Published in Int. Journal for Numerical Methods in Engineering Vol. 83 (2), pp.196227, 2010
doi: 10.1002/nme.2836
The paper describes a methodology for extending rotationfree plate and beam elements in order to accounting for transverse shear deformation effects. The ingredients for the element formulation are: a HuWashizu type mixed functional, a linear interpolation for the deflection and the shear angles over standard finite elements and a finite volume approach for computing the bending moments and the curvatures over a patch of elements. As a first application of the general procedure we present an extension of the 3noded rotationfree basic plate triangle (BPT) originally developed for thin plate analysis to accounting for shear deformation effects of relevance for thick plates and composite laminated plates. The nodal deflection degrees of freedom (DOFs) of the original BPT element are enhanced with the two shear deformation angles. This allows to computing the bending and shear deformation energies leading to a simple triangular plate element with 3 DOFs per node (termed BPT+ element). For the thin plate case the shear angles vanish and the element reproduces the good behaviour of the original thin BPT element. As a consequence the element is applicable to thick and thin plate situations without exhibiting shear locking effects. The numerical solution for the thick case can be found iteratively starting from the deflection values for the Kirchhoff theory using the original thin BPT element. A 2noded rotationfree beam element termed CCB+ applicable to slender and thick beams is derived as a particular case of the plate formulation. The examples presented show the robustness and accuracy of the BPT+ and the CCB+ elements for thick and thin plate and beam problems.
Keywords: Rotationfree triangle, rotationfree beam, thick and thin plates and beams, finite elements, shear deformation
In past years some authors have derived a number of thin plate and shell triangular elements free of rotational degrees of freedom (DOFs) based on Kirchhoff theory [1]–[24]. This idea goes back to the original attempts to solve thin plate bending problems using finite difference schemes with the deflection as the only nodal variable [25]–[27]. In essence all methods attempt to express the curvature field over an element in terms of the displacements of a collection of nodes belonging to a patch of adjacent elements. Oñate and Cervera [6] proposed a general procedure of this kind combining finite element and finite volume concepts for deriving thin plate triangles and quadrilaterals with the deflection as the only nodal variable and presented a simple and competitive rotationfree three DOFs triangular element termed BPT (for Basic Plate Triangle). These ideas were extended in [12] to derive a number of rotationfree thin plate and shell triangles. The basic ingredients of the method are a mixed HuWashizu formulation, a standard discretization into threenoded triangles, a linear finite element interpolation of the displacement field within each triangle and a finite volume type approach for computing constant curvature and bending moment fields within appropriate nonoverlapping control domains. The so called cellcentered and cellvertex triangular domains yield different families of rotationfree plate and shell triangles. Both the BPT plate element and its extension to shell analysis (termed BST for Basic Shell Triangle) can be derived from the cellcentered formulation. Here the control domain is an individual triangle. The constant curvature field within a triangle is computed in terms of the displacements of the six nodes belonging to the four element patch formed by the chosen triangle and the three adjacent triangles. In the cellvertex approach the control volume is the tributary domain of a node and this leads to a different family of rotationfree plate and shell triangles. The detailed derivation of the different rotationfree triangular plate and shell element families can be found in [12]. The threenoded BST element has been successfully extended to nonlinear shell problems involving frictionalcontact situations and dynamics [15,19,20]. Practical applications of the BST element to sheet stamping analysis are reported in [9,10,11,24].
The paper describes an extension of the original rotationfree thin BPT element to accounting for transverse shear deformation effects of relevance for thick plates and composite laminated plates. The nodal deflection DOFs of the original BPT element are enhanced with the two shear deformation angles. This allows to computing the bending and shear deformation energies leading to a simple triangular plate element with 3 DOFs per node (termed BPT+ element). For the thin plate case the shear angles vanish and the element reproduces the good behaviour of the original thin BPT element. As a consequence the element is applicable to thick and thin plate situations without exhibiting shear locking. It is interesting that the thick plate solution can be found iteratively starting from the deflection values obtained using the standard Kirchhoff theory and the original thin BPT element. The ingredients of the formulation are: a HuWashizu type mixed functional, a linear interpolation for the deflection and the shear angles over 3noded triangles and a finite volume approach for computing the bending moments and the curvatures over a patch of elements. Details of the element formulation are given in the paper.
The rotationfree formulation described for the BPT+ element is taken as the starting point for deriving a twonoded rotationfree beam element with shear deformation effects (termed CCB+ element). The examples presented in the last part of the paper show the robustness and accuracy of the BPT+ and the CCB+ elements for thick and thin plate and beam problems.
Let us consider the plate of Figure 1. We will assume ReissnerMindlin conditions to hold, i.e.

