\ AD LRA mtn nd mA MN pipe 3 1293 00991 5830 | I> 3 oe h ( Ly | ae os Uo p ) Sit dk 1 og ets LES PLEMENTARY Mat EK IAL IN BACK OF BOOK PLACE IN RETURN BOX to remove this heckout from your recor VOID FINES return on or before date due 11/00 c/CIRC/DateDue.p65-p.13 THE ANALYSIS OF A CONCRETE AROH OVER THORHAPPLE RIVER NEAR HASTINGS A THESIS Submitted to the Faculty of THE MIOHIGAN AGRICULTURAL COLLEGE BY 4 s i’ \ \ ALLIF L. HATOVSKY For the Degree of BACHELOR OF SCIENCE JUNE 1922 THESIS Lop. | PREFACE In this volume the author has set forth in detail what he considers te be the mest practical methed fer the analysis of a cencrete areh. The primary reason fer sempiling this volume was to become more fully acquainted with the dusign and analysis ef concrete arches. The author wishes to acknowledge his infebtuess to Heol and Johnson fer the diagrams repreduced frem their book "Concrete ingineers’ Handbook"; to Mr. C.A Melick, Bridge kugineer of the State Highway, for his kind assistance in furnishing plans of the structure and helpful suggestions; to Prof. Allen for his valuable instruction and guidance in the various courses in eoncrete design; and to Prof. Vedder for his aidfal criticiam and suggestions. 96524 TABLi, OF CONTENTS Deseription ef the Structure. Metheds of Analysis. Outline of the Graphical Analysis. Graphical Analysis Notations Analysis by Cechrane's Method.- Data. Live Lead. Deaf Load. Thrusts and Moments. squivalent Areas and Moments ef Inertia. Average Stresses. Summary for Max. ~ Mem. at Crown. Summary for Max. - MOm. at Crown. Summary for Max. - Mom. at Springing Line. Summary for Max. - Mon. at Springing Line Approximate Maximum Stresses. Correchion for Max. Stresses. Maximum Fiber Stresscs. Conclusion. Diagrams Blue Print of Structure. Plate 1 a I NWN 74 @ DESCRIPTION OF THE STRUCTURE This bridge is being built by the State Highway Depart- ment across Thernapple River just north the eity of Hastings. It replaces a steel bowstring bridge which has been condemned as inadequate forthe present traffie leads. It is a single span, reinforced concrete bridge designed by the Bridge Depart- ment of the State Highway in accoréance with their specifica- tions and requirements. The contract for its construction was let to w¥.G. Crebo, of Grand Rapids, during July 1921. It was te be completed by the end of last year; but, due to the fact that quick sand was encountered when exsavating for the abut- ments, only the abutements and the wings have been poured at the present tine. The superstructure censists of ten paneis ef 10'-10" over the arch ané three panels of 10'-3" over each abutment, making a total length ef 172'-4". The readway is 24° wide,whieh is quite an increase over that of the old bridge. The arch consists of twe areh rings, eaeh 5'-4" wide, connected by a cress brace of reinforeed concrete 22' on each side ef the erown. They are 2'-0" thick at the erown ané 6'-O" thick af the springing line. They are three centered circular arch rings with a clear span ef 100° and a rise of 18°. The springing line elevation is 1.7° above mean water level which is sufficient since the river is only 4’ deep. The superstructure consists of a 10" floor slab with eurbs and railings made up of spindles ané pilesters as shown by the accompening blue print. The superstructure - j] @ above the arch rests upon reinforeed cross beams which are supported on the arch rings by nine pair of spandrel columns. These columns are interconnected by small arches. The reinforcing is ef steel bars thru-out, laid en the principle of one-way reinforcing with just enough transverse to prevent checking