Vibration Study of Three-Span Continuous I-Beam Bridge

JOHN M . HAYES, Associate Professor, School of Civ i l Engineering, and JOHN A. SBAROUNIS, Research Assistant, Joint Highway Research Project Purdue University

This is a report on a vibration study of a three-span continuous I-beam highway bridge. The f i e ld study was made during the summer of 1953. Data f r o m static loadings as wel l as vibration data f r o m dynamic loadings are presented.

The slab was found to distribute to the individual stringer about the same magnitude of live load as provided f o r m the specifications of the American Association of State Highway Officials . Although the test struc­ture was not designed for composite action, considerable interaction ex­ists between the slab and the stringers i n the carrying of l ive load.

Curves are presented for use in computing the natural frequency of a three span continuous beam with constant cross-section. The frequency of application of axle loads at a point was considered as the frequency of application of the live load as a dynamic force at that point in the struc­ture. This IS a function of speed and axle spacing. Resonance occurs i f the frequency of application of the axle loads at a point coincides with the natural frequency of vibration of that point. I t is suggested that, to pre­vent undesirable vibrations, the natural frequency of vibration of the bridge, including the effect of the mass of the live load, should be greater than the frequency of application of axle loads based on a reasonable maximum speed and axle spacing.

A simplif ied analysis of the computation of moment of inert ia of the transformed section due to par t ia l interaction of the concrete slab with the steel stringer is presented.

# T H I S study was made to obtain information about the vibration characteristics of three span continuous I-beam highway bridges as constructed by the State Highway Depart­ment of Indiana. Considerable vibration had been observed in some structures of this type. Data were desired as to the frequency and seriousness of the possible vibration of this type of structure. Design methods would be proposed which would prevent un­desirable vibration.

Secondary studies would include the determination of the amount that the concrete slab participates with the steel stringer m resist ing the load and the amount of d i s t r i ­bution of the live load among the stringers supporting the slab.

Excessive vibrations of a structure on a heavily traveled highway might be the cause of the roadway slab cracking-up over a period at years. I t i s also a factor i n dr iver psychology.

DESCRIPTION OF TEST STRUCTURE

The test structure was a three span continuous I-beam highway bridge with r e in ­forced concrete substructure resting on piles. Figure 3 shows a general layout of the bridge.

The reinforced concrete roadway slab rested on six 33 WF 130 stringers. No pro­visions were made f o r composite action between the slab and stringers. The stringers were braced by 18 channel 42. 7 diaphragms on eleven foot centers. Figure 4 shows the slab details. The stringers were spliced at each interior support. The short flange plates at the splices were of varying lengths fo r the different stringers, but they were neglected in computing the moments of inert ia of the stringers.

The bridge was designed for the H20 loading in accordance with the AASHO standard specifications for highway bridges, fourth edition, 1944. The structure was completed m October 1949. I t was also designed f o r a one hundred percent increase i n l ive load

47

48

Figure 1. Side view of part of test structure. and impact with the allowable stresses increased fifty percent.

INSTRUMENTATION Strain, deflection, and slip measurements were taken at five stations as shown in

Figure 3. Stations 1, 3, and 5 were located at points of maximumdead load deflection.

Figure 2. Roadway of test structure.

\ \ ' -Concrete Piles

49

Exp Brg

—Timber P i l e s

E L E V A T I O N

2 0 0 ' - 0 c-c end beofings 6 2 ' - 6 c-c bearings 7 5 ' - 0 c-c beoringa 6 2 ' - 6 c-c bearings

26-6 ' /< 2 6 - 6 % 4. Stringers

Sym obout i Roadway

" Pref bit exp Jl I Bridge ^Toothed exp Jt 18 C 4 2 7 diaphragms l l ' - O " c - c

P L A N

Figure 3. Details of test structure anc" measuring stations.

except that Station 3 was offset one foot because of diaphragms at the center of the bridge. General details of the instrumentation at Station 1 are shown in Figure 6.

Strain The strains were determined with SR-4 electrical resistance strain gages. The l o ­

cations of the gages are shown in Figures 3, 4, and 5. The types of gages used are shown in Figure 5. The resistances to ground of the SR-4 gages were as fol lows: A l , 5, 000 to 10,000 ohms; A9, 100 to 1,000 ohms; A16, 10, 000 ohms.

The dynamic strains were recorded on f i l m f r o m a 12-channel cathode-ray strain oscillograph shown m Figure 8, except that a Brush Magnetic Oscillograph was used to record the strains measured at locations 1B4 and 1B6. The 12-channel oscillograph

3 0 ' - 0 2 8 ' - 0 Clear Roadway

11-0 Truck

Z'-S>/i\z'-8'/i.

I i

-Symmetrical t Roadway

^ ^ ^ T " Constr Jt

-SR-4 Gages ISC 4 2 7

- 3 3 « F I 3 0 '

5-10 5 ' -5 2 - 8 ' / i I 2 - 8 ' / a I 5 ' -5

3 3 V F I 3 0 -

5'-10

HALF SECTION SHOWING S R - 4 GAGES

® CD HALF SECTION

SHOWING DIAPHRAGMS

l'-0'/2

Figure 4. Typical section of test structure.

50

was designed and built by the Department of Structural Engineering, Purdue University. The static strains were measured with a Type K Portable Strain Indicator.

Deflection

Deflections were measured only on stringers B and C and at Stations 1, 3, and 5 as shown in Figure 3.

The static deflections were measured with Federal dial gages reading directly to one thousandths of an inch. The gages were fastened to 4-by 4-inch timbers, which were driven into the ground and braced entirely f ree f r o m the working platforms. The point­ers of the dial gages rested against bars clamped to the bottom flanges of the stringers.

The dynamic deflections were measured with Schaevitz linear differential trans­formers . The results were recorded on f i l m f r o m the 12-channel cathode-ray osci l lo­graph shown in Figure 8. A typical set-up of a transformer is shown in Figure 9.

^ T y p e A9 at Stringer IB

( 5 \ / D Type Gage

Location Stringer

A9 1,2 All Stringers Al 7,8,9,10 All Stringers Al 3,4,5,6 I A , I F , 4 B , 4 C , 5 B , 5 C Al 4,6 IB AI6 3,5 IB AI6 3,4,5,6 IC , I0 , IE ,2B,2C,3B,3C

Figure 5. Location of SR-4 gages. Slip

The dif ferent ia l s l ip between the concrete slab and the steel stringers was meas­ured with the device shown in Figure 10. A Federal dial gage was fastened to the stringer with i ts pointer resting against an a rm fastened to the slab directly i n line transversely with the connection of the dial gage to the stringer. These two points of connection were offset six inches longitudinally with respect to the stations shown in Figure 3. The dial gages were graduated to read directly to one thousandths of an inch, except that one gage used read directly to one ten-thousandths of an inch.

Event Marker

The positions of the test trucks were recorded fo r the dynamic runs with t r a f f i c

51

Figure 6. View at Station 1. tubes placed across the roadway at both ends and at the centerline of the structure. Figure U shows the device which picked up the air pulse from the traffic tube and transmitted it to the light bulb in the camera and the event marker in the Brush in­strument.

TEST LOADS The dimensions and weights of the test loads are shown in Figure 12.

the state test truck is shown in Figure 13. A picture of

TEST PROCEDURE Static Tests

Measurements were taken at Stations 1, 3, and 5 for six different positions of the state test truck and at Stations 2 and 4 for four different positions of the state test truck. See Figure 14 for position and sequence of loadings.

All traffic was stopped and zero readings were taken. The load was positioned and readings were taken. The load was removed and zero readings were immediately taken before traffic was resumed. All static measurements were made in the early morn­ing before the sun rose.

About thirty SR-4 gages were read at each placement of the test load so the zero readings would be close together. They averaged about twenty minutes apart. The individual SR-4 gage readings were averaged with the zero readings preceding and following each loading. These zero readings were, in general, within from zero to six microinches per inch of each other. The total drift in zero readings over a period of about two and one-half hours was usually not over ten microinches per inch. This

52

Figure 7. View at Station 3.

Figure 8. Twelve-channel oscillograph with cameras.

53

Figure 9. Typical set-up of Schaevitz transforme

Figure 10. Sl ip treasuring device.

Figure 11. Pick-up for t r a f f i c s ignal .

