This Matlab program gives the solution to problem 11.3-12 of the
textbook "Mechanics of Materials", by Roy R. Craig, Jr., 2nd. edition, 2000.
Units are US customary system, Lb and inch.
Areas of elements:
Element (1): 1 in^2
Element (2): 1 in^2
Element (3): 1.5 in^2
Element (4): 1.5 in^2
NODES OF STRUCTURE (TRUSS):
Node A = Node 1. Coordinates: (0, 0, 0)
Node B = Node 2. Coordinates: (40, 0, 0)
Node C = Node 3. Coordinates: (0, -30, 0)
Node D = Node 4. Coordinates: (70, -30, 0)
All elements are straight links of steel, with modulus of elasticity E = 30E6 Lb/in^2.
The structure is a truss, with pins joining the elements at the nodes A, B, C and D.
ANSYS SOLUTION FOR LOAD STEP (LS) 1:
Matlab program:
______________
% craig_11_3_12.m
% Ricardo E. Avila Montoya / Luis Adan Villa Chaparro
% University of Ciudad Juarez, Mexico . 9 May 2014
% e-mail: ricardo_avila@hotmail.com% Chihuahua, Mexico
% The method of virtual work is applied to solve a 2D
% structure, formed with elastic links pinned at their ends.
clc
clear
A = [1; 1; 1.5; 1.5]; % in^2, Area of link elements.
E = 30e6; % Lb/in^2, Elastic modulus of steel.
L = [40; 42.426; 50; 70]; % Lengths of link elements.
K = A * E./L; % Calculate element stiffness.
% Initialize the Stiffness Matrix
SM = zeros(4, 4);
% Build the Stiffness Matrix
SM(1, 1) = K(1) + .5 * K(2) + .64 * K(3);
SM(1, 2) = - .5 * K(2) + .48 * K(3);
SM(1, 3) = - .5 * K(2);
SM(1, 4) = .5 * K(2);
SM(2, 1) = SM(1, 2);
SM(2, 2) = .5 * K(2) + .36 * K(3);
SM(2, 3) = .5 * K(2);
SM(2, 4) = - .5 * K(2);
SM(3, 1) = SM(1, 3);
SM(3, 2) = SM(2, 3);
SM(3, 3) = .5 * K(2) + K(4);
SM(3, 4) = - .5 * K(2);
SM(4, 1) = SM(1, 4);
SM(4, 2) = SM(2, 4);
SM(4, 3) = SM(3, 4);
SM(4, 4) = .5 * K(2);
% Forces applied on the structure
RHS = [0; 0; 0; -2000];
% Calculate and display displacements of nodes B and D
disp('Displacement of nodes B and D, inch')
format long
% Solve the system of linear equations using Matlab algorithm
% (use the inverted backlash Matlab operator for matrix solution)
U = SM\RHS
% Calculation of forces
K2_sqrt_2 = K(2)/sqrt(2); % Auxiliary arithmetic operation
Force_Matrix = [ K(1) 0 0 0
-K2_sqrt_2 K2_sqrt_2 K2_sqrt_2 -K2_sqrt_2
.8*K(3) .6*K(3) 0 0
0 0 K(4) 0 ];
% Calculate and display vector of forces for the elements
Force_vector = Force_Matrix * U
% Calculate stress in link elements of 2D structure:
Stress_in_elements = Force_vector ./ A
disp('________________________________')
disp('Successful run of Matlab program')
_______________________________________________________________
ANSYS ANIMATION OF DEFORMED RESULTS:
___________________________________________________________________
ANSYS results for displacements of the nodes:
ANSYS results for displacements of the nodes:
PRINT U NODAL SOLUTION PER NODE
***** POST1 NODAL DEGREE OF FREEDOM LISTING *****
LOAD STEP= 2 SUBSTEP= 1
TIME= 2.0000 LOAD CASE= 0
THE FOLLOWING DEGREE OF FREEDOM RESULTS ARE IN GLOBAL COORDINATES
NODE UX UY UZ USUM
1 0.0000 0.0000 0.0000 0.0000
2 0.62222E-02-0.14469E-01 0.0000 0.15750E-01
3 0.0000 0.0000 0.0000 0.0000
4 -0.31111E-02-0.29459E-01 0.0000 0.29623E-01
MAXIMUM ABSOLUTE VALUES
NODE 2 4 0 4
VALUE 0.62222E-02-0.29459E-01 0.0000 0.29623E-01
_________________________________________________________________
ANSYS results of forces and stresses in the elements of the truss:
PRINT ELEMENT TABLE ITEMS PER ELEMENT
***** POST1 ELEMENT TABLE LISTING *****
STAT CURRENT CURRENT
ELEM FORCE STRESS
1 4666.7 4666.7
2 2828.4 2828.4
3 -3333.3 -2222.2
4 -2000.0 -1333.3
MINIMUM VALUES
ELEM 3 3
VALUE -3333.3 -2222.2
MAXIMUM VALUES
ELEM 1 1
VALUE 4666.7 4666.7
