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PRACTICE PROBLEMS FOR FINITE ELEMENT METHOD

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PRACTICE PROBLEMS FOR FINITE
ELEMENT METHOD

Exercise Sheets

Mahesh Chandra Luintel

 Table of Contents
EXERCISE 1 .................................................................................................................................................... 3

EXERCISE 2 .................................................................................................................................................... 9

EXERCISE 3 .................................................................................................................................................. 19

EXERCISE 4 .................................................................................................................................................. 25

EXERCISE 5 .................................................................................................................................................. 29

EXERCISE 6 .................................................................................................................................................. 31

EXERCISE 7 .................................................................................................................................................. 34

EXERCISE 8 .................................................................................................................................................. 40

2
 PRACTICE PROBLEMS FOR FINITE ELEMENT METHOD

EXERCISE 1
DISCRETE ONE DIMENSIONAL ELEMENTS: SPRING AND BAR

1. For the spring assemblages shown in Figures [P1.1a to P1.1d], determine the nodal
displacements, the forces in each element, and the reactions. Use the direct stiffness method
for all problems.

(a)

(b)

(c)

(d)

Figure P1.1
2. Number the elements and nodes for the spring system shown in Figure P1.2. Compute the
global stiffness and force vector, (b) Partition the system and solve for the nodal
displacements and (c) Compute the reaction forces. Take k = 5 kN/m and force in N.

Figure P1.2

3
3. For the spring assembly shown in Figure P1.3 solve for the displacements and the reaction
force at node 1 if k1 = 4 kN/m k2 = 6 kN/m k3 = 3 kN/m, F2 = − 30 N and F4 = 50 N.

Figure P1.3

4. Four rigid bodies, 1, 2, 3, and 4, are connected by four springs as shown in the Figure P1.4.
A horizontal force of 1,000 N is applied on Body 1 as shown in the figure. Using FE
analysis, (a) find the displacements of the two bodies (1 and 3), (b) find the element force
(tensile/compressive) of spring 1, and (c) the reaction force at the right wall (Body 2).
Assume the bodies can undergo only translation in the horizontal direction. The spring
constants are k1 = 400 N/mm, k2 = 500 N/mm, k3 = 500 N/mm, and k4 = 300 N/mm.

Figure P1.4

5. For the spring assemblage shown in Figure P1.5, obtain (a) the global stiffness matrix, (b)
the displacements of nodes 2, 3 and 4, (c) the global nodal forces, and (d) the local external
forces. Node 1 is fixed while node 5 is given a fixed known displacement d = 20 mm. The
spring constants are all equal to k = 200 kN/m.

Figure P1.5

6. The stepped bar shown in the Figure P1.6 is subjected to a force at the center. Use FEM to
determine the displacement at the center and reactions. Assume: E = 100 GPa, area of cross
sections of the three portions shown are, respectively, 10-4m2, 2 × 10-4m2, and 10-4m2, and F
= 10,000N.

4
 Figure P1.6

7. Determine the support reaction forces at the two ends of the bar shown in Figure P1.7, given
the following, P = 6.0 × 104 N, E = 2.0 × 104 N/mm2, A = 250 mm2, L = 150 mm, ∆ = 1.2
mm.

Figure P1.7

8. Determine the field values (nodal displacements and internal forces) for the following
structural system shown in Figure P1.8. Take F=50,000N, Aa = 1.5×10-4m2, Ab =1.0×10-4m2,
Ea = 203×109Pa, Eb = 69×109Pa, La = Lb = 1 m.

Figure P1.8

5
9. Derive the finite element equations for a torsion element and analyze the shaft shown in
Figure P1.9.

Figure P1.9
10. Consider a tapered bar of circular cross-section shown in Figure P.10. The length of the bar
is 1 m, and the radius varies as r(x) = 0.050 - 0.040x:, where r and x are in meters. Assume
Young's modulus = 100MPa. Both ends of the bar are fixed, and F = 10,000 N is applied at
the center. Determine the displacements, axial force distribution, and wall reactions using
four elements of equal length.
To approximate the area of cross-section of a bar element, use the geometric mean of the end
areas of the element.

Figure P.10
MATLAB Programming
11. Determine the field values (nodal displacements and internal forces) for the following
structural system shown in Figure P1.11. Take K = 40,000 N/m, F = 5000 N.

Figure P1.11

6
12. Find the displacements of the rigid bodies in Figure P1.12. Assume that the only non-zero
forces is F3 = 1000 N. Determine the element forces in the springs. What are the reactions at
the walls? Assume the bodies can undergo only translation in the horizontal direction. The
spring constants are k1 = 500 N/mm, k2 = 400 N/mm, k3 = 600 N/mm, k4 = 200 N/mm, k5 =
400 N/mm and k6 = 300 N/mm.

Figure P1.12

13. Determine the field values (nodal displacements and internal forces) for the following
structural system shown in Figure P1.13. Take k1 = 1300 N/m, k2 = 1400 N/m, k3 = 1500
N/m, k4 = 1600 N/m, k5 = 1700 N/m, F = 8000 N.

