ex741.m

The Finite Element Method Using MATLAB

%----------------------------------------------------------------------------%
% Example 7.4.1                                                              
% to solve static 2-d truss structure
%                                                                            
% Problem description                                                        
%   Find the deflection and stress of the truss made of two members           
%   as shown in Fig. 7.4.1.           
%                                                                            
% Variable descriptions                                                      
%   k = element stiffness matrix                                             
%   kk = system stiffness matrix                                             
%   ff = system force vector                                                 
%   index = a vector containing system dofs associated with each element     
%   gcoord = global coordinate matrix
%   disp = nodal displacement vector
%   elforce = element force vector
%   eldisp = element nodal displacement
%   stress = stress vector for every element
%   elprop = element property matrix
%   nodes = nodal connectivity matrix for each element
%   bcdof = a vector containing dofs associated with boundary conditions     
%   bcval = a vector containing boundary condition values associated with    
%           the dofs in 'bcdof'                                              
%----------------------------------------------------------------------------%            

%---------------------------
%  control input data
%---------------------------

clear
nel=2;           % number of elements
nnel=2;          % number of nodes per element
ndof=2;          % number of dofs per node
nnode=3;         % total number of nodes in system
sdof=nnode*ndof; % total system dofs  

%---------------------------
%  nodal coordinates
%---------------------------

gcoord(1,1)=0.0;  gcoord(1,2)=0.0;   % x, y-coordinate of node 1
gcoord(2,1)=10.0; gcoord(2,2)=0.0;   % x, y-coordinate of node 2
gcoord(3,1)=0.0;  gcoord(3,2)=10.0;  % x, y-coordinate of node 3

%------------------------------------------
%  material and geometric properties
%------------------------------------------

elprop(1,1)=30000000;  % elastic modulus of 1st element
elprop(1,2)=0.4;       % cross-section of 1st element
elprop(2,1)=30000000;  % elastic modulus of 2nd element
elprop(2,2)=0.5;       % cross-section of 2nd element

%-----------------------------
%  nodal connectivity
%-----------------------------

nodes(1,1)=1;  nodes(1,2)=2;   % nodes associated with element 1
nodes(2,1)=2;  nodes(2,2)=3;   % nodes associated with element 2

%-----------------------------
%  applied constraints
%-----------------------------

bcdof(1)=1;      % 1st dof (horizontal displ) is constrained
bcval(1)=0;      % whose described value is 0 
bcdof(2)=2;      % 2nd dof (vertical displ) is constrained
bcval(2)=0;      % whose described value is 0
bcdof(3)=5;      % 5th dof (horizontal displ) is constrained
bcval(3)=0;      % whose described value is 0 
bcdof(4)=6;      % 6th dof (vertical displ) is constrained
bcval(4)=0;      % whose described value is 0

%----------------------------
%  initialization to zero
%----------------------------

ff=zeros(sdof,1);              % system force vector
kk=zeros(sdof,sdof);           % system stiffness matrix
index=zeros(nnel*ndof,1);      % index vector
elforce=zeros(nnel*ndof,1);    % element force vector
eldisp=zeros(nnel*ndof,1);     % element nodal displacement vector
k=zeros(nnel*ndof,nnel*ndof);  % element stiffness matrix
stress=zeros(nel,1);           % stress vector for every element
%-----------------------------
%  applied nodal force
%-----------------------------

ff(4)=-1000;     % 2nd node has 1000 lb in downward direction

%--------------------------
%  loop for elements
%--------------------------

for iel=1:nel    % loop for the total number of elements

nd(1)=nodes(iel,1);   % 1st connected node for the (iel)-th element
nd(2)=nodes(iel,2);   % 2nd connected node for the (iel)-th element

x1=gcoord(nd(1),1); y1=gcoord(nd(1),2);  % coordinate of 1st node
x2=gcoord(nd(2),1); y2=gcoord(nd(2),2);  % coordinate of 2nd node

leng=sqrt((x2-x1)^2+(y2-y1)^2);  % element length

if (x2-x1)==0; 
if y2>y1;   
   beta=2*atan(1);       % angle between local and global axes
else
   beta=-2*atan(1);
end
else
beta=atan((y2-y1)/(x2-x1));
end

el=elprop(iel,1);               % extract elastic modulus
area=elprop(iel,2);             % extract cross-sectional area

index=feeldof(nd,nnel,ndof);  % extract system dofs for the element

k=fetruss2(el,leng,area,0,beta,1); % compute element matrix

kk=feasmbl1(kk,k,index);           % assemble into system matrix

end

%---------------------------------------------------
%  apply constraints and solve the matrix
%---------------------------------------------------

[kk,ff]=feaplyc2(kk,ff,bcdof,bcval);  % apply the boundary conditions

disp=kk\ff;   % solve the matrix equation to find nodal displacements

%--------------------------------------------------
%  post computation for stress calculation
%--------------------------------------------------

for iel=1:nel         % loop for the total number of elements

nd(1)=nodes(iel,1);   % 1st connected node for the (iel)-th element
nd(2)=nodes(iel,2);   % 2nd connected node for the (iel)-th element

x1=gcoord(nd(1),1); y1=gcoord(nd(1),2);  % coordinate of 1st node
x2=gcoord(nd(2),1); y2=gcoord(nd(2),2);  % coordinate of 2nd node

leng=sqrt((x2-x1)^2+(y2-y1)^2);  % element length

if (x2-x1)==0; 
if y2>y1;   
   beta=2*atan(1);       % angle between local and global axes
else
   beta=-2*atan(1);
end
else
beta=atan((y2-y1)/(x2-x1));
end

el=elprop(iel,1);               % extract elastic modulus
area=elprop(iel,2);             % extract cross-sectional area

index=feeldof(nd,nnel,ndof);  % extract system dofs for the element

k=fetruss2(el,leng,area,0,beta,1); % compute element matrix

for i=1:(nnel*ndof)           % extract displacements associated with
eldisp(i)=disp(index(i));     % (iel)-th element
end

elforce=k*eldisp;             % element force vector
stress(iel)=sqrt(elforce(1)^2+elforce(2)^2)/area; % stress calculation 

if ((x2-x1)*elforce(3)) < 0;
stress(iel)=-stress(iel);
end

end

%----------------------------
% print fem solutions
%----------------------------

num=1:1:sdof;
displ=[num' disp]          % print displacements

numm=1:1:nel;
stresses=[numm' stress]    % print stresses

%--------------------------------------------------------------------