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Dynamic interaction between heavy vehicles and highway bridges
Peqpmon PII:
DYNAMIC
Computers & Srrwtures Vol. 62, No. 2, pp. 253-264, 1997 Copyright0 19%Ehwier Sciemx Ltd Printed in Great Britain. All rights reserved 0045.7949/97 $17.00 + 0.00 soo4s7!34!y%)00198-8
INTERACTION BETWEEN HEAVY VEHICLES AND HIGHWAY BRIDGES M. F. Greent
and D. Cebon$
TDepartment of Civil Engineering, Queen’s University at Kingston, Ontario, Canada K7L 3N6 SUniversity Engineering Department, Trumpington Street, Cambridge CB2 lPZ, U.K. (Received 9 June 1995) Ahstraet-This paper discusses the importance of dynamic interaction (coupling) between heavy vehicles and highway bridges. A new method for calculating the dynamic response of bridges to dynamic wheel loads is presented, and used to investigate a simply-supported bridge traversed by a single degree of freedom vehicle model. Six non-dimensional parameters of the bridge-vehicle system are varied to assess the amount of interaction between the vehicle and the bridge. Guidelines are presented to ascertain when dynamic interaction is important and when it can be ignored. Copyright @ 1996 Efsevier Science Ltd.
NOTATION vehicle damping constant viscous damping operator with respect to the spatial variables vehicle damping matrix damping matrices for suspension and tire elements maximum dynamic response increment Young’s modulus of elasticity force exerted by the vehicle on the bridge vectors of suspension and tire forces _ product of mode shape value and wheel load vector of road profile heights under the tires impulse response function for force applied at position xr and response at position x modal impuhe response function normalized step height, H,/yst height of step up at bridge entry second moment of area of the bridge (beam model) vehicle stiffness matrix stiffness matrices for suspension and tire elements vehicle suspension stiffness wheel number bridge length self-adjoint linear differential operator with respect to the spatiai variables mass per unit surface area of the bridge mass per unit length of the bridge (beam model) inertia matrix of vehicle vehicle mass mode number number of tires on the vehicle dynamic wheel loads vector of generalized coordinates for the vehicle vector of generalized forces applied to the vehicle radius of level curves of normalized error o, (Appendix) constant t~nsfo~ation matrices relating the
t TV tu1 V x0 X
xr Xl
Yh
t)
Y(t) vi(t)* PO Y/(6 Y&
suspension and tire forces to the generalized coordinates time time for vehicie to completely cross the bridge constant vector of generalized gravitational forces applied to the vehicle vehicle speed bridge midspan position, L/2 vector describing a position on the bridge surface point of application of the force input position of lth wheel transverse deflection of the bridge displacement response of the bridge measured under the tires of the vehicle two oscillatory functions (Appendix} $h estimate for bridge displacement response initial estimate for bridge displacement response maximum dynamic response of the bridge maximum static response of the bridge amplitudes of the oscillatory functions y,(r), Y&I
QM=W
static deflection (Ap~ndix) normalized vehicle acceleration speed parameter, x V/oh L relative amplitude parameter, Yz/YI (Appendix) frequency ratio, wv/wb mass ratio, m,/mL mean square error (Ap~ndix) interaction error term nth bridge damping ratio first bridge damping ratio vehicle damping ratio amplitude error (Appendix) phase error (Appendix) modal mass ratio, 2 m,/mL standard deviation of interaction error integration variable nth mode shape value at position x angular frequency (rad s-l) first bridge natural frequency (rad s-‘) 8th bridge naturai frequency (rad s-l) vehicle natural frequency (rad s-l) 253
254
M. F. Green and D. Cebon 1. INTRODUCTION
Highway bridges are subjected to the dynamic tire forces of heavy vehicles. The resulting dynamic response can lead to bridge deterioration; eventually increasing maintenance costs, and decreasing service life. In order to assess and design bridges for dynamic wheel loads, or to design vehicles to reduce the dynamic response of bridges, a fundamental understanding of the nature of the dynamic interaction between vehicles and bridges is necessary. Figure 1 is a schematic diagram of bridge--vehicle interaction. The roughness input to the oehicle is the sum of the initial surface profile of the bridge and the dynamic deflection of the bridge. This input excites the vehicle and results in dynamic tire forces. These forces are in turn applied to the bridge and cause larger dynamic displacements of the bridge. This feedback mechanism of interaction forces couples the dynamic response of the bridge to that of the vehicle. The largest bridge vibrations are known to occur when the natural frequencies of the bridge and vehicle are close together [l]. It is not clear, however, under what circumstances large bridge responses are mainly due to excitation of the vehicle by the initial roughness profile, or due to excitation substantially by dynamic deflection of the bridge. The former case can be considered as the feedback loop in Fig. I being cut at point “A”. There has been little systematic study about when dynamic interaction is important, and when the two systems can be considered to be essentially uncoupled. 2. BACKGROUND
Walker and Veletsos [2] considered the response of a simply-supported bridge to a moving constant force and to a moving sprung mass. They compared the two responses by computing the maximum dynamic increments for each case. Their definition of the maximum dynamic increment was
where DZ is the maximum increment, yrnaris the maximum
dynamic response dynamic response of
the bridge, and y,, is the maximum static response of the bridge. They related the importance of interaction to the frequency ratio, 7, defined as
where w, is the natural frequency of the vehicle and (11~is the first natural frequency of bending of the bridge. Bridge-vehicle interaction was deemed unimportant when ;’ was less than 0.3 or greater than about 1.0 [2]. This criterion seems quite reasonable, but Walker and Veletsos did not quantify the errors incurred by neglecting bridge-vehicle interaction. Chiu et al. [3] conducted a parametric study similar to that of Walker and Veletsos for an elevated guideway traversed by a high-speed vehicle. They found that dynamic interaction was unimportant when the ratio of the vehicle mass to the mass of the guideway span was below l/S, when the frequency ratio, ;‘, was low. Most of their results were calculated for relative speeds far in excess of those usual on highway bridges. Their criteria are consistent with those of Walker and Veletsos, but do not provide enough detail for low speeds. Finally, Ting and Genin [4] outlined interaction criteria based on the quantity 2~2 where 6 is the vehicle/bridge mass ratio, and 2 is the second time derivative of the vehicle displacement normalized by the maximum static deflection of the bridge. They concluded that interaction can be ignored if 2c.?!
3. BRIDGE RESPONSE CALCULATION-THEORY AND VALIDATION
3. I
Theory
The dynamic response of most bridges to arbitrary heavy vehicle loads can be represented by the following equation of motion:
I
I
A
Dynamic Tire FOfCBD
t
1ay I + L{Y@, 0) =f(x,
I Dynamic
azy
m(x,,,i(x, t) + C -;i;(x, t)
Bridge Deflection
t)
(3)
Dynamica
Fig. 1. Schematic block diagram of dynamic bridge-vehicle interaction.