(1) 
with

(2.a) 
or

(2.b) 
with

(2.c) 
where are the cartesian displacements , are the rotations and , are angles coinciding with the transverse shear deformations as shown below.
Figure 1: Sign convenion for the deflection and the rotations in a plate 
The generalized bending and shear strain vectors are defined as [29,30]

(3.a) 

(3.b) 
where and denote the bending strains (typically called curvatures) and the transverse shear strains, respectively and

(4) 
Substituting Eq.(2.a) into (3.b) gives

(5) 
i.e. the angles , coincide (with opposite sign) with the transverse shear deformations. In the following and will be called“shear angles”.
The set of governing equations can be expressed in integral form starting from the following HuWashizu type functional [12,28]

(6) 
where is the distributed vertical loading, are distributed bending moments, is the vertical force acting at point , is the number of points with external concentrated forces and is the area of the plate. Concentrated bending moments have been excluded for simplicity.
In Eq.(6) is the bending moment vector, is the shear force vector and is the bending constitutive matrix given by (for the isotropy case)

(7.a) 
The shear forces and the shear strains are assumed to be related pointwise by the standard constitutive equation

(7.b) 
In Eqs.(7) is the plate thickness and and are the Young modulus, the Poisson's ratio and the shear modulus, respectively and is the shear correction factor; is taken for an isotropic plate.
Let us express Eqs.(3)–(6) in terms of the deflection and the shear angles and . The resulting ReissnerMindlin theory contains the standard expressions of Kirchhoff thin plate theory plus additional transverse shear deformation terms.
Substituting Eq.(2) into Eqs.(3) gives

(8) 
with

(9) 
Substituting Eqs.(2.b) and (8) into the functional of Eq.(6) gives

(10) 
Note that by making the shear angles equal to zero we recover precisely the variational form of standard Kirchhoff thin plate theory [29].
Variation of with respect to , , and leads to the following equations:

(11) 

(12) 

These integral expressions, completed with the constitutive equation for the shear forces (Eq.(7.b)), are the basis for deriving the element equations.
Let us consider an arbitrary discretization of the plate into standard threenoded triangles. We assume a linear interpolation of the deflection and the shear angles within each element in terms of the nodal values in the standard manner [28]

(14) 
where are the linear shape functions of the threenoded triangle [28,30], denotes nodal variables and

(15.a) 

(15.b) 
with

(15.c) 
and is the element area [28]. In the above expressions and in the following superindex denotes element values.
Note that the introduction of the linear approximation for into the variational form (13a) will give a vanishing of the virtual bending energy term involving second derivatives of . This problem is overcome by “relaxing” further the weak form using an assumed constant value for the curvatures and bending moments over the triangle and computing the curvatures from the integral of the slopes along the element sides as explained next.
We describe the bending moments and the curvatures (and their virtual values) by constant fields within the triangle, i.e.

(16) 
where denotes constant values within the triangle.
We define the term “patch of triangular elements”. This is typically formed by four elements: a central triangle and the three adjacent triangles. Central triangles adjacent to a boundary lack the element adjacent to the boundary side (Figure 2).
Eqs.(16) are introduced into the governing integral equations (11)–(13) which are modified as follows.

(17) 
Recalling that the virtual curvatures are arbitrary we obtain the following constitutive equations for the assumed (constant) bending moments for each element

(18) 
where is the average bending constitutive matrix for the triangular element.
Substituting Eqs.(16) into Eq.(12) gives

(19) 
(a)  (b) 
Figure 2: Triangular element patches. (a) Standard four element patch; (b) Three element boundary patch. Numbers 1,2,3 in brackets denote local node numbers for the element 
As the virtual bending moments are arbitrary and is constant within each element we get

