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DESIGN OFA REINFORCED-CONCRETE ARCH

BY

EARL RUNDLES

THESIS

FOR

DEGREE OF BACHELOR OF SCIENCE

IN

CIVIL ENGINEERING

COLLEGE OF ENGINEERING

UNIVERSITY OF ILLINOIS

1913

Ui\riVEiisiTY oy iLLiisruis

woliege of Engineering.

May 21+, 1913.

I r?coniinencl tnat the thesis prepared under rny

supervision by earl rukdL-cg entitled Design or a Reinrorced-.:)oncret

e

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.

L246514

Table of Contents.

Introduction Page 1

Reinforcement Page 1

Method Page 1

Dimensions Page 2

Amount of Heinforcement Page 2

Loading Page 2

V/eights of IJatf.rial Page 2

Design Page 2

Conclusion Page 7

Tables 1 Page a

II Page 3

III Page /O

Plate I Page //

Plate II

.-§3ros9 Sections of ArchT Pape/-2

UlUC

DESIGIJ 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 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

DIIJEIT3I01IS.

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. 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.

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 100.

Assuming one foot in width of arch, this reive the dead

load for each division.

Live Load:- The live load over each divi-

sion was found by scaling the distance between verticals

and multiplying by 150.

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 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 by

an application of Navier'a principle; T - pp (2.166 x 150.-f-

.50 X 100-1-200) 146. - 84,000 lb.

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-or-

dinates 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 129.75

and to act 0.944 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^ 20-1- 2

above it. This assumption simplified the subsequent work.

Taking monents about a point in Tl and

Tr (Plate I) givest-

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

^1 ""'1 ~ Tr"^ '^l 1^1'-'rial

^

_ 0^r —f- ^^ V ^

ri-|^ = 1.063 X Y-^ n^^1 ^1 ~ Xr = 6.26 ft.

5

and

^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 from

n,-|_to m22.

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,/ —156.08 IIa/(y

' GO. 000 X i337g^= 93, 680

True Equilibrium Polygon: The true pole is

located by measuring the true pole distance from Q then

beginning at Xr or K-j^ the true equilibrium polygon

can be drav.'n.

Stresses I>ue to Dead and Live Load:

Leta c = intercept between the neutral line

and the true equilibrium polygon.

b •= the breadth of the unit section

of the arch.

c ~ the distance of the most remote fiber

from the neutral line.

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 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 =~

T

^s d

These stresses are recorded in Tible II.

Effect of temperature Changes.

LetX - span of the neutral line,

e = the expansion of concrete per unit

of length per 1° Fahr.

t°- the difference in de^'rees hetv/een

the mean and the actual temperature of the arch ring,

E = 1,500,000

a = 92 ft.

^-.000,005,4

I= horizontal resisting force.

Then

Z\s = 1.165

Q - (1,500,000 X 144) 92 x ,0000054 x 20

133,28 X 1.17

7

a 1840'

^ = ^ y 1040 X 5. = 4930%' or 34^'3.352

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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