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Investigation of the lateral-torsional buckling behaviour of engineered wood I-joists with varying end conditions
Engineering Structures 187 (2019) 329–340
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Investigation of the lateral-torsional buckling behaviour of engineered wood I-joists with varying end conditions B. Pelletier, G. Doudak
T
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Department of Civil Engineering, University of Ottawa, K1N 6N5, Canada
A R T I C LE I N FO
A B S T R A C T
Keywords: Wood I-joists Lateral torsional buckling Full scale testing Lateral restraints
Beam members, especially those that have large depth/width ratios and long spans, are prone to lateral-torsional buckling as a possible mode of failure. Laboratory testing rarely takes into account actual end conditions and initial imperfection which might have a significant impact on the buckling behavior of beams. The current research project aims to investigate the lateral-torsional buckling of wooden I-joists with realistic boundary conditions used in construction. A total of 41 joists were tested using various commercial joist hangers and enhanced connections to represent different support conditions. A numerical 3D model was also developed using commercially available finite element program ABAQUS to determine the buckling loads and associated mode shapes of joists similar to those tested. Based on the results from the current study it is recommended that 20% reduction in the critical moment capacity be considered in order to take into account the use of joist hangers typically used in construction. Finite element analysis of the linear eigenvalue buckling load was found to be in reasonable agreement with the experimental results. The nonlinear behavior of the joists was found to be influenced by the initial imperfections. The ultimate critical load was attained at large lateral displacement of the joists due to their non-linear behaviour. This observation could have a significant implication on design and should be investigated further.
1. Introduction 1.1. General The development of engineered wood products, such as wood I-joists, has meant that structural bending members that are longer and deeper can be achieved without significantly increasing the overall building weight. However, long-span beams that have large depth/ width ratios are more prone to buckling failure. Similar to solid sawn joists, engineered wood I-joists are typically sheathed with wood panels bracing them at the top face and partially restraining the lateral displacement of the top flange. However, during or after construction, several conditions might lead to the wood I-joists not being adequately braced. Such cases include: workers walking on floor or roof joists during installation, wind uplift governing the design of a roof, and continuous or cantilever beams with negative moments. Established knowledge on the behavior of beams to lateral torsional buckling failure has addressed cases involving idealized boundary conditions. However, practical construction often involves non-ideal boundary conditions. For example, a variety of metallic hangers and connection details are used in construction, and the effect of such
⁎
connections has been investigated very scarcely. In this context, the current research aims at investigating the effect of such “realistic” boundary conditions on the lateral stability of the engineer wood Ijoists as well as assessing the ability of numerical models to predict the buckling behavior. 1.2. Constitutive behaviour of wood The mechanical properties in the tangential and radial axis are very close for a large array of species and therefore are commonly assumed to be identical [1]. Although wood can be modelled as an orthotropic material, not all the material properties have an important role in the lateral torsional buckling resistance. Xiao [2] conducted a sensitivity analysis on rectangular solid sawn lumber and concluded that the modulus of elasticity along the longitudinal direction (EL ) and the transverse shear modulus (GT ) are the two variables that have the strongest influence on the lateral torsional buckling capacity. St-Amour [3,4] presented a sensitivity analysis on the mechanical properties of wood I-joists and it was revealed that the most influential mechanical properties are the longitudinal modulus of elasticity and the torsional shear modulus of the compression flange. The longitudinal modulus of
Corresponding author at: Civil Engineering, University of Ottawa, 161 Louis-Pasteur, Room A115 (CBY), Ottawa, ON K1N 6N5, Canada. E-mail address: [email protected] (G. Doudak).
https://doi.org/10.1016/j.engstruct.2019.03.003 Received 27 November 2018; Received in revised form 18 February 2019; Accepted 1 March 2019 Available online 06 March 2019 0141-0296/ © 2019 Elsevier Ltd. All rights reserved.
Engineering Structures 187 (2019) 329–340
B. Pelletier and G. Doudak
elasticity and the torsional shear modulus of the tension flange, and the transverse modulus of elasticity of the web also impact the buckling response but to a lesser degree. Studies have also investigated the material properties of OSB, and their effect on the lateral torsional buckling capacity of wood I-joists [3–7].
a sensitivity analysis on the effect of orthotropic material properties on the lateral torsional buckling behavior of rectangular timber beams. The study showed that the numerical model developed was mostly affected by variation in the longitudinal modulus of elasticity and the transverse shear modulus of the timber member. Xiao et al. [20] conducted full-scale lateral torsional buckling tests on rectangular wood joist elements. Numerical buckling analysis was conducted and validated using full-scale joist test results. Du [21] and Du et al. [22] studied the behavior of wooden beam-deck systems, where models for sway and non-sway systems were formulated with analytical and energy-based solutions. In general, the non-sway model consistently predicted critical moments higher than those based on the sway model. The effects of bracing action of the deck board, the deck span between beams and the span type (continuous or single span) were investigated and discussed. Hu et al. [23,24] investigated the effect of bracing height on wooden beams subjected to lateral-torsional buckling. Energy based solutions were formulated for a simply supported beam with rigid or flexible bracing.
1.3. Research goals The current research aims at investigating the lateral torsional buckling behavior of engineered wood I-joists with idealized pinned ends as well as “realistic” boundary conditions using metallic joist hangers. The study also aims at investigating the effect of initial imperfection on the buckling behavior of wood I-joists. A finite element model was developed and validated by comparing the results against those obtained from full-scale tests. 2. Review of relevant literature Several studies have been conducted on the lateral torsional buckling instability of rectangular timber beams and engineered wood Ijoists products. Some studies investigated the ability of various code design provisions to estimate the buckling load, while others focused on the bracing action in order to prevent buckling failures. The topic of loading height has been investigated numerically in a number of studies, however, since this topic is not directly related to the present study, it was not covered in details. Very limited number of studies considered the impact of realistic boundary conditions on the LTB capacity of the beams. The initial geometric imperfection and nonlinear displacement of I-joists has not been adequately investigated in the literature. The present study aims to investigate the nonlinear behavior influence on the resistance during the entire displacement history of the joists.
2.2. Research on engineered wood I-joists Hindman et al. [25] investigated the material properties of structural composite lumber products using isotropic and orthotropic material properties. The torsional stiffness was also computed using mathematical models based on isotropic and orthotropic formulations. Hindman et al. [26] compared the buckling loads for two types of wood I-joists specimens of various lengths, with flanges constructed of structural composite lumber. It was found that both the design equation [27] and the elastic beam buckling prediction [28] provided conservative estimates when compared to the experimental wood I-joist buckling loads. Zhu et al. [29] studied the buckling behavior of wood Ijoists by comparing experimental tests to finite element simulations. Geometric imperfections were introduced in the model to conduct nonlinear buckling analysis. Burow et al. [30] performed lateral torsional buckling tests on cantilevered and simply supported wood I-joist specimens. Three mathematical models were developed based on elastic stability solution [28] as well as the American codes provisions [27,31]. It was shown that both the American timber design standard [31] and the modified Euler model consistently underestimated the buckling load. Villasenor [32] investigated the lateral buckling behavior of wood I-joists subjected to human walking loads. The author concluded that the buckling loads were influenced by the overall lateral stiffness of the end support conditions. St-Amour et al. [3,4] investigated the effect of the material properties of wood I-joists on the lateral torsional buckling behavior. Full-scale LTB experiments were conducted by subjecting wood I-joist with various span length and simply supported end conditions to a concentrate point load located on the top flange at midspan. The authors concluded that the lateral bending and rotational stiffness observed from whole joists were adequate material input in the numerical model to describe the joist behaviour. The FE linear and nonlinear predictions were found to be in reasonable agreement with the experimental buckling loads and displacements.
