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# Modelling and finite element analysis: Question Papers

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### Modelling and finite element analysis: Question Papers

1. 1. USN 06ME63 Sixth Semester B.E. f)egree Examination, December 2Ol2 Modeling and Finite Element Analysis Tirne: 3 hrs. Max. Marks:100 Note: Answer FIVE full questions, selecting at leust TWO questions from eoch part. ! .! PART _ A a u e la. Using Rayleigh-Ritz method, derive an expression for maximum det.lection of the simply = b. supported beam with point load P at centre. Use trigonometric function. (08 Marks) a Solve the following system of simultaneous equations by Gauss elimination method. O X-l Y l Z:9aX, x-2y+32:8=D- 2x+ Y - z:3 (08 Marks) c. Explain the principle of minimum potential energy and principle of virtual work. (04 Marks)3 otll =co.= a.l 2a. Explain the basic steps involved is FEM. (10 Marks) b. Explain the concepts of iso, sub and super parametric elements. (05 Marhs)Etf c. Define a shape function. What are the properlies that the shape functions should satisly?-O (05 Marks)=tsa2 3a. What are the convergence requirements? Discuss three conditions of convergence i6= requirements. (05 Marks) 1OO b. What are the considerations for choosing the order of the polynomial functions? (05 Marks) I Derive the shape functions for CST element. rl-1 c. (10 Marks) {boi :2G 4a. Derive the Hermite shape function tbr a 2-noded beam element. (10 Marks) b. Derive the shape functions fbr a four noded quadrilateral element in natural coordinates.3u (10 Marks)AE6X I :o --: PART _ B,i .9.6EoLE 5a. Derive an expression for stifthess matrix for a2-D truss element. (10 Marks)a,- b Derive the strain displacement matrix tbr 1-D linear element and show that o: E[B]{u}>(k (10 Marks)cno bIr=0=soF> 6a. Discuss the various steps involved in the finite element analysis of a one dirnensional heato transfbr problem with refbrence to a straight unifbrm fin. (10 Marks)(-) < b. Derive the element matrices, using Galerkin for heat conduction in one dimensional element with heat generation Q. (l0 Marks)-NoZ 7 a. A bar is having uniform cross sectional area of 300 mm2 and is subjected to a load6 P : 600kN as shown in Fig.Q7(a). Determine the displacement field, stress and supporto reaction in the bar. Consider two element and rise elimination method to handle boundary conditions. Take E :200 GPa. (10 Marks) I of 2
2. 2. aI , , , /. t Fig.Q7(a) b. For the two bar truss shown in Fig.Q7(b), determine the nodal displacements and stress in each number. Also find the support reaction. Take E :200 GPa. (10 Marks) , SotlN Fig.Q7(b) 8a. For the beam shown in Fig.Q8(a), determine the end reaction and deflection at mid span. Take E :200 GPa,I:4x106 mma. (10 Marks) TYc 24hNlm h,,|tl,lfi Fig.Q8(a) Fie.Q8(b) Determine the temperature distribution through the composite wall subjected to convection heat loss on the right side surface with convection heat transfer coefficient shown in Fig.Q8(b). The ambient temperature is *5oC. (10 Marks) **+*8 2 of2
3. 3. 06ME63 sixth semester B.E. Degree Examinatlon, December 2011 Modelling and Fisrite Elememt Analysis Time:3 hrs" Max. Marks:100 Note: Answer uny FIYE full questions, selecting at least TWO questions from each part PART-A equatio, for ffrtate of stress and state the terms involved" (04 Marks) oi I a. write the equilibrium o o b. solve the following system of equations by Gaussian elimination rnethod : (d a x1*x2*x:=6 (6 Xr-Xz*2x3=5 Marksi (08 rd x1* 2x2-x3=2. {) c. Determine the displacements of holes of the spring system shown in the figure using iE e) (08 Marks) principle of minimum potential en?rg{; _o? o t{" trln"* Srtcll.t"r Sorf (!u!., Fig.Q.1(c). 6-s rr lxt ll ao traP.=N d+ i. 60 number and otr 2a. Explain the discreti zationprocess of a given domain based on element shapes -o (06 Marks) slze. a structural o= b. Explain basic steps involved in FEM with the heip of an example involving Es member subjected to axial loads. (08 Marks) od vd 0. Why FEA is widely accepted in engineering? List various appiications of FEA in (06 Marks) 6o engineering oo boc 3a. Derive interpolation model for 2-D simplex element in global co - ordinate system "o! (10 Marks) 26 !s= d(g b. What is an interpolation function? Write the interpolation functions for: .a EO o€ (, i) 1 -Dlinearelement ; ii) 1 -Dquadraticelement 2O iiU 2-D linearelement ; iv) 2-Dquadraticelement tro. v) 3-Dlinearelement. (06 Marks) (04 Marks) oj c. Explain "complete" and "conforming" elements AE Derive shape function for 1 - D quadratic bar element in neutral co-ordinate tVttelm 5L) olE 4a. Marks) LO o.E >.9 b. Derive shape functions for CST element in NCS. (08 Marks) on- troo c. What ur. rhup. functions and write their properties. (any two). (04 Marks) qo :a) EE PART -B -h U< 5a. Derive the body force load vector for I - D linear bar etrement. (04 Marks) (06 Marks) --.; ci b. Derive the Jacobian matrix for CST element starting from shape function (10 Marks) o c. Derive stiffness matrix for a beam element starting from shape function o z d 6a, Explain the various boundary conditions in steady state heat transfer problems with simple o o, sketches. (06 Marks) b. Derive stiffness matrix for 1 - D heat conduction problem using either functional approach or Galerkins approach l (08 Marks) .j l I of Z ij ii
4. 4. 06M863rI c. For the composite wall shown in the figure, derive the global stifftress matrix. (06Marts) Take Ar:Az=A3:A Fie.Q.6(c) 7 a. The structured member shown in figure consists of two bars. An axial load of P:200 kN is loaded as shown. Determine the following : i) Element stiffness matricies. ii) Global stiffness matrix. iii) Global load vector. iv) Nodaldisplacements. i) Steel Ar = 1000 mm2 Er :200 GPa ii) Bronze Az:2000 mm2 Ez: 83 GPa (08 Marls) b. For the truss system shown, determine the nodal displacements. Assume E: 210 GPa and A = 500 mm2 for both elements. (I2 Marks) ;f =loovl.rt Fie.Q.7(b) 8 a. Determine the temperature distribution in 1 - D rectangular cross - section fin as shown in figure. Assume that convection heat loss occurs from the end of the fin. Take 3w K=-. CmoC - = 0.1w , T*:20oC. Consider two elements h "" (10 Marks) = CmoC fo v 5 E*fr.,r Fr z reg,il .f.y-tol nnll Fie.Q.8(a) b. For the cantilever beam subjected to UDL as shown in Fig.Q.8(b), determine the deflections of the free end. Consider one element. (10 Marks) fo;5s1. Fs z rD qlt .t-tot\$fltt Fie.Q.8(b) ,***** 2 of?
