FINITE ELEMENT SIMULATION OF IMPACTED FIBROUS COMPOSITE PANELS AND EFFICIENT PREDICTION OF TRANSVERSE SHEAR STRESSES

Transcription

1 VOL. 4, NO. 7, SEPTEMBER 9 ISSN Asian Research Publishing Newor (ARPN). All righs reserved. FINITE ELEMENT SIMULATION OF IMPACTED FIBROUS COMPOSITE PANELS AND EFFICIENT PREDICTION OF TRANSVERSE SHEAR STRESSES Umar Farooq and Karl Gregor Buil Environmen and Engineering Deparmen, Universi of Bolon, Unied Kingdom ufres@bolon.ac.u,.r.gregor@bolon.ac.u ABSTRACT In his paper mahemaical formulaions and simulaions were carried ou for he predicion of hrough-hicness sress disribuions of fibrous composie panels under variable shape impacors. Consideraions were given o selecive specimens from aerospace indusrial environmen o obain he degree of uniformi of sress disribuions hroughou he ou-of-plane geomer. The inerlaminar sresses and srains induced in hin laminaes hrough applicaion of membrane loads (i.e., in-plane loads) were also considered. Formulaion was also developed for Poisson s raios. Finie Elemen Mehod (FEM) has received a remendous aenion in engineering and indusr because of is diversi and fleibili as an analsis ool. The soluions o phsical problems can be obained ver effecivel and o a high degree of accurac using FEM sofware pacages. Therefore, he FEM was chosen o perform simulaion in commerciall available sofware ABAUS. In-plane sresses were compued from he model and Trapeium rule was applied o calculae ou-of-plane ransverse shear sresses. The procedure is simple and efficien o predic -D ransverse shear sresses from -D model. Resuls were compared wih he resuls from he available lieraure and found o be in good agreemen. Some of he resuls are shown herein in he form of ables and graphs. Kewords: model, fibrous composie panels, finie elemen analsis, hrough-hicness, ransverse shear Sresses, veloci impac.. INTRODUCTION The use of composies in ligh-weigh advanced srucures requires assessmen of heir damage olerance. Efficien analical ools, accurael predicing he behaviour of damage srucures, are needed a he preliminar design sage in order o ensure heir coseffecive performance. All laer-wise models wih he firs order shear deformaion heor depending on he number of laers are oo epensive for pracical applicaions and need coninuous shape funcions, a mied/hbrid approach, or lead o non-conforming approimaions. Plae deformaion heories can be divided in o wo groups: sress based and displacemen based heories. A brief review of displacemen based heories is given below: The firs shear deformaion heor was uniform proposed in [-4]. According o he heor, ransverse lines before deformaion would be line afer deformaion bu he were no normal o he mid-plane. The heor assumed consan ransverse shear sress and i needed a shear correcion facor in order o saisf he plae boundar condiions on he lower and upper surface. Differen higher order heories were proposed in order o saisf he plae boundar condiions. In [6-7] a ransverse shear sress funcion in order o eplain plae deformaion was proposed. Laer some new funcions were proposed in [,, and 8]. Differen shear deformaion heories were compared for dnamic and saic analsis of laminaed composies in [9]. A higher order compuaional model for he free vibraion analsis of ani-smmeric angle-pl plaes in []. Sudied saic and free vibraion and bucling of shear deformable composie laminaes using mesh-free radial basis funcion mehod []. In recen ears, he laer-wise heories and individual laer heories have been presened o obain more accurae informaion on he pl level in [-5]. However, hese heories require numerous unnowns for mulilaered plaes and were ofen compuaionall epensive o obain accurae resuls. The displacemen based plae finie elemen procedures are no capable of calculaing accurae inerfacial sresses. Three dimensional solid finie procedures are capable of calculaing a deailed sress field when a sufficien number of elemens are used. However, a hree dimensional analsis poses pracical problems in erms of compuer memor usage and compuaion ime requiremens. The alernaive approach, based on a hbrid finie elemen wih independen displacemen and sress approimaions, leads o a higher order ssem of equaions and resuls in pracical problems in numerical implemenaion. The need for hrough-hicness daa poses a major problem; wih he ecepion of shear, here are no recognied sandards available for generaing reliable design daa. As a consequence, designers and engineers rel on in-plane daa or ad hoc ess o deermine srucural performance. This approach is clearl unsaisfacor, as he use of he daa will resul in eiher under-designed or over-designed srucures. Therefore, here is a need for an efficien numerical analsis o calculae dnamic sress field in hrough-hicness in laminaed composie srucures. Inplane sresses from wo-dimensional (-D) plane analsis were used o predic ransverse shear sresses and o 7

