Composite Structures - San Diego State Universitydemasi/articles/GUF_PART... · 2014. 2. 25. · 16...

Composite Structures xxx (2008) xxx–xxx

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Composite Structures

journal homepage: www.elsevier .com/locate /compstruct

16 Mixed plate theories based on the Generalized Unified FormulationPart II: Layerwise theories

Luciano Demasi *

Department of Aerospace Engineering and Engineering Mechanics, San Diego State University, 5500 Campanile Drive, San Diego, CA, USA

a r t i c l e i n f o

Keywords:Generalized unified formulationLayerwise plate theories,Reissner’s mixed variational theorem

0263-8223/$ - see front matter � 2008 Elsevier Ltd. Adoi:10.1016/j.compstruct.2008.07.012

* Tel.: +1 (619) 594 3752; fax: +1 619 594 6005.E-mail address: [email protected]

Please cite this article in press as: Demas(2008), doi:10.1016/j.compstruct.2008.07.

a b s t r a c t

The generalized unified formulation introduced in Part I for the case of composite plates and Reissner’sMixed variational theorem is, for the first time in the literature, applied to the case of layerwise theories.Each layer is independently modeled. The compatibility of the displacements and the equilibrium of thetransverse stresses between two adjacent layers are enforced a priori. Infinite combinations of the ordersused for displacements ux, uy, uz and out-of-plane stresses rzx , rzy , rzz can be freely chosen.16 layerwisetheories are therefore presented. The code based on this capability can have all the possible16 theoriesbuilt-in, thus, making the code a powerful and versatile tool to analyze different geometries, boundaryconditions and applied loads. All 16 theories are generated by expanding 13 1� 1 invariant matrices(the kernels of the Generalized Unified Formulation). How the kernels are expanded and the theories gen-erated is explained. Details of the assembling in the thickness direction and the generation of the matri-ces are provided.

� 2008 Elsevier Ltd. All rights reserved.

1. Introduction

1.1. Layerwise models: main theoretical concepts

Equivalent single layer theories give a sufficiently accuratedescription of the global laminate response. However, these theo-ries are not adequate for determining the stress fields at ply level.Layerwise theories assume separate displacement field expansionswithin each layer. The accuracy is then greater but the price is inthe increased computational cost. Many layerwise plate modelshave been proposed in the past by applying classical plate theoryor higher order theories at each layer. Generalizations of these ap-proaches were also given, and the displacements variables wereexpressed in terms of Lagrange polynomials. Among the papers de-voted on the subject of layerwise theories, Refs. [1–5] give an ideaof the different approaches. Normally, displacement-based layer-wise models do not a priori take into account the continuity ofthe transverse stresses between two adjacent layers. The problemof satisfying the interlaminar continuity of the transverse stressesa priori led to the derivation of mixed layerwise theories [6–8].Layerwise theories could also be easily extended to the case ofcomposite beams as explained in Ref. [9]. The conceptual differ-ences between the displacement fields in layerwise and equivalentsingle layer theories are depicted in Fig. 1. Layerwise models arecomputationally more expensive than the less accurate equivalent

ll rights reserved.

i L, 16 Mixed plate theories012

single layer models. Therefore, layerwise models can be used in re-gions of the structure in which an accurate description is required[10], whereas equivalent single layer models are employed in otherparts of the structure. It is also possible to develop quasi-layerwisetheories in which some quantities are described using the layer-wise approach and some are described using the equivalent singlelayer approach. These two categories of theories will be describedin Part III Ref. [11] and Part IV Ref. [12] of this work.

1.2. What are the new contributions of this work

The generalized unified formulation (GUF) [13,14] is a new for-malism and a generalization of Carrera’s Unified Formulation (CUF)[15]. GUF was introduced in the case of displacement-based theo-ries. GUF was extended, for the first time in the literature, to thecase of mixed theories (see Part I, Ref. [16]). In particular, Reissner’smixed variational theorem (RMVT) (see [17,18]) was employed.The unknown variables were the displacements and transversestresses.

In the present work GUF will be extended to the case of com-posite multilayered structures analyzed with layerwise models.Each layer variable (a displacement or a transverse stress) will beindependently expanded along the thickness leading to a very widevariety of new theories. Since each variable can be expanded ininfinite different forms (by simply changing the order of the poly-nomial used in the expansion along the thickness), the case ofRMVT-based theories leads to the writing of 16 layerwise mixedtheories. All the possible theories generated using GUF can be

based on the Generalized Unified Formulation ..., Compos Struct

mailto:[email protected]://www.sciencedirect.com/science/journal/02638223http://www.elsevier.com/locate/compstructhttp://www.engineering.sdsu.edu/aerospace/index.htmlwww.lucianodemasi.com

Fig. 1. Layerwise theories vs equivalent single layer theories in a three layered structure.

2 L. Demasi / Composite Structures xxx (2008) xxx–xxx

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implemented in a single code without the requirements of newimplementations or theoretical developments.

This is a new powerful methodology to create layerwise theo-ries, and the details will be given in this work. In particular, it willbe shown how to generate the layer matrices from the fundamen-tal nuclei or kernels (introduced in Part I, Ref. [16]) and how toassemble these matrices. The interlaminar continuity of the dis-placements and transverse stresses is taken into account. Numer-ous examples clarify how to use GUF to generate a desiredlayerwise theory.

