buckling optimized shear panels

8/3/2019 Buckling Optimized Shear Panels

http://slidepdf.com/reader/full/buckling-optimized-shear-panels 1/17

Buckling of optimised curved composite panels under shearand in-plane bending

C.A. Featherston a,*, A. Watson b

a Cardiff School of Engineering, Cardiff University, Queen’s Buildings, Cardiff CF24 0YF, UK b Department of Aeronautical and Automotive Engineering, Loughborough University, Loughborough LE11 3TU, UK

Received 31 March 2005; received in revised form 9 February 2006; accepted 17 February 2006Available online 6 May 2006

Abstract

Due to the level of complexity of the governing equations for curved composite panels, few solutions, allowing the critical bucklingloads and postbuckling behaviour of such structures to be determined, exist. Those that do are for basic problems based on single loadtypes and simple boundary conditions. For other cases, such behaviour can only be determined by either grossly simplifying both the loadand boundary conditions to those which can be predicted using these simplified equations (which may lead to overestimations in bucklingload, and thereby premature failure or collapse), or by using alternative tools such as finite strip techniques or finite element analyses.

This paper details a series of tests carried out to determine the behaviour of a number of optimised fibre composite panels of differingradii of curvature and aspect ratio, simply supported along two edges and built in along the other two, subject to a varying combinationof shear and in-plane bending, for which no theoretical solution exists, and assesses the suitability of finite strip techniques and finiteelement analysis to predict this behaviour.Ó 2006 Elsevier Ltd. All rights reserved.

Keywords: A. Carbon fibres; C. Buckling; C. Finite element analysis; Optimisation

1. Introduction

Composite materials are used increasingly in a number of industries, most notably aerospace, due to their superiorspecific stiffness and strength. Many of the components con-structed from such materials take the form of curved panelssubject to combined loading regimes and complex bound-ary conditions, which can potentially fail due to buckling.

Due to the need to consider, even for small deflections,

the development of membrane stresses in addition to bend-ing stresses during deformation, shell theory, which can beused to determine the behaviour of such structures, is morecomplex than the plate theory applied to simpler designs.In the case of composites, where depending upon lay-up,shear, membrane-bending and bend–twist couplings mayalso exist, the governing equations are further complicated

and form a set of three eighth order differential equationswhich remain coupled even for symmetrical laminates.Due to the complexity of these equations, very few solu-tions for buckling exist, and those that do, cover only thesimplest of load and boundary conditions for very specificlay-ups.

One case of interest, for which no theoretical solutionsexists, is that of a curved panel built-in at one end and subjected to a shear load across the other as illustrated in

Fig. 1, leading to a combination of shear and in-planebending which vary across the structure. This arrangementis typical of that found in structures such as fan blades,which are formed from two curved panels of differing radiiof curvature joined together along their two axial edges,built-in at their hub, and loaded across their tip.

This paper presents the results of work carried out toinvestigate such a case using experimental, exact finite stripand finite element techniques, in order to improve under-standing of the structures buckling and postbuckling behav-iour and assess the suitability of analytical modelling

0266-3538/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved.

doi:10.1016/j.compscitech.2006.02.028

* Corresponding author. Tel.: +44 0 2920 875328; fax: +44 0 2920874597.

E-mail address: [email protected] (C.A. Featherston).

www.elsevier.com/locate/compscitech

Composites Science and Technology 66 (2006) 2878–2894

COMPOSITES

SCIENCE AND

TECHNOLOGY

mailto:[email protected]

mailto:[email protected]



techniques to predict this behaviour. Nine different geome-

tries of panel with three different radii of curvature andthree aspect ratios were investigated with a total of sixtypanels with two different radii of curvature and three aspectratios tested as detailed in Section 4.1. Each panel was man-ufactured from eight plies of carbon fibre epoxy pre-pregarranged in a balanced, symmetrical lay-up. In each casethe lay-up was designed to optimise the buckling strengthof the panel using a two stage, multi-level optimisation pro-cedure based on a combination of the exact finite strip codeVICONOPT and the commercially available finite elementanalysis (FEA) code ABAQUS/Standard. The experimen-tally determined buckling loads and the postbuckling

behaviour of each set of panels are compared with theresults of these two types of analysis. A number of recom-mendations based on the results obtained are then made.

2. Theory

2.1. Governing differential equations

As is the case for isotropic shells, shell theory for lami-nated composites is governed by a set of eighth order differ-ential equations whose solution requires four boundaryconditions to be defined along each edge of the shell. In

composites however, these equations become algebraically

more complex than those associated with isotropic shelltheory due to the large number of stiffness coefficientsinvolved. In addition to this added complexity, it can alsobe shown that in contrast to the case of laminated flatplates these equations remain coupled even for midplanesymmetric laminates.

Derivation of the shell buckling equations for laminatedcomposite shells is thus a complicated procedure. Depend-ing upon the assumptions made however, (for example theuse of classical shell theory (CST) or first order shear the-ory (FST)) various theories can be obtained. These alongwith the results of a range of experimental work are

described in a number of reviews including Tennyson [1],Leissa [2] and Kapania [3].

2.2. Plates under combined shear and in-plane bending

Due to the complexity of the shell buckling equationsonly a limited number of solutions are available, mainlyfor shells under single, simple types of loading. For the caseof a curved panel, such as that considered here therefore,most of the work that has been completed previouslyinvolves the use of analytical codes such as STAGS [4] todetermine critical buckling loads which are then comparedwith the results of experimental studies.

Shear – Studies on composite curved shear panels includethose by Bert et al. [5] in which a number of thin walledsandwich shells, cylindrical panels and full-scale cylindersmanufactured from two-ply epoxy fibre glass using a struc-ture analogous to a picture frame were examined. Correla-tion between test results and calculated values was similar tothat found with homogeneous isotropic specimens.

Wilkins and Olson [6] performed tests on eight-ply +45°specimens manufactured from graphite epoxy and boronepoxy. Strain gauges were used to monitor the behaviourof the panels (which were mounted in sets of four to forma cylinder to which a torsional load was applied), thereby

allowing the buckling loads to be calculated in a non-

Nomenclature

d displacement vector f load factorJ number of eigenvalues that lie between zero and

a trial valueJ 0 value of J when the components of the displace-ment vector corresponding to K are clamped

J m value of J for a particular memberkij member stiffness matrixK global stiffness matrixKD upper triangular matrixk half wavelengthm member of the structureN x,N y direct stress intensities

N xy shear stress intensity N x ratio of the critical compressive axial load with

shear to that without shear

N xy ratio of the critical shear load with compressionto that without compressionP perturbation force vectorR radius of curvatures{K} sign count of Ku,v in-plane displacementsw out-of-plane displacementh ply orientation anglex frequency

= +

Curved panel

Load

In-plane bending Shear

Fig. 1. Load case.

