July 2012 

NASA/TM–2012-217589 

Conditions for Symmetries in the Buckle 

Patterns of Laminated-Composite Plates 

Michael P. Nemeth 

Langley Research Center, Hampton, Virginia 
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July 2012 

NASA/TM–2012-217589 

Conditions for Symmetries in the Buckle 

Patterns of Laminated-Composite Plates 

Michael P. Nemeth 

Langley Research Center, Hampton, Virginia 
Available from: 

NASA Center for AeroSpace Information 

7115 Standard Drive 

Hanover, MD 21076-1320 

443-757-5802 
1

Abstract

The term "symmetry" is practically synonomous with the terms "reduction" and 

"simplification" in the field of structural mechanics. Conditions for the existence of certain 

symmetries to exist in the buckle patterns of symmetrically laminated composite plates are 

presented. The plates considered have a general planform with cutouts, variable thickness and 

stiffnesses, and general support and loading conditions. The symmetry analysis is based on 

enforcing invariance of the corresponding eigenvalue problem for a group of coordinate 

transformations associated with buckle patterns commonly exhibited by symmetrically laminated 

plates. The buckle-pattern symmetries examined include a central point of inversion symmetry, 

one plane of reflective symmetry, and two planes of reflective symmetry.

Symbols

B1( ), B2( ) differential operators associated with the plate boundary 

conditions (see equations (8) and (9))

C set of curves bounding the plate planform

Ci interior boundary curve of a cutout in the plate planform

(see figure 1)

CM portion of boundary curve C with Mn = 0 specified

Co exterior boundary curve of the plate planform (see figure 1)

CV portion of boundary curve C with Vn = 0 specified

Cδ portion of boundary curve C with specified

CΔ portion of boundary curve C with w = 0 specified

D coordinate domain associated with the plate planform (see

figure 1)

D11, D12, D16, D22, D26, D66 plate bending and twisting stiffnesses (see equations (4)), in.-lb

G( ) differential operator associated with the equation governing

buckling (see equation (6b))

G1( ) differential operator associated with the plate boundary 

conditions (see equation (8))

∂w

∂n = 0
2

k constant used to specify symmetry or antisymmetry 

(see equation (15))

L( ) differential operator associated with the equation governing

buckling (see equation (6a))

Mn, Mns bending stress resultant defined by equations (3), in-lb/in.

Mx, My, Mxy bending stress resultant defined by equations (4), in-lb/in.

Mn, Mns differential operators defined by equations (7)

n inplane direction normal to a plate edge, in.

nx, ny components of the unit-magnitude vector field normal to a plate

edge (see figure 1)

prebuckling stress resultants (see equation (1)), lb/in.

Pn, Ps

 applied inplane edge loads (see equation (1)), lb/in.

Pk points of boundary curve C with Mns = 0 specified

Qn transverse shear stress resultant defined by equation (3e), lb/in.

Qn differential operator defined by equation (7f)

s arc-length parameter used to defined the parametric coordinates 

of the plate boundary, in.

T[ ] symmetry transformation operator (see equation (11b))

w(x, y) buckling-displacement eigenfunction (see equation (1))

Vn transverse shear stress resultant defined by equation (2a), lb/in.

(x, y) plate coordinates (see figure 1), in.

transformed plate coordinates (see equations (11)), in.

λ loading parameter and eigenvalue

λcr value of the eigenvalue at buckling

Nx

0

, Ny

0

, Nxy

0

(x, y)
3

Introduction

In the pursuit of high-performance aerospace structures, tailoring of material properties, 

support conditions, load introduction, and geometry play an important role. For example, previous 

studies have been conducted to find ways of using curvilinear fiber placement and thickness 

tailoring to reduce the severity of stress concentrations caused by cutouts, stiffener terminations, 

and other local stress raisers. Moreover, similar tailoring approaches have been used to find ways 

to enhance load-carrying, vibration and buckling performance. Specific examples that are 

representative of these efforts, spanning nearly forty years, are given in references 1-49. These 

examples, particularly the later ones, illustrate the complexity of the analyses and the 

computational expense generally involved in optimizing structures with tailored pointwise elastic 

properties. Thus, ways of reducing computational expense, such as developing relatively simple 

special-purpose analyses that exploit symmetries are likely to provide more lattitude for 

optimization efforts conducted in a building-block setting. This scenario is one example that 

illustrates that even in this era of powerful, general purpose computational tools for structural 

analysis, there remains a need for special-purpose analysis tools with a limited scope. 

The term "symmetry" is practically synonomous with the terms "reduction" and 

"simplification" in the field of structural mechanics. To illustrate this point, consider a generally 

laminated, fiber-reinforced rectangular plate with spatially uniform plies. For this general class of 

plates; extension, shearing, bending, and twisting deformations are fully coupled through the plate 

constitutive equations (see reference 50, pp. 187-236). As a result of this coupling, the plate 

behavior, as modelled by classical plate theory, is governed by three fully coupled partial 

differential equations that generally cannot be solved in closed form. In contrast, for 

symmetrically laminated plates, the coupling between inplane and bending deformation modes is 

absent, and the partial differential equations governing the behavior reduce to two equations for 

the inplane response and one for the out-of-plane bending response. Moreover, when coupling 

between inplane shear and extension is absent, the two partial differential equations for the 

inplane response become significantly simpler. Likewise, when coupling between bending and 

twisting is absent, the corresponding partial differential equation governing the out-of-plane 

bending behavior reduces to a significantly simpler form. Both of these simplifications 

correspond to a symmetry exhibited by laminates with orthogonal ply directions that are aligned 

with the rectangular-plate edges. Thus, the symmetries of the plate lamination correspond to a 

tremendous simplification of the mathematical analysis that governs laminated-plate behavior. As 

a result of these symmetry-based simplications, clarity and insight into the physical behavior is 

gained and the complexity of the corresponding mathematical analyses is reduced. For example, 

symmetry-based simplications lead to finite element analyses that use only a portion of the 

original computational domain. Likewise, these simplifications lead to the elimination of 

unneeded basis functions in classical Rayleigh-Ritz and Bubnov-Galerkin analyses.

Exploiting symmetry in structural mechanics is not a new idea. In particular, using symmetry 

arguments to simplify the pointwise, local constitutive equations of elastic materials is a 

cornerstone of the Theory of Elasticity that began in the 19th century.51 These symmetry-based 

simplifications ultimately contributed to the development of analytical solutions for a variety of 

practical boundary-value problems that yielded significant advances in structural mechanics 
4

technology. In more recent times, Renton52 used global, or overall, symmetry arguments to 

simplify buckling analyses of symmetrical frames. In 1973, Glockner53 demonstrated how to use 

symmetry arguments systematically to simplify structural analyses and reduce computational 

expense. A key ingredient of Glockner’s work is his axiom of symmetry, which is based on the 

physical observations and experience base within the structural mechanics discipline up to that 

time. In essence, it is not unreasonable to expect that the inherent symmetry of a structural system 

results in a corresponding symmetry of its mechanical response. More specifically, Glockner’s 

axiom is paraphrased as follows. Given a structure whose geometry, support and loading 

conditions, elastic stiffness and mass distributions, and thermal expansion properties exhibit a 

common certain type of symmetry, the structural response will exhibit the same type of symmetry.

