Onate et al 2012a - Revision history

Move page script: Move page script moved page Draft Samper 624436055 to Onate et al 2012a

2018-10-10T14:14:37Z

Move page script moved page Draft Samper 624436055 to Onate et al 2012a

Cinmemj at 08:52, 9 October 2018

2018-10-09T08:52:36Z

Cinmemj at 08:51, 9 October 2018

2018-10-09T08:51:51Z

Cinmemj at 08:44, 9 October 2018

2018-10-09T08:44:04Z

Scipedia at 14:58, 8 October 2018

2018-10-08T14:58:38Z

Cinmemj at 10:46, 8 October 2018

2018-10-08T10:46:47Z

Cinmemj at 09:50, 8 October 2018

2018-10-08T09:50:16Z

Cinmemj at 09:43, 8 October 2018

2018-10-08T09:43:37Z

Cinmemj at 09:39, 8 October 2018

2018-10-08T09:39:32Z

Cinmemj at 09:37, 8 October 2018

2018-10-08T09:37:55Z

@@ Line 578: / Line 578: @@
 {| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
 |-
-|[[Image:Draft_Samper_624436055-test-Fig3.png|600px|]]
+|[[Image:Draft_Samper_624436055-test-Fig3.png|400px|]]
 |- style="text-align: center; font-size: 75%;"
 | colspan="1" | '''Figure 2:''' Cantilever beam under point  load

@@ Line 578: / Line 578: @@
 {| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
 |-
-|[[Image:Draft_Samper_624436055-test-Fig2.png|600px|Cantilever beam under point  load ]]
+|[[Image:Draft_Samper_624436055-test-Fig3.png|600px|]]
 |- style="text-align: center; font-size: 75%;"
 | colspan="1" | '''Figure 2:''' Cantilever beam under point  load
@@ Line 586: / Line 586: @@
 {| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
 |-
-|[[Image:Draft_Samper_624436055-test-Fig3.png|600px|Mesh of 27000 4-noded plane stress rectangular elements for analysis of cantilever and simple supported beams]]
+|[[File:Draft_Samper_624436055_3572_Fig3-con299.png|600px|]]
 |- style="text-align: center; font-size: 75%;"
 | colspan="1" | '''Figure 3:''' Mesh of 27000 4-noded plane stress rectangular elements for analysis of cantilever and simple supported beams

@@ Line 1: / Line 1: @@
-<!-- metadata commented in wiki content
+Published in ''Comput. Methods Appl. Mech. Engrg''. Vol. 213–216, pp. 362–382, 2012<br />
-==SIMPLE AND ACCURATE TWO-NODED BEAM ELEMENT FOR COMPOSITE LAMINATED BEAMS USING A REFINED ZIGZAG THEORY==
+doi: 10.1016/j.cma.2011.11.023
-−
-'''E. Oñate<math>^{1,2}</math>, A. Eijo<math>^{1}</math> and S. Oller<math>^{1,2}</math>'''
-−
-{|
-|-
-|<math>^{1}</math> International Center for Numerical Methods in Engineering (CIMNE)
-|-
-| <math>^{2}</math> Universitat Politecnica de Catalunya (UPC)
-|-
-| Campus Norte UPC, 08034 Barcelona, Spain
-|-
-| [mailto:onate@cimne.upc.edu onate@cimne.upc.edu], [http://www.cimne.com/eo www.cimne.com/eo]
-|}
--->
 ==Abstract==
@@ Line 20: / Line 6: @@
 In this work we present a new simple linear two-noded beam element adequate for the analysis of composite laminated and sandwich beams based on the combination of classical Timoshenko beam theory and the refined zigzag kinematics  proposed by Tessler ''et al.'' <span id='citeF-22'></span>[[#cite-22|[22]]]. The element has just four kinematic variables per node. Shear locking is eliminated by reduced integration. The accuracy of the new beam element is tested in a number  of applications to the analysis of composite laminated beams with simple supported and clamped ends under point loads and uniformly distributed loads. An example showing the capability of the new element for accurately reproducing delamination effects is also presented.
-'''keywords''' Two-noded beam element, zigzag kinematics, Timoshenko theory, composite, sandwich beams
+'''Keywords''': Two-noded beam element, zigzag kinematics, Timoshenko theory, composite, sandwich beams
 ==1 INTRODUCTION==

