Vibration and buckling of prismatic plate structures having intermediate supports and step thickness changes

Comput. Methods Appl. Mech. Engrg. 191 (2002) 2759–2784 www.elsevier.com/locate/cma

Vibration and buckling of prismatic plate structures having intermediate supports and step thickness changes D.J. Dawe *, D. Tan School of Civil Engineering, The University of Birmingham, Edgbaston, Birmingham, UK B15 2TT Received 8 February 2001; received in revised form 11 September 2001; accepted 9 January 2002

Abstract Description is given of the development and use of a general spline finite strip method for predicting natural frequencies and buckling stresses of complex prismatic plate structures. The component plates in general are thin composite laminates whose properties are based on the use of classical plate theory. The assumed strip displacement field allows for the arbitrary location of spline knots in the longitudinal direction and the efficient solution procedure incorporates multi-level substructuring techniques with the use of superstrips. This gives a versatile and powerful analysis capability which allows the solution of problems which involve plate structures with intermediate supports and/or step changes in the thickness of component plates at arbitrary locations lengthwise. The use of this capability is illustrated by description of the solution of a number of problems involving both single plates and complicated structural panels. Ó 2002 Published by Elsevier Science B.V. Keywords: Buckling; Free vibration; Plate structures; Composite laminates; Finite strip method; Spline functions

1. Introduction The finite strip method (FSM) is an efficient, useful and popular method for predicting, amongst other things, frequencies and buckling stresses of prismatic plate and shell structures. Such structures are important load-carrying components in various branches of engineering and, in aerospace and marine engineering particularly, may be made of composite laminated material. The FSM exists in a number of variants, the chief two of which may be referred to as the semi-analytical FSM (S-a FSM) and the spline FSM. These are distinguished one from another by the nature of the variation of the displacement quantities along the length of the strip. In the S-a FSM use is made of analytical functions, i.e. trigonometric functions, beam eigenfunctions, etc., whilst in the spline FSM use is made of polynomial spline functions, usually B-spline functions. A considerable body of work has been

*

Corresponding author. Tel.: +44-121-414-5062; fax: +44-121-414-3675. E-mail address: [email protected] (D.J. Dawe).

0045-7825/02/$ - see front matter Ó 2002 Published by Elsevier Science B.V. PII: S 0 0 4 5 - 7 8 2 5 ( 0 2 ) 0 0 2 1 2 - 8

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generated in the subject area of finite strip vibration and buckling analysis but this complete body of work will not be reviewed here since a quite recent review is available elsewhere [1]. The S-a FSM and spline FSM are complementary procedures, with relative advantages and disadvantages. The main advantage of the spline FSM, as compared to the S-a FSM, is its increased versatility, for example in accommodating a wide range of structure end conditions, and this increased versatility is of enhanced importance when considering composite laminated structures which may have generally anisotropic and unbalanced properties. It is the spline FSM, introduced by Cheung and Fan [2], which is used here as a means of predicting the natural frequencies and buckling stresses of complicated prismatic plate structures. For truly prismatic structures (i.e. with no change whatsoever of form or properties over the length) and for single spans (i.e. no intermediate supports along the structure) the spline FSM has been developed by Dawe and Wang for the analysis of the buckling and vibration of both plate [3–5] and shell structures [6,7]. Component plate (flat or curved) properties have been based on the use of both classical, thin theory and first-order shear deformation theory. In this earlier work it is B-spline functions that are used longitudinally, with equal section lengths and uniform spacing of the spline knots in what may be termed an equal spline FSM (E-s FSM). Clearly the versatility of spline functions can be increased further by making allowance for non-uniform spacing of the spline knots, as described in specialist texts on spline usage, such as those of de Boor [8] and of Bartels, Beatty and Barsky [9]. Here the phraseology ‘‘general spline’’ will be used to denote spline representation with non-uniform knot spacing and the term ‘‘general spline FSM’’ (G-s FSM) will denote the FSM when general spline functions are used. In the area of structural analysis such non-uniform knot spacing can clearly have its uses in dealing with practical situations involving step changes of properties, localised areas of stress concentration, intermediate supports at arbitrary locations, etc. In doctoral theses, Li [10] first presented the G-s FSM in the context of classical plate theory (CPT) whilst Wang [11] used general splines in a Rayleigh–Ritz analysis, in the context of first-order shear deformation plate theory (SDPT), of the buckling and vibration of single plates. Gutkowski et al. [12] studied the static CPT plate bending analysis of single plates using the G-s FSM, with cubic splines. Madasamy and Kalyanaraman [13] further considered the CPT bending problem using the cubic G-s FSM, but as well studied in-plane behaviour in the analysis of shear walls. A description of the G-s FSM is also included in the recent text of Cheung and Tham [14]. The present authors have developed the G-s FSM for the buckling and vibration analysis of single plates and shells [15] and of complicated, but single-span, composite laminated plate and shell structures [16]. This development has been made in the contexts of both classical theory and first-order shear deformation theory. A specific study has been made recently [17] of the use of the G-s FSM, in the contexts of both theories, in predicting buckling stresses and natural frequencies of single plates which have step changes of properties along their lengths. In the present work the aim is to describe the development and application of the G-s FSM in predicting the frequencies and buckling stresses of complicated plate structures which have intermediate line supports and/or step changes in thickness along their lengths. Various solution procedures for structures with intermediate line supports (i.e. multi-span structures) have been proposed in the past but generally their usage has been restricted to quite simple structural forms. In the realm of S-a FSM analysis Wu and Cheung [18] considered the frequency analysis of single rectangular, isotropic CPT plates continuous over rigid line supports and used continuous-beam functions along the strips. This work was later extended to the analysis of the buckling and vibration of multi-span prismatic plate structures [19], although the only application to plate structures involved a simple threestrip ‘‘tee’’ model of a small part of a blade-stiffened panel (as described here later in Section 3.3). Puckett et al. [20,21] used what they termed a ‘‘compound strip method’’, incorporating the use of a S-a FSM approach in conjunction with the presence of elastic stiffening beams and supporting columns, to determine natural frequencies and buckling loads of single continuous plates. Graves-Smith et al. have used a combined S-a FSM/finite element method (FEM) approach for analysing the vibration and local buckling of

