GTU BE 8th Semester 2181911 Finite Elements Method(Department Elective II) Summer 2018 Question Paper

1
Seat No.: ________ Enrolment No.___________
GUJARAT TECHNOLOGICAL UNIVERSITY
BE – SEMESTER ?VIII (NEW) – EXAMINATION ? SUMMER 2018
Subject Code: 2181911 Date: 30/04/2018
Subject Name: Finite Elements Method(Department Elective II)
Time: 10:30 AM to 01:00 PM Total Marks: 70
Instructions:
1. Attempt all questions.
2. Make suitable assumptions wherever necessary.
3. Figures to the right indicate full marks.
Q.1 (a) Can the FEM handle a wide range of problems, i.e., solve general PDEs?
Enlist two advantages of FEM.
03
(b) List four applications of FEM and computer programs used for the FEM. 04
(c) List and briefly describe the process of the Finite Element Method. 07

Q.2 (a) What are the characteristics of shape function? Why polynomials are
generally used as shape function?
03
(b) Draw three 2D and 3D types of finite element. 04
(c) Derive the Stiffness Matrix for a Spring Element. 07
OR
(c) (a) Formulate the global stiffness matrix and equations for solution of the
unknown global displacement and forces. The spring constants for the
elements are k1; k2, and k3; P is an applied force at node 2. (b) Using the direct stiffness method, formulate the same global stiffness
matrix and equation as in part (a).
Figure 1
07
Q.3 (a) Distinguish between essential boundary conditions and natural boundary
conditions. Give their examples.
03
(b) Discuss the penalty approach for FEM. 04
(c) A tapered bar 1200 mm long, having cross-sectional area 450 mm
2
at one
end and 150 mm
2
at other end is fixed at the larger end. It is subjected to an
axial load of 35 kN. Calculate the stress on a model bar having three finite
elements 400 mm long. Assume modulus of elasticity, E = 2 x 105 N/mm
2
circular cross section at both end.
07

1
Seat No.: ________ Enrolment No.___________
GUJARAT TECHNOLOGICAL UNIVERSITY
BE – SEMESTER ?VIII (NEW) – EXAMINATION ? SUMMER 2018
Subject Code: 2181911 Date: 30/04/2018
Subject Name: Finite Elements Method(Department Elective II)
Time: 10:30 AM to 01:00 PM Total Marks: 70
Instructions:
1. Attempt all questions.
2. Make suitable assumptions wherever necessary.
3. Figures to the right indicate full marks.
Q.1 (a) Can the FEM handle a wide range of problems, i.e., solve general PDEs?
Enlist two advantages of FEM.
03
(b) List four applications of FEM and computer programs used for the FEM. 04
(c) List and briefly describe the process of the Finite Element Method. 07

Q.2 (a) What are the characteristics of shape function? Why polynomials are
generally used as shape function?
03
(b) Draw three 2D and 3D types of finite element. 04
(c) Derive the Stiffness Matrix for a Spring Element. 07
OR
(c) (a) Formulate the global stiffness matrix and equations for solution of the
unknown global displacement and forces. The spring constants for the
elements are k1; k2, and k3; P is an applied force at node 2. (b) Using the direct stiffness method, formulate the same global stiffness
matrix and equation as in part (a).
Figure 1
07
Q.3 (a) Distinguish between essential boundary conditions and natural boundary
conditions. Give their examples.
03
(b) Discuss the penalty approach for FEM. 04
(c) A tapered bar 1200 mm long, having cross-sectional area 450 mm
2
at one
end and 150 mm
2
at other end is fixed at the larger end. It is subjected to an
axial load of 35 kN. Calculate the stress on a model bar having three finite
elements 400 mm long. Assume modulus of elasticity, E = 2 x 105 N/mm
2
circular cross section at both end.
07

2
OR

Q.3 (a) For the loading system as shown in Figure 2, determine the element
stiffness matrix and globle stiffness matrix. Assume modulus of elasticity as
80 x 10
3
N/mm
2

Figure 2
03
(b) For above Q. 3 (a) determine the displacements, stresses and support reaction
using penalty approach.
04
(c) Axial load P = 300 KN is applied at 20? C to the rod as shown in Figure 3.
The temperature is then raised to 60? C. The coefficient of thermal
expansion for Aluminium is 23×10
-6
per ?C and Steel is 11.7×10
-6
per ?C.
AAl = 900 mm
2
, ASteel = 1200 mm
2
, E Al = 70 x 10
9
N/m
2
, ESteel = 200 x 10
9
N/m
2
. Using FEM, Determine the nodal displacement and element stresses
and the reaction forces at the supports.
Figure 3
07
Q.4 (a) Write the shape function and stiffness matrix for one-dimensional finite
element formulation of the fluid-flow problem.
03
(b) Derive the element stiffness matrix of truss element and write the stress
calculation formula for truss.
04
(c) A three bar truss is shown in Figure 4. The modulus of elasticity of the
material is 300 x 10
3
N/mm
2
. The area of the bar used for the truss is 60
mm
2
for all the elements. The length L1 = 750 mm and L2 = 100 mm. The
load P = 20 kN and P2 = 25 kN. Determine the element stiffness matrix for
each element and the global stiffness matrix.
Figure 4
07

