J Wood Sc (2002) 48:14-19 9 The Japan Wood Research Socety 2002 Hrosh Yoshhara 9 Yoshtaka Kubojma Measurement of the shear modulus of wood by asymmetrc four-pont bendng tests Receved: December 8, 2000 / Accepted: February 28, 2001 Abstract We conducted asymmetrc four-pont bendng tests of wood and obtaned the shear modul on the bass of Tmoshenko's theory of bendng. Akamatsu (Japanese red pne, Pnus densflora D. Don) and shoj (Japanese ash, Fraxnus spaethana Lngelsh.) were used for the tests. Asymmetrc four-pont bendng tests were undertaken by varyng the depth/span ratos; and Young's modulus and the shear modulus were calculated by Tmoshenko's bendng theory. Independent of the asymmetrc bendng tests, we also conducted three-pont bendng tests, free-freeflexural vbraton tests, and numercal calculatons by the fnte element method. Young's and shear modul obtaned by these methods were compared wth those derved from the asymmetrc bendng tests. Based on these comparsons, we concluded that the shear modulus can be properly obtaned by the asymmetrc four-pont bendng tests when the span s 20 tmes larger than the depth. Key words Shear modulus - Asymmetrc four-pont bendng test 9 Tmoshenko's beam theory 9 Depth/span rato Introducton To develop a desgn methodology for wood and wood products, deformaton by shear makes the shearng propertes one of the mportant parameters n the desgn processes. In the beam wth an I-shaped cross secton, for example, the deflecton by shear s sgnfcant because of the slenderness along the neutral axs. If beams desgned wthout consderng the shearng deflecton are used for floorng materals, H. Yoshhara ([]) Faculty of Scence and Engneerng, Shmane Unversty, Nshkawazu-cho 60, Matsue, Shmane 690-804, Japan Tel. +81-82-32-608; Fax +81-82-32-6123 e-mal: yoshara@rko.shmane-u.ac.jp Y. Kubojma Forestry and Forest Products Research Insttute, Tsukuba 30-8687, Japan the deformaton of the floor s so serous t causes vbratons from whch the resdent may suffer. Thus, the shear modulus should be measured properly to predct the magntude of the shearng deformaton. Varous testng methods are avalable to determne the shear modulus, such as the torson test, I'a the Iospescu testy and the tenson or compresson test of a 4 ~ off-axs specmen. 1 These tests, however, have ther drawbacks. The torson and Iospescu tests requre specal equpment. For the tenson or compresson test of the off-axs specmen, the specmen's geometry s restrcted. If these testng methods were smpler and more convenent, a relable engneerng database for shearng propertes, ncludng the shear modulus, could be developed. The bendng test under varyng the span/depth ratos s smpler than these methods. In prevous studes the shear modul of several wood speces were measured by three-pont bendng tests based on Tmoshenko's bendng theory. The theoretcal shear modulus was estmated to be smaller than the real value because of the extra deflecton that cannot be predcted by the theoretcal construct. To obtan the proper shear modulus value, the orgnal Tmoshenko equaton was modfed takng nto consderaton the test results and numercal analyses. '6 Nevertheless, t s more convenent that the shear modulus s properly measured wthout modfyng the orgnal equaton because several expermental condtons (e.g,, the radus of the loadng nose and the measurement of deflecton) nfluence the modfcaton. 6'7 The asymmetrc four-pont bendng test whose detal s mentoned below s a promsng method for obtanng the shear modulus because the deflecton produced by the shearng force s emphaszed. In ths study we conducted asymmetrc four-pont bendng tests on two wood speces and examned ther valdty by comparng the results wth those obtaned by other testng methods. Theores Fgure la outlnes the asymmetrc four-pont bendng test. The dstance between the outer spans s I, and the drectons
