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Minimum weight design for stiffness in sandwich plates with rigid foam cores
Materials Science and Engineering, 85 (1987) 33-42
33
Minimum Weight Design for Stiffness in Sandwich Plates with Rigid Foam Cores L. A. DEMSETZ and L. J. GIBSON
Department of Civil Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139 U.S.A.) (Received February 3, 1986)
ABSTRACT
A new relationship between the shear modulus and density o f foams has been used to identify the core density as well as the core and face thicknesses which minimize the weight o f a sandwich plate for a given bending stiffness. The results o f bending tests on simply supported circular sandwich plates with alum i n i m u m skins and polyurethane foam cores confirm the results o f the analysis. The work should be useful in improving the design o f sandwich panels in many engineering applications. It also has relevance to the understanding o f the design o f natural sandwich structures, such as those found in the human skull and in the iris leaf.
1. INTRODUCTION Composite structural panels made up of two stiff strong faces separated by a lightweight core are known as sandwich panels. Separation of the faces by the core increases the m o m e n t of inertia of the panel with little increase in weight, giving an efficient member for resisting bending and buckling. For this reason, they are often used in applications where the weight of the component is critical: in structural components in aircraft, in portable buildings, in the decks and hulls of racing yachts and in modern downhill skis. Sandwich structures are found in nature, too, e.g. in the human skull and the iris leaf. In the skull, two thin outer layers of dense compact bone are separated by a lightweight core of sponge-like cancellous bone [1, 2]; in the iris leaf, the almost fully dense and relatively stiff fibre-composite-like ribs on the outer skin of the leaf are separated by weak low density cells in the core. 0025-5416/87/$3.50
Many materials can be used in a sandwich construction; usually, however, the faces are made of either aluminum or fibre-reinforced composites, while the cores are aluminum or paper-phenolic honeycombs or polymeric foams. H o n e y c o m b cores are used in applications where the weight of the panel is the only critical factor, e.g. in structural components of aircraft. Foam cores are usually preferred in applications where a low thermal conductivity is required in addition to a low weight, e.g. in portable buildings, mobile homes and refrigerated shipping containers. In this study, only sandwich panels with foam cores are investigated. Optimization analyses of sandwich plates aim to develop a m e t h o d of designing the lightest possible plate for some specified structural requirement. The requirement may be that the panel must have some given stiffness or strength, or a given combination of stiffness and strength. There may be dimensional constraints, too; the face and core may have to be within certain size limitations determined by the availability of materials, or the depth of the panel may have to be less than some m a x i m u m allowable dimension. Previous optimization studies have usually focused on determining the core and face thicknesses which minimize the weight of a panel assuming that the core and face materials are completely specified and that their densities, elastic moduli and strengths are known. Allen [3] shows how this can be done for a sandwich beam subject to a stiffness constraint. Huang and Alspaugh [4] and Ueng and Liu [5] have optimized the core and face thicknesses for more highly constrained members. Huang and Alspaugh minimized the weight of a sandwich beam subject to the conditions of maximum allowable bending stress, shear stress and deflection, as well as © Elsevier Sequoia/Printed in The Netherlands
34 minimum face and core thicknesses. Ueng and Liu solved a similar problem for a specific type of corrugated core. Because of the complexity of the constraints, the solutions for both of these problems were obtained using numerical techniques, as no analytical solution was possible. Little attention has been paid to optimizing the core density of sandwich panels to minimize their weight. Wittrick [6] and Ackers [ 7], working on sandwich beams with foamed calcium alginate cores, and Kuenzi [8], working on the use of wood cores in sandwich plates, all recognized that, to include the core density as a design variable in the optimization analysis, relationships between core properties and core density must be known. The tests of Wittrick and Ackers on foamed calcium alginate and Kuenzi's tests on wood showed that moduli and strength were linearly related to density for both of these materials. All these researchers then used this linear relationship in their analyses. A recent study of the mechanical behavior of cellular solids has shown that, while these linear relationships are valid for h o n e y c o m b cores loaded transversely in shear [9] and for wood loaded along the grain [10], they are not, in general, valid for foamed materials [11, 12]. By identifying and modelling the mechanisms of deformation in cellular solids, Gibson, Maiti and Ashby found that foam properties are related to three parameters: a constant related to the cell geometry, a property of the cell wall material and the relative density of the foam raised to some power. In particular, they found that Young's modulus and the shear modulus of rigid foams are related to their relative density squared and that the compressive, tensile and shear strengths are related to the related density raised to the power 3/2. Gibson [13] has recently used these new non-linear relationships between foam properties and foam density to find the core density as well as the core and face thicknesses which minimize the weight of a foam core sandwich beam subject to a stiffness constraint. In this paper the continuation of that work to the optimum design of sandwich plates with rigid foam cores is described. In it we develop the closed-form solution for the core density and the core and face thicknesses which minimize the weight of a circular sandwich plate subject to a stiffness constraint. The
results of tests on a series of sandwich plates of nominally the same stiffnesses but of different designs and weights confirm the analysis. The m e t h o d described should allow the design of sandwich plates with foam cores to be improved by optimizing the core density as well as the core and face thicknesses.
2. A N A L Y S I S
The sandwich plate to be optimized is shown in Fig. 1. It is a simply supported circular plate which spans a radius r and carries a distributed load q over a central circular portion of its area of radius a. The thickness of each face is t, while that of the foam core is c. The faces have a density pf, Young's modulus E~ and Poisson's ratio v~. The solid polymer of which the core is made has a density Ps and a modulus E s. The density of the foamed core is Pc, its Young's modulus is E c and its shear modulus is Gc. The analysis is to develop a technique for choosing the values of core density Pc, and core and face thicknesses c and t, which minimize the weight of the plate for a given stiffness. It is assumed that the material properties of the faces and of the solid polymer from which the core is foamed are known.
2o FACE pf ,Ef,~lf\
~ •
q o
.
• FOAM CORE ~P~'Ec'G c
.
.
.
o
.
c
Fig. 1. S a n d w i c h plate g e o m e t r y : a simply s u p p o r t e d circular s a n d w i c h plate o f radius r carrying a distrib u t e d load q over a central circular p o r t i o n o f its area o f radius a.
