ANALYSIS OF FLEXIBLE CLAMPING IN TENSILE TESTS OF MULTIDIRECTIONAL LAMINATES

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1 THE 19 TH INTERNATIONA CONFERENCE ON COOITE ATERIA 1 General Introdution Tenile tet in multidiretional laminate are uually applied in ae where oupling effet do not eit or when they are negligile. The prolem i related to the retraint aued y lamping devie: At one end the diplaement i impoed in the diretion of the ai of the teting mahine. Neverthele, the grip at oth end do not allow the free deformation of the peimen. In the ae of a unidiretional off-ai tet, hear train are prevented and onequently hear fore and ending moment appear. Figure 1 how a unidiretional off-ai peimen loaded in pure tenion and the ame peimen auming that oth end are lamped and only aial diplaement i allowed at one end. m ANAYI OF FEIBE CAING IN TENIE TET OF UTIDIRECTIONA AINATE n F. ujika*, N. Carajal, G. Varga Department of ehanial Engineering, aterial+tehnologie Group, Univerity of the Baque Country(UV/EHU), an eatian, pain. *Correponding author (fautino.mujika@ehu.e) Keyword: Tenile tet, multidiretional laminate, fleile lamping Fig. 1. Unidiretional off-ai peimen under tenile load: (a) ure tenion; () Clamped end and aial diplaement allowed at one end. The prolem of the unidiretional off-ai tenile tet ha een analyed y everal author, a it i ued a a tet method for otaining in-plane hear propertie of unidiretional ompoite [1,]. In the ae of multidiretional laminate with all type of oupling effet, memrane hear train and ending and twiting urvature appear. A the grip prevent thee deformation mode, hear fore, ending moment and twiting moment mut at at the end of the peimen. Figure how a n m multidiretional peimen under tenile loading at one end and lamped at the other end, howing ending and twiting urvature due to oupling effet. Fig.. ultidiretional peimen under tenile load lamped at one end. The tre and diplaement field of a multidiretional laminate ujeted to a tenile tet ha een reently otained y applying Engeer- Catigliano theorem [ 3 ], after determining the omplementary train energy a a funtion of fore and moment reultant of the laminate, inluding hygrothermal effet. The analyi ha een arried out auming that the gripping ytem and the ta allow elati rotation at the end of the peimen. An eperimental proedure for determining lamping ompliane aed on the meaurement of train and diplaement at ome point of the peimen ha een propoed. The aim of thi artile i to how the main apet of that analyti approah, to eplain the proedure for otaining the end ompliane and to analye y numerial eample the influene of the lamping ondition. Tenile tet in a multidiretional laminate Compoite laminate are ured in many ae at temperature greater than the room temperature at laoratory. Conidering the mot general ae of a multidiretional laminate, ompoite have normal and hear reidual train and ending and twiting urvature. Figure 3 how a aron/epoy laminate trip [80/-10] of two layer, where reidual ending and twiting urvature appear.

