Optimum mass design of prismatic assemblies of plates with longitudinal voids

Computers& SrrucruresVol. 44, No. 3, pp. 557-565, 1992 Printed in Great Britain.

0045-7949/92 fS.GU+ 0.00 0 1992 Pergamon Press Ltd

OPTIMUM MASS DESIGN OF PRISMATIC ASSEMBLIES OF PLATES WITH LONGITUDINAL F. W.

VOIDS

and YE JIANQIAO

WILLIAMS

Division of Structural Engineering, School of Engineering, University of Wales College of Cardiff, Cardiff CF2 IXH, U.K. (Receid

19 June 1991)

Abstract-This paper investigates a new family of fibre laminated plate assemblies. The assemblies covered are prismatic and consist of a series of thin, flat, rectangular plates that have a series of equally spaced voids, which run for the full length of the central layer of the plates and have constant rectangular cross-section. Optimization of an important sub-group of such plate assemblies, namely blade-stiffened panels, is studied parametrically with the objective of minimizing their mass subject to buckling constraint under pure longitudinal compression. Ply thicknesses, fibre orientation and height of the blade stiffeners are the design variables used when optimizing and the variables changed during the parametric study are the number of stiffeners, a nondimensional load parameter and the aspect ratio of the panels. The parametric study shows graphically the effects on the panel mass of the number of voids in the portion between stiffeners, the thickness of individual voids, the number of stiffeners and the effects of prescribing the values of the stiffener height or thickness. It is observed that a substantial, e.g. 16-20%, mass saving can be achieved by using voids in the way advocated. These percentages are very close to those obtained by using the same voids for an individual plate that is identical to the skin between blades and is simply supported.

INTRODUCTION

Stiffened laminated panels are widely used structures in the aerospace and other industries. In the majority of cases they can be entirely represented as a series of long thin rectangular flat plates rigidly connnected together (eccentrically where necessary) along their longitudinal edges. A minimum mass design of such panels is usually sought by optimizing their ply thicknesses, fibre orientations and stiffener sizes, etc. In [I], a new type of fibre laminated composite plate was proposed to achieve a high buckling strength to mass ratio compared with conventional laminates, such that a mass saving of about 19% was reported for a typical individual plate with simply supported edges. The next section [e.g. see Fig. l(a)] shows that the plate has a series of equally spaced voids in its central core layer, which run for the full length of the plate and have a constant rectangular cross-section. The present paper studies the use of plates with voids instead of solid ones as the components of stiffened panels. Detailed results are confined to panels consisting of a skin with blade stiffeners (e.g. see Fig. 3, which is described in a later section) that are optimized subject to a longitudinal compression buckling constraint, i. e. which are designed to have a prescribed buckling load. The design variables are the skin and stiffener thicknesses, fibre orientation and stiffener height. Such a study was needed because optimum design of panels involves both local and overall buckling and whereas the former is made less likely to occur (see the previous paragraph) by intro-

ducing voids, the overall buckling might be expected to be affected to only a minor extent. Thus, smaller mass savings might be expected when using voids for stiffened panels than were achieved for individual plates. However, it will be seen later that the savings for the panels were surprisingly close to those for individual plates, with the probable reason being given at the end of the Conclusions section. Optimization was achieved by using the exact theory [2] of the computer program VICONOPT, in which it is called VIPASA theory. This means that the end conditions of the plate assemblies are assumed to be such that every individual plate is simply supported and that the end cross-sections are free to warp longitudinally during buckling, so that the buckling displacements of every individual plate for all possible modes vary sinusoidally in the longitudinal direction with a common half-wavelength 1. In order to make the plots presented later applicable for different load levels, breadths and lengths, the nondimensional parameters P/( 10-6E, B*) and L/B were introduced for load and aspect ratio. Here P is the total axial load acting on the ends of the panel; E, is the elastic modulus in the longitudinal direction of fibres; L is the overall length of the panel; and B is the breadth of the panel. The main results presented are nondimensional values (defined later) of the panel mass, the stiffener height and the skin and stiffener thicknesses. These are plotted against load level for different aspect ratios and numbers of stiffeners. For typical panels, additional curves for optimum mass were also plotted against the number of stiffeners, number of voids in the portion between 557

F. W. WILLIAMSand YE JIANQIAO

558

stiffeners, the void ratio a, (defined later) and prescribed values of the thickness or height of the stiffeners. These additional results give a better understanding of the behaviour of the panels. The results in this paper show that the proposed type of stiffened panels can achieve about 16-20% mass savings compared with conventional ones for almost all the load levels, aspect ratios and numbers of stiffeners investigated.

