use of the finite strip method in predicting the behaviour

Use of the finite strip method in predicting the behaviourof composite laminated structures

D.J. Dawe *

School of Engineering, The University of Birmingham, Edgbaston, Birmingham B15 2TT, UK

Abstract

A description is given of the use of the finite strip method (FSM) in determining the behaviour of composite laminated, prismatic

plate and shell structures, with emphasis placed on relatively recent work conducted at The University of Birmingham. Both the

semi-analytical and the spline variants of the method are described, and some attention is also paid to ‘‘exact’’ strips. Consideration

is given to analyses conducted in the contexts of first-order shear deformation theory and of classical, or thin, theory. The calcu-

lation of buckling stresses and natural frequencies of vibration is discussed in detail for single span structures and then, using the

spline finite strip approach with variable knot spacings, for multi-span structures and stepped structures. An account is given of the

use of the FSM in predicting the post-buckling response of plate structures to progressive end-shortening strain. Brief description is

given of the use of the method in predicting the thermal buckling and the transient response to dynamic loading of flat plates.

Finally, the calculation of buckling stresses and natural frequencies of sandwich plate structures is considered, based on the

adoption of a three-zone plate theory. Numerous examples of the application of the FSM are included in the paper.

� 2002 Published by Elsevier Science Ltd.

1. Introduction

Plate and shell structures occur as important struc-tural components in a number of branches of en-gineering, including mechanical, civil, marine andaeronautical engineering. Often such structures are ofrectangular planform and are assemblies of componentflat and/or curved plates which are rigidly connectedtogether at their longitudinal junction lines such that thestructure is basically prismatic. The finite strip method(FSM) is an efficient and accurate method for theanalysis of this type of structure, of which Fig. 1 showstypical examples. In this figure the structures are mod-elled by a number of finite strips which run the length ofthe structure. The individual strips have reference lines(at which degrees of freedom are located) at their ex-ternal longitudinal edges and may also have other lon-gitudinal reference lines in their interiors. It is noted thatthe FSM can be used in the analysis of structures ofother than rectangular planform, such as skew paral-lelogrammic structures or structures curved in plan, buthere attention is restricted to rectangular prismatic

structures (except that change of thickness and physi-cal properties is allowed along the structure). Themethod can be used in the analysis of various types ofbehavioural response but the concern in this paper iswith calculating natural frequencies and critical buck-ling stresses (or critical temperature increases) and withpredicting post-buckling behaviour and response todynamic loading.

The FSM can be regarded generally as a specialisa-tion of the ubiquitous finite element method (FEM) and,of course, both methods can be viewed as multi-fieldforms of the traditional single-field Rayleigh–Ritzmethod. The properties of a strip are based on the use ofan assumed displacement field in conjunction with po-tential energy or virtual work principles. The first pub-lished FSM paper along these lines, by Cheung [1],appeared in 1968 and concerned the linear static analysisof single rectangular plates having a pair of oppositeends simply supported. In the intervening years up tothe present time, very considerable development and useof the method has taken place. The aim in this paper isnot to attempt to describe and discuss the full range ofFSM work that has been conducted by many investi-gators over the years: the reader is referred to the earlytexts of Cheung [2] and Loo and Cusens [3], and tothe recent text of Cheung and Tham [4] for general

* Tel.: +44-121-414-5062; fax: +44-121-414-3675.

E-mail address: [email protected] (D.J. Dawe).

0263-8223/02/$ - see front matter � 2002 Published by Elsevier Science Ltd.

PII: S0263-8223 (02 )00059-4

Composite Structures 57 (2002) 11–36

www.elsevier.com/locate/compstruct

mail to: [email protected]

descriptions. Rather, the approach adopted is to con-centrate attention on those particular, important typesof behavioural response of composite laminated struc-tures which have already been mentioned and further,and rather parochially perhaps, to focus the descriptionon work conducted in recent years by the author andhis co-workers at The University of Birmingham, al-though references arising from other sources are notedthroughout the paper. In analysing the behaviour ofprismatic structures the FSM is very competitive withthe general FEM in terms of matters such as accuracy,speed of solution and ease of data preparation. Theadvantages of the FSM are particularly pronounced insolving the eigenvalue-type problems of predicting nat-ural frequencies and buckling stresses.

As well as the mainstream development of ap-proximate energy-based or work-based finite strip ap-proaches, it is noted that, for the prediction of naturalfrequencies and buckling stresses of flat-plate structures,what may be termed an exact FSM may be used in somecircumstances, wherein the strip properties are based onthe direct solution of the governing differential equa-tions of classical plate theory, rather than on the use ofenergy or work principles. It happens that it was also in1968 that pioneering works in this area were publishedby Wittrick [5,6] (at The University of Birmingham) andby Smith [7].

The emphasis here is on the analysis of compositelaminated structures which may have the complicationof general material properties, including anisotropy andcoupling between in-surface and out-of-surface actions.A further complication when considering the out-of-surface behaviour of component plates of compactcomposite construction, of other than very thin geo-metry, is that it is often necessary to take account ofthrough-thickness shearing effects. The classical, thintheory ignores these effects, of course, whilst shear de-formation theories include them and provide improvedrepresentations of structural behaviour. A number of

shear deformation theories exist in the literature, ofwhich the simplest is the well-known first-order theory(the Reissner–Mindlin theory for flat plates) and in whatfollows it is only this theory which is considered whenincluding through-thickness shear effects. (Shear defor-mation theories of higher order than first order can alsobe used in conjunction with the FSM, of course, buttheir use has been limited and they are not consideredhere. Unlike the first-order theory they have the ad-vantage of not requiring the prescription of shear cor-rection factors but they also have some disadvantagesrelated to their relative complexity.) Structures of thetype considered here may also include component plateswhich are of sandwich construction, in the popularconfiguration of two outer faceplates and a thicker in-terior core. Then the variation of displacement quanti-ties through the thickness becomes potentially morecomplicated and use of a three-zone theory becomesappropriate.

The FSM exists in a number of variants, of which thechief two may be referred to as the semi-analytical finitestrip method (S-a FSM) and the spline finite stripmethod (spline FSM). These are distinguished, one fromanother, by the nature of the variation of the displace-ment quantities along the length of the strip. In the S-aFSM use is made of a multi-term series of analyticalfunctions, i.e. trigonometric functions, beam eigenfunc-tions etc, whilst in the spline FSM use is made of seriesof polynomial spline functions, usually B-spline func-tions. The S-a FSM and the spline FSM are com-plementary procedures, with relative advantages anddisadvantages. In comparatively simple situations, suchas for orthotropic structures with diaphragm ends, theS-a FSM can be the more accurate procedure: indeed,where the true mode shape of buckling or vibration ispurely sinusoidal longitudinally, the S-a FSM analysiscan be based on the use of a single-term representationof the displacements over one longitudinal half-wave. Inmore general situations, however, the spline FSM ismuch the more versatile procedure in incorporatingansotropic material properties and different end condi-tions.

In Section 2 a quite detailed description is given of thedevelopment and use of the S-a FSM and the splineFSM in predicting the buckling stresses and naturalfrequencies of single-span composite laminated plateand shell structures. In Section 3 this description is ex-tended in the context of the spline FSM to embrace platestructures which may be multi-span or may have stepchanges of thickness or other physical property alongtheir lengths. The use of both variants of the FSM inpredicting the post-buckling response of plate structuresis described in Section 4. In Section 5 some consider-ation is given to predicting the effect of thermal loadingon the buckling behaviour of rectangular plates, usingthe spline FSM. Finite strip work of both FSM variants

Fig. 1. Typical prismatic structures modelled with finite strips.

12 D.J. Dawe / Composite Structures 57 (2002) 11–36

in the area of transient dynamic response is describedbriefly in Section 6. The last technical area considered, inSection 7, looks at the spline FSM analysis of thebuckling and free vibration of sandwich plate structures,and incorporates a refined model of through-thicknessbehaviour. Finally, concluding remarks are given inSection 8.

In what follows the notations SDST, TST, SDPT andCPT are used to denote first-order shear deformationshell theory, thin shell theory, first-order shear defor-mation plate theory and classical plate theory, respec-tively.

2. Buckling and vibration of single-span plate and shell

structures

2.1. Background

It is in the prediction of buckling stresses and naturalfrequencies of single-span prismatic structures that theFSM has found its greatest usage and for which thevolume of literature is largest. Here, although a con-siderable number of references in this sphere of work isnoted, the list of references is nevertheless by no meanscomplete. Indeed the attention is almost exclusively di-rected to those references dealing with the FSM analysisof the buckling and/or vibration of plate and shellstructures, as distinct from with single flat or curvedplates. The reader is referred to an earlier work of theauthor [8] for details of further FSM work (up to 1995)involving single plates.

Although the so-called exact FSM, mentioned inSection 1, is not the approach pursued in this papersome remarks on this approach are appropriate here.Wittrick [5,6] derived properties for an istropic, flatcomponent plate of a plate structure by solving explic-itly the governing differential equations (of plane stressand of CPT) for the situation in which the mode shapeof buckling or vibration varies purely sinusoidally in thelongitudinal direction. By formulating the analysis interms of complex quantities [6] each component platecan accommodate applied shear stress as well as biaxialdirect stresses. The assumption of sinusoidal variationresults in a convenient single-term type of analysis overone half-wavelength of a mode, but where nodal linesare skewed such assumption is strictly true only for astructure of infinite length. In the absence of appliedshear stress, and of anisotropic material behaviour, thesingle-term analysis is perfectly satisfactory for dia-phragm-supported structures of any length, since thenthe mode shapes are such that the nodal lines across astructure are straight and parallel to the ends. Relateddevelopments to those of Wittrick have been describedby Smith [7] (for orthotropic material and applied bi-axial direct stresses only) and by Viswanathan et al.

[9,10] whose analysis embraces assemblies of curved aswell as flat component plates and anisotropic material[10]. The exact approach of Wittrick [5,6] has been ex-tended to incorporate a certain level of anisotropicmaterial behaviour (but not for membrane propertiesand with no bending–stretching coupling) in the VI-PASA analysis capability of Wittrick and Williams [11].This single-term capability also incorporates a sophis-ticated solution procedure with the use of multi-levelsubstructuring [12], to determine buckling stresses andnatural frequencies of complicated plate structures,corresponding to specified values of the half-wavelength.The scope of this type of analysis has been extendedlater by the use of Lagrangian multipliers to incorporatea set of constraints, such as are associated with regularlyrepeating interior supports or with closely matching theend conditions, in the programs VICON [13,14] andVICONPT [15]. Collectively, the exact FSM approachesare sophisticated, accurate and powerful but inevitablythey lack something of the versatility associated with theapproximate energy-based or work-based FSM ap-proaches.

