- Email: [email protected]
Numerical and experimental investigation of lateral torsional buckling of wood beams
Engineering Structures 151 (2017) 85–92
Contents lists available at ScienceDirect
Engineering Structures journal homepage: www.elsevier.com/locate/engstruct
Numerical and experimental investigation of lateral torsional buckling of wood beams Q. Xiao, G. Doudak ⇑, M. Mohareb Department of Civil Engineering, University of Ottawa, K1N 6N5, Canada
a r t i c l e
i n f o
Article history: Received 13 April 2017 Revised 27 June 2017 Accepted 11 August 2017
Keywords: Wood beams Lateral torsional buckling Finite element Full scale testing Lateral restraint Load position effect Bracing height Eurocode
a b s t r a c t The present study reports the results of a full-scale experimental and numerical investigation aimed at predicting the elastic lateral torsional buckling capacity of wooden beams. The experimental component consists of 18 Spruce-Pine-Fir (SPF) No. 1/No. 2 grade lumber joists consisting of five 38 mm 184 mm 4200 mm, six 38 mm 235 mm 3600 mm, and seven 38 mm 286 mm 4200 mm specimens. For each specimen, the shear and longitudinal elastic moduli are first determined experimentally through non-destructive tests. A full-scale bending test is then conducted on each specimen, to determine its elastic lateral torsional buckling resistance. A 3D finite element model is developed to predict the lateral torsional buckling resistance for each specimen based on the experimentally determined shear and longitudinal elastic moduli. The validity of the finite element analysis is assessed through comparisons with fullscale test results. The validated model was used to assess the Eurocode provisions and it was found that for simply supported end conditions the code equation seemed reasonable and slightly conservative. However, for cantilevered beams, the Eurocode provisions seem to be overly conservative for the case of bottom edge loading and non-conservative for the case of top edge loading. Changes have been proposed to the wording of the effective length adjustment and the results based on the revised definition provides critical moment predictions that are conservative and more consistent for cantilevers under top and bottom edge loading. Ó 2017 Published by Elsevier Ltd.
1. Objective 1.1. General Timber design standards and guidelines (e.g., [1–4]) recognize lateral torsional buckling as a likely governing failure mode when designing long span laterally unsupported wooden beams. Guidelines in [2] provide estimates for the lateral torsional buckling resistance, and account for common loading patterns, load height effect relative to the section centroid (i.e., load eccentricity), and presence of bracing. The current study is motivated by the lack of consistency and transparency in some of the design standards. For example, although the European standard [1] and the AFPA approach [2] both include the modulus of Elasticity (E) and shear modulus (G) in the critical moment expression, the Canadian standard [3] assumes a fixed value for the ratio of E/G. Also, whereas, the AFPA model [2] specifically includes a factor accounting for partial twist fixity condition that may occur in practical situations, the Canadian approach [3] embed the factor directly in the buck⇑ Corresponding author at: Civil Engineering, University of Ottawa, 161 Louis-Pasteur, Room A115 (CBY), Ottawa, ON K1N 6N5, Canada. E-mail addresses: [email protected] (Q. Xiao), [email protected] (G. Doudak), [email protected] (M. Mohareb). http://dx.doi.org/10.1016/j.engstruct.2017.08.020 0141-0296/Ó 2017 Published by Elsevier Ltd.
ling equation. The current paper aims to bring clarity and possibly consistency between the various approaches by understanding and highlighting the differences between the various standards through experimental and numerical analysis. A 3-D Finite Element Analysis (FEA) model that is able to reliably represent the experimental test results, is developed. In order to assess the reliability of the FEA model, the present study first reports a recent experimental investigation, involving both material testing and full-scale lateral torsional bucking tests on simply supported beams with no intermediate lateral bracing. The parameters obtained from material tests are then input into the FEA model and the buckling capacity obtained from the FEA model is compared with the full-scale beam buckling test results. The validated FEA model is then used to expand the scope of the experimental tests by investigating the accuracy of the lateral torsional buckling provisions in the current Eurocode [1] where comparison between Eurocode and FEA model is made for different boundary and loading conditions.
1.2. Constitutive behaviour of wood Wood can be considered as a cylindrical orthotropic material [5] with three orthogonal directions: longitudinal, radial, and tangential
86
Q. Xiao et al. / Engineering Structures 151 (2017) 85–92
(Fig.1a). As a matter of notation, the faces normal to these directions are denoted as the L, R, and T faces. The three directions correspond to three moduli of elasticity (EL ; ET ; ER ). The L face is associated with two distinct Poisson ratios (tLR and tLT ) and two shear moduli (GLR and GLT ), depending on whether shear stresses acting on the L face are oriented along the tangential or the radial directions. An additional Poisson’s ratio tRT and a shear modulus GRT are also required to fully characterize the mechanical behaviour of wooden beams, bringing the total number of constitutive parameters to nine independent constants. The differences between the mechanical properties in the radial and tangential directions are typically considered insignificant in wood (e.g., [5]). Therefore, one can assume ET ER , tLT tLR and GLT GLR ¼ GL , thus reducing the number of constitutive parameters to six independent constants. Under this assumption, the material can be considered as orthotropic (Fig.1b). For a wooden beam undergoing lateral torsional buckling, the predominant stresses are the normal stresses induced by flexure, and the associated shear stresses acting on the L face. Consequently, the most relevant constitutive parameters are the longitudinal elasticity modulus EL , and the shear modulus GL [6]. An accurate characterization of both parameters is important to reliably estimate the lateral torsional buckling capacity of wooden beams in the FEA model whereas the remaining constitutive parameters have a negligible effect [6]. Thus, both parameters will be meticulously measured in the present study. 1.3. Theory and analysis procedure
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pEL 2 Mcr ¼ Iy C w EL Iy GL J þ le le
p
The moment gradient factor approach (adopted by [2]) introduces a factor C b to account for non-uniform moment distributions and is given as:
Mcr ¼ C b M u
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pEL 2 EL I y GL J þ Mu ¼ Iy C w l l
p
ð1Þ
where l is the unbraced span of the beam, EL is the modulus of elasticity taken in the longitudinal direction and GL ¼ GLT ¼ GRT is the shear modulus, J is the Saint-Venant torsional constant, Iy is the weak axis moment of inertia (see Fig. 1), and C w is the warping constant. When formulating Eq. (1), the ends of member are assumed to be fully restrained in the lateral direction and fully restrained relative to twist, but entirely free to warp and rotate about the weak axis. For more general cases involving non-uniform moments two approaches (i.e., the effective length approach and the moment gradient factor approach) are commonly adopted in design standards to modify the elastic critical moment Mcr . The effective length approach (adopted by [1,3,4]) adjusts the unbraced length l in Eq. (1) to an effective length le based on types of boundary and loading conditions. The critical moment Mcr takes the following form
Mcr C b Mcr
p qffiffiffiffiffiffiffiffiffiffiffi
(Y) (X)
EIy GJ
l
p qffiffiffiffiffiffiffiffiffiffiffi le
EIy GJ
C b > 1:0
le < l
ð4Þ ð5Þ
The omission of the warping contributions means that the relationship between the moment gradient coefficient and effective length becomes le =l ¼ 1=C b . 1.4. Review of available standards The Eurocode [1] takes the effective length approach and the critical moment for rectangular beams takes the following form:
p qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi le
E0:05 Iy G0:05 J
ð6Þ
where E0:05 the fifth percentile value of modulus of elasticity parallel to grain, G0:05 is the fifth percentile value of shear modulus parallel to grain. The AFPA [2] is based on the moment gradient factor approach and the critical moment is given as
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pC b C e E0:05 Iy G0:05 J Mcr ¼ C fix l c
ð7Þ
where C b is the moment gradient factor accounting for different loading and boundary conditions, C e is the load eccentricity factor, C fix is the end fixity adjustment accounting for boundary imperfections in practical situations and is taken as 1.15 (as proposed by Hooley and Madsen [9]), and the cross-section slenderness factor c ¼ 1 Iy =Ix is intended to account for pre-buckling deformation effects. The Canadian timber design standard [3] also follows the effective length approach and introduces a factor, KL, to address the reduction in capacity when buckling failure occurs
M n ¼ Sx F b K L If C K ¼
ð8Þ qffiffiffiffiffi 0:97E 6 C B ¼ Lbe2d 6 50 the elastic lateral torsional F
qffiffiffiffiffiffiffiffi b
buckling governs and the corresponding stability factor is
K LðeÞ ¼ 1:2 (Z)
ð3Þ
where the C b factor is solely dependent on the moment distribution. For a rectangular section, the second term (involving the warping contribution) tends to be small compared to the first term (the Saint-Venant contribution) for practical spans. In such a case, one can conservatively approximate the above equations to
Mcr ¼
The lateral torsional buckling solution for a simply supported beam with a doubly symmetric cross-section subjected to uniform moment (e.g., [7]) is given in Eq. (1):
ð2Þ
E C 2B F b
ð9Þ
Here, the effective length approach incorporates multiple effects (moment gradient, safety factors from Hooley and Madsen, and an allowance for partial twist restraints at the end), which makes a comparison to experimental results non-evident. 1.5. Review of relevant literature
(a)
(b)
Fig. 1. Principal directions in wood (a) cylindrical orthotropic material representation, and (b) orthotropic material representation.