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simple single-box structures having one or more diaphragm-type supports along their length [22,23]. It is also noted that the finite layer approach has been used in analysing the free vibration of single, thick plates which may have intermediate line supports [24] or point supports [25]. In considering the analysis of plate structures which have step changes in thickness along their lengths, serious consideration has to be given to the nature of the continuity conditions required at a step change and as to how this relates to an assumed displacement field. At a step change the proper requirement, in the context of CPT, when considering out-of-plane behaviour is for continuity of deflection w and its first derivative ow=ox (where x is measured along the structure) but discontinuity of its second and third derivatives, i.e. the requirement is for C1 continuity. In the past when analyses based on the use of continuous functions have been conducted, it has not infrequently been the case that the proper conditions have not been met, because the assumed functions have had excessive continuity. It is noted that this is not the situation when using the FEM where generally the piecewise field is no more than C1 continuous. However, it is a potential difficulty when using the FSM, whether with a smooth, continuous field in the S-a FSM or with a piecewise-continuous polynominal field in the spline FSM. As an exception to this statement, Cheung et al. [26] have developed a FSM approach for single-plate vibration and buckling analysis in which step changes of thickness are accommodated properly in the longitudinal representation of w: the beam vibration modes appropriate to the end conditions are augmented with piecewise-cubic polynominals to produce C1 continuous functions. In related vein, but when considering in-plane static analysis of plates having step changes in rigidity at which only C0 continuity should be maintained, the same authors have presented a suitable displacement field in a FSM approach [27]: the longitudinal variation of each of the inplane displacements is represented by Fourier series augmented by piecewise-linear correction functions. In the present study using the versatile G-s FSM, cubic splines are employed in the longitudinal direction and the analysis of the buckling and vibration of composite laminated plate structures takes place in the context of CPT. The analysis incorporates the superstrip concept and associated advanced solution procedures [28,29] and will allow the solution of problems for which it would be impracticable to obtain accurate solutions using alternative procedures, other than using the general FEM which is unlikely to match the efficiency of the G-s FSM. The question of the suitability of the longitudinal spline representation in accommodating intermediate supports and step thickness changes is deferred to Section 2 where details are given of the whole analysis procedure. In Section 3 a number of applications of the developed G-s FSM approach is described, ranging from those involving single beams and plates to those concerning complex structural panels. Brief conclusions are recorded in Section 4.

2. Analysis 2.1. General remarks Fig. 1 shows a ‘‘uniform’’ finite strip of the sort that has been used in earlier studies of the buckling and vibration of prismatic plate structures [3–7]. The strip forms part of a component plate (of length A and width B P b) which is in general a laminate which has an arbitrary lay-up of a number of layers of unidirectional fibre-reinforced composite material. The quantity h shown in Fig. 1 indicates the clockwise angle measured from the x-axis to the fibre direction of a layer of the laminate. In the buckling problem the component plate is subjected to an in-plane stress system which comprises one or more of longitudinal stress r0x , transverse stress r0y and shear stress s0xy . In the vibration problem the component plate is undergoing harmonic motion whilst vibrating in a natural mode with circular frequency p. In the context of CPT the behaviour of the plate is characterised by the three fundamental displacement quantities u, v and w, the translational displacements at the middle surface in the x, y and z directions. In the buckling problem these displacements are, in fact, perturbation displacements that occur at the instant of buckling.

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Fig. 1. The ‘‘uniform’’ cubic finite strip.

In the present study the interest is with rectangular plates and prismatic plate structures which may have one or more intermediate line supports and/or step changes in plate thickness at locations x ¼ constant along the structure length. The typical finite strip is thus no longer a ‘‘uniform one’’ as shown in Fig. 1 but may incorporate intermediate line supports at which w ¼ 0 and/or step thickness changes with, for the buckling problem, consequent step changes in the applied stresses r0x , r0y and s0xy . Any step thickness changes will be made symmetrically about a common plate middle surface. 2.2. Constitutive and energy equations The linear constitutive equations assumed 9 2 8 Nx > A11 A12 A16 B11 B12 > > > > > > > 6 A12 A22 A26 B12 B22 N > > y > > = 6 < 6 A16 A26 A66 B16 B26 Nxy ¼6 B11 B12 B16 D11 D12 > 6 > Mx > > > 6 > > > 4 B12 B22 B26 D12 D22 > My > > > ; : Mxy B16 B26 B66 D16 D26

here for an arbitrary laminate are 9 38 ou=ox B16 > > > > > > > > ov=oy B26 7 > > > 7> = < 7 B66 7 ou=oy þ ov=ox D16 7 > o2 w=ox2 > > 7> > > 5 D26 > > > o2 w=oy 2 > > > ; : D66 2 o2 w=ox oy

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or F ¼ Le:

ð1Þ

Here, in column matrix F the Nx , Ny and Nxy are the membrane direct and shearing forces per unit length, and Mx , My and Mxy are the bending and twisting moments per unit length. In column matrix e the entries are the three linear in-plane direct and shearing strains at the middle surface and the three direct and twisting curvatures. In matrix L the laminate stiffness coefficients are defined in the usual way as Z h=2     ð2Þ Aij ; Bij ; Dij ¼ Qij 1; z; z2 dz; i; j ¼ 1; 2; 6 h=2

where Qij are the plane-stress reduced stiffnesses. These quantities are determined on the basis of the local thickness h. The form of the constitutive equation (1) assumed here is very general, allowing for anisotropic properties with regard to both in-plane and out-of-plane behaviours and for full coupling between in-plane and out-of-plane behaviours. The required properties of a finite strip, i.e. the elastic stiffness matrix k, the geometric stiffness matrix, kg , and the consistent mass matrix, m, are based on the use of expressions for the strain energy, U, the potential energy of applied in-plane stresses, Vg , and the maximum kinetic energy, Tmax , respectively. These expressions are as follows: Z Z 1 b=2 A T U¼ e Le dx dy; ð3Þ 2 b=2 0 1 Vg ¼ 2

Tmax

Z

b=2

b=2

p2 ¼ 2

Z

Z

A

"  2  2  2 #  2  2 # 2 ou ov ow ou ov ow 0 h þ þ þ þ ry þ ox ox ox oy oy oy  ! ou ou ov ov ow ow þ þ 2s0xy þ dx dy; ox oy ox oy ox oy r0x

0

b=2

b=2

Z

A

"

  qh u2 þ v2 þ w2 dx dy

ð4Þ

ð5Þ

0

where q is the mass density. It is noted from Eqs. (1) and (3) that the expression for strain energy contains first derivatives of u and v and second derivatives of w, and hence that C0 continuity is required for u and v and C1 continuity for w. Where a step change, or changes, of properties occurs at one or more locations x ¼ constant along the strip length the required overall integrals with respect to x, between 0 and A, are broken down into a number of discrete parts (with each part corresponding to a certain part-length of the strip over which the properties are constant) and then summed. 2.3. Strip displacement field The variation of each of the displacements u, v and w is represented spatially as a summation of products of spline functions in the longitudinal x-direction and polynomial shape functions in the crosswise y-direction. In the longitudinal direction each of u, v and w is expressed as a summation of a series of terms, each term being the product of an unknown coefficient and a local spline function of x of degree three (i.e. cubic splines are used here but splines of other degrees could be used [15,16]). The strip length A is divided into q spline sections which are generally of unequal lengths as shown in Fig. 2. There are q þ 1 spline knots

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Fig. 2. Spline sections and knots.

within the length A plus, for the cubic splines, one other knot outside each end of this length which is present for the purposes of completing the definition of a function and of prescribing appropriate end conditions. The basis cubic spline function b3i ðxÞ with knot sequence fxi g, i 2 Z, where Z is the set of integer knot numbers, is expressed [15,16] as a linear combination of the truncated power functions, which are typical cardinal spline functions [30], as b3i ðxÞ ¼

4 X

ali hx  xiþl i

3

ð6Þ

l¼0

where hx  xiþl i3 ¼



3

ð x  xiþl Þ for x > xiþl ; 0 for x 6 xiþl :

