rr420302-finite-element-methods.pdf

Code No: RR420302 Set No. 1

IV B.Tech II Semester Supplimentary Examinations, Apr/May 2009FINITE ELEMENT METHODS

( Common to Mechanical Engineering and Production Engineering)Time: 3 hours Max Marks: 80

Answer any FIVE QuestionsAll Questions carry equal marks

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1. Explain the following with neat sketches.

(a) Mathematical Finite Element method.

(b) Physical Finite Element Method. [8+8]

2. Explain the mathematical interpretation of finite element method for one dimen-sional field problems. [16]

3. Estimate the stiffness matrix and the deflection at the center of the simply sup-ported beam of length 3 m. A 50 kN of load is acting at the center of the beam.Take EI = 800 X 103 N-m2. [16]

4. Prove that a three nodded triangle element has a constant strain or stress state.

For the triangular element shown in figure4, find the consistent nodal load. [5+11]

Figure 4

5. Derive the element conductivity matrix and load vector for solving 1- D heat con-duction problems, if one of the surfaces is exposed to a heat transfer coefficient ofh and ambient temperature of T∞? [16]

6. Derive the elemental mass matrix for 1-D bar element and 1-D plane truss element?[16]

7. (a) A uniform pressure of 100 N/cm2 is applied on the face 234 of the tetrahedronelement shown in figure7b below. Determine the corresponding load vector ofthe element

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(b) The X,Y,Z coordinates of a hexahedron element are shown in figure below7b.Derive the Jacobian matrix [J]. [8+8]

Figure 7b

8. With an example, explain the procedure involved in solving an engineering problemin computational finite element analysis using computer software. [16]

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1. Explain the different approaches of getting the finite element equations. [16]

2. With a suitable example explain the formulation of finite element equations bydirect approach. Assume suitable data for the example. Use I-D analysis . [16]

3. Define and derive the Hermite shape functions for a two nodded beam element?[16]

4. (a) Show that the value of the shape function N1 of node i and node k of a simplextriangular element is zero and at node I is one.

(b) 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 locatedinside the element are given by N1 = 0.15 and N2 = 0.25. Determine the xand y coordinates of the point P. [6+10]

5. Compute the elemental conductivity matrix and load vector for the 2-D triangularelement as shown in figure5. The faces 1-3 and 2-3 are exposed to a convection andthere is an internal heat generation of 50 W/cm3. Assume thermal conductivity is60 W/m K. [16]

Figure 5

6. Consider the axial vibrations of a steel bar shown in the figure6:

(a) Develop global stiffness and mass matrices,

(b) determine the natural frequencies? [16]

Figure 6

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7. Explain the following semiautomatic mesh generation techniques:

(a) Conformal mapping approach.

(b) Mapped element approach. [8+8]

8. Explain and compare the different features of the following FEM software packages.NISA, ANSYS, NASTRON. [16]

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1. Write notes on the following:

(a) Engineering application of finite element method.

(b) Discretization process. [8+8]

2. Explain the mathematical interpretation of finite element method for one dimen-sional field problems. [16]

3. Find the deflection at the load of the steel shaft as shown in figure 3: take E = 200Gpa [16]

Figure 3

4. (a) Show that the shape function at node i (Ni ), for the simplex triangle is oneand zero at nodes j and k.

(b) The nodal displacements for the simplex two-dimensional element shown figure4bare u1 = 2 mm, u2 = 6 mm, u3 = -1 mm, v1 = 4 mm , v2 = 5 mm and v3 = 8mm. Determine the displacement components at an interior point B (10,10).The nodal coordinates (in mm) are given in parenthesis. [5+11]

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Figure 4b

5. Calculate the temperatures at nodal points in 1-D fin as shown in figure5: [16]

Figure 5




Figure 6

7. Write a short notes on:

(a) Convergence requirements

(b) Mesh preparation for tetrahedral element

(c) Shape Function for tetrahedral element. [6+4+6]

8. (a) Describe C0, C

1, C2 classes.

(b) Write features of the ANSYS, NISA, NASTRAN. [8+8]

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1. Determine the circumference of a circle of radius ‘r’ using the basic principles offinite element method. [16]

2. With a suitable example explain the formulation of finite element equations bydirect approach. Assume suitable data for the example. Use I-D analysis . [16]

3. Define and derive the Hermite shape functions for a two nodded beam element?[16]

4. 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-displacementrelation matrix B and determine the strains εx,εy and γxy. [16]

5. Derive the element conductivity matrix and load vector for solving 1- D heat con-duction problems, if one of the surfaces is exposed to a heat transfer coefficient ofh and ambient temperature of T∞? [16]




Figure 6

7. (a) Write about fully automatic and semi-automatic mesh generation concepts.[4+4]

(b) Explain 3-D eight nodded iso-parametric solid element. [8]

8. Explain the procedure for solving the following 3-D beam problem shown in Figure8 using ANSYS software. [16]

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Figure 8

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rr420302-finite-element-methods.pdf

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