Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

50, 4, pp. 933-941, Warsaw 2012
50th Anniversary of JTAM

FACE WRINKLING OF SANDWICH BEAMS UNDER PURE BENDING

Paweł Jasion, Krzysztof Magnucki
Poznan University of Technology, Institute of Applied Mechanics, Poznań, Poland

e-mail: pawel.jasion@put.poznan.pl; krzysztof.magnucki@put.poznan.pl

The paper is devoted to sandwich beams under pure bending. The local buckling problem
is analysed. The analytical description of the upper face wrinkling is proposed. From the
principle of stationary total potential energy, formulae describing critical stresses in the
faces of the beam are derived. The algorithm for determining the critical stresses is shown.
Two particular cases of the solution following the core properties arementioned. The finite
elementmodel of the sandwich beam is formulated. The comparison of the results obtained
fromtheproposedanalyticalmodel and fromFEManalysis is shown for a family of sandwich
beams with different thicknesses and core properties.

Key words: sandwich beam, local buckling, wrinkling, elastic foundation

1. Introduction

Atypical sandwich structure consist of thin stiff faces, upper and lower, and a light flexible core.
The monographs concerning theoretical investigation of such structures are these, for example,
by Plantema (1966), Allen (1969), Libove and Butdorf (1948) and Reissner (1948). The core of
a sandwich element can bemade of polyurethane foam, metal foam or thin shapedmetal sheet
– corrugated core. In the case of a soft core, for which Young’s modulus is at a level of 10MPa,
the stiff face may be treated as a beam on an elastic foundation. This phenomenon has been
described in many monographs, like those by Życzkowski (1988), Woźniak (2001) or Bažant
and Cedolin (1991). Stability problems of beams on an elastic foundation were presented by
Vlasov andLeontev (1960). Discretemodel of elastic-plastic problem of the beamon foundation
is analysed by Chen and Yu (2000).
The behaviour of sandwich members as well as their failure modes may be determined in

simple tests like axial compression or bending. As to the first load, the sandwich column may
buckle both globally and locally. An example of an analytical model describing global and local
buckling of sandwich columns can be found in Léotoing et al. (2002). Numerical investigation of
this problemwas presented by Hadi (2001).
The second load case, the bending,may be realised in two ways.When the beam supported

at both ends is loaded with one force, the load scheme is called the three point bending. The
collapse mechanism of sandwich beams under this kind of load was analytically described by
Steeves andFleck (2004). Thenumerical analysis andexperimental results of threepointbending
can be found in Bart-Smith et al. (2001).
The three point bending leads usually to the global mode of failure. If local phenomena

are under consideration the four point bending is a more adequate load. This kind of load
induces pure bending conditions between the applied forces. In sandwich structures it causes
local wrinkling of the upper compressed face. Since the local buckling-wrinkling is a small scale
phenomenon, the properties of the core are of high importance. When the core is a metallic
foam, the cell size and its homogeneity is important since it influences the shear stiffens of the



934 P. Jasion, K.Magnucki

core. Moreover some defects in the foam structure, like large holes, may induce the buckling
process (see Kesler and Gibson, 2002; Rakow andWaas, 2005).
In the present paper, the sandwich beam under pure bending is considered. The analytical

model of the face wrinkling is proposed. Thework is a continuation of two papers by Jasion and
Magnucki (2011) and Jasion et al. (2011). In the latter, the experimental results on sandwich
beams with a metal foam core under axial compression and pure bending are presented. Other
papers devoted to this subject are by Koissin et al. (2010), in which the influence of physical
nonlinearities on the wrinkling of the upper compressed face is analysed and by Stifinger and
Rammerstorfer (1997) who presented analytical and FEManalysis of the face wrinkling in shell
structures including post-buckling analysis.

2. Analytical analysis of local buckling-wrinkling of the face

2.1. Pre-buckling state

The sandwich beam considered in the paper is simply supported at both ends and loaded
with two equal transverse forces F placed symmetrically. The load and support scheme as well
as the dimensions of the beam are shown in Fig. 1a. Two forces applied to the beam induce a
pure bending state in the area between them.

