May 2012 

NASA/TM–2012-217570 

Cubic Zig-Zag Enrichment of the Classical 

Kirchhoff Kinematics for Laminated and 

Sandwich Plates 

Michael P. Nemeth 

Langley Research Center, Hampton, Virginia 
NASA STI Program . . . in Profile

Since its founding, NASA has been dedicated to the 

advancement of aeronautics and space science. The 

NASA scientific and technical information (STI) 

program plays a key part in helping NASA maintain 

this important role. 

The NASA STI program operates under the 

auspices of the Agency Chief Information Officer. 

It collects, organizes, provides for archiving, and 

disseminates NASA’s STI. The NASA STI 

program provides access to the NASA Aeronautics 

and Space Database and its public interface, the 

NASA Technical Report Server, thus providing one 

of the largest collections of aeronautical and space 

science STI in the world. Results are published in 

both non-NASA channels and by NASA in the 

NASA STI Report Series, which includes the 

following report types: 

 TECHNICAL PUBLICATION. Reports of 

completed research or a major significant phase 

of research that present the results of NASA 

Programs and include extensive data or 

theoretical analysis. Includes compilations of 

significant scientific and technical data and 

information deemed to be of continuing 

reference value. NASA counterpart of peerreviewed formal professional papers, but 

having less stringent limitations on manuscript 

length and extent of graphic presentations. 

 TECHNICAL MEMORANDUM. Scientific 

and technical findings that are preliminary or of 

specialized interest, e.g., quick release reports, 

working papers, and bibliographies that contain 

minimal annotation. Does not contain extensive 

analysis. 

 CONTRACTOR REPORT. Scientific and 

technical findings by NASA-sponsored 

contractors and grantees. 

 CONFERENCE PUBLICATION. 

Collected papers from scientific and 

technical conferences, symposia, seminars, 

or other meetings sponsored or cosponsored by NASA. 

 SPECIAL PUBLICATION. Scientific, 

technical, or historical information from 

NASA programs, projects, and missions, 

often concerned with subjects having 

substantial public interest. 

 TECHNICAL TRANSLATION. 

English-language translations of foreign 

scientific and technical material pertinent to 

NASA’s mission. 

Specialized services also include organizing 

and publishing research results, distributing 

specialized research announcements and feeds, 

providing information desk and personal search 

support, and enabling data exchange services. 

For more information about the NASA STI 

program, see the following: 

 Access the NASA STI program home page 

at http://www.sti.nasa.gov

 E-mail your question to help@sti.nasa.gov

 Fax your question to the NASA STI 

Information Desk at 443-757-5803 

 Phone the NASA STI Information Desk at 

443-757-5802 

 Write to: 

 STI Information Desk 

 NASA Center for AeroSpace Information 

 7115 Standard Drive 

 Hanover, MD 21076-1320 
National Aeronautics and 

Space Administration 

Langley Research Center 

Hampton, Virginia 23681-2199 

 

May 2012 

NASA/TM–2012-217570 

Cubic Zig-Zag Enrichment of the Classical 

Kirchhoff Kinematics for Laminated and 

Sandwich 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

A detailed analysis and examples are presented that show how to enrich the kinematics of 

classical Kirchhoff plate theory by appending them with a set of continuous piecewise-cubic 

functions. This analysis is used to obtain functions that contain the effects of laminate 

heterogeneity and asymmetry on the variations of the inplane displacements and transverse 

shearing stresses, for use with a {3, 0} plate theory in which these distributions are specified 

apriori. The functions used for the enrichment are based on the improved zig-zag plate theory 

presented recently by Tessler, Di Scuva, and Gherlone. With the approach presented herein, the 

inplane displacements are represented by a set of continuous piecewise-cubic functions, and the 

transverse shearing stresses and strains are represented by a set of piecewise-quadratic functions 

that are discontinuous at the ply interfaces.

Introduction

Many variants of refined theories for laminated-composite and sandwich plates have 

appeared in the technical literature for many years.1-10 Each of these theories generally have a 

different degree of complexity and a corresponding range of validity. Plates that exhibit a 

relatively large amount of transverse shearing flexibility typically require a refined theory in order 

to obtain accurate predictions of their structural response, particularly when the characteristic 

dimension of the response is on the order of the plate thickness. In contrast, for plate-response 

phenomena that consist of deformations with relatively large characteristic dimensions, simpler 

theories that are less complex and computationally expensive can be utilized to obtain useful 

results.

The most basic plate theory that accounts for transverse shearing deformations is known as 

the first-order shear-deformation theory (FSDT). With regards to its origins, it is also commonly 

referred to as the Reissner-Mindlin plate theory.11-13 In this theory, the deformation of a plate is 

expressed in terms of three translational displacements and two rotations associated with throughthe-thickness shearing, in contast to the classical plate theory (CLPT) conceived by Kirchhoff that 

uses only the three translational displacements. A drawback associated with the relative simplicity 

of FSDT is that the through-the-thickness distributions of the transverse shearing strains are 

approximated as uniform distributions. These approximate strain distributions correspond to 

transverse shearing stresses that violate the traction boundary conditions on the top and bottom 

bounding surfaces of a plate. As a result, a shear-correction factor must be used to obtain adequate 

results.

One step up from FSDT, is a group of relatively popular plate theories that satisfy the traction 

boundary conditions on the top and bottom bounding surfaces of a plate and avoid the need for a 

shear-correction factor.14-41 This boundary-condition problem is mitigated in these theories, at the 

expense of more complexity, by appending the kinematics of CLPT with terms that account for 

transverse shearing by using two additional kinemetic variables and presumed distributions of 

through-the thickness shearing strains. Typically, these distributions are presumed to be 

symmetric over the plate thickness, even for laminated plates with asymmetric ply distributions. 
2

Although, variations in actual shape of the transverse-shearing strain distributions for moderately 

thick laminated and sandwich plates may have a relatively small effect on the prediction of global 

response phenomena like buckling loads and fundamental frequencies, the mismatch between the 

distribution symmetry and ply lay-up symmetry is a fundamental issue that should be addressed. 

This issue is addressed in the present paper by enriching the CLPT kinematics with zig-zag 

variations to obtained transverse shearing strain distributions that are based on the actual throughthe-thickness construction of a plate and without introducing any aditional kinematic variables. 

To accomplish this task, the baseline plate kinematics are presented first. Then, the details of the 

procedure used to enrich the kinematics are given. Lastly, examples are presented for an 

unsymmetric laminate and a sandwich plate with an isotropic core and two identical laminated 

face plates.

