Bolted Timber Joints with Self-Tapping Screws

Revista EIA, ISSN 1794-1237 Número 8, p. 37-47. Diciembre 2007 Escuela de Ingeniería de Antioquia, Medellín (Colombia)

BOLTED TIMBER JOINTS WITH SELF-TAPPING SCREWS

César Echavarría*

ABSTRACT

The use of self-tapping screws with continuous threads in the joint area as a reinforcement to avoid splitting of timber members is studied. A theoretical model is developed to calculate the stress distribution around a pin-loaded hole in a timber joint, to predict brittle failure modes in bolted connections and to calculate the load in the reinforcing screws. Laboratory experiments on reinforced and non-reinforced timber joints with 15,9-mm bolts have shown good agreement with the model predictions. KEYWORDS: timber joints; brittle failure mode; reinforcement perpendicular-to-grain; analytical model.

RESUMEN

En este artículo se estudia el uso de tornillos autoperforantes como refuerzo para evitar rupturas frágiles en uniones de madera. Se presenta un modelo teórico para calcular la distribución de esfuerzos alrededor de un perno en una unión de madera, predecir las rupturas frágiles y evaluar el esfuerzo en los tornillos autoperforantes. Los experimentos�� de laboratorio con uniones de madera, con pernos de 15,9 mm de diámetro, reforzadas y no reforzadas mostraron la efectividad del modelo teórico propuesto. PALABRAS CLAVE: uniones de madera; ruptura frágil; refuerzo perpendicular a las fibras; modelo analítico.

* Ingeniero Civil, Universidad Nacional de Colombia; Master in Timber Structures and Docteur en Sciences, École Polytechnique Fédérale de Lausanne, Switzerland. Ph.D. Researcher, Département des sciences du bois et de la forêt, Université Laval, Québec. [email protected] Associate Professor, Faculty of Architecture, School of Construction, Universidad Nacional de Colombia. [email protected]

Artículo recibido 31-VIII-2007. Aprobado 11-XI-2007 Discusión abierta hasta junio de 2008 Bolted timber joints with self-tapping screws

1. INTRODUCTION and for shear stresses were compared with those from a fracture mechanics model to predict ultimate It is known that the problem of mechanically strength. Jorissen [18] found, however, that the tensile fastened joints in timber is difficult to analyze because stresses perpendicular-to-grain were underestimated of the anisotropic and heterogeneous nature of the and, to allow crack initiation to be detected by the material. Since there are no available analytical solu- fracture theory, added an assumed peak stress perpen- tions associated with loaded holes in wood, current dicular-to-grain at the bolthole location. The joint area design procedures for mechanical fasteners are based including the stable crack propagation was modelled on approximate or empirical solutions and exist only as a beam on elastic foundation. This assumption limits for the simplest types of joints. the robustness of the model. Due to the importance of the problem, bolted Moses [23] and Moses and Prion [24] proposed joint has been studied by using numerical and ex- a material model that is based on orthotropic elas- perimental [4, 7-10, 21, 22, 26, 29, 31, 32] methods ticity, anisotropic plasticity for non-linear behaviour in the past. of wood in compression, and the Weibull’s weakest link theory to predict brittle failure. Linear elastic Patton-Mallory et al. [27] developed and behaviour was assumed for tension and shear. The evaluated a three-dimensional numerical model of weakest link theory provides a probabilistic approach a bolted wood connection loaded parallel to grain. to predicting the failure based on the stressed volume Nonlinear parallel to grain compression of wood and of wood and can be used for cases when the ultimate degradation of shear stress stiffness were described strength of the single bolt connection is governed using a trilinear stress-strain relationship. The con- by brittle failure (such as shear and tension per- nection model also accounted for an elastic-perfectly pendicular-to-grain). This three-dimensional model plastic steel pin, oversized hole, and a changing con- was implemented using finite element analysis for a tact surface at the pin-hole interface. The numerically single-bolt connection specimen. predicted load-displacement curves were stiffer than the experimental curves. Regrettably, experiments and numerical methods do not produce open-form solutions as a In Kharouf et al. [19], a nonlinear numerical result of the high amount of possible combinations of model is developed to study the behaviour of timber involved parameters. In contrast, it would be practi- connections with relatively low member thickness- cal to have equations developed using the detailed to-fastener diameter ratios. A plasticity-based com- analytical basis. pressive constitutive material model is proposed to represent wood as elastoplastic orthotropic material The objective of this paper is to present in regions of biaxial compression. Linear elastic or- a comprehensive analytical method capable of thotropic material response was used otherwise with predicting the ultimate strength of reinforced and maximum stresses taken as the basis for predicting non-reinforced timber bolted joints. The method of failure criteria. Nonlinear geometry due to increased complex functions [20, 25] for anisotropic materials sliding contact between the bolt and the hole is mod- is used to obtain the stress distributions. The solution elled using the Lagrange multiplier algorithm. is compared directly to results of laboratory tests. Jorissen [18] attempted to account for brittle This study examined as well the technical feasi- fracture in timber joints using the European Yield bility of reinforcing the wood at bolted joints with self- Model by calculating stress distributions along poten- tapping screws. The purpose of local reinforcements tially critical load paths within the wood member. The in a joint is to improve its load-carrying capacity and average stresses for tension perpendicular-to-grain stiffness and to improve its ductility.

38 Revista EIA [32] Zhang K., Ueng C. Stresses around a pin-loaded hole in orthotropic plates with arbitrary loading direction. Composite Structures 1985; 3:119-143.