Hooley and Madsen [9] carried out 33 tests on glued-laminated beams with no intermediate lateral bracing. End conditions investigated included simply supported beams and cantilevers. The
Q. Xiao et al. / Engineering Structures 151 (2017) 85–92
study indicated that lateral torsional buckling capacity of rectangular beams is dependent upon the slenderness ratio. The authors also provided design proposals for the elastic and inelastic buckling of wooden beams. Hindman et al. [10] performed 12 lateral torsional buckling tests on cantilevers with rectangular crosssections of machine-stress-rated lumber, laminated veneer lumber (LVL), parallel strand Lumber (PSL), and laminated strand lumber (LSL). Their study outlined the importance of incorporating the modulus of elasticity to shear modulus ratio when predicting the critical loads. In another study, Hindman et al. [11] conducted 60 tests on I-joists where the flanges consisted of either Laminated Veneer Lumber (LVL) or Laminated Strand Lumber (LSL). Either plywood or oriented strand board (OSB) were used for the I-joist webs. The study concluded that the beam elastic buckling capacities based on design standards were, on average, 77% and 28% less than the measured buckling loads. Burow et al. [12] conducted 22 lateral torsional buckling tests for cantilevered and simply supported composite wood I-joists. The study compared the lateral stability formulation from Nethercot [13], the Euler model [7] and the moment gradient factor approach in [2]. The results suggest that Nethercot equation was overly conservative. In contrast, the Euler and moment gradient factor approach were found to be better able to predict the critical buckling moments. The average percent differences between the experimental and theoretical values were 14.1 and 5.7% for the Euler and moment gradient factor approach, respectively. Suryoatmono and Tjondro [14] performed lateral torsional buckling tests on three rectangular beams of the Bangkirai, Meranti and Albasia wood species. All beams were simply supported and were subjected to a mid-span concentrated load. The experimentally determined critical loads were compared with predictions based on Eq. (1) and with finite element models with isotropic and orthotropic constitutive features. Results from the isotropic model were in good agreement with Eq. (1). However, the isotropic models provided significantly higher buckling predictions than those experimentally observed. The orthotropic model was found to be able to accurately replicate the test results. Hindman [15] performed lateral torsional buckling tests on simply supported laterally unbraced I-joists under concentrated loads at midspan, aimed at simulating walking loads during construction. The study indicated the presence of a significant dynamic load that induces lateral torsional buckling. Bamberg [16] investigated the behaviour of wood I-joists loaded with human test subjects. The study found that braced I-joist exhibited negligible lateral acceleration but noticeable vertical displacement and angle of twist. The study suggested that an increase in bracing stiffness is effective in reducing the both lateral displacement and twist angle. Effects of continuous bracing provided by deck boards on the lateral torsional buckling resistance of wooden beams were investigated in Du et al. [17] and effects of mid-span brace height was investigated in Hu et al. [18].
87
2.2. Determination of shear modulus Tests aimed at determining the shear modulus GL were conducted in accordance with the ASTM D198 [19] standard. The test specimens were symmetrically clamped about the longitudinal axis by two wooden brackets at each end (Fig. 2). The load was applied by a hydraulic jack to the top of a wide flange steel beam. The applied load was measured using a load cell located between the jack and the steel bracket. The point of load application and the end supports were configured to include beveled wood cross sections in order to allow free twist of the specimens. Given the skew symmetry of the setup, specimens were subjected to a uniform twisting moment induced by two equal and opposite torques at each end. The angle of twist was determined by measuring the lateral displacements at the top and bottom of the beam at both ends using wire gauges. Inclinometers were also used as a supplementary measure to record the angles of twist. In order to ensure an safe level of loading to be applied without inducing material failure, a few destructive tests were first conducted on similar lumber joists and special attention was then taken not to exceed 40% of the ultimate twisting moment measured in the destructive tests, thus ensuring that the specimens remain within the linear elastic range. Given the applied torque T, the beam span L, the measured relative angle of twist Dh between both ends, and the Saint-Venant torsional constant J, the transverse shear modulus GL can be determined from GL ¼ TL=J Dh. The resulting shear modulus values are tabulated in Table 1. 2.3. Determination of elasticity modulus Each specimen was subjected to a four-point bending test about the weak axis. Test was conducted according to ASTM D198 standards [19]. Both strong axis and weak axis bending induce longitudinal stresses. This is also consistent with the Eurocode which defines E as the modulus of elasticity parallel to grain. Studies in the literature (e.g. [9]) have therefore proceeded to testing only the weak-axis bending modulus of elasticity for determining the lateral torsional buckling capacity. Weak axis bending was determined in the current project to provide material properties that are representative of lateral torsional buckling behaviour. During the test, the mid-span deflection was measured using a wire gauge mounted on the lab floor. The measured force, mid-span deflection, and the geometric parameters of the specimen, were used to backcalculate the longitudinal modulus of elasticity EL for each specimen (Table 1). Table 1 also includes the ratio E/G obtained from the specimens tested, where a significant spread is observed. A minimum and maximum value of E/G are found to be 8 and 21, respectively. Although the average value used in design standards (e.g. [3]) seem appropriate, the variation in the values need to be reflected in design standards. This is the case in the Eurocode [1] and AFPA [2], but is not the case in CAN/CSA [3].
2. Experimental program
2.4. Lateral torsional buckling test
2.1. Test matrix and specimen geometry
The boundary conditions for the classical lateral torsional buckling solution (Eq. (1)) are such that the end cross-sections are (a) fully restrained from moving along the lateral and vertical directions and from twisting along the longitudinal direction, (b) completely free to rotate about strong and weak axes, and (c) completely free to warp. The boundary conditions in the experimental program were detailed to accurately reflect the boundary conditions postulated in Eq. (1). The end supports were constructed using two steel rods welded to a bottom plate and fastened with one piece of HSS steel at the top to prevent twisting and lateral translation at each end (Fig. 3). A 19 mm (3/400 ) diameter hole
Eighteen Spruce-Pine-Fir (SPF) No. 1/No. 2 grade lumber joists were tested. The tests consisted of five 38 mm 184 mm 4200 mm (nominal dimensions of 200 800 140 ), six 38 mm 235 mm 3600 mm (200 1000 100 ), and seven 38 mm 286 mm 4200 mm (200 1400 140 ) specimens. The specimens were conditioned to a moisture content of 12% on average. Specimen geometries were selected to ensure an elastic lateral torsional buckling failure mode. Table 1 lists the geometric parameters for all 18 specimens tested.
88
Q. Xiao et al. / Engineering Structures 151 (2017) 85–92
Table 1 Comparison of critical moments between FEA and test results. Spec No.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 Minimum Maximum Average Standard deviation
Geometric parameters (mm)
Material properties (MPa)
Critical moment (kNm)
Width
Depth
Span
E
G
E/G
Test
FEA
EU code
AFPA
38 38 38 38 39 38 38 38 38 38 39 38 38 38 39 38 38 38
182 181 180 181 180 234 232 235 231 235 234 234 284 285 286 285 281 284
4200 4200 4200 4200 4200 3600 3600 3600 3600 3600 3600 3600 4200 4200 4200 4200 4200 4200
10434 9022 11185 10111 9947 11217 9745 5563 12339 7619 8092 9009 6668 7171 6469 5960 7880 5491
501 614 558 548 540 542 535 569 606 550 632 515 451 521 479 515 442 720
21 15 20 18 18 21 18 10 20 14 13 17 15 14 14 12 18 8
3.34 3.72 3.58 3.62 3.62 4.60 5.67 3.44 6.30 4.87 4.78 4.51 3.16 3.28 3.68 3.87 3.19 4.53 3.16 6.30 4.10 0.89
3.21 3.33 3.47 3.31 3.49 5.05 4.68 3.80 5.52 4.30 5.13 4.46 3.77 4.24 4.18 3.85 3.99 4.42 3.21 5.52 4.12 0.67
2.99 3.07 3.24 3.07 3.24 4.69 4.31 3.40 5.14 3.91 4.65 4.10 3.45 3.86 3.80 3.49 3.68 3.95
2.91 3.02 3.15 3.00 3.17 4.54 4.21 3.43 4.98 3.88 4.63 4.01 3.40 3.82 3.77 3.49 3.58 4.03
FEA/test
EC/FEA
AFPA/FEA
0.96 0.90 0.97 0.91 0.96 1.10 0.83 1.10 0.88 0.88 1.07 0.99 1.19 1.29 1.14 0.99 1.25 0.98 0.83 1.29 1.02 0.13
0.93 0.92 0.93 0.93 0.93 0.93 0.92 0.89 0.93 0.91 0.91 0.92 0.91 0.91 0.91 0.91 0.92 0.89 0.89 0.93 0.92 0.01
0.91 0.91 0.91 0.90 0.91 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.91 0.90 0.91 0.90 0.91 0.909 0.00
Mid-span load
Clamps Test joist (a)
(b)
Fig. 2. Twisting test setup (a) schematic, and (b) actual test setup.
