DEVELOPING A GRADE STAMP
FOR RECYCLED TIMBER
FINAL REPORT

Report No. CDL-99-2

 

Table 2

Knot Limits for Douglas Fir Structural Joists and Planks

per WCLIB Grade Rules (Applicable to Middle Third of Beam Length)

                                                                                               

 

3.0     BENDING TESTS

The timbers were tested in a 1,000,000 lb. capacity Universal Twin Screw testing machine under third point bending according to American Society of Testing Materials standards  (ASTM D198, 1994).  A span-to-depth ratio of approximately 2:1 was used.  The bolt holes were mapped and the influence of these defects on ultimate strength was monitored. 

Figure 8 indicates the test set-up used for the 166 beams.

 

Figure 8    Laboratory Set-up for Testing of Beams

For each beam, load and displacement were recorded continuously throughout the test.  From the load-displacement curve, modulus of elasticity (MOE) and modulus of rupture (MOR) were determined according to standard procedures.  Failure patterns were recorded for each beam.  Figure 9 shows load-displacement curves for Group C beams (control).

 

Figure 9

Experimental Load-Displacement Curve

Group C - Control, n=22

The failures exhibited by the beams varied depending on the location of the hole in the beam. Generally, if the hole was on the tension side of the beam, tension failure typically initiated at the hole (see Figure 10).  If the hole was on the compression side of the beam, local buckling occurred above the hole (Figure 11), often followed by a load increase and general tension side failure in the beam.

 

Figure 10

Failure at Hole on Tension Side of Beam

 

Figure 11

Localized Buckling at Hole on Compression Side of Beam

(stiffness) of the tested beams was relatively unaffected by the presence of the holes.

Figure 12

Box Plot of MOR by Group

 

Figure 13

Box Plot of MOE by Group

4.0     ANALYTICAL MODEL FOR RECYLCED TIMBERS

The development and calibration of an analytical model was performed by the Civil and Environmental Engineering Department at Washington State University.  The purpose of the proposed analytical work was to develop and calibrate a finite element model for recycled timbers, which would predict the effects of bolt holes on timber strength.  This model was used to establish critical hole sizes and locations, information necessary in the development of appropriate grade criteria.

4.1     Strength Theory Applied to Failure in Wood

In order to analytically predict the ultimate failure strength of each tested beam, a tensor polynomial (Tsai-Wu) strength theory for anisotropic materials was utilized.  The Tsai-Wu yield criterion is an operationally simple theory that is based on a scalar function of two strength tensors (Tsai 1971).  Originally developed to predict the failure of filamentary composite materials, the strength theory has several advantageous attributes that differ from other existing failure criteria.  Specifically, all of the invariant requirements of coordinate transformations are satisfied.  The general form for anisotropic materials can be specialized to account for different material symmetries, including orthotropic materials.  This takes into account differences between tensile and compressive stress and accounts for multi-axial states of stress.  

The general form of the failure surface proposed by Tsai-Wu (1971) has the scalar form:

F_is_i + F_ijs_is_j = 1 i,j = 1,2,…,6                                   (1)

where F_i and F_ij are strength tensors of the second and fourth rank, respectively, and s_i and s_j are the related applied stresses. 

Derived from the general form of equation (1), the specialized case for an orthotropic material such as wood becomes:

F₁s₁ + F₂s₂ + F₁₁s₁² + +2F₁₂s₁s₂ + F₂₂s₂² + F₆₆s₆² < 1              (2)

for the plane-stress state where s₃ = s₁₃ = s₂₃ = 0.  Furthermore, the 1-direction is the perpendicular orientation and the 2-direction is parallel to grain.  Note that failure is defined as the point at which the value of the scalar Tsai-Wu coefficient reaches or exceeds 1.0.  Liu (1984) and Hasebe et al. (1987) proved that the Tsai-Wu criterion is applicable to wood.  Liu determined that the Tsai-Wu theory is reasonably accurate when used to predict failure in solid sawn wood under combined states of stress.  This is found to be true when the normal stress interaction term, F₁₂, is based on the Hankinson formulas under the plane stress conditions.  Similarly, Hasebe and Seizou  (1987) successfully used the Tsai-Wu strength theory to determine mechanical properties of Japanese cedar.  Assuming that the stress perpendicular to grain is negligible, equation (2) was modified to a plane stress condition where the ultimate tensile, compressive and shearing strength values were investigated.

