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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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APPENDIX C. ANALYTICAL FORM OF CANDIDATE FAILURE CRITERIA

C.1 REVIEW OF CRITERIA

Numerous failure criteria available in the literature were reviewed and evaluated for modeling the yield strength of wood and composite materials. To our knowledge, a validated three-dimensional theory for modeling wood is not documented in the literature. However, numerous failure criteria have been documented for fiber-reinforced plastic (FRP) composites (e.g., see the survey by Nahas(28)). Composites are similar to wood because they are transversely isotropic materials with distinct failure modes in the parallel (fiber) and perpendicular (transverse fiber) directions. Therefore, many of the criteria originally developed for modeling composites were evaluated as candidates for modeling wood.

The functional form of each failure criterion that was evaluated is given in section C.2. These include one limit and six interactive criteria. Both orthotropic and transversely isotropic criteria are reported. All criteria are stress-based criteria. The stresses are transformed to the principal material axes (L-T-R axes) before application of the failure criteria. Strain-based criteria were not evaluated because failure strains are not reported in the literature for wood. One cannot derive failure strains from stresses if the stress-strain behavior is nonlinear, as it is for wood in compression. A brief summary of each criterion is given here:

Maximum Stress (commonly applied limit theory): Failure occurs when any component of stress exceeds its corresponding strength

Tsai-Wu (tensor polynomial theory that was originally developed for anisotropic materials): It contains linear and quadratic stress terms. Seven coefficients must be defined for transversely isotropic applications. The noninteraction coefficients are determined from measured uniaxial and pure shear strengths. By noninteraction, one means terms that contain one component of stress (e.g., F1s11). The interaction terms are determined from measured biaxial strengths. By interaction, one means terms that have two or more components of stress multiplied together (e.g., F12s11s22).

Hoffman: Hoffman extended Hill’s distortional energy criterion for orthotropic materials to account for different strengths in tension and compression. The criterion contains linear and quadratic stress terms. Six coefficients are determined from uniaxial stress and pure shear tests. Biaxial strengths are not needed.

Norris: Norris developed three yield criteria for mutually orthogonal planes. Each criterion contains quadratic stress terms (no linear terms). Nine coefficients are determined from uniaxial and pure shear tests. Tensile strengths are used when the corresponding stresses are tensile. Compressive strengths are used when the corresponding stresses are compressive.

Extended Yamada-Sun: Three yield criteria are reported for mutually orthogonal planes. Each criterion predicts that the normal and shear stresses are mutually weakening (the presence of shear stress reduces the strength below that measured in uniaxial stress tests). Nine coefficients are determined from uniaxial and pure shear tests.

Hashin: Hashin formulated a quadratic stress polynomial in terms of the invariants of a transversely isotropic material. Separate formulations are identified for parallel and perpendicular modes by assuming that failure is produced by the normal and shear stresses acting on the failure plane. In addition, the parallel and perpendicular modes are subdivided into tensile and compressive modes. Assumptions include: (1) biaxial compressive strength perpendicular to the grain is much greater than the uniaxial compressive strength and (2) shear stress does not contribute to compressive failure parallel to the grain. All coefficients are determined from six uniaxial and shear strengths.

Modified Hashin (extended form of Hashin’s criteria): More terms are retained in this modified form than in the original form because fewer assumptions are made regarding material behavior. All coefficients are determined from six uniaxial and shear strengths.

C.2 FORM OF CRITERIA

Here, we give the functional form of the various failure criteria that were evaluated in section C.1 for modeling the strength of wood. Both orthotropic and transversely isotropic criteria are reported.

The following notation is used for orthotropic criteria:
XT Tensile strength in longitudinal direction
XC Compressive strength in longitudinal direction
YT Tensile strength in tangential direction
YC Compressive strength in tangential direction
ZT Tensile strength in radial direction
ZC Compressive strength in radial direction
Sxy Shear strength parallel to the grain in L-T plane
Sxz Shear strength parallel to the grain in L-R plane
Syz Shear strength perpendicular to the grain in T-R plane

The following notation is used for transversely isotropic criteria:
XT Tensile strength parallel to the grain
XC Compressive strength parallel to the grain
YT Tensile strength perpendicular to the grain
YC Compressive strength perpendicular to the grain
S||  Shear strength parallel to the grain
S┴  Shear strength perpendicular to the grain

Here, X, Y, and Z are the strengths in the longitudinal, tangential, and radial directions, respectively, and S is the shear strength. The subscripts T and C refer to the tensile and compressive components, respectively.

