HANDBOOK 2 DESIGN OF TIMBER STRUCTURES ..., Study notes of Design and Analysis of Algorithms

Download HANDBOOK 2 DESIGN OF TIMBER STRUCTURES ... and more Study notes Design and Analysis of Algorithms in PDF only on Docsity! HANDBOOK 2 DESIGN OF TIMBER STRUCTURES ACCORDING TO EUROCODE 5 3 Preface This handbook makes specific reference to design of timber structures to European Standards and using products available in Europe. The handbook is closely linked to Eurocode 5 (EC5), the European code for the design of timber structures. This handbook is explaining the general philosophy of the Eurocode 5 and giving the basic background for its requirements and design rules. For better understanding of the Eurocode 5 design rules the worked examples are presented. The purpose of this handbook is to introduce readers to the design of timber structures. It is designed to serve either as a text for a course in timber structures or as a reference for systematic self-study of the subject. May 2008 Authors 6 2 Design of timber structures Before starting formal calculations it is necessary to analyse the structure and set up an appro- priate design model. In doing this there may be a conflict between simple, but often conserva- tive, models which make the calculations easy, and more complicated models which better reflect the behaviour but with a higher risk of making errors and overlooking failure modes. The geometrical model must be compatible with the expected workmanship. For structures sensitive to geometrical variations it is especially important to ensure that the structure is produced as assumed during design. The influence of unavoidable deviations from the assumed geometry and of displacements and deformations during loading should be estima- ted. Connections often require large areas of contact and this may give rise to local excentricities which may have an important influence. Often there is a certain freedom as regards the modelling as long as a consistent set of assumptions is used. The Eurocodes are limit state design codes, meaning that the requirements concerning structural reliability are linked to clearly defined states beyond which the structure no longer satisfies specified performance criteria. In the Eurocode system only two types of limit states are considered: ultimate limit states and serviceability limit states. Ultimate limit states are those associated with collapse or with other forms of structural failure. Ultimate limit states include: loss of equilibrium; failure through excessive deforma- tions; transformation of the structure into a mechanism; rupture; loss of stability. Serviceability limit states include: deformations which affect the appearance or the effective use of the structure; vibrations which cause discomfort to people or damage to the structure; damage (including cracking) which is likely to have an adverse effect on the durability of the structure. In the Eurocodes the safety verification is based on the partial factor method described below. 2.1 Principles of limit state design The design models for the different limit states shall, as appropriate, take into account the following: − different material properties (e.g. strength and stiffness); − different time-dependent behaviour of the materials (duration of load, creep); − different climatic conditions (temperature, moisture variations); − different design situations (stages of construction, change of support conditions). 2.1.1 Ultimate limit states The analysis of structures shall be carried out using the following values for stiffness properties: − for a first order linear elastic analysis of a structure, whose distribution of internal forces is not affected by the stiffness distribution within the structure (e.g. all members have the same time-dependent properties), mean values shall be used; − for a first order linear elastic analysis of a structure, whose distribution of internal forces is affected by the stiffness distribution within the structure (e.g. composite members 7 containing materials having different time-dependent properties), final mean values adjusted to the load component causing the largest stress in relation to strength shall be used; − for a second order linear elastic analysis of a structure, design values, not adjusted for duration of load, shall be used. The slip modulus of a connection for the ultimate limit state, Ku , should be taken as: u ser 2 3 K K= (2.1) where Kser is the slip modulus. 2.1.2 Serviceability limit states The deformation of a structure which results from the effects of actions (such as axial and shear forces, bending moments and joint slip) and from moisture shall remain within appropriate limits, having regard to the possibility of damage to surfacing materials, ceilings, floors, partitions and finishes, and to the functional needs as well as any appearance requirements. The instantaneous deformation, uinst, see Chapter 7, should be calculated for the characteristic combination of actions using mean values of the appropriate moduli of elasticity, shear moduli and slip moduli. The final deformation, ufin, see Chapter 7, should be calculated for the quasi-permanent combination of actions. If the structure consists of members or components having different creep behaviour, the final deformation should be calculated using final mean values of the appropriate moduli of elasticity, shear moduli and slip moduli. For structures consisting of members, components and connections with the same creep behaviour and under the assumption of a linear relationship between the actions and the corresponding deformations the final deformation, ufin, may be taken as: 1 ifin fin,G fin,Q fin,Qu = u u u+ + (2.2) where: ( )fin,G inst,G def1u = u k+ for a permanent action, G (2.3) ( )fin,Q,1 inst,Q,1 2,1 def1u = u kψ+ for the leading variable action, Q1 (2.4) ( )fin,Q,i inst,Q,i 0,i 2,i defu = u kψ ψ+ for accompanying variable actions, Qi (i > 1) (2.5) inst,Gu , inst,Q,1u , inst,Q,iu are the instantaneous deformations for action G, Q1, Qi respectively; ψ2,1, ψ2,i are the factors for the quasi-permanent value of variable actions; ψ0,i are the factors for the combination value of variable actions; kdef is given in Chapter 3 for timber and wood-based materials, and in Chapter 2 for connections. 8 For serviceability limit states with respect to vibrations, mean values of the appropriate stiffness moduli should be used. 2.2 Basic variables The main variables are the actions, the material properties and the geometrical data. 2.2.1 Actions and environmental influences Actions to be used in design may be obtained from the relevant parts of EN 1991. Note 1: The relevant parts of EN 1991 for use in design include: EN 1991-1-1 Densities, self-weight and imposed loads EN 1991-1-3 Snow loads EN 1991-1-4 Wind actions EN 1991-1-5 Thermal actions EN 1991-1-6 Actions during execution EN 1991-1-7 Accidental actions Duration of load and moisture content affect the strength and stiffness properties of timber and wood-based elements and shall be taken into account in the design for mechanical resistance and serviceability. Load-duration classes The load-duration classes are characterised by the effect of a constant load acting for a certain period of time in the life of the structure. For a variable action the appropriate class shall be determined on the basis of an estimate of the typical variation of the load with time. Actions shall be assigned to one of the load-duration classes given in Table 2.1 for strength and stiffness calculations. Table 2.1 Load-duration classes Load-duration class Order of accumulated duration of characteristic load Permanent more than 10 years Long-term 6 months – 10 years Medium-term 1 week – 6 months Short-term less than one week Instantaneous NOTE: Examples of load-duration assignment are given in Table 2.2 11 def def,1 def,22 = kk k (2.13) where kdef,1 and kdef,2 are the deformation factors for the two timber elements. 2.3 Verification by the partial factor method A low probability of getting action values higher than the resistances, in the partial factor method, is achieved by using design values found by multiplying the characteristic actions and dividing the characteristic strength parameters, by partial safety factors. 2.3.1 Design value of material property The design value Xd of a strength property shall be calculated as: k d mod M X X k γ = (2.14) where: Xk is the characteristic value of a strength property; γM is the partial factor for a material property; kmod is a modification factor taking into account the effect of the duration of load and moisture content. NOTE 1: Values of kmod are given in Chapter 3. NOTE 2: The recommended partial factors for material properties (γM) are given in Table 2.3. Information on the National choice may be found in the National annex of each country. Table 2.3 Recommended partial factors γM for material properties and resistances Fundamental combinations: Solid timber 1,3 Glued laminated timber 1,25 LVL, plywood, OSB, 1,2 Particleboards 1,3 Fibreboards, hard 1,3 Fibreboards, medium 1,3 Fibreboards, MDF 1,3 Fibreboards, soft 1,3 Connections 1,3 Punched metal plate fasteners 1,25 Accidental combinations 1,0 The design member stiffness property Ed or Gd shall be calculated as: 12 mean d M E E γ = (2.15) mean d M G G γ = (2.16) where: Emean is the mean value of modulus of elasticity; Gmean is the mean value of shear modulus. 2.3.2 Design value of geometrical data Geometrical data for cross-sections and systems may be taken as nominal values from product standards hEN or drawings for the execution. Design values of geometrical imperfections specified in this handbook comprise the effects of − geometrical imperfections of members; − the effects of structural imperfections from fabrication and erection; − inhomogeneity of materials (e.g. due to knots). 