15 - Impact Factors for Simple-Span Highway Girder Bridges

I M P A C T F A C T O R S F O R S I M P L E - S P A N H I G H W A Y G I R D E R BRIDGES

By Dongil Chang, 1 Member, ASCE, and Heehyun Lee 2

ABSTRACr: Vibrational behavior of simple-span highway girder bridges with rough surfaces due to heavy trucks is discussed. The causes of vibration and dynamic characteristics of bridge structures are investigated in the time and frequency do- mains. An appropriate vehicle model for vibrational analysis of bridges is found by comparing dynamic responses from four different vehicle models, and impact factors are calculated using the suggested vehicle model with different speeds, deck roughnesses, and span lengths. The results are compared with the values specified by several standard codes, and it is found that the current design codes tend to underestimate impact factors especially in long-span bridges with rough decks. Finally, empirical formulas for impact factors represented in terms of span length, vehicle speed, and surface roughness of a bridge deck are suggested from the multiple linear regressional using the data obtained from this study.

INTRODUCTION

Bridge structures that have long service years or long spans, or that are frequently subjected to heavier loadings than their design loads, are greatly affected by heavy traffic-induced vibrations. Many bridge engineers t reat such vibration problems by considering only impact factors specified in their current design codes, even though the vibrat ions may depend on such factors as vehicle and bridge dynamic characteristics, vehicle speeds, and deck conditions. In recent years, several questions have been proposed on whether the current specification on impact factors is adequate under the present traffic and road situations, because the specification was made under the conditions of several decades ago when vibrat ion problems were not of importance.

The main objectives of this paper are to investigate vibrat ional behavior of highway girder bridges induced by heavy trucks and to show a way to estimate impact factors appropr ia te ly .

VIBRATION ANALYSIS

Response in Time Domain The equations of mot ion of a bridge discretized by finite e lements can

be written in the form:

i ~ + C a + K u : P ( 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (1)

in which M, C, and K = the mass, damping and stiffness matrices, respectively, and are symmetric; and u and P(t) = the vectors of nodal displace- ments and loads, respectively. A dot represents a derivative with respect to time. The load vector is constructed by t ransforming an arbi t rary point

1prof., Dept. of Civ. Engrg., Hanyang Univ., Seoul, 133-791, South Korea. 2Mgr. of Dyn., KHRC, Seoul, 100-714, South Korea; formerly, Lect., Dept. of

Civ. Engrg., Hanyang Univ., Seoul, South Korea. Note. Discussion open until August l , 1994. To extend the closing date one month,

a written request must be filed with the ASCE Manager of Journals. The manuscript for this paper was submitted for review and possible publication on February 27, 1992. This paper is part of the Journal of Structural Engineering, Vol. 120, No. 3, March, 1994. �9 ISSN 0733-9445/94/0003-0704/$2.00 + $.25 per page. Paper No. 3561.

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load at arbitrary positions into joint loads by using the cubic Hermitian interpolation functions. Then, (1) can be solved in the time domain by the mode superposition method (Bathe and Wilson 1976; Biggs 1964).

Response in Frequency Domain The frequency response function S(f) of the time signal x(t) is defined as

S(f) = ~ x(t) exp(-j2"rrft) dt; j = V ' - - Z 1 " . . . . . . . . . . . . . . . . . . . . ( 2 )

in which T = measured time of response signal; and t and f = variables representing time and frequency, respectively. The fast-Fourier-transform (FFT) algorithm is used to solve (2). Then, the main causes inducing vibration of the bridge are observed from the frequency response function (Bendat and Piersol 1971, 1980).

Pavement Roughness A vehicle traveling over a highway containing surface irregularities ex-

periences vertical and horizontal motions. Associated with the vertical mo- tion are forces between the highway and the vehicle that are developed in addition to the dead load of the vehicle. These forces are frequently referred to as the dynamic reactions or the dynamic forces. They depend on the suspension characteristics, the condition of the pavement, and velocity of the vehicle. Based on these characteristics, the load imposed on a pavement can be double the static loads of vehicles (Quinn and Thompson 1962).

Road roughness usually refers to an uneven, impaired, or bumpy road- way. According to Balmer (1973), ASTM defines roughness as "the deviations of a pavement surface from a true planar surface with characteristic dimensions greater than 16 mm" to distinguish roughness from pavement texture. Highway elevation profiles are either periodic or nonperiodic. In case of highway profiles exhibiting well-defined periodicity, they can be directly analyzed using a Fourier series. However, other highway profiles do not display a well-defined period and therefore do not lend themselves to this type of analysis. In this case, it is convenient to assume that the highway elevations are random and to apply a statistical analysis commonly used in dealing with random phenomena. One method of characterizing a random function is by use of the power spectrum (Quinn and De Vries 1960).

