Effect of beam-deck connection flexibility on lateral torsional buckling strength of wooden twin-beams

Engineering Structures 207 (2020) 110226

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Effect of beam-deck connection flexibility on lateral torsional buckling strength of wooden twin-beams

T

Yang Du, Ghasan Doudak , Magdi Mohareb ⁎

Department of Civil Engineering, University of Ottawa, Ottawa, ON K1W6N5, Canada

ARTICLE INFO

ABSTRACT

Keywords: Lateral torsional buckling Timber beam-deck system Connection flexibility Lateral restraint Twist restraint Finite element formulation Experimental testing

Two finite element solutions are developed for the lateral torsional buckling analysis of timber beam-deck assemblies consisting of two beams braced by decking through fasteners. In contrast to past solutions, both solutions capture the rotational flexibility provided by the connections between the deck boards and the beams. The first solution is intended for systems with partial lateral restraint provided by the deck boards allowing lateral sway while the second solution is intended for systems that are restrained from lateral movement at the deck level. An experimental program is conducted to quantify the rotational stiffness of beam-deck connections for different types of fasteners and the results are input into the finite element formulations to evaluate the corresponding buckling capacities for beam-deck systems. The results indicate that the buckling capacity of beam-deck systems can be significantly increased with commonly used fasteners while high-capacity fasteners can achieve buckling capacities nearly identical to those where rigid rotational connections are assumed.

1. Introduction and literature review Modern timber construction involves the use of deep beams for bridging long spans as they provide efficient means of material usage in bending. When such beams are laterally unsupported, they become prone to lateral torsional buckling as a possible mode of failure. Current design standards (e.g., CAN/CSA O86 [1], NDS [2], Eurocode [3]) recognize this phenomenon and provide design guidance for (1) laterally unsupported beams, (2) beams with intermediate lateral supports, and (3) beams with continuous rigid lateral restraint at the compression edge. However, no provisions have been developed to assess the lateral torsional buckling capacity of floor and roof assemblies where the beams are continuously restrained against lateral movement and twist along their longitudinal axis by decking through fasteners. Depending on the decking and connection details, deck boards can be assumed to provide either full or partial lateral restraint and partial twist restraint to the beams. A recent study by Du et al. [4] proposed a non-sway model for the lateral torsional buckling of a beam-deck system comprised of two parallel beams braced at the top face by deck boards. Deck boards were considered to provide continuous rigid lateral restraint preventing the beams from lateral sway and partial twist restraint controlling the twist of the beams. In a subsequent study, Du et al. [5] presented a sway model for cases where the decking details are able to provide

⁎

continuous partial lateral and twist restraints. In both models, the beam-deck joints were assumed to provide rigid rotational connections such that the deck board twists by the same angle as that of the beam. In reality, such rigid connection may be difficult to achieve in timber. Within this context, the present study is aimed at incorporating the effects of rotational flexibility of connections when characterizing the lateral torsional buckling capacity of beam-deck systems, both in sway and non-sway scenarios. The stability of beams braced by discrete or continuous restraints has been researched extensively in the past half century. A detailed review of those studies has been provided in Du et al. [4] and Du et al. [5]. Pertinent to the subject of the present study, only research relating to the lateral torsional buckling of continuously restrained beams is reviewed and is presented in a chronological sequence. Vlasov [6] formulated the general differential equations for the buckling of a beam embedded in an elastic medium. The buckling load was determined for a beam continuously braced by elastic lateral and twist restraints and subjected to uniform moment. For the special case where the beam is continuously braced by rigid lateral restraint offset from the beam shear center, the lateral displacement was related to the angle of twist and the resulting equations were solved for the critical load under uniform moment. Pincus and Fisher [7] developed an energy-based solution for a structural system consisting of two beamcolumns continuously restrained by decking. The total potential energy

Corresponding author. E-mail addresses: [email protected] (Y. Du), [email protected] (G. Doudak), [email protected] (M. Mohareb).

https://doi.org/10.1016/j.engstruct.2020.110226 Received 25 November 2018; Received in revised form 25 November 2019; Accepted 12 January 2020 0141-0296/ © 2020 Elsevier Ltd. All rights reserved.

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Notation

distance between point of crushing and that of separation distance between point of loading and point of edge crushing Lb , Ld beam and deck span, respectively L (z ) T Hermitian polynomials l element length reference strong-axis moment applied to the beam M M1 moment applied to the beam-deck joint MA, MB , MC , MD nodal moment along longitudinal direction Mei deck board end moments O, O’ points of rigid lateral restraints P1,P2 ,P3 concentrated loads in the joint rotational models PA, PB, PC , PD nodal vertical force q (z ) reference beam transverse load R rotational stiffness of beam-deck joints R1 joint rotational stiffness of present model R2 joint rotational stiffness of Loferski and Gamalath model Ubi ,Ubi internal strain energy under sway model and non-sway model, respectively Ud ,Ud internal strain energy for deck bending under sway model and non-sway model, respectively Ul internal strain energy for the relative partial lateral restraint ui lateral displacement ui generalized nodal lateral displacement vector Vbi ,Vbi load potential energy under sway model and non-sway model, respectively v vertical displacement v nodal displacement vector vA, vB , vC , vD nodal vertical displacement in the joint rotational model z0 distance from a given deck board to the beam end A , B , C , D nodal rotation in the joint rotational model vertical deformation at the deck end angle of twist i generalized nodal angel of twist vector i load multiplication factor deck end rotation i total potential energy

L2 L3

The following symbols are used in this paper: distance between beam shear center and point of lateral restraint A, B, C , D nodes defined in the joint rotational model [B1], [B2], [B3], [B4], [B5] submatrices of the elastic and geometric stiffness matrices b deck board width bb beam width C sectoral inertia of the beam section compared to the imposed axis of rotation Cw beam warping constant Eb, Ed modulus of elasticity of beam and deck, respectively F nodal force vector Gb shear modulus of the beam h (z ) distance between loading point and beam shear center hb , hd depth of beam and deck, respectively Id moment of inertia of deck board Iy moment of inertia about beam weak-axis Jb beam Saint-Venant torsional constant [K1] stiffness matrix for the joint stiffness model [Kb],[Kb ] beam stiffness matrix under sway model and non-sway model, respectively [K d],[K d ] twist stiffness matrix under sway model and non-sway model, respectively [K e],[K e ] elastic stiffness matrix under sway and non-sway models, respectively [K g ],[K g ] geometric stiffness matrix under sway and non-sway models, respectively [Kl] stiffness matrix for the relative lateral restraint under the sway model k relative lateral stiffness provided by deck boards and fasteners k¯ relative lateral stiffness per unit length kc crushing stiffness of beam-deck joints ks separation stiffness of beam-deck joints L1 clear span of deck boards

a

was expressed and the Euler-Lagrange conditions were applied to formulate the equilibrium equations which were solved by assuming trigonometric functions for the buckling displacements, leading to expressions for the critical load. Zahn [8] analytically investigated the lateral torsional buckling of wooden floor systems by formulating the total potential energy of a single beam and its tributary strip of decking. Unlike Pincus and Fisher [7] whose model incorporated both shear and bending resistance of decking, Zahn [8] captured the deck shear stiffness while omitting its bending contribution. A closed-form solution was obtained for simply-supported beams under uniform moment. Power series solutions were presented for other loading and boundary conditions. Hancock and Trahair [9] formulated a finite element solution for beam-columns continuously restrained by elastic springs that partially restrain lateral displacement, twist along beam longitudinal axis, weak-axis rotation and warping. For the same problem, Trahair [10] presented a closed-form solution for beam-columns under uniform load. Li [11] formulated an energy solution for continuously restrained steel zed-purlins where the restraining effects of steel diaphragm were modelled by translational and rotational springs. The buckling loads were recovered by assuming the displacement fields as spline interpolation functions. The solution accounted for pre-buckling deformation effects and was able to solve general boundary and loading conditions. Li [11] further investigated the case of infinite translational stiffness (i.e., non-sway) and zero rotational stiffness. Larue et al. [12]

developed lateral torsional buckling solutions for beams continuously restrained by rigid lateral restraint. For a simply-supported beam under uniform moment, their solution was shown to coincide with that of Vlasov [6]. For beams under linear or parabolic moment distributions, the study proposed an eigen-value solution where the displacement fields were assumed as trigonometric functions. Khelil and Larue [13] developed an energy-based solution for simply-supported beams with continuous partial lateral restraint. The proposed solutions were able to predict the buckling capacity for uniform and non-uniform moment distributions. Zhang and Tong [14] formulated the total potential energy of C- and Z-shaped purlins considering the pre-buckling effects. Closed-form solutions were developed for purlins with continuous top flange lateral bracing and subjected to uniform wind uplift load. Du et al. [4] formulated a series of solutions for the lateral torsional buckling of twin-beam-deck assemblies where the decking was assumed to provide rigid lateral restraint and partial twist restraint. Closed-form solutions were developed for simply-supported beams with uniform and non-uniform moment distributions. A finite element formulation was also developed to tackle cases with arbitrary boundary and loading conditions. Lei and Li [15] developed a closed-form solution for the lateral torsional buckling of beams with continuous lateral and torsional bracing on the tension flange. The solution assumed sinusoidal functions for the displacement fields and incorporated the effects of web distortion. Du et al. [5] studied the lateral torsional buckling of 2

