May 2012 

NASA/TP–2012-217573 

Analytic and Computational Perspectives 

of Multi-Scale Theory for Homogeneous, 

Laminated Composite, and Sandwich Beams 

and Plates 

Alexander Tessler 

Langley Research Center, Hampton, Virginia 

Marco Gherlone, Daniele Versino, and Marco Di Sciuva 

Department of Mechanical and Aeronautics Engineering 

Politecnico di Torino, Torino, Italy 
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May 2012 

NASA/TP–2012-217573 

Analytic and Computational Perspectives 

of Multi-Scale Theory for Homogeneous, 

Laminated Composite, and Sandwich Beams 

and Plates 

Alexander Tessler 

Langley Research Center, Hampton, Virginia 

Marco Gherlone, Daniele Versino, and Marco Di Sciuva 

Department of Mechanical and Aeronautics Engineering 

Politecnico di Torino, Torino, Italy 
Available from: 

NASA Center for AeroSpace Information 

7115 Standard Drive 

Hanover, MD 21076-1320 

443-757-5802 

The use of t rademarks or names of manufacturers in thi s report is for accurate reportin g and does not 

constitute an official endorsement, either expressed or implied, of such products or manufacturers by the 

National Aeronautics and Space Administration. 
1

Abstract 

 

This paper reviews the theoretical foundation and computational 

mechanics aspects of the recently developed shear-deformation theory, 

called the Refined Zigzag Theory (RZT). The theory is based on a multiscale formalism in which an equivalent single-layer plate theory is 

refined with a robust set of zigzag local layer displacements that are free 

of the usual deficiencies found in common plate theories with zigzag 

kinematics. In the RZT, first-order shear-deformation plate theory is 

used as the equivalent single-layer plate theory, which represents the 

overall response characteristics. Local piecewise-linear zigzag 

displacements are used to provide corrections to these overall response 

characteristics that are associated with the plate heterogeneity and the 

relative stiffnesses of the layers. The theory does not rely on shear 

correction factors and is equally accurate for homogeneous, laminated 

composite, and sandwich beams and plates. Regardless of the number of 

material layers, the theory maintains only seven kinematic unknowns that 

describe the membrane, bending, and transverse shear plate-deformation 

modes. Derived from the virtual work principle, RZT is well-suited for 

developing computationally efficient, C0-continuous finite elements; 

formulations of several RZT-based elements are highlighted. The theory 

and its finite element approximations thus provide a unified and reliable 

computational platform for the analysis and design of high-performance 

load-bearing aerospace structures. 

 
2

1 INTRODUCTION 

Lightweight and high-performance characteristics of advanced composite materials have 

spurred a wide range of application of these materials in military and civilian aircraft, aerospace 

vehicles, and naval and civil structures. To realize the full potential of composite structures for 

primary load-bearing components, further advances in structural design, analysis methods, and 

failure prediction and progressive damage methodologies are necessary. 

 A wide variety of modern civilian and military aircraft use relatively thick laminated 

composite and sandwich laminates for primary load-bearing structures. Such structures 

commonly undergo relatively pronounced design-critical transverse-shear deformations and, 

under certain conditions, thickness-stretch deformations. Fail-safe designs require accurate 

stress-analysis methods, particularly in the regions of stress concentration. Moreover, 

computationally efficient progressive damage analysis necessitates accurate modeling of the 

interlaminar damage modes such as delamination. For these reasons, structural theories that 

account for higher-order deformation effects have attracted much attention in recent years. 

 Recently, Tessler and co-authors presented an improved structural theory for beams and 

plates, labeled the Refined Zigzag Theory (RZT), that offers substantial analytic and 

computational advantages for the analysis of homogeneous, laminated composite, and sandwich 

laminates [1-5]. This new theory is based on a multi-scale approach in which an equivalent 

single-layer plate theory is used to represent the overall (coarse) plate response characteristics 

and through-the-thickness zigzag kinematics are used to model the local (fine) layer-level 

behavior. In particular, the RZT uses First-order Shear-Deformation Theory (FSDT) as the 

equivalent single-layer theory, and uses sets of piecewise linear continuous functions to model 

the local behavior of the layers comprising a plate. The zigzag kinematic framework enables 

sufficiently accurate and computationally efficient modeling of a wide range of homogeneous 

and heterogeneous laminates without the use of shear correction factors. Novel zigzag functions, 

derived a priori from constitutive relations without enforcing debilitating stress-equilibrium 

constraints, are responsible for overcoming several critical shortcomings of the earlier zigzag 

theories (refer to [1-5] for the literature reviews and pertinent discussions). The stress resultants 

are obtained from the equilibrium equations and, as a result, are physically consistent with their 

definitions based on Hooke’s relations. The formulation, which maintains a fixed number of 

kinematic unknowns regardless of the number of material layers, does not enforce full continuity 

of the transverse shear stresses along material-layer interfaces, yet is robust. Using the principle 

of virtual work, equilibrium equations and consistent boundary conditions are derived in a 

variationally consistent manner. The variational framework, requiring relatively simple C0-

continuous kinematic interpolations, provides a convenient means for developing 

computationally efficient and robust finite elements. 

 The focus of this paper is to assess the current state of the art in the development of RZT and 

to highlight its recent advances in finite element analysis. To accomplish these objectives, this 

paper first reviews the theoretical foundation of RZT for laminated plates and highlights the 

essential aspects of the piecewise linear zigzag functions derived from transverse-shear 

constitutive relations. Related efforts of the zigzag-function enrichment using higher-order 

polynomials are also highlighted. In Section 2, the principle of virtual work is presented, which 

serves as the foundation for developing finite element approximations. Also highlighted in this 

section are (i) a simple and effective method of computing highly accurate interlaminar stresses, 

(ii) the unique modeling of homogeneous plates using the full kinematics of RZT, and (iii) a 
3

higher-order theory that includes odd and even zigzag kinematic terms in the inplane expansions 

and which also accounts for the thickness-stretch deformations. In Section 3, recent efforts to 

develop RZT-based finite element for laminated composite and sandwich beam, plate, and shell 

structures are discussed, focusing on their kinematic approximations, element nodal 

configurations, and modeling capabilities. In Section 4, finite element results for a laminated 

cylindrical shell undergoing large displacements are presented to demonstrate the latest RZTbased shell modeling capability. Finally, conclusions are presented which highlight the salient 

features of RZT and the finite elements developed on its foundation. 