was drilled into the wood support and a steel shaft with the same diameter was welded underneath the end support. A thrust bearing was placed between the steel rod and the wood support. Since the steel rod and the hole had the same diameter, the support was free to rotate while preventing lateral movement. The load application detail was designed to provide no lateral restraint to the beam at the point of application of the load, since such restraint would cause an artificial increase the lateral torsional buckling capacity of the member. To realize this objective, the specimen was loaded at the mid-span through a load transfer device (Fig. 4) where weight was added to a wood base and transferred to the top of the specimen. The loading frame consisted of three parts (Fig. 5): a HSS steel load transfer beam, a base to support the weight, and a steel rod that allows free rotation of the loading frame as the beam undergoes lateral torsional buckling. Two lateral wire gauges were mounted at the top and bottom of the beam mid-span section, respectively. During the test, the beam specimen underwent increased vertical displacement as the load was increased. At this stage, the lateral displacement and angle of twist were small. Once the load was above a critical value, significant lateral displacement and angle of twist were observed. Fig. 6 illustrates representative examples of such behaviour for dif-
ferent joist sizes and the moment corresponding to the emergence of non-negligible lateral displacement was deemed as the critical load. The figure shows large discrepancies in the maximum deflection, which can be attributed to the variability found in the material and possibly the initial out of straightness. Table 1 records the critical moment for all the 18 specimens tested.
3. Finite element model 3.1. Element and mesh The C3D8 brick element from the Abaqus library [20] was used to model the lateral torsional buckling of wooden beams. The C3D8 element has 8 nodes, each having three translational degrees of freedom (DOFs). Abaqus [18] suggests that high-level of computational accuracy is achieved when the element aspect ratio is close to unity. A mesh sensitivity analysis was conducted [6] and it suggests that reasonable results can be achieved when the element dimension in the longitudinal direction is twice as its width or depth. The FEA mesh is fully characterized by the numbers of elements ðm1 ; m2 ; m3 ; m4 Þ along the beam span m1 , each overhang
89
Q. Xiao et al. / Engineering Structures 151 (2017) 85–92
Fig. 5. Load application frame.
Fig. 3. Rotating end support for lateral torsional buckling SETUP (a) profile, (b) side view, (c) rotating end support, and (d) rotation during test.
Fig. 4. Loading apparatus for lateral torsional buckling test.
m2 , the cross-section height m3 , and its width m4 . The number of elements ðm1 ; m2 ; m3 ; m4 Þ were taken as (552,5,48,10), (472,5,62,10), and (552,6,74,10), respectively, for the 38 mm 184 mm 4200 mm, 38 mm 235 mm 3600 mm, and 38 mm 286 mm 4200 mm specimens. 3.2. Type of analysis A linearly elastic eigenvalue buckling analysis was used to determine the elastic buckling resistance in the numerical model. The eigenvalue buckling analysis is a linear perturbation procedure to estimate the critical loads by solving the system of equation:
ð½K 0 þ ki ½K G Þfv i g ¼ 0
ð10Þ
where ½K 0 is the elastic stiffness matrix with respect to the base state, ½K G is the geometric matrix, ki represent the eigenvalues, corresponding to the buckling mode shapes fv i g .
Fig. 6. Moment-displacement curves for the lateral torsional buckling test.
90
Q. Xiao et al. / Engineering Structures 151 (2017) 85–92
3.3. Constitutive model For the C3D8 element, the material was modelled as orthotropic. In a previous sensitivity analysis [6], the critical moments were observed to be sensitive to the magnitudes of the longitudinal elasticity moduli EL and GL . Therefore, input for EL and GL were based on those determined from non-destructive tests for each specimen. The rest of the mechanical properties were taken based on FPL [21] as ET ¼ ER ¼ 700 MPa, GRT ¼ 51:5 MPa, t ¼ tLR ¼ tLT ¼ 0:347, tRT ¼ 0:469 for all specimens. Xiao et al. [6] showed that varying ET , ER , GRT , tL , tRT within 50–150% of their listed reference values resulted in changes in the critical moments of less than 1%. Thus, an accurate experimental determination of ET , ER , GRT , tL , tRT was deemed unecessary. 3.4. Boundary conditions In the experiments, the boundary supports were located at the underside of the section at the end of the clear span and restrained at the edge of the cross-section (Fig. 3). The end boundary conditions in the FEA model should be configured to reflect the boundary conditions in the test. This involved two types of constraints: (a) Those related to displacements within the plane of the crosssection (Fig. 7): vertical displacements were restrained along horizontal lines DC and D0 C0 along the underside of the section. Also, the horizontal displacements were restrained along vertical lines AD, BC, A0 D0 , and B0 C0 . Vertical displacements were also restrained along DC and D0 C, and (b) those related to the longitudinal displacements at the location of the support: At one end (Point E), the longitudinal displacement was restrained but at the other end, Point E0 was set free to move longitudinally.
applied on a small area and remained vertical as the beam buckled laterally. Similarly, in the Abaqus model, the applied load was distributed over an area of 6 6 elements (Fig.8b) to approximately emulate the conditions of the experiment. In order to avoid premature localized buckling failure in the FEA model, no distributed loads were applied near the edge of the beam as shown in Fig.8b. 4. Discussion 4.1. Comparison of results and validation of FEA model A linearly elastic eigenvalue buckling analysis within the commercial software Abaqus was conducted to predict the buckling loads and the mode shapes of simply-supported beams under a mid-span point load. A comparison between critical moments measured in the experimental study and those based on the Abaqus FEA model is provided in Table 1. The test to FEA critical moment ratios are observed to range from 0.83 to 1.29 with an average value of 1.02 and a standard deviation of 0.13. Deviations from test results can be attributed, in part, to the spatial variability within the material and the geometric imperfections in the specimens; both effects not captured by the FEA model. However, on average, the FEA critical moments exhibit reasonable agreement with experimental results. Also, for all specimens investigated, the FEA buckling configuration (Fig. 9), are observed to be consistent with those experimentally tested, all exhibiting the classical mode of failure where both the lateral displacement and the angle of twist peak at the mid-span and vanish at the supports. The above comparison suggests that the present finite element model is capable of predicting the elastic lateral torsional buckling capacity and mode shape for wooden beams. In the following section, the validated FEA model is used to assess the buckling capacity of beams for cases beyond the scope of the experimental program.
3.5. Modelling of load application detail 4.2. Code considerations In the experiments, the beams were subjected to a mid-span load applied at the top edge of the beam (Fig.8a). The load was
Fig. 7. Boundary conditions (crossed arrows denote restrained DOFs): (a) experimental end detail, (b) elevation of experimental end detail, (c) DOFs restrained at one end, (d) DOFs restrained at the other end.
4.2.1. Mid-span concentrated load This section is aimed at assessing the lateral torsional buckling provision of the Eurocode [1] for the case of a simply-supported beam under mid-span point load, which is reflective of the test conditions. The Eurocode provisions [1] are evaluated against the FEA model previously developed and requirements found in AFPA [2]. The Eurocode is assessed through quantifying the critical moments for the 18 specimen whose material and geometric properties were tested. For calculating critical moments, the test values for E and G are used instead of E0:05 and G0:05 , and the results are summarized in Table 1. Although the approach adopted by the two codes is seemingly different, it is observed that the Eurocode predicts similar results as the AFPA. Both solutions predict a consistently smaller buckling capacity than the FEA, with Eurocode on average being 0.92 of FEA and AFPA being 0.90. The Eurocode accounts for the load eccentricity (denoted as C e in the AFPA equation) by increasing le by 2d for a load applied at the compression edge and decreasing it by 0.5d when the load is applied at the tension edge, where d is the beam cross-section depth. From Section 1.3, it is noted that for a rectangular section the relationship between the effective length factor and the moment gradient factor can be approximated as le =l ¼ 1=C b . The moment gradient factor provided in the AFPA provisions for a mid-span concentrated load is 1.35. This would yield an equivalent effective length factor of 0.74. However, the effective length factor reported in the Eurocode is 0.8. The higher Eurocode effective length factor is presumed to include an allowance to account for the partial end fixity in a manner similar to the C fix coefficient in the AFPA [3].
Q. Xiao et al. / Engineering Structures 151 (2017) 85–92
91
Fig. 8. Load application detail (a) experimental details and (b) corresponding FEA model detail.
Fig. 9. Buckled configuration (a) experimentally observed and (b) FEA predicted.