Discussions of Glockner’s work are found in references 54-58. 

In 1973, MacNeal et. al.59 presented a method for reducing the number of simultaneous 

equations to be solved in a linear finite element analysis of a general structure composed of two 

or more identical sub-regions. In this method, the sub-regions are arranged symmetrically with 

respect to an axis and, as a result, the symmetry is referred to in reference 59 as cyclic symmetry. 

Later, in 1976, Evensen60 showed how to exploit symmetry in the analysis of structural dynamics 

problems. In particular, Evensen showed how to identify the smallest subdomain of a structure, 

and the appropriate boundary conditions on the subdomain, that will yield all the vibration modes 

of the overall structure. Also in 1976, and in 1977, Noor and his colleagues61-64 identified the 

symmetries exhibited by fiber-reinforced laminated-composite plates and shells, as a function of 

ply orientation and stacking sequence. In addition, the conditions on the geometry, stiffness and 

mass coefficients, loading and support conditions, stress resultants, and kinematics quantities are 

deduced from Glockner’s axiom of symmetry. Futhermore, a procedure is presented that exploits 

a given symmetry to obtain the smallest computational domain that can be used in linear and 

nonlinear finite element analyses. Everstine65 presented similar ideas in 1977, and in 1979 

Williams66 presented quantitative structural analysis results that showed substantial benefits in 

reducing computational costs by exploiting symmetry. In 1992, Li and Reid67 presented a more 

detailed examination of the symmetry conditions exhibited by laminated-composite structures, 

similar to that given by Noor and his colleagues. Several years later, Balaji et. al.68 presented a 

computer program for the design of radial impellers that is based on exploiting cyclic symmetry.

The concepts and results presented in reference 59-68 are applications of the mathematical 

discipline known as Group Theory; as demonstrated by Miller69 in 1981 and by Zheng et.al.70 in 

1982, and elucidated by Lobry and Broche71 in 1994. Additional studies that use Group Theory to 

exploit symmetry in structural analyses are found in references 72-102. Furthermore, reviews of 

this topic have been given in 2002 and 2009 by Zingoni.103-104 

For the most part, the previous studies cited herein are focused on simplifying the numerical 

solution of complicated boundary-value or eigenvalue problems. The studies presented by Noor 

and his colleagues appear to be the only ones that address the conditions required on the geometry, 

loading and support conditions, and material properties of plates and shells for certain prescribed 

symmetries in the response quantities to exist. These conditions are based on Glockner’s 

symmetry axiom previously stated herein and are presented as a direct consequence of the axiom. 

In contrast, the conditions for certain symmetries to exist in the response quantities of anisotropic 
5

and orthotropic plates and shells are derived herein based on enforcing invariance of a given 

eigenvalue problem; that is, enforcing the primative concept of symmetry. Thus, the objective of 

the present study is to present the details of a direct procedure for obtaining the necessary and 

sufficient conditions on the problem parameters for prescribed symmetries to exist in the response 

quantities, in a simple manner. This procedure is based on the treatment of Schröedinger’s 

equation given by Humi and Miller105 (pp. 169-170), and on the treatment of functions and 

operators presented by McWeeny106 (pp. 166-202). Just as the existence of symmetries were 

shown to yield reduction of the computational domain in many of the previous studies cited herein 

that focus on finite element analyses, the existence of symmetries lead to simplification in the 

basis functions that are used to represent the response quantities in classical Rayleigh-Ritz and 

Galerkin methods. This simplification is manifested as the elimination of unneeded waveforms, 

which improves computational efficiency of a special-purpose buckling analysis.

To achieve the objective of the present study, the eigenvalue problem associated with 

buckling of symmetrically laminated plates is considered first, without becoming entangled in 

mathematical jargon and notation that are not typically encountered by structural engineers. 

Specifically, symmetrically laminated plates of general shape with variable stiffness and general 

support and inplane loading conditions are examined and convenient differential operators are 

introduced. Then, symmetries associated with plate buckling are discussed and coordinate 

transformations that are used to characterize the symmetries are presented. After this step, the 

transformed eigenvalue problem is derived and the conditions that are necessary and sufficient for 

the plate-buckling symmetries to exist are given. More specifically, these conditions are given for 

buckle patterns with a central point of inversion symmetry, one plane of reflective symmetry, and 

two planes of reflective symmetry. Although the analysis presented subsequently is focused on 

symmetrically laminated plates, its extension to more complicated eigenvalue problems of fully 

anisotropic shells is straightforward. 

Equations Governing Buckling Behavior

The plates considered in the present study are idealized as perfectly flat and have a midplane 

with the general planform shown in figure 1. Coordinates of points of the midplane are given by 

(x, y) with respect to the x and y axes and are collectively referred to herein as the domain D. 

In general, the planform domain D is bounded by a continuous exterior curve Co and by one or 

more interior curves Ci that correspond to cutouts. Each of these curves generally consist of a 

finite number of smooth, connected pieces with coordinates that are specified parametrically by 

x = x(s) and y = y(s), where s is an arc length parameter. Collectively, the plate boundary is 

denoted by C = Co ∪ Ci

. At any given point of a smooth segment of a boundary curve, the unitmagnitude vector field normal to the segment is given by the components nx(x, y) and ny

(x, y), as 

shown in figure 1, where x = x(s), y = y(s), nx

(x, y) = dy/ds, and ny(x, y) = - dx/ds. Traversal of a 

boundary curve, associated with increasing values of the arc length parameter s, and orientation 

of the unit-magnitude vector field are presumed to be consistent with the standard convention of 

Stokes Integral Theorem; that is, the exterior boundary is traversed counterclockwise and each 

interior boundary is traversed clockwise, and the unit-magnitude vector field points away from 
6

the interior of the plate domain. As depicted in figure 1, the plate is loaded by force-per-unitlength inplane tractions λPn

(x, y) and λPs(x, y) that are normal and tangent to the boundary curve 

at a given point, respectively. In general, the tractions are presumed to be distributed over one or 

more of the segments forming the boundaries of the planform. The symbol λ represents a loading 

parameter that is increased monotonically from a value of zero to a value λcr that corresponds to 

buckling. 

The equations governing the buckling behavior of plates are found in references 107 and 108. 

