Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

52, 4, pp. 971-980, Warsaw 2014

TRANSVERSE SHEAR MODULUS OF ELASTICITY FOR THIN-WALLED

CORRUGATED CORES OF SANDWICH BEAMS. THEORETICAL STUDY

Ewa Magnucka-Blandzi

Poznan University of Technology, Institute of Mathematics, Poznań, Poland

e-mail: ewa.magnucka-blandzi@put.poznan.pl

Krzysztof Magnucki

Poznan University of Technology, Institute of Applied Mechanics, Poznań, Poland

e-mail: krzysztof.magnucki@put.poznan.pl

The subject of the paper are four corrugated cores in form of circular arcs, a sin wave,
trapezoids and an odd function. Transverse shear modules of these corrugated cores are
analytically determined. A comparative analysis of these transverse shear modules is pre-
sented. Areas of cross sections of the corrugated cores are constant. The theoretical study
shows considerable sensitivity of the shear modulus to shape of the corrugation.

Keywords: shear modulus, corrugated core, sandwich beams

1. Introduction

Theoretical fundamentals for sandwich structures were initiated in themid of the 20th century.
The shear modulus of a core of a sandwich structure considerably affects its bending and buc-
kling. Libove and Hubka (1951) presented the primary elaboration related to analytical study
of elastic constants for corrugated cores of sandwich plates. Carlsson et al. (2001) reviewed and
adapted previous analytical approaches to the analysis of elastic stiffnesses of corrugated core
sandwich panels into the framework of the first-order shear deformation laminated plate the-
ory. Buannic et al. (2003) computed the effective properties of corrugated core sandwich panels
using homogenisation theory. Motivated by the results of numerical simulations, Aboura et al.
(2004) examined behaviour of the linear homogeneous cardboard analytically, and confirmed
good agreeement of the results. A numerical approach to evaluate the stiffness parameters for
corrugated boardwasdescribedbyBiancolini (2005). Cheng et al. (2006) proposedfinite element
analysis approach to evaluate equivalent elastic properties of complex sandwich structures. The
elastic bending of unstiffened and stiffened corrugated plates was studied by Peng et al. (2007),
where amesh-freeGalerkinmethodwas applied in the analysis. Amethod for themodelling of a
corrugated boardpanelwas outlined by Isaksson et al. (2007) – corrugated core sandwicheswere
homogenized and the panels transformed to equivalent homogeneous layers with effective equal
properties. Talbi et al. (2009) presented an analytical homogenization model for a corrugated
cardboard and its numerical implementation with a shell element. An analytical model for the
compressive and shear response of monolithic and hierarchical corrugated composite cores was
developed by Kazemahvazi and Zenkert (2009). Kress andWinkler (2010) studied the problem
of finding a substitutematerialmodel for describing the load response of globally flat corrugated
sheetsmade of multidirectional laminates. They determined themaximal possible deformations
of a corrugated sheet where the corrugation pattern consisted of two circular segments. Pan
et al. (2008) investigated the transverse shear mechanical behaviour and failure mechanism of
aluminum alloy honeycomb. He et al. (2012) presented a semi-analytical method for bending
analysis of the sandwich panel with a core of triangular-shape, honeycomb-shape and X-shape.



972 E.Magnucka-Blandzi, K.Magnucki

Aganovic et al. (1996) presented the equilibrium displacements corresponding to Koiter’s shell
model. The sequence of shells was considered as a slight periodical perturbation of the middle
surface of the plate was shown to converge to the equilibriumdisplacement of the classical plate
model. Corresponding corrector-type results were proved by the homogenization method. Mi-
chalak (2001) presented such a form of the mezo-shape function for a mezostructural model,
which is suitable for quantitative analysis of dynamic behaviour of awavy-plate. Governing equ-
ations of the averaged theory of wavy-plates were obtained for different forms of themezo-shape
functions for in-plane and out-of-plane displacements of the plate. The work does not address
the averaged values ofmodules determined, for example, with the use the averaged theory or the
asymptotic homogenization method, which are presented by Aganivic et al. and byMichalak.

The subject of the theoretical studypresented in this paper are four corrugated cores in form
of circular arcs, a sin wave, trapezoids and an odd function. The transverse shear modulus for
each core is analytically determined.

2. Analytical description of shear moduli for corrugated cores

2.1. Corrugation of the core in form of circular arcs

The corrugated core between two faces undergoes shearing as shown in Fig. 1.

