Beams

Abstract

Beams are structural members that are designed to support lateral forces and bending moments. Beams can be also subjected to combined bending, torsion as well as axial tensile or compressive loads. In the case of linear elasticity the laterally loaded beams, rods subjected to torque as well as axially loaded rods can be analyzed separately and the superposition principle can be applied to establish the resultant stress and deformation states. For nonlinear material behavior such a superposition is not possible and combined loadings should be considered. Furthermore, inelastic material response may be different for tensile and compressive loadings leading to a shift of the neutral plane under pure bending. Beams are also important in testing of materials. Three or four point bending tests are frequently used to analyze inelastic behavior experimentally. Examples are presented in Chuang (1986); Scholz et al (2008); Xu et al (2007) for homogeneous beams, in Weps et al (2013) for laminated beams and in Nordmann et al (2018) for beams with coatings. Beams are discussed in monographs and textbooks on creep mechanics (Boyle and Spence, 1983; Hult, 1966; Kachanov, 1986; Kraus, 1980; Malinin, 1975, 1981; Odqvist, 1974; Penny and Mariott, 1995; Skrzypek, 1993), where the Bernoulli-Euler beam theory and elementary constitutive equations, such as the Norton-Bailey constitutive law for steady-state creep are applied.

Chapter 3 presents examples of inelastic structural analysis for beams. In Sect 3.1 the classical Bernoulli-Euler beam theory is introduced. Governing equations and variational formulations for inelastic analysis are introduced. Closed-form solutions and approximate analytical solutions are derived for beams from materials that exhibit power law creep and stress regime dependent creep. Numerical solutions by the Ritz and finite element methods are discussed in detail. Creep and creep-damage constitutivemodels are applied to illustrate basic features of stress redistribution and damage evolution in beams. Furthermore, several benchmark problems for beams are introduced. The reference solutions for these problems obtained by the Ritz method are applied to verify user-defined creep-damage material subroutines and the general purpose finite element codes.

For many materials inelastic behavior depends on the kind of stress state. Examples for stress state effects including different creep rates under tension, compression and torsion are discussed in Sect. 3.2. For such kind of material behavior, the classical beam theory may lead to errors in computed deformations and stresses. Section 3.3 presents a refined beam theory which includes the effect of transverse shear deformation (Timoshenko-type theory). Based on several examples, classical and refined theories are compared as they describe creep-damage processes in beams.