5. 5. r- I I 06ME63 USN SixthSemesterB.E.DegreeExamination,December2010 Modeling and Finite Element Analysis Max. Marks:100 Time:3 hrs. selecting Note: Answer ony FIVE futl questions at least TWO questions from each part PART _ A ,9 o ffi"-" tt for two dimensions (06 Marks) H I a. Explain, with a sketch, plain stress p*"ii"f energy Explain the potential energy with usual "in a (g b. State the principles of minimu* (06 Marks) notations. c. t^hTT: the steps invotved in Ravleigh-Ritz methog? DeTnTl:,X ul?l]":::* d shown in Fig.l(c). use second degree ::#:: () d 0) ffii ilrffi:i:.1il;J*_d#;,I;J;;g-^ approximation, for the displacement (08 Marks)39 iolynomiald9-o ,, ao" Fm.=+E-f b?p Fig.l (c).Pfr method with finite element methoclo> (04 Marks)!1 a 2a. Bring out the four differences in continuum with example b. What do you underrtuod FEM? eri"ny e*piain the steps involved in FEMacd the generai node numbering *d t"l-ff:;Tl 5(J do Write properries of stiffness matrix K. Show - (06 Marks) the half bandwidth- 6d {tz Marks) }E 3a. What is an interpolation function? ,..r - of convergence , -r ^^--.^- requirements tr5 b. what are convergence requirements? Discuss three conditions (08 Marks) Write a shot notes on : !O oe c. - o- gt Eo. i1 C.o*etrical isotropy for 2D Passal triangle (CST) elernent with a sketch (lG Marks) ii) Shapg function for constant strain triangrilar si ^9 bar eiernent, in natural co-ordinates" @q 4a. Derive the shape functions for the one-dimensional (08 Marlcs) quadrilateral elerrent, in natural co-rdinates Derive the shape functions for a four-node L0 b. (08 Marks) >.k mo (04 Marksi c. Write four properties of shape functions g0 =(6 tr> PART - B 59 o- U< 5a. Derive the following : 1) Element stiffness matrix (K") il Element load vector (f) c.i -i () (12 Manlis) o Uy aire"t method for one-dimensional bar etrement Z (l-1) for constant strain triangle (csr) b. K:ff:"Iffi::f the Jocabian transformation matrix (08 Marks) (08 d o a (06 Marks) 6a.Explainwithasketch,one-dimensionalheatconduction. for heat conduction in one b.Derivetheelementmatrices,usingGalerkinapproach, (10 Marks) dimensional element (04 Marks) dimension c. Explain heat flux boundary condition in one
6. 6. I 06M863 7 a. Solve for nodal displacements and elemental stresses for the following. Fig.Q,7(a), shows a thin plate of uniform lmm thickness, Youngs modulus E : 200 Gfa, weighr aensity of the plate : 76.6 x 10-6 N/mm2. In addition to its weight, it is subjected to a point load of 1 kN at its mid point and model the plate with 2 bar elements. (10 Marks) r Fig.Q.7(a). I I + t b. For the pin-jointed configuration shown in Fig.e.7(b), formulate the stiffness matrix. Also determine the nodal displacements. (10 Marks) IKN fiIomrJ. {. Es = E2=26r6\$Pr. Fig-Q.7(b). 8a. Solve for vertical deflection and slopes, at points 2 and,3, using beam elements, for the structure shown in Fig.Q.8(a). Also determine the deflection at the centre of the the beam carrying UDL. ir"rtffi E:z.o06P( Fig.Q.8(a). J = 4x lob**ti b. Determine the temperature distribution through the composite wall, subjected to convection heat transfer on the right side surface, with convective heat transfer co-efficient shown in Fig.Q.8(b). The ambient temperature is -5oC. (10 Marks) a- t* [2 looo \$ly*tt Fig.Q.8(b). Kz=*** ****,r 2 of2
7. 7. 06M863 USN May/June 20L0 Sixth semester B.E. Degree Examination, Modeling and Finite Element Analysis Max Marks:100 Time: 3 hrs. d questions, seleeting atleast TIYO from each part o o Note: Answer any FIVE futl a PART - A of a simply supported beam with (g i ! a. Using Rayleigh Ritz method, find tt. ,il*l*--a"flection (10 Marks) o () point load at center . . -- ,--- ^^-.^+r^-- L., I],,,oci;rn eli method by Gaussian elimination b. Solve the following system of simultaneous equations (B oB9qp- 4xr f 2W+ 3x3:4 2xr * 3x2* 5x3:2 (10 Marks)=h Zxr * 7xz: 4aollt-6.= el 2a.ExplainthedescretizationprocesssketchthedifferenttypesofelementslD2D3D (06 M::Y) cdS c^ bI) !i {) elements used in the finite element analysis -..,-- L-- A:-^ !a otr rA b. considerinil;gton"rrt,.ou.oio it. "tl*.rrt stiffness matrix by direct stiffiress fir;111*;eE Comment on its characteristics tudti9,}*rr.ut o7 the properties that the shape flrnction should 8z a. De{ine " J# #ffi;. irh;;*" oid ?d 6o criteria with suitable examples and compatibil* *o*T#H:i; do 3 a. Explain the convergence boc .dd b. FEM. Explain simplex, complex and multiplex elements using element shapes (06 Marks) rk coordinates for one dimensionai }E !o= c. Explain linear interpolatiorr, potyrro*ials in terms "igilu"r - (06 Marks) !rg -2" ts simPlex element irO oe E3 ."t,,rtePr vr revr*r*. 4a,Explaintheconceptofisoparametric,subparametricandsuperparametricelementsanrj (06Marks) o9 tro theirurrrto a ,1 r:--ri^^^*a o-i b.DerivetheshapefunctionsforaCsTelementandalsothedisplacementmatrix.(08Marks) for abeam element (06 Marks) 9E A,E c. Derive the Hermite shape nn.ti* =9 LO PART. B shown in fig Qs(a) s a. Find the shape tunctions forpgintp ruiffi"lement o.< >.(I at g";o ^t11r11r*; (10 Marks) 6E area and Jacobian the eiement matrix AE 61 8) tr> =6J Ek P* _h C6rsJ o< I ..I e.i Fig.Qs(a) I (trt{ C C\$, o z (10 l![arks) (l b.Derivethestiffrressmatrixfota2_dimensionaltrusselement. o + analysis of a one dimensional heat 6 a. Discuss the various steps involvedain the finite element unjform fin (10 Marks) transfer problem with reference to straight for linear interpolation of b. Explain the finite element *oa"rirrg La rrrrp" functions (10 Marks) tieat trunsfeielement) temperature field (one - dimensional 1of?
