## Finite Element Methods - Dec 2013

### 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)** Differentiate between plane stress and plain strain problem with examples. Write the stress strain relation for both .(8 marks)
**1 (b)** Explain the node numbering scheme and its effect on the half band-width(6 marks)
**1 (c)** List down the basic steps involved in FEM for stress analysis of elastic solid bodies.(6 marks)
**2 (a)** State the principle of minimum potential energy. Determine the displacement at nodes for the spring system shown in fig Q2(a)
(8 marks)
**2 (b)** Determine the deflection of a cantilever beam of length 'L' subjected to uniformly distributed load (UDL) of P unit length, using the trail function $$y=a \sin \left (\dfrac{\pi x}{21} \right )$$. compare the result with analytical solution and comment on occupancy.(12 marks)
**3 (a)** Derive an expression for Jacobian matrix for a four-noded quadrilateral element.(10 marks)
**3 (b)** For the triangular element shown in Fig Q3(b). Obtain the strain-displacement matrix 'B' and determine the strains ε_{x}; ε_{y}; and γ_{xy}.

Nodal displacement {q}={2 1 1 -4 -3 7} × 10^{-2}mm.
(10 marks)
**4 (a)** An axial load P=300× 10^{3} N is applied at 20°C to the rod as shown in the FigQ4(a) the temperature is then raised to 60°C.

i) Assemble the global stiffness matrix(K) and global load vector(F)

ii)Determine the nodal displacement and element stresses.

E_{1}=70×10^{9} N/m^{2}, E_{2}=200×10^{9}N/m^{2}

A_{1}=900mm^{2}, A_{2}=1200mm^{2}

α_{1}=23×10^{-6}/°C, α_{2}=11.7×10^{-6}/°C
(12 marks)
**4 (b)** Solve the following system of equation of Gaussian-Elimination method

x_{1}-2x_{2}+6x_{3}=0

2x_{1}+2x_{2}+3x_{3}=3

-x_{1}+3x_{2}=2(8 marks)
**5 (a)** Using Lagrangian method, derive the shape function of three-noded one-dimension (1D) element [quadratic element](6 marks)
**5 (b)** Evaluate $$I=\int_{-1}^{+1}\left ( 3e^{x } +x ^{2}+\frac{1}{x +2}\right )dx$$ using one-point and two-point Gaussian quadrature.(6 marks)
**5 (c)** Write short notes on higher order element used in FEM.(8 marks)
**6 (a)** For the two bar truss shown in the figQ6(a). determine the nodal displacement and element stresses. A force of P=1000kN is applied at node 1. take E=210 Gpa and A=600mm^{2} for each element.
(12 marks)
**6 (b)** Derive an expression for stiffness matrix for a 2-D truss element(8 marks)
**7 (a)** Derive the Hermine shape function of a n beam element(8 marks)
**7 (b)** A simply supported beam of span 6m and uniform flexural rigidity EI=40000 kN-m^{2} is subjected to clockwise couple of 300 kN-m at a distance of 4m from the left end as shown in the Fig Q7(b). find the deflection at the point of application of the couple and internal loads.
(12 marks)
**8 (a)** Find the temperature distribution and heat transfer through an iron fin of thickness 5mm. Height 50mm and width 100mm. The heat transfer coefficient around the fin is 10 W/m^{2}. K and ambient temperature is 28°C. The base of fin is at 108°C. Take k=50 W/mK. Use two element.
(10 marks)
**8 (b)** Derive element matrices for heat conduction in one-dimension element using Galerkin's approach.(10 marks)