第三章MATLAB有限元分析與應(yīng)用§3-1彈簧元結(jié)構(gòu)分析編程及軟件應(yīng)用§3-2線性桿元§3-3二次桿元§3-4平面桁架元§3-5空間桁架元§3-6梁元8/1/20231第三章MATLAB有限元分析與應(yīng)用§3-1彈簧元結(jié)構(gòu)分§3-1彈簧元結(jié)構(gòu)分析編程及軟件應(yīng)用1、有限元方法的步驟：離散化域形成單剛矩陣集成整體剛度矩陣引入邊界條件求解方程后處理8/1/20232§3-1彈簧元結(jié)構(gòu)分析編程及軟件應(yīng)用1、有限元結(jié)構(gòu)分析編程及軟件應(yīng)用2、基本方程§3-1彈簧元彈簧元是總體和局部坐標(biāo)一致的一維有限單元每個(gè)彈簧元有兩個(gè)節(jié)點(diǎn)（node）單剛矩陣為：總剛矩陣：結(jié)構(gòu)方程：單元節(jié)點(diǎn)力：8/1/20233結(jié)構(gòu)分析編程及軟件應(yīng)用2、基本方程§3-1彈簧元結(jié)構(gòu)分析編程及軟件應(yīng)用3、MATLAB函數(shù)編寫§3-1彈簧元%SpringElementStiffnessThisfunctionreturnstheelementstiffness%matrixforaspringwithstiffnessk.%Thesizeoftheelementstiffnessmatrixis2x2.3.1單元?jiǎng)偠染仃嚨男纬蓎=[k-k;-kk];functiony=SpringElementStiffness(k)8/1/20234結(jié)構(gòu)分析編程及軟件應(yīng)用3、MATLAB函數(shù)編寫§3-1結(jié)構(gòu)分析編程及軟件應(yīng)用3、MATLAB函數(shù)編寫§3-1彈簧元%SpringAssembleThisfunctionassemblestheelementstiffness%matrixkofthespringwithnodesiandjintothe%globalstiffnessmatrixK.%ThisfunctionreturnstheglobalstiffnessmatrixK%aftertheelementstiffnessmatrixkisassembled.3.2整體剛度矩陣的形成K(i,i)=K(i,i)+k(1,1);K(i,j)=K(i,j)+k(1,2);K(j,i)=K(j,i)+k(2,1);K(j,j)=K(j,j)+k(2,2);y=K;functiony=SpringAssemble(K,k,i,j)8/1/20235結(jié)構(gòu)分析編程及軟件應(yīng)用3、MATLAB函數(shù)編寫§3-1結(jié)構(gòu)分析編程及軟件應(yīng)用3、MATLAB函數(shù)編寫§3-1彈簧元%SpringElementForcesThisfunctionreturnstheelementnodalforce%vectorgiventheelementstiffnessmatrixk%andtheelementnodaldisplacementvectoru.3.3節(jié)點(diǎn)載荷計(jì)算y=k*u;functiony=SpringElementForces(k,u)8/1/20236結(jié)構(gòu)分析編程及軟件應(yīng)用3、MATLAB函數(shù)編寫§3-1結(jié)構(gòu)分析編程及軟件應(yīng)用4、實(shí)例計(jì)算分析應(yīng)用§3-1彈簧元如圖所示二彈簧元結(jié)構(gòu)，假定k1=100kN/m，k2=200kN/m，P=15kN。求：系統(tǒng)的整體剛度矩陣；節(jié)點(diǎn)2、3的位移；節(jié)點(diǎn)1的支反力；每個(gè)彈簧的內(nèi)力解：步驟1：離散化域8/1/20237結(jié)構(gòu)分析編程及軟件應(yīng)用4、實(shí)例計(jì)算分析應(yīng)用§3-1結(jié)構(gòu)分析編程及軟件應(yīng)用4、實(shí)例計(jì)算分析應(yīng)用§3-1彈簧元步驟2：形成單元?jiǎng)偠染仃噆1=SpringElementStiffness(100);k1=

100-100-100100k2=SpringElementStiffness(200)；k2=