8. 8. 7 a. Determine the nodal displacement and stresses in the erement shown in fig. e7(a).1r0 Marks) Ar = 500 mm2 Fie.Q7(a) lokN Az = 2000mm2 : Er 100 GPa E2:200 GPa 300mm 300mm 3::::_1"":::rl-::tg::r^ *"gx oferementr (1);; b. truss etements shown in fig. of 200mmz and .the e7(b). Au the elements have an area irt;; sil,- f;*.-;:;fibil;: (10 Marks) go l.^1, Fie.Q7(b) f 6o*t A composite wall consists of three materials as shown in fig. eg. The outer temperature To = 200c convective heat transfe, tuk., place on the inner surface of the wall with Too = 8000c and h :25 wrmz o.[""ir. it . ,.rp.rature distribution on the wall. C. (20 Marks) f*: I, otc ^*Jli kr:20 WmoC kz:30 WmoC k3 = 50 WimoC Fig.Q8 h-25WlmzoC T*:8000C ***** 2 of2
9. 9. 06M863USN sixth semester B.E. Degree Examination, June-July 2009 Modeling and Finite Element AnalysisTime: 3 hrs. Max. Marks:100 Note: Answer any FIVE full questions, selecting at least TWO questions from each part. PART _ A L a. Explain the principle of minimum potential energy and principle of virtual work. (06 Marks) +l b. Evaluate the integral 1= J{fE +2z ++2F by using 2 point and 3 point Gauss -t (06 Marks) quadrature. c. Sotve the following system of simultaneous equations by Gauss Elimination method: x, -2x, * 6x, = Q Zxr+Zxr*3x, =l - Xr * 3x, = 0 (08 Marks) 2a. Explain briefly about node location system (06 Marks) Explain preprocessing and preprocessing in FEM. (06 Marks) b. Explain the basic steps involved in FEM. (08 Marks) c. 3a. What are the considerations for choosing the order of polynomial functions? (06 Marks) b. Explain convergence requirements of a polynomial displacement model. (06 Marks) c. Derive the linear interpolation polynomial in terms of natural co-ordinate for 2-D triangular elements. (08 Marks) 4a. What are Hermite shape functions of beam element? (06 Marks) b. Derive the shape function for a quadratic bar element using Lagrangian method. (06 Marks) c. Derive the shape function for a nine noded quadrilateral element. (08 Marks) PART _ B 5a. Derive the element stiffness matrix for truss element. (10 Marks) b. Derive the Jacobian matrix for 2D triangular element. (I0 Marks) 6a. Explain the types of boundary conditions in heat transfer problems. (r0 Marks) b. Discuss the Galerkin approach for l-D heat conduction problem. (10 Marks) la. Using the direct stiffness method, determine the nodal displacements of stepped bar shown in figure Q7 (a). (lo Marks) Er :200 GPa Ez:70 GPa Ar : 150 mm2 Az: i00 mm2 Fr:l0kW Fz:5 kW Fie. Q7 (a) I ofZ
10. 10. 06M8637 b. For the truss shown in figure Q7 (b), find the assembled stiffness matrix. (10 Marks) lkN T 5oo E1 : E2:200 GPa I L Fig. Q7 (b)8 a. Determine the temperature distribution through the composite wall subjected to convection heat loss on the right side surface with convective heat transfer coefficient shown if figure Q8 (a). The ambient temperature is -5"c. (r0 Marks) t:rdc k= 6 =j_ Kr: )*cr.l/*.r-lt , v+K prq6.b- -ri._o,94 F Fig. Q8 (a) b Determine the maximum deflection in the uniform cross section of Cantilever beam shown in figure Q8 (b) by assuming the beam as a single element. (10 Marks) loe i< Fj E:7x10e N/m2 i I:4x10-a ma ----* l Fig. Q8 (b) **{.** 2 of2
11. 11. u ME6Fl - USN OLD SCI{EME -;? l--/-- sixth semester B.E. Degree Examination, July 20A6 Mechanical Engineering Finite Element MethodsTime:3 hrs.l [Max. Marks:100 Note: Answer any FIVE full questions (03 Marks) Define functional. (10 Marks) Derive Eulers Langrangess equation (07 Marks) Expiain principle of minimum potential energy Briefly explain the steps involved in FEM (10 Marks) (10 Marks) Derive shape functions for CST triangular element in local co-ordinater. Explain Banded matrix. Write an algorithm for Guass elimination technique (10 Marks) Explain Raieighs Ritz method in detail (10 Marks)4 What do you understand by weak form of differential equation. (05 Marks) a*,..u_-, . 3j!.j ft, ="lS\$* tY,,Y.d.,c :1 u .j -"F, -lt ffi:gnr*:.et*r-** bar whose cross i) For the above problem compute [B] and [c] matrix. It is^tapered - section area decreases linearly from t-000 m*2 to 500 Take E:2x 10s N/mm2 mm2. ii) Use two elements and findthe nodal displacements. (15 Marks) a. Derive shape functions and stiffness matrix for beam element (15 Marks) b. Explain the need of Jacobian transformation matrix. (05 Marks) a. Explain in detail ISO - parametric, sub - parametric and Super - parametric (10 Marks) elements. b. Explain "penalty approach" for handling the boundary conditions (10 Marks) a. Discuss the requirements to be fulfilled for the convergence of FEM solution Marks) (10 b. Derive FEM equation by variational principle (10 Marks) Write short notes on anY four : a. Pascals triangle d. Truss element b. Local - co - ordinate sYstem e. Shell element c. Patch test. f. EliminationaPProach :t:t:k* *