2.3.3 Design resistances The design value Rd of a resistance (load-carrying capacity) shall be calculated as: k d mod M R R k γ = (2.17) where: Rk is the characteristic value of load-carrying capacity; γM is the partial factor for a material property, kmod is a modification factor taking into account the effect of the duration of load and moisture content. NOTE 1: Values of kmod are given in Chapter 3. NOTE 2: For partial factors, see Table 2.3. 3 Design values of material properties Eurocode 5 in common with the other Eurocodes provides no data on strength and stiffness properties for structural materials. It merely states the rules appropriate to the determination of these values to achieve compatibility with the safety format and the design rules of EC5. 3.1 Introduction Strength and stiffness parameters Strength and stiffness parameters shall be determined on the basis of tests for the types of action effects to which the material will be subjected in the structure, or on the basis of comparisons with similar timber species and grades or wood-based materials, or on well- established relations between the different properties. Stress-strain relations Since the characteristic values are determined on the assumption of a linear relation between stress and strain until failure, the strength verification of individual members shall also be based on such a linear relation. For members or parts of members subjected to compression, a non-linear relationship (elastic- plastic) may be used. Strength modification factors for service classes and load-duration classes The values of the modification factor kmod given in Table 3.1 should be used. If a load combination consists of actions belonging to different load-duration classes a value of kmod should be chosen which corresponds to the action with the shortest duration, e.g. for a combination of dead load and a short-term load, a value of kmod corresponding to the short- term load should be used. Table 3.1 Values of kmod Load-duration class Material Standard Service class Permanent action Long term action Medium term action Short term action Instanta- neous action 1 0,60 0,70 0,80 0,90 1,10 2 0,60 0,70 0,80 0,90 1,10 Solid timber EN 14081-1 3 0,50 0,55 0,65 0,70 0,90 1 0,60 0,70 0,80 0,90 1,10 2 0,60 0,70 0,80 0,90 1,10 Glued laminated timber EN 14080 3 0,50 0,55 0,65 0,70 0,90 1 0,60 0,70 0,80 0,90 1,10 2 0,60 0,70 0,80 0,90 1,10 LVL EN 14374, EN 14279 3 0,50 0,55 0,65 0,70 0,90 EN 636 Part 1, Part 2, Part 3 1 0,60 0,70 0,80 0,90 1,10 Part 2, Part 3 2 0,60 0,70 0,80 0,90 1,10 Plywood Part 3 3 0,50 0,55 0,65 0,70 0,90 16 0,2 h 150 min 1,3 h k      =     (3.1) where h is the depth for bending members or width for tension members, in mm. For timber which is installed at or near its fibre saturation point, and which is likely to dry out under load, the values of kdef, given in Table 3.2, should be increased by 1,0. Finger joints shall comply with EN 385. 3.3 Glued laminated timber Glued laminated timber members shall comply with EN 14080. NOTE: Values of strength and stiffness properties are given for glued laminated timber allocated to strength classes in EN 1194. Formulae for calculating the mechanical properties of glulam from the lamination properties are given in Table 3.3. The basic requirements for the laminations which are used in the formulae of Table 3.3 are the tension characteristic strength and the mean modulus of elasticity. The density of the laminations is an indicative property. These properties shall be either the tabulated values given in EN 338 or derived according to the principles given in EN 1194. The requirements for glue line integrity are based on the testing of the glue line in a full cross- sectional specimen, cut from a manufactured member. Depending on the service class, delamination tests (according to EN 391 “Glued laminated timber - delamination test of glue lines”) or block shear tests (according to EN 392 “Glued laminated timber - glue line shear test”) must be performed. Table 3.3 Mechanical properties of glued laminated timber (in N/mm2) Property Bending , ,m g kf ,0, ,7 1,15 t l kf= + Tension ,0, ,t g kf ,90, ,t g kf ,0, ,5 0,8 t l kf= + ,0, ,0, 2 0,015 t l kf= + Compresion ,0, ,c g kf ,90, ,c g kf 0,45 ,0, ,7,2 t l kf= 0,5 ,0, ,0,7 t l kf= Shear , ,v g kf 0.8 ,0, ,0,32 t l kf= Modulus of elasticity 0, ,g meanE 0, ,05gE 90, ,g meanE 0, ,1,05 l meanE= 0, ,0,85 l meanE= 0, ,0,035 l meanE= Shear modulus ,g meanG 0, ,0,065 l meanE= 17 Density ,g kρ ,1,10 l kρ= NOTE: For combined glued laminated timber the formulae apply to the properties of the individual parts of the cross- section. It is assumed that zones of different lamination grades amount to at least 1/6 of the beam depth or two laminations, whichever is the greater. The effect of member size on strength may be taken into account. For rectangular glued laminated timber, the reference depth in bending or width in tension is 600 mm. For depths in bending or widths in tension of glued laminated timber less than 600 mm the characteristic values for fm,k and ft,0,k may be increased by the factor kh ,given by 0,1 h 600 min 1,1 h k      =     (3.2) where h is the depth for bending members or width for tensile members, in mm. Large finger joints complying with the requirements of ENV 387 shall not be used for products to be installed in service class 3, where the direction of grain changes at the joint. The effect of member size on the tensile strength perpendicular to the grain shall be taken into account. 3.4 Laminated veneer lumber (LVL) LVL structural members shall comply with EN 14374. For rectangular LVL with the grain of all veneers running essentially in one direction, the effect of member size on bending and tensile strength shall be taken into account. The reference depth in bending is 300 mm. For depths in bending not equal to 300 mm the characteristic value for fm,k should be multiplied by the factor kh ,given by h 300 min 1,2 s h k      =     (3.3) where: h is the depth of the member, in mm; s is the size effect exponent, see below. The reference length in tension is 3000 mm. For lengths in tension not equal to 3000 mm the characteristic value for ft,0,k should be multiplied by the factor kℓ given by 18 / 2 3000 min 1,1 s k      =     l l (3.4) where ℓ is the length, in mm. The size effect exponent s for LVL shall be taken as declared in accordance with EN 14374. Large finger joints complying with the requirements of ENV 387 shall not be used for products to be installed in service class 3, where the direction of grain changes at the joint. For LVL with the grain of all veneers running essentially in one direction, the effect of member size on the tensile strength perpendicular to the grain shall be taken into account. 3.5 Wood-based panels Wood-based panels shall comply with EN 13986 and LVL used as panels shall comply with EN 14279. The use of softboards according to EN 622-4 should be restricted to wind bracing and should be designed by testing. 3.6 Adhesives Adhesives for structural purposes shall produce joints of such strength and durability that the integrity of the bond is maintained in the assigned service class throughout the expected life of the structure. Adhesives which comply with Type I specification as defined in EN 301 may be used in all service classes. Adhesives which comply with Type II specification as defined in EN 301 should only be used in service classes 1 or 2 and not under prolonged exposure to temperatures in excess of 50 °C. 3.7 Metal fasteners Metal fasteners shall comply with EN 14592 and metal connectors shall comply with EN 14545. 21 5 Durability Timber is susceptible to biological attack whereas metal components may corrode. Under ideal conditions timber structures can be in use for centuries without significant biological deterioration. However, if conditions are not ideal, many widely used wood species need a preservative treatment to be protected from the biological agencies responsible for timber degradation, mainly fungi and insects. 5.1 Resistance to biological organisms and corrosion Timber and wood-based materials shall either have adequate natural durability in accordance with EN 350-2 for the particular hazard class (defined in EN 335-1, EN 335-2 and EN 335-3), or be given a preservative treatment selected in accordance with EN 351-1 and EN 460. NOTE 1: Preservative treatment may affect the strength and stiffness properties. NOTE 2: Rules for specification of preservation treatments are given in EN 350-2 and EN 335. Metal fasteners and other structural connections shall, where necessary, either be inherently corrosion-resistant or be protected against corrosion. Examples of minimum corrosion protection or material specifications for different service classes are given in Table 5.1. Table 5.1 Examples of minimum specifications for material protection against corrosion for fasteners (related to ISO 2081) Service Classb Fastener 1 2 3 Nails and screws with d ≤ 4 mm None Fe/Zn 12ca Fe/Zn 25ca Bolts, dowels, nails and screws with d > 4 mm None None Fe/Zn 25ca Staples Fe/Zn 12ca Fe/Zn 12ca Stainless steel Punched metal plate fasteners and steel plates up to 3 mm thickness Fe/Zn 12ca Fe/Zn 12ca Stainless steel Steel plates from 3 mm up to 5 mm in thickness None Fe/Zn 12ca Fe/Zn 25ca Steel plates over 5 mm thickness None None Fe/Zn 25ca a If hot dip zinc coating is used, Fe/Zn 12c should be replaced by Z275 and Fe/Zn 25c by Z350 in accordance with EN 10147 b For especially corrosive conditions consideration should be given to heavier hot dip coatings or stainless steel. 22 5.2 Biological attack The two main biological agencies responsible for timber degradation are fungi and insects although in specific situations, timber can also be attacked by marine borers. Fungal attack This occurs in timber which has a high moisture content, generally between 20 % and 30 %. Insect attack Insect attack is encouraged by warm conditions which favour their development and reproduction. 