## Aa

Fig. 4.4. Lateral torsional buckling of a beam subjected to uniform end moments M applied about the major axis (lateral buckled position shown solid).

Km is the partial coefficient for material properties, given in Table 2.6. km is a modification factor that allows for an element of redistribution of stress (yield behaviour) in the stress block and also takes into account the effect of variation in material properties. Values for the factor are given in 6.1.6(2), EC5:

- For solid timber and LVL (and glued-laminated timber), km = 0.7 for a rectangular section, and km = 1.0 for other cross-sections.

- For other wood-based structural products, km = 1.0 for any cross-section.

Sections in which the second moment of area about the y-y and z-z axes has the same value (e.g. square or circular solid timber members) lateral torsional instability will not occur and /m,y,d will equal /m,z,d. When using a circular cross-section, the section modulus will be (nd3)/32, where d is the diameter of the member section.

4.5.1.2 Bending (where the relative slenderness ratio for bending about the major axis is >0.75)

When a perfectly straight beam is subjected to bending about its y-y axis, it can be shown by elastic buckling theory that there is an elastic critical load at which the beam will become unstable, failing suddenly by deflecting sideways and twisting about its longitudinal x-x axis. This mode of failure is termed lateral torsional buckling and is shown in section A-A in Figure 4.4 for a member subjected to pure bending about the y-y axis.

The bending moment at which elastic buckling will occur is termed the elastic critical moment and is a function of the nature of the loading on the beam, the length of the beam and its support conditions, the position of the beam loading relative to its shear centre, the shear modulus and modulus of elasticity of the beam as well as its section properties. For a member with a design span, I, restrained against torsional movement at its ends but free to rotate laterally in plan, subjected to a pure moment applied at its ends about the y-y axis, as indicated in Figure 4.4, it can be shown that the elastic critical moment of the beam, My crit, will be:

For a rectangular section of breadth b, depth h, and Iz = (1/12)hb3, Itor can be taken to approximately equal 1/3(b)3h (1 - 0.63(b/h)), and inserting these relationships in equation (4.6a) the elastic critical moment can be written as:

n(bWEofi5G0,05 ((1 - 0.63(b/h))/(1 - (b/h)2)) My, cit = ------(4.6b)

where Eq,q5 is the 5th-percentile value of the modulus of elasticity parallel to the grain and Gq,q5 is the 5th-percentile value of the shear modulus.

For all practical sizes of solid timber beam, the factor ((1 - 0.63(b/h))/(1 -(b/h)2))0 5 (referred to as factor a) is less than unity and will only influence the buckling strength by a small percentage. This is shown in Figure 4.5 where the effect of deleting the function, expressed as a percentage of the buckling strength, is plotted over a practical range of beam breadth to depth ratios. It is seen that the elastic buckling strength will be overestimated by 3-6% and the maximum value will occur when the breadth to depth ratio is between 0.3 and 0.4.

In EC5 factor a is ignored for solid rectangular timber softwood beams and the elastic critical moment becomes:

n(b3h)JE0 05G0 q5

For solid rectangular sections made with hardwood, LVL (or glued-laminated timber), in EC5 the full torsional rigidity is retained but function (1 - (b/ h)2) in factor a is ignored so that the elastic critical moment relationship for these materials is:

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