55

O OJ oJ

31.7' ' 3 .9 ' 11.9' 3.5', 12.4'

in 0 GO 00 0 q OJ — 0) — in

® ® ® ® ® A X L E SPACING AND LOADING IN KIPS

1.00' .98j |. 5.26' .

II II II II I I A X L E S 4a5 A X L E S 2 8 3 A X L E I

W H E E L SPACING

S T A T E T E S T TRUCK

o in oJ

10'-10'A o o

ro

14'-2 o to

2) ® ® ®

A X L E SPACING AND LOADING IN KIPS

6'-5 'A 4'-7V4

5'-6V8 7 - 0 % 4'-IP/4

II ill I II III I A X L E 2 A X L E I A X L E 2 A X L E I

W H E E L SPACING

HIS TRUCK H20 TRUCK Figure 12. Details of test loads.

56

Figure 13. State test truck.

drift did run up to thirty or forty microinches per inch for about five percent of the readings. The gages at Station 1 on stringers A, B, and C were read as a unit. The gages at Station 1 on stringers D, E, and F were read as a unit. Stations 2, 3, 4, and 5 were each read as a unit. The loadings were easy to duplicate and everything should be constant, except for a possible variation in the percentage of composite action of the slab with the stringers.

Dynamic Tests Strains were read on six channels and deflections were read on six channels of the

twelve channel oscillograph. The deflections were read continuously at Stations 1, 3, and 5 on stringers B and C for all dynamic runs. Two channels were always con­nected to gages at positions 3 and 5 on stringer B at Station 1. The four remaining channels measured strains consecutively on gages at positions 3, 4, 5, and 6 at sev­eral other station and stringer locations for the various dynamic runs.

RESULTS OF STATIC TESTS Static Strains

Figure 15 shows the strain distributions at all stations for the static loadings. Values showing the location of the neutral axis and the percentage participation of the concrete slab with the individual steel stringers, computed in accordance with the data presented in this paper, are given in Table 1 for Station 1. Figure 16 shows the strain distributions in stringer B at Station 1 where one Type A9 SR-4 gage was positioned on top of the concrete slab.

Distribution of Static Loads to Stringers The distribution of the static loads is given in Table 1 for Station 1. Here the load

distribution was obtained by direct use of the measured strains in each stringer. The value of total bending moment computed from the measured strains is compared with the total theoretical bending moment.

T A B L E 1

DISTRIBUTION OF LOAD TO STRINGERS - STATION 1

57

bo h e e a

Strain Percentage bo h e e a Participation

. sta

tic

. L

oad]

. S

trin

{

y e of Slab

. sta

tic

. L

oad]

. S

trin

{

Micro In.

. sta

tic

. L

oad]

. S

trin

{

Inches per In. n=8 n=10 n=12 n=8

1 1 A 12.6 +73. 93.5 100. 100. 108,300 B 6.9 +73. 34. 5 43 51. 94, 500 C 8.2 +72.5 46.5 58.5 70. 95,900 D 10.3 +59.5 71. 89. 100. 81,400 E 12.8 +32.5 100. 100. 100. 45,700 F 11.6 + 7. 77. 96. 100. 10, 200

29 T5 •I

Ft. Lb. n=10 n=12

Percent of Total Load

to Each Stringer n=8 n=10 n=12

1 4

1 3

1 6

107,100 94, 500 96,200 81,400 45, 000 10,200

105, 400 94, 000 96,200 81,000 44, 500 10,200

24.8 21.7 22.0 18.7 10.5 2.3

24.7 21.8 22.1 18.8 10.3 2.3

436,000 434,400 431,300 100.0 Theoretical Bending Moment = 435, 200 ft.

A 0. + 2. 0 0 0 2, 200 2, 200 2,200 0.5 B 14.5 +28.5 100. 100. 100. 40, 100 39, 400 38, 900 9.2 C 12.7 +64.5 100. 100. 100. 90, 100 89, 200 87,700 20.7 D 10.6 +84.5 75.5 94.5 100. 114, 600 114, 600 114,800 26.3 E 5.0 +87.5 21.5 27. 32. 108, 300 108, 000 107, 500 24.9 F 2.5 +68.0 8.5 11. 13. 80, 000 79, 300 79,000 18.4

100.0 lb.

0.5 9.1

20.6 26.5 25.0 18.3

435,300 432,700 430,100 Theoretical Bending Moment

A 17.0 -22.5 100. 100. 100. 33, 600 B 14.5 -22. 100. 100. 100. 30,900 C 16.5 -22. 5 100. 100. 100. 31,400 D 9.5 -14.5 61. 76. 91.5 19, 600 E 8.0 -13. 43. 54. 65. 17, 200 F 4.6 - 7. 18.5 22.5 27.5 8, 900

33,100 30, 500 31,200 19,600 17,200 8,700

32,400 30,000 30,700 19, 600 17,200 8,700

100.0 435, 200 ft.

23.7 21.8 22.2 13.8 12.1 6.3

100.0 lb.

23.6 21.7 22.2 14.0 12.2 6.2

141, 600 140,300 138, 600 Theoretical Bending Moment

A 13.5 - 6. 100. 100. 100. 8,900 B 6.0 - 7. 28. 35. 41.5 8,700 C 7.0 -13. 36. 45.5 54.5 16,900 D 8.5 -19. 49.5 62. 74.5 25,400 E 11.0 -20. 79.5 98. 100. 27, 600 F 2.5 -18. 8.5 11. 13. 21,300

8,700 8,700

16,900 25,400 27, 600 21,000

8,700 8,700

16,900 25,400 27,300 21,000

99.9 120,600 ft.

8.2 8.0

15.6 23.3 25.3 19.6

8.0 8.0

15.6 23.4 25.4 19.4

108,800 108,300 108,000 100.0 Theoretical Bending Moment = 120, 600 ft.

99.8 lb.

24.4 21.8 22.3 18.8 10.3 2.4

100.0

0.5 9.0

20.4 26.7 25.0 18.4

100.0

23.4 21.6 22.2 14.1 12.4 6.3

99.9 100. 0 lb.

8.1 8.1

15.7 23.5 25.3 19.5

100.2

i l l Inches

Strain

Micro In. per In.

DISTRIBUTION OF LOAD TO Percentage

Participation of Slab

n=8 n=10

T A B L E 2

STRINGERS - STATIONS 3 AND 5

Resisting Moment

Ft. Lb n=12 n=8 n=10 n=12

Percent of Total Load

to Each stringer

n=8 n=10 n=12

5 2

5 5 B 11. C 11.8

5 3 B 7.3 C 7.8

5 6 B 5. C 10.

3 3 B 6.6 C 9.5

3 6 B 6.9 C 10.9

+76.5 +73.

+27.5 +57.5

-15.5 -16.

-7. -10.5

+70:5 +69.5

+28. +52.

37.5 43.

21.5 67.

32.5 61.

34.5 80.

35. 68.5

28. 55.

79.5 97.5 100.

98.

46.5 53.5

27. 84.

40. 76.

42.5 100.

41.5 96,900 96,900 96,700 22.2 82. 98,100 98,100 98,100 22.6

Theoretical Bending Moment = 435,200 ft.

100. 38,200 37,900 37,700 8.8 100. 80,500 79,800 78,100 18.5

Theoretical Bending Moment = 435,200 ft.

56. 20,300 20,100 20,100 20.1 64.5 21,000 21,000 21,000 20.9

Theoretical Bending Moment = 100,800 ft. 32.5 8,700 8,600 8,600 8.6

100. 14,400 14,400 14,300 14.2 Theoretical Bendmg Moment = 100,800 ft.

47.5 90,900 90,400 90,100 21.4 91. 94,000 94,300 94,000 22.2

TheoreUcal Bending Moment = 424,200 ft.

51. 36,300 36,300 36,000 8.5 100. 72,000 72,000 70,600 17.0

Theoretical Bending Moment = 424, 200 ft.

22.2 22.6

lb. 8.7

18.3 lb.

19.9 20.9

lb.

8.6 14.3

lb.

21.3 22.2

lb.

8.5 17.0

lb.