Fig. 1. The scheme of the load and dimensions of the sandwich beam (a); displacements of the
particular faces (b)

Since the wrinkling of the upper face is analysed, the internal longitudinal force Nf acting
in that face will be determined first. The field of displacements is assumed in accordance with
the broken line hypothesis presented e.g. by Volmir (1967) (see Fig. 1b). According to that,
individual displacement components for three layers have the form

— lower face: −
(

1
2
+x1
)

¬ ζ ¬−1
2

u(x,ζ)=−tc
[

ζ
dw

dx
+ψ0(x)

]

γxz =0 (2.1)

— core: −1
2
¬ ζ ¬ 1

2

u(x,ζ)=−tcζ
[dw

dx
−2ψ0(x)

]

γxz =2ψ0(x) (2.2)

— upper face: 1
2
¬ ζ ¬ 1

2
+x1

u(x,ζ)=−tc
[

ζ
dw

dx
−ψ0(x)

]

γxz =0 (2.3)



Face wrinkling of sandwich beams under pure bending 935

where ζ = z/tc,x1 = tf/tc and ψ0(x)= u1(x)/tc aredimensionlessparameters and −
(

1
2
+x1
)

¬

¬ ζ ¬ 1
2
+x1.

The internal bendingmoment Mb(x) and longitudinal Nf and shear Q forces acting in the
beam are defined as follows:
— bendingmoment

Mb(x)=
∫

A

σz dA =−
1
12
Ecbt

3
c

{

[1+2(3+6x1+4x
2
1)x1e1]

d2w

dx2

−2[1+6(1+x1)x1e1]
dψ0(x)
dx

}

(2.4)

— longitudinal force

Nf =

−

1

2
∫

−

(

1

2
+x1

)

bσ dz = Efbt
2
c

[1
2
(1+x1)x1

d2w

dx2
+x1

dψ0(x)
dx

]

(2.5)

— shear force

Q =
∫

A

τ dA =2Gcbtcψ0(x) (2.6)

Since the pure bending of the beam is considered, the shear force Q equals zero. In such a case,
from equation (2.6) there is ψ0 =0. Then the equations (2.4) and (2.5) take the form

Mb(x)=−
1
12
Ecbt

3
c[1+2(3+6x1+4x

2
1)x1e1]

d2w

dx2
Nf = Efbt

2
c

1
2
(1+x1)x1

d2w

dx2
(2.7)

The bendingmoment induced by the couple of forces Nf acting in the upper and lower face can
be written

M
(Nf)
b
=−Nf(tc+ tf)=−

1
12
Ecbt

3
c6(1+x1)

2x1e1
d2w

dx2
(2.8)

From Fig. 1a there is M0 = Fa0. Assuming that Mb(x) = M
Nf
b
= M0, the normal force Nf

can be finally written

Nf =
6(1+x1)x1e1

1+2(3+6x1+4x21)x1e1

M0
tc

(2.9)

The longitudinal force acting in the lower face is expressed by the same equation, but its value
is opposite.

2.2. Local elastic buckling

The sandwich beam under pure bending may buckle locally. The buckling shape has the
form of short wrinkles appearing on the upper face. Here it is assumed that the deformation
of the upper compressed face has the shape of longitudinal waves of a constant amplitude. The
deformation of the core follows the upper face but its magnitude diminishes to zero near the
lower face. Additional assumption is that there is no longitudinal displacements. Then the field
of displacements can be defined as follows (see Fig. 2a):

u(x,z)≡ 0 w(x,z) = w1w(z)sin
mπx

a1
(2.10)



936 P. Jasion, K.Magnucki

The unknown displacements function w(z), which describes the core deformation has the follo-
wing boundary conditions (see Fig. 2b)

w
(

−
tc
2

)

=1 w
(tc
2

)

=0 (2.11)

Fig. 2. Deformation of the upper face and core (a); the unknown function w(z) (b)

The deflection of the upper face is then

w(x) = w1 sin
mπx

a1
(2.12)

Limiting considerations to the elastic range, the strains in the core have the form

εx =
∂u

∂x
≡ 0 εz =

∂w

∂z
γxz =

∂u

∂z
+
∂w

∂x
=
∂w

∂x
(2.13)

