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R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ ME6603 / VI / MECH / JAN – MAY 2017
FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) 3
UNIT – I – INTRODUCTION
PART – A
1.1) What is the finite element method?
1.2) How does the finite element method work?
1.3) What are the main steps involved in FEA. [AU, April / May – 2011]
1.4) Write the steps involved in developing finite element model.
1.5) What are the basic approaches to improve a finite element model?
[AU, Nov / Dec – 2010]
1.6) What are the methods generally associated with the finite element analysis?
[AU, May / June – 2016]
1.7) Write any two advantages of FEM Analysis. [AU, Nov / Dec – 2012]
1.8) What are the methods generally associated with finite element analysis?
1.9) List any four advantages of finite element method. [AU, April / May – 2008]
1.10) What are the applications of FEA? [AU, April / May – 2011]
1.11) Define finite difference method.
1.12) What is the limitation of using a finite difference method? [AU, April / May – 2010]
1.13) Define finite volume method.
1.14) Differentiate finite element method from finite difference method.
1.15) Differentiate finite element method from finite volume method.
1.16) What do you mean by discretization in finite element method?
1.17) What is discretization? [AU, Nov / Dec – 2010, 2015]
1.18) What is meant by node or joint? [AU, May / June – 2014]
1.19) What is meant by node? [AU, Nov / Dec – 2015, 2016]
1.20) List the types of nodes. [AU, May / June – 2012]
1.21) Define degree of freedom.
1.22) What is meant by degrees of freedom? [AU, Nov / Dec – 2012]
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ ME6603 / VI / MECH / JAN – MAY 2017
FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) 4
1.23) State the advantage of finite element method over other numerical analysis
methods.
1.24) State the fields to which FEA solving procedure is applicable.
1.25) What is a structural and non-structural problem?
1.26) Distinguish between 1D bar element and 1D beam element.
[AU, Nov / Dec – 2009, May / June – 2011]
1.27) Write the equilibrium equation for an elemental volume in 3D including the body
force.
1.28) How to write the equilibrium equation for a finite element? [AU, Nov / Dec – 2012]
1.29) Classify boundary conditions. [AU, Nov / Dec – 2011]
1.30) What are the types of boundary conditions?
1.31) What do you mean by boundary condition and boundary value problem?
1.32) Write the difference between initial value problem and boundary value problem.
1.33) What are the different types of boundary conditions? Give examples.
[AU, May / June – 2012]
1.34) List the various methods of solving boundary value problems.
[AU, April / May – 2010, Nov / Dec – 2016]
1.35) Write down the boundary conditions of a cantilever beam AB of span L fixed at A
and free at B subjected to a uniformly distributed load of P throughout the span.
[AU, May / June – 2009, 2011]
1.36) Briefly explain force method and stiffness method.
1.37) What is aspect ratio?
1.38) Write a short note on stress – strain relation.
1.39) Write down the stress strain relationship for a three dimensional stress field.
[AU, April / May – 2011]
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ ME6603 / VI / MECH / JAN – MAY 2017
FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) 5
1.40) If a displacement field in x direction is given by 𝑢 = 2𝑥2
+ 4𝑦2
+ 6𝑥𝑦 Determine
the strain in x direction. [AU, May / June – 2016]
1.41) State the effect of Poisson’s ratio.
1.42) Define total potential energy of an elastic body.
1.43) What is the stationary property of total potential energy? [AU, May / June – 2016]
1.44) 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.
[AU, April / May, Nov / Dec – 2008]
1.45) State the principle of minimum potential energy.
[AU, Nov / Dec – 2007, 2013, April / May – 2009]
1.46) State the principle of minimum potential energy theorem. [AU, May / June – 2016]
1.47) How will you obtain total potential energy of a structural system?
[AU, April / May – 2011, May / June – 2012]
1.48) Write down the potential energy function for a three dimensional deformable body
in terms of strain and displacements. [AU, May / June – 2009]
1.49) What should be considered during piecewise trial functions?
[AU, April / May – 2011]
1.50) What do you understand by the term “piecewise continuous function”?
[AU, Nov / Dec – 2013]
1.51) Write about weighted residual method. [AU, May / June – 2016]
1.52) Distinguish between the error in solution and Residual. [AU, April / May – 2015]
1.53) Name the weighted residual methods. [AU, Nov / Dec – 2011]
1.54) List the various weighted residual methods. [AU, Nov / Dec – 2014]
1.55) What is the use of Ritz method? [AU, Nov / Dec – 2011]
1.56) What is Rayleigh – Ritz method?
[AU, May / June – 2014, Nov / Dec – 2015, 2016]
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ ME6603 / VI / MECH / JAN – MAY 2017
FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) 6
1.57) Mention the basic steps of Rayleigh-Ritz method. [AU, April / May – 2011]
1.58) Highlight the equivalence and the difference between Rayleigh Ritz method and the
finite element method. [AU, Nov / Dec – 2012]
1.59) Distinguish between Rayleigh Ritz method and finite element method.
[AU, Nov / Dec – 2013]
1.60) Distinguish between Rayleigh Ritz method and finite element method with regard
to choosing displacement function. [AU, Nov / Dec – 2010]
1.61) Compare the Ritz technique with the nodal approximation method.
[AU, Nov / Dec – 2014]
1.62) Why are polynomial types of interpolation functions preferred over trigonometric
functions? [AU, April / May – 2009, May / June – 2013]
1.63) What is meant by weak formulation? [AU, May / June – 2013]
1.64) What are the advantage of weak formulation? [AU, April / May – 2015]
1.65) Define the principle of virtual work.
1.66) Differentiate Von Mises stress and principle stress.
1.67) What do you mean by constitutive law?[AU, Nov / Dec – 2007, April / May – 2009]
1.68) What are h and p versions of finite element method?
1.69) What is the difference between static and dynamic analysis?
1.70) Mention two situations where Galerkin’s method is preferable to Rayleigh – Ritz
method. [AU, Nov / Dec – 2013]
1.71) What is Galerkin method of approximation? [AU, Nov / Dec – 2009]
1.72) What is a weighted resuidal method? [AU, Nov / Dec – 2010]
1.73) Distinguish between potential energy and potential energy functional.
1.74) What are the types of Eigen value problems? [AU, May / June – 2012]
1.75) Name a few FEA packages. [AU, Nov / Dec – 2014]
1.76) Name any four FEA software
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ ME6603 / VI / MECH / JAN – MAY 2017
FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) 7
PART – B
1.77) Explain the step by step procedure of FEA. [AU, Nov / Dec – 2010]
1.78) Explain the general procedure of finite element analysis. [AU, Nov / Dec – 2011]
1.79) List and briefly describe the general steps of the finite element method.
