## Def

where the functions are as previously defined and f 2 is the factor for the quasipermanent value of the action causing the largest stress in relation to the strength. If this is a permanent action, a value of 1 should be used. If the determination of f 2 is assessed to be a complicated or difficult exercise, a safe result will be obtained by adopting a value of 1 for the factor (see Table 2.2 for the value of f 2).

(c) When undertaking a second-order linear elastic analysis of a structure (see 2.2.16.1.1(c)), in accordance with the requirements ofEC5,2.2.2(1)P, stiffness-related properties must be derived using the design values of the appropriate modulus of elasticity and/or shear modulus as defined in EC5, 2.4.1(2)P. For connections, the value used for the slip modulus will be Ku, as given in EC5, 2.2.2(2).

After derivation of the stiffness-related properties in accordance with the above requirements, the design value used in the ULS analysis will be Case (a)

Fd,ULS = Emean, G d,ULS = Gmean, Kd,ULS = Kser (2.41)

Case (b) (final condition)

Fd,ULS = ———--, Gd,ULS = . --, Kd,ULS = M . -"

Km Km where the functions are as previously defined and Fd,ULS is the design value of the modulus of elasticity at the ULS, Gd,ULS is the design value of the shear modulus at the ULS, Kd,ULS is the design value of the slip modulus at the ULS, and kM is the partial factor for a material property (or connection) given in NA.2.3 of the UKNA to EC5.

In the above equations, it has to be remembered that the value of kdef used for conditions must be as explained in 2.3.2.

2.3.5 Stress-strain relations (EC5, 3.1.2)

Although the actual stress-strain relationship for timber and wood-related products when loaded to failure is generally non-linear, the characteristic strengths of structural timbers and wood products are derived assuming that a linear relationship exists. Consequently, when calculating the design stress in a section, e.g. flexural, axial, shear, etc., it is to be assumed that elastic behaviour will apply up to the failure condition. Where EC5 considers that plastic behaviour can be taken into account to enhance member strength, this is incorporated into the relevant strength validation rules given in the code.

2.3.6 Size and stress distribution effects (EC5, 3.2,3.3, 3.4 and 6.4.3)

Timber is not a homogeneous material and due to the presence of defects, variability in strength across and along member lengths, as well as the loading configuration being used, the strength between and within members will vary. Although there has been considerable research and theoretical investigation into member size, length, volume, load configuration and stress distribution effects, there is not yet full agreement on the effect of these factors and how they should be incorporated into the design process.

Because timber and wood-related products are brittle materials, the most widely adopted theoretical approach used in investigations has been based on the application of weakest link theory, using a Weibull distribution. Although not valid for some timber species, it is assumed that the strength-degrading defects in timber and wood-related products are randomly distributed throughout the sample volume and are of random size, enabling the sample to be considered as a chain element comprising several small volumes of different strength when subjected to tension. In this condition, the sample strength will be dictated by the strength of the weakest volume. If there are two members of differing volumes, V1 and V2, and the strength distribution fits the Weibull distribution, it has readily been demonstrated (e.g. [20, 21]) that the theory will conclude that the ratio of the respective failure strength of the members (a 1 and a 2) can be written as where k is the shape factor of the Weibull distribution.

For bending stresses, equation (2.44) should be further adjusted to take account of the effect of the variation in stress distribution over the length of the member arising from the loading configuration being used; however, this effect has not been included for in EC5.

For timber sections, the volumes V1 and V2 can be represented by the member breadth, depth and length, i.e. b1 x h1 x L1 and b2 x h2 x L2, respectively, and equation (2.44) can be written as where the exponents 1/kb, 1/kh and 1/kLare the factors associated with each dimension.

Because the width of timber members does not vary significantly, the width effect is generally ignored and if members are strength tested at a constant span to depth ratio, the depth and length exponents can be combined to give a single factor, i.e.

This is the approach used in EC5 for bending, where bending strengths obtained in accordance with BS EN408:1995 [22] require all tests to be carried out using a two-point bending configuration on a beam having a span to depth ratio of 18 (plus a tolerance allowance).

The Weibull theory (as it is commonly referred to) has also been used to investigate volume effects as well as the effect of varying the types of loading configuration applied

to members. However, opinions vary among researchers on the application of these effects, and in particular loading configuration effects. Also, there is not full agreement on the values to be used for the relevant exponents. In such circumstances, EC5 has adopted a simplified approach and ignored certain of the effects. Factors have been included for bending and tensile strengths in solid timber, glued-laminated timber and LVL as well as a stress distribution and volume effects in double tapered, curved and pitched camber beams. The effects are not applicable to wood-based panel products, e.g. plywood, OSB, fibreboard, etc.

The consequence of the above is that unlike designs in structural steel and reinforced concrete, when using timber from the same strength class the effect of defects can result in members of different sizes having different characteristic strengths.

