l .
‘- .~‘.. b I.
II. -
tr... - - - “I
.. C O . I.‘|.O|l
. OOI 1 Or ‘
v. ....v.u.
H w I. . - ‘ ‘ - ‘ul :v.‘.(, n o".t
)n‘v-|,4tw..\.-.Vot¢‘.lkfll'u ltmiru.ll~.

.- ‘v. .IH O O u--. IVI -.¢.I n“ . . . '. n . _ . .‘
§§3.EQ§IN‘I.%§1.1Q.I 12349... .0. a u to \ . I72...) . {0.1! \ -.

.
\ . . o . . .
Uul a-.. .. o L! . . l-l. ‘ -.r 3 a.-. ‘.ro. .‘

 

| Ir

|l~

.’

I‘d

 

An Analysis of a Continuous

Beam Bridge

 

A Thesis Submitted to

The Faculty of
MICHIGAN STATE COLLEGE
of
AGRICULTURE AND APPLIED SCIENCE

by

M. L. Deimling

Candidate for the Degree of

Bachelor of Science

June 1949

ACKNOWLEDGEMENT

I wish to acknowledge my appreciation for the
assistance given to me by Mr. Takashi Nakamuraawithout

which this Thesis might never have been written.

$318981

INDEX

I Abbreviations & Symbols used in Thesis

II Allowable Stresses

III Introduction

IV Design of Railing
A Specifications
B Bolts
C Straps
D Railing Channels
E Railing Posts

V Design of Sidewalk & Curbs
A Specifications
B Loading
C Reinforcing Steel

VI Design of Floor Slab
A;Specifications
B Loading
C Reinforcing Steel

VII Design of Diaphragm
A Specifications
B‘Design
VIII Design of Girder

A Specifications E Maximum Moments
B Moment Distribution F Design of Girder
C Reactions
D Influence Diagrams

IX Conclusions

N105 OH CD 0

I. ABBREVIATIONS AND SYEBOLS USED IN THESIS

- Michigan State Highway Department

A.A.S.H.0.- American Association of State Highway Officials

# ..
sq.“
inc-

ft.-

fJ -
f0 -

f5 -

pounds

square

inch

foot

area

Moment of Inertia,
Section.Modulus
EXtreme fiber distance
Base dimension

Height

II. ALLOWABLE STRESSES
3,000 #/sq. in.

.4.£g #/sq. in.
18,000 #/sq. in.

Loading - H20-516-44

p.s.
p.s.
u

v

p.
Art.
Lft.

W‘ L. b
I

R -

i. -
f.

10
.867
.4
209

#/sq . in .
$®q.ft.

bond stress
unit shear
page
article
lineal feet
summation
less than

per

v- 60 #/sq. in.; u- 150 #/sq. in. with no special anchorage

v- 90 #/sq. in.; u-300 #/sq. in. with special anchorage

Shear- 13,500 p.s.i. for power driven rivets

Shear- 10,000 p.s.i. for unfinished bolts

Piles- 20 tons/sq. ft. supported.

III. INTRODUCTION

This thesis was primarily written to increase my own
knowledge, to crystalize half formed ideas, to better
understand the work of the designer, and most of all, to
integrate many of our individual courses such as reinforced
concrete, indeterminate structures, and contracts. Upon
completion of this thesis I understood much better how these
courses related to and depended upon one another.

Secondarily, this thesis was written to check the design
of the superstructure of the Michigan State Highway Depart-
ment bridge B1 of 32-23-13 on which I was an inspector last
summer.

The bridge is a 3—span continuous I-beam bridge having
two end spans of 42 feet each and a center span of 57 feet,
giving a total length of 141 feet. This bridge is located
in Sebewaing, Michigan over the Sebewaing River on highway 51.
The abutments and paers are supported by 12" H 74# bearing
piles. The clear distance of the roadway is 38'-0" curb to
curb.

Continuous girder bridges are best proportioned when
the interior spans are from 1.3 to 1.4 times the length of
the end spans. The interior span of this bridge is 1.36
times the end spans and therefore is of good design in this
respect.

During the spring thaws, ice had been piling and
Jamming on the old bridge causing a hazard to the structure

by backing up water like a dam thereby causing undue pressure.

