Info

Figure 10.3 Indentation mode of collapse for a three-point bend configuration f

Consider first the case of three-point bending. Then the collapse load F on the upper indenter can be derived by a simple upper bound calculation. Two segments of the upper face, of wavelength X, are rotated through a small angle B. The resulting collapse load is given by

where Mp = bt2/4 is the full plastic moment of the face-sheet section. Minimization of this upper bound solution for F with respect to the free parameter X gives an indentation load FI of

We note in passing that the same expression for Fj and X as given by equations (10.13) and (10.14) are obtained by a lower bound calculation, by f y considering equilibrium of the face sheet and yield of the face sheet and core. We assume that the bending moment in the face sheets attains a maximum local value of —Mp at the edge of the indenter and also a value of Mp at a distance X from the edge of the indenter. It is further assumed that the foam core yields with a compressive yield strength oy and exerts this level of stress on the face sheet, as shown in Figure 10.3. Then, force equilibrium on the segment of a face sheet of length (2X + a) gives

and moment equilibrium provides

Relations (10.15) and (10.16) can be rearranged to the form of (10.13) and (10.14), demonstrating that the lower and upper bound solutions coincide. We conclude that, for a rigid-perfectly plastic material response, these bounds give exact values for the collapse load and for the span length X between plastic hinges. The presence of two indenters on the top face of the sandwich beam in four-point bending results in a collapse load twice that for three-point bending, but with the same wavelength X.

Core shear

When a sandwich panel is subjected to a transverse shear force the shear force is carried mainly by the core, and plastic collapse by core shear can result. Two competing collapse mechanisms can be identified, as shown in Figure 10.4 for the case of a beam in three-point bending. Mode A comprises plastic hinge formation at mid-span of the sandwich panel, with shear yielding of the core. Mode B consists of plastic hinge formation both at mid-span and at the outer supports.

Consider first collapse mode A (see Figure 10.4). A simple work balance gives the collapse load FA, assuming that the face sheets on the right half of the sandwich panel rotate through an angle B, and that those on the left half rotate through an angle —B. Consequently, the foam core shears by an angle B. On equating the external work done FIB/2 to the internal work dissipated within the core of length (I + 2H) and at the two plastic hinges in the face sheets, we obtain

where r'y is the shear yield strength for the foam core. Typically, the shear strength of a foam is about two-thirds of the uniaxial strength, tC = 2oC/3.

Collapse mode B
Figure 10.4 Competing collapse modes A and B for core shear of a sandwich beam in three-point bending

We note from equation (10.17) that FA increases linearly with the length of overhang, H, beyond the outer supports.

Second, consider collapse mode B. As sketched in Figure 10.4, this collapse mechanism involves the formation of plastic hinges in the face sheets at both mid-span and at the outer supports. The core undergoes simple shear over the length, L, between the outer supports, with no deformation beyond the outer supports. A work calculation gives for the plastic collapse load FB,

4bt2 f c

Since the two calculations given above are upper bounds for the actual collapse load, the lower is taken as the best estimate for the actual collapse load. It is instructive to compare the collapse loads as a function of overhang length H, as sketched in Figure 10.5. Collapse mode A is activated for small lengths of overhang, whereas collapse mode B has the lower collapse load and is activated for large overhangs. The transition length of overhang, Ht, is determined by equating (10.17) and (10.18), giving

1 t2 ofy

Mode B

Mode B

Overhang, h

Figure 10.5 The competition between collapse modes A and B for core

In order to gage the practical significance of the overhang, let us take some representative values for a typical sandwich panel comprising aluminum skins and a metallic foam core, with c/l = 0.1, t/c = 0.1, r^/af = 0.005. Then, the transition overhang length, Ht, is given by Ht = 0.1l: that is, an overhang of length 10% that of the sandwich panel span l is sufficient to switch the collapse mode from mode A to mode B. Furthermore, the enhancement in collapse load due to plastic bending of the face sheets above the load required to shear the core is about 20% for a small overhang, H — Ht, and is about 40% for H > Ht. In much of the current literature on sandwich panels, a gross approximation is made by neglecting the contribution of the face sheets to the collapse load.