Since the critical loadwill bedeterminedwith theuseof theprincipleof stationary total potential
energy, the elastic strain energy has to be formulated for the buckled beam. The strain energy
of the core U

(c)
ε can be written as follows

U(c)ε =
Ecb

2(1−ν2c)

tc
2
∫

−

tc
2

a1
∫

0

(

ε2x+2νcεxεz +ε
2
z +
1−νc
2

γ2xz

)

dxdz (2.14)

After substitution function (2.10) into Eq. (2.14) and integrating through the length, Eq. (2.14)
takes the form

U(c)ε =
Eca1b

4(1−ν2c)
w21

tc
2
∫

−

tc
2

[(dw

dz

)2
+
1−νc
2

(mπ

a1

)2
w2
]

dz (2.15)

The strain energy of the compressed face U
(f)
ε for which z =−tc/2 is

U(f)ε =
1
2
EfJ

(f)
z

a1
∫

0

(d2w

dx2

)

dx =
Efa1bt

3
f

48

(mπ

a1

)4
w21 (2.16)

The work of load related to the deformation of the upper face caused by the force Nf is

W =
1
2
Nf

a1
∫

0

(dw

dx

)2
dx =

1
4
Nfa1

(mπ

a1

)2
w21 (2.17)



Face wrinkling of sandwich beams under pure bending 937

The equation of stationary total potential energy has the form

δ(U(f)ε +U
(c)
ε −W)= 0 (2.18)

Solving Eq. (2.18), the equation of equilibrium is obtained as follows

d2w(z)
dz2

−k2w(z) = 0 where k2 =
1−νc
2

(mπ

a1

)2
(2.19)

Taking into account boundary conditions (2.11), the solution to the above equation is

w(z) =
1

sinhCc
sinh
[

Cc
(1
2
−
z

tc

)]

where Cc = ktc = mπ

√

1−νc
2

tc
a1

(2.20)

Knowing the function of displacements w(z), the formulae for critical stresses in the compressed
face can be obtained from equation (2.18)

σ̃(f)cr =
σ
(f)
cr

Ec
=min
Cc

( α1
Cc tanhCc

+α2C
2
c

)

(2.21)

where

α1 =
1

2(1+νc)x1
α2 =

e1x
2
1

6(1−νc)
x1 =

tf
tc

e1 =
Ef
Ec

(2.22)

Equation (2.21) is a general formula for dimensionless critical stresses in the compressed face
of the sandwich beam. If the character of hyperbolic tangent is taken into account (see Fig. 3)
two particular cases of Eq. (2.21) can be distinguished:
— if Cc ­ 2 then tanhCc =1; the critical stress can be written as follows

σ̃
(f)
cr,I =

σ
(f)
cr

Ec
=
{ 1
2(1+νc)

[2(1+νc)
3(1−νc)

]

1

3

+
1

6(1−νc)

[3(1−νc)
2(1+νc)

]

2

3
}

e
1

3

1 (2.23)

— if Cc ≪ 1 then tanhCc = Cc; the formula for the critical stress reduces to the form

σ̃
(f)
cr,II =

√

e1x1
3(1−ν2c)

(2.24)

It can be shown that Eq. (2.24) is consistent with the classical solution of the beam on the
Winkler foundation.

Fig. 3. Hyperbolic tangent



938 P. Jasion, K.Magnucki

2.3. Numerical example

An example, in which the buckling analysis of one beam is presented, is shown below. The
results obtained from the proposed model are compared with those given by FEM analysis.
Broader investigation on a family of beams as well as details concerning the FEM model are
shown in the next section.
In this example, the sandwich beam has the following parameters:

• material properties of the faces: Ef =65600MPa, νf =0.33

• material properties of the core: Ec =100MPa, νc =0.3

• dimensions according to Fig. 1: a0 = 350mm, a = 300mm, b = 100mm, tf = 1mm,
tf =48mm.

