This paper presents a stress analysis of the off-axis tension test of clear wood specimens based on orthotropic elasticity theory. The effects of Poisson's ratio and shear coupling coefficient on stress distribution are analyzed in detail. The analysis also provides a theoretical foundation for the selection of a 10° grain angle in wood specimens for the characterization of shear properties. The Tsai-Hill failure theory is then applied to derive a formula for predicting shear strength. Existing strength data for Sitka spruce (Picea sitchensis Carr.) were used in a numerical analysis. Because of the large discrepancies in published test data from different sources, the accuracy of the formula is limited to the data used to derive it. However, the procedures are believed to be accurate. The off-axis tension test is attractive mainly because of its economy and ease of application. This research promises to pave the way for the adoption of the off-axis tension test for characterizing the shear properties of clear wood by the practicing engineer once representative input data become available

Liu, Jen Y.

English

Wood and Fiber Science (E-Journal)

ANALYSIS OF OFF-AXIS TENSION TEST OF WOOD SPECIMENS Jen Y. Liu Research General Engineer U.S. Department of Agriculture Forest Service Forest Products Laboratory1 One Gifford Pinchot Drive Madison, WI 53705-2398 (Received March 2001) ABSTRACT This paper presents a stress analysis of the off-axis tension test of clear wood specimens based on orthotropic elasticity theory. The effects of Poisson's ratio and shear coupling coefficient on stress distribution are analyzed in detail. The analysis also provides a theoretical foundation for the selection of a 10" grain angle in wood specimens for the characterization of shear properties. The Tsai-Hill failure theory is then applied to derive a formula for predicting shear strength. Existing strength data for Sitka spruce (Picea sitchensis Carr.) were used in a numerical analysis. Because of the large discrepancies in published test data from different sources, the accuracy of the formula is limited to the data used to derive it. However, the procedures are believed to be accurate. The off-axis tension test is attractive mainly because of its economy and ease of application. This research promises to pave the way for the adoption of the off-axis tension test for characterizing the shear properties of clear wood by the practicing engineer once representative input data become available. Keywords: Off-axis tension test, orthotropic elasticity, shear strength, tensile strength. Ihai-Hill fail ure theory. INTRODUCTION modulus and the necessity of including shear ~h~ off-axis tension test has become attrac- coupling in stress analyses. Pierron and Vau- tive for determining the shear properties of trin (1996) discussed the use of oblique tabs composite materials because of its relatively ~ r o ~ o x d  by Sun and Chung (1993). Pierron low cost and operational simplicity. Pagan0 et al. (1998) then performed a whole-field as- and Halpin (1968) studied the problem of SeSSment of the effects of end constraints on specimen end constraints and the effect of the the strain field and reported that oblique tab- length-to-width ratio. Wu and Thomas (1968) discussed a rotating clamp fixture design. The results of Pagano and Halpin (1968) were ver- ified and extended by Richards et al. (1969) and Rizzo (1969). Chamis and Sinclair (1977) proposed a 10" off-axis tensile specimen for characterizing fiber-composite intralaminar shear. More recently, Pindera and Herakovich ( 1  986) studied the relation between end con- straints and