Use of axiomatic/asymptotic approach to evaluate various refined theories for sandwich shells Daoud S. Mashat a , Erasmo Carrera a,b,⇑ , Ashraf M. Zenkour a,c , Sadah A. Al Khateeb a a Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia b Department of Mechanical and Aerospace Engineering, Politecnico di Torino, Italy c Department of Mathematics, Faculty of Science, Kafrelsheikh University, Kafr El-Sheikh 33516, Egypt article info Article history: Available online 6 November 2013 Keywords: Sandwich structures Shell theories Layer-wise theories Zig-zag theories Higher order theories Best shell theories abstract This paper evaluates refined theories for sandwich shells. Layer-wise and equivalent single-layer models (including zig-zag theories) are used with linear and higher order expansion in the thickness layer/shell direction for the displacement variables. So called asymptotic/axiomatic approach is employed to estab- lish the effectiveness of each displacements terms for a given problems: that is the initial axiomatic expansion with all the terms related to the assigned order N is asymptotically reduced to a ‘best’ displace- ment models which has the same accuracy of the full model but with less terms. The various sandwich theories are conveniently formulated by using the unified formulation by Carrera (CUF) that leads to gov- erning equations which forms are formally the same for the different sandwich shell theories. Accuracy of a given theory is established by fixing the sandwich shell in term of geometry, boundary conditions, lay- out of the face/core layers (including very soft-core cases) as well as by choosing a criteria to measure the errors. Two error criteria have been adopted which are related to a fixed point and to the maximum val- ues of displacement/stress variables in the thickness shell direction. A number of problems have been treated and the related ‘best’ displacement model have been obtained. The effectiveness of the asymp- totic/axiamotic problems is proved by comparing with available reference solutions. It has been found that the resulting reduced ‘best’ theories are very much subordinated to the considered problems. These changes by changing geometrically parameters as well as by adopting a different error criteria. Ó 2013 Elsevier Ltd. All rights reserved. 1. Introduction Many problems are of interest for a better understanding of the mechanical behavior of sandwich shells. The books by [1–4] dis- cuss the main aspects considered in the design, analysis, and con- struction of sandwich structures. Among these issues, our interest is in this work directed to compare shell theories able to provide accurate evaluations of stress/strain fields in each layer of the sandwich structures. On this respect, sandwich consists of one of the pioneering example of structure which demands amendments to classical theories originally developed for metallic one-layered structures. Two are the main reasons of that: sandwich structures are multilayered made (at least three layers); one of these layers (the core) has very high transverse shear deformability. Three-lay- ers configuration demands for accounting so-called zig-zag (ZZ) for the distribution of displacement fields in the thickness directions as well as fulfillment of interlaminar continuity (IC) conditions for transverse shear and normal stresses at the two interfaces. Most of the contributions made in the last century on refined theories for multilayered beam/plate/shell structures have been, in fact, originated by early work on sandwich structure analysis, see [5]. Accurate analysis of stress/strain fields could be acquired by three-dimensional (3D) elasticity solutions, see [6,7]. However the use of two-dimensional 2D plate/shell models is preferred in most of the applications related to sandwich structures. 2D models are, in fact, more convenient than 3D ones in terms of required computational efforts. Various 2D models have been developed in the literature. These have been discussed in many texbooks [8], Ambartsumian [9–11], Librescu [12], and Reddy [13] and more recentely by Carrera et al. [14]. Over 800 references on sandwich structures are considered in Noor et al. [15] and in Reddy and Rob- bins [16], Carrera [5,17,18]. For our purpose 2D models sandwich structures could be classified by accounting for manner in which the description of the kinematic in the layers is made. Layer-wise models (LWMs) are those in which the number of the variables is independent in each layer. ZZ effect is intrinsically considered in the LWMs. The number of the unknown variables is kept inde- pendent by the number of the layers in case of Equivalent single layer models (ESLMs). 0263-8223/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compstruct.2013.10.046 ⇑ Corresponding author. Address: Department of Mechanical and Aerospace Engineering, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy. Tel.: +39 011 090 6836; fax: +39 011 090 6899. E-mail address: [email protected](E. Carrera). URL: http://www.mul2.com (E. Carrera). Composite Structures 109 (2014) 139–149 Contents lists available at ScienceDirect Composite Structures journal homepage: www.elsevier.com/locate/compstruct
Use of axiomatic/asymptotic approach to evaluate various refinedtheories for sandwich shells
0263-8223/$ - see front matter � 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.compstruct.2013.10.046
⇑ Corresponding author. Address: Department of Mechanical and AerospaceEngineering, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy.Tel.: +39 011 090 6836; fax: +39 011 090 6899.
E-mail address: [email protected] (E. Carrera).URL: http://www.mul2.com (E. Carrera).