where L{o} is a self-adjoint differential operator with respect to the spatial variables, x is a two dimensional
Heavy vehicles and highway bridges position vector, t is time, m(x) is the mass per unit surface area, y(x, t) is the transverse deflection of the bridge, C(o) is a viscous damping operator with respect to the spatial variables which must satisfy certain orthogonality conditions [5], andf(x, t) is the force exerted by the vehicle on the bridge. The derivation of this differential equation assumes linear viscous damping, linear elasticity, small deflections, and neglects shear deformation and rotary inertia [5,6]. The solution to the equation of motion can be expressed by the following convolution integral:
s m
Yk t) =
h(x, xr, t - rMf(xr,r)dr
-a
(4)
255
dynamic vehicle response model through the following iterative procedure:
(0 Calculate the vehicle response to a specified bridge surface profile and hence, a first set of dynamic wheel loads. (ii) Calculate the displacement response of the bridge under each tire of the vehicle due to the first set of wheel loads. (ii) Add the bridge displacement response to the surface profile and use this as the new input to the vehicle model. Calculate the new dynamic wheel loads. (iv) Recalculate the displacement response, y(t), from the loads in (iii) by the procedure of (ii). Average successive displacement responses to facilitate convergence y
where h(x, x, t) is the impulse response function at position x for an impulse applied at position xf. Therefore, eqn (3) may be solved by determining the appropriate impulse response function, providing the vehicle loads, f(xl, t), are known. If the deflection, y(x, t) is expressed in terms of normal mode functions, &“)(x), it can be shown that the response to a moving vehicle with Nt wheels, is [5,6]
ct)
=
I
YCr) Yi2 +
I(l)
where y, is the jtb estimate for the bridge displacement response. (v) Repeat steps (iii) and (iv) until convergence is obtained. Convergence is deemed to occur when the following criterion is satisfied IYW - Y/- IWI
max(l
< tolerance,
for 0 S t G Tvpr (7)
y(x, t) =
-
t
f
,=I .=I
4’“‘(x)
h,(t - t)g’“*“(r)dt,
(5)
where h,(t - 7) is the modal impulse response function, and g97) = c#3xl(7)) x S(7). In the formula for g@*‘)(r),P,(t) is the dynamic wheel load for wheel I and x,(t) is the position of that wheel. With the general bridge response written in this convolution format, the equation of motion [eqn (3)] can be solved by evaluating the convolution integral in either the time or frequency domains. For this study, a frequency domain method using discrete Fourier transforms was chosen. By evaluating the discrete Fourier transform with the fast Fourier transform (FFT) algorithm [7], the integral can be solved significantly faster than by equivalent time domain convolution [5,6]. The modal formulation of this calculation procedure is efficient. Mode shapes can be obtained from a variety of sources (e.g. finite element models, measurements). Additionally, only a few mode shapes are normally required to model the dynamic characteristics of a bridge. Therefore, this method is much faster than an alternative finite element model which would have many more degrees of freedom. The calculation described above assumes that the wheel loads, P,(t), are specified and does not account for the dynamic interaction between the bridge and the vehicle. The interaction can be included for any
where T,, is the time for the vehicle to pass completely over the bridge. For the results presented in this paper, the tolerance was set to 0.02 (i.e. 2% of the maximum displacement of the bridge). Similar iterative methods have been used by other authors and the results have compared favourably against Runge-Kutta-Nystrom numerical integration [8]. Other demonstrations of the convergence of the iterative method can be found in Refs [S, 61. 3.2. Validation An experimental procedure was developed to validate the convolution calculation presented in the previous section. The procedure consisted of four main steps. The first step was the measurement of the transfer functions and mode shapes of the bridge. Step two consisted of single vehicle tests in which dynamic wheel forces and bridge responses were measured simultaneously. In step three, the measured dynamic wheel forces were combined with the modal properties of the bridge to predict bridge responses. Finally, these predicted responses were compared with the measured bridge responses to validate the model. It should be noted that this experimental procedure validated the bridge calculation, but not the iterative procedure for combining the bridge and vehicle models.
256
M. F. Green and D. Cebon Lower Earley bridge: Validation results South to north at 50 km/h
Time (s)
Fig. 2. Validation results; midspan bridge response; 50 km h-l, South
to north;
measured
-;
predicted
---
-.
Two typical highway bridges near Reading, England were tested. The first was a four span, prestressed concrete, box-girder bridge, and the second was a three-span, slab on girder, prestressed concrete bridge. The results of these tests are presented elsewhere [5, 61 and will not be included here in detail. However, for demonstration purposes, Fig. 2 shows a typical validation result for the Lower Earley bridge with the vehicle travelling at 50 km h- ‘. In this case, the response was measured at the midspan of the centre girder of this slab on girder bridge. The agreement between the calculation method and experiment is very good.
4. VEHICLE
SIMULATION
A lumped parameter vehicle model is assumed with the equation of motion given by [9]
[Ml(q) = [SIJK) + [Tl{F,J + iq,
(8)
where [M] is the inertia matrix of the vehicle, {q} is the vector of generalized coordinates for the vehicle, {F,, F,) are the vectors of “suspension” and “tyre” forces, {U} is the constant vector of generalized gravitational forces applied to the vehicle, and [S], [T] are the constant transformation matrices relating the suspension and tyre forces to the generalized coordinates. This formulation of the equations of motion is very versatile because it facilitates the analysis of vehicle models with nonlinear constitutive elements by numerical integration, or calculation of the conventional stiffness and damping matrices for linearized frequency domain analyses. For the results presented in this paper, the vehicle model only contains linear suspension elements and therefore the equation of motion can be rewritten as
Pfl{iij + [Cl{41 + Wl{ql = {Ql,
(9)
where{Q}= [TIKl{h} = [Gl{A}, [Cl = [Sl[CJSIT + [r][C,][T]’ is the damping matrix for the vehicle, [a = [S][KJS]’ + [T][K,][TIT is the stiffness matrix for the vehicle where superscript T denotes the matrix transpose and [KS], [K,] are the stiffness matrices for the suspension and tire elements, {Q} is the vector of generalized forces applied to the vehicle, {h} is a vector of the input road profile heights under the tires, and [C,], [C,] are the damping matrices for the suspension and tire elements. This vehicle simulation package has been validated extensively against experimental results [9, IO], and is combined with the bridge model using the method described in Section 3 to produce the results presented in Sections 6 and 7. 5. CRITERIA
FOR BRIDGE-VEHICLE
INTERACTION
The amount of dynamic interaction was assessed by calculating the dynamic response of the bridge when crossed by a vehicle model and comparing this with the response calculated by ignoring interaction (i.e. cutting the feedback loop in Fig. 1 at point “A”). The procedure was as follows: (i) an initial set of wheel loads was predicted using the specified bridge surface profile as the input to the vehicle model; (ii) the initial loads were applied to the bridge to calculate the initial midspan displacement, y,,(xO, t) where x0 = L/2; (iii) the iterative calculation of the previous section was performed to calculate the midspan bridge displacement, y(xO, t); (iv) y(xO, t) was compared with y,,(xO, t) by computing the error term
t,(t) =
Y(Xo,t) - Yl”(% t). YSl
(10)
(v) the maximum value of ItI (the “maximum error”) and the standard deviation, crc, were determined for the period during which the vehicle was on the bridge. The properties of 6, are examined briefly in the Appendix. 6. INTERACTION
STUDY METHODOLOGY
The simple bridge-vehicle model shown in Fig. 3 will be used for the remainder of this paper to investigate dynamic interaction. The model consists of a single degree of freedom vehicle crossing a simply-supported bridge. This simple vehicle model does not contain the detailed suspension nonlinearities and complexities of sprung mass motion that are typical of heavy vehicles. However, its dynamic characteristics are a good representation of the majority of heavy commercial vehicles which generate their dynamic
Heavy vehicles and highway bridges
251
V
Vehicle
*
Model
Euler beam
/’
7’
L
Fig. 3. Single degree of freedom vehicle model on a simply-supported
tire forces at low frequencies due to sprung mass bouncing and pitching motion [ 111. Although the simply supported bridge model is not representative of all types of bridges, it embodies many of the important dynamic characteristics of bridges. For example, Cantieni [12] has used such simple models to examine dynamic bridge-vehicle interaction, and has found good agreement with experiments on highway bridges. The bridge parameters were based on measurements of the Pirton Lane highway bridge in Gloucester by Wills [13], Leonard [14] and Eyre and Tilly [15]. They reported the bridge length as 40 m, and the first natural frequency as 3.2 Hz with a modal damping ratio of 0.02. The mass per unit length was estimated from drawings in the papers as 12,000 kg m-‘. Three vibration modes were used in the analysis, and the natural frequencies of the second and third modes were calculated from the following formula for the natural frequencies of a simply-supported beam
(11)
where EZ is the flexural rigidity of the bridge, m is the mass per unit length of the bridge, and L is the length of the bridge. The modal damping ratio, IF), was assumed to be constant for all three modes (see Table 1). The important properties of the bridge-vehicle system will be expressed by the following nondimensional parameters: the speed parameter, a; the
bridge.
frequency ratio, y; and the modal mass ratio, K. These parameters are defined as
h-c-
2rn” ml;’
(12)
where V is the speed of the vehicle, and m, is the vehicle mass. In addition to these three parameters, the bridge and vehicle damping ratios were identified by the symbols [b and cV, respectively. Figures 4 and 5 show normalized displacement responses of the bridge to the simple vehicle model (dotted curve) and to a constant force (solid curve). The normalization was performed by dividing the dynamic deflection response by the maximum midspan static deflection, ySt. The following parameters were used: y = 1.0, K = 0.32, cV= 0.05 and [b = 0.02. The bridge surface profile was smooth and therefore the initial bridge response to the vehicle loads, yi,(xo, t), was exactly the same as the response of the bridge to a constant force. In Fig. 4, the speed parameter, a, is 0.10 (corresponding to a speed of 25 m s-l), while in Fig. 5 a = 0.20. These two figures show that the dynamic response to the force (i.e. ignoring interaction) is considerably larger than the dynamic response to the
0.5t
Table 1. Natural freuuencies and damoina ratios Mode number 1 2 3
Natural frequency (rad s-l)
Damping ratio, lb
20.0 80.0 180.0
0.02 0.02 0.02
Time
(seconds)
Fig. 4. Midspan bridge deflection-interaction Q = 0.10, y = 1.0, K = 0.32, [. = 0.05, (b = 0.02; force -, vehicle - - - - - (y = 1.0).
study; constant
258
M. F. Green and D. Cebon
Time
(seconds)
Fig. 5. Midspan bridge deflection-interaction study; 2 = 0.20, ;’ = 1.0. K = 0.32. :, = 0.05. ih = 0.02: constant force -, vehicle - - - - - (7 = 1.0).