2 VOL. 4, NO. 7, SEPTEMBER 9 ISSN Asian Research Publishing Newor (ARPN). All righs reserved. evaluae response o hree-dimensional (-D) loading configuraions required for difficul bu real aspecs of design where he composies have o perform is funcion shaped (e.g. flanged) and conneced (e.g. bonded) o he remainder of he ssem.. MATHEMATICAL FORMULATION STRATEGY.. Displacemen based equaions Displacemen based shear deformaion heor in [7] for N-plies: The relaions of displacemens are given as, From Eq. () he srain-displacemen relaionship are rewrien here for componens of displacemen, u and v, confined o he - plane ("in-plane"). A similar epression for wis,, can be derived for wis is defined as an "angular change" per "uni lengh"., (), () Where u, v, w are he unnown displacemen funcions of middle surface of he plae while shape funcions deermining he disribuion of he ransverse shear srains and sresses along he hicness. Figure-. Configuraion of N-pl la-up. The ranslaional displacemen of he mid-plane could be seen from he Figure- below while Figure- shows how in-plane u o and v o displacemens are augmened b ou-of-plane (Z-direcion) displacemens and roaions. Figure-4. Schemaic of a laminaed plae under ou-of-plane blun nose impacor. Figure-. Translaional displacemen of he mid-plane. The classical plae equaions arise from a combinaion of four disinc subses of plae heor: he inemaics, consiuive, force resulan, and equilibrium equaions. The same analog applies in modeling fibrous composie panels. To relae he plae's ou-of-plane displacemen w o is pressure loading p, i combines he resuls of he four plae subcaegories in his order: Kinemaics -> Consiuive -> Resulans -> Equilibrium Plae Equaion - plane u θ w w v θ w w Tradiionall, volume force is denoed b p and line force b q. If forces are acing on a domain from an eernal source hese are called eernal forces. The general sress componens acing on an infiniesimal elemen are shown in Figure-. - plane Figure-. Bending displacemen of he mid-plane. 8

3 VOL. 4, NO. 7, SEPTEMBER 9 ISSN Asian Research Publishing Newor (ARPN). All righs reserved. p p p u v w.. Through-hicness properies: Poisson s raios We begin wih he definiion of his hroughhicness Poisson s raio υ and υ in erms of he hrough-hicness srain and he in-plane srain in he direcion of he applied load, i.e., ν () (4) For he loading N which wih all oher applied forces and momens being ero. The average (or overall) hrough-hicness srain can be wrien in erms of he oal change in hicness, w, divided b he laminae hicness H w H (5) The oal hicness change can be wrien as he inegral of he displacemen d over he hicness of he laminae: H w d (6) H For uniform plane sress in each laer of he laminae, he hrough-hicness srain consan in an h laer can be deermine from he -D Hooe s law: S σ Sσ Sσ S6 S σ S σ S σ S 6 S σ S σ Sσ S6 S S 45 S 45 S55 S6 σ S 6σ S6σ S66 S σ S σ S6 (7) The sresses in from he plane sress consiuive are: σ σ (8) For he h laer σ 6 σ 6 (9) Now, for a smmeric laminae subjeced o he in-plane loading N, NY N, and {M}, he srains in he h laer are A N () Combining he hrough-hicness srains in he h laer: σ σ Gives: N A A A A A A 6 N ( S ) S S6 ( ) ( ) S S S66 S S S 6 Prescribed loading hus o give he oal change in laminae hicness: 6 6 ( A F A F A ) 6 F6 66 () () w N () Where he Fi are defined as F i N [( S S S ) ] ( i,,6 ) i i 6 6i Finall, becomes ( A F A F A F ) (4) 6 6 ν (5) HA Or more simpl A i F i HA ( i,,6 ) ν (6) In a similar fashion, i can be shown ha he hroughhicness Poisson s raios υ for loading in he -direcion is 9

