BRIDGE-VEHICLE INTERACTION MODELS
3.3 Deck roughness
3.3.2 Random unevenness
The road surface roughness in Bridge-Vehicle interaction dynamics can be described as a stationary Gaussian random field characterized by its power spectral density (PSD) function (Schiehlen, 2009; Lombaert and Conte, 2011; ISO 8608: 1995). Therefore, this subsection includes a general introduction to random Gaussian processes with their stochastic characteristics, a description of roadway profiles as a random process, as well as an illustration of models for the realizations of roadway profiles.
In general, for a Gaussian stochastic process g0(t) with an autocorrelation function Rg0g0( ) and a two-sided power spectral density function Sg0g0( ), the following relations hold (Shinozuka and Deodatis, 1991)
0 )]
( [g0 t =
E (3.4) )
( )]
( ) (
[g0 t τ g0 t Rg0g0 τ
E + = (3.5) τ
ωτ π τ
ω R i d
Sgg gg ( )exp( )
2 ) 1
( 0 0
0
0 = ∞ −
∞
−
(3.6)
κ ωτ ω
τ S i d
Rg0g0( )= ∞ g0g0( )exp(− )
∞
−
(3.7) where denotes the temporal frequency
Dodds and Robson (1973) suggested a simplified way of describing the road surfaces to avoid a laborious procedure which is often obtained by measuring existing roadways. In the present research, road unevenness has been treated as a realization of a stationary Gaussian homogeneous random process described by its power spectral density function in the space domain Sg0g0( ) with as the wave number / spatial frequency. However, the dynamic analysis has been performed in the time domain, hence, a description of the road unevenness in the time domain is needed. Therefore, the temporal power spectral density function Sg0g0( ) is to be computed. Assuming a constant speed for the vehicle V, the relationship between Sg0g0 ( ) and Sg0g0 ( ) can be derived as follows:
(i) Since the vehicle is traveling at a constant speed of V, any instant of the travel time ‘ ’ can be expressed in terms of the distance travelled ‘x’ as
V
=x
τ (3.8) (ii) A cycle of wavelength = 2 / is covered in period T as
T =Vλ (3.9) Therefore, the temporal frequency can be written as
Ω
=
=
= V V
Tπ πλ
ω 2 2 (3.10) (iii) Substituting the above Eqs. (3.8), (3.9) and (3.10) in Eq. (3.6), we get
} ) exp(
) 2 (
{ 1 ) 1
( 0 0
0
0 R x i x dx
V V
Sgg = Ω = ∞ gg − Ω
∞
π −
ω (3.11)
) 1 (
0
0 Ω
= Sgg
V (3.12) Using the relationship given in Eq. (3.12) the temporal spectral density function Sg0g0( ) can be obtained from the spatial spectral density function Sg0g0 ( ). When performing the analysis in the time domain, one can deduce that the excitation of the vehicle due to road unevenness can be described as non-stationary when the vehicle speed is time dependent (Schiehlen, 2009).
In most of the engineering applications, the one-sided spectral density function is derived from measurements for which the following relation holds
) ( 2 )
(Ω = gg Ω
GG S
S (3.13)
in which SGG ( ) is one sided spectral density. Based on the work of past researchers (Rice,1954; Shinozuka and Deodatis, 1991), there are two main models for generating realizations of random processes. One consists of a series of sines and cosines with random amplitudes and the other consists of a series of cosine terms having random phase angles with a specified probability density function. In regard to Vehicle-Bridge interaction dynamics analysis, the latter is often adopted for the realization of road profiles (Olivia et. al., 2013), as shown below
=
+ Ω
= N
s s s s
r x A x
h
1
) 2
cos(
)
( π θ
(3.14) where hr(x) is the deck surface random unevenness which is a Gaussian process (Shinozuka, 1971) with an independent random phase angle θs uniformly distributed from 0 to 2 . N is the number of terms used to build up the road surface roughness, .As is the amplitude of cosine wave, s is the spatial frequency (c/m) within the interval [ L , U] where L , U are lower and upper cut-off frequencies of spatial unevenness . The parameters As and s are computed as
∆Ω Ω
= GG( s)
s S
A (3.15)
∆Ω
− + Ω
=
Ω 2
s 1
L
s (3.16)
where SGG( s) is the one-sided spatial power spectral density function (m2/c/m) of road surface unevenness as given in Eq. (3.13).
