Solve using MatLab For the rigid frames shown in Figure dete

Solve using MatLab

For the rigid frames shown in Figure, determine the displacements and rotations of the nodes, the element forces, and the reactions. The values of E;A, and I to be used are listed next to each figure.

Solution

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% Finite element program for truss analysis %%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
clear all;
clc;

%%%%%%%%%%%%% Edit the following data %%%%%%%%%%%%%%
numnode = 3; %number of nodes
numelem = 2; %number of elements
Nodes = [0 0; 7.071 7.071; 22.071 7.071]; %x, y coordinates
Elements = [1 2 30e6 5;2 3 30e6 5]; % first node, second node, E, A

% Boundary conditions
act = 1:2*numnode; % Total DOF
act([1 2 5 6]) = []; %Boundary conditions: here u1x, u1y, u3x, u3y are zero

% Load: Load needs to be defined for the DOF\'s where no BC is defined
F = zeros(2*numnode, 1); % Load vector
F(3) = 0;
F(4) = -15000; %Total pressure load of 30000 lb is divided into node 2 and 3
%%%%%%%%%%%%%%%%%%% Inputs completed %%%%%%%%%%%%%%%%

KG = zeros(2*numnode,2*numnode); % Global stiffness matrix
U = zeros(2*numnode, 1); % displacement vector
strain = zeros(numelem,1); % strain in members
stress= zeros(numelem,1); % stress in members
axialforce =zeros(numelem,1); % axial force in members
ElemStiff = zeros(numelem,1); %Basic element stiffness

%%%%%%%%%%%%%%%%%%% Stiffness Matrix Calculation %%%%%%%%%%%%%%%%%%%%
for i = 1:numelem
  
DOFs = [2*Elements(i, 1)-1, 2*Elements(i, 1), 2*Elements(i, 2)-1, 2*Elements(i, 2)]; %Holds element’s DOFs
X1 = Nodes(Elements(i,1), 1);
Y1 = Nodes(Elements(i,1), 2);
X2 = Nodes(Elements(i,2), 1);
Y2 = Nodes(Elements(i,2), 2);
L = sqrt((X2-X1)^2+(Y2-Y1)^2); %Length of each element
s = (Y2-Y1)/L;
c=(X2-X1)/L;
ms = (Y2-Y1)/L;

E = Elements(i,3); %Modulus of elasticiy of element
A = Elements(i,4); %Cross sectional area of element
ElemStiff(i) = A*E/L; % Basic element stiffness for each element
Trans1 = [c 0;s 0;0 c;0 s]; % Transformation matrix
Trans2 = (Trans1)\';
Kelem = Trans1*[ElemStiff(i) -ElemStiff(i);-ElemStiff(i) ElemStiff(i)]*Trans2;% Element stiffness matrix
KG(DOFs,DOFs) = KG(DOFs,DOFs) + Kelem; % Global stiffness matrix
end
%%%%%%%%%%%%%%%%%%%% End of Stiffness Matrix Calculation %%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%% Unknown displacement calculations %%%%%%%%%%%%%%%%%
disp(\'============= 2D-Truss analysis Outputs ============\');
disp(\'================ Nodal Displacements ===============\');
U(act) = KG(act,act)\\F(act)

%%%%%%%%%%%%%%%%% Elemental force calculation %%%%%%%%%%%%%%%%%%%%%
for i = 1:numelem
DOFs = [2*Elements(i, 1)-1, 2*Elements(i, 1), 2*Elements(i, 2)-1, 2*Elements(i, 2)]; %Holds element’s DOFs
X1 = Nodes(Elements(i,1), 1);
Y1 = Nodes(Elements(i,1), 2);
X2 = Nodes(Elements(i,2), 1);
Y2 = Nodes(Elements(i,2), 2);
L = sqrt((X2-X1)^2+(Y2-Y1)^2); %Holds length of element
s = (Y2-Y1)/L;
c=(X2-X1)/L;
d = [c s 0 0; 0 0 c s]*U(DOFs);
strain(i) = (d(2) - d(1))/L;
stress(i)= Elements(i, 3)*strain(i);
axialforce(i) = stress(i)*Elements(i,4);
end

%%%%%%%%%%%%%%% New position of nodes %%%%%%%%%%%%
X1 = Nodes (1,1) + U(1);
X2 = Nodes (2,1) + U(3);
X3 = Nodes (3,1) + U(5);
Y1 = Nodes (1,2) + U(2);
Y2 = Nodes (2,2) + U(4);
Y3 = Nodes (3,2) + U(6);

% Rotation of elements in degree
% As there is no nodal rotation DOF, so nodal rotation is not defined
disp(\'=========== Elemental Rotations in degree =========\');
theta1 = 45-(180/pi)*atan((Y2-Y1)/(X2-X1))
theta2 = (180/pi)*atan((Y3-Y2)/(X3-X2))

disp(\'============== Elemental Forces ====================\');
axialforce

Reaction = KG*U;
Reaction(3) = 0;
Reaction(4) = 0;
Reaction(6) = Reaction(6)+15000;
disp(\'========== Reaction at the supports =================\');
Reaction

Results/Outputs:

============= 2D-Truss analysis Outputs ============
================ Nodal Displacements ===============

U =

0
0
0.0015
-0.0035
0
0

=========== Elemental Rotations in degree =========

theta1 =

0.0203

theta2 =

0.0134

============== Elemental Forces ====================

axialforce =

1.0e+04 *

-2.1213
-1.5000

========== Reaction at the supports =================

Reaction =

1.0e+04 *

1.5000
1.5000
0
0
-1.5000
1.5000