[32] Zhang K., Ueng C. Stresses around a pin-loaded hole in orthotropic plates with arbitrary loading direction. Composite Structures 1985; 3:119-143.

This paper reports test results of various connection configurations with reduced end distances, with and without reinforcement.

2. STRESS DISTRIBUTION AROUND A PIN-LOADED HOLE IN AN ELASTICALLY ORTHOTROPIC PLATE

In this paper, the stress distribution around a pin-loaded hole in an elastically orthotropic plate is Fig. 1. Double-lap mechanical joint investigated for the main member of a double-lap Fig. 1. Double-lap mechanical joint Double-lap mechanical joint mechanically fastened joint. Consider a homoge- Figure 1. neous, orthotropic plate of width b with a circular hole of diameter d as is shown in Fig. 1. Let the x- and y-axes be the principal axes of the plate, and the direction of the pin load F be the same as the positive y-axis. The hole is at a distance e from the free end of the plate. The clearance between the pin and the hole is denoted as λ. The pin-hole problem is essentially a two- body-contact problem. In order to determine contact surfaces and stresses the inverse method is used, a value for the contact angle between the bolt and the timber is assumed and the corresponding load is calculated. The process is repeated for a series of prescribed contact angles. The assumption of a value for the contact angle has the advantage of simplicity. Fig. 2. Geometry for the joint and boundary conditions It is assumed that the hole is loaded without friction on a portion of its edge by an infinitely rigid Fig. 2. Geometry for the joint and boundary conditions pin of diameter d. The loading pin is represented Figure 2. Geometry for the joint and boundary by compressive edge loads distributed around the conditions hemispherical contact area. The resulting force F equals 2pRt, where p is the average bearing stress The method of complex functions (Lekhnitskii according to the classical definition, R is the radius [20], Muskhelishvili [25]) for anisotropic materials of the hole and t is the unit thickness of the plate. is used to obtain the stress distributions. For plane The analytical model with its boundary and loading stress situations in orthotropic plates the stresses can conditions is shown in Fig. 2. be expressed by means of derivatives of two stress

complex functions ϕ (z 1 ) and Ψ (z 2 ):