35
The properties of foams depend on three parameters: the geometry of the cells, the properties of the solid cell wall material and the relative density of the foam, i.e. the density of the foam normalized by that of the solid from which it is made. It has long been recognized that the properties of foams depend on their density raised to some power; recently, Gibson and Ashby [ 11] and Maiti e t al. [12] have modelled the deformation mechanisms in foams to estimate this power for m a n y properties. At small strains (less than about 5%), the cell walls deform mainly by bending; it can easily be shown that, if this is the case, then Young's modulus and the shear modulus of an open cell foam are related to the square of its relative density: ~
~cc
(1)
L)
(2)
while Poisson's ratio is independent of relative density and depends only on the geometry of the cells in the foam. These simple square laws have been found to describe Young's modulus and the shear modulus of m a n y foams reasonably well. In some cases, there is some deviation in the power from 2; this may be explained by several factors. If the cells are highly anisotropic, then axial deformations become more important for loading along the elongated direction of the cells; this tends to reduce the exponent towards a limiting value of 1. If isotropic cells are closed and have uniformly thick faces, then bending deformations in the cell faces lead to an exponent of 3; usually, however, surface tension draws most of the material away from the faces and into the edges of the cell during the foaming process, so that most closed-cell foams behave more like open-cell foams. Nevertheless, as the faces of closed-cell foams become more uniform, the exponent of relative density for the moduli tends towards an upper limit of 3. There is another source of deviation from the squared rule, too; if the nodes where cell edges intersect are enlarged, then the moduli should tend to be less sensitive to relative density, as less of the solid is subject to bending deformations. In general, then, we can write
E = C~
E~
(3)
G = Cg
Es
(4)
where n is about 2. In the analysis that follows, the shear modulus-relative density relationship will be used to introduce the core density into the constraint equation and thus into the optimization analysis. The mass of the sandwich plate to be optimized is given by (5)
M = 2 p f ~ r 2 t + pcTrr2c
This is the objective function which the analysis aims to minimize. The stiffness of the sandwich plate is the ratio of the applied load to its resulting midspan deflection. This deflection has both bending and shearing components; shear deflections, which are negligible in an isotropic plate, are significant in a sandwich plate because of the low shear modulus of the core. The midspan bending deflection is given by [3, 14] qa 4 wb - 16D gl
(6)
where gl
=,,,(/)' (/) +
1 + vf
in
--
4(1 + v~)
and Eftc 2
D~
2(1 _~f2)
The midspan shear deflection is given by [3] qa 2 Ws : ~ - g2
(7)
where g2=1 + 21n(r)
\a /
and S ~ cGc
The exact expressions for D and S are given in Appendix A. The approximations given here are valid if Young's modulus of the core is very much less than that of the face and if the face thickness is very much less than the
36 core thickness. Since this generally is the case for sandwich plates, the simpler approximate expressions for D and S are used in the analysis. Summing wb and w~ to get the total deflection w, and dividing by the uniformly distributed load gives the inverse of the stiffness:
W q
a4(1 -- Vf2) a2 8E~tc 2 gl + 4c~-~g2
-
(8)
Replacing the shear modulus of the core by eqn. (4) introduces the core density into the stiffness constraint equation. Solving the stiffness equation for the relative density of the core gives
P_A - [ a2g2 I w Ps - L 4CgcE~ [ q
a4(l_--~v~2)gl~-l~ vn 8E~tc 2 J J
(9)
Substituting this equation for relative density into the equation for the mass of the plate (eqn. (5)) and solving for aM~at = 0 and aM/ac = 0, we find the optimum solution for the plate design. It is .
( 1 (n + 1 ) ~ ( n - - 1) 2 Ps~ 256-C~ a n ( 1 - - P f 2 )
Pcopt
×
Co~
g2__3 Ps Ef (~)2}1/(3n-1) gl P~ Es a
(10)
__ a S4n+lc., n " ( n + 1 ) " - 1 ( 1 - v ~ 2 ) n 2( g (n -- i) 2~
g l n (~-~S] p ~ n ~n(~)n Es - 1}1](3n_ 1) X-~2
(11)
and if both second derivatives are greater than zero,
a2M -->0
at 2
a2M -->0 ac 2 This is the case for the mass objective function so that eqns. (10)-(12) describe the global optimum. For the optimum solution, it can be shown that the ratio of the weight of both of the faces to that of the core is always ( n - 1)/2n, independent of the given stiffness, the span, the radius of the load or the materials used. In addition, the ratio of the bending to the shearing deflections is always (n -- 1)/2, again independent of the given stiffness, the span, the radius of the load or the materials used. For m a n y foams, n -- 2; in this case, the faces weigh one-quarter of the weight of the core and the deflections arising from bending are one-half of those from shearing. Both of these ratios are identical with those found for the optimum design of a sandwich beam subject to a stiffness constraint [13]. If n -= 1 the solution becomes unrealistic; the face thickness and core density become equal to zero while the core thickness tends to infinity. In this case, a side constraint on the maximum allowable core thickness would control the optimum design, and a new analysis taking this side constraint into account would have to be performed to find it.
1 (n 2 -- 1)n+l(1--v~2) "-1 to~ = 4a 213.+1C~2
n2n
X
gl n-1 g22~-f/
--~-s2 ~ - - ]
3. EXPERIMENTAL METHOD ) (12)
It has been shown [15] that the optimum obtained by setting derivatives equal to zero is a global o p t i m u m if the objective function is convex, i.e. if the determinant of its hessian matrix is greater than zero,
a2M
a2M
~t 2
at ac
~2M
a2M
ac at
ac 2
T0
To test the analysis developed above, a series of sandwich plates with aluminum skins and polyurethane foam cores were made, each having the same design stiffnesses but with design parameters (the core density and the core and face thicknesses) that were varied about the optima found from the analysis. The density and moduli of the aluminum and the solid polyurethane from which the core was foamed are given in Table 1. Before the plates were designed, the shear moduli of the various densities of polyurethane foam to be used in the cores were
37 TABLE 1 Properties of aluminum and solid unfoamed polyurethane D e n s i t y (kg m -a)
Material
Aluminum Solid unfoamed polyurethane
I /
/
~
Pf = 2700 [16] ps = 1200 [17]
Young's modulus
Poisson's ratio
(GN m -2)
(--)
Ef = 70 [16] Es= 1.6 [18]
0.334 [16] --
100 P
~
I
ALUMINUM BAR~
(D
g o rr <~ 10-' 03
J w 8[
Fig. 2. Jig for measuring the shear modulus of the foam to be used in the cores of the sandwich plates.
m e a s u r e d using the shear jig s h o w n in Fig. 2. Because t h e m o d u l u s o f the f o a m is v e r y m u c h less t h a n t h a t o f the a l u m i n u m sidepieces, t h e f o a m is s u b j e c t e d to a l m o s t p u r e shear stress. T h e results o f these tests s h o w e d t h a t t h e shear m o d u l u s o f the p o l y u r e t h a n e f o a m was r e l a t e d to its relative d e n s i t y b y
iO-l iOo RELATIVE DENSITYD/fls(-)
Fig. 3. Relative shear modulus G/E s plotted against relative density PIPs for the foams used in the cores of the sandwich plates. The data fall on the line G/E s = 0 . 5 4 ( p / p s
)1.6.