From the equations in these references, and reference 50, the general form of the partial 

differential equation governing bifurcation buckling of perfectly flat plates with flexural-twist 

anisotropy, in terms of the Cartesian coordinates (x, y), is found to be given by

 (1)

for the set of points that form the domain D. The function w(x, y) is the out-of-plane buckling 

deflection and the functions denote the prebuckling, inplane 

stress resultants produced by the edge loads λPn(x, y) and λPs(x, y). Positive values of these stress 

resultants are shown in figure 1. The terms D11, D12, D22, and D66 are the plate bending stiffnesses 

that account for flexural orthotropy, and the terms D16 and D26 are the plate bending stiffnesses 

that account for flexural-twist anisotropy. For the general case of plates with variable thickness 

and nonuniform fiber placement, these stiffnesses are functions of the coordinates (x, y). Herein, 

these functions are presumed to be differentiable. Formulas for computing these bending 

stiffnesses for laminated-composite plates are found in reference 50. It is important to note that 

the determination of the prebuckling stress resultants generally requires the solution of a separate 

boundary-value problem. However, for several practical cases, they are statically determinant.

The general form of the boundary conditions that correspond to equation (1) are given by the 

conjugate pairs

 or (2a)

 or (2b)

evaluated on one or more segments of the boundary curves and 

 (2c)

∂

2

∂x

2 D11

∂

2

w

∂x2 + D12

∂

2

w

∂y2 + 2D16

∂

2

w

∂x∂y

+ ∂

2

∂y

2 D12

∂

2

w

∂x2 + D22

∂

2

w

∂y2 + 2D26

∂

2

w

∂x∂y

+ 2 ∂

2

∂x∂y

D16

∂

2

w

∂x

2 + D26

∂

2

w

∂y

2 + 2D66

∂

2

w

∂x∂y = λ Nx

0 ∂

2

w

∂x2 + Ny

0 ∂

2

w

∂y2 + 2Nxy

0 ∂

2

w

∂x∂y

λNx

0

(x, y), λNy

0

(x, y), and λNxy

0

(x, y)

w = 0 Vn ≡ Qn +

∂Mns

∂s

+ λ Pn

∂w

∂n

+ Ps

∂w

∂s = 0

∂w

∂n = 0 Mn = 0

Mns = 0
7

evaluated at points of the boundary curves with sharp corners (discontinuous derivatives). In these 

equations, 

 (3a)

 (3b)

 (3c)

 (3d)

 (3e)

where

 (4a)

 (4b)

 (4c)

are the bending stress resultants. Together, the homogeneous partial differential equation and the 

homogeneous boundary conditions, given by equations (1) and (2), constitute an eigenvalue 

problem in which λ is the eigenvalue and w(x, y) is the corresponding eigenfunction. An 

eigenfunction is referred to herein as a buckle pattern.

To facilitate further analysis presented herein, it is convenient to express equation (1) in 

operator form as

 (5)

where the linear differential operator L( ) is defined by

∂w

∂n = nx

∂w

∂x

+ ny

∂w

∂y

∂w

∂s = − ny

∂w

∂x

+ nx

∂w

∂y

Mn = Mxnx

2

+ 2Mxynxny + Myny

2

= 0

Mns = My − Mx nxny + nx

2

− ny

2

Mxy

Qn = ∂Mx

∂x

+

∂Mxy

∂y

nx +

∂Mxy

∂x

+

∂My

∂y

ny

Mx = – D11

∂

2

w

∂x

2 + D12

∂

2

w

∂y

2 + 2D16

∂

2

w

∂x∂y

My = – D12

∂

2

w

∂x2 + D22

∂

2

w

∂y

2 + 2D26

∂

2

w

∂x∂y

Mxy = – D16

∂

2

w

∂x2 + D26

∂

2

w

∂y2 + 2D66

∂

2

w

∂x∂y

L w(x, y) = λ G w(x, y)
8

 (6a)

and the linear differential operator G( ) is defined by

 (6b)

Similarly, the boundary conditions are expressed in operator from as follows. First, the boundary 

stress resultants are expressed in operator form as

 (7a)

 (7b)

 (7c)

where

 (7d)

L = ∂

2

∂x2 D11(x, y)

∂

2

∂x2 + D12(x, y)

∂

2

∂y2 + 2D16(x, y)

∂

2

∂x∂y

+ ∂

2

∂y2 D12(x, y)

∂

2

∂x

2 + D22(x, y)

∂

2

∂y

2 + 2D26(x, y)

∂

2

∂x∂y

+ 2 ∂

2

∂x∂y

D16(x, y)

∂

2

∂x

2 + D26(x, y)

∂

2

∂y

2 + 2D66(x, y)

∂

2

∂x∂y

G = Nx

0

(x, y)

∂

2

∂x

2 + Ny

0

(x, y)

∂

2

∂y

2 + 2Nxy

0

(x, y)

∂

2

∂x∂y

Mn = − Mn w(x, y)

Mns = − Mns w(x, y)

Qn = − Qn w(x, y)

Mn = nx

2

(x, y)D11(x, y) + ny

2

(x, y)D12(x, y) + 2nx(x, y) ny(x, y)D16(x, y)

∂

2

∂x2

+ nx

2

(x, y)D12(x, y) + ny

2

(x, y)D22(x, y) + 2nx(x, y) ny(x, y)D26(x, y)

∂

2

∂y2

+ 2 nx

2

(x, y)D16(x, y) + ny

2

(x, y)D26(x, y) + 2nx(x, y) ny(x, y)D66(x, y)

∂

2

∂x∂y
9

 (7e)

 (7f)

Then, equation (2a) is expressed in operator form as 

 (8a)

where for the displacement boundary condition w = 0,

 and (8b)

and for the conjugate force boundary condition Vn = 0, 

 (8c)

and

 (8d)

Likewise, equation (2b) is expressed in operator form as 

 (9a)

Mns = nx(x, y)ny(x, y) D12(x, y) − D11(x, y) + nx

2

(x, y) − ny

2

(x, y) D16(x, y)

∂

2

∂x2

+ nx(x, y)ny(x, y) D22(x, y) − D12(x, y) + nx

2

(x, y) − ny

2

(x, y) D26(x, y)

∂

2

∂y

2

+ 2 nx(x, y)ny(x, y) D26(x, y) − D16(x, y) + D66(x, y) nx

2

(x, y) − ny

2

(x, y)

∂

2

∂x∂y

Qn = nx(x, y)

∂

∂x

D11(x, y)

∂

2

∂x

2 + D12(x, y)

∂

2

∂y

2 + 2D16(x, y)

∂

2

∂x∂y

+ ny(x, y)

∂

∂y

D12(x, y)

∂

2

∂x2 + D22(x, y)

∂

2

∂y2 + 2D26(x, y)

∂

2

∂x∂y

+ nx(x, y)

∂

∂y

+ ny(x, y)

∂

∂x

D16(x, y)

∂

2

∂x

2 + D26(x, y)

∂

2

∂y

2 + 2D66(x, y)

∂

2

∂x∂y

B 1 w(x, y) = λ G1 w(x, y)

B 1 = G1 = 0

B 1 = Qn + nx(x, y)

∂

∂y − ny(x, y)

∂

∂x

Mns

G1 = nx(x, y)Pn(x, y) − ny(x, y)Ps

(x, y)

∂

∂x

+ nx(x, y)Ps

(x, y) + ny(x, y)Pn(x, y)

∂

∂y

B 2 w(x, y) = 0
10

where for the slope boundary condition 

 (9b)

and for the conjugate moment boundary condition Mn = 0, 

 (9c)

Furthermore, equation (2c) is expressed as 

 (10)

for points of the boundary curves with discontinuous derivatives.

