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

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) 79
UNIT – IV – TWO DIMENSIONAL VECTOR VARIABLE PROBLEMS
PART – A
4.1) Write a displacement function equation for CST element. [AU, May / June – 2016]
4.2) Give the stiffness matrix equation for an axisymmetric triangular element.
4.3) What is axisymmetric element?
4.4) Give examples of axisymmetric problems. [AU, May / June – 2012]
4.5) What is an axisymmetric problem? [AU, April / May – 2011]
4.6) Write short notes on axisymmetric problems.
[AU, Nov / Dec – 2007, April / May – 2009]
4.7) What is meant by axi-symetric field problem? Given an example.
[AU, Nov / Dec – 2009]
4.8) When are axisymmetric elements preferred? [AU, Nov / Dec – 2013]
4.9) State the situations where the axisymmetric formulation can be applied.
[AU, April / May – 2011]
4.10) Give four applications where axisymmetric elements can be used.
4.11) List the applications of axisymmetric elements. [AU, May / June – 2016]
4.12) State the applications of axisymmetric elements. [AU, Nov / Dec – 2010]
4.13) Write down the stress-strain relationship matrix for an axisymmetric triangular
element. [AU, May / June – 2016]
4.14) Write down the constitutive relationship for axisymmetric problem.
4.15) Write down the constitutive relationship for the plane stress problem.
[AU, Nov / Dec – 2010]
4.16) What do you mean by constitutive law and give the constitutive law for axi-
symmetric problems? [AU, April / May, Nov / Dec – 2008]

4.17) Specify the body force term and the body force vector for axisymmetric triangular
element. [AU, Nov / Dec – 2013]
4.18) Give one example each for plane stress and plane strain problems.
[AU, April / May - 2008]
4.19) Explain plane strain problem with an example. [AU, May / June – 2012]
4.20) Give a brief note on static condensation.
4.21) Prove that 2  0 for plane strain condition.
4.22) Differentiate axi – symmetric and cyclic –symmetric structures.
4.23) Differentiate axi-symmetric load and asymmetric load with examples.
4.24) State the condition for axi-symmetric problem.
4.25) List the required conditions for a problem assumed to be axisymmetric.
4.26) What are the four basic sets of elasticity equations? [AU, May / June – 2012]
4.27) Give examples for the following cases.
4.28) a) plane stress problem b) plane strain problem c) axi-symmetric problem
4.29) Define the following terms with suitable examples [AU, April / May – 2010]
i) Plane stress, plane strain ii) Node, element and shape functions
iii) Axisymmetric analysis iv) Iso – parametric element
4.30) Define the term initial strain.
4.31) State the effect of Poisson’s ratio in plane strain problem.
4.32) How will the stress field vary linearly?
4.33) Compare the changes in the D matrix evolved out of plane strain, plane stress and axi-
symmetric problem.
4.34) What are the types of shell element? [AU, May / June – 2016]

PART – B
4.35) Derive the element strain displacement matrix and element stiffness matrix of a CST
element. [AU, April / May – 2011]
4.36) Explain the terms plane stress and plane strain problems. Give the constitutive laws
for these cases. [AU, Nov / Dec – 2007, April / May – 2009]
4.37) Derive the equations of equilibrium in the case of a three dimensional system.
[AU, Nov / Dec – 2007, 2008, April / May – 2009]
4.38) Derive the expression for constitutive stress-strain relationship and also reduce it to
the problem of plane stress and plane strain. [AU, Nov / Dec - 2008]
4.39) Derive the constant-strain triangular element’s stiffness matrix and equations.
[AU, April / May - 2008]
4.40) Derive the linear – strain triangular element’s stiffness matrix and equations.
4.41) Derive the stiffness matrix and equations for a LST element.
[AU, Nov / Dec – 2012]
4.42) Derive the element strain displacement matrix and element stiffness matrix of a
triangular element. [AU, May / June – 2012]
4.43) Derive the expression for the element stiffness matrix for an axisymmetric shell
element. [AU, Nov / Dec – 2007, April / May – 2009]
4.44) Describe the step by step procedure of solving axisymmetric problem by finite
element formulation. [AU, May / June – 2012]
4.45) Derive an expression for the stiffness matrix of an axisymmetric element.
4.46) For an axisymmetric triangular element. Obtain the [B] matrix and constitutive matrix
[AU, Nov / Dec – 2010]
4.47) Derive the stress-strain relationship matrix (D) for the axisymmetric triangular
element. [AU, Nov / Dec – 2012]
4.48) Explain the modeling of cylinders subjected to internal and external pressure using
axisymmetric. [AU, Nov / Dec – 2011]

