Abstract
This paper deals with the presentation of a general, variational principle as theoretical support for the development of simple and efficient triangular elements having only one displacement and two rotations at the corner nodes to model thin to thick plates based on the first order Reissner- Mindlin plate theory. The functional is a modified Hellinger–Reissner mixed expression in terms of the kinematic variables (transverse displacement w and rotations βx and βy), and independent transverse shear strains (γx and γy). The approximations of the five independent variables of the mixed formulation take into account the accumulated knowledge on existing performing 2D Timoshenko beams and triangular elements such as T3γs, MITC3, DKT, DST, DKMT. The present mixed variational support is useful, not only to give a unique theoretical support to the above existing elements, usually based on assumed natural strain formulations, but it allows also to propose new simple and efficient elements, here called BAK1, BAK2 and BAK3. The paper includes a detailed presentation of results of patch tests for very thin and very thick plates, for convergence of displacements, bending moments and shear forces, for clamped circular and simply supported square plates and for s-norm convergence tests considering regular and irregular meshes. Shear force distribution is also considered for situations with boundary layer effects.
Similar content being viewed by others
References
Batoz JL, Dhatt G (1990) Modélisation des structures par éléments finis, vol 2. Hermès, Paris
Cen S, Shang Y (2015) Developments of Mindlin–Reissner plate elements. Math Probl Eng 2015, Article ID 456740. https://doi.org/10.1155/2015/456740
Lee SW, Pian THH (1978) Improvement of plate and shell finite elements by mixed formulations. AIAA J 16(1):29–34
Batoz JL, Bathe KJ, Ho LW (1980) A study of three-node triangular plate bending elements. Int J Numer Methods Eng IJNME 15:1771–1812
Lee SW, Wong SC (1982) Mixed formulation finite elements for Mindlin theory plate bending. IJNME 18:1297–1311
Malkus DS, Hughes TJR (1985) Mixed finite element methods—reduced and selective integration techniques: a unification of concepts. Comput Methods Appl Mech Eng CMAME 15(1):63–81
Shimodaira H (1985) Equivalence between mixed models and displacement models using reduced integration. IJNME 21:89–104
Hughes TJR, Hinton E (1986) Finite elements methods for plate and shell structures, vol. 1, Element technology. Pineridge Press International, Swansea
Hughes TJR (1987) The finite element method: linear static and dynamic finite element analysis. Prentice Hall, Hoboken
Saleeb AF, Chang TY (1987) An efficient quadrilateral element for plate bending analysis. IJNME 24:1123–1155
Saleeb AF, Chang TY, Yingyeunyong S (1988) A mixed formulation of C° linear triangular plate/shell element. The role of edge shear constraints. IJNME 26:1101–1128
Zienkiewicz OC, Lefevre D (1988) A robust triangular plate bending element of the Reissner–Mindlin type. IJNME 26:1169–1184
Gellert M (1988) A new method for derivation of locking free plate bending finite elements via mixed/hybrid formulation. IJNME 26:1185–1200
Pinsky P, Jasti R (1989) A mixed finite element formulation for Reissner–Mindlin plates based on the use of bubble functions. IJNME 28:1677–1702
Zienkiewicz OC, Taylor R, Papadopoulos P, Onate E (1990) Plate bending elements with discrete constraints: new triangular elements. Comput Struct CS 35:505–522
Papadopoulos P, Taylor R (1990) A triangular element based on Reissner–Mindlin theory. IJNME 30:1029–1049
Xu Z (1992) A thick-thin triangular plate element. IJNME 33:963–973
Batoz JL, Katili I (1992) On a simple triangular Reissner/Mindlin plate element based on incompatible modes and discrete constraints. IJNME 3:1603–1632
Zienkiewicz OC, Xu Z, Zeng LF, Samuelsson A, Wiberg NE (1993) Linked interpolation for Reissner–Mindlin plate elements: part I—a simple quadrilateral. IJNME 36:3043–3056
Taylor R, Auricchio F (1993) Linked interpolation for Reissner–Mindlin plate elements: part II—a simple triangle. IJNME 36:3057–3066
Katili I (1993) A new discrete Kirchhoff–Mindlin element based on Mindlin-Reissner plate theory and assumed shear strain fields—part I: an extended DKT element for thick-plate bending analysis. IJNME 36:1859–1883
Katili I (1993) A new discrete Kirchhoff-Mindlin element based on Mindlin-Reissner plate theory and assumed shear strain fields—part II: an extended DKQ element for thick plate bending analysis. IJNME 36:1885–1908
