![]() ![]() This can only be correct if Σ(y δa) or Σ(y.z. y) over the whole section must equal the externally applied moment. Assuming E is the same forĬompression and tension the relationship is the same.Īs the beam is in static equilibrium and is only subject to moments (no vertical shearįorces) the forces across the section (AB) are entirely longitudinal and the total compressive forces The compressive stress is also directly related to the distance below the neutral axis. Therefore, for the illustrated example, the tensile stress is directly related to the distance above the neutral axis. The accepted relationship between stress and strain is σ= E.e Therefore Let y be the distance(E'G') of any layer H'G' originally parallel to EF.ThenĮ = (H'G'- HG) / HG = (H'G'- HG) / EF = /R θ = y /R The development lines of A'B' and C'D' intersect at a point 0 at an angle of θ radians and the radius of E'F' = R This surface is called neutral surface and its intersection with Z_Z is called the neutral axis The line EF will be located such that it will not change in length. The beam material is the same for tension and compression ( σ= E.e )Ĭonsider two section very close together (AB and CD).Īfter bending the sections will be at A'B' and C'D' and are no longer parallel.ĪC will have extended to A'C' and BD will have compressed to B'D' The fixed relationship between stress and strain (Young's Modulus)for The traverse plane sections remain plane and normal to the longitudinal fibres after Z = section modulus = I/y max(m 3 - more normally cm 3)Ī straight bar of homogeneous material is subject to only a moment at one end and an equal and opposite I = Moment of Inertia (m 4 - more normally cm 4) Y = distance of surface from neutral surface (m). Loading problems allowing simplification of very complicated design problems.įor beams subjected to several loads of different types the resulting shearįorce, bending moment, slope and deflection can be found at any location by summing theĮffects due to each load acting separately to the other loads. The superposition principle is one of the most important tools for solving beam This theory relates to beam flexure resulting fromĬouples applied to the beam without consideration of the shearing forces. The stress, strain, dimension, curvature, elasticity, are all related, under certain assumption,īy the theory of simple bending. ![]()
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