In the context of CPT, and of the use of the ap-proximate S-a FSM, the analysis of the buckling andvibration of flat-plate structures has been considered by,amongst others, Cheung and Cheung [16], Turvey andWittrick [17], Przemieniecki [18], Plank and Wittrick[19], Petyt [20], Graves-Smith and Sridharan [21],Sridharan [22,23] and Mahendran and Murray [24].Some of these analyses are of the single-term type [17–19,22–24], with the works of Refs. [19] and [24] allowingskewed nodal lines, whilst others are of the more ver-satile multi-term type [16,20,21]. Extensions to embracethin curved-plate finite strips for the analysis of shellstructures have been described by Dawe [25], Morris andDawe [26,27], Petyt and Fleischer [28] and Mohd andDawe [29].

In a series of papers [30–37] the author and his co-workers have described the buckling and vibrationanalysis of complicated, composite laminated flat-platestructures using the S-a FSM in the contexts both ofSDPT and of CPT. The single-term approach is con-sidered in Refs. [30–32] and this has included adoptionof the complex-quantity philosophy to account for ap-plied shear stress and anisotropic material when dealingwith ‘‘long’’ structures [31,32]. Multi-term approachesfor plate structures of finite length with diaphragm ends,and again accounting for applied shear stress and an-isotropy, are described in Refs. [33,34]. General de-scriptions of the powerful and efficient FSM capabilitiesare given in Refs. [35–37]. Two computer programsdeveloped for the single-term complex algebra analysisof long structures are designated BAVPAS (bucklingand vibration of plate assemblies using SDPT) andBAVPAC (as for BAVPAS but using CPT). Two pro-grams developed for the multi-term analysis of finite

D.J. Dawe / Composite Structures 57 (2002) 11–36 13

length, diaphragm-supported structures are designatedBAVAMPAS (buckling and vibration analysis of multi-term plate assemblies using SDPT) and BAVAMPAC(as for BAVAMPAS but using CPT). All these pro-grams incorporate the capability of using multi-levelsubstructuring procedures, including the use of super-strips, as part of a highly efficient solution procedure. Arelated multi-term analysis capability has been devel-oped in similar fashion for the analysis of shell struc-tures [38], using both thin and shear-deformable curvedfinite strips. Returning consideration to shear deform-able flat-plate structures with diaphragm ends, it isnoted that Hinton et al. have described the calculationof natural frequencies of isotropic structures [39,40] andLaughlan has considered the buckling of compositestiffened panels [41,42], both using the multi-term ap-proach.

Where the S-a FSM has been used in the multi-termanalysis of structures of finite length, the assumedstructure end conditions are almost invariably dia-phragm supports and the longitudinal series terms usedin representing the displacements are sine and cosinefractions. One exception to this is Ref. [16] wherein, inthe context of CPT, other end conditions are accom-modated through the use of Bernouilli–Euler beamfunctions in the longitudinal series. For the analysis ofshear deformable structures the use of Timoshenkobeam functions can be considered, but although suchfunctions have been used successfully in analysing singleplates [43,44], their use in practice for plate structures isproblematical. To provide increased versatility in ac-commodating, amongst other things, a range of endconditions, we turn attention to the spline FSM.

The spline FSM, incorporating the use of cubic B-splines longitudinally, was first introduced by Cheungand Fan [45] in a static context and then used by theseauthors in vibration analysis [46]. Lau and Hancock

have used a similar B-spline FSM (or B-s FSM) to studythe buckling of flat plate structures subjected to applieddirect and shear stresses [47,48]. In these approaches thematerial is homogeneous and the analysis is in thecontext of CPT. Wang and Dawe [49] first used B-splinefunctions in the context of SDPT in studying the vi-bration of anisotropic laminated plates and identified aproblem associated with the shear locking of thin plateswhose solution was described in Refs. [50,51]. The B-sFSM has since been developed by these authors for theanalysis of the buckling and vibration of complicatedprismatic flat plate structures [52,53] and curved-shellstructures [54,55]. Component plate (flat or curved)properties have been based on both classical, thin theoryand first-order shear deformation theory. The associatedsoftware package, PASSAS [54], incorporates advancedanalysis features such as multi-level substructuring andsuperstrips. A further B-s FSM approach is reported byLuo and Edlund [56] who considered the buckling ofisotropic flat-plate trapezoidally corrugated panels inthe context of CPT.

The spline FSM approaches referred to in the pre-ceeding paragraph all use equal spacing of spline knotsalong the structure. Unequal spacing provides greaterversatility in dealing with, for instance, multi-spanstructures and is considered in Section 3.

2.2. The curved plate finite strip

An individual curved plate finite strip, which is as-sumed to form part of a prismatic structure, is shown inFig. 2(a). The finite strip has length A, uniform middle-surface radius of curvature R, uniform thickness h andcurved breadth b at the middle surface. The local axes x,y and z are surface ones, i.e. are axial, circumferentialand normal ones. The corresponding translational dis-

Fig. 2. A curved finite strip: (a) geometry and displacements, and (b) applied stress system.


placements at the middle surface are u, v and w. InSDST analysis the quantities wx and wy shown in Fig.2(a) are independent rotations of the middle-surfacenormal along the x and y directions, respectively.

The finite strip may be subjected to an applied stresssystem, comprising r0

x , r0y and s0

xy as shown in Fig. 2(b),leading to buckling, or it may be undergoing harmonicmotion whilst vibrating in a natural mode with circularfrequency p, or both these influences may be present.Each of the applied stresses will here be taken to haveuniform distribution throughout the strip, as illustrated,but it is possible to accommodate non-uniform distri-butions.

All displacement quantities (i.e. u, v, w, wx and wy inthe context of SDST, or just u, v and w in the context ofTST) are in fact perturbation quantities representingchanges that occur at the instant of buckling followingthe application of the applied stress system at its criticallevel, or representing changes that occur during vibra-tion about a datum state which corresponds to someprescribed value (including zero value) of the stresssystem.

2.3. Basic SDST shell equations [38,54,55]

In first-order SDST it is assumed that the displace-ments at a general point, namely u, v and w, are ex-pressed in terms of displacements u, v, and w at thecorresponding point on the middle surface and the in-dependent rotations wx and wy , by the equations

�uuðx; y; zÞ ¼ uðx; yÞ þ zwxðx; yÞ�vvðx; y; zÞ ¼ vðx; yÞ þ zwyðx; yÞ

�wwðx; y; zÞ ¼ wðx; yÞ

ð1Þ

The linear expressions for the five significant straincomponents of the enhanced Koiter–Sanders SDST,which are used as the basis for the strain energy ex-pression, are

ex ¼ouox

þ zowx

ox; ey ¼

ovoy

þ wRþ z

owy

oy;

cxy ¼ouoy

þ ovox

þ zowy

ox

�þ owx

oyþ 1

2Rovox

�� ou

oy

��;

cyz ¼owoy

þ wy �vR; czx ¼

owox

þ wx

ð2Þ

Here ex and ey are in-surface direct strains, cxy is the in-surface engineering shear strain, and cyz and czx are thethrough-thickness shear strains.

For an arbitrary lay-up the constitutive equations fora laminate are

or F ¼ Le ð3Þ

Here Nx;Ny and Nxy are the membrane direct andshearing forces per unit length; Mx;My and Mxy are thebending and twisting moments per unit length; and Qx

and Qy are the through-thickness shear forces perunit length. The laminate stiffness coefficients are de-fined as

ðAij;Bij;DijÞ ¼Z h=2

�h=2Qijð1; z; z2Þdz i; j ¼ 1; 2; 6

Aij ¼ kikj

Z h=2

�h=2Qij dz i; j ¼ 4; 5

ð4Þ

where Qij for i; j ¼ 1, 2, 6 are in-surface reduced stiffnesscoefficients and Qij for i; j ¼ 4, 5 are through-thicknessshear stiffness coefficients. The kikj are the prescribedshear correction factors of the first-order theory.

The strain energy density (i.e. energy per unit area ofthe middle surface) of the curved finite strip can now beexpressed as

dU ¼ 12eTLe ð5Þ

and is such that only first derivatives of the five funda-mental displacement quantities occur in it: hence onlyC0-type continuity is required for these quantities. Thislinear strain energy will be a quadratic function of thedisplacement quantities, of course.

NxNyNxyMx

My

Mxy

Qy

Qx

8>>>>>>>>>><>>>>>>>>>>:

9>>>>>>>>>>=>>>>>>>>>>;

¼

A11

A12 A22

A16 A26 A66 SymmetricB11 B12 B16 D11

B12 B22 B26 D12 D22

B16 B26 B66 D16 D26 D66

0 0 0 0 0 0 A44

0 0 0 0 0 0 A45 A55

266666666664

377777777775

ou=oxov=oy þ w=Rou=oy þ ov=ox

owx=oxowy=oy

owx=oy þ owy=oxþ ov=ox� ou=oyð Þ=2Row=oy þ wy � v=R

ow=oxþ wx

8>>>>>>>>>><>>>>>>>>>>:

9>>>>>>>>>>=>>>>>>>>>>;


The potential energy density of the applied stressesr0x , r0

y and s0xy is

dVg ¼1

2h r0

x

ouox

� �2"

þ ovox

� �2

þ owox

� �2#

þ r0y

ouoy

� �2"

þ ovoy

�þ wR

�2

þ owoy

�� vR

�2#

þ 2s0xy

ouox

ouoy

�þ ovox

ovoy

�þ wR

�þ ow

oxowoy

�� vR

��

þ h2

12r0x

owx

ox

� �2"(

þowy

ox

� �2#

þ r0y

owx

oy

� �2"

þowy

oy

� �2#

þ 2s0xy

owx

oxowx

oy

�þowy

ox

owy

oy

�)!ð6Þ

The kinetic energy density of the finite strip when vi-brating with circular frequency p (and with the fun-damental displacement quantities then regarded asamplitudes of the motion) is

dT ¼ 1

2p2qh u2

�þ v2 þ w2 þ h2

12w2x

�þ w2

y

��ð7Þ

where q is the material density (which is assumed here tobe uniform).