Daoud S. Mashat a, Erasmo Carrera a,b,⇑, Ashraf M. Zenkour a,c, Sadah A. Al Khateeb a
a Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabiab Department of Mechanical and Aerospace Engineering, Politecnico di Torino, Italyc Department of Mathematics, Faculty of Science, Kafrelsheikh University, Kafr El-Sheikh 33516, Egypt
a r t i c l e i n f o a b s t r a c t
Article history:Available online 6 November 2013
Keywords:Sandwich structuresShell theoriesLayer-wise theoriesZig-zag theoriesHigher order theoriesBest shell theories
This paper evaluates refined theories for sandwich shells. Layer-wise and equivalent single-layer models(including zig-zag theories) are used with linear and higher order expansion in the thickness layer/shelldirection for the displacement variables. So called asymptotic/axiomatic approach is employed to estab-lish the effectiveness of each displacements terms for a given problems: that is the initial axiomaticexpansion with all the terms related to the assigned order N is asymptotically reduced to a ‘best’ displace-ment models which has the same accuracy of the full model but with less terms. The various sandwichtheories are conveniently formulated by using the unified formulation by Carrera (CUF) that leads to gov-erning equations which forms are formally the same for the different sandwich shell theories. Accuracy ofa given theory is established by fixing the sandwich shell in term of geometry, boundary conditions, lay-out of the face/core layers (including very soft-core cases) as well as by choosing a criteria to measure theerrors. Two error criteria have been adopted which are related to a fixed point and to the maximum val-ues of displacement/stress variables in the thickness shell direction. A number of problems have beentreated and the related ‘best’ displacement model have been obtained. The effectiveness of the asymp-totic/axiamotic problems is proved by comparing with available reference solutions. It has been foundthat the resulting reduced ‘best’ theories are very much subordinated to the considered problems. Thesechanges by changing geometrically parameters as well as by adopting a different error criteria.
� 2013 Elsevier Ltd. All rights reserved.
1. Introduction transverse shear and normal stresses at the two interfaces. Most
Many problems are of interest for a better understanding of themechanical behavior of sandwich shells. The books by [1–4] dis-cuss the main aspects considered in the design, analysis, and con-struction of sandwich structures. Among these issues, our interestis in this work directed to compare shell theories able to provideaccurate evaluations of stress/strain fields in each layer of thesandwich structures. On this respect, sandwich consists of one ofthe pioneering example of structure which demands amendmentsto classical theories originally developed for metallic one-layeredstructures. Two are the main reasons of that: sandwich structuresare multilayered made (at least three layers); one of these layers(the core) has very high transverse shear deformability. Three-lay-ers configuration demands for accounting so-called zig-zag (ZZ) forthe distribution of displacement fields in the thickness directionsas well as fulfillment of interlaminar continuity (IC) conditions for
of the contributions made in the last century on refined theoriesfor multilayered beam/plate/shell structures have been, in fact,originated by early work on sandwich structure analysis, see [5].
Accurate analysis of stress/strain fields could be acquired bythree-dimensional (3D) elasticity solutions, see [6,7]. Howeverthe use of two-dimensional 2D plate/shell models is preferred inmost of the applications related to sandwich structures. 2D modelsare, in fact, more convenient than 3D ones in terms of requiredcomputational efforts. Various 2D models have been developedin the literature. These have been discussed in many texbooks[8], Ambartsumian [9–11], Librescu [12], and Reddy [13] and morerecentely by Carrera et al. [14]. Over 800 references on sandwichstructures are considered in Noor et al. [15] and in Reddy and Rob-bins [16], Carrera [5,17,18]. For our purpose 2D models sandwichstructures could be classified by accounting for manner in whichthe description of the kinematic in the layers is made. Layer-wisemodels (LWMs) are those in which the number of the variablesis independent in each layer. ZZ effect is intrinsically consideredin the LWMs. The number of the unknown variables is kept inde-pendent by the number of the layers in case of Equivalent singlelayer models (ESLMs).
In the last years Carrera and co-authors [21–26] have proposeda unified formulation (UF or CUF) for multilayered structuresanalysis. Classical models formulated were obtained as particularcases. LW and ESL models related to linear to fourth order expan-sion in the plate/shell thickness direction (z) were implemented.ZZ, IC, transverse shear, and normal strains effects were addressed.Navier-type closed form solutions as well as finite element solu-tion were obtained [5]. Extensive application to sandwich struc-tures plates and shells have been provided in Carrera et al.[18,27,19,20]. The application of UF permits an extensive evalua-tion of a large number of theories in both LW and ESLM framework.
In order to try to reduce the computational costs without losingaccuracy a so called axiomatic/asymptotic approach has beenrecently established by Carrera and Petrolo [28]. CUF formulatedtheories, as definition, should be classified as axiomatic theories,that is the order of the expansion for the displacement variablesis assumed ‘a priori’. It is well known that in contrast to axiomaticapproach, the asymptotic approach could be used. The latterexpands the governing equations in terms of a perturbationparameter p of the structures (e.g. the length-to-thickness ratio)by leading to class-of-problems related to a set of governingequations which contain the whole contribution with the same or-der of magnitude with respect to p. Reviews and analysis on thisapproach with applications to plates and shells can be found in[29–32,15,16,33].
