Experimental Study of Beam Stability Factor of Sawn Lumber Subjected to Concentrated Bending Loads at Several Points

: The beam stability factor ( C L ) is applied in construction practices to adjust the reference bending design value ( F b ) of sawn lumber to consider the lateral-torsional buckling. Bending tests were carried out on 272 specimens of four wood species, namely, red meranti ( Shorea sp.), mahogany ( Swietenia sp.), pine ( Pinus sp.), and agathis ( Agathis sp.), to analyze a simply supported beam subjected to concentrated loads at several points. The empirical C L value is a ratio of the modulus of rupture ( S R ) of a specimen to the average S R of the standard-size specimens. The non-linear regression estimated the Euler buckling coefﬁcient for sawn lumber beam ( K bE ) in this study as 0.413, with 5% lower and 5% upper values of 0.338 and 0.488. Applying the 2.74 factor, which represents an approximately 5% lower exclusion value on the pure bending modulus of elasticity ( E min ) and a factor of safety, the adjusted Euler buckling coefﬁcient ( K bE (cid:48) ) value for a timber beam was 1.13 (0.92–1.34), which is within the range approved by the NDS ( K bE (cid:48) = 1.20). This study harmonizes the NDS design practices of C L computation with the empirical results. Because agathis has the lowest ductility ( µ ), most natural defects (smallest strength ratio, S ), and highest E/S R ratio, the agathis beam did not twist during the bending test; instead, it failed before twisting could occur, indicating inelastic material failure. Meanwhile the other specimens (pinus, mahogany, and red meranti), which have smaller E/S R ratio, higher ductility, and less natural defects, tended to fail because of lesser beam stability. This phenomenon resulted in the C L curve of agathis being the highest among the others. The C L value is mathematically related to the beam slenderness ratio ( R B ) and the E / S R ratio. Because the strength ratio ( S ) and ductility ratio ( µ ) have signiﬁcant inverse correlations with the E / S R ratio, they are correlated with the C L value. Applying the C L value to adjust the characteristic bending strength is safe and reliable, as less than 5% of the specimens’ S R data points lie below the curve of the adjusted characteristics values.


Introduction
Timbers are often used as structural components such as beams, columns, and trusses in residential housing [1], multi-story buildings [2], cooling towers [3][4][5], and bridges [6]. Beam design involves analyzing member strength, stability, and stiffness for four essential criteria (e.g., bending (including stability), deflection, horizontal shear, and bearing). In addition to mechanical resistance and serviceability, a designer must verify other requirements (e.g., environmental condition, durability aspects, fire resistance, and fatigue) to build a timber construction. Timber beam strength is commonly determined through edgewise bending tests with the loads applied to the narrow face [7,8]. Edgewise bending stiffness and strength are the essential grade-determining properties (GDP) in wood strength After 1997, the NDS has chosen to report E min rather than E; thus, the K bE value of 0.438 is replaced with 1.20 of the adjusted Euler buckling coefficient (K bE ) (Equation (8)) to correspond to the factor of 2.74. The factor of 2.74 represents a 5% lower exclusion value of the modulus of elasticity (E min ) (1.03 times tabulated E values) and a safety factor of 1.66 [16]. The F bE calculation by K bE and E min follows Equation (8). The SNI 7973:2013 [33] and the current NDS currently designate this value.
Several reports on timber beam stability subjected to bending are available in the literature. Hindman et al. [18] studied lateral-torsional buckling on rectangular cantilever beams to validate the C L formula and reported that the C L formula was non-conservative for Laminated Veneer Lumber (LVL), although it was suitable for solid sawn lumber. Kimble and Bender [42] studied the stability of built-up timber beams with various E values in which the beam has a slenderness ratio value of 26.20 (unbraced long beams); their results showed that the obtained C L value was low. Du et al. [43,44] reported that the deck-board stiffness significantly affected the lateral-torsional buckling capacity of twin-beam deck systems. Hu et al. [45,46] developed an energy-based solution to analyze the lateraltorsional buckling of a wooden beam with a midspan lateral brace and indicated that such beams were prone to symmetric and antisymmetric mode buckling patterns. St-Amour and Doudak [47] conducted a sensitivity analysis on a wood I-joist and determined that the critical buckling load is affected by the longitudinal modulus of elasticity, the transverse shear modulus of the flanges, and the elastic modulus of the web. They determined that the equivalent moment factor theory is more adaptable than the Euler elastic buckling theory to predict the I-joist's critical buckling moment [48]. Sahraei et al. [49] proposed a simplified expression to account for moment gradient, load height, and pre-buckling deformation effects for elastic lateral-torsional buckling of wooden beams.
The available standards designate various methods for designing a beam able to resist lateral-torsional buckling, however, these approaches are significantly different and lack consistency [50]. Balaz [51] proposed formulas to harmonize the different rules in the Eurocode concerning critical moments and suggested replacing the warping constant of a solid rectangular cross-section beam. AFPA [52] described the basis of the current effective length approach used in the NDS and summarized the equivalent uniform moment factor approach, providing a comparison between the two approaches and proposing modifications to the NDS design provisions. The present study aims to analyze the response of beams made from several locally available commercial lumber varieties subjected to concentrated loads at several points in order to validate the beam stability factor (C L ) formula designated by NDS 2012 [35], which has been adopted by SNI 7973:2013 [33]. Because red meranti, mahogany, pinus, and agathis sawn lumber are abundantly available in the building material market and are commonly used for structural construction members, they were chosen as the specimens in this study. Center point loading bending tests of small clear specimens and several point loading bending tests of full-sized specimens were conducted to measure the empirical beam stability factor (C L,e ), then the non-linear regression was employed to estimate the K bE value. The empirical C L values, estimated best-fit curve, and 5% upper and 5% lower curves were plotted in Cartesian Diagrams together with the building code's designated graph to justify the harmony of the empirical and designated C L values.

