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 (EC5, equation (6.43)) (6.12) (EC5, equation (6.44)) (6.13) (EC5, equation (6.45)) (6.14) (EC5, equation (6.46)) (6.15) (EC5, equation (6.47)) (6.16) (EC5, equation (6.48)) (6.17)

r = rin + 0.5hap where rin is the inner radius of the beam, as shown in Figure 6.7c for a pitched cambered beam.

As an alternative to calculation, the value of kt can be obtained from Figure A6.3, where the function is plotted for various angles of aap over a range of hap/rin from 0 to 0.5.

6.4.2.1.2 Radial stress in the apex zone

The radial stress in a double tapered and a pitched cambered beam can be taken to be a maximum at the neutral axis position, as shown in the case of a double tapered beam in Figure 6.8d, and reduces to zero at the top and bottom faces.

Under the action of a design moment, Map, d, at the apex the critical radial stress condition will be the maximum tensile stress perpendicular to the grain, au90 , d, and in accordance with the requirement of NA.2.4 of the UKNA to EC5  the stress will be derived from

' ' bhlp where b and hap are the width and depth, respectively, of the section at the apex as shown in Figure 6.8b and for a double tapered beam kp = 0.2 tan(aap) and for a pitched cambered beam it is obtained from:

kp = k5 + k6 f + k7 f (EC5, equation (6.56)) (6.19)

k6 = 0.25 - 1.5 tan(aap) + 2.6 tan2(aap) (EC5, equation (6.58)) (6.21)

(a) Curved beam elevation h -

(a) Curved beam elevation

(b) Beam section (c) Bending stress

Fig. 6.9. Bending and radial stresses in a curved beam.

kj = 2.1 tan(«ap) - 4 tan2(aap) (EC5, equation (6.59)) (6.22)

As an alternative to calculation, the value of kp can be obtained from Figure A6.4, where the function is plotted for various angles of aap over a range of hap/rin from 0 to 0.5.

6.4.2.2 Bending and radial stresses in the apex zone - of a curved beam

Unlike the case of a uniform straight beam, under the action of a pure moment on a uniform section curved beam, as shown in Figure 6.9, the bending stress distribution across any section along the beam will not be linear. The position of the neutral axis will be below the centroidal axis and the bending stress distribution will be as shown in Figure 6.9c, with the maximum bending stress occurring at the inner radius position.

The radial tension stress induced in the section will be a maximum at the neutral axis position reducing in a non-linear manner to zero at the outer faces as indicated in Figure 6.9d.

6.4.2.2.1 Bending stress in the apex zone

The bending stress distribution in a curved beam can be derived using the Airy stress function written in terms of polar coordinates, and the maximum bending stress will be shown to occur at the inner radius position as indicated in Figure 6.9c. In EC5 a close approximation to the value of the maximum stress at the inner radius position in the apex zone of a curved beam of breadth b and depth hap when subjected to a design moment Map,d is obtained from EC5, equation (6.42), by setting the angle of taper, aap, equal to 0 in EC5, equation (6.43), as follows,

6Md tfm.Cl.d = kcurve,b7T-r (6.23) bhlp where kcurve.b = 1 + 0.35 ^^ + 0.6 ((6"24) and r = rin + 0.5hap as defined in equation (6.17).

As an alternative to calculation, the value of kcurve,b can be obtained from Figure A6.3, where the value of the function will equal the value of kt when taking aap = 0°.

6.4.2.2.2 Radial stress in the apex zone

As with double tapered and pitched cambered beams, the radial stress can be taken to be a maximum at the neutral axis position, and will reduce to zero at the top and bottom faces as shown in Figure 6.9d.

Under the action of a design moment, Map,d, at the apex, in EC5 the maximum tensile stress perpendicular to the grain, at,90,d, at the neutral axis position in the apex zone of a curved beam of breadth b and depth hap is obtained from EC5, equation (6.54), by setting aap (the angle of taper) equal to zero in EC5, equation (6.56). In accordance with the requirement of NA.2.4 in the UKNA to EC5 the stress will be:

6Map,d

Ot,90,d = kcurve.t ,,2 (EC5, equation (6.54)) (6.25)

bh ap where kcurve>t = 0.25(hap/r), and, as defined in equation (6.17), r = rin + 0.5hap.

6.4.2.3 Bending strength in the apex zone - for double tapered beams, curved beams and pitched cambered beams

When a curved or pitched cambered glulam beam is being formed, the laminates are bent to the required curvature for the beam and as a consequence are subjected to a bending stress. This effect is not relevant to double tapered beams as they are produced from conventional glulam beams formed using straight laminations.

To take the effect of this bending stress into account, a stress reduction factor derived from tests is applied to the bending strength of the glulam beam and the reduced design bending strength, /r,mj0jd, is given in EC5, equation (6.41), as follows:

where /m,o,d is defined in equation (6.2) and kr is a reduction factor. As there is no stress reduction in the case of double tapered beams, for this type the factor is taken to be 1. For curved and pitched cambered beams, the factor is based on the ratio of rin/t, and the EC5 requirement is

• when rin/1 > 240 no strength reduction is required,

• when rin/1 is less than 240 the modification factor is 0.76 + 0.001(rin/t).

Here the inner radius of the curved beam, rin, and the lamination thickness, t, are shown in Figure 6.7.

6.4.2.4 Radial tensile strength in the apex zone -for double tapered beams, curved beams and pitched cambered beams

Larsen  and Colling  have shown that with these beam types the tensile strength in the apex zone of the section will be affected by stress and volume effects. Factors for these effects have been developed, and in EC5 the design tensile strength of the beam member at right angles to the grain, fr,t,90,d, is obtained from

where ft,90,d is defined in equation (6.8) and kdis is a stress distribution factor in the apex zone. It is

Table 6.6 The stressed volume of the apex zone

Figure reference

Beam type

Stressed volumet (V)

Maximum allowable value of the stressed volume*

Figure 6.7a Curved beam -b("2p + 2«aprinj

180 p

Figure 6.7b Double tapered beam bh2p(1 — 0.25 tan(aap)) Figure 6.7c Pitched cambered beam b^sin(aap) cos(aap)(r;n + hap)

0 0