M = 2.51 kg
copt = 60 m m
T h r e e series o f plates were designed. In e a c h series, o n e o f the design variables was held c o n s t a n t at its o p t i m u m value while the o t h e r t w o variables were varied a b o u t t h e i r o p t i m a to give plates of c o n s t a n t stiffness b u t differe n t weights. U n f o r t u n a t e l y , the 3 0 8 kg m -3 f o a m was n o t available at the t i m e the plates were m a d e a n d so, instead, 320 kg m -s f o a m was s u b s t i t u t e d . T h e " o p t i m u m " plate for the t e s t series was t h e n slightly d i f f e r e n t f r o m the t r u e o p t i m u m d e s c r i b e d above. T o m a i n t a i n t h e design stiffness o f 1.17 X 104 N m m -1, it h a d
tom = 0.64 m m
Cop t =
Pcopt = 3 0 8 kg m -3
tom = 0.64 m m
/p \t.6
Go = o.54( )
Zs
(13)
\Ps / as s h o w n in Fig. 3 a n d in a g r e e m e n t w i t h the general r e l a t i o n s h i p o f eqn. (4). T h e plates were designed to have a stiffness o f 1.17 × 104 N m m -z, a span o f 381 m m a n d a c e n t r a l load d i a m e t e r o f 51 m m . F o r this p a r t i c u l a r l o a d i n g c o n f i g u r a t i o n the o p t i m u m design f o r a s a n d w i c h p l a t e was f o u n d t o be
58
mm
38 Pcopt = 320 kg m -3 M = 2.52 kg As its mass is only 0.01 kg greater than the true o p t i m u m design, this substitution was considered to be acceptable. The designs of all the plates tested are listed in Table 2. It was initially planned to test five plates for each of five different designs for each test series; unfortunately, the shortage of foam available from the supplier reduced both the number of plate designs in each test series and the number of specimens of each design in some cases. In the final set of experiments, four different plate designs were used for each test series, and between three and six specimens of each design were tested. The plate specimens were made by bonding two aluminum faces to a precut polyurethane foam core with polyester resin. The thickness of each face and core was measured before being bonded. The foam cores were cut to within 0.5 mm of their design values. The face thicknesses were designed to be standard gauge thicknesses for aluminum and so were
within +0.01 mm of their design values. The foam cores were weighed individually to measure the density of the foam used in each plate. The core densities were within 5% of the design values for all the plates except P1 which was within 10%. In view of the variability in foam density caused by the foaming process itself, these deviations in foam density from the design values are thought to be acceptable. Each face was sand blasted prior to being glued and the specimens were clamped for 24 h while the resin hardened to ensure a good bond. Each plate was weighed before testing. To measure the stiffness of the panels, each one was simply supported on eight cylindrical rollers 3 in long arranged in a circular pattern. A steel plate ¼ in thick, 3 in long and 1 in wide was placed on top of each roller to distribute the load at the support and to prevent the foam core from crushing. The load was applied with a 60 000 lb Baldwin hydraulictesting machine and measured with a pressure gauge connected to an X - Y recorder. A piece of polyurethane foam 51 mm in diameter was
TABLE 2 Plate b e n d i n g test results (design stiffness, 1.17 X 104 N m m -1)
Plate design identification
P9 P2 P10 Pll P1 P2 P3 P4 P6 P2 P7 P8
Number o f specimens tested
5 5 5 6 6 5 6 3 3c 5 4e 4
Design core thickness c
Design face thickness t
Design core density Pc ( k g
Measured plate stiffness q/Tra2w
(mm)
(ram)
m -3)
(X 104 N m m -1)
60 58 60 60 121 58 50 37 65 58 55 52
0.31 0.64 0.81 1.025 0.64 0.64 0.64 0.64 0.41 0.64 1.27 2.03
386 320 a 296 284 175 320 381 618 313 d 320 296 296
1.25 1.37 1.37 1.24 1.28 1.37 1.36 1.14 1.19 1.37 1.46 1,37
Weight W (N) Theoretical
27.8 24.7 24.8 25.2 27.4 24.6 25.1 29.6 25.2 24.6 25.9 29.5
Stiffness q/Tra2wW per unit weight Experimental
27.2 25.4 26.0 25.0 25.0 25.4 25.9 28.3 24.4 25.4 27.7 30.5
( m m -1 )
Theoretical
Experimental
420 473 472 465 426 476 466 395 465 476 452 397
460 539 527 496 512 539 525 403 488 539 527 449
a T h e original o p t i m u m f o a m d e n s i t y of 308 kg m -3 was r e p l a c e d b y f o a m o f d e n s i t y 320 kg m -3 because of t h e unavailability o f t h e 308 kg m -3 f o a m . b T h e m e a s u r e d t h i c k n e s s was 0.94 ram. c T w o o f these t h r e e s p e c i m e n s d e b o n d e d d u r i n g t e s t i n g a n d were n o t i n c l u d e d in t h e test results. d T h e original o p t i m u m f o a m d e n s i t y o f 308 kg m - a w a s replaced b y o t h e r densities as close to 308 kg m -3 as possible. e One o f these f o u r s p e c i m e n s d e b o n d e d d u r i n g testing a n d was n o t i n c l u d e d in t h e test results.
39
placed between the test machine platen and the plate specimen to distribute the load uniformly while the plate deflected. A d.c. differential transformer was placed under the centre of the plate to measure its midspan displacement. Dial gauges were placed on top of the plate over each support to check that the plate was not crushed at the supports; these deflections were found to be negligible. The stiffness of each plate was then calculated from the slope of the load-deflection curve plotted on an X- Y recorder. The stiffness of the supports was also measured; this was found to change the measured plate stiffnesses by about 5% and was taken into account in calculating the true plate stiffnesses.
600 o_
I _x- -x i ~x
f
-
IX
400 I _
tOp i
Pc
opt
u_ on
0
I 0
I
J
0.5 1.0 FACE THICKNESS t (ram)
I5
(a)
(b)
l(~O 2;0 5l~0 CORE DENSITY P c ( k g / m 3 )
400
Fig. 4. Plate bending test results for s a n d w i c h plates with the core thickness held c o n s t a n t at the o p t i m u m value o f 60 ram. The measured and theoretical stiffnesses q/Tm2wWper unit weight are p l o t t e d against (a) the face thickness t and (b) the core density Pc. The design stiffness for all the plates was 1.17 × 104 N m m -1.
4. R E S U L T S A N D D I S C U S S I O N
The designs of all the plates tested are listed in Table 2 together with the results of the plate stiffness tests. Deviations from the design value of the density of the foam used in the plates caused some variation in the theoretical stiffness of the plates. Revised theoretical stiffnesses were calculated on the basis of the actual foam densities used in the plate cores. The average revised stiffness was 1.17 × 1 0 4 N mm -1, in agreement with the original design value. The coefficient of variation for the revised theoretical plate stiffnesses was 7%, of the order that might be expected from the variation of about 5% from the design values of the foam densities. The measured values of plate stiffnesses, corrected for the support stiffness, are listed in Table 2 together with the theoretical and measured weights of the plates and the theoretical and m~asured stiffness per unit weight. The average measured stiffness of the plates was 1.31 X 1 0 4 N mm -1, 12% higher than the design value of 1.17 X 104 N mm -1. The coefficient of variation in the measured plate stiffnesses was 9%, again of the order that might be expected from the variation in the revised theoretical plate stiffness. Figures 4-6 show the theoretical and measured values of the stiffness per unit weight for all the plates plotted against the design variables of core thickness c, face thickness t and the core density Pc. Each pair of plots shows the results from one series of tests in which one of the design variables was held
6O0
x
I
,x.X-- - - _
I
,X
.x ~ - -
X--x
~ ~ ~oo i
~ ~2oo
Copl
Pc opt
u_ u_ m
0
(a)
510 I 0J0 CORE THICKNESS c(mm)
1 5I 0 0 (b)
200 400 600 CORE DENSITY ~(kg/rn ~1
800
Fig. 5. Plate bending test results for s a n d w i c h plates with the face thickness t held c o n s t a n t at the optim u m value o f 0 . 6 4 ram. The measured and theoretical stiffnesses q/Tra2wWper unit weight are p l o t t e d against (a) the core thickness c and (b) the core density Pc. The design stiffness for all the plates was 1.17 X 1 0 4 N m m -1.
600
/ x 'x
%
xl~l x~
X ~
~oo
z
a.%
~ zoo
m
C
O0
[a)
opt
40 60 CORE THICKNESS c(mm)
20 l
top t
I
80 (b)
I
0,5 I0 I!5 FACE THICKNESS timm)
Fig. 6. Plate bending test results for s a n d w i c h plates with the core d e n s i t y Pc held constant at the optim u m value o f 3 2 0 kg m -3. The measured and theoretical stiffnesses q/Tra2wWper unit weight are p l o t t e d against (a) the core thickness c and (b) the face thickness t. The design stiffness for all the plates was 1.17 × 10 4 N m m -1.
2.0
40 constant at its optimum value while the other two were varied about their optima. The plots confirm t h a t the optimum design is Copt = 60 mm topt = 0.64 mm Pcopt = 308 kg m -s as predicted by the analysis presented in Section 2. The weights of the plates did n o t vary as much as might be desirable in a series of tests such as this; there were several limitations on the design of the plates which prevented this. They were the following: the span of the loading frame was limited to 381 ram; the smallest available face thickness was 0.31 mm; the core thickness had to be large relative to the face thickness and so had to be at least 30 mm; the core-thickness-to-span ratio could n o t be too large or else the plate would no longer behave as a plate (the maximum feasible core thickness was 120 mm); the available range of foam densities was between 48 and 620 kg m-3; finally, only certain face thicknesses and core densities were available. The design stiffness had to be chosen so that the corresponding optimum values of core density and of core and face thicknesses were roughly in the middle of the available ranges for each variable. The design stiffness of 1.17 X 104 N m m -1 was chosen on this basis and allowed a reasonable variation in all the design variables. It appears that the only way of obtaining greater variation in the design variables and, hence, the weight of the plates would be to test larger plates; this was considered to be n o t feasible because of the time and cost involved in building a larger loading frame and in making the plates themselves. In spite of the limited variation in the weight of the test plates, the results of the tests do confirm that the analysis can be used to find the optimum core density as well as the core and face thicknesses which minimize the weight of a sandwich panel with a foam c ore.
