This is an electronic reprint of the original article. This reprint may differ from the original in pagination and typographic detail. Powered by TCPDF (www.tcpdf.org) This material is protected by copyright and other intellectual property rights, and duplication or sale of all or part of any of the repository collections is not permitted, except that material may be duplicated by you for your research use or educational purposes in electronic or print form. You must obtain permission for any other use. Electronic or print copies may not be offered, whether for sale or otherwise to anyone who is not an authorised user. Reinaldo Goncalves, Bruno; Romanoff, Jani Size-dependent modelling of elastic sandwich beams with prismatic cores Published in: International Journal of Solids and Structures DOI: 10.1016/j.ijsolstr.2017.12.001 Published: 01/04/2018 Document Version Publisher's PDF, also known as Version of record Published under the following license: CC BY-NC-ND Please cite the original version: Reinaldo Goncalves, B., & Romanoff, J. (2018). Size-dependent modelling of elastic sandwich beams with prismatic cores. International Journal of Solids and Structures, 136, 28-37. https://doi.org/10.1016/j.ijsolstr.2017.12.001
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This is an electronic reprint of the original article.This reprint may differ from the original in pagination and typographic detail.
Powered by TCPDF (www.tcpdf.org)
This material is protected by copyright and other intellectual property rights, and duplication or sale of all or part of any of the repository collections is not permitted, except that material may be duplicated by you for your research use or educational purposes in electronic or print form. You must obtain permission for any other use. Electronic or print copies may not be offered, whether for sale or otherwise to anyone who is not an authorised user.
Reinaldo Goncalves, Bruno; Romanoff, JaniSize-dependent modelling of elastic sandwich beams with prismatic cores
Published in:International Journal of Solids and Structures
DOI:10.1016/j.ijsolstr.2017.12.001
Published: 01/04/2018
Document VersionPublisher's PDF, also known as Version of record
Published under the following license:CC BY-NC-ND
Please cite the original version:Reinaldo Goncalves, B., & Romanoff, J. (2018). Size-dependent modelling of elastic sandwich beams withprismatic cores. International Journal of Solids and Structures, 136, 28-37.https://doi.org/10.1016/j.ijsolstr.2017.12.001
odels that include higher-order displacement gradients can sup-
lement the continuum with information from the unit cell scale.
n the modified couple stress Timoshenko beam (MCSTBT), size ef-
ects arising from the local scale stiffness are estimated through
he non-dimensional relation
=
√
S xy
D Q L 2 (21)
If face and core material properties are equal, the magnitude of
uch size effects for a given set of loading and boundary conditions
s uniquely dependent on the cell geometric parameters and length
etween supports L . For selected prismatic geometries in Section
.4 ( Figs. 5 and 6 ) the resulting ∇
2 are given in Table 2 .
To investigate size effects in the continuum modelling of sand-
ich beams with prismatic cores, selected topologies are studied
n the linear elastic range. The responses are compared to the
nes obtained with the classical Timoshenko beam theory (TBT)
hroughout the study. To emphasize the size effect, deflections are
resented relative to the maximum deflection obtained with the
BT. In all analyses, the material properties E = 206 GPa and ν = 0.3
32 B. Reinaldo Goncalves, J. Romanoff / International Journal of Solids and Structures 136–137 (2018) 28–37
Fig. 7. Example of three-dimensional finite element model used for validation throughout this work (web-core panel strip in three-point bending and mid-length symmetry).
Fig. 8. Discrete cell geometry and deflection lines obtained with classical and non-classical continua and discrete finite elements.
3
d
e
t
t
w
are assumed (steel). Discrete three-dimensional finite element (3D
FE) models are used for validation ( Fig. 7 ). The validation models
represent the actual panel geometry, finely discretized with Abaqus
S4R shell elements ( Dassault Systemes, 2013 ). Boundary conditions
are applied as to ensure transverse incompressibility at the cross-
sections under loading and supports. At the surfaces perpendicular
to the beam main axis, plane strain conditions guarantee beam-like
response.
A
s
.1. Discrete response vs. classical and non-classical continuum
escriptions
Consider prismatic sandwich panel strips with unit-cell prop-
rties shown in Fig. 8 . The lengthwise deflection distribution is
raced for limiting cases using classical and size-dependent con-
inuum theories. In all cases, the beams are in three-point bending,
ith total lengths L A = 2.4 m, L B = 0.64 m and L C = 0.72 m. The cases
-C are selected based on their shear-flexibility and sensitivity to
ize effects. In short, they can be summarized as follows,
B. Reinaldo Goncalves, J. Romanoff / International Journal of Solids and Structures 136–137 (2018) 28–37 33
Fig. 9. Maximum deflection predictions for web-core panel strip using beam and
reference models as function of (a) relative unit cell size s / L (b) unit cell aspect
ratio d / s (c) face-to-core thickness ratio t f / t c .
