Introduction to Finite Elements in Engineering. Tirupathi R. Chandrupatla Rowan University Glassboro, New Jersey

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1 Introduction to Finite Elements in Engineering Tirupthi R. Chndruptl Rown Universit Glssboro, New Jerse Ashok D. Belegundu The Pennslvni Stte Universit Universit Prk, Pennslvni Solutions Mnul Prentice Hll, Upper Sddle River, New Jerse 758 Introduction to Finite Elements in Engineering, Fourth Edition, b T. R. Chndruptl nd A. D. Belegundu. ISBN Person Eduction, Inc., Upper Sddle River, NJ. All rights reserved. This publiction is protected b Copright write to: Rights nd Permissions Deprtment, Person Eduction, Inc., Upper Sddle River, NJ 758.

2 CONTENTS Prefce Chpter Fundmentl Concepts Chpter Mtri Algebr nd Gussin Elimintion 8 Chpter One-Dimensionl Problems 6 Chpter Trusses 6 Chpter 5 Bems nd Frmes 86 Chpter 6 Two-Dimensionl Problems Using Constnt Strin Tringles Chpter 7 Aismmetric Solids Subjected to 5 Aismmetric Loding Chpter 8 Two-Dimensionl Isoprmetric Elements 8 nd Numericl Integrtion Chpter 9 Three-Dimensionl Problems in Stress 7 Anlsis Chpter Sclr Field Problems 8 Chpter Dnmic Considertions 6 Chpter Preprocessing nd Postprocessing 8 Introduction to Finite Elements in Engineering, Fourth Edition, b T. R. Chndruptl nd A. D. Belegundu. ISBN Person Eduction, Inc., Upper Sddle River, NJ. All rights reserved. This publiction is protected b Copright write to: Rights nd Permissions Deprtment, Person Eduction, Inc., Upper Sddle River, NJ 758.

3 PREFACE This solutions mnul serves s n id to professors in teching from the book Introduction to Finite Elements in Engineering, th Edition. The problems in the book fll into the following ctegories:. Simple problems to understnd the concepts. Derivtions nd direct solutions. Solutions requiring computer runs. Solutions requiring progrm modifictions Our bsic philosoph in the development of this mnul is to provide complete guidnce to the techer in formulting, modeling, nd solving the problems. Complete solutions re given for problems in ll ctegories stted. For some lrger problems such s those in three dimensionl stress nlsis, complete formultion nd modeling spects re discussed. The students should be ble to proceed from the guidelines provided. For problems involving distributed nd other tpes of loding, the nodl lods re to be clculted for the input dt. The progrms do not generte the lods. This clcultion nd the boundr condition decisions enble the student to develop phsicl sense for the problems. The students m be encourged to modif the progrms to clculte the lods utomticll. The students should be introduced to the progrms in Chpter right from the point of solving problems in Chpter 6. This will enble the students to solve lrger problems with ese. The input dt file for ech progrm hs been provided. Dt for problem should follow this formt. The best strteg is to cop the emple file nd edit it for the problem under considertion. The dt from progrm MESHGEN will need some editing to complete the informtion on boundr conditions, lods, nd mteril properties. We thnk ou for our enthusistic response to our first three editions of the book. We look forwrd to receive our feedbck of our eperiences, comments, nd suggestions for mking improvements to the book nd this mnul. Tirupthi R. Chndruptl P.E., CMfgE Deprtment of Mechnicl Engineering Rown Universit, Glssboro, NJ 88 e-mil: chndruptl@rown.edu Ashok D. Belegundu Deprtment of Mechnicl nd Nucler Engineering The Pennslvni Stte Universit Universit Prk, PA 68 e-mil: db@psu.edu Introduction to Finite Elements in Engineering, Fourth Edition, b T. R. Chndruptl nd A. D. Belegundu. ISBN Person Eduction, Inc., Upper Sddle River, NJ. All rights reserved. This publiction is protected b Copright write to: Rights nd Permissions Deprtment, Person Eduction, Inc., Upper Sddle River, NJ 758.

