1 W m H '1 \HH ALO—J IxJO ODD—4 . CONSTRUCTION or muwALmlumronM LOAD DIAGRAM ran HIGHWAY smoees. ‘THESISV‘FDR THE um 03 B. s; ' ‘ ’ P; A. 3611.. , M ngcmg .f Construction of Equivalent Uniform Load Diagram for Highway Bridges A Thesis Submitted to The Faculty of MICHIGAN STATE COLLEGE of AGRICULTURE AND APPLIED SCIENCE BY P.A.§§_11 M.Bogema Candidates for the Degree of Bachelor of Science June 1935 ACKNOWLEDGMENT We take this opportunity to express our appreciation for the very helpful suggestions offered by Mr. Neil Van Eenam of the Michigan State Highway Department. 83895 From the beginning of a bridge design, it is necessary for the engineer to decide upon the lead for which the bridge is to be designed to carry. Very seldom is it possible for hrm to obtain the actual weights of vehicles to be carried, and it therefore is necessary for him to deve10p types of loadings which will closely approach the actual ones. It has only been in recent years that a definite type of loading for highway bridge has been used. Previous to about 1924 a steam roller type was used as a bases of design -- that being the heaviest type of vehicle considered. At present we have various types of loadings, such as, the Cooper's E-loadings for railways, the n-loading for highways, and the elecuric railway loadings. In general, these various loadings consist of a series of concentrated loads spaced at definite intervals so as to represent the wheel loads of the train or truck as passes over the bridge. In working up the design, the particular loading chosen is moved back and forth over the bridge span until the position which gives the maximum stress is determined. ihe unfortunate thing about using tnese loadings is that there is no one particular position or the loading wnich will give the maximum Stress in all parts and members of the bridge Structure. Because of CfllS it becomes neCessary to determine a new position of the loading for practically every point Which is to be considered in the design. rhis pro- cess is very laborious and time consuming but is absolutely necessary if the design is to be of any value. It is need- less to state the necessity of knowing the capacity of the structure within reasonably close limits. As in the case of most laborious jobs and processes, certain short cuts and aids have been deve10ped. One of these is the moment diagram which finds its use in the determining of the stresses after the position of the concentrated load system.has been determined. This diagram gives the axle loads and their spacing, and also the sum of the loads and of the distances from the head of the train or vehicle procession to each load, and the moment about each load of all the loads that precede it. The method of using it can be found in any textbook on structural design or particularly in "Structural Theory" by Sutherland and Bowman. A second aid to the designer is in the form of equivalent loadings. These loadings may be of two types. The first one consists of a uniform load extending over the whole span along with a concentrated load so placed as to give the maximum stress. This type is illustrated in the Michigan State Highway Department Standard Road and Bridge Specifications which states in part,"A total load on each traffic lane com- posed of a uniform load of 450 pounds per linear foot and a single concentrated load of 21,000 pounds." This type it must be remembered is only an assumed equivalent, and there- fore, in many cases the results may vary guite a bit