Fea Module 1

Updated on 16.04.
2020/28092020
B. E. MECHANICAL ENGINEERING
Choice Based Credit System (CBCS) and Outcome Based Education (OBE)
SEMESTER - VI
FINITE ELEMENT METHODS
Course Code 18ME61 CIE Marks 40
Teaching Hours /Week (L:T:P) 3:2:0 SEE Marks 60
Credits 04 Exam Hours 03
Course Learning Objectives:
 To learn the basic principles of finite element analysis procedure
 To understand the design and heat transfer problems with application of FEM.
 Solve 1 D, 2 D and dynamic problems using Finite Element Analysis approach.
 To learn the theory and characteristics of finite elements that represent engineering structures.
 To learn and apply finite element solutions to structural, thermal, dynamic problem to develop the
knowledge and skills needed to effectively evaluate finite element analyses.
Module-1
Introduction to Finite Element Method: General steps of the finite element method. Engineering applications
of finite element method. Advantages of the Finite Element Method.
Boundary conditions: Homogeneous and non-homogeneous for structural, heat transfer and fluid flow
problems. Potential energy method, Rayleigh Ritz method, Galerkin’s method, Displacement method of finite
element formulation. Convergence criteria, Discretisation process, Types of elements: 1D, 2D and 3D, Node
numbering, Location of nodes. Strain- displacement relations, Stress-strain relations, Plain stress and Plain
strain conditions, temperature effects.
Interpolation models: Simplex, complex and multiplex elements, linear interpolation polynomials in terms of
global coordinates 1D, 2D, 3D Simplex Elements.
Module-2
Introduction to the stiffness (Displacement) method: Introduction, Derivation of stiffness matrix, Derivation
of stiffness matrix for a spring element, Assembly the total stiffness matrix by superposition. One-Dimensional
Elements-Analysis of Bars and Trusses, Linear interpolation polynomials in terms of local coordinate’s for1D,
2Delements. Higher order interpolation functions for 1D quadratic and cubic elements in natural coordinates,
, , Constant strain triangle, Four-Nodded Tetrahedral Element (TET 4), Eight-Nodded Hexahedral Element
(HEXA 3 8), 2D iso-parametric element, Lagrange interpolation functions.
Numerical integration: Gaussian quadrature one point, two point formulae, 2D integrals. Force terms: Body
force, traction force and point loads, Numerical Problems: Solution for displacement, stress and strain in 1D
straight bars, stepped bars and tapered bars using elimination approach and penalty approach, Analysis of
Module-3
Beams and Shafts: Boundary conditions, Load vector, Hermite shape functions, Beam stiffness matrix based
on Euler-Bernoulli beam theory, Examples on cantilever beams, propped cantilever beams, Numerical
problems on simply supported, fixed straight and stepped beams using direct stiffness method with
concentrated and uniformly distributed load.
Torsion of Shafts: Finite element formulation of shafts, determination of stress and twists in circular shafts.
Module-4
Heat Transfer: Basic equations of heat transfer: Energy balance equation, Rate equation: conduction,
convection, radiation, 1D finite element formulation using vibration method, Problems with temperature
gradient and heat fluxes, heat transfer in composite sections, straight fins.
Fluid Flow: Flow through a porous medium, Flow through pipes of uniform and stepped sections, Flow
through hydraulic net works.
Module-5
Updated on 16.04.2020/28092020
Axi-symmetric Solid Elements: Derivation of stiffness matrix of axisymmetric bodies with triangular elements,
Numerical solution of axisymmetric triangular element(s) subjected to surface forces, point loads, angular
velocity, pressure vessels.
Dynamic Considerations: Formulation for point mass and distributed masses, Consistent element mass matrix
of one dimensional bar element, truss element, axisymmetric triangular element, quadrilateral element, beam
element. Lumped mass matrix of bar element, truss element, Evaluation of eigen values and eigen vectors,
Applications to bars, stepped bars, and beams.
Course Outcomes: At the end of the course, the student will be able to:
CO1: Identify the application and characteristics of FEA elements such as bars, beams, plane and iso-
parametric elements.
CO2: Develop element characteristic equation and generation of global equation.
CO3: Formulate and solve Axi-symmetric and heat transfer problems.
CO4: Apply suitable boundary conditions to a global equation for bars, trusses, beams, circular shafts, heat
transfer, fluid flow, axi-symmetric and dynamic problems
Question paper pattern:
 The question paper will have ten full questions carrying equal marks.
 Each full question will be for 20 marks.
 There will be two full questions (with a maximum of four sub- questions) from each module.
 Each full question will have sub- question covering all the topics under a module.
 The students will have to answer five full questions, selecting one full question from each module.
Name of the
Sl No Title of the Book Name of the Publisher Edition and Year
Author/s
Textbook/s
1 A first course in the Finite Logan, D. L Cengage Learning 6th Edition
Element Method 2016
2 Finite Element Method in Rao, S. S Pergaman Int. Library of 5th Edition
Engineering Science 2010
3 Finite Elements in Engineering Chandrupatla T. R PHI 2nd Edition
2013
Reference Books
1 Finite Element Method J.N.Reddy McGraw -Hill
International Edition
2 Finite Elements Procedures Bathe K. J PHI
3 Concepts and Application of Cook R. D., et al. Wiley & Sons 4th Edition
Finite Elements Analysis 2003
E- Learning
• VTU, E- learning
Finite Element Analysis 18ME61
MODULE I
Syllabus
Introduction to Finite Element Method: General description of the finite element method.
Engineering applications of finite element method. Boundary conditions: homogeneous and
nonhomogeneous for structural, heat transfer and fluid flow problems. Potential energy
method, Rayleigh Ritz method, Galerkin‟s method, Displacement method of finite element
formulation. Convergence criteria, Discretization process, Types of elements: 1D, 2D and
3D, Node numbering, Location of nodes. Strain displacement relations, Stress strain relations,
Plain stress and Plain strain conditions, temperature effects.
Interpolation models: Simplex, complex and multiplex elements, Linear interpolation

