BIRLA INSTITUTE OF TECHNOLOGY, MESRA, RANCHI

Department of Civil and Environmental Engineering

CLASS: M.Tech. SESSION: SP/2024
Subject: Finite Element Method (CE506)

Tutorial Sheet

Module - I
1 Derive the stiffness matrix for the linear spring given in the figure using the force equilibrium approach.

2 For each assembly of springs shown in the accompanying figures determine the global stiffness matrix using
the system assembly procedure

3 For the spring assembly given in below figure, determine force F3 required to displace node 2 an amount 𝛿𝛿 =
0.75 in. to the right. Also compute displacement of node 3. Given k1 = 50 lb./in. and k2 = 25 lb./in.

4 A steel rod subjected to compression is modeled by two bar elements, as shown in the given figure. Determine
the nodal displacements and the axial stress in each element.

5 Given figure depicts an assembly of two bar elements made of different materials. Determine the nodal
displacements, element stresses, and the reaction force.

6 Derive the element stiffness matrix for an axially loaded bar using the force equilibrium approach.
7 Derive the element stiffness matrix for an axially loaded bar using the generalized approach (weak form and
shape function.
8 Derive the global stiffness matrix for an axially loaded bar, inclined to x axis by an angle 𝜃𝜃, of a plane truss
using the element transformation.
9 The plane truss shown in Figure 1 is subjected to a downward vertical load and a horizontal load at node 2.
Determine (via the direct stiffness method) the deflection of node 2 in the global coordinate system specified
and the axial stress in each element. For both elements, A = 0.2 m2, E = 200 GPa.

(0, 0) (4, 0)

1 2
(3, -1)

10 kN
15 kN

10 The plane truss shown in the given figure is composed of members having a square 15 mm × 15 mm cross
section and modulus of elasticity E = 69 GPa.
a. Assemble the global stiffness matrix.
b. Compute the nodal displacements in the global coordinate system for the loads shown.
c. Compute the axial stress in each element.
 Module - II
1 Derive the method of weighted residual statements (both strong form and weak form) axially loaded bar
show in the given figure

2 Derive the method of weighted residual statements (both strong form and weak form) the beam show in the
given figure

3 For a beam element (of length L) with a node in each end, as shown in the Figure, considered the lateral
displacement field v = N1v1 + N 2θ1 + N 3 v2 + N 4θ 2 , where N1, N2, N3 and N4 are the interpolation functions
and given by
3x 2 2 x3 2 x 2 x3
N1 = 1 − + , N 2 = x − + 2
L2 L3 L L
2 3 2 3
3x 2x x x
N3 = 2 − 3 , N4 = − + 2
L L L L

θ1 θ2
1 2
v1 v2
Figure 1

Show that this field includes the required capability to represent the rigid body rotational deformation of
beam.

4 Calculate the equivalent nodal load vector for the given beams. Use the interpolation functions of beam
elements given in Q3.
5 Use Galerkin’s method of weighted residuals to obtain an approximate solution of the differential equation
d2 y
− 10 x 2 = 5 0 ≤ x ≤1
dx 2
with boundary conditions y(0) = y(1) = 0.

6 For each of the following differential equations and stated boundary conditions, obtain solutions using
Galerkin’s method of weighted residuals

a) b)

c) d)

e)

Module - III
1 Derive the strong and weak form of method of weighted residual statement for three-dimensional elasticity
problem. From the weak form provide the expression for the followings (in integral form)
a) Element consistent mass matrix
b) Element stiffness matrix
c) Equivalent nodal load vector due to body force
d) Equivalent nodal load vector due to surface traction

2 Derive the shape functions for three nodded C0 element of one-dimensional problem using the polynomial
forms.

3 Explain the local coordinate/area co-ordinate in context of triangular elements. Derive the shape function of
constant strain triangle (CST) using local co-ordinates.

4 Derive the shape functions of 6 nodded linear strain triangle using the area co-ordinates.

5 Derive the polynomial form of the shape functions for 4 nodded rectangular elements.

6 Derive the polynomial form of the shape functions for 9 nodded rectangular elements of Lagrangian family.

7 Derive the polynomial form of the shape functions for 8 nodded rectangular elements of Serendipity family.

8 Describe the requirement of compatibility and completeness for finite element formulation.

9 Write a short note on plane stress, plane strain and axi-symmetric idealization.
 Module – IV & V
1 Describe different kind of parametric mapping in context of finite element method.

2 Derive then relation between the derivates with respected global Cartesian co-ordinates (x, y, z) and
local co-ordinates (ξ, η, ζ) for isoperimetric mapping of 3D brick element.

3 Derive then relation between the derivates with respected global Cartesian co-ordinates (x, y) and
local co-ordinates (ξ, η) for isoperimetric mapping of 2D rectangular element.

4 Explain iso-parametric formulation for quadrilateral element.

5 Derive the Jacobian matrix for the isoperimetric mapping of linear element (shown in the figure)

(7,7)
η (5,5)
(-1,1) (1,1)

ξ y (8,3)
(4,2)
(-1,-1) (1,-1)
x

6 Explain the following steps in context of any commercial FE Application:

a) Pre-Processing
b) Analysis
c) c) Post-processing
7 Use Gaussian quadrature to obtain exact values for the following integrals

8 Evaluate the following integral using 2-point Gauss quadrature:

1 1
∫∫
−1 −1
(1 + 2 x + 3x 2 y ) dx dy
9 Use Gaussian quadrature to obtain exact values for the following integrals in two dimensions