2.1. Research on solid sawn wooden beams Hooley and Madsen [8] conducted an experimental study on gluedlaminated beams with the objective to develop simple code provisions. The experimental results correlated well with the theory and the study concluded that the slenderness ratio governed the lateral stability of the beams. To account for the flexibility of the support in realistic conditions, a 15% reduction of the LTB load was suggested. Zahn [9] formulated the governing equilibrium conditions for wooden rectangular beams with continuous lateral bracing along the beam’s main axis through energy-based solutions. Bell and Eggen [10] investigated the capacities of beam-columns subjected to lateral instability as evaluated by the Norwegian [11] and European timber codes [12]. Linear and non-linear buckling models were developed, using simply supported rectangular wood beams. A reduction of 10% for the critical axial force and bending moment buckling loads was observed for the non-linear buckling analysis. AFPA [13] released a technical report to address the design of timber members prone to instability failure. The effective length and equivalent moment factor approaches were outlined and compared. Hindman et al. [14] studied the lateral torsional buckling behavior of rectangular cantilever beams composed of various structural composite lumber products. The elastic constant ratios and torsional rigidity of the structural composite lumber material resulted in a more constant buckling behavior than what was observed with solid sawn lumber members. Baláž [15] investigated the LTB of timber beams and proposed an expression to calculate the elastic critical moments. Baláž and Koleková [16,17] provided a generalized critical moment framework for critical moment expressions of beams with mono-symmetric cross-sections. Suryoatmono and Tjondro [18] performed lateral torsional buckling tests on simply supported rectangular wooden beams. Finite element models were also developed using isotropic and orthotropic material properties. The classical solution used was based on isotropic material and neglected warping. Xiao et al. [19] conducted
3. Experimental program 3.1. Specimen description and test setup A total of 41 I-joist specimens were tested, 18 of which consisted of 0.4 m deep and 4.6 m long joists (Group A) and 23 consisted of 0.6 m deep and 5.8 m long joists (Group B). The joists were manufactured using machine stress rated (MSR) lumber flanges and oriented strand board (OSB) webs connected together with a waterproof structural adhesive [33]. The geometry of the tested specimens is shown in Fig. 1. The specimens were stored in the laboratory so their temperature and relative humidity were acclimatized to the lab environment. The average moisture content for all joists was approximately 14% with a 330
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Table 2 Group A specimen names and boundary conditions. Sample group
Specimens #
Boundary condition
Hangers/Connection
A
1–6 7–10 11–12 13–14 15–16 17–18
Simply supported Top mounted Face mounted Face mounted Nailed Enhanced connection
LBV2.56/16 IUS2.56/16 HU316 Nails in End grain Angle
Table 3 Group B specimen names and boundary conditions.
Group A
Sample group
Specimen #
Boundary condition
Hangers/Connection
B
1–7 8–9 10–11 12–13 14–15 16–17 18–19 20–21 22–23
Simply supported Top mounted Top mounted Top mounted Face mounted Face mounted Nailed Enhanced connection Enhanced connection
HWI424 HB3.56/24 HIT424 IUS3.56/16 MIU3.56/16 Nails in End grain Side screwed Angle
Group B
Fig. 1. I-joists specimens’ cross-section geometry (all dimensions are in mm). Table 1 I-joists hanger types.
Face mounted Top mounted
Group A
Group B
IUS2.56/16 HU316 LBV2.56/16
MIU3.56/20 IUS3.56/16 HIT424 HWI424 HB3.56/24
Fig. 2 shows the types of hangers used. Tables 2 and 3 provides details on the joist specimens and their boundary conditions. The lateral-torsional buckling capacity was evaluated by applying a concentrated load at mid-span of the specimen’s bottom flange, as illustrated in Fig. 3. The load was applied using a cable and pulley system developed by the authors specifically for this project. One end of the cable was attached to an HSS frame anchored to the lab’s strong floor while the other was attached to a hydraulic jack via a load cell to record the tension force in the cable. The pulley was attached to the bottom flange by a series of shackles acting as hinges. This setup enabled continuous application of the vertical load while also allowing the bottom flange to displace freely. As the joist displaces laterally and twists, the angle of the cable will vary and adjustment to the vertical load must be included in the analysis. This was possible because the geometry of the test setup was fully known at any given time during the test. The vertical displacement, lateral displacement of the top and bottom flanges at mid-span were recorded using wire sensors. The angles of rotation of the top and bottom flanges were recorded using inclinometers.
coefficient of variation (COV) of 5.4%. Each joist end was supported by an elevated wood support, at which the chosen boundary conditions were constructed. Once the joist was installed in the test setup, its initial out-of-straightness was documented using a self-leveling laser. The end supports were divided in 3 categories depending on the boundary conditions to be tested: simply supported, commercially available joists hangers and enhanced fixity connections. Simpsons Strong-Tie products were selected due to their wide availability in the North American market. The hangers were classified into 2 categories; namely face-mounted and top-mounted hangers. Table 1 describes the hangers used for each joist section and
Fig. 2. Hanger type tested. 331
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Fig. 3. Schematic of the test setup showing elevation view and attachment details.
rotation and translation. The top flange was secured in place using two 76 mm × 76 mm × 6 mm angles on each panel and by gripping the flange between the angles using two steel threaded rods, as illustrated in Fig. 4b.
Three designs were evaluated to investigate the effect of enhanced fixity. The first design involved nailing the joists directly to the wooden rim-board [33]. The second design involved a modification to the HWI424 top mounted hanger. To prevent any lateral displacements, a steel plate was welded on each side of the hanger and aligned with the bottom flange. To reduce the rotation relative to the weak axis of the section, six holes (three on each sides) were installed in the hanger and aligned with each flange. 38 mm long screws were used to connect the hanger to the joists (Fig. 4a). The third connection was intended to create high level of fixity on the rotational degree of freedom of the top flange along its longitudinal axis. The bottom flange was therefore attached to the support by two 50 mm × 50 mm × 6 mm angles at each end. Six 38 mm screws (three per flange side) were used to limit the
3.2. Component and material level testing The web’s transverse modulus of elasticity and the flanges torsional modulus were determined based on standard test methods in conformance with ASTM [34] and [35], respectively. The flanges were tested to determine their longitudinal modulus of elasticity in accordance with the test method describe in [35]. The flange was bent about its strong axis in order to simulate the same bending action as 332
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Fig. 4. Enhanced connections using HWR with screwed flanges connection (a) and angles contained flanges connection (b). Fig. 6. Typical deformation of lateral-torsional buckling experiment for simply supported boundary conditions.
calculated stiffness is listed in Table 4. Similar tests to obtain the lateral stiffness of face-mounted hangers was not possible due to complications associated with the test set-up. 4. Experimental results 4.1. Lateral torsional buckling capacity From the experimental tests it was observed that all tested specimens exhibited lateral deflection almost immediately after the application of the load. A typical buckling mode shape of the I-joist specimens is shown in Fig. 6. Critical buckling moment is measured considering the maximum moment values resisted by the beam at midspan. Fig. 7 shows an example of the moment-displacement curves for the specimens with simply supported boundary conditions. The ultimate buckling moment for these experiments has been taken as the maximum bending force experience at the beam before or at failure. The average results obtained for each category of boundary conditions are presented in Tables 8 and 9 and discuss in Section 6: Discussion. Table 5 presents the critical moment capacities of all the I-joists tested, with different boundary conditions for all sample groups. A discussion of the results is presented in Section 6.
Fig. 5. Joist Hangers lateral rigidity test setup. Table 4 Lateral stiffness for top mounted hangers. Group
Hangers
Stiffness (N/mm)
A B
LBV2.56/16 HB3.56/24 HIT424 HWI424
10.7 38.7 13.7 6.7
5. Finite element model and numerical results 5.1. Model description The orientation of the principal axes of the wood I-joist used for this study is illustrated in Fig. 8. The lateral axis (axis 3) represents the strong bending axis of the joist and the lateral direction vector. Axis 2 represents the height of the section, weak bending axis of the section and also strong bending axis of each flange taken individually. The longitudinal axis (axis 1) represents the length of the joists and the torsional direction of each flange. A commercial finite element software ABAQUS [36], which is
that in lateral-torsional buckling. The lateral stiffness of the top mounted hangers was determined using a universal testing machine by applying a point load at the bottom of the hanger as shown in Fig. 5. A wooden support frame was constructed to allow the assembly to be installed with sufficient torsional and vertical support to resist the applied load. The resulting 333
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20 18
Applied Moment (kN*m)
16 14 12 10 8 B-3
6
B-4 4
B-5 B-6
2
B-7
0 0
20
40
60
80
100
120
140
160
180
200
220
240
Top flange lateral displacement (mm) Fig. 7. Moment-lateral displacement of top flange – simply supported boundary conditions. Table 5 Critical moment capacities of specimens tested (kN*m). Specimen #
Critical moment (kNm) Group A
Critical moment (kNm) Group B
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23
11.2 11.4 10.4 10.1 10.2 10.1 8.5 9.2 8.6 9.5 9.0 9.0 9.3 9.1 10.0 10.1 15.0 15.1 – – – – –
26.3 23.0 18.5 18.8 20.3 19.7 18.8 18.9 17.9 20.2 19.7 18.4 20.9 17.0 15.8 17.8 18.5 21.6 21.3 22.6 23.4 23.4 24.8
Fig. 8. Main axis directions used in the numerical model.
