Materials Science and Engineering, 67 (1984) 125135
125
Optimization of Stiffness in Sandwich Beams with Rigid Foam Cores L. J. GIBSON Department of Civil Engineering, University of British Columbia, Vancouver, British Columbia V6T 1 W5 (Canada) (Received October 5, 1983; in revised form February 6, 1984)
SUMMARY
A new m e t h o d for maximizing stiffness per unit weight in sandwich beams with foam cores is described. O p t i m u m values o f core thickness, face thickness and core density are obtained from the analysis. Measurements of the stiffness per unit weight have been made on sandwiches with foamed polyurethane cores. The theoretical analysis is in good agreement with the results o f these tests.
1. INTRODUCTION
1.1. Scope of paper Structural members made up of two thin stiff skins separated by a weak lightweight core are known as sandwich panels. Common skin, or face, materials are aluminium, wood and fibrereinforced plastics while balsa, honeycombs and foams are used in the core. The core serves as a filler to separate the stiff faces, increasing the m o m e n t I of inertia of the panel. This increase in the m o m e n t of inertia gives sandwich panels a high ratio of bending stiffness to weight. Because of this, they are used in applications where weight is critical, e.g. as structural components in aircraft. Sandwich construction is even found in nature where mechanical design is often optimized, as for example in the human skull where two outer layers of dense bone axe separated by a lightweight core of spongelike cancellous bone. The bending stiffness of a sandwich beam depends on two factors: its flexural rigidity and its shear rigidity. This is different from isotropic beams where shear deflections are usually negligible. The flexural rigidity is governed by the stiffness of the faces and their separation and the shear rigidity by the shear modulus and area of the core. To obtain 00255416]84]$3.00
the sandwich effect, the core must have sufficient shear stiffness to prevent the faces from sliding over one another. Recent work on the mechanical properties of foams has shown that their shear moduli are proportional to the square of their density [1]. With this knowledge, the bending stiffness of a sandwich beam with a foam core can be related to the core density as well as to the core and face thicknesses. This is new: because the modulidensity relationship for foams was not well understood, in previous optimization studies it has not been possible to minimize weight sandwich beam stiffness with respect to core density. In this paper an analytical method for finding the values of the core thickness, face thickness and core density which minimize the weight of a foam core sandwich beam of given stiffness and span length is described. Measurements of the stiffness per unit weight have been made on sandwich beams with foamed polyurethane cores. The theoretical analysis is in good agreement with the results of these tests.
1.2. Previous work Optimization analyses in sandwich beams minimize the weight, subject to one or more conditions. The condition may be that the beam has to have a given stiffness or strength, or some combination of stiffness and strength. There may be dimensional constraints also: the faces and core may have to be within certain size limitations determined by the availability of materials. Most of the work done until now has been aimed at finding the values of face and core thicknesses which minimize weight; in these studies it has been assumed that the face and core materials are completely specified and that their densities, moduli and strengths are known. Huang and Alspaugh [2] and Ueng and Liu [3] have © Elsevier Sequoia/Printed in The Netherlands
126
done this for highly constrained beams. Huang and Alspaugh minimized the weight subject to the conditions of maximum allowable bending stress, shear stress and deflection, as well as 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 by numerical techniques, and no analytical solution was possible. The goal of this paper is to minimize the weight not only with respect to the core and face thicknesses but also with respect to the core density. Wittrick [4] and Ackers [5], working on sandwich beams with foamed calcium alginate cores, both recognized that there is an o p t i m u m core density (and so stiffness) which minimizes the weight. Their interest was in minimizing the weight for a given strength of beam. They considered three possible failure modes: Euler buckling, skin wrinkling and face yielding. The first t w o of these are related to the stiffness of the core material. Tests on foamed calcium alginate showed that its Young's modulus varied in a roughly linear relationship with density. From this, Wittrick assumed that the shear modulus also varied linearly with density. By substituting these expressions into the equations for failure and requiring Euler failure and skin wrinkling to occur simultaneously, Wittrick was able to develop expressions for core density as well as for core and face thickness which minimized the weight for a given required strength. Kuenzi [6], working on the use of w o o d cores in a sandwich construction at the U.S. Forest Products Laboratory, has made a similar analysis. He optimized sandwich design for three separate cases: a given equivalent flexural rigidity requirement; a given bending m o m e n t capacity (strength) requirement; a given buckling resistance. In the last t w o cases he assumed, like Wittrick, that moduli are related linearly to density. This is a wellknown relationship for Young's modulus of w o o d loaded parallel to the grain [7]. A recent extensive study into the mechanics of cellular solids such as foams and woods has shown that these linear modulidensity relationships are n o t generally valid [1, 8, 9]. Instead it was found that, for foams, Young's modulus and the shear modulus are propor
tional to the density squared. Young's modulus for w o o d loaded parallel to the grain was found to be directly proportional to the density, in agreement with previous observations. However, Young's modulus for wood loaded across the grain was more sensitive to density than this; it varied as the density raised to a b o u t the power 5/2. The mechanisms of deformation within the cells which gave rise to these different dependences on density have been analysed and explained in detail in previous papers. Briefly, moduli are linearly related to density when the cell walls deform axially, but they are related to the density squared or cubed (depending on the symmetry of the cell structure) when the cell walls bend. Similar relationships exist for the strength of cellular materials. Using the fact that the moduli is related to the density squared in foams, we can n o w minimize the weight of a foam core sandwich with respect to core density as well as with respect to the core and face thicknesses.