2. Theoretical derivation of ‘6 layerwise mixed theories

For a generic layer k, the displacements ukx , uky, u

kz and out-of-

plane stresses rkxz ¼ skx , rkyz ¼ sky, rkzz ¼ skz are written in a compactnotation (the generalized unified formulation) as follows:

ukx ¼ xFtukxt þxFlukxl þ

xFbukxb ¼xFaux u

kxaux

aux ¼ t; l; b; l ¼ 2; . . . ;Nux

uky ¼ yFtukyt þyFmukym þ

yFbukyb ¼yFauy u

kyauy

auy ¼ t;m; b; m ¼ 2; . . . ;Nuy

ukz ¼ zFtukzt þzFnukzn þ

zFbukzb ¼zFauz u

kzauz

auz ¼ t;n; b; n ¼ 2; . . . ;Nuz

skx ¼ xF tskxt þxF pskxp þ

xF bskxb ¼xFasx s

kxasx

asx ¼ t;p; b; p ¼ 2; . . . ;Nsx

sky ¼ yF tskyt þyF qskyq þ

yF bskyb ¼yFasy s

kyasy

asy ¼ t; q; b; q ¼ 2; . . . ;Nsy

skz ¼ zF tskzt þzF rskzr þ

zF bskzb ¼zF asz s

kzasz

asz ¼ t; r; b; r ¼ 2; . . . ;Nsz

ð1Þ

The functions of the thickness coordinate are introduced in a gen-eral form. For example, xFt is a function of z and can be a polyno-mial, trigonometric, exponential or another function chosen apriori. To have the assembling process along the thickness directionimmediate and intuitive and indicated for the case of multilayeredstructures a convenient expansion along the thickness is introduced.

The displacements and out-of-plane stresses must be continu-ous functions along the thickness to ensure the compatibility ofthe displacements and the equilibrium between two adjacent lay-ers. Therefore, it is convenient for the axiomatic expansions alongthe thickness to have the following properties:

Please cite this article in press as: Demasi L, 16 Mixed plate theories(2008), doi:10.1016/j.compstruct.2008.07.012

� Property 1For z ¼ zbotk (the bottom surface of layer k; see Fig. 1 in Part I, Ref.[16]) all the functions along the thickness are zero except theone which multiplies the term corresponding to the bottom(subscript b). For example, in the case of the displacement ukx ,the functions calculated at the bottom of layer k should givethe following values:

xFtðz ¼ zbotk Þ ¼ 0xFlðz ¼ zbotkÞ ¼ 0

xFbðz ¼ zbotk Þ ¼ 1 ð2Þ

If the previous conditions are satisfied then from the first expres-sion of Eq. (1) it is possible to deduce:

ukxðz ¼ zbotk Þ ¼ 0 � ukxt þ 0 � u

kxlþ 1 � ukxb ¼ u

kxb

ð3Þ

Therefore, if the conditions reported in Eq. (2) are satisfied, ukxb is notjust a term in the thickness expansion of the variable ukx but as-sumes the meaning of the value that the displacement ukx takeswhen the bottom surface of layer k is considered (i.e., z ¼ zbotk ). Thisnow explains why the subscript ‘‘b” is introduced in the notation.

� Property 2For z ¼ ztopk (the top surface of layer k) all the functions alongthe thickness are zero except the one which multiplies the termcorresponding to the top (subscript t). For example, in the caseof the displacement ukx , the functions calculated at the top oflayer k should give the following values:

xFtðz ¼ ztopk Þ ¼ 1xFlðz ¼ ztopk Þ ¼ 0

xFbðz ¼ ztopk Þ ¼ 0 ð4Þ

If the previous conditions are satisfied then from the first expres-sion of Eq. (1) it is possible to obtain:

ukxðz ¼ ztopk Þ ¼ 1 � ukxt þ 0 � u

kxlþ 0 � ukxb ¼ u

kxt ð5Þ

Therefore, if the conditions reported in Eq. (4) are satisfied, ukxt is notjust a term in the thickness expansion of the variable ukx but as-sumes the meaning of the value that the displacement ukx takeswhen the top surface of layer k is considered (i.e., z ¼ ztopk ). Thisnow explains why the subscript ‘‘t” is introduced.

� Property 3It is known that polynomial functions of the type znk are respon-sible for ill conditioning (see a discussion of this problem in[19]) when n is increased. This can be avoided by using orthog-onal polynomials.

A good set of functions (for all the displacements and out-of-plane stresses) which satisfy the above mentioned propertiesshould be selected. It is possible to demonstrate that all the previ-ous properties are satisfied if particular combination of Legendre


L. Demasi / Composite Structures xxx (2008) xxx–xxx 3

ARTICLE IN PRESS

polynomials is used. Legendre polynomials are defined in the inter-val [�1,1]. Thus, a transformation is necessary:

fk ¼2

ztopk � zbotkz�

ztopk þ zbotkztopk � zbotk

� 1 6 fk 6 þ1 ð6Þ

where fk is a non-dimensional coordinate. The following formula isalso valid:

ztopk ¼ zbotk þ hk ð7Þ

The transformation is then

fk ¼2hk

z�ztopk þ zbotkztopk � zbotk

ð8Þ

The Legendre polynomial of order zero is P0ðfkÞ ¼ 1. The Legendrepolynomial of order one is P1ðfkÞ ¼ fk. The higher order polynomialscan be obtained by using Bonnet’s recursion [20]:

Pnþ1ðfkÞ ¼ð2nþ 1ÞfkPnðfkÞ � nPn�1ðfkÞ

nþ 1 ð9Þ

Bonnet’s formula is a convenient method to calculate the Legendrepolynomials in a practical code based on the generalized unifiedformulation.