OPTIMIZATION OF LAMINATED PLATES WITH LONGITUDINAL MID-SURFACE VOIDS

A full investigation of the buckling design of the new type of fibre laminated composite plate shown in Fig. l(a), which has equally spaced voids at the central layer of the cross-section, was reported in [l] which considered such plates in isolation, with their longitudinal edges either simply supported or clamped. The plate shown is constructed of two outer layers and a core layer with voids. The outer layers were made of eight plies of equal thickness and the angle-ply stacking sequence was symmetric with alternate plies at + 6 and - 0. The core consisted solely of zero degree fibres. The cross-section of the plate was specified by the plate thickness, the number of voids, the volume ratio cc, obtained by dividing the volume of the core material by that of the whole plate, and the voids ratio a, obtained by dividing the total breadth of the voids by the breadth of the cross-section. The plates with voids were represented as a prismatic structure which consists of a series of thin, flat, rectangular strips (which are themselves plates for analysis purposes) that are rigidly connected together, eccentrically where necessary, along their longitudinal edges and are simply supported at their ends. Thus, the plate with voids of Fig. l(a) was

assembled from a series of substructures of the type shown in Fig. l(b), with the crosses denoting two internal longitudinal line junctions that are connected together by two strips, with eccentric connections such that the void is formed between them. The other two parts of the substructure each consist of a single strip that does not have eccentric connections and is composed of the core layer and the two outer layers. The mass of such plates was minimized subject to buckling constraints consisting of either pure longitudinal compression or pure in-plane shear. Layer thicknesses and fibre orientation were the design variables and a thorough study was made of the effects of varying the boundary conditions, the loading conditions, a,, a, and the number of voids. The datum of comparison for plates with voids was obtained by calculating the optimum mass of solid plates which still had outer layers with the eight-layer stacking sequence just described and a core that consisted solely of zero degree fibres. The thickness of the zero fibre core was such that the mass of the core was the same fraction a, of the total mass as for the plates with voids. The comparison between these masses gave the mass saving ratio due to using voids. The results in [l] showed that significant percentage mass savings 6 were obtained by using voids in the cross-section. For example, Fig. 2 gives the curves of L against a, for different values of a, and of the number of voids N for a uniformly longitudinally compressed plate with simply supported longitudinal edges. Explicitly 6 = lOO(1 - Mu/M), where M, and M represent the optimum masses of, respectively, the plates with and without voids. The curves show that the mass saving increases as a,, a, and N increase. These curves are repeated here because the local buckling behaviour of a stiffened

core

outer

layers

h

(a) Plate

with N voids, such that it consists substructures of the type shown in Fig.l(b) edge to edge. hv= ach/( 1- %+ ac 01 )

$_,/2

I I

a;b/N

of N connected

bs/2 I

(b) Substructure bs=(l-a(v)b/N Fig. 1. Dimensions of plates and substructures used to assemble them.

Prismatic assemblies of plates with longitudinal voids

+j:&

559

!:kO (a) N=5

0

.

*

0.3

0.4

0.5

0.6

0.7

0

0.3

%

0.4

0.5

0.6

0.7

%

Fig. 2. Curves for simply supported plate under longitudinal compression. a, = 0.2, 0.3, 0.4 and 0.5 for, respectively, the symbols A, +, * and x

panel can be approximately investigated by taking the longitudinal edges of the intermediate plates formed by the skin between stiffeners to be simply supported. OPTIMIZATION

constructed of two [(f e),l, outer layers and a core made solely of zero degree fibres. The thickness of the core is determined from the total thickness by the volume ratio a, and the void ratio a, in the way given in the caption of Fig. l(a). In the present study, it is assumed that the values of tl, and LX,for the skin and stiffeners are the same. The values used for the main body of results are a, = 0.4 and ~1,= 0.5. These values of c(, and tl,, and the number of voids defined above, were chosen to avoid undue manufacturing difficulties, because Fig. 2 shows that a single plate with just five voids and with CL,= 0.4 and a, = 0.5 gives significant mass savings of around 19% when the edges are simply supported. The datum panels for comparison study are the solid panels which are made of the same material, have the same number of stiffeners, the same lay-up sequence, the same values of aspect ratio L/B, share

