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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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508 Compliance Captions for Figures

Figure 1. Wood material properties vary with orientation. The wood material coordinate system does not necessarily coincide with the board coordinate system. Source: American Society of Civil Engineers. Diagram.

This diagram shows two images. The first is a black line drawing of a tree stump, with a quarter section cut away. Within this quarter section is a cross section of timber, accentuated to show the location of the board. The second image is a closeup of the timber, the grain running up and down the length of the wood, with a large knot on the left-hand side of the image.

Figure 2. Ultimate tensile strength of Douglas fir measured in off-axis tests drops rapidly as the load is oriented at increasing angles to the grain. Source: Society of Wood Science and Technology. Diagram.

This diagram shows the relative compression strength of Douglas fir as a function of orientation angle relative to the grain and rings of the wood. The vertical axis is the relative strength in the longitudinal direction. It is labeled L and ranges from 0 to 1. The other two axes are the grain angles in the tangential and radial directions. They are labeled T and R and range from 0 to 90 degrees. The strength rapidly decreases from a non-dimensional value of 1 at 0 degrees to a value of 0.2 at 45 degrees in L-R plane, then decreases slowly to a value of about 0.1 at 90 degrees. A solid line is drawn to reflect this decrease in strength. The L-R plane corresponds to a ring angle of 0 degrees. The strength rapidly decrease from a non-dimensional value of 1 at 0 degrees to a value of 0.35 at 45 degrees in L-T plane, then decreases slowly to a value of about 0.15 at 90 degrees. The L-T plane corresponds to a ring angle of 90 degrees. A solid line is drawn to reflect this decrease in strength. The ring angles are shown between the L-R and L-T axes. Similar solid lines with rapid then slow decay are drawn at 15, 30, 45, 60, and 75 degrees of ring angle, and interpolate smoothly between the values cited at ring angles of 0 and 90 degrees. Solid lines showing the smooth interpolation betweem the L-R and L-T planes are drawn at 15, 30, 45, 60, and 90 degrees of grain angle.

Figure 3. Measured stress-strain relationships of southern yellow pine depend on load direction (parallel or perpendicular), load type (tensile or compressive), and moisture content. Graphs A and B.

(A) Tension parallel.  This graph shows five distinct lines. The first line is solid red and is labeled 4 percent moisture content. The second line is solid yellow and is labeled 8 percent MC while the third line is solid green and is labeled 12 percent MC. The fourth line is solid blue and is labeled 18 percent MC and the fifth and final line is solid black and is labeled saturated. The vertical axis of this graph ranges from 0 to 150 and represents stress (megapascals) while the horizontal axis of this graph ranges from negative 0.4 to 1.0 and represents strain (percent). The red line begins at the points of zero on both the vertical and horizontal axis and rises steeply across the graph to the points 125 on the vertical axis and 0.6 on the horizontal axis where it comes to an end. The yellow line begins at the points of zero on both the vertical and horizontal axis and rises steeply in a zigzag fashion across the graph to the points 130 on the vertical axis and 0.82 on the horizontal axis where it comes to an end. The green line begins at the points of zero on both the vertical and horizontal axis and rises steeply in a zigzag fashion across the graph to the points 149 on the vertical axis and 0.8 on the horizontal axis where it comes to an end. The blue line begins at the points of zero on the vertical and negative 0.25 on the horizontal axis and rises steeply across the graph to the points 130 on the vertical axis and 0.8 on the horizontal axis where it comes to an end. The black line begins at the points of zero on both the vertical and horizontal axis and rises steeply across the graph to the points 90 on the vertical axis and 0.98 on the horizontal axis where it comes to an end.
 
(B) Compression perpendicular.  This graph shows five distinct lines. The first line is solid red and is labeled 4 percent MC. The second line is solid yellow and is labeled 8 percent MC while the third line is solid green and is labeled 12 percent MC. The fourth line is solid blue and is labeled 18 percent MC, and the fifth and final line is solid black and is labeled saturated. The vertical axis of this graph ranges from 0 to 15 and represents stress (megapascals) while the horizontal axis of this graph ranges from 0.0 to 5.0 and represents strain (percent). All five lines begin at the points of zero on both the vertical and horizontal axes. The red line slopes upward to the points of 14.9 on the vertical axis and 4.6 on the horizontal axis where it comes to an end. The yellow line slopes upward to the points of 13 on the vertical axis and 3.7 on the horizontal axis where it comes to an end. The green line slopes upward to the points of 10 on the vertical axis and 4.7 on the horizontal axis where it comes to an end. The blue line slopes upward gradually to the points of 6 on the vertical axis and 4.9 on the horizontal axis where it comes to an end. The black line slopes upward gradually to the points of 4 on the vertical axis and 4.9 on the horizontal axis where it comes to an end.

Figure 4. Temperature affects the dynamic behavior of wood posts impacted by bogies at 9.4 meters per second. Graphs A, B, and C.

(A) Impact force.  This graph shows four distinct lines. The first line is solid red and is labeled DS-65 southern yellow pine. The second is dotted yellow and is labeled grade 1 southern yellow pine while the third line is solid blue and is labeled grade 1 Douglas fir. Finally, the fourth line is solid green and is labeled grade 1 frozen southern yellow pine. The vertical axis of this graph ranges from negative 50 to 100 and represents force (kilonewtons) while the horizontal axis of this graph ranges from 0 to 600 and represents deflection (millimeters). All four lines begin at the points of zero on both the vertical and horizontal axes and run across the length of the graph in waves. The red line first peaks at the points of 90 on the vertical axis and 80 on the horizontal axis. It then slopes down to the points of negative 40 on the vertical axis and 190 on the horizontal axis where it slopes back up to the points of 30 on the vertical axis and 270 on the horizontal axis. The red line then slopes down again to the points of negative 15 on the vertical axis and 370 on the horizontal where it then slopes up one last time to the points of 15 on the vertical axis 470 on the horizontal axis. The red line completes its wave-like format, sloping down to its end at the points of negative 5 on the vertical axis and 550 on the horizontal axis. The remaining three lines follow the same pattern as the red, only in much smaller waves. The yellow line first peaks at the points of 50 on the vertical axis and 70 on the horizontal axis where it slopes down to the points of negative 20 on the vertical axis and 190 on the horizontal axis. From there the yellow line slopes back up to the points of 15 on the vertical axis and 290 on the horizontal axis where it slopes down to the points of negative 5 on the vertical axis 360 on the horizontal axis. It then slopes back up gradually to the points of 5 on the vertical axis and 450 on the horizontal axis where it slopes back down to its end at the points of zero on the vertical axis and 600 on the horizontal axis. The blue line first peaks at the points of 45 on the vertical axis and 65 on the horizontal axis where it slopes down to the points of negative 15 on the vertical axis and 190 on the horizontal axis. From there the blue line slopes back up to the points of 15 on the vertical axis and 250 on the horizontal axis where it slopes down to the points of negative 5 on the vertical axis and 340 on the horizontal axis. It then slopes back up gradually to the points of 5 on the vertical axis and 450 on the horizontal axis where it slopes back down to its end at the points of zero on the vertical axis and 600 on the horizontal axis. The green line first peaks at the points of 30 on the vertical axis and 65 on the horizontal axis where it slopes down to the points of negative 20 on the vertical axis and 190 on the horizontal axis. From there the green line slopes back up to the points of 10 on the vertical axis and 250 on the horizontal axis where it slopes down to the points of negative 5 on the vertical axis 340 on the horizontal axis. It then slopes back up gradually to the points of 5 on the vertical axis and 450 on the horizontal axis where it slopes back down to its end at the points of zero on the vertical axis and 600 on the horizontal axis.
 
(B) Bogie velocity reduction.  This graph shows four distinct lines. The first line is solid red and is labeled DS-65 southern yellow pine. The second is dotted yellow and is labeled grade 1 southern yellow pine while the third line is solid blue and is labeled grade 1 Douglas fir. Finally, the fourth line is solid green and is labeled grade 1 frozen southern yellow pine. The vertical axis of this graph ranges from 0.0 to 1.0 and represents velocity (meters per second) while the horizontal axis of this graph ranges from 0.00 to 0.07 and represents time (seconds). All four lines begin at the points of zero on both the vertical and horizontal axes and run across the length of the graph in waves. The red line first peaks at the points of 0.95 on the vertical axis and 0.015 on the horizontal axis. It then slopes down to the points of 0.65 on the vertical axis and 0.025 on the horizontal axis where it slopes back up to the points of 0.85 on the vertical axis and 0.035 on the horizontal axis. The red line then slopes down again to the points of 0.75 on the vertical axis and 0.045 on the horizontal where it then slopes up one last time to the points of 0.9 on the vertical axis 0.06 to its end. The remaining three lines follow the same pattern as the red, only in much smaller waves. The yellow line first peaks at the points of 0.45 on the vertical axis and 0.015 on the horizontal axis where it slopes down to the points of 0.35 on the vertical axis and 0.025 on the horizontal axis. From there the yellow line slopes back up to the points of 0.45 on the vertical axis and 0.035 on the horizontal axis where it slopes down to the points of 0.4 on the vertical axis 0.04 on the horizontal axis. It then slopes back up gradually to the points of 0.45 on the vertical axis and 0.05 on the horizontal axis to its end. The blue line first peaks at the points of 0.43 on the vertical axis and 0.015 on the horizontal axis where it slopes down to the points of 0.35 on the vertical axis and 0.025 on the horizontal axis. From there the blue line slopes back up to the points of 0.5 on the vertical axis and 0.035 on the horizontal axis where it slopes down to the points of 0.45 on the vertical axis and 0.04 on the horizontal axis. It then slopes back up gradually to the points of 0.5 on the vertical axis and 0.05 on the horizontal axis to its end. The green line first peaks at the points of 0.3 on the vertical axis and 0.015 on the horizontal axis where it slopes down to the points of 0.2 on the vertical axis and 0.025 on the horizontal axis. From there the green line slopes back up to the points of 0.25 on the vertical axis and 0.032 on the horizontal axis where it slopes down to the points of 0.2 on the vertical axis 0.04 on the horizontal axis. It then slopes back up gradually to the points of 0.25 on the vertical axis and 0.05 on the horizontal axis to its end.
 
(C) Post displacement.  This graph shows four distinct lines. The first line is solid red and is labeled DS-65 southern yellow pine. The second is dotted yellow and is labeled grade 1 southern yellow pine while the third line is solid blue and is labeled grade 1 Douglas fir. Finally, the fourth line is solid green and is labeled grade 1 frozen southern yellow pine. The vertical axis of this graph ranges from 0 to 700 and represents displacement (millimeters) while the horizontal axis of this graph ranges from 0.00 to 0.07 and represents time (seconds). All four lines run diagonally across the graph in a straight-line beginning at the points of zero on both the vertical and horizontal axes. The red line finds its end at the points of 550 on the vertical axis and 0.062 on the horizontal axis. The yellow line finds its end at the points of 620 on the vertical axis and 0.062 on the horizontal axis. The green line mimics the yellow line completely. The blue line finds its end at the points of 580 on the vertical axis and 0.062 on the horizontal axis.

Figure 5. Wood exhibits progressive softening. Source: Forest Products Laboratory. Diagrams A and B.

(A) Softening curve.  This graph shows one distinct solid black line. The vertical axis ranges from 0 to 210   and represents force in newtons. The horizontal axis ranges from o to 20 centimeters and represents deflection in centimeters. The solid black line increases from 0 to 205 newtons over  0 to 1.8 centimeters. This portion of the curve is mainly linear to 170 newtons at 1.7 centimeters, with 205 newtons as the peak force in the wood. Then the solid line decreases from 205 to 18 newtons over 1.8 to 20 centimeters. This decrease is the softening portion of the curve. This decrease is nonlinear and its shape is similar to an expontential decay with a force of 120 newtons at 4 centimeters, 40 newtons at 8 centimeters ant 18 newtons at 16 centimeters.
 
(B) Testing apparatus.  This figure shows the testing apparatus used to generate the softening curve. It shows the rectangular piece of test wood with a 1 to 3 millimeter wide saw cut extending 60 percent up in height from the bottom of the cube. The cubes dimensions are `a` in width, `b` in depth, and `a` in height. Rectangular pieces of wood sit on each side of the test specimen. These pieces are `3a’ in width, `b` in depth and `a` in height. The phrase `surfaces glued` is listed above one piece of wood, with arrows pointing to the four surfaces: the right surface of one end piece, the left surface of the test speciment, the right surface of the specimen and the left surface of the other end piece. These arrows indicate the the test specimen is sandwiched between the end pieces and glued to them prior to testing.

Figure 6. Wood exhibits modulus reduction and permanent deformation (splitting test data for spruce wood from Stanzl-Tschegg, et al.). Source: Kluwer Academic Publishers, with the permission of Springer Science and Business Media. Diagrams A and B.

(A) Softening Curve.  This graph shows one distinct solid black line. The vertical axis of this graph shows four tic marks (no numerical range) and represents load P (unitless) while the horizontal axis of this graph shows four tic marks (no numerical range) and represents displacement u (unitless). The black line begins at the points of zero on both the vertical and horizontal axes and smoothly increases to a peak value of 4.5 tics on the vertical load axis and 0.5 tics on the horizontal displacement axis where it comes to a maximum load. Then the black line changes direction and decreases to 2 tics on the vertical load axis and 1.3 tics on horizontal displacement axis. This decrease in load with an increase in displacement represents softening. At this point the curve begins to unload and reload. The curve unloads to zero tics on the vertical load axis and 0.6 tics on the horizontal displacement axis. Then the curve reloads to its original starting point at 2 tics on the vertical load axis and 1.3 tics on the horizontal displacement axis. The words `unloading` and `reloading` are listing with arrows drawn to the respective unloading and reloading curves. The unloading and reloading curves are each smooth but are slightly nonlinear such that they do not lie on top of each other. Instead they form a loop about 0.1 tics wide. After reloading, the curve softens to 1.2 tics on the vertical load axis to 2 tics on the horizontal displacement axis. At this point another unloading and reloading loop is shown. The curve unloads to 0 on the vertical load axis and 1.1 on the horizontal displacement axis. Once again the loading and reloading curves form a loop about 0.1 tics wide on the horizontal axis. After reloading for the second time, the curve softens to 0.3 tics on the vertical load axis and 4.3 tics on the horizontal displacement axis.
 
