MANE 4240 & CIVL 4240
Introduction to Finite Elements
Prof. Suvranu De
Principles of minimum
potential energy and
Rayleigh-Ritz
Reading assignment:
Section 2.6 + Lecture notes
Summary:
Potential energy of a system
Elastic bar
String in tension
Principle
%this script will solve reactions and displacements for a space truss.
d = zeros([3*nodes,1]);
F = zeros([3*nodes,1]);
disp(' ')
disp(' ')
disp('This part of the program will solve for gloabl reactions and displacements.')
disp(' ')
disp('Label the forces
%Practice with symbols
syms L E I A
k = E*I/L*[1,-1;-1,1];
j = L/L;
r = L^2;
f = L*L;
g = L^2/L;
syms b1 b2
A = sym('a',[2 2]);
A(1,1)=A(1,1)-A(1,1)+L;
i=1;
while i<=2
while j<=2
if A(i,j) = 2;
A(i,j)=0;
else
r=7;
end
j=j+1;
end
i=i+1
%Alexander Tadla
%CEE2330 Homework No. 7
%This program solves Problem 4.22 for Homework No. 7
%This program solves a displacements, slopes, member end forces, and reactions
tart script
clear
clc
%Define Givens
w=4000; %lb/ft
L=10; 0t
E=29*10^6; %psi
Ep=29
%Practice with symbols
syms L E I A
k = E*I/L*[1,-1;-1,1];
j = L/L;
r = L^2;
f = L*L;
g = L^2/L;
syms b1 b2
A = sym('O',[2 2]);
A(1,1)=A(1,1)-A(1,1)+L;
i=1;
while i<=2
j=1;
while j<=2
if size(A(i,j) = 1;
A(i,j)=0;
else
r=7;
end
j=j+1;
%Alexander Tadla
%CEE2330 Homework No. 7
%This program solves Problem 4.22 for Homework No. 7
%This program determines the midspan deflection for the beam and the
%reactions.
tart script
clear
%Define Givens
syms w E I L
%create k_hat matrix
k_hat=E*I/(
%This program will create a global stiffness matrix
%And Calculate reactions and bar forces
%3D problem only with 3 DoF at each node
clear
disp(' ')
disp(' ')
0efine system
members = input('How many members?: ');
nodes = input('How many nodes?: ');
%creat
%Alexander Tadla
%CEE2330 Homework No. 7
%This program solves Problem 4.22 for Homework No. 7
%This program determines the midspan deflection for the beam and the
%reactions.
tart script
clear
%Define Givens
syms w E I L
%create k_hat matrix
k_hat=E*I/(
%Alexander Tadla
%CEE2330 Homework No. 7
%This program solves Problem 4.22 for Homework No. 7
%This program solves a displacements, slopes, member end forces, and reactions
tart script
clear
%Define Givens
w=4000; %lb/ft
L=10; 0t
E=29*10^6; %psi
Ep=29*10^
%This program will create a global stiffness matrix
%And Calculate reactions and bar forces
%3D problem only with 3 DoF at each node
clear
disp(' ')
disp(' ')
0efine system
members = input('How many members?: ');
nodes = input('How many nodes?: ');
%creat
%This program will solve reactions and displacements for a spring system,
%using the direct stiffness method, with known forces
clear
tart script
disp('Label the elements and nodes. Label the nodes with fdof first.')
disp(' ')
disp(' ')
%Define the system
%this script will solve reactions and displacements for a space truss.
d = zeros([3*nodes,1]);
F = zeros([3*nodes,1]);
disp(' ')
disp(' ')
disp('This part of the program will solve for gloabl reactions and displacements.')
disp(' ')
disp('Label the forces
%this script will solve for member forces in a space truss
counter=1;
while counter<=members
cx = eval(['cos_' num2str(counter) '(1,1)']);
cy = eval(['cos_' num2str(counter) '(2,1)']);
cz = eval(['cos_' num2str(counter) '(3,1)']);
T_star = [cx,cy,
%this script will solve for member forces in a space truss
counter=1;
while counter<=members
cx = eval(['cos_' num2str(counter) '(1,1)']);
cy = eval(['cos_' num2str(counter) '(2,1)']);
cz = eval(['cos_' num2str(counter) '(3,1)']);
0isp(cx) %
0isp
%CEE2330 Homework Set No. 3
%February 11, 2013
%This program will create a global stiffness matrix
%And Calculate reactions and bar forces
%2D problem only with 2 DoF at each node
clear
disp(' ')
disp(' ')
members = input('How many members?: ');
nodes = i
%this part of the program will find member forces
counter=1;
while counter<=members
c=cosd(theta_matrix(1,counter);
s=sind(theta_matrix(1,counter);
T = [c,s,0,0;-s,c,0,0;0,0,c,s;0,0,-s,c];
d1x = eval(['d_' num2str(counter) '(1,1)']);
d1y = ev
%this part of the program will find member forces
counter=1;
while counter<=members
c=cosd(theta_matrix(1,counter);
s=sind(theta_matrix(1,counter);
T = [c,s,0,0;-s,c,0,0;0,0,c,s;0,0,-s,c];
d1x = eval(['d_' num2str(counter) '(1,1)']);
d1y = ev
%CEE2330 Homework Set No. 3
%February 11, 2013
%This program will create a global stiffness matrix
%And Calculate reactions and bar forces
%2D problem only with 2 DoF at each node
clear
disp(' ')
disp(' ')
members = input('How many members?: ');
nodes = i
%Alexander Tadla
%CEE2330 Homework No. 6
%This program solves Number 3 for Homework No. 6
%This program solves a displacements, member end forces, and reactions for
%an indeterminate beam with a spring at midspan.
tart script
clear
clc
%Define Givens
w=50
0ind reactions and find displacements of a 2D truss
displacement_matrix = zeros([2*nodes,1]);
force_matrix = zeros([2*nodes,1]);
disp(' ')
disp(' ')
disp('This part of the program will solve for gloabl reactions and displacements.')
disp(' ')
disp('Label
%Alexander Tadla
%CEE2330 Homework No. 6
%This program solves Number 2 for Homework No. 6
%This program solves dispalcements, member end forces, and reactions for an
%indeterminate beam with fixed ends and two rollers along its span.
tart script
clear
clc
%Alexander Tadla
%CEE2330 Homework No. 6
%This program solves Number 1 for Homework No. 6
%This program solves a displacements, member end forces, and reactions for
%an indeterminate beam with a spring at midspan.
tart script
clear
%Define Givens
E=4500;