It has been previously shown that the introduction of high orderterms in a given axiomatic model offers a benefit in terms of im-proved structural response analysis, with higher computationalcosts. The possibility to obtain accurate high order theories withless computational cost could be offered by evaluating the impor-tance/effectiveness of each term of the expansion in the solutionprocess. With that information a decision could be taken and thecorresponding term could be retained (if relevant) or discarded(if not significant). By doing that the above axiomatic/asymptoticapproach in [28] is obtained: the effectiveness of each displace-ment variable of a model is compared to a reference solution andthe terms which do not influence the response are discarded. Thistechnique was proposed in the finite element framework in[34,22,23]. The genetic-like algorithms were used in [35] to evalu-ate the importance of each displacement variables for FE platemodels. The results of these works were presented in form of a dia-gram. That diagram was stated as ‘Best Plate/Shell Theoriescurves’; it gives the minimum number of displacement variablesversus the accuracy on a given stress or displacement parameter.Recent application to sandwich plates [36,37] have clearly shownthat the choice of the most appropriate theories is very muchdependent on the given problems, geometrical parameter of thesandwich structures, mechanical properties of faces and core,boundary conditions (mechanical and geometrical ones), to theunknowns variables (displacement, stress, strain components)used to measure accuracy as well as the criteria used to build theerrors (one-points, multi-points, etc.). This work extend the previ-ous finding to sandwich shell geometries.
2. Preliminary
In the following multilayered shells are considered. The shellshave Nl layers and each layer is identified by an index k, whichvaries from 1 to Nl starting from the bottom. The reference systemfor a such structure is reported in Fig. 1 (the particular case ofcylindrical shell has been drawn). ak and bk are the curvilinearorthogonal co-ordinate which coincides with the principalcurvature lines. zk denotes the rectilinear co-ordinate in the normaldirection to the reference surface Xk. In the following a furtherdimensionless coordinate is introduced for each layer: fk ¼ 2zk
hk
where hk is the thickness of the k-layer. For a given orthogonal sys-tem of curvilinear coordinates the infinitesimal length of a line ele-ment, the infinitesimal area of a rectangle lying on the surface Xk
and the infinitesimal volume can be expressed respectively:
ds2k ¼ Hk
ada2k þ Hk
b db2k þ Hk
z dz2k ; dXk ¼ Hk
aHkb dak dbk;
dV ¼ HkaHk
bHkz dak dbk dzk ð1Þ
where
Hka ¼ Ak 1þ zk
Rka
!; Hk
b ¼ Bk 1þ zk
Rkb
!; Hk
z ¼ 1 ð2Þ
Rka and Rk
b are the curvature radii of the a k layer along the directionsof ak and bk respectively. Ak and Bk are the coefficients of the firstfundamental form of Xk.
3. Carrera unified formulation for shell theories
According to CUF the displacement field of a 2D structure can bedescribed as:
u ¼ Fs � us s ¼ 1;2; . . . ;N þ 1 ð3Þ
where u is the displacement vector and N is order of the expansion.As already described in the previous section the development of Eq.(3) can follow two different approaches: Equivalent Single Layerand Layer Wise. According to the former approach the behavior ofa multilayered shell can be attributed to a single equivalent surfacewhich sums up all the properties of a multilayered shell. Accordingto the latter approach each layer of a multilayer shell presents in-stead its own displacement variables. It has to be highlight that asthe number of layers increases the number of displacement vari-ables required for Layer Wise approach increases as well but for aSingle Layer Approach it remains the same.
In Equivalent Single Layer approach Fs functions are polynomialfunctions of z defined as Fs ¼ zs�1. In the following the ESL modelsare synthetically indicated as EDN, where N is the expansion order.As example for an ED4 displacement field one has:
ua ¼ ua1 þ zua2 þ z2ua3 þ z3ua4 þ z4ua5
ub ¼ ub1 þ zub2 þ z2ub3 þ z3ub4 þ z4ub5
uz ¼ uz1 þ zuz2 þ z2uz3 þ z3uz4 þ z4uz5
ð4Þ
Classical Lamination Theory, CLT and First order Shear Deforma-tion Theory FSDT [12], can be considered as special cases of full lin-ear expansion (ED1). An improvement of an ESL model followsfrom the introduction of Murakami’s Zig-Zag function:
It has been proved in previous works [34] that the Murakami ZZfunction is very suitable for application to sandwich structures. Infact, it introduces the physical discontinuity at the interface core/skins for the derivatives of the displacement fields in the thicknessshell direction. These higher order theories are herein denoted byacronyms EDZ1, EDZ2, EDZ3.
In Layer-Wise approach, the displacement field can be de-scribed as:
uk ¼ Ft � ukt þ Fb � uk
b þ Fr � ukr ¼ Fsuk
s s ¼ t; b; r
r ¼ 2;3; . . . ;N k ¼ 1;2; . . . ;Nl ð8Þ
where k is the generic k-layer of a plate and Nl is the number of thelayers. Subscripts t and b correspond to the top and the bottom of alayer. Functions Fs depend on a coordinate fk, its range is�1 6 fk 6 1. Functions Fs derive from the Legendre polynomials Paccording to the following equations
Ft ¼P0 þ P1
2Fb ¼
P0 � P1
2Fr ¼ Pr � Pr�2 r ¼ 2;3; . . . ;N ð9Þ
The Legendre polynomials used for fourth order theory are:
P0 ¼ 1 P1 ¼ fk P2 ¼3f2
k � 12
P3 ¼5f3
k � 3fk
2
P4 ¼35f4
k
8� 15f2
k
4þ 3
8ð10Þ
LW models require the compatibility of displacement at theinterfaces, that is
ukt ¼ ukþ1
b k ¼ 1; . . . ;Nl � 1 ð11Þ
In the following LW models are indicated as LDN, where N is theexpansion order. As an example the LD4 displacement field of ashell is:
uka ¼ Ftuk
at þ F2uka2 þ F3uk
a3 þ F4uka4 þ Fbuk
ab
ukb ¼ Ftuk
bt þ F2ukb2 þ F3uk
b3 þ F4ukb4 þ Fbuk
bb
ukz ¼ Ftuk
zt þ F2ukz2 þ F3uk
z3 þ F4ukz4 þ Fbuk
zb
ð12Þ
More details about CUF can be found in the two recent books[14,39].