The explicit form of the Legendre polynomials of order 2, 3, 4and 5 are the following (but in practice these formulas are not con-venient and the recursive method previously introduced should beused):

P2ðfkÞ ¼ 3ðfkÞ2�1

2 P3ðfkÞ ¼5ðfkÞ3�3fk

2

P4ðfkÞ ¼ 35ðfkÞ4�30ðfkÞ2þ3

8 P5ðfkÞ ¼63ðfkÞ5�70ðfkÞ3þ15fk

8

ð10Þ

The same functions for all displacements are used. This is not nec-essary with the generalized unified formulation but it is more prac-tical. The following combination of Legendre functions is used:

xFt ¼ yFt ¼ zFt ¼ P0þP12 ; xFb ¼ yFb ¼ zFb ¼P0�P1

2xFl ¼ Pl � Pl�2; l ¼ 2;3; . . . ;NuxyFm ¼ Pm � Pm�2;m ¼ 2;3; . . . ;NuyzFn ¼ Pn � Pn�2;n ¼ 2;3; . . . ;Nuz

ð11Þ

in which Pj ¼ PjðfkÞ is the Legendre polynomial of j-order. The cho-sen functions have the following properties:

fk ¼þ1; xFt; yFt ; zFt ¼ 1; xFb; yFb; zFb ¼ 0; xFl; yFm; zFn ¼ 0�1; xFt; yFt ; zFt ¼ 0; xFb; yFb; zFb ¼ 1; xFl; yFm; zFn ¼ 0

�ð12Þ

Thus, the properties earlier mentioned are all satisfied and this setof functions is a good choice to build the mixed layerwise theories.It is convenient (but not necessary) to use the same type of func-tions for the thickness expansions of the stresses:

xF t ¼ yF t ¼ zF t ¼ P0þP12 ; xF b ¼ yF b ¼ zF b ¼P0�P1

2

xF p ¼ Pp � Pp�2; p ¼ 2;3; . . . ;NsxyF q ¼ Pq � Pq�2; q ¼ 2;3; . . . ;NsyzF r ¼ Pr � Pr�2; r ¼ 2;3; . . . ;Nsz

ð13Þ

With the generalized unified formulation other functions could beused without changing the formalism. However, combination ofLegendre’s polynomials has been proven effective and convenient(see Ref. [6]). It is then possible to create a class of theories bychanging the orders of displacements and stresses. Suppose, forexample, that a theory has the following data: Nux ¼ 3, Nuy ¼ 2,


Nuz ¼ 4, Nsx ¼ 5, Nsy ¼ 4, Nsz ¼ 6. The corresponding theory is indi-cated as LM546324. The first letter ‘‘L” means ‘‘Layerwise” theory, thesecond letter ‘‘M” means that a ‘‘mixed” variational theorem is used(Reissner’s variational theorem). The subscripts are the orders of theLegendre polynomials used for the displacements. The superscriptsare the orders of the Legendre polynomials used for the out-of-plane stresses. In general, the acronym is then built as follows:

LMNsx Nsy NszNux Nuy Nuz

.

Mixed theories based on RMVT and a Compact Notationwere also introduced by Carrera [6–8]. In particular, he formu-lated the problem with Carrera’s Unified Formulation (see thediscussions in [13,15]). The main differences between CUF andGUF and the notations for layerwise theories are shown inFig. 2. If the Static Condensation Technique (SCT) is not per-formed then the theory is indicated with the acronymLMF

Nsx Nsy NszNux Nuy Nuz

, where ‘‘F” means ‘‘full”. The other quantities havethe same meaning as before.

A more detailed discussion on the static condensation tech-nique is reported in another of author’s work [21]. In particu-lar, FEM applications of Carrera’s Unified Formulation areanalyzed, and it is shown that SCT is fundamental to reducethe CPU time for the (already computationally expensive) lay-erwise theories. In fact, using the ‘‘full” approach improvesthe calculation of the out-of-plane stresses but the price istoo high. Therefore, for the common engineering applicationsof mixed layerwise theories SCT should be performed at finiteelement level.

3. How ‘6 layerwise theories are generated

3.1. Kernels of the generalized unified formulation

In Part I Ref. [16] the governing equations (Navier-type solu-tion) were written with a Compact Notation: the generalized uni-fied formulation. All of the equations, including 16 differentcombinations, were written as function of kernels of the generalizedunified formulation. In particular, six pressure kernels of 1� 1matrices were introduced. Also, 21 kernels were used to generatethe matrices (but only 13 are really required). The fundamentalequations, invariant with respect to the orders used for the expan-sions of the different variables, are the following:

