IFET COLLEGE OF ENGINEERING
(An Autonomous Institution)
Department of Mechanical Engineering
COURSE DETAILS
Course Code : 19UMEPC702
Course Name : FINITE ELEMENT ANALYSIS
Semester/Year : B.E (MECH) VII SEMESTER/ IV YEAR
Offered By : MECHANICAL ENGG
Prepared By : Dr. B.ELAMVAZHUDI
Reviewed By : Mr.N.SETHURAMAN
Approved By : Dr. B.ELAMVAZHUDI
UNIT I INTRODUCTION
PART-A
4 Write the applications of polynomial type interpolation functions in solving governing A
equations of field problems.
It is easy to perform differentiation or integration.
accuracy of the results can be improved by increasing the order of the polynomial.
Orientation and integration of polynomials are quite easy.
accuracy of the results can be improved by increasing the order of the polynomial.
It is easy to formulate and computerize the finite element equations.
5 Brief about three phases of finite element method followed in FEA. A
The effectiveness of FEM has been shown when applied to problems with domains having
complicated boundaries, problems with micro-scales, and heat transfer problems. The three phases
of finite element methods are preprocessing, analysis and post processing.
Phase1: Preprocessing – specify material and boundary conditions
Phase2: Processing - solve the stiffness matrix and force applied
Phase3: Post processing -results output and stress, strain behavior and temperature distribution.
11 If a displacement field of a steel bar subjected to displacement in x direction is given by A
u=2x2+4y2+6xy. Determine the strain in x direction.
strain in x direction of axially loaded bar :
u
4x 6 y
x
The relationship that connects the displacement field with the strain is called as strain displacement
relationship. The rigid body displacements do not give raise to any stresses. Further stresses are
induced only when there are relative displacements of the material particles. Displacement is the
difference between the position vectors of a material at two different instances of time
12 Illustrate the various field problems with their governing equations followed in one dimensional A
problems.
i).longitudinal vibration of bar element
, d 2u
AE A 2u 0
dx 2
ii.)forced axial vibration of a rod
2u 2u
AE 2 A 2
x t
iii.)Governing equation for transverse vibration of beam
d4y
EI 4 q0 0
dx
14 Formulate governing equation of one dimensional steel rod subjected axial load. A
If the forces and displacements do not vary with time, we have a static analysis situation and the
equilibrium equation is an ordinary second-order differential equation as follows:
While formulating the governing equation let us consider the forces acting on the body and the
nodes where the load are acting and also the edges whether it is fixed or free to the load acting on it
to get the accurate or exact solution for the given problem.
AE +a x=0
The boundary conditions are u (0)=0, A E x=L =0
15 Express the governing equation for an iron rod is held at absolute zero condition at one end. A
The objective of most analyses is to determine unknown functions as called dependent variables,
that satisfy a given set of differential equation in a given domain or region and some boundary of
the domain. This differential equation is said to describe a boundary value problem if the dependent
variable and possibly its derivatives are required to take specific values on the boundary. The
following differential equation depicts boundary value problem.
d du
a f for 0 x l
dx dx
du
u( x ) 0; a x l g 0
dx
16 Formulate governing equation of cantilever beam extended from a balcony A
d 4v
The general governing equation for the beam is EI qo 0
dx 4
Since there is no distributed load q0 = 0
The governing equation becomes
, d 4v
EI 0
dx 4
dv
v(0)=0 and 0
dx x 0
d 2v d 3v
EI M 0 and EI p0
dx 2 x l dx 3 x l
24 Formulate the governing equation for a simple pendulum with suitable illustration. A
Initial value problem is one in which the dependent variable and possibly its derivatives are
specified initially (i.e., at time t=0).Initial value problems are generally time dependent problems.
