Summary of

FINITE ELEMENT METHOD

PRESENTED BY:

AUMAIR AZEEM MALIK (140919)

MS AEROSPACE ENGINEERING

INTRODUCTION TO FINITE

ELEMENT METHOD

Finite Element Analysis is a method for numerical

solution of field problems.

Individual finite elements can be visualized as

small pieces of a structure.

In each finite element a field quantity is allowed to

have only a simple spatial variation.

So, FEA provides approximate solutions.

Elements of a structure are connected at

points called nodes

The assemblage of elements is called a finite

element structure

The particular arrangement of elements is

called a mesh.

A finite element ‘mesh’ is represented be a system

of algebraic equations to be solved for unknowns at

nodes.

The solution of nodal quantities when combined with

the assumed field in any given element completely

determines the spatial variation of the field in that

element.

Field quantity over the entire structure is

approximated element by element.

ADVANTAGES OF FINITE ELEMENT METHOD1. Versatility and physical appeal

2. Applicable to any field problem

3. No geometric restriction

4. Boundary conditions and loadings are not restricted

5. No isotropic restrictions

6. An FE structure closely resembles the actual body or region

to be analyzed

7. Components with different behaviors and mathematical

descriptions can be combined

SOLUTION STEPS IN AN FINITE ELEMENT

PROBLEM 1. Classification

2. Mathematical Modeling

3. Discretization

4. Preliminary Analysis

5. Finite Element Analysis

6. Checking the results

7. Expect to revise

HISTORY OF FINITE ELEMENT ANALYSIS

The method originated from the need to solve

complex elasticity and structural analysis problems

in civil and aeronautical engineering.

Its development can be traced back to the work

by A. Hrennikoff and R. Courant.

In China, in the later 1950s and early 1960s, based on the computations of dam constructions, K. Fengproposed a systematic numerical method for solving partial differential equations.

The method was called the finite difference method based on variation principle, which was another independent invention of finite element method.

The finite element method obtained its real impetus in the 1960s and 1970s by the developments of J. H. Argyris with co-workers.

Further impetus was provided in these years

by available open source finite element

software programs i.e. with the development

of computers and software technology.

NASA sponsored the original version

of NASTRAN, and UC Berkeley made the finite

element program SAP IV widely available.

1-D Elements & Computational Procedures

Now we will discuss the entire computational

process of linear static FEA, which includes:

Formulation of element matrices

Their assembly into structural matrices

Application of loads and boundary conditions

Solution of structural equations

Extraction of gradients (element strains and

stresses)

INTRODUCTION:

One dimensional elements include a straight bar loaded axially, a

straight beam loaded laterally, bar that conducts heat or electricity,

and so on.

Here we will restrict our attention only to linear problems, which

means that material properties are essentially unchanged by loading.

We will exclude the non linear behavior such as yielding of steel,

crumbing of concrete, opening an closing of gaps and lateral gaps large

enough to generate membrane stretching action.

We will also consider only steady state problems which are called

quasistatic in structural mechanics

COMPUTATIONAL PROCEDURE:

Regardless of the number or types of elements used, the computational

procedure for time independent FEA is as follows:

Generate matrices that describe element behavior.

Connect elements together, which implies assembly of element matrices to

obtain a structural matrix.

Provide some nodes with loads.

Provide other nodes with boundary conditions.

The structure matrix and array of loads are parts of a system of algebraic

equations. Solve these to get nodal values of field quantities.

Compute gradients: strains, heat flux etc.

BAR ELEMENT:

Consider a uniform elastic bar element of length ‘L’ and elastic modulus ‘E’.

The element under consideration has a cross sectional area ‘A’.

A node is located at each end. For now, we allow nodes to displace only in axial direction.

Axial displacements at nodes are u1 and u2.

Internal axial stress σ can be related to nodal forces F1 and F2 by free body

diagrams, as shown in figure.

The element to be in equilibrium requires that F1 = - F2

So the matrix equation can be abbreviated as:

Structure Equations:

Consider a structure built of two uniform elastic bars attached

end to end as shown below:

Only axial displacement are allowed. Stiffness of the respective

elements are k1 and k2

The structural stiffness equation is:

Here [K] is the global stiffness matrix

BEAM ELEMENT

2D Beam Element:

Beam element is a very versatile line-element, it has 6 DOF at each node,

which include, translations and rotations along the x, y, and z directions,

respectively.

The stiffness constant of a beam element is derived by combining

the stiffness constants of a beam under pure bending, a truss

element, and a torsion bar.

In FEA it’s a common practice to use beam elements to represent

all or any of these three loads.

We will derive the element stiffness equation for a beam element

by first deriving the stiffness equation of a beam in bending, and

then superimposing the stiffness of a truss and a torsion bar

element.

Derivation of the stiffness equation for a beam element:

A beam, such as, a cantilever beam, under pure bending (without axial loads or torsional loads), has two-degrees of freedom at any point, transverse deflection v and rotation θ, as shown in figure.

A beam element has a total of four degrees of freedom, two at each node. Since there are four degrees of freedom, the size of the stiffness matrix of a beam element has the size 4 x 4.

We will derive the stiffness matrix equation using a simple

method, known as Stiffness Influence Coefficient Method.

In this procedure, a relationship between force and the

coefficients that influence stiffness is established.