2.4. Displacement fields for SDST finite strips

2.4.1. Multi-term analysis of finite length structuresIn analysing the buckling or vibrational behaviour of

actual, finite-length structures it is generally necessary toemploy a strip displacement field of the multi-term type,i.e. a field in which each of the displacement referencequantities is represented as a series of products of lon-gitudinal functions and crosswise, or circumferential,functions. With appropriate choice of the functions thiswill allow the analysis of structures which may have arange of different end conditions or may have aniso-tropic material properties or which may be subjected toapplied shear stress. Here description is given of thedisplacement fields for the multi-term S-a FSM and theB-s FSM. These displacement fields differ radically inthe longitudinal representation of the displacement

quantities but the crosswise representation is the same inboth approaches and is based on the use of standardshape functions of the type employed in finite elementanalysis.

In the S-a FSM analysis of structures of finite lengththe longitudinal representation of each of the five fun-damental quantities u, v, w, wy and wx is by a multi-termseries of analytical functions whose form is chosen so asto meet the prescribed boundary conditions at the endsof the finite strip as closely as possible. Thus the func-tions have to be changed when the end conditionschange. When considering single flat or curved plateswhich are homogeneous or are balanced laminates it isfeasible to employ Timoshenko beam functions in thelongitudinal series [43,44] but when considering plateor shell structures the use of these functions becomesdifficult for structures with general end conditions.For structures with diaphragm ends, however, the beamfunctions are, in fact, sine and cosine functions and itis very useful and practical to use the S-a FSM to pre-dict the behaviour of such structures. Then the as-sumed strip displacement field can be written in the form[33]

Here

Ci ¼ cos ipx=A; Si ¼ sin ipx=A; ð9Þthe row matrix / is defined as

/ðyÞ ¼ ½1 y y2 . . . yn� ð10Þand the A1 etc. are generalised displacement coefficientscorresponding to the ith terms of the longitudinal series.The definition of /ðyÞ allows for the generation of afamily of finite strip models with different degrees n ofcrosswise polynomial representation. The finite stripshown in Fig. 2(a), with four reference lines, corre-sponds to cubic interpolation (n ¼ 3).

The displacement field of Eq. (8) is written in terms ofgeneralised coefficients but can be transformed to readin terms of values of u, v, w, wy and wx at the referencelines. In doing this, Lagrangian interpolation is assumedsince the problem is one of C0-type continuity.

For the case of diaphragm ends the boundary con-ditions at x ¼ 0 and x ¼ A are that

v ¼ w ¼ wy ¼ Nx ¼ Mx ¼ 0 ð11Þ

and the chosen displacement field should satisfy explic-itly the kinematic conditions (v ¼ w ¼ wy ¼ 0) whilst

uvwwy

wx

8>>>><>>>>:

9>>>>=>>>>;

¼Xri¼1

Ci 0 0 0 00 Si 0 0 00 0 Si 0 00 0 0 Si 00 0 0 0 Ci

266664

377775

/ 0 0 0 0

0 / 0 0 0

0 0 / 0 0

0 0 0 / 0

0 0 0 0 /

266664

377775

A1

A2

::

A5nþ5

8>>>><>>>>:

9>>>>=>>>>;

i

ð8Þ


allowing implicit satisfaction, via the variational proce-dure, of the natural conditions (Nx ¼ Mx ¼ 0). Thefield of Eq. (8) does satisfy all these conditions for or-thotropic laminates but there is a difficulty with thenatural conditions when the material is anisotropic,leading to a degree of over-constraint, but the effect ofthis is generally small for structures of moderate an-isotropy.

In the B-s FSM the analytical longitudinal functionsof the S-a FSM are replaced with B-spline polynomialfunctions. The longitudinal B-spline functions can cor-respond to different orders of polynomial representationand use has been made of linear, quadratic, cubic,quartic and quintic B-splines which are designated as Bk-splines, with k ¼ 1–5. In employing the spline functionsthe length A is divided into q spline sections which arehere taken to be of equal length d as shown in Fig. 3(a).Corresponding to the q sections there are qþ 1 splineknots within the length A, plus other knots outside eachend of the length A which are required to complete thedefinition of a function and to prescribe appropriateboundary conditions. The number of these latter knotsvaries with the polynomial degree k but is two for k ¼ 3,to which Fig. 3 refers. Fig. 3(b) shows a local B3-splinefunction and Fig. 3(c) shows the combination of localfunctions which contribute to the complete variation ofeach of the displacement quantities, u etc., along thestrip. In algebraic terms we have

uðxÞ ¼ hdu; etc: ð12Þ

Here h is a row matrix which is called the modified B-spline function basis. It contains all the local B-splinefunctions, with some of these modified to take into ac-count any prescribed boundary conditions at the splineknots at x ¼ 0, A. The quantity du is a column matrixcontaining values of generalised knot coefficients cor-responding to u, and values of u (and perhaps ou=ox) atthe end knots.

A full description of one-dimensional spline func-tions, with equally spaced knots, is included in Ref. [50]wherein algebraic definitions of the local spline func-tions of different polynomial degrees (k ¼ 1–5) are given.A spline function of degree k has compact support overk þ 1 spline sections and is Ck�1 continuous.

The complete displacement field for a B-spline finitestrip is [52–55]

Here the quantity j denotes the number of a referenceline and n again is the order of polynomial representa-tion in the y-direction. The Nj ¼ NjðyÞ are Lagrangianshape functions. (Effectively the crosswise representa-tion is the same as in the S-a FSM.)

A particular and important point to note in Eq. (13)is that in the longitudinal direction wx is represented byB-splines of degree k � 1 whilst the other displacementquantities are represented by B-splines of degree k. Thiscorresponds to the so-called Bk;k�1-spline approach [50–55] and is introduced to avoid the detrimental effects ofshear-locking behaviour which would otherwise bepresent when using a common value of k.

The above displacement field of Eq. (13) is of generalapplicability with regard to the satisfaction of end con-ditions. Kinematic conditions can be applied directlyand explicitly whilst natural conditions are properly al-lowed to be approached indirectly as a result of thevariational procedure. Also, the same basic spline rep-resentation is used whatever the end conditions, withmodification made only local to structure ends whenaccommodating different end conditions. It follows thatthe B-s FSM is considerably more versatile than is the S-a FSM, but often it will be less accurate in the particularsituation of diaphragm ends.

Fig. 3. Spline representation: (a) spline sections and knots, (b) local

spline function, and (c) combination of local spline functions.

uvwwy

wx

8>>>><>>>>:

9>>>>=>>>>;

¼Xnþ1

j¼1

Nj 0 0 0 00 Nj 0 0 00 0 Nj 0 00 0 0 Nj 00 0 0 0 Nj

266664

377775

hk 0 0 0 0

0 hk 0 0 0

0 0 hk 0 00 0 0 hk 0

0 0 0 0 hk�1

266664

377775du

dv

dw

dwy

dwx

8>>>><>>>>:

9>>>>=>>>>;

j

ð13Þ


2.4.2. Single-term analysis of ‘‘long’’ structuresThere are situations in which the mode shape of

buckling or vibration of a prismatic structure can rea-sonably be assumed to be purely sinusoidal in the lon-gitudinal direction. Then it is only necessary to considerbehaviour over one prescribed half-wavelength using asingle term to represent the variation of each referencequantity in the longitudinal direction.

In circumstances where the ends of the structure aresupported by diaphragms, where the component platesare made of orthotropic material and where no shearstress is applied, the assumption of a sinusoidal mode inthe longitudinal direction is perfectly correct. Then thenodal lines of a buckling or vibrational mode arestraight and parallel to the ends of the structure and theanalysis simply becomes a special case of the multi-termS-a FSM of Section 2.4.1, with just the value i ¼ 1 usedin Eq. (8) and with ipx=A in Eq. (9) replaced by px=kwhere k is the prescribed half-wavelength of the mode.

In circumstances where one or more of the compo-nent plates is made of anisotropic material and/or whereapplied shear stress is present, the nodal lines are gen-erally curved and skewed across the structure. Withinthe restriction of a single-term approach this behaviourcan be accommodated if the displacement referencequantities are represented as complex quantities, in themanner first suggested by Wittrick [6]. When this is donethe conditions at the ends of the analysis half-wave-length do not equate to conditions that could apply atthe ends of a practical structure of finite length. Thus thesingle term, complex analysis is strictly correct only forstructures of infinite length. However, it will be ap-proximately correct in circumstances where the structureis much longer than is the half-wavelength of the modeof buckling or vibration, i.e. where the structure is‘‘long’’ and the mode is local in nature.

In the context of shear-deformation theory the com-plex displacement field is assumed to be [31,32]

Here j ¼ffiffiffiffiffiffiffi�1

p, n ¼ px=k and Re( ) denotes the real part

of the quantity inside the parentheses. As before, A1 etc.are generalised coefficients and /ðyÞ is defined byEq. (10), and so the crosswise representation of dis-placements is unchanged from that in the multi-termanalysis.

It is noted that the complex displacement field of Eq.(14) has been used in the finite strip analysis of flat plate

structures but not as yet in the analysis of curved shellstructures.

2.5. SDST strip models, substructuring and solution

The properties of a particular, individual finite stripcan be established by using the appropriate displace-ment field of Section 2.4 in conjunction with the energyexpressions of Section 2.3, and by integrating over theappropriate middle-surface area. The elastic stiffnessmatrix k, geometric stiffness matrix kg and consistentmass matrix m arise from the expressions for the den-sities of strain energy, Eq. (5), potential energy of ap-plied stresses, Eq. (6), and kinetic energy, Eq. (7),respectively. Full algebraic details of the formulation ofk, kg and m in the various finite strip approaches aregiven elsewhere and so only a few pertinent remarks willbe given here.

A family of finite strip models is available for each ofthe types of approach described in Sections 2.4.1 and2.4.2. The family is based upon the use of different de-grees, n, of crosswise polynomial (Lagrangian) interpo-lation. This order ranges from 1 to 5, corresponding tolinear to quintic interpolation.

In the multi-term approaches, for finite-length struc-tures, the evaluation of the strip matrices requires inte-gration to be carried out over the middle surface of astrip running the whole length of the structure, of course,i.e. for the limits 06 x6A and �b=26 y6 b=2. In thesingle-term approach the integration is carried out overone half-wavelength only, i.e. for 06 x6 k and �b=26 y6 b=2. In the longitudinal direction the integrationsare evaluated analytically in the S-a FSM approachesand numerically, but fully, in the B-s FSM. In thecrosswise direction the integration is performed numer-ically for all approaches, with full integration as standardbut with the option of reduced integration for the lower-order strip models corresponding to n ¼ 1 or 2 or 3.

It is noted that in the single-term approach of Sec-tion 2.4.2 the use of the complex displacement fieldof Eq. (14) results in k being complex Hermitian whenthe material properties are anisotropic and in kg be-ing complex Hermitian when applied shear stress ispresent.

