stress projection, layerwise-equivalent,...

February 23, 2001 9:43 WSPC/156-IJCES 00003

International Journal of Computational Engineering ScienceVol. 1, No. 1 (2000) 91–138c© Imperial College Press

STRESS PROJECTION, LAYERWISE-EQUIVALENT,

FORMULATION FOR ACCURATE PREDICTIONS OF

TRANSVERSE STRESSES IN LAMINATED PLATES AND SHELLS

JAY Z. YUAN∗

Graduate Research Assistant, Department of Civil Engineering,The University of Akron, Akron, OH 44325, USA

ATEF F. SALEEB

Department of Civil Engineering, The University of Akron,Akron, OH 44325, USA

ATEF S. GENDY

Structural Engineering Department,Cairo University, Giza, Egypt

Received 16 August 1999Accepted 14 December 1999

A two-phase scheme for accurate predictions of interlaminar stresses in laminated plateand shell structures has been addressed in this study. A modified superconvergent patchrecovery (MSPR) technique has been utilized to obtain accurate nodal in-plane stresseswhich are subsequently used with the thickness integration of the three-dimensionalequilibrium equations to evaluate the transverse shear and normal stresses. Remarkably,the continuity of the resulting interlaminar stresses is automatically satisfied. Such atwo-phase scheme has been applied successfully to a simple smeared layer model (SLM),i.e., a low-order quadrilateral hybrid/mixed element (HMSH5). This simple procedureis found to be completely equivalent to the far more computationally expensive alterna-tive approaches, e.g., sophisticated layerwise approach, for flat geometry. A fairly largenumber of numerical examples have been solved and the results have shown that theproposed scheme is fairly reliable and computationally cost effective.

Keywords: Laminates; Transverse Stresses; Stress Recovery; Plates and Shells

1. Introduction

1.1. General

In a great variety of complex composite structural applications, plate and shell finite

element models are now almost invariably utilized in many phases of the analysis

∗Stress Engineering Services, Inc. 5380 Courseview Dr., Mason, OH 45040, USA.

91


92 J. Z. Yuan, A. F. Saleeb & A. S. Gendy

and design processes. However, some of such complex applications, e.g., failure

studies and various damage mechanisms associated with composite materials, have

often posed great challenges for the analysts, in view of the fact that they are

beyond the capability of conventional shell element theory. Therefore, recent interest

has been focused on the theoretical developments and associated computational

algorithms for a number of refinements for the plate and shell models to be used in

the progressive failure analyses, design, and life predictions of composite laminated

structures. One of these refinements, which is addressed in this work, is associated

with the high gradients (or extreme discontinuity) of transverse shear strains in

highly-anisotropic composite laminates.

Being developed on the basis of hypotheses and assumptions, classical shell the-

ories and formulations are inherently approximate in nature. For instance, constant

transverse shear strains and zero normal stress in the thickness direction have been

often assumed. However, the transverse shear and normal stresses are usually non-

linear functions of thickness coordinates, due to dislike material in each of the layers.

By simply extending the ordinary shell formulations to laminated structures, only

the global responses; i.e., displacements, vibration frequencies, stress resultants,

and such, can be accurately captured. However, the local responses, such as inter-

laminar stresses, are only approximated in an integral sense. For instance, using

the kinematical assumptions of first-order, shear-deformable, shells, the resulting

interlaminar shear stresses become constant within each layer and the transverse

normal stress is implicitly taken to be zero through the whole thickness.

On the other hand, it is well known that laminated plate and shell structures

are more likely to fail due to “delamination”. Consequently, the determination of

“accurate” transverse shear stress variation through thickness, especially at lamina

interface and free edge, plays a vital role in the progressive failure study for such

structures.

To this end, since the classical single-layer shell theory is often inadequate to

determine the interlaminar stresses for the laminated plate and shell structures,

several refined laminate approaches have been proposed to date to extend the clas-

sical shell theory to account for such effects. As far as the shear deformations are

concerned, the “typical” kinematical assumptions for these extensions can be clas-

sified into several categories: (1) first-order (shear deformation) equivalent-single-

layer (FESL) approach; (2) higher-order (shear deformation) equivalent-single-layer

(HESL) approach; (3) layerwise approach. The spirit of these approaches is to re-

duce the three dimensional problem into a two-dimensional one, thereby reducing

the computational cost. A brief literature review of some of the research works

concerning these approaches is given in the sequel.

1.2. Literature review

Starting with the First-order (shear deformation) equivalent-single-layer (FESL)

approach, this approach extends the classical single-layer theory by accounting


Transverse Stresses in Laminated Plates and Shells 93

for the varying layer thickness and material properties in an integral sense. For

many applications, the FESL approach provides a sufficiently accurate description

of the global laminate response (e.g., transverse deflection, fundamental vibration

frequency, critical buckling load, force and moment resultants, etc.). The main ad-

vantages of the FESL models are their inherent simplicity and low computational

cost due to the relatively small number of dependent variables that have to be

solved. However, the FESL models are often inadequate in determining the three-

dimensional field at the ply level. The major deficiency of the FESL models in

modeling composite laminates is that the accuracy of transverse stresses mainly

depends on the accuracy of corresponding in-plane stresses when the transverse

stresses are computed through stress equilibrium. For example, for a displacement-

based shell (plate) element, in order to get a linear distribution for transverse shear

stresses along the thickness direction (instead of constant distribution carried out

by constitutive equations), at least a linear distribution for in-plane stresses in mid-

surface is required. This means that one has to utilize quadratic or even higher-order

elements; e.g., 9-node or 16-node quadrilateral element, where the surface displace-

ment interpolations of which (i.e., second or higher order polynomials) may lead

to the requirements for linear distribution of the in-plane stresses. Actually, this

is one of the main motivations for using higher-order elements in early attempts;

e.g. Chaudhuri, (1986), Tocher and Hartz, (1967). However, these elements suffer

from the following major deficiencies: (1) computational expensiveness, (2) high

distortion sensitivity.

For the Higher-order (shear deformation) equivalent-single-layer (HESL)

approach, the transverse stresses can also be carried out through the constitutive

equations instead of using the equilibrium. This leads to a higher-order (shear de-

formation) equivalent-single-layer approach. In this approach, a higher-order kine-

matics is assumed in thickness direction in order to capture the complex shear

deformation of laminated structures. The varying layer thickness and material prop-

erties are still considered in an integral sense. Since a higher-order approximation is

made in shear deformation and transverse strains, more accurate transverse stresses

(compared with those of FESL) can be expected. However, this approach has its

own major deficiencies: (1) the variables associated with the higher-order terms

have no physical meaning, (2) complex derivation leads to implementation difficul-

ties, (3) almost inextendable to the non-linear plasticity analysis, (4) the continuity

of interlaminar stresses is still not satisfied, and (5) computationally expensive.

