Evaluation of Strength the TensorPolynomial Theory for Wood J. Y. L IU* Forest Products Laboratory, Forest Service U.S. Department of Agriculture, Madison, Wisconsin 53705 (Received October 10, 1983) (Revised December 27, 1983) ABSTRACT The tensor polynomial (Tsai-Wu) strength theory with the interaction stress coefficient, F 12, determined from the Hankinson formula, has been applied in this study to predict the strength of wood under plane-stress conditions. The biaxial failure envelope for Sitka spruce, based on this choice of F 12 and the mechanical properties from the Wood Handbook, is presented. F 12 is very sensitive in biaxial loading tests and can be extremely difficult to obtain in such tests on wood. Since the Hankinson formula has been verified repeatedly and has been popular in the wood industry, it is recommended that F 12 be based on the Hankinson formula. Keywords: Strength theory; wood: biaxial loading; composite materials. INTRODUCTION HE STRENGTH OF WOOD UNDER PLANE-STRESS STATES HAS BEEN IN- Tvestigated using the quadratic form of the Tsai-Wu strength theory [I]. The mathematical model of the theory was developed to predict the occurrence of material failure without considering the physics or mode of failure. It obeys tensor transformation rules and is, therefore, convenient for engineering applications. The first formula for the compressive strength of wood in a direction inclined at an angle to the grain was proposed by Hankinson [2] in 1921: (1) where X' denotes the compressive strength along the grain or axis 1, Y' the compressive strength in a cardinal direction perpendicular to the grain or axis 2, and the compressive strength in a direction at an angle to the grain in *Research General Engineer. 216 Journal of COMPOSITE MATERIALS, Vol. 18-May 1984 0021-9983/84/03 0216-11 $4.50/0 1984 Technomic Publishing Co., Inc.
Evaluation of the Tensor Polynomial Strength Theory for Wood 217 the 1-2 plane. The Hankinson formula, although strictly empirical, has also been found by Rowse [3], Norris [4], and Goodman and Bodig [5] to fit their experimental data for wood reasonably well. Kollmann and Côté [6] reported that the Hankinson formula is also suitable for computing the tensile strength of wood in the modified form: (2) where X and Y are the tensile strengths in the grain direction and in a cardinal direction perpendicular to the grain, respectively, and n is a constant; values of n between 1.5 and 2 in Equation (2) are satisfactory [7]. In 1962, Norris [8] developed a theory for the strength of orthotropic material based on the Henky-von Mises theory for isotropic materials. The theory was verified by his experimental data on wood, plywood, and fiberglass laminate from off-axis compressive or tensile tests. The Hankinson and Norris formulas have enjoyed more popularity than any of the other strength criteria in the wood industry. Tsai and Wu [1] presented a general tensor polynomial theory of strength for anisotropic materials in 1971. Their theory is an operationally simple failure criterion from strength tensors. It includes many anisotropic failure criteria including the one by Norris [8] as special and restricted cases (see Wu [9]). However, the theory has failed to gain widespread acceptance because of the difficulty in evaluating an interaction stress coefficient, usually denoted by F 12. This coefficient has to be determined by a proper strength test characterized by a combined state of stress in addition to those tests required to define the usual uniaxial strengths of the material (see Narayanaswami and Adelman [10]). In 1979, Cowin [11] derived the Hankinson formula from the Tsai-Wu theory by using only the linear terms of the tensor polynomial. The expressions for the coefficients obtained by keeping only the linear terms contradicted with those when the quadratic terms of the theory were retained. Clearly, this represented an inconsistency in the application of the Tsai-Wu theory. Recently, van der Put [12] also applied the Tsai-Wu theory on wood and discussed several expressions for F 12. By decomposing the applied loads with respect to the material axes, he obtained an expression for F 12 that could reduce the Tsai-Wu theory into the Hankinson formulas, which was an objective Cowin did not fulfill in [11]. However, guided by the discussions on filamentary composite materials in [10], van der Put [12] recommended that F 12 be neglected for wood. In the present study, formulas for strength of wood have been derived from the Tsai-Wu theory by transforming the strength tensors with respect to the applied loads. This approach is a unique feature of the theory but was not followed by Cowin [11]. An expression for F 12 is derived in this paper using
218 J. Y. LIU the quadratic form of the theory so that the Hankinson formulas for both compressive and tensile strengths can be reproduced. Finally, numerical analysis was made on Sitka spruce to elucidate the choice of the value for F 12 in engineering applications. TENSOR POLYNOMIAL THEORY The quadratic strength theory for anisotropic materials proposed by Tsai- Wu [1] is: where F i and F ij are strength tensors of the second and fourth rank, respectively, following the contracted notation, and and are the stresses as shown in Figure 1. When the applied stresses do not agree with the axes of material symmetry, one can rotate the material axes from F i to F i ' and F ij to F i ' j in Equation (3) to obtain Alternatively, the applied stresses can be transformed from to obtain so that the applied stresses agree with the material axes. Most existing strength criteria can be applied only by transforming the external stresses as shown in Equation (5). This approach was followed by Cowin [ll] and van der Put [12]. In the present study, Equation (4) will be used with the transformation relations in [1] to obtain straight-forward solutions. (3) (4) (5) Figure 1. Positive normal and shear stresses.