polynomials in terms of global coordinates 1D, 2D, 3D Simplex Elements. 10
Hours
Introduction to Numerical Methods
The formulation for structural analysis is generally based on the three fundamental
relations: equilibrium, constitutive and compatibility. There are two major approaches to the
analysis: Analytical and Numerical. Analytical approach which leads to closed-form
solutions is effective in case of simple geometry, boundary conditions, loadings and material
properties. However, in reality, such simple cases may not arise. As a result, various
numerical methods are evolved for solving such problems which are complex in nature. For
numerical approach, the solutions will be approximate when any of these relations are only
approximately satisfied. The numerical method depends heavily on the processing power of
computers and is more applicable to structures of arbitrary size and complexity. It is common
practice to use approximate solutions of differential equations as the basis for structural
analysis. This is usually done using numerical approximation techniques. Few numerical
methods which are commonly used to solve solid and fluid mechanics problems are given
below.
• Finite Difference Method
3
Course Outcomes
CO1 Explain the general steps of the finite element method. Apply Potential energy
method, Rayleigh Ritz method, Galerkin’s method & Displacement method to solve
static structural problems. Derive the Strain- displacement relations & Stress-strain
relations of 3D elements.
CO2 Establish shape function for different types elements to formulate stiffness matrices
and load vectors to obtain global equilibrium equation. Development of Solution for
displacement, stress and strain in 1D straight bars, stepped bars and tapered bars using
various approach
CO3 Formulation of beam stiffness matrix and computation of displacement, stresses, slope
and support reactions based on Eular Bernoulli beam theory by applying suitable
boundary condition for cantilever simply supported and fixed beams subjected to
point load and UDL. finite element formulation and determination of stresses and
twist in circular shaft
CO4 derive basic equation of heat transfer and determine the temperature distribution in
composite wall and pin fin using finite element formulation for 1D steady state heat
transfer problems develop finite element formulation for or fluid flow problems
CO5 Derive stiffness matrix and calculate stresses in axisymmetric solid elements
subjected to surface forces and point load. Compute consistent element and lumped
mass matrix for 1D bar and truss element
• Finite Volume Method

• Finite Element Method
• Boundary Element Method
• Meshless Method.
Each method has noteworthy advantages as well as limitations. However, it is possible to
solve various problems by finite element method, even with highly complex geometry and
loading conditions, with the restriction that there is always some numerical errors. Therefore,
effective and reliable use of this method requires a solid understanding of its limitations.
Background of Finite Element Analysis
The finite element analysis can be traced back to the work by Alexander Hrennikoff (1941)
and Richard Courant(1942). Hrenikoff introduced the framework method, in which a plane
elastic medium was represented as collections of bars and beams.These pioneers share one
essential characteristic: mesh discretization of a continuous domain into a set of discrete sub-
domains, usually called elements.
• In 1950s, solution of large number of simultaneous equations became possible because of
the digital computer.
• In 1960, Ray W. Clough first published a paper using term “Finite Element Method”.
• In 1965, First conference on “finite elements” was held.
• In 1967, the first book on the “Finite Element Method” was published by Zienkiewicz and
Chung.
• In the late 1960s and early 1970s, the FEM was applied to a wide variety of engineering
problems.
• In the 1970s, most commercial FEM software packages (ABAQUS, NASTRAN, ANSYS,
etc.) originated.Interactive FE programs on supercomputer lead to rapid growth of CAD
systems.
• In the 1980s, algorithm on electromagnetic applications, fluid flow and thermal analysis
were
developed with the use of FE program.
• Engineers can evaluate ways to control the vibrations and extend the use of flexible,
deployable structures in space using FE and other methods in the 1990s. Trends to solve fully
coupled solution of fluid flows with structural interactions, bio-mechanics related problems
with a higher level of accuracy were observed in this decade. With the development of finite
4
element method, together with tremendous increases in computing power and convenience,
today it is possible to understand structural behavior with levels of accuracy. This was in fact
the beyond of imagination before the computer age.
Introduction to Finite Element Method – General Description
Finite Element Method (FEM) is a numerical technique to find approximate solutions of

partial differential equations. It was originated from the need of solving complex elasticity
and structural analysis problems in Civil, Mechanical and Aerospace engineering.
Figure 1.1 Finite Element meshing of an aircraft structure
The Finite Element Method (FEM) is a numerical technique to find approximate solutions of
partial differential equations. It was originated from the 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 and its cost of the structures.
5
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. Several modern FEM packages include specific components such as
fluid, thermal, electromagnetic and structural working environments. FEM allows entire
designs to be constructed, refined and optimized before the design is manufactured. This
powerful design tool has significantly improved both the standard of engineering designs
and the methodology of the design process in many industrial applications. The use of FEM
has significantly decreased the time to take products from concept to the production line.
One must take the advantage of the advent of faster generation of personal computers for the
analysis and design of engineering product with precision level of accuracy. Therefore FEM
can be broadly defined as,
Definition: The Finite Element Method (FEM) is a numerical analysis technique used to
obtain approximate solutions to the differential equations or partial differential equations
that describe a physical or non-physical problem.
Basic Steps in Finite Element Analysis
The following steps are performed for finite element analysis.

1. Discretisation of the continuum: The continuum is divided into a number of elements by
imaginary lines or surfaces. The interconnected elements may have different sizes and
shapes.
2. Identification of variables: The elements are assumed to be connected at their intersecting
points referred to as nodal points. At each node, unknown displacements are to be prescribed.
3. Choice of approximating functions: Displacement function is the starting point of the
mathematical analysis. This represents the variation of the displacement within the element.
The displacement function may be approximated in the form a linear function or a higher
order function. A convenient way to express it is by polynomial expressions. The shape or
geometry of the element may also be approximated.
4. Formation of the element stiffness matrix: After continuum is discretised with desired
element shapes, the individual element stiffness matrix is formulated. Basically it is a
minimization procedure whatever may be the approach adopted. For certain elements, the
form involves a great deal of sophistication. The geometry of the element is defined in
6
reference to the global frame. Coordinate transformation must be done for elements where it
is necessary.
5. Formation of overall stiffness matrix: After the element stiffness matrices in global
coordinates are formed, they are assembled to form the overall stiffness matrix. The assembly
is done through the nodes which are common to adjacent elements. The overall stiffness
matrix is symmetric and banded.
6. Formation of the element loading matrix: The loading forms an essential parameter in
any structural engineering problem. The loading inside an element is transferred at the nodal
points and consistent element matrix is formed.
7. Formation of the overall loading matrix: Like the overall stiffness matrix, the element
loading matrices are assembled to form the overall loading matrix. This matrix has one
column per loading case and it is either a column vector or a rectangular matrix depending on
the number of loading cases.
8. Incorporation of boundary conditions: The boundary restraint conditions are to be
imposed in the stiffness matrix. There are various techniques available to satisfy the boundary
conditions. One is the size of the stiffness matrix may be reduced or condensed in its final
form. To ease computer programming aspect and to elegantly incorporate the boundary
conditions, the size of overall matrix is kept the same.
9. Solution of simultaneous equations: The unknown nodal displacements are calculated by
the multiplication of force vector with the inverse of stiffness matrix.
10. Calculation of stresses or stress-resultants: Nodal displacements are utilized for the
calculation of stresses or stress-resultants. This may be done for all elements of the
continuum or it may be limited to some predetermined elements. Results may also be
obtained by graphical means. It may desirable to plot the contours of the deformed shape of
the continuum
The basic steps for finite element analysis are shown in the form of flow chart below
7
Figure 1.2 Steps in finite element method
Concepts of Elements and Nodes
Figure 1.3 Elements and nodes in finite element method
Any continuum/domain can be divided into a number of pieces with very small dimensions.
These small pieces of finite dimension are called „Finite Elements‟ (Fig. 1.3). A field quantity
8
in each element is allowed to have a simple spatial variation which can be described by
polynomial terms. Thus the original domain is considered as an assemblage of number of
such small elements. These elements are connected through number of joints which are called
„Nodes‟. While discretizing the structural system, it is assumed that the elements are attached
to the adjacent elements only at the nodal points. Each element contains the material and
geometrical properties. The material properties inside an element are assumed to be constant.
The elements may be 1D elements, 2D elements or 3D elements. The physical object can be
modelled by choosing appropriate element such as frame element, plate element, shell
element, solid element, etc. All elements are then assembled to obtain the solution of the
entire domain/structure under certain loading conditions. Nodes are assigned at a certain
density throughout the continuum depending on the anticipated stress levels of a particular
domain. Regions which will receive large amounts of stress variation usually have a higher
node density than those which experience little or no stress.
Advantages of finite element method
1. The physical properties, which are intractable and complex for any closed bound solution,
can be analyzed by this method.
2. It can take care of any geometry (may be regular or irregular).
3. It can take care of any boundary conditions.
4. Material anisotropy and non-homogeneity can be catered without much difficulty.
5. It can take care of any type of loading conditions.
6. This method is superior to other approximate methods like Galerkine and Rayleigh-Ritz
methods.
7. In this method approximations are confined to small sub domains.
8. 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.
9. Enable to computer programming.
Disadvantages of finite element method
1. Computational time involved in the solution of the problem is high.

