ateral-torsional buckling of orthotropic rectangular section beams Summary Bambang SUYOAMONO Professor Department of Civil ngineering, Parahyangan Catholic University Bandung, Indonesia Adhijoso JONDO Associate Professor Department of Civil ngineering, Parahyangan Catholic University Bandung, Indonesia he elastic lateral-torsional buckling of rectangular section beams made of wood under various loading conditions are investigated in this paper. hree dimensional finite element models considering the orthotropic behavior of wood are utilized to predict the elastic buckling load. he effects of depth/width ratios of the cross section, wood species, and cross grain of the beams on the predicted buckling loads are investigated. For all the cases considered, the critical loads predicted using finite element models assuming wood as orthotropic material are much smaller than those predicted assuming wood as isotropic material. he predicted critical loads of beams with cross grain of 45 o are approximately a third of the predicted critical loads of beams without any cross grain. he finite element results agree very well with the beam tests on several species of the hardwood category.. Introduction ateral-torsional buckling (B) is a limit state where beam deformation includes in-plane deformation, out-of-plane deformation, and twisting []. Consider, for example, a simply supported rectangular beam loaded by uniform moment M as seen in Figure. he beam is laterally supported at both ends so that the cross sections at both ends cannot rotate but free to warp. Assuming that the material is isotropic and linearly elastic, the critical moment, M cr, that causes lateral-torsional buckling to occur is Fig. ateral orsional Buckling (B) of a rectangular section beam. π M cr = I yj () u where = modulus of elasticity, I y = moment of inertia with respect to the weak (y) axis, = shear modulus, and J = torsional constant, and u is the unbraced length of the beam. For rectangular sections with depth d and width b, = b d (2) 2 I y J = kb d () where k is a cross section constant and can be obtained from able [2]. Note that warping, a deformation that occurs in a thin walled open section, is neglected in q. ().he derivation of q. () can be found in literature []. For nonuniform moment diagram, the critical moment can be obtained by multiplying the critical moment for uniform moment, q. (), by an B modification factor for nonuniform moment C b
[4]. For a simply supported beam laterally supported at its ends under uniform load and, C b =.4 and if the beam is loaded by concentrated load at midspan, C b =.2 [5]. herefore, the critical uniform load for a simply supported beam laterally supported at its ends is able Constant k required for computing torsional constant J of a rectangular section [2] d/b k 2 4 5.229.26.28.29 J I w y u u cr π (.4) 8 2 = (4) and the critical concentrated load at midspan for a simply supported beam laterally supported at its ends is ( ) J I P y u u cr π.2 4 = (5) It should be noted that q. (), (4), and (5) were derived by assuming that the beam material is isotropic. In this paper, the applicability of these equations for orthotropic material is investigated by using Finite lement Method (FM). 2. Finite lement Method If wood is assumed as an orthotropic and elastic material with three mutually perpendicular material principal axes (longitudinal, radial, and tangential), then the relationship between strain components and stress components can be expressed as [6] = (6) where i are moduli of elasticity, ij are Poisson s ratios, and ij are shear moduli, where i =,, and. For an isotropic material, the relationship reduces to = (7) where = /{2(+)}. In the subsequent analyses using a finite element program (SAP2 Advanced.. ), the beam is modelled by using solid (-D) elements. Boundary conditions of the finite element mesh are the same as the boundary conditions used in the derivation of q.(), i.e. no out-of-plane deformation
at both beam ends but both ends can rotate in the plane of the beam. By defining buckling analysis in the program, the critical load along with the buckling modes can be obtained. he first mode is the one that is associated with the critical (lowest) load. As seen in Fig. 2, the buckling mode shows in-plane and out-of-plane deformations indicating that it is B mode. Fig. 2 ypical lateral torsional buckling mode for simply supported beam laterally supported at its ends. q. (6) and (7) are incorporated in the program for orthotropic and isotropic materials, respectively. he hypothetical beam is assumed to be made of walnut. he static bending modulus of elasticity sb of the material is 6 MPa [7]. he longitudinal modulus of elasticity can be taken as ten percent higher than the static bending modulus of elasticity [7], i.e. =. x 6 MPa = 276 MPa. he other elastic properties are taken from the literature [7], namely = 5 MPa, = 75 MPa, =.495, =.62, =.78, = 85 MPa, =.79, and = 268 MPa. he same beam