Kkaux buxuxux

zfflfflfflffl}|fflfflfflffl{1�1kernelxUkbux

zffl}|ffl{unknownþKkaux buyuxuyzfflfflfflffl}|fflfflfflffl{1�1kernel

yUkbuy

zfflffl}|fflffl{unknownþKkaux bsxuxsxzfflfflfflffl}|fflfflfflffl{1�1kernel

xSkbsx

zffl}|ffl{unknownþKkaux bszuxszzfflfflfflffl}|fflfflfflffl{1�1kernel

zSkbsz

zffl}|ffl{unknown¼ xRkauxzffl}|ffl{known

Kkauy buxuyux


zffl}|ffl{unknownþKkauy buyuyuyzfflfflfflffl}|fflfflfflffl{1�1kernel

yUkbuy

zfflffl}|fflffl{unknownþKkauy bsyuysyzfflfflfflffl}|fflfflfflffl{1�1kernel

ySkbsy

zffl}|ffl{unknownþKkauy bszuyszzfflfflfflffl}|fflfflfflffl{1�1kernel

zSkbsz

zffl}|ffl{unknown¼ yRkauyzffl}|ffl{known

Kkauz bsxuzsx

zfflfflfflffl}|fflfflfflffl{1�1kernelxSkbsx

zffl}|ffl{unknownþKkauz bsyuzsyzfflfflfflffl}|fflfflfflffl{1�1kernel

ySkbsy

zffl}|ffl{unknownþKkauz bszuzszzfflfflffl}|fflfflffl{1�1kernel

zSkbsz

zffl}|ffl{unknown¼ zRkauzzffl}|ffl{known

Kkasx buxsxux


zffl}|ffl{unknownþKkasx buzsxuzzfflfflfflffl}|fflfflfflffl{1�1kernel

zUkbuz

zffl}|ffl{unknownþKkasx bsxsxsxzfflfflffl}|fflfflffl{1�1kernel

xSkbsx

zffl}|ffl{unknown¼0

Kkasy buysyuy

zfflfflfflffl}|fflfflfflffl{1�1kernelyUkbuy

zfflffl}|fflffl{unknownþKkasy buzsyuzzfflfflfflffl}|fflfflfflffl{1�1kernel

zUkbuz

zffl}|ffl{unknownþKkasy bsysysyzfflfflffl}|fflfflffl{1�1kernel

ySkbsy


Kkasz buxszux


zffl}|ffl{unknownþKkasz buyszuyzfflfflfflffl}|fflfflfflffl{1�1kernel

yUkbuy

zfflffl}|fflffl{unknownþKkasz buzszuzzfflfflffl}|fflfflffl{1�1kernel

zUkbuz

zffl}|ffl{unknownþ Kkasz bszszszzfflfflffl}|fflfflffl{1�1kernel

zSkbsz


ð14Þ

It will be shown that Eq. (14) is valid for the case of layerwise the-ories and also for higher order shear deformation theories (see Part


Fig. 2. Acronyms used to define the RMVT-based layerwise theories using Carrera’s unified formulation and generalized unified formulation.


ARTICLE IN PRESS

III, Ref. [11]) with or without Zig-Zag functions (see Part IV, Ref.[12]). The loads have been defined as follows:

xRkaux ¼ Dktaux buxuxux

zfflfflfflffl}|fflfflfflffl{1�1pressure kernelxPktbux

zffl}|ffl{inputþ Dkbaux buxuxux

zfflfflfflfflffl}|fflfflfflfflffl{1�1pressure kernelxPkbbux

zffl}|ffl{input


yRkauy ¼ Dktauy buyuyuy

zfflfflfflffl}|fflfflfflffl{1�1pressure kernelyPktbuy

zffl}|ffl{inputþ Dkbauy buyuyuy

zfflfflfflfflffl}|fflfflfflfflffl{1�1pressure kernelyPkbbuy

zffl}|ffl{input

zRkauz ¼ Dktauz buzuzuz

zfflfflfflffl}|fflfflfflffl{1�1pressure kernelzPktbuz

zffl}|ffl{inputþ Dkbauz buzuzuz

zfflfflfflffl}|fflfflfflffl{1�1pressure kernelzPkbbuz

zffl}|ffl{inputð15Þ



ARTICLE IN PRESS

The amplitudes of the type xPktbux are assigned by the user. This inputis given at multilayer level, as will be shown.

Eqs. 14 and 15 are invariant with respect to the theory. That is,theories LM546324 and LM

626334 (among the infinite possible theories) are

generated from Eqs. 14 and 15. Where is the difference betweenthe two theories? The difference is at layer level, after the kernelshave been expanded to have the layer matrices. These matricesthen have to be assembled at multilayer level.

3.2. Expansion of the 1� 1 kernels: matrices at layer level

First, it has to be pointed out that so far each layer is treatedusing the same functions. Therefore, the number of terms used todescribe the layer matrix is kept the same. This is not mandatory,but unless local effects have to be taken into account (e.g., delam-ination) it is a ‘‘natural” choice. Even in the case of local effects asufficiently high order can be used and the usage of different or-ders for the different layers can be avoided. In this paper theexpansion used in the different variables does not change and each

Fig. 3. Generalized unified formulation: example of expansion from a k


layer is treated in the same way. Thus, for example,Nkux ¼ N

kþ1ux ¼ Nux . The expansion of the kernels is the most impor-

tant part of the generation of one of the possible16 theories. Thisoperation is done at layer level. To explain how this operation isperformed, consider the case of theory LM546324, in which the numberof degrees of freedom, at layer level, is the following:

½DOF�kux ¼ Nux þ 1 ¼ 3þ 1 ¼ 4 ½DOF�kuy ¼ Nuy þ 1 ¼ 2þ 1 ¼ 3

½DOF�kuz ¼ Nuz þ 1 ¼ 4þ 1 ¼ 5 ½DOF�ksx ¼ Nsx þ 1 ¼ 5þ 1 ¼ 6

½DOF�ksy ¼ Nsy þ 1 ¼ 4þ 1 ¼ 5 ½DOF�ksz ¼ Nsz þ 1 ¼ 6þ 1 ¼ 7

ð16Þ

From the number of degrees of freedom it is possible to calculatethe size of the layer matrices. For example, when matrix Kkaux bszuxsz isexpanded then the final size at layer level will be½DOF�kux � ½DOF�

ksz . In the example relative to theory LM

546324, matrix

Kkaux bszuxsz at layer level (indicated as K

kuxsz ) is a 4� 7 matrix and ob-

tained as explained in Fig. 3.

ernel to a layer matrix. Case of theory LM546324. From Kkaux bszux sz to K

kux sz .


Fig. 4. Equilibrium between two adjacent layers.


ARTICLE IN PRESS

3.3. Assembling in the thickness direction: from layer to multilayerlevel

In addition to the compatibility of the displacements, the equi-librium between two adjacent layers implies that skxt ¼ s

ðkþ1Þxb ,

skyt ¼ sðkþ1Þyb and s

kzt ¼ s

ðkþ1Þzb (see Fig. 4). Therefore, the assembling

must consider this fact. Fig. 5 shows how the assembling of a typ-ical matrix is performed. The pressure matrices are obtained fromthe pressure kernels using the same method shown in Figs. 3 and 5.The use of combinations of Legendre polynomials ensures the con-tinuity of the functions, which are used to expand the displace-ments and stresses. Fig. 6 shows this concept for the case oftheory LM546324. The pressure amplitudes at multilayer level are inputof the problem. Some input examples are shown in Fig. 7.