The Baseline Plate Kinematics

Consider a uniform-thickness plate and the coordinate system shown in figure 1. Points of the 

plate are located by the orthogonal Cartesian coordinates (x, y, z), where (x, y, 0), 0 ≤ x ≤ a, and 

0 ≤ y ≤ b define points of the rectangular midplane. The plate lengths in the x- and y-directions 

are denoted by a and b, respectively. Points above and below the midplane are given by the 

nonzero z-coordinate values within -h/2 ≤ z ≤ h/2, where h is the plate thickness dimension. The 

principal material coordinate system at the point (x, y, z) is shown in figure 2. For laminated 

plates, the fiber orientation is denoted by the angle θ(z) shown in figure 2.

 The baseline plate kinematics used in the present study are defined by 

 (1a)

 (1b)

 (1c)

where U(x, y, z), V(x, y, z), and W(x, y, z) are the displacement-field components in the x-, y-, 

and z-coordinate directions, respectively, of the material point (x, y, z). The functions u(x, y) and 

v(x, y) are the corresponding inplane displacements of the material point (x, y, 0) of the midplane, 

and w(x, y) is the out-of-plane displacements of the material point (x, y, 0). The functions Λx(z) 

and Λy(z) are selected to satisfy the traction-free boundary conditions on the transverse shear 

stresses at the bounding surfaces of the plate given by the coordinates . In addition, 

Λx(z) and Λy(z) are required to satisfy the conditions U(x, y, 0) = u(x, y) and V(x, y, 0) = v(x, y), 

but are otherwise arbitrary. The unknown functions Hxz(x, y) and Hyz(x, y) account for the effects 

of transverse shear deformation. Inspection of equations (1) reveals that CPT is contained as a 

proper, well-defined subset that is obtained by setting Λx(z) and Λy(z) equal to zero. Several 

choices for Λx(z) and Λy(z) have appeared in the technical literature; for example, see reference 

32. When Λx(z) and Λy(z) are specified as cubic polynomials in z, the plate theory is sometimes 

U x, y, z = u x, y – z

∂w x, y

∂x

+ Λx(z) Hxz x, y

V x, y, z = v x, y – z

∂w x, y

∂y

+ Λy(z) Hyz x, y

W x, y, z = w x, y

(x, y, ± h

2)
3

referred to as a {3, 0} theory, which indicates that the inplane displacements U(x, y, z) and

V(x, y, z) are cubic and the out-of-plane displacement W(x, y, z) is constant across the plate 

thickness.

The linear strain-displacement relations used in the present study are obtained by substituting 

equations (1) into linear strain-displacement relations of the linear theory of elasticity, given by

 (2a)

 (2b)

 (2c)

 (2d)

 (2e)

 (2f)

First, consider the transverse shearing strains defined by equations (2e) and (2f). Substituting the 

presumed displacement field, given by equations (1), into these strain-displacement expressions 

gives

 (3)

At the plate midplane z = 0, the transverse shearing strains are given by

 (4)

Using equation (4), the inplane displacement field is expressed as

 (5a)

 (5b)

where

εxx x, y, z = ∂U

∂x

εyy x, y, z = ∂V

∂y

ε zz x, y, z = ∂W

∂z

γ xy x, y, z = ∂U

∂x

+ ∂V

∂y

γ xz x, y, z = ∂U

∂z

+ ∂W

∂x

γ yz x, y, z = ∂V

∂z

+ ∂W

∂y

γ xz x, y, z

γ yz x, y, z = Λx′(z) Hxz x, y

Λy′(z) Hyz x, y

γ xz

o

x, y

γ yz

o

x, y ≡ γ xz x, y, 0

γ yz x, y, 0 = Λx′(0) Hxz x, y

Λy′(0) Hyz x, y

U x, y, z = u x, y – z

∂w x, y

∂x

+ Fx(z) γ xz

o

x, y

V x, y, z = v x, y – z

∂w x, y

∂y

+ Fy(z) γ yz

o

x, y
4

 (5c)

 (5d)

In these equations, and are required, and Fx(0) = Fy(0) = 0 are required 

to obtain U(x, y, 0) = u(x, y) and V(x, y, 0) = v(x, y). Substituting equations (1c), (5a), and (5b) into 

the remaining strain-displacement expressions yields the following expressions for the nonzero 

strains

 (6)

and

 (7)

Next, to get the complete picture of the plate deformation, as shown in figures 3-7, consider 

the components of the linear rotation vector given by

 (8a)

 (8b)

 (8c)

Substituting equations (1c), (5a), and (5b) into these expressions yields

 

 (9a)

 (9b)

 (9c)

Fx(z) = Λx(z)

Λx′(0)

Fy(z) = Λy(z)

Λy′(0)

Λx′(0) ≠ 0 Λy′(0) ≠ 0

εxx x, y, z

εyy x, y, z

γ xy x, y, z

=

∂u x, y

∂x

∂v x, y

∂y

∂v x, y

∂x

+

∂u x, y

∂y

− z

∂

2

w x, y

∂x2

∂

2

w x, y

∂y2

2 ∂

2

w x, y

∂x∂y

+

Fx(z)

∂γxz

o

x, y

∂x

Fy(z)

∂γyz

o

∂y

x, y

Fy(z)

∂γyz

o

x, y

∂x + Fx(z)

∂γxz

o

x, y

∂y

γ xz x, y, z

γ yz x, y, z = Fx′(z) γ xz

o

x, y

Fy′(z) γ yz

o

x, y

ωx x, y, z = 1

2

∂W

∂y – ∂V

∂z

ωy x, y, z = 1

2

∂U

∂z – ∂W∂x

ωz x, y, z = 1

2

∂V

∂x – ∂U

∂y

ωx x, y, z = ∂w x, y

∂y − 1

2Fy′(z) γ yz

o

x, y

ωy x, y, z = 1

2Fx′(z) γ xz

o

x, y − ∂w x, y

∂x

ωz x, y, z = 1

2

∂v x, y

∂x – ∂u x, y∂y

+ Fy(z)

∂γ yz

o

x, y

∂x – Fx(z)

∂γ xz

o

x, y

∂y
5

At the plate midplane,

 (10a)

 (10b)

 (10c)

where Fx(0) = Fy(0) = 0 have been used and, as shown by equations (5c) and (5d), it is noted that 

.

The constitutive equations used in the present study are those for a plate made of one or more 

layers of linear elastic, specially orthotropic materials that are in a state of plane stress. These 

equations, referred to the plate (x, y, z) coordinate system are given by

 (11a)

 (11b)

where the superscript (k) denotes the kth ply of a laminated plate with a total of N plies. The 

 terms are the transformed, reduced (plane stress) stiffnesses of classical laminated-plate 

theory, for the kth ply, and the terms are the corresponding stiffnesses of a generally 

orthotropic ply.42 For a heterogeneous, laminated plate with N plies, both the and the 

terms are functions of the through-the-thickness coordinate z, as indicated by the superscript (k). 