[AU, May / June – 2014]
1.80) Briefly explain the stages involved in FEA.
1.81) Explain the step by step procedure of FEM. [AU, Nov / Dec – 2011]
1.82) List out the general procedure for FEA problems. [AU, May / June – 2012]
1.83) Compare FEM with other methods of analysis. [AU, Nov / Dec – 2010]
1.84) Define discretization. Explain mesh refinement. [AU, Nov / Dec – 2010]
1.85) Explain the various aspects pertaining to discretization, process in finite element
modeling analysis. [AU, Nov / Dec – 2013]
1.86) Explain the process of discretization of a structure in finite element method in
detail, with suitable illustrations for each aspect being & discussed.
[AU, Nov / Dec – 2012]
1.87) Discuss procedure using the commercial package (P.C. Programs) available today
for solving problems of FEM. Take a structural problem to explain the same.
[AU, Nov / Dec – 2011]
1.88) State the importance of locating nodes in finite element model.
[AU, Nov / Dec – 2011]
1.89) Write briefly about weighted residual methods. [AU, Nov / Dec – 2015]
1.90) Write a brief note on the following.
(a) isotropic material
(b) orthotropic material
(c) anisotropic material
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ ME6603 / VI / MECH / JAN – MAY 2017
FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) 8
1.91) What are initial and final boundary value problems? Explain.
[AU, Nov / Dec – 2010]
1.92) Explain the Potential Energy Approach [AU, Nov / Dec – 2010]
1.93) Explain the principle of minimization of potential energy. [AU, Nov / Dec – 2011]
1.94) Explain the four weighted residual methods. [AU, Nov / Dec – 2011]
1.95) Explain Ritz method with an example. [AU, April / May – 2011]
1.96) Explain Rayleigh Ritz and Galerkin formulation with example.
[AU, May / June – 2012]
1.97) Write short notes on Galerkin method? [AU, April / May – 2009]
1.98) Discuss stresses and equilibrium of a three dimensional body.
[AU, May / June – 2012]
1.99) Derive the element level equation for one dimensional bar element based on the
station- of a functional. [AU, May / June – 2012]
1.100) Derive the characteristic equations for the one dimensional bar element by using
piece-wise defined interpolations and weak form of the weighted residual method?
[AU, May / June – 2012]
1.101) Develop the weak form and determine the displacement field for a cantilever beam
subjected to a uniformly distributed load and a point load acting at the free end.
[AU, Nov / Dec – 2013]
1.102) Explain Gaussian elimination method of solving equations.
[AU, April / May – 2011]
1.103) Write briefly about Gaussian elimination? [AU, April / May – 2009]
1.104) The following differential equation is available for a physical phenomenon.
𝑑
𝑑𝑥
(𝑥
𝑑𝑢
𝑑𝑥
) −
2
𝑥2
= 0, 1 ≤ 𝑥 ≤ 2
Boundary conditions are, x = 1 u = 2
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ ME6603 / VI / MECH / JAN – MAY 2017
FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) 9
x = 2 𝑥
𝑑𝑢
𝑑𝑥
= −
1
2
Find the value of the parameter a, by the following methods.
(i) Collocation (ii) Sub – Domain (iii) Least Square
(iv) Galerkin
1.105) The following differential equation is available for a physical phenomenon.
𝑑2
𝑦
𝑑𝑥2
+ 50 = 0, 0 ≤ 𝑥 ≤ 10
Trial function is 𝑦 = 𝑎1 𝑥(10 − 𝑥)
Boundary conditions are, y (0) = 0 y (10) = 0
Find the value of the parameter a, by the following methods.
(i) Collocation (ii) Sub – Domain (iii) Least Square
(iv) Galerkin
1.106) Discuss the following methods to solve the given differential equation :
𝐸𝐼
𝑑
2
𝑦
𝑑𝑥2
𝑀( 𝑥) = 0
with the boundary condition y(0) = 0 and y(H) = 0
(i) Variant method (ii) Collocation method. [AU, April / May – 2010]
1.107) The differential equation of a physical phenomenon is given by
𝑑2
𝑦
𝑑𝑥2
+ 𝑦 = 4𝑥, 0 ≤ 𝑥 ≤ 1
The boundary conditions are: y(0)=0; y(1)=1; Obtain one term approximate solution
by using Galerkin's method of weighted residuals.
[AU, May / June – 2014, 2016, Nov / Dec – 2016]
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ ME6603 / VI / MECH / JAN – MAY 2017
FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) 10
1.108) Find the approximate deflection of a simply supported beam under a uniformly
distributed load ‘P‘ throughout its span. Using Galerkin and Least square residual
method. [AU, May / June – 2011]
1.109) Solve the differential equation for a physical problem expressed as
𝑑2 𝑦
𝑑𝑥2 + 100 =
0, 0 ≤ 𝑥 ≤ 10 with boundary conditions as y (0) = 0 and y (10) = 0 using
(i) Point collocation method
(ii) Sub domain collocation method
(iii) Least squares method and
(iv) Galerkin method. [AU, May / June – 2013]
1.110) Solve the differential equation for a physical problem expressed as
𝑑2 𝑦
𝑑𝑥2 + 50 =
0, 0 ≤ 𝑥 ≤ 10 with boundary conditions as y (0) = 0 and y (10) = 0 using the
trail function 𝑦 = 𝑎1 𝑥 (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. [AU, Nov / Dec – 2011]
1.111) Solve the following equation using a two – parameter trial solution by the
(a) Collocation method (𝑅 𝑑 = 0 𝑎𝑡 𝑥 =
1
3
𝑎𝑛𝑑 𝑥 =
2
3
)
(b) Galerkin method.
Then, compare the two solutions with the exact solution
𝑑𝑦
𝑑𝑥
+ 𝑦 = 0, 0 ≤ 𝑥 ≤ 1
y (0) = 1
1.112) Determine the Galerkin approximation solution of the differential equation
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ ME6603 / VI / MECH / JAN – MAY 2017
FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) 11
𝐴
𝑑2 𝑢
𝑑𝑥2 + 𝐵
𝑑𝑢
𝑑𝑥
+ 𝐶 = 0, 𝑢(0) = 𝑢( 𝑙) = 0
1.113) Solve the following differential equation using Galerkin’s method.
