By A. J. M. Ferreira

This e-book intend to provide readers with a few MATLAB codes for ?nite aspect research of solids and constructions. After a brief creation to MATLAB, the e-book illustrates the ?nite aspect implementation of a few difficulties via easy scripts and services. the next difficulties are mentioned: • Discrete structures, akin to springs and bars • Beams and frames in bending in 2nd and 3D • aircraft pressure difficulties • Plates in bending • unfastened vibration of Timoshenko beams and Mindlin plates, together with laminated composites • Buckling of Timoshenko beams and Mindlin plates The e-book doesn't intends to provide a deep perception into the ?nite point information, simply the fundamental equations in order that the consumer can adjust the codes. The ebook used to be ready for undergraduate technology and engineering scholars, even though it might be priceless for graduate scholars. TheMATLABcodesofthisbookareincludedinthedisk.Readersarewelcomed to exploit them freely. the writer doesn't ensure that the codes are error-free, even if a big e?ort used to be taken to ensure them all. clients should still use MATLAB 7.0 or better whilst working those codes. Any feedback or corrections are welcomed by means of an electronic mail to ferreira@fe.up.pt.

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Calculation 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 In the first line of the cycle, we inspect the degrees of freedom at each element, in a vector elementDof. For example, for element 1, elementDof =[1,2], for element 2, elementDof =[2 3] and so on. % elementDof: element degrees of freedom (Dof) elementDof=elementNodes(e,:) ; Next we state that the stiffness matrix for each element is constant and then we perform the assembly process by “spreading” this 2 × 2 matrix at the corresponding lines and columns defined by elementDof , stiffness(elementDof,elementDof)=...

M ref: D. Logan, A first couse in the finite element method, third Edition, page 121, exercise P3-10 direct stiffness method antonio ferreira 2008 8kN E = 70GPa A = 200mm2 k 1 2 3 1 2 2m 2m Fig. 5 Problem 3 45 % clear memory clear all % E; modulus of elasticity % A: area of cross section % L: length of bar % k: spring stiffness E=70000;A=200;k=2000; % generation of coordinates and connectivities % numberElements: number of elements numberElements=3; numberNodes=4; elementNodes=[1 2; 2 3; 3 4]; nodeCoordinates=[0 2000 4000 4000]; xx=nodeCoordinates; % for structure: % displacements: displacement vector % force : force vector % stiffness: stiffness matrix displacements=zeros(numberNodes,1); force=zeros(numberNodes,1); stiffness=zeros(numberNodes,numberNodes); % applied load at node 2 force(2)=8000; % computation of the system stiffness matrix for e=1:numberElements; % elementDof: element degrees of freedom (Dof) elementDof=elementNodes(e,:) ; L=nodeCoordinates(elementDof(2))-nodeCoordinates(elementDof(1)); if e<3 ea(e)=E*A/L; else ea(e)=k; end stiffness(elementDof,elementDof)=...

Logan, A first couse in the finite element method, third Edition, page 121, exercise P3-10 direct stiffness method antonio ferreira 2008 8kN E = 70GPa A = 200mm2 k 1 2 3 1 2 2m 2m Fig. 5 Problem 3 45 % clear memory clear all % E; modulus of elasticity % A: area of cross section % L: length of bar % k: spring stiffness E=70000;A=200;k=2000; % generation of coordinates and connectivities % numberElements: number of elements numberElements=3; numberNodes=4; elementNodes=[1 2; 2 3; 3 4]; nodeCoordinates=[0 2000 4000 4000]; xx=nodeCoordinates; % for structure: % displacements: displacement vector % force : force vector % stiffness: stiffness matrix displacements=zeros(numberNodes,1); force=zeros(numberNodes,1); stiffness=zeros(numberNodes,numberNodes); % applied load at node 2 force(2)=8000; % computation of the system stiffness matrix for e=1:numberElements; % elementDof: element degrees of freedom (Dof) elementDof=elementNodes(e,:) ; L=nodeCoordinates(elementDof(2))-nodeCoordinates(elementDof(1)); if e<3 ea(e)=E*A/L; else ea(e)=k; end stiffness(elementDof,elementDof)=...

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