2 FOUNDATION OF REFINED ZIGZAG THEORY 

Consider a multilayer composite plate of uniform thickness 

2h

 that is composed of perfectly bonded 

orthotropic layers, labelled with superscript (k), where k=1,..., N; furthermore, the plate undergoes smallstrain deformations under static loading while exhibiting negligible inertial effects (refer to Figure 1.) The 

inplane coordinates of the plate are defined by the vector 

1 2 ( , ) m

x   x x S

, where 

mS

 represents the set 

of points given by the intersection of the plate with the plane 

z  0

(the midplane); the symbolism 

z h h [- , ]

denotes the domain of the through-the-thickness coordinate. 

(a) (b)

Figure 1. (a) Plate of uniform thickness, 

2h

, subjected to transverse pressure loading, 

q

, and edge 

tractions, {

T ( 1,2)    , Tz

}; (b) through-thickness notation of 

N -layer laminate.

 

 The displacement vector of any material point 

1 2 P x x z ( , , )

 in the 

th k

 layer is defined by the three 

orthogonal components 

( ) ( )

1 2 ( , , ) k kz

uuu

that are expressed as [4-5], 

 

( ) ( )

1 1 1 1

( ) ( )

2 2 2 2

( , ) ( ) ( ) ( ) ( )

( , ) ( ) ( ) ( ) ( )

( , ) ( )

k k

k k

z

u z u z z

u z v z z

u z w

  

  

  

  



x x x x

x x x x

x x

 (1) 
4

where 

u( ) x

 and 

v( ) x

 are the inplane displacements in the 

1

x

 and 

2

x

 coordinate directions, 

respectively, and 

w( ) x

is the transverse deflection. The symbols 

1  ( ) x

 and 

2  ( ) x

 represent overall 

bending rotations of a transverse material line element, that is initially straight and normal to the 

midplane, about the positive 

2

x

 and the negative 

1

x

 directions, respectively. These overall rotations 

represent linearized weighted averages of the actual nonlinear through-the-thickness displacement 

variations that are produced when the displacement variations within each layer are pronounced. 

Collectively, these three displacements and two rotations form the basis of FSDT and, as such, are viewed 

herein as the quantities that define the overall, first-approximation, coarse displacement characteristics of 

a plate. 

 Refinements to the coarse displacement characteristics, and hence improved fidelity, are obtained by 

introducing additional displacement functions that model lamina-scale, or even sub-lamina-scale, 

responses adequately. In the RZT, the refined contributions to the “coarse” inplane displacements are due 

to the zigzag terms 

( )( ) ( ) ( 1,2) k

z      x 

 that appear in Eqs. (1). These refinement functions 

generally produce piecewise-continuous, nonlinear through-the-thickness distributions that are defined by 

the zigzag functions 

( )( ) k

z 

and their corresponding rotation (or amplitude) functions 

( )  x

. The 

simplest form for 

( )( ) k

z 

is given by a set of piecewise linear, continuous functions with derivatives 

( )

,

k

 z

 that are discontinuous at the layer interfaces. Herein, a comma followed by the subscript z denotes 

differentiation with respect to the through-the-thickness coordinate z. This particular class of functions is

denoted herein by 

(0) Cz

.

 The localized rotations of the line elements within each layer, which prior to deformation are straight 

and perpendicular to the reference midplane, are obtained from Eqs. (1) as 

       ( ) ( )

,

, ( 1,2) k k

z

u z  x x x         



 (2) 

where 

( ) ( )

,

k k    



z

are constant valued within the 

th k

layer and generally have different values for layers 

with different material properties; thus, they are discontinuous at the layer interfaces. Therefore, the 

rotations 

( )

,

( , ) k

z

u z  x are also uniform within the 

th k

layer and generally vary from layer to layer. 

 A key step within RZT is to identify its relationship with FSDT. This task is accomplished by 

representing the coarse rotations, 

( )  x

, as the weighted-average rotations, i.e., 

 

  ( )

,

1

( ) , ( 1,2)

2

h

k

z

h

u z dz

h

    

  x x

 (3) 

Substituting Eq. (2) into (3), results in 

 

( ) ( ) (1) ( ) ( ) 0 ( 1,2)

h

k N

h

dz h h       



     

 (4) 

thus guaranteeing equal values of the zigzag functions on the bounding surfaces, i.e., 

( ) (1) ( ) ( ) ( 1,2) N

h h     

  

. To correlate the displacement of FSDT and RZT at the bounding 

surfaces, 

z h 

, it becomes immediately apparent that the top and bottom values of the zigzag functions 

must vanish identically, i.e., 
5

 

( ) (1) ( ) ( ) 0 ( 1,2) N

h h    

    

 (5) 

Equations (3) also imply that the following weighted-average transverse shear strains are those which 

correspond to FSDT, i.e., 

 

( )

,

1

( 1,2)

2

h

k

z

h

dz w

h

   

   



    

 (6) 

where henceforth the notation 

,

( ) ( ) / x      denotes partial differentiation with respect to the 

midplane coordinate, 

x

. 

 It can now be readily ascertained that within RZT only the coarse kinematic variables define the 

values of the inplane displacements at top and bottom surfaces, i.e., 

 

(1) ( )

1 1 1 1

(1) ( )

2 2 2 2

( , ) ( ) ( ), ( , ) ( ) ( )

( , ) ( ) ( ), ( , ) ( ) ( )

N

N

u h u h u h u h

u h v h u h v h

 

 

    

    

x x x x x x

x x x x x x

 (7) 

With this insight, it becomes clear that the zigzag kinematics can indeed be regarded as local ply-level 

perturbations to the overall, coarse displacement fields. From Eqs. (7), the physical interpretations of the 

coarse variables {

u , v , 1

,  2} in terms of the top-surface inplane displacements 

( ) ( ) ( , ) N

t

u u h   x x 

and the bottom-surface inplane displacements 

(1) ( ) ( , ) b

u u h   x x  

also become apparent, i.e., 

 

1 1 1 1 1

1 1 ( ) ( ) ( ) , ( ) ( ) ( )

2 2 t b t b u u u u uh

        

   

x x x x x x

 (8) 

 

2 2 2 2 2

1 1 ( ) ( ) ( ) , ( ) ( ) ( )

2 2 t b t b v u u u uh

        

   

x x x x x x

 (9) 

These same relations hold for these kinematic variables used in FSDT. 