D
𝑑2 𝜑
𝑑𝑥2 + 𝑄 = 0, 0 ≤ 𝑥 ≤ 𝐿
subjected to 𝜑(0) = 𝜑0 𝑎𝑛𝑑 𝜑( 𝐿) = 𝜑1 [AU, April / May – 2011]
1.114) A physical phenomenon is governed by the differential equation
𝑑2 𝑤
𝑑𝑥2 − 10𝑥2
= 5 𝑓𝑜𝑟 0 ≤ 𝑥 ≤ 1 The boundary conditions are given by
𝑤(0) = 𝑤(1) = 0. By taking two-term trial solution as 𝑤( 𝑥) = 𝑐1 𝑓1( 𝑥) +
𝑐2 𝑓2(𝑥) with, 𝑓1( 𝑥) = 𝑥 ( 𝑥 − 1) 𝑎𝑛𝑑 𝑓2 = 𝑥2
(𝑥 − 1) find the solution of
the problem using the Galerkin method. [AU, Nov / Dec – 2009]
1.115) Determine the two parameter solution of the following using Galerkin method.
𝑑2 𝑦
𝑑𝑥2 = − cos 𝜋𝑥 , 0 ≤ 𝑥 ≤ 1, 𝑢(0) = 𝑢(1) = 0 [AU, Nov / Dec – 2012]
1.116) The following differential equation is available for a physical phenomenon.
𝑑2
𝑦
𝑑𝑥2
− 10𝑥2
= 5, 0 ≤ 𝑥 ≤ 1
With boundary conditions y (0) = 0, y (l) = 1. Find an approximate solution of the
above differential equation by using Galerkin's method of weighted residuals and
also compare with exact solution [AU, May / June – 2016]
1.117) A physical phenomenon is governed by the differential equation
𝑑2 𝑤
𝑑𝑥2 − 10𝑥2
=
5 𝑓𝑜𝑟 0 ≤ 𝑥 ≤ 1. The boundary conditions are given by w (0) = w (1) = 0.
Assuming a trail function 𝑤(𝑥) = 𝑎0 + 𝑎1 𝑥 + 𝑎2 𝑥2
+ 𝑎3 𝑥3
. Determine using
Garlerkin method the variation of “w” with respect to “x”. [AU, Nov / Dec – 2016]
1.118) Using Collocation method, find the maximum displacement of the tapered rod as
shown in Fig. E = 2 *107
N/cm2
ρ = 0.075N/cm3
[AU, Nov / Dec – 2014]
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ ME6603 / VI / MECH / JAN – MAY 2017
FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) 12
1.119) A cantilever beam of length L is loaded with a point load at the free end. Find the
maximum deflection and maximum bending moment using Rayleigh-Ritz method
using the function 𝑦 = 𝐴{ 1 − cos (
𝜋𝑥
2𝐿
)} Given: EI is constant.
[AU, April / May – 2008]
1.120) Compute the slope deflection and reaction forces for the cantilever beam of length
L carrying uniformly distributed load of intensity 'fo'. [AU, Nov / Dec – 2014]
1.121) A simply supported beam carries uniformly distributed load over the entire span.
Calculate the bending moment and deflection. Assume EI is constant and compare
the results with other solution. [AU, Nov / Dec – 2012]
1.122) 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 midspan and compare with exact solution using Rayleigh-Ritz
method. Use 𝑦 = 𝑎1 sin (
𝜋𝑥
𝑙
) + 𝑎2 sin (
3𝜋𝑥
𝑙
) [AU, Nov / Dec – 2008]
1.123) Compute the value of central deflection in the figure below by assuming
𝑦 =
𝑎𝑠𝑖𝑛𝜋𝑥
𝐿
The beam is uniform throughout and carries a central point load P.
[AU, Nov / Dec – 2007, April / May – 2009]
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ ME6603 / VI / MECH / JAN – MAY 2017
FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) 13
1.124) A concentrated load P = 50 kN is applied at the centre of a fixed beam of length 3
m, depth 200 mm and width 120 mm. Calculate the deflection and slope at the
midpoint. Assume E = 2 x 105
N/mm2
[AU, May / June – 2016]
1.125) A beam AB of span '1' simply supported at ends and carrying a concentrated load W
at the centre C as shown in fig. Determine the deflection at midspan by using
Rayleigh-Ritz method and compare with exact solution
[AU, May / June – 2016, Nov / Dec – 2016]
1.126) If a displacement field is described by
𝑢 = (−𝑥2
+ 2𝑦2
+ 6𝑥𝑦)10−4
𝑣 = (3𝑥 + 6𝑦 − 𝑦2)10−4
Determine the direct strains in x and y directions as well the shear strain at the point
x = 1, y =0. [AU, April / May – 2011]
1.127) In a solid body, the six components of the stress at a point are given by x= 40
MPa, y = 20 MPa, z = 30 MPa, yz = -30 MPa, xz = 15 MPa and xy = 10 MPa.
Determine the normal stress at the point, on a plane for which the normal is (nx, ny,
nz) = ( ½, ½, 2
1 )
1.128) In a plane strain problem, we have
x = 20,000 psi y = - 10,000 psi E = 30 x 10 6
psi,  = 0.3.
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ ME6603 / VI / MECH / JAN – MAY 2017
FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) 14
Determine the value of the stress z.
1.129) For the spring system shown in figure, calculate the global stiffness matrix,
displacements of nodes 2 and 3, the reaction forces at node 1 and 4. Also calculate
the forces in the spring 2. Assume, k1 = k3 = 100 N/m, k2 = 200 N/m, u1 = u4= 0 and
P=500 N. [AU, April / May – 2010]
1.130) Use the Rayleigh – Ritz method to find the displacement of the midpoint of the rod
shown in figure. [AU, April / May – 2011]
1.131) Consider the differential equation
𝑑2 𝑦
𝑑𝑥2 + 400𝑥2
= 0 𝑓𝑜𝑟 0 ≤ 𝑥 ≤ 1 subject
to boundary conditions 𝑦(0) = 0; 𝑦(1) = 0 The functional corresponding to this
problem, to be extremized is given by 𝐼 = ∫ {−0.5 (
𝑑𝑦
𝑑𝑥
)
2
+ 400𝑥2
𝑦
1
0
1.132) 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) [AU, Nov / Dec – 2009]
1.133) A bar of uniform cross section is clamped at one end and left free at the other end. It
is subjected to a uniform load axial load P as shown in figure. Calculate the
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ ME6603 / VI / MECH / JAN – MAY 2017
FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) 15
displacement and stress in the bar using three terms polynomial following Ritz
method. Compare the results with exact solutions. [AU, May / June – 2011]
1.134) A simply Supported beam subjected to uniformly distributed load over entire span
and it is subjected to a point load at the centre of the span. Calculate the deflection
using Rayleigh-Ritz method and compare with exact solutions.