(B) Testing Apparatus.  This figure shows the testing apparatus used to generate the softening curve. It shows two blocks of wood. One block is the wooden test specimen and the other block is a wedge used to generate the forces on the test specimen. The dimensions of the test specimen are unspecified but its shape is roughly cubic with an irregular shaped top surface due to a cut-out. The top surface is symmetric from left to right. The left and right sides of the top surface are flat for about 25 percent of its horizontal width. Then the cutout begins. The cutout drops vertically for about 15 percent of the block height, then slopes gradually towards the horizontal middle of the block for about 20 percent of the block height. At the horizontal mid-point lies a vertical rectangular cut. This cut is about 5 percent of the block width and 15 percent of the block height. All cuts are symmetric. The test specimen sits on a very small strip which is labeled as the Linear support area. The Wedge sits above the test specimen but is not in contact with the specimen due to the presence of two roll bodies on each side of the trapezoid wedge. The roll bodies sit on two angled bars which rest on the test specimen. The top portion of the wedge is rectangular while the bottom portion is trapezoidal. The smaller face of the trapezoid touches the test specimen while and the larger face coincides with the top rectangular block. The maximum and minimum widths of the wedge are about 20 percent and 15 percent of the width of the test specimen. The depths the wedge is equal to that of the test specimen. The height of the wedge is about 50 percent of that of the test specimen. The top left portion of the figure lists the phase `Force from testing machine.` Four forces are labeled with arrows indicating the directions of each force. The first force is labeled F subscript M with an arrow pointing downward on the top surface of the wedge. The second force is labeled F subscript H and points away from the block to the left on the left top surface of the test specimen. The subscript H indicates the horizontal direction. The third force is labeled F subscript H and points away from the block to the right on the right top surface of the test specimen. The fourth force is labeled F subscript V and points downward vertically from right top surface of the test specimen. The subscript V indicates the vertical direction. The figure also shows the dimensions `u/2` where `u` represents the displacement of the test specimen. This dimension lies beneath the roll bodies and angle bars to the right and left of the cut-out.

Figure 7. Variability of southern yellow pine clear wood data at 12-percent moisture content depends on the load direction and type. Graphs A and B. 

(A) Tension parallel.  This graph shows one distinct solid black line. The vertical axis of this graph ranges from 0 to 10 and represents load (kilonewtons) while the horizontal axis of this graph ranges from 0.0 to 0.8 and represents displacement (millimeters). The black line begins at the points of zero on both the vertical and horizontal axes and zigzags up to the points of 7 on the vertical axis and 0.4 on the horizontal axis where it comes to an end. The black line is shadowed by a multitude of red lines, all clustered together, shadowing the black line’s trail.
 
(B) Compression perpendicular.  This graph shows one distinct black line. The vertical axis of this graph ranges from 0 to 7 and represents load (kilonewtons) while the horizontal axis of this graph ranges from 0.0 to1.2 and represents displacement (millimeters). The black line begins at the points of zero on both the vertical and horizontal axes where it then slopes upward to the points of 4 on the vertical axis and 1.1 on the horizontal axis where it comes to an end. The black line is shadowed by a multitude of red lines, all clustered together, shadowing the black line's trail.

Figure 8. Wood material properties vary with position. Board strength depends on position and size of knot. Source: Society of Wood Science and Technology. Graph. 

This graph shows one distinct curve with variable points specifically marked along its path. The vertical axis of this graph ranges from 0.0 to 5000.0 and represents tensile strength (pounds per square inch) while the horizontal axis of this graph ranges from 0.0 to 0.5 and represents knot position (inches per inches). The curve begins at the points of 3200.0 on the vertical axis and 0.0 on the horizontal axis and curves downward and then up to the points of 4400.0 on the vertical axis and 0.5 on the horizontal axis where it comes to an end. The specific points marked are found in the following locations:

Point 1. 2500.0 vertical by 0.0 horizontal

Point 2. 3400.0 vertical by 0.002 horizontal

Point 3. 2800.0 vertical by 0.1 horizontal

Point 4. 2000.0 vertical by 0.14 horizontal

Point 5. 2300.0 vertical by 0.25 horizontal

Point 6. 2600.0 vertical by 0.3 horizontal

Point 7. 2100.0 vertical by 0.3 horizontal

Point 8. 1900.0 vertical by 0.3 horizontal

Point 9. 2500.0 vertical by 0.4 horizontal

Point 10. 4600.0 vertical by 0.42 horizontal

Figure 9. Dynamic strength of wood increases with impact velocity in Hopkinson bar tests and is most pronounced in the perpendicular direction. Source: Pergamon, Elsevier Science Ltd. Graphs A and B.

(A) Parallel.  This graph shows numerous points of different shapes. The solid black triangles represent Oak, the open squares represent Redwood, the open triangles represent Pine and the solid black circles represent Balsa. The vertical axis of this graph ranges from 0 to 6 and represents stress ratio S subscript r (unitless) while the horizontal axis of this graph ranges from 0.0 to 360 and represents impact velocity (meters per second). All data points plotted are for 0 degrees (parallel to the grain of the wood). The general trend of the data points is low stress ratio at low impact velocity increasing to high stress ratio at high impact velocity. The scatter in the data increases with increasing impact velocity. The scatter at 100 meters per second ranges from stress ratios of about 1.25 to 2.75. The scatter at 320 meters per second ranges from stress ratios of about 1.5 to 4.0. For each impact velocity, the stress ratios for the parallel direction are lower than those for the perpendicular direction.
 
(B) Perpendicular direction.  This graph shows numerous points of different shapes. The soild diamond represents Ekki, the solid black triangles represent Oak, the open squares represent Redwood, the open triangles represent Pine and the solid black circles represent Balsa. The vertical axis of this graph ranges from 0 to 6 and represents stress ratio S subscript r (unitless) while the horizontal axis of this graph ranges from 0.0 to 360 and represents impact velocity (meters per second). All data points plotted are for 90 degrees (perpendicular to the grain of the wood). The general trend of the data points is low stress ratio at low impact velocity increasing to high stress ratio at high impact velocity. The scatter in the data increases with increasing impact velocity. The scatter at 100 meters per second ranges from stress ratios of about 3 to 4. .The scatter at 320 meters per second ranges from stress ratios of about 8 to 22. For each impact velocity, the stress ratios for the perpendicular direction are higher than those for the parallel direction.

Figure 10. Organization of wood material model. Graphic with equations.
This figure shows a series of data segmented into blocks of equations cascading from the top of the page to the bottom of it. The first block of equations is labeled “Enter Wood Material Model” and reads viscid with damage stress tensor superscript N, backstress tensor superscript N, strain increment.

   An arrow from this block leads to the next which is labeled “Update Trail Elastic Stresses” on the left-hand side and reads trial elastic stress tensor superscript n plus 1 equals viscid with damage stress tensor superscript n over 1 minus D plus material stiffness tensor subscript kl, while the right hand side is labeled “Update Invariants” and reads I subscript 1, I subscript 2, I subscript 3, I subscript 4.

  An arrow from this block lead to the next which is labeled “Calculate Value of Each Initial Yield Function” and reads parallel trial elastic yield surface function, perpendicular trial elastic yield surface function.

  An arrow extends from this block to a line which in turn is extends to two separate lines. The first line is labeled parallel and reads parallel trial elastic yield surface function is less than or equal to zero, parallel trial elastic yield surface function is greater than zero while the other line is labeled Perpendicular and reads perpendicular trial elastic yield surface function is greater than zero, perpendicular trial elastic yield surface function is less than or equal to zero.

  The parallel labeled line extends, by arrow, to two separate blocks of data. The first block is labeled elastic and reads parallel plasticity consistency parameter equals zero, incremental backstress tensor subscript 11 equals zero. The second block of data is labeled plastic and reads parallel plasticity consistency parameter equals negative parallel elastic trial yield surface function divided by a quantity that is the sum of two terms. The first term is the partial derivative of the parallel yield surface function with respect to the first invariant times the partial derivative of the first invariant with respect to the parallel consistency parameter. The second term is the partial derivative of the parallel yield surface function with respect to the fourth invariant times the partial derivative of the fourth invariant with respect to the parallel consistency parameter. The backstress tensor subscript 11 equals the parallel hardening parameter c times the parallel hardening function G times left parenthesis stress subscript 11 minus backstress subscript 11 right parenthesis times the parallel effective strain rate times the time step.

  The perpendicular labeled line extends, by arrow, to two separate blocks of data. The first block is labeled elastic and reads perpendicular plasticity consistency parameter equals zero, incremental backstress tensor subscript 22 equals zero, incremental backstress tenors subscript 33 equals zero. The second block of information is labeled plastic and reads perpendicular plasticity consistency parameter equals negative perpendicular elastic trial yield surface function divided by a quantity that is the sum of two terms. The first term is the partial derivative of the perpendicular yield surface function with respect to the second invariant times the partial derivative of the second invariant with respect to the perpendicular consistency parameter. The second term is the partial derivative of the perpendicular yield surface function with respect to the third invariant times the partial derivative of the third invariant with respect to the perpendicular consistency parameter. The backstress tensor subscript 22 equals the perpendicular hardening parameter c times the perpendicular hardening function G times left parenthesis stress subscript 22 minus backstress subscript 22 right parenthesis times the parallel effective strain rate times the time step. The backstress tensor subscript 33 equals the perpendicular hardening parameter c times the perpendicular hardening function G times left parenthesis stress subscript 33 minus backstress subscript 33 right parenthesis times the parallel effective strain rate times the time step.

  An arrow extends from the latter four blocks of information to the next block of information which reads viscid stress tensor superscript N plus 1 equals trial elastic stress tensor superscript N plus 1 minus material stiffness tensor  plasticity consistency parameter times partial derivative of the yield surface function with respect to the stress tensor evaluated at the nth time step plus backstress tensor for hardening model.

  An arrow extends from the latter block of data to the next, which is labeled “Update Stresses with Viscoplastic Rate Effects” and reads viscid stress tensor superscript N plus 1 equals parenthesis 1 minus general viscoplastic interpolation parameter close parenthesis inviscid with backstress stress tensor superscript N plus 1 plus general viscoplastic interpolation parameter trial elastic stress tensor superscript N plus 1. General viscoplastic interpolation parameter equals parallel viscoplastic interpolation parameter for orthotropic stress component subscript 11, orthotropic stress component subscript 12, orthotropic stress component subscript 13. General viscoplastic interpolation parameter equals perpendicular viscoplastic interpolation parameter for orthotropic stress component subscript 22, orthotropic stress component subscript 33, orthotropic stress component subscript 23.

  An arrow extends from the latter block of data to the next block, which is labeled “Update Stresses With Damage” and reads viscid with damage stress tensor superscript N plus 1 equals parenthesis 1 minus D close parenthesis viscid stress tensor superscript N plus 1. D equals D for orthotropic stress component subscript 11, orthotropic stress component subscript 12, orthotropic stress component subscript 13. D equals max parenthesis D perpendicular, D parallel, close parenthesis, for orthotropic stress component subscript 22, orthotropic stress component subscript 33, orthotropic stress component subscript 23.

  An arrow extends from the latter block to the final block of data which is labeled “Exit Wood Material Model” and reads viscid with damage stress tensor superscript N plus 1 backstress tensor for hardening model superscript N plus 1.

Figure 11. Failure criteria for wood depend on four of the five invariants of a transversely isotropic material. Graphs A and B. 

(A) Parallel modes.  This graph shows one distinct line, colored solid green and running across the graph in a horizontal curve. The vertical axis of this graph ranges from negative 50 to 150 and represents parallel normal stress invariant, eye subscript one (megapascals) while the horizontal axis of this graph ranges from 0 to 300 and represents parallel shear stress invariant, eye subscript four (megapascals squared). The green curve begins at the points of negative 50 on the vertical axis and 0 on the horizontal axis and then curves outward horizontally to the points of 0 on the vertical axis and 300 on the horizontal axis. From there the curve continues to its end at the points of 150 on the vertical axis and 0 on the horizontal axis.
 
(B) Perpendicular modes.  This graph shows one distinct line, colored solid green and running across the graph in a horizontal curve. The vertical axis of this graph ranges from negative 15 to 10 and represents parallel normal stress invariant, eye subscript two (megapascals), while the horizontal axis of this graph ranges from negative 100 to 600 and represents parallel shear stress invariant, eye subscript three (megapascals squared). The green curve begins at the points of negative 10 on the vertical axis and negative 50 on the horizontal axis and then curves outward horizontally to the points of 0 on the vertical axis and 550 on the horizontal axis. From there the curve continues to its end at the points of 5 on the vertical axis and negative 50 on the horizontal axis.

Figure 12. Failure criteria for wood produce smooth surfaces in stress space. Diagrams A and B. 

(A) Parallel modes.  This diagram shows a cylindrical pod-like structure, narrow and long. The top of the structure is marked sigma subscript LR (stress components). The right hand side of the structure is marked with sigma subscript LT (stress components) and the very tip of the structure is marked with sigma subscript L (stress components).
 
(B) Perpendicular modes.  This diagram shows a cylindrical pod-like structure, narrow and long, and dissected in half lengthwise. The top of the structure is marked sigma subscript R (stress components). The right hand side of the structure is marked with sigma subscript T (stress components) and the very tip of the structure is marked with sigma subscript TR (stress components).

Figure 13. Prepeak nonlinearity is modeled in compression with translating yield surfaces that allow user to specify the hardening response. Diagrams A and B.

(A) Initial and ultimate yield surfaces.  This diagram shows a green curve running horizontally in a vacant graph. The vertical axis of this graph represents longitudinal stress while the horizontal axis of this graph represents square root of parallel shear invariant. The green curve begins at a point labeled minus X C and curves horizontally to a point labeled S parallel; this curve is labeled ultimate yield surface. It then continues its curve and end at a point labeled X T. The bottom half of the green curve is marked compression, while the top half of the curve is marked tension. There also exists a red line that begins just above the inside of the green curve and curves itself into the S parallel point on the green curve. This red line begins on the vertical axis at a point marked negative parenthesis one minus N parallel close parenthesis X C and is labeled initial yield surface.
 