2. ANALYSIS
The sandwich beam to be optimized is shown in Fig. 1. The face material has a density pf and Young's modulus E~, and the corresponding parameters for the unfoamed solid core material are Ps and Es; the values of these parameters are known. The foamed core has a density Pc which is to be determined in the optimization analysis. Gibson and Ashby [1 ] have found that core moduli and density are related by
Ec= CE\ps/ (P~)2Es
(1)
and
\Ps /
FACE p,,E,..~ '.
FOAM:CORE p ; : E i , 6
At
c "
c
d= f + C
Fig. 1. Sandwich beam geometry.
127 where the subscript c refers to the core, and CE and CG are constants of proportionality. Gibson and Ashby [1] found that CE = 1 for a wide range of foams. C6 was assigned a value of 0.4 based on a limited a m o u n t of data. For a given bending stiffness p/A requirement, we wish to minimize the weight of a sandwich beam spanning a length I with a width b. The properties of the face and solid u n f o a m e d core are k n o w n and we wish to solve for the o p t i m u m core density Pc opt, the o p t i m u m core thickness Copt and the optim u m face thickness topt. We assume that the faces are thin compared with the core (c ~ c + t ~ d). A trialanderror solution for thick faces is described in Appendix A. We also assume that the flexural rigidity D (or equivalent EI) of the sandwich is given by
btd 2 D = Ef  2
(3)
W = 2pfblt + pcblc
(4)
A = Ab + As
P13
PI + 

(5)
C2AeGc
where P is an applied concentrated load, D is the flexural rigidity of the faces, Ae (= bd2/c) is an equivalent core area, and C~ and C2 are constants relating to the load conditions. For example, C~ = 3 and C2 = 1 for an endloaded cantilever. Combining eqns. (5), (3) and (2) and letting c = d gives A
P
W = 2 p , b t l + ( 1 C1E'pps21atcab2)l/2 × CG C2 Es × (AC1 btc2Ef  2P13)1/2 This gives the weight as a function of the face and core thicknesses. We can simplify this equation and optimize by taking partial derivatives with respect to t and c. To simplify the weight equation, let
( 1 G C1E, pps21ab2)l/2 A=
(8)
C2 Es
213
C1E~btc 2
I
+
CG C2bc(pc/P~)2Es
(9)
Then
The deflection A is the sum of the bending deflection Ab of the faces and the shear deflection As of the core:
CID
(7)
When eqn. (7) is substituted in eqn. (4), we obtain
B = AC1 bEf
bt a btd 2 bc 3 ~  + Ef ~   + Ec 12
The first and third terms are the bending stiffnesses of the faces about their own centroidal axis and that of the core about its centroidal axis. The second term is the bending stiffness of the faces about the beam's centroidal axis and is usually much larger than the sum of the other two terms. The weight of the sandwich beam is given by

(laC1EfPps21tc;2 Pc = C2 Es AC1 btc2Ef  2Pl s
and
The exact expression for flexural rigidity is D = E,
for the stiffness constraint equation. Solving for the density of the core, we obtain
(6)
W = 2pfbtl + Atl12ca12(Btc 2  2PlS) 1/2 Letting 8W/St = 0 and aW/Sc = 0 and solving, we find that
Copt =4.3411CG(PfI2 Es C2 p~115 xPs /
E~2
C12~~
(10)
and P
la
topt = 6 Ab C1Efc 2
(11)
These values can then be substituted into eqn. (7) to give the o p t i m u m core density Pc opt. It has been shown that, if a function is convex, the o p t i m u m obtained by setting derivatives equal to zero is a global optimum [10]. The weight objective function, with two variables, is convex if the determinant of its hessian matrix and both second derivatives are greater than zero, i.e. 82W
82W
8c 2