vehicle model. This increase in response occurs for two reasons. Firstly, the damping of the vehicle is typically much larger than that of the bridge. Therefore, when the vehicle is combined with the bridge, the overall damping of the system is increased resulting in smaller dynamic responses. Secondly, the vehicle and the bridge tend to act in opposition to each other. If the bridge displaces downwards, then the springs in the vehicle suspension are extended resulting in lower dynamic wheel forces. These lower wheel forces then cause smaller displacements of the bridge. Because of the combination of these two effects, the vehicle often acts as a vibration absorber for the bridge, and the dynamic displacements calculated by ignoring interaction are conservative. For Figs 4 and 5, the errors incurred by ignoring interaction were calculated by the formula of eqn (10). The values of these errors are shown in Table 2. From an inspection of Fig. 4, a 5% standard deviation of error seems to be a reasonable threshold for ignoring interaction (as per the Appendix). At this level of error, the dynamic responses have approximately the same amplitude and the maximum response is calculated with approximately 11% accuracy. Although, a 5% average error may seem relatively small for such a complicated system, it should be noted that 5% average errors correspond to approximately 1 I % maximum error which is five times greater than the convergence tolerance of 2%. The responses of the bridge to vehicle loads with several combinations of frequency ratios and speed parameters were computed and errors were calculated. Figure 6 shows values of maximum error and standard deviation of the error plotted against the frequency ratio, y. Error estimates for each speed
Table Speed (m s-l) 25 50
2. Errors incurred bridge responses Speed parameter,
by ignoring interaction for the shown in Figs 4 and 5 z
0.1 (Fig. 4) 0.2 (Fig. 5)
Maximum error (%)
u< (%)
11.4 21.7
5.3 9.8
parameter are connected by solid lines. The errors increase with the speed parameter, CI. Generally, the maximum errors are approximately double the standard deviation values. The curves peak at y = I .O where the natural frequencies of bridge and vehicle are matched, except at the slowest speed (a = 0.05). For the slow speed, the quasi-static displacement response dominates and so there only a small amount of energy at the natural frequency of the bridge to be fed back into the vehicle. This results in a lower level of interaction, regardless of the frequency ratio. When the frequency ratio, ;‘, is less than or equal to 0.5. the errors are small for all speeds. In this case, interaction is seen to be important (i.e. 6, > 5%) if SI> 0.1 and ^J> 0.7. For the Pirton Lane bridge with length 40 m and natural frequency of 3.2 Hz, this threshold would be crossed for vehicles with natural frequencies above 2 Hz travelling at speeds greater than 25 m ss’ (90 km h-‘). Therefore interaction would only be significant at highway speeds. For the Lower Earley bridge (described in the validation study), the spans are approximately 20 m long and the lowest natural frequency is 5.7 Hz, giving a speed threshold of about 25 m SK’. However, vehicles with frequencies below 3 Hz would cause little interaction. Since the curves for maximum error and standard deviation, cr,, are similar in shape, only the plots for 0, will be presented for the rest of this paper. 7.
PARAMETRIC STUDY
A parametric study was performed using the bridge and vehicle models as described in the previous section. Six nondimensional parameters were varied as shown in Table 3. Most of these parameters have been discussed previously. The exception is the bump height, H,, which is the height of a step up in the road profile height at the entry to the bridge. The effect of vehicle mass (modal mass ratio K) on the interaction is demonstrated in Fig. 7. Curves for two values of the speed parameter, c(, are shown. Larger modal mass ratios cause more interaction, and this effect is pronounced at the higher speed. At the lowest speed (a = 0.05), changes in vehicle mass result in only minor alterations in the amount of interaction, which is small anyway. It is apparent that little dynamic interaction will occur for high frequency “wheel hop” modes of vehicle vibration because the modal mass ratio is very small. Damping is varied for the curves in Figs 8 and 9. Increased vehicle damping (Fig. 8) reduces the interaction, especially near a frequency ratio of unity. For very low values of vehicle damping (cV = O.Ol), interaction increases as the frequency ratio decreases below 1.0. This behaviour contradicts the general trends observed in the rest of the study. However, such low values of damping do not occur in practice and cy = 0.05 can be considered a practical minimum. Figure 9a shows the effect of bridge damping with
259
Heavy vehicles and highway bridges
m sy
0
a = 0.05
0
Q =
*
0.10 u f 0.15
A
0 = 0.20
-
8.0
0.5
1.0 Frequency
G -
1.5
2.0
1.5
2.0
Threshold
Rotio (7)
20
5 t w E E’ ‘f z0 10
8 .O
0.5
1 .o Frequency
Ratio (y)
Fig. 6. Error involved in ignoring bridge-vehicle interaction; effect of speed parameter, a i” = 0.05, [b = 0.02, h, = 0.
light vehicle damping (cV= 0.05), whereas Fig. 9b shows the effect of {b when [, = 0.2. Bridge damping is seen to influence the interaction over a larger frequency range than vehicle damping.
0.32,
Finally, the influence of bridge abutment roughness on the interaction is presented in Fig. IO. The vehicle model was excited by a step up at the entrance to the bridge. Initial excitation of the vehicle has an
Table 3. Parameters varied in interaction study Symbol
Definition
Parameter range
Speed
a
nV WbL
0.05-0.20
Frequency ratio
Y
2 Ob
0.5-2.0
Modal mass ratio
K
Vehicle damping ratio
i.
Bridge damping ratio
lb
Bump height
h,
Parameter
K =
2m Y
mL
L 2&K
0.16464
0.01-0.20 0.01-0.05
H > Ya1
O-10
260
M. F. Green
and D. Cebon
Kev a = 0.05 a = 0.20 K = 0.16 6 = 0.32 n = 0.64
Threshold
0.5
1 .o Frequency
Fig. 7. Error involved in ignoring
1.5 Ratio (y)
bridge-vehicle
interaction;
effect of mass ratio, K; i> = 0.05. [b = 0.02.
h, = 0.
overwhelming influence on the amount of bridge-vehicle interaction. Increased vehicle response results in more interaction and the standard deviation of the errors incurred by neglecting the interaction can be almost 60%. Nevertheless, for a frequency ratio of 0.5, the errors averaged less than 5% for all cases. Because of the large errors observed with the inclusion of surface roughness, it is useful to examine some of the typical responses. Figure 11 shows bridge responses corresponding to K = 0.32, y = 1.0 and h, = 10. Two values of c( are shown. The solid curves were obtained by including the interaction, while the dotted lines were the responses to the initial wheel loads. The discrepancies between the estimates are large, and the vibration is much larger when the interaction is ignored. This occurs because the vehicle
damping dissipates some of the energy of the bridge vibration. Another point to note about Fig. 11b is that the maximum dynamic deflection is substantially larger than the maximum static deflection (approximately double). Therefore, the errors normalized by the static deflections are more conservative than those normalized by maximum dynamic deflections. 7.1. Discussion This parametric study has revealed some interesting aspects of bridge-vehicle interaction. Within the normal range of parameters on highway bridges, the most important parameters are speed, frequency and initial vehicle excitation. Large mass ratios result in significant interaction at high speeds. Bridge and
1 = 0.05 a = 0.20
(I
C” = C” = (” = C” =
-
8.0
0.5
involved
in ignoring
Threshold
1.5
1 .o Frequency
Fig. 8. Error
0.01 0.05 0.10 0.20
Ratio (7)
bridge-vehicle interaction; lb = 0.02, h, = 0.
effect of vehicle damping,
iv; K = 0.32,
Heavy vehicles and highway bridges
*
Frequency (a) cv =
Threshold
Ratio (y)
0.05
t-
8.0
0.5
261
1.0 Frequency
1.5
Threshold
2.0
Ratio (y)
(b) cV = 0.20 Fig. 9. Error involved in ignoring bridge-vehicle interaction; effect of bridge damping, {b; K
vehicle damping are important when the frequency ratio is near unity. It should also be noted that the interaction or coupling of the bridge and vehicle systems generally tends to reduce the dynamics of the bridge compared to the case when interaction is ignored. For this study, interaction was deemed important if the standard deviation of the error, a,, were greater than 5%. Using this criterion, dynamic interaction can be ignored if (i) y < 0.5, or (ii) a < 0.1 and profiles.
K c 0.3
for
smooth
bridge
These guidelines are valid for the parameter ranges specified in the study: CV> 0.05, lb > 0.01 and K <
0.64.
=
0.32, h, = 0.