capable of conducting eigenvalue and nonlinear analysis, was used to develop the numerical model. At the nodes joining the web and flange elements, the degrees of freedom of the shell and solid elements are coupled using the ABAQUS feature “SHELL TO SOLID COUPLING”. This feature enables the transfer of the translational degrees of freedom of the web shell elements to the solid elements used for flanges. The web was modeled with S4R shell elements and the flanges with C3D8R solid elements. The meshing of the flanges and web was dependent on the size of the section. The ideal element aspect ratio would be as close as possible to unity. In the present models, the aspect ratio of the flange elements width to height are 1.08 and 1.11 for the 0.6 m and 0.4 m joist sections, respectively. The longitudinal aspect ratio of the elements is 4.0 for both section’s flanges. The aspect ratio of the shell elements used for the web was chosen as 1.0. The material properties used in the FE model were determined from
Table 6 Average input parameters of joist components. Group #
A B
Web parallel modulus of elasticity (MPa)
Bottom
Web transverse modulus of elasticity (MPa)
1070 1225
2123 1647
5308 4118
Flanges longitudinal modulus of elasticity (MPa)
Flanges torsional shear modulus (MPa)
Top
Bottom
Top
12,749 11,550
12,746 11,458
1044 1243
the experimental program described in Section 3.2. Table 6 lists the average mechanical values used as input. The joist specimens were tested experimentally with four distinct 334
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Fig. 9. Boundary conditions for FE analysis. Table 7 Results from critical eigenvalue buckling moment. Sample group
LTB Boundary condition
A
Simply supported Top mounted LBV2.56/16 Enhanced fixity connection Simply supported Top mounted HWI424 HIT424 HB3.56/24 Enhanced fixity connection
B
Hangers/ Connection
Buckling moment (kN*m)
Fig. 11. Initial imperfection of joists in FE model.
10.8 8.1 13.7 20.9 15.4 17.9 19.6 28.7
condition can be described by having lateral supports on the side of the top and bottom flange without restraining the rotation of the flanges (Fig. 9a). To simulate the top mounted hangers, the top flange was considered to be restrained by a lateral support at mid-height and the bottom flange lateral support was represented by a lateral spring (Fig. 9b). The experimental values of spring stiffness used in the analysis for these hangers type are presented in Table 4. As mentioned earlier, face-mounted hangers could not be modelled the same way due to lack of experimental information on the stiffness of the springs. The enhanced connections with rotational fixity were designed to restrain/ rotation of the flanges about the joist’s weak axis (axis 2) and twist
categories of boundary conditions: simply supported, top mounted and face mounted commercial joists hangers as well as hangers with enhanced rotational fixity. All boundary conditions are considered to be supported vertically at each joist extremity. The simply supported
Fig. 10. Critical buckling mode for the boundary types tested. 335
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Fig. 12. Linear and non-linear analysis (Specimen B-4).
software enables the user to include initial imperfections by using a preceding eigenvalue mode and scaling it to match the initial out-ofstraightness of the top flange of the specimen. The initial imperfections incorporated in the numerical model is illustrated in Fig. 11. The out-ofstraightness was measured during the LTB experiment of some specimens tested with simply supported end conditions. The RIKS method uses load increments to calculate the stiffness matrix of the structure. Every analysis required between 10 and 15 automatic load increments with an average of 55 iterations to converge. Fig. 12 shows an example of the lateral displacement of the top flange as a function of the internal moment.
Table 8 Commercial hangers’ ultimate buckling moment. Sample group
Hangers/ Connection
A
* TM LBV2.56/16 ** FM IUS2.56/16 ** FM HU316 Nails in End grain (NLD) * TM HWI424 * TM HIT424 * TM HB3.56/24 ** FM IUS3.56/16 ** FM MIU3.56/16 Nails in End grain (NLD)
B
Average COV % Average COV % Average COV % Average COV % Average COV % Average COV % Average COV % Average COV % Average COV % Average COV %
Buckling moment (kN*m)
Avg buckling moment with SS end conditions (kN*m)
% Difference
9.0 5.1 9.0 0.3 9.2 1.4 10.0 0.63 18.4 3.9 19.7 9.0 19.9 1.7 16.4 5.1 18.2 2.5 21.5 0.9
Average : 10.6 COV : 5.4%
−15.1 −14.5 −12.7 −5.0
Average : 19.8 COV : 8.4%
6. Discussion
−7.3
6.1. Influence of the connector’s rigidity
−0.9
This section discusses experimental results for “realistic” boundary conditions and compares the results to those obtained from idealized simply supported conditions. Joist specimens from each group were tested with end nailing, face mounted or top mounted commercial hangers. The inclusion of such realistic boundary connections generally had negative effect on buckling moments compared with those tested with simply supported end conditions. Comparison between the average buckling moment obtained for all realistic boundary conditions and those obtained from simply-supported conditions are summarized in Table 8. Selected results are also presented graphically in Fig. 13 for Group A specimens and in Fig. 14 for Group B specimens. For specimens tested with the top-mounted hangers, the experimental observations indicated that at the end joints, the top flange could be considered restrained from lateral movement. However, lateral displacement of the joists’ bottom flange was detected for some of the hangers. The displacement was most significant for the LBV2.56/16 and HWI424 hangers. Consistently, little to no lateral movement was observed in the bottom flange of the HIT424 and HB3.56/24 hangers. With the face mounted hangers, the reduction in buckling moment for the IUS2.56/16 and HU316 hangers used with specimen Group A was 14.5% and 12.7%, respectively. For group B specimens tested with the IUS3.56/16 hanger, the reduction was on average 17.3%. Although in the case of face-mounted hangers, the bottom flanges were generally held in place by the hanger side plate and the top flange was not directly supported, stiffeners were used to restrain the joist from laterally translating. For all cases, a maximum reduction in the joists’ buckling capacity of approximately 19% was observed.
0.4 −17.3 −8.4 8.2
* TM :Top Mounted Hangers. ** FM :Face Mounted Hangers.
about the longitudinal axis (axis 1). These rotational restraints for both flanges were modeled by adding a vertical and longitudinal support on each side of the flanges (Fig. 9c). To model the load application, a concentrated load was applied on a single node, positioned at the joist mid-span, on the bottom face of the bottom flange. 5.2. Analysis procedure and results An eigenvalue analysis was conducted to determine the buckling capacity of the I-joist specimens and their buckling mode. Table 7 presents the results for the linear buckling load for each model tested with the various I-joist specimens and end conditions. Fig. 10 shows an example of the critical buckling mode for each category of end conditions tested. The ABAQUS Software uses the modified RIKS method to conduct nonlinear geometric analysis [36]. The iterative procedure used by the 336
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Fig. 13. LTB Critical moment of group A specimens with various boundary conditions.
24
Experimental Buckling Load (kN*m)
23 22
Simply supported
SS_Average and Variation
HWI424
HIT424
HB3.56/24
IUS3.56/16
MIU3.56/16
Nails in End grain
21 20 19 18 17 16 15
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
Specimen # Fig. 14. LTB Critical moment of group B specimens with various boundary conditions. Table 9 Ultimate buckling moment of enhanced connections. Sample group
Hangers/connection
A
Angle Contained Flanges (ACF)
B
Side Screwed (HWR) Angle Contained Flanges (ACF)
Average COV % Average COV % Average COV %
Buckling moment (kN*m)
Avg buckling moment with SS end conditions (kN*m)
% Increase
15.0 0.3 23.0 2.4 24.1 3.9
Average : 10.6 COV : 5.4%
42.4
Average : 19.8 COV : 8.4%
15.8 21.5
When considering the implication of the reduction on design of floor joists, it is noteworthy to mention that the study by Hooley and Madsen [10] had suggested that 15% reduction would be appropriate to use for design when dealing boundary conditions found in construction. It should be noted that the selected value for reduction was not based on testing “realistic” boundary conditions but rather an estimate provided
by the researchers. Although this estimate seems reasonable on average, it is non-conservative in some cases. Based on the test configurations considered in the current study, it is recommended that design provisions for lateral torsional buckling be modified to include a reduction of 20% rather than 15% to account for non-idealized boundary conditions and to ensure better safety. 337
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30 28
Finite Element Numerical Predictions (kN*m)
26 24 22 20 18 16 Group A_SS
14
Group A_TM
12
Group A_NLD
10
Group A_EC
8
Group B_SS
6
Group B_TM Group B_NLD
4
Group B_EC 2
1:1 Ratio
0 0
2
4
6
8
10
12
14
16
18
20
22
24
26
28
30
Experimental Buckling Load (kN*m) Fig. 15. Comparison between numerical predictions and experimental observations for joists with simply-supported and realistic boundary conditions. Table 10 Average moment compared to linear numerical predictions.