2. For fluid dynamics problems some other methods of analysis may prove efficient than the
FEM.
Limitations of finite element method
9
1. Proper engineering judgment is to be exercised to interpret results.

2. It requires large computer memory and computational time to obtained results.
3. There are certain categories of problems where other methods are more effective, e.g.,
fluid problems having boundaries at infinity are better treated by the boundary element
method.
4. For some problems, there may be a considerable amount of input data. Errors may creep up
in their preparation and the results thus obtained may also appear to be acceptable which
indicates deceptive state of affairs. It is always desirable to make a visual check of the input
data.
5. In the FEM, many problems lead to round-off errors. Computer works with a limited
number of digits and solving the problem with restricted number of digits may not yield the
desired degree of accuracy or it may give total erroneous results in some cases. For many
problems the increase in the number of digits for the purpose of calculation improves the
accuracy.
Engineering applications of finite element method
1. Mechanical engineering: In mechanical engineering, FEM applications include steady

and transient thermal analysis in solids and fluids, stress analysis in solids, automotive design
and analysis and manufacturing process simulation.
2. Geotechnical engineering: FEM applications include stress analysis, slope stability

analysis, soil structure interactions, seepage of fluids in soils and rocks, analysis of dams,
tunnels, bore holes, propagation of stress waves and dynamic soil structure interaction.
3. Aerospace engineering: FEM is used for several purposes such as structural analysis for
natural frequencies, modes shapes, response analysis and aerodynamics.
4. Nuclear engineering: FEM applications include steady and dynamic analysis of reactor
containment structures, thermo-viscoelastic analysis of reactor components, steady and
transient temperature-distribution analysis of reactors and related structures.
5. Electrical and electronics engineering: FEM applications include electrical network

analysis, electromagnetics, insulation design analysis in high-voltage equipments, dynamic
analysis of motors and heat analysis in electrical and electronic equipments.
6. Metallurgical, chemical engineering: In metallurgical engineering, FEM is used for the

metallurgical process simulation, moulding and casting. In chemical engineering, FEM can
10
be used in the simulation of chemical processes, transport processes and chemical reaction
simulations.
7. Meteorology and bio-engineering: In the recent times, FEM is used in climate

predictions, monsoon prediction and wind predictions. FEM is also used in bio-engineering
for the simulation of various human organs, blood circulation prediction and even total
synthesis of human body.
Basic terminologies used in FEM
1. Discretization (Meshing): The process of dividing the model of the problem continuum
into a finite number of regular subdivisions
2. Elements: Each subdivision is called an “Element”
3. Nodes: The grid (connection) points at which the elements meet each other are called
“nodes”
4. Degrees of Freedom (DOF): The total number of variables (displacements) that are
associated with each node
5. Boundary Conditions: Known values of the variable at the continuum boundary
6. Displacement Function: It is an assumed polynomial expression which closely represents

the anticipated variation of the unknown variable over the element domain.
Problem Dimensions:
One Dimensional Problems: If only one independent coordinate axis is sufficient to

represent the displacement, material and geometry of the problem, then such problems are
known as One Dimensional Problems. Line Elements are used to model 1D problems.
Two Dimensional Problems: If two independent coordinate axes are required to represent
the displacement, material and geometry of the problem, then such problems are known as
Two Dimensional Problems. Area Elements are used to model 2D problems.
Three Dimensional Problems: If three independent coordinate axes are required to represent
the displacement, material and geometry of the problem, then such problems are known as
Three Dimensional Problems. Volume Elements are used to model 3D problems
Boundary conditions
Boundary conditions are constraints necessary for the solution of a boundary value problem.
A boundary value problem is a differential equation (or system of differential equations) to be
11
solved in a domain on whose boundary a set of conditions is known. It is opposed to the

“initial value problem”, in which only the conditions on one extreme of the interval are
known. Boundary value problems are extremely important as they model a vast amount of
phenomena and applications, from solid mechanics to heat transfer, from fluid mechanics to
acoustic diffusion.
Figure 1.4. Boundary condition
The main types of loading available in FEA include force, pressure and temperature. These
can be applied to points, surfaces, edges, nodes and elements or remotely offset from a
feature. The way that the model is constrained can significantly affect the results and requires
special consideration. Over or under constrained models can give stress that is so inaccurate
that it is worthless to the engineer. In an ideal world we could have massive assemblies of
components all connected to each other with contact elements but this is beyond the budget
and resource of most people. We can however, use the computing hardware we have
available to its full potential and this means understanding how to apply realistic boundary
conditions.
Basically, there are two types of boundary conditions: Homogeneous boundary condition and
Non - homogeneous boundary conditions.
Homogeneous boundary conditions: These are most commonly used boundary conditions
that occurs at the location that is completely prevented from the movement or the prescribed
basic variable values are zero. Eg. Elimination method of boundary condition.
12
Non - homogeneous boundary conditions: These boundary conditions occurs where finite
non zero values of variables are specified such as the settlement of the support. Eg. Penalty
method of boundary condition.
Basic types of Elements used in FEM:
Discretization is a process of engineering judgment. A selection of the shape or configuration

of the basic element in FEM is very important. The choice of the element depends upon the
geometry (structure) of the problem and the number of independent space coordinates
necessary to describe the system. A finite element has usually a One, Two- or Three-
dimensional configuration. The boundary of the element may be straight (in most of the
cases) or curvilinear.
Potential Energy Method
Potential energy method is one of the variational principles to determine approximate

solution for the problem using total potential energy concept.
POTENTIAL ENERGY ∏.
The total potential energy of an elastic body , is defined as the sum of total strain energy (U)
and the work potential (WP) .
∏ = U + WP
Potential energy method for a three - dimensional body:
Consider a 3-D elastic body of volume v, subjected to body force, surface force and point
loads. Let u, v and w be the displacement components in x, y and z directions respectively.
∏ = U + WP
The strain energy is given by the area under stress-strain curve
U = SE = ϵ
13
For a small elemental volume dv within a body, U = SE = ϵ dv
Total strain energy is U = SE = ∫
Where
Work potential
1. Work potential due to body force