In particular, the fiber orientation of the kth ply within a laminated plate is given by the angle θ(z), 

shown in figure 2, where the values of z correspond to points within the thickness of the kth ply. 

From equation (11b), it follows that the transverse shearing stresses on the top and bottom plate 

surfaces vanish provided that the corresponding transverse shearing strains vanish. Thus, from 

equation (7), it also follows that 

 (12)

is required for the transverse shearing stresses on the top and bottom plate surfaces to vanish.

ωx x, y, 0 ≡ ωx

o

x, y = ∂w x, y

∂y – 1

2γ yz

o

x, y

ωy x, y, 0 ≡ ωy

o

x, y = 1

2γ xz

o

x, y – ∂w x, y

∂x

ωz x, y, 0 ≡ ωz

o

x, y = 1

2

∂v x, y

∂x − ∂u x, y∂y

Fx′(0) = Fy′(0) = 1

σxx x, y, z

σyy x, y, z

σxy x, y, z

=

Q11

k

Q12

k

Q16

k

Q12

k

Q22

k

Q26

k

Q16

k

Q26

k

Q66

k

εxx x, y, z

εyy x, y, z

γ xy x, y, z

σyz x, y, z

σxz x, y, z = C44

k

C45

k

C45

k

C55

k

γ yz x, y, z

γ xz x, y, z

Qij

k

Cij

k

Qij

k

Cij

k

Fx′ ± h

2 = Fy′ ± h2 = 0
6

Equations (1c), (5a), and (5b); the conditions defined by equation (12); and Fx(0) = Fy(0) = 0 

define completely the kinemetics of the baseline theory used herein. The kinematics of CLPT are 

recovered from these equations by specifying Fx(z) = Fy(z) = 0. Likewise, the kinematics of FSDT 

are recovered from these equations by specifying Fx(z) = Fy(z) = z such that equations (5a) and 

(5b) become

 (13a)

 (13b)

The corresponding FSDT kinematics, in the x-z plane, are illustrated in figures 3 and 4. In 

contrast, the kinematics in the x-z plane of the baseline theory presented herein are illustrated in 

figures 5-7.

Zig-Zag Enrichment of the Kinematics

In the baseline kinematics, the functions Fx(z) and Fy(z) are typically specified as 

continuous functions, with continuous derivative, such as those given in reference 32. Moreover, 

for the most part, the functions are specified as antisymmetric functions; that is, Fx(- z) = - Fx(z) 

and Fy(-z) = - Fy(z). This antisymmetry with respect to the plate midplane appears to be 

inconsistent with laminates that exhibit a high degree of asymmetry in their ply constitutive 

properties. To obtain expressions for Fx(z) and Fy(z) that are based on actual laminate 

construction, particularly for asymmetric laminates, these functions are partitioned herein as 

follows

 (14a)

 (14b)

where Hx(z) and Hy(z) are continuous functions, with continuous derivatives, such as those given 

in reference 32 which include linear, cubic, trigonometric, hyperbolic, and exponential functions. 

The terms and are piecewise linear functions with piecewise constant 

derivatives that are referred to herein as zig-zag functions. In these equations, the superscript (k) 

also denotes the ply number of a laminate composed of N plies. Moreover, for the kth ply, it 

follows that z(k-1) ≤ z ≤ z(k), where z(k-1) and z(k) are the ply coordinates shown in figure 8 for the 

x-z plane. Substituting equations (14) into equations (5a) and (5b) yields the inplane 

displacements

 (15a)

U x, y, z = u x, y + z γ xz

o

x, y – ∂w x, y

∂x

V x, y, z = v x, y + z γ yz

o

x, y – ∂w x, y

∂y

Fx z = Hx z + Φx

k

z

Fy z = Hy z + Φy

k

z

Φx

k

z Φy

k

z

U x, y, z = u x, y – z

∂w x, y

∂x

+ Hx z + Φx

k

z γ xz

o

x, y
7

 (15b)

The user-specified functions Hx(z) and Hy(z) define a smooth distribution of displacements 

associated with transverse shearing deformations and and are corrections 

associated with laminate heterogeneity and asymmetry.

Following the refined zig-zag theory of Tessler et. al.,43 ply interface displacements 

(see figure 8) and are defined in terms of the zig-zag functions by 

 (16a)

 (16b)

for k = 0, 1, ..., N; with . Within the kth ply, z(k-1) ≤ z ≤ z(k) and linear 

interpolation gives

 (17a)

 (17b)

where 

 (18a)

 (18b)

and where h(k) = z(k) - z(k-1) is thickness of the kth ply, as shown in figure 8. For z(k-1) ≤ z ≤ z(k), it 

follows that the piecewise-constant derivatives of the zig-zag functions are given by

 (19a)

 (19b)

Thus, differentiating equations (14) gives

 (20a)

V x, y, z = v x, y – z

∂w x, y

∂y

+ Hy z + Φy

k

z γ yz

o

x, y

Φx

k

z Φy

k

z

Δx

k

Δy

k

Δx

k

= Φx

k

z k

Δy

k

= Φy

k

z k

Δx

0

= Δy

0

= Δx

N

= Δy

N

≡ 0

Φx

k

z = Δx

k−1

+ βx

k

z − z k−1

Φy

k

z = Δy

k−1

+ βy

k

z − z k−1

βx

k

≡ Δx

k

− Δx

k−1

h k

βy

k

≡ Δy

k

− Δy

k−1

h k

Φx

k

′ z = βx

k

Φy

k

′ z = βy

k

Fx′ z = Hx′ z + βx

k
8

 (20b)

Substituting these results into equation (7) yields

 (21)

for the transverse shearing strains.

At this point in the analysis, it is convenient to use equation (21) to express constitutive 

equations (11b) as

 (22)

for laminated plates, where are constant-valued for material points of the kth

ply located by z(k-1) ≤ z ≤ z(k). To proceed following the refined theory of Tessler et. al.,43 the 

transverse shearing stresses are partitioned into

 (23a)

where

 (23b)

Next, the ply parameters

 (24a)

 (24b)

are introduced to eliminate and from equation (23b). Because and appear as 

independent unknowns in the analysis, they can be normalized by requiring and 

 for k = 1, 2, ..., N such that 

Fy′ z = Hy′ z + βy

k

γ xz x, y, z

γ yz x, y, z = Hx′ z + βx

k

γ xz

o

x, y

Hy′ z + βy

k

γ yz

o

x, y

σyz x, y, z

σxz x, y, z = C44

k

C45

k

C45

k

C55

k

Hy′ z + βy

k

γ yz

o

x, y

Hx′ z + βx

k

γ xz

o

x, y

C44

k

, C45

k

, and C55

k

σyz x, y, z

σxz x, y, z = C44

k

C45

k

C45

k

C55

k

Hy′ z − 1 γ yz

o

x, y

Hx′ z − 1 γ xz

o

x, y

+

σyz

zz

σxz

zz

σyz

zz

σxz

zz = C44

k

C45

k

C45

k

C55

k

1 + βy

k

γ yz

o

x, y

1 + βx

k

γ xz

o

x, y

Gx

k

= C55

k

1 + βx

k

Gy

k

= C44

k

1 + βy

k

βx

k

βy

k

βx

k

βy

k

Gx

k

= Gx

Gy

k

= Gy
9

 (25a)