from the results obtained from the regular loading. From the second type of equivalent loading, known as an equivalent uniform load, more accurate results may be obtained and if used properly the results are equivalent to those obtained by the regular loading. Work with this type has only been done, as for as we can ascertain, with the railway loadings. A great share of this work on equivalent loads was done by Dr. Steinman and presented in the paper "Locomotive Loadings for Railway Bridges", Transactions American Society of Civil Engineers, 1923. The data compiled by Dr. Steinman is presented in the form of diagrams readily gives the equivalent load to use in any case after the influence diagram has been constructed. Its great value lies in the amount of time and labor it saves the designer. An equivalent loading chart of this type would be welcome to the highway bridge designer, so we propose in this thesis to present such a diagram which is suitable for determining the equivalent uniform load which when applied to the whole . span will give the maximum stress developed by the regular H-15 loading. Method used in deve10pment 23 chart: The H-15 loading is a concentrated load system which represents a fifteen ton truck followed and preceded by a con- tinuous procession of eleven and one-quarter ton trucks. The distance between axles of the same truck is taken as fourteen feet, and the distance from the rear axle of one truck to the front axle of the following truck is taken as thirty feet. The load of each truck is considered as having eight-tenths carried by the rear wheels and two-tenths by the front. In compiling the data for the construction of the dia- gram, this H-15 loading was first put in the form of a moment diagram to facilitate its use in determining maximum stresses. The complete diagram as we used it consisted of not just one, but of a series of diagrams so arranged that when considering a situation there was a diagram which could be used without having any load passing off of the span. This helped greatly in that it alleviated the work of subtracting the effects of the loads which had passed off. The first step in the actual computation of the equi- valent uniform load is the determination of the positions of the concentrated loads which will give the desired maximum stress. This may be done in any of the ways described in the texts on structural design, but because of the apparent uniformity of the loads this might more easily be done by direct application of the moment diagrams. It can easily be seen that with the H-type loadings the maximum moment will occur with the heaviest concentrated load at the peak of the influence line. In this case, the first step is eliminated leaving only the computation of the bending moment to be done in order to determine the stress. As an example, consider finding the equivalent uniform load for the sixty foot point of a two hundred foot span. Draw the influence line for the moment as shown in Fig.1. Next 42.0 apply the moment diagram to determine the maximum moment, 11» 12 remembering to test for the con- 60 I40 dition where the loads are pass- ing from the short segment to the long as well as from the long segment to the short. Long to short: ( 140-60 ) Select from the series the moment diagram which has its heaviest load at a distance of 60 feet or less. This is M.D. I which has its 24 kip load at a distance of 58 feet. Mom. = 11226'gbé120°0 x 8) 60 - 1157 = 2519k“ V// Short to long: ( 60-140 ) Use diagram