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Simultaneous cost and weight minimization of composite-stiffened panels under compression and shear
Composites Purt A 28A (1997) 419-435 IC’I997 Elsevier Science Limited
Printed in Great Britain. All rights reserved PII:S1359-835X(96)00141-8
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Simultaneous cost and weight minimization of composite-stiffened panels under compression and shear
Christos
Kassapoglou
Structural Technologies, Sikorsky Aircraft, Mail Stop S314A2, CT 06497, USA (Received 6 November 1995; revised 9 October 1996)
6900 Main Street, Stratford,
An approach is presented to determine the part configuration that minimizes the cost and weight of composite-stiffened panels under combined compression and shear loads. The stiffened panel is designed so that no overall, bay, or stiffener buckling occurs under the applied loads. Skin and stiffener material failure conditions and manufacturing constraints are also imposed. The stiffened panel cost and weight are minimized for a variety of stiffener cross-sectional shapes. A set of near-optimum (in a Pareto sense) configurations is determined. The final optimum configuration is selected from this set by minimizing a weight and cost penalty function. 0 1997 Elsevier Science Limited (Keywords: cost and weight optimization/minimization;
compression and shear: stiffened panel)
INTRODUCTION The usage of composite materials in aircraft structures has increased steadily over recent years but their application in primary structures is still limited. One of the main reasons is that the cost of composite structures is, in many applications, not competitive with the equivalent metal structure. While the development and implementation of new manufacturing technologies and materials have significantly reduced the cost of composite structures and, in several applications’, have made them competitive with metals, more improvements are necessary for further cost reductions. One of the approaches that the aerospace industry is exploring with promising results aims at incorporating cost as one of the design variables early in the design process’. This leads to design concepts that are efficient and compatible with the manufacturing technologies considered and the available material forms. In addition, it leads to studies where weight is traded against cost for various structural concepts and assists in the selection of a configuration (design concept, manufacturing process and material) that minimizes cost and/ or weight. Many studies have been published on optimizing the weight of stiffened panels3’4 but very few attempted to include cost as a design driver. Usually, the optimumweight designs based on different stiffener cross-sectional shapes or other initial assumptions are a posteriori compared with one another on the basis of cost estimates.
Recently, several attempts have been made to give cost a more prominent role in the design selection in composite fuselage studies5-‘. The results of a recent study in which cost was one of the two primary design drivers are presented in this paper. The design concept is that of a composite-stiffened panel under compression and shear loads representative of fuselage structure of civil transport aircraft. The stiffener cross-sectional shape, area and stiffness properties, as well as the skin thickness, are allowed to vary. The manufacturing process selected for this investigation is a combination of automated cutting and kitting of prepreg material with subsequent hand lay-up. Cost equations are developed for the various process steps and their predictions are summed over all steps to determine the total cost (in labour hours) of the stiffened panel. At the same time, the structure is sized to meet the applied loads (without buckling) and the corresponding weight is calculated. An approach to simultaneously minimize the cost and weight of the panel is presented. The effect of various geometry and stiffness parameters such as stiffener cross-sectional shape, spacing, skin thickness, stiffener area and stiffener stiffness on cost and weight is examined.
PANEL CONFIGURATION Consider the composite stiffened panel shown in Figure 1. The panel is flat with dimensions 76.2 cm x 76.2 cm and is
419
Cost and weight minimization: C. Kassapoglou
stiffeners
Nx
Figure 1
Configuration
of stiffened panel under consideration
loaded in compression (ultimate load 140 kNm_‘) and shear (ultimate load 140kNm-‘). The stiffeners are assumed to have any of the shapes: L, C, Z, T, J, I and hat (shown in Figure 2). The skin thickness and stiffness, and stiffener cross-sectional area, shape, stiffness, moment of inertia and spacing, are allowed to vary. The stiffeners are assumed to be co-cured with the skin.
COST EQUATIONS The fabrication process for the stiffened panel is divided into the following steps: (1) cut material and package in kits, (2) hand lay-up, (3) bag, (4) remove bag, and (5) trim. Several intermediate steps such as tool preparation and cleaning, transportation of materials and parts, and curing are treated as constants. They depend on specific factory procedures and equipment lay-out. They are not included since they do not affect the optimization process. The first and last steps (cutting, kitting and trimming) are assumed to be fully automated and the cost or time associated with their completion is treated as a percentage of the total cost. The most labour-intensive steps are those of hand layup, vacuum bagging and bag removal. A series of equations that estimate the time required to lay-up stiffened panels has been developed based on actual time studies performed on the factory floor. These equations account for the type of material used and the geometry and complexity of the part to be laid-up. They are made up of three modules that are repeated as required: 1) time to lay-up a flat ply C,=k,A,+kzP
(1)
where A, and P are the area and perimeter of the ply; 2) time to bend a ply over a radius (convex to the operator) C, = k3 L
420
(2)
where L is the length of the ply to be draped; and 3) time to locate a ply into a radius (concave to the operator) Ci = k4 L’
(3)
where L’is the length of the tool into which the ply is being positioned. The constants ki-k4 depend on the material used (tape versus fabric), fibre orientation (for tape mainly) and factory practices. It should be noted that for manuallabour-intensive processes, the constant k4 is significantly higher than k3 (typically about 3 times higher). The equations for bag placement and bag removal have the same form as equations (1) and (2) with different values for the constants k,-k3. For simplicity, the material used in this study was assumed to be typical plain-weave fabric for both the skins and the stiffeners. Then, the total labour (in minutes) associated with fabricating a rectangular, flat, stiffened panel is obtained by summing the appropriate equations (l)-(3) over all plies for the skin and stiffeners, and is described by the following equations: skin c, = [CIAB + &(A + B)]t/tp,y
(4)
st$Zener (individual) G = [GA, +
C4t1
+
Cd2
+
Gt:l~l~p,y
(5)
where A and B are the panel dimensions, t is the skin thickness, A, is the stiffener cross-sectional area, and tpiy is the ply thickness which may or may not be the same for skin and stiffeners. The constants Ci-C, are given in Table 1 for various panel configurations. They are generic and represent the process selected and not a specific factory or company performance. They were obtained from timed fabrication trials. The corresponding cross-sectional shapes are shown in
Cost and weight minimization:
C. Kassapoglou
tr
3 t2
h
t1
tr-c-
t2
f
b
4
12
1
D
“T” or blade
b Figure 2
Table 1
Skin L C Z T I J Hat
Stiffener
b II I, I
“HAT”
shapes under consideration
Cost equations
(hand
lay-up of stiffened panel)
(‘1
(‘2
c3
c4
0.01 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.017 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.01 0.01 0.01 0.01 0.01 0.01 0.01
0.0 0.083 0.15 0.25
C5
0.0 0.017 0.017 0.033 0.083-0.005b 0.017 0.17 0.033 0.2-0.005b 0.033 0.5 0.067
cb
0.0 -0.01 -0.02 -0.02 -0.005 -0.01 -0.01 -0.04
Figure 2. In all cases, the lowest thickness (tl) occurs in the web. The web plies branch out at the bottom (and top if required) to create a portion of the flange. Additional flat plies may be added at the top and bottom flanges as required. Thus the flange thickness, t2, is larger than or equal to the web thickness. The equations in Table I do not include the addition of roving material at the web ends where the web plies branch out into the flange. An equation similar to equation (3) can be used to add that contribution if necessary. It should be noted that. if more than one material is
used with different ply thickness and draping properties, equations (4) and (5) should be modified accordingly. These equations make an attempt at capturing the essence of manual labour during lay-up and, at the same time, are in a simple (mostly linear) form. This is in agreement with recent findings by Gutowski et al.’ where simple linear models with appropriate selections of the coefficients based on equipment parameters were found to be equally as accurate as other more involved power-law models. The coefficients in Table 1 are generic and represent the process selected, and not a specific factory or company performance. Specific applications are expected to have somewhat different cost values than the ones predicted by the above equations.
WEIGHT The second objective function is the total weight of the panel. The panel is designed not to buckle under combined ultimate compression and shear loads. In addition, no bay buckling (the skin portion between
421
Cost and weight minimization: C. Kassapoglou
adjacent stiffeners) is allowed below ultimate loads. It should be noted that the buckling analysis used here does not account for transverse shear effects or for coupling between skin and stiffeners. These may affect the minimum weight and should be added for a more accurate analysis. Finally, the skin and stiffeners are also checked for material failure at ultimate load. The weight of the panel includes the weight of the skin and stiffeners, and can be expressed in the form (see Appendix for derivation and Figure I): W = pA{Bt + [int(B/d,) + l]A,}
(6)
where p is the density of the material, A and B are the length and width of the panel (compression is along dimension A), t is the skin thickness, A, is the crosssectional area of the stiffener, and d, is the stiffener spacing. The function ‘int’ in equation (6) implies rounding down to the nearest integer. Equation (6) assumes that the stiffeners are co-cured with the skin and they are of the same material. If these assumptions are not valid, the expression can be modified accordingly.
Timoshenko”
as:
Dll =Dllskin +
Wsc 7 s
012
= D12skin
022
= D22skin
066
= D66skin
GstJst + 2d s
It is assumed that D16 = &6 = 0. Es, and G,, are the stiffener Young’s and shear moduli. Z,, is the moment of inertia of a stiffener with respect to the centroid of the entire cross-section. Jst is the stiffener polar moment of inertia (negligible for open cross-sections). The accuracy of these expressions will be examined below. For simply supported panels under applied shear, Whitney” gives the following expression for the critical panel shear load: 7r44[Dllm4+ 2(D12 f 2D66)m2n2R2 + D2*n4R4]Amn - 32 mn R3b2 N xycrit2 5MgA, = 0; i=l &I .. m+iodd
CONSTRAINTS
Mij =
The cost and weight expressions are to be minimized subject to the buckling, strength and producibility constraints. These constraints are discussed below.
