Question

The plane truss is made of three members and is loaded with force P as shown. Each member has the same E, A and L as shown below. Determine, a) the displacements at node 1, b) the reaction forces. Show all your work. Truss Element Stiffness Matrix [K]_XY^(e) = AE/L [cos^2 theta sin theta cos theta -cos^2 theta -sin theta cos theta sin theta cos theta sin^2 theta -sin theta cos theta -sin^2 theta -cos^2 theta -sin theta cos theta cos^2 theta sin theta cos theta -sin theta cos theta -sin^2 theta sin theta cos theta sin^2 theta] Recall that if A = [a b c d] then A^-1 = 1/|A| [d -b -c a] {R} = [K]^(G) {U} - {F}

Answer 1

function TrussElementStiffnessMatrixMethod
   elasticity = 10200000; % aluminium modulus elasticity
   area = 200; % area cross-section
   L = 1; % bar length
   elasticity_area = elasticity*area;
   nodes = [1 4; 2 4; 3 4; 4 6; 3 6; 3 5; 5 6; 5 7; 6 7];
   coordinates = [0 1 ;0 0 ;1 1 ;1 0 ;2 1 ;2 0 ;3 1 ];
   elements = size(nodes,1);
   totalNodes = size(coordinates,1);
   x_coord = coordinates(:,1);
   y_coord = coordinates(:,2);
   degree_freedom = 2 * totalNodes;
   u_displacement = zeros(degree_freedom,1); % displacement
   reactionForces = zeros(degree_freedom,1); % solution for (b)
   % gravitational force loads
   reactionForces(4)= -2500;
   reactionForces(5)= -2500;
   reactionForces(6)= -2500;
   reactionForces(7)= -5000;
   % stiffnessMatrix matrix
   [stiffnessMatrix]=formStiffness2Dtruss(degree_freedom,elements,nodes,totalNodes,coordinates,x_coord,y_coord,elasticity_area);
   % boundary-conditions
   prescribed_freedom = [0 -100 0 0 50 0 10]';
   % a) final displacements answer
   displacements = stiffnessMatrix\prescribed_freedom;
   disp('displacements: ')
   disp(displacements)
   fprintf('reactionForces: \n')
   disp(reactionForces)
end
function stiffnessMatrix = formStiffness2Dtruss(degree_freedom,elements,nodes,totalNodes,coordinates,x_coord,y_coord,elasticity_area)
   stiffnessMatrix = zeros(totalNodes);
   for e=1:elements;
       % elementDof: element degrees of freedom (Dof)
       degree_freedom=nodes(e,:) ;
       stiffnessMatrix(degree_freedom,degree_freedom)=stiffnessMatrix(degree_freedom,degree_freedom)+[1 -1;-1 1];
   end
end
%caling method
TrussElementStiffnessMatrixMethod();

OUTPUT