Investigation of vehicle - bridge interaction for highway bridges
R. CANTIENI, Dr se teehn, ETH, Head, Concrete Structures Section, EMPA, Swiss Federal
Laboratories for Materials Testing and Research, Switzerland
Experience shows that bridges with fundamental natural frequencies in the range f = 2.0.. .4.0 Hz respond more
strongly to the dynamic action of heavy commercial traffic than other bridges [1]. To achieve better insight into
the processes occurring during the passage of a vehicle over a highway bridge with a critical natural frequency
dynamic load tests have been performed. The parameters "natural behavior of the bridge" and ~Iongitudinal
pavement profile" have been carefully determined before the tests. During the tests the dynamic wheel loads
were measured as well as the bridge response. Data processing included analysis in the time and frequency
domains. The results showed that the frequency content of the dynamic wheel loads is distributed over a rather
broad band and that the vehicle/bridge interaction is a rather complicated non-stationary process. The basic
characteristics of the interaction process can nevertheless be explained with the help of a rather simple two-degree-of-freedom model. Subsequent more detailed analysis consisted of considering an MDOF-model for the
bridge and of dividing the analysis range into three parts according to the bridge spans.
1. THE BRIDGE
The DeibOel Bridge is a one-cell prestressed concrete box girder, continuous over three spans of
roughly 32, 41 and 37 m length and 11.75 m wide
(Fig. 1). Sinusoidal excitation with a servohydraulic
vibration generator yielded among other factors the
natural frequencies of the three lowest longitudinal
bending modes f1 = 3.03 Hz, f2 = 4.28 Hz and f3 =
5.49 Hz [2]. Comparison with calculations showed
that the bridge is not behaving as a three-span continuous girder but as a frame with clamped-in
columns and horizontally (in the longitudinal direction) fixed bearings at the abutments. The theoretical
static boundary conditions as given in Figure 1 do
not apply for the practical dynamic loading case of
one or more vehicles crossing the bridge. The horizontal force produced under these circumstances
e.g. is too small to overcome friction in the PTFE
pot-bearings. Hence, the bridge is "horizontally fixed"
at the abutments.
This is of special importance because it eliminates
the first mode given by a calculation with the condition "horizontally free" at the abutments, f "" 2.50 Hz.
Damping of the first bending mode f1 = 3.03 Hz was
determined at , = 0.8% (percentage of critical damping).
Deflection was measured with inductive transducers HBM W20 at several points of the bridge. Only
results for the midpoint of the 41 m span, WG 02,
and of the 37 m span, WG 22, will be discussed
here.
approach. To investigate the influence of pavement
unevenness on the behavior of the vehicle on a rigid
surface as well as of the vehicle/bridge system,
pavement A was smoothed by placing a thin layer of
epoxy resin mortar after having performed the first
part of the tests. This pavement is referred to as
"pavement BH. It can be seen from Figure 2 that
pavement A was smoother than pavement B above
all for large wavelengths.
3. THE VEHICLES
The vehicles dealt with in this paper are four twoaxle trucks loaded with concrete blocks up to the
legally admitted gross weight of roughly 160 kN. The
axle spacing was 3.5 m, 4.3 m, 4.7 m and 5.0 m for
the vehicles 11 R, 12R, 14R and 13R respectively.
The letter "R" indicates that all of the vehicles had
radial tires. The car body was supported by leaf
springs at both axles, additional viscous dampers
were installed at the front axle. Figure 3 gives photographs of the vehicles 12R and 14R.
The device used to measure the dynamic wheel
loads is shown in Figure 4. It is basically a contactless infrared sensor which measures the distance
between axle and pavement surface and hence the
tire deflection. Knowing the force/deflection relationship of the tires, the signals from these sensors can
be transformed into wheel loads. To ensure the reliability of this measurement method special measures had to be taken to keep the tire pressure constant during the tests.
2. THE PAVEMENT
The pavement was intended to have the same
longitudinal profile as a section of an existing rou~h
highway with known unevenness ("pavement A').
This was achieved by placing a concrete layer on the
bridge deck and using a special copying technique.
This technique also allowed the laying of a pavement
of equal roughness on an adjacent portion of the
130
4. TEST PROCEDURES
The vehicle was driven at constant speed along
the center line of the bridge over the approach and
structure. The bridge response and the signals from
the four wheel load sensors (transferred to the
measurement center by a 2.45 GHz radio-telemetry
Heavy vehides and roads: technology, safety and policy. Thomas Telford, London, 1992.