Escuela de Ingeniería de Antioquia 39 FRμ ⎧ μ ⎫ Ψ =ζ + Ξ ζ + 1 ⎧− 1 ⎫ (7) ()z2 BLn 2 0() 2 FRμ1⎨ μ1z 2 ⎬ (7) Ψ()z2 = BLnζ2 + Ξ 0() ζ 2 4 +b μFR− μμ ζ⎨⎧ μ− + μμ z 2 ⎬⎫ Ψ =ζ + Ξ ζ + 4FRb2μμ− 11⎩ μ⎧ 2 ζ− 1 μμ+ 21 μ ⎭ ⎫ (7) Ψ()()zz2 = BLn BLnζ2 + Ξ 0()() ζ 2 + 21 1⎩⎨ 2− 11 2 zz 2 ⎭⎬ (7) 2 2 0 2 4b μ2− μ 1⎨⎩ ζ 2 μ 1+ μ 2 2 ⎬⎭ FRμ ⎧ 4μb μ2−⎫ μ 1 ζ 2 μ 1+ μ 2 Ψz = BLnζ + Ξ ζ + FRμ1 − μ1 z ⎩ (7) ⎭ Ψ()z2 = BLnζ2 + Ξ 0() ζd 2ϕ + d ϕ d ζ1 ⎨ F − μ2 μ⎧z 2 ⎬⎧R d ζ1μ μ 2 ⎫ ⎫ ϕ ' z =d =ϕ4b μ d− ϕ μ d +ζ ζ F μFR+ μ μ 1 −⎧ R d −ζ1 μ ⎫ (7) (8) ()1 Ψ()z2 =4 BLnb μζ22− + μ Ξ 1 0⎩() ζ1 ζ 2 2 + μ 1+ μ2 2⎨ ⎭⎨ 2 − 1z 2 ⎬ 2⎬ (8) ϕ '()z 1 dz=dϕ d =ζ d dz ϕ d ζ4 +b μ F4−b μ μ− μζ⎨⎧− ζRdz μ d ζ+μ μ − + μ μ ⎬⎫ 1dϕ 1 d ϕ 1d ζ1 F 1 μ2 22⎩ 1⎧⎩1 2R2 1 d 1 ζ1 12 ⎭ μ 2 2⎭ ⎫ (8) ϕ '()z 1 =dz1 = dζ 1 dz1 1 + 4 b μ 1−2 μ 2 ⎩⎨−ζ 12 dz11 − μ 1+ 2 μ 2 ⎭⎬ (8) Bolted timber joints with self-tapping screws ϕ '()z 1 = = + ⎨− − ⎬ dϕ d ϕ d ζ Fdz1 μ dζ⎧ 1 dzR 1 d ζ4 b μ 1− μ μ 2 ⎩⎫ ζ 12 dz 1μ 1+ μ 2 ⎭ dϕ d ϕ d ζ1 Fdz1 μ2 dζ 1 dzR 1 d ζ4 b1μ 1− μ μ 2 2 ⎩ ζ 1 dz 1μ 1(8)+ μ 2 ⎭ ϕ ''()z 1 = = + dϕ d ϕ⎨− d ζ2 F − μ⎧ ⎬R d ζ μ (8)⎫ ()1 dΨΨ d d⎨ζ2 1 2 F μ12 ⎧ ⎬R dζ12 2 μ 1 ⎫ (8) dz1 dζ 1 dzϕ ' 1()z 1 4= b μ 1− = μ 2 ⎩ ζ 1 +dz 1μ 1+⎨ μ− 2 ⎧⎭ 2 − ⎬ ⎫ (9) 1Ψ'()z 12 = 1d =dzΨΨ 1 d 2ζ⎩ dz dζ +12 4 b F1μ− μ11⎨− 2 ζ⎭ 2dzR dζ −μ2+ μ μ 1⎬ Ψ'()z dz = d1 =ζ dz 1 14 +b μ 1− μ 2 ⎩ ζ⎧1−dz1 1μ − + 2 μ⎭ ⎫ (9) 2 2dΨΨ 2 d 2 dζ2 F 2 μ 11⎩ ⎧⎨ 2 R2 2 dζ2 1 μ 2 1⎭ ⎫⎬ (9) Ψ'()z 2(1) =ddzΨΨ2 = dζ 2 d dzζ2 2 + 4 F b μ 2 μ−1 μ 1 ⎨−ζR2 ddzζ22 − μ 1 μ+ 1 μ 2 ⎬ Ψ' z = = + ⎩−2 − (8) ⎭ (9) dΨΨ d() dζ2 Fdz μ dζ ⎧ dzR dζ4 b μ− μ μ ⎨⎫ ζ 2 dz μ+ μ ⎬ Ψ' z =dΨΨ = d dζ2 + FdzdΨΨ2 μ1 d dζ 2− ddzζR 2 d F4ζ b2μ − μ 2− μ μ⎧ 1 1 ⎩Rζ d2ζdz 2 μ μ(9) 1+⎫ μ 2 ⎭ where,Ψ'()z 2 = = + 2 2⎨−2 22 − 21 1 ⎩⎬ 2 22 1 1 2 ⎭ (9) dz dζ Ψ dz'()z 2 4 = b μ− = μ ζ 2 + dz μ+⎨ μ−2 − ⎬ where, dz2 dζ 2 dz 2(2) 4 bdzμ 2− μ dζ 1 ⎩ dzζ 2 dz4 b 2μ−μ μ 1+ μ 2ζ⎭ dz μ+ μ where, 2 2 2 2 1 ⎩ 2 2 1 2 ⎭ where, (9) where, dϕ A dΓ0 where, where, (3) d=ϕ + A dΓ (10) dζ ζ= d +ζ 0 (10) 1dϕ 1 A 1dΓ0 (10) where dζϕ1=ζ A 1 +d dΓ ζ0 1 dϕ A dΓ 0 dϕ = A +dΓ (10) = + 0 dζ 1=ζ+ 1 d0 ζ 1 (10) (10) where ϕ (z 1 ) and Ψ (z 2 ) are arbitrary functions of dζ 1ζ 1 d ζ 1 (10) dζ 1ζ 1 d ζ 1 dΨdζ Bζd dΞ ζ d1 1 d 1 =1 1+ 0 1 (11) the complex variables z 1 and z 2 : dΨ B dΞ0 dζ ζ= d +ζ (11) 2dΨ 2 B 2dΞ0 (11) dΨ B dΞ ddΨζΨ2= Bζ B 2 +dΞd dΞ ζ 0 2 (11) (4) dΨ B dΞ 0 = + 0 (11) (11) z= x + µ y = + dζ =ζ + d ζ (11) 1 1 = + dζζ 2ζζ 2 d ζ d ζ 2 The values of complex constantsdζ 2ζ 2 A dand ζ 2 B are22 found 2 2 from: 2 2 (5) 2 2 2 z= x + µ y The values of complex constants A and B are found from: 2 2 The values of complex constants AThe and values B are of found complex from: constants A and B are The values of complexThe valuesThe constants values of complex of A complex and B constantsare constants found Afrom: A andand B areare found found from: from: (12) found from:A = A" i (12) In general, the complex functions A = A" i ϕ (z 1 ) A = A" i (12) (12) A = A" i A = A ” iA = A" i (12)(12) (12) and Ψ z can be estimated from the boundaryA = A" i B =A B="Ai " i (13) ( 2 ) B = B" i (13) conditions of the problem. The pertinent solutions B = B ” iB ==B" i" (13) (13) (13) B = B" i B = B i (13) (13) for the pin-loaded hole in an elastically orthotropic B B" i u ⎛ F ⎞ ⎧ 2 ⎫ (14) B" = ⎜ −⎛ ⎟F÷⎞⎨⎧1 −⎧ u u⎬⎫2 ⎫ or isotropic plate problem have been obtained by B" =⎛ − F ⎞ ÷ 1 −2 (14) (14) (14) ⎛ F ⎞ ⎧ B "⎝u=2⎜⎜⎛⎫4−π ⎠F⎟ ÷⎟⎞⎩⎨1⎨⎧−u 1 u⎭⎬2 ⎬⎫ " = ⎛ − ⎞ ÷ 1 −" =2 ⎝⎛ −44πFπ⎠⎞ ÷⎧ 1u− uu2 ⎫ (14) (14) Echavarría [11] and Echavarría et al. [12]B and" = ⎜are− ⎟ ÷ ⎨1BB−" = ⎝⎜⎬− ⎠⎟÷⎩ ⎩⎨1 −1 ⎭1 ⎭⎬ (14) ⎝ 4π ⎠ u 1 ⎜⎝ 4π ⎟⎠ ⎨ u ⎬ summarized here. ⎝ ⎠ ⎩ u 1 ⎝⎭ 4π ⎠ ⎩ u 1 ⎭ ⎛ F ⎞ ⎩ 1 ⎭ = −⎛ F ⎞− (15) (15) A"A"⎜= ⎜⎛− ⎟F⎟ −⎞BB"" (15) (15) ⎛ F ⎞ A"⎝= ⎜⎛4−π4π⎠F ⎟⎞ − B" A"(6)= ⎜ − ⎟ − B" ⎝ F⎠ (15) (15) ⎜ ⎟ A" = ⎛⎝⎜ − 4π ⎞⎠⎟ − B" (15) ⎝ 4π ⎠ A" = ⎜⎝− 4π ⎟⎠− B" where, where,u and uu1 andareu constants2 are constants determined determined by: by:⎝ 4π ⎠ 1 2 where u 1 and u 2 are constants determined by: where, and where,are constantsu 1 and determinedu 2 are constants by: determined by: where, u 1 and u 2 are constants determined by: where, u 1 and u 2 are constants(7) determined 2by: where, u and u are constants determined=2μ by: + = 1 2 u= uj S Sμ11 + j S S 12(( j j = 1, 1, 2) 2) (16) (16) (16) 2 j11 j 2 12 u= Sμ2 + S( ju =j 1,= 2) S11 μ j + S 12 ( j = 1, 2) (16) (16) ujj= S11μ j j + S 12 ( j = 1, 2) 2 (16) uj= S11μ2 j + S 12 ( j = 1, 2) (16) Γ0 ()ζ 1 and Ξ 0 ()ζ 2 are the holomorphicu= Sμ parts + S of ϕ(() jz =1 1,and 2) Ψ ()z 2 : (16) Γ ()ζ and Ξ ()ζ are the holomorphicj11 parts j of 12 ϕ ()z and Ψ ()z : Γ ζ and Ξ0 Γζ1 ζ are and the0 holomorphicΞ 2 ζ are parts the holomorphicof ϕ z and partsΨ z of: ϕ1 z and Ψ2 z : Γ0 ()ζ 1 and Ξ 0 ()ζ0 2() 1are the holomorphic0 ()2 parts of ϕ ()z 1 and Ψ ()z 2 : ()1 ()2 Γ ζ and Ξ ζ areσ =− 1the holomorphic parts of ϕ and Ψ : Γ 0 ()ζ 1 and Ξ 0 ()ζ 2 are⎧ the holomorphic parts of ϕ ()zz 1 and Ψ ()zz 2 : 0 ()1 0 ()i σ2=−1⎪ ⎡ μ2F 2 2 Fω ()2 1 2 ⎤ σ()+ ζ21 d σ Γσ0=−()ζ 1 = ⎧ ⎨ 4Lnσ− σ + σ + σ + σ − 2 + ⎧σ =−1 σ =−1 ⎢ ( ) ( )⎥ ⎧ i 4π() μ1⎪− μ 2⎧ ⎪⎡ μ∫2F⎣ 4 πi 2 2 F4ωπ 2 2 ⎦⎤σσ−+ ζ1 ζ1 σd σ Γi ζ =⎪ ⎡μ 2F σ⎩=−σ2 =+1 1 2 4LnFσω − σ2 + σ2 +2⎤σ+ ζ σ1 d +σ σ − 2 + Γζ = 0() 1 ⎪ ⎡ 2 i4Ln⎨σ−⎪⎧σ =−⎢2 1 +⎡ σμ2F + σ2 + σ2 − 2 ⎤ Fω1 2+ ⎥ ⎤ σ+ ζ1 d σ (17) Γ0()ζ 1 = Γ()ζ⎨ 4π = ⎢ μ− μ4Lnσ−⎧ σ⎣ 4 + π σi ( +4Ln σσ− + σ σ +−) σ2 2⎥4 +π ( σ+ + σ 2 ) −⎦2σ− ζ σ + 4π() μ− μ0 1 ()⎢⎣ 41 πi (i 2 ⎪ ∫⎨⎪ σ =+1⎢⎡ μ2)F (4π ( 2 )⎥⎦)σ−F ζω ( σ 2 )⎥⎤ σ1+ ζ1 d σ 4π() μ1− μ 2 ⎪ ∫ ⎣4π4 πi μi − μσ =+⎪ μ4 πFi 4π 2⎦ σ−F4 ζωπ1 σ 2 σσ+− ζ ζd σ σ Γ0()ζ⎪ 1 ∫ = ()1⎩ 2 1 ⎡⎣ 2 4Lnσ− σ2 + σ + σ2 + σ − 2 ⎤⎦ 11 + (17) ⎩σ =+1 ⎪⎨ ∫ σ+ ζd σ μ F σ+ ζd σ ⎫ Γ0()ζ⎩ 1σ =+ =1 4π μ− μ⎨⎩σ =+1 ⎢ 4 πi (4Lnσ1 − σ +⎡ σ2 ) +4π ⎤ ( σ +1 σ − 2 )⎥ σ(17)− ζ σ + ()1σ 2=+1⎪ ∫ ⎢⎣[]μ2F ( − ) Lnσ ( ⎬ )⎥⎦ 1 (17) σ =+1 4π() μ− μ⎣ 4 πi ⎢4π ⎥ ⎦ σ− ζ σ σ =+1 1 2 ⎪⎩σσ=+∫=+∫1 1 σ− ζ1 σ ∫ ⎣2 πi ⎦ σ− ζ1 σ ⎭ 1 (17) ⎩σσ=+=−11 σ+ ζd σγ μ F σ+ ζd σ ⎫ (17) σ+ ζ1 d σσ =+1 ⎡ μ2 F 1 ⎤ σ+ ζ⎡1 d σ2 ⎫ ⎤ 1 (17) μ F 1 []μ−2F ⎡2 σLn+ ζσ ⎤ d− σ1 ⎡ μLn Fσ ⎤ σ+ ζ⎬d σ ⎫ []μ2F σ =+−1 ⎢Lnσ1 ⎥ ⎢⎬ 2 ⎥ 1 σ− ζ1 σ[]μ⎢2⎣σF2 π−iσ ζ1+ ζσ ⎥⎦ d σ σ−− ζ⎣1 2 σ πi μ F Ln ⎦ σ− σ ζ1+ ζσ ⎭d σ ⎬⎫ ∫σ− ζ1∫ σ ∫ ⎣2 πσiσ+− ζ ζ1 ⎦d σ σ σ− ∫ ζ1 σ⎢⎡ μ⎭2 π2 Fi ⎥⎤ σ σ+− ζ ζ1 d σ σ ⎫ σ =−1 σ =−1 ∫γ[]μ2F 11 γ − ∫ ⎡⎣2 Lnσ ⎤ ⎦ 11 ⎭⎬ σ =−1[]μ2F σ− ζ σ− γ ⎢2 πi Lnσ ⎥ σ− ζ σ ⎬ ∫σ− ζ1 σ ∫ ⎢⎣2 π ⎥ ⎦ σ− ζ1 σ ⎭ σ∫=−1 1 ∫γ ⎣i ⎦ 1 ⎭ σ =−1 γ ⎧σ =−1 i ⎪ ⎡ μ F 2 2 Fω 2 2 ⎤ σ+ ζd σ Ξζ = 1 σ− σ + σ + σ + σ − 2 + 0() 2 ⎨ ⎢ (4Ln ) ( 2)⎥ 4π() μ2− μ 1 ⎪ ∫ ⎣ 4 πi 4π ⎦ σ− ζ2 σ ⎩σ =+1 (18) σ =+1 (18)

