00694 Pouyan Zarnani - Timber Design Society

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ittle failure mode had better predictions in comparison to the ones from the Canadian code. P w = min (P a , P b , P c ), P i = 0.2 f t,k + 0.2 f v,k (3) (a) (b) (c) Figure 1: Proposed wood failure modes by Stahl (2004) 1.3 MOTIVATION In the models which have been proposed for the wood strength so far, the minimum, maximum or the summation of the tensile and shear capacities of the failed wood block resisting planes is considered as the wood strength of the connection. These methods might not be appropriate since the stiffness in tensile and shear planes differs and this results in uneven load distribution amongst the resisting planes [1,7]. For instance, in a plug shear failure (Fig. 1a), the contribution of the bottom or lateral shear planes to the wood resistance cannot simply be considered as a function of their respective area as the connection load is not shared uniformly among the resisting planes due to the unequal stiffness of the adjacent wood. In the proposed analysis, the shortcomings of the existing predictive models are taken into account. 2 NEW ANALYSIS FOR WOOD STRENGTH The proposed analysis for wood strength is based on a linear elastic spring system in which the applied load transfers from the main loaded wood block to the contact planes in conformity with the relative stiffness ratio of each resisting plane (Fig. 2). By predicting these stiffnesses, the contribution of each plane to lateral resistance and finally the ultimate load that cause brittle failure on these planes can be derived. Head tensile plane - h Bottom shear plane - b Lateral shear planes - l Figure 2: Proposed elastic spring model The difference in the stiffness of the tensile and shear planes is a function of the differences between the modulus of elasticity and modulus of rigidity, the wood surrounding the loaded block (bottom, end and edge distances-d z , d a and d e ) and also the connection geometry. 2.1 HEAD TENSILE PLANE STIFFNESS In the rivet connection, the load is transferred from the plate to the wood block through the rivets. The load which applied to the wood increases as it reaches the head of the joint (Fig. 3). The load distribution in the row of rivets is assumed to be linear. Johnsson and Stehn P w K h K b P w Main loaded block K l P h P b P l Δ [7], using a load distribution model based on a spring system, showed that the maximum variation from the linear assumption, on the distributed loads among fasteners in a nail connection with 10 nails in a row before yielding was approximately 12%. The head tensile plane stiffness can then be derived by considering the tensile deformation of the loaded block, Δ, which is given by L EA ( N (4) where E is the modulus of elasticity, L is the length subjected to the tensile stress and A th the area subjected to the tensile stress at the head of the block. Thus, the average tensile stiffness for the head plane would be K h th 2EA L th C (5) These equations use the connection geometry variables shown in Figure 4 and all dimensions are in mm. Figure 3: Simplified analytical model 2.2 BOTTOM SHEAR PLANE STIFFNESS By developing an FE model, Foschi and Longworth [2] studied the effect of the bottom distance d z on the bottom plane shear stress. They observed that the shear stresses vary when d z is less than 2 times the rivet penetration, L p . They considered L p as the thickness of the failed block, t ef . This observation is applied when considering the effective depth of the wood bottom block in contact with the main loaded block in order to simplify the model (Fig. 3) to estimate the distortion of the bottom block which is considered fixed at the bottom edge and is subjected to shear stress on the top surface. Dividing the sum of the bottom shear forces P b by the total area over which they act A sb defines the average shear stress τ sb : Fixed edge sb Pb A sb in which G is the modulus of rigidity, γ sb the shear strain and X b the maximum effective depth of the bottom block defined as X b =2t ef . Thus, the average pure shear stiffness would be K GA X . sb 1) Nc 1 Ph i i1 NC sbG G X sb b b Ph L 2EA da P l P Main loaded L t P h block ef P b P b X b=2t ef d z ≥ X b In contact bottom block Tensile force distribution L da However, as it is shown in Figure 3, the bottom block has a fixed edge at its head on its entire cross section which increases the stiffness and prevents deformation under the applied shear force. th X b Bottom block Fixed edge Shear force distribution (6)