4.2.2. Other loading and boundary conditions This section provides a review of the lateral torsional buckling provisions in the Eurocode and compares them to the FEA model. The two methods are evaluated for a wooden beam whose crosssection is 80 mm wide and 570 mm deep. The beam span is taken as 6 m. The modulus of elasticity parallel to grain is assumed to be 10300 MPa and the shear modulus is 474 MPa. Two types of boundary conditions are investigated; simply-supported and cantilever. Load types considered are uniform moments (to reflect reference point adopted in design standards), uniformly distributed load (UDL), and point load applied at beam’s top edge, centroid and bottom edge (to replicate practical situations). Table 2 shows the effective length factor le =l based on the Eurocode provisions and those based on the FEA model from:
le =l ¼
p
Mcr ðFEAÞl
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi EL Iy GL J
ð11Þ
where Mcr ðFEAÞ is the critical moment extracted from Abaqus FEA model. As expected, the value for uniform moments at beam centroid is an exact match between the Eurocode and the FEA model. According to the code provision, the effective length le shall be increased by 2d for compression edge loading and decreased by 0.5d for tension edge loading. This provision, which aims to incorporate the load position effect, works well for simply-supported beams under gravity loads where the beam top edge is in compression and the bottom edge is in tension. Table 2 shows that for simply supported beam cases the Eurocode equation provides values that are close to those obtained by the FEA model, while being slightly on the conservative side (le =l values larger than 1.0) especially for loads applied at the bottom edge. For cases involving simply-supported
beams under uniform moment and transverse loads applied at top and centroid, the conservativeness of the Eurocode is within 11%. For simply-supported beams under transverse loads applied at the bottom, the code exhibits higher conservativeness, with UDL effective length ratios of 1.16 and mid-span point load of 1.27. Two sets of values are provided in Table 2 for cases involving cantilevered beams under top and bottom edge loading. When applying the literal wording of the code, where adjustment for load position is based on tension and compression edge, a discrepancy in trends is observed between the FEA model predictions and the Eurocode results. Whereas, as expected, the FE analysis indicates that moving the load from bottom to top edge results in a decrease in the critical moments (and hence increases the effective length), an opposite trend is observed when applying the Eurocode values. This leads to the conclusion that the Eurocode wording is intended only for simply supported beams. Results from Table 2 also confirm this observation where the ratio between the Eurocode and the FEA model for cantilevered beams applied at beam bottom is 1.62 and 2.11 for point load and UDL, respectively. Furthermore, for top loading cases, the code provides non-conservative results. Based on the above observations, it is proposed that the code wording be modified for cantilevers to address adjustment for top and bottom edge loading rather than using tension and compression edges. The results based on the proposed code revisions are provided in the last two rows of Table 2. The table shows that, in general, the Eurocode predicts effective length factors that are greater than those from the FEA, thus providing conservative critical moments. For cantilevered beams, the ratio between Eurocode and FEA ranges from 1.18 to 1.38. The high level of conservativeness can account for the fact that a perfect fixity condition at the cantilever root is difficult to achieve in wood construction.
92
Q. Xiao et al. / Engineering Structures 151 (2017) 85–92
Table 2 Comparison between Eurocode and FEA for different boundary and loading conditions. Beam type
Load type
1
2
Effective length factor
Ratio
Eurocode
Simply supported
Cantilever (effective length based on present wording of Eurocode) Cantilever (effective length based on proposed wording)
Uniform moment UDL Mid-span point load UDL Point load at tip UDL Point load at tip
Computed from FEA
3
4
5
6
7
8
9
10
11
Top edge
Centroid
Bottom edge
Top edge
Centroid
Bottom edge
EC/FEA (Top edge)
EC/FEA (Centroid)
/
1
/
/
1
/
/
1.00
EC/FEA (Bottom edge) /
1.09 0.99
0.90 0.80
0.85 0.75
1.02 0.89
0.87 0.72
0.74 0.59
1.07 1.11
1.04 1.11
1.16 1.27
0.45 0.75
0.50 0.80
0.69 0.99
0.51 0.80
0.39 0.68
0.33 0.61
0.88 0.94
1.27 1.18
2.11 1.62
0.69 0.99
0.50 0.80
0.45 0.75
0.51 0.80
0.39 0.68
0.33 0.61
1.34 1.24
1.27 1.18
1.38 1.23
5. Conclusions References An experimental program was conducted to investigate the elastic lateral torsional buckling capacity of wooden beams. Eighteen full-scale tests were tested. For each specimen, nondestructive material tests were first conducted to determine the shear and longitudinal elastic moduli, followed by a lateral torsional buckling test. A 3D finite element model was developed to predict the elastic lateral torsional buckling resistance for each specimen. The model features orthotropic material representation and is based on material properties as determined in nondestructive tests. The model provides a realistic representation of boundary conditions and loading details of the tests. The following conclusions can be drawn from the current research project: – The FEA model is observed to reliably predict the elastic lateral torsional buckling capacity of the wooden beams. The test to FEA critical moment ratios were on average 1.02 with a standard deviation of 0.13. – The Eurocode was compared to the AFPA guidelines for simply supported beams with top edge loading and the results showed that although the approach adopted by the two codes is seemingly different, solutions from both codes were similar and predicted a consistently smaller buckling capacity than the FEA. Eurocode reports higher effective length factor than AFPA, which is presumed to include an allowance to account for the partial end fixity. – Comparison between the Eurocode and FEA model for simply supported end conditions seems to provide a reasonable fit with the Eurocode values being slightly conservative. However, when the cantilevered beam values are evaluated, the Eurocode provisions seem to be overly conservative for the case of bottom edge loading and non-conservative for the case of top edge loading. The observation suggests that current wording seems to be intended for simply-supported beams under gravity loads where the beam top edge is in compression and the bottom edge is in tension. – More accurate wording has been proposed regarding the effective length adjustment for top and bottom edge loading rather than using tension and compression. The results based on the proposed wording provides critical moment predictions that are more consistent for cantilevers under top and bottom edge loading.
[1] EN 1995-1-1 Eurocode 5 – design of timber structures, Part 1–1, general: common rules and rules for buildings. Brussels (Belgium): CEN; 2014. [2] AFPA. Technical report 14: designing for lateral-torsional stability in wood members. Washington, D.C.: American Forest and Paper Association; 2003. [3] CAN-CSA O86. Engineering design in wood. Mississauga (Ontario, Canada): Canadian Standard Association; 2015. [4] NDS. National design specification for wood construction, national design specifications. Washington, D.C.: American Forest and Paper Association; 2015. [5] Isopescu D, Stanila O, Asatanei I. Analysis of wood bending properties on standardized samples and structural size beams tests. Buletinul Insititutului Politehnic Din Iasi, Publicat de Universitatea Tehnica˘, Gheorghe Asachi” din Iasßi, Tomul LVIII (LXII), Fasc 1, Sectia Constructii Arhitectura; 2012. [6] Xiao Q, Doudak G, Mohareb M. Lateral torsional buckling of wood beams: FEAmodelling and sensitivity analysis. In: World conference in timber engineering, Quebec City, Quebec, Canada, August 10–14. [7] Timoshenko S. Theory of elastic stability. 2nd ed. McGraw-Hill Book Co; 1961. [9] Hooley RF, Madsen B. Lateral buckling of glued laminated beams. J Struct Eng Div ASCE 1964;90(ST3):201–303. [10] Hindman DP, Harvey HB, Janowiak JJ. Measurement and prediction of lateral torsional buckling loads of composite wood materials: rectangular section. For Product J 2005;55(9):42–7. [11] Hindman DP, Harvey HB, Janowiak JJ. Measurement and prediction of lateral torsional buckling loads of composite wood materials: I-joist sections. For Product J 2005;55(10):43–8. [12] Burow JR, Manbeck HB, Janowiak JJ. Lateral stability of composite wood I-joists under concentrated-load bending. Am Soc Agric Biol Eng 2006;49: 1867–80. [13] Nethercot DA. Elastic lateral buckling of beams. In: Narayanan R, editor. Beams and beam columns: stability in strength. Barking, Essex (UK): Applied Science Publishers; 1983. [14] Suryoatmono B, Tjondro A. Lateral-torsional buckling of orthotropic rectangular section beams. Bandung (Indonesia): Department of Civil Engineering, Parahyangan Catholic University; 2008. [15] Hindman DP. Lateral buckling of I-joist. VA (USA): Department of Wood Science and Forest Products, Virginia Tech Blacksburg; 2008. [16] Bamberg CR. Lateral movement of unbraced wood composite I-joists exposed to dynamic walking loads [M. A. Sc. thesis]. Blacksburg (Virginia, USA): Virginia Polytechnic Institute and State University; 2009. [17] Du Y, Mohareb M, Doudak G. Non-sway model for lateral torsional buckling of wooden beams under wind uplift. J Eng Mech 2016;142(12). [18] Hu Y, Mohareb M, Doudak G. Effect of eccentric lateral bracing stiffness on lateral torsional buckling resistance of wooden beams. Int J Struct Stabil Dyn 2017. [19] ASTM. D198 standard test method of static tests of lumber in structural sizes, Philadelphia, PA; 2009. [20] Simulia. ABAQUS analysis user’s manual (Version 6.11). Dassault Systèmes; 2011. [21] FPL (Forest Products Laboratory). Wood handbook-wood as an engineering material, Wisconsin; 2010.