5.3 Classification of hazard conditions The levels of exposure to moisture are defined differently in EC5 and EN 335-I “Durability of wood and wood-based products - Definition of hazard (use) classes of biological attack - Part 1: General”. EC5 provides for three service classes relating to the variation of timber performance with moisture content, see Chapter 2. In EN 335-1, five hazard (use) classes are defined with respect to the risk of biological attacks: Hazard (use) class 1, situation in which timber or wood-based product is under cover, fully protected from the weather and not exposed to wetting; Hazard (use) class 2, situation in which timber or wood-based product is under cover and fully protected from the weather but where high environmental humidity can lead to occasional but not persistent wetting; Hazard (use) class 3, situation in which timber or wood-based product is not covered and not in contact with the ground. It is either continually exposed to the weather or is protected from the weather but subject to frequent wetting; Hazard (use) class 4, situation in which timber or wood-based product is in contact with the ground or fresh water and thus is permanently exposed to wetting; Hazard (use) class 5, situation in which timber or wood-based product is permanently exposed to salt water. 5.4 Prevention of fungal attack It is possible to reduce the risk through careful construction details, especially to reduce timber moisture content. 5.5 Prevention of insect attack Initially, the natural durability of the selected timber species should be established with respect to the particular insect species to which it may be exposed. It is also necessary to establish whether the particular insect is present in the region in which the timber to be used. 23 6 Ultimate limit states Timber structures are generally analysed using elastic structural analysis techniques the world over. This is quite appropriate for the serviceability limit state (which is fairly representative of the performance of the structure from year to year). Even the ultimate limit state (which models the failure of structural element under an extreme loading condition) can be reasonably modelled using an elastic analysis. 6.1 Design of cross-sections subjected to stress in one principal direction This section deals with the design of simple members in a single action. 6.1.1 Assumptions Section 6.1 applies to straight solid timber, glued laminated timber or wood-based structural products of constant cross-section, whose grain runs essentially parallel to the length of the member. The member is assumed to be subjected to stresses in the direction of only one of its principal axes (see Figure 6.1). Key: (1) direction of grain Figure 6.1 Member Axes 6.1.2 Tension parallel to the grain Tension members generally have a uniform tension field throughout the length of the member, and the entire cross section, which means that any corner at any point on the member has the potential to be a critical location. However a bending member under uniformly distributed loading will have a bending moment diagram that varies from zero at each end to the maximum at the centre. The critical locations for tension are near to the centre, and only one half of the beam cross section will have tension, so the volume of the member that is critical for flaws is much less than that for tension members. The inhomogeneities and other deviations from an ideal orthotropic material, which are typical for structural timber, are often called defects. As just mentioned, these defects will cause a fairly large strength reduction in tension parallel to the grain. For softwood (spruce, fir) typical average value are in the range of ,0tf = 10 to 35 N/mm2. In EC5 the characteristic strength values of solid timber are related to a width in tension parallel to the grain of 150 mm. For widths in tension of solid timber less than I50 mm the characteristic values may be increased by a factor hk . For glulam the reference width is 600 mm and, analogously, for widths smaller than 600 mm a factor hk should be applied. 26 The value of kc,90 should be taken as 1,0, unless the member arrangements in the following paragraphs apply. In these cases the higher value of kc,90 specified may be taken, up to a limiting value of kc,90 = 4,0. NOTE: When a higher value of kc,90 is used, and contact extends over the full member width b, the resulting compressive deformation at the ultimate limit state will be approximately 10 % of the member depth. For a beam member resting on supports (see Figure 6.4), the factor kc,90 should be calculated from the following expressions: − When the distance from the edge of a support to the end of a beam a, ≤ h/3: c,90 2,38 1 250 12 h k   = − +      l l (6.4) − At internal supports: c,90 2,38 1 250 6 h k   = − +      l l (6.5) where: l is the contact length in mm; h is member depth in mm. Figure 6.4 Beam on supports For a member with a depth h ≤ 2,5b where a concentrated force with contact over the full width b of the member is applied to one face directly over a continuous or discrete support on the opposite face, see Figure 6.5, the factor kc,90 is given by: 0,5 ef c,90 2,38 250 k   = −      ll l (6.6) where: lef is the effective length of distribution, in mm; l is the contact length, see Figure 6.5, in mm. 27 Figure 6.5 Determination of effective lengths for a member with h/b ≤ 2,5, (a) and (b) continuous support, (c) discrete supports The effective length of distribution lef should be determined from a stress dispersal line with a vertical inclination of 1:3 over the depth h, but curtailed by a distance of a/2 from any end, or a distance of l1/4 from any adjacent compressed area, see Figure 6.5a and b. For the particular positions of forces below, the effective length is given by: - for loads adjacent to the end of the member, see Figure 6.5a ef 3 h= +l l (6.7) - when the distance from the edge of a concentrated load to the end of the member a, 2 3 h≥ ,see Figure 6.5b ef 2 3 h= +l l (6.8) where h is the depth of the member or 40 mm, whichever is the largest. For members on discrete supports, provided that a ≥ h and 1 2 ,≥ hl see Figure 6.5c, the effective length should be calculated as: ef s 2 0,5 3 h = + +    l l l (6.9) where h is the depth of the member or 40 mm, whichever is the largest. 28 For a member with a depth h > 2,5b loaded with a concentrated compressive force on two opposite sides as shown in Figure 6.6b, or with a concentrated compressive force on one side and a continuous support on the other, see Figure 6.6a, the factor kc,90 should be calculated according to expression (6.10), provided that the following conditions are fulfilled: − the applied compressive force occurs over the full member width b; − the contact length l is less than the greater of h or 100 mm: ef c,90k = l l (6.10) where: l is the contact length according to Figure 6.6; lef is the effective length of distribution according to Figure 6.6 The effective length of distribution should not extend by more than l beyond either edge of the contact length. For members whose depth varies linearly over the support (e.g. bottom chords of trusses at the heel joint), the depth h should be taken as the member depth at the centreline of the support, and the effective length lef should be taken as equal to the contact length l. 31 NOTE: The shear strength for rolling shear is approximately equal to twice the tension strength perpendicular to grain. Figure 6.7(a) Member with a shear stress component parallel to the grain (b) Member with both stress components perpendicular to the grain (rolling shear) At supports, the contribution to the total shear force of a concentrated load F acting on the top side of the beam and within a distance h or hef from the edge of the support may be disregarded (see Figure 6.8). For beams with a notch at the support this reduction in the shear force applies only when the notch is on the opposite side to the support. Figure 6.8 Conditions at a support, for which the concentrated force F may be disregarded in the calculation of the shear force 6.1.8 Torsion Torsional stresses are introduced when the applied load tends to twist a member. This will occur when a beam supports a load which is applied eccentric to the principal cross sectional axis. A transmission mast may be subjected to an eccentric horizontal load, resulting in a combination of shear and torsion. The following expression shall be satisfied: tor,d shape v,d k fτ ≤ (6.14) with shape 1,2 for a circular cross section 1+0,15 min for a rectangular cross section 2,0 h k b    =      (6.15) 32 where: τtor,d is the design torsional stress; fv,d is the design shear strength; kshape is a factor depending on the shape of the cross-section; h is the larger cross-sectional dimension; b is the smaller cross-sectional dimension. 6.2 Design of cross-sections subjected to combined stresses While the design of many members is to resist a single action such as bending, tension or compression, there are many cases in which members are subjected to two of these additions simultaneously. 6.2.1 Assumptions Section 6.2 applies to straight solid timber, glued laminated timber or wood-based structural products of constant cross-section, whose grain runs essentially parallel to the length of the member. The member is assumed to be subjected to stresses from combined actions or to stresses acting in two or three of its principal axes. 6.2.2 Compression stresses at an angle to the grain Interaction of compressive stresses in two or more directions shall be taken into account. The compressive stresses at an angle α to the grain, (see Figure 6.9), should satisfy the following expression: c,0,d c,α,d c,0,d 2 2 c,90 c,90,d sin cos f f k f σ α α ≤ + (6.16) where: σc,α,d is the compressive stress at an angle α to the grain; kc,90 is a factor given in 6.1.5 taking into account the effect of any of stresses perpendicular to the grain. Figure 6.9 Compressive stresses at an angle to the grain 6.2.3 Combined bending and axial tension The following expressions shall be satisfied: m,y,dt,0,d m,z,d m t,0,d m,y,d m,z,d 1k f f f σσ σ + + ≤ (6.17) 33 m,y,dt,0,d m,z,d m t,0,d m,y,d m,z,d 1k f f f σσ σ + + ≤ (6.18) The values of km given in 6.1.6 apply. 