Skip to contentUnited States Department of Transportation - Federal Highway Administration FHWA Home
Research Home
Report
This report is an archived publication and may contain dated technical, contact, and link information
Publication Number: FHWA-HRT-04-097
Date: August 2007

Measured Variability Of Southern Yellow Pine - Manual for LS-DYNA Wood Material Model 143

PDF Version (2.92 MB)

PDF files can be viewed with the Acrobat® Reader®

APPENDIX D. GRAPHICAL COMPARISON OF CANDIDATE FAILURE CRITERIA

D.1 COMPARISON RESULTS

Each failure criterion was evaluated by comparing it with available test data and with other criteria. Comparisons include:

  • Off-axis strength predictions.
  • Parallel-to-the-grain strength predictions.
  • Perpendicular-to-the-grain strength predictions.

Details of the comparisons are given in section D.2. The results of the comparisons are briefly given here:

  • The failure stresses for all candidate criteria agree for states of uniaxial stress in the principal material directions. The criteria disagree on what constitutes failure for biaxial and triaxial stress states.
  • Nearly sufficient uniaxial data are available for evaluating the candidate criteria. The exception is the lack of a perpendicular shear strength measurement for southern yellow pine. Insufficient biaxial and triaxial data are available to thoroughly evaluate each criterion. Biaxial and triaxial data are limited to off-axis test data for Douglas fir.
  • The Modified Hashin and Tsai-Wu criteria fit off-axis test data the best. All candidate failure criteria were compared with off-axis test data in two planes: the L-T and T-R planes. In the L-T plane, all criteria except Maximum Stress and Hashin accurately predict the trend observed in off-axis test data. In the R-T plane, no criterion accurately predicts the trend observed in off-axis test data. However, the transversely isotropic criteria (Hashin, Modified Hashin, and Tsai-Wu) provide the best fit. These comparisons suggest that the Modified Hashin and Tsai-Wu criteria are the best candidates.
  • The Modified Hashin criterion is more practical than the Tsai-Wu criterion because it identifies four distinct failure modes. The Hashin, Yamada-Sun, Norris, and Maximum Stress criteria also identify the mode of failure. On the other hand, the Tsai-Wu and Hoffman criteria predict the onset of failure, but not the mode of failure. This makes it difficult to model postpeak softening, because the extent of softening depends on the failure mode (tensile, compression, or shear) and failure direction (parallel or perpendicular).
  • Simultaneous and/or sequential failure in perpendicular and parallel modes is modeled by those criteria that predict distinct failure modes (Modified Hashin, Hashin, Yamada-Sun, Norris, and Maximum Stress). This is because these criteria are formulated with multiple equations and more than one failure equation can be satisfied at one time. Those criteria formulated with a single equation (Tsai-Wu and Hoffman) cannot make such predictions.
  • Most criteria produce smooth failure surfaces in stress or stress invariant space. Having a smooth failure surface is important when applying the plasticity algorithm because nonsmooth surfaces (with corners) require additional coding and add computational intensity. The only nonsmooth surface that was evaluated is the Hashin criterion, and it occurs in the perpendicular modes. Most criteria, such as the Modified Hashin, produce segmented surfaces when cross-plots are made involving both the parallel and perpendicular modes; however, this is not a computational issue because plasticity will be treated separately for each mode. The Modified Hashin criterion produces a smooth surface for the parallel modes and a separate smooth surface for the perpendicular modes.
  • Most criteria model little or no interaction between the parallel and perpendicular modes. This means that parallel strength is not significantly affected by perpendicular confinement. Five of the seven criteria predict little or no increase in parallel compressive strength with perpendicular confinement. The Modified Hashin criterion is one of the five criteria. The two exceptions are the Hoffman and Tsai-Wu criteria. Four of the seven criteria predict little or no increase in parallel tensile strength with perpendicular confinement. The Modified Hashin criterion is one of the four criteria. The three exceptions are the Hoffman, Tsai-Wu, and Norris criteria. Interaction terms could readily be added to the Modified Hashin criterion if test data become available that suggest that an interaction mechanism is appropriate.
  • The transversely isotropic criteria conservatively model parallel failure under parallel shear stress. These criteria (Modified Hashin, Hashin, Yamada-Sun, and Tsai-Wu) predict shear failure at lower shear stresses than the orthotropic criteria. The lower shear strength results from simultaneous application of two shear stress components.
  • The strength predictions of the transversely isotropic criteria (Modified Hashin, Hashin, Yamada-Sun, and Tsai-Wu) are realistic under transformation of stress in the isotropic plane. These criteria recognize sT = –sR as a state of pure shear stress; the orthotropic criteria do not.
  • The transversely isotropic criteria are more flexible in modeling failure and yielding in the perpendicular modes than the orthotropic criteria. This is because the transversely isotropic predictions are sensitive to the value of the perpendicular shear strength selected by the analyst.