 For certain classes of problems, modelling advantages can be exploited by using a different set of 

primary unknown functions instead of the seven appearing in Eqs. (1). For example, in finite element 

analyses that use the sub-laminate concept [6-9], the inplane displacements of the top and bottom 

bounding surfaces of a sub-laminate are particularly useful as primary unknowns. Thus, by treating a 

laminate as a collection of contiguous sub-laminates, the RZT equations presented herein can be modified 

as follows. From Eqs. (8) and (9), it is seen that an alternate description of the RZT displacement fields is 

obtained by expressing the four coarse variables {

u , v , 1

, 2} in terms of the four inplane 

displacements of the top and bottom surfaces {

( ) tu x , ( ) b

u x

} (

 1,2

), and then substituting the 

result into Eqs. (1). In this way, RZT is redefined in terms of the alternative set of seven displacement 

variables: {

( ) tu x , ( ) b

u x , ( )  x , w( ) x

} (

 1,2

) which can be applied on a laminate and a sublaminate level. Because this transformation of the primary unknowns is well defined, this version of RZT 

is expected to have the same analytic accuracy as the original RZT. 

 Using the linear strain-displacement relations of elasticity theory and the displacement assumptions 

given by Eqs. (1), the RZT strains are given as 
6

 

( ) ( )

11 ,1 1,1 1 1,1

k k        u z

 

 

( ) ( )

22 ,2 2,2 2 2,2

k k        v z

 

 

( ) ( ) ( )

12 ,2 ,1 1,2 2,1 1 1,2 2 2,1 ( ) k k k

       u v z      

 (10) 

 

  ( ) ( ) 1,2 k k

    z

       

 

where it is noted that the inplane strains, {

( )

11

k

 ,

( )

22

k

 , 

( )

12

k



}, are piecewise linear through the laminate 

thickness. The transverse shear strains, 

( ) k

z



, are piecewise constant , i.e., they are constant within each 

material layer but discontinuous along the layer interfaces, in contrast to those of FSDT which are 

constant across the total laminate thickness. 

 Each layer of the plates considered herein is presumed to be specially orthotropic with respect to a set 

of principal material coordinate directions. In addition, each set of layer principal material axes have two 

axes that reside in the plane of the layer and a third that is perpendicular to that plane. This third axis is 

parallel to the z-axis for the plate coordinates defined herein. The two principal axes that reside in the 

plane of the layer are generally noncoincident with the inplane coordinate axes of the plate. In terms of 

the plate coordinate system, the generalized Hooke's relations for the 

th k

layer are given herein by 

 

( ) ( ) ( )

11 11 12 16 11

( ) ( ) ( )

22 22 26 22

12 66 12

or

.

k k k

k k k

C C C

C C

sym C

 

 

 

                               

σ C ε

 (11.1) 

for the inplane stresses and by 

 

( ) ( ) ( )

2 22 12 2 ( ) ( ) ( )

1 11 1

or

.

k k k

z z k k k

z z

Q Q

sym Q

 

 

                   

τ Q γ

 (11.2) 

for the transverse shear stresses. The symbols 

( ) k Cij

 and 

( ) k Qij

 denote the transformed elastic-stiffness 

coefficients referred to the plate coordinate system. In RZT, these coefficients are also based on the 

presumption that the transverse-normal stresses are negligible. 

 The next step in formulating RZT is to provide a mathematical description of the zigzag functions. 

Thus, for the 

th k

material layer, which is located in the range 

( 1) ( ) [ , ] k k z z 

, the zigzag functions are given 

as (see Figure 1(b) depicting the lamination notation), 

     ( ) 1 1 ( ) ( )

2 2 ( 1) ( ) 1 1 ( 1,2) k k k

            k k   (12.1) 

 

where 

       ( ) ( )

( 1) / 1 1,1 1,..., k kk

         z z h k N  

 (12.2) 
7

with the first layer, 

k 1

, beginning at 

(0) z h  

, the last 

th N

 layer, 

k N

, ending at 

( ) N

z h 

, and 

the 

th k

 layer ending at 

( )

( ) ( 1) 2

k

k k z z h   

, where 

( ) 2k

h

 denotes the 

th k

layer thickness. Evaluating 

Eqs. (12) at the bottom 

( ) ( 1) k

  

 and top 

( ) ( 1) k

 

surfaces of the 

th k

layer, gives rise to the 

definitions of the interface displacements 

 

  ( ) ( ) ( ) ( )

( 1) ( ) ( 1), ( 1) 1,..., k k k k

k k               

  k N

 (13) 

whereas according to Eqs. (5) the zigzag functions vanish at the bottom 

(1) ( 1, 1) k    

and top 

( ) ( , 1) N

k N  

plate surfaces, i.e., 

(0) ( ) 0    

  N

.

(a) Layer notation 

 

(b) Zigzag functions 

( )

1

( ) k

 z

and 

( )

2

( ) k

 z .

Figure 2. Notation for a three-layer laminate and 

( ) ( ) k

z 

 zigzag functions defined in terms of interface 

displacements,

( ) k

.
8

The remaining interface displacements can be obtained from the simple expression 

 

( ) ( )

( )

1

2

k

i i

k

i

h    





 (14) 

In [5], two alternative expressions have been derived for 

( ) k



(

k N   1,..., 1

); they are given as 

 (a) 

( )

( ) ( )

( )

1

1

1

k

k N k

k

k

Q h

h Q











 



, (b) 

( )

( )

( ) ( )

1

1 ( 1,2)

1

k

k

N

k k

k

S

h S

h







 



  



 (15) 

where 

( ) k Q

 and 

( ) k

S

( 1,2)  

 denote, respectively, the diagonal transverse-shear stiffness and 

compliance coefficients of the k

th layer, respectively. Note that these expressions are only slightly 

different from each other for the general lay-up case, in which the transverse-shear constitutive matrix has 

a non-zero off-diagonal term (

( )

12 0

k Q 

). The two definitions, however, are identical for the special, 

decoupled case for which 

( )

12 0

k Q 

. From the predictive perspective, either form of 

( ) k



 yields excellent 

results, with the second form exhibiting slightly improved results, particularly for sandwich plates. 

