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The effect of stiffeners on the optimal ply orientation and buckling load of rectangular laminated plates
Computers and Structures 80 (2002) 2229–2239 www.elsevier.com/locate/compstruc
The effect of stiffeners on the optimal ply orientation and buckling load of rectangular laminated plates M. Walker
*
Center for Advanced Materials, Design and Manufacturing Research, Durban Institute of Technology, PO Box 953, 4001 Durban, South Africa Received 14 November 2000; accepted 4 July 2002
Abstract Optimal designs of symmetrically laminated rectangular plates with different stiffener arrangements are presented. The plates are subject to a combination of simply supported, clamped and free boundary conditions, and the design objective is the maximisation of the biaxial buckling load. This is achieved by determining the fibre orientations optimally with the effects of bending–twisting coupling taken into account. Square tubing is used as stiffeners, and the finite element method coupled with an optimisation routine is employed in analysing and optimising the laminated plate designs. The effect of the stiffener arrangement and boundary conditions on the optimal ply angles and the buckling load are numerically studied. It is demonstrated that in some cases, stiffeners can sometimes reduce, rather than increase the buckling strength of an optimally designed plate. Ó 2002 Civil-Comp Ltd. and Elsevier Science Ltd. All rights reserved.
1. Introduction An advantage of composite materials over conventional ones is the possibility of tailoring their properties to the specific requirements of a given application. The tailoring is mostly achieved by optimising the mechanical properties, thereby increasing the load carrying capacity of the structure. Optimisation of composite laminates with respect to ply angles to maximise the strength is necessary to realise the full potential of fiberreinforced materials. A common composite layup is the symmetrically laminated angle ply configuration that avoid bending– stretching effects by virtue of mid-plane symmetry. These structures often contain components which may be modeled as rectangular plates. An important failure mode for these plates is buckling under in-plane loading. The buckling resistance of fiber composite plates can be improved by using the ply angle as a design variable, and determining the optimal angles to maximise the buckling
*
Tel.: +27-31-204-2116; fax: +27-31-204-2139. E-mail address: [email protected] (M. Walker).
load, and also by the addition of stiffeners. It has long been recognised that the use of stiffeners in plates increases structural efficiency and reduces structural weight. One phenomenon associated with symmetric angleply configurations is the occurrence of bending–twisting coupling which may cause significantly different results as compared to cases in which this coupling is exactly zero [1]. The effect of bending–twisting coupling becomes even more pronounced for laminates with few layers. Due to this coupling, closed-form solutions cannot be obtained for any of the possible boundary conditions and thus many studies concerned with the buckling performance of laminates have neglected these effects [2]. The present study adopts a numerical approach to include the effect of bending–twisting coupling and to obtain the optimal design solutions for stiffened plates with a variety of boundary conditions. Results obtained using different approaches can be found in the literature. These studies established that symmetric lamination yields the highest buckling load compared to other stacking sequences. Additionally, Ref. [2] reported the effect of different boundary conditions of the optimal design solutions. In that study, bending– twisting coupling was included in the formulation, and
0045-7949/02/$ - see front matter Ó 2002 Civil-Comp Ltd. and Elsevier Science Ltd. All rights reserved. PII: S 0 0 4 5 - 7 9 4 9 ( 0 2 ) 0 0 2 6 5 - 1
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M. Walker / Computers and Structures 80 (2002) 2229–2239
it was demonstrated how misleading results can be when these effects are neglected. A more detailed review of the subject can be found in [3]. Analyses and formulations for rectangular isotropic plates with longitudinal or transverse stiffeners were developed at least 40 years ago, by scholars like Timoshenko and Gere [4]. More recently, some studies have dealt with the effect of different arrangements of stiffeners [5–7] on the mechanical integrity of structures. Also, various approaches have been used to account for their effect, including that of minimising the total energy of the stiffened panel, and of using Lagrange multipliers to accurately predict the transverse deflections [8]. Subsequently, some researchers applied these techniques to the study of laminated plates with stiffeners, and of those, a few have even dealt with stiffened plates subjected to buckling or vibrational loads. For example, Phillips and G€ urdal [9] studied the effects of geodesic stiffeners on the buckling load capacity of plates, and carried out the analysis using the finite element method, while Lee [10] published an article that dealt with the effects of torsional and bending restraints of intermediate stiffeners on the free vibration of rectangular plates. Some experimental results have also appeared in the literature, like that of Mahendran [11], which details the effect of web stiffeners on the buckling performance of hollow flange beams, a subject also examined by Avery [12] using the finite element method. As there is little reported on the optimal design of stiffened laminates with bending–twisting accounted for, and with different boundary conditions, the present study aims to fill this gap. The finite element formulation which is used is based on Mindlin type theory for laminated composite plates. Numerical results demonstrating the effect of the stiffener arrangement and boundary conditions on the optimal ply angles and the buckling load are presented. 2. Buckling of symmetric laminates Consider a symmetrically laminated rectangular plate of length a, width b and thickness h which consists of n orthotropic layers with fiber angles hk , k ¼ 1; 2; . . . ; K, as shown in Fig. 1 (top). The plate is defined in the Cartesian coordinates x, y and z with axes x and y lying on the middle surface of the plate, and is subjected to biaxial compressive forces Nx and Ny in the x and y directions, respectively. In the present study, a first-order shear deformable theory is employed to analyse the problem and the following displacement field is assumed: u ¼ u0 ðx; yÞ þ zwx ðx; yÞ v ¼ v0 ðx; yÞ þ zwy ðx; yÞ w ¼ wðx; yÞ
where u0 ; v0 and w0 are the displacements of the reference surface in the x, y and z direction, respectively, and wx ; wy are the rotations of the transverse normal about the x and y axes. The in-plane strain components can be written as a sum of the extensional and flexural parts and they are given as fg ¼ fp g þ zff g
ð2Þ
where fgT ¼ ðx ; y ; xy Þ and 0
1 u0;x A; fp g ¼ @ v0;y u0;y þ v0;x
0
1 wx ff g ¼ @ wy A wx þ wy
ð3Þ
Here, a subscript after the comma denotes differentiation with respect to the variable following the comma. The transverse shear strains are obtained from w0;x þ wx cxz ¼ ð4Þ fcg ¼ cyz w0;y þ wy The equations for in-plane stresses of the kth layer under a plane stress state may be written as 0 1 0 1 0 1 11 Q 12 Q 16 x rx Q @ ry A ¼ @ Q 12 Q 22 Q 26 A @ y A 16 Q 26 Q 66 rxy ðkÞ xy Q ðkÞ fg ¼ ½Q ðkÞ
ð5Þ
and similarly for the transverse shear stresses as cyz syz Q Q ¼ a½CðkÞ fcg ¼ a 44 45 sxz ðkÞ Q45 Q55 ðkÞ cxz
ð6Þ
ij are the where a is a shear correction factor [13], and Q transformed stiffnesses. Eqs. (5) and (6) may be written in compact form as k rk ¼ Q
ð7Þ
ij Þ , k refers to the full matrix with elements ðQ where Q k; and rk and represent in-plane and transverse stresses and strains, respectively. The resulting shear forces and moments acting on the plate are obtained by integrating the stresses through the laminate thickness, viz Z h=2 fV gT ¼ ðVx ; Vy Þ ¼ ðsxz ; syz Þdz h=2
fMgT ¼ ðMx ; My ; Mxy Þ ¼
Z
h=2
ðrx ; ry ; rxy Þz dz
ð8Þ
h=2
ð1Þ
The relations between V and M, and the strains are given by
M. Walker / Computers and Structures 80 (2002) 2229–2239
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Fig. 1. Configuration of the laminated plate.
fV g ¼ ½Sfcg;
fMg ¼ ½Dff g
ð9Þ
where the stiffness matrices ½S and ½D are computed from ½S ¼ a
K Z X k¼1
½D ¼
K Z X k¼1
hk
hk 1 hk
hk 1
½CðkÞ dz ð10Þ
z2 dz ½Q ðkÞ
From the condition that the potential energy of the plate is stationary at equilibrium, and neglecting the prebuckling effects, the equations governing the biaxial buckling of the shear deformable laminate are obtained as
Mx;xx þ 2Mxy;xy þ My;yy þ Nx w;xx þ Ny w;yy ¼ 0 Mx;x þ Mxy;y Vx ¼ 0 My;y þ Mxy;x Vy ¼ 0
ð11Þ
where Nx and Ny are the pre-buckling stress components which are shown in Fig. 1. As no simplifications are assumed on the elements of the ½D matrix, Eq. (11) include the bending–twisting coupling as exhibited by virtue of D16 6¼ 0, D26 6¼ 0.