12. 12. Poge No,,. I ME6FI Reg. No. Sixth Semesler B.E. Degree Exominolion, Jonuory/Februory 2006 Mechonicol Engineering (Old Scheme) Finite Elemenl MethodsTime: 3 hrs.) (Mox.Morks: 100 NOtg: Answer any FVE lull questions. l. (o) Find the inverse of [lt] (5 Morks) ror a: [3 ]] ,:l; {l Find : i) AB ii1 gT 4T (5 Morks) Solve by Gouss eliminotion 2q * * rJ: -7 3x2 5r1 * n2 * a3: Q (10 Morks) 321 *2x214x3:11 2. @t Whot is finite element method? Whot ore the odvontoges of FEM over finite difference method? (4 Morks) (b) Exploin boundory volue ond initiol volue problems using suitoble exomples. (8 Morks) (c) Exploin the steps involved in the finite element onolysis of solids ond structures. (8 Morks) . 3. (o) whot is meont by Bclnd width of o motrix? Give on exomple. Exploin why it should be minimized. (6 Morks) (b) Stote the principle of minimum potentiol energy, ond derive on expression for totol potentiol energy of o solid bor under compression. (6 Morks) (c) Exploin the Royleigh-Rit method with on exomple, (8 Morks) 4. (o) Exploin the Golerkins opprooch for obtoining stiffness motrix of o bor element, (10 Morks) (b) TwopointsPl(10,5)ondP2(80,10)onosolidbodydisplocesto PlOO.z,b.4)ond P;(80.5,10.2) ofter looding. Determine normol ond sheor stroins. (10 Morks) Confd.,.. 2
13. 13. Poge No,,, 2 ME6FI 5. A solid stepped bor os shown in fig.l is subjected to on oxiol force. Determine the following D Element ond ossembled stiffness motrix iD Displocement of eoch node iii) Reoction force ot fixed end (20 Morks) 2- A,=t0O mm. *r=1-Oo mm Lku E = 200G PaI tr= ro Q Po 6. (o) Whot is Jocobion Motrix? Derive o Jocobion motrix for Two-Dimensionol element. (10 Morks) ,, l] (b) Derive shope function CST triongulor element. (10 Morks) 7. @) Derive shope functions for o l-D quodrotic element with 3 nodes. (10 Mofts) . (b) Exploin convergence criterio ond potch test in brief, (10 Morks) 8. Write short note on ony FOUR: o) Voriotionol opprooch b) Hermition shope functions c) Penolty opprooch for hondling boundory conditions d) Logronge ond serendipity fomily of elements e) ISO porometric: elements (5x4 Morks)
14. 14. Page N0... 1 ME6F1 USN Sixth Semester B.E. Degree Examination, July/August 2005 Mechanical En gineering Finite Element MethodsTime: 3 hrs.I [Max.Marks : 100 Note: 1. Answer any FIVE full questions. 2. Missing data may be suitable assumed. 1. (a) Define positive definite matrix. (2 Marks) (b) Solve the system of simultaneous equations given below by Gaussian elimination method. 2c1 * 2n2 * ns :9 n1*n2+fry:6 (10 Marks) 2a1 * a2: 4 (c) Determine the inverse and eigen values of the given matrix A . I 4 -2.286 (8 Marks) ^: -z.zJG L 8 2. (a) Explain basic steps in FEM. (10 Marks) (b) Explain potential energy of an elastic body. (5 Marks) (c) Explain isoparametric, subparametric and superparametric element concept. (5 Marks) (a) Derive the shape functions for a three node 1-D element in natural coordinate. (EMarks) (b) Determine the displacemenl field, stress and support reactions in the body shown in fis.Q3(b). , P :60k/f E : 2O0kN lrrlnlZ : Et : Ez At :2000mm2 Az : l00Omrn2 (12 Marks) F tS , a.z ir. 4. (a) Explain steps involved in Galerkin method. (10 Marks) (b) Determine the defleetioh of canlilever beam of length I and loaded with a vertical load P at the free end by Rayleigh-Ritz method. (10 Marks) Contd.... 2
15. 15. Page N0... 2 ME6Fi 5. (a) For the one dimensional truss element, develop the element stiffness matrix in the global coordinate system. (10 Marks) (b) Determine the nodal displacement and stress by using truss element. (10 Marks) (a) Derive the stiffness matrix for a two node beam element. (10 Marks) (b) For the beam shown in fig,Q.No.6(b). Determine the maximum deflectlon and the reaction at the support. El is constant throughout the beam. (10 Marks) 7. (a) What is the significance of the band width? lllustrate best method of node numbering with an example. (5 Marks) (b) Evaluate the following by Gaussian quadrature i) /: /]i (s"* + *, + #)da by one point and two point formula. (3 Marks) ii) I : I: * OV 3-point formula. (8 Marks) 8. Write short nole on the following : (a) Coordinate systems (b) Convergence criteria (c) Variational method (d) Plane stress and plane strain conditions (e) Penalty approach for handling boundary conditions. (5x4=20 Marks) *****