The first guideline is important for bridges with high natural frequencies. Heavy vehicles generate most of their wheel loads in the 1.5 to 4.5 Hz frequency range [ 111, so bridges with natural fmquenties near 10 Hz will have little interaction with most vehicles. The second guideline applies for bridges with well maintained approaches, and relatively light vehicles travelling at low speeds. For the Pirton Lane bridge, interaction can be ignored for vehicles up to 40 tonnes as long as they are travelling below 90 km h-r. This is a fairly high speed for heavy vehicles, and so the bridge-vehicle interaction can be ignored in many cases as long as the approaches are smooth. Nevertheless, more research is needed on the importance of road roughness in determining the amount of bridge-vehicle interaction.
M. F. Green and D. Cebon
262
4-
Frequency
Fig. 10. Error involved in ignoring
Ratio
(7)
bridge-vehicle interaction; effect of step at bridge entrance, c5 = 0.20. [h = 0.02.
The frequency criterion is similar to that suggested by Walker and Veletsos [2], although they suggested that interaction could be ignored for y < 0.3 and 1’> 1.0. This study has verified some of their work, but also expanded it by estimating the errors involved in neglecting the interaction.
1
0
3
2 Time
4
Threshold
5
8. CONCLUSIONS
A new calculation method for predicting the dynamic response of a bridge to arbitrary wheel loads was presented and validated by field tests on a highway bridge. The calculation method was extended to calculate the response of a dynamically coupled bridge-vehicle system using any vehicle model and an iterative procedure. A parametric study was performed to investigate the importance of dynamic bridge-vehicle interaction. The errors involved in ignoring interaction were quantified. Criteria to assess whether or not interaction is important were developed and it was concluded that interaction can be ignored if: (i) 7 < 0.5, or (ii) c( < 0.1 and profiles,
6
h,; K = 0.32,
K < 0.3
for
smooth
bridge
(seconds)
(C%)u = 0.05 3
where y is the ratio frequency to the first speed parameter, and of the vehicle to that
““““‘,“...‘,..,““‘..,.,’
9.
-a.:,
0.5 ’
’
1 .o Time
’
1.5
1
(seconds)
of the lowest vehicle natural bridge natural frequency, c( is a K is the ratio of the modal mass of the bridge.
RECOMMENDATIONS
FOR FURTHER
WORK
Many of the important parameters governing bridge-vehicle dynamic interaction have been investigated in this paper. Nevertheless, further study is required to obtain a more complete understanding. The following future work is recommended:
(b) ~1 = 0.20
Fig. 11.Midspan initial response
bridge displacement. Iterative solution -, -- -- -, K = 0.32, & = 0.20, [h = 0.02, y = 1.0, h, = 10.
(i) The effects of pseudo random bridge surface roughness should be examined in a systematic manner.
263
Heavy vehicles and highway bridges
(ii) The simple vehicle model should be extended to include wheel-hop and pitch modes of vibration.
275, Transport and Road Research Crowthome, U.K. May (1977).
Laboratory,
APPENDIX
Acknowledgements-The
authors are grateful to the Director of the Transport Research Laboratory (TRL) and to the members of the Bridges Division and Vehicles and Environment Division for assistance with the experiments described in this paper. Any views expressed in this paper are those of the authors and not necessarily those of the U.K. Department of Transport or the TRL. Doctor Green would also like to thank the Association of Commonwealth Universities (U.K.) and the Natural Sciences and Engineering Research Council of Canada for supporting this research. REFERENCES 1. J.
Billing and R. Green, Design provisions for dynamic loading of highway bridges. TRB Second Bridge Engineering Conf., Transportation Research Record 950, September, pp. 94-103 (1984). 2. W. Walker and A. Veletsos, Response. of simple-span highway bridges to moving vehicle loads. Bulletin 486, University of Illinois, Engineering Experiment Station, Urbana, IL (1966). 3. W. S. Chiu, R. G. Smith and D. N. Wormley, Influence of vehicle and distributed guideway parameters on high speed vehicle-guideway dynamic interactions. J. Dyn. Systems, Measmt Control ASME, March, 25-34 (1971). 4. E. C. Ting and J. Genin, Dynamics of bridge structures.
PROPERTIES OF k
The properties of the error standard deviation, u,, can be examined by considering two oscillatory functions y,(t) and y*(t) that differ in magnitude and phase y,(t) = Y,, + Ylsin(ot)
(At)
y2(t) = Y,, + Yzsin(wt + 6)
where Y,, is the static deflection, Y, and YZ are the amplitudes of the oscillatory components of the functions, and 8 is the phase difference between the two functions. The mean square error F is defined by Zr
c? = g
s
O”[y,(t) - yz(#dt
[
=;
fl+Y:-2Y,Y*COSe
1
The standard deviation of error (normalized by the static deflection Y.,) is given by
Solid Mech. Arch. 5, 217-252 (1980). 5. M. F. Green and D. Cebon, Dynamic response
of highway bridges to heavy vehicle loads: theory and experimental validation. J. Sound Vibr. 170, 51-78
(A3)
(1994). 6. M. F. Green, Dynamic response of short-span highway
which can be simplified by defining the parameter j = Y,/ Y, as follows:
bridges to heavy vehicle loads. Ph.D. thesis, University of Cambridge, U.K. (1990). I. D. E. Newland, Mechanical Vibration Analysis and Computation. Longman, London (1989). 8. H. Hawk and A. Ghah. Dvnamic resnonse of bridges to multiple truck loading. Can. J. Civ.-Engng 8, 392-401 (1981). 9. D. Cebon, Theoretical road damage due to dynamic
tyre forces of heavy vehicles. Part 1: dynamic analysis of vehicles and road surfaces. Proc. Inst. mech. Engnrs 202(C2),
103-108 (1988).
10. D. J. Cole and D. Cebon, Validation of an articulated vehicle stimulation. Vehicle System Dyn. 21, 197-223 (1992). Il. D. Cebon, Vehicle-generated road damage: a review. Vehicle System Dyn. 18, 107-150 (1989). 12. R. Cantieni, Investigation of vehiclebridge interaction for highway bridges. Heavy vehicles and roads: technology, safety and policy. In Proc. Third Int. Symp. on Heavy Vehicle Weights and Dimensions, pp. 130-I 37. __ Thomas Telford, London (1992). 13. J. Wills. Correlation of calculated and measured dynamic’ behaviour of bridges. Symp. on Dynamic Behaviour of Bridges. Transport and -Road Research Laboratory, Crowthome, U.K. May, TRRL Supplementary Report 275 (1977). 14. D. Leonard, Damping and frequency measurements on eight box girder bridges. TRRL Laboratorv Report 682, Research _Laboratory; Transport and -Road Crowthome, U.K. (1975). 15. R. Eyre and G. P. Tilly, Damping measurements on steel and composite bridges. Symp. on Dynamic Behauiour of Bridges. TRRL Supplementary Report
Q< =
-
J
Yl Y,,
$1 + B* - 28 cos
e].
The surface described by eqn (A4) can be visualized by using polar coordinates with B representing the radial coordinate and 6 representing the angular coordinate. The level curves of the surface are circles centred at (/?,0) = (1,0) with radius, r, given by
Figure Al shows a contour plot of eqn (A4) with a, = 0.05 for different values of the ratio K/Y,,. The differences between the two functions y,(t) and y*(t) can be divided into two categories: phase errors characterized by 0, and amplitude errors characterized by the parameter rj as follows: (A6) Table Al shows values of 6 and g calculated for a normalized error, c,, of 5% for different values of the relative vibration amplitude, Yl/Y,. This table shows that for the same normalized error, u, , more phase shift, 6, can be tolerated with smaller amplitude errors. The phase shift error is also limited as the relative vibration amplitude,
264
M. F. Green
and D. Cebon Table Al. Phase and amplitude
errors
Phase error
voos -7--r
0.5
00
.o
15
20
2’5
of the normalized 0, = 0.05.
error
surface
1
r
0.05
1.41
0.0 45.0 90.0 180.0
7.1% 4.7% 0.0% 2.9%
0.10
0.71
0.0 30.0 45.0
7.1% 3.7% 2.9%
0.15
0.47
0.0 10.0 20.0 28.1
7.1% 6.3% 4.0% 1.5%
0.20
0.35
0.0 10.0 20.7
7.1% 5.9% 1.1%
p case
Fig. Al.
Level curves
for
VI/Y,,, increases. For example, if Y,/Y>, = 0.2 then the maximum phase error is approximately 20.7” (corresponding to an amplitude error, 1. of 1.I%). In addition the amplitude error, q, reaches a maximum when 0 = 0. In this case, the normalized error of eqn (A4) reduces to:
CT, =
-
YI
J
Y,,
$1 - 2/j + fi’] = +
u ”
&
- 4 - j2
(A7)
Amplitude error ‘I
YI Y,,
-
0.5
for u, = 0.05
(d:g)
which is in direct proportion to the amplitude error; the factor I/,,/2 converts peak to r.m.s. levels. Therefore a normalized error of 5% (u, = 0.05) corresponds to an amplitude error of approximately 7.1% (see Table Al). It is apparent that y,(r) and _r~(t) are appreciably different if 6, > 0.05 (5%).
