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Optimum design and evaluation of stiffened panels with practical loading
Compurm & S,r”rr”rlv Vol. 52, No. 6. pp. 1107-l 118. 1994 Copyright 1’. 1994 Elsevier Science Ltd Prinkd in Greal Briuin. All rights reserved 0045-7949194 $7.00 + 0.00
OPTIMUM DESIGN AND EVALUATION OF STIFFENED PANELS WITH PRACTICAL LOADING R. Butler, A. A. Tyler and W. Caot School of Mechanical Engineering, University of Bath, Claverton Down, Bath BA2 7AY, U.K. (Received 23 August 1993)
Ahatract-Optimum design of a blade-stiffened panel of composite/honeycomb sandwich construction and a metal T-stiffened panel is considered using the buckling and strength constraint program VICONOPT. Both panels have practical loadings which produce a nonlinear out-of-plane bending moment, calculated using beam-column expressions. Large deflection finite element analysis of the optima shows that modifications to these expressions are necessary when the panels are shear loaded. The use of integrally machined stiffeners, as opposed to a conventional, built-up panel designed using PANDA2, is shown to permit 20% mass saving when the latter has no postbuckling strength and 3% saving when postbuckling strength is allowed for.
INTRODUCTION
A number of computer programs for the optimum design of stiffened panels have been developed over the last l&15 years. These programs mostly use mathematical programming techniques to automatically produce low-mass dimensions, given a user input configuration, a set of design variables and a set of constraints (design criteria). Constraints include various buckling, material strength and stiffness criteria and design variables include plate breadths, plate thicknesses and, in some cases, where composites are used, ply angles of layers within laminated plates. Some programs allow for design in the postbuckled region [l-3] and some make approximations for the out-of-plane loads which arise when panels are imperfect or subject to normal pressure [I, 3-S]. One of these programs is the buckling and strength constraint program VICONOPT [5], which has been developed over the last 6 years to supersede the earlier and widely used PASCO [4] by including additional capabilities. In this paper, optimum design of a composite blade-stiffened panel and a metal T-stiffened panel using VICONOPT is evaluated. Both panels have practical loadings which produce a nonlinear out-ofplane bending moment. Design of the metal panel has already been considered [6], but the current paper includes important additional features which have been added to improve this design, Comparison is made with PANDA2 [l] design for this panel, although such comparison is limited to the VICONOPT restriction of design without including tcurrently at: Department of Mechanics and Electricity, Civil Aviation Institute of China, Tianjin 300300, People’s Republic of China.
any postbuckling strength. The merit, in terms of structural efficiency, of using a panel of integrally machined construction, as opposed to a more conventional, built-up panel with stiffeners rivetted or bonded to the skin, is also presented. Design feasibility of both panels is investigated, via the use of large deflection finite element analysis. OVERVIEWOF DESIGN CAPABILITIES
A detailed description of the VICONOPT design process has already been given [7]. Briefly, it proceeds as a series of design cycles in which efficiently calculated constraints and design sensitivities are used within the optimizer CONMIN [8] to produce a design of lower mass than previous cycles. Sequential Taylor series approximations are made to avoid excessive constraint calculation and feasibility is maintained throughout the design process by stabilization, which factors plate thicknesses to exactly satisfy the most critical of these constraints. The extent of such thickening is used to automatically adjust the move limits for subsequent calls to CONMIN within a design cycle, so that move limits are increased (reduced) when the adjustment is small (large), and the best use is made of constraints and sensitivities during each design cycle. Buckling constraints and sensitivities are calculated using the earlier analysis programs VIPASA [9] and VICON[lO] which both involve the exact solution of plate differential equations and therefore avoid the approximation associated with finite element methods. Analysis covers any prismatic assembly of anisotropic plates, but for the purpose of this paper only flat, stiffened panels are considered hereafter. Each plate can carry in-plane loads comprising any combination of longitudinal, transverse or shear
1107
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R. Butler et
stresses, which are invariant in the longitudinal x direction. These may he input directly or calculated from the applied loads N,, N,, and p and the initial imperfection e, shown in Fig. 1, and the load IV,
(applied along the longitudinal panel edges) and bending moments MJ and M; (applied along the transverse panel edges), where N,, Mj and M; are not shown in Fig. I because they are not considered here. In VIPASA analysis, modes of buckling are assumed to vary sinusoidally with half-wavelength ,l in the longitudinal direction and the theory includes coupling between in-plane and out-of-plane systems and uses a fully populated [A ] matrix. In addition, there exists the option to make classical thin plate assumptions or to consider transverse shear deformation [I I]. An overall stiffness matrix of order 4N is assembled for the complete structure, using the stiffness matrix method, and multi-level substructuring is used to reduce N, the number of nodes required to model the full structure. Longitudinal line supports may be prescribed to restrain displacements parallel to the x-. _Y-and z-axes of Fig. 1 and rotation about the x-axis. Transverse supports (parallel to the y-axis of Fig. 1) nearly always occur in practice (e.g. wing ribs) but cannot be prescribed in VIPASA analysis. However, when all plates are either isotropic or orthotropic and carry no shear, the lines of zero displacement of the buckled panel are straight and parallel to the y-axis; this is consistent with simply supported end conditions which are parallel to the y-axis. provided the buckling half-wavelength 1 divides exactly into the length I of the panel. Otherwise, i.e. if any plates are anisotropic or more especially, loaded in shear, the solutions obtained approximate the results for such end conditions. Here the stiffness calculations involve complex arithmetic which produces skewed lines of zero displacement, resulting from phase differences between displacements along
Fig. 1. Panel with applied axial load N,. shear load N,,., initial sinusoidal imperfection e and uniformly distributed normal pressure p; M is the bending moment calculated at governing sections along the panel length 1.
al.
transverse cross-sections. These results can, however, be expected to be realistic at short wavelength (e.g. i < 1/3) but will become excessively conservative as i approaches 1. PASCO, which contained VIPASA analysis. used an approximate method to model the panel as infinitely wide, in the above circumstances, rather than infinitely long. assuming better results would be obtained with boundary conditions satisfied transversely. Hence, to satisfy the VIPASA requirement of invariant loading and stiffness in the infinite direction, a ‘smeared stiffness’ solution was obtained with the panel represented by a single orthotropic stiffness. A subsequent study [12] showed that the method could be up to 50% inaccurate. VICON analysis was developed to overcome the above difficulties by using Lagrangian multipliers to couple VIPASA stiffness matrices for an appropriate set of half-wavelengths, 1, so that compatibility with an arbitrary supporting structure was achieved. Supports repeat at intervals of 1, hence a panel of finite length 1with simply supported ends may be modelled reasonably accurately by representing the simple supports by a line of rigid supports at x = 0. The results assume that the mode repeats over a length L = 21/5 for some value 0 < 5 < 1. Each value of 5 generates an infinite series of I, although a small finite number, chosen by selecting a suitable value of ny in the following equation, usually gives acceptable results: J =r/(< +2q),
where q =O. fl,
*2,..
., *n,,(l)
where the minus sign can be ignored for present purposes, its sole purpose being to cause VICON to use for half-wavelength -1 the complex conjugate of the stiffness matrix given by +l. During the design process, buckling modes are classified in a way which keeps track of critical and nearly critical buckling loads. Here local skin and stiffener buckling and torsional buckling are covered using VIPASA analysis, where each analysis involves a sinusoidal mode of suitable half-wavelength, i. In addition, overall buckling is predicted either by using VICON analysis, for values of 5 of 0, 0.25, 0.5, 0.75 and 1 in eqn (I), or by using VIPASA analysis at a suitable uncoupled value of 1, i.e. 1 = I is used for a panel which is simply supported at its ends and I = l/2 is used for a panel which is clamped at its ends. Here, for the reasons described above, the VICON analysis is more accurate than VIPASA analysis when the panel is shear loaded or contains anisotropic plates, but otherwise VIPASA is used, because it is more efficient than VICON since it involves only one half-wavelength. All VICONOPT results assume that the panel is one of an infinite set of identical panels connected rigidly end to end to form a panel of infinite length, thereby approximating longitudinal continuity with adjacent structure. Panels which repeat in the transverse direction can be
Design and evaluation of panels analysed by assuming infinite width and using recurrence relations [13] to enable analysis involving only a datum repeating portion.
II09
and at the panel ends MJ@cotj;-I).