OF STIFFENED PANELS WITH VOIDS

Figure 3 shows the cross-section of a bladestiffened panel which has voids in both the skin and the stiffeners. There are five equally spaced and identical voids in the skin between stiffeners, four voids identical to them in the edge portions of skin and two voids in stiffeners. These voids were all modelled by the substructuring procedure described in the previous section in connection with Fig. I(b) and the multi-level substructuring option of VICONOPT was used during this process to reduce solution times, data preparation times and computer memory usage. The skin and the stiffeners are both

I Ab12 4 voids

I

/I

5 voids 3

I

4

5:el I3

6

YE7 \I

I

h

Fig. 3. Blade-stiffened panels and detailed cross-section. b, = 0.2b(l -a,.), bf = 0.56’ (1 - a,,).

560

F. W. WILLIAMSand YEJIANQIAO

0.003 M/?B 3 0.002

0.001

0

Fig. 4. Mass versus number of stiffeners. The dashed and solid lines are for L/B = 0.3 and L/B = 0.5, respectively. The symbols 0, + and t denote, respectively, ~/(10-6~,~~) = 6, 14 and 22.

the same 01,.and c(,,and are subjected to the same level of loading. The four design variables of the present problems are the skin and blade thicknesses, the blade height and the shared fibre orientations +Q of the skin and stiffeners. For a given length of panel, VICONOPT used exact theory [3] to choose the optimum design [4] by calculating the critical buckling load of the uniformly longitudinally compressed blade stiffened panel at each of the values of 1 given by

~=Llj,

j=l,2,3

,....

(1)

Here the upper limit on j is an integer chosen such that the smallest value of i, calculated from eqn (I) is

0.006

sufficiently small compared with the smallest unsupported width of skin (i.e. 0.8b on Fig. 3) for local buckling to be detected. To ensure that this upper limit had been chosen appropriately the program was modified to give a warning if the lowest value of /z used governed the design. It should be noted that the design procedure just described would have been strictly exact if the plates had been isotropic or orthotropic, but involves very small (i.e. negligible) errors because the plates are anisotropic, but are nearly orthotropic. For buckling of a given panel, a,,/E, obviously remains unaltered if the length, breadth and thicknesses of the panel are scaled in proportion, where g,, is the average buckling stress in pure longitudinal compression. Therefore, P??lE, B” remains constant if L/I3 is kept constant, where P,, is the critical longitudinal buckling load of the panel. This nondimensional relationship between mass and design load for given aspect ratio can also be used during minimum mass optimization. Hence the nondimensional graphs which follow can be used to obtain results for many different panels with various different load levels and aspect ratios, particularly since interpoIation can be used without excessive loss of accuracy.

NUMERICAL RESULTS

Figure 4 gives curves of optimum mass against the number of stiffeners for the panel shown in Fig. 3. The curves show that for higher aspect ratios L/B or design loads P, the optimum number of stiffeners is less than for lower values of L!B or P. The curves

7

0.008 h/B _

7

6

18

22

2

6

18

22

0.008 h*/B 0.006

0.006-

0 .