Parallel expressions can be derived for the collapse of a sandwich beam in four-point bending by core shear. The collapse load for mode A becomes

The transition length of overhang at which the expected collapse mode switches from mode A to mode B is given by the same expression (10.19) as for a beam in three-point bending.

10.3 Collapse mechanism maps for sandwich panels shear

and that for mode B is

It is assumed that the operative collapse mechanism for a sandwich beam is the one associated with the lowest collapse load. This can be shown graphically

Figure 10.6 (a) Collapse mechanism map for three-point bending, with flat-bottom indenters. Contours of non-dimensional collapse load

F = F/biof are plotted, with c/i and t/c as axes. The map is drawn for the f selected values ocY/oJY = 0.005 and a/i = 0.1. (b) The effect of the size of the indenter a/i upon the relative dominance of the collapse mechanisms for three-point bending by plotting a non-dimensional measure of the collapse load F = F/biof on a diagram with the non-dimensional axes c/i and t/c, for selected values of a/i and ocy/of. An example is given in Figure 10.6(a) for the case of three-point bending, with a/i = 0.1 and oy/of = 0.005. It is assumed that the overhang H exceeds the transition value Ht so that core shear is by mode B, as depicted in Figure 10.4. The regimes of dominance for each collapse mechanism are marked: for example, it is clear that failure is by face yield for thin face-sheets (small t/c) and long beams (small c/i). The boundaries of the indentation regime are sensitive to the value taken for a/i: with diminishing a/i the magnitude of the indentation load drops and this regime enlarges on the map, as illustrated in Figure 10.6(b). It is striking that the boundary between the core shear and the indentation regimes has a large curvature, with the indentation mechanism operating at a large values of t/c as well as at small values. This is a consequence of plastic hinge formation within the face sheets in the core shear collapse modes: the collapse load for core shear increases quadratically with increasing t/c due to the contribution from face sheet bending, as seen by examining the first term on the right-hand side of relations (10.17) and (10.18). The contours of collapse load in Figure 10.6(b) show that the load increases along the leading diagonal of the map, with increasing c/l and t/c.

A similar map can be constructed for four-point bending; this is illustrated in Figure 10.7, for the same values a/l = 0.1 and oy/of = 0.005 as for three-point bending, but with the added parameter s/l = 0.5. A comparison with the map of Figure 10.6(a) reveals that the domain of face yield shrinks slightly for four-point bending, and indentation almost disappears as a viable mechanism. Core shear dominates the map for the values of parameters selected.

Figure 10.7 Collapse mechanism map for four-point bending, with flat-bottom indenters. Contours of non-dimensional collapse load F = F/blaf are plotted for the selected values o^of = 0.005, a/l = 0.1 and s/l = 0.5

Now, some words of caution. The collapse mechanisms described neglect elastic deformation and assume perfectly plastic behavior. Alternative failure modes are expected when the face sheets are made of a monolithic ceramic or composite layers, and behave in an elastic-brittle manner. Then, collapse is dictated by fracture of the face sheets, as analysed by Shuaeib and Soden (1997) and Soden (1996). The above treatment has been limited to the case of flat-bottomed indenters. An alternative practical case is the loading of sandwich beams by rollers, of radius R. This case is more complex because the contact size increases with increasing load. The failure modes of core shear, face yield and indentation have been observed for sandwich beams in a four-point bend with roller-indenters by Harte (1999), and good agreement between the measured strengths of the beams and theoretical predictions are observed, upon making use of the formulae given above but with a = 0.

