PK ä³ÖLñB–H mimetypetext/x-wxmathmlPK ä³ÖL>ÏàM M content.xml
kill(all);assume(EA>0,EJ>0);assume(l>0,dx>0);y,v A | | \\|| F \\|O==============O ----> ---> x,u \\| /_\ --------- ///////x,y global coordinatesu,v displacement according to the global reference systemtwo motion modes are considered.a) an axial elongation, in the absence of tranverse displacementsb) a tranverse motion, sinusoidal with respect to xmotion adefine( u(x) , (a + da)*x);motion b define(v(x) , (0+db) * sin(1*%pi*x/l) );derivatives of the two motion modesdefine( up(x) ,diff(u(x),x));define(vp(x) , diff(v(x),x) );define(vpp(x) , diff(vp(x),x) );auxiliary functions for the nonlinear calculations:- length of an A->B segmentl(XA,YA,XB,YB) := sqrt((XB-XA)^2 + (YB-YA)^2);- cosinus of the angle comprised between an A->B segment, and the x axisc(XA,YA,XB,YB):=(XB-XA)/l(XA,YA,XB,YB);- nonlinear definition for the beam axial straintmp: fullratsimp( sqrt( ( v(x+dx) -v(x))^2 + (x+dx+u(x+dx)-x-u(x))^2 )/dx -1);define(eps_nl(x) , limit(tmp,dx,0,plus) );
*dx2−dxdx(%o11) eps_nl
x
:=−l−%pi2*db2*cos
%pi*xl
2+
1+2*a+a2+
2*a+2
*da+da2*l2ldefine(eps_nl(x) , expand(taylor(eps_nl(x), [a,0,1], [da,0,2] , [db,0,2] )) );I cut&paste only the terms up to the second order in the "small" "a,da,db" variablesdefine(eps_nl(x) ,(%pi^2*db^2*cos((%pi*x)/l)^2)/(2*l^2)+da+a);define(eps_nl(x) ,taylor((sqrt(1+vp(x)^2)-1),db,0,2)+da+a);nonlinear energetic termUb_nl: integrate( 1/2 * EJ * vpp(x)^2 ,x,0,l);Un_nl: integrate( 1/2 * EA * eps_nl(x)^2 ,x,0,l);we calculate the hessian matrix in the neighborhood of the preloaded equilibrium condition, i.e. - da = db = 0;- a = F/EA;- b = 0; such hessian matrix becomes the tangential stiffness matrix, i.e.the matrix that describes the linearized relation between generalized load and displacement configurations. hessian( Ub_nl + Un_nl,[da,db]);K : ev(%,[da=0,db=0]);K : ev(K,a=F/EA);if the following equation holds, the tangential stiffness matrix is singular;such condition paves the way for the bending solution branches to emerge.linsolve(K[2,2]=0,F);the relation above defines the critical load for which solutions appearother than the linear elastic one that was in continuity with theunloaded/undeformed condition.Also, at the critical load the tangent stiffness matrix loses its positivedefiniteness; for loads beyond the critical value, the linear elastic solutiondefines an unstable equilibrium condition.we separate the stiffness matrix into the two following contributionsmaterial stiffness matrixK0:ev(K,F=0);preload related, "geometric" stiffness matrixK1:diff(K,F)*F;this second contribution disappears if the strain derivation from the generalizeddisplacements is linearized.The preload effect is the amplified by a lambda factor, and such condition is assumed equivalent to the effect of an amplified preload. Kt : K0 + lambda * K1;both the Kt terms are left-multiplied by 1/lambda*invert(K0), then the variable substitionxi = -1/lambda is performedto obtain the standard eigenproblem( A - xi*I).x = 0A : invert(K0).K1;I : invert(K0).K0;we now solve the eigenproblemvals : eigenvaluesmult : eigenvalue multiplicitiesvecs : eigenvectors[[vals,mult],vecs]:eigenvectors(A);we focus on the nonzero eigenvaluevals[1];the related eigenvector denotes the bending solution branch that suddenly appears,[da,db] = vecs[1][1];
PK ä³ÖLñB–H mimetypePK ä³ÖL>ÏàM M 5 content.xmlPK o tM