Both FESL and HESL approaches mentioned above belong to the so-called

“smeared layer model” (SLM), because in such a model the varying layer thickness

and material properties are only considered in an integral sense. On the other

hand, one can also consider each layer (or layer group) individually. This leads to

the so-called “discrete layer model” (DLM). The discrete layer model widely used in

current literature is the layerwise constant strain approach, layerwise approach in

short. In such approach, the transverse strains are discontinuous across the interface

of the layers, which leaves a room to satisfy the interlaminar continuity of transverse



shear stresses. Such a continuity condition is often enforced through the satisfaction

of equilibrium equations along the thickness directions. Unlike the smeared layer

models, the layerwise approach accounts for each layer individually which leads to

accurate transverse shear stresses. However, corresponding computational expense

is also greatly increased. Basically, an assembly process is required to form the

element, or laminate, stiffness from the individual layer stiffness. As pointed out

by Spilker, Jakobs and Engelmann (1985) the number of degree-of-freedom grows

rapidly as the number of layers increases and computer core storage may limit the

number of layers allowed. This has motivated a large number of efforts to modify

the “direct” layerwise approach, aiming at the reduction in computational expense;

e.g. see Averill and Reddy (1992), Reddy (1993), Barbero (1992) and Zinno and

Barbero (1994) among many others.

Because of their high computational cost, the layerwise and HESL approaches

are often applied in combination with less expensive schemes, such as FESL ap-

proach to reduce the total computational cost. In other words, since accurate trans-

verse shear stresses are often required only in a certain small region, the rest of the

problem domain can be analyzed by using less expensive approaches. The global-

local analysis technique [Ochoa and Reddy, 1992] and hierarchic models [Babuska

et al., 1992] are motivated by such a consideration. The global-local analysis refers to

the combination of layerwise approach and FESL approach in a practical problem.

This technique can be considered as an h-version refinement topology. In this, the

refinement is made in the thickness direction instead of in the in-plane dimensions.

However, transition elements or equivalent computational procedures are needed

to pass the variables from the global analysis to the local one. This often leads to

implementation complexity and application inconvenience. Parallel to the global-

local analysis, the hierarchic models have been proposed by Babuska and coworkers

(1992). The hierarchic models refer to the combination of HESL and FESL ap-

proaches. The hierarchic sequence of the models is classified based on the degree of

equilibrium equations satisfied by the model. Different from the global-local analy-

sis technique, for hierarchic models, in a particular application, the accuracy of the

region of most interest can be increased by choosing a high order model locally while

the whole mesh remains unchanged; thus transition elements are not needed. It is

specifically efficient for accounting for the boundary layer effects. This technique can

be considered as a p-version refinement topology. Again, in this case the refinement

is made in the thickness direction. However, this approach is somewhat restricted in

applications because some assumptions must be made to simplify the complicated

manual derivation; e.g., only a laminated strip is discussed in [Babuska et al., 1992].

1.3. Objective and outline

Based on the preceding literature review in the area of handling laminated struc-

tures by the two-dimensional analysis, it can be concluded that most, if not all, of

the proposed approaches are either too simple to capture the intrinsic behavior of



transverse shear and normal stresses at lamina interfaces and free edges (FESL)

or too expensive and/or restrictive to be applied independently (HESL, layerwise,

global-local technique and hierarchic models). The present effort focuses on the de-

velopment of a low-order quadrilateral model and layerwise equivalent finite element

approach for general laminate shell applications. As demonstrated in the applica-

tions section, the resulting model is not only computationally economical but also

capable of accurately predicting stress components at the layer level.

To this end, we are introducing a two-phase hierarchical approach to accurately

predict interlaminar stresses corresponding to localized regions of high transverse

strain fields across the thickness in laminates. In its first phase, a modified su-

per convergent patch recovery (MSPR) technique is utilized to obtain accurate,

through the thickness, distributions of nodal in-plane stresses. These are subse-

quently utilized, in the second phase, together with the thickness integration of the

three-dimensional (3D) equilibrium equations to evaluate the transverse shear and

normal stresses.

In summary, in our proposed methodology, the classical FESL approach has

been extended by exploiting an appropriate stress projection, i.e., modified super-

convergent patch recovery technique (MSPR), as post-processing. Since the super-

convergent property of in-plane stresses can be obtained by this MSPR technique,

the corresponding transverse stress can be carried out accurately by satisfying the

equilibrium through-the-thickness pointwise. The mathematical derivations prove

that such a simple approach is actually equivalent (in terms of the good quality and

accuracy of results) to the more sophisticated, layerwise, approach for the case of

plate structures. The proposed scheme is readily extendible to include both geomet-

ric and material nonlinear effects (since such an extension is made outside the main

finite element’s formulation). In addition, like the underlying FESL approach, it is

also applicable to thin as well as to moderately thick laminate plate- and shell-type

structures. The present methodology has been applied to a fairly large number of

sample problems using the modified HMSH5 quadrilateral element [Saleeb et al.,

1987, 1990] as a vehicle for analysis to assess its efficiency, accuracy and robustness.

2. Modified Superconvergent Patch Recovery (MSPR) Procedure

In this section, the Superconvergent Patch Recovery (SPR) scheme proposed by

Zienkiewicz and Zhu (1992 and 1994) is extended to the present Modified Super-

convergent Patch Recovery (MSPR). The major modification is that an isopara-

metric coordinates system is utilized in MSPR to perform the least square fitting.

Thus, MSPR can be applied to general curved shell applications, and the resulting

geometrical interpolatory scheme is thus completely devoid of any singularity in

general geometries of element patches, see also Sec. 7.1.2.

In the MSPR procedure, the nodal in-plane stress vector σ∗ is assumed to be a

polynomial expansion. Such a polynomial is valid over an element patch surrounding

the particular assembly node considered (see Fig. 1); where the element patch has



Fig. 1. A typical patch of HMSH5.

nine nodes defining its geometry of such a patch as shown in Fig. 1. The polynomial

expansion may be expressed for each ith component of σ∗, as

σ∗i = P a (1)

where P = [1, r, s, rs] and a is a set of unknown parameters. The polynomial

expansion in Eq. (1) is expressed in the isoparametric coordinate defined by the 9

nodes of the patch, that is

x(r, s, t) =

9∑k=1

Nkxk +t

2

9∑k=1

hkNkef(k)3 (2)

where x = (x, y, z) are the Cartesian coordinates of a point in the patch; xk is the

position vector of nodal point k on the reference surface; ef(k)3 are the components of

a unit vector emanating from node k in the fiber direction; hk is the fiber dimension

of shell thickness at node k; and Nk is the “familiar” two-dimensional Lagrangian

shape function in terms of (r, s) associated with node k.

The determination of the unknown parameters a of the expansion given in

Eq. (1) is best made by ensuring a least square fit of this to the set of superconvergent

(or at least high accuracy) sampling points existing in the patch considered, if such

points are available. Since the MSPR procedure is performed in an isoparametric

surface with constant thickness-coordinate t, it can be omitted in the following for

convenience. To perform the least square fit we minimize the following function:

F (a) =

4∑i=1

(σh(ri, si)− σ∗(r, s)

)2=

4∑i=1

(σh(ri, si)−P(r, s)a

)2(3)

where (ri, si) are the natural coordinates of a group of “optimal” sampling points,

σh(ri, si) are the components of the in-plane stress vectors calculated directly from

finite element analysis (where their components are resolved in the local coordi-

nates at centroid of each corresponding element). In addition to the basic form

of error terms in Eq. (3) above, several other modifications and enhancements

to the original SPR strategy have also been proposed in the literature; e.g., to



include tractions/displacement boundary terms, or to append additional, penatly-

like, curvature-control functions to achieve higher-order degrees of smoothing for

the recovered stress fields (Tessler, Riggs, and Macy, 1994).