Maximum Stress: This is one of the most common limit theories. Failure occurs when any component of stress in the principal material directions exceeds its corresponding strength. Its application to wood as an orthotropic material is:

This equation reads Orthotropic stress component subscript 11 is equal to or greater than parallel tension wood strength when orthotropic stress component subscript 11 is greater than zero.
This equation reads the absolute value of Orthotropic stress component subscript 11 is greater than or equal to parallel compression wood strength when orthotropic stress component subscript 11 is less than zero.
This equation reads Orthotropic stress component subscript 22 is greater than or equal to perpendicular tension wood strength when orthotropic stress component subscript 22 is greater than zero.
This equation reads the absolute value of Orthotropic stress component subscript 22 is greater than or equal to perpendicular compression wood strength when orthotropic stress component subscript 22 is less than zero.
This equation reads Orthotropic stress component subscript 33 is equal to or greater than Capital Z subscript Capital T when orthotropic stress component subscript 33 is less than zero.
This equation reads the absolute value of Orthotropic stress component subscript 33 is equal to or greater than Capital Z subscript Capital C when orthotropic stress component subscript 33 is less than zero.
This equation reads the absolute value of Orthotropic stress component subscript 12 is greater than or equal to shear strength subscript Capital XY.
This equation reads the absolute value of orthotropic stress component subscript 13 is greater than or equal to shear strength subscript Capital XZ.
This equation reads the absolute value of orthotropic stress component subscript 23 is greater than or equal to shear strength subscript capital YZ.

Nine independent modes of failure are predicted: tensile, compressive, and shear failure parallel to the grain; tensile, compressive, and shear failure in the tangential direction; and tensile, compressive, and shear failure in the radial direction. The number of failure modes reduces to six for a transversely isotropic material.

Tsai-Wu: Tsai and Wu developed a stress tensor component polynomial theory as a failure criterion for anisotropic materials.(29) A reduced form of their criterion is applicable to transversely isotropic materials. The criterion contains both linear and quadratic stress terms. Failure occurs when the following equation is satisfied:

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Seven coefficients must be defined for wood modeled as a three-dimensional transversely isotropic material. Six coefficients (F1, F2, F11, F22, F23, and F66) are determined from uniaxial and shear tests on unidirectional specimens. Each of the coefficients F1, F2, F11, F22, and F23 includes contributions from both tensile and compressive strengths:

This equation reads F subscript 1 equals the quotient of 1 divided by parallel wood strength tension minus the quotient 1 divided by parallel wood strength compression.
This equation reads F subscript 2 equals the quotient 1 divided by perpendicular wood strength tension minus the quotient 1 divided by perpendicular wood strength compression.
This equation reads F subscript 11 equals the quotient of 1 divided by the product of parallel wood strength tension times parallel wood strength compression.
This equation reads F subscript 22 equals the quotient of 1 divided by the product of perpendicular wood strength tension times perpendicular wood strength compression.
This equation reads F subscript 66 equals the quotient of 1 divided by parallel shear strength superscript 2.
This equation reads the product of 2 times F subscript 23 equals the quotient 1 divided by perpendicular shear strength superscript 2 minus the quotient of 2 divided by the product of perpendicular wood strength tension times perpendicular wood strength compression.

One coefficient, F12, must be determined from biaxial tests, a variety of which are available. Different biaxial tests produce different values of F12. The choice was to fit F12 to off-axis compression test data at 45 degrees:

This equation reads the product of 2 times Capital F subscript 12 equals the quotient of 1 divided by stress component superscript 2 minus the product of the quotient 1 divided by stress component times parenthesis Capital F subscript 1 plus Capital F subscript 2 parenthesis minus parenthesis Capital F subscript 11 plus Capital F subscript 22 plus F subscript 66 parenthesis.

Here, s is the biaxial strength measured in the principal material directions. It is equal to half the ultimate strength measured in off-axis tests at 45 degrees. If s = XcYc / (Xc + Yc), then the Tsai-Wu model will be in agreement with Hankinson’s two-dimensional formula plotted in ultimate stress versus grain angle space. The failure envelope is a smooth surface in stress space. Only the onset of failure is predicted, not the mode of failure.