8c 8t >0 82W
82W 8t 8c ~2W ~0 8c 2
8t 2
128 and a2W >0 ~t 2
2.0
All three of these conditions are satisfied for the weight function. The solution for the minimum weight design given by eqns. (7), (10) and (11) is therefore a global optimum. We now have a way of minimizing the weight of a sandwich beam for a given stiffness, once the loading conditions and the face and solid u n f o a m e d core materials are specified. Using this solution, we can develop an expression for the minimum weight of a sandwich beam of given stiffness by substituting the optimum values for the core and face thicknesses and for the core density (eqns. (7), (10) and (11)) in the equation for weight (eqn. (4)). When this has been rearranged, we obtain
Co,,
= K [ P ~~5 c
/ p \315 topt = Ktt~bJ
[ p \2/5
1.0
z
0.5
0
200 400 BENDING STIFFNESS
600 P/A(Nmm  j )
region for sandwich weight is bounded by the weight of a solid aluminium beam. For a given stiffness a polyurethane beam (either foamed or solid) will always weigh more than an aluminium beam as its ratio of (pZ/E)Z/2 is larger (see for example ref. 11, pp. 6667). Conversely, this expression m a y also be rearranged to give the maximum stiffness for a given weight. We can find the ratio of the weight of the faces to that of the core at the optimum design from the analysis. It can be shown that this ratio is always 1/4, independent of the stiffness, span, loading conditions or materials used.
\psi E~2C121
6l s
Kt = CzEtK¢ 2 and ( 1 CIE, P2KtKc)Zl2 4Ca C2 E, l2
Combining these equations gives
Wopt = 2pfbtoptl + Pc optbcopt l
=( p ~3/5 \~]
uJ
)
where
Kp¢
pr
Fig. 2. Minimum weight as a function of stiffness for a sandwich beam with aluminium faces (pf = 2700 kg m3 ; Ef = 70 GN m2) and a foamed polyurethane core (Ps = 1200 kg m3; Es = 1.60 GN m2) in threepoint bending (l = 762 mm; b = 50 ram).
and
!
c
0
!
poo , = Koot
1.5
bl(2p,Kt + KpcKc)
(12) 3. EXPERIMENTAL METHOD
Weight increases with stiffness to the threefifths power, as Kt, Kc and Kp¢ are dependent on the loading conditions and the solid face and core materials only. A plot of the minim u m weight against the bending stiffness (eqn. (12)) is shown in Fig. 2. The feasible
Simply supported sandwich beams with aluminium faces and foamed polyurethane cores were tested in threepoint bending. The properties of the aluminium faces and of the solid unfoamed polyurethane are given in
129
The face and core materials were cut to size and bonded with a polyester resin. The foam core was cut with the rise direction perpendicular to the plane of shearing. The bonded specimens were held in a press overnight while curing. The faces and cores were weighed separately before bonding and again after bonding to estimate the weight of the glue used. The beams were simply supported, at one end with a plate resting on a roller and at the other end with a plate on a slider. In initial tests, two nonsliding rollers were used as supports. These caused some indentation in the beams with thin faces (t = 1.3 mm, t = 0.8 mm and t  0.4 mm) and were replaced with the plate supports. Central deflections (measured using a linear variabledifferential transducer beneath the loading point) and the load {from the test machine load cell) were recorded on an x  y recorder, to give the beam stiffnesses. The beams were loaded, unloaded and then reloaded with the reload slope being used to calculate the beam stiffness. Four beams of each design were tested.