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wooden twin-beam-deck systems where the decking was characterized as continuous partial lateral and twist restraints. A spring mechanism was introduced to quantify the lateral stiffness provided by the combined resistance of deck boards and the beam-deck joints. Closed-form solutions were formulated for simply-supported beams under uniform and non-uniform moment distributions. A finite element formulation based on the Hermitian interpolation functions was also presented. Egilmez et al. [16] modelled a twin-beam-diaphragm system with commercial software ANSYS and studied the strength requirements for the shear diaphragms used as lateral bracing for steel beams. The bracing force at the connection between the beam and the shear diaphragm was also investigated. Among the studies reviewed, the lateral restraining effect of decking, either rigid or partial, was widely acknowledged. However, whether the deck boards are able to provide effective torsional bracing remains unclear, with Zahn [8], Li [11], Larue et al. [12], Khelil and Larue [13], Zhang and Tong [14] and Egilmez et al. [16] omitting the torsional bracing of the deck, and the remaining studies incorporating such effects. The main reason for ignoring deck torsional bracing was reportedly the uncertainty of the rigidity of beam-deck rotational connection (Zahn [8]). However, depending on the deck and connection configuration, ignoring deck torsional bracing may lead to gross underestimation of the buckling capacity. To the authors’ knowledge, no theoretical nor experimental studies in the literature provide buckling solutions for beams considering the effects of rotational flexibility of beam-deck connections. Within this context, the present study is aimed at providing lateral torsional buckling solutions for beam-deck systems that incorporate the effects of connection flexibility by: (1) quantifying the rotational stiffness of beam-deck joints through theoretical modelling and experimental tests, and (2) developing finite element formulations that are able to incorporate the joint rotational stiffness determined in (1) when quantifying the lateral torsional buckling capacity of beam-deck systems.

model developed by Loferski and Gamalath [17]. This is followed by an experimental program conducted to quantify the rotational stiffness of beam-deck connections joined with different types of fasteners. 2.1. Theoretical model Fig. 1a shows the cross-sectional view of a beam-deck system. For the joint connecting the left beam to the deck board, an idealized model is provided in Fig. 1b where the joint flexibility is modelled by Element 1 whose nodal points are each connected to an elastic spring. The spring attached to node A represents the separation stiffness ks that combines the embedment action of the fastener head and the withdrawal action of fastener shank as the connection deforms (Loferski and Gamalath [17]). The stiffness kc of the spring linked to node B represents the crushing action between the beam edge and deck board as the beam ¯ . As twists relative to the deck board. Element 1 has a length L2 = AB the system undergoes lateral torsional buckling, the right beam also rotates relative to the deck board and thus exerts an upward load P1 at node C induced by the crushing action, and a downward load P2 at node D induced by the separation action. The beam segment is represented ¯ = CD ¯ . Element 2 connects Elements by Element 3 with length L2 = AB 1 and 3 and has a length L1. Under the idealized beam-deck model in Fig. 1b, a structural matrix analysis is conducted to relate the nodal force vector {F }8 × 1 = PA MA PB MB PC MC PD MD T to the nodal displacement vector {v}8 × 1 = vA A vB B vC C vD D T , where P , M , v , respectively denote the vertical forces, moments, vertical displacements, and rotations and all subscripts denote the relevant nodes. Under external loads P1 and P2 (Fig. 1b), the nodal force vector P2 0 T can be related to the nodal dis{F }8 × 1 = 0 0 0 0 P1 0 placement vector {v}8 × 1 through {F }8 × 1 = [K1 ]8 × 8 {v}8× 1, where the stiffness matrix [K1] is defined as

2. Determination of rotational stiffness of beam-deck joints

One recovers the following expressions for the vertical displacements vA and vB as

This section aims at assessing the rotational stiffness of beam-deck joints within floor or roof systems. A theoretical model is developed to investigate parameters that influence the joint rotational stiffness through a matrix analysis for an idealized beam-deck joint model. The resulting rotational stiffness expression is then compared against a 3

vA =

P2 (L1 + L2) ks L2

P 1 L1

vB =

P1 (L1 + L2) P2 (L1 + 2L2 ) k c L2

(1)

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element formulations to study its effect on the lateral torsional buckling capacity. Five types of connectors, including 10d and 60d common nails, wood screws and two types of self-tapping screws, were chosen for the experiments. The geometric properties of the connectors are summarized in Table 1. The Spruce-Pine-Fir (SPF) 20f-EX grade glulam was chosen as the beam material with a dimension of 130 mm in width, 300 mm in depth and 140 mm in length. SPF Select grade deck boards with a cross-section of 38 mm by 140 mm and a span of 280 mm were used. The specific gravity for the beam material is 0.465 and that for the deck board is 0.444. The moisture contents for the beam and the deck are approximately 10%. The test setup is shown in Fig. 3 where a beam specimen was securely fastened to the base of the testing machine and joined by a deck board through connectors. A concentrated load was applied at the tip of the deck board. Fig. 4 shows the plan views of the joint configurations for a single-connector joint (Fig. 4a) and a two-connector joint (Fig. 4b). For 10d and 60d common nails and wood screws, two connectors were installed at the joint, each placed 65 mm away from the beam edge and 70 mm apart from the other connector (Fig. 4b). For self-tapping screws, a single connector was placed at the center of the connection (Fig. 4a). All connectors were carefully driven until their heads flushed with the deck board surface. During the test, the magnitude of the concentrated load P applied at the edge of the deck was recorded through a load cell. The corresponding vertical displacement at the deck’s tip was also recorded using a linear variable differential transducer (LVDT) attached to the bottom of the deck end. A relatively short deck span of 280 mm was chosen to ensure that the flexural and shear deformation of the deck board under the applied concentrated load is negligible and that only joint rotation contributes to the vertical displacement recorded at the deck end. The loading rate was controlled at 2.54 mm/min, as suggested by ASTM D1761 [18]. Two repetitions were conducted for each type of connector. The ten load-displacement curves obtained from the joint rotational tests of the aforementioned five types of fasteners are provided in Appendix B. The rotational stiffness of a beam-deck connection R is

Fig. 1. (a) Cross-sectional view of beam-deck system, (b) idealized joint model of beam-deck system.

