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1 Set No. 1 IV B.Tech. I Semester Regular Examinations, November 2010 FINITE ELEMENT METHODS (Mechanical Engineering) Time: 3 Hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ******* 1. In a plane strain problem, we have σ = Pa, σ = Pa E = ν = Determine the value of the stress. x 6 6 y Pa, Consider the bar in Fig.1. Cross-sectional area A e = cm 2, and Young s modulus E = Pa. If q 1 = 0.5mm and q 2 =0.635 mm, determine the following (by hand calculation): a) The displacement at point P, b) The strain and stress, c) The element stiffness matrix 3. Find the deflection at the load and the slopes at the ends for the steel shaft shown in Fig.2. Consider the shaft to be simply supported at bearings A and B. 1of 3

2 Set No For the triangular element shown in Fig. 3, obtain the strain-displacement relation matrix B and determine the strains and Fig The open-ended steel cylinder shown in Fig. 4 is subjected to an internal pressure of 1MPa. Find the deformed shape and the distribution of principal stresses. Fig Fig. 5 shows a four-node quadrilateral. The (x,y) coordinates of each node are given in the figure. The element displacement vector q is given as q = [0, 0, 0.20, 0, 0.15, 0.10, 0, 0.05] T Fig. 5 2 of 3

3 Set No. 1 Find the following: (a) the x-,y-coordinates of a point P whose location in the master element is given byξ = 0.5 and η = 0.5 and (b) the u, υ displacements of the point P. 7. Consider the shaft with a rectangular cross section shown in Fig.6. Determine, in terms of M and G, the angle of twist per unit length. Fig Consider axial vibration of the steel bar shown in Fig.7. (a) Develop the global stiffness and mass matrices. (b) By hand calculations, determine the lowest natural frequency and mode shape. Fig. 7 3 of 3

4 Set No. 2 IV B.Tech. I Semester Regular Examinations, November 2010 FINITE ELEMENT METHODS (Mechanical Engineering) Time: 3 Hours Answer any FIVE Questions All Questions carry equal marks ******* 1. If a displacement field is described by u = ( x + 2y Determine at the point x = 1, y = A finite element solution using one-dimensional, two-noded elements has been obtained for a rod as shown in Fig xy)10 v = (3x + 6y y ) Fig Displacements are as follows: Q = [ -0.2, 0, 0.6, -0.1] T mm, E = 1 N/mm 2, Area of each element = 1 mm 2, L 1-2 = 50 mm, L 2-3 = 80 mm, L 3-4 = 1000 mm. (a) Determine the B matrix for element 2-3. (b) Determine the strain energy in element 1-2 using U = T kq. 3. A three-span beam is shown in Fig. 2. Determine the deflection curve of the beam and evaluate the reactions at the supports. Fig. 2 1of 3 Max Marks: 80

5 Set No For a two-dimensional triangular element, the stress-displacement matrix DB appearing in = DBq is given by DB = N/mm If the coefficient of linear expansion is / 0 C, the temperature rise of the element is C, and the volume of the element is 25mm 3, determine the equivalent temperature load for the element. 5. Find the deformed configuration c and the stress distribution in the walls of the closed cylinder shown in Fig Using a 2 2 rule,, evaluate the integral A 2 2 ( x + xy ) ) dx dy Fig. 3 by Gaussian quadrature, where A denotes the region shown in Fig. 4. For 2 2 Gauss 1 rule sampling pointss are ± and the weights are unity. 3 Fig. 4 2 of 3

6 Set No Consider a brick wall (Fig. 5) of thickness L = 30 cm, k = 0.7 W/ /m 0 C. The inner surface is at 28 0 C and the outer surface is exposed to cold air at C. The heattransfer coefficient associated with the outside surface is h = 40 W/m 2. C. Determine the steady state temperature distribution within the wall and also the heat flux through the wall. Use a two-element model, and obtain the solution by hand calculations. Assume one-dimensional flow. 8. Determine all natural frequencies of the steel cantilever beam shown in Fig.6 using one-element model. 600 mm Fig. 6 3 of 3 Fig mm 20 mm