4 VOL. 4, NO. 7, SEPTEMBER 9 ISSN Asian Research Publishing Newor (ARPN). All righs reserved. A i F i HA ( i,,6 ) ν (7).. Formulaion of ransverse shear effecs The basic approaches as summaried b a smmeric laminaes wih orhoropic laminae having principal maerial direcions aligned wih he plae aes are reaed. The ransverse normal srain can be found from he orhoropic sress-srain relaions as Figure-5. Transverse shear sresses a edges. ( σ C C C ) (8) Which can be used o eliminae e from he sress-srain relaions for he h laer, leaving σ σ (9) Where, if σ is negleced as in classical laminaion heor, CiC j Cij, if i, j, ij C () Cij, if i, j 4,5,6 The ransverse shear sress disribuion is hen approimaed b [ 55 f ( ) a55] φ ( ) f ( ) a, [ ] φ (, ) () Where f () f (-) because of laminae smmer. Also, and are deermined from he equilibrium condiions ha he shearing sresses vanish a he op and boom surfaces of he plae [f (/) f (-/) ] and are coninuous as laer inerface. The shearing srains are obained from he sress-srain relaions as f f a 55 ( ) φ a 55 ( ) φ () () Then, inegraion of he srain-displacemen relaion, Eq. (), wih respec o (wih w assumed o be independen of ) ields u v Where w [ J ( ) g ( )] φ J ( ) g ( ) φ, [ ] w (4) ( ), J f ) d a g g b ( ( ) ( ) b (5) b and The consans b are found o from coninui condiions for u and v a he laer inerfaces and he smmer condiion ha u and v vanish a he laminae middle surface. Obviousl, because of he presence of Ф and Ф, u and v are no linear funcions of as in classical laminaion heor. The momen relaions are obained from inegraion of he srain relaions, Eq. () afer he sraindisplacemen relaions, Equaion and he displacemen relaions, equaion-are he displacemen relaions, Equaion are subsiued: M D w, D w, ( F H) φ, ( F H) φ, M D w, D w, ( F H) φ, ( F H) φ, M D66 w, ) ( F66 H66) φ, ( F66 H66 φ, Where he D ij are he usual bending siffness and ( ) a (6) F J d i,j,, 6 (7) ij ij ( ) H g d l, (8) ijl ij l

5 VOL. 4, NO. 7, SEPTEMBER 9 ISSN Asian Research Publishing Newor (ARPN). All righs reserved. The shear resulans are Where d K 55 φ d K φ (9) [ ii f ( ) aii ] K d i 4, 5 () ii The large-deflecion equilibrium equaions are M M, M,, M, p Nw N w, N w, (),,, Or, in erms of he presen variables, w, ( D D66) w, ( F H) φ, ( F66 H66) Φ, ( F F66 H H66) φ, K55φ ( D D66) w, Dw, ( F F66 H H66) φ, ( F H ) Φ ( F H ) φ K φ D K 66 φ 55, N w, 66 K, φ N, w, p N, w, () Then, he overall problem is deermined and reduces o he soluion of he following se of simulaneous algebraic equaions for A, B, and C from Eq. (6) becomes: mπ K55R S [ D m ( D D ) n R ] A ( F H ) m ( F H ) n R B ( F F H H ) mnrc a π nπ R KR S () D66 m Dn R A F F66 H H 66 mn RB F66 H 66 m F H n R C a π [( D ) ] ( ) ( ) ( ).4. Load sress relaionships Normal, angenial, and aial forces ma ac on he surface, and he bod forces ma ac parallel and normal o he longiudinal ais. In addiion, an aial force, momen, and orque ma be applied a he ends. The forces acing perpendicular and parallel o he longiudinal ais are referred o as in-plane and ou-of-plane. When wo or more pes of loads are applied, he sresses and srains can independenl be calculaed for each pe of load. The sresses and srains hus obained are hen superimposed o obain he final resuls. Plane-srain condiion requires ha he srains do no var along he longiudinal ais, hus, in he,, coordinae ssem we have ; () The following displacemens saisf hese condiions u v U (, ) C C (4) V (, ) C C w W, ( C C C ) ( ) Where U, V, and W are funcions ha depend on and, and C, C, C, and C 4 are consans. For small displacemens we have he following relaionships: v (5) w u 4 Where, is he srain along he longiudinal ais? Are he curvaures of he longiudinal ais in he - and - planes, respecivel? B virue of, i gives and v u ( ) ( ) u ϑ could be defined. v The ϑ represen he rae of wis of he cross-secion. The consans are deermined from v u ( ) ( ) C ϑ he srain equaions can be wrien as w u v C4 C ; ; C (6) And displacemens can be wrien as