The minimization condition of F (a) implies that the unknown parameter vector

a has to satisfy

4∑i=1

PT(ri, si)P(ri, si)a =

4∑i=1

PT(ri, si)σh(ri, si) . (4)

This can be solved in matrix form as

a = A−1 b (5)

where

A =

4∑i=1

PT(ri, si)P(ri, si) and b =

4∑i=1

PT(ri, si)σh(ri, si) . (6)

For boundary nodes, the patches are assembled and solved in the way shown in

Figs. 2 and 3, i.e., we simply recover the stresses at the edge and corner nodes for

the corresponding patch associated with the adjacent interior nodes. In addition to

its simplicity, this boundary treatment will also maintain the important property of

superconvergence for the present linear quadrilateral element. This latter property

has been confirmed in all our numerical experiments performed to date. The same

property was also demonstrated for the plane-stress/strain cases in Zienkiewicz and

Zhu (1992, 1994). See also related discussions in the later Sec. 7.1.2. Furthermore, we

have also performed a number of limited numerical experiments utilizing alternate

recovery schemes in which additional conditions are included; e.g., traction-free

conditions at free edges, specified boundary displacements, etc., but the results did

not exhibit any significant improvements. Extended work and results of this type

will be reserved for a future publication.

As in the SPR procedure precisely the same matrix A is needed for the solution

of each component of σ∗ and hence only a single evaluation of such a matrix is

necessary. Since only four equations need to be solved for each patch in the case

Fig. 2. Boundary nodal recovery (corner node).



Fig. 3. Boundary nodal recovery (edge node).

of the present linear quadrilateral element we considered, the whole procedure is

quite inexpensive. On the other hand, because the sample points are located in the

centroid of each element and the stress functions in Eq. (1) are complete bilinear

polynomials, the superconvergent property of recovered nodal stresses (both interior

and boundary) still remains.

It is also worthwhile repeating that the singularity problem in SPR is completely

eliminated here (see Sec. 7.1.2). This singularity problem, which is also noticed

by Zienkiewicz and Zhu (1994) is that the determinate of the matrix A defined

in Eq. (5) is not always nonzero if the recovery scheme is formed in the global

coordinate described in the SPR. Thus, the SPR will not be applicable for some

geometrical patch alignments (see Sec. 7.1.2).

The patch recovery technique provides not only the superconvergence of the in-

plane nodal stresses themselves but also their accurate (first) derivatives from the

bilinear polynomial assumptions on each patch. Therefore, the following step of ap-

plying 3D stress equilibrium equations becomes more straightforward. In addition,

applying the patch recovery technique to the hybrid/mixed formulation provides

the same superconvergent rate to all three in-plane stress components rather than

being treated differently in their original polynomial assumptions, i.e., linear as-

sumption for σx and σy, constant assumption for τxy; see references Saleeb et al.,

(1987, 1988, 1990). This equal importance of the rate of convergence of all three

recovered in-plane stresses benefits the following procedure of applying the equilib-

rium equations. In particular, the above three in-plane stresses have equal position

in equilibrium equations (see Sec. 3) and the final solutions for recovering inter-

laminar stresses can be disturbed by any lack of accuracy in any of the in-plane

components.

3. Transverse Stresses via Equilibrium Equations

The elasticity equilibrium equations for the ith layer of a multi-layered laminate

have the following general form

Divσ + ρb = 0 . (7)



Here, σ denotes the symmetric Cauchy stress tensor; and ρb are the body forces

(which will be neglected in the following equations), and Div is the divergence

operator.

The three differential equations of equilibrium, i.e, Eq. (7), give three relations,

namely,

∂τxz

∂z= −

(∂σx

∂x+∂τxy

∂y

)(8a)

∂τyz

∂z= −

(∂τxy

∂x+∂σy

∂y

)(8b)

and

∂2σz

∂z2=∂2σx

∂x2+∂2σy

∂y2+ 2

∂2τxy

∂x∂y(8c)

from which the transverse stresses τxz, τyz and σz can be evaluated through integra-

tion with respect to the laminate thickness z. As described previously, the in-plane

normal stresses σx and σy and the in-plane shear stress τxy utilized in Eq. (8) are

obtained by the modified superconvergent patch recovery scheme.

The final τxz and τyz for the ith layer whose z-dimension ranges between hi−1

at the bottom to hi at the top, of an N -layer laminated shell can be integrated

through the thickness either from the bottom surface, i.e.,

iτxz(z) = −i−1∑k=1

∫ hk

hk−1

(∂iσx

∂x+∂iτxy

∂y

)dζ −

∫ z

hi−1

(∂iσx

∂x+∂iτxy

∂y

)dζ (9a)

iτyz(z) = −i−1∑k=1

∫ hk

hk−1

(∂iσy

∂y+∂iτxy

∂x

)dζ −

∫ z

hi−1

(∂iσy

∂y+∂iτxy

∂x

)dζ (9b)

or, from the top surface, i.e.,

iτxz(z) = −N∑

k=i+1

∫ hk−1

hk

(∂iσx

∂x+∂iτxy

∂y

)dζ −

∫ z

hi

(∂iσx

∂x+∂iτxy

∂y

)dζ (9c)

iτyz(z) = −N∑

k=i+1

∫ hk−1

hk

(∂iσy

∂y+∂iτxy

∂x

)dζ −

∫ z

hi

(∂iσy

∂y+∂iτxy

∂x

)dζ . (9d)

Note that since ∂x, ∂y and τxy are linearly distributed along the thickness in each

layer, the transverse shear stresses computed from Eqs. (9) are quadratic variations

through the thickness.



4. Continuity of Interlaminar Transverse Shear Stresses

Integrating either Eqs. (9a) and (9b), starting from the bottom surface (or Eqs.

(9c) and (9d)), starting from the top surface) could, in principle, lead to different

values for transverse shear stresses at a given location, i.e., a nonunique solution

for transverse shear stresses can be obtained. However, it will be proved next that

the interlaminar continuity of the transverse shear stresses is actually automatically

satisfied utilizing the proposed approach.