DYNAMIC
Computers & Srrwtures Vol. 62, No. 2, pp. 253-264, 1997 Copyright0 19%Ehwier Sciemx Ltd Printed in Great Britain. All rights reserved 0045.7949/97 $17.00 + 0.00 soo4s7!34!y%)00198-8
INTERACTION BETWEEN HEAVY VEHICLES AND HIGHWAY BRIDGES M. F. Greent
and D. Cebon$
TDepartment of Civil Engineering, Queen’s University at Kingston, Ontario, Canada K7L 3N6 SUniversity Engineering Department, Trumpington Street, Cambridge CB2 lPZ, U.K. (Received 9 June 1995) Ahstraet-This paper discusses the importance of dynamic interaction (coupling) between heavy vehicles and highway bridges. A new method for calculating the dynamic response of bridges to dynamic wheel loads is presented, and used to investigate a simply-supported bridge traversed by a single degree of freedom vehicle model. Six non-dimensional parameters of the bridge-vehicle system are varied to assess the amount of interaction between the vehicle and the bridge. Guidelines are presented to ascertain when dynamic interaction is important and when it can be ignored. Copyright @ 1996 Efsevier Science Ltd.
NOTATION vehicle damping constant viscous damping operator with respect to the spatial variables vehicle damping matrix damping matrices for suspension and tire elements maximum dynamic response increment Young’s modulus of elasticity force exerted by the vehicle on the bridge vectors of suspension and tire forces _ product of mode shape value and wheel load vector of road profile heights under the tires impulse response function for force applied at position xr and response at position x modal impuhe response function normalized step height, H,/yst height of step up at bridge entry second moment of area of the bridge (beam model) vehicle stiffness matrix stiffness matrices for suspension and tire elements vehicle suspension stiffness wheel number bridge length self-adjoint linear differential operator with respect to the spatiai variables mass per unit surface area of the bridge mass per unit length of the bridge (beam model) inertia matrix of vehicle vehicle mass mode number number of tires on the vehicle dynamic wheel loads vector of generalized coordinates for the vehicle vector of generalized forces applied to the vehicle radius of level curves of normalized error o, (Appendix) constant t~nsfo~ation matrices relating the
t TV tu1 V x0 X
xr Xl
Yh
t)
Y(t) vi(t)* PO Y/(6 Y&
suspension and tire forces to the generalized coordinates time time for vehicie to completely cross the bridge constant vector of generalized gravitational forces applied to the vehicle vehicle speed bridge midspan position, L/2 vector describing a position on the bridge surface point of application of the force input position of lth wheel transverse deflection of the bridge displacement response of the bridge measured under the tires of the vehicle two oscillatory functions (Appendix} $h estimate for bridge displacement response initial estimate for bridge displacement response maximum dynamic response of the bridge maximum static response of the bridge amplitudes of the oscillatory functions y,(r), Y&I
QM=W
static deflection (Ap~ndix) normalized vehicle acceleration speed parameter, x V/oh L relative amplitude parameter, Yz/YI (Appendix) frequency ratio, wv/wb mass ratio, m,/mL mean square error (Ap~ndix) interaction error term nth bridge damping ratio first bridge damping ratio vehicle damping ratio amplitude error (Appendix) phase error (Appendix) modal mass ratio, 2 m,/mL standard deviation of interaction error integration variable nth mode shape value at position x angular frequency (rad s-l) first bridge natural frequency (rad s-‘) 8th bridge naturai frequency (rad s-l) vehicle natural frequency (rad s-l) 253
254
M. F. Green and D. Cebon 1. INTRODUCTION
Highway bridges are subjected to the dynamic tire forces of heavy vehicles. The resulting dynamic response can lead to bridge deterioration; eventually increasing maintenance costs, and decreasing service life. In order to assess and design bridges for dynamic wheel loads, or to design vehicles to reduce the dynamic response of bridges, a fundamental understanding of the nature of the dynamic interaction between vehicles and bridges is necessary. Figure 1 is a schematic diagram of bridge--vehicle interaction. The roughness input to the oehicle is the sum of the initial surface profile of the bridge and the dynamic deflection of the bridge. This input excites the vehicle and results in dynamic tire forces. These forces are in turn applied to the bridge and cause larger dynamic displacements of the bridge. This feedback mechanism of interaction forces couples the dynamic response of the bridge to that of the vehicle. The largest bridge vibrations are known to occur when the natural frequencies of the bridge and vehicle are close together [l]. It is not clear, however, under what circumstances large bridge responses are mainly due to excitation of the vehicle by the initial roughness profile, or due to excitation substantially by dynamic deflection of the bridge. The former case can be considered as the feedback loop in Fig. I being cut at point “A”. There has been little systematic study about when dynamic interaction is important, and when the two systems can be considered to be essentially uncoupled. 2. BACKGROUND
Walker and Veletsos [2] considered the response of a simply-supported bridge to a moving constant force and to a moving sprung mass. They compared the two responses by computing the maximum dynamic increments for each case. Their definition of the maximum dynamic increment was
where DZ is the maximum increment, yrnaris the maximum
dynamic response dynamic response of
the bridge, and y,, is the maximum static response of the bridge. They related the importance of interaction to the frequency ratio, 7, defined as
where w, is the natural frequency of the vehicle and (11~is the first natural frequency of bending of the bridge. Bridge-vehicle interaction was deemed unimportant when ;’ was less than 0.3 or greater than about 1.0 [2]. This criterion seems quite reasonable, but Walker and Veletsos did not quantify the errors incurred by neglecting bridge-vehicle interaction. Chiu et al. [3] conducted a parametric study similar to that of Walker and Veletsos for an elevated guideway traversed by a high-speed vehicle. They found that dynamic interaction was unimportant when the ratio of the vehicle mass to the mass of the guideway span was below l/S, when the frequency ratio, ;‘, was low. Most of their results were calculated for relative speeds far in excess of those usual on highway bridges. Their criteria are consistent with those of Walker and Veletsos, but do not provide enough detail for low speeds. Finally, Ting and Genin [4] outlined interaction criteria based on the quantity 2~2 where 6 is the vehicle/bridge mass ratio, and 2 is the second time derivative of the vehicle displacement normalized by the maximum static deflection of the bridge. They concluded that interaction can be ignored if 2c.?!
3. BRIDGE RESPONSE CALCULATION-THEORY AND VALIDATION
3. I
Theory
The dynamic response of most bridges to arbitrary heavy vehicle loads can be represented by the following equation of motion:
I
I
A
Dynamic Tire FOfCBD
t
1ay I + L{Y@, 0) =f(x,
I Dynamic
azy
m(x,,,i(x, t) + C -;i;(x, t)
Bridge Deflection
t)
(3)
Dynamica
Fig. 1. Schematic block diagram of dynamic bridge-vehicle interaction.
where L{o} is a self-adjoint differential operator with respect to the spatial variables, x is a two dimensional
Heavy vehicles and highway bridges position vector, t is time, m(x) is the mass per unit surface area, y(x, t) is the transverse deflection of the bridge, C(o) is a viscous damping operator with respect to the spatial variables which must satisfy certain orthogonality conditions [5], andf(x, t) is the force exerted by the vehicle on the bridge. The derivation of this differential equation assumes linear viscous damping, linear elasticity, small deflections, and neglects shear deformation and rotary inertia [5,6]. The solution to the equation of motion can be expressed by the following convolution integral:
s m
Yk t) =
h(x, xr, t - rMf(xr,r)dr
-a
(4)
255
dynamic vehicle response model through the following iterative procedure:
(0 Calculate the vehicle response to a specified bridge surface profile and hence, a first set of dynamic wheel loads. (ii) Calculate the displacement response of the bridge under each tire of the vehicle due to the first set of wheel loads. (ii) Add the bridge displacement response to the surface profile and use this as the new input to the vehicle model. Calculate the new dynamic wheel loads. (iv) Recalculate the displacement response, y(t), from the loads in (iii) by the procedure of (ii). Average successive displacement responses to facilitate convergence y
where h(x, x, t) is the impulse response function at position x for an impulse applied at position xf. Therefore, eqn (3) may be solved by determining the appropriate impulse response function, providing the vehicle loads, f(xl, t), are known. If the deflection, y(x, t) is expressed in terms of normal mode functions, &“)(x), it can be shown that the response to a moving vehicle with Nt wheels, is [5,6]
ct)
=
I
YCr) Yi2 +
I(l)
where y, is the jtb estimate for the bridge displacement response. (v) Repeat steps (iii) and (iv) until convergence is obtained. Convergence is deemed to occur when the following criterion is satisfied IYW - Y/- IWI
max(l
< tolerance,
for 0 S t G Tvpr (7)
y(x, t) =
-
t
f
,=I .=I
4’“‘(x)
h,(t - t)g’“*“(r)dt,
(5)
where h,(t - 7) is the modal impulse response function, and g97) = c#3xl(7)) x S(7). In the formula for g@*‘)(r),P,(t) is the dynamic wheel load for wheel I and x,(t) is the position of that wheel. With the general bridge response written in this convolution format, the equation of motion [eqn (3)] can be solved by evaluating the convolution integral in either the time or frequency domains. For this study, a frequency domain method using discrete Fourier transforms was chosen. By evaluating the discrete Fourier transform with the fast Fourier transform (FFT) algorithm [7], the integral can be solved significantly faster than by equivalent time domain convolution [5,6]. The modal formulation of this calculation procedure is efficient. Mode shapes can be obtained from a variety of sources (e.g. finite element models, measurements). Additionally, only a few mode shapes are normally required to model the dynamic characteristics of a bridge. Therefore, this method is much faster than an alternative finite element model which would have many more degrees of freedom. The calculation described above assumes that the wheel loads, P,(t), are specified and does not account for the dynamic interaction between the bridge and the vehicle. The interaction can be included for any
where T,, is the time for the vehicle to pass completely over the bridge. For the results presented in this paper, the tolerance was set to 0.02 (i.e. 2% of the maximum displacement of the bridge). Similar iterative methods have been used by other authors and the results have compared favourably against Runge-Kutta-Nystrom numerical integration [8]. Other demonstrations of the convergence of the iterative method can be found in Refs [S, 61. 3.2. Validation An experimental procedure was developed to validate the convolution calculation presented in the previous section. The procedure consisted of four main steps. The first step was the measurement of the transfer functions and mode shapes of the bridge. Step two consisted of single vehicle tests in which dynamic wheel forces and bridge responses were measured simultaneously. In step three, the measured dynamic wheel forces were combined with the modal properties of the bridge to predict bridge responses. Finally, these predicted responses were compared with the measured bridge responses to validate the model. It should be noted that this experimental procedure validated the bridge calculation, but not the iterative procedure for combining the bridge and vehicle models.