Symmetries Associated With Plate Buckling

A buckle pattern w(x, y) that is a legitimate solution to the eigenvalue problem for plate 

buckling defined herein may exhibit certain types of symmetries, depending on the circumstances 

of a given problem. These circumstances are determined by the geometry of the midplane domain 

D, the distributions of the edge loading and supports, and the stiffnesses associated with the plate 

material composition and construction. For example, the buckle pattern may exhibit one plane of 

reflective symmetry, as shown by the contour plots in figures 2 and 3 for a rectangular plate 

domain, or one plane of reflective antisymmetry, as shown in figures 4 and 5. Inspection of figure 

2 reveals that a buckle pattern with one plane of reflective symmetry, given by x = 0, satisfies the 

condition w(-x, y) = w(x, y). Similarly, in figure 3 the plane of reflective symmetry given by y 

= 0 satisfies the condition w(x, -y) = w(x, y). In contrast, the contour plots in figures 4 and 5 

reveal that a buckle pattern with one plane of reflective antisymmetry satisfies the conditions 

w(-x, y) = - w(x, y) or w(x, -y) = -w(x, y), respectively. In many cases, a buckle pattern may 

exhibit two planes of reflective symmetry, as shown in figure 6, for which w(-x, y) = w(x, y) and 

w(x, -y) = w(x, y). Similarly, a buckle pattern may exhibit one plane of reflective symmetry and 

one plane of reflective antisymmetry where w(-x, y) = -w(x, y) and w(x, -y) = w(x, y) are 

satisfied (figure 7) or where w(-x, y) = w(x, y) and w(x, -y) = -w(x, y) are satisifed (figure 8). 

Furthemore, a buckle pattern may exhibit two planes of reflective antisymmetry where w(-x, y) 

= -w(x, y) and w(x, -y) = -w(x, y) are satisifed, as shown in figure 9. Two other types of 

symmetries often encountered are depicted in figures 10 and 11. In figure 10, the buckle pattern 

exhibits a central point of inversion symmetry for which w(-x, -y) = w(x, y), and in figure 11 the 

buckle pattern exhibits a central point of inversion antisymmetry for which w(-x, -y) = -w(x, y). 

∂w

∂n = 0,

B 2 = nx(x, y)

∂

∂x

+ ny(x, y)

∂

∂y

B 2 = Mn

Mns w(x, y) = 0
11

It is significant to note that buckle patterns simlilar to those with the reflective symmetries 

described above have been observed in experiments conducted on actual plates that are not 

perfectly symmetric, but may be reasonbly close for engineering purposes. Examples of real 

buckle patterns with reflective quasi-symmetries are found in references 109 and 110. An 

example of a real plate buckle pattern similar to the idealized central point of inversion symmetry 

shown in figure 10, is shown in figure 12. This plate is a highly anisotropic [+606/-606]s laminatedcomposite plate made of a typical high-strength graphite-epoxy material and loaded in 

compression. This plate was tested at the NASA Langley Research Center by the author circa 

1990. 

Although the symmetries associated with plate buckling have been illustrated for a 

rectangular plate domain, they can exist for certain types of nonrectangular domains. For 

example, an elliptical domain with its principal axes coincident with the x and y axes can exhibit 

the symmetries described in this section, depending on the loading and support conditions and on 

the material properties and corresponding stiffness distributions.

Necessary and Sufficient Conditions for a Prescribed Symmetry

Each of the symmetries depicted in figures 2-11 are characterized by a relationship between 

the values of the eigenfunction w at the points (x, y) ∈ D and at a corresponding set of related 

points that are reckoned with respect to the same coordinate frame. This relationship 

between the sets of points is quantified by a general unique orthogonal coordinate transformation 

of the domain D onto itself. Herein, this transformation is denoted symbolically by 

 and (11a)

or by

 (11b)

where T[ ] denotes the transformed image of the quantity within the brackets. In this context, 

equations (11) are viewed as a mapping of points in which the coordinate axes remain fixed. In 

addition, because the transformations are orthogonal, the lengths of lines in D, and the angles 

between them remain unaltered; that is, invariant. Moreover, it follows that since D is bounded 

by the curves forming the boundary C, that for every (x, y) ∈ C, the corresponding point 

is also an element of C for this class of transformations. Thus, the actual form of the functional 

relationships given by equations (11) places restrictions on the shape of the domain and its 

boundary; that is, T[ D ] = D and T[ C ] = C. 

Three coordinate transformations are used in the present study to characterize symmetries 

commonly exhibited by the buckle patterns of idealized plates described herein previously. The 

first transformation corresponds to a reflection of the points forming the region D ∪ C about the 

(x, y) ∈ D

x = x(x, y) y = y(x, y)

(x, y) = T (x, y)

x, y
12

plane x = 0, and is given by

 (12)

For this transformation, the domain D and boundary C are required to be symmetric about the 

plane x = 0, as shown in figure 13. The second transformation corresponds to a reflection of the 

points comprizing D ∪ C about the plane y = 0, and is given by

 (13)

For this transformation, the domain D and boundary C are required to be symmetric about the 

plane y = 0. The third transformation corresponds to a reflection of the points comprizing D ∪ C

about the plane normal to the line passing through the point (x, y) and the origin. This 

transformation is given by

 (14)

and is depicted in figure 14. It is important to observe that the partial derivatives of and with 

respect to x and y are constants for these three very special transformations. Moreover, the 

derivatives and the values of and are equal to either 1 or -1. 

Furthermore, the inverses are given by and These simplifications are used 

subsequently to simplify the analysis.

Necessary Conditions

Each of the three transformations given by equations (12)-(14) corresponds to a reflective 

symmetry about a given plane that may be either symmetric or antisymmetric about the plane. If 

a buckle pattern exists that is symmetric about the symmetry plane, with respect to a given 

coordinate transformation, then the buckle pattern defined in both sets of coordinates satisfies the 

condition In contrast, if a buckle pattern exists that is antisymmetric about the 

symmetry plane, then the buckle pattern defined in both coordinate systems satisfies the condition 

For convenience, both of these cases are represented herein by

 (15)

where k = 1 and k = -1 for buckle patterns that are symmetric and antisymmetric about the 

plane of reflection symmetry, respectively. The conditions that are necessary for the existence of 

x(x, y) = − x and y(x, y) = y

x(x, y) = x and y(x, y) = − y

x(x, y) = − x and y(x, y) = − y

x y

∂y

∂x = 0, ∂x

∂y = 0, ∂x

∂x

∂y

∂y

∂x

∂x = ∂x

∂x

∂y

∂y = ∂y

∂y

.

w(x, y) = w(x, y).

w(x, y) = − w(x, y).

w(x, y) = k w(x, y)
13

equation (15) are found by examining the transformed eigenvalue problem as follows. 