or cracking. MuTHODS OF ANALYSIS There are a number of beth graphical and analytical methods, ocr a conbination of the two, which have been used in the past in arch analysis. The two main theories upon wuick they have been based are the Linc-of-Thrust Theory and the iklastio Theory. Simee the dead load usually sontrois the shape of the arch ring, an approximate graphical analysis based on the Line-of Thrast Theory is first made. The obdjeet of this analysis is te determine whether the line ef thrust falis within the middle third, which indicates that there is no tension in the con- crete. Since the elastic properties of concrete is not taken into consideration in this theory, it is useful only for Pre- liminary investigations as to the proper shape of the arch ring. | All of the methods of dusign and analysis based upon the elastic theory have been derived by making numerous asaup- tions. Also many unoertain factoés enter, some of which are the following: the approximate character og the flexure for mulas; the uncertainity of the tensile stresses in concrete; the variation in the live load; the effect of temperature - 2- variations; the effect of the shrinkage of eoncrete; and the effect of slight movement or distortion of the abut- ments. Thus conditions justify the usu of CAchrane's For mules and Diagrams in the analysis of a concrete arch ring. These formulas and diagrams were compiled by Mr. Cochrane fromthoroagh investigations of a great number of arch de- signs found in technical literature. He also constructed a curve giving the ratio of the thickness ef the arch ring at any point to that at the crown for various ratios of the thickness of the aroh ring at tke springing lime te its thickness at the crown. He alse determined that if an areh ring were designed in accofiance to this curve, the maximum stress eceured at the springing line or at the crown. Thus onjy these two points have to be investigated to determine the safety of the areh ring. This areh ring was not laid out by Cochrane's curves; but, since it is almost exactiy thru-out as specified by them, the use of Cochrane’s formulas ané diagrams are applicable. Also the stresses at only the springing line and at the crown will have to be investigated. | LIBK = OF - Sf RY 1. Draw one-half of the arch ring te as iarge & sea].e as cenvenicnt. 7 2e Divide the ~erohk ring into five seetionali divisions with the diviéd sections at the spanédrel eoclumns. 3. . Lecate the centers of gravity ef the trapezoidal sections. ‘ae Extend Di until UN = Rg db. sxtamd BC in the opposite direction until Mie Ty. ceThe intersection of MN and the median OP locates the center of gravity. 4. Compute the weight of each seotion. §. te theuwe % of the spandrel columns anf dead | Oo) EE Rneerea © ° 6 Drew in the lines of foree. 7. Tay off the load line. 8. Seleet a convenient pole 0 on a horisonutal thre EK and draw in the rays ef the feree polygon. 9. Ceonstract the correspond ing equilibrium pelygon start- ing at the center of springing line. 10. kExtené the first ani last rays, oa and ok, until they intersect at WV. This losates the canter of forees. 12. Draw the resultant of forces vertically thre the point W. 12. Extend a horizontal line from Z until it i&tersects the resultant at Y, which is the point of intersection of the first and last rays of the required equilibrium polygon. 