5. CONCLUSIONS
A new relationship between the shear modulus and density of foams based on a
recent model of the deformation of cellular solids has been used to identify the core density as well as the core and face thicknesses which minimize the weight of a foam core sandwich plate of a given bending stiffness. The results of bending tests on simply supported circular sandwich plates with aluminum skins and polyurethane foam cores confirm the results of the analysis. The work should be useful in improving the design of sandwich panels in many applications, ranging from portable buildings to racing yachts. It also has relevance to the understanding of the design of natural sandwich structures, such as those found in the human skull and in the iris leaf.
ACKNOWLEDGMENTS
We gratefully acknowledge the financial support of the U.S. A r m y Corps of Engineers Construction Engineering Research Laboratory and the Massachusetts Institute of Technology Sloan Fund. General Plastics Manufacturing Company, of Tacoma, Washington, generously supplied the polyurethane foam for the experimental portion of the work at cost.
REFERENCES
1 J. McElhaney, N. Alem and V. Roberts, A porous block model for cancellous bones, A S M E Publ. 70-WA/BHF-2, 1970, pp. 1-9 (American Society of Mechanical Engineers). 2 J. H. McElhaney, J. L. Fogle, J. W. Melvin, R. R. Haynes, V. L. Roberts and N. M. Alem, Mechanical properties of cranial bone, J. Biomech., 3 (1970) 495-511. 3 H. G. Allen, Analysis and Design o f Structural Sandwich Panels, Pergamon, Oxford, 1969. 4 S. N. Huang and D. W. Alspaugh, Minimum weight sandwich beam design, A I A A J., 12 (1974) 1617-1618. 5 C. E. S. Ueng and T. L. Liu, Least weight of a sandwich panel, in R. R. Craig (ed.), Proc. ASCE Engineering Mechanics Division 3rd Speciality Conf., University o f Texas at Austin, September 17-19, 1979, American Society of Civil Engineers, New York, 1979, pp. 41-44. 6 W. H. Wittrick, A theoretical analysis of the efficiency of sandwich construction under compressive end load, R & M 2016, 1945 (U.K. Aeronautical Research Council, Farnborough, Hants.). 7 P: Ackers, The efficiency of sandwich struts utilizing a calcium alginate core, R & M 2015,
41
8 9 10 11 12
13 14 15 16 17 18
1945 (U.K. Aeronautical Research Council, Farnborough, Hants.). E. W. Kuenzi, Minimum weight structural sandwich, U.S. For. Serv. Res. Note FPL-086, 1965 (Forest Products Laboratory, Madison, WI). S. Kelsey, R. A. Gellatly and B. W. Clark, The shear modulus of foil honeycomb cores, Aircr. Eng., 30 (1958) 294-302. J. M. Dinwoodie, Timber, its Nature and Behaviour, Van Nostrand, New York, 1981. L. J. Gibson and M. F. Ashby, The mechanics of three-dimensional cellular materials, Proc. R. Soc. London, Ser. A, 382 (1982) 43-59. S. K. Maiti, L. J. Gibson and M. F. Ashby, Deformation and energy absorption diagrams for cellular solids, Acta Metall., 32 (1984) 19631975. L. J. Gibson, Optimization of stiffness in sandwich beams with rigid foam cores, Mater. Sci. Eng., 67 (1984) 125-135. S. P. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells, McGraw-Hill, New York, 1959. R. M. Stark and R. L. Nicholls, Mathematical Foundations for Design: Civil Engineering Systems, McGraw-Hill, New York, 1972. R. M. Tennant (ed.), Science Data Book, Oliver and Boyd, Edinburgh, 1971. W.J. Roff and J. R. Scott, Fibres, Films, Plastic and Rubbers -- a Handbook of Common Polymers, Butterworths, London, 1971. M. R. Patel and I. Finnie, Structural features and mechanical properties of rigid cellular plastics, J. Mater., 5 (1970) 909-932.
APPENDIX A E x a c t expressions for the flexural rigidity D and the shear rigidity S T h e e x a c t e x p r e s s i o n f o r the flexural rigidity D is [ A l l E f t ( c + t) 2
Eft 3 D-
12(1 - Pt 2)
+
2(1 - - v~2)
Ec c3 +
12(1 - - v~2) (A1)
IfE¢ ~ E~, t h e n the third t e r m can be neglected. In a d d i t i o n , if t ~ c, the first t e r m can also be n e g l e c t e d and t h e a p p r o x i m a t e e x p r e s s i o n f o r D becomes Eftc 2
D ~
2(1
-
-
Vf2)
(A2)
T h e e x a c t e x p r e s s i o n f o r the shear rigidity S is [ A 1 ] (C + t) 2 S -
c
Gc
(A3)
This r e d u c e s to S = cG¢
(A4)
ift~c.
R e f e r e n c e for A p p e n d i x A A1 H. G. Allen, Analysis and Design of Structural Sandwich Panels, Pergamon, Oxford, 1969.
APPENDIX B: NOMENCLATURE radius o f the central, u n i f o r m l y distrib u t e d load (m) s a n d w i c h p l a t e c o r e t h i c k n e s s (m) c constant of proportionality between relative Y o u n g ' s m o d u l u s and relative d e n s i t y f o r a f o a m (--) Cg c o n s t a n t of p r o p o r t i o n a l i t y b e t w e e n relative shear m o d u l u s a n d relative density f o r a f o a m (--) E f t c 2 / { 2(1 -- pf2 )}, flexural rigidity o f D the sandwich p l a t e (N m ) Y o u n g ' s m o d u l u s o f a f o a m (N m -2) E Y o u n g ' s m o d u l u s o f t h e f o a m e d core m a t e r i a l (N m -2 ) E~ Y o u n g ' s m o d u l u s o f the face m a t e r i a l (N m -2) Es Y o u n g ' s m o d u l u s o f the solid u n f o a m e d cell wall m a t e r i a l in a f o a m (N m -2) gl f u n c t i o n of r/a related to p l a t e b e n d i n g d e f l e c t i o n [--] g2 f u n c t i o n o f r/a r e l a t e d t o p l a t e shearing d e f l e c t i o n [--] G shear m o d u l u s o f a f o a m (N m -2) G c shear m o d u l u s o f the f o a m e d core material (N m -2) M mass o f the s a n d w i c h p l a t e (kg) n e x p o n e n t in m o d u l u s - d e n s i t y relationship f o r f o a m s (--) q u n i f o r m l y d i s t r i b u t e d load on the p l a t e (N m -2 ) r radius of t h e s a n d w i c h p l a t e ( m ) S cGc, shear rigidity o f the s a n d w i c h p l a t e (N m -1 ) t t h i c k n e s s o f each face o f t h e s a n d w i c h p l a t e (m) w t o t a l d e f l e c t i o n o f the s a n d w i c h p l a t e (m) Wb b e n d i n g c o m p o n e n t o f d e f l e c t i o n o f the sandwich plate (m) a
42 We W
shearing c o m p o n e n t of deflection of the sandwich plate (m) weight of the sandwich plate (N)
Greek symbols vc
Poisson's ratio of the foamed core material (--)
v~ p pc pf Pe
Poisson's ratio of the face material (--) density of a foam (kg m -3) density of the foam core in a sandwich plate (kg m -s) density of the face material (kg m -3) density of the solid unfoamed cell wall material in a foam (kg m -3)
33
Minimum Weight Design for Stiffness in Sandwich Plates with Rigid Foam Cores L. A. DEMSETZ and L. J. GIBSON
Department of Civil Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139 U.S.A.) (Received February 3, 1986)
ABSTRACT
A new relationship between the shear modulus and density o f foams has been used to identify the core density as well as the core and face thicknesses which minimize the weight o f a sandwich plate for a given bending stiffness. The results o f bending tests on simply supported circular sandwich plates with alum i n i m u m skins and polyurethane foam cores confirm the results o f the analysis. The work should be useful in improving the design o f sandwich panels in many engineering applications. It also has relevance to the understanding o f the design o f natural sandwich structures, such as those found in the human skull and in the iris leaf.