4.49) For a thick cylinder subjected to internal and external pressure, indicate the steps of
finding the radial stress. [AU, Nov / Dec – 2010]
4.50) Derive the material property matrix for axisymmetric elasticity.
[AU, Nov / Dec – 2011]
4.51) Explain Galerkin’s method of formulation for determining the stiffness matrix for an
axisymmetric triangular element. [AU, Nov / Dec – 2013]
4.52) The (x, y) coordinates of nodes i, j and k of an axisymmetric triangular element are
given by (3, 4), (6, 5), and (5, 8) cm respectively. The element displacement (in cm)
vector is given as q = [0.002, 0.001, 0.001, 0.004, -0.003, 0.007]T
. Determine the element
strains. [AU, Nov / Dec – 2009]
4.53) Calculate the element stresses 21 and,,,  xyyx and the principle angle p for
the element shown below.
4.54) For the triangular element shown below, obtain the strain – displacement relation
matrix B and determine the strains x ,y and xy.

4.55) 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 co-
ordinates are xi =0, yi = 0, xj =0, yj = -1, xk =0, yk = -1 cm,. Determine the elemental
stiffness matrix. Assuming plane stress analysis. Take µ = 0.3 and E = 2.1*107
N/cm2
[AU, May / June – 2012]
4.56) For the CST element given below, assemble stain displacement matrix. Take t = 20
mm and E = 2*105
N/mm2
[AU, Nov / Dec - 2008]
4.57) For the triangular element as shown in figure. Determine the strain – displacement
matrix [B] and constitutive matrix [D]. Assume plane stress conditions. Take μ = 0.3, E =
30*106
N/m2
and thickness t = 0.1m. Also calculate the element stiffness matrix for the
triangular element. [AU, Nov / Dec – 2016]

4.58) A plate of dimensions 15cm * 6cm * 1cm is subjected to an axial pull of 5kN.
Assuming a typical element is of dimensions as shown in figure. Determine the strain
displacement matrix and constitutive matrix. E = 200GPa, µ = 0.3 and t = 10mm.
4.59) Obtain the global stiffness matrix for the plate shown in figure. Taking two triangular
elements. Assume plane stress condition. [AU, May / June – 2012]

4.60) For the constant strain triangular element shown in figure below, assemble the strain –
displacement matrix. Take t = 20 mm and E = 2 x 105
N/mm2
.
[AU, Nov / Dec – 2007, 2013, 2015, April / May – 2009]
4.61) For the plane strain element shown in the figure, the nodal displacements are given as
: u1= 0.005 mm, u2 = 0.002 mm, u3=0.0mm, u4 = 0.0 mm, u5 = 0.004 mm, u6 = 0.0
mm. Determine the element stresses. Take E = 200 Gpa and  = 0.3. Use unit thickness
for plane strain. [AU, April / May - 2010]
4.62) For the plane strain elements shown in figure, the nodal displacements are given as u1
= 0.005 mm, v1 = 0.002 mm, u2 = 0.0, v2 = 0.0, u3 = 0.005 mm, v3 = 0.30 mm. Determine
the element stresses and the principle angle. Take E = 70 GPa and Poision's ratio = 0.3
and use unit thickness for plane strain. All coordinates are in mm.
[AU, May / June – 2016]