4 CHAPTER FUNDAMENTAL CONCEPTS. We use the first three steps of Eq.. Adding the bove, we get ε ε ε σ σ σ ν ν E E E σ σ σ ν ν E E E σ σ σ ν ν E E E ε ε Adding nd subtrcting ( σ σ σ ) ν ε E σ ν from the first eqution, E ( σ σ σ ) ν ν ε σ E E Similr epressions cn be obtined for ε, nd ε. From the reltionship for γ nd Eq.., E τ γ etc. ( ν) Above reltions cn be written in the form σ Dε where D is the mteril propert mtri defined in Eq..5.. Note tht u () stisfies the ero slope boundr condition t the support. Introduction to Finite Elements in Engineering, Fourth Edition, b T. R. Chndruptl nd A. D. Belegundu. ISBN Person Eduction, Inc., Upper Sddle River, NJ. All rights reserved. This publiction is protected b Copright write to: Rights nd Permissions Deprtment, Person Eduction, Inc., Upper Sddle River, NJ 758.

5 . Plne strin condition implies tht σ σ ε ν ν E E which gives σ ν σ σ ( ) σ E 6 We hve, σ psi σ psi E psi ν. On substituting the vlues,. σ psi. Displcement field u v u v ε u t, ε ( 6) ( 6 ) ( 6) ( 6) u v v 6 9 u v ( 6 ).5 On inspection, we note tht the displcements u nd v re given b It is then es to see tht u. v Introduction to Finite Elements in Engineering, Fourth Edition, b T. R. Chndruptl nd A. D. Belegundu. ISBN Person Eduction, Inc., Upper Sddle River, NJ. All rights reserved. This publiction is protected b Copright write to: Rights nd Permissions Deprtment, Person Eduction, Inc., Upper Sddle River, NJ 758.

6 ε ε γ u v.6 The displcement field is given s u v. u 6 v 7 () The strins re then given b u ε 6 v ε u v γ (b) In order to drw the contours of the strin field using MATLAB, we need to crete script file, which m be edited s tet file nd sve with.m etension. The file for plotting ε is given below file probp5b.m [X,Y] meshgrid(-:.:,-:.:); Z..*X.^6.*Y.^; [C,h] contour(x,y,z); clbel(c,h); On running the progrm, the contour mp is shown s follows: Introduction to Finite Elements in Engineering, Fourth Edition, b T. R. Chndruptl nd A. D. Belegundu. ISBN Person Eduction, Inc., Upper Sddle River, NJ. All rights reserved. This publiction is protected b Copright write to: Rights nd Permissions Deprtment, Person Eduction, Inc., Upper Sddle River, NJ 758.

7 Contours of ε Contours of ε nd γ re obtined b chnging Z in the script file. The numbers on the contours show the function vlues. (c) The mimum vlue of ε is t n of the corners of the squre region. The mimum vlue is..7 (, ) (u, v) ). u u. v u v u v b) ε ε γ. Introduction to Finite Elements in Engineering, Fourth Edition, b T. R. Chndruptl nd A. D. Belegundu. ISBN Person Eduction, Inc., Upper Sddle River, NJ. All rights reserved. This publiction is protected b Copright write to: Rights nd Permissions Deprtment, Person Eduction, Inc., Upper Sddle River, NJ 758.

8 .8. σ τ σ MP MP n From Eq..8 we get T T T n σ 5.67 MP τ 6. MP τ.7 MP T n n n n τ σ τ σ τ.9 MP T n n n n MP 5 MP T τ τ σ T n.9 From the derivtion mde in P., we hve n n n σ τ MP MP σ σ E ( )( ) ( ν ) ν ν E [ ε νε νε ] which cn be written in the form ( )( ) [( ν ) ε ] νε v ν ν nd E τ γ ( ν) Lme s constnts λ nd µ re defined in the epressions σ τ λ λε µγ ( ν)( ν) E µ Eν ( ν) µε On inspection, v Introduction to Finite Elements in Engineering, Fourth Edition, b T. R. Chndruptl nd A. D. Belegundu. ISBN Person Eduction, Inc., Upper Sddle River, NJ. All rights reserved. This publiction is protected b Copright write to: Rights nd Permissions Deprtment, Person Eduction, Inc., Upper Sddle River, NJ 758.

9 µ is sme s the sher modulus G... ε ε. T E GP α ε α T σ E δ C du d L L 5-6 /.6 ( ε ε ) 69.6 MP du d C d L L. Following the steps of Emple., we hve ( 8 5) 8 8 q 8 q 6 5 Above mtri form is sme s the set of equtions: 7 q 8 q 6 8 q 8 q 5 Solving for q nd q, we get q. mm q.87 mm Introduction to Finite Elements in Engineering, Fourth Edition, b T. R. Chndruptl nd A. D. Belegundu. ISBN Person Eduction, Inc., Upper Sddle River, NJ. All rights reserved. This publiction is protected b Copright write to: Rights nd Permissions Deprtment, Person Eduction, Inc., Upper Sddle River, NJ 758.

