## Info Tse

Equation (5.23) is valid for the condition where lateral torsional buckling will not occur, i.e. A.rel,m < 0.75, at which £crit = 1. For this situation the boundary condition for equation (5.23), based on the use of solid timber, LVL (or glulam) rectangular sections, will be as defined by the solid line in Figure 5.13.

Where the relative slenderness ratio for bending exceeds 0.75, the EC5 strength criteria will be as given in 5.4.2.

See Example 5.7.3.

5.4.2 Lateral torsional instability under the effect of bending about the major axis

This situation will apply to members that are subjected to a combination of direct compression and bending about the major axis only, and where lateral torsional instability of the member can occur. This will apply to members in which the relative slenderness ratio for bending about the major axis, A.rel,m, is greater than 0.75. No condition is given for a member subjected to axial compression with bending about the y-y and the x-x axes and A.rel,m is greater than 0.75.

For the condition where a member is subjected to direct compression and bending about the major axis and lateral torsional instability of the member can occur, no plastic behaviour is allowed to occur under the effects of the axial load but is permitted under the effect of the moment. The interaction between axial load and moment at failure is based on a solution involving plastic behaviour similar to that shown in Figure 5.10b and the design requirement is:

kcrit fm, d/ kc,zfc,0,d where the terms are as previously described, and am,d is the design bending stresses about the strong axes y-y, and am,d = My,d/ Wy, where My,d is the design bending moment about the y-y axis and Wy is the associated section modulus; ac,d is the kc,z/c,0,d kc,z/c,0,d

Fig. 5.14. Comparison between equation (5.24) (when £crit = 1) and equation (5.23).

Oc d

Fig. 5.14. Comparison between equation (5.24) (when £crit = 1) and equation (5.23).

design compressive stress and equates to ac 0,d as defined in equation (5.10); kcrit is the factor that takes into account the reduced bending strength due to lateral buckling. It is discussed in Chapter 4 and defined in Table 4.3.

In applying equation (5.24) it is to be noted that if the relative slenderness ratio for bending of the member, A.rel,m, is close to 0.75, because there is no bending about the z-z axis, the member state can also be considered to approximate the same condition as addressed by equation (5.23).

A comparison of equations (5.23) and (5.24) is shown in Figure 5.14.

For such a condition, when am,y,d//m,y,d < 0.7, equation (5.23) will dictate the limiting design condition and when am,y,d//m,y,d > 0.7, equation (5.24) should be complied with.

See Example 5.7.4.

5.4.3 Members subjected to combined bending and axial tension

Although an element of plastic behaviour due to bending is permitted, because members in tension fail in a brittle mode, EC5 takes the approach that the ultimate load will be achieved when the material reaches its failure strength in the extreme fibre. This is in line with the elastic theory solution shown in Figure 5.10a.

From this the design requirement for members subjected to combined bending and axial tension, given in EC5, 6.2.3(1)P, is as follows:

ft°d + + km^.d < 1 (EC5, equation (6.17)) (5.25)

Aci.d fm,y,d fm,z,d where the functions remain as previously defined.

Equations (5.25) and (5.26) assume that lateral torsional buckling of the member when bent about its major axis (y-y) is prevented. Where this is not the case, and for the cases where the axial tensile stress is small, the members should also be checked to comply with the EC5 requirements for a beam subjected to bending, taking into account lateral torsional instability effects, as described in Chapter 4.

See Example 5.7.5.

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