(m2
_
i*~(n*
i=m,
_j*)
m+i
n+j
(9)
odd
even
=o j = n, n + j even
Buckling constraints Overall buckling of the panel under compression and shear is predicted by using an interaction equation of the form’: &+R;=l
(7)
where
NX
R, = NXM
R, = - NXY N xyxit
are the ratios of the applied compression and shear to the respective buckling loads when the loads are applied individually. The individual compression and shear buckling loads for a simply supported panel are determined by using the expressions given by Whitney lo.For compression alone, the panel as a whole buckles when the applied load N, equals N
xcnt
= n2[D,lm4 + 2(D12 + 2D&m2R2 + D22R4] (8) A2m2
where an infinite series in the out-of-plane deflection w of the form w = ~~Amnsin~sin~ is assumed. If the series is truncated after m = n = 3, the buckling load under shear can be shown to be the smallest of the two values:
and Nxy
=~
1
128R A2
X
__- _ d
E~~-%J%&I - 4 lE33 + E33& &E**+-zJ81
where m (the number of half-waves along A) is picked such that NXcct is minimized, R is the aspect ratio A/B, and D, are the bending matrix entries for the cross-section, including skin and stiffeners, given by
422
(11) + E22-b
-
2.5
where the constants Cij and J!$ are combinations of the bending stiffnesses and are given in the Appendix. This truncation is done to speed up the computer iterations
Cost and weight minimization: C. Kassapoglou
250
i
50
0
100
Nx Figure 3
Buckling
prediction
comparisons.
Present
+ WA2
+ 2Wm2R2
A*m*
--e-
present
200
150
vwsus finite element solutions
+ hR41
Using equation (12) and the fact that Fst =
(84
where R = A/d, is the aspect ratio of the bay and Dii = Dvskin(no contribution from the stiffeners) with analogous changes to equations (10) and (11) for the shear buckling load of the bay. When the bay buckles, the bay shear buckling load is equal to the shear load applied to the panel since the shear load is assumed to be uniform along the panel edge and all the shear load is taken by the skin. The compression buckling load, however, is not the same as the applied load since part of the load is taken by the skin and part by the stiffeners. The compressive load in the skin, Nxskin,is found as a function of the applied load N, by applying strain compatibility (the compressive strain in the stiffeners equals the compressive strain in the skin) and integrating the strain-displacement equations using simply supported boundary conditions and immovable edges along the panel edges parallel to the stiffener axis. This approach is valid for non-buckling panels which is the case here. The resulting expression is:
(12)
where aG are entries of the inverse of the membrane matrix Aii for the skin and Es, is the stiffener Young’s modulus. An additional buckling check is imposed for column buckling of each individual stiffener under compression.
(N, - Nrskin)B
(13)
i.e. the force in the stiffeners is equal to the total applied force minus the force in the skin, the standard (simply supported ends) Euler column-buckling equation gives: (~11 1 +
NX Nrskin= ~ 1 + (a11- ~:,l~,,KWs
ref[12,13]
(kN/m)
during the optimization process. The inaccuracies introduced by this truncation are alluded to by Whitney” and are also briefly discussed below. For bay buckling, the same expressions (7 )-(1 1) are used but are modified to represent bay properties and loads. For example: Nbay = r2V4,m4 xcr1t
_t_
(all
-
d2/a22b4,/4 -
~:2/~22)WsPs
7r *
=
N,A2
(14)
where 1, is the moment of inertia of each stiffener about the stiffener centroid. The accuracy of equations (7)-(12) was verified by comparing their predictions with published results’*‘” where very detailed finite element methods were used to predict buckling loads of stiffened panels (NASA Panel 1 of ref. 13 is used here) under combined compression and shear. The analysis in ref. 13 is based on the finite strip method, where the structure is divided in strips. Within each strip, deflections are represented as polynomials across the width and trigonometric functions along the length, resulting in eight degrees of freedom per strip. The predictions of equations (7)-(12) and the results of ref. 13 are compared in Figure 3. Excellent agreement is observed for N,,,/N, ratios between 0 and 3. For higher ratios, the present method becomes unconservative with the worst deviation being 19.5% for the case of pure shear (N,,/ N, = co). This deviation is primarily due to the series truncation after the third term discussed earlier and is considered adequate for preliminary design. The results of Figure 3 give confidence in the design equations chosen for buckling predictions. It should be pointed out that, in all loading cases of this example, the bay buckling load was significantly higher than the overall panel buckling load. Also, the moment of inertia in equations (7)-(12) was calculated about the centroid of the cross-section in two ways. In the first, the
423
Cost and weight minimization: C. Kassapoglou
exact centroid was found and 1, calculated using standard procedures. In the second, a portion of the skin equal to 30 times the skin thickness (standard procedure in design’) was assumed to be effective at each stiffener location and both the centroid and 1, were calculated for the combined cross-section. The latter approximate method was within 3% of the exact calculations and was selected for all subsequent optimization runs in order to decrease computation time. This approximation, while valid for post-buckling structures, will not always be accurate for prebuckled panels. The exact I,, value should be calculated in such cases.
portion of the stiffener in question (flange or web) and f,/ft (flange width to flange thickness ratio) is given in Table 2. The terminology one-edge-free and no-edge-free
refers to the edges of the portion in question15. If both edges are attached to a flange or web, or the stress changes sign, the portion is considered a no-edge-free type (in the case of sign change, f, is taken to be the length from the edge to the location of sign reversal). If one edge is free, the portion is of the one-edge-free type. Manufacturing constraints
Manufacturing constraints that reflect geometry or material limitations, usually established by experience, are also imposed. These are the following.
Strength constraints
By using equations (12) and (13), the stresses in the skin and stiffeners can be calculated. For the skin, a first-ply-failure criterion (such as the Tsai-Hill14) is applied to check for skin failure. For the stiffeners, a flange or web crippling check is done. Crippling here refers to the final failure of a member (flange or web) after it has buckled locally. The axial load on each stiffener is given by:
1) The stiffener spacing can be no less than 10.16 cm.
2) 3) 4)
The compression load at the panel edges is acting at the skin mid-plane. This implies that a moment is generated about the stiffener centroid, or about the centroid of the combined cross-section. Combining axial and bending loads, the maximum stresses in each of the stiffener components of Figure 2 (top flange, bottom flange, web) are given in Table 2. Since the stiffeners are assumed to be co-cured with the skin, the bottom flange stresses in Table 2 include the stresses in the skin below the flange. Each of the components in Table 2 (top flange, web, bottom flange) is checked separately. For the top or bottom flange check, the stress value in Table 2 (provided it is compressive) is compared with the crippling allowable. For the web check, the value of gaVsin Table 2 is compared with the crippling allowable. The quantity j in TubZe 2 is the distance between the centroid of the stiffener (alone) and the bottom of the stiffener (top of the skin). The crippling allowable is obtained from a variety of tests and is given as follows.
5)
6)
This is to facilitate locating tools and reaching with both hands between stiffeners during fabrication and assembly. The width of the stiffener flanges cannot exceed the stiffener spacing, to avoid stiffeners coalescing into each other. No thickness value can be less than the minimum gauge. The latter is set equal to 0.381 mm. No flange width or web height can be less than 1.27 cm. Otherwise tooling and lay-up become very expensive. The skin and stiffener thicknesses (t, tl, tl + t2) can only be an integral multiple of ply thickness. No intermediate ply thickness values are allowed. The flange width must be larger than the web thickness.
It should be noted that hard tooling is used for the stiffeners as shown in Figure 4. A nylon or semi-rigid silicone-based bag is used for vacuum bagging.
vaou”m
repeating between
tools stiffeners
bag
/
For one-edge-free: outer
1.6753 = F,,
KVlft)o.7245
mold toot
line
(16a) f&
> 2
For no-edge-free:
Fcrip= Fcu = Fm
fwlh < 8 9.8307
KV/ft)1~099
fwlh > 8
(16b)
where F,, is the compression strength allowable for the
424
fly-away mandrels
Figure 4
Tooling
(foam)
used during
stiffened panel fabrication
Cost and weight minimization: C. Kassapoglou
t
0
V
425
Cosr and weight
minimization:
C. Kassapoglou
OPTIMIZATION The complexity of this optimization problem is increased by the fact that there are two objective functions, cost and weight. It is desired to determine the configuration that will minimize both objective functions simultaneously. This is done in a Pareto sensei where a configuration is considered optimum if, for any other feasible configuration, at least one of the objective functions to be minimized has a higher value. In order to gain some insight into the process, the panel is first optimized for each objective function separately. The minimum weight is determined as a result of either the bay buckling requirement or a combination of manufacturing and strength constraints, given the minimum stiffener moment of inertia dictated by the overall panel buckling condition. It is important to note that, for the weight minimization, the exact cross-sectional shape of the stiffener is not needed as the stiffener is represented by its cross-sectional area A, and its moment of inertia Z,. This makes the approach very flexible and sets up a possible cost minimization approach where the shape that matches the required A, and Z, (from weight minimization) must be found such that the cost is minimized. The individual minimum weight and cost configurations are then used to determine a set of Pareto optimum configurations. These are configurations for which any excursion away from the corresponding design results in at least one of the two objective functions increasing. This set will, in general, have more than one configuration. The optimization flow is shown in Figure 5. It consists of eight major steps (marked in Figure 5) and a series of checks and secondary calculations. Steps 3, 5 and 8 are discussed below. The rest are self-explanatory or covered in a previous section.
