Analytical Solution for Bending Stress Intensity Factor from Reissner s Plate Theory

Transcription

1 Engineering, 0, 3, doi:0.436/eng Publised Online a 0 (ttp:// Analtial Solution for Bending Stress Intensit Fator from Reissner s Plate Teor Abstrat Lalita Cattopada National Aerospae Laboratories, Bangalore, India lalita@nal.res.in Reeived Deember 30, 00; revised Januar 7, 0; aepted April 7, 0 Plate-tpe strutural members are ommonl used in engineering appliations like airraft, sips nulear reators et. Tese strutural members often ave raks arising from manufature or from material defets or stress onentrations. Designing a struture against frature in servie involves onsideration of strengt of te struture as a funtion of rak size, dimension and te applied load based on priniples of frature meanis. In most of te engineering strutures te plate tikness is generall small and in tese ases toug te lassial plate teor as provided solutions, te neglet of transverse sear deformation lea to te limitation tat onl two onditions an be satisfied on an boundar wereas we ave tree psial boundar onditions on an edge of a plate. In tis paper tis inompatibilit is eliminated b using Reissner plate teor were te transverse sear deformation is inluded and tree psiall natural boundar onditions of vanising bending moment, twisting moment and transverse sear stress are satisfied at a free boundar. Te problem of estimating te bending stress distribution in te neigbourood of a rak loated on a single line in an elasti plate of varing tikness subjeted to out-of-plane moment applied along te edges of te plate is eamined. Using Reissner s plate teor and integral transform tenique, te general formulae for te bending moment and twisting moment in an elasti plate ontaining raks loated on a single line are derived. Te tikness depended solution is obtained in a losed form for te ase in wi tere is a single rak in an infinite plate and te results are ompared wit tose obtained from te literature. Kewor: Reissner Plate Teor, Integral Transform, Stress Intensit Fator, Singular Integral Equation.. Introdution In te lassial teor of bending of tin plates, it is possible to satisf stress-free onditions at an edge onl in an approimate wa, sine onl two boundar onditions ma be enfored in onnetion wit te bi-armoni differential equation. It is te purpose of tis paper to eamine te rak problem b using te teor of bending of elasti plates developed b Reissner[] in wi te tree psiall natural boundar onditions of vanising bending moment, twisting moment and transverse sear stress must be satisfied at a free boundar. Te present problem is onerned wit te problem of an infinite plate under uniform uniaial bending far from te rak (Figure ). In te present work te omplete solution is obtained for bending stresses in te viinit of a rak tip in a plate taking transverse sear deformation into aount troug te use of Reissner s plate teor. Using Reissner s teor and integral transform teniques, te general formulae for te bending moment,twisting moment and bending stress distribution in an elasti plate ontaining raks loated on a single line are derived. Te proedure emploed ere is to formulate te problem in terms of Reissner s equations. Te stress intensit fator is obtained for te ase in wi tere is a single rak in an infinite plate and te results are ompared wit tose given in te literature. Te meanial beavior near te rak tip is modeled using Reissner s plate teor in te ase of an elasti plate in [,3].Effet of plate tikness on te bending stress distribution is inluded b Hartranft and Si []. Te general solution for bending stress in te viinit of a rak tip in a plate taking sear deformation into aount troug te use of a sit order plate bending teor of Reissner s teor is developed b Viswanat [3]. Coprigt 0 SiRes.