The following equations depicts the initial vale problem
d 2u
2 au f for 0 x t 0
dt
du
u( x ) u 0 ; x l v 0
dt
If any of the initial condition of any of the specified are non-zero then the conditions are non-
homogeneous for example u ( x ) u 0 . If the initial condition is zero homogeneous condition for
example u(0) 0
31 Deduce two dimensional differential equation of axially loaded bar into weak form A
A weak form is a weighted integral statement of a differential equation in which the differentiation
is distributed among the dependent variable and the weight function and also includes the natural
boundary conditions of the problem
d 4u
EI 4 q 0
dx
Boundary conditions
Essential v(0) = 0; v(L) = 0 (zero displacement)
d 2v d 2v
Natural (0) 0 2 ( L) 0 (zero moments)
dx 2 dx
The solution of the higher order gets simplified since its order has been reduced
l l
d 2 v d 2 w( x )
0 EI dx 2 dx 2 0 Wqdx
35 Apply the stationary property of total potential energy functions for beam. A
To generalise, the “potential energy” of the applied loads for beam to be V W so that
𝜋=𝑈+𝑊
The external loads must be conservative, precluding for example any sliding frictional loading.
Taking the total potential energy to be a function of displacement u, one has
δ𝜋=𝑈+𝑊=0
Thus of all possible displacements u satisfying the loading and boundary conditions, the actual
displacement is that which gives rise to a stationary point d / du 0 and the problem reduces to
finding a stationary value of the total potential energy U W.
, 39 Formulate the minimum potential energy equation for a bridge girder with UDL load
Total potential
U W
2
1 l d 2v l
EI 2 dx q0 v dx
2 0 dx 0
40 Express the minimum potential equation for cylindrical pin-fin used in heat exchangers. A
A single fin is isolated, and assuming temperature is uniform over the width w of the fin, the
problems can be described in terms of the following differential equation.
PART-B
9 A tapered bar made of steel is suspended vertically with the larger end rigidly clamped and the 16 A
smaller end acted on by a pull of 105 N. The areas at larger and smaller end are 80 cm2 and
20cm2 respectively. The length of the bar is 3m.the bar weighs 0.075 N/cc. E = 2X107
N/cm2.obtain the approximate expression for the deformation of the rod using Ritz technique.
Determine the maximum displacement at the tip of the bar.
According to Rayleigh Ritz The total potential function for the taper bar when subjected to end
load and uniformly distributed load is
2
du
300 300
1
E A( x) dx qudx pu ( L) --------------------------- (1)
2 0 dx 0
Since Area is varying linearly throughout the stretch.
(An Autonomous Institution)
Department of Mechanical Engineering
COURSE DETAILS
Course Code : 19UMEPC702
Course Name : FINITE ELEMENT ANALYSIS
Semester/Year : B.E (MECH) VII SEMESTER/ IV YEAR
Offered By : MECHANICAL ENGG
Prepared By : Dr. B.ELAMVAZHUDI
Reviewed By : Mr.N.SETHURAMAN
Approved By : Dr. B.ELAMVAZHUDI
UNIT I INTRODUCTION
PART-A
4 Write the applications of polynomial type interpolation functions in solving governing A
equations of field problems.
It is easy to perform differentiation or integration.
accuracy of the results can be improved by increasing the order of the polynomial.
Orientation and integration of polynomials are quite easy.
accuracy of the results can be improved by increasing the order of the polynomial.
It is easy to formulate and computerize the finite element equations.
5 Brief about three phases of finite element method followed in FEA. A
The effectiveness of FEM has been shown when applied to problems with domains having
complicated boundaries, problems with micro-scales, and heat transfer problems. The three phases
of finite element methods are preprocessing, analysis and post processing.
Phase1: Preprocessing – specify material and boundary conditions
Phase2: Processing - solve the stiffness matrix and force applied
Phase3: Post processing -results output and stress, strain behavior and temperature distribution.
11 If a displacement field of a steel bar subjected to displacement in x direction is given by A
u=2x2+4y2+6xy. Determine the strain in x direction.
strain in x direction of axially loaded bar :
u
4x 6 y
x
The relationship that connects the displacement field with the strain is called as strain displacement
relationship. The rigid body displacements do not give raise to any stresses. Further stresses are
induced only when there are relative displacements of the material particles. Displacement is the
difference between the position vectors of a material at two different instances of time
12 Illustrate the various field problems with their governing equations followed in one dimensional A
problems.
i).longitudinal vibration of bar element
, d 2u
AE A 2u 0
dx 2
ii.)forced axial vibration of a rod
2u 2u
AE 2 A 2
x t
iii.)Governing equation for transverse vibration of beam
d4y
EI 4 q0 0
dx
14 Formulate governing equation of one dimensional steel rod subjected axial load. A
If the forces and displacements do not vary with time, we have a static analysis situation and the
equilibrium equation is an ordinary second-order differential equation as follows:
While formulating the governing equation let us consider the forces acting on the body and the
nodes where the load are acting and also the edges whether it is fixed or free to the load acting on it
to get the accurate or exact solution for the given problem.