10.4 Case study: the three-point bending of a sandwich panel

A set of three-point bend experiments on foam-cored sandwich panels has been performed by Bart-Smith (1999), using a foam core of Alporas aluminum alloy (relative density p/ps = 0.08) and 6061-T6 aluminum alloy face sheets. The sandwich beams were loaded by flat-bottom indenters of size a/l = 0.1; the pertinent geometric and material parameters are summarized in Table 10.1 for two specimen designs, 1 and 2. These designs are based on the collapse load predictions, such that Design 1 sits within the core shear regime, whereas Design 2 lies just within the indentation regime, as marked in Figure 10.6(a).

Table 10.1 Geometric and material parameters for Designs 1 and 2

Parameter Design 1 Design 2

Table 10.1 Geometric and material parameters for Designs 1 and 2

Parameter Design 1 Design 2

t, mm

79

40

t, mm

0.8

0.8

c, mm

10

10

b, mm

20

21

H, mm

18

21

a, mm

7.9

4.0

Ef, GPa

70

70

af, MPa

263

263

Ec, GPa

0.236

0.263

ay, MPa

1.5

1.5

The measured load F versus displacement S curves are shown in Figure 10.8; it was found that the Designs 1 and 2 failed by core shear and by indentation, respectively. For comparison purposes, the predicted elastic stiffness and collapse load have been added to each plot by employing relations (10.1), (10.13) and (10.18). Additionally, the load-deflection responses were calculated by a finite element procedure, using the constitutive model of

Design 1: Core shear

"5 400

Design 1: Core shear

"5 400

(Eq. 10.1)

Displacement (mm)

Design 2: Indentation

Displacement (mm)

Design 2: Indentation

1000

200 0

1000

200 0

(Eq. 10.1)

Experiment

^ —L :

■ \l X

Y •

- Lff

-jl 1

\ :

7 ! Í

Unload

j h j

reload

Displacement (mm)

Displacement (mm)

Figure 10.8 Load versus displacement curves for two designs of sandwich beams in three-point bending compared with the predictions of a finite-element simulation. In the first, the failure mechanism is core shear; in the second, it is indentation. The details of the geometry and material properties are listed in Table 10.1

Chapter 7. The constitutive model for the foam was calibrated by the uniaxial compressive response, whereas the uniaxial tensile stress-strain response was employed for the solid face sheets. Excellent agreement is noted between the analytical predictions, the finite element calculations and the measured response for both failure modes.

10.5 Weight-efficient structures

To exploit sandwich structures to the full, they must be optimized, usually seeking to minimize mass for a given bending stiffness and strength. The next four sections deal with optimization, and with the comparison of optimized sandwich structures with rib-stiffened structures. The benchmarks for comparison are: (1) stringer or waffle-stiffened panels or shells and (2) honeycomb-cored sandwich panels. Decades of development have allowed these to be optimized; they present performance targets that are difficult to surpass. The benefits of a cellular metal system derive from an acceptable structural performance combined with lower costs or greater durability than competing concepts. As an example, honeycomb-cored sandwich panels with polymer composite face sheets are particularly weight efficient and cannot be surpassed by cellular metal cores on a structural performance basis alone. But honeycomb-cored panels have durability problems associated with water intrusion and delam-ination; they are anisotropic; and they are relatively expensive, particularly when the design calls for curved panels or shells.

In what follows, optimized sandwich construction is compared with conventional construction to reveal where cellular metal sandwich might be more weight-efficient. The results indicate that sandwich construction is most likely to have performance benefits when the loads are relatively low, as they often are. There are no benefits for designs based on limit loads wherein the system compresses plastically, because the load-carrying contribution from the cellular metal core is small. The role of the core is primarily to maintain the positioning of the face sheets.

Structural indices

Weight-efficient designs of panels, shells and tubes subject to bending or compression are determined by structural indices based on load, weight and stiffness. Weight is minimized subject to allowable stresses, stiffnesses and displacements, depending on the application. Expressions for the maximum allowables are derived in terms of these structural indices involving the loads, dimensions, elastic properties and core densities. The details depend on the configuration, the loading and the potential failure modes. Non-dimensional indices will be designated by n for the load and by f for the weight. These will be defined within the context of each design problem. Stiffness indices are defined analogously, as will be illustrated for laterally loaded panels. The notations used for material properties are summarized in Table 10.2. In all the examples, Al alloys are chosen for which ef = off /Ef = 0.007.

