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I 3"ir ‘3_ LV- Bernard C.‘Steinbacher Candidate for the Degree of Bachelor or Science June 1942 THESIS C I /_ INTRODUCTION Since the removal of the old footbridge across the Red Cedar at the dam on the Michigan State College Campus, there has been no means of getting across the river on foot between the Farm Lane Bridge and the Railroad bridge. The students occupying the new N.Y.A. barracks in back of the Forestry Cabin will have no means of getting to the campus unless by the Farm Lane Bridge or the Railroad bridge. To solve this problem, I suggest a foot- bridge to be placed across the Red Cedar at a point near the Forestry Cabin and the N.Y.A. barracks. This is approximately 500 feet East of the dam. I suggest that a rigid frame bridge be used, both because of its beauty and its economy. The design I suggest is a modern and fairly simple one with a single span of 80 feet and a width of 10 feet. This gives a slenderness ratio of 8 which is large for this type bridge but on the judgement of Mr. Weber, Chief drafts- man of the Michigan State Highway Bridge Division, 143099 -page two- it is quite safe. He pointed out that the Chicago Parks Commission have built footbridges of similar width and span over the Chicago Outer Drive. Greater economy could undoubtedly have been obtained with two or three spans but the beauty of one span, in my Opinion, over- balances the greater economy of a multi-span bridge. I would line here to thank Mr. C.L. Allen and Mr. C.A. Miller for there suggestions and help in the design of this structure. -page three- THEORY OF DESIGN As mentioned in the introduction, an 80 foot single span was chosen for beauty, rather than a more economical multi-span structure The structure was designed as a rigid frame from footing to footing with "free end" conditions at each footing. The theory of the "free end" conditions is that due to the compressability of the soil, the footing will rotate about the center of its bottom when the structure bends. Since the flood stage of the Red Cedar is quite high on the south bank and overflows the north bank, it was necessary to place a fill on the north side such that the walk across the structure had no grade. Due to the soil conditions the footings for both the north and south abut- ments were put at the same level. Thus the structure is symmetrical in every way. The method of design used was that of Hayden's, described in his book, ”The Rigid Frame Bridge," in the chapter on symmetrical structures with ”free end" conditions. This method of design consists of assuming a structure -page four- of certain dimensions and then checking to see if it is economical, uneconomical, or unsafe. After the first time over, you can choose dimensions that will be more economical and safe from the knowledge you gained from the original check. The dimensions I chose