16. 16. Page N0,. 1 ME6F1 USN Sixth Semester B.E, Degree Examination, January/February 2005 Mechanical Engineering Finite Element MethodsTime: 3 hrs.l [Max.Marks : lO0 Note: Answer any FIVE full questions. 1. (a) Distinguish between : Symmetric and skew symmetric matrix, transpose and inverse of a matrix. (4 Marks) (b) What is a banded matrix? What are its merits? (4 Marks) (c) Solve the following system of simultanegus equations : 11l2t2lrt:4 3*t-4xz-2r3-2 5r1l3r2*5r3- -7 either by Gaussian elimination method or malrix inversion method. (6 Marks) (d) Find the eigen values of the matrix A lz B A- lr 4 -21 -2lr (6 Marks) Lz 10 ,r j 2. (a) What is the basis of the Finite Element Method? Explain the basic steps involved in the finite element method. (10 Marks) (b) Determine the true displacement field for a two noded one dimensional tapered elemenl shown in Fig.1. Also compute the stiffness matrix for this elemerit. o c.n^--*J I"t-eJ At= loo n;ro[, " q2&) , Ftq t At :700rnz -t12 :900mm2 A . 2 (10 Marks) An : (* #) Contd.... 2
17. 17. Pase N0... 2 MEOF1 3. (a) What are the principles of continuum method? Compare this method with finite element method clearly bringing out their relative merits. (6 Marks) (b) Stale the variational principle of minimum potential energy. (4 Marks) A uniform cantilever is subjected to a uniformly distributed load of W kN/m over ils span l. The displacement function is given as y: 4(*, +612a,2 - lrs)where A is fhe displacement at the free end, Compute the v"alue of the deflection A by the principle of minimum polential energy. Compare this with the exact value. (r0 Marks) 4. (a) Derive the strain displacement relations. (2 Marks) (b) b<plain the concepts of plane stress and plane strain with suitable examples, Also derive the corresponding equations. (8 Marks) (c) A uniform rod of lengh I fixed al both ends is subjected to a constant axial load of w kN/m. Establish the displacement field and compute the stresses at the fixed ends and rnidspan. What are lhe nalure and magnitude of the reaclions at the lwo ends? Use Rayleigh-Bitz method. (10 Marks) 5. (a) What are interpolation rnodels? Give reasons for choosing polynomial funclions for such npdels. (5 Marks) (b) Explain briefly the penalty approach for handling displacement boundary conditions. (5 Marks) Using the penalty approach, determine the nodal displacements and lhe stresses in each material in the axially loaded bar shown in Fig.2 A l,^v, i*1,^r" 3oo tttT 4 OO x^1^4 Area of (1):2400mm2 Area of (2) :6A0mm2 (10 Marks) EAL:o7 xTosNfrnrnz Esteel:2x705Nlmrnz 6. (a) Explain the concept of isoparametric formulation. (5 Marks) (b) Derive an elemenl stiffness matrix of a constant strain triangular element using the above concept. (15 Marks) Contd.... 3
18. 18. Pase N0... 3 MEOF1 7, (a) what is a higher order element? what is its importance? (4 Marks) (b) Derive the stiffness matrix for an element in the form K: IW)r t"l tBl d,a Show that the above matrix is symmetric. (10 Marks) (c) A beam element carries a concentrated load P af { from one end. Obtain nodal loads using the formulae of fixed beam. (6 Marks) 8. Write brief explanatory notes on any FOUR: (5x4=!Q [irs*s; D Advantages and disadvantages of finite element methods ii) Types of Finite Elements iii) Boundarycondifions iv) Principle of virtual work v) Cohvergence criteria ** * **
19. 19. Page No., 1 ME6Fl USN Sixth Semester B,E. Degree Examinatlon, July/August 2004 Mechanical Engineerlng Finite Element MethodsTime: 3 hrs.I lMax.Marks : lOO Note: 1, Answer any FIVE futt questions. 2. Assume suitable dak if necessiry. 1. (a) Explain with example. i) Symmetric matarix ii) Determinant of a matrix iii) Positive definite matrix iv) Half band width v) Partitioning of matrices. (10 Marks) (b) Give the aigorithm for fonruard elimination and back substitution of Gauss elimination for a general matrix, (10 Marks) 2, (a) With suitable examples explain. i) Essential (geometric) boundary condition ii) Natural (force) boundary condition. (5 Marks) (b) Outline the steps in finite element analysis. (5 Marks) (c) State the. principle. of minimum potential energy. Obtain the equilibrium equation of the system shown in fig 2.c using the principle of minimum potential energy. (10 Marks) 3. (a) Derive the equilibrium equation of 3D elastic body occupying a volume V and having a : surface S, subjected to body force and a concentiated lodd. (10 Marks) (b) An elastic bar of length L, modulus of elasticity E, area of cross section A, which is fixed at one end and is subjected to axial load at the other end. Obtain the Euler equation governing the bar, and natural boundary conditions. (10 Marks) 4. (a) For a two noded one dimensional element, show that the strain and stress are constant with in the element. (10 Marks) (b) Explain the criteria for monotonic convergence. (10 Marks) 5. (a) A component shown in fig. 5(a) is subjected to a load 5 kN. Detennine the lollowoing. ii Element stiffness matrices iD B - matrices iii) Dispiacemerrts and strains iv) Stresses and reactions. Obtain the stiffness matrix and load vector assuming two elements, (12 Marks) (b) What are characteristics of stiffness matrix ? (8 Marks) 6. (a) For a pin jointed configuration shown in Fig 6.a determine the stilfness matrix. Also determine g, interms of g,. (10 Marks) (b) Derive the Hermite shape functions of a beam. (r0 Marks) Contd.... 2