Hoffman: Hill generalized von Mises’ distortional energy criterion for isotropic materials to include orthotropic materials.(30) Failure occurs when the following equation is satisfied:

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The six coefficients (A, B, D, E, F, and G) are determined from uniaxial stress and pure shear tests. However, Hill’s orthotropic criterion is not directly applicable to wood materials because it does not model different strengths in tension and compression. Hoffman modified Hill’s quadratic criterion by adding linear stress terms that take into account different strengths in tension and compression.(31) Failure occurs when the following equation is satisfied:

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This equation reads Capital A equals the quotient of 1 divided by the product of 2 times parallel wood strength tension times parallel wood strength compression plus the quotient of 1 divided by 2 times perpendicular wood strength tension times perpendicular wood strength compression minus the quotient 1 divided by 2 times Z tension times Z compression.
This equation reads Capital B equals the quotient of negative 1 divided by the product of 2 times parallel wood strength tension times parallel wood strength compression plus the quotient of 1 divided by the product of 2 times perpendicular wood strength tension times perpendicular wood strength compression plus the quotient of 1 divided by 2 times Capital Z tension times Capital Z compression.
This equation reads Capital C equals the quotient of 1 divided by the product of 2 times parallel wood strength tension times parallel wood strength compression minus the quotient 1 divided by the product of 2 times perpendicular wood strength tension times perpendicular wood strength compression plus the quotient of 1 divided by the product of 2 times Z tension times Z compression.
This equation reads Capital D equals the quotient 1 divided by the product of shear strength times x times y superscript 2.
This equation reads Capital E equals the quotient of 1 divided by shear strength times x times z superscript 2.
This equation reads Capital F equals the quotient of 1 divided by shear strength times y times z superscript 2.
This equation reads Capital G equals the quotient of 1 divided by parallel wood strength tension minus the quotient 1 divided by parallel wood strength compression.
This equation reads Capital H equals the quotient of 1 divided by perpendicular wood strength tension minus the quotient 1 divided by perpendicular wood strength compression.
This equation reads Capital I equals 1 divided by Capital Z tension minus the quotient 1 divided by Capital Z compression.

This criterion predicts a parabolic increase in strength with confining pressure. The nine coefficients are determined from uniaxial stress and pure shear tests. The criterion is readily simplified for materials with transversely isotropic strength values. One advantage of this criterion is that the interaction terms are not based on biaxial data, so it is easier to fit than the Tsai-Wu criterion. One disadvantage of this criterion (and the Tsai-Wu criterion) is that the onset of failure is predicted, but not the mode of failure.

Norris: Tsai and Azzi simplified the Hill criterion to account for transverse isotropy and plane stress conditions of composite materials:(32)

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Tsai showed that the criterion is applicable to composites with different properties in tension and compression. Tensile strengths are used when the corresponding stresses are tensile; compressive strengths are used when the corresponding stresses are compressive. Tsai also developed two additional equations for mutually orthogonal planes (similar to equation 153) for failure analysis of three-dimensional materials.(27)

Similarly, Norris reports three yield criteria for mutually orthogonal planes. His criteria are similar to the Tsai-Azzi criteria except that the interaction terms are not biased toward one particular strength. In addition, he applied his criteria to wood materials, not composites. Norris’s criterion for the 1-2 (L-R) plane is as follows:

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Similar equations can be written for the 1-3 and 2-3 planes by proper interchange of subscripts. This criterion for modeling wood was evaluated by using tensile strengths when the corresponding stresses are tensile, and compressive strengths when the corresponding stresses are compressive. For each of the three equations, three combinations of tensile and compressive stresses are possible. Therefore, nine modes of failure are modeled.

Extended Yamada-Sun: Yamada and Sun developed a plane stress criterion for composites that is a degenerative form of the Tsai-Azzi and Norris criteria.(33) Failure occurs when the following equation is satisfied:

This equation reads the quotient of Orthotropic stress component subscript 11 superscript 2 divided by general parallel wood strength superscript 2 plus the quotient of orthotropic stress component subscript 12 superscript 2 divided by shear strength superscript 2 is greater than or equal to 1.

This criterion predicts that the normal and shear stresses are mutually weakening (the presence of shear stress reduces the strength below that measured in uniaxial stress tests). We extended this concept to three dimensions for application to wood as either an orthotropic or transversely isotropic material:

This equation reads the quotient of Orthotropic stress component subscript 11 superscript 2 divided by general parallel wood strength superscript 2 plus the quotient of orthotropic stress component subscript 12 superscript 2 divided by shear strength subscript XY superscript 2 plus the quotient of orthotropic stress component subscript 13 superscript 2 divided by shear strength subscript XZ superscript 2 is greater than or equal to 1.
This equation reads the quotient of Orthotropic stress component subscript 22 superscript 2 divided by general perpendicular wood strength superscript 2 plus the quotient of orthotropic stress component subscript 12 superscript 2 divided by shear strength subscript XY superscript 2 plus the quotient of orthotropic stress component subscript 23 superscript 2 divided by shear strength subscript YZ superscript 2 is greater than or equal to 1.
This equation reads the quotient of Orthotropic stress component subscript 33 superscript 2 divided by Capital Z superscript 2 plus the quotient of orthotropic stress component subscript 13 superscript 2 divided by shear strength subscript X lowercase Z superscript 2 plus the quotient of orthotropic stress component subscript 23 superscript 2 divided by shear strength subscript Y lowercase Z superscript 2 is greater than or equal to 1.