 Further accuracy enhancements within the present RZT methodology can be achieved by using higherdegree polynomial expansions for the zigzag functions 

( ) k



, that are piecewise continuous nonlinear 

functions, while keeping the basic kinematic assumptions, Eqs. (1), unchanged (e.g., refer to [10-13]). 

2.1 Variational framework 

The principle of virtual work, for the case of negligible body forces and zero shear tractions on the top 

and bottom bounding plate surfaces, may be written as 

 

 

( ) ( ) ( ) ( )

( ) ( )

1 1 2 2 0

m m

h

k T k k T k

S h S

h

k k

z z

C h

dz dS q wdS

T u T u T u ds dz



  

  





 

     

 

  

 

σ ε τ γ

 (16) 

where 



is the variational operator; 

q

 is the applied transverse pressure attributed to the middle reference 

surface, 

mS

; {

T ( 1,2)    , Tz

} are the inplane and transverse shear tractions that are prescribed along 

the cylindrical edge surface of the laminate, 

S C s    

. The bold-face vectors appearing in equation 

(16) are defined by Eqs. (11). 

 Integrating Eq. (16) across the laminate thickness, while accounting for the relationships in Eqs. (1) 

and (10), yields the corresponding two-dimensional statement of the principle of virtual work 
9

 [ ] 0

m

T

T T T

m m b b s s S C

q w dS ds



            N e M e Q e F u

 (17.1) 

where the membrane stress resultants and conjugate strain measures are defined as 

 

( ) ( ) ( )

1 2 12 11 22 12 ,1 ,2 ,2 ,1 { , , } { , , } , { , , }

h

T k k k T

m m h

N N N    dz u v u v



    N e 

 (17.2) 

Likewise, the bending stress resultants and conjugate strain measures are defined as 

 

 

 

1 1 2 2 12 12 21

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

11 1 11 22 2 22 12 1 12 2 12

, , , , , ,

, , , , , ,

T

b

h

k k k k k k k k k k k

h

M M M M M M M

z z z dz

   

          





 

M

 

 

 1,1 1,1 2,2 2,2 1,2 2,1 1,2 2,1 , , , , , , 

T

eb          

 (17.3) 

The transverse shear stress resultants and conjugate strain measures are defined as 

 

   

( ) ( ) ( ) ( ) ( ) ( )

2 2 1 1 2 2 2 1 1 1 , , , , , ,

h

T k k k k k k

s z z z z

h

Q Q Q Q dz        



  Q 

 

 ,2 2 2 ,1 1 1 , , , 

T

es    w w    

 (17.4) 

The force and moment resultants due to the prescribed tractions and their conjugate displacements are 

defined as 

 

1 2 1 2 1 2

( ) ( )

1 2 1 2 1 1 2 2

1 2 1 2

{ , , , , , , }

{ , , , , , , }

{ , , , , , , }

T

n n zn n n n n

h

k k

z

h

T

N N Q M M M M

T T T zT zT T T dz

u v w

 

 

   











F

u

 (17.5) 

All elements of Eqs. (17) with the superscript 



are associated with the zigzag functions. 

 The stress resultants are readily obtained in terms of the two-dimensional plate strain measures by 

integrating the expressions in Eqs. (17) through the laminate thickness, resulting in the following 

constitutive relations of RZT: 
10 

 

m m

T

b b

s s

                              

N A B 0 e

M B D 0 e

Q 0 0 G e

 (18) 

where 

3 3 [ ] A  Aij 

 denotes the membrane stiffness matrix, 

3 7 [ ] B  Bij 

 is the membrane-to-bending 

coupling stiffness matrix, 

7 7 [ ] D  Dij 

 is the bending stiffness matrix, and 

4 4 [ ] G  Gij 

 is the transverse 

shear stiffness matrix. The explicit forms of these stiffness coefficients are found in [4,5]. Note that for a 

general composite lay-up, the membrane-to-bending coupling stiffnesses are nonzero; that is, 

0 Bij  .

 2.1.1 Special case 

When the zigzag functions in the 

1

x

 and 

2

x

directions are identical, 

( ) ( ) ( )

1 2 ( ) ( ) ( ) ( 1,..., ) kkk  z z z k N   

. As a result, the two twisting moments associated with the 

zigzag kinematics are given by the same expression (refer to Eqs. (17.2) and [14]) as follows 

 

( ) ( )

12 21 12

h

k k

h

M M dz   



  

 (19) 

This simplification gives rise to the following reduced form of Eqs. (17.2) 

 

1 2 12 1 2 12

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

11 22 12 11 22 12

1,1 2,2 1,2 2,1 1,1 2,2 1,2 2,1

{ , , , , , }

{ , , , , , }

{ , , , , , }

T

b

h

k k k k k k k k k

h

T

b

M M M M M M

z z z dz



        

       







  



M

e

 (20) 

and the resulting simplifications in Eqs. (18). These constitutive relations provide a clear physical 

interpretation of the new quantities associated with the zigzag kinematics. Specifically, 

1 2 12 { , , } MMM 

represent the bending and twisting moments due to the zigzag related cross-sectional 

distortions of the layers, whereas 

1,1 2,2 1,2 2,1 { , , }    

 are their conjugate curvatures due to the normal 

zigzag rotations, 

1

and 

 2

.

2.2 Interlaminar stresses 

An important attribute of any plate theory that is formulated to be robust at the local level is the ability to 

predict adequately interlaminar stresses that can be used to assess laminate failures. To provide adequate 

estimates of interlaminar stresses, it is often customary to integrate the three-dimensional equilibrium 

equations of elasticity theory. In this approach, the interlaminar transverse shear stresses are obtained by 

integrating the 

1

x

and 

2

x

derivatives of the inplane stresses 
11 

 

( ) ( ) ( )

1 11,1 12,2

( ) ( ) ( )

2 22,2 12,1

( )

( )

k z k k

z

k k k

z h

dz

  

   

       

     

         

 (21.1) 

whereas the interlaminar transverse normal stresses are obtained by integrating the derivatives of the 

transverse shear stresses as

 

( ) ( ) ( )

1 ,1 2 ,2 ( )

z

k k k

zz z z

h

   dz



   

 (21.2) 

Although this post-processing procedure has been widely advocated for use with FSDT and many higherorder theories, the adequacy of this scheme can be seen to be directly linked to (a) the accuracy of the 

inplane stresses obtained from constitutive relations of the underlying theory, and (b) the accuracy of the 

derivatives of these stresses (or second derivatives of the kinematic variables, refer to Eq. (10)), with this 

issue particularly relevant when finite element analyses are performed. 