@@ Line 640: / Line 640: @@
 Figure [[#img-4|4]] shows the ratio <math display="inline">r</math> between the end node deflection  obtained with the LRZ element (<math display="inline">w_{zz}</math>) and with the plane stress quadrilateral (<math display="inline">w_{ps}</math>) (i.e. <math display="inline">r=\frac{w_{zz}}{w_{ps}}</math>) versus the beam span-to-thickness ratio <math display="inline">d = \frac{L}{h}</math>. Results for the LRZ element have been obtained using ''exact'' two-point integration for all terms of  matrix <math display="inline">{\boldsymbol K}^e_t</math> (Eq.(27)) and  a one-point ''reduced'' integration for the following three groups of matrices: <math display="inline">{\boldsymbol K}^e_s</math>; <math display="inline">{\boldsymbol K}^e_s</math> and <math display="inline">{\boldsymbol K}^e_{s\psi }</math>; and <math display="inline">{\boldsymbol K}_s</math>, <math display="inline">{\boldsymbol K}^e_{s\psi }</math> and <math display="inline">{\boldsymbol K}^e_{\psi }</math> (Eqs.(29b)).
-Labels ``''all'''', ``''S'''', ``''SPsi'''' and ``''Psi'''' in Figures [[#img-4|4]]&#8211;[[#img-7|7]] refer to matrices <math display="inline">{\boldsymbol K}^e_t</math>, <math display="inline">{\boldsymbol K}^e_s</math>, <math display="inline">{\boldsymbol K}^e_{s\psi }</math> and <math display="inline">{\boldsymbol K}^e_{\psi }</math> of Eq.(29a), respectively.
+Labels ``all'<nowiki/>''', ``S'<nowiki/>''', ``SPsi'<nowiki/>''' and `''`''Psi'''' in Figures [[#img-4|4]]&#8211;[[#img-7|7]] refer to matrices <math display="inline">{\boldsymbol K}^e_t</math>, <math display="inline">{\boldsymbol K}^e_s</math>, <math display="inline">{\boldsymbol K}^e_{s\psi }</math> and <math display="inline">{\boldsymbol K}^e_{\psi }</math> of Eq.(29a), respectively.
 Results in Figure [[#img-4|4]] clearly show that the exact integration of <math display="inline">{\boldsymbol K}^e_t</math> leads to shear locking as expected. Good (locking-free) results are obtained by one-point reduced integration of the three groups of matrices.
@@ Line 651: / Line 651: @@
 |[[Image:Draft_Samper_624436055-test-Fig4a.png|600px|r ratio \left(r = \fracw<sub>zz</sub>wₚₛ\right) versus L/h for cantilever under point load analyzed with the LRZ element. Labels ``''all'''', S, SPsi and Psi refer to matrices K<sup>e</sup>ₜ, Kₛ<sup>e</sup>,K<sub>sψ</sub><sup>e</sup> and K<sub>ψ</sub><sup>e</sup>, respectively]]
 |- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figure 4:''' <math>r</math> ratio <math>\left(r = \frac{w_{zz}}{w_{ps}}\right)</math> versus <math>L/h</math> for cantilever under point load analyzed with the LRZ element. Labels ``''all'''', <math>S</math>, <math>SPsi</math> and <math>Psi</math> refer to matrices <math>{\boldsymbol K}^e_t</math>, <math>{\boldsymbol K}_s^e,{\boldsymbol K}_{s\psi }^e</math> and <math>{\boldsymbol K}_{\psi }^e</math>, respectively
+| colspan="1" | '''Figure 4:'<nowiki/>'' <math>r</math> ratio <math>\left(r = \frac{w_{zz}}{w_{ps}}\right)</math> versus <math>L/h</math> for cantilever under point load analyzed with the LRZ element. Labels ``''all'''', <math>S</math>, <math>SPsi</math> and <math>Psi</math> refer to matrices <math>{\boldsymbol K}^e_t</math>, <math>{\boldsymbol K}_s^e,{\boldsymbol K}_{s\psi }^e</math> and <math>{\boldsymbol K}_{\psi }^e</math>, respectively
 |}