10 . When the wll is smooth, σ. T is the temperture rise. ) When the block is thin in the direction, it corresponds to plne stress condition. The rigid wlls in the direction require ε. The generlied Hooke s lw ields the equtions σ ε ν α T E σ ε α T E From the second eqution, setting ε, we get σ Eα T. ε is then clculted using the first eqution s ( ν) α T. b) When the block is ver thick in the direction, plin strin condition previls. Now we hve ε, in ddition to ε. σ is not ero. σ σ ε ν ν α T E E σ σ ε ν α T E E σ σ ε ν α T E E From the lst two equtions, we get Eα T ν σ σ Eα T ν ν ε is now obtined from the first eqution. Introduction to Finite Elements in Engineering, Fourth Edition, b T. R. Chndruptl nd A. D. Belegundu. ISBN Person Eduction, Inc., Upper Sddle River, NJ. All rights reserved. This publiction is protected b Copright write to: Rights nd Permissions Deprtment, Person Eduction, Inc., Upper Sddle River, NJ 758.

11 . For thin block, it is plne stress condition. Treting the nominl sie s, we m set the. initil strin ε α T in prt () of problem.. Thus σ.e..5 The potentil energ Π is given b Π du d EA d ugad Consider the polnomil from Emple., u du d ( ) ( ) ( ) g E A On substituting the bove epressions nd integrting, the first term of becomes nd the second term Thus ugad ud ( ) Π Π u this gives ( ). 5.6 E A f Introduction to Finite Elements in Engineering, Fourth Edition, b T. R. Chndruptl nd A. D. Belegundu. ISBN Person Eduction, Inc., Upper Sddle River, NJ. All rights reserved. This publiction is protected b Copright write to: Rights nd Permissions Deprtment, Person Eduction, Inc., Upper Sddle River, NJ 758.

12 We use the displcement field defined b u. u t u t We then hve u ( ), nd du/d ( ). The potentil energ is now written s Π 6 du d ( ) d ( ) 5 ( ) d ( ) 5 6 d fud d d Π This ields,. Displcemen u.( ) Stress σ E du/d.( ).7 Let u be the displcement t mm. Piecewise liner displcement tht is continuous in the intervl 5 is represented s shown in the figure. u u u 5 Introduction to Finite Elements in Engineering, Fourth Edition, b T. R. Chndruptl nd A. D. Belegundu. ISBN Person Eduction, Inc., Upper Sddle River, NJ. All rights reserved. This publiction is protected b Copright write to: Rights nd Permissions Deprtment, Person Eduction, Inc., Upper Sddle River, NJ 758.

13 u t u u t u / u (u /) du/d u / 5 u t 5 5 u u t u u / (5/)u u (5/)u (u /) du/d u / Π E l du A d d u u Π El A Est A El A Est A u u 5 E st du A d d u u Π u E A E A u l st Note tht using the units MP (N/mm ) for modulus of elsticit nd mm for re nd mm for length will result in displcement in mm, nd stress in MP. Thus, E l 7 MP, E st, nd A 9 mm, A mm. On substituting these vlues into the bove eqution, we get u.9 mm This is precisel the solution obtined from strength of mterils pproch.8 In the Glerkin method, we strt from the equilibrium eqution d du EA g d d Following the steps of Emple., we get Introducing du EA d dφ d d gφd Introduction to Finite Elements in Engineering, Fourth Edition, b T. R. Chndruptl nd A. D. Belegundu. ISBN Person Eduction, Inc., Upper Sddle River, NJ. All rights reserved. This publiction is protected b Copright write to: Rights nd Permissions Deprtment, Person Eduction, Inc., Upper Sddle River, NJ 758.

14 ( ) ( ) nd, φ φ u u where u nd φ re the vlues of u nd φ t respectivel, ( ) ( ) φ d d u On integrting, we get 8 φ u This is to be stisfied for ever φ, which gives the solution u.5.9 We use t t u u u This implies tht 8 nd ( ) ( ) ( ) ( ) d du u nd re considered s independent vribles in ( ) ( ) [ ] ( ) Π d on epnding nd integrting the terms, we get Π We differentite with respect to the vribles nd equte to ero. Introduction to Finite Elements in Engineering, Fourth Edition, b T. R. Chndruptl nd A. D. Belegundu. ISBN Person Eduction, Inc., Upper Sddle River, NJ. All rights reserved. This publiction is protected b Copright write to: Rights nd Permissions Deprtment, Person Eduction, Inc., Upper Sddle River, NJ 758.