C.A. Featherston, A. Watson / Composites Science and Technology 66 (2006) 2878–2894 2879



destructive manner, using the Southwell method. Resultswere compared with classical shell theory predictions andgood correlation was found. The results also highlightedthe existence of a postbuckled load-bearing capacity.

Viswanathan et al. [7,8] conducted wide-ranging para-metric studies on, in particular the influence of curvature

on the buckling of symmetrically laminated eight plyquasi-isotropic boron epoxy shell panels of infinite lengthunder a range of boundary conditions. The work showedresults obtained using Donnell-type shell theory deviatedsignificantly from those using more accurate analyses forlarge values of axial half wavelength of the buckling pattern/panel width. Extensive data was also provided by Fogg [9]who examined graphite epoxy panels under shear loading.

Zhang and Matthews [10,11] derived a set of governingequations for buckling for arbitrarily laminated cylindricalshells in terms of the transverse displacement and an Airystress function. These were solved for shells having alledges clamped and subjected to axial and/or shear loads

using the Galerkin method, with the work in particularillustrating the importance of using double precision inthe calculation of the beam functions. The resulting equa-tions were used to investigate the effect of fibre orientationon the buckling loads of a series of boron epoxy panelswith unidirectional lay-ups. Results for both positivelyand negatively directed shear loads were determined for arange of curvatures, and it was shown that a change in signfor this shear load (which may also be interpreted as achange from +h to Àh in fibre orientation for a given shearload) had considerable effect on the shear buckling loads. Itwas also found that this effect was less pronounced for

symmetric angle ply lay-ups, and decreased as the numberof plies increased. Finally, for antisymmetric lay-ups thedirection of shear loading was found to have almost noeffect on critical buckling load.

The results of a number of these pieces of work werereviewed by Wolf and Kossira [12], who also performedresearch, based on the use of a picture-frame device, todevise an improved method for investigating shear behav-iour, with particular attention to postbuckling behaviour[13]. This was carried out on two types of panels with fourply and eight ply ±45° lay-ups and a radius of curvature of 4000 mm. The results of the tests performed were com-pared with predictions made using a finite element codeFiPPS [12]. Good correlation was obtained.

Bending – A number of studies have been performed toinvestigate bending in cylindrical shells, however, no workhas been carried out to investigate in-plane bending incurved panels.

Combined loading – Most curved panels need to bedesigned to resist combined loads, however the majorityof analyses carried out are limited to single, simple typesof loading. There have however been a number of studieson cylindrical specimens subject to combined loads includ-ing those of Wilkins and Love [14], Herakovich and John-son [15], Viswanathan et al. [7,8] and Zhang and Matthews

[10,11] each of whom looked at the combination of shear

and compressive loading. Although not explicitly the sameload case as that considered here, this combined loading isequivalent to that found in the upper part of the panels tobe tested (see Fig. 6) and hence the results will be discussedbriefly here.

Wilkins and Love investigated the effects of material and

lay-up on behaviour, testing boron-epoxy and graphite-epoxy four ply ±45° and six ply 0°, ±45° configurations,both of which possess significant bend–twist coupling inter-actions. Comparison between test results and analyticalpredictions showed a knock down factor of approximately65%. One point of particular interest in these studies wasan almost linear relationship between shear and compres-sive stress over a wide range of compressive stresses, withtorsional buckling stress reducing by between 10% and20% of the increase in applied compressive stress.

Herakovich and Johnson again tested a combination of boron-epoxy and graphite-epoxy cylinders with a numberof different lay-ups (four ply (±45)s, eight ply (À452/

452)s, eight ply (0/±45/90)s, and an unsymmetric four ply(À82.5/30/20/À82.5)). The results of these tests were com-pared with those obtained using a computer programdeveloped by Wu (described in [15]) and good correlationwas observed. In particular it was noted that the shape of the interaction curve is extremely dependant upon thelay-up configuration.

Viswanathan et al. as mentioned earlier examined a ser-ies of symmetric, quasi-isotropic boron epoxy panels. Interms of combined loading, they investigated three loadingcases in which – N x = N xy,N y = À N xy and N x = N y = N xy.

Finally Zhang and Matthews produced interaction

curves showing critical combinations of axial and shearloads which cause buckling. In each case N x is plottedagainst N xy , where N x is the ratio of the critical compressiveaxial load with shear to that without shear and N xy is theratio of the critical shear load with compression to thatwithout compression.

As far as the authors are aware therefore, there is noexisting theory that would allow the buckling loads andpostbuckling behaviour of the panels to be tested to be pre-dicted. Data with which to compare experimental resultswill instead be determined using either VICONOPT (anexact finite strip program described in Section 3.1), or finiteelement analysis which is capable of allowing the bucklingloads of plates having any particular lay-up under a specificset of load and boundary conditions to be calculated.

3. Optimisation

In order to determine the optimum design for each of the panels tested, as discussed earlier, a two stage optimisa-tion process using the software VICONOPT (based on anexact finite strip method) in conjunction with a series of finite element analyses performed using the commerciallyavailable software ABAQUS was carried out. In the firststage of the optimisation the VICON analysis route in

VICONOPT was used to perform series of parametric

2880 C.A. Featherston, A. Watson / Composites Science and Technology 66 (2006) 2878–2894



studies to determine the effect of varying ply orientation onthe buckling load. These studies provided sufficient data togenerate response surfaces plotting buckling load againstlay-up for each geometry (i.e., combination of radius of curvature and aspect ratio), from which an initial estimateof the optimum design could be determined. Due to the

efficiency of the analysis, it was possible to run series of studies relatively quickly, making the software ideal for thisstage of the optimisation process. During the second stageof the process which used the results of the VICON analy-sis as a starting point, the FEA software ABAQUS wasused to carry out a series of more precise analyses (beingable to represent the load and boundary conditionsexactly), systematically investigating the effect of deviatingfrom this initial recommended lay-up to determine an accu-rate local optimum. (However, as the starting point of thisanalysis was based on a number of assumptions it was notpossible to ensure that this was a global optimum solution).