3.1. Fundamental nuclei and governing equations
The principle of virtual displacementstates:
XNl
k¼1
ZXk
ZAk
dekrk dAk dXk ¼ dLext ð13Þ
where Ak is the cross sectional area of the kth layer and Xk is theplan area of the kth layer. Nl defines the number of layers of theshell. The previous equation can be written as
XNl
k¼1
ZXk
ZAk
dekpr
kp þ dek
nrkn
� �dAk dXk ¼ dLext ð14Þ
where n defines the normal quantities (normal to a reference plane)and p defines the in-plane quantities. Stresses are evaluated accord-ing to Hooke law:
rkp ¼ eCpp e
kppG þ eCnpG ek
npG
rkn ¼ eCnp e
knpG þ eCnnG ek
nnG
ð15Þ
where the subscript G indicates the geometrical derivation of thedeformations e. Since orthotropic materials are considered thematrices of the elastic coefficients are described as
eCkpp ¼
eCk11
eCk12
eCk16eCk
21eCk
22eCk
26eCk16
eCk26
eCk66
26643775 C
�kpn ¼ C
�kTnp ¼
0 0 eCk13
0 0 eCk23
0 0 eCk33
26643775
eCknn ¼
eCk44
eCk45 0eCk
45eCk
55 0
0 0 eCk66
26643775 ð16Þ
The differential relation between displacement and deforma-tion can be expressed as:
The analysis herein reported are based on the closed-form solu-tion proposed by Navier for simply supported orthotropic shells.The following properties hold:
Cpp16 ¼ Cpp26 ¼ Cpn63 ¼ Cpn36 ¼ Cnn45 ¼ 0 ð24Þ
The terms uks for a LW approach are expressed as:
ukas¼Xm;n
eUkas� cos
mpkak
ak
� �sin
npbk
bk
� �k ¼ 1;Nl
ukbs¼Xm;n
eUkbs� sin
mpak
ak
� �cos
npbk
bk
� �s ¼ 1;N
ukzs ¼
Xm;n
eUkzs � sin
mpak
ak
� �sin
npbk
bk
� �ð25Þ
where eUkas; eUk
bsand eUk
zs are the amplitudes, m and n are the num-ber of waves (they range from 0 to 1) and ak and bk are thedimensions of the plate. The same solution can be applied toESL approach, in which displacement variables appears withoutsuperscript k. The detailed algebra is not reported here, readersare addressed to [38] for additional details. Numerical analysishas been restricted to cylindrical shells, with Ra ¼ 1 andAk ¼ Bk ¼ 1.
Table 1Symbols to indicate the status of a displacement variable.
Active term Inactive term Non-deactivable term
N � 4 � j
Table 2Representation of the reduced kinematics model with uz3 deactivated.
4. Axiomatic/asympotic evaluation of various shell theories
The possibility to perform an accurate analysis of a sandwichshell comes from the introduction of higher order terms. As adrawback the use of higher order theories leads to an increase ofcomputational costs with respect to classical formulations. Axiom-atic/asymptotic technique offers the capability to reduce ‘as muchas possible’ the computational cost of a model and at the sametime to preserve the accuracy of a high order model. This techniqueconsists of the following steps:
1. shell parameters such as the geometry, boundary condi-tions, loadings, materials and layer layouts, are fixed;
2. a set of output parameters is chosen, such as displacementand stress components;
3. a theory is fixed, that is the displacement variables to beanalyzed are defined;
4. a reference solution is defined; in the present work LD4approach is adopted, since this fourth order model offersan excellent agreement with the three-dimensional solu-tions as highlighted in [28];
5. CUF is used to generate the governing equations for thetheories considered;
6. each displacement variable effectiveness is numericallyestablished measuring the loss of accuracy on the chosenoutput parameters compared with the reference solution;
7. any displacement variable which does not alter themechanical response is considered not effective for thekinematics model;
8. the most suitable kinematics model for a specific parame-ter is then obtained discarding the non-effective displace-ment variables. The reduced combined models arecreated discarding all displacement variables which arenot crucial for all parameters at the same time.
The analysis of the relevance of each displacement variable canbe performed considering stress and displacement components atseveral points. The position of the points considered defines differ-ent criteria. In the following two criteria are used.
� Criterion C1 is used when stress and displacement componentsare evaluated at ½a=2; b=2;h=2� for uz;raa, and rzz, at ½0; b=2;0�for raz and at ½a=2;0;0� for rbz.� Criterion C2 is used when the evaluation of the effectiveness of
displacement variables is performed considering the maximumvalues of above mentioned stress and displacement variables. Inthis case the position of the points is not known a priori.