Following Eq. (2.22), the dimensionless parameters equal: e1 = 656, x1 = 1/48, α1 = 18.46,
α2 =0.0678. According to Eq. (2.21), the dimensionless critical stresses are

σ̃(f)cr =min
Cc

( 18.46
Cc tanhCc

+0.0678C2c
)

Theabove function is presented inFig. 4a. Itsminimumequals 5.383 at Cc =5.146. The critical
stress are then

σ(f)cr = σ̃
(f)
cr Ec =538.3MPa

Knowing that Nfcr = σ
f
crtcb, the critical bendingmoment can be determined from Eq. (2.9)

M0,cr =2669Nm

Agoodagreement canbe seen between the shapeof thedisplacement function w(z) obtained
analytically and these given by FEM analysis (Fig. 4b).

Fig. 4. Minimisation of σ̃(f)cr (a); shape of the displacement function w(z) (b)

3. FEM analysis of local buckling-wrinkling of the face

The finite element analysis has been performed with the use of ABAQUS software. The finite
elements and procedures available in this package have been used. The FEmodel consists of 3D
brick elements used for modelling of the core and 2D shell elements used to model the faces.
The tie constrains have been applied between the faces and the core. The boundary conditions
correspond to those shown inFig. 1a. The size of finite elements has been determined in analysis
of the mesh convergence. A family of sandwich beams has been analysed for which dimensions



Face wrinkling of sandwich beams under pure bending 939

and material properties were as follows: face thickness tf = 1mm, core thickness tc = 18, 28,
38, 48mm,widthof thebeam b =100mm,distancebetween the forceandsupport a0 =350mm,
distance between forces a1 =300mm, Poisson’s ratios and Young’s moduli of the faces and the
core νf = 0.3, νc = 0.33, Ef = 65600MPa, Ec = 10, 50, 100, 400, 800, 1200MPa. Buckling
analysis has been performed to determine the critical bending moment. The critical stresses
were derived from static analysis by applying the critical bending moments determined in the
buckling analysis.
The buckling shapes for two beams are shown in Fig. 5. The longitudinal waves appear in

the area between forces applied to the upper face. It can be seen that the amplitude of thewaves
is the highest in the mid-length of the beam and it diminishes moving to the point where the
force is applied.

Fig. 5. Buckling shapes of the sandwich beams (Ec =100MPa): tc =18mm (a); tc =48mm (b)

Thecritical stresses andthecritical bendingmoments for the familyofbeamsdescribedabove
are shown in Fig. 6. In the plots the analytical and numerical (FEM) results are compared. A
good agreement can be seen between the results obtained with both approaches. The biggest
discrepancy appear for beams with the stiffest core, but it does not exceed 7%.

Fig. 6. Critical stresses and bending moments for a family of sandwich beams

4. Conclusions

In the paper, a mathematical model of the face wrinkling of the sandwich beam has been
presented. It allows one to estimate the buckling load for the beam subjected to pure bending.
The results of calculations made on a family of beams show that the elastic local buckling
may appear only for beams with a soft core for which Young’s modulus is not higher than
about 100MPa.
The results obtained from the analytical model have been comparedwith those given by the

finite element method. The discrepancy as to the critical bendingmoment is less than 7%. The



940 P. Jasion, K.Magnucki

reason for this difference may be the buckling shape assumed in the analytical considerations.
It has the form of a sine function, whereas in the FEM results the amplitude of the waves
diminishes near the applied force.
An algorithm has been proposed, which gave the possibility to determine critical stresses in

the compressed face.Three cases are possible to occur dependingon theparameter Cc describing
properties of the core.

The research has been supported by the Ministry of Science and Higher Education in the frame of
Grant No. 0807/B/T02/2010/38.

References

1. Allen H.G., 1969,Analysis and Design of Structural Sandwich Panels, PergamonPress, London

2. Bart-Smith H., Hutchinson J.W., EvansA.G., 2001,Measurement and analysis of the struc-
tural performance of cellular metal sandwich construction,Mechanical Science, 43, 1945-1963

3. Bažant Z.P., Cedolin L., 1991,Stability of structures. Elastic, Inelastic, Fracture, and Damage
Theories, Oxford University Press, NewYork, Oxford

4. Chen X W, Yu T.X., 2000, Elastic-plastic beam-on-foundation under quasi-static loading, Int.
Journal of Mechanical Sciences, 42, 2261-2281

5. Hadi B.K., 2001,Wrinkling of sandwich column: comparison between finite element analysis and
analytical solutions,Composite Structures, 53, 477-482

6. Jasion P., Magnucka-Blandzi E., Szyc W., Wasilewicz P., Magnucki K., 2011, Global
and local buckling of a sandwich beam-rectangular plate with metal foam core, Proc. 6th Int.
Conference onThin-walled Structures,Vol. 2,DubinaD.,UngureanuV. (Eds.), ECCSPublication,
Printed inMulticomp Lda,MemMartins, Portugal, ISBN(ECCS) 978-92-9147-102-7, 707-714

7. Jasion P., Magnucki K., 2011, Buckling-wrinkling of a face of sandwich beam under pure ben-
ding,Modelowanie Inżynierskie, 41, 151-156, ISSN 1896-771X [in Polish]

8. Kesler O., Gibson L.J., 2002, Size effects in metallic foam core sandwich beams, Materials
Science and Engineering A, 326, 228-234

9. Koissin V., Shipsha A., Skvortsov V., 2010, Effect of physical nonlinearity on local buckling
in sandwich beams, Journal of Sandwich Structures and Materials, 12, 7, 477-494

10. Léotoing L., Drapier S., Vautrin A., 2002, First applications of a novel unified model for
global and local buckling of sandwich columns,European J. of Mech. A/Solids, 21, 683-701

11. Libove C., Butdorf S.B., 1948, A General Small-Deflection Theory for Flat Sandwich Plates,
NACATN 1526

12. Plantema F.J., 1966, Sandwich Construction: The Bending and Buckling of Sandwich Beams,
Plates and Shells, JohnWiley and Sons, NewYork

13. RakowJ.F.,WaasA.M., 2005,Size effects and the shear responseof aluminium foam,Mechanics
of Materials, 37, 69-82

14. Reissner E., 1948, Finite deflections of sandwich plates, Journal of the Aeronautical Science, 15,
7, 435-440

15. Steeves C.A., Fleck N.A., 2004,Collapsemechanisms of sandwich beamswith composite faces
and a foam core, loaded in three-point bending. Part I: analytical models and minimum weight
design, Int. Journal Mechanical Sciences, 46, 561-583

16. Stiftinger M.A., Rammerstorfer F.G., 1997, Face layer wrinkling in sandwich shells – The-
oretical and experimental investigations,Thin-Walled Structures, 29, 1/4, 113-127



Face wrinkling of sandwich beams under pure bending 941

17. VlasovV.Z., LeontevN.N., 1960,Beams, Plates and Shells on Elastic Foundation, Gosud. Izd.
Fiz-Mat-Lit., Moscow [in Russian]

18. VolmirA.S., 1967,Stability of Deformable Systems, Izd.Nauka,Fiz-Mat-Lit,Moscow [inRussian]

19. Woźniak M., 2001, Interaction of a plate with elastic foundation, [In:] Mechanika Techniczna,
Mechanics of Elastic Plates and Shells,VIII, C.Woźniak (Edit.),Wyd.NaukowePWN,Warszawa,
510-541 [in Polish]

20. Życzkowski M., 1988, Stability of bars and system bars, [In:]Mechanika Techniczna, Strength of
Structures Elements, IX, M. Życzkowski (Edit.), PWN,Warszawa, 241-380 [in Polish]

Marszczenie ściskanej okładziny belki trójwarstwowej poddanej czystemu zginaniu

Streszczenie

Wpracy omówiono zagadnienie lokalnej stateczności belki trójwarstwowej poddanej czystemu zgina-
niu. Zaproponowano analitycznymodel marszczenia górnej, ściskanej okładziny. Z zasady stacjonarności
energii potencjalnej wyprowadzono równanie opisujące naprężenia krytyczne. Zaproponowano algorytm
pozwalający określić wartość naprężeń krytycznych w zależności od własności rdzenia. Opracowanomo-
del numeryczny MES belki trójwarstwowej. Dla rodziny belek przeprowadzono analizę numeryczną na
wartości własne, a wyniki porównano z otrzymanymi z zaproponowanegomodelu analitycznego.

Manuscript received October 28, 2011; accepted for print November 30, 2011