[AU, May / June – 2013]
1.135) A simply Supported beam subjected to uniformly distributed load over entire span
as shown in figure. Calculate the bending moment and deflection at midspan using
Rayleigh-Ritz method. [AU, Nov / Dec – 2015, 2016]
1.136) A simply supported beam (span L and flexural rigidity EI) carries two equal
concentrated loads at each of the quarter span points. Using Raleigh – Ritz method
determine the deflections under the two loads and the two end slopes.
[AU, April / May – 2009]
1.137) Analyze a simply supported beam subjected to a uniformly distributed load
throughout using Rayleigh Ritz method. Adopt one parameter trigonometric
function. Evaluate the maximum deflection and bending moment and compare with
exact solution. [AU, Nov / Dec – 2010]
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ ME6603 / VI / MECH / JAN – MAY 2017
FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) 16
1.138) Solve for the displacement field for a simply supported beam, subjected to a
uniformly distributed load using Rayleigh – Ritz method. [AU, Nov / Dec – 2013]
1.139) Derive the governing equation for a tapered rod fixed at one end and subjected to its
own self weight and a force P at the other end as shown in fig. Let the length of the
bar be l and let the cross section vary linearly from A1 at the top fixed end to A2 at
the free end. E and γ represents the Young’s modulus and specific weight of the
material of the bar. Convert this equation into weak form and hence determine the
matrices for solving using Ritz technique. [AU, April / May – 2015]
1.140) Use the Rayleigh – Ritz method to find the displacement field u(x) of the rod as
shown below. Element 1 is made of aluminum and element 2 is made of steel. The
properties are
Eal = 70 GPa A1 = 900 mm2
L1 = 200 mm
Est = 200 GPa A2 = 1200 mm2
L2 = 300 mm
Load = P = 10,000 N. Assume a piecewise linear displacement.
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ ME6603 / VI / MECH / JAN – MAY 2017
FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) 17
Field u = a1 + a2x for 0  x  200 mm, and u = a3 + a4 x for 200  x  500 mm.
1.141) A fixed beam length of 2L m carries a uniformly distributed load of a w(in N / m)
which run over a length of ‘L’ m from the fixed end, as shown in Figure. Calculate
the rotation at point B using FEA. [AU, Nov / Dec – 2011]
1.142) A rod fixed at its ends is subjected to a varying body force as shown in Figure. Use
the Rayleigh-Ritz method with an assumed displacement field 𝑢( 𝑥) = 𝑎0 +
𝑎1 𝑥 + 𝑎2 𝑥2
to find the displacement u(x) and stress σ(x). Plot the variation of the
stress in the rod. [AU, Nov / Dec – 2012]
1.143) 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.
[AU, April / May – 2011]
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ ME6603 / VI / MECH / JAN – MAY 2017
FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) 18
1.144) A steel rod is attached to rigid walls at each end and is subjected to a distributed
load T(x) as shown below.
a) Write the expression for potential energy.
b) Determine the displacement u(x) using the Rayleigh – Ritz method.
Assume a displacement field u(x) = a0 + a1 x + a2 x2
.
1.145) Derive the stress – strain relation and strain – displacement relation for an element
in space.
1.146) Derive the equation of equilibrium in case of a three dimensional stress system.
[AU, Nov / Dec – 2008]
1.147) What is constitutive relationship? Express the constitutive relations for a linear
elastic isotropic material including initial stress and strain. [AU, Nov / Dec – 2009]
1.148) Give a detailed note on the following:
(a) Rayleigh Ritz method (b) Galerkin method
(c) Least square method and (d) Collocation method
1.149) Determine using any Weighted Residual technique the temperature distribution
along the circular fin of length 6cm and 1cm. the fin is attached to a boiler whose
wall temperature is 140ºC and the free end is insulated. Assume the convection
coefficient h = 10 W/cm2
ºC. Conduction coefficient K = 70 W/ cm ºC and T∞ =
40ºC. The governing equation for the heat transfer through the fin is given by
−
𝑑
𝑑𝑥
[𝐾𝐴(𝑥)
𝑑𝑇
𝑑𝑥
] + ℎ𝑝(𝑥)(𝑇 − 𝑇∞) = 0 Assume appropriate boundary conditions
and calculate the temperatures at every 1cm from the left end.
[AU, April / May – 2015]
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ ME6603 / VI / MECH / JAN – MAY 2017
FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) 19
1.150) Give a one – parameter Galerkin solution of the following equation, for the two
domain’s shown below. .1
2
2
2
2














y
u
x
u
1.151) Find the Eigen values and Eigen vectors of the matrix.
[
𝟑 −𝟏 𝟎
−𝟏 𝟐 −𝟏
𝟎 −𝟏 𝟑
]
1.152) Find the Eigen values and Eigen vectors of the matrix.
[
𝟑 𝟏𝟎 𝟓
−𝟐 −𝟑 −𝟒
𝟑 𝟓 𝟕
]
1.153) Find the Eigen value and the corresponding Eigen vector of 𝐴 = [
1 6 1
1 2 0
0 0 3
]
[AU, May / June – 2016]
1.154) Describe the Gaussian elimination method of solving equations.
[AU, April / May – 2011]
1.155) Explain the Gaussian elimination method for the solving of simultaneous linear
algebraic equations with an example. [AU, April / May – 2008]
1.156) Solve the following system of equations using Gauss elimination method.
[AU, Nov / Dec – 2010]
x1 – x2 + x3 = 1
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ ME6603 / VI / MECH / JAN – MAY 2017
FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) 20
-3x1 + 2x2 – 3x3 = -6
2x1 – 5x2 + 4x3 = 5
1.157) Solve the following system of equations by Gauss Elimination method.
2x1 – 2x2 – x4 = 1
2x2 + x3 + 2x4 = 2
x1 – 2x2 + 3x3 – 2x4 = 3 [AU, May / June – 2012]
x2 + 2x3 + 2x4 = 4
1.158) Solve the following equations by Gauss elimination method.