accurate determination of shear I The Forest Products Laboratory is maintained in co- operation with the University of Wisconsin. This article was written and prepared by U.S. Government employees on official tirne, and it is therefore in the public domain and not subject to copyright. bing can lead to a homogeneous strain field. Yoshihara and Ohta (2000) conducted an off- axis tension test for shear strength of Sitka spruce (Picea sitchensis Carr.) and other spe- cies. Their sample size was relatively small, nor did they apply any failure criteria to ana- lyze their data. Consequently, their results seem to be somewhat inconclusive. In the present study, we performed a de- tailed stress analysis for the off-axis tension test for clear Sitka spruce specimens to eval- uate the effects of Poisson's ratio and shear coupling coefficient on stress distribution. We then applied the Tsai-Hill failure criterion us- ing test data from Yoshihara and Ohta (2000), the Forest Products Laboratory (FPL 1999), M'or,<l <i,t<l Fih<,r .Y<rol<.e, 3-1(2) .  2002. pp. 205-21 1 WOOD AND FIBER SCIENCE, APRIL 2002, V.  34(2) FIG. 1. Schematic of off-axis tension specimen. (x,? arc geometrical axes; I ,? are material axes.) and Liu and Floeter (1984). A procedure to use the 10" off-axis tension test for the deter- mination of clear wood shear strength is sug- gested. STRESS-STRAIN RELATIONS Let the 1-2 coordinate system represent the principal material axes and the x-y coordinate system the geometrical axes with angle 0 from the x axis to the I axis, as shown in Fig. 1. The normal stresses a, and u2 in the 1 and 2 axes, respectively, and the shear stress T, or T,? in the 1-2 plane can be written as where m = cos 0 and n = sin 0 (2) The relations in Eq. (1 ) are independent of ma- terial properties; i.e., they are the same for iso- tropic or anisotropic materials. For an anisotropic or orthotropic material such as wood, the applied stress o, in Fig. 1 produces the following strains: In Eq. (3), the Poisson's ratio v,,, corre- sponding to stress in the x-direction and strain in the y-direction, is the negative ratio of the transverse strain E, to the axial stain E,. The shear coupling coefficient qx,, corresponding to normal stress in the x-direction and shear strain in the x-J plane, is the ratio of the shear strain y,  or y,, to the axial strain E,. The strain components referred to the 1 and 2 axes can be expressed in terms of those re- ferred to the x, y axes by the following trans- formation relations: where the transformation matrix [TI is given by m2 .z n2 m' -2nzn (51 -mn mn m2 - n' The stress-strain relations in the 1-2 coor- dinate system are Q , ,  Q12 L~N-OFF-AXIS TENSION TEST OF WOOD SPECIMENS The reduced stiffnesses Q11, Q22, Q12, and (266 = can be related to the basic engineering con- E, stants E l ,  E,, G I : ,  and v , ,  as follows: El E2 (1 1) (211 = Q22 = 1 - v12'/21 1 - V,zV,1 From Eqs. (3), (4), and (6) by means of Eqs. (5) and (7), we obtain expressions for a , ,  a,, and T, in terms of the engineering constants, which should be equal to the corresponding expressions in Eq. (1). These expressions can be put in dimensionless form as follows: TSAI-HILL FAILURE CRITERION The Tsai-Hill criterion for orthotropic ma- terials (Daniel and Ishai 1994) is In Eq. (8), the transformed elasticity mod- ulus E,,  the Poisson's ratio v,,, and the shear coupling coefficient q,, are as follows (Jones 1975): in which F , ,  F,, and F, are the values of a , ,  a,, and T~ at failure, respectively, and the nor- mal stress and strength in any one direction must be either tensile or compressive. Substi- tuting Eq. (1) into Eq. (12) with the applied stress a ,  replaced by F,, we obtain Note that F,  corresponding to any specified angle 0 can be predicted once F l ,  F2, and F, are known. The Tsai-Hill theory allows for considerable interaction among the stress com- ponents uI  u,, and T ~ .  