DESIGN or A REINFORCED I '

Thesis for the Degree oI B S I
MICHIGAN STATE. COLLEGE

Henry G Dunkererg
‘ ‘ I943 9'-"*

 

CONCRETE .QPEN SPA'NDRELl-BR‘IDCE' 7 f

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L .
Xx . . .
.. 303 [N ,h .

Design of a Reinforced Concrete

Open Spandrel Bridge

A Thesis Submitted to

The Faculty of
MICHIGAN STATE COLLEGE
of

AGRICULTURE AKD APPLIED SCIENUE

by

1’
r" (-7‘

.
\
.,-

e “.
Henry G..Qpnkelberg

Candidate for the Degree of

Bachelor of Science

June, 19h}

TABLE OF COET31.TS

Page
Acknowledgment.......................... ...... ..........
Introduction.......................,... .......... .......1
Design of floor slab, floor beam and spandrel column....
Design of floor sléb..................................

3
u
Design of floor beam....r: .......... ...... ......... ...5
Design of spandrel column.............................7

9

Design of arch rib. ................. ...........L......

Drawings and design of abutment......Pbcket on Back Cover

D' ‘ .
l‘ - _ . h .. ’ '5.

-H It .. . “-
4. a... ”.2 .....

lI-‘I eAcK OF BOOK

iibwd

 

matron-53mm:

The author wishes to express his appreciation to Professors
C. L. Allen, J. E. Meyer, and G. A. Miller of the Civil Engineer-
ing department of Michigan State College for their invaluable
suggestions and.aid in the following design.

Also, his appreciation is extended to Mr. Veigt, head of the
Michigan State Hiway Bridge Drafting Room and other members of his
department, who so willingdy gave information and suggestions as

to bridge design problems.

IIITROZ'MCTIOE

The importance of bridges is unquestionable when one reads the
history of the world. Important battles have been fought at bridge
sites, weary miles of travel have been saved by the spanning of tur-
bulent rivers and impassable canyons.

The very lives of many peOple have been altered by bringing
within easy reach routes to heretofore inaccessible places of market
and.trayel.

From.the ancient draw bridges over castle moats to the magnifi-
cent engineering structures spanning the San Francisco Bay and the
Golden Gate,bridges have been essential to civilization.

In the beginning, if a bridge was structurally sound enough to
support the traffic upon it, it served its purpose.

Ifith.the development of civilization, however, and the building
of arch bridges, the requirements for a bridge grew. no longer was
structural soundness sufficient. The structure must also present a
pleasing appearance. And some of the ancient ardh bridges which have
stood through the years with their graceful, pleasing, spans surpass
many of the modern attempts at artistic design.

With the advent of reinforced concrete as a building material,

. a new type of structural beauty has come into existence. The modern
builder wdth his modern materials and greatly accelerated construction
methods, must now vie with the ancient day builders for that illusive

gracefulness and artistic orginality in bridge design.
It is with these thoughts in mind that the author presents this

fdlowing arch design. First the bridge must be structurally sound

and then it must present a.pleasing appearance in harmony with the
setting.

The ground profile used in the design, while only assumed, ap-
proaches very nearly the conditions to be met with the building of a
new bridge at Berrien Springs, Michigan, to span the St. Joseph river.
The greatest difference in the design site and the actual site is that
three or more spans would be needed to cross the St. Joseph river. It
is the author's Opinion that the following arch design could be used

as one of the spans for this bridge.

Design of floor slab, floor beam
and spandrel column.

Fbr references see "The Design of Reinforced Concrete Structures"
by Dean Peabody, Jr. The pages will be noted below.
Notations and Equations.

b a width of section.

h a total depth of section.

d e distance from extreme fiber in compression to the center of

gravity of the steel in tension.
jd = moment arm of internal couple.
kd a distance from extreme fiber in compression to neutral axis.

EC 5 modulus of elasticity of concrete.

t
M.

modulus<3f elasticity of steel.

n a ”ratio. of moduli, or ES

30

fC I maximum intensity of fiber stress in concrete.
f8 5 average intensity of fiber stress in steel.
M = external moment at the section.
MD I positive moment of resistance exressed in concrete terms
Mn sznegatize moment of resistance expressed in concrete terms
As I area of steel
AP = area of positive tension steel
An 5 area of negative steel
v e'unit shear
R a least radius of gyration

H e axial column load

2 ._f .
‘p'_£2=. kada pageEu

1L

 

Min. (13/ __1-_'I page 2’4
Kb
AP .333. page 211,
f8 35.
v ‘3 page 36
b j d
R J E pa e 227
. .A

Design of Floor Slab
(pages 22-25)
Span between center of supports is 13'-u".