3. Finite element formulation We now consider the finite element formulation of the plate. Let the region S of the plate be divided into n sub-regions Sr ðSr 2 S; r ¼ 1; 2; . . . ; nÞ such that
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M. Walker / Computers and Structures 80 (2002) 2229–2239
PðuÞ ¼
n X
PSr ðuÞ
ð12Þ
r¼1
5.1. Verification
where P and PSr are potential energies of the plate and the element, respectively, and u is the displacement vector. Using the same shape functions associated with node i ði ¼ 1; 2; . . . ; nÞ, Si ðx; yÞ, for interpolating the variables in each element, we can write u¼
n X
Si ðx; yÞui
ð13Þ
i¼1
where ui is the value of the displacement vector corresponding to node i, and is given by ðiÞ
ðiÞ
5. Numerical results and discussion
ðiÞ
ðiÞ T u ¼ fu0 ; v0 ; w0 ; wðiÞ x ; wy g
ð14Þ
The static buckling problem reduces to a generalised eigenvalue problem of the conventional form, viz ð½K þ k½KG Þfug ¼ 0
ð15Þ
where ½K is the stiffness matrix and ½KG is the initial stress matrix. The lowest eigenvalue of the homogeneous system (15) yields the buckling load.
In order to verify the finite element formulation described above, some solutions are compared with those available in the literature. In particular, a single-layered simply supported square plate (having sides of length 1.066 m and thickness 0.0254 m) was modeled with properties specified as E ¼ 206:8 GPa and m ¼ 0:3: The plate has a single central solid beam stiffener (see Fig. 1), with a width of 0.254 m, and this was modeled using standard beam elements. The analytical solution for this problem is available in Ref. [4], and the buckling load Nx ¼ 39:2 MN/m. The use of 256 plate and shell elements (of the type described by the formulation above) for a square plate resulted in an error of <0.08% as compared to the analytical solution. This mesh density was accepted as providing sufficient accuracy [2]. Consequently, in the present study, a square plate is meshed with 256 elements. Plates of aspect ratios other than 1 are meshed with a corresponding proportion of 256 elements. 5.2. Numerical results
4. Optimal design problem The objective of the design problem is to maximise the buckling loads Nx and Ny for a given thickness h by optimally determining the fiber orientations given by hk ¼ ð 1Þkþ1 h for k 6 K=2 and hk ¼ ð 1Þk h for k P K=2 þ 1. Let Nx ¼ N and Ny ¼ kN , where 0 6 k 6 1 is the proportionality constant (and should not be confused with the eigenvalue represented similarly in Eq. (15)). The buckling load N ðhÞ is given by NðhÞ ¼ min½Nmn ðm; n; hÞ m;n
ð16Þ
where Nmn is the buckling load corresponding to the halfwave numbers m and n in the x and y directions, respectively. The design objective is to maximise N ðhÞ with respect to h, viz D
Nmax max½N ðhÞ; h
0° 6 h 6 90°
ð17Þ
where N ðhÞ is determined from the finite element solution of the eigenvalue problem given by Eq. (16). The optimisation procedure involves the stages of evaluating the buckling load N ðhÞ for a given h and improving the fiber orientation to maximise N. Thus, the computational solution consists of successive stages of analysis and optimisation until a convergence is obtained and the optimal angle hopt is determined within a specified accuracy. In the optimisation stage, the Golden section [14] method is employed.
Numerical results are given for a typical T300/5208 graphite/epoxy material with E1 ¼ 181 GPa, E2 ¼ 10:3 GPa, G12 ¼ 7:17 GPa and m12 ¼ 0:28. The symmetric plate is constructed of four equal thickness layers with h1 ¼ h2 ¼ h3 ¼ h4 ¼ h and the thickness ratio is specified as h=b ¼ 0:01: Three different stiffener arrangements are investigated, and these are illustrated in Fig. 1. Square tubing with wall thickness t is used as the stiffeners, and these have the same material properties as the plates (see above). For all the designs considered, g=b ¼ 0:01, and t=b ¼ 0:001. In addition, different combinations of free (F), simply supported (S) and clamped (C) boundary conditions are implemented at the four edges of the plate. In particular, five different combinations are studied, namely, (F; S; F; S), (F; S; C; S), (S; S; S; S), (C; S; C; S) and (C; C; C; C), where the first letter refers to the first plate edge, and the others follow in the anti-clockwise direction as shown in Fig. 1. At this point it may be important to repeat that convergence tests showed that the choice of the number of elements for modeling purposes was sufficient, irrespective of the boundary conditions implemented or the stiffener arrangement used (see Section 5.1). For each of the boundary conditions, three in-plane load cases are considered, namely, uniaxial compression (Ny ¼ 0), biaxial compression with Ny =Nx ¼ 0:5, and biaxial compression with Ny =Nx ¼ 1. The results presented in this section are obtained for rectangular plates with aspect ratios varying between 0.5
M. Walker / Computers and Structures 80 (2002) 2229–2239
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and 2. The non-dimensionalised buckling parameter Nb is defined as Nb ¼
Nb2 h3 E0
ð18Þ
where N is the critical buckling load, and E0 is a reference value having the dimension of YoungÕs modulus and is taken as E0 ¼ 1 GPa. The dependence of the buckling load Nb on the fiber angle has been investigated for the five cases of boundary conditions in Ref. [2] for unstiffened plates. It was demonstrated clearly that the maximum buckling load for a given boundary condition and aspect ratio occurs at a specific value of the fiber angle (referred to as the optimal fiber angle), and this value can be several times higher than the buckling load at other fiber angles. This fact emphasises the importance of carrying out optimisation in design work of this nature to obtain the best performance of fiber composite plates. The dependence of Nb on h for stiffened plates is demonstrated in Fig. 2, for (S; S; S; S) and (C; C; C; C) plates with a=b ¼ 1:25, and k ¼ 1. It is obvious from the curves that, as before, non-optimal values of the fiber orientation can lead to buckling loads which may be a small fraction of the maximum, and that also for plates with stiffeners, optimisation is important. It is also evident that (C; C; C; C) plates with two parallel stiffeners (viz type 2) give the highest buckling loads. Fig. 3 shows the influence of the loading ratio k on the value of Nb as h is varied, for (S; S; S; S) plates with a=b ¼ 1:25. It is obvious that as the magnitude of k increases, so Nb decreases; such that a type 1 plate with k ¼ 1 has the lowest buckling load through the whole range of fiber orientation. The effect of a=b on the buckling load is illustrated in Fig. 4, for type 1 plates with (S; S; S; S) or (C; C; C; C) boundary conditions, and k ¼ 1. As r decreases, so the
Fig. 2. Buckling load vs. ply angle for (S,S,S,S) and (C,C,C,C) stiffened plates with a=b ¼ 1:25 and k ¼ 1.
Fig. 3. Buckling load vs. ply angle for (S,S,S,S) stiffened plates with a=b ¼ 1:25 and k ¼ 0 or k ¼ 1.
Fig. 4. Buckling load vs. ply angle for (S,S,S,S) and (C,C,C,C) type 1 plates of varying aspect ratio with k ¼ 1.
plate becomes stiffer, and thus the magnitude of Nb increases, while (C; C; C; C) plates have greater buckling loads than those which are simply-supported. Finally, the effect of the boundary conditions on the buckling load is shown in Fig. 5, for type 1 plates with a=b ¼ 1:25 and k ¼ 1. Once again, it is obvious that as the number of degrees of freedom are increased, so the buckling load decreases. Consequently, (C; C; C; C) plates have the highest buckling loads, and (F; S; F; S) plates the lowest. Tables 1–5 give the optimal design solutions for the five boundary conditions implemented, and k ¼ 0: For each of the tables, the first column specifies the aspect ratio r, while columns 2 and 3 give the values of hopt and Nb for plates without stiffeners. Subsequently, the values of hopt and Nb are given for plates with one, two parallel and two perpendicular stiffeners, respectively. Table 1, for which the boundary condition is (F; S; F; S), illustrates that the optimal fiber angle is always
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M. Walker / Computers and Structures 80 (2002) 2229–2239
Fig. 5. Buckling load vs. ply angle for type 1 plates with varying boundary conditions and k ¼ 1, a=b ¼ 1:25.
0° [2] for plates without stiffeners, regardless of r. The buckling load decreases monotonically as the aspect ratio increases, from a maximum value of 596 (at r ¼ 0:5) to 37 (at r ¼ 2). Adding one stiffener increases the buckling load by 5.2% at r ¼ 0:5 and by 438% at r ¼ 2.
