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DEPARTMENT OF MECHANICAL ENGINEERING
QUESTION BANK
ME6603 FINITE ELEMENT ANLYSIS
UNIT I FINITE ELEMENT FORMULATION OF BOUNDARY VALUE
PROBLEMS
Part-A
1. What is the finite element method?
2.What are the main steps involved in FEA
3. What are the basic approaches to improve a finite element model?
4. List any four advantages of finite element method.
5. What are the applications of FEA?
6. What is the limitation of using a finite difference method?
7. Define finite difference method.
8. What is the limitation of using a finite difference method?
9.What is discretization?
10.Distinguish between 1D bar element and 1D beam element.
11.Classify boundary conditions
12. What are the different types of boundary conditions? Give examples.
13. List the various methods of solving boundary value problems.
14. Write the potential energy for beam of span L simply supported at ends,
subjected to a concentrated load P at mid span. Assume EI constant.
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15. State the principle of minimum potential energy.
16.What should be considered during piecewise trial function?
17. Mention the basic steps of Rayleigh-Ritz method.
18. Why are polynomial types of interpolation function preferred over
trigonometric function?
19. What do you mean by constitutive law?
20. What is meant be weak formulation?
part-B
1. Explain the step by step procedure of FEA.
2. Explain the process of discretization of a structure in finite element method in
detail, with suitable illustrations for each aspect being & discussed.
3.Explain Rayleigh Ritz and Galerkin formulation with example.
4.Explain Gaussian elimination method of solving equations.
5.Discuss the following methods to solve the given differential equation:
With the boundary condition y(0) = 0 and y (x) = 0 (i) Variant method (ii)
Collocation method
6. Determine the expression for deflection and bending moment in a simply
supported beam subjected to uniformly distributed load over entire span. Find the
deflection and moment at mid span and compare with exact solution Rayleigh-Ritz
method. Use
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7. Find the solution of the problem using Rayleigh Ritz method by considering a
two term solution as y (x) = C1 x (1-x) + C2 x2 (1-x) .
8. A uniform rod subjected to a uniform axial load is illustrated in figure. The
deformation of the bar is governed by the differential equation given below.
Determine the displacement using weighted residual method.
9. Solve the differential equation for a physical problem expressed as
d2y/dx2 + 50 = 0, 0≤x≤10 with boundary conditions as y (0) = 0 and y (10) = 0
using the trial function y = a1x (10-x) find the value of the parameters a1 by the
following methods. (i) Point collocation method (ii) Sub domain collocation
method (iii) Least squares method and (iv) Galerkin method.
10. Solve following system of equations using Gauss elimination method.
x1 – x2 + x3 = 1
-3 x1 + 2x2 – 3x3 = -6
2x1 – 5x2 + 4x3 = 5
UNIT II ONE DIMENSIONAL FINITE ELEMENT ANALYSIS
Part-A
1.What are the types of problems treated as one dimensional problem?
2. Highlight at least two rules to guide the placement of the nodes when obtaining
approximate solution to a differential equation.
3. Define stiffness matrix for 1D bar element.
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4. What is a shape function?
5. Draw the shape function of a two noded line element.
6. Draw the shape function of a two noded line element with one degree of
freedom at each node.
7. Draw the shape function for one dimensional line element with three nodes.
8. State the properties of stiffness matrix.
9. Draw a mode shape of beam element.
10. State the characteristics of shape functions.
11. When does the stiffness matrix of a structure become singular?
12. What are the properties of global stiffness matrix?
13. Write the properties of Global stiffness Matrix of a one dimensional element.
14. Why polynomial terms preferred for shape functions in finite element method?
15. States the assumption are made while finding the force in a truss.
16. What is an interpolation function?
17. Differential global and local coordinates.
18. Write the element stiffness matrix of a truss element.
19. Determine the load vector for the beam element in fig.
20. Sketch a typical truss element shown local global transformation.
Part-B
1. Consider a bar as shown in figure. Young’s Modulus E = 2 x 10 5 N/mm2.
A1 = 2cm2, A2 = 1cm2 and force of 100N. Determine the nodal displacement.
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2.Consider the bar shown in figure Axial force P =30 KN is applied as shown.
Determine the nodal displacement, stresses in each element and reaction forces.
3. Axial load of 500N is applied to a stepped shaft, at the interface of two bars. The
ends are fixed. Obtain the nodal displacement and stress when the element is
subjected to all in temperature of 100ºC. take
E1 = 70 x 103 N/mm2,
E2 = 200 x 103 N/mm2, A1 = 900mm2, A2 = 1200mm2, α1 = 23 x 10-6 / ºC,
α2 = 11.7 x 10-6 / ºC L1 = 200mm, L2 = 300mm.
4. For the beam and loading as shown in figure. Determine the slopes at the two
ends of the distributed load and the vertical deflection at the mid-point of the
distributed load. Take E=200 GPa. And I=4X106 mm4 .
