2. Theoretical and numerical approach

#### 2.1. Simply supported beam subjected to a moving constant force

The simplest calculation model of bridge vibration is based on a simply supported elastic beam with a mass moving across the beam at constant velocity. The moving mass is assumed negligible compared with the mass of the beam. This basic case of dynamic bridge response was solved by authors, e.g., [4, 6, 13, 15] and others. The vibration caused by a force moving across an elastic Bernoulli-Euler's beam (Figure 1) with viscous damping, is described by the equation

$$\pm i \frac{\partial^4 \nu \{\boldsymbol{x}\_1 \boldsymbol{l}\}}{\partial \boldsymbol{x}^4} + \mu \frac{\partial^2 \nu \{\boldsymbol{x}\_1 \boldsymbol{l}\}}{\partial \boldsymbol{t}^2} + 2 \mu \boldsymbol{\omega}\_{\boldsymbol{\vartheta}} \frac{\partial \nu \{\boldsymbol{x}\_1 \boldsymbol{l}\}}{\partial \boldsymbol{t}} = \delta (\boldsymbol{x} - \boldsymbol{c} \boldsymbol{t}) \boldsymbol{\nu} \tag{1}$$

with appropriate boundary and initial conditions.

The state of oscillation may be expressed as follows:

$$\begin{aligned} \nu(\alpha\_1 t) &= \nu\_0 \sum\_{j=1}^{\infty} \frac{1}{j^2 [j^2 \{j^2 - \alpha^2\} + 4\alpha^2 \beta^2]} [j^2 \{j^2 - \alpha^2\} \sin j\omega t \\ &- \frac{ja[j^2 \{j^2 - \alpha^2\} - 2\beta^2]}{(j^4 - \beta^2)^2} e^{-\alpha\_0 t} \sin \omega'\_{\bullet 0} t \end{aligned} \tag{2}$$

$$-2j\alpha\beta \Big(\cos\rho t \text{s} \text{ $t$ }-\text{e}^{-\omega\_b t} \cos\omega'\_{\text{(j)}} \text{t}\Big) \text{s} \text{in} \frac{j\pi\alpha}{l}$$

where

the middle of this century. The first report on this problem was published in 1931 by the American Society of Civil Engineers—ASCE [8] after which significant advances were made using analogue and digital computers, see also Biggs et al. [9], Looney [10], Huang and Valetsos [11], Tung et al. [12], Chaallal and Shahawi [13]. Czechoslovakia (till 1993), relevant studies were performed by Koloušek [14], Frýba [15], Baťa [16], Benčat [17], and others.

In the majority of studies, the bridge is considered as a one-dimensional beam, for which the differential equations of motion have been solved by numerical methods. The application of advanced calculation methods (finite element methods—FEM and other relevant numerical methods) enables two- and three-dimensional simulation models of bridge vibration to be solved. In the 1970s, Ting et al. [18] proposed an algorithm for solving this problem, based on an integral formulation of bridge vibration, which took into account the relations of kinematic bond of the moving vehicle-and-bridge interaction. The application of the above-mentioned methods in many cases was successful and led to the introduction of design standards and methods of assessing bridge structures. From a historical point of view, these solutions represent a gradual development in the understanding of bridge dynamic response, due to moving

The solution of bridge service life and reliability problems, as influenced by bridge vibration, which is mostly of random character, contributes to the complexity of this problem. However, in spite of all the complications of bridge vibration and the numerous parameters incorporated in regulations and standards in many countries, the natural frequencies and corresponding modes of vibration, the dynamic coefficient (dynamic increase of stress or deformation) and the damping are the basic bridge vibration characteristics, which can be verified by in situ experi-

Presently for evaluation of dynamic response and projected parameters, new and existing bridges are utilizing numerical and experimental bridges dynamic analysis. Full-scale bridges dynamic testing and monitoring give relevant information for projecting and assessment of real bridge behaviors [16–23]. This information consists of observed quantities obtained by experimental tests, theoretical analysis, and numerical computation and their comparison. Nowadays, the important role in the control of the bridge structures with bridge dynamic parameters (relative change of eigen-frequencies, damping parameters, fatigue parameters, vibration effective amplitudes value in time histories, etc.) plays monitoring of the structural parameters during normal bridge traffic on. Some results from bridge forced vibration tests (vibration is artificially induced (e.g., during the dynamic loading tests—DLT, etc.)) and also from bridge monitoring ambient vibration

The simplest calculation model of bridge vibration is based on a simply supported elastic beam with a mass moving across the beam at constant velocity. The moving mass is assumed

tests (input excitation is not under the control of the test engineer) are also used.

2.1. Simply supported beam subjected to a moving constant force

2. Theoretical and numerical approach

vehicles and their interactions.