NORMAL PRESSURE AND IMPERFECTION
The bending moment M of Fig. I occurs in most practical applications and has a very significant effect on design. It arises when the normal pressure p is applied or when the panel has an imperfection e and is calculated assuming that the entire panel acts as a beam-column. For the examples of the next section, governing values of M are found using the following expressions. The bending moment at the panel midlength for a panel with simply supported end conditions and an imperfection e is given by the standard result M=
N,e I - NJN,’
where N, is the Euler buckling load for the panel. For a panel with clamped end conditions and an applied normal pressure p, the bending moments are given by theory taken from Young [14]. At the panel midlength
where (4)
In each case M is assumed to act over the entire panel length and is distributed, assuming that plane sections remain plane, to give the longitudinal stress in each plate and hence to calculate the material strength constraints and the local and torsional buckling constraints; M does not effect overall buckling and so distribution of M does not occur for VICON analysis or for VIPASA overall buckling analysis. PRESENTATION AND DISCUSSION OF RESULTS
Composite blade stiffened panel
This problem illustrates VICONOPT design of an imperfect panel constructed from an efficient structural sandwich comprising a low density honeycomb core and high stiffness, carbon fibre composite covers. A previous study [I I], which did not consider imperfection, showed that use of the honeycomb core produced about 60% mass saving when compared with a solid composite panel. The panel is loaded in axial compression and acts as a wide column, simply supported at its ends, with an initial sinusoidal imperfection of f 0.03 in. Figure 2 shows the geometry, loading, design variables and material properties. Earlier designs, considering composite lay-ups of +45”/o”/S, led the optimizer to reduce the thickness of the k45” plies to zero and so all the composite material, designated by thicknesses t, and t, in Fig. 2(b), comprised 0” fibres. Allowable strain limits of 0.006 along the fibres and
Fig. 2. Composite blade stiffened panel. (a) Geometry and loading. (b) Repeating portion showing design variables and nodes used in the LUSAS finite element model. Material of thickness f, and t, has: E, = I9 x IO6 psi; Ez = 1.89 x IO6 psi; G,, = G,, = 2G2, = 0.93 x IO6 psi; viz = 0.31 and; p = 0.0571 lb/in.“. Material of thickness r2 and f, has: E, = IO’psi; Ez = G,, = Gz3 = Y,? = 0; G,, = 28,000 psi and; p = 0.00868 lb/in.).
R. Butler et
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0.004 normal to the fibres were imposed and, since the panel was not shear loaded and consists of orthotropic plates, the full range of buckling modes was covered using VIPASA analysis for the halfwavelengths i = 30. 15, IO, 7.5, 5.2.5 and 1.O in. Such analysis was performed having selected the VICONOPT option, which allows for the significant effect that transverse shear deformation has on this panel. Wide column behaviour was approximated during design by preventing rotation about the x-axis along the two longitudinal edges shown in Fig. 2(a). The initial design. with design variable values b = h = 5.0 in. and t, = tZ = t, = t, = 0.1 in., was governed by torsional buckling at load factor 1.80 and had a mass of 14.21 lb for a panel width of 24 in. The final design, with design variables b = 2.403 in.; 11= 2.204 in.; t, = 0.0401 in., t2= 0.333 in.; t,= 0.0110 in. and t4 = 0.173 in, was governed, at a load factor of 1.OO,by overall buckling, local skin buckling and torsional buckling, where the skin and torsional buckling occurred for the positive and negative imperfections, respectively. This design was achieved after 15 design cycles and had a final mass of 4.45 lb, again for a panel of width 24in. VIPASA infinite which was performed for this width analysis, final design to accurately model wide column behaviour. gave a critical buckling load factor of 1.06 for overall buckling, indicating that the approximate wide column model used in design was slightly conservative. The finite element software used to verify the VICONOPT optimum design in this case was LUSAS [ 151. The 3-stiffener wide panel was modelled using 4-node thin shell (QTS4) elements for the composite material and 8-node brick (HX8) elements for the honeycomb core. The model was six elements long and Fig. 2(b) shows the nodes used for each repeating portion. The material properties given in Fig. 2 were used, except, because of the difficulty in entering material properties of zero value, the core was modelled as an isotropic material with E = 72,800 psi, 1’ = 0.3 and Gu = G,, = G?3 = 28,000 psi.
Fig.
3. LUSAS
buckling
modes
al.
(The above VICONOPT critical load factors for the final design increased by less than 3% when these values were used.) Restraints were applied at the panel ends, where displacement in the z direction was prevented for all nodes and rotation about the .x-axis was prevented for the stiffener flange nodes, In addition, all nodes at the panel midlength were restrained in the x direction and, in a similar way to the model used during VICONOPT design, all nodes along the longitudinal edges were restrained from rotating about .Y. The axial load was applied to the nodes at the panel ends in proportion to the composite thickness between adjacent nodes, in order to achieve a constant strain across the panel width. LUSAS buckling analysis of the panel without imperfection was first carried out to validate the model; this gave a critical eigenvalue (load factor) of 0.96 for torsional buckling [Fig. 3(a)] and a load factor, corresponding to the 5th eigenvalue. of I. 10 for overall buckling [Fig. 3(b)]. The VICONOPT critical mode for the perfect panel, with the core material properties used in LUSAS. was. however, an overall mode which occurred at load factor 1.02. The discrepancy in these results was thought to be due to two features of the LUSAS model which approximate the VICONOPT result. Firstly, to keep the problem to a reasonable size, the LUSAS mesh was fairly coarse. A finer mesh, consisting of more elements in the longitudinal direction, would improve accuracy. Secondly, the LUSAS simple supports approximate the VICONOPT supports, since the latter completely restrain displacement of the stiffener flanges in the y direction, whereas the former restrain rotation about x at the four nodes of each stiffener. (The displacement in the )’ direction of the LUSAS stiffeners could not be restrained. since this would produce stresses in that direction.) It was, however, decided that the accuracy of the LUSAS model was adequate for the purpose of the following comparison. Large deflection analysis was then carried out for the above model with the positive and negative imperfections of Fig. 2. This analysis uses standard
for composite panel corresponding eigenvalue.
to: (a)
1st eigenvalue;
(b) 5th
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Design and evaluation of panels formulations ities which
to account for the geometric nonlineararise when significant changes in the
structural configuration occur during loading. Within each of the ten equal load steps that were applied, a linear prediction of the nonlinear response is made and subsequent interative corrections are performed, in order to restore equilibrium. These equilibrium iterations, which eliminate the residual (out-ofbalance) forces, are performed until a convergence criteria has been achieved. The LUSAS nonlinear solution was based on a Total Lagrangian approach to the standard Newton-Raphson procedure. Figure 4 shows the maximum compressive stress for each load step; this occurred at the centre of the panel for both imperfections and at the top of the skin for the positive imperfection and the tip of the stiffener for the negative imperfection. The figure also shows that the equivalent VICONOPT stresses are very similar up to load factor 0.9, but differ by about 3% and 13% at load factor 1.00 for the skin and stiffener respectively. Here the greater discrepancy in stiffener stresses was again thought to be due to the difference in stiffener boundary conditions explained above.
Metal T-stiffened panel
The results for VICONOPT design of a metal T-stiffened panel, which acts as a wide column and has a normal pressure loading are now presented and compared with the PANDA2 results obtained by Bushnell and Bushnell [I61 for the same problem. PANDA2 [I] is a buckling and strength constraint program for the optimum design of composite panels with a range of stiffener types. The program can handle both overall imperfections of the type shown in Fig. 1 and local imperfections in the form of a local buckling mode. Buckling loads are calculated with the use of either closed-form expressions or discretized models and local postbuckling of the skin may be accounted for. Figure 5 shows the geometry, loading, material properties and design variables for the problem. The very high allowable stress limit of IO5psi was used, since this value was used during PANDA2 design to enable design considering postbuckling strength. A selection of VIPASA uncoupled wavelengths in the range 0.25 in. < 1 < 10 in. were used for local buckling analysis and VICON overall buckling analysis,
Load Factor 0
0.2
0.4
0.6
Load Factor 0.8
C
1.0 I
0.2 0.4 0.6 0.8 D 1 I 1 , , I , I-
-20
-40
-60
-80
-100 x = VICONOPT o = LUSAS -120
stress (x103 psil)
(a)
(b)
Fig. 4. Comparison of VICONOPT and LUSAS predictions of maximum longitudinal compressive stress for composite panel at panel midlength. (a) Skin stress for positive imperfection. (b) Stiffener stress for negative imperfection.