Step j-bay
If the skin stiffness is high relative to the applied loads and the stiffener spacing is sufficiently small, the bay will not buckle. Then, the stiffener area A, is determined as the lowest possible A, value that satisfies the strength and overall panel buckling constraints. This value must be higher than or equal to the value imposed by the minimum gauge and minimum web and flange dimensions. The panel buckling constraint gives a condition for the stiffener moment of inertia (I,,) as shown in Figure 5. The selected value for A, must be such that the required Z,, is feasible.
of st$ener geometry
The panel and bay buckling conditions are used to determine the minimum acceptable values for Z, (stiffener moment of inertia about the centroid of the entire cross-section including the skin) and A, (stiffener cross-sectional area). For simplicity in the calculations, Z,, is translated to Z, the moment of inertia about the bottom
426
A, =2bt,+ht,
(17)
Z=;(h+2tz)3-(b-t,);+
(18)
From equation (17) b=
A, - htl 2t 2
(19)
and substituting into equation (18) and rearranging gives: t = A,(3h2 + 9ht2 + 8t;) - 61 1 h(h2 + 3ht2 + 2t;)
(20)
Substituting this expression to equation (19) yields b = A, Et2
1 _ 3h2 + 9ht2 + 81; - 61/A, h2 + 3ht, + 2t;
1 (21)
Since b must be positive, the right-hand side of equation (21) must be positive. This yields (after some manipulation): 0 <
h < ,/481/A,
- 12t; - 6t, (22)
4
Since tl must be positive, the right-hand side of equation (20) must be positive. Hence,
buckling
Step 5-determination
of the stiffener (top of the skin). The two requirements on A, and Zgive two conditions for b, tl, h and t2 (see Figure 2 for definition of these quantities). The cost equations are then used to calculate the panel cost for each set of feasible b, h, tl and t2 values. The values of b, h, t1 and t2 that satisfy the requirements on A, and Z and minimize the cost, define completely the stiffener geometry. As an example, consider the case of an ‘I’stiffener (see Figure 2). The cross-sectional area A, and moment of inertia Z about the bottom of the stiffener are given by:
-9t2 + \/721/A, h>
- 15t; (23)
6
Equations (22) and (23) are combined to give the condition on h:
-9t, + J_ 6
4
(24) The following procedure is used. Given A, and Z, successive t2 values are selected as integral multiples of ply thicknesses and no less than the minimum gauge requirement. For each t2 value, the limits dictated by equation (24) are calculated and h is varied between these limits, provided the manufacturing constraint 4 in the previous section is not violated. For each pair of acceptable t2 and h values, equations (20) and (21) are used to determine tl and b with a check that manufacturing
Cost and weight minimization:
INPUT
C. Kassapoglou
OUTPUT
I/
11 Panel
Bucklino
L
=firyLLb7@ Determine
Stiffener
Geometry
A impose Strength Constraints
6
1
Use Minimum Cost & Weight Curves to Approach Pareto Optimum 8
Figure 5
Optimization
flow chart
constraints 2, 3, 4 and 6 of the previous section are satisfied. If tl is not an integral multiple of ply thickness, it is set to the next highest integral multiple of ply thickness and b is recalculated from equation (24). With b, tl, t, and h determined, the corresponding cost is calculated from equations (4) and (5). The procedure is repeated for various t2 values until the minimum cost is determined. Throughout the procedure, the additional conditions t2 > t, /2 (for ‘T’ and ‘I’stiffeners) and t2 > tl (for ‘C’, ‘Z’, ‘J’ and ‘hat’ stiffeners) are also imposed. An analogous approach is used for all other stiffener cross-sectional shapes. The respective conditions on h, tl, t2 and b are summarized in Table 3. Step 8--Pareto
optimization: minimum weight and cost
The procedure described so far and shown in Figure 5 can be used to determine separately the configurations for the lowest weight or cost. The lowest cost configuration will, in general, be different from the one for lowest weight. By varying the stiffener cross-sectional area between its two values for the lowest cost and weight configuration, the boundary of ‘near-optimum’ configurations (the Pareto set) is obtained in cost-weight space. A penalty function is established as the sum of the
percentage difference of the weight and cost of a specific configuration from the individual minimum weight and cost points. A search is done among the configurations in this Pareto set to determine the one for which the penalty function is minimized. This corresponds to the optimum configuration.
RESULTS The procedure described in the previous section was applied to the first example panel of refs 12 and 13 (NASA Panel 1). This is a 762mm x 762mm panel with a skin thickness of 3.912mm. The membrane and bending stiffness matrices (A and D matrices) for the skin are given in refs 12 and 13. While in these references the stiffeners were of rectangular constant cross-section and the spacing was fixed to 127 mm, here only the stiffener axial modulus was kept constant at 74.6GPa, while the stiffener spacing was free to vary and the stiffener cross-sectional properties (area and moment of inertia) were also varied. In addition, seven different cross-sectional shapes were considered: L, C, Z, T, J, I and hat (see Figure 2). The panel was assumed to be under 0.14 MN m-l compression and 0.14 MN m-l shear.
427
Cost and weight minimization: C. Kassapoglou
The stiffener compression failure strength [F,, in equation (16)] was assumed to be 10.2 MPa. No post-buckling was allowed. The skin was assumed to consist of plies with ply thickness 0.152 mm and the stiffener of plies with ply thickness 0.19mm. Thus, for the stiffener thicknesses, only integral values of 0.19 mm were allowed. For each stiffener cross-sectional shape, the lowest weight and cost configurations were determined separately. Typical examples for a ‘T’ stiffener are shown in Figures 6 and 7. The optimum weight of the panel as a function of stiffener spacing is shown in Figure 6. Two curves are shown, one for the configuration corresponding to the minimum weight for the selected stiffener spacing and one corresponding to the minimum cost. In general, given a stiffener spacing, the minimum weight configuration does not coincide with the minimum cost configuration. It is seen from Figures 6 and 7 that the optimum weight and cost curves have sudden jumps that occur at stiffener spacings of 10.89, 12.70, 15.24, 19.05 and 25.40cm. These are the spacings at which the number of required stiffeners decreases by one. Since both the weight and cost are a function of the total number of stiffeners, these incremental decreases in the stiffener number translate to incremental decreases in the weight and cost of the panel. In between these points, the weight and cost increase slightly. This is due to the fact that the skin thickness (and therefore skin weight and cost) are held constant. Then, as the stiffener spacing increases without decreasing the number of stiffeners, the area and moment of inertia of the stiffeners must increase to satisfy the buckling and stiffener crippling conditions. This trend of decreasing cost and weight with decreasing number of stiffeners holds true as long as the bays of the panel do not buckle. For the applied loading selected, bay buckling coincides with panel buckling when the stiffener spacing equals 29.72cm. Any increase in stiffener spacing beyond this value requires substantial increases in the stiffener area in order to keep the load in the skin equal to the bay buckling load. These increases in the cross-sectional area of the stiffeners will tend to more than offset any decreases in the weight (and cost) due to further decreases in the number of stiffeners. It is therefore expected that, at large stiffener spacings, the weight and cost will be increasing monotonically and at a relatively high rate. As a result, the lowest weight and lowest cost configurations are expected to occur right after the last decrease in the number of stiffeners, before bay buckling. This occurs at a stiffener spacing of 25.4 cm and is shown in Figures 6 and 7. It is important to note that while both the cost and the weight of the panel are a minimum for that stiffener spacing, the corresponding configurations (in terms of stiffener area, and values for h, tl , t2 and b of Figure 2) will not be the same (for the same crosssectional shape). A comparison between all stiffener cross-sectional shapes is shown in Figures 8 and 9. The lowest weight
428
curves are shown in Figure 8 as a function of stiffener spacing. It is seen that the lowest weight is achieved at a stiffener spacing slightly larger than 25.4cm for a ‘J’ stiffener and is equal to 3.7 kg. The next ‘best’ stiffener shape is that of a ‘T’ stiffener with a weight of 3.72 kg. It should be pointed out that the lowest weight for a ‘C’ stiffener coincides with that for a ‘Z’ stiffener. This may be due to the fact that any coupling and transverse shear effects have been neglected in the analysis. The cost, however, will not coincide for these two stiffeners. The situation is significantly different when minimum cost is the objective. Figure 9 shows that the lowest cost occurs for a ‘T’ stiffener (302.5min) and the next lowest cost is achieved by the ‘L’ stiffener (307.7min). The ‘J’ stiffener configuration, which had the lowest weight, is rather costly and ranks fifth among the shapes considered. The cross-sectional shapes for the lowest weight and cost cases are shown in order of increasing weight or cost and approximately to scale in Figure 10. The corresponding weight and cost values for each case are also shown. As expected, the highest web heights occur for the T and L shapes that have no top flanges. Also, the ‘T’ stiffener has weight and cost values lower than those of the ‘L’ stiffener even though, for the same amount of material in the web and flange, the two shapes would have the same cost. The reason is that placing the web in the middle of the flange (rather than the end) for the ‘T’ stiffener decreases the unsupported flange length (b) to half the corresponding value for an ‘L’ stiffener, significantly increasing the flange crippling strength. As is seen from equation (16) lower flange widths result in higher crippling strengths. This is also the reason why the flange for the ‘L’ stiffener is thicker than for the ‘T’ stiffener. The results of Figure 10 pose the question of how to select the optimum stiffener shape and satisfy both the minimum weight and cost requirements. If weight and cost are weighted equally, there is no definite answer since the lowest weight (‘J’ stiffener) configuration does not coincide with the lowest cost (‘T’ stiffener) configuration. However, in this specific example, the second lowest weight configuration (‘T’ stiffener) differs only by 0.5% from the lowest weight configuration and is the lowest cost configuration. This would lead to a nearly optimal selection of a ‘T’ stiffener. The specific optimum geometry for the ‘T’ stiffener that gives the lowest weight and cost is still undetermined. As was pointed out earlier and is shown in Table 4, the lowest cost and weight configurations for a ‘T’ stiffener do not coincide. Using the procedure of Figure 5 and a ‘T’ stiffener, the stiffener area was varied between the two values in Table 4 and the corresponding lowest cost and weight configurations were determined. The stiffener spacing was kept constant at 25.654cm, which is the spacing that gave the lowest cost and weight configurations (see Figures 8
Cost and weight minimization: C. Kassapoglou
I
131 II
II 9
9
$1 II Q
lel
<:“1;3” II Q
Q
429
Cost and weight minimization: C. Kassapogfou
T STIFFENER
4.2
-
min weight min cost
3.2,.,.,.,.,.,.,,,.,.,.,.,. 10 12 14
16
18
20
STIFFENER Figure 6
22
24
26
SPACING
28
30
32
34
(cm)
Effect of stiffener spacing on panel weight. Weight curves for lowest cost and lowest weight configurations (‘T’stiffener)
T STIFFENER
370
330
-
minweight min cost
310
27C
25c lo
12
14
16
16
20
STIFFENER Figure 7
430
22
24
26
SPACING
(cm)
26
30
32
34
Effect of stiffener spacing on panel cost. Cost curves for lowest weight and lowest cost configurations (‘T’stiffener)
Cost and weight minimization:
C. Kassapoglou
4.2
-L -T -
z,c
-J -
3.6 -1 10
12
14
16
18
20
22
STIFFENER
Figure 8
Minimum panel weight as a function
of stiffener spacing.
24
26
SPACING
(cm)
All stiffener
and 9). These configurations trace the boundary curve shown in Figure 1 I in cost-weight space. The lowest cost and lowest weight points are shown in that figure. As can be seen from that figure, any point above the boundary line is sub-optimal, since there is always a point on the boundary for which at least one of the two
28
30
32
HAT
34
shapes
objective functions (weight or cost) can be improved upon without increasing the value of the other. Also, any point below the boundary line is not feasible since the stiffener fails by web or flange crippling. The segment of the boundary line between the minimum weight and the minimum cost points (segment
I C L T
J HAT
280!.,.,.,.,.,.,.,.,.,.,.,.,1 10 12 14
18
18
20
STIFFENER
Figure 9
Minimum
panel cost as a function
of stiffener spacing.
