On the equivalent flexural and shear moduli of laminated beams: Definition and determination by bending tests

Composi es: Pa A 175 (2023) 107802 A ailable online 21 Sep embe 2023 1359-835X/© 2023 The Au ho (s). Published by Else ie L d. This is an open access a icle unde he CC BY-NC-ND license (h p://c ea i ecommons.o g/licenses/by- nc-nd/4.0/). On he equi alen lexu al and shea moduli o lamina ed beams: De ini ion and de e mina ion by bending es s Faus ino Mujika a , * , Mi eia Ola e b , M. Asunci´ on Can e a a , Ugu z Ga i aonaindia a , Mi en Isasa a , Ainhoa A ese a a Mechanics o Ma e ials G oup, Depa men o Mechanical Enginee ing, Uni e si y o he Basque Coun y (UPV/EHU), Spain b Ike lan Technology Resea ch Cen e, Basque Resea ch and Technology Alliance (BRTA), P◦J.M. A izmendia ie a 2, 20500 A asa e-Mond ag´ on, Spain ARTICLE INFO Keywo ds: A. Hyb id composi es B. Sandwich ma e ial C. Equi alen modulus D Lamina ed beam ABSTRACT Equi alen lexu al and shea moduli o lamina ed beams o ec angula c oss-sec ion made o o ho opic laye s a e de ined by an analy ical app oach. Equi alen lexu al and shea modulus a e de ined as hose ha would co espond o a homogenous beam. In o de o check he sui abili y o he moduli de ined, h ee-poin bending es s a ying he span a e ca ied ou in a sandwich ma e ial and in hyb id lamina es. In he case o he sandwich ma e ial, i is shown ha a h ee-poin bending es could be conside ed a shea es om he s i ness poin o iew. In he case o hyb id ma e ials, i ual es s a e ca ied ou by he Fini e Elemen Me hod. The ag eemen be ween equi alen moduli ob ained om he analy ical app oach and he bending es s has been checked, being qui e good in all cases. Finally, he e ec o dimension unce ain y in h ee-poin bending es s is analyzed by a Mon e Ca lo simula ion. 1. In oduc ion Shea e ec s in he s i ness o beams o ec angula c oss sec ion depend on he a io be ween lexu al modulus and shea modulus. In iso opic ma e ials, his a io is less han 3 and shea con ibu ion in displacemen s is negligible. Ne e heless, shea e ec s can in luence s i ness p ope ies o some o ho opic ma e ials as wood, used o wing-beam ma e ials in he ea ly s ages o a ia ion [1]. Usually, Fi s - o de Shea De o ma ion Theo y (FSDT) is applied, named also Timo- shenko’s beam heo y [2], assuming ha shea s ains a e uni o m in he hickness o he beam. Shea co ec ion ac o s a e applied o ake in o accoun he ac ual a ia ion o shea s esses and s ains in he hickness. Be [3], de e mined he shea ac o o a non-homogeneous c oss-sec- ion using a simple mechanics-o -ma e ials app oach. Adams and Mille [4], applied Classical Lamina ed Pla e Theo y (CLPT) o he h ee-poin beam bending p oblem o de e mine he lexu al modulus and he en- e gy abso bed o bo h impac and s a ic loadings o hyb id lamina es. Be and Go daninejad [5], analyzed he ans e se shea e ec s in bimodula ma e ials, which ha e di e en elas ic moduli in ension and comp ession, based on equi alen shea s ain ene gy. Raman and Da alos [6], de i ed a gene al exp ession o he shea co ec ion ac o o lamina ed ec angula beams wi h an a bi a y lay-up con igu a ion. Resul s we e compa ed wi h exis ing esul s o composi e beams and pla es. He and Zhang [7], analyzed he bending o ec angula , simply suppo ed, an isymme ic angle-ply lamina ed pla es, using a e ined shea de o ma ion heo y o ob ain closed- o m solu ions. Pai and Schulz [8], ca ied ou a new de i a ion o shea co ec ion ac o s in he case o iso opic ma e ials, being ene gy-consis en and showing i s physical meaning. Al enbach [9], p oposed a me hod o de e mine shea s i ness o sandwich and lamina ed pla es and compa ed i wi h esul s om o he au ho s. Ghugal and Shimpi [10], p esen ed a e iew o displacemen and s ess based e ined heo ies o iso opic and aniso- opic lamina ed beams, unde lining some c i ical issues. Gibson [11], p oposed simpli ied mechanics o ma e ials equa ions o p edic ing bo h lexu al and shea componen s o ans e se de lec ions in com- posi e sandwich beams, concluding ha shea de lec ions we e g ea e han lexu al de lec ions. He compa ed p oposed equa ions wi h expe imen al measu emen s and nume ical esul s. Na a o e al. [12], de eloped an analy ical model o he s a ic inden a ion o sandwich beams wi h a oam co e. Good co ela ion was ob ained om he com- pa ison o he model wi h expe imen al and ini e elemen me hod (FEM) esul s. Gao e al. [13], in es iga ed analy ically and expe i- men ally he shea e ec s on he de