Abstract
This chapter covers the continuum mechanical description of beam members under the additional influence of shear stresses. Based on the three basic equations of continuum mechanics, i.e., the kinematics relationship, the constitutive law and the equilibrium equation, the partial differential equations, which describe the physical problem, are derived.
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- 1.
For a function f(x, z) of two variables usually a Taylor’s series expansion of first order is formulated around the point \((x_0,z_0)\) as follows: \(f(x,z)=f(x_0+{\text {d}}x,z_0 +{\text {d}}z)\approx f(x_0,z_0)+\left( \tfrac{\partial f}{\partial x}\right) _{x_0,z_0}\times (x-x_0)+\left( \tfrac{\partial f}{\partial z}\right) _{x_0,z_0}\times (z-z_0)\).
- 2.
A closer analysis of the shear stress distribution in the cross-sectional area shows that the shear stress does not just alter over the height of the beam but also through the width of the beam. If the width of the beam is small when 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: \(\tau _{xz}(y,z) \rightarrow \tau _{xz}(z)\). See for example [2, 15].
- 3.
It should be noted that the so-called form factor for shear is also known in the literature. This results as the reciprocal of the shear correction factor.
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Öchsner, A. (2020). Timoshenko Beams. In: Partial Differential Equations of Classical Structural Members. SpringerBriefs in Applied Sciences and Technology(). Springer, Cham. https://doi.org/10.1007/978-3-030-35311-7_4
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DOI: https://doi.org/10.1007/978-3-030-35311-7_4
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