Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

55, 2, pp. 433-446, Warsaw 2017
DOI: 10.15632/jtam-pl.55.2.433

STRENGTH OF A METAL SEVEN-LAYER RECTANGULAR PLATE WITH

TRAPEZOIDAL CORRUGATED CORES

Ewa Magnucka-Blandzi, Zbigniew Walczak, Leszek Wittenbeck

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

e-mail: ewa.magnucka-blandzi@put.poznan.pl, zbigniew.walczak@put.poznan.pl, leszek.wittenbeck@put.poznan.pl

Marcin Rodak

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

e-mail: marcin.rodak@put.poznan.pl

The subject of analytical and numerical investigations in this paper is a metal seven-layer
rectangular plate with a trapezoidal corrugated main core and two trapezoidal corrugated
cores of faces. The hypothesis of deformation of the normal to the middle surface of the
plate after bending and field of displacements is formulated. The plate is simply supported
on all its edges and subjected to a uniform pressure. Equations of equilibrium are derived
based on the theorem of minimum total potential energy and are solved with the use of
the Galerkinmethod. The influence of the trapezoidal corrugation pitch of the cores on the
deflection and the equivalent stress is analysed.

Keywords: rectangular layeredplate, corrugated cores, hypothesis of deformation, deflection,
stresses

1. Introduction

The basic theoretical models of sandwich structures were formulated in the mid of the 20th
century. Two decades later, Plantema (1966) and Allen (1969) elaborated first monographs de-
voted to bending, buckling and design problems of sandwich beams, plates and shells. Noor
et al. (1996), Vinson (2001), Carrera (2003), Carrera and Brischetto (2009) presented a re-
view of the problems related to computational models, applications and analysis of sandwich
structures. Kazemahvazi and Zenkert (2009) developed an analytical model for the compressive
and shear response of monolithic and hierarchical corrugated composite cores. Ji et al. (2010)
described design procedures and the construction process of a glass fiber reinforced polymer
corrugated-core sandwich bridge superstructure. Seong et al. (2010) introduced bi-directionally
corrugated cores in order to reduce anisotropic behaviour of sandwich plates with open channel
cores under the bending load. Magnucka-Blandzi (2011) described and solved analytically the
problemof a simply supported rectangular sandwich plate under compression in plane. Poirier et
al. (2013) proposed a methodology for designing lightweight laser-welded steel sandwich panels
with superior structural performance. Jha et al. (2013) presented static analysis of orthotro-
pic functionally graded elastic, rectangular and simply supported plates under transverse loads.
Zhang et al. (2013) investigated compressive strengths and the dynamic response of corruga-
ted sandwich plates with unfilled and foam filled sinusoidal plate cores. Magnucka-Blandzi and
Magnucki (2014) determined analytically transverse shear moduli of corrugated cores in four
different shapes. The influence of the corrugation shape on the shear modulus was studied. Le-
winski et al. (2015) studied transverse shear moduli of two thin-walled trapezoidal corrugated
cores of seven-layer sandwich plates. Magnucka-Blandzi et al. (2015) presented a mathematical
modelling of the transverse shearing effect for sandwich beamswith sinusoidal corrugated cores.



434 E.Magnucka-Blandzi et al.

The buckling and bending problems were solved. Magnucki et al. (2016) formulated two analy-
tical models of a seven-layer steel beam with a transverse sinusoidal corrugated main core and
two sandwich facings with steel foam cores, and solved the problem of bending and buckling.
Cheon andKim (2015) suggested an equivalent platemodel to analyze themechanical behavio-
ur of corrugated-core sandwich panels under tensile and bending loads. Mantari and Granados
(2015) presented a static analysis of functionally graded plates. In the paper, a simply supported
square sandwich plate was subjected to a bi-sinusoidal load. Vaidya et al. (2015) investigated
the response of sandwich steel beams with corrugated cores to quasi-static loading by employ-
ing experimental and computational approaches. A parametric study was also carried out on
large-scale structural size beams of a fewmeters in length.

The subject of this study is a metal seven-layer rectangular plate with a trapezoidal corru-
gated main core and two trapezoidal corrugated cores of facings. The plate is simply supported
and loaded with a uniformly distributed pressure.

2. Mathematical modelling of a seven-layer plate

2.1. Displacements and strains

A seven-layer rectangular plate with the trapezoidal corrugated main core, two inner flat
sheets, two trapezoidal corrugated cores of the facings and two outer flat sheets is shown in
Fig. 1. The plate is simply supported on all its edges and subjected to a uniform pressure p0.

