Beam with Shear Contribution
By this element the basic deformation bending under consideration of the shear influence will be described. First, several basic assumptions for the modeling of the Timoshenko beam will be introduced and the element used in this chapter will be distinguis
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Beam with Shear Contribution
Abstract By this element the basic deformation bending under consideration of the shear influence will be described. First, several basic assumptions for the modeling of the Timoshenko beam will be introduced and the element used in this chapter will be distinguished 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 with the definitions for load and deformation parameters which are commonly accepted at the handling via the FE method. The derivation of the stiffness matrix at this point also takes place via 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 introduced.
8.1 Introductory Remarks The general differences regarding the deformation and stress distribution of a bending beam with and without shear influence have already been discussed in Chap. 5. In this chapter the shear influence needs to be considered with the help of the Timoshenko beam theory. Within the framework of the following introductive remarks, first the definition of the shear strain and the connection between shear force and shear stress needs to be covered. For the derivation of the equation for the shear strain in the x − y plane, the infinitesimal rectangular beam element ABC D, shown in Fig. 8.1 will be considered, which deforms under exposure of shear stress. Here, a change of the angle of the original right angles as well as a change in the lengths of the edges occurs. The deformation of the point A can be described via the displacement fields u x (x, y) and u y (x, y). These two functions of two variables can be expanded in TAYLOR’s
A. Öchsner and M. Merkel, One-Dimensional Finite Elements, DOI: 10.1007/978-3-642-31797-2_8, © Springer-Verlag Berlin Heidelberg 2013
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8 Beam with Shear Contribution
Fig. 8.1 Definition of the shear strain γx y in the x − y plane at an infinitesimal beam element
series1 of first order around point A to calculate the deformations of the points B and D approximatively: ∂u x ∂u x dx + dy, ∂x ∂y ∂u y ∂u y dx + dy, = u y (x + dx, y) = u y (x, y) + ∂x ∂y
u x,B = u x (x + dx, y) = u x (x, y) +
(8.1)
u y,B
(8.2)
or alternatively ∂u x ∂u x dx + dy, ∂x ∂y ∂u y ∂u y dx + dy. = u y (x, y + dy) = u y (x, y) + ∂x ∂y
u x,D = u x (x, y + dy) = u x (x, y) +
(8.3)
u y,D
(8.4)
In Eqs. (8.1) up to (8.4) u x (x, y) and u y (x, y) represent the so-called rigid-body displacement, which does not cause a deformation. If one considers that point B has For a function f (x, y) of two variables usually a TAYLOR’s series expansion of first order is assessed around the point (x0, y0 ) as follows: f (x, y) = f (x0 + dx, y0 + dx) ≈ f (x0 , y0 ) + ∂f × (x − x0
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