(B) Stress-strain behavior.  The vertical axis of this graph is Longitudinal stress with no units and no range. The horizontal axis is Strain with no units and no range. Two horizontal dashed lines are drawn. The first is dotted green and is labeled ultimate yield strength. This line is at a constant stress of X subscript C, which represents the ultimate compressive strength of the wood. The second is dotted red and is labeled initial yield strength. This line is at a constant stress of  X subscript C multiplied by the quantity 1 minus N subscript parallel, which represents the initial strength of the wood. The difference in value between these two horizontal lines is marked on the graph as N subscript parallel times X subscript C. Four curves are shown which represent four possible stress versus strain behaviors of the wood. The first behavior is presented by a solid black line and runs linearly from zero stress and strain to strength X subscript C (the exact strain is not specified), and then the strength remains constant at X subscript c for all increasing values of strain. This curve represents of case of no hardening. The second, third, and fourth curves are represented by dashed lines. Each originates at a strength of  X subscript C multiplied by the quantity 1 minus N subscript parallel (no strain specified). Each of these three curves extends or asymptotes to a strength of X subscript C in a nonlinear manner. The strain at which each curve reaches X subscript C depends upon the value of the hardening parameter C subscript parallel. The larger the value of C subscript parallel, the lower the value of the strain. The notations C subscript parallel large and C subscript parallel small are shown next to the corresponding curves. The phrase `Increasing rate of Translation`  is shown on the figure with an arrow extending in the direction of increasing C subscript parallel.

Figure 14. Postpeak hardening is modeled in compression with positive values of the parameter G subscript hard. Graph. 

This graph shows three distinct lines. Line A is solid green and is labeled G subscript hard equals 1.00. Line B is solid blue and is labeled G subscript hard equals 0.05. Line C is solid red and is labeled G subscript hard equals 0.00. The vertical axis of this graph ranges from 0 to 50 and represents stress (megapascals) while the horizontal axis of this graph ranges from 0.0 to 2.0 and represents Strain (percent). All three lines begin at the points of zero on both the vertical and horizontal axes. Line A runs diagonally across the graph in a steep line to the points of 50 on the vertical axis and 1.0 on the horizontal axis where it leaves the graph. Lines B and C run atop each other as they slope upward to the points of 20 on the vertical axis and 0.8 on the horizontal axis, where the two lines split. Line B continues a gradual climb until it leaves the graph at the points of 23 on the vertical axis and 2.0 on the horizontal axis. Line B plateaus and runs straight off the graph at the points of 20 on the vertical axis and 2.0 on the horizontal axis.

Figure 15. Damage d accumulates with energy tau once an initial threshold tau subscript zero is exceeded. Graph. 

This graph shows four distinct lines. The first line A is solid red and is labeled C equals 3, D equals 20. The second line B is solid blue and is labeled C equals 10, D equals 20. The third line C is solid green and is labeled C equals 3, D equals 200. The fourth line D is solid yellow and is labeled C equals 10, D equals 200. The vertical axis of this graph ranges from 0.0 to 1.2 and represents damage parameter (italic d) while the horizontal axis of this graph ranges from 0.00 to 0.35 and represents energy (megapascals superscript one-half). All four lines begin at the points of zero on the vertical axis and 0.05 on the horizontal axis. The red line A slopes approximately 60 degrees upward to the points of 1.0 on the vertical axis and 0.25 on the horizontal axis where the line plateaus and runs off the graph at the points of 1.0 on the vertical axis and 0.35 on the horizontal axis. The blue line B slopes more sharply upward and meets the yellow line D, which also slopes sharply upward after a more gradual beginning, at the points of 1.0 on the vertical axis and 0.13 on the horizontal axis. From there both lines plateau and run off the graph at the points of 1.0 on the vertical axis and 0.35 on the horizontal axis. The green line C slopes up gradually and off the graph at the points of .0 on the vertical axis and 0.35 on the horizontal axis, where it meets the other three lines.

Figure 16. Softening depends on the values of the damage parameters C and D (calculated with dmax = 1). Graph. 

This graph shows four distinct lines. The first line A is solid red and is labeled C equals 3, D equals 20. The second line B is solid blue and is labeled C equals 10, D equals 20. The third line C is solid green and is labeled C equals 3, D equals 200. The fourth line D is solid yellow and is labeled C equals 10, D equals 200. The vertical axis of this graph ranges from 0.0 to 5.0 and represents tensile stress perpendicular to grain (megapascals) while the horizontal axis of this graph ranges from 0.0 to 3.5 and represents tensile strain perpendicular to grain (percent). All four lines begin at the points of zero on both the vertical and horizontal axes. All four lines rise steeply to the points of 4.5 on the vertical axis and 0.52 on the horizontal axis where they split and slope down towards the horizontal axis. The red line A slopes down and off the graph at the points of 0.2 on the vertical axis and 3.5 on the horizontal axis. The blue line B slopes down steeply until points it runs parallel with the horizontal axis and off the graph at the points of 0.1 on the vertical axis and 3.5 on the horizontal axis. The yellow line slopes down steeply to the points of 0.1 on the vertical axis and 1.5 on the horizontal axis where it runs off the graph overlapping the blue line. The green line slopes downward gradually and off the graph at the points of 1.0 on the vertical axis and 3.5 on the horizontal axis.

Figure 17. Softening response modeled for parallel modes of southern yellow pine. Graphs A, B, and C. 

(A) Tension softening.  The vertical axis of this graph ranges from 0 to 160 and represents tensile stress parallel to grain (megapascals) while the horizontal axis of this graph ranges from 0.0 to 1.5 and represents tensile strain parallel to grain (percent). The single blue line rises from the zero points on both axes diagonally across the graph peaking at the points of 150 on the vertical axis and just before 1.0 on the horizontal axis. From there, it begins to slope back down until it meets the horizontal axis at the points of zero on the vertical axis and 1.2 on the horizontal axis. This slope depicts brittle postpeak softening.
 
(B) Shear softening.  The vertical axis of this graph ranges from 0 to 20 and represents shear stress parallel to grain (megapascals) while the horizontal axis ranges from 0 to 6 and represents shear strain parallel to grain (percent). The single blue line rises from the points of zero on both axes diagonally, peaking at the points of 17 on the vertical axis and 2 on the horizontal axis. From there, it begins to gradually slope back down into the horizontal axis, where it leaves the graph at zero point on the vertical axis and 6 on the horizontal axis. This gradual slope depicts moderate postpeak softening.
 
(C) Compression yielding.  The vertical axis of this graph ranges from 0 to 60 and represents compressive stress parallel to grain (megapascals) while the horizontal axis of this graph ranges from 0.0 to 1.5 and represents compressive strain parallel to grain (percent). The single blue line rises at a sharp angle up the length of the graph where it peaks at the points of 52 on the vertical axis and 0.5 on the horizontal axis. Just before the line peaks, that curve is labeled prepeak hardening. From the peak onward, the line plateaus and runs straight off the graph at the points of 52 on the vertical axis and 1.5 on the horizontal axis; this plateau depicts no postpeak softening.

Figure 18. Softening response modeled for perpendicular modes of southern yellow pine. Graphs A, B, and C.

(A) Tension softening.  The vertical axis of this graph ranges from 0.0 to 5.0 and represents tensile stress perpendicular to grain (megapascals) while the horizontal axis of this graph ranges from 0.0 to 2.0 and represents tensile strain perpendicular to grain (percent). The single blue line rises from the zero points on both axes diagonally across the graph peaking at the points of 4.5 on the vertical axis and just after 0.5 on the horizontal axis. From there, it begins to slope back down until it meets the horizontal axis at the points of zero on the vertical axis and 1.5 on the horizontal axis and runs straight off the graph.
 
(B) Shear softening.  The vertical axis of this graph ranges from 0 to 25 and represents shear stress perpendicular to grain (megapascals) while the horizontal axis ranges from 0 to 15 and represents shear strain perpendicular to grain (percent). The single blue line rises from the points of zero on both axes diagonally, peaking at the points of 24 on the vertical axis and 8.5 on the horizontal axis. From there, it begins to gradually slope back down into the horizontal axis, where it leaves the graph at zero point on the vertical axis and 13 on the horizontal axis where it runs straight off the graph.
 
(C) Compression yielding.  The vertical axis of this graph ranges from 0 to 12 and represents compressive stress perpendicular to grain (megapascals) while the horizontal axis of this graph ranges from 0.0 to 3.0 and represents compressive strain perpendicular to grain (percent). The single blue line steadily rises at an approximate 60-degree angle to peak at the points of 10 on the vertical axis and 1.5 on the horizontal axis. From the peak, the blue line plateaus and runs straight off the graph at the points of 10 on the vertical axis and 3.0 on the horizontal axis.

Figure 19. Hopkinson bar tests indicate that the measured strength of pine increases with impact velocity. Source: Pergamon, Elsevier Science Ltd. Graph. 

This graph shows numerous points of different shapes accompanied by one solid black line and one dashed line running through the clusters of points. The solid black triangles represent pine at 90 degrees (perpendicular to the grain) while the solid black circles represent pine at 0 degrees (parallel to the grain). The vertical axis of this graph ranges from 0 to 15 and represents stress ratio S subscript r (unitless) while the horizontal axis of this graph ranges from 0.0 to 360 and represents impact velocity (meters per second). The general trend is low stress ratio at low impact velocity increasing to high stress ratio at high impact velocity, with the triangular data points for pine tested at 90 degrees lie above the data points for pine tested at zero degrees. The data at 90 degrees increase from a stress ratio of about 2.1 at 50 meters per second to 14.6 at 320 meters per second. The data at 0 degrees increase from a stress ratio of about 1.1 at 40 meters per second to 4 at 320 meters per second. Hence the stress ratios measured perpendicular to the grain are greater than the stress ratios measured parallel to the grain at a given impact velocity. Also shown on this graph are numerical points from Shock Theory. The open black triangles represent pine at 90 degrees (perpendicular to the grain) while the open black circles represent pine at 0 degrees (parallel to the grain). The trend in the theoretical points closely follows that of the measured data. The theoretical points at 90 degrees increase from a stress ratio of about 1.5 at 50 meters per second to 13 at 320 meters per second. The data at 0 degrees increase from a stress ratio of about 1.1 at 40 meters per second to 2.2 at 320 meters per second. One solid line is drawn through the theoretical points at 90 degrees. One dashed line is drawn through the theoretical points at 0 degrees.

Figure 20. Hopkinson bar data indicate that strength and stiffness increase with strain rate. Source: EDP Sciences. Graphs.

(A) Tested Parallel.  All data are generated by a split Hopkinson pressure bar test conducted parallel to the grain of the wood. The vertical axis is True Stress in megapascals and ranges from 0 to 100. The horizontal axis is True Strain (unitless) and ranges from 0 to 0.12. Two curves are shown. The solid black line represents data tested at a strain rate of 500 per second while the dashed black line represents data tested at a strain rate of 1000 per second. The solid black line increases from zero stress at zero strain to a peak value of 65 megapascals at a strain of 0.03. The increase is nonlinear with a gradual increase in slope at low strain and a gradual decrease in slop just before the peak. Following the peak there is a nearly vertical drop in stress with strain down to a value of 19 megapascals at a strain of 0.028. The dashed black line increases from zero stress at zero strain to a peak value of 95 megapascals at a strain of 0.04. The increase is nonlinear with a gradual increase in slope at low strain and a gradual decrease in slope just before the peak stress. The peak is a plateau that decreases to 73 megapascals at a strain of 0.102. Following the plateau there is a nearly vertical drop in stress with strain down to a value of 10 megapascals at a strain of 0.09.
 
(B) Tested Perpendicular.  All data are generated by a split Hopkinson pressure bar test conducted perpendicular to the grain of the wood. The vertical axis is True Stress in megapascals and ranges from 0 to 100. The horizontal axis is True Strain (unitless) and ranges from 0 to 0.12. Two curves are shown. The solid black line represents data tested at a strain rate of 500 per second while the dashed black line represents data tested at a strain rate of 1000 per second. The solid black line increases from zero stress at zero strain to a peak value of 9 megapascals at a strain of 0.04. The increase is nonlinear with a gradual increase in slope at low strain and a gradual decrease in slop just before the peak. The peak is a plateau that decreases slightly to 8.5 megapascals at a strain of 0.063. Following the plateau there is a nearly vertical drop in stress with strain down to a value of 4 megapascals at a strain of 0.07. The dashed black line increases from zero stress at zero strain to a peak value of 15 megapascals at a strain of 0.04. The increase is nonlinear with a gradual increase in slope at low strain and a gradual decrease in slope just before the peak stress. Following the peak there is a gradual decrease in stress to 7 megapascals at a strain of 0.12, followed by a plateau at 7 megapascals to a strain of 0.17.

Figure 21. These single-element simulations demonstrate the rate-effect behavior of the shifted surface formulation at 500 per second. Graphs A and B.

(A) Tension parallel.  This graph shows two distinct lines. Line A is solid red and is labeled with rate effect; line B is solid blue and is labeled without rate effects. The vertical axis of the graph ranges from 0 to 200 and represents tensile stress parallel to grain (megapascals) while the horizontal axis of this graph ranges from 0.0 to 2.0 and represents tensile strain parallel to grain (percent). Both lines begin at point zero on both the vertical and horizontal axes. The two lines rise steeply, the blue line B breaking from red line A at the points of 150 on the vertical axis and 1.0 on the horizontal axis, where B then slopes down and off the graph to the points of zero on the vertical axis and 1.1 on the horizontal axis. The red line A continues its rise to the points of 190 on the vertical axis and 1.3 on the horizontal axis. From there it slopes back down and off the graph to the points of zero on the vertical axis and 1.3 on the horizontal axis.
 
(B) Compression parallel.  This graph shows two distinct lines. Line A is solid red and is labeled with rate effect; line B is solid blue and is labeled without rate effects. The vertical axis of this graph ranges from 0 to 100 and represents compressive stress parallel to grain (megapascals) while the horizontal axis of this graph ranges from 0.0 to 1.5 and represents compressive strain parallel to grain (percent). Both lines begin at point zero on both the vertical and horizontal axes. Line A slopes upward where it peaks at the points of 88 on the vertical axis and 0.8 on the horizontal axis. From there, A plateaus and runs straight off the graph at the points of 88 on the vertical axis 1.5 on the horizontal axis. Line B runs much more shallowly, continuing a slight arc until it runs off the graph at the points of 50 on the vertical axis and 1.5 on the horizontal axis. The vertical distance between the two lines approximately at exit off the graph is shown as E subscript 11 followed by italic epsilon with small dot above it (to indicate rate) followed by eta subscript parallel.

Figure 22. Two-parameter viscoplastic model is flexible in fitting data. Graph. 