σ+ ζ2 d σ⎡ μ1 F ⎤ σ+ ζ2 d σ ⎫ []μ1F − ⎢Lnσ ⎥ ⎬ ∫σ− ζ2 σ ∫ ⎣2 πi ⎦ σ− ζ2 σ ⎭ σ =−1 γ

40 where, Revista EIA

1− Sin α ω = (19) Cos α

2 π⎧e−2 d ⎫ π ⎧ e − 2 d ⎫ α =⎨ ⎬ − ⎨ ⎬ 2d≤ e ≤ 5 d (20) 5⎩d ⎭ 30 ⎩ d ⎭

3π π ⎧e− 5 d ⎫ α = + ⎨ ⎬ 5d≤ e ≤ 14 d (21) 10 45⎩d ⎭

In this manner, stress functions ϕ ()z 1 and Ψ ()z 2 are completely determined. Evidently, with this analytical method, it is possible to estimate the stresses in any point of the orthotropic joint and to show the stress distribution around a pin-loaded hole in an elastically orthotropic plate. The effect of material properties and geometry of the joint can be determined analytically. Although it was developed for orthotropic composite materials, the presented approach is equally effective for analyzing mechanical joints involving solid wood and wood-based composites.

3. BRITTLE FAILURE OF A BOLTED TIMBER JOINT AND IMPROVING LOAD- CARRYING CAPACITY

The proposed model can be used to determine analytically the points of stress concentrations in the zone of contact between the fastener and the timber and predict the brittle modes of failure in dowel-type timber joints.

Let’s consider, for instance, the stress distributions calculated for a wood element of a unit thickness t and width b = 4d assuming the elastic properties of red spruce shown in Table 1. Table 2 summarizes the peak stresses xy and x at corresponding angles for red spruce calculated in this example. The calculated stresses are normalized by the average bearing stress p = F/d and are shown along the hole edge as a function of angle (as defined in Fig. 3).

Avoiding completely tension perpendicular-to-grain and shear is not ever possible. If not placed properly, bolts may cause undesirable brittle failure in timber due to excessive tension perpendicular-to-grain. A possibility to avoid splitting and to guarantee a plastic joint behaviour is to reinforce the timber in the joint area. The tension stresses are then transferred by a reinforcement perpendicular-to-grain. Known methods to prevent splitting of timber ⎧σ =−1 i ⎪ ⎡ μ F 2 2 Fω 2 2 ⎤ σ+ ζd σ Ξζ = 1 σ− σ + σ + σ + σ − 2 + 0() 2 ⎨ ⎢ (4Ln ) ( 2)⎥ 4π() μ2− μ 1 ⎪ ∫ ⎣ 4 πi 4π ⎦ σ− ζ2 σ ⎩σ =+1 (18) σ =+1

σ+ ζ2 d σ⎡ μ1 F ⎤ σ+ ζ2 d σ ⎫ []μ1F − ⎢Lnσ ⎥ ⎬ ∫σ− ζ2 σ ∫ ⎣2 πi ⎦ σ− ζ2 σ ⎭ σ =−1 γ where, where Table 1. Elastic constants (MPa) of red spruce 1 1–− sinSin aα (19) (Wood Handbook [30]) w =ω = (19) cosCos aα