Thus the effect of the stiffener is greatest at larger aspect ratios, and this can be explained by the change in the values of the optimal fiber angle, which jumps from 2.5° at r ¼ 0:9 (where the value is similar to that of the unstiffened plate) to 47.4° at r ¼ 1. It is this change in mode that is responsible for the vastly increased strength of the stiffened plates at higher values of r. Similarly, the effect of two parallel stiffeners is to further increase the plate buckling strength. As before, at smaller aspect ratios, the increase is small (compared to a plate with one stiffener), but increases with increasing r. In this case, the mode change occurs at r ¼ 0:7. It is interesting to note that the buckling strength increases from 450 (at r ¼ 0:7) to 467 (at r ¼ 2), viz with increasing r the strength of plates with two parallel stiffeners increases. Using two stiffeners which are perpendicular has the effect of increasing the strength, compared to plates having two parallel stiffeners, but only for r ¼ 0:5 ! 0:6. Mode changes in the case of plates with perpendicular stiffeners have the effect of decreasing the buckling strength of these plates by up to 52% compared to those with two parallel stiffeners. It is obvious that the choice
Table 1 Optimal values of laminate fibre angle vs. maximum buckling load for (F; S; F; S) plates with k ¼ 0 a=b
0.5 0.6 0.7 0.8 0.9 1.0 1.25 1.5 1.75 2.0
hopt (°)
Nb
hopt (°)
Nb
hopt (°)
Nb
hopt (°)
Nb
0 0 0 0 0 0 0 0 0 0
596 415 305 234 185 149 96 67 49 37
0 0.1 0.5 1.2 2.5 47.4 45.5 44.1 44.4 45.6
627 449 342 272 225 205 204 201 199 199
0.4 1.3 47.2 48.6 48.1 47.4 46.1 46.0 46.5 47.0
677 505 450 457 461 462 463 463 464 467
3.3 5.3 8.5 14.6 36.0 38.4 39.1 37.3 35.2 53.6
695 508 396 327 295 282 262 245 230 220
Table 2 Optimal values of laminate fibre angle vs. maximum buckling load for (S; S; S; S) plates with k ¼ 0 a=b
0.5 0.6 0.7 0.8 0.9 1.0 1.25 1.5 1.75 2.0
hopt (°)
Nb
hopt (°)
Nb
hopt (°)
Nb
hopt (°)
Nb
0 0 0 0 39.5 44.5 47.5 40.3 36.3 43.7
623 444 336 267 247 243 254 252 238 231
39.4 38.9 36.7 35.0 34.3 35.2 45.3 46.6 46.8 46.1
888 865 834 798 761 729 707 720 734 747
31.6 30.6 30.0 30.7 34.9 43.2 45.6 45.8 46.3 47.1
1491 1392 1301 1219 1159 1142 1155 1177 1199 1222
16.6 21.4 25.5 24.1 23.6 22.4 19.8 31.2 38.9 45.3
2371 1929 1650 1453 1304 1184 955 820 769 760
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Table 3 Optimal values of laminate fibre angle vs. maximum buckling load for (F; S; C; S) plates with k ¼ 0 a=b
0.5 0.6 0.7 0.8 0.9 1.0 1.25 1.5 1.75 2.0
hopt (°)
Nb
hopt (°)
Nb
hopt (°)
Nb
hopt (°)
Nb
0 0 0 0 0 0 0 0 0 48.0
603 423 313 242 294 159 107 78 62 57
0 0 0 0 48.6 49.3 48.6 48.4 48.7 49.1
629 452 347 280 215 215 215 215 215 215
0 0 49.9 50.2 50.0 49.8 49.2 49.6 50.4 51.8
687 521 462 465 469 472 474 478 481 485
6.7 8.3 9.9 11.6 13.2 14.7 16.3 14.0 10.6 57.4
1884 1376 1064 862 726 630 489 408 349 235
Table 4 Optimal values of laminate fibre angle vs. maximum buckling load for (C; S; C; S) plates with k ¼ 0 a=b
0.5 0.6 0.7 0.8 0.9 1.0 1.25 1.5 1.75 2.0
hopt (°)
Nb
hopt (°)
Nb
hopt (°)
Nb
hopt (°)
Nb
0 0 55.1 49.7 45.0 41.0 32.1 46.5 40.4 49.6
637 462 386 404 408 403 368 375 371 364
39.7 37.3 35.1 33.9 34.4 39.4 46.4 46.6 46.3 46.6
1048 1022 981 935 894 868 878 896 910 922
31.7 30.6 30.3 32.8 44.2 45.6 45.0 45.0 45.6 46.7
1668 1574 1485 1411 1390 1404 1429 1449 1469 1492
16.6 21.5 22.9 22.7 21.5 19.8 17.2 34.3 42.8 47.4
2585 2095 1792 1583 1424 1296 1061 942 908 915
Table 5 Optimal values of laminate fibre angle vs. maximum buckling load for (C; C; C; C) plates with k ¼ 0 a=b
0.5 0.6 0.7 0.8 0.9 1.0 1.25 1.5 1.75 2.0
hopt (°)
Nb
hopt (°)
Nb
hopt (°)
Nb
hopt (°)
Nb
0 0 0 0 0 0 41.7 34.3 41.7 42.2
2397 1698 1272 994 805 672 470 445 404 402
0 0 32.6 31.4 30.7 30.5 48.2 48.7 48.7 49.6
2554 1918 1567 1457 1365 1288 1202 1247 1284 1319
28.8 26.2 24.5 23.7 23.8 25.5 46.8 46.2 47.2 49.1
3312 2940 2616 2347 2134 1974 2015 2066 2111 2169
11.8 15.7 18.5 19.5 19.5 19.2 17.9 17.4 33.2 50.3
4930 3823 3123 2660 2332 2088 1691 1456 1314 1313
of stiffener arrangement for (F; S; F; S) plates with k ¼ 0 must be that of two, perpendicular to each other for 0:5 6 r 6 0:6, and parallel to each other for r P 0:7. These trends are verified in Fig. 6, which illustrates for three values of a=b the dependence of Nb on h. For each of the aspect ratios (r ¼ 0.5, 1 and 2), the three different stiffener arrangements are implemented. The
peaks of each of the nine curves correspond to the figures given in Table 1 (rows 1, 6 and 10), viz the values of hopt and maximum Nb . Similar trends are noted in Tables 2–5, although for the cases (S; S; S; S), (C; S; C; S) and (C; C; C; C), the value of the buckling load also increases at larger values of r for plates with a single stiffener, as r increases. As
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M. Walker / Computers and Structures 80 (2002) 2229–2239
Fig. 6. Buckling load vs. ply angle for (F,S,F,S) stiffened plates of varying aspect ratios and k ¼ 0.
for the (F; S; F; S) plates, this did not occur with the (F; S; F; S) plates (although they do hold their strength for r P 0:7), and the reason may lay with the increased degrees of freedom associated with free edges. In addi-
tion, as the number of degrees of freedom is restricted, so the buckling load increases, and thus the Nb values for (C; C; C; C) plates are greatest. Tables 6–10 give the optimal designs for plates with k ¼ 1. In the boundary condition case (F; S; F; S), it is immediately obvious that the biaxial loading results in hopt 6¼ 0° for unstiffened plates (Table 6), as was the case with k ¼ 0 (Table 1). Also, the values of Nb are reduced by approximately 40% throughout the range of a=b. The addition of a single stiffener reduces the buckling load by 10% at r ¼ 0:5, and this trend continues until r ¼ 0:7. At r ¼ 0:8 (when a mode change takes place, and hopt jumps from 32.1° to 54.6°) the buckling load is 6% greater than that of the unstiffened plate, and the strengthening effect increases as r increases until at r ¼ 2, there is 260% more. When two parallel stiffeners are used, the results (columns 6 and 7) show that increased strength is the result, compared to plates with either one or no stiffener, at all value of r, but this is not the case when two perpendicular stiffeners are used (columns 8 and 9). For 0:9 6 r 6 2, the value of Nb is reduced, compared to the plates with parallel stiffeners, although slightly greater
Table 6 Optimal values of laminate fibre angle vs. maximum buckling load for (F; S; F; S) plates with k ¼ 1 a=b
0.5 0.6 0.7 0.8 0.9 1.0 1.25 1.5 1.75 2.0
hopt (°)
Nb
hopt (°)
Nb
hopt (°)
Nb
hopt (°)
Nb
24.7 25.3 24.8 23.3 21.6 20.7 31.0 30.5 30.5 29.7
252 176 130 99 78 63 39 27 20 15
25.4 26.0 32.1 54.6 57.5 58.2 57.6 56.8 57.4 59.8
228 161 122 105 98 92 79 69 60 54
49.5 57.3 59.6 60.3 60.5 60.6 61.5 62.9 64.2 65.5
239 211 196 185 175 166 147 134 125 118
1.7 3.1 5.2 8.0 11.1 14.5 48.7 54.2 58.3 61.7
547 373 269 202 158 127 90 75 66 60
Table 7 Optimal values of laminate fibre angle vs. maximum buckling load for (S; S; S; S) plates with k ¼ 1 a=b
0.5 0.6 0.7 0.8 0.9 1.0 1.25 1.5 1.75 2.0
hopt (°)
Nb
hopt (°)
Nb
hopt (°)
Nb
hopt (°)
Nb
24.8 23.0 21.1 28.2 39.9 45.0 62.0 68.3 66.5 65.2
402 285 211 161 136 121 103 103 102 101
48.6 51.5 51.9 52.3 53.5 56.0 62.2 64.7 66.1 67.2