Group A
Group B
Span Ratio
L/600
L/360
L/240
L/120
L/60
Displ. (mm) Average critical moment capacity from experiments (kN*m) COV % Numerical solution for simply supported boundary conditions (kN*m) % Difference Displ. (mm) Average critical moment capacity from experiments (kN*m) COV % Numerical solution for simply supported boundary conditions (kN*m) % Difference
7.6 5.3 38.6 10.8 −50.93 9.7 10.1 25.9 20.9 −51.68
12.7 6.1 31.6
19.1 7.4 21.4
38.1 8.3 14.1
76.2 9.0 10.4
−43.52 16.1 12.4 18.8
−31.48 24.1 13.9 14.1
−23.15 48.3 16.1 9.2
−16.67 96.5 17.8 8.6
−40.67
−33.49
−22.97
−14.83
6.2. Proposed measures to enhance connection rigidity
6.3. Experimental vs numerical predictions
In order to enhance the rotational restraint in the weak direction, two connectors were investigated, namely: a modified HWI hanger (HWR) and a connection developed by the authors using two angle brackets. Lateral displacements at the bottom of the HWI hanger was prevented by welding a steel plate on each side of the hanger. The developed bracket connection was intended to create higher level of rotational restraint of the top flange. A summary of the results can be seen in Table 9. The modified HWI connection performed well and enhanced the performance compared to the unmodified HWI hanger. The increase was on average 24.9% compared to the unmodified connection and approximately 15% compared to the reference case with simply supported conditions. The connection with the angle bracket was capable of providing some rotational fixity. An increase of 42% and 21% was achieved for specimen groups A and B, respectively. Little to no rotation about the beam’s weak axis was observed at each end. The developed bracket connection was merely an attempt to show that by using simple considerations for the joist and beam connections, enhancement of the LTB capacity can be achieved rather easily. Although outside the scope of the current research project, development of connections that specifically targets maximizing the LTB capacity would be interesting to investigate.
6.3.1. Simply supported end conditions To validate the numerical model, the buckling moment obtained from specimens tested under simply supported end conditions, for both specimen groups, were compared to the FE model eigenvalue predictions. Material properties obtained from the components tests were used as input. Fig. 15 provides a graphical illustration of the comparison amongst all test specimens and includes cases for simply-supported (SS) boundary conditions. In general, it can be observed that the FE analysis overestimated the buckling moment on average by 4.6% with a COV of 6.1% for group A and slightly underestimated the buckling moment by 0.9% with COV 8.0% for the group B. The variability found in the results can possibly be attributed to the fact that the properties were obtained from specimens that were cut from I-joists tested for full-scale LTB and might have been affected by the loading history. Also, only a portion of the flange length was considered in the component test, however, the joist flange consists of several lumber pieces finger-jointed together.
6.3.2. Model prediction of joists with realistic boundary conditions To compare the numerical predictions with the test results for joists with hangers and nailed configurations, a finite element model where the flanges’ lateral restrains were replaced by springs was developed. 338
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Fig. 16. Example of non-linear load-displacement relationship for simply-supported boundary condition as measured and simulated by the FEA model.
depending on what the design displacement may be selected, the difference in capacity relative to the ultimate elastic capacity could be significant. For example only half of the elastic moment capacity is obtained at a span ratio of L/600. Even if the L/60 limit is adopted, the actual moment capacity is on average 10–15% less than that calculated based on elastic theory. This has significant implication on design and further research in this area is required.
The average material properties were used as input for each components of the joists. The results are shown in Fig. 15. It can be observed that the numerical model yielded results that predict the buckling moment with reasonable accuracy. Only the Group B with enhanced connections is showing greater variation and one possible explanation for this observation could be attributed to the fact that the clamping force used during the installation of the bracket connection was not measured and could have influenced the ultimate moment obtained. More research should be conducted to investigate the effect of the clamping action and how to develop numerical input parameters to predict greater accuracy in the results.
6.4.2. Modeling the nonlinear behaviour This section compares the nonlinear buckling behavior of the joists with the numerical predictions from the FE geometric nonlinear analysis. An example of the comparison is shown in Fig. 16. In general, the FE prediction of nonlinear behavior fits the experimental buckling displacements reasonably well. Some deviation was found between the model results and those obtained experimentally. This deviation could be attributed to measuring the initial imperfection. In general, it seems that knowing the initial out-of-straightness provided sufficient knowledge to be able to conduct a non-linear buckling analysis that would describe the joist behaviour adequately.
6.4. Non-linear lateral torsional buckling behavior 6.4.1. Geometric nonlinear analysis with simply supported end conditions. Although reasonable agreement between the elastic buckling moment analysis and the experimental results was achieved (Fig. 15), the observed behavior of the joists was non-linear and depended primarily on their initial out-of-straightness. Some of the joists reached their ultimate buckling capacity at lateral displacements ranging between 180 and 240 mm. The reported lateral displacement under load may be unacceptable from a serviceability point of view; it is therefore suggested that a limit be imposed on the design for lateral torsional buckling of timber joists and beams. Obviously, such limits depend on several factors which are mostly related to tolerances for non-structural components. Such tolerances are well established for vertical deflection of joists and beams. Whether similar, more or less stringent requirements are needed for lateral displacement is a larger subject than can be adequately addressed in this paper. In an attempt to investigate the implication of various limits, the average capacity curve was compared with the idealized linear solution for L/600, L360, L/240, L/120 and L/60 span ratios. The results are presented in Table 10. Fuller details can be found in [37]. It can be observed that within all the reference points, the lower displacement ratios are shown to have a higher variation in their resistance values, which could be attributed to the variation in the initial out-of-straightness. As the displacement of the top flange increases, the critical moment average associated with each ratio tends to increase and their standard deviation decrease. It can be clearly seen that
7. Conclusions The study’s main conclusions can be summarized as follows: – The experimental study showed that I-joists tested with top mounted hangers and nailed connections did not have significant decrease of the buckling moment when compared to the simply supported boundary condition. However, the I-joists tested with the face mounted hangers exhibited a significantly lesser buckling moment. To take into account the decrease in the buckling moment with the use of realistic boundary conditions, a reduction of 20% is recommended. All enhanced connections tested showed a significant increase of the buckling moment when compared to the simply supported end conditions. – Finite element analysis of the linear eigenvalue buckling moment was found to be in reasonable agreement with the experimental results for the simply supported end conditions as well as realistic boundary conditions. 339
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– The nonlinear behavior of the joists is influenced by the initial imperfections. Experimental results on simply supported I-joists have shown a general reduction in the load applied on the beam, varying from 50% to 10% with initial imperfection ratios ranging from L/ 600 to L/60, respectively.