Ex Ey Gxy νyx 2 π⎧e−2 d ⎫ π ⎧ e − 2 d ⎫ 470 (20) 11100 670 0,470 α = ⎨ ⎬ − ⎨ ⎬ 2d≤ e ≤ 5 d (20) 5⎩d ⎭ 30 ⎩ d ⎭

3π π ⎧e− 5 d ⎫ Table(21) 2. Predicted peak stresses at a joint α = + ⎨ ⎬ 5d≤ e ≤ 14 d (21) 10 45⎩d ⎭ of red spruce Shear stress at Perpendicular-to-grain In this manner, stress functions and ϕ (z 1 ) e/d θ = 60° stress at θ = 90° In this manner, stress functions ϕare()z 1completelyand Ψ ()z 2 determined. are completely Evidently, determined. Evidently, Ψ (z 2 ) τxy / p σx / p with this analytical method,with this it isanalytical possible method, to estimate it is possible the stresses to estimate in any point of the 2 0,80 0,52 orthotropic joint and to showthe stresses the stress in anydistribution point of thearound orthotropic a pin-loaded joint holeand in an elastically 3 0,50 0,20 orthotropic plate. The effectto show of the material stress distribution properties around and ge ometrya pin-loaded of the joint can be determined analytically. Although it was developed for orthotropic composite materials, the hole in an elastically orthotropic plate. The effect of 4 0,34 0,04 presented approach is equally effective for analyzing mechanical joints involving solid5 wood 0,31 0,01 and wood-based composites.material properties and geometry of the joint can be determined analytically. Although it was developed 7 0,28 -0,03 for orthotropic composite materials, the presented 10 0,18 -0,13 3. BRITTLE FAILUREapproach OF A BOLTED is equally TIMBER effective for JOIN analyzingT AND mechani IMPROVING- LOAD- CARRYING CAPACITYcal joints involving solid wood and wood-based composites. The proposed model can be used to determine analytically the points of stress concentrations in the zone of contact between the fastener and the timber and predict the brittle modes of failure in dowel-type timber3. joints. BRITTLE FAILURE OF A

Let’s consider, for instance, theBOLTED stress distributions TIMBER calculated JOINT for a wood element of a unit thickness t and width b = 4d assumingAND theIMPROVING elastic properties LOAD-of red spruce shown in Table 1. Table 2 summarizes the peak stressesCARRYING xy and x CAPACITY at corresponding angles for red spruce calculated in this example. The calculated stresses are normalized by the average bearing stress p = F/d and are shown alongThe proposedthe hole edge model as cana function be used of to angledetermine (as defined in Fig. Angle θ 3). analytically the points of stress concentrations in the FigureFig. 3. 3.Angle zone of contact between the fastener and the timber Avoiding completely tension perpendicular-to-grain and shear is not ever possible. Avoiding If not completely tension perpendicular- and predict the brittle modes of failure in dowel-type placed properly, bolts may cause undesirable brittle failure in timber due to excessiveto-grain tension and shear is not ever possible. If not placed timber joints. perpendicular-to-grain. A possibility to avoid splitting and to guarantee a properly, plastic joint bolts may cause undesirable brittle failure behaviour is to reinforce the timberLet’s in consider, the joint for area. instance, The tension the stress stresses distribu are- thenin transferredtimber due to excessive tension perpendicular- by a reinforcement perpendicular-to-grain. Known methods to prevent splitting of timber tions calculated for a wood element of a unit thickness to-grain. A possibility to avoid splitting and to guar- t and width b = 4d assuming the elastic properties of antee a plastic joint behaviour is to reinforce the red spruce shown in Table 1. Table 2 summarizes the timber in the joint area. The tension stresses are then transferred by a reinforcement perpendicular-to- peak stresses τxy and σx at corresponding angles θ for red spruce calculated in this example. The calculated grain. Known methods to prevent splitting of timber stresses are normalized by the average bearing stress members are reinforcements of the joint area with p = F/d and are shown along the hole edge as a func- glued-on wood-based panels, pressed-on punched tion of angle θ (as defined in Fig. 3). metal plates or glass fibre reinforcements. If wood Fig. 4. Reinforced bolted joint

Escuela de Ingeniería de Antioquia 41

Fig. 5. Load ~ displacement behaviour for reinforced and non-reinforced bolted joints. (d=15,9 mm, b/2=44,5 mm, t=38,0 mm, e/d=5). Bolted timber joints with self-tapping screws

joint could be reinforced (Haller et al. [14], Haller and Wehsener [15], Blass and Bejtka [5], Chen [6], Hansen [16], Hockey et al. [17], Soltis et al. [28]), end distance requirements could be re-evaluated, which would allow more compact joints and an easier installation.

