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Unformatted text preview: Section 4.3 Solid Mechanics Part I Kelly 111 4.3 One Dimensional Axial Deformations In this section, a specific simple geometry is considered, that of a long and thin straight component loaded in such a way that it deforms in the axial direction only. The x-axis is taken as the longitudinal axis, with the cross-section lying in the y x − plane, Fig. 4.3.1. Figure 4.3.1: A slender straight component; (a) longitudinal axis, (b) cross-section 4.3.1 Basic relations for Axial Deformations Any static analysis of a structural component involves the following three considerations: (1) constitutive response (2) kinematics (3) equilibrium In this Chapter, it is taken for (1) that the material responds as an isotropic linear elastic solid. It is assumed that the only significant stresses and strains occur in the axial direction, and so the stress-strain relations 4.2.8 reduce to the one-dimensional equation xx xx E ε σ = or, dropping the subscripts, ε σ E = (4.3.1) Kinematics (2) is the study of deformation, the subject of §3.6-3.8. In the theory developed here, known as axial deformation , it is assumed that the axis of the component remains straight and that cross-sections that are initially perpendicular to the axis remain perpendicular after deformation. This implies that, although the strain might vary along the axis, it remains constant over any cross section . As defined in the previous Chapter, the axial strain occurring over any section is given by L L L − = ε (4.3.2) This is illustrated in Fig. 4.3.2, which shows a (shaded) region undergoing a compressive (negative) strain. Note that individual particles/points undergo displacements whereas regions/line- elements undergo strain. In Fig. 4.3.2, the particle originally at A has undergone a displacement ) ( A u whereas the particle originally at B has undergone a displacement ) ( B u . From Fig. 4.3.2, another way of expressing the strain in the shaded region is x y z Section 4.3 Solid Mechanics Part I Kelly 112 ) ( ) ( L A u B u − = ε (4.3.3) Figure 4.3.2: axial strain; (a) before deformation, (b) after deformation Both displacements ) ( A u and ) ( B u of Fig. 4.3.2 are positive , since the particles displace in the positive x direction – if they moved to the left, for consistency, one would say they underwent negative displacements. Further, positive stresses are as shown in Fig. 4.3.3a and negative stresses are as shown in Fig. 4.3.3b. From Eqn. 4.3.1, a positive stress implies a positive strain (lengthening) and a compressive stress implies a negative strain (contracting) Figure 4.3.3: Stresses arising in the slender component; (a) positive (tensile) stress, (b) negative (compressive) stress Equilibrium, (3), will be considered in the individual examples below....
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This note was uploaded on 01/20/2012 for the course ENGINEERIN 1 taught by Professor Staff during the Fall '11 term at Auckland.
- Fall '11