ð7Þ

The real coefficients ali in Eq. (6) satisfy the equations [30] 4 X

 3 ali xg  xiþl ¼ 0;

g ¼ i þ 1; . . . ; i þ 4

ð8Þ

l¼0

to ensure appropriate continuity of the function at the knots together with the required compact support condition, and the identity 4 X

ali ðxiþ4  xiþl Þ4 ¼ 4

ð9Þ

l¼0

which normalizes the area under the curve. This set of equations can be solved to yield the coefficients ali and hence to complete the definition of b3i ðxÞ. It is noted that b3i ðxÞ has non-zero value over the range xi < x < xiþlþ3 and that within this range it is continuous up to the second derivative. Outside of this range, b3i ðxÞ and its derivatives have zero value. For a function f ðxÞ, where f represents each of u, v and w, the complete longitudinal spline representation is f ðxÞ ¼

qþ1 X

b3i ðxÞai ¼ b3 a

ð10Þ

i¼1

where ai are knot coefficients. The row matrix b3 contains all the local spline functions of x of degree three and the column matrix a contains all the knot coefficients. The expression for f ðxÞ can be modified to relate to physical quantities, i.e. to certain knot values of f ðxÞ and its derivatives, through an appropriate transformation. Here values of f and df =dx are used at the end knots 0 and q þ 1 and values of f are used at the other knots. The transformation has the form

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f ¼ Ra

or

a ¼ R1 f

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ð11Þ

where f is the column matrix of physical quantities and R is a square transformation matrix. Then Eq. (10) can be re-expressed as f ðxÞ ¼ b3 R1 f ¼ B3 f

ð12Þ

where B3 is a row matrix whose coefficients are functions of x. The crosswise representation of each of the displacement quantities is by standard shape functions of degree n. Although different values of n could be assumed [3,16,28], here attention is restricted to n ¼ 3, i.e. cubic interpolation is adopted in the y-direction. In representing u and v (with a requirement for only C0 continuity) Lagrangian shape functions Nj ðyÞ are used with degrees of freedom, related to values of u alone or of v alone, respectively, located at four reference lines equi-spaced across the strip at y ¼ b=2 and y ¼ b=6, as shown in Fig. 1. In representing w (with a requirement for C1 continuity) Hermitian shape functions NjH are used with degrees of freedom related to values of both w and ow=oy at the two external reference lines 1 and 4 only. Both types of shape function are well known, and are defined specifically in, for instance, Ref. [16]. With the above remarks and definitions in mind, the strip displacement field can be expressed finally in the form 8 9 2 32 38 9 Nj 0 0 0 < du = B3 0 4
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the shear force is discontinuous (and conditions on the u and v displacements are not of concern). Thus the expression for w longitudinally should have C2 continuity and this, of course, is precisely what it does have in the usual cubic spline representation used here: thus no special action is necessary to accommodate intermediate supports. Where a step change in thickness (and/or of material properties) occurs at any longitudinal location, the conditions that should apply with regard to w are that w should be only C1 continuous, to allow the possibility of continuity of bending moment and shear force (rather than of o2 w=ox2 and o3 w=ox3 ) across the step change. This, of course is at odds with the usual C2 continuity of cubic splines but this continuity could be reduced directly by locating a multiple knot at the step-change location, as discussed above. However, for practical reasons, linked to the superstrip approach (use of which is very highly advantageous), this is not what is done here. Rather, a less direct approach is taken by adopting the view that a close approximation to the proper condition can be achieved by locating a number of simple knots close to the step change, in place of a multiple knot at the change, i.e. by refining the spline spacing local to the change. This was the policy adopted in Ref. [17] where it was shown to be very successful in considering the out-of-plane vibration and buckling of single stepped plates. The policy fits in with the philosophy expounded in the texts of de Boor [8] and of Bartels, Beatty and Barsky [9] to the effect that a spline representation with multiple knots can be replaced with little difference by a spline representation involving only simple knots if each knot of multiplicity m > 1 is replaced by m simple knots nearby. The foregoing remarks have been made with regard only to out-of-plane plate behaviour. For in-plane effects some not dissimilar considerations apply at a step change, e.g. the longitudinal force should be properly continuous through the step, with ou=ox being discontinuous, i.e. u should be only C0 continuous at the step. The principle of spline refinement local to a step change continues to apply in approximating this in-plane consideration, although it is clear that in-plane effects are very much less important than are out-of-plane effects for the types of application considered here. In the presented applications that follow, the general policy adopted at a step change will be to locate a simple knot at the step itself and a simple knot to each side of, and close to, the step. The approach is a pragmatic one but is soundly based in spline philosophy and has been shown to be successful in predicting the overall behaviour of single plates, as exemplified by the calculation of natural frequencies and buckling stresses [17]. (For static analysis, especially if detailed stress distributions were to be determined local to a step change, the procedure would be less appropriate, but this is not the concern here.) 2.4. Strip matrices, substructuring and solution On substituting the strip displacement field of Eq. (13) into Eqs. (3)–(5) expressions can be derived for the strain energy, the potential energy of the applied in-plane stresses and the kinetic energy of a finite strip as quadratic functions of the complete set of reference-line displacements in column matrix d. Here d ¼ f d1

d2

d3

d4 g

ð14Þ

where the subscripts 1–4 refer to the reference lines. Ultimately the expressions have the familiar forms: U ¼ 12dT k d;

ð15Þ

Vg ¼ 12dT kg d

ð16Þ

Tmax ¼ 12p2 dT m d:

ð17Þ

and

The integrations involved in evaluating the stiffness matrix, k, geometric stiffness matrix, kg , and consistent mass matrix, m, are carried out analytically.

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Beyond the stage at which the properties of individual finite strips are established, the procedures used here in calculating natural frequencies or buckling stresses are much the same as have been described in earlier related works [7,15,28] and hence only need brief summary here. The superstrip concept is invoked, such that usually a plate, or each component plate of a prismatic structure, is represented by one superstrip which is an assembly of 2c identical individual strips (where c ¼ 0; 1; 2 . . .). This is done through an efficient repetitive substructuring scheme and the assembly of 2c strips is referred to as a superstrip of order c, or simply as a SuperstripC. The superstrip has degrees of freedom located only at its outside edges and if c is chosen typically to have the value 5, say, it provides a very accurate model of crosswise structural behaviour. A prismatic structure is generally modelled as an assembly of superstrips, with transformations applied as necessary at each superstrip edge to transform properties to a global co-ordinate system [28]. Beyond the superstrip level, higher levels of substructuring can often be invoked if the structural crosssection has a repetitive nature, and this helps further in reducing the number of effective degrees of freedom. In total the multi-level substructuring procedures that are used can easily reduce a problem having initially hundreds of thousands of freedoms to a problem which involves only of the order of a hundred effective freedoms. The final eigenvalue problem is non-linear and the determination of the eigenvalues is made using an extended Sturm sequence-bisection approach. The details of this are as given in Refs. [7,15,28], wherein description is also given of the manner in which the mode shapes of vibration or buckling can be obtained.