Specimen Preparation
The materials were 272 pieces of specimens consisting of 200 small pieces (2.5 × 2.5 × 41) cm 3 and 72 full-size pieces (4 × 10 × 130) cm 3 . The specimens were sawn lumber of red meranti (Shorea sp.), mahogany (Swietenia sp.), pine (Pinus sp.), and agathis (Agathis sp.) purchased from commercial timber markets in Bogor, West Java-Indonesia. All specimens were air-dried in indoor environmental conditions (27 • C and 80% RH) for a month to reach equilibrium moisture content. All experimental work was undertaken in the Wood Engineering Laboratory, Faculty of Forestry and Environment, IPB University (Bogor, Indonesia).

Small Specimen Bending Test
The small specimens of red meranti, mahogany, pine, and agathis, 50 pieces each, were prepared for the bending test. The length (L), width (b), and depth (d) of each specimen were measured employing a digital caliper with an accuracy of 0.01 mm. The masses were measured three times: before the bending test (m 0 ), after the bending test (m 1 ), and after oven-drying (m ot ). The oven-drying was carried out at (103 ± 2) • C for two days to evaporate water content in the specimens. Moisture content measurements were conducted soon after the bending test. The moisture content (M c ) was calculated following Equation (9).
The static bending test was carried out following the ASTM D143 secondary method [20] (Figure 1). The span of the specimen was 360 mm in order to maintain a span-to-depth ratio of 14. The specimens were then bending tested with center-point loading using a five-ton capacity Instron type 3369 Universal Testing Machine (UTM). Both supports were rollers with 2 cm diameter. The load was applied continuously throughout the test at the movable crosshead motion rate of 1.3 mm/min. The modulus of elasticity (E) and modulus of rupture (S R ) values were calculated following Equations (10) and (11). The E values were calculated within the proportional region of the load-deflection curve. (11) of rupture (SR) values were calculated follow were calculated within the proportional region The small specimens' E and SR statistics d (s), minimum value, maximum value, and coef The 5% exclusion limit (R0.05) values of E and S strength characteristic value (Rk) following A lognormal, and Weibull standard distribution and the best choice was determined through t Material ductility may affect the beam s undergo significant deformation in the plastic ten expressed as the ratio between ultimate dis (Equation (12)) [55]. Muñoz et al. [56] stated th Equation (12) as the ductility ratio (μ), while The small specimens' E and S R statistics data, including the mean, standard deviation (s), minimum value, maximum value, and coefficient of variance (CV), were summarized. The 5% exclusion limit (R 0.05 ) values of E and S R were calculated to determine the bending strength characteristic value (R k ) following ASTM D2915 [53] and D5457 [54]. Normal, lognormal, and Weibull standard distributions were applied to fit the experimental data, and the best choice was determined through the Anderson-Darling test [8,12,13].
Material ductility may affect the beam stability. Ductility is a structure's ability to undergo significant deformation in the plastic ranges before its collapse [55], which is often expressed as the ratio between ultimate displacement (∆ u ) and yield displacement (∆ y ) (Equation (12)) [55]. Muñoz et al. [56] stated the ultimate and yield displacement ratio in Equation (12) as the ductility ratio (µ), while Jorissen and Fragiacomo [57] termed it the ductility factor. Ultimate displacement (∆ u ) is the displacement at the ultimate load (F u ), while yield displacement (∆ y ) can be calculated following several methods suggested by researchers [58][59][60] and standard specifications [61][62][63]. The yield displacement (∆ y ) in this study is equal to deformation at 0.5F u following Karacabeyli and Ceccotti [58].

