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Note: For strength-related calculations, take Gg,05 = E0,g,05/16.

Mean density is taken to be the average of the mean density of the inner and outer laminates, based on BS EN 338.

* Based on the properties given in BS EN 1194:1999.

Note: For strength-related calculations, take Gg,05 = E0,g,05/16.

Mean density is taken to be the average of the mean density of the inner and outer laminates, based on BS EN 338.

* Based on the properties given in BS EN 1194:1999.

In BS EN 1194 examples are given of beam lay-ups complying with the requirements of Table 6.2, where the properties of the laminations in the section have been derived using the characteristic values in BS EN 338:2003 and these are summarised in Table 6.4.

The characteristic bending strength, /m,gjk, in Table 6.2 relates to members with a minimum depth of 600 mm and a minimum thickness of 150 mm, and the tensile strength parallel to the grain, /t,o,g,k, relates to members with a minimum width of 600 mm and a minimum thickness of 150 mm. These strengths can be increased when glulam sections with smaller sizes are used by the application of a size effect factor, referred to in 2.3.6 and Table 2.11. EC5 only permits the factor to be applied to the depth for bending and the width for tension and makes no adjustment for thickness. The value of the factor, kh, is:

Characteristic property of homogeneous glued laminated timber made from softwood laminations |
BS EN 1194:1999 relationship (based on the properties of the softwood laminations*) |

Bending strength fm,g,k (N/mm2) |
= 7 + 1.15 ft,o,l,k |

Tension strength ft,0,g,k (N/mm2) ft,90,g,k (N/mm2) |
= 5 + 0.8 ft,o,1,k = 0.2 + 0.015 ft,o,l,k |

Compression strength fc,0,g,k (N/mm2) fc,90,g,k (N/mm2) |
= 0.7 ft005l,k |

Shear strength fv,k (N/mm2) |
= 0.32 f0081,k |

Modulus of elasticity £o,g,mean (N/mm2) E0.g.05 (N/mm2) E90,g,mean (N/mm2) |
= L°5 E0,l,mean = 0.85 E0,l,mean = 0.035 E0,l,mean |

Shear modulus Gg,mean (N/mm2) Density = 1.10 Pi,k |

* The lamination properties are as follows: ft,o,l,k - the characteristic tensile strength of the softwood lamination in N/mm2; E0lmean - the mean value of the modulus of elasticity of the lamination parallel to the grain in N/mm2; pi,k - the characteristic density of the softwood lamination in kg/m3.

t Based on Table A.1 in BS EN 1194:1999.

* The lamination properties are as follows: ft,o,l,k - the characteristic tensile strength of the softwood lamination in N/mm2; E0lmean - the mean value of the modulus of elasticity of the lamination parallel to the grain in N/mm2; pi,k - the characteristic density of the softwood lamination in kg/m3.

t Based on Table A.1 in BS EN 1194:1999.

where h is the depth of the glulam member (in mm) when subjected to bending and the width of the glulam member (in mm) when subjected to tension.

When using homogeneous glued laminated timber or combined glued laminated timber compliant with the strength classes given in BS EN1194:1999, Table 2 (i.e. the combined glulam strength classes given in Table 6.2 above), conventional bending theory will apply. The design procedure for uniform section straight members subjected to bending, shear, torsion and axial loading will be as described in Chapters 4 and 5, respectively, and the relevant strength property given in Table 6.2 will be used in the strength condition being validated.

If combined glued laminated timber with properties that are not compliant with the GL 24c to GL 38c strength classes is to be formed, bending strength verification must be carried out at all relevant points in the cross-section, and because this will involve the analysis of members with different values of modulus of elasticity, the equivalent section approach (commonly referred to as the modular ratio approach) can be used for the analysis. In this method, the material used in the inner or the outer laminations is

Table 6.4 Examples of beam lay-ups compliant with Table 6.2, in which the characteristic properties are derived from the equations given in Table 6.3 using properties obtained from BS EN 338:2003 for the strength class being used for the laminations.

Glulam strength classes GL 24 GL 28 GL 32

Homogeneous glulam - lamination strength class C24 C30 C40

Combined glulam: outer/inner - lamination strength C24/C18 C30/C24 C40/C30 classes selected for use and the other material is replaced by an equivalent area of the selected material such that the force in the replaced material at any distance from the neutral axis caused by bending of the section will be the same as that to be taken by the original material at the same position. By this method, a section using only one material is formed and the theory of bending can be applied enabling the stress in the material selected for the section to be found directly. For the converted laminations, the stress is obtained by multiplying the calculated stress in these members by the ratio of the E value of the original material to the E value of the selected material (called the modular ratio). The mean value of the glulam stress class, E0 g mean, should be used in the analysis. Further information on the method is given in Chapter 7.

