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^-2mm. (10 marks) 4 (a) An axial load P=300× 10³ 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.
E₁=70×10⁹ N/m², E₂=200×10⁹N/m²
A₁=900mm², A₂=1200mm²
α₁=23×10^-6/°C, α₂=11.7×10^-6/°C (12 marks) 4 (b) Solve the following system of equation of Gaussian-Elimination method
x₁-2x₂+6x₃=0
2x₁+2x₂+3x₃=3
-x₁+3x₂=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=600mm² 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-m² 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/m². 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)