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TA YH02449 IRLF B 3 IDE MOD EXCHANGE *l Me GILL UNIVERSITY PAPERS FROM THE DEPARTMENT OF ENGINEERING. No. 6. — ON THE STRESSES DEVELOPED IN BEAMS LOADED TRANSVERSELY. BY HENRY T. BOVEY, LL.D. [Reprinted from the Trans, of the Royal Society of Canada, Section iii, pp. 3-28, May 27, 1902.] MONTREAL, 1902. ROYAL SOCIETY OF CANADA TRANSACTIONS SECTION III. MATHEMATICAL, PHYSICAL AND CHEMICAL SCIENCES PAPERS FOR 19O2 SECTION III., 1902 [3] TRANS. R. S. C. . — On the Stresses Developed in Beams Loaded Transversely. By HENRY T. BOVEY, R E. S. (Read May 27, 1902.) The present paper contains further results obtained with the new extensometer described in Section III., Yol. YIL, of the "Transactions " of the Eoyal Society of Canada, in the year 1901-02: The loading was of the same description and of two kinds, namely, (1) loads of increasing magnitude were placed at the centre; and (2) equal loads of increasing magnitude were concentrated at two points equidistant from the centre, the maximum B. M. in each case being the same as for the corresponding centrally placed load. In all measurements, the beams were placed on supports 60 ins. apart and the distance between the extensometer points was 8 ins. In each set of observations, the horizontal row at the top of the columns gives the loads successively placed on the beam, and the columns under the several loads give the corresponding changes of length between the extensometer points in millions of an inch, each change of length being the average obtained from five to ten observations. The decrements and increments of length are indicated by negative and positive signs respectively. A. — EXPERIMENTS WITH AN 8 IN. STEEL EOLLED JOIST WEIGHING 49| LBS. PER LINEAL YARD. An extensometer was placed horizontally at the centre of gravity of the section and extensometers were also placed on parallel lines at distances of 1-27 ins., 2 54 ins. and 3'81 ins., above and below the centre of gravity, the extensometers in the last case being on the flanges. LOADS CONCENTRATED AT 30-iN. CENTRES. 3,f 00 Ibs. 6,000 Ibs. 9,000 Ibs. 12,000 Ibs. 15,000 Ibs. - 935 - 1701 - 2415 - 3126 - 3824 - 641 - 1228 - 1778 - 2325 - 2840 - 323 - 617 - 903 - 1101 - 1460 6 + 2 + 13 + 22 + 25 + 311 + 627 + 937 + 1239 + 1536 + 631 + 1256 + 1863 + 2446 + 3008 + 891 + 1682 + 2478 + 3252 + 4038 By plotting the above results, diagram I. is obtained. 4 ROYAL SOCIETY OF CANADA Dlft&RflMS FOR Q/M ROLLED JOIST [BOVEY] STRESSES IN BEAMS LOADED TRANSVERSELY TABLE OF MAXIMUM BENDING MOMENTS AND DEFLECTIONS. Load in Ibs. Max. Bending Mo- ment in in.-lbs. Deflections in inches. 3000 45000 •on 6000 90000 •024 9000 135000 •038 12000 180000 •051 15000 225000 •063 LOADS CONCENTRATED AT 20-iN. CENTRES. 