22.2 22.6

8.7 17.9

19.9 20.9

8.6 14.1

21.2 22.2

8.5 16.6

58

The distribution of the static loads is given in Table 2 fo r Stations 3, and 5 and in Table 3 f o r Stations 2 and 4. Here, the bending moments carr ied by the two stringers, where readings were taken, were computed f r o m the measured strains and the distribution was obtained by comparing these computed bending moments to the total theoretical bending moment.

The distribution of the static loads, assuming simultaneous loading of both lanes, i s shown in Table 4.

Deflections

Table 5 gives the results of the static deflection measurements. The computed deflec­tions are based on the percentage participation of the slab obtained f r o m the strain measure­ments and the theory presented i n this paper. Slip

Values of relative slip between the slab and the stringers fo r the static loadings, as measured with the device pictured i n Figure 10, are given i n Table 6. These values did not repeat exactly fo r successive applications of the loadings. Neither did the readings always go back to the or iginal zero after the loading was removed. In general, there is r e l ­ative correlat ion between the slip readings and the percentage participation of the slab with the stringers as determinedby the strain measurements, which should give an accurate l o ­cation of the neutral axis.

Dynamic Strains RESULTS OF DYNAMIC TESTS

Figure 17 shows the dynamic strain distribution f o r a range of velocities of the State Test Truck f o r stringer B a t Station 1. These data locate the neutral axis about 4. 5 in . above the mid-depth of the stringer. Indications are that the percentage participation of the slab with the stringers was less under dynamic loading than i t was under static load­ing. The strain data is insufficient to permit any conclusions being drawn as to any d i f ­ference in the distribution of the l ive load among the stringers fo r dynamic loading as compared to static loading.

gOtf-O c-c tnd btorinot

4JAxle i Truck

Roodvo, FOR READINGS AT STATIONS 1 , 3 8 5

® A . I . ( 4 )

-t Truck

Roadway

FOR READINGS AT STATIONS Zi*

Figure 14. Sequence and location of s ta t i c loadings.

The maximum increase in dynamic stress over static stress was about 31 percent f o r the state test t ruck as a smoothly running load. This increased to about 75 percent when the state test truck ran over the 2^4 inch plank obstruction.

Dynamic Deflections

The dynamic deflections resulting f r o m the passage of the state test truck over the structure at various speeds are shown in Figures 18 and 19 f o r Station I B and in F ig ­ures 20 and 21 f o r Station 3B. The dynamic deflections resulting f r o m the passage of the H20 test t ruck over the structure are shown In Figure 22. These deflectioh-time graphs are f o r the test trucks running smoothly over the bridge and f o r the test trucks

• 3 I t

^Top of slab "t-^Top of stringer | t

4. stringer

bo t tom of stringer

/Top of slob ^Top of slob

The numbers indicate the location and sequence of l o a d i n g ^ (See Fig. 14) 3

j t /Top of stringer \i. | i ^ T o p of stringer I I I I I I I I I I I Scale of unit strain

microinches/inch _ + strain to right of 4 . I . - strain to left of t

K /'^ stringer

'^Bottom of stringer'^

^ t . stringer

J 1 » ^Top of stringer t

o N f ^

^ \ \ Stringer /

< Stringers v v \ \ \ \

/ \ \ 3 / » -^^^ i—

t - x 6 \ \

\ \ 3

\ \ 3B 3C

1* ^ ^Top of stringer y • 33 130

/ / l

iT '

Stringer ^

2 / / /

/ / 1

/ / ,t -

2 / /

I I I

/ f* — 1 V'fi 1 — 4B 4 C

CO

60

i l l

T A B L E 3 D I S T R I B U T I O N O F L O A D T O S T R I N G E R S

S t r a i n P e r c e n t a g e e

S T A T I O N S 2 A N D 4 P a r t i c i p a t i o n o f S l a b

R e s i s t i n g M o m e n t e l x l ? ^-•"15

Ct Q U

" . i " . I n c h e s M i c r o I n . p e r I n . n = 8 n = 1 0 n = 1 2

2 1 B 9 . 6 - 5 2 . 5 6 0 . 7 5 . 9 0 . C 1 0 . 6 - 5 0 . 7 5 . 5 9 4 . 1 0 0 .

2 3 B 8 . 6 - 1 5 . 5 4 9 . 6 1 . 7 3 . C 6 . 9 - 3 4 . 5 3 5 . 5 4 4 . 5 5 3 .

2 2 B 7 . 2 - 3 6 . 3 7 . 4 5 . 5 5 5 . C 9 . 6 - 3 7 . 5 6 2 . 7 7 . 5 9 3 .

2 4 B 9 . 6 - 1 7 . 5 6 0 . 7 5 . 9 0 . C 1 1 . 2 - 2 8 . 5 8 6 . 1 0 0 . 1 0 0 .

4 2 B 7 . 9 - 4 8 . 4 2 . 5 5 3 . 6 3 . 5 C 8 . 0 - 4 5 . 5 4 4 . 5 5 6 . 5 6 7 . 5

4 4 B 1 1 . 0 - 1 9 . 5 7 9 . 5 9 8 . 1 0 0 . C 7 . 9 - 3 4 . 4 3 . 5 5 5 . 6 6 .

4 1 B 7 . 6 - 3 3 . 5 4 0 . 5 0 . 6 0 . C 1 0 . 6 - 3 5 . 5 7 5 . 5 9 4 . 5 1 0 0 .

4 3 B 9 . 4 - 1 5 . 5 7 . 5 7 2 . 8 6 . 5 C 9 . 6 - 2 8 . 6 2 . 7 7 . 5 9 3 .

n = 8

y F t . L b . n = 1 0 n = 1 2 n = 8

P e r c e n t o f T o t a l L o a d t o E a c h S t r i n g e r n = 1 0 n = 1 2

l b

7 1 , 3 0 0 7 1 , 3 0 0 7 1 , 3 0 0 2 7 . 4 2 7 . 4 6 7 , 9 0 0 6 7 , 9 0 0 6 8 , 2 0 0 2 6 . 1 2 6 . 1 T h e o r e t i c a l B e n d i n g M o m e n t = 2 6 0 , 2 0 0 f t . l b . 2 0 , 7 0 0 2 0 , 7 0 0 2 0 , 7 0 0 7 . 9 7 . 9 4 4 , 5 0 0 4 4 , 7 0 0 4 4 , 7 0 0 1 7 . 1 1 7 . 2 T h e o r e t i c a l B e n d i n g M o m e n t = 2 6 0 , 2 0 0 f t . l b . 4 6 , 6 0 0 4 6 , 6 0 0 4 6 , 6 0 0 2 0 . 5 2 0 . 5 5 0 , 8 0 0 5 1 , 0 0 0 5 0 , 8 0 0 2 2 . 3 2 2 . 4 T h e o r e t i c a l B e n d i n g M o m e n t = 2 2 7 , 4 0 0 f t . 2 3 , 7 0 0 2 3 , 7 0 0 2 3 , 7 0 0 1 0 . 4 3 9 , 6 0 0 3 9 , 4 0 0 3 8 , 7 0 0 1 7 . 4 T h e o r e t i c a l B e n d i n g M o m e n t = 2 2 7 , 4 0 0 f t . 6 3 , 1 0 0 6 3 , 1 0 0 6 2 , 8 0 0 2 4 . 2 5 9 , 9 0 0 5 9 , 9 0 0 6 0 , 2 0 0 2 3 . 1 T h e o r e t i c a l B e n d i n g M o m e n t = 2 6 0 , 2 0 0 f t . 2 7 , 1 0 0 2 7 , 1 0 0 2 6 , 8 0 0 1 4 . 4 4 4 , 7 0 0 4 5 , 0 0 0 4 5 , 0 0 0 1 7 . 2 T h e o r e t i c a l B e n d i n g M o m e n t = 2 6 0 , 2 0 0 f t . 4 4 , 0 0 0 4 4 , 0 0 0 4 4 , 0 0 0 1 9 . 3 4 8 , 1 0 0 4 8 , 1 0 0 4 8 , 3 0 0 2 1 . 1 T h e o r e t i c a l B e n d i n g M o m e n t = 2 2 7 , 4 0 0 f t . 2 0 , 3 0 0 2 0 , 3 0 0 2 0 , 3 0 0 8 . 9 3 7 , 9 0 0 3 7 , 9 0 0 3 7 , 9 0 0 1 6 . 7 T h e o r e t i c a l B e n d i n g M o m e n t = 2 2 7 , 4 0 0 f t . l b .