3. Applications 3.1. Vibration of multi-span plates Wu and Cheung [18] have presented numerical results for some lower natural frequencies of rectangular, isotropic plates of uniform thickness which are continuous in one or two directions over intermediate line supports. Their method is a finite strip approach in the context of CPT in which analytical functions for continuous beams are used in representing the longitudinal variation of the plate deflection w. Here, results obtained using the present spline FSM are compared with some of the earlier results of Ref. [18], and also some extensions to other than isotropic plates are considered. The geometries of the three types of plate considered here are shown in Fig. 3. As well as the different geometrical forms, each of Plate I, Plate II and Plate III is considered in turn to be of isotropic material (as considered in Ref. [18]) and of composite laminated material. The laminated plates considered have a balanced five-layer 0°=90°=0°=90°=0° lay-up for Plate I (with each 0° ply of thickness h=6 and each 90° ply of thickness h=4), an unbalanced 0°=90° lay-up for Plate II with plies of equal thickness, and a single layer, with fibre angle h ¼ 30°, for Plate III. The x-direction for each plate is shown in Fig. 3, indicating the direction in which the finite strips run and from which the fibre angles are measured. The material of all plies is defined by EL =ET ¼ 30

GLT =ET ¼ 0:6 and mLT ¼ 0:25

where, as is usual, subscripts L and T denote directions along and perpendicular to the fibres, respectively. The external boundary conditions for the plates are indicated in Fig. 3 by the letters S, C and F, indicating simple support, fully clamped support and free edge, respectively. It is noted that for the unbalanced laminated version of Plate II (where there is a coupling between in-plane and out-of-plane behaviours) the in-plane conditions are that u ¼ v ¼ 0 at the clamped ends and that u 6¼ 0, v 6¼ 0 at the simply supported longitudinal edges. The numerical results for the first six frequencies of vibration are presented in Table 1 as convergence studies with respect to the number of spline sections and the comparative results of Wu and Cheung [18] are included for the isotropic plates (for which Poisson’s ratio m ¼ 0:3). The quoted number of spline sections,

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Fig. 3. Three types of multi-span rectangular plates (– – – denotes an interior line support with w ¼ 0, u 6¼ 0 v 6¼ 0).

q, is the number in each span of a plate and, since there is no need for knot refinement local to a support, the knots are equispaced within a span. The present spline FSM results relate to the use of one Superstrip5 (i.e. 32 strips) running in the x-direction for Plates I and II and two such strips for Plate III. The results of Table 1 reveal the good convergence properties, with respect to q, of the present approach, and for the isotropic plates there is close comparison with the earlier results of Ref. [18], with the present results for greatest q being slightly lower than the earlier results, as expected. 3.2. Vibration of simple stepped structures In Ref. [17] the authors have presented results obtained using the G-s FSM for natural frequencies and buckling factors of a number of single-span beams and plates having single or double step changes in thickness. The philosophy of refining spline knot spacings local to a step change has been explored in

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Table 1 Frequencies of various multi-span plates (values of q quoted are per span) Plate type I, isotropic

I, balanced orthotropic

II, isotropic

II, unbalanced orthotropic

III, isotropic SSSS

III, isotropic SCSF

III, balanced anisotropic SSSS

Solution type

Values of pL2 ðqh=D11 Þ1=2 Mode 1

Mode 2

Mode 3

Mode 4

Mode 5

Mode 6

Spline FSM: q¼2 q¼3 q¼4 q¼5

20.06 20.02 20.02 20.02

22.19 22.09 22.07 22.07

25.72 25.47 25.43 25.42

28.95 28.54 28.47 28.45

49.56 49.52 49.51 49.51

50.95 50.78 50.36 50.20

Wu and Cheung [18]

20.02

22.11

25.50

28.51

49.55

50.94

Spline FSM: q¼2 q¼3 q¼4 q¼5

12.80 12.75 12.74 12.74

15.43 15.31 15.30 15.30

19.61 19.36 19.31 19.31

23.38 23.11 23.03 23.02

29.07 29.05 29.04 29.04

30.38 30.32 30.31 30.31

Spline FSM: q¼2 q¼3 q¼4 q¼5

13.05 12.98 12.97 12.96

23.96 21.06 20.88 20.84

25.99 25.78 25.67 25.65

42.39 42.32 42.28 42.26

51.98 50.23 50.05 50.01

53.57 53.21 53.11 53.07

Wu and Cheung [18]

12.99

20.84

25.89

–

–

–

Spline FSM: q¼2 q¼3 q¼4 q¼5 q¼6 q¼7

6.740 6.642 6.625 6.621 6.619 6.618

13.27 10.36 9.960 9.825 9.818 9.809

15.67 13.51 12.92 12.78 12.74 12.74

23.70 15.79 14.43 14.03 13.90 13.87

26.98 23.65 22.56 19.82 18.84 18.47

28.53 25.71 23.65 23.64 23.64 23.64

Spline FSM: q¼2 q¼3 q¼4 q¼5

19.77 19.75 19.74 19.74

23.68 23.65 23.65 23.65

23.81 23.68 23.66 23.65

27.22 27.09 27.07 27.06

49.37 49.35 49.35 49.35

52.03 50.17 49.56 49.42

Wu and Cheung [18]

19.74

23.67

23.68

27.11

49.35

49.74

Spline FSM: q¼2 q¼3 q¼4 q¼5

15.65 12.18 12.16 12.16

19.39 17.17 17.15 17.15

32.68 27.13 26.10 25.09

32.85 29.86 28.42 28.41

37.73 32.94 31.69 31.66

38.16 35.40 34.67 34.64

Wu and Cheung [18]

12.17

17.17

25.13

28.48

31.76

34.81

Spline FSM: q¼2 q¼3 q¼4 q¼5

16.05 15.71 15.55 15.48

16.50 16.11 15.92 15.84

18.58 17.27 17.00 16.93

18.88 17.65 17.38 17.30

29.44 24.80 23.91 23.71

29.75 25.40 24.44 24.23

numerical studies and shown to be very appropriate. Before moving on to the consideration of the behaviour of plate structures of much greater complexity, it is pertinent here to examine further the behaviour of simpler, isotropic structures for which accurate comparative results are available.

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Fig. 4. A stepped cantilever beam.

In the earlier study [17] the calculation of the first four natural frequencies of an isotropic Bernoulli– Euler cantilever beam was considered and (close) comparison was made with accurate results available from the work of Balasubramanian and Subramanian [31]. However, this was done only for one beam having a particular size of step. To examine the effect of the magnitude of the step change on the accuracy of frequency calculations when using the G-s FSM we revisit this problem, but consider only the fundamental frequency. It is noted that in Ref. [31] a FEM approach is used, with each of the two beam sections represented by one highly-refined element (w represented as a polynominal of degree 7) at the end nodes of which the freedoms are w, dw=dx, the bending moment and the shear force. The cantilever beam of length A shown in Fig. 4 is of rectangular cross-section and has a step change of depth (and hence of cross-sectional area A and second moment of area I) at the mid-length position. There is also a difference in values of mass density q and Young’s modulus E in each of the two parts, a and b, of the beam. In Ref. [31] values of the fundamental frequency are recorded for a series of beams of different values of the step-thickness ratio H ¼ hb =ha (where the subscripts refer to parts a and b) and in each case the material properties of the two parts are such that Eb =Ea ¼ qb =qa ¼ H . In using the G-s FSM plate capability to predict the beam frequencies, Poisson’s ratio is set to zero, an arbitrary value is assigned to the width and a single finite strip runs along the full length of the beam. Numerical results are presented in Table 2, in a convergence study with q, for beams with H values ranging from 1.0 (a uniform beam) down to 0.4. It is noted that spline knots are located at the step change and very locally (0.5% of the beam length away) on either side of the change. The manner of the convergence of the G-s FSM results is good. Comparison of the q ¼ 12 solutions with the results of Ref. [31] shows excellent correspondence for the larger values of H and only a small decline in accuracy as the step becomes more severe. It is noted that when the knot spacing is further reduced local to the step, to become 0.2% of the beam length, the percentage error in the G-s FSM, q ¼ 12, solution is reduced from 0.44% to Table 2   Fundamental natural frequency of stepped cantilever beams: values of pA2 ½ qA a =ðEI Þa 1=2 Type of solution G-s FSM: q¼4 q¼6 q¼8 q ¼ 10 q ¼ 12 Ref. [31] a

