=R 1 2 HMeeM
y=1'2R1 X - X2.
3JIeMeHTapHbIM 06'beM 3nlOpbI ,lI.aBJleHHH Ha rpYHT Hero
YCJIOBHbIM CTaTHlleCKHH M OMeHT OTHO CHTeJlbHO OCH y paBHbI:
dV === 2YPx dx == 2 V 2Rt x - x 2 pmax aRt - x dx =
aRt
= 2pmax V 2Rt x - x 2 (1R. - x) dx;
aRt
dS y = 2Pmax y 2RtX - x 2 ("(' 1 -x) xdx,
aRt
119
)f
R ,
X
'i'R .,
x
!J
PHC. 4.12
OTK}'Jta
C) ,Rl
V = -Pmax \ 1/ 2R X- ' x 2 ('VR - X) dx.
,Rt t.I r -1 L. ,'1 , ,
o
( 4.92)
. -(R 1
S 2pmax , :. r
::..:= V 2 R x - X 2 ( R - X ) xdx
y IRt '! 1 \1 1 .
I)
( 4.93)
PellIHB HHTerpaJIbI (4.92) H (4.93),
BYIOlll.HX npeo6pa30BaHHH nOJIYIIHM:
V=Rt2pmaX I;
S y = R t 3 Pmax'r}t;
nOCJIe COOTBeTCT-
( 4. 94)"
( 4.95):
fJle
I == 2 [ (, 1 )2 V l' (2 - 1') + I - 1 arcsi n ( 1- 1 ) +
21 2,
+ ;, V .({ (2 - 1) 3 + 7t (j 1 ) ]; ( 4.9 6)
u = 2 - ( O,t5-2- ) \arcsin (1- 1 ) + ( 0,5- 5 ) ( 1-1) V 1(2 -1) +
_ 8, 8,
+ .5 - , ...:, (2-" ) 3 + 1t (4, -- .5) ] . (4.97)
I ,. (\ ( 1 ul .
120
OTKY,lI.a
113 YCJIOBHM paBHOBeCHH HMeeM:
N= V=Rt2Pmax t; (4.98)
Rt- ex=Sy/V=Rt('r}I/ t), (4.99)
Pmax=N/(Ri 2 :1); (4.100)
ex/R1=1-'r}1/:t. (4.101)
<f>YHJJ.8MeHTaM C KOJIbII.eBOH nO,lI.OllIBOH cooTBeTcTBYIOT 06-
JIaCTH 3nlOpbI ,lI.aBJIeHH5I H"a rpYHT B npe,lI.eJIax Kpyra pa,lI.I1Y-
COM R 2 , KOTOpbIe Ha30BeM BHYTpeHHeH 06JIaCTblO. 3,lI.eCb
B03MO}l{HbI ,U.Ba CJIYIIaH paCnOJIO>KeHHH HeHTpaJIbHOH JIH-
HHM.
CI/ yttau 1. HeMTpaJIbHaH JIHHHH nepeceKaeT nO,lI.OllIBY B
npe,AeJIax BHYTpeHHeH 06JIaCTH V 1 +a, r,lI.e a=RJR 1
(pRC. 4.13). 0603HalIRM ,lI.JIHHY C>KaTOH 30HbI nO,lI.OllIBbI B
npeJ{eJIax BHYTpeHHeM 30HbI v1R2=VtaRt. ¥l3 pHC. 4.13:
, 1 + a
P - P I - · ", N R - ( ", 1 + a. \ R
1 - max , '1.1. 1 --- .. , - . ) .,
"'(
OTKY,lI.a
'\'1= (V -1 +a)/a.
06'beM 3nlOpbI ,lJ.aBJIeHHH Ha rpYIJT B npe,lI.eJIax BHYT-
peHHeH 06JIaCTH Hero YCJIOBHbIH CTaTHqeCKHH MOMeHT
OTHOCHTeJIbHO OCH Y1 MO>KHO onpe,lJ.eJIHTb no cpopMYJIaM
(4.94-4.97), 3aMeHHB B HRX Pm ax Ha Pi, V Ha V1 H R1 Ha R. 2 .
nOCJIe YKa3aHHbIX 3aMeH H COOTBeTCTBYIOI..II.HX npe06pa30-
B8HMH nOJIyqHM ,lI.JIH qaCTH 3nlOpbI B npe,lI.eJIaX BHYTpeHHeH
06JIaCTH 3HaqeHHH H 112 H ,lI.JIH BCeM 3nlOpbI )J.aBJIeHHH Ha
rpYHT: ===t1- 2; 'r}='r}t-'r}2- (I-a).
OTHOCHTeJIbH H 3KCII.eHTpHCHTeT npO,lI.OJIbHOH CHJI on-
pe,lJ.eJIHM no cpOPMY JIe
eX/R1= 1 - 'r}/ . (4.102)
CAyttau 2. HeHTpaJIbHaH JIHHHH nepeceKaeT nO,lI.OIllBY
3a npe,lI.eJIaMH BHYTpeHHeM 06JIaCTH 1 + a:::;;;v 2 (pHC.
4.14). PaClIJIeHHM BHYTpeHHlO1O 06JIaCTb Ha ,U.Be qaCTH-
nOJIYIJ.HJIHH,U.P 1, 2, 3 pa,lI.HycoM OCHOBaHHH R2= aRt, 8bI-
COTO" P1 - P2 H IJ.HJIHH,lI.P 2, 3, 4, 5 C TeM }l{e pa,lI.HycoM
OCHOBaHHH BbICOTOM P2. 113 pHC. 4.14:
"'( - 1 - r:J.. _ 2a
P2 == Pmax , PI - P2 -- pmax-.
"'( "'(
CJIeJtoBaTeJIbHO, .lI.JIH ,lI.BYX YKa3aHHbIx lIaCTeM 3nlOpbI ,lI.aB-
neHHH Ha rpYHT B' npe,lI.enax'BHYTpeHHeA o6nacTH nonYQHM
COOTBeTCTBeHHO:
121
tt"
R 1
crt":
I I
4 1 I o
x
x r:<
--
PHC. 4.13
Y. Y
Puc. 4.14
C 3 = 0,5na 2 . 2a/v= 'Ita 3 /v;
4='Ita2(1' - 1 - a)/v;
1]3== 0, 75 3a (OTHOCHTeJIbHO OClI Yl); 1]4= 4,
r)1.e O,75 3 - OTHOCHTeJIbHaH npOeKl.lHH Ha nJ10llI.aJI.b nOJI.OlllBbI paCCTOH"
8MB OT ueHTpa TH1KeCTH nOJIYUHJIHH.n.pa BHYTpeHHeii 06JIaCTH )J.O OCH Yt.
,I:LJIH BceH 3IIIOpbI .n.aBJIeHH5I Ha rpYHT HMeeM
=. 1 - C 3 - :4; 11=1]1 -:3(0,75a+ 1 - a) - :4---.:
=-= rJI-C a (l - 0,25a) - 4.
3HaqeHHe e X /R I orrpe.n.eJIHIOT no <popMYJIe (4.102). 3a-
.u.aBa5lCb pa3JIHtIHbIMH peaJIbHbIMH 3HaIIeHHHMH V C HHTep-
BaJIaMH, paBHbIMlI 0,02, no nplIBe.n.eHHhIM cpopMYJIaM BbI-
IIHCJIeHbI cooTBeTcTBYIOJ.l{He 3HaIIeHHH H OTHOCHTeJIbHbIe
3KClI.eHTpHCHTeThI ext Rl, CBe.n.eHHbIe B Ta6JI. 4.3, C rrOMOlll.blO
KOTOpOH BeCbMa npocTo onpe,lJ.eJI5ITb Pmax H O .n...TIHHY C./KaTOH
30HbI rro.n.OllIBbI vR 1 no 3a,aaHHOMY OTHOCHTeJIbHOMY 3KC-
u;eHTpnCHTeTY e X /Rl H a=R2/Rl. LLJIH npOMeJKYTOtIHbIX
3HaqeHHH e X /R 1 H a BeJIH'-IHHbI H V onpeJI.eJI5IIOT no HHTep-
nOJI 51 I..I.HH.
Onpe,lI,eJIHM rpaHHIIHbIe 3HaIIeHHH Vrp, npH KOTOpbIX
Y OBJIeTBOp eTCH YCJIOBHe
F OTP =0,25F, (4.103)'
r)J.e F OTp - nJlOw.a)J.b OTpbIBaeMOH qaCTH nO.ll.OlllBbI; F - BCH nJIOllI.a)lb
nOJI.OlUBbl.
113 pnc. 4.12 3JIeMeHTapilaH nJIOJ.l{a,lJ.Ka C}KaTOH 30HbI
nOJJ.OllIBbI:
122
-
e x ' Rl
I .
0,25
0,26
0,27
0,28
0,29
0,30
0,31
0,32
o 33
,
0,34
0,35
0,36
0,37
0,38
o 39
,
0,40
0,41
0,42
0,43
0,44
0,45
0,46
0,47
0,48
0,49
0,50
0,51
0,52
0,53
0,54
0,55
0,56
0,57
0,58
0,59
0,60
0,61
0,62
o 63
,
0,64
1 ,233
1 , 233
1 ,213 I, 977 I, 164
1 , 193 1,946 I, 146
1 , 174 1,917 1, 128
1 ,154 I, 888 1, t 10
I ,135 I, 860 I, 092
] , I 16 1,832 1,075
I ,097 1,804 I, 057
1 ,078 1, 777 1, 040
1 ,059 t, 750 I ,023
1 ,OtO 1,7'23 1,006
1 ,022 I, 695 0 , 988
t ,003 I, 668 I, .971
1 ,641 0 , 954
t , 613 0 , 937
I ,584 0, 920
1 1,556 0,902
1,526 0,885
1 , 4 5 ° , 866
1 ,463 0, 848
1 ,430 0, 829
1 ,397 0 , 81 0
I ,363 ° , 791
1 ,328 0,772
1,293 0,752
1,258 0,732
1 , 222 ° , 712
1 ,186 0 , 691
1 , 150 0, 671
I , 1 1 4 0 , 650
1 ,077 0,629
1 ,041 0,607
] ,004 0,586
(X.::ZO
a=0.4
a=O.45
v
t
t
v
v
2,000 1 ,571
1 , 961 1,540
1 , 024 1 ,5] °
1 ,889 1 , 480
I 854 1 ,450 2 ,000 I ,319 .
1 ' 820 1 421 ] 967 1 ,297 2 ,000
1 :787 1 :392 1: 934 1,275 1 ,970
] , 755 1 ,364 1 ,903 I ,253 I, 938
1 723 1 336 1 873 1 2.32 I, 908
, , , ,
1 ,692 1 ,307 I ,843 1 ,211 1,878
1 ,66 I 1,279 ] ,813 1 , 190 1,849
1,63] 1,252],7841,169 1,821
] ,601 1,224 I ,755 1 ,148 I, 793
] ,571 I, I 97 I ,727 1 , I 28 I, 765
] ,542 I ,1.70 1 ,699 1,107 1,737
I ,512 I , 143 I ,6.71 1,087 I, 709
1 ,484 1 , 116 I , 643 1 ,067 I, 682
1 ,455 1 , 089 I ,615 I ,0 t6 1,654
1 ,427 I ,063 I ,587 1 ,926 1,626
I ,:399 I ,037
1,3711,010
I ,343 0, S 84
1 ,316 0, 9,)9
1 ,288 0, 933
1 ,261 0, 908
I 234 0,883
,
1 , 207 0, 858
1 , 181 0,833
I , 154 0 , 808
1 , 128 0, 784
I , 1 02 0, 759
1,0750,733
1 , 04-9 0, 712
1 ,023 0,688
I ,559 I ,006 1 ,598 0,984
I 0.31 o 985 1 ,570 0,965
, ,
1,5020,965 1 ,542 0,946
I ,47 .: 0,945 1 ,5]2 O, 27
I 44. J 0,924 1 , 483 0,908
,
1,413 0,903 ] ,452 0,888
1 , 381 0,881 I , 420 0,868
t ,31-9 0,860 1 , .388 0,848
1 ,316 0,8.38 I , 354 0,828
] , 283 0,816 1 , :321 0,807
1 , 249 0,793 1 ,286 0,786
I ,216 0,771 1 , 2,52 0,764
1 , 182 0,748 1,217 0, 743
I , 147 0,725 1 , 182 0,721
1,113 a,702 I , 147 0,699
1 ,079 0,678 1,111 0,617
1 ,043 0,655 1 ,076 0,655
1,011 0,632 1 , 041 0,632
I , 005 0,609
i I
ex. = 0.5
'V
'fa6nH a 4.3
a=O.55
v
I t
i
1 ,987 1,088
1 ,957 1,072
1 ,929 1, 056
1 , 90 I 1,040
1 ,873 1,024
I ,8!6 1, 008
I ,81 9 ° , 992
I ,792 0, 977
I ,766 0,961
I ,739 0, 946
1 ,712 0 , 930
1 ,685 0 , 9 I 5
I ,6 j8 0, 900
I ,630 0, 884
1 ,60] 0 , 868
1 ,572 0, 852
I ,540 0,835
1,510 0,819
I ,477 0,802
1 ,444 0, 785
1 ,41 ° 0 , 768
1 ,375 0, 750
1,340 0,732
1,304 0,714
1,268 0,696
I , 23 I 0 , 677
1 , 1 95 0, 658
I ,158 0 , 639
I ,120 0, 620
1 ,083 0,600
1 ,045 0,580
1 ,007 1 0, 560
123
npoiJoll eHue Ta6JL. 4.3
a = 0,6 a=O,65 a=0,7 1 a=-0,75 . I a.=0,8
e x 'R 1 I
I I
V t V V I V t V t
I
I
I
0,34 2,000 1 ,005
0,35 1 , 971 0,991
0,36 1 ,943 0,976 1 , 988 0,902
0,37 1,916 0,962 1 , 961 0,889
0,38 1 ,890 0,948 1 , 934 0,877 1 ,980 1 , 793
0,39 I ,863 0,934 1 , 908 0,865 1 , 954 0,783 2,000 0,687
0,40 1 ,837 0,920 1,883 0,852 1 , 929 D,772 1 , 977 0,679
0,41 1,811 0,907 1 ,857 0,840 1,904 0,762 1 , 952 0,671 2,000 0,566
0,42 1 , 785 0,893 I , 832 0,828 1 ,879 0, :752 1 ,928 0,662 1 , 976 0,559
0,43 1 , 758 0,879 1,806 0,816 1 , 854 0,741 1,903 0,654 1 ,953 0,552
0,44 t , 732 0,865 1 , 780 0,804 1 , 829 0,731 1 ,879 0,645 1 , 929 0,545
0,45 1 , 705 0,852 1 ,754 0,792 1 ,804 0,721 1 ,855 0,637 1 , 905 0,539
0,46 1 ,678 0,838 1 , 727 0,780 1 , 778 0,711 1 ,830 0,628 I ,882 0,532
0,47 I ,650 0,824 I , 700 0,768 1 ,752 0,700 1 ,804 0,620 1 ,857 0,526
° 48 I ,621 0,810 I , 672 0,756 1 ,.724 0,6£0 1 , 778 0,611 1,832 0,518
,
0,49 1 ,59 I ° ,.795 1 , 640 0,742 1 ,696 0,679 1 ,750 0,602 1 ,805 0,511
0,50 1 , 5 9 0,780 1,611 0,730 1 ,665 O,6E8 1 , 721 0,593 1 , 777 0,504
0,51 1 ,527 0,765 , ,5800,717 1 ,624 0,657 I ,691 0,584 1 ,748 0,497
0,52 t ,4-94 0,750 1 ,547 o ,70411 ,602 0, 646 1, 659 0,575 1,718 0,490
0,53 I ,460 0,735 1,513 ° ,691 1,569 0, 635 1 1,627 0,565 ] ,686 0,482
0,54 1 , 425 0,719 1 ,479 0,677 1,536 0, 62 '3 1, b 4 0,5561 1,654 0,475
0,55 1 ,390 0,703 1 ,444 0,663 1,501 0,611 1 ,560 0,546 1,621 0,467
0,56 1 ,355 0,686 1 ,40d 0,649 I ,466 0,599 I ,526 0,536 I ,587 0,459
0,57 1,318 0,670 1 ,373 0,635 1 ,430 0,587 1 ,490 0,526 1 , 553 0,452
0,58 1 ,282 0,654 1 ,336 0,620 J ,394 0,575 1 ,455 0,516 1,517 0,444
0,59 1 , 245 0,637 1 , 299 0,605 1 , 357 ° ,562 J ,418 0,506 1 ,482 0,436
0,60 I ,207 0,620 1 , 261 0,591 1 ,320 0,550 1 ,381 0,496 1 ,445 0,427
0,61 1 , 169 0,602 1 , 223 0,575 1 ,282 0,537 1 ,344 0,485 I , 408 0,419
0,62 1,131 0,585 1,185 0,560 1 ,243 0,524 1 ,306 0,474 1,371 0,411
0,63 1 ,093 0,567 1,146 0,544 1 ,204 0,510 I ,267 0,463 1 ,333 0,402
0,64 1 ,054 0,549 1,107 0,528 1,16E 0,497 1 , 228 0,452 I ,294 0,393
0,65 1 ,015 0,530 1 , 068 0,512 I , 1 26 ° , 483 1 , 189 0,441 1 , 255 0,385
0,66 1 , 028 0,496 1 ,086 0,469 1,149 0,429 1,215 0,376
0,67 1 ,046 0,455 1 , 1 09 0,418 1 , 175 0,366
0,68 I ,005 0 ,440 1 ,068 0,406 1,135 0,357
O,G9 1 ,027 0,394 1 ,094 0,348
0,70 1 ,053 0,338
0,71 1,011 0,328
124
liF c>K=2ydx=21'2R 1 x - x 2 dx,
CJIe,nOBaTeJIhHO,
,Rl
FC)K=2 J 1"2Rtx - X2 dx.
(J
( 4.104)
PeWHB HHTerpaJI (4.104) If npHpaBHHB F cm =0,75F,
nOCJIe pH,na npeo6pa30BaHHH nOJIYIIHM:
2Vrp - 5V 2 rp+4V 3 rp - V 4 rp - 0,6169+
-r- 1,5 708. arcsin (Vrp - 1) - arcsin 2 (yrp - 1) =0. (4.105)
113 BbIpa:>KeHHH (4.105) nOCJ1e,nOBaTeJIbHbIMH npH6J1H-
:>KeHHHMH onpe,neJIeHO rpaHHIIHOe 3HatJeHHe Vrp= 1,403 )(JlH
KpyrJIOro <PYH,naMeHTa II II3 pHC. 4.12 IIMeeM npH F OTP =
=0,5F; ,,=1.
AHaJIOflIlIHO HaH,neH rpaHHIIHWe 3HaqeHHH JIH KOJlh-
neBbIX <PYH,U.aMeHToB, CBe,U.eHHbIe B Ta6JI. 4.4.
Ta6J1HQa 4.4
a I (Xrp a 1'rp I 1'rp v rp
I I
0 1 ,403 0,55 1 ,535 0,75 1 ,625
0,4 1 ,474 0,60 1 ,557 0,8 1 ,652
0,45 1 ,493 0,65 1 ,579 - -
0,5 1,513 0,7 1 ,601 - -
,QaBJIeHHH Ha OCHOBaHHe B JLPYfHX npOl\1e>KYTOIIHbIX TOtJ-
Kax nOJI.OIllBbI -onpe,neJIHIOT no CPOPMY JIa M:
( -- 1 + rJ. "'( - 1 - a ( 4 1 06 )
PI == pmax ; P2 = pmax · \.
1 1
,LLJIH CJIyqaH 1 HMeeM P2==0. ,[{JIH KpyrJIbIX CPYH,U.aMeH-
TOB BO Bcex npHBe.n.eHHbIX cpopMYJIaX npHHHMalOT a=O.
npHMep 7. p'aHO: pa3MepbI nO.lI.OlllBbI KOJIbu.eBoro 4>YH.lI.3MeHT3
R 1 = 5 M, R 2 = 3 M; YCHJIHH, .n.eiiCTBYlOllI.He Ha ypOBHe nO.n.OIllBbI 4>YH.na-
MeHTa, N = 5750 KH, Mx = 11 500 KH. M. Tpe6yeTcH onpe.n.eJIHTb MaKCH-
M3JIbHOe .n.3BJIeHHe H3 rpYHT H JIHHY e*aTOH 30HM no.n.OIllBM.
Pac'teT: e X /Rl = Mx/NRI = 11 500/5750.5=0,4; a=3/5=0,6. no
Ta6JI. 4.3 npH eX/Rl =0,4 H a=O,6 H3XO.n.HM v= 1,837 H t=O,92. CJIe.no-
BaTeJIbHO, Pmax=5750/ (52 .0,92)-=250 Kna; VRI= 1,837.5=9,185 II.
CJIyqai1 2, TaK K3K 1 +a= 1 +0,6= 1,6<,\,= 1,837. TaK K8K IC03c1»cI»HUB-
eHTbI '\' H t p3CnOJIo*eHbI B T36JIHu.e BbIllle pa3rpaHHQHTeJIbHoit .1HHHH,
TO F OTP <0,2F. no 4>opMYJIaM (4.106):
125
p
"'x N
.
Ii
d.l
2
Xi. I
X2
i a
ex
:
I.c)
<:S
, ,
fY
PHC. 4.15
P _ 1,837-1+0,6 250 19 r- 6 n .
1 - · I 837 = . v, K a,
,
P = 1,837 - I - 0,6 250 :== 32 3 Kna.
2 1 837 '
,
4.8. YH,l(aMeHTbI c npaBHJlbHOH MHoroyroJlbHoH
v
nOAOWBOH
PaccMoTpHM CHaqaJIa Q>YH.n.aMeHTbI C llIecTHyrOJIbHOH
no ollIBoH. B 3aBHCHMOCTH OT HanpaBJIeHHH nJIOCKOCTH ,neH-
CTBHH H3fH6alOll.\erO MOMeHTa .n.Jl5I TaKHX Q>YH.naMeHToB
B03MO)KHbl ABa BapHaHTa.
BapuaHT I. nJIOCKOCTb .n.eHcTBHH H3fH6alOIII.erO MOMeH-
Ta nepeceKaeT cepe.n;HHbI ,lJ.BYX IIPOTHBOIIOJIO)KHbIX CTOPOH"
nO OWBbI (pHC. 4.15): Bce 06 3HaqeHHH - 06III.enpHHHTbIe
H no pHC. 4.15. PaCqJleHHM 3IIIOPY .n.aBJleHHH Ha fPYHT Ha
,nBe qaCTH ,lJ.JIHHOH OT x=O !-LO x=O,866a H OT x=O,866a
Jl.O x=vA. 113 pIfC. 4.15:
126
b t - _ a + 2 O,.5a + 1 1547
..\: t = a , X t ;
O,866a
pXt=[Pmax(V A - Xl)]!(V a ).
3 eMeHTapH M 06'beM nepBoH qaCTH 9nrop aB eHHH
H3 rpYHT M ero YC OBH H CTaTHqeCKHH MOMeHT OTHOCH-
Te bHO OCM y:
dV t === b t Px 1 dx l === (a + 1,1547 Xl) Pmax (la- Xt) dx t ;
,a
dS t = btPx1xtdx t = (a + 1, 1547xd p;;x ('la-xI) x1dx t ,
oTKy.na
0." 866a
VI === Pmax S (a -f- 1,1547.x 1 j (",a - Xl) dx 1; (4.107 )
.a
o
O,866a
Sl =Pmax! (VA) J (a+ 1,1547 x l) (Va - Xl) (V a -
o
-Xl)X1 dx t. (4.108J
AHanOfHqHO ,n H BTOpOM qaCTH 3nrop ,naBneHHH H3
rpYHT nOJIyqHM:
Ta
V === Pmax S ( 3a- x ) j ""a-x ) dx. ( 4.109 ) .
2 a . 0 866 \, - 2 2 ,
· 0, 866a \ ,
Ta
s=== Pmax S ( 3a- X2 ) (""a-X ) XdX. 14.110 )
2 -va 0 866 \, 2 2 2 \
I 0 , 866 a '
PeIlIHB. HHTerpaJIbI (4.107-4.110) II CnO>KHB COOTBeT-
CTBeHHO nOJIyqeHHble 06'beMbI H CTaTIIqeCKHe MOMeHTbI qac-
u
TeH 3nlOpbI aB eHHH Ha OCHOBaHHe, nonyqHM:
V= V 1 + V2=PmaXa ;
S =8 1 + S2= Pmaxa3r),
fn.e
= (0,25 - 0,866,\,+ 1,5,,2 - 0,19241'3) Iv;
11= (0,1082 - 0,25,\,+0,5,,3 - 0,0962,\,4)/".
yl3 ycnoBHH paBHoBecMH IIMeeM:
N = V = Pmaxa2C;
O,866a - ex ::= S / V == a'Y / ,
OTKY.lla
Pmax -::= IV I a 2 C;
(4.11-1)
(4.112J
(4.113)
(4.114}
(4.115)
(4.116)
(4.117)
127
'1) f!}( 0) ex N
N a}( J XJ
Pf
x
\Q.. 'x )(
't] tI
Q.£2. Q.El
Xi Xf
ti. 2 dJ( X2
2
i ya
i
t
d.. x"
"J
ex
I
.;: ...'
...,
i
X x !
(.C)
to
-
I
oQ
" "
!I
PRC. 4.18
e x / a == 0,866 - ( / C) . ( 4. 118)
BapuaHT II. nJIOCKOCTb AeHcTBHH H3rH6alOIl.{erO MOMeH-
Ta npoxo.nHT Qepe3 .n.Be npOTHBOnOJIO>KHble BepIIIMHbI no-
.nOllIBbI. 3.necb B93MO>KHbI .n.Ba CJIYtIaH, 3aBMCHIl.{He OT otIep-
T8HHH C>KaTOH 30HbI IIO.n.OIIIBbI. CJIYtIaii 1 npH a Va I,Sa
(pRC. 4.16, a). npMBOJI.HM paCtIeTHble CPOPMYJIbI B OKOHtIa-
TeJIbHOM BH,l(e:
C= (0,0722 - 0,4331'+.0,866V 2 ) Iv; (4.119)
1)= (0,0181 - O,07221'+0,2887V3)/V; (4.120)
ex / a === 1 - rJ / C. ( 4. 121 )
3HatIeHHe Pmax onpe.n.eJIHIOT no cpopMYJIe (4.117).
CJIyqaH 2 npR 1,5a Va 2a (pHC. 4.16,6):
C= (2,0207 - 4,33V+ 3 ,464'f - 0,5773V 3 ) Iv; (4.122)
1) = (1,4796 - 2, 0207y+ 1, 1547V3 - 0,, 2887 V"') Iv. (4.123)
3HaqeHHH Pmax II ex/a BblqHCJIHIOT COOTBeTCTBeHHO no
tl>opMY TIaM (4.117) Ii (4.122).
128
0) ex N I ex N
0)
N ,oJ
)( x
-
Q.>< . )(
' ;_.; .
0..:'
I dX1 IX(
! X2. d X 2
ya aXJ X3
I -..,.-- ya
I
j I
!
I !
a7071a fJ,5a b.5a O,Zfl71a
x
r
a. OJ
dXJ
!!
Pac. 4.17
AHaJIOfHqHO nOJIyqeHbI CJIe,nYIOInHe <POPMYJIhI ,nJIH CPYH-
,naMeHTOB C BocbMHyrOJIbHOH nO.nOIIIBOH.
BapuaHT I. nJIOCKOCTb eHCTBMH H3rH6alOInerO MOMeH-
T3 nepeceKaeT cepe,l(MHhI ,nBYX npOTHBOnOJIO>KHhIX CTOpOH
no,nOIIIBbI. CJIyqaH 1 npH 1,2071a Va 1,7071a (pHC.
4.17,a):
C= (1,2071y2 - 9,5V+6),1180)/,,; (4.124)
1)= (0,0417 - 0,1 179V+e,4024v3)/v; (4.125):
exla= 1,2071 - 'fI/C. (4.126J
3HaqeHMe Pmax onpe,neJIHIOT no cpopMYJIe (4.117).
CJIyqaH 2 IIPH 1,7071a Va 2,4142a (pRC. 4.17,6):
C = (1,7762 - 3,14421'+ 2,9142V2 - 0,3333V 3 ) Iv; (4.127}
1')= (1,4569 - 1 ,77!3QV+Q,9714Vs - Q, 16e7v ) tv. (4.128):
3HaqeHHH Pmax Hex/a BblqMCJIHIOT ceOTBeTCTBeHHO no
<popMYJIaM (4.117) H (4.126).
129
P
I"'X N
a)
0) dx
e
N x4
)(
y
fa
.: 'f
;,
PRC. 4.18
BapuaHT II. nJIOCKOCTb ,neHcTBHS1 M3rH6alOIU.erO MOMeH-
Ta npoxo,nHT Qepe3 .l(Be npoTHBOnOJIO>KHbIe BepWHHbI no.lI.O-
WBbl. C.nyqaH 1 npH 1,3066a Va 2,2332a (pHC. 4.18, a):
C = (0,6469 - 1,69301'+ 1,8424'f - 0, 1367V3)/V; (4.129)
1)= (0,4045 - 0,64691'+O,6141,\,3 - 0,0683V 4 )/,\,; (4.130)"
exla= 1,3066 - fll C . (4.131)
3HaqeHMe Pmax onpe,neJIS1IOT no QJopMYJIe (4.117).
CJIyqaH 2 npH 2,2332a Va 2,6132a (pHC. 4.1 , 6):
c= (8,1772 - 11,8087y+6,3721r - 0,81281'3) Iv; (4.132)
1}= (8,8139 - 8,1772y+2,124y3 - O,4064y )/V. (4.133)
3HatIeHHS1 Pmax Hex/a BbIQHCJIS1IOT COOTBeTCTBeHHO no
cpopMYJIaM (4.117) H (4.113).
B Ta6JI. 4.5 npHBe,neHbl rpaHHQHbIe 3HalleHHH V' COOT-
BeTcTBYIOIU.Me F OTP =0,25F II F OTP =0,5F ,nJIH Bcex pac-
CMOTpeHHbIX CJIyqaeB 01IepTaHHS1 CLKaTOH 30HbI no,nowBbI.
130
Taf)JlHQa 4.5
nO,:1.0mBa
weCTHyronbR8S1
BocbMHyronbHaJi
& ,
=
Cl,)G,)
s:I
. .
0. 0
t:
'\' npH F OTP' paBHOM
. .
-=:c
G)Q)
I::{
. .
0. 0
G) t::
'\' npH F OTP ' paBHOM
tJ" :'Iii ::r
...0 0$E:-a ....0 O;E:-a
g:21 O:c :c)1 c:O:c
aiD:: (1;:10:=02 O,25F 0,5F (1;:10:: (I;:Io::o:a O,25F 0,5F
=:s:: ::r':s:(I)a:\ =::s:: tJ"::s::(I)a:\
o.ca :c :3 Cl.1!I >':I: :3
tSlt-ra t:;(I;:IOO C';II-o(l;:l t:;C";Soo
!:QUi-< UI-ol-ot::i COUI-- UI--I-ol::{
I 1 1 ,229 . 0,866 I g 1 ,707 1 ,207
{ 1 1 ,325 1 1,707 FOTP
II 2 F OTP < O,25F <0,5F
- - I- II 0 1 ,811 1 ,307
F OTP <O,25F
3a aBaHCb peanbH MH 3HaqeHHHMH V C HHTepBanaMH,
paBHbIMH 0,02, no cooTBeTcTBYIOInHM <popMynaM BbIlIHCJIe-
HbI K03<p<PHIl.HeHTbI H OTHOCHTeJlbHbIe 9KCIl.eHTpHCHTeTbI
ex/a, CBe.n:eHHbIe B Ta6n. 4.6, n03BOnHIOIII-He BeCbMa npocTo
onpe.n:enHTb MaKCHMaJIbHOe aBJIeHHe Ha OCHOBaHHe Pmax
H ,llJIHHY 9IIIOpbI ,naBJIeHHH Ha OCHOBaHHe Va IIO 3a.n:aHHoMY
OTHocHTenbHoMY 3KCIJ.eHTpHcHTeTY ex/a. B Ta6J1. 4.6 3Haqe-
HHH V H , pacnOJIO}KeHHbIe BbIIlle pa3rpaHHlIHTenbHoH nu-
HMH, cooTBeTcTBYIOT ycnoBHIO FOTP 0,25F, HH:>Ke pa3rpa-
HH1JHTeJIbHOH JIHHHH - YCJIOBHIO FOTp 0,5F. AJISI rrpoMe-
>KYT01JHbIX 3HaqeHHH ex/a K03<P<PHIJ.HeHTbI H Va HaXO,lI.HT
HHTepnOnHIJ.HeH. ,aaBneHHH Ha OCHOBaHHe B npoM-e:>KYTOq-
HbIX TOqKaX nO,lI.OIllBbI onpeJ1.eJIHIOT no <popMynaM:
v
,llJISI wecTHyrOJIbHOH no,nOIliBbI:
l' - 0, 866
BapHaHT / Pt = pmax ;
1
BapH8H v T II, P === P max l' - 0,5 P o
CJJyqau 1 1 l' 2 = ; (4.134)
'Y - 0,5 . I - I ,5 .
CJIY1JaM 2 Pt = pmax , P2 === Pmax ,
'Y l'
)lnH BocbMHyrOJIbHOH nO,lI.OWBbI:
BapH3HT I, P - p 'Y - 0,.7071 P2 === 0 , . '
CJI V I 1-max
Y1JaH 1
CJIyqaH 2 p1=pmax 1- 0,7071
1
1 - I ,7071
P2 == pmax ;
1
131
BapM8HT II, P _ p 1- 0,38 .
CJIYIIaH 1 t - max I '
"{ - I ,3066
P2 = pmax
..
( 4.135 )
P3 == 0;
u 2 I - 0, 38 . I-I ,3066 .
CJIYIIaM P1=Pmax , P2=pmax ,
I I
1- 2 ,2332
P3 = pmax ·
"i
n pH F OTp 60JIbI1le 3a,naHHoro O,25F HJIM O,5F, a TaK>Ke
npM Pmax> 1,2R CJIe,nyeT YBeJIMqMTb ,nJIMHY CTOpOHbI no,no-
UIBbI cpYH,naMeHT3.
npHMep 8. P'aflo: 4>YHllaMeHT C lIIecTHyrOJIbHOH nO)J.olIIBoH C .n.JIH-
HOM CTOpOHbI a = 1,5 M. Y CHJIHSI, )J.eHcTBYlOlllHe Ha ypOBHe nOllOWBbl,
N=800 KH, M x =432 KH'M, R=300 Kna. nJIOCKOCTb .n.eHcTBHH MOMeH-
Ta nepeceKaeT .n.Be npOTHBOnOJIO>KHble CTOpOHbI nO)J.OIIIBbl, Ba'pHaHT I.
Tpe6yeTcH nocTpoHTb 3nlOpy )J.aBJIeHHH Ha OCHOBaHHe.
PaCtteT: ex/a = 432/ (800 .1,5) =0,36. no Ta6JI. 4.6 npH e.tl a =O,36
HaXO)J.HM 1'= 1,344; =0,989. no 4>opMYJIe (4.117) Pmax=800/(0,989X
x 1,5 2 ) =359 Kna<1,2R= 1,2.300=360 Kna. ,\,a= 1,344.1,5=2,02 M.
TaK KaK 3HaqeHHH K034>4>HUHeHTOB '\' H pacnOJIO>KeHbI B Ta6n. 4.6
BbIllie pa3rpaHH WreJIbHOH JIHHHH, TO F OTP <0,25F. no 4>opMYJIe (4.134):
PI == 359 1,344 - 0,866 === 127 7 Kna.
1,344 '
npHMep 9. ,QJIH Tex me YCJIOBHH, qTO H B npHMepe 8, HO Mx ne-
peceKaeT .n.Be npOTHBOnOJIO>KHble BeplIIHHbI no.n.OlliBbI, BapHaHT I I H R =
= 340 Kna, Tpe6yeTcH nocTpoHTb 3nlOpy )J.aBJIeHHH Ha OCHOBaHHe.. 113
npHMepa 8 CJIe)J.yeT, QTO e x la=0,36. no Ta6JI. 4.6 npH e x la=O,36 H8-
XOllHM '\'= 1,486 H =O,903. CJIyqaH I, TaK KaK a= 1,5<1'a= 1,486.1,5=
c:2,23<1,5=1,5a.l,5=2,25 M. no 4>opMYJIe (4.117):
800
Pmax === = 394 Kna < 340 . 1,2 === 408 Kna.
o ,903 . 1, 52
TaK KaK 3HaqeHHH K034>4>HUHeHToB '\' H pacnOJIO>KeHbI B Ta6.n. 4.6 BbI-
me pa3rpaHHQHTeJIbHOH JIHHHH, TO F OTp <0,25F. no 4>opMYJIaM (4.134):
394 1 ,486 - 0, 5 261 7 n . 0
P1 == === ,I{ a, P2 == ·
1 ,486
npHMep 10. lI.aflo: 4>YH)J.aMeHT C BocbMHyrOJIbHOH no.n.olIIBoiI C AJIH-
HOM CTOpOHbl a = 1 M. Y CHJIHH, )J.eMcTBYIOllI.He Ha ypoBHe nOllOWBH.
N = 1000 Kna, M x =600 KH. M, R=570 Kna. nJIOCKOCTb .n.eiicTBHSI MO-
MeHTa nepeceKaeT JJ.Be npOTHBOnOJIO}l{HbIe CTOpOHbI nO)J.OlliBbl, sapH-
3HT I. Tpe6yeTcH nocTpoHTb 9nlOpy llaBJIeHHH Ha OCHOBaHHe.
132
Ta6J1HQa 4.6
mecTHyronbHaB nO,D.OlllBa ,D.nSl CJIyqaB BocbMHyronhHaH nO.ll.OlllBa .lI.JIB CJIyqaR
J 2 J 2
e:JCla e Ja
V V V fo V t
Ie
I I
o 21
0'22
0'23
,
o 24
,
o 25
,
o 26
,
0,27
o 28
'29
0,
0,3
o 31
,
o 32
,
0,33
0,34
0,35
0,36
0,37
0,38
0,39
O,
0,41
° 42
,
0,43
0,44
° 45
,
0,46
0,47
0,48
0,49
0,5
0,51
0,52
0,53
0,54
J , 993 1 , 295
1,948 I ,264
1 , 905 1 ,235
I ,866 1 ,207
1 , 829 1 , 179
I , 793 1,152
1 , 759 1 , 125
1 , 726 1,099
1 694 1 ,074
,
1 662 1 ,049
,
1 ,632 1 ,024
1 ,602 0,999
1 ,573 0,915
1 ,544 0,951
1,515 0,927
1 ,486 0,903
1 , 458 0,879
1 429 0,855
,
1,401 0,832
1 373 0,808
,
] ,345 0,785
1 ,317 0,762
1 , 289 0,739
1 , 26 I 0,716
1 , 233 0,693
I ,205 0,671
1 178 0,648
, o 626
1,150 ,
1 , 123 0,604
1 096 ° 582
, ,
I ,069 0,560
1 ,042 0,538
1,015 0,517
!
