## 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=60×10^{3}N 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=250mm^{2}; 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=200m^{2}; 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=4×10^{6} mm^{4}.
(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)).

K_{1}=20W/m°C; k_{2}=30W/m°C; k_{3}; h=25W/m^{2}°C;T_{∞}=800°C
(14 marks)