3.1. Viconopt

The VICONOPT [16] computer code is an analysis anddesign-optimisation code for the buckling, postbucklingand free vibration analyses of prismatic assemblies of flator curved plate elements subjected to in-plane loads. Itincorporates the earlier program VIPASA (Vibration andInstability of Plate Assemblies including Shear and Anisot-ropy) [17] and VICON (VIpasa with CONstraints) whichhas substantial additional capability. In contrast to conven-tional finite element methods, the VICONOPT analyses usea transcendental stiffness matrix derived from an analytical

or numerical solution of the governing differential equa-tions of the component plates. For any longitudinallyinvariant loading combination (of longitudinal, transverse,in-plane shear and pressure loads), critical buckling loads orundamped natural frequencies can be found with certainty,using the Wittrick–Williams algorithm [18]. In the originalform of the software VIPASA, the modes of buckling orvibration are assumed to have a sinusoidal longitudinal var-iation with half-wavelength k, limiting the load cases whichcan be considered. In VICON the assumed modes are sumsof VIPASA modes, obtained by coupling different values of k thereby extending the original capability to allow end con-ditions to be modeled more accurately for overall modes of plate assemblies with substantial in-plane shear loads, forwhich VIPASA gives very conservative results.

3.2. ‘Exact’ strip method and Wittrick–Williams algorithm

As stated previously VICONOPT analyses are based onan exact finite strip method which assumes a continuousdistribution of stiffness over the structure rather than discre-tised stiffnesses at nodal points, as is the case for the morewidely used finite element method. The exact finite stripmethod is based on solutions to the partial differential equa-tions, which govern the in-plane and out-of-plane deforma-

tion of the component plates. Where possible, an analytical

solution procedure [17] is used to determine the memberstiffness matrices kij , which are subsequently assembled intothe global stiffness matrix K for the overall structure. If ananalytic solution of the member equations is not availablethe kij matrices can be found by solving the member equa-tions numerically [19]. The global stiffness matrix K relates

a finite set of displacements d at the nodes of the structure totheir corresponding perturbation forces P, by

Kd ¼ P. ð1Þ

The critical buckling load factors or natural frequencies of the structure correspond to the eigenvalues found bysolving

Kd ¼ 0; ð2Þ

where K consists of transcendental, and thus highly non-linear, functions of the load factor f or frequency x [20].The solution of this transcendental eigenvalue problem re-quires an iterative search for values of f or x at which Eq.

(1) is satisfied. VICONOPT analyses make use of the Wit-trick–Williams algorithm [18,21], to converge on eigen-values (i.e., critical buckling load factors or naturalfrequencies of free vibration) to any desired accuracy withthe guarantee that none are missed. The algorithm calcu-lates J , the number of eigenvalues, which lie between zeroand any trial value of f or x. Any change in J betweentwo trial values is equal to the number of eigenvalues lyingbetween these trial values.

In its general form the Wittrick–Williams algorithm canbe stated as

J ¼ J 0 þ sfKg; ð3Þ

where s{K} is known as the sign count of K, and is equal tothe number of negative leading diagonal elements of theupper triangular matrix KD obtained by applying conven-tional Gauss elimination, without pivoting, to the matrixK. J 0 is the value J would have if all the freedoms corre-sponding to K were clamped. Unless substructures areused, J 0 can be calculated as

J 0 ¼X

m

J m; ð4Þ

where the summation is over all members m of the struc-ture, and J m is calculated for each member as the numberof critical load factors or natural frequencies exceeded bythe trial value, when the member ends are clamped.

3.3. Optimisation step 1 – VICONOPT

As discussed earlier, for each of the panels examined, aseries of analyses using the VICON route in VICONOPTwere performed to create response surfaces representingthe variation of buckling load with lay-up.

Each panel was modelled using the curved plate optionin VICONOPT. In this option the geometry of the curvedpanel is approximated by discretising it into a series of flatplate members that are joined to form the complete curved

panel. In order to ensure that an adequate number of plate




members were used in the analysis a convergence study wasundertaken. McGowan and Anderson [22] show a mini-mum of 100 plate members are necessary for accuratelymodelling a complete cylinder and the authors found that50 plate members were sufficient for the panels presentedin this paper.

Each of the models used was constructed from 10 strips.Each strip was modelled as a curved panel made up of fiveflat plate elements as illustrated in Fig. 2. The panel wasclamped along its loaded edges by restricting out-of-planedisplacement and rotation. In order to speed up the processdue to the large number of analyses required, a small num-ber of point supports were used to represent the line sup-port and a small number of wavelengths were used torepresent the mode (Watson et al. [23]). The remainingedges were simply supported by restricting out-of-planedisplacement. A shear load was applied to the ends of eachof the strips, and in-plane bending was modelled by apply-ing a series of compressive and tensile loads, which varied

linearly from the top to the bottom edge of the panel. Ineach case the lay-up was assumed to be symmetric and bal-anced to minimise the possible alternatives. Individual plieswere then paired such that each pair had one ply at +h andone at Àh (where h is defined as illustrated in Fig. 3). As ineach case the laminate comprised eight plies three possiblecombinations of pairs existed; plies 1 and 2 and plies 3 and4; plies 1 and 3 and plies 2 and 4; and plies 1 and 4 and plies2 and 3, and three response surfaces were therefore gener-ated for each of the nine combinations of radius of curva-ture (100, 177 and 322 mm) and aspect ratio (1, 1.5, 2)investigated – a total of 27 response surfaces. The angle

of each pair was varied in steps of ±5°, thereby producing1369 results for each response surface. Using a Pentium 43 GHz 1 GB Ram computer, these results took 253 s togenerate for each surface demonstrating the efficiency of the programme.

3.4. Optimisation step 2 – finite element analysis

Once the optimum lay-up based on the VICON analyseshad been determined, a small number of finite elementanalyses where performed in the region of this design

(again by varying the ply angle by ±5°) to identify the opti-mum ply angle based on an exact representation of the testundertaken (since the VICON analyses had been based on

a number of assumptions which changed the loading andboundary conditions of the plates being analysed slightlyrelative to those being tested, such as the application of pure shear, where in the test shear varies across the panel,and the application of a fixed bending load, when in thetest the bending load varies with the moment arm andtherefore is maximum at the built-in end and zero at theend at which the load is applied). Thus the optimisationprocess was multi-level, with the quicker, more efficientcode VICONOPT used to identify the area of the optimumsolution, and the slower, but more accurate in terms of rep-resentation of the exact problem finite element analysis, toidentify the final optimum solution.

3.5. Results

A total of 27 response surfaces were generated using theresults of the VICON analyses as discussed previously.Study of these surfaces resulted in the optimum lay-upspresented in the third column of Table 1 being identified.As described above, a second procedure using the finite ele-ment analysis code ABAQUS/Standard was then under-taken, the results of which are provided in the fourthcolumn of the table. The results of this second stage of the optimisation procedure are also illustrated in the form

Fig. 2. Segmented representation of curved plate geometry used by

VICONOPT.

+

Fig. 3. Ply orientation.