256
M. F. Green and D. Cebon Lower Earley bridge: Validation results South to north at 50 km/h
Time (s)
Fig. 2. Validation results; midspan bridge response; 50 km h-l, South
to north;
measured
-;
predicted
---
-.
Two typical highway bridges near Reading, England were tested. The first was a four span, prestressed concrete, box-girder bridge, and the second was a three-span, slab on girder, prestressed concrete bridge. The results of these tests are presented elsewhere [5, 61 and will not be included here in detail. However, for demonstration purposes, Fig. 2 shows a typical validation result for the Lower Earley bridge with the vehicle travelling at 50 km h- ‘. In this case, the response was measured at the midspan of the centre girder of this slab on girder bridge. The agreement between the calculation method and experiment is very good.
4. VEHICLE
SIMULATION
A lumped parameter vehicle model is assumed with the equation of motion given by [9]
[Ml(q) = [SIJK) + [Tl{F,J + iq,
(8)
where [M] is the inertia matrix of the vehicle, {q} is the vector of generalized coordinates for the vehicle, {F,, F,) are the vectors of “suspension” and “tyre” forces, {U} is the constant vector of generalized gravitational forces applied to the vehicle, and [S], [T] are the constant transformation matrices relating the suspension and tyre forces to the generalized coordinates. This formulation of the equations of motion is very versatile because it facilitates the analysis of vehicle models with nonlinear constitutive elements by numerical integration, or calculation of the conventional stiffness and damping matrices for linearized frequency domain analyses. For the results presented in this paper, the vehicle model only contains linear suspension elements and therefore the equation of motion can be rewritten as
Pfl{iij + [Cl{41 + Wl{ql = {Ql,
(9)
where{Q}= [TIKl{h} = [Gl{A}, [Cl = [Sl[CJSIT + [r][C,][T]’ is the damping matrix for the vehicle, [a = [S][KJS]’ + [T][K,][TIT is the stiffness matrix for the vehicle where superscript T denotes the matrix transpose and [KS], [K,] are the stiffness matrices for the suspension and tire elements, {Q} is the vector of generalized forces applied to the vehicle, {h} is a vector of the input road profile heights under the tires, and [C,], [C,] are the damping matrices for the suspension and tire elements. This vehicle simulation package has been validated extensively against experimental results [9, IO], and is combined with the bridge model using the method described in Section 3 to produce the results presented in Sections 6 and 7. 5. CRITERIA
FOR BRIDGE-VEHICLE
INTERACTION
The amount of dynamic interaction was assessed by calculating the dynamic response of the bridge when crossed by a vehicle model and comparing this with the response calculated by ignoring interaction (i.e. cutting the feedback loop in Fig. 1 at point “A”). The procedure was as follows: (i) an initial set of wheel loads was predicted using the specified bridge surface profile as the input to the vehicle model; (ii) the initial loads were applied to the bridge to calculate the initial midspan displacement, y,,(xO, t) where x0 = L/2; (iii) the iterative calculation of the previous section was performed to calculate the midspan bridge displacement, y(xO, t); (iv) y(xO, t) was compared with y,,(xO, t) by computing the error term
t,(t) =
Y(Xo,t) - Yl”(% t). YSl
(10)
(v) the maximum value of ItI (the “maximum error”) and the standard deviation, crc, were determined for the period during which the vehicle was on the bridge. The properties of 6, are examined briefly in the Appendix. 6. INTERACTION
STUDY METHODOLOGY
The simple bridge-vehicle model shown in Fig. 3 will be used for the remainder of this paper to investigate dynamic interaction. The model consists of a single degree of freedom vehicle crossing a simply-supported bridge. This simple vehicle model does not contain the detailed suspension nonlinearities and complexities of sprung mass motion that are typical of heavy vehicles. However, its dynamic characteristics are a good representation of the majority of heavy commercial vehicles which generate their dynamic
Heavy vehicles and highway bridges
251
V
Vehicle
*
Model
Euler beam
/’
7’
L
Fig. 3. Single degree of freedom vehicle model on a simply-supported
tire forces at low frequencies due to sprung mass bouncing and pitching motion [ 111. Although the simply supported bridge model is not representative of all types of bridges, it embodies many of the important dynamic characteristics of bridges. For example, Cantieni [12] has used such simple models to examine dynamic bridge-vehicle interaction, and has found good agreement with experiments on highway bridges. The bridge parameters were based on measurements of the Pirton Lane highway bridge in Gloucester by Wills [13], Leonard [14] and Eyre and Tilly [15]. They reported the bridge length as 40 m, and the first natural frequency as 3.2 Hz with a modal damping ratio of 0.02. The mass per unit length was estimated from drawings in the papers as 12,000 kg m-‘. Three vibration modes were used in the analysis, and the natural frequencies of the second and third modes were calculated from the following formula for the natural frequencies of a simply-supported beam
(11)
where EZ is the flexural rigidity of the bridge, m is the mass per unit length of the bridge, and L is the length of the bridge. The modal damping ratio, IF), was assumed to be constant for all three modes (see Table 1). The important properties of the bridge-vehicle system will be expressed by the following nondimensional parameters: the speed parameter, a; the
bridge.
frequency ratio, y; and the modal mass ratio, K. These parameters are defined as
h-c-
2rn” ml;’
(12)
where V is the speed of the vehicle, and m, is the vehicle mass. In addition to these three parameters, the bridge and vehicle damping ratios were identified by the symbols [b and cV, respectively. Figures 4 and 5 show normalized displacement responses of the bridge to the simple vehicle model (dotted curve) and to a constant force (solid curve). The normalization was performed by dividing the dynamic deflection response by the maximum midspan static deflection, ySt. The following parameters were used: y = 1.0, K = 0.32, cV= 0.05 and [b = 0.02. The bridge surface profile was smooth and therefore the initial bridge response to the vehicle loads, yi,(xo, t), was exactly the same as the response of the bridge to a constant force. In Fig. 4, the speed parameter, a, is 0.10 (corresponding to a speed of 25 m s-l), while in Fig. 5 a = 0.20. These two figures show that the dynamic response to the force (i.e. ignoring interaction) is considerably larger than the dynamic response to the
0.5t
Table 1. Natural freuuencies and damoina ratios Mode number 1 2 3
Natural frequency (rad s-l)
Damping ratio, lb
20.0 80.0 180.0
0.02 0.02 0.02
Time
(seconds)
Fig. 4. Midspan bridge deflection-interaction Q = 0.10, y = 1.0, K = 0.32, [. = 0.05, (b = 0.02; force -, vehicle - - - - - (y = 1.0).
study; constant
258
M. F. Green and D. Cebon
Time
(seconds)
Fig. 5. Midspan bridge deflection-interaction study; 2 = 0.20, ;’ = 1.0. K = 0.32. :, = 0.05. ih = 0.02: constant force -, vehicle - - - - - (7 = 1.0).
