Thesis (B.S.)--University of Illinois, 1913.Typescript

Rundles, Earl

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Design of a reinforced-concrete arch

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THE UNIVERSITYOF ILLINOISLIBRARYDigitized by the Internet Archivein 2013http://archive.org/details/designofreinforcOOrundDESIGN OFA REINFORCED-CONCRETE ARCHBYEARL RUNDLESTHESISFORDEGREE OF BACHELOR OF SCIENCEINCIVIL ENGINEERINGCOLLEGE OF ENGINEERINGUNIVERSITY OF ILLINOIS1913Ui\riVEiisiTY oy iLLiisruiswoliege of Engineering.May 21+, 1913.I r?coniinencl tnat the thesis prepared under rnysupervision by earl rukdL-cg entitled Design or a Reinrorced-.:)oncreteArch \'^e approved as fulfilling this part of the requirements for thedegree of Facheior of Science in Jivii Engineering.Recommendation approvedHead of Department of Jivil Jing'g.L246514Table of Contents.Introduction Page 1Reinforcement Page 1Method Page 1Dimensions Page 2Amount of Heinforcement Page 2Loading Page 2V/eights of IJatf.rial Page 2Design Page 2Conclusion Page 7Tables 1 Page aII Page 3III Page /OPlate I Page //Plate II.-§3ros9 Sections of ArchT Pape/-2UlUCDESIGIJ OF A {EILFORGEI; C01ICR::TE /RCH.IIITRODUCTIOII.Ovi-iTiR to the fact that the regular Uni-versity curriculum for undergraduate v;ork does not pro-vide for instruction on the theory and design of arches,thesis work covering this field was chosen. The archherein designed is intended to carry heavy highway traffic.It has a clear span of ninety-two feet and rise of elevenfeet. '"he small rise was used so as to give ample roomfor the flow of flood waters near the haunches of the arch,REIMFORCEIOIT.The reinforcement consistfs of square twisted"bars. Three quarter inch hars , placed six inches apartwere used in the intrados and extrados. One half inch"bars placed two feet apart were used in the transversedirection. This system was chosen rather than the Lielantype in that tests show that a smaller unit gives greaterstrength, area for area.METHOD.Prof. Baker's graphical method according tothe elastic theory was used in the design. The generaldim.ensions were first assumed nnd then the design workedout to see v/hether the stresses obtained 7;ere within theallowable limit. The allowable stresses and the v;eightsof the m.aterial are those v;hich have been generally accepted.2DIIJEIT3I01IS.Span (clear) 92 feet.Rise 11 feet,i^arth fill over crown 6 inches.Width of Roadway 20 feet,REIKFOHC^MEKT.44 lines of 3/4 "a bars along intrados.44 lines of 3/4 bars along extrados.2 lines of 1/2" a bars in each parapet wall.LOADS.Live load on one half of bridge, 200 lb. sq.ft.Dead load of earth and concrete.WEIGHTS Oi' MATERIALS.,Concrete nnd steel 150 lb. per cu. ft.Earth filling 100 lb. per cu. ftDESIGN.The neutral axis was divided so that each lengthdivided by the moHient of inertia of concrete and steel gavea constant. The length of the first division was takenso as to get a convenient size and number of divisions.See Table I.3The lengths were laid off In suocecsion onthe neutral axis from the crovm to the springing lines.The centers of the divisions are marked a;L ®2 ^22as shown in Plate I,Dead lioad:- Vertical lines were drawn throughag - agg and the areas included between the successiveverticals, the intrados and the upper limit of the earthfilling, found. These areas were multiplied by 100.Assuming one foot in width of arch, this reive the deadload for each division.Live Load:- The live load over each divi-sion was found by scaling the distance between verticalsand multiplying by 150.Horizontal Pressure of Earth Filling:- Thetheory used does not take into consideration the horizontalpressure of the earth filling. On account of the flatarch this force could he safely neglected.Plate I gives the oead (earth and concrete)and live load for each division, ilie loads were laid offat Che center of gravity of the division between the verti-cals a^ ag - agg as shovm in Plate I.Construction of the trial equlibrum polygon,1st. The load line 1-21 was laid off.2nd. The trial pole v/as determined byan application of Navier'a principle; T - pp (2.166 x 150.