D.2 GRAPHICAL COMPARISONS

The criteria are plotted using the transversely isotropic strength values previously reported in table 4 for southern yellow pine at 12-percent moisture content.

In some plots, the various failure criteria are compared with Hankinson’s formula. Hankinson’s formula is an empirical equation that is frequently fit to off-axis compression tests of wood in two dimensions. Goodman and Bodig extended the formula to three dimensions.(7) Although Hankinson’s formula fits off-axis test data fairly well, it is not a general-purpose formulation that can be applied to other types of tests. This is because it is explicitly formulated in terms of the off-axis grain and the ring angles. Thus, it is not suitable for use in finite element codes. However, it is reported here so that it can be compared with the various criteria to help in their evaluation.

D.3 OFF-AXIS STRENGTH COMPARISONS

Off-axis strength is characterized by performing uniaxial tests with the symmetry axis (L‑T‑R axis) oriented at-angle to the loading axis, as schematically shown in figure 46. The measured strength depends on two off-axis angles―the angle q between the grain and loading axis, and the angle f between the rings and loading axis. A grain angle of q = 0 degrees means that the load is being applied in the longitudinal direction. A grain angle of q = 90 degrees means that the load is being applied in the perpendicular direction. If f = 0 degrees when q = 90 degrees, then the load is being applied in the tangential direction. If f = 90 degrees when q = 90 degrees, then the load is being applied in the radial direction.

The various failure criteria are compared here with off-axis test data (i.e., Hankinson’s formula). Two sets of comparisons are made. In the first, the off-axis strength in the L‑T plane (parallel and perpendicular to the grain) is examined. In the second, the off-axis strength in the T‑Rplane (perpendicular to the grain only) is examined.

click on the image for Section 508 compliancy text

Figure 46.

Geometry of an off-axis test specimen.

Source: Krieger Publishing Company (16).

D.3.1 Off-Axis Strength in the L-T Plane

Data:Off-axis compressive strength data for Douglas fir were previously shown in the three-dimensional plot of figure 2.(7) Also shown is the three-dimensional Hankinson formula. The comparison is good, although Hankinson’s formula tends to underestimate the measured strength, particularly for grain angles between 15 and 45 degrees. Hankinson’s formula also tends to underestimate the measured strength of Engelmann spruce, oak, and aspen (not shown), as reported by Goodman and Bodig.(7)

Although Hankinson’s formula is an empirical fit to wood data, the good fit does not validate the formula. This is because longitudinal, tangential, radial, and shear stresses all act on the wood in the material coordinate system; however, only longitudinal, tangential, and radial stresses are assumed to contribute to compressive failure in Hankinson’s formula. The relative effects of the normal and shear stresses are not established.