Table 1. The beams were designed to be of equal stiffness (p/A = 750 N m m 1) but have different combinations of face and core thicknesses and core densities (and thus weights). In designing the beams, CG was assumed to be 0.4. The unsupported length was 762 m m and the width of all the beams was 50 mm. Using the analysis of Section 2, the o p t i m u m design has a core thickness of 93 mm, a face thickness of 1.3 mm and a core density of 320 kg m 3. Systematic tests on the beams varied c, t and Pc about these o p t i m u m values. The details of the beams tested are given in Table 2. All beams satisfied the two assumptions made, namely that c ~ d and D ~ E~btd2/2. Shear lag was avoided in all the beams (see Appendix B). TABLE 1 P r o p e r t i e s of a l u m i n i u m a n d u n f o a m e d p o l y u r e t h a n e
Al a Unfoamed polyurethane b
Density
Young's modulus
(kg m 3)
( G N m 2)
pf = 2 7 0 0 Ps = 1 2 0 0
E f = 70 E s = 1.6
4. R E S U L T S A N D D I S C U S S I O N
a T h e d a t a for a l u m i n i u m were t a k e n f r o m ref. 12. b T h e d e n s i t y for solid u n f o a m e d p o l y u r e t h a n e was t a k e n f r o m ref. 13, a n d Y o u n g ' s m o d u l u s f r o m ref. 14.
Table 2 lists the details of the beams tested, together with the measured and calculated
TABLE 2 S u m m a r y of results
l
b
c
t
Pc
(mm)
(mm)
(mm)
(mm)
( k g m 3)
P/~ (Nmm1)
P/AW (N m m  1 kgf 1)
W (kgf)
Experiment
Theory a
Experiment
Theory
Experiment Theory
762 762 762 762 762 762
50 50 50 50 50 50
60 81 90 95 108 136
4.8 2.6 1.6 1.3 0.8 0.4
320 320 320 320 320 320
1060 1080 1120 1140 1060 1080
1070 1200 1120 1090 1020 960
1.76 1.57 1.48 1.50 1.49 1.80
1.72 1.52 1.43 1.43 1.48 1.74
602 690 752 756 713 603
620 787 782 764 691 553
762 762 762 762 762
50 50 50 50 50
71 79 95 134 190
1.3 1.3 1.3 1.3 1.3
480 400 320 240 192
940 990 1140 1300 1270
960 1020 1100 1200 1260
1.56 1.53 1.50 1.61 1.71
1.57 1.47 1.43 1.49 1.65
605 650 756 811 746
611 690 760 801 760
762 762 762
50 50 50
93 93 93
0.65 0.80 1.30
480 400 320
850 950 980
890 950 1060
1.82 1.66 1.44
1.83 1.58 1.40
464 570 678
486 602 757
a T h e t h e o r e t i c a l p / A value was calculated using C G = 0.7.
130
values of stiffness, weight and stiffness per unit weight. The measured values for stiffness are larger than the design value of 750 N mm1; they have a mean value of 1070 N m m 1. This discrepancy could be caused by an error in Ca, the constant of proportionality between the foam shear modulus and the density. To see whether this was the case, the shear modulus for the core of each beam was calculated from the measured beam stiffness using eqn. (5). The resulting shear moduli are plotted against the relative density of the foam core in Fig. 3. The data fit the following equation extremely well:
The shear modulus varies as the density squared, but the constant Ca of proportionality is 0.7 instead of 0.4. The theoretical values for the stiffness listed in Table 3 have been calculated using Ca = 0.7. As expected from the fitting of experimental data to eqn. (13), there is good agreement between t h e o r y and experiment; nevertheless it is significant that the shear moduli derived from the beam stiffness measurements did vary as density squared. The experiments were originally designed to test beams of equal stiffnesses but different weights; to do this, c, t and Pc were systematically varied about the predicted optimum values. Because of the discrepancy in Ca just discussed, several problems arise: the beams are much stiffer than expected; t h e y are not all of equal stiffnesses; the geometries tested do not vary systematically about the optimum for the measured stiffness. To determine how significant these problems are, we can look at how the experiments would have been de
signed had the given required beam stiffness been 1070 N m m 1. Ideally the beams would all have been designed to have this stiffness; instead, the test beams show a standard deviation of 125 N m m 1 about this mean value, leading to a coefficient of variation of 12%. This is small enough that the beams may be considered to be of equal stiffnesses. Had the given stiffness been 1070 N m m 1, the optim u m beam design would have been different; Table 3 lists this new optimum geometry (for p/A = 1070 N m m 1) together with the test " o p t i m u m " geometry (for p/A = 750 N mm1). This table also shows the percentage difference between the two geometries, and
i0 o
l
l
l
I
[ I[
I
I
I
I
o 10*
/
r
l
J IlL
x
I
102
I
I
r
i
I
~ iJ~
i0=
I0 ~
CORE
RELATIVE
I0 °
DENSITY
pc/p=
Fig. 3. Relative s h e a r m o d u l u s G/E s vs. relative d e n sity Pc/Ps for t h e f o a m core: o, t = c o n s t a n t = 1.3 m m ; ~, Pc = c o n s t a n t = 320 kg m3; ×, c = c o n s t a n t = 9 3 mm; ~ , G/E s = 0.70(Pc/Ps) 2. T h e values o f the s h e a r m o d u l u s were calculated f r o m t h e m e a s u r e d b e a m stiffness data.
