A DESlGN CHECK OF A TYHCM. 200 FOOT THROUGH RAILWAY TRUSS SPAN find: but flu Dean. of I. S. MICHIGAN STATE COLLEGE John 5. Carter 1949 ‘ .-' x. f " $5,; w-l‘. "Vt. V“’fio\'"_" A , ) i. V'- i .' 2’ .‘ ' 2-9.— . «yw A Design Check of a Typical 200 Foot Through Railway Truss Span A Thesis Submitted to The Faculty of MICHIGAN STATE COLLEGE of AGRICULTURE AND APPLIED SCIENCE by John S. Carter m Candidate for the Degree of Bachelor of Science June 1949 ACKNOWLEDGEMENT The author wishes to express his sincere appreciation to Mr. Marvin Leeper of the American Bridge Company, for supplying the plans used herein, and to those faculty members, whose timely aid and criticism proved so valuable in the completion of this thesis. £15935 Article I. II. III. IV. V. VI. VII. VIII. IX. X. XI. Figure Figure Figure Figure Figure Figure Figure Figure Figure Table of Contents IntrOdUCtion 000000000000000000000000000 General Features ....................... Loads 000000000000000000000000000.000000 Floor syStem 000000000000000000000000000 Stringers 000000000000 00000000000000000 Intermediate Floor Beams ............... End Floor Beam ......................... Truss 0000000000000000000000000000000000 Br301ng 000000000 0000000000000000000000 End Bearings ........................... conCIuS'ion 00000000000000000000000000000 l. 2. 5. 4. 5. 6. 7. 8. 9. Table #1. Diagrams and Tables Clearances Floor System Lateral Stringer Bracing Intermediate Floor Beam End Floor Beam-—Live Loading Truss--Dead Loading & Stresses Influence Lines Section of Member LO-Ul Bottom Lateral Bracing Summary of Stresses & Properties 30 34 55 I. INTRODUCTION The railway bridge, the design of which is to be checked herein, is a single track through Warren type truss with vertical members and riveted connections. It is a straight single span 200 feet long, consisting of eight 25 foot panels, and a height of 52 feet. The distance between center of trusses is 18 feet-2 inches and between stringer is 7 feet. The plans from which this analysis is made were fur- nished by the American Bridge Company, Gary, Indiana and consist of: D-2 : General Design (Sheet 2 of 4) L-l : 200' Truss Span Joints L-2 : 200' Truss Sean Joints L-S : Portal, TOp Chord Bracing, Sway Frames L-4 : Bottom Chord Bracing and Floor System As no information is at hand regarding the particular specifications used in the original design, the "Specifi— cations for Steel R ailway Bridges" as published in 1935 by the American Railway Engineering Association will be used as a guide. Any features found not to agree with the above mentioned specifications will be noted and, if pos- sible, an explanation presented to justify the discrepancy. _I_;. GE IEF. AL FEATURES MATERIALS: All structural steel has been used with inserts of USS 12 specified at expansion shoe. This is in agreement with specifications. TYPE OF BRIDGE: A riveted truss is the preferred type of bridge for scans 100 feet or longer. S?ACING OF TRUSEES AND GIRDERS: Trusses: Spaced 18'-2" c.c. Length of span is 200'-O" c.c. end bearing. R : 18.17 0 200 - 0.091 o.k.(0.05 min.) Stringers: 2 stringers used @ 7'-O" o.k.(6'-6" min.) DEPTH RATIO: Depth of truss is 32'-O" c.c. of chords Ratio = 52 O 200 a 0.16 O.ko(0.10 min.) CLEARANCE: Designed clearances are found to be either equal to or greater than specified clearances. (See fig. 1). L—m . ., / ,, t ' _ ‘¢-‘ " FVQJW )- t xi) V‘ . \ (— ‘ i 9. A- z! 2 r ~ "‘ 3 l '\ 3,_ “ c > I“ .a ‘4 ‘ > ‘- '. .‘ I‘ x. ”/0 ’73.?" cf ICU" of“ run/wrm: FO//5 C/eww :e p=45* I ~I" .“ Z ‘6 /u x/o”x/090” ____J .q £335" 7<-' dysfem III. LOADS DEAD LOAD: Track rails guard rails & fastenings 200 lbs./ft. Timber ties @ 60 lbs./cu.ft. = (0.75)(7)(60) 515 v " Steel; w = k(9L 4 700). (l.264)(1800 4 700) = 3160 " " Total dead load - 3675 lbs./ft. Assumed total dead load 5700 1bs./ft. 570 kips. Total dead load per truss : (%)(3700)(200) End reaction per truss : (%)(570) 185 kips. It should be noted that the formula, 0 : k(9L 4 700) as found in the "American Civil Engineers' Handbook," 5th edition, is based on an allowable unit stress of 16,000 psi. and therefore will produce a conservative result in that present day practice permits an unit stress of 18,000 psi. LIVE LOAD: The live load to be used is the COOper E-72 as is re- commended in the specifications. IMPACT: Numerical values for impact are computed by the use of the following formulas: (a) The lurching effect: A percentage of the static live load stress equal ......................... 100 S S = snacing, in feet, between centers of stringers or trusses; or length, in feet, of floor beams. (b) The direct vertical effect: A percentage of the static live load stress equal to: For "L" less than 100 feet............. 100-0.60L For "L" 100 feet or more...............l800 4 10 L-40 "L" : length, in feet, center to center of supports for stringers and trusses (chords and main members); or length of floor beams, in feet, for floor beams and floor beam hangers. g1. FLOOR SYSTEM The floor system consists of wooden ties placed on stringers as shown in fig. 2. Specifications state that the maximum wheel load on one rail is uniformly distributed over three ties, and is applied without impact. The re- commended maximum wheel load is 45,000 lbs. The moment due to this load is: M : 45000 x 7-5 x 12 = 180,000 in.lbs. 5 2 The moment due to weight of the floor is very small com- pared to the live load moment and is usually neglected. Cross ties are assumed to be of White oak with an allowable extreme fiber stress in bending equal to 1200 psi. The required section modulus of the tie is pp}: leo,ggg . 