Measurement and prediction of lateral torsional buckling loads of composite wood materials: rectangular sections.
Differences in the elastic constant ratios and torsional rigidity between solid wood and structural composite lumber may result in differences of the lateral torsional buckling behavior of these materials. The experimental critical buckling loads (CBLs) determined from a cantilever beam test method were compared with CBL predictions based upon the load resistance factor equations for wood design and modifications to the design equations incorporating measured E:G ratios and measured torsional rigidity terms. Experimental materials included machine-stress-rated lumber, laminated veneer lumber (LVL), parallel strand lumber (PSL), and laminated strand lumber (LSL) tested at three different lengths. The current LRFD equations best predicted the CBL values for solid wood and the PSL, but predicted nonconservative CBL values for the LVL. The current LRFD equation modified to incorporate measured E:G ratios was the best predictor for the LVL and LSL materials tested and represented no improvement in CBL prediction for the solid-sawn lumber and PSL. The incorporation of the measured torsional rigidity term did not significantly enhance any of the CBL predictions. The modeling of the test specimens as isotropic materials produced more agreeable CBL predictions than use of the measured torsional rigidity values. Therefore, the current LRFD equations are verified for solid-sawn lumber and recommended for use with PSL. The LRFD equations incorporating the material specific E:G ratios are recommended for LVL and LSL to provide more accurate CBL predictions.
Structural wood composites including structural composite lumber (SCL) and wood composite I-joists represent significant enhancements in the strength and stiffness of wood materials compared to solid-sawn lumber products. With the increased usage of these products, questions have arisen about potential differences between the elastic behavior of wood composites and solid-sawn lumber. Several recent papers including Hindman et al. (200_), Hindman et al. (2005a) and Hindman et al. (2005b) have noted differences in the elastic constant ratios and torsional rigidity terms between solid-sawn lumber and SCL materials.
One area of structural design affected by the elastic constant ratios and torsional rigidity differences is the design of beams for lateral torsional buckling as expressed in the National Design Specification (NDS) for wood construction (AF & PA 2001) and load resistance factor design (LRFD) provisions (AF & PA 1996). Lateral torsional buckling is an instability condition resulting from inadequate bracing of deep beams in the compression zone. As wood materials are used in longer span applications, the lateral torsional buckling becomes an increasingly important design consideration. Cantilever and continuous span beams with inadequate bracing can also be susceptible to lateral torsional buckling. Whether differences in the elastic constant ratios and torsional rigidity between solid wood and composite materials require changes in current design assumptions for SCL materials needs to be documented.
Experimental evaluation and concurrent agreement with load capacity predictions is needed to insure that current design specifications are conservative for these materials. The most recent published work on the lateral torsional buckling procedures for wood materials was from Hooley and Madsen (1964), who reported on a series of glued laminated beams tested in bending. More recent work based upon other wood composites is largely unavailable in the literature.
The current paper reports experimental and predicted critical buckling load (CBL) results for unbraced rectangular solid-sawn lumber (SSL) and SCL cantilevered beams. Current LRFD equations (AF & PA 1996) can be used to determine the corresponding CBL value. These equations use an assumed E:G ratio of 16:1 in the computation of CBL and are based upon the Ylinen equation to describe inelastic buckling behavior (Zahn 1992). Prediction efforts for the measured CBLs used current LRFD equations, a modified LRFD equation which allowed substitution of the material E:G ratio as determined from Hindman et al. (2005a) and another modified LRFD equation which allowed substitution of the torsional rigidity term as determined by Hindman et al. (200_). These different CBL equations explored whether differences in the E:G ratio or torsional rigidity affect the CBL calculation compared to current LRFD assumptions.
Materials and methods
Test materials included both SSL and SCL material for the experimental study. Southern yellow pine machine-stress-rated (MSR) lumber rated as 2250f-1.9E was chosen for the solid-sawn lumber comparison because of low property variability. SCL materials tested included laminated veneer lumber (LVL), parallel strand lumber (PSL), and laminated strand lumber (LSL). LVL and LSL products were sampled from material typically used for the fabrication of commercial I-joists. LVL and PSL materials were composed of southern pine (Pinus spp.) veneers and strands, respectively, while the LSL was uniquely composed of yellow-poplar strands (Liriodendron tulipifera). Table 1 lists the cross-sectional dimensions, the measured modulus of elasticity, and E:G ratio determined from Hindman et al. (200_) and the calculated lower fifth percentile modulus of elasticity used in the LRFD equations. Materials were tested at approximately nominal 2 by 10 dimensions with some differences in the PSL (4.4 cm [1.75 in] thick) and LSL (24.1 cm [9.5 in] high). PSL is typically not available at a 1.5-inch thickness as a standard product dimension. Table 1 also lists the unsupported lengths at which the materials were tested, i.e., 256 cm (101 in), 378 cm (149 in), and 569 cm (224 in), and the number of specimens for each material and length combination.