Fig. 1. Scheme of the seven-layer rectangular plate

The direction of the core facings corrugations is orthogonal to the one of the main core
corrugation. Trapezoidal corrugations of the main core and facings cores are shown in Fig. 2.

Fig. 2. Scheme of the corrugations of the (a) main core and (b) faces cores



Strength of a metal seven-layer rectangular plate... 435

Taking into account the layered structures of the plate, it is easy to notice that the straight
line normal to themiddle plane of the plate before bendingdoes not remain straight and normal
after bending. The hypothesis is assumed that the straight line – normal after bending – takes
a shape of a broken line (Fig. 3). The problem of the hypothesis for multi-layer structures was
described, e.g. by Carrera (2003) andMagnucki et al. (2016).

Fig. 3. Deformation of the normal to the middle plane of the plate

The displacements with consideration of the hypothesis are as follows:
1) outer flat sheets

• the upper sheet for−(0.5+2x1+x2)¬ ζ ¬−(0.5+x1+x2)

u(x,y,z)=−tc1
[
ζ
∂w

∂x
+ψ(x,y)

]
v(x,y,z) =−tc1

[
ζ
∂w

∂y
+x2φ(x,y)

]
(2.1)

• the lower sheet for 0.5+x1+x2 ¬ ζ ¬ 0.5+2x1+x2

u(x,y,z)=−tc1
[
ζ
∂w

∂x
−ψ(x,y)

]
v(x,y,z) =−tc1

[
ζ
∂w

∂y
−x2φ(x,y)

]
(2.2)

2) trapezoidal corrugated cores of the facings

• the upper core for −(0.5+x1+x2)¬ ζ ¬−(0.5+x1)

u(x,y,z) =−tc1
[
ζ
∂w

∂x
+ψ(x,y)

]

v(x,y,z) =−tc1
{
ζ
∂w

∂y
−
[
ζ +
(1
2
+x1
)]
φ(x,y)

} (2.3)

• the lower core for 0.5+x1 ¬ ζ ¬ 0.5+x1+x2

u(x,y,z) =−tc1
[
ζ
∂w

∂x
−ψ(x,y)

]

v(x,y,z) =−tc1
{
ζ
∂w

∂y
−
[
ζ −
(1
2
+x1
)]
φ(x,y)

} (2.4)



436 E.Magnucka-Blandzi et al.

3) inner flat sheets

• the upper sheet for−(0.5+x1)¬ ζ ¬−0.5

u(x,y,z)=−tc1
[
ζ
∂w

∂x
+ψ(x,y)

]
v(x,y,z) =−tc1ζ

∂w

∂y
(2.5)

• the lower sheet for 0.5¬ ζ ¬ 0.5+x1

u(x,y,z)=−tc1
[
ζ
∂w

∂x
−ψ(x,y)

]
v(x,y,z) =−tc1ζ

∂w

∂y
(2.6)

4) main corrugated core for−0.5¬ ζ ¬ 0.5

u(x,y,z) =−tc1ζ
[∂w
∂x
−2ψ(x,y)

]
v(x,y,z) =−tc1ζ

∂w

∂y
(2.7)

where x1 = ts/tc1, x2 = tc2/tc1 are dimensionless parameters, ξ = z/tc1 – dimensionless
coordinate, tc1, tc2, ts – thicknesses of the main core, facing cores and flat sheets (Fig. 2),
ψ(x,y) = u1(x,y)/tc1,φ(x,y)= v1(x,y)/tc2 –dimensionless functions of displacements,u1(x,y),
v1(x,y) – displacements in the x and y directions, respectively, w(x,y) – deflection (Fig. 3) – de-
flections of each layer are equal and referenced to the middle plate layer, so w(x,y,z)≡ w(x,y)
and εz ≡ 0.
Thus, the linear relations for strains are as follows:

1) outer flat sheets (upper/lower)

ε(u/l)x =
∂u

∂x
=−tc1

(
ζ
∂2w

∂x2
±
∂ψ

∂x

)
ε(u/l)y =

∂v

∂y
=−tc1

(
ζ
∂2w

∂y2
±x2

∂φ

∂y

)

γxz = γyz =0 γ
(u/l)
xy =

∂u

∂y
+
∂v

∂x
=−tc1

(
2ζ

∂2w

∂x∂y
±
∂ψ

∂y
±x2

∂φ

∂x

) (2.8)

2) trapezoidal corrugated cores of the facings (upper/lower)