In modelling a plate or shell structure any componentplate can be represented by one or more finite strips and

uvwwy

wx

8>>>><>>>>:

9>>>>=>>>>;

¼ Re ejf

j 0 0 0 00 1 0 0 00 0 1 0 00 0 0 1 00 0 0 0 j

266664

377775

0BBBB@

/ 0 0 0 00 / 0 0 0

0 0 / 0 0

0 0 0 / 0

0 0 0 0 /

266664

377775

A1

A2

::

A5nþ5

8>>>><>>>>:

9>>>>=>>>>;

1CCCCA ð14Þ


the structure matrices K, Kg and M could be assembledusing the normal direct stiffness procedure. However, itis much more efficient to make use of multi-level sub-structuring techniques, across the structure, to reducedrastically the number of effective degrees of freedomand the solution time.

Substructuring is first used at the level of each indi-vidual finite strip to eliminate the freedoms at all inter-nal reference lines. Then, at the level of a componentplate repetitive substructuring is used whilst creating anassembly of 2c identical strips (where c ¼ 0; 1; 2 . . .) torepresent the component plate by a process of ‘‘doublingup’’ [33]. The assembly of 2c strips is called a superstripof order c, or simply a SuperstripC. Typically we maytake c ¼ 5 so that each component plate is then mod-elled with one Superstrip5 which is an assembly of 32finite strips, but it is possible to use up to c ¼ 10 (1024strips) before beginning to meet numerical problems.Whatever the value of c, a superstrip ultimately has ef-fective freedoms located only at the outside edges of thecomponent plate. There is no loss of accuracy involvedin the substructuring procedure, i.e. the performance ofa Superstrip5 is precisely the same as that of an assemblyof 32 individual strips, and superstrips of high order canbe created without any great time penalty as comparedto using just one individual strip.

A prismatic structure is modelled as an assembly ofsuperstrips and this gives effectively an exact crosswisemodelling of the structure if cP 5, say. Before the as-sembly of superstrips a rotation transformation is ap-plied between local and global axes, and an eccentricitytransformation may be applied if deemed necessary (toaccount for off-set connections) [33]. Beyond the sup-erstrip level and following transformation to a globalconfiguration, higher levels of substructuring can beinvoked which involve progressive breakdown of thestructure into substructures which are assemblies of twoor more component plates [33,34]. The whole multi-levelsubstructuring technique makes it possible to solve ef-ficiently problems of very considerable complexity, withhundreds of thousands of freedoms.

Eigenvalues, i.e. buckling stress levels or naturalfrequencies of vibration, can be determined using anextended Sturm sequence-bisection approach to the non-linear eigenproblem, in the manner first proposed byWittrick and Williams [12]. Procedures have also beendeveloped to extract the eigenvectors and then to plotmode shapes using a three-dimensional graphics routine.

2.6. Reduction to TST Analysis

In thin shell analysis the Kirchhoff normalcy condi-tion is invoked and it follows that

wx ¼ � owox

; wy ¼vR� ow

oyð15Þ

This means, of course, that wx and wy are no longerindependent quantities and hence that u, v and w are theonly fundamental quantities of TST.

The basic equations of thin shell analysis are obtainedfrom those of the first-order shear deformation analysis,given in Section 2.3, by substituting for wx and wy , asdefined in Eq. (15), into Eqs. (1)–(3), by removing theexpressions for Qy and Qx from the constitutive rela-tionships of Eq. (3) and ignoring the second of Eq. (4),and by removing the wx and wy contributions from Eqs.(6) and (7). The SDST strip displacement fields of Eqs.(8), (13) and (14) are reduced for TST analysis by re-moving the expressions for wy and wx, of course. Also, inTST analysis there is a requirement for C1-type conti-nuity of w and hence the crosswise representation of whas to be by Hermitian rather than Lagrangian inter-polation.

In the context of TST, Fig. 2(a) can still represent anindividual finite strip of finite length. So far as crosswisemodelling is concerned, two strip models have beendeveloped for shell analysis. In a B-s FSM approach thecrosswise variations of u, v and w are each representedby cubic polynomial interpolation of such a type thatthe freedoms are located at the four reference linesshown in Fig. 2(a) [54,55]. In a S-a FSM approach thesecrosswise variations are taken to be quintic polynomials[38] and again the freedoms can be located at the fourreference lines of Fig. 2(a). It is noted that in the S-aFSM analysis of plate and shell structures in the contextof TST it is rather easier than it is in the context ofSDST to deal with end conditions that are other thanthose of diaphragm supports. In the TST approach theanalytical longitudinal functions can be the well-knownBernoulli–Euler beam functions which have been usedin buckling and vibration studies in reference [29].

In the complex, single-term S-a FSM for the analysisof ‘‘long’’ structures, only flat finite strips have beenconsidered in the context of classical theory and onlyone type of strip model has been generated [32]. In thismodel the crosswise representations of u, v and w are bycubic polynomial functions (Lagrangian for u and v, andHermitian for w). As with the corresponding SDSTapproach, k is complex Hermitian for anisotropic ma-terial and kg is complex Hermitian when shear stress isapplied.

Beyond the level of the individual finite strip, theprocedures used in the context of TST to obtain solu-tions to practical problems are broadly similar to thosedescribed for SDST analysis in Section 2.5.

2.7. Selected applications

The literature contains many examples of the appli-cation of the FSM to the prediction of buckling stressesand natural frequencies of prismatic structures. Appli-cations which are particularly pertinent to the analysis


described in Sections 2.2–2.6 are described in Refs. [25–27,29–32,34–38,52–55] and many of these involve theuse of the programs BAVPAS, BAVPAC, BAVAM-PAS, BAVAMPAC and PASSAS. Here, just three se-lected applications are described.

2.7.1. Buckling of a long plate structure under axial stressAs an example of the analysis of a ‘‘long’’ plate

structure we consider the orthotropic stiffened platewhose cross-section is shown in Fig. 4(a) and which hassimply supported longitudinal edges. Each componentplate is of the same thickness h ¼ 0:05B and is a fivelayer 0�/90�/0�/90�/0� laminate in which the thickness ofeach of the 0� plies is h=6 and that of each of the 90�plies is h=4. The material properties of all plies areidentical and are given by E1=E2 ¼ 30, G12=E2 ¼ 0:6,G13=E2 ¼ G23=E2 ¼ 0:5 and m12 ¼ 0:25.

The plate structure is subjected to uniform longitu-dinal compressive stress r0

x and calculations of bucklingstress (with BAVPAS) are made for many differentvalues of prescribed half-wavelength k using the single-term S-a FSM in the context of SDPT. In the crosswisedirection the modelling uses one quartic finite strip for

each component plate. For comparison purposes cal-culations have also been made using VIPASA in thecontext of CPT. A buckling factor K is defined as

K ¼ r0x

# $crB2h=p2D11 where D11 ¼ 0:0599629E1h3

The results for this application are presented graph-ically in Fig. 4(b) with k plotted on a logarithmic scale.It can be seen from Fig. 4(b) that, as expected, verysubstantial differences occur between the CPT andSDPT predictions of K at short wavelengths but that thedifferences become negligible at long wavelengths. Threekinds of buckling mode shape can be identified as kvaries. These are local, coupled and overall modes, andtypical mode shapes in these three categories are illus-trated in Fig. 4(c): the quoted values of Kc and Ks referto buckling factors K calculated through the CPT andSDPT approaches, respectively.

2.7.2. Buckling of a long shell panel under axial stressViswanathan and Tamekuni [9] have investigated the

buckling under uniform longitudinal compression of thecomplicated isotropic thin shell panel shown in Fig. 5(a),which has diaphragm ends. Their results are presented

Fig. 4. Buckling of stiffened plate: (a) geometry, (b) plot of buckling factor versus half-wavelength, and (c) typical buckling modes.


graphically as a plot of buckling stress versus half-wavelength of buckling, created by performing bucklingcalculations at 20 discrete values of half-wavelength andreproduced here in Fig. 5(b). The physical properties ofthe material are that E ¼ 71:02 GN/m2 and m ¼ 0:33.

FSM results for this problem could be generatedusing the single-term semi-analytical approach (withBAVPAC) but here the spline approach is used (withPASSAS in the context of TST). The B3-spline FSMresults are also shown in Fig. 5(b) and correspond to amodel in which one Superstrip5 represents each com-ponent plate and in which a series of different lengths ofthe structure is considered, with q ¼ 4 for each length. Itis seen from Fig. 5 that there is very close comparisonbetween the spline FSM results and those of Ref. [9].

2.7.3. Buckling of NASA example 6 panel under com-pressive and shear stress

The finite length corrugated panel under consider-ation here is referred to as the NASA Example 6 paneland the buckling of this panel was first considered byStroud et al. [57] as part of a study of seven panels. Fig.6(a) shows an overall view of the complete corrugatedpanel whilst Fig. 6(b) gives geometrical details of one ofthe six identical repeating elements. However, note thatthe intermediate supports indicated in Fig. 6(a) are to beignored completely for the present application in thissection. (These intermediate supports are brought intoconsideration in Section 3.2.)

The panel is simply supported all around its bound-ary and is of square planform with a side length ofA ¼ 762 mm. The panel material is a laminated graph-ite-epoxy composite with E1 ¼ 131 GN/m2, E2 ¼ 13GN/m2, G12 ¼ G13 ¼ G23 ¼ 6:41 GN/m2 and m12 ¼ 0:38.Component plate flats are symmetrically laminated,with thicknesses of 1.969 mm for plate flats 1, 3 and 5

(see Fig. 6(b) and 1.113 mm for plate flats 2 and 4).For flats 1, 3 and 5 the lay-up is ½þ45�=� 45�=� 45�=þ45�=0��s with ply thicknesses of 0.13917 mm except forthe 0� plies which have thicknesses of 0.42763 mm. Forflats 2 and 4 the lay-up is ½þ45�=� 45�=� 45�=45��s andall the ply thicknesses are 0.13917 mm. The laminatesare slightly anisotropic in bending.

The applied pre-buckling loadings defined by Stroudet al. [57] are combinations (six in total but only thecases of pure compression and pure shearing are con-sidered here) of longitudinal compressive force N 0

x andshearing force N 0

xy per unit width of panel. The appliedforces are distributed to give specific uniform stressstates in each of the component plates of the panel in themanner described in Ref. [57] (and also recorded in Ref.[34]). The most accurate procedure used by Stroud et al.to predict buckling is a FEM approach in which a veryfine mesh of 1728, four-node, rectangular, hybrid CPTelements is used.

