DESIGN OF SANDWICH BEAMS, STRUTS AND PANELS

DESIGN OF SANDWICH BEAMS, STRUTS AND PANELS

C H A P T E R 11 D E S I G N OF S A N D W I C H BEAMS, STRUTS A N D PANELS 11.1 Introduction The process of trial and error is often the most effecti...

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C H A P T E R 11

D E S I G N OF S A N D W I C H BEAMS, STRUTS A N D PANELS 11.1 Introduction The process of trial and error is often the most effective method o f designing sandwich panels. Elaborate methods o f optimum design have occasionally been proposed in which the proportions of a sandwich with specified loads and spans are adjusted with minute precision in order to save the last ounce o f weight. In reality, however, the choice of faces and cores is not infinite; face materials may be available in relatively few gauges or standard thicknesses ; core materials may be restricted in the choice of thickness and density. In such cases it can be expedient to use a computer (which would otherwise be required for the preparation o f optimum design charts) to check the strength and stiffness o f a selection o f practical sandwiches and to choose the lightest which will perform the desired task. This is particularly true when effects such as plasticity are to be taken into account and it avoids the embarrassment engendered by an optimum structure with impossibly thin faces. A l l the same, it is convenient to have methods o f design which can indicate roughly where the process o f trial and error should begin. In the nature o f things, these design methods need not be as precise as the final analysis or check calculation. Advantage can be taken o f this to take short cuts or to make approximations which are not acceptable in the check calculation. Indeed, unless 233

ANALYSIS OF STRUCTURAL SANDWICH PANELS

236

short cuts are taken, the design method is likely to be more cumbersome than the process of analysis

and its use will be corre-

spondingly restricted. One short cut is to ignore completely any effects due to the thickness of the faces. A sandwich with thick faces and a weak core is by definition an inefficient sandwich, because the faces are well on the way to working as two independent beams, struts or panels. Furthermore, the difference between "thin" and "very thin" faces as defined in Section 1 0 . 1 is merely the difference be2

tween Gd /c and Gd as the shear stiffness ; in terms of approximate design this difference is small enough to be neglected and the faces may be treated as "very thin". There are three main design processes. In the first and simplest the core and face materials are specified, as is the thickness of the faces. The problem is to determine the necessary core thickness. In many building structures and in semi-structural applications the loads are light and a truly optimum design (in terms of weight or cost) would lead to unpractically thin faces. Consequently it is convenient to begin by choosing the thinnest face which can

be used

(in terms of robustness,

fire-resistance,

weathering, etc.) and then finding the thinnest core which can be used with it. In the second design process the core and face materials are specified but the thicknesses of the faces and the core are to be found. Generally there is a whole range of combinations of face and core thicknesses which will provide adequate strength or stiffness and the problem is to choose the combination which provides the sandwich with the lowest weight (or cost). It is this process which is usually referred to as optimum design, or minimum-weight design. It is more likely to be used for aero structures where weight-saving is vital and where the fabricator is willing to take extra trouble to obtain non-standard sizes to fit the design. The third design process is similar to the second except that

DESIGN OF SANDWICH BEAMS

237

the core density is to be chosen as well as the face and core thicknesses. It is usual to assume that the strength and stiffness o f the core are directly proportional to the density. This type o f problem will not be considered here.

11.2. Determination of Core Thickness Beam with uniformly distributed load Consider a simply-supported sandwich beam which is required to support a uniformly distributed load q per unit length at failure. The load q is equal to the product o f the working load and a suitable load factor. The face thickness t and the face and core materials are predetermined and it is desired to find d (approximately the core thickness if the faces are very thin). 2

The maximum bending moment is qL /S and the maximum stress 2

in the faces is therefore qL /8bdt.

This stress must not exceed

the ultimate strength o f the material. A l s o , it must not exceed the wrinkling stress. F o r the present purpose the wrinkling stress may be calculated from equation (10.13) with the value B± = 0-55 ( F i g . 8.5). I f the lesser o f the ultimate strength and the wrinkling stress is denoted by a u then (11.1) The maximum shear force is qL/2 and the shear stress in the core is qLjlbd.

