Set No. 1
Code No: R05311402 R05311402
III B.Tec B.Tech h I Semest Semester er Regula Regularr Examin Examinati ations ons,, Nov Novem ember ber 2007 2007 FINITE ELEMENT METHODS (Mechatronics) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ⋆⋆⋆⋆⋆
1. Using the general approach of displacemen displacementt function, derive the force-displac force-displacement ement rel relatio ations nshi hip p an and d elem elemen entt sti stiffn ffnes esss matr matriix. for for a trus trusss ba barr elem elemen ent. t. [16] [16] 2. Derive stiffness equations for a bar element from the one dimensional second order equation by variated approach. [16] 3. The members (1) and (2) are circular in cross section with diameters of 10 cm and 20 cm respective respectively ly.. Determi Determine ne the displa displaceme cement nt at the node where where load is acting.{As shown in the Figure3} [16]
Figure 3 4. Consider a beam with uniform distributed load as shown in the figure4. Estimate the deflection at the centre of the beam. E = 200 Gpa ; A = 25 mm ×0 25 mm. [16]
Figure 4 5. (a) Derive Derive strain strain displac displacemen ementt [B] matrix matrix for a 3 noded Trian Triangul gular ar element? element? (b) The nodal coordinat coordinates es and the nodal displacem displacemen ents ts of a triangul triangular ar elemen element, t, under a specific load condition are given below: Xi = 0, Yi = 0, X j = 1 mm , Y j = 3 mm, Xk = 4 mm, Yk = 1, u1 =1 mm, u2 = -0.05 mm, u3 = 2 mm, v1 = 0.5 mm , v2 = 1.5 mm and v3 = -1 mm. If E = 2 × 105 N/mm2 and µ = 0.3. 0.3. fin find d the the stre stress sses es in the the elem elemen ent. t. [8+8 [8+8]] 1 of 2
Set No. 1
Code No: R05311402 R05311402
6. Derive the element stiffness matrix for torsion element in terms of modulus of rigidity, po pollar moment of inertia and the length of the shaft? [16] 7. Determine the natural frequencies of a simply supported beam of length 800 mm with the cross sectional area of 75 cm × 25 cm. Take E= 200 Gpa and density of 7850 kg/m3 . {As shown in the Figure7} [16]
Figure 7 8. (a) Explai Explain n the converge convergence nce criteri criteriaa in finite elemen elementt analysis analysis.. (b) Write about pre-processor, processor, processor, and post-process p ost-processor or in any FEM software. software. [8+8] ⋆⋆⋆⋆⋆
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Set No. 2
Code No: R05311402 R05311402
III B.Tec B.Tech h I Semest Semester er Regula Regularr Examin Examinati ations ons,, Nov Novem ember ber 2007 2007 FINITE ELEMENT METHODS (Mechatronics) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ⋆ ⋆ ⋆ ⋆ ⋆
1. Determine the circumference of a circle of radius ‘r’ using the basic principles of finite element metho d. [16] 2. With a suitable example explain the formulation of finite element equations by direct direct approac approach. h. Assume Assume suitabl suitablee data for the example example.. Use I-D analysi analysiss [16] [16] 3. Estimate the displacement vector, stresses and reactions for the truss structure as shown below Figure3:
Figure 3 Note: - Area is not given and assumed as A( ) = 1mm2 ‘E’ is not given. Assumed as E=2×105 N/mm2 [16] e
4. Define and derive the Hermite shape functions for a two nodded beam element? [16] 5. Derive the shape functions for a triangular element in global coordinator system. [16] 6. Derive the element conductivity matrix and load vector for solving 1-D heat conduction problems, if one of the surfaces is exposed to a heat transfer coefficient of h and ambient temperature of T∞? [16] 7. Explain the following with examples. (a) Lumped Lumped parameter parameter model. (b) Consistant mass matrix mo del.
