Stress distribution in structural composite lumber ... - Semantic Scholar
Stress distribution in structural composite lumber under torsion ✳
Rakesh Gupta Tobias S. Siller
Abstract Stress distribution in full-size, structural composite lumber (SCL) under torsion was analyzed using finite element analysis (FEA). The objective was to determine shear strength of SCL using orthotropic torsion theory and compare the results to experimental results. The commercial finite element program ANSYS was used to perform the analysis using orthotropic material properties. A three-dimensional, eight-node brick element was used to model the specimens. The FEA showed that SCL is more orthotropic in material behavior than solid-sawn lumber and orthotropic torsion theory should be used to determine the shear strength of SCL. The degree of orthotropicity depends upon two shear moduli in longitudinal-tangential and longitudinal-radial planes. The shear strength based on orthotropic FEA was within 2 percent of experimental results.
tructural composite lumber (SCL) is a generic name for a group of products within a wide variety of engineered wood products. SCL includes materials like laminated veneer lumber (LVL), parallel strand lumber (PSL), and laminated strand lumber (LSL). SCL is composed of either wood veneers or strands that are mainly oriented longitudinally, dried to a low moisture content, and bonded by a resin. As a result of the manufacturing process, defects, like knots and slope of grain found in solid sawn lumber (SSL), are either eliminated or at least dispersed. SCL is more homogeneous than SSL and shows less variability in measured properties, resulting in higher design values. The major advantage of SCL is availability of relatively large sizes in combination with high strength, uniformity, and straightness. In general, allowable stresses are highest for SCL, followed by glued-laminated (glulam) timber and SSL. However, the same may not be true for shear strength. FOREST PRODUCTS JOURNAL
Historically, shear strength of SSL and SCL have been determined from small specimens, known as shear blocks. However, Gupta et al. (2002a) showed that shear strength of SSL from shear blocks is lower than the shear strength of SSL based on full-size torsion tests. Even though the shear block specimen was designed as a pure shear specimen, previous studies (Radcliff and Suddarth 1955) have shown that it is not; it has stress concentration at the re-entrant corner and bending stresses. Several investigators (Rammer et al. 1996, Riyanto and Gupta 1998) have used various bending test setups to determine the shear strength of wood. However, in a bending test setup, shear stresses are affected by the presence of
other stresses, as has been shown by previous studies (Mandery 1969, Riyanto and Gupta 1998). Torsion is the only test method that produces pure shear stresses in the specimen and may be used to determine shear strength of various wood products. Therefore, the objective of this study is to perform a finite element analysis (FEA) of the full-size SCL subjected to torsion in order to enhance the understanding of torsion theory using orthotropic material properties, and to compare FEA results with experimental results.
Literature review Cofer et al. (1997) modeled different size SSL beams by means of FEA and applied the Tsai-Hill failure criterion to predict shear strength. The FEA results compared reasonably well with experimental results for small member sizes, whereas the reduction in shear strength with increasing beam size, as reported by their experimental results, was not reproduced by the FEA results. Therefore the authors concluded that the size effect might be caused by the higher probability of the presence of natural defects in larger members of wood, since the model did not account for any flaws. Leichti and Nakhata (1999) modeled 38- by 89-mm2 and 38- by 240-mm2 Douglas-fir beams under a five-point
The authors are, respectively, Associate Professor and Former Graduate Research Assistant, Dept. of Wood Science and Engineering, Oregon State Univ., Corvallis, OR 97331. The authors would like to thank Milo Clauson for his help and advice on instrumentation and testing. This paper was received for publication in September 2003. Article No. 9753. ✳Forest Products Society Member. ©Forest Products Society 2005. Forest Prod. J. 55(2):51-56.
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51
ing area to prevent crushing for smaller sizes of dimension lumber presented an immense problem. Consequently, the authors concluded that beams having smaller cross sections should not be tested in the five-point configuration or that a standard test method, which is applicable to all sizes of lumber, should be introduced. One such method, as shown by Gupta et al. (2002a), is a torsion test that is applicable to all sizes.
Figure 1. — Cross-sectional view of a specimen showing FE mesh.
Gupta et al. (2002b) performed FEA of full-size SSL under torsion using isotropic material properties and revealed that uniform shear stress occurs within the shear span, which begins and ends at a distance of approximately two times the depth plus the grip distance away from each end of the specimen. In addition, FEA verified that the shear failure plane in torsion is similar to the known shear failure mode in specimens subjected to transverse loads. They finally concluded that torsion is the best practical method for determining the pure shear strength of full-size, structural lumber. Aicher (1990) investigated torsional properties of structural size, vertically laminated glulam beams using FEA and compared isotropic with orthotropic material behavior. Two different models were evaluated, one simulating St. Venant torsion (free warping), the other one considering warping torsion (restrained warping). The consideration of warping torsion resulted in a significant difference between isotropic and orthotropic material behavior.
Figure 2. — Beam under torsion loads and boundary conditions.
bending load configuration with a spanto-depth ratio of five. The authors concluded that elementary beam theory did not accurately predict the complex state of shear and bending stresses within the specimen, but overestimated the results of FEA by about 30 to 40 percent. In addition, the provision of sufficient bear52
The torsion test has also been used to determine elastic constants of any material where pure shear stress state is needed. Janowiak and Pellerin (1992) determined the orthotropic shear moduli for three reconstituted wood panel products using torsion. The authors concluded that the torsion test is a more efficient test method to determine the orthotropic shear moduli compared to flexure tests. Hindman (1999) evaluated the elastic constants of selected engineered wood products using torsion, bending, compression, and tension tests. He concluded that SCL showed a different elastic response than did solid wood. Lathe checks in LVL and PSL were found to decrease transverse elastic stiffness as well as the shear resistance through the thickness. He strongly recommended not to use the set of elastic constants of solid wood for SCL manu-
factured of the same species, because of the obviously different elastic behavior.