Ten sets of spectrum-compatible pavement profiles are generated by using the power spectrum of road-surface roughness (Honda et al. 1982). One example of the simulated profiles is shown in Fig. 1. Table 1 shows maximum magnitudes of the generated profiles multiplied by 0.5, 1.0, 2.0, and 3.0 with different span lengths. The vibrational-analysis method of bridge structures considering surface roughness is explained in Lee (1985, 1989), Honda et al. (1986), Inbanathan and Wieland (1987), and Kim (1986).

RESULTS

Vehicle and Bridge Models The simple-span girder bridges can be idealized as simple beams; how-

ever, it is known that the suspension system and inertia forces of the vehicle should be incorporated when modeling the vehicle for bridge vibration analysis (Biggs 1964; Honda et al. 1986).

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0.02 ~ (unit:m) [

~' o.oo " ~'~ ~,f ~v~ "

'~ -0.02 / I I i I i I i I i i I i I o') 0 20 40 60 80 100 120

Distance

FIG. 1. Example of Artificially Generated Spectrum--Compatible Pavement Pro- file

(a) (b)

k•c kz ~ cL

(c) (d)

FIG. 2. Four Vehicle Models: (a) Moving Force; (b) Moving Mass; (c) Single DOF; and (d) Two DOFs

TABLE 1. Maximum Magnitudes of Generated Pavement Profiles with Different Span Lengths

Span Length (m)

Multiplier 25 50 75 100 (1) (2) (3) (4) (5)

0.5 0.00367 0.00423 0.00464 0.00464 1.0 0.00733 0.00845 0,00928 0.00928 2.0 0.01466 0.01690 0.01856 0.01856 3.0 0.02199 0.02535 0.02784 0.02784

To confirm this, vibration responses obtained from the four different vehicle models in Fig. 2 are compared in Fig. 3.

When modeling the vehicle, total weight is 210 kN, and first and second natural frequencies are 3.1 and 13.0 Hz, and their respective damping ratios are 3% and 9%. Deflect ion t ime histories i l lustrated in Fig. 3 are calculated at midspan when the vehicle is moving at a speed of 19.4 m/s over a bridge of 50-m span length (Table 2).

Fig. 3 and Tables 3 and 4 show that vibrat ion responses are very different depending on the vehicle model when the effect of pavement condit ion is included, but they are not different when the pavement condit ion is ignored.

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TABLE 2. Sectional Properties of Bridges of Various Span Lengths

Section property (1)

Beam depth (m) Cross-sectional area (m 2) Moment of inertia (m 4) Mass per unit length (kg/m) Flexural eigenfrequency (rad/s)

Span Length (m) 25 (2) 1.25 4.46 0.86

11,150 24.58

50 (3) 2.50 5.96 4.94

14,900 12.74

75 (4) 3.75 7.46

13.58 18,650

8.39

100 (5) 5.00 8.96

28.01 22,400

6.18

TABLE 3. Impact Factors of Bridges with Different Vehicle Speeds and Models Including Surface-Roughness Effects [Korean Code Value of 0.13 (Standard 1982)]

Model (1)

Moving force Moving force

Moving mass Moving mass

Single DOF Single DOF

Two DOFs Two DOFs

Statistical value 60 (2) (3)

(a) Deflection

Mean 0.16 Mean plus standard 0.21

deviation Mean 1.32 Mean plus standard 2.19



deviation

(b) Moment

Moving force Moving force

Moving mass Moving mass

Single DOF Single DOF

Two DOFs Two DOFs

Mean Mean plus standard

deviation Mean Mean plus standard



deviation

Speed (km/h)

1 7 0 1 8 0 ) 9 0 100 (4) (5) (6) (7)

0.38 0.36 0.30 0.50 0.56 0.41

1.51 0.66 0.66 2.28 0.95 1.05

0.23 0.24 0.25 0.32 0.35 0.35

0.08 0.05 0.10 0.09 0.06 0.10

0.30 0.36

0.99 1.35

0.29 0.41

0.13 0.13

1.49 1.07 3.06 2.56 2.35 1.99

2 . 8 0 1 2 . 3 8 ] 1.38 I 1 . 4 6 1 1 . 8 3 4.52 3.33 1.84 2.12 2.46

0.11 I 0.28 [ 0.19 I 0.18 I 0.25 0.17 0.37 0.33 0.29 0.35

0.03 [ 0.13 I 0.06 I 0.06 I 0.14

Table 3 shows the values of mean and mean plus standard deviation for impact factors obtained at midspan when including the effects of 10 sets of simulated profiles. Table 4 is for impact factors with different vehicle speeds when neglecting the pavement effect.

