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A
THESIS
SUBMITTED TO THE FACULTY OF
THE

MICHIGAN AGRICULTURAL COLLEGE

HAROLD KOOPMAN
A Candidate For The Degree Of
Batchelor Of Scieme.

JUNE 1922,
+
i a
\S -

u
0 |
ne
Foreword.

The writer undertook the analysis of the ecencrete
thru girder bridge in order to gain a mowledge of
the theory of design of this type of structure, being
already familiar with their construstion thru acting
as bridge inspector for the Michigan State Highway
Departaent in 1920 and {(92!.A description of this
type of bridge and its construction is given so that
the analysis will be clearer.

Asknowledgenent is made to Mr,.C.E.Melick and Mr.
Omans, Bridge Enginees’ and Assistant Bridge Engineer
respectively of the Michigan State Highway Dept., for
their assistance anf advice in this analysis.

May 1922. HK.

1021172
Outline.

A, Introdustien.

I. Description of standard sconorete $hru girder bridge,
II.Construction of standard concrete thru gimer bridge.
III.Methed of analysis of these bridges.

B. Body.
I.State Highway Dept. , specifications relating to the
design of these bridges.
1i.Computation for floor slab of the 35° bridge.
b. Computation for girder of the 3§' bridge.
c.Computation for the abutment of the 35° bridge.
d.Computation for floor slab of the 45° bridge.
&.Computation for girder of the 45’ bridge.
C,.Conelusion.
T.Actual stresses as compared to the theoretical, and
the economy of the designs.
Bibliography.

Conerete Mgineers' Handbook----Hool and Johnson,
Design of Highway Bridges------Ketchum.

Conerete and Reinforeed Concrete Construsction---Reid,
Lestures on Reinforced Concrete----Dunn,.

Mishigan State Highway Dept. , Specifications for Steel
and Comrete Highway Bridges.'920 Edition.
An Analysis of Two Concrete Thru Girder Bridges.

Description.

Figure No.! is a drawing te scale of a standard 35'
span soncrete thru girder bridge,designed and built by
the Michigan State Highway Department.A bridge of this
type was built in '920 ever the Pine River in Standish,
Michigan.Its description will apply as well te the
sesond bridge analyzed,a 45° span structure.It consists
ef two plain conorete abutments with battered faces,
two feet thick footings, and 45 degree wing-walls,ani a
superstructure consisting of two reinforced soncrete
girders supporting a reinforced floor slab.The girders
also act as railings en the bridge.The method of scon-
structing one of these bridges will now be given.

Construction.

The lines and grades are established and then the
exeavatiens are made for the abutaents.Test borings are
made te determine the quality of the supporting soil.
If 1% is found necessary,piles are ordered driven
aceerding to the plan.Then the footings are poured to
the required elevation, with an extra 6" of concrete if
piles are used —- the extra concrete being below the
pile heads.This conerete is plain and of a ':2 1/2:5
mix with the aggregate passing the Highway Dept., tests
as to the proportiens of the various sizes of stone.
After the footings have seasoned for several days the
abutaent foras are built on thea and wired and braced.
The abutments are then poured,using the sane mix as in
the foetings,and pouring each one continuously. After
the abutments have set for three days,more or less,
according to the weather, the forms are renoved ani the
surface is rubbed down with a carboruniua brick and any
holes are filled with a grout consisting of one part
cement and two parts sand,

The falsework for the support ef the floer and girders
is next constructed.Rows of eight piles are driven at
specified intervals between the abutaents and at right
angles to the read.These are cut off just abeve the ground
ani heavy timbers are laid on them.Short timbers are set
vertically on the horizontal ones,one over each pile head,
and another long timber is laid horizontally across,Floor
joists are laid en this falsework and the floering is
nailed te them.Wedges are used for leveling and the floer
is given a slight canber,both for appearance and to pro-
vide for slight settlement of the superstructure.The out-
side girder forns and panels are now erected, nailed, and
braced in place. The steel reinforcing for the floer and
girders is placed and wirédjthen the inside girder fora
is erected ani wired to the cutside form, with spreaders
te held the proper dimendions.¥Formwork is placed at each
end ef the bridge floer,and 1/4" of tarred felt is placed
over each abutment to provide for expansion and contract-
4on ef the conocrete.After a thoro inspection one half of
the floer and one girder is poured one day and the other
floer half and girder is poured on the succeeding day.
The floer construction joint is in the center of the slab
and parallel to the direction of traffic.The concrete mix
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

SG

Cross-Section.

