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| <math> \begin{array}{l} \displaystyle \mathbf{B}_w = \frac{1}{2 l^{a} l^{e} l^{b}} \left[ l^{b},-l^{b},-l^{a},l^{a} \right]\\[.3cm] \displaystyle \mathbf{B}_\phi = \frac{1}{l^{e}}[1,-1] \end{array}</math> | | <math> \begin{array}{l} \displaystyle \mathbf{B}_w = \frac{1}{2 l^{a} l^{e} l^{b}} \left[ l^{b},-l^{b},-l^{a},l^{a} \right]\\[.3cm] \displaystyle \mathbf{B}_\phi = \frac{1}{l^{e}}[1,-1] \end{array}</math> |
Published in Int. Journal for Numerical Methods in Engineering Vol. 83 (2), pp.196-227, 2010
doi: 10.1002/nme.2836
The paper describes a methodology for extending rotation-free plate and beam elements in order to accounting for transverse shear deformation effects. The ingredients for the element formulation are: a Hu-Washizu 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 3-noded rotation-free 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 2-noded rotation-free 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: Rotation-free triangle, rotation-free 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 rotation-free three DOFs triangular element termed BPT (for Basic Plate Triangle). These ideas were extended in [12] to derive a number of rotation-free thin plate and shell triangles. The basic ingredients of the method are a mixed Hu-Washizu formulation, a standard discretization into three-noded 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 non-overlapping control domains. The so called cell-centered and cell-vertex triangular domains yield different families of rotation-free 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 cell-centered 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 cell-vertex approach the control volume is the tributary domain of a node and this leads to a different family of rotation-free plate and shell triangles. The detailed derivation of the different rotation-free triangular plate and shell element families can be found in [12]. The three-noded BST element has been successfully extended to non-linear shell problems involving frictional-contact 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 rotation-free 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 Hu-Washizu type mixed functional, a linear interpolation for the deflection and the shear angles over 3-noded 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 rotation-free formulation described for the BPT+ element is taken as the starting point for deriving a two-noded rotation-free 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 Reissner-Mindlin conditions to hold, i.e.
|
(1) |
with
|
(2.a) |
or
|
(2.b) |
with
|
(2.c) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle u,v,w}
are the cartesian displacements Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \theta _x}
, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \theta _y}
are the rotations and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \phi _x}
, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \phi _y}
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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle [\kappa _x,\kappa _y,\kappa _{xy}]}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle [\gamma _{xz},\gamma _{yz}]} 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \phi _x} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \phi _y}
coincide (with opposite sign) with the transverse shear deformations. In the following Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \phi _x} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \phi _y} will be called“shear angles”.
The set of governing equations can be expressed in integral form starting from the following Hu-Washizu type functional [12,28]
|
(6) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle q}
is the distributed vertical loading, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol m}=[m_x,m_y]^T} are distributed bending moments, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle P_i} is the vertical force acting at point Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle i}
, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n_c}
is the number of points with external concentrated forces and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle A} is the area of the plate. Concentrated bending moments have been excluded for simplicity.
In Eq.(6) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol \sigma }_b = [M_x,M_y,M_{xy}]^T}
is the bending moment vector, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol \sigma }_s = [Q_x,Q_y]^T } is the shear force vector and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol D}_b} 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 point-wise by the standard constitutive equation
|
(7.b) |
In Eqs.(7) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t}
is the plate thickness and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle E,\nu } and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle G} are the Young modulus, the Poisson's ratio and the shear modulus, respectively and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \alpha } is the shear correction factor; Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \alpha =5/6} is taken for an isotropic plate.
Let us express Eqs.(3)–(6) in terms of the deflection Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle w}
and the shear angles Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \phi _x} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \phi _y}
. The resulting Reissner-Mindlin 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol \phi }}
equal to zero we recover precisely the variational form of standard Kirchhoff thin plate theory [29].
Variation of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Pi }
with respect to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol \varepsilon }_b}
, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol \sigma }_b} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle w}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol \phi }} 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 three-noded triangles. We assume a linear interpolation of the deflection Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle w}
and the shear angles Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol \phi }} within each element in terms of the nodal values in the standard manner [28]
|
(14) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N_i}
are the linear shape functions of the three-noded triangle [28,30], Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \bar{(\cdot )}} denotes nodal variables and
|
(15.a) |
|
(15.b) |
with
|
(15.c) |
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle A^e}
is the element area [28]. In the above expressions and in the following super-index Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle e} denotes element values.