Strength Comparisons: The measured off-axis strengths are not tabulated in the paper by Goodman and Bodig and are difficult to extract from figure 2.(7) In addition, no off-axis test data are available from FPL for southern yellow pine. Therefore in figure 47(a), the failure criteria are compared with Hankinson’s formula rather than test data (keep in mind that Hankinson’s formula tends to underestimate the measured strength). The comparisons are for the ultimate compressive strength of southern yellow pine calculated as a function of grain angle for f = 0 degrees.

All but two criteria, the Maximum Stress and Hashin criteria, are in good agreement with Hankinson’s formula. In fact, most criteria predict a slightly greater strength than Hankinson’s formula, a trend consistent with the measured data. On the other hand, the Maximum Stress criterion significantly overestimates the strength of the wood compared with Hankinson’s formula. The Maximum Stress criterion predicts an increase in off-axis strength with increasing grain angle between 0 and 23 degrees. This is opposite the trend predicted by Hankinson’s formula and observed in off-axis tests of various wood species. In addition, the Hashin criterion overestimates the strength of wood compared with Hankinson’s formula for small grain angles and underestimates the strength at moderate grain angles. These comparisons suggest that the Maximum Stress and Hashin criteria are not good candidates for modeling the off-axis strength of wood.

In addition to compressive strength, tensile strength is also plotted as a function of grain angle in figure 47(b). The primary reason these tensile comparisons were made was to check the Tsai-Wu criterion. One coefficient of the Tsai-Wu criterion must be fit to biaxial strength data. The criterion was fit to the off-axis compressive strength predicted by Hankinson’s formula at 45 degrees. Even though the fit was made in compression, the Tsai-Wu criterion is in agreement with Hankinson’s formula and the other interactive criteria in tension. Although Goodman and Bodig did not report comparisons of Hankinson’s formula with wood test data in tension, various authors suggest that Hankinson’s formula is a reasonable fit in tension as well.

click on the image for Section 508 compliancy text

Figure 47.

Most of the interactive failure criteria are in agreement with Hankinson's formula for the off-axis strength of southern yellow pine in the L-T plane.

Failure Mode Comparisons: Although most of the interactive criteria accurately predict the off-axis compressive strength, it is not known if the criteria predict the correct failure mode. This is because the measured failure mode was not reported by Goodman and Bodig.(7) We expect the failure to be in the parallel modes at low grain angles and in the perpendicular modes at high grain angles. Five of the failure criteria (Hashin, Modified Hashin, Yamada-Sun, Norris, and Maximum Stress) predict distinct failure modes.

Considering the Modified Hashin criterion, for grain angles less than about 24 degrees, the predicted mode is compression failure parallel to the grain. For grain angles greater than about 24 degrees, the predicted mode is compression failure perpendicular to the grain. This change in mode is evident by the discontinuity in slope in figure 47(a). Failure modes are readily identified because Hashin formulated separate criteria (equations) for the parallel and perpendicular modes. Two of the criteria, the Tsai-Wu and Hoffman criteria, do not distinguish the mode of failure because they are formulated with a single equation.

D.3.2 Off-Axis Strength in R-T Plane

Data: Off-axis test data are also useful for evaluating the assumption of transverse isotropy. Transversely isotropic strength values were previously reported in table 4 for pine and in table 5 for fir. This means that no distinction is made between the strengths in the tangential and radial directions, so only one measurement is made and labeled as perpendicular. The assumption of transverse isotropy is assessed by examining the off-axis strength data measured by Goodman and Bodig in the R-T (isotropic) plane. These data are shown in the two-dimensional plot of figure 48. Goodman and Bodig measured the ultimate strength at various ring angles, holding the grain angle constant at q = 90 degrees. These data are for four different wood species, one of which is Douglas fir.

If the wood species in figure 48 were transversely isotropic, then the tangential strength would be equal to the radial strength and the data would form a straight line between 0 and 90 degrees. The data at 0 degrees is the tangential strength. The data at 90 degrees is the radial strength. For Douglas fir, the radial strength is about 85 percent of the tangential strength, which is reasonably close. However, all of the data follow a similar pattern―the off-axis strength measured at 45 degrees is less than that measured at 0 or 90 degrees. For Douglas fir, the strength at 45 degrees is about 60 percent of the tangential strength.

click on the image for Section 508 compliancy text

Figure 48.

