Finite Element Methods Exam Questions

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Code No: R05410102
Set No. 1
IV B.Tech I Semester Regular Examinations, November 2008
FINITE ELEMENT METHODS IN CIVIL ENGINEERING
(Civil Engineering)
Time: 3 hours Max Marks: 80
Answer any FIVE Questions
All Questions carry equal marks

1. (a) Explain various considerations that are to be taken into account while choosing
the order and type of polynomial-type of interpolation function as a displace-
ment mo del in FEM
(b) What are the di erent methods available for solving problems of structural
Mechanics? name six di erent engineering applications of FEM. [8+8]

2. (a) What do you mean by axisymmetric loading? Explain
(b) Establish the di erential equations of equilibrium for a body subjected to two
dimensional stress systems.
[6+10]

3. A two span continuous Beam has each span t=2m and exural rigidity equal to
unity. The beam is simply supported on three rigid unyielding supports. Obtain
the structure sti ness matrix corresponding to the three rotational unrestrained
degrees of freedom after imposing the boundary conditions( gure 3). [16]

Figure 3

4. The plane truss shown in gure 4 is composed of members having a square 20 mm ×
20mm cross section and modulus of elasticity E= 2.5E5 N/mm2 Assemble global
sti ness matrix and Compute the Nodal displacements in global Coordinate system
for the loads shown in gure 4. [16]

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Code No: R05410102
Set No. 1

Figure 4
5. (a) Obtain the linear relation between Cartesian and natural volume coordinates.
(b) What is geometric invariance? Discuss the geometric invariance with an ex-
ample. [8+8]

6. (a) What is CST element? Show that why it is called as CST element with proof.
(b) Determine the Jacobian of the transformation J for the triangular element
shown in gure 6b. [10+6]

Figure 6b

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Code No: R05410102
Set No. 1
7. (a) Obtain the body force at typical node ?i? of an axisymmetric element.
(b) Derive the shape functions for a typical triangular element in solving axisym-
metric problem. [6+10]

8. (a) How the no de numbering scheme in uences the matrix sparsity in banded
sti ness matrix.
(b) Evaluate the function = cos p x 2 between x=-1 and x=1 using Gaussian two
and three point rule and check the answer with the exact solution. [6+10]

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Code No: R05410102
Set No. 2
(Civil Engineering)

1. (a) Using the principle of virtual displacement derive the expression for the sti -
ness matrix of any element
(b) Discuss the Engineering Applications of Finite element metho d? [10+6]

2. (a) What are the assumptions made in plane stress problems? Explain
(b) Develop strain - displacement relationship for a plane stress problem and ex-
press it in matrix form. [4+12]

3. (a) Prove that the structure sti ness matrix is always symmetric?
(b) Does the determinant of an element sti ness matrix exist? Explain. [8+8]

4. The plane truss shown in gure 4 is composed of members having a square 20 mm ×
20mm cross section and modulus of elasticity E= 2.5E5 N/mm2 Assemble global
sti ness matrix and Compute the Nodal displacements in global Coordinate system
for the loads shown in gure 4. [16]

Figure 4
5. Derive the element sti ness matrix for a plane rectangular bilinear element. [16]

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Code No: R05410102
Set No. 2
6. Using displacement formulation, derive the shape functions for the CST element.
[16]

7. (a) Derive the equilibrium equations for a two dimensional plane stress condition.
(b) Write the constitutive matrices for a plane stress and plane strain conditions.
[8+8]

8. List di erent nite element solution techniques, explain brie y one solution tech-
nique. [16]

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Code No: R05410102
Set No. 3
(Civil Engineering)

1. The potential energy for the linear Elastic one dimensional rod shown in gure 1
is given by
2
2
=1 -21
2
0
where
1 = ( = 1)

Find the value of stress at any point in the bar. Use Raleigh-Ritz method. Compare
the result with exact solution.
[16]

Figure 1
2. (a) What do you mean by axisymmetric loading? Explain
(b) Establish the di erential equations of equilibrium for a body subjected to two
dimensional stress systems.
[6+10]

3. (a) State and explain Local coordinate system and global coordinate system with
the examples.
(b) Discuss the necessity for adopting local co ordinate System for one dimensional
elements? [10+6]

4. Obtain the global sti ness matrix taking two elements 1 and 2 as beam elements
for planar structure shown in gure 4. The length of the element ‘1’ may be taken
as L, the values of E and I are same for both elements. [16]

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Code No: R05410102
Set No. 3

Figure 4
5. (a) Derive the shape functions to the rectangular bilinear element.
(b) Force F acts on one edge of the plane bilinear element at y=b/2, as shown in
gure 5b. What the element nodal load vector results? [8+8]

Figure 5b
6. Determine the Global sti ness matrix for a thin plate of thickness 10mm subjected
to the surface traction shown in Figure 6. Consider the plate is modeled with two
CST elements. [16]

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Code No: R05410102
Set No. 3

Figure 6
7. Obtain the strain displacement matrix for an axisymmetric triangular element. [16]

8. (a) Describe the Gaussian quadrature metho d.

(b) Evaluate 3 dx
x using Gaussian three point rule and check the answer with the
1
exact solution. [6+10]

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Code No: R05410102
Set No. 4
(Civil Engineering)

1. For a simply supported Beam of uniformly distributed load of Intensity Po per unit
length and a concentrated load P at centre, Find the Transverse de ection using
Raleigh-Ritz method of Functional Evaluation and compare the result with exact
Analytical solution.
[16]

2. (a) Derive the equations of equilibrium for two dimensional problems
(b) Determine the stresses x y xy in the case of plane stress problem if the

[16]
strains are x = 10 × 10-5 y2 5 × 10-5 xy 2 0 50 30
= = 7 × 10-5 = = × 10-4

3. A bar of length L has a cross-sectional area, which varied linearly from value 2A
at one end to A at other end . End 1 is held against any moment while the
bar is stretched by an axial force F applied at end 2. Obtain solutions for axial
displacements and axial stress distributions and the value of the potential energy
based on the following displacement elds:

(a) u = a1+a2x
(b) u = a1+a2x+a3x2.
[16]

4. Give a detailed method of nding the stresses in the frame shown in the gure 4
Take Cross section = 2cm × 1cm. [16]

Figure 4
5. (a) Obtain the linear relation between Cartesian and natural area coordinates.

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Code No: R05410102
Set No. 4
(b) Force F acts on one edge of the plane bilinear element at y=b/2, as shown in
gure 5b. What the element nodal load vector results? [8+8]

Figure 5b
6. (a) What is CST element? Show that why it is called as CST element with proof.
(b) Determine the Jacobian of the transformation J for the triangular element
shown in gure 6b. [10+6]

Figure 6b

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Code No: R05410102
Set No. 4
7. (a) Write the stress-strain relation for an isotropic material in solving axisymmet-
ric problem.
(b) Derive the shape functions for a typical triangular element in solving axisym-
metric problem. [6+10]

8. (a) Describe the Gaussian quadrature metho d.

(b) Evaluate 3 dx
x using Gaussian three point rule and check the answer with the
1
exact solution. [6+10]

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Finite Element Methods Exam Questions

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Finite Element Methods Exam Questions