 (25b)

and

 (26a)

 (26b)

Following Tessler et. al.,43 the constants Gx and Gy are found by first noting that 

 (27)

Then, using equations (18) gives

 (28)

and using gives the desired result 

 (29)

Therefore, the constants Gx and Gy are found by integrating equations (25) to get

 (30a)

βx

k

= Gx

C55

k − 1

βy

k

= Gy

C44

k − 1

σxz

zz

= Gxγ xz

o

x, y + Gy

C45

k

C44

k γ yz

o

x, y

σyz

zz

= Gx

C45

k

C55

k γ xz

o

x, y + Gyγ yz

o

x, y

βx

k

βy

k dz

− h

2

+ h

2

= βx

k

βy

k dz

z k−1

z k

Σk = 1

N

= βx

k

βy

Σ k h k k = 1

N

βx

k

βy

k dz

− h

2

+ h

2

= Δx

N

Δy

N − Δx

0

Δy

0

Δx

0

= Δy

0

= Δx

N

= Δy

N

≡ 0

βx

k

βy

k dz

− h

2

+ h

2

= 0

0

Gx = 1

h

dz

C55

k

− h

2

+ h

2

− 1

= 1

h

1

C55

Σ k k = 1

N

dz

z k−1

z k

− 1

= 1

h

h k

C55

Σ k k = 1

N − 1
10

 (30b)

With Gx and Gy known, equations (20) are written as

 (31a)

 (31b)

and constitutive equations (22) become

 (32)

The next step in the analysis is to obtain expressions for the zig-zag functions and 

in terms of the constants Gx and Gy. Toward this goal, equations (18) are re-written as 

 (33a)

 (33b)

Examining these equations for successive values of the index k, starting with k = 1 and using 

, it is seen by mathematical induction that 

 (33c)

 (33d)

Substituting these expressions into equations (17) gives

 (34a)

Gy = 1

h

dz

C44

k

− h

2

+ h

2

− 1

= 1

h

1

C44

Σ k k = 1

N

dz

z k−1

z k

− 1

= 1

h

h k

C44

Σ k k = 1

N − 1

Fx′ z = Hx′ z +

Gx

C55

k − 1

Fy′ z = Hy′ z +

Gy

C44

k − 1

σyz x, y, z

σxz x, y, z = C44

k

C45

k

C45

k

C55

k

Hy′ z +

Gy

C44

k − 1 γ yz

o

x, y

Hx′ z +

Gx

C55

k − 1 γ xz

o

x, y

Φx

k

z

Φy

k

z

Δx

k

= Δx

k−1

+ h kβx

k

Δy

k

= Δy

k−1

+ h kβy

k

Δx

0

= Δy

0

= 0

Δx

k

= βx

p

Σ h p p = 1

k

Δy

k

= βy

p

Σ h p p = 1

k

Φx

k

z = βx

p

Σ h p p = 1

k − 1

+ βx

k

z − z k−1
11

 (34b)

Now, consider h(k) = z(k) - z(k-1) for successive values of the index k, starting with k = 1. Noting 

that , mathematical induction yields 

 (35)

Thus, using equations (25) and (35), equations (34) are expressed as

 (36a)

 (36b)

Let M denote the ply that corresponds to the plate midplane, z = 0. At z = 0, equations (36) 

reduce to

 (37a)

 (37b)

Examination of equations (14) indicates that the constraints Fx(0) = Fy(0) = 0 require

 (38a)

 (38b)

Similarly, the constraints given by equations (12), which enforce the traction boundary conditions 

on the top and bottom plate surface, yield

 (39a)

 (39b)

Φy

k

z = βy

p

Σ h p p = 1

k − 1

+ βy

k

z − z k−1

z 0 = − h

2

z k = − h

2 +Σh p p = 1

k

Φx

k

z = z + h

2

Gx

C55

k − 1 + Gx

1

C55

p − 1

C55

k Σ h p p = 1

k − 1

Φy

k

z = z + h

2

Gy

C44

k − 1 + Gy

1

C44

p − 1

C44

k Σ h p p = 1

k − 1

Φx

M

0 = h

2

Gx

C55

M − 1 + Gx

1

C55

p − 1

C55

M Σ h p p = 1

M − 1

Φy

M

0 = h

2

Gy

C44

M − 1 + Gy

1

C44

p − 1

C44

M Σ h p p = 1

M − 1

Hx 0 + Φx

M

0 = 0

Hy 0 + Φy

M

0 = 0

Hx′ − h

2 +

Gx

C55

1 − 1 = 0

Hx′ + h

2 +

Gx

C55

N − 1 = 0
12

 (39c)

 (39d)

where equations (31) have been used. For convenience, let the continuous functions have the form

 (40a)

 (40b)

which satisfy equations (38) provided fx(0) = fy(0) = 0. Substituting equations (40) into equations 

(39) gives

 (41a)

 (41b)

 (41c)

 (41d)

Solving these equations yields

 (42a)

 (42b)

 (42c)

 (42d)

Substituting these results into equations (40) gives

Hy′ − h

2 +

Gy

C44

1 − 1 = 0

Hy′ + h

2 +

Gy

C44

N − 1 = 0

Hx z = fx z − Φx

M

0 + axz + bx

z

2

2

Hy z = fy z − Φx

M

0 + ayz + by

z

2

2

ax − h

2 bx = 1 − Gx

C55

1 − fx′ − h

2

ax + h

2 bx = 1 − Gx

C55

N − fx′ + h

2

ay − h

2 by = 1 − Gy

C44

1 − fy′ − h

2

ay + h

2 by = 1 − Gy

C44

N − fy′ + h

2

ax = 1 − 1

2

Gx

C55

1 +

Gx

C55

N + fx′ − h

2 + fx′ + h2

bx = 1

h

Gx

C55

1 − Gx

C55

N + fx′ − h

2 − fx′ + h2

ay = 1 − 1

2

Gy

C44

1 +

Gy

C44

N + fy′ − h

2 + fy′ + h2

by = 1

h

Gy

C44

1 − Gy

C44

N + fy′ − h

2 − fy′ + h2
13

 (43a)

 (43b)

In addition,

 (44a)

 (44b)

In the present study, the functional form of fx(z) and fy(z) are obtained by requiring Hx(z) and