555 507 470 439 414 396 377 372 370 370
45.8 46.3 47.4 50.0 54.3 58.1 63.4 65.9 67.4 68.5
909 802 719 655 614 590 565 559 558 558
29.6 32.3 33.8 36.1 46.6 45.3 41.6 49.0 54.6 59.2
1542 1206 992 845 750 684 550 459 419 399
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Table 8 Optimal values of laminate fibre angle vs. maximum buckling load for (F; S; C; S) plates with k ¼ 1 a=b
0.5 0.6 0.7 0.8 0.9 1.0 1.25 1.5 1.75 2.0
hopt (°)
Nb
hopt (°)
Nb
hopt (°)
Nb
hopt (°)
Nb
24.7 24.7 24.7 24.6 24.5 24.3 24.3 53.9 72.0 74.1
252 176 130 100 79 64 41 30 28 28
24.3 24.3 26.0 63.0 68.0 72.5 76.5 76.1 75.8 75.7
252 177 131 114 110 109 108 108 108 107
58.0 69.3 74.4 75.8 75.9 75.7 75.6 75.5 75.4 75.4
258 245 243 243 243 243 242 240 239 237
23.6 23.7 24.1 25.0 26.2 27.3 29.4 64.7 70.2 73.4
1016 718 532 410 326 265 172 125 117 114
Table 9 Optimal values of laminate fibre angle vs. maximum buckling load for (C; S; C; S) plates with k ¼ 1 a=b
0.5 0.6 0.7 0.8 0.9 1.0 1.25 1.5 1.75 2.0
hopt (°)
Nb
hopt (°)
Nb
hopt (°)
Nb
hopt (°)
Nb
22.6 19.5 65.4 61.4 58.5 56.2 64.1 59.0 55.4 60.7
411 300 248 251 249 244 230 230 225 224
50.2 49.0 48.3 50.1 55.9 59.0 61.6 62.8 63.7 64.6
728 685 641 604 585 580 579 581 584 588
42.5 42.3 44.8 54.3 58.3 59.8 60.9 61.6 62.2 63.0
1212 1102 1006 956 947 946 949 956 965 975
23.4 26.0 27.6 31.2 41.7 40.5 37.8 50.1 57.0 62.1
1831 1446 1201 1034 931 855 702 625 600 596
Table 10 Optimal values of laminate fibre angle vs. maximum buckling load for (C; C; C; C) plates with k ¼ 1 a=b
0.5 0.6 0.7 0.8 0.9 1.0 1.25 1.5 1.75 2.0 a
hopt (°)
Nb
hopt (°)
Nb
hopt (°)
Nb
hopt (°)
Nb
33.3 28.8 35.0 32.3 29.6 63.1a 57.5 53.9 59.6 56.5
902 648 494 396 322 265 253 239 235 230
45.3 51.2 51.2 51.3 53.8 61.3 66.2 68.3 70.0 71.6
1034 908 833 773 729 712 714 721 732 745
45.5 44.0 43.2 44.6 57.5 63.4 65.9 67.0 67.8 68.8
1875 1619 1413 1252 1161 1150 1154 1169 1186 1205
26.8 28.7 28.2 27.6 27.6 28.3 42.1 54.5 59.2 59.2
2856 2196 1783 1505 1301 1142 875 790 756 747
dual point.
strength is exhibited when compared to plates with a single stiffener. This phenomenon is difficult to explain in light of the fact that the loading Nx ¼ Ny , and perpendicular crossed stiffeners would seem to be the answer for increased strength.
The results reflected in Tables 7–10 are as expected, with decreases in strength for perpendicular stiffeners compared to parallel stiffeners at larger values of r. The highest value for Nb is naturally found for a (C; C; C; C) plate with r ¼ 0:5 and to two perpendicular stiffeners.
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M. Walker / Computers and Structures 80 (2002) 2229–2239
Table 11 Optimal values of laminate fibre angle vs. maximum buckling load for (F; S; C; S) plates with k ¼ 0:5 a=b
0.5 0.6 0.7 0.8 0.9 1.0 1.25 1.5 1.75 2.0
hopt (°)
Nb
hopt (°)
Nb
hopt (°)
Nb
hopt (°)
Nb
16.4 16.3 16.3 16.2 16.1 15.8 14.7 14.0 63.7 64.6
380 266 197 151 120 98 63 44 38 38
16.0 15.0 14.1 62.5 65.7 66.1 65.4 65.2 65.5 65.8
382 269 199 153 152 152 152 151 151 151
13.6 66.0 66.3 66.0 65.9 65.9 66.2 66.6 67.0 67.4
383 332 333 335 335 335 335 335 335 335
14.2 13.9 14.4 15.7 17.5 19.5 21.6 20.9 67.0 68.6
1579 1121 834 644 513 421 279 200 163 162
Table 12 Optimal values of laminate fibre angle vs. maximum buckling load for (C; C; C; C) plates with k ¼ 0:5 a=b
0.5 0.6 0.7 0.8 0.9 1.0 1.25 1.5 1.75 2.0
hopt (°)
Nb
hopt (°)
Nb
hopt (°)
Nb
hopt (°)
Nb
21.2 22.5 19.4 25.3 22.6 19.8 52.0 47.7 55.0 51.5
1307 947 709 566 463 385 325 305 293 290
16.9 44.0 44.3 44.3 45.2 52.2 59.0 60.2 61.4 63.0
1511 1182 1077 996 934 897 907 927 946 969
39.2 37.5 36.7 37.2 47.0 56.5 58.1 58.8 59.8 61.0
2407 2089 1832 1627 1459 1478 1495 1522 1551 1588
17.3 20.6 20.2 21.7 23.9 25.3 30.2 46.1 52.1 58.6
3777 2963 2458 2095 1808 1586 1202 1026 984 983
In order to demonstrate the influence of k taking a non-integer value, designs are given for (F; S; F; S) and (C; C; C; C) plates with k ¼ 0:5 (Tables 11 and 12). Similar trends prevail, and the results show that as k increases, so the values of Nb decrease. It is noted that the discontinuities that occur in hopt as the aspect ratio increases from 0.5 to 2 are due to changes in the buckling modes.
6. Conclusion The optimised designs of symmetric fibre-reinforced laminates with different stiffener arrangements are presented. The plate designs are optimised with the objective of maximising the buckling load, and the design variable used is the ply fiber orientation. During the optimisation procedure, the fitness of each laminate design is determined using the finite element method, and the formulation of the element used is presented. The numerical approach employed in the present study is necessitated by the fact that the inclusion of the bending–twisting coupling effect and the consideration
of various combinations of free, clamped and simply supported boundary conditions rule out an analytical approach. The results show that the difference in the buckling loads of optimal and non-optimal plates could be quite substantial, emphasising the importance of optimisation for fiber composite structures. Square tubing is used as stiffeners, and the effect of the stiffener arrangement and boundary conditions on the optimal ply angles and the buckling load are numerically studied and compared. It is demonstrated that in some cases, stiffeners can sometimes reduce, rather than increase the buckling strength of an optimally designed plate.
References [1] Jones RM. Mechanics of Composite Materials. New York: McGraw-Hill; 1975. [2] Walker M, Adali S, Verijenko V. Optimisation of symmetric laminates for maximum buckling load including the effects of bending–twisting coupling. Comput Struct 1995; 58:313–9.
M. Walker / Computers and Structures 80 (2002) 2229–2239 [3] Adali S. Lay-up optimisation of laminated plates under buckling loads. In: Turvey GJ, Marshall IH, editors. Buckling and Postbuckling of Laminated Plates. London: Chapman and Hall; 1994. [4] Timoshenko SP, Gere JM. Theory of Elastic Stability. New York: McGraw-Hill; 1961. [5] Gea HC. Automated optimal stiffener pattern design. Mech Struct Mach 1999;27(3):275–80. [6] Lui ZS. Eigenvalue sensitivity analysis of stiffened plates with respect to the location of stiffeners. Struct Optimisation 1998;16(2):155–61. [7] Bedair OK. Plate/stiffener assemblies under non-uniform edge compression. J Struct Eng 1995;121(11): 1603–10. [8] Al-Shareedah EM, Seireg AA. Use of undetermined multipliers in the design of stiffened plates. Comput Mech Eng 1986;4(5):57–64.
2239
[9] Phillips JL, G€ urdal Z. Analysis and optimum design of geodesically stiffened composite panels. In: Wilde WP, Blain WR (Eds.). Composite Materials Design and Analysis, Computational Mechanics, Southampton, 1990. [10] Lee HP. Effects of torsional and bending restraints of intermediate stiffeners on the free vibration of rectangular plates. Mech Struct Mach 1995;23(3):309–14. [11] Mahendren M. Buckling experiments on hollow flange beams with web stiffeners. J Struct Eng 1997;123(9): 1130–8. [12] Avery P. Finite-element analysis of hollow flange beams with web stiffeners. J Struct Eng 1997;123(9):1123–30. [13] Reddy JN. Energy and Variational Methods in Applied Mechanics. New York: John Wiley and Sons; 1984. [14] Haftka RT, G€ urdal Z. Elements of Structural Optimisation. 3rd ed. Dordrecht: Kluwer Academic Publishers; 1992.