28r1 + 6r2 = 1
6r1 + 24r2 + 6r3 = 0
6r2 + 28r3 + 8r4 = -1
8r3 + 16r4 = 10 [AU, Nov / Dec – 2010, 2012]
1.159) Use the Gaussian elimination method to solve the following simultaneous
equations:
4x1 + 2x2 – 2x3 – 8x4 = 4
x1 + 2x2 + x3 = 2
0.5x1 – x2 + 4x3 + 4x4 = 10
–4x1 – 2x2 – x4 = 0 [AU, April / May – 2009]
1.160) Solve the following system of equations using Gauss elimination method.
x1 + 3x2 + 2x3 = 13
– 2x1 + x2 – x3 = –3
- 5x1 + x2 + 3x3 = 6 [AU, Nov / Dec – 2009]

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ME6603 - FINITE ELEMENT ANALYSIS UNIT - I NOTES AND QUESTION BANK

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  • 73. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ ME6603 / VI / MECH / JAN – MAY 2017 FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) 3 UNIT – I – INTRODUCTION PART – A 1.1) What is the finite element method? 1.2) How does the finite element method work? 1.3) What are the main steps involved in FEA. [AU, April / May – 2011] 1.4) Write the steps involved in developing finite element model. 1.5) What are the basic approaches to improve a finite element model? [AU, Nov / Dec – 2010] 1.6) What are the methods generally associated with the finite element analysis? [AU, May / June – 2016] 1.7) Write any two advantages of FEM Analysis. [AU, Nov / Dec – 2012] 1.8) What are the methods generally associated with finite element analysis? 1.9) List any four advantages of finite element method. [AU, April / May – 2008] 1.10) What are the applications of FEA? [AU, April / May – 2011] 1.11) Define finite difference method. 1.12) What is the limitation of using a finite difference method? [AU, April / May – 2010] 1.13) Define finite volume method. 1.14) Differentiate finite element method from finite difference method. 1.15) Differentiate finite element method from finite volume method. 1.16) What do you mean by discretization in finite element method? 1.17) What is discretization? [AU, Nov / Dec – 2010, 2015] 1.18) What is meant by node or joint? [AU, May / June – 2014] 1.19) What is meant by node? [AU, Nov / Dec – 2015, 2016] 1.20) List the types of nodes. [AU, May / June – 2012] 1.21) Define degree of freedom. 1.22) What is meant by degrees of freedom? [AU, Nov / Dec – 2012]
  • 74. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ ME6603 / VI / MECH / JAN – MAY 2017 FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) 4 1.23) State the advantage of finite element method over other numerical analysis methods. 1.24) State the fields to which FEA solving procedure is applicable. 1.25) What is a structural and non-structural problem? 1.26) Distinguish between 1D bar element and 1D beam element. [AU, Nov / Dec – 2009, May / June – 2011] 1.27) Write the equilibrium equation for an elemental volume in 3D including the body force. 1.28) How to write the equilibrium equation for a finite element? [AU, Nov / Dec – 2012] 1.29) Classify boundary conditions. [AU, Nov / Dec – 2011] 1.30) What are the types of boundary conditions? 1.31) What do you mean by boundary condition and boundary value problem? 1.32) Write the difference between initial value problem and boundary value problem. 1.33) What are the different types of boundary conditions? Give examples. [AU, May / June – 2012] 1.34) List the various methods of solving boundary value problems. [AU, April / May – 2010, Nov / Dec – 2016] 1.35) Write down the boundary conditions of a cantilever beam AB of span L fixed at A and free at B subjected to a uniformly distributed load of P throughout the span. [AU, May / June – 2009, 2011] 1.36) Briefly explain force method and stiffness method. 1.37) What is aspect ratio? 1.38) Write a short note on stress – strain relation. 1.39) Write down the stress strain relationship for a three dimensional stress field. [AU, April / May – 2011]
  • 75. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ ME6603 / VI / MECH / JAN – MAY 2017 FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) 5 1.40) If a displacement field in x direction is given by 𝑢 = 2𝑥2 + 4𝑦2 + 6𝑥𝑦 Determine the strain in x direction. [AU, May / June – 2016] 1.41) State the effect of Poisson’s ratio. 1.42) Define total potential energy of an elastic body. 1.43) What is the stationary property of total potential energy? [AU, May / June – 2016] 1.44) 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. [AU, April / May, Nov / Dec – 2008] 1.45) State the principle of minimum potential energy. [AU, Nov / Dec – 2007, 2013, April / May – 2009] 1.46) State the principle of minimum potential energy theorem. [AU, May / June – 2016] 1.47) How will you obtain total potential energy of a structural system? [AU, April / May – 2011, May / June – 2012] 1.48) Write down the potential energy function for a three dimensional deformable body in terms of strain and displacements. [AU, May / June – 2009] 1.49) What should be considered during piecewise trial functions? [AU, April / May – 2011] 1.50) What do you understand by the term “piecewise continuous function”? [AU, Nov / Dec – 2013] 1.51) Write about weighted residual method. [AU, May / June – 2016] 1.52) Distinguish between the error in solution and Residual. [AU, April / May – 2015] 1.53) Name the weighted residual methods. [AU, Nov / Dec – 2011] 1.54) List the various weighted residual methods. [AU, Nov / Dec – 2014] 1.55) What is the use of Ritz method? [AU, Nov / Dec – 2011] 1.56) What is Rayleigh – Ritz method? [AU, May / June – 2014, Nov / Dec – 2015, 2016]
  • 76. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ ME6603 / VI / MECH / JAN – MAY 2017 FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) 6 1.57) Mention the basic steps of Rayleigh-Ritz method. [AU, April / May – 2011] 1.58) Highlight the equivalence and the difference between Rayleigh Ritz method and the finite element method. [AU, Nov / Dec – 2012] 1.59) Distinguish between Rayleigh Ritz method and finite element method. [AU, Nov / Dec – 2013] 1.60) Distinguish between Rayleigh Ritz method and finite element method with regard to choosing displacement function. [AU, Nov / Dec – 2010] 1.61) Compare the Ritz technique with the nodal approximation method. [AU, Nov / Dec – 2014] 1.62) Why are polynomial types of interpolation functions preferred over trigonometric functions? [AU, April / May – 2009, May / June – 2013] 1.63) What is meant by weak formulation? [AU, May / June – 2013] 1.64) What are the advantage of weak formulation? [AU, April / May – 2015] 1.65) Define the principle of virtual work. 1.66) Differentiate Von Mises stress and principle stress. 1.67) What do you mean by constitutive law?[AU, Nov / Dec – 2007, April / May – 2009] 1.68) What are h and p versions of finite element method? 1.69) What is the difference between static and dynamic analysis? 1.70) Mention two situations where Galerkin’s method is preferable to Rayleigh – Ritz method. [AU, Nov / Dec – 2013] 1.71) What is Galerkin method of approximation? [AU, Nov / Dec – 2009] 1.72) What is a weighted resuidal method? [AU, Nov / Dec – 2010] 1.73) Distinguish between potential energy and potential energy functional. 1.74) What are the types of Eigen value problems? [AU, May / June – 2012] 1.75) Name a few FEA packages. [AU, Nov / Dec – 2014] 1.76) Name any four FEA software