-f-.50 X 100-1-200) 146. - 84,000 lb.4Therefore the trial pole distance v;as taken at 80,000 lb,^nd the tri il equilibrium polygon drawn. It was necessarynext to find the re^ ultant of the positive forces and theclosing line of the trial equilibrium polygon.Table II gives the values of the co-or-dinates x and y to the points of intereection of thelines of action and neutral line of the arch ring andalso various intercepts and products employed in thework to follow. The resultant was found to be 129.75and to act 0.944 ft. to the left of the center line of arch.The trial closing line was assumed tobe parallel to Vo . Voo and to be 5 = 5.89 ft^ 20-1- 2above it. This assumption simplified the subsequent work.Taking monents about a point in Tl andTr (Plate I) givest-True Tr jq" — xTrial TandTrue Te ^ —Trial T ^Then if mi mgg is the true closing line,the following proportion .s true:I'rue Te^1 ""'1 ~ Tr"^ '^l 1^1'-'rial ^_0^r —f- ^^ V^ri-|^ = 1.063 X Y-^ n^^1 ^1 ~ Xr = 6.26 ft.5and^22- "'22 - — ^ vog. m-oy- e= 0.935 X 5.89= 5.51 ft.'•^'hc true closing line is obtained "by drav;ing a line fromn,-|_ to m22.True Pole "Distance; The true etjuilitriumpolygon must give ZTck, y * a (Plate I) hencethe trial pole must "be moved accordingly.True pole distance = Trial pole distance,/ —156.08 IIa/(y' GO. 000 X i337g^= 93, 680True Equilibrium Polygon: The true pole islocated by measuring the true pole distance from Q thenbeginning at Xr or K-j^ the true equilibrium polygoncan be drav.'n.Stresses I>ue to Dead and Live Load:Leta c = intercept between the neutral lineand the true equilibrium polygon.b •= the breadth of the unit sectionof the arch.c ~ the distance of the most remote fiberfrom the neutral line.d " the depth of the arch ringf - the unit fiber stress,h = the true pole distance.II = the component parallel to the radiusat any point of the neutral line of all the forces to oneside of the point,T the component parallel to the tangentat any point of the neutral line of all the forces toone side of the point,V » the unit shearing stress,b H a cfh =~T^s dThese stresses are recorded in Tible II.Effect of temperature Changes.LetX - span of the neutral line,e = the expansion of concrete per unitof length per 1° Fahr.t°- the difference in de^'rees hetv/eenthe mean and the actual temperature of the arch ring,E = 1,500,000a = 92 ft.^-.000,005,4I= horizontal resisting force.ThenZ\s = 1.165Q - (1,500,000 X 144) 92 x ,0000054 x 20133,28 X 1.177a 1840'^ = ^ y 1040 X 5. = 4930%' or 34^'3.352Concliision:- Table III shoY.'S the stresses dueto clead and live loads. The stresses use(? irj checkingthe' design of the arch ring are the maximum jtresses v/hichoccur at the points shovm in the tr.hle. The shearingstress Ft most points ras almost negligable being toosm.all to measure by graphical methods.X N\NO\ \X XM)XXXXXCMXXXX<\Am.NON CMnOsN CMCMCMX •MCOOM CMCMCM CN/f\ 1ooCM"X CM«)IV)^:^/^ rv\)x\CMXonX X X>NO\ \(5\1CMCMcm"CMCMCOCM1CM <0 0)31 1001X1NX1N1CM1 1 100X1\X1 r 1 1 1XtX11<<>>OQ111CO1 > > > >X1> >oo1 1X1111oo> >;qX1N NO>NO"1 1XiIs00rss)11 1IQ1\1 1CM11 1 1 1 1X1\-11X1WNO1X1X1^.1 1^'1 1 1X1x ?1XX1 11Istlxia;\iN\\<^ ho NQOXXNT)> >rsSJ«0X1csCO11fSX11>rs> 1 11NCM XXX CMOn UCOrsNj rsSJM03rss^K(fS rs\JCMX11MWMi<^ XX K?111111T11^ }11^11X11X111111X1MiX1 r r >CM> > >^X >sN 00 XX 5i X X ^5XX X X CMCM\^0 \N1s>^ 1 \NCM KnCM\k1CM\sinNT)T NT V VC3\lN \ KV1:11 Hi 1PO/NT^^ ^-^ 5UNIVf.92•I- 71-M-y/-V.<J<V-fl.7.-<l\\10:IJ,0717.'. 1I14it44

Typescript.Thesis (B.S.)--University of Illinois, 1913.Mode of access: Internet

Rundles, Earl.

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