1. INTRODUCTION Composite structural panels made up of two stiff strong faces separated by a lightweight core are known as sandwich panels. Separation of the faces by the core increases the m o m e n t of inertia of the panel with little increase in weight, giving an efficient member for resisting bending and buckling. For this reason, they are often used in applications where the weight of the component is critical: in structural components in aircraft, in portable buildings, in the decks and hulls of racing yachts and in modern downhill skis. Sandwich structures are found in nature, too, e.g. in the human skull and the iris leaf. In the skull, two thin outer layers of dense compact bone are separated by a lightweight core of sponge-like cancellous bone [1, 2]; in the iris leaf, the almost fully dense and relatively stiff fibre-composite-like ribs on the outer skin of the leaf are separated by weak low density cells in the core. 0025-5416/87/$3.50
Many materials can be used in a sandwich construction; usually, however, the faces are made of either aluminum or fibre-reinforced composites, while the cores are aluminum or paper-phenolic honeycombs or polymeric foams. H o n e y c o m b cores are used in applications where the weight of the panel is the only critical factor, e.g. in structural components of aircraft. Foam cores are usually preferred in applications where a low thermal conductivity is required in addition to a low weight, e.g. in portable buildings, mobile homes and refrigerated shipping containers. In this study, only sandwich panels with foam cores are investigated. Optimization analyses of sandwich plates aim to develop a m e t h o d of designing the lightest possible plate for some specified structural requirement. The requirement may be that the panel must have some given stiffness or strength, or a given combination of stiffness and strength. There may be dimensional constraints, too; the face and core may have to be within certain size limitations determined by the availability of materials, or the depth of the panel may have to be less than some m a x i m u m allowable dimension. Previous optimization studies have usually focused on determining the core and face thicknesses which minimize the weight of a panel assuming that the core and face materials are completely specified and that their densities, elastic moduli and strengths are known. Allen [3] shows how this can be done for a sandwich beam subject to a stiffness constraint. Huang and Alspaugh [4] and Ueng and Liu [5] have optimized the core and face thicknesses for more highly constrained members. Huang and Alspaugh minimized the weight of a sandwich beam subject to the conditions of maximum allowable bending stress, shear stress and deflection, as well as © Elsevier Sequoia/Printed in The Netherlands
34 minimum face and core thicknesses. Ueng and Liu solved a similar problem for a specific type of corrugated core. Because of the complexity of the constraints, the solutions for both of these problems were obtained using numerical techniques, as no analytical solution was possible. Little attention has been paid to optimizing the core density of sandwich panels to minimize their weight. Wittrick [6] and Ackers [ 7], working on sandwich beams with foamed calcium alginate cores, and Kuenzi [8], working on the use of wood cores in sandwich plates, all recognized that, to include the core density as a design variable in the optimization analysis, relationships between core properties and core density must be known. The tests of Wittrick and Ackers on foamed calcium alginate and Kuenzi's tests on wood showed that moduli and strength were linearly related to density for both of these materials. All these researchers then used this linear relationship in their analyses. A recent study of the mechanical behavior of cellular solids has shown that, while these linear relationships are valid for h o n e y c o m b cores loaded transversely in shear [9] and for wood loaded along the grain [10], they are not, in general, valid for foamed materials [11, 12]. By identifying and modelling the mechanisms of deformation in cellular solids, Gibson, Maiti and Ashby found that foam properties are related to three parameters: a constant related to the cell geometry, a property of the cell wall material and the relative density of the foam raised to some power. In particular, they found that Young's modulus and the shear modulus of rigid foams are related to their relative density squared and that the compressive, tensile and shear strengths are related to the related density raised to the power 3/2. Gibson [13] has recently used these new non-linear relationships between foam properties and foam density to find the core density as well as the core and face thicknesses which minimize the weight of a foam core sandwich beam subject to a stiffness constraint. In this paper the continuation of that work to the optimum design of sandwich plates with rigid foam cores is described. In it we develop the closed-form solution for the core density and the core and face thicknesses which minimize the weight of a circular sandwich plate subject to a stiffness constraint. The
results of tests on a series of sandwich plates of nominally the same stiffnesses but of different designs and weights confirm the analysis. The m e t h o d described should allow the design of sandwich plates with foam cores to be improved by optimizing the core density as well as the core and face thicknesses.
2. A N A L Y S I S
The sandwich plate to be optimized is shown in Fig. 1. It is a simply supported circular plate which spans a radius r and carries a distributed load q over a central circular portion of its area of radius a. The thickness of each face is t, while that of the foam core is c. The faces have a density pf, Young's modulus E~ and Poisson's ratio v~. The solid polymer of which the core is made has a density Ps and a modulus E s. The density of the foamed core is Pc, its Young's modulus is E c and its shear modulus is Gc. The analysis is to develop a technique for choosing the values of core density Pc, and core and face thicknesses c and t, which minimize the weight of the plate for a given stiffness. It is assumed that the material properties of the faces and of the solid polymer from which the core is foamed are known.
2o FACE pf ,Ef,~lf\
~ •
q o
.
• FOAM CORE ~P~'Ec'G c
.
.
.
o
.
c
Fig. 1. S a n d w i c h plate g e o m e t r y : a simply s u p p o r t e d circular s a n d w i c h plate o f radius r carrying a distrib u t e d load q over a central circular p o r t i o n o f its area o f radius a.