2 3 Analyti approah 3.1 oment and fore in a ro etion Figure 4 how the geometry of a tenile peimen with trip geometry lamped at oth end. The diplaement i impoed at end A, applying a fore. Fig. 3. Bending and twiting reidual urvature in a [80/- 10] trip of aron/epoy at room temperature. In a tenile tet, when the peimen i plaed in the grip, reidual normal and hear train are not prevented. Neverthele, reidual ending and twiting urvature are ontrained in order to adopt the plane geometry that orrepond to the gripping ytem. Then, when the peimen i plaed at the grip, ending and twiting moment at at peimen end in order to retraint reidual urvature. Therefore, the peimen i ated on y moment without applied tenile load. Neverthele, normal and hear fore do not appear efore applying load. When the tenile load i applied, oupling effet tend to generate in-plane hear train and ending and twiting urvature. In thi ae, in-plane hear train are retrained and then hear fore appear, eide twiting and flatwie ending moment. Inplane hear fore generate alo edgewie ending moment in the plane of the laminate. Therefore, the appliation of load generate flatwie and edgewie ending moment, twiting moment and in-plane hear fore. In the analytial approah of thi tudy, the end are aumed to e linearly fleile with repet to rotation. Three kind of end, depending on their fleiility will e treated through the work: 1. Clamped end: The fleiility at the end i null.. Fleile end: The fleiility at the end ha a finite value. 3. Free end: The fleiility at the end i infinite. It i equivalent to the load applied in Figure 1(a). A h Fig. 4. Tenile peimen fied at it end. Figure 5 how the fore and moment applied at the end of the peimen. owerae letter indiate that they orrepond to the whole ro etion. The inde notation orrepond to Daniel and Ihai [6]; q uript orrepond to y, r orrepond to and orrepond to y. ma ma y ma vra na Fig. 5. Fore and moment at the end of the peimen. There are five redundant unknown loated at the end A of the peimen: In-plane and out-of plane hear fore v ra and n A, repetively, flatwie and edgewie ending moment m A and m A, repetively, and twiting moment m A. By equilirium in any ro etion, it reult that v r, n and m are uniform and that m and m vary linearly along the length of the peimen n( ) = m( ) = m (1) n ( ) = n m( ) = ma + vr v ( ) = v m ( ) = m + n r r A It i aumed that rotation are partially prevented and that the diplaement related to n fore i totally prevented. It i alo aumed that ompliane are the ame at oth end. Then, the rotated angle at the end are given y θ = Cm θ = Cm θ = Cm θ = Cm θ = Cm θ = Cm A A B B A A B B A A B B nb B vrb nb mb mb mb () where θ i are rotated angle and C i are their repetive ompliane.

3 3. oment and fore per unit length With repet to fore and moment per unit length, It i aumed that V r, and are uniform along the width of the peimen. Aording to equilirium onideration, it reult that V q = y = 0. Due to m ending moment, it i aumed that in any ro etion normal fore per unit length N vary linearly along the width. For equilirium onideration, the ditriution of N i paraoli along the width. Reultant fore and moment per unit length in a ro etion at a ditane from A are ( ma + n) y N N = 0 y N 6n y ( ma + vr ) y = 0 m 0 Vq = vr Vr (3) 3.3 Engeer-Catigliano theorem Engeer firt theorem or the generalied form of Catigliano eond theorem tate that: U k * = U = δ Where k i a generalied load applied at point k and δ k i the generalied diplaement at point k in the diretion of k. Generalied load inlude load and moment and generalied diplaement inlude diplaement and rotation [4]. Otherwie, if i a redundant fore it aoiated diplaement i null. Then, Engeer eond theorem or the generalied form of Catigliano theorem of leat work tate that * *, k k (4) U * = U, = 0 (5) The derivative of the omplementary train energy of a