Note that the introduction of the linear approximation for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle w}
into the variational form (13a) will give a vanishing of the virtual bending energy term involving second derivatives of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle w}
. 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol \sigma }_b}
and the curvatures Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol \varepsilon }_b } (and their virtual values) by constant fields within the triangle, i.e.
|
(16) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (\cdot )^e}
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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol D}_b^e}
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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol \varepsilon }_b^e}
is constant within each element we get
|
(20) |
The term Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol L}_w w}
in Eq.(20) is integrated by parts to give
|
(21) |
where
|
(22) |
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n_x,n_y}
are the components to the outward unit normal Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol n} to the boundary of the element Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Gamma ^e} (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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol L}_b {\boldsymbol \phi }}
including the gradients of the shear deformation angles. This term is constant within each element for a linear interpolation of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol \phi }
.
Making Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol \varepsilon }_s= -{\boldsymbol \phi }}
(Eq.(8)) and substituting the approximation for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol \phi }} 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle w}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol \phi } 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol B}_w}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol B}_{\phi }} are generalized bending strain matrices and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \bar {\boldsymbol w}^e} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \bar{\boldsymbol \phi }^e } are nodal deflection and nodal shear angles vectors for the element. The expression for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \bar{\boldsymbol \phi }^e} is given in Eq.(15.b). The expression for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \bar {\boldsymbol w}^e} is given below (Eq.(31)).
The computation of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol B}_\phi }
is straightforward from the second integral of Eq.(21) as
|
(28) |
Recalling that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N_i}
are linear shape functions gives (using Eq.(15.c))
|
(29) |
The computation of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol B}_w^e}
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 3-noded triangle this is not so straightforward as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol \nabla }w} is discontinuous at the element sides for a linear approximation of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle w}
. 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle e} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol T}_j}
is the transformation matrix of Eq.(22) for side Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle j}
, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle l^e_j}
are the lengths of the element sides and superindex Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p} refers to each of the three triangles Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle a,b,c} adjacent to the central triangle Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle e} with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p=a,b,c} for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle j=1,2,3} (Figure 2). Sides are assigned the number of the opposite node. Thus, side Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle l^e_3} is opposite to node 3, etc.
Note that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol B}_w}
is a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 3\times 6} matrix relating the three bending strains with the deflections at the six nodes of the patch of triangles linked to element Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle e}
. The explicit form of matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol B}_w}
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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \bar {\boldsymbol w}}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \bar {\boldsymbol \phi }} 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol K}_{w}^e}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol K}_{w\phi }^e} can be explicitly computed from the strain and constitutive matrices for the element. The exact computation of the integral in the expression of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol K}_\phi } requires a 3 Gauss point quadrature. Excellent results have been obtained in all examples analyzed using a reduced one point integration rule for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol K}_{\phi }} which indicates no advantage in using a full quadrature for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{K}_\phi }
. 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle q}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol m}}
|
(37) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle b_i^e}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle c_i^e} are given in Eq.(15).
The vertical load Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle P_i}
acting at a node Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle i} contributes the following terms to the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle i}
th component of the global force vector Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol f}_w}
|
(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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \bar w_i}
and the two shear angles Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \bar {\phi }_{x_i}} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \bar {\phi }_{y_i}} at the node. The size of all the element stiffness matrices in Eqs.(35) is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 6\times 6}
. 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \bar{\boldsymbol w}}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \bar {\boldsymbol \phi }} variables can be found simultaneously by solving Eqs.(34), the following iterative algorithm is recommended for computing Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \bar{\boldsymbol w}} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \bar {\boldsymbol \phi }}
|
where superindex Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle i}
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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle L_2} norm of vectors Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \bar{\boldsymbol w}} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \bar {\boldsymbol \phi }} (Eqs.(58) and (59)). An advantage of the above iterative scheme is that for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle i=1} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol \phi }^0={\boldsymbol 0}} the value of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \bar{\boldsymbol w}^1} 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \bar{\boldsymbol \phi }} 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 ̄ϕe
|
(42) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \bar{\boldsymbol \phi }^{e}}
are mean shear angles for element Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle e} for the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle i}
th iteration and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle a,b,c}
are the global numbers of the three nodes of element Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle e}
.
Step 4 Compute the smoothed nodal shear angles ̃ϕₖ
The smoothed nodal values Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tilde{\boldsymbol \phi }_k^i}
are computed as
|
(43) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n_k}
is the number of elements sharing the node with global number Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N} 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle w=0}
is directly imposed at the edge nodes at the solution level when solving the system of equations for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \bar {\boldsymbol w}} 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle i}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle j}
) in the boundary element of Figure 2b. At the clamped edge
|
(45) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol \theta }\vert _3^e}
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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol B}_w}
in Eq.(30) gives
|
(47) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol B}_w}
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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \bar{\boldsymbol B}_\phi }
substitutes matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol B}_\phi } in the expression for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol K}^e_\phi } 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle w=0}
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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle w=0} at an edge node also automatically implies that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\partial w\over \partial s}} is zero along the SS edge direction Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle s} and this is the option taken for the standard rotation-free thin BPT element. Note that this is equivalent to assuming a “soft” simply support condition in general plate theory (i.e. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle w=M_n=M_{ns}=0} at the SS edge) [29,30].
The “hard” support condition requires prescribing the tangential rotation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \theta _s}
to a zero value. This means
|
(51) |
Prescribing Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle w}
at the SS nodes gives automatically Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \left(\frac{\partial w}{\partial s} \right)_i=0} and hence the nodal variable Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \bar \phi _{s_i}} 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle s,n} and making Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \bar \phi _{s_i}=0} at each simply supported boundary node.
Indeed, prescribing just Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle w_i =0}
at the support nodes and letting Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \bar \phi _{s_i}} free reproduces the soft support condition for Reissner-Mindlin 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 (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \theta _n=0} ) is imposed by neglecting the contributions from the normal rotation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \frac{\partial w}{\partial n}}
at the symmetry edge when computing Eq.(30). The condition Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \theta _n=\frac{\partial w}{\partial n}+\phi _n= 0} 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \bar{\phi }_{n_i}=0}
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 two-noded rotation-free 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 cell-centered 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 cell-centered finite-volume 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) |
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \boldsymbol{Constitutive~equations:}
|
(53) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle I}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle A} 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 Hu-Washizu functional for the beam has the same form as in Eq.(10) with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol m} = [m]} .
Similarly, the governing equations have identical expressions as for the plate problem Eqs.(11–13).
A linear interpolation is chosen for the deflection Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle w}
and the shear angle Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \phi } as
|
(54) |
![]() |
Figure 3: (a) Patch of three rotation-free CCB+ elements. (b) Patch of element adjacents to a clamped boundary |
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \begin{array}{l} \displaystyle \mathbf{B}_w = \frac{1}{2 l^{a} l^{e} l^{b}} \left[ l^{b},-l^{b},-l^{a},l^{a} \right]\\[.3cm] \displaystyle \mathbf{B}_\phi = \frac{1}{l^{e}}[1,-1] \end{array} |
Box II. Matrices Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\boldsymbol B}^e_{w}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\boldsymbol B}^e_{\phi }for the 2-noded rotation-free CCB+ element
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N_i = 1/2 (1+\xi \xi _i)}
with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \xi _1=-1} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \xi _2=1} are the standard linear shape functions for the 2-noded Lagrange element [28,30].
Following an identical process as for the 3-noded BPT+ element, the same system of Eqs.(34) is found. The expressions for the stiffness matrices Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol K}_w^e} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol K}_\phi ^e}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol K}_{w_\phi }^e} coincide with those given in Eq.(35) simply by noting that the integrals change from area to line ones and substituting the area Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle A^e} by the element length Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle l^e}
. The form of matrices Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol B}_w}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol B}_\phi } 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle q}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle m}
|
(55) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \xi _i}
is the value of the natural coordinate Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \xi } at node Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle i}
.
The procedure for imposing the boundary conditions follows precisely the lines described for the BPT+ element in Section 4. The Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Delta{\boldsymbol B}_\phi }
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 2-noded rotation-free CCB element based on Euler-Bernouilli 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle L}
and circular plates of diameter Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 2L} under a uniformly distributed loading and a central point load for different thicknesses ranging from to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t/L =10^{-3}} (very thin plate) to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t/L= 0.1} (thick plate). Simply supported and clamped boundary conditions have been considered. For rectangular plates the soft SS condition has been assumed. For circular plates Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \phi _s} 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) |
is the number of nodes in the mesh and an upper index denotes the iteration number. A value of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \bar w_j^0=0} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \bar {\boldsymbol \phi }_j^0=\mathbf{0}} has been taken. For the examples considered, the iterative scheme stops when Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle L_2^w< 10^{-3}}
.
for uniform load and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle w_c = \frac{D\bar w_c\cdot 10^4}{PL^2}} for point load with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle D=\frac{Et^3}{12(1-\nu ^2)}}
) for the range of thickness ratios Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 10^{-3}-10^{-1}}
for each of the meshes considered. Results are compared with analytical and series values for the thin and thick cases [26,29,30,31] when available, or, alternatively, with FEM results obtained for the deflection at the center of the midle plane using a mesh of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 40\times 40 \times 6}eight-noded hexahedra in a quarter of plate.
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 mid-point along a central line in this case.
and the shear force Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Q_x} along the central line for the thick case (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t/L=0.10}
) for each of the five meshes considered. The isovalues of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle M_x}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Q_x} over a quarter of the plate are also shown for the finer mesh.
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, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): t/L =0.10
| ||||||
Mesh 1 | Mesh 2 | Mesh 3 | Mesh 4 | Mesh 5 | Mesh 6 | |
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): w_c | -381,73 | -404,59 | -416,31 | -421,98 | -424,65 | -425,59 |
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): w_c/w_c^a | 0,893 | 0,947 | 0,974 | 0,988 | 0,994 | 0,996 |
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): w_c^a
(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 rotation-free 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 01/01/2010
DOI: 10.1002/nme.2836
Licence: CC BY-NC-SA license
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