Analysis of Cedar Street ECO RMe ORO R ICM ROAM ATO ex A. F. ZICKGRAF E. K. LOVELAGE 1913 FHESIS This thesis was contributed by dentin iiemguas E. K. Lovelace under the date indicated by the department stamp, to replace the original which was destroyed in the fire of Maroh 5, 1916. £. Lovelace. Mich. hansl7g: THESIS IN CIVIL ENGINZTIRING. -999000999- ANALYSIS OF . THE CHDAR STretT ReINFORCED CUNCRETE ARCH BRIDGE OVER .CaDAR KIiVas, LANSING, MICHIGAN, ~~090009090909-- INVSSTLGATID BY THH wLASTIC TTyRY. june re L d : o ct & Ae F. QZickeraf. Ee K. Lovelace. -1913- THESIS This thesis was chosen with the ijea of gaining more know- leige of the theory ani practice in the lesion of reinforced concrete arches. This bridge was built in 1909, by the Western Brijige Co. of Chicago, Ill., following the Jestruction of an 014 woolen briige at that place. The briize is 30 feet wide and consists of two equal spans of 70 feet, with the sprinsins line at the pier 2 feet rizher than that of the abutments at each end. It is on 2 skew of 30 degrees with the river which makes the prob- len of desiztn ani construction mere difficult. The arch is 16 inches thick at the crown ani hes 9 rise of 10 feet. It is back-filled with earth on wiich rests the base of the brick pavenent. Tie oriige seems well constricte? ani in 7003 corii- tion, no cracks of any magnitule beif®—% visible except in the railinz. The loagtlings and allowable stresses we usei were those given in the specifications ani for which the vriige was design 61. A live load of 200 # per sq. ft. of roaiway, 100 # per sa. ft. of walls, a conc@ntratei moving load of a 23 ton steam roll- er, ani a ieai loai of 90 #¥ per sq. ft. for earth backfiTl and 100 # per sq ft. for concrete. Tie stresses allowed are; Modulus of Hlas. of Concrete 1,800,000. lbs. Molulus of Hlas. of Steel 28,000,000. lbs. Maxe comp. per sg. in. on steel 19,000. lbs. Max. comp. per sq. in. On concrete 900. 1b6. 103995 Max. shear per sq. ine On concrete (plain) 100. lbs.» Max. tension per sa. in. on concrete OU. lbs. The theory followed in analyzing the Jesign is on plates No. 1 ani 2 anil is a copy of the Elastic Theory of Arches by Prof. C. Ae Melick. In our analysis most of the lata for the horizongal forces, jue to lateral earth oressure were calcul- atei but owines to a lack of time only the vertical forces were consijlered. The arching was diviiei into 20 ralial parts symmetrically about the center an’? the loa's spnlied at the points A,B,C,D,i, E,D,C,B, ani A, the location of which being Jeternined graphic- ally as shown on diagram. Tables No. 1 and 2 consist of values to be usel in the furtner calculation ani were constructed according to the pron- erties of the arch. Plate 1 shows the genersl data that was computed ani uselji in further work. The area of the arch ring where given is the equivalent area of concrete anj is equal to bh = 2pbh( n-1) = bh + Ag (n-1), Where b arfi h are respectively the bréaity and thickness of the ring at the section, A,the area of the steel, p the percentage of steel in the section, anin the ratio of the Modulus of tlas. of steel to that of concrete and taken as 15. Table 3 was made of the summations of the quanties in Table s 1 ani 2. Table 4 was calculate? from the formilae of the theory ana from wnich Table 3 was male from the summations which were use to construct this table. The laad used was a 1,000 # log? ot 4 on Plates 2 