2,250 Ibs. 4,500 Ibs. 6,750 Ibs. 9,000 Ibs. 11,250 Ibs. - 778 - 15S2 - 2384 - 3208 - 4075 - 645 - 1265 - 1860 - 2452 - 3027 - 337 - 679 - 1014 - 1350 - 1671 10 38 69 - 100 - 133 + 306 + 597 + 887 + 1172 + 1452 + 637 + 1257 + 1^70 + 2475 + 3068 + 943 + 1840 4 2664 + 3404 + 4268 By plotting the above results, diagram II. is obtained. TABLE OF MAXIMUM BENDING MOMENTS AND DEFLECTIONS. Load in Ibs. Max. Bending Mc- ment in in.-lbs. Deflections in inches. 2250 45000 •013 4500 90000 •015 6750 135000 •017 9000 180000 •019 11250 225000 •021 ROYAL SOCIETY OF CANADA LOADS CONCENTRATED AT 15-iN. CENTRES. 2,000 Ibs. 4,000 Ibs. 6,000 Ibs. 8,000 Ibs. 10,000 Ibs. - 879 - 1711 - 2482 — 3323 - 4153 — 656 — 1267 — 1865 - 2446 - 3029 — 352 - 694 — 1051 — 1419 — 1728 - 22 — 47 — 86 - 123 — 156 + 295 + 591 + 872 + 1156 + 1435 + 629 + 1243 + 1838 + 2429 + 3008 + 933 + 1859 + 2685 + 3502 + 4290 By plotting the above results, diagram III. was obtained. TABLE OF MAXIMUM BENDING MOMENTS AND DEFLECTIONS. \ Load in Ibs. Max. Bending Mo- ment in in. -Ibs. Deflections in inches. 2000 45000 •012 4000 90000 •023 6000 135000 •034 8000 180000 •047 10000 225000 •059 LOADS CONCENTRATED AT S-IN. CENTRES. 1,730 Ibs. 3,460 Ibs. 5,190 Ibs. 6,920 Ibs. 8,650 Ibs. - 913 - 1778 — 2632 — 3473 - 4318 — 599 - 1164 - 1707 — 2239 - 2751 — 290 — 559 - 811 - 1066 — 1310 + 15 + 40 + 75 + 109 + H5 + 327 + 659 + 995 + 1318 + 1645 + 636 + 1264 + 1886 + 2495 + 3066 + 1068 + 2014 + 2862 + 3683 + 4486 By plotting the above results, diagram IY. was obtained. [BOVEY] STRESSES IN BEAMS LOADED TRANSVERSELY TABLE OF MAXIMUM BENDING MOMENTS AND DEFLECTIONS. Load in Ibs. Max. Bending Mo- ment in in. -Ibs. Deflections in inches. 1730 44980 •on 3460 88960 •021 5190 134940 •032 6920 179920 •043 8650 224900 •054 LOADS CONCENTRATED AT 6 IN. CENTRES. 1,665 Ibs. 3,330 Ibs. 4,995 Ibs. 6,660 Ibs. 8,325 Ibs. — 1050 - 1924 — 2775 — 3587 - 4385 — 594 — 1147 - 1653 — 2149 - 2637 - 290 - 548 - 795 - 1038 - 1271 + 11 + 45 + 80 + 119 + 165 + 317 + 641 -?- 964 + 1289 + 1605 + 627 + 1235 + 1846 + 2423 + 2965 + 922 + 1777 + 2612 + 3413 + 4177 By plotting the above results, diagram V. was obtained. TABLE OF MAXIMUM "BENDING MOMENTS AND DEFLECTIONS. Load in Ibs. Max. Bending Mo- ment in in.-lbs. Deflections in inches. 1665 44955 •on 3330 89910 •021 4995 134865 •031 6660 179820 .042 8325 224775 •053 8 ROYAL SOCIETY OF CANADA LOAD CONCENTRATED AT CENTRE. 3,000 Ibs. 6,000 Ibs. 9,000 Ibs. 12,000 Ibs. 15,000 Ibs. QJ.7 O:r t - 1660 - 2488 - 3270 — 4077 - 471 - 927 - 1409 — 1873 - 2311 - 142 - 298 - 486 - 712 - 837 + 120 + 220 + 285 + 311 + 378 + 387 + 742 + 1063 + 1376 + 1687 + 635 + 1233 + 1796 + 2384 + 2907 + 1099 + 2040 + 2866 + 3707 + 4504 -By plotting tke above results, diagram VI. was obtained. TABLE OF MAXIMUM BENDING MOMENTS AND DEFLECTIONS. Load in Ibs. Max. Bending Mo- ment in in.