150 cu.in. e 1200 For a 10 x 10 in. tie, the section modulus equals 167 cu. in. which meets the requirements. Length of tie equals lO'-0" which also agrees with specifications. They are to be spaced not more than six inches apart and secured to prevent bunching. V. STEINGEES The stringers are designed as simply supported beams equal in length to the distance between centers of floor beams which, in this case, is 25'-0". In the design, the stringers are 550B240 with end connections to floor beams of 2 - 3% x 3% x 5/8 in. angles 2'-3%" long. DEAD LOAD: Assumed weight of stringers = 1.264((12.5)(25) + 100).... \‘Jeight or floor 00000000000000000000000: Total dead load.. Dead load per stringer ................ a Dead load per stringer used in design.. Dead load center moment = 600 x 625 ... Live load moment 3-72 = 72 x 381.5 .... =‘ 50 ImpaCt39903% Of 550 0.090000000000000: Total maximum moment Required section modulus, S - 1143 x 12 522 515 1057 518 600 47 550 546 lbs./ft. I! n 1bs./ft. lbs./ft. H " ft. kips 1143 ft. kips Section modulus of a 330B240 equals 811 cu. in., therefore, section meets requirements. END SHEAR: Dead loadgsooxzs 00000000000000...000000000 : 705 kips 2 Live 108d E-72=72x.71 ccoocoooocccocooococcc:10200 " 50 Impact=99.5“/30f102.0 ooooooccooooccooooccooc.10100 " Total end shear 5 210.5 kips Unit shearing stress in web - ‘;210500 .....= 8250 psi. 30.7 x ofES‘ o.k. (allowable-11,000 psi.) FLANGE BUCKLITG: Ratio L 3 25 x 12 = 18.9 o.k. (allowable a 40) 15.9 Allowable unit compressive stress in flange 18,000 - 5(18.9)2 16,210 psi. Effective area : (15.9)(1.4) 4 25.5 ...... .2 25.44 sq.in. 8 Actual unit compressive stress : 1145000 x 12 a 15,600 psi. 25.44 52.4 This shows a slight overstress in the compressive flange but hhis is negligible as the formula used to calculate allowable stress is empirical and conservative; and also the restraining effect of the timer ties has been neglected. DIAGOKAL WEB BUCKLING: Ratio h_s 50.7 = 57. This is less than 70 and is con- t 0.85 sidered very safe. END CONNECTION: The connecting angles are 6 x 8 x 5/8 in. angles 2'-s%" long with the 8 in, leg against the web of the string- er. The number of rivets through the floor beam web is controlled by single shear at 8.1 kips per rivets. The number of rivets required is giggg = 26 rivets. Detail 8.1 sheet shows 26 rivets which meets the requirements. LATERAL BRACING: The stringers are provided with lateral beacing con- sisting of 5% x 5% x 5/8 in. angles arranged as shown in figure 5. Each 25 foot panel is divided by cross frames consisting of 2-5% x 5% x 5/8 in. angles and a 24 x 5/8 plate. This agrees with specifications which limits the length of panel without cross frames to 20 feet. The lat- eral forces carried by this bracing is due to a wind force on the train of 500 lbs. per linear foot of bridge plus one-half of the wind force on the truss, or 840 lbs. per lin. foot. The total force carried'by stringer bracing is 1140 lbs. per lin. ft. As the same size angles are used throughout, only that member having the maximum compressive stress need be investigated. In this case, the greatest stressed member is the diagonal "ab". The force is con— sidered a moving load and is applied at the panel points. The load per panel point equals 1140 x 12.6 or 14.25 kips and the shear on section y-y equals 7.15 kios. Diagonal "ab" will have a compressive stress of 7.15 x 88.5/66 : 9.55 kips. The unsupported length of the member is 88.5 inches and the least radius of gyration of a 5% x 5% x 5/8 inch angle is 1.07 inches. The ratio L/r z 88.5/1.07 : 82.6. The allowable unit stress is 15000 -(O.25)(82.6)2= 15000 -1700 = 15,500 psi. The area required for member "ab" is 7.15/15.5 or 0.54 sq. inch. One 5% x 53 x 5/8 in. angle has an area of 2.48 sq. in. which is excessive but it is the smallest angle gener- ally used for this purpose. P; 7.1 9’ ’ 3 ' U \l c |. l \j A . '1 \ {/1 :5. w l ) ' I II ‘ >4 1/ x— , 1 (7 a r - y . .. < .3 . /-; 0 II“ (”I ll . {)llr'c-c; 4/0!) C/ 4/} id F13... {— a-f<.=z;‘.«:z/ 5:37: I321? g7: -‘ I “ - / ‘- I, ... _/ _- ., v 1) Acsw . 1/ 11*. ~ 3:” “1+ A ' .. y ” r“ /~ ,_ R F1? 4 f *7 1‘8 "Nu. ,2"st .4 6? F7.“ 5 ‘ *5 ; it x a- l; . x / 10. VI. INTLFEEDIATE FLOOR BEAMS At the panel points are located intermediate floor beams built up of 4 - 6 x 6 x 7/8 in. angles and a 52 x 5/8 in. web. In calculating the stresses, the length of beam is taken as the distance between centers of trusses which is 18'-2" in this cas e. The floor beams transmit to the trusses their own weight plus two concentrated loads from the stringers, as shown in figure 4. The dead load consists of the weight of the floor beam assumed at 225 lbs. per lin. ft. and the weight of one stringer equal to 15000 lbs. MAX III—U154 SHELAE 3 Dead load 18.17 x 225/2 + 15000 .......... = 17 kips I! Live load _ stress in hanger ............... = 156 ImpaCt ’ 95% Of 136 oocccoccoococccococccccc = 130 " Total shear . 285 kips MAXIMUM MOMENT: Dead load 95 ft.kips (225)(18.17) ; (15000)(5.58).... 8 Live load-156XS.58 ooooocccooooocooooccc 760 " " ImpaCt = 95% Of 760 cocoooocccccocccocccooco : 720 H N Total moment :1572 ft.kips ll. CHECK OF SECTION: Required web area : 285/11 ................ = 25.8 sq.in. Furnished web area = 52 x 0.625 ........... = 52.5 sq.in. o.k. Minimum thickness of web plate 2 40.5/170.. : 0.24 in. Actual thickness of web plate ............. = 0.625 in. o.k. Required section modulus - 1575 x 12 ...... 