H ¼ 1:0

H ¼ 0:9

H ¼ 0:8

H ¼ 0:7

H ¼ 0:6

H ¼ 0:5

H ¼ 0:4

3.5177 3.5162 3.5162 3.5161 3.5160 (0.00)a

3.8367 3.8342 3.8342 3.8340 3.8340 (0.00)

4.1938 4.1898 4.1898 4.1895 4.1895 (0.02)

4.5735 4.5669 4.5668 4.5664 4.5663 (0.06)

4.9219 4.9111 4.9110 4.9102 4.9101 (0.13)

5.1017 5.0855 5.0853 5.0840 5.0839 (0.27)

4.8686 4.8489 4.8487 4.8467 4.8466 (0.44)

3.5160

3.8340

4.1888

4.5637

4.9035

5.0703

4.8253

Values in parentheses are percentage errors in the G-s FSM, q ¼ 12 solutions. Knot positions for the G-s FSM are as follows: q ¼ 4: (0, 495, 500, 505, 1000) A/1000, q ¼ 6: (0, 250, 495, 500, 505, 750, 1000) A/1000, q ¼ 8: (0, 250, 400, 495, 500, 505, 600, 750, 1000) A/ 1000, q ¼ 10: (0, 125, 250, 375, 495, 500, 505, 625, 750, 875, 1000) A/1000, q ¼ 12: (0, 100, 200, 300, 400, 495, 500, 505, 600, 700, 800, 900, 1000) A/1000.

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Table 3   Natural frequencies of stepped, two-span beam: values of pA2 ½ qA a =ð EI Þa 1=2 Type of solution

Mode 1

Mode 2

Mode 3

Mode 4

G-s FSM: q¼4 q¼6 q¼8 q ¼ 10 q ¼ 12

43.66 39.63 39.58 39.49 39.47

82.14 62.54 62.48 61.96 61.93

198.7 167.0 165.3 158.8 158.3

272.8 245.7 223.8 202.8 201.4

Ref. [31] Ref. [32]

39.51 39.48

61.76 61.67

158.3 158.0

200.2 199.9

Knot positions for the G-s FSM are as specified in Table 2.

0.17%. However, it is not advocated generally that local knot refinement be overdone, because of the possibility of numerical instability. The combination of step change in properties with multi-span form has also been considered by Balasubramanian and Subramanian [31] and one particular example is considered here. The two-span beam is of the type shown in Fig. 4 but now with simple supports at points 1, 2 and 3, and with hb ¼ 0:50606ha , qb ¼ 0:256096qa and Eb ¼ Ea . The present capability has been used to predict the lowest four natural frequencies and these are recorded in Table 3, again in a convergence study with q and with spline knot spacings exactly as defined for the cantilever problem in Table 2. In Table 3 comparison is made both with the results of the FEM approach of Ref. [31] and with results (also quoted in Ref. [31]) from the text of Gorman [32] obtained from a different approach. It can be seen that the G-s FSM results converge well to values that correspond very closely with the comparative results. One example (reproduced from Ref. [17]) of a single-stepped rectangular plate is shown in Fig. 5. The plate is fully clamped at its boundary and is of homogeneous, isotropic material with Young’s modulus E and Poisson’s ratio m ¼ 0.3. In the present G-s FSM the finite strips run along the x-direction, with one Superstrip5 modelling the whole plate and, of course, a refinement of spline knot spacings is introduced local to the step change. The first four natural frequencies are considered and results corresponding to the use of up to twelve spline sections are recorded in Table 4, with details given of the precise knot locations.

Fig. 5. A stepped rectangular plate.

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Table 4 Natural frequencies of singly-stepped, isotropic clamped plate: values of pA2 ½12qð1  m2 Þ=Eh2 1=2 Type of solution

Mode 1

Mode 2

Mode 3

Mode 4

G-s FSM, x-dir. strips: q¼2 q¼4 q¼6 q¼8 q ¼ 10 q ¼ 12

5.640 5.357 5.303 5.300 5.299 5.299

8.042 7.667 7.517 7.513 7.509 7.505

10.77 8.872 7.632 7.622 7.616 7.597

13.59 10.09 9.242 9.239 9.226 9.211

E-s FSM, y-dir strips [17] LUSAS FEM [17]

5.297 5.272

7.505 7.438

7.589 7.581

9.204 9.114

For the x-direction strips, knot positions are as follows: q ¼ 2: (0, 0.5, 1) A, q ¼ 4: (0, 0.49, 0.5, 0.51, 1) A, q ¼ 6: (0, 0.24, 0.49, 0.5, 0.51, 0.76, 1) A, q ¼ 8: (0, 0.24, 0.46, 0.49,0.5, 0.51, 0.54, 0.76, 1) A, q ¼ 10: (0, 0.2, 0.4, 0.46, 0.49, 0.5, 0.51, 0.54, 0.6, 0.8, 1) A, q ¼ 12: (0, 0.1, 0.22, 0.34, 0.46, 0.49, 0.5, 0.51, 0.54, 0.66, 0.78, 0.9, 1) A.