A cantilever beam with an applied point load at the free end was used to determine the CBLs for the materials. Figure 1 shows an LVL beam in the cantilever arrangement. Three important parts of the test set-up used were the hold-down fixture, the loading apparatus, and the angular rotation measurement.
[FIGURE 1 OMITTED]
A fixed-end condition was simulated using a hold-down fixture in combination with a universal testing machine (UTM). The fixture held the specimen in the correct placement and applied a horizontal restraining force through four large bolts and a clamp plate to distribute the horizontal force. The UTM applied a vertical force of 5.28 kN (1200 lb) downward for each specimen. Dead weight loading of a beam specimen produced deflection results within 4 percent of those predicted by beam theory for a fixed end (Hindman 2003). Therefore, the hold-down fixture was assumed to approximate a perfectly fixed-end condition for critical load calculations.
For loading, a hydraulic jack with 107-kN (12-ton) capacity and attached load cell was used on the free end of the beam specimen. The load cell had a 22,200-N (5,000-lb) capacity and a sensitivity of 44.4 N (10 lb). The ram contacting the beam had a rounded profile so that specimen rotation was allowed. Load was applied 5.08 cm (2 in) away from the end of the specimen to allow space for angular rotation measurements and ensure that the beam would not slip off the ram. Beam length measurements were taken from the front of the hold-down fixture to the point of loading.
[FIGURE 2 OMITTED]
To determine the angle of rotation of the specimen, [phi], four potentiometers were attached to the corners of the specimen at the end of the beam (Fig. 2). Potentiometers were 5.08-cm (2-in) maximum deflection with sensitivity of 0.0025 cm (0.001 in). Because of the vertical travel of the beam during testing, two parallel high density polyethylene (HDPE) plates were used to provide a smooth contact surface for the potentiometers. The four potentiometers were labeled according to their placement on the specimen (TL = top left, TR = top right, BL = bottom left, and BR = bottom right). Equation  shows the calculation of the angle of rotation [phi] based upon the readings from the four linear displacement potentiometers.
[phi] = [tan.sup.-1]([(TR - BR) - (TL - BL)]/h) 
where [phi] = angle of rotation; TR = deflection of potentiometer located at top right; BR = deflection of potentiometer located at bottom right; TL = deflection of potentiometer located at top left; BL = deflection of potentiometer located at bottom left; h = distance between top and bottom potentiometers.
CBL evaluation included an applied load to the end of the specimen using the previously described hydraulic jack and load cell system. The load application rate was approximately 0.64 cm/min (0.25 in/min) to achieve the estimated time to failure given in ASTM D 198 (ASTM 2001) recommendations for bending tests. However, specimens were not tested to failure as per ASTM D 198 recommendation. Ten beams of each material and length were tested using three test repetitions to determine the average CBL. During testing, both load and rotation measurements specified above were continuously monitored through a computer data-acquisition system. When rotation of the beam became noticeable, the test was concluded. Load and deflection measurements were used to calculate the P-[phi] curve. Figure 3 shows a typical plot of the P-[phi] curve (load-rotation) for a 378-cm- (149-in-) long MSR beam. The irregular nature of the curve is due to incremental stepwise loading from the hydraulic jack and the potentiometers moving over the HDPE plates. Bifurcation is identified by a distinct change in slope of the P-[phi] curve and the CBL is the corresponding load at the point of bifurcation. Bifurcation was defined by a visual inspection of the P-[phi] curve.
[FIGURE 3 OMITTED]
Equation  shows the calculation of the slenderness ratio [R.sub.B], which originated in work by Hooley and Madsen (1964) as the term [C.sub.s] and was modified in subsequent design equations (AF & PA 1996). The slenderness ratio serves to incorporate the specimen dimensions into a single common factor, allowing comparison of equivalent spans, beam sizes, and end conditions. The effective length term, [l.sub.e], accounts for different end restraint conditions (AF & PA 1996). The effective length term defined in Equation  is for a cantilever beam with a concentrated load applied at the free end.
