ISOPARAMETRIC ELEMENTS

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) 93
UNIT – V – ISOPARAMETRIC FORMULATION
PART – A
5.1) What do you mean by Isoparametric formulations?
[AU, Nov / Dec – 2007, April / May – 2009]
5.2) What is the purpose of isoparametric elements? [AU, May / June, Nov / Dec – 2016]
5.3) Write the shape functions for a 1D quadratic iso parametric element.
[AU, Nov / Dec – 2014]
5.4) Express the shape functions of four node quadrilateral element.
[AU, May / June – 2012]
5.5) What do you understand by a natural co – ordinate system?
[AU, April / May – 2011]
5.6) What do you mean by natural co-ordinate system? [AU, May / June – 2011]
5.7) What are the advantages of natural co-ordinates?
[AU, Nov / Dec – 2007, 2014, April / May – 2009]
5.8) What are the advantages of natural coordinates over global co-ordinates?
[AU, Nov / Dec – 2008]
5.9) What is the difference between natural coordinates and local coordinates?
[AU, May / June – 2016]
5.10) Give a brief note on natural co-ordinate system.
5.11) Write the natural co-ordinates for the point “P” of the triangular element. The point
‘P’ is the C.G. of the triangle. [AU, Nov / Dec – 2008]
5.12) Show the transformation for mapping x-coordinate system onto a natural coordinate
system for a linear spar element and for a quadratic spar element.
[AU, Nov / Dec – 2012]
5.13) Define a local co – ordinate system. [AU, Nov / Dec – 2011]
5.14) What is area co – ordinates? [AU, Nov / Dec – 2011]
5.15) What do you understand by area co – ordinates? [AU, April / May – 2011]

5.16) State the basic laws on which Isoparametric concept is developed.
5.17) Differentiate: local axis and global axis. [AU, April / May – 2008]
5.18) Define super parametric element. [AU, April / May – 2009]
5.19) Explain super parametric element. [AU, Nov / Dec – 2010]
5.20) Define Isoparametric elements? [AU, Nov / Dec – 2008]
5.21) Define Isoparametric elements with suitable examples [AU, April / May – 2010]
5.22) Define Isoparametric element formulations. [AU, Nov / Dec – 2012]
5.23) What do you mean by Isoparametric formulation? [AU, April / May – 2011]
5.24) What is the purpose of Isoparametric elements?
5.25) What is the salient feature of an Isoparametric element? Give an example.
[AU, Nov / Dec – 2013]
5.26) What are the applications of Isoparametric elements? [AU, April / May – 2011]
5.27) Differentiate x – y space and - space.
5.28) Write the advantages of co-ordinate transformation from Cartesian co-ordinates to
natural co-ordinates.
5.29) Define Jacobian. [AU, Nov / Dec – 2013]
5.30) What is a Jacobian? [AU, Nov / Dec – 2010]
5.31) What is the need of Jacobian? [AU, April / May – 2011]
5.32) Write down the Jacobian matrix. [AU, Nov / Dec – 2010]
5.33) Write about Jacobian transformation used in co-ordinate transformation.
5.34) What is the significance of Jacobian of transformation? [AU, May / June – 2012]
5.35) Differentiate between sub-parametric, isoparametric and super – parametric
elements.
5.36) Give two examples for sub parametric elements. [AU, Nov / Dec – 2013]