and 3 were plottei anit besides being a comparative check on the calculations up to trat noint sive graphical val- ues of the moment, thrust, and shear for every point on the arm ring. These values were scaled off ani make up part of Table 5. The other values shown in Table 5 were used to compute Table 6. From Table 5 the influence lines for thrust, shear, an? moment were plottei (see Blate 4). After the thrust was calculated it's affect upon the sec- tions of the arch was calculatel but was so small it could not be noticed ani was neglected. The calculations for a chance in 4H due to this thrust denenjis on the following theory: 4 Average axial thrust —_ 2. T ~ T. n M = Hy or ax = D HyfA s ° EI If 4=1,000 then a=4x= 1,000 > y7As . E o I Change in 4x jue to axial thrust=be TA-x 1 > TAx. AE §& “ A H due to axial thrust-_ b yy 19090 = Q a a 2 or Q=/] 1 ae: Ax . 1099 ? y A |: E a. ° E ¢ I Q=)T { 7 “* _ dz s_| ; Actual H = Hy — Qe In Table 6 from values shown in Table o , we have firure4 the stresses at the diffrent noints for the ton ani bottom of the arch ring, for a 1,000 # load at the loaded points. The upper and lower are tabulated respectively. o The unoper is T 4 —iec sn2 the lower T _ Me » where Ais A I A I the transformed area of that section, c one half the thickness of the arch rinz at that point, ani I the moment of inertia of the transformed section. Plates 0 and 6 show respectively the influence lines for the stresses in the top anid bottom fibres for a 1,000 # loai, (movinz)e The stresses for ieai and live load and for combined loads are given for top and bottom in Tables 6 ani 7 respectively and give the maximum stresses in the ring and where they occur. In testing the archfor shear the value of the shear in Table 5 were usei. The shear for ieai lowd was first comnuted ani then the shears for live load compute] so as to give the maximum positive or negative values of the same sign as the jer load shear.These values of shear are given in Table 3. This shows the max. shear t9 be 25 # per sq. in. The bond strength of the shear hars Jid not have to he tested as this strencth does not have to be figured wiere the shear is less than 40 # per sq. in. In Table 8 the Desd an? Live loaj loadings are also given. A slijer of the axle loais of the roller was used on the shear curves of Plate 4 to vet the position for max. live loal shear. On Table 9 the diagrams andi Bables of the stress in the steel are given. The reinforcement was placei at a uniform distance of 2.5 inches from the outside ani is shown by the dotted lines. The top ani bottom lines are the stress at the top ani bottom from :ables 6 and ? ani the the ring thickness at that point. Fron the specifications the vier is supported by piles , 12 inches in diameter at the larve ent and 3 in. in dia. at the Stereo Sm91]1 Oni ani placed 3 foot cc. to c. They were jriven to such depth that the penetration unjer the last blow of 4a hammer weizhing 2,000 # ani falling 30 feet shoulli not be great er than one inch. From the Engineering New'’s Formula, the load one pile wil! support or Pj 2WH where Weweigit 32,u0u #f Nzheicht of s—] fall of hammer 230 feet, ani s =penetration =1 inch. Then P is founj equal to 60,000 #4 In this pier’, considering that one pile takes care of 8 sq. ft. of pier-base, Which is equal to a section 1 foot wide times the wiith of the nier which is 8 ft., the dJead loai on one pile equals 