VEHICLE - BRIDGE INTERACTION
32,35 m
WG 82
11030m
I
!
on
i
i
-+
~
)8( WG 22
WG 02 )8(
i
I
.. A ~
i
.Lt
)8( WG 82
Fig. 1
-+
I
I
l
11,75 m
f
I
I
I
I
~J
8t
3,22 5
3,22 5
5,30
~
WG
Geometry and theoretical static system of the Deibuel Bridge. WG 02: Inductive displacement transducer; A: Driving axis.
cnp
5
2
10 2
Frequency
m 20
10
5 3 2
Wavelength
Fig. 2
5
2
5
10- 1 1/cm
[1/cm]
1 0,5
1ff
0,2 0,1
[ m]
Unevenness Power Spectral Density of the
pavements A (1) and B (2).
Fig. 3
Test vehicles Nr. 12R and 14R (below).
131
HEAVY VEHICLES AND ROADS
link) were simultaneously stored on magnetic tape. A
PCM,.system TMrrD8K13 (Johne & Reilhofer) and a
Stellavox tape machine were utilized. Test identifica. tion consists of the vehicle number, the tire pressure
and the pavement: "12R10A" means that the vehicle
12R with a tire pressure of 10 bar was driven over
pavement A.
INSTR.
TEST SERIES
ASTAT
PHIMAX[%J
AT v=[km/hJ
0
WG
82
VEH NR 11
-1.85 MM
48.97
18.2
r - - -......
WG
82
VEH NR 12
-1.83 MM
45.65
15.1
+----+1
WG
82
VEH NR 13
-1.83 MM
52. t8
53.9
_ -.......
WG
82
VEH NR 14
-1.81 MM
57.18
52.6
..
5. TIME DOMAIN ANALYSIS
Data processing in the time domain consisted primarily of the determination of the dynamic increment
q, = A dm - A stat
Asta t
•
100 [%]
with Astat and ~yn being the maximum bridge response under static and dynamic loading with the
same vehicle at a given measurement pOint. Data
processing was performed by transferring the measured Signals into a Data Generai Eclipse S130 computer and using Fortran programs. As an example of
the results, Figure 5 gives the variation of q, with vehicle speed for four two-axle 160 kN vehicles, a tire
pressure of 10 bar and pavement A.
n
~
L.J
i
40+-~H*HH~---r---+--~~rM-+~~~
a..
/-
~
30
L
W
0:
o
~
20+-~~--~~-*~~~~~~~~r-~
....o
L
~
>o
10~~~--~~~~~~-+--~--~-4~
o
10
20
30
VEHICLE SPEED
Fig. 5
40
50
v
60
70
80
[km/h]
Dynamic Increments as a function of vehicle speed for two-axle 160 kN vehicles, a
tire pressure of 10 bar and pavement A.
6. FREQUENCY DOMAIN ANALYSIS
Axle of the Vehicle
E
E
E
E
o
..,
~
Ln
I
To determine the behavior of the vehicle on a rigid
pavement, the freely vibrating bridge and the system
vehicle/bridge during the passage, a modal analysis
system was utilized. This consists of an eight-channel DIFAlSCADAS front end with programmable filters and sampling rate, an HP 1000/A700 computer
and a software package delivered by LMS, Leuven
Measurements & Systems, Leuven, Belgium. The
standard application of this system yields the modal
parameters of a structure excited by a force xi(t) at
point i and responding with a signal Yk(t) in each
measurement pOint k. The basic data for the calculations are the frequency response functions Hik{iro) =
Sy;/,Sxx with Syx = cross power spectrum of Yk(t) vs.
Xi{t) and Sxx = auto power spectrum of Xi(t). For the
present application no input Signal Xi(t) was available. Therefore, this signal was replaced by an output signal yr(t) measured at a reference pOint r. The
cross power spectrum Syr,yk of yr(t) vs. Yk(t) for every
response point k was useCl instead of the frequency
response function H. Phase and amplitude relationships among all of the three bridge and four vehicle
signals could thus easily be established.
7. THE VEHICLE ON THE RIGID PAVEMENT
a; 30° Front Wheels
45° Rear Wheels
± 3° adjustable (Zeroing)
Fig. 4
132
Measurement of the dynamic wheel load:
Principle of operation; the device mounted
on the two wheels of a rear axle.