In the regular finite element formulation, the equilibrium through the thickness

is satisfied in an integral sense, that is,∫ hN

h0

(∂σx

∂x+∂τxy

∂y+∂τxz

∂z

)dz = 0 (10a)

∫ hN

h0

(∂τxy

∂x+∂σy

∂y+∂τyz

∂z

)dz = 0 (10b)

In addition, the transverse shear stresses are usually known at the top and

bottom surfaces of the laminate (e.g., zero for no surface tractions), i.e.,

τxz(hN ) = τxz(h0) = 0 (11a)

τyz(hN ) = τyz(h0) = 0 (11b)

Substituting from Eqs. (11a) and (11b) into Eqs. (10a) and (10b) respectively,

and noting that in the modified superconvergent recovery technique the in-plane

stress components satisfy the in-plane equilibrium equations in each layer, i.e.,

Eqs. (8a), (8b) and (10a) can be integrated through the thickness utilizing the

proposed procedure as

N∑k=1

∫ hk

hk−1

(∂σx

∂x+∂τxy

∂y

)dζ = 0 . (12a)

This can be expressed as (after exchanging the integration limits)

i∑k=1

∫ hk

hk−1

(∂σx

∂x+∂τxy

∂y

)dζ =

N∑k=i+1

∫ hk−1

hk

(∂σx

∂x+∂τxy

∂y

)dζ . (13a)

Similarly, Eq. (10b) can be expressed as

i∑k=1

∫ hk

hk−1

(∂τxy

∂x+∂σy

∂y

)dζ =

N∑k=i+1

∫ hk−1

hk

(∂τxy

∂x+∂σy

∂y

)dζ . (13b)

From Eqs. (13a), (13b) and Eqs. (9a) and (9b), one concludes that the interlaminar

shear stresses are thus continuous across each layer, i.e., the continuity of interlam-

inar shear stress is satisfied.



Fig. 4. A typical layer i and its interfaces.

5. Equivalent Layerwise Linear Kinematics

We now focus on a typical layer (Fig. 4). The stress resultants of such typical layer

i can be expressed according to Fig. 4 as

inα =

∫ jh2

− ih2σ eαdζ ,

imα =

∫ jh2

− ih2

(ix− ix0

)⊗ σ eαdζ (14)

where n and m are the resultant vectors of normal forces and bending moments,

respectively; σ eα is Cauchy stress vector on the plane whose normal is defined

by eα; α = 1, 2 for plane x-z and y-z, respectively; and, the symbol (⊗) indicates

cross product of vectors. Integrate equilibrium equations, i.e., Eq. (7), through the

thickness of the layer i yields the following set of static field equations:

inα∣∣∣α

+ iq− i−1q = 0 (15a)

imα∣∣∣α

+ ix0,α × inα +ih

2ia3 ×

(iq + i−1q

)= 0 (15b)

and boundary conditions,

ναinα = 0 , να

imα = 0 on ∂Ωσ

iu = iu , ia3 = ia3 on ∂Ωu(15c)

where (.)α∣∣α

is the covariant derivative of the quantity (indicated in Eq. (15b)) with

respect to lamina direction α; i−1q and iq are the traction force vectors, and the

superscript i− 1 and i refer to the lower and upper surfaces of layer i, respectively;

να (α = 1, 2) are the components of the normal vector of boundary Ωσ; and iakis the orthonormal basis system describing the deformed cross-section of layer i.



Again, due to the effect of transverse shear strains, the “director” vector ia3 is

generally not normal to the current middle surface of the ith layer.

An equivalent discrete weak formulation can be obtained by weighting the field

equations with test functions δix0 and δiw, i.e.,∫Ω

N∑i=1

[inα∣∣α

+ iq− i−1q]· δix0

dΩ

+

∫Ω

N∑i=1

[imα

∣∣α

+ ix0,α × inα +ih

2ia3 ×

(iq + i−1q

)]· δiw

dΩ

+

∫∂Ωσ

N∑i=1

(να

inα)· δix0 +

(να

imα)· δiw

d∂Ω = 0 (16)

where δiw is the axial vector following the variation of the basis system iak in the

current configuration, i.e.,

δiak = δiw× iak . (17)

Assuming that the axial displacement is linearly distributed along the thickness

within each layer, i.e.,

δix = δix0 + iζδia3 −ih

2≤ ζ ≤

ih

2(18)

where the variable δia3 in general is different for different layers, our objective is

to prove that the axial displacement is actually continuous across the interface of

the layers. To this end, it is known that the contribution of the interlaminar stressiq (i = 0, 1, . . . , N) within the variational formulation has to vanish, i.e.,

0 =

∫Ω

N∑i=1

[(iq− i−1q

)· δix0 +

ih

2ia3 ×

(iq + i−1q

)· δiw

]dΩ

=

∫Ω

N∑i=1

[iq ·

(δix0 +

ih

2δia3

)− i−1q ·

(δix0 +

ih

2δia3

)]dΩ . (19)

Since 0q and Nq are equal to zero, Eq. (19) can be rewritten (after employing

Eq. (18)) as follows:

=

∫Ω

N−1∑i=1

[iq ·

(δix0 +

ih

2δia3

)]−

N∑i=2

[i−1q ·

(δix0 −

ih

2δia3

)]dΩ

=

∫Ω

N−1∑i=1

[iq ·

(δix0 +

ih

2δia3

)]−N−1∑i=1

[iq ·

(δi+1x0 −

i+1h

2δi+1a3

)]dΩ

=

∫Ω

N−1∑i=1

iq ·

[(δix0 +

ih

2δia3

)−(δi+1x0 −

i+1h

2δi+1a3

)]dΩ . (20)



Thus, one finally gets

δix0 +ih

2δia3 = δi+1x0 −

i+1h

2δi+1a3 for i = 1, 2, . . . , N − 1 , (21)

that is, the continuity of interlaminar kinematics condition is obtained. This shows

equivalency in the sense of layerwise-constant transverse shear strain distribution

as with more sophisticated layerwise kinematics models Babuska et al., (1992),

Kant and Manjunatha, (1988, 1994), Reddy, (1993), Ren, (1986,1987), Seide and

Chaudhuri, (1987), Sciuva, (1987), Spilker et al., (1977, 1985) among several others.

Remark 1:

The mathematical arguments and proofs given above for interlaminar-stress-

continuity and layerwise constant strain equivalency are based on the requirement

that in-plane equilibrium satisfaction is approached. For flat geometry, this fact is

met simply because the superconvergency of in-plane stresses is achieved “exactly”.

For general spatially-curved shells, however, since the element patches can not fully

match the curved geometry, the superconvergency of in-plane stresses is only ap-

proached asymptotically through mesh refinement.

Remark 2:

As a corollary to the above remark, the asymptotic property of superconvergence

must also carry over to the interpretation of all the “equivalency” results given.

That is, this latter equivalency does not imply “equality” or “identity” in the strict

mathematical sense, but rather it indicates comparable accuracy and good quality

of the results. In particular, in practice, where relatively coarse meshes are utilized,

results from the present approach and those from alternatives based on truly layer-

wise formulations, will be different in their details. For example, direct evaluation

of interlaminar stresses (i.e., from constitutive equations) will not guarantee their

continuity. Furthermore, the degree of approximations (in-plane discretizations) of

the underlying plate/shell formulations (e.g. 4-noded, 9-noded, etc.) will also impact

the accuracy of respective results in any of these latter schemes.

