2D Scalar Variable FEA Problems

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) 62
UNIT – III – TWO DIMENSIONAL SCALAR VARIABLE PROBLEMS
PART – A
3.1) Name few 2-D elements along with a neat sketch.
3.2) State the differences between 2D element and 1D element.
3.3) How do you define two dimensional elements? [AU, May / June – 2014]
3.4) Define Lagrange’s interpolation.
3.5) What is geometric Isotropy? [AU, May / June – 2013, Nov / Dec – 2016]
3.6) Write the Lagrangean shape functions for a 1D, 2 noded elements.
[AU, Nov / Dec – 2008]
3.7) Write the relation to obtain the size of the stiffness matrix for a linear quadrilateral
element having Ux and Uy as dof.
3.8) Why is the 3 noded triangular element called as a CST element?
[AU, Nov / Dec – 2010]
3.9) Write down the interpolation function of a field variable for three-node triangular
element. - [AU, April / May – 2010]
3.10) What is a CST element? [AU, April / May – 2011, Nov / Dec – 2015]
3.11) Why a CST element so called? [AU, Nov / Dec – 2014]
3.12) Write down the shape functions associated with the three noded linear triangular
element and plot the variation of the same. [AU, April / May – 2015]
3.13) Draw the shape functions of a CST element. [AU, Nov / Dec – 2010]
3.14) Explain the important properties of CST elements. [AU, Nov / Dec – 2008]
3.15) Write a note on CST element. [AU, May / June – 2011]
3.16) What is QST (Quadratic strain Triangle) element? [AU, May / June – 2014]
3.17) Write briefly about LST and QST elements.
3.18) What are CST and LST elements? [AU, Nov / Dec – 2009]
3.19) What is an LST element? [AU, Nov / Dec – 2016]

3.20) Define LST element. [AU, Nov / Dec – 2012]
3.21) Write the displacement function equation for CST element.
3.22) Write the strain – displacement matrix for CST element. [AU, Nov / Dec – 2016]
3.23) Differentiate CST and LST elements. [AU, Nov / Dec – 2007, April / May – 2009]
3.24) Give the Jacobian matrix for a CST element and state its significance.
[AU, Nov / Dec – 2013]
3.25) Evaluate the following area integrals for the three node triangular element
∫ 𝑁𝑖 𝑁𝑗
2
𝑁𝑘
3
𝑑𝐴 [AU, May / June – 2012]
3.26) A triangular element is shown in Figure and the nodal coordinates are expressed in
mm. Compute the strain displacement matrix. [AU, Nov / Dec – 2012]
3.27) What do you mean by the terms : c0
,c1
and cn
continuity?
[AU, April / May – 2010]
3.28) Distinguish between C0, C1 and C2 continuity elements.
3.29) What are the different problems governed by 2D scalar field variables?
3.30) Use various number of triangular elements to mesh the given domain in the order of
increasing solution refinement.

3.31) Define Pascal triangle.
3.32) Write the significance of Pascal triangle in developing triangular elements.
3.33) Plot the variation of shape function with respect node of a 3 noded triangular
element.
3.34) Write down the nodal displacement equations for a two dimensional triangular
elasticity element. [AU, April / May – 2010]
3.35) Define a plane stress condition. [AU, Nov / Dec – 2011]
3.36) What is meant by plane stress analysis? [AU, Nov / Dec – 2016]
3.37) State the condition for plane stress problem.
3.38) Give one example each for plane stress and plane strain problems.
[AU, Nov / Dec – 2008]
3.39) Distinguish between plane stress and plane strain problems. [AU, Nov / Dec – 2009]
3.40) Distinguish plane stress and plane strain conditions. [AU, Nov / Dec – 2010]
3.41) Define plane strain with suitable example. [AU, Nov / Dec – 2012]
3.42) Define plane strain analysis. [AU, Nov / Dec – 2011, 2015]
3.43) Define a plane stress problem with a suitable example.
[AU, May / June – 2013, Nov / Dec – 2016]
3.44) Explain plane stress problem with an example. [AU, April / May – 2011]
3.45) Explain plane stress conditions with example. [AU, May / June – 2011]