112 Bridge Engineering

mental tests and monitoring.

$$\nu\_0 = \frac{Fl^3}{48EI} \approx \frac{2F}{\mu l \omega \rho\_{\text{f1}}^2} = \frac{2Fl^3}{\pi^4 EI} \tag{3}$$

is the static deflection at mid-span of the beam. The circular frequency of the damped beam vibration, with light damping, is

$$
\omega \circ\_{\triangleright}^2 = \omega \circ\_{\triangleright}^2 - \omega \circ\_{b}^2 \tag{4}
$$

and with heavy damping is

Figure 1. Simple beam subjected to a moving load.

$$
\omega' \, \_\circ^2 \mathbf{j} = \omega \mathbf{j} - \omega \, \_\circ^2 \tag{5}
$$

ð13Þ

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ð14Þ

ð15Þ

ð16Þ

ð17Þ

ð18Þ

Eq. (13) can be simplified for low vehicle speed, α ≪ 1, into the form

the dynamic deflection in the region of resonance is given by

where ω(1) = Ω and v<sup>0</sup> is as defined by Eq. (3).

deflection at the mid-span of the beam (Figure 2).

is expressed as

into account only the first term of the series

2.2. Moving harmonic force

Eq. (14) represents the influence line of beam deflection at point x, expressed as a Fourier series. Since the terms for j > l are negligible, for practical applications, it is sufficient to take

In the first half of twentieth century, dynamic bridges response analyses were focused mainly on studies regarding to the rail bridge vibration caused by steam traction. It has been the subject of much research (e.g., [6, 7, 14]). Inglis [7] modeled the so-called "hammer blows," due to unbalanced weights on the driving wheels of a locomotive, by a sinusoidal alternating force moving at a constant velocity across a beam. Expressing the time variation of the concentrated force by F(t) = Q sinΩt, and considering only the first mode of beam response,

The dynamic coefficient is often defined as the ratio of the maximum dynamic deflection to the static

The dependence of the dynamic coefficient on speed is sometimes called the resonance curve. The dynamic coefficient attains its maximum at resonance, e.g., when ω(1) = Ω and is given by

where Δ, after substitution of the speed and damping parameters α and β from Eqs. (6) and (7)

Parameters α and β are defined as

$$\alpha = \frac{\alpha}{\alpha\_1} = \frac{c}{2f\_{(1)}l} = \frac{T\_{(1)}}{2T} = \frac{cl}{\pi} \Big| \frac{\mu}{EI} \Big|^{\frac{1}{2}} = \frac{c}{c\_{cr}} \tag{6}$$

$$\beta = \frac{\alpha \nu\_{\oplus}}{\omega\_{\{1\}}} = \frac{\alpha \nu\_{\oplus} l^2}{\pi^2} \left(\frac{\mu}{EI}\right)^2 = \frac{\nu}{2\pi} \tag{7}$$

The circular frequency of the jth mode of vibration of a simply supported beam is denoted by

$$
\omega\_{\{\vec{l}\}} = \frac{j^4 \pi^4 \hbar l}{l^4 \cdot \mu} \tag{8}
$$

the corresponding natural frequency by

$$f\_{\circlearrowleft} = \stackrel{\epsilon \otimes \epsilon\_{\circlearrowleft}}{2\pi} = \stackrel{\epsilon^2 \pi}{2l^2} \begin{pmatrix} \mathbb{E}f \\ \mu \end{pmatrix}^{\frac{1}{2}} \tag{9}$$

and the circular frequency by

$$
\omega = \frac{\pi c}{l} \tag{10}
$$

The solution of Eq. (2) has been analyzed by Frýba [15], with regard to parameters α and β. The maximum dynamic deflection corresponds to values α ≈ 0.5–0.7. For large values of α, deflection tends to zero, while for small values of α, deflection is practically equal to the static deflection. The critical velocities defined as

$$c\_{cr} = 2f\_{\{1\}}l = \begin{matrix} \pi \{\frac{l\overline{c}l}{\mu} \} \\ \end{matrix} \tag{11}$$

are too high for practical cases. The critical velocity for the first natural frequency of steel bridges is

$$c\_{cr} = 2f\_{\langle 1 \rangle} l \approx 2l \frac{10^3}{4l} = 5000 \text{m/s} = 1800 \text{ km/h} \tag{12}$$

The results of the theoretical analysis given in this paragraph are applicable for large-span rail and highway bridges. These bridges have very low values of the first natural frequencies and the vehicle mass is negligible compared with the bridge mass, across which they are moving. Since damping of large-span bridges is light, the dynamic displacement may be calculated from

Bridges Subjected to Dynamic Loading http://dx.doi.org/10.5772/intechopen.73193 115

$$\nu\_{\{\iota\_{\mathbb{L}}\}} \approx \nu\_o \sum\_{l=1}^{\infty} \sin^{j\pi\chi} \times \frac{1}{j^2 \{j^2 - \alpha^2\}} \left(\sin j\omega t - \frac{\alpha}{j} e^{\omega\_0 t} \sin \omega\_{\mathbb{G}\uparrow} t\right) \tag{13}$$

Eq. (13) can be simplified for low vehicle speed, α ≪ 1, into the form

$$\nu\_{\{\iota\_{\mathbb{L}}\}} \approx \nu\_{\ o} \sum\_{l=1}^{\sim} \mathbf{1}\_{l} \sin \frac{j\pi\chi}{l} \sin j\omega t \tag{14}$$

Eq. (14) represents the influence line of beam deflection at point x, expressed as a Fourier series. Since the terms for j > l are negligible, for practical applications, it is sufficient to take into account only the first term of the series

$$
\upsilon\_{\{\chi\_i\}} \approx \upsilon\_\varphi \sin \cot \sin \frac{\pi \chi}{l} \tag{15}
$$