0,3 2,595 2,405
0,31 2,553 2,365
0,32 2,412 2,412 2,514 2,326
o 33 2,375 2,374 2,476 2,288
,
0,34 2,338 2,337 2,439 2,251
0,35 2,302 2,300 2,404 2,214
0,36 2,268 2,263 2,370 2 , 178
o 37 2,233 2,227 2,336 2, 142
,
0,38 2,200 2, 191 2,303 2,106
o 39 2, 167 2,155 2,271 2,071
,
0,4 2 , 134 2,120 2,239 2,036
0,41 2, 1 02 2,084 2,207 2,001
0,42 2,071 2,049 2, 176 1 , 966
0,43 2,040 2,015 2,145 1 , 932
0,44 2,010 1 , 980 2,115 1 ,898
0,45 1 , 980 1 , 946 2,084 1 ,863
0,46 1 ,950 1,912 2,054 1 ,829
0,47 1 , 920 1 ,878 2,024 1 , 795
0,48 1 , 891 1 ,844 1 ,994 1 ,762
° 49 1 ,862 1,811 1 ,965 1 , 728
,
0,.5 1 ,834 1,778 1 ,935 1 ,695
0,51 1 ,806 1 , 7 4.5 1 ,906 1 ,662
0,52 1 ,777 1,712 1 , 878 1 ,629
° 53 1 ,749 1, 679 1 ,849 1 ,596
,
0,54 I ,722 1,647 1,821 1 ,563
0,55 1 , 693 1,613 1 , 792 1 ,531
0,56 1 ,665 1,581 1 , 765 1 , 499
0,57 I ,638 1 ,549 1 ,7.17 1 , 467
0.58 1 ,610 1,.517 1 ,709 I ,435
,
0,59 1 ,583 1 , 485 1 , 682 1 ,404
0,6 1 , 5.55 1 , 453 1 , 655 1 ,373
0,61 1 , 528 1 ,422 1 , 628 I ,342
0,62 1 ,501 1 ,390 1 ,601 1 ,311
0,63 1 ,474 I ,359 1 ,575 1 , 280
0,64 1 ,447 I ,328 I ,548 1 ,250
° 65 1 ,420 1 ,297 1 ,522 1 219
, ,
0,66 1 ,393 1 ,266 1 ,.:1-96 1 190
,
0,67 1 ,366 I ,235 1 ,470 1,160
0,68 1 , 339 1 ,205 I , 445 1 ,131
0,69 1,313 I , 17.5 1,419 1,102
0,7 1 ,286 1,145 1 , 393 1 ,073
0,71 1 ,260 1,115 1 ,368 1,044
0,72 1 ,234 I ,085 1 ,343 1,016
0,73 1 , 208 1 ,0.56 1,317 0,988
1 ,734 1,301
1 ,698 1,274
I ,663 1,24.7
1,629 I ,220
1 ,595 I, I 94
1 , 562 1, 168
1 ,529 1, 142
1 , 497 1, I 16
I ,466 1 ,090
1 ,435 1,064
1 ,404 1,039
1 ,374 1,014
I ,344 0, 989
1 , 315 0, 964
1 ,286 0,939
1 ,258 ° , 915
1 ,230 0,891
1,202 0, 867 1
I ,175 0,843
I ,148 0 ,820
1 , 121 ° , 797
1 095 0,774
1 : 069 0,7.52
1 ,044 0,729
1 ,0 18 0 ,707
0,993 0,686
0,968 0,664
o ,943 0,643
° , 9 I 9 ° ,622
0,894 0,601
0,869 0,580
133
PaCtteT: ex/a = 600/ (1000. I) = 0,6. no Ta6JI. 4.6 npn ex/a = 0,6 Ha-
XO.nHM ,\,=1,555, =1,453. CJIyqaii 1, TaK KaK 1,207Ia=1,2071.1=
= 1,2071 < l,7071a= 1,7071.1 = 1,7071. no 4>opMYJIe (4.117):
pmax=IOOO/(1,453.1 2 )=688 KOa 1,2R=I,2.570=
=684 Kna.
TaK KaK K034><t>HLlHeHTbl V M pacnOJlO:H<eHbI B Ta6.'l. 4.6 HH>Ke
pa3rpaHHqHTeJIbHOH JIHHUH, TO 0,25F <F oTP<0,5F.
no 4>opMYJIaM (4.135):
- _ 688 1,555 - 0,7071 == 345 Kna . 0
P t I 555 ' , P 2 == ·
,
npHMep 11. LLJIff Tex >Ke YCJIOBHH, qTO H B npHMepe 10, HO 3J1.eCb
TIoilOCKOCTb .LleHCTBHH Mx = 500 KH. M nepeceKaeT JI.Be npOTHBOnOJIO>KHbIe
BeplllHHbI nO.lI.OlllBbI 11 R = 470 Kna, Ba pHaHT II, Tpe6yeTcH nocTpoHTb
3nJOpy .LlaBJIeHHH Ha OCHOBaHHe.
PaCtteT: e x /a=500j(1000.1) =0,5; npH e x /a=0,5 HaXOD.HM '\'=
= 1 ,834, = 1,778. npHHHMaeM npe.n.BapHTeJIbHO cJly-qai1 1. npoBepHeM
cJJYlJaii pacnOJlO>KeHHH HeHTpaJlbHOH OCH. npH cJlyqae 1: 1 ,3066a =
= 1,3066.1 = 1,3066; ,\,a= 1,834-) = 1,834; 2,2332a=2,2332; 1,3066<
< 1,834<2,2332. CJIYlJaH 1 nO.LlTBep.LlHJICH. no 4>opMYJIe (4.117):
P == 1000 == 562 < 470 . 1 2 === 564. Kna.
max 1 778 . 12 '
,
TaK KaK K03<t>4>HLlHeHTbI '\' H pacnOJIO}l{eHbI B Ta6JI. 4.6 BbIwe
pa3rpaHHlJHTeJIbHOH JIHHHH, TO F oTP<0,25F. no <t>opMYJIaM (4.135):
pt==562 1 ,834-0,38 ===442 Kna.
1,834 '
P == 562 t ,834 - t ,3066 == 190 Kna , . P3 == o.
2 1 834
,
fJlABA 5. CS-YHAAMEHTbl, PA60TAIOUJ.HE C OTPblBOM
nOAOIUBbl UPH AERCTBH" MOMEHTOB B .llBYX
83AHMHO nEPnEHAHKYJlSlPHblX HAnPABJlEHHSlX
5.1. npeAnOCbIJlKH, npHHHTble np" BbiBOAe
pacMeTHbix tjJOPMYJI
KaK H npH eHcTBHH MOMeHTa B O HOM HanpaBneHHH,
paC1JeTHbIe CPOPMYJIbI nOJIY1JeHbI 113 YCJIOBHH paBHOBeCH5I.
B OTJIH1JHe OT cnyqaH ,lI.eHCTBHH MOMeHTa B OJI.HOM HanpaB-
JIeHHH B paCCMaTpHBaeMOM CJIYlJae BMeCTO BYX YCJIOBHH
paBHOBeCHH nOTpe6oBaJIOCb TpH. TpeTbHM YCJIOBHeM paBHO-
seCHfl npHHHTO paBeHCTBO MOMeHTa npO,ll.OJIbHOH CHnbI N
OTHOCHTeJIbHO OCM Y YCJIOBHOMY CTaTHlIeCKOMY MOMeHTY
134
06'beM3 3nlOpbI Jl3BJIeHHH H3 OCHOB3HHe OTHOCHTeJIbHO TOM
JKe OCH. COBMecTHbIM perneHHeM Tpex YCJIOBHH p3BHOBeCHH
onpeJI.eJIeHbI M3KCHM3JIbHOe .lI.3BJIeHHe H3 OCHOB3HHe H paa-
MepbI C}K3TOH 30HbI rrO.lI.ornBbI.
33MeTHM, 4TO .3nlOpbI .lI.3BJIeHHH H3 OCHOBaHHe npH .lI.eH-
CTBHH MOMeHTOB B .lI.BYx B33HMHO neprreH.lI.HKYJIHpHMX Ha-
np3BJIeHH51X OK333JIHCb .lI.OCTaT04Ho CJIO}l{HbIMH, B CBH3l1 C
qeM YCJIO}l{HHJIHCb H p3CQeTHbIe <POPMYJIbI. no Mepe YCJIO}l{-
HeHHH <pOpMbI nO,ll.OIIIBbI COOTBeTCTBeHHO nOBbIIII3..1I3Cb
CJIO}KHOCTb 3nlOp ,lI.3BJIeHHH Ha OCHOB3HHe, npH 3TOM YBe-
JIH4HB3JIOCb KOJIH1.JeCTBO B03MOLKHbIX CJIY1.J,aeB 01.JepT3HHH
C}I{3TOH 30HbI nO.lI.OIIIBbI. T3K, JlJIH npocreHlllero CJIyqaH
npHMoyrOJIbHOH nO.lI.OlllBbI B03MOLKHbIe .3nlOpbI Jl3BJIeHHH Ha
OCHOBaHHe npe,lI.CT3BJIHIOT C060H rrHpaMH.lI.Y, yce1.JeHHYIO TIM-
paMH.lI.Y H .lI.B3)K,lI.bI yce1.JeHHYIO nHp3MH,lI.Y C COOTBeTCTBeHHO
TpeyrOJIbHbIM, TpaneUHeBH.lI.HbIM M nHTHyrOJIbHbIM 01.JepTaHH-
eM C}I{3TOH 30HbI nO.lI.OIIIBbI.
)lJIH ,llpyrHx <pOpM nO.lI.OlllB .3nlOpbI .lI.3BJIeHHH H3 OCHO-
o
BaHRe Ii C001DelCTBYIOmHe HM OtIepTaHHH C}I{3TOH 30Hbl
nO,lI,OlllBbI YCJIO>KHHIOTCH 3H3qHTeJIbHee. 003TOMY TOIl Hoe
peIIIeHMe nOJIyqeHO TORbKO RnH HaH60JIee pacnpOCTpaHeH-
HbIX Q>YH,lI.aMeHTOB C npHMoyrOJIbHOH nO.ll.OlllBOH. ,[{JIH <PYH-
.lI.3MeHTOB OCT3JIbHbIX THnOB npe.lI.JIO)KeH npHeMJIeMbIH .lI...TIH
npaKTIfQeCKHX paClleTOB npH6JIHLKeHHbIH cnoc06, CYI.llHOCTb
KOToporo H3JIO>KeH3 B n. 5.3.
5.2. YHAaMeHTbI c npHMoyroJlbHoH nOAowBoH
B 33BHCHMOCTH OT 3KCl.I.eHrpHCHTeToB npO,lI.OJIbHOH CHJIbI
B03MO}KHbI 1.JeTbIpe CJIY1l3H nOJIO>KeHHH HeHTp3JIbHOH JIH-
HIfH.
P3ccMorpHM K3LK,lI.bIH CJIY4aH B OT,l(eJIbHOCTH. DpH 9TOM
v
3a.lI.31.J3 CBO,l(HTCH K onpe,ll.eJIeHHIO pa3Me-poB CLK3TOH 30HbI
nO,ll.OIlIBbI <PYH.lI.3MeHT3 II BeJIH1.JHH ,naBJIeHHH H3 OCHOBaHHe
B ee xa paKTepHbIx TOllK3X. HCKoMbIe BeJIH1.JHHbI H3H.lI.eM H3
CJIe.lI.YIOI.llHX YCJIOBHH p3BHoBecHH Bcex CHJI, JI.eHcTBYIO HX
Ha YPoBHe nO.lI.OIIIBbI <PYH.lI.3MeHT3: np'OeKl.I.HH ueHTpa TH..
)KeCTH 06'beM3 3nlOpbI .lI.aBJIeHHH Ha nJIOCKOCTb no.rr.OllIBbI
<PYH.lI.3MeHT3 ,nOJI>KHa COBMeCTHTbCH C TOllKOH npHJIOlKeHHH
npOJlOJIbHOH CHJIbI; 06'beM 3nlOpbI ,lI.3BJIeHHH Ha OCHOBaHHe
JlOJI)KeH 6bITb p3BeH rrpOJI.OJIbHOH CHJIe.
CAyttau 1 (pHC. 5.1). 3nlOpa .lI.3BJIeHHH H3 OCHOBaHlie
npe,l(cT3BJIHeT C060H yce4eHHYlO nHpaMH.lI.Y 1, 2, 3, 4, 5, 6
C 06'beMOM, paBHbIM
135
c = S1 A
npo.kIlM" Ha MOCNOCTit )( V
14. T. 061teMa 3nfOptai 1.. 6
!I
:> I
PUC. 5.1
V = FtB ( l+_ + ) =-= FtB ( C2+Cd+d2 )
1 3 C C 2 3C 2 '
(5.1 )
r)le
F I =O,5Cpmax
113 PIlC. 5.1 bjd= (b+B)/C, OTKYAa
b=Bd/(C - d);
B + b =--= B + _Bd == BC ;
C ._- d C - d .
Pl!Pmax=b/(b +B),
( 5.2)"
(5.3)'
(5.4)
OTKYAa
p1=dpmax/C. (5.5)
3JIeMeHTapHbIH 06'beM yceqeHHOH llHpaMHJJ.bl
dV=Fidx, (5.6)
fJle F,- nJIOllI.a)lb nOnepe1.JHOrO CeqeHIUl, nHpaMHJI.bl, nepneHD.HKYJIHp-
Horo llJIOCKOCTH xy C 86cUHCCOH x.
136
113 pHC. 5.1
OTKy.na
Fi _.. (B + b - X)2
PI (B + b)2 '
F 1 (B + b -- X)2
F.== .
1 -.---- --- (B + b) 2
(,5.7)
CJIe.n.OBaTeJIbHO,
[ 2x X 2 ]
dV = F 1 1 - -+- dx.
B + b I (B + b)2
KpoMe Taro,
(5.8 )
C (B -t-b)-x
y === "3 -- B + b ·
( 5.9)
JIH onpe.n.eJIeHHH Koop HHaT npoeKUHH Ha rrnOCKOCTb
xy ueHTpa TH}KeCTH 06'beMa 3nlOpbI .n.aBJIeHHH Ha OCHosa..
HMe 3.n.eCb H B .n.3JIbHeHIIIeM 6y.n.eM BbIlIHCJIHTb He HCTHHHbIe
CTaTHlIeCKHe MOMeHTbI YKa3aHHoro 06'beMa, 3 YCJIOBHbIe,
paBHbIe npOH3Be.n.eHHIO 06'beMa Ha npoeKUHIO paCCTOHHHH
OT ero IJ.eHTpa T5I}KeCTH .n.o COOTBeTCTBYIOIneH OCM Ha TY >Ke
nJIOCKoCTb. B paCCM3TpHBaeMOM CJIYlIae YCJIOBHbIH CTaTH
qeCKHH MOMeHT OO'beMa .n.aBJIeHHH H3 OCHOBaHHe OTHOCH
TeJIbHO OCM x
S == S B dV= r F [ 1- 2x + x 2 ] C[(B+b)-X] dx. ( 5.10 )
x Y J 1 B+b (B+b)2 3(B+b)
o 0
OOCJIe pemeHHH HHTerp3J1a (5.10) H COOTBeTCTBYIOInHX
npeo6pa30BaHHH nOJlYQHM
S === FIB [C(C 2 +Cd+d 2 ) -1-d 3 ]. (5.11)
x 12C 2 I
Op,l1HH8T3 neHTpa TH>KeCTH 06'beMa 3nlOpbI ,naBJIeHHH
FB
-L (C(C2+Cd+d2)+d3]
= Sx ===1 2C2 - 02S ( C+ d3 ) . ( 5.12 )
Yc V F B ' C 2 +Cd+d 2
1 ---1-(C 2 +Cd+d 2 )
3C2
0603HaQHM d= .H C=t 1 A, TOT,n3
=== 0 25A ( C + 23 )
Y c , 1 l' 2 + l' ,.. + r 2 ·
1 1 2 2
lis BbIpa)l(eHHH (5.9)
(5.13 )
137
x= (B+b) (C - 3Y)/C. (5.14)
nO cTaBHM B (5.14) 3HaqeHHH B+b, C, Yc. H3 (5.4),
(5.13) H C= lA nOJIYQHM
x === BC [ C-O 75A ( c + :: ? )] ===
C (C-d)C ' t 12+ 1 +C22
_ 0,258 ( c _ a 23 )
-1' f' 1 1'2 1'1' + 1'2.
"1 - 2 1 + "':>1"2 1,2
COCTaBHM TPH YCJIOBHH paBHOBeCM5I BCeX CMJI, .n.eHCT-
BYlOllI.lIX Ha ypOBHe nO)I.OllIBbI cpYH.n.aMeHTa:
N === V === FIB ( 1 +- + ) === '1ABPmax X
13 'c C2 6
X ( 1 + 2 + ) ;
1 12
05B-e === 0,258 ( C 1 - 3:22 ) - (5.17)
, y C 1 - C 2 12 + C 1 C 2 + C 2 2 '
( ?2 )
O,5A - ex === O,25A C. + .. .
C 1 2 Cl 2 C 2 2
113 BbIpa}KeHHH (5.16)
pmax=a. N {(AB),
r)1e
[ ( C<) r <)2 )]
at === 6 / 1 1 + + .
C 1 C 1 2
v13.. CPOPMYJIbI (5.17):
e y === 0,5- 0,25 ( C t - 3':22 ) .
B C 1 - C 2 C 1 2 + C 1 C 2 + 22
113 BbIpa}KeHHH (5.18):
.
ex === 0,5-0,25 ( C t + IC 2 2 ) .
A C 1 2 + Cl + C 2 2
113 CPOPMYJIbI (5.5):
" .,
Pi = '...e2Pmax{ I t.
(5. t 5)
(5.16 )
(5.18 )
(5.19)
( 5 .20 )
( 5.21 )
( 5.22)
( 5.23 )
CAy'tau 2 (pRC. 5.2). 3nlOpy ,aaBJIeHHH Ha OCHOBaHHe
paCCMaTpHBaeM KaK pa3HOCTb 06'beMOB yceQeHHOH nHpa-
MH)J.hI 1, 5, 6, 7, 8, 9 H nHpaMH,l(bI 1, 2, 3, 4. llpH onpe,l(e-
JIeHHH reOMeTpHqeCKHX napaMeTpOB BOCnOJIb3yeMcH pe3YJIb-
TaTaMH, nOJIyqeHHbIMH ,l(JIH CJIyqa51 1. 113 pHC. 5.2 kjb t =
'= CI (B + b), OTKY,l(a
138
c
n eKI4 R H2 n.nOCKOCT" x V
q. T. 061t8M2 3n10pbl 1 4
y
'r
1./ lnpo.KII.... Ha nnoeK...,.. x V
y- &4. T. 061teM8 3nlOPirC
3,4 , 2, 5, 6, 7, - 8, 9
(J.5A
"
1----
A
Pac. 5.2
k=ll(C - d)JB; (5.24)'
nO,l(CT3BHB B (5.24) l1=1}2 B , 3 T3K:>Ke 3H3QeHHSI C=ttA
H d= H, 0603H3tJHB
1- 11]2=m1; 1 -1]2=nl H 1- C:a=r
nOJIYQHM:
k == 2A (1 - '2) = 2r A;
1 - 2 nl
C == A + k == A ( 1 + 2r ) === A ml ;
nl nl
P2 k 2rnl 2r
= - c ; P2 == - pm ax == - pmax.
Pmax nl m l ml
AHaJIOrHtIHO
C 2 n l
P. == - Pmax.
. ml
( 5.25 ),
( 5.26 )
( 5.27 )
( 5.28)
( 5.29)
0603HaQHM nJIOm3)1.H TpeyrOJIbHHKOB 1, 5, 6 H 1, 2, 3
COOTBeTCTBeHHO Qepe3 F 1 H F2' Tor)l.3
139
F 2 == === 11:! 2r2 111 2
FI C 2 1112ml
OTKYAa
F - 2 " F I
2 === 2 r- 1/ mt 2 . (5.30)'
OO'be!vt 3nlOpbI ,lI.aBJIeHHH H8 OCHOBaHHe
V== FIB ( 1+ + ) _ F ll .
3 C C 2 3
nOCJIe COOTBeTCTBYIOIUHX npeo6pa30B8HHH nOJIYQHM:
V FIB. 2 + r + r 2 2 3 2 ) . (5 31 )
=== 3m l 2 (m 1 2mln1 2 nt - 2 r :' .
S === O,25F 1 B ( C+d + + ' ) - F 2 l 1 ( A + ) .
x 3 C C2, 3 4
nOCJIe COOTBeTCTBYIOIUHX npeo6pa30BaHHH HMeeM:
S x = O.:5 )IAB [(mt 2 + t/n. 2 )(m 1 + 2nl) - '1/2 3 ,2 X
mr"'n 1
X (4n 1 + 2r)]; (5.32)
Y = Sx = O.25FIAB3m12[(mI2+ 22nI2){ml+:2nl)-1Jir2{4YI1+1J2r)] =
c V 3nlm12P1B [ml (ml + ::!11 1 ) + 22n12 - :?3r2]
=== 0 2.5A (m1 2 + 22n12) (ml + 2nl) - 11:!3r 2 (4 1 + 2r) (5 33 \
, "'Jl [ml (m 1 + ::!nt) + :!2n12 - ::?3r2] ..
AHaJIOfHtIHO:
S ,== F1B ( 1 + + ' ) O.25B( C_ 3d3 ) _ P'1.l12 ::;:::=
) 3 C C 2 C - d C 2 + Cd + d 2 12
Fl 2 [mt:\ + t 2 m,2 n1 + t22mtnt2 3C 2 3 n 1 3 :-
12 (mi + . nl) m 1 2
- "'12 4 r 2 (m 1 - 2nl) ] ; ( 5.34 )
x =.Sy FIB2Jml [m13+ m12nl +: 2mln12 -3 3n13_ 2tr2(ml-C2nl) 1
c V 12 (ml - .::!n 1 ) m 1 2 F 1 B (m1 2 * mlnl + 22n12 - 112 3 r 2 )
=== 0 2 r B m1 2 + :?nt2 + 2 2nt (in! + 2nl) - 112 tr2 ( 5 . 35 )
, Q ( + y )+ r2 2 32 ·
ml m 1 2n( 2 nl - YJ2 r
COCTaBHM TPH YCJIOBHB paBHOBeCHSI Bcex CRJI, ,neHCT-
BYIOIl1.HX Ha ypOBHe nO,l(OlliBbI CPYH,l(aMeHTa
N===V== FIB (m 1 2 + 2mlnl + 22n12_ 23r2) ===
3m12
ABPmax ( 'm4: . + r + r 2 nl 23r2 ) .
- - '-s2 2 -- - ,
6 n 1 . m I ml nl
( 5.36)
140
O 5 B _ e == 0 2.JB m1 2 + ':}1l 1 2 + 2'2 n l(ml + 2nl) - 2'lr2 .
, \" ( 1' ) + 2 I) :3 I) ,
ml ml + '-:.2 n l ''''2 n!," - YJ:! r...
A-e ==0 25A (m12+ :?2n12)(ml+:2nl)- 23r2(4nl+ 2r) .
O,t..- x' [ ( ;.- ) + r 2 2 <, ]
nl ml ml + :.?nl '-:.:! nl + :!3r2
113 BbIp8LKeHHH (5.36):
Pmax=al N /(AB) ,
r.ll.e
( 5 .37 )
( 5.38)
( 5.39)
( 5.40)
_ 6 ! ( ml + r + U' 2 _ :!3r2 )
CX I - / - 2 "'2 - ·
n 1 m. m.n.
113 Bblpa)KeHlfH (5.37):
e y =0 5-0 25 m12+ 22n12+ 2 2n. (m. + 2nl) -'l2 Ir2 . (5.411
B ' , ml (ml+ :!nl)+:::!2n12_ :!3r2
113 BbIpaLKeHHH (5.38):
ex c= 0,5-0,25 (m12+ 22n12)(mt+ 2nl)-1j23r2( 4nl+'l2r) . (5.42)
A nl [ml (m1+'2 n l) + 22n12 - ir2]
Cilyttau 8 (pHC. 5.3). STOT CJIYtJ8H aHaJIOrlftJeH CJIY.
tIalO 1. ,aJISI nOJIyqeHlfH cooTseTcTBYIeH{Hx t>actJeTHbIx cpop-
MYJI Heo6xo,l(llMO nOMeHflTb MeCTaMIf OCli X II y, a TaK>Ke
BeJIlflllfHbI C, d, A, ex, ey If Pi COOTBeTCTBeHHO H8 U, 1, B,
A, e y , ex If P2, r,ll.e
u= 1B. (5.43)
TaKH!\1 06p830M nOJIytIHM: H3 BbIpaLKeHlfH (5.19):
6
cx - ·
1 - ( 1 + 2 + :. ) . 'It'
"'4 1 "'4 I ..
Pmax=a1N/(AB) ;
H3 (5.21):
t y _ 0 5 _ 0,25 ( _ 3 22 ) .
--, 711 ,
B YJl - YJ:! YJ1 2 + 141 YJ:l
H3(5.22) :
== 0 5--0 25 ( + 1123 ) .
A ' , 1 "t + + 2 '
YJl" YJI 2 YJ2
H3 (5.23):
( 5 .44 )
(5.45 )
( 5.46 )
( 5.47)
'2 =fl2fJmax/Tlt. (5.48)
C./lyttaU 4 (pIiC. 5.4). 3nlOpa J(aBJIeHHH H8 OCHOBaHHe
l1.pe.acTaBJIH.eT co.ooK DHpaMHAY 1, 2, 3, 4 06'beMOM
141
np:> ,,", M nJlOCKCICT X Y
U. T 06 eMa 3mOplti 2 ... 7
2
.3
tJ.5A
,.,
A
PUC. 5.3
c
npo.te'''''' H8 nII oCTi, x Y
.... T. oheM8 3nNfpW 'J .t.. 4
!I
PRC. 5.4
142
t. I A
£:3 r--- :-
{}3j' 1- O. -L} I " 1-' . - l:..[l
I , ' .. I I I 1 _ __ -J-L ' .
i L;K I " 2:.-i. b . sLit _
0.3 1-+_" l-tTf--'- + --v +-- f-- --
"'., ' .>c: F-t-T i.1--- -,- f'C\ I I
0.79 . '1"6'.......... '- I '- .- \ T H. I ...." -- I ' _e-n 1-- -. -.- i I
0.28: ."", : " - i -'f.-L--,- - {- l fi=o.9 --t- i / ' I :
v Ki r ' -:-";"-' ;--Cfl1-H . -:{+ ..:,¥-=t H
o.2:J ' 'Dt-t..;; -, :. ---r-+-. -'. . .LL 7 - 1
/J.2 "......v ; -1\ ;--r-- ! .. . ' I t i-+--; \ -- -- / ..
0.23 a .../" ,\..,...r, - .r ',. ----;1 ')2=0.1 -H
.0.22 u;4--Ll - " --t--- ,.t--+--1-. - ::
0.21' i\' @ _I:_ _ --\- --f---+. l.._ / ?I- " ;
. :.tf "/ . I' : T ., \ i .:.... '/ U/I. " ; J
0.2 ?\/ \r t j__ -'-I-] jI\- -- ,' V J __. -- -- -9, ::0--= J-
:0,19 ;,....-:::s..-+ : ! . \L I.. .;\.- L-4- \ /1. .... .;.2......2 l '
,0':8 '\. I r-..: I '\ : I \ I! I / - ..... I I
an '\. I_ \ f .\: I--+': vr2: -+-' ..........J. - - -
a16 ::--........ ......L I : . rf1-17\- :"---- I ---
D.".. '"........ "'"- \_L--' 1 'k \.. _........J I 7......... 7..... __
O.l i '".... I/l.\ :::1'r---.' I........... / "'r-( C?J
0.13 I ""\.. \L \ . r-... - 7 '-J ) --
I \ 1£1 \ ,\r;\i::r i £ I /2==
D. 1 Ls-f V '\r"-.. \ , '\ .......... I J ...... J '''...... , ,__
0.09 i : "9T'....... i\ \ ..........: I '- / I ...... I J ,
0.08 ! '-r- .xT " \"\,. -- ".. I ...... . 7! '/... I '\.. t?,
0.07; I v: 1- ..... , ..... f'l / l... \. I i
0.06 / i-z L i ' .....' l \' ..../ J p, fI ,: I 'I... j ""
0.05 '/ ! I I ! '..... . '..\. /' "'...7 "A. \! VA'"
g ; v/ H t- , '" \ w l'''"' 1 //7-- /",v; ....., -- ..?'
9.02 y -rt- ':; Z'(9T7 /
o.Ot V. I I , I 7 \V \
o .
" ''''1 !'")""':g(C) r--.. '"'"::: ;! ;::; e y /8
I
!
PRC. 1.&
v = FlU = 0, 5Cup = AB'I-qtPmax
3 3 6'
rAe Ft - DnOllta,lJ.b TpeyroJlhHHKa 1, 2. 8.
(5.49 )
113 pHC. 5.4: C/(3yc)=u/(u-x c ), oTKy.na
Yc=C(u - xc)/(3u).
(5.59 )
xc=u(C - Yc)/(3C). (5.51)
B (5.50), nOJlYtIliM
AH3J10rHQHO
nO,nCTaBHB 3HaqeHHe Xc 113 (5.51)
( C - Yc )
u-u
2C
y-= C
c 3u
2C + Yc
9
,
QTKY,lla
Ye =O,25C =O,25t 1 A.
( 5.52)
143
""d-
1 I I I 1 -- ", _m-;'-l -'r--
T mf u+- -+---j- t !./
I -t-I I 1-1--;---+--+- '... -- .>-
1 --1---i- - I : I 1 I AJ./'-I --
! I I I I .. ,----- r--r -- - +-: J' _ /1 . ,; I : " " !
--- r-L I i -t- - I -- -1---!-_--i---- I I" /' ;--J
l _. :- I -1-1- I . I 'i, ' 'VoF"/ I J 10
I' ----<i I I i "1-- - _. --'" .- - : - -1- tr)
r--+-t- -t-. ! i r-. --'--f ) 4-- i-j-;
ri7_LJ I T- -.-- 'f>' f! / j : I :;}
I---T- t ' I : -----...- . ._ I - I Y , I
\.---t--+--- 1 -....- j--.-;__, .___ _-1___ _ _: I '-.-.!
I I ' I I I : I ' I , I ! t-.f'I; ;;;;-'
I ,,_ I I ' I .
'<')
f -+- J + Iii -i--: co! .- ; I J - . t _u' I __
I-+-----.-t 1---'- --I ---! (;)" _ -!- +-' ,
j +=!=j--1 ;j} ]--;.; :_-: i i- M,, ;
I ' , , '. I I I I ; I ' i f j' I --
I ! l--. I; ,! ' I"" fI,., : 10
... _....__ -t--. , +--1 --I _: . - .-- +- - .----! j i $
I ' i I i I i I ; , ; I : i : , r7 !
l - --r -- j - T I j ! - r -- .;- I - . ;; "" I I Y i i 1/ i -g..
L _-!___ ___ - --;--.J--4 _: __. ___ _:_ /! i
r ' : ! I ! . I I j I i ! ! ' I 1 I ! T *"
, ---4----i.",-.1-.-'- -r--1 c::i ... - I : +- - f " - --: ---- L ? C'-I
. I ' I III I ., I' I I
-
"".
[-r--:----i . -r .---- j "1 ---:- :'____.1_ I . "I _1___. - -r I
I : . _ .__1 ' I I : " ,i:'
j I ' *H ' : . ; / 'J r
tl--n I !-=lJ . -:+ 77 A L, ,.L
. ----r-r I I Iii ! . I ' / '
L j i i -----t-T- - -+--+--/ . .. -. I)
j I i I I t /1' 1'117 V // .
LilJ. , I iA //'/V Vi / I '
I i I i rJ I'll 7I7f i/ "v ' .
I _ _J !J,)V/ J V /1/
I i/!/[/II)' Vi //.
IlIJl/V Vi Yi - '\,.'t :: ;;
/lY/IVY /v .,.,"1
rl.l/ V V ...........V I I ' I I C\i
,--' i ' I V VI i .... I r1 O,1-_-
-4- ,,"/ / j........V" ! - Y'1 t'--
r I 1 . ... ' ! .L-.C-:r-
! I I 'r """,V' ...--,! I' , C\J
I I --1-- -/ ./ I I I
' v I I '2.' C\;
I f 1 = ;--- c-H- i ; I lit
l - I r T I 'h-o. g >-
I- .1 .
Wl I I
..... Ie) Q:) I'- lC';) to '-0. , C\J' _ (\L
0> r--.. co,.......,A ,.......,. r--.... A I..C') ..::2- r-...!' f'r:) ro-...'" "r-.!"' .......
r-'"'
t--
,......
-
\
144
-
La
-
.....
foot
c:J
:s
u
1.0
C,)
=:
0..
06naCTb
rpa41MK8
nonOMOHK8
HeitTpanbHoM .nMHMM
:
1
C()
I.l)
2
:.
L
/ P1
! l oW 2A
I 0. SA u i
--;.I'
J
Pmax
::
t:::)
PMY"hl
AJI onptA8.neHMR HsnpFnKeHMM
nOA n01\IJUH? MM £t>V H AaM8HToB
r7iQ'i == 0(,1
N
AB
OTcy"cnnt O btBa N ( 1 e e )
P = 6 A;£ - 6 !. A x B y I
nOAOLiJB cf!YH ai"ieitT n u
I
,...
Ij.
;\11 a lOrIIqHO
"
:; = L P
, 1 max
N
Pmax=- ci 1 AB
$2(1- 2)
P 1 = 1 - 2 7- 2 P ma x
P 2 =- '/,2 ( 1 - ';2) P
-1 - 212 max
II
r> = I -
,.0r- {;. x 0... 1 A B
'7 dt
D = c. ,..
r 2 -;.- I rnQX
[,1
Xc=O,25u=0,25T) I B.
(5.-53}
COCTaBHM TPH YCJIOBHSI paBHOBeCHSI BCeX CHJI, .lleHCTBY-
IOIltHX Ha ypOBHe nO,nOWBbl c}>YH,naMeHTa:
N = V = C 1 11 1 ABpmax/6;
0,5B - ey=xc=0,25111B;
0,5A - e x =Yc=O,25C 1 A.
Hs Bhlpa>KeHHH (5.54):
pmax = atNJ (ABl,
r rl O
-
at =-=6/ ('C t l1t)'.
(5.54 Y
( 5.55 J
( 5.56J,
(5.57)
(5.58
145
H3 ct>OPMYJIbI (5.55):
e"IB=0,5 - 0,25f)i. (5.59)
113 BbIpameHHH (5.56):
ex/A =0,5 - 0,25C te (5.60)
Ha OCHOBaHHH nOJIY4eHHbIX cpOPMYJI COCTaBJIeH rpacpHK,
npe.lI.CTaBJIeHHbIH Ha pHC. 5.5, 3aBHCHMOCTH pa3MepoB C)KaTOH:
30HbI nOJJ.OlIIBbI CPYH.lI.aMeHTa OT OTHOCHTeJIbHbIX .3KC,-"eHTpHcH-
TeTOB npO.lI.OJIbHOH CHJIbI ex/A H eylB, n03BOJIHIOllI.HM BeCbMa
npOCTO onpe,lI.eJIHTb IlOJIO)KeHHe HeHTpaJIbHOH JIHHHH (o6JIa-
CTH rpacpHI{a 1-3 cooTBeTcTBYIOT CJIyqaHM 1-3 nOJIOLKe-
HHH HeHTpaJIbHoH JIHHHH 3nlOpbI .lI.aBJIeHH5I Ha rpYHT; 06-
JlaCTb 4 - CJIY4alO OTCYTCTBHH oTpbIBa nO.lI.OlIIB CPYH.n.aMeH-
TOB) .
BenHqHHW aBJIeHHfi Ha OCHOBaHHe onpe.n.eJIHIOT B 3a-
BHCHMOCTH OT CJIYGaH 04epTaHHH C)KaTOM 30HbI no.nOIliBbI
no cpopMYJIaM (5.19), (5.23), (5.28), (5.29), (5.39), (5.45),
(5..48), (5.57) Ii no rpaQ>HKY, npHBe.n.eHHoMY H3 pHC. 5.6,
,C n0110lll.blO KOToporo MO)KHO BblqHCJIHTb KO.3<p<PH HeHT at,
BXOJI5ImHM B nepeqHCJIeHHbIe ct>OPMYJIbI (rpact>HK Ha pHC.
.5.6 nOCTpoeH TOJIbKO .lI.JIH CJIyqaeB 1, 2 nOJIOmeHH5I HeHT-
paJIbHOM .ryHHHH. ,llJIH CJIYtIaH 3 CJIe.lI.yeT OCb 2 ClIHTaTb
'OCbIO Y)2, a KpHBbIe 1 - KpHBbIMH f) 1) .
33MeTHM, 'ITO npH COCTaBJIeHHH rpa<pHKOB HaHMeHb1II3H
TIJIOlll.3Jlb C)KaTOH 30HbI no.n.OlllBbI Q>YH.lI.aMeHTa npHHHT3
p3BHOfI nOJIOBHHe BceM nJIOII.I.aJlH nOJlOIliBbI, T. e. C06JIIO-
J(eHO YCJIOBHe FOTP 0,5F. no .3THM C006pa}KeHHHM cJIYGaH:
4 nOJIO)KeHHH HeMTpaJIbHOH JIHHHH B rpa<pHK He BKJIIO'IeH
K3K He HMelOlll.HM npaKTHqeCKOrO npHMeHeHHH, 3 npHBe.n.eH
JIlIllib .n.JIH 06IIJ.HOCTH perneHHH.
npHMep 1. !laHo: pa3MepbI nO.ll.OlllBbI <}>YHJl8MeHTa A=3 M, B =
==2 M; YCHJIHH, JleHcTBYIOllJ,He H8 ypoBHe no,nOWBbI, N =400 KH, M x =
= 324 KH. M, My = 64 KH. M. Tpe6yeTcSI onpe.n.eJIHTb nOJIO)l{eHHe HeR-
'7paJIbHOH JIHHMH 3nlOp .n.aBJIeHHH Ha OCHOBaHHe H BeJIHIJHHN ,naBJIeHHH
:H8 OCHOB3Hue.
PacfLeT: ex/A=Mx/(NA) =324/400.3=0,27, ey/B=My/(NB) =
=64/(400.2)=0,08. no rpa<}>HKY 5.1 (CM. pHC. 5.5) npH e x /A=0,27 H
,e y /B=0,08 H8XO)J.HM l :z:::0,83 H 2=0,5 (CJIYQ8H 1 nOJIO}l{eHHH HeHT-
paJIbWOH JIMHHH, TaK KaK TOIJKa C KOOp.n.HH8TaMH e x /A=0,27 H e y /B=0,08
paCnOJIo:>KeH8 B 06JI8CTH 1 rp8<pHKa). no rpa<pHKY 5.2 (CM. pHC. 5.6)
np" I =0,83 H 2=0,5 Haxo.n.HM u=3,675. Pa3Mepbl C>KaTOR 30HbI noJJ.o-
lliBbl dA=0,83.3=2,49 M, 2/A=0,5.3=1,5 M. no c)?OPMYJI8M (5.19)
"II (5.23):
Pmax= 3,675. 4001 (3. 2) = 245 Kna;
P1=0,5.245/0,83= 147,6 Kna.
J46
O.pHMep 2. l1,aHo: pa3MepbI no.n.OllIBbl cl>YH.n.aMeHTa A= B = 2,5 M;
YCHJIHSI, J1.eUcTBYIOIlI.He Ha ypOBHe nO.ll.OllIBbl cl>YH)J.aMeHTa, N = 777 Kna,.
lW x = My = 227 KH. M. Tpe6yeTcH onpe.ll.eJIHTb nOJIO>KeHHe HeHTpaJIbHoi%"
JIHHH" 3nlOpbl )J.aBJIeHHH Ha OCHOBaHHe H BeJIHIJHHbI .n.aBJIeHHH Ha ac-
HOBaHHe.
Pac'ler: e x IA=e y /B=227/777 .2,5=0,117. no rpa<pHKY (CM. pRc.5.5)
npH e x IA=e y /B=O,l17 Haxo.n.HM 112= 2=0,7 (CJlYIJaH 2 nOJIO>KeHIUI"
HeHTpaJIbHOH JlHHHH, TaK KaK TOIJKa C KOOp.lI.HHaTaMH e x /A=e Il IB=O,l11
pacnOJIO)KeH3 B 06JIaCTH 2 rpa<pHKa). no rpa<pHKY (CM. pHC. 5.6) npH:
112= 2=0,7 HaXO.ll.HM a=2,417. Pa3MepbI C)KaTOH 30HbI nO)J.OllIBbI 2A=.
=11 2 8=0,7.2,5= 1,75 M.
no <}JopMYJIaM (5.39), (5.29) H (5.28):
Pmax=2,417-777/2,5 2 =300 Kna;
0,7(1 -0,7) 300 123 7 n
PI == P2 == == ,K 8.
1 - 0 7 2
,
5.3. npHfiJlH]KeHHWH MeTOA paClleTa 4»YHAaMeHTOB
CYI.IJ.HOCTb npe,lI.JIaraeMOro MeTO,lI.a paClIeTa COCTOHT B
TOM, qTO CHaqaJIa CTpOHT 3nIDpy ,lI.aBJIeHHH Ha OCHOBaHRe
OT .neiiCTBIiSi TOJIbKO N II Mx B COOTBeTCTBHH C peKOMeH.na-
UHSlMH, npHBe,neHHbIMH B r JI. 4. 3aTeM, HCnOJIb3YH nplIH-
lI,Hn He3aBHCHMOCTII ,lI.eHCTBHH CHJI, Ha nOJIyqeHHYID 3nlOp-y
HaKJIa,nbIBaIDT 3nIDpy ,naBJIeHHH OT ,lI.eHCTBHH My. OpH 00-
CTpoeHHH 3nIDpbI .lI.aBJIeHHH OT My B KaqeCTBe no,nOWBbl
q>YH,naMeHTa npHHHMalOT ee CLKaTYID 30HY OT ,neHCTBHH N
H Mx. ,QaBJIeHHSI Ha OCHOBaHHe OT ,lI.eHCTBIIH My. onpe,neJIH.'
lOT no cl>opMYJIaM:
M
py max=My/W xc ; py mln =- (5.61)
w xc
f.ne W xc - MOMeHT conpOTHBJIeHHH nJlOllta.n.H C}K8TOH 30HbI no,nOW8h1
OTHOeHTeJIbHO OeH x OT .neHCTBHSI N H Mx.
