Abstract
On the basis of the bar element, tension and compression as types of basic load cases will be described. First, the basic equations known from the strength of materials will be introduced. Subsequently the bar element will be introduced, according to the common definitions for load and deformation quantities, which are used in the handling of the FE method. The derivation of the stiffness matrix will be described in detail. Apart from the simple prismatic bar with constant cross-section and material properties also more general bars, where the size varies along the body axis will be analyzed in examples [1–9] and exercises.
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Notes
- 1.
The parlance tension bar includes the load case compression.
- 2.
The form \(\varPi _\text{int}=\tfrac{1}{2}\int _\varOmega \varvec{\varepsilon }^\text{T}\varvec{\sigma }\mathrm{d} \varOmega \) can be used in the general three-dimensional case, where \(\varvec{\sigma }\) and \(\varvec{\varepsilon }\) represents the column matrix with the stress and strain components.
- 3.
\((\varvec{A} \varvec{B})^\text{T}=\varvec{B}^\text{T}\varvec{A}^\text{T}\)
- 4.
A usual representation of the partial integration of two functions \(f(x)\) and \(g(x)\) is: \(\int fg^{\prime }\mathrm{d} x = fg -\int f^{\prime }g \mathrm{d} x\).
- 5.
Here the FE solution is shown in brief. A detailed derivation for the development of a total stiffness matrix, for the introduction of boundary conditions and for the identification of the unknown is introduced in Chap. 7.
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Öchsner, A., Merkel, M. (2013). Bar Element. In: One-Dimensional Finite Elements. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31797-2_3
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DOI: https://doi.org/10.1007/978-3-642-31797-2_3
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