Remark 3:

Considering the important practical situations involving free edges in laminated

plate/shell problems, additional important factors will also come to play. Here, the

characteristic sharp variations (singularities, localizations, etc.) in the gradients of

the in-plane stress fields must be captured, through local in-plane mesh refinement

near these edges, before ensuring the prerequisite (close) satisfaction of equilibrium

conditions in the present approach (of course, accuracy considerations demand sim-

ilar mesh refinements in other alternative approaches, irrespective of the degree of

approximations utilized in the thickness direction). Although not specifically ad-

dressed here, our numerical experiments with such problems have yielded results



that are quite positive. In particular, the recovery scheme utilized was found very

effective in capturing nearly-singular variations in stresses at edges, and an im-

portant case to the point is included in Sec. 7.4 for a strongly-anisotropic beam

problem.

6. Interlaminar Transverse Normal Stress

The transverse normal stress can be obtained by integrating the second-order dif-

ferential equation, i.e., Eq. (8c), twice. Two constants of integration result for each

layer. Two described stress conditions on the top and bottom surfaces plus conti-

nuity conditions at layer interface are insufficient to define all those constants of

integration for laminate problems. However, in a thin plate/shell problem, such a

transverse normal stress can be approximated as

iσz(z) =

∫ih

∫ih

(∂2σx

∂x2+∂2σy

∂y2+∂2τxy

∂x∂y

)dzdz︸︷︷︸

Compute locally

+ C3z + C4︸︷︷︸Compute globally

(22)

where the constants C3 and C4 are obtained from the known values of σz at

z = ±H/2 (H is the total thickness of the laminate). It is worthwhile to men-

tion that unlike in Eqs. (9a) and (9b), the second derivatives of the in-plane stress

components appear in Eq. (22). Because of the bilinear assumption in Eq. (1), only

a constant results for the terms between the brackets in Eq. (22). Thus the trans-

verse normal stress is expected to be less accurate than the corresponding transverse

shear stresses; (e.g., a bi-quadratic polynomial assumption has to be introduced to

get the same or comparable accuracy). In addition, such a normal stress can also

be discontinuous across the interface of the layers. In fact, the continuity of inter-

laminar transverse normal stress can not be generally satisfied until the equality of

the derivatives of axial displacements at interfaces are enforced.

7. Numerical Examples

The solution scheme developed above has been implemented in conjunction with

a 5-noded mixed shell linear-quadrilateral element, designated here as HMSH5,

(i.e., with independently assumed displacement/rotation and strain fields within

the element) for the analysis of composite laminates; e.g. see Saleeb et al., (1987 &

1990). In this, the mixed element possesses a multi-layered capability (e.g., see Wilt,

Saleeb and Chang (1990) in which each layer contains (3×3) integration points in

the lamina plane in addition to two integration points in the thickness direction.

All numerical results reported here have been obtained using this model. These are

selected to demonstrate the capabilities, as well as various aspects, of the proposed

scheme; e.g., robustness, accuracy and computational efficiency. In particular, an

emphasis is placed on laminated plate/shell problems with relatively small thick-

ness aspect ratios (i.e., moderate and thick cases) where greater effects of trans-

verse shear deformations exist, together with complex distributions of interlaminar



stresses. However, for completeness, we have also included the results of a compos-

ite shell problem with a range of thicknesses (i.e., from thick to thin regimes). In

these numerical simulations, comparisons with available “exact” solutions as well

as other independent numerical schemes are also presented.

7.1. Convergence of nodal in-plane stresses obtained by MSPR

7.1.1. A square plate subjected to specified displacement field

A square plate with unit length is subjected to a specified displacement field as

u(x, y) = x(1− x)y(1− y)(1 + 2x+ 7y) (23)

where the origin of the x- and y-axes is located at the bottom left corner of this

plate, i.e., x and y range between 0 and 1. The displacement field u(x, y) vanish

along the plate boundary. The local error of the recovery stress σx is examined at a

nodal point with coordinates of (0.25, 0.25) where the exact solution has the values

of σx = 0.375. Uniform subdivisions 4×4, 8×8 and 16×16 meshes are used in the

finite element analysis. The results of σx obtained using the proposed scheme are

listed in Table 1 for different mesh size. The error of stress obtained from the MSPR

scheme is depicted in Fig. 5. As shown from this figure, the theoretically expected

superconvergent rate of 2 is reached here.

Fig. 5. Rate of convergence of error in stresses σx of present HMSH5 at the point (0.25, 0.25).



Table 1. σx of present HMSH5

evaluated at the point (0.25, 0.25).

mesh σx

4×4 0.3488×8 0.368

16×16 0.373Exact solution of σx 0.375

Fig. 6. Infinite plate with a central circular hole subjected to unidirectional tensile loads and theportion of the infinite plate analyzed.

7.1.2. Infinite plate with central circular hole

The problem considered here is a portion of an infinite plate with a central circular

hole subjected to a unidirectional tensile stress equal to unity as shown in Fig. 6.

Plane strain conditions are assumed with a Poisson’s ratio ν = 0.3 and Young’s

modulus E = 1000. A small portion of the plate near the hole is considered for

analysis. By making use of symmetry, only a quarter of such a portion is analyzed.

The boundary conditions are prescribed such that symmetry conditions are imposed

on edges AB and ED; and the plate is loaded by tractions given by an analytic



solution on edges BC and DC. Such tractions are expressed as

σx = 1− 1

r2

(3

2cos 2θ + cos 4θ

)+

3

2r4cos 4θ

σy = − 1

r2

(1

2cos 2θ − cos 4θ

)− 3

2r4cos 4θ (24)

τxy = − 1

r2

(1

2sin 2θ + sin 4θ

)+

3

2r4sin 4θ

where (r, θ) are the usual polar coordinates.

Three mesh sizes are used for the analysis, i.e., (12×6), (24×12) and (36×18).

The recovered stresses are examined at the nodal point (1.5, 450) where the exact so-

lution has the values of σx = 1.1481481, σy = −0.14814810 and τxy = −0.22222229.

Results from the finite elements are listed in Table 2 along with those from the an-

alytical solution. The errors in stresses are also depicted in Fig. 7 for different mesh

size. Again the numerical results show that the superconvergent property is fully

preserved in the proposed MSPR scheme.

Fig. 7. Convergence of the recovered in-plane stresses at the point (1.5, 450).



Table 2. In-plane stresses of present HMSH5 evaluated at the

point (1.5, 450).

mesh σx σy τxy

1 1.03843 −0.0371982 −0.2421572 1.09170 −0.0967173 −0.2280293 1.15910 −0.1542700 −0.225927

Exact solution 1.1481481 −0.1481481 −0.22222229

Remark 4:

It is of interest to note the patch surrounding the point (1.5, 450) (see Fig. 8). If the

patch recovery scheme is formed in the global coordinates, the matrix P in Eq. (1)

takes the following form

P =

1 a b ab

1 b a ab

1 c d cd

1 d c cd

. (25)

It is easy to verify that matrix P is singular and thus matrix A is also singular

(see Eq. (6)), which leads to non-unique solutions of the unknown parameters a.