for my original design were based on the emperical formulae found in "Analysis of Rigid Frame Concrete Bridges", by the Portland Cement Association. (pg. 7). These first dimensions proved too small after the forst time over and were sub- stantially changed for the second try. I selected a parabolic curve for the intrados of the structure with a two foot rise in forty feet. The geometric axis of the profile was then determined and this axis divided into equal lengths. The length I chose was six feet with one of the points directly in the center of the span. It is then the idea to check the stresses at each of these points due to all the contributing factors which is done in the tables that follow. -page five- Mr. Hayden, in his design, depended on the concrete to take all the compressive stresses, while I found that by using steel to take part of these stresses, I was able to cut the total area of steel at some of the points almost in half. This work may be found directly after the tables. The earth pressure was determined according to Rankin's formula and the wingwalls were desig- ned accordingly. These wingwalls are to retain the fill for the sidewalk on the approaches to the bridge. Allowances were made for this fill to be ten feet wide at the top with 2 to 1 side lepes. The wingwalls cut these side slopes at a 45 degree angle making a fall of 1 foot in every 2'/.707 or 1 to 2.85. The walls were designed for their maximan height and these dimensions and this steel carried on through the lower part. ’ Due to the fact that the wingwalls were at a 45 degree angle with the face of the abutement, a difficulty arose in that a part of the heel of the wingwall footing must coincide with a part of the toe of the abutment footing. This could -page six- not be since the structure was originally designed assuming "free end" conditions and interposing these two footings would cause the abutment footing to be restrained. Thus, it was deemed best to place a con- struction joint between these two footings and use a counterfort to cupport the part of the wingwall that was without footing support in the heel. The details of this counterfort may be found on the sheet with the wall details. “Page seven- :TQGVQQ -11 I--- 1.- A-.- .iu k M~. nu \<.V\.V.Q —page eight- .QQQQ _ QQQ -..QQWN. H NHNQ Q.QQN.- Q.Q.N.- _ Q. NWN QQQ .QVQQN VII- .Ii. Ilil..| -. .91..¢.I QNQ . 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Q- 4-N4Q 4.44.4.4. mix-.4 ...4-4....- 4414-4» .-4- ..4 4444-4- .444. ..4-_.4,-.. .- -.4 4 4.4-4 4 4.4.4.4 Q--...4--N.“.444-41444-444-4. 1.44.4444 EQQ Q .4 Q. 4.4.4Q4 4.4m .4 ,Q- 44 --Q.- 4 4.44.4 4.4m.- Q4Q4 Q-Q 4.444. 4.4-4.4 4.4-4 Q:4 N 444 _\44 Q 4 Q 4»Q4, QQ4 Q .Q. QM Q§N.34 .N.-...! H...- ...--..-.. 4..., .- .m-..-.....-...----.H_T-.-.-. .-..-..-.-.H-..--n..- ...... .-.-..l-.--i-.----...I-H4 -4-.4QM4-4 -44.... 4.444.- 4......444- 4.44.4.4 4-44.4-.. 