Full-Size Specimen Bending Test
Full-size specimens of red meranti, mahogany, pine, and agathis, 18 pieces each, were prepared. Every sawn lumber piece was visually graded following ASTM D245 [64] ( Figure 2). Defects and imperfections in the wood were identified and measured. Three indicating defects were observed, namely, the slope of the grain, knot diameter, and the presence of shakes, checks, and splits, then converted into strength ratio (S) values.
The values of E and SR in the edgewise configuration bendin employing UTM SATEC/Baldwin. The span of the bending test mens were tested in simply-supported bending with various co figurations, namely, center point loading, third point loading, fo point loading, and sixth point loading ( Figure 3). Lateral suppor point of loading. Figure 3 shows the position of the five linear vari ducers (LVDTs) which were employed to measure the deflectio tests with a center-point loading configuration and without late out. The formulae of E and SR with various loading configurat Table 1.

Loading Configuration
Bending F er-point loading with or without lateral support The length (L) of each specimen was measured employing a tape measure, while width (b) and depth (d) were measured using a digital caliper with an accuracy of 0.01 mm. The mass was measured three times: before the bending test (m 0 ), after the bending test (m 1 ), and after oven-drying (m ot ). The specimens used for measuring m 1 and m ot were 4 × 4 × 4 cm 3 and were cut near the bending test specimen's failure position. The M c value was calculated according to Equation (9). The density (ρ) was calculated in air-dry conditions before the bending test of a full-size specimen (Equation (13)), while the relative density or specific gravity (G b ) was calculated according to Equation (14).
The values of E and S R in the edgewise configuration bending tests were determined employing UTM SATEC/Baldwin. The span of the bending test was 120 cm. The specimens were tested in simply-supported bending with various concentrated loading configurations, namely, center point loading, third point loading, fourth point loading, fifth point loading, and sixth point loading ( Figure 3). Lateral supports were applied at every point of loading. Figure 3 shows the position of the five linear variable displacement transducers (LVDTs) which were employed to measure the deflection. As a control, bending tests with a centerpoint loading configuration and without lateral support were carried out. The formulae of E and S R with various loading configurations are summarized in Table 1.
Note: P max = maximum load borne by beam or column loaded to failure (N), P = increment of applied load below the proportional limit (N), L = span of a beam (mm), ∆ = increment of deflection of beam's neutral axis measured at midspan over distance L and corresponding load P (mm), b = width of beam (mm), d = depth of beam (mm).

Beam Stability Factor (C L ) Value Calculation
The effective span length (l e ) of the beams was determined following Table 2. The beam slenderness ratio (R B ) of each specimen was calculated (Equation (1)), while the critical buckling design value of a beam (F bE ) was calculated per Equation (7). The empirical C L (C L,e ) values from the experimental study were calculated per Equation (15). S Ri is the modulus of rupture (MOR) value of each specimen. S R is the average MOR value of the small-size specimen (b = d = 2.5 cm), tested in bending at a standardized condition following ASTM D143 secondary methods [20]. Table 2. Effective span length (l e ) for bending members.

No Single Span Beam
Effective Span Length (l e ) l u /d < 7 l u /d ≥ 7 1. center-point loading without lateral support l e = 1.8 l u l e = 1.37 l u + 3d 2.
center-point loading with lateral support l e = 1.11 l u 3.
third point loading with lateral supports l e = 1.68 l u 4.
fourth point loading with lateral supports l e = 1.54 l u 5.
fifth point loading with lateral supports l e = 1.68 l u 6.
sixth point loading with lateral supports l e = 1.73 l u Note: l e is effective span length for bending members (mm), l u is the distance between such points of intermediate lateral support (mm).
The non-linear regression model was derived from Equation (6), with F bE substituted by Equation (7), and is presented in Equation (16). Then the non-linear parameter for beam (c) was set to 0.95, E was the small specimens' average modulus of elasticity, F b was the experimental value of S R , and the K bE value was estimated by the regression coefficient (a) (Equation (17)). The non-linear regression analysis with Levenberg-Marquardt algorithm iteration [65] was applied to estimate the empirical value of K bE from experimental results. The starting value of the iteration was 0.438, following Zahn [41].
whereŷ is the estimated empirical C L obtained through experimental study (C L,e ), x is E/(S R × R B 2 ), c is set to 0.95, and a is the regression coefficient. The regression coefficient (a) estimates the Euler buckling coefficient (K bE ).

Moisture Content (M c ), Density (ρ), and Specific Gravity (G b )
Many physical and mechanical properties of wood depend upon its moisture content. The specimens' average moisture content ranged from 13.45% to 16.98% (Figure 4a), the common air-dry moisture content in Bogor, West Java Indonesia [13,14,[66][67][68]. The specimens' moisture content is within the permissible range in the Indonesian Wooden Building Code (PKKI) regulation [69], which states that the average air-dry moisture content is 15% and its range is 12-18%. The wood density (ρ) depends on its moisture content because both the mass and volume of wood depend upon it. The density of air-dry wood varies significantly between species. The wood relative density or specific gravity (G b ) in this study was referenced based on air-dry volume and oven-dry mass. The average results of ρ and G b measured in this experimental study are presented in Figure 4, and conform to the ρ value stated in PKKI [69].
ts 2022, 13, x FOR PEER REVIEW 9 of Building Code (PKKI) regulation [69], which states that the average air-dry moisture co tent is 15% and its range is 12-18%. The wood density (ρ) depends on its moisture conte because both the mass and volume of wood depend upon it. The density of air-dry woo varies significantly between species. The wood relative density or specific gravity (Gb) this study was referenced based on air-dry volume and oven-dry mass. The average r sults of ρ and Gb measured in this experimental study are presented in Figure 4, and co form to the ρ value stated in PKKI [69].