(20) 
The term in Eq.(20) is integrated by parts to give

(21) 
where

(22) 
and are the components to the outward unit normal to the boundary of the element (Figure 2). The transformation of the integral of the curvature field over the element domain into the integral of the deflection gradient along the element boundary is a distinct feature of the BPT formulation [6,12,15,19,20].
Eq.(21) defines the bending strains as the sum of the integral of the deflection gradient along the boundary of the element and the integral over the element of the term including the gradients of the shear deformation angles. This term is constant within each element for a linear interpolation of .
Making (Eq.(8)) and substituting the approximation for of Eq.(14) into Eq.(7.b) gives

(23) 
Integrating by parts the first integral in Eq.(13.a) and recalling that the bending moments are constant within each element and the additive property of the element integrals in the FEM, allows us to write Eqs.(13) as

The sum in above expressions extends over all the elements in the mesh.
Substituting the linear interpolation for and of Eqs.(14) into the integral expressions for the constant bending strain field of Eq.(21) gives

(26.a) 

(26.b) 
and, therefore

(27) 
where and are generalized bending strain matrices and and are nodal deflection and nodal shear angles vectors for the element. The expression for is given in Eq.(15.b). The expression for is given below (Eq.(31)).
The computation of is straightforward from the second integral of Eq.(21) as

(28) 
Recalling that are linear shape functions gives (using Eq.(15.c))

(29) 
The computation of in Eq.(26.a) requires integrating the product of the components of the normal vector and the deflection slopes along the element sides. For the 3noded triangle this is not so straightforward as is discontinuous at the element sides for a linear approximation of . A simple way to overcoming this problem is to compute the deflection gradients at the



element sides as the average value of the gradients contributed by the two triangles adjacent to the side [6,12]. This gives

(30) 
with

(31) 
The sum in Eq.(30) extends over the three sides of an element , is the transformation matrix of Eq.(22) for side , are the lengths of the element sides and superindex refers to each of the three triangles adjacent to the central triangle with for (Figure 2). Sides are assigned the number of the opposite node. Thus, side is opposite to node 3, etc.
Note that is a matrix relating the three bending strains with the deflections at the six nodes of the patch of triangles linked to element . The explicit form of matrix is given in Box I. This matrix coincides with the curvature matrix of the original BPT element, as presented in [12].
Substituting Eq.(27) into (18) gives the relationship between the bending moments and the nodal values of the deflection and the shear angles for the element as

(32) 
Substituting Eqs.(14), (23), (26) and (32) into the equilibrium equations (24) and (25) yields

(33.a) 

(33.b) 
After simplification of the virtual nodal deflections and the virtual nodal shear angles we obtain finally the matrix system of equilibrium equations

(34) 
where vectors and contain the nodal deflections and the nodal shear angles for the whole mesh and the rest of the matrices and vectors are assembled from the element contributions given by

(35) 
The element stiffness matrices and can be explicitly computed from the strain and constitutive matrices for the element. The exact computation of the integral in the expression of requires a 3 Gauss point quadrature. Excellent results have been obtained in all examples analyzed using a reduced one point integration rule for which indicates no advantage in using a full quadrature for . The simple one point quadrature allows computing all the element stiffness matrices in Eq.(34) explicitly.
The equivalent nodal force vectors for the element are

(36) 
For a uniform distribution of and

(37) 
where and are given in Eq.(15).
The vertical load acting at a node contributes the following terms to the th component of the global force vector

(38) 
The reactions (the vertical force and the bending moment) can be computed at the prescribed nodes from the nodal displacement and rotations field in the usual manner.
Remark 1 . Each node has three DOFs: the deflection and the two shear angles and at the node. The size of all the element stiffness matrices in Eqs.(35) is . This means that the effective DOFs for each element are in fact the nodal deflections of the patch of four elements assigned to each element (typically six nodal deflections except for boundary elements) and the six nodal shear angles, i.e. two shear angles for each of the three nodes of the element.
Despite that the solution for the and variables can be found simultaneously by solving Eqs.(34), the following iterative algorithm is recommended for computing and

where superindex denotes the number of iterations. The iterative scheme of Eqs.(39) continues until convergence for the nodal deflection and the nodal shear deformation angles is achieved. Convergence is typically measured by the norm of vectors and (Eqs.(58) and (59)). An advantage of the above iterative scheme is that for and the value of corresponds to the thin plate solution of Kirchhoff theory which is accurate enough for many practical cases. The effect of shear deformation is introduced progressively with the number of iterations. Shear deformation effects are negligible for thin plates and hence the variables tend rapidly to zero in this case.
An enhanced iterative scheme can be devised by performing a smoothing of the shear angles field as follows.
Step 1 Computation of the nodal deflections ̄w¹