Auricchio F, Taylor R (1995) A triangular thick plate finite element with an exact thin limit. Finite Elem Anal Des 19:57–68
Ayad R, Dhatt G, Batoz JL (1998) A new hybrid-mixed variational approach for Reissner–Mindlin plates: the MiSP Model. IJNME 42:1149–1179
Ayad R, Rigolot A, Talbi N (2001) An improved three-node hybrid-mixed element for Mindlin/Reissner plates. IJNME 51:919–942
Brasile S (2008) An isostatic assumed stress triangular element for the Reissner–Mindlin plate bending element. IJNME 74:971–995
Lee Y, Yoon K, Lee P-S (2012) Improving the MITC3 shell finite element by using the Hellinger–Reissner principle. Comput Struct 110–111:93–106
Hughes TJR, Taylor RL (1982) The linear triangle bending elements. In: Whiteman JR (ed) The mathematics of finite element and application IV, MAFELAP 1981. Academic Press, London, New York. ISBN 13: 9780127472546
Bathe KJ, Dvorkin EN (1985) A four nodes plate bending element based on Mindlin-Reissner plate theory and a mixed interpolation. IJNME 21:367–383
Brezzi F, Bathe KJ, Fortin M (1989) Mixed interpolated elements for Reissner–Mindlin plates. IJNME 28:1787–1801
Batoz JL, Ben Tahar M (1982) Evaluation of a new thin plate quadrilateral element. IJNME 18:1655–1678
Batoz JL, Lardeur P (1989) A discrete shear triangular nine dof element for the analysis of thick to very thin plates. IJNME 28:533–560
Lee PS, Bathe KJ (2004) Development of MITC isotropic triangular shell finite elements. Comput Struct 82:945–962
Lee Y, Lee PS, Bathe KJ (2014) The MITC3+ shell element and its performance. Comput Struct 138:12–23
Lee Y, Jeon HM, Lee PS, Bathe KJ (2015) The modal behavior of the MITC3+ triangular shell element. Comput Struct 153:148–164
Jun H, Mukai P, Kim S (2018) Benchmark tests of MITC triangular shell elements. Struct Eng Mech 68:17–38
Katili I, Batoz JL, Maknun IJ, Lardeur P (2018) A comparative formulation of DKMQ, DSQ and MITC4 quadrilateral plate elements with new numerical results based on s-norm tests. Comput Struct 204:48–64
Katili I, Maknun IJ, Batoz JL, Katili AM (2018) Asymptotic equivalence of DKMT and MITC3 elements for thick composite plates. Compos Struct 206:363–379
Katili AM, Maknun I, Katili I (2019) Theoretical equivalence and numerical performance of T3γs and MITC3 plate finite elements. Struct Eng Mech 69(5):527–536
Katili I, Maknun IJ, Batoz JL, Katili AM (2019) A comparative formulation of T3γs, DST, DKMT and MITC3+ triangular plate elements with new numerical results based on s-norm tests. Eur J Mech A Solids 78:15
Bletzinger KU, Bischoff M, Ramm E (2000) A unified approach for shear-locking-free triangular and rectangular shell finite elements. Comput Struct 75:321–334
Razzaque A (1973) Program for triangular bending elements with derivative smoothing. IJNME 6:633–643
Arnold DN, Falk RS (1989) Edge effects in Reissner–Mindlin plate theory. In: Noor AK, Belytschko T, Simo JC (eds) Analytical and computational models of shells. ASME, New York, pp 71–89
Kant T, Hinton E (1983) Mindlin plate analysis by segmentation method. J Eng Mech ASCE 109:537–556
Jirousek J, Bouberguig A (1984) A contribution to evaluation of shear forces and reactions of Mindlin plates by using isoprametric elements. Comp Struct 19:793–800
Shang Y, Cen S, Li C-F, Huang J-B (2015) An effective hybrid displacement function element method for solving the edge effect of Mindlin–Reissner plate. IJNME. https://doi.org/10.1002/nme.4843
Acknowledgements
The first author wants to thank his former Ph.D students, Dr. P. Lardeur and Dr. S. Bouabdallah, for their encouragements and the Université de Technologie de Compiègne for the delivery of the Emeritus Status since 2015. The third author wants to gratefully acknowledge the financial support from the Universitas Indonesia, Depok, Indonesia through the PUTI-KI program (NKB-780/UN2.RST/HKP.05.00/2020).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendices
Appendix A: 2D Timoskenko beam mixed FE model for an exact stiffness matrix
Consider a 2D beam finite element with \(H_{b} = E \cdot I\) (the bending rigidity), \(H_{s} = k \cdot G \cdot A\) (the shear rigidity), Query ID="Q4" Text="Please note that the Appendix section equations and their citations are renumbered to ensure sequential ordering." with \(E\) and \(G\) the Young and shear modulus, \(I\) moment of inertia, A cross section.
