Abstract
This element describes the basic deformation mode of bending under the consideration of the shear influence. First, several basic assumptions for the modeling of the Timoshenko beam will be introduced and the element used in this chapter will be differentiated from other formulations. The basic equations from the strength of materials, meaning kinematics, the equilibrium as well as the constitutive equation will be introduced and used for the derivation of a system of coupled differential equations. The section about the basics is ended with analytical solutions. Subsequently, the Timoshenko 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 linear shape functions, a general concept for an arbitrary arrangement of the shape functions will be reviewed.
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Notes
- 1.
For a function f(x, y) of two variables usually a Taylors’ series expansion of first order is assessed around the point \((x_0,y_0)\) as follows: \(f(x,y)=f(x_0+\mathrm {d}x,y_0 +\mathrm {d}x)\approx f(x_0,y_0)+\left( \tfrac{\partial f}{\partial x}\right) _{x_0,y_0}\times (x-x_0)+\left( \tfrac{\partial f}{\partial y}\right) _{x_0,y_0}\times (y-y_0)\).
- 2.
A closer analysis of the shear stress distribution in the cross-sectional area shows that the shear stress does not just alter through the height of the beam but also through the width of the beam. If the width of the beam is small compared to the height, only a small change along the width occurs and one can assume in the first approximation a constant shear stress throughout the width: \(\tau _{xy}(y,z) \rightarrow \tau _{xy}(y)\). See for example [1, 2].
- 3.
One notes that in the English literature often the so-called form factor for shear is stated. This results as the reciprocal of the shear correction factor.
- 4.
Maple\(^\circledR \), Mathematica\(^\circledR \) and Matlab\(^\circledR \) can be listed at this point as commercial examples.
- 5.
Consider also the supplementary Problem 8.4.
- 6.
A numerical Gauss integration with two integration points yields the same results as the exact analytical integration.
- 7.
- 8.
One considers the definition of \(I_z\) and A in Eq. (8.123) and divides the fraction by \(h^3\).
- 9.
- 10.
The numerical integration according to the Gauss–Legendre method with n integration points integrates a polynomial, which degree is at most \(2n-1\), exactly.
- 11.
- 12.
For this see the supplementary Problem 8.5.
- 13.
At the so-called Lagrange interpolation, m points are approximated via the ordinate values with the help of a polynomial of the order \(m-1\). In the case of the Hermite interpolation the slope of the regarded points is considered in addition to the ordinate value. For this see Chap. 6.
- 14.
It needs to be remarked that the influence of distributed loads is disregarded in the derivation. If distributed loads occur, the equivalent nodal loads have to be distributed on the remaining nodes.
- 15.
A similar example is presented in [19].
- 16.
For this see the supplementary Problem 8.6.
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Öchsner, A., Merkel, M. (2018). Beam with Shear Contribution. In: One-Dimensional Finite Elements. Springer, Cham. https://doi.org/10.1007/978-3-319-75145-0_8
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DOI: https://doi.org/10.1007/978-3-319-75145-0_8
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