M.D. VI Mom. =_9711 igéél5'5 x 16) 140 - 5556 - 2555k' // Equivalent Load (q) Moment = 2535 ‘— q - Area of'influence triangle 3 x 200 x 42 = 605.6 lbs. per linear foot rhis load of 603.6 lbs. per linear foot when applied to a 200 foot span will produce the same bending moment at the 60 foot point as would the H-lS loading. The advantage of knowing this load when finding the bending moment is quite apparent after working backwards through the last problem. Given q I 605.6 lbs. Find the maximum bending moment at the 60 foot point of a 200 foot span. Solution: Draw the influence line as in Fig.1. Substitute in the formula B.M. = %q1112 11 and 12 are the segments of the span. B.M. 8 e x 605.6 x 60 x 140 = 2555.1 rhis is unquestionably a much shorter process than that used in first determining the bending moment from.the moment diagram. The only thing which now prevents the use of these various uniform loadings is a source from which to obtain the proper "q" for the particular situation under considera- tion. For this purpose We offer the accompanying diagram along with an explaination of the method of its construction 'with illustrations to prove its validity. The computation of the diagram consisted chiefly in com- puting a uniform loading for sufficiently large number of possible conditions. This of course could be extended indef- initely, so we set the limits at a 500 foot span. Most ordin- ary spans fall well within this limit. The basis of the computation of moments is the influence line for moment, so in selecting the points to be computed, we assumed various conditions of this influence line. The first condition considered was with the short segment of the influence line held constant at 10 feet and the long segment varied from 10 to 300 feet by small intervals. Next the short segment was held at 15 feet. This was continued until the short segment had been increased to 500 feet by the same intervals as the long segment had been increased. This data gives a concept of the range over which the uniform loads are spread as well as the points which have the same uniform load. This data may also be plotted upon the diagram in the form of lines through the points of equal uniform loads.To facilitate the selection of these points, and to reduce the errors of interpolation between the points, the computations are compiled in the form of graphs with the short span held constant using the long span as the abscissa and the "q" as the ordinate. From these graphs the desired points were taken and plotted upon the diagram, and lines of equal load sketched in. _ Upon inspecting the resulting diagram, it was found desirable to compute the "q" for a few additional points so as to more accurately locate the position of the load lines. These points were computed, graphed, and plotted as the others were, thus completing the diagram as here submitted. The actual results of the computations made may be found in the accompan- ing chart. The graphs for the major part of the work have also been shown. Use of the diagram consists of finding the point at which the long and short spans intersect on the diagram, and select- ing of the uniform load for this point. e.g. The uniform load for a short span of 80 and a long span of 110 is 602 lbs. per linear foot. As to the validity of the results obtained from the dia- gram we offer the following examples as proof. Example 1. Required the maximum.bending moment at the quarter and half points of a 120 foot span. For quarter point: leeeeti + : ‘ 3°. 