Once the matrices are all assembled, the system of equationsbecomes:

Kuxux Kuxuy 0uxuz Kuxsx 0uxsy KuxszKuyuy 0uyuz 0uysx Kuysy Kuysz

0uzuz Kuzsx Kuzsy KuzszKsxsx 0sxsy 0sxsz

Symm Ksysy 0syszKszsz

2666666664

3777777775�

xUyUzUxSySzS

2666666664

3777777775¼

xRyRzRx0y0z0

2666666664

3777777775ð17Þ

where

xR¼tDuxux � xPtþbDuxux � xP

b

yR¼tDuyuy � yPtþbDuyuy � yP

b

zR¼tDuzuz � zPtþbDuzuz � zP

b

ð18Þ

4. Theoretical examples

The generation of the infinite theories is not a very difficultproblem when the generalized unified formulation is used. To help


the readers create their own code based on this procedure, theauthor reports here several numerical examples. In particular, the-ory LM546324 will be analyzed in detail. Suppose that the goal is thegeneration of matrix Kuxsz .

The kernel associated with matrix Kuxsz (at layer level) is thefollowing:

Kkaux bszuxsz ¼ �

mpa

Zkaux bsz13uxsz ¼ �

mpa

Ck13

Z ztopkzbotk

xFaux ðzÞzFasz ðzÞdz ð19Þ

The functions used in the expansions along the thickness aredefined as a combination of Legendre polynomials. Therefore, itis convenient to transform the interval (see Eq. 8). The kernel isthen

Kkaux bszuxsz ¼ �

mpa

Ck13hk2

Z þ1�1

xFaux ðfkÞzFasz ðfkÞdfk ð20Þ

For the case in which aux ¼ bsz ¼ t:

Kkttuxsz ¼ �mp

a Ck13

hk2

Rþ1�1

xFtðfkÞ zF tðfkÞdfk¼ �mpa C

k13

hk2

R þ1�1

P0þP12 �

P0þP12 dfk

ð21Þ

or

Kkttuxsz ¼ �mpa

Ck13hk2

Z þ1�1

1þ fk2

� �2dfk ¼ �Ck13

mphk3a

ð22Þ

Similarly for the case in which aux ¼ t and bsz ¼ 2:

Kkt2uxsz ¼ �mp

a Ck13

hk2

Rþ1�1

xFtðfkÞ zF2ðfkÞdfk¼ �mpa C

k13

hk2

R þ1�1

P0þP12 � ðP2 � P0Þdfk

ð23Þ

or

Kkt2uxsz ¼ �mpa

Ck13hk2

Z þ1�1

1þ fk2� 3ðfkÞ

2 � 32

dfk ¼mpa

Ck13hk2

ð24Þ


Fig. 5. Generalized unified formulation: example of assembling from layer matrices to multilayer matrix. Case of theory LM546324. From Kkux sz and K

ðkþ1Þux sz to Kux sz .


ARTICLE IN PRESS

Thirteen independent matrices are required to solve the problem.Kkuxsz is just one of them. The other matrices can be calculated witha similar procedure. Another case is analyzed to clarify the proce-dure. Consider the kernel used to generate Kkuxsx :

Kkaux bsxuxsx ¼ þZ

kaux ;zbsxuxsx ¼

Z ztopkzbotk

d½xFaux ðzkÞ�dzk

zFasx ðzkÞdzk ð25Þ

changing the variables and using the thickness coordinate fk:

Kkaux bsxuxsx ¼

Z þ1�1

d½xFaux ðfkÞ�dfk

zF asx ðfkÞdfk ð26Þ

The other calculations are omitted for brevity. The 13 independentmatrices at layer level are reported in Appendix A.

The sizes of the layer pressure matrix tDkuxux andbDkuxux are the

same as the size of matrix Kkuxux . Similarly, the sizes of the matricestDkuyuy and

bDkuyuy are the same as the size of matrix Kkuyuy . Finally, the

sizes of tDkuzuz andbDkuzuz are the same as the size of matrix K

kuzuz . The

pressures can be applied only at the top or bottom surfaces of theplate. This means that the pressure matrices at layer level are cal-culated only for k ¼ Nl and k ¼ 1, the top and bottom layers respec-tively. In particular, tDkuxux ,

tDkuyuy andtDkuzuz are calculated only for

k ¼ Nl (for the other layers these matrices are set to be with only


zeros). Similarly, bDkuxux ,bDkuyuy and

bDkuzuz are calculated only fork ¼ 1 (for the other layers these matrices are set to be with onlyzeros). The assembling to multilayer level is then done as for thecorresponding matrices. For example, matrix tDuxux is built usingthe same procedure used for matrix Kkuxux .

The pressure matrices at layer level are obtained using the def-initions reported in Part I Ref. [16]. For example, for the top layerk ¼ Nl (notice that in any case the top surface of the layer is foundwhen fk ¼ þ1):

tDk¼Nl aux buxuxux ¼ xF

taux

xFtbux ¼xFaux ðfk ¼ þ1Þ

xFbux ðfk ¼ þ1Þ ð27Þ

Considering the properties of the functions along the thickness(combination of Legendre polynomials), the only term that is differ-ent than zero is the one corresponding to aux ¼ bux ¼ t:

tDk¼Nl ttuxux ¼xFtt

xFtt ¼ xFtðfk ¼ þ1Þ xFtðfk ¼ þ1Þ ¼ 1 ð28Þ

For the other pressure matrices of the top layer:

tDk¼Nl ttuyuy ¼yFtt

yFtt ¼ yFtðfk ¼ þ1Þ yFtðfk ¼ þ1Þ ¼ 1tDk¼Nl ttuzuz ¼

zFttzFtt ¼ zFtðfk ¼ þ1Þ zFtðfk ¼ þ1Þ ¼ 1

ð29Þ


Fig. 6. Case of theory LM546324. Multilayer unknown displacement and out-of-plane stresses for the case in which the number of layers is two.


