Presently, four unidirectional pre-preg materials containing from 1 to 16 plies are used. 2. Identify and calculate (or estimate) all laminate properties. This is itself ...
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Designing composite structures: lay-up selection P M Weaver Department of Aerospace Engineering, University of Bristol, Queen’s Building, University Walk, Bristol BS8 1TR, UK
Abstract: The use of high-performance composite materials of the sort used in the aerospace and Formula 1 industries is becoming more widespread. On one level they are exciting to work with because they give scope for designing the ‘material’ in addition to a ‘structure’ through judicious placement of the ply orientation. Layers are stiff and strong in the fibre direction while weak and compliant in the transverse direction. It is this ability to tailor material properties layer by layer that gives designers huge potential in design. One possible explanation for the prevalent use of quasiisotropic (‘black aluminium’) carbon composites in structures is the lack of available design tools. Analysis packages exist that will predict performance, but only for a given choice of fibre orientation. Here a design tool is presented that aids selection of fibre orientations. Optimization of laminate fibre angles is difficult for multiple-load cases and objectives since there are many local minima to assess. The alternative approach that is presented here, for flat plates and cylindrical shells, circumvents the need (in the early stages of design) for conventional optimization strategies that often prove difficult and complicated to implement. The basic idea is to build a database that stores appropriate properties of all permutations of lay-up angles for a laminate. Rather than access these properties by a question and answer, black-box technique, a graphical method is proposed. The designer can select viable laminates by first plotting a succession of two-dimensional charts containing relevant properties. Then, using simple on-screen techniques, the number of potential laminates is visually reduced by selecting those with desirable properties. Two case studies are presented to illustrate the selection method. The first concerns the optimization of a spar web, typically found in an aircraft wing structure, while the second concerns the optimization of a cylindrical shell, subject to axial compression, that undergoes simultaneous Euler-type buckling and local buckling. Keywords:
design, buckling, optimization, composite structures, laminate selection 1
NOTATION b Ii K l Mi Ni P R Ri t ym
s t
plate width second moment of inertia about the laminate mid-plane plate curvature tube length bending moment resultant in-plane force resultant applied compression load tube radius radius of curvature plate (laminate) thickness perpendicular distance from the midplane to the surface of the plate direct stress shear stress
The MS was received on 30 March 2001 and was accepted after revision for publication on 13 May 2002. G00901
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INTRODUCTION
Most laminate software calculates stiffnesses and strengths for a particular composite lay-up. Undoubtedly this approach is important from the analysis perspective. However, in design the opposite approach is often needed. Here, material orientations and ply thickness must be chosen to satisfy given stiffnesses and loads. In practice, most laminate design is based on the choice of just four fibre angles (08, 908 and +458). Even by restricting the choice of angles to four allows a huge number of possible lay-ups to be chosen. For an n-layer laminate there are 4n potentially different laminates; thus for a 16-layer laminate which may have a plate thickness of as little as 2 mm there are in excess of 46109 permutations—a staggeringly high number. In practice, designers restrict the actual number of viable alternatives, typically by eliminating undesirable coupling responses and so making the laminate balanced and symmetrical. Even so, the number of possible choices remains more than 12 000. There are plenty of rules for choosing laminate configurations used in industry. Two Proc Instn Mech Engrs Vol 216 Part G:
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examples of these include using a maximum of four consecutive plies of the same orientation so as to reduce the likelihood of transverse cracking and placing 458 plies on the surface to maximize in-plane buckling resistance. In addition, some designers will specify a minimum number of plies of a given orientation to give strength in directions that are not primary load paths. Such guiding rules coupled with experience help to reduce the number of viable laminate choices. Furthermore, there are analytical results that identify optimal fibre angles for various buckling cases [1–3]. Unfortunately, identifying optimal laminates for multiple-load cases is non-trivial and little guidance appears to be available to designers, who often appear to use quasiisotropic configurations which may lead to a potentially missed opportunity in terms of minimizing weight. With this in mind, a novel approach has been adopted that builds upon the technique of materials selection using databases as developed by Ashby and co-workers [4–7]. The basic idea is to plot a succession of two-dimensional material property charts and identify a subset of materials that perform well on each chart. The overall best materials are those that perform well on all selection charts. Instead of different materials, the same principle is applied here to laminates of differing stacking sequences. The proposed design tool is of most use in the early stages of design and can be used to identify a small subset of potential laminates that can be investigated in more detail. It builds upon the techniques of material selection and, as such, is used within the Cambridge Materials Selector software environment [8]. Initially, a description of the method is given paying particular attention to the number and type of laminate stacking sequences in the database and the material and structural properties stored. This is followed by a detailed description of the properties in the database and by two case studies. These examples are chosen to show the power of the selection methodology presented but do not use all of the properties in the database. This would be an impossible task because there are potentially an infinite number of such charts depending on the combination of properties plotted on an axis. In order to introduce the method, only single property axes are plotted. However, it should be noted that compound properties are easy to form. A simple compound property to plot is cost per unit width. This gives an indication of material costs in design and is obtained from the compound property of cost per unit kg, Cm, multiplied by mass per unit area, m, and again multiplied by thickness of laminate, t, to give an overall property of cost per unit width.