Figure P1.13

7
14. Model a tapered member of length L circular cross section as stepped shaft with several steps
as shown in Figure P1.14 and develop a discrete element methodology to evaluate the free
end elongation. Show how the elongation converges to the exact value as the number of
elements increase. Show a graph of the elongation value versus number of elements. D1 and
D2 are diameter. Take P = 10 kN, E = 200 GPa, D1 = 10cm, D2 = 5 cm, L= 50 cm.

Figure P1.14

8
 PRACTICE PROBLEMS FOR FINITE ELEMENT METHOD

EXERCISE 2
DISCRETE ELEMENTS: PLANE AND SPACE TRUSS

1. For each assembly of the bar elements shown in Figure P2.1, determine the global stiffness
matrix.

Figure P2.1
2. For the bar shwon in Figure P2.2, determine the axial stress. Take A = 10 × 10-4 m2, E = 210
GPa, and L = 3 m. Assume the global displacements have been previously determined to be
{0𝑚𝑚 2.5𝑚𝑚 5.0𝑚𝑚 3.0𝑚𝑚 }𝑇 .

Figure P2.2

9
3. The plane truss shown in the Figure P2.3 has two elements and three nodes. What are nodal
displacements and the element forces? Assume E = 1011 Pa, A = 10-4m2, L = 1 m, F = 14,142
N.

Figure P2.3
4. For the plane trusses shown in Figure P2.4, determine the horizontal and vertical
displacements of node 1 and the stresses in each element. All the elements have E = 210 GPa
and A = 10 × 10-4 m2.

(a) (b)
Figure P2.4
5. For the two bar truss shown in Figure P2.5, determine the horizontal displacement of node 1
and the axial force in each element. A force of P = 1000 kN is applied at node 1 while node 1

10
 settles an amount δ = 50 mm as shown. Let E = 210 GPa and A = 6 × 10-4 m2 for each
element.

Figure P2.5
6. Each joint of the structural system shown in Figure P2.6 is a pinned joint. The cross-
sectional are of each member in cm2 is underlined. Each member is made of steel, E = 20 ×
106 N/cm2. All lengths are given in centimeters. Calculate the unknown nodal displacements
and the axial force in each member.

(a) (b)

11
 (c) (d)

(e)

(f)
Figure P2.6

12
7. For the plane trusses supported by the spring at node 1, as shown in Figure P2.7, determine
the nodal displacements and the stresses in each element. Let E = 210 GPa and A = 6 × 10-4
m2 for both truss elements.

(a) (b)
Figure P2.7
8. The plane truss shown in Figure P2.8 is composed of members having a square 15 mm × 15
mm cross section and modulus of elasticity E = 69 GPa. Determine the nodal displacements
and the stresses in each element.

Figure P2.8
9. The displacements for the truss shown in Figure P2.9 are given below. The cross-sectional
are of each member in cm2 is underlined. Each member is made of steel, E= 20 × 106 N/cm2.
All lengths are given in centimeters. Calculate the axial force in each member.

13
 Figure P2.9
10. For the space truss elements shown in Figure P2.10, the global displacements at node 2 have
been determined to be u2 = 5 mm, v2 = 10 mm and w2 = 15 mm. Determine the displacement
along the local axis at node 2 of the elements. The coordinates shown in Figures are in
meters.

Figure P2.10
11. For the space truss shown in Figure P2.11, determine the displacement of node and stresses
in each element. Let E = 210 GPa and A = 10 × 10-4 m2 for all elements. A load of 20 kN is
applied at node 1 in the global X direction.
12. Determine the normal stress in each member of the truss structure shown in Figure P2.12.
All joints are ball-joint and the material is steel, whose Young's modulus is E = 210 GPa.

14
 Figure P2.11 Figure P2.12

Programming
13. Find the nodal displacements developed in the planar truss shown in Figure P2.13 when a
vertically downward load of 1000 N is applied at node 4. The pertinent data are given in
Table. Also find the stresses developed in each element.

Figure P2.13

15
14. Find the stresses of the truss structure shown in Figure P2.14. All members have elastic
modulus of 200 GPa and cross-sectional area of 2.5 × 10-3 m2.

Figure P2.14
15. Each joint of the structural system shown in Figure P2.15 is a pinned joint. The cross-
sectional are of each member in cm2 is underlined. Each member is made of steel, E = 20 ×
106 N/cm2. All lengths are given in centimeters. Calculate the unknown nodal displacements
and the axial force in each member.

Figure P2.15

16
16. For the space truss shown in Figure P2.16, determine the nodal displacements and the
stresses in each element. Let E = 210 GPa, A= 10 × 10-4 m2 for all elements. All supports are
ball and socket joints.

Figure P2.16
17. For the space truss shown in Figure P2.17, determine the deflection of upper joint, the stress
in each member and the reaction forces.

Figure P2.17
17
18. The space truss shown in Figure P2.18 has four members. Determine the displacement
components of node A and the force in each member. Assume AE = 106 N.