For a beam element, these coefficient consist of: the

modulus of elasticity, moment of inertia, and length of the

element. For a two-node beam element, there are two

deflections and two rotations, namely, v1, θ1, v2, and θ2.

The following deflection relationships for loading of figures can be found in any

Machine Design Handbook, and is given as:

Applying these relationships to the beam of figure we get:

Solved to get:

Writing the equations we got in matrix form:

Using similar procedure we will get the final matrix as:

Properties of Stiffness Matrix

Nonnegative Kii : Diagonal coefficients cannot be negative.

Symmetry: Stiffness matrix of any element or structure is

symmetric if loads are linearly related to displacements.

Sparsity: A global stiffness coefficient is Kij is zero unless at

least one element is attached to both DOF i and j.

Basic Elements

Since most elements in common use are

displacement based, so now we will discuss the

interpolation and simple elements based on

displacement fields.

An understanding of element displacement fields and

especially of shortcomings an element may have

because of its displacement fields, is needed in order

to prepare a good FE model and to properly check

the computed results.

Interpolation and Shape Functions

Interpolation means to devise a continuous function that

satisfies prescribed conditions at a finite number of points.

In FEA, those points are nodes of an element and the

prescribed conditions are nodal values of a field quantity.

In FEA, the interpolating function is almost always a

polynomial, which automatically provides a single valued

and continuous field.

In terms of generalized DOF ai , an interpolating polynomial with dependent variable φ and

independent variable x can be written in the form:

In which:

Where n=1 for linear interpolation, n=2 for quadratic interpolation and so on.

The ai can be expressed in terms of nodal values of φ, which appear at known values of ‘x’

The relation between nodal values φe and ai is symbolized as:

An individual Ni in matrix [N] is called a shape function, sometimes

called a basis function

Linear Triangle (CST)

A linear triangle is a plane triangle whose field quantity varies

linearly with Cartesian coordinates x and y.

In stress analysis, a linear displacement field produces a constant

strain field, so the element may be called a CST (Constant Strain

Triangle)

Formulation Techniques:

Variational Methods

Now we will discuss Integral expressions called ‘functional’.

We seek values of DOF that make these functional either

stationary or minimum.

Functional provide a powerful technique for generating

finite element approximations.

In structural mechanics, the most commonly used

functional is that of potential energy.

In preceding slides element stiffness matrices are formulated

either by direct physical argument or by using principle of

virtual work.

Direct argument is limited to simple problems and simple elements.

Virtual work is powerful and has physical appeal, but does not

provide a framework for producing more general FE approximations.

Rayleigh – Ritz method on the other hand is a systematic procedure

for producing FE approximations.

Rayleigh – Ritz Method:

A continuum such as an elastic solid, has an infinite number of DOF, namely

the displacements of every particle of the material.

Behavior of a continuum is described by partial differential equations.

The need to solve differential equations can be avoided by applying the

Rayleigh Ritz method to a functional that describes a mathematical model.

The result is a substitute model that has finite number of DOF and is

described by algebraic equations rather than by differential equations.

Formulation Techniques:

Galerkin & Other Weighted

Residual Methods

Approximate solutions, including FE solutions, can be

constructed from governing differential equations.

The Galerkin method is commonly used for this

purpose and summarizes related methods e.g.

Method of Mixed Formulation and nonstructural

problems.

Galerkin Method:

For some applications the functional needed for a variational

approach cannot be written.

A case in point is fluid mechanics, where, for some types of flow,

all that is available are differential equations and boundary

conditions

FE formulations of such problems can still be obtained using

Weighted Residual Methods of which Galerkin is most widely used.

Isoparametric Elements

The Isoparametric method leads to a simple computer program

formulation, and it is generally applicable for two and three-

dimensional stress analysis and for nonstructural problems.

The Isoparametric formulation allows elements to be created that

are nonrectangular and have curved sides.

Numerous commercial computer programs have adapted this

formulation for their various libraries of elements. Extensive

libraries are there in the latest version of PATRAN® and ANSYS®

Introduction

Why Isoparametric Method ?

The usual procedures to formulate the stiffness equations of the

linear triangle can be formally extended to quadrilateral elements

as well as higher order triangles. But one quickly encounters

technical difficulties:

The construction of shape functions that satisfy consistency

requirements for higher order elements with curved boundaries

becomes increasingly complicated.

Integrals that appear in the expressions of the element stiffness

matrix and consistent nodal force vector can no longer be

evaluated in simple closed form.

These two obstacles can be overcome through the concepts

of isoparametric elements and numerical quadrature,

respectively. The combination of these two ideas

transformed the field of finite element methods in the late

1960s.

Together they support a good portion of what is presently

used in production finite element programs.

The Linear Triangle:

The three-noded linear triangle, pictured in figure, may be

presented as an Isoparametric element:

The shape functions are simply the triangular coordinates:

The Quadratic Triangle:

The six node triangle shown in figure is the next complete-polynomial

member of the

Isoparametric triangle family. The isoparametric definition is:

Shape functions are:

Continued ……

The element may have parabolically curved sides

defined by the location of the mid nodes 4, 5 and 6. The

triangular coordinates for a curved triangle are no longer

straight lines, but form a curvilinear system as can be

observed in figure: