Bending Element
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
- PDF / 2,357,332 Bytes
- 54 Pages / 439.37 x 666.142 pts Page_size
- 11 Downloads / 313 Views
Bending Element
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.
5.1 Introductory Remarks In the following, a prismatic body will be examined, at which the load occurs perpendicular to the center line and therefore bends. Perpendicular to the center line means that either the line of action of a force or the direction of a momental vector are oriented orthogonally to the center line of the element. Consequently a different type of deformation can be modeled with this prismatic body compared to a bar (see Chaps. 3 and 4), see Table 5.1. A general element, which includes all these deformation mechanisms will be introduced in Chap. 6. Table 5.1 Differentiation between bar and beam element; center line parallel to the x-axis Force Unknown
Bar
Beam
Along the bar axis Displacement in or rotation around bar axis
Perpendicular to the beam axis Displacement perpendicular to and rotation perpendicular to the beam axis
A. Öchsner and M. Merkel, One-Dimensional Finite Elements, DOI: 10.1007/978-3-642-31797-2_5, © Springer-Verlag Berlin Heidelberg 2013
57
58
5 Bending Element
Generally in beam statics one distinguishes between shear rigid and shear flexible models. The classic, shear rigid beam, also called Bernoulli beam, disregards the shear deformation from the shear force. With this modeling one assumes that a crosssection, which was at the right angle to the beam axis before the deformation is also at right angles to the beam axis after the deformation, see Fig. 5.1a. Furthermore it is assumed that a plane cross-section stays plane and unwarped. These two assumptions are also called the Bernoulli’s hypothesis. Altogether one imagines the crosssections fixed on the center line of the beam,1 so that a change of the center line affects the entire deformation. Consequently, it is also assumed that the geometric dimensions of the cross-sections2 do not change. Regarding a shear flexible beam, also referred to as Timoshenko beam besides the bending deformation also the shear deformation is considered, and the cross-sections will be distorted by an angle γ compared to the perpendicu
Data Loading...
