国立中央大学土木工程学系硕士论文

DiscreteLagrangianMethod()(ü)()()()()()()::DiscreteLagrangianMethodoþ8832206190710IIIAbstractConventionaldesignofpilegroupsisbasedonthetrial-and-errorprocedures.Althoughthedesignresultscansatisfysafetyrequirements,itisnotaminimumcostdesign.ThepurposeofthisstudyistoapplytheDiscreteLagrangianMethodforfindingtheminimumcostdesignofprecastconcretepilegroupswithdiscretedesignvariables.Theobjectivefunctionoftheproblemisthetotalcostofapilegroup,whichincludesthecostsofsoilexcavation,cap,andpiles.Thedesignvariableswillbethediameterofpiles,lengthofpiles,distancesbetweenpilesanddimensionsofcap.Thestressanddeformationconstrainsofthepilegroupdesignareformulatedaccordingtothecodeprovisions.Sizeconstrains,suchasthelengthofpiles,thediameterofpiles,distancebetweenpilesarealsoconsideredintheoptimizationformulation.SeveralnumericalexamplesforminimumcostdesignofpilegroupsareusedtoillustratetheefficiencyoftheDiscreteLagrangianMethod.Theinfluencesofdesignvariablesonthecostofpilegroupsarealsodiscussedinthisreport.Keywords:PileGroups,DiscreteDesignVariables,DiscreteLagrangianMethod,MinimumCostDesignIIIIIIIIIVIVII………………………………………………………………………..11.1………………………………………………………...………….11.2………………………………………………………..…………..21.3………………………………………………...………….4……………………………………………….……...………….72.1LagrangianMethod……………………………………..………….72.2DLM(DiscreteLagrangianMethod)………………………………………102.2.1DMPD………………………….………………112.2.2……………….………………122.2.3………………………………….…………162.2.4…………………………………….………………182.3Lagrange…………………………………………..…………19………………………………...…………233.1………………………………………………..…………233.1.1………………………………………………………233.1.2………………………………………………………243.1.3………………………………….…………………………253.2………………………………………………..…………263.2.1……………………………………………….…………26IV3.2.2……………………………………………….…………273.2.3……………………………………………….…………293.2.4…………………………………………………………303.2.5…………………………………………….…………313.2.6………………………………………………….…………343.2.7……………………………………………….…………34………………………………………..………….354.1Lagrange……………………………………..…………354.2…………………………………………..…………364.2.1……………………………………………………374.2.2………………………………………………394.2.3……………………………………………………404.2.4………………………………………………414.2.5………………………………………………424.2.6……………………………………………………424.2.7…………………………………………………43…………………………………………………..…………445.1……………………………………………………….…….…………445.2……………………………………………………….…….…………45……………………………………………………….……...…………47……………………………………………………….……...………….…78V3-1………………….504-1cLagrange(Mx=My=3000t-m,P=1000t,Hy=Hx=300t)……………………………514-2cLagrange(Mx=My=6000t-m,P=2000t,Hy=Hx=300t)…………………………...514-3Lagrange…………….524-4………………………………………………………...524-5a………………………………………..534-5b……………………………………..544-6(Mx=My=3000t-m,P=1000t,Hy=Hx=100t)…………………………..554-7(Mx=My=3000t-m,P=3000t,Hy=Hx=100t)…………………………..564-8(Mx=My=3000t-m,P=1000t,Hy=Hx=300t)…………………………..574-9(Mx=My=8000t-m,P=1000t,Hy=Hx=100t)…………………………..584-10(,Mx=My=3000t-m,P=1000t,Hy=Hx=100t)……..594-11(,Mx=My=3000t-m,P=3000t,Hy=Hx=100t)……..604-12(,Mx=My=3000t-m,P=1000t,Hy=Hx=100t)……..61VI4-13(,Mx=My=3000t-m,P=3000t,Hy=Hx=100t)……..624-14…………..634-15…………………..644-16……………………………………………..654-17…………………………………….65…………………………………………….80………………………………………………………………………..81()………………………………………………………………………..82VII1-1………………………………………………………….662-1Lagragian……………………………………672-2…………………………………………………………….672-3XCLM……………………………………………………..682-4DMPD……………………………………………………………682-5DLM………………………………………………………692-6…………………………………………………………….703-1………………………………………………………………713-2…………………………………………724-1cLagrange…………..734-2Lagrange…………..744-3Lagrange………..754-4Lagrange………..754-5…………………………………………………………764-6………………………………………………………764-730…………………………………………..7711.1.21.2…:1)3:2):;3):GRG(GeneralizedReducedGradientMethod)LagrangeMultiplierMethod…(GRDMethod)(LagrangeMultiplierMethodPenaltyFunctionMethods)[9]4MultiplierMethod)[15]Lagrangeil1998DLM(screteLagrangeMutiplier)51.3:1)p-y;2);3)Focht-Koch;4)Hybird;5)p-y[]672.1LagrangianMethodminimizef(X)subjecttohi(X)=0,i=1~p(2-1)gj(X)0,j=1~nf(X)(objectionfunction)g(X)(inequalityconstrainfunction)h(X)(equalityconstrainfunction)X=(x1x2….xm)T(designvariables)0)X(gj≤(slackvariable)2js0s)X(g2jj=+(2-2):)X(f0)X(hi=i=1~m(2-3)0s)X(g2jj=+j=1~nλμ)s,,,x(Lμλ8∑∑==+++=n1j2jjjip1ii)s)X(g(ì)X(hë)X(f)s,ì,ë,X(L)s)X(g()X(h)X(f2TT+μ+λ+=(2-4))s,,,X(LμλX**λμ*:0x)X(gx)X(hx)X(fxLn1jk**jp1ik*i*ik*k=∂∂μ+∂∂λ+∂∂=∂∂∑∑==(2-5)0)X(hL*ii==λ∂∂i=1~p(2-6)0s)X(gL2j*jj=+=μ∂∂j=1~n(2-7)0ssLj*jj=μ=∂∂j=1~n(2-8)0*j≥μj=1~n(2-9)Kuhn-Tuckernecessaryconditionsfirst-ordernecessaryconditionsKuhn-Tucker0x)X(gx)X(hx)X(fxLn1jk**jp1ik*i*ik*k=∂∂μ+∂∂λ+∂∂=∂∂∑∑==(2-10)0)X(h*i=i=1~p(2-11)0)X(g*j≤j=1~n(2-12)0*j≥μj=1~n(2-13)9(2-5)*Xf(x)()X(f*x∇)()X(g*x∇)2-1:∑∑==∂∂+∂∂λ=∂∂-n1jk**jp1ik*i*ik*x)X(gux)X(hx)X(f(2-14)2-1(first-ordersearchmethod)[15],(Newton’smethod)(modifiedNewton’smethods)(quasi-Newton’smethods)[19]sequentialquadraticprogrammingmethod[20]Tkkxkk1k),x(Lxxλ∇α-=+(2-15))x(hkkk1kα+λ=λ+(2-16)kα(step-size)102.2DLM(DiscreteLagrangianMethod)LagrangianmethodLagrangianWuDiscreteLagrangianmethod(DLM)[2,3,4,5]SAT(Satisfiability)[6,7,8]DLMminimizef(X)(2-17)S.T.hi(X)=0i=1~pX=(x1,x2,….xm)T))X(h(H)X(f),X(LTdλ+=λ(2-18)H(X)H(X)=0X=00X0)X(H=⇔=Hh(X)DLM111.N(X)X(theneighborhoodofpointX)2-22