[R.sub.B] = [square root of ([[l.sub.e]d]/[b.sup.2])] 
[l.sub.e] = 1.44[l.sub.u] + 3d
where [l.sub.e] = effective length; d = specimen height; b = specimen width; [l.sub.u] = unbraced length or gage length of specimen.
The recommended approach for defining the design load, or CBL, for a wood beam using LRFD is given by Equations  to  (AF & PA 1996). These equations and accompanying commentary are located in section 5.2.3 (AF & PA 1996). These equations apply to both solid wood and SCL materials of rectangular shape. The CBL of the cantilever beam is equal to the [M*.sub.x], the maximum moment unadjusted for lateral stability, times [C.sub.L], the adjustment factor for lateral stability, divided by the beam length L. Equation  shows the calculation of the [C.sub.L] term using the Ylinen equation. The [c.sub.b] term is an empirical parameter equal to 0.95 and the [[alpha].sub.b] term is the ratio of the ultimate moment from elastic buckling, [M.sub.e], divided by the ultimate moment capacity unadjusted for lateral stability, [M*.sub.x].
CBL = [[C.sub.L][S.sub.x][F*.sub.b]]/L = [[C.sub.L][M*.sub.x]]/L 
[C.sub.L] = [[1 + [[alpha].sub.b]]/[2[c.sub.b]]] - [square root of ([[1 + [[alpha].sub.b]]/[2[c.sub.b]]][.sup.2] - [[[alpha].sub.b]/[c.sub.b]])] 
[[alpha].sub.b] = [[[phi].sub.s][M.sub.e]]/[[lambda][[phi].sub.b][M*.sub.x]] 
[M.sub.e] = [[2.40[E'.sub.y05][I.sub.y]]/[l.sub.e]] prismatic beams 
where [E'.sub.y05] = 1.03E'[1 - 1.645([COV.sub.E])]; [c.sub.b] = 0.95; [[phi].sub.s] = 0.85; [[phi].sub.b] = 0.86.
Equation  assumes that the E:G ratio of the wood material is 16:1 and that the torsional rigidity term can be approximated as an isotropic section. To explore how the measured E:G ratio or the measured torsional rigidity term affects the CBL predictions, the [M.sub.e] equation can be rewritten to include these factors. Equation  shows the [M.sub.e] term rewritten to accept different E:G ratios. Equation  can be applied to Equations  to  to determine the maximum critical load using the different E:G ratio. This first computational model is defined herein as the E:G LRFD prediction.
[M.sub.e] = [[9.6[E'.sub.y05][I.sub.y]]/[l.sub.e]](1/[square root of (E:G)]) 
Equation  shows the [M.sub.e] equation rewritten with the torsional rigidity term included. The LRFD document recommends using Equation  to determine the [M.sub.e] value of nonrectangular, nonI-joist sections. Equation  is based upon elastic beam buckling from Timoshenko and Gere (1961). This second model is defined herein as the Timoshenko LRFD prediction.
[M.sub.e] = [4.013 [square root of (E[I.sub.y]GJ)]]/[l.sub.e] 
Experimental and predicted CBL values were compared at corresponding slenderness ratio values for each material. The predictions use average values of specimen dimensions and properties. The E:G ratios and torsional rigidity terms were measured and reported previously (Hindman et al. 200_, 2005a).
Results and discussion
Table 2 shows the average measured CBL values from the test materials. The greatest CBLs for the 256 cm (101 in) and 378 cm (149 in) lengths were for PSL, while the greatest CBL for the 569 cm (249 in) length was for LSL. The coeeficients of variation for the measured CBLs for the study materials ranged from 4.5 to 14.8 percent. The low variation was considered excellent for measurement of an instability condition.
Determination of the "best fit" model is rather difficult with the collection of only three data points. The small number of data points prevents statistical goodness-of-fit tests from being used. To compare the three different prediction models with the measured CBL values, both a tabular format and a graphical format were employed. Table 3 shows the percentage difference values for the measured CBLs of the four materials compared to the three predicted CBLs applied to each measurement for the different materials, prediction models, and unsupported lengths. Figures 4 to 7 show the measured and predicted CBL values graphically for the MSR, LVL, PSL, and LSL materials, respectively. Both the tabular and graphical results were examined for each material.
