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Publication Number: FHWA-HRT-04-097
Date: August 2007
Measured Variability Of Southern Yellow Pine - Manual for LS-DYNA Wood Material Model 143
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1.4 FAILURE CRITERIA
Strength variations are readily modeled with failure criteria, which are also called yield criteria. Failure criteria relate critical combinations of stresses or strains to failure in a material. Two types of failure criteria are limit and interactive criteria. With limit criteria, like the Maximum Stress criterion, there is no interaction between the stresses, so failure depends on one component of stress or strain. With interactive criteria, like the Hashin criterion, the stresses interact, so failure depends on more than one component of stress or strain. The failure stresses/strains for the interactive and maximum stress/strain criteria typically agree in the material principal directions (uniaxial stress states). The criteria disagree on what constitutes failure in off-axis directions (biaxial and triaxial stress states).
1.4.1 Measured Clear Wood Strengths
Failure criteria are formulated with coefficients that are obtained from fits to measured strengths (peak strength in tension and shear, yield strength in compression). The clear wood strengths of southern yellow pine are given in table 4 in terms of the parallel, perpendicular, and shear directions. The shear strength refers to the parallel-to-the-grain direction. No shear strength was reported for the perpendicular-to-the-grain direction because it is difficult to measure and interpret. The modulus of rupture is calculated from the beam-bending test results, in which the grain runs parallel to the length of the beam. It is not an input parameter of the wood material model. These strengths were measured as a function of moisture content.(13) The saturated data are measured at the fiber saturation point of approximately 23-percent moisture content.
The clear wood strengths of Douglas fir are given in table 5. These strengths were obtained from a variety of sources. No single source provides a complete set of strengths. Some sources distinguish between the radial and tangential directions, while others report strengths in the perpendicular direction. Whenever perpendicular strengths were reported, they were listed under the subheading Tangential for the normal strengths and LT for the shear strengths.
1.4.2 Wood Model Failure Criteria
The strength of wood is modeled as transversely isotropic for a number of reasons. First, the data measured by FPL do not distinguish between the strengths in the tangential and radial directions. Second, the data from Goodman and Bodig suggest that Douglas fir is about 15 percent weaker (compressively) in the radial direction than in the tangential direction.(7) However, this difference in strength is small in comparison with the difference between the parallel and perpendicular directions. Table 4 indicates that the tensile strength measured parallel to the grain is about 30 to 50 times greater than that measured perpendicular to the grain. The compressive strength measured parallel to the grain is about five times greater than that measured perpendicular to the grain.
The wood model failure criterion is formulated from six ultimate strength measurements obtained from uniaxial and pure-shear tests on wood specimens:
Here, X and Y are the strengths parallel and perpendicular to the grain, respectively, and S is the shear strength.
Seven criteria were evaluated for modeling the failure of wood. The theoretical form of each candidate criterion and the graphical comparisons are given in appendix C. A reduced form of the Modified Hashin criterion was chosen for implementation for the following reasons:
The analytical form of the Hashin criterion is different for the parallel and perpendicular modes.
For the parallel modes, the failure criterion is composed of two terms involving two of the five stress invariants of a transversely isotropic material. These invariants are and . This criterion predicts that the normal and shear stresses are mutually weakening (i.e., the presence of shear stress reduces the strength below that measured in the uniaxial stress tests). This form is equivalent to that discussed in appendix C under Modified Hashin or Extended Yamada-Sun. Failure occurs when f|| ≥ 0, where:
For the perpendicular modes, the failure criterion is also composed of two terms involving two of the five stress invariants of a transversely isotropic material. These invariants are I2 = s22 + s33 and . This form is similar to that discussed in appendix C under Modified Hashin, except that two of the three reported terms are retained (the parallel shear stress invariant term (I4) in equation 172 is neglected). This is because its effect on perpendicular failure was not evaluated in appendix C and no test data are available to aid in the evaluation. It is desirable to keep the failure criterion as simple as possible unless measured data suggest otherwise. Failure occurs when f^ ≥ 0, where:
Failure Surface Plots
Four modes of failure are predicted: tension and compression failure parallel to the grain, and tension and compression failure perpendicular to the grain. Parallel shear failure is a subset of the parallel modes and perpendicular shear failure is a subset of the perpendicular modes.
Each failure criterion is plotted in two dimensions in figure 11 in terms of the stress invariants of a transversely isotropic material. Separate plots are drawn for failure or yielding in the parallel and perpendicular modes. Each failure criterion is plotted in three dimensions in figure 12 in terms of the parallel and perpendicular stresses. Each criterion is a smooth surface (no corners).
1.4.3 Default Strength Properties
Room-temperature clear wood strengths at fiber saturation are listed in table 6. Strengths for southern yellow pine are average values obtained from empirical fits to the data previously reported by FPL in table 4 and reproduced in appendix B. Those for Douglas fir are based on the U.S. Department of Agriculture (USDA) Wood Handbook strengths previously reported in table 6.(18) The shear strength perpendicular to the grain has been included as an input parameter even though it was not measured for southern yellow pine. This is because it is included in the failure criterion that was selected. Here it is assumed that the shear strength perpendicular to the grain is 140 percent of the shear strength measured parallel to the grain. This percentage was chosen because the perpendicular shear strength measured by Goodman and Bodig for Douglas fir is 140 percent greater than the parallel shear strength (average of sLT and sLR).(7)
The strength of graded wood posts is less than that of clear wood posts; therefore, the clear wood strengths in table 6 must be scaled down according to grade. One scale factor, QT, reduces the tensile and shear strengths as a function of grade. A second scale factor. QC, reduced the compressive strengths as a function of grade. Default scale factors for grade 1 are QT=0.43 and QC=0.63 for pine, and QT=0.40 and QC= 0.70 for fir. Default scale factors for DS-65 are QT=0.80 and QC=0.93 for both pine and fir. Scale factors by grade are more thoroughly discussed in section 1.12.