1 0 . 4 1 7 . 3 l b . 2 4 . 2 2 3 . 1

b. 1 4 . 4 1 7 . 3

l b .

l b . 1 9 . 3 2 1 . 1 8 . 9 1 6 . 7

l b .

2 7 . 4 2 6 . 2 7 . 9 1 7 . 2

2 0 . 5 2 2 . 3 1 0 . 4 1 7 . 0 2 4 . 1 2 3 . 1 1 4 . 3 1 7 . 3 1 9 . 3 2 1 . 2

8 . 9 1 6 . 7

t 3 3 \ ^ I 3 0

stringer

+ 100

3 3 WF 130

V 7 9 0 ft par s e c V - 3 0 5 3 f t per s e c V * 4 9 0 4 ft per s e c V 5 5 4 0 ft per sec

V - 3 2 0 9 ft per s e c 21/4 in plonk

i_ s t r i n g e r ^

- 5 0 0 + 5 0 Unit Strom in micro inches/ inch

+ 100

Figure 17. Dynairic s t r a i n d i s t r ibut ions . Station IB, state test truck.

0 + 5 0 Unit strain in microinches/inch

Figure 16. S t a t i c s t r a i n d i s t r i b u t i o n Station 113, state test truck,

running over 2)4 inch and 1% inch plank obstructions placed f i r s t at Station 1 and then at Station 3. The dynamic deflections resulting f r o m the passage of the H I 5 test truck over the structure are not plotted as they are too small . Both of the planks represent larger obstructions than what would ever be the actual case f o r normal operation.

The structure vibrated in the f i r s t mode when the state test t ruck ran smoothly over the bridge. This is shown in Figure 23 where the amplitudes of vibration at Stations 1, 3, and 5 are plotted with respect to the "c rawl" deflections f o r the state

61

-.1

.1 0

+.1

Distance of fr 0 50

L brg

ont axle from 4 . so 100

L brg ,

J t h bearing, feet 150

L brg

?o 250 * 1 1 1 1 I 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

J 11 1 1 1 1

^ ^ H

Truck

Z'7.90 f t per sec. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 . . 1

Time, seconds

-.1 c

u

•S +.1 > o

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

- / v » 2 3 . 4 5 f t . per sec , 1 , 1 . 1 , 1 , 1 , 1 , 1 , 1 1 . I . I .

Time, seconds

u

Q

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 T 1 1 1 1 1 1 1 1

•

/v'"30.53 ft . per sec. i , 1 , 1 I . I . 1 1 . 1

0 1 2 3 4 5 6 7 1 ' ' '

8 9

1 1 1 1 T r

Time,

1 I 1 T , , ,

seconds

1 1 1 1 1 1 1 . .

I 0 o

1 + ^ ' y«37.6l ft . per sec.

-.1

o Q +.2

0 1 2 3 Time,

4 5 seconds

6 7

< I . 1 1 1 , 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

" ' w ' V .—^ _^— -^^—

—''crawl" deflection --yv"49.04 f t per sec.

1 1 , 1 1 I 2 3

Time, seconds

Figure 18. Dynamic deHections, Station IB, state test truck.

62

i brg t brg t brg I

Distance of front axle from t soutb bearing, feet

brg

) 50 100 150 2C )0 250 1 1 1 1 I 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 . 1 1 .

Trucit

\ A / v-51.32 f t . per sec.

, 1 ^ 1 1 1 1 1 0 1 2 3 4 5

Time, seconds

m

- 0

•5 + .1 «

Q +.2

« - •' «

•E 0 « c o

•5 +.1 « «

Q +.2

» - .1 u •S 0

S +.1 H -

Q +.2

1 1 1 1 1 1 1 . 1 1 1 1 > 1 1 1 1 1 1 1 1 1

1 1 1 1

v"55.40 f t . per sec

1 1 1 1 1 2 3

Time, seconds

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 I 1

1 1 1 1

^v=37 66 f t . per se c , IVsin. plonk at .425L first span

1 1 , 1 1 1

Time, seconds -.1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1

r ^ 1 r \ 1 v=32.09 f t . per s

ec , 2 ' / 4 in. planit at 425L first span

, J . | V , 1 , 1 , 1 , 1 1 1 1 0 1 2 3 4 5 6 7 8

Time, seconds

.c o

•a +.1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1

/ ' v' '8.32 f t . per sec , IVs in planit at ^ second span

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 t 1

u 0

+.1

Time, seconds

Figure 19. Dynamic deflections, Station IB, state test truck.

63

t brg t brg t brg t brg 1 I Distance of f r 0 5 0

ont axle from 4 . 100

j th bearing, feet 150 2( )0 250

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 . , . 1

1 11 v-7.90 f t . per sec. \ . , Truck

1 1 1 1 1 1 1 1 1 1 1 1 1 1 T l 1 1 1 1 1 1 1 1 1 1 t 1 ! . . , , !

o q>

10 15 20 Time, seconds

25 30 35

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 I 1 1 1 1 1 1 1 1 1 1

v«23.45 f t . per sec. \

, 1 . 1 , 1 1 1 V m — 1 , 1 , 1 1 , 1 1 1 1

1 0

X +.1

0 I 2 3 4 5 6 7 8 9 10 II Time, seconds

.1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 I 1 1 1 1 1 1 1

^ —

v=30 53 f t . per sec

. 1 1 1 . 1 , ^ ^ 1 ^ 1 . 1 1 > 1 1 1

o

•5 +.1 a

4 5 Time, seconds

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

—

v"37.6l f t . per sec. ^ f 1 I . I , \ , 1 1 1

.2 0

- .1

• 0

+ j

+ .2

3 4 Time, seconds

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

v»49 .04 f t . pe ' sec. ^ ' ^ ^ , '^y—^crawl" deflection

1 1 ^ ^ - ^ . ^ 1 1 1 1 i 0 i 2 3 4 5

Time, seconds

Figure 20. I^ynamic deflections. Station 3B, state test truck.

64

-.1

.2 +.

I +.2

.b rg (

Distance of f r 0 50

L brg \

ont axle from i so 100

L brg \

Jth bearing, feet 150 2C

L brg

)0 250

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1

v 51.32 f t . per sec. \ a t

\ / ^ 13

Truck 0 . | l |2 Y |4 |5

-.1 Time, seconds

• . I 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1

—

v=55.40 f t . per sec. \ ^ ^

1 1 1 1 1 1 1 £ +.1 Q

2 3 Time, seconds

-.1 « « u _c 0

+J

S.

Def

+.2

-.1 CO o o ,c 0

ion.

+.1 +.1 »

Def

+.2

-.1 M O 0

A 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1

Si \

V"37.66 f t . per sec. \ ^ I'/s in. plonk at 425L 1st spon \ / \

0 . | l |2 . |3 , |4 1

Time, seconds

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1

V \ . . .

v=33.46 f t . per sec. W l / \ 2}h in plank at 4 . 2nd span ^ \ V 0 . | l . i2 . i3 ; F . 15 , 6 |7 . |8

Time, seconds

1 1 I 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1

— '

J i l l l f v=8.32 f t . per sec. m M

I'/s in. plonk at ^ 2nd span T 1 , 1 1 1 1 1 , 1 1 1 1 1 1

1

1 1 1 . 1 1 . . 1 1 . . 1 1 1 1 1 1

+.1

? +.2 a> O

+.3 10 15 20 25 30

Time, seconds Figure 21. Dynamic deflections, Station 3B, state test truck.

65

Lbrg <

Distance of f r 0 5 0

Lbrg <

snt axle from 4 . 100

L brg

th bearing, feet 150 2(

L br(

) 0

J

2 5 0 1 1 1 1 I 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 , . . ,

1 ' rucl t

Stringer B, Station , v = 4 8 . 3 6 f t . per se c.

1 1 1 1 1 , 1 1 1

u o ^ +. a> a 2 3

Time, seconds

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

\ A ft A Stringer B, Station 1, v 47.46 i t. per sec.

' 2V* in. plank at . 425L f i r s t s pan

1 1 , 1 1 1 3 1 2 3 4 5

u

E +.1

Time, seconds

-.1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1

0

+ .1 str inger B , Station 3 , ^ \ ^ > v=48 .36 f t . f )er sec.