15 Π Π On solving, we get.7856 nd.5. On substituting in the epression for u, t, u.79 This pproimtion is close to the vlue obtined in the emple problem.. () Π L L T σ εad T ( )ud σ E ε nd ε On substitution, Π Π 6 du EA d du d d du ( 6 ) d ud ud d Tud 6 Tud (b) Since u t nd 6, nd u, we hve u du d ( 6) ( 6) On substituting nd integrting, Π Setting dπ/d gives Introduction to Finite Elements in Engineering, Fourth Edition, b T. R. Chndruptl nd A. D. Belegundu. ISBN Person Eduction, Inc., Upper Sddle River, NJ. All rights reserved. This publiction is protected b Copright write to: Rights nd Permissions Deprtment, Person Eduction, Inc., Upper Sddle River, NJ 758.

16 .5 du σ E 6.95( 6) d Plots of displcement nd stress re given below: Displcement u Stress.. t 6 implies tht 6 ( 8 ) 6, which ields Substituting for k, h, L, nd in I, we get I I I ( ) d ( 5) [ ( 8 ) 8] ( ) d 5( 8 9) Introduction to Finite Elements in Engineering, Fourth Edition, b T. R. Chndruptl nd A. D. Belegundu. ISBN Person Eduction, Inc., Upper Sddle River, NJ. All rights reserved. This publiction is protected b Copright write to: Rights nd Permissions Deprtment, Person Eduction, Inc., Upper Sddle River, NJ 758.

17 di d di d On solving, Substituting into the epression for, we get Since u t, the displcement stisfing the boundr condition is u. Also the coordintes re, nd. The potentil energ for the problem is du π EA d P u Pu d du We hve u, u, E, A, nd d. Thus π ( ) d. dπ For sttionr vlue, setting, we get d, which gives.75. The pproimte solution is u.75. Introduction to Finite Elements in Engineering, Fourth Edition, b T. R. Chndruptl nd A. D. Belegundu. ISBN Person Eduction, Inc., Upper Sddle River, NJ. All rights reserved. This publiction is protected b Copright write to: Rights nd Permissions Deprtment, Person Eduction, Inc., Upper Sddle River, NJ 758.

18 . Use Glerkin pproch with pproimtion u b c to solve du u d u ( ) The week form is obtined b multipling b φ stisfing φ ( ). du φ u d d We now set u b c into the bove integrl, stisfing u ( ) nd φ. On introducing these ( )( b c b c ) d ( ) ( ) b b c c d b b c c d On integrting, we get b c c b b c c b b c b c 6 This must be stisfied for ever nd. Thus the equtions to be solved re 7 7 b c 6 b c The solution is b.957, c.8. Thus u Introduction to Finite Elements in Engineering, Fourth Edition, b T. R. Chndruptl nd A. D. Belegundu. ISBN Person Eduction, Inc., Upper Sddle River, NJ. All rights reserved. This publiction is protected b Copright write to: Rights nd Permissions Deprtment, Person Eduction, Inc., Upper Sddle River, NJ 758.

19 P The deflection nd slope t due to P re nd EI nd slope t L due to lod P re ( ) P L P v EI EI P v EI The deflection nd slope due to lod P re PL v EI PL v EI We then get v v v v v v P EI. Using this the deflection.5 (,) (,) (,) (,) () The displcement of B is given b (.,.) nd A, C, nd D remin in their originl position. Consider displcement field of the tpe u v b b b b The four constnts cn be evluted using the known displcements Introduction to Finite Elements in Engineering, Fourth Edition, b T. R. Chndruptl nd A. D. Belegundu. ISBN Person Eduction, Inc., Upper Sddle River, NJ. All rights reserved. This publiction is protected b Copright write to: Rights nd Permissions Deprtment, Person Eduction, Inc., Upper Sddle River, NJ 758.

20 At A (, ) At B (, ) At C (, ) At D (, ) b. b b. b b b b b b The solution is,.,,. b, b., b, b u.. This gives v.. (b) The sher strin t B is u v γ... B ( ) ( ) γ.... Introduction to Finite Elements in Engineering, Fourth Edition, b T. R. Chndruptl nd A. D. Belegundu. ISBN Person Eduction, Inc., Upper Sddle River, NJ. All rights reserved. This publiction is protected b Copright write to: Rights nd Permissions Deprtment, Person Eduction, Inc., Upper Sddle River, NJ 758.