1 A Specmen Load P c B D J r I where M s the bendng moment, Ex s Young's modulus n the length drecton, and I s the moment of nerta. By solvng ths equaton, the dsplacement lne caused by the bendng moment, y> s gven when (x, Yb) = (0, 0) as: Yb = EJ ~. 24 P ( 1 x3 _ 11x2 + -- 23 12 x ~f~//~ 8 432!13/ 216 j P. P (_ I..1._ x3 _}_!lx2-49 12 x E~I ~, 24 8 432 ] + 216 ) (a) 14--- l _[_ l 3 rl- 3 ~'X 3P P y 4 4 t The equaton represents that the beam s supported by A and C, and that the dsplacements of B and D are smlar to each other. Durng bendng the dsplacement caused by the shearng force, denoted y~, s always produced; and the slope of the deflecton lne by the shearng force dys/dx s represented by Tmoshenko's bendng theory as: s (2) dys_ sv dx GxyA (3) (b) P 3P 4 4 Fg. 1. Asymmetrc four-pont bendng test and the beam subjected to the lateral forces parallel and perpendcular to the length are defned as x and y, respectvely. The specmen s supported at the pont of x = 0 and 21/3, whch are denoted A and C, respectvely; and the load s appled at ponts x = 21/3 and l, denoted B and D, respectvely. When the dsplacements of B and D are smlar, the total load of P s dvded nto 3P/4 and P/4 for B and D, respectvely, whereas the reacton forces at A and C are P/4 and 3P/4, respectvely. Fgure lb shows the beam asymmetrcally subjected to the lateral forces. Under ths loadng condton, the bendng equaton s derved by the elementary bendng theory as follows, s where Gxy s the shear modulus n the xy-plane, V s the shearng force, A s the cross-sectonal area of the beam, and s s Tmoshenko's shear factor. Ths factor s 1. for a beam wth a rectangular cross secton when t s defned as the maxmum/average shear stress rato, whereas t s derved as 1.2 by calculatng the stran energy. The shearng force s obtaned by dfferentatng the bendng moment by x. Hence, from Eqs. (1) and (3), dys/dx s represented as follows: dy~ _ s dm dx GxyA dx 2G 4GxyA s p ~2l < - (T _x<-i 4G,,yA ) The value of Ys s 0 n x = 0 and l; hence (4) s Px 4GxyA 0_<x_< d2y M = ExI-j- Z = (1) Ys = s Px 2GKyA s PI 4GxyA l<x<2@) 3 () _ s Px + s PI 4G~yA 4GxyA '2l ~x<_l'~ ) 3
- + 16 AAC - 2~--~D C deflecton that cannot be predcted by Tmoshenko's bendng theory s so marked the shear modulus calculated by Tmoshenko's equaton tends be smaller than the real value. The emphaszed shearng effect durng asymmetrc bendng, however, mght obscure the extra deflecton. 3 3 3 Fg. 2. Defnton of vertcal dsplacements n asymmetrc four-pont bendng Fgure 2 llustrates the vertcal dsplacement wth asymmetrc bendng. In the elastc stran range, the vertcal dsplacement s much smaller than the span, and the deflecton at pont B, denoted d, s approxmated as follows: 1 A (~-Z~ABq- ~ AC (6) where AAB and Z~AC are the vertcal dsplacements of ponts A and C, respectvely, wth respect to pont B. By substtutng x = I/3 and 2l/3 nto Eqs. (2) and (), Z~AB and Aac are calculated as: AAB - and p/3 sp - - (7) 432ExI 12GxyA sp AAc -- (8) 12GxyA When the depth and breadth of the specmen are denoted h and b, respectvely, d s derved from Eqs. (6), (7), and (8) as; d - Pl 3 + s P _ Pl 3 + 4.s Ex (h21 432Ex 8GxyA 36Exbh 3 axy \ l ) J Accordng to the elementary bendng theory, the effect of shearng force s gnored, and the second term n the braces of Eq. (9) vanshes. When Young's modulus based on the elementary bendng theory s gven by Es, the deflecton d s represented as follows: pl 3 pl 3 - - - - - - () 432EsI 36Esbh 3 The followng relaton s obtaned from Eqs. (9) and (). 1 1 E~ G s (11) + 4. Gxy Ths equaton ndcates that the effect of shearng force durng asymmetrc four-pont bendng s 4. tmes that wth three-pont bendng. -s Wth three-pont bendng, the extra (9) Experment Materals and testng procedures Akamatsu (Japanese red pne, Pnus densflora D. Don) and shoj (Japanese ash, Fraxnus spaethana Lngelsh.) were used for the tests. The densty of akamatsu was 0.66 g/ cm 3, whereas that of shoj was 0.8g/cm 3. Specmens were condtoned at 20~ and 6% relatve humdty before and durng the tests. Sx specmens were used for each speces. These 12 specmens ntally had the dmensons of mm (tangental) depth, 20mm (radal) breadth, and 30mm (longtudnal) length. Young's and shear modul correspondng to the 12 specmens were determned by the flexural vbraton, asymmetrc four-pont bendng, and three-pont bendng tests. The depth of the specmen was then decreased to mm by a planer, and Young's modulus and the shear modulus for