Figure P2.18

18
 PRACTICE PROBLEMS FOR FINITE ELEMENT METHOD

EXERCISE 3
DISCRETE ELEMENTS: BEAM AND FRAME

1. For a cantilever beam subjected to a vertical force at its free end, how many elements should
be used to obtain the exact solution for the deflection of the beam?
2. Calculate the nodal displacements and the internal member forces for each of the beam
loadings shown Figures P3.2 (a-l). Construct the shear force and bending moment diagram
for each member. Take 𝐸 = 200 GPa and 𝐼 = 0.8 × 10−4 m4 .

Figure P3.2 (a) Figure P3.2 (b)

Figure P3.2 (c) Figure P3.2 (d)

Figure P3.2 (e) Figure P3.2 (f)

19
 Figure P3.2 (g) Figure P3.2 (h)

Figure P3.2 (i) Figure P3.2 (j)

Figure P3.2 (k) Figure P3.2 (l)

3. Determine the nodal displacements and rotations and the global and element forces for the
beam shown in Figure P3.3. Take 𝐸 = 210 GPa and 𝐼 = 2 × 10−4 m4.
4. For the beam shown in Figure P3.4, determine the displacements and the slopes at the nodes,
the forces in each element, and the reactions. Take E = 210 GPa and I= 4 × 10-4 m4.

20
 Figure P3.3 Figure P3.4
5. The bar element 2 is used to stiffen the cantilever beam element 1, as shown in Figure P3.5.
Determine the displacements at node 1 and the element forces. For the bar, let 𝐴 = 1 ×
10−3 m2 . For the beam, let 𝐴 = 2 × 10−3 m2 , 𝐼 = 5 × 10−5 m4 , and 𝐿 = 3 𝑚. For both the
bar and the beam elements, let 𝐸 = 210 GPa.
6. Assemble element stiffness matrix for the member of plane frame shown in Figure P3.6, if it
is oriented at angle 30o to the x-axis. Take E = 200 GPa, I = 4 × 10-6 m4 and A = 4 × 10–3 m2.
7. Calculate the displacement at the free end of the inclined beam shown in Figure P3.7.Repeat
the problem when the applied load is parallel to the x- axis and has a magnitude of 15000 N.

Figure P3.6

Figure P3.5 Figure P3.7

21
8. Determine the stress distribution in the two members of the frame shown in Figure P3.8. Use
one finite element for each member of the frame.

Figure P3.8
9. For the rigid frame shown in Figure P3.9, determine the displacements and rotations of the
nodes, the element forces, and the reactions. The values of E, A and I to be used are listed
next to each figure.

22
 Figure P3.9
Programming
10. For the beam shown in Figure P3.10, determine the displacements and the slopes at the
nodes, the forces in each element, and the reactions. Also, draw the shear force and bending
moment diagrams.

Figure P3.10
11. For the beam shown in Figure P3.11, determine the displacements and the slopes at the
nodes, the forces in each element, and the reactions. Also, draw the shear force and bending
moment diagrams.

Figure P3.11

23
12. For the rigid frame shown in Figure P3.12, determine the displacements and rotations of the
nodes, the element forces, and the reactions. Take E = 210 GPa, A = 1 × 10-2 m2 and I = 2 ×
10-4 m4.

Figure P3.12
13. For the rigid frames shown in Figure P3.13, determine the displacements and rotations of the
nodes, the element forces, and the reactions. Take 𝐸 = 210 GPa, 𝐴 = 1 × 10−2 m2 and
𝐼 = 1 × 10−4 m4 .
14. Analyze the grid shown in Figure P3.14. The grid consists of two elements, is fixed at nodes
1 and 3, and is subjected to a downward vertical load of 22 kN. The global coordinates axes
and element length are shown in the figure. Let 𝐸 = 210 GPa, 𝐺 = 84 GPa, 𝐼 = 16.6 ×
10−5 m4 and 𝐽 = 4.6 × 10−5 m4 .

Figure P3.13 Figure P3.14

24
 PRACTICE PROBLEMS FOR FINITE ELEMENT METHOD

EXERCISE 4
CONTINUUM PROBLEMS

1. Find the governing differential equation of a string under tension 𝑇 subjected to lateral load 𝑞
for which potential energy is given by
𝐿 𝐿
𝑇 𝑑𝜙 2
Π = � � � 𝑑𝑥 − � 𝑞𝜙𝑑𝑥
2 𝑑𝑥
0 0

with boundary conditions 𝜙(0) = 𝜙(𝐿) = 0.

2. A 1-Dimensional beam of length , is subjected to a lateral distributed load of intensity 𝑞 per
unit length. The governing differential equation is given by
𝑑4𝑤
𝐸𝐼 4 + 𝑞 = 0
𝑑𝑥
Derive the variational form of the problem.
3. Obtain an approximate displacement equation (two terms) for the simply supported shown in
Figure P4.3. Compare the deflection at the center with the theoretical value 𝑦 = 𝑃𝐻 3 /48𝐸𝐼.
The governing differential equation is
𝑑 2 𝑦 𝑃𝑥
𝐸𝐼 −
𝑑𝑥 2 2
=0 0≤𝑥
𝐻
≤
2
𝑑2𝑦
𝐸𝐼
𝑑𝑥 2
𝑃(𝐻 − 𝑥)
−
2
𝐻
=0 ≤𝑥
2
≤𝐻

Figure P4.3
Use (a) the Galerkin method, (b) the Petrov-Galerkin method, (c) the least squares method
and (d) the point collocation method.