#### 2.2. Moving harmonic force

ð5Þ

ð6Þ

ð7Þ

ð8Þ

ð9Þ

ð10Þ

ð11Þ

ð12Þ

Parameters α and β are defined as

114 Bridge Engineering

the corresponding natural frequency by

deflection. The critical velocities defined as

and the circular frequency by

bridges is

from

The circular frequency of the jth mode of vibration of a simply supported beam is denoted by

The solution of Eq. (2) has been analyzed by Frýba [15], with regard to parameters α and β. The maximum dynamic deflection corresponds to values α ≈ 0.5–0.7. For large values of α, deflection tends to zero, while for small values of α, deflection is practically equal to the static

are too high for practical cases. The critical velocity for the first natural frequency of steel

The results of the theoretical analysis given in this paragraph are applicable for large-span rail and highway bridges. These bridges have very low values of the first natural frequencies and the vehicle mass is negligible compared with the bridge mass, across which they are moving. Since damping of large-span bridges is light, the dynamic displacement may be calculated In the first half of twentieth century, dynamic bridges response analyses were focused mainly on studies regarding to the rail bridge vibration caused by steam traction. It has been the subject of much research (e.g., [6, 7, 14]). Inglis [7] modeled the so-called "hammer blows," due to unbalanced weights on the driving wheels of a locomotive, by a sinusoidal alternating force moving at a constant velocity across a beam. Expressing the time variation of the concentrated force by F(t) = Q sinΩt, and considering only the first mode of beam response, the dynamic deflection in the region of resonance is given by

$$\begin{aligned} \nu\_{\{\mathbf{x}\_1\mathbf{t}\}} &= \nu\_o \frac{Q \omega\_{\{1\}} \cos \omega\_{\{1\} \mathbf{t}}}{2F} \Big| \omega \{\cos \omega t - \exp^{-\alpha \omega \mathbf{t}}\} \\\\ &- \omega\_b \sin \omega t \Big| \sin \frac{\pi \mathbf{x}}{l} \end{aligned} \tag{16}$$

where ω(1) = Ω and v<sup>0</sup> is as defined by Eq. (3).

The dynamic coefficient is often defined as the ratio of the maximum dynamic deflection to the static deflection at the mid-span of the beam (Figure 2).

$$\delta = \frac{\max \nu \{ l/2, t \}}{\nu\_o} \tag{17}$$

The dependence of the dynamic coefficient on speed is sometimes called the resonance curve. The dynamic coefficient attains its maximum at resonance, e.g., when ω(1) = Ω and is given by

$$\delta = 1 + \frac{Q}{2F\_{\alpha b}} \frac{\omega\_{\{1\}}}{-\omega\_b^2} \left\{ \omega e^{-\omega\_b \mathbf{1}/2c} + \omega\_b \right\} = 1 + \frac{Q}{F} \Delta \tag{18}$$

where Δ, after substitution of the speed and damping parameters α and β from Eqs. (6) and (7) is expressed as

Figure 2. Mid-span deflection produced by a constant moving harmonic load.

$$\Delta = \frac{1}{2\{\alpha^{\gamma} + \beta^{\gamma}\}} \{\alpha \mathbf{e}^{-\imath(\beta\_{i2\ell} + \jmath)} + \beta\} \tag{19}$$

ð22Þ

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> v(a)/dt 2 , which

> > ð24Þ

ð25Þ

DAF ¼ 1 þ DA (23)

In addition to specifications of most codes, the dynamic effects of vehicles on bridges are considered by multiplying the static live loads by a dynamic load factor (DLF = δ) greater than one; DAF—there are many ways of interpreting this simple definition of the DAF from test data,

The problem of a beam loaded by a moving load with negligible mass has been discussed in Section 2.1. The other extreme problem of negligible mass beam, subjected to a moving load of finite mass, was solved in [2, 3]. Consider a simply supported beam with span l and negligible mass, traversed by a load F with mass m = F/g, moving with a constant velocity c (Figure 3). Since the instantaneous position of the mass and the beam deformation are defined by a vertical displacement v(a) at point "a," where the force is situated, the system has one degree of freedom.

The static deflection caused by force F is given by Eq. (3) and hence the approximate solution

From Eq. (25), it is evident that magnitude of δ decreases with increasing span l. Large-span bridge structures are heavy and their mass cannot be neglected compared with the mass of moving vehicles. However, the real dynamic action of vehicles moving across short-span bridges is not reliably described by Eqs. (24) and (25). The effect of the moving mass is fairly

The total acting force Y(a) consists of mass gravity F = mg and inertia force –md2

for the dynamic coefficient δ, given by Zimmerman [3], is as follows

where DAF is the dynamic amplification factor given by

2.3. Massless beam subjected to a moving load

depends on the vertical acceleration at point a=ct, i.e.,

Figure 3. Massless beam with a moving mass.

see also, e.g., [19, 24].

The dynamic deformation (or stress) increment may be defined as an alternative to Eq. (17), e.g., EMPA (Swiss Federal Laboratories for Material Testing and Research), for experimental tests of bridges, defined the dynamic increment ϕ [19]:

$$\boldsymbol{\phi} = \begin{array}{c} \boldsymbol{\upsilon}\_{\text{dyn}} - \boldsymbol{\upsilon}\_{\text{stat}} \\ \boldsymbol{\upsilon}\_{\text{stat}} \end{array} \tag{20}$$

where vdyn is the peak value of the bridge displacement measured during a passage of the test vehicle across the bridge and vstat is the peak value of the bridge deflection observed under static loading caused by the same vehicle.