6 VOL. 4, NO. 7, SEPTEMBER 9 ISSN Asian Research Publishing Newor (ARPN). All righs reserved. u U v V w W (, ) (, ) ϑ ϑ (, ) ( ) (7) Deformaions and displacemens under planesrain condiion. In he,, coordinae ssem u, v, w are replaced b u, u, u ; U, V, W b U, U, U ; and b, b,. The u, v, w componens of he displacemens corresponding o he each of he deformaions. Again use he,, coordinae ssem for generall anisoropic maerials. For monoclinic, orhoropic, ransversall isoropic, and isoropic maerials we use he,, coordinae ssem, wih being along he longiudinal ais of he bod. In he,, coordinae ssem he displacemens are u u U U u U (, ) (, ) ϑ ϑ (, ) ( ) (8) Where is he srain along longiudinal ais:, are he curvaures of his ais in he - and - planes, respecivel; υ represens he rae of wis of he cross secion. Shear sresses were calculaed from Eqs () and (5).. FINITE ELEMENT MODEL SETTING UP Composie laminaes of diameer.94 mm, wih sacing sequences [45//-45/9] S and [45//-45/9] S he effec of impac duraion, lengh and posiion on he impac loads were sudied wih.88 mm hicness of quasi-isoropic configuraion under cenrall locaed load a circle of 6. mm diameer was sudied for various daa inpu and sacing sequences and clamped boundar condiions. The developed model o predic he composie srucure behaviour and implemened ino sofware ABAUS are shown in Figure-7 below. Drop-weigh model Un-pariioned Pariioned Figure-6. Models for simulaion. The properies of a unidirecional lamina of specimens wih geomeries and daa are given in Table-. Table-. Maerial and geomerical inpu daa. Maerial properies Ulimae srenghs Sacing sequences E 5 GPa, T σ E E 5 GPa ( ) ul G 5.7 GPa G 5.7 Gpa G 7.6 GPa Poisson's Raios ν. ν. ν. T ( σ ) ul C ( ) ul σ ( ) ul 5 MPa 4 MPa MPa 5 MPa [45//-45/9] S, [45//-45/9] S The compuaional model was developed for his analsis: a saic analsis was performed o calibrae he finie elemen model. An arbirar load of 6 N was applied o he op surface and boh aial and roaional resrains were applied o he edge boundaries as illusraed in he Figure-7. The maerial properies were assigned o he impacor and he specimen, he local elemen orienaion was defined in erms of he global X, Y and Z aes. Each laer of elemens hrough he hicness was reaed as a single pl alhough he acuall as previousl menioned accouned for hree plies. Simulaion for overall laminae siffness of each pl in he model roaed o align he major pl mechanical properies wih he principal fibre direcions in he code. The composie la-up and impacor were assembled in he assembl module. In he Sep module Eplici Dnamic was chosen for his analsis and impac duraion was se o. µs.

7 VOL. 4, NO. 7, SEPTEMBER 9 ISSN Asian Research Publishing Newor (ARPN). All righs reserved. Seleced resuls for [45//-45/9] 6 la-up run in he hree models were considered and compared. Given he fied pl hicness for he maerial resuls for acceleraion, velociies, displacemens, sresses and srains were shown in he Table-. Shows ime seps and duraion for which ampliudes velociies or loads used o impac he models. Tables o 6 show he prediced resuls for ransverse sresses and heir calculaion from -D daa. Figure 7: Meshed model using finie elemens: (a) Shell elemen S4 and (b) Solid elemen CD8 for impacor. 4. NUMERICAL RESULTS AND DISCUSSIONS A compuaional model was developed in he commerciall available sofware ABAUS. In-plane sresses were prediced b he model and recorded for pos-processing calculaions of ransverse shear sresses. Three configuraions of 4, 5, and 8 plies were esed and heir resuls were compared. Resuls from he model consising of 4 plies were seleced o include herein. Transverse sresses from mahemaical formulaions were calculaed for hree poins, 9, and 8 were seleced for each and ever pl of he model o record he resuls and shown in Table- and 4 above. In-plane sresses were used along wih he Trapeoidal rule o calculae hrough he hicness sresses and were shown in able 5 above. Graphs of he ransverse shear sresses were also drawn and shown for he same daa. The abular resuls and graphs shown above prove ha he resuls obained from he compuaions are real wihin he limis and reliable. Compuer generaed resuls are presened in abular, graphs and conour plos. These resuls provided informaion as a basis o build analsis for he composie panels subjeced o impacs from various nose impacors. These resuls are for sacing sequences [45//-45/9] S, [45//-45/9] s and [45//-45/9] S. The effec of impacor s shape for impac duraion, lengh and posiion on he impac velociies and loads were sudied. Based on he illusraion and prior eperience, a shorened procedure was o selec resuls for one se-up for discussion wih siing and raning procedure is based on FPF. Table-. Time and 5 ampliudes of velociies and loads. Time (S d) Ampliudes (non-dimensional) E