multidiretional laminate ha een determined in a reent work [5] in term of reultant fore and moment of the laminate. The derivative orreponding to in-plane tree i U t t { N} [ a]{ N} + { N} [ ]{ } *( N ; ), k, k, = k t t y + { } [ ]{ N} + { } [ d]{ }, k, k ddy (6) Otherwie, the omplementary train energy that orrepond to the fleile end i * end 1 U = C( ma + mb) + C( ma + mb) + C( ma + mb) (7) Derivating Equation (7) with repet to k it reult ( ) (, ) k, k (,, ) k k U = C m m + m m * end, k A A, k B B, k + C m m + m m A A B B + C m m + m m A A B B With repet to out-of-plane tree, taking into aount Equation (3), the following ondition are atified, Vr y, 0 =, y, 0, yy, 0 = = y =, 0 y, 0 { } { } Then, the derivative of the omplementary energy an e written a [5] U V * t, V k, y k = [ K] V ddy (8) (9) (10) Where the omponent of the matri [K] are equivalent hear ompliane oeffiient. 3.4 Redundant unknown The redundant unknown are otained y applying Engeer eond theorem. Out-of-plane hear fore v r are null and in-plane hear fore and edgewie ending moment are related y ma n m = (11) Reultant fore and moment per unit length are 6 m N0 + N 0( ) y N Ny = 0 y = 0 N 6N 0 m y 4 (1) Where N 0 = and N 0 =. The unknown n, m and m are otained y applying Engeer firt theorem. They are related y the following ytem of equation n

4 6 1 a + a + C N + + = a N 5 C N0 + d + d N0 + = κ C N0 + d + d + = N0 κ 0 0 (13) Being the lengt-to-widht ratio; a ij, ij and ij the ompliane oeffiiente of the laminate; and κ i the hygrothemal urvature. 3.5 Diplaement and train The diplaement in diretion δ and the longitudinal train ε, otained y differentiation of the diplaement in diretion, are ued a eperimental meaurement for the determination of end ompliane. The diplaement field i otained y applying Engeer firt theorem with the unit load method. The diplaement in diretion i ( ε0 + κ ) + N0( a + ) 6 u = ( ) + N0 a + + a y + + d + + d 1 + C + CN0y ( ) ( ) (14) The diplaement field given in Equation (14) i valid for any value of N 0, and applied at the end A. In the ae that A i a fleile end, Equation (13) i atified and it reult that ε + a N u = + a + a 0y0 N h C + y C N 0 0 ( 0 1) ( 3 ) ( 1) (15) Where 0, y 0 and 0 are the following normalied oordinate y = y = = h1 = i the length-to-width ratio, 1 h h = ; 1 =. The longitudinal train ε an e otained y derivation from Equation (15) ε = ε0 + an0 + a + 3ay0( 0 1) N0 (16) h + + C 0 The diplaement in diretion i ( ξ ) ( ) ( ξ) w = + κ + N0 + N 0 + d + d ξηκ + N0 + N0 + d + d C + ηc (17) The diplaement given in Equation (17) i valid for any value of N 0, and applied at the end A. In the ae that A i a fleile end, Equation (13) and (13) 3 are atified and it reult that ( 1) ( 1) w= C + y C (18) Determination of end ompliane 4.1 otivation Two limit ae an e onidered with repet to end ompliane: If the end are lamped, ompliane are null. Otherwie, if the end are free with repet to the end rotation, the moment are null. Both limit ae an e ompared in order to analye end ontraint effet. If the differene i not negligile, it i neeary to inlude the ompliane C, C and C in the analyi. It i aumed that the end ompliane depend on the tiffne of the ta and the lamping ytem. Then, it i aumed that end ompliane do not depend on the laminate onfiguration of the peimen. An eperimental proedure i propoed in order to determine thoe end ompliane oeffiient. In order to illutrate the methodolgy propoed, numerial value for A4/ ompoite material have een onidered. The alulation proe i arried out y preadheet. In order to fi maimum load value for alulation, maimum tenile load that orrepond to a afety fator F = 0.99 aording to Tai-Wu riterion are determined. Otherwie, moiture onentration i aumed to e null and the variation of temperature i aumed to e ΔT = -150 ºC. In all ae, laminate have 8 plie of 0.15 mm thikne. Tale 1 and how material propertie [6].