1050 cu. in. 18 Total net moment of inertia of section I... 28050 in? Furnished section modulus : 1’: 28050 ..... 1065 cu.in. o.k. c 26.25 COMPRESSION FLANGE: EatiOL:15.59X12 coco-000.000.00.000...:14081n0 12.625 2 Allowable unit stress ; 18,000-5(l4.8) .... 3 16,900 psi. Effedtive depth : 52.2-5.64 ............... - 48.85 in. Total flange stress - 1575 x 12 ........... 3 587 kips 48.86 Total compressive area required - 587/16.9. 22.9 sq.in. Including 1/8 web area as effective flange area, the total effective area supplied is 19.46 4 4.06 = 25.52 sq.in. o.k. FLANGE RIVETS: The total flange stress is assumed to be the same as that at the center of the floor beam which is 587 kips as calculated above. Due to the assumption that the web car- ries a portion of this stress, the total flange stress may be reduced by the ratio of net flange area to total moment- carrying area. Therefore, the stress in the angles is s a 587 x 17.49 3 514 kips 2I.55 12. The number of rivets required between stringer and end is controlled by bearing on the 5/8" plate and is 514/14.75 or 22 rivets. Plans show 54 rivets being used which is more than enough. The shear between stringers is so small that the rivets are spaced at 6 inches which is the maxi- mum pitch. INTEFMEDIATE STIFFENE”S: Stiffeners are required where the depth of web between flange angles exceeds 60 times its thickness. In this case, the depth of web is 40.5 inches which is greater than the 60 x 0.625 or 57.5 inches. Therefore, stiffeners are requir- ed. The spacing of stiffeners as calculated by the formula Specified is found to be 94 inches but 72 inches is the maximum spacing permitted. Stiffeners consisting of 2 - 5% x 5 x 5/8- in. angles have been placed 5'-5" from each end and at the midpoint. These are conservatively placed but this is probably due to need to place one stiff- ener between the stringer and the end which would require that one be placed between stringers. The midpoint of span is the best place to locate the stiffener between stringers to preserve symetry. Width of outstanding leg of each angle is 5 inches, which is less than 16 times its thickness and greater than 2 4 52.5/50; therefore, size of angle is alright. 2ND CONNECTIONS: The connecting angles are 6 x 4 x 5/8-inches with the 6-inch leg riveted with a 10 x 7/8-in. fill plate to the web of the floor beam. The required number of rivets is. controlled by bearing on the 5/8-inch web and equals 285/14.75 or 20 rivets. Plans show 26 rivets being used which is sufficient. < .‘o. “\1 ..- 5. I‘d. u .5 : ‘5. I, (:7 1/ 4 . ......fl. 4 a «a x" ‘~ ‘ ' 1‘ - I‘I/ \\ ,/ .\ 1 I I ‘7 3 .' ‘94 “ Ix_ fl \4‘ 4 - ‘ I < '§ ‘4; «l, , n" \‘t. . _ .-, ’ r) " ‘2’. 4 r-v‘i a ...q I. . I , x ‘ \ 4 .v ’3 ! I" .. \ I 1‘ b 4. . \ / (j?) ...—..4 . -_‘.._ ., ....._. -.--— - -t //-6 I I 1‘,“ ‘ u (-1 (x :4; lief +35 ‘/£3 LT.“ 4.4-4,..- ‘7 ~- 7 ~ -+'—-- ‘ {L 4 A ‘x: .4 4 4 4 ..L L ' ‘- 4. r 14 A .‘5 f , - 3‘ ' .IC .— n / 1 mm 1.», . >M .JL . , ,, . 4 f f3 M II! ' ". ,vP-Z' "I f ['1‘ ,‘ _ 4’ r u. 1 [PM L ‘5 4.. C ' r I v— Eiei 5. sins ‘7 s 5 ~ : , J - 4 » as .L «<1 U/ A —‘ Fig: 4’2 9 11.61.10 .09/ L2 19 p Q! is . V‘ 7 _ s ”3 “2 ,. ' ., "I e? .2510” = 2000,07 J- Qi .t‘i.~:—‘.;ff;_§i:3_ in T/Zér’é ». J) 14. VIII END FLOOR BEAM The stringers in the end panels are supported by means of end floor beams consisting of 4 - 6 x 6 x 5/4-in. angles and a 52 x %-in. web plate. The end beam is connected by gusset plates on the foot of the end posts. Stringer brack- ets are designated for the end floor beams. The dead load is composed of the weight of the floor beam, assumed to be 5000 lbs. and the end reaction per stringer equal to 7500 lbs. flAXIMUM SHEAR: Dead load 1500*VSOO ......OOOOCOOOOOOOO 9.0 kips Live 108d - (SSS figure 5) 0000000000000... 2 10805 H ImpaCt 94.6% Of 108.5 ......OOOIOOOOOO £205 " Total shear... _ 220.0 kips MAXIMUM MOMENT: Dead load 7.5 x 5.6 + 18.17 x 5.0/8 ..... z 49 ft.-kips Live load = 108.5 x 5.6 00000000000 ooooo 000 609 n " ImpaCt = 94.6% Of 609 000.000.000.000... : 576 n " Total moment :1254 ft.-kips CHECK OF SECTION: Required web area : EHHLCUL................ 20.0 sq.in. Actual web area - 52 x 0.5 ................ 26.0 " " olk. Required web thicmesa.......OOOOOOOOOOOOO. 0.24 in. Actual web thickness ...................... 0.50 in. o.k. 15. Required section modulus : 1254 x 12/18 .... 822 cu. in. Total net moment or inertia . C O O O O O O C O I O O O O O : 25,962 in. Actual section modulus = %_= 225??? ........ : 912 cu.in. 0.k. Flange section ratio L/b = 18.17 x 12/12.5 . 17.45 in. Z Allowable stress - 18,000-5(17.45) ......... - 16,500 psi. Effective depth : 52.5 - 5.56 .............. = 48.94 in. Total flange stress 1234 x 12/4s.94 ...... = 303 kips Required total area 505/16.5 ............. - 18.4 sq.in. Including l/8 web area as effective in compression, the actual area is = 16.88 4 5.25 .............. : 20.15 sq.in. o.k. FLANGE RIVETS: Stress in angles between stringer and end is equal to 505 x 15.19/18.44 or 250 kips. The required number of rivets as controlled by bearing on the 4" web is 250/ll.8 or 22 rivets. Plans show 27 rivets are used. The shear between stringers is so small that the rivets are again spaced at 6 inches. IE 17‘34:»;431;113.1533 STIFFENERS: Stiffeners are required because the depth of web, 40.5", is greater than 60 times the thickness or 50 inches. The spacing of stiffeners as calculated by the formula specified is found to be 71". Plans