2

’

2

6

18

22

with voids; - - - without voids; . Fig. 5. Curves for the panel with seven stiffeners: skin only; + L/B =0.4;a L/B = 0.7.

with voids in

Prismatic assemblies of plates with longitudinal voids

561

0.06

I

b*/B

0

I 2

6

10

14

18

0.04

I

#S

0.02

-

*---

0

22

2

6

10

14

18

22

2

6

10

14

18

22

P~OBE,B~)

02

0

’

I

P/(l@E,B*)

pm VE,B~)

Fig. 6. Curves for the panel with nine stiffeners: -

with voids; --- without voids; skin only; + L/B = 0.4;x L/B = 0.6.

also show that using the optimum numbers of stiffeners, e.g. 15-20 for L/B = 0.5, or 25 or more for L/B = 0.3, does not always give significantly greater mass savings than using considerably fewer stiffeners. Therefore, only panels with 7-13 stiffeners are investigated in detail in this paper, because the advantages

of using a larger number of stiffeners are likely to be outweighed by the associated practical difficulties of manufacture and maintenance. Figures 5-8 give the optimum mass and configuration of blade-stiffened panels with different numbers of stiffeners. There are four graphs on each of these

0.004

0.04

M/m3

b”/B

0.002

0.02

0

0 2

6

18

2

22

6

0.006

0.006,

. with voids in

h’/B 0.004

7

_.::I-

_I--

_,:_-

.-• ___.-__ 18

22

18

22

,.-:. *’

g:.

0.002

0

1

’

2

6

18

22

Fig. 7. Curves for the panel wiht I 1 stiffeners: -

0

2

6

with voids; - - - without voids; only; A L/B = 0.3;* LIB =0.5.

CA.3 44,3---E

. with voids in skin

562

F. W. WILLIAMS and YEJIANQIAO

0.04

-*_---- _c---

__$...---

E!G ._-’

0

1

0

1

2

6

18

22

6

18

22

I

I

2

Fig. 8. Curves for the panel with 13 stiffeners: -with voids; - -- without voids; . with voids in skin only; A L/B =0.3;* L/B =0.5.

figures. The first one gives the dimensionless mass Comparison of the solid lines with the dashed ones M/pB’ versus the load parameter P/(IO-6E, If*),for the first graph of each of Figs 5-g shows that the where M is the optimum mass of a panel, p is density percentage mass saving achieved by using the voids of its material and P is its design load for pure shown on Fig. 3 is almost the same as for a simply longitudinal compression. The optimum mass of each supported plate which is identical to the portion of panel with voids is contrasted with that of the datum skin between stiffeners. For instance, Fig. 2 shows panel, which does not have voids. The remaining that for a Ion~tudinaIly compressed and simply three graphs give, dimensionlessIy, the total thickness supported plate with five voids, SL,= 0.4 and OL, = 0.5, h of the skin and the total thickness h * and height b * the mass saving is about 19%. This compares with of the blades, see Fig. 3. mass savings of about l&-20% given by the various On Figs 5-8 the points shown were given by the points on Figs 5-8. computer, i.e. they were obtained for each of the Comparison of the solid and dotted lines on the numbers given on the horizontal axis. For convenfirst graphs shows that not much additional mass ience, they are connected by straight lines, but for saving results from using voids in the stiffeners when brevity the resulting plots are referred to as curves the load and aspect ratio are both quite small. below. However, as the load level and aspect ratio increase In Figs 5-8, each solid line represents results when this additional mass saving becomes gradually more the panel has voids in both the skin and the blades, significant, in particular when a large number of as shown in Fig. 3, whereas each dashed line is for the stiffeners is involved. In some cases, the mass saving solid panel used as a datum for comparison, The is doubled by using voids in stiffeners. Nevertheless, dotted lines on the first graph of each of Figs 5-8 further curves, which are not presented, showed that show the optimum mass for panels with voids in the there was little additional advantage when more than skin but with solid stiffeners. Ahhough these results two voids [e.g. four) were used in the stiffeners. show that omitting the voids from the stiffeners often The optimum con~guration plots are the second, causes only a moderate increase in mass, the panels third and fourth graphs of Figs 5-8. They show that with voids in both the skin and the stiffeners are the second moment of area of the optimum crossnevertheless judged to be preferable in this paper, so section increases progressively as the aspect ratio increases, mainly due to the height of stiffeners that most results are given only for such panels. increasing. This confirms the expected result that the In the calculations, typical values for carbon fibre construction were used, namely El/E, = 0.0692,overall buckling behaviour of panels is more relevant Gj2/E1 =0.0369and p,, 8: 0.3. Each figure also gives when the aspect ratio is quite large. On the other hand, from the h/R plots of Figs 5-8, the skin of the results for two values of the aspect ratio L/B.

