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-
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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
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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.