I f the ultimate shear strength o f the core is T i ,

then (11.2) The maximum deflection is given by equation (10.10). There may be a restriction on the deflection which can be expressed as a limiting ratio AX\L

at failure. Then (11.3)

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ANALYSIS OF STRUCTURAL SANDWICH PANELS

The required core depth d is the least o f the values which can be obtained from these three equations, viz. : (11.4) (11.5) (11.6) Pin-ended strut Again the face thickness t and the face core materials are predetermined and it is desired to find d to enable the strut just to support a load Ρ over a length L. The load Ρ is the product o f the working load and a suitable load factor. ι 3 First, i f there is any chance o f (t/d) (Ef/Ec) falling in the range indicated in Fig. 8.6, the chosen thickness t must be adequate to support Ρ without wrinkling. For this purpose the wrinkling stress may be calculated initially from equation (10.13) with the value Bi = 0-5 ( F i g . 8.6). Obviously the chosen face thickness t must also be large enough to avoid the possibility o f crushing or yielding o f the face material. The critical load is given by equation (10.11) and it must be greater than the specified load P. Thus: (11.7) (11.8)

or Isotropic panel with uniform transverse load

Once again the face thickness t and the face and core thicknesses are predetermined. The minimum approximate core depth d is to be evaluated which will just enable the panel to support a

239

DESIGN OF SANDWICH BEAMS

uniform transverse pressure q. The pressure q is equal to the product of the working pressure and a suitable load factor. A s in the beam problem, the face stress at failure is limited to (Ti,

which may be the ultimate strength of the face material or

the wrinkling stress, whichever is the lower. The ultimate strength of the core material is t i and the deflection at failure is limited to Δ χ. It is assumed that Δ χ is equal to the limiting deflection at working load multiplied by the load factor. It may be assumed that a > b without loss o f generality, in which case the face stress ay is greater than the face stress σχ and the core stress xyz is greater than the core stress xzx.

Hence,

from equations (5.35a) and (5.39),

(

EtcP

/ ? 1 +

2(Γ=^

~G

•er/*»)*' * 11

01.9)

^-(ft + ^ & W i ,

(11.10)

•^-/fr-eti.

(11.11)

The coefficients β may be obtained from Fig. 5.5 for any particular ratio ajb. The required approximate core depth d is the least of the values which can be obtained from these three relationships, v i z : ,

qbWß 2GA

Μ ' - / ( ' + ^ Γ ^ ) } · « - >

2

ab d^^—(ßi+vfß3), Oit

(11.13)

d^^-β·;.

(11.14)

^1

I f the face material is weak in shear in the *y-plane an additional

ANALYSIS OF STRUCTURAL SANDWICH PANELS

240

condition can be obtained from equation (5.39c): (11.15) where txy

is the ultimate shear strength o f the faces.

Isotropic panel with uniform edge load The face thickness / and the face and core materials are predetermined. It is desired to select the minimum approximate core depth d which will just enable the panel to support a uniform edge load Ρ (per unit length) in #the x-direction. The load Ρ is the product o f the working load and a suitable load factor. A s in the strut problem it is necessary to check first that the face thickness is adequate to support Ρ without wrinkling or failure of the face material. The critical load is given by equation (5.29a) and it must be greater than the specified load. Before equation (5.29a) can be used it is necessary to know the appropriate value of m to be used in equation (5.29c). Provided it is known that ρ will be very small (i.e. the core will be quite stiff in shear) the value of m can be obtained from Fig. 5.4 for any particular value of a\b9 using the curve ρ = 0. For example, if a/b <
< a/b < ύ / 6 , m = 2; if Λ/6 < a/b < y/l29

m = 3 and so

forth. Once the value of m has been selected, equation (5.29a) may be written in the following form: 7t^ 2

b

Etd

2

(imb

2 ( 1 - » ï)'\ })'[ a π

a +

2

2(1-r?)

l

2

mb\ Ε G

td 6

(11.16)

^P.

2

Or, d2*

1 + 2G

n*EtP

a2 2 mb (11.17)

DESIGN OF S A N D W I C H

BEAMS

241

If ρ is not very small it is difficult to estimate the value of m from Fig. 5.4, at least near the intersections of the curves. I f ρ is large (ρ > 0-3, say) it is nearly impossible to do so.

11.3. Optimum Design: Determination of Core and Face Thickness for Minimum Weight (or Cost) A particularly good introduction to the problem of optimum design is presented by K u e n z i

( 2 1 1 )

and the remarks which fol-

low are based on his study. Suppose that the bending stiffness D of a sandwich beam (width b) is specified, as are the materials to be used for the faces and the core. The bending stiffness is defined by equation (11.18a) and the combined weight of the faces and core (per unit area) is given by equation (11.18b), in which ^ a n d μ£ are the densities of the face and core materials respectively. D =

Ebtd

2

(11.18a)

2

w =

μ^+2μ/ί.