The piers of this bridge were designed as ice breakers by
slanting the upstream ends about 6 feet at water level and
protecting this slanted pier nose with 6"x6"x%" steel angles.
It was thus hoped that the ice flows would break up on these
and flow under the bridge without stress or strain to the
bridge.

All the above mentioned conditions were taken into

consideration both in the analysis and design of the bridge.

IV. DESIGN OF THE RAILINGS

Roadway railings shall be designed to resist a lateral
horizontal force of 150#/lineal ft. together with a simul-
taneous vertical force of 100#/lineal ft. applied at the
top of the railing. When curbs are 10" or less in height,
the lower rails shall be designed to resist a lateral hori-
zontal force of 300#/lineal ft.

M.S.H.D. Spec. Art.35 p.14
RAILINGS:
(The bottom rail carries the maximum load and will be
investigated first; if it proves satisfactory, the
top rail will be also).

1. Bolts 3/4"? in single shear.
Capacity=(3.l4)(3/8)2(10,000)=4,420#
Load =(300)(7.874)42 =1,310#
Factor of Safety: 4,420;1,310=338%

2. Straps:
Shear Capacityfl1 3/4 ' 13/16)(5/8)(10,000)=5,860#
Load: (Using 50#/Lft. as the dead weight of the

railing).
L4 300%

50#l:/
304%

Load=(304)(7.874)=1.330#
Factor of Safety=586041330=4.41

Bolts control as they have the minimum capacity.

3. Bending Moment in Strap:

Distance of strap load application from post is 2"
f =Mo/I = M/z (where z = bh2/6 )

f = 1 10 2 6 = 8,200 p08 01.
h°r° (Lg25)(1.)§)2

f _ P 110 2 6 :3830p.s.i.
vert. 1.75 . 25 ’

ftotal = 12,030 p.s.i. actual

fallow.= 18,000 p.s.i. allowable 0.K.

Safety factor 3 1.5

Bending Moment is controlling factor.

4. Railing Channels:

0.44"
-—————+>-

l - (Only the two side

 

 

 

 

 

 

 

 

 

 

 

 

{ I B. channels will be used
nx i
“’ I ‘1 g and if they give enough
' . stren th it won't be
“S r:"""‘ l 8 ’
. 2.26" 5520:?“ __1 N 0.44 necessary to add the
. top channel).

Railing Channels

a. Length=(7.874) + (2x 1 5/8) = 7.60 ft.
12
b. M = w12 = (300)(1.60)2(12) 2 26,100 in.1bs.
—S_ 8

c. A: (2)(1.46) = 2.92 sq. in.
c = 2.5 e = 2.06" I8 = 2x.25 = 0.50"

Ae2 - I8 = 12.89 1n.3

I (l2.é9)

18,000 p.s.i. allowable

Q) 'p-I
. .

"5 H
II N

Factor of safety 3 356% (without top channel).

5 oifiafliae Peale;

 

Ia"

 

I

._I

71‘—
H—

Li

 

 

 

 

 

L~——L—a4

 

 

 

 

 

 

 

 

 

 

Bending Moments:

00

f

a. 150(7.874‘+ 1.333) 32
300(z.874 + 1.3331i5
Total B.M.

o = 16"; a: 9%"

AS 4>x 3/4 "¢ bars

4 2 3.88 sq.in.

1.76

nfc : 0.40

f8 + nfc
(1 - k/3 )=
a<rcka)- 208

0.867

.H c...
II II

44,220 1n0-1b80
13,830 in.-1bs.
58,050 ino-leo

 

8 0 0 u 4.2" required
3 §(1200;(.Z)(.S67)(16)

: M
S 333 '

7 9.5

9.5" furnished 0.K.
: 8,020 p.801. antual.

18,000 p.s.i. allowed
O.K.

The large safety factor (45% ) is desireable for
safety in case of accidents. Another reason for the
posts being overdesigned is for aesthetic reasons.
The massive architectural design lends beauty to the
structure. The modern bridge is certainly an object

of beauty.