-lbs. Deflections in inches. 3000 45000 •01 6000 90000 •02 9000 135000 •03 12000 180000 •04 15000 225000 •05 In diagram IV., in which the loads are concentrated at 8-in. centres, the stress in the material is almost directly proportional to the distance from the neutral axis which seems to be slightly above the centre of gravity. The diagrams for the 30-in., 20-in. and 15-in. concentrations indicate that the stress in the material increases more rapidly than the distance from the neutral axis, while the increase is not so rapid for the 6-in. concentration and for the beam loaded at the centre. In the last case, the neutral axis has moved very appreciably above the centre of gravity. [BOVKY] STRESSES IN BEAMS LOADED TRANSVERSELY SEC. III., 1902 [BOVEY] STRESSES IN BEAMS LOADED TRANSVERSELY 9 B. — EXPERIMENTS WITH A 7'85-iN. x 3'425-iN. CAST STEEL BEAM WEIGHING 496J- LBS. PER CUBIC FOOT. Five extensometers were used with this beam, one beiDg placed horizontally at the centre of gravity and the remainder on parallel lines at distances of 1-8125 ins. and 3-625 ins. above and below the centre of gravity. DM&ftflMS FOR CAST -STEEL BEflM W/TH LOADS CONCENTRATED f,T 30 IN. CENTRES 10 KOYAL S )CIETY OF CANADA SERIES I. LOADS CONCENTRATED AT 30-iN. CENTRES. 800 Ibs. 1,600 Ibs. 2,400 Ibs. 3,200 Ibs. 4,000 Ibs. 87 - 174 — 269 - 361 — 457 38 81 — 122 - 170 - 211 + 0 + 1 + 4 + 6 + 8 + 34 + 97 + 142 + 197 + 240 + 92 + 205 + 310 + 412 + 515 By plotting the above results, diagram VII. was obtained, TABLE OF MAXIMUM BENDING MOMENTS AND DEFLECTIONS. Load in Ibs. Max. Bending Mo- ment in in. Ibs. Deflections in inches. 800 120GO •001 1600 24000 •0015 2400 36000 •003 3200 48000 •0045 4000 60000 •0055 SERIES II. 1,600 Ibs. 3,200 Ibs. 4,800 Ibs. 6,400 Ibs. 8,000 Ibs. - 183 - 377 — 575 — 760 - 935 78 - 166 — 263 - 357 - 447 + 6 + 8 + 8 + 7 + 7 + 92 + 188 + 280 + 375 + 473 + 202 + 405 + 601 + 789 + 975 By plotting the above results, diagram VIII. was obtained. [BOVEY] STRESSES IN BEAMS LOADED TRANSVERSELY 11 TABLE OF MAXIMUM BENDING MOMENTS AND DEFLECTIONS. Load in Ibs. Max. Bending Mo- ment in in. Ibs. Deflections in inches. 1600 24000 •002 3200 48000 •004 4800 72000 •007 6400 96000 •0085 8000 120000 •on SERIES III. 2,400 Ibs. 4,800 Ibs. 7,200 Ibs. 8,60 Ibs. 12,000 Ibs. - 272 — 559 - 840 - 1120 - 1383 - 119 - 242 — 368 - 493 — 616 + 8 + 17 4- 26 + 37 + 49 + 145 + 297 + 447 + 602 + 757 + 304 + 605 + 901 + 1176 + 1440 By plotting the above results, diagram IX. was obtained. TABLE OF MAXIMUM BENDING MOMENTS AND DEFLECTIONS. Load in Ibs. Max. Bending Mo ment in in.-lbs. Deflections in inches. 