However, if the original axes are rotated by an angle of 450 and the SPR is performed

in such a new coordinate system (x′, y′) (see Fig. 8), the matrix P becomes:

P =

1 a′ b′ a′b′

1 a′ −b′ −a′b′1 c′ d′ c′d′

1 c′ −d′ −c′d′

. (26)

This matrix P is non-singular unless a′ = c′. Such a singularity-free treat-

ment of rotating the axes is particularly important for most general cases involving

shells curved in space. Here, we completely by-pass the above singularity problem

and adopt surface-(lamina)-based components for the physical stresses in the shell

elements (see details in Saleeb et al. (1987 & 1990), with polynomial expansions in

terms of the natural/isoparametric coordinates, in Eqs. 1 and 26, when performing

the patch recovery scheme.

7.2. Accuracy and efficiency of the projection-layerwise-equivalent

approach

7.2.1. Cross-ply plates

Numerical results of the improved stress predictions by the present approach

for a group of cross-ply plates are presented in this section. The three-dimensional



(a) A patch leads to singularity of matrix A.

(b) The patch in the new co-ordinate system.

Fig. 8. Alternative coordinates for singularity-free patch recovery.

elasticity solutions of Pagano, (1970) for simply supported rectangular plates un-

der sinusoidal loading are used to assess the accuracy and reliability of the present

approach for different span to thickness ratios (a/h). In addition, results from two

different sophisticated layerwise approaches, by Reddy, (1993), Ren, (1987) as well

as one higher-order approach by Kant and Pandya, (1988) are also included for

comparison. The following four laminated plate problems are investigated: (1) Two

layers (0/90) square (b = a) laminate; (2) Three layers (0/90/0) square (b = a) lam-

inate; (3) Three layers (0/90/0) rectangular (b = 3a) laminate; and (4) Four layers

(0/90/90/0) square (b = a) laminate. The four problems are subjected transverse

loading defined as

q = q0 sinπx

asin

πy

b. (27)



The material coefficients for each layer are

EL = 25× 106 Et = 106

GLT = 0.5× 106 GTT = 0.2× 106

VLT = VTT = 0.25

(28)

where the subscripts L and T signify the parallel and transverse directions, respec-

tively, to the fiber. The following normalized quantities are defined with respect to

the data

(σx, σy, τxy

)=(σx, σy, τxy

)/(q0S

2)(

τyz, τxz)

=(τyz, τxz

)/(q0S)

σz = σz/q0

S = a/h z = z/h .

(29)

By making use of symmetry conditions, only a quarter of a plate has been used

for the analysis in which a mesh of 8×8 HMSH5 elements is utilized. Comparisons

of the stresses obtained using the MSPR approach with those provided either by

analytical solutions or numerical simulations, e.g., layerwise approach, are given in

Tables 3–8 for the four plate problems. Typical distributions for transverse stresses

over plate thickness are depicted in Figs. 9 and 10 for first plate problem and in

Figs. 11 and 12 for the third one. It is worthwhile to mention that the accurate

transverse stresses provided by the present scheme are based on the more accurate

in-plane stresses coming from the patch projection technique.

Remark 5:

All the transverse shear stresses reported are evaluated at the corner point of the

discretized model (due to symmetry considerations) for the four plate problems. The

worst approximation of the transverse stresses using the patch recovery scheme may

occur in such corner nodes since the in-plane stresses are extrapolated from a patch

similar to that shown in Fig. 3. In this case, the derivatives of these stresses with

respect to x and y are evaluated at the “dimensional limits” (i.e., x = a/2 and

y = b/2). To improve accuracy, the entire or half of the problem can be modeled

instead of its quarter. In this case the transverse stresses are still evaluated at the

boundaries, but they are away from the corners thus at least one of the derivatives

(with respect to x or y) is not in the dimension limit. Certainly, computational cost

will increase by such a solution. However, this is the case if there is no symmetrical

condition in the problem, such as the angle-ply plate examples illustrated in the

following section. It is also worthwhile to emphasize that the best approximation

would be inside element patch.



Table

3.

Str

esse

sin

square

lam

inate

pla

tew

itha/h

=4(0/90).

Pagano,

1970

Pre

sent

HM

SH

5

zσx

σy

τ yz

τ xz

τ xy

σx

σy

τ yz

τ xz

τ xy

( a 2,b 2,z

)( a 2

,b 2,z

)( a 2

,0,z)

( 0,b 2,z

)(0,0,z

)

−0.5

−0.7

807

−0.0

955

0.0

000

0.0

000

0.0

591

−0.7

328

−0.0

864

0.0

000

0.0

000

0.0

537

−0.4

−0.3

988

−0.0

734

0.0

423

0.1

976

0.0

427

−0.4

608

−0.0

729

0.0

388

0.1

957

0.0

430

−0.3

−0.1

437

−0.0

553

0.0

738

0.2

919

0.0

307

−0.1

887

−0.0

594

0.0

703

0.3

056

0.0

322

−0.2

0.0

598

−0.0

398

0.0

968

0.3

127

0.0

209

0.0

834

−0.0

460

0.0

944

0.3

297

0.0

215

−0.1

0.2

792

−0.0

258

0.1

122

0.2

659

0.0

117

0.3

554

−0.0

325

0.1

112

0.2

680

0.0

107

−0.0

0.5

872

−0.0

122

0.1

202

0.1

353

0.0

012

0.6

275

−0.0

190

0.1

206

0.1

206

0.0

000

+0.0

0.0

247

−0.6

307

0.1

202

0.1

353

0.0

012

0.0

190

−0.6

257

0.1

206

0.1

206

0.0

000

+0.1

0.0

372

−0.3

100

0.2

632

0.1

244

−0.0

089

0.0

325

−0.3

554

0.2

680

0.1

112

−0.0

107

+0.2

0.0

508

−0.0

789

0.3

188

0.1

063

−0.0

188

0.0

460

−0.0

834

0.3

297

0.0

944

−0.0

215

+0.3

0.0

667

0.1

387

0.3

027

0.0

807

−0.0

282

0.0

594

0.1

887

0.3

056

0.0

703

−0.0

322

+0.4

0.0

859

0.4

152

0.2

075

0.0

461

−0.0

410

0.0

729

0.4

608

0.1

957

0.0

388

−0.0

430

+0.5

0.1

098

0.8

417

0.0

000

0.0

000

−0.0

588

0.0

864

0.7

328

0.0

000

0.0

000

−0.0

537



Table 4. Transverse normal stress (σz) for simply supported laminate under sinu-soidal loading (a/h = 4) (0/90).

Transverse normal stress (σz)

Thickness HOST9(16N) [Kant, 1994] HMSH5 Elasticity [Pagano, 1970]

−0.5 0.000000 0.000000 0.000000

−0.4 0.072587 0.094968 0.078940

−0.3 0.175074 0.192091 0.250000

−0.2 0.289113 0.293080 0.460526

−0.1 0.399094 0.396046 0.671050

0.0 0.491027 0.500000 0.789474

+0.1 0.584600 0.603954 0.868421

+0.2 0.698580 0.706920 0.921053

+0.3 0.817625 0.807909 0.960526

+0.4 0.924730 0.905932 0.973684

+0.5 1.000000 1.000000 1.000000

Fig. 9. Variation of transverse shear stress (τxz) through the thickness of a simply supportedlaminate under sinusoidal loading (a/h = 4) (0/90).