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DESCRIPTION OF DATABASE
Admirable laminate selection requires a degree of optimization. The fewer constraints and objectives placed upon a particular laminate the easier this process becomes. Indeed, as previously mentioned, there are analytical formulae and methods available that help the designer when there is only one objective and one constraint. A typical example might be the minimization of weight of a laminate subject to a specific stiffness. However, if there is also a need to trade weight with cost, strength and various buckling constraints, analytical methods become more difficult—though attempts have been made with multiple objectives [9]. While it is not intended for the end-user of the selection method to form their own database the strategy adopted needs justification and, as such, is outlined below: 1. Decide the materials of interest to populate the database. Presently, four unidirectional pre-preg materials containing from 1 to 16 plies are used. 2. Identify and calculate (or estimate) all laminate properties. This is itself an onerous task. Although easy for in-plane stiffness properties, a new strategy must be identified for out-of-plane properties that are a function of D matrix terms. Furthermore, there are, as yet, no universally agreed properties for predicting strength and toughness in a design situation. However, some of these data are thought to be a function of those properties stored in this database and so may be plotted as compound properties (i.e. a property that is a function of one or more other laminate properties). Of course, this function does need defining. 3. Determine all permutations of lay-up angles and calculate laminate properties to store in the database. This task is simplified by restricting use to balanced, symmetric laminates. Once the database has been completed, potential laminates may be identified using the selection methodology outlined here: 1. Plot the required number of two-dimensional property charts. The number of charts is given by the number of laminate properties of interest (as dictated by the design requirements) divided by 2, rounded up to the nearest integer. This number is not to be confused with the number of laminates in the database. 2. Identify, on each chart, a subset of ‘good performing’ laminates. 3. Identify a global subset of laminates for all charts. These are the laminates of interest. By relaxing or tightening objective-type properties on the appropriate chart, to a greater or lesser extent, laminates that are potentially nearly optimal may be identified. G00901 # IMechE 2002
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The pre-pregs included [10] are common unidirectional types based on two types of carbon fibre, Kevlar 49 and E-glass, all with epoxy resins. The laminate properties stored in the database and an appropriate brief description are give below.
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PROPERTIES IN THE DATABASE
3.1
General properties
Laminate thickness, t (m) The thickness of the laminate in metres. Mass per unit area, m (kg/m2) The mass in kilograms of one square metre of the laminate. Cost per unit mass, Cm ($/kg) The price, Cm, per kg of the laminate. The price per m2, Ca, can be found by forming the quantity Ca ¼ Cm m
Structural properties
In-plane stiffnesses A11, A22, A12, A66 [10] (MN/m) For a balanced, symmetric laminate, the linear-elastic response is described by 2 3 2 32 3 e1 A11 A12 N1 4 N2 5 ¼ 4 A12 A22 54 e2 5 ð1Þ A66 N6 e6 where A11, etc., are the in-plane elastic stiffnesses (MN/ m) and the directions of the stresses per unit thickness, N, and the strains, e, are shown in Fig. 1. The in-plane strains are related to the in-plane stresses per unit thickness by the compliance matrix 2 3 2 32 3 a11 a12 N1 e1 4 e2 5 ¼ 4 a12 a22 54 N2 5 ð2Þ a66 e6 N6 where A22 A66 A11 A66 , a22 ¼ jAj jAj A12 A66 A11 A22 A212 a12 ¼ , a66 ¼ jAj jAj with jAj ¼ A11 A22 A212 A66
a11 ¼
ð3Þ
Selection is usually based on the effective Young’s moduli Exx and Eyy and the in-plane shear modulus Gxy . They are defined in Section. G00901
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The flexural stiffnesses D11, D12, D22, D66 [10] (N m) The flexural stiffness of laminates depends on the stacking sequence of the plies. As before, it is assumed that the laminate is symmetric and balanced, and that ply and laminate strains are linear functions of the thickness coordinates z. Then the average bending moment, M, is related to average curvature, K, by 2 3 2 32 3 M1 D11 D12 D16 K1 4 M2 5 ¼ 4 D21 D22 D26 54 K2 5 ð4Þ M6 D16 D26 D66 K6 where D is the flexural stiffness matrix. Data for D11, D22, D12 and D66 are stored in the database. The other off-diagonal stiffness, D16, is captured by the anisotropy factors defined in Section 3.6. The strains are related to the stresses by 2 3 2 32 3 d11 d12 d16 M1 K1 4 K2 5 ¼ 4 d21 d22 d26 54 M2 5 ð5Þ K6 d16 d26 d66 M6 where D22 D66 D226 D11 D66 D216 , d22 ¼ , jDj jDj D16 D26 D12 D66 d12 ¼ jDj D11 D22 D212 D12 D26 D22 D16 , d16 ¼ , d66 ¼ jDj jDj D12 D16 D11 D26 d26 ¼ jDj with jDj ¼ D11 D22 D212 D66 þ 2D12 D26 D16 d11 ¼
using the ability to form a compound property.