2( Dead load on all points plus Pier weight) = 2( lese 1946 + 2768 ¢ 3965 + 7541 plus 150](8' x 5.5'x 1) + ( 5.5 x 5 2 x) | Or equals 35,5494 8,025 248,573 # . This shows the nier to be safe from sinking unjer the jeaqd loa. The one foot section cf tne arch analyzei was taken parall- el to the face of the arch. For Voussoir Arches the thrust, (lines of pressure), in an oblique arch are assumei to act in planes parallel to it's faces(( S. HZiward Warren on Stereotomy). A momolithec arch on a skew is considjere’ in this same way. The section 1 foot wide analyzea in this arch was taken unier that assumption ani tnerefore the affect of skew is not considered. No other references than that of Warren's book on otereotory on this subject coald be found as the theory of skew arches is one not taken into consiieration in most of the books on arches The bridge was fount to be safe. The stresses determine are within the allowable limits with a *actor of safty of about 2 for compression ani about 4 for shear. The only tension foun? in the arch was less than 3 pounds per sq in. NO account of forces due t9 the slight incline of the arch, 2 ft. in 70 ft., were considered as they were to9 small to affect the stresses by any amount. Theoretical Fundamental pn a WTS Cie de A fended by the fa Fe eT 1 { Bik hae | on . Vi) o} Pada vijay Be a Bares Mat Series tessa, sh Women? es 64 trng or. Arih - 2ymetrical=the fattowing | Conditions are Assumed - Fe) TK MME) ais Tame S72 HED, see rea ai cain ye) reek ate eae ea BIT Pl MeN i Taye Kinging Points ba See a 5s ae rom aM da Be On Ae sreperenreny fa es lade la PP, Te Pa Pb = et Ax= ae sa Pe ey since 2 ere Seen poate) oy aes pepe ie th ea an Te ae Tl 4 Be ea ha el et ae ue fs ar Zt ae ree oF Es cae 0 p i? T* PATA Loree Sa ee et Pa ae Pe Me a) gets TN LIT aa eed A tee ty Ae ESTE Se ne 4 7 =th.cos@ +V sin b+ Wacos Ox a = VWs. tnd cE even heise eS ye ns LR,ES. they ESS syereaqesy seeaeseuse oar Naseer a F523 ane) eat ee PE TL Ce aor sau) 50 3 Dia fh a ase, S5 sert seest fasst seersssesresaeaseeese : ; x | A diet LI ' [a oe > a Ca a LL Atha) IATA TS LE ASS ‘ é wing ee FOLP |SN2L00 \0fZ94L |6288000° Cree, CIA LAL Se AYA TE ED C/A TT MAYS " BOI'E +) 92h 216 Ss L60S48+| I TEA ad aE AD en ETE AT dl ede RLtRCA ecg eA Pe Cae Rae Boyd a gery set Rea ora EAST SOULS eae PTET ERE Ee PB Ere ph ich hdd KALMAR UT Me Mlidetehid Medd b23. 2d LOA La | ll eid ikl KAW dSLR CARTER Ce ToT Lier he ce deca Weld NhddecAPLA ILIA USER AL PAOLA TOT Ee A Ye i ea a) Ei2ieraeeeaea ea od ee oa ea eT AERA raed TTT chk bill ahd REO Re SOLE ict hide Mid EASELS dU KOLA SATE Ubi dice Me ee shld eed RLaeU eeepc UGS IL OCTANE ARTE sdadtcall WGA ELL I GLa a eee eld hich Behe LL dae LLL CR EEA Ted TT hints MARL MO Xe zea el Broa aid indie Uae ad ee REA a ERE EL STA ITI Lom RRA TELA TEA TIE AR RTT Te 9€120"| 06'Sh4| —» | 4h9ZOl| 08 96/h'| LHL E9E+| | sAce 2 eae ESATA IS AWTS TTY) eed ate x7 ieciel eidadcedl Lae Sid RAL Ue LR CT eae) FTA EUR XLA LL ESa dCTP idl all Mellel Mead zac aa Taam eTTT dtu ka suh'Ghe] ZI6E6/\h81EC |LmI6Yh| HI6LE| 192121 O8EL | a Te Rea aaa ee rd ACT oS, lille ddd Cade 2 ed LR CANE iN edad aca Lea OST od eA illite Lilith Maal ae decd Gack CK reac a ee Mehta ed MA WAETLA RT RT ALT SOLER TL aR T aU TET TT LPP 2 tee Re) nsszs't| ee6e's | ze9n'0l| 1xeee0 lassie! lezeeeo ls onaTeleeracel 165°90 5619 $2) $851'9h| BeLO6'| O&49h'| 08°16 | «| thete +| 8/4920 | 21699 +] 2 aida ed ace a apres aes Ir er a nr Ret Tea ee) : : 5 3 Tana el? y MO Aa ae de ¥y 4 ¢ oad] -— — i sad 09 LanT I I ‘4 f rf vi OCS ACE! $9(P-x) oe aR Won Le 22 ‘ PhZOZE LOAEPES 2 | E9ESEL | EVEELL | 9SEz20z2£| “CG ALAC IE 131! 