2400 36000 •003 4800 22000 •006 7200 108000 •010 9600 144000 •014 120000 180000 •017 12 ROYAL SOCIETY OF CANADA SERIES IV. 3,200 Ibs. 6,400 Ibs. 9,600 Ibs. 12,800 Ibs. 16,000 Ibs. — 387 - 763 - 1116 - 1438 - 1767 - 172 - 360 - 545 - 724 - 900 + 8 + 2 + 13 + 15 + 35 + 184 + 370 + 578 + 780 + 990 + 398 + 795 + 1177 + 1551 + 1913 By plotting the above results, diagram X. was obtained. TABLE OF MAXIMUM BENDING MOMENTS AND DEFLECTIONS. Load in Ibs. Max. Bending Mo- ment in in. -Ibs. Deflections in inches. 3200 48000 •004 6400 96000 •009 9600 144000 •013 12800 192000 •018 16000 240000 •028 [BOVEY] STRESSES IN BEAMS LOADED TRANSVERSELY 13 DIAGRAMS FOR CAST- STEEL BE3M W/TH LOADS CCHC£NTffflT£D flT 20 IN. CENTRES SERIES I. LOADS CONCENTRATED AT 20 IN. CENTRES. 600 Ibs. 1 200 Ibs. 1,800 Ibs. 2,400 Ibs. 3,000 Ibs. - 104 - 212 - 318 - 421 - 521 58 - 112 - 168 - 218 - 272 2 — 3 6 12 15 + 35 + 70 + 119 + 155 + 202 + 86 + 185 + 286 + 379 + 475 By plotting the above results, diagram XI. was obtained. 14 ROYAL SOCIETY OF CANADA TABLE OF MAXIMUM BENDING MOMENTS AND DEFLECTIONS. Load in Ibs. Max. Bending Mo- ment in in. Ibs. Deflections in inches. 600 12000 •001 1200 24000 •002 1800 36000 •0035 2400 48000 •005 3000 60000 •0055 SERIES II. 1,200 Ibs. 2,400 Ibs. 3,600 Ibs. 4,800 Ibs. 6,000 Ibs. - 188 - 378 - 577 - 765 - 942 81 - 168 — 265 - 361 - 451 + 7 + 3 + 4 + 2 + 4 + 89 + 187 + 276 + 368 + 467 + 195 + 403 + 599 + 799 + 980 By plotting the above results, diagram XII. was obtained. TABLE OF MAXIMUM BENDING MOMENTS AND DEFLECTIONS. Load in Ibs. Max. Bending Mo- ment in in. Ibs. Deflections in inches. 1200 24000 •001 2400 48000 •003 3600 72000 •0055 4800 96000 .0075 6000 120000 •0094 [BOTEY] STRESSES IN BEAMS LOADED TRANSVERSELY 15 SERIES III. 1,800 Ibs. 3,600 Ibs. 5,400 bs. 7,200 Ibs. 9,000 Ibs. - 279 — 566 - 851 - 1132 - 1406 126 - 254 - 383 - 512 - 638 + 2 + 10 + 16 + 24 + 32 + 143 + 296 + 445 + 596 + 747 + 299 + 603 + 889 + 1175. + 1453 By plotting the above results, diagram XIII. was obtained. TABLE OF MAXIMUM BENDING MOMENTS AND DEFLECTIONS. Load in Ibs. Max. Bending Mo- ment in in. -Ibs. Deflections in inches. 1800 36000 •003 3600 72000 •006 5400 108000 •009 7200 144000 •012 9000 180000 •015 SERIES IV. 2,400 Ibs. 4,800 Ibs. 7,200 Ibs. 9,600 Ibs. 12,000 Ibs. - 381 - 761 - 1116 - 1447 - 1781 - 168 - 361 - 549 - 742 - 919 + 9 + 3 + 8 + 6 + 10 + 190 + 369 + 572 + 77 + 969 + 406 + 80 + 1178 + 154 + 1906 By plotting the above results, diagram XIV. was obtained. 16 ROYAL SOCIETY OF CANADA TABLE OF MAXIMUM BENDING MOMENTS AND DEFLECTIONS. Load in Ibs. Max. Bending Mo- ment in in. -Ibs. Deflections in inches. 