Deemed University 2010 M.Tech Mechanical Engineering University: Lingayas University Term: III Title of the : Finite Element Methods - Question Paper

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Lingayas University, Faridabad

M.Tech (Part-Time) Mechanical Engg.

Term-III Examination May, 2010

Finite Element Methods (ME -504)

[Time: 3 Hours] [Max. Marks: 100]

 

Before answering the question, candidate should ensure that they have been supplied the correct and complete question paper. No complaint in this regard, will be entertained after examination.

 

Note: Question No. 1 Section A is compulsory. Attempt any two questions from Section B and any two questions from Section C. In all attempt five questions.

Section A

Q-1 (i) What is the difference between plane stress and plane strain problems? (4)

(ii) Explain the convergence criterion used in FEM. (4)

(iii) Derive the shape functions for a Beam element. (4)

(iv) Explain P refinement & H-refinement in FEM

(v) Explain in detail steps involved in analysis of a typical problem using FEM. (4)

Section B

Q-2. Develop (a) the weak form and (b) the finite element model for the governing differential equation of a Euler Bernouli Beam.

For O< x < L

Assume the Boundary conditions. Use Galerkin method. (20)

 

 

Q-3. Derive the stiffness matrices for

(a) Bar (8)

(b) Beam (12)

Using shape functions.

Q-4. For the two-bar truss shown in the figure (1) determine the displacements of node 1 and the stress in element 1-3. (20)

Section C

Q-5.Use the minimum number of Euler Bernouli beam finite elements to analyze the beam problem shown in Fig. (2) Obtain

(a) The assembled stiffness matrix and force vector.

(b) The condensed matrix equations for the primary unknowns. (20)

Q-6. An axail load P = 300 x 10³ N is applied to the rod as shown in figure (3) obtain

(a) The assembled stiffness matrix and force vector.

(b) The nodal displacements and element stiffness. (20)

Q-7. (a) Determine the Jacobian matrix and the transformation equations for the element given in figure (4) (10)

(b) Derive the shape functions (Interpolation) for a 9 Node square element. (10)

 

Q-8. (a) Using one and two Gauss point numerical integration formulae, evaluate the following integral

I = (12)

Compare the results with exact integration (refer to Table1).

(b) Derive the shape (Interpolation) functions for a 4 node bar element. (8)