Fig. 3 and Tables 3 and 4 also show that vibration responses are greatly overestimated if a moving force or moving mass model that includes the

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TABLE 4. Impact Factors of Bridges with Different Vehicle Speeds and Models Neglecting Surface-Roughness Effects [Korean Code Value of 0.13 (Standard 1982)]

Speed (kin/h)

Model 60 70 80 I 90 1 O0 (1) (2) (3) (4) I (5) (6)

(a) Deflection

Moving force 0.07 0.01 0.06 0.11 0.15 Moving mass 0.08 0.09 0.06 0.12 0.16 Single DOF 0.07 0.01 0.06 0.11 0.15 Two DOFs 0.07 0.01 0.06 0.11 0.15

(b) Moment

Moving force 0.00 0.07 0.00 0.00 0.11 Moving mass 0.02 0.05 0.00 0.03 0.13 Single DOF 0.00 0.07 0.00 0.00 0.11 Two DOFs 0.00 0.07 0.00 0.00 0.11

1.00

.~ 0.80

0 . 8 0

0.40

~ 0.20

0.00 0

FIG. 4.

0.70Hz,

2.09Hz

I0 20 30 40 Frequency(Hz)

Fourier Amplitude Spectrum of Fig. 3(d)

effect of surface condition is used. On the other hand, if the surface condition is neglected, the responses are almost the same regardless of the vehicle model used, So, in subsequent parts of this paper, the vehicle is modeled as a two-degree-of-freedom (DOF) vibrating system to analyze bridge vibration incorporating surface roughness.

The amplitude spectrum of Fig. 3 (d) i s shown in Fig. 4. In Fig. 4, the vibration components corresponding to 0.7, 2.09, and 17.67 Hz are related to vehicle speed, first-mode flexural eigenfrequency of bridge, and surface roughness, respectively. Therefore, it can be said that the main causes of bridge vibrations are these three parameters. Table 2 shows the sectional properties of simple two-lane bridges of various span lengths.

Impact Factors due to Three Parameters Impact factors are calculated by changing three parameters, i.e., surface

roughness condition, vehicle speed, and span length. Impact factor is defined as maximum dynamic response divided by maximum static response minus 1 at the midspan of bridge. The current Korean code (Standard 1982) spec- ifies impact factor in terms of span length only, i.e., i = 20/(50 + L), where L is span length in meters.

Tables 5 -8 show impact factors of deflection and bending moment at

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TABLE 5. Impact Factors for Bridge of 25-m Span Length

Speed (kin/h)

Multiplier 60 70 80 90 100 (1) (2) (3) (4) (5) (6)

(a) Deflection

0.5 0.05670 0 . 0 7 2 1 6 0 . 0 5 5 1 6 0 . 1 0 5 2 2 0.13629 1.0 0.05175 0 . 0 7 6 0 2 0 . 0 5 7 8 2 0 . 1 0 5 2 6 0.13823 2.0 0.04732 0 . 0 8 4 3 4 0 . 0 6 3 1 9 0 . 1 0 5 3 0 0.14246 3.0 0.05306 0 . 0 9 3 1 8 0 . 0 6 8 5 7 0 . 1 0 5 3 6 0.14706

(b) Moment

0.5 0.01493 0 . 0 5 5 9 7 0 . 0 1 3 0 2 0 . 0 4 4 7 8 0.13433 1.0 0.01119 0 . 0 7 0 9 0 0 . 0 1 2 5 6 0 . 0 6 3 4 3 0.17910 2.0 0.02985 0 . 0 9 7 0 1 0 . 0 2 2 3 9 0 . 1 1 1 9 4 0.26493 3.0 0.05224 0 . 1 2 6 8 7 0 . 0 6 3 4 3 0 . 1 7 1 6 4 0.35448


Multiplier (1)

6O (2)

Speed (km/h)

70 80 90 1 O0 (3) (4) (5) (6)

(a) Deflection

0.5 0.05258 0 . 0 8 1 6 1 0 . 0 4 3 5 9 0 . 0 9 4 5 4 0.12938 1.0 0.05426 0 . 0 9 2 2 9 0 . 0 5 0 0 5 0 . 0 9 8 8 7 0.13354 2.0 0.05858 0 . 1 1 3 5 3 0 . 0 6 2 5 0 0 . 1 0 7 9 1 0.14193 3.0 0.06470 0 . 1 3 4 8 9 0 . 0 7 5 1 2 0 . 1 1 6 8 4 0.15029