@@ Line 28: / Line 28: @@
 Indeed to achieve accurate computational results, 3D finite element analyses are often preferred over beam models. For composite laminates with hundred of layers, however, 3D modelling becomes prohibitely expensive, specially for non linear and progressive failure analyses.
-Improvements to the classical beam theories have been obtained by the so called equivalent single layer (ESL) theories that assume a priori the behaviour of the displacement and/or the stress through the laminate thickness [3,4]. Despite of being computational efficient, ESL theories often produce innacurate distributions for the stresses and strains (in particular the transverse shear stress) across the thickness.
+Improvements to the classical beam theories have been obtained by the so called equivalent single layer (ESL) theories that assume a priori the behaviour of the displacement and/or the stress through the laminate thickness <span id='citeF-3'></span>[[#cite-3|[3,4]]]. Despite of being computational efficient, ESL theories often produce innacurate distributions for the stresses and strains (in particular the transverse shear stress) across the thickness.
-The need for composite laminated beam theories with better predictive capabilities has led to the development of the so-called ''higher order'' theories. In  these theories higher-order kinematic terms with respect to the beam depth  are added to the expression for the axial displacement and, in some cases, to the expressions for the deflection. A review of these theories can be found in [3,4].
+The need for composite laminated beam theories with better predictive capabilities has led to the development of the so-called ''higher order'' theories. In  these theories higher-order kinematic terms with respect to the beam depth  are added to the expression for the axial displacement and, in some cases, to the expressions for the deflection. A review of these theories can be found in <span id='citeF-3'></span>[[#cite-3|[3,4]]].
 Accurate predictions of the correct shear and axial stresses for thick and highly heterogenous composite laminated and sandwich beams can be obtained by using ''layer-wise'' theory. In this theory the thickness coordinate is split into a number of ''analysis layers'' that may or not coincide with the number of laminate plies. The kinematics are independently described within each layer and certain physical continuity requirements are enforced <span id='citeF-3'></span><span id='citeF-4'></span>[[#cite-3|[3,4]]].
@@ Line 36: / Line 36: @@
 A drawback of layer-wise theory is that the number of kinematic variables depends on the number of analysis layers. The layer displacements  can be condensed at each section in terms of the axial displacement for the top layer during the equation solution process <span id='citeF-5'></span><span id='citeF-6'></span>[[#cite-5|[5,6]]]. The displacement condensation processes can be however expensive for problems involving many analysis layers.
-Discrete layer theories in which the number of unknowns in the model does not depend on the number of layers in the laminate are described in [7,8,9]. In this class of discrete layerwise theories (called zigzag theories) a piewise linear in-plane displacement function (the zigzag function) is superimposed over a linear displacement field [7,8], a quadratic displacement field [10,11] or a cubic displacement field [12,13,14] through the thickness of the laminate.
+Discrete layer theories in which the number of unknowns in the model does not depend on the number of layers in the laminate are described in <span id='citeF-7'></span>[[#cite-7|[7,8,9]]]. In this class of discrete layerwise theories (called zigzag theories) a piewise linear in-plane displacement function (the zigzag function) is superimposed over a linear displacement field <span id='citeF-7'></span>[[#cite-7|[7,8]]], a quadratic displacement field <span id='citeF-10'></span>[[#cite-7|[10,11]]] or a cubic displacement field <span id='citeF-12'></span>[[#cite-12|[12,13,14]]] through the thickness of the laminate.