Organization and rationale

Optimization procedures are difficult to express when performed in a general manner with non-dimensional indices. Accordingly, both for clarity of presentation and to facilitate comprehension, the remainder of this chapter is organized in the following sequence:

126 Metal Foams: A Design Guide Table 10.2 Material properties

Property

Face

Foamed

Solid

core

core

Density (kg/m3)

Pf

Pc

Ps

Young's modulus (GPa)

Ef

Ec

Es

Shear modulus (GPa)

-

Gc

-

Yield strength (MPa)

of

®y

oy

Yield strength in shear (MPa)

Notation b = width c = core thickness t = face sheet thickness l = span length W = weight P = load p = load per unit area

Notation b = width c = core thickness t = face sheet thickness l = span length W = weight P = load p = load per unit area

1. A specific example is given for the case of a sandwich plate subject to a uniformly distributed transverse load. This example illustrates issues and procedures related to designs that limit deflections, subject to strength criteria. It then demonstrates how optimization is achieved in terms of a dimensionless load index n and weight index f.

2. Following this example, generalized results are presented for stiffness-limited sandwich beams and plates. These results apply to a range of loadings and give non-dimensional strengths for the local and global weight minima at a specified stiffness.

3. Sandwich panel results are compared with results for waffle-stiffened panels, in order to establish domains of preference.

4. Strength-limited sandwiches are considered and compared with stringer-stiffened construction. Cylindrical sandwich shells are emphasized because these demonstrate clear weight benefits over conventional designs.

5. Overall recommendations regarding sandwich design are given.

10.6 Illustration for uniformly loaded panel

The design of a wide sandwich plate subject to a uniform distributed load will be used to illustrate how the thicknesses of the skins and the cellular metal core are chosen to produce a weight-efficient plate. The structure is shown in Figure. 10.9. The plate is simply supported along its long edges. The span l and the load per unit area p are prescribed. The cellular metal core material is assumed to be pre-selected, such that the analysis can be used to explore the

p

IHIIIIIHIIII

Figure 10.9 Sandwich panel under uniform load i

Figure 10.9 Sandwich panel under uniform load effect of different core densities. The objective is to choose the thickness of the core, c, and the thickness of each of the two face sheets, t, so as to minimize the weight of the plate. The width of the plate is assumed large compared with I and the design exercise focuses on a one dimensional, wide plate.

Deflection and failure constraints

The first step in the optimization process is identification of the constraints on failure and deflection. For the sandwich plate under uniform transverse pressure, face sheet yielding and wrinkling must be considered, as well as yielding of the core (Figure 10.10). Deflection constraints must also be imposed to ensure that deflections of an optimally designed panel do not exceed tolerable limits. Normal crushing (indentation) of the core will not be at issue because the loading pressure, p, will necessarily be small compared with the compressive yield strength of the core material.

In the example considered, the face sheet material is the same as that of the fully dense core material (i.e. ps = pf, asy = of and Es = Ef). The relation between the core modulus and density is taken to be

where a2 is a quality factor, with a2 = 1 applying to a core material having relatively inferior properties and a2 = 4 to a material with properties somewhat higher than those currently available. The core is assumed to be isotropic with

Core Indentation

_Face Yield

- Face Wrinkle

Core

Yield

Bond Failure

Failure Modes

Loading

Figure 10.10 Failure modes in sandwich panels

Gc = Ec/2(1 + vc). The uniaxial yield strain of the core, oy/Ec, is assumed to be no less that of the face sheets, of /Ef. Consequently, a constraint on face sheet yielding ensures that only shear yielding of the core need be explicitly considered. In this example, the yield strength of the core in shear is taken to be

where the quality factor, a3, is about 0.3.

