FEA 1

Question 1

FEA (Finite Element Analysis)

Answer 1

An analysis/simulation that uses the Finite Element Method to solve a problem.

Question 2

FEM (Finite Element Method)

Answer 2

A powerful numerical method to calculate approximate solutions to many types of problems.

The FEM formulation is a system of simultaneous algebraic equations for solution, rather than requiring the solution of differential equations.

Question 3

Discretization

Answer 3

The process of modelling a body by dividing it into an equivalent system of smaller bodies of units (finite elements) interconnected at points common to two or more elements (nodal points or nodes).

Question 4

The solution for structural problems

Answer 4

Typically refers to determining the displacements at each node and the stresses within each element making up the structure that is subjected to applied loads.

Question 5

Global stiffness equation (formulated in the stiffness or displacement method)

Answer 5

{F} = [K]{d}

{F} = global nodal forces
{d} = global nodal displacements
[K] = global stiffness matrix

Question 6

General Steps of the FEM (1)

Answer 6

Discretize and Select the Element Types

The elements must be made small enough to give usable results and yet large enough to reduce computational effort:

Small elements are generally desirable where the results are changing rapidly, such as where changes in geometry occur

Large elements can be used where results are relatively constant.

Question 7

Two-dimensional (or plane) elements

Answer 7

Usually loaded by forces in their own plane (plane stress or plane strain conditions)

Triangular or quadrilateral elements

The simplest two-dimensional elements have corner nodes only (linear elements) with straight sides

The elements can have variable thicknesses throughout or be constant

Question 8

Line elements: bar (or truss) and beam elements

Answer 8

They have a cross-sectional area but are usually represented by line segments

The simplest line element (a linear element) has two nodes, one at each end

Question 9

Three-dimensional elements

Answer 9

Used for a three-dimensional stress analysis

The basic three-dimensional elements have corner nodes only and straight sides

There exist higher-order elements with mid-edge nodes (and possible midface nodes)

Usually tetrahedral or hexahedral (brick) elements

Question 10

General Steps of the FEM (2)

Answer 10

Select a Displacement Function : Linear, quadratic, and cubic polynomials are frequently used functions

The function are expressed in terms of the nodal unknowns

The same general displacement function can be used repeatedly for each element

For one-dimensional spring and bar elements, the displacement function is a function of a single coordinate

Question 11

General Steps of FEM (3)

Answer 11

Define the Strain/Displacement and Stress/Strain Relationships

For small strains, Strain (epsilon) = du/dx (displacement)

Constitutive law (Hooke’s Law) :

Stress (sigma) = Modulus of Elasticity (E) x Strain (epsilon)

Question 12

General Steps of the FEM (4)

Answer 12

Derive the Element Stiffness Matrix and Equations

1. Direct method (restricted to one-dimensional elements)

Two general direct approaches:

Forces method: uses internal forces as the unknowns of problems

Governing equation = equilibrium equations and compatibility equations

Displacements (or stiffness) method:

(More desirable)

Displacements of the nodes as the unknowns of the problem.

Governing equation = nodal displacements using the equations of equilibrium & an applicable law
relating forces to displacements.