indicate a stiffener of 2-5% x 5 x 5/8-in. angle placed at center of span. Since the end floor beam is to be used as a jacking beam, the stiffener placed at bearing point which is 5'-6" from center line of truss, must be designed to transmit the concentrated load of 185 kips (see general design sheet D-2). The moment caused by this lead is 185 x 5.5 or 648 ft. kips. Specifi- cations allow that unit design stresses may be increased by 50% for such cases. Therefore, the required effective area is 185/27 or 6.85 sq.in. The design sheet indicates 4 - 5 x 5% x %—in. angles are to be used. These angles furnish a gross area of 4 x 4.00 or 16.00 sq.in. and an effective area of 4 x 4% x 3 or 9.00 sq.in. which is ample. The outstanding leg of the angles is 5-inches which.meets specifications in that it is less than 16 x g or 8 inches, and more than 2 4 52.5/50 or 5.75 inches. Here it should be noted that although the design sheet indicates that the stiffener at the jacking point should consist of 4 - 5 x 5% x %-in. angles, the detail sheet, L-4, shows only two such angles to be used. The reason for this discrepancy is not immediately apparent so the detail sheet will be assumed to be in error. The size of stiffener at center of beam is not checked because the shear value is so small. The size was chosen to match the other stiffener angles. END CONNiCTION: The end connection consists of 2 - 4 x 4 x 5/8-in. angles and 2 - 7 x 5/4-in. fill plates. The outstanding leg of the angle meets the specifications in that it is less than 16 times its thickness or 10" and more than 17. 2 4 52.5/50 or 5.75 inches. Number of rivets required is controlled by'bearing on the %-inch web and equals 220/1l.8 or 19 right. Detail plan shows 15 rivets in the angles and 9 rivets in the fill plate or a total of 22 rivets, which is considered sufficient. 18. VIII- 231155 The truss under consideration is of the Warren type with vertical members. The span is 200'-0" center to cen- ter of end bearings and is divided into 8 equal panels 25'-0" long. As the truss is symmetrical about the center vertical, stresses will be calculated for one-half of the truss only. DEADfiLOAD STRESSES: From the assumed dead load of 5700 lbs. per linear foot, the dead panel load per truss is 25 x 5700/2 or 46.2 kips. Generally, this panel load is assumed to be distributed two-thirds to each lower chord joint, and one-third to each top chord joint. However, in this case, it appears that this distribution has been assumed as three-fourths of the dead panel load or 54.6 kips at each lower chord joint and one-fourth, or 11.6 kips, at each top chord joint (see fig.6). The reaction at one end of the truss due to these loads is 46.2 x 7/2 or 162.0 kips. The stresses are determined by the general methods of statics. The chords stress is determined by dividing the algebraic sum of the moment on one side of the section about the Opposite chord point, "M", by the height of the truss, "h". The web stresses are found by multiplying the shear on the section, "V", by the secant of the angle which the member makes with horizontal. The secant of the angle in this case equals 40.6/52 or 1.27. 19. LO-Ul = v sec 0 = -152 x 1.27 .............. = -205. kips Ul-U5 = M/h = -5940/52 ..................... z -217 " U5-U5 = m/h : -9255/52 ..................... = -289. " L0-L2 = 152 x 25/52 ........................ : 127. " L2-L4 : M/h : {8670/52 ..................... : 272. " Ul-L2:VSOOO llSOVXlOZV 0000000000000” 1470 " L2’U3 = V 360 0 8 -6904 X 1.27 00.00.00.000. : -880 " U3-L4 : V 860 Q 2301 X 1.27 00000000000000 - 29.40 " Hangers : by method of joints .............. 3 54.6. " Verticals =‘by method of joints ............ : -1l.6. " Reaction ; see section III, Loads .......... = 185.0. " LIVE-LOAD STRESSES: The maximum stresses due to the assumed live loading of the COOper E-72 type are calculated by use of influence lines for stress (see Figure 7). The position of the load which would produce the maximum stress is first determined, then the stress equals the summation of the wheel load times the ordinate of the influence line under each load. The method of computation will be carried out for one member, after which only a summary will be presented for each member. Member LO-Ul: with train moving toward the left-- ;p1ace wheel #5 @ right of L1: 54 is less than 741 25 175 ;p1ace wheel #5 @ left of L1: 99 is less than 195 25 175 ;p1ace wheel #4 @ right of L1: ‘99 is less than 125. 25 175 ;place wheel #4 @ left of L1: 126 is greater than 687 25 H 01 }\ 0’ I 4 ‘”‘ LJ‘L I" g ‘/'.// 07-03 '7 / -/./ ,Q V- Ll +/.~'1(’)‘ 3.5." («l/v ; Z -O-/é 4C 24. n0... I -9 '\ 1’ .— 3-44 U -0.47 + /. :5" [,‘l-,_' / ~ Read/o." a] M. rv’ 7 r7 . .5113 2 C - I J ‘ J r . - . I — 51.5.2: :2 1n /‘ A ' I r\ a ' f K‘- ‘ ' l ‘1‘ .. ‘— E f I" i -‘ - ...