Finite Element Programming with MATLAB Lecture-13 MATLAB Codes
Learn about finite element analysis using MATLAB codes for spring elements in a structural problem. Understand the steps, boundary conditions, system stiffness matrix computation, and solution implementation. Explore the MATLAB code snippets provided for structure, displacements, forces, stiffness, and more.
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Finite element programming with Matlab Lecture-13 MATLAB Codes for Finite Element Analysis (Spring Elements) Dr. ReemAlsehnawi
we consider a problem, illustrated in figure 2.2 where the central bar is defined as rigid. Our problem has three finite elements and four nodes. Three clamped, being the boundary conditions defined as u1 = u3 = u4 = 0. In order to solve this problem, we set k = 1 for all springs and the external applied load at node 2 to be P = 10. nodes are 3
and then obtain the static global equilibrium equations in the form the boundary conditions u1 = u3 = u4 = 0, we may write 4
and then obtain the static global equilibrium equations in the form 5
MATLAB Code: % MATLAB codes for Finite Element Analysis % antonio ferreira 2008 % clear memory clear all % elementNodes: connections at elements elementNodes=[1 2;2 3;2 4]; In this problem, the number of nodes is the same as the number of degrees of freedom % numberElements: number of Elements numberElements=size(elementNodes,1); % numberNodes: number of nodes numberNodes=4; 6
MATLAB Code: % for structure: % displacements: displacement vector % force : force vector % stiffness: stiffness matrix displacements=zeros(numberNodes,1); force=zeros(numberNodes,1); stiffness=zeros(numberNodes); % applied load at node 2 force(2)=10.0; place the applied force at the corresponding degree of freedom: 7
MATLAB Code: % computation of the system stiffness matrix for e=1:numberElements; % elementDof: element degrees of freedom (Dof) elementDof=elementNodes(e,:) ; stiffness(elementDof,elementDof)=... stiffness(elementDof,elementDof)+[1 -1;-1 1]; end % boundary conditions and solution % prescribed dofs prescribedDof=[1;3;4]; % free Dof : activeDof activeDof=setdiff([1:numberNodes] ,[prescribedDof]) ; 8
MATLAB Code: % solution displacements=stiffness(activeDof,activeDof)\force(activeDof); % positioning all displacements displacements1=zeros(numberNodes,1); displacements1(activeDof)=displacements; % output displacements/reactions outputDisplacementsReactions(displacements1,stiffness,... numberNodes,prescribedDof) 9
displacements and reactions Function % to output displacements and reactions: %.............................................................. function outputDisplacementsReactions... (displacements,stiffness,GDof,prescribedDof) % output of displacements and reactions in tabular form % GDof: total number of degrees of freedom of the problem % displacements disp( Displacements ) %displacements=displacements1; jj=1:GDof; format [jj displacements] % reactions F=stiffness*displacements; reactions=F(prescribedDof); disp( reactions ) [prescribedDof reactions] 10
Results: for example displacements and reactions: Displacements ans = 1.0000 0 2.0000 3.3333 3.0000 0 4.0000 0 Reactions ans = 1.0000 -3.3333 3.0000 -3.3333 4.0000 -3.3333 11