20. 20. : Fage No... 2 ME6FT 7. (a) Evaluate I I [r,,* ;r*ffif* -1 Using two point Gauss quadrature. (5 Marks) (b) Derive the expression for shape functions of eight noded isoparametric element. (15Marks) 8, (a) Determine the Jacobian for the triangular element shown in lig Q8.a. (5 Marks) (b) Give the element number and mode numbers for the structure shown in Fig Q 8.b, so as matrix. to minimize the half band width of the resulting stilfness (5 Marks) (c) For the beam shown in fig Q.8c. obtain the global stiffness matrix. (10 Marks) F fi?. qL. c vf, ts* +ol )I o. g-- 7oxto3 ^/t",ol -/- I fZ A= l3oo ss +ozn I J CLo,P) L= S m * fi3. Q6.o l s0 ooo c+1) mm . / A.: 5oo ms , gnz Qoo c z 3.5) ri , too 6Pa Ct.gr.l L; zoo aoo- 63 QBo- FS E(") +R =l qe. b ng. 7* lrD ---+L t -o ----l |[---6--G-re-Z- /,r-----=------i{--a----v -7., I ,=2-ooePd Fs, Ee .c ?=- +^iie *"-4 ; - r -,r^O *****
21. 21. Page No... 1 ME6Fl Heg. No. Sixth Semester B.E. Degree Examination, January/February 2009 Mechanical Engineering Finite Elembnt MethodsTirne: 3 hrs.I lMax.Marks : IOO Note: Answer any FIVE questions. l 1. (a) Solve the following system of simultaneous equations by Gaussian Elimination Method. t1 -2n2 f 613 - 0 l l 2a1*2c2*3n3-3 -rr*3r2-2 (10 Marks) (b) Find the inverse of the following matrices l0 1 21 f1 2 , Ll?il ilL;:, ll -21 (5+5 Marks) 2. (a) Explain the theorem,of minimum potential energy. Distinguish between minimum potential - theorem and principle of virtual displacement" (10 Marks) (b) Explain the basic steps in the formulation of finite element analysis. (10 Marks) 3. (a) ,Flnd_thg.s!re!9 al w.:0 and displacement at the mid - point of the rod shown below. Use Raleigh Ritz method A) I Tq-Ke E = L1q6J- (Yo*,,.3S ^".U.*[-y A luniF (A tea fl(r.,1-r a-h.orr (10 Marks) (b) Explain plane stress and plane strain methods with rerevant equations. (10 Marks) 4. (a) Explain the penalty approach for handling the specified displacement boundary conditions. (10 Marks) Contd.... 2
22. 22. Page N0... 2 ME6F1 (b) For the {ollowing figure (bar), find the nodal displacements. The cross sectional area decreases linearly from 1000rnm2 lo 500mm2. Use two elements. Take E :2x1O5MPa,7:0.3 ,t 5ooss lbbo -, looo A1 k- J\$otwr 4 - (10 Marks) (a) Explain convergence criteria in detail, (10 Marks) to) Derive shape functions for CST element from generalized co-ordinates. (10 Marks) (a) Derive the stiffness matrix for a two noded beam element (12 Marks) i (b) Distinguish between isoparametric, sub-parametric and super-parametric elements.la uarxsl" l[^ 7. Consider the 4 -bur truss shown below, Determine. i) Element stiffness matrix for each elementII ii) Using eliminations approach to solve for the nodal displacements. (iiD Calculate stresses in each element. (20 Marks) +v 2-gooor..t (n.+i5 Ja-svr*,; ll Qg Ar t 3otv t"t @ I 20,0001.; >1 rQrC) 4-4O I * *r I Write shorl notes on any FOUR of the following. a) Eliminationapproach b) Patch test c) Galerkins approach d) Geometric isotropy e) Post Processing f) LST triangular element ** * ** (5x&20 Marks)
23. 23. a Poge No.,. I ME6FI Reg. No. Sixth Semesler B.E. Degree Exominolion, Jonuory/Februory 2006 Mechonicol Engineeilng (Old Scheme) Finiie Elemenl Methods 1. Time: 3 hrs.) :. (Mox.Morks: 100 NOle: Answer ony FIVE tuil queslions. I. (o) Find the inverse of [r ol lo rl (5 Morks) ,o, a: [3 1] ,:l; {l Find : i) AB ii1 BT ar (5 Morks) (c) Solve by Gouss eliminotion 2*t+3a2*nJ:-1 541*e2*rs:0 (10 Morks) 3rr + 2a2l4a3 -]".1 2. @, Whot is finite element method? Whot ore the odvontoges of FEM over finite difference method? (4 Morks) (b) Exploin boundory volue ond initiol volue problems using suitoble exomples. (8 Morks) (c) Exploin the steps involved in the finite element onolysis of solids ond structures. (S Morks) : . 3. tol whot is meont by Bcind width of o motrix? Give on exomple, Exploin why it should be minimized, (6 Morks) (b) Stote the principle of minimum potentiol energy, ond derive on expression for totol potentiol energy of o solid bor under compression. (6 Morks) (c) Exploin the Royleigh-Ritz method with on exompte. (8 Morks) 4. (o) Exploin the Golerkins opprooch for obtoining stiffness motrix of o bor element, (10 Morks) (b) TwopointsPl(10,8)ondP2(80,10)onosotidbodydisptocesto pl(Lo.z,b.4)ond Pj(80.5,10.2) ofter looding. Determine normol ond sheor stroins, (t0 Mql1s) Confd.... 2
