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truss1.m
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truss1.m
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% truss.m - October 15 2003 %
% author: Jean H. Prevost + David Luet %
% analyses of 1,2 and 3D elastic trusses %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
clear; % removes all variables from the workspace.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% DATA %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
nmat=2;
%%%%%%%%%%%%
% Material %
%%%%%%%%%%%%
E(1)=100.; % Young's modulus
E(2)=100.; % Young's modulus
%%%%%%%%%%%%%
% Geometric %
%%%%%%%%%%%%%
A(1)=1.; % Area
A(2)=1.; % Area
%%%%%%%%
% Mesh %
%%%%%%%%
nsd=2; % number of space dimension
ndf=nsd; % number of freedom per node
nen=2; % number of element nodes
nel=5; % number of elements/trusses
nnp=4; % number of nodal points
%%%%%%%%%%%%%%%%%%%%%
% Nodal coordinates %
%%%%%%%%%%%%%%%%%%%%%
% xn(i,N):= coordinate i for node N
% N=1,...,nnp
% i=1,...,nsd
xn=zeros(nsd,nnp);
xn(1,2)=5.0;
xn(2,2)=7.0;
xn(1,3)=11.;
xn(2,3)=7.;
xn(1,4)=16.;
%%%%%%%%%%%%%%%%
% Connectivity %
%%%%%%%%%%%%%%%%
% ien(a,e)=N
% N: global node number - N=1,...,nnp
% e: element number - e=1,...,nel
% a: local node number - a=1,...,nen
ien=zeros(nen,nel);
mat=zeros(nel);
ien(1,1)=1; ien(2,1)=2; mat(1)=1;
ien(1,2)=2; ien(2,2)=3; mat(2)=1;
ien(1,3)=3; ien(2,3)=4; mat(3)=2;
ien(1,4)=1; ien(2,4)=3; mat(4)=2;
ien(1,5)=4; ien(2,5)=2; mat(5)=2;
%%%%%%%%%%%%%%%%%%%%%%%
% Boundary conditions %
%%%%%%%%%%%%%%%%%%%%%%%
% prescribed displacement (essential boundary condition)
%
% idb(i,N)=1 if the degree of freedom i of the node N is prescribed
% =0 otherwise
%
% 1) initialize idb to 0
idb=zeros(ndf,nnp);
% 2) enter the flag for prescribed displacement boundary conditions
idb(1,1)=1;
idb(2,1)=1;
idb(1,4)=1;
idb(2,4)=1;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% prescribed nodal displacement boundary conditions %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% g(i,N): prescribed displacement for the dof i of node N
% initialize g
g=zeros(ndf,nnp);
% enter the values
%%%%%%%%%%%%%%%%%%%%%%%%%%%
% prescribed nodal forces %
%%%%%%%%%%%%%%%%%%%%%%%%%%%
% f(i,N): prescribed force for the dof i of node N
% initialize f
f=zeros(ndf,nnp);
% enter the values
f(2,2)=-1.;
f(2,3)=-1.;
%---------------------------------------------------------------
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% number the equations; build the id table %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
[id,neq]=number_eq(idb,nnp,ndf)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Compute the elemental quantities in the elemental coordinate system %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
for e=1:nel
[Ke(:,:,e),ke(:,:,e),Qe(:,:,e)]=Ke_truss(E(mat(e)),A(mat(e)),xn,ien(:,e),nen,ndf,nsd);
end;
% Contribution of the prescribed displacements to the elemental force vector
% fe=fe-Ke*Ue
fe=zeros(ndf*nen,nel); % fe may be non zero in general
Ue=zeros(ndf*nen,nel);
for e=1:nel
for n=1:nen
for i=1:ndf
Ue(i+(n-1)*ndf,e)=g(i,ien(n,e));
end
end
fe(:,e)=fe(:,e)-Ke(:,:,e)*Ue(:,e);
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Assembly operation %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%----------------
% build K and F
%----------------
K=zeros(neq,neq);
F=zeros(neq,1);
% input the prescribed nodal forces in F
for N=1:nnp
for i=1:ndf
if (id(i,N) > 0)
P=id(i,N);
F(P)=f(i,N);
end
end
end
% compute global K and F
if (neq > 0)
for e=1:nel
K = addstiff(K,id,Ke(:,:,e),ien(:,e),nen,ndf);
F = addforce(F,id,fe(:,e),ien(:,e),nen,ndf);
end
end
% Solve the system
if (neq > 0)
U=K\F;
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% post processing %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%
% complete U %
%%%%%%%%%%%%%%
Ucomp=zeros(ndf,nnp);
for N=1:nnp
for i=1:ndf
if (id(i,N) == 0)
Ucomp(i,N)=g(i,N);
else
P=id(i,N);
Ucomp(i,N)=U(P);
end
end
end
% print results
disp('Nodal Displacements:')
disp(' node d1 d2')
for N=1:nnp
disp(sprintf('%5d %7g %7g',N,Ucomp(:,N)))
end
disp(' ')
%%%%%%%%%%%%%
% REACTIONS %
%%%%%%%%%%%%%
% build the idb table; overwrite original idb table
% idb(i,N): equation number associated with dof i of node N
ineq=0; % number of equations
for i=1:ndf
for N=1:nnp
if (idb(i,N) > 0) % assign an equation number to all prescribed nodes
ineq=ineq+1;
idb(i,N)=ineq;
end;
end;
end;
% Contribution of the displacement to the elemental force vector
% fe=Ke*Ue
for e=1:nel
Ue(:,e)=zeros(ndf*nen,1);
for n=1:nen
for i=1:ndf
Ue(i+(n-1)*ndf,e)=Ucomp(i,ien(n,e));
end
end
fe(:,e)=Ke(:,:,e)*Ue(:,e);
end
% compute reactions R %
R=zeros(ineq,1);
for e=1:nel
R = addforce(R,idb,fe(:,e),ien(:,e),nen,ndf);
end
% Collect reactions
Rcomp=zeros(ndf,nnp);
for N=1:nnp
for i=1:ndf
if (idb(i,N) > 0)
Rcomp(i,N)=R(idb(i,N));
end
end
end
% print results
disp('Nodal Reactions')
disp(' node R1 R2')
for N=1:nnp
disp(sprintf('%5d %7g %7g',N,Rcomp(:,N)))
end
disp(' ')
%%%%%%%%%%%%%%%%%%%%%%%%%
% AXIAL FORCES/STRESSES %
%%%%%%%%%%%%%%%%%%%%%%%%%
for e=1:nel
Ue(:,e)=zeros(ndf*nen,1);
for n=1:nen
for i=1:ndf
Ue(i+(n-1)*ndf,e)=Ucomp(i,ien(n,e));
end
end
if (nsd > 1)
axial(:,e)=ke(:,:,e)*Qe(:,:,e)*Ue(:,e);
else
axial(:,e)=ke(:,:,e)*Qe(e)*Ue(:,e);
end;
stress(e)=axial(2,e)/A(mat(e));
strain(e)=stress(e)/E(mat(e));
end;
% print results
disp('Element Axial force/stress/strain')
disp(' elem force stress strain')
for e=1:nel
disp(sprintf('%5d %7g %7g %7g',e,axial(2,e),stress(e),strain(e)))
end
disp(' ')
%%%%%%%%%%%%%%%%%%%%
% plot the results %
%%%%%%%%%%%%%%%%%%%%
plot_results('truss',xn,f,idb,Ucomp,Rcomp,ien,nel,nen,nsd,ndf,nnp);