 Many zigzag theories  require <math display="inline">C^1</math> continuity for the deflection field, which is a drawback versus simpler <math display="inline">C^\circ </math> continuous FEM approximations. Also many zigzag theories run into theoretical difficulties to satisfy equilibrium of forces at a clamped support.
@@ Line 44: / Line 44: @@
 A 2-noded beam element based on Euler-Bernouilli beam theory and an extension of Averill's zigzag theory including a cubic in-plane displacement field within each layer has been recently proposed by Alam and Upadhyay <span id='citeF-18'></span>[[#cite-18|[18]]]. Good results are reported for cantilever and clamped composite and sandwich beams.
-An assessment of different zigzag theories for beam is reported in <span id='citeF-19'></span><span id='citeF-20'></span>[[#cite-19|[19,20]]]. A review of zigzag theory for plate analysis can be found in [21].
+An assessment of different zigzag theories for beam is reported in <span id='citeF-19'></span><span id='citeF-20'></span>[[#cite-19|[19,20]]]. A review of zigzag theory for plate analysis can be found in <span id='citeF-21'></span>[[#cite-21|[21]]].
-Tessler ''et al.'' [22,23] have developed a refined zigzag (RZ) theory  starting from the standard Timoshenko kinematic assumptions. This allows one using <math display="inline">C^\circ </math> continuous interpolation for all the kinematic variables. Timoshenko beam theory also introduces naturally shear deformation effect for the homogeneous material case, which is advantageous for many problems. The zigzag functions chosen have the property of vanishing on the top and bottom surfaces of a laminate. A particular feature of this zigzag theory is that the transverse shear stresses are not required to be continuous at the layer interfaces. This results in simple piewice-constant functions that approximate the true shear stress distribution. An accurate continuous thickness distribution of the transverse shear stress can be obtained “a posteriori” in terms of the axial stress by integrating the equilibrium equations. This theory also provides good results for clamped supports.
+Tessler ''et al.'' <span id='citeF-22'></span>[[#cite-22|[22,23]]] have developed a refined zigzag (RZ) theory  starting from the standard Timoshenko kinematic assumptions. This allows one using <math display="inline">C^\circ </math> continuous interpolation for all the kinematic variables. Timoshenko beam theory also introduces naturally shear deformation effect for the homogeneous material case, which is advantageous for many problems. The zigzag functions chosen have the property of vanishing on the top and bottom surfaces of a laminate. A particular feature of this zigzag theory is that the transverse shear stresses are not required to be continuous at the layer interfaces. This results in simple piewice-constant functions that approximate the true shear stress distribution. An accurate continuous thickness distribution of the transverse shear stress can be obtained “a posteriori” in terms of the axial stress by integrating the equilibrium equations. This theory also provides good results for clamped supports.
 Gherlone et al. <span id='citeF-24'></span>[[#cite-24|[24]]] have  developed two and three-noded <math display="inline">C^\circ </math> beam elements based on the RZ theory for analysis of multilayered composite and sandwich beams. Locking-free elements are obtained by using  ''anisoparametric'' interpolations that are adapted to approximate the four independent kinematic variables  that model the beam deformation. A family of beam elements is achieved by imposing different constraints  on the original displacement approximation. The constraint conditions requiring a constant variation of the transverse shear force provide an accurate 2-noded beam element <span id='citeF-24'></span>[[#cite-24|[24]]].