2 58 L. CHATTOPADHYAY Figure. Plate ontaining a single rak and subjeted to smmetri bending load. Te solution of te tin plate-bending problem was pioneered b Williams [4], wo made use of te eigen funtion epansion tenique and determined te stress distribution in te neigborood of a rak. Si et al. [5] applied a omple variable metod for evaluating te strengt of stress singularities at rak tips in plate etension and bending problems. Te general solution for finite number of raks using anisotropi elastiit is presented b Krenk[6]. Alwar and Ramaandran [7] sowed tat te stress intensit fator is nearl linear troug te tikness for tin plates, in te absene of rak losure. Using finite element metod, ark et al. [8], Alberto Zuini et al. [9] omputed stress intensit fators for tin raked plates. Using omple variable metod Zender et al. [0] alulated stress intensit fator for a finite rak in an infinite isotropi plate. Te present metod uses an integral transform tenique and does not assume an smmetr about te o-ordinate aes. Also te onstants appearing in te solution of te governing differential equations are obtained from te displaement boundar onditions b defining te derivative of te displaement disontinuities on te rak surfaes apart from te moment boundar onditions and ontinuit onditions. In te present stud, te general formulae for te stress distribution in an infinite elasti plate ontaining raks are derived and te stress intensit fator is determined in a losed form in te ase of a single rak wen te plate is subjeted to out-of-plane moments and te results are ompared wit tose from te literature.. Formulation of te Problem Let us onsider te ases of bending or twisting ations of an infinite plate b moments tat are uniforml distributed along te edges of te plate ontaining ollinear raks. We take -plane to oinide wit te middle plane of te plate before deformation. Te z-ais is assumed to be perpendiular to te middle plane. We denote te bending moment per unit lengt about -ais b and about -ais b and te twisting moment per unit lengt b. Let Q and Q be te sear fores omponents. Te tikness of te plate is and we onsider it to be small in omparison wit oter dimensions. Let us assume tat during bending, te plate undergoes te displaement w perpendiular to -plane. In te present analtial metod, we onsider te problem in wi an infinite elasti plate, ontains raks loated on a single line is ated upon b applied moments. Let te o-ordinate sstem be so osen tat te -ais oinides wit te line on wi te raks are loated. Let L denote te union of intervals oupied b te raks on te -ais and is te interval not oupied b te raks. Suppose tat a tin plate ontaining a rak is subjeted to uniform bending or twisting moments at infinit. Sine te rak surfae is tration-free te boundar onditions along te rak surfae permitting all of te free edge onditions for te present problem is given b te following equations: Te free boundar onditions on te rak surfae are given b,,0 0, L (),0 0, L () Q,0 0, L (3) Te solution to tis problem ma be obtained b superposing te simple solution of an unraked plate under uniform bending moment or twisting moment to tat of a raked plate wit bending or twisting moment applied to te rak surfaes. Tat is, te solution ma be obtained b using standard superposition tenique and tus for te purpose of evaluating te rak tip singular stresses it is suffiient to onsider te problem in wi self-equilibrating rak surfae loa are te onl eternal loa. Tus, it suffies to solve te problem of speifing uniform bending and twisting moment on te rak segment of te plate. Let te desired sstem be omposed of two parts, one te uniform moment field and te oter a perturbation field due to te rak wi dies out at infinit. Wile te boundar onditions along te free edges of te rak require tration free onditions, it is possible to formulate te problem as one of finding solutions for te perturbation solutions satisfing te field equations and te boundar onditions * G,0, L (4) Coprigt 0 SiRes.

3 * H,0, L (5) Q,0 0 Te equilibrium equations are given b, Q 0 Q 0 Q Q 0 Also, te stress omponents are te linear ombination of te variable z. σ z ;σ ;σ 3 z z 3 (9) 3 Te strain ompatibilit equation is given b, L. CHATTOPADHYAY 59 (6) (7) (8) (0) If we define te moment resultants in terms of te biarmoni funtion (, ) as given b ; ; () ten te governing Equations (6)-(8) are satisfied. From te ompatibilit onditions (0), te present problem redues to tat of solving te bi-armoni equation in (, ) 4 0 () were, (3) 4 4 Let G (, ) be te Fourier transform of, for 0. Ten i G,, e d, 0 (4) Taking Fourier transformation of te bi-armoni equation w. r. t te variable, we get te ordinar differential equation in G, as given b were G is te Fourier transformation of, for 0, and P, P, Q, Q are te unknown funtions to be determined. From te moment boundar onditions we ave te following equations,,0,0 0, (8),0,0 0, (9) Te bending and twisting moments in terms of G, for 0 are given b G, i, e d, 0 π (0) i, G, e d, 0 () π i G, π i, e d, 0 () Similarl we get te bending and twisting moments for 0 in terms of G, Te bending and twisting moments in terms of G, for 0 are given b G, i, e d, 0 π (3) i, G, e d, 0 (4) π i G, π i, e d, 0 (5) Te displaement omponents are given b te following epressions: w u z (6) u w z (7) Te displaement boundar onditions are given b 0, A (8) B 0, (9) d G, 0 (5) were te displaement disontinuities are defined b te d funtions A(), B() wose solutions are given b A u,0 u,0, L (30) G, P Qe 0 (6) G, P Qe 0 (7) B u,0 u,0, L (3) Coprigt 0 SiRes.