5.37) Represent the variation of shape function with respect to nodes for quadratic elements
in terms of natural co-ordinates.
5.38) Compare linear model, quadratic model and cubic model in terms of natural co-
ordinate system.
5.39) Write a brief note on continuity and compatibility.
5.40) Write down the element force vector equation for a four noded quadrilateral element.
5.41) Write down the Jacobian matrix for a four noded quadrilateral element
5.42) Write the shape function for the quadrilateral element in ,  space.
5.43) Why is four noded quadrilateral element is preferred for axi-symmetric problem than
three noded triangular element?
5.44) Sketch a four node quadrilateral element along with nodal degrees of freedom.
5.45) Write down the stiffness matrix for four noded quadrilateral elements.
[AU, May / June – 2011]
5.46) Distinguish between essential boundary conditions and natural boundary conditions.
[AU, Nov / Dec – 2009]
5.47) Write the advantages of higher order elements in natural co – ordinate system.
5.48) What are the types of non-linearity?
[AU, Nov / Dec – 2007, April / May – 2009, May / June – 2012]
5.49) State the advantage of Gaussian integration.
5.50) State the four-point Gaussian quadrature rule.
5.51) Briefly explain Gaussian quadrature. [AU, April / May – 2011]
5.52) What are the advantages of Gaussian quadrature? [AU, Nov / Dec – 2012]
5.53) What are the weights and sampling points of two point formula of Gauss quadrature
formula? [AU, May / June – 2012]

5.54) What are the advantages of gauss quadrature numerical integration for Isoparametric
elements? [AU, Nov / Dec – 2016]
5.55) Why numerical integration is required for evaluation of stiffness matrix of an
Isoparametric element? [AU, Nov / Dec – 2011]
5.56) When do we resort to numerical integration in 2D elements?
[AU, Nov / Dec – 2013]
5.57) Write the Gauss points and weights for two point formula of numerical integration.
5.58) Write down the Gauss integration formula for triangular domains.
5.59) Evaluate the integral ∫ (
1
−1
3𝜉2
+ 𝜉3
) 𝑑𝜉 using Gaussian quadrature method.
[AU, Nov / Dec – 2012]
5.60) Name the commonly used integration method in natural – co-ordinate system.
5.61) Write the relation between weights and Gauss points in Gauss-Legendre quadrature.
PART – B
5.62) Derive the Isoparametric representation for a triangular element.
[AU, Nov / Dec – 2010]
5.63) Derive element stiffness matrix for a linear Isoparametric quadrilateral element
5.64) Derive the shape functions for 4-noded rectangular element by using natural
coordinate system [AU, May / June – 2016]
5.65) Derive stiffness matrix for a linear Isoparametric element. [AU, Nov / Dec – 2012]
5.66) Explain the terms Isoparametric, sub parametric and super parametric elements.
[AU, Nov / Dec – 2013]
5.67) Distinguish between sub parametric and super parametric elements.
[AU, Nov / Dec – 2009, 2010]

5.68) Establish the shape functions of an eight node quadrilateral element and represent
them graphically. [AU, April / May – 2011]
5.69) Establish any two shape functions corresponding to one corner node and one mid –
node for an eight node quadrilateral element. [AU, Nov / Dec – 2013, 2016]
5.70) Derive the shape function for an eight noded brick element. [AU, April / May – 2009]
5.71) Derive the shape functions of a nine node quadrilateral Isoparametric element.
[AU, April / May – 2011, May / June – 2012]
5.72) Derive the element characteristics of a nine node quadrilateral element.
[AU, May / June – 2016]
5.73) Derive element stiffness matrix for linear Isoparametric quadrilateral element.
[AU, April / May – 2009, May / June – 2011]
5.74) Describe the element strain displacement matrix of a four node quadrilateral element.
[AU, May / June – 2012]
5.75) Derive the shape function for an eight – noded quadrilateral element in ,  space.
5.76) Establish the body force and traction force (uniformly distributed) vector for a lower
order quadrilateral element. [AU, Nov / Dec – 2013, 2016]
5.77) For the Isoparametric quadrilateral element as shown below, the Cartesian co-
ordinates of the point P are (6,4). The loads 10 KN and 12 KN are acting in X and Y
directions on that point P. Evaluate the nodal equivalent forces.