TABLE 3 O p t i m u m b e a m g e o m e t r y for m e a s u r e d m e a n b e a m stiffness
O p t i m u m g e o m e t r y for p / A = 1 0 7 0 N m m 1 T e s t o p t i m u m g e o m e t r y for
c (ram)
t (mm)
Pc (kg m 3)
p / A (N m m 1)
W (kgf)
P / A W (N m m 1 k g f 1)
109
1.41
280
1070
1.45
751
93
1.30
320
1030
1.40
735
p / A = 7 5 0 N m m 1 Difference
17%
8%
13%
4%
4%
2%
Range tested
60190
0.44.8
192480
8501300
1.441.82
464811
131
the range of geometries tested. The difference in the two optimum geometries is n o t large relative to the range of geometries tested. Finally, we can see h o w varying c, t and Pc about the optima for p/A = 1 0 7 0 N mm t affects the theoretical calculation of stiffness per unit weight. Figures 4  6 show theoretical stiffness per unit weight against c, t and Pc. In each figure, one variable is held constant at its optimum value while the other t w o are varied to give beams of equal stiffnesses but different weights. The heavy lines are the theoretical predictions of stiffness per unit weight for the test beams by which c, t and Pc were varied about the values c = 93 ram, t = 1.3 mm and Pc = 320 kg m ~. The thinner lines are the theoretical predictions of stiffness per unit weight had the given stiffness been 1 0 7 0 N mm t and had the geometry varied about the optimum for this stiffness (c = 109 mm, t = 1.41 mm and Pc = 280 kg ma). The t w o lines are close in each case. All these factors (the low coefficient of variation for stiffness, the test optimum geometry close to ideal for p / A = 1 0 7 0 N mm ~ and the test stiffness per unit weight close to ideal for p/A = 1070
~D
G ~" Boo
~ 'E 6OO E
..=,~ o_ ~= 4 0 0 uJ o
z
200
ul
I
0
(a)~_
50
I
I
I
L
IO0 150 200 250 CORE THICKNESS c (ram)
uJ _" BOO eL 
zT
6OO
5z o. ~ 4 0 0
~.
aoo I
0
(b)
o
IOO
I
I
I
I
200 300 400 500 CORE DENSITY Pc(kg Ms)
Fig. 5. S t i f f n e s s P / A W per u n i t w e i g h t for b e a m s o f c o n s t a n t face t h i c k n e s s vs. (a) core t h i c k n e s s a n d (b) core d e n s i t y (l = 7 6 2 ram; b = 50 ram): , theory for test g e o m e t r y , t = 1.3 ram;   , t h e o r y for p / A = 1 0 7 0 N m m 1, t = 1.4 m m .
IliD m T 800
,:, _ s 0 c
p
P .
'E 6 0 0 = E ~
z
.= ;
400
4oo m
z u. u.
(a)
200
= LIuI.
I 20
P
I I I I 40 60 80 I00 CORE THICKNESS c(mrn)
L 120
"I"
I 5 t(mm)
,"5,_ 8 0 0
"
7E 6 0 0
600
E
~Z 4 0 0
u. u..
I I I 2 3 4 FACE THICKNESS
0
(a)
;:,  Boo
=:
200
~ 4 0 0 Z 0. ~_ 200
200
c0
(b)
I ,
k I i 1 2 5 4 5 FACE THICKNESS t (ram)
L
6
Fig. 4. S t i f f n e s s P / A W per u n i t w e i g h t for b e a m s o f c o n s t a n t core d e n s i t y vs. (a) core t h i c k n e s s a n d (b) face t h i c k n e s s (l = 7 6 2 ram; b = 50 ram): t h e o r y for test g e o m e t r y , Pc = 3 2 0 kg m3;  t h e o r y for p / A = 1 0 7 0 N m m 1, Pc = 2 8 0 kg m ~.
