## Specification of boundary conditions for Reissner/Mindlin plate bending finite elements (1990)

Venue: | Int. J. Numer. Meth. Engng |

Citations: | 4 - 2 self |

### BibTeX

@ARTICLE{Haggblad90specificationof,

author = {Bo Haggblad},

title = {Specification of boundary conditions for Reissner/Mindlin plate bending finite elements},

journal = {Int. J. Numer. Meth. Engng},

year = {1990},

pages = {981--1011}

}

### OpenURL

### Abstract

Plate bending finite elements based on the Reissner/Mindlin theory offer improved possibilities to pursue reliable finite element analyses. The physical behaviour near the boundary can be modelled in a realistic manner and inherent limitations in the Kirchhoff plate bending elements when modelling curved boundaries can easily be avoided. However, the boundary conditions used are crucial for the quality of the solution. We identify the part of the Reissner/Mindlin solution that controls the boundary layer and examine the behaviour near smooth edges and corners. The presence of boundary layers of different strengths for different sets of boundary conditions is noted. For a corner with soft simply supported edges the boundary layer removes the singular behaviour of Kirchhoff type from the stress resultants. We demonstrate the theoretical results by some numerical studies on simple plate structures which are discretized by an accurate, higher-order plate element. The results provide guidance in choosing efficient meshes and appropriate boundary conditions in finite element analyses. 1.

### Citations

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The effect of transverse shear deformation on the bending of elastic plates
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(Show Context)
Citation Context ...ansverse shearing forces is revealed if we rewrite the relations (14a) and (14b) in the form a h2 af h2 Tx +--(v2w) - v) ax 12k and Ty + ax + a hZ af h2 6k(l - v) ay 12k ay + Using equations (12) and =-=(16)-=- and the right hand side of equations (17a) and (17b) we obtain Hence, the transverse shearing forces may be written and where Y is a stress function. Inserting equations (19a) and (19b) into equation... |

43 | Mixed-interpolated elements for ReissnerMindlin plates
- Bathe, Brezzi, et al.
(Show Context)
Citation Context ...onding Kirchhoff problem and, as such, independent of the plate thickness, since the factor h3 cancels out in the governing equation. Observing that 1 DJD2 = 6k(l - V) and using equations (2)-(3) and =-=(6)-=--(8) we obtain the following fundamental relations: aT, a?' ay __+.=-f ax aw h2 - 0 =--+ x ax 6k(i - v) Tx aw h2 - 0 =--+ ay 6k(i - v) TS - a h2 h2(i + v) aj T, = - -(V2w) + __ V2 Tx - ax 12k 12/41 - ... |

43 |
A four-node plate bending element based on Mindlin Reissner plate theory and a mixed interpolation
- Bathe, Dvorkin
(Show Context)
Citation Context ...he limit problem (h + 0) will produce solutions that are neither infinite nor zero. This is achieved by letting the prescribed pressure (p) and stress resultants be proportional to h3: M,, = D, h3Mxx =-=(10)-=- etc. where ?', n?,,, etc. are scaled stress quantities. We also suppose that the fixed function f (x, y) is smooth with well-defined derivatives of sufficiently high order. The interior part of the l... |

33 | Introduction to Singular Perturbations - O’Malley - 1974 |

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18 |
Finite elements based upon Mindlin plate theory with particular reference to the four-node bilinear isoparametric element element
- Hughes, Tezduyar
- 1981
(Show Context)
Citation Context ...of a corresponding Kirchhoff problem and, as such, independent of the plate thickness, since the factor h3 cancels out in the governing equation. Observing that 1 DJD2 = 6k(l - V) and using equations =-=(2)-=--(3) and (6)-(8) we obtain the following fundamental relations: aT, a?' ay __+.=-f ax aw h2 - 0 =--+ x ax 6k(i - v) Tx aw h2 - 0 =--+ ay 6k(i - v) TS - a h2 h2(i + v) aj T, = - -(V2w) + __ V2 Tx - ax ... |

17 | The Boundary Layer of the Reissner-Mindlin Plate Model: Soft Simply Supported, Soft Clamped and Free Plates
- Arnold, Falk
- 1990
(Show Context)
Citation Context ...ior' type. The transverse twist alone governs the edge-zone behaviour of rotations and stress resultants. This may be seen more clearly by substituting equations(19a) and (19b) into the formulas (13)-=-=(15)-=-. We use the natural co-ordinates (n, s) and limit ourselves to quantities that are employed to specify the most common boundary conditions (see Appendix). Thus, along any smooth (boundary) curve we h... |

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9 | A formulation of general shell elements––the use of mixed interpolation of tensorial components - Bathe, Dvorkin - 1986 |