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5.Determine the displacements and slopes at the nodes for the beam shown in
figure. Find the moment at the midpoint of element 1.
6. Determine the force in the members of the truss as shown in figure. Take
E=200GPa.
7. The loading and other parameters for a two bar truss element is shown in figure.
Determine (i) The element stiffness matrix for each element (ii) Global stiffness
matrix (ii) Nodal displacements (iv) Reaction Force (v) The stresses induced in the
elements. Assume E=200GPa.
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8. Derive the shape functions for a 2D beam and bar element.
9. Derive the interpolation function for the one dimensional linear element with a
length ‘L’ and two nodes one at each end, designed as ‘i’ and ‘j’. Assume the
origin of the coordinator systems is to the left of node ‘i’.
10. Determine the displacement and slopes at the nodes for the beam shown in
figure. Take k=200KN/m, E=70GPa. And I=2X10-4 m4.
Unit III TWO DIMENSIONAL FINITE ELEMENT ANALYSIS
Part-A
1. What is geometric Isotropy?
2. Write down the interpolation function of a field variable for three-node
triangular element.
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3. What is a CST element?
4. Draw the shape functions of a CST element.
5. What are CST and LST elements?
6. Write the Lagrange shape functions for a 1D, 2 noded elements.
7. A triangular element is shown in figure and the nodal coordinates are expressed
in mm. compute the strain displacement matrix.
8. Define a plane stress condition
9. Define plane strain with suitable example.
10. Write down the strain displacement relation.
11. State whether plane stress or plane strain elements can be used to model the
following structures. Justify you answer.
a. A wall subjected to wind load
b. A wrench subjected to a force in the plane of the wrench.
12. How to reduce a 3D problem into a 2D problem?
13. Give examples of axisymmetric problem.
14. State the situations where the axisymmetric formulation can be applied.
15. Write down the constitutive relationship for the plane stress problem.
16. Express the shape function of four node quadrilateral elements.
17.Show the transformation for mapping x-coordinate system into a natural
coordinate system for a linear spar element and for a quadratic spar element.
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18. What do you understand by area coordinates?
19. Define Isoparametric elements with suitable examples.
20. What is the significance of Jacobian transformation?
Part-B
1. What are shape functions? Derive the shape function for the three noded
triangular elements.
2. Derive the stiffness matrix and equations for a LST element.
3. A two noded line element with one translational degree of freedom is subjected
to uniformly varying load of intensity P1 at node 1 and P2 at node 2. Evaluate the
nodal load vector using numerical integration.
4. Consider the triangular element show in figure. The element is extracted from a
thin plate of thickness 0.5 cm. the material is hot rolled low carbon steel. The
Nodal coordinates are xi = 0, yi = 0, xj = 0, yj = -1 and kx = 0, ky = -1cm. determine
the elemental stiffness matrix. Assuming plane stress analysis. Take μ=0.3 and
E= 2.1 x 107 N/cm2 .
5.Obtain the global stiffness matrix for the plate shown in figure. Taking two
triangular elements. Assume plane stress condition
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6. Derive element force vector when linearly varying pressure acts on the side
joining nodes jk of triangular element shown in figure and body force of 25N/mm2
acts downwards. Thickness = 5mm.All dimensions are in mm.
7. Calculate nodal displacement and elemental stresses for the truss shown in
figure. E=70GPa cross-sectional area A=2cm2 for all truss members.
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8. Derive the stress-strain relationship matrix (D) for the axisymmetric triangular
element.
9. Determine the stiffness matrix for the axisymmetric element shown in fig take E
as 2.1 x 106 N/mm2 and Poisson’s ratio as 0.3 . All dimensions are in mm.
10. a.Determine the Jacobin matrix for the following quadrilateral element at x =
4.35mm and y = 3mm .
b. Use Gaussian quadrature rule (n=2) to numerically integrate.
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Unit IV DYNAMIC ANALYSIS USING FINITE ELEMENT METHOD
Part-A
1.What is dynamic analysis? Give examples.
2. What is mean by dynamic analysis?
3. List the types of dynamic analysis problems.
4. Define normal modes.
5. Sketch two 3D element exhibiting linear strain behavior.
6. What is the influence of element distortion on the analysis results?
7. Determine the element mass matrix for one-dimensional dynamic structural
analysis problems. Assume the two-node, linear element.
8. Comment on the accuracy of the values of natural frequencies obtained by using
lumped mass matrices and consistent mass matrices.
9. Explain consistent load vector.
10. What do you mean by Lumped mass matrix?
11. Write down the lumped mass matrix for the truss element.
12. What are the types of Eigen value problems?
13. What is mean by mode superposition technique?
14.Specify the consistent mass matrix for a beam element.
15.What is the principle of mode superposition technique.
16.What are the four basic sets of elasticity equations.
17.What is meant by orthogonal eigen vector.
18.What is meant by transverse vibration.
19.what is meant by longitudinal and torsional vibration .
20.What is meant by orthogonality.
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PART-B
1.Derive the equation of motion based on weak form for transverse vibration of a
beam .