200-200 -200200調(diào)用functiony=SpringElementStiffness(k)函數(shù)8/1/20238結(jié)構(gòu)分析編程及軟件應(yīng)用4、實(shí)例計(jì)算分析應(yīng)用§3-1結(jié)構(gòu)分析編程及軟件應(yīng)用4、實(shí)例計(jì)算分析應(yīng)用§3-1彈簧元步驟3：集成整體剛度矩陣調(diào)用functiony=SpringAssemble(K,k,i,j)函數(shù)n=3;K=zeros(n,n);K=SpringAssemble(K,k1,1,2)K=000000000K=SpringAssemble(K,k2,2,3)K=100-1000-1001000000K=100-1000-100300-2000-2002008/1/20239結(jié)構(gòu)分析編程及軟件應(yīng)用4、實(shí)例計(jì)算分析應(yīng)用§3-1結(jié)構(gòu)分析編程及軟件應(yīng)用4、實(shí)例計(jì)算分析應(yīng)用§3-1彈簧元步驟4：引入邊界條件已知邊界條件：8/1/202310結(jié)構(gòu)分析編程及軟件應(yīng)用4、實(shí)例計(jì)算分析應(yīng)用§3-1結(jié)構(gòu)分析編程及軟件應(yīng)用5、實(shí)例計(jì)算分析應(yīng)用§3-1彈簧元步驟5：解方程U=zeros(2,1);F=[0;15];K=K(2:3,2:3);U=K\FU=inv(K)*FK(1,:)=[];K(:,1)=[];U=0.15000.22508/1/202311結(jié)構(gòu)分析編程及軟件應(yīng)用5、實(shí)例計(jì)算分析應(yīng)用§3-1結(jié)構(gòu)分析編程及軟件應(yīng)用5、實(shí)例計(jì)算分析應(yīng)用§2-1彈簧元步驟6：后處理U=[0;U]U=00.15000.2250F=K*UF=-15.00000.000015.0000u1=U(1:2);f1=SpringElementForces(k1,u1);f1=-15.000015.0000u2=U(2:3);f2=SpringElementForces(k2,u2);f2=-15.000015.00008/1/202312結(jié)構(gòu)分析編程及軟件應(yīng)用5、實(shí)例計(jì)算分析應(yīng)用§2-1結(jié)構(gòu)分析編程及軟件應(yīng)用5、實(shí)例計(jì)算分析應(yīng)用§3-1彈簧元k1=SpringElementStiffness(100);k2=SpringElementStiffness(200);n=3;K=zeros(n,n);K=SpringAssemble(K,k1,1,2);K=SpringAssemble(K,k2,2,3);U=zeros(2,1);F=[0;15];K=K(2:3,2:3);KK=K;U=K\FU=[0;U];F=K*U;u1=U(1:2);f1=SpringElementForces(k1,u1)u2=U(2:3);f2=SpringElementForces(k2,u2)8/1/202313結(jié)構(gòu)分析編程及軟件應(yīng)用5、實(shí)例計(jì)算分析應(yīng)用§3-1結(jié)構(gòu)分析編程及軟件應(yīng)用1、基本方程§3-2線性桿元線性桿元也是總體和局部坐標(biāo)一致的一維有限單元，用線性函數(shù)描述每個(gè)線性桿元有兩個(gè)節(jié)點(diǎn)（node）單剛矩陣為：總剛矩陣：結(jié)構(gòu)方程：單元節(jié)點(diǎn)力：8/1/202314結(jié)構(gòu)分析編程及軟件應(yīng)用1、基本方程§3-2線性桿元結(jié)構(gòu)分析編程及軟件應(yīng)用2、MATLAB函數(shù)編寫%LinearBarElementStiffnessThisfunctionreturnstheelement%stiffnessmatrixforalinearbarwith%modulusofelasticityE,cross-sectional%areaA,andlengthL.Thesizeofthe%elementstiffnessmatrixis2x2.2.1單元?jiǎng)偠染仃嚨男纬蓎=[E*A/L-E*A/L;-E*A/LE*A/L];functiony=LinearBarElementStiffness(E,A,L)§3-2線性桿元8/1/202315結(jié)構(gòu)分析編程及軟件應(yīng)用2、MATLAB函數(shù)編寫%Linear結(jié)構(gòu)分析編程及軟件應(yīng)用2、MATLAB函數(shù)編寫%LinearBarAssembleThisfunctionassemblestheelementstiffness%matrixkofthelinearbarwithnodesiandj%intotheglobalstiffnessmatrixK.%Thisfunctionreturnstheglobalstiffness%matrixKaftertheelementstiffnessmatrix%kisassembled.2.2整體剛度矩陣的形成K(i,i)=K(i,i)+k(1,1);K(i,j)=K(i,j)+k(1,2);K(j,i)=K(j,i)+k(2,1);K(j,j)=K(j,j)+k(2,2);y=K;functiony=LinearBarAssemble(K,k,i,j)§3-2線性桿元8/1/202316結(jié)構(gòu)分析編程及軟件應(yīng)用2、MATLAB函數(shù)編寫%Linear結(jié)構(gòu)分析編程及軟件應(yīng)用2、MATLAB函數(shù)編寫%LinearBarElementForcesThisfunctionreturnstheelementnodal%forcevectorgiventheelementstiffness%matrixkandtheelementnodal%displacementvectoru.2.3節(jié)點(diǎn)載荷計(jì)算y=k*u;functiony=LinearBarElementForces(k,u)§3-2線性桿元8/1/202317結(jié)構(gòu)分析編程及軟件應(yīng)用2、MATLAB函數(shù)編寫%Linear結(jié)構(gòu)分析編程及軟件應(yīng)用2、MATLAB函數(shù)編寫%LinearBarElementStressesThisfunctionreturnstheelementnodal%stressvectorgiventheelementstiffness%matrixk,theelementnodaldisplacement%vectoru,andthecross-sectionalareaA.2.4節(jié)點(diǎn)應(yīng)力計(jì)算y=k*u/A;functiony=LinearBarElementStresses(k,u,A)§3-2線性桿元8/1/202318結(jié)構(gòu)分析編程及軟件應(yīng)用2、MATLAB函數(shù)編寫%Linear結(jié)構(gòu)分析編程及軟件應(yīng)用3、實(shí)例計(jì)算分析應(yīng)用如圖所示二線性桿元結(jié)構(gòu)，假定E=210MPa，A=0.003m^2，P=10kN，節(jié)點(diǎn)3的右位移為0.002m。求：系統(tǒng)的整體剛度矩陣；節(jié)點(diǎn)2的位移；節(jié)點(diǎn)1、3的支反力；每個(gè)桿件的應(yīng)力解：步驟1：離散化域§3-2線性桿元8/1/202319結(jié)構(gòu)分析編程及軟件應(yīng)用3、實(shí)例計(jì)算分析應(yīng)用如圖所示二線性桿元結(jié)構(gòu)分析編程及軟件應(yīng)用3、實(shí)例計(jì)算分析應(yīng)用步驟2：形成單元?jiǎng)偠染仃噆1=LinearBarElementStiffness(E,A,L1)k2=LinearBarElementStiffness(E,A,L2)調(diào)用functiony=LinearBarElementStiffness(E,A,L)函數(shù)§3-2線性桿元8/1/202320結(jié)構(gòu)分析編程及軟件應(yīng)用3、實(shí)例計(jì)算分析應(yīng)用步驟2：形成單元?jiǎng)偨Y(jié)構(gòu)分析編程及軟件應(yīng)用3、實(shí)例計(jì)算分析應(yīng)用步驟3：集成整體剛度矩陣調(diào)用functiony=LinearBarAssemble(K,k,i,j)函數(shù)n=3;K=zeros(n,n)K=LinearBarAssemble(K,k1,1,2)K=000000000K=LinearBarAssemble(K,k2,2,3)§3-2線性桿元8/1/202321結(jié)構(gòu)分析編程及軟件應(yīng)用3、實(shí)例計(jì)算分析應(yīng)用步驟3：集成整體剛結(jié)構(gòu)分析編程及軟件應(yīng)用3、實(shí)例計(jì)算分析應(yīng)用步驟4：引入邊界條件已知邊界條件：§3-2線性桿元8/1/202322結(jié)構(gòu)分析編程及軟件應(yīng)用3、實(shí)例計(jì)算分析應(yīng)用步驟4：引入邊界條結(jié)構(gòu)分析編程及軟件應(yīng)用3、實(shí)例計(jì)算分析應(yīng)用步驟5：解方程U=zeros(1,1);U3=0.002F=[-10];K=K(2,2)

105000K0=K(2,3);-630000U=K\(F-K0*U3)U=0.0012§3-2線性桿元8/1/202323結(jié)構(gòu)分析編程及軟件應(yīng)用3、實(shí)例計(jì)算分析應(yīng)用步驟5：解方程U=結(jié)構(gòu)分析編程及軟件應(yīng)用3、實(shí)例計(jì)算分析應(yīng)用步驟6：后處理U=[0;U;0.002]U=00.00120.0002F=K*UF=-500.0000-10.0000510.0000u1=U(1:2);f1=LinearBarElementForces(k1,u1)sigma1=LinearBarElementStresses(k1,u1,A)u2=U(2:3);f2=LinearBarElementForces(k2,u2)sigma2=LinearBarElementStresses(k2,u2,A)§3-2線性桿元8/1/202324結(jié)構(gòu)分析編程及軟件應(yīng)用3、實(shí)例計(jì)算分析應(yīng)用步驟6：后處理U=結(jié)構(gòu)分析編程及軟件應(yīng)用3、實(shí)例計(jì)算分析應(yīng)用E=210E6;A=0.003;L1=1.5;L2=1;k