5 Tale 1. Thermoelati propertie of A4/ E 1 (Ga) E (Ga) G 1 (Ga) ν 1 ν 1 α 1 (ºC -1 ) α (ºC -1 ) Tale. trength propertie of A4/ F 1t (a) F 1 (a) F t (a) F (a) F 6 (a) Unidiretional laminate and determination of C In a unidiretional off-ai peimen, normal-hear oupling our due to the ompliane oeffiient a. Otherwie, ij = ij = κ = κ = 0. Aording to Equation (13) and (13) 3 it reult that m = m = 0. From Equation (13) 1 the hear fore n i n = a a + 1.a + 0.5C (19) In the lamped ae, when C = 0, the value of n inreae when dereae. Otherwie, ompliane matrie are h h [ a] = [ ] [ ] = [ ] = [ 0] [ d] = [ ] Being [ ] = [ Q] 1 the ompliane matri of a unidiretional ply. Auming lamped end it reult n = + 1. (0) The reult of Equation (0) agree with that otained in ujika []. In thi ae, the unique elati rotation at the end i related to m. By two train gage loated at two point 1 and of normalied oordinate 1 (1,1,±1) and (0,1,±1) repetively, utrating the orreponding train otained from Equation (16) it reult ε a n 1 ε 6 From Equation (1) the value of n i n = (1) 1 ( ε ε ) = () 6a Equation () an e ued in inrement form, for two load vaule 1 and, eing Δ = 1. In thi way, a reidual train do not depend on the applied load, their effet i eliminated. Then, Equation () an e written a where 1 1 Δ ε = ε ε ε ε 1 Δε Δ n = (3) 6a C an e otained after writing Equation (13) 1 in inrement form, replaing Δn given in Equation (3) Δ 6 C = a + a + a Δn 5 (4) A laminate with dimenion = 00 mm, = 0 mm i onidered for imulating eperimental meaurement. A the maimum oupling oeffiient a our at θ =35º thi i the mot appropriate angle for otaining C. Otherwie, for the determination of in-plane hear propertieθ =10º i ued. For the lamped ae, the oupling fore n i given y Equation (0). Replaing it in Equation (1) it reult a Δ ε = 6 Δ 1. + (5) The maimum train differene our at the lamped ae, a in the free ae ε train are uniform. The maimum load i determined for F = 0.99 in the lamped ae at the point of oordinate 0 = 0 and y 0 =-1, where the effet of edgewie ending moment i maimum. Tale 3 preent the ompliane value otained for θ =35º with different degree of lamping, ranging from 0% (free end) to 100% (lamped end) orreponding to Δ = 1000 N and. Being f the lamping fator, value of Δ ε are otained y multiplying y f the value ( ) that orrepond to the lamped ae otained from Equation (5). The lamping fator i ued in order to otain intermediate value of C and it i not an aolute referene. Depending on peimen dimenion, train that orrepond to the lamped ae hange and end ompliane orreponding to different value of f.

6 Tale 3. Compliane oeffiient in a [35 8 ] laminate. Clamping fator, f(%) Δ ε (10-6 ) C (10-6 (N mm) -1 ) Tale 4 preent everal reult otained for θ = 10º with different degree of lamping, orreponding to Δ = 1000 N. The value of Δ ε are otained a in the previou ae and are leer than in Tale 3. Compliane oeffiient for the ame lamping degree are alo leer due to the lower value of a. Tale 4. Effet of the lamping fator in [10 8 ]. Clamping fator, f(%) Δ ε (10-6 ) Δn (N) C (10-6 (N mm) -1 ) = 0, y 0 = -1, =5490 N τ T (a) F (Tai-Wu) = 0.5, y 0 = 0, =5490 N ε (10-6 ) Tale 4 preent alo tre and train reult for the maimum load. Auming that failure our at the point of maimum edgewie moment, F value vary from 1 (lamped) to 1.5 (free end). Due to the main influene of hear tre τ T in the failure, the ratio etween tree i inverely proportional to the ratio etween F. Therefore, in-plane hear trength value otained y auming free-end are lower than thoe otained onidering lamping effet. Otherwie, the