13. Draw XY. 14. Draw the corresponding rays of the force polygon parallel to XY from point A, thus locating point 0 at its intersec- tion with the horizontal from K. 15. Construct the corresponding equilibrium polygon parallel to the rays ef the foree polygon drawn from the pole QO. 16. Draw in the areh axis. | ~ 4- DuAD LOADS Left Heseny Av. i Del. Force Fe. Se RS eS te ~ Lbs. 6.80 4.90 5.85 7.7 65.33 36,200 AB 30,080 BC 4.90 3.03 3.965 10.83 " 34,400 cD 33,875 Dé 5.03 2.28 2.655 " " 25', 000 SF Fa 40,440 2.28 2.05 2.165 " " 18, 780 GH 32,570 HI 2.05 2.00 2.025 " " 17,580 IJ 16,210 JK Total Dele --- 283 ,085 pE ) D.L., LoL, Temp. Var. & Areh S. THRUST = § _MOMENT AY. STRESSHS DeLe & Leb. 492,800 67,500 31,370 Temp. Var. - 48 , 350 134,100 ~ 2,670 Areh. 8S. = 281,000 $8,200 ~- 1,160 Tetals 423,450 259, 600 27,540 Computation of f. . Tc . & Mo for Arch 58. Let f, due to AroA SS. % x Then = . $1,570 - x “835800 — “hag ? X “= 26 70 ® xX 3 =2670 x 28,700 66,070 2 =] ,270 lbs. - -1,160 Lbs. Let fT. a a —1—". 30,100 248,35 63,400 - ~_48,560 x 30,100 . _ y 66,070 = -~22,990 lbs. y= WA8S50 x 27,540 21,000 66.070 1,000 lbs. - 14 - Let M. due to Arch-S. = g Then z = 30,000 and ~ _154,100 x 30,100 ff 63,400 63, 700-lbs. ff a2 cree S40 = 58,200"1be. SUMMARY FOR MAX. = AT CROWN (a) Debe, Lele, & Aroh-Shortening. THRUST Dead Load j#§ 430,000. Live Load 63,100 Arch-S. - 23,130 Total 479,970 MOMiNT AV. STRESSHS £7,600 ~ 57,2650 4,075 64,000 - 1,280 6,750 50,295 (bo) D.L., LLL, Temp. Var. & Arch-S. THRUST Dead L.& L.L. 493,100 Temp. Var. - 48,350 Arch-5. - 21,150 Total 423,700 MOMiNT AVY. STRESSES - §7,250 51,575 134,100 - 2,670 58 , 550 ~ 1,170 127,200 27,7355 Above summary shows that there is no negative moment ag the crown. - 16 - SUMMARY FOR MAX. # MOM. AT SPRINGING LIBy. (a) DL, bebe, & Arch- Shortening. THRUS® lbs. Dead Load 533,000 Live Load 88,400 Aroh-S. = 19,700 Total 601, 700 MOMENT AV. STRESSisS ° Be Be 27,500 508, 200 5,690 ~328,5300 - 1,340 179, 900 31,850 (b)_ D.L., L.L., Temp. Var. & Aroh-S, THRUST bs. Dead L. & L.L. 621,400 Temp. Var. 39,200 Arech-S. ~- 21,280 Total 639 , 520 - 17 < MOMENT AV. STRESSES “ft, ibs. ibs. 508, 200 33,190 653, 9000 2,670 ~ 354,600 - 1,450 807, 300 34,410 SU:MARY FOR MAX. = MOM. AT SPRINGING LINK. = i8 = (a) D odes Lele, & Arch-Shortening e THRUST MOMENT AV. SEB SiS lbs. ids. e Dead Load 633,000 27,500 Live Load 60,160 ~410, 500 2,220 Areh-8, - 17,620 -294,000 - 1,200 ‘etal 575,540 ~604,500 28,520 (bd) D. @ Lek. Var. _ZHRUST _MOMANT AV. S®R&SS5S Lbs. ft,ibs. ibs. Dele, LoL., 593,160 - 410,500 29,720 Temp. Var. - 39,200 - 653,900 -— 2,670 Areh-S. = 16,030. - £67,700 = 1,090 Total 637,930 ~1,332.100 25,960 ‘ APPROXIMATE MAXIMUM FIBER STRiSSES Max. ~ Mom. at Crown. f.« ; + =e ~ i é er (lb:. per sq. ft.) (a) 469.810 131,000 _ fo = “Tg73i * “B.596 = 31,800 + 24,730 = 59,980 = 59,980 / 144 = 417;'/sq.in. (>) . ~_ 423,450 . 259,600 f. ie-3i + Ets 7 31/800 + 49,000 = 80,800 - 60,800 / 144 = 661 #/sq.in. Max. - Mom. at Crown. —_ No megative moment at the crewn. Max. ~ Mom. at the Springing Line f.2 ie Hite ( lbs. per sq. ft. ) = 32,410 / 144 = 225 #/sq.in. (bd) 537,930 1,332,100 x 6 . . fo eae Stiga 725-520 - 34,820 =50,540, =z 50,340 / 144 = 350 #/sq.in. axe + Mom. at the spri Line. (a) f= 153 7#/aq.in. (bd) £.% 282 £¢/eqein. - 19 « » wh CORRECTION FOR MAX. PIBR: STRESSiS Hote; The results obtained by Cochrane's are a little low, s0 a correetion is made by comsidering the thrast of the dead load as having an eccentricity of ene-fortieth of the areh scetion. Correction for Stresses at Crown Corr, #(€.1.7 x 025 x%,x0) / 14 I, =(430,000 x ,085 x 2) / (144 x 5.296) = 28.2 ;/sq.in. Correction for streases at Springing Linc. | a Corr. =(4.1.0, x 028 xt.xo) / 14 I, =(533,000 x .025 x 6 x 38) / (144 x 13.31)@ 125,2;/eq.in. - £0 « OO eee MAXIMUM FIBER STRESSES Comition of Leeding _ Max. =~ Mom. Max. + liom. at crown at spring. i/sq.in. #/8q. in. (a) 4217 153 (b) 517 282 COrrection 28 125 fetal (a) 445 278 é Total {(b) 6589 407 Crowmm ------. Max. - Mom. at spring. #/sqein. 