Finite element modeling The model The experimental torsion test configuration used by Gupta and Siller (2005) was modeled using the finite element (FE) method to understand the state of stress under torsion using orthotropic material properties. The commercial FE program (ANSYS 5.6®) was used to perform the analysis. A three-dimensional, eight-node brick element (SOLID45 element in ANSYS) was used to model the specimen as an orthotropic continuum. The SOLID45 element offered 6 degrees of freedom per node, allowing translations and rotations in the three major directions. The analysis focused on the full-size specimen measuring 44 by 140 by 1372 mm3 (same size as used previously [Gupta and Siller 2005]) under torsion for all three composites: LSL, LVL, and PSL. The closed-form solutions of Trayer and March (1930) for isotropic material behavior, also given in ASTM D198 (ASTM 2001), and Lekhnitskii (1981) for orthotropic materials were compared to the results of FEA. Mesh size Based on the results of a previous study (Gupta et al. 2002b), the mesh size shown in Figure 1 was used in this study. It consisted of 6 × 22 × 108 = 14,256 elements and 7 × 23 × 109 = 17,549 nodes. This mesh is already very fine, especially in the area of interest, and further refinement would not be appropriate since the results are mainly influenced by the set of elastic constants. Boundary conditions As shown in Figure 2, one end of the beam was fixed while the opposite end had a twisting moment. At the fixed end, all nodes in the longitudinal direction within the gripping distance of 51 mm (2 in.) on the surface of both wide faces were fixed over the entire depth. All degrees of freedom were constrained for these nodes. The end face and both narrow faces were not restrained. This procedure best described the actual test conditions and was as close as possible to pure St. Venant torsion, i.e., unrestrained warping, while still securely clenching the specimen. Unrestrained warping was possible for the entire end face and both narrow faces. The conFEBRUARY 2005
Table 1. — Input values for FEA model.
Elastic range
Moment (M)
Force (F)
Pressure (p)
Slope (s)
(Nm)
(N)
(MPa)
(N/mm ) 0.0260
a
3
300
3221
1.82
LSL
850
9123
5.14
0.0736
LVL
570
6117
3.45
0.0494
PSL
460
4937
2.78
0.0398
aFor
all three composites.
Table 2. — Material properties for FEA model. MOE Material
E R = ET
Shear moduli EL
GRT
GRL
Poisson ratios GTL
QRT
QRL
QTL
- - - - - - - - - - - - - - - - (MPa) - - - - - - - - - - - - - - - LSL
1207
11000
94.5
318
782
0.496
0.054
0.022
LVL
397
14600
44.4
407
593
0.390
0.036
0.029
PSL
404
15000
72.8
310
398
0.390
0.036
0.029
Table 3. — FEA shear stresses. Elastic range Material
Results and discussion
At failure WLR
WLT
WLT
WLR
- - - - - - - - - - - - - - - - - - - - - - (MPa) - - - - - - - - - - - - - - - - - - - - - LSL
4.35
2.22
12.33
6.28
LVL
4.13
2.61
7.84
4.96
PSL
4.05
2.73
6.21
4.19
straints at the wide faces were necessary to provide model stability. A force couple was used to apply torque at the end receiving the twisting moment. It was decided to choose a triangular pressure distribution instead of a uniform one, or just point loads, to avoid high stress concentrations and to model the experimental tests more realistically. The tests consistently showed more impact on the corners of the wide faces compared to the center. Consequently, a triangular pressure distribution was applied to the gripping length of 51 mm (2 in.) and one half of the pressure couple loaded the lower depth, while the other one was imposed on the upper half on the opposite side as shown in Figure 2. The twisted end of the model better reflects the real test conditions, since the restraints in the longitudinal direction of the fixed end might slightly influence the state of stress. Gupta et al. (2002b) used similar boundary conditions while analyzing full-size SSL under torsion. Loads Two different load cases were investigated. For the first one, the applied torque was within the linear range of the FOREST PRODUCTS JOURNAL
moment - rotation curve obtained by the tests (Gupta and Siller 2005). A torque of 300 Nm was chosen for this case for all three composites. Second, the average ultimate torsional moment at failure, obtained from Gupta and Siller (2005), for each of the three composites was applied. The torque values were converted into equivalent pressure (p) values using simple statics. All necessary input values for the different load cases are summarized in Table 1 and shown in Figure 2. Material properties As shown in Table 2, the FE model used orthotropic material properties. Gupta et al. (2002b) used only isotropic material properties for solid- sawn Douglas-fir because they found out that it was not beneficial to perform an orthotropic analysis, since the results between an orthotropic FEA and isotropic theory differed only by 1.3 percent. However, it was believed that SCL might behave differently from solidsawn Douglas-fir because of large differences in material constants. Therefore, this study included the orthotropic material behavior for all three compos-
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ites. The assumption that the radial and the tangential moduli of elasticity (MOE) are equivalent had to be used, since no source was found that offered more detailed values. The longitudinal MOE values (Ez) were measured via third-point bending tests following ASTM D198 (ASTM 2001). The span-to-depth ratio was 31. The transverse MOE and all shear moduli values were calculated using elastic constant ratios for SCL given by Hindman (1999), based on the measured value of Ez. The Poisson Ratios are taken from the Wood Handbook (FPL 1999) for solid wood. All model input values are shown in Table 2. Shear stresses are evaluated at the middle of the long (wide face; points A in Fig. 1; WLT) and short (narrow face; points B in Fig. 1; WLR) sides because that is where the stresses are maximum.