The effect of stiffeners on the optimal ply orientation and buckling load of rectangular laminated plates M. Walker
*
Center for Advanced Materials, Design and Manufacturing Research, Durban Institute of Technology, PO Box 953, 4001 Durban, South Africa Received 14 November 2000; accepted 4 July 2002
Abstract Optimal designs of symmetrically laminated rectangular plates with different stiffener arrangements are presented. The plates are subject to a combination of simply supported, clamped and free boundary conditions, and the design objective is the maximisation of the biaxial buckling load. This is achieved by determining the fibre orientations optimally with the effects of bending–twisting coupling taken into account. Square tubing is used as stiffeners, and the finite element method coupled with an optimisation routine is employed in analysing and optimising the laminated plate designs. The effect of the stiffener arrangement and boundary conditions on the optimal ply angles and the buckling load are numerically studied. It is demonstrated that in some cases, stiffeners can sometimes reduce, rather than increase the buckling strength of an optimally designed plate. Ó 2002 Civil-Comp Ltd. and Elsevier Science Ltd. All rights reserved.
1. Introduction An advantage of composite materials over conventional ones is the possibility of tailoring their properties to the specific requirements of a given application. The tailoring is mostly achieved by optimising the mechanical properties, thereby increasing the load carrying capacity of the structure. Optimisation of composite laminates with respect to ply angles to maximise the strength is necessary to realise the full potential of fiberreinforced materials. A common composite layup is the symmetrically laminated angle ply configuration that avoid bending– stretching effects by virtue of mid-plane symmetry. These structures often contain components which may be modeled as rectangular plates. An important failure mode for these plates is buckling under in-plane loading. The buckling resistance of fiber composite plates can be improved by using the ply angle as a design variable, and determining the optimal angles to maximise the buckling
*
Tel.: +27-31-204-2116; fax: +27-31-204-2139. E-mail address: [email protected] (M. Walker).
load, and also by the addition of stiffeners. It has long been recognised that the use of stiffeners in plates increases structural efficiency and reduces structural weight. One phenomenon associated with symmetric angleply configurations is the occurrence of bending–twisting coupling which may cause significantly different results as compared to cases in which this coupling is exactly zero [1]. The effect of bending–twisting coupling becomes even more pronounced for laminates with few layers. Due to this coupling, closed-form solutions cannot be obtained for any of the possible boundary conditions and thus many studies concerned with the buckling performance of laminates have neglected these effects [2]. The present study adopts a numerical approach to include the effect of bending–twisting coupling and to obtain the optimal design solutions for stiffened plates with a variety of boundary conditions. Results obtained using different approaches can be found in the literature. These studies established that symmetric lamination yields the highest buckling load compared to other stacking sequences. Additionally, Ref. [2] reported the effect of different boundary conditions of the optimal design solutions. In that study, bending– twisting coupling was included in the formulation, and
0045-7949/02/$ - see front matter Ó 2002 Civil-Comp Ltd. and Elsevier Science Ltd. All rights reserved. PII: S 0 0 4 5 - 7 9 4 9 ( 0 2 ) 0 0 2 6 5 - 1
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M. Walker / Computers and Structures 80 (2002) 2229–2239
it was demonstrated how misleading results can be when these effects are neglected. A more detailed review of the subject can be found in [3]. Analyses and formulations for rectangular isotropic plates with longitudinal or transverse stiffeners were developed at least 40 years ago, by scholars like Timoshenko and Gere [4]. More recently, some studies have dealt with the effect of different arrangements of stiffeners [5–7] on the mechanical integrity of structures. Also, various approaches have been used to account for their effect, including that of minimising the total energy of the stiffened panel, and of using Lagrange multipliers to accurately predict the transverse deflections [8]. Subsequently, some researchers applied these techniques to the study of laminated plates with stiffeners, and of those, a few have even dealt with stiffened plates subjected to buckling or vibrational loads. For example, Phillips and G€ urdal [9] studied the effects of geodesic stiffeners on the buckling load capacity of plates, and carried out the analysis using the finite element method, while Lee [10] published an article that dealt with the effects of torsional and bending restraints of intermediate stiffeners on the free vibration of rectangular plates. Some experimental results have also appeared in the literature, like that of Mahendran [11], which details the effect of web stiffeners on the buckling performance of hollow flange beams, a subject also examined by Avery [12] using the finite element method. As there is little reported on the optimal design of stiffened laminates with bending–twisting accounted for, and with different boundary conditions, the present study aims to fill this gap. The finite element formulation which is used is based on Mindlin type theory for laminated composite plates. Numerical results demonstrating the effect of the stiffener arrangement and boundary conditions on the optimal ply angles and the buckling load are presented. 2. Buckling of symmetric laminates Consider a symmetrically laminated rectangular plate of length a, width b and thickness h which consists of n orthotropic layers with fiber angles hk , k ¼ 1; 2; . . . ; K, as shown in Fig. 1 (top). The plate is defined in the Cartesian coordinates x, y and z with axes x and y lying on the middle surface of the plate, and is subjected to biaxial compressive forces Nx and Ny in the x and y directions, respectively. In the present study, a first-order shear deformable theory is employed to analyse the problem and the following displacement field is assumed: u ¼ u0 ðx; yÞ þ zwx ðx; yÞ v ¼ v0 ðx; yÞ þ zwy ðx; yÞ w ¼ wðx; yÞ
where u0 ; v0 and w0 are the displacements of the reference surface in the x, y and z direction, respectively, and wx ; wy are the rotations of the transverse normal about the x and y axes. The in-plane strain components can be written as a sum of the extensional and flexural parts and they are given as fg ¼ fp g þ zff g
ð2Þ
where fgT ¼ ðx ; y ; xy Þ and 0
1 u0;x A; fp g ¼ @ v0;y u0;y þ v0;x
0
1 wx ff g ¼ @ wy A wx þ wy
ð3Þ
Here, a subscript after the comma denotes differentiation with respect to the variable following the comma. The transverse shear strains are obtained from w0;x þ wx cxz ¼ ð4Þ fcg ¼ cyz w0;y þ wy The equations for in-plane stresses of the kth layer under a plane stress state may be written as 0 1 0 1 0 1 11 Q 12 Q 16 x rx Q @ ry A ¼ @ Q 12 Q 22 Q 26 A @ y A 16 Q 26 Q 66 rxy ðkÞ xy Q ðkÞ fg ¼ ½Q ðkÞ
ð5Þ
and similarly for the transverse shear stresses as cyz syz Q Q ¼ a½CðkÞ fcg ¼ a 44 45 sxz ðkÞ Q45 Q55 ðkÞ cxz
ð6Þ
ij are the where a is a shear correction factor [13], and Q transformed stiffnesses. Eqs. (5) and (6) may be written in compact form as k rk ¼ Q
ð7Þ
ij Þ , k refers to the full matrix with elements ðQ where Q k; and rk and represent in-plane and transverse stresses and strains, respectively. The resulting shear forces and moments acting on the plate are obtained by integrating the stresses through the laminate thickness, viz Z h=2 fV gT ¼ ðVx ; Vy Þ ¼ ðsxz ; syz Þdz h=2
fMgT ¼ ðMx ; My ; Mxy Þ ¼
Z
h=2
ðrx ; ry ; rxy Þz dz
ð8Þ
h=2
ð1Þ
The relations between V and M, and the strains are given by
M. Walker / Computers and Structures 80 (2002) 2229–2239
2231
Fig. 1. Configuration of the laminated plate.
fV g ¼ ½Sfcg;
fMg ¼ ½Dff g
ð9Þ
where the stiffness matrices ½S and ½D are computed from ½S ¼ a
K Z X k¼1
½D ¼
K Z X k¼1
hk
hk 1 hk
hk 1
½CðkÞ dz ð10Þ
z2 dz ½Q ðkÞ
From the condition that the potential energy of the plate is stationary at equilibrium, and neglecting the prebuckling effects, the equations governing the biaxial buckling of the shear deformable laminate are obtained as
Mx;xx þ 2Mxy;xy þ My;yy þ Nx w;xx þ Ny w;yy ¼ 0 Mx;x þ Mxy;y Vx ¼ 0 My;y þ Mxy;x Vy ¼ 0
ð11Þ
where Nx and Ny are the pre-buckling stress components which are shown in Fig. 1. As no simplifications are assumed on the elements of the ½D matrix, Eq. (11) include the bending–twisting coupling as exhibited by virtue of D16 6¼ 0, D26 6¼ 0.
