Question Paper: Finite Element Methods : Question Paper Jun 2014 - Mechanical Engineering (Semester 6) | Visveswaraya Technological University (VTU)
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## Finite Element Methods - Jun 2014

### Mechanical Engg. (Semester 6)

TOTAL MARKS: 100
TOTAL TIME: 3 HOURS
(1) Question 1 is compulsory.
(2) Attempt any four from the remaining questions.
(3) Assume data wherever required.
(4) Figures to the right indicate full marks.
1 (a) What is Fem? Sketch the different types of elements used based on geometry in finite element analysis(1D,2D,and 3D).(4 marks) 1 (b) Explain with a sketch plane stress and plane strain.(6 marks) 1 (C) Derive the equilibrium equation in elasticity subjected to body force and traction force and traction force(10 marks) 2 (a) A cantilever beam of span 'L' is subjected to a point load at free end. Derive an equation for the deflection at free end by using RR method. Assume polynomial displacement function.(10 marks) 2 (b) Write the properties of stiffness matrix and derive the element stiffness matrix(ESM) for a 1D bar element.(10 marks) 3 (a) A modal co-ordinate of the triangular element is shown in Fig Q3(a). at the interior point 'P' the co-ordinate is 3.3 and N1=0.3 Determine 'N2'' and the y co-ordinate at point P. (5 marks) 3 (b) What is convergence requirement? Discuss the 3 conditions of convergence requirement.(5 marks) 3 (c) Derive the shape function of a 4 noded quadrilateral element.(10 marks) 4 (a) Consider the bar shown in FigQ4(a). using elimination method of handling boundary conditions. Determine the following:
i) Nodal displacements
ii)Stress in each element.
iii) Reaction forces
Take E=200GPa.
(10 marks)
4 (b) Consider the bar shown in figQ4(b).An axial load P=60×103N is applied at its midpoint. Using penalty method of handling boundary condition. Determine i) Nodal displacements; ii) Stress in each element; iii) Reaction at supports. Take A=250mm2; E=200GPa. (10 marks) 5 (a) Derive the Shape Function for a quadratic bar element using Lagrange's interpolation.(5 marks) 5 (b) Evaluate $$I=\int_{-1}^{+1}\left ( 3e^{\xi } +\xi ^{2}+\frac{1}{\xi +2}\right )d\xi$$using 1P and 2P Gaussian quadrature.(6 marks) 5 (c) Derive 1 arange quadratic quadrilateral (elements)(9 marks) 6 (a) List out the assumptions made in the derivation of truss element.(4 marks) 6 (b) For thr truss shown in Fig Q6(b), determine
i) Nodal displacement; ii) Stress in each element iii) Reaction supports.
A=200m2; E-70GPa.
(16 marks)
7 (a) Derive the Hermine shape function of a n beam element(8 marks) 7 (b) For the beam and loading shown in fig Q7(b) determine
i) the slopes at 2 and 3 and ii) the vertical deflection at the midpoints of the distributed load. Take E=200 Gpa, I=4×106 mm4.
(12 marks)
8 (a) Bring out the differences between continuum methods and FEM.(6 marks) 8 (b) Solve the temperature distribution in the composite wall using 1D heat elements, use penalty approach of handling boundary conditions. (Fig Q8(b)).
K1=20W/m°C; k2=30W/m°C; k3; h=25W/m2°C;T=800°C
(14 marks)