B) Shear flexible θQ = 1.530; moderate size-effect sensitivity
∇ = 0.035;
C) Shear flexible θQ = 1.589; high size-effect sensitivity ∇ = 0.109;
here θQ = D xx / L 2 D Q measures the shear flexibility of the beam
Zenkert, 1995 ). In addition to the TBT and MCSTBT models, we
lso present the curves obtained with the modified couple stress
uler–Bernoulli (MCSEBT) beam theory ( Park and Gao, 2006 ),
herein the non-classical parameter is taken the same as in
ections 2.3 and 2.4 .
The MCSTBT model is observed to predict the average static de-
ection distributions accurately in all cases. In case A ( Fig. 8 (a)),
bending-dominated problem, the three models render approxi-
ately the same deflection lines. Case B ( Fig. 8 (b)) shows a shear-
ensitive problem; the MCSEBT has no mechanism to include shear
eformations, thus predicts overly stiff response. The TBT and
CSTBT models behave similarly, and a kink is observed under
he concentrated force as the unit cell has little internal bending
tiffness. Local bending is observed only in the immediate vicin-
ty of the concentrated force and its effect is nearly negligible.
ase C ( Fig. 8 (c)) emphasizes the need of the MCSTBT for a size-
ependent continuum; the unit cell stiffness is high, thus local
ending is non-negligible in the entire cell near the concentrated
orce. This behaviour cannot be described with TBT and MCSEBT
heories due to the lack of length-scale parameter and shear defor-
ation measure respectively.
.2. Fundamental ratios governing the size effect: web-core case
tudy
Table 2 indicates that size effect in sandwich panels with pris-
atic cores is governed by certain structural ratios. To understand
heir influence in the response prediction with TBT and MCSTBT
odels, we select a simple geometry with weak coupling among
eometric parameters. Consider a web-core sandwich panel strip
f unit width in three-point bending. For this geometry, three ra-
ios govern the size effect, which are studied here separately: s / L,
/ s and t f / t c (see Fig. 5 (a)). The sandwich panel depth is taken con-
tant, d = 0.05 m for all analyses. In (a) and (b), t f = 0.004 m.
.2.1. Relative unit cell size s / L ( Fig. 9 (a))
Let d / s = 5/8 and t f / t c = 1.6 be constant and the relative unit cell
ize 1/36 ≤ s / L ≤ 1/6. As predicted in Table 2 , the local effects
ecome negligible as s / L → 0, hence the TBT model is incrementally
ccurate. The relative difference between beam models is a linear
unction of s / L , which accounts for the local effects magnitude.
.2.2. Unit cell aspect ratio d / s ( Fig. 9 (b))
In this analysis, t f / t c = 1 and s = L /6 are constant, while 1/10 ≤ / s ≤ 2. The parameter ∇ is here proportional to
√
d/s + 1 / 2 . As
/ s → 0, the local effects are no longer dependent on the cell aspect
atio, but still on the other ratios; the relative difference between
eam models becomes therefore nearly constant.
.2.3. Face-to-core thickness ratio t f / t c ( Fig. 9 (c))
To study the role of member thicknesses, we select s = L /6 and
/ s = 5/8, with 1/8 ≤ t f / t c ≤ 2. It is shown that the magnitude of
ocal effects does not depend substantially on this ratio if t f < t c ,
n fact, converging to a constant as t f / t c → 0. Conversely, if t f > t c ,
ncrementally increasing the ratio results in ∇ rapidly increasing
s a function of ( t f / t c ) 3/2 .
.3. Role of core topology and relative density
The position of corrugations and their relative density in re-
ation to the overall unit cell area affect the structural ratios of
able 2 and thus the size-effect sensitivity. The core relative den-
ity is defined in the general case
c =
1
sd
j ∑
i =1
t c l c,i (22)
here l c,i is the length of the i th to a total of j core members. We
tudy common prismatic cores with single and multiple orders of
34 B. Reinaldo Goncalves, J. Romanoff / International Journal of Solids and Structures 136–137 (2018) 28–37
Fig. 10. Core geometries and dimensions used for the core density analysis, n c = 1.
Fig. 11. Core geometries and dimensions for panels with varying corrugation
order n c .
u
t
c
a
i
o
t
o
3
f
n
i
t
n
i
d
e
c
d
i
t
e
l
g
c
e
c
f
l
4
o
s
U
w
r
t
M
s
I
v
b
m
i
l
o
s
i
g
corrugation to compare their behaviour and validate the MCSTBT
in terms of maximum deflection against three-dimensional finite
element models. Face thicknesses and depths are scaled to obtain
constant bending stiffness ( D xx = 848.9 kNm), while the thickness
of the corrugations is adjusted as function of the core relative den-
sity.