To make clear how the evaluation of the displacement variablesis carried out an example is given. Let’s consider an ED4 model fora two layer shell:
ua ¼ F0ua0 þ F1ua1 þ F2ua2 þ F3ua3 þ F4ua4
ub ¼ F0ub0 þ F1ub1 þ F2ub2 þ F3ub3 þ F4ub4
uz ¼ F0uz0 þ F1uz1 þ F2uz2 þ F3uz3 þ F4uz4
ð26Þ
If the effectiveness of uz3 term has to be analyzed the relative re-duced model is:
ua ¼ F0ua0 þ F1ua1 þ F2ua2 þ F3ua3 þ F4ua4
ub ¼ F0ub0 þ F1ub1 þ F2ub2 þ F3ub3 þ F4ub4
uz ¼ F0uz0 þ F1uz1 þ F2uz2 þ þF4uz4
ð27Þ
The response (Qi) given by this model reported is comparedwith the reference (Q ref ) solution according to the equation:
dQ ¼ 1� Q i
Q iref
� 100 > tolerance ð28Þ
where Qiref and Qi synthetically denote respectively the reference
value and the actual value of any variable under exam, as displace-ment uz or stress raz, evaluated at the generic i point denoted in theabove described criteria. A term is considered to be effective if thequantity dQ is greater than a specific level. The suppression of aterm is obtained by means of a penalty technique. In the followinganalyses the tolerance on error is considered as a parameter and it isequal to 0:05%. In order to present in a synthetic and clear way thereduced model representation in two Tables 1 and 2 are proposed.
As second example a LD4 model for a two layers shell isconsidered:
ua¼ Ftu1atþF2u1
a2þF3u1a3þF4u1
a4þFbu1abþFtu2
atþF2u2a2þF3u2
a3þF4u2a4þFbu2
ab
ub¼ Ftu1btþF2u1
b2þF3u1b3þF4u1
b4þFbu1bbþFtu2
btþF2u2b2þF3u2
b3þF4u2b4þFbu2
bb
uz¼ Ftu1ztþF2u1
z2þF3u1z3þF4u1
z4þFbu1zbþFtu2
ztþF2u2z2þF3u2
z3þF4u2z4þFbu2
zb
ð29Þ
It has to be remarked that when the axiomatic/asymptotictechnique is applied to LW model the terms uk
t and ukb cannot be
suppressed since they are fundamental to impose the continuity
b . If the effectivenessof uz3 term of the first layer has to be analyzed the relative reducedmodel is:
ua¼ Ftu1atþF2u1
a2þF3u1a3þF4u1
a4þFbu1abþFtu2
atþF2u2a2þF3u2
a3þF4u2a4þFbu2
ab
ub¼ Ftu1btþF2u1
b2þF3u1b3þF4u1
b4þFbu1bbþFtu2
btþF2u2b2þF3u2
b3þF4u2b4þFbu2
bb
uz¼ Ftu1ztþF2u1
z2þF4u1z4þFbu1
zbþFtu2ztþF2u2
z2þF3u2z3þF4u2
z4þFbu2zb
ð30Þ
As in the previous case the graphical representation of LD4 re-duced model for a two layer shell is proposed in Table 2 with cor-responding legend in Table 1.
5. Results and discussion
Reduced ESL and LW models for isotropic and multilayeredcylindrical shells are discussed in this section. The static responseis considered, an harmonic distribution of transverse pressure isassumed as:
pz ¼ pz � sinmaa
� �� sin
nbb
� �ð31Þ
Fig. 1 shows the reference system a and b and h are the length alonga; b and z, respectively.
A multilayered composite shell is considered to assess the pres-ent results with those in [41]. A three-layered laminated cylindri-cal shell is analyzed. m ¼ 1;n ¼ 8, pressure acts on the innersurface. Material properties, denoted with standard notation, areEL=ET ¼ 25, GLT=ET ¼ GTL=ET ¼ 0:5, GLz=ET ¼ 0:2, m ¼ 0:25. Ply stack-ing sequence is 90�=0�=90�. Geometric data of this shell are:Rb=h ¼ 10, a=h ¼ 40, b ¼ 2pRb. Results are given in Table 3 forthe transverse displacements and various stress components incase of thin and very-thick shells. The quoted values have been gi-ven in non-dimensional form as in the following: uz ¼ 10ELuz
pzhðRb=hÞ4,
raa=bb=ab ¼10raa=bb=ab
pzðRb=hÞ2, raz=bz ¼
10raz=bz
pzðRb=hÞ, rzz ¼ rzzpz
. 3D solution is com-
pared to LD4, ED4 and EDZ3 results. It is confirmed the excellentresults by LD4 analysis even in the very thick shell which permitsus to use LD4 results as reference solution for those problems inwhich elasticity solutions are not available (Benchmarks 1 and 2
Table 3Comparison of results offered by LD4, ED4 and EDZ4 models with the exact solutions in [
Rb=h uz rbb raa
z ¼ 0 z ¼ h=2 z ¼ h=2
Reference solution [41]2 10.11 �18.19 �0.8428
7.168 0.1761100 0.4715 �3.507 0.0018
3.507 0.0838
LD42 10.1017 �18.11 �0.8721
7.156 0.1735100 0.4715 �3.507 0.0018
3.507 0.0838
ED42 9.1582 �13.6060 �0.7264
6.3815 0.1617100 0.4708 �3.5054 0.0021
3.5062 0.0838
EDZ32 11.1063 �18.5015 2.8047
14.9212 3.7338100 0.4715 �3.5068 0.0016
3.5068 0.0839
below). EDZ3 shows that the use of Murakami zig-zag function isvery effective in layered structures; EDZ3 has the same numberof degree of freedom dof of ED4 analysis but it produces much bet-ter results.