0, SD' 50.. Using moment diagram M.D. I ,1 Iii-'0’ 5990 k' 2&‘_ M "-I§U- x 50 - 84 8 915.5 ' ' Using uniform load diagram 0 Z ' so 90' l q ‘- 6'79 36 ' r ; l/l\i M'%x6’79x50x90=916.6 For half point: Using moment diagram M.D. I M = é%%%-x so- 1137 = 1220.5 Using uniform 10ad diagram q 3 678 l x 678 x so x so 3 1380.4 4 Example 2. Required the maximum bending moment for the 40, 80 and 120 of a 240 foot span. At 40 foot point. .__ l 1 l ' I db- 1 r Using moment diagram 139391j-124.5 x 10 x 200 _10Q39 24o ‘ ‘ 2371 // Using uniform load. Jl__d f 14 6 @ 40 ' 240' rI M 65.3 Q= 581 40 ' 200 ' M = l x 581 x 40 x 200 = 2324 (3 At the 80 foot point. 53.5 Using moment diagram M : 14586 240 Using uniform load x 80 - 1137 3785 80 160 q = 584 60.0 M : MP x 594 x 80 x 1 (3) 0 5737.6 At the 120 foot point I Using moment diagram I 120 120 I] M: 1%E%§.x 120 - 3180 . 4193 Using uniform load q z 583 M : % x 583 x 120*x 120 = 4198 These results by the two methods vary less than 0.5 of one per cent either way, but are sufficiently close for any ord- inary design. The variation is probably due to errors in plOt- ting and in interpolating the results. In conclusion we wish to point out that this diagram is nOt limited to only the a-15 loading, but may be applied to any of the H loadings by using a conversion factor. We selected the H-15 loading as it is used in 50 per cent or more cases of highway bridge design. r. parry r hr _ a C Tu Y. .C of" D'lll .(f 1.. .4” \QSV m m m m uthkuh. ’4 W‘ 1 ‘ (or); Seymcnt J/Iort Seq men 6 300 30 4o 50 60 70 do 90 mo “—0 /:)"0 <90 70 60 JD om CURVES OF' 40 EQUIVALENT UNIFORM LOADS FOR MONENT5 ‘ FOR ,0 CLASS H—I5 LOADING Values are given in pounds per linear foot per lane. Class H'IS Loading 20 /0 e (unafrovct‘ed by: ”Joyema \ Pfie/l. > Equivalent Load Chart 10 15 2o 25 50 55 40 45 50' 55 60 500 558 555 559 579 571 555 557 557 5'51 552 562 280 514 504 594 584 575 570 554 550 554 555 565 250 521 508 599 588 579 572 555 552 555 558 568 240 552 525 507 595 585 578 572 557 572 575 575 220 559 527 512 550 588 581 574 559 574 575 575 200 555 540 525 512 598 590 585 577 582 584 584 180 557 557 554 518 505 595 585 580 585 587 588 150 590 570 552 554 518 508 598 591 595 598 597 150 701 578 559 541 525 515 502 594 500 502 602 140 711 589 555 545 551 517 505 597 505 505 604 150 722 597 574 552 554 520 508 599 505 507 605 120 745 718 595 559 548 554 520 511 517 519 617 110 758 759 709 572 552 545 550 519 525 527 625 100 787 755 722 595 575 551 555 524 550 525 529 95 797 751 728 598 575 554 555 525 552 529 631 90 805 755 752 700 575 555 555 525 552 538 652 85 825 785 745 712 585 555 544 525 558 650 658 80 850 805 757 729 700 575 557 555 550 551 649 75 880 850 787 747 712 588 570 545 550 501 659 70 905 848 804 751 729 702 580 557 570 570 558 55 952 878 825 777 747 717 589 570 578 672 574 50 955 895 842 792 752 729 599 578 582 700 578 55 987 915 855 802 747 759 707 585 588 701 50 1015 957 872 815 780 745 708 575 589~ 45 1058 952 878 814 775 759 700 578 40 1115 1015 950 857 798 751 725 55 1225 1104 1005 4920 849 807 50 1550 1210 1085 990 907 25 1522 1552 1184 1055 20 1720 1472 1290 15 1952 1550 10 2400 ‘ Equivalent Load Chart 55 70 475 80 *85 90 *95 100 110 120 150 500 551 '559 ‘557 555 '552 '552 f555 '55? ‘552 ‘556 '517 280 554 555 559 558 550 555 555 555 555 555 549 250 557 554 550 559 555 555 555 555 555 554 550 240 572 559 555 554 559 559 551 551 550 558 554 220 575 570 555 555 551 559 552 551 551 558 554 200 580 578 575 572 555 555 557 558 557 