![]() ![]() With similar considerations, the plastic modulus, of the rectangular section, under y-y bending, can be found through the following formula: In other words, the plastic neutral axes pass through the centroid of the rectangle. Rectangular section however, is a symmetrical one (indeed it features two axes of symmetry) and therefore its plastic neutral axes coincides with the elastic ones. The axis is called plastic neutral axis, and for non-symmetric sections, is not the same with the elastic neutral axis (which again is the centroidal one). Similarly, the tensile force would be A_t f_y, using the same assumptions. the material would have yielded everywhere) and that the compressive yield stress is equal to f_y. Indeed, the total compressive force, over the entire compressive area, would be A_cf_y, assuming plastic conditions (i.e. This is a result of equilibrium of internal forces in the cross-section, under plastic bending. For materials with equal tensile and compressive yield stresses, this leads to the division of the section into two equal areas, A_t, in tension, and A_c, in compression, separated by the neutral axis. In that case the whole section is divided in two parts, one in tension and one in compression, each under uniform stress field. The plastic section modulus is similar to the elastic one, but defined with the assumption of full plastic yielding of the cross section due to flexural bending. The dimensions of section modulus are ^3. For the latter, the normal stress is F/A. Absolute maximum \sigma will occur at the most distant fiber, with magnitude given by the formula:įrom the last equation, the section modulus can be considered for flexural bending, a property analogous to cross-sectional A, for axial loading. If a bending moment M_x is applied on x axis, the section will respond with normal stresses, varying linearly with the distance from the neutral axis (which under elastic regime coincides with centroidal x-x axis). One can observe, that the formula for S_y, becomes identical to that for S_x, if we swap b for h and vice versa. Application of the general formula leads to the following elastic modulus around the y axis: Similarly, for the section modulus S_y, defined around y axis, the most distant fibers from the axis, are the side edges h, with a distance equal to b/2. Therefore, application of the above formula for the rectangular cross-section results in the following expression for elastic section modulus around x axis: ![]() For the a rectangle, the most distant fibers, from the x axis, are those at the top and bottom edge b, with a distance equal to h/2. ![]() Typically, the more distant fiber(s) are of interest. Where I_x the moment of inertia of the section around x axis and Y the distance from centroid, of a section fiber, parallel to the x axis and measured perpendicularly from it. The elastic modulus S_x of any cross section around axis x (centroidal), describes the response of the section under elastic flexural bending. Where the I_x and I_y are the moments of inertia around axes x and y, which are mutually perpendicular with z and meet at a common origin. The calculation of the polar moment of inertia I_z around an axis z (that is perpendicular to the section plane), can be done with the Perpendicular Axes Theorem: The polar moment of inertia, describes the rigidity of a cross-section against torsional moments, likewise the planar moments of inertia, described above, are related to flexural bending. The dimensions of moment of inertia are ^4. Therefore, it can be seen from the former equation, that when a certain bending moment M is applied to a beam cross-section, the resulting curvature is reversely proportional to the moment of inertia I. Where E is the Young's modulus, a property of the material, and \kappa the curvature of the beam due to the applied load. The bending moment M, applied to a cross-section, is related with its moment of inertia with the following equation: The moment of inertia (second moment or area) is used in beam theory to describe the rigidity of a beam against flexure. The area A and the perimeter P of a rectangular cross-section, with sides b and h, can be found with the next formulas: ![]()
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