differene etween train at the entre of the peimen are mall. ε in the free ae i 5% greater than the value of the lamped ae. 4.3 Antiymmetri ro-ply laminate and determination of C In an antyimmetri ro-ply laminate [0/90] tenile-ending oupling our due to ompliane oeffiient. Otherwie, a = = = = = d = κ = 0. Therefore, aording to Equation (13) 1 and (13) 3 it reult that n = m = 0. From Equation (13) i N + κ = C d + 0 (6) When the peimen i plaed in the grip, auming C = 0, the reidual moment i κ = (7) d The moment due to the load i: = N (8) 0 d oment in Equation (7) and (8) do not depend on the dimenion and of the peimen. Thu, they do not depend on the length-to-width ratio. Otherwie they have oppoite ign. Then, the reidual moment ompenate in ome etent the moment generated y the load. By two train gage loated at point 1 and of oordinate 1 (½,0,-1) and (½,0,1), repetively, utrating the orreponding train otained from Equation (16), it reult From equation (9) ε h Cm 1 ε Cm = (9) 1 ( ε ε ) h = (30) m an e determined from Equation (13), replaing the value otained in Equation (30). Then 1 ( ε ε ) 1 m = + + κ d h (31) Applying Equation (31) at two different value of the load, 1 and and utrating them it reult Where 1 Δε Δ m = + Δ d h Δ ε = ε ε ε ε (3) (33) C i otained from the inremental form of Equation (30) eing Δε C = hδm (34) The ompliane C an e alo determined y a meaurement of the out-of-plane diplaement δ, aording to Equation (18), in the point of oordinate (½, 0, ±1) Cm δ 4 = (35) From Equation (13), after replaing the value otained in Equation (35)

7 m 1 8δ = κ d (36) In a imilar manner than in the ae of the meaurement y train, Equation (35) and (36) an e ued in inrement form in order to eliminate reidual effet. The inremental form of Equation (36) i where 1 8Δδ Δ m = Δ d Δ δ = δ δ 1 (37) (38) The method of train meaurement ha een ued in order to imulate eperiment. In the lamped ae Δ ε = 0 and in the free ae the differene i maimum. ε in the free ae an e otained derivating Equation (14) and impoing that n = m = m = 0. Then, the differene given in Equation (33) in the free ae i: Δ Δ ε = h (39) In order to avoid great tree indued y reidual ending urvature, a laminate [(0/90) ] a of 8 plie and dimenion = 00 mm and = 0 mm i onidered. The value of load for the free ae that orrepond to F = 0.99 i ma = 10 N and for the lamped ae ma = 688 N. Then, the end ontraint effet inreae the trength of the laminate. In the lamped ae, end moment given in Equation (6) and (7) are: =.5 N = 7.3 N mm From Equation (39), with Δ = 1000 N the -6 differene etween train i Δ = Tale 5. Effet of fleile end in [(0/90) ] a. Clamping fator, f(%) Δ ε (10-6 ) Δm (N mm) C (10-6 (N mm) -1 ) < 0 < 1, -1 < y 0 < 1, = 10 N σ T (a)( 0 = 1) ε (10-6 )( 0 = 0) F (Tai-Wu) Tale 5 preent the ompliane value otained for different degree of lamping, ranging from 0% (free end) to 100% (lamped end). Being f the ε lamping fator, value of Δ ε are otained multiplying the value that orrepond to the free ae ( ) y (1-f). In thi ae, failure i related to σ T tree. The maimum σ T of the free ae i 6% greater with repet to the value of the lamped ae. Otherwie, the longitudinal train ε in the middle plane in the free ae i 3% greater with repet to the value of the lamped ae. Therefore, the onideration of fleile lamping in thi ae i not ritial. 