3. Finite element formulation We now consider the finite element formulation of the plate. Let the region S of the plate be divided into n sub-regions Sr ðSr 2 S; r ¼ 1; 2; . . . ; nÞ such that
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M. Walker / Computers and Structures 80 (2002) 2229–2239
PðuÞ ¼
n X
PSr ðuÞ
ð12Þ
r¼1
5.1. Verification
where P and PSr are potential energies of the plate and the element, respectively, and u is the displacement vector. Using the same shape functions associated with node i ði ¼ 1; 2; . . . ; nÞ, Si ðx; yÞ, for interpolating the variables in each element, we can write u¼
n X
Si ðx; yÞui
ð13Þ
i¼1
where ui is the value of the displacement vector corresponding to node i, and is given by ðiÞ
ðiÞ
5. Numerical results and discussion
ðiÞ
ðiÞ T u ¼ fu0 ; v0 ; w0 ; wðiÞ x ; wy g
ð14Þ
The static buckling problem reduces to a generalised eigenvalue problem of the conventional form, viz ð½K þ k½KG Þfug ¼ 0
ð15Þ
where ½K is the stiffness matrix and ½KG is the initial stress matrix. The lowest eigenvalue of the homogeneous system (15) yields the buckling load.
In order to verify the finite element formulation described above, some solutions are compared with those available in the literature. In particular, a single-layered simply supported square plate (having sides of length 1.066 m and thickness 0.0254 m) was modeled with properties specified as E ¼ 206:8 GPa and m ¼ 0:3: The plate has a single central solid beam stiffener (see Fig. 1), with a width of 0.254 m, and this was modeled using standard beam elements. The analytical solution for this problem is available in Ref. [4], and the buckling load Nx ¼ 39:2 MN/m. The use of 256 plate and shell elements (of the type described by the formulation above) for a square plate resulted in an error of <0.08% as compared to the analytical solution. This mesh density was accepted as providing sufficient accuracy [2]. Consequently, in the present study, a square plate is meshed with 256 elements. Plates of aspect ratios other than 1 are meshed with a corresponding proportion of 256 elements. 5.2. Numerical results
4. Optimal design problem The objective of the design problem is to maximise the buckling loads Nx and Ny for a given thickness h by optimally determining the fiber orientations given by hk ¼ ð 1Þkþ1 h for k 6 K=2 and hk ¼ ð 1Þk h for k P K=2 þ 1. Let Nx ¼ N and Ny ¼ kN , where 0 6 k 6 1 is the proportionality constant (and should not be confused with the eigenvalue represented similarly in Eq. (15)). The buckling load N ðhÞ is given by NðhÞ ¼ min½Nmn ðm; n; hÞ m;n
ð16Þ
where Nmn is the buckling load corresponding to the halfwave numbers m and n in the x and y directions, respectively. The design objective is to maximise N ðhÞ with respect to h, viz D
Nmax max½N ðhÞ; h
0° 6 h 6 90°
ð17Þ
where N ðhÞ is determined from the finite element solution of the eigenvalue problem given by Eq. (16). The optimisation procedure involves the stages of evaluating the buckling load N ðhÞ for a given h and improving the fiber orientation to maximise N. Thus, the computational solution consists of successive stages of analysis and optimisation until a convergence is obtained and the optimal angle hopt is determined within a specified accuracy. In the optimisation stage, the Golden section [14] method is employed.
Numerical results are given for a typical T300/5208 graphite/epoxy material with E1 ¼ 181 GPa, E2 ¼ 10:3 GPa, G12 ¼ 7:17 GPa and m12 ¼ 0:28. The symmetric plate is constructed of four equal thickness layers with h1 ¼ h2 ¼ h3 ¼ h4 ¼ h and the thickness ratio is specified as h=b ¼ 0:01: Three different stiffener arrangements are investigated, and these are illustrated in Fig. 1. Square tubing with wall thickness t is used as the stiffeners, and these have the same material properties as the plates (see above). For all the designs considered, g=b ¼ 0:01, and t=b ¼ 0:001. In addition, different combinations of free (F), simply supported (S) and clamped (C) boundary conditions are implemented at the four edges of the plate. In particular, five different combinations are studied, namely, (F; S; F; S), (F; S; C; S), (S; S; S; S), (C; S; C; S) and (C; C; C; C), where the first letter refers to the first plate edge, and the others follow in the anti-clockwise direction as shown in Fig. 1. At this point it may be important to repeat that convergence tests showed that the choice of the number of elements for modeling purposes was sufficient, irrespective of the boundary conditions implemented or the stiffener arrangement used (see Section 5.1). For each of the boundary conditions, three in-plane load cases are considered, namely, uniaxial compression (Ny ¼ 0), biaxial compression with Ny =Nx ¼ 0:5, and biaxial compression with Ny =Nx ¼ 1. The results presented in this section are obtained for rectangular plates with aspect ratios varying between 0.5
M. Walker / Computers and Structures 80 (2002) 2229–2239
2233
and 2. The non-dimensionalised buckling parameter Nb is defined as Nb ¼
Nb2 h3 E0
ð18Þ
where N is the critical buckling load, and E0 is a reference value having the dimension of YoungÕs modulus and is taken as E0 ¼ 1 GPa. The dependence of the buckling load Nb on the fiber angle has been investigated for the five cases of boundary conditions in Ref. [2] for unstiffened plates. It was demonstrated clearly that the maximum buckling load for a given boundary condition and aspect ratio occurs at a specific value of the fiber angle (referred to as the optimal fiber angle), and this value can be several times higher than the buckling load at other fiber angles. This fact emphasises the importance of carrying out optimisation in design work of this nature to obtain the best performance of fiber composite plates. The dependence of Nb on h for stiffened plates is demonstrated in Fig. 2, for (S; S; S; S) and (C; C; C; C) plates with a=b ¼ 1:25, and k ¼ 1. It is obvious from the curves that, as before, non-optimal values of the fiber orientation can lead to buckling loads which may be a small fraction of the maximum, and that also for plates with stiffeners, optimisation is important. It is also evident that (C; C; C; C) plates with two parallel stiffeners (viz type 2) give the highest buckling loads. Fig. 3 shows the influence of the loading ratio k on the value of Nb as h is varied, for (S; S; S; S) plates with a=b ¼ 1:25. It is obvious that as the magnitude of k increases, so Nb decreases; such that a type 1 plate with k ¼ 1 has the lowest buckling load through the whole range of fiber orientation. The effect of a=b on the buckling load is illustrated in Fig. 4, for type 1 plates with (S; S; S; S) or (C; C; C; C) boundary conditions, and k ¼ 1. As r decreases, so the
Fig. 2. Buckling load vs. ply angle for (S,S,S,S) and (C,C,C,C) stiffened plates with a=b ¼ 1:25 and k ¼ 1.
Fig. 3. Buckling load vs. ply angle for (S,S,S,S) stiffened plates with a=b ¼ 1:25 and k ¼ 0 or k ¼ 1.
Fig. 4. Buckling load vs. ply angle for (S,S,S,S) and (C,C,C,C) type 1 plates of varying aspect ratio with k ¼ 1.
plate becomes stiffer, and thus the magnitude of Nb increases, while (C; C; C; C) plates have greater buckling loads than those which are simply-supported. Finally, the effect of the boundary conditions on the buckling load is shown in Fig. 5, for type 1 plates with a=b ¼ 1:25 and k ¼ 1. Once again, it is obvious that as the number of degrees of freedom are increased, so the buckling load decreases. Consequently, (C; C; C; C) plates have the highest buckling loads, and (F; S; F; S) plates the lowest. Tables 1–5 give the optimal design solutions for the five boundary conditions implemented, and k ¼ 0: For each of the tables, the first column specifies the aspect ratio r, while columns 2 and 3 give the values of hopt and Nb for plates without stiffeners. Subsequently, the values of hopt and Nb are given for plates with one, two parallel and two perpendicular stiffeners, respectively. Table 1, for which the boundary condition is (F; S; F; S), illustrates that the optimal fiber angle is always
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M. Walker / Computers and Structures 80 (2002) 2229–2239
Fig. 5. Buckling load vs. ply angle for type 1 plates with varying boundary conditions and k ¼ 1, a=b ¼ 1:25.
0° [2] for plates without stiffeners, regardless of r. The buckling load decreases monotonically as the aspect ratio increases, from a maximum value of 596 (at r ¼ 0:5) to 37 (at r ¼ 2). Adding one stiffener increases the buckling load by 5.2% at r ¼ 0:5 and by 438% at r ¼ 2.