The moment induced by P1 and P2 is M1 = P1 L1 P2 (L1 + L2) . For a beam-deck joint with small angle of twist, which is considered to be valid for elastic buckling analysis, the joint rotation is approximated tan = (vA vB )/ L 2 . Accordingly, the rotational stiffness R1 is as given by

R1 =

M1

=

L 22 [P1 L1 P2 (L1 + L 2 )] P2 (L1 + L2) P1 L1 P (L + 2L ) P (L + L ) + 2 1 2k 1 1 2 ks c

(2)

Loferski and Gamalath [17] proposed a simplified idealization to characterize the rotational stiffness R of the joint, in which a deck board is assumed to be connected to the beam at one end and subjected to a concentrated load P at the other end (Fig. 2a). Using a single beam element with elastic springs representing separation and crushing actions at the joint (Fig. 2b), the authors formulated an expression for the rotational stiffness as

R2 = L 22 L3 /

L3 L + L3 + 2 ks kc

R=

M

(4)

where M is the moment applied to the connection, and is the rotational angle at the joint. For a concentrated load P applied at the end of the deck board (Fig. 2b), the corresponding moment is M = PL3 , where L3 is the distance between the applied load and the connection

(3)

where L3 is the lever arm of the applied load P . While the rotational stiffness based on Eq. (2) is dependent on six parameters, i.e., L1, L2 , ks, kc , P1, P2 , that based on the Loferski and Gamalath [17] model depends upon four parameters L2 , L3 , ks, kc . A sensitivity analysis on the parameters identified in each model is provided in Appendix A. The results suggest that the joint rotational stiffness in both models depends primarily on the joint details (i.e., distance L2 between the point of separation and the point of crushing, separation stiffness ks and crushing stiffness kc ) and is nearly independent on distances L1, L3 or the load magnitudes P1, P2 . For the reference beam-deck joint defined in Appendix A, the rotational stiffness sought from the present model, i.e., Eq. (2), is found to be merely 1% higher than that from the Loferski and Gamalath model [17]. This observation provides the foundation of the design of the experimental program described in the following section where a simplified testing setup (Fig. 2a) is adopted instead of the more elaborate one (Fig. 1a) without compromising the accuracy of the obtained rotational stiffness. 2.2. Experimental program An experimental investigation is undertaken to study the rotational stiffness of beam-deck joints. The objective of the experimental program is to provide a basic estimation of the rotational stiffness of beamdeck joints with different types of connectors ranging from common nails and wood screws to the state-of-art connectors. The experimentally estimated joint rotational stiffness will later be fed into finite

Fig. 2. (a) Cross-sectional view of beam-deck joint under concentrated load, (b) idealized model of beam-deck joint under concentrated load. 4

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Table 1 Fastener geometry. Type of fastener

Shank length (mm)

Thread length (mm)

Shank diameter (mm)

Head diameter (mm)

10d common nail 60d common nail Wood Screw Self-tapping screw-type 1 Self-tapping screw-type 2

76.2 152 76.2 120 270

No threads No threads 51.0 100 244

3.76 6.05 4.83 8.00 8.50

7.94 13.5 9.78 18.0 12.0

(Fig. 2b). For small rotational angles, which is consistent with elastic /L3, where is the displacement at buckling, is approximated as the deck end induced by concentrated load P. Substituting M and into Eq. (4) yields

R=

PL32

(5)

Based on the load-displacement curves obtained from the tests, the rotational stiffness of the joints was computed by determining the slope of the tangent to the test curve. Since the present study is focused on the linear elastic buckling, the joint rotational stiffness R was quantified in the initial displacement region (displacement of 0.2 mm). For the connectors investigated, Table 2 provides the test results of joint initial rotational stiffness calculated based on Eq. (5). In Table 2, values of the joint rotational stiffness were presented based on a two-connector joint configuration. For the self-tapping screws tested under a single-connector configuration, the stiffness is hypothesized to be doubled of the test results. It is observed that joints with 10d nails have an average stiffness of 3665 Nm. This compares to average stiffness values of 9986 Nm and 7169 Nm for 60d nails and wood screws, respectively. Type 1 self-tapping screws have an average rotational stiffness of 11,760 Nm while the type 2 self-tapping screws average 35,552 Nm.

Fig. 4. Plan view of joints with (a) a single-connector connection, (b) a twoconnector connection. Table 2 Rotational stiffness based on two connectors at each beam-deck joint. Type of fastener

Test No.

Rotational stiffness (Nm)

Average rotational stiffness (Nm)

10d common nail

1 2 1 2 1 2 1 2 1 2

3946 3383 10,858 9114 7746 6592 12,536 10,984 31,069 40,035

3665

60d common nail

3. Overview of past lateral torsional buckling models for rigid rotational joints

wood screw Self-tapping screwtype 1 Self-tapping screwtype 2

The models sought in the present study aim at quantifying the lateral torsional buckling capacity of beam-deck systems where the rotational flexibility of beam-deck joints is incorporated. They can be regarded as modifications of those developed in Du et al. [4] for the nonsway model and Du et al. [5] for the sway model, where the beam-deck

Fig. 3. (a) Joint rotational stiffness test setup, (b) schematic of test setup. 5

9986 7169 11,760 35,552

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rotational connection was postulated to be fully rigid. A review of key features of both studies are thus provided in the following subsections.

sections with a depth-to-width ratio between 4 and 7, solutions which account for pre-buckling effects ([21;22]) show that the omission of pre-buckling effects would under-predict the critical moments by less than 3%.

3.1. Assumptions and limitations The following assumptions were adopted for beam-deck systems with rigid twist connections and are equally applicable to the models sought in the present study with partial twist connections:

3.2. Sway model For the sway model shown in Fig. 5, the beam-deck system attains the onset of buckling as the external load applied to both beams reaches its critical value of q (z ) , where is the unknown load multiplier under which the beam-deck system buckles, and q (z ) is the reference transverse load applied at a distance h (z ) from the beam shear center. Under the load q (z ) , the system undergoes lateral torsional buckling manifested by lateral displacements u1, u2 and angles of twist 1, 2 , where subscripts 1 and 2 represent the left and right beam, respectively. In the sway model, the beams are assumed to be continuously restrained by the partial lateral and twist restraints of the deck boards. The partial lateral restraint is provided by the combined resistance of deck axial stiffness and fastener shear stiffness. The partial twist restraint reflects the decking transverse bending action with the beam-deck joints providing rigid rotational connection, i.e., the beam’s angle of twist i is the same as the rotation of the adjoining deck board i . The total potential energy 1 is assembled by adding the energy contributions of each component, yielding

1. The lateral-torsional buckling of wood members is only characterized by the modulus of elasticity in longitudinal direction EL and shear modulus in the transverse direction GT . As was discussed in Xiao et al. [19], the orthotropic behavior of wood is characterized by six constants EL,ET = ER,GL = GLR = GLT , Buckling mode shape for the non-sway and sway solutions and µRT , where subscripts L, R and T refer to longitudinal, radial and tangential direction, respectively. Further, in the beam formulation sought, since the normal stresses in the radial and tangential directions are negligible compared to longitudinal stresses, the associated Young’s moduli ET = ER will not contribute to the internal strain energy expression. Also, given that the stress state in the beam is uniaxial, Poisson’s ratios µLR , µLT , µRT will not influence the internal strain energy. Finally, as the shear stresses acting on the radial and tangential directions are negligible, the associated shear modulus GRT does not appear in the strain energy expression. In summary, of the six orthotropic constitutive constants, only the modulus of elasticity in the longitudinal direction EL and shear modulus GT will appear in the internal strain energy expression, in a manner similar to isotropic materials. This observation is also confirmed by Xiao et al. [19]. 2. Throughout deformation, beam cross-sections remain rigid within their own plane, i.e., distortional effects are neglected. 3. Beams are perfectly straight and no initial imperfections are considered. 4. Beam and decking materials are linearly elastic. 5. Local warping effects are accounted in a manner consistent with the kinematics of the Gjelsvik theory [20]. 6. Shear deformation effects within the beams and deck boards are neglected. By comparing critical moments predicted by shear deformable to non-shear deformable solutions, it was shown that shear deformation has a negligible effect on the predicted lateral torsional buckling capacity of rectangular of wooden beams when the spanto-width ratio exceeds 20. 7. Pre-buckling deformation effects are omitted. For typical beam

1

= Ub1 + Vb1 + Ub2 + Vb2 + Ul + Ud Lb

1

= 2(

(

+ +

+

Lb 0

1 Lb 2( 0

(

+ k¯ 2

Lb 0 Lb 0

Lb

Eb Iy u1 2 dz +

0

0

2

M 2 u2 dz + u2

u1 +

0

Lb

2

Eb Iy u2 dz +

Lb

1 2

M 1 u1 dz + 0 1 2

hb ( 2 2

Gb Jb

0

2

qh 1)

dz+

2 1 dz

qh

Gb Jb Lb

1

2

2

dz+

2 2 dz 2

)

)

dz +

Lb

Eb Cw

1

Lb

Eb Cw

2

0

0

Ed hd3 6Ld

Lb 0

(

2

dz )

2

dz )