25
4. Derive the variational form of the Problem 3 and determine two terms approximate solution
by using Ritz method.
5. Using Ragleigh–Ritz method determine the expressions for deflection and bending moments
in a simply supported beam subjected to uniformly distributed load over entire span as shown
in Figure P4.5. Find the deflection and moment at midspan and compare with exact
solutions.

Figure P4.5
6. Find two parameter approximate solution of the equation
𝑑2𝜙 𝑑𝜙
+ 3 + 2𝜙 = 4𝑥 0≤𝑥≤1
𝑑𝑥 2 𝑑𝑥
subjected to boundary conditions
(𝐢) 𝜙(0) = 0, 𝜙(1) = 20
(𝐢𝐢) 𝜙(0) = 5, 𝜙′(1) = 2
(𝐢𝐢𝐢) 𝜙′(0) = 2, 𝜙′(1) = 20
Use (a) the Galerkin method, (b) the Petrov-Galerkin method, (c) the least squares method
and (d) the point collocation method. Compare solution of each case with exact
solution.
7. Find two parameter approximate solution of the equation
𝑑2𝜙 𝑑𝜙
2
−6 + 5𝜙 = 𝑥 2 0≤𝑥≤1
𝑑𝑥 𝑑𝑥
subjected to boundary conditions 𝜙(0) = 0, 𝜙′(1) = 1.
Use (a) the Galerkin method, (b) the Petrov-Galerkin method, (c) the least squares method
and (d) the point collocation method. Compare solution of each case with exact
solution.
8. Consider the following boundary value problem:
𝑑2𝜙 𝑑𝜙
2
sin(𝑥) + cos(𝑥) + 𝜙 sin(𝑥) = 0 𝜋/4 < 𝑥 < 𝜋/2
𝑑𝑥 𝑑𝑥
𝜙(𝜋/4) = 1 𝜙′(𝜋/2) = 2
Using the Galerkin weighted residual method, obtain two term approximate solution of the
boundary value problem.

26
9. Consider the following boundary value problem:
𝑑2𝜙 𝑑𝜙
2
+𝑥 − 2𝜙 = 1 0<𝑥<1
𝑑𝑥 𝑑𝑥
𝜙′(0) − 𝜙(0) = 1 and 𝜙′(1) = 2
Using the Galerkin weighted residual method, obtain two term approximate solution of the
boundary value problem.
10. Consider the following boundary value problem:
𝑑2𝜙 𝑑𝜙
+ 𝑥 −𝜙 =1 0<𝑥<1
𝑑𝑥 2 𝑑𝑥
𝜙′(0) − 𝜙(0) = 1 and 𝜙′(1) = 1
Derive a suitable weak form for use with the Galerkin method. Obtain two term
approximate solution of the problem.
11. The boundary value problem for a cantilevered beam can be written as
𝑑4𝑤
−𝑥 =0 0<𝑥<1
𝑑𝑥 4
𝑤(0) = 𝑤′(0) = 0, 𝑤′′(1) = 1, 𝑤′′′(1) = −1
Solve for the boundary value problem using the Galerkin method. Compare the
approximate solution to the exact solution by plotting the solutions on a graph.
12. Find the three parameter Galerkin solution of the equation
𝜕 2𝜙 𝜕 2𝜙
+ = −2 in Ω Ω: −3 ≤ x ≤ 3, −2 ≤ y ≤ 2
𝜕𝑥 2 𝜕𝑦 2
subjected to boundary conditions 𝜙 = 0 on Γ.
13. Solve the problem of steady state heat condition in a material of unit thermal conductivity
and occupying the square region : −1 ≤ x ≤ 1, −1 ≤ y ≤ 1. The sides 𝑦 = ±1 are
maintained at zero temperature, while heat is supplied at the rate cos(𝜋𝑦/2) per unit length
on the sides 𝑥 = ±1.
14. Set up the equations for 2 parameters Ritz approximation of the following equations
associated with a string under tension 𝑇 subjected to lateral load 𝑞 for which potential energy
is given by
𝐿 𝐿
𝑇 𝑑𝜙 2
Π = � � � 𝑑𝑥 − � 𝑞𝜙𝑑𝑥
2 𝑑𝑥
0 0

with boundary conditions 𝜙(0) = 𝜙(𝐿) = 0.

a) Use algebraic polynomials.
b) Use trigonometric functions.