3.46) Give at least one example each for plane stress and plain strain analysis.
3.47) Give the application of plane stress and plane strain problems.
[AU, May / June – 2016]
3.48) Write down the strain displacement relation. [AU, April / May – 2011]
3.49) State whether plane stress or plane strain elements can be used to model the
following structures. Justify your answer. [AU, Nov / Dec – 2012]
(a) A wall subjected to wind load
(b) A wrench subjected to a force in the plane of the wrench.
3.50) Define path line and streamline. [AU, May / June – 2016]
3.51) Write the assumptions used to define the given problem as plane stress problem.
3.52) Write the assumptions used to define the given problem as plane strain problem.
3.53) Using general stress - strain relation, obtain plane stress equation.
3.54) Beginning with general elastic stress-strain relation, derive the plane strain
condition.
3.55) What are the differences between 2 Dimensional scalar variable and vector variable
elements? [AU, Nov / Dec – 2009]
3.56) What are the ways by which a three dimensional problem can be reduced to a two
dimensional problem?
3.57) What are the ways by which a 3D problem can be reduced to a 2D problem?
[AU, Nov / Dec – 2014]
3.58) How to reduce a 3D problem into a 2D problem? [AU, Nov / Dec – 2012]
3.59) Write down the governing equation for two-dimensional steady state heat
conduction. [AU, May / June – 2014]
3.60) Write down the governing differential equation for a two dimensional steady-state
heat transfer problem. [AU, Nov / Dec – 2009]

3.61) Write down the conduction matrix for a three noded linear triangular element.
[AU, April / May – 2015, Nov / Dec – 2016]
3.62) Sketch a two dimensional differential control element for heat transfer and obtain
the heat diffusion equation. [AU, Nov / Dec – 2012]
3.63) Define element capacitance matrix for unsteady state heat transfer problems.
[AU, May / June – 2013]
3.64) Name a few boundary conditions involved in any heat transfer analysis.
3.65) Mention two natural boundary conditions as applied to thermal problems.
3.66) 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. [AU, April / May – 2009]
3.67) State the assumptions in the theory of pure torsion. [AU, Nov / Dec – 2016]
3.68) Give the governing equation of torsion problem. [AU, May / June – 2012]
3.69) Write the step by step procedure of solving a torsion problem by finite element
method. [AU, April / May – 2011]
3.70) Outline the step by step procedure of handling torsion problem using the finite
element method. [AU, May / June – 2012]
PART – B
3.71) Determine the shape functions for a constant strain triangular (CST) element in
terms of natural coordinate system. [AU, Nov / Dec – 2008, May / June – 2014]
3.72) Derive the shape function for the constant strain triangular element.
[AU, May / June – 2016]
3.73) What are shape functions? Derive the shape function for the three noded triangular
elements. [AU, Nov / Dec – 2011]

3.74) A two noded line element with one translational degree of freedom is subjected to a
uniformly varying load of intensity P1 at node 1 and P2 at node 2. Evaluate the
nodal load vector using numerical integration. [AU, Nov / Dec – 2012]
3.75) The temperature at the four corners of a four – noded rectangle are T1, T2 T3 and T4.
Determine the consistent load vector for a 2-D analysis, aimed to determine the
thermal stresses. [AU, Nov / Dec – 2007, April / May – 2009]
3.76) Establish the finite element equations including force matrices for the analysis of
two dimensional steady – state fluid flows through a porous medium using
triangular element. [AU, Nov / Dec – 2013]
3.77) Explain the potential function formulation of finite element equations for ideal flow
problems. [AU, May / June – 2013]
3.78) Develop stiffness coefficients due to torsion for a three dimensional beam element.
3.79) Determine the shape functions N1, N2 and N3 at the interior point p for the
triangular element shown in figure [AU, May / June – 2014]
3.80) The nodal co-ordinates of the triangular element is as shown below. At the interior
point P, the x- co-ordinate is 3.3 and N1 = 0.3. Determine N2, N3 and the y – co-
ordinate at point P.

