Abstract
This chapter introduces first the theory to derive the elemental stiffness matrix of plane elasticity elements, i.e., elements which can be loaded and deform only in-plane. Then, the principal finite element equation of such plane elements and their arrangements as plane element structures is covered. Four- and three-node elements are covered and the particularities between both formulations are highlighted. The chapter includes detailed Maxima examples which allow an easy transfer to other problems.
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Notes
- 1.
The plane stress element is known in literature as the constant strain triangle or CST element.
- 2.
Alternatively named as area coordinates or barycentric coordinates.
- 3.
If we skip the index ‘P’, then we simply have the relationship between the triangular and the Cartesian coordinates.
- 4.
Remember: \(0!=1\), \(1!=1\), \(2!=2\), and \(3!=6\).
- 5.
See [5] for further details.
- 6.
See [5] for further details.
- 7.
Remember: \(0!=1\), \(1!=1\), \(2!=2\), and \(3!=6\).
- 8.
This example is adapted from [2].
References
Akin JE (2005) Finite element analysis with error estimators: an introduction to the FEM and adaptive error analysis for engineering students. Elsevier Butterworth-Heinemann, Burlington
Fish J, Belytschko T (2013) A first course in finite elements. Wiley, Chichester
MacNeal RH (1994) Finite elements: their design and performance. Marcel Dekker, New York
Öchsner A (2016) Continuum damage and fracture mechanics. Springer, Singapore
Öchsner A (2020) Computational statics and dynamics - an introduction based on the finite element method. Springer, Singapore
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Öchsner, A., Makvandi, R. (2021). Plane Elements. In: Plane Finite Elements for Two-Dimensional Problems. Springer, Cham. https://doi.org/10.1007/978-3-030-89550-1_3
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DOI: https://doi.org/10.1007/978-3-030-89550-1_3
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