ε(u/l)x =−tc1
(
ζ
∂2w

∂x2
±
∂ψ

∂x

)
ε(u/l)y =−tc1

{
ζ
∂2w

∂y2
−
[
ζ ±
(1
2
+x1
)]∂φ
∂y

}

γxz =0 γyz = φ(x,y)

γ(u/l)xy =−tc1
{
2ζ

∂2w

∂x∂y
±
∂ψ

∂y
−
[
ζ ±
(1
2
+x1
)]∂φ
∂x

}
(2.9)

3) inner flat sheets (upper/lower)

ε(u/l)x =−tc1
(
ζ
∂2w

∂x2
±
∂ψ

∂x

)
ε(u/l)y =−tc1ζ

∂2w

∂y2

γxz = γyz =0 γ
(u/l)
xy =−tc1

(
2ζ

∂2w

∂x∂y
±
∂ψ

∂y

) (2.10)

The sign “+” refers to the upper facing (u), and the sign “−” refers to the lower facing (l).
4) main corrugated core

εx =−tc1ζ
[∂2w
∂x2
−2

∂ψ

∂x

]
εy =−tc1ζ

∂2w

∂y2

γxz =2ψ(x,y) γyz =0 γxy =−2tc1ζ
( ∂2w
∂x∂y

−
∂ψ

∂y

) (2.11)

Strains (2.8)-(2.11) make a basis for formulation of the elastic strain energy of the seven-layer
plate.



Strength of a metal seven-layer rectangular plate... 437

2.2. Total potential energy of the plate

The elastic strain energy of the plate is a sum of the energy of the individual layers

U(plate)ε = U
(s−o)
ε +U

(c−2)
ε +U

(s−i)
ε +U

(c−1)
ε (2.12)

Consecutive components of the sum are as follows:
1) energy of the outer flat sheets

U(s−o)ε =
tc1
2

a∫

0

b∫

0





−(1
2
+x1+x2)∫

−(1
2
+2x1+x2)

[Φ(u,s−o)σ,ε ] dζ +

1

2
+2x1+x2∫

1

2
+x1+x2

[Φ(l,s−o)σ,ε ] dζ




dxdy (2.13)

where

Φ(u/l,s−o)σ,ε = σ
(u/l)
x ε

(u/l)
x +σ

(u/l)
x ε

(u/l)
x + τ

(u/l)
xy γ

(u/l)
xy (2.14)

stresses (Hooke’s law)

σ(u/l)x =
E

1−ν2
(ε(u/l)x +νε

(u/l)
y ) τ

(u/l)
xy =

E

2(1+ν)
γ(u/l)xy (2.15)

and strains – expressions (2.8).
Integration of expression (2.13) with respect to the coordinate ζ provides

U(s−o)ε =
Et3c1
1−ν2

a∫

0

b∫

0

(
C
(s−o)
2 f

(s−o)
22 − c

(s−o)
1 f

(s−o)
12 +x1f

(s−o)
11

)
dxdy (2.16)

where

c
(s−o)
2 =

1

12
[28x21+18x1(1+2x2)+3(1+2x2)

2]x1 c
(s−o)
1 =(1+3x1+2x2)x1

f
(s−o)
22 =

(∂2w
∂x2

)2
+2ν

∂2w

∂x2
∂2w

∂y2
+
(∂2w
∂y2

)2
+2(1−ν)

( ∂2w
∂x∂y

)2

f
(s−o)
12 =

(∂2w
∂x2
+ν

∂2w

∂y2

)∂ψ
∂x
+x2
(
ν
∂2w

∂x2
+
∂2w

∂y2

)∂φ
∂y
+(1−ν)

(∂ψ
∂y
+x2

∂φ

∂x

) ∂2w
∂x∂y

f
(s−o)
11 =

(∂ψ
∂x

)2
+
1−ν

2

(∂ψ
∂y

)2
+x2
[
2ν
∂ψ

∂x

∂φ

∂y
+(1−ν)

∂φ

∂x

∂ψ

∂y

]

+x22

[1−ν
2

(∂φ
∂x

)2
+
(∂φ
∂y

)2]

2) energy of the corrugated cores of the facings

U(c−2)ε =
1

2

a∫

0

b∫

0

{
1

b02

∫

ATr

[Φ(u,c−2)σ,ε ] dA
(c−2)
Tr +

1

b02

∫

ATr

[Φ(l,c−2)σ,ε ] dA
(c−2)
Tr

}
dxdy (2.17)

where

Φ(u/l,c−2)σ,ε = σ
(u/l)
x ε

(u/l)
x +σ

(u/l)
y ε

(u/l)
y + τ

(u/l)
xy γ

(u/l)
xy +τ

(u/l)
yz γ

(u/l)
yz (2.18)

stresses

σ(u/l)x = Eε
(u/l)
x σ

(u/l)
y = E

(c−2)
y ε

(u/l)
y τ

(u/l)
xy = G

(c−2)
xy γ

(u/l)
xy

τ(u/l)yz = G
(c−2)
yz γ

(u/l)
yz

(2.19)

and strains – expressions (2.9).