Fig. 1. Scheme of the corrugated core of the circular arcs shape

Geometrical relations for the circular arc (Fig. 1) are as follows:

— radius of the circular arc

R0 =
tc
16Cca

(2.1)

— complementary angle of the circular arc

β=arccos(4xb0Cca) for 2(1−xt0)¬xb0 (2.2)

where: Cca =(1−xt0)/[x2b0+4(1−xt0)
2], xt0 = t0/tc, xb0 = b0/tc – dimensionless parameters,

b0 – corrugation pitch.

The basic system of forces for the half-pitch of the circular arc corrugation (Fig. 1) with the
reaction

R=
1

xb0
F (2.3)

enables one to formulate the bendingmoment

M
(ca)
b (ϕ)=

1

2
FR0
[
sinϕ− sinβ−

2

xb0
(cosβ− cosϕ)

]
(2.4)



Transverse shear modulus of elasticity for thin-walled corrugated cores... 973

The elastic strain energy

U(ca)ε =
12R0
Eat30

π/2∫

β

[M
(ca)
b (ϕ)]

2 dϕ (2.5)

where a is width of the corrugated core in the x-axis direction.

The displacement vB (Fig. 1) is determined on the basis of Castigliano’s second theorem

vB =
dU
(ca)
ε

dF
=
6FR30
Eat30

Sca (2.6)

where

Sca =Sca1+
4

xb0
Sca2+

4

x2b0
Sca3

Sca1 =
(π
2
−β
)(1
2
+sin2β

)
−
3

4
sin(2β)

Sca2 =1− sinβ+
1

2

(π
2
−β
)
sin(2β)−

3

2
cos2β

Sca3 =
(π
2
−β
)(1
2
+cos2β

)
−2
(
1−
3

4
sinβ
)
cosβ

The shear strain in the yz-plane is as follows

γ(ca)yz =
vB
tc
=
6FR30
Eatct

3
0

Sca (2.7)

From Hooke’s law

τ(ca)yz =
F

ab0
=G(ca)yz γ

(ca)
yz (2.8)

the shear modulus of elasticity for the circular arc corrugation is

G(ca)yz = G̃
(ca)
yz E (2.9)

where the dimensionless shear modulus is

G̃(ca)yz =
2048

3

x3t0C
3
ca

xb0Sca
(2.10)

The cross section area of the circular arc corrugation for one pitch (Fig. 1) amounts to

A
(ca)
0 =2(π−2β)R0t0 = Ã

(ca)
0 t

2
c (2.11)

where the dimensionless area

Ã
(ca)
0 =

1

8
(π−2β)

xt0
Cca

(2.12)



974 E.Magnucka-Blandzi, K.Magnucki

2.2. Corrugation of the core in form of a sin wave

The function of the corrugation is

f(y)=
1

2
tc(1−x0)sin(2πη) (2.13)

where η= y/b0 denotes the dimensionless coordinate.
The force system for the half-pitch of the sin wave corrugation (Fig. 2) is similar to that of

the circular arc corrugation with reaction (2.3). The bendingmoment is

M
(sin)
b (η)=

1

4
Ftc
[
(1−xt0)sin(2πη)−4η

]
(2.14)

Then, the elastic strain energy

U(ca)ε =
12b0
Eat30

1/4∫

0

[M
(sin)
b (η)]

2
√
1+ c20cos

2(2πη) dη (2.15)

where c0 =π(1−xt0)/xb0 is the dimensionless parameter.

Fig. 2. Scheme of the wave-shaped corrugated core

The displacement vB (Fig. 2) on the basis of Castigliano’s second theorem is as follows

vB =
dU
(sin)
ε

dF
=
3Fb0t

2
c

2Eat30
S
(sin)
1 (2.16)

where

S
(sin)
1 =(1−xt0)

2S
(sin)
11 +8(1−xt0)S

(sin)
12 +16S

(sin)
13

S
(sin)
11 =

1/4∫

0

sin2(2πη)
√
1+ c20cos

2(2πη) dη

S
(sin)
12 =

1/4∫

0

ηsin(2πη)
√
1+ c20cos

2(2πη) dη

S
(sin)
13 =

1/4∫

0

η2
√
1+ c20cos

2(2πη) dη

The shear strain in the yz-plane is as follows

γ(sin)yz =
vB
tc
=
3Fb0t0
2Eat30

S
(sin)
1 (2.17)



Transverse shear modulus of elasticity for thin-walled corrugated cores... 975

Thus, by analogy to expressions (2.8) and (2.9), the dimensionless shear modulus is