6 |
Skew plates and structures
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(Show Context)
Citation Context ...e Y is a stress function. Inserting equations (19a) and (19b) into equations (17a) and (17b) gives and "(Y-izkv2Y ax h2 ) = o Thus, Y is determined by an elliptic differential operator: From equation =-=(21)-=- we conclude that for small h (or large k) the stress function exhibits a strong boundary layer behaviour within a distance of the order of Jm from the boundary. Using the relations (21),(19a),(19b),(... |

6 |
A geometric and material nonlinear plate and shell element
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(Show Context)
Citation Context ...on the boundary curve. We also assume that all (higher-order) differentiations can be performed without difficulties. Using the relations (Al), (A2) and (16) several alternative forms of the formulas =-=(24)-=--(26) can be derived. Remark 2. The replacement of w, 0, and 0, by w and Q might look like a complication. However, in this way we obtain a formal decomposition of the solution in parts which we can c... |

3 | Aboundary layer theory for elastic plates - Friedrichs, Dressler - 1961 |

2 | A Family of Quadrilateral Mindlin Plate Elements with Substitute - Hinton, Huang - 1986 |

2 |
On the theory of transverse bending of elastic plates
- Reissner
- 1976
(Show Context)
Citation Context ...of the transverse shearing forces is revealed if we rewrite the relations (14a) and (14b) in the form a h2 af h2 Tx +--(v2w) - v) ax 12k and Ty + ax + a hZ af h2 6k(l - v) ay 12k ay + Using equations =-=(12)-=- and (16) and the right hand side of equations (17a) and (17b) we obtain Hence, the transverse shearing forces may be written and where Y is a stress function. Inserting equations (19a) and (19b) into... |

2 | Benchmark computation and performance evaluation for a rhombic plate bending problem - Babuˇska, Scapolla - 1989 |

2 |
Fraeijs de Veubeke, A Course in Elasticity
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(Show Context)
Citation Context ...(19b),(13a) and (13b) we obtainSPECIFICATIONS OF BOUNDARY CONDITIONS FOR PLATE BENDING 987 which means that the stress function Y is essentially equal to the local transverse twist R: Y = - (1 - v)R =-=(23)-=- In the following we prefer R to Y since R has a physical interpretation. Equations (16) and (21) are the two fundamental (decoupled) differential equations in the Reissner/Mindlin plate theory, l2 wh... |

1 |
A four node plate bending element based on Reissner/Mindlin plate theory and a mixed interpolation’, Inc. j. numer. methods eng
- Bathe, Dvorkin
- 1985
(Show Context)
Citation Context ... corresponding Kirchhoff problem and, as such, independent of the plate thickness, since the factor h3 cancels out in the governing equation. Observing that 1 DJD2 = 6k(l - V) and using equations (2)-=-=(3)-=- and (6)-(8) we obtain the following fundamental relations: aT, a?' ay __+.=-f ax aw h2 - 0 =--+ x ax 6k(i - v) Tx aw h2 - 0 =--+ ay 6k(i - v) TS - a h2 h2(i + v) aj T, = - -(V2w) + __ V2 Tx - ax 12k ... |

1 | The MITC7 and MIlC9 plate bending elements - Bathe, Brezzi, et al. - 1989 |

1 |
Approximation of three-dimensional models by two-dimensional models in plate theory
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(Show Context)
Citation Context ...ng Kirchhoff problem and, as such, independent of the plate thickness, since the factor h3 cancels out in the governing equation. Observing that 1 DJD2 = 6k(l - V) and using equations (2)-(3) and (6)-=-=(8)-=- we obtain the following fundamental relations: aT, a?' ay __+.=-f ax aw h2 - 0 =--+ x ax 6k(i - v) Tx aw h2 - 0 =--+ ay 6k(i - v) TS - a h2 h2(i + v) aj T, = - -(V2w) + __ V2 Tx - ax 12k 12/41 - v) a... |

1 | Timoshenko and Woinowsky-Krieger, Theory of Plates and Shells - P - 1959 |

1 |
On the analysis of first and second-order shear deformation effects for isotropic elastic plates
- Reissner
- 1980
(Show Context)
Citation Context ...interior' type. The transverse twist alone governs the edge-zone behaviour of rotations and stress resultants. This may be seen more clearly by substituting equations(19a) and (19b) into the formulas =-=(13)-=--(15). We use the natural co-ordinates (n, s) and limit ourselves to quantities that are employed to specify the most common boundary conditions (see Appendix). Thus, along any smooth (boundary) curve... |

1 | Mindlin plate analyses by segmentation method - Kant, Hinton - 1983 |