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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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1.3 ELASTIC CONSTITUTIVE EQUATIONS

1.3.1 Measured Clear Wood Moduli

Wood guardrail posts are commonly made of southern yellow pine or Douglas fir. The clear wood moduli of southern yellow pine are given in table 1 in terms of the parallel and perpendicular directions. The parallel direction refers to the longitudinal direction. The perpendicular direction refers to the radial or tangential direction, or any normal stress measurement in the R-T plane. These properties were measured as a function of moisture content.(13) Moisture content is more thoroughly discussed in section 1.9.

Table 1. Average elastic moduli of southern yellow pine.
Moisture Content (%) Modulus of Elasticity (MPa) Poisson’s Ratio
Tension Parallel Tension Perpendicular Compression Parallel Compression Perpendicular LT LR
4
16,469
959
18,345
848
0.291
0.158
7
15,559
1,000
17,469
766
0.270
0.133
12
15,503
-
16,572
593
0.260
0.126
18
12,283
552
11,690
414
0.183
0.078
Saturated
11,283
297
7,959
221
0.162
0.138

The clear wood moduli of Douglas fir are given in table 2 as a function of the longitudinal (parallel), tangential, and radial directions. These properties were measured as a function of moisture content.(15) Only compressive moduli were measured.

Table 2. Average elastic moduli of Douglas fir.
Moisture Content (%) Modulus of Elasticity (MPa) Poisson’s Ratio
Compression Longitudinal Compression Tangential Compression Radial LT LR TR
7
16,345
993
959
0.441
0.295
0.368
13
16,414
779
1,062
0.449
0.292
0.374
20
15,193
483
166
0.496
0.274
0.396

1.3.2 Review of Equations

Wood materials are commonly assumed to be orthotropic because they possess different properties in three directions―longitudinal, tangential, and radial. The elastic stiffness of an orthotropic material is characterized by nine independent constants. The nine elastic constants are E11, E22, E33, G12, G13, G23, v12, v13, and v23, where E = Young’s modulus, G = shear modulus, and v = Poisson’s ratio.

The general constitutive relationship for an orthotropic material, written in terms of the principal material directions(16) is:

click on the image for Section 508 compliancy text

Subscripts 1, 2, and 3 refer to the longitudinal, tangential, and radial stresses and strains (s1 = s11, s2 = s22, s3 = s33, e1 = e11, e2 = e22, and e3 = e33, respectively). Subscripts 4, 5, and 6 are a shorthand notation that refers to the shearing stresses and strains (s4 = s12, s5 = s23, s6 = s13, e4 = e12, e5 = e23, and e6 = e13). As an alternative notation for wood, it is common to substitute L (longitudinal) for 1, R (radial) for 2, and T (tangential) for 3. The components of the constitutive matrix, Cij, are listed here in terms of the nine independent elastic constants of an orthotropic material:

This equation reads C subscript 11 equals times the product of E subscript 11 and the quantity 1 minus Impact Velocity subscript 23  times  Impact Velocity subscript 32, all over delta.
This equation reads C subscript 22 equals E subscript 22 times the quantity 1 minus Impact Velocity subscript 31 times Impact Velocity subscript 13, all over delta.
This equation reads C subscript 33 equals E subscript 33 times the quantity 1 minus Impact Velocity subscript 12 times Impact Velocity subscript 21, all divided by delta.
This equation reads C subscript 12 equals  the quantity of the sum of Impact Velocity subscript 21 plus the product of  Impact Velocity subscript 31 times Impact Velocity subscript 23, end quantity, times E subscript 11, all divided by delta.
This equation reads C subscript 13 equals the quantity of the sum of Impact Velocity subscript 31 plus the product of Impact Velocity subscript 21times Impact Velocity subscript 32, end quantity, times E subscript 11, all divided by delta.
This equation reads C subscript23 equals  the quantity of the sum of Impact Velocity subscript 32 plus the product of Impact Velocity subscript 12 times Impact Velocity subscript 31, end quantity, times E subscript 22, all divided by delta.
This equation reads C subscript 44 equals G subscript 12.
This equation reads C subscript 55 equals G subscript 23.
This equation reads C subscript 66 equals G subscript 13.
This equation reads delta equals 1 minus Impact Velocity subscript 12 times Impact Velocity subscript 21 minus Impact Velocity subscript 23 times Impact Velocity subscript 32 minus Impact Velocity subscript 31 times Impact Velocity subscript 13 minus 2 times Impact Velocity subscript 21 times Impact Velocity subscript 32 times Impact Velocity subscript 13.