35
The properties of foams depend on three parameters: the geometry of the cells, the properties of the solid cell wall material and the relative density of the foam, i.e. the density of the foam normalized by that of the solid from which it is made. It has long been recognized that the properties of foams depend on their density raised to some power; recently, Gibson and Ashby [ 11] and Maiti e t al. [12] have modelled the deformation mechanisms in foams to estimate this power for m a n y properties. At small strains (less than about 5%), the cell walls deform mainly by bending; it can easily be shown that, if this is the case, then Young's modulus and the shear modulus of an open cell foam are related to the square of its relative density: ~
~cc
(1)
L)
(2)
while Poisson's ratio is independent of relative density and depends only on the geometry of the cells in the foam. These simple square laws have been found to describe Young's modulus and the shear modulus of m a n y foams reasonably well. In some cases, there is some deviation in the power from 2; this may be explained by several factors. If the cells are highly anisotropic, then axial deformations become more important for loading along the elongated direction of the cells; this tends to reduce the exponent towards a limiting value of 1. If isotropic cells are closed and have uniformly thick faces, then bending deformations in the cell faces lead to an exponent of 3; usually, however, surface tension draws most of the material away from the faces and into the edges of the cell during the foaming process, so that most closed-cell foams behave more like open-cell foams. Nevertheless, as the faces of closed-cell foams become more uniform, the exponent of relative density for the moduli tends towards an upper limit of 3. There is another source of deviation from the squared rule, too; if the nodes where cell edges intersect are enlarged, then the moduli should tend to be less sensitive to relative density, as less of the solid is subject to bending deformations. In general, then, we can write
E = C~
E~
(3)
G = Cg
Es
(4)
where n is about 2. In the analysis that follows, the shear modulus-relative density relationship will be used to introduce the core density into the constraint equation and thus into the optimization analysis. The mass of the sandwich plate to be optimized is given by (5)
M = 2 p f ~ r 2 t + pcTrr2c
This is the objective function which the analysis aims to minimize. The stiffness of the sandwich plate is the ratio of the applied load to its resulting midspan deflection. This deflection has both bending and shearing components; shear deflections, which are negligible in an isotropic plate, are significant in a sandwich plate because of the low shear modulus of the core. The midspan bending deflection is given by [3, 14] qa 4 wb - 16D gl
(6)
where gl
=,,,(/)' (/) +
1 + vf
in
--
4(1 + v~)
and Eftc 2
D~
2(1 _~f2)
The midspan shear deflection is given by [3] qa 2 Ws : ~ - g2
(7)
where g2=1 + 21n(r)
\a /
and S ~ cGc
The exact expressions for D and S are given in Appendix A. The approximations given here are valid if Young's modulus of the core is very much less than that of the face and if the face thickness is very much less than the
36 core thickness. Since this generally is the case for sandwich plates, the simpler approximate expressions for D and S are used in the analysis. Summing wb and w~ to get the total deflection w, and dividing by the uniformly distributed load gives the inverse of the stiffness:
W q
a4(1 -- Vf2) a2 8E~tc 2 gl + 4c~-~g2
-
(8)
Replacing the shear modulus of the core by eqn. (4) introduces the core density into the stiffness constraint equation. Solving the stiffness equation for the relative density of the core gives
P_A - [ a2g2 I w Ps - L 4CgcE~ [ q
a4(l_--~v~2)gl~-l~ vn 8E~tc 2 J J
(9)
Substituting this equation for relative density into the equation for the mass of the plate (eqn. (5)) and solving for aM~at = 0 and aM/ac = 0, we find the optimum solution for the plate design. It is .
( 1 (n + 1 ) ~ ( n - - 1) 2 Ps~ 256-C~ a n ( 1 - - P f 2 )
Pcopt
×
Co~
g2__3 Ps Ef (~)2}1/(3n-1) gl P~ Es a
(10)
__ a S4n+lc., n " ( n + 1 ) " - 1 ( 1 - v ~ 2 ) n 2( g (n -- i) 2~
g l n (~-~S] p ~ n ~n(~)n Es - 1}1](3n_ 1) X-~2
(11)
and if both second derivatives are greater than zero,
a2M -->0
at 2
a2M -->0 ac 2 This is the case for the mass objective function so that eqns. (10)-(12) describe the global optimum. For the optimum solution, it can be shown that the ratio of the weight of both of the faces to that of the core is always ( n - 1)/2n, independent of the given stiffness, the span, the radius of the load or the materials used. In addition, the ratio of the bending to the shearing deflections is always (n -- 1)/2, again independent of the given stiffness, the span, the radius of the load or the materials used. For m a n y foams, n -- 2; in this case, the faces weigh one-quarter of the weight of the core and the deflections arising from bending are one-half of those from shearing. Both of these ratios are identical with those found for the optimum design of a sandwich beam subject to a stiffness constraint [13]. If n -= 1 the solution becomes unrealistic; the face thickness and core density become equal to zero while the core thickness tends to infinity. In this case, a side constraint on the maximum allowable core thickness would control the optimum design, and a new analysis taking this side constraint into account would have to be performed to find it.
1 (n 2 -- 1)n+l(1--v~2) "-1 to~ = 4a 213.+1C~2
n2n
X
gl n-1 g22~-f/
--~-s2 ~ - - ]
3. EXPERIMENTAL METHOD ) (12)
It has been shown [15] that the optimum obtained by setting derivatives equal to zero is a global o p t i m u m if the objective function is convex, i.e. if the determinant of its hessian matrix is greater than zero,
a2M
a2M
~t 2
at ac
~2M
a2M
ac at
ac 2
T0
To test the analysis developed above, a series of sandwich plates with aluminum skins and polyurethane foam cores were made, each having the same design stiffnesses but with design parameters (the core density and the core and face thicknesses) that were varied about the optima found from the analysis. The density and moduli of the aluminum and the solid polyurethane from which the core was foamed are given in Table 1. Before the plates were designed, the shear moduli of the various densities of polyurethane foam to be used in the cores were
37 TABLE 1 Properties of aluminum and solid unfoamed polyurethane D e n s i t y (kg m -a)
Material
Aluminum Solid unfoamed polyurethane
I /
/
~
Pf = 2700 [16] ps = 1200 [17]
Young's modulus
Poisson's ratio
(GN m -2)
(--)
Ef = 70 [16] Es= 1.6 [18]
0.334 [16] --
100 P
~
I
ALUMINUM BAR~
(D
g o rr <~ 10-' 03
J w 8[
Fig. 2. Jig for measuring the shear modulus of the foam to be used in the cores of the sandwich plates.
m e a s u r e d using the shear jig s h o w n in Fig. 2. Because t h e m o d u l u s o f the f o a m is v e r y m u c h less t h a n t h a t o f the a l u m i n u m sidepieces, t h e f o a m is s u b j e c t e d to a l m o s t p u r e shear stress. T h e results o f these tests s h o w e d t h a t t h e shear m o d u l u s o f the p o l y u r e t h a n e f o a m was r e l a t e d to its relative d e n s i t y b y
iO-l iOo RELATIVE DENSITYD/fls(-)
Fig. 3. Relative shear modulus G/E s plotted against relative density PIPs for the foams used in the cores of the sandwich plates. The data fall on the line G/E s = 0 . 5 4 ( p / p s
)1.6.