vehicle model. This increase in response occurs for two reasons. Firstly, the damping of the vehicle is typically much larger than that of the bridge. Therefore, when the vehicle is combined with the bridge, the overall damping of the system is increased resulting in smaller dynamic responses. Secondly, the vehicle and the bridge tend to act in opposition to each other. If the bridge displaces downwards, then the springs in the vehicle suspension are extended resulting in lower dynamic wheel forces. These lower wheel forces then cause smaller displacements of the bridge. Because of the combination of these two effects, the vehicle often acts as a vibration absorber for the bridge, and the dynamic displacements calculated by ignoring interaction are conservative. For Figs 4 and 5, the errors incurred by ignoring interaction were calculated by the formula of eqn (10). The values of these errors are shown in Table 2. From an inspection of Fig. 4, a 5% standard deviation of error seems to be a reasonable threshold for ignoring interaction (as per the Appendix). At this level of error, the dynamic responses have approximately the same amplitude and the maximum response is calculated with approximately 11% accuracy. Although, a 5% average error may seem relatively small for such a complicated system, it should be noted that 5% average errors correspond to approximately 1 I % maximum error which is five times greater than the convergence tolerance of 2%. The responses of the bridge to vehicle loads with several combinations of frequency ratios and speed parameters were computed and errors were calculated. Figure 6 shows values of maximum error and standard deviation of the error plotted against the frequency ratio, y. Error estimates for each speed
Table Speed (m s-l) 25 50
2. Errors incurred bridge responses Speed parameter,
by ignoring interaction for the shown in Figs 4 and 5 z
0.1 (Fig. 4) 0.2 (Fig. 5)
Maximum error (%)
u< (%)
11.4 21.7
5.3 9.8
parameter are connected by solid lines. The errors increase with the speed parameter, CI. Generally, the maximum errors are approximately double the standard deviation values. The curves peak at y = I .O where the natural frequencies of bridge and vehicle are matched, except at the slowest speed (a = 0.05). For the slow speed, the quasi-static displacement response dominates and so there only a small amount of energy at the natural frequency of the bridge to be fed back into the vehicle. This results in a lower level of interaction, regardless of the frequency ratio. When the frequency ratio, ;‘, is less than or equal to 0.5. the errors are small for all speeds. In this case, interaction is seen to be important (i.e. 6, > 5%) if SI> 0.1 and ^J> 0.7. For the Pirton Lane bridge with length 40 m and natural frequency of 3.2 Hz, this threshold would be crossed for vehicles with natural frequencies above 2 Hz travelling at speeds greater than 25 m ss’ (90 km h-‘). Therefore interaction would only be significant at highway speeds. For the Lower Earley bridge (described in the validation study), the spans are approximately 20 m long and the lowest natural frequency is 5.7 Hz, giving a speed threshold of about 25 m SK’. However, vehicles with frequencies below 3 Hz would cause little interaction. Since the curves for maximum error and standard deviation, cr,, are similar in shape, only the plots for 0, will be presented for the rest of this paper. 7.
PARAMETRIC STUDY
A parametric study was performed using the bridge and vehicle models as described in the previous section. Six nondimensional parameters were varied as shown in Table 3. Most of these parameters have been discussed previously. The exception is the bump height, H,, which is the height of a step up in the road profile height at the entry to the bridge. The effect of vehicle mass (modal mass ratio K) on the interaction is demonstrated in Fig. 7. Curves for two values of the speed parameter, c(, are shown. Larger modal mass ratios cause more interaction, and this effect is pronounced at the higher speed. At the lowest speed (a = 0.05), changes in vehicle mass result in only minor alterations in the amount of interaction, which is small anyway. It is apparent that little dynamic interaction will occur for high frequency “wheel hop” modes of vehicle vibration because the modal mass ratio is very small. Damping is varied for the curves in Figs 8 and 9. Increased vehicle damping (Fig. 8) reduces the interaction, especially near a frequency ratio of unity. For very low values of vehicle damping (cV = O.Ol), interaction increases as the frequency ratio decreases below 1.0. This behaviour contradicts the general trends observed in the rest of the study. However, such low values of damping do not occur in practice and cy = 0.05 can be considered a practical minimum. Figure 9a shows the effect of bridge damping with
259
Heavy vehicles and highway bridges
m sy
0
a = 0.05
0
Q =
*
0.10 u f 0.15
A
0 = 0.20
-
8.0
0.5
1.0 Frequency
G -
1.5
2.0
1.5
2.0
Threshold
Rotio (7)
20
5 t w E E’ ‘f z0 10
8 .O
0.5
1 .o Frequency
Ratio (y)
Fig. 6. Error involved in ignoring bridge-vehicle interaction; effect of speed parameter, a i” = 0.05, [b = 0.02, h, = 0.
light vehicle damping (cV= 0.05), whereas Fig. 9b shows the effect of {b when [, = 0.2. Bridge damping is seen to influence the interaction over a larger frequency range than vehicle damping.
0.32,
Finally, the influence of bridge abutment roughness on the interaction is presented in Fig. IO. The vehicle model was excited by a step up at the entrance to the bridge. Initial excitation of the vehicle has an
Table 3. Parameters varied in interaction study Symbol
Definition
Parameter range
Speed
a
nV WbL
0.05-0.20
Frequency ratio
Y
2 Ob
0.5-2.0
Modal mass ratio
K
Vehicle damping ratio
i.
Bridge damping ratio
lb
Bump height
h,
Parameter
K =
2m Y
mL
L 2&K
0.16464
0.01-0.20 0.01-0.05
H > Ya1
O-10
260
M. F. Green
and D. Cebon
Kev a = 0.05 a = 0.20 K = 0.16 6 = 0.32 n = 0.64
Threshold
0.5
1 .o Frequency
Fig. 7. Error involved in ignoring
1.5 Ratio (y)
bridge-vehicle
interaction;
effect of mass ratio, K; i> = 0.05. [b = 0.02.
h, = 0.
overwhelming influence on the amount of bridge-vehicle interaction. Increased vehicle response results in more interaction and the standard deviation of the errors incurred by neglecting the interaction can be almost 60%. Nevertheless, for a frequency ratio of 0.5, the errors averaged less than 5% for all cases. Because of the large errors observed with the inclusion of surface roughness, it is useful to examine some of the typical responses. Figure 11 shows bridge responses corresponding to K = 0.32, y = 1.0 and h, = 10. Two values of c( are shown. The solid curves were obtained by including the interaction, while the dotted lines were the responses to the initial wheel loads. The discrepancies between the estimates are large, and the vibration is much larger when the interaction is ignored. This occurs because the vehicle
damping dissipates some of the energy of the bridge vibration. Another point to note about Fig. 11b is that the maximum dynamic deflection is substantially larger than the maximum static deflection (approximately double). Therefore, the errors normalized by the static deflections are more conservative than those normalized by maximum dynamic deflections. 7.1. Discussion This parametric study has revealed some interesting aspects of bridge-vehicle interaction. Within the normal range of parameters on highway bridges, the most important parameters are speed, frequency and initial vehicle excitation. Large mass ratios result in significant interaction at high speeds. Bridge and
1 = 0.05 a = 0.20
(I
C” = C” = (” = C” =
-
8.0
0.5
involved
in ignoring
Threshold
1.5
1 .o Frequency
Fig. 8. Error
0.01 0.05 0.10 0.20
Ratio (7)
bridge-vehicle interaction; lb = 0.02, h, = 0.
effect of vehicle damping,
iv; K = 0.32,
Heavy vehicles and highway bridges
*
Frequency (a) cv =
Threshold
Ratio (y)
0.05
t-
8.0
0.5
261
1.0 Frequency
1.5
Threshold
2.0
Ratio (y)
(b) cV = 0.20 Fig. 9. Error involved in ignoring bridge-vehicle interaction; effect of bridge damping, {b; K
vehicle damping are important when the frequency ratio is near unity. It should also be noted that the interaction or coupling of the bridge and vehicle systems generally tends to reduce the dynamics of the bridge compared to the case when interaction is ignored. For this study, interaction was deemed important if the standard deviation of the error, a,, were greater than 5%. Using this criterion, dynamic interaction can be ignored if (i) y < 0.5, or (ii) a < 0.1 and profiles.
K c 0.3
for
smooth
bridge
These guidelines are valid for the parameter ranges specified in the study: CV> 0.05, lb > 0.01 and K <
0.64.
=
0.32, h, = 0.
The first guideline is important for bridges with high natural frequencies. Heavy vehicles generate most of their wheel loads in the 1.5 to 4.5 Hz frequency range [ 111, so bridges with natural fmquenties near 10 Hz will have little interaction with most vehicles. The second guideline applies for bridges with well maintained approaches, and relatively light vehicles travelling at low speeds. For the Pirton Lane bridge, interaction can be ignored for vehicles up to 40 tonnes as long as they are travelling below 90 km h-r. This is a fairly high speed for heavy vehicles, and so the bridge-vehicle interaction can be ignored in many cases as long as the approaches are smooth. Nevertheless, more research is needed on the importance of road roughness in determining the amount of bridge-vehicle interaction.