Effect of ring angle variation at 90-degree grain angle on the relative compression strength of four wood species.

Source: Society of Wood Science and Technology.(7)

Although strength varies with the ring angle for these wood species, the variation is not great when compared with the compressive strength in the longitudinal direction. For Douglas fir, the tangential strength at 0 degrees is about 10 percent of the longitudinal strength. The off-axis strength at 45 degrees is about 6 percent of the longitudinal strength. Therefore, the variation in perpendicular strength is about 4 percent of the parallel strength. The assumption of transverse isotropy in strength is probably reasonable, especially if wood posts fail catastrophically in the parallel modes rather than in the perpendicular modes.

Strength Comparisons:Hankinson’s formula in the R-Tplane is an excellent fit to wood data, as previously shown in figure 48. All failure criteria are compared with Hankinson’s formula in the R-T plane in figure 49. None of the criteria is in agreement with Hankinson’s formula, except in uniaxial compression at 0 and 90 degrees. At 45 degrees, the ultimate strength predicted by the various failure criteria is greater than that predicted by Hankinson’s formula. Hankinson’s formula suggests a 40-percent reduction in strength at 45 degrees. The Tsai-Wu, Hashin, and Modified Hashin criteria predict no reduction in strength at 45 degrees and no variation with ring angle. This is because they were derived from the invariants of a transversely isotropic material and the strength of a transversely isotropic material does not vary with ring angle. All other orthotropic criteria predict an increase in ultimate strength at 45 degrees (in poor agreement with Hankinson’s formula and the trend observed in the measured data (figure 48)). It is interesting to note that the orthotropic failure criteria do not predict transversely isotropic behavior even though transversely isotropic strength values are used. These comparisons suggest that the best criteria for modeling biaxial compressive strength perpendicular to the grain are the transversely isotropic criteria (Tsai-Wu, Hashin, and Modified Hashin).

click on the image for Section 508 compliancy text

Figure 49.

Failure criteria comparison for perpendicular modes as a function of the ring angle.

Transformations:Failure criteria are applied to stresses in the principal material directions. The off-axis strength, sult, is transformed into stresses in principal material directions:

This equation reads bracket stress component subscript Capital T, stress component subscript Capital R, stress component subscript Capital TR bracket equals stress component subscript lowercase ULT bracket lowercase M superscript 2, lowercase N superscript 2, negative lowercase MN bracket.

where:

m = cosf

n = sinf

f = ring angle between the tangential direction and loading axis

The state of stress at 45 degrees is biaxial, with s T = s ult / 2, s R = s ult / 2, and s TR = s ult / 2. Referring back to the Douglas fir data in figure 48, one sees that the equal biaxial compressive stress perpendicular to the grain is 60 percent of the tangential strength, which, in turn, is half of the applied stress. Therefore, the biaxial compressive strength of Douglas fir is 30 percent of the uniaxial (tangential) compressive strength, at least in the presence of shear stress. On the other hand, the transversely isotropic failure criteria predict a biaxial compressive strength that is 50 percent of the uniaxial compressive strength, also in the presence of shear stress.

To accurately measure the biaxial compressive strength, one would need data from biaxial tests performed with and without the application of shear stress. If one argues that shear stress is mutually weakening, as it is for composites, then one would expect the biaxial compressive strength measured with shear stress to be less than that measured without shear stress. Some failure criteria predict such a trend, as discussed in section C.3.