is also analyzed assuming isotropic behaviour by using the modulus of elasticity = 276 MPa and Poisson s ratio =.525. he FM is used to analyze isotropic and orthotropic beams under three loading conditions, namely uniform moment, distributed load, and concentrated load. 2. Uniform Moment Fig. A simply supported beam under uniform bending moment diagram. Both ends of the beam are laterally supported. able 2 Critical moment M cr (knm) computed using isotropic equation (q.), FM assuming isotropic material, and FM assuming orthotropic material. Method Isotropic quation (q. ) FM, isotropic FM, orthotropic, cross grain o FM, orthotropic, cross grain 5 o FM, orthotropic, cross grain o FM, orthotropic, cross grain 45 o M cr (knm) 24.4 24.46 9.7 7.5 4.88.7 o investigate if FM is appropriate in predicting critical load, consider a simply supported beam under uniform moment as seen in Fig.. he beam length = u = 8 mm. he cross section is rectangular with the width b = 4 mm and depth d = 8 mm. Assuming the beam as isotropic material, it can be seen in able 2 that FM results in a critical moment M cr = 24.46 knm which is only.2% higher than the critical moment computed using q. (). his result shows that FM can be used in buckling analysis with a very high accuracy. o investigate if the isotropic equation (q. ) can be used in predicting the critical load of an orthotropic material, the same beam is further analyzed using FM assuming orthotropic material. he beam plane is - plane. he analysis is repeated for several angles of cross grain, namely o, 5 o, o, and 45 o. Cross grain is defined as the angle between the beam longitudinal axis and the material axis. As seen in able 2 and Fig. 4, the critical moment for orthotropic material is much lower than the critical moment computed using q. (). he difference is higher for greater cross grain angle. hese results show that it is unsafe to use the isotropic equation (q. ) for orthotropic material such as wood.
Fig. 4 he effect of cross grain on the critical end moments M cr for a simply supported beam under uniform bending moment diagram. 2.2 Distributed oad Fig. 5 A simply supported beam under uniform load. Both ends of the beam are laterally supported. able Critical load w cr (kn/m) computed using isotropic equation (q.4), FM assuming isotropic material, and FM assuming orthotropic material. Method Isotropic quation (q. 4) FM, isotropic FM, orthotropic, cross grain o FM, orthotropic, cross grain 5 o FM, orthotropic, cross grain o FM, orthotropic, cross grain 45 o w cr (kn/m) 48.9 28.8 9. 9.88 6.84 4.97 he same beam as described before is now loaded by uniformly distributed load w as seen in Fig. 5. As before, the beam is analyzed assuming as isotropic material and orthotropic material with several angles of cross grain. As seen in able and Fig. 6, the critical load w cr obtained from FM is only 5.54% lower than the critical load computed using q. 4. his result shows that, once again, FM can be used in buckling analysis with a very high accuracy. If the material is orthotropic, FM shows that the critical load w cr is much lower than the critical load for isotropic material. For an orthotropic beam with cross grain of 45 o, the critical load is only approximately 2% of the critical load of isotropic beam. he conclusion for this loading case is the same as in the case of uniform moment, i.e. it is unsafe to use the isotropic equation (q. ) for orthotropic material such as wood.
Fig. 6 he effect of cross grain on the critical load w cr for a simply supported beam under uniformly distributed load. 2. Concentrated oad Fig. 7 A simply supported beam loaded by concentrated load at midspan. Both ends of the beam are laterally supported. able 4 Comparison of critical loads P cr (kn) computed using isotropic equation (q. 5), FM assuming isotropic material, and FM assuming orthotropic material. d/ b 2 4 5 Isotropic quation (q. 5) 6.72 259.7 57.8 47.46 FM, isotropic 5.92 25.55.5 85.88 FM, orthotropic 47.47 68.64 85.49 98.2 o investigate the effect of cross section aspect ratio, d/b, on the critical buckling load, a hypothetical simply supported walnut beam under a concentrated load P as seen in Fig. 7 is analyzed using FM. he material properties are the same as described before. he beam length and width are still the same as before, i.e. = 8 mm and b = 4 mm. he depth of the cross section varies to give d/b ratios of 2,, 4, and 5. As seen in able 4 and Fig. 8 the difference between FM results for isotropic material and q. 5 are quite small. he largest difference is 7.98% (for d/b = 5). As in the loading cases analyzed before, the critical load obtained using FM for orthotropic material is much smaller than that for isotropic material. he difference between isotropic and orthotropic critical loads is larger as the ratio d/b increases. his is due to the fact that as d/b increases, the beam behaves more like a deep beam and the anisotropic nature of the material becomes more pronounced.