Evaluation of the Tensor Polynomial Strength Theory for Wood 219 In Equation (3), the quadratic terms define an ellipsoid in the stress space. Failure occurs when a stress vector reaches the failure surface. To be sure that the failure surface is ellipsoidal, the following stability conditions must be satisfied: where repeated indices are not summations for the inequality. (6) Strength of Orthotropic Materials When the Tsai-Wu theory is specialized to the case of an orthotropic material such as wood, in a general state of plane-stress, consider i, j = 1,2 and 6, (see Figure 1). In expanded form, Equation (3) becomes with the stability condition where (7) (8a) (8b) and is known to be an invariant. Equation (7) does not contain odd-order terms of because material strength remains unchanged when reverses its direction on a principal plane. Equations (4) and (5) can likewise be expanded. The coefficients in Equation (7) can be obtained from uniaxial strength data with some algebraic manipulations as follows: (9a) (9b) (9c) (9d) and
220 J. Y. LIU (9e) where X and Y are the tensile strengths and X' and Y' the compressive strengths in the 1 and 2 directions respectively, and S is the shear strength in the 1-2 plane. The only coefficient left to be determined is F 12 in Equation (7)* Strength in a Direction Not Parallel to Grain Let be the stresses in a direction oriented at an angle with respect to the grain or the material axis 1 as shown in Figure 2. Expressing the strength tensors with respect to the 1 '2' coordinate system, one obtains Using the transformation relations of the strength tensors in [1] and the stress coefficients in Equation (9), Equation (10) becomes (10) (11) which satisfies the strength conditions at = 0 and Equation (11) has two roots for that should be comparable with the Hankinson formulas, Equations (1) and (2). Let these two equations be grouped with n = 2 in Equation (2) as follows: Figure 2. Stresses at an angle to grain.
Evaluation of the Tensor Polynomial Strength Theory for Wood 221 It can be shown by comparing Equations (1 I) and (12) that the two equations are identical if Note that Equation (13) is different from the one obtained by Cowin [11] but agrees with the expression derived by van der Put [12] by transforming the applied loads rather than the strength tensors. Also, if the linear term of in Equation (11) is neglected, can be identically expressed as shown by Cowin [11]. Of course, the linear term can only vanish when the tensile and compressive strengths along an axis are the same. This is not the case for wood. NUMERICAL ANALYSIS The only requirement for F 12 in the Tsai-Wu theory is that it must satisfy the stability condition (Equation (8a)). It is, therefore, of interest to see how the failure envelope defined by Equation (7) will change when F 12 varies within its allowable range and how reliable Equation (13) can be in defining F 12 based on the Hankinson formulas. Consider the mechanical properties for Sitka spruce from the Wood Handbook [13] as shown in Table 1 with a specific gravity of 0.40 and a moisture content of 12%. The corresponding strength tensors are presented in Table 2. According to the stability condition (8a) and (8b), F 12 must fall between where = 2.6875 x 10-7 (psi) -2 (5.6535 x 10-3 (MPa) -2 ). It is obvious that zero is always an acceptable value for F 12. Table 1. Assigned values for mechanical properties for Sitka spruce 1 [13]. Property Value Psi Tensile strength parallel to grain, X 2 11,500 Tensile strength perpendicular to grain, Y 370 Compressive strength parallel to grain, X' 5,610 Compressive strength perpendicuiar to grain, Y' 580 Shear strength parallel to grain, S 1,150 1Specific gravity = 0.40; moisture content = 12%. 2Estimated at 10 times shear strength, see reference [6]. Note: 1 psi = 6.8947 kpa.