ARTICLE IN PRESS

Considering again the properties of the functions used for the thick-ness expansions it is deduced that for the bottom layer (k ¼ 1) theonly terms that are different than zero are the ones with masterindices equal to b:

bDk¼1 bbuxux ¼xFbb

xFbb ¼ xFbðfk ¼ �1Þ xFbðfk ¼ �1Þ ¼ 1bDk¼1 bbuyuy ¼

yFbbyFbb ¼ yFbðfk ¼ �1Þ yFbðfk ¼ �1Þ ¼ 1

bDk¼1 bbuzuz ¼zFbb

zFbb ¼ zFbðfk ¼ �1Þ zFbðfk ¼ �1Þ ¼ 1

ð30Þ

5. Calculation of the stresses

The system of equations with unknown amplitudes of displace-ments and stresses is represented by Eq. (17). This system caneither be directly solved or the static condensation technique(see part I, Ref. [16]) can be performed. When the Navier-type solu-tion is considered it is not really important if the static condensa-tion is performed or not, but when FEM computations areconcerned, the static condensation (if performed at element level)can significantly save CPU time. Suppose now the amplitudes areknown. The stresses need to be calculated. Once the amplitudesare known, it is possible to extract the vectors of amplitudes atlayer level (see Fig. 6). Form these vectors the displacements andout-of-plane stresses can be calculated immediately using the def-inition of generalized unified formulation. For the generic layer kthe following formulas are valid:

ukx ¼ xFaux ukxaux ¼xFaux

xUkaux Cmpxa S

npyb

uky ¼ yFauy ukyauy ¼yFauy

yUkauy Smpxa C

npyb

ukz ¼ zFauz ukzauz ¼zFauz

zUkauz Smpxa S

npyb

skx ¼ xFasx skxasx ¼xF asx

xSkasx Cmpxa S

npyb

sky ¼ yFasy skyasy ¼yFasy

ySkasy Smpxa C

npyb

skz ¼ zFasz skzasz ¼zF asz

zSkasz Smpxa S

npyb

ð31Þ

where, for example, Cmpxa ¼ cos mpxa and Smpxa ¼ sin mpxa .


5.1. In-plane stresses calculated using mixed form of Hooke’s Law(MFHL)

Since a mixed approach has been adopted, it is a natural choiceto use the MFHL (which is explicitly shown in Part I, Ref. [16]) tocalculate the in-plane stresses:

rkxx ¼ �mpa Ck11

xFauxxUkaux �

npb C

k12

yFauyyUkauy

hþCk13zF asz zS

kasz

iSmpxa S

npyb

rkyy ¼ �mpa Ck12

xFauxxUkaux �

npb C

k22

yFauyyUkauy

hþCk23zF asz zS

kasz

iSmpxa S

npyb

rkxy ¼ þ npb Ck66

xFauxxUkaux þ

mpa C

k66

yFauyyUkauy

h iCmpxa C

npyb

ð32Þ

The following can be observed:

� The stresses rkxx and rkyy do not have explicit dependence on theamplitudes zUkauz ,

xSkasx andySkasy .

� The stress rkxy does not have explicit dependence on the ampli-tudes zUkauz ,

xSkasx ,ySkasy and

zSkasz .

Even if there is no explicit dependence on some amplitudes, thisfact does not mean that the orders used for the corresponding vari-ables do not affect the result. In fact, the orders used for the othervariables change the solution of Eq. (17) which, therefore, affectsall the quantities.

5.2. In-plane stresses calculated using classical form of Hooke’s law(CFHL)

Even if the formulation is based on a mixed approach, it ispossible to use CFHL (see Part I, Ref. [16]) to calculate the in-plane stresses. If this approach is chosen, the stresses can be calcu-lated as


Fig. 7. Case of theory LM546324. Example of pressure amplitudes and inputs at multilayer level for the case in which the number of layers is two.


ARTICLE IN PRESS

rkxx ¼ �mpa eCk11xFaux xUkaux � npb eCk12yFauy yUkauyhþeC k13zFauz;z zUkauz

iSmpxa S

npyb

rkyy ¼ �mpa eCk12xFaux xUkaux � npb eCk22yFauy yUkauyhþeC k23zFauz;z zUkauz

iSmpxa S

npyb

rkxy ¼ þ npb eCk66xFaux xUkaux þ mpa eCk66yFauy yUkauyh i

Cmpxa Cnpyb

ð33Þ

The following can be observed:

� The stresses rkxx and rkyy do not have explicit dependence on theamplitudes xSkasx ,

ySkasy andzSkasz . This is a direct consequence of

the fact that the displacement-based formulas (the CFHL) havebeen used.

� The stress rkxy does not have explicit dependence on the ampli-tudes zUkauz ,

xSkasx ,ySkasy and

zSkasz . This property was found alsowhen MFHL was used.

Calculating the in-plane stresses by using the classical form ofHooke’s law (CFHL) instead of the mixed form of Hooke’s Law(MFHL) is in theory not consistent because a mixed approach isused. However, for a ‘‘converged” case using either MFHL or CFHLis practically equivalent.


5.3. Out-of-plane stresses calculated using classical form of Hooke’slaw (CFHL)

The stresses rkxz, rkyz, rkzz can be calculated a priori by using Eq.(31) (see in particular the last three expressions). However, it ispossible to use CFHL and calculate the stresses. If this procedureis chosen then the following expressions are valid:

rkzx ¼ eCk55ðmpa zFauz zUauz þ xFaux ;z xUaux ÞCmpxa Snpybrkzy ¼ eCk44ðnpb zFauz zUauz þ yFauy ;z yUauy ÞSmpxa Cnpybrkzz ¼ �mpa eC k13xFaux xUaux Smpxa Snpyb� npb eCk23yFauy yUauy Smpxa SnpybþeC k33zFauz ;z zUauz Smpxa Snpyb

ð34Þ

Strictly speaking this approach is not consistent because the mixedapproach (RMVT) is used and the out-of-plane stresses are calcu-lated a priori.