1=LinearBarElementStiffness(E,A,L1);k2=LinearBarElementStiffness(E,A,L2);n=3;K=zeros(n,n);K=LinearBarAssemble(K,k1,1,2);K=LinearBarAssemble(K,k2,2,3);U=zeros(1,1);U3=0.002;F=[-10];§3-2線性桿元KK=K;K=K(2,2);K0=K(2,3);U=K\(F-K0*U3);U=[0;U;U3];F=KK*Uu1=U(1:2);f1=LinearBarElementForces(k1,u1)sigma1=LinearBarElementStresses(k1,u1,A)u2=U(2:3);f2=LinearBarElementForces(k2,u2)sigma2=LinearBarElementStresses(k2,u2,A)8/1/202325結(jié)構(gòu)分析編程及軟件應(yīng)用3、實(shí)例計(jì)算分析應(yīng)用E=210E6;§結(jié)構(gòu)分析編程及軟件應(yīng)用1、基本方程§3-3二次桿元二次桿元也是總體和局部坐標(biāo)一致的一維有限單元，用二次方程描述每個(gè)線性桿元有三個(gè)節(jié)點(diǎn)（node）單剛矩陣為：總剛矩陣：結(jié)構(gòu)方程：單元節(jié)點(diǎn)力：8/1/202326結(jié)構(gòu)分析編程及軟件應(yīng)用1、基本方程§3-3二次桿元結(jié)構(gòu)分析編程及軟件應(yīng)用2、MATLAB函數(shù)編寫%QuadraticBarElementStiffnessThisfunctionreturnstheelement%stiffnessmatrixforaquadraticbar%withmodulusofelasticityE,%cross-sectionalareaA,andlengthL.%Thesizeoftheelementstiffness%matrixis3x3.2.1單元?jiǎng)偠染仃嚨男纬蓎=E*A/(3*L)*[71-8;17-8;-8-816];functiony=QuadraticBarElementStiffness(E,A,L)§3-3二次桿元8/1/202327結(jié)構(gòu)分析編程及軟件應(yīng)用2、MATLAB函數(shù)編寫%Quadra結(jié)構(gòu)分析編程及軟件應(yīng)用2、MATLAB函數(shù)編寫%QuadraticBarAssembleThisfunctionassemblestheelementstiffness%matrixkofthequadraticbarwithnodesi,j%andmintotheglobalstiffnessmatrixK.%Thisfunctionreturnstheglobalstiffness%matrixKaftertheelementstiffnessmatrix%kisassembled.2.2整體剛度矩陣的形成K(i,i)=K(i,i)+k(1,1);K(i,j)=K(i,j)+k(1,2);K(i,m)=K(i,m)+k(1,3);K(j,i)=K(j,i)+k(2,1);K(j,j)=K(j,j)+k(2,2);functiony=QuadraticBarAssemble(K,k,i,j,m)§3-3二次桿元K(j,m)=K(j,m)+k(2,3);K(m,i)=K(m,i)+k(3,1);K(m,j)=K(m,j)+k(3,2);K(m,m)=K(m,m)+k(3,3);y=K;8/1/202328結(jié)構(gòu)分析編程及軟件應(yīng)用2、MATLAB函數(shù)編寫%Quadra結(jié)構(gòu)分析編程及軟件應(yīng)用2、MATLAB函數(shù)編寫%QuadraticBarElementForcesThisfunctionreturnstheelementnodal%forcevectorgiventheelementstiffness%matrixkandtheelementnodal%displacementvectoru.2.3節(jié)點(diǎn)載荷計(jì)算y=k*u;functiony=QuadraticBarElementForces(k,u)§3-3二次桿元8/1/202329結(jié)構(gòu)分析編程及軟件應(yīng)用2、MATLAB函數(shù)編寫%Quadra結(jié)構(gòu)分析編程及軟件應(yīng)用2、MATLAB函數(shù)編寫%QuadraticBarElementStressesThisfunctionreturnstheelement%nodalstressvectorgiventheelement%stiffnessmatrixk,theelementnodal%displacementvectoru,andthe%cross-sectionalareaA.2.4節(jié)點(diǎn)應(yīng)力計(jì)算y=k*u/A;functiony=QuadraticBarElementStresses(k,