This graph shows four distinct lines. The first line A solid black and is labeled N equals 0.000. The second line B is solid red and is labeled N equals 0.100. The third line C is solid yellow and is labeled N equals 0.250. The fourth line D is solid blue and is labeled N equals 0.500. The vertical axis of this graph ranges from 0.0 to 4.0 and represents dynamic-to-static stress ratio while the horizontal axis ranges from 0.0 to 3.0 and represents log open bracket strain rate, open parenthesis seconds superscript minus 1, close parenthesis, close bracket. All four lines begin at the points of 1.0 on the vertical axis and zero on the horizontal axis where they run straight, overlapping one another to the points of 1.0 on the vertical axis and 1.0 on the horizontal axis, where they begin to split. The black line A bends upward from the break and runs off the graph at the points of 3.7 on the vertical axis and 3.0 on the horizontal axis. The red line B bends upward from the break and runs off the graph at the points of 2.4 on the vertical axis and 3.0 on the horizontal axis. The yellow line C bends upward ever so slightly from the break and runs off the graph at the points of 1.5 on the vertical axis and 3.0 on the horizontal axis. The blue line D bends upward ever so slightly from the break and runs off the graph at the points of 1.1 on the vertical axis and 3.0 on the horizontal axis.

Figure 23. These single-element simulations demonstrate the rate-effect behavior of the viscoplastic formulation at 500 per second. Graphs A and B.

(A) Tension parallel.  This graph shows two distinct lines. Line A is solid red and is labeled with rate effect. Line B is solid blue and is labeled without rate effects. The vertical axis of the graph ranges from 0 to 200 and represents tensile stress parallel to grain (megapascals) while the horizontal axis of this graph ranges from 0.0 to 2.0 and represents tensile strain parallel to grain (percent). Both lines begin at point zero on both the vertical and horizontal axes. The two lines rise steeply, with line B breaking from line A at the points of 150 on the vertical axis and 1.0 on the horizontal axis, where B then slopes down and off the graph to the points of zero on the vertical axis and 1.2 on the horizontal axis. Line B continues its rise to the points of 180 on the vertical axis and 1.5 on the horizontal axis. From there it slopes back down and off the graph to the points of zero on the vertical axis and 1.5 on the horizontal axis.
 
(B) Compression parallel.  This graph shows two distinct lines. Line A is solid red and is labeled with rate effect. Line B is solid blue and is labeled without rate effects. The vertical axis of this graph ranges from 0 to 100 and represents compressive stress parallel to grain (megapascals) while the horizontal axis of this graph ranges from 0.0 to 1.5 and represents compressive strain parallel to grain (percent). Both lines begin at point zero on both the vertical and horizontal axes. Line A slopes upward where it peaks at the points of 88 on the vertical axis and 0.8 on the horizontal axis. From there line A plateaus and runs straight off the graph at the points of 88 on the vertical axis 1.5 on the horizontal axis. Line B continues in a slight arc until it plateaus and runs off the graph at the points of 50 on the vertical axis and 1.5 on the horizontal axis. The vertical distance between the two lines is labeled E subscript 11, followed by italic epsilon with small dot above it (to indicate rate) followed by eta subscript parallel.

Figure 24. Effect of temperature and moisture interaction on the longitudinal modulus. Graphs A and B.

(A) Data reproduced from Bodig and Jayne 1993 (average of six species); source, Krieger Publishing Company.  This graph shows four distinct lines. All lines are solid black. The lines are labeled as follows: 20 percent, 12 percent, 8 percent, and 0 percent moisture content. The vertical axis of this graph ranges from 0.7 to 1.2 and represents relative modulus of elasticity while the horizontal axis of this graph ranges from minus10 to 60 and represents temperature (degrees Celsius). The four lines represented on this graph vary in position, but distinctly cross each other at one specific point, from which they fan out in either direction. The point in question is represented by two dotted lines, being drawn from their points of origin at 1.0 on the vertical axis and 20 on the horizontal axis.
 
(B) Quadratic fits to data.  This graph shows four distinct lines. All lines are solid black. The lines are labeled as follows: 20 percent, 12 percent, 8 percent, and 0 percent. The background for this graph is a dull blue. The vertical axis of this graph ranges from 0.7 to 1.2 and represents relative modulus while the horizontal axis of this graph ranges from negative 20 to positive 60 and represents temperature (degrees Celsius). The four lines represented on this graph vary in position, but distinctly cross each other at one specific point, from which they fan out in either direction. The point in question locates at approximately 1.0 on the vertical axis and 20 on the horizontal axis.

Figure 25. Temperature effects are more pronounced for the strength parallel to the grain than for the modulus parallel to the grain. Source: Forest Products Laboratory. Graphs A and B.

(A) Strength.  The vertical axis of this graph ranges from 0 to 250 and represents relative modulus of rupture (percent) while the horizontal axis of this graph ranges from negative 200 to positive 150 with temperature in degrees centigrade. This figure shows three regions of test data. One region is marked with crossed lines and represents data at 18 percent moisture content. The relative modulus of rupture ranges from an average of 220 percent at negative 175 degrees to 100 percent at zero degrees and down to 75 percent at 150 degrees. The scatter in the data bands ranges from 170 to 205 percent at negative 100 degrees and 55 to 85 percent at 150 degrees, with no scatter at zero degrees. Hence the band forms two triangular shapes extending from zero degrees towards both the positive and negative temperature directions. The second region is marked with horizontal lines and represents data at 12 percent moisture content. The relative modulus of rupture ranges from an average of 170 percent at negative 175 degrees to 100 percent at zero degrees. The scatter in the data bands ranges from 165 to 178 percent at negative 175 degrees with no scatter at zero degrees. This band also forms one triangular shape extending from zero degrees towards both the negative temperature direction. The third region is marked with vertical lines and represents data at 0 percent moisture content. The relative modulus of rupture ranges from an average of 115 percent at negative 35 degrees to 100 percent at zero degrees and down to 80 percent at 60 degrees. The scatter in the data bands ranges from 125 to 135 percent at negative 35 degrees and 75 to 85 percent at 50 degrees with no scatter at zero degrees. Hence the band also forms two triangular shapes extending from zero degrees towards both the positive and negative temperature directions. These data indicate that the relative modulus of rupture increases with an increase in moisture content at temperatures below zero degrees. Conversely the relative modulus of rupture decreases with an increase in moisture content at temperatures above zero degrees. These data for the modulus of rupture show changes with moisture content that are more pronounced that those measured for the modulus of elasticity.
 
(B) Stiffness.  The vertical axis of this graph ranges from 0 to 200 and represents relative modulus of elasticity (percent) while the horizontal axis of this graph ranges from negative 200 to positive 300 with temperature in degrees centigrade. This figure shows two regions of test data. One region is marked with horizontal lines and represents data at 12 percent moisture content. The relative modulus of elasticity ranges from an average of 140 percent at negative 165 degrees to 100 percent at zero degrees and down to 85 percent at 100 degrees. The scatter in the data bands ranges from 125 to 170 percent at negative 150 degrees and 75 to 95 percent at 100 degrees, with no scatter at zero degrees. Hence the band forms two triangular shapes extending from zero degrees towards both the positive and negative temperature directions. The second region is marked with vertical lines and represents data at 0 percent moisture content. The relative modulus of elasticity ranges from an average of 120 percent at negative 150 degrees to 100 percent at zero degrees and down to 65 percent at 250 degrees. The scatter in the data bands ranges from 125 to 135 percent at negative 150 degrees and 50 to 85 percent at 150 degrees, with no scatter at zero degrees. Hence the band also forms two triangular shapes extending from zero degrees towards both the positive and negative temperature directions. These data indicate that the relative modulus of elasticity increases with an increase in moisture content at temperatures below zero degrees. Conversely the relative modulus of elasticity decreases with an increase in moisture content at temperatures above zero degrees. These data for the modulus of elasticity show changes with moisture content that are less pronounced that those measured for the modulus of rupture.

Figure 26. Yield criteria for wood produce smooth surfaces in stress space.

(A) Parallel modes.  This diagram shows a cylindrical pod-like structure, narrow and long. The top of the structure is marked sigma subscript LR (stress components). The right hand side of the structure is marked with sigma subscript LT (stress components) and the very tip of the structure is marked with sigma subscript L (stress components).
 
(B) Perpendicular modes.  This diagram shows a cylindrical pod-like structure, narrow and long, and dissected in half lengthwise. The top of the structure is marked sigma subscript R (stress components). The right hand side of the structure is marked with sigma subscript T (stress components) and the very tip of the structure is marked with sigma subscript TR (stress components).

Figure 27. Prepeak nonlinearity in compression is modeled with translating yield surfaces that allow user to specify hardening response.

(A) Initial and ultimate yield surfaces.  The vertical axis of this graph is Longitudinal stress with no units and no range. The horizontal axis is the Square Root of Parallel Shear Invariant. Three curves are drawn in solid black lines. The first is elliptical in shape and runs from a tensile strength of X subscript T at zero invariant to zero strength at an invariant of S subscript parallel. This curve represents the strength of wood in tension and shows that the initial and ultimate surfaces coincide in tension. The second curve extends from a compressive strength of minus X subscript C at zero invariant to zero strength at an invariant of S subscript parallel. This curve represents the ultimate yield surface in compression. The third curve begins at a compressive strength of minus X subscript C multiplied by the quantity 1 minus N subscript parallel at zero invariant. It extends to zero strength at an invariant of S subscript parallel. This curve represents the initial yield surface in compression.
 
(B) Stress-strain behavior.  The vertical axis of this graph is Longitudinal stress with no units and no range. The horizontal axis is Strain with no units and no range. Two horizontal dashed black lines are drawn. One line is at a constant stress of X subscript C, which represents the ultimate compressive strength of the wood. The other line is at a constant stress of  X subscript C multiplied by the quantity 1 minus N subscript parallel, which represents the initial strength of the wood. The difference in value between these two horizontal lines is marked on the graph as N subscript parallel times X subscript C. Four curves are shown which represent four possible stress versus strain behaviors of the wood. The first behavior is presented by a solid black line and runs linearly from zero stress and strain to strength X subscript C (the exact strain is not specified), and then the strength remains constant at X subscript c for all increasing values of strain. This curve represents of case of no hardening. The second, third, and fourth curves are represented by dashed lines. Each originates at a strength of X subscript C multiplied by the quantity 1 minus N subscript parallel (no strain specified). Each of these three curves extends or asymptotes to a strength of X subscript C in a nonlinear manner. The strain at which each curve reaches X subscript C depends upon the value of the hardening parameter C subscript parallel. The larger the value of C subscript parallel, the lower the value of the strain. The notations C subscript parallel large and C subscript parallel small are shown next to the corresponding curves. The phrase `Increasing rate of Translation` is shown on the figure with an arrow extending in the direction of increasing C subscript parallel.

Figure 28. Softening response modeled for parallel modes of southern yellow pine.

(A) Tensile softening.  The vertical axis of this graph is Tensile Stress Parallel to Graph in units of megapascals with a range of 0 to 150. The horizontal axis is Tensile Strain Parallel to Grain in percent with a range of 0 to 1.5. One curve is drawn in a solid blue line. The curve extends linearly from 0 on both the horizontal and vertical axes to 0.98 percent strain on the horizontal axis and 145 stress on the vertical axis. Then the curve drops sharply in a nonlinear manner to 1.2 percent strain on the horizontal axis and near 0 stress on the vertical axis. The curve remains at 0 stress until the end of the graph at 1.5 percent strain.
 
(B) Shear softening.  The vertical axis of this graph is Shear Stress Parallel to Graph in units of megapascals with a range of 0 to 20. The horizontal axis is Shear Strain Parallel to Grain in units of percent with a range of 0 to 6. One curve is drawn in a solid blues line. The curve extends linearly from 0 on both the horizontal and vertical axes to 2 percent strain on the horizontal axis and 17 stress on the vertical axis. Then the curve drops gradually in a nonlinear manner to 6 percent strain on the horizontal axis and near 0 stress on the vertical axis.
 
(C) Compressive yielding.  The vertical axis of this graph is Compressive Stress Parallel to Graph in units of megapascals with a range of 0 to 60. The horizontal axis is Compressive Strain Parallel to Grain in units of percent with a range of 0 to 1.5. One curve is drawn in a solid blue line. The curve extends linearly from 0 on both the horizontal and vertical axes to 0.25 percent strain on the horizontal axis and 38 stress on the vertical axis. Then the curve continues increasing in a nonlinear manner to 0.7 percent strain on the horizontal strain axis and 52 stress on the vertical axis. The curve remains at 52 stress until the end of the graph at 1.5 percent strain.

Figure 29. Example wood model input for selection of default input parameters (option MAT_WOOD_PINE). Diagram. 

This figure presents sets of data in seven different columns. The first column reads MID, 143, MOIS, 0.0, AOPT, 2, XP, D1, 1.0. The second column reads RO, 6.73 minus 04, TEMP, 0.0, YP, D2, 0.0. The third column reads NPLOT, 0, QT, minus 2.0, ZP, D3, 0.0. The fourth column reads ITERS, 0, QC, 0.0, A1, 0.0. The fifth column reads IRATE, 0, UNITS, 1, A2, 0.0. The sixth column reads GHARD, 0, IQUAL, 0, A3, 1.0. The seventh columns reads IFAIL, 0.

Figure 30. Example wood model input for user specification of input parameters (option MAT_WOOD). 

This figure presents sets of data in 8 separate columns. The first column reads MID, 143, EL, 11350.0, XT, 85.2, GF1par, 4.27, FLPAR, 0, NPAR, 0.5, AOPT, 2, XP, D1, 1.0. The second column reads RO, 6.73 minus 04, ET, 246.8, XC, 21.2, GF2par, 8.83, FLPARC, 0, CPAR, 400.0, YP, D2, 0.0. The third column reads NPLOT, 0, GLT, 715.2, YT, 2.05, B, 30.0, POWPAR, 0, NPER, 0.4, ZP, D3, 0.0. The fourth column reads ITERS, 0, GTR, 87.5, YC, 4.08, DMAXpar, 0.9999, FLPER, 0, CPER, 100.0, A1, 0.0. The fifth column reads IRATE, 0, PR, 0.157, SXY, 9.1, GF1per, 0.04, FLPERC, 0, A2, 0.0. The sixth column reads GHARD, 0.0, SYZ, 12.7, GF2per, 0.83, POWPER, 0, A3, 1.0. The seventh line reads IFAIL, 0, D, 30.0. The eighth column reads DMAXper, 0.99.

