{"id":10726,"date":"2018-12-25T12:34:40","date_gmt":"2018-12-25T04:34:40","guid":{"rendered":"http:\/\/www.jdcui.com\/?p=10726"},"modified":"2018-12-28T23:55:47","modified_gmt":"2018-12-28T15:55:47","slug":"femmatlab%e6%9c%89%e9%99%90%e5%85%83%e7%bc%96%e7%a8%8b-buckling-analysis-of-trusses-structure-%e6%a1%81%e6%9e%b6%e5%8d%95%e5%85%83%e6%9b%b2%e5%b1%88%e5%88%86%e6%9e%90%e7%bc%96%e7%a8%8b","status":"publish","type":"post","link":"http:\/\/www.jdcui.com\/?p=10726","title":{"rendered":"[FEM][MATLAB][\u6709\u9650\u5143][\u7f16\u7a0b] FEM Buckling Analysis Programming with MATLAB (Truss Elements) (\u6841\u67b6\u5355\u5143\u66f2\u5c48\u5206\u6790\u7f16\u7a0b)"},"content":{"rendered":"

\u575a\u6301\u5b9e\u5e72\u3001\u575a\u6301\u4e00\u7ebf\u3001\u575a\u6301\u79ef\u7d2f\u3001\u575a\u6301\u601d\u8003\uff0c\u575a\u6301\u521b\u65b0\u3002<\/strong><\/span><\/p>\n

<\/p>\n

\u63a5\u7740\u535a\u6587\u300a[\u529b\u5b66][\u6709\u9650\u5143][FEM]Basics of Buckling Analysis [\u66f2\u5c48\u5206\u6790\u57fa\u7840]<\/strong><\/a>\u300b\u7ee7\u7eed\u4ecb\u7ecd\u6841\u67b6\u5355\u5143(Truss Element)\u7528\u4e8e\u5c48\u66f2\u5206\u6790<\/strong>\u7684\u65b9\u6cd5\u3002\u8be5\u90e8\u5206\u5185\u5bb9\u4e5f\u662f \u4e66\u672c\u00a0\u300a\u6709\u9650\u5355\u5143\u6cd5\uff1a\u7f16\u7a0b\u4e0e\u8f6f\u4ef6\u5e94\u7528\u300b<\/strong><\/a>\u5c48\u66f2\u5206\u6790<\/strong>\u7ae0\u8282\u7684\u90e8\u5206\u5185\u5bb9\u8282\u9009\u3002<\/p>\n

11.3 <\/span>\u5c48\u66f2\u5206\u67901\uff1a2D\u6841\u67b6<\/strong><\/span><\/p>\n

\u56fe 11\u20112\u662f\u672c\u7ae0\u7684\u7b97\u4f8b\u7ed3\u6784\uff0c\u662f\u4e00\u6980XZ\u5e73\u9762\u5185\u7684\u6841\u67b6\u7ed3\u6784\uff0c\u7ed3\u6784\u51e0\u4f55\u4fe1\u606f\u3001\u6784\u4ef6\u5c5e\u6027\u548c\u7ea6\u675f\u4fe1\u606f\u4e0e\u7b2c\u4e8c\u7ae0\u4e2d\u9759\u529b\u5206\u6790\u65f6\u76f8\u540c\uff0c\u8282\u70b95\u30016\u30017\u53d7\u5230-z\u65b9\u5411\u7684100kN\u96c6\u4e2d\u529b\u4f5c\u7528\u3002\u672c\u7ae0\u5c06\u57fa\u4e8e2D\u6841\u67b6\u5355\u5143\u5bf9\u8be5\u7ed3\u6784\u8fdb\u884c\u56fe\u4e2d\u8377\u8f7d\u6837\u5f0f\u4e0b\u7684\u5c48\u66f2\u5206\u6790\uff0c\u5e76\u5c06\u57fa\u4e8eMATLAB\u7f16\u7a0b\u8ba1\u7b97\u7684\u7ed3\u679c\u4e0eSAP2000\u3001midas Gen\u5206\u6790\u7ed3\u679c\u8fdb\u884c\u5bf9\u6bd4\u3002<\/p>\n

\"\"<\/p>\n

\u56fe 11\u20112 \u7ed3\u6784\u6a21\u578b\u793a\u610f<\/strong><\/p>\n

\u4ee5\u4e0b\u76f4\u63a5\u7ed9\u51fa\u5c0f\u53d8\u5f62\u60c5\u51b5\u4e0b\u6841\u67b6\u5355\u5143\u7684\u51e0\u4f55\u521a\u5ea6\u77e9\u9635<\/p>\n