Application of the theoretical analysis of bridge vibration caused by a moving harmonic force is presently not of practical significance, due to the decline in the use of steam engines. The given knowledge, however, serves to explain the bridge vibration concepts which developed from the literature of that period. The parameter that was expressed as the dynamic coefficient (δ) and its dependence on moving vehicle speed is still one of the most important parameters characterizing bridge stiffness.

From the previous sections, it follows that moving vehicle on a bridge generates deflection and stresses in the bridge structure that are greater than those generated by the same vehicle applied statically. In general, the dynamics amplification (DA) is defined by

$$DA = \frac{R\_{\text{dyn}} - R\_{\text{stol.}}}{R\_{\text{stat}}} \tag{21}$$

where Rdyn and Rstat are maximum dynamic and static response (deflection, stresses, etc.) of the bridge, see also Eq. (20). Therefore, dynamic response can be calculated as

$$R\_{dyn} = \text{DAF} \ge R\_{\text{start}} \tag{22}$$

where DAF is the dynamic amplification factor given by

$$\text{DAF} = 1 + \text{DA} \tag{23}$$

In addition to specifications of most codes, the dynamic effects of vehicles on bridges are considered by multiplying the static live loads by a dynamic load factor (DLF = δ) greater than one; DAF—there are many ways of interpreting this simple definition of the DAF from test data, see also, e.g., [19, 24].

#### 2.3. Massless beam subjected to a moving load

ð19Þ

ð20Þ

ð21Þ

The dynamic deformation (or stress) increment may be defined as an alternative to Eq. (17), e.g., EMPA (Swiss Federal Laboratories for Material Testing and Research), for experimental tests of

where vdyn is the peak value of the bridge displacement measured during a passage of the test vehicle across the bridge and vstat is the peak value of the bridge deflection observed under

Application of the theoretical analysis of bridge vibration caused by a moving harmonic force is presently not of practical significance, due to the decline in the use of steam engines. The given knowledge, however, serves to explain the bridge vibration concepts which developed from the literature of that period. The parameter that was expressed as the dynamic coefficient (δ) and its dependence on moving vehicle speed is still one of the most important parameters characterizing

From the previous sections, it follows that moving vehicle on a bridge generates deflection and stresses in the bridge structure that are greater than those generated by the same vehicle

where Rdyn and Rstat are maximum dynamic and static response (deflection, stresses, etc.) of the

applied statically. In general, the dynamics amplification (DA) is defined by

bridge, see also Eq. (20). Therefore, dynamic response can be calculated as

bridges, defined the dynamic increment ϕ [19]:

Figure 2. Mid-span deflection produced by a constant moving harmonic load.

static loading caused by the same vehicle.

bridge stiffness.

116 Bridge Engineering

The problem of a beam loaded by a moving load with negligible mass has been discussed in Section 2.1. The other extreme problem of negligible mass beam, subjected to a moving load of finite mass, was solved in [2, 3]. Consider a simply supported beam with span l and negligible mass, traversed by a load F with mass m = F/g, moving with a constant velocity c (Figure 3). Since the instantaneous position of the mass and the beam deformation are defined by a vertical displacement v(a) at point "a," where the force is situated, the system has one degree of freedom.

The total acting force Y(a) consists of mass gravity F = mg and inertia force –md2 v(a)/dt 2 , which depends on the vertical acceleration at point a=ct, i.e.,

$$Y\_{\{a\}} = mg - m \frac{\mathrm{d}^2 \nu(a)}{\mathrm{d}t^2} \tag{24}$$

The static deflection caused by force F is given by Eq. (3) and hence the approximate solution for the dynamic coefficient δ, given by Zimmerman [3], is as follows

$$\delta = 1 + \frac{16\nu\_0 c^2}{g l^2} \left( 1 + \frac{40\nu\_0 c^2}{g l^2} \right) \tag{25}$$

From Eq. (25), it is evident that magnitude of δ decreases with increasing span l. Large-span bridge structures are heavy and their mass cannot be neglected compared with the mass of moving vehicles. However, the real dynamic action of vehicles moving across short-span bridges is not reliably described by Eqs. (24) and (25). The effect of the moving mass is fairly

Figure 3. Massless beam with a moving mass.

small compared to that of other factors which produce high dynamic stresses in such bridges. For example, in short-span rail bridges, impact effects of flat wheels, rail joints, etc., predominate over those of the moving load. Thus, a vehicle cannot be represented adequately by a single moving point mass, even for short-span bridges.

#### 2.4. Beam subjected to a moving system with two degrees of freedom and two axle

The need to quantify dynamic bridge response induced by moving vehicles has led to the development of improved but more complex physical models. The use of modern computers and advanced numerical methods enables satisfactory solutions of such problems to be obtained.