4.2     Strength Prediction of Wood and Failure Modeling

Combining the techniques of Finite Element Analysis (FEA) and strength theory to predict the ultimate load capacity has been proven to be a valuable tool for both solid sawn wood and wood composites.  Leichti and Tang (1989) successfully developed a plane stress, linear elastic model that predicted the ultimate load capacity of wood composite I-beams with the use of FEA and the Tsai-Wu strength theory.  Using a stepwise technique that increased the applied load at intervals, ultimate load capacity was determined.  Different web materials and web joint configurations were evaluated and “weak” regions and areas of high stress concentration were identified.

Similarly, Cramer and Goodman (1983) developed a FEA model that predicted the ultimate tensile strength of solid lumber containing circular knots and cross grain.  The objective of the study was to determine the effect of knot location on the resulting stress field under a uniformly applied tension stress, parallel to the long axis of the member.  Using the stepwise analysis technique, the researchers were able to predict the location of the initial failure based upon a maximum stress theory.  Failure was assumed to take place when the maximum tensile stress in the parallel- or perpendicular-to-grain direction exceeded the clear wood strength value.  Also, the researchers utilized an “effective section technique” to model ultimate failure as a progressive series of initial failures.  It is assumed that, upon initial failure, a crack would form and propagate along the grain to the end of the modeled segment.  To account for the failed section of the member, the region below the knot was assumed to be ineffective in resisting any load.  Therefore, a net effective section was used in the subsequent analysis.  This process was used both for knots located near the edge and knots near the center.  The authors determined that the stepwise analysis was in accordance with the experimental data and that knots located near the edge of the member caused the largest longitudinal stress concentrations.  Of the entire set of knot locations considered, knots near the edge, but completely encased in the member, caused the highest stress concentrations.  Conversely, the lowest stress concentration occurred when the knot was located along the center of the member.  Similarly, Cramer (1986) developed a finite element/fracture mechanics algorithm that predicted the ultimate tensile capacity of solid lumber with knots located within the wide face of the member.  The objective of this research was to model the progressive fracture process of failure in wood members and to quantify the strength-reducing effects of knots located within the wide face of the member. 

4.3     Modeling Procedures

Seven member configurations were modeled corresponding to the tested beam groups A, B, C, D, E, H, and I.  For the purpose of this research, the fastener hole was located along the midspan of the beam to represent the connection pattern that would create the greatest section reduction relative to flexural resistance.  The fastener hole sizes were 25.4 mm  (1-in.) and 44.5 mm (1-3/4 in.) in diameter.  The 25.4 mm (1-in.) hole represents the largest allowable bolt hole diameter (NDS, 1997), whereas the 44.45 mm (1-3/4 in.) hole represents half of the largest allowable edge knot on the wide face of a No. 2 structural joist and plank (WCLIB, 1990).

Also, the supports and end conditions used in the model were the same as those used in the beam testing.  Each 101mm x 203mm (4 in. x 8 in.) specimen was 3.65 m (12 ft) long with a 3.35-m (11 ft.) clear span.  The shear span-to-depth ratio was a/h = 6.1, well within the allowable range of a/h = 5 to 12 (ASTM D198).

4.4     Model Generation  

In order to generate and analyze a model, two different finite element software packages were used.  The model geometry and mesh (nodes and elements) were created using COSMOS^Ò and then reformatted to use ABAQUS^Ò  (Hibbitt et al., 1997).  All material properties, boundary condition specifications, member thickness, and analysis commands were then defined by ABAQUS^Ò.  All analyses were performed in ABAQUS^Ò.  All analyses were two-dimensional, plane stress models.  A concentrated load was applied at the third point along the length of the beam, while boundary conditions were applied to each node located along the midspan to constrain the member in the longitudinal direction.  Using symmetry, a boundary condition was applied at the support to constrain the member in the vertical direction.  As a result of the presence of a fastener hole, there was a discontinuity in the stress flow.  Therefore, localized stress concentrations formed around the outer edges of the hole.  Thus, to increase the accuracy of results, a mesh was generated with a fine region around the hole and coarser meshes radiated outwards towards the support, Figure 14.

 

 

Figure 14

Typical Finite Element Mesh

4.5     Elements

All models consisted of eight-node biquadratic plane stress elements.  Due to the nature of the curved surface around the hole and the radiating mesh, the eight-node elements were very advantageous.  The isoparametric formulation made it possible to generate elements that were nonrectangular and had curved sides.  These shapes have obvious uses in grading a mesh from coarse to fine and in modeling curved boundaries (Cook et al., 1989).  To evaluate stresses within each element, numerical integration was performed using the Gauss quadrature procedure.  Nine sampling points, or integration points, were considered within each element.