4 50 L. CHATTOPADHYAY Solving te above equations we obtain te unknowns P, P, Q, Q as given b, P P sgn BD and te supersripts () and () denote te omponents in te upper alf plane 0 and lower alf plane 0 respetivel. From te ontinuit onditions and te moment boundar onditions we get te four simultaneous equations for solving,,, P P Q Q A D 3 P P Q Q P P (3a) Q P Q P (3b) (3) i P P i Q Q B D (3d) i (33a) 4 4 (33b) Q Asgn ib D 4 (33) Q Asgn ib D,, (34a) G G, (34b) G, i (34) Substituting te values of P, Q into (), we get te bending moment resultants in te upper alf plane 0, in te transformed o-ordinates as given b te following equation, D i, A Bisgn e 8π d, 0 (35a) Performing te inner integral in terms of A(s) and B(s) we get te bending moment resultants in te upper alf-plane 0, in terms of te unknown displaement funtions A(s) and B(s) as given b D, A s B s s s Substituting te values of P, Q into (4), we get te bending moment resultants in te lower alf plane s s 3 d 0 s s (35b) 0, in te transformed oordinates as given b te following equation D i, A Bisgn e 8π d, 0 (35) Performing te inner integral in terms of A(s) and B(s) we get te bending moment resultants in te lower alf-plane 0, in terms of te unknown displaement funtions A(s) and B(s) as given b s s 3 D, As Bs s s d 0, (35d) s s Combining te Equations (35b) and (35d) we get te epression for bending moment resultant, s s 3 D, As Bs s s d 0, (36) s s Coprigt 0 SiRes.

5 L. CHATTOPADHYAY 5 Similarl te epression for te twisting moment, as given b D, A s s B s s s 3 s d 0, s s (37) were A() s and Bs () are te unknown funtions to be determined from te given boundar onditions. Te limiting values as 0 and 0 of te bending and twisting moments along te rak line are given b, Bs D,0 (38a) s As D,0 (38b) s B using te onditions (4)-(5) in te above epressions, te interval of integration redues to L. From te boundar onditions (4) and te above relations we get te singular integral equations L A s G, L (39) s L L B s H, L (40) s L D were, for te determination of unknown funtions A and B on te interval L. One te π funtions A(s) and B(s), L are known, te bending and twisting moments for te rak problem are determined. 3. Single Crak Problem In order to illustrate te present proedure, we give te details in te ase of a single rak opened b te ation of smmetri bending load applied at te edges of te plate. In tis setion, we onsider te problem of determining te distribution of bending stress in te viinit of a Griffit rak of lengt, ouping te interval (-, ) on te -ais in an infinite isotropi elasti plate. Te bending moment resultants and te transverse sear fore omponents are given b, w w D 5 Q w w Q D 5 (4) (4) w w Q D (43) w w Q D (44) Taking Fourier transform of te above equations w. r. t., we get te displaement omponent in terms of te bending moment omponents in te transformed o-ordinate sstem as given b, w w (45) D Te transverse sear fore Q are given b, Q i (46) were is alulated from te equation, 0 0 (47) Taking Fourier Transform of te above Equation (47) we get d 0 (48) d e e (49) were 0 Sine Q 0 and Q 0 as we ave 0 e (50) Te onstant 0is determined from Q,0 0, Q Substituting, i,, and, (5) in (46) and using te rak surfae boundar ondition, Q,0 0, Coprigt 0 SiRes.

6 5 L. CHATTOPADHYAY te onstant is given b, D C π B (5) From (50), (5) te funtion, is given b, C π i s, Bs e e d (53) s (54) s D, B s K s d 8π, D K s Bs 5 8π 5 0 (55) s 0, D Bs K 5 8π 5 0 s D Bs (56) Lt, Lt 0 (57) 0 5 8π 5 0 s, D Bs Lt 0 5 8π s Te bending moment,0 along te rak line 0, as 0 is given b 0 (58) D B s D B s,0 d aking use of te boundar ondition (4) we get te singular integral equation for te determination of B(s) as given b, 3 s D B s H 8π (60) Solving for B(s), we get te following epression for B(s), * H 0 s 8π s s (59) Te bending moment resultant along te rak line from te above equation is given b, sgn,0 0 (63) 3 Te bending stress,0 along te rak line B s 4 3 D π a s H t s d (6) Substituting te value of B(s) in (38a), te bending moment for a single rak problem in te limiting ase as 0 is given b, sgn H t,0 dt π 3 t (6) 0, as 0 is given b 0zsgn,0, 3 3 (64) Te bending stress intensit fator K I due to bending moment at z = / is given b 6 0 KI Lim,0 3 (65) Coprigt 0 SiRes.

