弹性力学习题解答6.docx

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1、6-1试证三节点三角形单元内任意一点有：ìNi+Nj+Nm=1ïíNixi+Njxj+Nmxm=xï证：1xy1xiyi1xiyi1xjyj1xy1xjyjNi=1xmym;Nj=1xmym;Nm=1xy1xiyi1xiyi1xiyi1xjyj1xjyj1xjyj1xmym1xmym1xmym1xy1xiyi1xiyi1xjyj1xy1xjyjN+N+N=1xmym+1xmym+1xyijm1xiyi1xiyi1xiyi1xjyj1xjyj1xjyj1xmym1xmym1xmym01xy-11xy-11xiyi-01xiyi1xiyi11xjyj01xjyj1xjyj=11xmy

2、m2´(-=1)Å101xmym=1xmym1xiyi1xiyi1xiyi1xjyj1xjyj1xjyj1xmym1xmym1xmym=1Ni+Nj+Nm=11xy1xiyi1xiyi1xjyj1xy1xjyjNixi+Njxj+Nmxm=1xmymxi+1xmymxj+1xyxm1xiyi1xiyi1xiyi1xjyj1xjyj1xjyj1xmym1xmym1xmym-1-01xy-x1xy-xi1xiyi-01xiyi1xiyixj1xjyj01xjyj1xjyj=xm1xmym3´(-=1)Å101xmym=x1xmym1xiyi1xiyi1xiyi1xjyj1xjyj1x

3、jyj1xmym1xmym1xmym=xNixi+Njxj+Nmxm=x1xy1xiyi1xiyi1xjyj1xy1xjyjNiyi+Njyj+Nmym=1xmymyi+1xmymyj+1xyym1xiyi1xiyi1xiyi1xjyj1xjyj1xjyj1xmym1xmym1xmym01xy-y1xy-yi1xiyi-01xiyi1xiyiyj1xjyj01xjyj1xjyj=ym1xmym4´(-=1)Å101xmym=y1xmym1xiyi1xiyi1xiyi1xjyj1xjyj1xjyj1xmym1xmym1xmym=yNiyi+Njyj+Nmym=y证毕-2-6-2如图

4、所示平面应力状态三结点等边三角形单元（边长为a；m=16）。试求：1、应力转换矩阵S及单元劲度（刚度）矩阵k。2、单元劲度（刚度）矩阵k每行、每列之和，并说明其原因。3、当该单元体发生结点位移ìui=uj=um=1ïyí=vj=vm=0mïviî或发生结点位移ìui=uj=vi=0ïijí31ïvj=1;um=-;vm=-ox22î单元中的应力，并说明其原因。4、若该单元在jm边上受线性分布压力，其在j点m点的集度分别为qj和qm。试求其等效结点载荷。解：基本方程：ìuüéNi0Nj0Nmíý=ê0Nj0îvþë0Ni{e}=[B]{de}ore=Bd0úù{uiviujvju

5、mmm}TNmûeì¶uüé¶Niïïê¶x¶xìeüïïê¶vïxïïïêíeyý=íý=ê0¶yïïïïêîgxyþï¶u¶vïê¶Niï¶y+¶xïê¶yîþêëé¶Ni0¶Njê¶x¶xêê¶NiB=ê00¶yêê¶Ni¶Ni¶Njê¶y¶x¶yêë=11xy=N1xy;Ni2Ajjj1xmym0¶Nj0¶Nm0¶x¶x¶Ni0¶Nj0¶Nm¶y¶y¶y¶Ni¶Nj¶Nj¶Nm¶Nm¶x¶y¶x¶y¶x0¶Nm0ùú¶xú¶Nj0¶Nmúú¶y¶yú¶Nj¶Nm¶Nmúú¶x¶y¶xúû11xiyi=11xiyi1xy;N1xy2Am2Ajj1xmym1xyùìuiü

6、úúïïviïïúúïíujïýúïvjïúúïïumïïúûïîmmïþ-3-¶Ni=(yj-ym);¶Ni=(xm-xj)¶x2A¶y2A¶Nj=(ym-yi);¶Nj=(xi-xm)¶x2A¶y2A¶Nm=(yi-yj);¶Nm=(xj-xi)¶x2A¶y2Aé(yj-ym)0(ym-yi)0(yi-yj)0ù1êúB=ê0(xm-xj)0(xi-xm)0(xj-xi)ú2Aê(xm-xj)(yj-ym)(xi-xm)(ym-yi)(xj-xi)(yi-yj)úëûéùê1m0úE®EEêú1-m2D

7、Planestress=êm10ú;D

8、PlanestressÛD

9、Pl

10、anestrain1-m2ê1-múm®mê00ú1-mêú2ëûS=DB1、+a);2A=a2i:(x,y);j:(x+a,y);m:(x+a,y33iiiiii222éù-3a3aê0000ú22êúé1220ùêú1aa3EêúB=ê0-0-0aú;D=2120êú222353aêúê005úêúëûa3aa3aê---a0ú2222êúëûéù33ê-0000ú22é1220ùêúS=BD=3E1êúê-1-1úê2120úê0001ú35223aê005úêúëûêú1