 Regarding issue (a), it has been observed that FSDT and the vast majority of available structural 

theories often underestimate the inplane stresses; this is especially the case when highly heterogeneous 

and relatively thick laminates are analyzed, including the sandwich construction. Relative to issue (b), 

reasonably good improvements in estimating derivatives of the inplane stresses can be achieved by using 

smoothing techniques, e.g., [15]. 

 Application of Eqs. (21), using RZT inplane stresses, has been shown to be highly effective, yielding 

accurate interlaminar stresses that are comparable to those predicted by three-dimensional elasticity. This 

robustness exists because the piecewise linear inplane stresses of RZT are consistently accurate, even for 

highly heterogeneous and relatively thick composite and sandwich laminates [1-5].

2.3 Modeling of homogeneous plates 

The key property of a zigzag function is that it vanishes identically when the material has homogenous 

transverse shear properties across the total thickness. Consequently, when the zigzag terms vanish 

identically, the kinematic assumptions given by Eqs. (1) revert to those of FSDT, which require transverse 

shear correction factors. Tessler et al. [5] have found a simple and effective way to use the full power of 

RZT’s kinematic field to model homogenous plate (and beam) problems, without the need for transverse 

shear correction factors. In this approach, a homogeneous plate is modelled as a multilayer heterogeneous 

laminate, in which the transverse shear moduli, 

( )

3

( 1,2) k G  

, vary only slightly from layer to layer, i.e., 

   ( ) ( )

3 3 1 ( 1,2; 1,..., ) k k G G         k N

 (22.1) 

where 

3 G ( 1,2)   

denote the constant values of the shear moduli, and where 

  

( ) 2 2

2 ( 1) ( ) ( 1) ( ) 1 ( 1) k

k k k k

s

z z z z s

h

     

 (22.2) 

is a layerwise, dimensionless coefficient that is a function of the position of the 

th k

 layer within the 

laminate thickness. Thus, since 

( ) 1k



 (e.g., by setting 

4

s 10

), the values of 

( )

3

k G

 are only slightly 
12 

perturbed compared to the constant shear-modulus values 

G3

 of the corresponding homogeneous plate. 

Thus, in this approach, the homogeneous plate is replaced with a corresponding plate with an 

infinitesimal degree of heterogeneity, which for all practical purposes is homogeneous. This approach is 

effective even when the deviation from the constant value of the shear modulus of the homogeneous plate 

is less than 1/100 of a percent. The results in [5] have shown that as the number of material layers 

increases within this infinitesimally heterogeneous laminate, the solution approaches that of a 

homogeneous plate, exhibiting the well-known parabolic distribution of the transverse shear stresses 

across the plate thickness, as well as the cubic distribution of the inplane stresses, with the latter being 

particularly evident in relatively thick plates. Remarkably, the solution itself finds the correct transverse 

shear stress distributions, albeit in a piecewise constant manner, thus validating the lack of necessity for 

the transverse shear correction factors that are commonly used within FSDT and even within some 

theories of higher order. 

2.4 Inclusion of transverse normal deformations 

The RZT presented herein incorporates FSDT as the underlying baseline plate theory that models the 

overall, coarse plate behavior. Thus, it follows that higher order RZT theories can be formulated by using 

other underlying baseline plate theories with the zigzag kinematics used herein or with zigzag kinematics 

of higher fidelity. For example, Barut et al. [14] combined an underlying {1,2}-order baseline theory, that 

uses an overall parabolic transverse displacement assumption, with piecewise quadratic RZT-based zigzag 

inplane displacements to obtain a higher-order refined zigzag plate with eleven kinematic variables; four 

more than the RZT presented herein. The displacement assumptions for this theory, labeled as RZT{2,2}, 

are given as 

 

( ) ( ) ( )

1 1 1 11 1 12

( ) ( ) ( )

2 2 2 21 2 22

2

1 2 2

( , ) ( ) ( ) ( ) ( ) ( ) ( )

( , ) ( ) ( ) ( ) ( ) ( ) ( )

1

( , ) ( ) ( ) ( )

3

k k k

k k k

z

z

u z u z z z

h

z

u z v z z z

h

z z

u z w w w

h h

    

    

   

   

 

       

x x x x x

x x x x x

x x x x

 (23) 

where 

( ) k



 are the piecewise linear zigzag functions within the 

th k

 layer, adopted from the original RZT 

formulation presented herein; 

1

( )  x

 and 

2

( )  x

 (

 1,2

) are the four zigzag amplitudes that permit 

both non-symmetric, 

1

( )  x

, and symmetric, 

2

( )  x

, inplane zigzag deformation modes. In addition, 

an average transverse normal stress, 

 zz

, is independently assumed as a cubic function through the 

laminate thickness, in the form proposed by Tessler [16]; that is 

 

3

0 1 3

( , ) ( ) ( )

3

zz z z

z z

z

h h

  

 

      x x x

 (24.1) 

This representation of the normal through-the-thickness stress yields a parabolic expression for its 

derivative 

,

( , ) zz z  x z

, given by 
13 

 

2

1

, 2

( ) ( , ) 1 z

zz z

z

z

h h





 

     

x

x (24.2) 

 

,

( , ) 0 zz z  x z h   

 (24.3) 

Equations (24.3) constitute the exact conditions on 

 zz

 in the sense of three-dimensional equilibrium 

equations of elasticity theory for the case of zero-valued transverse-shear tractions on the bounding 

surfaces. The 

0

( )  z x

 and 

1

( )  z x

 terms in Eq. (24) are expressed in terms of kinematic variables of the 

theory by way of the least-square transverse strain compatibility relations originally proposed in [16]. 