+ .1 1 1 1 . 1 1

0 1 2 3 4 5

c o

«

Time, seconds

1 1 1 1 I 1 1 1 1 I 1 1 1 1 1 1 1 1 1 1 1 1 1

stringer B, Station 3 / " ^ A v=47.24 f t . per sec. " " A m N „ 2V4 in. planit at 4 . second span

1 V V w

1 1 • 1 1 3 1 2 3 4 5

u

Time, seconds

Figure 22. Dynamic deflections, H20 test truck, test t ruck going over the structure at 49.04 feet per second. The "c rawl" deflection is the same as the static deflection and is shovm on the bottom of both Figures 18 and 20.

Higher modes of vibration are induced in addition to the f i r s t mode when the plank obstructions were used. This is also shown m Figure 23 where the additional move­ment at Stations 1, 3, and 5 are plotted with respect to the f i r s t mode. The data are insufficient to give the exact mode of vibration when the plank obstructions were used. The application of an almost instantaneous Impact load to the structure fur ther com-

66

plicates the problem when the plank obstructions are used. The vibratory motions of stringers B and C were identical fo r a l l dynamic loadings.

The dynamic deflections of stringers B and C were identical when the live load was in the west lane and placed symmetrically with respect to these two stringers.

TABLE 4

DISTRIBUTION OF AXLE LOADS TO STRINGERS ASSUMING SIMULTANEOUS LOADING I N BOTH LANES

Sta- Simultaneous Percentage of One Axle Load to Each Stringer^ t ion Loading

A B

n = 10

C D E F 1 1 and 4 25.2 30.9 42.7 45.3 35.3 20.6 1 3 and 6 31.6 29.7 37.8 37.4 37.6 25.6

5 2 and 5 30.9 40.9 5 3 and 6 28. 5 35.2

3 3 and 6 29.8 39.2

2 1 and 3 35.3 43.3 2 2 and 4 30.9 39.7

4 2 and 4 38.6 40.4 4 1 and 3 28.2 37.8

Percentage of one axle load to each stringer in accordance with latest tentative re ­vised AASHO Specifications:

(5. 42) (100) / (5. 5) (2) = 49. 5 %

'Data f r o m Tables 1, 2, and 3.

TABLE 5

STATIC DEFLECTIONS

Deflections in Inches at Stations Load­ Type I B IC 5B 5C 3B 3C ing 1 Measured 0. 1710 0. 1583 0. 0104 0. 0085 -0. 0519 -0. 0453

Computed 0. 164 0. 153 -0. 0702 -0. 0618 2 Measured 0. 0103 0. 0090 0. 1727 0. 1533 -0. 0483 -0. 0426

Computed 0. 177 0. 149 3 Measured -0. 0569 -0. 0471 -0. 0483 -0. 0405 0. 2025 0.1818

Computed -0. 0548 -0 . 0566 -0. 0519 -0. 0526 0. 195 0.170 4 Measured 0. 0573 0. 1102 0. 0053 0. 0071 -0. 0246 -0.0364

Computed 0. 054 0. 123 -0. 0276 -0. 0487 5 Measured 0. 0056 0. 0076 0. 0617 0. 1105 -0. 0240 -0,0335

Computed 0. 052 0. 109 -0,0335

6 Measured -0, 0262 -0. 0381 -0. 0214 -0. 0296 0. 0736 0.1325 Computed -0. 0273 -0. 0498 -0. 0261 -0. 0318 0. 0766 0.121

Computed values are based on measured location of neutral axis f r o m strains as shown in Figure 15.

The magmtudes of the dynamic deflections are less than the magnitudes of the static deflections. This may be due, in part, to non-linearities i n the electric measuring and recording devices as they are used to measure small deflections. The deflections are

67

close to the null point of the Schaevitz differential transformers for which there was an appreciable null voltage, 0. 3 to 0. 5 volts. The amplitudes and measured frequen­cies are accurate within normal experimental tolerances.

PARTICIPATION OF CONCRETE SLAB WITH STEEL STRINGER

The natural frequency of vibration of the structure varies with the amount of par­ticipation of the concrete slab with the steel stringer, which depends upon the capacity fo r horizontal shear transfer between the slab and the stringer. This capacity f o r shear transfer depends upon bond and / or f r i c t i o n i f shear connectors are not used.

.425L,

-.05

I 0

i < .05

:-.05

I 0 a . E < .05

1.2 L, L,

-t=1.325 sec

•t»l.l75 sec

Firs1 Span Lo aded

t=2.70se<

t<2.52sec

Middle Span Loaded

v=49.04 ft . per sec.

Amplitude of vibration with respect to line of "crawl" deflection. Structure vibrating in f i r s t mode.

t=3.62sec

425 L,

t=3.55 sec. Middle Span Loaded, plonk at

middle span, v=33.46 f t . per sec.

t=3.588ec.' M=3.50sec.

First Span Loaded, plank at .425 first span, v=16.04 ft . per sec.

Additional modes of vibration with respect to f i r s t mode which occur only when truck passes over the plank obstruction.

The difference between the times shown is the time elapsed during one-half a cycle of the respective mode of vibration.

Figure 23. Modes of vibration.

The simplif ied analysis outlined in Figure 24 is based on the assumptions that the centroid of the area of the concrete slab remains at a constant distance above the centroid of the steel stringer and that the effective area of the slab is equal to the total slab area times the percentage participation with the stringer. The two l imi t s are no participation and f u l l participation. The centroid of the actual corrected trans­formed section varies between the centroids f o r these two l imi t ing conditions. The effective width of the slab acting as a T-flange with the stringer is taken mid-way to the adjoining stringers. Any bending moment carr ied by the slab acting along the span of the stringer is neglected.

The following is a complete notation for the simplif ied analysis shown in Figure 24: A = Corrected area of concrete slab acting with one steel stringer = t b p. a = Distance f r o m the centroid of the corrected transformed section to the

plane of separation between the concrete slab and the steel stringer, b = Corrected effective width of the concrete slab = w / n . c = Distance f r o m neutral axis of transformed section to extreme f iber . Ct = Distance f r o m the centroid of the corrected transformed section to the

top of the concrete slab.

68

Cb = Distance f r o m the centroid of the corrected transformed section to the bottom of the steel stringer,

f = Unit f lexura l stress at any point, f t = Unit f lexura l stress at the top of the concrete slab, fb = Unit f lexura l stress at the bottom of the steel stringer. I = Moment of inertia of the corrected transformed section about the centroid

of the corrected transformed section. M = Bending moment. n = Ratio of modulus of elasticity of steel to that of reinforced concrete, p = Effective participation of concrete slab in its composite action with the

steel stringer. pQ = Statical moment of effective concrete area, t b p, about centroid of cor­

rected transformed section, t = Thickness of concrete slab—the slab is assumed to be of constant thick­

ness and horizontal i n so fa r as one stringer is considered, w = Actual effective width of concrete slab acting with steel stringer, y = Distance f r o m the centroid of the corrected transformed section to any

point in the cross-section. - = Distance between the centroid of the steel stringer alone and the centroid

of the transformed section. EFS = Extreme fiber stress in f lexure.

Deflections may be computed by the usual theory using the moment of inertia of the corrected transformed section.

Using the theory presented here and the dimension of the test structure the graphs in Figures 25 and 26 were drawn. Figure 25 shows the variation in y, I , and I / y at the upper surface of the bottom flange of the steel stringer, SR-4 gage positions 5 and 6, with respect to the percentage interaction of the concrete slab with the steel stringers fo r the i n ­dividual stringers. Figure 26 shows the variation in y and I with respect to the percentage interaction of the entire con­crete slab with a l l six steel stringers.

feentroid /^L transformed

corrected sect ion

st r inger

NATURAL FREQUENCY OF VIBRATION

Computation of Natural Frequency of Vibration

Zh F lexura l s t r e s s e s r O

D I S T R I B U T I O N

0

O F F L E X U R A L S T R E S S

Zm obout section

r O

Unit s l ip ing

s t r inger " ^ f d

S t r o m

P)

The computation of the natural frequen­cy of vibration of a three span continuous beam with uniform cross-section may be solved by a general method presented by E. R. Darnley (1,2, 3). I t may also be de­termined by an unpublished method of successive approximations developed by Professor L . T. Wyly. This method is s imilar to the Rayleigh and Ritz pro­cedures. Considering any three span umt with equal end spans, L i , and mid -span, L 2 , the frequency of the fundamen tal mode of vibration may be computed after the determination of the smallest root, K i L i , of Darnley's frequency equation, as fol lows:

the centroid of the corrected transformed

"•/oM"" ^jC"^'''"" ^ / ' ' o ^ ^ ' "

/ ° y ' d A + / ' ' y « d A + / " ' p y « d A l . E F s 4 -L - ' - c , -'0 - ' a J

between concrete slob and steel

Hor izonta l s h e o r i n g f o r c e be tween

•• dM dx

concre te s l a b a n d

s t e e l s t r i n g e r ' J ^ ' j y d A - - ] ^ P y ^ ° ' y d A '

Figure 24. S i m p l i f i e d a n a l y s i s posite action.