For small element, WPf = -
For entire body, WPf = -∫
WPf = -∫
2. Work potential due to traction/surface force

For small element, WPT = -
For entire body, WPT = -∫
WPT = -∫
3. Work potential due to point loads

WPp = -
= -
If there are “ i ” number of point loads, then
WPp = - ∑
Total Work Potential due to external forces
WP = - ∫ +∫ +∑ ]
The total potential energy, ∏ = U + WP
∏=∫ - ∫ +∫ +∑ ]
Principle of Minimum Potential Energy:
The principle of minimum potential energy states that “Of all possible displacement
configurations a body can assume, which satisfy compatibility and constraints (boundary
14
conditions), the configuration which satisfies equilibrium makes the potential energy assume
a minimum value.”
The meaning of the above statement is that there are several values of displacements, which
satisfy compatibility conditions and boundary conditions, when substituted in the above
equation for gives some values for . But then the system may not be in equilibrium and
the corresponding value of may not be the minimum. Only that displacement which gives
minimum value for will keep the system in equilibrium. To apply Minimum Potential
Energy Principle one has to assume a displacement function for the given problem, which
satisfies the boundary and compatibility conditions, and substitute it in the equation for .
Then must be differentiated with respect to the displacement and equated to zero. The
resulting equation/s when solved yields the value of the unknown displacement.
Rayleigh – Ritz Method
Rayleigh-ritz method is a most widely used variational method to obtain the approximate
solution for the given problems.in this method, an approximate solution of the following type
is assumed for the field variable
u(x) = ∑
where, fi(x) are known linear independent functions are called trial function defined over a
entire domain and boundary and ai are unknown parameters to be determined
Let the potential energy functional
i=1,2,3,4, , , , , ,n
General steps in Rayleigh -Ritz method
1. Formulate the potential energy functional for the given problem.

2. Assume a trial displacement function for a given problem which should satisfy the
boundary conditions.
3. Substitute the displacement function in to potential energy functional.
4. Minimize the potential energy functional so as to obtain the equilibrium conditions.
5. Determine the unknown displacements and thus stresses and strains.
Weighted Residual Method
15
Virtual work and Variational method are applicable and adequate for most of the problems.
However, in some cases functional analogous to potential energy cannot be written because
of not having clear physical meaning. For some applications, such as in fluid mechanics
problem, functional needed for a variational approach cannot be expressed. For some types of
fluid flow problems, only differential equations and boundary conditions are available. For
Such problems weighted residual method can be used for obtaining the solutions.
Approximate solutions of differential equation satisfy only part of conditions of the problem.
For example a differential equation may be satisfied only at few points, rather than at each.
The strategy used in weighted residual method is to first take an approximate solution and
then its validity is assessed. The different methods in weighted Residual Method are
1. Collocation method
2. Least square method
3. Method of moment
4. Galerkin method
Galerkin’s Method
Galerkin method is the most widely used among the various weighted residual methods.
Galerkin method incorporates differential equations in their weak form, i.e., before starting
integration by parts it is in strong form and after by parts it will be in weak form, so that they
are satisfied over a domain in an integral. Thus, in case of Galerkin method, the equations are
satisfied over a domain in an integral or average sense, rather than at every point. The
solution of the equations must satisfy the boundary conditions. There are two types of
boundary conditions:
• Essential or kinematic boundary condition
• Non essential or natural boundary condition
Steps in Galerkin‟s method
1. Formulate the differential equation of equilibrium for the given problem.

2. Assume a trial displacement function which should satisfy the boundary conditions.
3. Substitute the displacement function in to differential equation of equilibrium and
calculate the residual R(x).
4. Use the Galerkin‟s formula, Determine the constants of the function.
5. Determine the unknown displacements and thus stresses and strains
16
Displacement method of finite element formulation

Displacement function is the beginning point for the structural analysis by finite element
method. This function represents the variation of the displacement within the element. On the
basis of the problem to be solved, the displacement function needs to be approximated in the
form of either linear or higher-order function. A convenient way to express it is by the use of
polynomial expressions.
Element Stiffness Matrix
The stiffness matrix of a structural system can be derived by various methods like variational
principle, Galerkin method etc. The derivation of an element stiffness matrix has already
been discussed in earlier lecture. The stiffness matrix is an inherent property of the structure.
Element stiffness is obtained with respect to its axes and then transformed this stiffness to
structure axes. The properties of stiffness matrix are as follows:
 Stiffness matrix issymmetric and square.

 In stiffness matrix, all diagonal elements are positive.
 Stiffness matrix is positive definite
Steps to form global stiffness

1. Initialize global stiffness matrix [K] as zero. The size of global stiffness matrix will be
equal to the total degrees of freedom of the structure.
2. Compute individual element properties and calculate local stiffness matrix [K] of that
element.
3. Add local stiffness matrix[K] to global stiffness matrix [K] using proper locations
4. Repeat the Step b. and c. till all local stiffness matrices are placed globally.
Convergence criteria
The Finite Element Analysis is an iterative procedure. We don‟t get a solution for a problem
being analyzed by FEM just in one go. We have to perform a number of trials or iterations
and record the solution for each iteration. In each iteration we modify the elements being
used for discretization. We then plot a chart known as “Convergence Chart” which is a plot
of solution on „y‟ axis and element parameter on the „x‟ axis. In this chart we see that the
FEM solution curve approaches the actual solution line as we continue with iterations. That
is, the error (difference between the actual solution and FEM solution) continuously
decreases. This is known as Convergence.
17
The FEM solution curve will finally go asymptotic (parallel) to the actual solution line.
Convergence is defined as the monotonic approach of FEM solution to the exact solution.
Monotonic means, with out change in sign or direction.
Methods of attaining Convergence:
There are two methods by which Convergence may be attained.
1. h-method &
2. p-method
h-method of Convergence:
In h-method, for each iteration, the number of elements is increased or in other words, the
element size is reduced, i.e., the element mesh fineness increases with iterations. But the
order of the displacement polynomial selected for the polynomial is maintained the same for
all iterations. In the above figure it may be noted that,
p-method of Convergence:
In p-method, number of elements and element size are kept constant at suitable values. The
number of nodes per element is increased with iterations. Considering more number of nodes
per element will cause the number of terms in the displacement polynomial as well as order
of the displacement polynomial to increase with iterations. (p stands for polynomial).
18
Convergence Requirements or Convergence Criteria to be met by Displacement