2 1

+

1 2

+

2 2 ) dz

(6) Lb

Lb

Lb

Ubi = (1/2) ( 0 Eb Iy ui dz+ 0 Gb Jb i dz+ 0 Eb Cw i dz ) where (i = 1, 2) is the internal strain energy stored in beam i , L L Vbi = [ 0 b M i ui dz + (1/2) 0 b qh i2 dz ] (i = 1, 2) is the load potential i, for external loads applied to beam L 2 Ul = (k¯ /2) 0 b [u2 u1 + hb ( 2 1)/2] dz is the energy stored in the lateral restraint provided by the combined effect of deck and beam-deck L connections, and Ud = (Ed hd3/6Ld ) 0 b ( 12 + 1 2 + 22 ) dz is the energy stored in deck boards undergoing transverse bending when rigid

Fig. 5. Buckling deformation under sway model. 6

2

2

2

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Fig. 6. Buckling deformation under non-sway model.

rotational connection is assumed. For the beams, Eb is the modulus of elasticity, Gb is the shear modulus, Iy is the moment of inertia about beam weak-axis, Jb is the Saint-Venant torsional constant and is computed by the expression Jb (b3b hb/3)[1 0.63b b /hb + 0.052(b b /hb )5] for rectangular sections (AFPA-TR14 [20]), Cw is the warping constant and is expressed as Cw = bb3 hb3/144 for rectangular sections, bb is the beam width, hb is the beam depth and Lb is the beam span. For the deck boards, Ed is the modulus of elasticity, hd is the deck thickness, k¯ = k / b is a parameter representing the stiffness of the lateral restraint, k is the combined stiffness of a single deck board and the beam-deck joint at each end of the deck, and b is the width of a deck board, and Ld is the deck span. M is the beam strong-axis moment induced by the reference load q. All primes denote differentiation with respect to coordinate z along the beam longitudinal axis.

expressed as the angle of twist i through ui = a i , where a is the distance between the beam shear center and the point of lateral restraint. From Eq. (6), by setting ui = a i , one obtains the total potential energy expression 2 for the non-sway model as 2

= Ub1 + V b1 + Ub2 + Vb2 + Ud Lb

1

= 2( + +

0

Eb C

Lb 1 2( 0

Eb C

Ed hd3

Lb

6Ld

0

(

1 2 2 1

2 2

dz + dz +

+

1 2

Lb 0 Lb

Gb Jb

0

Gb Jb

+

2 2 ) dz

1 2

2 2

dz ) + dz ) +

2 2

Lb

(hq

1

2aM

1

) 1 dz

Lb

(hq

2

2aM

2

)

0 0

2 dz

(7) L

L

where Ubi = (1/2) ( 0 b Eb C i 2 dz + 0 b Gb Jb i 2 dz ) (i = 1, 2) is the internal strain energy stored in beam i , C = Iy a2 + Cw represents the sectoral inertia of the section compared to the imposed axis of rotation, L and Vbi = ( /2) 0 b (hq i 2aM i ) i dz is the load potential for ex-

3.3. Non-sway model

L

ternal loads applied to beam i , Ud = (Ed hd3/6Ld ) 0 b ( 12 + 1 2 + 22 ) dz is the energy stored in the deck board and is identical to that provided in the sway model. Other properties have been previously defined under Eq. (6).

For beam-deck assemblies where the beams are restrained from swaying laterally, i.e., the lateral displacements at points O and O’ in Fig. 6 vanish, a non-sway model can be established as a special case of the sway model. In this case, the beam lateral displacement ui can be

Fig. 7. (a) Wooden beam-deck joint under deformation, (b) internal force within the joint. 7

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4. Incorporating the effect of rotational flexibility of connections

4.2. Total potential energy

This section aims at modifying past developments summarized in Section 3 to incorporate the effects of rotational flexibility of connections on the deck twist restraint to the beams.

For the sway model, by replacing the energy term Ud in Eq. (6) with Ud in Eq. (13), one obtains the total potential energy 3 for the sway model in which the rotational flexibility of the beam-deck joints is incorporated

4.1. Internal strain energy

1 (z 0 )

1 (z 0 )

2 (z 0 )

2 (z 0 )

+

Ld 3Ed Id

=

2 (z 0 )

Ld 6Ed Id

Ld 6Ed Id Ld 3Ed Id

+

1 R

1 Me1 (z 0 ) 2

1 (z 0 )

+

1 Me2 (z 0 ) 2

+

1 (z 0 )

2 (z 0 )

i (z )

(12)

Ld2 .

3

Ud (z 0) Lb

1 (z )

0

1 (z )

=

2Ed Id b

(L

d

+

3Ed Id R

2

)

2

dz )

dz

1 2

(

+ Ld +

3Ed Id R

)

2 2 dz

(14)

Eb C

Lb

Eb C

0

Lb

2Ed Id b

0

2

1 2

(L

d

2

Lb

dz +

Lb

dz +

+

Gb Jb

0

Gb Jb

0

3Ed Id R

)

2 1

+ Ld

1 2

2 2

dz ) + dz ) +

1 2

(

2 2

Lb

(hq

Lb

(hq

0 0

+ Ld +

3Ed Id R

1

2aM

2

2aM

)

1

)

1 dz

2

)

2 dz

2 2 dz

= L (z )

T 1 × 4 {ui }4 × 1 T 1 × 4 { i }4 × 1

(16)

T 1×4

=

3z 2/ l 2 (z

(1 +

2z 2/ l + z 3/l 2 ) (3z 2/ l 2

2z 3/ l3) (z 3/l 2

z 2/ l)

2z 3/l3)

1 = 2

u1

T 1×4

u2

T 1×4

T 1 1×4

T 2 1×4

[[K e]

[K g ]]16× 16

{u1 }4 × 1 {u2 }4 × 1 { 1 }4 × 1 { 2 }4 × 1

where the elastic stiffness matrix [K e] consists of three contributions, [K e] = [Kb] + [K d] + [Kl], [Kb] is the beam stiffness matrix, [K d] is the stiffness matrix for deck transverse bending considering joint rotational flexibility, [Kl] is the stiffness matrix for the lateral restraint, and [K g ] is the geometric stiffness matrix. Matrices [Kb], [Kl] and [K g ] are provided in Appendix C and the deck stiffness matrix [K d] is given by

Ed Id Ld 2Ed Id (Ld + 3Ed Id/ R)

dz Lb

0

+ Ld

Eb Cw

dz )

(17)

2 (z )

2Ed Id (Ld + 3Ed Id /R) Ed Id Ld 2 (z )

)

1) 2 1

Lb 0

2

is the vector of Hermitian interpolation functions, l is the element length, i T1 × 4 = 0 0 l l i is the generalized nodal angle of twist vector, ui T1 × 4 = u0 u0 ul ul i is the generalized nodal lateral displacement vector, and subscripts 0 and l denote the element nodal points. From Eq. (16), by substituting into the total potential energy in Eq. (14), one obtains

+ 8Ed Id Ld /R + The internal strain energy Ud where = for the whole decking is simply the summation of the internal strain energy stored in each deck board, which can be approximately expressed in an integration form as

1 b

+

3Ed Id R

2

1

where

1 (z 0 )

Ud =

Lb

ui (z ) = L (z )

2Ed Id (Ld + 3Ed Id /R) Ed Id Ld Ed Id Ld 2Ed Id (Ld + 3Ed Id/ R)

12Ed2 Id2/ R2

d

qh

)

dz+

2 2 dz

Eb Cw

4.3.1. Sway model For the sway model, the displacement fields associated with lateral torsional buckling are the lateral displacement ui and the angle of twist (i = 1, 2) . Applying the Hermitian polynomials to relate the disi placement fields to the nodal displacements yields

(i = 1, 2)

2 (z 0 )

(L

0

hb ( 2 2

2

2

Lb 0

4.3. Finite element formulations

L (z )

1

u1 +

Lb

dz+

2 1 dz

qh

Gb Jb

0

2

(15)

Ud (z 0) =

0

0

1

(11)

By inverting Eq. and substituting the resulting Mei (z 0 ) into Eq. (10), one obtains

Lb

0

1

= Ub1 + Vb1 + Ub2 + V b2 + Ud

+ 2(

(10)

2 (z 0 )

u2

0

1

The energy stored in the deck board accounting for connection flexibility is

Ud (z 0 ) =

+

2Ed Id b

Lb

Lb 1 2

M 2 u2 dz +

0

+

M 1 u1 dz +

Eb Iy u2 dz +

Lb

= 2(

(9)

Me1 (z 0 ) Me2 (z 0)

(

1 2

2

Lb

k¯ 2

4

1 (z 0 )

