r05410102 finite element methods in civil engineering

Code No: R05410102 Set No. 1

IV B.Tech I Semester Regular Examinations, November 2008FINITE ELEMENT METHODS IN CIVIL ENGINEERING

(Civil Engineering)Time: 3 hours Max Marks: 80

Answer any FIVE QuestionsAll Questions carry equal marks

1. (a) Explain various considerations that are to be taken into account while choosingthe order and type of polynomial-type of interpolation function as a displace-ment model in FEM

(b) What are the different methods available for solving problems of structuralMechanics? name six different engineering applications of FEM. [8+8]

2. (a) What do you mean by axisymmetric loading? Explain

(b) Establish the differential equations of equilibrium for a body subjected to twodimensional stress systems.

[6+10]

3. A two span continuous Beam has each span t=2m and flexural rigidity equal tounity. The beam is simply supported on three rigid unyielding supports. Obtainthe structure stiffness matrix corresponding to the three rotational unrestraineddegrees of freedom after imposing the boundary conditions(figure 3). [16]

Figure 3

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

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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 elementshown in figure 6b. [10+6]

Figure 6b

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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 node numbering scheme influences the matrix sparsity in bandedstiffness matrix.

(b) Evaluate the function = cos pix2between 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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1. (a) Using the principle of virtual displacement derive the expression for the stiff-ness matrix of any element

(b) Discuss the Engineering Applications of Finite element method? [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 stiffness matrix is always symmetric?

(b) Does the determinant of an element stiffness matrix exist? Explain. [8+8]

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

Figure 4

5. Derive the element stiffness matrix for a plane rectangular bilinear element. [16]

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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 different finite element solution techniques, explain briefly one solution tech-nique. [16]

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1. The potential energy for the linear Elastic one dimensional rod shown in figure 1is given by

=1

2

20

AE

[du

dx

]2dx 2u1

whereu1 = u(x = 1)

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

[16]

Figure 1

2. (a) What do you mean by axisymmetric loading? Explain

(b) Establish the differential equations of equilibrium for a body subjected to twodimensional stress systems.

[6+10]

3. (a) State and explain Local coordinate system and global coordinate system withthe examples.

(b) Discuss the necessity for adopting local coordinate System for one dimensionalelements? [10+6]

4. Obtain the global stiffness matrix taking two elements 1 and 2 as beam elementsfor planar structure shown in figure 4. The length of the element 1 may be takenas L, the values of E and I are same for both elements. [16]

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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 infigure 5b. What the element nodal load vector results? [8+8]

Figure 5b

6. Determine the Global stiffness matrix for a thin plate of thickness 10mm subjectedto the surface traction shown in Figure 6. Consider the plate is modeled with twoCST elements. [16]

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Figure 6

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

8. (a) Describe the Gaussian quadrature method.

(b) Evaluate31

dx

xusing Gaussian three point rule and check the answer with the

exact solution. [6+10]

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1. For a simply supported Beam of uniformly distributed load of Intensity Po per unitlength and a concentrated load P at centre, Find the Transverse deflection usingRaleigh-Ritz method of Functional Evaluation and compare the result with exactAnalytical solution.

[16]

2. (a) Derive the equations of equilibrium for two dimensional problems

(b) Determine the stresses x yand xy in the case of plane stress problem if the

strains arex= 10 10

5, y = 7 105 and xy = 0.5 10

4

E = 2.5 105N/m2 and = 0.30.[16]

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

(a) u = a1+a2x

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

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

Figure 4

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

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(b) Force F acts on one edge of the plane bilinear element at y=b/2, as shown infigure 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 elementshown in figure 6b. [10+6]

Figure 6b

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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 method.

(b) Evaluate3

1

dxxusing Gaussian three point rule and check the answer with the

exact solution. [6+10]

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r05410102 finite element methods in civil engineering

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