For the MSR lumber, Table 3 shows that the percentage differences of all three CBL comparisons for all lengths were at least 24 percent less than the measured CBL. The percentage difference of the LRFD prediction CBLs ranged from 24.2 to 49.2 percent less than the measured CBLs, while the E:G LRFD ranged from 34.3 to 56.1 percent less and the Timoshenko LRFD ranged from 37.4 to 58.2 percent. All of the predictions with the longer 569-cm (224-in) length beams had percent difference values greater than or equal to 49.2 percent. The CBL predictions appear to become invalid at the longer unbraced span length. The use of the measured E:G ratio of 21.5:1 did not provide better CBL predictions than the estimated E:G ratio of 16:1. The LRFD prediction had the smallest percent difference values for the three beam lengths tested. Figure 4 shows the difference between the measured CBLs and the three predictions graphically. The LRFD prediction was the most accurate for the MSR lumber material.
[FIGURE 4 OMITTED]
For the LVL, Table 3 shows a different trend in the predicted CBLs for the LRFD compared to the E:G LRFD and Timoshenko LRFD predictions. The LRFD-predicted CBL values ranged from 5.8 to 37.8 percent greater than the measured CBLs. Therefore, the LRFD prediction for the LVL materials predicted greater CBL values than the measured CBLs, especially at the higher [R.sub.B] values. For the E:G LRFD, the predicted CBLs for the 256-cm (101-in) and 378-cm (149-in) beam length are 14.9 and 14.4 percent less than the measured CBLs, respectively, while the 569-cm (224-in) predicted CBL was 10.2 percent greater than the measured CBL. The use of the LVL E:G ratio of 23.5:1 represented an improved CBL prediction compared to the assumed E:G ratio of 16:1. All of the the Timoshenko LRFD predicted CBLs are at least 16.9 percent less than the measured CBLs. Figure 5 confirms the observations of the LRFD predictions being greater than the measured CBLs. The E:G LRFD prediction corresponds with the LVL materials and provides a slightly conservative CBL value.
For the PSL material, the LRFD prediction and the E:G LRFD predictions are very similar. The LRFD predicted CBLs were between 6.4 and 12.6 percent less than the measured CBL values. The E:G LRFD predicted CBLs varied from 12.8 to 18.4 percent and the Timoshenko LRFD predicted CBLs varied from 27.6 to 32.0 percent less than the measured CBLs. The PSL E:G ratio of 18.8:1 does not appear to produce better CBL predictions than the assumed E:G ratio of 16:1. Figure 6 shows the graphical comparison of the measured and predicted CBLs. The PSL measured CBLs corresponded with the LRFD prediction better than the E:G LRFD or Timoshenko LRFD predictions.
For the LSL material, the E:G LRFD predicted CBLs were greater than the LRFD predicted CBLs. The LRFD predicted CBLs ranged from 16.9 to 43.1 percent less than the measured CBLs. The E:G LRFD predicted CBLs ranged from 12.8 to 18.4 percent less than the measured CBLs; the Timoshenko LRFD predicted CBLs were all at least 27 percent less than the measured CBLs. At the 569-cm (224-in) length, all of the CBL predictions were at least 34.9 percent less than the measured CBLs, indicating that the CBL predictions may not be valid for these longer lengths. The E:G ratio of LSL (11.4:1) from Hindman et al. (200_) was less than the E:G ratio of 16:1 assumed for the LRFD equation. Figure 7 shows the change in the trends of the LRFD and E:G LRFD predicted values for the LSL material.
[FIGURE 5 OMITTED]
[FIGURE 6 OMITTED]
The best prediction for MSR lumber and PSL materials were obtained with the current LRFD equations. Hindman (2003) reported an E:G ratio of 18.8:1 for PSL material, which was very similar to the 16:1 ratio used for solid wood. The best prediction for the LVL and LSL materials was the E:G LRFD equations, where the E:G ratios measured in Hindman (2003) were used in the calculation of the [M.sub.e] term. The differences in the E:G ratios of the LVL and LSL materials (23.5:1 and 11.4:1, respectively) compared to the assumed E:G ratio of 16:1 affected the prediction of the CBL values. In the case of LVL, which had a greater E:G ratio than 16:1, the use of the LRFD equations produced predicted CBLs greater than the measured CBLs.