G̃(sin)yz =
2x2t0

3x2b0S
(sin)
1

(2.18)

The cross section area of the circular arc corrugation for one pitch (Fig. 2) is

A
(sin)
0 =4b0t0S

(sin)
0 = Ã

(sin)
0 t

2
c (2.19)

where

S
(sin)
0 =

1/4∫

0

√
1+ c20cos

2(2πη) dη

and the dimensionless area

Ã
(sin)
0 =4xt0xb0S

(sin)
0 (2.20)

2.3. Corrugation of the core in form of trapezoids

Geometrical relations for the trapezoid (Fig. 3) are as follows

sinα0 =
2(1−xt0)√
Ct

cosα0 =
xb0−2kbxt0√

Ct
(2.21)

where kb = b1/t0,Ct =(xb0−2kbxt0)2+4(1−xt0)2 are dimensionless parameters.

Fig. 3. Scheme of the corrugated core of the trapezoid shape

The force system for the half-pitch of the sin wave corrugation (Fig. 3) is similar to that of
the circular arc corrugation with the reaction (2.3). The normal force and the bendingmoment
in the trapezoidal corrugated core are

N(trap)(s)=
1

2
F
(
cosα0+

2

xb0
sinα0

)

M
(trap)
b (s)=

1

2
F
(
sinα0−

2

xb0
cosα0

)
s

(2.22)

The elastic strain energy with consideration of the tension and bending energy is as follows

U(trap)ε =
1

Eat0

st∫

0

[N(trap)(s)]2 ds+
12

Eat30

st∫

0

[M
(trap)
b (s)]

2 ds (2.23)

where st = tc
√
Ct/4 is the length of the trapezoid arm.



976 E.Magnucka-Blandzi, K.Magnucki

The shear strain in the yz-plane, by analogy to expressions (2.6) or (2.16), is written

γ(trap)yz =
vB
tc
=
Fsttc
2Eat0b

2
0

S
(trap)
1 (2.24)

where

S
(trap)
1 =

1

Ct
xb0[xb0(xb0−2kbxt0)+4(1−xt0)]+(xb0−2kb)2

Thus, the dimensionless shear modulus is

G̃(trap)yz =
8xt0xb0

S
(sin)
1

√
Ct

(2.25)

The cross section area of the trapezoid corrugation for one pitch (Fig. 3) is

A
(trap)
0 = tct0

(
2kbxt0+

√
Ct
)
= Ã

(trap)
0 t

2
c (2.26)

where the dimensionless area is

Ã
(trap)
0 =xt0

(
2kbxt0+

√
Ct
)

(2.27)

Fig. 4. Scheme of the corrugated core in form of an odd function shape

The function of the corrugation is

f(y)=
1

2
tc(1−xt0)φ(η) (2.28)

where the odd function is in the following form

φ(η) = η
[
6−32η2+

kf
256
(1−32η2+256η4)

]
(2.29)

and kf is dimensionless parameter.
Thus, by analogy to the sin wave corrugation the dimensionless shear modulus is

G̃(odd−f)yz =
2x2t0

3x2b0S
(odd−f)
1

(2.30)

where

S
(odd−f)
1 =

1/4∫

0

[(1−xt0)φ(η)−4η]2
√
1+ c2fφ

2
1(η) dη

cf =
1−xt0
2xb0

φ1(η)= 6−96η2+
kf
256
(1−96η2+1280η4)



Transverse shear modulus of elasticity for thin-walled corrugated cores... 977

and the dimensionless area

Ã
(odd−f)
0 =4xt0xb0S

(odd−f)
0 (2.31)

where

S
(odd−f)
0 =

1/4∫

0

√
1+ c2fφ

2
1(η) dη

Expressions (2.10), (2.18), (2.25) and (2.30) for the dimensionless shear moduli and (2.12),
(2.20), (2.27) and (2.31) for dimensionless areas serve as a basis of comparative analysis of the
four shapes of the corrugated cores.

3. Comparative analysis of shear moduli of the corrugated cores

Values of deflections and critical loads of sandwich structures are related to the values of the core
shear moduli. Maximization of the value of the shear modulus results in the maximum value of
rigidity of the sandwich structure. In consequence, the qualitymeasure of the corrugated core is
the value of the dimensionless shearmodulus for a constant value of the dimensionless area of a
single pitch. The comparative analysis is carried out for the following example data: thickness
of the core tc = 12.2mm, corrugation pitch b0 = 28mm and dimensionless area of the single

corrugation pitch Ã
(c)
0 = 0.2. The geometric size and dimensionless transverse shear moduli of

the studied cores calculated based on the above data are as follows:

• the circular arc shape of the corrugation (Fig. 1)

(Thickness of the corrugated sheet t0 = 0.760mm, radius of the circular arcs (2.1)
R0 = 7.14mm, complementary angle (2.2) β = 0.2006rad, and dimensionless shear mo-

dulus (2.10) G̃
(ca)
yz =0.00170.)