Fig.2.

NotesThese figures apply to both the 35' and 45° spans.
is 1:23:34 with tested maderials.The girder forms are re-
moved in two or three days,and the surface is finished
with a carborundua brick and grout,if necessary,as was
the abutment.Sonetimes a coat of tar is placed on the
floer after the consrete is dry.Traffic is allowed on the
bridge after ten days, but the falsework and floor form
is not removed for a month or six weeks.

The method of analysis of these structures will now
be given.

Method of Analysis.

The floor slab was first taken and analyzed as a re-
inforeed concrete beam with a concentrated live load of
as many *8 ton trusks as could be placed on it.This is
the loading used in the 1920 specifications of the High -
way Dept.,and it gives a load on any cross-section of the
bridge of a maximum of 24 tons, spaced as show in fig.6
and spread over the slab in the direction of traffic
according to specifications.The tetal moment was next
Gonputed,and standard formulas for the design of re-
inforced concrete members were used to determine the
stresses and the amount of steel required.The actual
stresses were compared to the allowable and deduations
drawn as te the strength and economy of the design. The
web reinforcement for @iagenal tension near the ends of
the slab was computed next.This was compared with the
actual steel area.

Then the girder was analyzed as a simply supported
beam with the same truck loading as before.The total
monent was found,anid the azount ef steel ef actual use
in the girder was compared to the theoretical anount
required.The web reinforcement for diagonal tensien was
investigated, both the vertical stirrups am the inclined
bars,and the actual amount of steel used was compared to
the theoretical amount required.As before, dedus ti ons
were drawn as te the strength and economy of the design.

The abutments were casputed with Rankine's foruwlas
using a level sureharge.All stresses were computed as
for retaining walls.Drainage and scour were also in-
vestigated.

All measurenents were taken from blueprints of the
the Highway Dept.
Michigan State Highaay Dept.Specifications
Used In The Analysis.
1920 Edition.

Sect.200 Reinforced Webs.Fer reinforced webs five-sixths
ef the average vertical shear shall be considered as
being taken by the stirrups,diagonal web pieces, and bent-
Wp bars.

Seet. 201 Vertical Web Reinforcement.Shall be loaped
around the horizontal reinforcenent.

Seot.202 Inclined Web Reinforcement. Shall be securely
attached to the longitudinal reds to prevent slipping.
Sect.243 Stress in Reinforcing Steel.The tensile stress
shall not exceed 16,000 lbs.per sq.in.

Sect, 244 Modulus of Elasticity of Concrete, Shall be
taken as 1/15 that ef steel when the strength ef the
concrete is taken as 2,200 lbs.or less,per sq.in.

Sect, 252 LiveLoad. Whether concentrated or uniforn, shall
be considered as moving and shall be placed in the
position which will give the maximum stresses for each
menber considered,

Seet.253 The loading for the floor system and its
supports shall consist of 18 ton trucks, concentrated as
shown in sketeh,and distributed as described in 258.
8eet.258 Distribution of Live Loads.