The spline FSM has been used [52] in examining theconvergence with respect to q of solutions for the NASAExample 6 plate-structure problem in the contexts ofboth CPT and SDPT, though differences between thetwo approaches would be expected to be very small inview of the thinness of the component plate flats. Inmodelling the cross-section each plate flat of all six re-peating elements is represented by a cubic Superstrip5.Further substructuring is used to reduce the effectivedegrees of freedom to those at only seven reference lines,these being the outside edges and the junctions betweenthe six repeating elements of the cross-section. It bearsmentioning that clearly these various levels of sub-structuring reduce dramatically the number of effectivedegrees of freedom of the problem. For the case of theshear deformation theory analysis with q ¼ 8, the totalnumber of freedoms prior to any substructuring is in

Fig. 5. Buckling of NASA advanced structural panel: (a) cross-section, and (b) plot of buckling stress versus half-wavelength.


excess of 155,000, as compared to <350 after substruc-turing (with no loss of accuracy in the calculated ei-genvalues).

Numerical results for the predicted critical values offorce per unit length are presented in Table 1. In thistable the results generated by the equal-spline FSM (E-sFSM) described in this section are given in results col-umns (1), (2), (4), and (5). (The results given in columns(3) and (6) refer to the use of a general-spline FSMwhich will be discussed in Section 3.) It can be seen thatconvergence of the E-s FSM results in this difficultproblem is orderly and appears very satisfactory forboth the CPT and SDPT analyses. Very close agreementbetween the E-s FSM results, a set of S-a FSM results

(for r ¼ 7) and the FEM results is observed. There islittle difference in this application between forecastsbased on the classical theory and on the shear defor-mation theory.

3. Multi-span and stepped plate structures

3.1. Background

The presentation of the spline FSM in Section 2 hasbeen based on a B-spline representation in which thespline knots are equally spaced longitudinally. It is clearthat the E-s FSM is versatile (more so than is the S-a

Fig. 6. The NASA Example 6 panel: (a) full geometry and loading, and (b) details of a repeating element. (Note that the indicated intermediate

supports are not present for the example of Section 2.7.)

Table 1

Compressive and shearing buckling loads of the original NASA Example 6 panel

Solution

method

ðN 0x Þcr (kN/m) ðN 0

xyÞcr (kN/m)

E-s FSM SDPT (1) E-s FSM CPT (2) G-s FSM CPT (3) E-s FSM SDPT (4) E-s FSM CPT (5) G-s FSM CPT (6)

Spline FSM

q ¼ 2 264.57 264.66 276.66 240.16 242.15 255.26

q ¼ 3 261.28 261.36 – 226.64 228.79 –

q ¼ 4 260.77 260.87 268.44 219.25 221.47 235.73

q ¼ 5 260.68 260.77 – 216.55 218.81 –

q ¼ 6 260.66 260.75 264.30 215.69 217.97 228.09

q ¼ 7 260.65 260.75 – 215.39 217.67 –

q ¼ 8 260.65 260.75 260.76 215.25 217.55 218.54

q ¼ 10 – – 260.74 – – 217.51

S-a FSM

r ¼ 7 260.65 260.75 260.75 215.99 218.23 218.23

FEM [57] – 261.26 261.26 – 218.56 218.56


FSM) but this versatility can be further increased if therestriction to equally spaced knots is removed. Thespecialist texts on spline usage by de Boor [58] andBartels et al. [59] describe spline representation usingnon-uniform spacing of the spline knots. Here thephraseology ‘‘general spline’’ is used to denote suchspline representation and the notation G-s FSM denotesthe FSM when general spline functions are used, andeffectively incorporates within it the E-s FSM. In thearea of structural analysis, non-uniform knot spacingcan clearly have its uses in dealing with practical situa-tions involving intermediate supports at arbitrary loca-tions, step changes in properties, localised areas of stressconcentration, etc.

In doctoral theses, Li [60] first presented the G-s FSMin the context of CPT whilst Wang [61] used generalsplines in a Rayleigh–Ritz analysis in the context ofSDPT of the buckling and vibration of single plates.Gutkowski et al. [62] studied the static CPT bendinganalysis of single plates using the G-s FSM, with cubicsplines. Madasamy and Kalyanaraman [63] furtherconsidered the CPT bending problem using the cubicG-s FSM, but also studied in-plane behaviour in theanalysis of shear walls.

The G-s FSM has been developed recently in generalterms by Tan and Dawe [64,65] for the analysis of thebuckling and vibration of composite laminated plateand shell structures, in the contexts of both TST andSDST. The procedure is much the same as described inSection 2 for the E-s FSM except, of course, that thedefinitions of the local spline functions change to ac-commodate arbitrary knot spacings. These definitionsare given in Refs. [64,65] and hence are not repeatedhere. It is noted that for splines of degree k the functionsare still Ck�1 continuous throughout their range.

A specific study [66] has been made of the use of theG-s FSM in predicting the buckling stresses and naturalfrequencies of single plates which have step changes ofproperties (primarily of thickness) along their length.The presence of a step change has consequences on theappropriate level of continuity of displacement quanti-ties at a step change. For example, in a TST bendingproblem it is continuity of bending moment and shearforce that should apply at the step, rather than of secondand third derivatives of w, i.e. at the step w shouldstrictly be only C1 continuous, and it follows that w willbe over-continuous for splines of degree k > 2. Whilst itwould be possible to introduce the concept of knotmultiplicity at a breakpoint (step change) [58,59], forpractical reasons the philosophy adopted in Ref. [66] isto maintain the usual level of continuity of the splinefunctions but to refine the knot spacings local to a stepchange. This is not a rigorous solution but is a philos-ophy that is soundly based theoretically [58,59] and hasbeen shown numerically to give accurate results whenusing tight refinement local to a step change [66].

3.2. Applications

3.2.1. Vibration of square plates having a thinned zone[66]

The square plates considered here are of side length Aand have simply supported edges. The basic thickness ish but a thinner square zone is present in the interior, asshown in Fig. 7, of side length A=5 and thickness h=2.Two types of construction are considered. In the firsttype the plate is made of homogenous, isotropic material(m ¼ 0:3). In the second type the plate is an anisotropicangle-ply laminate with equal-thickness plies for each ofwhich the material properties are defined by E1=E2 ¼ 14,G12=E2 ¼ 0:533 and m12 ¼ 0:323. In the main the lami-nate is of four-layer ½þ45�=� 45��s lay up but in thethinner zone the outer two layers are removed. Theplates are thin, with h=A ¼ 0:004, and analysis is madein the context of CPT.

In applying the G-s FSM, the finite strips run in thex-direction with three cubic superstrips with (n ¼ 3,k ¼ 3) used in the model, namely a Superstrip3 in theregion containing the thinned zone and a Superstrip4 ineach of the other two regions. For the isotropic plate,comparative solutions have been generated using theLUSAS FEM commercial program, with a fine mesh of30 30 QSL8 elements. The G-s FSM and LUSASFEM results are recorded in Table 2 for the first fournatural frequencies. In applying the G-s FSM, differentnumbers of spline sections and different knot locationsare used: the knot locations are recorded beneath Table2 and are such that tight knot refinement is used local tothe two positions of step change in thickness. The pre-sented G-s FSM results show that the manner of con-vergence with increasing q is good and that there is very

Fig. 7. A square plate with a thinned zone.


close comparison with the FEM results for the isotropicplate.

3.2.2. Buckling and vibration of NASA Example 6 paneland modifications

Here attention is returned to the corrugated NASAExample 6 panel considered earlier in Section 2.7. Theoriginal panel, as defined in Section 2.7, is consideredfirst: it is recalled that results for this panel as generatedusing the E-s FSM for pure compression and pure shearloadings in the contexts of both CPT and SDPT, arerecorded in columns (1), (2), (4) and (5) of Table 1. Nowthe G-s FSM is used, in the context of CPT, to solve thesame two problems with variable knot spacings suchthat a knot refinement is made local to the longitudinalposition x ¼ 0:782A. This would not be expected to haveany beneficial effect as compared to using the E-s FSMwhen dealing with the original panel, as here, but it ispertinent to a situation considered in what follows inwhich a step change of thickness will be introduced atx ¼ 0:782A where A ¼ 762 mm (30 in.). The results ofthe application of the G-s FSM are recorded in columns(3) and (6) of Table 1 for q ¼ 2, 4, 6, 8 and 10, where theknot spacings are defined by

q ¼ 2: ð0; 23:46; 30ÞA=30

q ¼ 4: ð0; 22; 23:46; 25; 30ÞA=30

q ¼ 6: ð0; 10; 22; 23:46; 25; 27:5; 30ÞA=30

q ¼ 8: ð0; 8; 14; 19; 22; 23:46; 25; 27:5; 30ÞA=30

q ¼ 10: ð0; 4; 8; 12; 16; 19; 22; 23:46; 25; 27:5; 30ÞA=30

The manner of convergence of the G-s FSM results isgood, but is a little slower than that of the E-s FSM, as

anticipated in this application where there is no specificneed for, or gain expected from, a refinement of the knotspacings in any region.

Now consideration is given to the analysis of thepanel but with the intermediate supports shown in Fig.6(a) being applied to the outer two top-surface flats ateach side of the panel, and at the arbitrary locationx ¼ 431:8 mm. The crosswise FSM modelling of thepanel uses a Superstrip4 for each of component plates 1and 5, and a Superstrip5 for each of component plates 2,3 and 4. In the longitudinal direction the same number,q, of equal spline sections is used in each of the twospans defined by 0 < x < 431:8 mm and 431:8 < x < 762mm. Calculated values of the first six natural frequencies(with q ¼ 1600 kg/m3) and of the buckling loads in purecompression and pure shear are presented in Table 3.Rapid and orderly convergence of the FSM results, withincrease in q, is evidenced and there is little differencebetween predictions based on q ¼ 2 and 5. This reflectsthe fact that the mode shapes of vibration and buckling,as depicted in Fig. 8, have relatively simple shapes lon-gitudinally.