7 Set No. 3 IV B.Tech. I Semester Regular Examinations, November 2010 FINITE ELEMENT METHODS (Mechanical Engineering) Time: 3 Hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ******* 1. Develop a deformation field u(x,y), v(x,y) that describes the deformation of the finite element shown in fig. 1 From this determine (0, 10) (0, 0) (5, 10) (5, 10) (10, 10) (0, 10) (4, 0) Fig Consider the bar in Fig. 2 loaded as shown. Determine the nodal displacements, element stresses, and support reactions. Solve this problem by hand calculation, adopting the elimination method for handling boundary conditions. Fig. 2 1of 3 (9, 0)

8 Code No: M03222 /R07 Set No For the beam and loading shown in Fig.3, determine (a) the slopess at nodes 2 and 3 and (b) the vertical deflection at the midpoint of the distributed load. 4. For the configuration shown in Fig. 4, determine the deflection at the point of load application using a one-element model. If a mesh of several triangular elements is used, comment on the stress values in the elements close to the rip. 5. Determine the diameters after deformation and the distribution of principal stresses along the radius of the infinite cylinder subjected to internal pressure as shown in Fig.5. Fig (a) Derive the sampling points and weights for the 2 nd order gauss rule. (b) Supply the order of gauss rule to evaluate integrals of a cubic and quadratic polynomials respectively. 2 of 3 Fig. 3 Fig. 4

9 Set No Heat is entering into a large plate at the rate of q 0 = - 300W/m 2 as in Fig. 6. The plate is 25mm thick. The outside surface of the plate is maintained at a temperature of 10 0 C. Using two finite elements, solve for the vector of nodal temperatures T. Thermal conductivity k = 1.0 W/m. 0 C. 8. Determine all natural frequencies of the simply supported beam shown in Fig. 7 using one-element model. Fig. 7 3 of 3 Fig. 6

10 IV B.Tech. I Semester Regular Examinations, November 2010 FINITE ELEMENT METHODS (Mechanical Engineering) Time: 3 Hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ******* 1. A displacement fieldd is imposed on the square element shown in Fig u = 1+ 3x + 4x + 6xy v = xy 7x 2 (-1,-1) (a) Write down the expressions for and (b) Find where is a maximum within the square. 2. Consider the bar in Fig.2. Determine the nodal displacements, element stresses, and support reactions. Fig. 2 1 of 3 y Fig. 1 (1,1) Set No. 4 x

11 Set No Determine the displacements and rotations of the joints for the portal frame shown in Fig. 3. Fig Solve the plane stress problem in Fig.4 using triangular element CST. Compare your deformation and stress results with values obtained from elementary beam theory. Fig The steel sleeve of internal diameter 7.5 cm, is press fitted onto a rigid shaft of diameter cm, as shown in Fig.5. Determine (a) the outer diameter of the sleeve after fitting and (b) the stress distribution. Estimate the contact pressure by interpolating the radial stress in the neighboring elements. 7.5cm 10cm 2 of 3 2.5cm 7.525cm Rigid Shaft E = x10 6 pa υ = 0.3 Elastic sleeve Fig.5

12 Set No Consider a rectangular element as shown in Fig.6. Assume plane stress condition, E = MPa, υ = 0.3, and q = [0, 0, 0.05, 0.075, 0.15,0.8,0,0] cm. Evaluate Jacobian J, B, and σ at ξ = 0 and η = 0. Y 4(0,2.5) 1(0,0) C(2.5,1.25 Fig. 6 3(5,2.5) 2(5,0) X 7. A point P is locatedd inside the triangle as shown in Fig. 7. Assuming a linear distribution, determine the temperature at P. Coordinates of the various points are given in the table. 8. Determine all natural frequencies of the simply supported beam shown in Fig.8 using two-element model. Fig. 8 3 of 3 Fig. 7 3 of 3