&.Floor slabs with main reinforcement transverse to the
@irection ef traffic.One or more rear truck axles,as the
slab span may permit, shall be placed in such a pesition
as to give the maximum moment in the slab, considered as
& simple span.An equivalent soncentrated load at the
center of the span producing the sane maxinum moment
shall then be saloculated,This equivalent lead shall

then be considered as concentrated transverse to the
direction ef traffic, but as distributed uniformly in

the direction of traffic over a length equal te one

and one-half feet plus sixty percent (60%) ef the span.
The resulting equivalent load (concentrated) per inch
width of slab shall be used in calculating the live load
bending moment,and shall alse be ¢onsidered as neving
load which shall be placed so as to produce the maxima
live load shear,

Seet.265 Impast.An impact lead of 25% of the specified
concentrated loading shall be added to said cencentrated
loads for the design ef all slabs, stringers, floor-beans,
and hangers.
i, =
i, =
E, =
Eg =
n =
M =

i”
i

ew oo £¢$s§ g¢ Sd CBenone wR YD mS
u

Dele
Toe leo

Ww 8

Standard Notatien.

tensile unit stress in steel.
Gcoaupressive unit stress in sonocrete.
modulus of elasticity of steel,

_ * " * conerete.
Eg/Eo = 15 in this analysis.
bending moment.
steel area.
breadth of bean,
depth of beam to center of steel.
total depth of bean,

ratio of depth of neutral axis to depth d.

M/ba®,

depth below tep to resultant of the sonpressive stresses.
ratio of the lever arm ef resisting couple to depth d.

steel ratio = A,/bd.

total shear,

shearing unit stress,

bond stress per unit area of bar.
circumference of bar.

horizontal spacing of reinfercing nenbers.

zs dead load.
= live load.
weight.

1 # length.
Steel Reinforcing. (Cross-Section)
Fig.3.

Scale 1" = 2',

 

 

 

 

 

 

 

 

 

 

(Elevation)
Fig.4.

Scale 1" = 4',
Analysis of a 35° Concrete Thru Girder Bridge.
5 claps b span---clear span plus depth of beam = 20' plus 1.2'

Loadling---rear axle loads of two 16 ton trucks side bg
side = 12,000 lbs. plus 25% for impast = 15,000 lbs. for
each wheel. |

Di stzxibution---over 1.5' plus 60% of 21.2' = 14,2a2'
15,000/14,.2@ = 1,054 lbs.per sq.ft.

150 lbs. = wt. of one cu.ft. concrete.

wom 1.46'x1'x150¢ = alo#

Mom wi?/6.m alo x 21.282/8 = 12,300 ft.1bs.

Inia, = 2 x 1,054 x 8,6" = 18,130 ft.1bs.
~ 1,054 x5 = «5,270 ft.ibs.

— an aie a aes —

Fetal M sz 25, 160 ft. lbs,
Ds 1735" dm 15"

 

160 x 12
Km t/pa? = 259 geen = 111.7
12 x 15
A, z= 1.41 sq.in. actual.
p & A,/ba = .00787

 

 

 

© = Jepn plus (pn)® -- pn = ° 988

f, = 50,520 x 12 = 678 lbs.
 382x. 86321 2x15°

{, = a5, 160 x \2 . = 16,450 lbs.
1.41x.863x15 |

Allowable f, is 650 lbs. and f, is 16,000 lbs. so the
slab is seen te be slightly overstressed,
Alternate loading is 125 lbs. per sq.ft.Using the sane
formula for M as before and w= 125 plus 219 lbs.= 344 lbs.
we have M = 344 x 21.27/48 = 19,340 ft. 1bs.This is maller
than the other M so is net used,

Web Reinfercenent.
Inslined bars te take diagonal tension.
A, = 1.14 sq.in.

2 zy Astsi4@ 2 x ae = 17,500¢
s

 

A, = Z x oe a2x gTxIT, 500z8t = 1,027 sq.in. needed,
e834 = 96, 000, 87x15

We have '.14 sq.in. and need 1.027 sq.in.so is safe.

Region where no web reinforoeent is needed:

X, 4s distance from slab end.

ay = 1/2 - vibjd/w = 21.2/2 — 40x12x.67215/344= T.6'

The actual distame is 1.6! So that is correctly designed.

Bond stress.
u ® ¥/oJd = 17,500/14x.87x15 = 21.99

The allowable is 80# so that toe is safe.