Functions: For FEM to give a solution close to the exact solution, the displacement functions
assumed for the elements must satisfy certain requirements (conditions) called “Convergence
Requirements”.
The displacement functions should satisfy the following three conditions:
First Requirement: The assumed displacement function should be continuous with in the
elements and the displacements must be compatible between adjacent elements.
Example: Consider a horizontal bar fixed at one end and subjected to an axial point load „P‟
at the other end. It is required to find the displacement distribution along the length of the bar.
Let it‟s FEM model be made up of 3 elements. Then there will be a total of 4 nodes. Let Q1,
Q2, Q3 & Q4 be the global nodal displacements. Since left end is fixed, Q1= 0. Let us
assume that the displacement variation is linear. According to convergence requirements, the
19
displacement function must be continuous within each element and compatible between
adjacent elements.
Meaning of Continuity:
In the above figure we can understand the difference between continuous and dis-continuous
displacement. Dis-continuous displacement occurs if a crack or defect in the bar is present.
Such discrepancies should be properly specified as boundary conditions.
Second Requirement: The assumed displacement functions should be capable of

representing rigid body displacement of the element. The rigid body displacement is the most
elementary displacement that an element may undergo. i.e., during rigid body displacement
the element should not be strained.
Usually, the first term of the assumed displacement polynomial will be a constant which
represents the rigid body displacement.
For example, for a 2D element, the assumed displacement polynomial may be
U( x, y) = + + + ….
Discretization process
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 whole structure 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. 1.5. shows a finite element mesh of a
20
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.
Figure 1.5. Discretization of Continuum
The conditions to be taken in the discretization process are
1. Type of elements
2. Size of elements
3. Number of elements
4. Location of nodes
5. Node numbering scheme
Type of elements
One Dimensional Elements (Line Elements): When the geometry, material and
displacement can be expressed in terms of only independent space coordinate, a 1D element
may be chosen. The coordinate is measured along the axis of the element.
21
A 1D element can be represented by a straight line whose end points are nodes (nodal points).
The nodal points numbered 1 & 2 are called external nodes because they represent connecting
points with adjacent elements. Some applications may require additional nodes, such as node
3. It is called an internal node because it doesn‟t form a connecting point with adjacent
elements. Additional nodes are usually used to improve the FEM result.
Two Dimensional Elements (Area Elements): When the geometry, material and
displacement can be expressed by minimum two independent space coordinate, a 2D element
may be chosen. Problems that can be modeled with 2D elements are plane stress, plane strain
and axisymmetric problems.
A 2D element can be represented by a triangular or quadrilateral configuration.
In the triangular element, the corner nodes (1, 2 & 3) are called primary external nodes.
Additional nodes on the sides (like nodes 4, 5 & 6) are called secondary external nodes. An
internal node (like node 7) may be used in a triangular element.
Similarly, a quadrilateral configuration may have a minimum of 4 primary external nodes.

As well they may contain the additional secondary external nodes and the internal nodes. A
special case of the quadrilateral configuration is the rectangular configuration
22
Three Dimensional Elements (Volume Elements): When the geometry, material and
displacement can be expressed by minimum of three independent space coordinate, a 3D
element may be chosen.
A tetrahedron element is the basic 3D element that can be used in FEM. It has 4 primary
external nodes. 3D elements with 8 primary external nodes may be the hexahedron element
or the rectangular prism element. The rectangular prism element is a special case of
hexahedron element.
Axisymmetric Elements: Problems involving 3D axisymmetric solids or solids of

revolution, subjected to axisymmetric loading, can be reduced to simple 2D problems.
Because of the total symmetry about „z‟ axis, all deformations and stresses are independent of
the rotational angle „q‟. Thus the problem needs to be looked as a 2D problem in „r‟ & „z‟
coordinates only, which defines the revolving area.
Size of Elements
Mesh size is one of the most common problems in FEA. There is a fine line here: bigger
elements give bad results, but smaller elements make computing so long that we don‟t get the
results at all. So it is appropriate to choose smaller elements at higher stress concentration
23
areas as shown in the Figure 1.6. It is required to choose the correct size of mesh and estimate
at which mesh size accuracy of the solution is acceptable.
Figure 1.6. Size of elements in discretization process
Number of Elements
Selection of number of elements in FE model is related to the size of elements and number of
degrees of freedom. Increase in number of elements gives better result but there is a certain
limit beyond which there is no improvement but solution time is more. Therefor it is
important to choose the number of elements appropriately.
Figure 1.7. Number of elements in FE model
Location of nodes: Node is a point where properties are defined and determined for an
element. Therefore, location of node is important. If the body has no discontinuity, then there
can be equal number of nodes in an element distributed uniformly. However, if there is
24
discontinuity such as geometry and load etc, the nodes can be assigned as shown in the
Figure 1.8.
Figure 1.9. Location of nodes in FEA
Banded Matrix
The size of the stiffness matrix increases as the number of elements and degree of freedom is
more which further increases the solution time. Therefore, the width of the square stiffness
matrix is reduced so as to reduce the solution time and is called as banded matrix. The width
of the banded matrix is called half band width.
25
Figure 1.10. Banded stiffness matrix
Node numbering scheme
The numbering of nodes such as in horizontal and vertical in regular or zig-zag pattern affects
the size of banded matrix which further affects the solution time. Therefore, It is important to
follow certain node numbering scheme. The different node numbering schemes are shown
in the Figure 1.11.
Numbering along longer edge:
Numbering along shorter edge:
Figure 1.11. Node Numbering scheme.
Strain-Displacement Relations:
26
Consider a two dimensional element abcd which lies in the x-y plane as shown in figure.when
a force acts on the element .it undergoes deformation and it becomes a‟b‟c‟d‟.Displacement
in the x-direction is u and y-direction is v
In general,normal strain is defined as the ratio of change in length to original length of the
body.
Considering element ab in x-direction
Strain,
from figure ab=dx
[ ] [ ]
Using the binomial theorem and neglecting the higher order terms
We have * +
Substituting we get
Strain in x-direction
similarly considering the element ad in y-direction
Strain in y-direction
27
The shear strain represent the amount of distortion abcd. the angle dab is right andle in the
undeformed state but has been distorted to angle d‟ab‟ by shearing .the change in angle is
denoted composed of two parts denoted by
Shear strain
the above equations are the strain displacement relations for a two dimensional element
for three dimensional relations are obtained by extending two dimensional analysis
strain in z-direction
Shear strain
Stress – Strain Relations:
The equations between stress and strain applicable to a particular material are known as
constitutive equations for that material.
The modulus of elasticity is defined as the slope of the stress-strain curve in the elastic region
Poisons ratio is defined as
V= unit lateral contraction/ unit axial elongation
Thus, in the tension test,if represents the strain resulting from applied load, the induced
strain components are given by
The general stress-strain relations for a homogeneous ,isotropic. Linearly elastic material
subjected to a general three-dimensional deformation are as follows
Normal strain are given by
{ }
{ }
{ ( )}
28
Shear strains are given by
Similarly, we have
Normal stress are given by
[ ( )]
[ ]
[ ( )]
Shear stress are given by
Where, G is the shear modulus or modulus of rigidity is defined by
The stress strain relations can be easily be expressed in matrix by defining the material
propert matrix [D] as
29
( )
( )
( )]
[
Plane stress:
plane stress case is represented by a thin plate in x-y plane,plane subjected to in-plane loads
along x-and y-directions and no load along the normal to the olane as shown in figure
Figure 1.12. Plane stress
Definition : Plane stress is defined to be aa state of stress in which the normal stress and shear
stress directed perpendicular to the plane are assumed to be zero
For plane stress we have
A plate loaded in its mid-plane is said to be in a state of plane stress or a membrane state,if
the following assumptions hold:
1.All loads applied to the plate act in the mid-plane direction,and are symmetric with respect
to the mid-plane
2. All support conditions are symmetric bout the mid-plane
3. In-plane displacement, strain and stresses can be taken to be uniform through the thickness
4.The normal and shear stress components in the z-direction are zero or negligible
The governing equations are
1.Equilibrium equations
30
2.Strain displacement relation
3. stress - strain relation
( )
( )
( )
Plane Strain:
Plane strain case is represented by a thin plate in x-y plane ,which is constrained along the
normal to the plane in z-direction
Definition: plane strain is defined to be state of strain in which normal-strain and shear strain
normal to xy plane are assumed to be zero
Assumptions :
1.The length of the structure is very large in comparision with the other two dimensions
2.The loads are applied only normally to the longitudinal axis
3.the support conditions are the same along the z-axis
31
Figure 1.13. Plane strain
The governing equations are
1.Equilibrium equations
2.Strain displacement relation
3.stress-strain relation:
[ ]
And
Temperature Effect
32
When the free expansion is prevented in a structure, the change in temperature causes stress.
In general temperature, or temperature changes in a solid may induce the following effects:
1) Temperature increase will change material properties: Such as decrease the Young‟s
modulus (E) and yield strength of materials (σy).
2) Induce thermal stresses that will be added to mechanically induced stresses in solid
structures.
3) Induce creep of the material, and thereby make materials vulnerable for failure at high
temperature
The thermal stress is given by [σ] = E
The initial stress due to change in temperature { } for 2-D analysis.
Stress [σ] = [D] {ϵ} – { }
Interpolation Models
Displacement Models: In FEM after we discretize the problem continuum into finite number
of elements, we assume a displacement function for an element which closely represents the
expected displacement variation over the element domain. These simple functions which are
assumed to approximate the displacements for each element are called as “Displacement
Models” or “Displacement Functions” or “Displacement Fields” or “Displacement Patterns”.
Earlier many types of mathematical functions (like trigonometric functions, algebraic