Shear stresses The focus of the FE analysis was to get a better idea of the stress distribution within the specimen and to compare the behavior of the three different SCL products. In addition, the analysis was performed to further verify the orthotropic theory that was applied to evaluate the torsion test results. Results of the FEA are given in Table 3, which shows the shear stresses within the elastic range based on a constant torque value of 300 Nm for all three composites. The last two columns show shear stresses at failure, based on the different average ultimate torque values for each material. All shear stresses in Table 3 are based on the orthotropic material properties given in Table 2. Even though eight material constants were used, a parametric study showed that shear stresses were heavily dependent on the two shear moduli in the longitudinal plane, GTL and GRL. All other elastic constants, MOE as well as Poisson Ratios and the third shear modulus, were only of very minor importance. This observation matches orthotropic theory, where the two shear moduli in the longitudinal planes are the only material constants that are considered. The ratio of these two shear moduli, g = GTL/GRL, is also an indicator of the degree of orthotropicity of various materials. The farther g is from unity, the more orthotropic the material is. Therefore, it is quite obvious that LSL (g = 2.46) be53
Figure 3. — FE shear stresses.
haves differently from LVL (g = 1.46) and PSL (g = 1.28), since g for LSL is almost twice the g value for LVL and PSL. Note the fact that within the elastic range, the shear stress τLR on the short side is highest for PSL (2.73 MPa), the least orthotropic material. Whereas, the shear stress τLT on the long side is highest for LSL (4.35 MPa), the most orthotropic material. This would lead us to 54
believe that as torque increases toward failure, the higher g material would probably fail at the middle of the long side. While looking at shear strengths at failure, which are based on three different values of failure torque for three composites, the highest shear stress on any side (long or short) is for LSL, the most orthotropic material. Shear strength at failure would also indicate that all three composites would fail at
the middle of the long side. The study by Gupta and Siller (2005) showed that LSL failed at the middle of the short side, whereas the other two composites failed at the middle of the long side. The main reason for this is the vastly different values of shear moduli for LSL in two longitudinal planes. This difference is probably caused by the special design of the composite, for instance, the high FEBRUARY 2005
plots are depicted for each material. The first plot for each material (Fig. 3a, 3c, and 3e) shows the front view (wide face) of the specimen where the shear stress, WLT, is maximum. The second plot for each material (Fig. 3b, 3d, and 3f) shows an isometric top view (narrow face) of the specimen where the shear stress, WLR, is maximum. The constrained end is at the left end of the specimen. The loaded end, where the origin of the coordinate system lies, is at the specimen’s right end. There is a symmetric state of stress in the case of axial torsion, i.e., shear stresses on opposite sides (faces) of the specimen have the same distribution and absolute values.
Figure 4. — FE maximum shear stresses along the beam length.
degree of densification, the strand orientation, or the resin used. Moreover, this ratio, g, explains why the behavior of solid Douglas-fir is almost isotropic. The g for Douglas-fir is closer to unity and that is why its behavior is almost isotropic. Gupta et al. (2002b) reported only a 1.3 percent difference between orthotropic- and isotropic-based shear strengths, considering a rectangular beam subjected to torsion. Since the behavior of SCL is orthotropic, there is a need for accurate evaluation of its material constants in order to better predict its performance. Table 3 shows that WLT is always more than WLR. But for g values of less than 1, it is possible for WLR to be higher than WLT. For the given aspect ratio of 3.1, the shear stresses at the different locations FOREST PRODUCTS JOURNAL
were equal (WLR = WLT = 3.8 MPa) for a shear moduli ratio g of 0.74, and the shear stress WLR on the center of the short side was higher than the shear stress WLT on the long side for g less than 0.74. Hardwoods possess such shear moduli ratios. Softwoods, however, have g ratios close to unity. SCL, in contrast, is proprietary and has, mainly due to densification during manufacture, g ratios well above unity. This illustrates the importance of anisotropy in the case of axial torsion of specimens with a rectangular cross section. Stress distribution The shear stress distribution in the specimens is shown in Figures 3a to 3f. The results are shown for specimens subjected to mean torque values at failure. All stresses are given in MPa. Two
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Figure 4 depicts the maximum shear stresses (in MPa on y-axis) along the specimen length (in mm on x-axis). Note that the rotated end is now shown to the left and the constrained one to the right. The small kink in the plots of WLT at the fixed end is similar to the kink Gupta et al. (2002b) had reported. It is due to the longitudinal direction constraint of the nodes at the fixed end that were required to provide model stability. The graphs for WLT depict the shear stresses on the center of the wide face, and the graphs for WLR show the shear stresses on the center of the narrow face. All graphs have the same abscissa scale, but the first one (Fig. 4a) has a different ordinate scale. For all specimens, shear stress is constant in the middle of the specimen (about two times the depth away from both ends). The same results were obtained by Gupta et al. (2002b) for isotropic modeling of SSL under torsion. The values listed in Table 3 are taken at the center of the span at mid-depth (WLT) or mid-width (WLR). The state of stress is the same for all materials. The FEA plots (Figs. 3 and 4) depict clearly that the maximum shear stresses act on the center of each side. In the case of axial torsion, there is a parabolic shear stress distribution along the width and the depth for specimens with a rectangular cross section. There is a constant state of stress within the specimens, except for a very short part of the span where end effects occur. The state of stress is more affected by the constrained end than by the loaded end. Shear stress (WLT) on the center of the long side shows its maximum value close to the loaded end and a distinct decline towards the constrained end. On the contrary, shear stress (WLR) on the center of the short side shows its maxi55