3.2
3
D11 D226 D22 D216
ð6Þ
Selection is usually based on an effective flexural Young’s modulus and shear modulus. Values for these are given in Section 3.4.
3.3
Effective material properties, in-plane
Selection involves comparisons. Comparing laminate composites with conventional, uniform, monolithic materials is difficult because the ranking depends on how they are loaded. A composite might, for instance, be stiffer in tension but more flexible in bending than a comparable material like steel. For this reason, a number of effective properties are listed. The effective in-plane properties allow comparison when stretching or in-plane shear are dominant. Each record contains values for the following in-plane properties. Young’s modulus (longitudinal) Exx, (transverse) Eyy (GPa) The effective Young’s modulus, relating stress to strain for stretching is defined by Proc Instn Mech Engrs Vol 216 Part G:
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Fig. 1 Schematic of types of loading on a composite laminated plate
Eii ¼
these criteria, which are well known to composites designers, can be found in reference [10].
1 aii t
Shear modulus Gxy ¼ Gyx (GPa) The effective shear modulus for in-plane shear is defined by Gxy ¼
1 a66 t
Poisson’s ratios nxy , nyx The effective Poisson’s ratio for in-plane loading is given by nxy ¼
a12 , a22
nyx ¼
a21 a11
Uniaxial strength (Tsai–Hill) Sth, (maximum strain) Sme, (maximum stress) Sms, (Tsai–Wu) Stw (MPa) This is the stress at which failure starts (i.e. first-ply failure) by each of four standard criteria. Details of Proc Instn Mech Engrs Vol 216 Part G: J Aerospace Engineering
Ultimate strength (maximum strain) Sult (MPa) This is the uniaxial strain at which the composite fails completely. For angle-ply laminates +y first-ply failure strength is used. Shear strength (Tsai–Hill) Tth, (maximum strain), Tme, (maximum stress) Tms, (Tsai–Wu) Ttw (MPa) This is the shear stress at which fracture starts (i.e. firstply failure) by each of four standard criteria.
3.4
Effective material properties, bending
Flexural modulus (longitudinal) Exxb (transverse) Eyyb (GPa) These moduli allow bending stiffnesses to be calculated. For bending about an axis normal to the longitudinal G00901 # IMechE 2002
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direction (subscript 1) the stiffness is M1 R1 ¼ Exxb I1
3.5 ð7Þ
where M1 is the moment, R1 the radius of curvature and I1 the appropriate second moment of area. For bending about an axis normal to the transverse direction the stiffness is defined by M2 R2 ¼ Eyyb I2
ð8Þ
5
Quasi-isotropic properties
Quasi-isotropic Young’s modulus, Eiso (GPa) and quasiisotropic Poisson’s ratio niso The quasi-isotropic Young’s modulus and Poisson’s ratio are characteristic of the fibre and matrix, and independent of lay-up. They are the properties of a homogeneous ð0, +45, 90Þn composite. These properties give an accurate first indication of the average stiffness properties of the composite laminate system. Efficient design will aim to improve upon these properties.