functions, etc.) were assumed for the displacement functions. But presently mathematical
polynomials of a specific order are used to represent the displacement functions.
Depending upon the geometry of the element and the order of the polynomial used in the
interpolation model, finite elements can be classified as
 Simplex model
 Complex model
 Multiplex model
Simplex model: The element for which the order of the assumed polynomial interpolation
model consist of constants and linear terms only are called Simples elements.
Thus the following polynomial interpolation models can be used to represent the simplex
elements
33
One dimensional case:
U(x) =
Two dimensional case:
U(x,y) =
Three dimensional case:
U(x,y,z) = +
Complex elements: The elements for which the assumed polynomial interpolation model
consists of quadratic ,cubic, and higher order terms in addition to the constant and linear
terms are called complex elements
Quadratic model
One dimensional case:
U(x,) =
Two dimensional case:
U(x,y) =
Three dimensional case:
U (x y z) =
Multiplex elements: The multiplex elements are those, for which assumed polynomial
interpolation function consists of higher order terms.in multiplex elements, boundaries are
parallel to the coordinate axis to achieve inter element continuity.
Reasons for using polynomials for displacement functions:
1. It is easy to handle the mathematics of polynomials in formulating the element equations

and in performing digital computations. Differentiation and integration of such a polynomial
is easy.
2. The solution can attain a particular level of approximation by selecting a particular order of
the polynomial. So a polynomial of infinite order represents an exact solution. Hence, by
truncating the polynomials at different orders, we can vary the solution approximation.
For example, suppose we want to represent the displacement „u‟ as a function
34
Greater the number of terms used in the polynomial, closer will be the solution to the exact
solution.
So, it can be concluded that a displacement polynomial with larger number of terms will
approximate the solution close to the exact solution. To represent the 1D problems (where the
displacement is along the element axis), the displacement polynomial should have only one
variable in it. If the 1D element axis is oriented along the horizontal direction (parallel to „x‟
axis),
If the 1D element axis is oriented along the vertical direction (parallel to „y‟ axis),
35
To represent 2D problems, where the displacement has components „u‟ & „v‟ along „x‟ and
„y‟ directions respectively, the displacement polynomial should have two variables
How to select the order of the assumed displacement polynomial?
The convergence requirements that
· Rigid body displacement should be represented, and
· Constant strain rates must be represented
upto some extent decide the order of the displacement polynomial.
An additional consideration in the selection of the polynomial order is that the selected
pattern (polynomial) should be independent of the orientation of the local coordinate system
in the global coordinate system. This property of the displacement function is called
“Geometric Isotropy” or “Spatial Isotropy” or “Geometric Invariance”.
For polynomials of higher orders, only those higher order coordinates are considered which
satisfy Geometric Isotropy. The best method to select the higher order coordinates which
represent Geometric Isotropy is to make use of the PASCAL‟S TRIANGLE of coordinates.
Pascal‟s triangle has symmetric coordinates which take care of Geometric Isotropy.
36
Using the Pascal‟s Triangle, suppose we want to write the displacement polynomial for a 8
noded quadrilateral element, keeping symmetry in mind, we can include the following terms.
It can be noted that since there are 8 nodes, the displacement polynomial has 8 terms
Serendipity Elements: Serendipity quadrilateral elements have additional nodes only on the
element edges and nodes within the element are absent. Figure (2) shows the Pascal‟s
Triangle which highlights the method of choosing higher order terms to be included and
number of terms or nodes in the displacement polynomial for Serendipity elements.
Other Serendipity Family higher order elements can similarly be defined.
37
Higher Order Elements:
The minimum number of DOFs (number of terms in the displacement polynomial) for a
given element is determined by
 The completeness requirements of convergence