22
24
26
SPACING
(cm)
All stiffener
28
30
32
34
shapes
431
Cost and weight minimization: C. Kassapoglou
r
4.85cm
J:Wt4.7Okg Cost-366.2min
TWb3.72Kg Cost-308.3mm
LzWtr3.74kg Cost-31 1 .Omin
Iwt-3.75kg C Wt-3.79kg Cost-381 .lmm Cost-340.9m
HAT: Wt-3.96 kg Cost-474.2 mms
Z:Wt-3.79kQ Coat-393.4 mins
(a) Order of increasing weight
T: Wt- 3.77 kg Cost- 302.5 mins
L: Wt- 3.76 kg Cost - 307.7 mins
C: Wt-3.79kg I: Wt.3.75kg Cost-340.9 mms Cost-349.3 mms
c
1
J: Wt-3.69kg Cost-3662 mms
1. 1
Z.Wt-3.79kg Cost-393.4mins
HAT: Wt-3.96 kg Cost- 474.2 mlns
(b) Order of increasing cost Figure 10
432
Lowest weight and cost stiffener configurations
Cost and weight minimization:
Table 4 ____..
Optimum
configuration
geometry
Lowest weight
Lowest cost
h (cm) b (cm) tl (mm) t? (mm) Area (cm’
4.86 1.32 0.76 1.33 0.547
5.32 2.48 0.57 1.52 0.68 1
)
(point A in Figure 11) and Cmin (point B) are the individual weight and cost absolute minimum points; and el and e2 are weight factors, the value of which should be deter-mined by the requirements of the specific application in question and other mission considerations. If the absolute minimum cost and weight points coincide, the penalty function from equation (25) is zero and, as expected, this point is the sought-for minimum. If e1 = e?, the weight and cost penalties are equally weighted. This approach is used here and a point along segment AB is sought that minimizes equation (25). By evaluating the penalty function along segment AB it can be shown that, for the specific example used here, the lowest penalty value occurs at point A (which also happens to be the point of lowest weight). Therefore, the configuration that optimizes cost and weight simultaneously is one using ‘T’ stiffeners at 25.654cm spacing with the dimensions shown in the second column of Tuble 4. The cross-sectional shape is also shown in Figure IOU (‘T’ stiffener). It should be pointed out that the optimum configuration found above is a function of the assumptions and loading used in this example. The optimum configuration would be different if the skin thickness and lay-up, stiffener modulus, and manufacturing constraints were different. It is also important to note that the cost and weight results presented here are a strong function of the manufacturing process and tooling approach selected.
for ‘T’ stiffener
Quantity
AB) corresponds to configurations that are the set of Pareto optimum points for the problem at hand. These are Pareto optimum points in the sense that any excursion away from these points, but not along the boundary line itself, will increase the value of at least one of the two objective functions. The question is how to select an optimum configuration among these points. This is done by assigning to each point in that segment a penalty value and finding the point that minimizes the penalty. This penalty value is a measure of how far away the specific point is from the lowest cost and weight values. In order to combine cost and weight in a single penalty function, a decision must be made on the relative merit of a cost or weight penalty. A penalty function is adopted in the following form:
PF =
(>I(WI -
C. Kassapoglou
Wmin)/ Wmm + e2(Ci- Cmin)/Cmin (25)
where Wi and Ci are the cost and weight of the specific point in question along segment AB; Wmin
T STIFFENER 310 309 308 307 306 (min
weight)
305 304 -
stiffeners and/or in this
fail by web flange crippling region
303 -
B-Riii 302 -
(mln
cost)
301 -
I
300 3.70
3.72
I 3.74 PANEL
Figure 1 I
Boundary
of lowest cost-weight
WEIGHT
I
1
3.76
3.78
3
IO
(Kg)
points (‘T’ stiffener)
433
Cost and weight minimization: C. Kassapoglou
Automated fabrication processes such as tape or fibre placement for the skins and pultrusion for the stiffeners, or use of resin-transfer moulding for the entire stiffened panel, will significantly reduce the cost figures given here and may alter the order of ‘best’ to ‘worst’ panel configurations. The present study aims at highlighting the approach and giving a flavour of the results rather than exhaustively examining all possibilities.
10.
CONCLUSIONS
13.
An approach to determine the configuration that simultaneously minimizes the cost and weight of composite-stiffened panels under compression and shear, under buckling, strength and manufacturing constraints was presented. A variety of stiffener crosssectional shapes was examined. It was found that the individual weight and cost minimum configurations do not, in general, coincide. In such cases, a set of ‘nearoptimum’ configurations was determined and a penalty function established. The optimum configuration is determined from this set as the one that minimizes this penalty function. For the examples examined, it was found that ‘J’ stiffeners give the lowest weight configurations while ‘T’ stiffeners give the lowest cost configurations. The optimum configuration for both cost and weight was obtained for a panel with ‘T’ stiffeners.
14.
for advanced composites fabrication.
Compos.
Manuf,
1994,
5(4), 231-239. 9.
11. 12.
Bruhn, E.F., Analysis and Design of Flight Vehicle Structures. S.R. Jacobs & Associates Inc., Indianapolis, IN, 1973, Section c5.11. Whitney, J.M., Structural Analysis of Laminated Anisotropic Plates. Technomic Publishing Co., Lancaster, PA, 1987, Ch. 5.7. Timoshenko, S. and Woinowsky-Krieger, S., Theory of Plates and Shells. McGraw Hill. New York. NY. 1959. Ch. 11. Peshkam, V. and Dawe, D.J., Buckling and vibration of finite length composite prismatic plate structures with diaphragm ends, Part II: computer programs and buckling applications. Comput. Meth. Appl. Mech. Eng., 1989, 77. 2277 252.
15.
Loughlan, J., The buckling performance of composite stiffened panel structures, subjected to combined in-plane compression and shear loading. Compos. Struct., 1994, 29, 197-212. Tsai, S.W., Mechanics of composite materials. AFML-TR66-149, 1966, p. 15. Needham, R.A., The ultimate strength of aluminum alloy formed structural shapes in compression. J. Aeronaut. Sci., 1954, 21, 217-229.
16.
Pareto, B., Cours d’Economie Politique. Switzerland, 1896.
F. Rouge, Lausanne,
APPENDIX The coefficients in the buckling equations (10) and (11) are given below: c,, =D,i + 18(Di2 + 2Dh6)R2 + 81D22R4 c,, = 81 Dl, + 18(Di2 + 2D&R2 + Dz2R4 cs3 = 81 Dll + 162(D12 + 2De6)R2 + 81 Dz2R4 El1 =D11 + 8(D12 + 2D66)R2 + 16D22R4
REFERENCES 1.
2.
3.
4.
McGettrick, M. and Abbott, R., To MRB or not to be: intrinsic manufacturing variabilities and effects on load carrying capacity. In Proc. 10th DOD/NASA/FAA Conf. on Fibrous Comvosites in Structural Design. Hilton Head, SC, l-4 November -- r-~ 1993. Vol. I. p. v-39. Pinckney, R.L., Fabrication of the V-22 composite aft fuselage using automated fiber placement. In Proc. First NASA Advanced Composites Technology Conf, Seattle, WA, 29 October-l November 1990, pp. 3855397; NASA CP 3104. Swanson, G.D., Gurdal, Z. and Starnes, J.H., Structural efficiency study of graphite-epoxy aircraft rib structures. J. Aircraft, 1990, 27, 101l-1020. Bushnell, D. and Bushnell, W.D., Optimum design of composite stiffened panels under combined loading. In Proc. 34th AIAA/ ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Con& La Jolla, CA, 19-22 April 1993, pp. 2194-
5.
6.
2228; AIAA Paper 93-1562. Swanson, G.D., Metschan, S.L.. Morris, M.R. and Kassapoglou, C., The effects of design details on cost and weight of fuselage structures. In Proc. 3rd NASA Advanced Composites Technology Conf, Long Beach, CA, June 1992, pp. 601-622. Mabson, G.E., Flynn, B.W., Ilcewicz, L.B. and Graesser, D.L., The use of COSTADE in developing composite commercial aircraft fuselage structures. In Proc. 35th AIAA/ASME/ASCE/ AHS/ASC
7.
8.
434
Structures,
Structural
Dynamics
and Materials
,f?22= 16 Dll + 8(D12 + 2De6)R2 + Dz2R4 &
= 16 D,, + 72(D,2 + 2Dh6)R2 + 81 D22R4
E44 = 81 Dl, + 72(D,2 + 2D&R2
+ 16 D22R4
Equation (6) in the main text is derived as follows. The total weight of the panel is equal to the sum of the skin weight and stiffener weight: w = Wskin + wstiff
(Al)
The skin weight can be written as: W&in= p A B t
(A2)
where p is the density, A and B are the panel dimensions, and t is the skin thickness. Assuming the same density for the stiffeners, the stiffener weight is given by:
Conf
Hilton Head, SC, 1994; AIAA Paper 94-1492. Olson, J.T., Smith, P.J. and Ilcewicz, L.B., Advanced composite fuselage technology. In Proc. 10th DOD/NASA/FAA Conference on Fibrous Comvosites in Structural Design, Lake Tahoe, NV, 1991. Gutowski, T., Hoult, D., Dillon, G., Neoh, E., Muter, S., Kim, E. and Tse. M., Development of a theoretical cost model
Ws,iff=npA,A
(A3)
where n is the number of stiffeners, A, is the stiffener cross-sectional area and A is the panel width (or stiffener length).