-.. .4 20. This indicates that the maximum stress in member LO-Ul is when wheel #4 is at Ll with the train moving toward the left. The total stress is -s -_- 1.11[(18)(7) +(56)(55)] 4 1.11 (84)(~52)(5.e)] 4 I75 175 [ 1.;gfi25.4)(1000) 4 (56)(851) 4 (18)(157)]= -497 Maps 1 Maximum stress for other members are as follows: MEMBER TRAIN HEADED WHEEL @ JOINT STRESS LO-Ul Left #4 @ Ll -497.0 kips Ul-U5 " #7 8 L2 -506.0 " U5-U5 " #15 @ L4 ~665.0 " L0-L2 " #4 @ Ll 505.5 " L2-L4 " #11 @ L5 650.0 " Ul-L2 " #5 @ L2 569.0 " L2-U5 " #5 @ L5 -270.4 " Right #5 3 L2 48.9 " U5-L4 Left #5 0 L4 178.0 " Right #2 @ L5 -102.0 " Hangers Left #4 @ L1 156.0 " Verticals -- ------ none Reaction Left #2 @ L0 470 " Reversal of stresses occurs in the diagonals but in member Ul-L2, the compressive stress is so small compared to dead load stress that it is neglected. The stress of member L2-U5 in tension is computed with wheel #1 omitted. IMPACT STRESSES: Using the formulas for impact as stated in section III, Loads, the following values for stress due to impact are Obtained: 21. Member Stress 110.111 = 26.75% Of -4970000000900000000 I “15500 kips Ul'U3 : H N ‘50600 0000000000000 : -15500 " U5-U3 8 " " -66500 0000000000000 : ”17800 " LO-L2 : " " 505.5 ............. = 81.7 " LZ’L4 : " W 63000 000.000.0000. = 169.0 " Ul-LZ = fl " 369.0 eoeoeeeoeoooo : 99.0 " L2-U3 = " N -27000 000.000.0000. : " 7200 n a " " 48.0 ............. = 13.0 " U3'L4 = " " 178.0 0 e e o o o e o e e o o = 748.0 " = " " -102.o ............. = - 27.0 " Hangers : 94.6% of 156 ............. = 129.0 " Reaction = 26.75% " 47o ............. = 126.0" " HAXIMUM DIRECT STRESSES: The total maximum stress in any member is the sum of the stresses due to dead load, live load, and impact as cal- culated above. The design stress of a member is equal to the total maximum stress for that member except in the case of reversal of stress. In that case, both the maximum ten- sile and the maximum compressive stress are increased by 50 per cent of the smaller, and the member proportioned so that it will resist either increased resultant stress. The values obtained by the author compass very favorable with those in- dicated on the design sheets. See table #1 for a summary of stresses. CHECK OF MEMBERS: Member L0-Ul is to be checked for a design stress of 855 kips compression. The section is shown on figure 8. To locate the "x-x" axis, the summation of moments about the base or "a—a" axis is made as follows: p1 .— I .... A <70 4 22. SECTION AREA ARM MOMENT 3 Cover Plate 26 x 2 15.00 sq.in. x 22.75 295.75 in. a = 2 angles 4 x 4 x 5/8 = 5.72 " " x 21.56 a 122.18 " 2 webs 22 x 5/4 = 55.00 " " x 11.25 : 571.25 " 2 angles 6 x x 5/4 : 15.88 " " x 2.08 3 28.87 " 818705 in? Distance from "a-a" axis to the "x-x" axis equals 818/65.6, Total A 65.60 sq.in. Total M or 12.45 inches. Moment of inertia about the "x-x" axis: Cover Plate : (15)(10. 25) ................ = 1567. inf‘ 4 x 4 angles 8 (2)(4. 4) 5 (2)(2. 86)(8. 86):. 3 460. " Web : (2)(665. .5)?(2)(16. .5)(l. 25):.. = 1582. " 6 x 4 angles : (2)(24.5)§ 2 (6 94)(10. 42). : 1557. " TOtal 1.: 47660 ino‘ Radius of gyration r, about the "x-x" axis is the square root of the total moment of inertia divided by the total area which is equal to 8.55 inches. Moment of inertia about the "y-y" axis: ‘ Cover olate - (l/l2)(% )(26)............... = 752.5 in. 4 x 4 angles : (2)(4. 4)}(2)(2. 86)(9. 95).... = 578.8 " Web : (16.5)(142)coo-0000.00.00.00 : 254300 N 6 x 4 angles . (2)(8. 7)+(2)(6. 94)(9. seyi... : 1575.4 " Total I 5027.5 1n.‘ Radius of gyration, r u 8.76 in. Ratio L/r : 487/8.55 s 57. Allowable unit stress : 15,OOO-O.25(57f1- 14.2 kips/sq.in. Rqud. area : 855/14.2 ... ...... ....... = 58.8 sq.in. Furnished gross area .................. 3 65.6 sq.in. o.k. Lacing consists of 2 5/4 x %-in. bars double laced at 45 degree angles. This agrees with specifications which state that where distance across member between rivet lines ex- ceeds 15 inches and a bar with single rivet connection is used, the lacing must be doubled and riveted at intersections. The width of bar exceeds the minimum which is 5 times the diameter of the rivet, or, in this case, 2.625 inches. The (\3 DJ 0 thickness of the bar exceeds the minimum which is one-six- tieth of its length between connections, or in this case, 0.486 inches. Using the radius of gyration about the "y-y" axis, the shearing stress normal to the member is 17,200 lbs. 7 = P 100 4 Lgr'] ldfi'fZF‘I"16 100 Actual 3 : §%?§172002§1.4142 A 2.75 0.5 2 Allowable g = 15000-; 0.7 x 29.2] A 4 0.29 x 0.5 8,840 psi. o.k. 10,050 psi. The least ratiOLL/r of flange between the lacing bar connections is 18.4 which is less than the maximum of 40 and less than the minimum of two-thirds of the L/r of the member, or 58, in this case. Tie Plates: Minimum length of tie plates : 20.6 x 1.25 - 25.8 in. Actual length of tie plates ............... 27.0 in. o.k. Minimum thickness 3 20.6 x 1/50 ........... 0.41 in. Actual thickness .... ..................... 0.44 in. o.k. Number of rivets required at end connections is con- trolled by single shear and equals 855/8.1 or 105 rivets. Actual number of rivets at joint L0 is 106 and at joint U1 is 122. Therefore, end connection is satisfactory. Note: As all of the compression members are similar in design to that of LO-Ul, the checking of the remaining compression members will be summarized here, without repeat- ing the details illustrated with.member LO-Ul. Member U1-U5: (See figure 8) Actual gross area ......COICOOOOIOOOIOOIO- 60.10 804.0111. Moment of inertia about "a—a" axis ......_ 756.80 cu.in. Distance from "a-a" axis to "x-x" axis.. 12.6 in. Moment of inertia about "x-x" axis...... _ 4556 in. Radius of gyration about "x-x" axis .... _ 8.68 in. a Moment of inertia about "y-y" axis ..... 4609.5 in. Radius of gyration about "y-y" axis .... : 8.76 in. Ratio L/r:SOO/8.68 ......OOOCOOOOCOOOO = 35 Allowable unit stress = 15000-1225/4 ... 