Hotes j = 1 -— 1/3 kin all computations.
Girder ef 35° Bridge.
Span---33.3° center to center of supports.
Average b = 235" er 1,9'
D = 5'7 1/2" or 5.625!
d, = depth to reinf.steel; = 4,96!
d= " * " steels = 5.30!
Average d = 5.13' or 61.6"
ws 1.9' x 5.625! x 1504 = 1,603 lbs.
Eash girder carries 1/2 ef the fldcr slab and 1/2 of the
live load.
M = wi?/8 = 1,603 x 33.3°/8 = 222,000 ft.1bs,
LeleMe = 33.3 x 12,580 = 419,000 ft.1bs.
Total M = 641,000 ft. lbs,
- 649,000 x 12 = 7,670,000 ineh lbs,
7,670,000/107.4 = 71,400 = ba®
a? = 1,6 x 71,400 = 114,200 and a = 48.5" or 4,041!
We have 5.13" and need 4,041' so this actual depth is seen
to be very unecononical.
A, = 8 x 1.5625 sa.in. = 12.5 sq.in, actual steel.

 

 

 

 

p = Ag/pa m __ 12.5 = ,00883
23 x 61.6
x =/30x, 0088- ( 15x.0088) @ — 15x.0088 = .400
f, = 2x re = 506 lbs,
04x .87x23%61.6
f, = 12870, 000 = 11,450 lbs,

— 12,.5x.87x61.6

The allowable f, is 650 lbs. and f, 18 16,000 lbs. so it
can be seen that the design is uneconomical.
14

Steel actually needed for f£, = 650,f, = 16,000,b =23",

and d = 61,6" is A, = phd = ,0077x23%5 1.6 = 10.91 sq. in.

There are 12.5 sq.in. so there is enough steel area.
Web Reinforcenent.

Aatual shearing stress fer inolined bars is:

 

 

y = 2 x 4-69x11,450x.87x61 .6 = 360,000 lbs.
12
A, = 2 x 275360 ,000x12 — = 2.94 eq.in.
16 ,000x, 87x6 1.6

There are 4.69 sq.in. and we need 2.94 sd.in. soe this
part of the web reinforcing is safe but wmeconomical,
Actual shearing stress for vertical stirrups is:

2

Ay = 2 x 42,800x12 __ = .582 #d.in.
16, 000x. 87x6 1.6

There is .65 sd.in. ami we need .582 sq.in. so it is safe.

The tensien reinforcement bars, vertical stirrups, and in-

elined bars in the girders all have hooked enis, thus pre-

venting slippage and increasing boni.All steel is thoroly

wired besides.All this is good practise.

 
/
YUL MMM MMMM >

WNL

 

 

 

Surcharge

 

 

 

 

 

 

 

Abutment.

o5-e
12
Abutment of 35' Bridge.
@ = angle of repose of filling.
h = vertical height of wall in feet.
h’ = vertical height of surcharge in feet.
w= weight of filling per cu.ft.
w= °* “ masonry per ft.of length.
W,= * " earth wedge one ft. long.
W= " " Woplus ¥,,
P = resultant earth pressure per ft.length of wall.
y = vertical distance from base of wall to point where
the resultant strikes the wall.
The abutment must be safe against overturning, sliding,
and crushing the masonry or foundatien.
Rankine's formula was used for & wall with a loaded
surcharge.
g= 1 1/2 to 1 or 33 42!
waz 110 lbs,
h = 13°,

hn’ = 1.84 from h' = 2 = 72000

ww | 25x110x14,22

P = 1/2 wh(h plus 2h')x1-sino/t-sine

P = ,143x110x13(13-3,68) = 35,410 lbs.
= n@ Plus 3n'h

3(h plus 2h‘)

W= 4,280 lbs. from 28.6 x t x 1507

W, = 2,402 lbs, from 21.834 x 1 x 110#

Wo = 6,682 lbs.

From fig.5 the resultant comes within the middle third

ef the foeting.This is where it should come. the con-

struction lines of this figure are not shown.