8 VOL. 4, NO. 7, SEPTEMBER 9 ISSN Asian Research Publishing Newor (ARPN). All righs reserved. Pl No. Table-. Prediced sresses in - direcion. Z-Dis Node Node Difference Diff sresses Mm 8 Z-Dir sresses Trap rule..57e8 5.65E8.8E8.E 6.6E.97E7.4.57E8 5.84E8.6E8.4E 4.E.5E7.6.69E7.4E8 7.77E7 8.E.4E 8.59E E7 9.6E7 5.95E7 6.E.84E.E E8 5.65E8.8E8.E 6.6E.97E E8 5.84E8.6E8.4E 4.E.5E E7.4E8 7.77E7 8.E.4E 8.59E E7 9.6E7 5.95E7 6.E.84E.E E8 5.65E8.8E8.E 6.6E.97E7..57E8 5.84E8.6E8.4E 4.E.5E7..69E7.4E8 7.77E7 8.E.4E 8.59E6..65E7 9.6E7 5.95E7 6.E.4E 7.45E E7 9.6E7 5.95E7 6.E -.8E -.E E7 -.46E8-7.77E7-8.E -4.E -.5E E8-5.84E8 -.6E8 -.4E -6.6E -.97E E8-5.65E8 -.8E8 -.E -.8E -.E E7-9.6E7-5.95E7-6.E -.4E -8.59E E7 -.46E8-7.77E7-8.E -4.E -.5E E8-5.84E8 -.6E8 -.4E -6.6E -.97E E8-5.65E8 -.8E8 -.E -.8E -.E E7-9.6E7-5.95E7-6.E -.4E -8.59E E7 -.4E8-7.77E7-8.E -4.E -.5E E8-5.84E8 -.6E8 -.4E -6.6E -.97E E8-5.65E8 -.8E8 -.E -.E -.9E7 4

9 VOL. 4, NO. 7, SEPTEMBER 9 ISSN Asian Research Publishing Newor (ARPN). All righs reserved. Table-4. Prediced sresses in - direcion. Pl Z-Dis Node : 8 Node : 9 Difference Diff sresses Z-Dir sresses Trap rule..5e7.9e7 5.69E6 6.97E9 7.9E9 4.76E5.4.4E7.4E7 7.8E5 9.57E8.48E9 8.88E4.6 6.E7 6.8E7 4.7E5 5.E8 7.6E9 4.E E7 6.57E7 5.E6 6.5E9.5E 8.E E7.9E7 5.69E6 6.97E9 7.9E9 4.76E E7.4E7 7.8E5 9.57E8.48E9 8.88E E7 6.8E7 4.7E5 5.E8 7.6E9 4.E E7 6.57E7 5.E6 6.5E9.5E 8.E E7.9E7 5.69E6 6.97E9 7.9E9 4.76E5..4E7.4E7 7.8E5 9.57E8.48E9 8.88E4. 6.E7 6.8E7 4.7E5 5.E8 7.6E9 4.E5. 6.4E7 6.57E7 5.E6 6.5E9.E 7.84E E7 6.57E7 5.E6 6.5E9 6.E9.6E E7-6.8E7-4.7E5-5.E8 -.4E9-8.88E E7 -.4E7-7.8E5-9.56E8-7.9E9-4.76E E7 -.9E7-5.69E6-6.97E9 -.E -8.E E7-6.57E7-5.E6-6.5E9-7.E9-4.E E7-6.8E7-4.7E5-5.E8 -.4E9-8.88E E7 -.4E7-7.8E5-9.56E8-7.9E9-4.76E E7 -.9E7-5.69E6-6.97E9 -.E -8.E E7-6.57E7-5.E6-6.5E9-7.E9-4.E E7-6.8E7-4.7E5-5.E8 -.4E9-8.88E E7 -.4E7-7.8E5-9.56E8-7.9E9-4.76E E7 -.9E7-5.69E6-6.97E9-6.9E9-4.8E5 5