1 R

+

0

Gb Jb

0

Similarly, substituting the deck bending term Ud of Eq. (7) with Ud yields the total potential energy 4 of the non-sway model where the rotational flexibility of beam-deck connections is incorporated

where Id is the deck moment of inertia. By inverting Eqs. (8) and (9), and adding the resulting expressions, one obtains 1 (z 0 )

Lb

Lb 1 2( 0

+

(8)

2 (z 0 )

(

Lb

Eb Iy u1 2 dz +

0

+

where Me1 (z 0) and Me2 (z 0) are the moments at the left and right ends of deck board, respectively. Also, the deck end rotations 1 (z 0) , 2 (z 0 ) are related to the end moments Me1 (z 0) and Me2 (z 0) through

Me1 (z 0 ) 2Ed Id 2 1 = 1 2 Me2 (z 0 ) Ld

Lb

1

= 2(

As a beam-deck system buckles, beam i (i = 1, 2) is assumed to undergo a twist angle i (z ) (Fig. 7a). As a result of the connection flexibility, the adjoining deck board rotates by a milder angle i (z ) . The internal forces within the beam-deck joint is shown in Fig. 7b where the joint can be modelled as an elastic rotational spring transferring the moment Mei (i = 1, 2) from the beam to the adjoining deck board. Given the stiffness R of the rotational spring, one can obtain the moment-rotation relationship at each beam-deck joint at a distance z 0 from the beam end as

Me1 (z 0 ) = R 0 0 R Me2 (z 0 )

= Ub1 + Vb1 + Ub2 + Vb2 + Ul + Ud

3

)

2 1

+ Ld

1 2

(

+ Ld +

3Ed Id R

)

2 2 dz

(13) For beam-deck joints with rigid rotational connection (i.e., R = Eq. (13) reverts to the deck energy term Ud in Eqs. (6) and (7).

[K d] =

),

where 8

[0]8 × 8 [0]8 × 8 [0]8 × 8 [K d ]8 × 8

16 × 16

(18)

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[K d ] =

5.1.1. Sway model The sway model adopts one dimensional B31OS elements (along Line 1 and 4 in Fig. 8) within the Abaqus library to model the two beams. The B31 elements between Lines 3 and 6 represent the deck boards. The B31 element has two nodes with six degrees of freedom per node (i.e., three translations, and three rotations) whereas the B31OS element has an additional degree of freedom for warping deformation. A rigid link is defined between each beam node (Line 1 or 4) and each beam top node (Line 2 or 5) through the BEAM type multi-point constraint (*MPC) feature in Abaqus. This enforces the assumption that beam cross-section remains rigid throughout deformation. To incorporate the lateral restraint, two-node SPRING2 spring elements are employed between the beam top nodes in Line 2 and 5. Then, another set of SPRING2 elements are introduced between the beam top node (Line 2 or 5) and the deck node (Line 3 or 6) to represent the rotational flexibility of beam-deck joints.

Ld [B3 ] 2Ed Id 2(Ld + 3Ed Id/ R)[B3 ] Ld [B3 ]T 2(Ld + 3Ed Id/ R)[B3] b

and submatrix [B3] has been defined in Appendix C. From Eq. (17), by evoking the stationarity of the total potential energy, i.e., 3 = 0 , one obtains

{u1 }4 × 1 {u2 }4 × 1 [K g ]16× 16 ) = {0}16× 1 { 1 }4 × 1 { 2 }4 × 1

([K e ]16 × 16

(19)

From Eq. (19), one recovers the load multiplier sponding buckling modes.

and the corre-

4.3.2. Non-sway model For the non-sway model, the displacement fields for lateral torsional buckling include only the twist angles 1 and 2 . By applying the Hermitian polynomials interpolation functions, one relates the twist angles to the generalized nodal displacements though i (z )

= L (z )

5.1.2. Non-sway model Fig. 9 shows the beam-deck assembly under the non-sway model. Similar to the sway model, the B31OS and B31 elements are respectively chosen to model the two beams and the deck boards. The SPRING2 elements are also introduced to model the rotational flexibility of the beam-deck connections. In addition, the beam top nodes along Line 2 and 5 are rigidly restrained in lateral direction to ensure the non-sway condition.

(20)

T 1 × 4 { i }4 × 1

where the vector of Hermitian interpolation functions L (z ) T1 × 4 and the generalized nodal angle of twist vector i T1 × 4 have been defined under Eq. (16). From Eq. (20), by substituting into Eq. (15), one rewrites the total potential energy 4 as 4

=

1 2

T 1 1×4

T 2 1×4

([K e ]8 × 8

[K g ]8× 8 )

{ 1 }4 × 1 { 2 }4 × 1

5.2. Model verification

(21)

in which the elastic stiffness matrix [K e ] = [Kb ] + [K d ], [Kb ] is the beam stiffness matrix, [K d ] is the deck stiffness matrix considering the rotational flexibility of beam-deck joints, [K g ] is the geometric stiffness matrix. Matrices [Kb ] and [K g ] are also presented in Appendix C while the deck stiffness matrix [K d ] has been defined after Eq. (18). By evoking the stationarity conditions of the total potential energy 4 = 0 to the discretized system, one obtains

([K e ]8× 8

[K g ]8 × 8 )

{ 1 }4 × 1 = {0}8 × 1 { 2 }4 × 1

from which one can extract the unknown load multiplier corresponding buckling modes.

The results based on the finite element formulations developed in the present study, i.e., Eqs. (19) and (22), are compared against the Abaqus results and against solutions found in the literature. The verification is performed for a reference beam-deck system where both beams are simply-supported and are each under the same magnitude of uniform moment. The beam and deck spans are 9 m and 3 m, respectively. Beams cross-sections are 130 mm in width and 874 mm in depth. The deck boards have a 38 mm by 140 mm cross-section. The beam and deck materials are chosen to be consistent with those from the experimental tests. The SPF 20f-EX grade glulam beam has a modulus of elasticity Eb = 9500 MPa along its longitudinal direction and a transverse shear modulus Gb = 437 MPa . The SPF Select grade deck board has a modulus of elasticity Ed = 10500 MPa along its longitudinal direction. The aforementioned mechanical properties are obtained from CAN/CSA O86 [1] and FPL [23]. The dimensions and material properties chosen for the beams and deck boards ensure that the beam-deck assembly buckles in the elastic range (based on CAN/CSA O86 [1] and Hooley and Madsen [24]). For the sway model, the lateral stiffness provided by the decking is assumed to be identical to that of deck boards with two 10d common nails at each beam-deck joint. This is

(22) and the

5. Verification 5.1. Abaqus models To assess the validity of the present sway model, i.e., (19), and nonsway model, i.e., (22), two finite element models under commercial software Abaqus are developed.

Fig. 8. Abaqus sway model. 9

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Fig. 9. Abaqus non-sway model.

because the lateral stiffness based on two 10d nails at each joint is able to attain maximum buckling capacity (Du et al. [5]) and thus an additional stiffness increase associated with stiffer connectors yields no buckling capacity increase. The verification of the present non-sway solution in Eq. (22) is conducted for a beam-deck system with varying joint rotational stiffness where the critical uniform moments obtained are compared with those from Du et al. [4], Vlasov [6] and the Abaqus non-sway model. The results are shown in Fig. 10(a) where the critical moments are plotted against joint rotational stiffness per unit length (kNm/m). For the entire stiffness range investigated, the present finite element

solution (FEA)predicts buckling capacities that are marginally higher than the Abaqus non-sway model. For beam-deck joints with relatively low rotational stiffness (i.e., stiffness per unit length below 1 kNm/m), the non-sway finite element solution reports a critical moment below 353 kNm, which compares with a critical moment of 344 kNm estimated from the Vlasov [6] solution where the effect of rigid lateral restraint of decking was captured and the twist restraint effect was ignored. For beam-deck connections with rotational stiffness greater than 103 kNm/m, the present formulation reports a constant buckling capacity of 581 kNm, which is identical to that predicted by Du et al. [4] where rigid lateral restraint and rigid beam-to-deck rotational

1

700

0.8

500 Angle of twist

Critical moment (kNm)

600

400 300 Du et. al. (2016) FEA non-sway model Abaqus non-sway model Vlasov (1961)

200 100 0

0.1

0.6 0.4 FEA (angle of twist)

0.2 0

1 10 100 1000 10000 Rotational stiffness per unit length (kNm/m)