D.4 PARALLEL-TO-THE-GRAIN STRENGTH COMPARISONS

Wood posts are observed to fail in tension or shear parallel to the grain. Therefore, it is particularly important to accurately model the critical combinations of stresses that produce failure in the parallel modes. Although ultimate stress versus grain angle plots from off-axis tests include assessment of parallel failure, they reveal few differences among the various criteria. A more exacting assessment is attained with biaxial stress plots. The biaxial stress plots discussed in subsequent paragraphs indicate that significant differences in strength are predicted by the various criteria. These differences are revealed in three sets of stress plots:

  • Longitudinal stress versus perpendicular stress.
  • Longitudinal stress versus parallel shear stress.
  • Longitudinal stress versus combined stress (perpendicular plus parallel shear stress).

D.4.1 Biaxial Comparisons of Longitudinal Versus Confining Stress

Strength Comparisons: The combinations of longitudinal and confining stresses that satisfy the various failure criteria are compared in the biaxial strength plot of figure 50 for southern yellow pine. By confining stress, we mean the sum of the tangential and radial stresses. This sum is one of the invariants of a transversely isotropic material. The sum was obtained for the specific case of equal tangential and radial stresses. This figure is plotted with the stresses positive in tension.

Stress states that lie on the vertical axes (s T + s R = 0) are uniaxial stress states. When the confining stress is zero, each curve intersects the vertical axis twice―once in tension (positive) and once in compression (negative). These intersections are the longitudinal strengths in uniaxial tension and compression. Stress states that lie on the horizontal axes (s L = 0) are biaxial stress states. When the longitudinal stress is zero, each curve (except Hoffman) intersects the horizontal axis twice―once in tension and once in compression. These intersections are twice the perpendicular strengths in equal biaxial tension and compression. Stress states that do not lie on the vertical or horizontal axes are triaxial stress states. All criteria are in agreement for states of uniaxial stress. Uniaxial strength measurements are available for fitting the criteria. The criteria disagree on what constitutes failure or yielding for states of biaxial and triaxial stress. Biaxial and triaxial strength measurements are not currently available for fitting the criteria. Biaxial and triaxial test data are needed to validate one or more of the proposed criteria.

click on the image for Section 508 compliancy text

Figure 50.

Predicted effect of perpendicular confinement and extension on the longitudinal strength of southern yellow pine in tension and compression.

Failure Mode Comparisons: All criteria plotted in figure 50 are closed, but not necessarily smooth surfaces.6 The failure surfaces that are smooth are the Tsai-Wu and Hoffman criteria. This is because failure is modeled by a single formula or equation. The drawback of these criteria is that only the onset of failure is predicted, not the mode of failure. This means that when a particular combination of stresses indicates that failure or yielding has occurred, the criteria do not indicate the type of failure (tensile, compressive, or shear) or its direction (parallel or perpendicular). It is important to know the type and direction of failure so that the correct softening response (brittle or ductile) and fracture energy are modeled, as discussed in section 1.7.

6 Although the Hoffman criterion appears to model infinite biaxial compressive strength perpendicular to the grain, it actually models an extremely large biaxial compressive strength (off the scale of the figure).

The nonsmooth criteria indicate both the type and direction of failure. This is because failure is modeled by more than one equation. For example, the Modified Hashin criterion models four modes of failure because four different failure equations are satisfied on different portions of the failure surface, which is shaped like a square. These modes are compressive yielding perpendicular to the grain (left side of square), tension failure parallel to the grain (top side of square), tension failure perpendicular to the grain (right side of square), and compressive yielding parallel to the grain (bottom side of square).

All criteria, except Hoffman and Tsai-Wu, predict no increase, or a small increase in compressive strength with compressive confinement (refer to the lower left-hand quadrant of figure 50). The Hoffman criterion predicts a large increase in compressive strength with confinement. The Tsai-Wu criterion predicts a reduction in strength with confinement. The only strength data available for comparison that include confinement are the off-axis test data; however, such data also include contributions from shear stress. The effect of shear stress on longitudinal strength is discussed in subsequent paragraphs.