Designing slab as simply supported for positive moment and.using

2
I¥%_ for negative moment is on the safe side.

Assume 7“ slab

Live load 8 200 lbs. per sq. ft.

Dead u g 88 u u n u
Total '288 " u u ..

Using 2500 lb. concrete
fc I 1200
fs .L 18000 lbs. per sq. in.
n =.12
k = .h
,j -= 1- pg - .867

. 3 . 2
no a 3.33 - 288 x (13.33) = 76,800 in lbs.
‘ . 8 , 8 .

’p‘f kad2 =76,800inlbs.

mlo

d2 = __»Zb,SOC x 2 = 5.21
1200 x .857 x .h x 12

Minimum d = 2.29
Assuming 7/8" steel, 1—1/2" covering, 1/2" wearing surface on t0p
from Kichigan State Hiway Specifications.
h 8 d + clearances
e 2.29 + 1/2 + 1-1/2 + 1/2 = n.79
Hence 7" slab satisfactory

Commercial d = 7 - 2.5 8 6.5

Positive steel

Ab . i = 78,800 a , 6 . 1 .
‘ ring a ,18,000 x .867 x 6.5 75 Sq n

 

Use 2 — 3/M” round bars 6" on center

Plot 5, page #36 gives point of inflection for positive bending
moment at 10% of clear span or lhh x .l 8 1h".

Point of inflection for negative bending moment is at 21.5% of clear
span or 31". Hence steel is bent up between these points.

Design of Floor Bean
(pages 55-60)

A'beam with two equal overhangs, supporting a uniform distributed
load. The most favorable condition is when the bending moment at the
center equals the bending moment at the supports. f the distance between
the supports 8 d and the length of the beam = L, equal bending moments
are obtained at the center and supports When d = .5x6 L. In this de-
sign L = M5' giving d 8 .586xu5 = 25.u'.

d is taken as 27"the distance between the centers of the arch ribs.

Assume a 16" x 32" beam.

Live load 8 200 lbs. per sq. ft.

ON

Dead load = _§_§ lbs. per sq. ft.

Total floor load a 288 x 13.3 = 3830 lbs. per lin. ft.

Wt. of beam 8 15x32 x 150 =' :3” u u u u
. 1 ' .
Total uniform lad 3 36” lbs. per lin. ft.

4364 7 x 45': 146,300.

JIIHIIT [[UJll III,
9' 27' I 4'

48.6.50" Q8,650

 

 

 

 

 

 

‘

 

SHEAHL fl

 

 

 

 

 

 

'BEhflMLH§ l

moment W ~

4376.800 476,300 "

 

 

2500 lb. concrete

fC- = 1000

fS 8.18000

K a 185 plot page uju

Minimum d =‘/ 'g 8 J 226;000 x 12 ‘ 30-3"
. Kb , 185 x 16 .

 

Assume one TOW'Of l—l/u " bars.

Min. clearance to stirrup 2.00
Stirrup diameter .50
Distance to center of bars .63

Tbtal 3.13

Hence use 16"x3h" beam.

 

 

An 3 M = 226,000 x 12 = 5.7% sq. in.
* _f‘Dj d 18,000 x .867 x 30.3 -
6

Use h l-l/h" square bars.

An.‘L§E__ = 176.800 x 12 3.21 sq. in.

.fs 3 d I 18,000 x .857 x 30.3
Use M one inch square bars.

Shear

v = 59.350

v s ,_1_ = 59.330 a 130 lbs. per sq. in.
, bjd lox.807x32 .

Allowable v using web reinforcement without ordering anchorrage of
longitudinal steel 8 150 lbs. per sq. in.

Eence beam is satisfactory.

Design of Spandrel Column

(pages 223-228)

The longest unsupported column will be 18'.
Assume column l6"x18" x 18' long.

R =.f__""

 

= J 18 x (16)3 = n.63
12 x 18 x 16 .

PIH

Limiting lengfih - h = to R - 185".
Actual length a 18x12 8 216“ giving a long column with radius of

gyration of h.b3.

