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Stuff for the logging industry:

Doyle's Log Table.

Doyle's Log Table is a method of computing the usable board footage of a log given the log's general size. The diameter is measured in inches is for the small end of the log, inside the bark. The length is measured in feet..

Diameter of log
(inches)		 Length of log (feet)
6 7 8 9 10 11 12 13 14 15 16 17
11 18 21 24 28 31 34 37 40 43 46 49 52
12 24 28 32 36 40 44 48 52 56 60 64 68
13 30 35 40 45 50 55 61 66 71 76 81 86
14 37 44 50 56 62 69 75 81 88 94 100 106
15 45 53 60 68 75 83 91 98 106 113 121 128
16 54 63 72 81 90 99 108 117 126 135 144 153
17 63 74 84 95 106 116 127 137 148 158 169 180
18 73 85 98 110 122 135 147 159 171 183 196 208
19 84 98 112 127 141 155 169 183 197 211 225 239
20 96 112 128 144 160 176 192 208 224 240 256 272
21 108 126 144 162 181 199 217 235 253 271 289 307
22 121 141 162 182 202 223 243 263 283 303 324 344
23 135 156 180 203 226 248 271 293 313 336 361 383
24 150 175 200 225 250 275 300 325 350 375 400 425
25 165 193 220 248 276 303 331 358 386 413 441 469
26 181 211 242 272 302 334 363 393 423 453 484 514
27 198 231 264 297 330 363 397 430 463 496 529 563
28 216 252 288 324 360 396 432 468 504 540 576 612
29 234 273 312 352 391 430 469 508 547 586 625 664
30 253 295 338 380 422 465 507 549 591 633 676 718
31 273 319 364 410 456 502 547 592 638 683 729 774
32 294 343 392 441 490 539 588 637 686 735 784 833
33 315 368 420 473 526 578 631 684 736 789 841 894
34 337 393 450 506 562 619 675 731 787 844 900 956
35 360 420 480 540 601 661 721 781 841 901 961 1021
36 384 448 512 576 640 704 768 832 896 960 1024 1088
37 408 476 544 613 681 749 817 884 953 1021 1089 1157
38 433 505 578 650 723 795 867 939 1011 1083 1156 1228
39 459 536 612 689 765 842 918 996 1072 1149 1225 1302
40 486 567 648 729 810 891 972 1053 1134 1215 1296 1377
41 513 599 684 770 856 941 1027 1112 1198 1284 1369 1455
42 541 632 722 812 902 993 1083 1173 1264 1354 1444 1534
43 570 665 761 856 951 1046 1141 1236 1231 1426 1521 1616
44 600 700 800 900 1000 1100 1200 1300 1400 1500 1600 1700

The Mathematics of Doyle's Log Table.

The formula for Doyle's table is very easy. It is given by (r-2)^2*L/4, where r is the radius in inches and L is the length in feet. Sometimes this is given as (d-4)^2*L/16 where d is the diameter in inches and L is the length in feet. Note that the table will not reflect these numbers in some entries. This is because of incorrect results computed in the original table that are still reproduced because the table has become a legal standard.

A C Function to Compute Board Footage Using Doyle's Log Table.