5.78) Consider the quadrilateral element as shown below using the linear interpolation
functions of a rectangular element, transform the element to the local co-ordinate system
and sketch the transformed element.
5.79) Derive the Jacobian matrix for triangular element with the (x, y) coordinates of the
nodes are (1.5, 2), (7, 3.5) and (4,7) at nodes i, j, k. [AU, Nov / Dec – 2014]
5.80) For the four noded element shown in figure determine the Jacobian and evaluate its
value at the point (1/2, 1/3). [AU, April / May – 2015]
5.81) Evaluate· the Cartesian coordinate of the point P which has local coordinates ε = 0.6
and η = 0.8 as shown in fig. [AU, May / June – 2016]

5.82) A four noded rectangular element is shown in figure. Determine the following: (i)
Jacobian Matrix (ii) Strain Displacement Matrix (iii) Element Stress. Take E =
20*105
N/mm2
, δ = 0.5.u = [ 0 0 0.003 0.004 0.006 0.004 0
0]T
ε = 0, η = 0. Assume plane stress condition. [AU, May / June – 2012]
5.83) A four nodal quadrilateral plane stress Isoparametric element is defined by nodes 1
(0,0), 2 (40,0), 3 (40, 15) and 4 (0,15). Determine the Jacobian matrix corresponding to
the Gauss point (0.57735, 0.57735) for the above element. [AU, Nov / Dec – 2013]
5.84) In a four-noded rectangular element, the nodal displacements in mm are given by
u1 = 0 u2 = 0.127 u3 = 0.0635 u4 = 0
v1 = 0 v2 = 0.0635 v3 = -0.0635 v4 = 0
For b = 50 mm, h = 25 mm, E = 2*105
N / mm2
and Poisson's ratio = 0.3,
determine the element strains and stresses at the centroid of the element and at
the corner nodes. [AU, Nov / Dec – 2012]
5.85) Find the Jacobian matrix for the nine-node rectangular element as shown below.
What is the determinant of the Jacobian matrix?

5.86) Determine the Jacobian for the (x, y) – (, ) transformation for the element shown
below. Also find the area of triangle using determinant method.
5.87) Compute the element and force matrix for the four noded rectangular elements as
shown below.
5.88) The Cartesian (global) coordinates of the corner nodes of a quadrilateral element are
given by (0,-1), (-2, 3), (2, 4) and (5, 3). Find the coordinate transformation between the
global and local (natural) coordinates. Using this, determine the Cartesian coordinates of

the point defined by (r,s) = (0.5, 0.5) in the global coordinate system.
[AU, Nov / Dec – 2009]
5.89) The Cartesian (global) coordinates of the corner nodes of an Isoparametric
quadrilateral element are given by (1,0), (2,0), (2.5,1.5) and(1.5,1). Find its Jacobian
matrix. [AU, Nov / Dec – 2009]
5.90) Find the jacobian transformation for four noded quadrilateral element with the (x,y)
coordinates of the nodes are (0, 0), (2, 0), (2, 1) and (0, l) at nodes i, j, k, l. Also find the
jacobian at point whose natural coordinates are (0,0) [AU, Nov / Dec – 2014]
5.91) Evaluate the Jacobian matrix for the linear quadrilateral element as shown in figure.
[AU, Nov / Dec – 2016]
5.92) A rectangular element has its nodes at the following points in Cartesian coordinate
system (0, 0), (5, 0), (5, 5), and (0, 5). Obtain the expressions for the shape functions of
the corresponding Isoparametric element. Using them obtain the elements if Jacobian
matrix of transformation. [AU, Nov / Dec – 2011]
5.93) If the coordinates of the quadrilateral are (1, 2), (10, 2), (8, 6) and (2, 10). Obtain the
Jacobian and hence, find the area of the element. [AU, Nov / Dec – 2011]
5.94) Determine the Jacobian matrix for the following quadrilateral element at x = 4.35 mm
and y = 3 mm. [AU, April / May – 2011]

5.95) For the isoparametric four noded quadrilateral element as shown in figure. Determine
the Cartesian coordinates of point P which has local coordinates ξ = 0.5 and η = 0.5.
[AU, Nov / Dec – 2015]
5.96) Consider the quadrilateral element as shown in figure. Evaluate
𝛿𝑁 𝑖
𝛿𝑦
and
𝛿𝑁 𝑖
𝛿𝑦
at
(ξ, η), (0, 0) and (
1
2
,
1
2
) using Isoparametric formulation.