14.7 kips/so.in. Required area - 859/14.7 ............... : 58.5 sq.in. Furnished area of 60.10 sq.in. is sufficient. The lacing is the same as in member LO-Ul except that the normal shearing force equals 22,500 lbs. and the actual P/A equals 11,500 psi. With an allowable of 10,050 psi., as in LO-Ul, this appears to be a slight overstress but this is satisfactory as the cover plate has not been con- sidered. The tie plates are the same as in member LO-Ul and are therefore satisfactory. The number of rivets in the and connection at joint Ul is controlled by the rivet value in single shear. The number of rivets required is 859/8.1 or 106 rivets which is less than the 152 rivets furnished. Member U5-U5: (See figure 8.) As the location of the "x-x" has been calculated twice and found to compare favor- ably with that used on the detail sheets, the location of this axis will be assumed hereafter to be that as given by the plans. 25. Actual gross area ........................ - 76.60 sq.in. 4 Moment of inertia about "x-x" axis ....... 5222.6 in. Radius of gyration about "x-x" axis ...... z 8.5 in. Moment of inertia about "y-y" axis ....... : 6001.5 in. Radius of gyration about "y-y" axis....... 8.85 in. Ratio L/r : 590/805 00000000000000.0000... : 36 Allowable unit stress : 15000-15/4 ....... 14.7 kips/sq.in. Required area : l152/l4.7 ................ 3 77.0 sq.in. Furnished area of 76.6 sq.in., while slightly less than that required, is not considered dangerous. The lacing of this member consists of 5 x 5/8-in. bars single laced. With a normal shear equal to 29,700 lbs., the actual P/A equals 5,500 psi. The allowable P/A as cal- culated for axial compression is 10,680 psi. This indicates that the lacing is greatly understressed . The reason why such a long bar was indicated for use here is not apparent. moreover, the specification requiring double lacing for mem- bers this wide has not been satisfied. The ratio L/r for the flange is found to be less than 56 and is therefore satisfactory. Tie plates are the same size as those prev— iously examined. Diagonal L2-U5: Least radius of gyration, r .............. = 6.29 in. Ratio L/r a 487/6.29 : 77.5 ..............o.k. (140 max.) Allowable unit stress .................... : 15,500 psi. R equired area 450/15.5 .................. = 51.8 sq.in. II Furnished gross area is o.k. ............. : 55.96 26. The lacing and tie plates are checked as above and are considered satisfactory. The number of rivets required is 450/8.l or 55 rivets at each end. Plans indicate 60 rivets use at L2 and U5 which is satisfactgry. Member L0-L2 is designed for tension and the net area will be the effective area. 28.6 sq.in. Required net area - 5l5/18 .................. Furnished gross area .............. ........ 59.0 sq.in. Furnished net area (deduct for 2 rivets in each angle and 2 rivets in each plate) 0000000000000000000000000 3 5300 sq.in. o.k. Req'd no. rivets at end : 515/8.1 ........... a 64 rivets Furnished number of rivets = (2)(52) ........ 64 rivets 0.k. Member L2-L4: The required net area is 1072/18 or 59.6 sq.in. The member as designed supplies a gross area of 75.84 sq.in. In calculating the supplied net section, it appears that two rivet holes have been deducted from each angle and four rivet holes from each plate, leaving a net area of 60.54 sq.in. This is very conservative as the rivets through the plates are staggered. The stitch rivets have a pitch of five inches which is less than the maximum specified of 24 times the thickness of the thinnest outside plate, or 16.5 inches in this case. 27. Member Ul-L2: R8Q'd0 net area : 627/18 00000-0-000000000000: 5409 sq.in. Furnished gross area ...... . ...............= 45.94 " " Furnisted net area (deducting 5 holes from each angle and 2 holes from web).. 54.94 sq.in. o.k. Req'd no. rivets at end = 627/8.1 .......... : 78 rivets Furnished rivets at joint U1 ............... 3 84 " o.k. Furnished rivets at joint L2 ...... ........ ; 92 " o.k. Member U5-L4 is designed for tension as the tensile stress is more than twice the compressive stress. The re- quired net area is 506/18 or 17 sq.in. The furnished gross area is 25.4 sq.in. Deducting 4 rivets from each web, the furnished net area is 19.2 sq.in. which is satisfactory. The number of rivets required at the end connection is 456/8.1 or 57 rivets. Each end has 68 rivets which is more than enough. Hangers: Required net area : 501/18 ................. = 16.7 sq.in. Furnished gross area ....................... = 20.00 " " Furnished net area (deduct 4 rivet holes) .. = 17.25 " " o.k. Req'd n01 rivets at end : 501/8.1 .......... = 58 rivets Furnished no. rivets ....................... : 40 " o.k. The vertical members are not designed for their small stress but rather as a minimum section which will keep the L/r ratio less than 140. The least radius of gyration of a 14CB78 member is 5.0. Hence the L/r ratio is 584/5.0 or 128 which is satisfactory. 28. The and reaction per truss will require a bearing area on masontry of 781/O.6 or 1500 sq.in. Table #1 shows the above calculation in tabular form for comparison purposes. SPLICES: The top chord splice at joint U5 is a compression splice. The plans indicate the ends of the members are to be finished for bearing, therefore, the splice material need only be designed to transmit one-half of the stress, or 450 kips. The main splice plate alone has an area of 51.9 sq. in. and would be stressed less than 8.5 kips per sq.in. so there is ample material provided. Required number of rivets in single shear is 450/8.1 or 55. Plans show about 87 rivets which make the splice very safe. The bottom chord splice near joint L4 is a tension splice and must develop the full net strength of the member. This splice is made up of two parts so each part will act spearately. Part A: Net area of member : 18.44 4 21.90 ... 