A further assessment is made by considering the sandwichcylindrical shell problem for which recent elasticity 3D resultshave been given in [40]. The shell is made by eleven layers withtwo cores. The skins properties are EL ¼ 172:5 GPa, ET ¼ 6:9 GPa,ET ¼ 6:9 GPa, GLT ¼ 3:45 GPa, GTL ¼ 3:45 GPa, Gz ¼ 2:76 GPa;m ¼ 0:25 q ¼ 2710 kg=m3 hs ¼ 0:015. The core properties are:EL ¼ 0:276 GPa, EL ¼ 0:276 GPa, EL ¼ 3:45 GPa, GLT ¼ 0:1104 GPa,GLT ¼ 0:414 GPa, GLT ¼ 0:414 GPa, mLT ¼ 0:25 mLT ¼ 0:02 mLT ¼ 0:02,q ¼ 1600 kg=m3;hc ¼ 0:4325. The stacking sequence is:0�=90�=0�=core=0�=90�=0�=core=0�=90�=0�. The pressure is sinusoi-dally distributed over the external surface (m ¼ n ¼ 1). The usedgeometric parameters of the shell are: a=h ¼ 5;10;Rb=a ¼ 2 andb=a ¼ 1;2. For sake of brevity comparison has been in Table 4.The following non-dimensional values are given: uz ¼ 100 uzET
hpzðR=aÞ4
raa=bb ¼raa=bb
pzðR=aÞ2, rab ¼ 10rab
pzðR=aÞ2raz=bz ¼
raz=bz
pzðR=aÞ Attention has been re-
stricted to LD4 analysis which confirms its capability to producequasi-3D-solutions for laminated sandwich shells and it can be,therefore, used as a reference solution in the following asymp-totic/axiamatic numerical investigations.
Axiomatic/asymptotic analysis has been conducted comparingthe two different error criteria C1 and C2 that have been discussedearlier. The geometrical and mechanical problem data are those in[20]. The first problem, herein denote as Benchmark 1, is consti-tuted of two external isotropic skins (aluminum) and an internalNomex core. Skin properties: E ¼ 73 � 109 Pa, m ¼ 0:34, h1=hTOT ¼0:1. Core properties EL ¼ ET ¼ 0:01 � 106 Pa, Ez ¼ 75:85 � 106 Pa,m ¼ 0:01;G ¼ 22:5 � 106 Pa, h2=hTOT ¼ 0:8. The ratio Ef =Ec , i.e. the ra-tio of the Young modulus of the skin over the Young modulus of thecore, is equal to 7:3 � 106. The second problem (Benchmark 2) differsfrom Benchmark 1 only for the core properties. The core is assumedto be 100 times more deformable: EL ¼ ET ¼ 0:0001 � 106 Pa, Ez ¼0:7585 � 106 Pa, m ¼ 0:01;G ¼ 0:225 � 106 Pa, h2=hTOT ¼ 0:8; and theEf =Ec is equal to 730 � 106. The following non-dimensional formare considered for both Benchmarks: uz ¼ 10 uzEskinh3
pzR4b
, rbz ¼ 10rbz
pzðRb=hÞ,rzz ¼ rzz
pz.
41] for cylindrical panels cross-ply laminated in cylindrical bending.
Results related to the full models are given in Table 5. Thin andthick shells are considered to evaluate transverse displacementsand transverse shear stresses. Me is the ratio between the numberof the used terms and number of terms related to the full model.The accuracy differs for the different variables. ESLMs such asED4 leads to quite poor evaluation of transverse stresses (both nor-mal and shear components), these are indeed very much improvedby the use of Murakami function along with parabolic/cubicexpansions.
Analysis of reduced models via axiomatic/asymptotic tech-nique is first given in Tables 6–8 which only refer to ESLM
analysis and LD4 is used as reference solution. Thick and thingeometries are considered in both cases for criteria C1 and C2.
Table 10LD3 reduced model for sandwich benchmark 1, LD4 model as reference solution. Tolerance on error: 0:05%;Rb=h ¼ 4.
Table 11LD4 reduced model for sandwich benchmark 1, Rb=h ¼ 4.
Cases ED1–ED4 require full models for both C1 and C2 as well asin thick and thin shell cases; to be noticed that the fixed accuracyis quite severe. EDZ type models appear more sensitive to shellgeometry as well as to the selected error criteria. Different re-duced models are, in fact, obtained in different cases as in Tables7 and 8. The crucial role of Murakami’s function should be under-lined; it coincides with the last column of the displacementmodels which is almost ‘always’ present in the ‘best’ reducedmodel.
Reduced models related to LW results for Benchmarks 1 and 2are given in Tables 9–12 and in Figs. 2, 3. Benchmark 1 is first con-sidered. LD2 reduced models are first considered in Table 9. Theevaluation of transverse displacements and in-plane stresses forthick and thin shells are accounted for. Four different models areobtained, Me ranges from 14/21 (stress and displacement evalua-tions in thin shell) to 21/21 (stress evaluation in thick shell). Re-duced models related to layer-wise cubic case LD3 are discussed
in Table 10. Four displacement and in-plane and transverse stresscases are considered for both thick and thin shell as well as C1and C2 error criteria used to build the reduced models. Different‘best’ reduced models are obtained in the different cases. Com-bined cases (that model that perform well for the uz and for thethree stresses) are depicted in the last row. Results confirm thatin thick shell geometries a full layer-wise analysis is required toget accurate stress and displacement fields. Some terms can beomitted without losing accuracy in the thin shell geometry case.To be underlined the important role played by the used criteria,C1 and C2 cases could lead to quite different ‘best’ reduced model.LD4 case is investigated in Tables 11 and 12. The remarks made forLD3 discussion are confirmed.