V. CURB AND SIDE WALK

Substantial curbs shall be built on each side of the
roadway and they shall have a width not less than 6" and
a height of not less than 9" measured above the wearing
surface at a point adjacent to the curb.
M.S.H.D. Spec. Art. 20, p. 8
Curbs shall be designed to resist a minimum force of
not less than 500 #/Lft. of curb applied at top of curb.
M.S.H.D. Spec. Art. 36, pi 14
Sidewalk floors, stringers and their immediate supports
shall be designed for a live load of not less than 100 pounds
per square foot of sidewalk area.
"$F_“ M.S.H.D. Spec. Art. 34, p.14

Pa 31'

(r wam
”““‘”1T‘~ve1’ T“! 7 i

l jiab 7 V 7 2:5-

tproaa-Section-of031abandofiidszelkei

 

 

 

 

 

(D

The sidewalk and curb are fully supported and there-
fore could fail only by crushing. But its strength by
crushing is more than adequate. Therefore temperature
steel is the only steel necessary. The hooked structural

steel bars serve to keep the curb in place.

VI. FLOOR SLAB

Bending moments shall be calculated by the following
methods. A.A.S.H.O. Spec. Art. 3.32, pal38-l40

In this bridge the main reinforcing steel is perpen-
dicular to the center line of roadway.
Distribution of wheel loads:

E I 0.68 + 2.5
Bending moment for freely supported spans:

M = 0.25 x P/E x S(lOO + I + 10% for longitudinal

forces.)

Bending moment for continuous spans::

M = 0.2 x P/E x 5(100 + I + 10%)

In the above formulas, the symbols used are:

E 8 Width of slab over which.whee1 load is distributed.

I = Impact coefficient as determined by the formula:

I . L + 20 a M.S.H.D. Spec. Art. 37. P.14
‘65‘3‘55‘

L = Span length

M-= Bending moment

P I Load on one wheel (maximum)

S 8 Distance between flanges of girders plus one-half

width of girder flanges. (s - 4.397')

The forces due to traction or sudden braking of vehicles
shall be considered as longitudinal forces having a magnitude
of 10% of the gross live load that can be placed in one

traffic lane. This load shall be assumed as acting in the

direction of traffic movement and applied at the top of

the pavement.

M.S.H.D. Spec. Art. 38, p.15

Design of Slab:
a. E = (.6 x 4.397) + 2.5 = 5.14' (P = 16,000, 8 = 4.397)

b. 11: 42 + 20 5 a 22.8% 12- 51 + 20 . 21.3%
(6Ix—42) + 20 x 5 + 2

c. M = (0.2)(16,000/5.l4)(4.397)(l.328) 3 43,600 in.-1bs.
d. d q/m/Rb =¢43,560/208 x 12 = 4.2 in.
Specifications for slabs: the distance from the

 

 

surface of the concrete either top or bottom, to the
center of the nearest bar shall be not less than one
and one-half times the diameter of the bar nor less
than one and one-half inches.

Required thickness:

1.1 1.5 therefore use 1.5"

t = 1.5 x 3/4

t = 4.2 + 1.5 5.7" required; 7.0" furnished. 0.K.

therefore d t 7.0 4‘1-5,= 5-5"

9. AS= M/Tsjd 43,600/18,000 x 0.875 x 5.5

= 0.504 sq. in. required; 0.612 sq.in. furnished.

f. Steel at bottom of slab for lateral distribution:
Per cent of main steel required 3 100/ St
= 100/ 4.397 = 47.5%
As = .504 x .475 8 .24 sq.in./ft. required. 0.K.
A.A.S.H.0. Spec. Art. 3.2.2, p.140

g. Bond and Shear:

Slabs designed for bending moment in accordance
with the foregoing shall be cnsidered satisfactory in
bond and shear.

A.A.S.H.0. Spec. Art. 3.2.2, p.140

VII. DIAPHRAGM DESIGN

Specifications:

The forces due to wind and lateral vibrations shall
consist of a horizontal moving load equal to 30 pounds
per square foot on 1% times the area of the structure as
seen in elevation, including the floor system and railing
and on 1/2 the area of all trusses and girders in excess
of 2 in the span.

M.S.H.D. Spec. Art.38, p.15
Rivets:

a- Size: Rivets shall be of the size specified but
generally shall be 3/4 or 7/8 inch diameter.

b. Pitch of Rivets: The minimum distance between
centers of rivets shall be three times the diameter
of the rivet but preferably shall be not less than
the following:

For 3/4 " diameter rivets --- 2% "

M.S.H.D. Spec. Art.92, p.39-40
Diaphragms shall be provided at the third points of all
I-beam spans of forty or more feet.