R. Butler et al.
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Fig. 5. T-stiffened panel. (a) Geometry and loading. (b) Repeating portion showing design variables. Crosses denote position of point supports used in VICONOPT analysis and the material is isotropic with properties:
E, = IO7 psi; v = 0.3 and p = 0.1 lb/in.‘.
which was necessary because the panel was shear loaded, used n4 = 2 in eqn (1) to couple 1 values of I 2 6 in. The initial PANDA2 results, which did not allow for any postbuckling strength, were obtained with a factor of safety of 1.1 applied to local skin buckling calculations and to permit comparison, all VIPASA analyses were carried out with this factor of safety. Wide column behaviour was again approximated during design by preventing rotation about the x-axis along the longitudinal edges. The panel had clamped transverse supports which were obtained by restraining rotation about the x- and y-axes for all the nodes shown as crosses in Fig. 5 and restraining displacement in the z direction for the 5 nodes on the skin at each support. Figure 6 shows the pane1 mass and the proportion of mass in the skin at the end of each of the 15 design cycles that were performed for VICONOPT design. using a conventionally stiffened panel of built-up construction, and VICONOPT design, using an integrally stiffened panel. In the latter, the skin of thickness t, was omitted, since unbounded VICONOPT optimization tended to reduce its width to zero. Hence, as can be seen from Fig. 7, the stiffener spacing b is reduced dramatically during design, which improves the local skin buckling capacity of the panel. The skin thickness t, and, therefore, the proportion of the total mass required in the skin, is thereby reduced, see Figs 7 and 6 respectively. This integrally stiffened VICONOPT optimum is about 25% lighter than the unbuckled, conventionally stiffened design produced by PANDA2 and about 6% lighter than the second PANDA2 optimum reported by Bushnell and Bushnell [16], which permitted local postbuckling. In contrast, the VICONOPT design using conventional stiffeners, which were maintained by placing practical lower bounds of 2.5
and 0.5 in. on plate widths 6 and b,, respectively, achieved a similar final mass to the PANDA2 optimum, with 75% of this mass in the skin (Fig. 6). Table 1 shows the design variable values for VICONOPT and PANDA2 design. using conventional stiffeners, and VICONOPT design, using integral stiffeners. The PANDA2 optimum was, according to PANDA2 analysis, governed by local skin buckling at the pane1 midlength, torsional buckling at the panel ends and wide column buckling. VIPASA analysis of this optimum, with rotation about x restrained along the longitudinal edges, gave buckling load factors of 1.I0 and 1.04 for the local skin and torsional modes respectively. VICON infinite width analysis, using n,, = 2 in eqn (I), was performed to accurately model wide column behaviour. The critical buckling load factor in this case was 1.58. Figure 8 shows that the governing modes of the VICONOPT optimum with conventional stiffeners were skin buckling at the panel midlength, which occurred at half-wavelength, I = 2.14 in., and local stiffener and torsional buckling at the pane1 ends, at J. = 0.75 and 10.0 in., respectively. These modes all corresponded to a buckling load factor of 1.10. Accurate wide column analysis of this optimum gave a critical buckling load factor of 1.24 for partialoverall buckling. The governing VIPASA modes for the integral design are, again, skin buckling at the panel midlength. at E. = 0.75 in., plus two local stiffener modes at i, = 0.5 and 1.76 in. and the torsional mode at J = 10.0 in. Again, all of these occur at load factor 1.10. The wide column analysis in this case gives a factor of 1.27 for the critical partialoverall buckling. Large deflection finite element analysis of the VICONOPT integral design and the PANDA2 design
1113
Design and evaluation of panels
-
Integral panel
l Unbuckled PANDA2 0
Buckled PANDA2
A
Mcdified Integral Panel
0.8
msk iii
0.6
0.4
I
0
I
2
I
I
4
I
I 6
I
I
8
I
10
12
14
Design Cycle Fig. 6. Mass of conventional and integral T-stiffened panel during VICONOPT design and final mass of unbuckled and buckled PANDA2 designs and modified integral panel design. m is the total panel mass, corresponding to a panel of width 24 in. and mSLis the mass of the skin.
of Table 1 was carried out with the ANSYS [ 171finite element code to check the validity of the designs and, in particular, the validity of using eqns (3) and (5) to give the longitudinal stress distributions that govern VICONOPT design for this panel. The load was applied in six steps, the first five of which were equally spaced up to the loading of Fig. S(a), whilst the sixth step was at loads 10% above the fifth, i.e. at the local skin buckling design load. A maximum of 20 equilibrium iterations were used per step and the full Newton-Raphson option was selected. This analysis, like the LUSAS analysis described earlier, updates the overall stiffness matrix at each equilibrium iteration to account for large deflections. The transverse supports were modelled by preventing all displacements in the x direction at one end and coupling all displacements in the x direction at the other end, whilst preventing rotation of the skin about y and rotation of the stiffeners about x at both ends. The longitudinal supports were modelled in the same way
as Bushnell and Bushnell’s finite element comparison [ 161to approximately simulate a panel of infinite width. Thirty lengthwise and two widthwise quadrilateral shell elements (STIF63) were used for each of the plates of Fig. 5. Thus a total of 1080 and 720 elements were required to model the PANDA2 and VICONOPT optima respectively. The ANSYS results produced for both optima were well converged and the maximum lateral deflection decreased for successive equilibrium iterations in each case, indicating that, according to ANSYS, the configuration remained stable up to the design load. Figure 9 shows the maximum compressive stresses predicted by ANSYS at the panel ends, compared with those predicted using the beam column theory of VICONOPT. These stresses are compared since they cause stiffener buckling and give a greater discrepancy than the skin compressive stresses at the midlength, which only differ by about 7% at the design load factor of 1.10. For the VICONOPT
R. Butler et
1114
I
I
I
2
I
I
4
6
,
al.
I
8
I
I
10
I
,
12
14
Design Cycle Design variable values for integral T-stiffened panel during VICONOPT design.
Table I. Initial design variable values and optimum VICONOPT and PANDA2 values for panel with conventional and integral T stiffeners. All demensions are in inches and masses are in Ibs for a panel width of 24in. Integral
Conventional Design variable
Initial
VICONOPT
PANDA2
Initial
VICONOPT
8.000 2.000
2.500 0.500 0.779 0.572 0.0441 0.0873 0.0356 0.027 1
2.500 0.582 0.925 0.615 0.0437 0.0766 0.0279 0.0355
2.000
0.854
1.400 1.200 0.2000 0.1000 0.1000
0.770 0.648 0.0246 0.01 IO 0.0289
5.04
5.07
1.400 1.200 0.1000 0.2000 0.1000 0.1000 Mass
II.3
23.8
4.07
1115
Design and evaluation of panels I I CONVENTIONAL I
4.4
I
I
8
I
4.4
I
1
I
INTEGRAL
I
I
I
4
I
6
I
I
4
7.5 in
I
I
2
I
PANEL
t-
0
1
I
I
I
I
8
I
I
I
I h (in)
,
PANEL
3.3 b
2.56 ‘x.
)c’
$ tj iL” 2.:
.
in 4
\
J 1.1 C
I
I
2
I
I
1
4
6
I
8
h (in)
Fig. 8. VIPASA buckling curves for T-stiffened VICONOPT designs at panel midlength and ends
optimum, the ANSYS stress is 19% greater than the VICONOPT stress for a load factor of 1.O and for the PANDA2 optimum it is 14% greater. The ANSYS stresses vary across each stiffener flange (Fig. IO), whereas the VICONOPT flange stresses are constant. However, although ANSYS gives an accurate stress distribution for this panel, which is shear loaded and restrained from warping at its ends, the total compressive force on the flange predicted by VICONOPT is very similar to that given by ANSYS, i.e. within 2% for both optima. A further VICONOPT analysis of both optima was then carried out to approximate the more accurate ANSYS stress distribution. Here the ANSYS longitudinal stresses for the worst stiffener/skin portion at the panel ends were applied to all three stiffeners in the VICONOPT model. The stresses used in the stiffeners were the plate edge stresses, with VICONOPT assuming linear stress distribution between each edge, whereas the stress used in the skin was the average ANSYS value. These stresses were only used in VICONOPT to show whether the panel was stable or unstable at load factor 1.00. Otherwise VICONOPT would simply factor such plate stresses to
find the critical buckling load and would therefore not take into account the normal pressure loading, which has a nonlinear effect on longitudinal stress. The results of this more accurate analysis were that the VICONOPT optimum, using ANSYS stresses, was unstable, whereas the PANDA2 optimum was stable. The accuracy of the VICONOPT stresses was then improved in an ad hoc way, similar to the method [ 161 which makes the above PANDA2 optimum conservative by taking into account the additional longitudinal stresses that arise for panels with out-of-plane and shear loading. Here the t of eqn (4) was factored by Nc/(F,),,N,), where N,-, the axial buckling load of a column with the same flexural rigidity as the panel, is N,(4N,) for a panel which is simply supported (clamped) at its ends; F,:,, is the load factor of the design loading (including shear) which causes overall buckling; N, is the axial design load. This factored value of L was used to give a more realistic nonlinear bending moment M for the final integral design of Table I. It was applied during an additional stabilization step for which Nc and F,, ,, were not updated as the design changed, whereas the N, of eqn (4) was
R. Butler et al. Load
Factor
-50 stress (x103psi)
(b)
(a)
Fig. 9. Comparison of VICON~PT and ANSYS predictions ofm~imum longitudinal compressive stress in stiffener flanges at panel ends for: (a) VICONOPT integral design; (b) unbuckled PANDA2 design.
