55 15

.106,553,458,402

110.007,590,967

55 30 .111,165,491,707

.092,386,171,311 .355,093.253.347

.248,440,399.551

.004,092,369,079 .004,030,387.998

251,728,465,144

.004,172,725,725

.004,39,482,253

.255,042,445,228

.004,254,083,480

.004,250.950,944

.258,382,374,657

.004,336,443,111

.004,364,832,478

261,748,287.205.004,419,805,114

.004,481,165,636

54 30
54 45 .107,701,469,365 .094,003,084,727 .357.501.536,091 .265,140,215,568
55 00 .108,852,881,645 .095,639,871,638 .359,905,949,952 .268,558,191,349
.097,296,660,379 .362,306,313,266 .272,002,245,068
.098,973,578,959 .364,702,444,032 .275,472,406,153
.367,094.159,912 .278,968,702,941
102,388,315,957 .369.481,278,245
104,126,388,661 371,863,616,050
105,885.099,744 374,240.990.026

289,614,674,434

.005,122,754,003

005 215,113,143 .005.645.072,173 .005,308,463,818 .005,788,137,292 .005,402,803,245

304,176,729,781 005,498,128,313

307,883,000,392.005,594,435,575
311 615,612,180.005.691.721,246

315,374,582,433

319,159,927.317

.322,971,661,884

.326,809,800,064

005,934,105,461

.006,083,019.967

.006,234,924,476

.006.389.863.027

.130,034,506,488.128,644,954,271

.131,229,191,120 .130,683,296,376

60 00 .132,424,934,960 60 15 .133,621,606,768 60 30 .134,819,073,959 60 45 .136,017,202,614 61 00 .137,215,857,476 61 15 .138,414,901,952

139,061,517,903

141,212,881,225

143,387,203,630

145,584,602,052

.413,717,869,558

415,974,229,382

418,222,091,224

.420,461,266,797

.132,744,004,308

.134,827,197,823

397,701,181,533

400,011,772,063

402,315,176,924

330,674,354,664 .006,192,668,538
334,565,337,368 006 295,725,299

.404,611,209,291.338,482,758,731.006.399,723,934

406,899.682.116.342,426,628,186 006.504657 720

409,180,408,128 346,396,954,030 006,610,519,552

136,932,996, 134 411,453, 199,840

.350,393,743,430 .006,717,301,922

354,417,002,420.006.824,996,923

.007,211,633,846 007 385,722,294 .007,563,163,448

007,744,004,308

.007,928,292,238

.008,116,074,965

,008,307,400.579

.366,645,640,254 .CO7,153,472,437.008,903,121,066 370,774,815,234

374,930,472,923

427,124,790,370 379,112,612,522 .429,327,334,473 383,321,232,086 .431.520,248,273 387,556,328,530

007,264,730,597

009,109.106.349

.433,703,342,481 391,817,897,614

173,795,032,784

.446,585,078,789

64 00

64 15 .152,782,293,309 64 30 .153,973,369,314 64 45 .155,162,783,641 65 00 156,350,379,927 65 15 .157,536,000.514 65 30 .158,719,486,447 65 45 .159,900,677,477 66 00 .161,079,412,068 66 15 .162.255.527.396 66 30 .163,428,859,354 66 45 .164,599,242,556 67 00 .165,766,510.339

67 15

67 30

197,260,839,884 .465,142,301,062
199,994,176,068 467,142,440,202

.166,930,494,766 .202,753,091,660.469,129,524,956

.168.091,026,631.205,537,686,334 .471,103,364,417

454,254,378.942

009,529,186,662 013,801,641,533

/o, cos do,

SHEET 3 OF 4

422,389,517,875

426,862,577,519

43362,019,848.008,907,986,552

012,414,053,411

454,952,468,535 .435,887,829,084

.440.439.988,291

74 00 196.469.083.288
74 15 .197,467,905,024 .290,767,763,092
74 30 .198,458.037.451 .294,308,087.644
74 45 .199,439,276,400 .297,876,568,087
75 00 200,411,416,563 .301,473.273.950
75 15 .201,374,251,510

.610,693,665,641
.616,148,331,716 013,722,249,337 .025,838.288,898
.621,627,959.425.013.853,351 244 .026,315.072.655

.627,132,488,495 .013.984,059,772

305,098,273.604 .524.681,656,025 .632,661.857,481 75 30 202.327,573,695 308,751,634,242 ,526,130,983.789 .638.216.003,760 75 45 .203,271,174,458 312,433,421,876 .527,560,789,955 .643,794.863.538 76 00 .204,204,844,031 .316,43,701,321 .528,970,886,357 .649,398,37,849 76 15 .205,128,371,545 319,882,536,189 .530,361,085,030 .655,026,462,558 76 30 .206,041,545.036 .323,649,988,883 .531,731,198,215 .660,679,068,354

76 45 .206,944,151,454 .327,446,120,575 .533,081,038.356 .666,356,120,757 77 00 .207,835,976,665 .331,270,991,212 .534,410,418,123 .672,057,550,117 .335,124,659,490 .535,719,150,401 .677,783,285,614 .339,007,182,855 ,537,007,048,307 .683,533,255,260 .342.918.617,494 .538,273,925,199 .689,307,385,897

77 15 -208,716,805,460
77 30 .209,586,421,558
77 45 210,444,607,620
78 00 21.291,145,249
78 15 212,125,814,996
78 30 .212,948,396,370
78 45
.213,758,667,847
79 00 -214,556,406.870
79 15
79 30
79 45