u,A)§3-3二次桿元8/1/202330結(jié)構(gòu)分析編程及軟件應(yīng)用2、MATLAB函數(shù)編寫%Quadra結(jié)構(gòu)分析編程及軟件應(yīng)用3、實(shí)例計(jì)算分析應(yīng)用如圖所示雙二次桿元結(jié)構(gòu)，假定E=210MPa，A=0.003m^2求：系統(tǒng)的整體剛度矩陣；節(jié)點(diǎn)2、3、4、5的位移；節(jié)點(diǎn)1的支反力；每個(gè)桿件的應(yīng)力解：§3-3二次桿元8/1/202331結(jié)構(gòu)分析編程及軟件應(yīng)用3、實(shí)例計(jì)算分析應(yīng)用如圖所示雙二次桿元結(jié)構(gòu)分析編程及軟件應(yīng)用3、實(shí)例計(jì)算分析應(yīng)用E=210E6;A=0.003;L=2;k1=QuadraticBarElementStiffness(E,A,L);k2=QuadraticBarElementStiffness(E,A,L);n=5;K=zeros(n,n);K=QuadraticBarAssemble(K,k1,1,3,2);K=QuadraticBarAssemble(K,k2,3,5,4);U=zeros(4,1);F=[5;-10;-7;10];KK=K;K=K(2:n,2:n);U=K\F;U=[0;U];F=KK*U;u1=[U(1);U(3);U(2)];f1=QuadraticBarElementForces(k1,u1);sigma1=QuadraticBarElementStresses(k1,u1,A);u2=[U(3);U(5);U(4)];f2=QuadraticBarElementForces(k2,u2);sigma2=QuadraticBarElementStresses(k2,u2,A);§3-3二次桿元8/1/202332結(jié)構(gòu)分析編程及軟件應(yīng)用3、實(shí)例計(jì)算分析應(yīng)用E=210E6;K結(jié)構(gòu)分析編程及軟件應(yīng)用1、基本方程§3-4平面桁架元平面桁架元是既有局部坐標(biāo)又有總體坐標(biāo)二維有限元，用線性函數(shù)描述每個(gè)平面桁架元有二個(gè)節(jié)點(diǎn)（node）單剛矩陣為：總剛矩陣：結(jié)構(gòu)方程：單元節(jié)點(diǎn)力：8/1/202333結(jié)構(gòu)分析編程及軟件應(yīng)用1、基本方程§3-4平面桁架結(jié)構(gòu)分析編程及軟件應(yīng)用2、MATLAB函數(shù)編寫%PlaneTrussElementLengthThisfunctionreturnsthelengthofthe%planetrusselementwhosefirstnodehas%coordinates(x1,y1)andsecondnodehas%coordinates(x2,y2).2.1計(jì)算單元長度y=sqrt((x2-x1)*(x2-x1)+(y2-y1)*(y2-y1));functiony=PlaneTrussElementLength(x1,y1,x2,y2)§3-4平面桁架元8/1/202334結(jié)構(gòu)分析編程及軟件應(yīng)用2、MATLAB函數(shù)編寫%PlaneT結(jié)構(gòu)分析編程及軟件應(yīng)用2、MATLAB函數(shù)編寫%PlaneTrussElementStiffnessThisfunctionreturnstheelement%stiffnessmatrixforaplanetruss%elementwithmodulusofelasticityE,%cross-sectionalareaA,lengthL,and%angletheta(indegrees).%Thesizeoftheelementstiffness%matrixis4x4.2.2單元?jiǎng)偠染仃嚨男纬蓌=theta*pi/180;C=cos(x);S=sin(x);y=E*A/L*[C*CC*S-C*C-C*S;C*SS*S-C*S-S*S;-C*C-C*SC*CC*S;-C*S-S*SC*SS*S];functiony=PlaneTrussElementStiffness(E,A,L,theta)§3-4平面桁架元8/1/202335結(jié)構(gòu)分析編程及軟件應(yīng)用2、MATLAB函數(shù)編寫%PlaneT結(jié)構(gòu)分析編程及軟件應(yīng)用2、MATLAB函數(shù)編寫%PlaneTrussAssembleThisfunctionassemblestheelementstiffness%matrixkoftheplanetrusselementwithnodes%iandjintotheglobalstiffnessmatrixK.