Column load = 98,650 lbs.
wt. of col. '= 5,000 " 001. 18x16 x 150x18

114$;
Total . n a 10u,050 lbs.

By article 306. page M01 for column with lateral ties, fc ‘ 553.

By article 1108 b

 

fe.A = 1.33 - ;L_
, 120 R

fc‘A

fe = 563 (1.33 — 18x12 ) a 563.
A 120 X1Lb3 ,

U = fe x A (1 x (n~1)p) ' fe x A (l x 11p)
10h,050 a 530 x 18 x 16 (1+llp)

p ='-029
As 8jp A 8 .029x288 = 8.35 sq. in.

Use 12-l"'round.bars and 3/8" ties 12" on center.

Design of the Arch Rib
Fbr references and derivations of formulae used see
"Bridge Engineering" Vbl. I by J. A. L. waddell, pages 783-
877. The diagrams noted are to be found within these pages.
Rotations
The Calculation of Stresses in Arch Ribs with Fixed.Ends.
L 8 length of span of arch,
r s rise of arch,
x,ly I co-ordinates of any point with reference to the
crown C as an origin, y being positive when measured down-
ward, and x being positive in each half when measured from
the crown toward the springing,
a 8 angle of inclination of the axis at any point, Fig. 37t
B = angle of inclination of the axis at the springing,
L i length of rib, measured along the axis ( = 2 ds),
b e width of rib at any point,
b 4- width of rib at crown,

0

b8 8 width of rib at springing.

h Bthickness of rib at any point, measured.normally to axis,
ho" thickness of rib at crown, measured normally to axis,
hS 5 thickeness of rib at springing, measured normally to
axis,

A = area of rib at crown, or area of concrete plus n times
area of steel,

A0 8 area of rib at crown, or area of concrete plus n times

area of steel,

10

I : moment of inertia or rib at any point, or moment of
inertia of concrete plus n times moment of inertia of steel,

Io = moment of inertia of rib at crown, or moment of

inertia of concrete plus n times moment of inertia of steel,

Fig. 17t

J

Iq 8 moment of inertia of rib at springing, or moment of

inertia of concrete plus n times moment of inertia of steel,
E = coefficient of elasticity of the concrete, to be
taken as 2,000,000 when dimensions are in inches, and as
288,000,0C0 when they are in feet,
w = coefficient of linear expansion of concrete ( = 0.000006),
t e change of temperature in degrees Fahrenheit, positive
when the temperature rises, negative when it falls,
P1’ P2, P3, etc., = any loads on the arch, acting in any direc-
tion,
Po 8 equivalent uniform load at crown,

pS 8 equivalent uniform load at springing,

naps,
1Do

I J-

p 8 equivalent uniform load at any point,

B V , Mo = thrust, shear, and moment at crown,

0’ O

H Ma 8 thrust and moment at crown from arch shortening,

a!
Ht’ Ht 3 thrust and moment at crown from temperature
change,

C = coefficient of temperature thurst, Fig. 37ff

t
yo 8 vertical distance from crown to plane of contraflexure
for arch shortening and temperature change, Fig. 37ff

T1 3 normal thrust at any section in left half of rib,

positive when compressive,

11

Tr 8 similar quantity for right half of rib,
T a Hormal thrust in general,

T '= normal thrust at the springing,

M 8 Actual beniing moment in general,

M In actual bending moment at springing,

on = moment coefficient Fig. 37ee

Equations

 

sec B = leT (hr)?

 

I”
h 3 ho +.§§ (Es - hg) sec a
L sec. D
u 3 p8
. 53

Thrust due to loading

. hSr

T 8 H see a
o

m '2 sec B

As Ho

Thrust due to arch shortening

Ia
Ha=-(1+o.3_:. Hoh2

Io "2"Q"B

r sec

Moment at or near a.spandre1 col. due to loading

Moments at other points due to loading

M = + DOLE
3000

Positive and negative live load moments (exclusive of
the effect of arch shortening) at the crown, springing

and various intermediate points.

12

M = 0m pL3
Moments due to arch shortening
Ma 3 -Ha yo
1.: a sa (y-yo)
MS 8 Ha (r- '0)

Moments due to temperature changes

i: = Ht (If-3'0)
MS a H.c (r-yo)
The following procedure is used to calculate the arch rib.
1. Assume values of ho, hS and calculate p0, p3. and.u,
for dead load.plus one half live load.