3.81) The (x, y) co-ordinates of nodes i, j and k of a triangular element are given by (0, 0),
(3, 0) and (1.5, 4) mm respectively. Evaluate the shape functions N1, N2 and N3 at an
interior point P (2, 2.5) mm for the element. For the same triangular element, obtain
the strain-displacement relation matrix B. [AU, Nov / Dec – 2009]
3.82) The (x, y) co-ordinates of nodes i, j and k of a triangular element are given by (0, 0),
(3, 0) and (1.5, 4) mm respectively. Evaluate the shape functions N1, N2 and N3 at
an interior point P (2, 2.5) mm for the element. For the same triangular element,
obtain the strain-displacement relation matrix B for the above same triangular and
explain how stiffness matrix is obtained assuming scalar variable problem
[AU, Nov / Dec – 2016]
3.83) Derive the interpolation function 14 for the quadratic triangular element as shown
below.
3.84) Derive the interpolation function of a corner node in a cubic serendipity element.
3.85) Find the temperature at a point P(1,1.5) inside the triangular element shown with
the nodal temperatures given as T1 = 400
C, TJ = 340
C, and TK = 460
C. Also
determine the location of the 420
C contour line for the triangular element shown in
figure below. [AU, April / May - 2008]

3.86) Calculate the temperature at the point for a three noded triangular element as shown
in figure. The nodal values are T1 = 40˚C, T2 = 34˚C and T3 = 46˚C. Point A is
located at (2, 1.5). Assume the temperature is linearly varying in the element. Also
determine the location of 42˚C contour line. [AU, May / June – 2011]
3.87) Calculate the value of pressure at the point A which is inside the 3 noded triangular
elements as shown in figure. The nodal values are φ1 = 40 MPa, φ2 = 34 MPa and φ3
= 46 MPa, Point A is located at (2, 1.5) Assume pressure is linearly varying in the
element. Also determine the location of 42 MPa contour line.
[AU, May / June – 2013]
3.88) A bilinear rectangular element has coordinates as shown in figure and the nodal
temperatures are T1 = 100ºC, T2= 60ºC, T3 = 50ºC and T4 = 90ºC. Compute the
temperature at the point whose coordinates are (2.5, 2.5). Also determine the 80ºC
isotherm. [AU, April / May – 2015]

3.89) A CST element has nodal coordinates 1 (0,0), 2 (5,0), 3 (0,4). The element is
subjected to a body force f = x3 N/m3. Determine the nodal force vector. Take the
element thickness as 0.3m. [AU, Nov / Dec – 2013]
3.90) Compute the elemental stress vectors for the following element, assuming plane
stress conditions. The nodal displacements in ‘mm’ [q] = [0 1 1 0 1
1]T
. The temperature increase in the element is 5˚C. Take E = 200 GPa and µ = 0.3.
The thermal coefficient of expansion is 11 * 10-6
/˚C. The thickness of the material
is 1 mm. [AU, April / May - 2011]
3.91) Calculate the element stiffness matrix and thermal force vector for the plane stress
element shown in figure below. The element experiences a rise of 100
C.
[AU, April / May - 2008]

3.92) Calculate the element stiffness matrix and the temperature force vector for the plane
stress element shown in fig. The element experiences a 20°C increase in
temperature. Assume coefficient of thermal expansion is 6*10-6
/ °C. Take E = 2 x
105
N/mm2
, v = 0.25, t = 5 mm. [AU, May / June – 2016]
3.93) Determine the element stiffness matrix and the thermal load vector for the plane
stress element shown in figure. The element experiences 20o
C increase in
temperature [AU, April / May - 2010]