(b) Moment

0.5 0.00000 0 . 1 4 2 5 9 0 . 0 0 1 8 8 0 . 0 6 0 0 4 0.14259 1.0 0.01689 0 . 1 9 3 2 5 0 . 0 1 8 7 6 0 . 1 2 0 0 8 0.18386 2.0 0.06567 0 . 3 8 0 8 6 0 . 1 4 4 4 7 0 . 2 6 0 7 9 0.28705 3.0 0.11069 0 . 5 3 8 4 6 0 . 2 7 4 9 0 0 . 3 9 5 8 7 0.38274

midspan with different three parameters. From these tables, impact factors increase with surface roughness and vehicle speed, but do not decrease with span length as specified in the current code (Standard 1982). In addition, comparing impact factors for deflection and bending moment of various span lengths, impact factors for deflections are higher than those for mo- ments when deck pavement is smooth and lower for rough pavement. There- fore, it can be imagined that pavement roughness would make the bridge damaged by fatigue because the fatigue phenomenon of the bridge is closely related to the bending movement. Furthermore, the current Korean code underestimates impact factors in particular for long rough bridges. Based on the results mentioned here, a way to estimate the impact factor appro- priately is discussed next.

Reasonable Estimation of Impact Factor The multiple linear regressional analysis (Ang and Tang 1975) is con-

ducted using the data obtained before to derive empirical formulas for impact factors. The regressional equations for deflection and bending mo-

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Speed (kin/h)

O) (2) (3) (4) (5) (6) (a) Deflection

0.5 0.05628 0.07282 0.04105 0.09916 0.13097 1.0 0.05718 0.07339 0.04294 0.10304 0.13244 2.0 0.05898 0.07454 0.04739 0.11117 0.14044 3.0 0.06087 0.07568 0.05183 0.11945 0.14943

(b) Moment

0.5 1.0 2.0 3.0

0.01197 0.02476 0.05033 0.07592

0.05540 0.06687 0.08983 0.11278

0.09753 0.02002 0.04375 0.06976

0.02222 0.03627 0.06497 0.09389

0.11456 0.12701 0.15232 0.17763


Speed (kin/h) Multiplier 60 70 80 90 100

(1) (2) (3) (4) (5) (6) (a) Deflection

0.5 1.0 2.0 3.0

0.06073 0.06153 0.06342 0.06532

0.07310 0.07609 0.08217 0.08825

0.04886 0.04836 0.04846 0.04986

0.10351 0.10560 0.10969 0.11408

0.13492 0.13672 0.14051 0.14659

(b) Moment

0.5 1.0 2.0 3.0

0.01999 0.02581 0.03857 0.05153

0.04280 0.04927 0.06241 0.07555

0.01642 0.02196 0.03105 0.04871

0.07048 0.11947 0.21933 0.33731

0.19578 0.28212 0.47198 0.66466

TABLE 9. Coefficients Resulting from Multiple Linear Regressional Analysis

Conditional Conditional s t a n d a r d Correlation

Response Variance variance deviation coefficient (1) (2) (3) (4) (5)

Deflection 0.00113 0.00042 0.02059 0.79267 Moment 0.01844 0.01043 0.10211 0.65908

m e n t are as fol lows, a n d T a b l e 9 shows t h e r e s u l t i n g v a r i a n c e s a n d s t a n d a r d devia t ions .

�9 D i s p l a c e m e n t :

i _ 644 6 3 7

104 107

185 6 4 8 - - L + - ~ - V + ~ - R . . . . . . . . . . . . . . . (3a)

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�9 Bending moment:

i - 327 898 441 748 103 + ] - ~ L + - ~ V + ~ R . . . . . . . . . . . . . . . (3b)

in which i, L, V and R = impact factor, span length (meters), vehicle speed (kilometers per hour), and maximum magnitude of surface roughness (meters), respectively. From Table 9, impact factor has high correlation with vehicle speed, span length and surface roughness. In this paper, the following is suggested for the conservative estimate, which is three times the condi-

FIG. 5.