Table 1Optimised lay-ups

Radius of curvature (mm)

Aspect ratio VICON analysis FE analysis

100 1 (À60/À30/60/30)s (À60/À20/60/20)s

1.5 (À65/À30/65/30)s (À65/À20/65/20)s

2 (À60/À30/60/30)s (À75/À30/75/30)s

177 1 (À60/À30/60/30)s (À65/À30/65/30)s

1.5 (À60/À25/60/25)s (À65/À20/65/20)s

2 (À65/À30/65/30)s (À65/À20/65/20)s

322 1 (À60/À30/60/30)s (À65/À20/65/20)s

1.5 (À65/À40/65/40)s (À60/À15/60/15)s

2 (À55/À30/55/30)s (À65/À20/65/20)s




of response surfaces in Figs. 4–6. For each combination of radius of curvature and aspect ratio, the surface illustratedis that obtained by considering plies 1 and 2 and plies 3 and4 as pairs and varying ply angle accordingly as discussed inSection 3.4. The lay-ups determined during the secondstage of the procedure then became the ones used during

the manufacture of the test specimens.

4. Testing

4.1. Specimens

The test specimens used were manufactured from thecarbon fibre epoxy unidirectional pre-preg T300 914. Each

specimen comprised 8 lamina, 0.125 mm thick, giving an

Aspect ratio 1

-55 -60 -65 -70-30

-25

-20

-15

Ply 1

Ply 2

10300-10350

10250-1030010200-10250

10150-10200

10100-10150

10050-10100

10000-10050

9950-10000

9900-9950

9850-9900

9800-98509750-9800

9700-9750

9650-9700

9600-9650

Aspect ratio 1.5

-60 -65 -70 -75-30

-25

-20

-15

Ply 1

Ply 2

7100-7150

7050-7100

7000-7050

6950-7000

6900-6950

6850-6900

6800-6850

6750-6800

6700-6750

6650-6700

6600-6650

6550-6600

6500-6550

Aspect ratio 2

-60 -65 -70 -75 -80-35

-30

-25

-20

-15

Ply 1

Ply 2

5100-5150

5050-5100

5000-5050

4950-5000

4900-4950

4850-4900

4800-4850

4750-4800

4700-4750

4650-4700

4600-4650

4550-4600

4500-4550

Fig. 4. ABAQUS buckling load versus lay-up – radius of curvature R = 100 mm.




overall panel thickness of 1 mm. Tests were carried out onsets of panels with two different radii of curvature (100 and322 mm). These radii were selected to give specimens of areasonable size for testing and to represent curvatures typ-ical of those found in the aerospace industry (as was the

additional radii of curvature of 177 mm examined during

the optimisation process). For each radius, three differentaspect ratios of panel were tested: aspect ratio 1 – 100 mm long · 100 mm wide; aspect ratio 1.5 – 150 mmlong · 100 mm wide and aspect ratio 2 – 200 mmlong · 100 mm wide giving a total of six different panel

geometries. Whilst the initial aim of the experiment was

Aspect ratio 1

-55 -60 -65 -70-35

-30

-25

-20

Ply 1

Ply 2

7100-7150

7050-7100

7000-7050

6950-7000

6900-6950

6850-69006800-6850

6750-6800

6700-6750

6650-6700

6600-6650

6550-6600

6500-6550

Aspect ratio 1.5

-55 -60 -65 -70-25

-20

-15

-10

Ply 1

Ply 2

4825-4850

4800-4825

4775-4800

4750-4775

4725-4750

4700-4725

4675-4700

4650-4675

4625-4650

4600-4625

Aspect ratio 2

-60 -65 -70 -75-30

-25

-20

-15

Ply 1

Ply 2

3800-3825

3775-3800

3750-3775

3725-3750

3700-3725

3675-3700

3650-3675

3625-3650

3600-3625

3575-3600

3550-3575





to examine panels with the two axial edges simply sup-ported, it proved too complicated to design a rig thatwould maintain this boundary condition throughout preand postbuckling. For this reason an attempt was madeto introduce similar conditions by modifying the shape of the test specimens. To do this, an additional 10 mm of

material was folded over at 90° along the longitudinal

edges (as illustrated in Fig. 8), thus allowing the panel torotate about this edge but reducing out-of-plane displace-ment. This was obviously not ideal, but could at least berepresented exactly using finite element analysis to allowdirect comparison.

To create the desired radii of curvature, each specimen

was laid-up by hand onto a male former, with a female

Aspect ratio 1

-55 -60 -65 -70-30

-25

-20

-15

-10

Ply 1

Ply 2

5075-5100

5050-5075

5025-5050

5000-5025

4975-5000

4950-4975

4925-4950

4900-4925

4875-4900

4850-4875

4825-4850

4800-4825

4775-4800

Aspect ratio 1.5

-55 -60 -65 -70 -75-40

-35

-30

-25

-20

-15

10

Ply 1

Ply 2

3425-3475

3375-3425

3325-3375

3275-3325

3225-3275

3175-3225

3125-3175

3075-3125

3025-3075

2975-3025

2925-2975

2875-2925

2825-2875

Aspect ratio 2

-55 -60 -65 -70 -75-30

-25

-20

-15

Ply 1

Ply 2

2550-25752525-2550

2500-2525

2475-2500

2450-2475

2425-2450

2400-2425

2375-2400

2350-2375

2325-2350

2300-2325

2275-2300

2250-2275





former then positioned on top. The whole assembly wascured in a hot press at 170 °C and 7 barg for 60 min andpost-cured also in a hot press at 190 °C and atmosphericpressure for 4 h. Each treatment process was monitoredwith pressure and temperature measurements taken every5 s to ensure the integrity of the specimen.

For each geometry, a total of ten tests were performed,allowing the mean and standard deviation of each set of results to be calculated, thus giving some indication of imperfection sensitivity.

4.2. Test rig

The test rig used was as illustrated in Figs. 7 and 8. Pan-els were attached along one curved edge thus representing abuilt-in boundary condition, using clamps bolted togetherthrough the specimen and then to the rig. In order to pre-vent slipping of the specimen, sheets of glass paper wereinserted between the specimen and the clamps (as per

ASTM D4255/D [24]). The two straight edges were leftfree, the shape of the panel then modelling a simply sup-ported edge condition as described in Section 4.1. Theremaining curved edge was held between two clamps again

bolted through the specimen, which were in this case thenfixed to a loading plate. The arrangement for the loadedend of the rig can be seen in detail in Fig. 8. In order tofacilitate the application of a bending and shear force,the end of the panel was allowed to move vertically andto rotate in the plane about its clamped end (i.e., about

the x axis); however, lateral displacement was not permit-ted to prevent twisting of the panel about the y axis. Toensure this, the loading plate was positioned between twouprights attached to the baseplate, thus preventing move-ment in the x direction and rotation about the y axis(Fig. 8). Ball-bearings located between the loading plateand the curved edges of the two vertical spacers or slidesallowed rotation about the y axis. The flat external sideof these slides and the inside surfaces of the uprights werehardened and ground, thus allowing them to slide againstone another to facilitate movement in the z direction androtation about the x axis.