3.3.1. Common prismatic cores with n c = 1
Consider prismatic sandwich panels with dimensions shown in
Fig. 10 . Fig. 12 (a) shows the parameter ∇ ( Table 2 ) in terms of ρc
for the geometries selected. An association of size-effect sensitiv-
ity with density and shear-carrying mechanism is observed. Cores
whose main mechanism involves bending of the corrugations dis-
play dramatic increases in sensitivity as their slenderness is re-
duced. In contrast, cores that rely on membrane stretching have
lower sensitivity overall, which only marginally depends on ρc .
Fig. 13 shows the maximum deflection predicted by the mod-
els as function of the core density, 0.01 ≤ ρc ≤ 0.12, for a unit-
width panel with L = 0.8 m in three-point bending. In size-effect
sensitive cores, the TBT exhibits overly soft response as the den-
sity is reduced. In contrast, the MCSTBT is able to predict deflec-
tions very satisfactorily for all cases studied with maximum rela-
tive error under 2%. Differences between MCSTBT and 3D FE are
attributed to internal cell fluctuations due to higher-order deriva-
tives of displacement, which are averaged to a constant curvature
as discussed in Section 2 .
3.3.2. Multi-layered prismatic cores
Consider unit-width panels with hexagonal honeycomb and di-
amond cores and varying core volume ratio as shown in Fig. 11 .
Fig. 12 (b) shows the sensitivity parameter ∇ as function of the cor-
rugation order and core relative density. In hexagonal honeycomb
panels, the sensitivity increases with the corrugation order in low-
density cores. The increments are increasingly smaller as n c grows,
tending to a constant as the cross-section becomes homogeneous,
n c → ∞ . In contrast, in diamond core panels, the corrugation order
plays a minimal role in the panel stiffness and size-effect sensitiv-
ity. The behaviour is consistent with the shear-carrying mechanism
dominant in each core: bending of cell walls in hexagonal honey-
combs and corrugation stretching in diamond cores.
Fig. 14 validates the MCSTBT against three-dimensional finite
elements in terms of maximum deflections. The analyses are un-
dertaken for a doubly-clamped unit-width 0.4 m long panel strip
nder centred vertical force. It is shown that the maximum deflec-
ion increases with the corrugation order in hexagonal honeycomb
ores due to the increased shear-flexibility as the wall thicknesses
re reduced ( Fig. 14 (a)). The increase is considerably more subtle
n diamond core panels ( Fig. 14 (b)). It is shown that the response
f multi-layered sandwich panels can be accurately predicted with
he MCSTBT regardless of their geometry, density or corrugation
rder.
.4. Role of boundary conditions
Besides the sensitivity parameter, the magnitude of size ef-
ects in prismatic sandwich structures depends on the loading sce-
ario, as concentrated loads and restrictive boundary conditions
nduce the size-dependent behaviour. In the context of concen-
rated forces, it is of particular interest how severe is the disconti-
uity in the shear force diagram. We compare three common load-
ng cases in which size effect is observable: simply supported and
oubly clamped beams under mid-span concentrated force and
nd-loaded cantilever.
Consider panels with web-core, Y-core and hexagonal honey-
omb core ( n c = 1) and dimensions shown in Fig. 10 . The relative
ifference in maximum deflections obtained with TBT and MCSTBT
s shown in Fig. 15 as function of ∇ . The difference highlights
he local effects associated with the second displacement gradi-
nt and average cell stiffness. Overall, the relative difference is a
inear function of ∇ when local effects are moderate, with slope
overned by the discontinuity type. It is largest for the doubly
lamped beam, as multiple discontinuities are present, and low-
st for the cantilever beam, where the only discontinuity is the
lamped end. The results of previous sections can be readily scaled
or other load conditions, with the size effect becoming more or
ess severe.
. Discussion and conclusions
In this study, the modified couple stress Timoshenko beam the-
ry ( Ma et al., 2008; Reddy, 2011 ) has been applied to study the
ize-dependent response of sandwich beams with prismatic cores.
nlike previous works ( Dai and Zhang, 2008; Liu and Su, 2009 ),
e consider solely local stiffening causing disturbances in the pe-
iodic shear field as size effect, as changes in flexural rigidity due
o material positioning can be modelled using classical continua.
oreover, the size effects considered are only due to the length-
cale interactions, and not to other local shear-field disturbances.
t has been assumed that the structures are a repetition of trans-
ersely rigid complete cells. In reality, prismatic structures might
e transversely flexible ( Frostig and Baruch, 1990 ) and the edges
ay contain incomplete cells ( Anderson and Lakes, 1994 ) promot-
ng other types of size effects, whose influence in the response of
ightweight sandwich structures is left for future work.