Figure 6 gives an example of the frequency spectra for the four wheel load signals. Relative peaks
occur with frequencies between f = 1 Hz and f = 5
Hz. The highest peak is referred to as "peak frequency". Peak frequencies occur between f = 1.2 Hz
and f = 3.2 Hz. This frequency band is much broader
than was assumed from EMPA standard tests. The
shape of the vehicle's motion was determined for
VEHICLE - BRIDGE INTERACTION
14R 10A, v= 23.25 km/h
A SPECTR
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14R10A, v=63.0km/h
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A SPECTR
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'AVG. T:
OE+OD
0
1. ST
SE+OO
~
REAL
OE+OO
0
A SPECTR
BE-02
REAL
OE+OO
0
BE-02
REAL
OE+OO
0
Fig. 6
o
'AVG. T:
'AVG. T:
A6
OE+OD
0
A BPECTR
1. ST
I~
\1.95
[195
REAL
\3.71
4E-GS
REAL
OE+OO
0
2E-GS
REAL
OE+OO
1.76
Fig. 7
K1
0
'AVa, T:
OIFt
3
HZ
S,9T
BLK •
S9.!IlI
2
PNT:
K2
2
OIFt
3
~AVa,
T:
'AVa, T:
HZ
BLK ,
S,9T
S9.!IlI
3
PNT:
K3
"'"
s. 9T
~~3.91
3
OIFt
3
HZ
BLK ,
SlJ.!IlI
4
PNT:
K4
"'"
""
"'"
4
OIFt
3
HZ
S9.!IlI
Examples of auto power spectra of wheel load signals for a moderate and a high vehicle speed. Vehicle
14R, tire pressure 10 bar, pavement A. Frequency resolution Llf = 0.125 Hz and Af = 0.25 Hz respectively. Peak frequencies are indicated. K1 to K4: Rear left, rear right, front left and front right wheel.
relative and absolute peaks with the help of the modal analysis system. This showed that in 90% of all
cases the vehicle mode shape is equal to one of the
five shapes between pure heave and pure pitch motion given in Figure 7. The "digitized" frequency
spectra for rear axles are plotted against vehicle
speed in Figure 8. The vehicle mode shapes are
indicated with a symbol, the peak frequencies are
linked with a solid line. Taking into account the results for both front and rear axles but only for the
peak frequencies simplifies the discussion (Fig. 9).
The behavior of the vehicle with increasing speed
can be divided into three parts. Region I: The peak
frequency is between 2.5 Hz and 3.5 Hz, no mode
shape is predominant. Region 11: The peak frequency decreases to f = 2.5... 2.0 Hz, mode shapes of
heave type are predominant. Region Ill: The peak
frequency reaches values of f "" 1.5 Hz and then increases again to f = 2.0 ... 2.5 Hz, only pitch motions
occur.
Heave
+
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PNT:
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1~10_
BLK'
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Pitch
o
Undef.
Vehicle mode shapes. The five modes
shown cover about 90% of all shapes observed. "Undefined" mode shapes are indicated with a circle symbol.
This behavior can be interpreted as follows: In
Region I, the suspension leaf springs are blocked
and the vehicle vibrates on its tires only. In Region 11,
the leaf springs are more and more activated and in
Region Ill, this activation is complete. The increase
of frequency during Region III may be due to the
progressive stiffness of the fully activated leaf
springs.
It may be noted that the boundaries between the
various regions are not the same for front and rear
axles: Front axles always reach the next higher region at a lower speed than rear axles. This may be
due to the fact that the force to activate a leaf spring
is smaller for front axle springs than for rear axle
springs. In addition, these boundaries clearly depend
on pavement roughness: For the rear axles and the
rough pavement A, these boundaries are v = 25 ...30
km/h for Regions 1111 and v = 50 km/h for Regions
111,.111. A Region 11 does not exist for the smooth pa'l\~­
ment B. The boundary occurring at v = 50 km/h
therefore separates Region I from Region III in this
case.
133
HEAVY VEHICLES AND ROADS
5
Frequency [Hz]
.
.
5
o
Frequency [Hz]
:
:
0:
Vehicle 12R1 aA, Front Axle
:0
.......+ ····:··n
4
o
o
o
0:
+ :0
..... 9 ...... .+....... +
4
···~-D
o
o
o
Vehicle 12R1 aA, Rear Axle
o·
t:.
1::.0
o
~
0
+
0:
o
.
o ... ~.::
3
0
.0. 9. . .... L!>
t:.
