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 ebook illustrates the ?nite aspect implementation of a few difficulties by way of uncomplicated scripts and features. the next difficulties are mentioned: • Discrete structures, equivalent to springs and bars • Beams and frames in bending in second and 3D • aircraft tension difficulties • Plates in bending • loose 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 offer a deep perception into the ?nite aspect info, simply the fundamental equations in order that the consumer can adjust the codes. The booklet was once ready for undergraduate technological know-how and engineering scholars, even though it will be priceless for graduate scholars. TheMATLABcodesofthisbookareincludedinthedisk.Readersarewelcomed to exploit them freely. the writer doesn't be sure that the codes are error-free, even if a tremendous e?ort was once taken to make sure them all. clients may still use MATLAB 7.0 or larger while working those codes. Any feedback or corrections are welcomed by way of an e-mail to firstname.lastname@example.org.
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Extra resources for MATLAB Codes for Finite Element Analysis - Solids and Structures
We need to impose essential boundary conditions before solving the system Ka = f . The lines and columns of the prescribed degrees of freedom, as well as the lines of the force vector will be eliminated at this stage. 5 First problem and ﬁrst MATLAB code 27 First we deﬁne vector prescribedDof, corresponding to the prescribed degrees of freedom. Then we deﬁne a vector containing all activeDof degrees of freedom, by setting up the diﬀerence between all degrees of freedom and the prescribed ones. The MATLAB function setdiff allows this operation.
Stiffness(elementDof,elementDof)+ea(e)*[1 -1;-1 1]; end % boundary conditions and solution 46 3 Analysis of bars % prescribed dofs prescribedDof=[1;4]; % free Dof : activeDof activeDof=setdiff([1:numberNodes]’,[prescribedDof]); % solution GDof=4; displacements=solution(GDof,prescribedDof,stiffness,force); % output displacements/reactions outputDisplacementsReactions(displacements,stiffness,... m. %................................................................ m ref: D. 0; % computation of the system stiffness matrix for e=1:numberElements; % elementDof: element degrees of freedom (Dof) elementDof=elementNodes(e,:) ; if e<3 % bar elements nn=length(elementDof); length_element=nodeCoordinates(elementDof(2))...
NaturalDerivatives=[-1;1]/2; end % end function shapeFunctionL2 Results are placed in structure p1, as before. 6. m, using direct stiﬀness method. %................................................................ 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)=...