updated for each new design. VICON analysis using 4 = 0.0 and au = 2 in eqn (1) was performed to give Foi,, since the overall mode for this clamped ended panel will be dominated by the half-wavelength 1 = I/2. The panel had Fo,, = 2.59, NC =4 x 832 lb/in. and N, = 500Ib/in. Hence a value of ~~/(F~,~~~) of 2.57 was applied; this increased the stiffener stress to a value 4% below the maximum ANSYS stress of Fig. 9(a). Stabilization required a thickening of all plates by 3.6%, thereby increasing the panel mass from 4.07 to 4.22 lb, which is 20% below the unbuckIed PANDA2 design mass and 3% below the buckled PANDA2 design mass, see
the modified integral panel masses of Fig. 6. This new design was re-analysed, using large deflection ANSYS analysis, and again the governing stresses at the panel end were used for further VICONOPT buckling analysis. The design was stable at load factor 1.OO. CONCLUDING REMARKS
The optimum design of a metal panel and a composite panel, both subject to buckling and material strength constraints with practical loadings, has been considered using the program VICONOPT. The
A=
-34369
B=
-25602
c=
-16835
D=
-8068
E=
stress (psi)
698 i
Fig. IO. ANSYS plot of longitudinal stress in stiffener flanges for VICONOPT integral design at load factor 1.00.
1117
Design and evaluation of panels
designs, which have been evaluated using large dellection finite element analysis, illustrate the following points. 1. A critical requirement of the analysis is that a means of identifying critical and potentially critical buckling modes must be possible. The analysis considered here assumes a sinusoida buckling mode along the panel Iength, which results in separate buckling calculations being made at different values of longitudinal half-wavelength. Thus local, intermediate and overall modes can be retained during optimization. 2. The analysis assumes that the loading and configuration of the panel does not vary longitudinally and in so doing requires substantially fewer degrees of freedom to achieve acceptable accuracy, compared with the number required using finite elements. For example, VICONOPT analysis of the honeycomb/composite sandwich panel required 16 degrees of freedom, whereas finite element analysis required 1848. Furthermore, the finite element model, which was reasonably coarse in this case and therefore somewhat inaccurate, gave a different critical buckling mode for the perfect panel to the result given by VICONOPT. The latter was close to the exact result for the panel, since the only approximations involved were those which allow for transverse shear deformation and these have been shown to lead to errors of only 1 or 2% [ll]. 3. Design
can allow for the important practical features of manufacturing im~rfection and normal pressure, where the effect of both is to introduce a nonlinear out-of-plane bending moment. VICONOPT uses simple beam-column expressions to calculate this bending moment, which are sufficiently accurate in the absence of any shear load. However, for one of the designs considered, which included shear load, the expressions were found to be unconservative and were therefore modified in a semi-automated way. The resulting stresses were acceptable when compared with those found using large deflection finite element analysis. The full automation of this modification is a matter of future work. 4. The designs produced by VICONOPT and PANDA2, where the latter did not permit any postbuckling, are very similar for a shear loaded Tstiffened metal panel of conventional built-up construction. Here the VICONOPT design did not include the above modifications for shear. Furthermore, VICONOPT analysis of the unbuckled PANDA2 design gives results similar to those predicted by PANDA2. 5. VICONOPT design of the T-stiffened panel using integrally machined stiffeners presents substantial mass saving, compared with the conventionally constructed pane& when buckling of the latter is prevented, and small savings when postbuckling of the latter is permitted.
Acknowledgements-The first author would like to thank Professor F. W. Williams and Mr D. Kennedy of the University of Wales College of Card% (UWCC) and Dr M. S. Anderson of the Old Dominion University, Virginia, for their helpful advice. VICONOPT has been developed at UWCC under the suo~ort of NASA. British Aerosuace and the Science and E&ineering Research Council. a
REFERENCES
1. D. Bushnell, PANDAZ-program for minimum weight design of stiffened, composite, locally buckled panels. Comput. Struct. 25, 469-605 (1986). 2. P. Arendson and J. F. M. Wiggenraad, PANOPT user manual. NLR-report CR 91255L, National Aerospace Laboratory, Amsterdam (1991). 3. J. N. Dickson and S. B. Bigers, POSTOP: Postbuckled open-stiffened optimum panels, theory and capability. NASA Contra&or Report from NASA Contract NASI-15949. NASA-Lannlev Research Center. Hamn. I . ton, Virginia (1982). 4. W. J. Stroud and M. S. Anderson, PASCO: structural panel analysis and sizing code, capability and analytical foundations. NASA TM-80181 (1981). [Supersedes NASA TM-80181 (1980).] 5. F., W. Williams, D. Kennedy, R. Butler, M. S. Anderson, VICONOPT: Proaram for exact vibration and buckling analysis or design of prismatic plate assemblies. AIAA J. 29, 1927-1928 (1991). 6. R. Butler, Evaluation of the stiffened panels produced by VICONOPT. Structural Optimization 93, World Congress on OFfima~ Degisn of Structurai Systems, Rio de Janeiro, 1, 189-196 (1993). 7. R. Butler and F. W. Williams, Optimum design using VICONOPT, a buckling and strength constraint program for prismatic assemblies of -anisotropic plates. Cornput. Struct. 43, 699-708 (1992). FORTRAN pro8. G. N. Vanderplaats, CONMIN-\ gram for constrained function miminization. NASA TM-X-62-282, (version updated in March. 19751Ames Research Centre, Moffei Field, CA (Aug. 1973). See also G. N. Vanderplaats and F. Moses, Structural optimization by methods of feasible directions. Cornput. Sirucr. 3, 7399755 (1973). 9. W. H. Wittrick and F. W. Williams, Buckling and vibration of anisotropic or isotropic plates assemblies under combined loadings. Inr. J. Me& Sci. 16,209-239 (1974). 10. M. S. Anderson, F. W. Williams and C. J. Wright, Buckling and vibration of any prismatic assembly of shear and compression loaded anisotropic plates with an arbitrary supportin structure. Int. J. Mech. Sci. 25, 5855596 (1983). 11 M. S. Anderson and D. Kennedy, Inclusion of transverse shear deformation in exact buckling and vibration analysis of composite ptate assemblies. 33rd AIAA~AS~E~ASCH~IAHS~ASC Struct, Sfrucr. Dynam. and Mat. Cor$, Dallas, Texas. AIAA Paper 92-2287~CP, 283-291 (1992). 12 W. J. Stroud, W. H. Greene and M. S. Anderson, Buckling loads of stiffened panels subjected to combined longitudinal compression and shear: results obtained with PASCO, EAL and STAGS computer programs. NASA TP-221.5 (19841. 13. F. W. Williams and M. S: Anderson, Buckling and vibration analysis of shear-loaded prismatic plate assemblies with supporting structures, utilising symmetric or repetitive cross-sections. In Aspects of the Analysis of PIare Structures-A Volume in Honour of W. H. Wittriek (Edited by D. J. Dawe, R. W. Horsington, A. G. Kamtekar and G. H. Little), pp. 51-71. Oxford University Press, Oxford (1985).
III8
R. Butler ef al.
14. W. C. Young, Roark’s formulas for stress and strain, 6th Edn, (esp. Formula 2(d), Table 10. pp. 166). McGraw-Hill. New York (1989). 15. LUSAS user manual. Finite Element Analysis Ltd, Kingston-upon-Thames. Surrey (1990). 16. D. Bushnell and W. D. Bushnell, Minimum weight design of a stiffened panel via PANDA2 and evaluation
of the optimized panel via STAGS. 33rd AIAAIASMEIASCHE/AHS/ASC Smxt., Sirucr. Dynuns. and Mat. Cor$. Dallas, Texas, AIAA Paper 92-2316-CP. 25862618 (1992). 17. G. J. DeSalvo and R. W. Gorman. ANSYS engineering analysis system user’s manual. Swanson Analysis Systerns Inc., Houston. Pennsylvania (1989).