Also recorded in the table are two sets of comparative results. The first set also relates to the use of the spline FSM but with the strips running now in the y-direction (using one Superstrip5 in each of the two physical sections of the plate) and with 10 equal spline sections in the y-direction (i.e. using the E-s FSM approach). The second set relates to the use of the FEM, employing the LUSAS commercial finite element software package with an 8  4 mesh of square, non-conforming (and hence no bound conditions apply) QLS8 elements. It is seen from the table that the G-s FSM results converge in very good fashion and that the predictions compare very closely indeed with those of the two alternative approaches. 3.3. Buckling of a bridge panel Cheung and Delcourt [19] have studied the buckling of an isotropic stiffened panel of a box girder bridge over the River Danube in Vienna, using a CPT finite strip approach which incorporates continuous beam eigenfunctions along the strip length in the displacement field. The basic idealization of the twospan panel, stiffened with twelve blade stiffeners, is shown in Fig. 6(a). The thicknesses of the main plate and the stiffeners are 10 and 12 mm, respectively. The material properties are that E ¼ 206 kN/mm2 and m ¼ 0:3. It is stated in Ref. [19] that the buckling mode, under uniform longitudinal compressive stress r0x , is a repetitive local one in which the crosswise shape is as shown in Fig. 6(b) and there is a large number of longitudinal half-waves. In view of the assumed nature of the mode, Cheung and Delcourt analyse only a simple cross-sectional model of the type shown in Fig. 6(c) with conditions that w ¼ 0 at point B and ow=oy ¼ 0 at points A and C. In the longitudinal direction this model is taken to be a single span of length 2770 mm, this choice being made so that comparison is possible between their FSM results (using nine series terms and three strips) and those of a finite element analysis which is limited to ten elements in the  longitudinal direction. The FSM and FEM predictions of the critical stress r0x cr when using this model are 256.14 and 267.52 N/mm2 , respectively, with five longitudinal half-waves, i.e. the half-wavelength   k ¼ 554 mm. Cheung and Delcourt also quote an analytical solution (for a panel of infinite length) of r0x cr ¼ 251:1 N/mm2 corresponding to a pure sinusoidal half-wavelength of 580 mm, and their FSM solution (over a  length of 2900 mm with five half-waves) of r0x cr ¼ 255:75 N/mm2 . In using the present spline FSM approach we firstly consider the analysis of the simple model shown in Fig. 6(c) over a number of discrete lengths, each of which in effect corresponds to an assumed single longitudinal half-wavelength of local buckling, with appropriate applied end conditions (u 6¼ 0, v ¼ w ¼ 0).

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Fig. 6. The stiffened bridge panel: (a) geometrical details; (b) local buckling mode; (c) simple model for analysis.

Table 5 Buckling of the two-span bridge panel using a simple model over one half-wavelength k (mm) 300 400 500 554 580 600 700 800

Values of (r0x Þcr , N/mm2 Spline FSM

Comparative methods [19]

357.81 283.44 258.78 254.88 254.49 254.74 261.58 275.38

– – – 256.14a , 267.52b 251.1c , 255.75a – – –

a

S-a FSM result. FEM result. c Analytical result. b

The FSM analysis is conducted with q ¼ 4 and uses one Superstrip4 for each of the three component plates of the model. The results of this exercise are given in Table 5. It is clear that for the simple model the use of the present spline FSM predicts values of buckling stress and of associated longitudinal half-wavelengths which are very close to the analytical and semi-analytical FSM predictions. Secondly, to show the power of the present approach, we consider the analysis of the whole panel of plan area 8000  7540 mm2 . The applied boundary conditions are that the main plate is simply supported all

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Table 6 Buckling of the two-span bridge panel using full and part modelling values of (r0x Þcr , N/mm2 q per span 2 4 6 8 10 12 16 20

Full-panel model

Part model

Mode 1

Mode 2

Mode 3

229.84 214.54 214.32 214.30 214.29 214.28 214.28 214.26

239.90 222.70 222.48 222.46 222.45 222.44 222.43 222.42

255.00 239.91 239.69 239.67 239.66 239.65 239.63 239.62

Mode 4 (278.97)a (263.98) (263.77) 251.60 250.34 248.99 248.04 247.77

562.62 290.62 271.51 257.17 255.41 254.03 253.04 252.75

a The ‘‘Mode 4’’ values in parentheses for q ¼ 2, 4 and 6 correspond to the calculated fourth buckling stress but relate to a different mode to that shown in Fig. 7(d).

around its boundary, and at the intermediate line support half-way along the length, whilst the ends of the stiffeners are free. The complete cross-section is subjected to uniform longitudinal stress and is modelled using 25 Superstrip5, i.e. one superstrip for each stiffener and each component part of the main plate shown in the view of the cross-section in Fig. 6(a). Calculations have been made in a convergence study using between two and 20 equal spline sections per span, and the first four buckling stresses have been determined using this full-panel model (which has about a quarter of a million initial freedoms for the largest value of q). Calculations have also been made using the simple model described above, and shown in Fig. 6(c), but now with this model running the full length of 8000 mm of the panel. The results of both sets of calculations are presented in Table 6 and the buckled mode shapes for the full-panel model are shown in Fig. 7. This figure reveals that the mode corresponding to the lowest buckling stress in fact has an overall, rather than local, nature with two half-waves longitudinally and one crosswise half-wave. The second and third modes also have two longitudinal half-waves, with two and three crosswise half-waves, respectively. The fourth mode does have a local nature, with fourteen longitudinal half-waves, but in the crosswise direction the nature of this mode is quite different from that indicated in Fig. 6(b), since when using the full-panel model the deformation is concentrated close to the longitudinal edges of the panel. The numerical results of Table 6 reveal rapid convergence of buckling stress values with increase in q for modes 1, 2 and 3 of the full-panel model, and good manner of convergence for mode 4 and for the part-model mode when taking into account the relative complexity of these modes. When using the realistic full-panel model the local mode is seen to occur at a buckling stress value which is over 15% higher than that at which initial, overall buckling occurs. 3.4. Buckling and vibration of NASA Example 1 panel and modifications Stroud et al. [33] have presented a detailed study of the buckling under combined longitudinal compression and shear of seven stiffened square panels with diaphragm ends, designated as NASA Examples 1–7. They have presented accurate results for buckling factors generated from a FEM approach, using the engineering analysis language (EAL) program and employing very fine meshes of four-node, rectangular, hybrid CPT elements. These NASA panel problems have since been used as test cases by Peshkam and Dawe [29] and by Wang and Dawe [3,5] to study the performance of the S-a FSM and the equal-spline FSM, respectively, in the contexts of both CPT and SDPT. Here, consideration is given to the analysis of the buckling and vibration of variants of the NASA Example 1 panel which include the incorporation of interior supports and of step changes in thickness. The

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Fig. 7. Buckling mode shapes of the stiffened bridge panel: (a)–(d) mode shapes 1–4, respectively.

original NASA Example 1 panel, with the possible pre-buckling loading, is shown in Fig. 8(a) whilst Fig. 8(b) gives details of one of the six repeating elements of the cross-section, with identifying numbers of the symmetrically-laminated component plates. The material of the panel is a laminated graphite-epoxy composite with EL ¼ 131 GN=m2 ;

ET ¼ 13 GN=m2 ;

GLT ¼ 6:41 GN=m2 ;

mLT ¼ 0:38:

Each of component plates 1 and 3 is of 12-layer ð45°=  45°=  45°=45°=0°=90°ÞS construction with the two 90°-plies each of thickness 1.2573 mm and all other plies of thickness 0.1397 mm. Each of the stiffener component plates 2 is of 10-layer ð45°=  45°=  45°=45°=0°ÞS construction with the two 0°-plies each of thickness 0.2794 mm and all other plies of thickness 0.1397 mm. The component plates have orthotropic in-plane properties but are anisotropic in bending to a very small extent [29,33]. Before considering modifications to the original panel, we examine the calculation, using the present G-s FSM, of buckling loads of the original panel when acted upon by pure compressive force Nx0 and by pure shear force Nxy0 per unit width of panel (i.e. the total forces are ANx0 and ANxy0 ). The compressive force is