Contents lists available at ScienceDirect
Engineering Structures journal homepage: www.elsevier.com/locate/engstruct
Numerical and experimental investigation of lateral torsional buckling of wood beams Q. Xiao, G. Doudak ⇑, M. Mohareb Department of Civil Engineering, University of Ottawa, K1N 6N5, Canada
a r t i c l e
i n f o
Article history: Received 13 April 2017 Revised 27 June 2017 Accepted 11 August 2017
Keywords: Wood beams Lateral torsional buckling Finite element Full scale testing Lateral restraint Load position effect Bracing height Eurocode
a b s t r a c t The present study reports the results of a full-scale experimental and numerical investigation aimed at predicting the elastic lateral torsional buckling capacity of wooden beams. The experimental component consists of 18 Spruce-Pine-Fir (SPF) No. 1/No. 2 grade lumber joists consisting of five 38 mm 184 mm 4200 mm, six 38 mm 235 mm 3600 mm, and seven 38 mm 286 mm 4200 mm specimens. For each specimen, the shear and longitudinal elastic moduli are first determined experimentally through non-destructive tests. A full-scale bending test is then conducted on each specimen, to determine its elastic lateral torsional buckling resistance. A 3D finite element model is developed to predict the lateral torsional buckling resistance for each specimen based on the experimentally determined shear and longitudinal elastic moduli. The validity of the finite element analysis is assessed through comparisons with fullscale test results. The validated model was used to assess the Eurocode provisions and it was found that for simply supported end conditions the code equation seemed reasonable and slightly conservative. However, for cantilevered beams, the Eurocode provisions seem to be overly conservative for the case of bottom edge loading and non-conservative for the case of top edge loading. Changes have been proposed to the wording of the effective length adjustment and the results based on the revised definition provides critical moment predictions that are conservative and more consistent for cantilevers under top and bottom edge loading. Ó 2017 Published by Elsevier Ltd.
1. Objective 1.1. General Timber design standards and guidelines (e.g., [1–4]) recognize lateral torsional buckling as a likely governing failure mode when designing long span laterally unsupported wooden beams. Guidelines in [2] provide estimates for the lateral torsional buckling resistance, and account for common loading patterns, load height effect relative to the section centroid (i.e., load eccentricity), and presence of bracing. The current study is motivated by the lack of consistency and transparency in some of the design standards. For example, although the European standard [1] and the AFPA approach [2] both include the modulus of Elasticity (E) and shear modulus (G) in the critical moment expression, the Canadian standard [3] assumes a fixed value for the ratio of E/G. Also, whereas, the AFPA model [2] specifically includes a factor accounting for partial twist fixity condition that may occur in practical situations, the Canadian approach [3] embed the factor directly in the buck⇑ Corresponding author at: Civil Engineering, University of Ottawa, 161 Louis-Pasteur, Room A115 (CBY), Ottawa, ON K1N 6N5, Canada. E-mail addresses: [email protected] (Q. Xiao), [email protected] (G. Doudak), [email protected] (M. Mohareb). http://dx.doi.org/10.1016/j.engstruct.2017.08.020 0141-0296/Ó 2017 Published by Elsevier Ltd.
ling equation. The current paper aims to bring clarity and possibly consistency between the various approaches by understanding and highlighting the differences between the various standards through experimental and numerical analysis. A 3-D Finite Element Analysis (FEA) model that is able to reliably represent the experimental test results, is developed. In order to assess the reliability of the FEA model, the present study first reports a recent experimental investigation, involving both material testing and full-scale lateral torsional bucking tests on simply supported beams with no intermediate lateral bracing. The parameters obtained from material tests are then input into the FEA model and the buckling capacity obtained from the FEA model is compared with the full-scale beam buckling test results. The validated FEA model is then used to expand the scope of the experimental tests by investigating the accuracy of the lateral torsional buckling provisions in the current Eurocode [1] where comparison between Eurocode and FEA model is made for different boundary and loading conditions.
1.2. Constitutive behaviour of wood Wood can be considered as a cylindrical orthotropic material [5] with three orthogonal directions: longitudinal, radial, and tangential
86
Q. Xiao et al. / Engineering Structures 151 (2017) 85–92
(Fig.1a). As a matter of notation, the faces normal to these directions are denoted as the L, R, and T faces. The three directions correspond to three moduli of elasticity (EL ; ET ; ER ). The L face is associated with two distinct Poisson ratios (tLR and tLT ) and two shear moduli (GLR and GLT ), depending on whether shear stresses acting on the L face are oriented along the tangential or the radial directions. An additional Poisson’s ratio tRT and a shear modulus GRT are also required to fully characterize the mechanical behaviour of wooden beams, bringing the total number of constitutive parameters to nine independent constants. The differences between the mechanical properties in the radial and tangential directions are typically considered insignificant in wood (e.g., [5]). Therefore, one can assume ET ER , tLT tLR and GLT GLR ¼ GL , thus reducing the number of constitutive parameters to six independent constants. Under this assumption, the material can be considered as orthotropic (Fig.1b). For a wooden beam undergoing lateral torsional buckling, the predominant stresses are the normal stresses induced by flexure, and the associated shear stresses acting on the L face. Consequently, the most relevant constitutive parameters are the longitudinal elasticity modulus EL , and the shear modulus GL [6]. An accurate characterization of both parameters is important to reliably estimate the lateral torsional buckling capacity of wooden beams in the FEA model whereas the remaining constitutive parameters have a negligible effect [6]. Thus, both parameters will be meticulously measured in the present study. 1.3. Theory and analysis procedure
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pEL 2 Mcr ¼ Iy C w EL Iy GL J þ le le
p
The moment gradient factor approach (adopted by [2]) introduces a factor C b to account for non-uniform moment distributions and is given as:
Mcr ¼ C b M u
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pEL 2 EL I y GL J þ Mu ¼ Iy C w l l
p
ð1Þ
where l is the unbraced span of the beam, EL is the modulus of elasticity taken in the longitudinal direction and GL ¼ GLT ¼ GRT is the shear modulus, J is the Saint-Venant torsional constant, Iy is the weak axis moment of inertia (see Fig. 1), and C w is the warping constant. When formulating Eq. (1), the ends of member are assumed to be fully restrained in the lateral direction and fully restrained relative to twist, but entirely free to warp and rotate about the weak axis. For more general cases involving non-uniform moments two approaches (i.e., the effective length approach and the moment gradient factor approach) are commonly adopted in design standards to modify the elastic critical moment Mcr . The effective length approach (adopted by [1,3,4]) adjusts the unbraced length l in Eq. (1) to an effective length le based on types of boundary and loading conditions. The critical moment Mcr takes the following form
Mcr C b Mcr
p qffiffiffiffiffiffiffiffiffiffiffi
(Y) (X)
EIy GJ
l
p qffiffiffiffiffiffiffiffiffiffiffi le
EIy GJ
C b > 1:0
le < l
ð4Þ ð5Þ
The omission of the warping contributions means that the relationship between the moment gradient coefficient and effective length becomes le =l ¼ 1=C b . 1.4. Review of available standards The Eurocode [1] takes the effective length approach and the critical moment for rectangular beams takes the following form:
p qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi le
E0:05 Iy G0:05 J
ð6Þ
where E0:05 the fifth percentile value of modulus of elasticity parallel to grain, G0:05 is the fifth percentile value of shear modulus parallel to grain. The AFPA [2] is based on the moment gradient factor approach and the critical moment is given as
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pC b C e E0:05 Iy G0:05 J Mcr ¼ C fix l c
ð7Þ
where C b is the moment gradient factor accounting for different loading and boundary conditions, C e is the load eccentricity factor, C fix is the end fixity adjustment accounting for boundary imperfections in practical situations and is taken as 1.15 (as proposed by Hooley and Madsen [9]), and the cross-section slenderness factor c ¼ 1 Iy =Ix is intended to account for pre-buckling deformation effects. The Canadian timber design standard [3] also follows the effective length approach and introduces a factor, KL, to address the reduction in capacity when buckling failure occurs
M n ¼ Sx F b K L If C K ¼
ð8Þ qffiffiffiffiffi 0:97E 6 C B ¼ Lbe2d 6 50 the elastic lateral torsional F
qffiffiffiffiffiffiffiffi b
buckling governs and the corresponding stability factor is
K LðeÞ ¼ 1:2 (Z)
ð3Þ
where the C b factor is solely dependent on the moment distribution. For a rectangular section, the second term (involving the warping contribution) tends to be small compared to the first term (the Saint-Venant contribution) for practical spans. In such a case, one can conservatively approximate the above equations to
Mcr ¼
The lateral torsional buckling solution for a simply supported beam with a doubly symmetric cross-section subjected to uniform moment (e.g., [7]) is given in Eq. (1):
ð2Þ
E C 2B F b
ð9Þ
Here, the effective length approach incorporates multiple effects (moment gradient, safety factors from Hooley and Madsen, and an allowance for partial twist restraints at the end), which makes a comparison to experimental results non-evident. 1.5. Review of relevant literature
(a)
(b)
Fig. 1. Principal directions in wood (a) cylindrical orthotropic material representation, and (b) orthotropic material representation.