M = 2.51 kg
copt = 60 m m
T h r e e series o f plates were designed. In e a c h series, o n e o f the design variables was held c o n s t a n t at its o p t i m u m value while the o t h e r t w o variables were varied a b o u t t h e i r o p t i m a to give plates of c o n s t a n t stiffness b u t differe n t weights. U n f o r t u n a t e l y , the 3 0 8 kg m -3 f o a m was n o t available at the t i m e the plates were m a d e a n d so, instead, 320 kg m -s f o a m was s u b s t i t u t e d . T h e " o p t i m u m " plate for the t e s t series was t h e n slightly d i f f e r e n t f r o m the t r u e o p t i m u m d e s c r i b e d above. T o m a i n t a i n t h e design stiffness o f 1.17 X 104 N m m -1, it h a d
tom = 0.64 m m
Cop t =
Pcopt = 3 0 8 kg m -3
tom = 0.64 m m
/p \t.6
Go = o.54( )
Zs
(13)
\Ps / as s h o w n in Fig. 3 a n d in a g r e e m e n t w i t h the general r e l a t i o n s h i p o f eqn. (4). T h e plates were designed to have a stiffness o f 1.17 × 104 N m m -z, a span o f 381 m m a n d a c e n t r a l load d i a m e t e r o f 51 m m . F o r this p a r t i c u l a r l o a d i n g c o n f i g u r a t i o n the o p t i m u m design f o r a s a n d w i c h p l a t e was f o u n d t o be
58
mm
38 Pcopt = 320 kg m -3 M = 2.52 kg As its mass is only 0.01 kg greater than the true o p t i m u m design, this substitution was considered to be acceptable. The designs of all the plates tested are listed in Table 2. It was initially planned to test five plates for each of five different designs for each test series; unfortunately, the shortage of foam available from the supplier reduced both the number of plate designs in each test series and the number of specimens of each design in some cases. In the final set of experiments, four different plate designs were used for each test series, and between three and six specimens of each design were tested. The plate specimens were made by bonding two aluminum faces to a precut polyurethane foam core with polyester resin. The thickness of each face and core was measured before being bonded. The foam cores were cut to within 0.5 mm of their design values. The face thicknesses were designed to be standard gauge thicknesses for aluminum and so were
within +0.01 mm of their design values. The foam cores were weighed individually to measure the density of the foam used in each plate. The core densities were within 5% of the design values for all the plates except P1 which was within 10%. In view of the variability in foam density caused by the foaming process itself, these deviations in foam density from the design values are thought to be acceptable. Each face was sand blasted prior to being glued and the specimens were clamped for 24 h while the resin hardened to ensure a good bond. Each plate was weighed before testing. To measure the stiffness of the panels, each one was simply supported on eight cylindrical rollers 3 in long arranged in a circular pattern. A steel plate ¼ in thick, 3 in long and 1 in wide was placed on top of each roller to distribute the load at the support and to prevent the foam core from crushing. The load was applied with a 60 000 lb Baldwin hydraulictesting machine and measured with a pressure gauge connected to an X - Y recorder. A piece of polyurethane foam 51 mm in diameter was
TABLE 2 Plate b e n d i n g test results (design stiffness, 1.17 X 104 N m m -1)
Plate design identification
P9 P2 P10 Pll P1 P2 P3 P4 P6 P2 P7 P8
Number o f specimens tested
5 5 5 6 6 5 6 3 3c 5 4e 4
Design core thickness c
Design face thickness t
Design core density Pc ( k g
Measured plate stiffness q/Tra2w
(mm)
(ram)
m -3)
(X 104 N m m -1)
60 58 60 60 121 58 50 37 65 58 55 52
0.31 0.64 0.81 1.025 0.64 0.64 0.64 0.64 0.41 0.64 1.27 2.03
386 320 a 296 284 175 320 381 618 313 d 320 296 296
1.25 1.37 1.37 1.24 1.28 1.37 1.36 1.14 1.19 1.37 1.46 1,37
Weight W (N) Theoretical
27.8 24.7 24.8 25.2 27.4 24.6 25.1 29.6 25.2 24.6 25.9 29.5
Stiffness q/Tra2wW per unit weight Experimental
27.2 25.4 26.0 25.0 25.0 25.4 25.9 28.3 24.4 25.4 27.7 30.5
( m m -1 )
Theoretical
Experimental
420 473 472 465 426 476 466 395 465 476 452 397
460 539 527 496 512 539 525 403 488 539 527 449
a T h e original o p t i m u m f o a m d e n s i t y of 308 kg m -3 was r e p l a c e d b y f o a m o f d e n s i t y 320 kg m -3 because of t h e unavailability o f t h e 308 kg m -3 f o a m . b T h e m e a s u r e d t h i c k n e s s was 0.94 ram. c T w o o f these t h r e e s p e c i m e n s d e b o n d e d d u r i n g t e s t i n g a n d were n o t i n c l u d e d in t h e test results. d T h e original o p t i m u m f o a m d e n s i t y o f 308 kg m - a w a s replaced b y o t h e r densities as close to 308 kg m -3 as possible. e One o f these f o u r s p e c i m e n s d e b o n d e d d u r i n g testing a n d was n o t i n c l u d e d in t h e test results.
39
placed between the test machine platen and the plate specimen to distribute the load uniformly while the plate deflected. A d.c. differential transformer was placed under the centre of the plate to measure its midspan displacement. Dial gauges were placed on top of the plate over each support to check that the plate was not crushed at the supports; these deflections were found to be negligible. The stiffness of each plate was then calculated from the slope of the load-deflection curve plotted on an X- Y recorder. The stiffness of the supports was also measured; this was found to change the measured plate stiffnesses by about 5% and was taken into account in calculating the true plate stiffnesses.
600 o_
I _x- -x i ~x
f
-
IX
400 I _
tOp i
Pc
opt
u_ on
0
I 0
I
J
0.5 1.0 FACE THICKNESS t (ram)
I5
(a)
(b)
l(~O 2;0 5l~0 CORE DENSITY P c ( k g / m 3 )
400
Fig. 4. Plate bending test results for s a n d w i c h plates with the core thickness held c o n s t a n t at the o p t i m u m value o f 60 ram. The measured and theoretical stiffnesses q/Tm2wWper unit weight are p l o t t e d against (a) the face thickness t and (b) the core density Pc. The design stiffness for all the plates was 1.17 × 104 N m m -1.
4. R E S U L T S A N D D I S C U S S I O N
The designs of all the plates tested are listed in Table 2 together with the results of the plate stiffness tests. Deviations from the design value of the density of the foam used in the plates caused some variation in the theoretical stiffness of the plates. Revised theoretical stiffnesses were calculated on the basis of the actual foam densities used in the plate cores. The average revised stiffness was 1.17 × 1 0 4 N mm -1, in agreement with the original design value. The coefficient of variation for the revised theoretical plate stiffnesses was 7%, of the order that might be expected from the variation of about 5% from the design values of the foam densities. The measured values of plate stiffnesses, corrected for the support stiffness, are listed in Table 2 together with the theoretical and measured weights of the plates and the theoretical and m~asured stiffness per unit weight. The average measured stiffness of the plates was 1.31 X 1 0 4 N mm -1, 12% higher than the design value of 1.17 X 104 N mm -1. The coefficient of variation in the measured plate stiffnesses was 9%, again of the order that might be expected from the variation in the revised theoretical plate stiffness. Figures 4-6 show the theoretical and measured values of the stiffness per unit weight for all the plates plotted against the design variables of core thickness c, face thickness t and the core density Pc. Each pair of plots shows the results from one series of tests in which one of the design variables was held
6O0
x
I
,x.X-- - - _
I
,X
.x ~ - -
X--x
~ ~ ~oo i
~ ~2oo
Copl
Pc opt
u_ u_ m
0
(a)
510 I 0J0 CORE THICKNESS c(mm)
1 5I 0 0 (b)
200 400 600 CORE DENSITY ~(kg/rn ~1
800
Fig. 5. Plate bending test results for s a n d w i c h plates with the face thickness t held c o n s t a n t at the optim u m value o f 0 . 6 4 ram. The measured and theoretical stiffnesses q/Tra2wWper unit weight are p l o t t e d against (a) the core thickness c and (b) the core density Pc. The design stiffness for all the plates was 1.17 X 1 0 4 N m m -1.
600
/ x 'x
%
xl~l x~
X ~
~oo
z
a.%
~ zoo
m
C
O0
[a)
opt
40 60 CORE THICKNESS c(mm)
20 l
top t
I
80 (b)
I
0,5 I0 I!5 FACE THICKNESS timm)
Fig. 6. Plate bending test results for s a n d w i c h plates with the core d e n s i t y Pc held constant at the optim u m value o f 3 2 0 kg m -3. The measured and theoretical stiffnesses q/Tra2wWper unit weight are p l o t t e d against (a) the core thickness c and (b) the face thickness t. The design stiffness for all the plates was 1.17 × 10 4 N m m -1.
2.0
40 constant at its optimum value while the other two were varied about their optima. The plots confirm t h a t the optimum design is Copt = 60 mm topt = 0.64 mm Pcopt = 308 kg m -s as predicted by the analysis presented in Section 2. The weights of the plates did n o t vary as much as might be desirable in a series of tests such as this; there were several limitations on the design of the plates which prevented this. They were the following: the span of the loading frame was limited to 381 ram; the smallest available face thickness was 0.31 mm; the core thickness had to be large relative to the face thickness and so had to be at least 30 mm; the core-thickness-to-span ratio could n o t be too large or else the plate would no longer behave as a plate (the maximum feasible core thickness was 120 mm); the available range of foam densities was between 48 and 620 kg m-3; finally, only certain face thicknesses and core densities were available. The design stiffness had to be chosen so that the corresponding optimum values of core density and of core and face thicknesses were roughly in the middle of the available ranges for each variable. The design stiffness of 1.17 X 104 N m m -1 was chosen on this basis and allowed a reasonable variation in all the design variables. It appears that the only way of obtaining greater variation in the design variables and, hence, the weight of the plates would be to test larger plates; this was considered to be n o t feasible because of the time and cost involved in building a larger loading frame and in making the plates themselves. In spite of the limited variation in the weight of the test plates, the results of the tests do confirm that the analysis can be used to find the optimum core density as well as the core and face thicknesses which minimize the weight of a sandwich panel with a foam c ore.
