Axially Loaded Bars

Abstract

Axially loaded bars are used in many applications, especially as members of truss structures. A prismatic bar is one whose cross section is uniform throughout its length. Suppose that such a bar with cross-sectional area A is subjected to axial loads P at the ends. On planes perpendicular to the bar’s axis, the material is subjected to a uniform normal stress σ = P/A. By passing an oblique plane through the bar and applying the equilibrium equations to the part of the bar to one side of the plane, expressions for the normal and shear stresses on oblique planes are obtained. Those results show that the maximum tensile or compressive normal stress on any plane through an axially-loaded prismatic bar is σ = P/A, and the magnitude of the maximum shear stress is ∣ σ/2 ∣ . The axial strain in a bar of isotropic elastic material is e = σ/E, where E is the modulus of elasticity. The lateral strain, or strain perpendicular to the bar’s axis, is elat = − νσ/E, where ν is Poisson’s ratio. The change in length of an axially-loaded prismatic bar of length L is δ = PL/AE. By analyzing truss and frame structures, the axial loads in bars, and thereby their changes in length, can be obtained. Using this information, changes in the geometries of the structures can be determined. Statically indeterminate problems involving axially-loaded bars are considered next. A compatibility condition is a relationship between the deformations of different parts of an object or structure. Solutions of statically indeterminate problems are based on three elements: (1) equilibrium; (2) relations between the axial forces in bars and their changes in length, or deformations; and (3) compatibility. In addition to prismatic bars, analyses are presented for the stress distributions and changes in length of bars with gradually varying cross sections and bars subjected to axial loads distributed along their lengths. The chapter concludes with a discussion of bars subjected to changes in temperature. The thermal strain of an unconstrained material of temperature T that is subjected to a change in temperature ΔT is eT = α ΔT, where α is a material constant called the coefficient of thermal expansion. Examples of bars subjected to both axial loads and changes in temperature are discussed.