TAILIEUCHUNG - Ebook Mechanics and strength of materials: Part 2
(BQ) Part 2 book “Mechanics and strength of materials” has contents: Shear force, bending deflections, torsion, structural stability, energy theorems, heorems of virtual displacements and virtual forces, considerations about the total potential energy. | VIII Shear Force General Considerations Pure bending is a very rare loading condition. In fact, slender members are very often under the action of shear forces caused by transversal loading or by end moments. The presence of the shear force V implies that the bending moment cannot be constant, since V = dM dz (non-uniform bending: M = 0 and V = 0). The shear force is balanced by shearing stresses τzx and τzy , acting on the cross-section of the bar. Denoting by Vx and Vy the components of the shear force in the reference axes x and y, the shearing stress distribution in the cross-section must obey the conditions ' ' τzx dΩ = Vx and τzy dΩ = Vy . (184) Ω Ω A supplementary condition is furnished by the reciprocity of shearing stresses in perpendicular facets, which is also an equilibrium condition (see Subsect. ). According to this condition, if there are no shear forces with a component in the longitudinal direction, applied in the lateral surface of the bar, the shearing stress will be zero in that direction and, as a consequence, in the points of the cross-section which are close to the boundary, the component of the shearing stress which is perpendicular to it will also be zero (Fig. 101). Thus, in the points of the cross-section at an infinitesimal distance to its boundary, the shearing stress will be tangent to the border line. It is obvious that there are infinite stress distributions which obey this condition and also satisfy (184). We have, therefore, a problem with an infinite degree of indeterminacy. The law of conservation of plane sections cannot be used to solve the problem, since, as explained in Sects. and , the shear force is not a symmetrical internal force. Besides, the superposition principle cannot be used to analyse the effects of the bending moment and of the shear force separately. In fact, this principle refers to distinct sets of external loads and it is not possible to find a system of transversal .
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