6.2.4 Combined bending and axial compression The following expressions shall be satisfied: 2 m,y,dc,0,d m,z,d m c,0,d m,y,d m,z,d 1k f f f σσ σ    + + ≤     (6.19) 2 m,y,dc,0,d m,z,d m c,0,d m,y,d m,z,d 1k f f f σσ σ    + + ≤     (6.20) The values of km given in 6.1.6 apply. NOTE: To check the instability condition, a method is given in 6.3. 6.3 Stability of members When a slender column is loaded axially, there exists a tendency for it to deflect sideways (see Figure 6.10). This type of instability is called flexural buckling. The strength of slender members depends not only on the strength of the material but also on the stiffness, in the case of timber columns mainly on the bending stiffness. Therefore, apart from the compression and bending strength, the modulus of elasticity is an important material property influencing the load-bearing capacity of slender columns. The additional bending stresses caused by lateral deflections are taken into account in a stability design. When designing beams, the prime concern is to provide adequate load carrying capacity and stiffness against bending about its major principal axis, usually in the vertical plane. This leads to a cross-sectional shape in which the stiffness in the vertical plane is often much greater than that in the horizontal plane.Whenever a slender structural element is loaded in its stiff plane (axially in the case of the column) there is a tendency for it to fail by buckling in a more flexible plane (by deflecting sideways in the case of the column). The response of a slender simply supported beam, subjected to bending moments in the vertical plane; is termed lateral-torsional buckling as it involves both lateral deflection and twisting (see Figure 6.11). Figure 6.10 Two-hinged column buckling in compression 36 G0,05 is the fifth percentile value of shear modulus parallel to grain; Iz is the second moment of area about the weak axis z. Itor is the torsional moment of inertia; lef is the effective length of the beam, depending on the support conditions and the load configuration, acccording to Table 6.1; Wy is the section modulus about the strong axis y. For softwood with solid rectangular cross-section, σm,crit should be taken as: 2 m,crit 0,05 ef 0,78b E h σ = l (6.32) where: b is the width of the beam; h is the depth of the beam. In the case where only a moment My exists about the strong axis y, the stresses should satisfy the following expression: m,d crit m,d k fσ ≤ (6.33) where: σm,d is the design bending stress; fm,d is the design bending strength; kcrit is a factor which takes into account the reduced bending strength due to lateral buckling. Table 6.1 Effective length as a ratio of the span Beam type Loading type lef/l a Simply supported Constant moment Uniformly distributed load Concentrated force at the middle of the span 1,0 0,9 0,8 Cantilever Uniformly distributed load Concentrated force at the free end 0,5 0,8 a The ratio between the effective length lef and the span l is valid for a beam with torsionally restrained supports and loaded at the centre of gravity. If the load is applied at the compression edge of the beam, lef should be increased by 2h and may be decreased by 0,5h for a load at the tension edge of the beam. For beams with an initial lateral deviation from straightness within the limits kcrit may be determined from expression (6.34) 37 rel,m crit rel,m rel,m rel,m2 rel,m 1 for 0,75 1,56 - 0,75 for 0,75 1, 4 1 for 1, 4 k λ λ λ λ λ  ≤  = < ≤   <  (6.34) The factor kcrit may be taken as 1,0 for a beam where lateral displacement of its compressive edge is prevented throughout its length and where torsional rotation is prevented at its supports. In the case where a combination of moment My about the strong axis y and compressive force Nc exists, the stresses should satisfy the following expression: 2 m,d c,d crit m,d c,z c,0,d 1 k f k f σ σ  + ≤     (6.35) where: σm,d is the design bending stress; σc,d is the design compressive stress; fc,0,d is the design compressive strength parallel to grain; kc,z is given by expression (6.26). 6.4 Design of cross-sections in members with varying cross-section or curved shape Due to the range of sizes, lengths and shapes available, glulam is frequently used for different beams. It is rare for sawn timber to be used as tapered or curved beams because of the difficulty obtaining large sized cross section material and difficulties in bending it about its major axis to give a curved longitudinal profile. 6.4.1 Assumptions The effects of combined axial force and bending moment shall be taken into account. The relevant parts of 6.2 and 6.3 should be verified. The stress at a cross-section from an axial force may be calculated from N N A σ = (6.36) where: σN is the axial stress; N is the axial force; A is the area of the cross-section. 38 6.4.2 Single tapered beams The influence of the taper on the bending stresses parallel to the surface shall be taken into account. Key: (1) cross-section Figure 6.12 Single tapered beam The design bending stresses, σm,α,d and σm,0,d (see Figure 6.12) may be taken as: d m, ,d m,0,d 2 6 M b hασ σ= = (6.37) At the outermost fibre of the tapered edge, the stresses should satisfy the following expression: m,α,d m,α m,d k fσ ≤ (6.38) where: σm,α,d is the design bending stress at an angle to grain; fm,d is the design bending strength; km,α should be calculated as: For tensile stresses parallel to the tapered edge: m,α 2 2 m,d m,d 2 t,90,dv,d 1 1 tan tan 0,75 k f f f f α α =     + +         (6.39) For compressive stresses parallel to the tapered edge: m,α 2 2 m,d m,d 2 c,90,dv,d 1 1 tan tan 1,5 k f f f f α α =     + +         (6.40) 6.4.3 Double tapered, curved and pitched cambered beams This section applies only to glued laminated timber and LVL. The requirements of 6.4.2 apply to the parts of the beam which have a single taper. 41 with 0,2 vol 0 1,0 for solid timber for glued laminated timber and LVL with all veneers parallel to the beam axis k V V  =       (6.51) dis 1,4 for double tapered and curved beams 1,7 for pitched cambered beams k  =   (6.52) where: kdis is a factor which takes into account the effect of the stress distribution in the apex zone; kvol is a volume factor; ft,90,d is the design tensile strength perpendicular to the grain; V0 is the reference volume of 0,01m³; V is the stressed volume of the apex zone, in m3, (see Figure 6.13) and should not be taken greater than 2Vb/3, where Vb is the total volume of the beam. For combined tension perpendicular to grain and shear the following expression shall be satisfied: t,90,dd dis volv,d t,90,d 1 k k ff στ + ≤ (6.53) where: τd is the design shear stress; fv,d is the design shear strength; σt,90,d is the design tensile stress perpendicular to grain; kdis and kvol are given in expressions (6.51) and (6.52). The greatest tensile stress perpendicular to the grain due to the bending moment should be calculated as follows: ap,d p 2t,90,d ap 6 M k b h σ = (6.54) or, as an alternative to expression (6.54), as ap,d d p 2t,90,d ap 6 0,6 M p k bb h σ = − (6.55) where: pd is the uniformly distributed load acting on the top of the beam over the apex area; b is the width of the beam; Map,d is the design moment at apex resulting in tensile stresses parallel to the inner curved edge; 42 with: 2 ap ap p 5 6 7 h h k k k k r r    = + +        (6.56) 5 ap0, 2 tank α= (6.57) 2 6 ap ap0,25 - 1,5 tan 2,6 tank α α= + (6.58) 2 7 ap ap2,1 tan - 4 tank α α= (6.59) 6.5. Notched members It is not uncommon for the ends of beams to be notched at the bottom to increase clearance or to bring the top surface of a particular beam, level with other beams or girdes. Notches usually create stress concentrations in the region of the re-entrant cornes. 6.5.1 Assumptions The effects of stress concentrations at the notch shall be taken into account in the strength verification of members. The effect of stress concentrations may be disregarded in the following cases: − tension or compression parallel to the grain; − bending with tensile stresses at the notch if the taper is not steeper than 1:i = 1:10, that is i ≥ 10, see Figure 6.14a; − bending with compressive stresses at the notch, see Figure 6.14b. a) b) Figure 6.14 Bending at a notch: a) with tensile stresses at the notch, b) with compressive stresses at the notch 6.5.2 Beams with a notch at the support For beams with rectangular cross-sections and where grain runs essentially parallel to the length of the member, the shear stresses at the notched support should be calculated using the effective (reduced) depth hef (see Figure 6.15). It should be verified that d v v,d ef 1,5 V k f b h τ = ≤ (6.60) where kv is a reduction factor defined as follows: 43 − For beams notched at the opposite side to the support (see Figure 6.15b) v 1,0k = (6.61) − For beams notched on the same side as the support (see Figure 6.15a) v 1,5 n 2 1 min 1,1 1 1 (1 - ) 0,8 - k i k h x h h α α αα       =    +      +    (6.62) where: i is the notch inclination (see Figure 6.15a); h is the beam depth in mm; x is the distance from line of action of the support reaction to the corner of the notch; efh h α = n 4,5 for LVL 5 for solid timber 6,5 for glued laminated timber k  =    (6.63) Figure 6.15 End-notched beams 6.6 System strength When several equally spaced similar members, components or assemblies are laterally connected by a continuous load distribution system, the member strength properties may be multiplied by a system strength factor ksys. 46 − winst is the instantaneous deflection; − wcreep is the creep deflection; − wfin is the final deflection; − wnet,fin is the net final deflection. Figure 7.1 Components of deflection The net deflection below a straight line between the supports, wnet,fin, should be taken as: net,fin inst creep c fin cw w w w w w= + − = − (7.2) NOTE: The recommended range of limiting values of deflections for beams with span l is given in Table 7.2 depending upon the level of deformation deemed to be acceptable. Table 7.2 Examples of limiting values for deflections of beams winst wnet,fin wfin Beam on two supports l/300 to l/500 l/250 to l/350 l/150 to l/300 Cantilevering beams l/150 to l/250 l/125 to l/175 l/75 to l/150 7.3 Vibrations In general there are many load-response cases where structural vibrations may constitute a state of reduced serviceability. The main concern, however, is with regard to human discomfort. People are in most cases the critical sensor of vibration. Among different dynamic actions, human activity and installed machinery are regarded as the two most important interval sources of vibration in timber-framed buildings. Human activity not only includes footfall from normal walking, but also children’s jumping, etc. Two critical load response cases are finally identified: - Human discomfort from footfall-induced vibrations. - Human discomfort from machine-induced vibrations. 