554 559 180 582 581 578 574 570 570 570 570 554 555 551 150 595 590 584 582 575 575 575 577 575 572 555 150 598 595 589 585 579 579 580 582 579 575 559 140 501 597 592 588 582 580 582 582 580 5/5 510 150 505 598 595 589 582 584 582 582 580 575 570 120 515 508 598 597 588 587 587 589 587 582 110 521 515 505 505 594 595 594 595 595 100 525 521 514 505 501 598 598 500 95 525 522 515 509 502 598 599 90 626 619 609 596 85 652 625 619 611 605 80 642 654 619 75 651 645 654 70 659 650 65 665 * Segments in this group are five feet longer, 15,105, 115, etc. 500 280 260 240 220 200 180 160 150 140 150 140 550 550 554 554 559 561 566 569 569 150 552 555 557 557 565 565 571 574 quivalent Load Chart 160 551 552 556 556 562 564 569 180 5'43 546 546 549 649 555 554 200 5'45 545 546 549 549 555 220 542 545 545 545 240 542 545 545 250 5'39 540 541 280 558' 540 500 5'57 Moment Diagrams and Computation Graphs MOD. Mom. SD SL for fi-l5 Loading Key to symbols used Moment diagram rMoment of all preceding loads about point Distance in feet from first load Dietance between loads in feet COHCenEfaodd load in kips Sum of loads up to that point in kips Equivalent uniform load M.D. 1 ;.D. 11 _Mom. 5D D, L 5L, 3 SD 9, L 3L 0, 0 4-5 4-0, 0 0 18 18-0 55, 14114 18 22.5 540 5 50 5 24.0 875 44 14 ,24 48-0 758 4 50 5.0 23.0 ‘-11L7 5 11* 24 52‘9 ,1020 .24, 80 4-5 ,52-5 1755 88 14 18, 70.5 2712 8 50 4-5 57.0 5510 102 14, 18 75-0 5870 118 50 ,4-5 75-0 4920 152, 14 18 95.0 5760 ,152 50 4.5, 79.5 6873 l45 14 18 97*5 710,152 50 4.5 97.5 9075 175 14 18 115-5 9798 17 50 4.5 102.0 112 r 0 . 2Q 19 *14 18 *1“0 Q _1254 205 50 4.5 120.0 14220 220 14 18 158-0 4825 220 50 4.5 124.5 5554 254514 18 14245 18550 250 50 4-5 142-5 20555 25 ,14 ,18 ,150.5 20844 26 50 4.5 147.0 2902 278 14 18 165.0 25170 20 50 ,4-5 155-0 .21480 508 14 18 180.0. 97852, 50 50 4-5 159-5 0225 ,52 14, 18 187-5 , , 52970 558 50 4.5 187.5 55595 552 14 ,18 205.5 5250 55 50 4.5 192.0 7988 ,55 14, 18 210.0 59750 582 50 4.5 210-0 BOP“ B. M.D. IV M.D.III L SD 0 4 415. 758 _1.4_ 44 .144 50 .18— 4.5 SD ‘.0 540 30 50 4.5 22.5 855 44 14 18 40.5 . 1116 58 14 18 2466 88 30 207 50 46.5 3180 102 14 24 5450 132 50 4.5 4555 - 50 75.0 5886 9468 176 30 1.14.318 4.5 8676 162 30 4.5 97.5 10041 176 14 18 115.5 10896 190 14 18 14496 220 30 13506 206 50 4.5 120.0 15186 220 14 18 158.0 16259 254 14 20514 264 30 19526 250 30 4.5 142.5 21521 264 14 18 160.5- 7 27522 78 308 50 14 26156 ‘294: 50 4.5 165.0 28446 308 14 18 ;;83.0 _22895 522 -14 55520 552 30 192.0 35956 338 50 4.5 187-5 36561 352 14 58208 366 14 210.0 44508 596 50 4.5 214.5 42726 582 30 205 5‘ 4.5 210.0 47511 410 14 18 252.5 54656 440 50 4.5 237.0 _45555 396 18 52506 426 50 4.5 252.5 55761 440 14 18 250.5 18 55.0 .63276 50 255.0 M.D.V Mo Fir VI Mom. 63 14 14 18 22.5 SL 18 18.0 738 44 30 4.5 27.0 540 4.5 22.5 855 18 40.5 58 14 18 45.0 1116 2466 30 4.5 49.5 2070 74 45.0 2700 88 63.0 3159 5187 9141 6216 181, 67.5 30 6,0 73.5 4590 118 69.0 14 24.0 97.5 30 4.5 102.0 5556 8346 132 162 4.5 gas—£3. 97.5 ‘ 971; 1056 14169 18 30 4.5 124.5 1251.9. 13176 176 206 4.5 18 115.5 120.0 15912 20187 264 18 412.5 30 4.5 147.0 18996 250 138.Q_ 4.5 142.5 20991 264_ £2385 278 14 18 165.0 27335 308 30 4.5 169.5 25806 294 160.5 30 165.0 _29708 322 14, 187.5 35333 352 30 4.5 192.0 28116 33606 Egoa 14 338 30 4.5 187.5 6231 352 18 29515.. 