Hy(z) to reduce to the distributions expected for a homogenous, orthotropic plate. For this very 

special class of plates, and , where G13 and G23 are the shear moduli. Thus, 

equations (30) give Gx = G13 and Gy = G23. In addition, Equations (43) 

reduce to 

 (45a)

 (45b)

and equations (44) reduce to

 (46a)

 (46b)

Moreover, the transverse sheaing stresses given by equation (32) become

 (47)

For a homogenous plate with traction-free shear-stress boundary conditions on the top and bottom 

Hx z = fx z − Φx

M

0 + z − z

2

Gx

C55

1 + fx′ − h

2 1 − z

h − z

2

Gx

C55

N + fx′ + h

2 1 + z

h

Hy z = fy z − Φy

M

0 + z − z

2

Gy

C44

1 + fy′ − h

2 1 − z

h − z

2

Gy

C44

N + fy′ + h

2 1 + z

h

Hx′ z = fx′ z + 1 − 1

2

Gx

C55

1 + fx′ − h

2 1 − 2z

h − 1

2

Gx

C55

N + fx′ + h

2 1 + 2z

h

Hy′ z = fy′ z + 1 − 1

2

Gy

C44

1 + fy′ − h

2 1 − 2z

h − 1

2

Gy

C44

N + fy′ + h

2 1 + 2z

h

C44

k

= G23 C55

k

= G13

Φx

M

0 = Φy

M

0 = 0.

Hx z = fx z − z

2 fx′ − h

2 1 − z

h + fx′ + h

2 1 + z

h

Hy z = fy z − z

2 fy′ − h

2 1 − z

h + fy′ + h

2 1 + z

h

Hx′ z = fx′ z − 1

2 fx′ − h

2 1 − 2z

h + fx′ + h

2 1 + 2z

h

Hy′ z = fy′ z − 1

2 fy′ − h

2 1 − 2z

h + fy′ + h

2 1 + 2z

h

σyz x, y, z

σxz x, y, z = G23 0

0 G13

Hy′ z γ yz

o

x, y

Hx′ z γ xz

o

x, y
14

surfaces, the distributions of transverse-shear stresses are symmetric about the plate midplane. 

Thus, and Enforcing these symmetry requirements yields 

 (48a)

 (48b)

for all values of -h/2 ≤ z ≤ h/2. Therefore, the functions fx(z) and fy(z) are required to satisfy:

fx(0) = fy(0) = 0 (49a)

 (49b)

 (49c)

for all values of -h/2 ≤ z ≤ h/2. Additional simplifications are obtained by specifying

 (50)

Enforcing these conditions in equations (43) and (44) yields, without a loss in generality,

 (51a)

 (51b)

and

 (52a)

 (52b)

Using these expressions with equations (36), equations (14) are expressed as

 (53a)

 (53b)

Hx′ − z = Hx′ z Hy′ − z = Hy′ z .

fx′ z − fx′ − z − 2z

h fx′ + h

2 − fx′ − h2 = 0

fy′ z − fy′ − z − 2z

h fy′ + h

2 − fy′ − h2 = 0

fx′ − z = fx′ z

fy′ − z = fy′ z

fx′ ± h

2 = fy′ ± h2 = 0

Hx z = fx z + z

2 2 − Gx

C55

1 1 − z

h − Gx

C55

N 1 + z

h − Φx

M

0

Hy z = fy z + z

2 2 − Gy

C44

1 1 − z

h − Gy

C44

N 1 + z

h − Φy

M

0

Hx′ z = fx′ z + 1

2 2 − Gx

C55

1 1 − 2z

h − Gx

C55

N 1 + 2z

h

Hy′ z = fy′ z + 1

2 2 − Gy

C44

1 1 − 2z

h − Gy

C44

N 1 + 2z

h

Fx z = fx z + Ψx

k

z

Fy z = fy z + Ψy

k

z
15

where fx(z) and fy(z) are continuous functions with continuous derivatives that satisfy equations 

(47) and (48). The functions and are the zig-zag enrichment functions given by

 (54a)

 (54b)

where and are given by equations (36), and are given by 

equations (37), and Gx and Gy are given by equations (30). The derivatives of Fx(z) and Fy(z) 

are obtained by substituing equations (52) into equations (31); which yields,

 (55a)

 (55b)

where

 (56a)

 (56b)

For the parabolic distribution of transverse shearing stresses commonly found in the technical 

literature for homogeneous plates,

 (57a)

 (57b)

These functions are illustrated in figure 9.

Ψx

k

z Ψy

k

z

Ψx

k

z = Φx

k

z − Φx

M

0 + z

2 2 − Gx

C55

1 1 − z

h − Gx

C55

N 1 + z

h

Ψy

k

z = Φy

k

z − Φy

M

0 + z

2 2 − Gy

C44

1 1 − z

h − Gy

C44

N 1 + z

h

Φx

k

z Φy

k

z Φx

M

0 Φy

M

0

Fx′ z = fx′ z + Ψx

k

′ z

Fy′ z = fy′ z + Ψy

k

′ z

Ψx

k

′ z = Gx

C55

k − 1

2

Gx

C55

1 1 − 2z

h +

Gx

C55

N 1 + 2z

h

Ψx

k

′ z = Gy

C44

k − 1

2

Gy

C44

1 1 − 2z

h +

Gy

C44

N 1 + 2z

h

fx z = fy z = z 1 − 1

3

2z

h

2

fx′ z = fy′ z = 1 − 2z

h

2
16

Examples

Results obtained from equations (53) and (55), based on equations (57), are presented for two 

examples in this section. The first example is a [-15/30/0/90]T laminated composite plate that 

exhibits the full extent of plate anisotropy. The second example is a sandwich plate with a 0.42-

inch-thick isotropic core and identical [±454] laminated composite face plates, denoted by [±454/

core]S. For both examples, the principal transverse-shear moduli of a ply are given by G13 = 0.864 

x 106 psi and G23 = 0.368 x 106 psi, and the thickness of each ply is 0.005 in. For the core of the 

sandwich plate, G13 = G23 = Gcore = 0.100 x 106

 psi.