10 VOL. 4, NO. 7, SEPTEMBER 9 ISSN Asian Research Publishing Newor (ARPN). All righs reserved. Table-5. Sresses calculaed from Table- and Table-4. Pl Z-Dis Node : 8 Node : Difference Diff sresses Z-Dir sresses Trap rule. -.4E7-4.5E7 -.47E7 -.58E -.5E9 -.E5.4.E7 4.7E7.E7.E 4.8E.88E6.6.4E7 4.5E7.47E7.58E.5E9.E E7-4.7E7 -.E7 -.E -4.8E -.88E E7-4.5E7 -.47E7 -.58E -.5E9 -.E5 6.7.E7 4.7E7.E7.E 4.8E.88E E7 4.5E7.47E7.58E.5E9.E E7-4.7E7 -.E7 -.E -4.8E -.88E E7-4.5E7 -.47E7 -.58E -.5E9 -.E5..E7 4.7E7.E7.E 4.8E.88E6..4E7 4.5E7.47E7.58E.5E9.E5. -.E7-4.7E7 -.E7 -.E -4.4E -.67E E7-4.7E7 -.E7 -.E -4.8E -.88E E7-4.5E7 -.47E7 -.58E -4.8E -.88E E7-4.7E7 -.E7 -.E.5E9.E E7 4.5E7.47E7.58E 4.8E.88E6 7.4.E7 4.7E7.E7.E -.5E9 -.E E7-4.5E7 -.47E7 -.58E -4.8E -.88E E7-4.7E7 -.E7 -.E.5E9.E5.4.4E7 4.5E7.47E7.58E 4.8E.88E6.5.E7 4.7E7.E7.E -.5E9 -.E E7-4.5E7 -.47E7 -.58E -4.8E -.88E E7-4.7E7 -.E7 -.E.5E9.E E7 4.5E7.47E7.58E.58E.54E6 6

11 VOL. 4, NO. 7, SEPTEMBER 9 ISSN Asian Research Publishing Newor (ARPN). All righs reserved. Table-6. Calculaed ransverse shear sresses. S s S s s s 6.5E7.67E6 6.77E7 5.65E5-4.8E5.6E5.9E7.9E6.7E7 5.E5.7E5 8.8E5.6E7 -.67E6.89E7.E6 4.8E5.66E6 6.8E7 -.9E6 5.97E7.8E6 -.7E5 9.6E5 6.5E7.67E6 6.77E7 5.65E5-4.8E5.6E5.9E7.9E6.7E7 5.E5.7E5 8.8E5.6E7 -.67E6.89E7.E6 4.8E5.66E6 6.8E7 -.9E6 5.97E7.8E6 -.7E5 9.6E5 6.5E7.67E6 6.77E7 5.65E5-4.8E5.6E5.9E7.9E6.7E7 5.E5.7E5 8.8E5.6E7 -.46E6.5E7.E6 8.7E5.E6 6.E6-5.56E6 7.55E5.4E6 4.56E5.6E6 -.64E7-5.77E6 -.E7.7E5 5.7E4.9E5-6.5E7 -.67E6-6.77E7-5.64E5 4.8E5 -.6E5-6.8E7.9E6-5.97E7 -.8E6.7E5-9.4E5 -.6E7.67E6 -.89E7 -.E6-4.5E5 -.65E6 -.9E7 -.9E6 -.7E7-5.E5 -.69E5-8.8E5-6.5E7 -.67E6-6.77E7-5.64E5 4.8E5 -.6E5-6.8E7.9E6-5.97E7 -.8E6.7E5-9.5E5 -.6E7.67E6 -.89E7 -.E6-4.8E5 -.66E6 -.9E7 -.9E6 -.7E7-5.E5 -.7E5-8.8E5-6.5E7 -.67E6-6.77E7-5.64E5 4.7E5 -.7E5-5.9E7.75E6-5.7E7-8.94E5 5.85E5 -.9E5 -.9E7.54E6 -.77E7 -.6E5.85E5-5.6E4 Graph-. Sresses in - direcion..e8 Through-Thicness Sresses (Z) Sresses 5.E7.E -5.E7 -.E Thicness.9..6 Z-Sresses Error! No e of specified sle in documen.- 7