The baseplate of the rig was bolted to a Howden univer-

sal testing machine and load was applied through a loadingarm attached to the crosshead. This arm was connected toa pin in the loading plate via a spherical bearing, therebyensuring the loading arm was always in tension. Thisresulted in a combination of shear and in-plane bendingloads being introduced into the specimen. The testmachine’s computer control software was used to programthe test therefore ensuring consistency between individualexperiments. The software also recorded the applied shearload using a 10 kN load cell and in-plane displacementusing a built-in displacement transducer. Load versus in-plane displacement plots for each test were then used to

determine the buckling loads by taking the intersection of tangents to the pre- and postbuckling gradients in amethod similar to that presented by Zaal [25] and used suc-cessfully in earlier work [26]. In addition to recording thisin-plane data, an analogue to digital (A/D) card wasinstalled to provide two further data input channels. Thisallowed two displacement transducers positioned to mea-sure the out-of-plane displacement at points coincidingwith the anticipated peaks of the first eigenmode to be con-nected via a conditioning unit to the software and loggedalso. The load was increased by moving the crossheadupwards at a speed of 2 mm/min. The results were sampledat a rate of 10 points/s.

The shadow moire technique (Featherston and Lester[27]) was used to provide a full-field representation of the out-of-plane displacement of the panel. This methodis non-contact and therefore does not affect the bucklingof the plate. A grating of 8 lines/cm was projected onto the surface of the panel (which was painted white toimprove contrast) using an ASK Impression 8300 DLPSVGA slide projector. The panel was then videoedthroughout the test and the video processed using soft-ware written in C++ with the Microsoft DirectShow soft-ware developer’s kit. This software captured the initialframe of the video, representing the grating projected

onto the undeformed specimen and superimposed it onFig. 7. Test rig.




all subsequent frames recording the displaced grating onthe progressively deformed specimen, thereby producinga real time series of fringes corresponding to points of equal lateral displacement. Thus full field out-of-planedisplacement of the panel could be monitored throughoutthe period of the test.

4.3. Results

The experimentally found buckling loads for specimenswith each combination of aspect ratio and radius of curva-ture are presented in Table 3, with their mean and standarddeviation. These are compared in Figs. 11 and 12 withthose predicted by ABAQUS using both linear eigenvalueand nonlinear Riks analyses. In-plane load versus displace-ment plots (with displacement measured at the point and inthe direction of loading) again for each combination of cur-vature and aspect ratio, measured experimentally and pre-dicted using nonlinear analysis are presented in Figs. 13and 14. Finally experimental and FEA out-of-plane dis-placement at points corresponding to the peaks of eacheigenmode for panels with aspect ratio 1 and radii of cur-

vature R = 100 and 322 mm, respectively, can be foundin Figs. 15 and 16.

5. Finite element analysis

5.1. Model

The panels investigated were modelled using 5 mmsquare shell elements for the curved section, and5 mm · 10 mm shell elements for the folded over edges,as shown in Fig. 9. (The mesh density being selected follow-ing a convergence study to produce errors of less than 1%

at loads equivalent to twice the buckling load.) The clamps

and loading plate were also modelled to ensure accurateload application, using brick elements.

The elements used for the panel were quadrilateral S8R5elements, which behave in a manner consistent with thinshell theory (i.e., they are based on the Kirchoff–Love the-ory). These elements are eight noded elements in whichedge behaviour is modelled using quadratic equations,which prevent the occurrence of the hourglass effect (inwhich spurious displacements occur perpendicular to theshell surface). Reduced integration, with four integrationpoints instead of the standard eight is used, and each nodehas five degrees of freedom, namely three displacementsand two in-surface rotations.

The solid elements used to the model the end clamps andloading plate were of type C3D20R. These are 20 nodedelements also with edge behaviour described by quadraticequations. Nodes have three degrees of freedom (three dis-

placements), and reduced integration is again employed

x

y

z

test panel

clamps

end guides

end plate

slides

directionof loading

bearings

foldedover edge

Fig. 8. Detail of loading arm.

Fig. 9. Finite element mesh.




giving 12 integration points. These elements were selectedfor compatibility with the shell elements.

The boundary conditions applied to the model can bedescribed by reference to Fig. 10, with three degrees of free-dom representing displacements, and three rotations.Along edge 1, all five degrees of freedom were restrained

thus representing the clamped end condition. Edges 2 and3 remained free. Along edge 4, to which the load wasapplied, boundary conditions were applied to preventout-of-plane displacement x and rotation about the y-axis,but movement and rotation in all other directions were per-mitted, thus allowing shear and bending to be transmittedthrough the panel.

5.2. Buckling analysis

An estimate of the initial, critical buckling load of eachgeometry of panel was calculated using the classical lineareigenvalue solver in ABAQUS/Standard. Since due to the

lay-up optimisation, the structures are weaker when loadedin the same sense but in the opposite direction, a number of negative eigenvalues (corresponding to a load applied inthe opposite direction) exist which are smaller in magnitudethan the eigenvalues corresponding to the initial failure inthe required direction. To prevent these being calculatedfirst, thus increasing the time to reach a solution, a preloadwas applied. The buckling loads corresponding to the firstpositive eigenvalue calculated by ABAQUS for each opti-mised panel are presented in Table 2, and are comparedwith those found experimentally in Figs. 11 and 12. Theseanalyses were also used to provide eigenmodes to be used

in modelling geometric imperfections in the subsequentnonlinear analysis.