5.4. Out-of-plane stresses calculated by integrating the indefiniteequilibrium equations

The out-of-plane stresses can be obtained from the indefiniteequilibrium equations as follows:



ARTICLE IN PRESS

orxxox þ

oryxoy þ

orzxoz ¼ 0)

orzxoz ¼ �

orxxox þ

oryxoy

� �orxyox þ

oryyoy þ

orzyoz ¼ 0)

orzyoz ¼ �

orxyox þ

oryyoy

� �orxzox þ

oryzoy þ

orzzoz ¼ 0)

orzzoz ¼ �

orxzox þ

oryzoy

� � ð35Þ

and integrating along the thickness of the plate. Two different optionsare available for the calculation of the shear stresses rzx and rzy.

� Option #1.The out-of-plane shear stresses are calculated by integrating thederivatives of the in-plane stresses obtained by using CFHL. SeeEqs. (35) and (33).

� Option #2.The out-of-plane shear stresses are calculated by integrating thederivatives of the in-plane stresses obtained by using MFHL. SeeEqs. (35) and (32).

For the calculation of stress rzz the following two methods canbe used:

� Method #1.rzz can be obtained by integrating the derivatives of the out-of-plane shear stresses calculated using CFHL.

� Method #2.rzz can be obtained by integrating the derivatives of the out-of-plane shear stresses calculated a priori using the stressesamplitudes.

As for the integration of the indefinite equilibrium equations,this work will use Option #1 for the out-of-plane shear stressesand Method #1 for the stress rzz. In this case, then, the stressesrzx, rzy and rzz depend explicitly only on the displacementamplitudes.

6. Conclusion

For the first time in the literature, the extension of the general-ized unified formulation to the cases of mixed variational state-ments (in particular Reissner’s mixed variational theorem) andlayerwise theories is presented. The displacements ux, uy, uz andthe stresses rzx, rzy, rzz are expanded along the thickness of eachlayer by using Legendre polynomials. Each variable can be treatedseparately from the others. This allows the writing, with a singleformal derivation and software, 16 theories. The new methodol-ogy allows the user to freely change the orders used for the expan-sion of the unknowns and to experiment the best combination thatbetter approximates the structural problem under investigation.The proposed approach for layerwise theories is very general andallows to enforce a priori the compatibility of the displacementsand the equilibrium between two adjacent layers. These a priorirequirements are met by using a particular assembling procedurefrom layer to multilayer level.

All the theories are generated by expanding 1� 1 matrices (thekernels of the generalized unified formulation), which are invariantwith respect to the theory. Thus, with only 13 matrices (the ker-nels) 16 theories can be generated without difficulties.

The numerical performances and properties of mixed layerwisetheories and generalized unified formulation will be discussed inPart V (see [22]) of the present work. In particular, the mixed lay-erwise theories will be compared against mixed higher order the-ories and mixed zig-zag theories. Several discussions on numericalstability and the effect of the relative orders used for the stressesand displacements will be discussed. It will be demonstrated thatthe lessons learned in the layerwise case can be used to interpretthe numerical performances of the other types of theories.


Acknowledgement

The author thanks his sister Demasi Paola who inspired himwith her strong will.

Appendix A. Expanded matrices at layer level

With the assumption of Navier-type solution and theory LM546324,the 13 independent matrices at layer level can be obtained byexpanding the corresponding kernels. The resulting matrices arethe following:

Kkuxux ¼hkp2ðCk11b

2m2 þ Ck66a2n2Þa2b2

þ 13 � 12 � 16 þ 16� 12 þ 65 0 � 12� 16 0 þ 1021 þ 16þ 16 � 12 þ 16 þ 13

26664

37775 ð36Þ

Kkuxuy ¼ðCk12 þ C

k66Þhkmnp2ab

þ 13 � 12 þ 16� 12 þ 65 � 12� 16 0 þ 16þ 16 � 12 þ 13

26664

37775 ð37Þ

Kkuxsx ¼

þ 12 �1 0 0 0 þ 12þ1 0 �2 0 0 �10 þ2 0 �2 0 0� 12 þ1 0 0 0 � 12

26664

37775 ð38Þ

Kkuxsz ¼Ck13hkmp

a

� 13 þ 12 þ 16 0 0 0 � 16þ 12 � 65 0 þ 15 0 0 þ 12þ 16 0 � 1021 0 þ 17 0 � 16� 16 þ 12 � 16 0 0 0 � 13

26664

37775 ð39Þ

Kkuyuy ¼ðCk66b

2m2 þ Ck22a2n2Þhkp2

a2b2

þ 13 � 12 þ 16� 12 þ 65 � 12þ 16 � 12 þ 13

264

375 ð40Þ

Kkuysy ¼þ 12 �1 0 0 þ 12þ1 0 �2 0 �1� 12 þ1 0 0 � 12

264

375 ð41Þ

Kkuysz ¼Ck23hknp

b

� 13 þ 12 þ 16 0 0 0 � 16þ 12 � 65 0 þ 15 0 0 þ 12� 16 þ 12 � 16 0 0 0 � 13

264

375 ð42Þ

Kkuzsx ¼hkmp

a

þ 13 � 12 � 16 0 0 þ 16� 12 þ 65 0 � 15 0 � 12� 16 0 þ 1021 0 � 17 þ 16þ0 � 15 0 þ 1445 0 0þ 16 � 12 þ 16 0 0 þ 13