%Thisfunctionreturnstheglobalstiffness%matrixKaftertheelementstiffnessmatrixkisassembled.2.3整體剛度矩陣的形成K(2*i-1,2*i-1)=K(2*i-1,2*i-1)+k(1,1);K(2*i-1,2*i)=K(2*i-1,2*i)+k(1,2);K(2*i-1,2*j-1)=K(2*i-1,2*j-1)+k(1,3);K(2*i-1,2*j)=K(2*i-1,2*j)+k(1,4);K(2*i,2*i-1)=K(2*i,2*i-1)+k(2,1);K(2*i,2*i)=K(2*i,2*i)+k(2,2);K(2*i,2*j-1)=K(2*i,2*j-1)+k(2,3);K(2*i,2*j)=K(2*i,2*j)+k(2,4);functiony=PlaneTrussAssemble(K,k,i,j)K(2*j-1,2*i-1)=K(2*j-1,2*i-1)+k(3,1);K(2*j-1,2*i)=K(2*j-1,2*i)+k(3,2);K(2*j-1,2*j-1)=K(2*j-1,2*j-1)+k(3,3);K(2*j-1,2*j)=K(2*j-1,2*j)+k(3,4);K(2*j,2*i-1)=K(2*j,2*i-1)+k(4,1);K(2*j,2*i)=K(2*j,2*i)+k(4,2);K(2*j,2*j-1)=K(2*j,2*j-1)+k(4,3);K(2*j,2*j)=K(2*j,2*j)+k(4,4);y=K;§3-4平面桁架元8/1/202336結(jié)構(gòu)分析編程及軟件應(yīng)用2、MATLAB函數(shù)編寫%PlaneT結(jié)構(gòu)分析編程及軟件應(yīng)用2、MATLAB函數(shù)編寫%PlaneTrussElementForceThisfunctionreturnstheelementforce%giventhemodulusofelasticityE,the%cross-sectionalareaA,thelengthL,%theangletheta(indegrees),andthe%elementnodaldisplacementvectoru.2.4節(jié)點(diǎn)載荷計(jì)算x=theta*pi/180;C=cos(x);S=sin(x);y=E*A/L*[-C-SCS]*u;functiony=PlaneTrussElementForce(E,A,L,theta,u)§3-4平面桁架元8/1/202337結(jié)構(gòu)分析編程及軟件應(yīng)用2、MATLAB函數(shù)編寫%PlaneT結(jié)構(gòu)分析編程及軟件應(yīng)用2、MATLAB函數(shù)編寫%PlaneTrussElementStressThisfunctionreturnstheelementstress%giventhemodulusofelasticityE,the%thelengthL,theangletheta(in%degrees),andtheelementnodal%displacementvectoru.2.5節(jié)點(diǎn)應(yīng)力計(jì)算x=theta*pi/180;C=cos(x);S=sin(x);y=E/L*[-C-SCS]*u;functiony=PlaneTrussElementStress(E,L,theta,u)§3-4平面桁架元8/1/202338結(jié)構(gòu)分析編程及軟件應(yīng)用2、MATLAB函數(shù)編寫%PlaneT結(jié)構(gòu)分析編程及軟件應(yīng)用3、實(shí)例計(jì)算分析應(yīng)用如圖所示平面桁架結(jié)構(gòu)，假定E=210MPa，A=0.0004m^2求：系統(tǒng)的整體剛度矩陣；節(jié)點(diǎn)2的水平位移；節(jié)點(diǎn)3的水平豎向位移；節(jié)點(diǎn)1、2的支反力；每跟桿件的應(yīng)力§3-4平面桁架元8/1/202339結(jié)構(gòu)分析編程及軟件應(yīng)用3、實(shí)例計(jì)算分析應(yīng)用如圖所示平面桁架結(jié)結(jié)構(gòu)分析編程及軟件應(yīng)用1、基本方程§3-5空間桁架元空間桁架元是既有局部坐標(biāo)又有總體坐標(biāo)三維有限元，用線性函數(shù)描述。各單元之間通過鉸接系統(tǒng)連接，只能傳遞力，而不能傳遞彎矩每個(gè)桁架元有二個(gè)節(jié)點(diǎn)（node）8/1/202340結(jié)構(gòu)分析編程及軟件應(yīng)用1、基本方程§3-5空間桁架結(jié)構(gòu)分析編程及軟件應(yīng)用1、基本方程§3-5空間桁架元總剛矩陣：結(jié)構(gòu)方程：單元節(jié)點(diǎn)力：單剛矩陣為：8/1/202341結(jié)構(gòu)分析編程及軟件應(yīng)用1、基本方程§3-5空間桁架結(jié)構(gòu)分析編程及軟件應(yīng)用2、MATLAB函數(shù)編寫%SpaceTrussElementLengthThisfunctionreturnsthelengthofthe%spacetrusselementwhosefirstnodehas%coordinates(x1,y1,z1)andsecondnodehas%coordinates(x2,y2,z2).2.1計(jì)算