Table 4. — Experimental shear stresses from Gupta and Siller (2005). Isotropic Material
Long WLT-I
Orthotropic Short WLR-I
Long WLT-O
Short WLR-O
- - - - - - - - - - - - - - - - - - - - - - (MPa) - - - - - - - - - - - - - - - - - - - - - - LSL
11.59
8.83
12.69
6.43
LVL
7.66
5.83
7.96
4.90
PSL
6.65
5.02
6.82
4.55
mum value close to the fixed end. However, the difference between maximum shear stresses and the values in the middle part of the span are in both cases negligible. For both the FEA model and experimental tests, the specimens were constrained by clamps only on the long side (Fig 2). Only the intensity of the shear stresses varies. The FEA model was subjected to different torque loads since the three composites were able to bear different torque loads in experimental testing. LSL showed the biggest difference in shear stress between the two faces. All three composites showed markedly higher shear stresses on the long side. Comparison between torsion test and FE model Shear strengths based on the torsion test are listed in Table 4, which shows stresses based on isotropic theory (WLT-I and WLR-I) and orthotropic theory (WLT-O and WLR-O). Test values based on the orthotropic approach (WLT-O and WLR-O in Table 4) are closer to the values obtained from orthotropic FEA (WLT and WLR at failure in Table 3). The orthotropic shear stresses are within 2 percent of the ones derived by FEA. The FEA model confirms the test results and simultaneously proves the distinct orthotropic nature of the composites. The maximum shear stresses are on the center of the long side for LSL, LVL, and
56
PSL. In general, the difference between the stresses at the two locations (center of long side and short side) is highest for LSL and much lower for LVL and PSL. The trend is the same for experimental results.
Conclusions and recommendations In this study, orthotropic finite element analysis was used to examine stress distribution in SCL under torsion. The objective was to determine the “pure” shear strength of SCL and compare it to the previously published experimental results. It was shown that orthotropic analysis predicts the shear strength of SCL closer to the experimental results because two shear moduli in the longitudinal planes have an effect on shear stress distribution. The degree of orthotropicty depends on the ratio of the two moduli, with a higher ratio showing more orthotropicity. Shear stresses were maximum at the middle of the two sides (faces) and constant in the middle (along the length) of the specimen. Literature cited American Society for Testing and Materials (ASTM). 2001. Test methods of static tests of lumber in structural sizes. Standard D 198. Annual Book of ASTM Standards, Vol. 04. 10 ASTM, West Conshohocken, PA. Aicher, S. 1990. Investigation on the torsion properties of vertical glulam including warping torsion of orthotropic materials. Otto Graf J. 1:9-35.
Cofer, W.F., F.D. Proctor, Jr., and D.I. McLean. 1997. Prediction of the shear strength of wood beams using finite element analysis. In: Mechanics of Cellulosic Materials. R. Perkins, ed. AMD-Vol. 221, MD-Vol. 77. Am. Soc. of Mechanical Engineers, NY. pp. 69-78. Gupta, R. and T.S. Siller. 2005. Shear strength of structural composite lumber using torsion test. J. of Testing and Eval. 33(2). __________, L.R. Heck, and T.H. Miller. 2002a. Experimental evaluation of the torsion test for determining shear strength of structural lumber. J. of Testing and Eval. 30(4): 283-290. __________, __________, and __________. 2002b. Finite-element analysis of the stress distribution in a torsion test of full-size, structural lumber. J. of Testing and Eval. 30(4): 291-302. Hindman, D. 1999. Elastic constants of selected engineered wood products. M.S. thesis, Pennsylvania State Univ., University Park, PA. Janowiak, J. and R. Pellerin. 1992. Shear moduli determination using torsional stiffness measurements. Wood and Fiber Sci. 24(4): 392-400. Leichti, R.J. and T. Nakata. 1999. The role of bearing plates in the five-point bending tests of structural-size lumber. J. of Testing and Eval. 27(3):183-190. Lekhnitskii, S.G. 1981. Theory of Elasticity of an Anisotropic Body. MIR Publishers, Moscow, Russia. Mandery, W. 1969. Relationship between perpendicular compressive stress and shear strength of wood. Wood Sci. 1(3):177-182. Radcliff, B.M. and S.K. Suddarth. 1955. The notched beam shear test for wood. Forest Prod. J. 5(2):131-135. Rammer, D., L. Soltis, and P. Lebow. 1996. Experimental shear strength of unchecked solidsawn Douglas-fir. FPL-RP-553. USDA Forest Serv., Forest Prod. Lab., Madison, WI. Riyanto, D.S. and R. Gupta. 1998. A comparison of test methods for evaluating shear strength of structural lumber. Forest Prod. J. 48(2):83-90. Trayer, G. and H. March. 1930. The torsion of members having sections common in aircraft construction. FPL Report No. 334. USDA Forest Serv., Forest Prod. Lab., Madison, WI. USDA Forest Service, Forest Products Laboratory (FPL). 1999. Wood Handbook: Wood as an Engineering Material. Forest Prod. Soc., Madison, WI.