• the sin wave shape of the corrugation (Fig. 2)

(The tickness of the corrugated sheet t0 =0.799mm, and the dimensionless shearmodulus

(2.18) G̃
(sin)
yz =0.00851.)

• the trapezoid shape of the corrugation (Fig. 3)

(Themaximum value of the dimensionless shearmodulus G̃
(trap)
yz,max =0.1755 occurs for the

thickness of the corrugated sheet t0 =0.815mmand the length of trapezoid parallel sides
b1 =0.9271mm.)

• the odd function shape of the corrugation (Fig. 4).

The maximum value of the dimensionless shear modulus G̃
(odd−f)
yz,max = 0.2707 occurs for the

thickness of the corrugated sheet t0 =0.816mmand the dimensionless parameter kf =−507.9.
It canbenoticed that for core corrugations in sinwave andodd function shapes, the values of the

shear moduli are equal to G̃
(sin)
yz = G̃

(odd−f)
yz = 0.00851 for the sheet thickness t0 = 0.799mm.

Moreover, for the trapezoidal corrugation and sheet thickness t0 = 0.780mm, the value of the

shearmodulus G̃
(trap)
yz =0.00849 approximates the above values of G̃

(sin)
yz = G̃

(odd−f)
yz =0.00851.

In this case, the graph of the sinusoidal shape of core corrugation coincides with the graph of
the odd function (Fig. 5).
The shear moduli of the corrugated cores with circular arcs or sin wave shapes for any data

are constant (G̃
(ca)
yz = 0.00170, G̃

(sin)
yz = 0.00851 are valid for the examplary data). The shapes

of these corrugations are uniquely defined for the assumed data. However, the trapezoid or the



978 E.Magnucka-Blandzi, K.Magnucki

Fig. 5. Comparison of three corrugation shapes for the trapezoid (b1 =3.710mm), odd function and sin
wave

odd function shapes are not uniquely defined for the assumed data as their shapes may be
controlled by varying the length of the trapezoid parallel sides b1 (Fig. 3) or the dimensionless
parameter kf of the function (2.29). The results of numerical calculations for these shapes of
corrugations (Table 1 and Table 2) are shown in Fig. 6.

Table 1.Values of t0, b1 and dimensionless shear modulus (2.25)

t0 [mm] 0.780 0.790 0.800 0.805 0.810 0.8125 0.8150 0.8175 0.8251

b1 [mm] 3.710 2.973 2.192 1.783 1.362 1.1465 0.9271 0.7041 0

G̃
(trap)
yz 0.00849 0.0152 0.0351 0.0622 0.1199 0.1574 0.1755 0.1552 0.0550

Table 2.Values of t0, kf and dimensionless shear modulus (2.30)

t0 [mm] 0.799 0.810 0.812 0.813 0.814 0.815 0.8155 0.816 0.8164

kf [mm] 71.36 −199.3 −270.0 −311.4 −359.4 −418.7 −456.7 −507.9 −607.6
G̃
(odd−f)
yz 0.00851 0.0264 0.0397 0.0523 0.0748 0.1245 0.1769 0.2707 0.2526

Fig. 6. Shear moduli for the trapezoid and odd function core shapes

These graphs reveal the shear moduli sensitivity to variation of the corrugated sheet thick-
ness t0 for the above two shapes of corrugations. A minor change in the of corrugated sheet

thickness t0 results in a significant change in the shear moduli values G̃
(trap)
yz and G̃

(odd−f)
yz .

This sensitivity arises in the neighbourhood of the extremum.Thus,manufacturing of sandwich
beamswith corrugated cores of trapezoid or odd function shapesmay be impossible, taking into
account the maximum values of the shear moduli. The profile and dimensional tolerance is of
high importance in this case. The trapezoid and odd function core shapes for the extremum (the



Transverse shear modulus of elasticity for thin-walled corrugated cores... 979

trapezoid: t0 =0.815mm, b1 =0.9271mm, G̃
(trap)
yz,max =0.1755, the odd function: t0 =0.816mm,

kf =−507.9, G̃
(odd−f)
yz,max =0.2707) are shown in Fig. 7.