27
MAPLE Application
15. Find one, two, three and four parameter approximate solution of the equation
𝑑2 𝜙 𝑑𝜙
2𝑥 2 + 3 =4 1<𝑥<2
𝑑𝑥 𝑑𝑥
𝜙′(1) = 1 and 𝜙(2) = 2
Use (i) the Galerkin method, (ii) the Petrov-Galerkin method, (iii) the least squares method
and (iv) the point collocation method.
(a) Compare one, two, three and four parameter solution obtained by the Galerkin method
with the exact solution.
(b) Compare one, two, three and four parameter solution obtained by the Petrov-Galerkin
method with the exact solution.
(c) Compare one, two, three and four parameter solution obtained by the least squares
method with the exact solution.
(d) Compare one, two, three and four parameter solution obtained by the point collocation
method with the exact solution.
(e) Compare one term solution of the Galerkin method, the Petrov-Galerkin method, the least
squares method and the point collocation method with the exact solution.
(f) Compare two term solution of the Galerkin method, the Petrov-Galerkin method, the least
squares method and the point collocation method with the exact solution.
(g) Compare three term solution of the Galerkin method, the Petrov-Galerkin method, the
least squares method and the point collocation method with the exact solution.
(h) Compare four term solution of the Galerkin method, the Petrov-Galerkin method, the
least squares method and the point collocation method with the exact solution.
16. Find the four parameter Galerkin solution of the equation for the deflection of a membrane
subjected to uniform transverse load q and tension T
𝜕 2𝜙 𝜕 2𝜙 𝑞
2
+ 2 =− in Ω Ω: −2 ≤ x ≤ 2, −1 ≤ y ≤ 1
𝜕𝑥 𝜕𝑦 𝑇
𝜙=0 on Γ.
a) Use algebraic polynomials.
b) Use trigonometric functions.

28
 PRACTICE PROBLEMS FOR FINITE ELEMENT METHOD

EXERCISE 5
INTERPOLATION FUNCTIONS FOR FINITE ELEMENT
FORMULATION

1. The nodal temperatures of nodes i and j (same as local nodes 1 and 2) of an element in a one
dimensional fin are known to be Ti = 120°C and Tj = 80°C with the x-coordinates xi = 30 cm
and xj = 50 cm. Find the following:
(a) Shape functions associated with the nodal values Ti and Tj.
(b) Interpolation model for the temperature inside the element, T(x).
(c) Temperature in the element at x = 45 cm.
2. A one-dimensional tapered fin element has the nodal coordinates xi = 20 mm and xj = 60 mm
with the area of cross section changing linearly from a value of Ai = 20 mm2 at xi to a value
of Aj = 10 mm2 at xj as shown in Figure P5.2.
(a) Determine the shape functions.
(b) Express the area of cross section of the fin element in terms of the shape functions.

Figure P5.2
3. The nodal coordinates and nodal temperatures of a triangular simplex element follow:
Node i: (xi, yi) = (10, 10) mm, Ti =100°C
Node j: (xj, yj) = (20, 50) mm, Tj = 80°C
Node k: (xk, yk) = (–10, 30) mm, Tk = 130°C
Express the temperature variation in the element, T(x, y), in terms of the shape functions.
4. The nodal coordinates and its functional value of a triangular linear element is given below.
Calculate the value at (20, 6).
Node Coordinate Value
(x,y)
1 (13,1) 190
2 (25,6) 160
3 (13,13) 185
5. Two-dimensional triangular elements have been used to determine the stress distribution in a
machine part. The nodal stresses and their corresponding positions for a triangular element

29
 are shown in Figure P5.5. What is the value of stress at x = 2.15 cm and y = 1.1 cm? Plot the
8.0 GPa and 7.86 GPa stress contour lines for an element.

Figure P5.5
6. The (x, y) coordinates of the nodes i, j and k of a triangular element are (1,1), (4, 2) and (3, 5)
respectively. The shape functions of a point P located inside the element are given by N1 =
0.15 and N2 = 0.25. Determine the x and y coordinates of the point P.
7. The nodal coordinates of the triangular element are (1,2), (5,3) and (4,6). At the interior point
P, the x-coordinate is 3.3 and the shape function at node 1 is 0.3. Determine the shape
functions at nodes 2 and 3 and also the y-coordinate at the point P.
8. A bilinear rectangular element has coordinates as shown in Figure P5.8 and the nodal
temperatures are T1 = 1000C, T2 = 600C, T3 = 500C and T4 = 900C. Compute the temperature
at the point (2.5, 2.5). Also determine the 800C isotherm.