Table 5. Stresses in square laminate plate with a/h = 10 (0/90).

Pagano, 1970 Present HMSH5

z σx σy τxy σx σy τxy(a

2,b

2, z

) (0,b

2, z

)(0, 0, z)

(a

2,b

2, z

) (0,b

2, z

)(0, 0, z)

−0.5 −0.730 0.000 0.0538 −0.733 0.000 0.0537

−0.4 −0.442 0.198 0.0422 −0.461 0.196 0.0430−0.3 −0.177 0.307 0.0314 −0.189 0.306 0.0322−0.2 0.077 0.331 0.0210 0.083 0.330 0.0215−0.1 0.335 0.271 0.0107 0.355 0.268 0.0107−0.0 0.610 0.125 0.0001 0.627 0.121 0.0000+0.0 0.020 0.125 0.0001 0.019 0.121 0.0000+0.1 0.033 0.115 −0.0104 0.033 0.111 −0.0107+0.2 0.046 0.098 −0.0207 0.046 0.094 −0.0215+0.3 0.060 0.073 −0.0311 0.060 0.070 −0.0322+0.4 0.074 0.041 −0.0420 0.073 0.039 −0.0430+0.5 0.089 0.000 −0.0536 0.087 0.000 −0.0537

Fig. 10. Variation of transverse shear stress (τyz) through the thickness of a simply supportedlaminate under sinusoidal loading (a/h = 4) (0/90).



Table 6. Normalized transverse shear stresses for simply-supported

square 0/90/0 plate with sinusoidal load.

S = a/h Source τyz(a/2, 0, 0) τxz(0, a/2, 0)

4 Elasticity [Pagano, 1970] 0.2170 0.2560

Present HMSH5 0.1947 0.3328





CLT 0.0823 0.3950

Fig. 11. Variation of the transverse shear stress (τxz) through thickness for 0/90/0 square plate(a/h = 4).



Table 7. Normalized transverse shear stresses for simply-supported rect-

angular (b = 3a) 0/90/0 plate with sinusoidal load.

S = a/h Source τyz(a/2, 0, 0) τxz(0, b/2, 0)









CLT 0.0108 0.440

Fig. 12. Variation of the transverse shear stress (τyz) through thickness for 0/90/0 square plate(a/h = 4).



Table 8. Normalized transverse shear stresses for simply-supported

square 0/90/90/0 plate with sinusoidal load.

S = a/h Source τyz(a/2, 0, 0) τxz(0, a/2, 0)









Fig. 13. Variation of transverse shear stress (τyz) through the thickness of a simply supportedlaminate under sinusoidal loading (a/h = 10) (15/−15).



7.2.2. Angle-ply plates

Two square angle-ply laminated plates are considered: (1) two layers of equal thick-

ness with each ply alternately oriented at +150 and −150 to the x-y axes of the

plate; (2) eight layers of equal thickness with each ply alternately oriented at +450

and −450 to the x-y of the plate. The plates are simply supported by smooth pins

allowing tangential displacements along the boundaries and subjected to sinusoidal

load as the one given in Eq. (27). The lamina properties are similar to those given

in Eq. (28) for the first case, while they are defined by the following properties for

the second case

EL = 40× 106 Et = 106

GLT = 0.5× 106 GTT = 0.6× 106

VLT = VTT = 0.25 .

(30)

The whole plate has been analyzed using a 16×16 uniform mesh. The results for

these two particular problems are compared only with those provided by other

finite element solutions since there is no exact three-dimensional solution in a finite

domain for the angle-ply plate. The variation of transverse shear stress τyz through

the plate thickness of the first case is depicted in Fig. 13 along with that given

by Kant and Manjunatha, (1994). The variation of shear stress τxz is predicted in

Fig. 14 with that predicted by a higher order element as reported by Kant and

Pandya, (1988).

7.2.3. Cross-ply shell problems

Two laminated cylindrical-type shells are considered in this section. The first one

is an infinitely long partially cylindrical simply supported shell under transverse

loading as shown in Fig. 15. The shell is analyzed for a single-layer (0), two-layer

(0/90) and three-layer (0/90/0) with three different radius-to-thickness ratios, i.e.,

4, 10 and 100. A (4×4) mesh size is adapted for such a shell. The transverse shear

stresses obtained utilizing the HMSH5 element for different number of layers and

different radius-to-thickness ratios are depicted in Figs. 16–24 along with those

provided by exact solutions and other results given in Ren, (1987).

The second problem is a simply supported complete cylinder with ten layers

(0/90/0 . . .) under internal pressure as shown in Fig. 25. Because of symmetry,

only a quarter of the shell has been analyzed utilizing (6×4) HMSH5 elements. The

variation of the transverse shear stress through the shell thickness is depicted in

Fig. 26 along with the ‘exact’ one reported in Noor and Peters, (1989). Again, the

proposed approach shows very good agreement with exact solution and is equivalent

to the layerwise approach.



Fig. 14. Variation of the transverse shear stress (τxz) through thickness for (45/−45/. . .8 layers)square plate (a/h = 10).

Fig. 15. Simply supported partially laminated cylinder shell.



Fig. 16. Variation of transverse shear stress (τθz) through the thickness of a simply supportedcylindrical shell in cylindrical bending (R/h = 4) (0).



Fig. 17. Variation of transverse shear stress (τθz) through the thickness of a simply supportedlaminate cylindrical shell in cylindrical bending (R/h = 10) (0).



Fig. 18. Variation of transverse shear stress (τθz) through the thickness of a simply supportedlaminate cylindrical shell in cylindrical bending (R/h = 100) (0).



Fig. 19. Variation of transverse shear stress (τθz) through the thickness of a simply supportedlaminated cylindrical shell in cylindrical bending (R/h = 4) (90/0).



Fig. 22. Variation of the transverse shear stress (τθz) through thickness of a simply supportedlaminated cylindrical shell in cylindrical bending (R/h = 4) (0/90/0).




Fig. 25. Ten-layered (0/90/0 . . .) laminated cylinder subjected to internal pressure p.



Fig. 26. Variation of transverse shear stress (τrθ) through thickness for simply supported ten-layered composite cylinder subjected to internal pressure p = p0 sinπξ cos 8θ.

Remark 6:

Extending the superconvergence patch recovery algorithm from plate to curved

shell structures required more consideration into the curvature effects. Therefore,

the natural coordinates (r, s) have been adopted in each individual patch instead

of global Cartesian coordinates (x, y) in the case of plate. Note that the generated

patch, in general, does not coincide with any of the individual element geometry in

the patch. Therefore, the in-plane stresses at the optimal points provided by the

normal finite element procedure are generally not on the patch surface. As a result,

the accuracy of the superconvergence patch recovery scheme might be somewhat

disturbed. This is demonstrated in the proceeding shell problems in which the

transverse stresses are integrated from the bottom to top surfaces through the

thickness and their values at top surface are all shown to slightly deviate away

from the expected correct values of zero. However, this disturbance can approach

zero with further refined meshes.



