Derivation of the bearing strength perpendicular to the grain of locally loaded timber blocks.

Transcription

1 Delft Wood Science Foundation Publication Serie 2006, nr. 3 - ISSN X Derivation of the bearing trength perpendicular to the grain of locally loaded timber block. Baed on the equilibrium method of platicity, the theoretical explanation of the bearing trength of locally loaded block i given in the Appendice. The reult of the numerical contruction of the lip-line can preciely be repreented by an analytical function a logarithmic piral that can be hown to be the exact olution. Thi function can be given in the power law form leading to a theoretical and experimental value of the power of 0.5. Thi power repreentation of the tre preading model of confined dilatation provide a imple deign method that preciely matche to the data in all circumtance and loading cae and explain the apparent contradictory tet reult of Suenon, the Eurocode, the French rule, Graf, Korin and Augutin et al dr. ir. T.A.C.M. van der Put of the Delft Wood Science Foundation em. aoc. Reearch Profeor of the Technical Univerity of Delft DelftWoodScienceFoundation@x4all.nl Wielengahof 16 NL 2625 LJ Delft Tel

2 Bearing trength perpendicular to the grain of locally loaded block The local compreion trength perpendicular to the grain may increae due to confined dilatation perpendicular to the loading direction. Thi i explained in Appendix A by the equilibrium method of the theory of platicity. A derived, the increae of trength i proportional with L / according to Eq.(1). fc, cfc,90 L/ 1.08 fc,90 L/ (1) The definition of L and i given in Fig. 2. Fig. 1 - Bearing trength f c, perpendicular to the grain. Specimen 15x15 cm, length: L = 15, 30, 45, 60, 75 cm, of curve a to e, [1] Suenon. = 15 cm. The trength value f c, are the top-value of the meaured curve of Fig. 1. The compreion trength f c,90, at the top of curve a at 15% train, i here 3.6 MPa. Table 1 Bearing trength perpendicular to the grain of locally loaded block Curve L/ L / fc,90 Theory Meaurement ultimate fc, 1.1 fc,90 L/ f c, train MPa MPa MPa a fc,90 = % b c d e limit a curve d % 13% 15% 10% The meaured maximal trength value, given in Table 1, are preciely according to the theory. Fig. 1 how the trength increae with the increaing poibility of 2

3 preading of the load. It further how that there i a maximal preading of about 4H becaue the trength of block e and d are equal. The trength of pecimen e with L = 5 = 5H, i a trong a pecimen d with L = 4 = 4H. The definition of L, and H i given in Figure 2. The maximal preading-length thu i 4H, or better i: 3H +. Becaue = H, the preading i 3H, thu 2 time 1.5 H of both ide. Thu L = H + = 3H + H = 4H. The preading thu i 1.5:1, a i applied in fig.5. When the ultimate tate i choen at a mall platic deformation, a often done, the preading lope i cloe to 1:1 of the elatic tate. Thi alo i to be expected when there i no friction at the bearing plate or when not the height H i limiting but the preading length L i limiting being equal then to the length of the block. On thi determining cae the derivation of Eq.(A.17) from Eq.(A.13) i baed in Appendix A. The ame maximal value of the preading lope of 1.5:1 alo follow from other invetigation a e.g. given in Table 2 of the French deign rule where alo for higher value of a above a/h 1.5 there i no trength increae. Eq.(1) provide a imple deign rule and i able to explain all mutual trongly different empirical reult, a will be dicued here. The rule of the Eurocode, given in [5], Eq.(4.20), follow from Eq.(1). Becaue fc, fc,90 L / and f c,,0 f c,90 L/ 0, i: c, c,, f / f / /. (2) Thi equation i choen to apply for 0 = 100 mm and in [5], the exponent 0.5 i replaced by 0.4, to better follow the exiting afe Code rule of Canada, Denmark, Norway Sweden and the UK. Thi however, only i the cae for mm. For 50 mm, the curve lie increaingly above thee Code value. Thi wa corrected for mall value of in the CIB Timber Code by chooing a power of 0.25 while wrongly f c,,0 wa taken to be equal to f c,90 for = 0 = 150 mm. Becaue L and thu H are eliminated in the derivation of Eq.(2), the equation i not general applicable. For very mall value of H for intance, there i no preading at all and the equation doen t apply. Therefore the right rule, baed on the theoretical Eq.(1) wa propoed for the Timber Code everal time in the pat a e.g. in [8] and [9]. Table 2 Value of kc fc c, / fc,90 /H a/h Fig. 2 Locally loaded Block 3