2400 48000 •004 4800 96000 •009 7200 144000 •013 9600 192^00 •017 12000 240000 •021 DlfiGRflMS FOR CflST-STEEL BEfiM WITH LOADS CONCENTRATED flT 15 IN. CENTRES _2T [BOVEY] STKESSES IN BEAMS LOADED TRANSVERSELY 17 SERIES I. LOADS CONCENTRATED AT 15-iN CENTRES. 535 Ibs. 1,070 Ibs. 1,605 Ibs. 2,140 Ibs. 2,675 Ibs. — 92 — 191 — 281 — 380 - 475 - 47  229 + 94 + 196 + 307 + 407 + 506 By plotting the above results, diagram XV. was obtained. TABLE OF MAXIMUM BENDING MOMENTS AND DEFLECTIONS. Load in Ibs. Max. Bending Mo- ment in in. -Ibs. Deflections in inches. 535 13138 •001 1070 26276 •002 1605 39414 •003 2140 52552 •004 2675 65690 •005 SERIES II. 1,070 Ibs. 2,140 Ibs. 3,210 Ibs. 4,280 Ibs. 5,350 Ibs. - 190 — 387 — 581 - 771 — 946 — 84 - 172 — 272 - 369 — 460 + 6 + 2 + 1 + 1 + 0 + 86 + 183 + 275 + 365 + 464 + 196 + 406 + 602 + 797 + 982 By plotting the above results, diagram XVI. was obtained. Sec. III., 1902. 2. 18 ROYAL SOCIETY OF CANADA TABLE OF MAXIMUM BENDING MOMENTS AND DEFLECTIONS. Load in Ibs. Max. Bending Mo- ment in in. -Ibs. Deflections in inches. 1070 26276 •001 2140 52552 •003 3210 78828 •0055 4280 105104 •0075 5350 131380 •010 SERIES III. 1,605 Ibs. 3,210 Ibs. 4,815 Ibs. 6,420 Ibs. 8,025 Ibs. — 284 — 572 — 850 — 1110 - 1359 — 129 - 167 — 416 — 555 — 692 + 3 + 6 + 5 + 4 + 10 + 144 + 285 + 425 + 577 + 726 + 312 + 614 + 912 + 1189 + 1470 By plotting the above results, diagram XYII. was obtained. TABLE OF MAXIMUM BENDING MOMENTS AND DEFLECTIONS. Load in Ibs. Max. Bending Mo- ment in in. -Ibs. Deflections in inches. 1605 39414 •003 3210 78828 •006 4815 118242 •009 6420 157656 •013 8025 197070 •016 [BOVEY] STRESSES IN BEAMS LOADED TRANSVERSELY 19 DlflGRflM5 FOR C ft ST- STEEL BE SIM WITH LOflDS CCMCEMTRflTED flT QlN. CENTRES ~mr SERIES I. LOADS CONCENTRATED AT S-IN. CENTRES. 460 Ibs. 920 Ibs. 1,380 Ibs. 1,840 Ibs. 2,300 Ibs. - 86 — 181 — 268 - 365 - 456 — 41 - 86 — 123 — 171 — 213 + 3 + 4 + 6 + 7 + 8 + 40 + 89 + 138 + 194 + 238 + 101 + 206 + 316 + 413 + 515 By plotting the above results, diagram XYIII. was obtained. 2O ROYAL SOCIETY OF CANADA TABLE OF MAXIMUM BENDING MOMENTS AND DEFLECTIONS. Load in Ibs. Max. Bending Mo- ment in in.-lbs. Deflections in inches. 460 11960 •001 920 23920 •0015 1380 35880. •002 1840 47840 •003 2300 59800 •004 SERIES II. 925 Ibs. 1,840 Ibs. 2,760 Ibs. 3,680 Ibs. 4,600 Ibs. — 185 — 364 — 557 — 746 — 920 - 77 - 156 — 248 — 336 — 434 + 4 + 15 + 12 + 18 + 13 + 90 + 199 + 287 + 381 + 476 + 200 + 413 + 608 + 801 + 990 By plotting the above results, diagram XIX. was obtained. TABLE OF MAXIMUM BENDING MOMENTS AND DEFLECTIONS. Load in Ibs. Max. Bending Mo- ment in in.-lbs. Deflections in inches. 