D.4.2 Biaxial Comparisons of Longitudinal Versus Combined Stress

The combinations of longitudinal and tangential stresses that satisfy the various failure criteria are compared in the biaxial strength plot of figure 51(a) for southern yellow pine. These curves include a contribution from the parallel shear stress and were calculated for the specific case of s LT = s T, with s LR = 0 and s R = 0. Also included in this figure is one point from Hankinson’s formula that represents off-axis test data measured at q = 45 degrees. Almost all criteria, expect the Maximum Stress criterion, are in good agreement with the Hankinson point, as expected from previous comparisons with off-axis test data.

The failure criteria differ most in the upper left-hand quadrant of figure 50. This is the quadrant that predicts the longitudinal tensile strength as a function of perpendicular confinement and parallel shear. The two criteria that differ most are the Tsai-Wu and Norris criteria. The Tsai-Wu criterion predicts an increase in longitudinal tensile strength with compressive confinement and shear, while the Norris criterion predicts a large decrease in longitudinal tensile strength with compressive confinement and shear. The remaining criteria predict little or no reduction in strength with combined confinement and shear. The small reduction in strength predicted by the Hashin, Modified Hashin, and Yamada-Sun models is a result of the application of shear. This is evident by comparing the failure criteria in figure 51(a), which were calculated with shear, with the failure criteria in figure 51(b), which were calculated without shear. No data are available in this quadrant for evaluating each model. More clear wood data are needed to understand the relative contributions of shear stress and perpendicular confining stress on the longitudinal strength in tension.

D.4.3 Biaxial Comparisons of Longitudinal Versus Shear Stress

All clear wood strength data available for southern yellow pine or Douglas fir are a measurement of either s LT or s LR, or, more generally, of parallel shear strength. The combined effect of applying two parallel shear stress components simultaneously has not been measured. Wood posts have been observed to fail in a shear mode, so it is important to establish the correct shear strength, particularly if failure is caused by simultaneous application of two shear components. The effects of unilateral and bilateral shear stresses on longitudinal strength are examined here.

The combinations of longitudinal and shear stresses that satisfy the various failure criteria are compared in the biaxial strength plot of figure 52 for southern yellow pine. By shear stress, we mean the sum of the squares of the L-T and L-R parallel shear stress terms the square root of sigma superscript 2 subscript LT plus sigma superscript 2 subscript LR . This sum is one of the invariants of a transversely isotropic material. The square root is taken to retain units of stress. Two plots are shown. In one plot, the sum was obtained for the specific case of s LT = s LR. In the other plot, the sum was obtained for the specific case of s LR = 0.

click on the image for Section 508 compliancy text

Figure 51.

Predicted effect of parallel shear and tangential stresses on the longitudinal strength of southern yellow pine in tension and compression.

click on the image for Section 508 compliancy text

Figure 52.

Predicted effect of parallel shear invariant on the longitudinal strength of southern yellow pine in tension and compression.

In figure 52(a), the transversely isotropic and orthotropic criteria predict different shear strengths. The orthotropic criteria (Maximum Stress and Norris) predict a combined shear strength that is 40 percent greater than that predicted by the transversely isotropic criteria.

The transversely isotropic criteria predict a value of 16.8 MPa, which is equal to the shear strength measured parallel to the grain for either the L-T or L-R shear stress component. The individual values of s LT and s LR do not affect the parallel shear strength of a transversely isotropic material, only the sum of their squares. This is because this sum is an invariant of a transversely isotropic material and the parallel shear stress components are not included in the other invariants used in the transversely isotropic models. However, this sum is not an invariant if the criterion is orthotropic, so the specific values of each stress component affect the predicted strength. For the specific case of equal shear stress in the L-T and L-R planes, this sum is equal to 23.7 MPa.

These comparisons demonstrate that the combined shear strength predicted by transversely isotropic criteria is lower than that predicted by orthotropic criteria. Use of transversely isotropic criteria in guardrail post calculations could result in a lower shear strength than that predicted by orthotropic criteria. Planned guardrail post calculations will examine the value of each shear stress component and check the sensitivity of the results to shear strength.