Thus the effect of the stiffener is greatest at larger aspect ratios, and this can be explained by the change in the values of the optimal fiber angle, which jumps from 2.5° at r ¼ 0:9 (where the value is similar to that of the unstiffened plate) to 47.4° at r ¼ 1. It is this change in mode that is responsible for the vastly increased strength of the stiffened plates at higher values of r. Similarly, the effect of two parallel stiffeners is to further increase the plate buckling strength. As before, at smaller aspect ratios, the increase is small (compared to a plate with one stiffener), but increases with increasing r. In this case, the mode change occurs at r ¼ 0:7. It is interesting to note that the buckling strength increases from 450 (at r ¼ 0:7) to 467 (at r ¼ 2), viz with increasing r the strength of plates with two parallel stiffeners increases. Using two stiffeners which are perpendicular has the effect of increasing the strength, compared to plates having two parallel stiffeners, but only for r ¼ 0:5 ! 0:6. Mode changes in the case of plates with perpendicular stiffeners have the effect of decreasing the buckling strength of these plates by up to 52% compared to those with two parallel stiffeners. It is obvious that the choice
Table 1 Optimal values of laminate fibre angle vs. maximum buckling load for (F; S; F; S) plates with k ¼ 0 a=b
0.5 0.6 0.7 0.8 0.9 1.0 1.25 1.5 1.75 2.0
hopt (°)
Nb
hopt (°)
Nb
hopt (°)
Nb
hopt (°)
Nb
0 0 0 0 0 0 0 0 0 0
596 415 305 234 185 149 96 67 49 37
0 0.1 0.5 1.2 2.5 47.4 45.5 44.1 44.4 45.6
627 449 342 272 225 205 204 201 199 199
0.4 1.3 47.2 48.6 48.1 47.4 46.1 46.0 46.5 47.0
677 505 450 457 461 462 463 463 464 467
3.3 5.3 8.5 14.6 36.0 38.4 39.1 37.3 35.2 53.6
695 508 396 327 295 282 262 245 230 220
Table 2 Optimal values of laminate fibre angle vs. maximum buckling load for (S; S; S; S) plates with k ¼ 0 a=b
0.5 0.6 0.7 0.8 0.9 1.0 1.25 1.5 1.75 2.0
hopt (°)
Nb
hopt (°)
Nb
hopt (°)
Nb
hopt (°)
Nb
0 0 0 0 39.5 44.5 47.5 40.3 36.3 43.7
623 444 336 267 247 243 254 252 238 231
39.4 38.9 36.7 35.0 34.3 35.2 45.3 46.6 46.8 46.1
888 865 834 798 761 729 707 720 734 747
31.6 30.6 30.0 30.7 34.9 43.2 45.6 45.8 46.3 47.1
1491 1392 1301 1219 1159 1142 1155 1177 1199 1222
16.6 21.4 25.5 24.1 23.6 22.4 19.8 31.2 38.9 45.3
2371 1929 1650 1453 1304 1184 955 820 769 760
M. Walker / Computers and Structures 80 (2002) 2229–2239
2235
Table 3 Optimal values of laminate fibre angle vs. maximum buckling load for (F; S; C; S) plates with k ¼ 0 a=b
0.5 0.6 0.7 0.8 0.9 1.0 1.25 1.5 1.75 2.0
hopt (°)
Nb
hopt (°)
Nb
hopt (°)
Nb
hopt (°)
Nb
0 0 0 0 0 0 0 0 0 48.0
603 423 313 242 294 159 107 78 62 57
0 0 0 0 48.6 49.3 48.6 48.4 48.7 49.1
629 452 347 280 215 215 215 215 215 215
0 0 49.9 50.2 50.0 49.8 49.2 49.6 50.4 51.8
687 521 462 465 469 472 474 478 481 485
6.7 8.3 9.9 11.6 13.2 14.7 16.3 14.0 10.6 57.4
1884 1376 1064 862 726 630 489 408 349 235
Table 4 Optimal values of laminate fibre angle vs. maximum buckling load for (C; S; C; S) plates with k ¼ 0 a=b
0.5 0.6 0.7 0.8 0.9 1.0 1.25 1.5 1.75 2.0
hopt (°)
Nb
hopt (°)
Nb
hopt (°)
Nb
hopt (°)
Nb
0 0 55.1 49.7 45.0 41.0 32.1 46.5 40.4 49.6
637 462 386 404 408 403 368 375 371 364
39.7 37.3 35.1 33.9 34.4 39.4 46.4 46.6 46.3 46.6
1048 1022 981 935 894 868 878 896 910 922
31.7 30.6 30.3 32.8 44.2 45.6 45.0 45.0 45.6 46.7
1668 1574 1485 1411 1390 1404 1429 1449 1469 1492
16.6 21.5 22.9 22.7 21.5 19.8 17.2 34.3 42.8 47.4
2585 2095 1792 1583 1424 1296 1061 942 908 915
Table 5 Optimal values of laminate fibre angle vs. maximum buckling load for (C; C; C; C) plates with k ¼ 0 a=b
0.5 0.6 0.7 0.8 0.9 1.0 1.25 1.5 1.75 2.0
hopt (°)
Nb
hopt (°)
Nb
hopt (°)
Nb
hopt (°)
Nb
0 0 0 0 0 0 41.7 34.3 41.7 42.2
2397 1698 1272 994 805 672 470 445 404 402
0 0 32.6 31.4 30.7 30.5 48.2 48.7 48.7 49.6
2554 1918 1567 1457 1365 1288 1202 1247 1284 1319
28.8 26.2 24.5 23.7 23.8 25.5 46.8 46.2 47.2 49.1
3312 2940 2616 2347 2134 1974 2015 2066 2111 2169
11.8 15.7 18.5 19.5 19.5 19.2 17.9 17.4 33.2 50.3
4930 3823 3123 2660 2332 2088 1691 1456 1314 1313
of stiffener arrangement for (F; S; F; S) plates with k ¼ 0 must be that of two, perpendicular to each other for 0:5 6 r 6 0:6, and parallel to each other for r P 0:7. These trends are verified in Fig. 6, which illustrates for three values of a=b the dependence of Nb on h. For each of the aspect ratios (r ¼ 0.5, 1 and 2), the three different stiffener arrangements are implemented. The
peaks of each of the nine curves correspond to the figures given in Table 1 (rows 1, 6 and 10), viz the values of hopt and maximum Nb . Similar trends are noted in Tables 2–5, although for the cases (S; S; S; S), (C; S; C; S) and (C; C; C; C), the value of the buckling load also increases at larger values of r for plates with a single stiffener, as r increases. As
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M. Walker / Computers and Structures 80 (2002) 2229–2239
Fig. 6. Buckling load vs. ply angle for (F,S,F,S) stiffened plates of varying aspect ratios and k ¼ 0.
for the (F; S; F; S) plates, this did not occur with the (F; S; F; S) plates (although they do hold their strength for r P 0:7), and the reason may lay with the increased degrees of freedom associated with free edges. In addi-
tion, as the number of degrees of freedom is restricted, so the buckling load increases, and thus the Nb values for (C; C; C; C) plates are greatest. Tables 6–10 give the optimal designs for plates with k ¼ 1. In the boundary condition case (F; S; F; S), it is immediately obvious that the biaxial loading results in hopt 6¼ 0° for unstiffened plates (Table 6), as was the case with k ¼ 0 (Table 1). Also, the values of Nb are reduced by approximately 40% throughout the range of a=b. The addition of a single stiffener reduces the buckling load by 10% at r ¼ 0:5, and this trend continues until r ¼ 0:7. At r ¼ 0:8 (when a mode change takes place, and hopt jumps from 32.1° to 54.6°) the buckling load is 6% greater than that of the unstiffened plate, and the strengthening effect increases as r increases until at r ¼ 2, there is 260% more. When two parallel stiffeners are used, the results (columns 6 and 7) show that increased strength is the result, compared to plates with either one or no stiffener, at all value of r, but this is not the case when two perpendicular stiffeners are used (columns 8 and 9). For 0:9 6 r 6 2, the value of Nb is reduced, compared to the plates with parallel stiffeners, although slightly greater
Table 6 Optimal values of laminate fibre angle vs. maximum buckling load for (F; S; F; S) plates with k ¼ 1 a=b
0.5 0.6 0.7 0.8 0.9 1.0 1.25 1.5 1.75 2.0
hopt (°)
Nb
hopt (°)
Nb
hopt (°)
Nb
hopt (°)
Nb
24.7 25.3 24.8 23.3 21.6 20.7 31.0 30.5 30.5 29.7
252 176 130 99 78 63 39 27 20 15
25.4 26.0 32.1 54.6 57.5 58.2 57.6 56.8 57.4 59.8
228 161 122 105 98 92 79 69 60 54
49.5 57.3 59.6 60.3 60.5 60.6 61.5 62.9 64.2 65.5
239 211 196 185 175 166 147 134 125 118
1.7 3.1 5.2 8.0 11.1 14.5 48.7 54.2 58.3 61.7
547 373 269 202 158 127 90 75 66 60
Table 7 Optimal values of laminate fibre angle vs. maximum buckling load for (S; S; S; S) plates with k ¼ 1 a=b
0.5 0.6 0.7 0.8 0.9 1.0 1.25 1.5 1.75 2.0
hopt (°)
Nb
hopt (°)
Nb
hopt (°)
Nb
hopt (°)
Nb
24.8 23.0 21.1 28.2 39.9 45.0 62.0 68.3 66.5 65.2
402 285 211 161 136 121 103 103 102 101
48.6 51.5 51.9 52.3 53.5 56.0 62.2 64.7 66.1 67.2