225 gE 8 FINAL SUMMARY AND COMPARISIOB Sprimging Line 475 By Cechrane's By Melick‘'s Allowable ____ Method 9/sqein. z/S8qeine «/8qein. -- §869 650 650 SONCLUS ION In the graphical analysis, the lime ef thrust fellowed very clesely the exis of the areh ring, whieh shews that the arch ring is acceptable fer farther analysis. Since the falls BO line of thrustiabove the arch axis,it indieates that the arokh ring is a trifle heavier than mecessary. The analysis by Cochrane's method shows that the max- imam fiber stress at the crown is 9.4% below the allowable, ané the maxim fiber strees at the springing line is 26.9% below the allowable. These results eheek very elosely with those obtained at the State Highway office by the use of Mr. Melick's method. Their maximum stress at the crown was 7.1% above the allowable, and the maximum fiber stress at the springing line was 17.4% below the allowabie. Thos, from the preceding results,it is evidenttthat the areh ig safe and well designed. The only correction that might be suggested is that the thickness be reduced slightly — from the haunch to the springing line; but the change is not advisable for it would introduce a fractional U which would be very bothersome and undesirable in analysis by Cechrane's method. = 22 - DIAGRAMS Coefficients of “wt” t9 ‘7 16 ~ te) ay -_ — - °o 07 0.10 | ie For open spandre! arches y~ B= (3c? +l0c4#r) For filled spandrel arches y=AgE (c7+4cr) ais Q20 . 0.25 0.30 Values of rise—ratio “r” Approximate dead-load thrusts and moments. 0.35 Coefficients of “wl?” for moments 0.0065 MOMENTS, Nie 00060 0.0055 Coefficients of ‘wl?’ for moments ° Cooffironts of i far thst 0.25 — MOMENTS: (For ail values of r) 0.20 Us Moment 15 0.00434 wt* 20 0.00404 wt? 0.15 25 0.00378 wt? 30 0.00357 wt* 0.10 O15 0.20 0.25 Q30 035 Values of rise —ratior’ Live-load thrusts and moments at crown; open spandrel arches. Coefficients of wt 5 & § 8 S 2 a 0.10 5 —~-020 0.25 Values. of rise —ratio”r” Live-load thrusts and moments at springing; open spandrel arches. 030 0.35 IIL Coefficients of wf Coefficients of “wt?” for moments QO 0.30 0.20 0.25 Values of rise -ratio’r” 0.10 a5 030 Live-load thrusts and moments at crown; filled spandrel arches. Coefficients of “we” for thrusts Coefficients of Wl?” for © °o Ww on 5 5 ° ° on 8 © 3S Live-load thrusts and moments at springing; filled spandrel arches. “Z ole azo ozs 0.30 035 Values of rise —ratio “r” Coefficients of ‘wt’ for thrusts — 4s <_ ~ > wy #->) your> ov 4s tO ews Ss a0 Yyvomr > an For a fallof % in E=-G 40 Mc=— Cg Ms =M.+ hie N © Ts =(1.09-175r-) Te 5 35 c 5 © 30 ‘6 © 25 - G ? 20 15 i FOR OPEN SPANDREL FOR FILLED SPANDREL 0.15 Q25 0.30 0.20 Values of rise —ratio*r” Temperature gpd rib shortening. & 8 a Values of Cy $ pam aad ri i pnereens — wT) : — > 4uU2s! 245807 FOR DEAD LOAD 120 | fa" Average direct stress throughout the arch in /b per sq. ft. fac™ Direct stress at crown section in Ib. ft * FOR LIVE PRODUCING c 110] MAX.4MOMENT AT CROWN t Values of coefficie FOR LIVE LOAD PRODUCING MAX.— MOMENT AT CROWN 0.20 0.25 ae Values of rise —-ratio Tr Average stresses. Values of ‘‘C” in formula fa = Cfae. — vi °C Values of coefficient FOR LIVE LOAD PRODUCING MAX. MOMENT AT SPRINGING 0.80 FOR LIVE LOAD PRODUCING MAX.=MOMENT AT SPRINGI FOR TEMPERATURE AND 0.90 ARCH SHORTENING’ 080 TO 060 0.50 0.30 0.35 O15 020 0.25 Values of rise —ratio"r” Average stresses. Values of ‘“‘C” in formula fa = Cfae. i wa Ee A MICHIGAN STATE UNIV. LIBR idan 312930099 1583@