222 J. Y. LIU Table 2. Values of strength tensor components. Component Value -0.9130 x 10-4 0.1550 x 10-7 0.9786 x 10-3 0.4660 x 10-5 -0.6223 x 10-7 0.7561 x 10-6 1 From Equation (9) and Table 1. 2From Equation (13) and Table 1. Note: 1 psi = 6.8947 kpa. Before considering the biaxial loading situations, it would be appropriate to discuss the strength in a direction intersecting the grain in a plane of material symmetry. Both Cowin [11] and Narayanaswami and Adelman [10] followed this approach to estimate their proper value for F 12. Figure 3 plots the compressive strength vs. grain angle for Sitka spruce from Equations (1) and (11). Equation (1), the original Hankinson formula, and Equation (11) with F 12 = 0 show very close agreement for > 15. The limit values for F 12 are also used in Equation (1 1) for comparison. This indicates that F 12 is not too insensitive in an off-axis test. The tensile strength variation with grain angle is presented in Figure 4. The curve for F 12 = 0 in Equation (11) is close to that of Equation (2) with n = 2. The curve with n = 1.5 [7] (Figure 4) falls below the one with F 12 = in Equation (I l)-le., it corresponds to a value of F 12 larger than its upper limit, However, as pointed out by Kollmann and Côté [6], the tensile test data are limited in number and reliability in the literature, as the failure in the wood specimens is often not entirely tensile. It would thus appear reasonable to take n = 2 in Equation (2). The high sensitivity of F 12 in a biaxial loading situation becomes evident in Figure 5. Obviously, for wood, F 12 must be only a small fraction of its limit Figure 3. Compressive strength vs. grain angle for Sitka spruce (1 psi = 6.8947 kpa).
Evaluation of the Tensor Polynomial Strength Theory for Wood 223 Figure 4. Tensile strength vs. grain angle for Sitka spruce (1 psi = 6.8947 kpa). value 4 so that the failure envelope will not be too elongated. It would be physically impossible for to increase or decrease very much with when X and Y or Y' have both been exceeded. Narayanaswami and Adelman [10] performed a numerical experiment to determine a proper choice of F 12 for composite materials: they solved Equation (7) for by setting = or for estimated values of F 12. The obtained values of were always close to one another, which led them to conclude that F 12 = 0 is a reasonable choice. Their results can easily be seen in Figure 5. For = or the point on any failure envelope is close to the vertical axis. Therefore, the obtained values of for different estimates of F 12 cannot differ very much. Some of the important biaxial testing methods in the literature such as those by Daniel [14,15], if successfully applied on wood, will lead to the same results, because in them the applied loads are either = or = Wood is a material less homogeneous than the composite materials considered by Narayanaswami and Adelman [IO], and therefore the test data of wood can be expected to contain more scatter.. However, the Hankinson formula as shown in Equation (1) has generally been considered to predict the mean compressive strength for wood very well. Therefore, it is of interest to compare the failure envelopes of Equation (7) with F 12 = 0, which was Figure 5. Normal stresses vs with = 0, F12 = 0, and from Equation (8b) in Equation (7) (1 psi = 6.8947 kpa).
224 J. Y. L IU Figure 6. Normalstresses vs. with = 0, F 12 = 0, and F 12 from Equation (13) in Equation (7) (1 psi = 6.8947 kpa). recommended in [10] and (12), and F 12 expressed by Equation (13), which reduces Equation (11) into the Hankinson formulas, Equations (1) and (2). Such a comparison is shown in Figure 6. It is seen that the differences between the two envelopes in the first and fourth quadrants are somewhat significant, although they are small in the second and third quadrants. These differences can, of course, vary when other wood species are considered. This observation should support the adoption of Equation (13) for F 12 in Equation (7) for wood since the Hankinson formula has enjoyed widespread acceptance by the wood industry. The effect of shear on wood failure in a combined stress state is shown in Figure 7. When the shear stress is high, the failure envelope shrinks at a higher rate, as expected. CONCLUSION The Tsai-Wu theory has been applied to predict the strength of wood under plane-stress states. The interaction stress coefficient, F 12, in the theory has been defined analytically so that the theory can be made identical to the Hankinson formula in an off-axis loading situation. With F 12 so defined, it can be evaluated using existing mechanical properties for wood in the Wood Handbook [ 13]. Figure 7. Normal stresses vs. in presence of shear stress o., Equation (7) with F12 from Equation (13), S = 1,150 psi (7.93 MPa) (1 psi = 6.8947 kpa).
