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When the section modulus about the strong y-y axis is Wy, from equation (4.7b) the stress in the section will be:

Equation (4.7c) relates to a hardwood, LVL (or glued-laminated timber) rectangular section beam subjected to a uniform moment at each end and will equate to the solution derived from equation (6.31) in EC5. With softwood rectangular sections, the ratio of E0,05/ G0,05 for timber is taken to be approximately 16, and by applying this to equation (4.7a), the critical bending stress (i.e. the buckling strength), am,crit, of a rectangular softwood beam bent about its strong axis can be written as:

Wy hi where E0 05 is the 5th-percentile value of the modulus of elasticity parallel to the grain, b is the breadth of the beam, h is the depth of the beam, i is the design span of the simply supported beam between lateral supports at the ends of the beam, Wy is the section modulus of the beam about the y-y axis.

Equations (4.7a) and (4.7b) are only valid for a pure moment condition applied to a simply supported beam where the beam ends are prevented from moving laterally, are free to rotate in plan, and are fully restrained against torsional rotation. For situations where different end fixing conditions exist and moment is induced by other types of loading, as well as the cases where load is applied at the compression (or tension) face rather than the centroidal axis of the beam, the elastic critical moment can be obtained by using the same expression but replacing the design span i by what is termed the 'effective length', ief, of the beam. The effective length is obtained by adjusting the design span to take account of the effect of the change in loading and end fixing conditions and values for commonly used cases in timber design are given in Table 4.2.

As it is extremely difficult to achieve full restraint against lateral rotation in plan at the ends of a single span beam, with the exception of cantilever beams (where full restraint is assumed to exist), all of the other cases given in Table 4.2 assume that the beam is fully restrained against torsional rotation but able to rotate in plan at its ends.

When designing solid softwood rectangular beams, based on the use of the effective length of the beam, equation (4.8) can be written as:

0.78b2

hief and when designing for hardwood, LVL (or glued-laminated) rectangular beams, equation (4.7c) will be as given in (4.9b), which is equivalent to equation (6.31) in EC5:

where lef is the effective length obtained from Table 4.2 for the loading configuration being used.

Table 4.2 The ratio of the 'effective length' to the design span of the beam (incorporating the cases in Table 6.1, EC5)

Beam end condition: restrained in position laterally; restrained torsionally; free to rotate in plan

Applied loading

other cases

Simply supported

Constant moment

0 0

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