438 E.Magnucka-Blandzi et al.

The area of one pitch of the trapezoidal corrugated cross section (Fig. 2)

A
(c−2)
Tr =2t

2
c2x02(xf2xb2+ s̃a2) (2.20)

where x02 = t02/tc2, xf2 = bf2/b02, xb2 = b02/tc2 are dimensionless parameters, s̃a2 – dimension-
less length of one pitch – trapezoid

s̃a2 =

√

(1−x02)2+x
2
b2

(1
2
−xf2

)2

Integration of expression (2.17) provides

U(c−2)ε = Et
3
c1

a∫

0

b∫

0

[
f
(c−2)
22 + Ẽ

(c−2)
y f

(c−2)
12 + G̃

(c−2)
xy f

(c−2)
11 + G̃

(c−2)
yz f

(c−2)
10

]
dxdy (2.21)

where

f
(c−2)
22 = c

(c−2)
2x

(∂2w
∂x2

)2
−2c

(c−2)
1x

∂2w

∂x2
∂ψ

∂x
+c
(c−2)
0x

(∂ψ
∂x

)2

Ẽ(c−2)y =
xb2x

3
02

2(3xf2xb2+ s̃a2)(1−x02)
2

f
(c−2)
10 = x2

φ2(x,y)

t2c1

f
(c−2)
12 = C

(c−2)
2y

(∂2w
∂y2

)2
−C

(c−2)
1y

∂2w

∂y2
∂φ

∂y
+C

(c−2)
0y

(∂φ
∂y

)2

f
(c−2)
11 =4C

(c−2)
2y

( ∂2w
∂x∂y

)2
+C

(c−2)
0y

(∂φ
∂x

)2
−
(
2C
(c−2)
1y

∂φ

∂x
+ c
(c−2)
1xy

∂ψ

∂y

) ∂2w
∂x∂y

+x2
(
x2
∂φ

∂x
+
∂ψ

∂y

)∂ψ
∂y

C
(c−2)
2x =

1

2
x2x02

[
x22(1−x02)

2
(
xf2+

s̃a2
3xb2

)
+(1+2x1+x2)

(
xf2+

s̃a2
xb2

)]

G(c−2)xy =
x02
2(1+ν)

C
(c−2)
1x = x2x02(1+2x1+x2)

(
xf2+

s̃a2
xb2

)

C
(c−2)
0x =2x2x02

(
xf2+

s̃a2
xb2

)
C
(c−2)
1y =

1

2
x22

(
1+2x1+

4

3
x2
)

C
(c−2)
0y =

1

3
x32 C

(c−2)
2y = x2

[
x21+x1(1+x2)+

1

4

(
1+2x2+

4

3
x22

)]

C
(c−2)
1xy =2x2(1+2x1+x2) G̃

(c−2)
yz =

2

(1−ν2)xb2fv

(x02
s̃a2

)3

details in Lewinski et al. (2015)
3) energy of the inner flat sheets

U(s−i)ε =
tc1
2

a∫

0

b∫

0





−

1

2∫

−(1
2
+x1)

[Φ(u,s−i)σ,ε ] dζ +

1

2
+x1∫

1

2

[Φ(l,s−i)σ,ε ] dζ




dxdy (2.22)

where

Φ(u/l,s−i)σ,ε = σ
(u/l)
x ε

(u/l)
x +σ

(u/l)
x ε

(u/l)
x + τ

(u/l)
xy γ

(u/l)
xy (2.23)

stresses (Hooke’s law)

σ(u/l)x =
E

1−ν2

(
ε(u/l)x +νε

(u/l)
y

)
τ(u/l)xy =

E

2(1+ν)
γ(u/l)xy (2.24)

and strains – expressions (2.10).