~,
o (b)
I00
I I I I 200 500 400 500 CORE DENSITY Pc(kg m5)
Fig. 6. S t i f f n e s s P I A W per u n i t w e i g h t for b e a m s o f c o n s t a n t core t h i c k n e s s vs. (a) face t h i c k n e s s a n d (b) core d e n s i t y (l = 7 6 2 ram; b = 50 ram): , theory for test g e o m e t r y , c = 9 3 ram;   , t h e o r y for P / A = 1 0 7 0 N m m 1, c = 1 0 9 ram.
132
N mm 1) indicate that the experiments are acceptable in their present form. Figures 7  9 show experimental and calculated values of stiffness per unit weight plotted against c, t and Pc. Figure 7 shows the data when Pc is held constant and c and t are varied; in Fig. 8, t is constant, and c and Pc are varied and, in Fig. 9, c is held constant for varying values of t and Pc. Figure 9 shows that only three beam geometries were tested with constant c; this was because appropriate materials for other geometries were unavailable. The graphs show several things. First, the theory for the test geometry (eqn. (6)) predicts the measured values of stiffness per unit weight well when G = 0.70(pJps)2Es. Second, the values of c, t and Pc which give the minimum weight axe reasonably well predicted by the new theory presented here. The measurements indicate a range of c, t and Pc values giving the minimum weight. This is because, with the test beams employed, c, t and Pc were not varied about the true optimum design values predicted by the calculations, but the range of values found experimentally does coincide with the calculated optimum
F
/
soo
b
~, 7 6 0 0 z ~ 400
z
200
I 50
0
(a) ~=
7 6oo E E
=400 z a200 u.
(b)
o
2~
300
400
.500
f..

~ z w ~
z
400
o. ~: 4 0 0
200
W a. Z U.
w
200
k01
I 20
(a) 
zoo
Fig. 8. S t i f f n e s s P / A W per u n i t w e i g h t for b e a m s o f c o n s t a n t f a c e t h i c k n e s s t vs. (a) c o r e t h i c k n e s s c a n d ( b ) c o r e d e n s i t y Pc (l = 7 6 2 ram; t = 1 . 3 r a m ) :  t h e o r y ; ×,    , e x p e r i m e n t a l .
G ' 8OC
z v..
~oo
CORE DENSITY pc(kg ~ ' )
x
;
I I I I I00 ISO 200 250 CORE THICKNESS c(mm)
800
Z, , 8 0 0
~
fx
~800
J I I I 40 60 80 I00 CORE THICKNESS c (ram)
I 120
ii
f I 3 4 FACE THICKNESS
2

i
J
0
(a)
I 5 t(mm)
800
.~ ,~ 60o
7 600
=400
~
400
ca
200 0
(b)
200 u.
I
J
r
I
2
3
I 4
f
I
S
6
FACE THICKNESS t(mm)
Fig, 7. S t i f f n e s s P / A W per unit w e i g h t for b e a m s w i t h c o n s t a n t c o r e d e n s i t y Pc vs. (a) c o r e t h i c k n e s s c a n d ( b ) f a c e t h i c k n e s s t (l = 7 6 2 m m ; Pc = 3 2 0 kg m  3 ) : , t h e o r y ; ×,    , e x p e r i m e n t a l .
t
o
(b)
I00
I
I
I
200 300 400 500 CORE DENSITY p c ( k g rR~)
Fig. 9. S t i f f n e s s P / A W per u n i t w e i g h t for b e a m s o f c o n s t a n t c o r e t h i c k n e s s c vs. (a) f a c e t h i c k n e s s t a n d ( b ) c o r e d e n s i t y Pc (l = 7 6 2 ram; c = 9 3 r a m ) : t h e o r y ; ×,    , e x p e r i m e n t a l .