M.S.H.D. Spec. Art.l24, p.48
Design: Lateral, longitudinal, and transverse bracing
shall be composed of angles or other shapes and shall have
riveted connections.

M.S.H.D. Spec. Art.l23, p.48

VII. DIAPHRAGM DESIGN (Con't.)

Area of structure as seen in elevation:

Beam 2.5 x 137 = 342
Railings 1.5 x 137 = 205
Sidewalk 1.0 x 137 = 137

Int. Posts 14(1.3 x 3.7) . 67

Ext. Posts 2(4.0 x 8.3) = 66

Piers 2(2.5 x 9.0) = 45

Abut. 2(4.15 x 11.2)’ =__93_

Total Area. = 955 sq.ft.

Total Effective Area = 1.5 x 955 = 1433 sq.ft.

Wind Force = 30 p.s.f. x 1433 42,990 #

204 sq .111.

Area required = 42,990/18,000
Area furnished by 4" x 4" x 3/8 " angles:
A 8 502 sq.in. 00K.

The intermediate diaphragms meet all the necessary
specifications provided by the M.S.H.D. for depth of
web, size of angles, pitch of rivets, depth of hitch-

angles, and number of stiffeners.

VIII. GIRDER DESIGN

Specifications:

Main trusses and girders shall be spaced a sufficient
distance apart center to center to be secure against overé
turning by the assumed lateral and other forces.

M.S.H.D. Spec. Art.76, p.36

For the calculation of stresses, span lengths shall
be assumed as follows: Beams and girders --- distance
between centers of bearings.

M.S.H.D. Spec. Art.77, p.36

For structures with concrete slab floors without sep-
arate wearing surface, a minimum allowance of 20 p.s.f.
of roadway shall be made in addition to the weight of any
monolithically placed concrete wearing surface, to provide
for future wearing snface.

M.S.H.D. Spec. Art.30, p.12

When provision is made for three or more lanes of
traffic, the design shall provide for the following per-v
centages of the simultaneous maximum loading of all lanes:
For four or more traffic lanes ------- 80%.

M.S.H.D. Spec. Art.33. P.13 & 14

Use H20 - Sl6 - 44 loading from appendix A.

A.A.S.H.0. Spec. p.229

Design of Girder:

SpanAB--- Mba-P.ab a L whereL=42',P=-'lk
‘12—1‘442

 

 

 

a Mba (ft.-kips) "
O O
6 2.94
__£; 5.51
18 7.35
23 8.04
24% 8.075
_g5g 8.08 max.
25 8.05
27 7.91
29 ' 7 .58
30 7.5Z_
36 4.77
42 0

2L 1 .88

 

 

 

Span BC --- Mbc . P L b2 --- (a,= distance to rt. of a)

a Mh3(ft.-kips) Mcb (ft.—kips)
0 0 0
6 -4 .80 0 .57

12 -7.46 1.99

17 -8 .36 ----

18 -8 .41 3 .89

19 -8 .43, max . 4 .22

20 -8 .42 ----

21 -8 .36 ----

’24 -8.04 5.84

28.5 -7.13 7.13

30 -6 .74 ----

33 -5 .84 8 .04

39 -3 .89 8 .41

45 -l .99 7 .46

51 -O.57 4.80

57 O 0

Moment Distribution:

Opposite end of member hinged: C =
Opposite end of member fixed: C =
4.

K1: I"4/L" : 8.84 Cin. K2: I"/L"
CK = 3 x 8.84 = 26.5»

b8 920K = 52.6
0be = 4 x 6.52 = 26.1-)
CK = 4 x 6.52 = 26.11

Cb )‘ZCK = 52.6
0ch = 3 x 8.84 a 26.51
rba = rcd = 26.5/52.6 = 0.504
rho = rcb = 26.1/52.6 = 0.496

U

4461 in.4
L1: 504"

I

6.52 Cin. (L23684")

.83

 

—.~

_ _. __.

 

 

At

 

1 93

21,

i,__—_—_--—~—~—

 

[350441996]

5051

M = 5.51 ft.-klps

_._.._—....—.——_.