For the [-15/30/0/90]T laminated composite plate, equations (30) yield Gx/G13 = 0.72 and 

Gy/G23 = 1.29. In addition, and defined by equations (37) are given by -0.0021 

and 0.0008 in., respectively, where M = 2 and 3 contain the plate midplane and give identical 

results. Graphs of Fx(z) and Fy(z) given by equation (53a) and (53b) are shown in figures 10 and 

11, respectively. These graphs clearly show the effects of laminate asymmetry and inplanecoordinate direction on the inplane-displacement distributions. Moreover, both graphs depict 

piecewise-cubic curves that are continuous at the ply interfaces. Similarly, graphs of the 

derivatives of Fx(z) and Fy(z) given by equation (55a) and (55b) are also shown in figures 10 and 

11, respectively. These two derivative graphs depict piecewise-quadratic curves that are 

discontinuous at the ply interfaces, consistent with the refined zig-zag theory of Tessler et. al.43

For the seventeen-ply [±454/core]S sandwich plate, the core comprises 84% of the plate 

thickness. Equations (30) yield Gx/Gcore = Gy/Gcore = 1.16. In addition, and 

defined by equations (37) are equal to zero, where M = 9 corresponds to the plate midplane. For 

this plate, Fx(z) and Fy(z) given by equation (53a) and (53b) are identical, and a graph of Fx(z) is 

shown in figure 12. This graph which clearly shows the presence of laminate symmetry is 

composed piecewise-cubic curves that are also continuous at the ply interfaces, with the most 

variation occurring in the more flexible core, as expected. Note that no variation in Fx(z) is shown 

across the faceplates, consistent with the fact that the and are identical for two 

adjacent +45 and - 45 degree plies. The corresponding graph of the derivatives of Fx(z) is also 

shown in figure 12. This derivative graph also depicts piecewise-quadratic curves that are 

discontinuous at the ply interfaces. Moreover, the distribution of the curves is symmetric across 

the plate thickness.

CONCLUDING REMARKS

A detailed anaylsis has been presented that shows how to enrich the kinematics of classical 

Kirchhoff plate theory by using a set of continuous piecewise-cubic functions referred to in the 

technical literature as zig-zag functions. This work was motivated by the desire to obtain realistic 

estimates of the effects of laminate heterogeneity and asymmetry on the variations of the inplane 

displacements and transverse shearing stresses, for use with a {3, 0} plate theory in which these 

Φx

M

0 Φy

M

0

Φx

M

0 Φy

M

0

C44

k

, C45

k

, C55

k
17

distributions are specified apriori. The enrichment is based on the improved zig-zag functions, 

and the corresponding analysis methodology, presented recently by Tessler, Di Scuva, and 

Gherlone. With the approach that has been presented herein, the inplane displacements are 

represented by a set of continuous piecewise-cubic functions. In contrast, the transverse shearing 

stresses and strains are represented by a set of piecewise-quadratic curves that are discontinuous 

at the ply interfaces. Corresponding results have also been presented for a general asymmetric 

laminate and a sandwich plate with identical laminated composite face plates and an isotropic 

core. These results demonstrate clearly that the zig-zag enrichment captures the expected effects 

of laminate heterogeneity and asymmetry.

References

1. Kreja, I.: A Literature Review on Computational Models for Laminated Composite and 

Sandwich Panels. Central European Journal of Engineering, vol. 1, no. 1, 2011, pp. 59-80.

2. Ghugal, Y. M. and Shimpi, R. P.: A Review of Refined Shear Deformation Theories of 

Isotropic and Anisotropic Laminated Plates. Journal of Reinforced Plastics and Composites, 

vol. 21, no. 9, 2002, pp. 775-813.

3. Carrera, E.: Theories and Finite Elements for Multilayered, Anisotropic, Composite Plates and 

Shells. Archives of Computational Methods in Engineering, vol. 9, no. 2, 2002, pp. 87-140.

4. Kant, T. and Swaminathan, K.: Estimation of Transverse/Interlaminar Stresses in Laminated 

Composites - a Selective Review and Survey of Current Developments. Composite Structures, 

vol. 49, 2000, pp. 65-75.

5. Altenbach, H.: Theories for Laminated and Sandwich Plates. Mechanics of Composite 

Materials, vol. 34, no. 3, 1998, pp. 243-252. 

6. Reddy, J. N.: Mechanics of Laminated Composite Plates - Theory and Analysis. CRC Press, 

1997.

7. Liu, D. and Li, X.: An Overall View of Laminate Theories Based on Displacement Hypothesis. 

Journal of Composite Materials, vol. 30, no. 14, 1996, pp. 1539-1561.

8. Burton, W. S. and Noor, A. K.: Assessment of Computational Models for Sandwich Panels and 

Shells. Computer Methods in Applied Mechanics and Engineering, vol. 124, 1995, pp. 125-

151.

9. Reddy, J. N. and Robbins, D. H.; Jr.: Theories and Computational Models for Composite 

Laminates. ASME Applied Mechanics Reviews, vol. 47, no. 6, part 1, 1994, pp. 147-169.

10. Reddy, J. N.: A General Non-Linear Third-Order Theory of Plates with Moderate Thickness. 

International Journal of Non-Linear Mechanics, vol. 25, no. 6, 1990, pp. 677-686.
18

11. Reissner, E.: On the Theory of Bending of Elastic Plates. Journal of Mathematics and Physics, 

vol. 23, 1944, pp. 184-191.

12. Reissner, E.: The Effect of Transverse Shear Deformation on the Bending of Elastic Plates. 

ASME Journal of Applied Mechanics, vol. 12, 1945, pp. 69-77.

13. Mindlin, R. D.: Influence of Rotatory Inertia and Shear on Flexural Motions of Isotropic, 

Elastic Plates. ASME Journal of Applied Mechanics, vol. 18, 1951, pp. 31-38.

14. Stein, M.: Nonlinear Theory of Plates and Shells Including the Effects of Transverse Shearing. 

AIAA Journal, vol. 24, no. 9, 1986, pp. 1537-1544. 

15. Soldatos, K. P.: On Certain Refined Theories for Plate Bending. ASME Journal of Applied 

Mechanics, vol. 55, December, 1988, pp. 994-995. 

16. Sun, L. X. and Hsu, T. R.: Thermal Buckling of Laminated Composite Plates with Transverse 

Shear Deformation. Computers and Structures, vol. 36, no. 5, 1990, pp. 883-889.

17. Touratier, M.: An Efficient Standard Plate Theory. International Journal of Engineering 

Science, vol. 29, no. 8, 1991, pp. 901-916.

18. Soldatos, K. P.: Nonlinear Analysis of Transverse Shear Deformable Laminated Composite 

Cylindrical Shells - Part I: Derivation of Governing Equations. ASME Journal of Pressure 

Vessel Technology, vol. 114, no. 1, 1992, pp. 105-109. 

19. Soldatos, K. P.: Nonlinear Analysis of Transverse Shear Deformable Laminated Composite 

Cylindrical Shells - Part II: Buckling of Axially Compressed Cross-Ply Circular and Oval 

Cylinders. ASME Journal of Pressure Vessel Technology, vol. 114, no. 1, 1992, pp. 110-114. 

20. Hadian, J. and Nayfeh, A. H.: Free Vibration and Buckling of Shear-Deformable Cross-Ply 

Laminated Plates Using the State-Space Concept. Computers and Structures, vol. 48, no. 4, 

1993, pp. 677-693.