3H811eHHe W xc onpe,neJISIIOT B Ka>KAOM KOHKp-eTHOM CJlY-
u
qae B 3aBHCHMOCTH OT pa3MepoB 11 ollepTaHHSI CLKaTOIl 30-
HbI no,nOlliBbI.
HeHTpaJIbHOH JIHHHeH CYMMa pHOH 9nlOpbI ,lI.aBJIeHHSI Ha
OCHOBaHHe SlBJIHeTCH JIHHHSI nepeCeqeHHSI BepXHeH nJIOCKO-
CTH 3TOH 9llIDpbI C llJIOCKOCTblO no,nolliBbI <PYHJI.aMeHTa.
npH cpaBHHTeJIbHO 60JIblllHX 3HaqeHHSlX py MO>KeT OKa-
3aTbCH, liTO HeHTpaJIbHaH JIHHHH nepeceKaeT O,lI.HY H3 CTO-
pOH no,nOlllBbI 3a ee npeJI.eJIaMH. B TaKHX CJIY1l3RX CJIe,nyeT
Ollpe,lI.eJIHTb OO'beM V t lIaCTH 3nIDpbI ,naBJIeHHSI, pacnOJIO-
:>KeHHOH 3a npe,neJIaMH no,nOlliBbI (3aIIITpHXOBaH Ra-
pHC. 5.7) H cpaBHHTb ero co BceM 06'beMOM 3llIDpbI, T. e.
C N. ECJIH y,noBJIeTBopSJeTcSI YCJIOBHe
147
I..C)
.
I'
,
._1
- I
-,.-"t
.>i.-
,
! _ )\ / a 2 :: #--+ 1i
/0, - 1,02 i: I A t' = J .'
r-- - --.;---
!; lAM = 'r.62 "
1,;----1------- -----'I
i I o.5A I '7 .
) · ,; J
/ A=J,6
--- - ---------
..I "J !
'Jl. t-r-f' !
-r-'
.. I C'\,j :
L-II
Cli C)
ct
PUC. 5.7
V1 O,05N,
(5.62)
TO BJIH}IHHeM V t npeHe6peralOT H yqHTbIBalOT TOJlbKO qaCTb
3n: pbI, paCnOJlO}KeHHYIO B npe,l(eJIaX nO,l(OIliBbI. B npOTHB-
HOM CJIyqae CJle,l(yeT YBeJIHqHTb pa3MepbI nO,l(OIliBbI TaK,
IIT06bI YCJIOBHe (5.62) Y,l(OBJleTBOpHJIOCb.
KaK Y}Ke OTMetlaJlOCb, npH Onpe,l(eJIeHHH py B KaqeCTBe
nJIOma.ll.H IIO,l(OIllBbI <PYHJJ.aMeHTa npHHHM aJIaCb ee C}KaTaH
30Ha OT .rr:eHCTBHH TOJlbKO N H Mx_ CTpOrO rOBOpH, CJIe.ll.O-
BaJIO 6bI npHHHMaTb <paKTHqeCKYIO C}KaTYIO 30HY nO,l(OUIBbI
c yqeTOM BJIHHHHSI My, KOTopaH Ha paCCMaTpHBaeMOH CTa-
.llHH paCqeTa eme HeH3BeCTHa. ToqHoe pellIeHHe 3a,l(aqH no-
yqeHO TOJIbKO ,lI. SI <PYH.lI.aMeHTOB C 'npHMoyrOJIbHOH nOAo-
llIBOH, ,lI.JI5I <PYH,lI.aMeHTOB Me ,l(pyrHx THnOB pellIeHHH
CJIHIlIKOM rpOM03,lI.KH H HenpHeMJIeMbI ,l(JIH npaKTHqeCKOrO
npHMeHeHHH. ConOCTaBJIeHHe TOqHOrO H npH6JIH eHHOro
perpeHHH .lI.JIH npSIMoyrOJIbHbIX <pYH.lI.aMeHTOB nOKa3aJIO He-
3H 1JHTeJIbHOe paCXO ,l(eHHe pe3YJIbTaTOB, He npeBbIllialO-
mee ,lI.OnYCTHMoro .lI.JIH HH eHepHbIX paClleTOB. TaK pa3Me-
pbI . c aTOH 30HbI nO.ll.OlIIBbI H ,l(aBJIeHHH Ha OCHOBaHHe OT-
JIMQaIDTCSI OT pe3YJIbTaTOB TOqHOrO paC1JeTa He 60JIee qeM
'H3 10%, n03ToMY onHcaHHbIH cnoco6 paCqeTa pacnpocTpa-
HeH M Ha APyrHe THnbI <PYH.lI.aMeHToB. n pHBeneM npHMepbI
npaKTHqeCKOrO npHMeHeHHH npe,l(JIaraeMOro MeTOD.a pac-
'qeTa.
148
npHMep 3. J1.aHo: pa3MepbI no.n.OIlIBbI npHMoyrOJlbHOrO cpYJII.n.aMeH.
Ta 1= 3,6 M; B =2,5 M; YCHJIHH, .n.eHcTBYIOllI.He Ha ypoBHe no.n.OIlIBbl,
N= OO KH, M x =640 KH'M, M y =120 KH.M. Tpe6yeTcH onpe.n.eJIHTb
pa3MepbI C:>KaTOH 30HbI no.u.OIllBbI H .n.aBJIeHHH Ha OCHOBaHHe.
PaCtteT: e x =640j800=0,8 M; e y = 120/800=0,15 M. no cpopMYJIaM
{4.5) H (4.6):
V A =3(0,5-3,6 - 0,8) ===3 M;
Pmax =2 - 8001 (3 - 2,5) = 213,3 Kf1a;
W xc= 3 - 2,5 2 /6=3,125 M 3 ; py= 120/3,125=== 38,4 Kna;
pmax + py ===213,3 + 38,4=251 ,7 KfIa;
Pmax - py=213,3 - 38,4= 174,9 KfIa;
A _ l' A (Pmax + P y) __ 3 . 2.) 1 , 7 __ .
1'1 - - 213 3 - 3,54 M,
Pmax ,
12 A == 'fA (Pmax - Py> _ 74. = 2 46 M.
Pmax - 213,3 '
3nlOpa J].aBJIeHHH H8 OCHOBaHHe npHBe.n.eH8 Ha pHC. 5.7. nJlOW.aJ].b C)Ka-
TOM 30HbI nOlI.OIlIBbI
F == 3,54 + 2, 46 2 5 === 7 5 2
cm " M.
2
flpH TOqHOM pellIeHHH: "e x /A=0,8j3,6=0,222; e y /B=0,15j2,5=O,06. no
rpacpHKY (CM. pHC. 5.5) npH e x /A=0,222 H e y /B=0,06 H8XOJ].HM " 1=0.96
Ii 2=O,666. no rpacpHKY (CM. pHC. 5.6) npH l =0,96 H 2=O,666 Haxo-
JI.RM Ul =2,91. Pa3MepbI H nJIow.a.n.b C:>K8TOH 30HbI nOJlOIllBbI. lA=O,96X
X3,6=3,46 M; 2A=0,73.3,6=2,4 M;
FC>K= 2,4 -:; 3,46 2,5 = 7,3) M 2 .
PaCXD)KJ].eHHe COCTaBJIHeT 2 0 /0.
P - 2 91 800 === 258 Kna.
max-, 3,6.2,5
P aCXO:>KJ].eHHe COCTaBJIHeT 2,4 °10.
npHMep 4. ,LlJIH YCJIOBHH npHMepa 3. 3a HCKJUOQeHHeU M II -
=360 KH. M, nOCTpOHTb 3nlOpy J].aBJleHHH Ha OCHOB8HMe (pHC. 5.8). 113
IIpHMepa 3: VA=3 M; Pmax=213,3 Kna; W xc =3,125 MS.
py=360/3,I25= 115,2 Kna.
pmax+py=213,3+ 115,2=328,5 Kna;
Pmax - py=213,3 - 115,2=98,1;
'\'tA=3-328,5/213,3=4,62 M > A=3,6 M;
V1=4,62/3,6= 1,283; V2A=3.98,1/213,3= 1,38 M,
V2= 1,38/3,6=0,384.
149
i
-. ..;
P-) _ 1
II C'-.I I
. I::l - " r
t,
;
Pac. 5.8
<.0/
II /
CCJ,
/
i
lea/
/7
U - t
-
I
I
Pac. 5.9
qacTb SDlOphI, pacnOJlO1KeHHaSl a8. npeJ1enaMH n0J10mBbI, npeACTIB-
nseT co60if TpeyrOJIbH}'IO DHpaMH.l1Y. Hs pac. 5.8:
a1=4,62 - 3,6= 1,02 M; a2= 1,02+0,6+0,6=2,22 M;
b - aiB _ 1,02. 2,5 -- 0 787 .
1 - - -, M,
at+ a2 ] ,02 + 2,22
1qO
_ (Pmax+PY)(At- l ) = 328,5(1,283-1) ==725 Kn.
p - 1 283 ' a
Al '
CJ1eJI.OBaTeJIbHO, VI =0,5.0,787. 72,5.},02/3=9,7 KH<0,05.800=40 ({H.
.YcoiloBHe (5.62) Y.LlOBJIeTBOpHeTcH, H 3alllTpHxoBaHHoH qaCTblO 3nlOpbl
MO)KHO npeHe6peQb. nJlOllla.llb C>KaTOH 30HbI nO,LlOIliBbI F c>K = 3,6.2,5 -
- 0,5.2,22.1,71 =7,1 M 2 . npH TOqHOM pellleHHH e x /A=O,222; ey/B=0,18.
no rpa<t>HKY (CM. pUC. 5.5) npH e x /A=0,222 H e y /B=O,18 HaXO.llHM
.t2=0,27; 1)2=0,12. no rpa<t>HKY (CM. pHC. 5.6) npH 2=0,27 H 1]2=0,12
HaXO.llHM at = 4,18. Pa3MepbI II nJlOllla.n.b C>KaTOH 30HbI nO.llOWBbI:
C =O,27.3,6=O,975 M; 1]28=0,12-2,5=0,3 M;
FC)K=3,6.2,5- (3,6-0,91)(2,5-0,3) =9-2,38=6,12 M 2 .
2
P aCXO>K,LleHHe COCTaBJlHeT 16,28 0/0,
Pmax=4,18-800f(3,6-2,5) =373 KIla.
PaCXO}l{.n.eHHe COCTaBJIHeT 11,8 % .
npHMep 5. p'aHo: pa3MepbI no.n.OlllBbI JIeHTOqHOrO 3aMKHYToro npH-
!doyroJlbHoro OlJepTaHHH B nJlaHe <t>YH.lI.aMeHTa A= 8 M, B = 6 M, a =
= 5,6 M, b = 3,6 M, YCHJlHH, .n.eHcTBYIOllI.He Ha ypoBHe nO,LlOIliBbI, N =
=2450 KH, Nlx=5527 KH.M, M y = 1027 KH.M. Tpe6yeTcH nocTpoHTb
9nlOpy ,LlaBJIeHHH Ha OCHOBaHHe (pHC. 5.9).
PaCtteT: ex/A=5527/(2450.8) =0,282; a=5,6/8=0,7; [3=3,6/6=0,6.
no Ta6J1. 4.2 np" a=O,7; =O,6 H e x /A=O,282 HaXO,LlIiM =O,254 H
1'= 0,86; "A=O,86. 8= 6,88 M;
Pmax=2450/(0,254-8-6) =201 KIla.
113 pHC. 5.9:
W = 6,88.6 3 - 5,6 · 3,6 3 = 34 02 M 3 .
xc I 2 . 3 "
py= 102/34,02=302 KfIa;
Pmax+py=201 +30,2=231,2 KIla;
Pmax -Py::= 201-30,2 = 170,8 Kna;
VtA =6,88- 231,2/201 =7,95 M; '\'1 =7,95/8=0,994;
'\'2A =6,88 - (7,95 - 6,88) =5,71 M; '\'2=5,71/8=0,714.
3aMeTHM, qTO pe3YJIbTaTbI paCqeTa no npe,LlJIaraeMOMY npH6JIH>KeH-
HOMY cnoc06y MO>KHO YTOqHHTb. ,[lJIH 3Toro CJIe.llyeT, nOJIyqHB OqepTa-
HHe H p33MepbI C>KaTOH 30HbI n0.l10WBbI no 9TOMY cnoc06y, CqHTaTb
ox nepBbIM npH6JIH>KeHHeM. 3aTeM onpe,LleJIHTb MOMeHTbI conpOTHBJIe-
HHH W xc (max) H W xc (m In) C>KaTOH 30HbI nO,LlOIliBbI, COOTBeTCTBYIOllI.He ee
pa3MepaM Ii OqepTaHHIO. TaK ,LlJIB YCJIOBHH npHMepa 3 C>KaTaH 30Ha
no.n.OlllBbI npe.n.CTaBJIHeT C060H Tpanen.HIO BbICOTOH 2,5 M, pa3MepaMH
OCHOBaHHH 3,54 H 2,46 M. ,[laJIbHeHlllHH paCqeT C yqeTOM YTOqHeHHbIX
3H8qeHHH Wxc(max)H W xc (mln)npOH3Bo.n.HT B COOTBeTCTBHH C BbIllienpH-
Se,LleHHbIMH peKOMeH,lI.8ltHHMH. .
npHMep 6_ P.aHO: p83MepbI TaBpOBOH no.nOWBbI A=4 M, B=3 M,
0=2,4 M, b=I,5 M; YCHJIHH, ,LleHCTBYIOUJ.He Ha ypOBHe nO.lJ.OlllBbI, N=
151
... Q-)
I" . . t-rf'
"
..
ct
co ". ::!9 ..
r::::J- . -
t t:
.. .....
,.-I c:c
r--.,..
..._ CO'
Q: Cr
.,
..
..;:t..
.
Pac. 5.10
=800 KH, Mx=896 'KH. , M y = 125 KH.M. Tpe6yeTcH nOCTpOHTb 9010-
PY .B.aBJIeHHH Ha OCHOBaHHe npH eHCTBHH Mx no BapHaHTY 1, T. e. B Ha-
npaBJIeHHH pe6pa no.n.OlllBbI (pHC. 5.10).
PaCtteT: exIA=896/(800.4) -:- 0,28; a=2,4/4=0,6;
= 1,5/3=0,5.
no Ta6JI. 4.4 npH e x /A=0,28, a=O,6 H =0,5 HaXO HM ,\,=0,832 H
t=0,224, "A=0,832.4=3,33 M. CJIyqaH 1, TaK KaK y=0,832>a=O,6;
Pmsx=800/0,224.4.3=297,6 Kna.
TIo <popMYJIe (4.88):
1- a 97 6 0,832 - 0,6 83
Pi = pmax == 2 , == KOa.
I 0,832
MOMeHT HHepUHH nJIOlll.a H C}KaTOH 30HbI no.n.OlllBbI
Ixc=2,4.1,5 3 /12+0,93.3 3 /12=2,7675 M'.
MOMeHT conpOTHBJIeHHH nJIorua.n.H C}KaTOH 30HbI nO OIllBbI:
W xc (pe6pa) = 2,7675/(0,5. 1,5) =3,69 M 3 ;
W XC (UOJIKH)=2,7675/1,5= 1,845 M 3 .
.n.8BJIeHHe H8 OCHOB8HHe OT .n.eHcTBHH My:
py(pe6pa)= 125/3,69=33,9 KOa; PY(UOJIRH)= 125/1,845=
=67,8 Kna.
no <popMYJIe (5.88):
_ "'( - a = 297 6 0,832 - 0, 6 83 n
Vt-pmax 1 ..' 0,832' == K . a.
152
113 pnc. 5.1 0:
a l = A Py(nOJIKH> ('y - a) - PI (1 - T)
PI
67,8 (0,832 - 0,6) - 83 (I - 0,832) === 0 086 M;
83 '
P === alPl __ 0,086. 8,3 === 7,7 Kna.
A (1' - a) 4 (0,832 - 0,6) ,
--.-.: " 4
a2=A (V - a) =4(0,832 - 0,6) ===0,93 M;
_ (PI - P y (nO.IJKM) ) a 2 _ (83 - 67 , 8) 0, 93 = 0 17M.
a 3 -. - , ,
Pi 83
V2=,2,57/4=0,643,
b 1 === a l b 0,086 · 3 = 0, 17M.
A (1-1'2) +al 4 (1- 0,643) + 0,086
06'beM qaCTH 3nlopbI, pacnOJIO>KeHHOH: 3a npe.n.eJIaMH no.n.OlllBbI <PYH-
.!J.aMeHTa,
1
V t =0,5.7,7.0,17-0,086=0,019 KH<O,05.800=40 KH.
3
YCJIOBHe (5.62) y.u.oBJIeTBopHeTcH. AHaJIOPHqHO paCCqUTbIBalOT <PYH-
.naMeH'rbI c .n.pyrHMH OqepTaHHHMH nO,lJ.OlllBbI.
rJlABA 6. OnPEAEJIEHHE OCAAOK OCHOBAHHSI
H BEPTHKAJlbHblX HAnpSI)I(EHHH 8 TOJlII.{E
rpYHTOBOrO OCHOBAHHSI
Cpe,l(H pa3JIHQHbIX cnoco6oB no onpeAeJIeHHIO ocaAoK
OCHOBaHHH H BepTHKaJIbHbIX HanpHLKeHHH B TOJIJ..l{e rpYH-
TOBoro OCHOBaHHH HaH60JIbillee pacnpocTpaHeHHe nOJIY H-
nH ,ilBa: C HCnOJII:>30BaHHeM paCqeTHOH CxeMbI OCHOBaHHH B
BHJ].e ynpyroro JIHHeHHoro ,l(ecpopMHpyeMoro nOJIynpocT-
paHCTBa H C HCnOJIb30BaHHeM pacqerHoH cxeMH OCHOBaHHH
B BH,ne JIHHeHHO J].e opMHpyeMoro (ynpyroro) CJIOH KO
HelIHoH TOJIIll.HHbI, KOTopbIe 6YJJ.eM B JJ.aJIbHeHilleM COOTBeT
CTBeHHO Ha3bIBaTb cnoc06 1 H cnoco6 2.
CJIeJJ.yeT nOAqepKHYTb, lITO 3TH JJ.Ba cnoco6a paCqeTa
pa3pa60TaH JJ.JIH OrpaHHqeHHOrO KOJIHqeCTBa cxeM 3arpy-
JKeUHH H OrpaHHqeHHOrO KOJIHqeCTBa OqepTaHHH nJIOma,lI.eH,
B npe,l(eJIax KOTOpHX npHJIO>KeHa HarpY3Ka. nOCJIe,lI.Hee 06-
CTORTeJIbCTBO BbIHYLKJJ.aeT npH6eraTb K cyry60 npH6JIH>KeH-
HbIM, HeJJ.OCTaTOQHO 060cHoBaHHbIM pellIeHHHM.
153
r \
I .
1\
h
M
(H)Z't'
.01
PRC. 6. t
,QaJIee 6Y YT rrpHBe eHbI npe JIO}KeHHH, HeKoTopbIe H3
HHX pa3pa6oTaHbI aBTopoM, a pSl H3BeCTHbIX crroco6oB
paCqeTa 3HaqHTeJIbHO ycoBepll1eHCTBOBaH, 6JIarO apH qeMY
cY1l.l,eCTBeHHo yrrp01.I.{eHbI paCqeTbI, a CJIe OBaTeJIbHO, co-
KpaIIleHa HX Tpy oeMKocTb.
6.1. BepTHKaJlbHbIe HanpH1KeHHH B TOJllI(e OCHOBaHHH
u
OT HarpY3KH, paBHoMepHo pacnpe,ll.eJleHHOH
no nJlOlI(a,ll.H MHoroyrOJIbHHKa
Cnoco6 1. CHaqaJIa paccMoTpHM npocTeHllIHH CJIyqaH
pacnpe,lI.eJIeHHH HarpY3KH HHTeHCHBHOCTblO p no nJIor.ua,QH
npHMoyrOJIbHOrO TpeyrOJIbHHKa ACB [(pHC. 6.1); OCb Z H
.cooTBeTcTBYI01.I.{aSi KOop,lI.HHaTa npHBe,l(eHbl 6e3 cKo6oK].
Onpe eJIHM HanpH2KeHHe (Jz B TO"4Ke M, paCnOJIOLKeHHOH Ha
r JIy6HHe z noJI. BepllIHHoH A OJI.HorO H3 OCTpbIX yr JIOB Tpe-
yrOJIbHHKa.
Hanp5l}KeHHe (Jz BbIqHCJIHM H3 BbIpa}KeHHH az=aP, rJI.e
a - K03cpcpHII.HeHT, yqHTbIBalO1.I.{HH H3MeHeHHe no rJIy6MHe
JI.aBJIeHHH B rpYHTe. ,L::{JIH ynp01.I.{eHHH BbIBO,lI.OB COBMe[l!a-
eM Ha qaJIO ll.HJIHHJI.pHqeCKOM CHCTeMbI KOOp,l(HHaT f), QJ, Z C
TOqKOH A, a OTCqeT yr JIa BeJI.eM OT JIHHHH AC no qaCOBOH
154
CTpeJIKe. qMTbIBaH, qTO rrJIOIIJ.3.lI.b 3JIeMeHTapHOH rrJIOIIJ.3.rr.-
KH dF = QdcpdQ, a paCCT OHHHe O T JII060H TOqKH nJIOCKOCTH
ACB .no TOqKH M R=y Q2 + Z2, H3 OCHOBaHHH H3BeCTHOH
4>OPMYJIbI ByccHHecKa [25] 33rrHIIIeM
3z 3 S S pd<pdp (6.1 )
a = 21t F (p2 + %2)6/2 '
fAe F - nJIow.8.nb paCCMaTpHBaeMoro TpeyrOJIbHHKa ACB.
ECJIH B TpeyrOJIbHHKe ACB OCTpbIH yrOJI npH BepllIHHe
A 0603H3qHTb qepe3 '1', npOTHBOJIeLKaIIJ.HH eMY KaTeT tIe-
pea i, rrpHJIeLKaIIJ.HH KaTeT qepe3 b, TO BbIpa}KeHHe 6.1 C
ooTBeTCTBYIOmHMH npe,lI.eJIaMH HHTerpHpoBaHHH rrpHMeT
BM)!
3z 3 S d b I crs 'f> pdp
rx-- q> ,
21t 0 r. (p2 + z2)5/2
nOcJIe perneHHH HHTerpaJIa 6.2 H pH.na rrpe06p330B3HHH
nOJIyqHM 3HaqeHHe K03cpcpHlI.HeHTa a
1 r mn +
a = 21t (1 + m2) VI +m2+ n 2
11 V 1 + m 2 + n 2 - mn J '
+ arctg
V 1 + m 2 + n 2 + mn 2
(6.2)
(6.3 )
f.n.e rll=z{b, n=tg '$=l{b.
no cpopMYJIe 6.3 BbIqHCJIeHbI K03cpcpHlI.HeHTbI a ,lI.JIH pa3-
JIHqHbIX 3HaqeHHH m H n, CBe.u.eHHbIe B Ta6JI. 6.1, aH3JIO-
rHQHYIO Ta6JI. I, rrpHBe.u.eHHoH B rrpHJI. 2, CHHD 2.02.01-83.
TaKHM 06pa30M, Ta6JIMIJHbIe K03QJQJHlI.HeHTbI a n03BOJIHIOT
onpe)!eJIHTb 3HaIJeHHH (Jz B JI1060H" TOIIKe OCHOB3HHH, pac-
nOJIOLKeHHOH Ha rJIy6HHe z no,lI. BepllIHHoH O Horo H3 OCT-
pwx yrJIOB npHMoyrOJIbHOrO TpeyrOJIbHHK3 OT H3rpY3KH
MHTeHCHBHOCTblO p, pacnpe,lI.eJIeHHOH no nJIOIIJ.a.u.H 3Toro
TpeyrOJIbHHKa.
BepTHK3JIbHbIe HanpH}KeHHH (Jz B TOIIKe M H3 rJIy6HHe
Z OT HarpY3KH, paBHoMepHo pacnpe.neJIeHHOH no nJIorua,lI.H
MHoroyrOJIbHHKa npOH3BOJIbHOrO OqepTaHHH, onpe.neJIHIOT
aJ1re6paHqeCKHM CYMMHpoBaHHeM HanpH}KeHHH B yrJIOBbIX
TOQKaX (BepIlIHHax) npHMoyrOJIbHbIX TpeyrOJIbHHKOB, H3
KOTopbIe pa36HBalOT 3a,lI.aHHbIH MHoroyrOJIbHHK. ,aJIH 3TO-
ro He33BHCHMO OT Toro, pacnOJIO:>KeHa JIH TOqKa M B npe-
.n.eJIax HJIM 3a IIpe eJIaMH MHoroyrOJIbHHKa, ee CoeJI.HHH-
155
Ta6J1Hl.{a 6.1
OTHoweHHe K8TeTOB TpeyrOJIbHHKa n=-l/b
m=zJb
J 1,4 1.8 2,4 3,2 5 n 10
0 o , J 25 0,151 0, 169 0, f 87 0,202 0,219 o 234"
,
° J 125 0,099 0,081 0,063 0,048 0,031 0,016
0,4 0,120 0, 146 0, 164 0, 181 0, 196 0,213 0,228
0, 120 0,097 0,080 0,063 0,048 0,031 0,016
0,8 0,100 0, 124 0,141 0, 158 0, 172 0, 189 0,204
° 100 0,088 0,076 0,061 0,048 0,031 0,016
,
1,2 0,076 0,096 0,111 ° , 127 0,141 0, 158 0,173
° 076 0,075 0,068 0,058 0,046 0,031 0,016
,
1 ,6 0,056 0,073 0,086 0,100 0,113 0, 129 0, 145
0,056 0,060 0,059 0,053 0,044 0,031 0,016
2 0,042 0,055 0,066 0,019 0,091 0,106 0, 122
° 042 0,048 0,050 0,047 0,041 0,030 0,016
,
2,4 ° 032 0,043 0,052 0,063 0,074 0,089 0,104
,
° 032 0,038 0,041 0,042 0,038 0,029 0,016
,
2,8 0,025 0,034 0,041 0,051 0,061 0,075 0,089
° 025 0,031 0,035 0,036 0,035 0,028 0,016
,
3,2 0,020 ° 027 0,034 0,042 0,051 0,063 0,078
,
0,020 0,025 0,029 0,032 0,032 0,027 0,015
3,6 ° 016 ° 022 0,028 0,035 0,043 0,056 0,069
, ,
0,016 0,02i 0,024 0,028 0,028 0,025 0,015
4 0,013 o 018 0,023 0,029 0,036 0,047 0,061
,
0,013 0,018 0,021 0,024 0,026 0,024 0,015
4,4 0,011 0,015 0,020 0,025 0,031 0,041 0,055
0,011 0,015 0,018 0,021 0,023 0,023 0,015
4,8 0,010 0,013 ° 017 0,022 0,027 0,037 0,049
,
0,010 0,013 0,016 0,019 0,021 0,021 0,015
5,2 0,008 ° OIl 0,014 0,019 0,024 0,032 0,045
,
0,008 . 0,011 0,014 0,016 0,<'19 0,020 0,015
5,6 0,007 0,010 0,013 0,016 0,021 0,029 0,041
0,007 0,010 0,012 0,015 0,017 0,018 0,015
6 0,006 ° 009 0,011 0,015 0,019 ° 026 0,038
, ,
0,006 0,009 ' 0,01 1 0,013 0,015 0,017 0,014
6,4 0,006 ° 008 0,010 0,013 0,017 0,023 0,035
,
0,006 ° 008 0,010 0,012 0,014 0,016 0,014
,
6 8 0,005 0,007 0,009 0,011 0,015 0,021 0,032
, 0,005 ° 007 0,008 0,011 0,013 0,015 0,01'1-
,
7,2 0,004 0,006 0,008 0,010 0,013 0,019 0,030
0,004 0,006 0,008 0,010 0,012 0,014 0,014
7,6 0,004 0,006 0,007 0,009 0,012 0,017 0,028
0,004 0,005 0,007 0,009 .0,011 0,013 0,013
8 0,004 0,005 0,007 0,009 0,011 0,016 0,026
0,004 0,005 0,006 0,008 0,010 0,012 0,013
8,4 0,003 0,005 0,006 0,008 0,010 0,015 0,024
0,003 0,005 0,006 0,007 0,009 0,012 0,013
8,8 0,003 0,004 0,005 0,007 0,009 0,014 0,023
0,003 0,004 0,005 0,007 0,008 0,011 0,013
156
n pOaOIl eflUe Ta6A. 6.1
-
OTHOllIeHHe K8TeTOB TpeyrOJIbHHKa n=-l/b
m= zj b
1.4
1.8
2,4
3,2
5
n lO
0,005 0,007 10,009 O,OI;i 0,022
. 0,005 0, 006 0,008 0,010 ° , ° 12
0,005 0,006 0,008 <\012 0,021
0,004 0,006 0,007 0,010 0,012
0,004 0, 006 0,007 0, ° II 0 ,020
° ,004 0,005 0,007 0, 009 0, 012
0,004 0,005 0 ,006 0, 009 0, ° 18
° , 034 0,005 ° ,OOG ° , 008 0,011
0,003 0,004 ° ,005 0, 008 0, ° 15
0, 003 0, 004 ° ,00t) I 0,007 0,011
. I
n p H M e q a H H Sf: 1. BepxHHe 3HaqeHHH K03Q>Q>HUHeHToB a co.
oTBeTcTBYIOT OCTPOMY yr JIY TpeyrOJIbHRKa, 60JIbWeMY 45°, npHJIe}Ka e..
My K KaTeTY b, HH}KHHe - OCTPOMY yrny TpeyrOJIbHHKa, MeHbweMy 45°,
npOTHBOJIe}KameMY KaTeTY b'. 2. l, b - COOTBeTCTBeHHO .lI.nHHbI 60JIbllle..
ro H MeHblUero KaTeTOB TpeyrOJIbHHKa.
lOT CO BCeMH BepllIHH3MH MHoroyrOJIbHHKa Ii OnYCK31OT Ha
HMX nepneH,lI.HKYJIHpbI H3 BCe ero CTOpOHbI. ,QJIH Bcex no-
JIYIIeHHbIX npHMoyrOJIbHbIX TpeyrOJIbHMKOB on peLI.eJIH lOT
Ta6JIHIIHbIe 3HaIIeHHH a, COOTBeTCTBYIOIIJJle TOIIKe M, KOTO.
pbIe aJIre6paHtIeCKH CYMMHPYIOT. TaK, HanpHMep, npH
pacnOJIO eHHM TOIIKM A1 B npe,lI.enax MHoroyrOJIbHHKa
ACEG (pHC. 6.2, a) 3HaIIeHHH a CYMMHPYIOT LI.JIH .6ABM +
+ .6BCM + .6DEM + .6CDM + .6EFM + .6FGM + .6GKM+
+.6KAM, a npH pacnOJIO}l{eHHH TOqKH A1 3a npe-
,lI.eJIaMH MHoroyrOJIbHHKa ACEG (pMC. 6.2, 6) ,lI.JIH
.6NEM+!:lEFA1+.6FGM+.6GKM+.6KAM - .6NCM--
- .6CLM - .6LAM.
AHaJIOrHQHO onpe.n.eJI5IIOT 3HaIIeHHH a ,lI.JIH JII060H TOQKH
M OT HarpY3KH, pacnpe,lI.eJIeHHOH no nJIOma,lIJI MHoroY OJIb-
HHKa, npH HaJIHqHH B HeM BXO,lI.HIll.HX yrJIOB.
3aMeTHM, IITO eCJIH npHMeHHTb npHBe,lI.eHHbIe peKOMeH-
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OCaJI.KH S TOqeK nOBepXHOCTH OCHOBaHHH OT HarpY3KH,
paBHOMepHO pacnpe.ll.eJIeHHOM no nJIOma.ll.H MHoroyrOJIbHHKa
npOH3BOJIbHOrO OqepTaHHH, onpe.ll.eJIHIOT no aHaJIOrHH co
163
cnoc060M 1, aJIre6paHQeCKHM CYMMHpOBaHHeM ocaAOK yr-
JlOBbIX TOIIeK (BepIlIMH) npHMoyrOJIbHbIX TpeyrOJIbHHKOB,
Ha KOTopble pa36HBalOT 3a.n.aHHbIH MHoroyrOJIbHHK. ,aJIH
SToro He3aBMCHMO OT paCnOJIOLKeHHH TOIIKH M, .n.JIH KOTO-
pOM onpe.n.eJIHIOT OCa,lJ.KY, B npeAeJIaX "JIH 3a npe,lJ.eJIaMH
MHoroyrOJIb HKa ee COe,lJ.HHHIOT ,co BceMH BeplllHHaMH MHO-
royrOJIbHHKa H onYCKalOT M3 Hee nepneH,lJ.HKYJIHp Ha Bce
CTOpOHbI nOCJIe,lJ.Hero. ,LLJIH Bcex nOJIYIIeHHbIX npHMoyrOJIb-
HbIX TpeyrOJIbHHKOB BbIQHCJI5IIOT 3HalleHHH S, COOTBeTCTBY-
IOlI.{He TOIIKe M, KOTopble aJlre6paHlleCKH CYMMHPYIOT.
TaK, HanpHMep, npH paCnOJIOLKeHHH TOIIKH M B npe,lJ.e-
J1ax MHoroyrOJIbHHKa ACEG (CM. pHC. 6.2, a) OCa,lI.KH CYM-
MMPYIOT ,lI.JI5I.6ABM +.6BCM +.6CDM +.6DEM+6.EFM+
+.6FGM+6.GKM+6.KAM, a npH pacnOJIO./KeHHH TOIIKH
lw 3a npe,neJlaMH MHoroyrOJIbHHKa ACEG (CM. pHC. 6.2,6)-
JIJ1H .6NEM + 6.EFM +6.FGM +6.GKM +6.KAM-
- 6.NCM - 6.CLM - LAM.
AHaJIOrHQHO BbIQRCJl5IIOT Oca,lJ.KI1 S ,lJ.JI5I JII060H TOIIKM M
OT HarpY3KH, pacnpe.n.eJIeHHOH no nJIO a,lJ.H MHoroyrOJIbHH-
Ka, npM HaJIMIIHM B HeM BXO,nH RX yrJIOB.
,LLoKa3aHO B 06I.I.J.eM BH.LI.e, liTO OCaD.Ka JII060H xa paKTep-
HOM TOIIKH npHMoyrOJIbHMKa, onpe,lJ.euleHHaH no peKOMeH-
,lI.aII.HHM, M3JIOLKeHHbIM B [12], paBHa oca,nKe TOll }Ke TOIIKH,.
BbIIIHCJleHHOH no npHBe,lJ.eHHbIM peKOMeH)I.aII.HHM. ,LLJI5I KpaT-
KOCTH H3JIO)KeHMH .9TO ,nOKa3aTeJIbCTBO 3,neCb He npHBe)J.eHO,.
O,nH3KO OHO nOJI.TBepLK,naeTCH CJIe,nYIOIUMM npHMepOM.
npHMep 2. npHMoyrOJIbHbIH <tJYH.naMeHT HMeeT pa3MepbI nO.llOWBbI
It =30 M; b 1 =20 M (CM. pHC. 6.3). Tpe6yeTcH onpe.n.eJIHTb OCa.lI.KY ueH-
Tpa nO.lI.OlllBbI npH TOJIllI.HHe JIHHeHHO .lI.e4>oP.MHpyeMoro CJIOH H = 10 M.
Y1JHTbIBaH n. 1 npHMeqaHHH K Ta6JI. 6.2, K03d>4>HI.J.HeHTbI k 1 H k 2 .lI.JIH
.lI.BYX O,lI,HHaKOBbIX Tpeyro.l1bHHKOB BMC H ABM npH m=2H/b=2.101l0=
=2 H n=l/b= 15/10= 1,5 paBHbl k 1 =O,145 H k 2 =0,089. CJIe.lI.OBaTeJIbHO
p H npHMoyro bHHKa
S==4 (kt +k 2 ) ==4 P10 (0,145+0,089)===9,36 L .
E np E np E np
no Ta6.H. 60 nOC0611H [12] np" m'=2H/b 1 =2. 10j20=1 H n=
=lJ/b 1 =30/20= 1,5; k o =0,469 H
So =-= pb l ko P · 20 0 ,469 == 9,38 L. 9,36 L.
E np Eop E np E np
6.3. BepTHKaJlbHbIe HanpH1KeHHH B TOJl e OCHOBaHHH
u
OT HarpY3KH, pacnpe,ll.eJleHHOH paBHoMepHo no nJlOlQaAH
paBH06e,ll.peHHOro TpeyroJlbHHKa
Cnoco6 1. B npaKTHKe npoeKTHpoBaHHH 3AaHMH II co-
opy:meHHH BCTpeIIalOTCSI CJlYllaH, Kor,na npH paCIIeTe OCHO-
164
BaHHH no ,necpopMal1.HHM npHXO,lI.HTCH onpe,lJ.eJI.HTb H3npH:>Ke-
HHH B rpYHTe OT HarpY3KH, pacnpe.n.eJIeHHOH paBHOMepHO
no nJIOma,lJ.H paBHo6e.n.peHHoro TpeyrOJIbHHK3. T3KOH pac-
qeT Heo6xo,lJ.HM npH H3rpY3Kax, npHJIOJKeHHbIX Henocpe.n.CT-
BeHHO H3 nOBepXHOCTH rpYHT3 HJIII Ha JIID60H rJIy611He OT
ero nOBepXHOCTH. B Q3CTHOCTH, p3CqeT CB3HHbIX <PYH,naMeH-
TOB H3 BHCHqHX CB3H II IIX OCHOB3HHH no .n.e<POPM3UIIRM
npOH3BO,lJ.HTCH K3K ,lJ.JIH YCJIOBHOrO <pYH.n.3MeHT3 H3 eCTecT-
BeHHOM OCHOBaHIIH C rp3HHU3MH, onpe.n.eJIHeMbIMH B COOT-
BeTCTBHH C YKa33HHHMII, npHBe.n.eHHbIMH B n. 6 CHHn
2.02.03-85.
npll Tpex CB3HX B KYCTe, p3CnOJIO}l{eHHbIX B BeplIlHH3X
paBH06e,lJ.peHHOro TpeyrOJIbHHK3, OCHOB3HHeM YCJIOBHOrO
<PYHJ1.3MeHTa C nplleMJIeMOH CTeneHbID npIl6JIH>KeHHH MO}KHO
CQHT3Tb TpeyrOJIbHHK, nO,lJ.06HWH TpeyrOJIbHHKY H3 ypOBHe
HII33 pOCTBepKa. ,aJIH onpe.n.eJIeHHH OC3.lI.OK T3KHX QJYH.lI.a-
MeHTOB ,lJ.OJ[}KHW 6bITb H3BeCTHbI 3HaqeHHH K03<l><l>HUHeHToB
a, yqHTbIB3IOlUHX H3MeHeHHe no r JIy6HHe ,nonOJIHHTeJIbHOrO
,naBJIeHH.H B rpYHTe. - B TeXHllqeCKOII }Ke JIHTep3Type OTCYT-
CTBYIOT YK33aHHH no onpe,neJIeHHID K03<l><P1lI1.HeHTOB a .lI.JlR
paCCM3TpHB3eMoro CJIYQ3H. ,.QaJIee 6YJI.YT JI.aHbI npe.n.Jlarae-
Mwe aBTopOM peKOMeHJI.aI1.HH no BWQHCJIeHHID 3Toro K03 -
<p M l1.HeHTa.
,aJIH HarpY3KH, paBHOMepHO pacnpe.n.eJIeHHOH no nJlorna-
JIM np.HMoyrOJIbHOrO TpeyrOJIbHIIKa, H3 633e perneHHH Eyc-
CMHeCKa nOJIYQeH3 (6.3) .n.JIH 3HaQeHH.H a H3 rJIy6HHe Z no,n
BepIIIIIHoH A npHMoyrOJIbHOrO TpeyrOJIbHHKa (CM. pllC. 6.1)".