47 7.3.1 Assumptions It shall be ensured that the actions which can be reasonably anticipated on a member, component or structure, do not cause vibrations that can impair the function of the structure or cause unacceptable discomfort to the users. The vibration level should be estimated by measurements or by calculation taking into account the expected stiffness of the member, component or structure and the modal damping ratio. For floors, unless other values are proven to be more appropriate, a modal damping ratio of ζ = 0,01 (i.e. 1 %) should be assumed. 7.3.2 Vibrations from machinery Vibrations caused by rotating machinery and other operational equipment shall be limited for the unfavourable combinations of permanent load and variable loads that can be expected. For floors, acceptable levels for continuous vibration should be taken from Figure 5a in Appendix A of ISO 2631-2 with a multiplying factor of 1,0. Residential floors For residential floors with a fundamental frequency less than 8 Hz (f1 ≤ 8Hz) a special investigation should be made. For residential floors with a fundamental frequency greater than 8 Hz (f1 > 8 Hz) the following requirements should be satisfied: mm/kN w a F ≤ (7.3) and 1( -1) m/(Ns²)fv b ζ≤ (7.4) where: w is the maximum instantaneous vertical deflection caused by a vertical concentrated static force F applied at any point on the floor, taking account of load distribution; v is the unit impulse velocity response, i.e. the maximum initial value of the vertical floor vibration velocity (in m/s) caused by an ideal unit impulse (1 Ns) applied at the point of the floor giving maximum response. Components above 40 Hz may be disregarded; ζ is the modal damping ratio. NOTE: The recommended range of limiting values of a and b and the recommended relationship between a and b is given in Figure 7.2. 48 Key: 1 Better performance 2 Poorer performance Figure 7.2 Recommended range of and relationship between a and b The calculations in should be made under the assumption that the floor is unloaded, i.e., only the mass corresponding to the self-weight of the floor and other permanent actions. For a rectangular floor with overall dimensions l × b, simply supported along all four edges and with timber beams having a span l, the fundamental frequency f1 may approximately be calculated as 1 2 ( ) 2 EI f m π= l l (7.5) where: m is the mass per unit area in kg/m²; l is the floor span, in m; (EI) l is the equivalent plate bending stiffness of the floor about an axis perpendicular to the beam direction, in Nm²/m. For a rectangular floor with overall dimensions b×l, simply supported along all four edges, the value v may, as an approximation, be taken as: 404(0,4 0,6 ) 200 n v mb += +l (7.6) where: v is the unit impulse velocity response, in m/(Ns2); n40 is the number of first-order modes with natural frequencies up to 40 Hz; b is the floor width, in m; 51 8.1 Basic assumptions There is a huge variety of configurations and design loadings of connections. 8.1.1 Fastener requirements Unless rules are given in this chapter, the characteristic load-carrying capacity, and the stiffness of the connections shall be determined from tests according to EN 1075, EN 1380, EN 1381, EN 26891 and EN 28970. If the relevant standards describe tension and compression tests, the tests for the determination of the characteristic load-carrying capacity shall be performed in tension. 8.1.2 Multiple fastener connections The arrangement and sizes of the fasteners in a connection, and the fastener spacings, edge and end distances shall be chosen so that the expected strength and stiffness can be obtained. It shall be taken into account that the load-carrying capacity of a multiple fastener connection, consisting of fasteners of the same type and dimension, may be lower than the summation of the individual load-carrying capacities for each fastener. When a connection comprises different types of fasteners, or when the stiffness of the connections in respective shear planes of a multiple shear plane connection is different, their compatibility should be verified. For one row of fasteners parallel to the grain direction, the effective characteristic load- carrying capacity parallel to the row, Fv,ef,Rk, should be taken as: v,ef,Rk ef v,RkF n F= (8.1) where: Fv,ef,Rk is the effective characteristic load-carrying capacity of one row of fasteners parallel to the grain; nef is the effective number of fasteners in line parallel to the grain; Fv,Rk is the characteristic load-carrying capacity of each fastener parallel to the grain. NOTE: Values of nef for rows parallel to grain are given in 8.3.1.1 and 8.5.1.1. For a force acting at an angle to the direction of the row, it should be verified that the force component parallel to the row is less than or equal to the load-carrying capacity calculated according to expression (8.1). 8.1.3 Multiple shear plane connections In multiple shear plane connections the resistance of each shear plane should be determined by assuming that each shear plane is part of a series of three-member connections. To be able to combine the resistance from individual shear planes in a multiple shear plane connection, the governing failure mode of the fasteners in the respective shear planes should be compatible with each other and should not consist of a combination of failure modes (a), (b), (g) and (h) from Figure 8.2 or modes (c), (f) and (j/l) from Figure 8.3 with the other failure modes. 52 8.1.4 Connection forces at an angle to the grain When a force in a connection acts at an angle to the grain, (see Figure 8.1), the possibility of splitting caused by the tension force component FEd sin α, perpendicular to the grain, shall be taken into account. To take account of the possibility of splitting caused by the tension force component, FEd sin α, perpendicular to the grain, the following shall be satisfied: v,Ed 90,RdF F≤ (8.2) with v,Ed,1 v,Ed v,Ed,2 max= F F F    (8.3) where: F90,Rd is the design splitting capacity, calculated from the characteristic splitting capacity F90,Rk according to 2.3.3; Fv,Ed,1, Fv,Ed,2 are the design shear forces on either side of the connection (see Figure 8.1). For softwoods, the characteristic splitting capacity for the arrangement shown in Figure 8.1 should be taken as: e 90,Rk e 14 1 h F b w h h =   −    (8.4) where: 0,35 pl max for punched metalplate fasteners100 1 1 for all other fasteners w w         =       (8.5) and: F90,Rk is the characteristic splitting capacity,  in N; w is a modification factor; he is the loaded edge distance to the centre of the most distant fastener or to the edge of the punched metal plate fastener, in mm; h is the timber member height, in mm; b is the member thickness, in mm; wpl is the width of the punched metal plate fastener parallel to the grain, in mm. 53 Figure 8.1 Inclined force transmitted by a connection 8.1.5 Alternating connection forces The characteristic load-carrying capacity of a connection shall be reduced if the connection is subject to alternating internal forces due to long-term or medium-term actions. The effect on connection strength of long-term or medium-term actions alternating between a tensile design force Ft,Ed and a compressive design force Fc,Ed should be taken into account by designing the connection for (Ft,Ed + 0,5Fc,Ed) and (Fc,Ed + 0,5Ft,Ed). 8.2 Lateral load-carrying capacity of metal dowel-type fasteners The failure of laterally loaded fasteners include both crushing of the timber and bending of the fastener. 8.2.1 Asumptions For the determination of the characteristic load-carrying capacity of connections with metal dowel-type fasteners the contributions of the yield strength, the embedment strength, and the withdrawal strength of the fastener shall be considered. 