Design values of glued laminated members are derived in the same way as for solid timber sections, using those factors in EC5 that are applicable to the material. For example, to derive the design bending strength of a glulam member compliant with BS EN 1194:1999,

Km where kmod is the modification factor for load duration and service classes as given in Table 2.4, kh is the modification factor for member size effect, referred to in Table 2.5, ksys is the system strength factor discussed in 2.3.7, fm,g,k is the characteristic bending strength of the glulam strength class being used, and km is the partial coefficient for material properties, given in Table 2.6.

6.4 DESIGN OF GLUED-LAMINATED MEMBERS WITH TAPERED, CURVED OR PITCHED CURVED PROFILES (ALSO APPLICABLE TO LVL MEMBERS)

Glued laminated (and LVL) beams can be tapered or curved in order to meet architectural requirements, to provide pitched roofs, to obtain maximum interior clearance and/or to reduce wall height requirements at end supports. The most common forms used in timber structures are double tapered beams and curved beams having a rectangular cross-section, as shown in Figure 6.4.

Because of the sloping surface, with these members the distribution of bending stress is non-linear across any section and, in the apex zone of the beam types shown

(c) Curved beam (d) Pitched cambered beam

Fig. 6.4. Single and double tapered, curved and pitched cambered beams.

in Figures 6.4b-6.4d, radial stresses perpendicular to the grain are also induced. If the bending moment tends to increase the radius of curvature, the radial stresses will be in tension perpendicular to the grain and if it tends to decrease the radius of curvature, the radial stresses will be in compression perpendicular to the grain.

With these types of beams it is recommended that the laminations are set parallel with the tension edge such that the tapered sides will be on the compression face when subjected to normal loading conditions.

### 6.4.1 Design of single tapered beams

These beams are rectangular in section and slope linearly from one end to the other as shown in Figure 6.5. No upper limit is set in EC5 for the angle of slope, a, but in practice it would normally be within the range 0-10°.

These types of beams are used in roof construction and the critical design checks will relate to the maximum shear stress and the maximum bending stress condition at the ULS and the deflection behaviour at the SLS.

Effective span = L

Fig. 6.5. A single tapered beam: (a) elevation; (b) section A-A; (c) bending stress; and (d) bending stress distribution used in EC5 at section A-A.

Effective span = L

Fig. 6.5. A single tapered beam: (a) elevation; (b) section A-A; (c) bending stress; and (d) bending stress distribution used in EC5 at section A-A.

With regard to the maximum shear stress, because of the taper in the beam, the distribution of shear stress will vary across the depth of any section and along the beam length. When subjected to uniformly distributed loading or to a point load at mid-span, the maximum shear stress will occur at mid-height of the beam at the end where the beam depth is hs and will be:

bhs where Vd is the design shear force at the end of the beam.

If there is a requirement to determine the shear stress at any other position along the beam, a reasonably accurate approximation can be obtained using the approach adopted by Maki and Keunzi [5].

The design shear strength of the tapered rectangular beam is derived in the same way as described in 4.5.2 for a uniform rectangular timber section, and is:

Km where the factors are as previously defined and fv g k is the characteristic shear strength of the glulam strength class being used. Where beam lay-ups not compliant with Table 6.2 are used, the shear strength should be based on the characteristic shear strength of the inner laminations.

Also, because of the taper, when subjected to a bending moment, M, the bending stress distribution across the section will be non-linear as indicated in Figure 6.5c and, on the basis of analysis by Riberholt [6], at any cross-section along the tapered beam the maximum stress parallel to the tapered face at the tapered face and the maximum horizontal tensile stress on the horizontal face can be approximated to:

Tapered face bending stress = (1 - 3.7 tan2 a)M/ Wy (6.5a)

Parallel face bending stress = (1 + 3.7 tan2 a)M/ Wy (6.5b)

where Wy is the section modulus about the y-y axis.

In EC5, to simplify the design process, stresses in the section are derived using conventional elastic bending theory ignoring the taper effect. The bending stress at the tapered face acting parallel to the tapered face and the bending stress at the horizontal surface on the bottom face of the beam, as shown in Figure 6.5c, are taken to equal the bending stresses derived assuming there is no taper, as shown in Figure 6.5d. As is seen from equations (6.5) this will be a safe approximation for the stress at the tapered face but will slightly underestimate the stress at the bottom face. For beam tapers up to about 10% the stress on the bottom parallel face will be underestimated by a maximum of around 11%. When the taper increases beyond 10% the underestimate in the value of the stress on the bottom face increases relatively rapidly and it is recommended that in such circumstances equation (6.5b) be used to derive the maximum bending stress at this position and that the stress be validated against the bending strength of the section.

On the basis of the EC5 approximation, when subjected to a design moment Md at a position x measured from the end where the beam depth is hs, as shown in Figure 6.5, the maximum bending stress at the tapered face of the beam at an angle a to the grain and at the outermost fibre on the beam face parallel to the grain will be:

&m,a,d = tfm.o.d = — (EC5, equation (6.37)) (6.6)

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