The requirement that each normal stress and the corresponding strength should be either tensile or compres- sive poses no problem because an off-axis ten- sion test involves only tensile stresses and strengths. RESULTS AND DISCIJSSlON We obtained numerical results to demon- strate the effects of Poisson's ratio and shear coupling coefficient on stress distribution in defect-free Sitka spruce based on Eq. (8). Identifying the longitudinal axis with the 1 axis and the radial axis with the 2 axis, the basic engineering constants for Sitka spruce (Liu and Ross 1998) are E l  = 11,800 MPa, E, = 2,216 MPa, G I 2  = 910 MPa, and v , ,  = 0.37. Using these values as input, Ex, v,?, and q .,,, which are needed in Eq. (8): are first calculat- ed. Figure 2 presents the variation of the ratio 208 WOOD AND FIBER SCIENCE, APRIL 2002, V. 34(2) 0 30 60 90 0 (degrees) FIG. 2. Elasticity modulus of Sitka spruce as a func- tion of grain angle. El = 1 1,800 MPa, E2 = 2,216 MPa, G I >  = 910 MPa, and v12 = 0.37. E,IEl with 0. The maximum value occurs at 0 = 0". It decreases drastically to 0 = 30" and reaches a constant value of E,IEl at about 0 = 55". At 0 = lo0, the decrease in E, is about 23% of El.  Figure 3 shows that qx, is negative at 0" 5 0 5 69" with a minimum value of - 1.42 at 0 = 15". At 69" < 0 5 90°, q, ,  is positive with a maximum value of 0.04 at 0 = 79". The Poisson's ratio starts from 0.37 at 0 = 0°, reaches a maximum value of 0.47 at 0 = 25", and drops to 0.07 at 0 = 90". Using these data, we found that all of equa- tions (8) are satisfied for any value of 0. We then plotted from Eq. (8a) the sum of terms containing q,,, the sum of terms containing v,,, and the total of all terms, i.e., m', versus 0 (Fig. 4). As Fig. 4 shows, q, ,  has a stronger effect than v, ,  on ul/u,  at 3" 5 0 r 45". Be- yond that range, the reverse is true. Figure 5 shows the results from Eq. (8b). Here the sum of terms containing q,, has the same sign as the total, n2, while that of v,, has the opposite sign in the range of 0 < 67". Be- -0.2 1 I I I 0 30 60 90 0 (degrees) FIG. 3. Shear coupling coefficient and Poisson's ratio of Sitka spruce as a function of grain angle. yond that range, both have insignificant effects on the total. The ratio of T,/u, in Eq. (8c) plotted in Fig. 6 reveals that up to 0 = lo0, the total mn is only slightly above the sum of terms contain- ing rlY, and the two vary linearly with 0, in- dicating that qlT has a predominant effect on T~ when 0 - 10". Assuming that when a, in- creases to cause material to fail and the same FIG. 4. Variations of crl/u., with H showing effects of qx, and v,. (Eq. 8a). a = m2, b = terms containing q, c = terms containing v,,.. Li~i-OFF-AXIS TENSION TEST OF WOOD SPECIMENS 209 (degrees) -0.41 FIG. 5.  Variations of (T?/(T, with 0 showing effects of 4 0 qx, and v,, (Eq. 8h). a = n', b = terms containing T,, c = terms containing v,,. 20 - Tsai-Hill Theory @ Test data (Yoshihara and Ohta, 2000) 0 1  0 30 60 90 0 (degrees) 0.6 - X FIG. 7. Failure curve based on Tsai-Hill theory for F I  = 141 MPa, F2 = 7.4 MPa, and F, = 1 1.3 MPa and test data for Yoshihara and Ohta (2000). 