Flexural Properties of the Small Specimens
The average modulus of elasticity (E) and modulus of rupture (MOR, SR) values the small specimens of red meranti, mahogany, pine, and agathis are summarized in Tab 3; they were similar to previous reports [70][71][72][73]. Based on their respective E values, r meranti is graded in the E16-E27 strength class, mahogany is E10-E20 strength class, pi is E6-E21 strength class, and agathis is E12-E17 strength class, per SNI 7973:2013 [3 Similar to Firmanti et al. report on tropical timber [8], the Weibull distribution was t best fit among others for most of the experimental data. The ASTM D5457 [54] standa distribution for wood strength is assumed to be the Weibull distribution, while Europe countries commonly designate a log-normal distribution. In this study, the Anderso Darling tests evaluated the goodness of fit of experimental data (modulus of elasticity ( and modulus of rupture (SR)), and the best-fit distribution was chosen among three stan ard distribution (Weibull, lognormal, and normal) ( Table 4). The characteristic bendi values (Emin and Fb) were determined by estimating their 5% exclusion limits. The 5% e clusion limits (R0.05) of E and SR are presented in Table 4, with the best-fit values we shown in bold font.

Flexural Properties of the Small Specimens
The average modulus of elasticity (E) and modulus of rupture (MOR, S R ) values of the small specimens of red meranti, mahogany, pine, and agathis are summarized in Table 3; they were similar to previous reports [70][71][72][73]. Based on their respective E values, red meranti is graded in the E16-E27 strength class, mahogany is E10-E20 strength class, pine is E6-E21 strength class, and agathis is E12-E17 strength class, per SNI 7973:2013 [33]. Similar to Firmanti et al. report on tropical timber [8], the Weibull distribution was the best fit among others for most of the experimental data. The ASTM D5457 [54] standard distribution for wood strength is assumed to be the Weibull distribution, while European countries commonly designate a lognormal distribution. In this study, the Anderson-Darling tests evaluated the goodness of fit of experimental data (modulus of elasticity (E) and modulus of rupture (S R )), and the best-fit distribution was chosen among three standard distribution (Weibull, lognormal, and normal) ( Table 4). The characteristic bending values (E min and F b ) were determined by estimating their 5% exclusion limits. The 5% exclusion limits (R 0.05 ) of E and S R are presented in Table 4, with the best-fit values were shown in bold font.  Ductility is a material's ability to attain high displacement without losing too much strength. For clear wood, the stress distribution in a bending member is related to the compression zone's plasticization. For structural timber-containing defects, this only occurs when the defects are located mainly in the compression zone [57]. The bending test's loaddisplacement curves ( Figure 5) proved that each wood species has a different proportional limit and maximum load. The ductility ratio (µ) of timbers from the experimental results are presented in Table 5. Because the ductility ratio ranges were 2.12-5.32, timber subjected to a bending load may be classified from low to medium ductility according to Eurocode 8 [74]. Agathis had the lowest µ value, which indicates that agathis subjected to bending is the most brittle among other timbers in this study. The agathis specimens were classified as low ductility because their ductility ratio was equal to or less than 4 (µ ≤ 4). At the same time, red meranti, mahogany, and pines varied from low to medium ductility. The µ value is thought to affect the beam stability factor (C L ), as the µ is inversely related to E/S R . In this study, agathis had the highest E/S R value (Table 3). Ishiguri et al. [73] previously reported that agathis had a high modulus of elasticity (E) and a low modulus of rupture (S R ) value. Ductility is a material's ability to attain high displacement without losing too mu strength. For clear wood, the stress distribution in a bending member is related to t compression zone's plasticization. For structural timber-containing defects, this only curs when the defects are located mainly in the compression zone [57]. The bending tes load-displacement curves ( Figure 5) proved that each wood species has a different p portional limit and maximum load. The ductility ratio (μ) of timbers from the expe mental results are presented in Table 5. Because the ductility ratio ranges were 2.12-5. timber subjected to a bending load may be classified from low to medium ductility cording to Eurocode 8 [74]. Agathis had the lowest μ value, which indicates that agat subjected to bending is the most brittle among other timbers in this study. The agat specimens were classified as low ductility because their ductility ratio was equal to or l than 4 (μ ≤ 4). At the same time, red meranti, mahogany, and pines varied from low medium ductility. The μ value is thought to affect the beam stability factor (CL), as th is inversely related to E/SR. In this study, agathis had the highest E/SR value (Table  Ishiguri et al. [73] previously reported that agathis had a high modulus of elasticity and a low modulus of rupture (SR) value.