lec ion o a building loo panel made o Ca bon Fib e Rein o ced Polyme (CFRP), including analy ical * Co esponding au ho . E-mail add ess: [email p o ec ed] (F. Mujika). Con en s lis s a ailable a ScienceDi ec Composi es Pa A jou nal homepage: www.else ie .com/loca e/composi esa h ps://doi.o g/10.1016/j.composi esa.2023.107802 Recei ed 3 May 2023; Recei ed in e ised o m 8 Sep embe 2023; Accep ed 17 Sep embe 2023 Composi es Pa A 175 (2023) 107802 2 and expe imen al esul s. Sayyad and Ghugal [14], ca ied ou a c i ical e iew o he li e a u e conce ning bending, buckling and ee ib a ion o lamina ed composi e and sandwich beams and p oposed u u e esea ch di ec ions. Mujika e al. [15], de e mined ou -o -plane elas ic p ope ies in honeycomb sandwich panels by expe imen al, nume ical and analy ic me hods. The esul s ob ained by h ee-poin bending a di e en spans in ha a icle a e analyzed wi h a no el pe spec i e in he cu en s udy. Lim and Kim [16], deal wi h he he mo-elas ic e - ec s on shea co ec ion ac o s o unc ionally g aded beams, assuming empe a u e dependence o ma e ial p ope ies. They concluded ha he mal e ec s ha e in luence on he shea co ec ion ac o s. Bisheh and Wu [17], in es iga ed he e ec s o he ans e se shea and o a y ine ia in wa e p opaga ion in piezoelec ic coupled lamina ed composi e cylind ical shells, conside ing di e en s acking sequences and ib e o ien a ions. Cao and Niu [18], de eloped a new model o he buckling analysis o composi e sandwich panels ha ook in o accoun in e laye shea e ec s. By compa ing o FEM, he solu ion p oposed led a be e accu acy han p e ious analy ical solu ions. Gio dano e al. [19], ca ied ou h ee-poin bending es s in a composi e sandwich s uc u e, wi h ca bon ib e wo en ace shee s and a ela i ely complian oam co e. Displacemen s and s ains we e ob ained by Dig- i al Image Co ela ion (DIC). Compa ing expe imen al alues wi h p e- ious models, he bes ag eemen was ob ained wi h he Fi s O de Shea model [20]. Ga g e al. [21] ca ied ou a e iew o he analysis o sandwich s uc u es including heo ies and analysis me hods o p ob- lems ela ed o s a ics, ib a ion and buckling. Zhou e al. [22] in es- iga ed he bending beha iou o no el magnesium alloy aceshee and 3D-p in ed PLA la ice co e sandwich panels, by compa ing expe i- men al esul s ob ained by h ee-poin bending wi h nume ical analysis. Bela bi e al. [23] de eloped new ini e elemen models o he analysis o symme ic and assymme ic sandwich panels, based on a laye wise app oach and ex ended he o mula ion [24] o he analysis o he ee ib a ion o mul ilaye sandwich pla es. The bending beha iou o unc ionally g aded single-laye ed, symme ic and non-symme ic sandwich beams by he ini e elemen analysis, using a no el pa a- bolic shea de o ma ion heo y was also analyzed by hem [25]. Mo e- o e , hey de eloped also a wo-node beam elemen [26] o in es iga e he case o cu ed beams o unc ionally g aded sandwich beams. Ga g e al [27–29] de eloped a C 0 ini e elemen -based highe -o de zigzag o mula ion o sol e he bending, ee ib a ion and buckling p oblems o sandwich pla es and beams. In [30] hei model, inco po- a ed ans e se no mal and shea s ess condi ion a in e aces. Mei e al. [30] s udied he bending beha io o oam illed composi e X-co e sandwich panels by h ee-poin bending es s, by analy ical and nu- me ical models. Vinh e al. [31] de oloped a no el, enhanced i s -o de mixed pla e elemen , o s a ic bending and ee ib a ion analysis o unc ionally g aded sandwich pla es. Sayyad e al. in es iga ed he e ec o concen aded loading in lamina ed sandwich a ches [32] and in lamina ed composi es shells [33], using a ious equi alen single laye shell heo ies. Sayyad and Ghugal [34] p esen ed a igonome ic shea de o ma ion heo y aking in o accoun ans e se shea de o ma ion as well as ans e se no mal s ain e ec in hei o mula ion, o dealing wi h he s a ic lexu e o symme ic and an i-symme ic c oss-ply lami- na ed beams. In hei conclusions, hey s a e ha he esul s o ans- e se shea s esses ob ained by he in eg a ion o equilib ium equa ions a e be e han hose ob ained by cons i u i e ela ions. The analy ic app oach o he cu en s udy is based on he main hypo hesis o a linea s ain ield, which co esponds o Eule -Be nouilli hypo hesys, and a plane s ess s a e, wi hou he assump ion o any displacemen ield. Then, he complemen a y s ain ene gy o coene gy co esponding o he beam is de e mined and he displacemen in h ee- poin bending is ob ained by he Engesse -Cas