OPTIMUM DESIGN AND EVALUATION OF STIFFENED PANELS WITH PRACTICAL LOADING R. Butler, A. A. Tyler and W. Caot School of Mechanical Engineering, University of Bath, Claverton Down, Bath BA2 7AY, U.K. (Received 23 August 1993)
Ahatract-Optimum design of a blade-stiffened panel of composite/honeycomb sandwich construction and a metal T-stiffened panel is considered using the buckling and strength constraint program VICONOPT. Both panels have practical loadings which produce a nonlinear out-of-plane bending moment, calculated using beam-column expressions. Large deflection finite element analysis of the optima shows that modifications to these expressions are necessary when the panels are shear loaded. The use of integrally machined stiffeners, as opposed to a conventional, built-up panel designed using PANDA2, is shown to permit 20% mass saving when the latter has no postbuckling strength and 3% saving when postbuckling strength is allowed for.
INTRODUCTION
A number of computer programs for the optimum design of stiffened panels have been developed over the last l&15 years. These programs mostly use mathematical programming techniques to automatically produce low-mass dimensions, given a user input configuration, a set of design variables and a set of constraints (design criteria). Constraints include various buckling, material strength and stiffness criteria and design variables include plate breadths, plate thicknesses and, in some cases, where composites are used, ply angles of layers within laminated plates. Some programs allow for design in the postbuckled region [l-3] and some make approximations for the out-of-plane loads which arise when panels are imperfect or subject to normal pressure [I, 3-S]. One of these programs is the buckling and strength constraint program VICONOPT [5], which has been developed over the last 6 years to supersede the earlier and widely used PASCO [4] by including additional capabilities. In this paper, optimum design of a composite blade-stiffened panel and a metal T-stiffened panel using VICONOPT is evaluated. Both panels have practical loadings which produce a nonlinear out-ofplane bending moment. Design of the metal panel has already been considered [6], but the current paper includes important additional features which have been added to improve this design, Comparison is made with PANDA2 [l] design for this panel, although such comparison is limited to the VICONOPT restriction of design without including tcurrently at: Department of Mechanics and Electricity, Civil Aviation Institute of China, Tianjin 300300, People’s Republic of China.
any postbuckling strength. The merit, in terms of structural efficiency, of using a panel of integrally machined construction, as opposed to a more conventional, built-up panel with stiffeners rivetted or bonded to the skin, is also presented. Design feasibility of both panels is investigated, via the use of large deflection finite element analysis. OVERVIEWOF DESIGN CAPABILITIES
A detailed description of the VICONOPT design process has already been given [7]. Briefly, it proceeds as a series of design cycles in which efficiently calculated constraints and design sensitivities are used within the optimizer CONMIN [8] to produce a design of lower mass than previous cycles. Sequential Taylor series approximations are made to avoid excessive constraint calculation and feasibility is maintained throughout the design process by stabilization, which factors plate thicknesses to exactly satisfy the most critical of these constraints. The extent of such thickening is used to automatically adjust the move limits for subsequent calls to CONMIN within a design cycle, so that move limits are increased (reduced) when the adjustment is small (large), and the best use is made of constraints and sensitivities during each design cycle. Buckling constraints and sensitivities are calculated using the earlier analysis programs VIPASA [9] and VICON[lO] which both involve the exact solution of plate differential equations and therefore avoid the approximation associated with finite element methods. Analysis covers any prismatic assembly of anisotropic plates, but for the purpose of this paper only flat, stiffened panels are considered hereafter. Each plate can carry in-plane loads comprising any combination of longitudinal, transverse or shear
1107
1108
R. Butler et
stresses, which are invariant in the longitudinal x direction. These may he input directly or calculated from the applied loads N,, N,, and p and the initial imperfection e, shown in Fig. 1, and the load IV,
(applied along the longitudinal panel edges) and bending moments MJ and M; (applied along the transverse panel edges), where N,, Mj and M; are not shown in Fig. I because they are not considered here. In VIPASA analysis, modes of buckling are assumed to vary sinusoidally with half-wavelength ,l in the longitudinal direction and the theory includes coupling between in-plane and out-of-plane systems and uses a fully populated [A ] matrix. In addition, there exists the option to make classical thin plate assumptions or to consider transverse shear deformation [I I]. An overall stiffness matrix of order 4N is assembled for the complete structure, using the stiffness matrix method, and multi-level substructuring is used to reduce N, the number of nodes required to model the full structure. Longitudinal line supports may be prescribed to restrain displacements parallel to the x-. _Y-and z-axes of Fig. 1 and rotation about the x-axis. Transverse supports (parallel to the y-axis of Fig. 1) nearly always occur in practice (e.g. wing ribs) but cannot be prescribed in VIPASA analysis. However, when all plates are either isotropic or orthotropic and carry no shear, the lines of zero displacement of the buckled panel are straight and parallel to the y-axis; this is consistent with simply supported end conditions which are parallel to the y-axis. provided the buckling half-wavelength 1 divides exactly into the length I of the panel. Otherwise, i.e. if any plates are anisotropic or more especially, loaded in shear, the solutions obtained approximate the results for such end conditions. Here the stiffness calculations involve complex arithmetic which produces skewed lines of zero displacement, resulting from phase differences between displacements along
Fig. 1. Panel with applied axial load N,. shear load N,,., initial sinusoidal imperfection e and uniformly distributed normal pressure p; M is the bending moment calculated at governing sections along the panel length 1.
al.
transverse cross-sections. These results can, however, be expected to be realistic at short wavelength (e.g. i < 1/3) but will become excessively conservative as i approaches 1. PASCO, which contained VIPASA analysis. used an approximate method to model the panel as infinitely wide, in the above circumstances, rather than infinitely long. assuming better results would be obtained with boundary conditions satisfied transversely. Hence, to satisfy the VIPASA requirement of invariant loading and stiffness in the infinite direction, a ‘smeared stiffness’ solution was obtained with the panel represented by a single orthotropic stiffness. A subsequent study [12] showed that the method could be up to 50% inaccurate. VICON analysis was developed to overcome the above difficulties by using Lagrangian multipliers to couple VIPASA stiffness matrices for an appropriate set of half-wavelengths, 1, so that compatibility with an arbitrary supporting structure was achieved. Supports repeat at intervals of 1, hence a panel of finite length 1with simply supported ends may be modelled reasonably accurately by representing the simple supports by a line of rigid supports at x = 0. The results assume that the mode repeats over a length L = 21/5 for some value 0 < 5 < 1. Each value of 5 generates an infinite series of I, although a small finite number, chosen by selecting a suitable value of ny in the following equation, usually gives acceptable results: J =r/(< +2q),
where q =O. fl,
*2,..
., *n,,(l)
where the minus sign can be ignored for present purposes, its sole purpose being to cause VICON to use for half-wavelength -1 the complex conjugate of the stiffness matrix given by +l. During the design process, buckling modes are classified in a way which keeps track of critical and nearly critical buckling loads. Here local skin and stiffener buckling and torsional buckling are covered using VIPASA analysis, where each analysis involves a sinusoidal mode of suitable half-wavelength, i. In addition, overall buckling is predicted either by using VICON analysis, for values of 5 of 0, 0.25, 0.5, 0.75 and 1 in eqn (I), or by using VIPASA analysis at a suitable uncoupled value of 1, i.e. 1 = I is used for a panel which is simply supported at its ends and I = l/2 is used for a panel which is clamped at its ends. Here, for the reasons described above, the VICON analysis is more accurate than VIPASA analysis when the panel is shear loaded or contains anisotropic plates, but otherwise VIPASA is used, because it is more efficient than VICON since it involves only one half-wavelength. All VICONOPT results assume that the panel is one of an infinite set of identical panels connected rigidly end to end to form a panel of infinite length, thereby approximating longitudinal continuity with adjacent structure. Panels which repeat in the transverse direction can be
Design and evaluation of panels analysed by assuming infinite width and using recurrence relations [13] to enable analysis involving only a datum repeating portion.
II09
and at the panel ends MJ@cotj;-I).