Figure P5.8

30
 PRACTICE PROBLEMS FOR FINITE ELEMENT METHOD

EXERCISE 6
ONE DIMENSIONAL APPLICATION

1. Solve the problem given below using Galerkin’s method and piecewise linear functions. Use
two elements.
𝑑2 𝜙
=1 0<𝑥<2 subjected to boundary conditions 𝜙(0) = 0 and 𝜙(2) = 0
𝑑𝑥 2
2. Consider the following boundary value problem:
𝑑2𝜙
2 + 3𝜙 = 0 0<𝑥<2 subjected to boundary conditions 𝜙(0) = 1 and 𝜙′(2) = 1
𝑑𝑥 2
Using equal-length two finite elements, calculate unknown 𝜙 (x) and its derivative.
Compare the finite element solution with the exact solution.
3. Consider the following boundary value problem
𝑑 𝑑𝜙 2
�𝑥 � = 2 1<𝑥<2 subjected to boundary conditions 𝜙(1) = 2 and 𝜙′(2) = −1/4
𝑑𝑥 𝑑𝑥 𝑥
Calculate the approximate nodal values using two equal-length finite elements.
4. Use three elements of equal length, solve the differential equation given below.
𝑑2𝜙
+𝑥 =0 0<𝑥<1 subjected to boundary conditions 𝜙(0) = 0 and 𝜙(1) = 0
𝑑𝑥 2
5. A differential equation with boundary conditions is given below:
d2 ϕ dϕ
x2
+ 2x +2=0 1<x<4 subjected to boundary conditions ϕ(1) = 1 and ϕ(4) = 0
dx 2 dx
a) Derive the weak formulation.
b) Compute element matrices and vectors for the given discretization using three linear
elements.
c) Assemble them into the global matrix and vector.
d) Solve for the unknown nodal values.
6. Solve the differential equation for the mixed boundary conditions
d2 ϕ
+ ϕ − x 2 = 0 0 < x < 1 subjected to boundary conditions ϕ(0) = 0 and ϕ′(1) = 1
dx 2
Use the uniform mesh of three linear elements.
7. Solve the differential equation for the natural boundary conditions
d2 ϕ ϕ′(0) = 1 and ϕ′(1) =
+ cosπx = 0 0<x<1 subjected to boundary conditions
0
dx 2
31
 Use the uniform mesh of three linear elements.

8. A vertical rod shown in Figure P6.8 of elastic material is fixed at both ends with constant
cross-sectional area A, Young's modulus E, and height of L under the distributed load 𝑓 per
unit length. The vertical deflection u(x) of the rod is governed by the following differential
equation
𝑑2𝑢
𝐴𝐸 2 + 𝑓 = 0
𝑑𝑥
Using three elements of equal length, solve for u(x) and compare it with the exact
solution. Use the following numerical values: A = 10-4 m2, E = 10 GPa, L = 0.3 m, f =
106 N/m.

Figure P6.8

Programming
9. Write a program to use the finite element method to solve the problem described by the
following differential equation and boundary conditions
d2 ϕ
+ ϕ − x2 = 0 0 < x < 1 subjected to boundary conditions ϕ(0) = 0 and ϕ(1) = 0
dx 2
(a) Solve using two linear elements.
(b) Solve using four linear elements.
(c) Compare solution obtained in (a) and (b) with the exact solution by plotting them.
10. A tapered bar with circular cross-section as shown in Figure P6.10 is fixed at x = 0, and an
axial force of 0.3 × 106 N is applied at the other end. The length of the bar (L) is 0.3 m, and
the radius varies as r(x) = 0.03 – 0.07 x, where r and x are in meters. Use E = 1010 Pa.
(a) Solve using two linear elements.
(b) Solve using four linear elements.
(c) Compare solution obtained in (a) and (b) with the solution obtained by Galerkin
method by plotting them.

32
 Figure P6.10

33
 PRACTICE PROBLEMS FOR FINITE ELEMENT METHOD

EXERCISE 7
APPLICATIONS IN HEAT TRANSFER

1. A wall of an industrial oven consists of three different materials, as shown in Figure

P7.1. The first layer is composed of 10 cm of insulating cement with a thermal
conductivity of 12 W/m· K. The second layer is made from 20 cm of S-ply asbestos board
with a thermal conductivity of 0.071 W/m.K. The exterior consists of 12-cm cement
mortar with a thermal conductivity of 0.72 W/m2.K. The inside wall temperature of the
oven is 250°C, and the outside air is at a temperature of 35°C with a convection
coefficient of 45 W/m2.K. Determine the temperature distribution along the composite
wall.

Figure P7.1
Replace the temperature boundary condition of the inside wall of the oven with air
temperature of 400°C and an associated convection coefficient of 100 W/m2.K. Show
the contribution of this boundary condition to the conductance matrix and the forcing
matrix of element (1). Also determine the temperature distribution along the
composite wall.
2. Consider a heat chamber in which the temperature inside is maintained at a constant value of
200oC (see Figure P7.2). The chamber is covered by a 1.0-m-thick metal wall and outside is
insulated so that no heat flows to the outer surface of the wall. There is a uniform heat source
inside the wall generating Q = 400 W/m3. The thermal conductivity of the wall is k = 25
W/m0C. Assuming that the temperature only varies in the x-direction, find the temperature
distribution in the wall.
3. Use the system analysis procedure described in this chapter and constructs the discrete
system for heat conduction through the composite wall shown in Figure P7.3. Also, from the
following data, calculate the temperature distribution in the composite wall.
Areas: A1 = 2.0 m2, A2 = 1.0 m2 and A3 = 1.0 m2.
Thermal conductivity: k1 = 2.00 W∕mK, k2 = 2.5 W∕mK and k3 = 1.5 W∕mK.
Heat transfer coefficient: h = 0.1 W∕m2K.