Different models are assessed in Table 13 in case of Benchmark2. The very soft core reduces the effectiveness of EDZ analysiswhile ED analysis leads to quite inaccurate results. For sake ofbrevity only few results on reduced models analysis are quoted.
Table 12LD4 reduce model for sandwich benchmark 1, Rb=h ¼ 100.
Table 14 investigates the ESLM that use Murakami Zig-Zag func-tions while Tables 15 and 16 investigate reduced ‘best’ models re-lated to LD4 analysis. Comments made for Benchmark 1 areconfirmed. The softer core case related to Benchmark 2 makes, insome of the quoted results, the reduced models more costly thanthose obtained for Benchmark 1 case. rbz and rzz distribution in
the thickness shell direction are given in Figs. 2 and 3, referringto LD4 analysis. C1 and C2 ‘best’ reduced model results are com-pared to the related full model. It appears that C1 and C2 error cri-teria could present some difficulties to get reduced model thatcorrectly traces the stress fields in any point of the thickness ofthe shell.
This paper has evaluated a number of shell theories for thebending analysis of sandwich structures. Attention has been re-stricted to simply supported cylindrical shells made by orthotropiclayer in which case closed form Navier-type solutions are available.Soft core and very soft case have been investigated in term of stressand displacements. Refined theories, including layer-wise and
zig-zag ones have been generated by using the Carrera Unified For-mulation (CUF). The various models have been evaluated by socalled axiomatic–asymptotic approach. That is, the contributionto the solutions is evaluated for each term of the displacementmodels of a given theories and the ‘best’ theories is obtained byremoving those terms that do not contribute according to a definederror criteria. The ‘best’ corresponds to theories that lead to thebest accuracy with minimum number of terms in displacementmodels; the number Me is used to denote the ratio between thenumber of the displacement variables in the reduced and full mod-els. Two different error criteria are used, C1 and C2. The followingmain conclusions can be drawn.
1. The axiomatic/asymptotic methods consists of a reliable tech-nique to build ‘best shell theory’ BST for a given problem.
2. BST is problem dependent, BST changes by changing the geo-metrical and mechanical parameters of the structures, in partic-ular BST is very much sensitive to thickness ratio and ratiobetween the stiffness of skin and core.
3. BST is very much influenced by the adopted error criteria.4. Murakami zig-zag functions is very effective; it is almost always
present in the BST.5. Full LW analyses are required to catch 3D stress field in thick
sandwich shell structures.6. Me values related to LW analysis could results very much lower
than Me values related to ESLM analysis (both ED and EDZones).
Future investigations could consider dynamic problems as wellas application to approximated method of solution such as finiteelement method.
Acknowledgements
This paper was funded by the Deanship of Scientific Research(DSR), King Abdulaziz University, Jeddah, under HiCi 1433130-21Grant. The authors, therefore, acknowledge with thanks DSR tech-nical and financial support.
References
[1] Plantema FJ. Sandwich construction. New York, USA: John Wiley & Sons, Inc.;1966.
[2] Allen HG. Analysis and design of structural sandwich panels. London,UK: Pergamon Press; 1969.
[3] Zenkert D. An introduction to sandwich structures. London, UK: ChamelonPress; 1995.
[4] Vinson JR. The behaviour of sandwich structures of isotropic and compositematerials. Lancaster, USA: Technomic Publishing Company, Inc.; 1999.
[5] Carrera E. Historical review of Zig-Zag theories for multilayered plates andshells. Appl Mech Rev 2003;56:287–308.
[6] Pagano NJ. Exact solutions for rectangular bidirectional composites andsandwich plates. J Compos Mater 1970;0(0):20–34.
[7] Meyer-Piening HR. Application of the elasticity solution to linear sandwichbeams, plates and shells analysis. J Sandwich Struct Mater 2004;6(4):295–312.
[8] Lekhnitskii GS. Anisotropic plates. 2nd ed. Bordon and Breach: SW Tsai andCheron; 1968.
[9] Ambartsumian SA. Theory of anisotropic shells. Moskwa: Fizmatzig; 1961.Translated from Russian, NASA TTF-118.
[10] Ambartsumian SA. In: Cheron T, Ashton JE, editors. Theory of anisotropicplates. Moskwa: Technomic Publishing Company; 1969. Translated fromRussian.
[11] Ambartsumian SA. Fragments of the theory of anisotropicshells. Singapore: World Scientific Publishing Co.; 1991. TechnomicPublishing Company, Translated from Russian, Moskwa..
[12] Librescu L. Elasto-statics and kinetics of anisotropic and heterogeneous shell-type structures. Leyden, Netherland: Noordhoff International; 1975.
[13] Reddy JN. Mechanics of laminated plates, theory and analysis. Boca Raton: CRCPress; 1997.