0.5 - relation between a, and 7, still holds, the value 0.4 - of T~ at that instant must have a definite rela- C C tion with its value at the yield limit. a, c The Tsai-Hill failure criterion in Eq. (13) is g 0.3 - based on assumptions of homogeneity (Per- E 0 kins, Jr. 1972) and linear stress-strain behavior 0 u to failure as are almost all other macrome- C 0.2 - m chanical failure theories. It shows that for the 6 prediction of failure load F ,  corresponding to . w 1; 0.1 - any specified value of 0, only the values of F , ,  F,, and F, are needed. Yoshihara and Ohta (2000) recently published these values for Sit- 0 ka spruce as F ,  = 141 MPa, F, = 7.4 MPa, gb and F, = 1 1.3 MPa. Using these values in Eq. 0 (degrees) (13), the failure curve obtained is shown in -0.1 - Fig. 7 together with the test data of F ,  at sev- FIG. 6.  Variations of T , / u ,  with 0 showing effects of era1 0 values. Within the range 110" < 0 < qr, and v,, (Eq. 8c). a = mn, b = terms containing ?,,, 75", the test data are significantly above the = terms containing v,,.. predicted values. With the F ,  values from the 210 WOOD AND FIBER SCIENCE, APRIL 2002. V. 34(2) - Tsai-Hill Theory o O Test data (Yoshihara o and Ohta, 2000) 120 1 4 ~ 1  a a X 0 (degrees) FIG. 8. Variation of with H at occurrence of F, Tsai-Hill theory and the test data in Fig. 7 representing the corresponding value of cr, in Eq. (Ic), the shear stresses 17,1 at failure are plotted in Fig. 8. (Note: In the Tsai-Hill the- ory, the sign for T ,  makes no difference. We therefore use its absolute value.) At 0 = lo0,  IT,^ is only about 1.2% less than its maximum value, which occurs at 0 .= 13", based on the Tsai-Hill theory. However, the test data are more prominently above the theoretical pre- dictions at 10" < 0 < 45". It is against any theory for IT,[ to be larger than F, in an off- axis tension test. When that happens, the test facilities, specimen conditions, sample size, and other variables may all need to be reex- amined. However, if we use the predicted val- ue of F, = 58.57 MPa at 0 = 10" and substi- tute it into Eq. ( l c )  for q,, we obtain  IT,^ = 10 MPa, which is about 13% less than F, = 11.3 MPa, and seems to be reasonable. So, the im- portant question is how reliable are the F,, F,, and F ,  values that were used in Eq. (13). In the Wood Handbook (FPL 1999), F , ,  F2, 0 30 60 90 0 (degrees) FIG. 9. Failure curve based on Tsai-Hill theory for two sets of F, values: (a) F, = 78.3 MPa, F2 = 2.55 MPa, and F6 = 7.93 MPa (FPL 1999). (b) FI = 78.3 MPa, FZ = 2.55 MPa, and F6 = 6.25 MPa (Liu and Floeter 1984). and F, values for Sitka spruce are 78.3, 2.55, and 7.93 MPa, respectively, which are consid- erably lower than the values found by Yoshi- hara and Ohta (2000).2 Using these values in Eq. (13), we obtain F ,  = 36.43 MPa at 0 = 10" (Fig. 9). The corresponding value of  IT,^ from Eq. ( lc)  is 6.23 MPa, which is about 27% less than F, = 7.93 MPa. Liu and Floeter (1984) reported that F, = 6.25 MPa for Sitka spruce. Using F ,  = 78.3 MPa, F, = 2.55 MPa, and F, = 6.25 MPa, and following the same steps as described in the preceding text, we obtain F, = 3 1.05 MPa at 0 = 10" and I T , [  = 5.31 MPa, which is about ?The Wood Hundbook F, value is adjusted based on data from dry specimens. Lfu-OFF-AXIS TENSION TEST OF WOOD SPECIMENS 21 1 18% less than F, = 6.25 MPa. Note the close proximity of the two failure curves in Fig. 9. We are now ready to discuss the procedures to determine F, from the 10" off-axis tension test. Assuming the set of data F ,  = 141 MPa, F, = 7.4 MPa, and F, = 11.3 MPa is valid, we have found that F,(8 = 10") = 3 1.05 MPa and IT,[ = 5.31 MPa, which is about 18% less than F ,  = 6.25 MPa. Since 13% and 18% are close to one another, we select their mean, which is about 15%. We therefore obtain F ,  - F,(H = 10") X sin 10°cos 10" X 1.15 Also, from these two sets of data, we find F,(0 = 10") = 0.4 X F, .  