If the vehicle mass as well as the bridge mass is taken into account, the problem is more complicated than the problems analyzed in Sections 2.1 pending 2.3. The actual problem is described by the differential equation

$$\begin{bmatrix} \partial^4 \nu \langle \mathbf{x}\_1 \mathbf{t} \rangle & \partial^2 \nu \langle \mathbf{x}\_1 \mathbf{t} \rangle \\ \partial \mathbf{f} & \partial \mathbf{x}^\dagger \end{bmatrix} + \mu \begin{bmatrix} \partial^2 \nu \langle \mathbf{x}\_1 \mathbf{t} \rangle \\ \partial t^2 \end{bmatrix} + \mathbf{2} \mu \iota \iota\_\mathbf{b} \quad \overset{\text{(}\mathcal{U}=\mathcal{E}t\text{)}}{\partial \iota} = \delta(\mathbf{x} - \iota t) \begin{bmatrix} \partial^2 \nu \langle \mathbf{c}t \mathbf{t} \rangle \\ \mathbf{F} - m \mathbf{f} \end{bmatrix} \tag{26}$$

The right-hand side of Eq. (26) expresses the motion of the force F with mass m, including the inertia effect. This problem has been discussed by [7, 9, 11–13] and others. Ting et al. [18] proposed a solution which takes into account a kinematic bond of the vehicle-bridge system. Many other solutions of this problem are published in contemporary works. These cannot be described in the context of this chapter. Therefore, only the basic formulation of the problem has been introduced to identify the parameters which influence the bridge-vehicle system vibration. The specific characteristics of the kinematic bond of rail bridges and highway bridges should be taken into account. A vehicle is a complex mechanical system. For the purpose of axle load calculation, it can be represented by a plane model consisting of mass points, material planes, and connecting elements. It is possible to idealize the physical model of a vehicle as a one-, two- or multi-axle system, with or without damping (Figure 4).

The bridge is modeled as a simply supported Bernoulli-Euler beam, with a continuously distributed mass, or as a discrete system with n-lumped masses. The surface of the beam may be assumed perfectly smooth or to have irregularities. The beam stiffness can be assumed constant or variable, based on a layered system with variable stiffness in each of the elastic layers mainly for application to rail bridges (Figure 5). The real behavior of the bridge-vehicle system can be described, more or less successfully, with the combination of physical vehicle and bridge models shown in Figures 4 and 5. Satisfactory results have been obtained using the vehicle models in Figure 4(A)–(C) for bridges with spans l > 30 m. The two- or multi-axle models of the vehicle system are more appropriate for short-span bridge investigation. The following simplifying assumptions are made in relation to the physical models:

• the springing and damping of the tires are not taken into account (highway bridges);

spacing effect).

Figure 5. Physical models of bridge.

Figure 4. Physical models of a vehicle.

• variable stiffness of elastic layers is taken into account for steel railway bridges (sleeper

Bridges Subjected to Dynamic Loading http://dx.doi.org/10.5772/intechopen.73193 119

The mathematical formulation of the problem of synchronous bridge-vehicle system vibration, taking into account the above simplifying assumptions, leads to the set of three simultaneous differential equations with variable coefficients (because of variable stiffness of elastic layers and track irregularities) describing, respectively, the vertical displacements of sprung and unsprung masses and beam vibration. The set of differential equations may be solved by

numerical integration utilizing a digital computer with relevant software package.


Figure 4. Physical models of a vehicle.

ð26Þ

small compared to that of other factors which produce high dynamic stresses in such bridges. For example, in short-span rail bridges, impact effects of flat wheels, rail joints, etc., predominate over those of the moving load. Thus, a vehicle cannot be represented adequately by a

The need to quantify dynamic bridge response induced by moving vehicles has led to the development of improved but more complex physical models. The use of modern computers and advanced numerical methods enables satisfactory solutions of such problems to be

If the vehicle mass as well as the bridge mass is taken into account, the problem is more complicated than the problems analyzed in Sections 2.1 pending 2.3. The actual problem is

The right-hand side of Eq. (26) expresses the motion of the force F with mass m, including the inertia effect. This problem has been discussed by [7, 9, 11–13] and others. Ting et al. [18] proposed a solution which takes into account a kinematic bond of the vehicle-bridge system. Many other solutions of this problem are published in contemporary works. These cannot be described in the context of this chapter. Therefore, only the basic formulation of the problem has been introduced to identify the parameters which influence the bridge-vehicle system vibration. The specific characteristics of the kinematic bond of rail bridges and highway bridges should be taken into account. A vehicle is a complex mechanical system. For the purpose of axle load calculation, it can be represented by a plane model consisting of mass points, material planes, and connecting elements. It is possible to idealize the physical model of a vehicle as a one-, two- or multi-axle

The bridge is modeled as a simply supported Bernoulli-Euler beam, with a continuously distributed mass, or as a discrete system with n-lumped masses. The surface of the beam may be assumed perfectly smooth or to have irregularities. The beam stiffness can be assumed constant or variable, based on a layered system with variable stiffness in each of the elastic layers mainly for application to rail bridges (Figure 5). The real behavior of the bridge-vehicle system can be described, more or less successfully, with the combination of physical vehicle and bridge models shown in Figures 4 and 5. Satisfactory results have been obtained using the vehicle models in Figure 4(A)–(C) for bridges with spans l > 30 m. The two- or multi-axle models of the vehicle system are more appropriate for short-span bridge investigation. The

• the bridge and vehicle damping is proportional to the velocity of vibration (viscous

following simplifying assumptions are made in relation to the physical models:

• the load remains in contact with the surface of the bridge;

2.4. Beam subjected to a moving system with two degrees of freedom and two axle

single moving point mass, even for short-span bridges.

described by the differential equation

system, with or without damping (Figure 4).