 The requirement of geometrical isotropy
 The necessity of adequate representation of terms in the potential energy functional
By providing additional DOFs, the FEM solution accuracy (nearness to exact solution) can be
improved. Additional DOFs beyond the minimum number may be included for an element by
adding secondary external nodes. So, the elements with additional DOFs are called “Higher
Order Elements”
Higher Order Quadrilateral Elements:
These can be classified as two types
(1) Lagrange Elements
(2) Serendipity Elements
Lagrange Elements: Lagrange quadrilateral elements have the additional nodes on the
element edges as well as within the element.
Figure (1) shows the Pascal‟s Triangle which highlights the method of choosing higher order
terms to be included and number of terms or nodes in the displacement polynomial for
Lagrange element.
Other Lagrange Family higher order elements can similarly be defined
38
Higher Order Triangular Elements:
Higher Order Triangular Elements have the additional nodes on the element edges as well as
within the element. Figure (3) shows the Pascal‟s Triangle which highlights the method of
choosing higher order terms to be included and number of terms or nodes in the displacement
polynomial for Higher Order Triangular Elements.
Loading Conditions:
There are multiple loading conditions which may be applied to a system. The load may be
internal and/or external in nature. Internal stresses/forces and strains/deformations are
developed due to the action of loads. Most loads are basically “Volume Loads” generated due
to mass contained in a volume. Loads may arise from fluid-structure interaction effects such
as hydrodynamic pressure of reservoir on dam, waves on offshore structures, wind load on
buildings, pressure distribution on aircraft etc. Again, loads may be static, dynamic or quasi-
static in nature. All types of static loads can be represented as:
• Point loads
• Line loads
• Area loads
• Volume loads
The loads which are not acting on the nodal points need to be transferred to the nodes
properly using finite element techniques.
Support Conditions:
In finite element analysis, support conditions need to be taken care in the stiffness matrix of
the structure. For fixed support, the displacement and rotation in all the directions will be
restrained and accordingly, the global stiffness matrix has to modify. If the support prevents
translation only in one direction, it can be modelled as „roller‟ or „link supports‟. Such link
supports are commonly used in finite element software to represent the actual structural state.
39
Sometimes, the support itself undergoes translation under loadings. Such supports are called
as „elastic support‟ and are modelled with „spring‟. Such situation arises if the structures are
resting on soil.
40
Question bank and solution
Theory questions
1. Define the following terms: Normal stress, Shear stress, Longitudinal stress.
(DEC.07-JAN.08)
2. Explain plane stress and plane strain problem with suitable examples.
(DEC.06-JAN.07, JUNE-JULY 2009, JUNE- JULY 2018)
3. With the example, explain node numbering and element connectivity for 1D bar.
4. Explain briefly about node location system.
5. Explain basic steps involved in FEM. (DEC07-JAN.08, DEC-JAN.2018)
6. What is FEM? List the advantages and disadvantages of FEM over other numerical
method FDM. (MAY/JUNE2010 JUNE- JULY 2017)
7. Explain pre-processing and post processing in FEA software.
8. Explain discretization process. sketch the different types of elements 1D,2D,3D,

Elements used in finite element analysis. (JUNE.JULY 2009, DEC-JAN.2017)
9. State and prove principle of minimum potential energy and principle of virtual work.
10. Explain banded matrix and node numbering scheme.
41
Problems
1. Find the Eigen values & Eigen vectors of the matrix

(DEC.06-JAN.07, JUNE-JULY 2009
A=* +
Solution: The characteristic equation is:
|* + * +|=0
|* +|=0
12 + 27 = 0
The eigen values are
To find the Eigen vectors: Using the characteristic equation:
[ ]{ }=0
* +{ }=0
{ } , -
Then,
[ ]{ }=0
* +{ }=0
{ } , -
42
2. Find the eigen values & eigen vectors of the matrix (DEC07-JAN.08, DEC-
JAN.2018)
A=* +
Solution: The characteristic equation is:
|* + * +|=0
|* +|=0
The eigen values are
To find the Eigen vectors: Using the characteristic equation:
[ ]{ }=0
* +{ }=0
{ } , -
Then,
[ ]{ }=0
* +{ }=0
{ } , -
3. Evaluate the integral using 1 point and 2 point integral rule and check the solution
by exact integration. (MAY/JUNE 2010 ’DEC2009)
I=∫ * +
Solution: The given integral is a 1D integral. (Integral is with respect to only one variable x)
43
By One-Point Integration Rule: f (x) =
One-point rule for 1D integral is: I = ∫ = W1 f (x) = 2 f (0)
By Two-Point Integration Rule:
Two point rule for 1D integral is:
I=∫ = W1 f (x) + W1 f (x) = (1) f ( ) + (1) f ( )

√ √
I = (1) 0 √ ( ) 1 + (1) 0 √ ( ) 1
√ √
√ √
By exact integration:
∫ * + = * +
= ) + ( 1+1) + (ln 3- ln1) = 8.816486
4. Determine the nodal displacements by using the principle of Minimum Potential

Energy (DEC’2013/2014, JUNE-JULY 2017)
Solution: There are 4 springs in the system shown above. K1, K2, K3 & K4 are the spring
constants (stiffnesses). Each spring (element) has 2 nodes (inter-connectivity) points. So
totally there are 5 nodes in the system. Each node is free to move in the +ve x direction (right
ward direction). But nodes 4 & 5 are completely constrained (fixed) and hence can‟t have any
displacement. So the system is free to move only at nodes 1, 2 & 3. Hence it has a total of 3
degree of freedom (dof).
Let δ be the spring displacement and u is the nodal displacement.
44
The relation between spring displacement and nodal displacement is given by
Element No. Nodes

i j
1 1 2
2 2 3
3 2 4
4 4 5
∏ = U + WP
U = SE spring =
SE system = +
SE system = +
SE system = –2 + –
WP =
∏ = U + WP
∏ = – 2 + –
For Potential energy functional to be minimum,
= 0, = 0, = 0,
= =0
or = …..(1)
= =0
45
or + ) …….(2)
+ =0
or ………….(3)
Substituting the values ,
, in equations 1, 2 and 3
60=
+ )
5. Determine the nodal displacements for the given spring system by using the
principle of Minimum Potential Energy. ( JUNE-JULY 2015, 2016)
Let δ be the spring displacement and u is the nodal displacement.
The relation between spring displacement and nodal displacement is given by
Element No. Nodes

