Abstract
By this element the basic deformation bending will be described. First, several elementary assumptions for modeling will be introduced and the element used in this chapter will be outlined towards other formulations. The basic equations from the strength of materials, meaning kinematics, equilibrium and constitutive equation will be introduced and used for the derivation of the differential equation of the bending line. Analytical solutions will conclude the section of the basic principles. Subsequently, the bending element will be introduced, according to the common definitions for load and deformation parameters, which are used in the handling of the FE method. The derivation of the stiffness matrix is carried out through various methods and will be described in detail. Besides the simple, prismatic bar with constant cross-section and load on the nodes also variable cross-sections, generalized loads between the nodes and orientation in the plane and the space will be analyzed.
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Notes
- 1.
More precisely this is the neutral fiber or the bending line.
- 2.
Consequently the width \(b\) and the height \(h\) of a, for example, rectangular cross-section remain the same.
- 3.
For this see the explanations in Chap. 8.
- 4.
The sum of all points with \(\sigma =0\) along the beam axis is called neutral fiber.
- 5.
Note that according to the precondition regarding the Bernoulli beam the length 01 and \(0^{\prime }1^{\prime }\) remain unchanged.
- 6.
The positive cutting surface is defined by the surface normal on the cutting plane which features the same orientation as the positive \(x\)-axis. It should be regarded that the surface normal is always directed outwardly. Regarding the negative cutting surface the surface normal and the positive \(x\)-axis are oriented antiparallel.
- 7.
If the axis is grasped with the right-hand in a way so that the spread out thumb points in the direction of the positive axis, the bent fingers then show the direction of the positive rotational direction.
- 8.
Just for the case that an external moment \(M^\text{ext}\) would act at position \(x=L\), the internal moment would result in: \(M_z(x=L)=M^\text{ext}\). Hereby it was assumed that the external moment \(M^\text{ext}\) would be positive in a mathematical sense.
- 9.
- 10.
Alternatively the expression interpolation or form function is used.
- 11.
In the general three-dimensional case the form \(\varPi _\text{int}=\frac{1}{2}\int _\varOmega \varvec{\varepsilon }^\text{T} \varvec{\sigma }\mathrm{d} \varOmega \) can be applied, whereat \(\varvec{\sigma }\) and \(\varvec{\varepsilon }\) represent the column matrix with the stress and strain components.
- 12.
The principle of virtual work encompasses the principle of the virtual displacements and the principle of the virtual forces [13].
- 13.
Castigliano’s theorems were formulated by the Italian builder, engineer and scientist Carl Alberto Castigliano (1847–1884). The second theorem signifies: the partial derivative of the stored potential energy in a linear-elastic body with regards to the displacement \(u_i\) yields the force \(F_i\) in the direction of the displacement at the considered point. An analog coherence also applies for the rotation and the moment.
- 14.
A common representation of the partial integration of two functions \(f(x)\) and \(g(x)\) is: \(\int f^{\prime }g\,\mathrm{d} x=fg -\int fg^{\prime } \mathrm{d} x\).
- 15.
See Sect. 5.2.2 with the executions for the internal reactions and external loads.
- 16.
Also see Table 5.6 and supplementary problems 5.6.
- 17.
Since the transformation matrix \(\varvec{T}\) is an orthogonal matrix, the following applies: \(\varvec{T}^\text{T}=\varvec{T}^{-1}\).
- 18.
If point loads appear between nodes, the discretization can of course be further sub-divided, so that a new node is positioned on the location of the loading point. However within this chapter the case of no further subdivision of the mesh ought to be regarded.
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Öchsner, A., Merkel, M. (2013). Bending Element. In: One-Dimensional Finite Elements. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31797-2_5
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DOI: https://doi.org/10.1007/978-3-642-31797-2_5
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