3 40.54 sq.in. Net area of splice plates ............ a 41.4 " " o.k. Required load : 40.54 x 18 ........... = 750 kips Load transmitted by rivets ........... : 756 kips o.k. 29. Part B: Net area of member ................. 20.00 sq.in. Net area of splice plates .......... 57.50 sq.in. o.k. Required load = (20.0)(18) ......... 3 560 kips Load transmitted by rivets ......... = 719 kips Section between parts A&B will transmit 766 kips and is therefore satisfactory. This type splice appears to be e- ffiective in keeping size and number of splicing plates at a minimum and also in producing a rigid connection. The splice near joint L2 is also a tension spltce. The net area of splicing plates is 56.0 sq.in. which is suffi- cient. The required loading will be 55 x 18 or 594 kips. Number of rivets furnished is 88 which will transmit a load of 715 kips in single shear. Therefore, Splice is satisfactory. GUSSET PLATES: The size of gusset plates is generally governed by the space required for the placing of rivets. As most end con- nections are in single shear, the minimum thickness for a gusset plate would be that thickness which would develop a bearing value equal to the shearing value of a rivet. In this case, the value of a 7/8-in. rivet in single shear is 8,100 lbs. and the minimum thickness would be 5/8-inches. However, specifications state that g-in. is the minimum thick- ness and the design sheet designates 5/8-in. Upon checking the gusset plate where U1-L2 is connected, it is found that more than twice the required area is furnished. As this is the most highly stressed plate, the other plates are assumed to be safe. 3w. - , otms Us. .. 2 . mm; s : xi ..r, s u i...\ t i z 1 l es... 6, ..., i i i its _ o .4 . . a...) r: E a . 1 . . 1 _ . 1 n .. ....-.wwl .i i . a d -... is... .- i ...-..ER . m . _ _ . v .. . . i . . . 1. x... r\ QC \ I. i\ W.\ “sax i. .i . -..? \,.,. i 1 x i - \ .. .1. t in . t :34?va l -_ . _ x. . . \s. r x l. \J\ | i .) Kuhn. -! .\ \ \\ . CJI I . 4 11 i... .. r... i. \ 44.. «XI . \u Ki u . . u . .. . . .4... N: t xv... so. was... new- as; v.1... . - a _ .,\ .\ -.- x: it i. i i .v .i . . -.. a N...» I .m x. .1 - i i .. i . ... i- «.4. We 1...... L t. x _ k ii .. l. t. .2.va 1 «4.7.. v... i i m... - .i l 334..“ 4 5.1.2 _.\m Jeremy»..- ..- Wm fix . ii _ i M k .....ch x31 ...... 1.... My...“ Tr... 2». . ..\ s. . \ midis. .. z . .7 .. ”a 1.. - t. .a . . . _ . . s. e , ssh...” .. has 3 i a i .. so... . «514%-... m , ...- .-sw. 1.. o... unexmm 1 004mm; ..QWN m3 . i i- KMM 13W 4. in.,?” New... sf. 7 1 i . + m J. A i. .-.. w \k 0N? P w...\ - “1.. . mime @er .. NMxNi mini}. ...:N\1 l kw.» m...» .41. ...:‘Qflib I. J\ 1% )Qifl k Vex. my. 3%..W f 0.0%. _. 4.41%.. ashes..." but..- .QUNMI \lxNi WV» 3% [I I’ll i I 1.1““... v...» $.11 25 i ... M. l i i. T ....L .I. ) . ... 1.x; . l )\l .I \J . . \I i .. . . WWW. 0 WW.“ NV» l kit . Who .U\ Names. 1 we «u. w MQiM smxi tmwi DUN. 5.0-94 . . 1 . 4 A .. _ ..9 1 _ 906.1191 9 C. W . . . .. M . l n w / VWMMQWHVdH/xfll .1.“ ..lndwmnvlifl Ill 1w]. 3 .aflsmieshawj. “a. :3, mos/.11 2% e7 ..7 w 33333735.].1 I . w , 1.19va :4... J. 3. .9 VJYWVSVW V I Calm a 1 mks/LPJ/ "r8119 C 7. 7; 7w 0 .19 a W W ”Fifi." C. was/1 Q C. a 30. 9;. ppm ING The bracing to be checked here consists of the upper and lower lateral, the sway, and the portal bracing. UPPER LATERAL BRACING: The top lateral bracing is composed of struts at each panel point of 4 - 5 x 5% x 5/8" angles, and two diagonals in each panel of 4 - 5% x 5% x 5/8" angles. It is assumed that the diagonals take tension only. Specifications state that lateral bracing of compression chords shall be proportioned for a transverse shear in any panel equal to 2% per cent of the total axial stress in both.members in that panel, in add- ition to the shear from the specified lateral forces. Diagonals of Panel Ul-U5 a 1.7 x 2 x 859 x 0.025 3 475 kips Diagonals of Panel U5-U5 3 1.7 x 2 x 1152 x 0.025 = 496.5 kips As these loads agree with those indicated on the design sheet, the above values will be used for use here. However, it appears that the stress due to the specified lateral forces has been omitted. The diagonals are all alike, therefore only the diagonal with the highest stress will be checked. If it is safe, all diagonals will be safe. The diagonal of panel U5-U5 will require an area of 96/18 or 5.55 sq.in. The designed mem- bers are composed of 4 - 52 x 5% x 5/8" angles with one rivet hole per angle. These give a net area of 4(2.48-0.575) . 8.4 sq.in. which is satisfactory. The number of rivets 51. required by stress in the end connection is 96/8.1 3 12 rivets in single shear. Number of rivets required to de- velOpe full strength of member is 18 x 8.4/8.1 3 19 rivets. However, as large excess of area has been provided, due to minimum size of angles, the 16 rivets provided are recog- nized as adequate. The struts are built up of 4 - 5 x 55 x 5/8". angles and the highest compressive stress is 56.6 kips. The un- supported length is in the horizontal direction and equals 218-27 or 191 inches. The least permissable radius of gyration is 191/120 or 1.59 inches. The angles are placed with short legs back to back and have a horizontal radius of gyration of 2.27 inches. This permits a working stress of 15,250 psi. The required area is then 56.6/15.25 or 4.5 sq.iin. The angles as designed supply 12.2 sq.in. Number of rivets required in the end connection is 56.6/8.1 or seven rivets. The design uses twelve rivets and is alright. BOTTOM LATERAL BRACING: The maximum stress is produced in the bottom lateral bracing with a wind force of 50 lbs. per square foot on the area formed by the vertical projection of the trusses. It is assumed that one-half of the total force is carried by this bracing, and is an uniform moving load concentrated at the panel points on the windward side. The stresses in the floor beams are so small, compared to their previously designed section, that they are neglected. It is also assumed 52. that the diagonals act at the same time with each diagonal taking one-half of the shear in a given panel. (See Fig.9). Load per interior panel point ............. 