The approach presented in this study is to use self-tapping screws oriented perpendicular-to- grain as internal reinforcement. The reinforcement is shown in Fig. 4. Compared to the reinforcement methods mentioned, self-tapping screws are easier to apply and less expensive. members are reinforcementsIn of this the jointpaper, area splitting with glued-on failure wood-baseis assumedd panels, to pressed-on punched metal platesoccur or glass as fibre soon reinforcemen as the perpendicular-to-graints. If wood joint could be stress reinforced (Haller et al. [14, 15], Blass and Bejtka [5], Chen [6], Hansen [16], Hockey et al. [17], Soltis et al. Figure 4. Reinforced bolted joint [28]), end distance requirementsreached the could strength be re perpendicular-to-grain-evaluated, which would of allow the more compact joints and an easier installation.wood. Basically, the perpendicular-to-grain strength can be improved by reinforcement. Reinforcement The approach presentedcould in this be usedstudy locallyis to us ine self-tappingthe vicinity screws of a bolt. oriented Conse perpendicular-to-- grain as internal reinforcement. The reinforcement is shown in Fig. 4. Compared4. to EXPERIMENTAL the reinforcement methodsquently, mentioned the cracks before may self-t beapping stopped screws at the are position easier to of apply and less expensive. the reinforcements and the bearing strength of the VERIFICATION wood could be attained. In this paper, splitting failure is assumed to occur as soon as when the perpendicular-to-grainLaboratory tests on reinforced and non-rein- stress reached the strength perpendicular-to-grain of the wood. Basically, the perpendicular- The analytical model presented allows pre- forced bolted timber joints loaded parallel to grain to-grain strength can dictingbe improved the load by reinforcemen in the reinforcingt. Reinforcement screw. The could force be used locally in the vicinity of a bolt. Consequently, the cracks may be stopped at the positionby a of single the bolt representing the geometry shown reinforcements and theFs bearing acting strengin theth screw of the iswood equal could to thebe attained. perpendicular in Fig. 1 were performed to verify the predictions stress σ multiplied by the area As on which the x of the proposed analytical model. The bolts were The analytical model presented previously allows predicting the load in the reinforcing screw. force acts. 15,9‑mm (5/8‑in.) in diameter made of low carbon The force Fs acting in the screw is equal to the perpendicular stress x multiplied by the area As on which the force acts. The equation that describes the area As, con- steel conforming to ASTM A307. Bolt lengths were sidering the stresses between y=R and y=2R, has selected to ensure that threads were excluded from The equation that describes the area As, considering the stresses between y=R and y=2R, has the following form: bearing against the wood. The ratio of the wood the following form: member thickness to bolt diameter was small enough to induce failure in the wood, with minimum bending As = t R As= t R (22) (22) deformation of the bolt. Wood plates for the joints The force acting in the screwThe is: force acting in the screw is: were cut from 38 by 89-mm (nominal 2 by 4-in.) red spruce kiln-dry lumber so that the joint area was free ⎪⎧⎛ ()4 +πFn ω ⎞ ⎛Fk3 Fk ⎞ ⎛υxy F ⎞⎪⎫ As (23) Fs = − + − of defects.(23) Tabla 3 and table 6 show the single-dowel ⎨⎜ 2 ⎟ ⎜ ⎟ ⎜ ⎟⎬ ⎩⎪⎝ 2Rπ ⎠ ⎝2b 2 Rπ ⎠ ⎝ 2 R π ⎠⎭⎪ t joint geometry.

In these circumstances localIn thesereinforcement circumstances can be very local effective reinforcement in ensuring a reasonablePrior to testing, the specimens were condi- load-carrying capacity and stiffness and in providing the necessary ductility. can be very effective in ensuring a reasonable load- tioned to attain 12 % equilibrium moisture content. carrying capacity and stiffness and in providing the Specific gravity based on oven-dry mass and volume 4. EXPERIMENTALnecessary VERIFICATION ductility. at 12 % moisture content of the specimens varied

Laboratory tests on reinforced and non-reinforced bolted timber joints loaded parallel to grain by a single bolt representing the geometry shown in Fig. 1 were performed to verify the predictions of the proposed analytical model. The bolts were 15,9-mm (5/8-in.) in diameter Revista EIA made of low carbon42 steel conforming to ASTM A307. Bolt lengths were selected to ensure that threads were excluded from bearing against the wood. The ratio of the wood member thickness to bolt diameter was small enough to induce failure in the wood, with minimum bending deformation of the bolt. Wood plates for the joints were cut from 38 by 89-mm (nominal 2 by 4-in.) red spruce kiln-dry lumber so that the joint area was free of defects.

Prior to testing, the specimens were conditioned to attain 12 % equilibrium moisture content. Specific gravity based on ovendry mass and volume at 12 % moisture content of the specimens varied from 0,37 to 0,41 as determined per ASTM D2395-02 [2]. Material shear Table 3. Non-reinforced single-dowel joint geometry

Bolt diameter Edge distance Thickness End distance Number of d (mm) b/2 (mm) t (mm) e replications

2d, 3d, 4d, 5d, 15,9 44,5 38,0 53 7d and 10d

Table 4. Mechanical properties of red spruce

Strength average (MPa) Plate thickness t (mm) σx σy τxy Perpendicular-to-grain Embedding Shear

38,0 3,80 29,7 8,80

from 0,37 to 0,41 as determined per ASTM D2395- Fig. 4 shows the reinforcing screw used in this 02 [2]. Material shear strength parallel-to-grain and study. A screw (GRK fastener 1/4” by 3½”) with 90 tensile strength perpendicular-to-grain were deter- mm of length, 6 mm of outer diameter and 70 mm mined using ASTM D143-94 [1]. The shearing sur- of threaded length was used. The reinforcing screw face dimensions were identical for all shear strength is at a distance s=d from the centre of the hole. parallel-to-grain tests. The dowel embedding strength Tests were normally conducted on single-hole for each bolt diameter was determined according to specimens which had the geometry described in ASTM D5764-97a [3]. The material properties are Tables 5 and 7. During the course of this experimenta- summarized in Table 4. tion, 37 specimens were tested using reinforced joints The joints were tested with static load applied with self-tapping screws. For comparison, 53 joints in tension parallel-to-grain using a universal testing were tested without reinforcement perpendicular- machine in accordance with EN 26891:1991 [13]. to-grain.