each specmen were obtaned by the asymmetrc four-pont and three-pont bendng tests. Thus, the fve values for Young's modulus and the fve values for the shear modulus were derved from one specmen. Asymmetrc four-pont bendng tests Asymmetrc four-pont bendng tests were undertaken by the followng procedure. The specmen was settled on supports that correspond to ponts A and C n Fg. la. The dstances between supports were 318, 222, 183, 19, 141, 129, 120, 111,, and 99mm. By determnng the dstance between the supports as above, the value of (h/l) 2 vared from approxmately 0.001 to 0.01 at an nterval of 0.001 and a depth of mm, whereas t vared from approxmately 0.0002 to 0.002 at an nterval of 0.0002 and a depth of mm. Wth loadng noses whose rad were 1 mm, a vertcal load was appled asymmetrcally at ponts B and D n Fg. la at a loadng speed of mm/mn. To reduce the extra deflecton produced by the stress concentraton around the loadng noses and the machne complance, the deflecton was measured at the bottom of the loadng pont, whch corresponds to pont B n Fg. la by the cantlevertype dsplacement gauge. The load was carefully appled so as not to exceed the elastc lmt of the specmen, whch was used repeatedly n ths experment. From the load (P)-dsplacement (6) relaton, the apparent Young's modulus E~ correspondng to the depth/span rato h/i was calculated by Eq. (). The 1/E~-(h/l) 2 relaton was then regressed nto Eq. (11) by the method of least squares, and Young's modulus Ex and the shear modulus G~y were obtaned.
17 Three-pont bendng tests Three-pont bendng tests had been conventonally undertaken to determne the shear modulus of wood. ~'ga~ Here we conducted the three-pont bendng tests wth the same specmens used for the asymmetrc bendng tests and compared the obtaned shear modul wth those obtaned by the asymmetrc bendng tests. The testng condtons such as span lengths, loadng speed, and loadng nose radus were smlar to those of the asymmetrc bendng tests. The load was appled at the center of the specmen, and the deflecton was measured by the dsplacement gauge set behnd the loadng pont. Based on the load-deflecton dagram, the apparent Young's modulus E~ correspondng to the span/depth rato h/l was calculated by the followng equaton: 13 ~ l 3 AP Es - 48/ Ad 4bh 3 Ad (12) where AP/Ad s the ntal nclnaton of the load-deflecton dagram. Young's modulus Ex and the shear modulus G v were calculated by regressng the 1/E~-(h/l) 2 relaton nto the followng equaton by the method of least squares. -7,9, A o / (( ((t (a) Asymmetrc bendng 300 l!ll rll 2 Ill (b) Three-pont bendng Pg. 3. Fnte element meshes used n the numercal calculatons. Meshes are unformly dvded to the dmensons of.0 1.2 mm 1 _ 1 + s (13) Es Ex axy Flexural vbraton tests Pror to the statc bendng tests, Young's and shear modul were obtaned by the flexural vbraton tests. The specmen was suspended by two threads at the nodal postons of the free-free vbraton correspondng to ts resonance mode and was excted n the drecton of the depth at one end by a hammer. The resonance frequences whose mode was from frst to fourth were measured by the fast Fourer transform (FFT) dgtal sgnal analyzer; and Young's modulus and the shear modulus were obtaned from the Tmoshenko-Goens-Hearmon method whose detals were descrbed n several prevous papers. 11 The values obtaned were compared wth those obtaned by the statc bendng tests mentoned above. Young's modul n the length and depth drectons, Gx~ for the shear modulus, and Vxy for Posson's rato; the values of these constants were Ex = GPa, Ey = Gxy = 1.2 GPa, and ~v = 0.4. The outer-span length l vared as 90, 120, 10, 180, 2, and 300mm. For the asymmetrc bendng smulatons, loads of 3 N and 1N were appled at the top of ponts B and D n Fg. 3a, respectvely. The deflecton d was measured at the bottom of pont B. The load-deflecton relaton was substtuted nto Eq. (), and the apparent Young's modulus E~ correspondng to the depth/span rato h/l was obtaned. For the three-pont bendng smulatons, a load of 4N was appled at the top of the center of the beam, and the deflecton was measured at the pont behnd the loadng pont. The