D.5 PERPENDICULAR-TO-THE-GRAIN STRENGTH COMPARISONS

Wood guardrail posts fail catastrophically in the parallel modes (tension and shear). However, failure and yielding in the perpendicular modes could precede parallel failure without causing catastrophic failure of the guardrail post. Failure in the perpendicular modes occurs at much lower strengths than failure in the parallel modes. It is important to accurately model the perpendicular failure criteria in order to limit the perpendicular stresses.

This section demonstrates that significant differences exist in the perpendicular strength predicted by the various failure criteria. In addition, the transversely isotropic criteria have an advantage over the orthotropic criteria in that they are more flexible in fitting data. This is demonstrated by comparing the candidate failure criteria with each other and with Hankinson’s formula. Two sets of biaxial stress plots are evaluated:

  • Radial versus tangential stress.
  • Shear versus tangential stress.

The biaxial stress plots provide a more exacting assessment of the perpendicular failure criteria than the off-axis strength predictions previously discussed.

D.5.1 Biaxial Comparisons of Radial Versus Tangential Stress

The biaxial strength of southern yellow pine calculated without the application of shear stress is examined here. The combinations of radial and tangential stresses that satisfy the various failure criteria are compared in the biaxial strength plot of figure 53 for southern yellow pine. Stress states that lie on the horizontal (s R = 0) and vertical (s T = 0) axes are uniaxial stress states. Stress states that do not lie on the horizontal and vertical axes are biaxial stress states. When the radial stress is zero, each curve intersects the horizontal axis twice―once in tension (positive) and once in compression (negative). These intersections are the tangential strengths in uniaxial tension and compression. When the tangential stress is zero, each curve intersects the vertical axis twice―once in tension and once in compression. These intersections are the radial strengths in uniaxial tension and compression. The failure stresses for all criteria are in agreement for states of uniaxial stress. The criteria disagree on what constitutes failure for states of biaxial stress. Here, three biaxial states are examined: equal biaxial compression, equal biaxial tension, and pure shear.

click on the image for Section 508 compliancy text

Figure 53.

Predicted strength of southern yellow pine perpendicular to the grain (no perpendicular shear stress applied).

Equal Biaxial Compression: States of equal biaxial compression and tension (s T = s R) are visualized as a diagonal line extending from the lower left-hand to the upper right-hand corners of the plot. There are three main clusters of curves in biaxial compression (lower left-hand quadrant). First, the Maximum Stress, Norris, and Yamada-Sun criteria predict a biaxial compressive strength equal to the uniaxial compressive strength. Second, the Hoffman criterion predicts a biaxial compressive strength that is much greater (off the scale of the plot) than the uniaxial compressive strength. Third, the Hashin, Modified Hashin, and Tsai-Wu criteria predict biaxial compressive strengths that are less than the uniaxial compressive strength. No strength data are available in biaxial compression for comparison with the failure criteria that do not include contributions from shear stress.

Equal Biaxial Tension: There are two main clusters of curves in biaxial tension (upper right-hand quadrant). First, the Maximum Stress, Norris, and Yamada-Sun criteria predict a biaxial tensile strength that is equal to the uniaxial tensile strength. In addition, the Hoffman criterion predicts a biaxial tensile strength that is approximately equal to the uniaxial tensile strength. The biaxial tensile strength predicted by the Hoffman criterion is sensitive to the strengths used to fit the model. Second, the Hashin, Modified Hashin, and Tsai-Wu criteria predict a biaxial tensile strength that is less than the uniaxial tensile strength. No data are available in biaxial tension for comparison with the failure criteria.