The weight (in units of kg) of the sandwich plate is

Under the uniform lateral pressure, p, the maximum stress in the face sheets is:

This maximum occurs at the middle of the plate, with tension on the bottom and compression at the top. The maximum shear stress in the core, which occurs near the supports, is:

The face sheet wrinkling stress is:

p c y with k = 0.58 (Allen, 1969). For a given core density, W is to be minimized with respect to t and c subject to three strength-related constraints. These constraints follow directly from the two expressions for the maximum stresses and the wrinkling stress:

(face sheet yielding, of ^ o) ct ^ (1/8)l2(p/of) (10.24a)

(face sheet wrinkling, of ^ a) ct ^ (1/8)l2 [p/k(EfE2c)1/3] (10.24b) (core shear yielding, rcy ^ r) c ^ (1/2)l(p/rcy) (10.24c)

From the above it is clear that failure is restricted to either face sheet yielding (10.24a) or wrinkling (10.24b), depending on the smaller of of and of. Note that, with (10.22a), off = kEfa22/3(pc/ps)4/3. Consequently, for an aluminum alloy (Table 10.2) and a core with a quality factor a2 = 1, the wrinkling stress exceeds the yield strength of the face sheets when pc/ps > 0.036. We proceed by considering only core materials with relative densities larger than 0.036, with the consequence that face sheet wrinkling (10.24b) is eliminated as a constraint. Were the core density below 0.036, face sheet wrinkling would supersede face sheet yielding.

The deflection S at the centre of the panel is given by (Allen, 1969; Gibson and Ashby, 1997) (see 10.1).

B-¡Eftc B2Gcc where the first term is due to face sheet stretching and the second is the contribution due to core shear. For the present case, the coefficients are B1 = 384/5 and B2 = 8. More results for a selection of loadings and boundary conditions are given in Table 10.3. The maximum deflection will be required to be no greater than 8.

Based on the dimensionless load and weight indices, n = p/of, f = W/(pfbl2) (10.26)

the two operative constraints in equation (10.24) can be re-expressed as

(core shear yielding) (c/l) ^ (1/2)(of /rcy)n (10.27b)

The corresponding deflection constraint is n (af/Ef)

Table 10.3 Coefficients for laterally loaded panels

Loading

Bi B2 B3 B4

Cantilever

Both ends clamped

Both ends simply supported

Uniform

Central

Uniform

Central

Uniform

Design diagrams

The remaining constraints on the design variables t and c, expressed by equation (10.27), are plotted as three curves on a design diagram (Figure 10.11(a)), using a particular case wherein the load index is n = 10 4 and the maximum allowable deflection is 1/1 = 0.02. The face sheets are assumed to be aluminum and the foam core is also aluminum with pc/pf = 0.1 and a2 = 1. Note that the face yielding constraint is independent of the material, whereas that for core shear is independent of the face sheet thickness, but is strongly affected by the core properties.

The relative core thickness cannot lie below the line for core yielding, nor to the left of the curve for face sheet yielding, thereby excluding solutions within the shaded areas of the figure. Similarly, because of the deflection constraint, the solution cannot be to the left of the maximum deflection curve. Consequently, the optimum resides somewhere along the solid curve ADC. (Any combination of c/l and t/i lying to the right of these constraint curves would have a larger weight than the combination with the same value of c/l lying on the closest constraint curve.) Note that, for this example, failure by core yield does not limit the design because the other two are more stringent. That is, the optimum design is limited either by face sheet yielding, above D, or deflection constraint, below D. Evaluating f along the two segments AD and DC and then determining its minimum gives the optimum. It is found to reside along DC at location X, where t/l = 0.001 and c/l = 0.032. The corresponding weight minimum is obtained from equation (10.6d) as f = 0.00638. That is, for load index n = 10~4, the design is deflection limited. This conclusion changes at larger n, for reasons explained below.