igliano heo em. By compa ison wi h he equa ion ha co espond o a homogeneus beam, equi alen lexu al and shea moduli a e de ined. Then, he sui abili y o hose moduli is checked by p e ious expe imen al esul s [15] o sandwich specimens, whe e shea e ec s a e dominan , and by nume - ical analysis o symme ic and asymme ic hyb id lamina es, whe e bending is dominan . The esul s o bending es s a di e en spans in he same specimen a e used o de e mine he equi alen moduli in bo h cases, using a p e iously de ined p ocedu e [35], and p oposing new eg ession schemes. Finally, an e o analysis is included, aking in o accoun he e ec o he unce ain y o di e en dimensions o he specimen and he es . The main no el ies o he cu en s udy a e: 1. The analy ical app oach is based on he Eule -Be nouilli hypo hesis conce ning he linea i y o axial no mal s ains h ough he hick- ness, wi hou he assump ion o any displacemen ield. Mo eo e , i is no necessa y o de ine any shea co ec ion ac o . 2. Equi alen lexu al and shea moduli ha would co espond o an equi alen homogeneous beam a e de ined in he gene al case o a lamina ed beam cons i u ed o o ho opic laye s o ec angula c oss-sec ion. 3. The me hodology based on he span a ia ion es ing he same specimen is used o de e mine he equi alen moduli p e iously de ined. I is applied o sandwich and hyb id lamina es. 4. I is shown ha in he case o sandwich lamina es, o small spans, bending es could become a shea es om he s i ness poin o iew. Then, he equi alen shea modulus can be de e mined by a single es a low span. I cons i u es he coun e pa o de e mine he equi alen lexu al modulus o la ge spans, when bending is dominan . 5. The impo ance o he accu a e de e mina ion o he span leng h is shown, using a Mon e Ca lo simula ion. 2. Analy ical app oach o de ine equi alen lexu al and shea moduli 2.1. No mal s esses The e e ence sys em is loca ed in he mid-plane o he beam, ac- co ding o Fig. 1. Assuming ha he lamina e has n laminae, he naming is made om op o bo om. The analysis de eloped is alid o a lamina ed beam o ec angula c oss sec ion, he beam axis being one o he p incipal di ec ions o o ho opy o each lamina. The main assump ions o he analy ic app oach a e: 1. No mal s ains a e linea ly dis ibu ed h ough he hickness. 2. Plane s ess s a e in he zx plane and σ z =0. 3. The s ess–s ain beha iou o each ply is linea elas ic and o ho opic. 4. Residual he mal s esses a e no conside ed. Acco ding o he i s assump ion: Fig. 1. Lamina ed beam cons i u ed by n laminae. F. Mujika e al. Composi es Pa A 175 (2023) 107802 3 ε x= ε 0+κz (1) Eq. (1) is he s ain ield ha esul s o m he Eule -Be nouilli Beam Theo y, whe e i is assumed ha sec ions o he beam emain plane and pe pendicula wi h espec o he beam axis. ε 0 is he s ain o he middle su ace o he beam and κ is he bending cu a u e o he beam. Ac- co ding o he second assump ion σ y= σ z=0. Then, no mal s esses in he k lamina, a e: σ k x=Ek x ε x⇒ σ k=Ek ε =Ek( ε 0+κz)(2) Whe e E k is he elas ic modulus o he lamina k. The no mal o ce and he bending momen in a sec ion a e he esul an and he esul an momen o no mal s esses in he sec ion, espec i ely: N=∑ n k=1∫zk zk−1 Ek( ε 0+κz)wdz =A ε 0+Bκ M=∑ n k=1∫zk zk−1 Ek( ε 0+κz)zwdz =B ε 0+Dκ (3) whe e w is he specimen’s wid h. In ma ix o m Eq. (3) is: {N M}=[A B B D ]{ ε 0 κ}(4) S i ness coe icien s in Eq. (4) a e: A=w∑ n k=1 Ek(zk−zk−1)B=w 2∑ n k=1 Ek(z2 k−z2 k−1)D =w 3∑ n k=1 Ek(z3 k−z3 k−1)(5) A and D a e he axial and bending s i ness coe icien s, espec i ely, and B is he no mal-bending coupling s i ness coe icien . The in e se o m o Eq.(4) is: { ε 0 κ}=[a b b d ]{N M}(6) Whe e he no mal, coupling and bending compliance coe icien s o he beam a e, espec i ely: a=D AD −B2b=−B AD −B2d=A AD −B2(7) Replacing Eq. (6) in Eq. (2), no mal s esses a e: σ k=Ek( ε 0+zκ) = Ek[N(a+bz) + M(b+zd) ] (8) 2.2. In e lamina shea s esses Acco ding o he assump ions ca ied ou , he ollowing equilib ium equa ions mus be sa is ied in each lamina: σ k x,x+ τ k zx,z=0 τ k zx,x+ σ k z,z=0(9) Di e en ia ing Eq. (8) and aking in o accoun ha M,x=V, being V he shea o ce, he de i a i e o he no mal s ess is: σ k x,x=VEk(b+zd)(10) Replacing Eq. (10) in Eq. (9) 1 and a e in eg a ing, shea s esses a e: τ k zx(z) = V[−Ek(bz +dz2 2)+ck](11) Bounda y condi ions o shea s esses in each lamina a e: τ 1 zx(z0) = τ n zx(zn) = 0 τ k−1 zx (zk−1) = τ k zx(zk−1)(12) Imposing he i s condi ion o Eq. (12) in Eq. (11), c 1 is ob ained: c1=E1(bz0+dz2 0 