Pure Shear: If wood is assumed to be transversely isotropic, then the R-T plane is the isotropic plane. Mohr’s circle indicates that equal normal stresses of opposite sign (s 22 = –s 33), calculated in the material principal directions, can be transformed into a state of pure shear stress. States of shear can be visualized as a diagonal line extending from the upper left-hand to the lower right-hand corners of the plot. There are three main clusters of curves in shear. First, the Maximum Stress and Yamada-Sun criteria predict a shear strength that is equal to the uniaxial tensile strength perpendicular to the grain. Second, the Hashin, Modified Hashin, and Tsai-Wu criteria predict a shear strength that is equal to the shear strength measured perpendicular to the grain (S^). A value for S^ is not reported in table 4; therefore, it is assumed that S^ is equal to 1.4 times the parallel shear strength (S||) and is thus greater than the uniaxial tensile strength. These transversely isotropic criteria explicitly include s T = –s R as a state of shear with strength S^. Third, the Norris and Hoffman criteria predict shear strengths that are less than the uniaxial tensile strength and less than S^. These criteria do not explicitly recognize s T = –s R as a state of shear stress upon transformation. No strength data are available fors T = –s R for comparison with the failure criteria.

Assessment: One advantage of the transversely isotropic criteria is that the shape of the failure surface is readily modified as a function of S^. Figure 54 displays the Hashin, Modified Hashin, and Tsai-Wu criteria for three different values of S^. Although S^ is hard to measure, its inclusion in the failure criteria is not only realistic, but it provides us with flexibility for modeling failure and yielding in the perpendicular modes. The orthotropic criteria, plotted in this plane, do not vary with S^ and, therefore, lack the flexibility of the transversely isotropic criteria.

click on the image for Section 508 compliancy text

Figure 54.

Shape of the failure surface is sensitive to perpendicular shear strength if the criteria are transversely isotropic.

D.5.2 Biaxial Comparisons of Shear Versus Tangential Stress

The purpose of this section is to determine if the biaxial strength is affected by the application of shear stress. The comparisons indicate that most criteria predict a reduction in biaxial strength with increasing shear stress. No test data are available to validate this trend.

The combinations of shear and tangential stresses that satisfy the various failure criteria are compared in the biaxial strength plots of figure 55. One plot was calculated with S^ = 1.4 MPa and S|| = 23.7 MPa, while the other was calculated with S^ = 0.33 MPa and S|| = 5.58 MPa. This figure allows us to examine the effect of shear stress on perpendicular compressive strength. The only data available to the author for evaluating the failure criteria under states of biaxial stress with shear are off-axis test data, such as that previously shown in figure 2 for Douglas fir. As previously discussed, off-axis strength data include a contribution from shear stress.

The stress combinations are plotted for the specific case of equal biaxial compression (s R = s T). Stress states that lie on the horizontal (s RT = 0) and vertical (s T = 0) axes are biaxial and pure shear stress states, respectively. When the shear stress is zero, each curve intersects the horizontal axis twice―once in tension (positive) and once in compression (negative). These intersections are the biaxial strengths in tension and compression. When the tangential stress is zero, each curve intersects the vertical axis twice―once in tension and once in compression.

These intersections are the pure shear strengths. All of the criteria, except Maximum Stress and Hoffman, predict a reduction in biaxial strength with the application of shear stress.

Also plotted is a solid black dot in the lower left-hand quadrant of each plot where the tangential stress is 30 percent of the uniaxial compressive strength and s TR = s R = s T. This is the biaxial stress state attained in off-axis tests at 45 degrees and predicted by Hankinson’s formula. Recall that Hankinson’s formula is in excellent agreement with test data in the T-R plane for a variety of woods. As expected, all failure criteria predict biaxial strengths that are greater than that predicted by Hankinson’s formula.

click on the image for Section 508 compliancy text

Figure 55.

Combinations of perpendicular and shear stresses that satisfy the failure criteria in the isotropic plane.

Previous | Table of Contents | Next

ResearchFHWA
FHWA
United States Department of Transportation - Federal Highway Administration