Abaqus (angle of twist) 0

0.2

0.4

1

1

0.5

0.5

0

-0.5

0.2

0.4

z/Lb

1

0.6

0.8

1

FEA (angle of twist)

-0.5

FEA (angle of twist)

0

0.8

0

Abaqus (angle of twist)

Abaqus (angle of twist) -1

0.6

(b)

Angle of twist

Angle of twist

(a)

z/Lb

0.6

0.8

-1

1

(c)

0

0.2

0.4

z/Lb

(d)

Fig. 10. (a) Critical moments for non-sway model, mode shapes for non-sway model for rotational stiffness of (b) 1 kNm/m, (c) 100 kNm/m, (d) 10,000 kNm/m. 10

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connection were assumed. The buckling mode shapes associated with the beam angle of twist are plotted in Fig. 10b, c and d for the rotational stiffness values of 1, 100 and 10,000 kNm/m, respectively. Since the mode shapes for the two beams in the beam-deck system are observed to have the identical magnitude, only mode shape for one beam is provided. In Fig. 10b, for rotational stiffness of 1 kN/m, the present finite element solution predicts a single-curvature mode shape which is matched by that from the Abaqus non-sway model. In Fig. 10c, for rotational stiffness value of 100 kNm/m, the angle of twist exhibits a double-curvature mode as predicted by both the present solution and the Abaqus model. For rotational stiffness value of 10,000 kNm/m, Fig. 10d shows nearly identical buckling mode shape as that in Fig. 10c with rotational stiffness value of 100 kNm/m. The present sway finite element solution is assessed for the same beam-deck system where the critical uniform moments from Eq. (19) are compared with those from Timoshenko and Gere [25], the Abaqus sway model developed in the present study and the solution from Du et al. [5] where rigid rotational connections were assumed at beamdeck joints. Similar to the non-sway solution, the present sway solution predicts buckling capacities (Fig. 11a) slightly higher than those from the Abaqus sway model in the entire stiffness range. For connections with low joint rotational stiffness (i.e., stiffness per unit length below 1 kNm/m), the present sway model reports a critical moment below 224 kNm, which compares with 222 kNm from Timoshenko and Gere [25] where the effects of both the lateral and torsional restraints of decking were not captured. This corroborates an observation from Du et al. [5]

Fig. 12. Boundary conditions for (a) simply-supported beams, (b) type 1 twospan beams, (c) type 2 two-span beams, (d) simply-supported beams with overhang.

that the presence of partial lateral restraint alone (without torsional restraint) leads to no additional buckling capacity increase. For beamdeck joints with rotational stiffness beyond 103 kNm/m, an identical critical moment of 441 kNm is obtained from the present sway model and Du et al. [5].

400 300

Du et. al. (2017) FEA sway model Abaqus sway model Timoshenko and Gere (1961)

100 0

0.1

1

10

100

1000

1.5

1.0

0.6

0.5

0.3

0.0

10000

Rotational stiffness per unit length (kNm/m)

0

0.2

0.4

(a)

1.2

0.5

0.3

0.2

0.4

z/Lb

0.6

0.8

1

0.0

Lateral displacement (m)

0.9

0.6

0

0.8

1

0.0

1.2

2.0

Angle of twist

Lateral displacement (m)

FEA (lateral displacement) ABAQUS (lateral displacement) FEA (angle of twist) ABAQUS (angle of twist)

1.0

0.0

z/Lb

0.6

(b)

2.0

1.5

0.9

1.5

0.9

1.0

0.6

0.5

0.3

0.0

(c)

FEA (lateral displacement) ABAQUS (lateral displacement) FEA (angle of twist) ABAQUS (angle of twist)

0

0.2

0.4

z/Lb

0.6

0.8

1

Angle of twist

200

1.2

FEA (lateral displacement) ABAQUS (lateral displacement) FEA (angle of twist) ABAQUS (angle of twist)

Angle of twist

2.0

Lateral displacement (m)

Critical moment (kNm)

500

0.0

(d)

Fig. 11. (a) Critical moments for sway model, mode shapes for sway model with rotational stiffness of (b) 1 kNm/m, (c) 100 kNm/m, (d) 10,000 kNm/m. 11

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gravity load. The difference in buckling capacity is attributed to the load position effect where the uplift load at beam top is associated with a stabilizing effect that increases the buckling capacity. Fig. 13 also shows that the present non-sway and sway models are sensitive in the rotational stiffness range of 10 kNm/m to 103 kNm/m, which covers all the connector types tested. Another observation from Fig. 13 is that the buckling capacity corresponding to the stiffest fastener is close to the maximum value with rigid rotational connection. For example, the buckling capacity for the non-sway model based on type 2 self-tapping screws is 867 kNm and attains 99.2% of the maximum capacity. The buckling capacity for type 2 self-tapping screw based on the sway model is able to achieve 88.4% of maximum capacity for gravity loading and 90.3% for wind uplift loading. Table 3 reproduces the buckling capacity values of all fastener types from Fig. 13 and is aimed at comparing the relative efficiency of fasteners in enhancing the buckling capacity of beam-deck systems. Provided in Table 3 also are the increase in buckling capacity of a given type of fastener compared to that of non-rotational-connected beams. It is observed that a connection of two 10d nails at each beam-deck joint (a connection configuration permitted by CAN/CSA O86 [1]) yields an increase in buckling capacity of 21% from non-rotational-connected beams for the non-sway model, a 33% increase for the sway model under gravity load, and a 23% increase for sway model under wind uplift. For beam-deck joints with stiffer fasteners (e.g., wood screws, 60d nails and self-tapping screws), the non-sway model is observed to have insignificant buckling capacity increase with the critical moment enhanced by less than 28% from that of non-rotational-connected beams. In contrast, the sway model is found more sensitive to the increase of joint rotational stiffness. It reports a buckling capacity increase between 50% (wood screws) and 90% (type 2 self-tapping screws) for gravity loading and between 35% (wood screws) and 64% (type 2 self-tapping screws) for uplift loading.

The buckling mode shapes for the beam angle of twist and lateral displacement are plotted in Fig. 11b, c and d for rotational stiffness values of 1 kNm/m, 100 kNm/m and 10,000 kNm/m, respectively. In all three figures, the mode shapes sought in the Abaqus model are nearly identical to those from the present solution. The mode shapes for angle of twists and lateral displacements in all three figures are singlecurvature. The magnitude of the twist angle is reduced as the rotational stiffness is increased. The above comparisons verify the validity of the present sway and non-sway models. Subsequently, the models are employed to further explore the buckling capacities of wooden beam-deck systems with other boundary and loading conditions. 6. Examples This section provides examples for beam-deck systems and is aimed at: (1) showcasing the practical applications of the present formulations, i.e., Eqs. (19) and (22), and (2) assessing the adequacy of different types of fasteners tested in achieving desirable buckling capacities. Unless otherwise specified, the examples below are based on the same material properties and dimensions presented in the previous section. 6.1. Example 1: Simply-supported beams under uniformly distributed load at the top of the beams This example considers a beam-deck system consisting of two beams braced at the top by deck boards. The beams are considered to be simply-supported (Fig. 12a) and are under identical magnitude of uniformly distributed load (UDL) applied at the top face of each beam. Fig. 13 illustrates the buckling capacity, represented by the moment at the beam’s mid-span, for each type of fastener under the sway model and the non-sway model. The experimentally determined joint rotational stiffness (Table 2), is divided by the deck width to obtain the unit rotational stiffness and is plotted as vertical lines in Fig. 13. It is observed that the buckling capacity of the non-sway model is consistently higher than that from the sway model. For the sway model, the critical moment of wind uplift loading is observed to be higher than that of

6.2. Example 2: Two-span beams under UDL at the top of the beams For a beam-deck assembly with two two-span beams braced continuously by decking, each beam has a total length of 18 m and an intermediate support is provided at half-length to provide either

1000 900

Critical moment (kNm)

800 700 600 500 400 non-sway model (wind uplift) sway model (wind uplift) sway model (gravity load) stiffness-10d nail stiffness-wood screw stiffness-60d nail stiffness-Self-tapping screw (type 1) stiffness-Self-tapping screw (type 2)

300 200 100 0

0.1

1

10

100

1000

10000

Rotational stiffness per unit length (kNm/m) Fig. 13. Buckling capacity for simply-supported beams with different fastener types. 12