apparent Young's modulus E~ correspondng to the depth/span rato h/l was obtaned by substtutng the load-deflecton relaton nto Eq. (12). The smulaton results were compared wth those obtaned from the statc bendng tests. Fnte element analyses Asymmetrc four-pont and three-pont bendng tests were smulated by the fnte element method (FEM), and the calculated results were compared wth those obtaned from the asymmetrc and three-pont bendng tests. The program used was "ISAS-II," whch s a lbrary program of the Computer Center of The Unversty of Tokyo. Fgure 3 shows the fnte element mesh and the boundary condtons used here. The fnte elements were dvded by the dmensons of mm length and 1.2mm depth; the breadth of the element was 20ram. The elastc constants used n the smulatons were defned as Ex and Ey for Results and dscusson Fgure 4 shows the 1/Es-(h/l) 2 relatons for the asymmetrc four-pont and three-pont bendng smulatons by the fnte element method. As mentoned above, the nfluence of the depth/span rato was more sgnfcant n the asymmetrc bendng test than n the three-pont bendng test. Accordng to the bendng theory, the nclnaton of the 1/Es-(h/l) 2 relaton for asymmetrc bendng s 4. tmes that for three-pont bendng. From the numercal calculatons, the nclnaton for asymmetrc bendng was 4.9 tmes that for three-pont bendng. Young's modulus Ex and the shear modulus a:~y
18 were calculated by regressng the numbercal calculaton results nto Eqs. (11) and (13). The value of Ex was 11.1 GPa for asymmetrc bendng and 11.0 for three-pont bendng. When Tmoshenko's shear factor s was determned as 1.2, the value of Gxy was derved as 1.22GPa for asymmetrc bendng, whereas t was 1.34 GPa for three-pont bendng. We thought that the valdty of Tmoshenko's bendng 2-" v 0.1 0.14-0.13 0.12 O.ll - o,........... oo9 - -00 1 9 ~ E s 0 I I 0 0.00 0.01 0.01 (Depth/span rato) 2 (h/l) 2 Fg. 4. Value of 1/E s correspondng to the (depth/span rato) a obtaned by the fnte element method. Open andfiled crcles are obtaned from the smulatons of asymmetrc four-pont and three-pont bendng tests, respectvely; and sold and dashed lnes are obtaned from the regressons of the 1/E<(h/l) z relatons of asymmetrc and three-pont bendng smulatons, respectvely, nto lnear relatons theory for asymmetrc bendng would be verfed by these smulaton results. Tables 1 and 2 are Young's modulus E, and the shear modulus G,y, respectvely, obtaned by the flexural vbraton, asymmetrc four-pont bendng, and three-pont bendng tests. Dfferently from the fnte element calculatons, the shear modulus concded well wth that obtaned by the vbraton test when Tmoshenko's shear factor of 1. was used. Wth the bendng tests, nonlnear deformaton caused by frctonal forces and stress concentratons around the loadng and supportng ponts, whch cannot be predcted by lnear fnte element analyss, produced the extra deflecton; and the shear modulus calculated usng s -- 1.2 was small. Thus, Tmoshenko's shear factor of 1. mght be applcable to the expermental data. As for three-pont bendng, the shear modul could not be obtaned properly despte the proper values of Young's modul. For asymmetrc four-pont bendng, we thought that Young's and shear modul were properly determned when the depth of the specmen was mm, although the shear modulus of akamatsu was somewhat smaller than that obtaned by the vbraton tests. Fgure shows the nverse value of apparent Young's modulus 1/E~ correspondng to the squares of the depth/span rato (h/l) 2 obtaned from the asymmetrc fourpont and three-pont bendng tests. Wth three-pont bendng, the deflecton caused by the shearng force s relatvely small. For akamatsu wth a thckness of mm, ths deflecton was too small to vary the apparent Young's modulus by the depth/span rato, and the shear modulus tended to be evaluated as extremely large. For the other results of threepont bendng tests, the extra deflecton caused by the stress concentraton was not reduced effectvely, and the shear modulus was evaluated to be small. In contrast, ths extra Table 1. Young's modulus obtaned by each method Depth (mm) Vbraton test (GPa) Statc bendng test (GPa) Asymmetrc four-pont Three-pont Akamatsu Shoj 17.3 -+ 1.4 13.4 -+ 1.1 Results are averages _+ standard devaton 18.3 _+ 1.4 18.0 -+ 1.1 23.3 + 4.9 16.9 1.6 13.8 _+ 1.8 13.0 -+ 2.1 16.8 _+ 1. 13.7 +_ 1.3 Table 2. Shear modulus obtaned by each method Depth (mm) Vbraton test (GPa) Statc bendng test (GPa) Asymmetrc four-pont Three-pont Akamatsu Shoj 1.2 _+ 0.14 0.91 +- 0.11 Results are averages _+ standard devaton 1.07 _+ 0.13 4.70 +- 2.06 0.6 +_ 0.17 0.82 -+ 0.03 0.89 _+ 0.2 0.26 -+ 0.06 0.6 0.11 0.44 +_ 0.06
