An Overview of the Finite Element Method

1.1 What Are Finite Elements?

Since the differential equations describing the displacement field of a structure are difficult (or impossible) to solve by analytical methods, the domain of the structural problem can be divided into a large number of small subdomains, called finite elements (FE). The displacement field of each element is approximated by polynomials, which are interpolated with respect to prescribed points (nodes) located on the boundary (or within) the element. The polynomials are referred to as interpolation functions, where variational or weighted residual methods are applied to determine the unknown nodal values.

1.2 Why Finite Element Method Is Very Popular?

The concept of the finite element method (FEM) was described in 1956, when Turner et al. used pin-jointed bars and triangular plates to calculate aircraft structures. However, as the method is based on the solution of systems of algebraic equations with large number of unknowns, in past few decades, FEM has become very popular due to the development of high-speed digital computers.

After 1980, new commercial software packages were developed, boosting the application of FEM to structural engineering, heat transfer, fluid mechanics, aerodynamics, and electrostatics.

Among the pioneers who founded and developed FEM are Przemieniecki, Zienkiewicz and Cheung, Gallagher, Argyris, etc.

1.3 Main Advantages of Finite Element Method

1. Analyzes problems with complex geometry.

2. Analyzes problems with complex loading (point loads, pressure, inertial forces, thermal loading, fluid-structure interactions, etc.).

3. Analyzes a wide variety of engineering problems (structural engineering, heat transfer, fluid mechanics, aerodynamics, and electrostatics).

1.4 Main Disadvantages of Finite Element Method

1. FEM results are approximate. Their accuracy depends on the number of elements, the type of elements, the adopted assumptions, etc.

2. The accuracy of the results of FEM depends on the experience of the software user, for example, the use of the wrong type or distorted elements, insufficient supports to prevent all rigid body motions, and different units for the same quantity yields mistakes.

3. FEM has inherent errors (e.g., the geometry of the structure is approximate, the field deformation is assumed to be a polynomial over the element, the computer carries only a finite number of digits, the combination of elements with very large stiffness differences yields numerical difficulties).

1.5 What Is Structural Matrix?

Structural matrix is a matrix correlating the forces and displacements in the nodal points of the elements. For a structural FE, the structural matrix contains the geometric and material behavior information that indicated the resistance of the element to deformation when subjected to loading. The primary characteristics of an FE are embodied in the element structural matrix. There are two types of structural matrices: stiffness matrices, and transfer matrices (Figure 1.1).

Taking into account the nomenclature of Figure 1.2, the stiffness and the transfer matrix for a simple beam are:

1.5.1 Stiffness Matrix

aMaVbMb=12EJℓ3−6EJℓ2−12EJℓ3−6EJℓ2−6EJℓ24EJℓ6EJℓ22EJℓ−12EJℓ36EJℓ212EJℓ36EJℓ2−6EJℓ22EJℓ6EJℓ24EJℓwaθawbθb

1.5.2 Transfer Matrix

aθaVbMb=1−ℓ−ℓ3/6EJ−ℓ2/2EJ01ℓ2/2EJℓ/EJ00100001waθaVaMa

1.6 What Are the Steps to be Followed for Finite Element Method Analysis of Structure?

1.6.1 Step 1. Discretize or Model the Structure

The structure is divided into FEs. This step is one of the most crucial in determining the solution accuracy of the problem.

1.6.2 Step 2. Define the Element Properties

At this step, the user must define the element properties and select the types of FEs that are the most suitable to model the physical problem.

1.6.3 Step 3. Assemble the Element Structural Matrices

The structural matrix of an element consists of coefficients that can be derived, for example, from equilibrium. The structural matrix relates the nodal displacements to the applied forces at the nodes. Assembling of the element structural matrices implies application of equilibrium for the whole structure.

1.6.4 Step 4. Apply the Loads

At this step, externally applied concentrated or uniform forces, moments, or ground motions are provided.

1.6.5 Step 5. Define Boundary Conditions

At this step the support conditions must be provided, that is, several nodal displacements must be set to known values.

1.6.6 Step 6. Solve the System of Linear Algebraic Equations

The sequential application of the above steps leads to a system of simultaneous algebraic equations where the nodal displacements are usually the unknowns.

1.6.7 Step 7. Calculate Stresses

At the users discretion, the commercial programs can also calculate stresses, reactions, mode shapes, etc.

1.7 What About the Available Software Packages?

Some of the important FEM packages that are available today include Ansys, Abaqus, Nastran, and Lusas. Their structure is based on pre-processor, solution process, post-processor.

Pre-processor stage: data preparation takes place, that is, selection of elements, selection of material properties, discretization of the structure, definition of boundary conditions, and definition of loadings. With these data, the computer algorithm creates the structural equations for every element. Since the data input takes place during this stage, the user interacts with the software only during the pre-processing step.

Solution process stage: computer software solves the system of algebraic equations derived at the pre-processor stage.

Post-processor stage: numerical results obtained by the previous stage are demonstrated graphically in order to represent the displacement, the strain, and the stress field.

1.8 Physical Principles in the Finite Element Method

As it is already mentioned, FEM is based on small subdomain elements. In order to derive the displacement field for the whole structure, a structural matrix and the associated element equation for each element should be derived first. This fundamental equation correlates the nodal displacements and the forces of each element according to the following abbreviated form

=kd

where [k] is the element stiffness matrix (or local matrix), {r} the forces at the nodes, and {d} the displacements at the nodes.

The derivation of the element stiffness matrix is very important for the accurate numerical simulation of the structure. To achieve this target, the methods for elements stiffness matrix derivation are classified in three categories: direct equilibrium method, energy methods, and weighted residual methods.

The characteristics of each of the above categories are shown in following figures.

In the following chapters, focus will be given on the direct equilibrium method for analyzing springs, bars, trusses, beams, and frames, and the minimum potential energy (MPE) for analyzing springs, bars, trusses, beams, frames, plane stress, and three-dimensional (3D) problems.

It should be noted that most of the FEM applications on structural engineering are based on the MPE method. This method uses the calculus of variations. Therefore, a functional should be initially identified. For solid mechanic problems, the total potential energy Π is the functional to be minimized. Since this functional is expressed in terms of nodal displacements {d}, the condition of its minimization Π/∂d yields the required element equation =kd.

1.9 From the Element Equation to the Structure Equation

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