8.2.2 Timber-to-timber and panel-to-timber connections The characteristic load-carrying capacity for nails, staples, bolts, dowels and screws per shear plane per fastener, should be taken as the minimum value found from the following expressions: − For fasteners in single shear h,1,k 1 h,2,k 2 2 2 h,1,k 1 ax,Rk2 32 2 2 2 1 1 1 1 y,Rkh,1,k 1 ax,Rkv,Rk 2 h,1,k 1 h,1 2 1 1 1 4 4 (2 )min 1,05 2 (1 ) 2 4 1,05 f t d f t d f t d Ft t t t t t t t Mf t d FF f d t f β β β β β β β β β β β          + + + + − + +      +            += + + − +  +    (a) (b) (c) (d) y,Rk,k 2 ax,Rk2 2 h,1,k 2 ax,Rk y,Rk h,1,k 4 (1 2 ) 2 (1 ) 1 2 4 2 1,15 2 (f) 1 4 Mt d F f d t F M f d β β β β β β β β               +  + + − +  +      + + (e) (8.6) 56 h,k 1 v,Rk ax,Rk y,Rk h,k 0,4 min 1,15 2 4 f t d F F M f d  =  +  (a) (b) (8.9) − For a thick steel plate in single shear: y,Rk ax,Rk h,k 1 2 h,k 1 ax,Rk v,Rk y,Rk h,k h,k 1 4 2 1 4 min 2,3 4 M F f t d f d t FF M f d f t d     + − +      =  +     (c) (d) (e) (8.10) − For a steel plate of any thickness as the central member of a double shear connection: h,1,k 1 y,Rk ax,Rk v,Rk h,1,k 1 2 h,1,k 1 ax,Rk y,Rk h,1,k 4 min 2 1 4 2,3 4 f t d M F F f t d f d t F M f d     = + − +        +  (f) (g) (h) (8.11) − For thin steel plates as the outer members of a double shear connection: h,2,k 2 v,Rk ax,Rk y,Rk h,2,k 0,5 min 1,15 2 4 f t d F F M f d  =  +  (j) (k) (8.12) − For thick steel plates as the outer members of a double shear connection: h,2,k 2 v,Rk ax,Rk y,Rk h,2,k 0,5 (l) min 2,3 (m) 4 f t d F F M f d  =  +  (8.13) where: Fv,Rk is the characteristic load-carrying capacity per shear plane per fastener; f h,k is the characteristic embedment strength in the timber member; t1 is the smaller of the thickness of the timber side member or the penetration depth; t2 is the thickness of the timber middle member; d is the fastener diameter; My,Rk is the characteristic fastener yield moment; Fax,Rk is the characteristic withdrawal capacity of the fastener. NOTE 1: The different failure modes are illustrated in Figure 8.3. 57 Figure 8.3 Failure modes for steel-to-timber connections For the limitation of the rope effect Fax,Rk 8.2.2 applies. It shall be taken into account that the load-carrying capacity of steel-to-timber connections with a loaded end may be reduced by failure along the perimeter of the fastener group. 8.3 Nailed connections Nails are the most commonly used fasteners in timber construction. 8.3.1 Laterally loaded nails The failure of laterally loaded nails include both crushing of the timber and bending of the nail. 8.3.1.1 Asumptions The symbols for the thicknesses in single and double shear connections (see Figure 8.4) are defined as follows: t1 is: the headside thickness in a single shear connection; the minimum of the head side timber thickness and the pointside penetration in a double shear connection; t2 is: the pointside penetration in a single shear connection; the central member thickness in a double shear connection. Timber should be pre-drilled when: − the characteristic density of the timber is greater than 500 kg/m3; − the diameter d of the nail exceeds 8 mm. For square and grooved nails, the nail diameter d should be taken as the side dimension. For smooth nails produced from wire with a minimum tensile strength of 600 N/mm2, the following characteristic values for yield moment should be used: 2,6 u y,Rk 2,6 u 0,3 for round nails 0, 45 for square nails f d M f d =   (8.14) where: My,Rk is the characteristic value for the yield moment, in Nmm; d is the nail diameter as defined in EN 14592, in mm; 58 fu is the tensile strength of the wire, in N/mm2. For nails with diameters up to 8 mm, the following characteristic embedment strengths in timber and LVL apply: − without predrilled holes 2-0,3 kh,k 0,082 N/mm f d ρ= (8.15) − with predrilled holes 2 kh,k 0,082 (1- 0,01 ) N/mm d f ρ= (8.16) where: ρk is the characteristic timber density, in kg/m³; d is the nail diameter, in mm. Figure 8.4 Definitions of t1 and t2 (a) single shear connection, (b) double shear connection For nails with diameters greater than 8 mm the characteristic embedment strength values for bolts according to 8.5.1 apply. In a three-member connection, nails may overlap in the central member provided (t - t2) is greater than 4d (see Figure 8.5). Figure 8.5 Overlapping nails 61 Table 8.2 Minimum spacings and edge and end distances for nails Spacing or distance (see Figure 8.7) Angle α Minimum spacing or end/edge distance without predrilled holes with predrilled holes ρk ≤420kg/m3 420 kg/m 3<ρk≤500 kg/m3 Spacing a1 (parallel to grain) 0° ≤ α ≤ 360 ° d < 5 mm: (5+5│cosα│)d d ≥ 5 mm: (5+7│cos α│)d (7+8│cos α│) d (4+│cos α│) d Spacing a2 (perpendicular to grain) 0° ≤ α ≤ 360 ° 5d 7d (3+│sin α│) d Distance a3,t (loaded end) -90° ≤ α ≤ 90° (10+5 cos α)d (15 + 5 cos α) d (7+ 5cos α) d Distance a3,c (unloaded end) 90° ≤ α ≤ 270° 10d 15d 7d Distance a4,t (loaded edge) 0° ≤ α ≤ 180° d < 5 mm: (5+2 sin α) d d ≥ 5 mm: (5+5 sin α) d d < 5 mm: (7+2 sin α) d d ≥ 5 mm: (7 + 5 sin α) d d < 5 mm: (3 + 2 sin α) d d ≥ 5 mm: (3 + 4 sin α) d Distance a4,c (unloaded edge) 180°≤ α≤ 360 ° 5d 7d 3d Timber should be pre-drilled when the thickness of the timber members is smaller than ( ) k 7 max 13 30 400 d t d ρ  =  − (8.18) where: t is the minimum thickness of timber member to avoid pre-drilling, in mm; ρk is the characteristic timber density in kg/m³; d is the nail diameter, in mm. Timber of species especially sensitive to splitting should be pre-drilled when the thickness of the timber members is smaller than ( ) k 14 max 13 30 200 d t d ρ  =  − (8.19) Expression (8.19) may be replaced by expression (8.18) for edge distances given by: a4 ≥ 10 d for ρk ≤ 420 kg/m3 a4 ≥ 14 d for 420 kg/m3 ≤ ρk ≤ 500 kg/ m3. 62 Note: Examples of species sensitive to splitting are fir (abies alba), Douglas fir (pseudotsuga menziesii) and spruce (picea abies).. Key: (1) Loaded end (2) Unloaded end (3) Loaded edge (4) Unloaded edge 1 Fastener 2 Grain direction Figure 8.7 – Spacings and end and edge distances (a) Spacing parallel to grain in a row and perpendicular to grain between rows, (b) Edge and end distances 8.3.1.3 Nailed panel-to-timber connections Minimum nail spacings for all nailed panel-to-timber connections are those given in Table 8.2, multiplied by a factor of 0,85. The end/edge distances for nails remain unchanged unless otherwise stated below. Minimum edge and end distances in plywood members should be taken as 3d for an unloaded edge (or end) and (3 + 4 sin α)d for a loaded edge (or end), where α is the angle between the direction of the load and the loaded edge (or end). For nails with a head diameter of at least 2d, the characteristic embedment strengths are as follows: 63 − for plywood: 0,3 h,k k0,11f dρ −= (8.20) where: fh,k is the characteristic embedment strength, in N/mm2; ρk is the characteristic plywood density in kg/m³; d is the nail diameter, in mm; − for hardboard in accordance with EN 622-2: 0,3 0,6 h,k 30f d t−= (8.21) where: fh,k is the characteristic embedment strength, in N/mm2; d is the nail diameter, in mm; t is the panel thickness, in mm. − for particleboard and OSB: 0,7 0,1 h,k 65f d t−= (8.22) where: fh,k is the characteristic embedment strength, in N/mm2; d is the nail diameter, in mm; t is the panel thickness, in mm. 8.3.1.4 Nailed steel-to-timber connections The minimum edge and end distances for nails given in Table 8.2 apply. Minimum nail spacings are those given in Table 8.2, multiplied by a factor of 0,7. 8.3.2 Axially loaded nails Smooth nails shall not be used to resist permanent or long-term axial loading. For threaded nails, only the threaded part should be considered capable of transmitting axial load. Nails in end grain should be considered incapable of transmitting axial load. The characteristic withdrawal capacity of nails, Fax,Rk, for nailing perpendicular to the grain (Figure 8.8 (a) and for slant nailing (Figure 8.8 (b)), should be taken as the smaller of the values found from the following expressions: − For nails other than smooth nails, as defined in EN 14592: ax,k pen ax,Rk 2 head,k h (a) (b) f d t F f d =  (8.23) 66 For staples produced from wire with a minimum tensile strength of 800 N/mm², the following characteristic yield moment per leg should be used: 2,6 y,Rk 240 M d= (8.29) where: My,Rk is the characteristic yield moment, in Nmm; d is the staple leg diameter, in mm. For a row of n staples parallel to the grain, the load-carrying capacity in that direction should be calculated using the effective number of fasteners nef according to 8.3.1.1- expression (8.17). Minimum staple spacings, edge and end distances are given in Table 8.3, and illustrated in Figure 8.10 where Θ is the angle between the staple crown and the grain direction. Key: (1) staple centre Figure 8.9 Staple dimensions Figure 8.10 Definition of spacing for staples 67 Table 8.3 Minimum spacings and edge and end distances for staples Spacing and edge/end distances (see Figure 8.7) Angle Minimum spacing or edge/end distance a1 (parallel to grain) for Θ≥ 30° for Θ<30° 0° ≤ α ≤ 360° (10 + 5│cos α│) d (15 + 5│cos α│) d a2 (perpendicular to grain) 0° ≤ α ≤ 3600° 15 d a3,t (loaded end) -90° ≤ α ≤ 90° (15 + 5│cos α│) d a3,c (unloaded end) 90° ≤ α ≤ 270° 15 d a4,t (loaded edge) 0° ≤ α ≤ 180° (15 + 5│sin α│) d a4,c (unloaded edge) 180° ≤ α ≤ 360° 10 d 8.5 Bolted connections Bolts are installed into pre-drilled clearance holes in the timber. 8.5.1 Laterally loaded bolts The failure of laterally loaded bolts include both crushing of the timber and bending of the bolt. 8.5.1.1 General and bolted timber-to-timber connections For bolts the following characteristic value for the yield moment should be used: 2,6 y,Rk u,k0,3 = f dM (8.30) where: My,Rk is the characteristic value for the yield moment, in Nmm; fu,k is the characteristic tensile strength, in N/mm²; d is the bolt diameter, in mm. For bolts up to 30 mm diameter, the following characteristic embedment strength values in timber and LVL should be used, at an angle α to the grain: h,0,k h,α,k 90 sin cos2 2 f = f + k α α (8.31) kh,0,k 0,082 (1- 0,01 ) = d f ρ (8.32) where: 90 1,35 0,015 for softwoods 1,30 0,015 for LVL 0,90 0,015 for hardwoods d dk d + = +  + (8.33) and: fh,0,k is the characteristc embedment strength parallel to grain, in N/mm2; 68 ρk is the characteristic timber density, in kg/m³; α is the angle of the load to the grain; d is the bolt diameter, in mm. Minimum spacings and edge and end distances should be taken from Table 8.4, with symbols illustrated in Figure 8.7. Table 8.4 Minimum values of spacing and edge and end distances for bolts Spacing and end/edge distances (see Figure 8.7) Angle Minimum spacing or distance a1 (parallel to grain) 0° ≤ α ≤ 360° (4 + │cos α│) d a2 (perpendicular to grain) 0° ≤ α ≤ 360° 4 d a3,t (loaded end) -90° ≤ α ≤ 90° max (7 d; 80 mm) a3,c (unloaded end) 90° ≤ α < 150 ° 150° ≤ α < 210° 210° ≤ α ≤ 270° max [(1 + 6 sin α) d; 4d] 4 d max [(1 + 6 sin α) d; 4d] a4,t (loaded edge) 0° ≤ α ≤ 180° max [(2 + 2 sin α) d; 3d] a4,c (unloaded edge) 180° ≤ α ≤ 360° 3 d For one row of n bolts parallel to the grain direction, the load-carrying capacity parallel to grain, see 8.1.2(4), should be calculated using the effective number of bolts nef where: ef 0,9 14 min 13 n n = a n d      (8.34) where: a1 is the spacing between bolts in the grain direction; d is the bolt diameter n is the number of bolts in the row. For loads perpendicular to grain, the effective number of fasteners should be taken as efn n= (8.35) For angles 0° < α < 90° between load and grain direction, nef may be determined by linear interpolation between expressions (8.34) and (8.35). Requirements for minimum washer dimensions and thickness in relation to bolt diameter are given in 10.4.3. 