THE UNIVERSITY OF ILLINOIS LIBRARY

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2 THE UNIVERSITY OF ILLINOIS LIBRARY

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5 DESIGN OF A REINFORCED-CONCRETE ARCH BY EARL RUNDLES THESIS FOR DEGREE OF BACHELOR OF SCIENCE IN CIVIL ENGINEERING COLLEGE OF ENGINEERING UNIVERSITY OF ILLINOIS 93

$e Ui\riVEiisiTY oy illiisruis woliege of Engineering. May 2+, 93. I r?$ $:)oncret Arch \'^e approved as fulfilling this part of the requirements for the degree of$

7 e Ui\riVEiisiTY oy illiisruis woliege of Engineering. May 2+, 93. I r?coniinencl tnat the thesis prepared under rny supervision by earl rukdl-cg entitled Design or a Reinrorced-.:)oncret Arch \'^e approved as fulfilling this part of the requirements for the degree of Facheior of Science in Jivii Engineering. Recommendation approved Head of Department of Jivil Jing'g. L 24654

Amount of Heinf orcement Page 2 Loading Page 2 V/eights of IJatf.

9 Table of Contents. Introduction Page Reinforcement Page Method Page Dimensions Page 2 Amount of Heinf orcement Page 2 Loading Page 2 V/eights of IJatf.rial Page 2 Design Page 2 Conclusion Page 7 Tables Page a II Page 3 III Page /O Plate I Page // Plate II.- 3ros9 Sections of ArchT Pape/-2

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The arch herein designed is intended to carry heavy highway traffic. It has a clear span of ninety-two feet and rise of eleven feet.

11 DESIGIJ OF A {EILFORGEI; C0ICR::TE /RCH. IIITRODUCTIOII. Ovi-iTiR to the fact that the regular University curriculum for undergraduate v;ork does not provide for instruction on the theory and design of arches, thesis work covering this field was chosen. The arch herein designed is intended to carry heavy highway traffic. It has a clear span of ninety-two feet and rise of eleven feet. '"he small rise was used so as to give ample room for 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 apart were used in the intrados and extrados. One half inch "bars placed two feet apart were used in the transverse direction. This system was chosen rather than the Lielan type in that tests show that a smaller unit gives greater strength, area for area. METHOD. Prof. Baker's graphical method according to the elastic theory was used in the design. The general dim.ensions were first assumed nnd then the design worked out to see v/hether the stresses obtained 7;ere within the allowable limit. The allowable stresses and the v;eights of the m.aterial are those v;hich have been generally accepted.

2 DIIJEIT3I0IS. Span (clear) 92 feet. Rise feet, i^arth fill over crown 6 inches. Width of Roadway 20 feet, REIKFOHC^MEKT. 44 lines of 3/4 "a bars along intrados.

Dead load of earth and concrete. WEIGHTS Oi' MATERIALS., Concrete nnd steel 50 lb. per cu. ft. Earth filling 00 lb. per cu. ft DESIGN.

13 2 DIIJEIT3I0IS. Span (clear) 92 feet. Rise 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 /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 50 lb. per cu. ft. Earth filling 00 lb. per cu. ft DESIGN. The neutral axis was divided so that each length divided by the mohient of inertia of concrete and steel gave a constant. The length of the first division was taken so as to get a convenient size and number of divisions. See Table I.

and the upper limit of the earth filling, found. These areas were multiplied by 00. Assuming one foot in width of arch, this reive the dead load for each division.

15 3 The lengths were laid off In suocecsion on the neutral axis from the crovm to the springing lines. The centers of the divisions are marked a;l 2 ^22 as shown in Plate I, Dead lioad:- Vertical lines were drawn through ag - agg and the areas included between the successive verticals, the intrados and the upper limit of the earth filling, found. These areas were multiplied by 00. Assuming one foot in width of arch, this reive the dead load for each division. Live Load:- The live load over each division was found by scaling the distance between verticals and multiplying by 50. Horizontal Pressure of Earth Filling:- The theory used does not take into consideration the horizontal pressure of the earth filling. On account of the flat arch this force could he safely neglected. Plate I gives the oead (earth and concrete) and live load for each division, ilie loads were laid off at Che center of gravity of the division between the verticals a^ ag - agg as shovm in Plate I. Construction of the trial equlibrum polygon, st. The load line -2 was laid off. 2nd. The trial pole v/as determined by an application of Navier'a principle; T - pp (2.66 x 50.-f ) ,000 lb.