NORMAL PRESSURE AND IMPERFECTION
The bending moment M of Fig. I occurs in most practical applications and has a very significant effect on design. It arises when the normal pressure p is applied or when the panel has an imperfection e and is calculated assuming that the entire panel acts as a beam-column. For the examples of the next section, governing values of M are found using the following expressions. The bending moment at the panel midlength for a panel with simply supported end conditions and an imperfection e is given by the standard result M=
N,e I - NJN,’
where N, is the Euler buckling load for the panel. For a panel with clamped end conditions and an applied normal pressure p, the bending moments are given by theory taken from Young [14]. At the panel midlength
where (4)
In each case M is assumed to act over the entire panel length and is distributed, assuming that plane sections remain plane, to give the longitudinal stress in each plate and hence to calculate the material strength constraints and the local and torsional buckling constraints; M does not effect overall buckling and so distribution of M does not occur for VICON analysis or for VIPASA overall buckling analysis. PRESENTATION AND DISCUSSION OF RESULTS
Composite blade stiffened panel
This problem illustrates VICONOPT design of an imperfect panel constructed from an efficient structural sandwich comprising a low density honeycomb core and high stiffness, carbon fibre composite covers. A previous study [I I], which did not consider imperfection, showed that use of the honeycomb core produced about 60% mass saving when compared with a solid composite panel. The panel is loaded in axial compression and acts as a wide column, simply supported at its ends, with an initial sinusoidal imperfection of f 0.03 in. Figure 2 shows the geometry, loading, design variables and material properties. Earlier designs, considering composite lay-ups of +45”/o”/S, led the optimizer to reduce the thickness of the k45” plies to zero and so all the composite material, designated by thicknesses t, and t, in Fig. 2(b), comprised 0” fibres. Allowable strain limits of 0.006 along the fibres and
Fig. 2. Composite blade stiffened panel. (a) Geometry and loading. (b) Repeating portion showing design variables and nodes used in the LUSAS finite element model. Material of thickness f, and t, has: E, = I9 x IO6 psi; Ez = 1.89 x IO6 psi; G,, = G,, = 2G2, = 0.93 x IO6 psi; viz = 0.31 and; p = 0.0571 lb/in.“. Material of thickness r2 and f, has: E, = IO’psi; Ez = G,, = Gz3 = Y,? = 0; G,, = 28,000 psi and; p = 0.00868 lb/in.).
R. Butler et
1110
0.004 normal to the fibres were imposed and, since the panel was not shear loaded and consists of orthotropic plates, the full range of buckling modes was covered using VIPASA analysis for the halfwavelengths i = 30. 15, IO, 7.5, 5.2.5 and 1.O in. Such analysis was performed having selected the VICONOPT option, which allows for the significant effect that transverse shear deformation has on this panel. Wide column behaviour was approximated during design by preventing rotation about the x-axis along the two longitudinal edges shown in Fig. 2(a). The initial design. with design variable values b = h = 5.0 in. and t, = tZ = t, = t, = 0.1 in., was governed by torsional buckling at load factor 1.80 and had a mass of 14.21 lb for a panel width of 24 in. The final design, with design variables b = 2.403 in.; 11= 2.204 in.; t, = 0.0401 in., t2= 0.333 in.; t,= 0.0110 in. and t4 = 0.173 in, was governed, at a load factor of 1.OO,by overall buckling, local skin buckling and torsional buckling, where the skin and torsional buckling occurred for the positive and negative imperfections, respectively. This design was achieved after 15 design cycles and had a final mass of 4.45 lb, again for a panel of width 24in. VIPASA infinite which was performed for this width analysis, final design to accurately model wide column behaviour. gave a critical buckling load factor of 1.06 for overall buckling, indicating that the approximate wide column model used in design was slightly conservative. The finite element software used to verify the VICONOPT optimum design in this case was LUSAS [ 151. The 3-stiffener wide panel was modelled using 4-node thin shell (QTS4) elements for the composite material and 8-node brick (HX8) elements for the honeycomb core. The model was six elements long and Fig. 2(b) shows the nodes used for each repeating portion. The material properties given in Fig. 2 were used, except, because of the difficulty in entering material properties of zero value, the core was modelled as an isotropic material with E = 72,800 psi, 1’ = 0.3 and Gu = G,, = G?3 = 28,000 psi.
Fig.
3. LUSAS
buckling
modes
al.
(The above VICONOPT critical load factors for the final design increased by less than 3% when these values were used.) Restraints were applied at the panel ends, where displacement in the z direction was prevented for all nodes and rotation about the .x-axis was prevented for the stiffener flange nodes, In addition, all nodes at the panel midlength were restrained in the x direction and, in a similar way to the model used during VICONOPT design, all nodes along the longitudinal edges were restrained from rotating about .Y. The axial load was applied to the nodes at the panel ends in proportion to the composite thickness between adjacent nodes, in order to achieve a constant strain across the panel width. LUSAS buckling analysis of the panel without imperfection was first carried out to validate the model; this gave a critical eigenvalue (load factor) of 0.96 for torsional buckling [Fig. 3(a)] and a load factor, corresponding to the 5th eigenvalue. of I. 10 for overall buckling [Fig. 3(b)]. The VICONOPT critical mode for the perfect panel, with the core material properties used in LUSAS. was. however, an overall mode which occurred at load factor 1.02. The discrepancy in these results was thought to be due to two features of the LUSAS model which approximate the VICONOPT result. Firstly, to keep the problem to a reasonable size, the LUSAS mesh was fairly coarse. A finer mesh, consisting of more elements in the longitudinal direction, would improve accuracy. Secondly, the LUSAS simple supports approximate the VICONOPT supports, since the latter completely restrain displacement of the stiffener flanges in the y direction, whereas the former restrain rotation about x at the four nodes of each stiffener. (The displacement in the )’ direction of the LUSAS stiffeners could not be restrained. since this would produce stresses in that direction.) It was, however, decided that the accuracy of the LUSAS model was adequate for the purpose of the following comparison. Large deflection analysis was then carried out for the above model with the positive and negative imperfections of Fig. 2. This analysis uses standard
for composite panel corresponding eigenvalue.
to: (a)
1st eigenvalue;
(b) 5th
IIll
Design and evaluation of panels formulations ities which
to account for the geometric nonlineararise when significant changes in the
structural configuration occur during loading. Within each of the ten equal load steps that were applied, a linear prediction of the nonlinear response is made and subsequent interative corrections are performed, in order to restore equilibrium. These equilibrium iterations, which eliminate the residual (out-ofbalance) forces, are performed until a convergence criteria has been achieved. The LUSAS nonlinear solution was based on a Total Lagrangian approach to the standard Newton-Raphson procedure. Figure 4 shows the maximum compressive stress for each load step; this occurred at the centre of the panel for both imperfections and at the top of the skin for the positive imperfection and the tip of the stiffener for the negative imperfection. The figure also shows that the equivalent VICONOPT stresses are very similar up to load factor 0.9, but differ by about 3% and 13% at load factor 1.00 for the skin and stiffener respectively. Here the greater discrepancy in stiffener stresses was again thought to be due to the difference in stiffener boundary conditions explained above.
Metal T-stiffened panel
The results for VICONOPT design of a metal T-stiffened panel, which acts as a wide column and has a normal pressure loading are now presented and compared with the PANDA2 results obtained by Bushnell and Bushnell [I61 for the same problem. PANDA2 [I] is a buckling and strength constraint program for the optimum design of composite panels with a range of stiffener types. The program can handle both overall imperfections of the type shown in Fig. 1 and local imperfections in the form of a local buckling mode. Buckling loads are calculated with the use of either closed-form expressions or discretized models and local postbuckling of the skin may be accounted for. Figure 5 shows the geometry, loading, material properties and design variables for the problem. The very high allowable stress limit of IO5psi was used, since this value was used during PANDA2 design to enable design considering postbuckling strength. A selection of VIPASA uncoupled wavelengths in the range 0.25 in. < 1 < 10 in. were used for local buckling analysis and VICON overall buckling analysis,
Load Factor 0
0.2
0.4
0.6
Load Factor 0.8
C
1.0 I
0.2 0.4 0.6 0.8 D 1 I 1 , , I , I-
-20
-40
-60
-80
-100 x = VICONOPT o = LUSAS -120
stress (x103 psil)
(a)
(b)
Fig. 4. Comparison of VICONOPT and LUSAS predictions of maximum longitudinal compressive stress for composite panel at panel midlength. (a) Skin stress for positive imperfection. (b) Stiffener stress for negative imperfection.