133 TABLE 4 Optimum values of core thickness, face thickness and core density for p/A = 1070 N mm 1
Measured Theoretical
Copt (ram)
top t (ram)
Pc opt (kg m 3)
95130 109
1.31.6 1.4
240320 280
values o f c, t a n d pc ( T a b l e 4). Finally, t h e m a x i m u m stiffness per u n i t w e i g h t m e a s u r e d in all t h e tests was 811 N m m 1 k g f 1 in c o m parison w i t h t h e c a l c u l a t e d value o f 751 N m m 1 k g f ~. This d i s c r e p a n c y is d u e t o t h e v a r i a t i o n in m e a s u r e d b e a m stiffness. 5. CONCLUSIONS A n e w analysis f o r m i n i m u m w e i g h t design o f s a n d w i c h b e a m s w i t h f o a m e d cores is presented. I t is b a s e d o n t w o o b s e r v a t i o n s : (1) t h e t o t a l d e f l e c t i o n o f a s a n d w i c h b e a m in b e n d i n g is t h e s u m o f t h e b e n d i n g a n d shear c o m p o n e n t s ; (2) t h e shear m o d u l u s o f t h e f o a m c o r e is p r o p o r t i o n a l t o t h e s q u a r e o f its d e n s i t y . T h e analysis gives values o f t h e c o r e d e n s i t y as well as t h e core a n d face thicknesses w h i c h m i n i m i z e t h e weight f o r a given stiffness. M e a s u r e m e n t s s h o w t h a t t h e basis for t h e analysis is valid a n d t h a t t h e analysis w o r k s well in o p t i m i z i n g s a n d w i c h b e a m design f o r stiffness. T h e analysis also s h o w s t h a t t h e m i n i m u m w e i g h t increases w i t h stiffness t o t h e t h r e e  f i f t h s p o w e r f o r given m a terials and span a n d t h a t at t h e o p t i m u m design t h e w e i g h t o f t h e faces is o n e  q u a r t e r t h a t o f t h e core.
ACKNOWLEDGMENTS We g r a t e f u l l y a c k n o w l e d g e t h e financial s u p p o r t o f t h e N a t i o n a l Science a n d Engineering R e s e a r c h Council o f Canada. G e n e r a l Plastics M a n u f a c t u r i n g o f T a c o m a , WA, g e n e r o u s l y supplied the p o l y u r e t h a n e f o a m for t h e e x p e r i m e n t s .
REFERENCES L. J. Gibson and M. F. Ashby, The mechanics of threedimensional cellular materials, Proc. R. Soc. London, Ser. A, 382 (1982) 4359.
2 S. N. Huang and D. W. Alspaugh, Minimum weight sandwich beam design, A I A A J., 12 (1974) 16171618. 3 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 Specialty Conf., University o f Texas at Austin, September 1719, 1979, American Society of Civil Engineers,
New York, 1979, pp. 4144. 4 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.). 5 P. Ackers, The efficiency of sandwich struts utilizing a calcium alginate core, R & M 2015, 1945 (U.K. Aeronautical Research Council, Farnborough, Hants. ). 6 E. W. Kuenzi, Minimum weight structural sandwich, U.S. For. Serv. Res. Note FPL086, 1965 (Forest Products Laboratory, Madison, WI). 7 J. M. Dinwoodie, Timber, its Nature and Behaviour, Van Nostrand, New York, 1981. 8 L.J. Gibson, M. F. Ashby, G. S. Schajer and C. I. Robertson, The mechanics of twodimensional cellular materials, Proc. R. Soc. London, Ser. A, 382 (1982) 2542. 9 K. E. Easterling, R. Harrysson, L. J. Gibson and M. F. Ashby, On the mechanics of balsa and other woods, Proc. R. Soc. London, Ser. A, 383 (1982) 3141. 10 R.M. Stark and R. L. Nicholls, Mathematical Foundations for Design: Civil Engineering Systems, McGrawHill, New York, 1972. 11 M. F. Ashby and D. R. H. Jones, Engineering Materials: an Introduction to their Properties and Applications, Pergamon, Oxford, 1980. 12 R.M. Tennant (ed.), Science Data Book, Oliver
and Boyd, Edinburgh, 1971. 13 W. J. Roff and J. R. Scott, Fibres, Films, Plastic and Rubbers  a Handbook o f Common Polymers, Butterworths, London, 1971.
14 M. R. Patel and I. Finnie, Structural features and mechanical properties of rigid cellular plastics, J. Mater., 5 (1970) 909932.