Tensile strengths are used when the corresponding stresses are tensile; compressive strengths are used when the corresponding stresses are compressive. Six independent modes of failure are predicted: tensile and compressive failure in the longitudinal direction, tensile and compressive failure in the tangential direction, and tensile and compressive failure in the radial direction. The application of shear stress contributes to failure in each of these modes and is mutually weakening.

Hashin: The Tsai-Wu and Hoffman interactive failure criteria predict when a given set of stresses will produce failure, but they do not predict the mode of failure. Hashin developed a set of interactive failure criteria in which distinct failure modes are modeled. He applied his failure criteria to fiber composite materials. Since most fiber composites are transversely isotropic (e.g., wood), Hashin defined a general failure criterion in terms of the stress invariants of a transversely isotropic material (27). The five stress invariants (I1, I2, I3, I4, and I5) are:

This equation reads Capital I subscript 1 equals orthotropic stress component subscript 11.
This equation reads Capital I subscript 2 equals orthotropic stress component subscript 22 plus orthotropic stress component subscript 33.
This equation reads Capital I subscript 3 equals orthotropic stress component subscript 33 superscript 2 minus the product of orthotropic stress component subscript 22 times orthotropic stress component subscript 33.
This equation reads Capital I subscript 4 equals orthotropic stress component subscript 12 superscript 2 plus orthotropic stress component subscript 13 superscript 2.
This equation reads Capital I subscript 5 equals the product of 2 times orthotropic stress component subscript 12 times orthotropic stress component subscript 23 times orthotropic stress component subscript 13 minus the product of orthotropic stress component subscript 22 times orthotropic stress component subscript 13 superscript 2 minus the product of orthotropic stress component subscript 33 times orthotropic stress component subscript 12 superscript 2.

Hashin’s three-dimensional failure criterion is a quadratic stress polynomial of the general form:

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The I5 invariant does not appear in the criterion because only linear and quadratic terms are retained in this polynomial.

To identify distinct failure modes, Hashin argued that failure is produced by the normal and shear stresses acting on the failure plane. For failure parallel to the grain, the failure plane is the 2-3 plane, acted on by stresses s11, s12, and s13. The perpendicular stresses (s22, s23, and s33) do not contribute to parallel failure. The implicit assumption here is that the perpendicular stresses do not impede compression bucking; thus, an interaction mechanism is not required (the term C12I1I2 is neglected).

In perpendicular-to-the-grain failure, failure occurs in any plane with axes parallel and perpendicular to the grain. The failure plane is acted on by stresses s22, s33, s23, s12, and s13. The implicit assumption here is that the stress parallel to the grain (s11) does not contribute to perpendicular failure because this stress is carried almost entirely by the fibers.

By applying these assumptions to the general criterion in equation 164, Hashin developed specific yield criteria for the parallel and perpendicular modes:

Parallel Mode

This equation reads the product of Capital A subscript F times orthotropic stress component subscript 11 plus the product of parallel softening parameter subscript F times orthotropic stress component subscript 11 superscript 2 plus the quotient of parenthesis orthotropic stress component subscript 12 superscript 2 plus orthotropic stress component subscript 13 superscript 2 parenthesis divided by parallel shear strength superscript 2 equals 1.

Perpendicular Mode

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Failure mechanisms are different for tensile and compressive modes, so Hashin further divided each criterion into tensile and compressive modes.

Tension Parallel: Hashin assumed that tensile and shear stresses are mutually weakening; therefore, both contribute to tensile failure. Data from direct pull and biaxial tests are needed to solve for both coefficients (Af and Bf). If data from a direct pull test are the only data available, then one can solve for either Af or Bf. The failure surface remains smooth and elliptical if one solves for Bf and neglects Af. Failure occurs when the following equation is satisfied:

This equation reads the quotient of Orthotropic stress component subscript 11 superscript 2 divided by parallel wood strength tension superscript 2 plus the quotient of parenthesis orthotropic stress component subscript 12 superscript 2 plus orthotropic stress component subscript 13 superscript 2 parenthesis divided by parallel shear strength superscript 2 is greater than or equal to 1 when orthotropic stress component subscript 11 is greater than zero.

This criterion is the same as our extension of the Yamada-Sun criterion.