← Older revision	Revision as of 14:14, 10 October 2018
(No difference)

@@ Line 1,664: / Line 1,664: @@
 <div id="cite-1"></div>
-'''[[#citeF-1|[1]]]'''  Timoshenko S.P. and Woinowsky-Krieger S., ''Theory of Plates and Shells''. McGraw-Hill, New York, 3rd Edition, 1959.
+[[#citeF-1|[1]]]  Timoshenko S.P. and Woinowsky-Krieger S., ''Theory of Plates and Shells''. McGraw-Hill, New York, 3rd Edition, 1959.
 <div id="cite-2"></div>
-'''[[#citeF-2|[2]]]'''  Timoshenko S. On the correction for shear of differential equations for transverse vibrations of prismatic bars. Philosophical Magazine Series, Vol. 41, pp. 744-746, 1921.
+[[#citeF-2|[2]]]  Timoshenko S. On the correction for shear of differential equations for transverse vibrations of prismatic bars. Philosophical Magazine Series, Vol. 41, pp. 744-746, 1921.
 <div id="cite-3"></div>
-'''[[#citeF-3|[3]]]'''  Liu D. and Li X. An overall view of laminate theories based on displacement hypothesis. Journal of Composite Materials, Vol. 30(14), pp. 1539-1561, 1996.
+[[#citeF-3|[3]]]  Liu D. and Li X. An overall view of laminate theories based on displacement hypothesis. Journal of Composite Materials, Vol. 30(14), pp. 1539-1561, 1996.
 <div id="cite-4"></div>
-'''[[#citeF-4|[4]]]'''  Reddy, J.N., Mechanics of laminated composite plates and shells. Theory and analysis. 2nd Edition, CRC Press, Boca Raton, 2004.
+[[#citeF-4|[4]]]  Reddy, J.N., Mechanics of laminated composite plates and shells. Theory and analysis. 2nd Edition, CRC Press, Boca Raton, 2004.
 <div id="cite-5"></div>
-'''[[#citeF-5|[5]]]'''  Owen D.R.J.  and  Li Z.H., A refined analysis of laminated plates by finite element displacement methods. Part I. Fundamentals and static analysis. II Vibration and stability.  Comp. Struct, Vol. 26, pp. 907&#8211;923, 1987.
+[[#citeF-5|[5]]]  Owen D.R.J.  and  Li Z.H., A refined analysis of laminated plates by finite element displacement methods. Part I. Fundamentals and static analysis. II Vibration and stability.  Comp. Struct, Vol. 26, pp. 907&#8211;923, 1987.
 <div id="cite-6"></div>
-'''[[#citeF-6|[6]]]'''  Botello, S. Oñate, E. and Canet, J.M., A layer-wise triangle for analysis of laminated composite plates and shells. Computers and Structures, Vol. 70, 635&#8211;646, 1999.
+[[#citeF-6|[6]]]  Botello, S. Oñate, E. and Canet, J.M., A layer-wise triangle for analysis of laminated composite plates and shells. Computers and Structures, Vol. 70, 635&#8211;646, 1999.
 <div id="cite-7"></div>
-'''[7]'''  Di Sciuva M. Bending, vibration and buckling of simply supported thick multilayered orthotropic plates: an evaluation of a new displacement model. Journal of Sound and Vibration, Vol. 105, pp. 425-442, 1986.
+[[#citeF-7|[7]]]  Di Sciuva M. Bending, vibration and buckling of simply supported thick multilayered orthotropic plates: an evaluation of a new displacement model. Journal of Sound and Vibration, Vol. 105, pp. 425-442, 1986.
 <div id="cite-8"></div>
-'''[8]'''  Murakami H. Laminated composite plate theory with improved in-plane responses. ASME Journal of Applied Mechanics, Vol. 53, pp. 661-666, 1986.
+[[#citeF-8|[8]]]  Murakami H. Laminated composite plate theory with improved in-plane responses. ASME Journal of Applied Mechanics, Vol. 53, pp. 661-666, 1986.
 <div id="cite-9"></div>
-'''[[#citeF-9|[9]]]'''  Aitharaju V.R. and Averill R.C., An assessment of zig-zag kinematic displacement models for the anlaysis of laminated composites. Mech. Composite Mat. and Struct., 6, 1&#8211;26, 1999a.
+[[#citeF-9|[9]]]  Aitharaju V.R. and Averill R.C., An assessment of zig-zag kinematic displacement models for the anlaysis of laminated composites. Mech. Composite Mat. and Struct., 6, 1&#8211;26, 1999a.
 <div id="cite-10"></div>
-'''[[#citeF-10|[10]]]'''  Aitharaju V.R. and Averill R.C., C<math display="inline">^\circ </math> zig-zag finite element for analysis of laminated composite beams. J. Engineering Mechanics, ASCE, 323&#8211;330, 1999b.
+[[#citeF-10|[10]]]  Aitharaju V.R. and Averill R.C., C<math display="inline">^\circ </math> zig-zag finite element for analysis of laminated composite beams. J. Engineering Mechanics, ASCE, 323&#8211;330, 1999b.