5.3. Postbuckling analysis

A nonlinear postbuckling analysis was carried out foreach geometry of specimen using the Riks method alsoavailable in ABAQUS/Standard [28,29], which is suitablefor unstable problems in which the load or the displace-ment may decrease as well as increase during the loadingprocess. In each case, a geometric imperfection was intro-duced in the form of the first positive eigenmode, with amaximum amplitude equal to the thickness of the panel,i.e., 1 mm. This method was originally promoted by Spei-cher and Saal [30], and has been adopted by most of the

commercial finite element codes to allow the calculationof a lower limit for any experimentally found bucklingloads. The buckling loads calculated using this methodfor panels with each combination of radius of curvatureand aspect ratio are compared with those predicted bythe earlier linear analyses and the experimental results inFigs. 11 and 12. Plots of load versus the in-plane displace-ment in the direction of loading predicted by nonlinearanalysis are compared with that measured experimentallyin Figs. 13 and 14, along with similar graphs comparingout-of-plane displacement for a limited number of speci-mens (Figs. 15 and 16). Finally Fig. 17 illustrates the devel-

opment of the moire´

fringes representing out-of-planedisplacement with the corresponding FEA contours at

Table 2Optimised buckling loads

Aspectratio

Radius of curvature (mm)

100 177 322

Step 1 – VICONanalysis

Step 2 – FEanalysis





1 9945 10341 7041 7149 4887 50921.5 7041 7150 4771 4846 3159 3465

2 4317 5135 3765 3815 2248 2552

Table 3Experimental buckling loads

Buckling load (N)

Radius of curvature (mm): 100 322

Aspect ratio: 1 1.5 2 1 1.5 2

Test 1 3300 3000 2500 3550 2500 1860

Test 2 3100 3100a

2900 2500 1650Test 3 3300 a a 3600 2400 1750Test 4 3900 3300 2600 3400 2220 1790Test 5 4500 3200 a 3600 2100 1850Test 6 2700 2900 a 3300 2100 1720Test 7 3500 2800 2300 3000 2280 1730Test 8 3500 2600 2450 3100 2080 1700Test 9 3900 2700 2300 3200 2230 1700Test 10 3650 2900 2100 3600 2000 1850

Mean 3535 2944 2375 3325 2241 1761

Standard deviation 496 230 178 266 178 74

a Denotes premature material failure.

Edge 4

Edge 1y

z

x

ForceEdge 2

Edge 3

Fig. 10. Boundary conditions.




0

2000

4000

6000

8000

10000

12000

1 1.5

Aspect ratio

a o L

) N ( d

2

ABAQUS eigenvalue analysis

ABAQUS Riks analysis

Experimental results

Mean experimental result

Fig. 11. Comparison of ABAQUS and experimental buckling loads for panels with radius of curvature R = 100 mm.

0

1000

2000

3000

4000

5000

6000

1 1.5

Aspect ratio

) N ( d a o L

2

ABAQUS eigenvalue analysis

ABAQUS Riks analysis


Mean experimental result

Fig. 12. Comparison of ABAQUS and experimental buckling loads for panels with radius of curvature R = 322 mm.

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10

Displacement (mm)

) N k ( d a o L

Experimental Results

Riks analysis

AR 1.5

AR 1

AR 2

Fig. 13. Comparison of FEA and experimental in-plane displacement for panels with radius of curvature R = 100 mm.




0

1

2

3

4

5

6

0 1 2 3 4 5 6

Displacement (mm)

a o L

d

( k N )


Riks analysis

AR 1.5

AR 1

AR 2

Fig. 14. Comparison of FEA and experimental in-plane displacement for panels with radius of curvature R = 322 mm.

0

1

2

3

4

5

6

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Displacement (mm)

L o a d ( k N )

Experimental resultsRiks analysis

LVDT 2LVDT 1

Fig. 15. Comparison of FEA and experimental out-of-plane displacement for panels with radius of curvature R = 100 mm and aspect ratio 1.

0

1

2

3

4

5

-0.5 0 0.5 1 1.5 2 2.5

Displacement (mm)

L o a d ( k N

)


Riks analysis

LVDT 1 LVDT 2

Fig. 16. Comparison of FEA and experimental out-of-plane displacement for panels with radius of curvature R = 322 mm and aspect ratio 1.




the point of buckling for a panel with radius of curvatureR = 322 and aspect ratio 1.

6. Discussion

6.1. Optimisation

The optimum lay-ups calculated using the two stageprocess involving multiple VICONOPT analyses in thefirst instance, followed by a much smaller number of finiteelements analyses, are given in Table 1. These illustrate thesuccess of the technique in determining complex optimumsolutions to a problem that is insolvable using simpledesign equations. Study of Figs. 4–6 illustrates a maximumof 35 finite element analyses, following an initial 1369VICON analyses required to provide an optimum design.Thus the majority of the optimisation is carried out bythe substantially quicker exact finite strip method, withthe final result determined using the much slower, but more

capable of modelling complex load and boundary condi-tions, finite element analysis, leading to a further increasein buckling load for the same thickness specimen of up to13%.

As anticipated, the optimum solutions align the externalplies with the direction of the compressive stress due to theshear load in all cases. In terms of trends between radius of curvature, aspect ratio and optimum lay-up however thereis little correlation, illustrating the complexity of the prob-lem with many factors influencing the design. For panelswith radius of curvature R = 100 mm there is a linkbetween optimum lay-up angle and aspect ratio, with theangle increasing with increased aspect ratio, as was foundin earlier studies on flat plates [26]. This is not supportedby the other two sets of panels, i.e., those with radius of curvature R = 177 and 322 mm where there is little varia-tion in lay-up with changing aspect ratio. There is alsosome correlation between ply orientation and relative pro-portion of compressive stress with orientation angledecreasing as the relative proportion of compressive loadincreases. This can be illustrated by examining the differ-ence between the VICON and FEA analyses. In the FEA(as in the experiment), the plates are built in at one endand a load is applied across the other introducing shearwhich varies across the plate and in-plane bending, which

increases from zero at the point of load application, to a

maximum at the built-in end. As noted earlier, it is not pos-sible to model this varying in-plane bending in VICON-OPT, since only longitudinally invariant loading can beintroduced, so an average load representative of that foundhalfway along the plate was used. This results in the com-pressive load in the model being too great at the loaded endof the plate, and too low in the region near the built-in end

where the plate buckles. Thus the effects of the compressiveloading are underestimated in the most critical regionresulting in optimum lay-ups with fibres aligned moretowards the transverse direction, than would otherwise bethe case. Study of the optimum lay-ups calculated byVICON and FEA show this to be the case, with the finiteelement results recommending a lay-up in which the fibresare more aligned with the h = 0° direction. This trend isalso in agreement with design guides such as Niu [31].