21. Shu, X. and Sun, L.: Thermomechanical Buckling of Laminated Composite Plates with 

Higher-Order Transverse Shear Deformation. Computers and Structures, vol. 53, no. 1, 1994, 

pp. 1-7.

22. Touratier, M. and Faye, J.-P.: On a Refined Model in Structural Mechanics: Finite Element 

Approximation and Edge Effect Analysis for Axisymmetric Shells. Computers and 

Structures, vol. 54, no. 5, 1995, pp. 897-920.

23. Shu, X.: An Improved Simple Higher-Order Theory for Laminated Composite Shells. 

Computers and Structures, vol. 60, no. 3, 1996, pp. 343-350.

24. Ziyaeifar, M. and Elwi, A. E.: Degenerate Plate-Shell Elements with Refined Transverse 

Shear Strains. Computers and Structures, vol. 60, no. 6, 1996, pp. 1079-1091.
19

25. Cheng, Z.-Q. and Kitipornchai, S.: Exact Connection Between Deflections of the Classical 

and Shear Deformation Laminated Plate Theories. ASME Journal of Applied Mechanics, vol. 

66, March, 1999, pp. 260-262. 

26. Aydogdu, M. and Timarci, T.: Vibration Analysis of Cross-Ply Laminated Square Plates with 

General Boundary Conditions. Composites Science and Technology, vol. 63, 2003, pp. 1061-

1070.

27. Karama, M. and Mistou, S.: Experimental Validation of Refined Models of Composite Plates. 

Science and Engineering of Composite Materials, vol. 11, no. 1, 2004, pp. 1-7.

28. Zenkour, A. M.: Buckling of Fiber-Reinforced Viscoelastic Composite Plates Using Various 

Plate Theories. Journal of Engineering Mathematics, vol. 50, 2004, pp. 75-93.

29. Zenkour, A. M.: Analytical Solution for Bending of Cross-Ply Laminated Plates Under 

Thermo-Mechanical Loading. Composite Structures, vol. 65, 2004, pp. 367-379.

30. Zenkour, A. M.: Thermal Effects on the Bending Response of Fiber-Reinforced Viscoelastic 

Composite Plates Using a Sinusoidal Shear Deformation Theory. Acta Mechanica, vol. 171, 

2004, pp. 171-187.

31. Kim, J.-S..: Reconstruction of First-Order Shear Deformation Theory for Laminated and 

Sandwich Shells. AIAA Journal, vol. 42, no. 8, 2004, pp. 1685-1697.

32. Timarci, T. and Aydogdu, M.: Buckling of Symmetric Cross-Ply Square Plates with Various 

Boundary Conditions. Composite Structures, vol. 68, 2005, pp. 381-389.

33. Aydogdu, M.: Comparison of Various Shear Deformation Theories for Bending, Buckling, 

and Vibration of Rectangular Symmetric Cross-Ply Plate with Simply Supported Edges. 

Journal of Composite Materials, vol. 40, no. 23, 2006, pp. 2143-2155.

34. Girish, J. and Ramachandra, L. S.: Thermomechanical Postbuckling Analysis of Cross-Ply 

Laminated Cylindrical Shell Panels. ASCE Journal of Engineering Mechanics, vol. 132, no. 

2, 2006, pp. 133-140.

35. Shi, G.: A New Simple Third-Order Shear Deformation Theory of Plates. International 

Journal of Solids and Structures, vol. 44, 2007, pp. 4399-4417.

36. Akavci, S. S. and Tanrikulu, A. H..: Buckling and Free Vibration Analyses of Laminated 

Composite Plates by Using Two New Hyperbolic Shear-Deformation Theories. Mechanics of 

Composite Materials, vol. 44, no. 2, 2008, pp. 145-154.

37. Aydogdu, M.: A New Shear Deformation Theory for Laminated Composite Plates. 

CompositeStructures, vol. 89, 2009, pp. 94-101.
20

38. Fu, Y.; Li, S.; and Jiang, Y.: Nonlinear Free Vibration Analysis of Piezoelastic Laminated 

Plates with Interface Damage. Applied Mathematics and Mechanics (English Edition), vol. 

30, no. 2, 2009, pp. 129-144.

39. Xiang S. and Wang, K.: Free Vibration Analysis of Symmetric Laminated Composite Plates 

by Trigonometric Shear Deformation Theory and Inverse Multiquadric RBF. Thin-Walled 

Structures, vol. 47, 2009, pp. 304-310.

40. Panda, S. K. and Ramachandra, L. S.: Buckling and Postbuckling Behavior of Cross-Ply 

Composite Plate Subjected to Nonuniform In-Plane Loads. ASCE Journal of Engineering 

Mechanics, vol. 137, no. 9, 2011, pp. 589-597.

41. Barut, A.; Madenci, E.; and Nemeth, M. P.: Stress and Buckling Analyses of Laminates with 

a Cutout Using a {3, 0}-Plate Theory. Journal of Mechanics of Materials and Structures, vol. 

6, no. 6, 2011, pp. 827-868.

42. Jones, R. M.: Mechanics of Composite Materials. Second ed., Taylor & Francis, 1999.

43. Tessler, A.; Di Sciuva, M.; and Gherlone, M.: A Consistent Refinement of First-Order Shear 

Deformation Theory for Laminated Composite and Sandwich Plates Using Improved Zigzag 

Kinematics. Journal of Mechanics of Materials and Structures, vol. 5, no. 2, 2010, pp. 341-

367.

a

y-axis

z-axis

x-axis

b

h

2 h

Fig. 1 Plate geometry and coordinate system. 
21

Plate midplane, z = 0

x-axis

y-axis

z-axis 

Fig. 2 Principal material coordinate system at z = constant. 

Fiber angle, θ(z)

x ́ axis

Major principal

material direction

of a lamina

Plane, z = constant

(x, y, 0)

(x, y, z)

z ́ axis

y ́ axis

Lamina

fiber
22

z

U(x,y,z)

w

(x,y,0)

(x,y,z)

x axis

z axis

u(x,y)

∂w

∂x

∂w

∂x

γxz

o

z γxz

o

z ∂w

∂x

z

π

2

Undeformed

configuration

 Deformed

configuration

Fig. 3 Kinematics of first-order shear-deformation plate theory in the x-z plane. 

∂w

∂x

∂w

∂x

γxz

o

Fig. 4 Deformation of a differential element in the x-z plane, based

 on first-order shear-deformation plate theory. 