Strength of a metal seven-layer rectangular plate... 439

Integration of expression (2.22) with respect to the coordinate ζ provides

U(s−i)ε =
Et3c1
1−ν2

a∫

0

b∫

0

(
C
(s−i)
2 f

(s−i)
22 −C

(s−i)
1 f

(s−i)
12 +x1f

(s−i)
11

)
dxdy (2.25)

where

C
(s−i)
2 =

1

4

(
1+2x1+

4

3
x21

)
x1 C

(s−i)
1 =(1+x1)x1 f

(s−i)
22 = f

(s−o)
22

f
(s−i)
12 =

(∂2w
∂x2
+ν

∂2w

∂y2

)∂ψ
∂x
+(1−ν)

∂2w

∂x∂y

∂ψ

∂y
f
(s−i)
11 =

(∂ψ
∂x

)2
+
1−ν

2

(∂ψ
∂y

)2

4) energy of the main corrugated core

U(c−1)ε =
1

2b01

a∫

0

b∫

0

{ ∫

ATr

[Φ(c−1)σ,ε ] dA
(c−1)
Tr

}
dxdy (2.26)

where

Φ(c−1)σ,ε = σxεx +σyεy +τxyγxy + τxzγxz (2.27)

stresses

σx = E
(c−1)
x εx σy = Eεy τxy = G

(c−1)
xy γxy τxz = G

(c−1)
xz γxz (2.28)

and strains – expressions (2.11).
The area of one pitch of the trapezoidal corrugated cross section (Fig. 2)

A
(c−1)
Tr =2t

2
c1x01(xf1xb1+ s̃a1) (2.29)

where x01 = t01/tc1, xf1 = bf1/b01, xb1 = b01/tc1 are dimensionless parameters, s̃a1 – dimension-
less length of one pitch – trapezoid

s̃a1 =

√

(1−x01)2+x
2
b1

(1
2
−xf1

)2

Integration of expression (2.30) provides

U(c−1)ε = Et
3
c1

a∫

0

b∫

0

( 1
24
Ẽ(c−1)x f

(c−1)
22 +

1

24
Ẽ(c−1)y f

(c−1)
12 +

1

6
G̃(c−1)xy f

(c−1)
11 +2G

(c−1)
xz f

(c−1)
10

)
dxdy

(2.30)

where

f
(c−1)
22 =

(∂2w
∂x2

)2
−4

∂2w

∂x2
∂ψ

∂x
+4
(∂ψ
∂x

)2
f
(c−1)
12 =

(∂2w
∂y2

)2

f
(c−1)
10 =

ψ2(x,y)

t2c1
f
(c−1)
11 =

( ∂2w
∂x∂y

)2
−2

∂2w

∂x∂y

∂ψ

∂y
+
(∂ψ
∂y

)2

Ẽ(c−1)x =
xb1x

3
01

2(xf1xb1+ s̃a1)
G̃(c−1)xy =

x01
2(1+ν)

Ẽ(c−1)y =2
x01
xb1
(1−x01)

2(3xf1xb1+ s̃a1) G̃
(c−1)
xz =

1−x01
4(1−ν2)xb1fu

(x01
s̃a1

)3

detail in Lewinski et al. (2015).



440 E.Magnucka-Blandzi et al.

The work of the load, a uniformly distributed pressure p0, is in the following form

W =

a∫

0

b∫

0

p0w(x,y) dxdy (2.31)

The total potential energy is a sum of elastic strain energy (2.12) and work (2.31).

3. Equations of equilibrium and its solution

The principle of minimum total potential energy

δ(U(plate)ε −W)= 0 (3.1)

whereU
(plate)
ε is the elastic strain energy of the plate (2.12) andW is thework of the load (2.35).

The system of the equations of equilibrium – three partial differential equations derived
based on principle (3.1) is in the following form

ℜ
(s−o−i)
w +ℜ

(c−2)
w +ℜ

(c−1)
w =

p0
Et3c1

(3.2)

where

ℜ
(s−o−i)
w =

1

1−ν2

{
2(C

(s−o)
2 +C

(s−i)
2 )∇

4w−C
(s−o)
1

[ ∂
∂x
(∇2ψ)+x2

∂

∂y
(∇2φ)

]

−C
(s−i)
1

∂

∂x
(∇2ψ)

}

ℜ
(c−2)
w =2ℜ

(c−2)
w,w −ℜ

(c−2)
w,ψ −ℜ

(c−2)
w,φ

ℜ
(c−2)
w,w = C

(c−2)
2x

∂4w

∂x4
+C

(c−2)
2y

(
4G̃(c−2)xy

∂4w

∂x2∂y2
+ Ẽ(c−2)y

∂4w

∂y4

)

ℜ
(c−2)
w,ψ =

∂

∂x

(
2C
(c−2)
1x

∂2ψ

∂x2
+C

(c−2)
1xy G̃

(c−2)
xy

∂2ψ

∂y2

)