I
2. Compute Io’ 1; .2 and 3
Io L

3. Figure the dead and live load stresses at the crown and
springing and at one point in the haunch. Use Fig. 37ee

H. Compute the stresses for arch shortening at the same
sections. Use Fig. 37ff.

5. Compute the stresses from temperature changes at the
same section. Use Figs. 37 ff and 37 gg.

6. Test the various sections for the calculated stresses.

Calculations of Arch Rib

L e span a 120'

_3_'. 50 5 .hl6
L 120

13

 

sec B = (1 + (Pi)? = 1.91'r
L
Assume each rib 3‘-0" thick x MJ—h" wide at crown
6l_on n X hP—M" u n springing

hsuho) SEC 9.

 

h=h0 +_2_:5(

sec B

u 36 + £3 ( 2—76) sec a = 35 + .309 x sec a
120 1.9fi .

Po and pg for dead load + 1/2 live load
p0 for one rib

D. L. floor - paving, spandrel cols, etc.
6,lOO#/1in ft.

 

Arch rib 3'-0" x h'—u" x 150 lbs. 1,950 " "
Live Load - 60' span ~200#/sq ft including impact
2,250 II II
1/2 (L. L.) 200 x 22.5 x 1/2
Total 10,300#/lin ft.

pa for one rib

D. L. Floor - paving spandrel cols, etc. 7,275fi/lin. ft.
Arch rib 6'u0" x u'-M" x 150 lbs. X 1.0h 7,5“0

1/2 (L. L.) as for pO 2,250

17,065#/1in. ft.

u '25 - 11,065 = 1.66
.po , 10,300

. ,4
Assume reinforcement as 0.5a in eadh face throughout located

at a distance of‘E from the surface.
10

:. I0 = bh3 — From tables d . .106
e .106 x n.33 x 3 x 3 x'3 - 12.h

IS = .106 x u.;; x 6 x 6 x 6 a 99.25

a???”

1h

Stesses from dead load + 1/2 live load

Crown
9

I...

2
Ho ‘ Po1 (n+5) = 10,300 x 12 #

 

 

,
, 1*
o 9
Mo - p01“ = 10,300 x 130“ = 98,000
‘1500 ' 1500
Raunch——say at a spandrel col. located at a distance of
_JL from the crown
10 ,
#
T I: 58;; + l408,000
. 1#
M a + 98,000
Springing
#
Ts B M08,000 x 1.9M = + 792,000
\1 .. 2 ' ' ,1?
. 3000 . 3000
Live Load Moments
Crown
Mo 8 f Cmpl2 - 0m from tables at crown haunnh
.springing
1#
Mo = 3 .00u3 x M500 x‘1202 = 3 278,500
Haunch 1
9 a
MO . 3 .007 x M500 x‘IEU = z 1“511.000
Springing
___2 1?
M0 - f .023 x MSOO x 120 I 1 1,H90,000
Stresses from.Arch Shortening
Crown
H'-(1+.113)Hh2 2
a J in o o B a -1+(.3x8) 3 Ho
. ‘5‘“;
0 I Be . ‘ Egg; 1.9M
#

s -.00hu5 Ho - - 1820

15

IS = 8 from tables

‘Zg = .225 when __
I0

"3

y = .225 x r 8 .225 x 50 = 11.25

 

o . 1%
34a a -Ha yo = 1820 x 11.25 = + 20,050
Eaunch' .
7."
T 3 say - 1,800
H?
1: = Ha (or-yo) = —1820 (6-11.25) = + 10,!+70
Springing . #
g = Ha x coo = Ha 4 en E = -1820 4 1.9U = - 9h0
'4.
1:3 = Ha (r-y) = ~1820 (50-11.23) = - 70,500
Stresses for Temperature Changes
Crown 50° 28.11 30° Rise
Ht g Gt £13.32; Cc from tables
. r2
- -le.000006 x50 3, 288,000 x 12.1; - -12,850# +. 7,700“
5°" ' Ia _ .18
Mt = -Ht yo = 12,850 x 11.25 “1110.500 - 86,500
Haunch .
'zl.‘ at
T = say - 12,850 + 7,500
. 15 P!
2-: - 12.550 x 5.25 + 67.500 + 140.900
Springing
H # at
T3 ' coo .-.. - 12,850 x 1 a - 6,620 3,970
' . t . 131+ , '..# 3.3%
ms - Ht (r-yo) a - 12.850 (50.11.25) = 497,500 ~298,000