Table II gives the values of the co-ordinates x and y to the points of intereection of the lines of action and neutral line of the arch ring and also various intercepts and products employed in the

17 4 Therefore the trial pole distance v;as taken at 80,000 lb, ^nd the tri il equilibrium polygon drawn. It was necessary next to find the re^ ultant of the positive forces and the closing line of the trial equilibrium polygon. Table II gives the values of the co-ordinates x and y to the points of intereection of the lines of action and neutral line of the arch ring and also various intercepts and products employed in the work to follow. The resultant was found to be and to act ft. to the left of the center line of arch. The trial closing line was assumed to be parallel to Vo. Voo and to be 5 = 5.89 ft ^ above it. This assumption simplified the subsequent work. Tr (Plate I) givest- Taking monents about a point in Tl and True Tr jq" x Trial T and True Te ^ Trial T ^ Then if mi mgg is the true closing line, the following proportion.s true: I'rue Te ^ ""' ~ Tr"^ '^l ^ '-'rial ^ _ 0^ r f- ^^ V ^ ri- ^ =.063 Y-^ n^ ^ ^ ~ r = 6.26 ft.

5 and ^22- "'22 - ^ vog. m-o y- e = 0.935 5.89 = 5.5 ft. ' ^'hc true closing line is obtained "by drav;ing a line from to m22.

000 i337g^ = 93, 680 True Equilibrium Polygon: The true pole is located by measuring the true pole distance from Q then beginning at r or K-j^ the true equilibrium polygon can be drav.'n.

19 5 and ^22- "'22 - ^ vog. m-o y- e = = 5.5 ft. ' ^'hc true closing line is obtained "by drav;ing a line from to m22. n,- _ True Pole "Distance; The true etjuilitrium polygon must give ZTck, y * a (Plate I) hence the trial pole must "be moved accordingly. True pole distance = Trial pole distance,/ IIa/(y ' GO. 000 i337g^ = 93, 680 True Equilibrium Polygon: The true pole is located by measuring the true pole distance from Q then beginning at r or K-j^ the true equilibrium polygon can be drav.'n. Stresses I>ue to Dead and Live Load: Let a c = intercept between the neutral line and the true equilibrium polygon. of the arch. from the neutral line. b = the breadth of the unit section c ~ the distance of the most remote fiber d " the depth of the arch ring f - the unit fiber stress, h = the true pole distance.

II = the component parallel to the radius at any point of the neutral line of all the forces to one side of the point, T the component parallel to the tangent at any

of length per Fahr. Effect of temperature Changes.

21 II = the component parallel to the radius at any point of the neutral line of all the forces to one side of the point, T the component parallel to the tangent at any point of the neutral line of all the forces to one side of the point, V» the unit shearing stress, b H a c ~ fh = ^s T d These stresses are recorded in Tible II. of length per Fahr. Effect of temperature Changes. Let - span of the neutral line, e = the expansion of concrete per unit t - the difference in de^'rees hetv/een the mean and the actual temperature of the arch ring, E =,500,000 a = 92 ft. ^-.000,005,4 Then I = horizontal resisting force. Z\s =.65 Q - (,500,000 44) 92 x, x 20 33,28.7

7 a 840' ^ = ^ y 040 5. = 4930%' or 34^' 3.352 Concliision:- Table III shoy.

irj checking the' design of the arch ring are the maximum jtresses v/hich occur at the

23 7 a 840' ^ = ^ y = 4930%' or 34^' Concliision:- Table III shoy.'s the stresses due to clead and live loads. The stresses use(? irj checking the' design of the arch ring are the maximum jtresses v/hich occur at the points shovm in the tr.hle. The shearing stress Ft most points ras almost negligable being too sm.all to measure by graphical methods.

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The Islamic University of Gaza Department of Civil Engineering ENGC Design of Spherical Shells (Domes)

The Islamic University of Gaza Department of Civil Engineering ENGC 6353 Design of Spherical Shells (Domes) Shell Structure A thin shell is defined as a shell with a relatively small thickness, compared