(40) 
Step 2 Compute ̄ϕⁱ, i ≥ 1

(41) 
Step 3 Compute the element shear angles ̄ϕ^{e}

(42) 
where are mean shear angles for element for the th iteration and are the global numbers of the three nodes of element .
Step 4 Compute the smoothed nodal shear angles ̃ϕₖ
The smoothed nodal values are computed as

(43) 
where is the number of elements sharing the node with global number and is the total number of nodes in the mesh.
Step 5 Compute ̄wⁱ, i > 1

(44) 
Return to step 2.
Convergence of the improved iterative scheme for the deflection field is quite fast (2–4 iterations), even for thick plates (see Section 6).
The accuracy and efficiency of the so called BPT+ triangle is shown in the examples presented in the paper.
A BPT element with a side along a boundary edge has one of the triangles belonging to the patch missing (Figure 2). This is taken into account by ignoring the contribution of this element when performing the average of the deflection gradient in Eq.(30) [12,15,19,20].
The condition is directly imposed at the edge nodes at the solution level when solving the system of equations for in the standard manner.
The condition of zero rotations at a clamped edge introduces additional terms in the system of Eqs.(34).
In order to explain the process, let us consider for example a clamped edge corresponding to side 3 linking nodes 1 and 2 (with global numbers and ) in the boundary element of Figure 2b. At the clamped edge

(45) 
where are the rotations at the edge mid point.
The shear angles at the edge mid point are approximated by the average of the nodal values along the edge, i.e.

(46) 
Introducing Eqs.(45) and (46) into the definition of matrix in Eq.(30) gives

(47) 
where is obtained by disregarding the contributions from the clamped side in the sum along the element sides in the expression of Box I and

(48) 
The bending strain field of Eq.(27) is now modified as

(49) 
where

(50) 
The new matrix substitutes matrix in the expression for of Eq.(35).
We recall that the above modifications are only needed for elements with a clamped edge.
The process is repeated twice if the element has two clamped edges.
The condition at the nodes laying on a simply supported (SS) edge is prescribed when solving the global system of equations as for the clamped case. Prescribing at an edge node also automatically implies that is zero along the SS edge direction and this is the option taken for the standard rotationfree thin BPT element. Note that this is equivalent to assuming a “soft” simply support condition in general plate theory (i.e. at the SS edge) [29,30].
The “hard” support condition requires prescribing the tangential rotation to a zero value. This means

(51) 
Prescribing at the SS nodes gives automatically and hence the nodal variable must be also prescribed to a zero value at these nodes. This condition can be imposed by transforming the cartesian shear strains to the boundary axes and making at each simply supported boundary node.
Indeed, prescribing just at the support nodes and letting free reproduces the soft support condition for ReissnerMindlin theory. This is the approach chosen in the examples presented in the paper ( Figures 5, 6, 9, 10).
It is interesting that the BPT element does not suffer from the difficulties associated to prescribing the tangential rotation at SS nodes which occur in standard thin plate elements [29,32,33]. This is due to the fact that the rotations do not appear explicitly as variables in the BPT formulation and the SS condition is directly imposed by prescribing the deflection at the support nodes.
The condition of zero normal rotation () is imposed by neglecting the contributions from the normal rotation at the symmetry edge when computing Eq.(30). The condition for the thick case introduces additional terms in the stiffness equations. The procedure is identical as explained for the clamped edge.
Remark 2 . The condition can also be imposed at free edge nodes. This however has not shown to be improve the quality of the solution or to lead to any computational advantage, other that the reduction in the number of DOFs, in those cases.
A twonoded rotationfree beam element with shear deformation effects can be simply derived as a particular case of the formulation for the BPT element previously described. The resulting beam element is termed CCB+ as a reference to the cellcentered approach used to compute the curvature at the element center.
Figure 3 shows the patch of three beam elements needed for computing the constant curvature at the central element using a cellcentered finitevolume type scheme. The starting point in the formulation of the CCB+ element is the standard Timoshenko beam theory [29,30]. The relevant expressions are:

(52) 