4.6     Material Properties

Material properties for the dry Douglas-fir larch (coastal) wood were obtained from the Wood Handbook  (FPL, 1991).  All analyses were completed with the same properties as: 

 

E_L    = 13.4 x 10⁹ Pa                  G_LT  = 1.05 x 10⁹ Pa                 n_LT   = 0.45

E_R    = 0.8 x 10⁹ Pa                   G_LR  = 0.0                                 n_LR   = 0.0

E_T    = 0.8 x 10⁹ Pa                   G_RT  = 0.0                                 n_RT  = 0.0

Note that 1.0 psi = 6894.8 Pa.  The values of E_T and E_R were computed as the average of the published modulus of elasticity (MOE) values in the tangential (T) and radial (R) directions, respectively.  Also, note that the magnitude of G_LR, G_RT, n_LR and n_RT were taken as zero, due to the plane stress configuration of the analytical model.

Failure stress limits for Douglas-fir were also obtained from the Wood Handbook (FPL, 1991).  The stress limits are: 

                                               

s_Lt = 85.5 x 10⁶ Pa

                                                            s_Lc = 49.8 x 10⁶ Pa

                                                            s_Rt = 2.3 x 10⁶ Pa

                                                            s_Rc = 5.5 x 10⁶ Pa

                                                            t₁₂ = 7.8 x 10⁶ Pa

where:

            s_Lt = ultimate tensile stress in the longitudinal direction;

            s_Lc = ultimate compressive stress in the longitudinal direction;

            s_Rt = ultimate tensile stress in the radial direction;

            s_Rc = ultimate compressive stress in the radial direction;

            t₁₂ = ultimate shear stress in the longitudinal-radial direction

4.7    Initial Models

Initial analyses completed on ABAQUS^Ò did not utilize the load-stepping method.  Hence, results obtained from the original tests provided strength or capacity values that were approximately 20-30% lower than results using the load-stepping method.  Similarly, failure was defined when the Tsai-Wu coefficient equaled or exceeded 1.0 when evaluated at the integration points.  However, it was assumed that load-carrying capacity ended when only one of the nine integration points in an element exceeded the allowable Tsai-Wu coefficient.  Thus, the initial results were too conservative compared to actual response.

4.8     Failure Prediction

In order to analytically predict the ultimate flexural capacity of each beam, a load-stepping method was utilized.  Inherent in this technique was an underlying assumption:  the failure process occurred incrementally, as a progressive series of small, localized failures.  Utilizing the principal of superposition, this method approximately modeled the propagation of cracks and load redistribution throughout the member during the loading event. 

A complete analysis was a multi-step process.  Every “step” included generating a finite element mesh and then analyzing the behavior of the model when subjected to a specific magnitude of load.  The initial model was analyzed to determine the maximum load capacity when there were no elements removed (Figure 15).

Figure 15

Typical Initial Model with No Elements Removed

 

A visual inspection of initial model results determined if and where the intensified stress regions were located.  Failure of an element was defined as the point at which all nine integration points in an element had exceeded the failure coefficient of 1.0, as defined by the Tsai-Wu failure theory (Tsai-Wu, 1970).  Thus, if an element or elements failed, that portion of the model was removed from the mesh to create a resultant net “effective section”.  The effective section was then used in the subsequent analysis step (Figure 16).  The maximum load that produced the initial failure response was assumed to be the linear response.  Responses beyond that elastic limit, in which elements were removed, were considered to be the non-linear response.  The “load stepping” method was repeated until entire failure of the member occurred or the deflections became excessive.  

 

Figure 16

Typical “Effective Section” after Element Removal

4.9     Model Verification

Using the load-stepping method as described earlier, load-deflection curves were generated (for each beam group).  These results were compared to the experimental load-deflection curves generated from the experimental testing.  If the load capacity and deflection response was for the model, and test results were similar throughout the loading event, then it was assumed that the numerical model accurately predicted the response. 