4 The French rule, given in Table 2, mentioned in [5], correct for the omiion of H in the CIB Code by howing the dependence of the trength on H. The table how the boundary value of a/h = 1.5, mentioned above. When a/h = (L - )/2H 1.5, thu when L 3H +, the maximal preading i reached according to Fig. 1. An other boundary of the table i given for /H 3. It then i aumed that in the middle of the pecimen the ame condition appear a in the cube tet. Thi applie for fully flexible, frictionle bearing plate. Fig. 3 -Cube tet condition in the middle when there i no friction. The ame condition i aumed to apply for a = 0 in fig. 2. Without friction, preading i not poible at the edge and the trength i equal to the trength of the cube tet. With friction along the plate, the confined preure may e.g. be build up, even for = L, according to Fig. 4. Fig. 4 Slip line of failure between two plate by friction along the plate. The influence of no friction along the bearing plate in the trong direction (and thu full friction in the width direction) can be aeed a lower bound by auming that only ymmetrical preading i poible. Thu for Table 2 and 3: L = 2a +. According to Eq.(1) then i: Table 3 Value of kc f c, / fc,90 kc L/ 1 (2a / H)/( / H) in Table 3. /H a/h = (L )/2H

5 Thee value are cloe to the value of Table 2 and are comparable when: Thu when, outer c = 1 in the firt column, c = 0.9 in column 2 and 3 i ued, indicating the afe lower bound given by the French rule. In [6], tet reult are given of bearing in the range where H i not limiting for preading becaue: L < 2H + in the central loaded pecimen. The determination of f i done on the ame pecimen, thu on the pecimen of fig. 5 with an upper c,90 loading plate of length L, the ame length a the bottom plate, giving by thi form a higher trength than follow from the common compreion tet. The ultimate train wa choen to be 2.5 %. Fig Spreading 1:1 in a central loaded block and end-loaded block of [6]. Table 4 Value of k c according to the tet-pecimen of fig.5. meaurement Theory /L central Loaded end loaded central loaded end loaded k c k c Eq.(1): k c = L / / / / / / / / ( )/ / ( )/ / ( )/ L/ 0.125L 2.8 ( )/ Thi compreion trength i compared in [6] with the trength of the ASTM-bearing tet, being the ame tet a given by the central loaded pecimen of fig. 5, however 5

6 with a length of the upper plate of L/3. Thi explain why in the graph in [6] of the ASTM value are L / 3 time higher than according to the compreion trength of [6] done on the ame pecimen with = L. In table 4 the tet reult (of erie of 3 pecimen) are compared with Eq.(1) and it i een that alo non-ymmetrical preading i poible of end loaded block becaue of the friction between plate and pecimen According to the Eurocode a limiting value occur at /L due to a local mechanim. The reult here however don t how uch an empirical reduction of the trength with repect to the theoretical value. Alo the theoretical limit value of the local mechanim how much higher value of k c. In table 5, the empirical value of c of eq.(1) i given, baed on the tet of [6]. Table 5 Value of f / f c,90 k c and of c = k c / L /, according to Table 4. Meaurement Theory kc L / /L central end - central end loaded loaded loaded loaded c-value of Eq.(1) k c k c k c k c c = k c / L / = k c / limit mean of c: In Fig. 6, the reult are given of tet on two ided locally loaded long block. From the figure it follow that: + 3αH = L + 3.(1 - α)h. Thu: L 0.5 6H and thu the equivalent preading factor (of the trength determining plate) i: L' 3H 3H L 3H L H 2 With H = 17.9; L = 35 and b = 18.1 cm according to the meaurement of O. Graf i: L' 3H L kc,90 c or: kc, ( )/ / leading to the value of f at 5 mm deformation (ee figure) of the curve: 1: 16-2: 30-3: 36-5: 43-6: 52 kgf/cm 2, about the ame a the meaurement a can be een in Table 6. L / 6

