MATLAB Codes for Finite Element Analysis

Frontmatter

Chapter 1. Short Introduction to MATLAB

Abstract

This chapter introduces MATLAB by presenting programs that investigate elementary mathematical problems. The primarily objective is to learn quickly the first steps. The emphasis here is “learning by doing”. Therefore, the best way to learn is by trying it yourself. Working through the examples will give you a feel for the way that MATLAB operates. In this introduction we will describe how MATLAB handles simple numerical expressions and mathematical formulas.

Antonio J. M. Ferreira, Nicholas Fantuzzi

Chapter 2. Discrete Systems

Abstract

In this chapter some basic concepts of the finite element method are illustrated by solving basic discrete systems built from springs and bars. Generation of element stiffness matrix and assembly for the global system is performed. First basic steps on finite element programs are described.

Antonio J. M. Ferreira, Nicholas Fantuzzi

Chapter 3. Bars or Trusses

Abstract

In this chapter, we analyze axially loaded structural elements termed bars or trusses. A truss is connected to other elements only through pins which are connections that do not constrain rotations. Trusses are modeled as discrete elements (or springs) because only axial force (traction or compression) and elongation is evaluated. In the present chapter, an isoparametric finite element formulation is considered for the bar/truss problem.

Antonio J. M. Ferreira, Nicholas Fantuzzi

Chapter 4. Trusses in 2D Space

Abstract

This chapter deals with the static and free vibration analyses of two dimensional trusses, which are basically bars oriented in two dimensional Cartesian systems. A transformation of coordinate basis is necessary to translate the local element matrices into the structural coordinate system. Trusses support compressive and tensile forces only, as in bars. All forces are applied at the nodes. After the presentation of the element formulation, some examples are solved by MATLAB codes.

Antonio J. M. Ferreira, Nicholas Fantuzzi

Chapter 5. Trusses in 3D Space

Abstract

The present chapter generalizes the 2D truss model of the previous chapter as trusses in 3D Cartesian space. Static and free vibration problems are solved transforming the local stiffness into global 3D quantities. Some simple problems are solved in MATLAB and verified with reference codes.

Antonio J. M. Ferreira, Nicholas Fantuzzi

Chapter 6. Bernoulli Beams

Abstract

Bernoulli theory is a classical beam theory where the transverse shear deformation is neglected and the deflection of the beam indicated by w is the only degree of freedom of the model and the in-plane rotation si given by the derivative of the transverse deflection with respect to the beam axis. In this chapter we perform the static, vibration and buckling analysis of Bernoulli beams in bending configuration. Results will be compared to analytical and reference results from the literature.

Antonio J. M. Ferreira, Nicholas Fantuzzi

Chapter 7. Bernoulli 2D Frames

Abstract

In this chapter two-dimensional frames under static loading and free vibrations are analyzed. The present formulation is a generalization of the previous Bernoulli beam in local coordinates. The stiffness and mass matrices are given by transformation of the same matrices in local coordinates by a matrix of rotation which is a function of the beam slope with respect to the horizontal axis.

Antonio J. M. Ferreira, Nicholas Fantuzzi

Chapter 8. Bernoulli 3D Frames

Abstract

The analysis of three dimensional frames is quite similar to the analysis of 2D frames. In the 2-node 3D frame finite element we now consider in each node three displacements and three rotations with respect to the three global cartesian axes. However, the complexity in such structures is due to the orientation of the beam in space other than in 2D plane. Before introducing the stiffness and mass matrices in the global reference system rotation matrices for vectors in 3D space are firstly introduced.

Antonio J. M. Ferreira, Nicholas Fantuzzi

Chapter 9. Grids

Abstract

In this chapter we perform the static analysis of grids, which are planar structures where forces are applied normal to the grid plane. In other words, the grid element is analogous to the 2D frame element where the axial stiffness is replaced by the torsional one.

Antonio J. M. Ferreira, Nicholas Fantuzzi

Chapter 10. Timoshenko Beams

Abstract

Unlike the Bernoulli beam formulation, the Timoshenko beam formulation accounts for transverse shear deformation. It is therefore capable of modeling thin or thick beams. In this chapter we perform the analysis of Timoshenko beams in static bending, free vibrations and buckling. We present the basic formulation and show how a MATLAB code can accurately solve this problem.

Antonio J. M. Ferreira, Nicholas Fantuzzi

Chapter 11. Plane Stress

Abstract

This chapter deals with the static and dynamic analysis of 2D solids, particularly in plane stress. Plane stress analysis refers to problems where the thickness is quite small when compared to other dimensions in the reference plane x–y. The loads and boundary conditions are applied at the reference or middle plane of the structure. In this chapter we consider isotropic, homogeneous materials four-node (Q4), eight-node (Q8) and nine-node (Q9) quadrilateral elements.

Antonio J. M. Ferreira, Nicholas Fantuzzi

Chapter 12. Kirchhoff Plates

Abstract

In the present chapter finite element implementation of Kirchhoff plates in bending is discussed using the so-called conforming and not conforming Hermite shape functions. Note that Hermite shape functions other than more common Lagrange functions, that consider nodal parameters only, use more kinematic parameters than the ones representing the displacement field of the mathematical differential problem that is currently in use.

Antonio J. M. Ferreira, Nicholas Fantuzzi

Chapter 13. Mindlin Plates

Abstract

This chapter considers the static, free vibration and buckling problem of Mindlin plates in bending. Many implementation codes will be taken from the previous chapters such as mesh generation, Gauss integration and field representation. The theory of Mindlin plates is firstly presented and several applications are described.

Antonio J. M. Ferreira, Nicholas Fantuzzi

Chapter 14. Laminated Plates

Abstract

In this chapter we consider a first order shear deformation theory for the static, free vibration and buckling analysis of laminated plates. We introduce a computation of the shear correction factor and solve some examples with MATLAB codes. The main difference between the present chapter and the previous one related to Mindlin plates is that due to lamination there might be a coupling between membrane and bending behaviors.

Antonio J. M. Ferreira, Nicholas Fantuzzi

Chapter 15. Functionally Graded Structures

Abstract

In the present chapter functionally graded materials (FGMs) and structures are presented. In particular, the static and free vibration problems of Timoshenko beams and Mindlin plates are studied. The buckling problem for both structures can be developed following analogous problems presented in Chap. 10 for Timoshenko beams and in Chap. 14 for laminated FSDT plates.

Antonio J. M. Ferreira, Nicholas Fantuzzi

Chapter 16. Time Transient Analysis

Abstract

In the present chapter time transient analysis is presented for Timoshenko beams and laminated FSDT plates. The theoretical background mainly focuses on how to implement linear time transient analysis in numerical methods, therefore the reader should refer to chapters 10 and 14 for the beam and plate theories and implementation, respectively.

Antonio J. M. Ferreira, Nicholas Fantuzzi

Backmatter