21 CHAPTER MATRIX ALGEBRA AND GAUSSIAN ELIMINATION., 8 d A () [ ] T d d I (b) det A 8 [ ()() ( ) ( ) ] ( ) [( )() ()( )] (c) The chrcteristic eqution is det (A - λ I ), or 8 det λ λ λ which ields 55 5 λ λ λ Hndbooks (e.g., CRC Mthemticl Hndbook) give eplicit solutions to cubic equtions. Here equtions given in Chpter 9 re used, which give formuls for finding the eigenvlues of the () smmetric stress tensor. Referring to Section 9. in the tet, we hve I A A A 5, I 55, I Thus,, b, c 5.6, θ 7. whence λ.5, λ 5.665, λ 9. Note: Since ll λ i >, A is positive definite. Now, eigenvector i corresponding to eigenvlue λ i is obtined from ( - λ i I ) i, i,, Thus, is obtined s Thus, Introduction to Finite Elements in Engineering, Fourth Edition, b T. R. Chndruptl nd A. D. Belegundu. ISBN Person Eduction, Inc., Upper Sddle River, NJ. All rights reserved. This publiction is protected b Copright write to: Rights nd Permissions Deprtment, Person Eduction, Inc., Upper Sddle River, NJ 758.

22 Onl two of the bove three equtions re independent. We hve Letting, we get [.7,.9, ] T The length of the vector is T. 8. Normliing to be unit vector ields [.7,.668,.7] T. Similrl, [.95,.577, -.65] T, [.85, -.7,.] T. (d) Solution to A b using Algorithm for generl mtri: n First Step (k ) i ( nd row) 8 A c / -/8 -/ () (-/)(-) 7/, () -, d () --(-/)() -/ i ( rd row) c () -, (), d () 8 () Thus () A 7 /, d / Second Step (k ) i ( rd row) c -6/7, () (-6/7)(-) /7, d () (-6/7)(-/)8/7 Thus A () 8 7 /, d / 7 () / 8/ 7 Introduction to Finite Elements in Engineering, Fourth Edition, b T. R. Chndruptl nd A. D. Belegundu. ISBN Person Eduction, Inc., Upper Sddle River, NJ. All rights reserved. This publiction is protected b Copright write to: Rights nd Permissions Deprtment, Person Eduction, Inc., Upper Sddle River, NJ 758.

23 Bck-Substitution 8 7 / / / 7 8/ 7 Third row gives: /7 8/7 whence 6 nd row then gives 5, st row gives.5 Thus, solution is [.5, 5, 6 ] T. Solution to A b using Algorithm for smmetric, bnded mtri 8 A is stored s n, nbw First Step (k ) nbk min (,) nd row (i ): i c / -/ j j (-/)(-) 7/ 8 Thus A () 7 / Second Step (k ) nbk rd row (i ): c -6/7 j j /7 8 Thus A () 7 / / 7 reduction of right-hnd-side vector d nd bck-substitution is sme s in Algorithm bove, resulting in the the sme solution [.5, 5, 6 ] T. Introduction to Finite Elements in Engineering, Fourth Edition, b T. R. Chndruptl nd A. D. Belegundu. ISBN Person Eduction, Inc., Upper Sddle River, NJ. All rights reserved. This publiction is protected b Copright write to: Rights nd Permissions Deprtment, Person Eduction, Inc., Upper Sddle River, NJ 758.

24 . N [ ξ ξ ] () (b) N dξ ξ ξ ( ξ ) T / N N dξ ( ) ( ) 6/ 5 ξ ξ ξ. q ( ) ( ) 6.5 T Q c T where.5 Q nd c.5 6. The detiled lgorithm is given in the tet. This is n ecercise in computer progrmming. The solutions re () (.5,.5,.5) (b) (.55, 5., 6.).5 A M M M det det, M i j C ij ( ) M ij The minors re, M X X X X.., M X X.... X X X. X., M, M, M Introduction to Finite Elements in Engineering, Fourth Edition, b T. R. Chndruptl nd A. D. Belegundu. ISBN Person Eduction, Inc., Upper Sddle River, NJ. All rights reserved. This publiction is protected b Copright write to: Rights nd Permissions Deprtment, Person Eduction, Inc., Upper Sddle River, NJ 758.