i j
1 1 2
2 2 3
3 2 4
46
∏ = U + WP
U = SE spring =
SE system = +
SE system = +
SE system = + ( – )
WP =
∏ = U + WP
∏ = + ( – )
For Potential energy functional to be minimum,
= 0, =0
= - =0
or = …..(1)
= =0
or …….(2)
Substituting the values ,
, in equations 1and 2
47
6. Consider a bar, fixed at both ends, loaded axially at the center with 2 units. The
length of the bar is 2 units. Determine stress and strain at the centre of the bar by
Rayleigh – Ritz method. Given that u x=1 = u1. (DEC’08-JAN.09, 2014)
Solution: The potential energy of the 1D bar can be written as:
Step 1: Formulate Potential energy functional
.∏ = U + WP
∏=∫ - ∫ +∫ +∑ ] ……..(1)
Body force ‘ ’ and traction force ‘T’ are not acting on the bar,
U = SE = ∏ = U + WP
WP = - P -2
Therefore, ∏ = ∫ -2 ……..(2)
∏= ∫ ( ) –2
It is required to solve the equation (4) to get „u‟ Let us assume an approximation function „u‟
such that it satisfies the boundary conditions and it is having the same degree as equation (4).
To evaluate stress or strain at any point on the bar, 3 a should be evaluated.
∏= ∫ ( ) –2
Step 2 : Assume trial displacement function
u= + ………(1)
48
Step 3 : Apply boundary conditions
u = 0 at x = 0
Therefore, from (1)
And u = 0 at x = 2
So (10 reduces to
0=
= -2
Now at the centre, u = at x = 1
Therefore -2 =-
∏= ∫ ( ) –2
Step 4 : Substitute displacement function in to PE functional
∏= ∫ ( ) –2
A= 1, E = 1
u= +
u=
∏= ∫ –2
Step 5: Minimize Potential energy functional
Stress σ = E = 1
σ=
at x= 0, σ = -1.5 units
49
at x= 1, σ = 0 units
at x= 2, σ = 1.5 units
Displacement variations over the length is shown below
x u=
0.00
0.00
0.33
0.25
0.56
0.50
0.7
0.75
0.75
1.00
0.7
1.25
0.56
1.50
0.33
1.75
0.00
2.00
7. Determine the stress and strain at the bar centre and expression for displacement by
Rayleigh – Ritz method.(JAN/FEB.2006,DEC.06/JAN.07, 2010, 2012)
Solution:
The potential energy of the 1D problem is,
.∏ = U + WP
∏=∫ - ∫ +∫ +∑ ] ……..(1)
U = SE = ∏ = U + WP
50
WP = body force is uniformly distributed over the length
Therefore, ∏ = ∫ -∫ ……..(2)
∏= ∫ ( ) –∫
It is required to solve the equation (4) to get „u‟ Let us assume an approximation function „u‟
such that it satisfies the boundary conditions and it is having the same degree as equation (4).
To evaluate stress or strain at any point on the bar, 3 a should be evaluated.
∏= ∫ ( ) –2
u= + ………(1)
Step 3 : Apply boundary conditions
= -200
WP = ∫
Step 4 : Substitute displacement function in to PE functional
∏= ∫ ( ) –2
∏= ∫ – A∫
∏ = 20000 AE + -200 AE + 100 A -
Step 5: Minimize Potential energy functional
2666667x103
Therefore u =
The displacement and stress variation is shown as
51
x u=
0.00 1.09 x 10-2

25 1.09 x 10-2
50 1.09 x 10-2
75 1.09 x 10-2
100 1.09 x 10-2
125 1.09 x 10-2
150 1.09 x 10-2
175 1.09 x 10-2
200 1.09 x 10-2
8. For the problem given, considering only body force, using Raleigh–Ritz method, find
the expression for displacement. (JAN/FEB 2006)
Solution:
Solution:
The potential energy of the 1D problem is,
.∏ = U + WP
∏=∫ - ∫ +∫ +∑ ] ……..(1)
U = SE = ∏ = U + WP
WP = body force is uniformly distributed over the length
52
Therefore, ∏ = ∫ -∫ ……..(2)
∏= ∫ ( ) –∫
u= +
For 1D problem this can be written as:
Boundary conditions are:
At x = 0, u = 0
At x = L, u =
The strain is given as: = = - ( L-2x)
. / ( )
=( )
Substitute in the expression for WP,
WP = - ∫ =-∫ ( )
=-( )
= * +
∏ = Strain energy (U) + Work potential (WP)
∏=* + - +
Substituting: U =
53
9. A beam of uniform cross-section area ‘A’, length ‘L’ of Young’s modulus ‘E’ and
moment of inertia ‘I’ is subjected to a U.D.L. of ‘p0’ per unit beam length and a
concentrated load ‘P’ acting at beam centre as shown in figure. Using Rayleigh – Ritz
method and using trigonometric approximation function, find the maximum deflection
in the beam. (DEC06-JAN.07)
Solution: The potential energy P of a beam is given by:
∏ = Strain energy (U) + Work potential (WP)
U= ∫ ( )
WP = ∫ -∑ -∑
∏= ∫ ( ) ∫ -∑ -∑
In the given problem, moments are not applied.
∏= ∫ ( ) ∫
The assumed displacement function is
sin( ) + sin( )
The boundary conditions of the given problem are:
At x = 0, w = 0
At x = L, w = 0
At x = L/2, w =
54
We can write:
( ) cos( )
( ) cos( )
sin( ) sin( )=- ( ( ) ( ))
( ) ( )( ( ))+( )( ( )) +
( ). ( ( ) ( ))/
Substituting (6) in expression for strain energy ‘U’
∏= ∫ ( ) ∫
∏ = ( )–p(
We get:
( )– ( )
55

Fea Module 1

Uploaded by

Copyright:

Available Formats

Fea Module 1

Uploaded by

Document Information

Original Description:

Original Title

Copyright

Available Formats

Share this document

Share or Embed Document

Sharing Options

Did you find this document useful?

Is this content inappropriate?

Copyright:

Available Formats

Fea Module 1

Uploaded by

Copyright:

Available Formats

Updated on 16.04.

Interpolation models: Simplex, complex and multiplex elements, Linear interpolation

Introduction to Numerical Methods

• Finite Volume Method

Background of Finite Element Analysis

Introduction to Finite Element Method – General Description

Finite Element Method (FEM) is a numerical technique to find approximate solutions of

Figure 1.1 Finite Element meshing of an aircraft structure

Basic Steps in Finite Element Analysis

The following steps are performed for finite element analysis.

Figure 1.2 Steps in finite element method

Concepts of Elements and Nodes

Figure 1.3 Elements and nodes in finite element method

Advantages of finite element method

Disadvantages of finite element method

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

1. Proper engineering judgment is to be exercised to interpret results.

1. Mechanical engineering: In mechanical engineering, FEM applications include steady

2. Geotechnical engineering: FEM applications include stress analysis, slope stability

5. Electrical and electronics engineering: FEM applications include electrical network

6. Metallurgical, chemical engineering: In metallurgical engineering, FEM is used for the

7. Meteorology and bio-engineering: In the recent times, FEM is used in climate

Basic terminologies used in FEM

2. Elements: Each subdivision is called an “Element”

5. Boundary Conditions: Known values of the variable at the continuum boundary

6. Displacement Function: It is an assumed polynomial expression which closely represents

One Dimensional Problems: If only one independent coordinate axis is sufficient to

solved in a domain on whose boundary a set of conditions is known. It is opposed to the

Figure 1.4. Boundary condition

Basic types of Elements used in FEM:

Discretization is a process of engineering judgment. A selection of the shape or configuration

Potential Energy Method

Potential energy method is one of the variational principles to determine approximate

Potential energy method for a three - dimensional body:

The strain energy is given by the area under stress-strain curve

For a small elemental volume dv within a body, U = SE = ϵ dv

Total strain energy is U = SE = ∫

1. Work potential due to body force

2. Work potential due to traction/surface force

3. Work potential due to point loads

Total Work Potential due to external forces

The total potential energy, ∏ = U + WP

Principle of Minimum Potential Energy:

Rayleigh – Ritz Method

Let the potential energy functional

General steps in Rayleigh -Ritz method

1. Formulate the potential energy functional for the given problem.

Weighted Residual Method

Steps in Galerkin‟s method

1. Formulate the differential equation of equilibrium for the given problem.

Displacement method of finite element formulation

Element Stiffness Matrix

 Stiffness matrix issymmetric and square.

Steps to form global stiffness

Methods of attaining Convergence:

There are two methods by which Convergence may be attained.

Convergence Requirements or Convergence Criteria to be met by Displacement

The displacement functions should satisfy the following three conditions: