nonlinear analysis of a curved sandwich beam joined with a straight sandwich beam
TRANSCRIPT
Nonlinear analysis of a curved sandwich beam joined
with a straight sandwich beam
Anders Lyckegaard *, Ole Thybo Thomsen
Institute of Mechanical Engineering, Aalborg University, Pontoppidanstræde 105, DK-9220 Aalborg Øst, Denmark
Received 6 May 2005; revised 10 August 2005; accepted 22 August 2005
Available online 29 September 2005
Abstract
The buckling behaviour of straight sandwich beams joined with curved sandwich beams loaded in pure bending is investigated using two
different models. One is based on a high order sandwich beam theory and the other model is based on finite element analysis. The analyses are
applied to a numerical example and the results are compared with experimental results.
q 2005 Elsevier Ltd. All rights reserved.
Keywords: B.Buckling; Sandwich
1. Introduction
Light-weight sandwich structures are of great interest for the
design and manufacture of spacecraft, aircraft and marine
vehicles, because of their high specific strength and stiffness.
Furthermore, sandwich structures offer excellent damage
tolerance in addition to low life cycle costs, see Lonno [1].
In the development towards further weight savings modern
sandwich structures are made with thin laminated composite
face sheets and light-weight compliant cores such as polymeric
foams or aramid nomex honey-comb.
This trend makes the structure sensitive to local buckling of
the face sheets due to in-plane compression, and this will limit
the load carrying capacity of the entire sandwich structure.
The influence of curvature upon the local buckling
behaviour of sandwich panels has been studied very little,
see the review by Noor et al. [2], but an assessment made by
Skvortsov [3] indicates that the influence is significant.
Meanwhile, Smidt [4] has published a number of experimental
results for pure bending of curved sandwich panels joined with
straight sandwich panels, Fig. 1. These experimental results
show that local buckling occur in the curved part of the beam,
but no calculations were included, that take into account the
curvature of the test specimen.
1359-8368/$ - see front matter q 2005 Elsevier Ltd. All rights reserved.
doi:10.1016/j.compositesb.2005.08.001
* Corresponding author. Tel.: C45 9635 9325; fax: C45 9815 1675.
E-mail address: [email protected] (A. Lyckegaard).
A higher order sandwich theory originally proposed by
Frostig [5] has previously been used to study buckling by e.g.
Frostig et al. [6–8], Li et al. [9], and Karyadi [10]. The accuracy
of the buckling loads that are predicted using this high order
sandwich theory have been considered by Reimerdes and
Schermann [11], who found that the high order theory
predicted buckling loads that were comparable with analytical
calculations, but somewhat lower than predictions found using
the finite element method.
Recently, Frostig et al. [12] compared the high order
sandwich beam theory to finite element analyses in the case of
nonlinear three point bending of a sandwich beam. It was found
that the high order sandwich beam theory gave similar result to
the finite element analysis.
In the present paper, sandwich structures consisting of
straight beams joined with curved beams will be analysed using
a high order sandwich theory. The theory will be applied to a
numerical example based on data from Ref. [4]. The governing
equations are solved using a numerical procedure at increasing
loads thereby mapping the load/displacement path. Finally, the
results from numerical calculations will be compared with the
experimental results from Ref. [4].
2. High order sandwich theory
The high order sandwich beam theory is formulated the same
way that Bozhevolnaya and Frostig [13] and Frostig [6] did for
curved and straight sandwich beams, respectively, hence this
section only states the basic assumptions of the theory.
The theory is derived for two separate cases namely
circularly curved and straight beams, although the assumptions
Composites: Part B 37 (2006) 101–107
www.elsevier.com/locate/compositesb
M M
Fig. 1. A sketch of the test specimen used by Smidt [4].
A. Lyckegaard, O.T. Thomsen / Composites: Part B 37 (2006) 101–107102
stated in the following are in a form, that applies to both the
straight and curved sandwich beams, Figs. 2 and 3.
The high order theory divides the sandwich beams into three
layers. The face sheets are treated as beams that follow the
Kirchoff assumptions, and allow deformations of the von
Karman class. The assumed displacement field is expressed as:
uiðx; zÞ Z ~uiðxÞCzbiðxÞ (1)
wiðx; zÞ Z ~wiðxÞ
biðxÞ Z~uiðxÞ
ri
K ~wi;xðxÞ
where ui, wi and bi are the in-plane displacement, transverse
displacement and the rotation of the either face sheet iZ{t, b},
respectively, Figs. 2 and 3. Note that (.),x denotes the
derivative with respect to x, which is defined as ddx
Z 1ri
ddf
for
curved beams, and that the terms which include 1/r are omitted
for straight sandwich beams.
The expression for the in-plane strains takes into account
moderate rotations, while shear angles and elongations are
assumed to be negligible and is defined as:
3xðx; zÞ Z ~3xðxÞCzkiðxÞ (2)
~3xðxÞ Z~wiðxÞ
ri
C ~ui;xðxÞC1
2b
2i ðxÞ
ki Z bi;x
x
uc
ub
t t
t c
tb
zc,wczb,wb
zt,w
Fig. 2. Symbols and definitions for the
The constitutive equations for the face-sheets are defined as:
Ni Z Ai ~3i (3)
Mi Z Diki
where Ai and Di are the extensional and bending stiffnesses,
respectively.
The governing equation for the core are derived using the
linear equilibrium equations, while it is assumed that body
forces are nil, and the stresses parallel to the faces, sxx, are
negligible. Thus, by making these assumption it is possible to
find the transverse distribution of stresses and displacements
analytically from the remaining part of the equilibrium
equations.
szz;z Ctxz;x Cszz
rZ 0 (4)
txz;z Ctzx
rZ 0
where the terms that are proportional to 1/r are neglected in
case of the straight beam. Then by employing the linear
definition of strain and in addition, Hooke’s constitutive law,
the displacement functions can be found.
3zz Zvwc
vzc
Zszz
Ec
(5)
gxz Zvuc
vzc
Cvwc
vzc
Ztxz
Gc
where Ec and Gc are the Young’s and shear modules of the
core. Furthermore, it is assumed that the core layer is perfectly
bonded to each of the face-sheets, i.e. it is assumed that the
displacements are continuous.
fuc;wcgðzc Z 0Þ Z fub;wbgðz Z hb=2Þ (6)
fuc;wcgðzc Z hcÞ Z fub;wbgðz Z hb=2Þ
These assumptions are used in conjunction with the
principle of minimum potential energy to derive the governing
τsz
Qt MtNt
Qb Mb Nb
ut
top face (t)core (c)
bottom face (b)
t
straight part of the sandwich beam.
τsz
Qb
Qt
Mb
Nb
Nt
Mt
rct rrcb
rb
rt
zb,wb
tb
tc
tt
core (c)
top face (t)
bottom face (b)φ
zt,wt
zc,wc x
xc,uc
xb,ub
xt,ut
Fig. 3. Symbols and definitions for the curved part of the sandwich beam.
A. Lyckegaard, O.T. Thomsen / Composites: Part B 37 (2006) 101–107 103
equations for the sandwich beams.
dP Z
ðsijd3ijdv CdV (7)
Finally, the governing equations can be formulated as a set
of differential equations and a set of boundary conditions,
which govern the straight and curved sandwich beams.
2.1. System of equations
The governing equations are restated as two separate
systems of 14 first order ordinary differential equations. (see
details in the Appendix).
d
dxy Z fðyðxÞ; x; lÞ a!s!b (8)
with boundary conditions
gðyðaÞ; yðbÞ; lÞ Z 0 (9)
where l is a parameter controlling the load and x is the arc
length coordinate of the sandwich, Figs. 2 and 3. The vector y is
defined as
fYg Z fub; ut;wb;wt;bb;bt;Nb;Nt;Mb;Mt;Qb;Qt; tc;jgT
(10)
In Eq. (10) the variables Ni, Mi and Qi are the normal,
bending moment and transverse shear stress resultants in the
faces, Figs. 2 and 3. The variable tc represents the shear stress
(txz or tfz) of the core at zcZ0, see Figs. 2 and 3. The variable
j is equal to the derivative of the shear stress jZtc,x for the
straight beam and jZtc,f for the curved beam.
2.2. Continuity and boundary conditions
At the internal boundary, where the straight panel is joined
with the curved panel continuity must be fulfilled. This is
trivial for the variables concerning the face sheets, because
these are modelled using beam theory, thus
fui;wi;bi;Ni;Mi;Qigcurved Z fui;wi;bi;Ni;Mi;Qig
straight (11)
However, it was pointed out by Thomsen and Vinson [14,
15], that the continuity conditions for the core needed special
attention. This was further investigated by Lyckegaard and
Thomsen [16], who formulated a set of approximate continuity
conditions based on the principle of minimum potential energy,
which will be used in the present study.
The data given in Ref. [4] indicate that buckling develops in
the curved beam part near the interface between the curved and
the straight part of the panel, and that it is localised near this
area. It is, therefore, considered safe to exploit symmetry in
order to limit the calculations. Thus, the following boundary
conditions are used for the left part of the curved beam, see
Fig. 4.
ui Z 0
wb Z 0
45˚
100 mm
τ = 0
R = 1
66 m
m
2 mm
50 mm
2 mmθ
Insert
Fig. 4. Dimensions and boundary conditions for the test specimen.
A. Lyckegaard, O.T. Thomsen / Composites: Part B 37 (2006) 101–107104
bi Z 0
Qt Z 0
tc Z 0 (12)
In the experiment described in Ref. [4] wooden inserts were
used to apply the load to the straight parts of the sandwich
panels. In these calculations this insert is considered infinitely
rigid, thereby assuming that the boundary conditions are such
that the panel thickness is assumed to be constant, the rotations
of both skins are the same as the insert, and the in-plane
displacements are given by the rotation of the insert.
Furthermore, the global normal stress resultant and the global
transverse stress resultant are both assumed to be zero, in
addition to the gradient of the shear stresses, j. The boundary
conditions prescribed at the end of the straight sandwich beam
part are given explicitly in Eq. (13).
wt Kwb Z 0 (13)
ðubKutÞðtc C tt=2 C tb=2Þ Z q
bb Z q
bt Z q
Nt CNb Z 0
tctc CQt CQb Z 0
j Z 0
The incremental load control parameter l is taken as the
rotation, q, of the rigid insert, Fig. 4.
Table 1
The material properties of the sandwich constituents
Ef 13.0 GPa
nf 0.19
Ec 60 MPa
Gc 22 MPa
2.3. Solution procedure
The governing equations are solved numerically using a
continuation method such that the load/displacement path can
be mapped. The two systems of 14 ordinary differential
equations are restated as one two point boundary value problem
with a system of 28 first order differential equations using the
technique described by Ascher and Russell [17].
The differential equations are discretized using a finite
difference scheme based on a three-stage Lobatto formula
described by Ascher et al. [18]. This results in a system of non-
linear algebraic equations that is solved using a predictor-
corrector approach, where a secant-predictor is used for the
predictor step and a standard Newton–Raphson algorithm is
used for the corrector step, see e.g. Seydel [19] for a detailed
description of this approach.
3. Finite element model
A finite element model was developed using ANSYS ver.
9.0 [20]. The faces were modelled using beam elements
(BEAM188), while the core was modelled using 2D-solid
elements (PLANE182). The core was extended with a rigid
insert at the end of the straight part to facilitate introduction of
the load, and the thickness of the core was increased artificially
to the centreline of the faces. Furthermore, the symmetry of the
problem assumed in this model, such that only half of the beam
was modelled. Hence, symmetry boundary condition was
applied at the middle of the beam, and the load was applied via
prescribed bending moments at the straight end of the face
sheets.
This model was produced after numerous attempts to
produce a working model and it was found that the choice of
beam elements for the face sheets was essential. Similar
difficulties with nonlinear modelling of three point bending of a
sandwich beam were found by Frostig et al. [12].
4. Numerical example
The numerical example is based on data from the
experiment that was reported by Smidt [4], who tested
sandwich panels comprised of glass-fibre face-sheets and
polymeric foam.
The material properties are given in Table 1, and the width
of the sandwich panels is fixed to 95 mm as given in the
reference.
5. Results and discussion
Initially the moment/rotation history is considered, where
the global moment versus the rotation of the endpoint is
plotted, Fig. 5. Both models predict an almost linear
relationship between moment and rotation in the beginning.
A small discrepancy between the models is found in the
linear range, because the high order beam model disregard
the bending stiffness of the core material and because the
–1600
–1400
–1200
–1000
–800
–600
–400
–200
0
0 0.05 0.1 0.15 0.2 0.25
M [N
m]
Rotation θ [rad]
Global momentFE
HOSBT
Fig. 5. The load global moment with respect to the prescribed deformation
found using high order sandwich beam theory (HOSBT) approach (dashed line)
and finite element model (solid line).
Table 2
Comparison of failure moments for curved panels were the experimental values
are given in Ref. [4] and the numerical value was obtain using the high order
analysis from the present paper
Core Experimental values
(Nm)
Numerical (HOSBT)
Sectioned core 840–980 1615 Nm
Precut core 570–650 –
A. Lyckegaard, O.T. Thomsen / Composites: Part B 37 (2006) 101–107 105
thickness of the core is artificially increase in the finite element
model.
When the rotation angle is about qZ0.1 [rad] and the
bending moment is MZ1270 Nm the FE-curves starts to
diverge from the linear relationship, while the non-linearity is
0
50
100
150
200
250
0 50 100 150 200 250 300
y [m
m]
x [mm]
Deformed geometry
Buckle
Undeformed top face
Deformed top face
Undeformed bottom face
Deformed bottom face
Fig. 6. The geometry of the mid surfaces of the face sheets in the undeformed
shape (dashed lines) and the deformed shape at qZ0.16 (full line).
not visible until the rotation angle is MZ0.14 [rad] and the
moment is MZ1555 Nm for the high order model.
The discrepancy between these predictions is rather large,
and the reason for this has not yet been found.
The buckling loads that were reported in Ref. [4] included
two types of panels. These are distinguished by the
manufacturing method of the curved part of the core. One set
of panels were fabricated by bonding together pre-cut curved
core blocks and are labelled pre-cut cores. Another set of
panels were fabricated by bonding together a sequence of
straight blocks into a curved shape and are labelled sectioned
cores. The buckling loads are repeated in Table 2 along the
numerical results. Note that the different between the buckling
loads found using the two types of panels is quite large.
The buckling loads that were found experimentally are
much lower than the values found numerically. However,
analysis of the stress state in the core shows that compressive
strength of the core material (Ref. [21]) is reached at a load
level of about MZ570 Nm next to the inner face sheet near the
junction between the straight and the curved part of the beam.
Hence, the buckling of the faces in the experimental
investigation is caused by crushing of the core material. The
panels in the pre-cut group generally show a higher failure
moment, which is probably due to the bonding material which
increases the strength of the core material.
5.1. Buckling modes
The undeformed geometry of the mid surfaces of the face
sheets are given together with the deformed shape at the load,
qZ0.16 (MZK1615 Nm or K1283 Nm), Fig. 6. The
buckling shape found using the finite element model is
shown in Fig. 7. The buckling mode found with the High
order model is a local buckle of the inner skin of the curved part
adjacent to the junction, Fig. 6, which is also the case for the
finite element model in Fig. 6.
Fig. 7. The deformed shape found using FE analysis where MZ1325 Nm and
qZ0.108.
A. Lyckegaard, O.T. Thomsen / Composites: Part B 37 (2006) 101–107106
6. Conclusion
The nonlinear behaviour of curved sandwich panels joined
with straight sandwich panels loaded in pure bending is
investigated using a high order sandwich beam theory and
finite element analysis. The two analyses show similar
buckling modes, but the correlation of buckling loads is less
that satisfactory, and is a matter of further research.
The comparison with the experimental results obtained by
Smidt [4] it was found that crushing of the core materials was
the cause of buckling.
Acknowledgements
The work reported was co-sponsored by the Danish
Research Agency under the ‘Program on Materials Research,
’ and by the U.S. Navy, Office of Naval Research (ONR) under
Grant/Award No. N00014001034, ‘Research on Advanced
Composite Sandwich Construction for Naval Surface Vessels:
Analysis, Design and Optimisation of Sandwich Shells.’ The
ONR supervisor was Dr Yapa Rajapakse. The financial support
is gratefully acknowledged. The authors would also like to
acknowledge the valuable suggestions for FE-modelling that
were provided by Professor Frostig and Ms H. Schwarts-Givli
Appendix
The governing equations of the straight sandwich beam are
given here in explicit form.
ui;x Z1
A11i
NiK1
2b2
i (14)
wi;x ZKbi (15)
bi;x Z1
D11i
Mi (16)
Nb;x ZKtc Knb (17)
Nt;x Z tcKnt (18)
Mb;x Z QbKtc
tb
2Cmb KNbbb (19)
Mt;x Z Qt Ktc
tt
2Cmt KNtbt (20)
Qb;x ZEcðwt KwbÞ
tcKtc;x
tc
2Kqb (21)
Qt;x ZEcðwt KwbÞ
tc
Ktc;x
tc2
Kqt (22)
tc;x Z j (23)
jx Z12Ec
t2c Gc
tc C6Ec
t3c
ðtc C ttÞbt C6Ec
t3c
ðtc C tbÞbb
C12Ec
t3c
ubK12Ec
t3c
ut (24)
where iZt, b and A11i and D11i are the extension and bending
stiffnesses respectively. ni, mi and qi are the distributed normal,
moment and transverse loads on the faces, respectively. Ec and
Gc are the Young’s modulus and shear modulus of the core.
The equations governing the curved sandwich beam in
explicit form are given below.
ui;f ZKwi Cri
A11i
Ni K1
2rib
2i (25)
wi;f Z ui0Kbir
bi;f Zri
D11i
Mi (26)
Nb;f ZKQb KtcrcbKnbrb (27)
Nt;f ZKQt Ktc
r2cb
rct
Kntrt (28)
Mb;f Z QbrbKtc
tb2
rcb Cmbrb CNbbbrb (29)
Mt;f Z Qtrt Ktc
tt
2
r2cb
rct
Cmtrt CNtbtrt (30)
Qb;f Z NbKjrcbKtc;fr2cbg0Kg1wb Cg1wt Kqbrb (31)
Qt;f Z Nt Cjr2
cb
rct
ð1 Crctg0ÞCg1wbKg1wt Kqtrt (32)
tc;f Z j (33)
jf Z 2rctðrcbKrctÞub
g7
K2rctðrcb KrctÞut
g7
Kr2ctln
rct
rcb
� �ð2rb C tbÞ
bb
g7
K2rbrctðrct KrcbÞbb
g7
Krctrcblnrct
rcb
� �ðtt K2rtÞ
bt
g7
C2rtrctðrcbKrctÞbt
g7
Crcblnðrct=rcbÞðr
2cbKr2
ctÞ
g7Gct1 (34)
where
g0 Zðrct KrcbÞ
rcbrctlnðrcb=rctÞ(35)
g1 ZEc
lnðrcb=rctÞ(36)
A. Lyckegaard, O.T. Thomsen / Composites: Part B 37 (2006) 101–107 107
g3 Z 1 C lnðrct=rcbÞ (37)
g7 Zrcblnðrct=rcbÞðr
2cbKr2
ctÞ
Ec
C2rcbðrcbKrctÞ
2
Ec
(38)
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