4.4.4.4- 4 4.444. 4.4.44-4 444-4- 4.4.4.4.»...3 4 4 4. 4 4 --.-QW -.4Q.-\-Q-o‘.4.-Q-Q-4.4QQ\.\- 44\4 ~443- 4Q \QWQ...» Qu-w-QQ\.\44\4~\%-4c how-Q .QQ.QQQ.\\4«\. 41:314- - 44-R0-4 «44.44448Q Q Q QQQR -page twelve- EXPLANATION AND SAMPLE COMPUTATIONS FOR TABLE 2 This is a table of the moments and thrusts developed at all points due to placing an influence load of 14/6 at each point individually. 14/6 was chosen as the influence lead as there are 14 spaces, each 6' in length across the tOp of the span; thus the moments are simplified. If! Taken from.Tab1e l y - Taken from.Table 1 Mom. 8 14/6 x 6/14 = l (for point 4)(inf. load at 4) H = (EMy/I)/(EY2/I) . 552.7/455.6 :- .774 (inf. load at pt. 4) Hy ‘ .774 x 2.8 = 2.2 (for pt. l)(inf. load at pt. 4) Note:- All Hy moments are negative unless otherwise noted. Total M ' Mom. + Hy = O + 2.2 = 2.2 ~page thirteen- 72h8££’g3 sunmmy or 7.4.97: .2 7-: x 7.9.7 Mam. ram} 9%? {Id 6R y: 7R 8/? T 972 L 70—7 [1. .9 Lg ‘ 33:3; __37L 9.0T 9.2} 2L029'5,7TT306/0.Z //6§125;/3.7‘ r- _T—“T___i ~ —-#—-—.~«- ~~—+— —~4~—3-—— a 5 3H 9-7 93' 1311/1: _L?~__3i_:_é/x l !_7_/L__3 L7_7§_9;g:83‘727L/9a 778 5790TL 9.7 337% 4.3 :____82 773:19531/53 L Z/ i767 §.___ .-.-.. .--- --..__ ___ .. i61‘39 17.5 No.9 73.7 §79.3 {797 €790; .... . -._ ---. _, 2--.“---1}- .. . - _N-.- ___..- I ' 2 7 i .3 f , L77.43376719547/5 412.3172 .777 '51. 3755.9 30:57.9 i70.3 9.9 3 8.3 ~‘-— —--—--:—— -——~—r—--—~7 ~——--—*+-—~—- — 3 ------- 7—5 Ljf-T 2.7 .il . 628 i 8H3 2,693 i 7.5’ 3 70' 2.3 ; 9.3 ‘ 3.3 6.6 ‘6./ f 9.9 ..__ ;-. _. ___ --_ _. --_- “___.-- _ __ __ .-.._____ +-__ 9. 0_ a. 3 E ______-T_.___-...._.__J- “ -. - .1 .-_ -_.T.-__. .... l 7 :t 5 .6 l “*4 7RL .9jJL6 LII /J.jr54 E‘s? i-HJ j I T I l ___.-. ... -- _____________ .--:. - *1“ ___- - .- $- 1 l wall...“ 59 o +3.I g9: _L72.7 L733 §/6.7 i ' " T 273.2 :737 i78.2 379.7 _ +1 1 i392 4,7 9,; 3,3,5 T772 379.3 “27.2 Law g 27‘? 2,9 5,7 i 5.0 70.2 7/. 6 72.5 ' 72.9 'l /R ,7 7.3 2.5 L 3.7:__L_3_._:_L_ 9.0 9.2L} -page fourteen- EXPLANATION AND SAMPLE COMPUTATIONS FOR TABLE 5 This is a summary table of Table 2. The results were obtained by multiplying the total moments obtained in Table 2 by 6/14 to convert the influence load back to a load of unity. This table might be used to draw influence lines such as the following sheet which were obtained in the first design with slightly diff- erent arch thicknesses and y distances. It is not necessary to draw these lines, however, since all values needed may be taken directly from this table. Thus, due to limited time, I did not draw influence lines for this table. These lines are valuable in that they tell you pictorially just how the bending moment at each point varies as a live load moves over the structure. INF! 0be1' L //Vt'5 _ _ :page.--fifte_en:_- I I I I I I I I I I , I '/ -'v—— —— ——-—¢—- ——————-+-- I I I I I I I I I L O ~20+-+-~—+— - — - I —.