Strength Ratio of the Full-Size Specimens
Wood, as a natural material, has imperfections and defects, which may reduce its strength. The presence of knots [75], the slope of the grain [76], and shakes, checks, and splits can all reduce the E and S R of timber. Imperfections and defects in wood are often used to determine the wood quality class using visual grading. Visual grading is conducted by converting wood imperfections and defects into a strength ratio (S) to be applied to adjust the strength of a wooden member [77]. S is the ratio of wood strength with defects on its surface to the clear specimen wood strength, and is expressed as a percentage (%) [78]. The S value of the full-size specimens in this study is presented in Table 6. Red meranti had the highest S value, followed by mahogany and pine, while the lowest S value belonged to agathis. A higher S value indicates that the wood has fewer defects. Red meranti and mahogany, which are hardwoods, have a few small knots and their grain direction is relatively straight, however, they have many shakes, checks, and splits. Pine and agathis, which are softwoods, have many knots with large diameter, and the slope of the grain direction means that the S value is low. Each timber species has a different defect variation. The difference in S values is influenced by the wood-working process as well [77]. The process by which logs are converted to sawn lumber affects the slope of the grain. In addition, cutting wood affects the variations in the grain direction. In this study, each specimen's grain direction varied. Distortions of grain direction occur around knots as well, causing large variations in the wood's mechanical properties. The knots and the slope of the grain are factors that significantly affect the S value, especially the slope of grain through the knots. The knots and the slope of the grain have an inverse relationship to wood strength. Timber boards with a 1 in 10 slope of the grain have a 24% lower strength than straight-grained wood [76]. The E value of wood with knots is lower than the E value of wood without knots [79]. Knot size has a negative correlation with the modulus of elasticity (E) and the modulus of rupture (S R ) [80].

Flexural Properties of the Full-Size Specimens in Various Loading Configurations
Six loading configurations were programmed in the full-size bending test. Both supports were 38 mm diameter rollers. The modulus of elasticity (E) and modulus of rupture (S R ) value of full-size specimens resulting from the experimental study are shown in Figure 6. Red meranti had the highest E and S R values, followed by mahogany, pine, and agathis. Red meranti and mahogany, which are hardwoods, have higher E and S R values than pine and agathis (softwood).
ts 2022, 13, x FOR PEER REVIEW 12 o agathis. Red meranti and mahogany, which are hardwoods, have higher E and SR val than pine and agathis (softwood). When more loading points are applied, the measured values of E and SR are hig because when the load is spread more evenly, the load received by the beam is m widely distributed, meaning that the beam can withstand a higher load. In the cent point loading configuration, the beam receives all loads from one source; thus, it reac its critical load and failure more quickly. These results strengthen the findings Brancheriau et al. [81], who reported that the timber beam in the third point loading c figuration produced an E value 19% higher than center-point loading. This shows t more points of loading applied to the bending test lead to higher the E and SR values.

Beam Stability Factor (CL)
The small specimens' beam slenderness ratio (RB) ranged from 4.4 to 5.6, while the full-size specimens it ranged from 4.5 to 11.1. The full-size specimens with appl center-point loading without lateral support were categorized as intermediate beams = 10.6-11.1) in this study, while the other specimens with a slenderness ratio less than (RB ≤ 10) were categorized as short beams. The RB value was affected by the specimen s irrespective of the loading configuration and lateral support. When more points of load and lateral supports are applied, the unsupported length (lu) is shorter, and thus the value is lower.
The ratio between the modulus of elasticity and the modulus of rupture (E/SR) is n essary to calculate the CL value. The E/SR value is an inherent property of wood species is assumed to be constant, as the modulus of elasticity (E) is significantly correlated w the modulus of rupture (SR) (e.g., eucalyptus wood [82], teak and ebony timber [83], tr When more loading points are applied, the measured values of E and S R are higher because when the load is spread more evenly, the load received by the beam is more widely distributed, meaning that the beam can withstand a higher load. In the center-point loading configuration, the beam receives all loads from one source; thus, it reaches its critical load and failure more quickly. These results strengthen the findings of Brancheriau et al. [81], who reported that the timber beam in the third point loading configuration produced an E value 19% higher than center-point loading. This shows that more points of loading applied to the bending test lead to higher the E and S R values.