3.94) Obtain the finite element equations for the following element. The thermal
conductivity (k) of the material of the element is 2 W/ mK. The convective heat
transfer coefficient (h) is 3 W/m2
K. The ambient temperature (Tf) is 25˚ C. The
thickness (t) of the material is 1mm. Assume convection along the edge ‘jk’ alone.
[AU, April / May - 2011]
3.95) Compute the elemental stress vectors for the following element, assuming plane
stress conditions. The nodal displacements in ‘mm’ [q] = [0 1 1 0 1
1]T
. The temperature increase in the element is 5˚C. Take E = 200 GPa and µ =
0.3. The thermal coefficient of expansion is 11 * 10-6
/˚C. The thickness of the
material is 1 mm. [AU, April / May - 2011]

3.96) Determine the temperature and heat fluxes at a location (2, 1) in a square plate as
shown in figure. Draw the isothermal for 125°C. T1 = 100°C, T2 = 150°C, T3 =
200°C, T4 = 50°C [AU, Nov / Dec – 2010]
3.97) A long bar of rectangular cross section having thermal conductivity of 1.5 W/m˚
C is
subjected to the boundary condition as shown below. Two opposite sides are
maintained at uniform temperature of 180 0
C. One side is insulated and the
remaining side is subjected to a convection process with T = 85˚
C and h = 50
W/m2˚
C. Determine the temperature distribution in the bar.
3.98) Compute the steady state temperature distributions in the plate shown in Fig. by
discretizing the domain of interest using triangular elements. Assume Thermal
Conductivity k = 1.5W/m°C. [AU, Nov / Dec – 2014]

3.99) The plane wall shown below is 0.5 m thick. The left surface of the wall is
maintained at a constant temperature of 2000
C and the right surface is insulated.
The thermal conductivity K = 25 W/Mo
C and there is a uniform heat generation
inside the wall of Q = 400 W/m3
. Determine the temperature distribution through
the wall thickness using linear elements.
3.100) Compute the steady state temperature distribution for the plate shown in the figure
below. A constant temperature of T0 = 1500
C is maintained along the edge y = w
and all other edges have zero temperature. The thermal conductivities are Kx = Ky
= 1. Assume w = L = 1 and thickness t = 1

3.101) A two dimensional fin is subjected to a heat transfer by conduction and convention.
It is discretized as shown in figure into two elements using linear triangular
elements. Derive the conduction, and thermal load vector. How is convection
accounted for solving the problem using finite element method?
3.102) Compute the element stiffness matrix and vectors for the element shown in figure
when the edge 2 – 3 and 3 – 1 experience heat loss. [AU, May / June – 2012]

e
3.103) Compute the element matrices and vectors for the element shown below, when the
edges jk and ik experience convection heat loss.
3.104) Compute element matrices and vectors for the elements shown in figure when the
edge kj experiences convection heat loss. [AU, Nov / Dec – 2009]

3.105) The triangular element shown in figure is subjected to a constant pressure 10
N/mm2
along the edge ij. Assume E = 200 Gpa, Poisson’s ratio  = 0.3 and
thickness of the element = 2 mm. The coefficient of thermal expansion of the
material  = 2 x10-6
/ o
C and T = 50o
C. Determine the constitutive matrix (stress-
strain relationship matrix D) and the nodal force vector for the element.
[AU, Nov / Dec - 2009]
3.106) Derive element force vector when linearly varying pressure acts on the side joining
nodes jk of a triangular element shown in Figure and body force of 25N/mm2
acts
downwards. Thickness = 5mm. [AU, April / May – 2011]
3.107) Find the expression for nodal vector in a CST element shown in figure subjected to
pressures Px1 on side 1. [AU, Nov / Dec – 2008]

3.108) Derive the expression for nodal vector in a CST element subjected to pressures Px1,
Py1 on side 1, Px2, Py2, on side 2 and Px3, Py3 on side 3 as shown in figure.
[AU, Nov / Dec – 2013,
2016]

2D Scalar Variable FEA Problems

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