The capacity of the recycled timbers was reported as a strength ratio.  A strength ratio was derived as the ratio of moment-carrying capacity of a member, with cross-section reduced by the largest fastener hole to the moment-carrying capacity of the member without the defect.  This procedure was identical to the strength ratios published in ASTM D245-93, “Establishing Structural Grades and Related Allowable Properties for Visually Graded Lumber.”  The strength ratio provided an accurate and efficient method of assessing the change in capacity when a defect such as a fastener hole was present.  The strength ratio provided a measure of the relative effect the different hole configurations had on beam capacity.  Thus, a member with a strength ratio = 0.80, had 80% of the capacity of a beam with no hole. 

4.10     Analytical Results and Discussion

Results obtained from the analytical tests and the results of the verification of the model are presented.  Failure prediction results were revised using the new definition of failure, and the results of the load-stepping method are presented.  A comprehensive comparison of the analytical and experimental data is given, and the results of the calibration are also outlined.

4.11     Final Models

One of the main goals of this research project was to develop an analytical model that could successfully and accurately predict the ultimate flexural capacity of recycled, or pseudo-recycled, No. 2 Douglas-fir 4 x 8.  To accomplish this, finite element models were analyzed using the finite element software ABAQUS^Ò.  The results of each failure load obtained by the models are listed in Table 3.

Table 3

Comparison of Model Results and Experimental Test Results

 

Although the analytically derived failure load values were lower than the mean of the experimental values, all analytical values were within one standard deviation of the mean value.  Failure in cases B, D and H were all very similar.  Localized stress concentration developed at the top and bottom of each hole due to the discontinuity in the stress flow around the hole.  The maximum load values obtained from the analytical model were lower than the mean value from the experimental test program; however, the behavior of the failure around the hole and the overall effect of the hole were successfully modeled.  In addition, there was only a 13% average margin of error between the numerical results and experimental results.  The trends of the strength ratio from the analytical tests were similar to what was reported in the experimental tests.  The lowest reduction in capacity was seen in cases A, B and I; whereas, the largest reductions were in Cases D, E and H.  Note that the value of the predicted maximum load for the control model is very close to the experimental value.

The main goal of the load-stepping technique was not only to predict accurate failure capacities, but also to attempt to model the material behavior during a loading event.  Therefore, with the new finite element results, load-deflection curves were plotted to compare the behavior of the model with the experimental load-deflection curves.  Figures 17-23 illustrate the load-deflection curves obtained from the finite element analyses.  Each load-deflection curve was plotted with the load-deflection data obtained from the entire sample of specimens tested for each respective case.

Figure 17

Load-Deflection Curves for Group C (Control)

 

Figure 18

Load-Deflection Curves for Group A

Due to the fact that all analyses were linear elastic models, the control model did not use the load-stepping method.  The initial failure of the model is the ultimate failure.

 

Figure 19

Load-Deflection Curves for Group B

Figure 20

Load-Deflection Curves for Group D

 

Figure 21

Load-Deflection Curves for Group E

 

Figure 22

Load-Deflection Curves for Group H

 

Figure 23

Load-Deflection Curves for Group I

Figure 24

Graph of the Strength Ratios vs. Size and Location

which the first element/elements were removed for the next load-step.  When the hole was within the compression region, failure initiated at the top of the hole.  The larger fastener holes, 44.5 mm, produced a greater reduction in capacity than the 25.4 mm hole (Figures 24-25).

In summary, the behavior of the analytical model was similar to that of the experimental data. This was evident due to the fact that the load-deflection curve from the analytical test was located within the middle of the experimental data.  Even though the magnitude of the predicted failure load was slightly lower than experimental values, the model was able to accurately provide a conservative estimate of the ultimate flexural capacity with the presence of a fastener hole. 

Figure 25

Typical Stress Contour around a Hole

5.0     AMENDMENT TO EXISTING GRADING RULES

The WCLIB has taken the lead in establishing appropriate grade criteria and developed a grade stamp that indicates if a piece of lumber was recycled.  While this effort is listed as the first task of the study, chronologically it was the last step to be performed.  Decisions regarding the grading criteria could not be made until both the experimental testing and analytical modeling were completed

As a first step, a draft supplement to existing WCLIB grading rules, specifically accommodating recycled timber, was written.  The intent of this task was to verify the appropriateness of the currently assigned allowable properties of Select Structural and No. 1 Beams & Stringers (B&S) and Post & Timbers (P&T).  The information developed in Tasks 2 and 3 on the effects of bolt holes on timber performance were incorporated into the developed grade criteria.

The test results indicated that, for the hole configurations tested, the effects on bending strength when compared to the control group, effected the average bending strength by approximately 10 - 30%.