1
Seat No.: ________ Enrolment No.___________
GUJARAT TECHNOLOGICAL UNIVERSITY
BE – SEMESTER ?VIII (NEW) – EXAMINATION ? SUMMER 2018
Subject Code: 2181911 Date: 30/04/2018
Subject Name: Finite Elements Method(Department Elective II)
Time: 10:30 AM to 01:00 PM Total Marks: 70
Instructions:
1. Attempt all questions.
2. Make suitable assumptions wherever necessary.
3. Figures to the right indicate full marks.
Q.1 (a) Can the FEM handle a wide range of problems, i.e., solve general PDEs?
Enlist two advantages of FEM.
03
(b) List four applications of FEM and computer programs used for the FEM. 04
(c) List and briefly describe the process of the Finite Element Method. 07

Q.2 (a) What are the characteristics of shape function? Why polynomials are
generally used as shape function?
03
(b) Draw three 2D and 3D types of finite element. 04
(c) Derive the Stiffness Matrix for a Spring Element. 07
OR
(c) (a) Formulate the global stiffness matrix and equations for solution of the
unknown global displacement and forces. The spring constants for the
elements are k1; k2, and k3; P is an applied force at node 2. (b) Using the direct stiffness method, formulate the same global stiffness
matrix and equation as in part (a).
Figure 1
07
Q.3 (a) Distinguish between essential boundary conditions and natural boundary
conditions. Give their examples.
03
(b) Discuss the penalty approach for FEM. 04
(c) A tapered bar 1200 mm long, having cross-sectional area 450 mm
2
at one
end and 150 mm
2
at other end is fixed at the larger end. It is subjected to an
axial load of 35 kN. Calculate the stress on a model bar having three finite
elements 400 mm long. Assume modulus of elasticity, E = 2 x 105 N/mm
2
circular cross section at both end.
07

2
OR

Q.3 (a) For the loading system as shown in Figure 2, determine the element
stiffness matrix and globle stiffness matrix. Assume modulus of elasticity as
80 x 10
3
N/mm
2

Figure 2
03
(b) For above Q. 3 (a) determine the displacements, stresses and support reaction
using penalty approach.
04
(c) Axial load P = 300 KN is applied at 20? C to the rod as shown in Figure 3.
The temperature is then raised to 60? C. The coefficient of thermal
expansion for Aluminium is 23×10
-6
per ?C and Steel is 11.7×10
-6
per ?C.
AAl = 900 mm
2
, ASteel = 1200 mm
2
, E Al = 70 x 10
9
N/m
2
, ESteel = 200 x 10
9
N/m
2
. Using FEM, Determine the nodal displacement and element stresses
and the reaction forces at the supports.
Figure 3
07
Q.4 (a) Write the shape function and stiffness matrix for one-dimensional finite
element formulation of the fluid-flow problem.
03
(b) Derive the element stiffness matrix of truss element and write the stress
calculation formula for truss.
04
(c) A three bar truss is shown in Figure 4. The modulus of elasticity of the
material is 300 x 10
3
N/mm
2
. The area of the bar used for the truss is 60
mm
2
for all the elements. The length L1 = 750 mm and L2 = 100 mm. The
load P = 20 kN and P2 = 25 kN. Determine the element stiffness matrix for
each element and the global stiffness matrix.
Figure 4
07

3
OR

Q.4 (a) List out the application of axisymmetric elements. 03
(b) Discuss the terms ?plain stress? and ?plain strain? problems. 04
(c) Evaluate the stiffness matrix for the element shown in Figure 5. The
coordinates are shown in units of inches. Assume plane stress conditions.
Let E = 30 x 10
6
psi, = 0.25, and thickness t =1 in. Assume the element
nodal displacements have been determined to be u1= 0, v1 = 0.0025 in., u2 =
0.0012 in., v2 = 0, u3 = 0, and v3 = 0.0025 in. Determine the element
stresses.
Figure 5 Plane stress element for stiffness matrix evaluation
07
Q.5 (a) Write the point force, body force and surface traction force using natural
coordinate system.
03
(b) Write the four shape function equations for a beam element. 04
(c) Using the direct stiffness method, solve the problem of the propped
cantilever beam subjected to end load P in Figure 6. The beam is assumed
to have constant EI and length 2L. It is supported by a roller at mid length
and is built in at the right end. Propped cantilever beam shown in below
Figure 6
Figure 6
07
OR
Q.5 (a) Write the consistent and lumped mass matrices for 1D element. 03
(b) List out applications of the axisymmetric elements. 04
(c) For the smooth pipe shown discretized in Figure 7 with uniform cross
section of 1 in
2
, determine the flow velocities at the center and right end,
knowing the velocity at the left end is vx = 2 in./s.
Figure 7 Discretized pipe for fluid-flow problem
07
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