Hooley and Madsen [9] carried out 33 tests on glued-laminated beams with no intermediate lateral bracing. End conditions investigated included simply supported beams and cantilevers. The
Q. Xiao et al. / Engineering Structures 151 (2017) 85–92
study indicated that lateral torsional buckling capacity of rectangular beams is dependent upon the slenderness ratio. The authors also provided design proposals for the elastic and inelastic buckling of wooden beams. Hindman et al. [10] performed 12 lateral torsional buckling tests on cantilevers with rectangular crosssections of machine-stress-rated lumber, laminated veneer lumber (LVL), parallel strand Lumber (PSL), and laminated strand lumber (LSL). Their study outlined the importance of incorporating the modulus of elasticity to shear modulus ratio when predicting the critical loads. In another study, Hindman et al. [11] conducted 60 tests on I-joists where the flanges consisted of either Laminated Veneer Lumber (LVL) or Laminated Strand Lumber (LSL). Either plywood or oriented strand board (OSB) were used for the I-joist webs. The study concluded that the beam elastic buckling capacities based on design standards were, on average, 77% and 28% less than the measured buckling loads. Burow et al. [12] conducted 22 lateral torsional buckling tests for cantilevered and simply supported composite wood I-joists. The study compared the lateral stability formulation from Nethercot [13], the Euler model [7] and the moment gradient factor approach in [2]. The results suggest that Nethercot equation was overly conservative. In contrast, the Euler and moment gradient factor approach were found to be better able to predict the critical buckling moments. The average percent differences between the experimental and theoretical values were 14.1 and 5.7% for the Euler and moment gradient factor approach, respectively. Suryoatmono and Tjondro [14] performed lateral torsional buckling tests on three rectangular beams of the Bangkirai, Meranti and Albasia wood species. All beams were simply supported and were subjected to a mid-span concentrated load. The experimentally determined critical loads were compared with predictions based on Eq. (1) and with finite element models with isotropic and orthotropic constitutive features. Results from the isotropic model were in good agreement with Eq. (1). However, the isotropic models provided significantly higher buckling predictions than those experimentally observed. The orthotropic model was found to be able to accurately replicate the test results. Hindman [15] performed lateral torsional buckling tests on simply supported laterally unbraced I-joists under concentrated loads at midspan, aimed at simulating walking loads during construction. The study indicated the presence of a significant dynamic load that induces lateral torsional buckling. Bamberg [16] investigated the behaviour of wood I-joists loaded with human test subjects. The study found that braced I-joist exhibited negligible lateral acceleration but noticeable vertical displacement and angle of twist. The study suggested that an increase in bracing stiffness is effective in reducing the both lateral displacement and twist angle. Effects of continuous bracing provided by deck boards on the lateral torsional buckling resistance of wooden beams were investigated in Du et al. [17] and effects of mid-span brace height was investigated in Hu et al. [18].
87
2.2. Determination of shear modulus Tests aimed at determining the shear modulus GL were conducted in accordance with the ASTM D198 [19] standard. The test specimens were symmetrically clamped about the longitudinal axis by two wooden brackets at each end (Fig. 2). The load was applied by a hydraulic jack to the top of a wide flange steel beam. The applied load was measured using a load cell located between the jack and the steel bracket. The point of load application and the end supports were configured to include beveled wood cross sections in order to allow free twist of the specimens. Given the skew symmetry of the setup, specimens were subjected to a uniform twisting moment induced by two equal and opposite torques at each end. The angle of twist was determined by measuring the lateral displacements at the top and bottom of the beam at both ends using wire gauges. Inclinometers were also used as a supplementary measure to record the angles of twist. In order to ensure an safe level of loading to be applied without inducing material failure, a few destructive tests were first conducted on similar lumber joists and special attention was then taken not to exceed 40% of the ultimate twisting moment measured in the destructive tests, thus ensuring that the specimens remain within the linear elastic range. Given the applied torque T, the beam span L, the measured relative angle of twist Dh between both ends, and the Saint-Venant torsional constant J, the transverse shear modulus GL can be determined from GL ¼ TL=J Dh. The resulting shear modulus values are tabulated in Table 1. 2.3. Determination of elasticity modulus Each specimen was subjected to a four-point bending test about the weak axis. Test was conducted according to ASTM D198 standards [19]. Both strong axis and weak axis bending induce longitudinal stresses. This is also consistent with the Eurocode which defines E as the modulus of elasticity parallel to grain. Studies in the literature (e.g. [9]) have therefore proceeded to testing only the weak-axis bending modulus of elasticity for determining the lateral torsional buckling capacity. Weak axis bending was determined in the current project to provide material properties that are representative of lateral torsional buckling behaviour. During the test, the mid-span deflection was measured using a wire gauge mounted on the lab floor. The measured force, mid-span deflection, and the geometric parameters of the specimen, were used to backcalculate the longitudinal modulus of elasticity EL for each specimen (Table 1). Table 1 also includes the ratio E/G obtained from the specimens tested, where a significant spread is observed. A minimum and maximum value of E/G are found to be 8 and 21, respectively. Although the average value used in design standards (e.g. [3]) seem appropriate, the variation in the values need to be reflected in design standards. This is the case in the Eurocode [1] and AFPA [2], but is not the case in CAN/CSA [3].
2. Experimental program
2.4. Lateral torsional buckling test
2.1. Test matrix and specimen geometry
The boundary conditions for the classical lateral torsional buckling solution (Eq. (1)) are such that the end cross-sections are (a) fully restrained from moving along the lateral and vertical directions and from twisting along the longitudinal direction, (b) completely free to rotate about strong and weak axes, and (c) completely free to warp. The boundary conditions in the experimental program were detailed to accurately reflect the boundary conditions postulated in Eq. (1). The end supports were constructed using two steel rods welded to a bottom plate and fastened with one piece of HSS steel at the top to prevent twisting and lateral translation at each end (Fig. 3). A 19 mm (3/400 ) diameter hole
Eighteen Spruce-Pine-Fir (SPF) No. 1/No. 2 grade lumber joists were tested. The tests consisted of five 38 mm 184 mm 4200 mm (nominal dimensions of 200 800 140 ), six 38 mm 235 mm 3600 mm (200 1000 100 ), and seven 38 mm 286 mm 4200 mm (200 1400 140 ) specimens. The specimens were conditioned to a moisture content of 12% on average. Specimen geometries were selected to ensure an elastic lateral torsional buckling failure mode. Table 1 lists the geometric parameters for all 18 specimens tested.
88
Q. Xiao et al. / Engineering Structures 151 (2017) 85–92
Table 1 Comparison of critical moments between FEA and test results. Spec No.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 Minimum Maximum Average Standard deviation
Geometric parameters (mm)
Material properties (MPa)
Critical moment (kNm)
Width
Depth
Span
E
G
E/G
Test
FEA
EU code
AFPA
38 38 38 38 39 38 38 38 38 38 39 38 38 38 39 38 38 38
182 181 180 181 180 234 232 235 231 235 234 234 284 285 286 285 281 284
4200 4200 4200 4200 4200 3600 3600 3600 3600 3600 3600 3600 4200 4200 4200 4200 4200 4200
10434 9022 11185 10111 9947 11217 9745 5563 12339 7619 8092 9009 6668 7171 6469 5960 7880 5491
501 614 558 548 540 542 535 569 606 550 632 515 451 521 479 515 442 720
21 15 20 18 18 21 18 10 20 14 13 17 15 14 14 12 18 8
3.34 3.72 3.58 3.62 3.62 4.60 5.67 3.44 6.30 4.87 4.78 4.51 3.16 3.28 3.68 3.87 3.19 4.53 3.16 6.30 4.10 0.89
3.21 3.33 3.47 3.31 3.49 5.05 4.68 3.80 5.52 4.30 5.13 4.46 3.77 4.24 4.18 3.85 3.99 4.42 3.21 5.52 4.12 0.67
2.99 3.07 3.24 3.07 3.24 4.69 4.31 3.40 5.14 3.91 4.65 4.10 3.45 3.86 3.80 3.49 3.68 3.95
2.91 3.02 3.15 3.00 3.17 4.54 4.21 3.43 4.98 3.88 4.63 4.01 3.40 3.82 3.77 3.49 3.58 4.03
FEA/test
EC/FEA
AFPA/FEA
0.96 0.90 0.97 0.91 0.96 1.10 0.83 1.10 0.88 0.88 1.07 0.99 1.19 1.29 1.14 0.99 1.25 0.98 0.83 1.29 1.02 0.13
0.93 0.92 0.93 0.93 0.93 0.93 0.92 0.89 0.93 0.91 0.91 0.92 0.91 0.91 0.91 0.91 0.92 0.89 0.89 0.93 0.92 0.01
0.91 0.91 0.91 0.90 0.91 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.91 0.90 0.91 0.90 0.91 0.909 0.00
Mid-span load
Clamps Test joist (a)
(b)
Fig. 2. Twisting test setup (a) schematic, and (b) actual test setup.