Figure 31. Example single-element stress-strain results for clear wood pine. Graphs A and B.

(A) Parallel.  This graph shows two connected lines. Line A is solid red and represents tension; line B is solid green and represents compression. The vertical axis of this graph ranges from negative 20 to positive 90 and represents Z-stress (megapascals) while the horizontal axis of this graph ranges from negative 0.05 to positive 0.125 and represents Z-displacement (centimeters). Line B begins at the points of negative 20 on the vertical axis and negative 0.05 on the horizontal axis where it runs parallel with the horizontal axis for a short time before sloping upward to the zero points of both the vertical and horizontal axes. From this point the green line B turns into red line A and continues to climb to points 85 on the vertical axis and positive 0.02 on the horizontal axis where it peaks. The red line then slopes downward and off the graph at points 1 on the vertical axis and 0.125 on the horizontal axis.
 
(B) Perpendicular.  This graph shows two connected lines. Line A is solid red and is labeled tension; line B is solid green and is labeled compression. The vertical axis of this graph ranges from negative 4 to positive 2 and represents Z-stress (megapascals) while the horizontal axis of this graph ranges from negative 0.0125 to 0.05 and represents Z-displacement. Line B begins at the points of negative 4 on the vertical axis and negative 0.125 on the horizontal axis where it runs parallel with the horizontal axis for a short time before sloping upward to the points of 0 on the vertical axis and 0 on the horizontal axis. From this point the green line turns into red and continues its climb to the points of 2 on the vertical axis and positive 0.025 on the horizontal axis where it peaks. Line A then slopes downward and off the graph at the points of 0 on the vertical axis and 0.0525 on the horizontal axis.

Figure 32. Deformed configuration of post at 40 milliseconds, including fringes of damage. Diagram. 

This diagram shows two distinct images. The first image is a colored scale of fringe levels, ranging from red to blue. The scale reads as follows: red equals 9.900e minus 0.01 to 8.910e minus 01; orange equals 8.910e minus 01 to 7.920e minus 01; yellow equals 7.920e minus 01 to 6.930e minus 01; pale green equals 6.930e minus 01 to 5.940e minus 01; light green equals 5.940e minus 01 to 4.950e minus 01; dark green equals 4.950e minus 01 to 3.960e minus 01; aqua equals 3.960e minus 01 to 2.970e minus 01; pale blue equals 2.970e minus 01 to 1.980e minus 01; light blue 1.980e minus 01 to 9.900e minus 02; and dark blue equals 9.900e minus 02 to 0.000e plus 00.

The second image is that of a rectangular column. The majority of the structure is colored a dark blue and is composed of a grid of small squares, roughly forty rows high and eight columns wide. The upper top three-quarters of this structure tilts to the right, forced by a cylindrical apparatus attached to a rectangular arm. The tilting of the structure has caused a severe gap to form nine rows from the bottom and seven columns in. The exact point where the gap has formed is dark red, both above and below the gap, which fades to yellow and then dark green. The last column on the structure, behind the gap, is colored dark red. The points where the cylindrical apparatus strikes the structure are colored dark red, which fades to light green.

Figure 33. Post deflection and cross-sectional force histories. Graphs, A and B.

(A) Deflection.  This graph shows one line colored solid red and labeled A 22. The vertical axis of this graph ranges from 0 to 400 and represents X-displacement (millimeters) while the horizontal axis of this graph ranges from 0 to 40 and represents deflection (milliseconds). The line begins at the zero points of both the vertical and horizontal axes. It runs diagonally across and off the graph at the points of 380 on the vertical axis and 40 on the horizontal axis.
 
(B) Force.  This graph shows one line colored a solid red and labeled A 1. The vertical axis of this graph ranges from negative 5 to positive 50 and represents X-force (kilonewtons) while the horizontal axis of this graph ranges from 0 to 40 and represents time (milliseconds). This jagged line, which begins at the zero points of both the vertical and horizontal axes, moves sharply upward at first and peaks at 48 on the vertical axis and 4 on the horizontal axis, followed by a precipitous drop in several quick steps to 2 on the vertical axis and 6 on the horizontal axis and then a zigzag course off the graph that ranges between negative 5 and positive 11 on the vertical axis.

Figure 34. Measured load displacement curves of southern yellow pine exhibit variability in tension parallel to the grain. Source: Forest Products Laboratory. Graphs, A, B, C, and D.

(A) 4 percent moisture content.  This graph shows one distinct, solid black line among a multitude of red lines clustered about the black line’s trail. The vertical axis of this graph ranges from 0 to 10 and represents load (kilonewtons) while the horizontal axis of this graph ranges from 0 to 7.5 and represents displacement (millimeters). The black line begins at the zero points of both the vertical and horizontal axes, runs diagonally across the graph to the points of 5.8 on the vertical axis and 3.5 on the horizontal axis, and there comes to an end.
 
(B) 8 percent moisture content.  This graph shows one distinct black line among a multitude of red lines clustered about the black line’s trail. The vertical axis of this graph ranges from 0 to 10 and represents load (kilonewtons) while the horizontal axis of this graph ranges from 0.0 to 0.75 and represents displacement (millimeters). The black line begins at the zero points on both the vertical and horizontal axes, zigzags up to the points of 6 on the vertical axis and 0.45 on the horizontal axis, and there comes to an end.
 
(C) 12 percent moisture content.  This graph shows one distinct, solid black line among a multitude of red lines clustered about the black line’s trail. The vertical axis of this graph ranges from 0 to 10 and represents load (kilonewtons) while the horizontal axis of this graph ranges from 0 to 12 and represents displacement (millimeters). The black line begins at the points of 1.75 on the vertical axis and zero on the horizontal axis, runs diagonally across the graph to the points of 6 on the vertical axis and 4 on the horizontal axis, and there comes to an end.
 
(D) Saturated.  This graph shows one distinct, solid black line among a multitude of red lines clustered about the black line’s trail. The vertical axis of this graph ranges from 0 to 10 and represents load (kilonewtons) while the horizontal axis of this graph ranges from 0 to 18 and represents displacement (millimeters). The black line begins at the zero points of both the vertical and horizontal axes, slopes upward in the slightest arc to the points of 4.75 on the vertical axis and 5 on the horizontal axis, and there comes to an end.

Figure 35. Measured load displacement curves of southern yellow pine exhibit variability in compression perpendicular to the grain. Graphs, A, B, C, and D.

(A) 4-percent moisture content.  This graph shows one distinct, solid black line among a multitude of red lines clustered about the black line’s trail. The vertical axis of this graph ranges from 0 to 10 and represents load (kilonewtons) while the horizontal axis of this graph ranges from 0.0 to 2.0 and represents displacement (millimeters). The black line begins at the zero points of both the vertical and horizontal axes, slopes upward to the points of 5.75 on the vertical axis and 1.2 on the horizontal axis, and there comes to an end.
 
(B) 8-percent moisture content.  This graph shows on distinct black line among a multitude of red lines clustered about the black line’s trail. The vertical axis of this graph ranges from 0 to 8 and represents load (kilonewtons) while the horizontal axis of this graph ranges from 0.0 to 1.5 and represents displacement (millimeters). The black line begins at the zero points of both the vertical and horizontal axes, slopes upward to the points of 5 on the vertical axis and 1.0 on the horizontal axis, and there comes to an end.
 
(C) 12-percent moisture content.  This graph shows one distinct, solid black line among a multitude of red lines clustered about the black line’s trail. The vertical axis of this graph ranges from 0 to 6 and represents load (kilonewtons) while the horizontal axis of this graph ranges from 0.0 to 2.5 and represents displacement (millimeters). The black line begins at the zero points of both the vertical and horizontal axes, slopes upward to the points of 2.75 on the vertical axis and 1.3 on the horizontal axis, and there comes to an end.
 
(D) Saturated.  This graph shows one distinct, solid black line among a multitude of red lines clustered about the black line’s trail. The vertical axis of this graph ranges from 0.0 to 2.5 and represents load (kilonewtons) while the horizontal axis of this graph ranges from 0.0 to 2.0 and represents displacement (millimeters). The black line begins at the zero points of both the vertical and horizontal axes, slopes upward to the points of 1.5 on the vertical axis and 1.3 on the horizontal axis, and there comes to an end.

Figure 36. Effect of moisture content on tensile modulus parallel to the grain. Source: Forest Products Laboratory. Graphs A and B.

(A) Data.  The vertical axis of this graph ranges from 0 to 4 and represents modulus parallel parenthesis times 10 superscript 6 measured in pounds per square inch, close parenthesis,while the horizontal axis of this graph ranges from 0 to 25 and represents moisture content (percent). A vertical cluster of points at 4.5 percent moisture content ranges between 1.3 and 3.6 million pounds per square inch. A vertical cluster of points at 7 percent moisture content ranges between 1.4 and 3.8 million pounds per square inch. A vertical cluster of points at 11 percent moisture content ranges between 1.5 and 3.2 million pounds per square inch. A cluster of points between 12 and 13 percent moisture content ranges between 1.1 and 3.0 million pounds per square inch. A cluster of points between 16 and 20 percent moisture content ranges between 1.0 and 3.1 million pounds per square inch. A vertical cluster of points at 23 percent moisture content ranges between 0.9 and 2.4 million pounds per square inch.
 
(B) Fit.  This graph shows one distinct, solid red line. The left-hand vertical axis of this graph ranges from 0.0 to 4.0 and represents modulus parallel parenthesis times 10 superscript 6 measured in pounds per square inch, close parenthesis, while the horizontal axis of this graph ranges from 0 to 25 and represents moisture content. The right-hand vertical axis of this graph ranges from 0.0 to 20.7 that represents modulus parallel in gigapascals. The red line begins at the points of 2.4 on the left-hand vertical axis and zero on the horizontal axis where it arcs over to the points of 1.5 per the left-hand vertical axis and 25 on the horizontal axis.

Figure 37. Effect of moisture content on tensile modulus perpendicular to the grain. Source: Forest Products Laboratory. Graphs A and B.

(A) Data.  The vertical axis of this graph ranges from 0 to 400 and represents modulus perpendicular parenthesis times 10 superscript 6 measured in pounds per square inch, close parenthesis,while the horizontal axis of this graph ranges from 0 to 25 and represents moisture content (percent). A vertical cluster of points at 4 percent moisture content ranges between 100 and 200 thousand pounds per square inch. A vertical cluster of points at 7 percent moisture content ranges between 110 and 210 thousand pounds per square inch. A cluster of points between 17 and 20 percent moisture content ranges between 60 and 110 thousand pounds per square inch. A vertical cluster of points at 23 percent moisture content ranges between 10 and 100 thousand pounds per square inch.
 
(B) Fit.  This graph shows one distinct, solid red line. The left-hand vertical axis of this graph ranges from 0.00 to 0.40 and represents modulus parallel parenthesis times 10 superscript 6 measured in pounds per square inch, close parenthesis, while the horizontal axis of this graph ranges from 0 to 25 and represents moisture content (percent). The right-hand vertical axis of this graph ranges from 0.00 to 2.07 and represents modulus parallel in gigapascals. The red line begins at the points of 0.14 on the left-hand vertical axis and zero on the horizontal axis, and it arcs over to the points of 0.01, per the left-hand vertical axis, and 25 on the horizontal axis.

Figure 38. Effect of moisture content on tensile strength parallel to the grain. Source: Forest Products Laboratory. Graphs A and B.

(A) Data.  The vertical axis of this graph ranges from 0 to 40 and represents tensile stress parallel parenthesis times 10 superscript 3 measured in pounds per square inch, close parenthesis, while the horizontal axis of this graph ranges from 0 to 25 and represents moisture content (percent). A vertical cluster of points at 4.5 percent moisture content ranges between 5 and 30 thousand pounds per square inch. A vertical cluster of points at 7 percent moisture content ranges between 10 and 33 thousand pounds per square inch. A cluster of points between 11 and 13.5 percent moisture content ranges between 11 and 29 thousand pounds per square inch. A cluster of points between 16 and 20.5 percent moisture content ranges between 10 and 29 thousand pounds per square inch. A vertical cluster of points at 23 percent moisture content ranges between 9 and 22 thousand pounds per square inch.
 
(B) Fit.  This graph shows one distinct solid red line. The left-hand vertical axis of this graph ranges from 0 to 40,000 and represents tensile strength parallel measured in pounds per square inch while the horizontal axis of this graph ranges from 0 to 25 and represents moisture content. The right-hand vertical axis of this graph ranges from 0.00 to 0.21 and represents tensile strength parallel measured in gigapascals. The red line begins at the points of 12,000 on the left-hand vertical axis and zero on the horizontal axis, and it arcs over to the points of 0.06 per the left-hand vertical axis and 25 on the horizontal axis.

Figure 39. Effect of moisture content on tensile strength perpendicular to the grain. Source: Forest Products Laboratory. Graphs A and B.

(A) Data.  The vertical axis of this graph ranges from 0 to 40 and represents tensile stress perpendicular  parenthesis times 10 superscript 3 measured in pounds per square inch, close parenthesis,while the horizontal axis of this graph ranges from 0 to 25 and represents moisture content (percent). A vertical cluster of points at 4.5 percent moisture content ranges between 0.4 and 0.83 thousand pounds per square inch. A vertical cluster of points at 6.5 percent moisture content ranges between 0.45 and 0.91 thousand pounds per square inch. A cluster of points between 10 and 13 percent moisture content ranges between 0.48 and 0.98 thousand pounds per square inch. A cluster of points between 16 and 20.5 percent moisture content ranges between 0.3 and 7 thousand pounds per square inch. A vertical cluster of points at 23 percent moisture content ranges between 0.1 and 0.5 thousand pounds per square inch.
 
(B) Fit.  This graph shows one distinct, solid red line. The left-hand vertical axis of this graph ranges from 0 to 1,000 and represents tensile strength perpendicular (in pounds per square inch) while the horizontal axis of this graph ranges from 0 to 25 and represents moisture content. The right-hand vertical axis of this graph ranges from 0.00 to 6.89 and represents tensile strength perpendicular measured in megapascals. The red line begins at the points of 400 on the left-hand vertical axis and zero on the horizontal axis, and it arcs over to the points of 180 per the left-hand vertical axis and 25 on the horizontal axis.

Figure 40. Effect of moisture content on compressive strength parallel to the grain. Source: Forest Products Laboratory. Graphs A and B.