Core shear

Figure 10.11 (a) A design map based on panel dimensions for Al alloy sandwich panels at specified load index (Tl = 10~4) and core density (Pc/Ps = 0.1), subject to an allowable displacement (E/l = 0.02). (b) A schematic design map using modified coordinates suggested by equation (10.7), showing trends with core properties and allowable stiffness. Line (3) refers to core shear, line (1) to the stiffness constraint and line (2) to face yielding

Pc/Ps

Core shear

Figure 10.11 (a) A design map based on panel dimensions for Al alloy sandwich panels at specified load index (Tl = 10~4) and core density (Pc/Ps = 0.1), subject to an allowable displacement (E/l = 0.02). (b) A schematic design map using modified coordinates suggested by equation (10.7), showing trends with core properties and allowable stiffness. Line (3) refers to core shear, line (1) to the stiffness constraint and line (2) to face yielding

The example has been used to illustrate the process used to minimize the weight subject to constraints on failure and deflection. In the present instance, the process can be carried out analytically. Often, however, a straightforward numerical approach is the simplest and most effective way to determine the optimal design. The preceding example may be used to illustrate the numerical methodology:

1. Form a rectangular grid of points (f/£, c/£) covering the potential design space, such as that in Figure 10.11(a).

2. Determine whether each point (f/£, c/£) satisfies all the constraints (10.27): if it does not, reject it; if it does, evaluate f.

3. The point which produces the minimum f will be close to the optimum design. The grid of points can be further refined if necessary.

Dependence on load index and core density

The above procedures can be used to bring out the regimes within which the design is limited by stiffness or strength. Some assistance is provided by reorganizing equation (10.27) and replotting the design diagram (Figure 10.11(b)). The constraints are:

f 2 ofJEf ofJGf 1

By plotting (1/n)(c/£) against f/€ using logarithmic axes (Figure 10.11(b)) the respective roles of the relative properties for the core, af /r£ and the stiffness constraint, 1/1, become apparent. The minimum weight design lies along DCE. The precise location depends on the specifics of the core properties and the allowable stiffnesses. As the core properties deteriorate, at given allowable stiffness, line (3) moves up and the minimum weight is now likely to reside along segment CE, being controled by the core. Correspondingly, for a given core material, as the stiffness allowable increases, line (1) displaces downward towards the origin, again causing the minimum weight to reside along CE and be core-controled. Conversely, improvements in the core properties and/or a lower allowable stiffness cause the minimum weight design to reside along DC, such that the panel is stiffness-controled.

Specific results are plotted in Figure 10.12 for four values of the relative core density, with 8/1 = 0.01. Again, the face sheet material and cellular core are taken to be aluminum with a2 = 1. Note that the designs are indeed stiffness-limited at low values of the load index and strength-limited at high load indices. The lower (solid) portion of each curve, below the change in slope, coincides with the former; the two strength constraints being inactive. Moreover, for pc/ps = 0.3, the deflection constraint is active over the entire range. At higher load indices (dotted above the change in slope), either or both strength constraints (10.27(a)) and (10.27(b)) are active, with the deflection at the design load being less than 8.

Strength limited

Strength limited

Figure 10.12 Weight index as a function of load index for optimally designed Al alloy sandwich panels subject to an allowable displacement (S/i = 0.01)

Figure 10.12 Weight index as a function of load index for optimally designed Al alloy sandwich panels subject to an allowable displacement (S/i = 0.01)

At a low load index, the core with the lowest relative density among the four considered gives the lowest weight structure. At a higher load index, a transition occurs wherein higher core densities produce the lowest weight. Plots such as this can be used to guide the optimal choice of core density.

10.7 Stiffness-limited designs

Sandwich structures

Panels subjected to lateral loads are often stiffness-limited, as exemplified by the panels in the previous section subject to lower load indices. The optimum configuration lies away from the failure constraints and corresponds to a fixed ratio of deflection to load, i.e. to a prescribed stiffness. Stiffness also affects the natural vibration frequencies: high stiffness at low weight increases the resonant frequencies. Minimum weight sandwich panels designed for specified stiffness are investigated on their own merits in this section. The results can be presented in a general form.