Beam Stability Factor (C L )
The small specimens' beam slenderness ratio (R B ) ranged from 4.4 to 5.6, while for the full-size specimens it ranged from 4.5 to 11.1. The full-size specimens with applied center-point loading without lateral support were categorized as intermediate beams (R B = 10.6-11.1) in this study, while the other specimens with a slenderness ratio less than 10 (R B ≤ 10) were categorized as short beams. The R B value was affected by the specimen size, irrespective of the loading configuration and lateral support. When more points of loading and lateral supports are applied, the unsupported length (l u ) is shorter, and thus the R B value is lower.
The ratio between the modulus of elasticity and the modulus of rupture (E/S R ) is necessary to calculate the C L value. The E/S R value is an inherent property of wood species; it is assumed to be constant, as the modulus of elasticity (E) is significantly correlated with the modulus of rupture (S R ) (e.g., eucalyptus wood [82], teak and ebony timber [83], tropical timber [8], Guadua bamboo [12], Gigantochloa apus bamboo [29,30]). Because E is strongly correlated with the modulus of rupture (S R ), E is a common indicating predictor for strength grading [8,84]. The average E/S R values of small clear specimens of red meranti, mahogany, pine, and agathis in this study were 152, 114, 131, and 336, respectively (Table 3). This E/S R was applied for calculating the C L value of both small and full-size specimens. The higher the E/S R , the higher the C L value, meaning that the beam can more stably resist lateral-torsional buckling. The C L,e values of the small specimens of red meranti, mahogany, pine, and agathis obtained in this study ranged from 0.61-1.25, 0.55-1.35, 0.60-1.39, and 0.85-1.31, respectively, while those of the full-size specimens ranged from 0.41-1.32, 0.22-1.36, 0.32-1.22, and 0.77-1.22, respectively. The variation of the C L,e value is affected by the beam slenderness ratio (R B ) and the ratio of the modulus of elasticity to the modulus of rupture (E/S R ).
The parameter estimations of non-linear regression analysis under Equation (17) are presented in Table 7, and the best-fit formula is Equation (18). The estimated Euler buckling coefficient (K bE ) value obtained in this study was 0.413, with 5% lower and 5% upper values of 0.338 and 0.488. These values were not significantly different from the K bE value proposed by Zahn (K bE = 0.438) [41]. The C L,e value of this experimental study was curve fitted using non-linear estimation (Figure 7). Zahn's C L curve [41] almost coincides with the estimated empirical C L curve obtained in this study, and is between the 5% lower and 5% upper estimation value of K bE . The parameter estimations of non-linear regression analysis under Equation (17 presented in Table 7, and the best-fit formula is Equation (18). The estimated Euler b ling coefficient (KbE) value obtained in this study was 0.413, with 5% lower and 5% up values of 0.338 and 0.488. These values were not significantly different from the KbE v proposed by Zahn (KbE = 0.438) [41]. The CL,e value of this experimental study was cu fitted using non-linear estimation (Figure 7). Zahn's CL curve [41] almost coincides w the estimated empirical CL curve obtained in this study, and is between the 5% lower 5% upper estimation value of KbE.
, R 2 = 32.90%  Multiplying KbE by the factor of 2.74 results in the adjusted Euler buckling coeffic (KbE′), for which the value found in this study was 1.13. The obtained KbE′ in this st was similar to that of the NDS (KbE′ = 1.20) [35]. The 5% lower and 5% upper values of were 0.92 and 1.33, respectively, which includes the KbE′ value proposed by the NDS cause the empirical KbE′ is similar to that of the NDS, this experimental study appr the NDS CL equation, and we therefore propose to continue using it. The NDS CL equa adopted by SNI 7973:2013 [33] proves reliable for adjusting the reference bending stren of red meranti, mahogany, pines, and agathis sawn lumber.  Multiplying K bE by the factor of 2.74 results in the adjusted Euler buckling coefficient (K bE ), for which the value found in this study was 1.13. The obtained K bE in this study was similar to that of the NDS (K bE = 1.20) [35]. The 5% lower and 5% upper values of K bE were 0.92 and 1.33, respectively, which includes the K bE value proposed by the NDS. Because the empirical K bE is similar to that of the NDS, this experimental study approves the NDS C L equation, and we therefore propose to continue using it. The NDS C L equation adopted by SNI 7973:2013 [33] proves reliable for adjusting the reference bending strength of red meranti, mahogany, pines, and agathis sawn lumber.
The obtained K bE values differ among wood species because of the variation of E/S R ; thus, the estimated C L formula on all R B ranges from this experiment can be inferred (Figure 8) for each timber species, then compared to the NDS formula. The C L curve positions of red meranti, mahogany, and pine were below that of the NDS, while the C L curve of agathis was above it. Agathis has the highest C L value among the other timbers. The parameter that mostly affects C L value is E/S R . The E/S R value of agathis is the highest due to its low modulus of rupture (S R ) value ( Table 3). The order of the C L curve positions ( Figure 8) proportionally corresponds to the E/S R values. A higher E/S R value means that the C L curve will be in the upper position. The C L value curve of agathis was the highest among the others because the agathis beam did not twist during the bending test; it failed before twisting could occur, indicating inelastic material failure. While red meranti, mahogany, and pine twisted when applying the edgewise mid-point loading bending test, failure occurred when twisting caused the specimen to turn flatwise or slip from the supports, indicating elastic lateral buckling failure. Inelastic failure of the beam happened with small deformation, while elastic member failure was indicated by the long deformation. The deformation value at the yield load is related to the ductility. s 2022, 13, x FOR PEER REVIEW 14 the others because the agathis beam did not twist during the bending test; it failed be twisting could occur, indicating inelastic material failure. While red meranti, mahog and pine twisted when applying the edgewise mid-point loading bending test, failur curred when twisting caused the specimen to turn flatwise or slip from the supports dicating elastic lateral buckling failure. Inelastic failure of the beam happened with s deformation, while elastic member failure was indicated by the long deformation. deformation value at the yield load is related to the ductility. The strength ratio (S) is thought to affect the beam stability factor. The variabili wood mechanical properties is increased in structural timber members due to defects imperfections (e.g., knots, the slope of the grain, and shakes, checks, and splits) Agathis contains the most defects among the others, and as such its S value is the low If ascendingly sorted according to their S value, agathis was the lowest, followed by p mahogany, and red meranti ( Table 5). The order of the S value was inversely proporti to the CL curve position because it is negatively correlated with E/SR (Figure 9a). A lo S value indicates that wood has many defects, in which case the CL curve lies at the u position. Pelletier and Doudak [86] investigated the lateral-torsional buckling of woo joists to account for the effect of the actual end condition and initial imperfection beam, and reported that the initial imperfection influences the nonlinear behavior. The strength ratio (S) is thought to affect the beam stability factor. The variability of wood mechanical properties is increased in structural timber members due to defects and imperfections (e.g., knots, the slope of the grain, and shakes, checks, and splits) [85]. Agathis contains the most defects among the others, and as such its S value is the lowest. If ascendingly sorted according to their S value, agathis was the lowest, followed by pine, mahogany, and red meranti ( Table 5). The order of the S value was inversely proportional to the C L curve position because it is negatively correlated with E/S R (Figure 9a). A lower S value indicates that wood has many defects, in which case the C L curve lies at the upper position. Pelletier and Doudak [86] investigated the lateral-torsional buckling of wood I-joists to account for the effect of the actual end condition and initial imperfection of a beam, and reported that the initial imperfection influences the nonlinear behavior. mahogany, and red meranti ( Table 5). The order of the S value was inversely proport to the CL curve position because it is negatively correlated with E/SR (Figure 9a). A l S value indicates that wood has many defects, in which case the CL curve lies at the u position. Pelletier and Doudak [86] investigated the lateral-torsional buckling of wo joists to account for the effect of the actual end condition and initial imperfection beam, and reported that the initial imperfection influences the nonlinear behavior. Figure 9. Correlation between the strength ratio (S) and E/SR of the small specimen (a) and bet the ductility ratio (μ) and E/SR of the full-size specimen (b). Both strength ratio (S) and ductility (μ) are negatively correlated with E/SR. Jorrissen and Fragiacomo [57] reported that wooden tension members, most ben members, and most connections are brittle; columns (buckling) and certain connec Jorrissen and Fragiacomo [57] reported that wooden tension members, most bending members, and most connections are brittle; columns (buckling) and certain connections are semi-ductile, while compression (both parallel and perpendicular to the grain), connection controlled by embedment, and/or steel failure can be categorized as ductile. In this study, the specimens subjected to a mid-point loading test without lateral support (the intermediate beams) mostly experienced twisting before reaching their failure points. Several agathis intermediate beams did not twist until failure. Agathis subjected to bending moment is the most brittle among others, as its ductility ratio (µ) value is the smallest. The failures of the full-size specimens of agathis occurred before twisting, and only a small deformation occurred before failure; thus, agathis subjected to bending can be categorized as brittle. The maximum load of red meranti, mahogany, and pine occurred when they began to twist. The applied load of red meranti, mahogany, and pine decreased as twisting occurred, and the failure load was much lower than the ultimate load. The beams of red meranti, mahogany, and pine can be categorized as low to medium ductile according to Eurocode 8 [74]. The correlation value between µ and E/S R is statistically significant (r = −0.701). Because the ductility value (µ) ( Table 5) has a significant inverse correlation to the E/S R value (Figure 9b), it is inversely correlated to the C L value as well. A brittle beam has a higher C L value than a ductile beam. A brittle beam produces only a small deformation, and inelastic failure happens before beam twisting. In contrast, a ductile beam produces significantly larger deformation, and elastic failure commonly occurs.