 <div id="cite-11"></div>
-'''[11]'''  Li X. and Liu D. An interlaminar shear stress continuity theory for both thin and thick composite laminates. J. Appl. Mech., 59, 502&#8211;509, 1992.
+[[#citeF-11|[11]]]  Li X. and Liu D. An interlaminar shear stress continuity theory for both thin and thick composite laminates. J. Appl. Mech., 59, 502&#8211;509, 1992.
 <div id="cite-12"></div>
-'''[12]'''  Di Sciuva M. An improved shear-deformation theory for moderately thick multilayered anisotropic shells and plates. J. Appl. Mech. 54, 589&#8211;594, 1987.
+[[#citeF-12|[12]]]  Di Sciuva M. An improved shear-deformation theory for moderately thick multilayered anisotropic shells and plates. J. Appl. Mech. 54, 589&#8211;594, 1987.
 <div id="cite-13"></div>
-'''[13]'''  Toledano A and Murakami H. A higher-order laminate plate theory with improved in-plane response. Int. J. Solids and Struct., 23, 111&#8211;131, 1987b.
+[[#citeF-13|[13]]]  Toledano A and Murakami H. A higher-order laminate plate theory with improved in-plane response. Int. J. Solids and Struct., 23, 111&#8211;131, 1987b.
 <div id="cite-14"></div>
-'''[14]'''  Cho M. and Parmerter R.R. Efficient higher order composite plate theory for general laminations configuration. AIAA J., 31, 1299&#8211;1306, 1993.
+[[#citeF-14|[14]]]  Cho M. and Parmerter R.R. Efficient higher order composite plate theory for general laminations configuration. AIAA J., 31, 1299&#8211;1306, 1993.
 <div id="cite-15"></div>
-'''[15]'''  Kapuria, S., Dumir, P.C., Ahmed, A. and Alam, N. Finite element model of efficient zigzag theory for static analysis of hybrid piezoelectric beams, Computational Mechanics,  Vol. 34 (6), pp. 475&#8211;483, 2004.
+[[#citeF-15|[15]]]  Kapuria, S., Dumir, P.C., Ahmed, A. and Alam, N. Finite element model of efficient zigzag theory for static analysis of hybrid piezoelectric beams, Computational Mechanics,  Vol. 34 (6), pp. 475&#8211;483, 2004.
 <div id="cite-16"></div>
-'''[[#citeF-16|[16]]]'''  Averill R.C. Static and dynamic response of moderately thick laminated beams with damage. Composites Engineering, Vol. 4(4), pp. 381-395, 1994.
+[[#citeF-16|[16]]]  Averill R.C. Static and dynamic response of moderately thick laminated beams with damage. Composites Engineering, Vol. 4(4), pp. 381-395, 1994.
 <div id="cite-17"></div>
-'''[[#citeF-17|[17]]]'''  Averill R.C. and Yuen Cheong Yip. Development of simple, robust finite elements based on refined theories for thick laminated beams. Computers & Structures, Vol. 59(3), pp. 529-546, 1996.
+[[#citeF-17|[17]]]  Averill R.C. and Yuen Cheong Yip. Development of simple, robust finite elements based on refined theories for thick laminated beams. Computers & Structures, Vol. 59(3), pp. 529-546, 1996.
 <div id="cite-18"></div>
-'''[[#citeF-18|[18]]]'''  Alam N.M. and Upadhyay N.Kr. Finite element analysis of laminated composite beams for zigzag theory using MATLAB. Int. J. of Mechanics and Solids, Vol. 5(1), pp. 1&#8211;14, 2010.
+[[#citeF-18|[18]]]  Alam N.M. and Upadhyay N.Kr. Finite element analysis of laminated composite beams for zigzag theory using MATLAB. Int. J. of Mechanics and Solids, Vol. 5(1), pp. 1&#8211;14, 2010.
 <div id="cite-19"></div>
-'''[[#citeF-19|[19]]]'''  Savoia M. On the accuracy of one-dimensional models for multilayered composite beams. Int. J. Solids Struct.,  Vol. 33, pp. 521-44, 1996.
+[[#citeF-19|[19]]]  Savoia M. On the accuracy of one-dimensional models for multilayered composite beams. Int. J. Solids Struct.,  Vol. 33, pp. 521-44, 1996.
 <div id="cite-20"></div>
-'''[[#citeF-20|[20]]]'''   Kapuria S, Dumir P.C. and Jain N.K. Assessment of zigzag theory for static loading, buckling, free and forced response of composite and sandwich beams. Composite Structures,  Vol. 64, pp. 317-27, 2004.
+[[#citeF-20|[20]]]   Kapuria S, Dumir P.C. and Jain N.K. Assessment of zigzag theory for static loading, buckling, free and forced response of composite and sandwich beams. Composite Structures,  Vol. 64, pp. 317-27, 2004.