First, equations (5)-(10) are expressed in terms of the coordinates by simply replacing 

x with and y with to obtain expressions for the transformed eigenvalue problem. Then, 

the coordinate transformation given by equations (11) is introduced and the chain rule of 

differentiation is applied to get the transformed eigenvalue problem. 

 (16a)

for every (x, y) ∈ D and by

 (16b)

and

 (16c)

for every . Likewise,

 (16d)

for points of C where the derivatives are discontinuous. In these transformed equations,

 (17a)

 (17b)

where the arguments of the stiffnesses and the prebuckling stress resultants are given in terms of 

(x, y) by equations (11). In obtaining these equations, it is noted that even powers of the 

(x, y)

x y

T L w(x, y) = λ T G w(x, y)

T B 1 w(x, y) = λ T G1 w(x, y)

T B 2 w(x, y) = 0

(x(s), y(s)) ∈ C.

T Mns w(x, y) = 0

T L = ∂

2

∂x2 D11(x, y)

∂

2

∂x

2 + D12(x, y)

∂

2

∂y

2 + 2D16(x, y) ∂x

∂x

3

∂y

∂y

∂

2

∂x∂y

+ 2 ∂

2

∂x∂y

D16(x, y) ∂x

∂x

3

∂y

∂y

∂

2

∂x

2 + D26(x, y)∂x

∂x

∂y

∂y

3

∂

2

∂y2 + 2D66(x, y)

∂

2

∂x∂y

+ ∂

2

∂y

2 D12(x, y)

∂

2

∂x2 + D22(x, y)

∂

2

∂y2 + 2D26(x, y)∂x

∂x

∂y

∂y

3

∂

2

∂x∂y

T G = Nx

0

(x, y)

∂

2

∂x

2 + Ny

0

(x, y)

∂

2

∂y

2 + 2Nxy

0

(x, y)∂x

∂x

∂y

∂y

∂

2

∂x∂y
14

derivatives and are equal to unity for the particular class of transformations considered 

in the present study. Likewise, the operators in equations (16b)-(16c) are transformed in two steps 

as follows. First, the partial derivatives in these operators are transformed to get 

 (18)

 (19)

(20)

For the displacement boundary condition w = 0,

 and (21a)

For the conjugate force boundary condition Vn = 0,

∂x

∂x

∂y

∂y

T Mn = nx

2

(x, y)D11(x, y) + ny

2

(x, y)D12(x, y) + 2nx(x, y) ny(x, y)D16(x, y)

∂

2

∂x2

+ nx

2

(x, y)D12(x, y) + ny

2

(x, y)D22(x, y) + 2nx(x, y) ny(x, y)D26(x, y)

∂

2

∂y2

+ 2 nx

2

(x, y)D16(x, y) + ny

2

(x, y)D26(x, y) + 2nx(x, y) ny(x, y)D66(x, y) ∂x

∂x

∂y

∂y

∂

2

∂x∂y

T Mns = nx(x, y)ny(x, y) D12(x, y) − D11(x, y) + nx

2

(x, y) − ny

2

(x, y) D16(x, y)

∂

2

∂x2

+ nx(x, y)ny(x, y) D22(x, y) − D12(x, y) + nx

2

(x, y) − ny

2

(x, y) D26(x, y)

∂

2

∂y

2

+ 2 nx(x, y)ny(x, y) D26(x, y) − D16(x, y) + D66(x, y) nx

2

(x, y) − ny

2

(x, y) ∂x

∂x

∂y

∂y

∂

2

∂x∂y

T Qn = nx(x, y)

∂x

∂x

∂

∂x

D11(x, y)

∂

2

∂x

2 + D12(x, y)

∂

2

∂y

2 + 2D16(x, y)

∂x

∂x

∂y

∂y

∂

2

∂x∂y

+ ny(x, y)

∂y

∂y

∂

∂y

D12(x, y)

∂

2

∂x

2 + D22(x, y)

∂

2

∂y

2 + 2D26(x, y)∂x

∂x

∂y

∂y

∂

2

∂x∂y

+ nx(x, y)

∂y

∂y

∂

∂y

+ ny(x, y)∂x

∂x

∂

∂x

D16(x, y)

∂

2

∂x

2 + D26(x, y)

∂

2

∂y

2 + 2D66(x, y)

∂x

∂x

∂y

∂y

∂

2

∂x∂y

T B 1 = T G1 = 0
15

 (21b)

and

 (21c)

For the slope boundary condition 

 (22a)

and for the conjugate moment boundary condition Mn = 0,

 (22b)

In these equations for the transformed boundary conditions, and are the 

components of the unit-magnitude vector field normal at points of plate boundary C given by 

 and The second step in the transformation process 

deals with transforming these components. Specifically, the relationships between the 

components of the unit-magnitude normal-vector fields are obtained by applying the chain rule of 

differentiation to the definitions nx(x, y) = dy/ds, and ny(x, y) = - dx/ds. This process gives 

or (23a)

or (23b)

for the special class of coordinate transformations considered herein. Applying equations (23) to 

equations (18)-(20) yields 

T B 1 = T Qn + nx(x, y)

∂y

∂y

∂

∂y − ny(x, y)∂x

∂x

∂

∂x

T Mns

T G1 = nx(x, y)Pn(x, y) − ny(x, y)Ps

(x, y) ∂x

∂x

∂

∂x

+ nx(x, y)Ps

(x, y) + ny(x, y)Pn(x, y)

∂y

∂y

∂

∂y

∂w

∂n = 0,

T B 2 = nx(x, y)∂x

∂x

∂

∂x

+ ny(x, y)

∂y

∂y

∂

∂y

T B 2 = T Mn

nx(x, y) ny(x, y)

x =x(x(s), y(s)) = x(s) y = y(x(s), y(s)) = y(s).

nx(x, y) = nx(x, y)

∂y

∂y

nx(x, y) = nx(x, y)

∂y

∂y

ny(x, y) = ny(x, y)

∂x

∂x

ny(x, y) = ny(x, y)

∂x

∂x
16

 (24)

 (25)

 (26)

For the force boundary condition Vn

 = 0, equations (21b) and (21c) become

 (27a)

and

T Mn = nx

2

(x, y)D11(x, y) + ny

2

(x, y)D12(x, y) + 2nx(x, y)ny(x, y)

∂x

∂x

∂y

∂y

D16(x, y)

∂

2

∂x2

+ nx

2

(x, y)D12(x, y) + ny

2

(x, y)D22(x, y) + 2nx(x, y)ny(x, y)

∂x

∂x

∂y

∂y

D26(x, y)

∂

2

∂y2

+ 2 nx

2

(x, y)

∂x

∂x

∂y

∂y

D16(x, y) + ny

2

(x, y)

∂x

∂x

∂y

∂y

D26(x, y) + 2nx(x, y)ny(x, y)D66(x, y)

∂

2

∂x∂y

T Mns = nx(x, y)ny(x, y)

∂x

∂x

∂y

∂y

D12(x, y) − D11(x, y) + nx

2

(x, y) − ny

2

(x, y) D16(x, y)