ℜ
(c−2)
w,φ = C

(c−2)
1y

∂

∂x

(
2G̃(c−2)xy

∂2φ

∂x2
+ Ẽ(c−2)y

∂2φ

∂y2

)

ℜ
(c−1)
w =

1

12
ℜ
(c−1)
w,w −

1

6
ℜ
(c−1)
w,ψ ℜ

(c−1)
w,w = Ẽ

(c−1)
x

∂4w

∂x4
+4G̃(c−1)xy

∂4w

∂x2∂y2
+ Ẽ(c−1)y

∂4w

∂y4

ℜ
(c−1)
w,ψ =

∂

∂x

(
Ẽ(c−1)x

∂2ψ

∂x2
+2G̃(c−1)xy

∂2ψ

∂y2

)
∇
4w =

∂4w

∂x4
+2

∂4w

∂x2∂y2
+
∂4w

∂y4

and

ℜ
(s−o−i)
ψ +ℜ

(c−2)
ψ +ℜ

(c−1)
ψ =0 (3.3)

where

ℜ
(s−o−i)
ψ =

1

1−ν2

{
(C
(s−o)
1 +C

(s−i)
1 )

∂

∂x
(∇2w)−2x1

[
2
∂2ψ

∂x2
+(1−ν)

∂2ψ

∂y2

]

−x1x2(1+ν)
∂2φ

∂x∂y

}

ℜ
(c−2)
ψ =

∂

∂x

(
2C
(c−2)
1x

∂2w

∂x2
+C

(c−2)
1xy G̃

(c−2)
xy

∂2w

∂y2

)
−2
(
C
(c−2)
0x

∂2ψ

∂x2
+x2G̃

(c−2)
xy

∂2ψ

∂y2

)

−x22G̃
(c−2)
xy

∂2φ

∂x∂y

ℜ
(c−1)
ψ =

1

6

∂

∂x

(
Ẽ(c−1)x

∂2w

∂x2
+2G̃(c−1)xy

∂2w

∂y2

)
−
1

3

(
Ẽ(c−1)x

∂2ψ

∂x2
+ G̃(c−1)xy

∂2ψ

∂y2

)
−4G̃(c−1)xz

ψ(x,y)

t2c1



Strength of a metal seven-layer rectangular plate... 441

and

ℜ
(s−o−i)
φ +ℜ

(c−2)
φ =0 (3.4)

where

ℜ
(s−o−i)
φ =

1

1−ν2

{
x2C

(s−o)
1

∂

∂y
(∇2w)−x1x

2
2

[
(1−ν)

∂2φ

∂x2
+2

∂2φ

∂y2

]
−x1x2(1+ν)

∂2ψ

∂x∂y

}

ℜ
(c−2)
φ =ℜ

(c−2)
φ,w −x

2
2G̃
(c−2)
xy

∂2ψ

∂x∂y
−ℜ

(c−2)
φ,φ

ℜ
(c−2)
φ,w = C

(c−2)
1y

∂

∂y

(
2G̃(c−2)xy

∂2w

∂x2
+ Ẽ(c−2)y

∂2w

∂y2

)

ℜ
(c−2)
φ,φ =2C

(c−2)
0y

(
G̃(c−2)xy

∂2φ

∂x2
+ Ẽ(c−2)y

∂2φ

∂y2

)
−2x2G̃

(c−2)
yz

φ(x,y)

t2c1

∇
2w =

∂2w

∂x2
+
∂2w

∂y2

Three equations of equilibrium (3.2), (3.3) and (3.4) with three unknown functions w(x,y),
ψ(x,y) and φ(x,y) are approximately solved assuming three unknown functions in the forms

w(x,y) = wa sin
πx

a
sin

πy

b
ψ(x,y)= ψacos

πx

a
sin

πy

b

φ(x,y) = φa sin
πx

a
cos

πy

b

(3.5)

where wa, ψa, φa are parameters of the functions, a, b – sizes of the plate (Fig. 1).
Substituting these functions into equations (3.2), (3.3) and (3.4) and using the Galerkin

method, three algebraic equations are obtained

α11wa −α12
b

π
ψa −α13

a

π
φa =

16

π3
a2b2

t3c1

p0
e

α21
π

a
wa −α22ψa −α23φa =0

α31
π

b
wa −α32ψa −α33φa =0

(3.6)

where the dimensionless elements

α11 = α
(1)
11 +α

(2)
11 +α

(3)
11 α

(1)
11 =

2

1−ν2
(C
(s−o)
2 +C

(s−i)
2 )