was drilled into the wood support and a steel shaft with the same diameter was welded underneath the end support. A thrust bearing was placed between the steel rod and the wood support. Since the steel rod and the hole had the same diameter, the support was free to rotate while preventing lateral movement. The load application detail was designed to provide no lateral restraint to the beam at the point of application of the load, since such restraint would cause an artificial increase the lateral torsional buckling capacity of the member. To realize this objective, the specimen was loaded at the mid-span through a load transfer device (Fig. 4) where weight was added to a wood base and transferred to the top of the specimen. The loading frame consisted of three parts (Fig. 5): a HSS steel load transfer beam, a base to support the weight, and a steel rod that allows free rotation of the loading frame as the beam undergoes lateral torsional buckling. Two lateral wire gauges were mounted at the top and bottom of the beam mid-span section, respectively. During the test, the beam specimen underwent increased vertical displacement as the load was increased. At this stage, the lateral displacement and angle of twist were small. Once the load was above a critical value, significant lateral displacement and angle of twist were observed. Fig. 6 illustrates representative examples of such behaviour for dif-
ferent joist sizes and the moment corresponding to the emergence of non-negligible lateral displacement was deemed as the critical load. The figure shows large discrepancies in the maximum deflection, which can be attributed to the variability found in the material and possibly the initial out of straightness. Table 1 records the critical moment for all the 18 specimens tested.
3. Finite element model 3.1. Element and mesh The C3D8 brick element from the Abaqus library [20] was used to model the lateral torsional buckling of wooden beams. The C3D8 element has 8 nodes, each having three translational degrees of freedom (DOFs). Abaqus [18] suggests that high-level of computational accuracy is achieved when the element aspect ratio is close to unity. A mesh sensitivity analysis was conducted [6] and it suggests that reasonable results can be achieved when the element dimension in the longitudinal direction is twice as its width or depth. The FEA mesh is fully characterized by the numbers of elements ðm1 ; m2 ; m3 ; m4 Þ along the beam span m1 , each overhang
89
Q. Xiao et al. / Engineering Structures 151 (2017) 85–92
Fig. 5. Load application frame.
Fig. 3. Rotating end support for lateral torsional buckling SETUP (a) profile, (b) side view, (c) rotating end support, and (d) rotation during test.
Fig. 4. Loading apparatus for lateral torsional buckling test.
m2 , the cross-section height m3 , and its width m4 . The number of elements ðm1 ; m2 ; m3 ; m4 Þ were taken as (552,5,48,10), (472,5,62,10), and (552,6,74,10), respectively, for the 38 mm 184 mm 4200 mm, 38 mm 235 mm 3600 mm, and 38 mm 286 mm 4200 mm specimens. 3.2. Type of analysis A linearly elastic eigenvalue buckling analysis was used to determine the elastic buckling resistance in the numerical model. The eigenvalue buckling analysis is a linear perturbation procedure to estimate the critical loads by solving the system of equation:
ð½K 0 þ ki ½K G Þfv i g ¼ 0
ð10Þ
where ½K 0 is the elastic stiffness matrix with respect to the base state, ½K G is the geometric matrix, ki represent the eigenvalues, corresponding to the buckling mode shapes fv i g .
Fig. 6. Moment-displacement curves for the lateral torsional buckling test.
90
Q. Xiao et al. / Engineering Structures 151 (2017) 85–92
3.3. Constitutive model For the C3D8 element, the material was modelled as orthotropic. In a previous sensitivity analysis [6], the critical moments were observed to be sensitive to the magnitudes of the longitudinal elasticity moduli EL and GL . Therefore, input for EL and GL were based on those determined from non-destructive tests for each specimen. The rest of the mechanical properties were taken based on FPL [21] as ET ¼ ER ¼ 700 MPa, GRT ¼ 51:5 MPa, t ¼ tLR ¼ tLT ¼ 0:347, tRT ¼ 0:469 for all specimens. Xiao et al. [6] showed that varying ET , ER , GRT , tL , tRT within 50–150% of their listed reference values resulted in changes in the critical moments of less than 1%. Thus, an accurate experimental determination of ET , ER , GRT , tL , tRT was deemed unecessary. 3.4. Boundary conditions In the experiments, the boundary supports were located at the underside of the section at the end of the clear span and restrained at the edge of the cross-section (Fig. 3). The end boundary conditions in the FEA model should be configured to reflect the boundary conditions in the test. This involved two types of constraints: (a) Those related to displacements within the plane of the crosssection (Fig. 7): vertical displacements were restrained along horizontal lines DC and D0 C0 along the underside of the section. Also, the horizontal displacements were restrained along vertical lines AD, BC, A0 D0 , and B0 C0 . Vertical displacements were also restrained along DC and D0 C, and (b) those related to the longitudinal displacements at the location of the support: At one end (Point E), the longitudinal displacement was restrained but at the other end, Point E0 was set free to move longitudinally.
applied on a small area and remained vertical as the beam buckled laterally. Similarly, in the Abaqus model, the applied load was distributed over an area of 6 6 elements (Fig.8b) to approximately emulate the conditions of the experiment. In order to avoid premature localized buckling failure in the FEA model, no distributed loads were applied near the edge of the beam as shown in Fig.8b. 4. Discussion 4.1. Comparison of results and validation of FEA model A linearly elastic eigenvalue buckling analysis within the commercial software Abaqus was conducted to predict the buckling loads and the mode shapes of simply-supported beams under a mid-span point load. A comparison between critical moments measured in the experimental study and those based on the Abaqus FEA model is provided in Table 1. The test to FEA critical moment ratios are observed to range from 0.83 to 1.29 with an average value of 1.02 and a standard deviation of 0.13. Deviations from test results can be attributed, in part, to the spatial variability within the material and the geometric imperfections in the specimens; both effects not captured by the FEA model. However, on average, the FEA critical moments exhibit reasonable agreement with experimental results. Also, for all specimens investigated, the FEA buckling configuration (Fig. 9), are observed to be consistent with those experimentally tested, all exhibiting the classical mode of failure where both the lateral displacement and the angle of twist peak at the mid-span and vanish at the supports. The above comparison suggests that the present finite element model is capable of predicting the elastic lateral torsional buckling capacity and mode shape for wooden beams. In the following section, the validated FEA model is used to assess the buckling capacity of beams for cases beyond the scope of the experimental program.
3.5. Modelling of load application detail 4.2. Code considerations In the experiments, the beams were subjected to a mid-span load applied at the top edge of the beam (Fig.8a). The load was
Fig. 7. Boundary conditions (crossed arrows denote restrained DOFs): (a) experimental end detail, (b) elevation of experimental end detail, (c) DOFs restrained at one end, (d) DOFs restrained at the other end.
4.2.1. Mid-span concentrated load This section is aimed at assessing the lateral torsional buckling provision of the Eurocode [1] for the case of a simply-supported beam under mid-span point load, which is reflective of the test conditions. The Eurocode provisions [1] are evaluated against the FEA model previously developed and requirements found in AFPA [2]. The Eurocode is assessed through quantifying the critical moments for the 18 specimen whose material and geometric properties were tested. For calculating critical moments, the test values for E and G are used instead of E0:05 and G0:05 , and the results are summarized in Table 1. Although the approach adopted by the two codes is seemingly different, it is observed that the Eurocode predicts similar results as the AFPA. Both solutions predict a consistently smaller buckling capacity than the FEA, with Eurocode on average being 0.92 of FEA and AFPA being 0.90. The Eurocode accounts for the load eccentricity (denoted as C e in the AFPA equation) by increasing le by 2d for a load applied at the compression edge and decreasing it by 0.5d when the load is applied at the tension edge, where d is the beam cross-section depth. From Section 1.3, it is noted that for a rectangular section the relationship between the effective length factor and the moment gradient factor can be approximated as le =l ¼ 1=C b . The moment gradient factor provided in the AFPA provisions for a mid-span concentrated load is 1.35. This would yield an equivalent effective length factor of 0.74. However, the effective length factor reported in the Eurocode is 0.8. The higher Eurocode effective length factor is presumed to include an allowance to account for the partial end fixity in a manner similar to the C fix coefficient in the AFPA [3].
Q. Xiao et al. / Engineering Structures 151 (2017) 85–92
91
Fig. 8. Load application detail (a) experimental details and (b) corresponding FEA model detail.
Fig. 9. Buckled configuration (a) experimentally observed and (b) FEA predicted.
