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Methods Materials

By W M C McKenzie

ISBN-10: 1403912246

ISBN-13: 9781403912244

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41. 41 Stress diagram The magnitude of the bending stresses at any vertical cross-section can be determined using the simple theory of bending from which the following equation is derived: Structural Analysis Techniques M E σ = = I R y ∴ σ = My I 37 Equation (2) where: M the applied bending moment at the section being considered, E the value of Young’s modulus of elasticity, R the radius of curvature of the beam, the bending stress, σ y the distance measured from the elastic neutral axis to the level on the cross-section at which the stress is being evaluated, I the second moment of area of the full cross-section about the elastic neutral axis.

These can be determined using: σtop = M Z top and σbottom = M Z bottom where: σ and M are as before, Ztop is the elastic section modulus relating to the top fibres and defined as Zbottom is the elastic section modulus relating to the top fibres and defined as I xx y top I xx y bottom If a cross-section is symmetrical about the x−x axis then Ztop = Zbottom. In asymmetric sections the maximum stress occurs in the fibres corresponding to the smallest Z value. For a rectangular cross-section of breadth B and depth D subject to a bending moment M about the major x−x axis, the appropriate values of I, y and Z are: D BD 3 BD 2 Zminimum = ymaximum = I = 12 6 2 In the case of bending about the minor y−y axis: DB 3 B I = ymaximum = Zminimum = 12 2 DB 2 6 38 Design of Structural Elements The maximum stress induced in a cross-section subject to bi-axial bending is given by: My Mx + σmaximum = Z x min imum Z y min imum where Mx and My are the applied bending moments about the x and y axes respectively.

18. 1 Shear Force Diagrams In a statically determinate beam, the numerical value of the shear force can be obtained by evaluating the algebraic sum of the vertical forces to one side of the section being considered. 19. 19 The calculation carried out to determine the shear force can be repeated at various locations along a beam and the values obtained plotted as a graph; this graph is known as the shear force diagram. The shear force diagram indicates the variation of the shear force along a structural member.

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Design of structural elements by W M C McKenzie

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