 

-2078 -2073

-0.69

1.36

{-0067
l
g 0.33

“00171-0016

-0.04

-0.0l

1.82

Mba = 7.88 ft.-kips

 

0.08
“0004
0.02

-0001

-l.82

 

[35%

7.88
-3-97
-0098
“0024

-0.06

“0002
2.61

#41957

 

 

“3091
.94
-0.96
0.48

'-O.24

0.12
-0.06

I 0.03

00.01
-2.61

 

0.34

0.08

,0.02

 

 

 

-1.81 1.81

[71897315 0.4.1

'3-91

1.911117
.0 95)

0.48!0. 48
-O.24I
0.12 0.12
-0.06

0003‘0003

 

 

 

"2060;2060

 

 

 

 

__._..—.__\_.__ ;

 

At ao= 24, Mba I 8.08 ft.-kips

 

 

 

 

 

 

 

 

A
Y

 

 

 

 

 

 

 

[.504 1 .496 1 [496415041
8.08
-4.07 ' -4.01 —» -4.01
1.99 +- 1.99 2.02
-1.00 -0.99 __4. -0.99
0.49 +— 0.49 0.50
-0.25 -0.24 —+ -0.24
0.12 «~— 0.12 1 0.12
-0.06 -0.06 -—> -0.06 '
0.03 «— 0.03 0.03
-0.02 -0.01 -—-
2.68 -2 .68 -2 .67 2.67
1.2138”: 30? ,Mpa =_7'37
. [.504 .496] ”:96 .504“)
1‘ 7.37 k
) ~3.71. -3.66 ——»— -3.66
I 1.82 «_— 1.82 1.84
-0 .92 -0 .90 —>— -0 .90
0.45 +— 0.45 0.45
-0.23 -0.22 —- -0.22
E 0.11 «— 0.11 0.11
4 -0.06 -0.05 —-——>— -0.05
‘ 0.02 «.— 0.02 0.03
49.221 .29191. ——~>
) 2.44 -2.44 -2.43 2.43

 

 

T391312! Mbc, =__"7 046. Mcb ’3 1.99 ft .-kips

r-e-v \\

 

 

 

 

 

h.-

 

 

 

 

 

 

 

 

 

 

i L 5047796) I .496 .5047
L -7.46 1.99
3.76 3.70 -—-»~ 3.70

-2 .83 +— -2 .83 -2 .86
1.43 1.40 —-—.— 1.40

-0.69 +— -0.69 -0.71
0.35 0.34 ——.- 0.34

—0.17 «— -0.17 -0.17
0.09. 0.08 _._, 0.08

-0.04 .1.— -0.04 -0.04
0.02 0.02 —->- 0.02

5.65 -5.65 -0.01 -0.01

3.79 -3 .79

{with _= 19, Mbc {78:43, Mcb =_ 4.22 ft.-ki;i

_ F504 .496j F496 .5041
f , -8.43 4.22
4.24 4.19 ——> 4.19

-4 .17 «— -4 .17 .4 .24
2.10 2.07 -——> 2.07

-1 .03 _._—— -1 .03 -1 .04
0.52 0.51 _.- 0.51

1 -0.25 _._— -0.25 -0.26
0.13 0.12 —-— 0.12

-0 .06 «— -0 .06 -0 .06
0.03 0.03 _._. 0.03

-0.01 -0.02

7.02 -7.02 5.62 -5.62

 

 

 

 

 

 

 

 

 

 

 

_At 82:: 28 059 Mbc :3 '4'ch 8 -7 .13 ?t°-kj:-Es

 

33M“— " M*—-————-————.~1

 

_faa.___- #W114

3 ~59

2.63

0.66

 

I
i
5
i
I
g

0.16 '

0 .04

0.01

7 ~09

I
1

 

 

 

 

 

 

 

__ r7932] 1 0495 ;5_Q_4:1_-,_i“__ __
[—3.13 7 013 1
3 .54 ——+- 3 .54
-5 .28 *——- -5 .28 -5 .39
2 .63 ——*— 2 .63
-1 .31 -< —— -1 .31 -1 .32
0 .65 ——*~ 0 .65
-0 .32 +— -0 .32 -0 .33
0 .16 -—-———~- 0 .16
‘ -0 .08 4— -0 .08 -O .08
0 .04 ——>- 0 .04
-0 .02 -<~———— -0.02 -0.02
0.01
-7 .09 7 .14 -7 .14

 

4.62.