[FIGURE 7 OMITTED]
The Timoshenko LRFD prediction using the torsional rigidity terms (GJ) previously determined by Hindman et al. (2005a) did not provide significant improvements to the model prediction compared to the LRFD and E:G LRFD predictions. The LRFD and E:G LRFD predictions contained the assumption of an isotropic material. Hindman et al. (2005a) noted that the torsional rigidity terms of the PSL and LSL materials related to an orthotropic model best. The CBL predictions seemed to produce accurate results when the torsional rigidity was calculated assuming the material to be isotropic, as in the LRFD equations, rather than by using the measured torsional rigidity values. The measured torsional rigidity terms were measured from a torsion test method, whereas the E:G ratios were measured from a bending test. The E:G ratios measured from a bending loading may be more applicable to the stress and deformation profile of lateral torsional buckling testing.
This study demonstrates that the CBL values measured from different SCL materials appear to deviate from CBL predictions using current wood design equations. The use of measured E:G ratios from individual SCL materials improved the CBL predictions. Further research including a more comprehensive study of different unbraced lengths and loading conditions related to SCL materials should be conducted before specific changes to the wood design equations are proposed.
A test method to examine the CBL of an unbraced cantilever beam was used to measure the CBLs of solid-sawn lumber and SCL materials. Results from CBL measurement yielded coefficient of variation (COV) values less than 15 percent for all materials. The best prediction for MSR lumber and PSL CBL values was the current LRFD equations published by AF & PA (1996). The best prediction for the CBL values of the LVL and LSL materials was the LRFD equation modified to incorporate individual E:G ratios. The current LRFD equations overestimated the CBL of LVL by at least 5.8 percent. The inclusion of the measured torsional rigidity terms in CBL predictions did not provide significant model enhancement. The assumption of an isotropic torsional rigidity term for the torsional rigidity was a valid assumption for the prediction of lateral torsional buckling. Use of individual E:G ratios from five-point bending tests for predicting CBLs of rectangular LVL and LSL materials is recommended in the CBL prediction equations.
American Forest and Paper Association. (AF & PA). 1996. Load factor resistance design (LRFD) manual for engineered wood construction. AF & PA, Washington, DC. 124 pp.
__________. (AF & PA). 2001. Allowable stress design (ASD) manual for engineered wood construction. AF & PA, Washington, DC. 97 pp.
American Society for Testing and Materials (ASTM). 2001. Standard test methods of static tests of lumber in structural sizes. ASTM D 198. ASTM, West Conshohocken, PA.
Hindman, D.P. 2003. Torsional rigidity and lateral stability of structural composite lumber and I-joist members. PhD diss. The Pennsylvania State Univ., University Park, PA. 222 pp.
__________, J.J. Janowiak, and H.B. Manbeck. 200. Comparison of ASTM D 198 and five-point bending for elastic constant ratio determination. Forest Prod. J. Submitted for publication.
__________, H.B. Manbeck, and J.J. Janowiak. 2005a. Torsional rigidity of rectangular wood composite materials. Wood and Fiber Sci. 37(2):283-291.
__________, __________, and __________. 2005b. Torsional rigidity of wood composite I-joists. Wood and Fiber Sci. 37(2):292-303.
Hooley, R.F. and B. Madsen. 1964. Lateral buckling of glued laminated beams. ASCE J. of the Structural Division 90:(ST3)201-218.
Timoshenko, S. and J.M. Gere. 1961. Theory of Elastic Stability. 2nd ed. McGraw-Hill, New York. 541 pp.
Zahn, J.J. 1992. Re-examination of Ylinen and other column equations. ASCE J. of Structural Engineering 118(10):2716-2728.
Daniel P. Hindman*
Harvey B. Manbeck*
John J. Janowiak*
The authors are, respectively, Assistant Professor, Dept. of Wood Science and Forest Products, Virginia Tech, Brooks Forest Products Center, Blacksburg, VA (email@example.com); Distinguished Professor, Agricultural and Biological Engineering, Pennsylvania State Univ., University Park, PA (firstname.lastname@example.org); and Professor, School of Forest Resources, Pennsylvania State Univ. (email@example.com). The authors would like to thank Trus Joist, a Weyerhaeuser company, Rigidply Rafters, the Pennsylvania Housing Research Center, and the Pennsylvania Agri. Expt. Sta. for material donations and support for this project. This paper was received for publication in March 2004. Article No. 9855.
*Forest Products Society Member.