with similar definitions. Alternatively, these moduli may be defined from the d matrix as 3.6 Exxb ¼
12 , t3 d11
Eyyb ¼
12 t3 d22
Shear modulus Gxyb (GPa) The effective shear modulus in bending is defined from the d matrix as Gxyb ¼
12 t3 d
66
Poisson’s Ratio (Longitudinal) nxyb , (transverse) nyxb These are the values of Poisson’s ratio associated with bending about axes normal to the longitudinal and transverse directions: nxyb ¼
d12 , d22
nyxb ¼
d21 d11
ð9Þ
Failure of composites in bending can be treated in the same way, using one of four alternative definitions of strength, listed in the heading of this section. As a warning, it should be borne in mind that because strength properties, in general, vary layer by layer, failure may not occur in the outermost ply. The failure criterion for loading about an axis normal to the longitudinal direction (which is generally the one of interest) is then M1 S ¼ ym I1 where S is the chosen failure strength. G00901
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Flexural stiffness cannot, in general, be orthotropic if off-axis plies, balanced and symmetric (or not), are present. The first two anisotropy factors characterize the coupling between flexural and twisting deformation (and are present even for balanced, symmetric laminates). They give a relative indication of the detrimental effect that flexural/twist anisotropy has on buckling loads. Numerical studies have shown that this coupling is negligible on flat plates for values of the anisotropy factor of less than 0.18 [11[. Note that this cannot be assumed for curved panels or cylindrical shells where the effect is more marked for similar values [12]. Isotropic (or quasi-isotropic) plates have zero values. They are defined [11] as Anisotropy factor AF1 ¼
Bending Strength (Tsai–Hill) Sthb, (maximum strain) Smeb, (maximum stress) Smsb, (Tsai–Wu) Stwb (MPa) A uniform beam with material of isotropic properties fails in bending when the stress in the ligaments furthest from the neutral axis (at ym) first exceeds the yield or fracture strength, sf , of the material of which it is made: M sf ¼ I ym
Anisotropy factors AF1, AF2 and AF3
ð10Þ
Anisotropy factor AF2 ¼
D16 D311
D22 D26
D322
D11
1=4
ð11Þ
1=4
ð12Þ
The twisting coefficient gives an effective measure of the torsional rigidity of a plate. Its values typically range from 0.25 to approximately 4, with larger values associated with greater torsional rigidity. Isotropic plates have a value of 1. In general, flat plates with larger values of AF3 are more buckling resistant [11]: Twisting coefficient AF3 ¼
3.7
D12 þ 2D66 ðD11 D22 Þ1=2
ð13Þ
Buckling constants
The buckling of isotropic plates and tubes is characterized by simple formulae [13]. Buckling of composite plates and tubes can be described by similar formulae if the quasi-isotropic Young’s modulus is multiplied by an appropriate orthotropic buckling constant. These buckling constants reflect the departure from isotropy by varying lay-up. Proc Instn Mech Engrs Vol 216 Part G:
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3.7.1
with
Buckling of a plate, under uniaxial, bending and shear loads
These three buckling constants allow prediction of the stress at which a long plate, simply supported at its edges, will buckle when loaded. (a) Uniaxial loads parallel to the long direction of the plate The buckling stress of a long flat, simply supported isotropic plate subject to uniaxial compression loading [13] is saxial ¼
p2 E t 2 3 1 n2 b
ð14Þ
kp2 ¼ 109:6
i 2p2 hpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D ð þ 2D Þ D þ D 11 22 12 66 tb2
saxial ¼ 4:4
p2 Eiso t 2 3 1 n2iso b
kp1 ¼ 6
1 n2iso
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D11 D22 ð1 þ AF3Þ Eiso t3
1=4 4 D11 D322 ð8:125 þ 5:05AF3Þ 2 tb for AF3 < 1
saxial ¼
ð16aÞ
ð16bÞ
ð17Þ
where E is the Young’s modulus and t and b the plate thickness and width respectively. The corresponding formula for orthotropic plates [14] is
109:6 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1:25 þ 1 sbend ¼ D D ð18Þ 11 22 tb2 AF3 which is rearranged into the following more convenient form: sbend ¼ 20:5kp2
Eiso t 2 1 n2iso b
Proc Instn Mech Engrs Vol 216 Part G: J Aerospace Engineering
1=4 1=2 4 D11 D322 AF 2 tb
0:532 0:938 þ 6 11:7 þ AF3 AF32
ð21Þ
for AF3 > 1 which is rearranged into the following more convenient form:
(b) In-plane bending of the plate The buckling stress of a long flat, simply supported isotropic plate subject to in-plane bending [13] is E t 2 1 n2 b
ð20Þ
or
A value of kp1 ¼ 1 corresponds to a quasi-isotropic layup. Note that larger values are correspondingly better. Therefore its value is always a measure of the buckling resistance of a laminate relative to the standard quasiisotropic laminate.