Based upon the testing and the modeling results, two important questions arose:   (1) Should there be separate grading provisions for recycled wood added to Standard Grading Rule No. 17; and (2) Should there be hole allowances for Select Structural and No. 1 Beams & Stringers and Post & Timbers added to Standard Grading Rule No. 17?

5.1     Current Hole Limits for Beams and Stringers and Post and Timbers

Currently, timbers can not be grade stamped as Select Structural (SS) or No.1 Structural (No. 1) under paragraphs 130 (Beams & Stringers) or 131 (Post & Timbers) of Standard Grading Rules No. 17, unless all holes are plugged or filled according to paragraph 736 of Standard Grading Rules No. 17.  One exception to this rule is cedar species, which are limited to holes one-half the size of allowable knots for grade stamping as No. 1 B&S and P&T.  No. 2 Structural B&S and P&T are allowed to have holes the same size as knots.

The current design values for B&S and P&T grades were developed by WCLIB and approved by the American Lumber Standard Committee in 1970.  These design values were based on two ASTM standards: ASTM D 2555 and ASTM D 245.  ASTM D 2555 contains tables with a summary of the clear wood property values and species distribution data.  ASTM D 245 developed allowable stress values based on tests of small clear-wood lumber that have been sampled from various growing regions for the species.  These stresses were then grouped and weighted by region and species (in the case of species groups such as Hem-Fir) and then further adjusted by various volume factors, moisture content, stress ratios, and other factors to derive allowable design values.  One of these factors involved determining the minimum stress ratio for the grade.

For bending, stress ratios were based on the maximum slope of grain, allowable edge, narrow-face, and wide-face knots.  Holes were treated the same as knots.  These stress ratios are expressed in percentages that are multiplied by the adjusted clear wood stresses.  For SS and No. 1, these are:

Table 4

Stress Ratios Used to Develop Allowable Bending Values

for Beams & Stringers and Posts & Timbers*

*The stress ratio in parentheses have been adjusted for size per ASTM D 245.

Holes were limited in dimension grades to the equivalent size of knots or smaller.  The smaller hole limit in some grades are for appearance purposes as explained in Standard Grading Rule No. 17, paragraph 2-f.

Table 5

A Comparison of the Modulus of Rupture (MOR) of

Tested Beam Groups at Selected Percentiles

The percent difference was calculated by dividing the test group MOR by the group “C” MOR.  For example, the 10^th percentile MOR for group “A” was 5,272 psi, while the 10^th percentile for group “C” was 5,227 psi.  The percent difference was:

(5,272 / 5,227) x 100 = 100.9%

When these percentages were plotted, the plot indicated a wide dispersion at the lower percentiles that tended to tighten as the percentiles increased.  This dispersion was a function of material variability and sample size.

If equivalent stress ratios were calculated for the test groups, based only on holes, they would be as follows:

Table 6

Stress Ratios for Test Groups Based of Hole Size and Location

¹The stress ratio in parentheses

  have been adjusted for size per

  ASTM D 245.

5.2     WCLIB Board of Director’s Actions

A proposal, including revisions to portions of paragraphs 130 and 131 of Standard Grading Rule No. 17, was submitted for approval to the WCLIB Board of Directors.  Appendix A includes the proposed revision.  This revision permits the grade stamping of B&S and P&T as “recycled” and limits holes to one-half the allowable knot in recycled SS and No.1 grades. The WCLIB Board of Directors approved this revision at the September 1998 meeting.

5.3     ALSC Board of Review Actions

Following acceptance by the WCLIB Board of Directors, a proposed a revision was sent to the National Grading Rules of American Lumber Standard reflecting the change in grading criteria.  The proposed revision was submitted for the fall American Lumber Standards Committee (ALSC) Board of Review Meeting (See September 14, 1998 letter, Appendix A).  The revision was then sent out to ALSC members for review.  The Western Wood Products Association (WWPA) offered the only negative response to this revision (See September 28, 1998, Appendix A).  The WCLIB's letter in response to this criticism is given in Appendix A (dated September 30, 1998).  The WWPA expressed reservations about how the load history of the recycled wood might impact the residual strength of the pieces and declined acceptance.  The ALSC's Board of Review official response to the submittal is also included in Appendix A (dated October 27, 1998).

