By W M C McKenzie

ISBN-10: 1403912246

ISBN-13: 9781403912244

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**Extra resources for Design of structural elements**

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During bending, the stresses induced in such sections are shared among all the component parts. The extent to which sharing occurs is dependent on the method of connection at the interfaces. This connection is normally designed such that no slip occurs between the different materials during bending. The resulting structural element is a composite section which is nonhomogeneous. ) A useful technique often used when analysing such composite sections is the transformed section method. When using this method, an equivalent homogeneous section is considered in which all components are assumed to be the same material.

Consider the beam AB to be subject to a variable bending moment along its dθ length. The beam is assumed to deflect as indicated. R is the radius of curvature, L is the span, I is the second moment of area about the axis of bending, E is the modulus of elasticity, ds is an elemental length of beam measured a distance of x from the left-hand end M is the value of the bending moment at position x. 31 dy 26 Design of Structural Elements Differentiating the slope with respect to x gives: M d2y and hence: = 2 EI dx d2y EI 2 = M dx Integrating Equation (1) with respect to x gives dy EI = Mdx dx ∫ Integrating Equation (2) with respect to x gives M EI y = dx dx EI ∫∫ Equation (1) – bending moment Equation (2) – slope Equation (3) – deflection Equations (1) and (2) result in two constants of integration C1 and C2; these are determined by considering boundary conditions such as known values of slope and/or deflection at positions on the beam.

9. 46. 45. Determine the maximum timber and steel stresses induced in the cross-section when the beam is subjected to a bending moment of 70 kNm. 45 150 mm (a) Transformed section based on timber Equivalent width of timber to replace the steel plate = (n × 150) mm where: Esteel 205 × 10 3 = 25 nB = (25 × 150) = 3750 mm n = = E timber 8200 The maximum stresses occur in the timber when y = 150 mm, and in the steel (or equivalent replacement timber) when y = 160 mm. 5 N/mm2 (b) Transformed section based on steel Equivalent width of steel to replace the timber beam = (n × 150) mm where: E timber 1 1× 250 n = = 10 mm = nB = Esteel 25 25 10 mm x x 10 mm 300 mm 10 mm The maximum stresses occur in the timber (or equivalent replacement steel) when y = 150 mm, and in the steel when y = 160 mm.

### Design of structural elements by W M C McKenzie

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