100000

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Table 3 Buckling capacity for beam-deck systems with simply-supported beams. Type of fastener

No connection 10d common nails wood screws 60d common nails Self-tapping screws-type 1 Self-tapping screws-type 2

Non-sway model

Sway model

Wind Uplift

Gravity load

Wind uplift

Critical moment (kNm)

Capacity increase

Critical moment (kNm)

Capacity increase

Critical moment (kNm)

Capacity increase

677 822 839 847 850 867

0% 21% 24% 25% 26% 28%

212 281 317 337 346 403

0% 33% 50% 59% 63% 90%

296 364 399 418 428 484

0% 23% 35% 41% 45% 64%

vertical support (type 1) shown in Fig. 12b or a combination of vertical, lateral and twist supports (type 2) in Fig. 12c. Identical magnitude of UDL is assumed to be applied to the top face of each beam along its entire length. All other material and dimensional properties are consistent with those presented in the reference case. Table 4 provides the buckling capacities, obtained from the present non-sway model, for two-span beam-deck systems with different types of fasteners. The values presented are based on critical moments at the intermediate support at which maximum moments occur. For beamdeck systems with two-span beams, the non-sway model is associated with buckling under both gravity and upward loadings. From Table 4, it is observed that the elastic buckling capacities of Cases 2, 3 and 4 for all fastener types exceed the beam material resistance of 1271 kNm (based on material strength provided in CAN/CSA O86 [1] multiplied by a factor of 3 to obtain mean resistance value, according to Hooley and Madsen [24]). In such cases, the load carrying capacity is governed either by inelastic buckling or material failure and is thus beyond the scope of present study. For Case 1 (type 1 intermediate support and gravity load), elastic buckling governs since the critical moments are below the elastic/inelastic limit of 848 kNm (evaluated based on Hooley and Madsen [24]). In this case, the buckling capacity is found to be increased from that of non-rotational-connected beams by 154% (10d nails) to 279% (type 2 self-tapping screws). Thus, fasteners are found very effective in enhancing the buckling capacity. The buckling capacities for the sway model are presented in Table 5. It is observed that the buckling capacity can be effectively enhanced in Case 5 where the critical moment is increased by 146% for 10d nails and by 275% for type 2 self-tapping screws. Cases 6 and 7 demonstrate moderate levels of sensitivity to fasteners, with the critical moment increasing from that of non-rotational-connected beam by 41% to 92% for Case 6 and 21% to 83% for Case 7. The effectiveness of fasteners is found to be relatively low in Case 8 where the critical moments increases by less than 41%.

6.3. Example 3: Simply supported beams with single overhang under UDL at the top of the beams In this example, each of the two beams within the beam-deck system consists of a simply supported beam with an overhang extended from one end (Fig. 12d). The overhang lengths considered are 1.5 m (Case 1), 3 m (Case 2) and 4.5 m (Case 3) with the simply-supported beam spanning 9 m. Same magnitude of UDL is assumed to be applied to each beam at its top face along the entire beam length. All other material and dimensional properties are consistent with those presented in the reference case. The buckling capacities, represented by the maximum moment along the beam length, are provided in Table 6 for different fastener types. Gravity and wind uplift loadings are considered for the sway model while only wind uplift is considered for the non-sway model. Theoretically, gravity loading is able to induce lateral torsional buckling under the non-sway scenario. Nevertheless, the elastic buckling capacity under gravity loading was found to far exceed the beam material resistance and is therefore not presented in the table. Table 6 also provides the ratios (within bracket) of the critical moment to that of a simply-supported beam-deck system with no overhangs (Example 1 and Table 3) for the same model (sway or non-sway) and same type of fasteners. For Case 1 with a 1.5 m overhang, the buckling capacities obtained are found to be marginally higher (by a factor of 1.01–1.05) than those of simply-supported systems with no overhangs. In Case 2 where the overhang is 3 m, the buckling capacity of is found to increase by a factor of 1.01–1.13 from non-overhang systems. In Case 3 where the overhang is 4.5 m and attains half of the simply supported span, the buckling capacity is observed to be 1.75 to 2.40 times higher than that based on simply-supported systems. 7. Conclusions The present study develops two finite element models for the elastic buckling of timber beam-deck assemblies consisting of two beams

Table 4 Buckling capacity for beam-deck systems with two-span beams under non-sway model. Type of fastener

Type 1 intermediate support

Type 2 intermediate support

Gravity load (Case 1)

No connection 10d common nails wood screws 60d common nails Self-tapping screws-type 1 Self-tapping screws-type 2

Wind uplift (Case 2)

Gravity load (Case 3)

Critical moment (kNm)

Capacity increase

Critical moment (kNm)

Critical moment (kNm)

173 439 525 562 579 655

0% 154% 203% 225% 235% 279%

1661 1809 1853 1872 1880 1919

1334 1521 1577 1602 1612 1661

13

Wind uplift (Case 4)

1642 1804 1851 1871 1879 1919

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Table 5 Buckling capacity for beam-deck systems with two-span beams under sway model. Type of fastener

No connection 10d common nails wood screws 60d common nails Self-tapping screws type I Self-tapping screws type II

Type 1 intermediate support

Type 2 intermediate support

Gravity load (Case 5)

Wind uplift (Case 6)

Gravity load (Case 7)

Wind uplift (Case 8)

Critical moment (kNm)

Capacity increase

Critical moment (kNm)

Capacity increase

Critical moment (kNm)

Capacity increase

Critical moment (kNm)

Capacity increase

126 310 391 417 427 472

0% 146% 210% 231% 239% 275%

395 555 622 655 671 758

0% 41% 57% 66% 70% 92%

363 441 493 525 542 663

0% 21% 36% 45% 49% 83%

681 752 800 830 845 957

0% 10% 17% 22% 24% 41%

Table 6 Buckling capacity for beam-deck systems with beams with overhangs. Type of fastener

No connection 10d common nails wood screws 60d common nails Self-tapping screw-type 1 Self-tapping screw-type 2

Critical maximum moment along beam span (kNm) Case 1 (9 m + 1.5 m)

Case 2 (9 m + 3 m)

Non-sway model

Sway model

Non-sway model

Sway model

Uplift

Gravity

Uplift

Uplift

Gravity

703 858 881 888 892 908

219 286 321 341 350 406

307 374 409 428 437 492

768 889 917 928 933 954

216 284 321 340 350 407

(1.04) (1.04) (1.05) (1.05) (1.05) (1.05)

(1.03) (1.02) (1.01) (1.01) (1.01) (1.01)

(1.04) (1.04) (1.02) (1.02) (1.02) (1.02)

(1.13) (1.08) (1.09) (1.10) (1.10) (1.10)

Case 3 (9 m + 4.5 m)

(1.02) (1.01) (1.01) (1.01) (1.01) (1.01)

Non-sway model

Sway model

Uplift

Uplift

Gravity

Uplift

326 393 428 447 457 512

1625 1779 1824 1843 1852 1891

370 499 565 601 619 723

679 (2.29) 796 (2.19) 858 (2.15) 891 (2.13) 908 (2.12) 1005 (2.08)

(1.10) (1.08) (1.07) (1.07) (1.07) (1.06)

(2.40) (2.16) (2.17) (2.18) (2.18) (2.18)

(1.75) (1.77) (1.78) (1.78) (1.79) (1.80)

Note: the bracketed values are ratios of the critical moment to that of a simply-supported beam-deck system with no overhangs.

connected to deck boards through fasteners. The non-sway model is aimed for situations where decking can be assumed to provide continuous rigid lateral restraint while the sway model reflects the continuous partial lateral restraint provided by the deck boards. Both models capture the continuous twist restraint of deck boards by incorporating the effects of rotational flexibility of beam-to-deck fasteners. Experimental testing is conducted to quantitatively assess the rotational stiffness of beam-deck joints with different fastener types, which is then used as inputs to the finite element models to evaluate the effect of each fastener type on the lateral torsional buckling resistance of the system. The validity of the present finite element models is confirmed through comparing the predicted buckling capacities against those from Abaqus and other studies. The finite element models are then used to model timber beam-deck assemblies with practical material properties and dimensions. For the examples considered, the key engineering observations are as follows:

capacity based on the rigid rotational connection assumption. 3. For beam-deck systems with two-span beams, depending on the loading and boundary conditions, the contribution of fastener stiffness to the overall buckling capacity can vary significantly. Buckling capacity for beams with intermediate vertical support and under downward UDL is associated with 146–154% capacity increase for low stiffness connections (10d nails) and 275–279% increase for high stiffness connections (type 2 self-tapping screws). 4. For beams with overhang length not exceeding half of the simply supported span, the buckling capacity for cantilevered beam-deck system is found to be greater than that of simply-supported system having the same support distance and the same fastener type. CRediT authorship contribution statement Yang Du: Methodology, Software, Validation, Formal analysis, Data curation, Investigation, Writing - original draft, Visualization. Ghasan Doudak: Conceptualization, Resources, Writing - review & editing, Supervision, Funding acquisition. Magdi Mohareb: Conceptualization, Resources, Writing - review & editing, Supervision, Funding acquisition.