7 For long block with repect to the bearing plate the maximal preading will occur at both plate according to the figure 6.. Fig. 6 - Local loading perpendicular to the grain [1] Graf Poible preading Table 6 - kc,90 f / fc, / Curve k Theory cm f c,90 MPa 1.6 c, f MPa or local limit Meaurement f MPa > 7.5 ultimate train 6/178 or: 3.4% 3.4% 3.4% 3.4% 3.4% 3.4% > 1% The highet maximum i not hown (of line 7 of fig.6). Predicted according to the lat formula i: f = 10 MPa. However thi may be cut off by a local mechanim. Becaue f 7.5 MPa i meaured, the maximum value of k c,90 i at leat 7.5/1.6 = 4.7, near the 7

8 theoretical value obtained from a local failure mechanim (giving an upper bound value) of about 5.5 to 6). The meaurement of fig. 6 ugget a contant loading rate with a udden intability of the tet at the end. Therefore the curve 2, 5 and 6 end early at about 6 mm or 3.4% train. For thi reaon all trength were defined at thi train. The theoretical explanation of the tet reult of [7] i dicued next. Thi till appear to be neceary although the theory wa publihed long ago and i applied in many report of the Stevin Laboratory a e.g. in [10], where it i hown to be the only poible theory to explain the very high embedding trength of particle board in compreion. The theory alo i publihed in CIB-paper e.g. in [8] and in a report for the CIB- Stability Committee and more recently in [9], where it a wa hown that the theory fully and preciely explain the data of Ballerini of [11] and the Karlruher data of joint with one dowel. According to the theory Eq.(1) applie for the compreion trength perpendicular to the grain, of a locally loaded bearing block: Fig. 7 Tet pecimen of [7] The factor of the increae of the compreion trength by local loading kc lc,90 c,90 f / f L/ k c thu i: Becaue the 1% permanent train ( 3% total train) i choen a ultimate train, the tre ditribution will be cloe to the elatic one and a preading of about 1 to 1, or 45 0 can be aumed (ee Fig. 7). The maximal preading at higher train will be 1.5 to 1. The length L thu will be for cae 1 of Fig. 1, L = 200α For cae 2 i: L = 200α , and for cae 3: L = 2α mm, where α = 1 to 1.5. The length = 150. For the pecimen height of 480 mm, all value of 200 in the expreion of L hould be replaced by 480. Thu: cae 1: kc L/ ( )/ to ( )/ , etc. 8