[18] [19]
[20]
References
[21] [1] FPL, Forest Product Laboratory. Wood handbook: wood as an engineering material. Wisconsin; 2010. [2] Xiao Qiuwu. Lateral torsional buckling of wood beams. Monograph: University of Ottawa; 2014. [3] St-Amour Rémi. Lateral torsional buckling of wood I-joist. University of Ottawa; 2016. [4] St-Amour Rémi, Doudak Ghasan. Experimental and numerical investigation of lateral torsional buckling of wood I-joists. Can J Civ Eng 2017;45:41–50. [5] Thomas HW. Mechanical properties of structural-Grade oriented strand board. Holz als Roh- und Werkstoff 2001;59:405–10. [6] Grandmond JF, Cloutier A, Gendron G, et al. Effect of density on the properties of oriented strandboard web stock used in wood I-joists. For Prod J 2010;60(7–8):592–8. [7] Grandmond JF, Cloutier A, Gendron G, et al. Wood I-joist model sensitivity to oriented standboard web mechanical properties. Wood Fiber Sci 2010;42(3):352–61. [8] Hooley RF, Madsen B. Lateral buckling of glued laminated beams. J Struct Eng Div, ASCE 1964;90:201–303. [9] Zahn JJ. Lateral stability of wood beam-and-deck systems. J Struct Div, ASCE 1973:1391–408. [10] Bell K, Eggen TE. Stability of timber beams and columns. IABSE conference. Lahti: International Association for Bridge and Structural Engineering; 2001. p. 155–62. [11] NS3470-1. Design of timber structures. Design Rules. Part 1: Common rules, 5th ed.; 1999. [12] CEN. Eurocode 5: Design of timber structures - Part 1–1: General rules and rules for buildings. Belgium: Comité Européen de Normalisation, Brussels; 2004. [13] AFPA, American Forest & Paper Association. Designing for lateral torsional buckling in wood members. In: Technical report. American Wood Council, 43. American Forest & Paper Association; 2003. [14] Hindman DP, Manbeck HB, Janowiak JJ. Measurement and prediction of lateral torsional buckling loads of composite wood materials: Rectangular sections. For Prod J 2005;55:42–7. [15] Baláž I. Lateral Torsional buckling of timber beams. Wood Res 2005;50(1):51–8. [16] Baláž I, Koleková Y. Resistance of timber beams to out-of-plane buckling. Zborník VI. Sympózia Devro v stavebných konštrukciách so zahraničnou účast’ou 2004. 2829.10.2004. Str.35-42. [17] Baláž I, Koleková Y. Lateral Torsional buckling of timber beams. 6th International
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Contents lists available at ScienceDirect
Engineering Structures journal homepage: www.elsevier.com/locate/engstruct
Investigation of the lateral-torsional buckling behaviour of engineered wood I-joists with varying end conditions B. Pelletier, G. Doudak
T
⁎
Department of Civil Engineering, University of Ottawa, K1N 6N5, Canada
A R T I C LE I N FO
A B S T R A C T
Keywords: Wood I-joists Lateral torsional buckling Full scale testing Lateral restraints
Beam members, especially those that have large depth/width ratios and long spans, are prone to lateral-torsional buckling as a possible mode of failure. Laboratory testing rarely takes into account actual end conditions and initial imperfection which might have a significant impact on the buckling behavior of beams. The current research project aims to investigate the lateral-torsional buckling of wooden I-joists with realistic boundary conditions used in construction. A total of 41 joists were tested using various commercial joist hangers and enhanced connections to represent different support conditions. A numerical 3D model was also developed using commercially available finite element program ABAQUS to determine the buckling loads and associated mode shapes of joists similar to those tested. Based on the results from the current study it is recommended that 20% reduction in the critical moment capacity be considered in order to take into account the use of joist hangers typically used in construction. Finite element analysis of the linear eigenvalue buckling load was found to be in reasonable agreement with the experimental results. The nonlinear behavior of the joists was found to be influenced by the initial imperfections. The ultimate critical load was attained at large lateral displacement of the joists due to their non-linear behaviour. This observation could have a significant implication on design and should be investigated further.
1. Introduction 1.1. General The development of engineered wood products, such as wood I-joists, has meant that structural bending members that are longer and deeper can be achieved without significantly increasing the overall building weight. However, long-span beams that have large depth/ width ratios are more prone to buckling failure. Similar to solid sawn joists, engineered wood I-joists are typically sheathed with wood panels bracing them at the top face and partially restraining the lateral displacement of the top flange. However, during or after construction, several conditions might lead to the wood I-joists not being adequately braced. Such cases include: workers walking on floor or roof joists during installation, wind uplift governing the design of a roof, and continuous or cantilever beams with negative moments. Established knowledge on the behavior of beams to lateral torsional buckling failure has addressed cases involving idealized boundary conditions. However, practical construction often involves non-ideal boundary conditions. For example, a variety of metallic hangers and connection details are used in construction, and the effect of such
⁎
connections has been investigated very scarcely. In this context, the current research aims at investigating the effect of such “realistic” boundary conditions on the lateral stability of the engineer wood Ijoists as well as assessing the ability of numerical models to predict the buckling behavior. 1.2. Constitutive behaviour of wood The mechanical properties in the tangential and radial axis are very close for a large array of species and therefore are commonly assumed to be identical [1]. Although wood can be modelled as an orthotropic material, not all the material properties have an important role in the lateral torsional buckling resistance. Xiao [2] conducted a sensitivity analysis on rectangular solid sawn lumber and concluded that the modulus of elasticity along the longitudinal direction (EL ) and the transverse shear modulus (GT ) are the two variables that have the strongest influence on the lateral torsional buckling capacity. St-Amour [3,4] presented a sensitivity analysis on the mechanical properties of wood I-joists and it was revealed that the most influential mechanical properties are the longitudinal modulus of elasticity and the torsional shear modulus of the compression flange. The longitudinal modulus of
Corresponding author at: Civil Engineering, University of Ottawa, 161 Louis-Pasteur, Room A115 (CBY), Ottawa, ON K1N 6N5, Canada. E-mail address: [email protected] (G. Doudak).
https://doi.org/10.1016/j.engstruct.2019.03.003 Received 27 November 2018; Received in revised form 18 February 2019; Accepted 1 March 2019 Available online 06 March 2019 0141-0296/ © 2019 Elsevier Ltd. All rights reserved.
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elasticity and the torsional shear modulus of the tension flange, and the transverse modulus of elasticity of the web also impact the buckling response but to a lesser degree. Studies have also investigated the material properties of OSB, and their effect on the lateral torsional buckling capacity of wood I-joists [3–7].
a sensitivity analysis on the effect of orthotropic material properties on the lateral torsional buckling behavior of rectangular timber beams. The study showed that the numerical model developed was mostly affected by variation in the longitudinal modulus of elasticity and the transverse shear modulus of the timber member. Xiao et al. [20] conducted full-scale lateral torsional buckling tests on rectangular wood joist elements. Numerical buckling analysis was conducted and validated using full-scale joist test results. Du [21] and Du et al. [22] studied the behavior of wooden beam-deck systems, where models for sway and non-sway systems were formulated with analytical and energy-based solutions. In general, the non-sway model consistently predicted critical moments higher than those based on the sway model. The effects of bracing action of the deck board, the deck span between beams and the span type (continuous or single span) were investigated and discussed. Hu et al. [23,24] investigated the effect of bracing height on wooden beams subjected to lateral-torsional buckling. Energy based solutions were formulated for a simply supported beam with rigid or flexible bracing.
1.3. Research goals The current research aims at investigating the lateral torsional buckling behavior of engineered wood I-joists with idealized pinned ends as well as “realistic” boundary conditions using metallic joist hangers. The study also aims at investigating the effect of initial imperfection on the buckling behavior of wood I-joists. A finite element model was developed and validated by comparing the results against those obtained from full-scale tests. 2. Review of relevant literature Several studies have been conducted on the lateral torsional buckling instability of rectangular timber beams and engineered wood Ijoists products. Some studies investigated the ability of various code design provisions to estimate the buckling load, while others focused on the bracing action in order to prevent buckling failures. The topic of loading height has been investigated numerically in a number of studies, however, since this topic is not directly related to the present study, it was not covered in details. Very limited number of studies considered the impact of realistic boundary conditions on the LTB capacity of the beams. The initial geometric imperfection and nonlinear displacement of I-joists has not been adequately investigated in the literature. The present study aims to investigate the nonlinear behavior influence on the resistance during the entire displacement history of the joists.
2.2. Research on engineered wood I-joists Hindman et al. [25] investigated the material properties of structural composite lumber products using isotropic and orthotropic material properties. The torsional stiffness was also computed using mathematical models based on isotropic and orthotropic formulations. Hindman et al. [26] compared the buckling loads for two types of wood I-joists specimens of various lengths, with flanges constructed of structural composite lumber. It was found that both the design equation [27] and the elastic beam buckling prediction [28] provided conservative estimates when compared to the experimental wood I-joist buckling loads. Zhu et al. [29] studied the buckling behavior of wood Ijoists by comparing experimental tests to finite element simulations. Geometric imperfections were introduced in the model to conduct nonlinear buckling analysis. Burow et al. [30] performed lateral torsional buckling tests on cantilevered and simply supported wood I-joist specimens. Three mathematical models were developed based on elastic stability solution [28] as well as the American codes provisions [27,31]. It was shown that both the American timber design standard [31] and the modified Euler model consistently underestimated the buckling load. Villasenor [32] investigated the lateral buckling behavior of wood I-joists subjected to human walking loads. The author concluded that the buckling loads were influenced by the overall lateral stiffness of the end support conditions. St-Amour et al. [3,4] investigated the effect of the material properties of wood I-joists on the lateral torsional buckling behavior. Full-scale LTB experiments were conducted by subjecting wood I-joist with various span length and simply supported end conditions to a concentrate point load located on the top flange at midspan. The authors concluded that the lateral bending and rotational stiffness observed from whole joists were adequate material input in the numerical model to describe the joist behaviour. The FE linear and nonlinear predictions were found to be in reasonable agreement with the experimental buckling loads and displacements.
