## MATLAB Codes for Finite Element Analysis: Solids and StructuresThis book intend to supply readers with some MATLAB codes for ?nite element analysis of solids and structures. After a short introduction to MATLAB, the book illustrates the ?nite element implementation of some problems by simple scripts and functions. The following problems are discussed: • Discrete systems, such as springs and bars • Beams and frames in bending in 2D and 3D • Plane stress problems • Plates in bending • Free vibration of Timoshenko beams and Mindlin plates, including laminated composites • Buckling of Timoshenko beams and Mindlin plates The book does not intends to give a deep insight into the ?nite element details, just the basic equations so that the user can modify the codes. The book was prepared for undergraduate science and engineering students, although it may be useful for graduate students. TheMATLABcodesofthisbookareincludedinthedisk.Readersarewelcomed to use them freely. The author does not guarantee that the codes are error-free, although a major e?ort was taken to verify all of them. Users should use MATLAB 7.0 or greater when running these codes. Any suggestions or corrections are welcomed by an email to ferreira@fe.up.pt. |

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### Contents

1 | |

2 | |

3 | |

4 | |

5 | |

6 | |

19 Scalar functions | 7 |

110 Vector functions | 8 |

53 A second 3D truss example | 73 |

Bernoulli beams | 78 |

62 Bernoulli beam problem | 81 |

63 Bernoulli beam with spring | 85 |

2D frames | 89 |

73 Another example of 2D frame | 95 |

Analysis of grids | 113 |

92 A ﬁrst grid example | 116 |

111 Matrix functions | 9 |

112 Submatrix | 10 |

113 Logical indexing | 12 |

114 Mﬁles scripts and functions | 13 |

1151 2D plots | 14 |

1152 3D plots | 15 |

116 Linear algebra | 16 |

Discrete systems | 19 |

23 Equilibrium at nodes | 20 |

24 Some basic steps | 21 |

Analysis of bars | 33 |

34 Problem 2 using MATLAB struct | 41 |

35 Problem 3 | 44 |

Analysis of 2D trusses | 50 |

43 Stiffness matrix | 52 |

44 Stresses at the element | 53 |

Trusses in 3D space | 69 |

93 A second grid example | 119 |

Analysis of Timoshenko beams | 122 |

Plane stress | 143 |

113 Boundary conditions | 144 |

114 Potential energy | 145 |

117 Element energy | 146 |

118 Quadrilateral element Q4 | 147 |

plate in traction | 149 |

beam in bending | 152 |

Analysis of Mindlin plates | 161 |

1221 Strains | 162 |

1222 Stresses | 163 |

a square Mindlin plate in bending | 165 |

Laminated plates | 202 |

231 | |

233 | |

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MATLAB Codes for Finite Element Analysis: Solids and Structures A. J. M. Ferreira No preview available - 2014 |

### Common terms and phrases

2D frame 2D truss A.J.M. Ferreira antonio ferreira 2008 boundary conditions buckling CCCF clear memory clear codes for Finite conditions and solution coordinates and connectivities cycle for element cycle for Gauss deﬁned deformed shape degrees of freedom derivatives shapeFunction,naturalDerivatives]=shapeFunctionQ4(xi,eta derivatives w.r.t. x,y displacement vector element degrees elementDof elementNodes eta=GaussPoint(2 ﬁnite element Finite Element Analysis ﬁrst fixedNodeW force vector force=zeros(GDof,1 free vibrations freedom Dof freedom GDof=3*numberNodes functions and derivatives Gauss point end Gauss quadrature GDof GDof,prescribedDof GDof=2*numberNodes global number illustrated in ﬁgure indice=elementNodes(e inverse of Jacobian Jacobian matrix Jacobian(nodeCoordinates(indice,:),naturalDerivatives length of bar MATLAB codes modulus of elasticity natural frequencies ndof=length(indice number of degrees number of elements numberElements numberNodes obtained output displacements/reactions outputDisplacementsReactions(displacements,stiffness plate with h/a Q4 elements shape functions shear modulus simply-supported Solid Mechanics Solids and Structures solution displacements=solution(GDof,prescribedDof,stiffness,force Springer Science+Business Media stiﬀness stiffness(elementDof,elementDof strain energy stresses system stiffness matrix Timoshenko beam w.r.t. x,y Jacob,invJacobian,XYderivatives xi=GaussPoint(1 xx=nodeCoordinates(:,1 XYderivatives(:,2 yy=nodeCoordinates(:,2