2)(13) Applying ecu si ely he o he bounda y condi ions, he in eg a ion cons an s c k a e gi en by: ck=ck−1+ (Ek−Ek−1)[bzk−1+dz2 k−1 2](14) Eq. (14) can be exp essed as: ck=E1(bz0+dz2 0 2)+∑ k i=2 (Ek−Ek−1)[bzi−1+dz2 i−1 2](15) Acco ding o Eq. (11), i V is uni o m along he leng h o he beam, as i is in h ee-poin bending, i esul s ha τ k zx,x=0. Then, conside ing Eq. (9) 2 , i esul s ha σ k z,z=0, being σ z uni o m h ough he hickness o each lamina. As σ z=0 a he op and bo om su aces o he beam, hey a e null in he i s and las laminae and consequen ly, in he whole sec ion. The e o e, in he case o h ee-poin bending, σ z=0, is a consequence o equilib ium equa ions. Ne e heless, i has been assumed as ini ial hypo hesis o be able o ela e di ec ly σ x wi h ε x in Eq. (2). The zones o load applica ion and eac ions, wi h pa icula s ess s a es ha include ans e se no mal s esses, a e no conside ed in he cu en s udy. 2.3. Engesse -Cas igliano heo em The displacemen o he load applica ion poin is de e mined by he Engesse -Cas igliano heo em, whe e he complemen a y s ain ene gy o s ain coene gy is used. The name coene gy has i s o igin in magne ic o ces [36], bu i has been used also in he ield o mechanics [37]. The de i a ion o he coene gy o a lamina ed beam is de eloped in Ap- pendix A. Acco ding o Engesse -Cas igliano’s heo em, he gene alized displacemen δ k o he applica ion poin o he gene alized o ce F k is: δk=C,Fk=C ′ (16) In Eq. (16) p ime means de i a i e wi h espec o F k . When F k is a o ce, δ k is he displacemen in he same di ec ion. When F k is a momen , δ k is he o a ion in he same di ec ion. Then, di e en ia ing Eq. (A.16), i esul s: δk=C,Fk=C ′ =a∫Lx NN ′ dx +d∫Lx MM ′ dx +s∫Lx VV ′ dx +b(∫Lx N ′ Mdx +∫Lx NM ′ dx)(17) 2.4. Displacemen in h ee-poin bending Fig. 2 shows a h ee-poin bending es con igu a ion o a lamina ed beam. In he case o a h ee-poin bending es whe e he span is L, wi h an Fig. 2. Th ee-poin bending con igu a ion o a lamina ed beam. F. Mujika e al. Composi es Pa A 175 (2023) 107802 4 applied o ce F in C, sec ion o ces and momen s and hei de i a i es wi h espec o F, indica ed by p imes, a e: N=0N ′ =0 V=1 2F V ′ =1 2 M=1 2Fx M ′ =1 2x (18) I is wo h poin ing ou ha in a h ee-poin bending es , no mal o ces and hei de i a i es a e null, e en in he case o no mal-bending coupling, whe e he coupling compliance coe icien is b∕= 0. I is due o he eedom o he specimen o axial de o ma ion, as i is suppo ed on wo olle s. Ne e heless, in a ensile es wi h b∕= 0, bending momen s a he g ips would a ise, p e en ing bending cu a u es. The displacemen o he load applica ion poin is: δC=2⎛ ⎜ ⎝d∫1 2L 0 1 4Fx2dx +s∫1 2L 0 1 4Fdx⎞ ⎟ ⎠=FdL3 48 +FsL 4(19) Eq. (19) can be exp essed in e ms o an equi alen lexu al modulus E eq and an equi alen shea moduli G eq , which a e de ined as: d=(EeqI)−1whe e I=1 12wh3 s=(5 6Geqwh)−1(20) Whe e h is he o al hickness; w is he wid h; and I is he momen o ine ia. I he displacemen is de e mined by he es ing machine, he displacemen due o he lexibili y o he es ing sys em and he de o - ma ion o he specimen h ough he hickness, has o be included. Those con ibu ions a e included in he e m de ined as sys em-s i ness and i is assumed o be linea wi h espec o he applied o ce [36]. Being K s he coe icien o he sys em-s i ness, i depends on he de o mabili y o he specimen h ough he hickness, and on he s i ness o he es ing ame, he ix u es and he load cell. Replacing he equi alen moduli o Eq. (20), he displacemen δ Cs including he sys em-s i ness is w i en as: δCs =FL3 4Eeqwh3+FL 4(5 6Geq)wh +F Ks (21) In Eq. (21), i is assumed ha he e ec o he span a ia ion due o he change o con ac be ween he specimen and he suppo olle s is negligible [38]. Eq. (21) is simila o ha esul ing in he case o ho- mogeneous o ho opic ma e ials. In ha case, being E he lexu al modulus in he beam di ec ion and G he ou o plane shea modulus, om Eqs. (5) and (7) wi h n =1 and om Eq. (20), i esul s ha Eeq = E. Mo eo e , om Eq. (A.14) wi h n =1 and om Eq. (20), Geq =G is ob ained. Then, E eq and G eq a e he moduli ha co espond o an equi alen homogeneous ma e ial. The e o e, knowing he elas ic p ope ies and he hicknesses o he laminae, he equi alen moduli can be de e mined. The explici o m o he equi alen lexu al modulus, aking in o accoun Eqs. (5), (7) and (20), is: Eeq =12 wh3 AD −B2 A(22) On he o he hand, he explici o m o he equi alen shea modulus, aking in o accoun Eqs. (15), (A.11), (A.14) and (20) is: Geq =(5 6swh)−1 (23) 3. De e mina ion o E eq and G eq by h ee-poin bending es s Eq. (21) can be exp essed in di e en o ms, aking in o accoun he con ibu ion o bending, shea and sys em-s i ness: δCs =FL3 4Eeqwh3⎡ ⎢ ⎢ ⎣ 1+Eeq (5 6Geq)(h L)2 +4wEeq Ks(h L)3⎤ ⎥ ⎥ ⎦ δCs =FL 4(5 6Geq)wh ⎡ ⎢ ⎢ ⎣ 1+(5 6Geq) Eeq (L h)2 +4w(5 6Geq) Ks(h L)⎤ ⎥ ⎥ ⎦ δCs =F Ks ⎡ ⎢ ⎢ ⎣ 1+Ks 4wEeq (L h)3 +1 4w Ks (5 6Geq)(L h)⎤ ⎥ ⎥ ⎦ (24) The sub-index s indica es ha he e ec o he sys em-s i ness is included. Being ms=F δCs he expe imen al slope o he load–displace- men cu e, he appa en alues o he lexu al modulus E 3ps , shea modulus G 3ps and sys em s i ness K 3ps ob ained expe imen ally a e de ined as: E3ps =msL3 4wh3G3ps =msL 4wh K3ps =ms(25) Replacing Eq. (25) in Eq.(24) i esul s: (E3ps)−1=[(Eeq)−1+(5 6Geq)−1 x2+4w(Ks)−1x3] (G3ps)−1=[(Eeq)−1x−2+(5 6Geq)−1 +4w(Ks)−1x] 4w(K3ps)−1=[(Eeq)−1x−3+(5 6Geq)−1 x−1+4w(Ks)−1] (26) Whe e x=h L.Depending on he ela i e in luence o bending, shea and sys em s i ness, i one o hose e ms is dominan , he appa en alues become he ac ual alues o moduli and s i ness: E3ps =Eeq G3ps =5 6Geq K3ps =Ks(27) In he cu en s udy, in he case o composi e lamina es, when g ea spans a e used, shea and sys em e ms could be negligible and E eq can be de e mined. In he case o sandwich beams, shea e m could be dominan and in ha case, G eq can be ob ained di ec ly. In he case o an inden a ion es , he span can be conside ed nea ze o and hen K s could be de e mined di ec ly. Ne e heless, i is necessa y o ake in o accoun ha he slope o he load–displacemen cu e has o be de e mined in he linea zone, once he con ac a ea be ween he load olle and he specimen emains cons an [36]. Eq. (26) can be w i en as: yE(x) = A+Bx2+Cx3 yG(x) = Ax−2+B+Cx yK(x) = Ax−3+Bx−1+C (28) whe e: yE=(E3ps)−1yG=(G3ps)−1yK=4w(K3ps)−1 A=(Eeq)−1B=(5 6Geq)−1 C=4w(Ks)−1(29) Doing N bending es s a di e en spans in he same specimen, A, B and C can be de e mined by eg ession o he expe imen al da a in any o Eqs. (28). The sum o squa es o esiduals is de ined as [39]: F. Mujika e al. Composi es Pa A 175 (2023) 107802 5 I(A,B,C) = ∑ N i=1 [yI(xi) − yIi ]2(30) whe e I is E, G o K; yI(xi)co espond o Eq. (28) and yIi co espond o expe imen al alues. Minimizing I(A,B,C)in Eq. (30) wi h espec o A, B and C, he ollowing sys em o equa ions is ob ained, whe e he co- e icien ma ix is symme ic: ⎡ ⎣ a11 a12 a13 a12 a22 a23 a13 a23 a33 ⎤ ⎦I ⎧ ⎨ ⎩ A B C⎫ ⎬ ⎭ =⎧ ⎨ ⎩ b1 b2 b3⎫ ⎬ ⎭I (31) Using any o Eq. (28) o ob aining A, B and C, he o he wo equa ions can be used o analyse he end o he o he a iables. Fo ins ance, i y E is used o ob aining A, B and C, eplacing hose alues in y G and y K we can see he a ia ion o G 3ps and K 3ps , espec i ely, and he loca ion o he expe imen al da a wi h espec o he global cu e. The coe icien alues o each eg ession cu e a e epo ed in Appendix B. The coe - icien o de e mina ion R 2 is de ined as [41]: R2=1−∑N i=1[yIi −yI(xi) ]2 ∑N i=1[yIi −yI]2(32) whe e yI=1 N∑N i=1yIi is he mean alue o he expe imen al da a. The coe icien gi es a measu e o he i o expe imen al poin s o he eg ession cu e. On he o he hand, i he sys em-s i ness has been de e mined by an inden a ion es , he ac ual displacemen due o bending and shea δ C can be de e mined as: δC=δCs −F Ks (33) In his case, Eq. (26) becomes: (E3p)−1=[(Eeq)−1+(5 6Geq)−1 x2] (G3p)−1=[(Eeq)−1x−2+(5 6Geq)−1](34) Whe e E 3p and G 3p a e he appa en lexu al and shea moduli dis- ega ding sys em-s i ness. In Eqs. (34), he expe imen al pa ame e s ha depend on he slope m=F δC a e: E3p=mL3 4wh3G3p=mL 4wh (35) The ela ion be ween he moduli a ec ed by he sys em s i ness gi en in Eq. (25) and hose no a ec ed by i gi en in Eq.(35), depend on he ela ion be ween he slopes m and m s , being: ms m=F δCs δC F=δCs −F Ks δCs =1−F δCsKs =1−ms Ks (36) Acco ding o Eqs. (25) and (36), he ela ion be ween moduli a ec ed and no a ec ed by he sys em-s i ness a e: E3ps =E3p(1−ms Ks)G3ps =G3p(1−ms Ks)(37) The e o e, E 3p and G 3p can be expe imen ally de e mined by wo me hods: •De ining a channel o δ C in he so wa e o he es ing machine, acco ding o Eq. (33), and de e mining he moduli based on he slope m. •De e mining E 3p and G 3p om Eq. (37), a e ha ing de e mined alues o E 3ps and G 3ps based on expe imen al alues o m s . Eq. (34) can be w i en as: zE=A+B zG=A −1+B(38) whe e: zE=(E3p)−1zG=(G3p)−1 =x2=(h L)2 In he case o Eq. (38), A and B can be ob ained by linea eg ession. 