FEBRUARY 2005
Rakesh Gupta Tobias S. Siller
Abstract Stress distribution in full-size, structural composite lumber (SCL) under torsion was analyzed using finite element analysis (FEA). The objective was to determine shear strength of SCL using orthotropic torsion theory and compare the results to experimental results. The commercial finite element program ANSYS was used to perform the analysis using orthotropic material properties. A three-dimensional, eight-node brick element was used to model the specimens. The FEA showed that SCL is more orthotropic in material behavior than solid-sawn lumber and orthotropic torsion theory should be used to determine the shear strength of SCL. The degree of orthotropicity depends upon two shear moduli in longitudinal-tangential and longitudinal-radial planes. The shear strength based on orthotropic FEA was within 2 percent of experimental results.
tructural composite lumber (SCL) is a generic name for a group of products within a wide variety of engineered wood products. SCL includes materials like laminated veneer lumber (LVL), parallel strand lumber (PSL), and laminated strand lumber (LSL). SCL is composed of either wood veneers or strands that are mainly oriented longitudinally, dried to a low moisture content, and bonded by a resin. As a result of the manufacturing process, defects, like knots and slope of grain found in solid sawn lumber (SSL), are either eliminated or at least dispersed. SCL is more homogeneous than SSL and shows less variability in measured properties, resulting in higher design values. The major advantage of SCL is availability of relatively large sizes in combination with high strength, uniformity, and straightness. In general, allowable stresses are highest for SCL, followed by glued-laminated (glulam) timber and SSL. However, the same may not be true for shear strength. FOREST PRODUCTS JOURNAL
Historically, shear strength of SSL and SCL have been determined from small specimens, known as shear blocks. However, Gupta et al. (2002a) showed that shear strength of SSL from shear blocks is lower than the shear strength of SSL based on full-size torsion tests. Even though the shear block specimen was designed as a pure shear specimen, previous studies (Radcliff and Suddarth 1955) have shown that it is not; it has stress concentration at the re-entrant corner and bending stresses. Several investigators (Rammer et al. 1996, Riyanto and Gupta 1998) have used various bending test setups to determine the shear strength of wood. However, in a bending test setup, shear stresses are affected by the presence of
other stresses, as has been shown by previous studies (Mandery 1969, Riyanto and Gupta 1998). Torsion is the only test method that produces pure shear stresses in the specimen and may be used to determine shear strength of various wood products. Therefore, the objective of this study is to perform a finite element analysis (FEA) of the full-size SCL subjected to torsion in order to enhance the understanding of torsion theory using orthotropic material properties, and to compare FEA results with experimental results.
Literature review Cofer et al. (1997) modeled different size SSL beams by means of FEA and applied the Tsai-Hill failure criterion to predict shear strength. The FEA results compared reasonably well with experimental results for small member sizes, whereas the reduction in shear strength with increasing beam size, as reported by their experimental results, was not reproduced by the FEA results. Therefore the authors concluded that the size effect might be caused by the higher probability of the presence of natural defects in larger members of wood, since the model did not account for any flaws. Leichti and Nakhata (1999) modeled 38- by 89-mm2 and 38- by 240-mm2 Douglas-fir beams under a five-point
The authors are, respectively, Associate Professor and Former Graduate Research Assistant, Dept. of Wood Science and Engineering, Oregon State Univ., Corvallis, OR 97331. The authors would like to thank Milo Clauson for his help and advice on instrumentation and testing. This paper was received for publication in September 2003. Article No. 9753. ✳Forest Products Society Member. ©Forest Products Society 2005. Forest Prod. J. 55(2):51-56.
Vol. 55, No. 2
51
ing area to prevent crushing for smaller sizes of dimension lumber presented an immense problem. Consequently, the authors concluded that beams having smaller cross sections should not be tested in the five-point configuration or that a standard test method, which is applicable to all sizes of lumber, should be introduced. One such method, as shown by Gupta et al. (2002a), is a torsion test that is applicable to all sizes.
Figure 1. — Cross-sectional view of a specimen showing FE mesh.
Gupta et al. (2002b) performed FEA of full-size SSL under torsion using isotropic material properties and revealed that uniform shear stress occurs within the shear span, which begins and ends at a distance of approximately two times the depth plus the grip distance away from each end of the specimen. In addition, FEA verified that the shear failure plane in torsion is similar to the known shear failure mode in specimens subjected to transverse loads. They finally concluded that torsion is the best practical method for determining the pure shear strength of full-size, structural lumber. Aicher (1990) investigated torsional properties of structural size, vertically laminated glulam beams using FEA and compared isotropic with orthotropic material behavior. Two different models were evaluated, one simulating St. Venant torsion (free warping), the other one considering warping torsion (restrained warping). The consideration of warping torsion resulted in a significant difference between isotropic and orthotropic material behavior.
Figure 2. — Beam under torsion loads and boundary conditions.
bending load configuration with a spanto-depth ratio of five. The authors concluded that elementary beam theory did not accurately predict the complex state of shear and bending stresses within the specimen, but overestimated the results of FEA by about 30 to 40 percent. In addition, the provision of sufficient bear52
The torsion test has also been used to determine elastic constants of any material where pure shear stress state is needed. Janowiak and Pellerin (1992) determined the orthotropic shear moduli for three reconstituted wood panel products using torsion. The authors concluded that the torsion test is a more efficient test method to determine the orthotropic shear moduli compared to flexure tests. Hindman (1999) evaluated the elastic constants of selected engineered wood products using torsion, bending, compression, and tension tests. He concluded that SCL showed a different elastic response than did solid wood. Lathe checks in LVL and PSL were found to decrease transverse elastic stiffness as well as the shear resistance through the thickness. He strongly recommended not to use the set of elastic constants of solid wood for SCL manu-
factured of the same species, because of the obviously different elastic behavior.