920 23920 •002 1840 47840 •0045 2760 71760 •0065 3680 95680 •0085 4600 119600 •010 [BOVBY] STRESSES IN BEAMS LOADED TRANSVERSELY 21 SERIES III. 2,380 Ibs. 2,760 Ibs. 4,140 Ibs. 5,520 Ibs. 6,900 Ibs. — 274 — 559 - 836 — 1090 — 1333 — 117 — 248 — 389 — 525 — 662 + 7 + 12 + 10 + 16 + 15 + 145 + 289 + 424 + 575 + 720 + 307 + 610 + 903 + 1176 + 1451 By plotting the above results, diagram XX. was obtained. TABLE OF MAXIMUM BENDING MOMENTS AND DEFLECTIONS, Load in Ibs. Max. Bending Mo- ment in in. Ibs. Deflections in inches. 2380 35880 •004 2760 71760 •007 4140 107640 •010 5520 143520 •012 6900 179400 •014 22 ROYAL SOCIETY OF CANADA DlAGRflMS FOR CflST- STEEL BEAM W/TH LO/JDS CONCENTRATED flT 6/N. CENTRES ~ZXL C.orCr. J71JT SERIES I. LOADS CONCENTRATED AT 6-iN. CENTRES. 445 Ibs. 890 Ibs. 1,335 Ibs. 1,780 Ibs, 2,220 Ibs. — 93 — 189 - 283 — 379 — 470 — 41 — 86 — 126 — 171 - 216 + 1 + 2 + 4 + 6 + 8 + 40 + 87 + 137 + 189 + 335 + 95 + 195 + 299 + 401 + 501 By plotting the above results, diagram XXI. was obtained. [BOVEY] STRESSES IN BEAMS LOADED TRANSVERSELY 23 TABLE OF MAXIMUM BENDING MOMENTS AND DEFLECTIONS. Load in Ibs. Max. Bending Mo- ment in in. Ibs. Deflections in inches. 445 12015 •001 890 24030 •002 1335 36045 •003 1780 48060 •004 2225 60075 •005 SERIES II. 890 Ibs. 1,780 Ibs. 2,670 Ibs. 3,560 Ibs. 4,450 Ibs. — 182 — 365 — 556 — 743 - 917 - 75 — 151 — 239 — 331 — 419 + 7 + 17 + 21 + 19 + 24 + 94 + 97 + 291 + 383 + 480 + 201 + 415 + 612 + 806 + 992 By plotting the above results, diagram XXII. was obtained. TABLE OF MAXIMUM BENDING MOMENTS AND DEFLECTIONS. Load in Ibs. Max. Bending Mo- ment in in. -Ibs. Deflections in inches. 890 24030 •003 1780 48060 •005 2670 72090 .007 3560 96120 •009 4450 120150 •on 24 ROYAL SOCIETY OF CANADA SERIES III. 1,335 Ibs. 2,670 Ibs. 4,050 Ibs. 5,340 Ibs. 6,675 Ibs. — 274 — 553 — 836 — 1078 — 1325 — 115 — 244 — 381 — 514 - 645 + 10 + 17 + 19 + 23 + 29 + 146 + 288 + 431 + 578 + 723 + 305 + 613 + 897 + 1167 + 1445 By plotting the above results, diagram XXIII. was obtained. TABLE OF MAXIMUM BENDING MOMENTS AND DEFLECTIONS. Load in Ibs. Max. Bending Mo- ment in in. -Ibs. Deflections in inches. 1335 36045 •004 2670 72090 •007 4050 108135 •009 5340 144180 •013 6675 180225 •016 SERIES IV, 1,665 Ibs. 3,330 Ibs. 4,895 Ibs. 6,660 Ibs. 8,325 Ibs. — 340 - 683 - 997 — 1285 — 1573 — 166 — 314 — 493 — 645 — 778 + 9 + 21 + 30 + 47 + 58 + 197 + 374 + 570 + 733 + 893 + 389 + 732 + 1002 + 1294 + 1594 By plotting the above results, diagram XXIY. was obtained, but only .one series of observations was made in this case. [BOVEY] STRESSES IN BEAMS LOADED TRANSVERSELY TABLE OF MAXIMUM BENDING MOMENTS AND DEFLECTIONS. Load in Ibs. Max. Bending Mo- ment in in. -Ibs. Deflections in inches. 1665 36045 •007 3330 72090 •010 4995 108135 •013 6660 144180 •017 8325 180225 .020 DIAGRAMS rtiR CAST- STEEL BEflK? WITH LO/JDS CONCENTRATED flT CENTRE 77V JXVJT 26 ROYAL SOCIETY OF CANADA SERIES I. LOAD CONCENTRATED AT CENTRE. 