(A) Data.  The vertical axis of this graph ranges from 0 to 40 and represents compressive stress parallel parenthesis times 10 superscript 3 measured in pounds per square inch, close parenthesis,while the horizontal axis of this graph ranges from 0 to 25 and represents moisture content (percent). A vertical cluster of points at 4.5 percent moisture content ranges between 6 and 16 thousand pounds per square inch. A vertical cluster of points at 7 percent moisture content ranges between 7 and 14 thousand pounds per square inch. A cluster of points between 8 and 13 percent moisture content ranges between 6 and 12 thousand pounds per square inch. A cluster of points between 16.5 and 20.5 percent moisture content ranges between 3.4 and 7.4 thousand pounds per square inch. A vertical cluster of points at 23 percent moisture content ranges between 3 and 4 thousand pounds per square inch.
 
(B) Fit.  This graph shows one distinct solid blue line. The left-hand vertical axis of this graph ranges from 0 to 40,000 and represents compressive strength parallel parenthesis as measured in pounds per square inch, close parenthesis, while the horizontal axis of this graph ranges from 0 to 25 and represents moisture content (percent). The right hand vertical axis of this graph ranges from 0.00 to 27.6 and represents compressive strength parallel as measured in megapascals. The blue line begins at the points of 14,000 on the left-hand vertical axis and zero on the horizontal axis, and it runs across and off the graph diagonally at the points of 25 on the horizontal axis and 2,000 per the left-hand vertical axis.

Figure 41. Effect of moisture content on compressive strength perpendicular to the grain. Source: Forest Products Laboratory. Graphs A and B.

(A) Data.  The vertical axis of this graph ranges from 0 to 4 and represents compressive stress perpendicular parenthesis times 10 superscript 3 as measured in pounds per square inch, close parenthesis, while the horizontal axis of this graph ranges from 0 to 25 and represents moisture content (percent). A vertical cluster of points at 4.5 percent moisture content ranges between 1.2 and 3.2 thousand pounds per square inch. A vertical cluster of points at 7 percent moisture content ranges between 1.1 and 2.7 thousand pounds per square inch. A cluster of points between 8 and 13 percent moisture content ranges between 1.1 and 2.1 thousand pounds per square inch. A cluster of points between 16 and 20.5 percent moisture content ranges between 0.75 and 1.3 thousand pounds per square inch. A vertical cluster of points at 23 percent moisture content ranges between 0.5 and 0.85 thousand pounds per square inch.
 
(B) Fit.  This graph shows one distinct, solid blue line. The left-hand vertical axis of this graph ranges from 0 to 4,000 and represents compressive strength perpendicular as measured in pounds per square inch, while the horizontal axis of this graph ranges from 0 to 25 and represents moisture content (percent). The right-hand vertical axis of this graph ranges from 0.00 to 27.6 and represents compressive strength as measured in megapascals. The blue line begins at the points of 2,500 on the left-hand vertical axis and zero on the horizontal axis, and it runs across and off the graph diagonally at the points of 25 on the horizontal axis and 400 per the left-hand vertical axis.

Figure 42. Effect of moisture content on shear strength parallel to the grain. Source: Forest Products Laboratory. Graphs A and B.

(A) Data.  The vertical axis of this graph ranges from 0 to 4 and represents shear strength parallel to the grain parenthesis times 10 superscript 3 as measured in pounds per square inch, close parenthesis,while the horizontal axis of this graph ranges from 0 to 25 and represents moisture content (percent). A cluster of points between 3.5 and 5 percent moisture content ranges between 2.3 and 3.8 thousand pounds per square inch. A vertical cluster of points between 6 and 7 percent moisture content ranges between 2.2 and 3.7 thousand pounds per square inch. A cluster of points between 9 and 13 percent moisture content ranges between 2 and 3 thousand pounds per square inch. A cluster of points between 16 and 20.5 percent moisture content ranges between 1.6 and 2.4 thousand pounds per square inch. A vertical cluster of points at 23 percent moisture content ranges between 1 and 1.6 thousand pounds per square inch.
 
(B) Fit.  This graph shows one distinct, solid green line. The left-hand vertical axis of this graph ranges from 0 to 4,000 and represents shear strength parallel to grain as measured in pounds per square inch while the horizontal axis of this graph ranges from 0 to 25 and represents moisture content. The right-hand vertical axis of this graph ranges from 0.00 to 27.6 and represents shear strength parallel to grain as measured in megapascals. The green line begins at the points of 2,900 on the left-hand vertical axis and zero on the horizontal axis, and it arcs across to the points 6.9 on the right-hand vertical axis and 25 on the horizontal axis.

Figure 43. Effect of moisture content on mode I fracture intensity. Source: Forest Products Laboratory. Graphs A and B.

(A) Data.  The mode one (roman numeral one) fracture intensity in compression (c) in the tangential-longitudinal (TL) orientation is designated as K subscript IcTL. The vertical axis of this graph ranges from 0 to 700 and represents K subscript IcTL parenthesis times 10 superscript 3 as measured in pounds per inch superscript minus 3 divided by 2, close parenthesis, while the horizontal axis of this graph ranges from 0 to 25 and represents moisture content (percent). A vertical cluster of points at 4.5 percent moisture content ranges between 290 and 580 thousand pounds per inch to the power of 3 divided by 2. A vertical cluster of points at 7 percent moisture content ranges between 310 and 680 thousand pounds per inch to the power of 3 divided by 2. A cluster of points between 10 and 13 percent moisture content ranges between 290 and 600 thousand pounds per inch to the power of 3 divided by 2. A cluster of points between 15 and 19.5 percent moisture content ranges between 280 and 500 thousand pounds per inch to the power of 3 divided by 2. A vertical cluster of points at 23 percent moisture content ranges between 190 and 350 thousand pounds per inch to the power of 3 divided by 2.
 
(B) Fit.  This graph shows one distinct, solid red line. The left-hand vertical axis of this graph ranges from 0 to 700 and represents mode I (roman numeral one) fracture intensity parenthesis as measured in pounds per square inch superscript 3 divided by 2, close parenthesis, while the horizontal axis of this graph ranges from 0 to 25 and represents moisture content (percent). The right-hand vertical axis of this graph ranges from 0.0 to 6.6 and represents mode I (roman numeral one) fracture intensity as measured in megapascals minus centimeter superscript one-half. The green line begins at the points of 400 on the left-hand vertical axis and zero on the horizontal axis, and it arcs across to the points of 2.3 on the right-hand vertical axis and 25 on the horizontal axis.

Figure 44. Effect of moisture content on mode II fracture intensity. Source: Forest Products Laboratory. Graphs A and B.

(A) Data.  The mode two (roman numeral two) fracture intensity in compression (c) in the tangential-longitudinal (TL) orientation is designated as K subscript IIcTL. The vertical axis of this graph ranges from 0 to 3.0 and represents K subscript IIcTL parenthesis times 10 superscript 3 as measured in pounds per inch superscript minus 3 divided by 2, close parenthesis, while the horizontal axis of this graph ranges from 0 to 25 and represents moisture content (percent). A vertical cluster of points at 4.5 percent moisture content ranges between 1.3 and 2.2 thousand pounds per inch to the power of 3 divided by 2. A vertical cluster of points at 7 percent moisture content ranges between 1.3 and 2.6 thousand pounds per inch to the power of 3 divided by 2. A cluster of points between 10 and 13 percent moisture content ranges between 1.5 and 2.5 thousand pounds per inch to the power of 3 divided by 2. A cluster of points between 15 and 20 percent moisture content ranges between 1.2 and 2.6 thousand pounds per inch to the power of 3 divided by 2. A vertical cluster of points at 23 percent moisture content ranges between 0.9 and 1.7 thousand pounds per inch to the power of 3 divided by 2.
 
(B) Fit.  This graph shows one distinct solid blue line. The left-hand vertical axis of this graph ranges from 0 to 3,000 and represents mode II (roman numeral two) fracture intensity (pounds per square inch superscript 3 divided by 2) while the horizontal axis of this graph ranges from 0 to 25 and represents moisture content (percent). The right-hand vertical axis of this graph ranges from 0.00 to 33.0 and represents mode II fracture intensity as measured in megapascals minus centimeter superscript one-half. The green line begins at the points of 1,400 on the left-hand vertical axis and zero on the horizontal axis, and it arcs across to the points of 11.0 on the right-hand vertical axis and 25 on the horizontal axis.

Figure 45. Compressive strength variation of clear wood is readily modeled by a sinusoidal correction in the R-T plane. Source: Krieger Publishing Company. Graph. 

This graph shows the sinusoidal shape of the Hankinson sine curve relative to a straight line. The vertical axis is labeled Relative Compression Strength and is unitless. The mid-point of the axis is labeled F subscript T. The horizontal axis is labeled Ring Angle in degrees and ranges from 0 to 90 degrees, where 0 degrees is the tangential direction designated by T and 90 degrees is the radial direction designated by R. The Hankinson sine curve is drawn as a solid black line. The straight line is dashed. At zero degrees, the value of both the sine curve and straight line is F subscript T. At 90 degrees the value of both the Hankinson model and straight line is F subscript R. The difference between these two points is F subscript R minus F subscript T. This difference is shown at ninety degrees by a line with arrows on each end. Also shown at 90 degrees is the quantity F subscript T plus F subscript R all divided by 2. The sine curve drops below the straight line between 0 and 90 degrees. At 45 degrees, the value of the sine curve is labeled F subscript 45 degrees. The equation shown is F subscript 45 degrees equals left parentheses quantity F subscript T plus F subscript R all divided by 2 right parentheses multiplied by the quantity 1 minus k.

Figure 46. Geometry of an off-axis test specimen. Source: Kreiger Publishing Company. Diagram. 

This diagram shows a cross section of timber in three-dimensional view with the top and side transparent. The diagram also shows two different sets of axes. One set of axes is labeled X subscript 1, X subscript 2, and X subscript 3. These axes correspond with the edges of the timber with is rectangular in shape. The other set of axes is labeled  L and R and T, with L for longitudinal, R for radial, and T for tangential. The longitudinal and radial axes line up with the grain and rings of the wood. The grain and rings are drawn as lines (slightly nonlinear) on the timber. The longitudinal axis is drawn at an angle theta from the X subscript 1 axis. The radial axis is drawn at an angle Psi from the X subscript 2 axis. The tangential axis is drawn at an angle phi from the X subscript 3 axis.

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. Graphs A and B.

(A) Compressive strength.  This graph shows eight distinct lines. Line A is solid pink and is labeled Hoffman. Line B is solid red and is labeled Norris. Line C is solid yellow and is labeled Yamada-Sun. Line D is solid dark green and is labeled Mod Hashin. Line E is solid aqua and is labeled Hashin. Line F is solid blue and is labeled Maximum Stress. Line G is solid gray and is labeled Tsai-Wu. Line H is solid black and is labeled Hankinson. The vertical axis of this graph ranges from 0 to 70 and represents compressive strength (megapascals) while the horizontal axis of this graph ranges from 0 to 90 and represents grain angle (degrees). Lines A, B, C, D, G, and H all begin at the points of 54 on the vertical axis and zero on the horizontal axis; they slope down and off the graph at the points of 10 on the vertical axis and 90 on the horizontal axis. Line E begins at the points of 54 on the vertical axis and zero on the horizontal axis, and it slopes upward to peak at the points of 55 on the vertical axis and 18 on the horizontal axis. Line E then slopes back down and off the graph at the points of 10 on the vertical axis and 90 on the horizontal axis. Line F begins at the points of 54 on the vertical axis and zero on the horizontal axis, and it slopes upward to peak at the points of 64 on the vertical axis and 24 on the horizontal axis. Line F then slopes back down and off the graph at the points of 10 on the vertical axis and 90 on the horizontal axis.
 
(B) Tensile strength.  This graph shows eight distinct lines. Line A is solid pink and is labeled Hoffman. Line B is solid red and is labeled Norris. Line C is solid yellow and is labeled Yamada-Sun. Line D is solid dark green and is labeled Mod Hashin. Line E is solid aqua and is labeled Hashin. Line F is solid blue and is labeled Maximum Stress. Line G is solid gray and is labeled Tsai-Wu. Line H is solid black and is labeled Hankinson. The vertical axis of this graph ranges from 0 to 200 and represents tensile strength (megapascals) while the horizontal axis of this graph ranges from 0 to 90 and represents grain angle (degrees). Lines A, B, C, D, E, G, and H all begin at the points of 145 on the vertical axis and zero on the horizontal axis; they slope down and off the graph at the points of 5 on the vertical axis and 90 on the horizontal axis. Line F begins at the points of 145 on the vertical axis and zero on the horizontal axis, and it slopes upward and peaks at the points of 155 on the vertical axis and 12 on the horizontal axis. Line F then slopes back down and off the graph at the points of 5 on the vertical axis and 90 on the horizontal axis.

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. Graph. 

The vertical axis is labeled Relative Compression Strength with no units and ranges from 0 to 0.3. The horizontal axis is labeled Ring Angle in degrees and ranges from zero to ninety. The graph shows four sets of measured data points and four corresponding predicted lines. The solid black predicted line fits through the open circle data and is labeled Englemann Spruce. The data have values of approximately 0.12 at 0 degrees, with a low of 0.086 at 45 degrees and a maximum of 0.17 at ninety degrees. The long dash black predicted line fits through the open square data and is labeled Douglas Fir. The data have values of approximately 0.11 at 0 degrees, with a low of 0.057 at 45 degrees and 0.1 at ninety degrees. The short dash black predicted line fits through the open triangle data and is labeled Oak. The data have values of approximately 0.23 at 0 degrees, with a low of 0.19 at 45 degrees and a maximum of 0.26 at ninety degrees. The dotted black predicted line fits through the solid black circle data and is labeled Aspen. The data have values of approximately 0.11 at 0 degrees, 0.1 at 45 degrees, and a maximum of 0.15 at ninety degrees.

Figure 49. Failure criteria comparison for perpendicular modes as a function of the ring angle. Graph. 