The following identity, relating the dependent (minor Poisson’s ratios v21, v31, and v32) and independent elastic constants, is obtained from symmetry considerations of the constitutive matrix:

This equation reads Poisson's Ratios over Normal Moduli subscript I equals Poisson's Ratios over Normal Moduli subscript j for I, when j equals 1,2, 3.

Another common assumption is that wood materials are transversely isotropic. This means that the properties in the tangential and radial directions are modeled the same (i.e., E22 = E33, G12 = G13, and v12 = v13). This reduces the number of independent elastic constants to five: E11, E22, v12, G12, and G23. Furthermore, the Poisson’s ratio in the isotropic plane, v23, is not an independent quantity. It is calculated from the isotropic relationship v  = (E 2G)/2G, where E = E22 = E33 and G = G23. Transverse isotropy is a reasonable assumption if the difference between the tangential and radial properties is small in comparison with the difference between the tangential and longitudinal properties.

The wood model formulation is transversely isotropic because the clear wood data in table 1 for southern yellow pine do not distinguish between the tangential and radial moduli. In addition, the clear wood data for Douglas fir in table 2 indicate that the difference between the tangential and radial moduli is less than 2 percent of the longitudinal modulus.

1.3.3 Default Elastic Stiffness Properties

Room-temperature moduli at saturation (23-percent moisture content) are listed in table 3. The same stiffnesses are used for graded wood as for clear wood. For southern yellow pine, the default Young’s moduli are average tensile values obtained from empirical fits to the clear wood data given in table 1 and shown in appendix B. These fits were published by FPL.(13) For Douglas fir, the default Young’s moduli and Poisson’s ratios are also obtained from empirical fits, made by the contractor, to the clear wood data given in table 2. The shear moduli were not measured. For both wood species, the parallel shear modulus is estimated from predicted elastic parameter tables for softwoods found in Bodig and Jayne.(16) The perpendicular shear modulus, G23, is calculated from the isotopic relationship between G23, E22, and v23. For both wood species, the value of the perpendicular Poisson’s ratio used to estimate G23 is that measured for Douglas fir (because no values are listed in table 1 for pine).

Table 3. LS-DYNA default values for the room-temperature moduli (graded or clear wood) of southern yellow pine and Douglas fir at saturation.
Southern Yellow Pine Douglas Fir
E11 or EL
11,350 MPa      
15,190 MPa      
E22 or ET
247 MPa      
324 MPa      
G12 or GLT
715 MPa      
784 MPa      
G23 or GTR
88 MPa      
116 MPa      
v12 or vLT
0.16      
0.39      

1.3.4 Orientation Vectors

Because the wood model is transversely isotropic, the orientation of the wood specimen must be set relative to the global coordinate system of LS-DYNA. The transversely isotropic constitutive relationships of the wood material are developed in the material coordinate system (i.e., the parallel and perpendicular directions). The user must define the orientation of the material coordinate system with respect to the global coordinate system. Appropriate coordinate transformations are formulated in LS-DYNA between the material and global coordinate systems. Such coordinate transformations are necessary because any differences between the grain axis and the structure axis can have a great effect on the structural response.

Keep in mind that the wood grain axis may not always be perfectly aligned with the wood post axis because trees do not always grow straight. It is up to the analyst to set the alignment of the grain relative to the wood post in LS-DYNA simulations.

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