34
Atmospheric temperature: Ta = 300C.
Temperature at the left face of wall: T1 = 75.0 0C.
4. A metallic fin, with thermal conductivity 360 W/m.K, 0.1 cm thick and 10 cm long extends
from a plane wall whose temperature is 235°C. Determine the temperature distribution along
the fin if heat is transferred to ambient air at 20°C with heat transfer coefficient of 9 W/m2K.
Take width of the fin as 1 m. Use three elements.

Figure P7.2 Figure P7.3

5. Find the temperature distribution in the one-dimensional fin shown in Figure P7.5
(a) using one finite element.
(b) using two finite elements.

Figure P7.5
6. The fin shown in Figure P7.6 is insulated on the perimeter. The left end has a constant
temperature of 1000C. A positive heat flux of q = 5000W/m2 acts on the right end. Let K = 6
W/ m.0C and cross-sectional area A = 0.1m2. Determine the temperatures at L/4, L/2, 3L/4,
and L; where L = 0.4 m.

35
 Figure P7.6
7. Aluminum fins with rectangular profiles shown in Figure P7.7 are used to remove heat from
a surface whose temperature is 150°C. The temperature of the ambient air is 20°C. The
thermal conductivity of aluminum is 168 W/m.K. The natural convective coefficient
associated with the surrounding air is 35 W/m2.K. The fins are 150 mm long. 5 mm wide, and
1 mm thick. (a) Determine the temperature distribution along a fin using five equally spaced
elements. (b) Approximate the total heat loss for an array of 50 fins. (c) Determine the
temperature of a point on the fin 45 mm from the base. Also compute the fraction of the total
heat that is lost through this section of the fin.

Figure P7.7
8. A rectangular aluminum fin is used to remove heat from a surface whose temperature is
80°C. The temperature of the ambient air varies between 18°C and 25°C. The thermal
conductivity of aluminum is 168 W/m.K. The natural convective coefficient associated with
the surrounding air is 25 W/m2. K. The fin is 100 mm long, 5 mm wide, and 1 mm thick. (a)
Determine the temperature distribution along a fin using five equally spaced elements for
both ambient conditions. (b) Approximate the total heat loss for an array of 50 fins for each
ambient condition. (c) Compare your finite element results to the exact results.
9. Find the temperature distribution in the tapered fin shown in Figure P7.9
(a) using one finite element.
(b) using two finite elements.

36
 Figure P7.9
10. Compute the element matrices and vectors for the element shown in Figure P7.10 when the
edges jk and ki experience convection heat loss.

Figure P7.10
11. A line source Q* = 52 W/cm is located at (5,2) in the element shown in Figure P7.11.
Determine the Q* allocated to each node.
12. For the square plate shown in Figure P7.12, determine the temperature distribution. Let k =
10 W/(m 0C) and h = 20 W/(m2 0C). The temperature along the left side is maintained at 100
0
C and that along the top side is maintained at 200 0C.
(a) Use four triangular elements.
(b) Use two rectangular elements.

37
 Figure P7.11 Figure P7.12

Programming
13. For the two-dimensional body shown in Figure P7.13, determine the temperature
distribution. The temperature of the top side of the body is maintained at 1000C. The body is
insulated on the other edges. A uniform heat source of Q = 1000 W/m3 acts over the whole
plate. Assume a constant thickness of 1 m. Let k = 25 W/(m 0C).
14. For the two-dimensional body shown in Figure P7.14, determine the temperature
distribution. The top and bottom sides are insulated. The right side is subjected to heat
transfer by convection. Let k = 10 W/(m °C).

Figure P7.13 Figure P7.14

ANSYS
15. Aluminum fins with triangular profiles, shown in the accompanying Figure P7.15, are used
to remove heat from a surface whose temperature is 150°C. T he temperature of the
surrounding air is 20°C. T he natural heat transfer coefficient associated with the surrounding
air is 30 W/m2.K. The thermal conductivity of aluminum is k = 168 W/m.K. Using manual
calculations, determine the temperature distribution along a fin. Approximate the heat loss
for one such fin.

38
 Figure P7.15
16. For the two-dimensional body shown in Figure P7.16, determine the temperature
distribution. The left and right sides are insulated. The top surface is subjected to heat
transfer by convection. The bottom and internal portion surfaces are maintained at 300 °C.

Figure P7.16

39
 PRACTICE PROBLEMS FOR FINITE ELEMENT METHOD

EXERCISE 8
APPLICATIONS IN SOLID MECHANICS

1. Determine the nodal forces for (a) a linearly varying pressure px on the edge of the triangular
element shown in Figure P8.1(a) and (b) the quadratic varying pressure shown in Figure
P81.1(b). Assume the element thickness is equal to t.

Figure P8.1

2. A two-dimensional triangular plane stress element made of steel, with modulus of elasticity E
= 200 GPa and Poisson's ratio v = 0.32, is shown in Figure P8.2. The element is 3 mm thick,
and the coordinates of nodes i, j and k are given in centimeters in Figure. Determine the
stiffness and load matrices under the given conditions.