Prismatic assemblies of plates with longitudinal voids 0.06

563

-

b*/B 0.04

:

&

Q.i32

-

+--*--

_+--

2

6

18

22

_;,. -__hyB 1N

0.008 ,

0.008 ::;:; h/B

P

_A-----

0.006

___-..+----. --

_a--

0.002

-

/

0.004

*’

0.002

-

.*::’ *--

a.--

2

6

0.

0

2

6

18

22

18

22

Fig. 9. Effect of using low convergence when obtaining the curves of Fig. 5.

for low values of L/B than for high values. It should be mentioned that Figs S-8 were obtained by requiring the optimum mass to exceptionally high accuracy. Figure 9 illustrates why this was necessary. It shows curves :vhich correspond exactly to those of Fig. 5, except that the mass was found to more usual accuracy. Clearly, the curves for the optimum panel configuration, i.e. the curves for 6*, h and h+, are not all as smooth as those of Fig. 5. This is especially true for h*. However, the h/B plot is not affected much and, of course, the mass plot is virtually unaffected. These facts suggest that the design surface in the vicinity of the optimum point is quite ‘flat’ as h* or b* vary in an appropriately inter-related way. This was confirmed by redesigning the panels for a series of fixed values of b* or h* close to their optimum values, i.e. the values given by interpolation on Fig. 5. Figure 10 gives a typical example, involving seven panel is thicker

stiffeners and an aspect ratio of L/B = 0.5, that relates to optimum designs of Fig. 5. It is clear that h+ and b* can vary significantly in the neighbourhood of their optimum values without causing significant mass changes, especially for low design loads. Hence small changes in the convergence factor used when optimizing mass in Figs 5-8 can cause significant changes of the optimized configuration, which explains the lack of smoothness of low accuracy optimization results such as those of Fig. 9. In practical design, it is unnecessary to require high accuracy and so relatively low accuracy would be used to effect substantial savings of computing time. An important consequence of Fig. 10 is that designers can often modify an optimum design so that it meets practical design requirements, such as lower stiffener height, with negligible mass penalty. The results presented above all had u, = 0.4, a, = 0.5 and five voids in the skin portion between the

I 0.006 M/d

0.004

\\k 0.002

0.02

0.03

0.04

0.05

b*/B

0.06

0.07

0

t t

1

___--___--

.

’

0.002 0.003 0.004 0.005 0.006 0.007 h*/B

Fig. 10. Mass versus stiffener height and stiffener thickness. The symbols A, + and s denote, respectively, P/(10-6E,B2) = 6, 14 and 22. There are seven stiffeners and L/B = 0.5. The circles denote values for optimum designs, obtainable by interpolation from Fig. 5.

F. W. WILLIAMS and YE JIANQIAO

564

0.004 M/P33 0.002 .

0.004 -*-..-___* 5-------,-_____.,______ . -‘---~-*--_.._*u.______-““-.__x x

M/p9 . a.002 .

L

i

x

X

(b)

(al 0

0 0.3

CL4

0.6

0.7

5

7

Q

Fig. Il. Mass versus void ratio a,. and the number of voids between stiffeners. Solid and dashed lines are a, = 0.4 and 0.25. respectively, when L/B = 0.5. The symbols x and * denote

the results for

P/(10m6E, B2) = 10 and 22, respectively. There are seven stiffeners.

stiffeners. However, Fig. 2 indicates that further mass savings can be achieved by increasing any of a?, a, and the number of voids. Figure 11 confirms this prediction. It plots the optimum mass of the same panel as discussed in Fig. 10 against a, and the number of voids in the skin between stiffeners, for different values of a, and load level. This figure shows that the optimum mass gets lighter as the values of CC, and a, are increased. In Fig. 1l(b), the results were obtained by using 5,7 and 9 voids in the skin between stiffeners, combined with, respectively, 4, 5 and 7 voids in edge plates and 2, 3 and 4 voids in the stiffeners. [Since identical substructures of the kind shown in Fig. l(b) were used for all skin portions the edge plates were no longer always of width 0.8b.l The figure shows that the optimum mass decreases as the number of voids increases although this decrease is very slow. Figure 11 shows how to get greater mass savings when manufacturing a panel. For instance, comparison with interpolation from Fig. S shows that the mass savings can reach 20-25% for the panel shown in Fig. 3 with seven stiffeners, a, = 0.4, L/B = 0.5 and a, = 0.7. Alternatively, results that are not presented show that about 30-35% savings can be obtained for this panel if the numbers of voids in the skin between stiffeners and in stiffeners are increased to 10 and 4, respectively, while still using a,=0.7. However, the extra manufacturing difficulties caused by using a large number of voids, or high values of a,. or a,, , may be prohibitive. The optimum fibre orientation of the panels studied above, i.e. for Figs S-8, was always found to be about &4S”, both for panels with voids and for those without. CONCLUSIONS