4.4 Antiymmetri angle-ply laminate and determination of C In an antyimmetri angle-ply laminate [θ/-θ] tenile-twiting oupling our due to ompliane oeffiient. Otherwie, a = = d = = = d = κ = 0. Therefore, aording to Equation (13) 1 and (13) it reult that that n = m = 0. From Equation (13) 3 i N + κ = C d + 0 (40) When the peimen i plaed in the grip, auming C = 0, the reidual moment i The moment due to the load i κ = (41) d = N (4) 0 d oment in Equation (41) and (4) do not depend on the dimenion and of the peimen. Thu, they do not depend on the length-to-width ratio. Otherwie, they have oppoite ign. Then, the reidual moment ompenate, in ome etent, the moment generated y the load. By the meaurement of the δ diplaement at the point 1 (0,-1,±1) and (1,-1,±1), aording to Equation (18) Cm 1 δ δ = (43) From Equation (13) 3, after replaing the value otained in Equation (43)

8 1 ( δ δ ) 1 m = + + κ d (44) Equation (43) and (44) an e ued in inrement form, utrating equation that orrepond to two load value 1 and, in order to eliminate the effet of reidual diplaement. Then the inremental form of Equation (44) i where 1 Δδ Δ m = + Δ d Δ δ = δ δ δ δ (45) (46) C i otained from the inremental form of Equation (43) eing C Δδ = (47) Δ m In the lamped ae Δ δ = 0 and in the free ae the differene i maimum. δ in the free ae i otained from Equation (17), impoing that n = m = m = 0. Then, the differene given in Equation (46) in the free ae i Δ δ = Δ (48) peimen of the type [(θ/-θ) ] a are onidered. Dimenion are = 0 mm and = 00 mm. A the maimum of orrepond to θ = 15º, thi orientation i eleted in order to maimie oupling effet. In the free ae the maimum load that orrepond to F = 0.99 i ma = N. In the lamped ae it i ma = 3130 N. Then, the end ontraint effet inreae the trength of the laminate, a in the ae of the ro-ply laminate. In the lamped ae, end moment given in Equation (41) and (4) are = 0.98 N mm = 16.3 N mm Auming that end are free, the differene etween diplaement with Δ = 000 N i Δ δ = -.34 mm. In the lamped ae, Δ δ = 0. Tale 6 preent everal reult otained for different degree of lamping, ranging from 0% (free end) to 100% (lamped end). Being f the lamping fator, value of Δ are otained δ mutiplying the value that orrepond to the free ae (-.34 mm) y (1-f). Tale 6. Effet of fleile end in [(15/-15) ] a Clamping fator, f(%) Δ δ (mm) Δm (N mm) C (10-3 (N mm) -1 ) < 0 < 1, -1 < y 0 < 1, = N τ T (a)( 0 = ±1) ε (10-6 )( 0 = 0) F (Tai-Wu) The failure i due to τ T tree at top and ottom lamina. The tre in the free ae i 45% greater with repet to the value of the lamped ae. Neverthele, ε of the middle plane in the free ae i only % greater than in the lamped ae. 5 Eample: [0 4 /50 4 ] A laminate [0 4 /50 4 ] of the ame material and dimenion a in the previou ae ha een analyed in order to ee the influene of the lamping end. Three different oundary ondition are analyed: Free end, fleile end and lamped end. In the ae of fleile end, end ompliane oeffiient orrepond to 60% of lamping, aording to Tale 3, 5 and 6. Then: C = C = C = Two point of the peimen are onidered for alulation: The entral point A( 0 = 0.5, y 0 = 0) and B( 0 = 0, y 0 = -1) of maimum N. The applied load orrepond to a F = 0.99 at point A in the lamped ae, eing = 85 N. A oupling effet inreae when the numer of layer dereae [7], thi ae an e onidered ritial. Tale 7 preent end fore and moment per unit length and afety fator. The fleiility of the end redue the value of the end fore and moment. Tale 7. End fore and moment in [0 4 /50 4 ] laminate Clamping N 0 (N/mm) (N) (N) F(A) F(B) Free Fleile Clamped At point A the maimum F i for the lamped ae and at point B the maimum F i for the fleile ae. oreover, F at point B i very imilar for the free and lamped ae. Figure 6-8 how tree in the prinipal diretion of orthotropy, σ, σ T and τ T at the ritial point B. The value of F at point B in Tale 7 orrepond to the ottom part of ply 8 in oth, the free ae (0.73)