25 Thus, the co fctor mtri is C A nd A det A.6 Are det.7 A det A det A det A A A A.65 A / A.8, A / A.5, A / A.7 C T ields.8 A i,j B i, j-i for j i Thus, A, corresponds to B, nd B 6, A 6,6.9 Full () mtri BANDED n, nbw No. of storge loctions (n) (nbw) SKYLINE Introduction to Finite Elements in Engineering, Fourth Edition, b T. R. Chndruptl nd A. D. Belegundu. ISBN Person Eduction, Inc., Upper Sddle River, NJ. All rights reserved. This publiction is protected b Copright write to: Rights nd Permissions Deprtment, Person Eduction, Inc., Upper Sddle River, NJ 758.

26 No. of storge loctions no. of column entries... () () / 55. A 6 Cholesk decomposition follows the steps s per Eq. (.): k ( st row) k ( nd row), 6 (/ ). 969 k ( rd row) (/ )(/ ), Thus, L We m see tht A L L T. Also, progrm implementtion is given in Subroutine CHOLESKI within Progrm GENEIGEN.. B epnding the mtri equtions, we obtin set of simultneous equtions for the nd u coefficients, resulting in the solution 5 L., U Introduction to Finite Elements in Engineering, Fourth Edition, b T. R. Chndruptl nd A. D. Belegundu. ISBN Person Eduction, Inc., Upper Sddle River, NJ. All rights reserved. This publiction is protected b Copright write to: Rights nd Permissions Deprtment, Person Eduction, Inc., Upper Sddle River, NJ 758.

27 . D (, 7) C (6, 6) B (7, ) A (, ) We split the qudrilterl into two tringles s shown. Using the epression from Problem.6, the re is then given b Are det 7 det det 6 det ( 5) ( ).5 At the second step, we subtrct the first row from second row nd third row, which does not chnge the determinnt vlue.. Setting cosθ sinθ T sinθ cosθ, it is es to show tht T TT.. A 5 Introduction to Finite Elements in Engineering, Fourth Edition, b T. R. Chndruptl nd A. D. Belegundu. ISBN Person Eduction, Inc., Upper Sddle River, NJ. All rights reserved. This publiction is protected b Copright write to: Rights nd Permissions Deprtment, Person Eduction, Inc., Upper Sddle River, NJ 758.

28 () Minors M 7 M 9 M 5 5 M M 7 M M M 5 M (b) Cofctors C M 7 C M 9 C M C M C M 7 C M 7 C M C M 5 C M (c) Adjoint 7 7 T AdjA C (d) Determinnt deta (7)-(9)() (e) Inverse A 7 AdjA det A 7 Introduction to Finite Elements in Engineering, Fourth Edition, b T. R. Chndruptl nd A. D. Belegundu. ISBN Person Eduction, Inc., Upper Sddle River, NJ. All rights reserved. This publiction is protected b Copright write to: Rights nd Permissions Deprtment, Person Eduction, Inc., Upper Sddle River, NJ 758.

29 CHAPTER ONE-DIMENSIONAL PROBLEMS. q.in P q.5in 5in in in A. in E 6 psi () At point P, (b) N N u Nq N q ε in du d ε Bq q q B N q [ ] B [ ] 8. ε [ ] σ Eε 875 psi 8.75 ksi (c) k EA L 6.5 (d) U e ½ q T kq. U [..5].5 6 e 56.5 lb-in.5. NBW m [(-), (-), (5-), (5-)] Introduction to Finite Elements in Engineering, Fourth Edition, b T. R. Chndruptl nd A. D. Belegundu. ISBN Person Eduction, Inc., Upper Sddle River, NJ. All rights reserved. This publiction is protected b Copright write to: Rights nd Permissions Deprtment, Person Eduction, Inc., Upper Sddle River, NJ 758.

30 . (i) Plotting the displcement is stright forwrd.6 u mm -. 5 mm (ii) Strins re constnt in ech of the three elements Since the displcements re liner within the elements, the strins re constnt over ech element. Strin in Element -: L - 5 mm ε 5 (.). Strin in Element -: L - 8 mm.6 ε.75 8 Strin in Element -: L - mm (.6). ε.7 The plotted curve represents the slope of the curve shown in section (i). (iii) The B mtri for element - is given b (iv) B The stiffness mtri of element - is given b [ ] [.5.5] k EA L ( )( ) 5 N/mm Introduction to Finite Elements in Engineering, Fourth Edition, b T. R. Chndruptl nd A. D. Belegundu. ISBN Person Eduction, Inc., Upper Sddle River, NJ. All rights reserved. This publiction is protected b Copright write to: Rights nd Permissions Deprtment, Person Eduction, Inc., Upper Sddle River, NJ 758.