 Thus, in this new theory, both transverse shear and transverse normal (thickness-stretch) deformations 

are included. The present theory models accurately the in-plane and transverse stress components through 

the thickness. The addition of the even (symmetric) inplane zigzag modes, which are not included in the 

seven-variable RZT, contributes to some improvements of the inplane displacement, strain and stress 

response, with further improvements in the interlaminar transverse shear stresses. Furthermore, RZT{2,2}

appears to be an excellent candidate for developing efficient and accurate C0-continuous finite elements. 

3 FINITE ELEMENT APPROXIMATIONS 

The variational statement given by Eq. (17.1) can be integrated by parts to produce a consistent set of 

equilibrium equations and boundary conditions, resulting in seven partial differential equilibrium 

equations in terms of seven kinematic variables [4-5]. The equilibrium equations can be solved exactly or 

approximately depending on the complexity of the material lay-up, boundary conditions, and loading. 

Refer to [1-5] for details of analytic and approximate solutions of simply supported and cantilevered 

beams and plates made of laminated composite and sandwich construction. 

Alternatively, to enable large-scale analysis of complex aerospace structures, the variational statement, 

Eq. (17.1), can be discretized using finite elements. As in FSDT, the strain measures within RZT are 

given as first-order partial derivatives. The direct implication is that computationally efficient C

0

-

continuous beam, plate, and shell finite elements can be developed, and that the legacy finite element 

technology that has been developed for FSDT can also be used with RZT. 

3.1 RZT beam elements 

Recently, Gherlone et al. [17,18] and Onate et al. [19] derived planar beam finite elements based on the 

RZT beam formulation in Tessler et al. [1]. When the beam deformations are restricted only to the 

1

x - z

plane, the second equation in Eqs. (1) is identically zero, and the theory reduces to the two displacement 

components, 

( )

1

k

u

 and 

z u

. Consequently, there remain four independent kinematic variables, 

1 1 1 1 1 1 u x w x x x ( ), ( ), ( ), and ( )  

, that describe the membrane, bending, transverse shear, and zigzag 

deformations. 

 Gherlone et al. [17,18] derived several low-order RZT beam finite elements with the aim of achieving 

the best compromise between accuracy and computational efficiency. The four kinematic variables use 

the anisoparametric (aka interdependent) interpolations, where the polynomial degree of 

1 w x( )

 is one 

order higher than those approximating the 

1 u x( ), 1 1  ( ) x

, and 

1 1  ( ) x

 variables. Such interpolation 

strategy enables free of shear locking element behaviour for slender beams. With an initial assumption of 

a parabolic distribution for 

1 w x( )

, the authors explored several shear-related constraint conditions that 

gave rise to two-node elements and a coupled form for the 

1 w x( )

 interpolation. The constraint condition 
14 

requiring a constant variation of the transverse shear force gave rise to a remarkably accurate two-node 

beam element. For further details of this formulation and for elastostatic solutions of simply supported 

and cantilevered beams of various slenderness ratios and lamination properties, the reader is referred to 

[18]. 

 Onate et al. [19] derived a two-node, RZT beam finite element using linear, isoparametric 

interpolations for the four kinematic variables, 

1 u x( ), 1 w x( ) , 1 1  ( ) x

, and 

1 1  ( ) x

. To achieve an 

element capable of modeling slender beams without incurring the shear locking effect, the authors used 

reduced integration (a one point Gaussian quadrature rule) of the transverse-shear strain energy of the 

beam element. One of the most interesting numerical solutions presented was for a laminated beam with 

an imbedded delamination that was modeled as a compliant layer. It was shown that the RZT beam 

element is computationally efficient and is capable of high-fidelity modeling of imbedded delaminations. 

3.2 RZT plate and shell elements 

Versino et al. [20,21] developed six- and three-node triangular RZT-based plate finite elements. Adopting 

linear shape functions for the in-plane displacements, bending rotations, and zigzag amplitudes, and a 

quadratic shape function for the transverse deflection, (i.e., using the Tessler-Hughes anisoparametric 

interpolation strategy [22]), the element interpolations in terms of the linear area-parametric coordinates 

Li

have the form 

 

3 3

1 2 1 2 1 2

1 1 1

3 3

1 1 2 1 2 1 2 2

1 1

3 3

1 1 2 1 2 1 2 2

1 1

6

( , ,

( ,

( ,

, ) ( , ) ( , )

, ) ( , )

, ) ( , )

i i i i

i i k

i i i i

i i

i i

k

i

i

k

i

i

x u L v x x v L w x x P

x L x x L

x L x x

u x w

x

x L

   

   

  

 

 

  

 

 

  

 

 

 (25) 

where 

i{1,2,3}

 is an index ranging over the three corner nodes; 

12 23 31 k m m m {1, ,2, ,3, }

 ranges over 

the corner and mid-edge nodes; and 

Pk

 defines the standard set of quadratic shape functions (see the 

nodal configurations in Figure 3.) The element topology includes six nodes; however, the mid-edge 

nodes have only degrees-of-freedom (dof) associated with the transverse deflection. Using these 

interpolations, the six-node element, called unconstrained, is readily derived by introducing Eqs. (25) in 

the variational principle, Eq. (17), while carrying out the necessary matrix and variational operations to 

obtain the element stiffness equations. 

 In addition, the authors derived a three-node, constrained anisoparametric element whose interpolation 

functions for the deflection variable in Eqs. (25) are constrained by suitable edge constraints, analogous to 

those which have been explored by Tessler and Hughes [22] for plates modeled with FSDT, and by 

Gherlone et al. [17,18] for beams modeled with RZT. The explicit edge-constraint procedure enforces the 

mid-edge w dof to be dependent on the corner-node dof of the rotation variables of the corresponding 

edges. By imposing one-dimensional constraint relations, requiring the two shear strain measures to be 

constant along the element edges, there results a coupled-form, parabolic deflection that is compatible 

along the element edges given by [21] 

    

3 3

1 2 1 1 1 2 2 2

1 1

( , ) i i i i i i i i

i i

w x x L w c     L c L

 

    



   

 (26.1) 
15 

with 

 

1 2

1 1 2 2 ,

( ), ( )

2 2

i i

i k j j k i

i k j i j

j k k j

k

L L L b L b L a L a L

a x x x

L

b x

   

   

 (26.2) 

where the subscripts are given by the cyclic permutation of 

i{1,2,3}, j{2,3,1}

, and 

k {3,1,2}

; 

and where 

c

is either 0 or -1, depending on the constraint strategy used in the element formulation. 