We can replace Eq. (14) by Our discussion is based on the Tsai-Hill theory and limited test data for Sitka spruce from different sources. Although there are considerable discrepancies between the two sets of test data, the formulas seem to be valid for both. The discrepancies may need to be reconciled, but the procedures presented are believed to be useful in applying the off-axis tension test for the characterization of wood shear properties. CONCLUSIONS 1. In an off-axis tension test of clear wood specimens, at a 10" grain angle, the shear stress in the principal material plane is due mainly to the shear coupling coefficient. 2. In applying the Tsai-Hill theory for tensile strength prediction of wood at any grain angle, the strength data in the principal ma- terial axes and plane must be reliable to produce accurate results. 3. The 10" off-axis tension test can be used to predict clear wood shear strength in a prin- cipal material plane of wood. 4. A formula has been derived for shear strength prediction of Sitka spruce based on input data in the literature. REFERENCES CHAMIS, C. C., AND J. H. SINCI.AIR. 1997. Ten degree off- axis test for shear properties in fiber composites. Exp. Mech. 17(9):339-346. DANIEL, I. M., AND 0. ISHAI. 1994. Engineering mechanics of composite materials. Oxford Univ. Press, Oxford, UK. 395 pp. FOREST PRODUCTS LABORATORY (FPI). 1999. Wood hand- book. Wood as an engineering material. Gen Tech Rep. FPL-GRT-I 13. USDA, Forest Scrv., Forest Products Lab., Madison, W1. 463 pp. JONES, R. M. 1975. Mechanics of composite materials. Scripta Book Co., Washington, DC. 344 pp. Llu, J. Y., AND L. H. FLOETER. 1984. Shear strength in principal plane of wood. J. Eng. Mcch. 110(6):930-936. , AND R. J. ROSS. 1998. Wood property variation with grain slope. Pages 1351-1354 in Proc. 12th En- gineering Mechanics Conf., American Society of Civil Engineers, La Jolla, CA. PAGANO, N. J., AND J. C. HALPIN. 1968. Influence of end constraint in the testing of anisotropic bodies. J. Comp. Mat. 2( 1): 18-3 1. PERKINS, JR., R. W. 1972. On the mechanical response of materials with cellular and tinely layered internal struc- tures. In B. A. Jayne, ed. Theory and design of wood and fiber composite materials. Syracuse University - Press, Syracuse, NY. PIERRON, E, AND A. VAUTRIN. 1996. The 10" off-axis ten- sile test: A critical approach. Comp. Sci. Technol. 56(4): 483-488. , E. ALI~OBA, Y. SURREL, AN[) A. VAUTKIN. 1998. Whole-field assessment of the effects of boundary con- ditions on the strain field in off-axis tensile testing of unidirectional composites. Comp. Sci. Technol. 58(12): 1939-1947. PINDERA, M.-J., A N D  C. T. HERAKOVI(:H. 1986. Shear char- acterization of unidirectional composites with the off- axis tension test. Exp. Mech. 26(1): 103-1 12. RICHARDS, G. L., T. I? AIRHART, AND J. E. ASHTON. 1969. Off-axis tensile testing. J. Comp. Mat. 3586-589. RIZZO, R. R. 1969. More on the influence of end con- straints on oo-axis tensile tests. J. Comp. Mat. 3:202- 219. SUN, C. T., AND I. CHUNC;. 1993. An oblique end-tab de- sign for testing off-axis composite specimens. Compos- ites 24(8):6 19-623. WU, E. M., A N D  R. L. THOMAS. 1968. Off-axis test of a composite. J. Comp. Mat. 2(4): 523-526. YOSHIHARA, H. AND M. OHTA. 2000. Estimation of the shear strength of wood by uniaxial-tension tests of off- axis specimens. J. Wood Sci. 46(2): 159-163. 

Analysis Of Off-Axis Tension Test Of Wood Specimens

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Analysis Of Off-Axis Tension Test Of Wood Specimens

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