/**FUN*****************************************************************
* Compute the BF of a log.  The language is obviously C.
*
* If the log size is appropriate then this function will use
* Doyle's table.
*
* If the log size is not appropriate then this function will
* compute the BF using a formula that approximates Doyle's table.
*
* Note that this formula doesn't match the table.  This is due to
* errors made in the original computation of the table.  These errors
* have been used traditionally in the industry and thus must be propagated
* here.
***************************************/
long int BFlog(long int diam, long int length)
{
  int BFDoil[][12] =
  {
    { 18,  21,  24,  28,   31,   34,   37,   40,   43,   46,   49,   52},
    { 24,  28,  32,  36,   40,   44,   48,   52,   56,   60,   64,   68},
    { 30,  35,  40,  45,   50,   55,   61,   66,   71,   76,   81,   86},
    { 37,  44,  50,  56,   62,   69,   75,   81,   88,   94,  100,  106},
    { 45,  53,  60,  68,   75,   83,   91,   98,  106,  113,  121,  128},
    { 54,  63,  72,  81,   90,   99,  108,  117,  126,  135,  144,  153},
    { 63,  74,  84,  95,  106,  116,  127,  137,  148,  158,  169,  180},
    { 73,  85,  98, 110,  122,  135,  147,  159,  171,  183,  196,  208},
    { 84,  98, 112, 127,  141,  155,  169,  183,  197,  211,  225,  239},
    { 96, 112, 128, 144,  160,  176,  192,  208,  224,  240,  256,  272},
    {108, 126, 144, 162,  181,  199,  217,  235,  253,  271,  289,  307},
    {121, 141, 162, 182,  202,  223,  243,  263,  283,  303,  324,  344},
    {135, 156, 180, 203,  226,  248,  271,  293,  313,  336,  361,  383},
    {150, 175, 200, 225,  250,  275,  300,  325,  350,  375,  400,  425},
    {165, 193, 220, 248,  276,  303,  331,  358,  386,  413,  441,  469},
    {181, 211, 242, 272,  302,  334,  363,  393,  423,  453,  484,  514},
    {198, 231, 264, 297,  330,  363,  397,  430,  463,  496,  529,  563},
    {216, 252, 288, 324,  360,  396,  432,  468,  504,  540,  576,  612},
    {234, 273, 312, 352,  391,  430,  469,  508,  547,  586,  625,  664},
    {253, 295, 338, 380,  422,  465,  507,  549,  591,  633,  676,  718},
    {273, 319, 364, 410,  456,  502,  547,  592,  638,  683,  729,  774},
    {294, 343, 392, 441,  490,  539,  588,  637,  686,  735,  784,  833},
    {315, 368, 420, 473,  526,  578,  631,  684,  736,  789,  841,  894},
    {337, 393, 450, 506,  562,  619,  675,  731,  787,  844,  900,  956},
    {360, 420, 480, 540,  601,  661,  721,  781,  841,  901,  961, 1021},
    {384, 448, 512, 576,  640,  704,  768,  832,  896,  960, 1024, 1088},
    {408, 476, 544, 613,  681,  749,  817,  884,  953, 1021, 1089, 1157},
    {433, 505, 578, 650,  723,  795,  867,  939, 1011, 1083, 1156, 1228},
    {459, 536, 612, 689,  765,  842,  918,  996, 1072, 1149, 1225, 1302},
    {486, 567, 648, 729,  810,  891,  972, 1053, 1134, 1215, 1296, 1377},
    {513, 599, 684, 770,  856,  941, 1027, 1112, 1198, 1284, 1369, 1455},
    {541, 632, 722, 812,  902,  993, 1083, 1173, 1264, 1354, 1444, 1534},
    {570, 665, 761, 856,  951, 1046, 1141, 1236, 1231, 1426, 1521, 1616},
    {600, 700, 800, 900, 1000, 1100, 1200, 1300, 1400, 1500, 1600, 1700}
  };

  if ((diam > 44) || (diam < 11) || (length > 17) || (length < 6))
    return ((diam - 4) * (diam - 4) * length / 16);
  else
    return (BFDoil[diam - 11][length - 6]);
}

Converting True Volume to/from Board Foot.

One board foot is equal to 1/12 of a cubic foot and is equal to 144 cubic inches.

To convert from board feet to cubic feet, multiply by 1/12=0.08333333333333333333333333333333. Or to do the same conversion divide by 12.

To convert from board feet to cubic inches, multiply by 144.

To convert from cubic inches to board feet, multiply by 1/144=0.006944444444444444. Or to convert from cubic inches to bard feet, divide by 144.

To convert from cubic feet to board feet, multiply by 12.

Computing Log Volume.

First lets assume that all measurements are taken in inches or feet, but not both. Thus we compute real volumes.

As a first guess we might just assume that a log is a mathematical cylinder. Cylinders have a length and a radius. The volume of a cylinder is equal to Pi*L*r^2, where r is the radius and L is the length.

The cylinder idea is a bit to rough for our work because a log generally doesn't have the same radius at each end. The next best approximation is to take the average radius, (r1+r2)/2, and use the same formula for a cylinder: Pi*L*(r1+r2)^2/4.

The above formula is NOT the area of a "truncated cone". It is rather an approximation to it. Our next exercise is to compute the volume of a truncated cone. That is to say a cylinder that has a radius, r1, at one end and a radius, r2, at the other. Let's still call the length L. So the volume of such an object is given by 1/3*Pi*L*(r1^2+r1*r2+r2^2). Some, non-trivial, algebra gives this formula as 1/3*Pi*L*(r1^3-r2^3)/(r1-r2); r1 must not be equal to r2 in this last formula of course. As a practical matter, the Pi and 1/3 cancel out and this leaves L*(r1^2+r1*r2+r2^2). If r1=r2-r, then we get L*r^2 as expected from above.

Lastly we consider the fact that the "length" we have been using is the "height" given in traditional geometry texts. Often we measure the side length. This gives a different answer in the last computation. The formula is thus: sqrt(l^2*(2*r2-r1)^2/(r1-r2)^2-r1^2)-r2*sqrt(L^2/(r1-r2)^2-1). This formula is very complicated, and yields little extra accuracy.

Log Volume Compared to Doyle's Log Table.

The easiest way to measure log volume is to use Pi*L*r^2. Subtract Doyle's approximation and one gets a difference of about L*d/2-L. The interpretation is that L*d/2-L board feet are wasted in the cutting of a log into lumber.

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