5.97) Establish the strain – displacement matrix for the linear quadrilateral element as
shown in figure below at Gauss point r = 0.57735 and s = -57735.
5.98) Write short notes on [AU, Nov / Dec – 2008]
 Uniqueness of mapping of Isoparametric elements.
 Jacobian matrix.
 Gaussian Quadrature integration technique.
5.99) Derive the Gauss points and weights in case of one point formula and two point
formula of Gauss numerical integration. [AU, April / May – 2011]
5.100)Derive the weights and Gauss points of two point formula of Gauss quadrature rule.
[AU, May / June – 2012]
5.101)Integrate 𝑓( 𝑥) = 10 + 20𝑥 −
3𝑥2
10
+
4𝑥3
100
+
5𝑥4
1000
+
6𝑥5
10000
between 8 and 12.
Use Gaussian quadrature rule. [AU, April / May – 2008]
5.102)Evaluate the integral I = ∫ (2 + 𝑥 + 𝑥2) 𝑑𝑥
1
−1
and compare with exact results.
[AU, Nov / Dec – 2009]
5.103)Numerically evaluate the following integral and compare with exact one.
∫ ∫ (3
1
−1
𝑥2
+ 4𝑦2
) 𝑑𝑥 𝑑𝑦
1
−1
5.104)Using natural coordinates derive the shape function for a linear quadrilateral
element. [AU, Nov / Dec – 2008]
5.105)Use Gauss quadrature rule (n=2) to numerically integrate [AU, Nov / Dec – 2008]

∫ ∫ 𝑥𝑦𝑑𝑥𝑑𝑦
𝑎
0
𝑏
0
5.106)Use Gaussian quadrature rule (n = 2) to numerically integrate
∫ ∫ 𝑥𝑦𝑑𝑥𝑑𝑦
1
−1
1
−1
[AU, Nov / Dec – 2012]
5.107)Evaluate ∫ (2𝑥2
+ 3𝑥 + 4) 𝑑𝑥
5
1
using Gauss quadrature formula.
[AU, May / June – 2012]
5.108)Evaluate the integral ∫ 𝑒 𝑥 + 𝑥2 +
1
𝑥+7
𝑑𝑥
1
−1
using one, two and three point Gauss
quadrature formula. [AU, Nov / Dec – 2014]
5.109)Evaluate the integral ∫ 3𝑒 𝑥
+ 𝑥2
+
1
𝑥+2
𝑑𝑥 using one point and two point Gauss
quadrature formula. [AU, April / May – 2011]
5.110)Evaluate the integral ∫ 𝑥2
+ cos (
𝑥
2
) 𝑑𝑥
1
−1
using three point Gauss quadrature and
compare with exact solution. [AU, Nov / Dec – 2011]
5.111)Evaluate ∫ ∫ (𝑥2
+ 𝑦2
+ 2𝑥𝑦)
1
−1
𝑑𝑥 𝑑𝑦
1
−1
using Gauss numerical integration.
5.112)Evaluate the integral by two point Gauss quadrature. 𝐼 = ∫ ∫ (2𝑥2 +
1
−1
1
−1
4𝑦2
+ 3𝑥𝑦) 𝑑𝑥 𝑑𝑦. Gauss points are +0.57735 and – 0.57735 each of weight 1.000
[AU, Nov / Dec – 2016]
5.113)Use Gaussian quadrature evaluate the following integral 𝐼 =
∫ ∫
3+ 𝜉2
2+𝜂2 𝑑𝜉 𝑑𝜂
1
−1
1
−1
5.114)Use Gaussian quadrature to obtain an exact value of the integral.
∫ ∫ ( 𝑟3
− 1)(𝑠 − 1)2
𝑑𝑟 𝑑𝑠
1
−1
1
−1

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