 <div id="cite-21"></div>
-'''[21]'''  Carrera, E. Historical review of zigzag theories for multilayered plate and shell. Applied Mechanics Review, 56(3), 287&#8211;308, 2003.
+[[#citeF-21|[21]]]  Carrera, E. Historical review of zigzag theories for multilayered plate and shell. Applied Mechanics Review, 56(3), 287&#8211;308, 2003.
 <div id="cite-22"></div>
-'''[[#citeF-22|[22]]]'''  Tessler, A., Di Sciuva M. and Gherlone, M. A refined zigzag beam theory for composite and sandwich beams. J. of Composite Materials, Vol. 43, 1051&#8211;1081, 2009.
+[[#citeF-22|[22]]]  Tessler, A., Di Sciuva M. and Gherlone, M. A refined zigzag beam theory for composite and sandwich beams. J. of Composite Materials, Vol. 43, 1051&#8211;1081, 2009.
 <div id="cite-23"></div>
-'''[23]'''  Tessler, A., Di Sciuva M. and Gherlone, M. A consistent refinement of first-order shear-deformation theory for laminated composite and sandwich plates using improved zigzag kinematics. J. of Mechanics of Materials and Structures, 5(2), 341&#8211;367, 2010.
+[[#citeF-23|[23]]]  Tessler, A., Di Sciuva M. and Gherlone, M. A consistent refinement of first-order shear-deformation theory for laminated composite and sandwich plates using improved zigzag kinematics. J. of Mechanics of Materials and Structures, 5(2), 341&#8211;367, 2010.
 <div id="cite-24"></div>
-'''[[#citeF-24|[24]]]'''  Gherlone, M., Tessler, A. and Di Sciuva M. C<math display="inline">^\circ </math> beam element based on the refined zigzag theory for multilayered composite and sandwich laminates. Composite Structures. Accepted for publication, May 2011.
+[[#citeF-24|[24]]]  Gherlone, M., Tessler, A. and Di Sciuva M. C<math display="inline">^\circ </math> beam element based on the refined zigzag theory for multilayered composite and sandwich laminates. Composite Structures. Accepted for publication, May 2011.
 <div id="cite-25"></div>
-'''[[#citeF-25|[25]]]'''  Oñate, E., Eijo, A. and Oller, S. Two-noded beam element for composite and sandwich beams using Timoshenko theory and refined zigzag kinematics. Publication CIMNE PI346, CIMNE, Barcelona, October 2010.
+[[#citeF-25|[25]]]  Oñate, E., Eijo, A. and Oller, S. Two-noded beam element for composite and sandwich beams using Timoshenko theory and refined zigzag kinematics. Publication CIMNE PI346, CIMNE, Barcelona, October 2010.
 <div id="cite-26"></div>
-'''[[#citeF-26|[26]]]'''  Di Sciuva M. and Gherlone, M. A global/local third-order Hermitian displacement field with damaged interfaces and transverse extensibility: FEM formulation. Composite Structures, 59(4), 433-444, 2003.
+[[#citeF-26|[26]]]  Di Sciuva M. and Gherlone, M. A global/local third-order Hermitian displacement field with damaged interfaces and transverse extensibility: FEM formulation. Composite Structures, 59(4), 433-444, 2003.
 <div id="cite-27"></div>
-'''[[#citeF-27|[27]]]'''  Di Sciuva M. and Gherlone, M. Quasi-3D static and dynamic analysis of undamaged and damaged sandwich beams. Journal of Sandwich Structures & Materials, 7(1), 31&#8211;52, 2005.
+[[#citeF-27|[27]]]  Di Sciuva M. and Gherlone, M. Quasi-3D static and dynamic analysis of undamaged and damaged sandwich beams. Journal of Sandwich Structures & Materials, 7(1), 31&#8211;52, 2005.
 <div id="cite-28"></div>
-'''[[#citeF-28|[28]]]'''  Oñate, E. ''Structural Analysis with the Finite Element Method''. Vol. 2 Beams, Plates and Shells. CIMNE-Springer, 2011.
+[[#citeF-28|[28]]]  Oñate, E. ''Structural Analysis with the Finite Element Method''. Vol. 2 Beams, Plates and Shells. CIMNE-Springer, 2011.
 <div id="cite-29"></div>
-'''[[#citeF-29|[29]]]'''  Oñate, E., Eijo, A. and Oller, S. Modeling of delamination in composite laminated beams using a two-noded beam element based in refined zigzag theory. Publication CIMNE, PI367, November 2011.
+[[#citeF-29|[29]]]  Oñate, E., Eijo, A. and Oller, S. Modeling of delamination in composite laminated beams using a two-noded beam element based in refined zigzag theory. Publication CIMNE, PI367, November 2011.