0.1 0.2!9 Shoj Depth = mm o Shoj Depth = mm O 0.09 O O,--., 0.1 _ 0 ~176 0.08-9 9 9 9-0.1... 0.07 I I I 0 0. 1.0 1. 2.0 (Depth/span rato) 2, (x -3) 2. 0.0 I I I I I I m 0 2 4 6 8 l0 12 (Depth/span rato) 2, (x -3) Fg.. Examples of the nverse values of apparent Young's modul correspondng to the squares of depth/span ratos. Blank and sold crcles are obtaned from the asymmetrc four-pont and three-pont bendng tests, respectvely. Sold and dashed lnes were obtaned by substtutng Young's and shear modul gven by the flexural vbraton tests nto the Tmoshenko's bendng equatons for the asymmetrc four-pont and three-pont bendngs, respectvely deflecton was effectvely obscured wth asymmetrc fourpont bendng when the specmen had a small depth/span rato. Nevertheless, t was sgnfcant wth an ncreasng depth/span rato. As shown n Fg., the 1/E<(h/l) 2 relaton obtaned by the asymmetrc bendng tests of the specmens wth a depth of mm was concave; and the ntercept of ths relaton, whch equals 1/Ex, was evaluated as small, whereas the slope, whch corresponds to 4.s/Gxy, was evaluated as large. We thought ths concave tendency was due to the ndentatons at the loadng and supportng ponts. By reducng these ndentatons, the depth/span rato range where Young's and shear modul are effectvely measured would be wder than the expermental results. Nevertheless, plural dsplacement gauges should be used, complcatng the measurement method. Ths method s effectve for homogeneous materal, and there s a concern that t cannot be appled to a large specmen n whch nhomogenety such as knots, gran nclnaton, and varaton of annual rng wdth are contaned. The applcablty should be examned for the materal wth these nhomogenetes. When the method adopted here s undertaken usng a homogeneous specmen, however, we recommend that the span s larger than 20 tmes the depth. The shear modulus can then be obtaned effectvely by the asymmetrc four-pont bendng tests as can Young's modulus. Concluson We conducted asymmetrc four-pont bendng tests of wood and concluded that the shear modulus can be properly ob- taned by the asymmetrc four-pont bendng tests when the span s 20 tmes larger than the depth. Acknowledgment We thank Prof. Masamtsu Ohta at The Unversty of Tokyo for hs advce n wrtng ths artcle. References 1. Kon T (1948) The comparatve study upon the modulus of rgdty of wood by the methods of compresson and torson. Bull Hokkado Unv Dept Eng 1:144-16 2. Yoshhara H, Ohta M (1993) Measurement of the shear modul of wood by the torson of a rectangular bar. Mokuza Gakkash 39:1993-997 3. Yoshhara H, Ohsak H, Kubojma Y, Ohta M (1999) Applcablty of the Iospescu shear test on the measurement of the shear propertes of wood. J Wood Sc 4:24-29 4. Dumal JF, Olofsson K, SalmEn L (2000) An analyss of rollng shear of spruce wood by the Iospescu method. Holzforschung 4:420-426. Yoshhara H, Kubojma Y, Nagaoka K, Ohta M (1998) Measurement of the shear modulus of wood by statc bendng tests. J Wood Sc 44:1-20 6. Yoshhara H, Ohta M (1998) Feasblty of Tmoshenko's bendng theory on the measurement of shear modulus of wood. Mem Fac Sc Eng Shmane Unv Ser A 32:177-184 7. Yoshhara H, Fukuda A (1998) Influence of loadng pont on the statc bendng test of wood. J Wood Sc 44:473-481 8. Tmoshenko SP (19) Strength of materals. Part 1. Elementary theory and problems, 3rd edn. Van Nostrand, New York, pp 16-3 9. Wangaad FF (1964) Elastc deflecton of wood-fberglass composte beams. For Prod J 14:26-260. Bbls EJ (196) Shear deflecton of wood beams. For Prod J 1:492-498 11. Hearmon RFS (198) The nfluence of shear and rotatory nerta on the free flexural vbraton of wooden beams. Br J Appl Phys 9:381-388