= 4,82"
13
The coefficient of masonry on sand for frietion is .4
The safety factor is derived from the formula:
{f, = F/P x .4
P = 3,410 lbs.
F = 15,262 lbs. from 1/2 x 66 cu.yds.x 27 cu.ft.x 1507
= 133,500 lbs.load on one abutment. (D.L.)Plus 36,000# for
one truck as live lead = 169,500 lbs.total lead. This equals
6,780 lbs.per one ft.of width.
6 , 780+6 ,682+1, 800 for footing = 15,262 lbs.= F,
£, = 15,262/3,410 x .4 = 1.8
The wall is safe against sliding.
Fer stability against crushing the masonry er foundation:
15,262-1,800 = 13,462 lbs.
a = width of base of wall.
b= * " "f * foeeting from center of footing to
point where the resultant strikes the base.
d = 4.5* and b = 1,4?
py = B/d = 135,462/4.5 = 3,000,
Po = +6Fb/d° = +5,630 lbs.
yp = 5,630-3,000 = 8,630 lbs.per sa.ft.
The allowable is from 4 to 6 tons per sq.ft. so this
is safe.
For maximum pressure at toe of walls
Pp, = (41-6a)¥/12 where 1 = width of base = 6" and a =
6/2 - 1.5' = 1,5° and F = 13,462 lbs.
P, = 5,615 lbs.
The allowable is from 4 to 6 tons so the actual p, is
well within the allowable.
 

 

Fig. 6.
Loading Diagran,.

 

 

 

 

 

 

 

 

 

 

 

 

 

yt and >
z=’ tl. fo’ ol. Zs" gt

f Tv Cc [rye
[ —- L_ TN

2 g

ry tl

N 9
t LN: C I Y

 

 

 

 

 

Space and weight distribution of 16 ton truck,

7

45900
435, O90 *
1F IAGO wt
4S 000 *

 

 

 

 

 

 

 

 

 

 

JF’ 5’ gf’ | _ IS 2’
RO’

 

 

 

 

 

Loading for Maxinum Moment.

 

 

 
14
Analysis ef a 45' Concrete Thru Girder Bridge.

Span---clear slab span plus depth of beam = 18' plus .9° —
= 18.9'

Loading---rear axle loads of two 16 ton trucks side by
side = 12,000 lbs. plus 25% for impast = 15,000 lbs. for
each wheel. |
Distribution---ever 1.5' plus 60% of 18.9' = 12.84°

wt. of one cu.ft. ef concrete = 150 lbs.

15,000/12.84 = 1,168 lbs.per sq.ft.

wm i,t25* x 1" x 1507 m 168,75 lbs.

M = wl?/6 = 168,75 x 18,92/8 = 7,530 ft.lbs.

Lele M, = 2% 9,168 x 7.45* = 17,400 ft.1lbs.
- 1,168 x5' «= -5,840 ft.1bs,

Total M = 19,090 ft. ibs.
Ds 13,5" a= 11,"

19,090 x 12
4 M/pa? — 7 °
x - 12 x 11 = 15767

 

Ag = 1.333 aq.in.actual,
p ™ A,/bd = .010! ns 45,

 

« =/oon plus (pn)? ~ pn = .480

- 19,090x2x'2_

 

 

f, = 874 lbs.
» 4ax,66xtaxi2 t
19 ,090x12

ty 8 eat = 18,180 lbs.

As the allowable f, is 650 and f, is 16,000 lbs., the floor
slab is seen to be considerably everstressed,
15

Alternate loading is 125 lbs. per sq.ft.

w= 168.75 plus 125 = 295.75 lbs.

Mom wl?/8 = 293.75 x 18.92/8 = 13,130 ft.1bs.

This is less than the naxinwm mement used so is not used.
Web Reinforcement.

Fer inclined bars:

Ag = .556 sd.in.

 

 

 

 

v= Z X 556x188, 160x. 86x11 z= $1,940 lbs,
| 12 . . 8
A, s Z x o Tait, 40x12 . = ,546 #q.in.
16, 000x, 87x11

There is 556 sqein. and we need .546 sd.in. of steel
so itis safe.
Regien where no web reinforcement is needed:

x, mh. YAP] 2. 18,9/2 - 40xlax.67x11/294 = 5.19!
1 "20 ow

distansve from end.
Actual distance is 5.4' seo is safe.
16

Gisder of 45° Bridge.