Evaluation of the Tensor Polynomial Strength Theory for Wood 225 Although F 12 is shown to be very sensitive in a biaxial loading test, such a test is very difficult to perform on wood. Even if a biaxial loading test can be successfully designed for wood, the material inhomogeneity will result in so much data scatter that the determination of F 12 will still involve a considerable amount of uncertainty. This indicates that the relative insensitivity of F 12 in an off-axis test may not reduce its merits as a means for the determination of F 12. In the numerical analysis of the present study, the biaxial failure envelope with F 12 based on the Hankinson formulas appears to be reasonable. Therefore, it is recommended that the Tsai-Wu theory with F 12 defined by Equation (13) be used to predict the strength of wood under combined stress states. This will be consistent with using the Hankinson formulas to predict failure in an off-axis loading condition. F i, F i ' F ij, F i ' j 01 X, Y X ', Y' NOMENCLATURE Strength tensor of the 2nd rank Strength tensor of the 4th rank Stress components Tensile strengths in 1 and 2 directions, respectively Compressive strengths in 1 and 2 directions, respectively Strength along angle Grain angle REFERENCES 1. Tsai, S. W., and Wu, E. M., A General Theory of Strength for Anisotropic Materials, Journal of Composite Materials, Vol. 5, pp. 58-80 (1971). 2. Hankinson, R. L., Investigation of Crushing Strength of Spruce at Varying Angles of Grain, Air Service Information Circular No. 259, U.S. Air Service (1921). 3. Rowse, R. C., The Strength of Douglas-Fir in Compression at Various Angles to the Grain, Thesis for Degree of Bachelor of Civil Engineering at Washington University, St. Louis, Mo. (1923). 4. Norris, C. B., The Elastic Theory of Wood Failure, ASME Transactions, Vol. 61, No. 3, pp. 259-261 (1939). 5. Goodman, J. R., and Bodig, J., Orthotropic Strength of Wood in Compression, Wood Science, Vol. 4, No. 2, pp. 83-94 (1971). 6. Kollmann, F. F. P., and Côte, W. A., Jr., Principles of Wood Science and Technology, I. Solid Wood, Springer-Verlag Inc., New York (1968). 7. Kollmann, F. F. P., Untersuchungen an Kiefern- und Fichtenholz aus der Rheinpfalz, Forstwissenschaften Centralblatt, Vol. 56, No. 6, pp. 181-189 (1934). 8. Norris, C. B., Strength of Orthotropic Materials Subjected to Combined Stress, U.S. Forest Products Lab. Rep. 1816, FPL, Madison, Wis. (1962). 9. Wu, E. M., Phenomenological Anisotropic Failure Criterion, in Mechanics of Composite Materials (G. P. Sendeckyj, ed.), Academic Press, New York (1974). 10. Narayanaswami, R., and Adelman, H. M., Evaluation of the Tensor Polynomial and Hoffman Strength Theories for Composite Materials, Journal of Composite Materials, Vol. 11, pp. 366-377 (1977).
226 J. Y. LIU 11. Cowin, S. C., On the Strength Anisotropy of Bone and Wood, J. Applied Mechanics, ASME Transactions, Vol. 46, No. 4, pp. 832-837 (1979). 12. van der Put, T. A. C. M., A General Failure Criterion for Wood, IUFRO Timber Engineering Group Meeting, Paper 23 [Sweden, 1982], IUFRO, Vienna. 13. U.S. Department of Agriculture, Forest Service, Forest Products Laboratory, Wood Handbook: Wood as an Engineering Material, Agric. Handb. 72, Gov t Printing Office, Washington, D.C. (1974). 14. Daniel, I. M., Behavior of Graphite/Epoxy Plates with Holes Under Biaxial Loading, Experimental Mechanics, Vol. 20, No. 1, pp. 1-8 (1980). 15. Daniel, I. M., Biaxial Testing of [02/±45]s Graphite/Epoxy Plates with Holes, Experimental Mechanics, Vol. 22, NO. 5, pp. 188-195 (1982).