4. Compa ison be ween de ined and expe imen al alues The equi alen moduli de ined in Eqs. (22) and (23) a e de e mined based on he elas ic p ope ies o he o ho opic laye s o he lamina e. Those moduli can be also ob ained by h ee-poin bending es s, a ying he span in he same specimen. To check he sui abili y o he de ined equi alen moduli, he alues ob ained om Eqs. (22) and (23)a e compa ed wi h alues ob ained by eg ession o bending es esul s. Tha compa ison is ca ied ou in he case o a sandwich specimen, and wo hyb id lamina es, one symme ic and one asymme ic. 4.1. Sandwich specimen 4.1.1. Ma e ial p ope ies In he case o a sandwich specimen, bending s esses a e suppo ed by he skins and shea s esses a e suppo ed by he co e. The da a o a p e ious s udy ca ied ou by some o he au ho s a e used in o de o ob ain new conclusions [15]. Sandwich specimens made o Poly- p opylene Honeycomb Co e - OpenThe mHex® and skins o aluminium Mangeal ok 30-H111 we e used. The o al hickness o he sandwich was 16.9 mm and he skins we e 1 mm hick. Equa ions o sandwich s anda d ASTM C393 [40] we e used in o de o ob ain he equi alen lexu al modulus and es a di e en spans we e ca ied ou in o de o ob ain G. In e e ence [15], G eq was he alue ha has been called 5 6Geq in he p esen s udy. This nomencla u e was due o he ac ha shea s esses in he co e o a sandwich ma e ial ha e a uni o m dis ibu ion and consequen ly, he shea ac o 5/6 associa ed o he pa abolic dis ibu- ion in a homogeneous ec angula sec ion becomes 1. In o de o a oid con usion, he sandwich shea modulus G san is de ined as: Gsan =5 6Geq (39) The elas ic p ope ies used o calcula ions a e gi en in Table 1. The elas ic modulus o he co e has been ob ained om a o mula ha akes in o accoun he geome y and p ope ies o he honeycomb [41]. The Table 1 P ope ies o he skins and co e o he sandwich. h (mm) E G Skins 1 70 GPa 26.9 GPa Co e 15 0.344 MPa 20.7 MPa Table 2 Equi alen moduli ob ained om di e en heo e ical app oaches. E eq (MPa) G san (MPa) ASTM C365-03 21,913 20.8 Cu en app oach 21,914 20.8 F. Mujika e al. Composi es Pa A 175 (2023) 107802 6 shea modulus o he honeycomb co e is he alue ob ained by FEM [15]. 4.1.2. Resul s ob ained om he analy ical app oach Table 2 shows ha he equi alen moduli ob ained om he cu en app oach, eplacing he alues o Table 1 in Eqs. (22) and (23), and he simpli ied equa ions gi en in he s anda d ASTM C365-03 [42] ag ee. Fig. 3 shows no mal and shea s esses no malized wi h espec o he maximum alues, ob ained om he cu en analy ical model, showing ha no mal s esses ac on he aces and shea s esses ac on he co e. 4.1.3. Expe imen al esul s The expe imen al da a co esponding o a specimen o e e ence [15] a e included in Table 3. G and K a e de ined as he ela i e in luence o shea and sys em-s i ness wi h espec o bending, in he displacemen o he load applica ion poin . F om Eq. (24) 1 hey a e: G=Eeq Gsan (h L)2 K=4wEeq Ks(h L)3 (40) Shea e ec s a e much g ea e han bending e ec s in all cases, e en o he la ges span, whe e he span- o- hickness a io is 20. The e ec o he sys em-s i ness dec eases as span inc eases, being less han he shea e ec . Table 4 shows he alues o E eq , G san and K s ob ained using y E and y G as eg ession unc ions. The alues o E eq and G san ob ained by he eg ession o y E ag ee be e wi h hose o Table 2. Mo eo e , he de e mina ion coe icien is g ea e in he case o y E . The di e ences o E eq and K s alues in Table 4 a e much g ea e han hose ob ained o G san . Reg ession cu es a e analyzed in o de o explain he sou ce o hese di e ences. Fig. 4 shows he eg ession cu es and expe imen al da a ha co espond o E 3ps and G 3ps , wi h alues o A, B and C ob ained om he eg ession o y E . The cu es ob ained wi h he alues o A, B and C coming om he eg ession o y G and y K a e quali a i ely simila . Acco ding o Fig. 4(a), spans needed o ob ain he equi alen lexu al modulus a e oo g ea o es specimens in h ee-poin bending. Mo e- o e , as expe imen al alues a e a om he asymp o ic alue, small a ia ions in hose alues can lead o di e en E eq alues. This could be he eason o he di e ence be ween alues o E eq ob ained om di e en eg ession unc ions in Table 4. In he case o es ing wi h spans Fig. 3. S ess dis ibu ion o a sandwich in bending: (a) No mal s esses; (b) Shea s esses. Table 3 Expe imen al alues o bending modulus and in luence ac o s o shea and sys em-s i ness. L(mm) E 3ps (MPa) G (%) K (%) 120 837 2180 332 160 1516 1226 140 200 2227 785 72 240 3192 545 41 300 4672 349 21 340 5769 272 15 Table 4 Equi alen alues