Undeformed

configuration

 Deformed

configuration

x

z

∂w

∂x − γxz

o

≈ ∂w

∂x − 1

2γxz

o

= − ωy

o
23

Fig. 5 Deformation of plate cross-section, based on the kinematics used in the

x

z

h

Plane y = constant

Undeformed

configuration

 Deformed

configuration

 present study. 

z

U(x,y,z)

w

(x,y,0)

(x,y,z)

x axis

z axis

u(x,y)

∂w

∂x

∂w

∂x

γxz

o

z ∂w

∂x

z

π

2

Undeformed

configuration

 Deformed

configuration

Fig. 6 Kinematics used in the present study. 

Fx(z) γ xz

o
24

∂w

∂x

∂w

∂x

γxz

o

∂w

∂x

∂w

∂x

γxz

Fig. 7 Deformation of a differential element in the x - z plane, based

 on the kinematics of the present study. 

(a) Differential element at z = 0 (b) Differential element at z = constant

x

z z

x

≈ ∂w

∂x − 1

2γxz

o

= − ωy

o

∂U

∂z z = 0

= γxz

o

− ∂w

∂x ∂U

∂z = γxz − ∂w

∂x ≈ 1

2γxz − ∂w

∂x = ωy

kth ply

(k+1)1st ply

(k-1)1st ply

z = z(k-2)

z = z(k-1)

z = z(k)

z = z(k+1)

h(k)

Fig. 8 Ply coordinates and zig-zag functions in the x-z plane.

x axis

z axis

Δ(k+1)

Φx

k

z

Φx

k+1

z

Φx

k−1

z

x

Δ(k)

x

Δ(k-1)

x

Δ(k-2)

x

k = N

Φx

N

z

Δ(N-1)

x

Δ(N)

x = 0

h

2 z = 

k = 1 Φx

1

z

Δ(1)

x

Δ(0)

x = 0

h

2 z = - 
25

fx(x)

h

z-axis

Plate

z

0.5

- 0.25

- 0.5

0.5

1.0

1.5

0.25

z

- 0.5 - 0.25 0.25 0.5 h

z

- 0.5 - 0.25 0.25 0.5 h

fx

′(z)

Fig. 9 Graphs of the function fx(z) and its derivative. 

h

2

h

2
26

Fig. 10 Graphs of the function Fx(z) and its derivative for [-15/30/0/90]T laminate. 

Fx(x)

h

Fx

′(z)

0.5

- 0.25

- 0.5

0.5

1.0

1.5

0.25

-15 deg ply 30 deg ply 0 deg ply 90 deg ply

z

- 0.5 - 0.25 0.25 0.5 h

-15 deg ply 30 deg ply 0 deg ply 90 deg ply

z

- 0.5 - 0.25 0.25 0.5 h

z-axis

Plate

z

h

2

h

2
27

Fig. 11 Graphs of the function Fy(z) and its derivative for [-15/30/0/90]T laminate. 

Fy(x)

h

Fy

′(z)

0.5

1.0

1.5

-15 deg ply 30 deg ply 0 deg ply 90 deg ply

-15 deg ply 30 deg ply 0 deg ply 90 deg ply

0.5

- 0.25

- 0.5

0.25

z

- 0.5 - 0.25 0.25 0.5 h

z

- 0.5 - 0.25 0.25 0.5 h

z-axis

Plate

z

h

2

h

2
28

Fig. 12 Graphs of the function Fx(z) and its derivative for [±454/core]S sandwich plate. 

Fx(x)

h

Fx

′(z)

1.0

- 0.5

- 1.0

0.5

1.0

1.5

0.5

Core

2.0

±454 face plate

Core

z

- 0.5 - 0.25 0.25 0.5 h

z

- 0.5 - 0.25 0.25 0.5 h

±454 face plate

±454 face plate

±454 face plate

z-axis

Plate

z

h

2

h

2
REPORT DOCUMENTATION PAGE Form Approved

OMB No. 0704-0188

2. REPORT TYPE 

Technical Memorandum

 4. TITLE AND SUBTITLE

Cubic Zig-Zag Enrichment of the Classical Kirchhoff Kinematics for 

Laminated and Sandwich Plates

5a. CONTRACT NUMBER

 6. AUTHOR(S)

Nemeth, Michael P.

 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

NASA Langley Research Center

Hampton, VA 23681-2199

 9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES)

National Aeronautics and Space Administration

Washington, DC 20546-0001

 8. PERFORMING ORGANIZATION

 REPORT NUMBER

L-20143

10. SPONSOR/MONITOR'S ACRONYM(S)

NASA

13. SUPPLEMENTARY NOTES

12. DISTRIBUTION/AVAILABILITY STATEMENT

Unclassified - Unlimited

Subject Category 39 

Availability: NASA CASI (443) 757-5802

19a. NAME OF RESPONSIBLE PERSON

STI Help Desk (email: help@sti.nasa.gov)

14. ABSTRACT

A detailed anaylsis and examples are presented that show how to enrich the kinematics of classical Kirchhoff plate theory by 

appending them with a set of continuous piecewise-cubic functions. This analysis is used to obtain functions that contain the 

effects of laminate heterogeneity and asymmetry on the variations of the inplane displacements and transverse shearing 

stresses, for use with a {3, 0} plate theory in which these distributions are specified apriori. The functions used for the 

enrichment are based on the improved zig-zag plate theory presented recently by Tessler, Di Scuva, and Gherlone. With the 

approach presented herein, the inplane displacements are represented by a set of continuous piecewise-cubic functions, and the 

transverse shearing stresses and strains are represented by a set of piecewise-quadratic functions that are discontinuous at the 

ply interfaces. 

15. SUBJECT TERMS

Refined plate theory; layerwise theories; plate kinematics; sandwich plates; zig-zag theories

18. NUMBER

 OF 

 PAGES

33

19b. TELEPHONE NUMBER (Include area code)

(443) 757-5802

a. REPORT

U

c. THIS PAGE

U

b. ABSTRACT

U

17. LIMITATION OF 

 ABSTRACT

UU

Prescribed by ANSI Std. Z39.18

Standard Form 298 (Rev. 8-98)

3. DATES COVERED (From - To)

5b. GRANT NUMBER

5c. PROGRAM ELEMENT NUMBER

5d. PROJECT NUMBER

5e. TASK NUMBER

5f. WORK UNIT NUMBER

869021.04.07.01.13 

11. SPONSOR/MONITOR'S REPORT

 NUMBER(S)

NASA/TM-2012-217570

16. SECURITY CLASSIFICATION OF:

The public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, 

gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this 

collection of information, including suggestions for reducing this burden, to Department of Defense, Washington Headquarters Services, Directorate for Information Operations and 

Reports (0704-0188), 1215 Jefferson Davis Highway, Suite 1204, Arlington, VA 22202-4302. Respondents should be aware that notwithstanding any other provision of law, no person 

shall be subject to any penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number.

PLEASE DO NOT RETURN YOUR FORM TO THE ABOVE ADDRESS.

1. REPORT DATE (DD-MM-YYYY)

01- 05 - 2012