6.2. Experimental results

The experimentally found load versus in-plane displace-ment plots for panels with radii of curvature R = 100 and322 mm illustrated in Figs. 13 and 14 respectively, demon-strate a stable initial buckling process in which the stiffnessof the panels gradually decreases with increasing load, fol-lowed by a sudden decrease in load carrying capacity at thepoint at which the axial edges buckle. These results illus-trate a reduction in buckling load with increasing aspectratio for panels with both R = 100 mm and R = 322 mm.This is due to a reduction in the stiffness of the panels asaspect ratio increases, which can be seen by examiningthe results of Figs. 13 and 14, and an increase in the max-imum in-plane bending load as the moment arm, i.e., thelength of the panel increases. The results also show anincrease in buckling load with increased curvature, i.e.,panels with R = 100 mm have higher initial buckling loadsthan those with R = 322 mm. This is as a result of theincreased stiffness due to the curvature of the panel, whichcan be illustrated by examining the relative gradients of panels having the same aspect ratio but different radii of curvature in Figs. 13 and 14. It is interesting to note how-ever, that this increase is not as pronounced as that foundearlier for metallic plates [32], due to the difference inlay-up between each set of specimens giving different mate-rial properties. Finally, it is apparent from examination

of the standard deviations of the experimental results for

Fig. 17. Comparison of Moire fringes and Riks out-of-plane displacement contours for a panel with radius of curvature R = 322 and aspect ratio 1.




panels with different geometries that the degree of scatterdecreases with increased aspect ratio and decreased curva-ture. This is as a result of the differing levels of imperfectionsensitivity. Specifically, we would anticipate a marked dif-ference in imperfection sensitivity between panels withR = 100 mm which are much more highly curved and

therefore behave in a manner similar to a cylinder whichis highly imperfection sensitive, particularly under com-pressive loading, and panels with R = 322 mm which areflatter and behave in a manner similar to flat plates, whichare relatively non-imperfection sensitive. These observa-tions are again consistent with earlier findings [32].

6.3. FEA results

6.3.1. Buckling analysis

The buckling loads calculated using linear eigenvalueanalysis, provided in Table 2 and compared with the resultsdetermined experimentally in Figs. 11 and 12 for panels

with radii of curvature R = 100 and 322 mm respectively,can be seen to overestimate those found in the actual panelsin all cases, although they can be seen to follow the sametrends, i.e., decreased bucking load with increased aspectratio and increased radius of curvature. This overestimatecan be seen to be greater in the panels with higher curva-ture. This is due to the fact that these values are basedon a linear analysis of a perfect structure. Since curved pan-els are imperfection sensitive, and this sensitivity is greaterfor panels with higher curvature, such analyses will alwaysoverestimate the actual buckling values as they do not takeinto account the effect of any imperfections present in the

‘as manufactured’ panels. In addition coupling betweenin-plane membrane stresses and out-of-plane bending stres-ses introduced due to the nature of the loading introducenonlinearity which is not modelled in this type of analysis.Again this effect is greater for panels with higher curvature.

6.3.2. Imperfection sensitivity and postbuckling analysis

The buckling loads calculated using a series of nonlinearRiks analyses which incorporated in each case an imperfec-tion in the form of the first eigenmode, scaled such that itsmaximum amplitude was equal to the thickness of thepanel, can be seen by reference to Figs. 11 and 12 to muchmore accurately predict those found experimentally, witherrors between the mean experimental result, and that pre-dicted using FEA being limited to less than 3%. This,greatly increased accuracy in comparison with that pro-vided by the linear eigenvalue analysis is due to a numberof factors. As discussed above, the structures being consid-ered are imperfection sensitive and any imperfections ineither the panels themselves (due to variations in lay-upand curvature and manufacturing defects) or the experi-mental set-up (due to misalignment of the plate, boundaryconditions and load eccentricities) will act to reduce thebuckling load by a ‘knock down’ factor. Models whichdo not incorporate these imperfections will therefore over-

estimate the behaviour of the panels. Due to the curvature

of the panels and the nature of the loading, an element of out-of-plane bending is introduced which when coupledwith the in-plane stresses causes nonlinear deformationbehaviour which can only be modelled accurately using afully nonlinear analysis and finally, as the panel moves intothe postbuckling area, deflections become larger and initial

assumptions regarding small deflections become invalid,requiring nonlinearity to be taken into account.The load versus in-plane displacement plots presented in

Figs. 13 and 14 can also be seen to correlate with experi-mental results, particularly prior to buckling and in theearly stages of postbuckling. Although this is true for allgeometries of test specimen, agreement is better for thepanels with R = 322 mm. The increased deviation betweenpredicted and actual results in the deep postbuckling regionare possibly due to the onset of other failure modes such asmatrix cracking, which were not incorporated into themodels used.

Correlation between the FEA and experimental load

versus out-of-plane displacements plots, with out-of-planedisplacement being monitored at the point of greatestdeflection, i.e., the centre of the first buckle, show less con-sistent correlation. Although agreement between the twosets of results for panels with radius of curvature R =100 mm is excellent as illustrated by Fig. 15, actual behav-iour is much less accurately predicted for panels withradius of curvature R = 322mm (Fig. 16). Regardless of the geometry of the specimen, variation in out-of-plane dis-placement as load increases, depends very much on theshape and position of any initial geometric imperfections,particularly prior to buckling, where deflection at a certain

point may follow a path totally dependant on its initialposition out-of-plane, and only ‘snap’ or ‘jump’ to theanticipated path in the postbuckled region. Unless theactual form of each specimen is known therefore prior tocommencing the test, and this form is introduced into thefinite element model, it is likely that the correlation maynot be a good as one might wish.

In addition to providing a comparison for the experi-mental results in terms of in-plane and out-of-plane dis-placement profiles, the Riks analyses also allow the effectof introducing a range of geometric imperfections to bequantified, as demonstrated in Fig. 18 where the percentagereduction in buckling load when an imperfection of maxi-mum amplitude 2t is introduced is plotted against aspectratio. From this figure it can be seen that this reductionin load varies with radius of curvature, panels with greatercurvature being more sensitive to the introduction of geo-metric imperfections, and as the aspect ratio increases withgreater sensitivity to imperfections being displayed in loweraspect ratio plates. Again this supports the experimentalresults discussed earlier. It is interesting to compare theseresults with those given in an earlier paper (Featherstonand Watson [26]) and presented again in this figure, for flatplates. Although again it can be seen that imperfection sen-sitivity decreases with an increase in aspect ratio, this effect

is much less marked.




7. Conclusions

The results presented in this paper have illustrated that arange of geometries of composite panels can be accuratelyand efficiently optimised using a multi-level approach,based on the initial application of an extremely fast exactfinite strip technique to a simplified version of the problem,followed by a series of more accurate finite element analy-ses, capable of modelling the specific conditions required.They have also shown nonlinear finite element analysis

techniques to be suitable for predicting the buckling loadsand early postbuckling behaviour of a range of geometriesof carbon fibre composite curved panels under a variablecombination of shear and in-plane bending. More specifi-cally it has been shown:

• There is no direct correlation optimum between plyangle and geometry for the specimens considered, dueto the complexity of the problem. It is therefore neces-sary in such cases to apply the type of techniquedescribed here to obtain an optimum result.

• There is a correlation, although small, between ply angleand the relative proportion of compressive stress withangle decreasing as the relative proportion of compres-sive load increases.

• Initial buckling loads are higher for panels with smalleraspect ratios, and those with greater curvature.