1. Buckling capacities of beam-deck assemblies with commonly used fasteners, e.g., common nails and wood screws, are found to increase by at least 1.2 times from those without rotational connections. 2. Buckling capacities of beam-deck assemblies with state-of-art fasteners are observed to achieve 88.4–99.2% of the maximum

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Appendix A. Sensitivity analysis for joint rotational stiffness This appendix is intended to conduct a sensitivity analysis of the present joint rotational stiffness model, i.e., Eq. (2), and the Loferski and Gamalath model [17], i.e., Eq. (3), through numerical examples. Example 1. As a reference case, consider a beam with 130 mm in width and with a specific gravity of 0.465. The deck board is 19 mm in thickness, 140 mm in width and 2130 mm in length and has a specific gravity of 0.444. Two 10d nails are assumed to be installed at the beam-deck joint with the joint configuration specified as Fig. 4b. These values correspond to geometric values L2 = 0.065 m , L3 = 2 m defined in Fig. 2b. For the 10d nails and the beam and deck materials chosen, the joint separation stiffness ks = 4080000 N/ m and crushing stiffness kc = 8140000 N/m are estimated based on the study of Loferski and Gamalath [17]. A sensitivity analysis of parameters L2 , L3 , ks, kc is conducted by varying one parameter at a time (whereby each parameter magnitude is doubled and halved) while keeping the other parameters constant. The results are shown in Table A.1. It is observed that the joint rotational stiffness is insensitive to the deck overhang distance L3 as the stiffness remains nearly constant when L3 is doubled or halved. In contrast, the rotational stiffness is found sensitive to changes of the other three parameters L2 , ks, kc where significant deviations from the reference stiffness are observed. Table A1 Sensitivity analysis for the joint stiffness based on Loferski and Gamalath model. Case No.

L2 / L 2

Reference case 1 2 3 4 5 6 7 8

1 2 0.5 1 1 1 1 1 1

L3/ L3

ref

ks/ks

ref

1 1 1 2 0.5 1 1 1 1

k c /k c

ref

1 1 1 1 1 2 0.5 1 1

ref

1 1 1 1 1 1 1 2 0.5

R (Nm)

R /Rref

11,359 44,955 2855 11,421 11,239 16,941 6847 13,694 8471

1.00 3.96 0.251 1.01 0.989 1.49 0.603 1.21 0.746

Example 2. This example is provided to assess the sensitivity of parameters L1, L2 , ks, kc , P1, P2 identified in Eq. (2) for the joint rotational stiffness R1 within a beam-deck system. Similar to the previous example, the rotational stiffness for a reference case is quantified and compared with other cases where, in each case, the magnitude of one parameter is changed while the remaining parameters are kept unchanged. For the reference case, the distance L1 = 2.74 m and the applied loads P1 = P2 = 10 kN are introduced while the values for L2 , ks, kc are identical to those in the previous example (all parameters are defined in Fig. 1b). It is observed from Table A.2 that the joint rotational stiffness is only sensitive to L2 , ks, kc , in a manner similar to the Loferski and Gamalath model [17]. Also, the joint rotational stiffness for the reference case under the present model is 11,483 Nm, which is 1% higher than the corresponding value of 11,359 Nm obtained from the Loferski and Gamalath model [17]. The above findings suggest that the joint rotational stiffness depends solely on the joint details (i.e., the distance L2 between the nail and point of crushing, separation stiffness ks and crushing stiffness kc ) and is nearly independent of other parameters.

Table A2 Sensitivity analysis for the joint stiffness within a beam-deck system. Case No.

L1/L1

Reference case 1 2 3 4 5 6 7 8 9 10 11 12

1 2 0.5 1 1 1 1 1 1 1 1 1 1

ref

L2 / L 2 1 1 1 2 0.5 1 1 1 1 1 1 1 1

ref

ks/ks 1 1 1 1 1 2 0.5 1 1 1 1 1 1

ref

k c /k c 1 1 1 1 1 1 1 2 0.5 1 1 1 1

ref

P1/P1 1 1 1 1 1 1 1 1 1 100 0.01 1 1

ref

P2/P2 1 1 1 1 1 1 1 1 1 1 1 100 0.01

ref

R (Nm)

R /Rref

11,483 11,483 11,483 45,930 2871 17,217 6892 13,784 8608 11,392 11,394 11,394 11,394

1.00 1.00 1.00 4.00 0.25 1.50 0.60 1.20 0.75 0.992 0.992 0.992 0.992

Appendix B. Load-displacement relationship from joint rotational tests This appendix provides the load–displacement curves from the joint rotational tests for the five types of fasteners described in Table 1, each fastener type with two repetitive tests. As was described in Section 2.2, the concentrated load and the corresponding vertical displacement at the right end of the deck board were measured and are plotted as Figs. B.1 and B.2 below. Fig. B.1 shows the load-displacement curves for the 10d, 60d common nails and wood screws and Fig. B.2 provides the load-displacement curves for the Type 1 and 2 self-tapping screws.

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Magnitude of Concentrated Load (N)

2500

2000

1500

1000

10d Nail-test 1 10d Nail-test 2 60d Nail-test 1 60d Nail-test 2 Wood screw-test 1 Wood screw-test 2

500

0

0

1

2

3

4

5 6 7 8 Vertical displacement (mm)

9

10

11

12

13

Fig. B1. Load-displacement curve for 10d, 60d common nails and wood screws.

Magnitude of Concentrated Load (N)

3000

2500

2000

1500

1000

Self-tapping screw (Type1)-test 1 Self-tapping screw (Type1)-test 2 Self-tapping screw (Type2)-test 1

500

0

Self-tapping screw (Type2)-test 2

0

1

2

3

4 5 Vertical displacement (mm)

6

7

8

9

Fig. B2. Load-displacement curve for type 1 and 2 self-tapping screws.

Appendix C. Stiffness and geometric matrices under finite element formulations This appendix provides stiffness and geometric matrices of the sway model and non-sway model. Sway model For the sway model, the beam stiffness matrix is

(C1) and the lateral restraint stiffness matrix [Kl] is

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Y. Du, et al.

(C2) and the geometric stiffness matrix [K g ] takes the following form

(C3) and submatrices [B1], [B2], [B3], [B4], [B5] are defined as

[B1] =

l 0

{L (z )}4 × 1 L (z )

[B2] =

l 0

{L (z )}4 × 1 L (z )

[B3] =

l 0

{L (z )}4 × 1 L (z )

[B4] =

l 0

M (z ){L (z )}4 × 1 L (z )

T 1 × 4 dz

T 1 × 4 dz

T 1 × 4 dz

=

=

=

1 l3

12 6l 6l 4l 2 12 6l 6l 2l 2

12 6l 6L 2l 2 , 12 6l 2 6l 4l

1 30l

36 3l 3l 4l2 36 3l 3l l2

36 3l 3l l2 , 36 3l 2 3l 4l

l 420

T 1 × 4 dz ,

156 22l 54 22l 4l2 13l 54 13l 156 13l 3l2 22l [B5] =

l 0

13l 3l 2 , 22l 2 4l

q (z ){L (z )}4× 1 L (z )

T 1 × 4 dz

(C4)

Non-sway model For the non-sway model, the beam stiffness matrix is

[Kb ] =

Eb C [B1] + Gb Jb [B2 ] [0] [0] Eb C [B1] + Gb Jb [B2 ]

(C5)

8×8

and the geometric stiffness matrix is

[K g ] =

2a [B4] h [B5 ] [0] [0] 2a [B4] h [B5 ]

(C6)

8×8

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