APPENDIX A A. 1. T h i c k f a c e s
I f t h e faces are t h i c k c o m p a r e d w i t h the core, we c a n n o t allow d t o e q u a l c in t h e analysis. I f c a n d d  c + t are k e p t as separate variables, eqn. (5) b e c o m e s A
2l 3
lc
P
CiE~bt(c } t) 2
CGC2b(pc/ps)2Es(c~ t) 2 (A1)
We again solve f o r Pc f r o m this e q u a t i o n , subs t i t u t e this e x p r e s s i o n i n t o t h e w e i g h t objective f u n c t i o n (eqn. (4)) a n d solve ~ W / ~ t = 0
134 and aW/ae = 0 for Copt and tout. This leads to a trialanderror solution for topt with
2[3t+{9t2 1
4
16
_4(t 2 
B 1/2 = 1.09ptblt a / 2 A
2P/at]l/2 ] Bt I, J pl 8
t
(A2)
Bt 2
As before, A and B are given by eqns. (8) and (9). The equation for optimum core thickness c is found to be 4
10
(A3) The optimum core density can be found by solving eqn. (A1) for core density and substituting the optimum core and face thickness values from eqns. (A2) and (A3) for c and t. As this is a trialanderror solution, no expression for minimum weight as a function of stiffness can be developed.
This depends only on the loading geometry (C1 and C2), the given beam stiffness P/Ab per unit width, the face properties pf and E~, and the solid unfoamed core properties Ps and Es. The face material and the unfoamed core material determine the value of 0 for a given load configuration; the designer should select materials which do not lead to shear lag. For threepoint bending, Allen suggests that 0 should be greater than 20 to ensure that shear lag does not occur. 0 was varied between 80 and 1430 for the tests described in the present study. Reference for A p p e n d i x B B1 H. G. Allen, Analysis and Design of Structural Sandwich Panels, Pergamon, Oxford, 1969.
APPENDIX C: NOMENCLATURE
A Ae
APPENDIX B
B.1. Shear lag In designing sandwich beams, the core should be stiff enough to avoid shear lag, i.e. plane sections should remain plane and the faces should not slide over one another when the beam is loaded. Allen [B1] discusses the problem of shear lag in some detail. He concludes that shear lag will not occur if the relative stiffness of the core to the face is sufficiently large. This relative stiffness is a function of the ratio of the core shear modulus to the face Young's modulus and of the ratio of the core thickness to face thickness. Allen develops a nondimensional parameter 0 to give a measure of the relative face and core stiffnesses. For a simply supported beam in threepoint bending,
,t c c(
0 = 1+ 3 c 12E~ t t 2 ]J
(B1)
Letting c ~ d and substituting for Go, Cout and tout (eqns. (2), (10) and (11)) in eqn. (B1), we find that
[p,I
•,
,p,1
e Cout
CE CG
C1
C2 d D Ee E~
Es Gc
1 [C'~2[1 +
I
b B
Kc E,
Cl
J
J
Kt
constant in eqn. (8) used to simplify analysis (kg N ~/2 m 1/2) equivalent area of the beam (m 2) width of beam (m) constant in eqn. (9) used to simplify analysis (N) core thickness (m) optimum value of the core thickness to minimize the weight (m) constant of proportionality for the relative Young's modulus and the relative density constant of proportionality for the relative shear modulus and the relative density loading constant for bending deflection loading constant for shear deflection distance between centroids of faces (m) equivalent flexural rigidity of the beam (MN m 2) Young's modulus of the foamed core (MN m 2) Young's modulus of the face material (MN m 2) Young's modulus of the unfoamed core material (MN m 2) shear modulus of the foamed core (MN m 2) constant relating the optimum core thickness to the stiffness (m 7/5 N 1/5) constant relating the optimum face
135 thickness to the stiffness (m zz/~ N 3/5) constant relating the optimum core density to the stiffness (kg m lz/5 N2/5) span length of a sandwich beam (m) l p~ applied load (N) P / A W stiffness per unit weight (N mm 1 kgf 1) face thickness (m) t optimum value of the face thickness topt to minimize the weight (m) W weight of the beam (kgf) Wopt o p t i m u m weight (kgf) Kpc
Greek symbols A total beam deflection (m) Ab bending deflection (m) A~ shear deflection (m) 0 parameter relating face and core stiffnesses Pc core density (kg m 3) Pc opt optimum value of the core density to minimize the weight (kg m 3) p~ face density (kg m 3) Ps unfoamed core material density (kg m 3)