• the vehicle speed is constant;

damping);

obtained.

118 Bridge Engineering

Figure 5. Physical models of bridge.


The mathematical formulation of the problem of synchronous bridge-vehicle system vibration, taking into account the above simplifying assumptions, leads to the set of three simultaneous differential equations with variable coefficients (because of variable stiffness of elastic layers and track irregularities) describing, respectively, the vertical displacements of sprung and unsprung masses and beam vibration. The set of differential equations may be solved by numerical integration utilizing a digital computer with relevant software package.

Figure 6. Model of a beam with an elastic layer and irregularities subjected to a moving system with two degrees of freedom and force Q(t).

Consider the physical model of a rail bridge (Figure 6) with the following assumptions [15]:

1. The moving vehicle is idealized by a system with two degrees of freedom. An unsprung mass m<sup>1</sup> is in direct contact with the beam; m<sup>2</sup> denotes the sprung parts of the vehicle and the total weight of the vehicle is

$$F = F\_1 + F\_2 = g(m\_1 + m\_2) \tag{27}$$

where

is the interactive force by which the moving system acts on a beam at the point of contact x1, and

Eqs. (29)–(31) should satisfy the boundary conditions of a simply supported beam as well as the appropriate initial conditions. These equations provide a very general statement of the problem of vibrations excited by a system of masses moving along a beam. Simpler sets of differential equations, which describe dynamic bridge response with sufficient accuracy, can be derived with

Various individual bridge and vehicle parameters, as well as the interaction between them, were included in the theoretical analysis of the bridge dynamic response. The parameters and effects considered were: the vehicle speed, the frequency parameter of unsprung and sprung masses, variable stiffness of elastic layers, the ratio between the weights of the vehicle and the beam, the ratio between the weights of the unsprung and sprung parts of the vehicle, the beam damping, the vehicle spring damping, the initial conditions and others. Many contributions and solutions of this problem can be found in the literature on bridge vibrations. Wen [25] was the first author to solve this problem with application to highway bridges. In [15], the influence of the individual bridge parameters and

Rail bridges—remarks. The theoretical results have been verified by experimental tests on more than 50 rail bridges in Slovak Republic and Czech Republic and also in former Czechoslovakia (Research Rail Institute, Prague; Department of Structural Mechanics, University of Transport and Communications (UTC) Žilina; University of Žilina (1993–2017) and others).

1. (a) for large-span bridges with spans over 30 m, it is appropriate to consider the physical vehicle model as a moving system with two degrees of freedom (see Figure 4(A)–(C)); (b) for short-span bridges with spans less than 30 m, it is necessary to idealize the vehicle as a

2. The greatest influence on the dynamic increment of deflection or stress (ϕ, DA) is the

3. It is necessary to include in the theoretical calculations the influence of the cross beams, uniform sleeper spacing and other regular unevenness that enlarges the local peaks in the

simplifying assumptions. These equations have been solved numerically in [15].

two-axle systems on the dynamic response of steel rail bridges is also analyzed.

The following conclusions can be made from the results:

two-axle or multi-axle system.

dynamic coefficient (δ)-velocity diagram.

vehicle speed.

ð31Þ

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ð32Þ

The coordinate of the contact point is x1= c.t, because of the constant speed c along the beam.


$$p(\boldsymbol{x}) = \frac{1}{2}\overline{a}\{1-\cos^{2}\pi\boldsymbol{x}/l\_{a}\}\tag{28}$$

where ā is the maximum depth of track unevenness and l<sup>a</sup> is the length of track irregularity. The equations of motion of the synchronous vehicle-bridge system within the interval 0 ≤ t can be written

$$-m\_2 \frac{d^2 \nu\_2(t)}{dt^2} - k\_v[\nu\_2(t) - \nu\_1(t)] - c\_v[\frac{d\nu\_2(t)}{dt} - \frac{d\nu\_1(t)}{dt}] = 0\tag{29}$$

$$\begin{aligned} F + Q(t) - m\_1 \frac{d^2 \nu\_1(t)}{dt^2} + k\_v[\nu\_2(t) - \nu\_1(t)] \\ + c\_\nu \left[ \frac{d \nu\_2(t)}{dt} - \frac{d \nu\_1(t)}{dt} \right] - R\_{\{t\}} = 0 \end{aligned} \tag{30}$$

$$\frac{\partial^4 \nu \{\boldsymbol{x}\_1 \boldsymbol{l}\}}{\partial \boldsymbol{x}^4} + \mu \frac{\partial^2 \nu \{\boldsymbol{x}\_1 \boldsymbol{l}\}}{\partial \boldsymbol{t}^2} + 2\mu \omega\_0 \frac{\partial \nu \{\boldsymbol{x}\_1 \boldsymbol{l}\}}{\partial \boldsymbol{t}} = \overline{\boldsymbol{\varepsilon}} \boldsymbol{\delta} (\boldsymbol{x} - \boldsymbol{x}\_1) \boldsymbol{\kappa}\_{\{\boldsymbol{t}\}} \tag{31}$$

where

ð27Þ

ð28Þ

ð29Þ

ð30Þ

Consider the physical model of a rail bridge (Figure 6) with the following assumptions [15]: 1. The moving vehicle is idealized by a system with two degrees of freedom. An unsprung mass m<sup>1</sup> is in direct contact with the beam; m<sup>2</sup> denotes the sprung parts of the vehicle and