The internal strain energy, for an element of length \(L\) can be expressed in a mixed form as [1]:
with
\(w\): normal displacement along z; \(\beta\): rotation of the cross section (in \(xz\) plane).
\(\gamma = w_{^{\prime}x} + \beta\): transverse shear (TS) strain; \(T\): shear force (along \(z\)).
Without distributed load, the equilibrium equation along the \(z\) direction can be expressed as:
Hence \(T = T_{0}\) constant is sufficient as a \(C^{ - 1}\) approximation at the element level. Consider a quadratic expression for \(\beta \left( x \right)\), of the form:
with \(N_{1} = 1 - \frac{x}{L}\); \(N_{2} = \frac{x}{L}\); \(\Delta \beta\) is the quadratic contribution of rotation at mid-node.
The curvature \(\chi = \beta_{^{\prime}x}\) is then:
with \(\left\langle {B_{1}^{b} } \right\rangle = \left\langle {\begin{array}{*{20}c} 0 & { - \frac{1}{L}} & {\begin{array}{*{20}c} 0 & \frac{1}{L} \\ \end{array} } \\ \end{array} } \right\rangle\); \(B_{2}^{b} = \frac{4}{L} \cdot \left( {1 - 2 \cdot \frac{x}{L}} \right)\); \(\left\langle {u_{n} } \right\rangle = \left\langle {\begin{array}{*{20}c} {w_{1} } & {\beta_{1} } & {\begin{array}{*{20}c} {w_{2} } & {\beta_{2} } \\ \end{array} } \\ \end{array} } \right\rangle\).
Eq. A2 can be expressed in matrix form as:
with \(\left[ {k_{11}^{b} } \right] = \left\{ {B_{1}^{b} } \right\} \cdot H_{b} \cdot \left\langle {B_{1}^{b} } \right\rangle \cdot L\); \(k_{22}^{b} = \frac{16}{{3 \cdot L}} \cdot H_{b} ; \left\{ {k_{12}^{b} } \right\} = \left\{ 0 \right\}\). or
The internal shear energy can be expressed as:
with
with: \(\left\langle a \right\rangle = \left\langle {\begin{array}{*{20}c} { - 1} & \frac{L}{2} & {\begin{array}{*{20}c} 1 & \frac{L}{2} \\ \end{array} } \\ \end{array} } \right\rangle\).
In matrix form:
The internal strain energy is then:
Elimination of \(\Delta \beta\) (internal variable) leads to:
Then the expression of \({\Pi }_{int}^{e}\) is:
with
Elimination of \(T_{0}\) leads to:
with
The above stiffness is exact (it can represent the exact strain energy) and will lead to exact nodal values for \(w\) and \(\beta\) if an “exact” consistent load vector is used for distributed load along \(x\). See [1] for more details. We add that an alternative formulation can be proposed, ending with the same stiffness matrix if an assumed transverse shear strain \(\underline {\gamma }\) is used instead of the shear force \(T\), but using \(\underline {\gamma } = \frac{T}{{H_{s} }}\).