Cost and weight minimization:
The number of stiffeners 12can be determined as a function of the panel width B and the stiffener spacing dS: n = int[(B/d,) + l]
(A4)
where the function ‘int’ implies rounding down to the nearest integer and accounts for the fact that the quantity in parentheses will, most likely, be a non-integer. Depending on the values of B and d,, the value of IZwill be such that
C. Kassapoglou
there may or may not be stiffeners at the edges of the panel. In the latter case, there will be an equal distance between the outermost stiffeners and the panel edges less than or equal to dJ2. Using equation (A4) to substitute for 12in equation (A3) and equations (A3) and (A2) to substitute in (Al), the final expression for W is obtained: W = pA{Bt
+ int[(B/d,)
+ l]A,}
435
Printed in Great Britain. All rights reserved PII:S1359-835X(96)00141-8
ELSEVIER
1359~835Xi97/$17.00
Simultaneous cost and weight minimization of composite-stiffened panels under compression and shear
Christos
Kassapoglou
Structural Technologies, Sikorsky Aircraft, Mail Stop S314A2, CT 06497, USA (Received 6 November 1995; revised 9 October 1996)
6900 Main Street, Stratford,
An approach is presented to determine the part configuration that minimizes the cost and weight of composite-stiffened panels under combined compression and shear loads. The stiffened panel is designed so that no overall, bay, or stiffener buckling occurs under the applied loads. Skin and stiffener material failure conditions and manufacturing constraints are also imposed. The stiffened panel cost and weight are minimized for a variety of stiffener cross-sectional shapes. A set of near-optimum (in a Pareto sense) configurations is determined. The final optimum configuration is selected from this set by minimizing a weight and cost penalty function. 0 1997 Elsevier Science Limited (Keywords: cost and weight optimization/minimization;
compression and shear: stiffened panel)
INTRODUCTION The usage of composite materials in aircraft structures has increased steadily over recent years but their application in primary structures is still limited. One of the main reasons is that the cost of composite structures is, in many applications, not competitive with the equivalent metal structure. While the development and implementation of new manufacturing technologies and materials have significantly reduced the cost of composite structures and, in several applications’, have made them competitive with metals, more improvements are necessary for further cost reductions. One of the approaches that the aerospace industry is exploring with promising results aims at incorporating cost as one of the design variables early in the design process’. This leads to design concepts that are efficient and compatible with the manufacturing technologies considered and the available material forms. In addition, it leads to studies where weight is traded against cost for various structural concepts and assists in the selection of a configuration (design concept, manufacturing process and material) that minimizes cost and/ or weight. Many studies have been published on optimizing the weight of stiffened panels3’4 but very few attempted to include cost as a design driver. Usually, the optimumweight designs based on different stiffener cross-sectional shapes or other initial assumptions are a posteriori compared with one another on the basis of cost estimates.
Recently, several attempts have been made to give cost a more prominent role in the design selection in composite fuselage studies5-‘. The results of a recent study in which cost was one of the two primary design drivers are presented in this paper. The design concept is that of a composite-stiffened panel under compression and shear loads representative of fuselage structure of civil transport aircraft. The stiffener cross-sectional shape, area and stiffness properties, as well as the skin thickness, are allowed to vary. The manufacturing process selected for this investigation is a combination of automated cutting and kitting of prepreg material with subsequent hand lay-up. Cost equations are developed for the various process steps and their predictions are summed over all steps to determine the total cost (in labour hours) of the stiffened panel. At the same time, the structure is sized to meet the applied loads (without buckling) and the corresponding weight is calculated. An approach to simultaneously minimize the cost and weight of the panel is presented. The effect of various geometry and stiffness parameters such as stiffener cross-sectional shape, spacing, skin thickness, stiffener area and stiffener stiffness on cost and weight is examined.
PANEL CONFIGURATION Consider the composite stiffened panel shown in Figure 1. The panel is flat with dimensions 76.2 cm x 76.2 cm and is
419
Cost and weight minimization: C. Kassapoglou
stiffeners
Nx
Figure 1
Configuration
of stiffened panel under consideration
loaded in compression (ultimate load 140 kNm_‘) and shear (ultimate load 140kNm-‘). The stiffeners are assumed to have any of the shapes: L, C, Z, T, J, I and hat (shown in Figure 2). The skin thickness and stiffness, and stiffener cross-sectional area, shape, stiffness, moment of inertia and spacing, are allowed to vary. The stiffeners are assumed to be co-cured with the skin.
COST EQUATIONS The fabrication process for the stiffened panel is divided into the following steps: (1) cut material and package in kits, (2) hand lay-up, (3) bag, (4) remove bag, and (5) trim. Several intermediate steps such as tool preparation and cleaning, transportation of materials and parts, and curing are treated as constants. They depend on specific factory procedures and equipment lay-out. They are not included since they do not affect the optimization process. The first and last steps (cutting, kitting and trimming) are assumed to be fully automated and the cost or time associated with their completion is treated as a percentage of the total cost. The most labour-intensive steps are those of hand layup, vacuum bagging and bag removal. A series of equations that estimate the time required to lay-up stiffened panels has been developed based on actual time studies performed on the factory floor. These equations account for the type of material used and the geometry and complexity of the part to be laid-up. They are made up of three modules that are repeated as required: 1) time to lay-up a flat ply C,=k,A,+kzP
(1)
where A, and P are the area and perimeter of the ply; 2) time to bend a ply over a radius (convex to the operator) C, = k3 L
420
(2)
where L is the length of the ply to be draped; and 3) time to locate a ply into a radius (concave to the operator) Ci = k4 L’
(3)
where L’is the length of the tool into which the ply is being positioned. The constants ki-k4 depend on the material used (tape versus fabric), fibre orientation (for tape mainly) and factory practices. It should be noted that for manuallabour-intensive processes, the constant k4 is significantly higher than k3 (typically about 3 times higher). The equations for bag placement and bag removal have the same form as equations (1) and (2) with different values for the constants k,-k3. For simplicity, the material used in this study was assumed to be typical plain-weave fabric for both the skins and the stiffeners. Then, the total labour (in minutes) associated with fabricating a rectangular, flat, stiffened panel is obtained by summing the appropriate equations (l)-(3) over all plies for the skin and stiffeners, and is described by the following equations: skin c, = [CIAB + &(A + B)]t/tp,y
(4)
st$Zener (individual) G = [GA, +
C4t1
+
Cd2
+
Gt:l~l~p,y
(5)
where A and B are the panel dimensions, t is the skin thickness, A, is the stiffener cross-sectional area, and tpiy is the ply thickness which may or may not be the same for skin and stiffeners. The constants Ci-C, are given in Table 1 for various panel configurations. They are generic and represent the process selected and not a specific factory or company performance. They were obtained from timed fabrication trials. The corresponding cross-sectional shapes are shown in
Cost and weight minimization:
C. Kassapoglou
tr
3 t2
h
t1
tr-c-
t2
f
b
4
12
1
D
“T” or blade
b Figure 2
Table 1
Skin L C Z T I J Hat
Stiffener
b II I, I
“HAT”
shapes under consideration
Cost equations
(hand
lay-up of stiffened panel)
(‘1
(‘2
c3
c4
0.01 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.017 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.01 0.01 0.01 0.01 0.01 0.01 0.01
0.0 0.083 0.15 0.25
C5
0.0 0.017 0.017 0.033 0.083-0.005b 0.017 0.17 0.033 0.2-0.005b 0.033 0.5 0.067
cb
0.0 -0.01 -0.02 -0.02 -0.005 -0.01 -0.01 -0.04
Figure 2. In all cases, the lowest thickness (tl) occurs in the web. The web plies branch out at the bottom (and top if required) to create a portion of the flange. Additional flat plies may be added at the top and bottom flanges as required. Thus the flange thickness, t2, is larger than or equal to the web thickness. The equations in Table I do not include the addition of roving material at the web ends where the web plies branch out into the flange. An equation similar to equation (3) can be used to add that contribution if necessary. It should be noted that. if more than one material is
used with different ply thickness and draping properties, equations (4) and (5) should be modified accordingly. These equations make an attempt at capturing the essence of manual labour during lay-up and, at the same time, are in a simple (mostly linear) form. This is in agreement with recent findings by Gutowski et al.’ where simple linear models with appropriate selections of the coefficients based on equipment parameters were found to be equally as accurate as other more involved power-law models. The coefficients in Table 1 are generic and represent the process selected, and not a specific factory or company performance. Specific applications are expected to have somewhat different cost values than the ones predicted by the above equations.
WEIGHT The second objective function is the total weight of the panel. The panel is designed not to buckle under combined ultimate compression and shear loads. In addition, no bay buckling (the skin portion between
421
Cost and weight minimization: C. Kassapoglou
adjacent stiffeners) is allowed below ultimate loads. It should be noted that the buckling analysis used here does not account for transverse shear effects or for coupling between skin and stiffeners. These may affect the minimum weight and should be added for a more accurate analysis. Finally, the skin and stiffeners are also checked for material failure at ultimate load. The weight of the panel includes the weight of the skin and stiffeners, and can be expressed in the form (see Appendix for derivation and Figure I): W = pA{Bt + [int(B/d,) + l]A,}
(6)
where p is the density of the material, A and B are the length and width of the panel (compression is along dimension A), t is the skin thickness, A, is the crosssectional area of the stiffener, and d, is the stiffener spacing. The function ‘int’ in equation (6) implies rounding down to the nearest integer. Equation (6) assumes that the stiffeners are co-cured with the skin and they are of the same material. If these assumptions are not valid, the expression can be modified accordingly.
Timoshenko”
as:
Dll =Dllskin +
Wsc 7 s
012
= D12skin
022
= D22skin
066
= D66skin
GstJst + 2d s
It is assumed that D16 = &6 = 0. Es, and G,, are the stiffener Young’s and shear moduli. Z,, is the moment of inertia of a stiffener with respect to the centroid of the entire cross-section. Jst is the stiffener polar moment of inertia (negligible for open cross-sections). The accuracy of these expressions will be examined below. For simply supported panels under applied shear, Whitney” gives the following expression for the critical panel shear load: 7r44[Dllm4+ 2(D12 f 2D66)m2n2R2 + D2*n4R4]Amn - 32 mn R3b2 N xycrit2 5MgA, = 0; i=l &I .. m+iodd
CONSTRAINTS
Mij =
The cost and weight expressions are to be minimized subject to the buckling, strength and producibility constraints. These constraints are discussed below.
