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IV B.Tech. I Semester Regular Examinations, November 2010

FINITE ELEMENT METHODS IN CIVIL ENGINEERING

(Civil Engineering)

Time: 3 Hours Max Marks: 80

Answer any FIVE Questions

All Questions carry equal marks

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1. What are the various steps involved in finite Element method and explain them

through an Example. [16]

2. (a) What is the basic principle of Rayleigh-Ritz method and explain it in detail.

(b) A cantilever beam of span, L and flexural rigidity EI , is subjected to uniformly

distributed load mw / throughout the span. Determine the deflection and slope at

its free end using ‘Rayleigh -Ritz method”. Compare the deflection with exact

solution and comment. [4 +12 ]

3. (a) What are the “convergence requirements” to be fulfilled in an assumed

displacement function of an element? Discuss them in detail.

(b) What do you mean by an interpolation function? Derive the interpolation

functions for a simple beam element from basic principles. [8+8]

4. A three member truss is loaded as shown in Fig. 1. Assume ��

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L

AE is same for all

the members. Analyze the truss using finite element method and determine the

a) Joint displacements

b) Member forces and tabulate them.

c) Reactions at the supports. [16]

Code No: M0122 /R07 Set No. 1

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θ

5. A stepped column of length 2L, as shown in Fig. 2, is loaded with point loads at its

centre and end. Analyze the column using the Finite Element Method. Assume

areasscareAandA /2 of the column. Length of each stepped is L. Determine

Strains in both top and bottom portion of the column.

Stresses in both top and bottom portion of the column.

Forces in both top and bottom portion of the column.

Reaction. [16]

2.Fig

P

045=θ

1.Fig

A

B

C

P2

P

Code No: M0122/R07

Set No. 1

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6. What is an “Axi-Symmetric problem” and explain how the problem is solved using

Finite element method. [16]

7. A 200 mm x 120 mm rectangular plate (2D) is subjected to an ”in-plane point

load” as shown in 3.Fig . Young’s modulus & poison’s ratio of the plate material

are 3.0&200 == νGPaE respectively. Determine [16]

a) The displacement field (b) Strain vector

(c) Stress vector (d) Reactions. (Explain the procedure only)

8. Write short notes on the following [16]

(a)Weighted residual methods.

(b)Simple elements and Higher order elements

(c)Serendipity elements and Legrangian elements.

(d)Sub-parametric element, isoperimetric element,elemet.

kN5

3.Fig

Code No: M0122/R07 Set No. 1

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(Civil Engineering)




*******

1. Compare and contrast the “Rayleigh-Ritz method” with the “Finite element

method” and comment on both the methods. [16]

2. (a) Explain the terms – “Local axis and Global axis” in the context of Finite

Element Method.

(b) Derive “shape functions” of a simple triangular element in terms of global

axes. If coordinates of vertices of the triangular element are

( ) ( ) ( ),,,,,, 111111 yxCandyxByxA [4+12]

3. A structural member, of length, 2L is subjected to axial loads at its centre and

end as shown in 1.Fig . Assume AE is constant throughout. Analyze it using

finite element method and determine the following

(a) Deflections at the load points.

(b) Strains & Stresses in the bar structure.

(c) Forces in the two metals.

(d) Reaction at the support. [16]


P P2

1.Fig

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4. (a) State and explain the principle of minimum potential energy and explain.

(b) Write Finite element formulation for a 2 Dimensional structure. [6+10]

5. (a) Derive equilibrium equations for 2 D solid mechanics problems.

(b) Explain “plane stress problem and plane strain problem”. Give examples for

each case. [8+8]

6. A simply supported beam of span L, is loaded with uniformly distributed load,

w/m throughout the span. its flexural rigidity, EI is constant. The beam is

propped at its centre with an axial spring. The stiffness of the spring is “k”

Explain the finite element procedure in solving the problem. [16]

7. (a) What are the advantages and disadvantage of finite element method?

(b) Derive materials stiffness matrix for plane stress problem. [16]

8. Write short notes on the following.

(a) Degree of freedom and boundary conditions.

(b) How temperature effect is taken in to account in the finite element

procedure.

(c) Static condensation.

(d) Guassian-quadrature. [16]

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(Civil Engineering)




*******

1. What are the various approximate methods of analysis and explain them. [16]

2. (a) Derive equilibrium equations for 2D elastic solid mechanics problem.

(b) Strains at a point in plate under in-plane loading are

455 103,106,1010 −−− === xandxx xyyyxx γεε . For plane stress case, determine

stresses xyyyxx and τσσ ,, at the same point. [8+8]

3. (a) What are the “convergence requirements” to be fulfilled in an assumed

displacement function of an element? Discuss them in detail.

(b) What are the properties of shape functions? Using the properties of the shape

functions, derive the shape functions of a four noded bar element. [8+8]

4. (a) What do you mean by plane stress and plain strain solid mechanics problems

and explain them with examples?

(b) Derive the material stiffness matrix for plane stress case. [8+8]

5. Explain how stiffness matrix and load vector of a three noded triangular element

using shape functions. [16]

6. (a) What do you mean by an isoparametric element? Explain its significance.

(b) Derive the interpolation functions of a simple quadrilateral isoparametric

element. [8+8]


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7. Explain how axi-symmetric problem are analyzed using finite element method?

[16]

8. Write short notes on the following [16]

(a) Numerical Integration

(b) Static condensation

(c) Area and volume coordinates

(d) Legrangian element

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(Civil Engineering)




*******

1. (a) Explain the advantages and disadvantages of Finite Element Method.

(b) What is meant by total potential of elastic structure? Write the

expression for total potential of a cantilever beam with uniformly

distributed load though out the span. [6+10]

2. A simply supported beam is subjected to lineally varying distributed load with zero

intensity at one end and w/m at other end. Analyze the beam using Raleigh –Ritz

method and find the following at centre of the beam.

(a) Deflection

(b) Slope

(c) Shear force

(d) Bending moment [4+4+4+4]

3. (a) Derive equations of equilibrium for a 2D elastic structure under external loading.

(b) Write constitutive relationships for plane stress and plane strain

cases. [8+8]

4. Explain how stiffness matrix and nodal load vector are developed from shape

functions through an example. [16]


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5. A fixed beam is loaded with uniformly distributed load of intensity w/m. Assume EI

is constant throughout. Analyze the beam by dividing it into two elements and find

the following at quarter span.

(a) Deflection

(b) Slope

(c) Shear force

(d) ending moment [4+4+4+4]

6. (a) Explain what is meant by an isoparametric element and its significance.

(b) What is meant by constant strain triangle (CST) element and explain it? Write

shape functions of CST in terms isoparametric element. [6+10]

7. What do you mean by axi-symmetric problem. Explain how axi-symmetric problems

are solved using finite element method. [16]

8. Write a short note on the following [16]

(a) Simple element and Higher order element

(b) Legrangian element serendipity element

(c) Assembling of elements’ stiffness matrices in to global stiffness matrix

(d) Static condensation.


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