+
+
t:. :
+
2.0 . ' : 0
2
x+o
+
xo
C:>o;9.
x
o
0
0:
10
0
?<
0
0
x+o
xo
20
9
+
.
:0
o
0:
x
30
40
50
60
70
80
o
20
10
30
Vehicle Speed [km/h)
Fig. 8
3,5
3,5
70
60
80
Frequency [Hz]
~ Ve~icle
Vehicle
12Rl
,
,
, aA,
Rear and Front Axles
3
3
~~r- ~
X
b
fi>
<)
2
"l!'.
30
40
.x
50
L- /
x
x···
0
60
70
0
o
80
20
10
30
40
60
70
80
I
I
Rear and Front Axles
~
3
x
,Vehicle
, 14R
, lOB,
1\ \
01
~..
IY Cl
x
X
2,5
V\
8: . .
r\
~
*
2
I
Rear and Front Axles
i!J
1.5r--1---+--~-~--+--+_-~-_4
50
3,5
Vehicle 14Rl aA,
I
1\
1.5
3,5 ,---,----,---,------,---,---,----,---,
I
~
~.
...
20
t V\
0
~ ..
10
/ \0
2
y
\d
(;
.X·
~
-J
.I
,12R19B,
: Rear and Front Axl as
p
~
1,5
~
2,5
.X JV
<>
..
o
50
"Digitized" wheel load spectra versus vehicle speed. The peak frequencies are connected with a solid
line. The symbols give the vehicle mode shape according to Figure 7.
Frequency [Hz]
2,5
40
Vehicle Speed [km/h]
x
V
Ax ..
K···
X
1,5
X
1
o
Fig. 9
134
10
20
30
40
50
Vehicle Speed [km/h]
60
70
80
o
10
20
30
40
50
60
70
80
Vehicle Speed [km/h]
Peak frequencies and mode shapes versus vehicle speed for front (dotted line) and rear (solid line)
axles.
HEAVY VEHICLES AND ROADS
Bridge Response PeakfrequeRcy
Vehicles 12R1 OA/12R1 OB. WG 02
Bridge/Vehicle Interaction Peakfrequency
Vehicles 12R1 OA/12R1 OB, WG 02
Frequency [Hz]
6r---~--~----~--~--~----~--~---.
5
........... .
.*
4
Frequency [Hz]
6.---~--~----~--~~~----~--~---.
5
: *Ilf . (J .**:
. .....
I!J
o :
1(
Q
1(
I!J
o : 0* .
.. ····0·······0
0
:0
3
*:
:0
0
3
0:
JlI:
.......... ,. .......... P··u
2
o
--e-
10
Primary A
-.... Primary B
20
30
40
50
Vehicle Speed [km/h)
o
*
60
70
80
o 0
o
10
Secondary A
--e-
Secondary B
-k- Primary B
Fig. 10 Peak response frequencies of the loaded
bridge.
0: [l\'
..... 0
2
Fig. 11
Primary A
20
:
*
o I!J:
0
*1(
·····u·:·····;:.. ·O··
0
: 0
40
30
50
Vehicle Speed [km/h)
o
*
60
1(
80
70
Secondary A
Secondary B
Vehicle/bridge interaction peak frequencies.
8. THE VEHICLE ON THE BRIDGE
The behavior of the vehicle during its passage
over the bridge is basically the same as over the
rigid pavement. Bridge vibrations do not seem to
influence the motion of the vehicle in a significant
manner.
5431 WF8
MODE
FAEG
3
(Hz)
3.13
ZETA
(X)
2.0000
9. THE FREELY VIBRATING BRIDGE
Frequency domain analysis shows that the first
three natural vibrations of longitudinal bending are
excited by the vehicle to a greater or lesser extent.
The relative magnitude of these modes depends on
the measurement point and on vehicle speed. The
first mode f1 = 3.03 Hz usually dominates. An example of the importance of higher modes for the bridge
behavior is given later (see Paragraph 14).
10. THE VIBRATIONS OF THE LOADED BRIDGE
It can be seen from Figure 10 that the peak frequency of the bridge response mainly varies in a
range of .M = ± 0.2 ... 0.3 Hz around the fundamental
frequency f1 = 3.03 Hz.
