节点应力和单元应力的关系

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1、:用三结点三角形平面单元计算平面结构的应力和位移1，设计说明书计算简图，网格划分，单元及结点的编号如下图所示。由于结构对称,去四分之一结构分析。其中E=2e10pa,mu=0.167,h=1m.变量注释:Node节点定义gElement-单元定义gMaterial-材料定义，包括弹性模量，泊松比和厚度gBC1约束条件gNF集中力gk总刚gDelta结点位移子程序注释：PlaneStructualModel定义有限元模型SolveModel求解有限元模型DisplayResults显示计算结果k=StiffnessMatrix(ie)计算单元刚度AssembleStiffnessMatrix(i

2、e,k)形成总刚es=ElementStress(ie)计算单元应力functionexam1%输入参数：无%输出结果：节点位移和单元应力PlaneStructualModel;%定义有限元模型SolveModel;%求解有限元模型DisplayResults;%显示计算结果return;functionPlaneStructualModel%定义平面结构的有限元模型%输入参数：无%说明：%该函数定义平面结构的有限元模型数据：%gNode节点定义%gElement单元定义%gMaterial-材料定义，包括弹性模量，泊松比和厚度%gBC1约束条件%gNF集中力globalgNodegEleme

3、ntgMaterialgBC1gNF%节点坐标%xygNode=0.0,2.0%节点10.0,1.0%节点21.0,1.0%节点30.0,0.0%节点41.0,0.0%节点52.0,0.0;%节点6%单元定义%节点1节点2节点3材料号gElement=3,1,2,1%单元5,2,4,1%单元22,5,3,1%单元36,3,5,1;%单元4%材料性质%弹性模量泊松比厚度gMaterial=1e0,0,1;%材料1%第一类约束条件%节点号自由度号约束值gBC1=1,1,0.02,1,0.04,1,0.04,2,0.05,2,0.06,2,0.0;%集中力%节点号自由度号集中力值gNF=1,2,-1

4、;returnfunctionSolveModel%求解有限元模型%输入参数：无%说明：%该函数求解有限元模型，过程如下%1.计算单元刚度矩阵，集成整体刚度矩阵%2.计算单元的等效节点力，集成整体节点力向量%3.处理约束条件，修改整体刚度矩阵和节点力向量%4.求解方程组，得到整体节点位移向量globalgNodegElementgMaterialgBC1gNFgKgDelta%step1.定义整体刚度矩阵和节点力向量node_number,dummy=size(gNode);gK=sparse(node_number*2,node_number*2);f=sparse(node_number*

5、2,1);%step2.计算单元刚度矩阵，并集成到整体刚度矩阵中element_number,dummy=size(gElement);forie=1:1:element_numberk=StiffnessMatrix(ie);AssembleStiffnessMatrix(ie,k);end%step3.把集中力直接集成到整体节点力向量中nf_number,dummy=size(gNF);forinf=1:1:nf_numbern=gNF(inf,1);d=gNF(inf,2);f(n-1)*2+d)=gNF(inf,3);end%step4.处理约束条件，修改刚度矩阵和节点力向量。采用乘大

6、数法bc_number,dummy=size(gBC1);foribc=1:1:bc_numbern=gBC1(ibc,1);d=gBC1(ibc,2);m=(n-1)*2+d;f(m)=gBC1(ibc,3)*gK(m,m)*1e15;gK(m,m)=gK(m,m)*1e15;end%step5.求解方程组，得到节点位移向量gDelta=gKf;returnfunctionDisplayResults%显示计算结果%输入参数：无globalgNodegElementgMaterialgBC1gNFgKgDeltafprintf(节点位移n);fprintf(节点号x方向位移y方向位移n);n

7、ode_number,dummy=size(gNode);fori=1:node_numberfprintf(%6d%16.8e%16.8en,.i,gDelta(i-1)*2+1),gDelta(i-1)*2+2);endfprintf(nn单元应力n);fprintf(X-STRY-STRXY-STRn);element_number,dummy=size(gElement);forie=1:element_numberes=ElementStress(ie);fprintf(单元号6d%16.8e%16.8e%16.8en,.ie,es(1),es(2),es(3);endreturnf