—--"' ‘Q l I I I 4 I i I I -—~ -9 I I ’ '1 ! I I _. -_J QR. Q.» .5 Ox «S Q3» Q9 (ANS/EV 3% MGEW VK\<\ -page seventeen- 7/4 7915 4 DEAD 8 L/VE £07405 ;_ Po m 7 Y I/o/uTm 33.57954} Tie/73.} 7575“} T 20.7 3,02 T o g 3.02. I T2 {.3 I____3.aa 0 I 3.82. T _33. o = 9. 9.3T I 7TaTTTTET57T 2 7.8 I T» --_— _—+.---.-. “Ha”--. _1___...__, I 89.6 ' 32.3 L eg'r";“3.757 ,7; s 3.82. I __792TI 2.235 T72 T3155 __ I__/?:~?L-Ta-T7% I73. 3-4?“ (9 %___/_3--9_I_-é-7_9 _I_____._77-_ r3331 -page eighteen- EXP LANATION AND smarLE COMPUTATIONS FOR TABLE 4 This is a summary of the dead and live loads at each point. The live load was considered to be 120#/sq.' with no impact because of the use to which the bridge is to be put with no high speed heavy vehicles passing over it. Thus the live load was added to the dead and the total used as deas load. The volumes were computed by the average end area method. The end areas were measured from the daigram at the half-way points between he points over the Span. The load was computed at 150#/cu. ft. for reinforced concrete. The live load equaled 6' X 120 #/sq.' = 720 # or .72 kips at each point. I6 .666 6.66.6-6 66-6 I- 6.6-6.6.- , -6.-p 6-6..- -, w \ 6.6-3m; 6.6-666.666 6.66 6.6 -page nineteen- w -II--I..II------III--I-.II-.I--.----._-II-I---._.III III-.-.--.II- . :66 6.6 _6 .66_ 6.6 6 6:6 6.6 I._TI..., _ -.6- 6.6 ; 6. .6.-l _ 6.66I N..: .663. -666 0266 .7- . _ _ ..II. 1 -- II I- T III-II- 66-6-6- 616: \6.6w 6.16.266 _ _ LII-II III--.-.III-o ..I-III-L- - IIfi-II- I I 6..-.6T6-66 6.66 \.66_ 66:66 _ _ III-III; II III. I- III III . D n; O: '- 0'} \ 1 I 7-“T .m SoI N . LT3'06: IQHOI‘" V‘ ! -.- .'____ -_ I I 6.6-I66 :6- -_ _- ,__..___-__ 4 —— —— $- — I I I . . I ___+-.__+- \ I I I I I ; I I ' I I I I I I,_ - II III II .N.-III.III- 1.6 II 47. _ 6.66 6.6: 6.66 6.6: W 6666\6\h66\ 6.\M 6:66. 66\T616\ - I - -. - I- - - - .I-II I4 _ III-I III-II. - I- I- ” _ 6.66 6.6: 6.66 6 ..6.\ 6.66 6.6\ $6.6 66 . .. 6--.: .....o S I ......... II-- .....-.1 - :1 66-? -..-6.6L 6-6.6- 6:666 .---W.6-\..-I.-6-..6MIN-I6.- k . M I”- Ih. N . .66 .6.-.666 .6.-Q 6.66---.- 6-6- 3:- . 6.. I.- % NW.- uni-6. - \HWNQ .6 ..Q .u6I 6:. NM: WI N Q IT 6 .6 WI «:6 .I...-..I-I N.- I II I- I. ..IILI - . . .- I --. -..! ..I.- I I. .II...IIuIIIIIII h - ---I.II-I-.II. . “III III- ...-.I-II-II ...u-HI-UII I III-IIHIIIII. ...-I II I 3‘- 6.6-XI“. Imm- --I-6-6KI -..-6.6-- 6-...6<- -Ifim kkgfsk .22.} 6 65 6 _ 6.63.6 _ 6. 6566 a 6 6566 .6 6 666.312.;6661 6.626363 6666 I63: 6 66x66 6 .6 6% 6K -puge twenty- I; 6 6....6,6--_.66 .6 ‘ ..--m-.mm.. . 6-6 - ..Nm. I. \- 66I 6.6- :66. T. I.+YI-I -. N .WII-I IQ.»- 11! .6.-m ... IIJvu'I 'D.-. .. III; II I I IIII ILrI IILTI IllIl’lIII III .LYI I _ III II.-- III-+- -IIIIII KKK 6.6 ...- 6me 6.6 6.6 .6.? T .6.-6 rIIIIII I III-II II I- III III. I II IIII IRRI- 66 Ink-WW-.. I 6 .6 II II. 'I T -_ 6.6 . N .QN T ..6. .: III II. IIII III-.. II 114T . I'IIIIIIII1IILYI III 65% le I I III .6.