(m2
_
i*~(n*
i=m,
_j*)
m+i
n+j
(9)
odd
even
=o j = n, n + j even
Buckling constraints Overall buckling of the panel under compression and shear is predicted by using an interaction equation of the form’: &+R;=l
(7)
where
NX
R, = NXM
R, = - NXY N xyxit
are the ratios of the applied compression and shear to the respective buckling loads when the loads are applied individually. The individual compression and shear buckling loads for a simply supported panel are determined by using the expressions given by Whitney lo.For compression alone, the panel as a whole buckles when the applied load N, equals N
xcnt
= n2[D,lm4 + 2(D12 + 2D&m2R2 + D22R4] (8) A2m2
where an infinite series in the out-of-plane deflection w of the form w = ~~Amnsin~sin~ is assumed. If the series is truncated after m = n = 3, the buckling load under shear can be shown to be the smallest of the two values:
and Nxy
=~
1
128R A2
X
__- _ d
E~~-%J%&I - 4 lE33 + E33& &E**+-zJ81
where m (the number of half-waves along A) is picked such that NXcct is minimized, R is the aspect ratio A/B, and D, are the bending matrix entries for the cross-section, including skin and stiffeners, given by
422
(11) + E22-b
-
2.5
where the constants Cij and J!$ are combinations of the bending stiffnesses and are given in the Appendix. This truncation is done to speed up the computer iterations
Cost and weight minimization: C. Kassapoglou
250
i
50
0
100
Nx Figure 3
Buckling
prediction
comparisons.
Present
+ WA2
+ 2Wm2R2
A*m*
--e-
present
200
150
vwsus finite element solutions
+ hR41
Using equation (12) and the fact that Fst =
(84
where R = A/d, is the aspect ratio of the bay and Dii = Dvskin(no contribution from the stiffeners) with analogous changes to equations (10) and (11) for the shear buckling load of the bay. When the bay buckles, the bay shear buckling load is equal to the shear load applied to the panel since the shear load is assumed to be uniform along the panel edge and all the shear load is taken by the skin. The compression buckling load, however, is not the same as the applied load since part of the load is taken by the skin and part by the stiffeners. The compressive load in the skin, Nxskin,is found as a function of the applied load N, by applying strain compatibility (the compressive strain in the stiffeners equals the compressive strain in the skin) and integrating the strain-displacement equations using simply supported boundary conditions and immovable edges along the panel edges parallel to the stiffener axis. This approach is valid for non-buckling panels which is the case here. The resulting expression is:
(12)
where aG are entries of the inverse of the membrane matrix Aii for the skin and Es, is the stiffener Young’s modulus. An additional buckling check is imposed for column buckling of each individual stiffener under compression.
(N, - Nrskin)B
(13)
i.e. the force in the stiffeners is equal to the total applied force minus the force in the skin, the standard (simply supported ends) Euler column-buckling equation gives: (~11 1 +
NX Nrskin= ~ 1 + (a11- ~:,l~,,KWs
ref[12,13]
(kN/m)
during the optimization process. The inaccuracies introduced by this truncation are alluded to by Whitney” and are also briefly discussed below. For bay buckling, the same expressions (7 )-(1 1) are used but are modified to represent bay properties and loads. For example: Nbay = r2V4,m4 xcr1t
_t_
(all
-
d2/a22b4,/4 -
~:2/~22)WsPs
7r *
=
N,A2
(14)
where 1, is the moment of inertia of each stiffener about the stiffener centroid. The accuracy of equations (7)-(12) was verified by comparing their predictions with published results’*‘” where very detailed finite element methods were used to predict buckling loads of stiffened panels (NASA Panel 1 of ref. 13 is used here) under combined compression and shear. The analysis in ref. 13 is based on the finite strip method, where the structure is divided in strips. Within each strip, deflections are represented as polynomials across the width and trigonometric functions along the length, resulting in eight degrees of freedom per strip. The predictions of equations (7)-(12) and the results of ref. 13 are compared in Figure 3. Excellent agreement is observed for N,,,/N, ratios between 0 and 3. For higher ratios, the present method becomes unconservative with the worst deviation being 19.5% for the case of pure shear (N,,/ N, = co). This deviation is primarily due to the series truncation after the third term discussed earlier and is considered adequate for preliminary design. The results of Figure 3 give confidence in the design equations chosen for buckling predictions. It should be pointed out that, in all loading cases of this example, the bay buckling load was significantly higher than the overall panel buckling load. Also, the moment of inertia in equations (7)-(12) was calculated about the centroid of the cross-section in two ways. In the first, the
423
Cost and weight minimization: C. Kassapoglou
exact centroid was found and 1, calculated using standard procedures. In the second, a portion of the skin equal to 30 times the skin thickness (standard procedure in design’) was assumed to be effective at each stiffener location and both the centroid and 1, were calculated for the combined cross-section. The latter approximate method was within 3% of the exact calculations and was selected for all subsequent optimization runs in order to decrease computation time. This approximation, while valid for post-buckling structures, will not always be accurate for prebuckled panels. The exact I,, value should be calculated in such cases.
portion of the stiffener in question (flange or web) and f,/ft (flange width to flange thickness ratio) is given in Table 2. The terminology one-edge-free and no-edge-free
refers to the edges of the portion in question15. If both edges are attached to a flange or web, or the stress changes sign, the portion is considered a no-edge-free type (in the case of sign change, f, is taken to be the length from the edge to the location of sign reversal). If one edge is free, the portion is of the one-edge-free type. Manufacturing constraints
Manufacturing constraints that reflect geometry or material limitations, usually established by experience, are also imposed. These are the following.
Strength constraints
By using equations (12) and (13), the stresses in the skin and stiffeners can be calculated. For the skin, a first-ply-failure criterion (such as the Tsai-Hill14) is applied to check for skin failure. For the stiffeners, a flange or web crippling check is done. Crippling here refers to the final failure of a member (flange or web) after it has buckled locally. The axial load on each stiffener is given by:
1) The stiffener spacing can be no less than 10.16 cm.
2) 3) 4)
The compression load at the panel edges is acting at the skin mid-plane. This implies that a moment is generated about the stiffener centroid, or about the centroid of the combined cross-section. Combining axial and bending loads, the maximum stresses in each of the stiffener components of Figure 2 (top flange, bottom flange, web) are given in Table 2. Since the stiffeners are assumed to be co-cured with the skin, the bottom flange stresses in Table 2 include the stresses in the skin below the flange. Each of the components in Table 2 (top flange, web, bottom flange) is checked separately. For the top or bottom flange check, the stress value in Table 2 (provided it is compressive) is compared with the crippling allowable. For the web check, the value of gaVsin Table 2 is compared with the crippling allowable. The quantity j in TubZe 2 is the distance between the centroid of the stiffener (alone) and the bottom of the stiffener (top of the skin). The crippling allowable is obtained from a variety of tests and is given as follows.
5)
6)
This is to facilitate locating tools and reaching with both hands between stiffeners during fabrication and assembly. The width of the stiffener flanges cannot exceed the stiffener spacing, to avoid stiffeners coalescing into each other. No thickness value can be less than the minimum gauge. The latter is set equal to 0.381 mm. No flange width or web height can be less than 1.27 cm. Otherwise tooling and lay-up become very expensive. The skin and stiffener thicknesses (t, tl, tl + t2) can only be an integral multiple of ply thickness. No intermediate ply thickness values are allowed. The flange width must be larger than the web thickness.
It should be noted that hard tooling is used for the stiffeners as shown in Figure 4. A nylon or semi-rigid silicone-based bag is used for vacuum bagging.
vaou”m
repeating between
tools stiffeners
bag
/
For one-edge-free: outer
1.6753 = F,,
KVlft)o.7245
mold toot
line
(16a) f&
> 2
For no-edge-free:
Fcrip= Fcu = Fm
fwlh < 8 9.8307
KV/ft)1~099
fwlh > 8
(16b)
where F,, is the compression strength allowable for the
424
fly-away mandrels
Figure 4
Tooling
(foam)
used during
stiffened panel fabrication
Cost and weight minimization: C. Kassapoglou
t
0
V
425
Cosr and weight
minimization:
C. Kassapoglou
OPTIMIZATION The complexity of this optimization problem is increased by the fact that there are two objective functions, cost and weight. It is desired to determine the configuration that will minimize both objective functions simultaneously. This is done in a Pareto sensei where a configuration is considered optimum if, for any other feasible configuration, at least one of the objective functions to be minimized has a higher value. In order to gain some insight into the process, the panel is first optimized for each objective function separately. The minimum weight is determined as a result of either the bay buckling requirement or a combination of manufacturing and strength constraints, given the minimum stiffener moment of inertia dictated by the overall panel buckling condition. It is important to note that, for the weight minimization, the exact cross-sectional shape of the stiffener is not needed as the stiffener is represented by its cross-sectional area A, and its moment of inertia Z,. This makes the approach very flexible and sets up a possible cost minimization approach where the shape that matches the required A, and Z, (from weight minimization) must be found such that the cost is minimized. The individual minimum weight and cost configurations are then used to determine a set of Pareto optimum configurations. These are configurations for which any excursion away from the corresponding design results in at least one of the two objective functions increasing. This set will, in general, have more than one configuration. The optimization flow is shown in Figure 5. It consists of eight major steps (marked in Figure 5) and a series of checks and secondary calculations. Steps 3, 5 and 8 are discussed below. The rest are self-explanatory or covered in a previous section.