∂

2

∂x2

+ nx(x, y)ny(x, y)

∂x

∂x

∂y

∂y

D22(x, y) − D12(x, y) + nx

2

(x, y) − ny

2

(x, y) D26(x, y)

∂

2

∂y2

+ 2 nx(x, y)ny(x, y) D26(x, y) − D16(x, y) + ∂x

∂x

∂y

∂y

D66(x, y) nx

2

(x, y) − ny

2

(x, y)

∂

2

∂x∂y

T Qn = nx(x, y)

∂x

∂x

∂y

∂y

∂

∂x

D11(x, y)

∂

2

∂x

2 + D12(x, y)

∂

2

∂y

2 + 2D16(x, y)∂x

∂x

∂y

∂y

∂

2

∂x∂y

+ ny(x, y)

∂x

∂x

∂y

∂y

∂

∂y

D12(x, y)

∂

2

∂x

2 + D22(x, y)

∂

2

∂y

2 + 2D26(x, y)∂x

∂x

∂y

∂y

∂

2

∂x∂y

+ ∂x

∂x

∂y

∂y

nx(x, y)

∂

∂y

+ ny(x, y)

∂

∂x ×

∂x

∂x

∂y

∂y

D16(x, y)

∂

2

∂x

2 + D26(x, y)

∂

2

∂y

2 + 2D66(x, y)

∂

2

∂x∂y

T B 1 = T Qn + nx(x, y)

∂

∂y − ny(x, y)

∂

∂x

T Mns
17

 (27b)

For the slope boundary condition equation (22a) becomes

(28)

The necessary conditions that correspond to the existence of a buckle pattern that is either 

symmetric or antisymmetric about a symmetry plane are obtained by enforcing invariance of the 

eigenvalue problem under the transformations given by equations (11). In other words, if equation 

(15) is valid, as presumed, the two corresponding eigenvalue problems must be identical. 

Specifically, invariance of the differential equation is given by 

 (29)

and yields the necessary conditions

 (30a)

 (30b)

 (30c)

 (30d)

 (30e)

 (30f)

on the stiffnesses and 

T G1 = nx(x, y)

∂y

∂y

∂x

∂x

Pn(x, y) − ny(x, y) Ps

(x, y)

∂

∂x

+ nx(x, y) Ps

(x, y) + ny(x, y)

∂x

∂x

∂y

∂y

Pn(x, y)

∂

∂y

∂w

∂n = 0,

T B 2 = nx(x, y)

∂y

∂y

∂x

∂x

∂

∂x

+ ny(x, y)

∂x

∂x

∂y

∂y

∂

∂y

T L w(x, y) − λG w(x, y) = L w(x, y) − λ G w(x, y)

D11(x, y) = D11(x, y)

D12(x, y) = D12(x, y)

D16(x, y) ∂x

∂x

3 ∂y

∂y = D16(x, y)

D22(x, y) = D22(x, y)

D26(x, y)∂x

∂x

∂y

∂y

3

= D26(x, y)

D66(x, y) = D66(x, y)
18

 (31a)

 (31b)

 (31c)

on the prebuckling stress resultants, for all values of (x, y) in the domain D. 

As for invariance of the boundary conditions, consider the general case in which w(x, y) = 0 

for an arbitrary subset of the plate boundary denoted by CΔ ⊂ C, as depicted in figures 13 and 14. 

In general, CΔ consists of a finite number of smooth disjoint boundary segments. Applying the 

coordinate transformation given by equations (11) to this displacement constraint, and enforcing 

equation (15), yields the necessary condition that w(x, y) = 0 on points of the plate boundary given 

by T[CΔ]. Similarly, if is specified on a given subset of the plate boundary Cδ ⊂ C, 

then the transformation of that is obtained from equations (16c) and (27), and 

enforcement of equation (15), yields the necessary condition that on T[Cδ

], for the 

class of transformations considered herein. 

Next, consider invariance of the boundary condition Mn(x, y) = 0, given by equations (7c) 

and (9c), for an arbitrary subset of the plate boundary denoted by CΜ ⊂ C. Invariance of the 

corresponding transformed boundary condition, given by equations (16c), (22b) and (24), yields 

the necessary conditions given by equations (28a), (28b), (28d), (28f), and

 (32a)

 (32b)

Equations (32a) and (32b) are effectively the same as equations (30c) and (30e), respectively, for 

the class of transformations considered herein. Applying these necessary conditions to equation 

(24) gives

 (33a)

Thus, the transformed boundary condition yields the additional necessary condition that 

Nx

0

(x, y) = Nx

0

(x, y)

Ny

0

(x, y) = Ny

0

(x, y)

Nxy

0

(x, y) ∂x

∂x

∂y

∂y = Nxy

0

(x, y)

∂w(x, y)

∂n = 0

∂w(x, y)

∂n = 0

∂w(x, y)

∂n = 0

D16(x, y)

∂x

∂x

∂y

∂y = D16(x, y)

D26(x, y)

∂x

∂x

∂y

∂y = D26(x, y)

T Mn = Mn
19

Mn(x, y) = 0 be specified on T[CΜ]. 

Now, consider the invariance of the operator that is given by equations (7b). 

Substituting the necessary conditions given by equations (28a), (28b), (28d), (28f), and (30) into 

the corresponding transformed quantity, given by equation (25), yields

 (33b)

Likewise, equation (26) becomes

 (34)

Next, substituting equations (33b) and (34) into equation (27a) yields

 (35)

Moreover, comparing equations (8d) and (27b) indicates that invariance of the operator 

yields the following necessary conditions on the applied loads

 (36a)

and

 (36b)

Enforcing these necessary conditions in equation (27b) yields

 (37)

Now, consider the boundary condition Vn(x, y) = 0, given by equations (8a), (8c), and (8d), 

for an arbitrary subset of the plate boundary denoted by CV ⊂ C. Invariance requirements for this 

boundary condition are obtained by substituting equation (15) into equation (16b) and making use 

of equations (35) and (37). This process yields the additional necessary condition that Vn(x, y) =

0 on T[CV].

Lastly, consider invariance of the boundary condition Mns(x, y) = 0, given by equations (7b) 

Mns

T Mns = ∂x

∂x

∂y

∂y

Mns

T Qn = ∂x

∂x

∂y

∂y

Qn

T B 1 = ∂x

∂x

∂y

∂y

B 1

G1

Pn(x, y) = Pn(x, y)

Ps

(x, y) = ∂x

∂x

∂y

∂y

Ps

(x, y)

T G1 = ∂x

∂x

∂y

∂y

G1
20

and (7e), for N points of the boundary with sharp corners, denoted by Pκ ∈ C, for k = 1, 2, ..., 

N. From equation (33), it is seen that invariance of this boundary condition produces the 

additional necessary condition that Mns(x, y) = 0 on T[Pκ], for k = 1, 2, ..., N.