3503 WFB
MODE
FAEG
7
(Hz)
5.47
ZETA
(X)
2.0000
11. VEHICLE/BRIDGE INTERACTION
The cross power spectrum for wheel load vs.
bridge response is used here to indicate the intensity
of frequency dependent interaction between vehicle
and bridge. A similar observation as for the spectra
of the bridge response can be made for the peak
frequencies of the interaction spectra: They occur in
a range of .M = ± 0.2 ... 0.3 Hz on both sides of the
bridge natural frequency 11 = 3.03 Hz, on the lower
side .M even reaches values of up to 0.7 Hz (Fig.
11). Figure 12 gives examples of vibrational shapes
of the vehicle/bridge system.
Fig. 12 Vibrational shapes of vehicle/bridge interaction.
135
HEAVYVEHICLES AND ROADS
12. MODELlNG
A simple model was sought after in an attempt to
explain these frequency variations. It was found that
• it does not suffice to merely take the vehicle's mass
effect into account. This would yield a maximum frequency shift of only .M = 0.06 Hz. Investigation of a
two-degree-of-freedom system built up by coupling
the 1DOF-systems representing the vehicle and
bridge respectively yielded a maximum frequency
shift of .M = ± 0.25 Hz for the case of egual natural
frequencies of the 1DOF-systems (Fig. 13).
Vehicle:
Results of the detailed analysis are presented in
Figures 15 and 16. These figures give a) the simultaneously measured time signals of the wheel load K1
and the bridge response WG 02, b) the frequency
spectra of the wheel loads K1 V and K1 W for driving
over the approach and bridge respectively, c) the
frequency spectra of wheel load K1W and bridge
response WG 02 during the vehicle's passage over a
given bridge span (calculated from the abovementioned time signals) and d) the cross power spectrum
XP of WG 02 vs. K1W.
The uncoupled 1DOF-Systems
with equal natural frequencies:
= 442.6
Mbr
kbr
fbr
Vehicle:
-i
Further investigations were based on the cross
power spectra of bridge response vs. wheel load .
The above mentioned model shows that it may not
be accurate to extend the data processing range on
the process of the vehicle's passage over the bridge
as a whole. Therefore, the range was divided into
three parts according to the bridge spans. It was
also noted that the modal system will always yield a
phase angle equal to zero or to 11:. To investigate the
exact phase between load and response in greater
detail it was therefore necessary to stop the automatic data processing at the spectrum level and to continue with manual methods.
In addition, an MDOF-model was utilized for the
bridge (Fig. 14).
Bridge:
Bridge:
13. DETAILED INVESTIGATION
. 103 kg
= 160.2 . 106 N/m
= 3.03 Hz
This cross power spectrum allowed investigation of
amplitude and phase relationships between wheel
load and bridge response in detail and yielded the
basis for the conclusions given below (see [2] for
more information).
Mve = 16.5 . 103 kg
fve
=3.03 Hz
kve = 6.0 . 106 N/m
The coupled system:
(o:fl/- (kbr + kve + k ve). co2 + k br · kve = 0
Mbr
~
Mbr·M ve
Frequency of the Coupled System
...; ......;. ..... :...... ;...... ;...... : ..... : .......... .
· .. .. .. .. .. ..
···
. . .
. . .
·· ... ... ... ... ... ...
· . . . . . .
4,3
····lo·····:·
3.8
····:·····1·· .. ~ ..... ~ ......:......:......:...... :...... :...... ~.
·
·····
··
·
3.3
.·
..
.....
.
.....
...
.
..
.....
.
..
..
.
..
.....
..
.
..
....
..
....
..
.
14. CONCLUSIONS
The most important parameter for the behavior of
a bridge span crossed by a vehicle is the relationship
between the peak frequency of the dynamic wheel
loads and the frequency of the dominating bridge
mode. Vehicle/bridge interaction and hence also
bridge response are governed by a superposition of
the following effects:
~
2,8
2.3
1.8
····z·····.··
..
··
.. ·.......·:.......:. .. , ...:. ..... .; ..... .;...... .: .. ... .... .
··: .:. ..: .:. ..: ..: ..:
·,
.. .. ..
.. .. ..
· ,
·
.
..
..
..
..
..
.: ..... :..... ..... :' .... ..... ,;, .... ': ..... :..... :..... :... .. . ....
· .
. . . . . . . .
:
:
:
:
:
:
:
:
:
:
··· ... ... ... ... ... ... ... ... ...
·· .. .. .. .. .
. ..
.
. ..
.
~
~
..
32.35m
I
I
I
I
I
'r
41.00m
I
I
• ••• •
=-. J. · "r'" ·...
·T' ···"·.:
:;;:. ····"r" ··....