7 L. CHATTOPADHYAY 53 For small, te bending moment on te rak line a = 0 is alulated as follows:, 0 D B s K 5 8π s were K 0 ln ; ; (66) Hene te bending moment along te rak line =0 is given b, D B s D B s,0 ln s 8π s (67) Substituting te value of B(s) from (6) into te above equation and performing te inner integral we get te bending moment along te rak line = 0 as given b, sgn 0,0 ln, 3 43 (68) Te bending stress,0 along te rak line 0, is given b sgn z 0,0 ln, (69) Stress intensit fator K I at z = /,due to bending mo- ment is given b 60 KI lim [ ],0 ln 3 43 (70) Te grap of non-dimensional stress intensit fator vs. tikness for 0.3 is plotted in Figure and te stress intensit fator is in good agreement wit te results in [] and [3]. Te stress variation near te rak tip, alulated from Equation (69) is plotted in Figure 3. For eample, a value of / = 5.0 is assumed for alulation. 4. Results and Disussion Te variation of non-dimensional stress intensit stress intensit fator wit tikness of te plate is sown in Figure. Te small differenes between te present results and in te referenes [] and [3] ma be due to two Figure. Variation of non-dimensional stress intensit wit tikness of te plate. Figure 3. Stress,0 distribution near te rak tip for, / = 5.0. Coprigt 0 SiRes.

8 54 L. CHATTOPADHYAY different approaes being used in [] and [3]. Hartranft and Si [] used more rigorous metod using eigenfuntion epansions for plate bending problem introduing te effet of plate tikness on rak-tip stress distribution. Te approimate metod based partl on finite element analsis and partl on a ontinuum analsis using Irwin s [] solution for an infinite plate is used in [3]. Figure 3 sows te eponential variation of normal stress omponent near te rak tip for, z = /. It dereases awa from te rak tip as epeted. Future work in tis diretion is planned to solve omposite plate problems wit delamination. 5. Conlusions A simple metod for determining te analtial epression for te bending stress distribution, in te viinit of a rak in an infinite elasti plate using Reissner plate teor is eplained. Te general formulae for te bending moment and twisting moment in an elasti plate ontaining raks loated on a single line are derived. Te solution is obtained in a losed form for te ase in wi tere is a single rak in an infinite plate and te stress intensit fator is alulated as a funtion of plate tikness, wen te plate is subjeted to smmetri bending loa. Te stress intensit fator is ompared wit tat obtained in te literature. 0. Referenes [] E. Reissner, Te Effet of Transverse Sear Deformation on te Bending of Elasti Plates, ASE Journal of Applied eanis, Vol., 945, pp. A68-A77. [] R. J. Hartranft and G. C. Si, Effet of Plate Tikness on te Bending Stress Distribution around Troug Craks, Journal of atematis and Psis, Vol. 47, 76-9,968 [3] S. Viswanat, On te Bending of Plates wit troug Craks from Higer Order Plate Teories, P. D Tesis, Indian Institute of Siene, 985. [4]. L. Williams, Te Bending Stress Distribution at te Base of a Stationar Crak, ASE Journal of Applied eanis, Vol. 8, 96, pp [5] G. C. Si, P. C. Paris and F. Erdogan, Crak-Tip, Stress-Intensit Fators for Plate Etension and Plate Bending Problems, ASE Journal of Applied eanis, Vol. 9, 96, pp [6] S. Krenk, Te Stress Distribution in an Infinite Anisotropi Plate wit Collinear Craks, International Journal of Soli and Strutures, Vol., No. 4, 975, pp doi:0.06/ (75) [7] R. S. Alwar and K. N. Ramaandran, Influene of Crak Closure on te Stress Intensit Fator for Plates Subjeted to Bending A 3-D Finite Element Analsis, Engineering Frature eanis, Vol. 7, No. 4, 983, pp doi:0.06/ (83) [8]. J. Viz, D. O. Potond, A. T. Zender, C. C. Rankin and E. Riks, Computation of embrane and Bending Stress Intensit Fators for Tin, Craked Plates, International Journal of Frature, Vol. 7, No., 995, pp doi:0.007/bf [9] A. Zuini, C.-Y. Hui and A. T. Zender, Crak Tip Stress Fiel for Tin Plates in Bending, Sear and Twisting: A Tree Dimensional Finite Element Stud, International Journal of Frature, Vol. 04, No. 4, 000, pp [0] A. T. Zender and C.-Y. Hui, Stress Intensit Fators for Plate Bending and Searing Problems, Journal of Applied eanis, Vol. 6, No. 3, 994, pp doi:0.5/.905 [] G. R. Irwin, Analsis of Stresses and Strains near te End of a rak Traversing a Plate, Journal of applied eanis, 957, Vol. 4, pp Coprigt 0 SiRes.