P3CCMOTPHM p3BH06e,lJ.peHHbIH TpeyrOJIbHHK ABC C ueHT-
pOM THJKeCTH B TOQKe 0 H pa306beM ero H3 IlIeCTb npHMO-
yrOJIbHbIX, nonapHO p3BHbIX TpeyrOJIbHJIKOB 1, 2 II 3, ,nJI5I
KOTOpbIX BblqHCJIHM ni II mi (pIIC. 6.6). n pe,ltB3 pHTeJIbHO
onpe.neJIHM ,nJIHHbI HCnOJIb3yeMhlx B .na JIbHeHlIIeM OTpeSKOB:
AE=O,5b ; OE=I/ 3BE= I/31a2 -1/4b2=
= b/61/4a 2 / b 2 - 1 =b/ 6-y4n 2 - 1 =bc/6,
rAe c="}'4n 2 - I; BE = V a 2 - IJ4b 2 = be /2;
BK= OB · BE = bcbc = b 2 c 2 ;
AB 3 · 2a 6a
OK=YO/32 - BK2=-yb 2 c 2 /9 - b c4/36a2=b2c/6a;
AK=AB - BK=a - b 2 c 2 /6a= (6a 2 - b 2 c 2 ) {6a.
3H3qeHHH ni II mi p3BHbI:
165
£
0,5/; 0.50
b
Pac. 6.6
AE 6b 3 .
n 1 = -=-=-,
OE 2bc c
2z 12z 8m .
mt==- =-=-,
OE bc C'
_ AK _ (6a 2 - b 2 e 2 ) 6a _ 2n 2 + 1
nC) - - - -
- OK 6ab 2 c c
2z 12za 6zn 6mn.
m 2 === m 3 == - = - = - =-,
OK b 2 e be c
(6.8)
( 6.9)
(6.10)
( 6. 11 )
BK b 2 e 2 6a
n 3 == -= =c. (6.12)
OK 6ab 2 e
3a.naBaSicb 3HaqeHHHMH n 8 m, IIO cpopMYJIaM (6.8-6.12)
onpe.u.eJIHM K03cp<pHu.HeHTbI ni 8 mi, a 3aTeM no cpopMYJIe
(6.3) COOTBeTCTBYIO He 3HatIeHHSI at, <12 H <13 B TOtIKe 0
( eHTpe TSI}KeCTH TpeyrOJIbHHKa ABC) LlJIH Tpex TpeyrOJIb-
HHKOB 1, 2, 3.
K03cpcpHu.HeHT a LlJISI paBHo6e,npeHHoro TpeyrOJIbHHKa
ABC, CBe,neHHble B Ta6JI. 6.3, nOJIyqeHbI no cpopMYJIe
a=2(a1 +a2+(3). (6.13)
PaCIIeT CBaHHbIX <PYH,lJ.aMeHTOB, COCTOSlIll.8X 83 Tpex B8-
CHIIHX CBaM, paCIIOJI02KeHHbIX B BepllIHHax paBHo6e,npeHHO-
ro TpeyrOJIbHHKa, pa3pa6oTaH B ,nBYX BapHaHTax.. nepBbIM
BapHaHT IIpe,nycMaTpHBaeT C06JIIOJl.eHHe YKa3aHHM II. 6
CHHn 2.02.03-85, npH 3TOM YCJIOBHbIM cpYH,naMeHT IIpe,n-
166
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CT3BJIHeT C060H llleCTHrp3HHYIO npH3MY (pHC. 6.7). BTOpOH
B3 pH3HT npe.l1.ycM3TpHB3eT C06JIIO.lI.eHHe YK33aHHoro rrYHKTa
CHHn3 TOJIbKO .lI.JIH O.l1.HOH rp3HH YCJIOBHOrO CPYH,Q3MeHT3,
3 paCrrOJIO.lKeHHe ,QBYX .lI.pyrHx rp3HeH npMHHTO M3 YCJIOBHH
nO,Q06HH TpeyrOJIbHOH rrO.ll.OlIIBbI cpYH,lI.3MeHT3 TpeyrOJIbHH-
- u
KY, BeplIIMH3MH KOTOpOrO HBJIHIOTCH u.eHTpbI CeqeHHH CB3H.
BTOpbIM YCJIOBHeM BTOpOrO B3PH3HT3 HBJIHeTCH p3CnOJIO}Ke-
Hue H3 O.ll.HOH BepTHK3JIbHOH OCM ueHTpOB TH}KeCTH ,QBYX
YK333HHbIX TpeyrOJIbHHKOB, rrpM 3TOM YCJIOBHbIH cpYH.lI.a-
MeHT npe.ll.CTaBJIHeT C060H TpeyrOJIbHYIO rrpM3MY C noJJ.o-
IlIBOH B BH.lI.e p3BHo6eLlpeHHoro TpeyrOJIbHHKa (pHC. 6.8l.
167
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PHC. 6.8
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oC)
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n = at fJ
0,6 0,8 1 1,2 1,4 I 1,6 1,8 2 2,2
I
I
° 1 ,000 1,000 I, 000 1,000 I, 000 1,000 1,000 1,000 1,000
0,4 0,62530,81430,87330.89930.91270.92030,92490,9279 0,9300
0.8 0, 3047 0,. 4860 0,5791 0.6351 0.6706 0.6940 0 , 7099 0 ,7209 0 , 7287
1,2 0,16930.29190.3691 0,4236 0,4634 0,4928 0.5148 0.53140,5439
1 ,6 0, 10530, 1880 0,24540,28980.32500;3531 0.37570.39390,4086
2 0,0711 0,1291 0,17170,20640.23550.25990.28050.29790,3125
2.4 0,05090,09340,12570,15290,17640.19700.21490,23050,2440
2,8 0, 0381 0,0705 0,0955 0, 1171 0, 1362 0, 1533 O. 1686 O. 1823 0, 1944
3,2 0,0296 0.05490,07470,09220,10790.12220.13520,14700,1578
3,6 0,02360.04390,06000,0743 0,08730.09930,11040.12070, 1301
168
npOaOIl'ICe1iUe Ta61l. 6.3
n = al b
..t:>
-.
N
" 0.6 0.8 1 1.2 1.4 1,6 1.8 2 2,2
E
4 0,01920,03590,0491 ° ,OGIO 0,0720 0,0822 0,0917 0,1006 0, 1089
4,4 ° , ° 1 GO 0,0298 ° ,041 ° ° ,051 ° ° ,0603 0,0690 ° ,0772 0,0850 ° , 0922
4,8 0,01350,0252 0,03470,04320,0512 0,0587 0,06590.0727 0,0791
5, 2 0, ° 1 15 ° ,0216 0,0297 0,0371 0,0440 0,0506 0,0568 ° ,0628 0,0684
5,6 0,01000,01860,02570,0321 0,03820,04390,04940,0547 O,05!!8
6 0,00870,01630,02250,0281 0,03340,03850,04340,0481 0,0526
6,4 0,00760,01430,01980,02480,02950,03400,0384 0,0426 0,0467
6,8 0,00680,01270,0176 0,02200,0262 0,0303 0,03420,03800,0416
7 ,2 0,0061 0, ° I 14 ° , ° 157 0, ° 197 0,0235 0,0271 0,0306 0,0341 0,0374
7,6 0,00540,01020,0141 0,01770,0211 0,02440,02760,03070,0337
8 0,00490,00920,0128 O,OIEO 0,0191 0,0221 0,02500,02780,0306
8 I 4 0,0045 0,0084 ° , ° 116 ° , ° 1 45 ° , ° 174 0.0201 0,0227 0,0253 ° ,0278
8,8 0,0041 0,00760,01060,01330,01590,01830,02080,0231 0,0255
9,2 0,00370,00700,00970,01220,01450,01680,0191 0,02120,0234
9,6 0,00340,00640,00890,01120,01340,01550,0175 0,0195 0,0215
1 ° ° ,0032 0,0059 0,0082 ° , ° 103 ° , ° 123 ° ,0143 ° , ° 162 0, ° 181 0, ° 199
10,4 0,00290,00550,00760,00950,01140,01320,0150 0,01670,0184
10,8 0,00270,0051 0,00700,00890,01060,01230,01390,01550,0171
11,2 0,00250,00470,00660,00820,00990,01140,0130 0,0145 0,0159
11,6 0,00240,00440,0061 0,00770,00920,01070,0121 0,01350,0149
12 0,00220,0041 0,00570,00720,00860,01000,01130,01260,0139
n p H M e q 3 H 1I e. B CJIyq3 p3CrrOJIO:>KeHlIH cBaM B BepIIIHHax
paBHocTopOHHero TpeyrOJIbHlIK3, KorJI.3 a= b, CJIeJI.yeT rrOJIb30B3TbCB
KOJlOHKOH T80JIHlI.bI JIH n = 1.
B TeOp"H paCqeTa CTpOHTenbH@X KOHCTPYKnHH H3BeCT-
. HbI CJlyqaH, KOr,ll.a CJlOJKHbIe H rpOM03,l1.KHe paCqeTbI SaMe-
H IOTCH 60JIee npOCTbIMH npH ycnoBHH, qTO nonyqaeMbIe pe-
3ynbTaTbI B3aHMOOTnHqalOTCH He3HaQHTenbHO. KaK 6YJleT
nOKa3aHO ,lI.aJIee, BTOpOH npH6JIH}KeHHbIH Ba pHaHT JlaeT
Y,lI.OBneTBOpHTeJIbH@e ,lI.nH npaKTHKH pe3YJIbTaT@.
PaCCMOTpHM Ka)K,lI.bIH H3 Ba pHaHTOB. BapuaHT nepBblu.
1 CPII cp
0603HaqHM c= CBtg , t - paCCTOHHHe Me.>K,lI.Y CBaHMH
-!
B,lI.OJIb OCH x; 2u - paCCTOHHHe Me}K,lI.y CBaHMH B,lI.Onb OCH y.
B 3aBHCHMOCTH OT BenHqHH@ C ,lI.JIH 3Toro BapHaHTa B03-
MO)KHbI ,lI.Ba C.HyqaH. CnyqaH 1, Kor,lI.a 13<18 H T01.JKa 7 pac-
nOnO}KeHa npaBee OCH y (CM. pHC 6.7, a) H CJIyqaH 2, Kor-
,lI.a 13>18 H TOqKa 5 pacnOnO}KeHa neBee OCH Y (CM. pHC.
6.7, 6).
169
,L{JIH BbIQHCJIeHHH K034>4>HI.I.HeHT3 a P336MB3eM ruecTH-
yrOJIbHYIO rrO,ll.OllIBY H3 rrpHMoyrOJIbHbIe TpeyrOJIbHHKH B
COOTBeTCTBHH C YK333HHSIMH, npHBe,ll.eHHbIMH B n. 6.2 3TOH
rJI3BbI. IlpHBe.n.eM CPOPMYJIbI .n.JIH onpe.n.eJIeHHH HCnOJIb3ye-
MbIX B ,lI.3JIbHeHllleM BeJIHIIHH npH CJIYlIae 1 (CM. pHC. 6.5, a):
tg a=u/t; ll=U+C+O,5d; 1'2=c tg a/2; 13=c+d+ !
+ [2; 1 4 =2c+d+t; 15=/4 -13; 16=15 tg a; 17=
=1 1 -1 6 ; 1 8 =tj3+c+O,5d; 19=18 -13; 110=19 tg a;
1u=16/ s ina; 1 12 =[4-/ 8 ; 1 13 =1 1 -[10; 1 1 4= } t 6 . 14 )
= 1 12 /cos a-1 13 sin a; 1 15 =1 11 -1 14 ; 1 16 =1 14 tg a+
+ 17/cos a; F = 211/4 - 1516; S = /114 2 - 132/6/3;
x=S/F,
r.n.e F - nJIOIlla)J.b no.n.oIllBbI; S - CTaTHqeCKHH MOMeHT nJIOlI!a.n.H no.lI.O-
IlIBbI OTHOCHTeJIbHO ee CTOpOHbI 3' -3; x - paccToHHHe OT ueHTpa TH-
.>KeCTH no.n.OlIIBbI .n.o CTOpOHbI 3'-3.
,QJIH CJIYQ3H 2 4>OprvlYJIbI (6.14) COXp3HHIOTCSI 3a HC-
KJIlOlIeHHeM:
1 13 =1 1 ; 11 =l12 cos a -1 7 sin a. (6.15)
):LJIH BToporo BapH3HT3 He33BHCHMO OT 3HalleHHH c:
l1=3c+t+ 1,5d; b=21 1 tg a; a=O,5bjsin a;
F =O,5bI 1 . (6.16)
IloK3jKeM H3. npHMep3X npHeMJIeMOCTb p3ClIeT3 no BTO-
POMY B3PH3HTY.
npHMep 3. P.uno: CB31lHbiti YHAaMeHT 113 Tpex cBaR CeqeHHeM
30X30 CM, .lI.JIHHOH l cB =5 N, paccToHHHHMH Me:>K.n.y HHMH t= 1,2 M, 2u=
=1,2 M H <Pucp=17°. Tpe6yeTcH onpe.n.eJIHTb K03<P<PHUHeHT a Ha rJIY-
6HHe z=2,4 M OT HH33 cBaH.
P3CqeT: C=lCB tg Q>lIcp =5t.g 4°15' =0,375, tg a=0,6/1,2=0,5. no
4
4>OPMYJI3M (6.14) nqJIyqHM: 1 1 = 1,125 M, 12=0,089 M, 1 3 =0,764 M, 11,.=
=2,2'5 M, 1 5 = 1,486 M, 1 6 =0,743 M, 1 7 =0,382 M, 1 8 =0,925 M, 19=0,162 M,
1 10 =0,081 M, 1 11 = 1,662 M, 1 12 = 1,325 M, 113= 1,044 M, 1}1j,= 1,015 M, l15=
=0,647 M, 1 1 6=0,935 M, F=3,958 M 2 , S=5,148 M 3 , x=1,301 M.
T3K KaK / 3 =0,764<1 8 =0,925, TO HMeeM c.nyqaH 1. no Ta6JI. 6.1
C yqeTOM npHMeqaHHM K HeM nOCJIe rpOM03AKHX )J.BOMHbIX HHTepnOJISu.U1H
nOJIyqeHbI .cJ1e.n.YIOlI!He 3U3QeHHH a .n.JIH Ka:>K.n.oro npHMoyrOJIbHOrO Tpe-
yrOJIbHHKa, H3 KOTopbIe pa36HT3 no.n.OlllBa: 6123,1=1 12 =1,325, b=17=
=0,382, m=6,28, n=3,47, a=0,0146;
134, 1=1 14 = 1,015, b=lI6=0,935, m=2,57, n= 1,09, a=O,0304;
147, 1=1 16 =0,935, b=1 13 =0,647, m=3,71, n=l,44, a=O,0205;
168, 1=1 1 = 1,125, b=18=0,925, m=2,59, n= 1,22, a=O,0320;
189, 1=1 1 = 1,125, b=l8=0,925, m=2,59, n= 1,22, a=0,0347;
170
.6167,1=1 1 =1,125, b=lg=O,162, m=14,81, 1l=6,94, a=0,010;
a=2 (0,0146+ 0,0304+0,0205+0,0320+0,0347 + 0,010) =0,2446.
Onpe.n.eJIHM )J.JIH TCX :>Ke YCJIOBHH 3HaqeHHH a H F .n.JIH BapHaHTa 2
C " TpeyrOJIbHOH no.n.olllBoH. no <l>opMYJIaM (6.16) 1 1 =2,775 M, b=2,775,
a=3,103, F=3,85, m=2z/b=2.2,4/2,775= 1,73, n=a/b=3,103/2,775=
= 1,12.
TIo Ta6JIH e 6.3 a=0,2452. TaKHM 06pa30M 3HaqeHHH a .n.JIH 060HX
BapHaHToB npaKTHqeCKH COBna)J.aIOT, a paCXO:>K.n.eHHe nJIOw.a.n.eH no.n.o-
wabI COCTaBJI5.leT
3,85 - 3,96 100 == -2 78 % .
3 96 ' 0
,
PaCXO:>K)J.eHHe Koop.n.HHaT neHTpOB TH:>KeCTH no.n.OlllB )J.JIH llleCTH-
yrOJIbHOfr no.n.OlllBbI h - x=2,25 - 1,3=0,95 M, AJIH TpeyrOJIbHOH no-
.nO IllBbI It!3=2,775/3=O,925 M. 3Kc eHTpHcHTeT e=0,95 - 0,925=
= 0,025 M = 2,5 CM.
npHMep 4. j],aHo: cBaMHbIH <PYH)J.aMeHT H3 Tpex cBaM CeqeHHeM
40X40 CM, )J.JIHHOH l cB = 12 M, paCCT05.lHH5.lMH Me:>K.n.y HHMH t= 1,2 M,
2u= 1,2 M, Q)ucp=30o. Tpe6yeTc5.I onpe.n.eJIHTb K03<p<pH HeHT a Ha r.ny-
6HHe z = 2,4 M OT HH3a cBaH.
Pac'ler: c= 12tg CPllcp = 12.tg 7°30'= 12.0,135= 1,62 M. tg a=
4
=0,6;.1,2=0,5. TIo <popMYJIaM (6.14) H (6.15) rrOJIYIlHM:
11 =2,42 M, 1 2 =0,383 M, [3=2,403 M, h.=4,84 M, 1 5 =2,438 M, 16= 1,219 M,
1 7 = 1,201 1, 18=2,22 M. TaK KaK 1 3 >1 8 , TO HMeeM CJIyqaH 2. I g =0,183 M,
1 10 =0,091 M, 111=2,725 M, 1 12 =2,62 M, 1 13 =2,42 M, 1 1 4= 1,806 M, 1 1 5=
=0,919 M, 1 16 =2,246 M, F=20,455 M 2 , S=54,277 M 3 , x=2,654 M.
no Ta6JI. 6.1 C yqeTOM npHMeqaHHH K HeH nOCJIe rpoM03.n.KHX .n.BOH-
HbIX HHTepnOJIH lIH nOJIyqeHbI cJIe)J.YIOll1.He 3Ha qeHH5.I a )J..nH Ka:>K.lI.OrO
np5.lMoyro.nbHoro Tpeyro.nbHHKa, Ha KOTopbIe pa36HTa no.nOlllBa:
6123, 1=1 12 =2,62, b=17= 1,201, m=2, n=2,18, a=0,0481;
6.134, 1=1 16 =2,24.6, b=l14= 1,806, m= 1,33, n= 1,24, a=0,0674;
6.145, 1=1 16 =2,246, b=I 15 =O,919, m=2,61, n=2,44, a=O,0388;
6.167, 1=1 1 =2,42, b=18=2,22, m= 1,08, n= 1,09, a=0,0822,
6.178, 1=1 1 =2,42, b=18=2,22, m= 1,08, n= 1,09, a=0,0880;
6.156, 1=1 1 =2,42, b=1 g =0,183, m= 12,8, n= 13,26, a=O,OII;
cx=2 (0,0481 +0,0674+0,0388+0,0822+0,0880+0,011) =0,6512.
Onpe)J.eJIHM .n.JIH Tex :>Ke YCJIOBHH 3HaqeHHH a H F .n.JIH BToporo BapH-
aHTa C TpeyrOJIbHOH no.n.olllBoH. TIo <popMYJIaM (6.16): II = 6,66; b =
=6,66 M; d=7,45 M; F=22,18 M 2 , m=2z/b=2.2,4/6,66=0,72, n=d/b=
=7,45/6,66= 1,12, a=0,6519.
TaKHM 06pa30M 3HalleHH5.I a .n.JIH 060HX BapllaHTOB npaKTHqeCKH co-
Bna.n.aIOT, a paCXO:>K.n.eHHe nJIOw.a.n.eH rro.nOIllBbI COCTaB.nHeT
22,18 - 20,46 100 == 8 4%.
20,40 '
PaCXO:>K.n.eHHe Koop.n.HHaT eHTpOB T5.I}KeCTlI no.n.OlllB - )J.JIH llleCTH-
yrOJIbHOH no.n.OlllBbI h - x=4,84 - 2,65=2,19 M, .n.JIH TpeyrOJIbHOH
no.n.OlllBbI 11/3= 6,66/3=2,22 M. 3Kc eHTpHcHTeT e=2,22 - 2,19=0,03 M.
113 rrpHBe eHHbIX npHMepOB CJIe yeT, liTO JIH BYX Ba-
pHaHTOB YCJIOBHbIX <PYH aMeHTOB 3HaqeHHH K03<P<PHU.HeH-
TOB H nOJIOLKeHHe ueHTpOB TH}I{eCTH rrO.lI.OlIIB rrpaKTHIIeCKH
171
COBna.n.aIOT. 4TO KaCaeTC5I He60JIblilOrO paCXO}K.l1.eHH5I fiJIO'"
InaJI.eH no,nOlI1B, KOTopoe MO}l{HO B Cpe.lI.HeM npHHHTb 5 °/0,
TO OHO Ha pe3YJIbTaTbI paCqeTa 3aMeTHO He nOB.TIHHeT. CJIe-
O aTeJIbHO, Tpy,noeMKHH paCqeT no nepBOMY BapHaHTY 1
MOLKeT 6bITb Ge3 y ep6a ,.QJISI TOqHOCTH pe3Y,..JIbTa TOB 3a Me-
HeH GOJIee npOCTbIM BTOpbIM BapHaHTOM.
6.4. OcaAKH OCHOBaHHSI OT HarpY3KH,
u
pacnpeAeJleHHOH paBHoMepHo no nJIOl1(aAH
paBHooeApeHHoro TpeyroJlbHHKa
Cnoco6 2. B pa3BHTHe II. 6.3 npHBe.n.eHbI aHaJIOrHIIHbIe
peKOMeH.n.alI.HH no onpe.n.eJIeHHIO oca,nOK OCHOBaHHH OT
HarpY3KH, pacnpe)(eJIeHHOH paBHOMepHO no IIJIO aJI.H paB-
HOOeJI.peHHOro TpeyrOJIbHHKa. PaccMoTpHM paBH06e,.QpeH-
HbIM TpeyrOJIbHHK ABC C lI.eHTpOM T5I}KeCTH B TOllKe 0 I{
pa306beM ero Ha llleCTb np5lMoyrOJIbHbIX, nonapHO paBHblX
TpeyrOJIbHHKOB 1, 2 H 3 (CM. pHC. 6.6).
OCaJI.KY OCHOBaHHH no.l1. TOqKOM 0 Ha rJIYOHHe If IIony-
lIHM no cpopMYJIe
bp
S=-ko. (6.17)
E np
,I:{JIH HarpY3KH, paBHOMepHO pacnpe)(eJIeHHOH no nJIO-
maJI.H np5lMoyrOJIbHOrO TpeyrOJIbHHKa, Ha OCHOBe perneHHH
K. E. Ero'poBa [5] nOJIyqeHa cpopMYJIa (6.7) JI.JIfI onpe,aeJIe-
HHH ocaJ(KH Ha rJIy6HHe H noJI. BepwHHoH A IIpHMO-
yrOJIbHOrO TpeyrOJIbHHKa (CM. pHC. 6.1), B KOTOpOH m=
=2H/b, n=a/b, Ao=O, A 1 =0,8, A 2 = 1,4, A3=2, A4=2,6,
B o = 1, B 1 =0,426, B 2 = -6,051, B 3 =7,395, B4= -2,77.
,I:{JIHHbI HCIIOJIb3yeMbIx B .n.aJIbHeMWeM OTpe3KOB, a TaK-
}Ke 3HaqeHHH ni H mi JI.JI5I TpeyrOJIbHHKOB 1, 2, 3 He OTJIH-
QaIOTC5I OT npHBe,lI.eHHbIX B II.. 6.3 C 3aMeHOM Z Ha H.
nO,ncTaBHB nOJIyqeHHbIe 3HaqeHH5I ni H mi .l1.JIH Ka>K.rr.o-
ro TpeyrOJIbHHKa 1, 2, 3 B CPOPMYJIY (6.7), onpe.rr.eJIHM co-
OTBeTCTBYIOInHe K03<P<PHlI.HeHTbI k i H ,lI.JI5I Bcero TpeyrOJIb-
HHKa ABC
1 - "II t,
ko= -[kl +n(k 2 +ks)].
1t 3
(6.18 )
3HaqeHHH ko B 3aBHCHMOCTH OT n=a/b H m=2H/b CBe.
,.QeHbI B Ta6JI 6.4. npH COCTaBJIeHHH Ta6JIH bI npHHflTO 3Ha-
qeHHe K03cpcpHlI.HeHTa nyaccoHa '\'=0,3. ,QJIH .npyrHX 3Ha-
lIeHHM '\' CJIe,lI.yeT TaOJIHqHOe ko YMHO>KaTb Ha (1 - ,,2)/0,91.
172
Ta6J1Hl\a 6.4
oC) n=alb
:t:: I I
0.6 , I
n 0,8 1 1.2 1.4 1,6 1,8 2 2,.2
o 0,00000,00000,00000,00000,00000,00000,00000,00000,0000
0,4 0,10760,14590,1890 0,2451 0,31740,40430,50950,62700,7628
0,8 0,15670,24820,34470,46290,60990,78320,9856 1,2158 1,4792
1 ,2 0, 179 I 0,2998 0,4339 0 ,5957 0,7949 1 ,0303 1 , 3037 1,6226 I , 9175
1 ,6 0, 1909 0,3291 0,4857 0,6777 0, 9102 I , 1926 1 , 5186 1 ,8897 2,3145
2 0,19870,34850,5176 0,73050,9883 I ,3006 1,6641 2,0801 2,5488
2,4 0,20400,36040,54100,7670 1,0400 1,3772 1,.76792,21682,7246
2,8 0,20770,3701 0,55660,7937 1,0843 1,4307 1,84082,31452,8583
3,2 0,21000,37730,57080,8140 t, 1147 1,4775 1,90482,39262,9590
3,6 0 ,2119 0,3818 0,580 I 0, 8299 1 , 1377 I, 5088 1 , 9482 2,4615 3,037 t
4 0,2139 0, 3870 0,5879 0,8398 I , 1572 1 , 5381 1, 9873 2, 5098 3, 1078
4,4 0 , 2158 0,3896 0,5950 0,8533 I , 1740 1 , 5576 2, 0215 2 ,5488 3, 1543
4,8 0,2171 0,39360,59960,8621 1,1865 1,57722,04102,58793,2089
5,2 0,21780,39550,60350,8695 1, 1963 1,59672,0709 2,62043,2483
5,6 0,21900,39750,60740,8760 I ,2061 1,61192,09032,6464 3,2822
6 0,21970,40020,61130,8789 1,2158 1,6241 2,09962,66603,3106
6 , 4 0,2197 0, 4004 0 ,6152 0,8864 I , 2243 1 ,6348 2, 1191 2,6856 3,3301
6,8 0,2210 0,4033 0,6181 0,8907 1 ,2308 1 ,6442 2, 1350 2, 7051 3,3607
7 , 2 0,2215 0,4043 0,6191 0,8945 I , 2354 1 ,6526 2, 1387 2, 7221 3,3812
7,6 0,22170,40430,62260,8945 1,2418 1,65532,15722,7361 3,3996
8 0,2217 0.4068 0, 6230 0, 9011 1, 2451 1,6650 2, 1582 2, 7441 3,4082
8,4 0,22280,40770,62620,9023 I ,2508 1,67302,1751 2,7603 3,4277
8,8 0,2231 0,40820,62700,9064 1,2546 1,67482,17772,7637 3,4448
9,2 0,22340,40820,62920,9088 1,2549 1,68372, 1899 2,7802 3,4573
9 ,6 0, 2236 0,4082 0, 6305 0 ,9102 1 ,2614 1 ,6846 2, 1964 2, 7832 3 ,4668
10 0,22360,41080,63090,9102 1,2644 1,69272, 19732,79703,4793
10,4 0,22360,41140,63280,9147 1,2647 1,69432,20792,8027 3,4863
10,8 0,22360,41190,63390,9164 t ,2697 1,70042,21302,8027 3,4981
11 ,2 0,2246 0,4121 0,6348 0,9179 1 , 2721 1, 7038 2, 2168 2,8177 3,5059
11 ,6 0,2248 0,4121 0.6348 0 ,9180 1 , 2743 1 , 7041 2,2168 2 ,8223 3,5059
12 0,2250 0,4121 0,6348 0, 9180 t , 2744 1 , 7100 2,? 168 2, 8223 3, 5216
n p H M e q a H H e. a - ,lJ.JIHHa Ka1K.n.oA 03 paBHblX CTOpOH; b -
.n.JIHHa TpeTbeH CTOpOHbl paBH06e,apeHHoro TpeyrOJlbHHKa.
npHMep 5. p'aHo: d=12 M, b=15 M, p=0,3 Mna, E np =20 Mna,
H = 7,5 M. Tpe6yeTcH onpe.n.eJIHTb Oca,D.KY no.n. eHTpoM TSDKeCTH 3a.
rpY}KeHHOrO paBH06e.n.peHHoro TpeyrOJIbHHKa.
PaCtleT: n=a/b= 12/15=0,8; m=2H/b=2.7,5/15= 1. no Ta6J1. 6.4
npH n=O,8 H m= 1 HaXO)J.HM no HHTepnOJIHUHH ko = 0,274. no 4>OPMY-
JIe (7.17):
s= bp k o == 1500.0,3 0 ,274=6,2 CM.
E np 20
173
6
6
a fULL
l4-5
() D
PRC. 6.9
6.5. K onpeAeJleHHIO ocaAOK BaHHbIx 4JYHAaMeHToB
Cnoco6 1. B npaKTHKe npoeKTHpoBaHHH cBaHHbIx <PYH-
,lI.aMeHTOB BCTpeqalOTCH CJIyqaH, KOr,lI.a KOHTYP paCnOJIOLKe-
HHH cBaM B KYCTe OTJIHqaeTCH OT npHMoyrOJIbHoro. ,QJIH Ta-
KHX CJIyqaeB paCqeT YH,lI.aMeHTOB H3 BHCHqHX cBaM H HX
oCHoBaHHM no ,lI.ecpopMa HHM YCJIO)l,(HHeTCH, TaK KaK B CO-
OTBeTCTBHH C YKa3aHHHMH TI. 6 CHHn 2.02.03-85 nO,LI.OlIlBa
YCJIOBHOrO cpYH.lI.aMeHTa nOJIyqaeTCH MHoroyrOJIbHoH. DpH-
Be,lI.eM peKOMeH,lI.a HH no paCqeTY TaKHX CPYH,lI.aMeHToB.
Pa3MepbI TIO,lI.OlIlBbI YCJIOBHOrO <PYH,lI.a MeHTa 3aBHCHT OT
CJIe,lI.YIOInHX HCXO,lI.HbIX .naHHbIX: KOJIHqeCTBa cBaM B KycTe,
pacnOJIO)KeHHH CBaM, paccTOHHHM e)K,lI.Y HHMH, pa3MepoB
u 1 Cfllcp
HX nOnepeqHbIX CeqeHHH H BeJIHqHHbI CBtg · DpH H3-
4
MeHeHHH XOT.H 6bI O,lI.HOrO H3 nepeqHCJIeHHbIX ,llaHHbIx H3-
MeHHIOTCH pa3Mepw nO,llOlliB YCJIOBHOrO CPYH,lI.aMeHTa npH
coxpaHeHHH KOHTypa ero OqepTaHHH. D03TOMY ,llJIH <PYH-
,llaMeHTOB paccMaTpHBaeMblx THTIOB HMeeTCH B03MO}KHOCTb
nOJIyqHTb paCtJeTHbIe <POPMYJIbI B o6llI.eM BH,lle, Y.lI.oBJIeTBO-
pHIOII.{He BceM Heo6xo,llHMbIM Tpe60BaHHHM He3aBHCHMO OT
cpaKTHqeCKHX HCXO,llHbIX ,lI.aHHblx.
CHaqaJIa paCCMOTpHM CBaMHbIH CPYH,llaMeHT, CO,lI.ep}Kalll.HH
KYCT H3 ceMH cBaM (pHC. 6.9, a). OrpaHH1J:HMCH TOJIbKO on-
pe,ll.eJIeHHeM K03cpcpH HeHTa a, C nOMOllI.blO KOToporo yqH-
TWBaeTCH H3MeHeHHe no rJIy6HHe ,lI.OnOJIHHTeJIbHOrO ,113BJIe-
HHH rpYHTa. T3K K3K TIPH H3BeCTHbIX a ,lI.aJIbHeMlIlHH paCqeT
npOH3BO,ll.HT no YK33aHHHM CHHn 2.02.01-83. BblqHCJIHB
174
SHatIeHHe a ,lI.JIH tIeTBepTH nOJ[.OllIBbI YCJIOBHOrO CPYH,lI.aMeHTa
1, 2, 4, 5, 6 H YBeJIHtIHB ero B 4 pa3a, nOJIYtIHM 3HalleHHe a
v
AJJSI BceH no,nOWBhI.
,LLJIH onpe,ll.eJIeHHH K03cpcpHll.HeHTa a .lI.JIH nHTHyrOJIbHHKa
1, 2, 4, 5, 6 BOCnOJIb3yeMcH peKOMeH,lI.aIl.HHMH, npHBe.ll.eHHbI-
Mil B n. 6.3. Ilpe,ll.BapHTeJIbHo COe,ll.HHHM Il.eHTp nO,ll.OlIlBbI-
TOqKY 0 co BceMH BepllIHHaMH nHTHyrOJIbHHKa H onycTHM
H3 Bee nepneH,lI.HKYJIHpbI Ha Bce ero CTOpOHbI. ,QJIH Bcex
nO YlleHHWX npHMoyrOJIbH X TpeyrOJIbHHKOB HaXO,ll.HM KO-
3<pqJl1Il.HeHTbI a no Ta6JI. 6.2. 3aTeM aJIre6paHtIeCKH CYMMH-
pyeM BbIQHCJIeHHbIe 3HatIeHHH a H HX CYMMY YMHO:>KaeM Ha
lIeTblpe. TaK, HanpHMep, J[.JISI paccMaTpHBaeMoH nO,ll.OIlIBbI
a = 4 ( a126 + a236 + a346 + a456). (6.19)
I1H.n.eKcbI y a 03HaqalOT BepWHHbI COOTBeTCTSYlOlltHX TpeyrOJIbHHKOB.
IlpHBe,ll.eM oe3 npOMe)KYTOIIHbIX BbIKJIa,ll.OK 3HatIeHHH
,lI.JII1H OTpe3KOB npHMbIX, HCnOJIb3yeMbIx B .n.aJIbHeHllIeM npH
BbltIHCJIeHHH K03<P<1>HnueHToB a:
a === O,5d --1- lCB tg CJ>IICP ; C == O,7071d cos (45° - );
4
e === c + l'B tg q>IICP ; 1 5 - 6 .- .A ==a + m; 57_:= a + /+ u;
4
( 6.20 )
k===lisin'f-a/tg ; S==a+/-k; ll_2--==A-Stg ;
12- =S/cos 1J'; 13-6=e+ (t + u) sin "l'; 14-5 == 0- S;
12-3= l3-6/tg 11'; 13- = 1 2 -4, - [2-3, r.ll.e tg "l' = mlt.
BbIllie 6bIJI paCCMOTpeH cJIYtIaii, KOr,ll.a TOllKa 3 paCllO-
JIO}KeHa Ha CTopOHe 2, 4 nHTHyrOJIbHHKa. B 3aBHCHMOCTH
OT paHee nepellHCJIeHHbIX HCXO.ll.HbIX ,lI.aHHbIX TOIIKa 3 MO-
LKeT ObITb pacnOJIO)KeHa Ha npO)I.OJI}KeHHH CTOpOHbI 2, 4
(CM. pHC. 6.9, 6). npH 3TOM Bce CPOPMYJIbI (6.20) coxpa-
HRIOTCR 3a HCKJIlOtIeHHeM
134 === 1 23 - 1 24 , (6.21 )
a K03cp<pHIl.HeHTbI a CYMMHPYIOT no cpopMYJIe
a === 4 (a126 + rJ..236-rJ..436 + (1456). (6.22)
3HaK «MUHYC» y K03<pcpunHeHTa a43603HallaeT, liTO ero
CJIe,ll.yeT npHHHMaTb OTpHnaTeJIbHbIM. KpHTepHeM ,lI.JIH OTHe-
ceHHH K CJIYllalO 2 paCtIeTa CJIY:>KHT YCJIOBHe
1 24 < 1 23 - (6.23)
175
PaCCMOTpHM aHaJIOrHqHbIe QJYH.lI.aMeHTbI, CO.ll.ep}KamH
KYCTbI H3 .lI.eCHTH CBaH. PaCqeT TaKHX cpYH,lI.aMeHTOB He 01-
JIHllaeTCH OT H3JIO}l{eHHOrO, YBeJIHqHBaeTCfl TOJIbK-O codr-
BeTCTBeHHO pa3Mep B H, CJIe,lI.OBaTeJIbHO, 136 H 1 45 . 06bI -HO
IlJar CBaH B npe.ll.eJIax O,lI.HOrO <PYH,lI.aMeHTa npHHHMalOT OJI.H-
HaKOBbIM B Ka}K,lI.OM HanpaBJIeHHH. B HeKOTopbIX CJIyqaHX
no KOHCTPYKTHBHbIM C006pa}KeHHflM OH MO}KeT 6bITb pa3-
HbIM. n03TOMY cpOPMYJIbI nOJIyqeHbI B 06IIleM BH.lI.e, KOr,lLa
liaf f MO}KeT OTJIHqaTbCH OT lIlara U.
npHMep 6. )J.JIH CBaHHoro CPYH.n.aMeHTa, co.n.epJKall{ero KYCT H3 ce-
MIl CBaM CeqeHHeM 30X30 CM, ,l1J1HHOH 10 M onpe.n.eJIHTb 3HaqeHHe a
Ha rJ1y611He z= 4,5 M HlILKe riO,nOlliBbI YCJIOBHOrO <PYH.n.aMeHTa, r.n.e 3a-
JleraeT rpYHT C CPucp=30o; f=u=O,475 M; m=0,8 M.
PaCtteT: no <popMYJIaM (7.20):
tg = 0,8 = 1,6842; = 59°18'; sin IjJ = 0,8599;
0,475
cos = 0,5105; tg (fllcp === tg 30° === 0,1317;
4 4
a=O)5. 0,3+ 10.0,1317=0,15+ 1,317= 1,467 M;
cos(45° - '\P) =cos(-14°18') =0,969;
c=0,7071.0,3. 0,969=0,206 M;
e=0,206+ 1,317= 1,523 M; A= 1,467 +0,8=2,261 M;
5=1,467+2.0,475=2,417 "M; k=I,523/0,8599-
- 1,467/1,6842=1,771-0,871=0,9 M; S=I,467+
+0,475 - 0,9= 1,042 M;
1 2 - 4 = 1,042/0,5105=2,041 M;
1 1 - 2 =2,267 - 1,042 · 1,6842=2,267 - 1,755=0,512 M;
13-6= 1,523+2. 0,475.0,8599= 1,523+0,817 =2,34 M;
12_3=2,34/1,6842 - 0,512/0,8599= 1,389. 0,595=0,794 M;
l4-5=B-S=2,417-1,042=1,375M; lS-6=A=2,267 M.
TIpoBepfleM YCJIOBHe (6.23): 12-4=2,041 >12_3=0,794, CJ1e.n.OBa-
TeJ1bHO, paCqeT OTHOCHTCH K cJIyqalO 1 H
l3-4=2,041 - 0,794= 1,247 M.
no HaM.neHHbIM pa3MepaM onpe.neJIHeM TaOJUlqHble 3HaqeHHfI a,
npn6eraH B Heooxo.n.HMbIX CJIyqaHX K HHTepnOJIHIJ.HH.
,l!JIH TpeyrOJlbHHKa -1, 2, 6 b=11-2=O,512; 1=B=2,417 M; n=
=2,417/0,512=4,72; m=4,5/0,512=8,8; (Xl.2.S=O,OI05.
)J.JIH TpeyrOJIbHHKa 2, 3, 6: b=12.3=O,794 M; 1=1 3 ,6=2,34 M; n=
= 2,34/0,794 = 2,95 3; m = 4,5iO,794 = 5,67; U2.3.6 = 0,0 165.
176
a 1
PHC. 6. to
.l!.JUI TpeyrOJIbHHKa 3, 4, 6: b=1 3 ,4= 1,247 M; 1=13.6=2,34 M; n::
=2,34/1,247= 1,877 1,8;
m=4,5/1,247 =3,608 3,6; as,4,6=0,028.
.lI.JIH TpeyrOJIbHHKa 4, 5, 6: b=[,.,s= 1,375 M; 1=1 5 . 6 =2,267 M; n::a
=2,267/1,375= 1,649; m=4,5/I,375=3,2; a4.5,6=O,0275.
no 4>opMYJIe (6.22):
a=4 (0,0105+ 0,0165 +0,028 + 0,0275) =0,33.
Onpe.n.eJIHM 3HaqeHHe K03«p<PHUHeHTa a .n.JIH npHMoyrOJIbHlIKa C pas-
MepaMH: b=2A=2.2,267=4,534 M; 1=26=2.2,417=4,834 M, T. e. 6es.
