ex8102.m

The Finite Element Method Using MATLAB

%----------------------------------------------------------------------------%
% Example 8.10.2                                                             %
% to solve the natural frequencies of a beam using beam elements with        %
% displacement degrees of freedom only                                       %
%                                                                            %
% Problem description                                                        %
%   Find the natural frequencies of a clamped beam whose length is 1 m.      %
%   long. The beam has the cross-section of 0.02 m by 0.02 m and the mass density            %
%   is 1000 Kg/m^3. The elastic and shear modulus is 100 GPAa and         %
%   40 GPa, respectively. Use 4 elements.                                %
%                                                                            %
% Variable descriptions                                                      %
%   k = element stiffness matrix                                             %
%   kk = system stiffness matrix                                             %
%   m = element mass matrix                                                  %
%   mm = system mass matrix                                                  %
%   index = a vector containing system dofs associated with each element     %
%   bcdof = a vector containing dofs associated with boundary conditions     %
%----------------------------------------------------------------------------% 

clear
nel=4;           % number of elements
nnel=2;          % number of nodes per element
ndof=3;          % number of dofs per node
nnode=(nnel-1)*nel+1;   % total number of nodes in system
sdof=nnode*ndof; % total system dofs  

el=100*10^9;     % elastic modulus
sh=40*10^9;      % shear modulus
tleng=1;         % total beam length
leng=tleng/nel;  % same size of beam elements
heig=0.02;       % height (or thickness) of the beam
width=0.02;      % width of the beam
rho=1000;        % mass density of the beam

bcdof(1)=1;      % bottom inplane displ. at node 1 is constrained
bcdof(2)=2;      % top inplane displ. at node 1 is constrained
bcdof(3)=3;      % transverse displ. at node 1 is constrained

kk=zeros(sdof,sdof);  % initialization of system stiffness matrix
mm=zeros(sdof,sdof);  % initialization of system mass matrix
index=zeros(nel*ndof,1);  % initialization of index vector

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

index=feeldof1(iel,nnel,ndof);  % extract system dofs associated with element

[k,m]=febeam3(el,sh,leng,heig,width,rho); % compute element stiffness matrix

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

mm=feasmbl1(mm,m,index); % assemble each element matrix into system matrix

end

[kn,mn]=feaplycs(kk,mm,bcdof);  % apply the boundary conditions

fsol=eig(kn,mn);   % solve the matrix equation and print
fsol=sqrt(fsol)


%_____________________________________________________________________