Beam Stability Factor (C L ) Application in Beam Design
Hayatunnufus reported bending tests of timber short beams with a slenderness ratio of 7.40-7.66, resulted in no twisting [87]. The beam stability factor (C L ) is the adjustment factor used to consider the effect of lateral-torsional buckling in intermediate and long beams. C L is applied as a reducing factor for the allowable bending stress (F b ) in combination with the safety factor, load duration factor (C D ), wet service factor (C M ), temperature (C t ), size factor (C F ), flat use factor (C fu ), incising factor (C i ), and repetitive member factor (C r ). The characteristic strength is the 5% exclusion limit (R 0.05 ) value from the experimental result. The result of the R 0.05 value of MOR multiplied by C L is presented in Figure 10. As shown in Figure 10, less than 5% of specimens lie below the characteristic strength curve multiplied by C L (R 0.05 × C L ) (e.g., three red meranti specimens, two mahogany specimens, a pine specimen, and an agathis specimen. Because very few data points lie below the curve, the designer can confidently use the C L value as an adjustment factor for the reference bending design value. The beam stability factor (C L ) formula obtained in this study is similar to that of NDS. We suggest that designers continue to use the NDS procedures to adjust the reference bending design value in order to safely and reliably design a beam. Figure 10, less than 5% of specimens lie below the characteristic stre curve multiplied by CL (R0.05 × CL) (e.g., three red meranti specimens, two mahogany imens, a pine specimen, and an agathis specimen. Because very few data points lie b the curve, the designer can confidently use the CL value as an adjustment factor fo reference bending design value. The beam stability factor (CL) formula obtained in study is similar to that of NDS. We suggest that designers continue to use the NDS p dures to adjust the reference bending design value in order to safely and reliably d a beam.