8.5.1.2 Bolted panel-to-timber connections For plywood the following embedment strength, in N/mm2, should be used at all angles to the face grain: 71 − the tension strength of the screw; − for screws used in conjunction with steel plates, failure along the circumference of a group of screws (block shear or plug shear); Minimum spacing and edge distances for axially loaded screws should be taken from Table 8.6. Table 8.6 Minimum spacings and edge distances for axially loaded screws Screws driven Minimum spacing Minimum edge distance At right angle to the grain 4d 4d In end grain 4d 2,5d The minimum pointside penetration length of the threaded part should be 6d. The characteristic withdrawal capacity of connections with axially loaded screws should be taken as: 0,8 ax,α,Rk ef ef ax,α,k( )F n d l f= π (8.38) where: Fax,α,Rk is the characteristic withdrawal capacity of the connection at an angle α to the grain; nef is the effective number of screws; d is the outer diameter measured on the threaded part; lef is the pointside penetration length of the threaded part minus one screw diameter; fax,α,k is the characteristic withdrawal strength at an angle α to the grain. The characteristic withdrawal strength at an angle α to the grain should be taken as: ax,k ax,α,k 2 2sin 1,5cos f f α α = + (8.39) with: 3 1,5 ax,k k3,6 10f ρ−= × (8.40) where: fax,α,k is the characteristic withdrawal strength at an angle α to the grain; fax,k is the characteristic withdrawal strength perpendicular to the grain; ρk is the characteristic density, in kg/m3. NOTE: Failure modes in the steel or in the timber around the screw are brittle, i.e. with small ultimate deformation and therefore have a limited possibility for stress redistribution. 72 The pull-through capacity of the head shall be determined by tests, in accordance with EN 1383. For a connection with a group of screws loaded by a force component parallel to the shank, the effective number of screws is given by: 0,9 efn n= (8.41) where: nef is the effective number of screws; n is the number of screws acting together in a connection. 8.7.3 Combined laterally and axially loaded screws For screwed connections subjected to a combination of axial load and lateral load, expression (8.28) should be satisfied. 73 9 Components 9.1 Glued thin-webbed beams If a linear variation of strain over the depth of the beam is assumed, the axial stresses in the wood-based flanges should satisfy the following expressions: f,c,max,d m,dfσ ≤ (9.1) f,t,max,d m,dfσ ≤ (9.2) f,c,d c c,0,dk fσ ≤ (9.3) f,t,d t,0,dfσ ≤ (9.4) where: σf,c,max,d is the extreme fibre flange design compressive stress; σf,t,max,d is the extreme fibre flange design tensile stress; σf,c,d is the mean flange design compressive stress; σf,t,d is the mean flange design tensile stress; kc is a factor which takes into account lateral instability. Key: (1) compression (2) tension Figure 9.1 Thin-webbed beams The factor kc may be determined (conservatively, especially for box beams) according to 6.3.2 with 76 Table 9.1 Maximum effective flange widths due to the effects of shear lag and plate buckling Flange material Shear lag Plate buckling Plywood, with grain direction in the outer plies: − Parallel to the webs 0,1l 20hf − Perpendicular to the webs 0,1l 25hf Oriented strand board 0,15l 25hf Particleboard or fibreboard with random fibre orientation 0,2l 30hf Unless a detailed buckling investigation is made, the unrestrained flange width should not be greater than twice the effective flange width due to plate buckling, from Table 9.1. For webs of wood-based panels, it should, for sections 1-1 of an I-shaped cross-section in Figure 9.2, be verified that: v,90,d w f 0,8 mean,d f v,90,d w f w for 8 8 for 8 f b h h f b h b τ ≤  ≤   >    (9.14) where: τmean,d is the design shear stress at the sections 1-1, assuming a uniform stress distribution; fv,90,d is the design planar (rolling) shear strength of the flange. For section 1-1 of a U-shaped cross-section, the same expressions should be verified, but with 8hf substituted by 4hf. The axial stresses in the flanges, based on the relevant effective flange width, should satisfy the following expressions: f,c,d f,c,d fσ ≤ (9.15) f,t,d f,t,d fσ ≤ (9.16) where: σf,c,d is the mean flange design compressive stress; σf,t,d is the mean flange design tensile stress; ff,c,d is the flange design compressive strength; ff,t,d is the flange design tensile strength. It shall be verified that any glued splices have sufficient strength. The axial stresses in the wood-based webs should satisfy the expressions (9.6) to (9.7) defined in 9.1.1 77 Figure 9.2 Thin-flanged beam 9.1.3 Mechanically jointed beams If the cross-section of a structural member is composed of several parts connected by mechanical fasteners, consideration shall be given to the influence of the slip occurring in the joints. Calculations should be carried out assuming a linear relationship between force and slip. If the spacing of the fasteners varies in the longitudinal direction according to the shear force between smin and smax (< 4smin), an effective spacing sef may be used as follows: ef min max 0,75 0, 25 s s s= + (9.17) NOTE: A method for the calculation of the load-carrying capacity of mechanically jointed beams is given in Chapter 10. 9.1.4 Mechanically jointed and glued columns Deformations due to slip in joints, to shear and bending in packs, gussets, shafts and flanges, and to axial forces in the lattice shall be taken into account in the strength verification. NOTE: A method for the calculation of the load-carrying capacity of I- and box-columns, spaced columns and lattice columns is given in Chapter 11. 78 10 Mechanically jointed beams 10.1 Simplified analysis 10.1.1 Cross-sections The cross-sections shown in Figure 10.1 are considered. 10.1.2 Assumptions The design method is based on the theory of linear elasticity and the following assumptions: − the beams are simply supported with a span ℓ. For continuous beams the expressions may be used with ℓ equal to 0,8 of the relevant span and for cantilevered beams with ℓ equal to twice the cantilever length − the individual parts (of wood, wood-based panels) are either full length or made with glued end joints − the individual parts are connected to each other by mechanical fasteners with a slip modulus K − the spacing s between the fasteners is constant or varies uniformly according to the shear force between smin and smax, with smax < 4 smin − the load is acting in the z-direction giving a moment M = M(x) varying sinusoidally or parabolically and a shear force V = V(x). 10.1.3 Spacings Where a flange consists of two parts jointed to a web or where a web consists of two parts (as in a box beam), the spacing si is determined by the sum of the fasteners per unit length in the two jointing planes. 10.1.4 Deflections resulting from bending moments Deflections are calculated by using an effective bending stiffness (EI)ef ,determined in accordance with 10.2. 81 11 Built-up columns 11.1 General 11.1.1 Assumptions The following assumptions apply: − the columns are simply supported with a length l; − the individual parts are full length; − the load is an axial force Fc acting at the geometric centre of gravity, (see 11.2.3). 11.1.2 Load-carrying capacity For column deflection in the y-direction (see Figure 11.1 and Figure 11.3) the load-carrying capacity should be taken as the sum of the load-carrying capacities of the individual members. For column deflection in the z-direction (see Figure 11.1 and Figure 11.3) it should be verified that: c,0,d c c,0,d k fσ ≤ (11.1) where: c,d c,0,d tot F A σ = (11.2) where: Atot is the total cross-sectional area; kc is determined in accordance with 6.3.2 but with an effective slenderness ratio λef determined in accordance with sections 11.2 - 11.4. 11.2 Mechanically jointed columns 11.2.1 Effective slenderness ratio The effective slenderness ratio should be taken as: tot ef ef A I λ = l (11.3) with ef ef mean ( ) EI I E = (11.4) where (EI)ef is determined in accordance with Chapter 10.. 11.2.2 Load on fasteners The load on a fastener should be determined in accordance with Chapter 10, where 82 c,d ef c c,d ef d ef c c,d ef c for 30 120 for 30 60 3600 for 60 60 V F k F k F k λ λ λ λ =  <   ≤ <   ≤  (11.5) 11.2.3 Combined loads In cases where small moments (e.g. from self weight) are acting in adition to axial load, 6.3.2 applies. 