The concepts can be found in several literature sources (Allen, 1969; Gerard, 1956; Gibson and Ashby, 1997; Budiansky, 1999). The key results are reiterated to establish the procedures, as well as to capture the most useful results. For laterally loaded flat panel problems, non-dimensional coefficients (designated B) relate the deflections to the applied loads. Details are given in Section 6.3. The key results are repeated for convenience in Table 10.3. Their use will be illustrated throughout the following derivations.

An analysis based on laterally loaded sandwich beams demonstrates the procedure. The results are summarized in Figure 10.13, wherein the weight is minimized subject to prescribed stiffness. The construction of this figure, along with the indices, will be described below. The global weight minimum has the core density pc as one of the variables. This minimum is the lower envelope of the minimum weight curves obtained at fixed pc: three of which are shown.

Minimum weight panels

TO ei

Global minimum Y = 3.19X3/5 Pc/Ps = C.59X275

- Global minimum

---Constant pc minima

C.C2

C.C4

C.C6

C.C8

x= sj~b1l3a2b2l8)312 Stiffness index

Global minimum Y = 3.19X3/5 Pc/Ps = C.59X275

- Global minimum

---Constant pc minima

C.C2

C.C4

C.C6

C.C8

x= sj~b1l3a2b2l8)312 Stiffness index

Figure 10.13 Minimum weight panels

Figure 10.13 Minimum weight panels

To construct such plots, the stiffness S and weight W are first defined. With S as the deflection and P as the transverse load amplitude, the compliance per unit width of panel can be obtained directly from equation (10.25) as (Allen, 1969):

5 P/S BxEftc2 B2cGc

The first term is the contribution from face sheet stretching and the second from core shear. The result applies to essentially any transverse loading case with the appropriate choice for B1 and B2 (Table 10.3). The weight index is:

These basic results are used in all subsequent derivations.

The global minimum

In the search for the global optimum, the free variables are t, c, and pc, with due recognition that Gc in equation (10.29) depends on pc. If the core density pc is taken to be prescribed so that Gc is fixed, the optimization proceeds by minimizing ty with respect to t and c for specified stiffness. Inspection of equation (10.29) reveals that the most straightforward way to carry out this process is to express t in terms of c, allowing equation (10.29) to be re-expressed with c as the only variable. For this problem, the expressions are sufficiently simple that the minimization can be carried out analytically: the results are given below. In other cases, the minimization may not lead to closed-form expressions. Then, the most effective way to proceed is to create a computer program to evaluate f (or W itself) in terms of c and to plot this dependence for specified values of all the other parameters over the range encompassing the minimum. Gibson and Ashby (1997) have emphasized the value of this graphical approach which can be extended to consider variations in core density simply by plotting a series of curves for different pc, analogous to what was done in Figure 10.12.

If the global minimum is sought, the dependence of the shear modulus of the core must be specified in terms of its density. In the following examples, the material comprising the face sheets is assumed to be the same as the parent material for the core (Ef = Ec = Es, when pc = ps = pf). The dependence of Young's modulus of the core on pc is again expressed by equation (10.22a). Taking the Poisson's ratio of the cellular core material to be | (Gibson and Ashby, 1997) then Gc/Ef = (§)(pc/ps)2.

Although it seems paradoxical, the search for the global optimum gives rise to simpler expressions than when the core density is fixed. The result of minimizing f with respect to t, c and pc with S prescribed is:

Note that t, c and pc are explicitly defined at the global minimum. It is readily verified that the globally optimized beam has the following two characteristics: (1) the compliance, S-1 = 6Û/(B1Eftc2), has exactly twice the contribution from the core as from the face sheets (the second term in equation (10.29) is twice the first term); (2) perhaps more surprisingly, the weight of the core is exactly four times that of the combined weight of the two face sheets.