Table 1. -- Properties and dimensions of engineered wood products used in experimentation. Width Height Material Description b h E (a) (cm [in]) (GPa) (psi) (COV) MSR 2250f-1.9E 3.81 23.5 16.2 2.35 X 18.3 lumber (southern pine) [1.5] [9.25] [10.sup.6] LVL Flange grade 3.81 23.5 17.9 2.60 X 4.6 (southern pine) [1.5] [9.25] [10.sup.6] PSL 2.0E grade 4.44 24.1 13.5 1.96 X 10.8 (southern pine) [1.75] [9.5] [10.sup.6] LSL Flange grade 3.81 24.1 12.6 1.83 X 5.1 (yellow-poplar) [1.5] [9.5] [10.sup.6] E:G Lengths Material Description ratio (a) [E'.sub.y05] tested (c) (GPa [psi]) (b) (cm [in]) MSR 2250f-1.9E 21.5 11.7 256  lumber (southern pine) [1.70 X [10.sup.6]] 378  569  LVL Flange grade 23.5 17.0 256  (southern pine) [2.47 X [10.sup.6]] 378  569  PSL 2.0E grade 18.8 11.4 256  (southern pine) [1.66 X [10.sup.6]] 378  569  LSL Flange grade 11.4 11.9 256  (yellow-poplar) [1.73 X [10.sup.6]] 378  569  (a) Modulus of elasticity and E:G ratios were measured and reported in Hindman et al. (200_). (b) [E'.sub.y05] = 1.03E'[1-1.645(COV)] (AF & PA 1996). (c) Ten specimens were tested for each different combination of material and length. Table 2. -- Measured CBLs for MSR lumber and SCL cantilever beams. 256 cm (101 in) 378 cm (149 in) 569 cm (224 in) Material Land COV Land COV Land COV (N [lb]) (%) (N [lb]) (%) (N [lb]) (%) MSR lumber (a) 3750  7.1 1680  14.5 1170  6.6 LVL (a) 3670  4.7 1780  7.4 636  14.8 PSL (b) 4150  4.5 1910  8.3 934  8.5 LSL (c) 3680  5.4 1790  5.6 1210  4.9 (a) [R.sub.B] values are 26.7, 31.6 and 37.9, respectively, for the MSR Lumber and LVL lengths. (b) [R.sub.B] values are 22.9, 27.1 and 32.5, respectively, for the PSL and LSL I-joist lengths. (c) [R.sub.B] values are 27.1, 32.0 and 38.5, respectively, for the LSL and LVL I-joist lengths. Table 3. -- Predicted CBLs with percent differences from measured CBLs. Material LRFD model (a) E:G LRFD model (b) (cm [in]) (N [lb]) (% diff.) (d) (N [lb]) (% diff.) MSR (e) 256  2590  -31.0 2260  -39.9 378  1280  -24.2 1110  -34.3 569  600  -49.2 511  -56.1 LVL (e) 256  3890  +5.79 3130  -14.9 378  1900  +6.73 1530  -14.4 569  880  +37.7 707  +10.2 PSL (f) 256  3630  -12.6 3390  -18.4 378  1780  -6.43 1660  -12.8 569  827  -11.2 774  -17.3 LSL (g) 256  3030  -17.7 3450  -6.32 378  1490  -16.9 1690  -5.17 569  689  -43.1 787  -34.9 Timoshenko Material LRFD model (c) (cm [in]) (N [lb]) (% diff.) MSR (e) 256  2150  -42.7 378  1050  -37.4 569  489  -58.2 LVL (e) 256  2370  -35.6 378  1150  -35.4 569  534  -16.9 PSL (f) 256  2820  -32.0 378  1380  -27.6 569  641  -31.5 LSL (g) 256  2710  -26.3 378  1330  -25.8 569  614  -49.2 (a) LRFD model CBLs predicted using Equations  to . (b) E:G LRFD model CBLs predicted using Equation  for [M.sub.e]. (c) Timoshenko LRFD model CBLs predicted using Equation  for [M.sub.e]. (d) % difference = (prediction - (measured)/measured. (e) [R.sub.B] values are 26.7 for the 256 cm, 31.6 for the 378 cm, and 37.9 for the 569 cm lengths. (f) [R.sub.B] values are 22.9 for the 256 cm, 27.1 for the 378 cm, and 32.5 for the 569 cm lengths. (g) [R.sub.B] values are 27.1 for the 256 cm, 32.0 for the 378 cm, and 38.5 for the 569 cm lengths.
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|Author:||Hindman, Daniel P.; Manbeck, Harvey B.; Janowiak, John J.|
|Publication:||Forest Products Journal|
|Date:||Sep 1, 2005|
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