Figure 6. Model of a beam with an elastic layer and irregularities subjected to a moving system with two degrees of

The coordinate of the contact point is x1= c.t, because of the constant speed c along the beam.

where ā is the maximum depth of track unevenness and l<sup>a</sup> is the length of track irregularity. The equations of motion of the synchronous vehicle-bridge system within the interval 0 ≤ t can be

3. The top surface of the beam is covered with an elastic layer of variable stiffness k(x).

4. Track irregularities are assumed to vary harmonically along the bridge span as

the total weight of the vehicle is

freedom and force Q(t).

120 Bridge Engineering

written

2. The unsprung mass is acted upon only by harmonic force.

$$R(\mathfrak{r}) = k(\chi\_1)[\nu\_1(\mathfrak{r}) - \overline{\varepsilon}\nu\_{(\chi\_{1,\mathfrak{r}})} - p(\chi\_1)] \cong 0 \tag{32}$$

is the interactive force by which the moving system acts on a beam at the point of contact x1, and

$$
\overline{\varepsilon} = \begin{cases} 1 & \text{for} \\ 0 & \text{for} \end{cases} \begin{array}{ll} 0 \le x\_1 \le 1 \\ x\_1 < 0; x\_1 > 1. \end{array}
$$

Eqs. (29)–(31) should satisfy the boundary conditions of a simply supported beam as well as the appropriate initial conditions. These equations provide a very general statement of the problem of vibrations excited by a system of masses moving along a beam. Simpler sets of differential equations, which describe dynamic bridge response with sufficient accuracy, can be derived with simplifying assumptions. These equations have been solved numerically in [15].

Various individual bridge and vehicle parameters, as well as the interaction between them, were included in the theoretical analysis of the bridge dynamic response. The parameters and effects considered were: the vehicle speed, the frequency parameter of unsprung and sprung masses, variable stiffness of elastic layers, the ratio between the weights of the vehicle and the beam, the ratio between the weights of the unsprung and sprung parts of the vehicle, the beam damping, the vehicle spring damping, the initial conditions and others. Many contributions and solutions of this problem can be found in the literature on bridge vibrations. Wen [25] was the first author to solve this problem with application to highway bridges. In [15], the influence of the individual bridge parameters and two-axle systems on the dynamic response of steel rail bridges is also analyzed.

Rail bridges—remarks. The theoretical results have been verified by experimental tests on more than 50 rail bridges in Slovak Republic and Czech Republic and also in former Czechoslovakia (Research Rail Institute, Prague; Department of Structural Mechanics, University of Transport and Communications (UTC) Žilina; University of Žilina (1993–2017) and others).

The following conclusions can be made from the results:


Figure 7 shows a comparison of the computed and measured deflections at mid-span of a bridge, and the dynamic coefficients δ at different locomotive speeds.

Highway bridge—remarks. The preceding discussion was directed primarily toward railway bridge vibration. Highway bridge vibration analysis should incorporate the specific features which are associated with highway bridge structures and vehicle construction, which result in different interaction of the bridge-vehicle system. In the case of highway bridges, the load bearing system of modern bridge structures consists mainly of prismatic and non-prismatic beams of box, open or partly closed cross section. In the majority of cases, the bridge structure approximates to the typical linear structure model. At the formulation stage of the physical model of the bridgevehicle system, it is necessary to take into account the effect of variable stiffness of the roadway, which may be replaced by the effect of track irregularities. The real bridge, as well as vehicle response, can be described adequately by the physical model of the vehicle-bridge system shown in Figure 8. Theoretically, the problem of forced vibration of a system consisting of a moving vehicle and a bridge structure (Figure 8) can be described generally by operator relations, e.g. [15]

$$\begin{aligned} L\_1[\{r\_q(t)\}, \{r\_s(t)\}, h\_q(t), \{\nu(\chi\_q, \mathbf{z}\_q, t)\}, a\_q(t)] &= 0\\ L\_2[\{\nu(\mathbf{x}, \mathbf{z}, t)\}, \{r\_q(t)\}, h\_q(t), a\_q(t), \{\nu(\chi\_q, \mathbf{z}\_q, t)\}] &= 0 \end{aligned} \tag{33}$$

structure displacements at the points with coordinates x, z at time t; {v(xq, zq, t)} is a vector of bridge structure displacements at the qth lower link between the vehicle and the bridge structure; aq(t) is the law defining the vehicle movement along the longitudinal bridge axis; hq(t) is the function describing the irregularities of the road surface; and xq, z<sup>q</sup> are the coordinates of the lower link between the vehicle and the bridge structure. Eq. (33) by the differential operators, together with boundary and initial conditions, defines the motion of a system

Bridges Subjected to Dynamic Loading http://dx.doi.org/10.5772/intechopen.73193 123

Figure 8. Physical model of a system consisting of a bridge structure and a moving vehicle.