"I"······
~
1.3~~-L~--~~-L~~~~-L~~--~~
1.5 1,7 1,9 2.1 2.3 2.5 2.7 2.9 3.1 3.3 3,5 3.7 3.9 4,1 4,3
Vehicle Frequency
...... fl Coupled 5y••
-e-
-&- fO Bridge
~ fO Vehicle
f2 Coupled 5ys.
Fig. 13 Modeling of the closely coupled vehicle and
bridge as a two-degree of freedom system.
Influence of the variation of the vehicles
natural frequency on the frequencies of the
coupled system (the bridge's natural frequency is constant fl = 3.03 Hz).
136
'r
• • • "0:
'Jt
;:,..... ·I·········T···~···';lt
Fig. 14 Modeling of the bridge as an MDOF-systern; coupling with the vehicle SDOF-model
according to its location on the bridge.
VEHICLE - BRIDGE INTERACTION
• Close coupling of the vehicle and bridge occurs
when their natural frequencies are similar and
when the vehicle is near the mid-point of a dynamically significant bridge span. The frequency shift
of the resulting in-phase movement of the coupled
system is determined by the parameters of the
vehicle and bridge as 1DOF-systems. In the case
of the DeibOel Bridge, a vehicle mass of 3% of the
bridge mass caused a shift of df =± 0.25 Hz.
• No coupling occurs when the vehicle is passing
over supports (or when the above mentioned frequencies differ significantly from each other).
• During these phases of decoupling, the bridge will
store energy in its fundamental modes.
• When the vehicle passes over the second or third
span, a superposition of vibrations occurs corresponding to both the coupled and uncoupled states.
As the frequencies of these vibrations are relatively close together this results in a beating phenomenon of considerable intensity (Fig. 15). Large
dynamic increments occur when the maximum
beating amplitude coincides with the maximum
static deflection of the bridge.
• All bridge modes with frequencies f = 1...5 Hz
have to be taken into consideration. For the DeibOel Bridge, the significant decrease in the dynamic increment for v = 25 ...30 km/h (Fig. 5) is due
to the fact that the wheel load frequency reached
values of f = 2 Hz here and therefore excited not
only the first but also the second bridge mode (f2 =
4.28 Hz; see Fig. 16).
REFERENCES
[1] Cantieni, R., Dynamic Load Tests on Highway
Bridges in Switzerland - 60 Years Experience
of EMPA. EMPA Report 211 (1983), Swiss
Federal Laboratories for Materials Testing and
Research, DObendorf, Switzerland
[2] Cantieni, R., Beitrag ?our Dynamik von StrassenbrOcken unter der Uberfahrt schwerer Fahrzeuge. Diss. ETH Nr. 9505 (1991);
English Translation: Dynamic Behavior of Highway Bridges under the Passage of Heavy Vehicles. EMPA Report No. 220 (1992)
A15
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SE-02
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1.95
2.54
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K1 W
Amp!.
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K1 W
OE+OO
OE+OO
OE..oo ~J..L---------=-__:_!!_~~
6E-02
2E+OO
K1V
Amp!.
HZ
0
7.S1
8E-02
SE-Ol
2.73
K1 W
WG02
Amp!.
K1 W
WG22
Ampl.
OE+OO
OE+OO
OE+OO o!-J..J..==-~=-=:......-...::::,.-====~~~
OE+OO
o!-L---------'><::..-""....H:l'7Z~7;;;-.81;--'
4E-02 , - - - - - - - - - - - - - - _
8E-021
-
2.73
'~Jvr
OE+OO L.-----.b.!...-~~--___,-:-:;--=:___l
o
12
HZ
S.S6
O~
72
Phase
Amp!.
XP
WG02
K1
131
180
O~
o
Fig. 15 Vehicle 12R10, Pavement A, v = 15 km/h,
WG 02 = bridge deflection in the 41 m
span, K1 = left rear wheel load. (see paragraph 13 for the detailed description of the
diagrams Shown.)
4.3
OE+OO o.L--c:...::::~-L----'''--'''---''>----H:7-::Z,.--:7;-;;-.Sl:--'
O~18
3
5
48
15 53
Phase
XP
WG22
K1
180
O~_ _ _=_~-------~
Fig. 16 Vehicle 12R10, Pavement A, v = 30 km/h,
WG 22 = deflection in the 37 m span, K1 =
left rear wheel load (see paragraph 13 for
the detailed description of the diagrams
137
shown).