-m. 6.6. ITIIIII III I I I \- 6 IIIISIIAW .. 9.. \...N.- \ .6. INN-66--. . .6- 6.. .6 -66 .6 \ I6 .Io, w .mI IIII IILTI ILIIAfiI-I-IIL..- ®.m. Ts I'lll. Ill] 6.6. 6.6. mekl 6.6 6 $qu m \ “ink 6 66-36 30.6686 6 66666-6 Vacu- -page twenty-one- EXPLANATION AN SAMPLE COMPUTATIONS FOR TABLE 5 This table gives the total maximum possible moments from both the dead and live loads.. MF = Moment Factor = Taken from.the influence Table 3 (or from influence lines when used) M = MF x load 3 2.9 x 4.9K = 14.2k (for point 2 with the load at point 4) . T {at . an. .. .. . -\ '.£_———» -page twenty-two- TABLE 6 TEMPERATURE STRESSES Pomf 7 +H)’ -h’)’ Pamf Y +h')’ -/'/}’ 2 8.6 I2./ 6./ 7 [5.6 Z/.8 10.9 3 I458 20.4- /0.£ 8 [.529 2.2.0 ”.0 4- 15.1 20.8 10.4 9 /6.0 22.0 Ma 5 15.5 21.4- [0.7 I0 /6.o 2.2.0 [1.0 6 I57 2A6 Ida To = Coeff. of thermal expansion 3 .0000065 t I temp. change = +55 or - 70 1 = Span length in ft. I 80 E = Modulus of elasticity of concrete. E a 144 x 2,000,000 = 288,000,000 length of axis divisions = 6' g/I) = 288,000,000x.0000065x35x80 E t1 ( ° )/(3 sum'y 6 x 1265 + esofi or - 1380# EARTH PRESSURE The earth pressure in this case is nearly balanced Y water pressure inside the bridge and there are no unequal moments set up since the earth fill on each end is of the same height. The small excess produces positive moments and are thus on the safe side and may be neglected. . r . -page twenty-thre e- 2fl 2 . -. .42 222 24421-22242 2222- 42-- 2.2 -2422 22 224-221-222 :22 24422 a . . . cN ewe 4 c efiNNmNNN 4 Q eN4-eNN-N 0 MN. HNN 2.--2-4422-- 442222- 24 e eN 44344 .2 . . 4 4 . . . .- -. 2.22 e. . ll «44f T814 0 _fmeeNN ..mN N 4 Q .43 4%NNN 0 mm, WNN 22...... -2-- -. #2. -224 oweN 4r: .2 - 2 4..- ----.2.. - .---2 -. N. 42224.... 4 mm . Nem: o Nam-44 N N, 4 m MN WWNN o QM. eiNN --2- . . 2..-.224 f2 2-4. N. 54.: T. 2-2-2 . .- 2-- 4.--- -4- .2. 2 2222222 .22. N. --4 2. «N . N..eNN4 e N.e.NN4 me .24 954-314 0 em 4 eNN --.-4 _ -, . 43-4 .qN _ . . _ . T- . 4 I .24 -4 4 .2- . 4- 4 .- :4 22 2 -. x 4222 222222 4 me ewe: o MeNe-N me-n .---4. N MN4MNNN O m: 4H“: 4.2- :22.- -4 44222-2. N5 4~Ne .-. -.-2 h ..2 4. 4 2 4. - I 2 T2 .22 2 N. 4.22- .212. 4+ 2 Nee-n _ m5: 0 4 QMNNN. me. e 4 2e .43 MNNN o NN 4%.NN 4.2- - 222L 4-2 ..... .4 Ewe-N454 QN N ewe. c 4 e ..eNN4 me E N.2eN 4 RNNN o w 4 2N,.NN +2-222-44-z2-4 T222224 N-eN 434 - -. - .. - 24 r... . -. -24 - .. - _ 2 . 2 4 2.4 4-2- -2.. M1. 2 4 4 . _ ._ . . . 4 . . 4 . T .. «m. NNNN e . NNN N433.h e N: LYNNN o e 4..e.NN4 . - .eeNNe - - . . 4 4 .22.- 4 . . 4 .2 -....24. 4 2 2- Q m, 4- 2-22. 4 4 _ 4 c o 4eNN 4- - . e. N. -.--.4 thN Ne. .22 .. . - 4 . .. 4 .4 -2 .N 4 - . 4 mc\ 4 w. 4 .244 Q Q..-- ..--24 24. QN NNxN. Q MNNNN.X\N 4 m.m4N-WNN Q 0 4N2: 4 4 4 T ...24m.me4tw4 . . . . 4 .4 .4%.N4-m.. -..-4QM2, .-.-4.4 . 4 4 _ 4 c 0 N54 4 miN fi- . me 4 .e4 . . . . 4 .H ._ - . 4 . 4 W N .V .N 4 M) M14: 22. «2 4. NNN NeN. e .mN.2eN_eN.m4 Neg-e4 ee24 c c 4ee4 2:: T - .4 ha 434 - -. _- 4 . . . 