(1)

of

69

where: f = Natural frequency of vibration. L i = Length of end span. I = Moment of inertia, g = Acceleration due to gravity, w = Weight per unit length.

Taking the value of the modulus of elasticity of steel as 29 mi l l ion p s i . , the fo l low­ing e^qpression fo r the natural frequency of vibration may be developed:

f= c J X = F J T L i - " w ^ w

(2) where:

f = Natural frequency of vibration in cycles per second. L i = Length of end span in feet. C = A coefficient depending upon the rat io L ? / L i . L2 = Length of mid-span in feet.

I = Moment of inert ia in inches fourth. w = Weight of beam in pounds per foot. F = A coefficient depending upon the rat io L z / L i and the magnitude of L i .

The variation in the coefficient C fo r various ratios of mid-span to end span are shown in Figure 27. Knowing the moment of inert ia and the weight per foot the natural frequency for any span ratio and length of end span may be determined by taking the value of coefficient F f r o m Figure 28.

Stringer A a F Stringer BSE Stringer CSD

0 2 4 6 8 10 IZ 14 y distance from t. stringer, inches

8 10 12 14 Total I, thousands

4 0 0 5 0 0 6 0 0 I/> at upper surface of bottom flange, In*

Figure 25. Properties of individual stringers.

70

Total section SIX stringers slab a curbs

0 2 4 6 8 10 12 14 y distance from centroid of stringers, inches

I I I I I I I I

4 0 0 0

3500

.S "5

S 3000

2500

unirs: T cycles per sec.

- L, f t . I in." w lbs per f t .

T cycles per sec. - L, f t .

I in." w lbs per f t .

T cycles per sec. - L, f t .

I in." w lbs per f t .

1.0 1.1 1.2 1.3 1.4 15

r = -40 50 60 70 8 0 ^ 0 ^ 100

Total I, thousands in* Figure 26. Proper t i e s of e n t i r e cros s -

section.

Effect of Live Load on Natural Frequency of Vibration

The natural frequency of vibration varies as the mass of the live load mo\^es across the structure. This variation in frequency was determined by the unpublished method of successive approximations developed by Professor L . T, Wyly. The effect of live load may be approximately evaluated by adding an equivalent uniform load to the dead weight of the structure in Equation 2.

Effect of Material Properties and Com­posite Action on Natural Frequency of Vibration

Figure 27. Variation in Coefficient C

Units f cycles par sec I in«

lbs per f t

r - L , / L ,

40 60 so Length of side span, L„ feat

Figure 28. Values of Coeff icient F.

The natural frequency of vibration is proportional to the square root of the mo­ment of inert ia of the span. The moment of inert ia of either the bridge as a unit or the stringers acting separately depends upon the rat io of the modulus of elasticity of steel to that of concrete and the percentage par­ticipation of the concrete slab i n composite action with the steel stringer.

The percentage participation of the concrete slab with the steel stringers is deter­mined in the test structure, which was not designed fo r composite action, f r o m the measured strains as shown m Figure 15 f o r the static loads and i n Figure 17 f o r the dynamic loads. An average value of y = 4. 5 inches is determined fo r the dynamic loads f r o m Figure 17. Using the curves shown in Figures 25 and 26 of this paper, computed f r o m the composite action theory developed there and with n = 10, 63, 000 inches fourth is obtained fo r the value of the moment of inert ia fo r the entire cross-section of the bridge with part ial composite action. The values of I f o r the individual stringers A or

71

TABLE 6

RELATIVE SLIP BETWEEN SLAB AND STRINGER STATIC LOADINGS

LOADING 1 2 3 4 5 6

1 A 0 0 0 0 0 0 B -0.00272 0 0 0 0 0 C +0.00321 0 0 0 0 0 D -0.00033 +0.00015 -0.00019 -0.00155 0 -0.00013 £ +0. 00043 0 +0. 00137 -0. 00078 -0.00002 +0.00002 F +0.00075 0 +0.00125 +0.0045 0 +0. 00335

3 B -0.00071 +0.00027 +0.00017 -0.00001 -0.00001 -0.00002 C -0.00003 0 -0. 00057 +0.00001 -0.00001 -0.00001

5 B -0.00001 +0.00041 -0.00084 -0.00015 +0. 00019 +0.00005 C -0.00003 +0.00012 -0.00032 0 +0.00001 0

2 B -0. 00023 +0. 00001 -0.00006 +0.00001 C +0.00042 +0. 00023 -0.00015 -0. 00005

4 B +0. 00018 -0.00001 +0.00005 0 C +0.00011 +0.00002 +0.00001 0

TABLE 7

NATURAL FREQUENCY OF TEST STRUCTURE EFFECT OF LIVE LOAD NEGLECTED

Stringer

Percentage Participation of Slab with Stringer

Zero Percent

Varying* Percent

n = 10

100 Percent

n = 10

Frequency, Cycles per Second

A or F 2.87 B or E 2.80 C or D 2. 84

Total Cross-Section 2.84

3.60 3.46 3. 51

3.56

4. 52 4. 24 4. 29

4. 40

' y = 4. 5 inches

F, B or E, and C or D are respectively 10, 500, 10, 200, and 10, 200 inches fourth, which gives a total value of 61, 800 inches fourth as the total value of the moment of inertia.

These curves also show that the magnitude of the moment of inert ia changes l i t t l e f o r n values varying f r o m eight to twelve. The greatest variation in moment of inert ia along the length of the structure would be due to changes in the percentage part icipa­tion of the concrete slab with the steel stringer, which depends upon the bond and / or f r i c t i on between the slab and stringer.

Natural Frequency of Vibration of Test Structure

The natural frequencies of vibration of the test structure, neglecting the effect of l ive load, are shown in Table 7. The natural frequencies of vibration of the test struc­ture, including the effect of the mass of the state test t ruck are shown in Table 8. These values are computed f r o m the data given m Sections A, B, and C above.

72

brg brg t brg t brg

Truck ii U I

-lOO I . slab part icipat lon V fith t tringe r X

X

( a A

v .

A

v . / 9

X

6

A

c o o

o -

easur pai

ed ticipa Hon

X

• • • 0 /

X ••• ^ i

0 • X

i X

0 • X

-0% with

slab strin

partic jer

ipatio n Leger d: • 7.90 49.04

ft. pe f t . pe

r sec. r sec.

X 51. • 55

32 f t 40 ft.

. per per 1

sec. sec.

o.

u >< u

>< u

100 200 Distance from ^ south bearing, feet

Figure 29. Var ia t ions in natural frequency as s ta te tes t truck moves over test structure.

TABLE 8

NATURAL FREQUENCY OF TEST STRUCTURE MASS OF STATE TEST TRUCK CONSffiERED

3 0 0

Location of Truck Percentage Participation of Slab with Stringer

0% Varies 100% n = 10 n =10

Frequency, Cycles per Second No Live Load

Axle 1, 52.8 f t . f r o m centerline S. Brg .

Axle 1, 94. 2 f t . f r o m centerline S. Brg .

Axle 1, 120. 7 f t . f r o m centerline S. B rg .

2.84

2. 72

2.75

2. 60

3. 56'

3.37»

3 .41 '

J. 23''

4. 40

4.16

4. 20

3.98

' y = 4. 5 inches ' y = 4. 2 inches

The variations m the natural frequency of vibration of the test structure as the state test t ruck passes over the bridge are determined f r o m the test data and shown in Figure 29. The theoretical frequencies for no participations and 100 percent par­ticipation of slab with stringer are shown. The percentage participation may be de­termined by noting where the actual frequency of vibration lies with respect to these two extremes. The solid line shows the theoretical frequency of vibration based on amount of participation of slab with stringer as determined f r o m the strain measure­ments.

