3.1 Introduction to FEA | unit 3 finite element analysis

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Unit 3

Finite Element Analysis

3.1 Introduction to FEA

The Finite Element Method (FEM) is a numerical technique to find approximate solutions of partial differential equations. It was originated fromthe need of solving complex elasticity and structural analysis problems in Civil, Mechanical and Aerospace engineering. In a structural simulation, FEM helps in producing stiffness and strength visualizations. It also helps to minimize material weight andits cost of the structures. FEM allows for detailed visualization and indicates the distribution of stresses and strains inside the body of a structure. Many of FE software are powerful yet complex tool meant for professional engineers with the training and education necessary to properly interpret the results.

3.1.1 Need of FEA

Design geometry is a lot more complex; and the accuracy requirement is a lot higher. We need

To understand the physical behaviors of a complex object (strength, heat transfer capability, fluid flow, etc.)

To predict the performance and behavior of the design

To calculate the safety margin; and to identify the weakness of the design accurately; and

To identify the optimal design with confidence.

3.1.2 Terminologies in FEA

Continuum

A continuous sequence in which adjacent elements are not perceptibly different from each other, but the extremes are quite distinct. A continuum is a complex region which defines a geometry. Any continuum/domain can be divided into a number of pieces with very small dimensions.

Element

In FEA, first step is to divide each model into discrete parts. Each part of a discretized body is called as element.

Node

Each element has one or more node. Each element is connected to each other at these nodes.

Object or Continuum

Element

Nodes

Domain

The original domain is considered as an assemblage of number of such small elements.

Load

Loads are used to represent inputs to the system. They can be in the forms of forces, moments, pressures, temperature, or accelerations. There are three types of load

The distributed force per unit volume is called as body force.

The distributed force per unit area is called as surface.

The load acting at a point is called as point load.

Field Variables

The field variables are the dependent variables of interest governed by the differential equation.

Boundary Conditions

The boundary conditions are the specified values of the field variables (or related variables such as derivatives) on the boundaries of the field.

Constraints

Displacement boundary conditions and surface loading conditions are the constraints.

3.1.3 Degree of Freedom

Consider a two-dimensional case with a single field variable φ(x, y). The triangular element described is said to have 3degrees of freedom, as three nodal values of the field variable are required to describe the field variable everywhere in the element (scalar).

In general, the number of degrees of freedom associated with a finite element is equal to the product of the number of nodes and the number of values of the field variable(and possibly its derivatives) that must be computed at each node.

3.1.4 Discretization Technique

The need of finite element analysis arises when the structural system in terms of its either geometry, material properties, boundary conditions or loadings is complex in nature. For such case, the whole structure needs to be subdivided into smaller elements. The wholestructure is then analyzed by the assemblage of all elements representing the complete structure including its all properties. The subdivision process is an important task in finite element analysis and requires some skill and knowledge. In this procedure, first, the number, shape, size and configuration of elements have to be decided in such a manner that the real structure is simulated as closely as possible. The discretization is to be in such that the results converge to the true solution. However, too fine mesh will lead to extra computational effort. Fig. shows a finite element mesh of a continuum using triangular and quadrilateral elements. The assemblage of triangular elements in this case shows better representation of the continuum. The discretization process also shows that the more accurate representation is possible if the body is further subdivided into some finer mesh.

3.1.5 General Procedure of FEA

To summarize in general terms how the finite element method works we list main steps of the finiteelement solution procedure below.

Discretize the continuum: The first step is to divide a solution region into finite elements. Thefinite element mesh is typically generated by a preprocessor program. The description of mesh consists of several arrays main of which are nodal coordinates and element connectivities.

Select interpolation functions: Interpolation functions are used to interpolate the field variables over the element. Often, polynomials are selected as interpolation functions. The degree of the polynomial depends on the number of nodes assigned to the element.

Find the element properties: The matrix equation for the finite element should be established which relates the nodal values of the unknown function to other parameters. For this task different approaches can be used; the most convenient are: the variational approach and the Galerkin method.

Assemble the element equations: To find the global equation system for the whole solution region we must assemble all the element equations. In other words we must combine local element equations for all elements used for discretization. Element connectivities are used for the assembly process. Before solution, boundary conditions (which are not accounted in element equations) shouldbe imposed.

Solve the global equation system:The finite element global equation system is typically sparse,symmetric and positive definite. Direct and iterative methods can be used for solution. The nodal valuesof the sought function are produced as a result of the solution.

Compute additional results: In many cases we need to calculate additional parameters. Forexample, in mechanical problems strains and stresses are of interest in addition to displacements, whichare obtained after solution of the global equation system.

3.1.6 Application of FEA in various field.

Evaluate the stress or temperature distribution in a mechanical component.

Perform deflection analysis.

Analyze the kinematics or dynamic response.

Perform vibration analysis.