單元長度y=sqrt((x2-x1)*(x2-x1)+(y2-y1)*(y2-y1)+(z2-z1)*(z2-z1));functiony=SpaceTrussElementLength(x1,y1,z1,x2,y2,z2)§3-5空間桁架元8/1/202342結(jié)構(gòu)分析編程及軟件應(yīng)用2、MATLAB函數(shù)編寫%SpaceT結(jié)構(gòu)分析編程及軟件應(yīng)用2、MATLAB函數(shù)編寫%SpaceTrussElementStiffnessThisfunctionreturnstheelement%stiffnessmatrixforaspacetruss%elementwithmodulusofelasticityE,%cross-sectionalareaA,lengthL,and%anglesthetax,thetay,thetaz%(indegrees).Thesizeoftheelement%stiffnessmatrixis6x6.2.2單元?jiǎng)偠染仃嚨男纬蓌=thetax*pi/180;u=thetay*pi/180;v=thetaz*pi/180;Cx=cos(x);Cy=cos(u);Cz=cos(v);w=[Cx*CxCx*CyCx*Cz;Cy*CxCy*CyCy*Cz;Cz*CxCz*CyCz*Cz];y=E*A/L*[w-w;-ww];functiony=SpaceTrussElementStiffness(E,A,L,thetax,thetay,thetaz)§3-5空間桁架元8/1/202343結(jié)構(gòu)分析編程及軟件應(yīng)用2、MATLAB函數(shù)編寫%SpaceT結(jié)構(gòu)分析編程及軟件應(yīng)用2、MATLAB函數(shù)編寫%SpaceTrussAssembleThisfunctionassemblestheelementstiffness%matrixkofthespacetrusselementwithnodes%iandjintotheglobalstiffnessmatrixK.%Thisfunctionreturnstheglobalstiffness%matrixKaftertheelementstiffnessmatrix%kisassembled.2.3整體剛度矩陣的形成K(3*i-2,3*i-2)=K(3*i-2,3*i-2)+k(1,1);K(3*i-2,3*i-1)=K(3*i-2,3*i-1)+k(1,2);K(3*i-2,3*i)=K(3*i-2,3*i)+k(1,3);K(3*i-2,3*j-2)=K(3*i-2,3*j-2)+k(1,4);K(3*i-2,3*j-1)=K(3*i-2,3*j-1)+k(1,5);K(3*i-2,3*j)=K(3*i-2,3*j)+k(1,6);K(3*i-1,3*i-2)=K(3*i-1,3*i-2)+k(2,1);K(3*i-1,3*i-1)=K(3*i-1,3*i-1)+k(2,2);K(3*i-1,3*i)=K(3*i-1,3*i)+k(2,3);K(3*i-1,3*j-2)=K(3*i-1,3*j-2)+k(2,4);K(3*i-1,3*j-1)=K(3*i-1,3*j-1)+k(2,5);K(3*i-1,3*j)=K(3*i-1,3*j)+k(2,6);functiony=SpaceTrussAssemble(K,k,i,j)§3-5空間桁架元8/1/202344結(jié)構(gòu)分析編程及軟件應(yīng)用2、MATLAB函數(shù)編寫%SpaceT結(jié)構(gòu)分析編程及軟件應(yīng)用2、MATLAB函數(shù)編寫2.3整體剛度矩陣的形成§3-5空間桁架元K(3*j-1,3*i-2)=K(3*j-1,3*i-2)+k(5,1);K(3*j-1,3*i-1)=K(3*j-1,3*i-1)+k(5,2);K(3*j-1,3*i)=K(3*j-1,3*i)+k(5,3);K(3*j-1,3*j-2)=K(3*j-1,3*j-2)+k(5,4);K(3*j-1,3*j-1)=K(3*j-1,3*j-1)+k(5,5);K(3*j-1,3*j)=K(3*j-1,3*j)+k(5,6);K(3*j,3*i-2)=K(3*j,3*i-2)+k(6,1);K(3*j,3*i-1)=K(3*j,3*i-1)+k(6,2);K(3*j,3*i)=K(3*j,3*i)+k(6,3);K(3*j,3*j-2)=K(3*j,3*j-2)+k(6,4);K(3*j,3*j-1)=K(3*j,3*j-1)+k(6,5);K(3*j,3*j)=K(3*j,3*j)+k(6,6);y=K;K(3*i,3*i-2)=K(3*i,3*i-2)+k(3,1);K(3*i,3*i-1)=K(3*i,3*i-1)+k(3,2);K(3*i,3*i)=K(3*i,3*i)+k(3,3);K(3*i,3*j-2)=K(3*i,3*j-2)+k(3,4);K(3*i,3*j-1)=K(3*i,3*j-1)+k(3,5);K(3*i,3*j)=K(3*i,3*j)+k(3,6);K(3*j-2,3*i-2)=K(3*j-2,3*i-2)+k(4,1);K(3*j-2,3*i