80 00

215,341,389,862

216,113,392,229
216,872,188,371

217,617,551,685

80 15 .218,349,254,574 80 30 .219,067,068,458 80 45 219.770.763,771 81 00 220,460,109,982

.221134,875,594

-221,794,828,150

.354,826,931,736

.541,946,567,022

.543,127,498.368

.358,854,547,708

362 91,336,593 366 997,346,796 .371,112,625,389

.544.286,479,258

.695105,603.197

.700,927,831,672

014,114,333,685

.014,244,131,006

026,798.669.999

.027,289,145,795

.027,786,565,081

.706,773,994,661 .015,748,205.279

.712,644,014,339.015,867,638,508

.718,537,811,720 .015,985,926,314 .035.516.295.612

.545,423,324,603.724,455,306,651 .016,103,014.920.036125,774,801

.546,537,849,601 .730,396,417,816 .016,218,849,719 036,743,257,505 .375,257,218, 109 .547,629,869,740.736,361,062,738 .016,333,375,272 .037,368,810,895 379,431,169,344.548,699,200.805 .742,349,157,782.016,446,535,302.038.002,502,207 383,634,522,119.549,745,658,880

.748,360,618,148.016.558,272,687 .038,644,398,741 016.668,529,463 .039,294,567,858

392 129,597,559.551.769,221,965.760,453,289,865.016,777,246,805 039,953,076,964

81 15

81 30

8 1 45

82 00

82 15

82 30

82 45

.222.439,734,246

22 3,069,359,537 .223.683,468,746 .224,281,825,667 .224,864,193,177 83 00 225,430.333.244 83 15 .225,980,006,934 83 30

409,474,307,197.555,533,815,253
413,884,559,480.556,415,039,667

418,324,507,508 .557,271,931,483

422,794,181,667

.558,104,309,813

427,293,610.891

.558.911.994,137

431,822,822,655

.559,694,804,308

.816,003,383,769.0 17.676,295,080.046,266,097,629

227.527,827,034 228.009,228,1 16 228,472,954.908 228,918,763,234 229,346,408 079 229,755.643.589 230146.223.090 230,5 17,899.079 -230 870.423,258

.226,512.974,418

83 45 84 00 84 15 84 30 84 45 85 00 85 15

227,028,994,981

85 30

85 45

86 00

86 15

86 30

86 45

231,203,546,516 231,517,018,959 -231,810.589.908

87 00 232,084,007,911 87 15 232,337,020,747

87 30 .232.569.375.452

450,237,993,031 .562.573,714,085 .841,275.164.298
454,916,477,915 .563,229,464,923 .847 .647,995,474

459,624,879,043.563,859,268,992

.464,363,213,425 .564,462,948,960

87 45 232,780,818,303 523,562,097,749

88 00

232,971,094,849 528.690.404.690

38 5 233139,949,907

569,024,224,088 .569,328,961,985

017.939 184,516.048,530,282,490

018,102,640,872 050,085,715.683

.886.337,948,836.0 18.529,983.555.054.978.626.006

854,042,574,125

.860,458,794,795

88 30

.233,287,127,578

539,037,032,489

.570,264,034,865

.959,223,981,631

O18,995.026.396
019,021,985,670 064,871.364.486

063,920,931.359

88 45

233,412.371,257

-544,255,338.053.570,425,980,955

.965.971,212,760

019,045,046,591 065,832,278,484

LOAD CONSTANTS FOR CIRCULAR ARCHES

203. General Statement. With the aid of tables 11 to 15, load constants, which are the trigonometric parts of the expressions for deflections due to applied loads, as stated in chapter V, may be computed and tabulated for use in trial load analyses. These quantities are designated D-terms, D, being used for twist deflections, D2 for radial deflections, and D, for tangential deflections. D-terms, multiplied by proper factors, give deflections at an arch point due to applied load between the point and the abutment. The method of using tabulated load constants. in arch analyses is described in chapter VI.

204. Evaluation. The formulas in chapter V, together with the tabular values of trigonometric functions and integrals in tables 11 to 15, inclusive, give all quantities needed for computing trigonometric parts of load formulas, or D-terms. The tabular load constants are the trigonometric parts of the D-terms with P1000 included in the uniform Р 1000 included in the triangular loads. Forms and Φι Φι sample calculations for D-terms of several unit loads are shown in table 16.

loads and

Sheet 1 of table 16 gives values of trigonometric functions and integrals needed in evaluating load constants for an arch with a 48° half central angle, proper sine and cosine multipliers being shown beneath the quantities where needed. The values were copied from tables 11 to 15, inclusive, and are tabulated on sheet 1 for convenience. Since D-terms are determined for the quarter-points of the half arch, the quarter-values of are listed in descending order, and the quantities for these angles are tabulated.

Sample computations for uniform and triangular radial, tangential, and twist loads are shown on sheets 2, 3, and 4 of table 16. In computing values for uniform loads, columns having integrals underneath are 1000 times the values for the integrals on sheet 1, except that some values are multiplied by three as indicated. Formulas for calculating the remaining columns are given on the computation sheets.

Computations for triangular loads are complicated, since some points are outside the loaded portion of the arch while other points are under the load. For points outside the load, values are obtained from sheet 1, using 1 as the angle between the abutment and the end of the load, and 4, as the angle between the end of the load and the point considered. As ex