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Fig. 8. The original NASA Example 1 panel: (a) full geometry and loading; (b) details of a repeating element.

distributed between the component plates on the assumption of uniform longitudinal strain which implies that the values of hr0x in the component plates are proportional to values of fA11  ðA212 =A22 Þg [29,33]. The shearing force is carried entirely by the main plate and so s0xy ¼ Nxy0 =h in component plates 1 and 3 and s0xy ¼ 0 in component plates 2. The panel ends are diaphragm supported (u 6¼ 0, v ¼ 0, w ¼ 0, ow=ox 6¼ 0) and the longitudinal edges are simply supported (u ¼ 0, v 6¼ 0, w ¼ 0, ow=oy 6¼ 0). In applying the spline FSM the full panel is modelled with one Superstrip5 representing each component plate. In the longitudinal direction the number of spline sections, q, is varied in two convergence studies, relating to the use firstly of equal spline sections (the E-s FSM approach) and secondly of variable spline sections (the G-s FSM approach). In the latter case a refinement of the knot spacings is introduced local to the longitudinal position x ¼ 0:782A. This would not be expected to have any beneficial effect (rather the reverse, in fact) as compared to using equal spline sections when dealing with the original panel as here, but is pertinent to a situation considered in what follows in which  astep change of thickness will be introduced at x ¼ 0:782A. The results of the convergence studies, for Nx0 cr and ðNxy0 Þcr , are given in Table 7 and the knot spacings used in the G-s FSM are also recorded. It can be seen that convergence of calculated buckling loads to values very close to the FEM values [33] does occur satisfactorily when using the G-s FSM, albeit somewhat more slowly than when using the E-s FSM. We now consider the situation in which the panel length is increased to 1397 mm (55 in.) and an interior line support is present at x ¼ 762 mm (30 in.) across the full width of the main plate. All other details are as before and calculations have been made of the first six natural frequencies, with an assumed mass density, q, of 1600 kg/m3 , and of the buckling loads in pure longitudinal compression and pure shear. The longitudinal modelling uses the same number, q, of spline sections in each of the two unequal spans, and within each span the knot spacings are uniform. The crosswise modelling uses a Superstrip4 for each of component plate types 1 and 3 and a Superstrip3 for each component plate type 2. The numerical results generated by the spline FSM as q is increased are recorded in Table 8. A very good manner of convergence is exhibited in

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Table 7 Buckling loads of original NASA Example 1 panel ðNx0 Þcr , kN/m

q 2 4 6 8 10

ðNxy0 Þcr , kN/m

E-s FSM

G-s FSM

E-s FSM

G-s FSM

178.93 175.34 175.22 175.20 –

185.97 180.39 177.67 175.23 175.21

313.69 274.53 271.52 271.29 –

327.18 304.10 292.17 272.19 271.29

FEM [33]

175.66

271.89

Knot position in the x-direction in the G-s FSM approach are as follows: q ¼ 2: (0, 23.46, 30) A/30, q ¼ 4: (0, 22, 23.46, 25, 30) A/30, q ¼ 6: (0,19, 22, 23.46, 25, 27.5, 30) A/30, q ¼ 8: (0, 8, 14, 19, 22, 23.46, 25, 27.5, 30) A/30, q ¼ 10: (0, 4, 8, 12, 16, 19, 22, 23.46, 25, 27.5, 30) A/30.

Table 8 Frequencies and buckling loads of modified two-span NASA Example 1 panel q per span 2 3 4 5 6

Natural frequencies, Hz

Buckling loads, kN/m

Mode 1

Mode 2

Mode 3

Mode 4

Mode 5

Mode 6

ðNx0 Þcr

ðNxy0 Þcr

120.82 117.12 116.59 116.51 116.51

151.28 148.14 147.66 147.58 147.57

222.36 199.06 193.24 192.10 191.70

236.19 220.04 217.12 215.98 215.56

256.60 222.59 219.66 219.58 219.57

308.17 280.09 275.45 274.42 274.03

210.21 200.79 199.86 199.79 199.86

425.15 335.92 312.08 304.32 301.91

general, although that for ðNxy0 Þcr is, not surprisingly, less good than for the other quantities. The mode shapes corresponding to the first six natural frequencies and to the two types of buckling are illustrated in Fig. 9 and are seen to be not unduly complicated. Finally, we return consideration to the single-span panel of length A ¼ 762 mm but now introduce thickness changes at the location x ¼ 0:782A. Two types of thickness change are introduced separately. In what will be referred to as the Type I panel the main plate is thinned down over the length of 0.218A by removing all plies except the two relatively-thick 90° plies. In the Type II panel it is all six stiffeners which are thinned down over the same length of 0.218A by removing all plies except the two 0° plies, resulting in locally very thin stiffeners. The crosswise modelling uses a Superstrip5 in representing each component plate and the longitudinal spline modelling uses refinement local to the thickness change. Numerical results for the first six natural frequencies and for the shear buckling load are given in Table 9, together with details of knot locations. (For the Type I panel, with the thinned-down main plate, the pre-buckling shear stress levels in component plates type 1 and 3 change in value at x ¼ 0:782A in inverse proportion to the change in thickness.) The manner of convergence of the results exhibited in Table 9 again appears to be good. It is noted that the predicted values of ðNxy0 Þcr for the two panel types differ only slightly, and that for the Type II panel the natural frequencies of modes 4, 5 and 6 lie very close together. 3.5. Buckling and vibration of NASA Example 6 panel and modifications In its original form, the second panel considered here from the work of Stroud et al. [33] is as shown in Fig. 10(a) but with the intermediate supports omitted. Details of one of its six repeating elements are shown

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Fig. 9. Mode shapes for the modified, two-span NASA Example 1 panel: (a)–(f) vibrational mode shapes 1–6, respectively; (g) compressive buckling mode shape; (h) shear buckling mode shape.

in Fig. 10(b). The material used is the same graphite-epoxy composite as defined for the NASA Example 1 panel. The component plates are symmetrically laminated. Each of plates 1, 3 and 5 is of 10-layer ð45°=  45°=  45°=45°=0°ÞS construction with the two 0°-plies each of thickness 0.42763 mm and all other plies of thickness 0.13917 mm. Each of plates 2 and 4 is of 8-layer ð45°=  45°=  45°=45°ÞS construction

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Table 9 Frequencies and shear buckling load of modified NASA Example 1 panel with two types of thickness change Panel type

q

ðNxy0 Þcr , kN/m

Natural frequencies Hz Mode 1

Mode 2

Mode 3

Mode 4

Mode 5

Mode 6

Type I

2 4 6 8 10

105.72 104.45 102.74 102.69 102.65

135.44 134.59 133.40 133.35 133.33

207.61 207.12 206.45 206.39 206.37

324.23 323.85 323.45 323.36 323.35

440.64 424.60 379.78 376.60 375.03

456.72 441.28 398.94 395.95 394.50

295.08 273.68 251.61 249.33 248.33

Type II

2 4 6 8 10

103.17 102.43 100.86 100.82 100.79

136.07 135.49 134.25 134.22 134.19

211.39 211.00 210.13 210.10 210.09

330.69 313.63 312.20 290.30 290.29

404.41 313.81 312.37 290.34 290.33

423.84 313.89 312.44 290.40 290.39

287.97 272.75 251.71 249.68 248.72

Knot positions in the x-direction are as follows: q ¼ 2: (0, 23.46, 30) A/30, q ¼ 4: (0, 22.5, 23.46, 24.5, 30) A/30, q ¼ 6: (0, 10, 19, 22.5, 23.46, 24.5, 30) A/30, q ¼ 8: (0, 8, 16, 19, 22.5, 23.46, 24.5, 27, 30) A/30, q ¼ 10: (0, 4, 8, 12, 16, 19, 22.5, 23.46, 24.5, 27, 30) A/30.