58 kips Load per end panel point .................. : 27.5 " 1. Shear in panel x 2 x sec 0 or 1.7 Stress in diagonal O-l: 122.5 - 104 kips 1.2: 9805 : 8506 " 2-3: '70 = 5905 M 5-4: 46.2 : 59.4 " Unsupported length - diagonal O-l approx. 9.0 ft. Minimum radius of gyration . 108/120 .... a .90 in. 0.99 Least radius of gyration of angles ...... Allowable compressive stress ............ l2. kips/sq.in. Required area 104/12 .................... 8.66 sq.in. Furnished area 2 - 5 x 5% x 5/8 angles .. = 9.84 " " Req'd. no. of rivets = IO4/8.1 .......... 15 rivets Number of rivets furnished .............. 3 l4 rivets o.k. Diagonals in panel O-l apiear to be satisfactory. The diagonals of the other panels are checked in a similar manner using their respective stresses. SWAY BRACING: The sway bracing at each panel point consists of diagon- als of 5% x 5% x 5/8-in. angles and a bottom horizontal member of 2 - 5 x 5% x 5/8-in. angles. These members are not usually designed for definite stresses but the general form and dimensions are governed by the minimum clearance required. PORTAL BRACING: As in the sway bracing the general form and dimensions are-governed by the required clearances. In this case the mem- bers are built-up of 4 - 5% x 5% x 5/8" angles and a 5/8" 55. plate. Due to limited time, an analysis 6f the portal effect is not presented here. This presents an interesting problem in indeterminate structures and for a thorough treatment of the subject, the reader is referred to the text, "Modern Framed Structures", Part III, by Johnson, Bryan, and Turneaure. 54. g, END BEARING The and bearings consist of a fixed shoe and an ex- pansion shoe. They have adjustable top plates to keep the vertical deflection of the bridge from creating excessive moments in the bearing plate. Both shoes are approximately of the same dimensions and have the same load, so one check applies to both shoes. Maximum reaction ........................... n 781 kips Base area required: 781/600 ................ = 1500 sq.in. Base area furnished 28 x 48 ............... = 1544 " " o.k. Allowable unit stress ...................... 3 18 kips/sq.in. Minimum cross section 4 x 18 .............. 3 72 sq.in. Unit stress 781/72 ......................... - 10.9 kips/sq.in Req'd. bearing area at exs. shoe 781/75 ... 10.4 sq.in. Length of insert ... ........................3 50 in. Width of insert in bearing 10.4/50 ......... 3 0.547 in. The plans show an intereaing feature which is not in- tended to be of the typical bridge plans. It is a specially designed hold down which is intended to restrain flood waters from lifting the span off its footings. 55. 3g. 001101051011 The foregoing check of the design of a typical bridge span indicates the design to be more than adequate through- out. The only inadequacy noted was in the detailing of the stiffeners at the jacking point on the end floor beam. In the course of completing this analysis, many refer- ences were made to A.R.E.A. Specifications of different years. The author was impressed with the amount of variations between these specifications. One difference particularly noteworthy, was the many formulas proposed for the calculation of stress due to impact. The use of formulas given in the 1948 Specification results in a much higher impact stress than do those used herein. The author regrets that the limited time allocated for the purpose of writing a thesis prevented a more thorough and detailed analysis. It is believed that much can'be learn- ed by the undergraduate civil engineering student in the making of an analysis such as this one. The fundamentals of engineering become more than isolated facts when it is seen how they are applied in actual practice. BIBLIOGRAPHY A.I.S.C., "Steel Construction," 5th ed., American Institute of Steel Construction, New York, 1947. American Railway Engineering Association, "Specifications for Steel Railway Bridges," A.R.E.A., Chicago, 1948. Burr & Falk, "The Design and Construction of Metalic Bridges," John Wiley & Sons, New Ybrk, 1905. Fuller & Kereckes, "Analysis and Design of Steel Struc- tures," D. Van Nostrand Company, Inc., New Ybrk, 1956. Grinter, Linton 3., "Design of Modern Steel Structures," The MacMillan Company, New York, 1941. Johnson, J.B., "The Theory and Practice of Modern Framed Structures," Parts I & III, 10th ed., John Wiley & Sons, Inc., New York, 1926. Matthews & Soneson, "Analysis of Framed Structures," McGraw- Hill Book Company, Inc., 1955. Merriman, Thaddeus, "American Civil Engineers' Handbook," 5th.ed., John Wiley & Sons, Inc., New York, 1944. Sutherland & Bowman, "Structural Theory," 5rd ed., John Wiley & Sons., Inc., New Ybrk, 1948. / MiCHiGAN STATE UNW . i i it i‘ Ii "ii i i iiii ii .. 3 EPSKTV’ LiBRARiES . ,1 ‘7 ‘3‘,“ . iii‘iuliii‘i‘lil i iwi i. 82 6998 1293 030