11.3 Spaced columns with packs or gussets 11.3.1 Assumptions Columns as shown in Figure 11.1 are considered, i.e. columns comprising shafts spaced by packs or gussets. The joints may be either nailed or glued or bolted with suitable connectors. The following assumptions apply: − the cross-section is composed of two, three or four identical shafts; − the cross-sections are symmetrical about both axes; − the number of unrestrained bays is at least three, i.e. the shafts are at least connected at the ends and at the third points; − the free distance a between the shafts is not greater than three times the shaft thickness h for columns with packs and not greater than 6 times the shaft thickness for columns with gussets; − the joints, packs and gussets are designed in accordance with 11.2.2; − the pack length l2 satisfies the condition: l2/a ≥ 1,5; − there are at least four nails or two bolts with connectors in each shear plane. For nailed joints there are at least four nails in a row at each end in the longitudinal direction of the column; − the gussets satisfies the condition: l2/a ≥ 2; − the columns are subjected to concentric axial loads. For columns with two shafts Atot and Itot should be calculated as tot 2A A= (11.6) ( )3 3 tot 2 12 b h a a I  + −  = (11.7) For columns with three shafts Atot and Itot should be calculated as 83 tot 3A A= (11.8) ( ) ( )3 3 3 tot 3 2 2 12 b h a h a h I  + − + +  = (11.9) Figure 11.1 – Spaced columns 11.3.2 Axial load-carrying capacity For column deflection in the y-direction (see Figure 11.3) the load-carrying capacity should be taken as the sum of the load-carrying capacities of the individual members. For column deflection in the z-direction 11.1.2 applies with 22 ef 1 2 nη λλ λ= + (11.10) where: λ is the slenderness ratio for a solid column with the same length, the same area (Atot) and the same second moment of area (Itot), i.e., tot tot /A Iλ = l (11.11) λ1 is the slenderness ratio for the shafts and has to be set into expression (11.10) with a minimum value of at least 30, i.e. 86 2 Beam with solid cross-section Simply supported timber beam with cross-section 50 x 200 mm, clear span l = 3500 mm. Timber of strength class C22 according to EN 338 ( fm,k = 22 MPa, fv,k = 2,4 MPa, E0,05 = 6 700 MPa ). Design uniformly distributed load of 2 kNm-1 ( medium-term ). Service class 1. Design bending and shear strength m,k m,d mod M 22,0 0,8 13,5 MPa 1,3 f f k γ = = = v,k v,d mod M 2,4 0,8 1,48 MPa 1,3 f f k γ = = = a) Bending ( beam is assumed to be laterally restrained throughout the length of its compression edge ) Verification of failure condition m,d m,dfσ ≤ 2 2 d d m,d 2 1 1 2 3500 6 9,2 MPa 8 8 50 200 M q W W σ ⋅ ⋅= = = = ⋅ l < 13,5 MPa b) Bending ( beam is not assumed to be laterally restrained throughout the length of its compression edge ) Buckling resistance 2 2 0,05 m,crit ef 0,78 0,78 50 6700 18,4 MPa 200 (0,9 3500 400) b E h σ ⋅ ⋅= = = ⋅ ⋅ +l m,k rel,m m,crit 22 1,06 18,4 f λ σ = = = crit rel,m1,56 0,75 1,56 0,75 1,06 0,76k λ= − = − ⋅ = crit m,d 0,76 13,5 10,3 MPak f⋅ = ⋅ = Verification of failure condition m,d crit m,dk fσ ≤ ⋅ 2 2 d d m,d 2 1 2 3500 6 9,2 MPa 8 8 50 200 M q W W σ ⋅ ⋅= = = = ⋅ ⋅ l < 10,3 MPa 87 c) Shear cr v,d 0,67 1,48 0,99 MPak f⋅ = ⋅ = cr 0,67k = is taking into account cracks caused by too rapid drying Verification of failure condition v,d cr v,dk fτ ≤ ⋅ d v,d 3 3 1 2 3500 0,53 MPa 0,99 MPa 2 2 2 50 200 V A τ ⋅ ⋅ ⋅= = = < ⋅ ⋅ ⋅ 3 Step joint Joint of a compression member with cross-section 140 x 140 mm, see Figure below ( cutting depth is 45 mm, shear length in chord 250 mm and β = 45° ). Design values of timber properties are fc,0,d = 11,03 MPa, fc,90,d = 2,21 MPa, fv,d = 1,32 MPa ). Design compressive force Nd = 55 kN. Design compressive strength at an angle to the grain c,0,d c,α,d c,0,d 2 2 c,90 c,90,d sin cos f f f k f α α = + = oo 5,22cos5,22sin 81,2 03,11 03,11 22 + = 7,72 MPa 88 Verification of failure conditions σc,α,d = z 2 d tb cosN α = 45140 5,22cos1055 23 ⋅ ⋅ o = 7,45 MPa < 7,72 MPa τv,d = z d b cosN l β = 250140 45cos1055 3 ⋅ ⋅ o = 1,11 MPa < 1,32 MPa 4 Timber-framed wall The walls assembly presented in Fig. 1 is subjected to the total design horizontal force FH,d,totx=25 kN (short-term) acting at the top of the wall assembly. FH,d,tot FH,d h ⇒ Fi,t,Ed Fi,c,Ed b ntot·b b y n F F tot,d,H d,H = zt timber frame sheathing board Figure 1: Example of the wall assembly. The single panel wall element of actual dimensions h = 263.5 cm and b = 125 cm is composed of timber studs (2x9x9 cm and 1x4.4x9 cm) and timber girders (2x8x9 cm). The plywood sheathing boards of the thickness t=15 mm are fixed to the timber frame using staples of Φ1.53 mm and length l = 35 mm at an average spacing of s = 75 mm (Fig. 2). 91 5 Single tapered beam Assessment of a single tapered beam (Fig. 10.1). Material: glue laminated timber (GL 24h), service class 1. Characteristic values: Dead load gk = 4,5 kNm-1, snow sk = 4,5 kNm-1. Materials and geometrical characteristics of the beam: Fig. 10.1 Scheme of the single tapered beam Span: L = 12 m Depth of the beam at the apex: hap = 1200 mm Angle of the taper: α = 30 Width of the beam: b = 140 mm Precamber of the beam: wc = 30 mm fm,g,k = 24 MPa fv,g,k = 2,7 MPa fc,90,g,k = 2,7 MPa ft,90,g,k = 0,4 MPa E0,mean,g = 11600 MPa The beam is prevented against lateral-torsional buckling. Design bending strength MPa f kf kgm dgm 28,17 25,1 24 9,0 M ,, mod,, === γ Design shear strength MPa f kf kgv dgv 94,1 25,1 7,2 9,0 M ,, mod,, === γ Design compressive strength perpendicular to the grain MPa f kf kgc dgc 94,1 25,1 7,2 9,0 M ,,90, mod,,90, === γ Basic combination of the load qd = 1,35gk + 1,5pk = 1,35⋅4,5 + 1,5⋅4,5 = 12,825 kNm-1 Shear force at a support kN L qV dd 95,76 2 12 825,12 2 === Depth of the beam at the support mtgLtghh apS 571,01232,1 0 =⋅−=⋅−= α 92 Verification of failure conditions a) Shear at support MPaMPa bh Vd dv 94,144,1 5711402 1095,763 2 3 3 0 , <= ⋅⋅ ⋅⋅==τ b) Bending at critical cross-section Critical cross-section position m h h L x s ap 87,3 571,0 2,1 1 12 1 = + = + = Depth of the beam at critical cross-section m h h h h s ap ap x 774,0 571,0 2,1 1 2,12 1 2 = + ⋅= + ⋅ = Bending moment at critical cross-section kNm xq xVM d dd 76,201 2 87,3825,12 87,395,76 2 22 =⋅−⋅=−= Stress at critical cross-section 2,,,0, 6 x d dmdm bh M == ασσ dgmdom f ,,,, ≤σ allowedMPaMPadm ⇒<= ⋅ ⋅⋅= 28,1743,14 774140 1076,2016 2 6 ,0,σ dgmmdm fk ,,,,, ⋅≤ αασ 9112,0 3 94,1 28,17 3 94,15,1 28,17 1 1 5,1 1 1 2 02 2 0 2 2 ,,90, ,, 2 ,, ,, , =      +      ⋅ + =         +         + = tgtgtg f f tg f f k dgc dgm dgv dgm m αα α allowedMPaMPadm ⇒=⋅<= 74,1528,179112,043,14,,ασ c) Deflection 0wkw um ⋅= Coefficient ku – see Fig. 10.2. m hh h aps 886,0 2 2,1571,0 20 =+= + = 1166,110,2 571,0 2,1 =⇒== u s ap k h h 6,0=defk 93 Fig. 10.2 Coefficient ku c1) Instantaneous deflection mm IE Lg kw y uginst 42,14 88614011600384 1210125,45 1166,1 384 5 3 1244 , = ⋅⋅⋅ ⋅⋅⋅⋅⋅= ⋅⋅ ⋅⋅⋅= mm IE Ls kw y usinst 42,14 88614011600384 1210125,45 1166,1 384 5 3 1244 , = ⋅⋅⋅ ⋅⋅⋅⋅⋅= ⋅⋅ ⋅⋅⋅= allowed LL mmwww sinstginstinst ⇒<==+=+= 400416 84,2842,1442,14,, c2) Final deflection ( ) ( ) mmkww defginstgfin 07,236,0142,141,, =+⋅=+⋅= ( ) ( ) mmkww defsinstsfin 42,146,00142,141 2,, =⋅+⋅=⋅+⋅= ψ allowed LL mmwww sfingfinfin ⇒<==+=+= 250320 49,3742,1407,23,, c3) Net final deflection allowed LL mmwww cfinfinnet ⇒<==−=−= 3001602 49,73049,37, 96 In other words, , /155net finw L= , which is well above the recommended value of table 7.2. Conclusion: The displacement may, depending on the type of building, be too large. It may be considered to produce the beam with a precamber of, say 100 mm. 7 Moment resisting joint Design and assessment of moment resisting joint in the corner of the three-hinged plane frame. Material: glued laminated timber (GL 24h), service class 1. Geometrical characteristics of the frame: 13,5º 25 000 4 5 0 0 3 0 0 0 Span: L = 25 m Depth of the rafter: hR = 1480 mm Width of the rafter: bR = 200 mm Depth of the column: hC = 1480 mm Width of the column: bC = 2×120 mm Angle of the rafter: α = 13.50 Material properties (characteristic values): fm,g,k = 24 MPa fv,g,k = 2,7 MPa ρk = 380 kg/m3 Design bending strength MPa f kf kgm dgm 28,17 25,1 24 9,0 M ,, mod,, === γ Design shear strength MPa f kf kgv dgv 94,1 25,1 7,2 9,0 M ,, mod,, === γ Dowels: Steel grade S235 ∅24 mm (4.6): fu,k = 400 MPa Internal forces at the corner: Column: Md = 676.8⋅106 Nmm Vd,C = 150.4⋅103 N Nd,C = 178.1⋅103 N Rafter: Md = 676.8⋅106 Nmm Vd,R = 138.1⋅103 N Nd,R = 187.8⋅103 N Design of dowel joints: 97 Outer circle: mmdhr 64424414805.045.01 =⋅−⋅=−≤ ⇒ r1 = 644 mm Inside circle: mmdrr 524245644512 =⋅−=−≤ ⇒ r2 = 524 mm Number of dowels in circles: ks d r n 1.28 246 6442 6 2 1 1 = ⋅ ⋅⋅=≤ ππ ⇒ n1 = 28 ks d r n 8.22 246 5242 6 2 2 2 = ⋅ ⋅⋅=≤ ππ ⇒ n2 = 22 Load of dowels: Load of dowel in column and rafter of the frame due to bending moment: N rnrn r MF dM 3 22 6 2 22 2 11 1 1069.24 5242264428 644 108.676 ⋅= ⋅+⋅ ⋅= + = Load of dowel in column of the frame due to shear and normal force: N nn V F Cd CV 3 3 21 , , 1000.3 2228 104.150 ⋅= + ⋅= + = N nn N F Cd CN 3 3 21 , , 1056.3 2228 101.178 ⋅= + ⋅= + = Load of dowel in rafter of the frame due to shear and normal force: N nn V F Rd RV 3 3 21 , , 1076.2 2228 101.138 ⋅= + ⋅= + = N nn N F Rd RN 3 3 21 , , 1076.3 2228 108.187 ⋅= + ⋅= + = Total load of dowel in the axis of the rafter and column of the frame: ( ) ( ) ( ) NFFFF CNCVMCd 3232332 , 2 ,, 1092.271056.31000.31069.24 ⋅=⋅+⋅+⋅=++= ( ) ( ) ( ) NFFFF RNRVMRd 3232332 , 2 ,, 1071.271076.31076.21069.24 ⋅=⋅+⋅+⋅=++= 98 Shear force in column and rafter in joint: N rnrn rnrnM V d M 3 22 6 2 22 2 11 2211 1074.360 5242264428 5242264428108.676 ⋅=      ⋅+⋅ ⋅+⋅⋅=      + + = ππ N V VF Cd MCdV 3 3 3, ,, 105.285 2 104.150 1074.360 2 ⋅=⋅−⋅=−= N V VF Rd MRdV 3 3 3, ,, 107.291 2 101.138 1074.360 2 ⋅=⋅−⋅=−= The mechanical properties of dowels: Embedding strength in fibres direction (characteristic value): ( ) ( ) MPadf kkh 68.233802401.01082.001,01082,0,0, =⋅⋅−⋅=−= ρ a) Carrying capacity of dowel in column axis: Angle between load and timber fibres: °=      ⋅ ⋅+⋅=       + = 7.82 1056.3 100.31069.24 3 33 , , 1 arctg F FF arctg CN CVMα ( ) °=−−=      −−= 2.67.82905.13 2 12 απαα Embedding strength (characteristic value): 71.124015.035.1015.035.190 =⋅+=+= dk MPa k f f kh kh 94.13 7.82cos7.82sin71.1 68.23 cossin 22 1 2 1 2 90 ,0, ,1, = +⋅ = +⋅ = αα MPa k f f kh kh 49.23 2.6cos2.6sin71.1 68.23 cossin 22 2 2 2 2 90 ,0, ,2, = +⋅ = +⋅ = αα 685.1 94.13 49.23 ,1, ,2, === kh kh f f β Yield moment (characteristic value): NmmdfM kuRky 36.26.2 ,, 103.465244003.03.0 ⋅=⋅⋅== mmt 1201 = , mmt 2002=