Set No. 1

Code No: RR410104

in

IV B.Tech I Semester Supplimentary Examinations, May/Jun 2009 FINITE ELEMENT METHODS (Civil Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ⋆⋆⋆⋆⋆ 1. Discuss in detail the general steps involved in Finite Element Methods.

ld .

2. (a) What do you understand by Functional approximation?

[16]

(b) Explain how the Raleigh-Ritz method can be used to approximate the solution of a problem. [6+10] 3. (a) Explain the types of forces that are encountered by a deformable body?

or

(b) Derive the differential equations of equilibrium for a body subjected to two dimensional stress system. [4+12] 4. (a) Compare and contrast Element Stiffness Matrix and structure stiffness matrix. (b) Does the determinant exists for the above matrices? Explain.

[12+4]

nt

uW

5. For the three stepped bar shown in figure 5.

Determine

Figure 5

(a) The displacements at Nodes 2 and 3

Aj

(b) Stresses in three sections (c) Reactions at the ends.

[16]

6. Derive the element stiffness matrix for a three Node Triangular element.

[16]

7. Show that Volume Coordinates are Related to shape functions for a Four Node quadrilateral element. [16] 8. Write short notes on: (a) Lagrangian Elements 1 of 2

Set No. 1

Code No: RR410104 (b) Serendipity Elements.

[8+8]

Aj

nt

uW

or

ld .

in

⋆⋆⋆⋆⋆

2 of 2

Set No. 2

Code No: RR410104

1. (a) Enumerate the Applications of Finite Element Method?

in

IV B.Tech I Semester Supplimentary Examinations, May/Jun 2009 FINITE ELEMENT METHODS (Civil Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ⋆⋆⋆⋆⋆

ld .

(b) Discuss the commercial packages (P.C.programs) available today for solving problems by FEM. [8+8] 2. (a) What do you understand by Functional approximation?

(b) Explain how the Raleigh-Ritz method can be used to approximate the solution of a problem. [6+10]

or

3. Develop the stress-strain relationships for a two dimensional Isotropic body in equilibrium. [16] 4. (a) Prove that structure stiffness matrix is always symmetric?

(b) Does the determinant of an element stiffness matrix exist? Explain.

[8+8]

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5. (a) Explain the static condensation of stiffness matrix.

(b) Formulate the condensed stiffness matrix 2×2 from the stiffness matrix 3×3 given below. [6+10]  12 −6 6 [K] =  −6 4 2  −6 2 4 

nt

6. Derive the strain-displacment [B] matrix for plane stress analysis using a Four Noded Rectangular element. [16]

Aj

7. Show that Area Coordinate are related to shape functions for a three Node Triangular element and that area Coordinates lead to a different but equivalent formulation for the shape function. [16] 8. Evaluate the

(a) Jacobian [J] matrix

(b) Geometric matrix for Isoparametric quadrilateral element shown in figure 8 for plane stress conditon. [16] Take E=200Gpa, υ=0.3, t=25mm.

1 of 2

in

Set No. 2

Code No: RR410104

Figure 8

Aj

nt

uW

or

ld .

⋆⋆⋆⋆⋆

2 of 2

Set No. 3

Code No: RR410104

in

IV B.Tech I Semester Supplimentary Examinations, May/Jun 2009 FINITE ELEMENT METHODS (Civil Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ⋆⋆⋆⋆⋆ 1. Discuss in detail the general steps involved in Finite Element Methods.

[16]

or

ld .

2. A uniform rod fixed at one end and free at the other end is subjected to a uniform axial load qo per metre run as shown in figure 2. Find the value of stress at any point and compare with exact result. Use two generalized coordinates (Ritz parameters). Comment on the result. [16]

Figure 2

uW

3. For a state of strain, if a displacement field is described by u=(-x2 + 2y 2 + 6xy)10−4 v= (3x+6y-y 2 )10−4 Determine the εx , εy and yxy at the point x=1, y=0 also find the stresses σx σy and τxy , if E=200 Gpa, υ =0.30. [16]

Aj

nt

4. A Two span continuous Beam has each span l=1m and flexural rigidity equal to unity. The Beam is simply supported on three rigid unyielding supports. Obtain the structure stiffness matrix corresponding to the three rotational unrestrained degrees of freedom after imposing the boundary conditions. as shown in figure 4 [16]

Figure 4

5. Using the Finite Element procedure with Two axial elements, Obtain the stress value at salient points for the stepped bar shown in figure 5. [16]

1 of 2

Set No. 3

Code No: RR410104

in

Figure 5

or

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6. For a three Noded Triangular element shown in figure 6. Derive the shape functions in terms of global coordinates. [16]

Figure 6

nt

uW

7. For the two element plate shown in figure 7. Determine the B-Matrices for the two elements. [16]

Figure 7

Aj

8. (a) What is meant by an Isoparametric Element? (b) Mention the Advantages of Isoparametric formulation. (c) Differentiate between local and global Coordinates? ⋆⋆⋆⋆⋆

2 of 2

[4+6+6]

Set No. 4

Code No: RR410104

in

IV B.Tech I Semester Supplimentary Examinations, May/Jun 2009 FINITE ELEMENT METHODS (Civil Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ⋆⋆⋆⋆⋆ 1. Discuss in detail the general steps involved in Finite Element Methods.

[16]

ld .

2. A uniform axial rod of length “L” is clamped at both ends. It can Resist only axial loads. The load varies uniformly with qo per metre run. Assuming one generalized coordinate (one Ritz paramenter). Establish the displacement field, also find the stresses at salient points. Exploiting the symmetry, show that your answer is the correct one. [16]

or

3. Develop stress-strain relationship for Axisymmetric solids subjected to axially symmetric loading. [16]

nt

uW

4. A Two span continuous Beam has each span l=1m and flexural rigidity equal to unity. The Beam is simply supported on three rigid unyielding supports. Obtain the structure stiffness matrix corresponding to the three rotational unrestrained degrees of freedom after imposing the boundary conditions. as shown in figure 4 [16]

Figure 4

5. (a) Generate the shape functions for the Two Noded one dimensional element in terms of Natural Coordinate system.

Aj

(b) Describe the characteristics of shape functions.

[10+6]

6. For a three Noded Triangular element shown in figure 6. Derive the shape functions in terms of global coordinates. [16]

1 of 2

in

Set No. 4

Code No: RR410104

Figure 6

ld .

7. Compare and contrast the area coordinates and volume coordinates is plane elasticity problems. [16]

or

8. Express the displacement field interms of Nodal displacement and express it in Matrix form for a 8 Node quadrilateral Isoparametric Element. [16]

Aj

nt

uW

⋆⋆⋆⋆⋆

2 of 2