UNIVERSITY OF BOLTON RAK ACADEMIC CENTRE BENG(HONS) MECHANICAL ENGINEERING SEMESTER TWO EXAMINATION 2017/2018 FINITE ELEMENT AND DIFFERENCE SOLUTIONS
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1 OCD0 UNIVERSITY OF BOLTON RAK ACADEMIC CENTRE BENG(HONS) MECHANICAL ENGINEERING SEMESTER TWO EXAMINATION 07/08 FINITE ELEMENT AND DIFFERENCE SOLUTIONS MODULE NO. AME6006 Date: Wednesda 0 Ma 08 Tme: 0:00 :00 INSTRUCTIONS TO CANDIDATES: Tere are FIVE questons on te paper. Answer an FOUR questons All questons carr equal marks. Marks for parts of questons are sown n brackets. Electronc calculators ma be used provded tat data and program storage memor s erased or cleared pror to te examnaton. CANDIDATES REQUIRE: A Formula Seet (attaced)
2 Page of 9 Q. Te rectangular beam sown below n Fgure Q s rgdl fxed at end A and smpl supported at end B. Pont load P s appled at L from support A along te lengt of te beam. Te beam span AB s L n total. A Note B s a smple roller support, wlst end A s bult n. Te followng data s gven: Young s modulus, E = 00 GNm -, Moment of nerta, I = 0 x 0-9 m, Beam lengt, L =.6 m and Load, P = 0 kn In answerng te questons below, ou sould splt te beam nto four equal sectons and use te Fnte Dfference metod of soluton, were: d dx d dx d dx d dx L L Fgure Q, Beam 6 a) State te Boundar Condtons for te beam. ( marks) b) Establs te Bendng Moment equatons for eac node on te beam. (6 marks) c) Establs te Fnte Dfference equatons for eac node. (7 marks) d) Determne te value of te reacton (R B )at te smple support B (6 marks) e) Determne te deflecton at te md-pont of te beam. ( marks) P B Total 5 marks Please turn te page
3 Page of 9 Q Te plane wall sown n Fgure Q below s m tck. Te left surface of te wall (x=0) s mantaned at a constant temperature of 00 o C, and te rgt surface (x = L = m) s nsulated. Te termal conductvt of te wall n te drecton of X axs s Kx =5 W/(m o C), tere s a unform generaton of eat nsde te wall of Q =00 W/m. Evaluate te followng. a) Dscretzed te model of te wall and calculate te element stffness matrx. (0 Marks) b) Determne te temperature dstrbuton troug te wall tckness. (5 Marks) Fgure Q, Conducton n a plane wall subjected to unform eat generaton Total 5 marks Please turn te page
4 Page of 9 Q For te bar sown n te Fgure Q below, wt lengt L, modulus of Elastct, E, mass denst ρ, and cross sectonal area A, usng te lumped mass matrx determne te followng. a) Dscretse te element nto two elements. (05 Marks) b) Usng drect stffness matrx develop te global stffness matrx. (05 Marks) c) Develop te global mass matrx. (05 Marks) d) Te frst two natural frequences of te sstem. (0 Marks) Fgure Q, One-dmensonal bar used for natural frequenc determnaton Total 5 marks Please turn te page
5 Page 5 of 9 Q Te beam sown n Fgure Q below as been dscretzed nto two parts as sown b te node numberng. Te beam s fxed at node, as a roller support at node, and as an elastc sprng support at node. A downward vertcal force of P=50 kn s appled at node. Gven te Modulus of elastct, E = 0 GPa, second moment of area, I = X 0 - m trougout te beam, stffness of sprng, K=00 kn/m. Determne te followng. a) Te nodal dsplacements (0 Marks) b) Rotatons at te nodes (0 Marks) c) Global forces (5 Marks) Fgure Q, Beam wt sprng combnaton. Total 5 marks Please turn te page
6 Page 6 of 9 Q5. For te sprng assemblage sown n te Fgure Q5 below, usng te compatblt condton at node for te tree sprngs attaced, were K, K and K are te stffness of tese sprngs, Obtan te global stffness matrx usng te followng metods. Gven node, and are fxed support and node te roller support wc can move n one drecton onl, P s a load appled at node n te drecton of x- axs. a) Drect stffness metod. (0 Marks) b) Drect Equlbrum metod. (5 Marks) Fgure Q5, sprng assemblage sstem. Total 5 marks END OF QUESTIONS
7 Page 7 of 9 Please turn te page FORMULA SHEET FINITE ELEMENT AND DIFFERENCE SOLUTIONS Fnte Dfference Equatons for Beam Deflecton: dx d dx d dx d 6 dx d Element conducton matrx. Elemental Force Matrx. Stffness matrx for bar element
8 Page 8 of 9 Please turn te page Lumped Mass Matrx Frequenc. Local stffness matrx for sprng element Beam Element Stffness matrx. Stffness matrx n D space.
9 Page 9 of 9 Please turn te page Elemental stress. END OF FORMULA SHEET
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