Remark 7:

From the numerical results provided in Sec. 7.2, one can observe that a good overall

accuracy is generally reached by the proposed scheme, indeed its accuracy is compa-

rable to the other layerwise and higher-order elements for both thin and moderately

thick plates. Furthermore, in terms of computational efficiency, the proposed ap-

proach has shown a greater advantage, since the cost of the post-processing is only

a small fraction of the total analysis cost. That is, the total cost of the proposed ap-

proach is only proportional to the total number of degrees of freedom (5 DOF/node)

of the problem and is independent of the number of the layers considered. For exam-

ple, if the total number of nodes in a problem is N , the total computational cost of

the proposed approach will be proportional to 5N . On the other hand, considering

a “direct” layerwise approach, and in the case of a problem containing a total of N

nodes and a total of M layers, the total computational expense is correspondingly

proportional to N(2M + 3). For the eight-layer laminated problem illustrated in

Sec. 7.2.2, for instance, the total computational cost of the layerwise approach is

almost four times as much as that of the present one.

7.3. Robustness of the projection-layerwise-equivalent approach

A ten-layer cross-ply square plate panel with L = 1, h/L = 0.05 proposed by

Noor and Kim, (1995) has been adopted in this section. The fiber orientation of

the laminates is arranged in the [0/90/0/90/0]s scheme. The panel is subjected to

transverse loading p = sinπx sin πy. The material properties are as follows:

EL = 15 ET = 1

GLT = 0.5 GTT = 0.3378

νLT = 0.3 νTT = 0.48 .

(31)

To assess the effect of mesh distortion on the accuracy of the distribution of the

transverse shear stresses through the thickness, the finite element mesh is distorted

such that the distortion ratio of a1/a varied from 0.5 to 0.2 as shown in Fig. 27. The

transverse shear stresses are evaluated at point A, B and C. The results from the

present approach along with those provided by Noor and Kim, (1995) are presented

in Figs. 28 and 29. As evident from these figures, both approaches perform very

well for this problem. The present approach shows better performance even though

the total number of degrees of freedom used in the present approach is much less

than those in Noor and Kim, (1995) (i.e., 39 versus 54 dof per element).

Different from the model proposed in Noor and Kim, (1995) in which the ac-

curacy of the transverse stresses on the three points (point A, B and C) are com-

parable, the accuracy of such stresses varies in the present approach, as shown in

Fig. 30. It is worthy to note that the patch surrounding point A always contains

the most distorted 4 elements (no matter how the mesh is refined) in the distorted



configurations. Thus, the transverse stresses at this point are expected to be much

more affected by distortion than the other points (i.e., point B and C). However,

all the results are still comparable with those of Noor and Kim, (1995).

Fig. 27. Simply supported laminated plate for the distortion sensitivity test.

7.4. Hyper-anisotropic cantilever beam under bending loads

This problem was proposed by P. Simacek, V. N. Kaliakin and R. B. Pipes (1993).

It was intended to describe numerical phenomena associated with the analysis of

hyper-anisotropic materials (namely, fiber reinforced thermoplastic composites –

FRTC) possessing a direction of near inextensibility. It poses several challenges in

dealing with element-locking potential (dependent on fiber orientation), spurious

tension modes, etc. Several cases demonstrating some of these numerical difficulties,

for both low-order (four-noded) and higher-order (nine-noded) shell elements, have

been well documented in the above reference.



Fig. 28. Distortion effect on the accuracy of the maximum σxz (point A).



Fig. 29. Distortion effect on the accuracy of the maximum σyz (point A).



Fig. 30. Distortion effect on the accuracy of the maximum transverse stresses for the presentHMSH5 (8 × 8 mesh).

The problem description is shown in Fig. 31; i.e., shear beam under in-plane

loading. The material properties, µ11, µ22 and µ12, represent the elastic rigidities in

the fiber and transverse directions, and shear rigidity, respectively. The constitutive

equation, under plane stress condition, isσ11

σ22

σ12

=

4µL + β 2µT 0

2µT 4µT 0

0 0 2µL

.ε11

ε22

ε12

(32)

In Eq. (32), the three independent material constants µL, µT and β characterize the

elastic moduli/rigidities (or their counterpart viscosities of the fluid-like FRTC in

processing modeling with strain rate replacing strains in Eq. (32)). These constants

are related to the more conventional moduli µ11, µ22 and µ12 in the following

manner:

µL = µ12 ; β =4(µ11)2

4µ11 − µ22− 4µ12 ;

1

µT=

4

µ22− 1

µ11. (33)



Fig. 31. A hypar-anisotropic cantilever beam under bending.

It is worthy of note that the ratio between longitudinal, µ11, and transverse, µ22,

moduli in the present case (see Fig. 31) is extremely high, i.e., roughly 108. The

exact solution for this problem gives a constant, in-plane, shear stress of value 1.0,

and a zero normal stress almost everywhere except for two very narrow singularity

regions at upper and lower fibers. This is the well-known phenomenon of “stress-

channeling”. Note that, although the beam is under a bending load, all deformations

are basically of the pure-shear type.

Figure 32 shows the oscillation of “raw” results for the normal stress distribution

through the thickness at the free end. Note also that this oscillation is greatly

reduced after MSPR, but the important feature of two “thin” boundary layers of

stress singularity are still well reproduced.

8. Conclusions

Utilizing a direct two-phase, hierarchical, scheme for laminated plate and shell mod-

els, the in-plane stresses with high accuracy have been recovered on the basis of the

modified superconvergent patch recovery approach. In this, the equilibrium equa-

tions can be satisfied accurately enough to carry out the subsequent calculations

of transverse stresses. In particular, the continuity of the inter-laminar transverse

shear stresses has been shown to be automatically satisfied in the asymptotic limit

of mesh refinements.



Fig. 32. Variation of normal stress σx through the thickness at free end.

The proposed approach can be considered as an extension of the equivalent-

single-layer approach. As such, this patch-recovery/stress-equilibrium strategy has

been applied to a mixed shell formulation of the simplest, linear-quadrilateral, type.

The results of the extensive number of test cases and comparisons included here

have clearly demonstrated its effectiveness. Combined together this hierarchical

approach has also led to a formal mathematical proof of the “equivalency” of the

resulting model to the more elaborate descriptions using the layer-wise technique,

at least for the case of flat geometry (and also with sufficient mesh refinement for

shells with curved geometry).

Overall the presented approach possesses a number of advantages. First, being

still built on the underlying simple surface kinematics of the shell element, all such

desirable attributes of low-order models (e.g., reduced distortion sensitivity, inher-

ent simplicity, and their appeal in nonlinear applications) are preserved. Secondly,

with its present simple post-processing format, the approach is quite computation-

ally inexpensive for both linear and nonlinear problems. Finally, the same approach

presented can, of course, be applied to other plate/shell models.



Acknowledgements

This work was partially supported by the following research grants: National Science

Foundation NSF Grant no. MSS-8816088 and NASA Glenn Research Center, Grant

no. NAG3-1747 to the University of Akron. This financial support is gratefully

acknowledged.

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