## Bwz

Emean,f

(b) At the condition associated with the final mean value of stiffness:

(i) where the permanent action produces the design condition:

/ Emeanw \ ( 1 + kdef,f \ Ief,fin = If + —- M ——- Iw (7.6)

(ii) where a variable action condition Qi produces the design condition:

Here the symbols are as previously defined and Af is the total flange area, and for Figure 7.3a Af = (b - 2bw) (f + h^), and for Figure 7.3b Af = (b - bw) (hf,c + h^); Aw is the area of the web, and for Figure 7.3a Aw = 2bw(hw + hfc + hfjt), and for Figure 7.3b, Aw = bw(hw + hf,c + h^);

f2i is the f2 factor for the quasi-variable value of the variable action Q¡; If is the second moment of area of both flanges about the neutral axis;

Iw is the second moment of area of the untransformed web about the neutral axis.

When subjected to the design bending moment, Md, and the modulus of elasticity of the web is less than that of the flange, the bending stresses in the flange will increase and those in the web will decrease with time. For this condition, it will only be necessary to check the stresses in the flanges at the final mean value condition and those in the web at the instantaneous condition. If, however, the modulus of elasticity of the web is greater than that of the flange, the bending stresses should be checked at the instantaneous condition in the flanges and at the final mean value condition in the web. When dealing with symmetrical sections, there will not be a significant difference between the respective values of the geometric properties at both conditions, and consequently the differences in stress will also be relatively small.

In the following sections, the stress equations for the flanges and the web at both the instantaneous and the final mean value condition are given for sections subjected to moment and shear conditions, together with the design condition to be satisfied in EC5.

7.3.1.1.1 Stresses in the flanges

(a) Bending stresses.

The maximum stress due to bending will arise at the extreme fibre locations at a distance of yi or y2 from the neutral axis, as shown in Figure 7.3c. When the section is symmetrical about the y-y axis, yi = y2 and when it is not, yi and y2 will have different values. Also, when the section is not symmetrical about the y-y axis, the position of the neutral axis will be different at the instantaneous and final mean value conditions and the respective values of yi and y2 at these states will also differ. These are referred to in the text as yi,inst and y2,inst for the instantaneous state and yijfin and y2,fin for the final mean value condition.

The maximum design compressive stress due to bending will be at yi, and: at the instantaneous condition it will be:

and at the final mean value condition it will be:

The maximum design tensile stress will be at y2, and at the instantaneous condition it will be:

and at the final mean value condition it will be: ( Md \

The design requirement in Section 9 of EC5 is that the design stress in equations (7.9) and (7.10) be less than or equal to the design bending strength, /m d, i.e.:

where the functions areas described above and /m,d — k mod ■ kh ■ kSyS ■ /m,k/KM, where kmod is the modification factor for load duration and service classes as given in Table 2.4, ksys is the system strength factor discussed in Chapter 2, kh is the modification factor for member size effect, referred to in Table 2.5 and discussed in 2.3.6. The effect applies to solid timber (as well as glulam and LVL, when bent flatwise). /m,kis the characteristic bending strength of the flange material. Strength information for timber and LVL is given in Chapter 1 and for glulam in Chapter 6. ym is the partial coefficient for material properties, given in Table 2.6, noting that the value will be dependent on the material being used.

(b) Compression stresses.

Although a full analysis into the lateral torsional instability behaviour of a glued thin webbed section can be undertaken, in EC5 it is conservatively assumed that lateral stability of the section is provided solely by the buckling strength of the compression flange. The design requirement is that the compression stress in the flange must be shown to be less than or equal to the compression strength. The compression stress is taken to be the average value of the compressive stress in the flange due to bending and for this requirement: at the instantaneous condition it will be:

Of,inst,c,d = Oc,2 — I —- I J1,inst--— II (/.12a)

and at the final mean value condition it will be

( Md ( hf,c\\ Of,fin,c,d = Oc,2 = I J- I J1,fin--^ M (7.12b)

The EC5 design requirement is that the design stress be less than or equal to the modified design compressive strength, i.e. kc /c,0,d, as follows:

Ym where/c,0,d is the design compressive strength of the flange material and /c,0,k is the characteristic compressive strength of the flange material parallel to the grain. Strength information for timber and LVL is given in Chapter 1 and for glulam in Chapter 6. kc is a factor that takes into account lateral instability, and is derived assuming that the compression flange behaves as a column between adjacent positions of lateral restraint. The section is conservatively equated to a solid rectangular section of depth, b, resulting in a radius of gyration about the z-z axis of the composite beam of b /VI2. On this basis, the slenderness ratio of the section will be ^/V2(lc/b), where lc is the length of the section between the adjacent positions of lateral support. Factor kc is then derived using the expressions in 5.4.1. Where full lateral restraint is provided by the floor structure, i.e. kc = 1, it is essential that sufficient fixings be used and located to prevent any lateral movement of the beams.

If a special investigation is made with respect to the lateral torsional instability of the beam as a whole, EC5, 9.1.1(3), allows kc to be assumed to be unity.

(c) Tensile stresses.

The flange must also be checked to ensure that the mean design tensile stress in the tension flange will be less the design tension strength: at the instantaneous condition it will be:

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