9 For cae 3 with H = 480, L can not be higher than the length of the pecimen of 980 mm and thu thi length i the real preading length giving kc 980 / Table 1 Empirical verification of the theoretical value of kc L / h = 200 mm cae 1 cae 2 cae 3 h = 480 mm cae 1 cae 2 cae 3 theory, 1% train = 1 k c = 1.53 = 1.73 = 1.91 k c = 2.05 = 2.21 = 2.56 meaurement 1% train k c = 1.58 = 1.94 = 1.94 k c = 1.82 = 2.12 = 2.46 k c. theory, prediction for high train = 1.5 k c = 1.73 = 1.92 = 2.24 k c = 2.41 = 2.54 = 2.56 It can be een that the meaurement are cloe to the applied low train prediction of the theory with the preading lope of 45 0, giving a very good explanation of the data at the different configuration. The higher train prediction of the theory hould be verified. It can be concluded that the theory give an excellent explanation and precie fit of all the apparent contradictory tet reult of Suenon, the Eurocode, the French rule, Graf, Korin and Augutin et al. in all circumtance and loading cae. Therefore the propoal of the pat remain to ue the right deign rule a for the Code a given below. A propoal for the Eurocode the following rule are poible for bearing block: k f, where: c,90,d c,90 c,90,d kc,90 L / with: L a + + l 1 / 2 ; L 3H + and: kc, when /L for central load; kc,90 2 when /L 0.25 for end load. For afe rule (when friction i only in the width direction), the condition are: L 2a + ; L + l 1 ; L 2H +, k c,90 = 2.8 when /L For the bearing trength of a middle ection of a beam between two plate of length L and i: 3H L kc, Thee rule for bearing block don t apply for upport tree of beam. For the combined tree in the beam, the failure criterion of [4] ha to be applied. A long 9

10 thi exact approach i not followed, the compreion trength perpendicular to the grain at a middle upport hould afely be limited to f c,90 / 2 in order to maintain the ultimate compreion tre of the bending trength of the beam. Appendix A Derivation of the bearing trength perpendicular to the grain or locally loaded block and of the preading equation by the method of characteritic The dependence of the trength upon preading can be explained by the equilibrium method of the theory of platicity. In the platic region, a tre field can be contructed in the pecimen that atifie the equilibrium condition: x y 0 and 0 (A.1) x y x y and the boundary condition and nowhere urmount the failure criterion. For thi motly determining failure criterion an incribed Treca criterion, Eq.(A.2), can afely be ued / 2 k f (A.2) 1 2 v Thi failure criterion applie after a flow and hardening tage in the weak direction until a quai iotropic flow behaviour occur followed by further hardening ([2], [4]) and flow. In the figure below, a Mohr-circle of the failure condition i given with the general tre tate x, y,. In Fig. A.1 i: p ( 1 2)/ 2 and k ( 1 2)/ 2. (A.3) Fig. A.1 Treca failure condition In general i: p y kco2 x kco2 and kin2 Subtitution of thee equation of x, y, in the equilibrium equation give p 2kin 2 2kco2 0 x x y (A.4) 10

11 p 2kco2 2kin 2 0 y x y Multiplication of Eq.(5) by tan(ψ π/4) and then addition with Eq.(4) give: a a tan( / 4) 0 x y where a p 2k. Thu along the characteritic with lope dy/dx = tan(ψ π/4), i a = contant. The ame can be done by multiplication of tan(ψ + π/4), leading to b b tan( / 4) 0 x y giving b p 2k = contant along the characteritic with dy/dx = tan(ψ + π/4), To how that thee line are characteritic, Eq.(4) and (5) are combined with their correponding equation of variation: (A.5) (A.6) (A.7) 1 0 2k in 2 2k co k co 2 2k in 2 p / x dy p / y dx / x dy / y dx = 0 0 dp / dx d / dx In thi region in the failable tate, line can be given along which failure i initiated correponding to the initiation of motion. Accordingly thee are line, called characteritic, acro which derivative may become dicontinuou, or along which dicontinuitie in derivative may propagate. On thee line, in the characteritic direction the derivative thu have no determinate value and the direction can be found by equating all determinant to zero. A zero value of the nominator determinant give, after ubtraction of the third row from the firt: dy 2kin 2 2kco 2 dx det 1 2kco 2 2kin 2 0 dy 0 1 dx or 2 dy dy co2 2 in 2 co2 0 or: dy tan 2 ec 2 dx dx dx or: dy tan dy and tan dx 4 dx 4 Thi thu are the lope of both orthogonal characteritic. A zero value of the denominator determinant give: (A.8) 11