\"\"\u00a0 \u00a0 \u00a0 (11.3\u20111)<\/strong><\/p>\n

\u5176\u4e2dP\u4e3a\u5355\u5143\u8f74\u529b\uff0cL\u4e3a\u5355\u5143\u957f\u5ea6\uff0c\u5355\u5143\u7684\u51e0\u4f55\u521a\u5ea6\u77e9\u9635\u4e0e\u5355\u5143\u7684\u5185\u529b\u6709\u5173\u3002<\/p>\n

11.3.1\u00a0<\/span>MATLAB\u4ee3\u7801\u4e0e\u6ce8\u91ca<\/strong><\/span><\/p>\n

\u5c48\u66f2\u5206\u6790\u7684Matlab<\/strong>\u4ee3\u7801\u4e0e\u524d\u9762\u7ae0\u8282\u4e2d\u9759\u529b\u5206\u6790\u7684Matlab\u4ee3\u7801\u7565\u6709\u4e0d\u540c\uff0c\u8fd9\u91cc\u7ed9\u51fa\u8fdb\u884c\u5c48\u66f2\u5206\u6790\u6240\u9700\u7684\u4e3b\u8981\u4ee3\u7801\u3002<\/p>\n

% Truss 2D Buckling Analysis<\/em><\/span><\/strong><\/code><\/p>\n

% Author <\/em>\uff1a<\/em> JiDong Cui<\/em>\uff08\u5d14\u6d4e\u4e1c\uff09\uff0c<\/em>Xuelong Shen<\/em>\uff08\u6c88\u96ea\u9f99\uff09<\/em><\/span><\/strong><\/code><\/p>\n

% Website : www.jdcui.com<\/em><\/span><\/strong><\/code><\/p>\n

% 20170609<\/em><\/span><\/strong><\/code><\/p>\n

\u2026\u2026<\/span><\/strong><\/code><\/p>\n

\u6b64\u524d\u4ee3\u7801\u4e0e\u9759\u529b\u5206\u6790\u4ee3\u7801\u76f8\u540c\uff0c\u4e0d\u518d\u8d58\u8ff0\u3002<\/p>\n

% Loading<\/em><\/span><\/strong><\/code><\/p>\n

force(10)=-100000;<\/strong><\/span><\/code><\/p>\n

force(12)=-100000;<\/strong><\/span><\/code><\/p>\n

force(14)=-100000;<\/strong><\/span><\/code><\/p>\n

\u2026\u2026<\/strong><\/span><\/code><\/p>\n

\u6839\u636e\u56fe 11\u20112\uff0c\u4e3a\u76f8\u5e94\u7684\u81ea\u7531\u5ea6\u65bd\u52a0\u8377\u8f7d\uff0c\u4f5c\u4e3a\u53c2\u8003\u8377\u8f7d\uff0c\u7528\u4e8e\u540e\u7eed\u6c42\u89e3\u5355\u5143\u7684\u51e0\u4f55\u521a\u5ea6\u77e9\u9635\u3002<\/em><\/p>\n

% Solute balance function<\/em><\/strong><\/span><\/code><\/p>\n

disp('Displacement')<\/strong><\/span><\/code><\/p>\n

displacement(activeDof) = stiffness( activeDof , activeDof)\\force(activeDof)<\/strong><\/span><\/code><\/p>\n

% Element force<\/em><\/strong><\/span><\/code><\/p>\n

InForce = zeros( numEle , 1 );<\/strong><\/span><\/code><\/p>\n

for i=1:numEle;<\/strong><\/span><\/code><\/p>\n

\u00a0\u00a0\u00a0 noindex = EleNode(i,:);<\/strong><\/span><\/code><\/p>\n

\u00a0\u00a0\u00a0 deltax = xx( noindex(2)) - xx( noindex(1));<\/strong><\/span><\/code><\/p>\n

\u00a0\u00a0\u00a0 deltay = yy( noindex(2)) - yy( noindex(1));<\/strong><\/span><\/code><\/p>\n

\u00a0\u00a0\u00a0 L = sqrt( deltax*deltax + deltay*deltay );<\/strong><\/span><\/code><\/p>\n

\u00a0\u00a0\u00a0 % Element stiffness matrix<\/span><\/em><\/strong><\/span><\/code><\/p>\n

\u00a0\u00a0\u00a0 C = deltax \/ L;<\/strong><\/span><\/code><\/p>\n

\u00a0\u00a0\u00a0 S = deltay \/ L;<\/strong><\/span><\/code><\/p>\n

\u00a0\u00a0\u00a0 % Corresponding freedom of the element nodes<\/span><\/em><\/strong><\/span><\/code><\/p>\n

\u00a0\u00a0\u00a0 eleDof = [noindex(1)*2-1 noindex(1)*2 noindex(2)*2-1 noindex(2)*2];<\/strong><\/span><\/code><\/p>\n