R. Butler et al.
1112
Fig. 5. T-stiffened panel. (a) Geometry and loading. (b) Repeating portion showing design variables. Crosses denote position of point supports used in VICONOPT analysis and the material is isotropic with properties:
E, = IO7 psi; v = 0.3 and p = 0.1 lb/in.‘.
which was necessary because the panel was shear loaded, used n4 = 2 in eqn (1) to couple 1 values of I 2 6 in. The initial PANDA2 results, which did not allow for any postbuckling strength, were obtained with a factor of safety of 1.1 applied to local skin buckling calculations and to permit comparison, all VIPASA analyses were carried out with this factor of safety. Wide column behaviour was again approximated during design by preventing rotation about the x-axis along the longitudinal edges. The panel had clamped transverse supports which were obtained by restraining rotation about the x- and y-axes for all the nodes shown as crosses in Fig. 5 and restraining displacement in the z direction for the 5 nodes on the skin at each support. Figure 6 shows the pane1 mass and the proportion of mass in the skin at the end of each of the 15 design cycles that were performed for VICONOPT design. using a conventionally stiffened panel of built-up construction, and VICONOPT design, using an integrally stiffened panel. In the latter, the skin of thickness t, was omitted, since unbounded VICONOPT optimization tended to reduce its width to zero. Hence, as can be seen from Fig. 7, the stiffener spacing b is reduced dramatically during design, which improves the local skin buckling capacity of the panel. The skin thickness t, and, therefore, the proportion of the total mass required in the skin, is thereby reduced, see Figs 7 and 6 respectively. This integrally stiffened VICONOPT optimum is about 25% lighter than the unbuckled, conventionally stiffened design produced by PANDA2 and about 6% lighter than the second PANDA2 optimum reported by Bushnell and Bushnell [16], which permitted local postbuckling. In contrast, the VICONOPT design using conventional stiffeners, which were maintained by placing practical lower bounds of 2.5
and 0.5 in. on plate widths 6 and b,, respectively, achieved a similar final mass to the PANDA2 optimum, with 75% of this mass in the skin (Fig. 6). Table 1 shows the design variable values for VICONOPT and PANDA2 design. using conventional stiffeners, and VICONOPT design, using integral stiffeners. The PANDA2 optimum was, according to PANDA2 analysis, governed by local skin buckling at the pane1 midlength, torsional buckling at the panel ends and wide column buckling. VIPASA analysis of this optimum, with rotation about x restrained along the longitudinal edges, gave buckling load factors of 1.I0 and 1.04 for the local skin and torsional modes respectively. VICON infinite width analysis, using n,, = 2 in eqn (I), was performed to accurately model wide column behaviour. The critical buckling load factor in this case was 1.58. Figure 8 shows that the governing modes of the VICONOPT optimum with conventional stiffeners were skin buckling at the panel midlength, which occurred at half-wavelength, I = 2.14 in., and local stiffener and torsional buckling at the pane1 ends, at J. = 0.75 and 10.0 in., respectively. These modes all corresponded to a buckling load factor of 1.10. Accurate wide column analysis of this optimum gave a critical buckling load factor of 1.24 for partialoverall buckling. The governing VIPASA modes for the integral design are, again, skin buckling at the panel midlength. at E. = 0.75 in., plus two local stiffener modes at i, = 0.5 and 1.76 in. and the torsional mode at J = 10.0 in. Again, all of these occur at load factor 1.10. The wide column analysis in this case gives a factor of 1.27 for the critical partialoverall buckling. Large deflection finite element analysis of the VICONOPT integral design and the PANDA2 design
1113
Design and evaluation of panels
-
Integral panel
l Unbuckled PANDA2 0
Buckled PANDA2
A
Mcdified Integral Panel
0.8
msk iii
0.6
0.4
I
0
I
2
I
I
4
I
I 6
I
I
8
I
10
12
14
Design Cycle Fig. 6. Mass of conventional and integral T-stiffened panel during VICONOPT design and final mass of unbuckled and buckled PANDA2 designs and modified integral panel design. m is the total panel mass, corresponding to a panel of width 24 in. and mSLis the mass of the skin.
of Table 1 was carried out with the ANSYS [ 171finite element code to check the validity of the designs and, in particular, the validity of using eqns (3) and (5) to give the longitudinal stress distributions that govern VICONOPT design for this panel. The load was applied in six steps, the first five of which were equally spaced up to the loading of Fig. S(a), whilst the sixth step was at loads 10% above the fifth, i.e. at the local skin buckling design load. A maximum of 20 equilibrium iterations were used per step and the full Newton-Raphson option was selected. This analysis, like the LUSAS analysis described earlier, updates the overall stiffness matrix at each equilibrium iteration to account for large deflections. The transverse supports were modelled by preventing all displacements in the x direction at one end and coupling all displacements in the x direction at the other end, whilst preventing rotation of the skin about y and rotation of the stiffeners about x at both ends. The longitudinal supports were modelled in the same way
as Bushnell and Bushnell’s finite element comparison [ 161to approximately simulate a panel of infinite width. Thirty lengthwise and two widthwise quadrilateral shell elements (STIF63) were used for each of the plates of Fig. 5. Thus a total of 1080 and 720 elements were required to model the PANDA2 and VICONOPT optima respectively. The ANSYS results produced for both optima were well converged and the maximum lateral deflection decreased for successive equilibrium iterations in each case, indicating that, according to ANSYS, the configuration remained stable up to the design load. Figure 9 shows the maximum compressive stresses predicted by ANSYS at the panel ends, compared with those predicted using the beam column theory of VICONOPT. These stresses are compared since they cause stiffener buckling and give a greater discrepancy than the skin compressive stresses at the midlength, which only differ by about 7% at the design load factor of 1.10. For the VICONOPT
R. Butler et
1114
I
I
I
2
I
I
4
6
,
al.
I
8
I
I
10
I
,
12
14
Design Cycle Design variable values for integral T-stiffened panel during VICONOPT design.
Table I. Initial design variable values and optimum VICONOPT and PANDA2 values for panel with conventional and integral T stiffeners. All demensions are in inches and masses are in Ibs for a panel width of 24in. Integral
Conventional Design variable
Initial
VICONOPT
PANDA2
Initial
VICONOPT
8.000 2.000
2.500 0.500 0.779 0.572 0.0441 0.0873 0.0356 0.027 1
2.500 0.582 0.925 0.615 0.0437 0.0766 0.0279 0.0355
2.000
0.854
1.400 1.200 0.2000 0.1000 0.1000
0.770 0.648 0.0246 0.01 IO 0.0289
5.04
5.07
1.400 1.200 0.1000 0.2000 0.1000 0.1000 Mass
II.3
23.8
4.07
1115
Design and evaluation of panels I I CONVENTIONAL I
4.4
I
I
8
I
4.4
I
1
I
INTEGRAL
I
I
I
4
I
6
I
I
4
7.5 in
I
I
2
I
PANEL
t-
0
1
I
I
I
I
8
I
I
I
I h (in)
,
PANEL
3.3 b
2.56 ‘x.
)c’
$ tj iL” 2.:
.
in 4
\
J 1.1 C
I
I
2
I
I
1
4
6
I
8
h (in)
Fig. 8. VIPASA buckling curves for T-stiffened VICONOPT designs at panel midlength and ends
optimum, the ANSYS stress is 19% greater than the VICONOPT stress for a load factor of 1.O and for the PANDA2 optimum it is 14% greater. The ANSYS stresses vary across each stiffener flange (Fig. IO), whereas the VICONOPT flange stresses are constant. However, although ANSYS gives an accurate stress distribution for this panel, which is shear loaded and restrained from warping at its ends, the total compressive force on the flange predicted by VICONOPT is very similar to that given by ANSYS, i.e. within 2% for both optima. A further VICONOPT analysis of both optima was then carried out to approximate the more accurate ANSYS stress distribution. Here the ANSYS longitudinal stresses for the worst stiffener/skin portion at the panel ends were applied to all three stiffeners in the VICONOPT model. The stresses used in the stiffeners were the plate edge stresses, with VICONOPT assuming linear stress distribution between each edge, whereas the stress used in the skin was the average ANSYS value. These stresses were only used in VICONOPT to show whether the panel was stable or unstable at load factor 1.00. Otherwise VICONOPT would simply factor such plate stresses to
find the critical buckling load and would therefore not take into account the normal pressure loading, which has a nonlinear effect on longitudinal stress. The results of this more accurate analysis were that the VICONOPT optimum, using ANSYS stresses, was unstable, whereas the PANDA2 optimum was stable. The accuracy of the VICONOPT stresses was then improved in an ad hoc way, similar to the method [ 161 which makes the above PANDA2 optimum conservative by taking into account the additional longitudinal stresses that arise for panels with out-of-plane and shear loading. Here the t of eqn (4) was factored by Nc/(F,),,N,), where N,-, the axial buckling load of a column with the same flexural rigidity as the panel, is N,(4N,) for a panel which is simply supported (clamped) at its ends; F,:,, is the load factor of the design loading (including shear) which causes overall buckling; N, is the axial design load. This factored value of L was used to give a more realistic nonlinear bending moment M for the final integral design of Table I. It was applied during an additional stabilization step for which Nc and F,, ,, were not updated as the design changed, whereas the N, of eqn (4) was
R. Butler et al. Load
Factor
-50 stress (x103psi)
(b)
(a)
Fig. 9. Comparison of VICON~PT and ANSYS predictions ofm~imum longitudinal compressive stress in stiffener flanges at panel ends for: (a) VICONOPT integral design; (b) unbuckled PANDA2 design.