4.2.2. Other loading and boundary conditions This section provides a review of the lateral torsional buckling provisions in the Eurocode and compares them to the FEA model. The two methods are evaluated for a wooden beam whose crosssection is 80 mm wide and 570 mm deep. The beam span is taken as 6 m. The modulus of elasticity parallel to grain is assumed to be 10300 MPa and the shear modulus is 474 MPa. Two types of boundary conditions are investigated; simply-supported and cantilever. Load types considered are uniform moments (to reflect reference point adopted in design standards), uniformly distributed load (UDL), and point load applied at beam’s top edge, centroid and bottom edge (to replicate practical situations). Table 2 shows the effective length factor le =l based on the Eurocode provisions and those based on the FEA model from:
le =l ¼
p
Mcr ðFEAÞl
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi EL Iy GL J
ð11Þ
where Mcr ðFEAÞ is the critical moment extracted from Abaqus FEA model. As expected, the value for uniform moments at beam centroid is an exact match between the Eurocode and the FEA model. According to the code provision, the effective length le shall be increased by 2d for compression edge loading and decreased by 0.5d for tension edge loading. This provision, which aims to incorporate the load position effect, works well for simply-supported beams under gravity loads where the beam top edge is in compression and the bottom edge is in tension. Table 2 shows that for simply supported beam cases the Eurocode equation provides values that are close to those obtained by the FEA model, while being slightly on the conservative side (le =l values larger than 1.0) especially for loads applied at the bottom edge. For cases involving simply-supported
beams under uniform moment and transverse loads applied at top and centroid, the conservativeness of the Eurocode is within 11%. For simply-supported beams under transverse loads applied at the bottom, the code exhibits higher conservativeness, with UDL effective length ratios of 1.16 and mid-span point load of 1.27. Two sets of values are provided in Table 2 for cases involving cantilevered beams under top and bottom edge loading. When applying the literal wording of the code, where adjustment for load position is based on tension and compression edge, a discrepancy in trends is observed between the FEA model predictions and the Eurocode results. Whereas, as expected, the FE analysis indicates that moving the load from bottom to top edge results in a decrease in the critical moments (and hence increases the effective length), an opposite trend is observed when applying the Eurocode values. This leads to the conclusion that the Eurocode wording is intended only for simply supported beams. Results from Table 2 also confirm this observation where the ratio between the Eurocode and the FEA model for cantilevered beams applied at beam bottom is 1.62 and 2.11 for point load and UDL, respectively. Furthermore, for top loading cases, the code provides non-conservative results. Based on the above observations, it is proposed that the code wording be modified for cantilevers to address adjustment for top and bottom edge loading rather than using tension and compression edges. The results based on the proposed code revisions are provided in the last two rows of Table 2. The table shows that, in general, the Eurocode predicts effective length factors that are greater than those from the FEA, thus providing conservative critical moments. For cantilevered beams, the ratio between Eurocode and FEA ranges from 1.18 to 1.38. The high level of conservativeness can account for the fact that a perfect fixity condition at the cantilever root is difficult to achieve in wood construction.
92
Q. Xiao et al. / Engineering Structures 151 (2017) 85–92
Table 2 Comparison between Eurocode and FEA for different boundary and loading conditions. Beam type
Load type
1
2
Effective length factor
Ratio
Eurocode
Simply supported
Cantilever (effective length based on present wording of Eurocode) Cantilever (effective length based on proposed wording)
Uniform moment UDL Mid-span point load UDL Point load at tip UDL Point load at tip
Computed from FEA
3
4
5
6
7
8
9
10
11
Top edge
Centroid
Bottom edge
Top edge
Centroid
Bottom edge
EC/FEA (Top edge)
EC/FEA (Centroid)
/
1
/
/
1
/
/
1.00
EC/FEA (Bottom edge) /
1.09 0.99
0.90 0.80
0.85 0.75
1.02 0.89
0.87 0.72
0.74 0.59
1.07 1.11
1.04 1.11
1.16 1.27
0.45 0.75
0.50 0.80
0.69 0.99
0.51 0.80
0.39 0.68
0.33 0.61
0.88 0.94
1.27 1.18
2.11 1.62
0.69 0.99
0.50 0.80
0.45 0.75
0.51 0.80
0.39 0.68
0.33 0.61
1.34 1.24
1.27 1.18
1.38 1.23
5. Conclusions References An experimental program was conducted to investigate the elastic lateral torsional buckling capacity of wooden beams. Eighteen full-scale tests were tested. For each specimen, nondestructive material tests were first conducted to determine the shear and longitudinal elastic moduli, followed by a lateral torsional buckling test. A 3D finite element model was developed to predict the elastic lateral torsional buckling resistance for each specimen. The model features orthotropic material representation and is based on material properties as determined in nondestructive tests. The model provides a realistic representation of boundary conditions and loading details of the tests. The following conclusions can be drawn from the current research project: – The FEA model is observed to reliably predict the elastic lateral torsional buckling capacity of the wooden beams. The test to FEA critical moment ratios were on average 1.02 with a standard deviation of 0.13. – The Eurocode was compared to the AFPA guidelines for simply supported beams with top edge loading and the results showed that although the approach adopted by the two codes is seemingly different, solutions from both codes were similar and predicted a consistently smaller buckling capacity than the FEA. Eurocode reports higher effective length factor than AFPA, which is presumed to include an allowance to account for the partial end fixity. – Comparison between the Eurocode and FEA model for simply supported end conditions seems to provide a reasonable fit with the Eurocode values being slightly conservative. However, when the cantilevered beam values are evaluated, the Eurocode provisions seem to be overly conservative for the case of bottom edge loading and non-conservative for the case of top edge loading. The observation suggests that current wording seems to be intended for simply-supported beams under gravity loads where the beam top edge is in compression and the bottom edge is in tension. – More accurate wording has been proposed regarding the effective length adjustment for top and bottom edge loading rather than using tension and compression. The results based on the proposed wording provides critical moment predictions that are more consistent for cantilevers under top and bottom edge loading.
[1] EN 1995-1-1 Eurocode 5 – design of timber structures, Part 1–1, general: common rules and rules for buildings. Brussels (Belgium): CEN; 2014. [2] AFPA. Technical report 14: designing for lateral-torsional stability in wood members. Washington, D.C.: American Forest and Paper Association; 2003. [3] CAN-CSA O86. Engineering design in wood. Mississauga (Ontario, Canada): Canadian Standard Association; 2015. [4] NDS. National design specification for wood construction, national design specifications. Washington, D.C.: American Forest and Paper Association; 2015. [5] Isopescu D, Stanila O, Asatanei I. Analysis of wood bending properties on standardized samples and structural size beams tests. Buletinul Insititutului Politehnic Din Iasi, Publicat de Universitatea Tehnica˘, Gheorghe Asachi” din Iasßi, Tomul LVIII (LXII), Fasc 1, Sectia Constructii Arhitectura; 2012. [6] Xiao Q, Doudak G, Mohareb M. Lateral torsional buckling of wood beams: FEAmodelling and sensitivity analysis. In: World conference in timber engineering, Quebec City, Quebec, Canada, August 10–14. [7] Timoshenko S. Theory of elastic stability. 2nd ed. McGraw-Hill Book Co; 1961. [9] Hooley RF, Madsen B. Lateral buckling of glued laminated beams. J Struct Eng Div ASCE 1964;90(ST3):201–303. [10] Hindman DP, Harvey HB, Janowiak JJ. Measurement and prediction of lateral torsional buckling loads of composite wood materials: rectangular section. For Product J 2005;55(9):42–7. [11] Hindman DP, Harvey HB, Janowiak JJ. Measurement and prediction of lateral torsional buckling loads of composite wood materials: I-joist sections. For Product J 2005;55(10):43–8. [12] Burow JR, Manbeck HB, Janowiak JJ. Lateral stability of composite wood I-joists under concentrated-load bending. Am Soc Agric Biol Eng 2006;49: 1867–80. [13] Nethercot DA. Elastic lateral buckling of beams. In: Narayanan R, editor. Beams and beam columns: stability in strength. Barking, Essex (UK): Applied Science Publishers; 1983. [14] Suryoatmono B, Tjondro A. Lateral-torsional buckling of orthotropic rectangular section beams. Bandung (Indonesia): Department of Civil Engineering, Parahyangan Catholic University; 2008. [15] Hindman DP. Lateral buckling of I-joist. VA (USA): Department of Wood Science and Forest Products, Virginia Tech Blacksburg; 2008. [16] Bamberg CR. Lateral movement of unbraced wood composite I-joists exposed to dynamic walking loads [M. A. Sc. thesis]. Blacksburg (Virginia, USA): Virginia Polytechnic Institute and State University; 2009. [17] Du Y, Mohareb M, Doudak G. Non-sway model for lateral torsional buckling of wooden beams under wind uplift. J Eng Mech 2016;142(12). [18] Hu Y, Mohareb M, Doudak G. Effect of eccentric lateral bracing stiffness on lateral torsional buckling resistance of wooden beams. Int J Struct Stabil Dyn 2017. [19] ASTM. D198 standard test method of static tests of lumber in structural sizes, Philadelphia, PA; 2009. [20] Simulia. ABAQUS analysis user’s manual (Version 6.11). Dassault Systèmes; 2011. [21] FPL (Forest Products Laboratory). Wood handbook-wood as an engineering material, Wisconsin; 2010.