0.80

0.60

-~ 0.40

0.20

0.00

c c c c c K o r e a n Standard C o d e ~ e ~ - ~ S u g g e s t e d E q . c o n s i d e r i n g s u r f a c e roughness ~ , ~ - ~ a S u g g s s t e d E q , n e g l e c t i n g s u r f a c e roughness ~ a ~ O n t a r i o B r i d g e C o d e : : : : : British Standards

0 .

i i i i i I i i i ~ I i i i i I i i i i I i i i i

25 50 75 100 125 Span l e n g t h ( m )

Impact Factors for Deflection of Bridges with Different Span Lengths

FIG. 6,

0.30"

0.20

~o.io

0.00

===== Sugges ted e q u a t i o n

S

�9 I I

, i J i t i i q i i i i t i i i i i t i i i i i i i i I d I I

50 60 70 80 90 I00 110 Veloe i ty (km/ /h)

Impact Factors for Deflection of Bridges with Different Vehicle Speeds

0.30

o.2o

~ 0 . i0

0.00

oooo= Sugges ted e q u a t i o n

* ~

i j i i i i i ] J t l ) i i i i i i i ] i 1 . 1 i l l i i i

0.00 0.01 0.02 0.03 Magni tude of s u r f a c e r o u g h n e s s ( m )

FIG. 7. Impact Factors for Deflection of Bridges with Different Surface Bough- nesses

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1.00 c c c c c K o r e a n S t a n d a r d Code

Suggested Eq. considering surface roughness S u g g e s t e d Eq. n e g l e c t i n g s u r f a c e r o u g h n e s s

o ~ 0 . 7 5 ~ O n t a r i o B r i d g e Code c:::: British Standards

,~ 0 .50 *

,

0 .25 t

i 0 .00 , , , , , , , , , 0 25 50 75 100 125

S p a n l e n g t h ( m )

FIG. 8. Impact Factors for Bending Moment of Bridges with Different Span Lengths

FIG. 9. Speeds

1.00

0 .75

o.50 g~

0 .25

c = = = = S u g g e s t e d e q u a t i o n

J

i i I 0 0 0 . . . . i, . . . . . . . . , . . . . . . . . . . . .

5 0 6 0 7 0 8 0 9 0 1 0 0 1 1 0 V e l o c i t y ( k m / h )

Impact Factors for Bending Moment of Bridges with Different Vehicle

1 . 0 0

o ~ 0 .75

0.50

0.25

0.00

o [] [] [] Suggested e q u a t i o n

�9 ' ; ! ,:,,,I 0.00 0.01 0 .02 0 .03

M a g n i t u d e o f s u r f a c e r o u g h n e s s ( m )

FIG. 10. Impact Factors for Bending Moment of Bridges with Different Surface Roughnesses

tional standard deviations in (3) and is obtained by adding the normal distribution of impact factors.

Displacement:

i - 265 637 185 648 10 5 10 ~ L + ~ V + ~ R . . . . . . . . . . . . . . . (4a)

�9 Bending moment:

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i = . . . . . . . . . . . . . . . (4b) 202 898 441 748

- 10----~ + -]-0-7 L +-~V+-~R Figs. 5-7 show calculated impact factor of displacement with different

span lengths, vehicle speeds, and maximum magnitudes of surface roughnesses, respectively; Figs. 8-10 are for bending moment.

Also in Figs. 5 and 8, various bridge-code values are plotted together with (4a) and (4b). From these figures, almost all of the calculated values are under the empirical values, and current codes underestimate impact factors especially for vehicles running with high speeds over a long rough bridge. Impact factors neglecting surface-roughness terms in (4a) and (4b) are also given in Figs. 5 and 8 for comparison. Among the codes, the Ontario bridge code (OBC) (Csagoly and Dorton 1977) gives a reasonably high and constant values for impact factors regardless of span length like the British standard ("Specification" 1966). Unlike the other codes, the OBC impact factors were not formulated considering span length only but on the basis that impact effects due to roughness and frequency ratio between bridge and vehicle may amplify the dynamic response. The results of this paper coincide comparatively with the OBC specification.

CONCLUSIONS

Because impact effects due to running heavy vehicles are caused mainly by bridge and vehicle characteristics and by surface roughness, impact factors should be represented in terms of these three parameters.

Impact factors increase with vehicle speed and surface roughness, however, are almost constant with span length. So it may be reasonable to use constant impact factor regardless of span length as in the Ontario and British codes. In addition, impact factor of moment is higher than that of deflection when the bridge deck is deteriorated.

If the effect of surface condition could be neglected as in the construction stage, any vehicle models among the four used in this paper can be adopted to estimate vibration response of the bridge. However, if surface condition could not be neglected, the single- or two-DoF vehicle model has to be used.

APPENDIX. REFERENCES

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Documents

15 - Impact Factors for Simple-Span Highway Girder Bridges