Fig. 10. The NASA Example 6 panel with intermediate supports: (a) full geometry and loading; (b) details of a repeating element.

with each ply of the same thickness of 0.13917 mm. Here again the component plates are orthotropic in their plane and slightly anisotropic out of their plane. The boundary conditions are the same as for the original NASA Example 1 panel. In a similar fashion to the approach adopted earlier when considering the NASA Example 1 panel, we first examine the use of the present spline FSM in calculating the buckling loads of the original NASA Example 6 panel when acted upon in turn by direct longitudinal compression and by pure shear. For the compression case, the distribution of applied pre-buckling stress is on the basis of uniform longitudinal

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Table 10 Buckling loads of original NASA Example 6 panel ðNx0 Þcr , kN/m

q 2 4 6 8 10

ðNxy0 Þcr , kN/m

E-s FSM

G-s FSM

E-s FSM

G-s FSM

264.66 260.87 260.75 260.75 –

276.66 268.44 264.30 260.76 260.74

242.15 221.47 217.97 217.55 –

255.26 235.73 228.09 218.54 217.51

FEM [33]

261.26

218.56

G-s FSM knot positions as defined in Table 7.

strain, as for NASA Example 1 panel. For the shear case, all the component plates have the same value of hs0xy . The adopted crosswise modelling uses a Superstrip4 for each of component plates 1 and 5, and a Superstrip5 for each of component plates 2, 3 and 4. Again, calculations are made using equal knot spacings and using variable knot spacings with refinement local to x ¼ 0:782A. The FSM results for various q are represented in Table 10 together with the FEM predictions of Ref. [33]. The manner of convergence of the FSM results is good and there is very close comparison with the FEM results. Now consideration is given to the analysis of the basic NASA Example 6 panel but with intermediate support along the panel length. The full panel length remains unchanged at A ¼ 762 mm but line supports are introduced across part of the panel width at the arbitrary location x ¼ 431:8 mm (17 in.), as shown in Fig. 10(a). These supports are applied only to the outer two top-surface flats at each side of the panel. The crosswise FSM modelling of the panel remains as described in the previous paragraph whilst in the longitudinal direction an equal number of spline sections is used in each of the two spans defined by 0 < x < 431:8 mm and 431:8 mm < x < 762 mm. Calculated values of the first six natural frequencies (q ¼ 1600 kg/m3 ) and of the buckling loads in pure compression and pure shear (for both of which the applied pre-buckling stress distribution is unchanged from the original panel) are presented in Table 11. Rapid and orderly convergence of the FSM results, with increase in q, is evidenced, and there is little difference between predictions based on q ¼ 2 and 5 (per span). This reflects the fact that the mode shapes of vibration and buckling, as depicted in Fig. 11, have a relatively simple shape longitudinally. Finally, we revert to considering again the original NASA Example 6 panel with no intermediate supports but now with the introduction of step change of thickness, at x ¼ 0:782A once more. The step change is achieved by removing all plies except the two central 0° plies from component plates 1, 3 and 5 of all six repeating elements of the panel cross-section, leaving these component plates with thickness 0.85526 mm over the region 0 < x < 0:782A. In the longitudinal modellings of this panel the adopted knot positions are the same as those used in considering earlier the NASA Example 1 panel, modified with thickness changes,

Table 11 Frequencies and buckling loads of modified, two-span NASA Example 6 panel q per span 2 3 4 5

Natural frequencies, Hz

Buckling loads, kN/m

Mode 1

Mode 2

Mode 3

Mode 4

Mode 5

Mode 6

ðNx0 Þcr

ðNxy0 Þcr

209.59 209.40 209.31 209.26

266.65 266.01 265.73 265.58

368.06 366.54 365.85 365.53

492.91 491.55 490.92 490.62

532.40 529.71 528.58 528.12

579.96 576.59 575.45 575.03

294.94 294.32 294.06 293.89

237.53 232.61 231.84 231.61

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Fig. 11. Mode shapes for the modified, two-span NASA Example 6 panel: (a)–(f) vibrational mode shapes 1–6, respectively; (g) compressive buckling mode shape; (h) shear buckling mode shape.

as defined in Table 9. The FSM results generated for the lowest six natural frequencies and the shear buckling force of the modified NASA Example 6 panel are presented in Table 12. (Note that the prebuckling shear stress levels again differ in the two regions of the panel, of course.) The results of Table 12 indicate rapid convergence of frequency values with increase in q but rather slower convergence of the shear buckling force. The mode shape for shear buckling is shown in Fig. 12 and reveals quite complicated behaviour in the thinner part of the panel.

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Table 12 Frequencies and shear buckling load of modified NASA Example 6 panel with thickness change q 2 4 6 8 10

ðNxy0 Þcr , kN/m

Natural frequencies, Hz Mode 1

Mode 2

Mode 3

Mode 4

Mode 5

Mode 6

226.95 221.89 218.36 218.28 218.22

244.11 238.69 234.94 234.86 234.80

273.90 267.82 263.66 263.58 263.51

317.58 310.34 305.53 305.42 305.35

354.36 343.45 337.07 336.84 336.72

358.40 347.53 341.00 340.76 340.64

117.47 84.246 66.105 63.681 61.942

Knot positions in x-direction are as specified in Table 9.

Fig. 12. Mode shape for shear buckling for the NASA Example 6 panel with thickness change.

4. Conclusions The general spline FSM has been developed for the analysis of the buckling and vibration of complicated, composite laminated plate structures. The approach provides greater versatility than do previous FSMs since the spline knot positions can be arbitrary, and this has allowed consideration of structures with intermediate supports and with step changes of properties along their length. Alongside this enhanced versatility, the powerful multi-level substructuring and solution procedures of previous related studies have been retained, thus allowing the accurate solution of problems of very considerable complexity which may have hundreds of thousands of freedoms initially. The use of the efficient developed software package has been demonstrated in a range of applications which encompasses both simple rectangular plates and complicated plate structures. Convergence with the number of spline sections used has been shown to be generally very good and comparison of the present predictions with the results of other studies is close. Such comparison has only been possible in limited situations but it is hoped that the results presented here for more-complicated situations will serve as benchmarks for other investigators. In this study the component plate properties have been based on the use of CPT, since the structures considered have thin geometry. For relatively thicker geometry the potentially significant effects of throughthickness shearing action could be included by employing first-order SDPT in a similar study to the one described here. Acknowledgements The authors are pleased to acknowledge that the work reported herein was supported financially by the Engineering and Physical Sciences Research Council and by the Ministry of Defence. They are grateful to

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