This graph shows eight distinct lines. Line A is solid pink and labeled Hoffman. Line B is solid red with diamonds and is labeled Norris. Line C is solid yellow and is labeled Yamada-Sun. Line D is solid dark green, dotted, and is labeled Mod Hashin. Line E is solid aqua, dotted, and is labeled Hashin. Line F is solid blue and is labeled Maximum Stress. Line G is solid gray and is labeled Tsai-Wu. Line H is solid black and is labeled Hankinson. The vertical axis of this graph ranges from 0 to 22 and represents compressive strength (megapascals) while the vertical axis of this graph ranges from 0 to 90 and represents ring angle (degrees). Line A begins at the points of 10 on the vertical axis and zero on the horizontal axis, and it slopes up significantly and off the graph at the points of 22 on the vertical axis and 26 on the horizontal axis. Line A then reappears at the points of 22 on the vertical axis and 62 on the horizontal axis, and from there, it slopes down and off the graph at the points of 10 on the vertical axis and 90 on the horizontal axis. Line B begins at the points of 10 on the vertical axis and zero on the horizontal axis, and it curves upwards to the points of 18 on the vertical axis and 43 on the horizontal axis. From there, line B curves back down and off the graph at the points of 10 on the vertical axis and 90 on the horizontal axis. Line F begins at the points of 10 on the vertical axis and zero on the horizontal axis, and it curves upwards to the points of 20 on the vertical axis and 43 on the horizontal axis. From there, line F curves back down and off the graph at the points of 10 on the vertical axis and 90 on the horizontal axis. Line C begins at the points of 10 on the vertical axis and zero on the horizontal axis, and it curves upwards to the points of 18 on the vertical axis and 43 on the horizontal axis. From there, line C curves back down and off the graph at the points of 10 on the vertical axis and 90 on the horizontal axis. Lines D, E, and G all begin at the points of 10 on the vertical axis and zero on the horizontal axis, and they run straight across and off the graph at the points of 10 on the vertical axis and 90 on the horizontal axis. Line H begins at the points of 10 on the vertical axis and zero on the horizontal axis, and it arcs downward to the points of 6 on the vertical axis and 42 on the horizontal axis. From there, line H arcs back upward and off the graph at the points of 10 on the vertical axis and 90 on the horizontal axis.

Figure 50. Predicted effect of perpendicular confinement and extension on the longitudinal strength of southern yellow pine in tension and compression. Graph. 

This graph shows seven distinct lines. Line A is solid pink and is labeled Hoffman. Line B is red, dotted, and labeled Norris. Line C is yellow, dotted, and is labeled Yamada-Sun. Line D is green, dotted, and is labeled Mod Hashin. Line E is pale green and is labeled Hashin. Line F is solid blue and is labeled Max Stress. Line G is solid gray and is labeled Tsai-Wu. (They are presented out of alphabetical order in the key.) The vertical axis of this graph ranges from negative 100 to positive 200 and represents stress parallel to grain (megapascals). The horizontal axis of this graph ranges from negative 25 to positive 15 and represents tangential stress plus radial stress (megapascals). The Hoffman line is originates at minus 24 on the vertical axis and 200 on the vertical axis. It is shaped like a parabola with an apex at 11 on the horizontal axis and 50 on the vertical axis. It goes off the graph at minus 12 on the horizontal axis and minus 100 on the vertical axis. The modified Hashin line is a rectangular box with corners at minus 10.4 and 4.5 on the horizontal axis and minus 50 and 145 on the vertical axis. The Hashin line is a rectangular box with corners at minus 10.6 and 4.5 on the horizontal axis and minus 50 and 145 on the vertical axis. The maximum stress line is a rectangular box with corners at minus 20 and 9 on the horizontal axis and minus 50 and 145 on the vertical axis. The Yamada-Sun line coincides with the maximum stress line. The Norris line is a closed irregular shape with points at minus 20 on the horizontal axis and 0 on the vertical axis, increasing to 0 on the horizontal axis and 145 on the vertical axis,  increasing to 5 on the horizontal axis and a peak of 170 on the vertical axis, decreasing to 9 on the horizontal axis and 145 on the vertical axis, dropping vertically to 0 on the vertical axis,  dropping to minus 20 on the horizontal axis and minus 50 on the vertical axis, and finally increasing vertically to 0 on the vertical axis to close the loop. The Tsai-Wu line is an ellipse with axes orientated at angle to the horizontal and vertical axis of the graph. The low defining point is zero on the horizontal axis with minus 50 on the vertical axis. The high defining point is minus 9.5 on the horizontal axis with 200 on the vertical axis.

Figure 51. Predicted effect of parallel shear and tangential stresses on the longitudinal strength of southern yellow pine in tension and compression. Graphs A and B.

(A) Calculated with parallel shear stress equal to tangential stress.  This graph shows seven distinct lines and one specific point. Line A is solid pink and is labeled Hoffman. Line B is solid red and is labeled Norris. Line C is yellow, dotted, and is labeled Yamada-Sun. Line D is solid green and is labeled Mod Hashin. Line E is pale green, dotted, and is labeled Hashin. Line F is solid blue and is labeled Max Stress. Line G is solid gray and is labeled Tsai-Wu. And point H is black in color and is labeled Hankinson. (They are presented out of alphabetical order in the key.) The vertical axis of this graph ranges from negative 100 to positive 200 and represents longitudinal stress (megapascals). The horizontal axis of this graph ranges from negative 15 to positive 7 and represents tangential stress (megapascals). The Hoffman line is shaped like an ellipse with major and minor axes parallel to the horizontal and vertical axes of the graph. The horizontal maximum and minimum points are 5.5 and minus 10 at 50 vertical. The vertical  maximum and minimum points are 150 and minus 45 at minus 3.5 horizontal. The modified Hashin line is nearly a rectangular box with corners at minus 8.7 and 4.5 on the horizontal axis and minus 55 and 130 on the vertical axis. The Hashin line is nearly a rectangular box with corners at minus 8 and 4.5 on the horizontal axis and minus 50 and 135 on the vertical axis. The maximum stress line is a rectangular box with corners at minus 10 and 4.5 on the horizontal axis and minus 50 and 145 on the vertical axis. The Yamada-Sun line coincides with the modified Hashin line. The Norris line is a closed irregular shape with points at minus 10 on the horizontal axis and 0 on the vertical axis, increasing to 0 on the horizontal axis and 145 on the vertical axis,  remaining constant at 145 until 4 on the horizontal axis, decreasing vertically to 0, decreasing in a nonlinear manner to 0 on the horizontal axis and minus 50 on the vertical axis, remaining constant at minus 50 until minus 8 on the horizontal axis, increasing to minus 9.5 on the horizontal axis and minus 25 on the vertical axis, and finally returning to the starting point at  minus 10 on the horizontal axis and 0 on the vertical axis. The Tsai-Wu line is an ellipse with axes orientated at angle to the horizontal and vertical axis of the graph. The low defining point is minus 1 on the horizontal axis with minus 50 on the vertical axis. The high defining point is minus 8.5 on the horizontal axis with 190 on the vertical axis.
 
(B) Calculated without parallel shear stress.  This graph shows seven distinct lines. Line A is solid pink and is labeled Hoffman. Line B is red, dotted, and is labeled Norris. Line C is yellow and is labeled Yamada-Sun. Line D is green, dotted, and is labeled Mod Hashin. Line E is pale green, dotted, and is labeled Hashin. Line F is solid blue and is labeled Max Stress. Line G is solid gray and is labeled Tsai-Wu. (They are presented out of alphabetical order in the key.) The vertical axis of this graph ranges from negative 100 to positive 200 and represents stress parallel to grain (megapascals). The horizontal axis of this graph ranges from negative 25 to positive 15 and represents tangential stress (megapascals). The Hoffman line is an ellipse with major and minor axes parallel to the horizontal and vertical axes of the graph. The horizontal maximum and minimum points are 11 and minus 5.5 at 50 vertical. The vertical  maximum and minimum points are 155 and minus 45 at minus 2.5 horizontal. The modified Hashin line is a rectangular box with corners at minus 10 and 4.5 on the horizontal axis and minus 50 and 145 on the vertical axis. The Hashin, maximum stress, and Yamada lines coincide with the modified Hashin line. The Norris line is a closed irregular shape with points at minus 10 on the horizontal axis and 0 on the vertical axis, increasing to 0 on the horizontal axis and 145 on the vertical axis, remains constant at 145 until 4.5 on the horizontal axis, decreases vertically to 0, decreases in a nonlinear manner to 0 on the horizontal axis and minus 50 on the vertical axis, remains constant at minus 50 until minus 10 on the horizontal axis, and finally returns to the starting point at minus 10 on the horizontal axis and 0 on the vertical axis. The Tsai-Wu line is an ellipse with axes oriented at angle to the horizontal and vertical axis of the graph. The low defining point is 0 on the horizontal axis with minus 50 on the vertical axis. The high defining point is minus 10 on the horizontal axis with 200 on the vertical axis.

Figure 52. Predicted effect of parallel shear invariant on the longitudinal strength of southern yellow pine in tension and compression. Graphs A and B.

(A) Calculated with parallel shear stress equal to tangential stress.  This graph shows seven distinct lines. Line A is solid pink and is labeled Hoffman. Line B is solid red and is labeled Norris. Line G is solid gray and is labeled Tsai-Wu. Line C is yellow, dotted, and is labeled Yamada-Sun. Line D is solid green and is labeled Mod Hashin. Line E is pale green dotted and is labeled Hashin. Line F is solid blue and is labeled Max Stress. (They are presented out of alphabetical order in the key.) The vertical axis of this graph ranges from negative 70 to positive 170 and represents longitudinal stress (megapascals). The horizontal axis of this graph ranges from 0 to 25 and represents square root of parallel shear invariant (megapascals). The Hoffman line is not shown. The modified Hashin line is shaped like half of an irregular ellipse. It originates at 0 on the horizontal axis and minus 52 on the vertical axis. It reaches a maximum horizontal value at 16.8 with 0 on the vertical axis. Then it curves back to 0 on the horizontal axis and 145 on the vertical axis. The Hashin line is shaped like a portion of a rectangle below 0 on the vertical axis and like a portion of an ellipse above 0 on the vertical axis. It originates at 0 on the horizontal axis and minus 52 on the vertical axis. It reaches a maximum horizontal value at 16.8 with 0 on the vertical axis before extending vertically to 0. Then it curves back to zero on the horizontal axis and 145 on the vertical axis. The Yamada-Sun line coincides with the modified Hashin line. The maximum stress line is half of a rectangular box with corners at 0 and 24 and on the horizontal axis and minus 52 and 145 on the vertical axis. The Norris line is shaped like half of an irregular ellipse. It originates at 0 on the horizontal axis and minus 52 on the vertical axis. It reaches a maximum horizontal value at 23.5 with 0 on the vertical axis. Then it curves back to 0 on the horizontal axis and 145 on the vertical axis. The Tsai-Wu line is half of an ellipse. It originates at 0 on the horizontal axis and minus 52 on the vertical axis. It reaches a maximum horizontal value at 19 with 50 on the vertical axis. Then it curves back to 0 on the horizontal axis and 145 on the vertical axis.
 
(B) Calculated without parallel shear stress.  This graph shows seven distinct lines. Line A is solid pink and is labeled Hoffman. Line B is solid red and is labeled Norris. Line C is yellow, dotted, and is labeled Yamada-Sun. Line D is solid green and is labeled Mod Hashin. Line E is pale green, dotted, and is labeled Hashin. Line F is solid blue and is labeled Max Stress. Line G is solid gray and is labeled Tsai-Wu. (They are presented out of alphabetical order in the key.) The vertical axis of this graph ranges from negative 70 to positive 170 and represents longitudinal stress (megapascals). The horizontal axis of this graph ranges from 0 to 25 and represents square root of parallel shear invariant (megapascals). The Hoffman line is not shown. The modified Hashin line is shaped like half of an irregular ellipse. It originates at 0 on the horizontal axis and minus 52 on the vertical axis. It reaches a maximum horizontal value at 16.8 with 0 on the vertical axis. Then it curves back to 0 on the horizontal axis and 145 on the vertical axis. The Hashin line is shaped like a portion of a rectangle below 0 on the vertical axis and like a portion of an ellipse above 0 on the vertical axis. It originates at 0 on the horizontal axis and minus 52 on the vertical axis. It reaches a maximum horizontal value at 16.8 with 0 on the vertical axis before extending vertically to 0. Then it curves back to zero on the horizontal axis and 145 on the vertical axis. The Yamada-Sun and Norris lines coincide with the modified Hashin line. The maximum stress line is half of a rectangular box with corners at 0 and 17 and on the horizontal axis and minus 52 and 145 on the vertical axis. The Tsai-Wu line is half of an ellipse. It originates at 0 on the horizontal axis and minus 52 on the vertical axis. It reaches a maximum horizontal value at 19 with 50 on the vertical axis. Then it curves back to 0 on the horizontal axis and 145 on the vertical axis.

Figure 53. Predicted strength of southern yellow pine perpendicular to the grain (no perpendicular shear stress applied). Graph. 

This graph shows seven distinct lines. Line A is solid pink and is labeled Hoffman. Line B is red, dotted, and is labeled Norris. Line C is yellow, slashed, and is labeled Yamada-Sun. Line D is green, dotted, and is labeled Mod Hashin. Line E is solid pale green and is labeled Hashin. Line F is solid blue and is labeled Max Stress. Line F is solid gray and is labeled Tsai-Wu. The vertical axis of this graph ranges from negative 27 to positive 27 and represents radial stress (megapascals). The horizontal axis of this graph ranges from negative 28 to positive 28 and represents tangential stress (megapascals). The Hoffman line is a half of an ellipse with major axis at a positive 45 degree angle to the horizontal axis of the graph. It originates at minus 28 on the horizontal axis and minus 28 on the vertical axis, then continues horizontally to  minus 12 on the horizontal axis and minus 28 on the vertical axis. It curves upwards to an apex at 4.5 on the horizontal axis and 4.5 on the vertical axis, curves back down to minus 28 on the horizontal axis and minus 12 on the vertical axis, then drops vertically to its starting point at 0 on the horizontal axis and minus 28 on the vertical axis. The modified Hashin line is an ellipse with major axis at a negative 45 degree angle to the horizontal axis of the graph and is centered at zero on the horizontal axis and zero on the vertical axis. It ranges from minus 24 to positive 24 on the horizontal axis, and minus 24 to positive 24 on the vertical axis. Its maximum width at 0 on the horizontal axis is from minus 10 to 5 on the vertical axis. The Hashin line nearly coincides with the modified Hashin line, except it is vertically narrower on each horizontal end. The Yamada-Sun line is a rectangular box with corners at minus 10 and 5 and on the horizontal axis and minus 10 and 5 on the vertical axis. The maximum stress line coincides with the Yamada-Sun line. The Norris line coincides in part with the Hoffman line beyond minus 10 on the horizontal and 0 on the vertical, and with the Yamada-Sun line for the remainder of the curve. The Tsai-Wu line is line is an ellipse with major axis at a negative 45 degree angle to the horizontal axis of the graph and is centered at zero on the horizontal axis and zero on the vertical axis. It ranges from minus 27 to positive 24.5 on the horizontal axis, and minus 27 to positive 24.5 on the vertical axis. It maximum width at 0 on the horizontal axis is from minus 10 to 5 on the vertical axis

Figure 54. Shape of the failure surface is sensitive to the perpendicular shear strength if the criteria are transversely isotropic. Graph. 