AE +a x=0
The boundary conditions are u (0)=0, A E x=L =0
15 Express the governing equation for an iron rod is held at absolute zero condition at one end. A
The objective of most analyses is to determine unknown functions as called dependent variables,
that satisfy a given set of differential equation in a given domain or region and some boundary of
the domain. This differential equation is said to describe a boundary value problem if the dependent
variable and possibly its derivatives are required to take specific values on the boundary. The
following differential equation depicts boundary value problem.
d du
a f for 0 x l
dx dx
du
u( x ) 0; a x l g 0
dx
16 Formulate governing equation of cantilever beam extended from a balcony A
d 4v
The general governing equation for the beam is EI qo 0
dx 4
Since there is no distributed load q0 = 0
The governing equation becomes
, d 4v
EI 0
dx 4
dv
v(0)=0 and 0
dx x 0
d 2v d 3v
EI M 0 and EI p0
dx 2 x l dx 3 x l
24 Formulate the governing equation for a simple pendulum with suitable illustration. A
Initial value problem is one in which the dependent variable and possibly its derivatives are
specified initially (i.e., at time t=0).Initial value problems are generally time dependent problems.
The following equations depicts the initial vale problem
d 2u
2 au f for 0 x t 0
dt
du
u( x ) u 0 ; x l v 0
dt
If any of the initial condition of any of the specified are non-zero then the conditions are non-
homogeneous for example u ( x ) u 0 . If the initial condition is zero homogeneous condition for
example u(0) 0
31 Deduce two dimensional differential equation of axially loaded bar into weak form A
A weak form is a weighted integral statement of a differential equation in which the differentiation
is distributed among the dependent variable and the weight function and also includes the natural
boundary conditions of the problem
d 4u
EI 4 q 0
dx
Boundary conditions
Essential v(0) = 0; v(L) = 0 (zero displacement)
d 2v d 2v
Natural (0) 0 2 ( L) 0 (zero moments)
dx 2 dx
The solution of the higher order gets simplified since its order has been reduced
l l
d 2 v d 2 w( x )
0 EI dx 2 dx 2 0 Wqdx
35 Apply the stationary property of total potential energy functions for beam. A
To generalise, the “potential energy” of the applied loads for beam to be V W so that
𝜋=𝑈+𝑊
The external loads must be conservative, precluding for example any sliding frictional loading.
Taking the total potential energy to be a function of displacement u, one has
δ𝜋=𝑈+𝑊=0
Thus of all possible displacements u satisfying the loading and boundary conditions, the actual
displacement is that which gives rise to a stationary point d / du 0 and the problem reduces to
finding a stationary value of the total potential energy U W.
, 39 Formulate the minimum potential energy equation for a bridge girder with UDL load
Total potential
U W
2
1 l d 2v l
EI 2 dx q0 v dx
2 0 dx 0
40 Express the minimum potential equation for cylindrical pin-fin used in heat exchangers. A
A single fin is isolated, and assuming temperature is uniform over the width w of the fin, the
problems can be described in terms of the following differential equation.
PART-B
9 A tapered bar made of steel is suspended vertically with the larger end rigidly clamped and the 16 A
smaller end acted on by a pull of 105 N. The areas at larger and smaller end are 80 cm2 and
20cm2 respectively. The length of the bar is 3m.the bar weighs 0.075 N/cc. E = 2X107
N/cm2.obtain the approximate expression for the deformation of the rod using Ritz technique.
Determine the maximum displacement at the tip of the bar.
According to Rayleigh Ritz The total potential function for the taper bar when subjected to end
load and uniformly distributed load is
2
du
300 300
1
E A( x) dx qudx pu ( L) --------------------------- (1)
2 0 dx 0
Since Area is varying linearly throughout the stretch.