  • 77. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ ME6603 / VI / MECH / JAN – MAY 2017 FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) 7 PART – B 1.77) Explain the step by step procedure of FEA. [AU, Nov / Dec – 2010] 1.78) Explain the general procedure of finite element analysis. [AU, Nov / Dec – 2011] 1.79) List and briefly describe the general steps of the finite element method. [AU, May / June – 2014] 1.80) Briefly explain the stages involved in FEA. 1.81) Explain the step by step procedure of FEM. [AU, Nov / Dec – 2011] 1.82) List out the general procedure for FEA problems. [AU, May / June – 2012] 1.83) Compare FEM with other methods of analysis. [AU, Nov / Dec – 2010] 1.84) Define discretization. Explain mesh refinement. [AU, Nov / Dec – 2010] 1.85) Explain the various aspects pertaining to discretization, process in finite element modeling analysis. [AU, Nov / Dec – 2013] 1.86) Explain the process of discretization of a structure in finite element method in detail, with suitable illustrations for each aspect being & discussed. [AU, Nov / Dec – 2012] 1.87) Discuss procedure using the commercial package (P.C. Programs) available today for solving problems of FEM. Take a structural problem to explain the same. [AU, Nov / Dec – 2011] 1.88) State the importance of locating nodes in finite element model. [AU, Nov / Dec – 2011] 1.89) Write briefly about weighted residual methods. [AU, Nov / Dec – 2015] 1.90) Write a brief note on the following. (a) isotropic material (b) orthotropic material (c) anisotropic material
  • 78. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ ME6603 / VI / MECH / JAN – MAY 2017 FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) 8 1.91) What are initial and final boundary value problems? Explain. [AU, Nov / Dec – 2010] 1.92) Explain the Potential Energy Approach [AU, Nov / Dec – 2010] 1.93) Explain the principle of minimization of potential energy. [AU, Nov / Dec – 2011] 1.94) Explain the four weighted residual methods. [AU, Nov / Dec – 2011] 1.95) Explain Ritz method with an example. [AU, April / May – 2011] 1.96) Explain Rayleigh Ritz and Galerkin formulation with example. [AU, May / June – 2012] 1.97) Write short notes on Galerkin method? [AU, April / May – 2009] 1.98) Discuss stresses and equilibrium of a three dimensional body. [AU, May / June – 2012] 1.99) Derive the element level equation for one dimensional bar element based on the station- of a functional. [AU, May / June – 2012] 1.100) Derive the characteristic equations for the one dimensional bar element by using piece-wise defined interpolations and weak form of the weighted residual method? [AU, May / June – 2012] 1.101) Develop the weak form and determine the displacement field for a cantilever beam subjected to a uniformly distributed load and a point load acting at the free end. [AU, Nov / Dec – 2013] 1.102) Explain Gaussian elimination method of solving equations. [AU, April / May – 2011] 1.103) Write briefly about Gaussian elimination? [AU, April / May – 2009] 1.104) The following differential equation is available for a physical phenomenon. 𝑑 𝑑𝑥 (𝑥 𝑑𝑢 𝑑𝑥 ) − 2 𝑥2 = 0, 1 ≤ 𝑥 ≤ 2 Boundary conditions are, x = 1 u = 2
  • 79. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ ME6603 / VI / MECH / JAN – MAY 2017 FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) 9 x = 2 𝑥 𝑑𝑢 𝑑𝑥 = − 1 2 Find the value of the parameter a, by the following methods. (i) Collocation (ii) Sub – Domain (iii) Least Square (iv) Galerkin 1.105) The following differential equation is available for a physical phenomenon. 𝑑2 𝑦 𝑑𝑥2 + 50 = 0, 0 ≤ 𝑥 ≤ 10 Trial function is 𝑦 = 𝑎1 𝑥(10 − 𝑥) Boundary conditions are, y (0) = 0 y (10) = 0 Find the value of the parameter a, by the following methods. (i) Collocation (ii) Sub – Domain (iii) Least Square (iv) Galerkin 1.106) Discuss the following methods to solve the given differential equation : 𝐸𝐼 𝑑 2 𝑦 𝑑𝑥2 𝑀( 𝑥) = 0 with the boundary condition y(0) = 0 and y(H) = 0 (i) Variant method (ii) Collocation method. [AU, April / May – 2010] 1.107) The differential equation of a physical phenomenon is given by 𝑑2 𝑦 𝑑𝑥2 + 𝑦 = 4𝑥, 0 ≤ 𝑥 ≤ 1 The boundary conditions are: y(0)=0; y(1)=1; Obtain one term approximate solution by using Galerkin's method of weighted residuals. [AU, May / June – 2014, 2016, Nov / Dec – 2016]
  • 80. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ ME6603 / VI / MECH / JAN – MAY 2017 FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) 10 1.108) Find the approximate deflection of a simply supported beam under a uniformly distributed load ‘P‘ throughout its span. Using Galerkin and Least square residual method. [AU, May / June – 2011] 1.109) Solve the differential equation for a physical problem expressed as 𝑑2 𝑦 𝑑𝑥2 + 100 = 0, 0 ≤ 𝑥 ≤ 10 with boundary conditions as y (0) = 0 and y (10) = 0 using (i) Point collocation method (ii) Sub domain collocation method (iii) Least squares method and (iv) Galerkin method. [AU, May / June – 2013] 1.110) Solve the differential equation for a physical problem expressed as 𝑑2 𝑦 𝑑𝑥2 + 50 = 0, 0 ≤ 𝑥 ≤ 10 with boundary conditions as y (0) = 0 and y (10) = 0 using the trail function 𝑦 = 𝑎1 𝑥 (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. [AU, Nov / Dec – 2011] 1.111) Solve the following equation using a two – parameter trial solution by the (a) Collocation method (𝑅 𝑑 = 0 𝑎𝑡 𝑥 = 1 3 𝑎𝑛𝑑 𝑥 = 2 3 ) (b) Galerkin method. Then, compare the two solutions with the exact solution 𝑑𝑦 𝑑𝑥 + 𝑦 = 0, 0 ≤ 𝑥 ≤ 1 y (0) = 1 1.112) Determine the Galerkin approximation solution of the differential equation