(53) 
where and are the inertia modulus and the area of the transverse cross section of the beam and the rest of the terms have been defined previously.
The HuWashizu functional for the beam has the same form as in Eq.(10) with .
Similarly, the governing equations have identical expressions as for the plate problem Eqs.(11–13).
A linear interpolation is chosen for the deflection and the shear angle as

(54) 
Figure 3: (a) Patch of three rotationfree CCB+ elements. (b) Patch of element adjacents to a clamped boundary 
where with and are the standard linear shape functions for the 2noded Lagrange element [28,30].
Following an identical process as for the 3noded BPT+ element, the same system of Eqs.(34) is found. The expressions for the stiffness matrices , and coincide with those given in Eq.(35) simply by noting that the integrals change from area to line ones and substituting the area by the element length . The form of matrices and is given in Box II.
The expressions for the equivalent force vector are similar to Eqs.(36)–(38), substituting the area integrals by integrals along the element length. For a uniform distribution of and

(55) 
where is the value of the natural coordinate at node .
The procedure for imposing the boundary conditions follows precisely the lines described for the BPT+ element in Section 4. The matrix for an element with a clamped node is (Eq.(48) and Figure 3b)

(56) 

(57) 
A similar expression is obtained for an element with a node on the symmetry axis.
The CCB+ element is an extension of the 2noded rotationfree CCB element based on EulerBernouilli beam theory. A description of the CCB element can be found in 11,30.
The efficiency and accuracy of the BPT+ element has been tested in the analysis of a number of square plates of side and circular plates of diameter under a uniformly distributed loading and a central point load for different thicknesses ranging from to (very thin plate) to (thick plate). Simply supported and clamped boundary conditions have been considered. For rectangular plates the soft SS condition has been assumed. For circular plates is automatically zero at the SS boundary for the problems studied due to symmetry and hence results correspond to the hard SS case. Figure 4 shows some of the different meshes used for the analysis.
Results for the cases studied using the enhanced iterative scheme of Section 3.4 are presented in Figures 5–12. Each figure shows:

(58) 

(59) 
For the point load case the analytical value for the deflection under the load given by thick plate theory is infinity. Hence, results for the deflection for thick plates are compared at the midpoint along a central line in this case.
The following conclusions are drawn from the examples:
Figure 4: Some of meshes of BPT+ elements used for analysis of square and circular plates 
SS (hard) square thick plate. Uniform load,  
Mesh 1  Mesh 2  Mesh 3  Mesh 4  Mesh 5  Mesh 6  
381,73  404,59  416,31  421,98  424,65  425,59  
0,893  0,947  0,974  0,988  0,994  0,996  
(Series solution): 427.28 [29,30]  
Convergence achieved in a maximum of 4 iterations for each mesh 
Indeed for problems with a discontinuous shear force field (such as the case of internal point forces), the nodal continuity of the shear strains which is intrinsic to the element formulation is a drawback to accurately capturing shear force jumps. This can be overcome by computing the shear forces at the center of the elements adjacent to the point load and extrapolating the solution within each element so as to reproduce the shear force jump at the common node.
The accuracy of the CCB+ beam element of Section 5 was tested in the analysis of simple supported and cantilever thick beams under distributed and point loads. Results of the study plotted in Figures 13–16 show the normalized distribution of the deflection along the beams and the normalized value of the central deflection (for the clamped beam) and the end deflection (for the cantilever beam) with the number of elements. Good results (error less than 10%) are obtained with a relatively coarse mesh (8 elements).
The distribution of the bending moment and the shear force along the beam is also plotted for the 40 element mesh. Results are practically coincident with the analytical values. Similar good behaviour was obtained for other thick and thin beam problems studied with the CCB+ element.
A methodology for extending the rotationfree plate and beam elements initially designed for thin/slender situations so as to account with shear deformation effects has been presented. The method allows one to introduce the effect of shear deformation in a progressive (iterative) manner starting from the initial thin solution.
The formulation is useful for analysis of plates and beams of a variety of thickness and materials. A particular interesting application of the new elements is the analysis of composite laminated plates and beams for which shear deformation effects are relevant.
The formulation is also suitable for implementing an adaptive solution scheme where the shear angles are introduced in structures (or zones of a structure) where the effect of shear deformation is relevant.
This research was partially supported by project SEDUREC of the Consolider Programme of the Ministerio de Educación y Ciencia of Spain.
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Published on 18/12/18
Submitted on 18/12/18
Licence: CC BYNCSA license