Figure 3. Six-node (unconstrained) and three-node (constrained) RZT-based anisoparametric plate 

elements. 

 An extensive numerical study was carried out in [21] on various laminated composite and sandwich 

plates undergoing elasto-static deformations. The results of this study demonstrate that the RZT-based 

anisoparametric elements possess superior accuracy and excellent convergence characteristics over a wide 

range of the span-to-thickness ratio. In addition, it was shown that the three-node elements provide the 

best compromise between computational efficiency and accuracy. Furthermore, the RZT models provide 

superior through-the-thickness predictions of the inplane and interlaminar stresses over comparable FSDT 

models, especially for relatively thick and highly heterogeneous laminated plate and sandwich plate 

constructions. 

The latest finite element implementation of RZT, by Versino [23], is focused on developing 

robust anisoparametric shell finite elements that include the drilling dof (i.e., the dof associated 

with a rotation about the normal to the element’s planar surface) and geometrically nonlinear 

deformations. In the next section, the application of RZT to an elastic cylindrical shell 

undergoing large displacements under quasi-static loading is demonstrated. 

4 NONLINEAR DEFORMATIONS OF A LAMINATED CYLINDRICAL SHELL 

In this demonstration problem, a cylindrical shell is subjected to a quasi-static concentrated force, F, 

applied at point C (Figure 4). The global shell dimensions are L=254 mm, r=2540 mm, and θ=0.1 rad. 

The shell wall is a three-layer laminate with a total thickness 2h=24 mm and with each layer having the 
16 

same thickness. The prescribed boundary conditions are such that the straight edges are fully clamped and 

the curved edges are free. The mechanical material properties of isotropic material layers A and B are 

summarized in Table 1, and the laminate stacking sequence is given by [A/B/A]. Note that the elastic 

modulus of the middle layer is two orders of magnitude less than those of the top and bottom layers. This 

aspect, typical of sandwich construction, presents a significant challenge for any structural theory, since 

the through-the-thickness distributions of the inplane displacements exhibit interface slope discontinuities 

that are not accounted for in most structural theories. 

 To establish a viable reference solution for this problem, a high-fidelity, three-dimensional 

geometrically nonlinear FEM analysis was performed using the ABAQUS commercial code [24]. 

Depicted in Figure 4(a) is an FEM model which is arrived at from a convergence study. The model is 

based on C3D20 brick elements, discretizing a symmetric quadrant of the shell, using a 64x64x6 mesh 

(64 elements along each edge and 2 elements across the thickness of each layer). In addition, two shellbased geometrically nonlinear solutions were obtained: (a) an ABAQUS solution using the S3 (threenode, FSDT) shell elements, and (b) an RZT-based solution using RZT3 – a three-node anisoparametric 

element with drilling dof. Both shell models are based on the 32x32 mesh subdivisions that span the 

shell’s symmetric quadrant, as shown in Figure 4(b). 

 Comparisons of the three nonlinear FEM solutions are shown in terms of the load-deflection curve at 

point C in Figure 5. It is evident from these results that the three-dimensional FEM solution and the 

corresponding RZT3 solution are in very close agreement over the entire range of the applied loading. In 

contrast, the S3-model predictions are considerable less accurate, particularly over the range of larger 

displacements. 

Table 1. Mechanical material properties for isotropic layers. 

Material

Young’s 

modulus, 

E (MPa)

Poisson 

ratio,



A 31



102

B 0.3 31
17 

(a) Solid model (b) Shell model

Figure 4. Finite element discretization for cylindrical shell. 

Figure 5. Applied force vs. deflection at point C. 

5 CONCLUSIONS 

In this paper the latest advances in the development of a shear-deformable, Refined Zigzag Theory (RZT) 

for homogeneous, laminated composite, and sandwich beams and plates have been reviewed and the 

salient features highlighted. This theory maintains four kinematic unknowns for beams and seven for 

plates, regardless of the number of layers, material composition, or layer stacking sequence. In addition, it 

has been shown that the theory can be expressed in terms of displacement quantities that are amenable to 

sub-laminate modelling techniques. A higher-order plate theory that incorporates the RZT presented 

herein has also been described, which includes the thickness-stretch deformations in addition to the 

transverse shear deformations, and has eleven kinematic variables. This higher-order theory illustrates 

how the basic RZT presented herein can be adapted to obtain results with a very high degree of fidelity. 

 Analytic and computational aspects of the RZT and several new finite elements for beams, plates, and 

shells have also been highlighted. The unique characteristics of RZT and its finite elements is that they 
18 

permit efficient modelling of a wide range of problems including homogenous beams and plates, 

composite laminates, sandwich construction, and compliant-layer delamination modelling. Such 

solutions do not require shear correction factors or any increase in model complexity or computational 

effort. In addition, RZT-based finite elements are amenable to much of the legacy finite element 

technology, especially that for FSDT-based elements. Because of these attributes, RZT has considerable 

promise for becoming the theory of choice for many practical applications including the computationally 

challenging problems of progressive damage modelling in composite structures. 

ACKNOWLEDGEMENTS 

The authors would like to express deep appreciation to Dr. Michael Nemeth, NASA Langley Research 

Center, and Prof. Erdogan Madenci and Dr. Atila Barut, University of Arizona, for their involvement in 

technical discussions and their thorough editorial review of this paper. 

REFERENCES 

[1] A. Tessler, M. Di Sciuva, and M. Gherlone, “A refined zigzag beam theory for composite and 

sandwich beams”, Journal of Composite Materials, 43(9), 1051-1081, 2009. 

[2] M. Di Sciuva, M. Gherlone, and A. Tessler, “A robust and consistent first-order zigzag theory for 

multilayered beams,” In Advances in Mathematical Modelling and Experimental Methods for 

Materials and Structures: The Jacob Aboudi Volume (Eds. R. Gilat, L. Banks-Sills) Springer 

(USA), ISBN, 255-268, 2009.

[3] A. Tessler, M. Di Sciuva, and M. Gherlone, “Recent advances toward computationally desirable 

zigzag bending theories for laminated composite and sandwich plates,” 13th Conference on the 

Mathematics of Finite Elements and Applications Brunel University (MAFELAP), UK, 9 - 12 

June, 2009 (keynote lecture). 