At the minimum, equation (10.31) enables the weight index, Y, to be expressed in terms of the stiffness index X,

These are the two non-dimensional quantities plotted as the global minimum in Figure 10.13. They contain all the information needed to characterize the support and load conditions encompassed by Table 10.3, inclusive of the coefficient determining the stiffness of the cellular metal, a2. The core density at the global minimum can also be expressed as a function of X:

Fixed core density

Return now to the minimization of weight with pc fixed. This is done for the same class of sandwich beams: parent core material the same as the face sheet material. Minimization of weight at prescribed stiffness now relates c and t to a free parameter £ such that,

Each value of £ generates a minimum weight beam for the fixed core density, with the stiffness specified by the index X, defined in equation (10.11), now given by:

The associated non-dimensional weight index defined in equation (10.11) is

Curves of Y as a function of X are included in Figure 10.13 for three values of the relative core density, pc/ps. Each curve necessarily lies above the global optimum, touching that curve only where its core density happens to coincide with that of the globally optimized sandwich beam. Note, however, that the minimum weight beams with fixed core density exhibit substantial stiffness and weight ranges over which they are close to optimal. For example, the beams with pc/ps = 0.2 have weights which are only slightly above the global minimum over the stiffness range, 0.02 < X < 0.1.

The results presented in Figure 10.13 have the merit that they are universal, encompassing a variety of support and load conditions through the dimension-less axes. The graphical approach (Gibson and Ashby, 1997) could have been used to produce the same results for specific sets of support and load conditions. For problems with greater geometric complexity, the graphical approach might well be the most effective way to seek out the lowest weight designs.

Failure limits

Application of these weight diagrams is limited by the occurrence of the various failure modes sketched in Figure. 10.10: yielding either of the face sheets or in the core, and face wrinkling. These phenomena govern the maximum load at which the beam responds elastically. The illustration in Section 10.6 has elaborated the consequences. Some additional considerations are given in this section.

Reiterating from equation (10.24a), face yielding commences when the maximum tensile or compressive stress caused by bending reaches the yield strength, af. For a given set of support and loading conditions, the maximum stress in the face sheet is determined by the coefficient B3 in Table 10.3, such that yielding in the face sheets commences when the transverse load satisfies bB3ct f

This condition can be re-expressed in terms of the stiffness index X defined in equation (10.32) and superimposed on Figure 10.13. That is, the condition that the face sheets of the globally optimal beam remain elastic will be violated if

Stiffness index, X

Figure 10.14 The relationship between X and Y showing the region bounded by core and face yield

Stiffness index, X

Figure 10.14 The relationship between X and Y showing the region bounded by core and face yield

For specific conditions, the equality of (10.38) corresponds to a point on the curve of weight versus stiffness (Figure 10.14) below which the elastic predictions are no longer valid. For values of the stiffness index below this point, weights in excess of the global minimum would be needed to ensure that the beam remains elastic. Logarithmic axes have been used in Figure 10.14 to highlight the inadmissible range.

Analogous conditions exist for core yielding (equation 10.24(c)). The coefficient B4 is defined such that the maximum shear stress in the core is P/(B4bc), and thus yielding occurs when

This condition can also be written in terms of X for the case of the globally optimized beams. Core yielding invalidates the results based on the elastic optimization if

48 [B4\a2B2J \bit/y Elastic wrinkling of the face sheets may also occur (equation (10.24b))

Note that, as the stiffness index increases, the face sheet thicknesses needed to achieve minimum weights increase substantially, relative to core thickness and density. Consequently at lower stiffnesses yielding is more likely to intervene because of the thinner face sheets and lower core densities at the global weight minimum (equation (10.31)). For yielding to be avoided, the loads on the structure must by limited by equations (10.37) and (10.39).

Stiffened panels

The principal competitors for sandwich systems subject to biaxial bending are waffle-stiffened panels (Figure 10.15). For comparison, it is convenient to re-express the result for the globally optimized sandwich beam (equation (10.32)) in the form:

For a waffle panel subject to bending about one of the stiffener directions, the weight and stiffness are related by:

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