 

 

”487. 1.1-"Z.

 

o 12” 1.32

(f ‘D
I

R = 067 R6: 013 Rfi‘ .064

9.
Va=.670k

(4

- k
vb- .394

"1.09, ,2.Ql

1121' '

 

 

 

Rae.437 Rb;.563 35:.092
van-.437k v5=.655k

“—2414 2,98

12.478

 

 

”L? 1) £1

Ra=.365 Rbp.635 Rb;.094

- k
va=.365k vb-.729

0‘ 2.3.01.1.) 2'43

-1,44

C _

 

 

 

I

 

Ra: 0228 RE: 0772 Rb":+ 0086

k 1 k
va-.228 Vb-.858

5.65 —St§ 2'J

 

 

 

 

°£ -) (-
l 1 T

RaB- .134- Rb”: .134- E8: .821

vba.955k

. , '
1.61 .7.C/Z.f"19_~_
/ _

.. 1:
Va“- 0134

 

 

 

 

*1 l) *
Rail-.167 lib-3.167 1153.690

 

 

 

Va" .167k Vb8 '857k
7.07 £7.09 38.9.5. '
I T

Raf-.170 R5=.l70 Rfi=.500

va=-.17ok vb=.670k

 

0
1‘) C ~>
Rcl=-0064 R0,,3-0043 Rd: 0043
k _, k
veg- .107 Va. 0043
-114" z. bl
‘3 6 1°
Ref-.092 Rd=-.062 Rd=.062
k _, k
v0.“ .154 Vd- 0062
-166 2'08 0
13 (l 1
chzofi4 Rd=-.O64 Rd-.064
k .. k
Vet-.158 Vd-.O64
—z.44 1.44 O
1
Rc'="0086 Rd=" 0058 Rd'goOSB
vc--.144k vd=.058k
3.74) -369 )0
Rd=0179 RC": 0090 Rd=-0090
Vc=.269k vac-.090k
5,67— 4.64 0
(‘f 1)
Rd=o310 1363.134 Rdfl- .134
. k g- k
Va .444 V'C1 .134
7.14 .41/4 0
Rc’= 0500 RC": 0170 Rde- 0170
vc=.670k van-.170k

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t . .. _ ..
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v ..u I a 1 r}.- n h. n .‘ ..- . v
‘ u .A .4 O ‘ I u . l . I

u . IV \I o
u a .. L... - . .
O b .

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BEAM FLEXURE

L.L.M0m.=(-243)+(-144)= 387 ft.-k1ps = 4,644,000 in.-lbs.

I = 4461 1n.4, c = 29.82/2 = 14.91", rmax= Mc/I

rmax.(4,644,ooo x 14.91) / 4461 = 15,550 p.s.i. actual
(18,000 p.s.i. allowable, beam 0.K. in flexure.)

FLANGE BUCKLING

flange width = b = O.872”ft. L 3 19 ft.

f8 = 22E500 - 22,500 :-
'1+L ,I'lOOb 1+ 19 100x0.72

rs = 17,800 p.s.i. actual (18,000 p.s.i. allowable. 0.K.)

VERTICAL BUCKLING AT REACTION

rs - R . 203000 = 8,900 p.s.i.
(a-+ d74ft 11 + 29. 2 .5 actual

(18,000 p.s.i. allowable. 0.K.)

 

DIAGONAL WEB BUCKLING
h/t a 28.3"/0.548" = 51.6 > 50

a = 15,000 - 100 h/t = 9,840 p.s.i. actual

8
(11,000 p.s.i. allowable. 0.K.)

VERTICAL WEB CRIMPLING AT REACTION

r8 = R where k = flange thickness = 0.760"
(a + kit

f8 3 20,000 = 14,000 p.s.i. actual
l +.700.5

(18,000 p.s.i. allowable. 0.K.)

IX. CONCLUSIONS

The superstructure of this bridge was very well
designed both for stress and architectural beauty. It
is neither over—designed nor under-designed, but designed
closely to the allowable specification stress limits.
This means that the design costs and material costs have
been kept to a minimum.

It was a pleasure to work with the Michigan State

Highway Department's plans and specifications.

 

 

 

   

1.11 H1JAN S1HTEUN1VERSIT( Ll BIAR

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