Effects of Pavement Roughness and Dynamic Tank Load on the Bridge Response

The dynamic force generated by the interaction of the bridge caused by the passage of vehicles plays an important role in the design of the bridge structure. In practice, usually, a moving static load is used to model the vehicle force on bridges, and the static design loads are increased by a dynamic load allowance (DLA) to account for the dynamic effects from the vehicle vibrations due to the interaction between the vehicle and bridge [1].

Many codes specify the DLA as a function of span length only which might not lead to a reliable prediction of the dynamic loads on the bridge due to avoiding dynamic response analysis of the vehicle vibration that is moving along the bridge allow for a straightforward estimation of the vehicle loads in the bridge design [2]. The dynamic load that is caused by the interaction between the vehicle and the bridge is a difficult problem that is impacted by a wide variety of parameters including, the level of roughness of the road surface, the speed of the vehicle, and the dynamic properties of both the vehicle and the bridge. Therefore, there has been an increasing interest in and concern about bridge design forces [3].

Thus, determining the vehicle's dynamic load due to the vehicle's passage through the span of the bridge is a problem of great interest to bridge engineers. Therefore, an approach was developed, and a simple closed-form solution was derived to predict a vehicle's dynamic load for bridge design applications. Two sets of equations of motion were written to solve the problem of vehicle-bridge interaction, one for the vehicle and the other for the bridge, and in order to mathematically relate the motion of the vehicle and the bridge, the interactive force existing at the contact point between the vehicle and bridge is considered [4, 5].

In this study, numerical analysis has been carried out to calculate the dynamic load allowance and the stochastic response of the bridge because of the passage of a vehicle moving at constant speed along a rough bridge surface. The vehicle has been replaced by a simple, linear, damped spring-mass system which moves on a bridge at a constant speed. The bridge has been represented as simply support with a single degree of freedom.

Suppose a vehicle's equation of motion is expressed in the static-equilibrium position. In that case, it will be referenced to the vehicle's displacements in future discussions when the vehicle's dynamic "random" response will be calculated [1, 8]. The analytical model of the dynamic vehicle-bridge interactive system is shown in Figure 1 (c). The total response of the vehicle, such as spring force, displacement, etc., is achieved by including the static response to the dynamic analysis results. The static responses that cause by gravity force from vehicle weight. The static force of the vehicle also represents the mean of the stochastic load, and it's considered the reference position when the system vibrates. Then to study the dynamic response separately from the static, the static force was ignored when considering the zero mean gaussian process. Then, Eq. (7) becomes [9]:

$m_1 \ddot{z}(t)+c_0 \dot{z}(t)+k z(t)=f(t)+c_0 \dot{y}(d, t)+k y(d, t)$         (8)

If y(d, t) and f(t) have, respectively, power spectral density function S_yy(d, ω) and S_ff(ω) with respect to time, the relation of the power spectral density function of the vehicle response S_zz(d, ω) and of the inputs is then given by:

$S_{z z}(d, \omega)=\sum_r \sum_s H_r{ }^*(\omega) H_s{ }^*(\omega) S_{r s}(d, \omega)$$r, s=y(d, t), f(t)$            (9)

where, $H_r{ }^*(\omega)$ is the complex conjugate of H_r(ω).

In general, the engine-induced force f(t) exhibits a harmonic form and little correlative with the pavement roughness and the moving distance of a vehicle. For uncorrelated inputs, S_zz (d, ω) can be expressed by the relation:

$S_{z z}(d, \omega)=\left|H_y(\omega)\right|^2 S_{y y}(d, \omega)+\left|H_f(\omega)\right|^2 S_{f f}(\omega)$          (10)

where,

$H_y(\omega)=\frac{k+i c_0 \omega}{\left(k-m_1 \omega^2\right)+i c_0 \omega}$         (11)

and

$H_f(\omega)=\frac{1}{\left(k-m_1 \omega^2\right)+i c_0 \omega}$         (12)

If the influence of engine motion on vehicle vibration is ignored as in this study, the S_zz (d, ω) become:

$S_{z z}(d, \omega)=\left|H_y(\omega)\right|^2 S_{y y}(d, \omega)$         (13)

The dynamic vehicle load F(t) is calculated empirically for a specific stretch of pavement, and it is supposed to have the properties of a stationary Gaussian random process with zero mean value. Considering that the power spectral density function was used to characterize the F(t), and that the F(t) is defined [15]:

$F(t)=C_0(\dot{Z}-\dot{y})+k(Z-y)=f(t)-m_1 \ddot{Z}$            (16)

The spectral density function of F(t) is approximately given by Eq. (17):

$S_{F F}(\omega)=T_f(\omega) S_{f f}(\omega)+T_r(\omega) S_{y_r y_r}(\omega)$         (17)

The T_f and T_r are expressed by Eq. (18) and Eq. (19), respectively:

$T_f(\omega)=1+m_1{ }^2 \omega^4\left|H_f(\omega)\right|^2$           (18)

$T_r(\omega)=m_1^2 \omega^4\left|H_y(\omega)\right|^2$          (19)

Then, the mean square and Root mean square of F(t) is related to S_FF(ω) by Eq. (20), and Eq. (21), respectively:

$\sigma_F^2=\int_0^{\infty} S_{F F}(\omega) d \omega$          (20)

$\sigma_F=\sqrt{\sigma_F^2}$         (21)

7.1 Equation of motion of bridge

The transverse deflection y_b (x, t) of the bridge at time t and distance x satisfies the partial differential equation given by Eq. (23) because the bridge with length L and mass per unit length $\bar{m}$ is elastic uniform straight and subjected to a viscous damping force c per unit length per unit velocity and a transverse force p(x, t) per unit length [1, 16].

$\bar{m} \frac{\partial^2 \tilde{y}_b}{\partial t^2}+c \frac{\partial \tilde{y}_b}{\partial t}+E I \frac{\partial^4 \tilde{y}_b}{\partial x^4}=p(x, t)$         (23)

where, EI is the constant bending stiffness of the bridge. $p(x, t)$ replaced by Eq. (26) for constant vehicle speed [17].

$p(x, t)=\delta(x-d) p(t)=\delta(x-d)\left(m_1 g+F(t)\right)$$=\delta(x-d) m_1 g+\delta(x-d) F(t)$    (24)

where, $\delta(\cdot)$ is the Dirac distribution, $F(t)$ is a stationary Gaussian random process with zero mean; m₁g is the vehicle gravity; the total vehicle load on the bridge P(t) is a stationary Gaussian random process with a mean value of m₁g. If F(t) assumed is independent of the mean deterministic deflection of the bridge. Eq. (23) is written as two separate equations as follows:

$\bar{m} \frac{\partial^2 y_{b_1}}{\partial t^2}+c \frac{\partial y_{b_1}}{\partial t}+E I \frac{\partial^4 y_{b_1}}{\partial x^4}=\delta(x-d) m_1 g$    (25)

$\bar{m} \frac{\partial^2 y_b}{\partial t^2}+c \frac{\partial y_b}{\partial t}+E I \frac{\partial^4 y_b}{\partial x^4}=\delta(x-d) F(t)$    (26)

Eq. (25) for deterministic mean values of random function y_b(x, t), and p(x, t). while the second, Eq. (26), is valid for their centered "random" components. The total deflection $\tilde{y}_b(x, t)$ of the bridge due to the moving vehicle is the sum of the deflection $y_{b_1}$ of the bridge subject to the constant force $m_1 g$ and the deflection y_b due to dynamic vehicle load F(t) (see Figure 1 (c)). $y_b=y_{b_1}+y_b, y_{b_1}$ is a deterministic function, and y_b is a random function.

7.2 Response of bridge to dynamic vehicle load

The statistical properties of the second-order (variation of $y_b$) that can be derived from Eq. (26) [1, 18]. One form of a solution of Eq. (26) obtained by separation of variables, assuming that the solution has the form:

$y_b(x, t)=\sum_{i=1}^{\infty} \psi_j(x) Y_j(t)$    (27)

It is believed that the free-vibration motions consist of a series of constant shapes ψ_j(x), and that the amplitude of these motions varies with time by Y_j(t). When analyzing undamped free vibration, it is possible to estimate the undamped angular frequencies readily ω_j of the bridge as well as the mode shapes ψ_j(x) of the bridge by considering the boundary conditions at the ends of the bridge segment. The ω_j and ψ_j(x) values of the bridge can be calculated as follows for a simply supported bridge:

$\omega_j=\left(j \frac{\pi}{L}\right)^2 \sqrt{\frac{E I}{\bar{m}}}$   (28)

or

$f_j=\left(j \frac{\pi}{2 L^2}\right) \sqrt{\frac{E I}{m}}$    (29)

and

$\psi_j(x)=\sqrt{2} \sin \left(j \frac{\pi x}{L}\right)$    (30)

The modes ψ_j(x) satisfy the orthogonal conditions:

$\int_0^L \psi_j(x) \psi_k(x) d x=L \delta_{j k}$   (31)

where, δ_jk is the Kronecker delta function.