800 Ibs. 1,600 Ibs. 2,400 Ibs. 3,200 Ibs. 4,000 Ibs. 81 - 169 — 245 — 330 - 417 — 33 — 64 93 - 124 - 164 + 5 + 6 + 13 + 28 + 35 + 38 + 89 + 141 + 203 + 255 + 87 + 185 + 287 + 39H + 507 By plotting the above results, diagram XXV. was obtained. TABLE OF MAXIMUM BENDING MOMENTS AND DEFLECTIONS. Load in Ibs. Max. Bending Mo- ment in in. -Ibs. Deflections in inches. 800 12000 •001 1600 24000 •002 2400 36000 •003 3200 48000 •004 4000 60000 •005 SERIES II. 1,200 Ibs. 2,400 ibs. 3,600 Ibs. 4,800 Ibs. 6,000 Ibs. - 113 — 253 — 381 - 517 — 657 - 52 - 100 - 158 — 206 — 273 + 3 + 13 + 28 + 28 + 30 + 61 + 140 + 228 + 294 + 356 + 133 + 286 + 450 + 586 + 715 By plotting the above results, diagram XXYI. was obtained. [BOVBY] STRESSES IN BEAMS LOADED TRANSVERSELY 27 TABLE OF MAXIMUM BENDING MOMENTS AND DEFLECTIONS. Load in Ibs. Max. Bending Mo- ment in in. -Ibs. Deflections in inches. 1200 1800 •002 2400 3600 •004 3600 5400 •005 4800 7200 •006 6000 9000 •008 SERIES III. 1,600 Ibs. 3,200 Ibs. 4,800 Ibs. 6,400 Ibs. 8,000 Ibs. - 172 — 338 - 519 — 706 — 886 71 - 133 — 211 — 296 — 376 + 3 + 22 + 29 + 27 + 29 + 83 -f 194 + 292 + 376 + 468 + 183 + 396 + 586 + 760 + 933 By plotting the above results, diagram XXVII. was obtained. TABLE OF MAXIMUM BENDING MOMENTS AND DEFLECTIONS. Load in Ibs. Max. Bending Mo- ment in in. -Ibs. Deflections in inches. 1600 24000 •002 3200 48000 •003 4800 72000 •005 6400 96000 •006 8000 120000 •008 28 KOYAL SOCIETY OF CANADA SERIES IV. 2,400 Ibs. 4,800 Ibs. 7,200 Ibs. 9,600 Ibs. 1,200 Ibs. - 251 — 514 — 790 — 1060 — 1330 — 103 — 213 - 342 - 460 - 581 + 14 + 35 + 36 + 41 + 46 + 141 + 297 + 432 + 567 + 706 + 290 + 588 + 853 + 1111 + 1352 By plotting the above results, diagram XXVIII. was obtained. TABLE OF MAXIMUM BENDING MOMENTS AND DEFLECTIONS. Load in Ibs. Max. Bending Mo- ment in in.-lbs. Deflections in inches. 2400 36000 . '003 4800 72000 •006 7200 108000 •010 9600 144000 •014 12000 180000 •017 An inspection of diagrams VII. to XXIV., for the cast steel beam, shows that in all cases the stresses in the material are very approxi- mately proportional to the distance from the neutral surface and that this neutral surface very approximately coincides with the centre of gravity, indicating that, in the case of the concentrated loads, the varia- tion of stress in the beam in question is closely in accordance with theory. In diagrams XV. to XXVIII., plotted from the results obtained for a beam loaded at the centre, the neutral surface has very appreciably moved towards the compression side, but the stress in the material is still proportional to the distance from the neutral surface in its changed position. 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