A new type of blade-stiffened laminated pane1 with voids in both the skin portion between stiffeners and the blades has been suggested to achieve a high buckling strength to mass ratio. The skin and stiffener thicknesses, fibre o~en~tion and blade height have been used as design variables. A plate with five voids in the core was used for the skin between blades.

Either a solid plate or a plate with two voids was used for the blades. The blades and the skin portions between them both have a, = 0.4 and a, = 0.5. The computer results presented graphically for panels governed by buckling in pure longitudinal compression show that such panels are lighter than ones without voids by about 14-20%. Further results show that greater mass savings can be obtained by using more voids, or by making them bigger by choosing higher values of 51,.or as,. The results include the nondimensional optimum configuration of such panels for different load levels P/(10-6E,B2), aspect ratios L/B and numbers of stiffeners n. Interpolation will usually give adequate results for cases that are not explicitly coveted. Additional results illustrated that altering the convergence factor can lead to quite different configurations of stiffeners. The reason has been shown to be that, in the region of the optimum mass, the mass is very insensitive to suitably inter-related changes of the stiffener height and thickness. Thus, designers can meet requirements such as restricted blade heights at low cost. Stiffeners without voids can be used to avoid manufacturing difficulties, particularly when the aspect ratio and the load level are small. The optimum fibre orientation for all of the results discussed in this paper was always about +45”. The optimization procedure used designed panels with coincident overall and local buckling loads. The computer program could have separated these two loads by any chosen amount, to reduce the possible adverse effects of post-buckling behaviour. However, it was decided not to judge how large a separation was advisable, since the results presented are completely sufficient for their main purpose, which is to indicate the size of mass savings that can be obtained by using voids. The percentage mass savings obtained by using voids for the panels were very similar to those obtained by introducing the same voids into individual simply supported plates identical to the skin between blades. This important conclusion is ptesumed to be due to the voids in the blades enabling them to be higher without violating the local buckling

Prismatic assemblies of plates with longitudinal voids constraint, so that the overall buckling constraint is more easily satisfied. The same conclusion was drawn from results for panels with blade stiffeners with flanges, which are not shown, and so it may be true for all practical types of stiffeners. Acknowledgement-The second author is grateful for an award from the Overseas Research Students Awards Scheme of the Committee of Vice-Chancellors and Principals. REFERENCES

1. F. W. Williams and Ye Jianqiao, Optimum mass design of laminated plates with longitudinal mid-surface voids. Compur. Srrucl. 43, 265-212 (1992).

565

2. F. W. Williams, D. Kennedy and M. S. Anderson,

Analysis features of VICONOPT, an exact buckling and vibration program for prismatic assemblies of anisotropic plates. Proceedings of the 3lst AIAA/ASME/ ASCEIAHSIASC Structures, Structural Dynamics and Materials Conference, Long Beach, California, pp. 920-929 (1990). 3. W. H. Wittrick and F. W. Williams, Buckling and vibration of anisotropic or isotropic plate assemblies under combined loadings. Inr. J. Mech. Sci. 16.209-239 (1974). 4. R. Butler and F. W. Williams, Optimum design features of VICONOPT, an exact buckling oroaram for Urismatic assemblies of anisotropic pl%. Proceedin& of the 31~1 AIAA/ASME/ASCE/AHSIASC Structures, Structural Dynamics and Materials Conference, Long Beach, California, pp. 1289-I 299 (1990).