2.1. Research on solid sawn wooden beams Hooley and Madsen [8] conducted an experimental study on gluedlaminated beams with the objective to develop simple code provisions. The experimental results correlated well with the theory and the study concluded that the slenderness ratio governed the lateral stability of the beams. To account for the flexibility of the support in realistic conditions, a 15% reduction of the LTB load was suggested. Zahn [9] formulated the governing equilibrium conditions for wooden rectangular beams with continuous lateral bracing along the beam’s main axis through energy-based solutions. Bell and Eggen [10] investigated the capacities of beam-columns subjected to lateral instability as evaluated by the Norwegian [11] and European timber codes [12]. Linear and non-linear buckling models were developed, using simply supported rectangular wood beams. A reduction of 10% for the critical axial force and bending moment buckling loads was observed for the non-linear buckling analysis. AFPA [13] released a technical report to address the design of timber members prone to instability failure. The effective length and equivalent moment factor approaches were outlined and compared. Hindman et al. [14] studied the lateral torsional buckling behavior of rectangular cantilever beams composed of various structural composite lumber products. The elastic constant ratios and torsional rigidity of the structural composite lumber material resulted in a more constant buckling behavior than what was observed with solid sawn lumber members. Baláž [15] investigated the LTB of timber beams and proposed an expression to calculate the elastic critical moments. Baláž and Koleková [16,17] provided a generalized critical moment framework for critical moment expressions of beams with mono-symmetric cross-sections. Suryoatmono and Tjondro [18] performed lateral torsional buckling tests on simply supported rectangular wooden beams. Finite element models were also developed using isotropic and orthotropic material properties. The classical solution used was based on isotropic material and neglected warping. Xiao et al. [19] conducted
3. Experimental program 3.1. Specimen description and test setup A total of 41 I-joist specimens were tested, 18 of which consisted of 0.4 m deep and 4.6 m long joists (Group A) and 23 consisted of 0.6 m deep and 5.8 m long joists (Group B). The joists were manufactured using machine stress rated (MSR) lumber flanges and oriented strand board (OSB) webs connected together with a waterproof structural adhesive [33]. The geometry of the tested specimens is shown in Fig. 1. The specimens were stored in the laboratory so their temperature and relative humidity were acclimatized to the lab environment. The average moisture content for all joists was approximately 14% with a 330
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Table 2 Group A specimen names and boundary conditions. Sample group
Specimens #
Boundary condition
Hangers/Connection
A
1–6 7–10 11–12 13–14 15–16 17–18
Simply supported Top mounted Face mounted Face mounted Nailed Enhanced connection
LBV2.56/16 IUS2.56/16 HU316 Nails in End grain Angle
Table 3 Group B specimen names and boundary conditions.
Group A
Sample group
Specimen #
Boundary condition
Hangers/Connection
B
1–7 8–9 10–11 12–13 14–15 16–17 18–19 20–21 22–23
Simply supported Top mounted Top mounted Top mounted Face mounted Face mounted Nailed Enhanced connection Enhanced connection
HWI424 HB3.56/24 HIT424 IUS3.56/16 MIU3.56/16 Nails in End grain Side screwed Angle
Group B
Fig. 1. I-joists specimens’ cross-section geometry (all dimensions are in mm). Table 1 I-joists hanger types.
Face mounted Top mounted
Group A
Group B
IUS2.56/16 HU316 LBV2.56/16
MIU3.56/20 IUS3.56/16 HIT424 HWI424 HB3.56/24
Fig. 2 shows the types of hangers used. Tables 2 and 3 provides details on the joist specimens and their boundary conditions. The lateral-torsional buckling capacity was evaluated by applying a concentrated load at mid-span of the specimen’s bottom flange, as illustrated in Fig. 3. The load was applied using a cable and pulley system developed by the authors specifically for this project. One end of the cable was attached to an HSS frame anchored to the lab’s strong floor while the other was attached to a hydraulic jack via a load cell to record the tension force in the cable. The pulley was attached to the bottom flange by a series of shackles acting as hinges. This setup enabled continuous application of the vertical load while also allowing the bottom flange to displace freely. As the joist displaces laterally and twists, the angle of the cable will vary and adjustment to the vertical load must be included in the analysis. This was possible because the geometry of the test setup was fully known at any given time during the test. The vertical displacement, lateral displacement of the top and bottom flanges at mid-span were recorded using wire sensors. The angles of rotation of the top and bottom flanges were recorded using inclinometers.
coefficient of variation (COV) of 5.4%. Each joist end was supported by an elevated wood support, at which the chosen boundary conditions were constructed. Once the joist was installed in the test setup, its initial out-of-straightness was documented using a self-leveling laser. The end supports were divided in 3 categories depending on the boundary conditions to be tested: simply supported, commercially available joists hangers and enhanced fixity connections. Simpsons Strong-Tie products were selected due to their wide availability in the North American market. The hangers were classified into 2 categories; namely face-mounted and top-mounted hangers. Table 1 describes the hangers used for each joist section and
Fig. 2. Hanger type tested. 331
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Fig. 3. Schematic of the test setup showing elevation view and attachment details.
rotation and translation. The top flange was secured in place using two 76 mm × 76 mm × 6 mm angles on each panel and by gripping the flange between the angles using two steel threaded rods, as illustrated in Fig. 4b.
Three designs were evaluated to investigate the effect of enhanced fixity. The first design involved nailing the joists directly to the wooden rim-board [33]. The second design involved a modification to the HWI424 top mounted hanger. To prevent any lateral displacements, a steel plate was welded on each side of the hanger and aligned with the bottom flange. To reduce the rotation relative to the weak axis of the section, six holes (three on each sides) were installed in the hanger and aligned with each flange. 38 mm long screws were used to connect the hanger to the joists (Fig. 4a). The third connection was intended to create high level of fixity on the rotational degree of freedom of the top flange along its longitudinal axis. The bottom flange was therefore attached to the support by two 50 mm × 50 mm × 6 mm angles at each end. Six 38 mm screws (three per flange side) were used to limit the
3.2. Component and material level testing The web’s transverse modulus of elasticity and the flanges torsional modulus were determined based on standard test methods in conformance with ASTM [34] and [35], respectively. The flanges were tested to determine their longitudinal modulus of elasticity in accordance with the test method describe in [35]. The flange was bent about its strong axis in order to simulate the same bending action as 332
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Fig. 4. Enhanced connections using HWR with screwed flanges connection (a) and angles contained flanges connection (b). Fig. 6. Typical deformation of lateral-torsional buckling experiment for simply supported boundary conditions.
calculated stiffness is listed in Table 4. Similar tests to obtain the lateral stiffness of face-mounted hangers was not possible due to complications associated with the test set-up. 4. Experimental results 4.1. Lateral torsional buckling capacity From the experimental tests it was observed that all tested specimens exhibited lateral deflection almost immediately after the application of the load. A typical buckling mode shape of the I-joist specimens is shown in Fig. 6. Critical buckling moment is measured considering the maximum moment values resisted by the beam at midspan. Fig. 7 shows an example of the moment-displacement curves for the specimens with simply supported boundary conditions. The ultimate buckling moment for these experiments has been taken as the maximum bending force experience at the beam before or at failure. The average results obtained for each category of boundary conditions are presented in Tables 8 and 9 and discuss in Section 6: Discussion. Table 5 presents the critical moment capacities of all the I-joists tested, with different boundary conditions for all sample groups. A discussion of the results is presented in Section 6.