Y'leTa cpe38HHbIX yrJIOB n=4,834/4,534=I,066; m=2.4,5/4,534 2.
no Ta6JI. 1 npHJI. 2 CHHn 2.02.01-83 a=0,387>O,33. Pacxo>K.lI.e-
EHe COCTaBJISIeT
0,387 -0,33 100
- 17 3 OL O .
o 33 ' 7(J
,
CJIe.n.OBaTeJIbHO, 3aMeHa cpaKTHQeCKOH no.n.OlllBbI YCJlOBHOrO <PYHJ1aMeH-
Ta rrpflMoyrOJIbHlIKOM pa3MepaMH 2AX26 He.n.onycTlIMa. Oca.nKH npH
Tpex CBaHX, pacnOJIOJKeHHbIX B BepwHHax paBH06e.npeHHoro TpeyrOJIb-
HHKa, onpe.neJIHIOTCH B COOTBeTCTBHM C peKoMeH)J.aI1.HHMH, H3JIO:>KeHHbIMH.
II n. 6.3.
6.6. K onpeAeJleHHIO ocaAOK OCHOBaHHSI Hacblneii
Cnoco6 2. Onpe,lleJIeHHe oca,nOK OCHoBaHHH cnoco60M 1
OT BepTHKa bHoH TpaneQHeBH,llHOM HarpY3KH, pacnpe,neneH-
HOM no rrOJIoce (npHBe,lleHO B pa60Te [23]-). B npaKTHKe
npoeKTHpOBaHHH pa3JIHQHWX coopy*eHHH rrpHxoAHTCH pe-
llJaTb 8HaJIOrHQHYIO 3a,nalIY ,nJIH CJIY1JaH paCQeTHOH cxeMhI
OCHOBaHHH B BHAe CJIOH KOHelIHOH TOJIIll.HHbI (cnocoooM 2)'..
Pe3YJIbTaTW paCqeTOB no 3TOH cxeMe GOJIee 6JIH3KH K pe-
3YJIbTaTaM 3KcrrepHMeHTaJIbHbIX HCCJIe.n.oBaHHH [3]. ,QaJIee
npHBe.n.eM COOTBeTCTBYIOInHe peKOMeH.n.au.HH. 113 TpeyrOJIb-
HOM HarpY3KH BbI.n.eJIHM 3JIeMeHTa PHYIO nOJIOCKY HHTeH-
CHBHOCTblO dpl H IlIHpHHOM a1=ap1/p (pHC. 6.10).
177"
Onpe.n.eJIHM OC8}1KY OCHOBaHHSI no.n. BepIIIHHOH: M npH-
Moro yrJIa TpeyrOJIbHOH HarpY3KH, .n.JISI qero BOCnOJIb3yeMcH
CPOPMYJIOH, npHBe,lJ.eHHOH B pa60Te [5], }1J1S1 nOJlOCOBOH paB-
HOMepHO pacnpe,lJ.eJIeHHOH HarpY3KH.
,LLJIH TOIIKH, pacnOJIO}KeHHOH no,lJ. KpaeM T8KOH HarpY3-
KH, oca.n.K8
S = 2ap (I - ..,2) (-B;) [ -In ( 1 + A, 2 m 2 ) +
1CE n p " 4 I
t=O
- 2 ]
+ Aim arctg - ,
A;m
rn.e Ao=O, A I =0,8, A2=l,4, A 3 =2, A4=2,6, B o =0,426, B 2 = -6,051.
B 3 =7,395, B 4 = -2,770, m=H la, a - nOJIYllIHpHH3 nOJIOCOBOH HarpY3KH
ECJIH pa36HTb TpeyrOJIbHYIO HarpY3KY Ha 3JIeMeHTap-
HbIe nOJIOCKH HHTeHCHBHOCTblO dp1 H nepeMeHHOH llIHpHHOH
ai, TO ,lJ.JIH KaJK,lJ.OH T8KOH nOJIOCKH COXp8HHTCH CPOPMYJl8
(6.24) npH YCJIOBHH 38MeHbI B HeH P, 2a"H m COOTBeTCTBeH-
HO H8 dPt, at H
( 6. 24 )
m=2Hp/ (apt) =2 m p/Pt, r.n.e m =H/a.
B pe3YJIbTaTe ,lJ.JIH K8JK.n.OH 3JIeMeHT8pHoH nOJIOCKH no-
JIyqHM
4
dS 1 == 1 -- 'J2 V (-B i ) [ In ( 1 + Ai2 2P2 ) +
1tE n p PI
1=0
+ 2Ai m L arctg Pt ] a PI dpl.
Pt A.mp p
LlJIH BceH TpeyrOJIbHOH HarpY3KH npOHHTerpHpyeM BM-
paJKeHHe (6.25) B npe,lJ.eJlax OT 0 }10 p, npe,nBapHTeJIbHO
3aMeHHB nepeMeHHYIO. npHHHMaeM P1/P=U, TOr,lJ.a dp1=
= pdu, npH 3TOM npe,lJ.eJIbI HHTerpHpOB8HHH 6y.n.YT OT 0 .n.o
1. CJIe,lJ.OB8TeJlbHO, OC8,nKa OCHOBaHHSI no,n TOqKOH M OT
BceH TpeyrOJIbHOH HarpY3KH OKaJKeTCSI
51 = 1 -- ..,2 (-Bd 5 1 [ In ( 1 + Ai2 2 ') +
,. E n p U J
1=0 0
2A . m U 1
+ 1 arctg apudu.
u A,m
nocJle perneHHSI HHTerpaJla H cooTBeTcTBYIOll{HX npe06-
p830BaHHH nOJIyqHM
( 6.25 )
( 6.26 )
pa
8 1 == - k 2 ,
Eop
(6.27 )
r.ll.e
178
5
1
2
PUC. 6.11
4
k.. = I 'II! (-B,) [0.5 (l- c 1l 2 ) In (1 + C1l 2 ) +
I=:{)
+O,5cl121nc1l2 + 2c 1 t arctg I ] , Ct' = Atm. (6.28)
C 4 i
I1cnOJIb3YH QJOPMYJIY (6.28), MOLKHO nOJlYllHTb OCa.lI.KY
OT Tpanel(HeBH)I.HOH HarpY3KH 1, 2,4, 5 (HacbInH) KaK pa3-
HOCTb oca OK OT TpeyrOJIbH X HarpY30K 1,2,Q H 3,4,5
(pHC. 6.11).
8=8 1 - S2.
( 6.29)
0603HallHM
1;,=a/ H, =bIH,
( 6.30 )
Tor,na
I 1 I
m 1 = , m2=I/ , Clt=A f , C 2 1=A 1 -. ( 6.31 )
C+t - + ..
npH npHHHTbIX 0603HaqeHHHX nOCJIe COOTBeTCTBYIOI..I.{HX
BbIKJIa)].OK nOJIYQIIM
ph
S = -k 1 ,
Eop
( 6.32)
f.ll.e
k 1 = 0.28966+ [( 1 + tr k 2 - ka} (6.33)
3HaQeHHH k 2 II kg onpe,neJIeHbI no cpopMYJIe (6.28), npII
3TOM ,!1JIH k 2 npHHHTbI CH, a )].JIH kg - C2i no cpopMYJIe
(6.31). 3aTeM no cpopMYJIe (6.33) BbIQHCJIeHbI K03<P<PHUH-
eHTbI k i , CBe eHHble B Ta6JI. 6.5. IlpH COCTaBJIeHIIH Ta6JIH-
lI.bI npHHSlTbI HHTepBaJIbI 3HaQeHHH H '\P, HCKJIIOQalOw.He
Heo6xo.lI.HMOCTb CJIO HOH .lI.BOHHOH HHTepnOJIHUHH npH 0.-
179
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0 .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. " .
0 0 .0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
CD CD r..... t.... 00 00 Q') Q') Q') 00 0 - -
CD (;0 CD CD CD CD t.... t.... ('-. 00 00 00
oq, I.C 1Q 1Q 1Q I.Q I.C I.C I.C I.C I.C I.C I.C I.Q LQ I.Q I.C I.Q
0 .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Q') Q') - - C'\1 I.C CD t.... I',. Q') CV') CD 00 0
r-- C'\1 C'\1 CV') CV') M M CV') CV') CV') I.C I.C I.C CD
CD CD CD CD CD CD CD CD CD CD CD CD (Q CD CD
= .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..
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0 .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..
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<.0 CD <.0 <.0 LQ If) LC lC <.0 CD CO to
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Pac. 6.13
pe,lleJIeHHH kt. ,QJIH HeKOTOpbIX CJIyqaeB, KOr.u.a TaKaH HH-
'fepnOJIHll.HH HeH36eLKHa, COCTaBJIeH rpacpHK (pHC. (3.12 H
6.13), aHaJIOrH1:IHbIH rpacpHKY H. OCTep6epra, npHBe,QeHHO-
MY B [23]. KpoMe Toro, npHHHTO 3Ha1:leHHe K03cpcpHll.HeHTa
nyaccoHa ,,=0,3. ,L:LJIH JJ.pyrHx 3Ha1:leHHll " Ta6JIH1:IHbIe KO-
3cpcpHll.HeHTbI k 1 CJIe.u.yeT YMHO}KaTb Ha (1 - v 2 ) /0,91.
YCTaHOBJIeHO, 1:ITO npH 11' 1 K03cpcpHu.HeHTbI k i He 3a-
BHCHT OT , n03TOMY Ta6JIHu.a COCTOHT H3 ,QBYX 1:IaCTeH, r,fte
BTopaH 1:IaCTb npe.u.CTaBJIeHa O,QHOH CTpOKOH co 3HaqeHHHMH
k i , 33BHCHlllHrvlH TOJIbKO OT ,p 1.
TaK KaK 3HaqeHHH =aJH H \p=b/H H3MeHHIOTCH B
60JIbillOM '.u.Hana30He, TO nOJIyqHTb rpacpHK B O,QHOM Mac-
lllTa6e C pa3MepaMH, npHeMJIeMbIMH ,QJIH npaKTH1:IeCKOrO HC-
nOJIb30BaHHH, He npeJJ.CTaBHJIOCb B03MO>KHbIM. 113 3THX
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eHHbIX B pa3HbIX MaCllITa6ax. KpOMe Toro, H3 Hero HCKJIIO-
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cpe CTBeHHO no Ta6JIHu.e JIH60 npOCTOM HHTepnOJIHlI.HeM no
TOM }Ke Ta6J1Hne.
B Ta6JI. 6.5 H rpacpHKax Ha pllC. 6.12, 6.13 3HaQeHHlO
t==O cooTBeTcTByeT paBHOMepHO pacnpe,QeJIeHHaH nOJIOCO-
BaH HarpY3Ka, a ,p=O - TpeyrOJIbHaH nOJIOCOBaH HarpY3Ka.
,Q H onpe,QeJIeHHH oca,QKH OT TpeyrOJIbHOH HarpY3KH HC-
nOJIb3yeTcH cpopMYJIa (4), B KOTOpOM k 2 .npHHHMaeTcH paB-
HbI l1 kt. OOJIY1:leHHbIe CPOPMYJIbI (6.27) H (6.32) aHaJIOrHQ-
HbI cpopMYJIe (67) [12].
BXO,lI.Hlllee B 3TH c}JOPMYJIbI npHBe.u.eHHoe B npe,QeJIax
..HHHeHHO-,fLecpopMl1pyeMoro CJIOH (no 3a.u.aHHOM BepTHKaJlH)
3Ha1:leHHe MO.u.YJIH ,QecpopMall.llH rpYHTOB OCHOBaHHH onpe-
}l,e,lJ5!eTCH aH3.HorHQHO QJopMYJIe (79) [12]:
kn'
E np ---=
n' k 1 - k. 1
l-
i=l E i
r.n.e k i - K03<P<PHI.J.HeHT, BbIQHCJISIeMbIH no Ta6JI. 6.5 HJIH rpacpHKaM 6.12,
6.13 3HaqeHHH k i .n.JISI i-ro CJIOH rpYHTa, npH rJIy6HHe 3aJIeraHHH no.n.o-
IllBbI 3Toro CJIOH Zi; Ei - MO.n.YJIb .n.eq,opMaUHH i-ro CJIOH rpYHTa, nf-
tIHC.rIO CJIOeB, pa3JIHqalOIU.HxCH no C}I{HMaeMOCTH B npe.n.eJIax C>KHMae-
MOrl TOJIllI.H H. I1cnoJIb30BaHHe paCtIeTHOH cxeMbI OCHOBaHHH B BH.n.e
JIHHeI1Ho- .n.ecpopMHpyeMoro CJIOH KOHetIHOH TOJIIU.HHbI, a TaK>Ke onpe.n.e-
JIeHHe rJIy6HHbI C>KHMaeMOH TOJIllI.H oroBopeHbI B [12].
npHMep 7. Onpe.n.eJIHTb oca.n.KY S B TOtIKe M .n.JIH Tpex CJIyqaeB,
H306pa)l{eHHbIX H3 pHC. 6.11, npH p= 0,3 Mna H E np = 20 Mna (pa3Me-
pbI npHBe.n.eHbl Ha pHcYHKax).
3HaqeHIIH K034>4>HUHeHToB k l npH He06xo.n.HMOCTH onpen.eJIHJIHCb no
HHTepnOJISII.J.HH. CAY'l,ar1 1 (pHC. 6.14, a). 3HaqeHHHM =aIH=O H =
= b I H = 5/7 = 0,714 cooTBeTcTByeT k l = 0,620.
183
aJ biPJM " .
M
K
II
6)
b'=5M
a 3M
I
I
M
6) b=2M a-::.2M b'=OM 0=2101
l ---
k r - -
l n
.
N
11
M
PUC. 6.14
S === 2 . 500 . 0,3 0,620==9,3 CM.
20
CAyttau 2 (pHC. 6.14, 6). ,aJIH HarpY3KH CJIeBa OT T01JKH M 3Ha qe-
HHjlM =3/7,2=0,417 H :c:: 1/7,2=0,139 cooTBeTcTByeT k 1 =2,473. .QJIJI
HarpY3KH cnpaBa OT TOtJKH M 3HaqeHHHM =3j7,2=0,417 H 1¥=5/7,2==
==0,694 cooTBeTcTByeT k 1 =O,66. S=0,3(2,473.100+0,660.500)j20=
== 8,67 CM.
CAY'laU 3 (pHC. 6.14, 8). ,lI.JIH onpe.ll.eJIeHHH oca.n.KH npHKJIa.ll.bIBaeM
4>HKTHBHYIO OTpHI.J.aTeJIbHYIO HarpY3KY no snlOpe klmn. ,[{JIH nOJIHOM Ha-
rpY3KH 3HaqeHHSIM =2j7,2=0,278 H 1¥= 12/7,2= 1,667 cooTBeTcTByeT
k 1 = 0,28. ,l1JIH cpHKTHBHOH HarpY3KH 3HaqeHHHM =1¥=2/7,2=O,278 co-
oTBeTcTByeT k 1 == 1,342.
S=0,3(1200.0,280 - 200.1,342)/20=0,978 CM.
6.7. Bbl1JHCJleHHe ocaAOK xapaKTepHblx T01JeK
nOAOWBbI npHMoyroJlbHoro 4JYHAaMeHTa
Cnoco6 2. B pa60Te [5] npHBe,neHB 4JopMYJIa ,nJIH on-
pe.n.eJIeHHH oca,nKH JII060H TOqKH no,nOllIBhI npHMoyrOJIbHO-
ro <PYHJ1.3MeHT3, 3 T3K>Ke BbITeK3101ItHe H3 Hee <POp MYJIbI,
BbIpa)K3lOmHe OC3)].KH Xap3KTepHbIX TOqeK nO,nOlliBbI (II.eH-
Tp3JIbHOH, CepeJ1.HH CTOpOH H yrJIOBbIX). B 3THX cpopMYJIaX
0603H3QeHHH npHHHTbI B COOTBeTCTBHH C lloco6HeM [12]:
11 Ao=O, A 1 =O,8, A 2 = 1,4, A3=2, A,,=2,6, B o = 1, B 1 =
=1,4, B 2 = -6,051, B 3 =7,395, B,,= -2,77, m'=t'=
=2H/b, n=f)=llb.
YCT3HoBneHo, QTO JIH Bcex nepellHCneHHHX xapaKTep
HbIX T01.IeK nOJ].OllIBhI npHMoyrOJIbHOrO Q>YH.D.aMeHT3 coxpa-
HHeTCH OCHOBHOH BH,lt paCqeTHOH Q>OPMYJIbI (67) OOC06HSI
[12]. npH 3TOM 3HaqeHHe K03Q>c}>HUHeHTa k' CJIe.n.yeT npH-
184
HHMaTb: .lI.JIH II.eHTpaJIbHOM TOqKH k' =ko, AJISI Cepe)J.HHbI
60JlbllIeH H MeHbllIeH CTOpOH nO.ll.OWBbI COOTBeTCTBeHHO
k' = k i Ii k' = O,5k 2 H ,lI.JIH yr JIOBOH TOlIKH k' = O,5k 3 . AHa-
JIH3 nOKa3aJI, lITO 3HalIeHHH kOI k i , u kg MOLKHO onpe]J.e..
JIHTb no cpopMYJIe (51) pa60TbI [5] C H3MeHeHHeM TOJlbl\O
BXO.n.HIUHX B Hee K094><PHI..(HeHTOB m' H n, cor JIaCHO Ta6JI.
6.6.
TaOJlHQa 6.6
TOqKH nO,lJ.OlllBbI cl>YH,lJ.aMeHTa.
KOTOpMX onpeAen eTCB
ocaAKa
K034>4>HltReHTbl
f
I m'
t&
k
UeHTpaJIbHaH
Cepe.n.HHbI 60JlblllHX CTOpOH
Cepe.n.HHbI MeHbWHX CTOpOH
YrJIOBble
ko
k 1
k 2
k3
2Hlb
Hlb
2Hlb
Hlb
lIb'
0,5llb
21/b
lIb
CKa3aHHoe n03BOJIHJIO OrpaHHqHTbCH TOJIbKO onpeAene-
HHeM K03cpcpHII.HeHTOB ko, BbtlIHCJIeHHbIX no Q>opMYJIe (51)
pa60TbI [5] H CBe.lI.eHHbIX B Ta6JI. 6.7. 3HaqeHHH k i , H k J
HaXO.lI.HT no TOM LKe Ta6JIHI1e, HO ,lJ.JIH m' H n, npHBe,l{eH..
HbIX B Ta6JI. 6.6.
B Ta6JI. 60 lloco6HH [12] npHBe.n.eHhI Bce 3HalIeHHH KO"
acpcpHI..(HeHToB k, a B npHJIaraeMOH Ta6JI. 6.7 OHH cOBMew.e-
Bbl. IlpH 9TOM Ta6JI. 6.7 COCTaBJIeHa C MeHbllIHMH HHTep-
BaJIaMH 3HaqeHHH m' H n, liTO npH }IX onpe)leJIeHHH HCKJIIO-
qaeT He06xo)(HMOCTb B HHTepnOJIHllHH H 06ecnellHBaeT 3Ha-
liIiTeJIbHO 66J1blllYIO TOIIHOCTb YKa3aHHbIX K09Q>cpHI..(HeHTOB.
npH m' H n, OTJIHlIalOInHXCH OT npHHHTbIX B Ta6JI. 6.7,
JI.OnYCKaeTCH C .lI.OCTaTOlIHOH TOlIHOCTblO J(JIH HH>KeHepHbIX
paCqeTOB npHHHMaTb 6JIH}KaHIlIHe HX Ta6JIHtlHbIe 3HalIeHHH.
LlJIH pe,lJ.KHX CJIYlIaeB, Kor.na HHTepnOJIHII.HH Bce LKe HeH3-
Oe}KHa, npHBe.lI.eH rpaQ>HK (pHC. 6.15).
npHMep 8. ,.QJHI <PYH.n.aMeHTHoH nJIHTbI pa3MepaMH l = 37,5 M H
b = 15 M npH paCqeTHOH TOJIllI.HHe JIHHeHHO )J.e opMHpyeMoro (ynpyroro)
CJIOH H =2,25 M Tpe6yeTcH onpe.n.eJIHTb 3HaqeHHH k. no Ta6JI. 60 noco-
6HH [12] npH m'=2.2,25/15=0,3 H n=37,5/15=2,5 nOCJIe .n.BOHHOH HH-
TepnOJIHUHH nOJIyqeHbI ko = 0, 137, k 1 = 0,068, k2 = 0,069 H k3 = 0,034. no
Ta6JI. 6.7 npH m" =0,3 H n=2,5, k o =O,136; npH m' =0,15 H n= 1,25,
k 1 =0,091; npH m'=0,3 H n=5; 0,5 k2=0,5.0,137=O,069; npH m'=O,15
H n=2,5; 0,5 k 3 =O,5.0,092=O,046.
185
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.LI.JIH MHorHX 3HaqeHHH m' H n npH HCnOJIb30BaHHH Ta6JI. 60
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TOqKH npHMoyrOJIbHOrO cpYHJlaMeHTa C npHMeHeHHeM pac-
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.nyeT co6JIIO,Lla Tb Bce Tpe6oBaHHH, npe)1.ycMoTpeHHbIe !loco-
6HeM [12], OTHOCHlll.HeCH K paCIIeTY Oca)1.0K C npHMeHeHHeM
paClIeTHOH cxeMbI O'CHOBaHHSI no cnoco6y 2.
6.8. Onpe.ll.eJleHHe OCa.ll.OK OCHOBaHHH OT HarpysKH,
u
pacnpe.ll.eJIeHHOH paBHoMepHo no nJIOll(aAH KOJIbQa
B [6] )1.aHbI BbIpa}KeHHH )1.JIH onpe.n.eJIeHHSI BepTHKaJIb-
HbIX HanpH}KeHHH B JII060M TOIIKe OCHOBaHHSI KOHellHOH TOJI-
Ill.HHbI HJIH nOJIynpocTpaHCTBa OT paBHOMepHO pacnpe,ne-
u 0
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BepxHeM nOBepXHOCTH OCHOBaHHH. HCnOJIb30BaHHe 3THX 3a-
BHcHMocTeM BBH)1.Y HX CJIO>KHOCTH B03MO}KHO JIHWb C npHBJIe-
qeHHeM 3BM JIH60 C nOMOlll.blO OrpaHHlIeHHbIX Ta6JIHI.(, npH-
Be)1.eHHbIX B TOM }Ke CTaTbe. DpH 9TOM ,nJISI onpe,neJIeHHH
oca.n.oK OCHOBaHHSI npHMeHSIIDT MeTo)1. 9JIeMeHTapHoro CYM-
MHpoBaHHH.
B nOpH)1.Ke )1.aJIbHeMllIero COBeplIleHCTBOBaHHH H 3HatJ:H-
TeJIbHOrO ynpOIll.eHHH paClIeTa B HaCTOHlll.eH rJIaBe npHBO-
,llHTCH npOCToe pellleHHe, n03BOJIHIDIlI.ee onpe,neJISlTb oca.qKH
B JII060M TOIlKe OCHOBaHHSI nOJ{ rH6KHMH KOJIbueBbIMH WTaM-
naMH C YIIeToM HeBbI.lI.ep>KaHHOCTH CJIOeB rpYHTa, a TaK>Ke
B3aHMOBJIHSlHHSI COCe.lI.HHX <PYH,naMeHTOB.
n pHBeJ{eHHble ,naJIee peKOMeH.lI.all.HH n03BOJIHIOT onpe-
.lI.eJIHTb OCa,LlKH no)1. YKa3aHHbIMH QJYH.lI.aMeHTaMH B nlO6bIX
TOQKaX OCHOBaHHH 6e3 npe,nBapHTeJIbHOrO B tJ:HCJIeHHSlHa-
189
np5I:iKeHHH, He npH6era5I K MeTO.lI.Y nOCJIOHHOrO CYMMHpOBa-
HH5I. B pe3YJIbTaTe .lI.OCTHraeTC5I pe3Koe COKpall.{eHHe TPY.lI.O-
eMKOCTH paCqeTa. 5Y.lI.eM HCXO.lI.HTb H3 Toro, tJ:TO OCa.lI.KH TO-
qeK, pacnOJIO>KeHHbIX no.lI. HapY>KHOH H BHYTpeHHeH OKPY>K-
HOCTHMH no,nOWBbI KOJIb eBOro <PYH.lI.aMeHTa, H3BeCTHbI.
Tor.lI.a OcaJI.KH npOH3BOJIbHbIX TOlleK M, pacnOJIO>KeHHbIX KaK
B npe.lI.eJIax, TaK H 3a npe.lI.eJIaMH nJIOlll.a.lI.H nO.lI.OIlIBbI MO>K-
HO onpe,LleJI5ITb, HCXO.lI.H H3 CJIe.lI.YIOll.{HX C006pa>KeHHH.
CAyttau 1. TOQKa M pacnOJIO>KeHa B npe.lI.eJIax no.lI.O-
IlIBbI <PYH.lI.aMeHTa Ha paccToHHHH r OT lI.eHTpa nO)l.OIlIBbI,
npH 9TOM R1 <r<R2 (pHC. 6.16, a). PaccMoTpHM )l.Ba KOJIb-
lI.eBbIX <PYH.lI.aMeHTa C HapY>KHbIM H BHYTpeHHHM paJI.Hyca-
MH, paBHbIMH COOTBeTCTBeHHO r, R 1 H R2.J r. TaK KaK TOIlKa
M pacnOJIO>KeHa Ha HapY>KHOH OKPY>KHOCTH nepBoro H3
u
HHX H Ha BHYTpeHHeH OKPY}KHOCTH BToporo, TO ee OCa.lI.Ka
SM paBHa CYMMe H3BeCTHbIX OCa)l.OK 51 H S2 YKa3aHHblx
JI.BYX YH.lI.aMeHTOB
SM=S1+ S 2.
(6.34)
CAyttau 2. TotJ:Ka M pacnOJIO}l{eHa 3a npe)l.eJIaMH no.lI.O-
llIBbI <PYH,naMeHTa, npH 9TOM r<R l (pHC. 6.16, 6). Pac-
CMOTpHM .lI.Ba KOJIb eBbIX <PYH.lI.aMeHTa C pa.lI.HycaMH, paB-
HbIMH COOTBeTCTBeHHO R r H R1, r. TaK KaK TOqKa M pac-
nOJIO}l{eHa Ha BHYTpeHHHX OKPY}I{HOCTHX YKa3aHH X .lI.BYX
YH.lI.aMeHTOB, TO ee OCa.lI.Ka SM paBHa pa3HOCTH H3BeCT-
HbIX OCa.lI.OK 52 H S 1 9THX YH.lI.aMeHTOB
SM=S2 - 51. (6.35)
CAyttau 3. TOqKa M pacnOJIO}KeHa 3a npe.lI.eJIaMH no)l.O-
WBbl <PYH.lI.aMeHTa, npH 9TOM r> R2. (pHC. 6.16, B). Pac-
C 10TpHM .lI.Ba KOJIb eBbIX <PYH.lI.aMeHTa C pa.lI.HycaMH, paB-
HbIMH COOTBeTCTBeHHO r, R1 H r, R2. TaK KaK TOIlKa M pac-
nOJIO>KeHa Ha HapY}l{HbIX OKPY>KHOCTHX YKa3aHHbIX .lI.BYX
YH.lI.aMeHTOB, TO ee OCa.lI.Ka SM paBHa pa3HOCTH H3BeCTHbIX
Oca)l.oK 51 H 52 9THX CPYH)l.aMeHTOB
SM=5- t - S2. (6.36)
OCa)l.KH 52 H S1 TOlleK KOJIb eBOro YHJI.aMeHTa, pacno-
JIO:iKeHHbIX Ha ero HapY}KHOH H BHYTpeHHeH OKpY>KHOCTHX
pa)l.HycaMH R2 H R1, onpe)l.eJI5IIOT no CJIe.lI.YIOll.{eH cpopMYJIe,
aHaJIOrHqHOH opMYJIe 6.4 PYKOBO.lI.CTBa [12]:
n'
k. - k. 1
S - R t i.-
-p 2 '
- l Ei
l=
(6.37 )
190
r.n.e p - Cpe)J.Hee .naBJIeHHe no.n. nO.n.OlllBOH <PYH.n.aMeHTa; n' - KOJIHtJe-
CTBO CJ10eB rpYHTa, pa3JIHtJalOw.nXCH no C:>KHMaeMOCTH; ki - Koa<P<PHI1.I1-
eHTbI, npHHHMaeMbIe no Ta6JI. 6.8 B 3aBHCHMOCTH OT m=HIR2 H n=
=R 1 /R2. 3HaqeHHH K034><PHl.{HeHTOB k i B Ta6JUIUe BbIqHCJIeHbI no <POPMY-
JIaM, npHBe.n.eHHbIM B pa60Tax [5, 6].
,LlJIH ynpyroro nOJIynpocTpaHCTBa K03<P<PHlI.HeHT npH-
HHT paBHbIM e,lI.HHHue B COOTBeTCTBHH C peKOMeH,lI.aUHHMH,
npIlBe,lI.eHHbIMH B pa60Te [3].
Ta6J1Hu.a 6.8
k
cot n=R.IR 2
I
:i: a=zlR 2
n 0.2 0,4 0,5 0,6 0,7 0,8
£:
0,5
0,25 0,154 1 0, 138 0,135 I 0,131 0, 128 0,114
0,122 0, 122 0,122 0, 121 0. 118 0,11
0,5 0,349 0,295 0,282 0,268 0,248 0,209
0,241 0,241 0,239 0,235 0,222 0, 196
0,25 0, 155 0, 139 0,134 0,129 0,123 0,112
0, 122 0,121 0,12 0,119 0,115 0, 108
0,75
0,5
0,358
0,238
0,299
0,234
0,278 0,258 0,233 0,197
0, 231 0, 223 0,21 0, 184
0,75 0,382 0,466 0,422 0,378 0,328 0,263
0,351 0,342 0,333 0,317 0,289 0,243
0,25 0, 153 0, 138 0, 134 0,129 0, 123 0, 11
0, 121 0,121 0,12 0,118 0,115 0,10
0,5 0,348 0,293 0,274 0,254 0,23 [ 0, 19-
0,236 - 0,232 0,227 0,225 0,206 0, 18
1
0,75 0,559 0,452 0,41 0,366 0,317 0,25
0,346 0,335 0,323 0,311 0,278 0,23
1 0,772 0,611 0,541 0,47 0.3951 0,306
0,453 0,432 0,413 0,389 0,342 0,28
2
7
b
2
5
5
1 ,5
0,25 0, 150 0, 136 I 0, 132 0, 128 0, 122 0, 112
0,121 0,12 0,119 0,118 0, 115 0, 107
0,5 0,335 0,284 0,267 1 0,249 0,227 0,193
0,233 0,229 0,225 0,218 0,205 0, 181
0,75 0,526 I 0,430 I 0,392 I 0, 354 1 0,309 0,25
° 337 0,326 0,316 0,299 ° 273 0 231
I .
191
n pOaOA3ICeftUe Ta6A. 6.8
k
n-RJ/R2
a=-zlR 2
I 0,2 0,4 0,5 0,6 0.7 0.8
E ,
1 0,704 0,566 0,505 l 0,444 0,377 0,295
0,434 0,414 0,395 0,368 0,329 0,271
1 ,5 0,945 0,751 0,713 0,607 0,496 0,373
0,619 0,576 o 53 0,492 0,407 0,339
,
0,25 I 0, 149 0, 1 35 0,131 0,127 0, 1 22 0,112
0,12 0,119 0,119 0,118 0,114 0, 107
0,5 1 0,331 0,281 0,264 0,247 0,225 ° , 192
0,231 0,227 0,223 0,216 0,204 0,18
0,75 0,514 0,491 , 0,385 0,348 0,305 0,247
0,331 0,325 0,311 0, 295 0,271 0, 229
2
1,0 0,68 0,547 0,491 0,433 0, 3691 0,29
0,423 0,408 0,387 0,361 0,3,24 0,267
1,5 0,956 0,761 0,669 I 0,575 0,414 0,359
0,586 0,552 0,515 0,471 0,411 0,328
2 1 , 185 0,944 0,822 0,696 0,565 0,417
0,735 0,68 0,629 0,567 0,486 0,38
0,25 0, 149 0,135 0,131 I 0, 127/ 0, 122 0, 112
0,12 0,119. 0,119 0, 1 17 0, 114 0, 107
0,5 0,329 0,279 0,263 0,246/ 0,225 0,192
0,229 0,226 0,222 o ,215 0, 203 0 J 18
0,75 0,509 0,416 0,381 0,345 0,303 0,246
0, 321 0,318 0,308 0,293 0,269 0,228
1 I 0,669 0,537 0,483 0,427 0,364 0,287
0,414 0,397 0,38 0,356 0,32 0,264
3
1,5 0,919 0,731 0,644 / 0,555/ 0,459 0,349
0,562 0,527 0,497 0,456 0, 399 ° , 32
2 1 ,097 I 0,874 I 0,764 0,651\ 0,531 0,397
0,683 0,631 0,59 0,534 0,461 0,363
3 1 ,365 I I ,094 J 0,953 I 0, 805 1 0, 646 1 0,473
0,884 0,804 0,744 0, 664 0, 563 0,434
192
n pOaOJl eHUe Ta6/l. 6.8
Ie
..
n-RI/RJ
ell
a=zlR a
:t::
II
s::
....
0,2 0,4 0,5 0,6 0,7 0,8
5
0,25 0, 149 0, 135 0,131 0,127 0,121 0,111
0,12 0,119 0,119 0,117 0, 114 0,101
0,5 0,329 0,279 0,263 0.246\ 0,224 0,192
0,229 0,225 0,222 0,215 0, 203 0,18
0,75 0,508 0,416 0,381 0.344/ 0,302 0,245
0,326 0,317 0,307 0,292 0,268 0,228
1 0,667 0,536 0,481 0,425 0,363 0,286
0,412 0,394 0,378 0,354 0,318 0,264
I ,5 0,91 0, 724 0,637 0,549 0,455 0,346
0,554 0,52 0,491 0,451 0,395 0,317
2 1 ,076 0,856 0,749 0,638 0,521 0,389
0,664 0,616 0,575 0,522 0,451 0,357
3 1 , 287 0,029 0,897 0,758 0,612 0,45
0,822 0,752 0,696 0,625 0,531 0,413
4 1,414 1 , 1 37 0,992 / 0.836 0.672\ 0,491
0,931 0,846 0,78 0,696 0,586 0,452
5 1,513 1,221 1 ,066 0,898 0,72 0,524
1,019 0,923 0,848 0,754 0,632 0,484
0,25
0, 149
0,12
0,135
0,119
0,131 0,127 0,122 0,111
0, 119 0, 117 ° ,114 0, 107
00
0,5 0,329 0,28 0,263 0,246 0,225 0,191
0,229 0,225 0,222 0,215 0,203 0,18
0,75 0,509 0,417 0,381 0,345 0,303 0,24
0,326 0,316 0,308 0,292 0,268 0,228
5
1
0,667
0,412
0,481 0,425 0,364 0,285
0,379 0,354 0,318 0,264
0,537
0,394
1,5 0,91 0,725 0,637 0,549 0,456 0,34
0,554 0,519 0,491 0,45 0,394 0,31
5
8
193
n pOaO/l eHUe ra6/l. 6.8
k
cot
-
:x::
II
£:
n=RI/R 2
a=zlR 2
0,2 1 0,7
I 0,4 0,5 0.6 0,8
00
2 1 ,074 0,856 0,747 0,637 0,521 0,388
0,662 0,613 0,574 0,52 0,449 0,357
3 1 , 2-76 I 1 ,022 I 0,889 0,752 0,608 0,446
0,812 0,743 0,689 0,617 0,525 0,41
4 1 , 38.7 1 , f16 0,97 0,82 0,66 0,482
0,905 0,824 0,761 0,678 0,573 0,443
5 1 ,456 1,175 1_,022 0,864 0,693 0,505
0,967 0,877 0,808 0,718 0,605 0,465
n p H M e q a H H H: 1. BepxHHe 3HaqeHHH K03<p HuHeHToB k COOT-
B TCTBYIOT BHYTpeHHeH rp8HH KOJIbua, HH:>KHHe - Ha PY:>KHOH rp8HH
KOJIbI.{8. 2. K03cl><PHIJ.HeHTbI k npH 3HaqeHHHX m OT 0,5 .n.o 5 COOTBeT-
CTBYIOT paCqeTHOH cxeMe OCHOBaHHH B BH e nHHeHHo .n.e<popMHpyeMoro
CJIOH KOHeqHOH TOJIIIJ.HHbI, npH 3HaqeHHH m= 00 B BH.n.e JIHHeHHO .n.ecl>op-
MHpyeMoro nOJIynpocTpaHcTB8. 3. B Ta6JIHIJ.Y He BKJIlOqeHbI K03<p<}JH-
UHeHTbI k = 0, JIH Bcex 3HaqeHHH m H n npH a = O.
n peJI.JIaraeMbIH cnoco6 OnpeJI.eJIeHHH OCaJI.OK KOJIb eBbIX
<PYHJI.aMeHTOB XOpOlliO COrJIacyeTcH C pe3YJIbTaTaMH COOT-
BeTCTBYlO HX 3KCnepHMeHTaJIbHbIX HCCJIeJI.OBaHHH (CM. - pa-
60TbI [3, 7]). flpHBeJI.eHHbIe peKOMeH,lI.aIl.HH MorYT 6bITb TaK-
}Ke HCnOJIb30B3HbI ,lI.JIH HaXO}K,lI.eHHH HanpH}KeHHH B JII060H
TOqKe OCHOBaHHH, npH 3TOM K03cp<pHIl.HeHTbI al H a2 ,lI.JIH
BbIQHCJIeHHH HanpH)KeHHH COOTBeTCTBeHHO Ha BHYTpeHHeii
Ii HaPY}KHOH OKPY}KHOCTHX KOJIbIl.a npHHHMalOT no Ta6JI. 2
pa60TbI [6].
,LLJIH npHMepa conOCTaBHM onpeJI.eJIeHHe OCaJI.KH KOJIbIl.e-
Boro CPYH,lI.aMeHTa H3 ynpyroM nOJIynpocTpaHcTBe C npH-
MepOM, npHBe,lI.eHHbIM B pa60Te [5] (pHC. 6.17).
npHMep 9. .l1.JIH CJIOH O.n.HOpO.n.HbIX rpYHToB H = 10 1\1 onpe.n.eJIHM
oca.n.KY TOqeK, pacnOJIO:>KeHHbIX H8 OKPY:>KHOCTH pa.n.HYCOM r=6 M. OK-
PY:>KHOCTb pa.n.HYCOM r = 6 M .n.eJIHT 3a.n.aHHbIH <PYH.n.8MeHT Ha BHYT-
peHHee KOJIbUO pa.n.HYC8MH Rl=4 M H R2=6 M H HapY:>KHOe KOJIbI.{O
pa.n.HycaMH R 1 =6 M H R2=8 M.
,lI.JIH BHYTpeHHero KOJIbIJ.a n2=4/6=0,667; a= 10/6= 1,667. no
Ta6JI. 6.8 npH n2=0,667 H a= 1,667 Haxo.n.HM k=O,439. ,lI.JIH HapY:>KHOrO
KOJIbIJ.a nl=6/8=O,75; a=10j8=1,25. no Ta6JI. 6.8 npH nl=O,75 M
a = 1,25 HaXO)J.HM k=O,362. I1cKoMaH oca.n.Ka
194
R f "=4
r=o
/?2"=8
PHC. 6.t7
S =5 1 +S2=p/E (B. 0,362+ 6.0,439) :- 5,53p / E.
,[{JIH conOCTaBJIeHHfI onpe.ll.eJIHM OCa.ll.KY no ,lI.aHHbIM npHMepa pa-
:i
60TbI [5]: S= pdld E , r.ll.e 3HaqeHHfI pi=pa npHHflTbI no Ta6.n. 2 pa6o-
i=l
TbI [5], a hi=2 M
S =pfE (1.0,5+0,BOB+0,55B+0,423+0,330+0,273X
X 0,.5) · 2=5,511PIE 5,53PfE.
OpeHMYIlI.eCTBO IIpe JIaraeMOro cnoco6a paCIIeTa CTa-
HeT OIIeBH HbIM, eCJIH YIIeCTb, IITO npH BbIQHCJIeHHH ocaJ{KH
no .n.aHHbIM npHMepa pa60TbI [5] HCnOJIb30BaHbI rOTOBble pe-
3YJIbTaT rpOM03 KHX BWIIHCJIeHHH, npHBe.n.eHHWX B 3TOM
npHMepe.