5. CONCLUSIONS
A new relationship between the shear modulus and density of foams based on a
recent model of the deformation of cellular solids has been used to identify the core density as well as the core and face thicknesses which minimize the weight of a foam core sandwich plate of a given bending stiffness. The results of bending tests on simply supported circular sandwich plates with aluminum skins and polyurethane foam cores confirm the results of the analysis. The work should be useful in improving the design of sandwich panels in many applications, ranging from portable buildings to racing yachts. It also has relevance to the understanding of the design of natural sandwich structures, such as those found in the human skull and in the iris leaf.
ACKNOWLEDGMENTS
We gratefully acknowledge the financial support of the U.S. A r m y Corps of Engineers Construction Engineering Research Laboratory and the Massachusetts Institute of Technology Sloan Fund. General Plastics Manufacturing Company, of Tacoma, Washington, generously supplied the polyurethane foam for the experimental portion of the work at cost.
REFERENCES
1 J. McElhaney, N. Alem and V. Roberts, A porous block model for cancellous bones, A S M E Publ. 70-WA/BHF-2, 1970, pp. 1-9 (American Society of Mechanical Engineers). 2 J. H. McElhaney, J. L. Fogle, J. W. Melvin, R. R. Haynes, V. L. Roberts and N. M. Alem, Mechanical properties of cranial bone, J. Biomech., 3 (1970) 495-511. 3 H. G. Allen, Analysis and Design o f Structural Sandwich Panels, Pergamon, Oxford, 1969. 4 S. N. Huang and D. W. Alspaugh, Minimum weight sandwich beam design, A I A A J., 12 (1974) 1617-1618. 5 C. E. S. Ueng and T. L. Liu, Least weight of a sandwich panel, in R. R. Craig (ed.), Proc. ASCE Engineering Mechanics Division 3rd Speciality Conf., University o f Texas at Austin, September 17-19, 1979, American Society of Civil Engineers, New York, 1979, pp. 41-44. 6 W. H. Wittrick, A theoretical analysis of the efficiency of sandwich construction under compressive end load, R & M 2016, 1945 (U.K. Aeronautical Research Council, Farnborough, Hants.). 7 P: Ackers, The efficiency of sandwich struts utilizing a calcium alginate core, R & M 2015,
41
8 9 10 11 12
13 14 15 16 17 18
1945 (U.K. Aeronautical Research Council, Farnborough, Hants.). E. W. Kuenzi, Minimum weight structural sandwich, U.S. For. Serv. Res. Note FPL-086, 1965 (Forest Products Laboratory, Madison, WI). S. Kelsey, R. A. Gellatly and B. W. Clark, The shear modulus of foil honeycomb cores, Aircr. Eng., 30 (1958) 294-302. J. M. Dinwoodie, Timber, its Nature and Behaviour, Van Nostrand, New York, 1981. L. J. Gibson and M. F. Ashby, The mechanics of three-dimensional cellular materials, Proc. R. Soc. London, Ser. A, 382 (1982) 43-59. S. K. Maiti, L. J. Gibson and M. F. Ashby, Deformation and energy absorption diagrams for cellular solids, Acta Metall., 32 (1984) 19631975. L. J. Gibson, Optimization of stiffness in sandwich beams with rigid foam cores, Mater. Sci. Eng., 67 (1984) 125-135. S. P. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells, McGraw-Hill, New York, 1959. R. M. Stark and R. L. Nicholls, Mathematical Foundations for Design: Civil Engineering Systems, McGraw-Hill, New York, 1972. R. M. Tennant (ed.), Science Data Book, Oliver and Boyd, Edinburgh, 1971. W.J. Roff and J. R. Scott, Fibres, Films, Plastic and Rubbers -- a Handbook of Common Polymers, Butterworths, London, 1971. M. R. Patel and I. Finnie, Structural features and mechanical properties of rigid cellular plastics, J. Mater., 5 (1970) 909-932.
APPENDIX A E x a c t expressions for the flexural rigidity D and the shear rigidity S T h e e x a c t e x p r e s s i o n f o r the flexural rigidity D is [ A l l E f t ( c + t) 2
Eft 3 D-
12(1 - Pt 2)
+
2(1 - - v~2)
Ec c3 +
12(1 - - v~2) (A1)
IfE¢ ~ E~, t h e n the third t e r m can be neglected. In a d d i t i o n , if t ~ c, the first t e r m can also be n e g l e c t e d and t h e a p p r o x i m a t e e x p r e s s i o n f o r D becomes Eftc 2
D ~
2(1
-
-
Vf2)
(A2)
T h e e x a c t e x p r e s s i o n f o r the shear rigidity S is [ A 1 ] (C + t) 2 S -
c
Gc
(A3)
This r e d u c e s to S = cG¢
(A4)
ift~c.
R e f e r e n c e for A p p e n d i x A A1 H. G. Allen, Analysis and Design of Structural Sandwich Panels, Pergamon, Oxford, 1969.
APPENDIX B: NOMENCLATURE radius o f the central, u n i f o r m l y distrib u t e d load (m) s a n d w i c h p l a t e c o r e t h i c k n e s s (m) c constant of proportionality between relative Y o u n g ' s m o d u l u s and relative d e n s i t y f o r a f o a m (--) Cg c o n s t a n t of p r o p o r t i o n a l i t y b e t w e e n relative shear m o d u l u s a n d relative density f o r a f o a m (--) E f t c 2 / { 2(1 -- pf2 )}, flexural rigidity o f D the sandwich p l a t e (N m ) Y o u n g ' s m o d u l u s o f a f o a m (N m -2) E Y o u n g ' s m o d u l u s o f t h e f o a m e d core m a t e r i a l (N m -2 ) E~ Y o u n g ' s m o d u l u s o f the face m a t e r i a l (N m -2) Es Y o u n g ' s m o d u l u s o f the solid u n f o a m e d cell wall m a t e r i a l in a f o a m (N m -2) gl f u n c t i o n of r/a related to p l a t e b e n d i n g d e f l e c t i o n [--] g2 f u n c t i o n o f r/a r e l a t e d t o p l a t e shearing d e f l e c t i o n [--] G shear m o d u l u s o f a f o a m (N m -2) G c shear m o d u l u s o f the f o a m e d core material (N m -2) M mass o f the s a n d w i c h p l a t e (kg) n e x p o n e n t in m o d u l u s - d e n s i t y relationship f o r f o a m s (--) q u n i f o r m l y d i s t r i b u t e d load on the p l a t e (N m -2 ) r radius of t h e s a n d w i c h p l a t e ( m ) S cGc, shear rigidity o f the s a n d w i c h p l a t e (N m -1 ) t t h i c k n e s s o f each face o f t h e s a n d w i c h p l a t e (m) w t o t a l d e f l e c t i o n o f the s a n d w i c h p l a t e (m) Wb b e n d i n g c o m p o n e n t o f d e f l e c t i o n o f the sandwich plate (m) a
42 We W
shearing c o m p o n e n t of deflection of the sandwich plate (m) weight of the sandwich plate (N)
Greek symbols vc
Poisson's ratio of the foamed core material (--)
v~ p pc pf Pe
Poisson's ratio of the face material (--) density of a foam (kg m -3) density of the foam core in a sandwich plate (kg m -s) density of the face material (kg m -3) density of the solid unfoamed cell wall material in a foam (kg m -3)