@@ Line 1,657: / Line 1,657: @@
 We have presented a simple and accurate 2-noded beam element based on the combination of the refined zigzag beam theory proposed by Tessler ''et al.'' <span id='citeF-22'></span>[[#cite-22|[22]]]. The element has four degrees of freedom per node (the axial displacement, the deflection, the rotation and the amplitude of the zigzag function). A standard <math display="inline">C^\circ </math> linear interpolation  is used for all variables. The resulting LRZ beam element is shear locking-free and has shown an excellent behaviour for analysis of thick and thin composite laminated beams with clamped and simple supported conditions. Numerical results  agree in practically all cases with  those obtained with a two-dimensional plane-stress FEM using a far larger number of degrees of freedom. It is remarkable that the zigzag distribution of the axial displacement and the axial stress across the thickness, typical of composite laminated beams, is very accurately captured with the basic approximation chosen. The possibilities of the new LRZ beam element for predicting delamination effects has been demonstrated in a simple but representative example of application.
-==9 ACKNOWLEDGEMENTS==
+==ACKNOWLEDGEMENTS==
 This research was partially supported by project SEDUREC of the Consolider Programme of the Ministerio de Educación y Ciencia of Spain.
-===BIBLIOGRAPHY===
+==REFERENCES==
 <div id="cite-1"></div>

@@ Line 1,047: / Line 1,047: @@
 {| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
 |-
-|[[File:Draft_Samper_624436055_2824_Fig8.jpg]]
+|[[File:Draft_Samper_624436055_2824_Fig8.jpg|420px|]]
 |- style="text-align: center; font-size: 75%;"
 | colspan="1" | '''Figure 8:''' Non symmetric 3-layered cantilever thick beam under end point load (<math>\lambda =5</math>). Distribution of the vertical deflection <math>w</math> for different theories and meshes