2666664

3777775 ð43Þ

Kkuzsy ¼hknp

b

þ 13 � 12 � 16 0 þ 16� 12 þ 65 0 � 15 � 12� 16 0 þ 1021 0 þ 160 � 15 0 þ 1445 0þ 16 � 12 þ 16 0 þ 13

2666664

3777775 ð44Þ

Kkuzsz ¼

þ 12 �1 0 0 0 0 þ 12þ1 0 �2 0 0 0 �10 þ2 0 �2 0 0 00 0 þ2 0 �2 0 0� 12 þ1 0 0 0 0 � 12

2666664

3777775 ð45Þ

Kksxsx ¼ Ck55hk

� 13 þ 12 þ 16 0 0 � 16þ 12 � 65 0 þ 15 0 þ 12þ 16 0 � 1021 0 þ 17 � 160 þ 15 0 � 1445 0 00 0 þ 17 0 � 1877 0� 16 þ 12 � 16 0 0 � 13

2666666664

3777777775

ð46Þ



ARTICLE IN PRESS

Kksysy ¼ Ck44hk

� 13 þ 12 þ 16 0 � 16þ 12 � 65 0 þ 15 þ 12þ 16 0 � 1021 0 � 160 þ 15 0 � 1445 0� 16 þ 12 � 16 0 � 13

2666664

3777775 ð47Þ

Kkszsz ¼ Ck33hk

� 13 þ 12 þ 16 0 0 0 � 16þ 12 � 65 0 þ 15 0 0 þ 12þ 16 0 � 1021 0 þ 17 0 � 160 þ 15 0 � 1445 0 þ 19 00 0 þ 17 0 � 1877 0 00 0 0 þ 19 0 � 22117 0� 16 þ 12 � 16 0 0 0 � 13

26666666664

37777777775

ð48Þ

Now consider a numerical example – a plate consists of one layer,with the following properties:

m ¼ 2; n ¼ 3; a ¼ 10; b ¼ 15; h ¼ 3; # ¼ 0 ð49ÞE11 ¼ 25 E22 ¼ 4 E33 ¼ 3G12 ¼ 12 G13 ¼ 35 G23 ¼ 15t12 ¼ 14 t13 ¼ 27100 t23 ¼ 29100

8><>: ð50ÞSome of the matrices are numerically calculated and their expres-sions are reported below:

Kuxux ¼

10:17 �15:25 �5:08 5:08�15:25 36:60 0 �15:25�5:08 0 14:52 5:085:08 �15:25 5:08 10:17

26664

37775 ð51Þ

Kuxuy ¼

0:60 �0:89 0:30�0:89 2:15 �0:89�0:30 0 0:300:30 �0:89 0:60

26664

37775 ð52Þ

Kuxsz ¼

�0:22 0:33 0:11 0 0 0 �0:110:33 �0:78 0 0:13 0 0 0:330:11 0 �0:31 0 0:09 0 �0:11�0:11 0:33 �0:11 0 0 0 �0:22

26664

37775 ð53Þ

Kuyuy ¼1:79 �2:69 0:90�2:69 6:45 �2:690:90 �2:69 1:79

264

375 ð54Þ

Kuysz ¼�0:19 0:29 0:10 0 0 0 �0:100:29 �0:69 0 0:11 0 0 0:29�0:10 0:29 �0:10 0 0 0 �0:19

264

375 ð55Þ

Ksxsx ¼

�1:67 2:50 0:83 0 0 �0:832:50 �6:00 0 1:00 0 2:500:83 0 �2:38 0 0:71 �0:83

0 1:00 0 �1:56 0 00 0 0:71 0 �1:17 0

�0:83 2:50 �0:83 0 0 �1:67

26666666664

37777777775

ð56Þ

Ksysy ¼

�5:00 7:50 2:50 0 �2:507:50 �18:00 0 3:00 7:502:50 0 �7:14 0 �2:50

0 3:00 0 �4:67 0�2:50 7:50 �2:5 0 �5:00

26666664

37777775

ð57Þ


Kszsz ¼

�0:31 0:46 0:15 0 0 0 �0:15

0:46 �1:11 0 0:18 0 0 0:46

0:15 0 �0:44 0 0:13 0 �0:15

0 0:18 0 �0:29 0 0:10 0

0 0 0:13 0 �0:22 0 0

0 0 0 0:10 0 �0:17 0

�0:15 0:46 �0:15 0 0 0 �0:31

26666666666666664

37777777777777775

ð58Þ

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http://dx.doi.org/10.1016/j.compstruct.2008.07.011http://dx.doi.org/10.1016/j.compstruct.2008.07.010http://dx.doi.org/10.1016/j.compstruct.2008.07.010http://dx.doi.org/10.1016/jcompstruct.2007.08.004http://dx.doi.org/10.1016/j.compstruct.2008.07.013http://dx.doi.org/10.1016/j.compstruct.2008.07.013http://dx.doi.org/10.1016/j.compstruct.2008.07.009http://dx.doi.org/10.1016/j.compstruct.2008.07.009

atl1IntroductionLayerwise models: main theoretical conceptsWhat are the new contributions of this work

Theoretical derivation of {\infty}^{6} layerwise mixed theoriesHow {\infty}^{6} layerwise theories are generatedKernels of the generalized unified formulationExpansion of the 1 \times 1 kernels: matrices at layer levelAssembling in the thickness direction: from layer to multilayer level

Theoretical examplesCalculation of the stressesIn-plane stresses calculated using mixed form of Hooke ' s Law (MFHL)In-plane stresses calculated using classical form of Hooke ' s law (CFHL)Out-of-plane stresses calculated using classical form of Hooke ' s law (CFHL)Out-of-plane stresses calculated by integrating the indefinite equilibrium equations

ConclusionAcknowledgementExpanded matrices at layer levelReferences

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