6.9. OnpeAeJleHHe ocaAOK B pa3J1HIJHbIX TOIJKaX
. v
OCHOBaHHSI OT paBHoMepHoH HarpY3KH,
v
pacnpeAeJleHHOH no nJIOl..l.{aAH npHMoyrOJIbHHKa
B npaKTHKe npoeKTHpoBaHHH 3.n.aHHH H cooPYLKeHHH
B03HHKaeT Heo6xo.n.HMOCTb B onpe.n.eJIeHHM oca.n.OK TOIIeK oc-
HOBaHHH Ha JIIDGOH r JIy6HHe He TOJIbKO B npe.n.eJIax 3arpy-
}KeHHOrO YIIaCTKa, HO H Ha pa3JIHIIHbIX y.n.aJleHHHX 3a ero
npe.n.eJlaMH. B lIaCTHOCTH, TaKaH Heo6xo.n.HMOCTb B03HHKaeT
npM paCIIeTe QCa.n.OK YH.lI.aMeHTOB C YII TOM BJIHHHHH CO-
ce.n.HHX YH.n.aMeHTOB. B 3TOM CJIyqae He06xoJI.HMO npe>KJI.e
Bcero YCTaHOBHTb, npH KaKHX paCCTOHHHHX Me)K.n.y lI.eHT-
paMH no.n.OIllB YH.n.aMeHTOB H HX pa3Mepax CJIeJlyeT YIIH-
TbIBaTb B3aHMHoe BJIHHHHe OCa)l.OK.
no Mepe YJI.aJIeHHH OT 3arpY>KeHHOrO YIIaCTKa HanpH)Ke-
HHH nOCTeneHHO yGbIBalOT H MO>KeT OKa3aTbCH, IITO 3Ha-
193
qJ., lll.J
CP, 2 I ,
ILL
aJb( aJl)
PUC. 6.18
tIHTeJIbHble HanpSl}KeHHH Y 6JIH>KaHlllerO KpaSi coce.nHero
q)YH.naMeHTa 3aMeTHO CHH3HTCH B HanpaBJIeHHH npOTHBono-
JIO>KHOrO KpaH H )J.a>Ke Ha HeKOTopOM Y)J.aJIeHHH MorYT .no-
CTHrHYTb HYJIH. TaKoe HepaBHOMepHoe JI.aBJIeHHe no)J. no-
.llOIliBOM coce)J.Hero Q:>YHJI.aMeHTa MO)KeT TaKLKe cymecTBeH-
HO nOBJIHHTb Ha BeJIHqHHY ero KpeHa, qTO )J.OJI)KHO 6bITb
yqTeHO paCqeTOM.
B Cf-IHfI 2.02.01-83 H B floc06HH [12] CO.ll.ep}l{aTCH HC-
tIep nbIBalOIlI.He YKa3aHHH no on peJI.eJIeHHIO MeTOJI.OM yr JIO-
B X TOqeK pacnpeJI.eJIeHHH no rJIy6HHe HOpMaJIbHWX JI.aB-
JIeHHM B JI1060M TOqKe C, pacnOJIO)KeHHOH B npe)J.eJIa x HJIli
3a npeJI.eJIaMH paccMaTpHBaeMoro <PYHJI.aMeHTa. O)J.HaKO 3TH
YKa3aHHH npHBOJI.5IT K rpOM03JI.KOMY paCqeTY, KOTOpbIH 1\10-
lKeT 6bITb 3HaqHTeJIbHO ynpOIl.{eH HCnOJIb30BaHHeM JIHlllb
OJI.HOrO- )J.BYX Ta6JIHqHbIX K03<P<PHlI.HeHTOB JI.JISI Henocpe.nCT-
BeHHoro BbIqHCJIeHHH oca)J.OK. PaCCMOTpHM Bce B03MO>KHbIe
JIyqaH pacnOJIO)KeHHH TOqKH C OTHOCHTeJIbHO paCCMaTpH-
BaeMoro Q:>YHJI.a"MeHTa.
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<1> 1 C lllHpHHOM nO)J.OlIIBbI b )J.JIHHOM 1 H .rionOJIHHTeJIbHbIM
n.aBJIeHHeM Ha ypOBHe nOJI.OlllBbI po (pHC. - 6.18, 0603Halle-
HHH 6e3 cK060K). Tpe6yeTcH onpeJI.eJIHTb oca)J.KY CJIOH OCHO-
BaHHH rJIy6HHoH Z B TOq-Ke C, pacnOJIO)KeHHOM 3a npe.neJIa-
MH <pYH)J.aMeHTa Ha ero OCH, napaJIJIeJIbHOM CTopOHe b, "
Ha YAaJIeHHH L OT ero lI.eHTpa. 3TOT CJIyqaH - OCHOBHOH J
H3 KOToporo nOJIyqeHbI pellIeHHH JI.JIH Bcex JI.pyrHx CJIYllaeB
pacnOJIO)KeHHH TOqKH C.
BBeJI.eM TPH yqaCTKa 3arpY}I{eHHH C AOnOJIHHTeJIbHbIM
,lJ.aBJIeHHeM Po. 113 HHX CP 1', OJI.HHaKOBbIH no pa3MepaM C
<1> 1 H CHMMeTpHIIHO pacnOJIO}KeHHbIH C HHM OTHOCHTeJIhHO
TOqKH C, CP2 IlIHpHHOH b i = 2L + b, JI.JIHHOM 1 H <1>3 IlIHpHHOH
b 2 =2L - b H AJIUHOH I. OnpeJI.eJIHM oca.nKY Sc B TOIlKe c
OT BJIHHHHH <pYH.naMeHTa CP 1 KaK nOJIypa3HocTb oca)J.OK S2
H Sg B 3TOM TOqKe OT HarpY3KH yqaCTKOB <1>2 " <1>3
196
Sc= (S2 - S3)/2.
0603Ha1lHM: ii=L/b "JIM L=iib, nt=l/b 1 ,
m=2zlb, mt=2z/b 1 H m2=2z/b 2 . YIIHTbIBaH,
=2L+b= (2ii+ l)b II b 2 =2L - b= (2ii - l)b,
( 6.38)
n2=l/b 2 ,
IITO b 1 =
nOJIYIIHM:
I I
n - -- -
1 - b l - 2L + b
1 I
n 2 =-= -
b 2 2L - b 2
I
(2n + I) b
I
( 2n - I) b
n .
= - ,
2n+ 1
n .
= - ,
2n-1
2z 2z 2z _ m .
m 1 = b l = 2L + b - ( 2n + 1) b - 2 n + l'
m
m 2 = - = - - - J
b 2 2L - b (2n - 1) b 2n - 1
COrJIaCHO _CHHn 2.02.01-83, oca,nKa OCHOBaHHH nO)1.0-
IllBbI cpYH)1.aMeHTa MeTO)1.0M nOCJIOHHOrO CYMMHpoBaHHH on-
pen.eJIHeTCH no cpopMYJIe
n' n'
S = 1: P1h 1 = 1: alPJ!1 . (6.39)
i=1 El 1=1 Ei
f.ae n' - tIHCJIO CJIOeB, H8 KOTopoe pa3.u.eJIeH8 no rJIy6HHe C}K11MaeMaSJ
TOJIw.a OCHOB8HHH.
B cpopMYJIe (6.39) npOH3Be)1.eHHe aih i paBHO lIaCTH
nJIOI.l.{a.n.H 3nlOpbI a B npe.n.eJIax CJIOH TOJII.l.{HHOH hi. 060-
3HallHM lIaCTb nJIOI.l.{a,nH 3nlOpbI a OT no,nOllIBbI cpYH.n.aMeHTa
,ao HH}KHeH rpaHHlI. bI paCCMaTpHBaeMoro CJIOH Qepe3
z
iio= S adz.
o
( 6.40)
ECJIH no.n.CTaBHTb B cpOPMYJIY (6.40) 3HaQeHHe a )1.JIH
npHMoyrOJIbHOrO YH.lI.aMeHTa H npOH3BeCTH 3aMeHY nepe-
MeHHoH Z Ha mi, TO nOJIYIIHM
ko = J adz=-;- r [ arci g _ n +
o 0 m / m 2 + n 2 + I
+ mn (2m 2 + n 2 + I) ]
dmt =
(I +m 2 )( n 2 +m 2 )V m 2 +n 2 + 1
= ( In V I + n 2 + n + n In V I + n 2 + I _
1t V 1 ,- n 2 - 11 J t + n 2 - I
197
v I + 11 2 + mi 2 + 11 I v I + 11 2 + ml 2 + I
- In - n n
V I + 11 2 + mt 2 - 11 V I + 11 2 + mi 2 - 1
+mi arctg 11 ) .
ml V I + n 2 + mi 2 I
CJle.QOBa TeJlbHO,
+
(6.41 )
- - -
k 02 - kot
S = Po ( kot +
£1 E 2
- -
k On ' -- ko (11' - I) )
+...+
En'
UJIM
n'
S - b kOi-ko(i-l)
- Po '
. l Ei
1=
(6.42)
r.n.e
k Oi = kOi/b. (6.43)
B03BpaIll.aHCb K pHC. 6.18 M cpopMYJle (6.38) t YCTaHaB-
JIHBaeM, lITO OCa,lI.KH 52 H 53 YIIacTKOB 3arPYLKeHHH <1>2 II
<1>3 10rYT 6bITb BbIIIHC.neHbI no cpopMYJle (6.42), TaK KaK
TOIIKa c paCnOJIOLKeHa no.Q HX u.eHTpaMH. TaKHM o6pa30M
Oca,lJ.Ka 5c TOIIKH C OT HarpY3KII CPYH,lI.aMeHTOM <I> 1 onpe.ne-
JIHeTCH no <jJopMYJle
n' k" - k
S === b' l (i-I)
c Po E"
i=l l
k - [b (1 V I -t- 11 1 2 + 111 + I v I + 111 2 + 1
i - - 1 n n 1 n
21t V I +1112 -111 VI +1112 _1
" 1 VI +1112+ m 1 2 + 11) 1 VI + 1112+ m)2 + I
- n - n 1 n +
VI'+ 1112Lf- m1 2 - 111 VI + 11 1 2 + m 1 2 - I
+m 1 arctg 11 1 ) -b 2 ( ln Vl+n22+1 +
ml VI+ nI2+mt 2 VI+ 1122_1
1 VI +1122+ I 1 VI+1122+m22+n2
,- n 2 n - n -
VI +112 2 -- I VI + 1122+ m2 2 - 112
-n2 ln Vl+n22+m22+1 +m 2 arctg n2 ' )] . (6.44)
VI + 1122+ m22-1 m2Vt+ 112+ m 2
198
no cpopMYJIe (6.44) ,nJIH pa3JIHqHbIX 3HaqeHUH nt, mt,
n2 H m2, 3aBUCHll.{UX OT n, m H ii, BbIQHCJIeHbI K03cp<p II l.{UeH-
TbI k, CBe eHHble B Ta6JI. 4 npliJI. 3.
TaKUM o6pa30M oca Ka TOqKH C
B npe eJIax CJIOH OCHOBaHHH C no-
CTOHHHbIM E (O,lI.HHaKOBoH CLKUMae-
MOCTU) MO)KeT 6bITb orrpe,lI.eJIeHa no
cpopMYJIe (6.44) C nOMOll.{blO O,lI.HOrO
Ta6JIuqHOrO K03cpcpul.{HeHTa.
CAY"laU 2. TOQKa c pacrrOJIO>KeHa
3a npe eJIaMH cpYH aMeHTa Ha ero
OCM, rrapaJ1J1eJ1bHOH CTopOHe 1, H Ha
YJI.aJIeHHH L OT ero lI.eHTpa (puc. 6.18,
0603HaQeHHH B cK06Kax). 3TOT CJIY-
lIaH aHaJIOrHQeH CJIYQalO 1 C 3aMeHoH
n, m H ii COOTBeTCTBeHHO Ha n2=b/1= l/n, m2=2z/1=
=2z/nb=m/n H ii2=L/1=L/nb=ii/n, T. e. rrpH 3TOM CJIe-
JJ.yeT 3aMeHHTb MeCTaMII b Ii l.
CAyttau 3. TOQKa c pacrrOJIO)l{eHa 3 rrpe eJIaMH rrOAO-
lllBbI cpYH aMeHTa Ha y aJIeHHH L OT ero lI.eHTpa B HarrpaB-
JIeHHM CTOpOHbI b H Ha y aJIeHHH L 1 B HarrpaBJIeHHII CTOpO-
HbI I, rrpH 3TOM L>O,5b (puc. 6.19, 0603HaIIeHIIH 6e3 CKO-
60K). B 3TOH cHTyall.HH MO)l{eT 6bITb HCnOJIb30BaH CJIYQaH 1.
OpH 3TOM oca Ka TOQKII C paBHa rrOJIypa3HocTH ocaAOK OT
3arpY3KH YQaCTKOB <1>2 C pa3MepaMH b 1 =2L+b H 1 1 =1 II
<1>3 C pa3MepaMH b 2 =2L - b II 1 2 =1, TaK KaK TOIIKa C
pacrrOJIO)l{eHa Ha ocu 3THX YQaCTKOB
8c= (S2 - 8 3 )/2. (6.45)
CAY"laU 4. TOQKa C pacnOJIO)l{eHa 3a rrpe eJIaMH no o-
lllBbI Ha y.n.aJIeHHH L OT ero l.{eHTpa B HanpaBJIeHHII CTOpO-
RbI I H Ha Y.ll.aJIeHIIH L 1 B HarrpaBJIeHIIII CTOpOHbI b, npll
3TOM L>O,5b (pHC. 6.19, 0603HaQeHIIH B cK06Kax). no aHa-
199
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WBbI CPYH aMeHTa Ha YJI.aJleHHH L OT ero lI.eHTpa B HarrpaB-
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HbI I, rrpH 3TOM L<0,5b (pHC. 6.20, 0603HaqeHHH 6e3 CKO"
60K). 3TOT CJIyqaH aHaJlOrHqeH CJIyqalO 3 C 3aMeHoH b 2 =
=2L - b Ha b2.=b - 2L.
CAytlaU 6. TOqKa c paCnOJIOLKeHa 3a rrpe,neJlaMH no,lI.O-
WBbI cpYHJI.aMeHTa Ha y.u.aJIeHHH L OT ero lLeHTpa B HanpaB"
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MH b i H 1 1 , a TaK>Ke b 2 II 1 2 .
CAytlaU 7. TOqKa c pacrrOJIO>KeHa B npe eJIax rrO OllIBbI
cpYHJI.aMeHTa Ha y.u.aJleHHH L B HanpaBJleHHH CTOpOHW b H
Ha y.u.aJIeHHH L 1 B HarrpaBJleHHH CTOpOHbI 1 (pIlC. 6.21, 060"
3HaqeHHH 6e3 cKo60K). 3TOT cJlyqaH aHaJlOrHqeH cJlyqalO 5.
CAyttau 8. TOqKa c paCnOJIOLKeHa B rrpeJI.eJlax noJI.O-
WBbI cpYH aMeHTa Ha y aJleHHH L B HarrpaBJleHHH CTOpOHbl
I H Ha y aJIeHHH L 1 B HarrpaBJIeHHH CTOpOHbI b (pHC. 6.21,
0603HaqeHHH B cK06Kax). 3TOT cJIyqaH: aHaJlOrllqeH CJIyqalO
6. ,QJIH y o6cTBa rrOJIb30BaHHH HCXO HbIMH aHHbIMH CBO-
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flpHBeJI.eHHble peKOMeH,llall.HH OTJIHqalOTCH OT COOTBeTCT"
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1) TpYJI.OeMKOe rrOCJIOHHOe CYMMHpOBaHHe ,nJIH CJIOeB
He60JIbWOH TOJIIll.HHbI 3aMeHHeTCH CYMMHpOBaHHeM JIH CJlO-
eB C O,llHHaKOBbIM 3HaqeHHeM E He3aBHCHMO OT TOJlI.l.lHHbl
CJIOH;
2) TpY,llOeMKOe orrpe,lleJIeHHe OCaJI.OK MeTo,nOM yrJIOB@X
TOqeK 3aMeHHeTCH BbIqHCJIeHHeM oca OK C nOMOI.l.lblO He 60-
JIee .u.BYX Ta6JIHqHbIX K03cpcpHlI.HeHToB. ,LLJlH HaH60JIee qac-
TO BCTpeqalOIll.HXCH CJIyqaeB rrpH onpeJI.eJleHHH OCaJI.OK TOqeK,
pacnOJIO)KeHHbIX Ha OCH CPYH aMeHTa, HcnoJlb3yeTcH TOJIbKO
O.ll.HH Ta6JIHqHbIH K03cpcpHlI.HeHT (CM. CJIyqaH 1 H 2);
3) Ta6JI. 4 npHJI. 3 rr03BOJIHeT BeCbMa npOCTO B 3aBH-
CHMOCTH OT 3HaqeHIiH ii, n H m orrpe eJIHTb paCCTOHHHH L
Ii L 1 OT lI.eHTpa 3arpY.>KeHHOrO npHMoyrOJIbHHKa .lI.0 JII060H
TOqKH OCHOBaHHH, rrpH KOTOpbIX YKa3aHHaH HarpY3Ka BJlHHeT
Ha OCa,llKY 3TOH TOQKH. B pe3YJIbTaTe pewaeTCH Bonpoc 0
BJIHHHHH 3arpY)l{eHHOrO <PYH,llaMeHTa Ha COCe,lI.HHe. npH 3TOM
201
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PUC. 6.22
MorYT 6bITb BbIqHCJIeHbI HepaBHOMepHbIe Oca,ll.KH nO,ll.OllIBbI
CPYH aMeHTa H eOOTBeTCTBYIOIII.HH HM KpeH.
nplIBe eM npHMep onpe eJIeHHH OCa,ll.KH CPYH.lI.aMeHTa
e yqeTOM II 6e3 yqeTa BJIHHHHH COCe.ll.HerO <PYH aMeHTa Me-
TO.ll.OM nOCJIOHHOrO CYMMHpOBaHHH H npe JIaraeMbIM cnoeo-
60M JIH conOCTaBJIeHHH KOJIHQeCTBa B QHCJIeHHH.
npHMep 10. Tpe6yeTCH' onpe.n.eJIHTb oca.n.KY <PYH.n.aMeHTa <1>2 c yqe-
TOM H 6e3 yqeTa BJIHHHHH <PYH.n.aMeHTa <I> 1 (pHC. 6.22).
J],aflO: 3arJIy6JIeHHe <PYH.n.aMeHTOB OT nOBepXHOCTH rpYHTa h' =2 M,
VII=17,8 KHfM 3 , E=10 Mna, P-Po=0,276 Mna.
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CYMMY .n.aBJIeHHH B yrJI9BOH TOqKe M qeTbIpex 3arpYLKeHHbIX npHMO-
yrOJIbHHKOB (<pHKTHBHbIX <PYH.n.aMeHTOB) MLA! H MNDL, B3HTbIX co
3HaKOM «+», H MKB! H MNCK, B3HTbIX co 3HaKOM «-».
COOTHorneHHH CTOpOH YKa3aHH X npHMoyrOJIbHHKOB paBHW:
,lJ.JIH EFGH (<1>2) n= 4/4= 1;
,lJ.JIH MLA! H MNDL n=7/2=3,5;
.n.JIH MKBJ H MNCK n=3j2= 1,5.
al - K03<p<p H U H eHT H3MeHeHHH .n.onOJIHHTeJIbHbIX .n.3BJIeHHH B rpYH-
Te no BepTHKaJIH, npOXO.ll.HllI.eH Qepe3 TOtJKY M OT HarpY3KH no npHMO-
yrOJIbHHKY EFG H (<1>2);
CX2 - TO >Ke, no npHMoyrOJIbHHKaM MLA! H MNDL;
a3 - TO >Ke, no npHMoyrOJIbHHKaM MKB! H MNCK;
a4 - TO >Ke, no npHMoyrOJIbHHKY ABCD a4= (2/4) (a2 - a3);
a - TO >Ke, no npHMoyrOJIbHHK3M ABCD 11 EFGH a=al +a4.
Onpe.n.eJIHM 3HatJeHI1H K03<p<PHu.HeHT3 a H3 rpaHHu.ax 3JIeMeHTapHbIx
CJ10eB no BepTHKaJIH, npoxo.n.HllI.eH qepe3 TOqKY M, nOJIb3YHCb T36JI. 5.5
nOC06HH [12]. Bce BblqHCJIeHHH CBe.n.eHbI B Ta6JI. 6.10.
Oca.n.Ka <PYH.n.aMeHTa <1>2 6e3 yqeTa BJIHHHHH <PYH.n.aMeHT3 <I> 1.
10
8 1 10 = M E!.. = 80.0,8 [2,76 + 2 (2,65 + 2,21 + 1,67 +
Ei 2.10
1.
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64
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200
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203
no npe JIaraeMOH TaGJI. 4 npHJI. 3 npH n=O, m=4, n= 1 H b/2=20()
H3XO.lI.HM k= 1,426 H SIIO=pbk/E=O,276.200.1,426/IO=7,871 CM. To
)Ke .n.JIH o.n.HHHa.u. aTH CJIOeB n=O, m=4,4, n= 1, k= 1,458 H S2 1l =O,276X
X200.I,458/IO=8,048 CM.
Oca,QKa 4>YH.n.aMeHTa <1>2 C yqeTOM BJIHHHHH <l>YH.n.aMeHTa <1> 1
S2 11 =[80. 0,8/(2.10)] [2,76+2(2,65+2,24+ 1,75+ 1,36+
I
+ 1,07 +0,87 +0,71 +0,6+0,51 +0,44) +0,38]10 =
= (64/200)27,54=8,813 CM.
no npe.n.JIaraeMOH Ta6JI. 4 npHJI. 3 npH n=L/b=500/400=1,25. m=4,4
H n= 1 HaXO,QHM k=O,1448.
S2 11 =0,276. 200. 0, 1448/10 =0,799 CM.
8111+S211=8,048+0,799=8,847 CM 8,813 CM.
113 npHBe.n.eHHoro npHMepa CJIe,lI.yeT, qTO npe.n.JIaraeMbIH
cnoco6 onpe,lI.eJIeHHH OCa,lI.OK n03BO HeT 3HaqHTeJIbHO co-
KpaTHTb KOJIHqeCTBO BbIqHCJIeHHH, eCJIH K TOMY LKe yqeCTb
qTO npH COCTaBJIeHHH Ta6JI. 6.10 K03cpcpHlI.HeHTbI a npHxo-
HTCH BbIQHCJIHTb HHTepnOJIHpOBaHHeM. OpH 3TOM rJIy6HHa
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,nOM nOCJIOHHOrO CY IMHpOBaHHH oca OK. B ,lI.eHCTBHTeJIbHO-
CTH npH perneHHH npe,lI.JIaraeMbIM cnoco60M YKa3aHHaH rJIY-
6HHa 3apaHee HeH3BeCTHa H nO,lI.JIe2KHT onpe eJIeHHID.
fJIy6HHa C}KHMaeMOH TOJIlltH OCHOBaHHH OrpaHHqHBaeTCH
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ECJIH HaH,lI.eHHaH H3 YCJIOBHH (6.46) HH}KHHH rpaHHI1.a!
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He3aBHcHMo OT cpOpMbI OqepTaHHH nO,lI.OllIBbI CPYH,naMeH-
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JIeHHH z' npaKTHqeCKH He npe,llCTaBJISIeTCH B03MOLKHbIM. B.
CBH3H C 3THM npe,llJIaraeTCSI BblqHCJIHTb Z' MeTO.ll.OM nOCJIe-
.lI.OBaTeJIbHWX rrpH6JIHLKeHHH C HCrrOJIb30BaHHeM YCJIOBHR
(6.50) HJIH (6.51) H Ta6JI. 1 npHJI. 2 CHHIl 2.02.01-83. Co-
rJIaCHO 3TOMY rrpe,nJIOLKeHHIO 3a aIOTCSI 3HaqeHHeM Zl, 3a-
TeM H3 YCJIOBHH (6.50) HJIH (6.51) onpe,neJISIIOT a HJIM
a' H no Ta6JI. 1 npHJI. 2 CHHIl 2.02.01-83 COOTBeTCTBYIO-
IlI.ee eMY 3HaqeHHe m ' , a CJIe,lI.OBaTeJIbHO, HZ', KOTopoe
cpaBHHBalOT C rrpe,lI.BapHTeJIbHO rrpHHSIThIM. B CJIyqae pac-
XOLK.lI.eHHH YTOqHHIOT 3Ha qeHHe z' H rrOBTopSiIOT rrpou.eec
BhIqHCJIeHHH ,lI.O Tex rrop, rrOKa nOCJIe Hee 3HaqeHHe Z' CO-
Bna aeT C npe bI,llYIII.HM.
B rrepBOM rrpH6JIHLKeHHH peKoMeH yeTcH npHHHMaTb 3Ha-
qeHHe Z' C HCnOJIb30BaHHeM Ta6JI. 6.11 H rpacpHKa, rrpHBe-
.rr.eHHoro Ha pHC. 6.24, 3aHMCTBOBaHHbIX H3 PYKOBO,lI.CTBa [14]..
IlopH,lI.oK onpe.rr.eJIeHHH OTHorneHHH z'/b YKa3aH Ha rpacpHKe
CTpeJIKaMH. B qaCTHOCTH, JISI (p-yh)/k=19,2 Kna, b=
=35 M H l / b= 1 4 HaxoTTHM. z' / b= 1 96. z'= 1 96. b=6 9
, ,/-J..."",
(3Ha qeHHe k onpe eJISleTCSI no Ta6JI. 6.11).
TaoJuu-la 6.11
3HaQeHH.H K034>4>H HeHTa k npH h. ..
b,
.. 1 2 3 4 5
I
2
3
4
5
I
I
I
1
1
1 ,3
I ,2
t , 15
1 , 1
1 , I
1,6
1 ,4
1 ,3
1 ,25
t ,2
1,9
1,65
1,45
1,4
1,35
2,2
t ,9
1 ,65
1,55
t ,45
3aMeTHM, qTO Bee BeJIHqHHbI, KpOMe z', BXO,lI.SlInHe B YC-
JIOBHH (6.50) H (6.51), nOCTOHHHbI ,lI.JISI KOHKpeTHO paCCMaT-
206
pHBaeMoro cnyqaH, qTO B COqeTaHHH C npOCTOTOH YKa3aH.
HbIX YCJIOBI1H 06eCnelIHBaeT 6bICTPYlO CXO,lI.HMOCTb npo ecca
nOCJIe,nOBaTeJIbHbIX npH6nH}I{eHHH. KaK npaBHJIO, ,lI.JISI Ha..
XO}l{.neHHH Z' ,lI.OCTaTOqHO OrpaHHqHTbCH ,lI.BYMH-TpeMSI npH"
6JIHJKeHHHMH.
npHMep 11. JI.aHo: b=2 M, l=2,8 M, h'==I,5 M, Po=P-Vh'=
=0,3 Mna, hI %:3 M, VI = 16 KH/M 3 , h2= 1,5 M, '\'2= 17 KH/M 3 , h3=2 M.
V3= 19 KH/M 3 , h4=2,5 M, V4=20 KH/M 3 . Tpe6yeTcH onpe.u.eJIHTb rJIyoH"
HY C>KHMaeMOH: TOJIllI.H z'.
no Ta6J1. 6.11 H rpacpHKY, npHBe.neHHoMY Ha pHC. 6.21, npH h':=
= 1,5 M Ii b=2 M onpe.n.eJIHIOT k= 1,1 (no HHTepnOJIHUHH) H npH n=
=2,82 = 1,4, (p - Vlh') /k=O,3/0,11 =2,727 HaXO,lIJIT z, /b =2,7; oTKy,na
z'=2.2,7=5,4 M. CJIe.lI.OBaTeJIbHO, HH}KHHH rpaHHII.a C>KHMaeMOM TOJI-
ll.{H pacnOJIO>KeHa B qeTBepToM CJIoe, TaK KaK h'l = hi - h' = 3 - 1,5 =
= 1,5 M H hi +h2+h3= 1,5+ 1,5+2=5<z' =5,4 M, npH 3TOM OHa 3arJIy6-
JIeHa B qeTBepTbIM CJIOM Ha 5,4 5=0,4 M: Ct.=3.16+ 1,5.17 +2.19=
= 111,5 KH/M2, C 2 = 1,5 - 3 - 1,5 - 2= -5 M. Do <popMYJIe (6.50):
a' = (0,2/300) [111,5+ (5,4 - 5) 20] =0,0797.
nOJIyqeHHOMY 3HaqeHHIO a' no Ta6JI. 1 npHJI. 2 CHHD' 2.02.01-83
cooTBeTcTByeT m'=5,58 (no HHTepno.nHUHH) H z'= (2.5,58)/2=5,58=
=5,4 M. npHHHMaeM BO BTOpOM npH6JIH>KeHHH z' = (5,4+5,58) /2==
=5,49 M, npH 3TOM a'= (0,2/0,3) [11,15+ (5,49-5)20]=0,0807. AHaJIo"
rHqHO Haxo.n.HM no HHTepnOJISlI.{HH z, = 5,53 H npHHHMaeM OKOHqaTeJIbHO
z, = (5,49+5,53) /2=5,51 M.
MOLKHO TaKLKe peKOMeH,lI.OBaTb ,lI.pyroH cnoco6 onpeJ.I.e..
JleHHSI rpy6HHbI C}I{HMalOIII.eii TOJIIII.H OCHOBa HSI. C yqeTOM,
lITO
poz' =:= a (P-P6)
ii-1
II P6z'=V1h1+V2h2+ ... +Vii(Z'+h'-. hi),
- l=1
ycnoBHe (6.46) npHHHMaeT BH,lI.
n-l
a (p - P6) =0,2[,\,l h l + V2 h 2 + . . . +Vii (z' +h' - .:E hi)],
1=1
oTKy.na
0,21- (z' + h' -
a. =-= 0,211 h l -t- 0,212 h 2 + . . . + n .
p - P6 P - P6 P - P6
HJ1H a=al +a2+ ... +an,
it-I
hi)
i=1
r.n.e
0,21,h i
cxi =
P-P6
I1PH i, paBHOM OT 1 .no ii - 1, H
(6.52 )
207
n-1
O,2-y-(z' + h' - hi)
n i==1
P-P6
,L{JIH ynpoIUeHHH paCqeTOB no 4>opMYJle (6.52) COCT8sne-
Ha Ta6JI. 5 npHJI. 4 K03cpcpHUHeHToB ai AJISI pa3JIHQHbIX 3Ha-
qeHHH P - P6, hi H 'Vi.
HH}KHIOIO rpaHHUY C>KHMaeMOM TOJIIUH onpe,lI.eJISlIOT B Ta-
KOM nOCJIe,lI.OBaTeJIbHOCTH. Do Ta6JI. 5 npHJI. 4 HaXO,lI.HT 3Ha-
qeHHSI ai Ha rpaHHuax 3a,lI.aHHbIX CJIOeB OCHOBaHHH, a TaK-
}l{e cooTBeTcTBYIOIUHe HM 3HaqeHHH a no Ta6JI. 1 npHJI. 2
CHHn 2.02.01-83, npH 3TOM rpaHHua C}KHMaeMOM TOJIIUH
pacnOJIO}KeHa B CJIoe n, ,lI.JIH KOToporo C03,lI.aeTCH YCJIOBHe
a-=
n
a- 1 < a <: 0:-,
n- n
rJle an-l Ii an paBHbI CYMMe 3HaqeHHH ai no Ta6JI. 5 npHJI. 4 JI.nJ1 Bcex
CJIOeB rpYHTa, BKJIlOqaH COOTBeTCTBeHHO HH}KHIOIO rpaHHUY .n.JISI CJIOH
fi - 1 Ii BepxHlO1O H HH}KHIOIO rpaHHLT.bI .n.JISI CJIOH fi, a a - cooTBeTcTBY-
IO HH K03£}J£}JHLT.HeHT no Ta6JI. 1 npHJI. 2 CHHn 2.02.01-83.
B CJIyqae onpe,lI.eJIeHHH r .ny6HHbI C}KHMaIOIUeH TOJIIUHH3
YC.nOBHH (6.47) HMeeM
O,5 ii_1 < a < O, 5a ii.
OpH HaJIHqHH 06BO,lI.HeHHbIx rpYHTOB Y,lI.eJIbHbIM Bec 1'B3B
npHHHMaIOT, cor JIaCHO n. 2.217 [12].
,LLJIH onpe,lI.eJIeHHH z' (CM. pHC. 6.21) -,lI.JIH CJIOH rpYHTa
ii, B KOTOpOM pacnOJIOmeHa HH}I{HHH rpaHHua C}KHMaeMOH
_ --I
TOJlIUH, BblqHC.nHIOT an a H an===a. - ai, r,lI.e an npHHH-
i= 1
MalOT ,lI.JIH BCeX CJIOeB OT i = 1 ,lI.O i = ii, npHBe,lI.eHHbIX no
y.neJIbHOMY Becy K CJIOIO ii. 3aTeM no H3BeCTHOMY an onpe-
.n.eJIHIOT no Ta6JI. 5 npHJI. 4 h'ii (CM. pHC. 6.21) H r JIy6HHY
C}I{HMaeMOH TOJIlll.H no <popMYJIe
(6.53 )
( 6)54 )
n-l
z'= h. + h'- - h'
l n ,
i==1
(6.55)
r.n.e h'n - 3arJIy6JIeHHe HH}KHeH rpaHHUbI C}KHMaeMOH TOJIllI.H 8 cnoe fi.
no ,lI.aHHbIM npe.llbI,lI.YIUero npHMepa onpe,lI.eJIHeM r JIy6HHY
C)KHMaeMOH To.nIUH. HH)KHHH rpaHHua C}KHMaeMOH TOJIIUH
pacnOJIO}KeHa B IJeTBepTOM CJIoe, TaK KaK no Ta6JI. 5 npHJI.
4 H Ta6JI. 1 npHJI. 3 CHHD 11-15-74 .n. H cooTBeTcTBYIOIUHX
3HaqeHHH p - Po, Vi, hi H n= 1,4, a3= ai=0,032+0,017 +
1=1
208
Yo .
OTMeTl<a n08epXHOCTM
npMpoAHOrO penlt a
..
b
OTMeTKa nOAowsla.
IPVHAaMeHTa
N
"-c:J
1<.
...
....c:. N
N
0, 2P 6z' POZ'
HM*H"" rpaHM",a CMCMMaeMO"
Tonw.1II C y"eTOM BJ1MAHMR
O,2'(f1z COCeAHerO c1JVHA8MeHTI
. PUC. 6.25
4
+O,025=O,074<0,098<a4 ai=O,074+0,033=O,107, r,lJ.e,
i=l
HanpHMep, ,lI.JIH nepBOrO CJIOH npH p - P6=0,3 MDa, VI =:
= 16 KH/M3 H h' +h'l =3 M, al =0,032. AHaJIOrHqHO nOJIY-
qeHbI 3HaqeHHH ai ,lJ.JIH OCTaJIbHbIX CJIOeB. Do Ta6JI. 5 npHJI.
4 npH 1'=20 KH/M3 3HaqeHHIO a3=0,074 COOTBeTcTByeT
h' +z' =5,5 M, a CJIe.u.YIOIlI.ee 3HaqeHHe ai=0,08, npHMep-
HO COBna,lI.aIOIlI.ee co 3HaqeHHeM a, no Ta6JI. 1 npHJI. 2 CHHn
2.02.01-83, KOTOPOMY COOTBeTcTByeT h' + z' = 6 M. TaKHM
06pa30M HH)KHHH rpaHHll.a C)KHMaeMOH TOJII.l.(H pacnoJIo}Ke-
Ha B qeTBepTOM CJIoe Ha rJIy6HHe h'4=6 - 5,5=0,5 M H
3
z'=h'n-h'+Lh i =0,5-1,5+{3+1,5+2)=5,5 M. DpH
i=l
paCqeTe no cnoc06y 1 z' = 5,51 5 5 M.
DpHBe.u.eM peKOMeH.n.all.HH no onpe,lJ.eJIeHHIO rJIy6HHbl C)KH-
MaeMOH TOJII.l.(H OCHOBaHHH C yqeTOM BJIHHHHH COce,lJ.HHX
cpYHJI.aMeHToB. ,LLJIH 3Toro O,lI.HHM H3 ,lI.BYX npHBe,lJ.eHHbIX
cnoc060B onpe.u.eJIHIOT r JIy6HHY C)KHMaeMOH TOJII.l.(H z' 6e3
yqeTa BJIHHHHH COCe,lI.HHX <PYH,lI.aMeHTOB H ,lI.JIH Hee no Ta6JI.
4 npHJI. 3 HaXOJI.HT 3Ha qeHHH k H k B (pHC. 6.25). BeJIHQHHbI,
yqHTbIBaIOI1.I-He BJIHHHHe coce.n.HHX CPYH,lI.aMeHTOB, ,naHbI C
HH;:leKCOM «B».
209
Ha pHC. 6.2 npaBe,neitbI 3niOpbi npHpO.llttoro .naBJteHHB,
OnOnHHTenbHOrO ,naBneHHH, 3 T3K*e OnOnHHTenbHOrO
.naBneHHH, YIIHTbIBalOmerO BJIHHHHe COCe,nHHX cl>YH,naMeHTOB.
YCTaHOBneHO, lITO Ha CpaBHHTenbHO He60nblllOM YIIaCTKe
OCHOBaHHH BbICOTOH Z BenHIIHHbI k H k B H3MeMHIOTCH He-
3HaIIHTenbHO H. npH 3TOM npHMepHO no npHMonHHeHHOMY
3aKOHY. aTO n03BOnHJIO C ,nOCTaTOIIHOH JIH noaKTHQeCKHX
lleneH TOIIHOCTblO npHHHTb ycnoBHe
z' + L1z z'
z::z:: -
,
bPok + b bPo,}t,b bpJl
oTKy.a.a
b P k
z -= z' b 06 b ·
b pJ '
( 6.56 )
z,/ = z' + l1z.
( 6.57 )
DpH b=bb H Po=POb
6.z = z'k b / k. (6.58)
npHMep 12. ,aJlH yeJlOBH" npHMepa 10 onpeJI.eJlHtb rJly6HHY e}KHMaeMOH
- TOJIllI.H Z, b. I1MeeM: k= 1,426, z, =800 eM, m=4, n= 1 H n= 1,25. no
Ta6JI.4 npHJI. 3 npH m=4. n= 1 fI n= ..25 HaX0,!!...I!M k":=O..! 9. no
4>opMYJIaM (6.57) H (6.58):
l\z = 8)0.0,129/1,426 = 72,4 CM; Zb' -', 8jO + 72,4 =
=872,4cM 830 CM.
880-872,4
PaCXO>K,lJ.eHHe COCTaBJIHeT 880 100==0,87%.
6.10. K onpe,l(eJleHHIO ocaAOK oCHoBaHHH
OT n0J10COBOH HarpY3KH 1
flpH npoeKTHpOBaHHH pa3JIHIIHbIX cOOPY}I{eHHH npHXO-
AHTCH onpe,neJIHTb oca,nKH OCHOBaHHH OT BepTHKaJIbHOH no-
JIOCOBOII HarpY3KH (paBHOMepHOH, TpeyronbHoii, TpanellHe-
BH,nHOH H npOH3BOJIbHOif). YKaaaHHH no onpeJ1.eneHHIO Ha-
npH*eHHH H oca,nOK OCHOBaHHH MeTOJ1.0M nOC OHHoro CYM-
MHposaHHH OT YKa3aHHWX cxeM 3arpy*eHHH npHBe,neHbI B
CHHn 2.02.01-83, CnpaBOIIHHKe [23], pa60Tax [24], [12] M p.
.OJ1.HaKO Bce OHM MMelOT CYI.I..{eCTBeHHbIH He,nOCTaTOK - npH-
XOJ1.HTCH BbI1JHCJIHTb 3Ha lIeHHH K03<P<PHllHeHTOB a ,nJIH CJIO-
1 OCHOB3HHe paCCMaTpHBaeTCH K3K JlHHeHHO Ae$opMHpyeMoe ynpy-
roe nOJIynpocTpaHCTBO.
210
eB OCHOBaHHH cpaBHHTeJIbHO He6oJIbUI0H TOJI HHbI, npH
3TOM .ll.ITH Ka}l{1J.OrO TaKoro CJIOH H806xo.l1HMa ,nBoAHaH HH-
TeprrOJIHUHH. B pe3YJIbTaTe paCqeT CTaHOBHTCS1 rpOM03,D.-
KHM H Tpy.n.oeMKHM.