To take these effects into account, the characteristic values of strength properties that are influenced by the effect are derived using reference sizes (which for depth, width and length effects are the sizes above which the effect can be ignored) and the characteristic strength of the property used for design is obtained by multiplying the characteristic strength given in the relevant British Standard by a factor, k, derived from the member size, the reference size and the factor 1/khL or 1/kL, as appropriate. The characteristic strengths in bending and tension given in BS EN 338:2003 have been derived using a reference depth of 150 mm for solid timber, 600 mm for glulam (EN 14080) and in BS EN 14374:2004, the reference depth for LVL bent edgewise is 300 mm. No size factor is applicable for LVL when bent flatwise. The tensile strength of LVL is also affected by the length of the member and the reference length used in EN 14374:2004 is 3000 mm.

In EC5 factor kh, which relates to depth, and kt, which relates to length, are derived from exponential functions of the reference size divided by the member size, where the exponent is the value derived for 1/ khL or 1/ kL, as appropriate. For sizes greater than the reference size the factor is less than unity, but, as the reduction is relatively small, it is ignored in EC5 and taken to equal 1. When sizes are less than the reference value the factor will be greater than unity, resulting in an increase in the property strength. Also, as the member size decreases, an upper limit of 1.3, 1.1 and 1.2 has been set for kh for solid timber, glulam and LVL, respectively, to derive the characteristic bending strength and 1.1 for kt when deriving the characteristic tensile strength of LVL along the grain direction. The factor kh will apply to bending about the strong or the weak axis when dealing with solid timber but for horizontally laminated glued laminated timber, it will only apply to the beam depth where the section is loaded perpendicular to the plane of the wide faces of the laminations. For LVL, kh will only apply when a section is subjected to edgewise bending.

As an example, the theoretical value of factor kh to be used to determine the characteristic bending and tensile strength of glulam beams is (600/h)01, where h is the depth for a member in bending or the width (the maximum size of the cross-section) for a member in tension and 1/khL = 0.1. The relationship is given in EC5, equation (3.2) and a comparison with the theoretical value is shown in Figure 2.8. Above the 600 mm reference size kh is taken to be 1 and as the size of the beam decreases it follows the theoretical function until it reaches a maximum value of 1.1, which, for glulam members, occurs around a depth of 230 mm.

Relationships for the size effects used in EC5 for timber, glued-laminated timber and LVL as well as the volume and stress distribution effects in the apex zone of double

Depth of glulam beam

Fig. 2.8. EC5 factor (kh) for glulam beams in bending and tension compared with the theoretical value.

Depth of glulam beam

Fig. 2.8. EC5 factor (kh) for glulam beams in bending and tension compared with the theoretical value.

tapered, curved and pitched cambered glued-laminated timber and LVL beams are summarised in Table 2.11 together with the associated reference criteria.

Where a continuous load distribution system laterally connects a series of equally spaced similar members, the distribution system will enable load sharing to take place between the members. This allows the member strength properties to be increased in value and is achieved by multiplying the relevant properties by a system strength factor, ksys. The factor is only relevant where the system is able to redistribute load. It takes advantage of the fact that stiffer members will take a greater share of the applied load than weaker members and that there will be a low probability that adjacent members in the system will have the same strength and stiffness characteristics.

The continuous load distribution system must be able to transfer the loads on the system from one member to the neighbouring members and for this condition ksys shall be taken to equal 1.1. This can be taken to apply where the load distribution system is as follows:

(a) Structural flooring connected to floor beams where the flooring is continuous over at least two spans, necessitating at least four members, and any joints in the flooring are staggered.

(b) Stud walling connected by sheathing fixed to the studs in accordance with the fixing manufacturers recommendations or as required by the design. The maximum spacing of studs should be taken to be 610 mm c/c.

(c) Tiling battens, purlins or structural panels connected to roof trusses where the load distribution members are continuous over at least two spans, necessitating at least four trusses, with any joints being staggered. The spacing of the trusses must not be greater than 1.2 m.

 Material Factor Definitions/conditions Characteristic or design value Solid timber For bending and tension: IVh) J Characteristic density <700 kg/m3 (i) Bending: reference depth ft = 150 mm. (ii) Tension: reference width (maximum cross sectional dimension) h = 150 mm. (ii) Tensile strength parallel to the grain: = fch/t,o,k Glued- laminated timber For bending and tension and stress distribution: In the apex zone of a double tapered, curved and pitched cambered beam with all veneers parallel to the beam In the apex zone of double tapered and curved beams: kas = 1.4 In the apex zone of pitched cambered beams: kas = 1.7 For the evaluation of k^ (i) Bending: reference depth h = 600 mm. (ii) Tension: reference width (maximum cross sectional dimension) h = 600 mm. For the evaluation of volume factor kY01 The stressed volume of the apex zone (in m3) as defined in (NB: the value used for Vshould not be greater than 21^/3 where Vt, is the volume of the beam.) (ii) Tensile strength parallel to the grain: (i) Tensile strength perpendicular to the grain: = fcdiAoi /t,90,d
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