sbend ¼ 20:5
E t 2 1 n2 b
saxial ¼
ð15Þ
with
ð19bÞ
where E is the Young’s modulus and t and b the plate thickness and width respectively. The corresponding formula for orthotropic plates [14] is
which is rearranged into the following more convenient form: saxial ¼ kp1
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D11 D22 ð1:25 þ 1=AF3Þ Eiso t3
(c) Shear loads applied along the long edges of a long plate The buckling stress of a long flat, simply supported isotropic plate subject to shear [13] is
where E is the Young’s modulus and t and b the plate thickness and width respectively. The corresponding formula for orthotropic plates [14] is saxial ¼
1 n2iso
ð19aÞ
saxial ¼ kp3 4:4
Eiso t 2 1 n2iso b
ð22aÞ
with
1=4 1 n2iso D11 D322 ð8:125 þ 5:05AF3Þ kp3 ¼ 0:91 Eiso t3 for AF3 < 1 1=4 1=2 AF 1 n2iso D11 D322 6 1:7 þ 0:532=AF3 þ 0:938=AF32 kp3 ¼ 0:91 Eiso t3 for AF3 > 1 ð22bÞ A value of kp3 ¼ 1 corresponds to a quasi-isotropic layup with larger values correspondingly better. Therefore its value is always a measure of the buckling resistance of a laminate relative to the standard quasi-isotropic laminate.
3.7.2 Buckling of circular tubes Similarly to flat plates, three constants allow the stress at which buckling will take place to be calculated from isotropic formulae, replacing Young’s modulus by the G00901 # IMechE 2002
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quasi-isotropic modulus multiplied by the appropriate buckling constant.
form: sbend ¼ 0:3kc2
(a) axial compression of circular tubes The theoretical buckling stress of an isotropic circular cylindrical shell subject to axial compression [13] is saxial
E t ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 R 3ð 1 n Þ
ð23Þ
where R is the radius. The shell is assumed to be thin walled and to be of proportions such that shallow-shell theory applies [12] and boundary conditions exert a minimal influence. It is noted that the presence of imperfections reduces the buckling load and some design codes suggest using a knockdown factor of 50 per cent [15]. The corresponding formula for orthotropic shells [16] is " # l4n 2 2 saxial ¼ t D11 lm þ 2ðD12 þ 2D66 Þln þ D22 2 lm " # Al2m ð24Þ þ R2 A11 l4m þ A6 l2m l2n þ A22 l4n which is rearranged into the following, more convenient, form: Eiso t ffi saxial ¼ kc1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 1 n2iso R
ð25aÞ
7
Eiso t 1 n2iso R
ð28aÞ
with pffiffiffiffiffi 12 1 n2iso pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kc2 ¼ 2 A11 D22 t Eiso
ð28bÞ
(c) Torsion of circular tubes The approximate buckling stress of a moderately long isotropic circular cylindrical shell subject to torsion [13, 19] is ttorsion ¼ 0:76
E t5=4 2 3=4 1 n R l 1=2
ð29Þ
The corresponding formula for orthotropic shells [19] is 5=8
ttorsion ¼ 3:6
3=8
D22 A11 tR3=4 l 1=2
ð30Þ
which is rearranged into the following more convenient form: ttorsion ¼ 0:76kc3
Eiso t5=4 1 n2iso R3=4 l 1=2
ð31aÞ
with 5=8
kc3 ¼ 4:74
3=8
1 n2iso D22 A11 Eiso t9=4
ð31bÞ
with h i ðA11 A22 A2 Þl2 l4 D11 l2m þ 2ðD12 þ 2D66 Þl2n þ D22 l2n þ R2 A l4 þ A A A2 =A 122Am ðl2 =l2 ÞþA l4 f 11 m ½ð 11 22 12 Þ 66 12 m n 22 n g m " # kc1 ¼ t R
ð25bÞ
Eiso pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 3ð1n2iso Þ
and lm ¼ mp=l and ln ¼ n=R, where m and n are the numbers of longitudinal half-waves and circumferential waves respectively. Integer numbers of m and n are determined such that kc1 is minimized.
Note that all of the buckling constants have a value of unity for isotropic lay-ups and their values give a direct measure of the improvement (or otherwise) as compared with black aluminium (quasi-isotropic lay-ups).
(b) Brazier collapse of circular tubes The theoretical buckling stress of a long isotropic circular cylindrical shell subject to pure bending [17] is
3.8
sbend ¼ 0:3
E t 1 n2 R
ð26Þ
The corresponding formula for orthotropic shells [18] is sbend ¼
3:42 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A11 D22 ptR
ð27Þ
which is rearranged into the following more convenient G00901
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Hygro-thermal properties
Coefficient of thermal expansion, CTE ð106 =KÞ The longitudinal coefficient of linear thermal expansion, CTE1, is the longitudinal strain associated with a change of temperature of 1 K. The quantity CTE2 characterizes thermal expansion in the transverse direction. Note that, for certain fibres and lay-ups, one or other CTE can be near zero or, occasionally, negative. The CMS system is unable to store negative values for material properties, so the default value 0:0016106 =K is used. Proc Instn Mech Engrs Vol 216 Part G:
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Coefficient of Moisture Expansion, CME (%/%) The moisture content, c, is the weight of water taken up by the composite divided by the weight of the dry composite. The longitudinal coefficient of moisture expansion, CME1, is the strain associated with unit change of moisture content c. The quantity CME2 characterizes moisture expansion in the transverse direction.
