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*l is the length of the compression member and a is the length between lateral supports.

*l is the length of the compression member and a is the length between lateral supports.

until they have reached a value that will effectively hold the member in position at each support.

Based on classical elastic stability theory, assuming a perfectly straight member, Timoshenko and Gere [2] have shown that for such a condition the minimum spring stiffness, C, to be provided by each bracing member will be:

ly where:

• C is the minimum axial stiffness of each spring;

• m is the number of supported elements of the compression member along its length (see Figure 9.3b where, for the example used, m = 4);

• Ne is the critical buckling load of the member, i.e. the Euler buckling load for a member of length l/ m, and for a timber or wood-based product Ne = n 2EI/(l/m)2, where EI is the flexural rigidity about the z-z axis of the member and E is the 5th-percentile value of the modulus of elasticity parallel to the grain;

• l is the overall length of the member;

• y is a numerical factor that depends on the number of supported sections of the laterally supported member and = (1/2)(1 + cos(n/m)).

Substituting for y, and letting l/m = a, as shown in Figure 9.3b, equation (9.2) can be written as:

For loads less than Ne the assumption is made that the stiffness can be allowed to reduce linearly and under the action of the mean design load in the member at the ULS, Nd, the minimum spring stiffness required is written as:

a where ks = 1/y , is a modification factor and the other functions are as described above.

The value of ks for increasing values of m is given in Table 9.2, achieving a maximum value of 4 when m is essentially greater than 10. A range of values is given for ks in EC5, however as it is a nationally determined parameter, the requirement in the UKNA to EC5 [3] is that ks = 4, i.e. the largest theoretical value.

In determining the stiffness of the bracing, the effect of deviation from straightness of the compression member has been ignored, however in determining the force to be taken by each bracing member, EC5 takes this into account. For the buckling mode shown in Figure 9.3d, EC5 assumes that the member will have an initial deviation from straightness, a0, as shown in Figure 9.4b.

Fig. 9.4. Buckling mode adopted in EC5 for the strength analysis of the bracing member.

Fig. 9.4. Buckling mode adopted in EC5 for the strength analysis of the bracing member.

The axial force in the bracing, Fd, arising from the design compression force in the member, Nd, will be increased due to the initial deviation and from a second-order linear elastic analysis, as stated in STEP 1 [4], a conservative value of Fd can be shown to be:

Adopting the maximum value of initial deviation from straightness between supports permitted in EC5,10.2, i.e. (2a)/300 and (2a)/500 for solid timber and for glulam or LVL, respectively, and applying these to equation (9.5), the design force in each bracing member will be as follows:

In EC5 the factors applied to the design force Nd are defined as 1/ f for solid timber and 1/ f for glued-laminated timber or LVL, and the design force is obtained as follows:

kf,2

As with ks, referred to above, kf,i and kf>2 are nationally determined parameters and the requirement in NA.2.10 of the UKNA to EC5 is that they shall be taken to be 60 and 100 respectively.

Applying the principle of static equilibrium to the beam section shown in Figure 9.4b, when the stiffness of the bracing, C, is as given in equation (9.4) the following relationship will exist between the design compression force in the member, Nd, and the lateral force, Fd, in each bracing member:

a0 4Nda0

The lateral force in a bracing member when using equation (9.7) will be greater than that obtained from equation (9.35) in EC5, the difference arising because deviation from straightness has not been taken into account when determining the bracing stiffness equation (9.4). It is to be presumed that taking into account other inherent conservatisms in the analysis methodology, EC5 does not consider this omission to be significant for practical design situations.

As the force in the bracing member can be a tensile or a compressive action, depending on the bracing system being used the bracing member and its end connections must be designed to resist both conditions.

See Example 9.7.1.

9.3.3 Bracing of single members (subjected to bending) by local support

The method is applicable to the design of bracing members when fitted to a single member subjected to compression forces arising from bending, e.g. the compression force in the compression zone of a beam. The function of the bracing is to prevent the single member from buckling laterally at the bracing positions and by so doing increase the lateral torsional buckling strength of the member. Consequently, when determining the lateral torsional buckling strength of the braced member, its effective length should be taken to be the distance between adjacent bracing members.

With beams, the relationships given in equations (9.4) and (9.6) for laterally braced single members subjected to direct compression will still apply. However, the value to be used for Nd, i.e. the mean design compressive force in the member, must be determined as described in 9.3.1.

When dealing with a rectangular member of depth h, Nd is derived in accordance with equation (9.1), i.e.:

h where kcA is derived from EC5,6.3.3(4), for the unbraced member length, as described in Chapter 4, Md is the maximum design moment acting on the beam, and h is the depth of the rectangular beam.

If kcrit is unity, function (1 - kcrit) will be zero, meaning that the beam will not buckle and there will be no requirement for lateral bracing along its length. When kcrit is less than unity, lateral torsional buckling can arise and Nd will be derived from equation (9.1). It is to be noted that for beams the design compressive force is derived using the maximum value of the design moment in the section, whereas for members primarily subjected to direct compression (referred to in 9.3.2) the mean value of the design compressive force is used. Also, to satisfy the theory, the bracing must be positioned such that the lateral support is provided at the compression edge of the member.

Having determined the design force, Nd, the bracing stiffness and the design axial force in each bracing member is derived as described in 9.3.2.

See Example 9.7.2.

9.3.4 Bracing for beam, truss or column systems

Where a bracing system is required to provide lateral stability to a series of compression or bending members (e.g. columns, trusses or beams), this is effectively achieved by providing lateral stiffness using truss or plate action within the plane of the bracing structure.

For the general case of a series of similar compression members that require to be braced at positions along their length, the approach used in EC5 is to assume that the deflected shape of each compression member under load will be a sinusoidal form between its supports and will include for the maximum initial out of straightness permitted in EC5, 10.2(1). Although the lateral stiffness of the structure will be a combination of the lateral stiffness of the members and of the bracing system, in EC5 the member stiffness is ignored and also the effect of shear deformations is not taken into account.

Each member in the structure, including the bracing system, is assumed to have an initial sine-shaped deformed profile, z = a0 sin(^x/I), as shown in Figure 9.5, where a0 is the maximum deviation from straightness at mid-length of the members as well as the bracing system. Where n members are to be braced and each member is subjected to a compression force Nd, assuming that all members contribute to the loading to be taken by the bracing system, the total compression load will be nNd. Taking the deflection of the bracing system under this load to be z1 at mid-length, the bending moment along the system will also be a sinusoidal function and the maximum value will be nNd(ao + zi).

As has been shown by Timoshenko and Gere [2], the above problem can alternatively be analysed by replacing the effect of the initial deviation from straightness on the deflection behaviour of the bracing system by the effect of an equivalent lateral load acting on the bracing system when in an initially straight condition, such that the bending moment diagram in each case will be the same. As the equivalent lateral load must also be a sine function, it can be expressed as:

nx z = ao sin — - is the assumed initial deformed shape of all members nx

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