555 507 470 439 414 396 377 372 370 370
45.8 46.3 47.4 50.0 54.3 58.1 63.4 65.9 67.4 68.5
909 802 719 655 614 590 565 559 558 558
29.6 32.3 33.8 36.1 46.6 45.3 41.6 49.0 54.6 59.2
1542 1206 992 845 750 684 550 459 419 399
M. Walker / Computers and Structures 80 (2002) 2229–2239
2237
Table 8 Optimal values of laminate fibre angle vs. maximum buckling load for (F; S; C; S) plates with k ¼ 1 a=b
0.5 0.6 0.7 0.8 0.9 1.0 1.25 1.5 1.75 2.0
hopt (°)
Nb
hopt (°)
Nb
hopt (°)
Nb
hopt (°)
Nb
24.7 24.7 24.7 24.6 24.5 24.3 24.3 53.9 72.0 74.1
252 176 130 100 79 64 41 30 28 28
24.3 24.3 26.0 63.0 68.0 72.5 76.5 76.1 75.8 75.7
252 177 131 114 110 109 108 108 108 107
58.0 69.3 74.4 75.8 75.9 75.7 75.6 75.5 75.4 75.4
258 245 243 243 243 243 242 240 239 237
23.6 23.7 24.1 25.0 26.2 27.3 29.4 64.7 70.2 73.4
1016 718 532 410 326 265 172 125 117 114
Table 9 Optimal values of laminate fibre angle vs. maximum buckling load for (C; S; C; S) plates with k ¼ 1 a=b
0.5 0.6 0.7 0.8 0.9 1.0 1.25 1.5 1.75 2.0
hopt (°)
Nb
hopt (°)
Nb
hopt (°)
Nb
hopt (°)
Nb
22.6 19.5 65.4 61.4 58.5 56.2 64.1 59.0 55.4 60.7
411 300 248 251 249 244 230 230 225 224
50.2 49.0 48.3 50.1 55.9 59.0 61.6 62.8 63.7 64.6
728 685 641 604 585 580 579 581 584 588
42.5 42.3 44.8 54.3 58.3 59.8 60.9 61.6 62.2 63.0
1212 1102 1006 956 947 946 949 956 965 975
23.4 26.0 27.6 31.2 41.7 40.5 37.8 50.1 57.0 62.1
1831 1446 1201 1034 931 855 702 625 600 596
Table 10 Optimal values of laminate fibre angle vs. maximum buckling load for (C; C; C; C) plates with k ¼ 1 a=b
0.5 0.6 0.7 0.8 0.9 1.0 1.25 1.5 1.75 2.0 a
hopt (°)
Nb
hopt (°)
Nb
hopt (°)
Nb
hopt (°)
Nb
33.3 28.8 35.0 32.3 29.6 63.1a 57.5 53.9 59.6 56.5
902 648 494 396 322 265 253 239 235 230
45.3 51.2 51.2 51.3 53.8 61.3 66.2 68.3 70.0 71.6
1034 908 833 773 729 712 714 721 732 745
45.5 44.0 43.2 44.6 57.5 63.4 65.9 67.0 67.8 68.8
1875 1619 1413 1252 1161 1150 1154 1169 1186 1205
26.8 28.7 28.2 27.6 27.6 28.3 42.1 54.5 59.2 59.2
2856 2196 1783 1505 1301 1142 875 790 756 747
dual point.
strength is exhibited when compared to plates with a single stiffener. This phenomenon is difficult to explain in light of the fact that the loading Nx ¼ Ny , and perpendicular crossed stiffeners would seem to be the answer for increased strength.
The results reflected in Tables 7–10 are as expected, with decreases in strength for perpendicular stiffeners compared to parallel stiffeners at larger values of r. The highest value for Nb is naturally found for a (C; C; C; C) plate with r ¼ 0:5 and to two perpendicular stiffeners.
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M. Walker / Computers and Structures 80 (2002) 2229–2239
Table 11 Optimal values of laminate fibre angle vs. maximum buckling load for (F; S; C; S) plates with k ¼ 0:5 a=b
0.5 0.6 0.7 0.8 0.9 1.0 1.25 1.5 1.75 2.0
hopt (°)
Nb
hopt (°)
Nb
hopt (°)
Nb
hopt (°)
Nb
16.4 16.3 16.3 16.2 16.1 15.8 14.7 14.0 63.7 64.6
380 266 197 151 120 98 63 44 38 38
16.0 15.0 14.1 62.5 65.7 66.1 65.4 65.2 65.5 65.8
382 269 199 153 152 152 152 151 151 151
13.6 66.0 66.3 66.0 65.9 65.9 66.2 66.6 67.0 67.4
383 332 333 335 335 335 335 335 335 335
14.2 13.9 14.4 15.7 17.5 19.5 21.6 20.9 67.0 68.6
1579 1121 834 644 513 421 279 200 163 162
Table 12 Optimal values of laminate fibre angle vs. maximum buckling load for (C; C; C; C) plates with k ¼ 0:5 a=b
0.5 0.6 0.7 0.8 0.9 1.0 1.25 1.5 1.75 2.0
hopt (°)
Nb
hopt (°)
Nb
hopt (°)
Nb
hopt (°)
Nb
21.2 22.5 19.4 25.3 22.6 19.8 52.0 47.7 55.0 51.5
1307 947 709 566 463 385 325 305 293 290
16.9 44.0 44.3 44.3 45.2 52.2 59.0 60.2 61.4 63.0
1511 1182 1077 996 934 897 907 927 946 969
39.2 37.5 36.7 37.2 47.0 56.5 58.1 58.8 59.8 61.0
2407 2089 1832 1627 1459 1478 1495 1522 1551 1588
17.3 20.6 20.2 21.7 23.9 25.3 30.2 46.1 52.1 58.6
3777 2963 2458 2095 1808 1586 1202 1026 984 983
In order to demonstrate the influence of k taking a non-integer value, designs are given for (F; S; F; S) and (C; C; C; C) plates with k ¼ 0:5 (Tables 11 and 12). Similar trends prevail, and the results show that as k increases, so the values of Nb decrease. It is noted that the discontinuities that occur in hopt as the aspect ratio increases from 0.5 to 2 are due to changes in the buckling modes.
6. Conclusion The optimised designs of symmetric fibre-reinforced laminates with different stiffener arrangements are presented. The plate designs are optimised with the objective of maximising the buckling load, and the design variable used is the ply fiber orientation. During the optimisation procedure, the fitness of each laminate design is determined using the finite element method, and the formulation of the element used is presented. The numerical approach employed in the present study is necessitated by the fact that the inclusion of the bending–twisting coupling effect and the consideration
of various combinations of free, clamped and simply supported boundary conditions rule out an analytical approach. The results show that the difference in the buckling loads of optimal and non-optimal plates could be quite substantial, emphasising the importance of optimisation for fiber composite structures. Square tubing is used as stiffeners, and the effect of the stiffener arrangement and boundary conditions on the optimal ply angles and the buckling load are numerically studied and compared. It is demonstrated that in some cases, stiffeners can sometimes reduce, rather than increase the buckling strength of an optimally designed plate.
References [1] Jones RM. Mechanics of Composite Materials. New York: McGraw-Hill; 1975. [2] Walker M, Adali S, Verijenko V. Optimisation of symmetric laminates for maximum buckling load including the effects of bending–twisting coupling. Comput Struct 1995; 58:313–9.
M. Walker / Computers and Structures 80 (2002) 2229–2239 [3] Adali S. Lay-up optimisation of laminated plates under buckling loads. In: Turvey GJ, Marshall IH, editors. Buckling and Postbuckling of Laminated Plates. London: Chapman and Hall; 1994. [4] Timoshenko SP, Gere JM. Theory of Elastic Stability. New York: McGraw-Hill; 1961. [5] Gea HC. Automated optimal stiffener pattern design. Mech Struct Mach 1999;27(3):275–80. [6] Lui ZS. Eigenvalue sensitivity analysis of stiffened plates with respect to the location of stiffeners. Struct Optimisation 1998;16(2):155–61. [7] Bedair OK. Plate/stiffener assemblies under non-uniform edge compression. J Struct Eng 1995;121(11): 1603–10. [8] Al-Shareedah EM, Seireg AA. Use of undetermined multipliers in the design of stiffened plates. Comput Mech Eng 1986;4(5):57–64.
2239
[9] Phillips JL, G€ urdal Z. Analysis and optimum design of geodesically stiffened composite panels. In: Wilde WP, Blain WR (Eds.). Composite Materials Design and Analysis, Computational Mechanics, Southampton, 1990. [10] Lee HP. Effects of torsional and bending restraints of intermediate stiffeners on the free vibration of rectangular plates. Mech Struct Mach 1995;23(3):309–14. [11] Mahendren M. Buckling experiments on hollow flange beams with web stiffeners. J Struct Eng 1997;123(9): 1130–8. [12] Avery P. Finite-element analysis of hollow flange beams with web stiffeners. J Struct Eng 1997;123(9):1123–30. [13] Reddy JN. Energy and Variational Methods in Applied Mechanics. New York: John Wiley and Sons; 1984. [14] Haftka RT, G€ urdal Z. Elements of Structural Optimisation. 3rd ed. Dordrecht: Kluwer Academic Publishers; 1992.