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DESCRIPTION OF PROPERTY CHARTS
The database, in its present form, contains in excess of 48 000 laminates. In practice, many similar laminates have near-identical material properties. By acknowledging this fact it is possible to reduce the number of laminates in the database quite considerably. This is achieved by storing properties in a simple range—from a lower to an upper bound. On a selection chart the material will be displayed as an ellipse that is drawn through the pair of upper and lower bound values of both properties. Each ellipse identifies a particular class of laminate. The reduction of the number of laminates in the database is a powerful means of aiding the designer. This requires some justification. Many nearidentical laminates will have near-identical laminate properties and for most design applications the designer will not be able to distinguish between them. Thus, by storing a general class of such laminates in the database the designer will be able to choose a particular lay-up that lies within that class. Presently, three laminates are stored in the database for a given number of ply angles. This is explained in more detail by way of an example. Consider a quasi-isotropic laminate with 25 per cent 08, 25 per cent 908 and 50 per cent +458 angle plies. The three laminates stored, all symmetric and balanced, would be (a) the most homogeneous and (b) and (c), the two extreme cases of inhomogeneity, i.e. (0, 45, 45, 90, 45, 90, 0, 45)s, (02, 452, 452, 902)s and (902, 452, 452, 02)s respectively. The method presented herein has similarities to carpet plots [20], but instead of plotting one material property (such as Young’s modulus) against the percentage number of 458 plies (with superimposed contours of the percentage number of 908 plies), one material property is plotted against another. There is no restriction on the type of engineering properties that may be plotted. A particular strength of this database is thought to be the consideration of properties that are based on out-of-plane bending (i.e. D terms), a feature that is not contained in carpet plots where the properties are based on homogeneous laminates. This feature allows designers to take advantage of non-homogeneous stacking sequences in structures subject to buckling and out-of-plane bending, for example. Another virtue of the database technique is that the Proc Instn Mech Engrs Vol 216 Part G: J Aerospace Engineering
need for complex mathematical analysis is circumvented while determining optimal lay-ups. Instead, these are obtained directly from the material property charts. The case studies highlight these features. Finally, the use of database and graphical selection techniques based on the properties in the database is well suited to computerization. This would allow a number of selection charts to be formulated such that those layups that pass each selection stage (property charts) are readily identified. The next two sections consider design examples that illustrate the selection strategy. While these examples do not use all of the properties in the database (a task that would require many more case studies and is beyond the remit of the current paper) they do give valuable insight into the merits of the selection methodology.
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STACKING SEQUENCE OPTIMIZATION OF A SPAR WEB—A CASE STUDY
Aircraft are typically built from longitudinal spars and transverse ribs. The function of the spars is to transmit bending loads. As such, a spar web is subjected to both flexural and shear loads. Figure 2 shows the position and origin of bending and shear loads in the main spar web. Depending on the ratio of bending moment to shear force carried by the web a different selection of laminates is optimal. Figure 3 plots the buckling constant for bending of a plate, kp2, against the buckling constant for shear of a plate, kp3, and is appropriate for selecting laminates that are used in spar webs or I-beam webs—only some of the laminate have been labelled for reasons of clarity. Figure 3 shows the potential buckling modes. It is apparent that laminates based on (+458 and 08) perform best. Note that the homogeneous quasiisotropic lay-up has buckling factors equal to unity and that relatively few laminates exceed its performance. Also note that in the software the ellipses are coloured to distinguish differing pre-preg systems.