FORCED VIBRATIONS RESULTING FROM TEST TRUCK Frequency of Application of Axle Loads

The maximum measured amplitude at Station 1 (point of maximum static deflection

73

in the end span) occurred when the f r e ­quency of the passage of the axles of the test truck over Station 1 coincided with the natural frequency of the loaded struc­ture. The moment of inert ia was com­puted f r o m the curves in Figures 25 and 26, using the measured percentage par­ticipation of slab with stringer.

Resonance occurs at Station 1 when the frequency of the passage of the axles of the test truck is in phase with the v i ­bratory motion of the structure at that point. At slow speeds this condition re ­sults f r o m the passage of the individual axles of a tandem set. At higher speeds each tandem set of axles acts as a single load applied at the point.

Let f i = frequency in cycles per second of passage of truck axles of each tandem set of axles, di = distance in feet between each tandem set, and v = velocity of test truck in feet per second, then:

- 5 0

V

dl (3)

2 3 4 5 F r e q u e n c y , c y c l e s p e r s e c

Figure 30. Frequency of app l i ca t ion axle loads.

of

with fz = frequency in cycles per second of passage of truck axles with each tandem set acting as a unit and da between each tandem set or single axles, there results:

distance in feet

V (4)

The c r i t i ca l velocities of the truck may be obtained fo r each condition by equating the frequency with which the axles or tandem set of axles pass over Station 1 to the natural frequency of the bridge, corrected for the mass of the truck at that point and considering the percentage participation of slab with stringers.

Figure 30 shows the frequencies of ap-

.05

0 4

: . 0 3

E <

.02

.01

no damping— 1 1

1

\ \ A \ \ 1

h —w-

plication of axle loads for various axle spacings and truck velocities.

Magnification Curve—State Test Truck

Figure 31 is a magnification curve show­ing both measured and theoretical values of amplitudes of vibration at Station 1 fo r various velocities of the state test truck. The theoretical amplitudes are computed as follows (3) and (4):

10 2 0 3 0 4 0 Velocity, ft per

_ I _ J .5 10 1.5 w / p (tandem

\ L set )

5 0 sec

±

6 0 7 0

J

(5)

25

Figure 31.

.50 75 u / p

Magnification curve truck.

1.00 125

state test

where: A = amplitude of vibration in inches. S = a function of the applied load, the

stiffness and length of the structure, and the velocity of the truck. I t may be expressed as 8 = Kv, where v is the velocity in feet per second

74

and K is a function of the physical properties of the applied load and the structure.

« = circular frequency of the forcing function (application of axle loadings), p = natural circular frequency of combined masses of structure and load, q = relaxation constant (a frequency parameter) = k /c , where k is the spring

constant of the structure and c is the damping coefficient. At resonance:

_ 8

q Expressing s in terms of kv gives:

Kv

(6)

A

which at resonance gives: „ A

P . q

TABLE 9

q

(8)

FREQUENCY OF APPLICATION OF AXLE LOADINGS FOR VARIOUS TRUCK SPEEDS AND AXLE SPACINGS

Velocity miles per hour

Frequencies, Cycles per Second

f2 f2/0. 8' fz/O. 9'

Axle spacings, dt, i n feet

14 30 14 30 14 30

3.66 4.71 6.81

1.71 2.20 3.18

4. 58 5.89 8.51

2.14 2.75 3.97

4.07 1,90 5. 24 2. 44 7. 57 3. 53

35 45 65

1

2 " = 0.8

0.9

The measured values on the magnification curve shows resonance at Station 1 r e ­sulting f r o m the passage of the state test truck over the structure at velocities of about 11. 4 f t . per sec. and 49. 0 f t . per sec. The f i r s t resonant velocity results f r o m an ap­plication of each axle in a tandem set, where fi = v / d i . With the state test truck in the end span the least measured natural frequency of the structure is 3.30 cycles per sec­ond. The value of di is 3. 5 f t . and an average value of d2 is 14. 875 f t . per sec. This gives average computed values of velocities of 11. 4 f t , per sec. and 49. 0 f t . per sec. f o r theoretical velocities of applied load which are in resonance with the natural period of the structure with the mass of the t ruck in place. These values are very close to the measured values.

With resonance at 49, 0 f t . per sec, and the natural frequency of 3, 30 cycles per second p = 2 ir f2 = 20,80 radians per second. With u /p values of 0. 6 and 1, 0, the measured values of A are 0. 003 inches and 0. 040 inches. Substituting these values in the formulas fo r amplitude gives two equations in terms of K and 1/q. Their s imul ­taneous solution gives K = 6. 549 x 10"' and 1/q = 3. 792 x 10" *. Using these values of K and 1/q, the theoretical magnification curve is drawn as shown in Figure 31. Ap­proximately the same value of q was obtained f r o m the logarithmic decrement, which was about 0.84 per cycle.

Taking the measured value of A as 0. 01 inch, resonant velocity of 11. 4 f t . per sec, and 1/q as determined above gives K = 7, 030 x 10"', This value d i f fers f r o m the pre-

75

vious value due to the accuracy of a value of A of 0. 01 inch. Using this value of K, the magnification curve giving the effect of the application of a tandem set of axles Is shown as a dotted line in Figure 31.

The slope of the theoretical magnification curve depends on the value of q, which varies with the damping characteristics of the structure. Both the superstructure and the substructure affect the damping characteristics of a bridge. There is no practical way to compute these damping characteristics. They must be determined experimen­tal ly fo r each type and length of structure. The eiqjerimental determination of the damping characteristics of a sufficient number of bridges of each type and length would provide a means for setting-up l imi t ing values of q for use m design. The value of q determined in this study applies only to the test structure. The damping character­istics of continuous I-beam highway bridges w i l l change as the percentage participation of the concrete slab with the steel stringers changes, since any relative movement between the slab and the stringers serves as a vibration damper.

Effect of Vibration of State Test Truck on Vibration of Test Structure

The natural frequency of vibration of the state test truck was determined by meas­uring the relative motion between the axles and the body of the truck with Schaevitz linear different ial t ransformers. Vibratory motion was induced by running the wheels over a 3 )4 - i nch plank obstruction and recording the resulting oscillations on a Brush Magnetic Oscillograph.

The natural frequency of vibration of the t ra i le r with respect to the two real axles, axles 4 and 5, is 3.3 cycles per second. The natural frequency of vibration of the tractor with respect to the two axles under the rear of the tractor, axles 2 and 3, is 4. 9 cycles per second.

A l l visible vibratory motion recorded on the oscillograph had disappeared within one and one-half seconds after the motion had been induced. The rate of damping is very high.

The roadway approach slabs and the bridge roadway slab of the test structure were in a very smooth condition. It is believed that the state test truck acted, in general, as a smoothly running load, except when the plank obstructions were used on the test structure. The natural frequency of vibration of the load carr ied on axles 2 and 3, 4. 9 cycles per second, is higher than the measured natural frequencies of the test struc­ture . The natural frequency of vibration of the load carr ied on axles 4 and 5, 3. 3 cycles per second, is in phase with the natural frequency of vibration of the test struc­ture fo r several positions of the state test truck. The variation in the natural frequency of vibration of the test structure is shown in Figure 29. The high damping rate of the state test truck minimizes the effect of truck vibrations on the test structure. The vibratory motion of axles 2 and 3 would counteract that of axles 4 and 5, except fo r the one instant in which they might be in phase.

The bridge vibrated in two modes when the state test truck ran over the plank ob­struction placed at the center of the bridge. The fundamental frequency was 2.80 cycles per second. There was also excited a super-imposed mode with a frequency of 7. 5 cycles per second. Neither of these frequencies coincide with the natural f r e ­quencies of vibration of the state test truck of 4. 9 and 3.3 cycles per second.

CONCLUSIONS AND RECOMMENDATIONS

Equivalent Uniform Mass fo r Live Loadings

The effect of the mass of the state test truck on the natural frequency of vibration of the test structure is shown in Table 7 and Figure 29. This effect may be closely approximated by the use of an equivalent uniform loading, which is added to the dead weight of the structure. This changes Equation 2 as follows:

F ^ w + 2 W j ^ (9)

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where: = Total weight of concentrated live load on structure in pounds—use truck

loadings. L = Total length of structure in feet, distance between abutment bearings.