M. F. Green and D. Cebon
262
4-
Frequency
Fig. 10. Error involved in ignoring
Ratio
(7)
bridge-vehicle interaction; effect of step at bridge entrance, c5 = 0.20. [h = 0.02.
The frequency criterion is similar to that suggested by Walker and Veletsos [2], although they suggested that interaction could be ignored for y < 0.3 and 1’> 1.0. This study has verified some of their work, but also expanded it by estimating the errors involved in neglecting the interaction.
1
0
3
2 Time
4
Threshold
5
8. CONCLUSIONS
A new calculation method for predicting the dynamic response of a bridge to arbitrary wheel loads was presented and validated by field tests on a highway bridge. The calculation method was extended to calculate the response of a dynamically coupled bridge-vehicle system using any vehicle model and an iterative procedure. A parametric study was performed to investigate the importance of dynamic bridge-vehicle interaction. The errors involved in ignoring interaction were quantified. Criteria to assess whether or not interaction is important were developed and it was concluded that interaction can be ignored if: (i) 7 < 0.5, or (ii) c( < 0.1 and profiles,
6
h,; K = 0.32,
K < 0.3
for
smooth
bridge
(seconds)
(C%)u = 0.05 3
where y is the ratio frequency to the first speed parameter, and of the vehicle to that
““““‘,“...‘,..,““‘..,.,’
9.
-a.:,
0.5 ’
’
1 .o Time
’
1.5
1
(seconds)
of the lowest vehicle natural bridge natural frequency, c( is a K is the ratio of the modal mass of the bridge.
RECOMMENDATIONS
FOR FURTHER
WORK
Many of the important parameters governing bridge-vehicle dynamic interaction have been investigated in this paper. Nevertheless, further study is required to obtain a more complete understanding. The following future work is recommended:
(b) ~1 = 0.20
Fig. 11.Midspan initial response
bridge displacement. Iterative solution -, -- -- -, K = 0.32, & = 0.20, [h = 0.02, y = 1.0, h, = 10.
(i) The effects of pseudo random bridge surface roughness should be examined in a systematic manner.
263
Heavy vehicles and highway bridges
(ii) The simple vehicle model should be extended to include wheel-hop and pitch modes of vibration.
275, Transport and Road Research Crowthome, U.K. May (1977).
Laboratory,
APPENDIX
Acknowledgements-The
authors are grateful to the Director of the Transport Research Laboratory (TRL) and to the members of the Bridges Division and Vehicles and Environment Division for assistance with the experiments described in this paper. Any views expressed in this paper are those of the authors and not necessarily those of the U.K. Department of Transport or the TRL. Doctor Green would also like to thank the Association of Commonwealth Universities (U.K.) and the Natural Sciences and Engineering Research Council of Canada for supporting this research. REFERENCES 1. J.
Billing and R. Green, Design provisions for dynamic loading of highway bridges. TRB Second Bridge Engineering Conf., Transportation Research Record 950, September, pp. 94-103 (1984). 2. W. Walker and A. Veletsos, Response. of simple-span highway bridges to moving vehicle loads. Bulletin 486, University of Illinois, Engineering Experiment Station, Urbana, IL (1966). 3. W. S. Chiu, R. G. Smith and D. N. Wormley, Influence of vehicle and distributed guideway parameters on high speed vehicle-guideway dynamic interactions. J. Dyn. Systems, Measmt Control ASME, March, 25-34 (1971). 4. E. C. Ting and J. Genin, Dynamics of bridge structures.
PROPERTIES OF k
The properties of the error standard deviation, u,, can be examined by considering two oscillatory functions y,(t) and y*(t) that differ in magnitude and phase y,(t) = Y,, + Ylsin(ot)
(At)
y2(t) = Y,, + Yzsin(wt + 6)
where Y,, is the static deflection, Y, and YZ are the amplitudes of the oscillatory components of the functions, and 8 is the phase difference between the two functions. The mean square error F is defined by Zr
c? = g
s
O”[y,(t) - yz(#dt
[
=;
fl+Y:-2Y,Y*COSe
1
The standard deviation of error (normalized by the static deflection Y.,) is given by
Solid Mech. Arch. 5, 217-252 (1980). 5. M. F. Green and D. Cebon, Dynamic response
of highway bridges to heavy vehicle loads: theory and experimental validation. J. Sound Vibr. 170, 51-78
(A3)
(1994). 6. M. F. Green, Dynamic response of short-span highway
which can be simplified by defining the parameter j = Y,/ Y, as follows:
bridges to heavy vehicle loads. Ph.D. thesis, University of Cambridge, U.K. (1990). I. D. E. Newland, Mechanical Vibration Analysis and Computation. Longman, London (1989). 8. H. Hawk and A. Ghah. Dvnamic resnonse of bridges to multiple truck loading. Can. J. Civ.-Engng 8, 392-401 (1981). 9. D. Cebon, Theoretical road damage due to dynamic
tyre forces of heavy vehicles. Part 1: dynamic analysis of vehicles and road surfaces. Proc. Inst. mech. Engnrs 202(C2),
103-108 (1988).
10. D. J. Cole and D. Cebon, Validation of an articulated vehicle stimulation. Vehicle System Dyn. 21, 197-223 (1992). Il. D. Cebon, Vehicle-generated road damage: a review. Vehicle System Dyn. 18, 107-150 (1989). 12. R. Cantieni, Investigation of vehiclebridge interaction for highway bridges. Heavy vehicles and roads: technology, safety and policy. In Proc. Third Int. Symp. on Heavy Vehicle Weights and Dimensions, pp. 130-I 37. __ Thomas Telford, London (1992). 13. J. Wills. Correlation of calculated and measured dynamic’ behaviour of bridges. Symp. on Dynamic Behaviour of Bridges. Transport and -Road Research Laboratory, Crowthome, U.K. May, TRRL Supplementary Report 275 (1977). 14. D. Leonard, Damping and frequency measurements on eight box girder bridges. TRRL Laboratorv Report 682, Research _Laboratory; Transport and -Road Crowthome, U.K. (1975). 15. R. Eyre and G. P. Tilly, Damping measurements on steel and composite bridges. Symp. on Dynamic Behauiour of Bridges. TRRL Supplementary Report
Q< =
-
J
Yl Y,,
$1 + B* - 28 cos
e].
The surface described by eqn (A4) can be visualized by using polar coordinates with B representing the radial coordinate and 6 representing the angular coordinate. The level curves of the surface are circles centred at (/?,0) = (1,0) with radius, r, given by
Figure Al shows a contour plot of eqn (A4) with a, = 0.05 for different values of the ratio K/Y,,. The differences between the two functions y,(t) and y*(t) can be divided into two categories: phase errors characterized by 0, and amplitude errors characterized by the parameter rj as follows: (A6) Table Al shows values of 6 and g calculated for a normalized error, c,, of 5% for different values of the relative vibration amplitude, Yl/Y,. This table shows that for the same normalized error, u, , more phase shift, 6, can be tolerated with smaller amplitude errors. The phase shift error is also limited as the relative vibration amplitude,
264
M. F. Green
and D. Cebon Table Al. Phase and amplitude
errors
Phase error
voos -7--r
0.5
00
.o
15
20
2’5
of the normalized 0, = 0.05.
error
surface
1
r
0.05
1.41
0.0 45.0 90.0 180.0
7.1% 4.7% 0.0% 2.9%
0.10
0.71
0.0 30.0 45.0
7.1% 3.7% 2.9%
0.15
0.47
0.0 10.0 20.0 28.1
7.1% 6.3% 4.0% 1.5%
0.20
0.35
0.0 10.0 20.7
7.1% 5.9% 1.1%
p case
Fig. Al.
Level curves
for
VI/Y,,, increases. For example, if Y,/Y>, = 0.2 then the maximum phase error is approximately 20.7” (corresponding to an amplitude error, 1. of 1.I%). In addition the amplitude error, q, reaches a maximum when 0 = 0. In this case, the normalized error of eqn (A4) reduces to:
CT, =
-
YI
J
Y,,
$1 - 2/j + fi’] = +
u ”
&
- 4 - j2
(A7)
Amplitude error ‘I
YI Y,,
-
0.5
for u, = 0.05
(d:g)
which is in direct proportion to the amplitude error; the factor I/,,/2 converts peak to r.m.s. levels. Therefore a normalized error of 5% (u, = 0.05) corresponds to an amplitude error of approximately 7.1% (see Table Al). It is apparent that y,(r) and _r~(t) are appreciably different if 6, > 0.05 (5%).