Conclusions
The range of sawn lumber beam slenderness ratio in this study was 4.37-11.1, and resulted in an empirical beam stability factor (C L ) ranging widely from 0.32 to 1.39. Curve fitting through nonlinear regression resulted in an estimated Euler buckling coefficient (K bE ) value of 0.413. By multiplying the factor of 2.74 by K bE , the adjusted K bE value obtained in this study was 1.13, with the 5% lower and 5% upper values of 0.92 and 1.33, respectively, which agrees with that of the NDS (K bE = 1.20). Because agathis has the lowest ductility (µ), most natural defects (smallest strength ratio, S), and highest E/S R ratio among others, the agathis beam did not undergo twisting during the bending test; rather, it failed before twisting could occur, indicating inelastic material failure. Meanwhile the other specimens (pinus, mahogany, and red meranti), which have a smaller E/S R ratio, higher ductility, and fewer natural defects, tended to fail because of their lesser beam stability. This phenomenon resulted in the C L curve of agathis being the highest among others. The C L value mathematically relates to the beam slenderness ratio (R B ) and the E/S R ratio. Because the strength ratio (S) and ductility ratio (µ) have significant inverse correlation with the E/S R ratio, they are inversely correlated with the C L value. Applying the C L value to adjust the characteristic bending strength is safe and reliable, as less than 5% of the specimens lay below the characteristic strength curve multiplied by the C L (R 0.05 × C L ). This study shows that designers can safely apply the C L formula to adjust the red meranti, mahogany, pines, and agathis sawn lumber reference bending design values.

Institutional Review Board Statement:
This study is not involving humans or animals and the ethical approval is not required.

Informed Consent Statement: Not applicable.
Data Availability Statement: The raw and processed data required to reproduce these findings are available for download from http://dx.doi.org/10.17632/t9k5rh4zy4.3 [88] since 8 January 2021.

Acknowledgments:
The authors would like to express many thanks to IPB University (Bogor Agricultural University) (ID), the Ministry of Education and Culture (ID), and the Ministry of Research and Technology (ID) for the finance, permit, facilities, and opportunity to conduct this research.

Conflicts of Interest:
The authors declare no conflict of interest.

Abbreviations
a regression coefficient to estimate the K bE b specimen width (mm) c non-linear parameter for beam = 0.95 C D load duration factor C F size factor C fu flat use factor C i incising factor C k beam stability factor limit C L beam stability factor C Le empirical beam stability factor C M wet service factor C r repetitive member factor C t temperature factor d specimen depth (mm) E modulus of elasticity (MPa) E min 5% lower exclusion value of modulus elasticity (MPa) F b reference bending design value (MPa) F b ' adjusted reference bending design value F bE critical buckling design (MPa) F u ultimate load (N) G b specific gravity K bE Euler buckling coefficient K bE ' adjusted Euler buckling coefficient L specimen length (mm) l e effective span length (mm) l u unsupported length (mm) m 0 mass before the bending test (g) m 1 mass after the bending test (g) M c moisture content (%) m ot mass after oven-drying (g) R 0.05 5% exclusion limit value R B beam slenderness ratio R k characteristic value (MPa) S strength ratio (%) S R modulus of rupture (MPa) Greek symbol µ ductility ratio ∆ u ultimate displacement (mm) ∆ y yield displacement (mm) ρ density (g/cm 3 )