Fig. 8 Critical load P cr (kn) for various cross section aspect ratio d/b. he width b is kept constant while depth d varies.. Static Bending ests o verify the results of FM, static bending tests on simply supported beams were performed. he experimental setup is shown in Fig. 9. As seen in the figure, both ends of the beam are laterally supported. he loading given by the Universal esting Machine (UM) was displacement controlled with a displacement rate of 2 mm/minute. A dial gauge was attached at midspan to monitor the lateral deformation. hree species in hardwood category grown in Indonesia were tested, namely albasia (albizzia falcata, batai), meranti = shorea spec. div.), and bangkirai (hopea spec. div.). For each species, three specimens were tested. he specific gravity of each species were.9,.45, and.8 for bangkirai, meranti, and albasia, respectively. he moisture content of all specimens were approximately 5%. he dimensions of all specimens were as follows: total length total = 26 mm, beam span = 6 mm, width b = 2 mm, and depth d = mm. Fig.9 est setup for simply supported beam under a concentrated load at midspan. By using the computer attached to the UM, the load versus midspan deflection for each test was recorded. he typical load - midspan deflection curve is shown in Fig.. he static bending modulus of elasticity sb can be computed using the elementary equation sb P = 48 ΔI x (8) where load P and midspan deflection Δ are the values during the linearly elastic stage of the curve. Static bending moduli of elasticity for each species computed using q. (8) are 22 MPa, 526 MPa, and 9 MPa, for bangkirai, meranti, and albasia, respectively.
Fig. ypical load (N) deflection (mm) curve obtained from static bending test. he critical load was taken as the load at which the beam started to deflect laterally as indicated by the dial gauge. he results for each species are shown in able 5. As expected, the critical load P cr for the species with the highest specific gravity is the largest. hese experimental results are compared with the FM results. able 5 Comparison of critical loads P cr (N) obtained from FM and static bending test. Species Bangkirai Meranti Albasia FM, orthotropic 2529.9 26. 27. Static bending test 9. 8. 7. Difference -5.7% 7.8%.8% o perform buckling analysis using FM, material properties for each species are needed. As before, longitudinal modulus of elasticity =. sb. Since there is no data of all other orthotropic elastic properties for the three species tested, the properties were taken as shown in able 6. he critical buckling loads P cr obtained from FM are shown in able 5. It is interesting to note that although the elastic properties in FM are approximation, the difference between experimental result and FM result is relatively small, especially for albasia. able 6 lastic properties used in finite element analyses of three species, namely bangkirai, meranti, and albasia. Bangkirai Meranti Albasia (MPa) (MPa) (MPa) (MPa) (MPa) (MPa) 457 5529 4 272 85 524 45 279.5.5.5.45.45.45.56.56.56 9 448 275 98 492 28 6 7 4. Discussion As shown in the analyses of beams subject to various loading conditions using FM, orthotropic material always results in a lower critical load than isotropic material. he difference between the two materials becomes larger if the beam has cross grain. While investigating various angles of cross grain using real beam specimen is difficult, FM is a versatile tool to perform such task.
Investigation on various depth/width ratio of the beam cross section shows that as the ratio increases, the difference between isotropic and orthotropic materials become larger. 5. Conclusions Since orthotropic assumption always results in lower critical load for the loading cases considered in this paper, then isotropic equations for B should not be used for orthotropic material such as wood. One possible approach is to use reduction factors in the isotropic equations. Such factor should depend on beam dimensions, specific gravity, cross grain, and perhaps all other possible factors that have not been considered in this paper. 6. Acknowledgements he authors gratefully acknowledge financial support of Parahyangan Catholic University, Indonesia. he authors also gratefully acknowledge the assistance of staffs at the Structural aboratory at the university. 7. eferences [] American Wood Council (AWC), Designing for lateral-torsional stability in wood members, echnical eport 4, Washington, DC, American Forest & Paper Association (AF&PA), 2, p.. [2] imoshenko, SP and oodier, heory of lasticity, McrawHill, New York, 97, p. 2. [] Simitses., Hodges, D.H., Fundamentals of Structural Stability, Butterworth-Heinemann, Burlington, 26, p. 25. [4] American Institute of Steel Construction (AISC). Specification for structural buildings. AISC, Chicago, 25, p. 46. [5] Segui, W, FD Steel Design. rd dition. homson Brooks/Cole, 2, pp. 55. [6] Bodig, J and Jayne, Mechanics of Wood and Wood Composites, Krieger Publishing Company, Malabar, 99, p. 98. [7] Forest Product Society, Wood Handbook, Wood as an ngineering Material,. Madison, 999, pp. 4-2, 4-.