24. 24. Poge No,,, 2 ME6FI 5. A solid stepped bor os shown in fig.l is subjected to on oxiol force. Determine the following i) Element ond ossembled stiffness motrix iD Displocement of eochnode I iii) Reoction force of fixed end (20 Morks) 2- A,= tOo hm , *r=LOo mhn- h-k u g = 2,00G Pa rt"= lo q Pq 6. (o) Whot is Jocobion Motrix? Derive o Jocobion motrix for Two-Dimensionol element. (10 Morks) (b) Derive shope function CST triongulor element, (10 Morks) 7. @| Derive shope functions for o l-D quodrotic element with 3 nodes. (t0 Morks) (b) Exploin convergence criterio ond potch test in brief. (10 Morks) 8. Write short note on ony FOUR: o) Voriotionol opprooch 6) Hermition shope functions c) Penolty opprooch for hondling boundory conditions d) Logronge ond serendipity fomily of elements e) ISO porometric elements (5x4 Mqrks)
25. 25. Page No.., 1 ME6Fl USN \$ixth sernester B"E. Degree Examination, July/August 2004 Mechanical Engineering Finite Element Methods 3 hrs.l [Max.Marks : 10O Note: 1. Answer any F|VE full questions. 2. Assume suitable data if necessary. 1. (a) Explain with example, i) Syrnmetric matarix ii) Determinant of a matrix iii) Pcsitive definite matrix iv) Half band width v) Partitioning of matrices. (10 Marks) (b) Give the algorithm for forurard elimination and back substitution of Gauss elimination for a general matrix. (io Marks) 2. (a) With suitable examples explain. i) Essential (geometric) boundary mndition ii) Ndtural (force) boundary condition. (5 Marks) (b) outline the steps in finite element analysis. (5 Marks) (c) State the principle of minimum potential energy. Obtain the equilibrium equation ol the system shown in fig 2.c using the principle of-minimum potentidl energy. (10 Marks) 3. (a) Derive the equilibriqm equation ol 3D.elastic body oc.cypyt"ng a volume V and having a surface s, subjected to body force and a concentrated lddd. (r0 Marks) (b) ry elastic bar of length.L, modulusof elasticity E, area of cross section A, which is fixed at one end and is subjected to axial load at-the other end. Obtain theEuler equation governing the bar, and natural boundary conditions. t10 Marks) 4. (a) Fo1 a two noded one dimensional element, show that the strain and stress are constant with in the element" (ro Marks) (b) Explain the criteria for monotonic convergence. (,l0 Marks) 5. (a) A component shown in fig. 5(a) is subjected to a load 5 kN. Determine the foilowoing. i) Element stiffness matrices ii) B - matrices iii) Displaeements and strains iv) Stresses and reactions. Obtain the stiffness matrix and load vector assurning two eiements. (t2 Marks) (b) What are characteristics of stiffness matrix ? (8 Marks) (a) For a.pin jointed configuration shown in Fig 6.a detennine the stiffness matrix. Also determine qt interms of g,. (10 Marks) (b) Derive the Hermite shape functions of a beam. (10 Marks) Contd.... 2
26. 26. Page Nor, 2 illE6F1 7. (a) Evaluate 1 -1 Using two point Gauss quadrature. (5 Marks) {b) Derive the expression for shape lunctions of eight noded isoparametric element. (15 Marks) 8. (a) Determine the Jacobian for the triangular element shown in fig eg.a, (5 Marks) (b) Give thp element number and mode numbers for the structure shown in Fig Q 8.b, so as to minimize the half band width of the resulting stiffness matrix. (5 Marks) (c) For the beam shown in fig Q.8c. obtain the global stiffness matrix. (10 Marks)iI fi?. qL. e-rt vf, cto+c> )t o, -/" Et 7oxto3^l/tol I {/Clo,rs) A= l3oo ss m"n I J. * V-- S n t fr3. Q6.a le ooo , ),+;) mm / A.; 5oo mw , gn: QOO C z ss) c : 0o GPa L1.51) L; r-oo aOo. 63 Qeo- F3 5(a1 s +R ft. q8. b ,@ h- /l Iro nD I .tlo -, ^^ ,-2oold. .qc.c i= "^lo6+nYo* t , *****
27. 27. a Page No... 1 ME6F1 USN Sixth Semester B.E. Degree Examination, January/February 2004 Mechanical Engineering Finite Element Methods Time: 3 hrs.l [Max.Marks : IO0 Note: 1. Answer any FIVE full questions. 2. Missing data may be suitably assumed, 1. (a) Find the eigen values of A- 4 -{51 (5 Marks) -,/3 a l (b) Solve the following system of simultaneous equations by Gaussian elimination method. 2e1*12!3rs:t\$ 4r1*r21.a3:\$ 3n1*2r2 * rs:3 (10 Marks) (c) Define the following with example i) Skew matrix ii) Symmetric banded matrix. (5 Marks) (a) Explain difference between continuum method and finite element method, (5 Marks) (b) Explain basic steps involved in FEM. (10 Marks) (c) Explain principle of minimum potential energy and virlual work. (5 Marks) (a) Expain steps involved in Rayleigh - Ritz method. (B Marks) (b) Determine the deflection at the free end of a cantilever beam of length 1, carrying a vertical load P at its free end by Rayleigh Ritzmethod (i0 Marks) (c) List the demerits of cantinuum methods. (2 Marks) 4 (a) Derive strain displacement matrix, stiffness matrix for one dimentional bar element. (8 Marks) (b) Solve for stresses and strains for the following problem by using bar element. (12 Marks) ? = loco l.J /t<_ E:2.7xlA5Nfrrurnz At :5Omm2 Az :25mm2 P : 100011 Contd.... 2