• Imperfection sensitivity is also greater for panels withsmaller aspect ratios, and those with greater curvature.

• A linear eigenvalue analysis will overestimate the initialbuckling load of a curved panel regardless of geometry.This overestimation however, is greatest for panels withsmaller aspect ratios and greater curvature.

• A fully nonlinear Riks analysis, incorporating initialgeometric imperfections in the form of the first eigen-

mode, scaled such that its maximum amplitude is equal

to the thickness of the plates will accurately predict theinitial buckling load and the load versus in-plane dis-placement behaviour of the panel up to the point of buckling and into the early postbuckling range.

References

[1] Tennyson RC. Buckling of laminated composite cylinders: a review.Composites 1975:17–24.

[2] Leissa AW. Buckling of laminated composite plates and shell panels,

Report AFWAL-TR-85-3069, AF Wright Aeronautical Laboratories;1985.

[3] Kapania RK. A review on the analysis of laminated shells. ASME JPress Vess Technol 1989;11:88–96.

[4] Almroth BO, Brogan FA, Stanley GM. Structural analysis of generalshells, 1,2, Report No. LMSC-D6333873. Palo Alto (CA): AppliedMechanics Laboratory, Lockheed Palo Alto Research Laboratory;1982.

[5] Bert CW, Crisman WC, Nordby GM. Fabrication and full-scalestructural evaluation of glass-fiber reinforced plastic shells. J Aircraft1968;5:27–33.

[6] Wilkins DJ, Olson F. Shear buckling of advanced composite curvedpanels. Exp Mech 1974;14:326–30.

[7] Viswanathan AV, Tamekuni M, Baker LL. Buckling analysis foranisotropic laminated plates under combined inplane loads. In: 25th

international astronautical congress, Amsterdam, Netherlands, 30thSeptember–5th October; 1974. p. 11.

[8] Viswanathan AV, Tamekuni M, Baker LL. Elastic stability of laminated, flat and curved, long rectangular plates subjected tocombined inplane loads, Contractor Report, NASA CR-2330; June1974. p. 68.

[9] Fogg L. Stability analysis of laminated materials, state of the artdesign and analysis of advanced composite materials. LockheadCalifornia Company; 1981. p. 162 [Sessions I and II].

[10] Zhang Y, Matthews FL. Initial buckling of curved panels of generallylayered composite materials. Compos Struct 1983;1:3–30.

[11] Zhang Y. PhD Thesis, University of London; 1982.[12] Wolf K, Kossira H. The buckling and postbuckling behavior of

curved CFRP laminated shear panels. In: Proceedings, ICAS 88, 16thcongress of the international council of the aeronautical sciences,

Jerusalem, Israel, August 28th–September 2, 1988;1:920–30.

0

5

10

15

20

25

30

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2

Aspect Ratio

%

D e c r e a s e i n B u c k

l i n g L o a d

R=100mm

R=322mm

Flat Plate

Fig. 18. Imperfection sensitivity of plates and panels with differing radii of curvature and aspect ratios.




[13] Wolf K, Kossira H. An efficient test method for the experimentalinvestigation of the postbuckling behavior of curved composite shearpanels. In: Proceedings, ECCM-CTS, European conference oncomposite testing and standardisation, Amsterdam, September 8th– 10th; 1992.

[14] Wilkins DJ, Love TS. Combined compression-torsion buckling testsof laminated composite cylindrical shells. J Aircraft 1975;12(11):885– 889.

[15] Herakovich CT, Johnson ER. Buckling of composite cylindersunder combined compression and torsion-theoretical/experimentalcorrelation, test methods and design allowables for fibrous com-posites. In: Chamis CC, editor. ASTM STP, vol. 743. WestConshohocken (PA): American Society for Testing and Materials;1981. p. 341–60.

[16] Anderson MS, Williams FW, Wright CJ. Buckling and vibration of any prismatic assembly of shear and compression loaded anisotropicplates with an arbitrary supporting structure. Int J Mech Sci1983;25(8):585–96.

[17] Wittrick WH, Williams FW. Buckling and vibration of anisotropic orisotropic plate assemblies under combined loading. Int J Mech Sci1974;16(4):209–23.

[18] Wittrick WH, Williams FW. An algorithm for computing criticalbuckling loads of elastic structures. J Struct Mech 1973;1(4):497–518.

[19] Anderson MS, Kennedy D. Transverse shear deformation in exactbuckling and vibration analysis of composite plate assemblies. AIAAJ 1993;31(10):1963–5.

[20] Plank RJ, Wittrick WH. Buckling under combined loading of thin,flat-walled structures by a complex finite strip method. Int J NumerMeth Eng 1974;8(2):323–39.

[21] Wittrick WH, Williams FW. A general algorithm for computingnatural frequencies of elastic structures. Q J Mech Appl Math1971;24(3):263–84.

[22] McGowan DM, Anderson MS. Development of curved plateelements for the exact buckling analysis of composite plate assembliesincluding transverse shear effects. In: 38th AIAA SDM conference,AIAA-1997-1305; 1997.

[23] Watson A, Kennedy D, Williams FW, Featherston CA. Buckling andvibration of stiffened panels or single plates with clamped ends. AdvStruct Eng 2003;6(2):135–44.

[24] ASTM D 4255/D 4255M – 01. Standard test method for in-planeshear properties of polymer matrix composite materials by the railshear method.

[25] Zaal K. Buckling and postbuckling of a square plate subjected touniform edge loads – an experimental investigation. ASL-138Technion Israel Institute of Technology, Department of AeronauticalEngineering; 1998.

[26] Featherston CA, Watson A. Buckling of optimised flat compositeplates under shear and in-plane bending. Compos Sci Technol2005;65:839–53.

[27] Featherston CA, Lester W. The use of automated projectioninterferometry to monitor the buckling behaviour of a simpleaerofoil. Exp Mech 2002;42(3):253–60.

[28] ABAQUS Theory Manual, Hibbitt, Karlsson and Sorenson, Inc.;1994.

[29] ABAQUS Users Manual, Hibbitt, Karlsson and Sorenson, Inc.; 1994.[30] Speicher G, Saal H. Numerical calculation of limit loads for shells of

revolution with particular regard to the applying equivalent initialimperfection. In: Julien JF, editor. Buckling of shell structures, onland, in the sea, and in the air. London: Elsevier Applied Science;1991. p. 466–75.

[31] Niu MCY. Composite airframe structures: practical design informa-tion and data. Hong Kong: Conmilit; 1992.

[32] Featherston CA, Ruiz C. Buckling of curved panels under combinedshear and bending. Proc Inst Mech Engr Part C J Mech Eng Sci Proc1998;212:183–96.


buckling optimized shear panels

Documents