Fig. 7. Comparison of the two corrugation shapes for the trapezoid (b1 =0.9271mm) and the odd
function at extremum

Fig. 8. Comparison of the three corrugation shapes for the trapezoid (b1 =0.9271mm) odd function
and sine wave form

It can benoticed that in the extreme case, the shapes of core corrugations in of the trapezoid
and odd function type are similar.

4. Conclusions

The theoretical studies of four corrugated cores allows one to draw the following conclusions:

• the core in form of circular arc is the most susceptible to shearing when the value of
dimensionless transverse shear modulus of elasticity is the lowest: G̃

(ca)
yz =0.00170,

• the core in form of the sin wave is more resistant to shearing than the circular arc core
when the transverse shear modulus of elasticity is higher: G̃

(sin)
yz =0.00851,

• the trapezoidal core is muchmore resistant to shearing than the two above, themaximum
value of the transverse shearmodulus is G̃

(trap)
yz,max =0.1755, nevertheless, the shearmodulus

is sensitive to variation of the corrugated sheet thickness t0,

• the core having shape of an odd function is distinguished by the greatest resistance to she-
aring, themaximum value of the transverse shearmodulus is G̃

(odd−f)
yz,max =0.2707, however,

it is very sensitive to the change of the corrugated sheet thickness t0.



980 E.Magnucka-Blandzi, K.Magnucki

The theoretical studies show significant differences between the four shapes, including the three
basic ones: circular arcs, sin wave and trapezoid.

Acknowledgements

The research was conducted within the framework of Statutory Activities No. 04/43/DSPB/0079

and 02/21/DSPB/3452.

References

1. Aboura Z., Talbi N., Allaoui S., Benzeggagh M.L., 2004, Elastic behavior of corrugated
cardboard: experiments andmodeling,Composite Structures, 63, 53-62

2. Aganovic I., Marusic-Paloka E., Tutek Z., 1996, Slightly wrinkled plate, Asymptotic Ana-
lysis, 13, 1-29

3. Biancolini M.E., 2005, Evaluation of equivalent stiffness properties of corrugated board, Com-
posite Structures, 69, 322-328

4. Buannic N., Cartraud P., Quesnel T., 2003, Homogenization of corrugated core sandwich
panels,Composite Structures, 59, 299-312

5. CarlssonL.A.,NordstrandT.,WesterlindB., 2001,On the elastic stiffnesses of corrugated
core sandwich, Journal of Sandwich Structures and Materials, 3, 253-267

6. Cheng Q.H., Lee H.P., Lu C., 2006, A numerical analysis approach for evaluating elastic con-
stants of sandwich structures with various core,Composite Structures, 74, 226-236

7. HeL.,ChengY.-S., Liu J., 2012,Precise bending stress analysis of corrugated-core,honeycomb-
core and X-core sandwich panel,Composite Structures, 94, 1656-1668

8. Isaksson P., Krusper A., Gradin P.A., 2007, Shear correction factor for corrugated core
structures,Composite Structures, 80, 123-130

9. Kazemahvazi S., Zenkert D., 2009, Corrugated all-composite sandwich structures. Part 1:Mo-
deling,Composites Science and Technology, 69, 913-919

10. KressG.,WinklerM., 2010,Corrugate laminate homogenizationmodel,Composite Structures,
92, 795-810

11. Libove C., Hubka R.E., 1951, Elastic constants for corrugated-core, Sandwich plates, Technical
Note 2289, NACA,Washington

12. Michalak B., 2001, Analysis of dynamic behaviour of wavy-type plates with a mezzo-periodic
structure, Journal of Theoretical and Applied Mechanics, 39, 947-958

13. Pan S.-D., Wu L.-Z., Sun Y.-G., 2008, Transverse shear modulus and strength of honeycomb
cores,Composite Structures, 84, 369-374

14. Peng L.X., Liew K.M., Kitipornchai S., 2007, Analysis of stiffened corrugated plates based
on the FSDTvia themesh-freemethod, International Journal of Mechanical Sciences, 49, 364-378

15. Talbi N., Batti A., Ayad R., Guo Y.Q., 2009, An analytical homogenizationmodel for finite
element modeling of corrugated cardboard,Composite Structures, 88, 280-289

16. Winkler M., Kress G., 2010, Deformation limits for corrugated cross-ply laminates,Composite
Structures, 92, 1458-1468

Manuscript received January 6, 2014; accepted for print May 12, 2014