Figure P8.2

40
3. Evaluate the stiffness matrix for the elements shown in Figure P8.3. The coordinates are
given in units of millimeters. Assume plane stress condition. Take E= 210 GPa, υ = 0.25 and
t = 10 mm.

(a) (b) (c)

Figure P8.3

4. The nodal displacements for the simplex two-dimensional element shown in the Figure
P8.4 given below are u1 = 2 mm, u2 = 6 mm, u3 = -1 mm, v1 = 4 mm , v2 = 5 mm and v3
= 8 mm. Determine the displacement components at an interior point B (10,10). The nodal
coordinates (in mm) are given in parenthesis.

Figure P8.4

5. For the elements shown in Figure P8.5, the nodal displacements are given as u1 = 2 mm, u2 =
0.5 mm, u3 = 3 mm, v1 = 1 mm , v2 = 0 mm and v3 = 1 mm. Determine the element
stresses. Take E = 210 GPa, υ = 0.25 and t = 10 mm.

41
 (a) (b) (c)
Figure P8.5

6. The nodal coordinates and the nodal displacements of a triangular element, under a specific
load condition are given below: xi = 0, yi = 0, xj = 1 mm , yj = 3 mm, xk = 4 mm, yk = 1, u1=1
mm, u2 = - 0.05 mm, u3 = 2 mm, v1 = 0.5 mm , v2 = 1.5 mm and v3 = -1 mm. If E = 2 × 105
N/mm2 and μ = 0.3. Find the stresses in the element.

7. The coordinates of the nodes 1, 2 and 3 of a triangular element are (1, 1), (8, 4) and (2, 7) in
mm. The displacements at the nodes are u1 = 1 mm, u2 = 3 mm, u3 = -2 mm, v1 = -4 mm, v2 =
2 mm and v3 = 5 mm. Obtain the strain-displacement relation matrix B and determine the
strains εx, εy and γxy.

8. For the plane strain elements shown in Figure P8.8, the nodal displacements are given as u1
= 0.005 mm v1 = 0.002 mm u2 = 0.0 mm v2 = 0.0 mm u3 = 0.005 mm v3 = 0.0 mm
Determine the element stresses σx; σy; τxy; σ1, and σ2 and the principal angle θp. Let E = 70
GPa and υ = 0.3, and use unit thickness for plane strain. All coordinates are in millimeters.

Figure P8.8

42
9. Consider a one-element triangular mesh shown in Figure P8.9. The boundary conditions are
as follows. The edge BC is constrained in y and traction free in x, whereas the edge AB is
constrained in x and traction free in y. The edge AC is subject to traction normal to the edge.
Assume Young’s modulus E = 3 × 1011Pa and Poisson’s ratio υ = 0.3.
(a) Construct the weak form corresponding to the generalized boundary conditions.
(b) Construct the stiffness matrix.
(c) Calculate the global force matrix.
(d) Solve for the unknown displacement matrix and calculate the stress at (1.5, 1.5).

Figure P8.9

10. For the configuration shown in Figure P8.10, determine the deflection at the point of load
application using a one element model. If a mesh of several triangular elements is used,
comment on the stress values in the elements close to the tip.

Figure P8.10

11. Find the nodal displacements in Figure P8.11. Use two linear triangular elements.

43
 Figure P8.11

12. Consider a rectangular panel as shown in Figure P8.12. The panel is modeled using a plane
stress linear elastic material with the following properties: Young’s modulus E = 3 × 1011 Pa
and Poisson’s ratio υ = 0.3. The essential boundary conditions are uAx = uAy = uBy = 0.
The natural boundary conditions are as follows. Along each edge of the panel, the prescribed
traction consists of normal and lateral components, both equal to 103N/m.
Discretize the panel using a single rectangular element as shown below. For convenience, use
identical global and local numberings as shown in Figure. Calculate nodal displacements and
stresses at the element Gauss points.

Figure P8.12

13. On a four nodal quadrilateral plane stress element the nodes are (0, 0), (6, 2), (6, 6) and (1,
5). A concentrated load whose x and y components are 10 kN, 16 kN respectively is applied
at a point (4, 5). Find the equivalent nodal forces and the displacement of nodes.

14. Solve the plane stress problem in Figure P8.14.

44
 Figure P8.14

15. Determine the nodal displacements and the element stresses, including principal stresses, due
to the loads shown for the thin plates in Figure P8.15. Use E = 210 GPa, υ = 0.3, and t = 5
mm. Assume plane stress conditions apply. The recommended discretized plates are shown
in the figures.

Figure P8.15

16. Solve the stress problem shown in Figure P8.16.

45
 Figure P8.16

17. Determine the stresses in the plate with the hole subjected to the tensile stress shown in
Figure P8.17. Graph the stress variation σx versus the distance y from the hole. Let E = 200
GPa, υ = 0.25, and t = 25 mm. Use symmetry as appropriate.

Figure P8.17

18. Determine the stresses in the plates with the round and square holes subjected to the tensile
stresses shown in Figure P8.18. Compare the largest principal stresses for each plate. Let E =
210 GPa, υ = 0.25, and t = 5 mm.

46
 Figure P8.18

47

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