\u00a0\u00a0\u00a0 % Element force<\/em><\/strong><\/span><\/code><\/p>\n

\u00a0\u00a0\u00a0 InForce(i) = (E\/L)*[ -C -S C S ]*displacement(eleDof)*A;<\/strong><\/span><\/code><\/p>\n

end\u00a0\u00a0\u00a0<\/strong><\/span><\/code><\/p>\n

disp('Element Stress')<\/strong><\/span><\/code><\/p>\n

InForce;<\/strong><\/span><\/code><\/p>\n

\u8be5\u6bb5\u4ee3\u7801\u4e0e\u666e\u901a\u9759\u529b\u5206\u6790\u7684\u4ee3\u7801\u57fa\u672c\u76f8\u540c\uff0c\u4e3b\u8981\u7528\u4e8e\u8ba1\u7b97\u53c2\u8003\u8377\u8f7d\u4f5c\u7528\u4e0b\u5355\u5143\u7684\u5185\u529b\uff0c\u7528\u4e8e\u540e\u7eed\u8ba1\u7b97\u5355\u5143\u7684\u51e0\u4f55\u521a\u5ea6\u77e9\u9635\uff0c\u4ee5\u4e0b\u5c06\u4ecb\u7ecd\u5c48\u66f2\u5206\u6790\u90e8\u5206\u7684\u4e3b\u8981\u4ee3\u7801\u3002<\/em><\/p>\n

% Geometric Stiffness Matrix<\/em><\/span><\/strong><\/code><\/p>\n

stiffnessM = zeros(numDOF);<\/span><\/strong><\/code><\/p>\n

stiffnessG = zeros(numDOF);<\/span><\/strong><\/code><\/p>\n

% Traverse all elements<\/em><\/span><\/strong><\/code><\/p>\n

for i=1:numEle;<\/span><\/strong><\/code><\/p>\n

\u00a0\u00a0\u00a0 % Index of the element nodes<\/em><\/span><\/strong><\/code><\/p>\n

\u00a0\u00a0\u00a0 noindex = EleNode(i,:);<\/span><\/strong><\/code><\/p>\n

\u00a0\u00a0\u00a0 % Element length<\/em><\/span><\/strong><\/code><\/p>\n

\u00a0\u00a0\u00a0 deltax = xx( noindex(2)) - xx( noindex(1));<\/span><\/strong><\/code><\/p>\n

\u00a0\u00a0\u00a0 deltay = yy( noindex(2)) - yy( noindex(1));<\/span><\/strong><\/code><\/p>\n

\u00a0\u00a0\u00a0 L = sqrt( deltax*deltax + deltay*deltay );<\/span><\/strong><\/code><\/p>\n

\u00a0\u00a0 \u00a0% Element g<\/span><\/em>eometric <\/em>stiffness matrix<\/em><\/span><\/span><\/strong><\/code><\/p>\n

\u00a0\u00a0\u00a0 C = deltax \/ L;<\/span><\/strong><\/code><\/p>\n

\u00a0\u00a0\u00a0 S = deltay \/ L;<\/span><\/strong><\/code><\/p>\n

\u00a0\u00a0\u00a0 T=[ C,S, 0,0;<\/span><\/strong><\/code><\/p>\n

\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 -S,C, 0,0;<\/span><\/strong><\/code><\/p>\n

\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0,0, C,S;<\/span><\/strong><\/code><\/p>\n

\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0,0,-S,C<\/span><\/strong><\/code><\/p>\n

\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 ];<\/span><\/strong><\/code><\/p>\n

\u00a0\u00a0\u00a0 eleKm = EA\/L*[ C*C C*S -C*C -C*S;<\/span><\/strong><\/code><\/p>\n

\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 C*S S*S -C*S -S*S;<\/span><\/strong><\/code><\/p>\n

\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 -C*C -C*S C*C C*S;<\/span><\/strong><\/code><\/p>\n

\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 -C*S -S*S C*S S*S];<\/span><\/strong><\/code><\/p>\n

\u00a0\u00a0\u00a0 eleKg0=InForce(i)\/L*[0,0,0,0;<\/span><\/strong><\/code><\/p>\n

\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0,1,0,-1;<\/span><\/strong><\/code><\/p>\n

\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a00,0,0,0;<\/span><\/strong><\/code><\/p>\n

\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0,-1,0,1];<\/span><\/strong><\/code><\/p>\n

\u00a0\u00a0\u00a0 eleKg=T'*eleKg0*T;<\/span><\/strong><\/code><\/p>\n

\u00a0\u00a0\u00a0 % Corresponding freedom of the element nodes<\/span><\/em><\/span><\/strong><\/code><\/p>\n

\u00a0\u00a0\u00a0 eleDof = [noindex(1)*2-1 noindex(1)*2 noindex(2)*2-1 noindex(2)*2];<\/span><\/strong><\/code><\/p>\n

\u00a0\u00a0\u00a0 % Integrate element g<\/em>eometric <\/em>stiffness matrix to the global g<\/em>eometric <\/em>stiffness matrix<\/em><\/span><\/strong><\/code><\/p>\n

