# Bars and Bar Systems

• Konstantin Naumenko
• Holm Altenbach
Chapter
Part of the Advanced Structured Materials book series (STRUCTMAT, volume 112)

## Abstract

Bars are structural members that support uni-axial tensile or compressive loadings. Chapter 1 presents elementary examples for bars and bar systems and gives an introduction to inelastic stress analysis. Basic features of inelastic structural responses including creep, relaxation, stress redistribution and others are discussed. In Sect. 1.1 governing equations a for two-bar system are introduced and initial value problems for one-dimensional inelastic stress analysis are formulated. Section 1.2 presents elementary solutions for stresses in linear thermo-elastic bars subjected to non-uniform heating. Closed-form solutions for a two-bar system under assumption of linear viscous material behavior are presented in Sect. 1.3. Various force controlled and displacement-controlled loading profiles are discussed. In addition to analytical solutions, examples for numerical time-step methods are introduced. They include one-step explicit and implicit time integration methods. Results are compared with closed form solutions to conclude on numerical accuracy and stability. Section 1.3 gives an overview of uni-axial constitutive models describing idealized non-linear inelastic behavior. Hardening, softening and damage processes are neglected to make the analysis transparent. Solutions are presented for different loading paths illustrating stress-range dependent creep, creep recovery, relaxation and tensile behaviors. Finally time-step methods are discussed to show basic features of numerical analysis for non-linear inelasticity problems.

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