The Three Sections are now Tested as Follows:

Section at Crown

 

 

Temperature not Temperature
Thrust Considered Considered
Dead + 1/2 live + u08,000 +‘u08,000
Arch Shortening . 1,820 - 1,820
Temperature - 50° Fall ------ - 12,850
Total + h06,l80 + 393,330
Koment
Deed + 1/2 live + 98,000 + 98,000
Live Load - Positive + 278,500 + 278,500
Arch Short + 20,h50 + 20,u50
Temp. 50° Fall ------ T IMMI50O
1# 1%
Total + 396,950 + 5ui,u5o
«9 u#
0r h,760,000 6,500,000
e - 16m 4 Thrust = 11.72" 16.51"
Dimensions h x b a 36"x52" : 1872 sq. in.
g ~71;§ a 3.6 = 2.18
e .11.?2
R/fc I U 60. 00 a .118 00 000 _
52% 600 . 52x%§5€7‘80— $236
d’ = .6 . .l .l
h '%3’ ,
p per face Fig. 37g.
This is satisfactory.
Section in Haunch
Thrust
# #
Dead + l/2 Live + U08,000 + h08,000
# #
Arch Short - 1,800 - 1,800
Temp. 50° Fall ------ - 12,850
1 ' # , #
“30.909 393.390

17

 

 

 

 

Temperature not Temperature
Considered Considered
Moment
1% 1#
Dead + 1/2 Live + 98,000 + 98,000
Live Load Positive + h5u,000 + 05%,000
Arch Short + 10,070 + 10,n70
Temperature 50° Fall —————— + 6[,500
u# 1#
+ 562,H70 + 7,560,000
n# n#
or 6,750,000 7.560,000
e = Mom a 16.6" 19.25"
,.mnmt
Thickness Hbunch Section 8 h = hO + .309 x sec d
sec =: J 1,+(§¥§FE = 1.3
h-36+.309;_ x1.3=56+52 1+1"
. 10 .
Dimensions Ml x 52t a 2132 sq. in.
h . hl - ' = 2.u7 = 2.13
_. .7 __
e , 10.6 19.25
p per face .55 x‘%g 8 .h83fl .3839
l
R/fc Fig. 37g

fc ' ._§§__ = 6,£50,000 B
.bh- X .128 '52 X -1- x .128 60k

Section at Springing

Thrust
Dead + 1/2 Live + 792,000
Arch Short - 900
Temp. 50° Fall .....
Total + 791,050

2, 60,000

52 x .12 z .129
= 671

+ 792,000

- 900

- 6,620

+ 78M,hh0

18

Temperature not

 

Considered
Moment
Dead + 1/2 Live - h9,000
Live Load - negative —1,h90,000
Arch Short - 70,500
Temp. 50° Fall .....
'#
-l,609,500
"#
or 19,300,000
e = Mom 3 2h,h"

 

. Thrust

Dimensions hxb = 72x52 8 37th sq. in.
.2 a ' 2 ' = 2.95
e , '%E.h .
R/fc 8 19,300;000 = .1193
. 52i72 00 .

i=2 =-05
h , 72 .

p per face Fig. 37g .39
Steel for the arch rib
Crown
Dimensions hzb a 36" x 52" = 1872 sq. in.
9 per face = .55% '
Use 12 - 1“ sq. bars in each face

raking p = 1221 = .6Mfl
.1872

Eaunch
Dimensions hib = M1" x 52" = 2132 sq. in.

0 per face = .h83fi

J.

Temperature
Considered

 

- M9,000
-l,h90,000
- 70,500

- 298,000
I
-199079500 #

I
22,860,000

29.2'

= 2.h65

§§§§32999 . .1086
52x72 x780 .

_ ...._Cl».

._......——._4 A.) A

 

 

«b-

' J‘A'

 

 

19

Use 12 — 1" sq. bars in each face maaing

p 8 122:1 = .55;3
. 2132
Springing

Dimensions hxb = 72' x 52" = 37MM sq. in.

J
p per face = .35

Use 1° - 1" sq. bars in each face making

p = 12:21 = .3256
. 3753

 

 

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MICHIGAN STATE UNIVERSITY LIBRARIES

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193030126