In this study, we have chosen the modified couple stress Tim-
shenko beam theory (MCSTBT) given its simplicity; it includes a
ingle non-classical stiffness parameter in its formulation, whose
nterpretation in the context of prismatic sandwich panels is tan-
ible. A downside is that the local stiffness must be considered
B. Reinaldo Goncalves, J. Romanoff / International Journal of Solids and Structures 136–137 (2018) 28–37 35
Fig. 12. Size-effect sensitivity parameter as function of the core geometry and relative density ratio (a) cores with order of corrugation n c = 1 (b) hexagonal and diamond
cores with varying n c .
Fig. 13. Maximum deflection predictions for (a) web-core (b) triangular corrugated core (c) hexagonal honeycomb core (d) Y-core sandwich panels with varying core relative
volume ratio using beam and validation models.
c
s
e
m
i
(
w
w
s
o
d
o
l
onstant, while in reality the cells are highly discrete. Further con-
ecutive scales could be introduced through higher-order gradi-
nts of displacement improving the cell stiffness description. The
zes the works of Romanoff and Reddy (2014) and Romanoff et al.
2015) in studying web-core panels with semi-rigid joints; in their
ork, equivalence with the classical sandwich theory ( Allen, 1969 )
as invoked to determine the non-classical stiffness. It has been
hown that the modified couple stress Euler–Bernoulli beam the-
ry ( Park and Gao, 2006 ) is not suitable to model the size-
ependent behaviour of these structures, as the size effects arise
nly when the shear flexibility is high.
Size effects in prismatic sandwich beams are shown to be re-
ated to certain structural ratios involving cell and overall beam
36 B. Reinaldo Goncalves, J. Romanoff / International Journal of Solids and Structures 136–137 (2018) 28–37
Fig. 14. Maximum deflection predictions for doubly-clamped (a) hexagonal honeycomb core (b) diamond core sandwich panels with two orders of corrugation as function
of the core relative density.
Fig. 15. Relative difference between TBT and MCSTBT models for different sets of
loading and boundary conditions as function of the parameter ∇ .
Table A.1
Transverse shear stiffness D Q and local bending stiffness S xy of hexagonal
honeycomb core beams with selected orders of corrugation n c .
n c D Q S xy
2 E ( 1 −υ2 )
54 t 3 c ( 281 t 6 f +255 t 3
f t 3 c +8 t 6 c )
s 2 ( 693 t 6 f +708 t 3
f t 3 c +88 t 6 c )
E ( 1 −υ2 )
14 t 6 f +15 t 3
f t 3 c +2 t 6 c
21 t 3 f +5 t 3 c
3 E ( 1 −υ2 )
432 t 3 c ( 668 t 6 f +788 t 3
f t 3 c +31 t 6 c )
s 2 ( 9398 t 6 f +11801 t 3
f t 3 c +1200 t 6 c )
E ( 1 −υ2 )
52 t 6 f +82 t 3
f t 3 c +14 t 6 c
78 t 3 f +19 t 3 c
5 E ( 1 −υ2 )
10800 t 3 c ( 273242 t 6 f +391311 t 3
f t 3 c +17223 t 6 c )
s 2 ( 60753591 t 6 f +89833161 t 3
f t 3 c +7281992 t 6 c )
E ( 1 −υ2 )
724 t 6 f +1866 t 3
f t 3 c +372 t 6 c
1086 t 3 f +265 t 3 c
A
c
p
s
R
A
A
A
B
C
D
E
F
G
H
dimensions, being the exponent of each ratio a measure of its sen-
sitivity. Such ratios have been studied in an example involving a
simple geometry, for which the effects are easily isolated. The in-
fluence of core density and topology in the size-dependency has
been then investigated. Overall, it has been shown that size ef-
fect is more pronounced in cores where corrugation bending is the
dominating shear-carrying mechanism. The core relative density
governs the size effect magnitude, which is also influenced by the
corrugation order in certain prismatic panels such as the hexag-
onal honeycomb type. Lastly, the influence of loading scenario in
the size effect magnitude has been investigated; three simple cases
have been selected, and the doubly clamped under mid-length ver-
tical load seen to be critical as it introduces the most severe set of
discontinuities. While the material properties of faces and core are
taken for simplicity equal in this study, the size effect magnitude
should otherwise also dependent on their ratio.
Acknowledgements
The authors gratefully acknowledge the financial support from
the Graduate School of Aalto University School of Engineering and
the partners of the Finland Distinguished Professor programme
“Non-linear response of large, complex thin-walled structures”:
Tekes, Napa, SSAB, Deltamarin, Koneteknologiakeskus Turku and
Meyer Turku.
ppendix A. Stiffness parameters for hexagonal honeycomb
ore panel
The coefficients for the stiffness of hexagonal core sandwich
anels ( Fig. 6 (b)) with selected orders of corrugation n c are pre-
ented in Table A.1 .
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