This graph shows eleven distinct lines. Line A is solid pink and is labeled Hoffman. Line B is solid purple and labeled Max Stress. Line C is solid green and labeled Mod Hashin, 23.7 megapascals. Line D is solid blue and labeled Hashin, 23.7 megapascals. Line E is solid gray and labeled Tsai-Wu, 23.7 megapascals. Line F is green, dotted, and labeled Mod Hashin, 10.1 megapascals. Line G is blue, dotted, and labeled Hashin, 10.1 megapascals. Line H is gray, dotted, and labeled Tsai-Wu, 10.1 megapascals. Line I is green, dashed, and labeled Mod Hashin 5.6 megapascals. Line J is blue, dashed, and labeled Hashin 5.6 megapascals. Line K is gray, dashed, and is labeled Tsai-Wu, 5.6 megapascals. The vertical axis of this graph ranges from negative 28 to positive 28 and represents radial stress (megapascals). The horizontal axis of this graph ranges from negative 28 to positive 28 and represents tangential stress (megapascals). The Hoffman line is a half of an ellipse with major axis at a positive 45 degree angle to the horizontal axis of the graph. It originates at minus 28 on the horizontal axis and minus 28 on the vertical axis, then continues horizontally to minus 12 on the horizontal axis and minus 28 on the vertical axis. It curves upwards to an apex at 4.5 on the horizontal axis and 4.5 on the vertical axis, curves back down to minus 28 on the horizontal axis and minus 12 on the vertical axis, then drops vertically to its starting point at minus 28 on the horizontal axis and minus 28 on the vertical axis. The maximum stress line is a rectangular box with corners at minus 10 and 5 and on the horizontal axis and minus 10 and 5 on the vertical axis. The modified Hashin line labeled 23.7 megapascals is an ellipse with its major axis at a negative 45 degree angle to the horizontal axis of the graph and centered at zero on the horizontal axis and zero on the vertical axis. It ranges from minus 23.7 to positive 23.7 on the horizontal axis, and minus 23.7 to positive 23.7 on the vertical axis. Its maximum width at 0 on the horizontal axis is from minus 10 to 5 on the vertical axis. The Hashin line labeled 23.7 megapascals nearly coincides with the modified Hashin line, except it is vertically narrower on each horizontal end. The Tsai-Wu line labeled 23.7 megapascals is an ellipse with major axis at a negative 45 degree angle to the horizontal axis of the graph and is centered at zero on the horizontal axis and zero on the vertical axis. It ranges from minus 27 to positive 24.5 on the horizontal axis, and minus 27 to positive 24.5 on the vertical axis. It maximum width at 0 on the horizontal axis is from minus 10 to 5 on the vertical axis. The modified Hashin line labeled 10.1 megapascals is an ellipse with its major axis at a negative 45 degree angle to the horizontal axis of the graph and centered at zero on the horizontal axis and zero on the vertical axis. It ranges from minus 10.1 to positive 10.1 on the horizontal axis, and minus 10.1 to positive 10.1 on the vertical axis. Its maximum width at 0 on the horizontal axis is from minus 10 to 5 on the vertical axis. The Hashin line labeled 10.1 megapascals nearly coincides with the modified Hashin line, except it is vertically narrower on each horizontal end. The Tsai-Wu line labeled 10.1 megapascals is an ellipse with major axis at a negative 45 degree angle to the horizontal axis of the graph and is centered at zero on the horizontal axis and zero on the vertical axis. It ranges from minus 12 to positive 10 on the horizontal axis, and minus 12 to positive 10 on the vertical axis. It maximum width at 0 on the horizontal axis is from minus 10 to 5 on the vertical axis. The modified Hashin line labeled 5.6 megapascals is a smooth closed curve. It ranges from minus 12.5 to positive 6.5 on the horizontal axis, and minus 12.5 to positive 6.5 on the vertical axis. Its maximum width at 0 on the horizontal axis is from minus 10 to 5 on the vertical axis. The Hashin line labeled 5.6 megapascals is nearly an ellipse with its major axis at 45 degrees to the horizontal axis. It ranges from minus 26 to positive 6.5 on the horizontal axis, and minus 26 to positive 6.5 on the vertical axis. Its maximum width at 0 on the horizontal axis is from minus 10 to 5 on the vertical axis. The Tsai-Wu line labeled 5.6 megapascals is an ellipse with its major axis at a 45 degree angle to the horizontal axis of the graph. It ranges from minus 10 to positive 5 on the horizontal axis, and minus 10 to positive 5 on the vertical axis. Its maximum width at 0 on the horizontal axis is from minus 10 to 5 on the vertical axis.

Figure 55. Combinations of perpendicular and shear stresses that satisfy the failure criteria in the isotropic plane. Graphs A and B.

(A) Calculated with S subscript perpendicular equals 23.7 megapascals.  This graph shows seven distinct lines and one specific point. Line A is solid pink and is labeled Hoffman. Line B is red, dotted, and is labeled Norris. Line C is solid yellow and is labeled Yamada-Sun. Line D is solid green and is labeled Mod Hashin. Line E is blue, dotted, and is labeled Hashin. Line F is solid blue and is labeled Max Stress. Line G is solid gray and labeled Tsai-Wu. (These are listed out of alphabetical order in the key.) The specific point is solid black and is labeled Hankinson. The vertical axis of this graph ranges from negative 28 to positive 28 and represents perpendicular shear stress (megapascals). The horizontal axis of this graph ranges from negative 15 to positive 7 and represents tangential stress (megapascals). The Hoffman line is similar to a parabola open to the left and symmetric about the horizontal axis at zero. Its apex is at 4 on the horizontal axis and zero on the vertical axis. It extends off of the graph at minus 1.8 on the horizontal axis and plus or minus 28 on the vertical axis. The maximum stress line is nearly a rectangular box with corners at minus 10 and 4.5 and on the horizontal axis and minus 23.7 and 23.7 on the vertical axis. The Yamada-Sun line is a closed smooth line similar to an irregular circle. It ranges from minus 10 to 4.5 on the horizontal axis and from minus 23.7 to positive 23.7 on the vertical axis. Its maximum width at 0 on the horizontal axis is from minus 23.7 to 23.7 on the vertical axis. The Norris line coincides with the Yamada-Sun line. The Tsai-Wu line is an ellipse with major axis parallel to the vertical axis of the graph and centered at minus 1.5 on the horizontal axis and zero on the vertical axis. It ranges from minus 5 to positive 2 on the horizontal axis, and minus 25 to positive 25 on the vertical axis. It maximum width at 0 on the horizontal axis is from minus 10 to 5 on the vertical axis. The modified Hashin line is similar to an ellipse with its major axis parallel to the vertical axis of the graph. It ranges from minus 5 to positive 2 on the horizontal axis, and minus 23.7 to positive 23.7 on the vertical axis. Its maximum width at 0 on the horizontal axis is from minus 10 to 5 on the vertical axis. The Hashin line nearly coincides with the modified Hashin line, except it is horizontally narrower on each vertical end. The Hankinson point is located at minus 3 on the horizontal axis and minus 3 on the vertical axis.
 
(B) Calculated with S subscript perpendicular equals 5.58 megapascals.  This graph shows seven distinct lines and one specific point. Line A is solid pink and is labeled Hoffman. Line B is red, dotted, and is labeled Norris. Line C is solid yellow and is labeled Yamada-Sun. Line D is solid green and is labeled Mod Hashin. Line E is blue, dotted, and labeled Hashin. Line F is solid blue and is labeled Max Stress. Line G is solid gray and is labeled Tsai-Wu. The specific point is colored black and is labeled Hankinson. (These are out of alphabetical order in the key.) The vertical axis of this graph ranges from negative 28 to positive 28 and represents perpendicular shear stress (megapascals). The horizontal axis of this graph ranges from negative 15 to positive 7 and represents tangential stress (megapascals).

The Hoffman line is similar to a parabola open to the left and symmetric about the horizontal axis at zero. Its apex is at 4 on the horizontal axis and zero on the vertical axis. It extends off of the graph at minus 14 on the horizontal axis and plus or minus 11.8 on the vertical axis. The maximum stress line is nearly a rectangular box with corners at minus 10 and positive 4.5 and on the horizontal axis and minus 5.5 and positive 5.5 on the vertical axis. The Yamada-Sun line is an ellipse with major axis parallel to the horizontal axis of the graph and centered at minus 2.75 on the horizontal axis and 0 on the vertical axis. It ranges from minus 5.5 to positive 5.5 on the horizontal axis and from minus 10 to positive 4.5 on the vertical axis. Its maximum width at 0 on the horizontal axis is from minus 10 to positive 5 on the vertical axis. The Norris line coincides with the Yamada-Sun line. The Tsai-Wu line is an ellipse with major axis parallel to the vertical axis of the graph and centered at minus 1.5 on the horizontal axis and zero on the vertical axis. It ranges from minus 7 to positive 2.5 on the horizontal axis, and minus 6 to positive 6 on the vertical axis. The modified Hashin line is similar to an ellipse with its major axis parallel to the vertical axis of the graph. It ranges from minus 11.7 to positive 2.5 on the horizontal axis, and minus 5.5 to positive 5.5 on the vertical axis. The Hashin line is similar to an ellipse except the ellipse extends off of the left side of the graph at minus 14 on the horizontal axis and plus and minus 4 on the vertical axis. It coincides with the modified Hashin curve at 2.5 on the horizontal axis and zero on the vertical axis. Its width at 0 on the horizontal axis is from minus 6 to positive 6 on the vertical axis. The Hankinson point is located at minus 3 on the horizontal axis and minus 3 on the vertical axis.

Figure 56. Single-element input file. Screen capture. 

This file is an example input file for an LS-DYNA single element simulation. A star precedes every LS-DYNA keyword command and is a required symbol. Every keyword command is followed by the input required for that particular command. The word input is used to designate the required input. A dollar sign precedes every comment. The title of the input file is Tpainn and is displayed by Notepad. A listing of the keywords and associated input begins here. All blank comment lines are ignored because they are irrelevant.

star KEYWORD

star TITLE Uniaxial Tension in Parallel Direction

Comment  Units: MPa cm

Comment  Control Ouput

star CONTROL_TERMINATION

comment endtim   

input 50.00

star CONTROL_TIMESTEP

comment dtinit  scft  isdo  tslimt  dtms  lctm  erode  ms1st

no input

star CONTROL_OUTPUT

comment  npopt  neecho  nrefup  iaccop  opifs  ipnint  ikedit

input        0        0

star CONTROL_ENERGY

comment   hgen   rwen   slnten   rylen

input         1        1

star DATABASE_BINARY_D3PLOT

comment      dt     lcdt

input      0.20

star DATABASE_BINARY_D3THDT

comment      dt     lcdt

input      0.20

star DATABASE_EXTENT_BINARY

comment   neiph   neips   maxint   strflg   sigflg    epsflg   rltflg   engflg

no input

comment  cmpflg  ieverp  beamip

input                  0

Figure 57. First continuation of single-element input file. Screen capture. 

This file is a continuation of an example input file for an LS-DYNA single element simulation. A star precedes every LS-DYNA keyword command. Every keyword command is followed by the input required for that particular command. A dollar sign precedes every comment. A listing of the keywords and associated input begins here.

star DATABASE_GLSTAT

comment       dt

input     0.2000

star DATABASE_MATSUM

comment       dt

input     0.2000

comment  Define Parts, Sections, and Materials

star PART

comment      pid       sid       mid     eosid      hgid    adpopt

input Southern Yellow Pine

input          1         1       143

star SECTION_SOLID

comment      sid    elform

input          1         1

star HOURGLASS

input          1         5

star MAT_WOOD_PINE

comment      MID        RO     NPLOT     ITERS     IRATE      HARD     IFAIL

input        143   6.73-04         0         0         0         0         0

comment     MOIS      TEMP        QT        QC     UNITS     IQUAL

input          0         0        -2         0         1         0

comment     AOPT

input          2

comment       XP        YP        ZP        A1        A2        A3

input                                      0.0       0.0       1.0

comment       D1        D2        D3

input        1.0       0.0       0.0

Figure 58. Second continuation of single-element input file. Screen capture. 

This file is a continuation of an example input file for an LS-DYNA single element simulation. A star precedes every LS-DYNA keyword command. Every keyword command is followed by the input required for that particular command. A dollar sign precedes every comment. A listing of the keywords and associated input begins here

comment  Define Nodes and Elements

star NODE

comment   node               x               y               z      tc      rc

input        1    0.000000E+00    0.000000E+00    0.000000E+00       7       7

input        2    2.5400000000    0.000000E+00    0.000000E+00       5

input        3    2.5400000000    2.540000E+00    0.000000E+00       3

input        4    0.0000000000    2.540000E+00    0.000000E+00       6

input        5    0.0000000000    0.000000E+00    2.540000E+00       4

input        6    2.5400000000    0.000000E+00    2.540000E+00       2

input        7    2.5400000000    2.540000E+00    2.540000E+00       0

input        8    0.0000000000    2.540000E+00    2.540000E+00       1

star ELEMENT_SOLID

comment    eid     pid      n1      n2      n3      n4      n5      n6      n7      n8

input         1       1       1       2       3       4       5       6       7       8

comment  Define Loads

star BOUNDARY_PRESCRIBED_MOTION_NODE

comment      nid       dof       vad      lcid        sf       vid

input          5         3         0         1 1.000E+00

input          6         3         0         1 1.000E+00

input          7         3         0         1 1.000E+00

input          8         3         0         1 1.000E+00

star DEFINE_CURVE

comment     lcid      sidr      scla      sclo      offa      offo

input          1

comment           abscissa            ordinate

input                0.000              0.00254

input            500.00000           0.00254

star END

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United States Department of Transportation - Federal Highway Administration