For the truss shown in Figure 1 use the numbering scheme ind

For the truss shown in Figure 1 use the numbering scheme indicated to solve steps (a)-(g). Use the element properties tabulated in the figure and take the origin to be at node 3. Please clearly state the answers to each part. The solutions should be obtained using MATLAB and hand calculations where appropriate. Append the printed text of your MATLAB m-files with your solutions. If your algorithm doesn\'t compile, or compiles with errors or warnings, please explain what troubleshooting steps you took. For the truss shown in Figure 1 answer (a)-(g): (a) Compute and report the basic element stiffness K^(e) for each of the elements in the truss in N/m. (b) Derive the Boolean transformation matrices A_bf^T and A_bd^T by inspection for each element using the exploded joint-and-element equilibrium equations. (c) Compute the element stiffness matrices using MATLAB, in global coordinates K^(e) in N/m. Be sure to apply the correct rotation and rigid body mode transformations. (d) Assemble and report the global stiffness matrix K_ff in N/m, then compute and report the unknown displacements u_f in meters using the Boolean matrix method in MATLAB. (e) Compute (in MATLAB) and report the bar forces in each clement F^(e) in Newtons. (f) Compute (in MATLAB) and report the reaction forces P_d in Newtons. (g) Report the element forces F^(e) in global coordinates and verify equilibrium at the joints by hand with free-body diagrams referring to your exploded equilibrium sketch.

Solution

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% Finite element program for truss analysis %%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
clear all;
clc;
Nodes = [9000 10000;9000 0;0 0]; %x, y coordinates
Elements = [1 2 200000 500;2 3 70000 2500; 3 1 200000 500]; % first node, second node, E, A

numnode = 3; %number of nodes
numelem = 3; %number of elements

KG = zeros(2*numnode,2*numnode); % Global stiffness matrix
F = zeros(2*numnode, 1); % Load vector
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
act = 1:2*numnode; %Holds active DOFs
act([4 5 6]) = [];
F(1) = 32000;
F(2) = 5000;

ElemStiff = zeros(numelem,1) %Basic element stiffness

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;
ms = (Y2-Y1)/L;
ls=(X2-X1)/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];
Trans2 = (Trans1)\';
Kelem = Trans1*[ElemStiff(i) -ElemStiff(i);-ElemStiff(i) ElemStiff(i)]*Trans2
KG(DOFs,DOFs) = KG(DOFs,DOFs) + Kelem; % Calculates the element stiffness matrix and assembles it to the global stiffness matrix
%K(DOFs,DOFs) = K(DOFs,DOFs) + ElemStiff(i)*[ls^2 ls*ms -ls^2 -ls*ms; ls*ms ms^2 -ls*ms -ms^2; -ls^2 -ls*ms ls^2 ls*ms; -ls*ms -ms^2 ls*ms ms^2]; % Calculates the element stiffness matrix and assembles it to the global stiffness matrix
end

U(act) = KG(act,act)\\F(act)

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