Step j-bay
If the skin stiffness is high relative to the applied loads and the stiffener spacing is sufficiently small, the bay will not buckle. Then, the stiffener area A, is determined as the lowest possible A, value that satisfies the strength and overall panel buckling constraints. This value must be higher than or equal to the value imposed by the minimum gauge and minimum web and flange dimensions. The panel buckling constraint gives a condition for the stiffener moment of inertia (I,,) as shown in Figure 5. The selected value for A, must be such that the required Z,, is feasible.
of st$ener geometry
The panel and bay buckling conditions are used to determine the minimum acceptable values for Z, (stiffener moment of inertia about the centroid of the entire cross-section including the skin) and A, (stiffener cross-sectional area). For simplicity in the calculations, Z,, is translated to Z, the moment of inertia about the bottom
426
A, =2bt,+ht,
(17)
Z=;(h+2tz)3-(b-t,);+
(18)
From equation (17) b=
A, - htl 2t 2
(19)
and substituting into equation (18) and rearranging gives: t = A,(3h2 + 9ht2 + 8t;) - 61 1 h(h2 + 3ht2 + 2t;)
(20)
Substituting this expression to equation (19) yields b = A, Et2
1 _ 3h2 + 9ht2 + 81; - 61/A, h2 + 3ht, + 2t;
1 (21)
Since b must be positive, the right-hand side of equation (21) must be positive. This yields (after some manipulation): 0 <
h < ,/481/A,
- 12t; - 6t, (22)
4
Since tl must be positive, the right-hand side of equation (20) must be positive. Hence,
buckling
Step 5-determination
of the stiffener (top of the skin). The two requirements on A, and Zgive two conditions for b, tl, h and t2 (see Figure 2 for definition of these quantities). The cost equations are then used to calculate the panel cost for each set of feasible b, h, tl and t2 values. The values of b, h, t1 and t2 that satisfy the requirements on A, and Z and minimize the cost, define completely the stiffener geometry. As an example, consider the case of an ‘I’stiffener (see Figure 2). The cross-sectional area A, and moment of inertia Z about the bottom of the stiffener are given by:
-9t2 + \/721/A, h>
- 15t; (23)
6
Equations (22) and (23) are combined to give the condition on h:
-9t, + J_ 6
4
(24) The following procedure is used. Given A, and Z, successive t2 values are selected as integral multiples of ply thicknesses and no less than the minimum gauge requirement. For each t2 value, the limits dictated by equation (24) are calculated and h is varied between these limits, provided the manufacturing constraint 4 in the previous section is not violated. For each pair of acceptable t2 and h values, equations (20) and (21) are used to determine tl and b with a check that manufacturing
Cost and weight minimization:
INPUT
C. Kassapoglou
OUTPUT
I/
11 Panel
Bucklino
L
=firyLLb7@ Determine
Stiffener
Geometry
A impose Strength Constraints
6
1
Use Minimum Cost & Weight Curves to Approach Pareto Optimum 8
Figure 5
Optimization
flow chart
constraints 2, 3, 4 and 6 of the previous section are satisfied. If tl is not an integral multiple of ply thickness, it is set to the next highest integral multiple of ply thickness and b is recalculated from equation (24). With b, tl, t, and h determined, the corresponding cost is calculated from equations (4) and (5). The procedure is repeated for various t2 values until the minimum cost is determined. Throughout the procedure, the additional conditions t2 > t, /2 (for ‘T’ and ‘I’stiffeners) and t2 > tl (for ‘C’, ‘Z’, ‘J’ and ‘hat’ stiffeners) are also imposed. An analogous approach is used for all other stiffener cross-sectional shapes. The respective conditions on h, tl, t2 and b are summarized in Table 3. Step 8--Pareto
optimization: minimum weight and cost
The procedure described so far and shown in Figure 5 can be used to determine separately the configurations for the lowest weight or cost. The lowest cost configuration will, in general, be different from the one for lowest weight. By varying the stiffener cross-sectional area between its two values for the lowest cost and weight configuration, the boundary of ‘near-optimum’ configurations (the Pareto set) is obtained in cost-weight space. A penalty function is established as the sum of the
percentage difference of the weight and cost of a specific configuration from the individual minimum weight and cost points. A search is done among the configurations in this Pareto set to determine the one for which the penalty function is minimized. This corresponds to the optimum configuration.
RESULTS The procedure described in the previous section was applied to the first example panel of refs 12 and 13 (NASA Panel 1). This is a 762mm x 762mm panel with a skin thickness of 3.912mm. The membrane and bending stiffness matrices (A and D matrices) for the skin are given in refs 12 and 13. While in these references the stiffeners were of rectangular constant cross-section and the spacing was fixed to 127 mm, here only the stiffener axial modulus was kept constant at 74.6GPa, while the stiffener spacing was free to vary and the stiffener cross-sectional properties (area and moment of inertia) were also varied. In addition, seven different cross-sectional shapes were considered: L, C, Z, T, J, I and hat (see Figure 2). The panel was assumed to be under 0.14 MN m-l compression and 0.14 MN m-l shear.
427
Cost and weight minimization: C. Kassapoglou
The stiffener compression failure strength [F,, in equation (16)] was assumed to be 10.2 MPa. No post-buckling was allowed. The skin was assumed to consist of plies with ply thickness 0.152 mm and the stiffener of plies with ply thickness 0.19mm. Thus, for the stiffener thicknesses, only integral values of 0.19 mm were allowed. For each stiffener cross-sectional shape, the lowest weight and cost configurations were determined separately. Typical examples for a ‘T’ stiffener are shown in Figures 6 and 7. The optimum weight of the panel as a function of stiffener spacing is shown in Figure 6. Two curves are shown, one for the configuration corresponding to the minimum weight for the selected stiffener spacing and one corresponding to the minimum cost. In general, given a stiffener spacing, the minimum weight configuration does not coincide with the minimum cost configuration. It is seen from Figures 6 and 7 that the optimum weight and cost curves have sudden jumps that occur at stiffener spacings of 10.89, 12.70, 15.24, 19.05 and 25.40cm. These are the spacings at which the number of required stiffeners decreases by one. Since both the weight and cost are a function of the total number of stiffeners, these incremental decreases in the stiffener number translate to incremental decreases in the weight and cost of the panel. In between these points, the weight and cost increase slightly. This is due to the fact that the skin thickness (and therefore skin weight and cost) are held constant. Then, as the stiffener spacing increases without decreasing the number of stiffeners, the area and moment of inertia of the stiffeners must increase to satisfy the buckling and stiffener crippling conditions. This trend of decreasing cost and weight with decreasing number of stiffeners holds true as long as the bays of the panel do not buckle. For the applied loading selected, bay buckling coincides with panel buckling when the stiffener spacing equals 29.72cm. Any increase in stiffener spacing beyond this value requires substantial increases in the stiffener area in order to keep the load in the skin equal to the bay buckling load. These increases in the cross-sectional area of the stiffeners will tend to more than offset any decreases in the weight (and cost) due to further decreases in the number of stiffeners. It is therefore expected that, at large stiffener spacings, the weight and cost will be increasing monotonically and at a relatively high rate. As a result, the lowest weight and lowest cost configurations are expected to occur right after the last decrease in the number of stiffeners, before bay buckling. This occurs at a stiffener spacing of 25.4 cm and is shown in Figures 6 and 7. It is important to note that while both the cost and the weight of the panel are a minimum for that stiffener spacing, the corresponding configurations (in terms of stiffener area, and values for h, tl , t2 and b of Figure 2) will not be the same (for the same crosssectional shape). A comparison between all stiffener cross-sectional shapes is shown in Figures 8 and 9. The lowest weight
428
curves are shown in Figure 8 as a function of stiffener spacing. It is seen that the lowest weight is achieved at a stiffener spacing slightly larger than 25.4cm for a ‘J’ stiffener and is equal to 3.7 kg. The next ‘best’ stiffener shape is that of a ‘T’ stiffener with a weight of 3.72 kg. It should be pointed out that the lowest weight for a ‘C’ stiffener coincides with that for a ‘Z’ stiffener. This may be due to the fact that any coupling and transverse shear effects have been neglected in the analysis. The cost, however, will not coincide for these two stiffeners. The situation is significantly different when minimum cost is the objective. Figure 9 shows that the lowest cost occurs for a ‘T’ stiffener (302.5min) and the next lowest cost is achieved by the ‘L’ stiffener (307.7min). The ‘J’ stiffener configuration, which had the lowest weight, is rather costly and ranks fifth among the shapes considered. The cross-sectional shapes for the lowest weight and cost cases are shown in order of increasing weight or cost and approximately to scale in Figure 10. The corresponding weight and cost values for each case are also shown. As expected, the highest web heights occur for the T and L shapes that have no top flanges. Also, the ‘T’ stiffener has weight and cost values lower than those of the ‘L’ stiffener even though, for the same amount of material in the web and flange, the two shapes would have the same cost. The reason is that placing the web in the middle of the flange (rather than the end) for the ‘T’ stiffener decreases the unsupported flange length (b) to half the corresponding value for an ‘L’ stiffener, significantly increasing the flange crippling strength. As is seen from equation (16) lower flange widths result in higher crippling strengths. This is also the reason why the flange for the ‘L’ stiffener is thicker than for the ‘T’ stiffener. The results of Figure 10 pose the question of how to select the optimum stiffener shape and satisfy both the minimum weight and cost requirements. If weight and cost are weighted equally, there is no definite answer since the lowest weight (‘J’ stiffener) configuration does not coincide with the lowest cost (‘T’ stiffener) configuration. However, in this specific example, the second lowest weight configuration (‘T’ stiffener) differs only by 0.5% from the lowest weight configuration and is the lowest cost configuration. This would lead to a nearly optimal selection of a ‘T’ stiffener. The specific optimum geometry for the ‘T’ stiffener that gives the lowest weight and cost is still undetermined. As was pointed out earlier and is shown in Table 4, the lowest cost and weight configurations for a ‘T’ stiffener do not coincide. Using the procedure of Figure 5 and a ‘T’ stiffener, the stiffener area was varied between the two values in Table 4 and the corresponding lowest cost and weight configurations were determined. The stiffener spacing was kept constant at 25.654cm, which is the spacing that gave the lowest cost and weight configurations (see Figures 8
Cost and weight minimization: C. Kassapoglou
I
131 II
II 9
9
$1 II Q
lel
<:“1;3” II Q
Q
429
Cost and weight minimization: C. Kassapogfou
T STIFFENER
4.2
-
min weight min cost
3.2,.,.,.,.,.,.,,,.,.,.,.,. 10 12 14
16
18
20
STIFFENER Figure 6
22
24
26
SPACING
28
30
32
34
(cm)
Effect of stiffener spacing on panel weight. Weight curves for lowest cost and lowest weight configurations (‘T’stiffener)
T STIFFENER
370
330
-
minweight min cost
310
27C
25c lo
12
14
16
16
20
STIFFENER Figure 7
430
22
24
26
SPACING
(cm)
26
30
32
34
Effect of stiffener spacing on panel cost. Cost curves for lowest weight and lowest cost configurations (‘T’stiffener)
Cost and weight minimization:
C. Kassapoglou
4.2
-L -T -
z,c
-J -
3.6 -1 10
12
14
16
18
20
22
STIFFENER
Figure 8
Minimum panel weight as a function
of stiffener spacing.
24
26
SPACING
(cm)
All stiffener
and 9). These configurations trace the boundary curve shown in Figure 1 I in cost-weight space. The lowest cost and lowest weight points are shown in that figure. As can be seen from that figure, any point above the boundary line is sub-optimal, since there is always a point on the boundary for which at least one of the two
28
30
32
HAT
34
shapes
objective functions (weight or cost) can be improved upon without increasing the value of the other. Also, any point below the boundary line is not feasible since the stiffener fails by web or flange crippling. The segment of the boundary line between the minimum weight and the minimum cost points (segment
I C L T
J HAT
280!.,.,.,.,.,.,.,.,.,.,.,.,1 10 12 14
18
18
20
STIFFENER
Figure 9
Minimum
panel cost as a function
of stiffener spacing.