Therefore, it is the decision of the WCLIB technical staff that the proposed revisions to Standard Grading Rule No. 17 will not be made at this time.  This leaves the remanufacturers of recycled lumber with the option of filling the holes in SS and No. 1 B&S and P&T if the pieces are to be grade stamped as SS or No. 1.  The filling of holes is allowed according to paragraph 736, Plugs and Fillers, of Standard Grading Rule No. 17.  It is hoped that when sufficient data is collected on the engineering performance of recycled lumber strength,  the proposal can be resubmitted to the Board of Review.

6.0     CONCLUSION

Though immediate approval for a recycled grade stamp was not granted at this time, this project, and the results obtained are important strides towards national acceptance of recycled lumber and timber.  As stated above, questions were raised about the structural adequacy of the broad population of recycled lumber and timber.  This project answered some of those questions, however additional work to further quantify the engineering performance of recycled lumber and timber is needed.  Eventually, when enough technical data is collected to allay the concerns of the ALSC membership, approval will be granted.

REFERENCES

American Society for Testing and Materials, 1994, ASTM Standard D198, Standard Methods  of Static Tests of Timbers in Structural Sizes, American Society for Testing and Materials, Philadelphia, PA.

American Society for Testing and Materials, 1992, D245-92, Standard Practice for Establishing Structural Grades and Related Allowable Properties for Visually Graded Lumber, American Society for Testing and Materials, Philadelphia, PA.

American Softwood Lumber Standard, 1994, Voluntary Product Standard PS 20-94, United States Department of Commerce, Technology Administration, National Institute of Standards and Technology, 40 pg.

Dolan, P., 1995, Unpublished Calculations, United States Army Corps of Engineers, Construction Engineering Research Laboratory, Urbana, IL.

Falk, R. H., DeVisser, D., Cook, S., D. Stansbury, 1999a, Military Building Deconstruction: Lumber Grade Yield from Recycling, Submitted for publication to the Forest Products Journal.

Falk, R. H., Green, D. W., Rammer, D., Lantz, S. F., 1999b, Engineering Evaluation of 55 Year Old  Timber Columns Recycled From an Industrial Military Building, Submitted for publication to the Forest Products Journal.

Falk, R. H., Green, D. W., Lantz, S. F., 1999c, An Evaluation of Lumber Recycled from an Industrial Military Building, Accepted for publication in the Forest Products Journal.

Falk, R. H., Green, D., Lantz, S. F., and M. R. Fix, 1995, Recycled Lumber and Timber In Restructuring America and Beyond: Proceedings of American Society of Civil Engineers Structures Congress 13, April 2-5, Boston, MA, Vol. 1, 1065-1068.

Fridley, K., Mitchell, J., Hunt, M., J. Senft, 1994, Evaluation of Load-Duration Effects from 85 Year Old Timbers, Proceedings of the Pacific Timber Engineering Conference, Gold Coast, Australia, July 11-15.

Green, D. W., Falk, R. H., Lantz, S. F., 1999, The Effect of Heat Checks on the Flexural Properties of Recycled Douglas-fir 6x8 Timbers, In review for publication in the Forest Products Journal.

Hill, R. (1950), The Mathematical Theory of Plasticity, Oxford Press, London.

McAlister, E. H., 1930, Strength Tests of Old Douglas Fir Timbers, University of Oregon Publication, Mathematics Series, Vol. 1, No. 2, January.

Proctor, F. D. (1996). “Finite Element Analysis of Shear Strength in Timber Beams,” Thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Civil Engineering, Washington State University, Pullman, Washington.

Santa Fe System, 1921, Comparative Test of New and Old Bridge Timbers to Determine the Effect of Age and Service on Physical Properties, Report of Test 84640,  Santa Fe System Office Engineer Tests.

West Coast Lumber Inspection Bureau, 1993, Standard No. 17, Grading Rules for West Coast Lumber, published by the West Coast Lumber Inspection Bureau, Portland, OR.

Wood, L. W., 1954, Tests of Old Floor beams from St. Raphaels Cathedral, Madison, WI.  Unpublished Wood Engineering Report, Forest Products Laboratory, Madison, WI.

Tsai, S. W., N. J. Pagano, (1968). “Invariant Properties of Composite Materials,” Composite Materials Workshop, J. C. Halpin and N. J. Pagano, eds., Technomic Publishing Co., Westport, CT, pp. 233-253.

Tsai, S. W., E. M. Wu, (1971). “A General Theory of Strength for Anisotropic Materials,” Journal of Composite Materials, Vol. 5, pp. 58-80.

Appendix A - Correspondence