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STACKING SEQUENCE OPTIMIZATION OF A CYLINDRICAL TUBULAR STRUT—A CASE STUDY
Detailed analysis shows that the expression for minimum weight of a cylindrical shell subject to axial compression may be given by [21]
2=3 m r P 1 ¼ C ð32Þ 1=3 l 2 1=3 l3 ð k E Eiso xx c1 Þ where the mass m is given in terms of the length l of cylinder subject to load P. The material properties are grouped into two terms: those that are indicative of the G00901 # IMechE 2002
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Fig. 2 Location of the spar web in an aeroplane wing
Fig. 3 Example of a selection chart generated using the Cambridge Materials Selector—composites database
composite, i.e. the density r and the quasi-isotropic Young’s modulus Eiso and those indicative of the lay-up, i.e. Exx and kc1 , which are measures of the effect of orthotropy on in-plane Young’s modulus and local buckling of thin-walled composite cylinders. C is a constant. This expression becomes valid for bucklingdriven designs. Note that it is independent of wall thickness and diameter. Therefore to minimize the weight, by optimizing lay-up, of a given composite cylindrical shell subject to axial compression it is necessary to maximize the material coefficients grouping G00901
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Exx kc1 only. To achieve this a plot of kc1 against Exx is created. Superimposed on the chart are contours of either constant mass (given load) or of constant load (given mass). Since these charts have been drawn for 16ply laminates they are of the same mass and so contours of the latter measure are plotted. However, the interpretation of the chart is the same for constant mass or load contours and as such the chart has been labelled with constant mass. The contour indicates better performance towards the top right-hand side of the chart and its shape derives its form directly from Proc Instn Mech Engrs Vol 216 Part G:
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Fig. 4 Effect of the stacking sequence on buckling of a composite circular strut: (a) Cross-ply and ð+452 0x 9012x Þ only; (b) ð+454 0x 908x Þ only; (c) all laminate families Proc Instn Mech Engrs Vol 216 Part G: J Aerospace Engineering
G00901 # IMechE 2002
DESIGNING COMPOSITE STRUCTURES: LAY-UP SELECTION
Table 1 Composite pre-preg material Property Exx (GPa) Eyy (GPa) G12 (GPa) n12 Compressive strength (08) (MPa) Compressive strength (908) (MPa) Shear strength (MPa)
CFRP AS 3501 Epoxy 138 8.96 7.1 0.3 1447 206 93
equation (32). Lay-ups that lie towards the top right of such a chart are more favourable, as shown in Fig. 4* . These plots contain a lot of information. To simplify them, firstly, these charts have been drawn for one composite pre-preg material only (see Table 1) for clarity purposes. Furthermore, Fig. 4c is the complete selection chart (containing approximately 12 000 laminates) while the other two have been plotted for selected laminate families only—once again for reasons of clarity. The charts are now examined in more detail. Because a particular laminate family (as indicated by the same percentages of each fibre angle) plots on a vertical line for a particular value of Exx, the individual data points have been replaced by vertical lines with arrowheads showing the extent of an achievable local buckling factor. It is of interest to note that on these charts individual data points are so closely spaced that they merge into a continuous line. The vertical spread of values of the local buckling factor for a given laminate family arises as a direct consequence of stacking sequence effects on flexural stiffness terms, Dij. For example, in Fig. 4a there are nine families of cross-ply laminates so represented. They are ð9016 Þ, ð02 9014 Þ, ð04 9012 Þ, ð06 9010 Þ, ð08 908 Þ, ð010 906 Þ, ð012 904 Þ, ð014 902 Þ, ð016 Þ. A number of interesting features are observed. Firstly, all cross-ply laminates perform relatively poorly in local buckling of cylindrical shells, as indicated by the maximum value of kc1 of 0.58. Secondly, we the scope can be indicated of values of the local buckling factor and longitudinal modulus obtainable if cross-ply laminates with different percentages of fibre angles were available, noting that this could only occur if more than 16 plies were used. For instance, if families of laminates with 32 plies present had been plotted, twice the number of equally spaced vertical lines would be present in each family. In fact, the hatched envelope shows the extent of properties achievable if an infinite number of plies were used—such an envelope is hereafter referred to as a class. An important conclusion of such studies is that by increasing the number of plies the boundary of the hatched region does not extend by a significant amount.
In contrast to Fig. 3, the chart in Fig. 4 was generated using a spreadsheet, not the selection software. G00901
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From such an analysis it is evident that stacking sequence effects have a large influence on local buckling, as shown by the vertical spread of values for a given family of laminates. A similar envelope is plotted for laminates containing 25 per cent +45 angle plies. Note that all of these laminates perform better than cross-plies. Figure 4b has been drawn for laminates containing 50 per cent +45 angle plies. As expected [20], the optimal fibre angle for local buckling is one that is quasi-isotropic in nature ð04 +454 904 Þ. The particularly wide spread of values for local buckling factor for this laminate family ð0.65 1Þ is of interest in design and reflects stacking sequence effects, and in particular the level of homogeneity in the laminate. Figure 4c shows all laminate families based on 16-ply balanced symmetrical laminates based on 08, 908 and +45 angle plies. Note that other families are present but are hidden beneath those shown. The best lay-up in a particular family is the most homogeneous one, as shown by the lay-up with the highest buckling coefficient within the family. For the 16-ply laminates shown, the best lay-up is [(+45), 03, 90, 02]s, which should be checked for interlaminar failure and other behaviour before a final selection is made. This optimization would prove difficult to obtain via a direct optimization method. Furthermore, the current method has further advantages: it is both quick and allows the designer to check other constraints (by plotting further charts) before a final selection is made.
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CONCLUSIONS
A graphical method for laminate selection, including the effects of the stacking sequence, is presented. The charts rely upon plotting material properties as axes and may be viewed as an extension to carpet plots for use in the earlier stages of design. The technique lends itself particularly well to computerization. Material properties of particular interest in the database include a number of buckling coefficients for various plates and cylindrical shells subject to various loading types, as well as other properties based on flexural stiffness terms. To demonstrate the selection technique two examples are studied. Firstly, candidate laminates are identified to resist shear and flexural buckling of shear webs, an example being an aircraft wing spar. Secondly, the optimal lay-up for cylindrical tubular strut is identified from the family of [0, +45, 90] laminates, example applications of which include truss and space frame structures. This latter example clearly identifies the best laminate, the solution of which is thought to be novel. Furthermore, it establishes an optimal lay-up that otherwise would be difficult to find from conventional direct optimization processes. Proc Instn Mech Engrs Vol 216 Part G:
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P M WEAVER
ACKNOWLEDGEMENTS The author would like to acknowledge the influence of Mike Ashby, with his invaluable insight into the science of materials selection. Thanks are also extended to Ian Farrow for valuable discussions. The author also wishes to thank the Nuffield Foundation for their support of this work under NUF NAL 99.
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REFERENCES 1 Fukunaga, H. Stiffness optimization of orthotropic laminated composites using lamination parameters. Am. Inst. Aeronaut. Astronaut. J., 1991, 29(4), 641–646. 2 Grenestedt, J. L. Lay-up optimisation against buckling of shear panels. Structural Optimisation, 1991, 3(7), 115– 120. 3 Adali, S. Lay-up optimisation of laminated plates. In Buckling and Postbuckling of Composite Plates (Eds G. J. Turvey and I. H. Marshall), 1996 (Chapman and Hall, London). 4 Ashby, M. F. Materials Selection in Mechanical Design, 1992 (Pergamon Press, Oxford). 5 Ashby, M. F. On the engineering properties of materials. Acta Metall. Mater., 1989, 37(5), 1273–1293. 6 Weaver, P. M. and Ashby, M. F. The optimal selection of material and section-shape. J. Engng. Des., 1996, 7(2), 129– 150. 7 Weaver, P. M. and Ashby, M. F. Material limits for shape efficiency. Prog. Mater. Sci., 1997, 41, 61–128. 8 Granta Design Limited, Cambridge Materials Selector, www.granta.co.uk, 1995. 9 Walker, M., Reiss, T., Adali, S. and Weaver, P. M. Application of mathematica to the optimal design of laminated cylindrical shells under torsional and axial
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buckling loads. Engng Comput.: Int. J. Computer-Aided Engng and Software, April 1998, 15(2). Tsai, S. W. Theory of Composite Design, 1990 (Dayton: Think Composites). Nemeth, M. P. Importance of anisotropy on buckling of compression-loaded symmetric composite plates. Am. Inst. Aeronaut. Astronaut. J., 1986, 24, 1831–1835. Weaver, P. M. Anisotropy induced spiral buckling in compression-loaded cylindrical shells. In Am. Inst. Aeronaut. Astronaut. J., 2002 (accepted). Young, W. C. Roark’s Formulas for Stress and Strain, 1989 (McGraw-Hill, New York). Lekhnitskii, S. G. AnisotropicPlates, translated from the second Russian edition by S. W. Tsai and T. Cheron, 1968 (Gordon and Breach, London). Allen, H. G. and Bulson, P. S. Background to Buckling, 1980 (McGraw-Hill, London). Vasiliev, V. V. Mechanics of Composite Materials, 1993 (Taylor and Francis, London). Brazier, L. G. The flexure of thin cylindrical shells and other ‘thin’ sections. Late of the Royal Aircraft Establishment, Reports and Memoranda 1081 (M.49.), 1926, pp. 1– 30. Kedward, K. T. Nonlinear collapse of thin-walled composite cylinders under flexural loading. In Proceedings of the 2nd International Conference on Composite Materials (ICCM2), 1978, pp. 353–365. Simitses, G. J. Instability of orthotropic cylindrical shells under combined torsion and hydrostatic pressure. Am. Inst. Aeronaut. Astronaut. J., 1967, 5(8), 1463–1469. Onoda, J. Optimal laminate configuration of cylindrical shells for axial buckling. Am. Inst. Aeronaut Astronaut J., 1985, 23(7), 1093. Bader, M. G. Materials selection, preliminary design and sizing for composite laminates. Composites Part A, 1996, 27A(1), 65–70. Weaver, P. M. Design of laminated composite cylindrical shells under axial compression. Composites Part B, 2000, 31(8), 669–679.
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