4 .4. 4 . 4 .- 4 Qw .N..m..4 2224 2a N 4222.22 24. . . 4 4 4 4 4 . .4e.m4- .2_ o __eNN .2 .2-Wx24 - 4. \A. .4 -. 2N- . .N 4 X: 4% .4 . .. 4 . - - 4 2 _ --4 2222 2 I.-.» 2 22.224.42-2-22222-42 2.2-“2442.2“-- . _ x- _ N _ ...: . s- . sm- 4 . “N . IKWXJAKI2222- X\ 22. 2.2-N22R24 22NI- -.24 .2--2 2242mWWWMWM-44i mu m .xNWNQ-n4 “\cekcuanWNflhx h\ {2W knew-wow .X “NW4 .XwN-M4 ANN 4 M HN.O-2Q..o-..N: \4 .N X.x- .NNK - Uxbhwuxnuw VXKEHN .42 Soy-N ..XOxx 4.N.Nooox2.. xx 2.x2.N.- «Cameo-NON .Nxxxaw ..NNNUL 4.x\0\4. M 4 .mcxxthfiwoNoN xkaxvfiw N MN-NQWN-N -page twenty-four- EXPLANATION AND SAMPLE COMPUTATIONS FOR TABLE 7 This table assembles the moments at each point caused by the earth pressure against the abutment. The earth pressure was computed acc- ording to Rankin's theory with C8 - .27. This pressure was broken up into 2 forces and placed as shown on the diagram. The table is self explanitory through the headings of the columns. — e V .1 .4- . . . - - I-II .-.4Ii-II-III III-II .IIIII- I4 W 4.44.. .m-VmIQ... “.444 4. m .\4\-_.4 44444 2%... 4 4.1.4.44? 444... 4N4M-.4444444I3 .5444” 44.44 4 4 4.3 4 69-4.:I 45-? 4.4. 44.4- 44\ we? . I .I-4I + III .I-.1I.III IIIII. II m 4.74.4? 4444me 4.7.4384 4444.344. 4.4-. 54.44.44 4.7 34.44 474444.: 4.4-4444... 4.4-4 «SEIREK w. I 4.- . . 4-I .I . . . w . Ii 4 - ----I4_Il -- th 544$; \.444\\I_\.4.Q\\I \.44m§\4 .4 4444:. 4\%I.N4 447 4.:N3I4x-I414I44I4I {...4-uh evII 4 II _ III44T- ..--IIII- L 44- _r -.--I . 4.I I- [4* III +.I II fiI:III-lI4_I-I IIII. 4.!Il I..9II.I-III|II44YI Mu NQW-x QQ\\.I .-.N..Qu..x. m SEQ _Nme JMQQFI ~1th _m-vaL NLQIV-x 4Q.\.IWM:IH NQVx- .\ .Q\VI4_ NQmI. \. I444.“ «YQQfimMNNI 4- NwIm.1Qm\$I_4 \nK- Q.--4W4 pI- L .4 _? 4 I . 1-7. I I 4 .I .I- a I II-MIIII- 4 .I- I I 4 II I. -lII .II-III 1 -IIIlfII II _ Q mum-mi. Q 4Q§%\* Q CfiQxfl O ..-._;.\.4\*. Q “mex Q _Q\44L4 Q 4.3%.“ b _\I\\-x_ 4.5 Qxfibq. I. I 4. .. . + .rl _. I II 4 + -. I . - I _r- - I+ .I-IIII I -I_-I. III-IL- III-III .III- .II I444... 4.44.4-4 4444 N 4.44-. , ~44. 4.45.4-4 4.44: 4:4- 4.444. .4444: 4.44.4 4.44.4-4 N44. .4 344 .4 4.4. 44.1-4 3? 4.442 _ 2..-4 4. 4. . 4 - ..I .4. a. w - . IIM . ..I ...k. III. I 4 .I4-II III-II II -I \2\IJ._ IIIIIIIIII..III4. -.\..\ I.._ \\4\ _ ...\ . II.)\ 4 >\ \\4\ . \< 4 \-\,\ >\ _I }\I 4. >-\ . Q1 4 >\ 4 \2\I >\ .._ E . .>\ . 4 . 4. III 4 -I. 4 - 4..- - -- - 4 . .I 4- I4 I .4 I. . 4--- .II-_II.-IIII. ___QcQuubu _N....Cmien: 4 Eton. 4445444 N 454% 4 .4 2:444 4 .4. 43444 4.I4 Esq .44..-- 4.43444 4 m ESQ. 4 4r . I 4 I - , . .I». I.» I. I .- I I .r I. - .III. .I III- III .I ..I-IIIIIIII IIIIII III LII I..IIIII-I III F bkhbtxxk 4.448344%? S434 ICE-“4&4: {\VECAVE KQ XQTtxbh Q QuQVR “page twenty-six- EXPLANATION AND SAMrLE CCMlUTATIOHS FOR TABLE‘S This table is a summary of the maximum moments and normal thrusts at each point. 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