[4] A. Tessler, M. Di Sciuva, and M. Gherlone, “A consistent refinement of first-order sheardeformation theory for laminated composite and sandwich plates using improved zigzag 

kinematics,” Journal of Mechanics of Materials and Structures, 5(2), 341-367, 2010. 

[5] A. Tessler, M. Di Sciuva, and M. Gherlone, “A homogeneous limit methodology and refinements 

of computationally efficient zigzag theory for homogeneous, laminated composite, and sandwich 

plates,” Numerical Methods for Partial Differential Equations, 27(1), 208-229, 2011. 

[6] Y. B. Cho and R. C. Averill, “First-order zig-zag sublaminate plate theory and finite element model 

for laminated composite and sandwich panels,” Composite Structures, 50, 1-15, 2000. 

[7] M. Di Sciuva and M. Gherlone, “A global/local third-order Hermitian displacement field with 

damaged interfaces and transverse extensibility: FEM formulation,” Composite Structures, 59(4), 

433-444, 2003. 

[8] M. Gherlone and M. Di Sciuva, “Thermo-mechanics of undamaged and damaged multilayered 

composite plates: a sub-laminates finite element approach,” Composite Structures, 81(1), 125-136, 

2007. 

[9] M. Gherlone and M. Di Sciuva, “Thermo-mechanics of undamaged and damaged multilayered 

composite plates: assessment of the FEM sub-laminates approach,” Composite Structures, 81(1), 

137-155, 2007. 

[10] M. Di Sciuva, “Multilayered anisotropic plate models with continuous interlaminar stresses,”

Composite Structures, 22(3), 149 – 167, 1992. 

[11] M. Di Sciuva, M. Gherlone, and L. Librescu, “Implications of damaged interfaces and of other nonclassical effects on the load carrying capacity of multilayered composite shallow shells,”

International Journal of Non-Linear Mechanics, 37, 851-867, 2002.

[12] M. Gherlone, “Refinement of cubic zigzag: preliminary results,” NASA-NIA (National Institute of 

Aerospace) Lecture, August 2007. 
19 

[13] M. P. Nemeth, “Cubic zig-zag enrichment of the classical Kirchhoff kinematics for laminated and 

sandwich plates,” NASA/TM-2012-217570, May 2012. 

[14] A. Barut, E. Madenci, and A. Tessler, “A refined zigzag theory for laminated composite and 

sandwich plates incorporating thickness stretch deformation,” Proceedings of 53rd AIAA/SDM 

Conference, Honolulu, Hawaii, Paper No. AIAA-2012-1705, 23-26 April 2012. 

[15] A. Tessler and H. R. Riggs, “Accurate interlaminar stress recovery from finite element analysis,” 

NASA-TM 109149, 1994. 

[16] A. Tessler, “An improved plate theory of {1,2}-order for thick composite laminates”, International 

Journal of Solids and Structures, 30, 981–1000, 1993. 

[17] M. Gherlone, A. Tessler, and M. Di Sciuva, “A C0-continuous two-node beam element based on 

refined zigzag theory and interdependent interpolation,” MAFELAP 2009 Conference, Brunel 

University, London (UK), June 2009. 

[18] M. Gherlone, A. Tessler, and M. Di Sciuva, “C

0

 beam elements based on the refined zigzag theory 

for multilayered composite and sandwich laminates,” Composite Structures, 93, 2882-2894, 2011. 

[19] E. Oñate, A. Eijo, and S. Oller, “Simple and accurate two-noded beam element for composite 

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362-382, 2012. 

[20] D. Versino, M. Mattone, M. Gherlone, A. Tessler, and M. Di Sciuva, “An efficient, C

0

-continuous 

triangular element for laminated composite and sandwich plates with improved zigzag kinematics,”

MAFELAP 2009 Conference, Brunel University, London (UK), June 2009. 

[21] D. Versino, M. Gherlone, M. Mattone, M. Di Sciuva, and A. Tessler, “C

0

 triangular elements based 

on the refined zigzag theory for multilayered composite and sandwich plates,” Composites: Part B 

(to appear, 2012). 

[22] A. Tessler and T. J. R. Hughes, “A three-node Mindlin plate element with improved transverse 

shear,” Comput. Methods Appl. Mech. Engrg., 50, 71–101, 1985. 

[23] D. Versino, “Refined theories and discontinuous Galerkin methods for the analysis of multilayered 

composite structures,” Ph.D. thesis, Politecnico di Torino, Italy, 2012. 

[24] ABAQUS/Standard User’s Manual, Version 6.10, Dassault Systems Simulia Corp., Providence, RI, 

2011. 
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Technical Publication

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Analytic and Computational Perspectives of Multi-Scale Theory for 

Homogeneous, Laminated Composite, and Sandwich Beams and Plates

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Tessler, Alexander; Gherlone, Marco; Versino, Daniele; Di Sciuva, Marco

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14. ABSTRACT

This paper reviews the theoretical foundation and computational mechanics aspects of the recently developed shear-deformation theory, called the Refined 

Zigzag Theory (RZT). The theory is based on a multi-scale formalism in which an equivalent single-layer plate theory is refined with a robust set of zigzag 

local layer displacements that are free of the usual deficiencies found in common plate theories with zigzag kinematics. In the RZT, first-order 

shear-deformation plate theory is used as the equivalent single-layer plate theory, which represents the overall response characteristics. Local piecewise-linear 

zigzag displacements are used to provide corrections to these overall response characteristics that are associated with the plate heterogeneity and the relative 

stiffnesses of the layers. The theory does not rely on shear correction factors and is equally accurate for homogeneous, laminated composite, and sandwich 

beams and plates. Regardless of the number of material layers, the theory maintains only seven kinematic unknowns that describe the membrane, bending, and 

transverse shear plate-deformation modes. Derived from the virtual work principle, RZT is well-suited for developing computationally efficient, C0-continuous 

finite elements; formulations of several RZT-based elements are highlighted. The theory and its finite elements provide a unified and reliable computational 

platform for the analysis and design of high-performance load-bearing aerospace structures. 

15. SUBJECT TERMS

C0-continuous finite elements; Laminated composites; inter-laminar damage; multi-scale; refined zigzag theory; sandwich 

structures

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