Substituting Eq. (27) into Eq. (26), multiplying through by ψ_j(x), integrating over $x$, and using the orthogonal conditions, here the ψ_j(x) is considered a filter to choose the mode required, all mods that are j≠k are disappearing. Hence, transforming from a multi-degree of freedom to a single degree of freedom leads to the uncoupled equation of motion for Y_j(t):

$\ddot{Y}_j+\beta \dot{Y}_j+\omega_j^2 Y_j=G_j(t)$    (32)

where,

$\beta_j=\frac{c}{\bar{m}}$   (33)

and

$G_j(t)=\frac{1}{\bar{m} L} \int_0^L \psi_j(x) \delta(x-d) F(t) d x$    (34)

The convolution integral gives the formal solution to Eq. (32):

$Y_j(t)=\int_0^t G_j(t-\theta) h_j(\theta) d \theta$    (35)

where, the impulse response function is:

$h_j(t)=\left\{\frac{e^{-0.5 \beta_j t}}{\omega_j \sqrt{1-\frac{\beta_j^2}{4 \omega_j^2}}} \sin \left(\omega_j \sqrt{1-\frac{\beta_j^2}{4 \omega_j^2}} t\right)\right\}$, t≥0          (36)

Substituting Eq. (34) into Eq. (35), the modal amplitude Yj(t) may then be written as:

$Y_j(t)=\int_0^t \frac{1}{\overline{m L}} \int_0^L \psi_j(x) \delta(x-d) F(t-\theta) d x h_j(\theta) d \theta$$=\frac{\psi_j(d)}{\overline{m L}} \int_0^L F(t-\theta) h_j(\theta) d \theta$    (37)

Thus,y_b(x, t) can be obtained in the form:

$y_b(x, t)=\sum_{j=1}^{\infty} \psi_j(x) Y_j(t)$    (38)

$y_b(x, t)=\sum_{j=1}^{\infty} \frac{\psi_j(x) \psi_j(d)}{\overline{m L}} \int_0^t F(t-\theta) h_j(\theta) d \theta$    (39)

In according to random vibration theory, the spectral density function of y_b(x, t) give:

$S_{y_b y_b}(x, \omega)=\frac{1}{2 \pi} \int_{-\infty}^{\infty} R_{y_b y_b}(x, \tau) e^{-i \omega \tau} d \tau$    (40)

where,

$R_{y_b y_b}(x, \tau)=E\left[y_b(x, t) y_b(x, t+\tau)\right]$           (41)

The power spectrum density of bridge deflection is given by Eq. (42):

$S_{y_b y_b}(x, \omega)$$=S_{F F}(\omega) \sum_{j=1}^{\infty} \sum_{k=1}^{\infty} \frac{\psi_j(x) \psi_j(d) \psi_k(x) \psi_k(d)}{(\bar{m} L)^2} H_j(\omega) H_k(-\omega)$    (42)

where, SFF (ω) is the power spectral density function of F(t) and Hj(ω) is the transfer function of the bridge.

$H_j(\omega)=\frac{1}{\left(\omega_j^2-\omega^2\right)+i \beta_j \omega}$    (43)

and

$B(x, \omega)$$=\sum_{j=1}^{\infty} \sum_{k=1}^{\infty} \frac{\psi_j(x) \psi_j(d) \psi_k(x) \psi_k(d)}{(\bar{m} L)^2} H_j(\omega) H_k(-\omega)$     (44)

Finally, the power spectrum density for single degree of freedom system of a bridge may be written as:

$S_{y_b y_b}(x, \omega)$$=S_{F F}(\omega) \frac{1}{(\bar{m} L)^2} \frac{1}{\left(\omega_j^2-\omega^2\right)+i 2 \xi \omega_j \omega}$    (45)