updated for each new design. VICON analysis using 4 = 0.0 and au = 2 in eqn (1) was performed to give Foi,, since the overall mode for this clamped ended panel will be dominated by the half-wavelength 1 = I/2. The panel had Fo,, = 2.59, NC =4 x 832 lb/in. and N, = 500Ib/in. Hence a value of ~~/(F~,~~~) of 2.57 was applied; this increased the stiffener stress to a value 4% below the maximum ANSYS stress of Fig. 9(a). Stabilization required a thickening of all plates by 3.6%, thereby increasing the panel mass from 4.07 to 4.22 lb, which is 20% below the unbuckIed PANDA2 design mass and 3% below the buckled PANDA2 design mass, see
the modified integral panel masses of Fig. 6. This new design was re-analysed, using large deflection ANSYS analysis, and again the governing stresses at the panel end were used for further VICONOPT buckling analysis. The design was stable at load factor 1.OO. CONCLUDING REMARKS
The optimum design of a metal panel and a composite panel, both subject to buckling and material strength constraints with practical loadings, has been considered using the program VICONOPT. The
A=
-34369
B=
-25602
c=
-16835
D=
-8068
E=
stress (psi)
698 i
Fig. IO. ANSYS plot of longitudinal stress in stiffener flanges for VICONOPT integral design at load factor 1.00.
1117
Design and evaluation of panels
designs, which have been evaluated using large dellection finite element analysis, illustrate the following points. 1. A critical requirement of the analysis is that a means of identifying critical and potentially critical buckling modes must be possible. The analysis considered here assumes a sinusoida buckling mode along the panel Iength, which results in separate buckling calculations being made at different values of longitudinal half-wavelength. Thus local, intermediate and overall modes can be retained during optimization. 2. The analysis assumes that the loading and configuration of the panel does not vary longitudinally and in so doing requires substantially fewer degrees of freedom to achieve acceptable accuracy, compared with the number required using finite elements. For example, VICONOPT analysis of the honeycomb/composite sandwich panel required 16 degrees of freedom, whereas finite element analysis required 1848. Furthermore, the finite element model, which was reasonably coarse in this case and therefore somewhat inaccurate, gave a different critical buckling mode for the perfect panel to the result given by VICONOPT. The latter was close to the exact result for the panel, since the only approximations involved were those which allow for transverse shear deformation and these have been shown to lead to errors of only 1 or 2% [ll]. 3. Design
can allow for the important practical features of manufacturing im~rfection and normal pressure, where the effect of both is to introduce a nonlinear out-of-plane bending moment. VICONOPT uses simple beam-column expressions to calculate this bending moment, which are sufficiently accurate in the absence of any shear load. However, for one of the designs considered, which included shear load, the expressions were found to be unconservative and were therefore modified in a semi-automated way. The resulting stresses were acceptable when compared with those found using large deflection finite element analysis. The full automation of this modification is a matter of future work. 4. The designs produced by VICONOPT and PANDA2, where the latter did not permit any postbuckling, are very similar for a shear loaded Tstiffened metal panel of conventional built-up construction. Here the VICONOPT design did not include the above modifications for shear. Furthermore, VICONOPT analysis of the unbuckled PANDA2 design gives results similar to those predicted by PANDA2. 5. VICONOPT design of the T-stiffened panel using integrally machined stiffeners presents substantial mass saving, compared with the conventionally constructed pane& when buckling of the latter is prevented, and small savings when postbuckling of the latter is permitted.
Acknowledgements-The first author would like to thank Professor F. W. Williams and Mr D. Kennedy of the University of Wales College of Card% (UWCC) and Dr M. S. Anderson of the Old Dominion University, Virginia, for their helpful advice. VICONOPT has been developed at UWCC under the suo~ort of NASA. British Aerosuace and the Science and E&ineering Research Council. a
REFERENCES
1. D. Bushnell, PANDAZ-program for minimum weight design of stiffened, composite, locally buckled panels. Comput. Struct. 25, 469-605 (1986). 2. P. Arendson and J. F. M. Wiggenraad, PANOPT user manual. NLR-report CR 91255L, National Aerospace Laboratory, Amsterdam (1991). 3. J. N. Dickson and S. B. Bigers, POSTOP: Postbuckled open-stiffened optimum panels, theory and capability. NASA Contra&or Report from NASA Contract NASI-15949. NASA-Lannlev Research Center. Hamn. I . ton, Virginia (1982). 4. W. J. Stroud and M. S. Anderson, PASCO: structural panel analysis and sizing code, capability and analytical foundations. NASA TM-80181 (1981). [Supersedes NASA TM-80181 (1980).] 5. F., W. Williams, D. Kennedy, R. Butler, M. S. Anderson, VICONOPT: Proaram for exact vibration and buckling analysis or design of prismatic plate assemblies. AIAA J. 29, 1927-1928 (1991). 6. R. Butler, Evaluation of the stiffened panels produced by VICONOPT. Structural Optimization 93, World Congress on OFfima~ Degisn of Structurai Systems, Rio de Janeiro, 1, 189-196 (1993). 7. R. Butler and F. W. Williams, Optimum design using VICONOPT, a buckling and strength constraint program for prismatic assemblies of -anisotropic plates. Cornput. Struct. 43, 699-708 (1992). FORTRAN pro8. G. N. Vanderplaats, CONMIN-\ gram for constrained function miminization. NASA TM-X-62-282, (version updated in March. 19751Ames Research Centre, Moffei Field, CA (Aug. 1973). See also G. N. Vanderplaats and F. Moses, Structural optimization by methods of feasible directions. Cornput. Sirucr. 3, 7399755 (1973). 9. W. H. Wittrick and F. W. Williams, Buckling and vibration of anisotropic or isotropic plates assemblies under combined loadings. Inr. J. Me& Sci. 16,209-239 (1974). 10. M. S. Anderson, F. W. Williams and C. J. Wright, Buckling and vibration of any prismatic assembly of shear and compression loaded anisotropic plates with an arbitrary supportin structure. Int. J. Mech. Sci. 25, 5855596 (1983). 11 M. S. Anderson and D. Kennedy, Inclusion of transverse shear deformation in exact buckling and vibration analysis of composite ptate assemblies. 33rd AIAA~AS~E~ASCH~IAHS~ASC Struct, Sfrucr. Dynam. and Mat. Cor$, Dallas, Texas. AIAA Paper 92-2287~CP, 283-291 (1992). 12 W. J. Stroud, W. H. Greene and M. S. Anderson, Buckling loads of stiffened panels subjected to combined longitudinal compression and shear: results obtained with PASCO, EAL and STAGS computer programs. NASA TP-221.5 (19841. 13. F. W. Williams and M. S: Anderson, Buckling and vibration analysis of shear-loaded prismatic plate assemblies with supporting structures, utilising symmetric or repetitive cross-sections. In Aspects of the Analysis of PIare Structures-A Volume in Honour of W. H. Wittriek (Edited by D. J. Dawe, R. W. Horsington, A. G. Kamtekar and G. H. Little), pp. 51-71. Oxford University Press, Oxford (1985).
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R. Butler ef al.
14. W. C. Young, Roark’s formulas for stress and strain, 6th Edn, (esp. Formula 2(d), Table 10. pp. 166). McGraw-Hill. New York (1989). 15. LUSAS user manual. Finite Element Analysis Ltd, Kingston-upon-Thames. Surrey (1990). 16. D. Bushnell and W. D. Bushnell, Minimum weight design of a stiffened panel via PANDA2 and evaluation
of the optimized panel via STAGS. 33rd AIAAIASMEIASCHE/AHS/ASC Smxt., Sirucr. Dynuns. and Mat. Cor$. Dallas, Texas, AIAA Paper 92-2316-CP. 25862618 (1992). 17. G. J. DeSalvo and R. W. Gorman. ANSYS engineering analysis system user’s manual. Swanson Analysis Systerns Inc., Houston. Pennsylvania (1989).