Boan---42' center te center of supports.
krerage b = 24.8" or 2,07°
d, = depth to reinf,steel, = 4.96
a= * "  *  steelg = 5.30'
Average 4 3 5,%3' or 61.5"
D = 5,695" or 67.5"
w= 2,07’ x 5.625" x 150# = 1,748 lbs.
Rach girder carries 1/2 of the floor slab and 1/2 of the
live lead.
Ms wi2/s = 1,748 x 427/8 = 385,000 ft.1bs.
Lela Me @ 42 x 19,090/2 = 401,000 f%.lbs.

Total x = 786,000 ft.1bs,
12 x 786,000 ft.1bs. = 9,430,000 inch lbs.
9,430,000/107.4 = 87,800 = ba®
a? = 1.6 x 87,800 = 140,500 and d = 52" or 4,333!
There is 5.13" and we need 4,333' so the design is safe
but uneconomical.
A, = 10 x (1.25")? = 15.625 sq.in.of actual steel area,

p= Ag/bd = 15.645 = ,010385
24,8x6%.5
n= 5 as before. |

 

k */om plus (pn)’ —- pn = 425

2 x 9,450,000

= : = 552 lbs.
C  6423x. 86X24, 8x6 1.5

= 9,450,000 . = 11,450 Lda.
®  $5,.625x.66x61.5

As the allowable fg is 650 ami fg is 16,000 lbs., the design

4s seen to be safe but wuneconcaical.
17

The steel area actually required for f, = 650 and f,

= 16,000 48 Ag = pbd = .0077 x 24.8 x 61.5 = 11.75 sa.in.
There are 15.625 sqd.in.ef steel and we need 11.75 sq.in.
so this teo is aafe,

. Web Reinforcement.

For inclined bars:

A, = '5.625/2 = 7.813 sq.in.

v =2 x 7.81311, 450x.86x61,5/12 = 591,000 lbs.

5 o 7X59 1, 000x122
A
0 ” © * 46, 000x.87x61.5

fheve are 7.813 sq.in. and we need 4,64 sq.in. So this
48 all rvight.

Vertical stirrups:

A, = pbd = .61 sq.in. actual stee] arena.

= 4, 84 BQ. in.

vs = x ,61x11,450x.86x61.5/12 = 46,250 lbs.

A, = 2 x 46,250 x 12 = .541 sq.in.

There is .6% 8qa.in. and we need .541 sq.in. so this part
of the design is safe,The ents of the steel are heoked
end all the steel is well wired,This is goed practise.
18
Gonclusion,.

In suamarizing the results ef the analysis of these
two bridges the reader's notice is called to the fact
that these bridges were designed in 1915 under no
printed speci fications, and with a smaller loading than
now used. These bridges were built in 1920 and 1921, how-
ever,so the 1920 specificatiens were used in the analysis.

In the 35° bridge the floer slab is found to be So
slightly overstressed that the anount is negligible, while
the web reinforcement is within the allowabls.The girder
stresses are so far winder the allowable that the design
is uneconomical in all details.The abutment is safe ani
of economical design.

The floor slab of the 45° bridge is considerably over-
stressed and shows poor design in the light of present day
knowledge.It is safe as the safety factor is between 35
and 4,and the maximum loading will probably never come
upen it.The web reinforcement is within the allowable.

The stresses in the girders are so far under the allowable
that their design,like the girders of the 35' bridge,are
uneconomical,The girder web reinforcement is all right.
The floers of both bridges are well drained by weep holes,

The abutments of both bridges are safe and protected
from scour,but are insufficiently drained. Several 3" or 4"
pipes or tile should be used to drain each abutment instead
of the 3/4" or 1" pipes used,

The latest designs of the State Highway Dept., show
improvenent in all respects over the above standards.
uf

RUUIA USE Ur.
“=,
NOOR
~ o.

a2 86—LISRARY

 
_ we