4.63) For the two-dimensional loaded plate as shown in Figure. Determine the nodal
displacements and element stress using plane strain condition considering body force.
Take Young’s modulus as 200 GPa, Poisson’s ratio as 0.3 and density as 7800 kg/m3
.
4.64) Determine the stiffness matrix for the triangular elements with the (x, y) coordinates
of the nodes are (0,-4), (8,0) and (0,4) at nodes i, j, k. Assume plane stress condition. E =
200 GPa, Poisson's ratio = 0.35. [AU, Nov / Dec – 2014]
4.65) A thin plate a subjected to surface fraction as shown in Figure. Calculate the global
stiffness matrix. Table t = 25 mm, E = 2 *105
N/mm2
and γ = 0.30. Assume plane stress
condition. [AU, Nov / Dec – 2011]

4.66) Determine the deflection of a thin plate subjected to extensional load as shown.
4.67) Determine the nodal displacements and the element stresses for the two dimensional
loaded plate as shown in figure. Assume plane stress condition. Body force may be
neglected in comparison to the external forces. Take E = 210GPa, µ = 0.25, Thickness t =
10mm. [AU, May / June – 2011]
4.68) Calculate nodal displacement and elemental stresses for the truss shown in Figure. E
= 70 GPa cross-sectional area A = 2 cm2
for all truss members. [AU, Nov / Dec – 2012]

4.69) A thin elastic plate subjected to uniformly distributed edge load as shown below.
Find the stiffness and force matrix of the element.
4.70) For the configuration as shown in figure determine the deflection at the point load
applications. Use one model method. Assume plane stress condition.
4.71) A long cylinder of inside diameter 80 mm and outside diameter 120 mm snugly fits in
a hole over its full length. The cylinder is then subjected to an internal pressure of 2

MPa. Using two elements on the 10 mm length shown, find the displacement at the inner
radius.
4.72) The nodal co-ordinates for an axisymmetric triangular element are given in figure.
Evaluate strain-Displacement matrix for that element. [AU, May / June – 2016]
4.73) Determine the stiffness matrix for the axisymmetric element shown in fig, Take E as
2.1* 105
N/mm2
and Poisson's ratio as 0.3. [AU, Nov / Dec – 2012, 2016]
4.74) Determine the element stresses for the axisymmetric element as shown below. Take
E = 2.1 x 105
N/mm2
and = 0.25.

Use the nodal displacements as
u1 = 0.05 mm w1 = 0.03 mm
u2 = 0.02 mm w2 = 0.02 mm
u3 = 0 mm w3 = 0 mm
4.75) Compute the strain displacement matrix for the following axisymmetric element.
Also calculate the element stress vectors. If [AU, April / May – 2011]
[q] = [ 3.484 0 3.321 0 0 0]T
* 10-3
cm
[D] = [
27 12 12 0
12 27 12 0
12 12 27 0
0 0 0 8
]
4.76) Calculate the element stiffness matrix and the thermal force vector for the
axisymmetric triangular element as shown below. The element experiences a 150º C
increase in temperature. Take  = 10 x 10-6
/ ºC, E = 2 x 105
N/mm2
and µ= 0.25
All dimensions are mm
(9,10)
(6,7) (8,7)
X
Y
1 2
3

4.77) Calculate the element stiffness matrix for the axisymmetric triangular element shown
in Fig. The element experiences a 15 °C increase in temperature. The coordinates are in
mm. Take α= 10 x 10-6
/ °C, E = 2 x 105
N/mm2
, v = 0.25. [AU, May / June – 2016]
4.78) An open ended steel cylinder has a length of 200mm and the inner and outer
diameters as 68mm and 100mm respectively. The cylinder is subjected to an internal
pressure of 2MPa. Determine the deformed shape and distribution of principle stresses.
Take E = 200GPa and Poisson’s ratio = 0.3 [AU, April / May – 2011]
4.79) The steel flywheel as shown below rotates at 3000 rpm. Find the deformed shape of
the fly wheel and give the stress distribution.
4 in
12 in
2 in
1 in
1 in
E = 30 x 10
0.3
6
Psi
v =
4.80) Solve the axi-symmetric field problem as shown below, for the mesh shown there. T0
= 1000ºC R0= 0.02 m, g0 = 107
x 2  W/ m3
= internal heat generation, K = 20
W/mºC

L
4.81) The open –ended steel cylinder as shown below is subjected to an internal pressure of
1MPa. Find the deformed shape and the distribution of principal stresses.

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

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