9 and in the lamped ae (0.7). σ T i greater in the lamped ae (76.5 a) than in the free ae (66.8 a). Otherwie, τ T i leer in the lamped ae (9.6 a) than in the free ae (49.1 a). Then, the imilarity in F i not due to imilar tre value, ut for the omined effet of σ T and τ T Total tre (a)free Total tre ()Fleile Total tre σ(a) σ(a) σ(a) the lamped ae are 40% in the free ae and 15% in the fleile ae Total tre (a)free Total tre ()Fleile Total tre 55.8 ()Clamped σt(a) σt(a) σt(a) Figure 7. σ T tre in a [0 4 /50 4 ] laminate at point B. ()Clamped Figure 6. σ tre in a [0 4 /50 4 ] laminate at point B. Figure 15 how ε train along the thikne of the peimen at the entral point A. In the ae of the lamped ae, the train ditriution i uniform. train are minimum in the lamped ae and the relative differene of mean train with repet to

10 Total tre (a)free Total tre ()Fleile Total tre ()Clamped τt(a) τt(a) τt(a) N 0 depend on the geometry of the peimen, inreaing when dereae. 969 Total train (a)free Total train ()Fleile Total train ε(με) ε(με) ε(με) Figure 8. τ T tre in a [0 4 /50 4 ] laminate at point B. The effet of the peimen dimenion ha een diued in the partiular ae of unidiretional, antiymmetri ro-ply and angle-ply laminate. In the preent eample, for a given value of N 0, if the length of the peimen dereae N 0 inreae and the hange in and i very mall. Inreaing the width for the ame value of N 0, the effet i imilar: N 0 inreae and the hange in and i not appreiale. Figure 10 how end fore and moment for an applied load N 0 = 100 N/mm. Therefore, for a given laminate onfiguration, only ()Clamped Figure 9. ε train in a [0 4 /50 4 ] laminate.

11 N0 (N/mm) - (N) -(N) = 15 mm = 0 mm = 5 mm = 30 mm Figure 10. Variation of the end fore and moment with repet to the width for a [0 4 /50 4 ] laminate. = 00 mm; N 0 = 100 N/mm. 6 Conluion The ompliane relative to edgewie and flatwie ending and to twiting at the end an e determined y train and diplaement meaurement in unidiretional off-ai, antiymmetri ro-ply and angle-ply laminate, repetively. Conidering unidiretional laminate, the end hear fore depend on the length-to-width ratio. In the ae of ro-ply and angle-ply laminate, end ending and twiting moment, repetively, do not depend on the length and the width of the peimen. Otherwie, reidual effet ompenate in ome etent the end effet produed y the appliation of the load. With repet to multidiretional laminate with general oupling, only the end hear fore depend on the length-to-width ratio, a in the partiular ae eplained previouly. Bending and twiting moment depend only on the laminate onfiguration. Otherwie, mean train at the entre of the peimen depend on the fleiility of the end. 7 Referene end. Journal of Compoite aterial, DOI: / hame I. H. and Dym C.. Energy and Finite Element ethod in trutural ehani, Taylor & Frani, [5] ujika F. (01). A novel approah for the three point fleure tet of multidiretional laminate. Journal of Compoite aterial, Vol. 46, pp [6] Daniel, I.. and Ihai, O. Engineering ehani of Compoite aterial. Oford Univerity re, 006. [7] Tai,.W. (1964). trutural Behavior of Compoite aterial. NAA CR-71. [1] N.J. agano and J.C. Halpin. Influene of End Contraint in the Teting of Aniotropi Bodie. Journal of Compoite aterial, Vol., pp , [] F. ujika. New Conideration on the tre Field in an Off-ai Tenile Tet. Journal of Compoite aterial, Vol. 39, pp , 005. [3] F. ujika. tre and diplaement field in tenile tet of multidiretional laminate with fleile lamped