Appendix B: General expressions of shear strains using equilibrium equations: DST and DKMT revisited
a) The DST element proposed in 1989 [32] is based on the approximations for the rotations (Eq. 6) but the shear strains are defined using the equilibrium equations. Those independent cartesian shear strains are:
Therefore:
with
For DST the shear energy is then given by:
On a side \(k\) the assumed tangential shear strain is:
leading to:
with
We impose the following constraints on each side:
Hence: \(\gamma_{{{\text{sk}}}} = \frac{1}{{L_{k} }}\cdot\left( {w_{j} - w_{i} } \right) + \frac{1}{2} \cdot C_{k} \cdot \left( {\beta_{{x_{i} }} + \beta_{{x_{j} }} } \right) + \frac{1}{2} \cdot S_{k} \cdot \left( {\beta_{{y_{i} }} + \beta_{{y_{j} }} } \right) + \frac{2}{3} \cdot {\Delta }\beta_{sk}\).
and for the element:
Considering A22 and A23 we obtain:
with \({\varvec{A}}_{{\Delta }}^{DST}\) a 3 × 3 full matrix: \(\left( {{\varvec{A}}_{{{\Delta } }}^{DST} } \right)^{ - 1} = - \frac{2}{3}\cdot{\varvec{I}}_{3} + {\varvec{CS}}\cdot{\varvec{B}}_{{s{\Delta \beta}}}^{{{\varvec{D}}ST}}\).
The order of magnitude of \({\varvec{B}}_{{s{\Delta \beta}}}^{{{\varvec{D}}ST}}\) is \({\varvec{O}}\left( \phi \right)\) and \({\varvec{O}}\left( {\left( {{\varvec{A}}_{{{\Delta } }}^{DST} } \right)^{ - 1} } \right) = - \frac{2}{3}\cdot{\varvec{I}}_{3} + {\varvec{O}}\left( \phi \right)\).
Hence
as for DKT, so that shear locking is avoided for DST. On the other hand:
as for \(T3\gamma\). It was the same situation for BAK1 (§6). Since \(\phi\) is never infinite, the risk is that some spurious bending \(\left( {{\Delta }{\varvec{\beta}}_{{{\varvec{sn}}}} \ne 0} \right)\) exist for thick plates. This was observed and reported in [21, 40] for DST.
The stiffness matrix of a DST element is then:
with \({\varvec{k}}_{s}^{DST}\) given by Eq. A21.
b) Another possibility would have been to use the same shear strain energy as for \(T3\gamma\) (Eq. 23)
but with full considerations of the equilibrium equations as for DST (A24) and (A25).
After programming it appears that using A28 instead of A21 we obtain the same stiffness matrix as for DST.
c) In [21], Katili proposed the DKMT element where he considers the equilibrium equation on a side k limited to (as for a beam):
With the expression of \(D_{b} , D_{s}\) and the quadratic expression of \(\beta_{s}\) on a side k, one obtains:
Hence for an element:
with the constraint (eq. A23), so that:
with
The conclusion is the same as for DST but the fact that \({\varvec{A}}_{{{\Delta } }}^{DKMT}\) is diagonal severely reduces the risk of bending locking for thick plates and in fact it is found that the constant curvature kinematical patch tests are exactly satisfied. This is not the case for DST for \(L_{k} /h \ll 1\) (Figs. 4, 5). Despite that eq. A29 is an exact expression for beam, it is not a complete expression for a triangular domain. However, eq. A29 is statically consistent with the kinematical constant shear strain constraint on each side (eq. A23a, 23b).
The stiffness matrix of a DKMT element is:
with \({\varvec{A}}_{\phi }\) and \({\varvec{A}}_{{\Delta }}^{DKMT}\) given by Eqs. A31 and A33.
Rights and permissions
About this article
Cite this article
Batoz, JL., Antaluca, E. & Katili, I. On the formulation and evaluation of old and new efficient low order triangular plate bending elements with shear effects. Comput Mech 68, 69–96 (2021). https://doi.org/10.1007/s00466-021-02020-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00466-021-02020-6