(11.18b)

The faces are assumed to be very thin ; it is therefore in order to take d as the core thickness and to neglect the local bending stiffnesses of the faces. The thicknesses t and d are to be adjusted to satisfy equation (11.18a) and, at the same time, to provide a minimum value for w. The weight o f the adhesive is constant and so omitted from the calculation. Elimination of t from equations (11.18a) and (11.18b) permits the expression of w in terms o f d: w =

μ0ά+

4μ/Ό 2

Ebd

'

For a minimum value of w with respect to d the following condi-

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ANALYSIS OF STRUCTURAL SANDWICH PANELS

tions must be satisfied :

or

(11.19)

This is the optimum core depth. The ratio o f the weight o f the core to the combined weight o f the faces is (11.20) This fact can be used as a quick check on the efficiency o f any given sandwich; for example, a construction in which the core is only a quarter o f the weight of the faces is not likely to be very efficient in terms o f bending stiffness. I f the bending strength is specified instead o f the bending stiffness, the face and core thicknesses must satisfy this equation: (11.21) Here M is the bending moment which the beam must carry at failure and σχ is a limiting stress such as the ultimate strength o f the face material or the wrinkling stress (whichever is the lower). For the pressent θ\ is treated as a constant. Elimination o f t from equations (11.18b) and (11.21) expresses the weight o f the sandwich as a function o f d: (11.22) Minimization of w with respect to dyields the optimum core depth and shows that the weight o f the core should equal the combined weight o f the faces: (11.23a, b ) I f it is known or suspected that failure of the face will occur as a result o f local instability (e.g. by buckling into the cells o f a

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DESIGN OF SANDWICH BEAMS

honeycomb of fixed cell dimensions) the limiting stress ax is not 2

constant but proportional to t : d

2

(11.24)

= kt .

Equation (11.21) is now (11.25)

M = kt*db.

Elimination of d from equations (11.18) and (11.25) and minimization o f the weight w with respect to t provides the optimum face thickness and shows that the core should weigh one-third of the combined weight of the faces: t

i

3 JJL^ Μ

μ£ά

2

2μ/ΐ

μ/

hb

The original report by K u e n z i

( 2 1 1 )

_

\ 3

(11.26a, b )

deals with faces of un-

equal thickness and dissimilar materials and also with cores in which the density is allowed to vary. Many permutations of the procedure are possible and Kuenzi uses it for an investigation of the optimum proportions o f a simply-supported panel which will just support a given edge load in the x-direction. I f the core is stiff in shear the critical load o f a panel with a fixed a/b ratio is directly proportional to the flexural rigidity. It has already been shown that the most efficient sandwich in terms of flexural rigidity is one in which the weight of the core is twice the combined weight o f the faces. This is also true of the panel with the prescribed critical load. The conclusion is less straightforward when the core is not very stiff in shear but in that case the weight of the core in the optimum sandwich is somewhat less than twice the combined weight o f the faces. A different approach is adopted by A l l e n .

( 2 5 , )3

The weight o f

the sandwich, as defined by equation (11.18), can be viewed as the height o f an inclined plane above horizontal axes d, t (Fig. 11.1). Each of the curves A, B9 C i n the horizontal plane represents some relationship between d and t imposed by a requirement such as the limitation o f face stress, buckling load, core shear stress,

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ANALYSIS OF STRUCTURAL SANDWICH PANELS

deflection, etc. The curves a, b, c in the inclined plane are vertically above A, B, C. Points such as Ρ (on the side o f a, b, c remote from the origin) represent possible sandwiches; points such as Q (on the side o f a, b9 c near the origin) represent sandwiches which are inadequate in some way. The object o f the analysis is to select a point Ρ as near to the origin and as far down the inclined plane as possible without crossing the curves a, b, c. Much depends on

d

C

FIG. 11.1.

the way the curves intersect (if they intersect at a l l ) ; this determines which combinations o f physical limitations are likely to govern the design. The analysis is straightforward and it can cope with a large variety of design restrictions, but it is not really suitable for calculations by hand. A selection of papers on other aspects of optimum design is included in the list o f references under an appropriate heading. It is worth noting that any analysis which provides a minimum weight design can also be used to provide a minimum cost design if the costs of the face and core materials (per unit volume) are substituted for the densities o f these materials.