(b
a
+
a

b

)2

α23 = x2
( x1
1−ν

+x2G̃
(c−2)
xy

)
α
(3)
11 =

1

12

[
Ẽ(c−1)x

(b
a
)2+4G̃(c−1)xy + Ẽ

(c−1)
y

(a
b
)2
]

α
(2)
11 =2

[
C
(c−2)
2x

(b
a
)2+C

(c−2)
2y

(
4G̃(c−2)xy + Ẽ

(c−2)
y

(a
b
)2
)]

α12 =
1

1−ν2
(C
(s−o)
1 +C

(s−i)
1 )

(b
a
+
a

b

)
+2C

(c−2)
1x

b

a
+C

(c−2)
1xy G̃

(c−2)
xy

a

b

+
1

6

(
Ẽ(c−1)x

b

a
+2G̃(c−1)xy

a

b

)

α13 =
x2
1−ν2

C
(s−o)
1

(b
a
+
a

b

)
+C

(c−2)
1y

(
2G̃(c−2)xy

b

a
+ Ẽ(c−2)y

a

b

)

α21 = α12 α31 = α13 α32 = α23

α22 =
2x1
1−ν2

[
2
b

a
+(1−ν)

a

b

]
+2
(
C
(c−2)
0x

b

a
+x2G̃

(c−2)
xy

a

b

)
+
1

3

(
Ẽ(c−1)x

b

a
+ G̃(c−1)xy

a

b

)

+
4

π2
G̃(c−1)xz

ab

t2c1

α33 =
x1x
2
2

1−ν2

[
(1−ν)

b

a
+2

a

b

]
+2C

(c−2)
0y

(
G̃(c−2)xy

b

a
+ Ẽ(c−2)y

a

b

)
+
2x2
π2

G̃(c−2)yz
ab

t2c1



442 E.Magnucka-Blandzi et al.

Solving equations (3.6) one obtains

wa =
16

π6αw

a2b2

t3c1

p0
E

ψa =
16

π5
αψ
αw

a2b

t3c1

p0
E

φa =
16

π5
αφ
αw

a2b

t3c1

p0
E

(3.7)

where

αw = α11− (αψα12+αφα13) αψ =
bα21α33−aα31α23
a(α22α33−α

2
23)

αφ =
aα31α22− bα21α32
b(α22α33−α

2
23)

The stresses on the outer sheets and in the middle of the plate, for ζo =∓(0.5+2x1+x2) and
x = a/2, y = b/2 are

σx =
1

1−ν2

[(b
a
+ν

a

b

)
ζo ± (αψ +νx2αφ)

] 16
π4αw

ab

t2c1
p0

σy =
1

1−ν2

[(a
b
+ν

b

a

)
ζo ± (ναψ +x2αφ)

] 16
π4αw

ab

t2c1
p0

(3.8)

and the equivalent stress (Huber-Mises-Hencky)

σeq =
√
f2σx −fσxfσy +f

2
σy

16

π4(1−ν2)αw

ab

t2c1
p0 (3.9)

where

fσx =
(b
a
+ν

a

b

)
ζo ± (αψ +νx2αφ) fσy =

(a
b
+ν

b

a

)
ζo ± (ναψ +x2αφ)

4. Finite element model of the seven-layer plate

A family of simply supported rectangular plates of dimensions 2024mm×2000mm subjected
to a uniform load of 0.01MPa has been considered. The linear static analysis was carried out
using the finite element software ABAQUS. A quarter of the rectangular plate was modeled.
The linear S4R shell elements were placed at the mid-surface of the plate layers (Fig. 4).

Fig. 4. Themeshing scheme of a simply supported plate

The mesh density study was carried out to refine the global mesh size to 4mm. The mesh
convergence plot for the maximum deflection in the middle of the top face sheet is presented in
Fig. 5.



Strength of a metal seven-layer rectangular plate... 443

Fig. 5. The mesh convergence plot

Perfect bonding between the cores and the flat sheets was assumed.The interaction between
flanges of the cores and the flat sheets was provided with the use of the tie constraint. The
flanges of the cores were slave surfaces and the flat sheets were master surfaces.

The boundary conditions were imposed only to edges of the flat sheets (master surfaces) –
each edgewas simply supported.The implementation of the symmetry and the simply supported
boundary conditions on a quarter of the plate is schematically shown in Fig. 6.