Finally, attention is returned once more to the orig-inal NASA Example 6 panel with no intermediate sup-ports but now with the introduction of a step change inthickness at x ¼ 0:782A. The step change is achieved byremoving all plies except the two central 0� plies fromcomponent plates 1, 3 and 5 of all six repeating elementsof the panel cross-section, leaving these componentplates with thickness 0.85526 mm over the region 0 <x < 0:782A. Numerical results obtained using the G-sFSM for the first six natural frequencies are recorded inTable 4. In obtaining these results the crosswise FSMmodelling remains as described in the previous para-

Table 2

Frequencies of two square plates with a thinned zone: values of frequency factor X for the first four modes. X ¼ p A2

h

ffiffiffiffiqE2

qSolution method Isotropic plate Angle-ply laminate

Mode 1 Mode 2 Mode 3 Mode 4 Mode 1 Mode 2 Mode 3 Mode 4

G-s FSM

q ¼ 3 5.889 14.78 16.18 24.72 14.09 30.07 38.89 51.77

q ¼ 7 5.778 14.69 15.19 24.18 13.83 29.45 36.72 50.45

q ¼ 12 5.766 14.35 14.66 23.49 13.72 28.61 35.48 48.13

q ¼ 14 5.744 14.34 14.66 23.49 13.67 28.60 35.45 48.10

LUSAS FEM 5.768 14.34 14.66 23.50 – – – –

Knot positions in the x-direction are as follows: q ¼ 3: ð0; 80; 160; 400ÞA=400, q ¼ 7: ð0; 75; 80; 85; 155; 160; 165; 400ÞA=400, q ¼ 12: ð0; 40; 75; 80; 85;

120; 155; 160; 165; 220; 280; 340; 400ÞA=400, q ¼ 14: ð0; 40; 75; 80; 85; 120; 155; 160; 165; 200; 240; 280; 320; 360; 400ÞA=400.

Table 3

Frequencies and buckling loads of modified two-span NASA Example 6 panels

q per span Natural freqencies (Hz) Buckling loads (kN/m)

Mode 1 Mode 2 Mode 3 Mode 4 Mode 5 Mode 6 ðN 0x Þcr ðN 0

xyÞcr2 209.59 266.65 368.06 492.91 532.40 579.96 294.94 237.53

3 209.40 266.01 366.54 491.55 529.71 576.59 294.32 232.61

4 209.31 265.73 365.85 490.92 528.58 575.45 294.06 231.84

5 209.26 265.58 365.53 490.62 528.12 575.03 293.89 231.61


graph whilst in the longitudinal direction the chosenknot positions are specified at the foot of the table. Theresults demonstrate rapid convergence of frequencyvalues with increase in q.

4. Post-buckling of plate structures

4.1. Background

As well as in linear analysis of the sort discussed thusfar, the FSM can be very usefully employed in the non-

linear analysis of prismatic structures. In this section theconcern is with geometrically non-linear behaviour ofthe large deflection and post-buckling kinds, with em-phasis on the latter.

In early geometrically non-linear analyses Graves-Smith and co-workers [67–69], Hancock [70], Bradfordand Hancock [71] and Cusens and Lengyel [72] haveused the S-a FSM in the context of CPT to predict theresponse of homogeneous single plates and prismaticplate structures, usually for post-buckling behavi-our. Dawe et al. [73–79] have used the S-a FSM inconsidering the post-buckling behaviour of composite

Fig. 8. Mode shapes for the two-span NASA Example 6 panel: (a)–(f) vibrational mode shapes 1–6, respectively, (g) compressive buckling mode

shape, and (h) shear buckling mode shape.


laminated structures in the contexts of both CPT andthe more refined SDPT. These analyses relate to singleplates [73–77], to the post-local-buckling of plate struc-tures [78], and to the post-overall-buckling of dia-phragm-supported plate structures [79].

The equal-spline FSM has been used in the elasticpost-buckling analysis of shells by Zhu and Cheung[80]. The geometrically non-linear analysis of flat-platestructures using the spline FSM has been developed byHancock et al. [48] and Kwon and Hancock [81]. Theseapproaches use cubic B3 splines and are limited to ho-mogeneous material and thin geometry. The B-s FSManalysis of the post-buckling behaviour of single com-posite laminated rectangular plates has been describedby Dawe and Wang in the context of CPT [82] and ofSDPT [83]. The same authors have extended these worksto embrace the analysis of complicated prismatic platestructures of composite laminated construction (in-cluding also shell panels when treated as faceted struc-tures) again in the contexts of both plate theories [84].

4.2. Some theoretical details

The geometric non-linearity is introduced in thecontext of a total Lagrangian approach by way of avon Karman-type modification to the linear strain–displacement equations recorded earlier in Eq. (2).Restricting attention to flat geometry with no imper-fections, and in the context of SDPT, the in-plane non-linear strains are assumed to be given by [79,84]

ex ¼ouox

þ zowx

oxþ 1

2

owox

� �2

þ 1

2

ovox

� �2

ey ¼ovoy

þ zowy

oyþ 1

2

owoy

� �2

þ 1

2

ovoy

� �2

cxy ¼ouoy

þ ovox

þ zowx

oy

�þowy

ox

�þ ow

oxowoy

þ ovox

ovoy

ð16Þ

whilst the expressions for the through-thickness shearstrains cyz and czx remain linear, as recorded in Eq. (2). Itis noted that non-linear contributions in the crosswisedisplacement v are present in the expressions for ex, eyand cxy so as to allow consideration of plate structureproblems that exhibit post-overall-buckling response as

well as those that exhibit post-local-buckling response.(On the other hand, non-linear terms in the longitudinaldisplacement u are quite insignificant and are omitted.)As a consequence of the enhancement of the strain–displacement equations, as compared to the linearequations, the strain energy density dU can be expressedas a summation of quadratic (dU2), cubic (dU3) andquartic (dU4) functions of the displacement-type quan-tities and their derivatives with respect to x and y. Thus

dU ¼ dU2 þ dU3 þ dU4 ð17Þ

The strip displacement field in non-linear analysisbroadly has the form of Eq. (8) for the S-a FSM or ofEq. (13) for the spline FSM, but possibly with someadditions up and vp to the expressions for u and v, re-spectively, depending upon the type of problem beingconsidered. As a particular case in point, consider aprismatic plate structure which is subjected to a pro-gressive uniform end-shortening strain e, applied to thecomplete end cross-sections or applied in a particularhorizontal plane, such as in the plane of the main plateof a stiffened structure. The additions up and vp areapplied only to any finite strip to which the shorteningstrain e is applied, and have the form

up ¼ e A=2ð � xÞ; vp ¼ eby ð18Þ

This represents explicitly a progressive uniform endshortening of a strip, with the up term, and a possiblePoisson expansion, through the vp term, in a simple pre-buckled state. If the strip ends are allowed free in-planelateral expansion, in what is referred to as a type Aproblem [74], then b ¼ m, the Poisson’s ratio, for iso-tropic material, or b ¼ A12=A22 for balanced orthotropicmaterial. However, the presence of the vp term is notessential or even particularly advantageous for the ma-jority of type A problems involving structures (as dis-tinct from single plates). If the strip ends are completelyprevented from expanding laterally, in what is referredto as a type B problem [74], then b ¼ 0 anyway.

Detailed description of the nature of the non-linearanalysis is available elsewhere [74–79], [82–84] and is notrepeated here. Ultimately a set of structure equationsis obtained of the form

Table 4

Frequencies of modified NASA Example 6 panel with thickness change

q Natural frequencies (Hz)

Mode 1 Mode 2 Mode 3 Mode 4 Mode 5 Mode 6

2 226.95 244.11 273.90 317.58 354.36 358.40

4 221.89 238.69 267.82 310.34 343.45 347.53

6 218.36 234.94 263.66 305.53 337.07 341.00

8 218.28 234.86 263.58 305.42 336.84 340.76

10 218.22 234.80 263.51 305.35 336.72 340.64

Knot positions in the x-direction are as follows: q ¼ 2: ð0; 23:46; 30ÞA=30, q ¼ 4: ð0; 22:5; 23:46; 24:5; 30ÞA=30, q ¼ 6: ð0; 10; 19; 22:5; 23:46; 24:5;

30ÞA=30, q ¼ 8: ð0; 8; 16; 19; 22:5; 23:46; 24:5; 27; 30ÞA=30, q ¼ 10: ð0; 4; 8; 12; 16; 19; 22:5; 23:46; 24:5; 27; 30ÞA=30.


�eVþ K

�� eK

þ 1

2K1 þ

1

3K2

�d ¼ 0 ð19Þ

where V is a column matrix of constants whilst K, K, K1

and K2 are square symmetric stiffness matrices whoseindividual entries are constants for K and K

, and are

linear and quadratic functions of the freedoms for K1

and K2, respectively. The K and K

are associated withthe quadratic strain energy density dU2 and K1 and K2

are associated with the cubic and quartic energy densi-ties dU3 and dU4, respectively. The equations are solvedfor the structure freedoms �dd using the Newton–Raphsoniterative method in which use is made of the tangentmatrix KT, defined as

KT ¼ K� eK þ K1 þ K2 ð20Þ

4.3. Applications

4.3.1. Six-blade, flat stiffened panelThe panel considered here has the cross-section

shown in Fig. 9(a) and its length is A ¼ 762 mm so thatthe panel is square in plan. It is one of the seven panelswhose buckling was considered by Stroud et al. [57]: it isreferred to as the NASA Example 4 panel. It happensthat the panel is made of aluminium, with E ¼ 72:4 GN/m2 and m ¼ 0:32, rather than being of laminated con-struction. The analysis is conducted in Ref. [57] and herein the context of CPT.

In their bifurcational buckling analysis Stroud et al.assumed that the panel ends are diaphragm supported(with v ¼ w ¼ 0) and the longitudinal edges are simplysupported (w ¼ 0). Of the loading cases considered [57]the one of concern here is that of pure compressionbased on the assumption of uniform longitudinal pre-buckling strain. The buckling of this panel has also beenconsidered by Peshkam and Dawe [34] using the S-aFSM. For the pure compression case the bifurcationalbuckling load is calculated as 39.47 kN using the S-aFSM, whereas the calculated FEM value from Ref. [57]is just 0.24% higher. The corresponding buckled modeshape is a local one, with six longitudinal half-waves.

The finite strip analysis of the post-buckling behav-iour of the NASA Example 4 panel under progressiveend shortening has been conducted in two ways, usingthe S-a-FSM in a simplified local analysis [78] and laterusing the spline FSM in a general analysis [84]. In theearlier S-a FSM approach, assumptions are made thatallow the analysis to be conducted over only a specifiedhalf-wavelength of buckling, rather than over the wholepanel length. Since it was found that the bifurcationalbuckling load of the panel corresponding to seven lon-gitudinal half-waves is close to that corresponding to sixhalf-waves, two separate non-linear analyses have beenperformed for these two distinct numbers of half-waves

[78]. In the later E-s FSM approach the full length of thepanel is considered, with no pre-conception of any likelylongitudinal form in the buckling or post-buckling re-sponse [84]. The panel is assumed to be subjected toprogressive uniform end shortening over the wholecross-section, and the prescribed end conditions, interms of local displacements, are that at x ¼ 0, A, wehave u ¼ �Ae=2 and w ¼ 0, v 6¼ 0, that is the componentplates are simply supported for out-of-plane behaviourand free to expand in their planes. The longitudinaledges of the main plate are simply supported, that isw ¼ 0, and are free to move in plane. Both the S-a-FSMand the spline FSM models of the panel take note of thesymmetry of the structure and the response about thelongitudinal centre line and use nine strips across halfthe panel width, with one strip for each of the threestiffeners and six strips of equal width for half of themain plate (with u and v interpolated quadratically andw cubically across a strip). For the spline FSM model(with k ¼ 3), 16 spline sections are used along the panellength.