Table 5. Experimental results and analytical predictions for non-reinforced single-bolted joints

Experimental Predicted Comparison Bolt Average Number of Average Standard (F-Fexp) diameter e/d Failure load bearing Prevalent replications failure load deviation Fexp d (mm) F (kN) stress failure mode Fexp (kN) (%) (%) p (MPa) 2 10 4,43 13 4,42 7,30 Splitting -0,40 3 8 12,1 7 10,6 17,6 Shear-out -12,5 4 10 17,3 4 15,6 25,9 Shear-out -9,60 15,9 5 10 18,2 9 17,2 28,4 Shear-out -5,60 7 7 20,5 7 17,9 29,7 Bearing -12,5 10 8 20,3 10 17,9 29,7 Bearing -11,4

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Table 6. Reinforced single-dowel joint geometry

Bolt diameter Edge distance Thickness End distance Number of d (mm) b/2 (mm) t (mm) e replications

15,9 44,5 38,0 2d, 3d, 4d and 5d 37

Fig. 3. Angle Table 7. Experimental results and analytical predictions for reinforced single-bolted joints

Experimental Predicted Comparison Bolt Number of Average Standard (F-Fexp) diameter e/d Failure load Screw load Prevalent replications failure load deviation Fexp d (mm) F (kN) Fs (kN) failure mode Fexp (kN) (%) (%) 2 10 9,76 10 6,65 1,10 Shear-out -31,9 3 9 12,7 25 10,6 1,10 Shear-out -16,5 15,9 4 9 17,4 5 15,6 0,30 Shear-out -10,3 5 9 18,4 10 17,2 0,10 Shear-out -6,5 Fig. 4. Reinforced bolted joint

The number of replications, the summary of test results and comparison with the analytical predictions for each test configuration are given in Tables 5 and 7. The load-carrying capacities were predicted using the stresses obtained by the analytical model and the failure criteria presented above.

The effectiveness of reinforcement methods is studied along with the possibility of reducing the end-distance requirements. Test results in Table 7 showed that reinforcement had positive effects on the load-carrying capacity when the end distance e is shortest. The predominant failure mode in specimens loaded parallel to grain was in shear without plug-shear-out. Little evidence of wood crushing was observed. The screws prevent the Fig. 5. Load ~ displacement behaviour for reinforced and non-reinforced bolted joints. cracks from further growing. Failure modes were Figure 5. Load ~ displacement behaviour for reinforced(d=15,9 mm,and b/2 non-reinforced=44,5 mm, t=38,0 bolted mm, e/d= joints.5). affected by reinforcement; the propagation of crack was reduced. Generally, the crack propagation in (d = 15,9 mm, b/2 = 44,5 mm, t = 38,0 mm, e/d = 5) non-reinforced joints was more pronounced than in the reinforced ones. The reinforced specimens Fig. 5 shows the load-displacement curves with e=2d, 3d, 4d and 5d failed mainly in shear as for a non-reinforced and a reinforced joint. Table 8 predicted by the model. shows the relation between load-carrying capacity of reinforced and non-reinforced joints. Particularly,

44 Revista EIA the load-carrying capacity increases 120 % for the ment with the calculated load-carrying capacities e=2d reinforced specimens with 15,9-mm bolts. It and predicted failure modes. is clear that the increase in load-carrying capacity is significant with the use of the reinforcing screws Nomenclature enabling smaller joints and significant savings in A complex constant timber volume. B complex constant Table 8. Relation between load-carrying capacity of b width of plate reinforced and non-reinforced joints d diameter of the hole Ratio of reinforced and e end distance Bolt diameter e/d non-reinforced load-carrying d (mm) E perpendicular-to-grain modulus of elasticity capacity (%) x E longitudinal modulus of elasticity 15,9 2 120 y F resultant force 15,9 3 4,68 Fs screw load 15,9 4 0,78 Gxy shear modulus 15,9 5 1,03 p average bearing stress R radius of the hole 5. CONCLUSIONS s screw distance

Sij elastic compliances of the plate material The solution proposed here is entirely ana- λ clearance lytical and the most important results are compiled constants clearly in tables. u 1 , u 2 z complex variable This study dealt with the analytical and k µ , µ complex parameters of the first order experimental investigation of the effectiveness of 1 2 self-tapping screws as means of reinforcing bolted νyx coefficient of Poisson timber connections loaded parallel-to-grain. Several σx perpendicular-to-grain stress equations are presented to calculate the load-carry- σ longitudinal stress ing capacity of reinforced and non-reinforced timber y shear stress joints. τxy complex stress functions The reinforced specimens showed a less cata- ϕ (z 1 ), Ψ (z 2 ) strophic failure mode whereas the non-reinforced specimens failed in a brittle way. In reinforced joints it ACKNOWLEDGMENTS is observed some increase in embedding and ultimate strength, when compared with non-reinforced joints. I thank CIBISA as well as the Université Laval It is also concluded that spacing and end distances for financial support of this project. The constructive can be reduced. comments of Daniela Blessent (Université Laval) and Ultimate loads from tests on wood plates an anonymous reviewer are greatly appreciated and loaded with a single bolt show a very good agree- have helped to improve the manuscript.

Escuela de Ingeniería de Antioquia 45 Bolted timber joints with self-tapping screws

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