2. Derive the consistent mass matrix for a truss element in its local coordinate
system .
3. Derive the finite equation for the time-dependent stress analysis of one
dimensional bar.
4. Find the natural frequencies of transverse vibrations of the cantilever beam
shown in figure using one beam element.
5. Determine the natural frequencies and mode shapes of transverse vibration for a
beam fixed at both ends. The beam may be modeled by tow elements each of
length L and cross-sectional area A. Considers Lumped mass matrix approach.
6.Formulate the mass matrix for two-dimensional rectangular element depicted in
figure. The element has uniform thickness 5mm and density ρ = 7.83 x 10 -6
kg/mm3.All dimensions are in mm.
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7. Consider the undamped 2 degree of freedom system as shown in figure. Find the
response of the system when the first mass alone is given an initial displacement of
unity and realized form rest. The methodical reoperation of the system for free,
Harmonic vibration is given by
8. Consider a uniform cross section bar, as shown in figure of length L made up of
material whose young’s modulus and density is given by E and ρ. Estimate the
natural frequencies of axial vibration of the bar using both consistent and lumped
mass matrices.
9. Determine the Eigen values and Eigen vectors for the stepped bar as shown in
figure.
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10.Use iterative procedures to determine the first and third Eigen values for the
structure shown in figure. Hence determine the second Eigen value and the natural
frequencies of building. Finally, establish the Eigen vectors and check the rest by
applying the orthogonality properties of Eigen vectors.
Unit V APPLICATIONS IN HEAT TRANSFER & FLUID MECHANICS
Part-A
1. Write down the one dimensional heat conduction equation.
2. Write down the expression of shape function and temperature function for one
dimensional heat conduction.
3. Write down the governing g differential equation for the steady state one
dimensional conduction heat transfer.
4. Write down the governing differential equation for a two dimensional steadystate heat transfer problem.
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5. Write down the stiffness matrix equation for one dimensional head conduction
element.
6. Sketch a two dimensional differential control element for heat transfer and
obtain the heat diffusion equation.
7. Define element capacitance matrix for unsteady state heat transfer problems.
8. Name a few boundary conditions involved in any heat transfer analysis.
9. Mention two natural boundary conditions as applied to thermal problems.
10. Consider a wall of a tank containing a hot liquid at a temperature T0 with an air
stream of temperature Tx passed on the outside, maintaining a wall temperature of
TL at the boundary. Specify the boundary conditions.
11. Define static condensation.
12. Give the governing equation of torsion problem.
13. Write the step by step procedure of solving a torsion problem by finite element
method.
14. Outline the step by step procedure of handling torsion problem using the finite
element method.
15.Define streamline.
16. Define the stream function for a one-dimensional incompressible flow.
17. List the application of the potential flow.
18. List the method of describing the motion of fluid.
19. State the relation between the velocity of fluid flow and the hydraulic gradient
according to Darcy’s law, explaining the terms involved.
20. Define the stream function for a two dimensional incompressible flow.
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Part-B
1.Write the mathematical formulation for a steady state heat transfer conduction
problem and derive the stiffness and force matrices for the same.
2. Derive a finite element equation for one dimensional heat conduction with free
end convection.
3. The temperature at the four corners of a four-noded rectangle arte T1, T2, T3
and T4. Determine the consistent load vector for a 2D analysis, aimed to determine
the thermal stresses.
4. Derive the stiffness matrix and load vectors for fluid mechanics in two
dimensional finite element.
5. A furnace wall is made up of three layers, inside layer with thermal conductivity
8.5 W/mK, the middle layer with conductivity 0.25 W/mK, the outer layer with
conductivity 0.08 W/mK. The respective thicknesses of the inner, middle and outer
layer are 25 cm, 5 cm and 3 cm respectively. The inside temperature of the wall is
600
and outside of the wall is exposed to atmospheric air at 30
with heat
transfer coefficient of 45 W/m2K. Determine the nodal temperatures.
6. Derive the shape function for fluid mechanics in two dimensional finite
elements?
7. Derive the force vector for one dimensional heat convection and internal heat
generation with free end convection.
8. For the two dimensional fluid flow shown in fig. , determine the potentials at the
center and right edge. Take kx= ky = 25 x 10-5 m/s.
Two-dimensional fluid flow.
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9. A wall of 0.6 m thickness having thermal conductivity of 1.2 W/mK. The wall is
to be insulated with a material of thickness 0.06 m having an average thermal
conductivity of 0.3 W/mK. The inner surface temperature is 1000
the insulated is exposed to atmospheric air at 30
and outside of
with heat transfer coefficient of
35 W/m2K. Calculate the nodal temperatures.
10.(i) Find the integral I =
dx using Gaussian quadrature method
with 2 point scheme. The Gauss points are
0.5774 and the weights at the two
points are equal to unity.
(ii) Evaluate the integral I=
(a1 + a2 x + a3 x2 + a4 x3) dx using the three point
Gauss interation.
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