ob ained by eg ession o y E and y G unc ions. Reg ession E eq (MPa) G san (MPa) K s (MPa) Coe . R 2 (%) y E (MPa −1 ) 22,282 20.3 6002 99.9 y G (MPa −1 ) 28,413 18.6 13,277 94.5 Fig. 4. Expe imen al alues and eg ession cu es including he s i ness o he sys em: (a) Bending modulus E 3ps ; (b) Shea modulus G 3ps . F. Mujika e al. Composi es Pa A 175 (2023) 107802 7 g ea e han 2.5 m, he bending o he specimen would be p obably a ec ed by i s own weigh . In he case o Fig. 4(b), G 3ps shows a maximum alue and he eg ession cu e does no each G san , as he e ec o he sys em-s i ness inc eases as span dec eases. Since expe i- men al alues a e nea his maximum alue, alues o G san a e less a ec ed by small a ia ions o expe imen al da a, as seen in Table 4. Acco ding o Fig. 5, K 3ps alues inc ease asymp o ically as span de- c eases. As K s alue co esponds o 0 span, his asymp o ic beha iou indica es ha i is e y sensi i e o small a ia ions o expe imen al alues. The e o e, i seems easonable o ob ain K s by an inden a ion es [36] and use Eq. (38) o he educ ion o expe imen al da a. The expe imen al alue ob ained by inden a ion es s was K s =4800 N/mm. As shea e ec s ha e mo e in luence han bending e ec s, Table 5 shows he alues o G 3p ob ained om Eq. (35), whe e he slope m does no include he in luence o he sys em-s i ness. E being he ela i e in luence o bending wi h espec o shea in he displacemen o he load applica ion poin , om Eq. (24) 2 , conside ing ha Ks→∞, i is: δCs =FL 4Gsanwh [1+ E]whe e E=Gsan Eeq (L h)2 (41) Table 5 shows ha he in luence o shea (100 %) is mo e impo an han bending in all cases. E en o he la ges span, whe e he span- o-dep h a io is L/h =20, he bending e ec is he 38 % o he shea e ec . The e o e, in a sandwich specimen, in spi e o he ac ha inc easing he dis ance be ween aces inc eases he lexu al s i ness, he global s i - ness o he specimen dec eases no iceably due o he g ea alue o he a io Gsan Eeq . In his pa icula case, his a io is a ound 1000. In he case o an iso opic ma e ial, i is less han 3 in all cases. Table 6 shows he alues o E eq and G san ob ained by z E and z G o Eq. (38)as eg ession unc ions. The di e ences in alues o equi alen modulus in Table 6 a e less han in Table 4. Mo eo e , he alue o G 3p ob ained o he minimum span o 120 mm in Table 5 is e y close o G san o Table 6, as he in luence o bending is 5 % o he shea in luence. Fig. 6 shows he cu es ha co espond o E 3p and G 3p , ha ing ob- ained A and B om z G . Fig. 6(a) is simila o Fig. 4(a), being E3p>E3ps. In he case o Fig. 6(b), o small spans, he cu e ends asymp o ically o G san . Consequen ly, G san could be ob ained di ec ly pe o ming a low span h ee-poin bending es , whe e he in luence o shea is g ea enough. I would be simila o ob ain E eq o g ea spans when bending e ec s a e dominan . The e o e, om a s i ness poin o iew, a h ee- poin bending es becomes a shea es when he shea is dominan . Ne e heless, as he inden a ion e ec inc eases as span dec eases, i is necessa y o de e mine he slope o he load–de lec ion cu e in he linea zone a e inden a ion, when he con ac a ea be ween he spec- imen and he load olle does no change any mo e [36]. Acco ding o Table 6 and Fig. 6, he de e mina ion o E eq and G san ha ing de e mined p e iously K s , gi es mo e accu a e esul s han he simul aneous de e - mina ion o E eq , G san and K s . Fig. 5. Expe imen al and eg ession alues o he s i ness K 3ps . Table 5 G 3p expe imen al alues dis ega ding he e ec o sys em-s i ness. L(mm) G 3p (MPa) E (%) 120 20.1 5 160 19.6 8 200 17.8 13 240 17.4 19 300 16.0 29 340 15.2 38 Table 6 Equi alen alues ob ained by eg ession o z E and z G unc ions. Reg ession E eq (MPa) G san (MPa) Coe . R 2 (%) z E (MPa −1 ) 20,001 21.2 99.7 z G (MPa −1 ) 22,301 20.8 95.4 Fig. 6. Expe imen al alues and eg ession cu es dis ega ding he s i ness o he sys em: (a) Bending modulus E 3p ; (b) Shea modulus G 3p . Table 7 Elas ic p ope ies o unidi ec ional ma e ials. E 1 (GPa) G 13 (GPa) E-Glass/Epoxy 41 4.3 Ca bon ib e/Epoxy 147 7.0 F. Mujika e al. Composi es Pa A 175 (2023) 107802 8 4.2. Hyb id lamina es 4.2.1. Ma e ial p ope ies Hyb id lamina es made up o plies o expoxy ma ix ein o ced wi h ca bon ib es and glass ib es, ha e been analyzed by FEM. The ma e ial p ope ies a e gi en in Table 7 [42]: A symme ic and an asymme ic hyb id lamina e made o ca bon/ epoxy and glass/epoxy laye s in h ee poin bending ha e been modelled in he so wa e Abaqus. Incompa ible plane-s ess