Sufficient Conditions

In the previous section, a prescribed symmetry was presumed to exist and the conditions that 

are necessary for the eigenvalue problem to be invariant under a corresponding coordinate 

transformation were detemined. In this section, the following question is addressed: "If the 

eigenvalue problem is invariant under a specified coordinate transformation, does the invariance 

imply that the eigenfunctions exhibit the symmetry inherent to the coordinate transformation?" If 

the answer to this question is yes, then the necessary conditions for invariance of the eigenvalue 

problem are also sufficient conditions for the existence of the corresponding symmetry.

To obtain the sufficient conditions for a buckle pattern to possess one of the symmetries 

described herein, it is presumed that the eigenvalue problem is invariant under the action of a 

coordinate transformation of the general form given by equations (11). Let the buckle pattern 

w(x, y) be a solution to the original eigenvalue problem for a given eigenvalue, and let be 

the solution to the corresponding transformed eigenvalue problem, for the same eigenvalue. 

Because of the presumed invariance, both eigenfunctions are solutions to the original eigenvalue 

problem, for a given eigenvalue. Moreover, both eigenvalue problems possess the same spectrum 

of eigenfunctions and, as a result, there exists a pair of eigenfunctions from the two identical 

eigenvalue problems, for a given eigenvalue, that possesses a unique correspondance. 

Additionally, since the amplitude of an eigenfunction is indeterminate, it follows that every 

corresponding pair of eigenfunctions w(x, y) and can differ only in sign and magnitude. 

These differences are represented by a multiplicative real constant; that is,

 (38)

To have complete equivalence of the eigenfunctions appearing in this equation, kw(x, y) and 

 are scaled to have the same maximum amplitude. Thus, the constant k has admissible 

values of +1 and -1 and, as a result, equation (38) corresponds to equation (15) that was used to 

determine the necessary conditions. This correspondance between equations (15) and (38), 

implies that the necessary conditions given previously for the existence of symmetries are also 

sufficient conditions. 

Up to this point in the analysis, the functions that define the bending stiffnesses have been 

presumed to be differentiable on the domain D. However, for laminated plates that are tailored in 

a patchwork fashion, the bending-stiffness functions are sectionally differentiable. For these 

cases, the necessary and sufficient conditions for a buckle pattern to possess one of the 

symmetries described herein are found by partitioning the domain D into a finite number of 

subdomains and then enforcing the symmetry requirements for each subdomain and its boundary. 

w(x, y)

w(x, y)

w(x, y) = k w(x, y)

w(x, y)
21

This process places additional conditions on the distribution of the patchwork bending stiffnesses 

that must be satisfied. 

The Three Specific Cases

The general necessary and sufficient conditions for the existence of a buckle pattern that is 

either symmetric or antisymmetric about a plane of reflective symmetry were derived in the 

previous section of the present study. The corresponding results for the three specific cases 

defined by equations (12)-(14) are presented subsequently.

One Plane of Reflective Symmetry at x = 0

Consider the case in which the plane x = 0 is a proposed plane of reflective symmetry, as 

depicted in figure 2. The coordinate transformation used herein to characterize this type of 

symmetry is given by equation (12). For this transformations, equations (30) give 

 (39a)

 (39b)

 (39c)

 (39d)

 (39e)

 (39f)

on the stiffnesses and 

 (40a)

 (40b)

 (40c)

on the prebuckling stress resultants. Likewise, the inplane loads given by equations (36) must 

D11(− x, y) = D11(x, y)

D12(− x, y) = D12(x, y)

D16(− x, y) = − D16(x, y)

D22(− x, y) = D22(x, y)

D26(− x, y) = − D26(x, y)

D66(− x, y) = D66(x, y)

Nx

0

(− x, y) = Nx

0

(x, y)

Ny

0

(− x, y) = Ny

0

(x, y)

Nxy

0

(− x, y) = − Nxy

0

(x, y)
22

obey

 (41a)

 (41b)

In addition, the transformation requires that for all the condition 

must hold, as illustrated in figure 13. As shown in this figure, a cutout placed 

on the y-axis must be symmetric about this axis, and a cutout located off the y-axis must have a 

correspond cutout on the other side of the axis with the appropriate symmetry. Moreover, for any 

boundary condition specified over all or parts of C, with coordinates it must also be 

specified over the corresponding parts of C, with coordinates It is also important to 

note than when the bending stiffnesses are constants, D16 = D26 = 0 must hold. Each of these 

necessary and sufficient conditions are completely consistent with Glockner’s symmetry axiom 

as applied in references 61-64.

Two Planes of Reflective Symmetry

Consider the case in which y = 0 is a reflective-symmetry plane, in addition to the plane x 

= 0. Because the plane x = 0 is taken as a pre-existing symmetry plane, the corresponding 

necessary and sufficient conditions given in the previous section must be enforced prior to 

applying the transformation used to characterize the plane y = 0 as a reflective-symmetry plane. 

The coordinate transformation used herein to characterize symmetries with respect to the plane y 

= 0 is given by equation (13). For this transformations, and the previous set of necessary and 

sufficient conditions, equations (30) give 

 and (42a)

 and (42b)

 and (42c)

 and (42d)

 and (42e)

 and (42f)

Pn(− x, y) = Pn(x, y)

Ps

(− x, y) = − Ps(x, y)

(x(s), y(s)) ∈ C,

(− x(s), y(s)) ∈ C,

(x(s), y(s)),

(− x(s), y(s)).

D11(− x, y) = D11(x, y) D11(x, − y) = D11(x, y)

D12(− x, y) = D12(x, y) D12(x, − y) = D12(x, y)

D16(− x, y) = − D16(x, y) D16(x, − y) = − D16(x, y)

D22(− x, y) = D22(x, y) D22(x, − y) = D22(x, y)

D26(− x, y) = − D26(x, y) D26(x, − y) = − D26(x, y)

D66(− x, y) = D66(x, y) D66(x, − y) = D66(x, y)
23

on the stiffnesses and 

 and (43a)

 and (43b)

 and (43c)

on the prebuckling stress resultants. Likewise, the inplane loads given by equations (36) must 

obey

 and (44a)

 and (44b)

In addition, the transformation requires that for all the conditions 

and must hold. Thus, a cutout placed on the x-axis must be 

symmetric about this axis, and a cutout located off the x-axis must have a correspond cutout on 

the other side of the axis with the appropriate symmetry. Moreover, for any boundary condition 

specified over all or parts of C, with coordinates it must also be specified over the 

corresponding parts of C, with coordinates and Each of these 

necessary and sufficient conditions are also completely consistent with Glockner’s symmetry 

axiom as applied in references 61-64.

