![]() It is verified that the stress is compressive ( σ x 0) when the bending moment M is positive, and tensile ( σ x > 0) when M is negative. Substituting for σ m from Eq9 into Eq3, the normal stress σ x at any distance y from the neutral axis is obtained:Įq9 and Eq10 are called the elastic flexure formulas, and the normal stress σ x caused by the bending or "flexing" of the member is often referred to as the flexural stress. Recalling that in the case of pure bending the neutral axis passes through the centroid of the cross section, it is noted that I is the moment of inertia, or second moment, of the cross section with respect to a centroidal axis perpendicular to the plane of the couple M. Specifying that the z-axis should coincide with the neutral axis of the cross section, σ x from Eq3 is substituted into Eq6: ![]() Recall Eq3 from the lesson Symmetric Member in Pure Bending with respect to an arbitrary horizontal z axis: In other words, for a member subjected to pure bending, and as long as the stresses remain in the elastic range, the neutral axis passes through the centroid of the section. This equation shows that the first moment of the cross section about its neutral axis must be zero. Substituting first for σ x from Eq3 into Eq1 from the lesson Symmetric Member in Pure Bending: Both can be found by utilizing Eq1 and Eq3 from the lesson Symmetric Member in Pure Bending. It should be noted that, at this point, the location of the neutral surface is not known, nor is the maximum value σ m of the stress. This result shows that, in the elastic range, the normal stress varies linearly with the distance from the neutral surface as shown in Fig1. Where σ m denotes the maximum absolute value of the stress. Recalling Eq7 of the lesson Deformations in a Symmetric Member in Pure Bending, and multiplying both members of that equation by E: Assuming the material to be homogeneous, and denoting by E its modulus of elasticity, the following results in the longitudinal x direction: ![]() There will be no permanent deformation, and Hooke's law for uniaxial stress applies. This means that, for all practical purposes, the stresses in the member will remain below the proportional limit and the elastic limit as well. The case is now considered when the bending moment M is such that the normal stresses in the member remain below the yield strength σ Y. Stresses and Deformations in the Elastic Range EngArc - L - Stresses and Deformations in the Elastic Range
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