The general response of the panel as predicted by thelocal S-a FSM (with the two prescribed values of half-wavelength) and by the spline FSM is shown in Fig. 9(b)in the form of plots of Nav (the average longitudinalforce) versus e (the end-shortening strain). Close com-parison between the two approaches is revealed. Thelocal S-a FSM approach shows that buckling and initialpost-buckling corresponds to six longitudinal half-wavesbut that at a fairly early stage of post-buckling there is achange to seven half-waves. The spline FSM approachshows no bifurcational buckling as such, but rather acontinuous progressive change throughout the defor-mation process, although the knee of the curve in Fig.9(b) is quite sharp and occurs at a force which is close tothe bifurcational buckling force. The progressive buildup of deformation in a symmetric half of the panel, aspredicted by the spline FSM, is shown for six particularvalues of shortening strain in Fig. 9(c). Initially there issome small deformation of an overall type, followed bylocal deformation with six longitudinal half-waves andthen with seven half-waves at higher values of shorten-ing strain.

4.3.2. Curved panel with edge stiffenersSome details of the physical testing of a composite

laminated curved panel considered here have been sup-plied by DERA Farnborough by private communica-tion. The cross-section of the edge-stiffened panel isshown in Fig. 10(a) and its length is 460 mm. The plythickness is 0.125 mm and the 32-ply lay up isof ½45�=� 45�=0�=90�=� 45�=45�=0�=0�=45�=� 45�=0�=0�=� 45�=45�=90�=90��s balanced construction. The plymaterial properties are E1 ¼ 135 GN/m2, E2 ¼ 9:2 GN/m2, G12 ¼ G13 ¼ G23 ¼ 5:4 GN/m2 and m12 ¼ 0:28. In thetest the panel was subjected to a progressive uniform


end shortening up to beyond the buckling level. Thelongitudinal edges were completely free and the curvedends were notionally fully clamped. The only test in-formation available is that the buckling load was esti-

mated at 880 kN and that the post-buckled shape wasphotographed as shown in Fig. 10(b).

The B-s FSM, in the context of shear deformationtheory, has been applied first to predicting the bifurca-

Fig. 9. The six-blade stiffened panel: (a) panel cross-section, (b) plot of average longitudinal force versus applied shortening strain, and (c) deformed

shapes of one half of the panel at six applied strain levels.

Fig. 10. The curved panel with edge stiffeners: (a) panel cross-section, (b) experimental deformed shape, (c) plot of average longitudinal force versus

applied shortening strain, and (d) calculated deformed shapes on pp, sp1 and sp2.


tional buckling of the panel, under uniform longitudinalcompressive stress, using PASSAS with 80 finite stripsand q ¼ 16. For fully clamped ends the calculated lowestbuckling load is 971 kN and it is found that numerousother buckling modes occur in close proximity to thisload. Changing the end conditions to diaphragm sup-ports reduces the lowest predicted buckling load to 795kN and thus the test value of load lies about midwaybetween the FSM predictions based on clamped endsand on diaphragm ends. When considering non-linearresponse to progressive uniform end shortening theFSM model is a faceted one of 12 flat strips (eight in thecurved portion), with q ¼ 6. The predicted general re-sponse of the panel (Nav versus e) is shown in Fig. 10(c)and reveals the stable parts of a primary path (pp) andtwo secondary paths (sp1 and sp2). The deformedshapes of the panel in these paths are shown in Fig.10(d) and it can be seen that the shape for the predictedlowest equilibrium path (sp2) is very similar to thatshown for the test panel in Fig. 10(b).

5. Thermal loading effects

Thermal loading is sometimes an important feature inanalysing the behaviour of structures, particularly in therealm of aerospace, whether acting on its own or inconjunction with applied mechanical loading. The FSMprocedures described above clearly can be extended toinclude thermal effects but developments in this areaseem to have been limited thus far to thermal andthermo-mechanical buckling and post-buckling of singlerectangular flat plates.

The thermal buckling of shear deformable compositelaminated plates by the E-s FSM is described recently byDawe and Ge [85] wherein the analysis takes place intwo distinct phases. In the first phase an in-plane ther-mal stress analysis is conducted for the pre-bucklingstage in which the plate is assumed to remain flat underthe action of a non-uniform temperature field (althoughthe temperature does not vary through the thickness). Theconstitutive equations for in-plane behaviour are mod-ified to incorporate a temperature increase T ðx; yÞ andare:

NxNyNxy

8><>:

9>=>; ¼

A11 A12 A16

A12 A22 A26

A16 A26 A66

264

375

ou=ox

ov=oy

ou=oy þ ov=ox

8><>:

9>=>;

0B@

� T

axay

2axy

8><>:

9>=>;1CA ð21Þ

where ax, ay and axy are thermal expansion coefficients.The pre-buckling u and v displacements are representedin the same way as are u and v in Eq. (13). The solution

of the plane stress problem yields the distributions of thepre-buckling stresses r0

x , r0y , s0

xy in terms of a datum valueof temperature, throughout the finite strips making upthe plate. In the second phase this stress distribution istaken forward to an eigenvalue buckling analysis inmuch the same way as described in Section 2, using thedisplacement field of Eq. (13) (where now the displace-ments are perturbations) and a somewhat less generalform of the constitutive equations recorded in Eq. (3)(with the B12 coefficients absent). It is noted that thesuperstrip approach cannot be used since, in general, thepre-buckling stress distribution is non-uniform.

The analysis of the buckling of single rectangularlaminates under the action of combined thermal andmechanical loading, i.e. thermo-mechanical loading, hasalso been considered using the spline FSM in the contextof SDPT [86,87]. Furthermore the extension to post-buckling behaviour in the presence of temperaturechange is described by Ge [86].

As one example of thermal buckling we consider thebuckling of symmetric five-layer þh=� h=þ h=� h=þ hangle-ply square laminates when subjected to uniformtemperature rise. This problem was considered by Pra-bhu and Dhanaraj [88] who generated results in graph-ical form based on a FEM approach in the contextof SDPT, using a 6 6 mesh of nine-node Lagran-gian isoparametric elements. The material properties arespecified as

E1=E2 ¼ 40; G12=E2 ¼ G13=E2 ¼ 0:6; G23=E2 ¼ 0:5;

m12 ¼ 0:25; a1=a0 ¼ 0:02;

a2=a0 ¼ 22:5; E2 ¼ 105; a0 ¼ 10�6:

For a laminate with S2 boundary conditions on allsides (i.e. v ¼ w ¼ wy ¼ 0 at edges x ¼ 0, A for instance)the pre-buckling stress distribution is highly non-uni-form: Fig. 11 shows this in the form of contour plots ofN 0x and N 0

xy when using the spline FSM (with n ¼ 3,k ¼ 3) for a thin square plate with A ¼ B ¼ 200,A=h ¼ 100, h ¼ 45�. (The plot for N 0

y is virtually iden-tical to that for N 0

x ).Numerical results for the B-s FSM calculation of

critical temperature rise Tcr are recorded in Table 5wherein NS is the number of finite strips used. Nocomparative numerical solution is available but themanner of convergence of the spline FSM results ap-pears to be very satisfactory despite the complexity ofthe pre-buckling stress distribution. Graphical compar-ison of the spline FSM results with the FEM results ofRef. [88] is made in Fig. 12 for the full range of fibreangles, h ¼ 0�–90�, and for both thin (A=h ¼ 100) andthick (A=h) geometries, and two sets of boundary con-ditions, namely S2 and C1 (fully clamped). It is seen thatthere is generally close comparison between the predic-tions of the two approaches, except perhaps for theS2-supported thick laminate with h ¼ 0� and h ¼ 90�.


6. Dynamic response

Another type of structural response which can be ofimportance for composite laminated structures is tran-sient response to dynamic loading. Again, only a verylimited body of work is available to date in this subjectarea when considering the application of the FSM. Forhomogeneous isotropic material, work has been con-

ducted on the use of the S-a FSM in predicting the dy-namic response of stiffened plate structures to pressureloading, by Khalil et al. [89] and by Houlston [90]. Non-linear behaviour is included in these studies which areconducted in the context of CPT. In considering singlerectangular laminates Chen and Dawe [91] have con-sidered the linear transient response to normal loadingthrough the use of the S-a FSM in the context of SDPTand in conjunction with a modal superposition proce-dure. Later, the spline FSM has been used in a similarway [92]. These approaches are limited to small-deflec-tion behaviour but more recently the S-a FSM approachhas been upgraded to the realm of geometrically non-linear analysis [93], with the non-linearity being intro-duced in the strain–displacement equation in the mannerof the von Karman assumptions, in similar fashion tothat described in Section 4. Solution in the time domainis made using the implicit Newmark method and at anytime step convergence towards the equilibrium state is

Fig. 11. Pre-buckling stress resultant distributions at uniform temperature rise of T ¼ 100 for a square five-layer symmetric angle-ply plate: (a)

106N 0x , and (b) 106N 0

xy .

Table 5

Critical temperatures of a five-layer, angle-ply, thin, square laminate

with S2 boundary, by spline FSM

NS ¼ 8 q ¼ 8

q Tcr NS Tcr

2 389.8 2 387.9

4 385.8 4 384.4

6 384.0 6 383.7

8 383.5 8 383.5

10 383.3 10 383.4

Fig. 12. Variation of critical temperature with fibre angle for square, symmetric angle-ply (h ¼ 45�) plates with C1 and S2 edges: (a) A=h ¼ 100, and

(b) A=h ¼ 10.


achieved using the Newton–Raphson iteration tech-nique. No further analysis details will be given here, buta typical application from Ref. [93] is now presented.

An orthotropic square plate of dimensions A ¼ B ¼250 mm and h ¼ 5 mm is considered. The plate is of0�=90�=90�=0� construction, with the plies being of equalthickness. The material property set is E1 ¼ 525 GN/mm2, E2 ¼ 21 GN/mm2, G12 ¼ G13 ¼ G23 ¼ 10:5 GN/mm2, m12 ¼ 0:25, q ¼ 800 kg/m3.