28. 28. Page N0... 2 ME6F1 5. (a) Derive stiffness matrix for a truss element. (8 Marks) (b) For a pin jointed configuration shown in figure, determine nodal displacements and stress by using truss elemenls. f : looo;? T 5oo r Ar : LAAmmz t :lSovnr Az:125Amm,2 E:200GPa (12 Marks) 6. (a) Compute.the deflection of simply supported beam carrying concentrated load at its centre, Use two beam elments. (16 Marks) (b) ls FEM analysis applicable for highly elastic materials? Explain. (4 Marks) 7. Find the displacement of node 1 in the triangurar element shown using one triangular element. Also find stress and strain in the elefient. . 1+----- 3o n (-3o,o ) loo l,/ l r.-__ 5o I 2o I.(,2,o ) I E:70GPa L 7:0.3 c 3o, Le : lAmm (20 Marks) Write short notes on any FOUR of the following : a) Static condensation b) lsoparametric, super parametric and subparametrlc element c) Static and kinematic boundary condition d) Lagrangian and Hermite shape functions e) Convergencecriterion (4x5=2Q fYl2Y[s) *****
29. 29. a ----- --t/ Page N0,,. I ME6F1 USN Sixth Semester B.E. Degree Examination, July/August 2000 Mechanical Engineering Finite Element Methods Time: 3 hrs.I [Max.Marks : 10O Note: Answer any FIVE futt questions. 1. (a) Given o:l; i], ort.,*in. i) Inverse of matrix ii) Eigen values. (10 Marks) (b) lf ,7"r: [€, 1-(2], evaluate /, wT Nag (5 Marks) (c) Explain symmetric banded matrix. (5 Marks) 2. (a) With an example explain Rayleigh -Ritz method. (10 Marks) (b) State the principle of minimum potential energy. (4 Marks) (c) Sketch the quadratic and Hermite shape functions. (6 Marks) 3. (a) Derive the following characteristics of three noded l-D element. i) Strain displacement matrix [B] ii) Stiffness matrix [frr] (10 Marks) (b) Solve for nodal displacements and stresses for the structure shown in fig 1. Use penality approach to apply boundary csnditions. (10 Marks) h t"laao n{ 2"17o frrn* .,€ r 2lo\$ pa *1,€=zo\$fo" ?JaoN 4. (a) Derive an expression for i) Jacobian matrix ii) Stiffness matrix for axisymmetric element. (10 Marks) Contd.... 2
30. 30. _ _ , ___:_Page N0... 2 ME6F1 (b) 0onsider a rectangular element as shown in Fig.2. Evaluate J and B matrices at (=0, =0, (10 Markr) + + C1i,o,{) cv>- t A,> L -t a) (0, ,) 5. (a) Explain with neat sketches the library of elements used in FEM. (10 Marks) (b) Using Gaussian quadrature, evaluate the following integral by two point formula d, /], (€2 + zrt€ + rf) dt drt (10 Marks) 6, (a) For the pin jointed_ configuration shown in Fig.3 determine the stiflness values of kn, l*e and,-k2, of global stiffness matrix. (10 Marks) O hratgroivl"nL /L L I I "l/ b MvY vjup ln7 >}lac?", , E- (b) Derive an expression lor stiffness matrix ol a two noded beam element. (10 Marks) 7. (a) Explain in detail the leatures of any one commercial FEA software package. (l0Marks) (b) Bring out the differences between continuum methods and FEM. (10 Marks) Write short notes on any FOUR : a) State functions b) Galerkin methods c) Elimination method of handling boundary conditions. d) Temperature effects e) Convergence criteria. ** * ** (4x5=20 Marks)
31. 31. IPage No... l ME6Fl Reg. No. sixth serrester B.E. Degree Examflnatlon, Februar5r zooz Mechanical Englneering Ftntte Element MethodsTime: 3 hrs.l [Max.Marks : I0O Note: Answer any FIVE full questions, 1. (a) What is a banded matrix and state its advantage? (b) Calculate the eigen values of the matrix A. o:lt ?,1 lz 0 1l (c) Evaluate .4.-1 when -d. : lo 4 ol fr o 2l (d) Drptain Gauss-elimination method to solve a set of simultaneous equations. (4X6=20 Marks) 2. (a) What is finite element method? Drplain the basic steps in the formulation of finite element analysis. (12 Marks) (b) Differentiate between continuum method and finite element mettrod. (8 Marks) 3. (a) A rectangular bar in subjected to an axial load P as shown in fig.l. Derive an expression for potential energr and hence determine the extreme value of the potential 9le-1ry forthe-following data. Modutus of elasticity E :200Gpa, load P - SkNr length of the bar I : L00mm, width of the ba; b :20mm arrd thickness of the bar t : Llmm. Also state its equilibrium stability. . , l_ { T -+ L Fta, I iff (b) use Rayleigh-Ritz method to find the disptacement and the stress .,tilIill point of the rod as shown in fig.2. The area of cross section of the bar is 4OO mmz and. the modulus of elasticity of the material is 7O GPa. Assume the displacement to be second degree polynomial. (to Marks) 4. (a) -Explain the elimination approach for handling the specified displacement boundary conditions (5 Marks) Contd.... 2