Compression Parallel: If compressive and shear stresses are assumed to be mutually weakening, then one can develop a compression parallel criterion similar to that for tension. However, Hashin argued that there is no physically reasonable method for including the effect of shear stress, at least for composites. Therefore, he represents parallel compressive failure in simple maximum stress form:

This equation reads Orthotropic stress component subscript 11 is greater than or equal to parallel wood strength compression when orthotropic stress component subscript 11 is less than zero.

Tension Perpendicular: Data from direct pull and biaxial tests are needed to solve for both coefficients (Am and Bm). If data from a direct pull test are the only data available, then one can solve for either Am or Bm. The failure surface remains smooth and elliptical if one solves for Bm and neglects Am. Failure occurs when the following equation is satisfied:

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Compression Perpendicular:Hashin argued that the biaxial compressive strength (Ycc) of composites is much greater than the uniaxial compressive strength (Yc). Therefore, he solved for both coefficients (Am and Bm) and retained only first-order terms in Yc /Ycc. Failure occurs when the following equation is satisfied:

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The Modified Hashin criteria predict four independent modes of failure: tensile and compressive failure parallel to the grain, and tensile and compressive failure perpendicular to the grain. Although Hashin applied these criteria to fiber composites, these criteria were also evaluated for wood. A plane stress version is currently implemented in model 22 of LS-DYNA, along with an augmentation by Chang that takes into account nonlinear shear stress-strain behavior.

Modified Hashin: In addition to the composite criteria proposed by Hashin, a simple modification of the Hashin criteria was evaluated.

Tension and Compression Parallel: Here, it is assumed that shear stress weakens wood in compression as well as in tension. In this case, the tensile and compressive yield criteria have the following form:

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Tension and Compression Perpendicular: It is not clear whether the biaxial compressive strength of wood is greater or lesser than the uniaxial compressive strength. The only available data for southern yellow pine are uniaxial stress data, so no assumptions are made regarding the relative strengths in biaxial and uniaxial compression. In this case, the tensile and compressive yield criteria have the following form:

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The Modified Hashin criteria predict four independent modes of failure: tensile and compressive failure parallel to the grain, and tensile and compressive failure perpendicular to the grain.

Hankinson: Hankinson developed an empirical formula that is frequently applied to off-axis wood tests in two dimensions.(16,34) Goodman and Bodig extended the formula to three dimensions.(7) Their three-dimensional formula predicts the ultimate compressive strength (sult) of wood relative to the both the grain (q) and ring angles (f). The three-dimensional formula has the following form:

This equation reads Stress component subscript ULT equals the product of general parallel wood strength times the quotient of Capital F subscript lowercase phi divided by the sum of sine superscript 2 Capital theta plus F subscript lowercase phi cosine superscript 2 lowercase theta.

where Ff varies sinusoidally with the ring angle between the relative strength in the tangential direction (FT = Y/X) and the relative strength in the radial direction (FR = Z/X). Ff has the following form:

This equation reads F subscript lowercase phi equals F subscript transverse plus the product of lowercase phi times the quotient of parenthesis F subscript radial minus F subscript transverse parenthesis divided by 90 plus the product of Fracture intensity times parenthesis negative sine of the product of 2 Capital Theta parenthesis times the quotient of parenthesis F subscript radial plus F subscript transverse parenthesis divided by 2.

Here, K is an empirical constant that is typically 0.4 for softwoods. The last term on the right of 172 is a sinusoidal correction to the straight line interpolation between the tangential (Y) and radial (Z) strengths. This correction is illustrated in figure 45. It can be compared with the data shown later in figure 48.

Comparisons of Hankinson’s three-dimensional formula with Douglas fir test data were previously shown in figure 2. Hankinson’s formula predicts large reductions in strength when the load is inclined at a small angle to the grain or ring, in agreement with test data.

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Figure 45.

Compressive strength variation of clear wood is readily modeled by a sinusoidal correction in the R-T plane.

Source: Krieger Publishing Company (16).

For a transversely isotropic material with no variations in strength with the ring angle (FT = FR = Ff), the three-dimensional formula reduces to its two-dimensional form:

This equation reads Stress component subscript ULT equals the quotient of the product of general parallel wood strength times perpendicular wood strength divided by the sum of general parallel wood strength sine superscript 2 Capital Theta plus general perpendicular wood strength cosine superscript 2 Capital Theta.

Although Hankinson’s formula has been shown to provide good fits to off-axis test data, it is not a general-purpose formulation that can be applied to other types of tests. Thus, it is not suitable for use in finite element codes. However, it is reported here so that it can be compared with other criteria to help in their evaluation.

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