\u00a0\u00a0\u00a0 stiffnessM( eleDof , eleDof) = stiffnessM( eleDof , eleDof) + eleKm;<\/span><\/strong><\/code><\/p>\n

\u00a0\u00a0\u00a0 stiffnessG( eleDof , eleDof) = stiffnessG( eleDof , eleDof) + eleKg;<\/span><\/strong><\/code><\/p>\n

end<\/span><\/strong><\/code><\/p>\n

\u672c\u6bb5\u4ee3\u7801\u4e3b\u8981\u4f5c\u7528\uff1a\u5229\u7528\u53c2\u8003\u8377\u8f7d\u4f5c\u7528\u4e0b\u7684\u5355\u5143\u5185\u529b\uff0c\u6c42\u89e3\u5355\u5143\u7684\u51e0\u4f55\u521a\u5ea6\u77e9\u9635\uff0c\u8fdb\u800c\u91c7\u7528\u201c\u5bf9\u53f7\u5165\u5ea7\u201d\u7684\u65b9\u5f0f\uff0c\u5c06\u5355\u5143\u7684\u51e0\u4f55\u521a\u5ea6\u77e9\u9635\u53e0\u52a0\uff0c\u83b7\u5f97\u7ed3\u6784\u6574\u4f53\u7684\u51e0\u4f55\u521a\u5ea6\u77e9\u9635 \u3002<\/p>\n

% Solution<\/em><\/span><\/code><\/strong><\/p>\n

A=-stiffnessM(activeDof,activeDof)\\stiffnessG(activeDof,activeDof);<\/span><\/code><\/strong><\/p>\n

[Vo,Do] = eig(A); % Get the eigenvalues and eigenvectors<\/em><\/span><\/code><\/strong><\/p>\n

fo=diag(Do);<\/span><\/code><\/strong><\/p>\n

posneg=1.\/fo.*abs(fo);\u00a0<\/span><\/code><\/strong><\/p>\n

Vf=sortrows([Vo',abs(fo),posneg],size(Vo,1)+1);<\/span><\/code><\/strong><\/p>\n

V=Vf(:,1:size(Vo,1))';\u00a0<\/span><\/code><\/strong><\/p>\n

f=Vf(:,size(Vo,1)+1).*Vf(:,size(Vo,1)+2);<\/span><\/code><\/strong><\/p>\n

fac=1.\/f<\/strong><\/span><\/code><\/p>\n

\u672c\u6bb5\u4ee3\u7801\u901a\u8fc7\u6c42\u89e3\u6807\u51c6\u7279\u5f81\u503c\u95ee\u9898\uff0c\u6c42\u5f97\u7279\u5f81\u503c\u548c\u7279\u5f81\u5411\u91cf\u3002Matlab\u5185\u7f6e\u51fd\u6570eig([A])<\/strong>\u53ef\u76f4\u63a5\u6c42\u89e3\u7279\u5f81\u503c\u548c\u7279\u5f81\u5411\u91cf\uff0c\u6c42\u5f97\u7684\u7279\u5f81\u503c\u653e\u5728\u5411\u91cf{Do}<\/strong>\u4e2d\uff0c\u76f8\u5e94\u7684\u7279\u5f81\u5411\u91cf\u653e\u5728\u77e9\u9635[Vo]<\/strong>\u4e2d\u3002\u5176\u540e\u7684\u4ee3\u7801\u7528\u4e8e\u5904\u7406\u6c42\u89e3\u5f97\u5230\u7684\u6570\u636e\uff0c\u5176\u4e2d\u4ee3\u7801\u201cV=\u201d\u7528\u4e8e\u5c06\u7279\u5f81\u5411\u91cf\u6392\u5e8f\uff0c\u4ee3\u7801\u201cf=\u201d\u7528\u4e8e\u5c06\u7ed3\u6784\u81ea\u632f\u9891\u7387\u8fdb\u884c\u6392\u5e8f\uff0c\u4ee3\u7801\u201cfac=\u201d<\/strong>\u7528\u4e8e\u6c42\u5f97\u5c48\u66f2\u56e0\u5b50<\/strong>\uff0c\u5373\u7279\u5f81\u503c\u7684\u5012\u6570\u3002\u56fe 11\u20113\u6240\u793a\u662f\u8be5\u6841\u67b6\u7ed3\u6784\u7684\u524d\u4e09\u9636\u5c48\u66f2\u6a21\u6001\u3002<\/p>\n

\"\"<\/p>\n

(a) Buckling Mode 1<\/strong><\/p>\n

\"\"<\/p>\n

(b) Buckling Mode 2<\/strong><\/p>\n

\"\"<\/p>\n

(c) Buckling Mode 3<\/strong><\/p>\n

\n