elemen s o 0.2 mm ha e been used. The o ce and displacemen o he load applica ion poin ha e been de e mined, assuming small displacemen s. In his case, he coe icien K s is ela ed o he local de o ma ion o he mesh nea he load applica ion and eac ion poin s. Fig. 7 shows he c oss sec ion o bo h lamina es. Th ee-poin bending models ha e been un a di e en spans. The alues o E eq and G eq ob ained by i ual es ing ha e been compa ed wi h hose ob ained by he analy ical model. The hickness in bo h lamina es is 3 mm and spans used in he simula ion a e (mm): 34, 40, 48, 60, 80. Fig. 7. C oss sec ions o he hyb id lamina es: (a) Symme ic; (b) Asymme ic. Table 8 Equi alen moduli o he cu en app oach o symme ic and asymme ic lamina es. E eq (MPa) G eq (MPa) Symme ic 115,593 4796 Asymme ic 67,213 4513 Fig. 8. S ess dis ibu ion o he symme ic lamina e: (a) No mal s esses; (b) Shea s esses. Fig. 9. S ess dis ibu ion o he asymme ic lamina e: (a) No mal s esses; (b) Shea s esses. F. Mujika e al. Composi es Pa A 175 (2023) 107802 9 4.2.2. Resul s ob ained om he analy ical app oach Table 8 shows he equi alen moduli ob ained by he cu en model, eplacing he alues o Table 7 and he dimensions o Fig. 7 in Eqs. (22) and (23). Fig. 8 and Fig. 9 show no mal and shea s esses no malized wi h espec o he maximum alues o he symme ic and asymme ic lam- ina es, espec i ely, ob ained wi h he model o he p esen wo k. 4.2.3. Nume ical esul s Table 9 and Table 10 show nume ical esul s o he appa en modulus E 3ps and in luence ac o s o shea and sys em-s i ness de ined in Eq. (40), o he symme ic and asymme ic lamina es, espec i ely. In spi e o in bo h cases bending is dominan , shea e ec s a e g ea e in he symme ic lamina e and he sys em-s i ness in luence is simila in bo h cases. Table 11 shows he alues o E eq , G eq and K s ob ained using y E as eg ession unc ion. In his case, he esul s ob ained by y G a e he same, as he e a e no expe imen al e o s. Taking as e e ence he alues o Table 8 ob ained om he analy ic app oach o his s udy, he di e ence in E eq is less han 1 % o alues in Table 11. In he case o G eq , he di e ence is 2.3 % in he case o he symme ic lamina e and 6.2 % in he case o he asymme ic lamina e. Those di e ences a e ela ed o he dominan ole o bending. Fig. 10 shows he eg ession cu es and expe imen al da a ha co espond o E 3ps and G 3ps in he case o he asymme ic beam. The alues o he symme ic lamina e a e quali a i ely simila , so hey a e no included. As bending is dominan , expe imen al esul s o E 3ps a e nea he ho izon al line ha co espond o E eq . On he o he hand, expe imen al alues o G 3ps and he maximum o he eg ession cu e a e a om he alue o G eq . Acco ding o Fig. 11, as in he case o Fig. 5 o he sandwich spec- imen, K s alue ela ed o he local de o ma ion o he mesh is e y sensi i e o small a ia ions o da a. As in his case expe imen al e o s do no exis , he da a o K s ob ained by eg ession ha e been in oduced o ob ain esul s o equi alen moduli, dis ega ding he e ec o he local mesh de o ma ion. Inden a ion simula ions ha e been done o e i y ha K s ob ained by eg ession and by inden a ion ag ee. Fu he mo e, i has been e i ied ha K s depends on he local de o ma ion o he model a load applica ion and suppo s poin s. Fig. 12 shows he cu es ha co espond o E 3p and G 3p , ha ing ob ained A and B om z E . Fig. 12(a) is simila o Fig. 10(a). In he case o Fig. 12(b), he cu e ends o G eq when he span goes o ze o. Ne e - heless, expe imen al alues a e a o om his alue e en in he case o he smalles span. The e o e, small a ia ions in expe imen al da a could a ec he esul o G eq . Table 9 Nume ical alues o bending modulus and in luence ac o s o shea and sys em s i ness in he symme ic lamina e. L(mm) E 3ps (MPa) G (%) K (%) 80 110,522 4 1 60 106,511 7 1 48 101,489 11 3 40 95,661 16 5 34 88,885 22 8 Table 10 Nume ical alues o bending modulus and in luence ac o s o shea and sys em s i ness in he asymme ic lamina e. L(mm) E 3ps (MPa) G (%) K (%) 80 65,396 2 0 60 63,892 4 1 48 61,935 7 2 40 59,563 9 3 34 56,665 13 6 Table 11 Equi alen alues ob ained by eg ession o y E . Specimen E eq (MPa) G eq (MPa) K s (MPa) Coe . R 2 (%) Symme ic 115,599 4906 59,260 100 Asymme ic 67,218 4810 49,772 100 Fig. 10. Nume ical alues and eg ession cu es including local de o ma ion in he asymme ic lamina e: (a) Bending modulus E 3ps ; (b) Shea modulus G 3ps . Fig. 11. Asymme ic lamina e: expe imen al and eg ession alues o he s i ness K 3ps . F. Mujika e al.