PelliHM 3a,naqy onpe.n.eJIeHHH oca.n.OK OT pa3JIHqHWX
u
cxeM nOJIOCOBOH HarpY3KH CYMMHpOBaHHeM OCa,D.OK ,nRH
CJIOeB C oJiHHaKoBbIM Mo,nYJleM .n.ecPopMau.HH E He3aBHCHMO
OT HX TOJII.UHHbI. TaKoH cnoc06 paCqeTa n03BOJIHeT ,nJIH 0.11-
HOCJIOHHOrO OCHOBaHHH 06XOlIJtTbCH TOJIbKO O.n.HHM Tao-
JIHqHbIM K03c)Jc)JH HeHTOM, a ,D.JIH MHOrOCJIOHHOrO OCHOBaHHSI
KOJIHQeCTBO HCnOJIb3yeMbIx K03<PcfJHUl1eHTOB paBHO KOJiHqe-
CTBY CJIOeB, pa3JIHqalOl1I.HXCH no ClKHMaeMOCTH. npH OM
pe3KO COKpal1I.aeTCH KOJIHIJeCTBO BbIQHCJIeHHH H, CJIeJ10Ba-
TeJIbHO, Tpy.n.oeMKOCTb Bcero paCqeTa.
K03cp<PHUHeHTbI .11.JIH OIIpe,neJIeHHH epTHI{aJIbHbIX HOp-
MaJIbHbIX HanpH:lKeHHH H oca,D.OK OCHOBaHHH OT pa3JIHQHblX
cxeM nOJIOCOBbIX HarpY30K B3aHMOYBH3aHbI. Onpe,neJIHM KO-
3<P<PHUHeHTbI .11JIH BbIIJHCJIeHHH HanpH}KeHHH H OCa,llOK OT
paBHOMepHO pacnpe.n.eJIeHHOH nOJIOCOBOH HarpY3KH, 6eCKO-
ReqHOH B O.n.HOM HanpaBJIeHHH H KOHeIJHOH B JIpyrOM. By-
n.eM Ha3bIBaTb TaKYIO HarpY3KY IIoJIy6ecKOHeqHOH. 3aMeTIIM,
qTO B pa60Te [8] A. Ke.D.3H npHBo,nHT cpOPMYJIY ,nJIH onpe-
JIeJIeHHH OCaJIOK OCHOBaHHH npH JIeHCTBHH paBHOMepHO pac-
npe,neJIeHHOH nOJIOCOBOH HarpY3KH. ,[{JIH HepaBHOMepHbIX
u u
JIHHeH O H3MeHHIOl1I.HXCH IIOJIOCOBbIX HarpY30K B TOH }Ke
paooTe co.n.ep)l{HTCH JIHllIb nporpaMMa ,nJIH paCqeTa H3
3BM, TaK KaK no MHeHHIO aBTopa pa60TbI paCqeT CTaHO-
BHTCH CTOJIb Tpy.n.oeMKHM, qTO MO)l{eT 6bITb BbInOJIHeH TOJIb"
KO Ha 3BM. . i
lJacTHqHbIe pellieHHH nOCTaBJIeHHbIX 3a,naq He .lI.OBe,l1eHbI
11.0 COCTOHHHH, npHeMJIeMOrO .11JIH npaKTHqeCKOrO HCnOJIb30-
BaHHH. TaK B PYKOBo.n.cTBe [13] CO.D.ep)l{aTCH YKa3aHHH lIO
onpe.n.eJIeHHIO oca.n.OK OCHOBaHHH OT .n.eHcTBHH nOJIOCOBOH
paBHoMepHoH H TPeyrOJIbHOH HarpY30K 6e3 OrpaHHqeHHH
TOJII.UHHbI CJIOeB. OJIHaKO 3TH YKa3aHHH pa3pa60TaHbI J{JIH
OrpaHHQeHHOrO KOJIHqeCTBa TOQeK OCHOBaHHH (nOA ueHT-
pOM H KpaHMH HarpY3KH), qTO CJIHlliKOM CY}KaeT 06JIaCTb
HX npHMeHeHHH, OTCYTCTByeT CJIytJaH TpaneUHeBH.D.HOH Ha."
rpY3KH, a opMYJIa ,nJIH BblqHCJIeHHH oca,n.OK .naHa TOJIbKO
JI.JIH Mep3JI@X rpYHTOB. CHaqaJIa npHBe.neM peKOMeH.naUHH
no onpe,neJIeHHIO npe.lI.JIaraeMbIM cnoc060M oca.n.OK ,nJ1H npo..
u
CTeHlliHX cxeM 3arpY}KeHHSI, a saTeM nOK3>KeM KaK pacnpo-
CTpaHHTh HX H3 oOJlee CJlO>KHble cxeMhI.
211
z ,.
PHC. 6.26
OCa,lI.KY Ka)K,lI.OrO C.nOR OCHOB3HHR C O)].HHaKOBbIM MO.ay-
.neM J{ecpopMa HH E BblqHCJIRlOT no. cpopM:YJle
Si= (bpfE i ) (k j , i - k j , i-i). (6.59)
11cKOMYIO oca,.QKY nOJIyqalOT CYMMHpOBaHHeM Oca,l1.QK JI5I
Ka)K)].oro CJI051 B npe.n.eJIax C}KHMaeMOM TOJIIlI.H no cpOPMYc!Ie
n
S= Si.
i=1
( 6.60 )
B cpop IYJ1aX (6.59) H (6.60): i - HOMep paccMaTpHBaeMoro CJIOH; fl-
KOJIljqeCTBO CJIOeB B npeJ\eJIax C}KHMaeMOH TOJIllI.H, pa3JIUqalO1ll.HXCSI no
C}I{HMaeMOCTH; 3HaqeHH51 k i , i H Il j , i-I npHHHMalOT no Ta6JIHII.aJ\J COOT-
BeTCTBeHHO JJ.JIH mi=2zdb; mi-l=2zi-tlb Ii cxeMbI 3arpY}KeHHH j; rJIY-
6IIHY t - npHHHMalOT OT 0 AO HfDKHeH rpaHHII.bI i-ro CJIOH; 8 .lI.J1fI no-
CJIeJI.HerO n-ro CJ10H -.lJ.0 HH>KHeH rpaHHu.bI C}KHMaeMOH TOJIllUI.
Ta6JIHI.I.bI COCTaB.neHbI .n.JIR = 1. flpH .n.pyroM 3Haqe-
HHH CJle.n.yeT BbIqHCJIeHHe oca.n.KH YMHO)KHTb Ha B.
CxeMa 1. Oca)].KH TOqeK OCHOBaHHH OT BepTHKaJIbHOH
TpeyrOJIbHOH nOJIOCOBOH HarpY3KH (pHC. 6.26). ,LLJIH 3TOH
cxeMbI' no cpopMYJIe (5.7) CnpaBOqHHKa [23] BbIQHCJIeHbI. H
YTOtIHeHbI 3HaQeHHH K03cpcpHIJ.HeHTOB a, ,l1..H5I qero YKa3aH-
HaR cp.opMYJIa rrpHBe,l1.eHa K BH)].Y
at == ( arct g c - I - arctg ) __
1t O,5m O,t_,m
O,5m c - 1
,
'it (c - 1) 2 + 0, 25m 2
(6.61 )
rJI.e
c=x/b; m=2zfb.
rIpoHHTerpHpoBaB BblpaX{eHl:fe (6.61) B npeAe ax Q.T
o o Z, nOJIyqHM cpOPMYJIY A.ryR c6oTBeTCTBYlO H.X _nJlQ -
JI.eH 3nlOp a KOTOpy nOCJIe pR,lI.':l npeo6pa30BaHHH, ony_-.
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0 0,25 0,5 0,75 1 1,5 2 2,5
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0,20 0,500 0,988 0,996 0,988 0,500 0,001 0,000 0,000
0,30 0,500 0,969 0, 990 0,969 0,500 0,005 0,000 0,000
0,40 0,498 0,937 0,978 0,937 0,498 0,011 0,001 0,000
0,50 0,497 0,902 0, 960 0,902 0,497 0,019 0,003 0,000
0,60 0,495 0,867 0,936 0,867 0,495 0,030 0,004 0,001
0,70 0,492 0,831 o ,910 0,831 0,492 0,043 0,007 0,002
0,80 0,489 0,797 0,880 0,797 Q,489 0,055 0,010 0,003
0,90 0,484 0,765 0,850 0,765 0,484 0,069 0,013 0,004
1,00 0,479 0,735 ° , 818 ° , 735 0,479 0,084 0,017 0,005
] ,20 0,468 0,679 0,756 0,679 ° ,4"68 0, 11 I 0,026 0,009
1,40 0,456 0,629 . ° , 696 ° , 629 ° ,456 ° , 135 0,037 0,013
I ,60 0,435 . 0, 586 0,642 0,586 0,435 0, 155 0,048 0,017
1 ,80 0,42.5 0,546 0,594 0,546 0,425 0, 172 0,059 0,023
2,00 0,409 0,510 0,550 0,510 ° ,409 ° , 185 0,071 0,029
2,50 0,370 0,436 0,462 0,436 0,370 0,205 0,096 o 044
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3,00 0,334 0,379 0,396 0,379 ° , 334 0, 21 1 0,114 0,059
3,50 0,302 0,334 ° , 346 0, 334 0,302 0,210 '0, I 27 0,072
4,00 0,275 0,298 0,306 0,298 0,275 0 ,204 0, 135 0,083
4,50. 0,251 0,268 0,274 0,268 0, 251 0 , 197 0, I 38 0,091
5,00 0,231 0,24 0,248 0,244 0,231 0, 189 0,140 0,097
6,00 0, 197 0,206 0, 208 ° ,206 0, 197 0, 171 o , 136 0, 103
7,00 0,173 ° , 178 0, 180 0, 178 ° , 173 ° , 154 0, 129 0,104
8,00 0, 153 0, 156 0, 158 0, 156 ° , 153 0, 140 o , 122 0, 102
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0,40 0,100 0, I 96 0, 198 0, 1 96 0, 100 0,000 0,000 0,000
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0,60 0,150 0,286 0,294 0,286 0, 150 0,003 0,000 0,000
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1 40 0,340 0,582 0,622 0,582 0,340 0,035 0,007 0,003
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2,00 0,470 0,752 0,808 0,752 0,470 0,084 0,024 0,008
2,50 0,568 0,870 0,934 0,870 0,568 0, 133 0,044 0,018
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3,50 o , 7 35 1 , 061 1,132 1 , 061 0,735 0,239 0,101 0,047
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4,50 0,873 1,210 I , 286 I ,210 0,873 0,341 o , I 68 0,088
5,00 0,933 I ,274 I , 352 1,2.74 0,933 0,289 0,202 0,112
6,00 1 , 040 I ,386 1 ,464 I , 386 1 ,040 0,479 0,271 0, 162
7,00 1 , 132 I , 482 1 ,562 I ,482 1 , 132 0,560 0,338 0,213
8,00 1 ,214 I , 565 I , 646 I ,565 1,214 0,6.14 0,402 0,265
9,00 I ,286 I , 638 I , 720 1 , 638 1 ,286 0,700 0,461 0,316
10,00 1 ,352. 1.704 1 , 786 1 , 704 I ,352 0,762 0,515 0,364
11,00 1,411 1 , 765 1 ,846 1 , 765 1,411 0,818 0,567 0,410
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- - - - - - - - - - - - - - - - - - - - -
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11 pOaOJl'JICeHUe Ta6Jl. 6.17
ks
== biz 1,00 1,20 /1.40 1.60 1.80 2.00 3.00 5.00 10.00
I
= (0...10) 0,4880,4080,352 0,309 0,275 0,248 0,166 0,099 0,049
Sf =-= - ap ( kg i - k's i-I ) . ( 6.67)
E i' ,
npHBe,ll.eM IJHCJIOBOH npHMep 13. Onpe JlHM Oca.lI.KY TOQKH M oc-
HOBaHHH Ha rJIy6HHe z=2 M OT BepTHK8JIbHOH nOJlOCOBOH Tp8ne HeBH.u.-
HOM HarpY3KH npH x=4 M, a=2 M, b=2 M C .lI.OnOJIHHTeJIbHbIM .lI.aBJIe-
HHeM Ha ypOBHe OCHOBaHHH HaCbInH p=0,25 Mila H E=10 Mila (CM.
pHC. 6.28).
Pact/,eT: CIIoc06 1 C HCnOJIb30BaHHeM k l no Ta6JI. 6.13: bI =a+
+b=2+2=4 M; p'l=pb/a=O,25.2/2=0,25 Mna, PI=P+P'I=0,25+
+0,25=0,5 Mila.
,lJ,JIH HarpY3KH 1,3 1 5 m=2z/b I =2.2/4= 1. no Ta6JI. 6.13 npH m= 1,
c=xjb l =l. k 1 . 1 =k t . 1 =0,212, k j ,i-l=k 1 . 0 =0. AJIH HarpY3KH 2 1 4 1 3 C=l,
m=2.2/2=2; no TOH }Ke Ta6JIHue k l . 1 =0,360, k1.0=0. .
no cpopMYJIe (6.59) SM= (0,212.4.0,5 - 0,360.2.0,25)/10=
= 0,0244 M = 2,44 CM.
Cnoc06 2 C HCnOJIb30BaHHeM k3 no Ta6JI. 6.17:
=a/z=2/2= 1; =b/z=2/2= 1.
no Ta6JI. 6.17 npH = = 1 Haxo.n.HM k3.1=O,488;
SM== 2.0,25 0,488 = 0,0244 M = 2,44 CM.
10
Cnoc06 3 C HCnOJIb30BaHHeM a3 no T86JI. 6.16. ,l{JIH nOCJIOHHOrO
CYMMHpoBaHHH pa36HBaeM OCHOBaHHe rJIy6HHoH z=2 M H8 qeTbIpe
CJIOH TOJIllI.HHOM 0,5 M Ka}KJ].bIH. Onpe.neJIHM .nJIH Ka}KJ].oro CJIOH 3Ha-
qeHHH , , COOTBeTCTBYIOllI.He HM K03<P<PH HeHTbI a3 no Ta6JI. 6.16 H
Poz = ap. BblqHCJIeHMH CBO;lJ.HM B Ta6JI. 6.18.
Ta6JIHI.\a 6.18
a.. no
0 t i=a/zi='\lJi=blz i '1. Mna
Zi' M Ta6JI. Poz= a.i P ,
C,) 6.16
1
2
3
4
o I =1l'1 =2/0= 00
0,5 2=1l'2=2/0,5=4
1 3=1l'3=2/1 =2
1,5 4=1l'4=2/1,5= 1,33
2 5 =1l's::;:: 2/2= 1
0,500
0,499
0,482
0,476
0,455
0,5 . 0,25=0,125
0,499 . 0,25=0,1242
0,482 . 0,25=0,1205
0,476 .0,25=0,1196
0,455 · 0,25=0,1133
223
no cJ»opMYJIe (1) npHJI. 2 CHHn 2.02.04-83
n
8 M = ? Pi h [ = ;5 [0,125 + 2 (0,1242 + 0,1205 +
i=1 E l
+ 0,1196) + 0,1133] = 0,0242 M = 2,42 CM 2,44 CM.
He6oJIbillOe paCXOLKJI.eHHe B 0,8 % CBH3aHO C 3aMeHoH
KpHBOJIHHeHHOH 3nlOpbI a JIOMaHOiL
113 npHBeJI.eHHOrO npHMepa CJIe.u.yeT, qTO no KOJIHqeCTBY
B qHCJIeHHH HaH60JIee npHeMJIeM cnoc06 2, oc06eHHo no
cpaBHeHHIO co cnoc060M 3. B npHMepe paCCMOTpeHO O,nHO-
CJIOHHOe OCHOBaHHe C nOCTOHHHbIM E, npH MHOrOCJIOHHOM >Ke
OCHOBaHHH npeHMYIII.eCTBO cnoc06a 2 OKa3aJIOCb 6w
eIII.e 66JIblllHM. OCa.lI.KH onpe)I.eJIeHbI )I.JIH nOJIYHaCbInH - ,llO
OCH CHMMeTpHH, ,llJIH BceR HacwnH OCa.lI.KH 6YJI.YT B)I.BOe
66JIhIlIHMH.
CxeMa 4. Y CJIOBHH TaKHe 2Ke, KaK H JI.JIH cxeM 1-3, HO
HarpY3Ka nOJIy6ecKOHelIHaH. ,LlJIH 3acpHKcHpoBaHHoro KpaH
nOJIy6ecKOHeqHOM HarpY3KH a4=0,5uJ H k,.=0,5k J , r,lI.e j-
1, 2. HJIH 3.
TaK KaK BeJIHqHHa OCa)I.KH 3acpHKcHpoBaHHoro KpaH no-
JIy6ecKOHeQHOH JIeHTW B Boe MeHbllIe OCa)I.KH ee yqaCTKOB,
Y.lI.aJleHHbIX OT KpaH, TO TaKaH HepaBHOMepHaH Oca)I.Ka 10-
LKeT BbI3Ba Tb nOB pe}K.lI.eHHe Ha3eMHbIX KOHCTPYKl1.HM. ,LLJIH
BwpaBHHBaHHH YKa3aHHWX Oca.lI.OK peKOMeH.lI.yeTcH yqaCTOK
JIeHTW y 3a<pHKCHpOBaHHoro KpaH npHHHMaTb nepeMeHHOM
lllHpHHbI OT 0,5b )I.O b. ,LlJIHHY 3Toro yqaCTKa, COrJIaCHO pe-
KOMeH)I.al1.HHM pa60T [20] H [24], MOLKI:I0 npHHHTb paBHoH
(2 . . . 3) b (pHC. 6.30). PerneHHe .lI.JIH BepTHKaJlbHOM paBHO-
MepHO paCnpeAeJIeHHOH nOJIOCOBOH nOJIy6ecKOHeqHOR Ha-
rpY3KH HYLK.lI.aeTCH B .lI.OnOJIHHTeJIbHbIX HCCJIe.lI.OBaHHHX.
,LlJIH nOJIy6ecKOHeQHOfI nOJIOCOBOH HarpY3KH Ta6JIHl1.bI He
COCTaBJIHJIHCb, TaK KaK .llJIH 3TOH HarpY3KH MO>KHO nOJIb
30BaTbCH Ta6"TJHuaMH 6.12 H 6.17 C K03cpcpHlI.HeHTOM 0,5.
LLJIH BbIQHCJIeHHH nOJIHOM OCa)I.KH OCHOBaHHH npe)I.JIaraeMbIM
cnoco60M He06xo.lI.HMO npe.lI.Ba pHTeJIbHO onpe.lI.eJIHTb r JIY-
6HHY CLKHMaeMOM TOJIII H. r JIy6HHY C>KHMaeMOH TOJIIII.H Ha-
XO.ll.HT H3 YCJIOBHH
POz' = 0,2p6z'. (6.68)
ECJIH HaH.lI.eHHaH H3 YCJIOBHH (6.68) HHLKHHSI rpaHHl.(a
C>KHMaeMOH TOJIIII.H paCnOJIOLKeHa B CJIoe rpYHTa C MO)I.Y-
JIeM .u.ecpopMal1.HH E <5 MIla HJIH eCJIH TaKOH CJIOH 3aJIera-
eT HenOCpe.lI.CTBeHHO HHme 3TOH rpaHHl.(bI, OH ,nOJI>KeH 6hITb
224
b...0,50
z
f 1---
Ld
""I
I"r)!
: I
C\J;
'-I
I
,I'
,
b
7
PRC. 6.30
BKJIlOqeH B COCTaB C}KHMaeMOH TOJIIIlH. B 9THX CJIyqaHX
rpaHHlI.a cLKHMaeMoH TOJIIIlH OrpaHHqHBaeTCH YCJIOBHeM
poz' === 0, Ip6z'. (6.69)
):LJIH fH.lI.pOTeXHHqeCKHX cooPY>KeHH", COrJIaCHO [20]:
POZ' == O,5p6z'. (6.70)
C yqeTOM, qTO poz'=a(p-po) H P6z' =Vlht+V2h2+ ... +
n-l
+l'n (z' +h' - hi) YCJIOBHSI (6.68)... (6.70) npHHHMalOT
i=l
BH,lI.
ii-I
a(p-p'6)=t[v t h 1 +Y2 h 2+ ... +yn(z'+h'- hd],
i=1
OTKY,lI.a
a.. ===
n-l
till! t'v h 2 t,n(z'+h'- hi)
+ . 2 + . . . + 1=1
P - P6 P - P6
P-P6
HJIH
n-I
a=f)t +f)2+ . . · +f)ii = ll i +'r)ii,
i=1
(6.71 )
225
r,ne
t l "h.
"r. 1 t .
'u == ,
P-P6
(6.72)
n-l
t"{_(z'+h'- hi)
n i=1
-
i"{ -h-
n n.
,
( 6. 73 )
ii
P - P6 P - P6
t= 0,1; 0,2 HJIH 0,5 B 3aBHCHMOCTH OT paCCMaTpHBaeMoro CJ1YQaH
(6.68), (6.69) HJIH (6.70); hi - TOJ1llI.HHa i-ro CJIOH rpYHTa C oAHHaKo.
BbIM y,neJIbHbIM BeCOM 'Vi;
n-l
h-=z' +h' - hi
n i=1
(6.74}
- To.nI.U.HHa GaCTH n- ro CJIOH rpYHTa OT HH}KHeH rpaHHIJ.bI C 10H
n - 1 ,no rpaHHIl.bI C}KHMaeMOH TOJII.U.H Hero y,neJIbHbIH Bec Y ii ; h'-
r .ny6HHa 3aJIO}KeHHH cpYH,naMeHTa OT OTMeTKH nOBepXHOCTH npHpO.ll.HOro
peJIbecpa. ,LLJIH HaCbIIIH h' = O.
HH}KHIOIO rpaHHUY CLKHMaeMOH TOJIIII.H Onpe.ll.eJIHIOT B
TaKOH nOCJIe.ll.OBaTeJIbHOCTH: no O.ll.HOH H3 Ta6JI. 6.12, 6.13
HJIH 6.14 B 3aBHCHMOCTH OT THna HarpY3KH BbIqHCJIHlOT 3Ha-
qeHHH Ui Ha rpaHHuax CJIOeB OCHOBaHHH pa3HbIX no C}KH-
n-l
MaeMOCTH, a TaKLKe COOTBeTCTBYlOllI.He HM 3Hat}eHHH L 11i
l=1
r.ll.e l1i HaXO.ll.HT no cpopMYJle (6.72).
rpaHHUa C}KHMaeMOH TOJllll.H 6Y.lI.eT pacnOJIO}KeHa B CJIoe
ii, .lI.JIH KOToporo C06JIIO,lI.aIOTCH CJIe,lI.YIOll.{He YCJIOBHSI:
n -l n
f1i:::;; uii B ; aii H :::;; f1i, (6.75)
i=1 i=1
_8 H a_ H onpe.u.eJIHIOT COOTBeTCTBeHHO .l!.JIH BepXHeH H HH:>KHeH rpa-
r,ne an n
HHIl. CJIOH n.
r JIy6HHY C}KHMaeMOH TOJlJ..l(H HaXOJI.HM MeTO.ll.OM nOCJIe-
.lI.OBaTeJIbHbIX npH6JIHLKeHHH. B nepBOM npH6JIH}KeHHH npH-
HHMaeM
n-l Ii-I Ii
. fli +11 n l = (aii B + Uii H .+. 11i + . f)i) 0,25; (6.76)
1==1 L==1 l=1
H3 paBeHCTBa (6.76) onpe.ll.eJIHeM 3HaqeHHe 11iil H "Ha OCHO-
BaHHH cpOPMYJlbI (6.75) h n1 . no cpopMYJIe (6.74) HaXO,lI.HM
3HaqeHHe Z'2 H no Ta6JI. 6.14, 6.16 HJIH 6.18 BbIQHCJlHeM at.
,LLJIH BToporo npH6JIHLKeHHH npHHHMaeM
n-l n-l
. f1 i +f)n2= (U1+ f)i+11 n l) 0,5. (6.77)
1=1 i=1
rpaHHll.Y C}KHMaeMOH TOJIll.{H YTOtIHHeM .lI.O Tex nop, nOK3
He 6Y.ll.eT C06JIlO.lI.a TbCH YCJIOBHe
226
n-l
+ [a - <. lli+lln)]. O,005/po. (6.78)
l=l
B q>opMYJIe (6.78) 3HaK «MHHYC» npHHHMaeTCH B CJIY-
qae oTpHuaTeJIbHOrO 3HaqeHHH B KBa paTHwx cKo6Kax.3a-
MeTHM, qTO B COqeTaHHH C npoCToTOH onpe eJIeHHH rJIy6H-
HbI C)KHMaeMOH TOJIIII.H o6eCneqHBaeTCSI ero 6bIcTpaH CXO,nH-
MOCTb, OrpaHHQHBaeMaH BYMH-TpeMH npH6JIH>KeHHHMH.
npHMep 14. Onpe)J.eJIeHH5I r JIy6HH C}KHMaeMOH TOJIll1.H )J.JI5I nOJIO-
COBOH TpaneUHeBH,lJ.HOH HarpY3KlI (CM. pHC. 6.28).
p'aHo: a=2 M; b=2 M; hi =2 M; 1'1 = 18 KH/M 3 ; h 2 =4,5 M; 1'2=
= 19 KHfM 3 ; h 3 = 11 M; 1'3=20 KH/M 3 ; po=P - p6=O,25 Mlla. rpaHH-
uy C}KHMaeMOH TOJIllI.H onpe,lI.eJIHeM ,D.JIH Tpex CJIyqaeB: t = 0,5; 0,2 II
n
0,1. HaXO,lI.HM 3HaqeHHH lli H lli ,D.JIH CJIyqaH t=O,I, .l!.JUI .lI.pyrHx 3Ha-
:"=1
qeHHH t BbI1JHCJIeHIHI a HaJIOrH1JHbI:
1 = 0,1.1,8.2 =0,0144; = 0,1.1,9. 4,5 ==0,0342
° 25 . 100 'YJ 2 0, 25 . 100
,
° I . 2 ° . 11 0 , 088 + 0 04 C 6 .
'Yl3 = ' , =,; "i11 "i12 == , c,
0, 25 · too
111 +t12+'r}3=O,1366 Bce BbI1JHCJIeHHH CBO)J.HM B Ta6JI. 6.19.
Ta6J1HQa 6.19
n
[T}i}
hit M ti=,pi= cx 3 i no i=l
z, M =ajz-b/z Ta6n. 6.16 no 4>opMYJIe (6..62)
t=0,5 I t=0,2 I tICSO.1
-
2
6,5
17,5
2
4,5
11
1
0,308
0,114
0,455
0,261
° , 102
0,072
0,243
0,029
0,097
0,273
0,014
0,049
0, 137
2, M
h- M
n'
i='lJi
CX3
ii-I
T}i+T}n
i=l
11 ,9 5,4 0, 168 0, 151 - 0,183 -
10,9 4,4 0, 183 ° , ) 65 - 0, t 67 -
-
16,5 10 0,121 0,111 - - 0, 137
15,9 9,4 0, 126 0,116 - - o , 124
227
BBH.n.y TOrO, tJTO .lI.JIH CJIyq8H 1=0,5 no <j>opMYJIe (6.78) a32-
2
- lli=O,261 - 0,243=0,018<0,005/0,25=0,02 rp8HHua C>KHMaeMOH
i=1
TOJIllI.H COBn8.l1.8eT C HH>KHeH rpaHHueH BToporo CJIOH rpYHT8, z, = 6,5 M.
,lI.JIH OCTaJIbHbIX 3HaqeHHH t rp8HHua C>KHMaeMOH TOJIllI.H H8XO.ll.HTCH B
TpeTbeM CJIoe, TaK K8K C06JIIO.lI.8eTCH YCJIOBHe (6.75): .lI.JIH 1= 0,20,097 <
<0,261 H 0,102<0,273, .lI.JIH 1=0,1 0,049<0,261 H 0,102<0,137.
HH>Ke npHBO.ll.HM onpe.neJIeHHe r JIy6HHbI C>KHMaeMOH TOJIJ..U.H .nJIH
CJIyq8H 1=0,2:
223
'r)i+fl1il= (aiiB+aiiB+. fli+ lli)0,25= (0,261 +
i=1 £=1 1=1
+0,102+0,097 +0,273)0,25=0,183,
OTKY.lI. 8
2
fl n l=0,183 - lli=0,183 - 0,097=0,086;
i=l
h lit = 0,086 . 0,25 100 = 5,4 M; Z = 6,5 + 5,4 = 11,9 M,
° 2 . 2
,
t='i' =2/11,9=0,168, a3=0,151.
TaK KaK -(0,151 - 0,183) =0,032>0,02, TO npOH3B0.l1HM BTopoe
2
npH6JIH>KeHHe lli+1l- 2 =0,5(0,151 +0,183) =0,167.
l=1 n
l1ii2=0,167 - 0,097=0,070
h- == 0,07.0,25 100 = 4,4 M,
n2 ° 2 . 2
,
z=6,5+4,4= 10,9 M; t=1l'=2/10,9=0,183 a3=0,165.
B CBH3H C TeM, tJTO -(0,165 - 0,183) =0,018<0,02, rp8HHIJ.a C>KHM8e-
MOH TOJIJ..U.H H8XO.ll.HTCH H8 rJIy6HHe z, = 10,9 M. ,lI.JIH CJIyq8H 1=0,1 BbI-
qHCJIeHHH 8HaJIOrHqHbI, a HX pe3YJIbTaTbI, KaK H .lI.JIH CJIyqaH 1 = 0,2,
CBe.ll.eHbI B Ta6JI. 6.19.
6. t t. K onpeAeJleHHIO ocaAOK OCHOBaHHH
OT oceCHMMeTpH1JHbIX pacnpeAeJleHHbIX HarpyaoK 1
B TeXHHqeCKOH JIHTepaType H, B qaCTHOCTH B Doco6HH
[12], npHBe,neHbI YKa3aHHH no onpe.n.eJIeHHIO HanpH}I{ HHH H
oca,nOK OT HarpY3KH, paBH-oMepHo pacnpe,neJIeHHOH no nJIO-
ma,nH Kpyra. TaKoH BH,n HarpY3KH BCTpeqaeTCH B cpYH.n.aMeH-
Tax ,nbIMOBbIX Tpy6, cpYH,narvteHTax no,n o6opy,nOBaHHe H
.npyrHx CJIyqaHX.
1 no p8CqeTHOH cxeMe OCHOBaHHH B BH.ne JIHHeHHO .lI.e<j>opMHPye-
Moro nOJIynpocTpaHCTBa.
228
f
Q) r;
r:::t
R
zt
Puc. 6.31
B KHHre 1V\. E. Xappa «OCHOBbI TeOpeTHqeCKOH l\1eXa-
HHKli rpYHToB» (M., 113.n.aTeJIbcTBo JIHTepaTypbI no CTPOH-
TeJIbCTBY, 1971) HMelOTCH rpacpHKH ,nJIH onpe.n.eJIeHHH Ha-
npH)KeHHH OT napa60JIHqeCKOH H KOHHQeCKOH HarpY30K, pac-
npe.n.eJIeHHbIX no nJIOllI.a.n.H Kpyra [no Xa PPY H JIoYBeJIJIY
( 1963)]. ABTOpOM npe.n.JIaraeTCH perneHHe 3a,naQ no BblqHC-
.neHHIO HanpH)KeHHH H oca.n.OK OT HarpY30K, pacnpe.n.eJIeH-
HbIX no Kpyry C HHTeHCHBHOCTblO' MeHHIOllI.eHCH no yceQeH
HOMY KOHYCY, H KOHHQeCKoi1, a TaK)Ke npHBe.n.eH 60JIee npo-
CTOH cnoc06 .n.JIH onpe eJIeHHH oca.n.KH OT HarpY3KH, paB-
HOMepHO pacnpe.n.eJIeHHOH no nJIOllI.a.n.H Kpyra.
PaCCMOTpHM peKOMeH.n.a HH no onpe.n.eJIeHHIO HanpH./Ke-
HHll H oca.n.OK OT KOHHqeCKOH HarpY3KH. 113 nOCJIe.n.HeH BbI-
JI,eJIHM 3JIeMeHTa PHYIO nOJIOcy HHTeHCHBHOCTbIO dp1 B BH.n.e
K Pyra paJI,HYCOM (pHC. 6.31, a).
rl=Rp1/P.
Onpe,neJIHM BepTHKaJIbHbIe HOpMaJIbHbIe HanpH./KeHHH
(Jzp B TOqKaX OCHOBaHHH, pacnOJIO./KeHHbIX Ha OCH CHMMeT-
pHH Z KOHHqeCKOH HarpY3KH, ,l(JIH Qero BOCnOJIb3yeMcH cpop-
MYJIOH (5.8), npHBe eHHoH B CnpaBOQHHKe 1. ,lLJIH TOQeK,
pacnOJIO){{eHHbIX Ha OCH CHMMeTpHH paBHOMepHO pacnpe.n.e-
JIeHHOH no nJIOllI.a)lH Kpyra HarpY3KH, BepTHKaJIbHbIe HOp-
MaJIbHbIe HanpH)KeHHH paBHbI
{ I [ I ] 3i'J }
(Jzp P - 1 + (r/z)2 . '
(6.79 J
rAe r - pa,llHYC Kpyra; z - KOopJUiHaTa TOqKH.
I CnpaBotJHHK _ npoeKTHpOBllI.HK3 npOMbllllJIeHHbIX )KHJIbIX H o611.1e-
CTBeHHblX 3 aHHH H CooPY>KeHldi.- OCHO"atUHI H <PYH aMeHTbI. - JI. - M.:
CTpoHH3 aT, 1964.
229
EcnH KOHHQeCKYro HarpY3KY pa30HTb Ha aneMeHTapHMe
HarpY3KH B BH e KpyroB HHTeHCHBHoCTbro dpl H nepeMeH-
v v - - '-
HbIM pa HYCOM '1, TO JIH Ka>KJI.OH TaKOH HarpY3KH coxpa-
HHTCH 4>opMYJIa (6.79) npH YCJIOBHH 3aMeHbI B HeM PH'
COOTBeTCTBeHHO Ha dpt--H- 't. B pe3YJIbTaTe ,lJ.JIH Ka>K,lI.OH
v
3JIeMeHTapHOH HarpY3KH B BH,lI.e Kpyra nOJIyqHM
d o = { I - [ 1 ] 3/'2 } d ( 6 )
zp 1 + (R/(pz»'PI' PI. .80
,aJIH BceM KOHUqeCKOM HarpY3KH npOHHTerpHpyeM Bblpa-
}I{eHHe (6.80) B npe eJIax OT 0 ,lI.O P
- p { I [ 1 ] 3/'1 } d
(Jzp- J - I + (R/(pZ»2 PI2 PI.
G
OOCJIe HHTerpHpoBaHHH H cooTBeTCTBYIOIll.HX npeo6pa-
30BaHHH nOJIyqeHa opMYJIa JIH onpe eJIeHHH BepTHKaJIb-
HbIX HanpH>KeHHH OT KOHHqeCKOH HarpY3KH
(Jzp == P [ 1 - I _ ] == p ( 1 _ c ), (6.82)
VI+<R!z)2 . VI+ 2/
r.l1.e = z / R - OTHOCHTeJIbH3H r JIYOHH3.
113 CPOPMYJIbI (6.82) CJIe yeT, qTO BepTHKaJIbHbIe Ha-
npH}I{eHHH B rpYHTe Ha O HOH H TOM }l{e rJIYOHHe Z OT KO-
HHqeCKHX HarpY30K C O HHaKoB M pa HYCoM OCHOBaHHH
npHMo nponopUHOHaJIbH HX HHTeHCHBHOCTH p.
11CnOJIb3YH 4>OPMYJIY (6.82), MO>KHO nOJIyqHTb c1JOPMYJIY
v
JIH HanpH}I{eHHH OT HarpY3KH B BHJI.e yceqeHHOrO KOHyca
1,2,4,5 KaK pa3HOCTb HanpH}I{eHHH OT KOHHqeCKHX Harpy-
30K 1,3,5 H 2,3,4 (CM. pHC. 6.31, 6)
C1 1.'2.4.5===0' 1.3.5_0' 2,3.4
zp zp zp.
BbIpa3HM HHTeHCHBHOCTH KOHHqeCKHX HarpY30K p2,3,!j, 1J
pl.S,S qepe3 HHTeHCHBHOCTb HarpY3KH B BHJI.e yceqeHHOrO vn-
Hyca pl,2,Q,5=p. H3 .pHC. 6.28, 6 BH HO, lITO
P I, 3. 5 pI I 3. 5 _ P
_ H pl.3.5r-==p1.3.5R_pR,
R
(6.81 )
(6.8.1 )
r
OTKY,lI.a
5 pR n
p1.3, = =p I
R-r 1l-1
n I
r.n.e n=R/r, a p2,3"=pl,3,5 - p= p - p = p .
n-I n-t
TaK KaK r=R/n, a tt,3,5=z/R, nOJIYQHM. qTO 2,3,,,=
=z/r=znIR= 1,3,5.n.
230
no nOJtyttelttIbiM Bbtpa>KenlistM IittteItCIfBliocteH H otltO-
CHTeJIbHbIX rJIy6HH KOHHqeCKHX HarpY30K nOCJIe no.n.CTaHOB-
KH COOTBeTCTBYIOI..l.(HX 3HaqeHHH, onpe.n.eJIeHHbIX no CPOPMY-
JIe (6.82), B CPOPMYJIY (6.83) H npe06pa30BaHHH nOJIYQHM
BbIpa}l{eHHe .lI.JIH BbIqHCJIeHHH BepTHKaJIbHbIX HanpSl>KeHHH OT
HarpY3KH B BH.lle yceqeHHOrO KOHyca
n [ I ( cn ) ] (
ozp=p 1- - - 1- . 6.8t)
n-l V1+C 2 n V l+C 2 n 2
OCa,llKIf OCHOBaHIfH S B COOTBeTCTBHIf co CHlfn 2.02.01-
-83 onpe,lleJISIIOTCSI MeTO,llOM nOCJIOHHOrO CYMMlfpOBaHHH no
opMyne \
azpih,
S= .
I-I E l
PaccMoTpHM CJIYtlaH O)1HopOAHoro rpYHTa E,=E=
n
=const. 3HaqeHlfe (JzPih i B <popMYJIe (6.85) npeACTaBnS-
'-I
eT C060R nplf6JIIf}l{eHHYIO nJIOLUa,nb 3nIOpbI HanpH>KeHHH
(Jzp, TaK KaK nJIOlI1.a,lJ.h KpHBOJIHHeHHOR 3nlOpbl aaMeHeHa
nJIow.a.lI.blO JIOMaHOH. Onpe.n.eJIeHHe TOqHOrO 3HaqeHHSI nJIO-
I..l.(a.n.H 3nlOpbI HanpH}I{eHHH (Jzp OT KOHHqeCKOH HarpY3KH
npOH3Be.11.eM HHTerpHpOBaHHeM ypaBHeHHSI (6.82) B npe.lle-
Jlax OT 0 .11.0 He, r.11.e He -r JIy6HHa C>KIfMaeMOH TOJII..l.(H
l!c
j p ( I ) dz ( 6.86 )
o - }I I + C 2 ·
( 6.85 )
BbInOJIHHM 3aMeHY nepeMeHHbIX HHTerpHpOBaHHH Z Ha ,
a dz Ha Rd , TaK KaK d =dz/R. Ope.n.eJIbI HHTerpHpOBaHHH
H3MeHHTCH OT 0 .11.0 c, a cpOPMYJI3 (6.86) npHMeT BH.11.
,
RPf ( l ' ) dC===RP(1+Cc- V l+C c 2 ).(6.87)
o V 1 + I
nO cTaBHB nOJIyqeHHOe 3HaqeHHe B CPOPMYJIY (6.85),
nOJIyqHM BbIpa)KeHHe .11.JIH onpe.11.eJIeHHH oc:a.ll.KH OT KOHHlle-
CKOH narpY3KH
s = pR (1 +C c - }/ 1 + c2).
E
AH3JIOrHQHbIM 06pa30M BbIBe.lleHbI CPOPMYJIbI J].JIH onpeJ1,e-
neHHSI OcaJ1,KH OT HarpY3KH B BHJ1,e yceqeHHOrO KOHyca
(6.88 )
\ J
231
pk { n [ - ..... i
S = - ( 1 + c - V I + C2) - - (1 + Ii -
E n - I n2 c
- -V-I + c 2 n 2)] (6.89)
H HarpY3KH, paBHOMepHO pacnpe e eHHoH no nROll{a H
Kpyra
S == pR ( 2 -j- c- VI + c2- I ) . (6.90)
E VI +Cc 2
OpHBe.n.eM CPOPMYRbI .n.RH BblqHC eHHH BepTHKa bHbIX
HanpHLKeHHH (6.79), (6.82), (6.84) H ,ll H onpeJI.e eHHH oca-
JI.OK (6.88), (6.89), (6.90) K o6meMY BH,llY
C1zp=aJp;
S=p.p.R.k,IE.
,Q H HeO.AHOpO Horo rpYHTa Q>opMYJIa
3yeTcH c e.nYIOI.UHM o6pa30M:
S= PRi: kjt-kjt-I .