the lateral blocks and is Atl 2tef Xl . Consequently, the average lateral shear planes stiffness can be defined by Kl ( 1 F)( Ksl Ktl) (11) in which F is the reduction factor for lateral shear planes stiffness [1]. The reduction factor for the lateral shear plane, F, is determined using: F=0 , If d e ≥ 1.25X l F=0.16 (2.5- d e / (X l /2)) 2 , If d e < 1.25X l (12) Figure 4: Definition of connection geometry variables If Δ is the whole deformation of the bottom block at the top, then it decreases rapidly in a nonlinear form as it reaches the bottom. Adjacent to the bottom fixed edge, the deflection curve would be tangential as the stress diminishes and approaches zero. It is assumed that the deformed shape is a polynomial curve with order nine (n=9). Therefore, the average deflection for the whole cross section can be theoretically considered as Δ/(n+1) equal to Δ/10 in case of tensile deformation. Thus, the additional average tensile stiffness for the whole cross section, K tb , can be estimated as K tb (7) where A tb is the effective tensile area of the bottom block given as A tb =S q X b (N R -1). Therefore, the average bottom shear plane stiffness can be defined by K b (8) Foschi and Longworth [2] observed that when the bottom distance d z becomes less than X b ,the bottom shear stress decreases and the load thus released is transferred almost in its entirety to the tensile plane. To take this effect into account, a factor H is proposed [1]. This factor can be considered as the reduction rate of the bottom shear plane stiffness, H (Eq. 9), as a result of decreasing the bottom distance d z less than X b . H=0 , If d z ≥ X b H=0.25 (2- d z / t ef ) 2 , If d z < X b Thus, EAtb 10L K sb K 2.3 LATERAL SHEAR PLANES STIFFNESS (10) Assuming that the mechanical properties of the wood for lateral and bottom shear planes are the same, the correspondent equations for the two side lateral shear planes can be developed similarly. The average pure shear stiffness for the lateral planes would become Ksl AslG X l where A sl is the summation of the areas subjected to the lateral shear stress and X l the maximum effective edge distance (equal to 2 times the half of the distance between the first and the last rows, which is comparable to X b =2t ef for the bottom shear plane). The additional average tensile stiffness can be given as K EA L in which A tl is the effective tensile area of tl tl 10 tb Kb ( 1 H)( Ksb Ktb) (9) 2.4 CLOSED-FORM APPROACH By predicting the stiffness of the wood surrounding each of the failure planes (K h , K b and K l ), one can predict the proportion of the total connection load applied to each plane, R K K i i . By further establishing the resistance of each of the failure planes as a function of a strength criterion, one can verify which of the failure planes governs the resistance of the entire connection. Thus, the wood load carrying capacity of the connection (Eq. 13) is the load which results in the earlier failure of one of the resisting planes due to being overloaded and equals to the minimum of P wh , P wb and P wl . In other words, the wood strength of the connection is the total capacity of the one plane which fails first plus a portion of strength capacity of the other planes. Moreover, while one plane fails then the load transfers to the rest of the planes in accordance with their relative stiffness ratios. It could be possible that the occurrence of the first failure of one plane does not correspond with the maximum load of the connection. Kb Kl Pwh ft, m Ath (1 ) K h K h Kh Kl P w = N p .min Pwb fv, mCab Asb (1 ) (13) Kb Kb Cal Asl Kb Pwl Pwb C A K ab sb Here f t,m is the wood mean strength in tension parallel to the grain (MPa) and f v,m is the wood mean strength in shear along the grain (MPa). Also, C ab and C al are the ratios of the average to maximum stresses on the bottom and lateral shear planes respectively given by Equations 14 and 15. These coefficients are derived based on the increasing load distribution on the shear planes (Fig. 3). The factor of k e is applied to C al to account for the reduction of the resisting area due to the cracks formation on the lateral planes, estimated at 20% of the failed block thickness while d e is less than 1.25X l . C ab S p( N C k C al e ab C ( N C C 1)/ 2 1) d N ( L d ) k e k e 1 0.8 a (14) (15) For a connection having only one plate, N p =1, the member thickness value, b, to be used to determine d z = b/2-t ef is twice the thickness of the wood. l a , If d e 1.25X l , If d e
Page 1: PREDICTVE ANALYTICAL MODEL FOR WOOD
Page 5 and 6: Grain 1200 mm Load F [kN]nt mm thic
Page 7 and 8: Test groups with tightly spaced riv
Page 9 and 10: Predicted Average Strength [kN] Pre

00694 Pouyan Zarnani - Timber Design Society

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