The plate edges are simply supported for out-of-planebehaviour, whilst in the plane of the plate normalmovement of an edge is allowed but tangential move-ment is prevented. A suddenly applied step loading actswith uniform intensity 1 N/mm2 over the plate surface.In applying the SDPT S-a FSM to predict the transientresponse, four cubic (n ¼ 3) strips are used to model theplate and four longitudinal series terms are used inrepresenting each displacement-type quantity. The FSManalysis has been conducted in the realms both of linear

theory and of geometrically non-linear theory (using atime step of 10 ls). For comparison purposes, resultshave been generated using LUSAS, with a 6 6 mesh ofQSL8 elements in the geometrically non-linear realm.The results for this application are presented graphicallyin Fig. 13 in the form of plots of dimensionless centraldeflection �wwc ¼ wc=h and dimensionless central surfacestress �rryc ¼ 525ryc=E1 against time. It is clear that thepredictions of the non-linear S-a FSM are broadly ver-ified by comparison with the FEM predictions. Thelinear FSM predictions are markedly different from thenon-linear predictions at the level of loading consideredhere.

7. Buckling and vibration of sandwich plate structures

Because of their advantageous stiffness-to-weight ra-tios, sandwich plates have been used as load-carrying

Fig. 13. Dynamic response of a four-layer, antisymmetric angle-ply plate to step pressure of 1 N/mm2: (a) central deflection versus time, and

(b) central stress versus time.


structures over many years, especially in the realms ofaeronautical and marine engineering. The most usualsandwich construction consists of a relatively thick coreof low-density material which separates top and bottomfaceplates which are relatively thin but stiff. The mate-rials used over the years have been numerous and variedbut interest in sandwich construction has increased inquite recent times with the introduction of new materialsfor use in the core (e.g. solid foams) and in the faceplates(e.g. fibre reinforced composite laminated material).

It appears that only a little work has been conductedon the analysis of the buckling and vibration of sand-wich plate structures when using the FSM. Chan andCheung [94] used the S-a FSM in considering the staticand dynamic behaviour of multi-layer orthotropic sand-wich plates. This work was extended to include bucklinganalysis by Chan and Foo [95]. Later Chong et al. [96]used a combined finite prism (for the core) and finitestrip (for the faceplates) approach to the free vibrationproblem. Recently Dawe and Yuan have used the B-sFSM in studying the buckling [97,98] and free vibration[99] of single sandwich plates with laminated faceplates.Some extensions to sandwich plate structures are de-scribed by Yuan [100].

The analysis of the structural behaviour of sandwichplates is naturally more complicated than is that ofhomogeneous plates or compact laminated plates, be-cause of the increased variability of the through-thick-ness properties and the potential increase in complexityof through-thickness behaviour, as met in particular inbuckling applications where local modes, such as face-plate wrinkling, can occur as well as overall modes. Forgood accuracy the analysis of conventional sandwichplates generally requires the use of a three-zone platetheory, with distinct but inter-related assumptions made

for the variation of displacement quantities through thethickness of each of the two faceplates and of the core.

In Refs. [97–100] the core is treated as a three-di-mensional body, with quadratic variation of in-planedisplacements and linear variation of out-of-plane dis-placement through its depth. In two variants, each of thefaceplates is modelled in effect in the context of SDPT orin the context of CPT. For SDPT faceplates Fig. 14shows a view of a partial cross-section of a sandwichplate in the x–z plane, with co-ordinates and displace-ment quantities indicated [97], and obviously a similarview could be drawn in the y–z plane. At a general pointin the core the displacements �uuc, �vvc and �wwc are defined as

�uuc ¼ uc þzctcu2ð � u1Þ �

zc2tc

t1wx1ð þ t2wx2Þ

þ z2ct2c

2u1ð þ 2u2 � 4uc þ t1wx1 þ t2wx2Þ

�vvc ¼ vc þzctcv2ð � v1Þ �

zc2tc

t1wy1

#þ t2wy2

$þ z2ct2c

2v1

#þ 2v2 � 4vc þ t1wy1 � t2wy2

$�wwc ¼

w1 þ w2

2þ zctcðw2 � w1Þ

ð22Þ

For a faceplate f, where f ¼ 1 or 2, the displacements ata general point are given by

�uuf ¼ uf þ zfwxf ; �vvf ¼ vf þ zfwyf ;

�wwf ¼ wf ; f ¼ 1; 2 ð23Þ

In the context of SDPT faceplates the through-thickness displacement variations are seen to involve atotal of 12 fundamental displacement quantities (asagainst the five involved in the SDPT analysis of com-pact laminates, as in Section 2). In the context of CPT

Fig. 14. Sandwich plate displacements in the x–z plane.


faceplates, simplification of the above displacement ex-pressions exists [97,99,100] which involve a total of eightfundamental displacement quantities (as against thethree involved in the CPT analysis of compact lami-nates). The reader is referred to Refs. [97,99] for allfurther details of the B-spline analyses of the bucklingand free vibration of sandwich plates.

Many applications of the FSM capabilities aredescribed in Refs. [98] and [99] for buckling and freevibration, respectively. The buckling problem is of par-ticular interest because of the possibility of highly loca-lised modes as well as overall modes. Here, two relatedbuckling applications, exhibiting localised wrinklingbehaviour, are presented [98].

The sandwich plates considered are simply supportedand square, with A ¼ B ¼ 228 mm and their core is 25mm thick with elastic properties defined as E ¼ 109 N/mm2, Gxz ¼ 26:6 N/mm2 and Gyz ¼ 15:5 N/mm2. Thesedetails are taken from a study by Pearce and Webber[101]. The core has flexibility in the direction normal tothe faceplates but is assumed to have zero stiffness in itsplane [101]. The two sandwich plates differ in theirfaceplates. For plate I each of the faceplates is of two-layer 90�=0� construction with each layer being ofthickness 0.125 mm; for plate II each of the faceplates isof three-layer 45�=0�=45� construction with the 45�layers each being 0.125 mm thick and the 0� layer being0.25 mm thick. The layer properties are that E1 ¼ 142GN/m2, E2 ¼ 9:8 GN/m2, G12 ¼ G13 ¼ G23 ¼ 4:3 GN/m2

and m12 ¼ 0:34. The applied in-plane loading is uniformuniaxial compressive stress on the faceplates. The pre-dictions of the critical values of this stress by Pearce andWebber are based on the analysis of one half-wave ofthe buckled mode, on the assumption of orthotropicmaterial properties. This assumption is satisfactory forplate I but would not be expected to be suitable for plateII since then the faceplates are heavily anistropic,through the presence of the D16 and D26 coefficients,leading to skewing of the buckled mode shape.

In applying the B-s FSM to these applications it isassumed that the applied uniaxial stress acts in the y-direction. Each of the complete plates is modelled withone superstrip running along the x-direction whosepower c is varied in a convergence study, up to c ¼ 8(256 strips). Quintic splines (k ¼ 5) are used and thenumber of equal spline sections used is q ¼ 8 for plate Iand q ¼ 30 for plate II (where the x-direction behaviouris more complicated). The results of the convergencestudies are recorded in Table 6 in the contexts of bothCPT and SDPT faceplates. For both plates it is neces-sary to use many strips to achieve convergence becauseof the extreme complexity of the highly localised buck-led mode shapes, as shown in Fig. 15. Some small effectof the through-thickness shear deformation of thefaceplates is apparent from the differences in Table 6between the sets of results corresponding to CPT andSDPT faceplates. For the orthotropic plate I there isa very close comparison of the FSM result with thatof Ref. [101], whilst for the anisotropic plate II the

Table 6

Critical direct loads per unit length ðN 0y Þcr (N/mm) for uniaxially loa-

ded, simply supported, square sandwich plates I and II

Solution

method

Plate I Plate II

CPT

faceplates

SDPT

faceplates

CPT

faceplates

SDPT

faceplates

B-s FSM

c ¼ 1 674.0 674.0 713.2 713.1

c ¼ 2 673.9 673.9 707.5 707.3

c ¼ 3 343.4 343.4 424.4 421.6

c ¼ 4 109.4 109.0 292.8 277.1

c ¼ 5 87.42 85.36 240.7 232.2

c ¼ 6 78.23 76.92 232.9 228.0

c ¼ 7 77.24 76.19 232.2 227.8

c ¼ 8 77.14 76.16 232.2 227.8

Single half-

wave analysis

[101]

77 – 285 –

Fig. 15. Buckled mode shapes of two sandwich plates: (a) 90�/0� faceplates, and (b) 45�/0�/45� faceplates.


prediction of Ref. [101] is seen to be considerablyoverstiff as compared to the FSM prediction.

8. Conclusions

A review has been presented of aspects of the use ofthe FSM in predicting a number of types of linear andnon-linear behavioural response of composite lami-nated, prismatic plate and shell structures. The use ofthe two main variants of the method, i.e. the semi-ana-lytical and spline variants has been discussed. Thesevariants are complementary to one another: the semi-analytical approach is capable of greater accuracy andefficiency in some relatively simple situations, includingparticularly in the single-term analysis of ‘‘long’’ struc-tures, whilst the spline approach is the more versatile,especially when employing variable knot spacing. Theuse of different plate/shell theories has been described,including the first-order shear deformation theory forcompact laminates and a three-zone theory for sandwichplate structures.

For the analysis of prismatic structures the FSM hasattractive qualities relative to other approaches. As aninitial design tool it compares favourably with the gen-eral FEM from considerations of computational econ-omy and ease of modelling. Its advantage is particularlymarked in the solution of the eigenvalue problems ofdetermining buckling stresses and natural frequencies ofvibration of complicated structures. In this sphere theuse of a solution procedure which incorporates multi-level substructuring, including the use of superstrips,provides a highly efficient analysis capability. The ex-amples of applications presented herein hopefully showsomething of the scope and efficiency of the FSM.

There is good reason to expect continued develop-ment of the FSM in the future. More work needs to beconducted in the areas of non-linear dynamic responseand thermomechanical behaviour, for plate and shellstructures. There is scope for extending the study of thebuckling and vibration of single sandwich plates toembrace plate and shell structures, and further to em-brace non-linear behaviour. With the use of generalspline functions, consideration could be given to thepresence of supporting structures at arbitrary locationsalong a structure in buckling and vibration applications.Also, other types of behavioural response could usefullybe considered using the method, such as dynamicbuckling behaviour [102].

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