[8+8]
8. (a) Sketc Sketch h any three 3-D structu structural ral elemen elementt showing showing their their degrees degrees of freedom freedoms. s. (b) Deri Derive ve the shape shape fun functi ction on of of any any one one of the the 3-D 3-D stru structu ctura rall elem elemen ent. t. [8+8 [8+8]] ⋆ ⋆ ⋆ ⋆ ⋆
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Set No. 3
Code No: R05311402 R05311402
III B.Tec B.Tech h I Semest Semester er Regula Regularr Examin Examinati ations ons,, Nov Novem ember ber 2007 2007 FINITE ELEMENT METHODS (Mechatronics) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ⋆ ⋆ ⋆ ⋆ ⋆
1. (a) Explain Explain the signific significance ance of node numbering numbering and element element numbering numbering during during the discretization discretization process. (b) (b) Explain the na natu turral and geom geomeetri tric bo bou unda darry con ond ditions. ns.
[8+8 8+8]
2. With a suitable example explain the formulation of finite element equations by direct direct approac approach. h. Assume Assume suitabl suitablee data for the example example.. Use I-D analysi analysiss [16] [16] 3. Consider the truss element with the coordinates i(10,10) & q(50,40) If the displacement vector is q=[15 10 21 43]T mm, then determine (a) The trace trace vector vector F (b) Stress Stress in each element element (c) Stiffnes Stiffnesss matrix if E= 70 GPA GPA and A= 200 mm2 .
[6+4+6]
4. Estimate the stiffness matrix and the deflection at the center of the simply supported ported beam of length length 3 m. A 50 kN of load load is actin actingg at the center center of the beam. beam. 3 2 Take EI = 800 × 10 N-m . [16] 5. (a) Discuss Discuss the signifi significanc cancee and applicati applications ons of triangul triangular ar elements elements.. (b) Two dimensional dimensional simplex simplex elements are used to find the pressure distribution distribution in a fluid fluid medi medium um.. The (x, y) coordinat coordinates es of nodes nodes i, j and and k of an elemen elementt are given given by (2, 4), (4, 0) and (2, 6) respective respectively ly.. Find Find the shape function functionss Ni , N j and Nk of the element. [10+6] 6. Compute the elemental conductivity matrix and load vector for the 2-D triangular element as shown in figure6. The faces 1-3 and 2-3 are exposed to a convection and there is an internal heat generation of 50 W/cm3 . Assume thermal conductivity is 60 W/m K. [16]
Figure 6 1 of 2
Set No. 3
Code No: R05311402 R05311402
7. Derive Derive the elemental jumped and consistant consistant mass matrices for 1-D bar element and 1-D plane truss element? [16] 8. Write a procedure for model creation and mesh generation generation for aerofoil shape turbine blades. [16] ⋆ ⋆ ⋆ ⋆ ⋆
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Set No. 4
Code No: R05311402 R05311402
III B.Tec B.Tech h I Semest Semester er Regula Regularr Examin Examinati ations ons,, Nov Novem ember ber 2007 2007 FINITE ELEMENT METHODS (Mechatronics) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ⋆ ⋆ ⋆ ⋆ ⋆
1. With a help of a neat block diagram, diagram, explain the model based simul simulatio ation n process of finite element method. [4+12] 2. With a suitable example explain the formulation of finite element equations by direct direct approa approach ch.. Assume suitab suitable le data data for for the exampl example. e. Use I-D analys analysis is [16] [16] 3. Determine Determine the stiffness matrix, matrix, stresses stresses and reactions reactions in the truss structure shown shown in Figure3:[16]
Figure 3 4. A cant cantile ileve verr beam of 1 m length length carries carries a single single point point verti vertica call load load at the end of the beam of 10 kN. Calculate the deflection at the end of the beam using FEM, if E = 70 Gpa, A=500 mm 2 and I = 2500 mm4 . [16] 5. Explain Explain in detail detail how the element element stiffness stiffness matrix and load vector vector are evaluated evaluated in isoparametric formulations. [16] 6. Derive the element stiffness matrix for torsion element in terms of modulus of rigidity, pol polar moment of inertia and the lengt ngth of the shaft? [16] 7. Find the natural frequencie frequenciess and the corresponding corresponding mode shapes for the longitudilongitudinal vibrations for the stepped bar. Assume A 1 = 2A and A2 = A ;I1 = I2 = I & ; E1 = E2 = E. [8+8] 8. Explain Explain the Topology Topology decomposition decomposition approach approach and Node connectio connection n approac approach h for mesh generation [16] ⋆ ⋆ ⋆ ⋆ ⋆
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