3.1.7 Mesh

Mesh is a way of communicating geometry to the solver, the accuracy of the solution is primarily dependent on the quality of the mesh. The better the mesh looks, the more accurate the solution is. A good-looking mesh should have well-shaped elements, and the transition between densities should be smooth and gradual without skinny, distorted elements.

The mesh transition from .05 to .5 element size without control of transition (a) creates irregular mesh around the hole which will yield disappointing results.

3.1.8 P and h formulation

In finite element analysis, solution accuracy is judged in terms of convergence as the element “mesh”is refined. There are two major methods of mesh refinement.

In the first, known as h-refinement, mesh refinement refers to the process of increasing the number of elements used to model a given domain, consequently, reducing individual element size.

In the second method, p-refinement, element size is unchanged but the order of the polynomials used as interpolation functions is increased.

The objective of mesh refinement in either method is to obtain sequential solutions that exhibit asymptotic convergence to values representing the exact solution.

3.1.9 Advantages of FEA

The physical properties, which are intractable and complex for any closed bound solution, can be analyzed by this method.

It can take care of any geometry (may be regular or irregular).

It can take care of any boundary conditions.

Material anisotropy and non-homogeneity can be catered without much difficulty.

It can take care of any type of loading conditions.

This method is superior to other approximate methods like Galerkine and Rayleigh-Ritz methods.

In this method approximations are confined to small sub domains.

In this method, the admissible functions are valid over the simple domain and have nothing to do with boundary, however simple or complex it may be.

3.1.10 Disadvantages of FEA

Computational time involved in the solution of the problem is high.

For fluid dynamics problems some other methods of analysis may prove efficient than the FEM.

3.2 One Dimentional Finite Element Modelling

3.2.1 Shape Function

The values of the field variable computed at the nodes are used to approximate the values at non-nodal points (that is, in the element interior) by interpolation of the nodal values. For the three-node triangle example, the field variable is described by the approximate relation

φ(x, y) = N1(x, y) φ1+ N2(x, y) φ2+ N3(x, y) φ3

where φ1, φ2, andφ3are the values of the field variable at the nodes, and N1, N2, and N3are the interpolation functions, also known as shape functions or blending functions.

In the finite element approach, the nodal values of the field variable are treated as unknown constants that are to be determined. The interpolation functions are most often polynomial forms of the independent variables, derived to satisfy certain required conditions at the nodes.

The interpolation functions are predetermined, known functions of the independent variables; and these functions describe the variation of the field variable within the finite element.

Let be the natural coordinate of 1D element, then the shape functions are given by

3.2.2 Stiffness Matrix

The primary characteristics of a finite element are embodied in the element stiffness matrix. For a structural finite element, the stiffness matrix contains the geometric and material behavior information that indicates the resistance of the element to deformation when subjected to loading. Such deformation may include axial, bending, shear, and torsional effects. For finite elements used in nonstructural analyses, such as fluid flow and heat transfer, the term stiffness matrix is also used, since the matrix represents the resistance of the element to change when subjected to external influences.

3.2.3 Steps to solve a FEA Problem

Step I: Discretization

Discretization is a process of dividing a body into finite number of elements.

Step II: Formulation of Global Load Vector

The elemental force vectors in the global coordinate system for all elements are assembled to form the global load vector {F} for the entire body.

The global Load Vector is given by,

Where F1, F2, F3, …, FN are the loads acting at nodes 1,2,3,…,N respectively

Step III: Formulation of Global Nodal Displacement Vector

The global nodal displacement vector {UN} is formed for the entire body.

The global nodal displacement vector is given as

Where U1, U2, U3,…, UN are the displacements acting at nodes 1,2,3,…,N respectively

Step IV: Formulation of elemental stiffness matrices

After the body is discretized, the elemental stiffness matrix is formulated for all discretized elements.

Consider a one-dimensional rod element as shown in fig.

It has two nodes. Each node has one degree of freedom

Let

l = length of element

A = Cross-sectional area

E = modulus of Elasticity

f1 = Force acting at node 1

f2 = Force acting at node 2

u1 = displacement of node 1

u2 = displacement of node 2

Stiffness of rod element is

From fig,

And

The above equations in matrix form can be written as

Where,

{f} = = elemental force vector

[k] = = elemental stiffness matrix

{uN} = = elemental nodal displacement vector

Step V: Formulation of Global Stiffness Matrix

The global stiffness matrix is formed from the elemental nodal stiffness matrices.

Such as

Step VI: Assembly of Global Stiffness – Nodal Displacement – Load Equations

The relationship between the global stiffness matrix [K], global nodal displacement vector {UN} and global load vector {f} is expressed as

{f} = [K]{UN}

This equation is called as Finite element equation.

If N is the total degree of freedom of the body.

Then dimension of

Global load vector is N x 1.

Global stiffness matrix is N x N.

Global displacement vector is N x 1.

Step VII: Specify boundary conditions

The specified boundary conditions are incorporated in equilibrium equation by using elimination approach or penalty approach.