22
24
26
SPACING
(cm)
All stiffener
28
30
32
34
shapes
431
Cost and weight minimization: C. Kassapoglou
r
4.85cm
J:Wt4.7Okg Cost-366.2min
TWb3.72Kg Cost-308.3mm
LzWtr3.74kg Cost-31 1 .Omin
Iwt-3.75kg C Wt-3.79kg Cost-381 .lmm Cost-340.9m
HAT: Wt-3.96 kg Cost-474.2 mms
Z:Wt-3.79kQ Coat-393.4 mins
(a) Order of increasing weight
T: Wt- 3.77 kg Cost- 302.5 mins
L: Wt- 3.76 kg Cost - 307.7 mins
C: Wt-3.79kg I: Wt.3.75kg Cost-340.9 mms Cost-349.3 mms
c
1
J: Wt-3.69kg Cost-3662 mms
1. 1
Z.Wt-3.79kg Cost-393.4mins
HAT: Wt-3.96 kg Cost- 474.2 mlns
(b) Order of increasing cost Figure 10
432
Lowest weight and cost stiffener configurations
Cost and weight minimization:
Table 4 ____..
Optimum
configuration
geometry
Lowest weight
Lowest cost
h (cm) b (cm) tl (mm) t? (mm) Area (cm’
4.86 1.32 0.76 1.33 0.547
5.32 2.48 0.57 1.52 0.68 1
)
(point A in Figure 11) and Cmin (point B) are the individual weight and cost absolute minimum points; and el and e2 are weight factors, the value of which should be deter-mined by the requirements of the specific application in question and other mission considerations. If the absolute minimum cost and weight points coincide, the penalty function from equation (25) is zero and, as expected, this point is the sought-for minimum. If e1 = e?, the weight and cost penalties are equally weighted. This approach is used here and a point along segment AB is sought that minimizes equation (25). By evaluating the penalty function along segment AB it can be shown that, for the specific example used here, the lowest penalty value occurs at point A (which also happens to be the point of lowest weight). Therefore, the configuration that optimizes cost and weight simultaneously is one using ‘T’ stiffeners at 25.654cm spacing with the dimensions shown in the second column of Tuble 4. The cross-sectional shape is also shown in Figure IOU (‘T’ stiffener). It should be pointed out that the optimum configuration found above is a function of the assumptions and loading used in this example. The optimum configuration would be different if the skin thickness and lay-up, stiffener modulus, and manufacturing constraints were different. It is also important to note that the cost and weight results presented here are a strong function of the manufacturing process and tooling approach selected.
for ‘T’ stiffener
Quantity
AB) corresponds to configurations that are the set of Pareto optimum points for the problem at hand. These are Pareto optimum points in the sense that any excursion away from these points, but not along the boundary line itself, will increase the value of at least one of the two objective functions. The question is how to select an optimum configuration among these points. This is done by assigning to each point in that segment a penalty value and finding the point that minimizes the penalty. This penalty value is a measure of how far away the specific point is from the lowest cost and weight values. In order to combine cost and weight in a single penalty function, a decision must be made on the relative merit of a cost or weight penalty. A penalty function is adopted in the following form:
PF =
(>I(WI -
C. Kassapoglou
Wmin)/ Wmm + e2(Ci- Cmin)/Cmin (25)
where Wi and Ci are the cost and weight of the specific point in question along segment AB; Wmin
T STIFFENER 310 309 308 307 306 (min
weight)
305 304 -
stiffeners and/or in this
fail by web flange crippling region
303 -
B-Riii 302 -
(mln
cost)
301 -
I
300 3.70
3.72
I 3.74 PANEL
Figure 1 I
Boundary
of lowest cost-weight
WEIGHT
I
1
3.76
3.78
3
IO
(Kg)
points (‘T’ stiffener)
433
Cost and weight minimization: C. Kassapoglou
Automated fabrication processes such as tape or fibre placement for the skins and pultrusion for the stiffeners, or use of resin-transfer moulding for the entire stiffened panel, will significantly reduce the cost figures given here and may alter the order of ‘best’ to ‘worst’ panel configurations. The present study aims at highlighting the approach and giving a flavour of the results rather than exhaustively examining all possibilities.
10.
CONCLUSIONS
13.
An approach to determine the configuration that simultaneously minimizes the cost and weight of composite-stiffened panels under compression and shear, under buckling, strength and manufacturing constraints was presented. A variety of stiffener crosssectional shapes was examined. It was found that the individual weight and cost minimum configurations do not, in general, coincide. In such cases, a set of ‘nearoptimum’ configurations was determined and a penalty function established. The optimum configuration is determined from this set as the one that minimizes this penalty function. For the examples examined, it was found that ‘J’ stiffeners give the lowest weight configurations while ‘T’ stiffeners give the lowest cost configurations. The optimum configuration for both cost and weight was obtained for a panel with ‘T’ stiffeners.
14.
for advanced composites fabrication.
Compos.
Manuf,
1994,
5(4), 231-239. 9.
11. 12.
Bruhn, E.F., Analysis and Design of Flight Vehicle Structures. S.R. Jacobs & Associates Inc., Indianapolis, IN, 1973, Section c5.11. Whitney, J.M., Structural Analysis of Laminated Anisotropic Plates. Technomic Publishing Co., Lancaster, PA, 1987, Ch. 5.7. Timoshenko, S. and Woinowsky-Krieger, S., Theory of Plates and Shells. McGraw Hill. New York. NY. 1959. Ch. 11. Peshkam, V. and Dawe, D.J., Buckling and vibration of finite length composite prismatic plate structures with diaphragm ends, Part II: computer programs and buckling applications. Comput. Meth. Appl. Mech. Eng., 1989, 77. 2277 252.
15.
Loughlan, J., The buckling performance of composite stiffened panel structures, subjected to combined in-plane compression and shear loading. Compos. Struct., 1994, 29, 197-212. Tsai, S.W., Mechanics of composite materials. AFML-TR66-149, 1966, p. 15. Needham, R.A., The ultimate strength of aluminum alloy formed structural shapes in compression. J. Aeronaut. Sci., 1954, 21, 217-229.
16.
Pareto, B., Cours d’Economie Politique. Switzerland, 1896.
F. Rouge, Lausanne,
APPENDIX The coefficients in the buckling equations (10) and (11) are given below: c,, =D,i + 18(Di2 + 2Dh6)R2 + 81D22R4 c,, = 81 Dl, + 18(Di2 + 2D&R2 + Dz2R4 cs3 = 81 Dll + 162(D12 + 2De6)R2 + 81 Dz2R4 El1 =D11 + 8(D12 + 2D66)R2 + 16D22R4
REFERENCES 1.
2.
3.
4.
McGettrick, M. and Abbott, R., To MRB or not to be: intrinsic manufacturing variabilities and effects on load carrying capacity. In Proc. 10th DOD/NASA/FAA Conf. on Fibrous Comvosites in Structural Design. Hilton Head, SC, l-4 November -- r-~ 1993. Vol. I. p. v-39. Pinckney, R.L., Fabrication of the V-22 composite aft fuselage using automated fiber placement. In Proc. First NASA Advanced Composites Technology Conf, Seattle, WA, 29 October-l November 1990, pp. 3855397; NASA CP 3104. Swanson, G.D., Gurdal, Z. and Starnes, J.H., Structural efficiency study of graphite-epoxy aircraft rib structures. J. Aircraft, 1990, 27, 101l-1020. Bushnell, D. and Bushnell, W.D., Optimum design of composite stiffened panels under combined loading. In Proc. 34th AIAA/ ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Con& La Jolla, CA, 19-22 April 1993, pp. 2194-
5.
6.
2228; AIAA Paper 93-1562. Swanson, G.D., Metschan, S.L.. Morris, M.R. and Kassapoglou, C., The effects of design details on cost and weight of fuselage structures. In Proc. 3rd NASA Advanced Composites Technology Conf, Long Beach, CA, June 1992, pp. 601-622. Mabson, G.E., Flynn, B.W., Ilcewicz, L.B. and Graesser, D.L., The use of COSTADE in developing composite commercial aircraft fuselage structures. In Proc. 35th AIAA/ASME/ASCE/ AHS/ASC
7.
8.
434
Structures,
Structural
Dynamics
and Materials
,f?22= 16 Dll + 8(D12 + 2De6)R2 + Dz2R4 &
= 16 D,, + 72(D,2 + 2Dh6)R2 + 81 D22R4
E44 = 81 Dl, + 72(D,2 + 2D&R2
+ 16 D22R4
Equation (6) in the main text is derived as follows. The total weight of the panel is equal to the sum of the skin weight and stiffener weight: w = Wskin + wstiff
(Al)
The skin weight can be written as: W&in= p A B t
(A2)
where p is the density, A and B are the panel dimensions, and t is the skin thickness. Assuming the same density for the stiffeners, the stiffener weight is given by:
Conf
Hilton Head, SC, 1994; AIAA Paper 94-1492. Olson, J.T., Smith, P.J. and Ilcewicz, L.B., Advanced composite fuselage technology. In Proc. 10th DOD/NASA/FAA Conference on Fibrous Comvosites in Structural Design, Lake Tahoe, NV, 1991. Gutowski, T., Hoult, D., Dillon, G., Neoh, E., Muter, S., Kim, E. and Tse. M., Development of a theoretical cost model
Ws,iff=npA,A
(A3)
where n is the number of stiffeners, A, is the stiffener cross-sectional area and A is the panel width (or stiffener length).
Cost and weight minimization:
The number of stiffeners 12can be determined as a function of the panel width B and the stiffener spacing dS: n = int[(B/d,) + l]
(A4)
where the function ‘int’ implies rounding down to the nearest integer and accounts for the fact that the quantity in parentheses will, most likely, be a non-integer. Depending on the values of B and d,, the value of IZwill be such that
C. Kassapoglou
there may or may not be stiffeners at the edges of the panel. In the latter case, there will be an equal distance between the outermost stiffeners and the panel edges less than or equal to dJ2. Using equation (A4) to substitute for 12in equation (A3) and equations (A3) and (A2) to substitute in (Al), the final expression for W is obtained: W = pA{Bt
+ int[(B/d,)
+ l]A,}
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