. l Ei
1=
(6.91 )
(6.92)
(6.92) npeo6pa-
(6.93 )
B cpopMYJIaX (6.91)- (6.93) j= 1,2 HRn 3 COOTBeTCT-
ByeT HJIHH.n.pHqeCKOH, B BH.n.e yceqeHHOrO KOHyca HJIH KO-
HHqeCKOH HarpY3KaM, a i - HOMep paC-CMaTpHBaeMoro CROH
rpYHTa. 3HaqeHHH k ji H k ji - 1 COOTBeTCTBYIOT HH}I{HeH H
BepxHeH rpaHHl.I.aM paCCMaTpHBaeMoro CJIOH. <l>OPMYJIbI ,lI. H
Ta6J1Hl.\a 6.20
I
j
k.
3
1-
l3
V < 1 + 2)3
1
2 + c - V 1 + ---
V 1 + 2
n [ I
1 --x
"l - 1 -- 1/) + 2 fl
X ( I -- 'n ) J
-J; 1 + 2n2
n [ < I +, _ /I + C2 ) --
n.- I
- (I + Il -- V I - 21l ) 1
n j
2
3
1-
1 1 + C 2
,
I + -1-/ 1 + 2
232
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8 C'1 l"'- e") 11') 0 I"'- LQ M CV) C'1 C'1 -
to "" C'1 - - 0 0 0 0 0 0 0 0
.. ... .. .. .. .. .. ... ... ... .. ..
- 0 0 0 0 0 0 0 0 0 0 0 0 0
8 I.Q co t-. Q') r...... 1C If') - 0 C":) 00 "" 0
Q') - If') to - 00 to LQ C":) C'1 C'--1 C'1
to "'d'" - - 0 0 0 0 0 0 0 0
... .. .. .. .. .. ... ... ... ...
- 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0') C\1 CD 00 - LQ C'1 CD LC - CD C'1
0 CD to 00 00 "" 0) r...... l{) C"') M C'1 C'1
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0 ........ tC C\1 0') 00 - c.o 0 00 0) C"':) 00
0 00 00 0 Q) C":) 0 r...... to CV) CV) C'1 C'1
0 ...... C") - - 0 0 0 0 0 0 0
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0 00 to C\1 M 00 00 C'1 - C'1 11') 0 1C
0 - - C'\I - 0 00 c.o LC C") CV) C'1
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0 00 If') C"':) C'.1 0 0 0 0 0 0 c
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0 If') CD LQ C":) to C'1 0) ...... 1.1') oo:::r C"':) C"':) C'1
0 90 lI) C":) C'J - - 0 0 0 0 0 0 0
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0 00 0) 00 LQ 00 C") 0 c.o 11') C") C"')
0 00 1..Q CV) C'1 - - - 0 0 0 0 0 0
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8 LC - CD "" C'1 - ........ ...... I.Q CD OJ ""
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0 00 CD C'J - - - 0 0 0 0 0 0
.. .. .. ... .. .. ... ... ...
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0 00 00 0 - CD - 11') 0 C"':) 0 0 c.o
0 0 CV) Q') 0 11') - Q') t--- CD LQ oo:::r C")
0 0) CD C'1 C\1 - - 0 0 0 0 0 0
... ... .. .. .. ... .. ... .. .. .. ...
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0 C'1 00 Q) I.Q oo:::r LQ to 0) 0 to LQ c.o 0
0 C\1 ....... 1..Q - C'1 CD C'1 0) 00 to LQ
0 Q') to oo:::r C'() C'1 - - 0 0 0 0 0 0
.. .. .. ... ... .. ... .. ... ... .. ... ..
- 0 0 0 0 0 0 0 0 0 0 0 0 0
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0 "" - 0) 00 ........ to 11')
0 0; t-. oo:::r C") C'1 - - - 0 0 0 0 0
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I.Q oo:::r C) 00 --- CD "" 0 00 t--- CD LQ
0 Q') ........ 1..Q "" C'1 C'1 - - 0 0 0 0
.. .. .. .. .. ... .. .. ..
- 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 00 C'1 CD 0 00 C'1 to 0 oo:::r 00 C'1
.. .. ... .. .. .. ... .. .. ... .. .. .. ...
0 0 0 - C'.J C'1 C'J C"':) C":) oo:::r L1,)
233
234
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a A a _
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o
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- - - - - - - - - - - - - - -
00000000000000000
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-------0000000000
00000000000000000
- - - - - - - - - - - - - - -
00000000000000000
OOO -o OOOO
------OOOOOOOOOo
00000000000000000
- - - - - - - - - - - - - - - -
00000000000000000
-o OOOO
-------000000000
00000000000000000
- - - - - - - - - - - - - - - -
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- - - - - - - - - - - - - - - -
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10
C'\I
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- - - - - - - - - - - - - - - - -
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- - - - - - - - - - - - - - - -
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00000000000000000
_ - _ _ _ _ _ _ _ a _ _ a _ _
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0 r-... C'1 ...... -00 C'1 11") r-..., 00 0 - C\1
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C\)oo-Mc.oO')O') c.o t_...C'OMLQ t_...o,)O') OOC"'J
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o,)C\) c.D MCD o,)Ot_...OOt_...-MC>':)O
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00 C') 00 C\J CD 00 C'-J CD 00 C\:t CD 00 C'_.1 C\J
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283
Ta6J1HLla 5. Koa4>4>HLlHeHTbI ai .lI.JUI onpe.ll.eJleHlut!rJly6HHbI
C1KHMaeMOH TOJlIU.H
,,=-1 KH/M 3
(p - P6)' Mna
... 6 10 15 20 25 3 35 40 45 50
I , 5 - 0,060 0,030 0,020 0,015 0,012 0,0 I ° 0,009 0,008 0.007 0,006
2 ,0 0,080 0,040 0,027 0,020 0,016 0 ,013 0,011 0,010 0,009 0,008
2, 5 0, 100 0,050 0,033 0,025 0, 020 0 ,0 17 ° , ° 14 0, ° 13 0, ° 11 0, ° 10
3,0 0,120 0,060 0,040 0,030 0,024 0,020 0,017 0,0150,0130,012
3,5 0,140 0,070 0,047 0,035 0,028 0,023 0,020 0,0180,0160,014
4, ° 0, 160 0,080 0, 053 0,040 0,032 ° ,027 0, 023 0.020 0,018 0,016
4,5 0,180 0,090 0,060 0,045 0,036 0,030 0,026 0,0230,0200,018
5, ° 0,200 0, 100 0, 067 0, 050 0,040 ° , 033 0,029 0,025 0,022 0,020
5,5 0,220 0,110 0,073 0,055 0,044 0,037 0,031 0,0280,0240,022
6,0 0,240 0,120 0,080 0,060 0,048 0,040 0,.034 0,0300,0270,024
() , 5 0,260 0, 130 0,087 0, 065 0 ,052 ° , 043 0, 037 0,033 0,029 0,026
7,0 0,280 0,140 0,093 0,070 0,056 0,047 0,040 0,0350,031 0,028
7,5 0,300 0,150 0,100 0,075 0,060 0,050 0,043 0,0380,0330,030
8,0 0,320 0,160 0,107 0,080 0,064 0,053 0,046 0,0400,0360,032
8 ,5 0,340 0, 170 0, 113 0,085 0,068 0 ,057 0,049 0,043 0,038 0,034
9,0 0,360 0,180 0,120 0,090 0,072 0,060 0,051 0,0450,0400,036
9,5 0,380 0,190 0,127 0,095 0,076 0,063 0,054 0,0480,0420,038
10,0 0,400 0,200 0,133 0,100 0,080 0,067 0,057 0,0500,0440,040
10,5 0,420 0,210 0,140 0,105 0,084 0,070 0,060 0,0530,0470,042
1 1,0 0,440 0,220 0,147 0,] 10 0,088 0,073 0,063 0,0550,0490,044
11 , 5 0,460 0,230 0, 153 0,] 15 0,092 0, 077 0,066 0,058 0,051 0,046
12,0 0,480 0,240 0,160 0,120 0,096 0,080 0,068 0,060,0,0530,048
l' = 1.5 KH1M s
1,5 0,090 0,045 0,030 0,023 0,018 0,0]5 0,013 0,011 0,0100,009
2, ° 0,] 20 0,060 0,040 0,030 0,024 0 , 020 0, ° 17 0, ° 15 0, ° 13 0, ° 12
2,5 0,150 0,075 0,050 0,038 0,030 0,025 0,021 0,0190,0170,015
3,0 0,180 0,090 0,060 0,045 0,036 0.030 0,026 0,0230,0200,018
3,5 0,210 0,105 0,070 0,053 0,042 0,035 0}030 0,0260,0230,021
4,0 0,240 0, 120 0,080 0,060 0, 048 0 , 040 0,034 0,030 0,027 0, 024
4,5 0,270 0, 135 0,090 0, 068 0, 054 0, 045 0,040 0, 034 0, 03] 0,027
5,0 0,300 0,150 0,100 0,075 0,060 0,050 0,0+3 0,0380,033 0,030
5, 5 0,330 0, 165 0, 110 0,083 0, 066 0,055 0,047 0, 041 0, 037 0,033
6 _ 0,360 0,180 0,120 0,090 0,072 0,060 0,051 0,045 0,0400,036
6,5 0,390 0,195 0,130 0,098 0,078 0,065 0,056 0,0400,0430,039
7 0,420 0,210 0,140 0,105 0,084 0,070 0,060 0,0530,047 0,042
:, ,5 0,450 0, 225 0, 150 0, 113 0,090 0,075 0,064 0,036 0,050 0,045
.8 0,480 0,240 0,160 0, '120 0,096 0,080 0,068 0,0600,0530,048
8,5 0, 510 0, 255 0, 170 0, 128 0, 102 0, 085 0, 07.3 0,064 0,057 0,051
.9 0,540 0,270 0,180 0,135 0,108 0,090 0,077 0,0680,0600,054
9,5 0,570 0,285 0,190 0,143 0,] 14 0,095 0,081 0,071 0,0630,057
10 0,.600 0, 300 0, 200 0, 150 0, 120 0, 100 0,086 0,075 0, 067 0,060
10,5 0,630 0,315 0,210 0,158 0,126 0,105 0,090 0,0790,0700,063
284
n pOaOA eH,Ue Ta6Jt. 5
v== 1,5 KH/M 3
1
I
1
(p - P6)' Mna
-
.:. 5 10 15 20 25 30 35 40 45 50
.c:
1 0,660 0,330 0,220 0,165 0, 132 0,110 0,094 0,083 0,073 0,066
1,5 0,690 0,345 0,230 0, 173 0, 138 0, I 15 0,099 0,086 0,077 0,069
2,0 0,720 0,360 0,240 0, 180 0, 144 0, 120 ° , 103 0,090 0,080 0,072
'\'==1,6 KH/M 8
I ,5 0,OJ6 0,048 0,032 ° 024 0,019 0,016 0,014 0,012 0,011 0,010
,
2,0 0, 128 0,064 0,043 0,032 0,026 0,021 0,018 0,016 0,014 0,013
2,5 0, 160 0,08 0,053 0,040 0,032 0,027 0,023 0,020 0,018 0,016
3,0 0, 192 0,096 0,064 0,048 0,038 0,032 0,027 0,024 0,021 0,019
3,5 0,224 0,112 0,075 0,056 0,045 0,037 0,032 0,028 0,025 0,022
4,0 0,256 0, 128 0,085 0,064 0,051 0,043 0,037 0,032 0,028 0,026
4,5 0,288 0,144 0,096 0,072 0,058 0,048 0,041 0,036 ° , 0:32 0,029
5,0 0,320 0, 160 0, 107 0,080 0,064 0,053 0,046 0,040 0,036 0,032
5,5 0,352 0, 176 0,117 0,088 0,070 0,059 0,050 0 044 0,039 0,035
6,0 0,384 0, 192 0, I 28 0,'096 0,077 0,064 0,055 0,048 0,043 0,038
6,5 0,416 0,208 0, 139 0, ] 04 0,OB3 0,069 0,059 0,052 0,046 0,042
7,0 0,448 0,224 0, 149 0,112 0,090 0,075 0,064 0,056 0,050 0,045
7,5 0,480 0,240 0, I 60 0, 120 0,096 0,080 0,069 0,060 0,053 0,048
8,0 0,512 0,256 0, 171 0, I 28 0, 102 0,085 0,073 0,064 0,057 0,051
8,5 0,544 0,272 0,181 0, 136 0,109 0,091 0,078 0,068 0,060 0,054
'9 ° 0,576 0,288 0, 192 0, 144 0,115 0,096 0,082 0,072 0,064 0,058
,
'9 5 0,608 0,304 0,203 0, 152 0,122 0, 101 0,087 0,076 0,068 0,061
,
10 0,640 0,320 0,213 0,160 0, 1 28 0, 107 0,091 0,080 0,071 0,064
to,5 0,672 0,336 0,224 0, 1 68 0,134 0, 112 0,096 0,084 0,075 0,067
11 0,704 0,352 0,235 0, 17G 0,141 0, 117 0, t OJ 0,088 0,078 0,070
I I ,5 0, .756 0,B68 0,245 0, 184 0,147 0, 123 0, 105 0,092 0,082 0,074
12 0,768 0,384 0,256 0, 192 0, 154 0, I 28 0,;110 0,096 0,085 0,077
,\,::::1 1,7 KHfy8
1,5 0, 102 0,051 0,034 0,026 0,020 0,017 0,015 0,0130,011 0,010
2,0 0, 136 0,068 0,045 0,034 0,027 0,023 0,019 0,0170,015 0,01
2,5 0, 170 0,085 0,057 0,043 0,034 0,028 0,024 ° ,021 0, ° 1 9 0,017
.3 ° 0,204 0, 102 0,068 0,051 0,041 0,034 0,029 0,0260,023 0,020
,
3,5 0,238 0,119 0,079 0,060 0,048 0,040 0,034 0,0300,026 0,024
40 0,272 0, 136 0,091 0,068 0,054 0,045 0,039 0,034 0, 030 0,027
,
4,5 0,306 0, I 53 0, 102 0,077 0,061 0,051 0,044 0,0380,034 0,031
5,0 0,340 0, 170 0,113 0,085 0,068 0,057 0,049 0,0430,038 0,034
5,5 0,374 0, 187 0, 125 0,094 0,015 0,062 0,053 0,0470,042 0,037
6,0 0,408 0, 204 0, 136 0, 102 0,082 0,068 0, 058 0,05 I 0, 045 0,041
6,5 0,442 0,22 I 0, 147 0, 111 0, OBB 0, 074 ° ,063 0,053 0,049 0,044
7,0 0,476 0,238 0,159 0,119 0,095 0,079 0,068 0,060 0,053 0,048
75 ° 510 ° 255 ° 170 ° 128 ° 102 ° 085 0,073 0,0640,057 0,051
4
285
n pOaOA3ICeflUe Ta6A. 5
v= 1,7 KH/M 4
(p - P6)' Mila
::E I
.;. 5 10 15 20 25 30 35 40 45 50
I
8,0 0,544 0,272 0,183 0,136 0,109 0,091 0,078 0,0680,0600,054
8,5 0,578 0,289 0,193 0,145 0,116 0,096 0,083 0,0720,0640,058
9 0,612 0,306 0,204 0,153 0,122 0,102 0,087 0,0770,0690,061
9 , 5 0, 646 0, 323 0,215 0, 162 0, 129 0, 108 0,092 0,08 I 0,072 0,065
10 0, 680 0, 340 0,227 0, 170 0, 136 0, 113 0,097 0,085 0,076 0,068
10,50,714 0,357 0,238 0,179 0,143 0,119 0,1020,0890,0790,071
II 0,748 0,374 0,249 0,187 0,150 0,125 0,107 0,0940,0830,075
11,5 0,782 0,391 0,261 0,196 0,156 0,130 0,112 0,0980,0870,018
12 0, 816 0,408 0,272 0, 204 0, 163 0, 136 0, 117 0, 102 0,091 0,082
v -= 1,8 KH/M'
1,5 0,108 0,054 0,036 0,027 0,022 0,018 0,015 0,0140,0120,011
2,0 0,144 0,072 0,048 0,036 0,029 0,024 0,021 0,0180,0160,014
2,5 0,180 0,090 0,060 0,045 0,036 0,030 0,026 0,0230,0200,018
3,0 0,216 0,108 0,072 0,054 0,043 0,036 0,031 0,0270,0240,022
3,5 0,252 0,126 0,084 0,063 0,050 0,042 0,037 0,0320,0280,025
4,0 0,288 0,144 0,096 0,072 0,0,53 0,048 0,041 0,0360,0320,029
4,5 0,324 0,162 0,108 0,081 0,061 0,054 0,046 0,041 0,0360,032
5,0 0,360 0,180 0,120 0,090 0,068 0,060 0,051 0,0450,0400,036
5,5 0,396 0,198 0,132 0,099 0,079 0,066 0,057 0,0500,0440,040
6,0 0,432 0,216 0,144 0, 108 0,086 0,072 0,062 0,0540,0480,043
6,5 0,468 0,234 0,156 0,117 0,094 0,078 0,067 0,0590,0520,047
7 ,0 0,504 0,252 0, 168 0, 126 0, I ° I 0,084 0,072 0, 063 ° ,056 0,050
7 ,5 0,540 0,270 0, 180 0, 135 0, 108 0,090 0,077 0,068 0,060 0, 054
8,0 0,576 0,288 0,192 0,144 0,115 0,096 0,082 0,0720,0640,058
8,5 0,612 0,306 0,204 0,153 0,122 0,102 0,087 0,0770,0680,061
9,0 0,648 0,324 0,216 0,162 0. t30 0,108 0,093 0,081 0,0720,065
9,5 0,684 0,342 0,228 0,171 0,137 0,114 0,095 0,0860,0760,068
10,0 0,720 0,360 0,240 0,180 0,144 0,120 0,103 0,0900,0800,072
10,5 0,756 0,378 0,252 0,189 0,151 0,126 0,108 0,0950,0840,075
11,0 0,192 0,396 0,264 0,198 0,158 0,132 0,113 0,0990,0880,079
11,5 0,828 0,414 0,276 0,217 0,165 0,138 0,118 0,1040,0920,083
12,0 0,864 0,432 0,288 0,226 0,172 0,144 0,123 0,1080,0960,086
",-1,9 KH/II'
-
1,5 0, I 14 0,057 0,038 0,029 0,023 0,019 0,016 0,014 0,013 0,011
2 0, 152 0,076 0,051 0,038 0,030 0,025 0,022 0,019 0,017 0,01
2,5 0,190 0,095 0,063 0,048 0,038 0,032 0,027 0,024 0,021 0,01
3 0,228 0,]14 0,076 0,057 0,046 0,038 0,033 0,029 0,025 0,02
3,5 0,266 0, 133 0,089 0,067 0,053 0,044 0,038 0,033 0,030 0,02
4 0,304 0, J 52 0,101 0,076 0,061 0,051 0,043 0,038 0,034 0,03
4,5 0,342 0,171 0,114 0,086 0., 068 0,0.57 0,049 0,043 0,038 ° 03
,
5
9
3
7
o
4
286
n po80A.3ICenue Ta6/l. 5
'\' == 1.9 KH/M s
(p - p ). Mna
....
-oS:
5
I 16 I 15
I I
20
25
30
35
40
45
50
5,0 0,380 0,190 0,127 0,095 0,076 0,063 0,054 0,0480,0420,038
5,5 0,418 0,209 0.139 0,105 0,084 0,070 0.060 0,0520,0460,042
() 0,456 0, 228 ° 152 0, 114 0,09 t I 0,076 0,065 0,057 ° ,051 0,046
{5,5 0,494 0,247 0,165 0,124 0,098 0,082 0,071 0,0620,0550,049
7 0,532 0,266 0,177 0,133 0,106 0,089 0,076 0,0670,0590,053
7 ,5 0,570 0,285 0, 190 0, 143 0, 114 0,095 0,081 0,071 0,063 ° ,057
8 0,608 0,304 0,203 0,152 0,122; 0,101 0,081 0,0760,0680,061
8,5 0, 645 0, 323 0,2 I 5 0, 162 0, 129 0, 108 0,092 0,081 0,072 0,065
9 0,684 0,342 0,228 0, 171 0, 137 0, 1 14 0,098 0.,086 0,076 0,068
9,5 0, 722 0, 36 I 0,241 0, 181 0, 144 0, 120 0, 103 0,090 0,080 0 , 072
10 0, 760 9, 380 0, 253 0, 190 0, 152 0, t 27 0, 109 0,095 0,084 0,076
10,5 0,798 0,399 0,266 0,200 0, 160 0, 133 0, 114 0, 100 0,089 0,080
11 0,836 0 , 418 0,279 0, 209 0, 167 0, 139 0, t 19> 0, 105 0,093 0,084
11 ,5 0, 874 0,437 0,291 0,219 0, 175 0, 146 0, 125 0, 109 0,097 0 , 087
t 2 0, 912 0,456 0,304 0,228 0, 182 0 , 152 0, 131 0, 1 14 0 , 101 0,091
,\,=2,0 KH!MI
1
I
1
1
1
1,5 0, 120 0,060 0,040 0,030 0,024 0,020 0,017 0,015 0,013 0,012
2,0 0, 160 0,080 0, Q53 '),040 0,032 0,027 0,023 0,020 0,018 0,016
2,5 0,200 0, 100 0,067 0,050 0,040 0,033 0,029 0,025 0,022 0,020
3,0 0,240 0, 120 0,080 0,060 0,048 0,040 0,034 0,030 0,027 0,024
3,5 0,280 0.140 0,093 0,070 0,056 0,047 0,040 0,035 0,031 0,028
4 0,320 0,160 0, 107 0,080 0,064 0,053 0,046 0,040 0,036 0,032
4,5 0,360 0.180 0,120 0,090 0,072 0,060 0,051 0,045 0,040 0,036
5 0,400 0,200 0, 133 0,100 0,080 0,067 0,057 0,050 0,044 0,040
5,5 0,440 0,220 0,147 0,110 0,088 0,073 0,063 0,055 0,049 0,044
.6 0,480 0,240 0,160 0, 120 0,096 0,080 0,069 0,060 0,053 0,048
6,5 0,520 0,260 0, 173 0, 130 0,104 0,087 0,074 0,065 0,058 0,052
7,0 0, -560 0,280 0, 187 0,140 0,112 0,Q93 0,080 0,070 0,062 0,056
7,5 0,600 0,300 0,200 0, 150 0, 120 0,100 0,086 0,075 0,067 0,060
8 0,640 0,320 0,213 0,160 0, 128 0, 107 0,091 0,080 0,071 0,064
8,5 0,680 0,340 0,227 0, 170 0, 136 0,113 0,097 0_,085 0,076 0,086
9 0,720 0,360 0240 0, 180 0, 144 0, 120 0, 103 0,090 0,080 0,072
,
9,5 0.760 0,380 0,253 0, 190 0, 152 o t 127 0. 109 0,095 0,084 0,076
° 0,800 0,400 0,267 0,200 0, 160 0, I 33 0,114 0, 100 0,089 0,080
0,5 0,840 0,420 0,280 0,210 0, 168 0,140 0,120 0, 105 0,093 0,084
I 0,880 0,440 0,293 0,220 0,176 0,147 0, 126 0,110 0,098 0,088
1,5 0,920 0,460 0,307 0,230 0, 184 0, 1 53 0,131 0,115 0, 102 0,092
2 0,960 0,480 0,320 0,240 0,192 o , 160 0, 137 0, 120 0, 107 0,096
287
cnHCOK J1HTEPATYPbl
1. fJla3ep C. H., WKOJlbHHK C. W. K paCqeTY 3aMKHYTbIX JIeHTOq-
HbiX npSlMoyrOJIbHOro OqepT8HHSl B nJIaHe <PYH.naMeHToB, pa60TalOJ..U.Hx
C OTpbIBOM nO.ll.OWBbI/ /OCHOB8HHSl, <PYH.naMeHTbI, MeX8HHK8 rpYHTOB.-
1980. -.N 6. PYKonHcb .nenoHHpyeTcH B UHI1HHC rOCCTpOSl CCCP.-
30 c.
2. fJla3ep C. H. <l>YH.n.8MeHTbI C TaBpoBoH no.n.oIllBoH, paooTalOJ..U.He
C qaCTHqHbIM OTpbIBOM no.n.olllBbI//OcHoBaHIIH, <PYH.naMeHTbI H MexaHH-
Ka rpYHToB. - 1981. - N2 6. PYKonHcb .n.enoHHpyeTcSl B UI1I1HI1C roc-
CTpOH CCCP. - C. 30.
3. fOJIb,lJ.WTeHH M. H., KywHep C. f., WeBIJeHKO M. H. P8CQeTbI
ocaAOK H npOQHOCTH OCHOB8HHH 3.n.aHHH H cooPYJKeHHH. - KHeB; By-
,zdBeJIbHHK) 1977. - C. 208.
4. fop6YHOB- nOCaJl,OB M. H., MaJlHKOBa T. A., COJlOMHH B. H.
P8CIJeT KOHCTPYKlI.HH Ha ynpyroM OCHOB8HHH. - M.: CTpOHH3.l1.aT,
1984. - C. 679.
5. EropoB K. E. K Bonpocy .n.e<popMaI1.HH OCHOB8HHH KOHeqHOH TOJI-
llJ.HHbI//Mex8HHK8 rPYHToB. Tp. HH-Ta HHI1 OCHOBaHHH H nO.ll.3eMHbIX
coopYJKeHHH, 1958. - BbIn. 34. - C. 5-33.
6. 3HHoBbeB A. B. Pacnpe.neJIeHHe HanpSl}l{eHHH B OCHOBaHHH KO-
HeIJHOM TOJIIIJ.HHbI no.n. KOJIblI.eBbIM <PYH.naMeHToM//Tp. HH-T8 Hl-II1
OCHOB8HHM H nO.ll.3eMHbIX cooPY}l{eHHM. - 1980. Bbin. 71 - C. 30-49.
7. 3HHoBbeB A. B., KHTaHKHHa 0_ B. 0 .ne<popM8lI.HH oCHoBaHHH
no.n. KOJIblI.eBbIMH <PYH.lI.aMeHTaMH cooPY}I{eHHH o8weHHoro THna//Tp.
HH-Ta HI1H OCHOBaHHH H no.n.3eMHblX cooPY}I{eHHH. - 1980.-
Bbln. 72. - C. 79-95. .
8. KeA3H A. PYKOBO.n.CTBO no MexaHHKe rPYHTOB. T. IV. npHMeHe-
HHe MexaHHKH B npaKTHKe CTpOHTeJIbCTB8. - M.: CTpOHH3.l1.aT, 1978.-
239 c.
9. KOHOBaJlOB n. A. OCHoBaHHSl H (jJYH.n.aMeHTbI peKoHcTpYHpyeMbix
3.l1.aHHM. - M.: CTPOHH3.n.8T, 1980. -- 134 c.
10. MaJlHKOBa T. A., Pbl6HH B. C. Onpe.n.eJIeHHe oca.nOK OCHOB8-
HHSl OT H8rpY3KH, p8cnpe.n.eJIeHHOH p8BHOMepHO -no nJIOJ..U.a.n.H KOJIbI1.all
HCCJIe.ll.OB8HHe H pa3p860TKa MeTO.ll.OB paCqeTa oCHoBaHHM H npOqHO-
CTH <pYH.n.8MeHToB C npHMeHeHHeM HeJIHHeHHblx TeopHH .lI.e<p0pMHpOB8-
HHSl. - Hnl1) 1984. - C. 116-122.
11. MeToAHKa 06CJIe.ll.oB8HHH H npoeKTHpOBaHHSl OCHOB8HHH H (jJYH-
,n8MeHTOB npH KanHT8JIbHOM peMOHTe, peKoHcTpYKlI.HH H Ha.ncTpoHKe
3.n.aHHH. - M.: CTpoHH3.n.aT, 1972. - 111 c.
12. nOCo6He no npoeKTHpOBaHHIO oCHoBaHHH 3.n.aHHH H cooPY}I{e-
HHM. HI1110Cn HM. H. M. repceBaHoB8. - M.: 1986. - 415 c.
13. PYKOBOJl.CTBO no npoeKTHpOB8HHIO OCHOBaHHH H (jJYH.naMeHToB
Ha BetJHoMep3JIblX rpYHTax. - M.: CTpoHH3.n.aT, 1980. - 303 c.
14. PYKOBOJl,CTBO no npoeKTHpoBaHHIO ct>YH.lI.8MeHTOB Ha eCTeCTBeH-
HOM OCHOB8HHH no.n. KOJIOHHbI 3.n.aHHH H coopYJKeHHH npOMblWJIeHHbIX
3,n8HHH. - M.: 1978. - 109 c.
15. Pbl6HH B. C., 3aXapQeHKO B. E. BepTHKaJIbHbIe HanpH}I{eHHSI
B TOJIllI.e OCHOB8HHSl OT H8rpY3KH, pacnpe.ll.eJIeHHOM paBHOMepHO no nJIO-
w.a.n.H MHoroyrOJIbHHKaIIOcHoBaHHSl, <PYH.lI.8MeHTbl H MexaHHKa rpYH-
TOB. - 1981. - Ng 5. - C. 26-27.
16. Pbl6HH B. C. BepTHKaJIbHbIe HanpSl}l{eHHH B TOJIw.e OCHOB8HHH
OT HarpY3KH, pacnpe.n.eJIeHHOH p8BHOMepHO no nJIoIIJ.8AH paBHooe.n.peH-
Horo TpeyrOJIbHHKa//CTpOHTeJIbHOe npoeKTHpOBaHHe npoMbIw eHHbIx
npe.ll.npHHTHH. - Sbln. 2. - 1982. - C. 24-29.
2BB
17. Pbl6HH B. C. BbIqHCJIeHHe oca.nOK xapaKTepHblX TOqeK n0.(10-
WBbI npHMoyrOJIbHOrO <PYH.lI.aMeHT8//0cHoBaHHH, cpYH.n.aMeHTbI " Mexa..
HHKa rpYHToB. - 1983. - N2 1. - C. 18-19.
18. Pbl6HH B. C. K onpe.ll.eJIeHHIO oca.n.OK CBaHHbIX 4>YH.lI.aMeHToB.
SKcnpecc-HH<popMalI.HH. CTpOHTeJIbCTBO H 8pxHTeKTYP8/lnpOMblllIneH-
HbIe KOMnneKCbI; 3.n.aHHH H coopY)KeHHH. - CepHH 29.53. BbID. 2.-
1983. - C. 28-32.
19. CHH n 2.02.0 t -83. OCHoBaHHH 3J1aHHH H coopY}I{eHHH:.
20. CHHn 11-16-76. OCHoBaHHH rH.n.poTexHHtJecKHx coopY>KeHHH.
21. CHnn 2.02.03-85. CB8HHbIe (jJYH.lI.8MeHTbi.
22. COTHHKOB C. H., CHMarHH B. f., BepmHHHH B. n_ npoeKTH-
pOB8HHe H B03Be.ll.eHHe (jJYH.n.8MeHToB BOJIH3M cYIIJ.eCTBYIOllI.HX coopY-
>KeHHH:. - M.: CTpoHH3,naT, 1986. - 93 c.
23. CnpaBOQHHK npoeKTHpOBJ..U.HKa. OCHoBaHHH, <PYH.n.aMeHTbl H
nO,ll3eMHble coopY>KeHHH. - M.: CTpOHH3.n.aT, 1985. - 450 c.
24.. llbiTOBHQ H. A. MexaHHK8 rpYHToB. - M.: BbIC1lI8H llIKOJI8,
1983. - 288 c.
25. UlBeu B. 6., $eKJlHH B. H., fHH36ypr J1. K. Y CHJIeHHe H pe';
KOHCTPYKlI.HH <l>YH.n.aM HToB. - M.: CTpOHH3.l1.aT, 1985. - 203 c.
26. Boussinesq J. Application des potentiels a l'etude de l'equilibre
et du mouvement des solides elastiques. Paris - 1885.
OrJ1ABJ1EHHE
npe.nHCJlOBHe . . . 3
OCHoRHhle YCJIOBHbIe 0603H8QeHHH H r OMeTpHqeCKHe xapaKTe-
pHCTHKH no.n.OllIB q>YH.naMeHToB 4
r JlaBa t. OOIl{He CBe,n.eHHH. . . . . . . .. 5
r JlaBa 2. 4»YH,n.aMeHTbI, pa60TalOIl{He C OrpaHH'IeHHblMH KpaeBbI-
MH ,n.aBJleHHJlMH Ha OCHOBaHHe npH ,lI.eHCTBHH MOMeHTa B 0)1.-
O II
H M HanpaBJleHHH . . . . . .
2.1. 06ll1.He paCtletHbIe CPOPMY JlbI . II
2.2. lip HMoyrOJIbHhle <pYH aMeHTbI 14
2.3. <I>YH.n.aMeHTbI C cHMMeTpHqHOM .n.BYTaBpoBoH no.n.ollIBoH 15
2.4. <I>YH.n.aMeHTbl C 3aMKHYToH JIeHTOQHOH npsrMoyroJlbHoro
OqepTaHHH B nJI8He no.u.olIIBoH . 18
2.5. <I>YH.n.aMeHTbI C KpecToBoH no.n.olllBoH. . .. 21
2.6. <I>YH.u.aMeHTbI C T8BpOBOH II r -06p83HOH no.nollIBc'H . 24
2.7. <I>YH.n.8MeHTbl C KOJIbu.eBoH II Kpyr JlOM no.n.ollIBoH . 32
2.B. <I>YHJ1aMeHTbI C rnecTHyrOJIbHOH no.n.ollIBoH . 35
2.9. cI>YH.n.8MeHTbI C BocbMHyrOJIbHOH nO,lJ.OllIBoH.. 38
2.10. <I>YH.n.aMeHTbI C HeJ10J1HblM KOHTaKTOM no.u.OllIBbI H oc-
'HOBaHHH . . . . . . . _ . .. . 41
r JlaBa 3. YH.lI.aMeHTbI, pa60TalOIl{He C OrpaHHQeHHblMH KpaeBb1-
MH ,lI.aBJleHHHMH Ha oCHoBaHHe, npH ,lI.eHCTBHH MOMeHTOB B ,lI.BYX
B3aHMHO nepneH,lI.HKYJlJlpHblX HanpaBJleHHHX 46
3.1. PaCqeTHbIe CPOPMYJlbI . . 46
3.2. npHMoyro.nbHbIe QJYH.n.8MeH:Fbl . . . 49
3.3. <l>YH.n.aMeHTbI C cHMMeTpHqHOH .n.BYTaBpoBoH no.n.ollIBoA 52
3.4. <I>YH.n.8MeHTbI C 3aMKHYToH JIeHTOqHOH npHMoyroJlbHoro
OqepTaHHH B nJI8He nOJ1011IBoH. . 56
3.5. <I>YH.n.8MeHTbI C KpecToBoH no.n.olllBoH. . .. 59
3.6. C1JYH.n.aMeHTbI C TaBpoBoM H r -06p83HOH no.n.OIIIBoH . 67
3.7. <I>YH.n8MeHTbI C rnecTHyrOJIbHOH no.n.ollIBoH . 80
3.8. <l>YH.n.aMeHTbI C BocbMHyroJlbHoH no.n.oIlIBoH . 83
r JlaBa 4. 4»YH,n.aMeHTbI, paOOTalOIl{He C OTpblBOM nO,ll.OWBbI "PH
,lI.eHCTBHH MOMeHTa B O,ll.HOM HanpaBJleHHH. . . . . . . 89
4.1. npe.n.noCbIJIKH, npHHHTbIe npH BblBo.n.e paCqeTHblX QJOPMY n
4.2. np5IMoyrOJIbHble cpYH.n8MeHTbI . . . . . . . 91
4.3. C1JYH.lI.aMeHTbl C cHMMeTpHqHOH .n.BYT8BPOBOH no.n.ollIBoH 92
4.4. <I>YHJJ.aMeHTbI C 38MKHYTOH JleHTOqHOH npHMoyrOJIbHOrO
OttepT8HHH' B nJI8He no.n.ollIBoH. . 102
4.5. <I>YH.u.aMeHTbI C KpecToBoH no.n.ollIBoH. . . .. 112
4.6. <I>YHJJ.aMeHTbI C TaBpoBoH H r -06p83HOH no.n.oIlIBoH . 115
4.7. <I>YH.n.aMeHTbI C KOJIbu.eBoH H Kpyr JIOR nO,lJ.owBoA. . . 119
4.8. <I>YH.n.aMeHTbI c npaBHJIbHOH MHoroyrOJIbHOH no.noIlIBoft 126
r JlaBa 5. 4»YHAaMeHTb1, pa60TalOII.\He C OTpblBOM nOAomBLI, npH
.u.eHcTBHH MOMeHTOB B ,n.BYX B3aHMHO nepneHAHKYJlHpHblX Ha-
npaBJleHHHX. . . . . . . . . . . . . . . 134
5.1. npe.nnoCbIJIKH, npHHHTble npH BblBo.u.e paCqeTHblX q>OPMYJI 134
5.2. <I>YH.n.8MeHTbl C npSIMoyrOJIbHOH no.n.owBoH.. 135
5.3. npH6JIHJKeHHblH MeTO.n. p8CqeT8 4>YH.n.8MeHToB . 141
rJl8Ba 6. Onpe,n.eneHHe Oca,ll.OK OCHOBaHHH H BepTHKaJlbHYX Ha-
npH1KeHHH B TOJlll{e rPYHToBoro OCHOBaHHH. . . . . . 153
6. . BepTHK8JIbHble HanpH}I{eHHfI 8 TOnlll.e OCHOB8HHH OT H8-
rpY3KH, p8BHOMepHO pacnpe.n.eJIeHHOH no nJIOma.n.H MHO-
royrOJIbHHK8 . 154
290
6.2. Onpe.u.eJIeHHe oca.u.OK OCHOBaHHH OT HarpY3KH, pacnpe-
.neJleHHOH no nJIOllI.8.nH MHoroyroJlbHHK8. . . . .
6.3. BepTHK8JIbHble HanpsDKeHHH' B TOJIllI.e OCHOBaHHH OT H8-
rpY3KH, p8cnpe.u.eJleHHoH paBHoMepHo no nJIOm8.u.H paB-
H06e.n.peHHoro TpeyrOJlbHHKa ... ..
6.4. Oca.n.KH OCHOB8HHH OT HarpY3KH, pacnpe,aeJIeHHOA pas-
HOMepHo no nJIOllI.a.n.H paBH06e.n.peHHoro TpeyrOJIbHHKa
6.5. K onpe.n.eJleHHIO oca.n.OK CB8HHbIX <pYH.n.aMeHToB .
6.6. K onpe.n.eJIeHHIO oca.n.OK OCHOBaHHH HaCbIneH .
.6.7. BblQHCJIeHHe oca.n.OK xap8KTepHblx TOqeK no.n.OllIBbI
np HMoyrOJIbHOrO <PYH.n.aMeHTa. .....
6.8. Onpe.n.eJIeHHe OC8.n.OK OCHOBaHHH OT HarpY3KH, pacnpe-
.n.e.neHHoH paBHOMepHO no nJIOma.n.H KOJIbu.a. . . .
6.9. Onpe.ll.eJIeHHe oca.n.OK B pa3JIHtlHbIX TOQKaX OCHosaHHH
OT paBHoMepHoH HarpY3KH, pacnpe.u.eJleHHOH no nnow.a-
H npHMoyrOJIbHHK8
6.10. K onpe.n.eJleHIHO OC8.n.OK oCHoBaHHH OT nOJIOCOSOH
H8rpY3KH. . . . . . . . . . . . .
6.11. K onpe.n.e.neHHIO OC8.n.OK OCHOB8HHH OT ocecHMMeTpHQ-
HbIX p-acnpe.n.eneHHblx HarpY30K
n pU/I,03ICeHUe 1 .
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npOH3BOACTBeHHoe H3AaHBe
PYSHH BAnEPHR CEMEHOBHq
npOEKTHPOBAHHE YH AMEHTOB PEKOHCTPYHPYEMWX 3AAHHA
Mn. pe.u.aKTop HI D. BoJt/CoBa
TeXHHqeCKHA pe.u.aKTop M. B.. naBJtoBa
KoppeKTop H. B. MeoBeob
HB 16 4848
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BYMara THO. Nt 2. fapHHTypa cJIuTepaTypHaSl. .
Ycn. nell. Jr. 15,54. Ycn. Kp.-OTT. 15,54. YQ.-H3J1,. n. 16,21.
Ha.n. Ng A-VI-2542. 3aKa3 2383.
opMaT 84X 108 1 / 32 .
nellaTb BLiCOKaSl.
THpa>K 20700 3K3.
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CTpoAH3J1,aT. 101442. MocKBa, KanSleBCKaSl. 23a.
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