Finite element modeling The model The experimental torsion test configuration used by Gupta and Siller (2005) was modeled using the finite element (FE) method to understand the state of stress under torsion using orthotropic material properties. The commercial FE program (ANSYS 5.6®) was used to perform the analysis. A three-dimensional, eight-node brick element (SOLID45 element in ANSYS) was used to model the specimen as an orthotropic continuum. The SOLID45 element offered 6 degrees of freedom per node, allowing translations and rotations in the three major directions. The analysis focused on the full-size specimen measuring 44 by 140 by 1372 mm3 (same size as used previously [Gupta and Siller 2005]) under torsion for all three composites: LSL, LVL, and PSL. The closed-form solutions of Trayer and March (1930) for isotropic material behavior, also given in ASTM D198 (ASTM 2001), and Lekhnitskii (1981) for orthotropic materials were compared to the results of FEA. Mesh size Based on the results of a previous study (Gupta et al. 2002b), the mesh size shown in Figure 1 was used in this study. It consisted of 6 × 22 × 108 = 14,256 elements and 7 × 23 × 109 = 17,549 nodes. This mesh is already very fine, especially in the area of interest, and further refinement would not be appropriate since the results are mainly influenced by the set of elastic constants. Boundary conditions As shown in Figure 2, one end of the beam was fixed while the opposite end had a twisting moment. At the fixed end, all nodes in the longitudinal direction within the gripping distance of 51 mm (2 in.) on the surface of both wide faces were fixed over the entire depth. All degrees of freedom were constrained for these nodes. The end face and both narrow faces were not restrained. This procedure best described the actual test conditions and was as close as possible to pure St. Venant torsion, i.e., unrestrained warping, while still securely clenching the specimen. Unrestrained warping was possible for the entire end face and both narrow faces. The conFEBRUARY 2005
Table 1. — Input values for FEA model.
Elastic range
Moment (M)
Force (F)
Pressure (p)
Slope (s)
(Nm)
(N)
(MPa)
(N/mm ) 0.0260
a
3
300
3221
1.82
LSL
850
9123
5.14
0.0736
LVL
570
6117
3.45
0.0494
PSL
460
4937
2.78
0.0398
aFor
all three composites.
Table 2. — Material properties for FEA model. MOE Material
E R = ET
Shear moduli EL
GRT
GRL
Poisson ratios GTL
QRT
QRL
QTL
- - - - - - - - - - - - - - - - (MPa) - - - - - - - - - - - - - - - LSL
1207
11000
94.5
318
782
0.496
0.054
0.022
LVL
397
14600
44.4
407
593
0.390
0.036
0.029
PSL
404
15000
72.8
310
398
0.390
0.036
0.029
Table 3. — FEA shear stresses. Elastic range Material
Results and discussion
At failure WLR
WLT
WLT
WLR
- - - - - - - - - - - - - - - - - - - - - - (MPa) - - - - - - - - - - - - - - - - - - - - - LSL
4.35
2.22
12.33
6.28
LVL
4.13
2.61
7.84
4.96
PSL
4.05
2.73
6.21
4.19
straints at the wide faces were necessary to provide model stability. A force couple was used to apply torque at the end receiving the twisting moment. It was decided to choose a triangular pressure distribution instead of a uniform one, or just point loads, to avoid high stress concentrations and to model the experimental tests more realistically. The tests consistently showed more impact on the corners of the wide faces compared to the center. Consequently, a triangular pressure distribution was applied to the gripping length of 51 mm (2 in.) and one half of the pressure couple loaded the lower depth, while the other one was imposed on the upper half on the opposite side as shown in Figure 2. The twisted end of the model better reflects the real test conditions, since the restraints in the longitudinal direction of the fixed end might slightly influence the state of stress. Gupta et al. (2002b) used similar boundary conditions while analyzing full-size SSL under torsion. Loads Two different load cases were investigated. For the first one, the applied torque was within the linear range of the FOREST PRODUCTS JOURNAL
moment - rotation curve obtained by the tests (Gupta and Siller 2005). A torque of 300 Nm was chosen for this case for all three composites. Second, the average ultimate torsional moment at failure, obtained from Gupta and Siller (2005), for each of the three composites was applied. The torque values were converted into equivalent pressure (p) values using simple statics. All necessary input values for the different load cases are summarized in Table 1 and shown in Figure 2. Material properties As shown in Table 2, the FE model used orthotropic material properties. Gupta et al. (2002b) used only isotropic material properties for solid- sawn Douglas-fir because they found out that it was not beneficial to perform an orthotropic analysis, since the results between an orthotropic FEA and isotropic theory differed only by 1.3 percent. However, it was believed that SCL might behave differently from solidsawn Douglas-fir because of large differences in material constants. Therefore, this study included the orthotropic material behavior for all three compos-
Vol. 55, No. 2
ites. The assumption that the radial and the tangential moduli of elasticity (MOE) are equivalent had to be used, since no source was found that offered more detailed values. The longitudinal MOE values (Ez) were measured via third-point bending tests following ASTM D198 (ASTM 2001). The span-to-depth ratio was 31. The transverse MOE and all shear moduli values were calculated using elastic constant ratios for SCL given by Hindman (1999), based on the measured value of Ez. The Poisson Ratios are taken from the Wood Handbook (FPL 1999) for solid wood. All model input values are shown in Table 2. Shear stresses are evaluated at the middle of the long (wide face; points A in Fig. 1; WLT) and short (narrow face; points B in Fig. 1; WLR) sides because that is where the stresses are maximum.