The above equations shows that the power spectral density can be discretized into spatial frequency bands of a width of , and the corresponding discretized frequencies are used in the realization of the process as shown in Fig. 3.3. However, the entire frequency domain of the power spectral density cannot be used in the realization due to mathematical and physical reasons (Shinozuka, 1971). Cut-off frequencies are needed for the realizations of road surfaces. The discretizing frequency band is defined as:
N
L
U )
(Ω −Ω
=
∆Ω (3.17) As mentioned earlier L, U are lower and upper cut-off frequencies respectively and N is the number of terms used to build up the road surface roughness.
Fig. 3.3 Discretized one-sided power spectral density function
The long wavelength irregularities correspond to low frequency components and short wavelength irregularities correspond to high frequency components (Newland, 1993). The different wavelengths and their corresponding frequencies excite different vibrational modes of the heavy vehicle. The bouncing mode of the sprung mass is more of low frequency mode while the axle hop and pitching modes are of higher frequencies (Cebon, 1999). Furthermore, when the wavelengths of the irregularities are too small compared with the dimensions of the contact patch between the tire and the roadway, the tyre absorbs these irregularities due to its flexibility. This phenomenon is referred to as “tyre envelopment”, which reduces the excitations of the axle of the vehicle. Therefore, filtering or smoothing algorithms are recommended (Newland 1986) depending on the dimension of the contact patch. Often a moving averaging filter is used for such purposes (Sayers and Karamihas, 1996). However, the effects of tyre envelopment of short wavelengths irregularities can be neglected for normal highway speeds, whereas for low speeds tyre envelopment becomes important and more detailed models for the contact patches may be significant (Captain et. al. 1979; Cebon, 1999).
The power spectral density function SGG( s) (m2/c/m) of road surface unevenness can be expressed as (Huang and Wang, 1992)
m GG s
s
GG S
S
−
Ω
× Ω Ω
= Ω
0 0) ( )
( (3.18)
where 0 is referred as discontinuity frequency and is taken as 1/2 (c/m). m and SGG( 0), are the roughness magnitude coefficient and the roughness shape coefficient respectively. They are indicators of roughness intensity. The ISO suggested that for m=2, lower cut off frequency L=0.1 c/m and upper cutoff frequency U = 2.0 c/m (Sun and Kennedy, 2002)
may be considered to determine dynamic load on the wheel induced by pavement unevenness.
On examination of the Eq. (3.18), it is revealed that at very low spatial frequency ( s 0), the power spectral density becomes unbounded, i.e., SGG( s) . In view of this, Yin et al (Yin et. al. 2010) suggested an improved equation as follows,
2 2
20 0)
( ) (
L s GG
s
GG S
S Ω +Ω
× Ω Ω
=
Ω
(3.19) The PSD function given by Eq.(3.19) has been adopted in the present study. Using the relationship given in Eq. (3.12), the temporal spectral density function SGG(ωs) has been obtained from the spatial spectral density function SGG( s) given in Eq. (3.19). ISO classified road into five classes, depending on different road surface conditions. Table 3.1 gives ISO specified values for this classification (ISO 8608: 1995).
Table 3.1 Road classification based on roughness magnitude coefficient (ISO 8608: 1995)
Road class Road roughness coefficient
SGG( 0) (10-6 m2/c/m) A (very good) 2 < SGG( 0) ≤ 8 B (good) 8 < SGG( 0) ≤32 C (average) 32 < SGG( 0) ≤128
D (poor) 128 < SGG( 0) ≤512
E (very poor) 512 < SGG( 0) ≤2028