Fig. 5. Joist Hangers lateral rigidity test setup. Table 4 Lateral stiffness for top mounted hangers. Group
Hangers
Stiffness (N/mm)
A B
LBV2.56/16 HB3.56/24 HIT424 HWI424
10.7 38.7 13.7 6.7
5. Finite element model and numerical results 5.1. Model description The orientation of the principal axes of the wood I-joist used for this study is illustrated in Fig. 8. The lateral axis (axis 3) represents the strong bending axis of the joist and the lateral direction vector. Axis 2 represents the height of the section, weak bending axis of the section and also strong bending axis of each flange taken individually. The longitudinal axis (axis 1) represents the length of the joists and the torsional direction of each flange. A commercial finite element software ABAQUS [36], which is
that in lateral-torsional buckling. The lateral stiffness of the top mounted hangers was determined using a universal testing machine by applying a point load at the bottom of the hanger as shown in Fig. 5. A wooden support frame was constructed to allow the assembly to be installed with sufficient torsional and vertical support to resist the applied load. The resulting 333
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20 18
Applied Moment (kN*m)
16 14 12 10 8 B-3
6
B-4 4
B-5 B-6
2
B-7
0 0
20
40
60
80
100
120
140
160
180
200
220
240
Top flange lateral displacement (mm) Fig. 7. Moment-lateral displacement of top flange – simply supported boundary conditions. Table 5 Critical moment capacities of specimens tested (kN*m). Specimen #
Critical moment (kNm) Group A
Critical moment (kNm) Group B
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23
11.2 11.4 10.4 10.1 10.2 10.1 8.5 9.2 8.6 9.5 9.0 9.0 9.3 9.1 10.0 10.1 15.0 15.1 – – – – –
26.3 23.0 18.5 18.8 20.3 19.7 18.8 18.9 17.9 20.2 19.7 18.4 20.9 17.0 15.8 17.8 18.5 21.6 21.3 22.6 23.4 23.4 24.8
Fig. 8. Main axis directions used in the numerical model.
capable of conducting eigenvalue and nonlinear analysis, was used to develop the numerical model. At the nodes joining the web and flange elements, the degrees of freedom of the shell and solid elements are coupled using the ABAQUS feature “SHELL TO SOLID COUPLING”. This feature enables the transfer of the translational degrees of freedom of the web shell elements to the solid elements used for flanges. The web was modeled with S4R shell elements and the flanges with C3D8R solid elements. The meshing of the flanges and web was dependent on the size of the section. The ideal element aspect ratio would be as close as possible to unity. In the present models, the aspect ratio of the flange elements width to height are 1.08 and 1.11 for the 0.6 m and 0.4 m joist sections, respectively. The longitudinal aspect ratio of the elements is 4.0 for both section’s flanges. The aspect ratio of the shell elements used for the web was chosen as 1.0. The material properties used in the FE model were determined from
Table 6 Average input parameters of joist components. Group #
A B
Web parallel modulus of elasticity (MPa)
Bottom
Web transverse modulus of elasticity (MPa)
1070 1225
2123 1647
5308 4118
Flanges longitudinal modulus of elasticity (MPa)
Flanges torsional shear modulus (MPa)
Top
Bottom
Top
12,749 11,550
12,746 11,458
1044 1243
the experimental program described in Section 3.2. Table 6 lists the average mechanical values used as input. The joist specimens were tested experimentally with four distinct 334
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Fig. 9. Boundary conditions for FE analysis. Table 7 Results from critical eigenvalue buckling moment. Sample group
LTB Boundary condition
A
Simply supported Top mounted LBV2.56/16 Enhanced fixity connection Simply supported Top mounted HWI424 HIT424 HB3.56/24 Enhanced fixity connection
B
Hangers/ Connection
Buckling moment (kN*m)
Fig. 11. Initial imperfection of joists in FE model.
10.8 8.1 13.7 20.9 15.4 17.9 19.6 28.7
condition can be described by having lateral supports on the side of the top and bottom flange without restraining the rotation of the flanges (Fig. 9a). To simulate the top mounted hangers, the top flange was considered to be restrained by a lateral support at mid-height and the bottom flange lateral support was represented by a lateral spring (Fig. 9b). The experimental values of spring stiffness used in the analysis for these hangers type are presented in Table 4. As mentioned earlier, face-mounted hangers could not be modelled the same way due to lack of experimental information on the stiffness of the springs. The enhanced connections with rotational fixity were designed to restrain/ rotation of the flanges about the joist’s weak axis (axis 2) and twist
categories of boundary conditions: simply supported, top mounted and face mounted commercial joists hangers as well as hangers with enhanced rotational fixity. All boundary conditions are considered to be supported vertically at each joist extremity. The simply supported
Fig. 10. Critical buckling mode for the boundary types tested. 335
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Fig. 12. Linear and non-linear analysis (Specimen B-4).
software enables the user to include initial imperfections by using a preceding eigenvalue mode and scaling it to match the initial out-ofstraightness of the top flange of the specimen. The initial imperfections incorporated in the numerical model is illustrated in Fig. 11. The out-ofstraightness was measured during the LTB experiment of some specimens tested with simply supported end conditions. The RIKS method uses load increments to calculate the stiffness matrix of the structure. Every analysis required between 10 and 15 automatic load increments with an average of 55 iterations to converge. Fig. 12 shows an example of the lateral displacement of the top flange as a function of the internal moment.
Table 8 Commercial hangers’ ultimate buckling moment. Sample group
Hangers/ Connection
A
* TM LBV2.56/16 ** FM IUS2.56/16 ** FM HU316 Nails in End grain (NLD) * TM HWI424 * TM HIT424 * TM HB3.56/24 ** FM IUS3.56/16 ** FM MIU3.56/16 Nails in End grain (NLD)
B
Average COV % Average COV % Average COV % Average COV % Average COV % Average COV % Average COV % Average COV % Average COV % Average COV %
Buckling moment (kN*m)
Avg buckling moment with SS end conditions (kN*m)
% Difference
9.0 5.1 9.0 0.3 9.2 1.4 10.0 0.63 18.4 3.9 19.7 9.0 19.9 1.7 16.4 5.1 18.2 2.5 21.5 0.9
Average : 10.6 COV : 5.4%
−15.1 −14.5 −12.7 −5.0
Average : 19.8 COV : 8.4%
6. Discussion
−7.3
6.1. Influence of the connector’s rigidity
−0.9
This section discusses experimental results for “realistic” boundary conditions and compares the results to those obtained from idealized simply supported conditions. Joist specimens from each group were tested with end nailing, face mounted or top mounted commercial hangers. The inclusion of such realistic boundary connections generally had negative effect on buckling moments compared with those tested with simply supported end conditions. Comparison between the average buckling moment obtained for all realistic boundary conditions and those obtained from simply-supported conditions are summarized in Table 8. Selected results are also presented graphically in Fig. 13 for Group A specimens and in Fig. 14 for Group B specimens. For specimens tested with the top-mounted hangers, the experimental observations indicated that at the end joints, the top flange could be considered restrained from lateral movement. However, lateral displacement of the joists’ bottom flange was detected for some of the hangers. The displacement was most significant for the LBV2.56/16 and HWI424 hangers. Consistently, little to no lateral movement was observed in the bottom flange of the HIT424 and HB3.56/24 hangers. With the face mounted hangers, the reduction in buckling moment for the IUS2.56/16 and HU316 hangers used with specimen Group A was 14.5% and 12.7%, respectively. For group B specimens tested with the IUS3.56/16 hanger, the reduction was on average 17.3%. Although in the case of face-mounted hangers, the bottom flanges were generally held in place by the hanger side plate and the top flange was not directly supported, stiffeners were used to restrain the joist from laterally translating. For all cases, a maximum reduction in the joists’ buckling capacity of approximately 19% was observed.
0.4 −17.3 −8.4 8.2
* TM :Top Mounted Hangers. ** FM :Face Mounted Hangers.
about the longitudinal axis (axis 1). These rotational restraints for both flanges were modeled by adding a vertical and longitudinal support on each side of the flanges (Fig. 9c). To model the load application, a concentrated load was applied on a single node, positioned at the joist mid-span, on the bottom face of the bottom flange. 5.2. Analysis procedure and results An eigenvalue analysis was conducted to determine the buckling capacity of the I-joist specimens and their buckling mode. Table 7 presents the results for the linear buckling load for each model tested with the various I-joist specimens and end conditions. Fig. 10 shows an example of the critical buckling mode for each category of end conditions tested. The ABAQUS Software uses the modified RIKS method to conduct nonlinear geometric analysis [36]. The iterative procedure used by the 336
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Fig. 13. LTB Critical moment of group A specimens with various boundary conditions.