Shear stresses The focus of the FE analysis was to get a better idea of the stress distribution within the specimen and to compare the behavior of the three different SCL products. In addition, the analysis was performed to further verify the orthotropic theory that was applied to evaluate the torsion test results. Results of the FEA are given in Table 3, which shows the shear stresses within the elastic range based on a constant torque value of 300 Nm for all three composites. The last two columns show shear stresses at failure, based on the different average ultimate torque values for each material. All shear stresses in Table 3 are based on the orthotropic material properties given in Table 2. Even though eight material constants were used, a parametric study showed that shear stresses were heavily dependent on the two shear moduli in the longitudinal plane, GTL and GRL. All other elastic constants, MOE as well as Poisson Ratios and the third shear modulus, were only of very minor importance. This observation matches orthotropic theory, where the two shear moduli in the longitudinal planes are the only material constants that are considered. The ratio of these two shear moduli, g = GTL/GRL, is also an indicator of the degree of orthotropicity of various materials. The farther g is from unity, the more orthotropic the material is. Therefore, it is quite obvious that LSL (g = 2.46) be53
Figure 3. — FE shear stresses.
haves differently from LVL (g = 1.46) and PSL (g = 1.28), since g for LSL is almost twice the g value for LVL and PSL. Note the fact that within the elastic range, the shear stress τLR on the short side is highest for PSL (2.73 MPa), the least orthotropic material. Whereas, the shear stress τLT on the long side is highest for LSL (4.35 MPa), the most orthotropic material. This would lead us to 54
believe that as torque increases toward failure, the higher g material would probably fail at the middle of the long side. While looking at shear strengths at failure, which are based on three different values of failure torque for three composites, the highest shear stress on any side (long or short) is for LSL, the most orthotropic material. Shear strength at failure would also indicate that all three composites would fail at
the middle of the long side. The study by Gupta and Siller (2005) showed that LSL failed at the middle of the short side, whereas the other two composites failed at the middle of the long side. The main reason for this is the vastly different values of shear moduli for LSL in two longitudinal planes. This difference is probably caused by the special design of the composite, for instance, the high FEBRUARY 2005
plots are depicted for each material. The first plot for each material (Fig. 3a, 3c, and 3e) shows the front view (wide face) of the specimen where the shear stress, WLT, is maximum. The second plot for each material (Fig. 3b, 3d, and 3f) shows an isometric top view (narrow face) of the specimen where the shear stress, WLR, is maximum. The constrained end is at the left end of the specimen. The loaded end, where the origin of the coordinate system lies, is at the specimen’s right end. There is a symmetric state of stress in the case of axial torsion, i.e., shear stresses on opposite sides (faces) of the specimen have the same distribution and absolute values.
Figure 4. — FE maximum shear stresses along the beam length.
degree of densification, the strand orientation, or the resin used. Moreover, this ratio, g, explains why the behavior of solid Douglas-fir is almost isotropic. The g for Douglas-fir is closer to unity and that is why its behavior is almost isotropic. Gupta et al. (2002b) reported only a 1.3 percent difference between orthotropic- and isotropic-based shear strengths, considering a rectangular beam subjected to torsion. Since the behavior of SCL is orthotropic, there is a need for accurate evaluation of its material constants in order to better predict its performance. Table 3 shows that WLT is always more than WLR. But for g values of less than 1, it is possible for WLR to be higher than WLT. For the given aspect ratio of 3.1, the shear stresses at the different locations FOREST PRODUCTS JOURNAL
were equal (WLR = WLT = 3.8 MPa) for a shear moduli ratio g of 0.74, and the shear stress WLR on the center of the short side was higher than the shear stress WLT on the long side for g less than 0.74. Hardwoods possess such shear moduli ratios. Softwoods, however, have g ratios close to unity. SCL, in contrast, is proprietary and has, mainly due to densification during manufacture, g ratios well above unity. This illustrates the importance of anisotropy in the case of axial torsion of specimens with a rectangular cross section. Stress distribution The shear stress distribution in the specimens is shown in Figures 3a to 3f. The results are shown for specimens subjected to mean torque values at failure. All stresses are given in MPa. Two
Vol. 55, No. 2
Figure 4 depicts the maximum shear stresses (in MPa on y-axis) along the specimen length (in mm on x-axis). Note that the rotated end is now shown to the left and the constrained one to the right. The small kink in the plots of WLT at the fixed end is similar to the kink Gupta et al. (2002b) had reported. It is due to the longitudinal direction constraint of the nodes at the fixed end that were required to provide model stability. The graphs for WLT depict the shear stresses on the center of the wide face, and the graphs for WLR show the shear stresses on the center of the narrow face. All graphs have the same abscissa scale, but the first one (Fig. 4a) has a different ordinate scale. For all specimens, shear stress is constant in the middle of the specimen (about two times the depth away from both ends). The same results were obtained by Gupta et al. (2002b) for isotropic modeling of SSL under torsion. The values listed in Table 3 are taken at the center of the span at mid-depth (WLT) or mid-width (WLR). The state of stress is the same for all materials. The FEA plots (Figs. 3 and 4) depict clearly that the maximum shear stresses act on the center of each side. In the case of axial torsion, there is a parabolic shear stress distribution along the width and the depth for specimens with a rectangular cross section. There is a constant state of stress within the specimens, except for a very short part of the span where end effects occur. The state of stress is more affected by the constrained end than by the loaded end. Shear stress (WLT) on the center of the long side shows its maximum value close to the loaded end and a distinct decline towards the constrained end. On the contrary, shear stress (WLR) on the center of the short side shows its maxi55