8、unctionk=StiffnessMatrix(ie)%计算单元刚度矩阵%输入参数:%ie单元号globalgNodegElementgMaterialk=zeros(6,6);E=gMaterial(gElement(ie,4),1);mu=gMaterial(gElement(ie,4),2);h=gMaterial(gElement(ie,4),3);xi=gNode(gElement(ie,1),1);yi=gNode(gElement(ie,1),2);xj=gNode(gElement(ie,2),1);yj=gNode(gElement(ie,2),2);xm=gNode(gE

9、lement(ie,3),1);ym=gNode(gElement(ie,3),2);ai=xj*ym-xm*yj;aj=xm*yi-xi*ym;am=xi*yj-xj*yi;bi=yj-ym;bj=ym-yi;bm=yi-yj;ci=-(xj-xm);cj=-(xm-xi);cm=-(xi-xj);A=(ai+aj+am)/2;B=bi0bj0bm00ci0cj0cmcibicjbjcmbm;B=B/2/A;D=1mu0mu1000(1-mu)/2;D=D*E/(1-muA2);k=transpose(B)*D*B*h*abs(A);returnfunctionAssembleStiffne

10、ssMatrix(ie,k)%把单元刚度矩阵集成到整体刚度矩阵%输入参数:%ie-单元号%k-单元刚度矩阵globalgElementgKfori=1:1:3forj=1:1:3forp=1:1:2forq=1:1:2m=(i-1)*2+p;n=(j-1)*2+q;M=(gElement(ie,i)-1)*2+p;N=(gElement(ie,j)-1)*2+q;gK(M,N)=gK(M,N)+k(m,n);endendendendreturnfunctiones=ElementStress(ie)%计算单元的应力%输入参数%ie节点号%es单元应力globalgElementgNodegDe

11、ltagMateriales=zeros(1,6);de=zeros(6,1);forj=1:1:3de(2*j-1)=gDelta(2*gElement(ie,j)-1);de(2*j)=gDelta(2*gElement(ie,j);endE=gMaterial(gElement(ie,4),1);mu=gMaterial(gElement(ie,4),2);h=gMaterial(gElement(ie,4),3);xi=gNode(gElement(ie,1),1);yi=gNode(gElement(ie,1),2);xj=gNode(gElement(ie,2),1);yj=gNo

12、de(gElement(ie,2),2);xm=gNode(gElement(ie,3),1);ym=gNode(gElement(ie,3),2);ai=xj*ym-xm*yj;aj=xm*yi-xi*ym;am=xi*yj-xj*yi;bi=yj-ym;bj=ym-yi;bm=yi-yj;ci=-(xj-xm);cj=-(xm-xi);cm=-(xi-xj);A=(ai+aj+am)/2;B=bi0bj0bm00ci0cj0cmcibicjbjcmbm;B=B/2/A;D=1mu0mu1000(1-mu)/2;D=D*E/(1-muA2);S=D*B;es(1:3)=S*de;es(6)=

13、0.5*sqrt(es(1)-es(2)A2+4*es(3)A2);es(4)=0.5*(es(1)+es(2)+es(6);es(5)=0.5*(es(1)+es(2)-es(6);return3，数据文件：输入数据：gNode=0.0,2.00.0,1.01.0,1.00.0,0.01.0,0.02.0,0.0;gElement=3,1,2,15,2,4,12,5,3,16,3,5,1gMaterial=1e0,0,1;gBC1=1,1,0.02,1,0.04,1,0.04,2,0.05,2,0.06,2,0.0;gNF=1,2,-1;输出数据：节点位移节点号x方向位移y方向位移1-8.7

14、9120879e-016-3.25274725e+0028.79120879e-017-1.25274725e+000-3.73626374e-001-8.35164835e-016-2.93040293e-0162.63736264e-016Y-STR3 -8.79120879e-0024 1.17216117e-0165 1.75824176e-0011.75824176e-001单元应力X-STRXY-STR单元号1-8.79120879e-002-2.00000000e+0004.39560440e-001单元号21.75824176e-001-1.25274725e+0002.56410256e-016单元号3-8.79120879e-002-3.73626374e-0013.07692308e-001单元号40.00000000e+000-3.73626374e-001-1.31868132e-001