  • 81. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ ME6603 / VI / MECH / JAN – MAY 2017 FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) 11 𝐴 𝑑2 𝑢 𝑑𝑥2 + 𝐵 𝑑𝑢 𝑑𝑥 + 𝐶 = 0, 𝑢(0) = 𝑢( 𝑙) = 0 1.113) Solve the following differential equation using Galerkin’s method. D 𝑑2 𝜑 𝑑𝑥2 + 𝑄 = 0, 0 ≤ 𝑥 ≤ 𝐿 subjected to 𝜑(0) = 𝜑0 𝑎𝑛𝑑 𝜑( 𝐿) = 𝜑1 [AU, April / May – 2011] 1.114) A physical phenomenon is governed by the differential equation 𝑑2 𝑤 𝑑𝑥2 − 10𝑥2 = 5 𝑓𝑜𝑟 0 ≤ 𝑥 ≤ 1 The boundary conditions are given by 𝑤(0) = 𝑤(1) = 0. By taking two-term trial solution as 𝑤( 𝑥) = 𝑐1 𝑓1( 𝑥) + 𝑐2 𝑓2(𝑥) with, 𝑓1( 𝑥) = 𝑥 ( 𝑥 − 1) 𝑎𝑛𝑑 𝑓2 = 𝑥2 (𝑥 − 1) find the solution of the problem using the Galerkin method. [AU, Nov / Dec – 2009] 1.115) Determine the two parameter solution of the following using Galerkin method. 𝑑2 𝑦 𝑑𝑥2 = − cos 𝜋𝑥 , 0 ≤ 𝑥 ≤ 1, 𝑢(0) = 𝑢(1) = 0 [AU, Nov / Dec – 2012] 1.116) The following differential equation is available for a physical phenomenon. 𝑑2 𝑦 𝑑𝑥2 − 10𝑥2 = 5, 0 ≤ 𝑥 ≤ 1 With boundary conditions y (0) = 0, y (l) = 1. Find an approximate solution of the above differential equation by using Galerkin's method of weighted residuals and also compare with exact solution [AU, May / June – 2016] 1.117) A physical phenomenon is governed by the differential equation 𝑑2 𝑤 𝑑𝑥2 − 10𝑥2 = 5 𝑓𝑜𝑟 0 ≤ 𝑥 ≤ 1. The boundary conditions are given by w (0) = w (1) = 0. Assuming a trail function 𝑤(𝑥) = 𝑎0 + 𝑎1 𝑥 + 𝑎2 𝑥2 + 𝑎3 𝑥3 . Determine using Garlerkin method the variation of “w” with respect to “x”. [AU, Nov / Dec – 2016] 1.118) Using Collocation method, find the maximum displacement of the tapered rod as shown in Fig. E = 2 *107 N/cm2 ρ = 0.075N/cm3 [AU, Nov / Dec – 2014]
  • 82. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ ME6603 / VI / MECH / JAN – MAY 2017 FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) 12 1.119) A cantilever beam of length L is loaded with a point load at the free end. Find the maximum deflection and maximum bending moment using Rayleigh-Ritz method using the function 𝑦 = 𝐴{ 1 − cos ( 𝜋𝑥 2𝐿 )} Given: EI is constant. [AU, April / May – 2008] 1.120) Compute the slope deflection and reaction forces for the cantilever beam of length L carrying uniformly distributed load of intensity 'fo'. [AU, Nov / Dec – 2014] 1.121) A simply supported beam carries uniformly distributed load over the entire span. Calculate the bending moment and deflection. Assume EI is constant and compare the results with other solution. [AU, Nov / Dec – 2012] 1.122) 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 midspan and compare with exact solution using Rayleigh-Ritz method. Use 𝑦 = 𝑎1 sin ( 𝜋𝑥 𝑙 ) + 𝑎2 sin ( 3𝜋𝑥 𝑙 ) [AU, Nov / Dec – 2008] 1.123) Compute the value of central deflection in the figure below by assuming 𝑦 = 𝑎𝑠𝑖𝑛𝜋𝑥 𝐿 The beam is uniform throughout and carries a central point load P. [AU, Nov / Dec – 2007, April / May – 2009]
  • 83. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ ME6603 / VI / MECH / JAN – MAY 2017 FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) 13 1.124) A concentrated load P = 50 kN is applied at the centre of a fixed beam of length 3 m, depth 200 mm and width 120 mm. Calculate the deflection and slope at the midpoint. Assume E = 2 x 105 N/mm2 [AU, May / June – 2016] 1.125) A beam AB of span '1' simply supported at ends and carrying a concentrated load W at the centre C as shown in fig. Determine the deflection at midspan by using Rayleigh-Ritz method and compare with exact solution [AU, May / June – 2016, Nov / Dec – 2016] 1.126) If a displacement field is described by 𝑢 = (−𝑥2 + 2𝑦2 + 6𝑥𝑦)10−4 𝑣 = (3𝑥 + 6𝑦 − 𝑦2)10−4 Determine the direct strains in x and y directions as well the shear strain at the point x = 1, y =0. [AU, April / May – 2011] 1.127) In a solid body, the six components of the stress at a point are given by x= 40 MPa, y = 20 MPa, z = 30 MPa, yz = -30 MPa, xz = 15 MPa and xy = 10 MPa. Determine the normal stress at the point, on a plane for which the normal is (nx, ny, nz) = ( ½, ½, 2 1 ) 1.128) In a plane strain problem, we have x = 20,000 psi y = - 10,000 psi E = 30 x 10 6 psi,  = 0.3.
  • 84. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ ME6603 / VI / MECH / JAN – MAY 2017 FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) 14 Determine the value of the stress z. 1.129) For the spring system shown in figure, calculate the global stiffness matrix, displacements of nodes 2 and 3, the reaction forces at node 1 and 4. Also calculate the forces in the spring 2. Assume, k1 = k3 = 100 N/m, k2 = 200 N/m, u1 = u4= 0 and P=500 N. [AU, April / May – 2010] 1.130) Use the Rayleigh – Ritz method to find the displacement of the midpoint of the rod shown in figure. [AU, April / May – 2011] 1.131) Consider the differential equation 𝑑2 𝑦 𝑑𝑥2 + 400𝑥2 = 0 𝑓𝑜𝑟 0 ≤ 𝑥 ≤ 1 subject to boundary conditions 𝑦(0) = 0; 𝑦(1) = 0 The functional corresponding to this problem, to be extremized is given by 𝐼 = ∫ {−0.5 ( 𝑑𝑦 𝑑𝑥 ) 2 + 400𝑥2 𝑦 1 0 1.132) 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) [AU, Nov / Dec – 2009] 1.133) A bar of uniform cross section is clamped at one end and left free at the other end. It is subjected to a uniform load axial load P as shown in figure. Calculate the
  • 85. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ ME6603 / VI / MECH / JAN – MAY 2017 FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) 15 displacement and stress in the bar using three terms polynomial following Ritz method. Compare the results with exact solutions. [AU, May / June – 2011] 1.134) A simply Supported beam subjected to uniformly distributed load over entire span and it is subjected to a point load at the centre of the span. Calculate the deflection using Rayleigh-Ritz method and compare with exact solutions. [AU, May / June – 2013] 1.135) A simply Supported beam subjected to uniformly distributed load over entire span as shown in figure. Calculate the bending moment and deflection at midspan using Rayleigh-Ritz method. [AU, Nov / Dec – 2015, 2016] 1.136) A simply supported beam (span L and flexural rigidity EI) carries two equal concentrated loads at each of the quarter span points. Using Raleigh – Ritz method determine the deflections under the two loads and the two end slopes. [AU, April / May – 2009] 1.137) Analyze a simply supported beam subjected to a uniformly distributed load throughout using Rayleigh Ritz method. Adopt one parameter trigonometric function. Evaluate the maximum deflection and bending moment and compare with exact solution. [AU, Nov / Dec – 2010]
  • 86. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ ME6603 / VI / MECH / JAN – MAY 2017 FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) 16 1.138) Solve for the displacement field for a simply supported beam, subjected to a uniformly distributed load using Rayleigh – Ritz method. [AU, Nov / Dec – 2013] 1.139) Derive the governing equation for a tapered rod fixed at one end and subjected to its own self weight and a force P at the other end as shown in fig. Let the length of the bar be l and let the cross section vary linearly from A1 at the top fixed end to A2 at the free end. E and γ represents the Young’s modulus and specific weight of the material of the bar. Convert this equation into weak form and hence determine the matrices for solving using Ritz technique. [AU, April / May – 2015] 1.140) Use the Rayleigh – Ritz method to find the displacement field u(x) of the rod as shown below. Element 1 is made of aluminum and element 2 is made of steel. The properties are Eal = 70 GPa A1 = 900 mm2 L1 = 200 mm Est = 200 GPa A2 = 1200 mm2 L2 = 300 mm Load = P = 10,000 N. Assume a piecewise linear displacement.