Table 4. — Experimental shear stresses from Gupta and Siller (2005). Isotropic Material
Long WLT-I
Orthotropic Short WLR-I
Long WLT-O
Short WLR-O
- - - - - - - - - - - - - - - - - - - - - - (MPa) - - - - - - - - - - - - - - - - - - - - - - LSL
11.59
8.83
12.69
6.43
LVL
7.66
5.83
7.96
4.90
PSL
6.65
5.02
6.82
4.55
mum value close to the fixed end. However, the difference between maximum shear stresses and the values in the middle part of the span are in both cases negligible. For both the FEA model and experimental tests, the specimens were constrained by clamps only on the long side (Fig 2). Only the intensity of the shear stresses varies. The FEA model was subjected to different torque loads since the three composites were able to bear different torque loads in experimental testing. LSL showed the biggest difference in shear stress between the two faces. All three composites showed markedly higher shear stresses on the long side. Comparison between torsion test and FE model Shear strengths based on the torsion test are listed in Table 4, which shows stresses based on isotropic theory (WLT-I and WLR-I) and orthotropic theory (WLT-O and WLR-O). Test values based on the orthotropic approach (WLT-O and WLR-O in Table 4) are closer to the values obtained from orthotropic FEA (WLT and WLR at failure in Table 3). The orthotropic shear stresses are within 2 percent of the ones derived by FEA. The FEA model confirms the test results and simultaneously proves the distinct orthotropic nature of the composites. The maximum shear stresses are on the center of the long side for LSL, LVL, and
56
PSL. In general, the difference between the stresses at the two locations (center of long side and short side) is highest for LSL and much lower for LVL and PSL. The trend is the same for experimental results.
Conclusions and recommendations In this study, orthotropic finite element analysis was used to examine stress distribution in SCL under torsion. The objective was to determine the “pure” shear strength of SCL and compare it to the previously published experimental results. It was shown that orthotropic analysis predicts the shear strength of SCL closer to the experimental results because two shear moduli in the longitudinal planes have an effect on shear stress distribution. The degree of orthotropicty depends on the ratio of the two moduli, with a higher ratio showing more orthotropicity. Shear stresses were maximum at the middle of the two sides (faces) and constant in the middle (along the length) of the specimen. Literature cited American Society for Testing and Materials (ASTM). 2001. Test methods of static tests of lumber in structural sizes. Standard D 198. Annual Book of ASTM Standards, Vol. 04. 10 ASTM, West Conshohocken, PA. Aicher, S. 1990. Investigation on the torsion properties of vertical glulam including warping torsion of orthotropic materials. Otto Graf J. 1:9-35.
Cofer, W.F., F.D. Proctor, Jr., and D.I. McLean. 1997. Prediction of the shear strength of wood beams using finite element analysis. In: Mechanics of Cellulosic Materials. R. Perkins, ed. AMD-Vol. 221, MD-Vol. 77. Am. Soc. of Mechanical Engineers, NY. pp. 69-78. Gupta, R. and T.S. Siller. 2005. Shear strength of structural composite lumber using torsion test. J. of Testing and Eval. 33(2). __________, L.R. Heck, and T.H. Miller. 2002a. Experimental evaluation of the torsion test for determining shear strength of structural lumber. J. of Testing and Eval. 30(4): 283-290. __________, __________, and __________. 2002b. Finite-element analysis of the stress distribution in a torsion test of full-size, structural lumber. J. of Testing and Eval. 30(4): 291-302. Hindman, D. 1999. Elastic constants of selected engineered wood products. M.S. thesis, Pennsylvania State Univ., University Park, PA. Janowiak, J. and R. Pellerin. 1992. Shear moduli determination using torsional stiffness measurements. Wood and Fiber Sci. 24(4): 392-400. Leichti, R.J. and T. Nakata. 1999. The role of bearing plates in the five-point bending tests of structural-size lumber. J. of Testing and Eval. 27(3):183-190. Lekhnitskii, S.G. 1981. Theory of Elasticity of an Anisotropic Body. MIR Publishers, Moscow, Russia. Mandery, W. 1969. Relationship between perpendicular compressive stress and shear strength of wood. Wood Sci. 1(3):177-182. Radcliff, B.M. and S.K. Suddarth. 1955. The notched beam shear test for wood. Forest Prod. J. 5(2):131-135. Rammer, D., L. Soltis, and P. Lebow. 1996. Experimental shear strength of unchecked solidsawn Douglas-fir. FPL-RP-553. USDA Forest Serv., Forest Prod. Lab., Madison, WI. Riyanto, D.S. and R. Gupta. 1998. A comparison of test methods for evaluating shear strength of structural lumber. Forest Prod. J. 48(2):83-90. Trayer, G. and H. March. 1930. The torsion of members having sections common in aircraft construction. FPL Report No. 334. USDA Forest Serv., Forest Prod. Lab., Madison, WI. USDA Forest Service, Forest Products Laboratory (FPL). 1999. Wood Handbook: Wood as an Engineering Material. Forest Prod. Soc., Madison, WI.
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