371927391390 EA Sal dS Le z TTL TT TAL Le AS Eh Ae 9 2S PERM ART £5 R Ad ae SAL PAR LA bed ATL TL EEL HELLLI | AhL6L00 \6ALL9£0 |Eh2Z9£0° as] US TTC eh ed 4 ed + A 9 DATE ASOT L124 21 TTA PA 9 € FAP ATM UT ALS ALY AA CAAA $14902 \G0L NOOO \Lh16400 | L4/6h00"| 8S£902'| 01 ‘(%- 1) 0001-7 + WH HW : AA CIT PMA LUA AL el nL, vr) ee He = "Nig -W Les va FSA TM Tn AT ML al i ba aa Te TTR Led Craze Cee eC Ph. Tot, Tea Ah UL£0S00"\EPKEE20' FOREEZO" Te pp ee ee Pw = "9 Pay TEMA a aa Ob eo 25 ’ a a Tonys See ee A An as E/9IST \LISZSIT \99CILAO 99E9L40'| 99/6 AI - Zz 7 4 ae OL al UT ATLA ALIS L384 ASS € i Z 2 PATA SL EP TA LAE + ‘i (o-x%) & °7 000/- = 'p URAL Le TAME LSS PAA PAO 2 ones oe aS A OS dl Ry XV "asd CTS aa Dea Le aA of at a Shed, pe gecee ns wax | i at ca 3 : i aiiitecsamaii ae i ' | | / ron | : | | t aie i 4 # ini ce Sane weSes Gases Sees Peessees VIM) oe REL Leste re weal ead Ps Joo lid) bh ? as Of lar ts ABCDS &; - adi < beset setter agree | Se eee ree a ae ayaa eames! ag me aS edt fede) | eeuas Feat Laua7s ae tada,/ Maude Be sameaieuaee aueee peenS cunae | eens peapnust ees } : | t if i | / } 4 i +t —_—-* ale asa el 4 9 4) Voi eas ae 173 Ysa | ; } : — } | Dri Se ae AL kms mT Te ee a abe hee, Petar DOT per rat earl) ee 5% | on nie } } t- Fer Tt of Barret at — i. ore assy : ee, bel PECOEERDEE EA: ; — 1 | | T ; | | ae | | ; , Besat hess: sonedessss| a pepes teres seese| Baad sabes sisss sence csGasuanas sassy t+ Fides sat sbersseat | | Vt Se itera ae mat 4 4 } } ; | GiBiertiser 2 9.7,’ eett, attarade Ya(0) Jie! 7 aoe eee Pe LTT] IZ 7 ie x ce 2 2d OZ 2 2 a SO TT Met 17 22 ee a A) Ea a g Ti, i 1 a 2 701 2 ee 2 7 ny al TY eS ed TY YY 8) ‘9 oT ii to a? ei oe 2 ed 2) a 2 2 YY Ke) hae eie Lee. eed Ls en yey ey ey 7 a YS | G > ed =% "9 ei i 2 Pace aT val ea A ir Wl ae) EZ A * 9482 Ta TS al ae 9 cal AL’) -|998hE +/)68h'sl +| “ods oan ees aa] ae) Tole 65 SH'9E - | ISS'RD +| £6IGHY +| “1256 + #0 OIE +| J yar) ee VEE NS eee a ae 27 [Te eo) he eee i 2 ee eT 7 ae, eas eM ar4 LE aia 98582 eT) ae Ee) 2 2 a 2 Ye ol 24 2 9£6%+| C7 Le ded 2d aT] ad LV°£L9 -| 19°L80' +| 029'FS6'£ +) C9L'x0h + The ef Fe £ = i) £ F a? Ay aX a 0 if ao if A i ty he alte CAAT PATH 90x! 94£100" cw a ee Teh aes Uae ra Tle ‘ a + iI) Ds oo Tee rr TE PTT TT PTT Pe Pa PCL Ce ec ce ca x/AFb | SOL |,0962 | 961 \ghS$! |ASS/ | 0/08 | 998% |,S06£ |, B00. | ¥*°7 preg CAL SEES APSR ees: i ar) *SONIGUOT a > id od) | ee Oe TT OT ree TL ae a ee 3 ~eaall ; PAE B- | a L9'SIE TTT ae ae) l go9e+ |, eraxe | yoret+ | 95 he a Em hi 00'S%+ | a $I'E9S a ar 3 oa 5 Tee Wey as 94F06+ [oy ; Then Wale ee ereie+ | 9 a4 | yas HI RIE 9'S2- iii ha eta ids oi2% yaad aT 7 Stress at §' Stress at 0 ee Stress at £° Plate 9. Maximum. Stresses in Stee/ NRA LM MLL Scales: I'=70" /"= 100* Cree ee stress at C' att HLL La acer he ae Fi P-re 0° | enn | 57/7" A’ | 63.7" | ons nT el A C’ loa * | eRe D’ | 129.9""| 133.5" ee coe led ————— A ae ae. Bap eZ) pe a a ws Bae fam oe Pi NIV | MICHIGAN STATE U TT 3 1293 0 Il ul 3 6 ERSITY | 9 | 7 LIBRARI ij 059 mete: amines oS Pe So _ fet — p ; — = i ae BRAS Si — — air. EEE my ae ; yi n/a ; joa . ' wf : — ° rr, - > * a —— ti il _& MICHIGAN STATE UNIVERSITY LIBRARIES || I! | | | Hl HANA OA A I I 169 7059