A Central Point of Inversion Symmetry or Antisymmetry

Consider the case in which the origin of the coordiinates is a proposed point of inversion 

symmetry or antisymmetry, as depicted in figures 10 and 11. The coordinate transformation used 

herein to characterize this type of symmetry is given by equation (14). For this transformation, 

equations (30) give 

 (45a)

 (45b)

 (45c)

 (45d)

Nx

0

(− x, y) = Nx

0

(x, y) Nx

0

(x, − y) = Nx

0

(x, y)

Ny

0

(− x, y) = Ny

0

(x, y) Ny

0

(x, − y) = Ny

0

(x, y)

Nxy

0

(− x, y) = − Nxy

0

(x, y) Nxy

0

(x, − y) = − Nxy

0

(x, y)

Pn(− x, y) = Pn(x, y) Pn(x, − y) = Pn(x, y)

Ps

(− x, y) = − Ps(x, y) Ps(x, − y) = − Ps(x, y)

(x(s), y(s)) ∈ C,

(− x(s), y(s)) ∈ C (x(s), − y(s)) ∈ C

(x(s), y(s)),

(− x(s), y(s)) (x(s), − y(s)).

D11(− x, − y) = D11(x, y)

D12(− x, − y) = D12(x, y)

D16(− x, − y) = D16(x, y)

D22(− x, − y) = D22(x, y)
24

 (45e)

 (45f)

on the stiffnesses and 

 (46a)

 (46b)

 (46c)

on the prebuckling stress resultants. Likewise, the inplane loads given by equations (36) must 

obey

 (47a)

 (47b)

In addition, the transformation requires that for all the condition 

must hold, as illustrated in figure 14. As shown in this figure, a cutout placed 

at the origin must have a central point of inversion symmetry, and a cutout located away from the 

origin must have a correspond cutout that produce a domain with the appropriate inversion 

symmetry. Moreover, for any boundary condition specified over all or parts of C, with 

coordinates it must also be specified over the corresponding parts of C, with 

coordinates Each of these necessary and sufficient conditions are also completely 

consistent with Glockner’s symmetry axiom as applied in references 61-64.

Concluding Remarks

Necessary and sufficient conditions for the existence of certain symmetries to exist in the 

buckle patterns of symmetrically laminated composite plates have been derived. The plates 

considered have a general planform with cutouts, variable thickness and stiffnesses, and general 

support and loading conditions. The specific form of the symmetry conditions are based on the 

invariance properties of the corresponding eigenvalue problem for a group of coordinate 

transformations associated with buckle patterns commonly exhibited by symmetrically laminated 

plates. The buckle-pattern symmetries examined include a central point of inversion symmetry, 

one plane of reflective symmetry, and two planes of reflective symmetry. The necessary and 

D26(− x, − y) = D26(x, y)

D66(− x, − y) = D66(x, y)

Nx

0

(− x, − y) = Nx

0

(x, y)

Ny

0

(− x, − y) = Ny

0

(x, y)

Nxy

0

(− x, − y) = Nxy

0

(x, y)

Pn(− x, − y) = Pn(x, y)

Ps

(− x, − y) = Ps(x, y)

(x(s), y(s)) ∈ C,

(− x(s), − y(s)) ∈ C

(x(s), y(s)),

(− x(s), − y(s)).
25

sufficient conditions presented herein are consistent with Glockner’s axiom of symmetry. 

Although only symmetrically laminated plates have been considered in the present study, the 

analysis presented herein is applicable to generally laminates plates and shells.

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y-axis

x-axis

Figure 1. Plate middle surface, coordinate system, prebuckling stress resultants, and

Exterior boundary, C o

Interior boundary, Ci nxi + nyj

i

j

λPn

Nx

0

Ny

0

Nxy

0

edge loads.

λPs

Domain, D
34

y-axis

x-axis

w(-x, w(x, y) y)

Figure 2. Buckle pattern with one plane of reflective symmetry given by

w(-x, y) = w(x, y).

y-axis

x-axis

w(x, y)

w(x, -y)

Figure 3. Buckle pattern with one plane of reflective symmetry given by

w(x, -y) = w(x, y).
35

y-axis

x-axis

w(-x, w(x, y) y)

Figure 4. Buckle pattern with one plane of reflective antisymmetry given by

w(-x, y) = -w(x, y).

y-axis

x-axis

w(x, y)

w(x, -y)

Figure 5. Buckle pattern with one plane of reflective antisymmetry given by

w(x, -y) = -w(x, y).
36

y-axis

x-axis

w(-x, w(x, y) y)

Figure 6. Buckle pattern with two planes of reflective symmetry given by

w(-x, y) = w(x, y) and w(x,-y) = w(x, y).

w(x, -y)

w(-x, -y)

y-axis

x-axis

w(-x, w(x, y) y)

Figure 7. Buckle pattern with one plane of reflective symmetry given by w(x,-y) = w(x,y) 

and one plane of reflective antisymmetry given by w(-x, y) = -w(x, y).

w(x, -y)

w(-x, -y)
37

y-axis

x-axis

w(-x, w(x, y) y)

Figure 8. Buckle pattern with one plane of reflective symmetry given by w(-x, y) = w(x, y) 

and one plane of reflective antisymmetry given by w(x, -y) = -w(x, y).

w(x, -y)

w(-x, -y)

y-axis

x-axis

w(-x, w(x, y) y)

Figure 9. Buckle pattern with two planes of reflective antisymmetry given by

w(-x, y) = -w(x, y) and w(x, -y) = -w(x, y).

w(x, -y)

w(-x, -y)
38

y-axis

x-axis

w(x, y)

w(-x, -y)

Figure 10. Buckle pattern with a central point of inversion symmetry 

given by w(-x, -y) = w(x, y).

y-axis

x-axis

w(x, y)

w(-x, -y)

Figure 11. Buckle pattern with a central point of inversion antisymmetry 

given by w(-x, -y) = -w(x, y).
39

Figure 12. Buckle pattern of a compression-loaded [+606/-606]s laminated-composite 

plate made of a typical high-strength graphite-epoxy material. 

Load

Simply

supported

edge

Simply

supported

edge

Clamped edge

Clamped edge

Moveable loading platten

Fixed loading platten

y-axis

x-axis

Figure 13. Geometry of domain D and boundary C imposed by the coordinate

transformation (x, y) = (− x, y).

Domain, D

CΔ

T[CΔ]
40

y-axis

x-axis

Figure 14. Geometry of domain D and boundary C imposed by the coordinate

transformation 

Domain, D

(x, y) = (− x, − y).

CΔ

T[CΔ]
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Technical Memorandum

 4. TITLE AND SUBTITLE

Conditions for Symmetries in the Buckle Patterns of Laminated-Composite 

Plates

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Nemeth, Michael P.

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NASA Langley Research Center

Hampton, VA 23681-2199

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National Aeronautics and Space Administration

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14. ABSTRACT

Conditions for the existence of certain symmetries to exist in the buckle patterns of symmetrically laminated composite plates 

are presented. The plates considered have a general planform with cutouts, variable thickness and stiffnesses, and general 

support and loading conditions. The symmetry analysis is based on enforcing invariance of the corresponding eigenvalue 

problem for a group of coordinate transformations associated with buckle patterns commonly exhibited by symmetrically 

laminated plates. The buckle-pattern symmetries examined include a central point of inversion symmetry, one plane of 

reflective symmetry, and two planes of reflective symmetry.

15. SUBJECT TERMS

Buckling; Laminated composites; Plates; Symmetry

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