580g; 366 14 18 210.0 44321 396 4.5 214.5 42396 382 30 4.5 210.0 47324 410 14 18 232.5 +35.356 396 14 18 228.0 52176 426 30 4.5 232.5 55431 440 18 25015 B . ('5': B. MID. VII SD D SL O 0 L 4425 14 14 18 22.5 30 4.5 271d 14 18 45.0 49.5 .192 67.5 72.0 176 146 90.0 96.0 l 220 190 ___ 120.0 19857 26865 308 41334 34863 352 E L: 755 43851 F” 396 5 _142 30 4.5 214.5 53829 440 JE k: 118 252.9. 30 4.5 237.0 64797 57147 7:? E .18 39 255.9. 259.5 M.D. VIII fiom.. SD D L SL 0 0 0* 18 1870“ 540 80 80 4.5 22.5 855 44 14 18 40.5 207 74 80 4.5 45. 2700 88 14 18 63.0 4590 118 80 4.5 87.5 585 182 14 .18 85.5 8100 182 80 8 91.5 4938117; 14 24 115.5 12840 208 80 4.5 120.0 4526£ZO__1_2_7_L_£5_§;0_ 18888 250 80 4.5 142.5 20881 284 14 18 180.5 25478 294 80 4.8 185.9 _21788 308. 14, 18 18810 88278 888 80 4.5 187.5 .22291_§§2__1A,18 205.5_ 42088 882 80 4.5 210.0 45008 898 14 18 228.0 81848 428 80 4.5 282.5 55101 449 14 18 250.5 82818 470 80 4.5 255.0 88188 484 14 _1§ 278.0 8.6: B. M.D. IX M.D. X Mom. 8D D L SL Mom. SD D L SL 0 U* 0 4-5 4-9 O 0 0 18* *‘18. 55 14 14 18 32-5 540 80 80 4.5 22.5 855 44 14 18‘ 40.5 1116 58 14 18 45.0 2070 74 30 4.5 45.0 2700 88 14 18 83.0 2488 88 80 4.5 49.5 5159 103 14 18 b7°5 4590 118 80 4.5 87.5 5585 182 14 18 85.5 5187 132 30 4.5 72.0 6195 145 ii: 18 9920 8100 182 80 4.5 90.0 ' 9880 178 14 18 108.0 8895 178 80 4.5 94.5 10218 190 14 18. 112.8 12600 206 50 6 114 0 14176 220 1;: 24 122.9 13593 220 30 6 118.5 mgéLg 184355 250 50 4.6 142.9 20881 284 14 18 188.5 19527 234 30 4.5 147.0 21583 278 14 18 65.0 25146 234 30 4.5 165.0 27436 308 14 18 183.0, 28585 808 80 4.5 188.5 38908 523 l4~ 18 ' ‘7'8 52948 888 80 4.5 187.5 88871 852 14 18 208.5 84588 852 80 4.8 192.0 77221,.36 C) 14 18. 110.0. 41758 882 80 4.5 210.0 44878 898 14 18 228.0 _1 43521 396 30 4.5 214.5 46524 410 14, 18 232.5 51516 426 30 4.5 232.5 54771 440 14 18 250.5 58499 440 80 4.5 287.0 85818 454 14 15 “00-0 82288 470 80 4.5 258.0 65856 484 14 18 273-01 04467 484 80 4.5 259.5 88100 498 14 18 r,7,; 74045 514 30 4.5 273.5 76425 528 30 4.5 282.0 P"? I D O XI Mom. SD d L SL ,2___92--_O--_Q--1§2§1-22§!51 55 1 14 +14 1§__.-.22.51 788 44 059_ “4.5 27.0 1115 58 14 1f 4500 2466 88 J_§_Q__ 111.1%. 5 1..- 49 o 5__1 r2239 .11- .l__2.__.. ii“ .1 .113. 67 Q 5 5187 152 50 4.5 72.0 ._ -11. 1 8195 148 14 18 90.0- 1421221174 .80,- 4.9 94.5.- rl921§11120.1141-18_222112221 1&2012229-1229 211-9.-- .lfigél .Zéé-fil§-118 1 l§§~Q1 19281 284 80 8 141.0- 21258 278 112 24 185.0 .25395. 508 50 4o§2.12995. .28529-1522_114.11822-1182.52 54204 1552-299” 1§-§1119329- 88892 888 14 18 210.9 43192 595 §9111§1§-121295- 48195 410+l§~413 -252.2- 22179129.- 291- -4.- 2 241-0 - 58488 $454-,14 l§1_ -22§.O- 24138 484 5.9- -. 22 25.9.5- szjl_1498- 14 18, 1277.54 78098 528 80 4.53880+ 80044 542 14 18 800.0_ M.D. XII Mom. 5D D I 3L 0 0 0 418 _17180Q 540-.30_1§9_1-§25_ _§?°Q 855 44.-14111§___ 40.5 L247i1HY-é. 1. 17991.1 1.231112? ° 9.1-1 L 2700 88 14 18 mug”j 4590 118 80 4.5 87.5 5555 152 14 18 85.5 8100 162 50 4.5 90.0 2462115214 18 108-2911 _12600 205 50 4.5 112.5” .14115-220_-14 18 180.81- 18090 250“ 30 _6 r100.5* ‘20001 284 :14 24 180.5 .24§1§1§§4_1§Q-11§25 -1$5¢9_ 2212§1§9§_114 18 428-0 _22216-§3§_1§911249§-118795 _fiégél1§§§;114 18 29525. _41406 882 50 4.5 210.g_fi L44548-898-k14-_18 228.9- 51186 426 50 13.52 252.911 84441 442- 14 18 250.02 .91222141911491 25(21 225-Q.- 2228-484114- 18-221272 . 0. y8718 814 80 1 4.5-1217.5. 11291-528 14 18 295.5 88458 558 8g_fi_4.5 fi809.g_- 8.8 B. 240 230 220 210 2000 190 leoo 170- _ 1300 1200 100 000 900 800 700 600 q 500 ml “col O 20 4O 60 80 100 120 140 160 180 200 220 240 260 280 300 long segment in feet Short Segment 10 B. & B. 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