  • 87. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ ME6603 / VI / MECH / JAN – MAY 2017 FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) 17 Field u = a1 + a2x for 0  x  200 mm, and u = a3 + a4 x for 200  x  500 mm. 1.141) A fixed beam length of 2L m carries a uniformly distributed load of a w(in N / m) which run over a length of ‘L’ m from the fixed end, as shown in Figure. Calculate the rotation at point B using FEA. [AU, Nov / Dec – 2011] 1.142) A rod fixed at its ends is subjected to a varying body force as shown in Figure. Use the Rayleigh-Ritz method with an assumed displacement field 𝑢( 𝑥) = 𝑎0 + 𝑎1 𝑥 + 𝑎2 𝑥2 to find the displacement u(x) and stress σ(x). Plot the variation of the stress in the rod. [AU, Nov / Dec – 2012] 1.143) 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. [AU, April / May – 2011]
  • 88. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ ME6603 / VI / MECH / JAN – MAY 2017 FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) 18 1.144) A steel rod is attached to rigid walls at each end and is subjected to a distributed load T(x) as shown below. a) Write the expression for potential energy. b) Determine the displacement u(x) using the Rayleigh – Ritz method. Assume a displacement field u(x) = a0 + a1 x + a2 x2 . 1.145) Derive the stress – strain relation and strain – displacement relation for an element in space. 1.146) Derive the equation of equilibrium in case of a three dimensional stress system. [AU, Nov / Dec – 2008] 1.147) What is constitutive relationship? Express the constitutive relations for a linear elastic isotropic material including initial stress and strain. [AU, Nov / Dec – 2009] 1.148) Give a detailed note on the following: (a) Rayleigh Ritz method (b) Galerkin method (c) Least square method and (d) Collocation method 1.149) Determine using any Weighted Residual technique the temperature distribution along the circular fin of length 6cm and 1cm. the fin is attached to a boiler whose wall temperature is 140ºC and the free end is insulated. Assume the convection coefficient h = 10 W/cm2 ºC. Conduction coefficient K = 70 W/ cm ºC and T∞ = 40ºC. The governing equation for the heat transfer through the fin is given by − 𝑑 𝑑𝑥 [𝐾𝐴(𝑥) 𝑑𝑇 𝑑𝑥 ] + ℎ𝑝(𝑥)(𝑇 − 𝑇∞) = 0 Assume appropriate boundary conditions and calculate the temperatures at every 1cm from the left end. [AU, April / May – 2015]
  • 89. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ ME6603 / VI / MECH / JAN – MAY 2017 FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) 19 1.150) Give a one – parameter Galerkin solution of the following equation, for the two domain’s shown below. .1 2 2 2 2               y u x u 1.151) Find the Eigen values and Eigen vectors of the matrix. [ 𝟑 −𝟏 𝟎 −𝟏 𝟐 −𝟏 𝟎 −𝟏 𝟑 ] 1.152) Find the Eigen values and Eigen vectors of the matrix. [ 𝟑 𝟏𝟎 𝟓 −𝟐 −𝟑 −𝟒 𝟑 𝟓 𝟕 ] 1.153) Find the Eigen value and the corresponding Eigen vector of 𝐴 = [ 1 6 1 1 2 0 0 0 3 ] [AU, May / June – 2016] 1.154) Describe the Gaussian elimination method of solving equations. [AU, April / May – 2011] 1.155) Explain the Gaussian elimination method for the solving of simultaneous linear algebraic equations with an example. [AU, April / May – 2008] 1.156) Solve the following system of equations using Gauss elimination method. [AU, Nov / Dec – 2010] x1 – x2 + x3 = 1
  • 90. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ ME6603 / VI / MECH / JAN – MAY 2017 FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) 20 -3x1 + 2x2 – 3x3 = -6 2x1 – 5x2 + 4x3 = 5 1.157) Solve the following system of equations by Gauss Elimination method. 2x1 – 2x2 – x4 = 1 2x2 + x3 + 2x4 = 2 x1 – 2x2 + 3x3 – 2x4 = 3 [AU, May / June – 2012] x2 + 2x3 + 2x4 = 4 1.158) Solve the following equations by Gauss elimination method. 28r1 + 6r2 = 1 6r1 + 24r2 + 6r3 = 0 6r2 + 28r3 + 8r4 = -1 8r3 + 16r4 = 10 [AU, Nov / Dec – 2010, 2012] 1.159) Use the Gaussian elimination method to solve the following simultaneous equations: 4x1 + 2x2 – 2x3 – 8x4 = 4 x1 + 2x2 + x3 = 2 0.5x1 – x2 + 4x3 + 4x4 = 10 –4x1 – 2x2 – x4 = 0 [AU, April / May – 2009] 1.160) Solve the following system of equations using Gauss elimination method. x1 + 3x2 + 2x3 = 13 – 2x1 + x2 – x3 = –3 - 5x1 + x2 + 3x3 = 6 [AU, Nov / Dec – 2009]