The use of 2 W L / L f o r an equivalent uniform loading is substantiated by Rayleigh's method (3) where a single load is used to replace uniformly distributed load. The designer would have to determine fo r each structure the size and number of trucks to use in correcting the natural frequency of vibration of the structure fo r the effect of the mass of the live load. Usually the effect of one t ruck on the structure w i l l be c r i t i ca l , since the probability of multiple loadings being m phase with each other is small .

Considering w^ as 4 9 , 3 7 0 pounds, the total weight of the state test truck, 2 W L / L = (2) (49, 370) / 200 = 494 pounds per foot. Taking the moment of inertia with the location of the neutral axis as determined f r o m the strain readings, 63, 300 inches fourth, ob-taimng a Coefficient F of 0 . 8 6 9 7 f r o m Figure 28 , and using Equation 9, the natural frequency of vibration of the test structure corrected f o r the mass of the state test t ruck is 3 . 3 5 cycles per second. This value may be compared with the values shown in Figure 29.

Conditions in Test Structure

The frequency of application of axle loadings f o r various t ruck speeds and axle spacings are shown in Table 9. This gives the frequency of application of the live load at a point in a structure. The speed of the design load and the axle spacing must be determined by the design engineer after consideration of these factors, which are, i n part, dependent upon the location of the structure in the highway system.

The variation of the natural frequency of vibration of the test structure, using two H 2 0 trucks f o r the mass of the design live load, with various stringer sizes is shown in Table 10. These frequencies may be compared with those in Table 9 to see which size stringer and type of design would give protection against the occurrence of reso­nance of vibration. The test structure as designed with zero percent participation has ample margin against vibration only fo r the thi r ty foot axle spacing at fo r ty - f i ve miles per hour. More stiffness would be required fo r shorter axle spacings. Had the struc­ture originally been designed fo r f u l l composite action i t would have been safe by 3 . 3 3 cycles per second compared to 2. 7 5 cycles per second for the th i r ty foot axle spacing at fo r ty - f ive miles per hour and an 0 . 8 safety margin.

The proper axle spacing and velocity to be used must be determined by the judg­ment of the Bridge Engineer for each individual structure. To a certain extent this w i l l have to be based on a statistical study of the variations of the axle spacings and speeds of the vehicles using the portion of the highway on which the bridge is located. Excessive repetitions of heavy axle loadings may conceivably accelerate the break-up of the concrete roadway slab if the frequency of application of the axle loadings is near or in resonance with the natural frequency of vibration of the structure. There is also the matter of driver psychology fo r those in passenger automobiles and pedestrians who may be using the bridge, i f the amplitude of vibration is appreciable.

Recommendations

It is recommended that the design procedure for the three span continuous I-beam highway bridge should consider the vibration problem. The recommended procedure I S as follows:

1. Design the structure as usual to keep the stresses within the allowable l imi t s . 2. Determine the natural frequency of vibration of the structure.

a. The value of Coefficient F may be obtained f r o m charts as shown in Figure 28. Then the natural frequency of vibration may be computed f r o m Equation 9, repeated below:

L

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TABLE 10

NATURAL FREQUENCY OF VIBRATION OF TEST STRUCTURE CONSIDERING MASS OF DESIGN LIVE LOAD WITH VARIOUS SIZE STRINGERS FOR

STRINGER B, AS TYPICAL

Percentage Participation of Slab with Stringer

Stringer Make-Up 100 0, n = 10

Frequency, Cycles per Second

L 33 WF 130' 2. 59 3.99 n. 36 WF 150 2. 92 4.31

ni. 36 WF 194 3.30 4. 67 IV. 36 WF 300 4.03 5.40 V, Built-up Section

4 - Angles 6 x 6 x 3.07 4. 56 1 - Web Plate 39^2 x fe

132.8 lb. per f t . V I . Built-up Section

4 - Angles 6 x 6 x 3.43 5.04 1 - Web Plate 43^2 x fs

137. 9 lb . per f t . vn. 30 WF 108 3.33

' stringer size used in the structure. ^ Stringer size required based on design with f u l l composite action.

b. The designer w i l l have to determine the size and number of trucks to use to compute wi^ depending upon circumstances at each individual bridge site, usually one truck is c r i t i ca l .

3. Choose the c r i t i ca l axle spacing and vehicle velocity fo r the site of the struc­ture and determine the frequency of application of axle loadings to a point in the structure. See Figure 29 fo r the variations in this frequency fo r various axle spacings and speeds.

4. Increase this frequency of application of axle loadings fo r a suitable safety margin with respect to resonance. A division factor of about 0.8 should give an ade­quate safety margin. The magnification curve in Figure 31 w i l l aid the judgment in this matter.

5. If the natural frequency of vibration of the structure, considering the mass of the live load, is greater than this corrected frequency of application of axle loads, the vibration w i l l not be serious.

6. If the natural frequency of vibration of the structure, considering the mass of the live load, is less than this corrected frequency of application of axle loads, the vibrations may be serious and the stiffness of the structure may have to be increased.

The stiffness of the structure may be increased materially by providing shear con­nectors to give f u l l participation of the concrete slab with the steel stringers, i f the structure has not been designed fo r composite action. Otherwise larger stringers must be used. Shear connectors may lower the damping characteristics of the struc­ture, since the absorption of energy by the slipping between the concrete and the steel w i l l be eliminated. This would increase the dynamic stresses and amplitudes of vibration.

Vibration w i l l not be serious in a l l structures, but the procedure recommended herein gives a simple way to check into the vibration problem. As allowable unit stresses are increased and thinner slabs and shallower stringers are used, the v i ­bration problem may become more important.

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ACKNOWLEDGMENTS

This study was made under the direction of the Joint Highway Research Project, Engineering Experiment Station, Purdue University. The original impetus fo r the study came f r o m J. R. Cooper, Engineer of Bridges, State Highway Department of Indiana, and Professor L . T. Wyly, Head, Department of Structural Engineering, School of C iv i l Engineering, Purdue University. Professor Wyly gave counsel throughout the entire study.

The f i e l d study was made with the assistance of the Division of Maintenance, State Highway Department of Indiana. W. H. Sorrel l is Superintendent of Maintenance. H. M . Sullivan is Dis t r ic t Maintenance Engineer in the dis t r ic t in which the work was done. Fred Stock is Sub-District Superintendent m the sub-district where the bridge is located. Rex Webb and C. F. Hotler of the Division of Tests of the State Highway Department of Indiana drove the state test t ruck during the f i e l d study. The f i e ld study could not have been made without the whole-hearted cooperation of these men.

August L . Flassig, J r . , now at the Engineering Research Laboratories of the Association of American Railroads, was the electronic technician on the project dur­ing the time data was taken on the dynamic action of the test loads.

Both the f i e l d study and the digest of the data have been under the direct super­vision of John M . Hayes with John A. Sbarounis serving as principal assistant.

References 1. Darnley, R .R . , "The Transverse Vibrations of Beams and the Whir l ing of

Shafts Supported at Intermediate Points", Philosophical Magazine, Vol . 41, 1921, p. 81. 2. Vandegritt, L . E . , "Vibration Studies of Continuous Span Bridges", Ohio State

University Engineering Experiment Station Bullet in No. 119, July 1944, pp. 38-57. 3. Timoshenko, S., Vibration Problems in Engineering, New York, 1937, D. Van

Nostrand Company, Inc . , p. 85 and pp. 345-348. 4. Hansen, H. M . and Chenea, P. F . , Mechanics of Vibrations, 1952, New York,

John Wiley and Sons, Inc . , pp. 78-91. 5. Wise, J. A . , "Dynamics of Highway Bridges", Highway Research Board Pro­

ceedings, 1953, pp. 180-187. 6. Inghs, C. E . , A Mathematical Treatise on Vibrations in Railway Bridges, 1934,

London, Cambridge University Press. 7. The American Association of State Highway Officials , Standard Specifications

f o r Highway Bridges, Sixth Edition, 1953. 8. "Distribution of Load Stresses in Highway Bridges", Highway Research Board

Research Report 14-B, publication 253.