12 dy dp 2k in 2 dx dx dy d d 1 dp det 1 2k co or co2 in 2 0 d dx dx dx 2k dx 0 1 dx or with the found equation above of dy/dx : d 1 dp d 1 dp tan 2 ec 2co2 in 2 0 or: 0 or: dx 2k dx dx 2k dx p 2k a contant (A.9) p 2k b contant (A.10) along the firt repectively the econd characteritic (a found before). Calculation of the network of thee lipline i done numerically. From two Fig. A.2 - Contruction of the lip line. known point, known from the boundary condition or previou calculation, x 1,y 1 and x 2,y 2, the next point x, y follow from: y y 1 (x x 1) tan( / 4) and y y 2 (x x 2) tan( / 4) 12

13 or after elimination of the unknown: x tan( / 4) y x tan( / 4) y x tan( / 4) tan( / 4) and y y 2 (x x 2)tan( 2 / 4) The value of ψ follow from (b a)/2 of point 1 and 2, and p follow from (a + b)/2. The reult of the numerical contruction of the lip-line, given in [3] and Fig. A.2, can be preciely approximated by the function: θ 0.62 ln(2h/) (A.11) Thi can be explained a follow. At the end of the outer curved lip-line over a length Rdφ i according to the chain equation Ndφ = σrdφ or N = σr, where N i the normal force along the lip-line. Further i alo dn = τrdφ, or σ dr = τrdφ, or d(lnr)/dφ = τ/σ = μ and thu R R0 exp( ), what i a logarithmic piral. Now i: RL H exp c( L ) exp 1.61 t or: t 0.62 ln2h/. R / 2 It thu i probable that Eq.(A.11) i not an approximation but the true olution for the end point of the outer lip-line. Triangle ABD of Fig. 2 i a region of contant tate, where the maximum hear line, or characteritic, are everywhere at 45 0 to the principal direction becaue of the uniform compreion load on plane AB. Becaue the pole of the plane in the Mohr circle now i at point 2 in Fig. A.1, i /2. Thi direction of the plane with the minor principle tre i alo the direction of the highet principle compreion tre. Fig. A.3 Determination of p and ψ in the p 2kψ plane From point D, or point 11 in Fig.A.2, to point 2., i: p 2k p2' 2k 2 2. Thu: p2' p 2k From point 2 to 22 i: p22' 2k p2' 2k 2 2. Thu: p22' p2' 2k p 4k and: p po 4k. The ame relation follow for point 33, when the angle between line BD, and BC (at 13

14 point B) i 2α: p po 4k(2 ). Thu in general i: p p 4k (A.12) o Inerting Eq.(A.11) and with p ( 2k)/ 2 k and po o k, thi i: o 2.48 k ln(2h/) (A.13) and becaue o L (ee Fig. A.4) i: (1 / L) 2.48 k ln(2h/ ). Further elatic preading will be at an angle of 45 0, thu for firt flow, L 2H +, or: H (L - )/2 when H >, thu: L/ > 3 Subtitution of the value for 0 and H in Eq.(A.13) give: L L / 2.48 kln 1 (A.14) L / 1 L L / and becaue from the power law approximation follow that ln 1 i L / 1 proportional to L /, (ee Appendix B and C), Eq.(A.14) become: 2.48 kc L/ (A.15) where C i a about Thu: k L/ 2k L/ (A.16) The value of k follow from the compreion tet (cube tet) with 1 fc,90 and 2 0 k f / 2. Thu Eq.(6) become: or: c,90 f cfc,90 L/ fc,90 L/ (A.17) The higher experimental value of c given in Eq.(1) how the lower bound approach of the choen method (the real lip/line mut give a higher value). Thu c give the poibility to adapt the model to tet reult. Fig. A.4. - "Slip-line" determining the direction of the main tree 14