Step VIII: Solution of Equations

After including the boundary conditions, the modified equations are solved for the unknown nodal displacements.

Step VIII: Computation of Elemental stresses and strains

Elemental strains are calculated from nodal displacements by

Where,

= Element strain

= element strain-nodal displacement matrix

And from the strain calculated, stresses can be calculated as

Where, = element stress

3.2.4 Properties of stiffness matrix

The dimension of global stiffness matrix is (N x N), where N is the d.o.f. of the body.

It is a symmetric matrix

It is a singular matrix

Sum of any tow row or column is zero.

All diagonal elements are positive and non-zero.

It is a banded matrix. All elements outside the matrix are non-zero.

3.2.5 Numerical on Spring element in series and parallel

Fig shows a cluster of three springs. Using the finite element method, determine:

1) The deflection of each spring

2) The reaction force at support

Solution:

Given: k1 = 4N/mm

k2 = 8N/mm

k3 = 6N/mm

U1 = 0

P3 = 2000 N

Discretization

Three springs can be treated as three 1D spar element

Element Number	Global Node number ‘n’ of
Element Number	Local Node 1	Local Node 2
1	1	3
2	1	2
3	2	3

2. Element stiffness matrix

For Element 1,

For Element 2,

For Element 3,

3. Global stiffness matrix

4. Global load vector

5. Global nodal displacement vector

6. Global stiffness-nodal displacement relationship

Hence after the assembly, the equilibrium equation is,

At node 1, U1=0, there is rigid support.

Using elimination approach, first rows and columns are eliminated.

7. Nodal Displacements

Solving two equations for two unknowns, we get,

U2 = 115.38 mm and U3 = 269.23 mm

Deflection of Spring 1 = U3 – U1 = 269.23 – 0 = 269.23 mm

Deflection of Spring 2 = U2 – U1 = 115.38 – 0 = 115.38 mm

Deflection of Spring 3 = U3 – U2 = 269.23 – 115.38 = 153.84 mm

8. Reaction force at support

We have,

12U1 – 8U2 – 4U3 = R

12 x 0 – 8 x 115.38 – 4 x 269.23 = R

R = -2000 N

3.2.6 Numerical on Composite bar element

An axial step bar is shown in figure. Determine the deflection, stresses in element and reaction force.

Solution:

Given: l1 = 200 mm

l2 = 100 mm

A1 = 200 mm2

A2 = 100 mm2

E1 = 2 x 105 N/ mm2

E2 = 1.5 x 105 N/ mm2

P = 10000 N

Discretization

Fig shows an assemblage of two 1D spar element. Its elemental connectivity is given below

Element Number	Global Node number ‘n’ of
Element Number	Local Node 1	Local Node 2
1	1	2
2	2	3

2. Element stiffness matrix

For Element 1,

For Element 2,

3. Global stiffness matrix

4. Global load vector

5. Global nodal displacement vector

6. Global stiffness-nodal displacement relationship

Hence after the assembly, the equilibrium equation is,

At node 1, U1=0, there is rigid support.

Using elimination approach, first rows and columns are eliminated.

7. Nodal Displacements

Solving two equations for two unknowns, we get,

U2 = 0.05 mm and U3 = 0.117 mm

8. Stresses in elements:

Stresses are given by,

In element 1,

In element 2,

9. Reaction force at support

We have,

104 (20U1 – 20U2 ) = R

104(20 x 0 – 20 x 0.05) = R

R = -10 x 103kN

3.2.7 Temperature Effects on 1D Element

Sometimes in a body, in addition to the stresses induces due to applied loads, the thermal stresses are also induced due to change in temperature.

Including the temperature effect, global load vector {F} is given by,

The initial strain due to change in temperature, in two noded one-dimensional element is given by,

Where,

= Change in temperature

= coefficient of thermal expansion

The stresses is given by,

3.3 Trusses

3.3.1 Introduction to 2D trusses

Figure shows a typical 2D plane truss. Such trusses can be analyzed using the method of joints and the method of sections.

FEM can be effectively used for the analysis of trusses and to determine displacements of joints.

3.3.2 Elemental Stiffness matrix for 2D trusses.

Elemental stiffness matrix for 2D trusses are given by

Where,

and are the coordinates of nodes.

And

3.3.3 Global Stiffness Matrix

All elemental Stiffness matrices are assembled to form Global Stiffness Matrix such as

3.3.4 FEA equation for 2D truss

The relationship between the global stiffness matrix [K], global nodal displacement vector {UN} and global load vector {f} is expressed as

{f} = [K]{UN}

3.3.5 Element Stress Calculation

The stress at any point within the 2D truss element is given by

3.3.6 Numerical on 2D Trusses

The two bar truss is shown in fig. The modulus of elasticity for four bar material is 70 x 103 N/mm2 and cross sectional area of each element is 200 mm2. Determine

1) The element stiffness matrix

2) The global stiffness matrix

3) The nodal displacements

4) The stresses in each elements

5) The reaction forces