12 VOL. 4, NO. 7, SEPTEMBER 9 ISSN Asian Research Publishing Newor (ARPN). All righs reserved. Graph-. Sresses in - direcion. Through-Thicness Sresses (Z) Sresses (Pascals).E6.E6.E6.E -.E6 -.E6 Z-Sresses Thicness (m) Error! No e of specified sle in documen.- Graph-. Sresses in direcion. Through-he-hicness Disribuion of Sresses Sresses (Pascals).E8 5.E7.E -5.E7 -.E Aial Disance (mm).64 sresses - sress - sresses - sress - sress - sresses - 5. CONCLUSIONS The wor presens predicion of hrough-hehicness sresses from -D model. These sresses are required o sud primar damage mode (delaminaion) on he siffness parameers and response on inerfaces due fibre orienaions and Poisson s raios. Iner-laminar sresses combined wih inherenl low hrough-hicness srengh properies in ension responsible for damage iniiaion and evenual srucural failure. Throughhicness properies are mari dominaed and significanl lower han he in-plane siffness and srengh properies of he maerial from he numerical eamples wored ou he following observaions are made: Transverse shear sresses prediced b presen model when compared wih hree-dimensional elasici soluions found o be reliable. The proposed procedure ma be checed for oher pe of advanced maerials and oher problems or oher classical boundar condiions. Efficien and reliable compuaions are epeced o improve he raio of design sress o maerial srengh, bu greaer advances are liel o come from improvemens in fibres (higher failure srains) and marices (greaer oughness). I is necessar o assess he impac damage olerance o improve i. REFERENCES [] J.N. Redd A simple higher-order heor for laminaed composie plaes. J. Appl. Mech. 5: [] M. Touraier. 99. An efficien plae heor. Inl. J. Eng. Sci. 9(8): [] R.D. Mindlin. 95. Influence of shear on fleural moions of isoropic plaes. J. Appl. Mech. 8: A- A8. [4] E. Reissner Reflecion on he heor of elasic plaes. J. Appl. Mech. 8: [5] K.P. Soldaos. 99. A ransverse shear deformaion heor for monoclinic plaes. Aca Mech. 94: 95-. [6] Z. Kacowsi Plaes-saisical calculaions. Arad, Warsaw. [7] V. Panc Theories of elasic plaes. Academia. Prague. 8

13 VOL. 4, NO. 7, SEPTEMBER 9 ISSN Asian Research Publishing Newor (ARPN). All righs reserved. [8] E. Reissner On he effecs of ransverse shear deformaion. Inl. J. Solids Sruc. 5: [9] C.T. Heraovich Mechanics of fibrous composies. John Wile and Sons. [] Whine J.M Srucural analsis of laminaed plaes. Lancaser, PA, Technomic. [] M. Adogdu. 6. Bucling analsis of cross-pl laminaed beams wih general b. condiions b Ri mehod. Compos Sci. Technol. 66: [] Fung Y.C Foundaions of Solid Mechanics. Prenice-Hall, Englewood Cliff, NJ, USA. [] Jones R.M. 99. Mechanics of Composie Maerials. Talor and Francis, Inc., Philadelphia, USA. [4] James R. A. 6. Impac Damage Resisance and Damage Tolerance of Fibrous Composies. PhD Thesis. Submied a Universi of Bolon, UK in April. [5] Tia I., Carvalho, J.D. and Vanepie D. 8. Failure analsis of composie laminaes: Eperimenal and numerical approaches. Composie and Srucures. 8: [6] M.K. Pandi, S. Haldar and M. Muhopadha. 7. Free vibraion analsis of laminaed composie recangular plae using finie elemen mehod. J. Reinf Plas Compos. 6(): [7] Meiwen Guo, E. Hari Issam and Wei-Xin Ren.. Free vibraion analsis of siffened laminaed plaes using laered finie elemen mehod. Sruc Eng. Mech. 4(): [8] Kumar Khare Raesh, Tarun Kan and Kumar Garg Aja. 4. Free vibraion of composie and sandwich laminaes wih a higher-order face shell elemen. Compos Sruc. 65(-4): [9] Y.M. Desai, G.S. Ramear and A.H. Shah.. Dnamic analsis of laminaed composie plaes using a laer-wise mied finie elemen model. Compos Sruc. 59():