[14] Carrera E, Brischetto S, Nali P. Plates and shells for smart structures classicaland advanced theories for modeling and analysis. New Delhi: Wiley; 2011.
[15] Noor AK, Burton WS, Bert CW. Computational model for sandwich panels andshells. Appl Mech Rev 1996;49(3):155–99. http://dx.doi.org/10.1115/1.3101923.
[16] Reddy JN, Robbins DH. Theories and computational models for compositelaminates. Appl Mech Rev 1994;47(7):147–65. http://dx.doi.org/10.1115/1.3111076.
[17] Carrera E. Theories and finite elements for multilayered plates and shells: aunified compact formulation with numerical assessment and benchmarking.Arch Comput Meth Eng 2003;10(3):215–96.
[18] Carrera E, Ciuffreda A. A unified formulation to assess theories of multilayeredplates for various bending problems. Compos Struct 2005;69(3):271–93.http://dx.doi.org/10.1016/j.compstruct.2004.07.003.
[19] Carrera E, Briscotto S. A survey with numerical assessment of classical andrefined theories for the analysis of sandwich paltes. Appl Mech Rev2009;62(20):1–17. http://dx.doi.org/10.1115/1.3013824.
[20] Carrera E, Briscotto S. A comparison of various kinematic models for sandwichshell panels with soft core. J Compos Mater 2009;43(20):2201–21. http://dx.doi.org/10.1177/0021998309343716.
[21] Carrera E. A class of two-dimensional theories for anisotropic multilayeredplates analysis. Atti Acc. Sci. Torino, Mem. Sci. Fis. 1995;19–20:49–87.
[22] Carrera E. Layer-wise mixed models for accurate vibration analysis ofmultilayered plates. J Appl Mech 1998;65(4):820–8.
[23] Carrera E, Demasi L. Classical and advanced multilayered plate elements basedupon PVD and RMVT. Part 1: Derivation of finite element matrices. Int J NumerMethods Eng 2002;55(2):191–231.
[24] Carrera E, Demasi L. Classical and advanced multilayered plate elements basedupon PVD and RMVT. Part 2: Derivation of finite element matrices. Int J NumerMethods Eng 2002;55(3):253–96.
[25] Carrera E, Brischetto S. Analysis of thickness locking in classical, refined andmixed multilayered plate theories. Compos Struct 2008;82(4):549–62.
[26] Carrera E, Brischetto S. Analysis of thickness locking in classical, refined andmixed theories for layered shells. Compos Struct 2008;85(1):83–90.
[27] Carrera E, Demasi L. Two benchmarks to assess two-dimensional theories ofsandwich, composite plates. AIAA J 2003;41(7):1356–62.
[28] Carrera E, Petrolo M. Guidelines and recommendation to construct theories formetallic and composite plates. AIAA J 2010;48(12):2852–66. http://dx.doi.org/10.2514/1.J050316.
[29] Demiray S, Becker W, Hohe J. A triangular finite element for sandwich platesaccounting for transverse core compressibility. In: Proceedings of the 7thinternational conference on sandwich structures, Sandwich Structures 7:Advancing with Sandwich Structures and Materials, Aalborg University,Aalborg, Denmark; 2005.
[30] Frostig Y. Bending of curved sandwich panels with a transversely flexible coreclosed-form high-order theory. J Sandwich Struct Mater 1999;1(1):4–10.
[31] Hohe J, Librescu L, Oh SY. Dynamic buckling of flat and curved sandwich panelswith transversely compressible core. Compos Struct 2006;74(1):10–24.
[32] Hohe J, Librescu L. A nonlinear theory for doubly curved anisotropic sandwichshells with transverse compressible core. Int J Solids Struct 2003;40(5):1059–88.
[33] Reissner E. n a certain mixed variational theory and a proposed application. IntJ Numer Methods Eng 1984;20:1366–8.
[34] Carrera E. Developments, ideas and evaluations based upon the Reissner‘smixed variational theorem in the modeling of multilayered plates and shells.Appl Mech Rev 2001;54(4):301–29.
[35] Carrera E, Miglioretti F. Selection of appropriate multilayered plate theories byusing a genetic like algorithm. Compos Struct 2012;94(3):1175–86. http://dx.doi.org/10.1016/j.compstruct.2011.10.013.
[36] Mashat DS, Carrera E, Zenkour AM, Al Khateeb SA. Axiomatic/asymptoticevaluation of multilayered plate theories by using single and multi-pointserror criteria. Compos Struct 2013;106(0):393–406.
[37] Petrolo M, Lamberti A. Axiomatic/asymptotic analysis of refined layer-wise theoriesfor composite and sandwich plates. Mech Adv Mater Struct 2013;0 [in press].
[38] Carrera E. Theories and finite elements for multilayered plates and shells: aunified compact formulation with numerical assessment and benchmarking.Arch Comput Meth Eng 2003;10(3):215–96.
[39] Carrera E, Giunta G, Petrolo M. Beam structures, classical and advancedtheories. New Delhi: Wiley; 2011.
[40] Yaqoob Yasin M, Kapuria S. An efficient layerwise finite element for shallowcomposite and sandwich shells. Compos Struct 2013;98:202–14.
[41] Varadan TK, Bhaskar K. Bending of laminated orthotropic cylindrical shells –an elasticity approach. Compos Struct 1991;17:141–56.
