Introduction The Finite Element Method (FEM) has developed into a key, indispensable technology in the modelling and simulation of advanced engineering systems in various fields like housing, transportation, communications, and so on. In building such advanced engineering systems, engineers and designers go through a sophisticated process of modelling, simulation, visualization, analysis, designing, prototyping, testing, and […]

# The Finite Element Method

## Introduction to Mechanics for Solids and Structures (Finite Element Method) Part 1

Introduction This topic tries to introduce these basic concepts and classical theories in a brief and easy to understand manner. Solids and structures are stressed when they are subjected to loads or forces. The stresses are, in general, not uniform, and lead to strains, which can be observed as either deformation or displacement. Solid mechanics […]

## Introduction to Mechanics for Solids and Structures (Finite Element Method) Part 2

Constitutive Equations Hooke’s law for 2D solids has the following matrix form with σ and ε from Eqs. (2.24) and (2.25): where c is a matrix of material constants, which have to be obtained through experiments. For plane stress, isotropic materials, we have To obtain the plane stress c matrix above, the conditions ofare imposed […]

## Introduction to Mechanics for Solids and Structures (Finite Element Method) Part 3

Moments and Shear Forces Because the loading on the beam is in the transverse direction, there will be moments and corresponding shear forces imposed on the cross-sectional plane of the beam. On the other hand, bending of the beam can also be achieved if pure moments are applied instead of transverse loading. Figure 2.11 shows […]

## Fundamentals for Finite Element Method Part 1

Introduction In each of these elements, the profile of the displacements is assumed in simple forms to obtain element equations. The equations obtained for each element are then assembled together with adjoining elements to form the global finite element equation for the whole problem domain. Equations thus created for the global problem domain can be […]

## Fundamentals for Finite Element Method Part 2

Properties of the Shape Functions1 Property 1. Reproduction property and consistency The consistency of the shape function within the element depends upon the complete orders of the monomial Pi (x) used in Eq. (3.12), and hence is also dependent upon the number of nodes of the element. If the complete order of monomial is k, […]

## Fundamentals for Finite Element Method Part 3

Formation of FE Equations in Local Coordinate System Once the shape functions are constructed, the FE equation for an element can be formulated using the following process. By substituting the interpolation of the nodes, Eq. (3.6), and the strain-displacement equation, say Eq. (2.5), into the strain energy term (Eq. (3.4)), we have where the subscript […]

## Fundamentals for Finite Element Method Part 4

Transient Response Structural systems are very often subjected to transient excitation. A transient excitation is a highly dynamic, time-dependent force exerted on the solid or structure, such as earthquake, impact and shocks. The discrete governing equation system for such a structure is still Eq. (3.96), but it often requires a different solver from that used […]

## FEM for Trusses (Finite Element Method) Part 1

Introduction A truss is one of the simplest and most widely used structural members. It is a straight bar that is designed to take only axial forces, therefore it deforms only in its axial direction. A typical example of its usage can be seen in Figure 2.7. The cross-section of the bar can be arbitrary, […]

## FEM for Trusses (Finite Element Method) Part 2

Worked Examples Example 4.1: A uniform bar subjected to an axial force Consider a bar of uniform cross-sectional area, shown in Figure 4.3. The bar is fixed at one end and is subjected to a horizontal load of P at the free end. The dimensions of the bar are shown in the figure, and the […]