15 A imilar olution i poible for the rotational ymmetrical cae, leading to the 2 2 extenion of Eq.(A.17) to the urface A ( / 4 ) and A L ( L / 4). Thu generalized to every urface form: f cfc,90 A L / A (A.18) Appendix B Derivation of the power law. Any function f(x) alway can be written in a reduced variable x/x0 f(x) = f1(x/x0) and can be given in the power of a function: f(x) = f1(x/x0) = [{f1(x/x0)} 1/n ] n and expanded into the row: f(x) = f(x0) + x x 0 1! giving:. f'(x0 ) (x x 0 )2 2!. f''(x0 )... n 1/ n x x 1 1/ n1. x x0 n x0 0 f (x) f (1) f (1) f '(1)... f (1) when: (f1(1)) 1/n = (f1(1)) 1/n-1 f1 (1)/n or: n = f1 (1)/f1(1) where: f1 (1) = [f1(x/x0)/(x/x0)] for x = x0 and f1(1) = f(x0) n x Thu: f(x) f(x 0 ). x 0 n with n f '(1) 1 f 1 (1) f'(x ) 0 f(x 0 ) (B.1) It i een from thi derivation of the power law, Eq.(B.1), uing only the firt two expanded term, that the equation only applie in a limited range of x around x 0. Appendix C Derivation of the power of the preading equation. The in the Appendix A found part of Eq.(A.14): (1.24 (L/) ln(l/-1))/(l/ 1), appear to follow the form of L /. Thi follow from the power law approximation of Eq.(A.14) (according to Appendix B) giving a power 0.5. It thu i poible to plit Eq.(A.14) into: L/ (1.24 ( L/) ln(l/-1))/(l/ - 1)= L/ C, becaue the econd part hould be about contant. The pecial value of 0.5 of the power can be explained a follow. In the following derivation, the trength of the upper and bottom plane will be related to the trength of an intermediate plane m e, having a trength according to the power law repreentation: 15

16 n 1n Lt m c me. Thu from: me me mme LLt L m c Lt Lt n n1 plane. Alo i for the upper plane: for the bottom me Lt me Lt L m c c. t me t me 1n 1n me 1n L c c With: me = αt i: Lt L n 1n L and i: c In general i Eq.(A.1): f(x) f(x ). x 0 x0 m, for x x0, equal to: L 1n 1n L c L L 1n L and for for x x0, equal to: c Becaue the exponent give the lope of the curve and the curve hould not be kinked at x 0, the exponent hould be the ame and: m = 1 - n = n, or n = 1/2. For α =1, the intermediate plane i the determining upper plane n Literature [1] F. Kollmann, Principle of wood cience and technology, vol. I, 1984, Springer- Verlag, Berlin. [2] T.A.C.M. van der Put, A general failure criterion for wood, Proc. IUFRO S5.02 paper 23, 1982, Bora, Sweden. [3] H. Schwarty, PhD diert. Stuttgart, [4] T.A.C.M. van der Put, The tenor-polynomial failure criterion for wood polymer, Delft, [5] H.J. Laren, The deign of timber beam, CIB-W18/5-10-1, Karlruhe, [6] U. Korin, Compreion perpendicular to grain, CIB-W18/23-6-1, Libon [7] M. Augutin et al. Behaviour of glulam in compreion perpendicular to grain in different trength grade and load configuration, CIB-W18/ , Florence, [8] T.A.C.M van der Put, Dicuion of the failure criterion for combined bending hear and compreion. CIB-W18/24-6-1, Oxford, [9] T.A.C.M van der Put, Leijten, A.J.M., Evaluation of perpendicular to the grain failure of beam by concentrated load of joint. CIB-W18/33-7-7, NL, [10] T.A.C.M van der Put, Explanation of the embedding trength of particle board, Stevin Report TU-Delft /09-HSC-6 or: EC-project MA1B-0058-NL, [11] M. Ballerini, A new et of tet on beam loaded perpendicular to the grain by dowel joint, CIB-W18/32-7-2, Graz, Autrie,