24
Experimental Buckling Load (kN*m)
23 22
Simply supported
SS_Average and Variation
HWI424
HIT424
HB3.56/24
IUS3.56/16
MIU3.56/16
Nails in End grain
21 20 19 18 17 16 15
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
Specimen # Fig. 14. LTB Critical moment of group B specimens with various boundary conditions. Table 9 Ultimate buckling moment of enhanced connections. Sample group
Hangers/connection
A
Angle Contained Flanges (ACF)
B
Side Screwed (HWR) Angle Contained Flanges (ACF)
Average COV % Average COV % Average COV %
Buckling moment (kN*m)
Avg buckling moment with SS end conditions (kN*m)
% Increase
15.0 0.3 23.0 2.4 24.1 3.9
Average : 10.6 COV : 5.4%
42.4
Average : 19.8 COV : 8.4%
15.8 21.5
When considering the implication of the reduction on design of floor joists, it is noteworthy to mention that the study by Hooley and Madsen [10] had suggested that 15% reduction would be appropriate to use for design when dealing boundary conditions found in construction. It should be noted that the selected value for reduction was not based on testing “realistic” boundary conditions but rather an estimate provided
by the researchers. Although this estimate seems reasonable on average, it is non-conservative in some cases. Based on the test configurations considered in the current study, it is recommended that design provisions for lateral torsional buckling be modified to include a reduction of 20% rather than 15% to account for non-idealized boundary conditions and to ensure better safety. 337
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30 28
Finite Element Numerical Predictions (kN*m)
26 24 22 20 18 16 Group A_SS
14
Group A_TM
12
Group A_NLD
10
Group A_EC
8
Group B_SS
6
Group B_TM Group B_NLD
4
Group B_EC 2
1:1 Ratio
0 0
2
4
6
8
10
12
14
16
18
20
22
24
26
28
30
Experimental Buckling Load (kN*m) Fig. 15. Comparison between numerical predictions and experimental observations for joists with simply-supported and realistic boundary conditions. Table 10 Average moment compared to linear numerical predictions.
Group A
Group B
Span Ratio
L/600
L/360
L/240
L/120
L/60
Displ. (mm) Average critical moment capacity from experiments (kN*m) COV % Numerical solution for simply supported boundary conditions (kN*m) % Difference Displ. (mm) Average critical moment capacity from experiments (kN*m) COV % Numerical solution for simply supported boundary conditions (kN*m) % Difference
7.6 5.3 38.6 10.8 −50.93 9.7 10.1 25.9 20.9 −51.68
12.7 6.1 31.6
19.1 7.4 21.4
38.1 8.3 14.1
76.2 9.0 10.4
−43.52 16.1 12.4 18.8
−31.48 24.1 13.9 14.1
−23.15 48.3 16.1 9.2
−16.67 96.5 17.8 8.6
−40.67
−33.49
−22.97
−14.83
6.2. Proposed measures to enhance connection rigidity
6.3. Experimental vs numerical predictions
In order to enhance the rotational restraint in the weak direction, two connectors were investigated, namely: a modified HWI hanger (HWR) and a connection developed by the authors using two angle brackets. Lateral displacements at the bottom of the HWI hanger was prevented by welding a steel plate on each side of the hanger. The developed bracket connection was intended to create higher level of rotational restraint of the top flange. A summary of the results can be seen in Table 9. The modified HWI connection performed well and enhanced the performance compared to the unmodified HWI hanger. The increase was on average 24.9% compared to the unmodified connection and approximately 15% compared to the reference case with simply supported conditions. The connection with the angle bracket was capable of providing some rotational fixity. An increase of 42% and 21% was achieved for specimen groups A and B, respectively. Little to no rotation about the beam’s weak axis was observed at each end. The developed bracket connection was merely an attempt to show that by using simple considerations for the joist and beam connections, enhancement of the LTB capacity can be achieved rather easily. Although outside the scope of the current research project, development of connections that specifically targets maximizing the LTB capacity would be interesting to investigate.
6.3.1. Simply supported end conditions To validate the numerical model, the buckling moment obtained from specimens tested under simply supported end conditions, for both specimen groups, were compared to the FE model eigenvalue predictions. Material properties obtained from the components tests were used as input. Fig. 15 provides a graphical illustration of the comparison amongst all test specimens and includes cases for simply-supported (SS) boundary conditions. In general, it can be observed that the FE analysis overestimated the buckling moment on average by 4.6% with a COV of 6.1% for group A and slightly underestimated the buckling moment by 0.9% with COV 8.0% for the group B. The variability found in the results can possibly be attributed to the fact that the properties were obtained from specimens that were cut from I-joists tested for full-scale LTB and might have been affected by the loading history. Also, only a portion of the flange length was considered in the component test, however, the joist flange consists of several lumber pieces finger-jointed together.
6.3.2. Model prediction of joists with realistic boundary conditions To compare the numerical predictions with the test results for joists with hangers and nailed configurations, a finite element model where the flanges’ lateral restrains were replaced by springs was developed. 338
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Fig. 16. Example of non-linear load-displacement relationship for simply-supported boundary condition as measured and simulated by the FEA model.
depending on what the design displacement may be selected, the difference in capacity relative to the ultimate elastic capacity could be significant. For example only half of the elastic moment capacity is obtained at a span ratio of L/600. Even if the L/60 limit is adopted, the actual moment capacity is on average 10–15% less than that calculated based on elastic theory. This has significant implication on design and further research in this area is required.
The average material properties were used as input for each components of the joists. The results are shown in Fig. 15. It can be observed that the numerical model yielded results that predict the buckling moment with reasonable accuracy. Only the Group B with enhanced connections is showing greater variation and one possible explanation for this observation could be attributed to the fact that the clamping force used during the installation of the bracket connection was not measured and could have influenced the ultimate moment obtained. More research should be conducted to investigate the effect of the clamping action and how to develop numerical input parameters to predict greater accuracy in the results.
6.4.2. Modeling the nonlinear behaviour This section compares the nonlinear buckling behavior of the joists with the numerical predictions from the FE geometric nonlinear analysis. An example of the comparison is shown in Fig. 16. In general, the FE prediction of nonlinear behavior fits the experimental buckling displacements reasonably well. Some deviation was found between the model results and those obtained experimentally. This deviation could be attributed to measuring the initial imperfection. In general, it seems that knowing the initial out-of-straightness provided sufficient knowledge to be able to conduct a non-linear buckling analysis that would describe the joist behaviour adequately.
6.4. Non-linear lateral torsional buckling behavior 6.4.1. Geometric nonlinear analysis with simply supported end conditions. Although reasonable agreement between the elastic buckling moment analysis and the experimental results was achieved (Fig. 15), the observed behavior of the joists was non-linear and depended primarily on their initial out-of-straightness. Some of the joists reached their ultimate buckling capacity at lateral displacements ranging between 180 and 240 mm. The reported lateral displacement under load may be unacceptable from a serviceability point of view; it is therefore suggested that a limit be imposed on the design for lateral torsional buckling of timber joists and beams. Obviously, such limits depend on several factors which are mostly related to tolerances for non-structural components. Such tolerances are well established for vertical deflection of joists and beams. Whether similar, more or less stringent requirements are needed for lateral displacement is a larger subject than can be adequately addressed in this paper. In an attempt to investigate the implication of various limits, the average capacity curve was compared with the idealized linear solution for L/600, L360, L/240, L/120 and L/60 span ratios. The results are presented in Table 10. Fuller details can be found in [37]. It can be observed that within all the reference points, the lower displacement ratios are shown to have a higher variation in their resistance values, which could be attributed to the variation in the initial out-of-straightness. As the displacement of the top flange increases, the critical moment average associated with each ratio tends to increase and their standard deviation decrease. It can be clearly seen that
7. Conclusions The study’s main conclusions can be summarized as follows: – The experimental study showed that I-joists tested with top mounted hangers and nailed connections did not have significant decrease of the buckling moment when compared to the simply supported boundary condition. However, the I-joists tested with the face mounted hangers exhibited a significantly lesser buckling moment. To take into account the decrease in the buckling moment with the use of realistic boundary conditions, a reduction of 20% is recommended. All enhanced connections tested showed a significant increase of the buckling moment when compared to the simply supported end conditions. – Finite element analysis of the linear eigenvalue buckling moment was found to be in reasonable agreement with the experimental results for the simply supported end conditions as well as realistic boundary conditions. 339
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– The nonlinear behavior of the joists is influenced by the initial imperfections. Experimental results on simply supported I-joists have shown a general reduction in the load applied on the beam, varying from 50% to 10% with initial imperfection ratios ranging from L/ 600 to L/60, respectively.
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