Fig. 6. The scheme of boundary conditions

5. Results of numerical calculations of deflection and stresses of the plate

The aim of these calculations was to verify the results obtained through the linear finite element
analysis with those obtained through an analytical method. The maximum deflection and the
equivalent stress of the family of seven-layer rectangular plates, using both analytical and finite
elementmethods, was evaluated. The results of the parametric studies for changes of b02 and b01
are collected in Case 1 and Case 2, respectively.

Case 1. The study for constant area of the trapezoidal corrugation of the facing core

A
(c−2)
Total = nA

(c−2)
Tr , where n is the number of the corrugations and A

(c−2)
Tr (2.20) is the

area of one pitch of the trapezoidal corrugated cross section. The numerical calcula-
tions are carried out for the rectangular plate with the following sizes: a = 2024mm,
b = 2000mm, ts = 0.8mm, tc1 = 11.2mm, t01 = 0.8mm, b01 = 46mm, bf1 = 10mm,

tc2 =9.2mm, bf2 =8mm, A
(c−2)
Total =1811.83mm

2, p0 =0.01MPa, and material constants



444 E.Magnucka-Blandzi et al.

E =2 ·105MPa, ν =0.3. The results of the calculations are presented in Table 1. The va-
lues in theABAQUS columns inTable 1 enclosed in parentheses are percentage differences
with respect to the analytical ones (the absolute value of the relative deviation).

Table 1.Thedeflection and the equivalent stresses of the plate for the first case A
(c−2)
Total = const

b02
[mm]

t02
[mm]

wa σeq
n Analytical ABAQUS Analytical ABAQUS

[mm] [mm] [MPa] [MPa]

50 40.0 0.8 5.30 5.41 (2.1%) 59.4 60.01 (1.0%)

60 33.333 0.751 5.33 5.43 (1.9%) 59.7 60.86 (1.9%)

70 28.571 0.6968 5.38 5.48 (1.8%) 60.1 62.40 (3.8%)

80 25.0 0.6409 5.43 5.53 (1.7%) 60.5 63.44 (4.7%)

90 22.222 0.5867 5.51 5.61 (1.8%) 60.9 64.94 (6.4%)

100 20.0 0.5363 5.61 5.72 (1.9%) 61.2 66.29 (8.0%)

Case 2. The study for constant area of the trapezoidal corrugation of the main core

A
(c−1)
Total = mA

(c−1)
Tr , where m is the number of the corrugations and A

(c−1)
Tr (2.29) is the area

of one pitch of the trapezoidal corrugated cross section. The numerical calculations are
carried out for the rectangular plate with the following sizes: a =2024mm, b =2000mm,
ts = 0.8mm, tc1 = 11.2mm, bf1 = 10mm, tc2 = 9.2mm, t02 = 0.8mm, b02 = 40mm,

bf2 =8mm, A
(c−1)
Total =1876.03mm

2, p0 =0.01MPa, E =2 ·10
5MPa, ν =0.3. The results

of the calculations are presented inTable 2.Thevalues in theABAQUScolumns inTable 2
enclosed in parentheses are percentage differences with respect to the analytical ones (the
absolute value of the relative deviation).

Table 2. The deflection and the equivalent stresses of the plate for the second case

A
(c−1)
Total = const

b01
[mm]

t01
[mm]

wa σeq
n Analytical ABAQUS Analytical ABAQUS

[mm] [mm] [MPa] [MPa]

44 46.0 0.8 5.30 5.41 (2.1%) 59.34 60.01 (1.1%)

54 37.481 0.7349 5.33 5.43 (1.8%) 59.44 60.07 (1.1%)

64 31.625 0.6652 5.41 5.49 (1.5%) 59.73 60.47 (1.2%)

74 27.351 0.5973 5.65 5.65 (0.0%) 60.55 61.33 (1.3%)

84 24.095 0.5245 6.48 5.98 (8.0%) 63.90 63.00 (1.4%)

6. Conclusions

Asa conclusion, it canbe said that the results obtained through the analytical and thenumerical
method are consistent with each other. It proves that the broken-line hypotheses assumed for
deformation of the cross-section in x and y directions are sufficient for evaluating the plate
deflection and equivalent stresses. It can also be seen that decreasing of the parameter b01 has
much more influence on the increase of the maximum deflection of the plate. This effect is due
to a significant change in the shear rigidity of the plate.

Acknowledgements The project was funded by the National Science Centre allocated on the basis of

the decision number DEC-2013/09/B/ST8/00170.



Strength of a metal seven-layer rectangular plate... 445

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Manuscript received May 19, 2016; accepted for print September 8, 2016