-1)=K(3*j-2,3*i-1)+k(4,2);K(3*j-2,3*i)=K(3*j-2,3*i)+k(4,3);K(3*j-2,3*j-2)=K(3*j-2,3*j-2)+k(4,4);K(3*j-2,3*j-1)=K(3*j-2,3*j-1)+k(4,5);K(3*j-2,3*j)=K(3*j-2,3*j)+k(4,6);8/1/202345結(jié)構(gòu)分析編程及軟件應(yīng)用2、MATLAB函數(shù)編寫2.3整體剛結(jié)構(gòu)分析編程及軟件應(yīng)用2、MATLAB函數(shù)編寫%SpaceTrussElementForceThisfunctionreturnstheelementforce%giventhemodulusofelasticityE,the%cross-sectionalareaA,thelengthL,%theanglesthetax,thetay,thetaz%(indegrees),andtheelementnodal%displacementvectoru.2.4節(jié)點(diǎn)載荷計(jì)算x=thetax*pi/180;w=thetay*pi/180;v=thetaz*pi/180;Cx=cos(x);Cy=cos(w);Cz=cos(v);y=E*A/L*[-Cx-Cy-CzCxCyCz]*u;functiony=SpaceTrussElementForce(E,A,L,thetax,thetay,thetaz,u)§3-5空間桁架元8/1/202346結(jié)構(gòu)分析編程及軟件應(yīng)用2、MATLAB函數(shù)編寫%SpaceT結(jié)構(gòu)分析編程及軟件應(yīng)用2、MATLAB函數(shù)編寫%SpaceTrussElementStressThisfunctionreturnstheelementstress%giventhemodulusofelasticityE,the%lengthL,theanglesthetax,thetay,%thetaz(indegrees),andtheelement%nodaldisplacementvectoru.2.5節(jié)點(diǎn)應(yīng)力計(jì)算x=thetax*pi/180;w=thetay*pi/180;v=thetaz*pi/180;Cx=cos(x);Cy=cos(w);Cz=cos(v);y=E/L*[-Cx-Cy-CzCxCyCz]*u;functiony=SpaceTrussElementStress(E,L,thetax,thetay,thetaz,u)§3-5空間桁架元8/1/202347結(jié)構(gòu)分析編程及軟件應(yīng)用2、MATLAB函數(shù)編寫%SpaceT結(jié)構(gòu)分析編程及軟件應(yīng)用3、實(shí)例計(jì)算分析應(yīng)用如圖所示空間桁架結(jié)構(gòu)，假定E=210MPa，A14=0.001m^2

A24=0.002m^2，A34=0.001m^2，P=12kN求：系統(tǒng)的整體剛度矩陣；節(jié)點(diǎn)4的水平位移；節(jié)點(diǎn)3的水平豎向位移；節(jié)點(diǎn)1、2、3的支反力；每跟桿件的應(yīng)力§3-5空間桁架元8/1/202348結(jié)構(gòu)分析編程及軟件應(yīng)用3、實(shí)例計(jì)算分析應(yīng)用如圖所示空間桁架結(jié)結(jié)構(gòu)分析編程及軟件應(yīng)用1、基本方程§3-6梁元梁元是總體坐標(biāo)與局部坐標(biāo)一致的二維有限元，用線性函數(shù)描述。各單元之間通過鉸接系統(tǒng)連接，只能傳遞力，而不能傳遞彎矩每個(gè)梁元有二個(gè)節(jié)點(diǎn)（node）單剛矩陣為：總剛矩陣：結(jié)構(gòu)方程：單元節(jié)點(diǎn)力：8/1/202349結(jié)構(gòu)分析編程及軟件應(yīng)用1、基本方程§3-6梁元結(jié)構(gòu)分析編程及軟件應(yīng)用2、MATLAB函數(shù)編寫%BeamElementStiffnessThisfunctionreturnstheelement%stiffnessmatrixforabeam%elementwithmodulusofelasticityE,%momentofinertiaI,andlengthL.%Thesizeoftheelementstiffness%matrixis4x4.2.1單元?jiǎng)偠染仃嚨男纬蓎=E*I/(L*L*L)*[126*L-126*L;6*L4*L*L-6*L2*L*L;-12-6*L12-6*L;6*L2*L*L-6*L4*L*L];functiony=BeamElementStiffness(E,I,L)§3-6梁元8/1/202350結(jié)構(gòu)分析編程及軟件應(yīng)用2、MATLAB函數(shù)編寫%BeamEl結(jié)構(gòu)分析編程及軟件應(yīng)用2、MATLAB函數(shù)編寫%BeamAssembleThisfunctionassemblesth