Research into highway bridges has not been systematic, either in Europe or elsewhere, and in some countries, it has been limited mainly to random tests of extraordinary bridge structures, before being put into operation. This has led to a variety of methods for calculating dynamic effects in individual countries, particularly in the provisions concerning dynamic coefficients (δ, DA, DLA) in respective standards or bridge regulations. At the present time, much information is available on the forced vibration of highway bridges, which should be taken into account during the formulation of the vehicle-bridge physical model. The results of theoretical analysis and parametric studies of highway bridge response, as well as the results of experimental bridge investigations, performed by the relevant research divisions of the UTC Žilina or University of Žilina (UZ Žilina) on more than 60 highway and road bridges, are now

1. As for railway bridges, vehicle speed has the greatest influence on the dynamic increments

2. The first mode natural frequency of vibration of the bridge in the vertical plane (bending) and the natural frequency of vibration of the sprung vehicle mass in the vertical direction. The theoretical dynamic coefficient δ (DLA) and also experimental δobs (DAF) are maximum when ω(1) <sup>≈</sup> ωv. It was noted also that the influence of frequency ratio ω(1)/ω<sup>v</sup>

discussed in detail. The dynamic bridge response is influenced primarily by:

consisting of a moving vehicle and a bridge structure.

of stress and deflection of highway bridges.

diminishes with increasing mass ratio (mv/mbr).

where L<sup>1</sup> and L<sup>2</sup> are linear or non-linear operators; {rq(t)} and {rs(t)} are displacement vectors of vehicle elements, conditioned by upper and lower links; {v(x, z, t)} is a vector of bridge

Figure 7. Deflection at the center of beam with span l = 34.8 m, traversed by an electric locomotive E 469 at speed c = 40.7 km/h [15]: (a) theory, (b) experiment (c) theoretical and experimental dependence of the dynamic coefficient δ.

Figure 8. Physical model of a system consisting of a bridge structure and a moving vehicle.

4. The dynamic effect of railway vehicles increases approximately in proportion to the fre-

5. The dynamic stresses in short-span railway bridges are affected primarily by the impact

6. For short-span rail bridges, the effects of sprung and unsprung vehicle masses that have

7. The periodic irregularities (sleeper effects) when multi-axle vehicle systems cross the bridge can cause their vibration with resonance, especially at velocities of 100–200 km/h. Figure 7 shows a comparison of the computed and measured deflections at mid-span of a

Highway bridge—remarks. The preceding discussion was directed primarily toward railway bridge vibration. Highway bridge vibration analysis should incorporate the specific features which are associated with highway bridge structures and vehicle construction, which result in different interaction of the bridge-vehicle system. In the case of highway bridges, the load bearing system of modern bridge structures consists mainly of prismatic and non-prismatic beams of box, open or partly closed cross section. In the majority of cases, the bridge structure approximates to the typical linear structure model. At the formulation stage of the physical model of the bridgevehicle system, it is necessary to take into account the effect of variable stiffness of the roadway, which may be replaced by the effect of track irregularities. The real bridge, as well as vehicle response, can be described adequately by the physical model of the vehicle-bridge system shown in Figure 8. Theoretically, the problem of forced vibration of a system consisting of a moving vehicle and a bridge structure (Figure 8) can be described generally by operator relations, e.g. [15]

where L<sup>1</sup> and L<sup>2</sup> are linear or non-linear operators; {rq(t)} and {rs(t)} are displacement vectors of vehicle elements, conditioned by upper and lower links; {v(x, z, t)} is a vector of bridge

Figure 7. Deflection at the center of beam with span l = 34.8 m, traversed by an electric locomotive E 469 at speed c = 40.7 km/h [15]: (a) theory, (b) experiment (c) theoretical and experimental dependence of the dynamic coefficient δ.

ð33Þ

resulting from track or wheel irregularities (rail joints, flat wheels, etc.)

been set in vibration prior to crossing the bridge are important.

bridge, and the dynamic coefficients δ at different locomotive speeds.

quency of sprung masses and the vehicle weight.

122 Bridge Engineering

structure displacements at the points with coordinates x, z at time t; {v(xq, zq, t)} is a vector of bridge structure displacements at the qth lower link between the vehicle and the bridge structure; aq(t) is the law defining the vehicle movement along the longitudinal bridge axis; hq(t) is the function describing the irregularities of the road surface; and xq, z<sup>q</sup> are the coordinates of the lower link between the vehicle and the bridge structure. Eq. (33) by the differential operators, together with boundary and initial conditions, defines the motion of a system consisting of a moving vehicle and a bridge structure.

Research into highway bridges has not been systematic, either in Europe or elsewhere, and in some countries, it has been limited mainly to random tests of extraordinary bridge structures, before being put into operation. This has led to a variety of methods for calculating dynamic effects in individual countries, particularly in the provisions concerning dynamic coefficients (δ, DA, DLA) in respective standards or bridge regulations. At the present time, much information is available on the forced vibration of highway bridges, which should be taken into account during the formulation of the vehicle-bridge physical model. The results of theoretical analysis and parametric studies of highway bridge response, as well as the results of experimental bridge investigations, performed by the relevant research divisions of the UTC Žilina or University of Žilina (UZ Žilina) on more than 60 highway and road bridges, are now discussed in detail. The dynamic bridge response is influenced primarily by:


3. The vehicle vibration at the moment the vehicle enters the bridge, since the vehicle's energy of vibration is the primary source of the dynamic bridge response. The vertical amplitude of the vehicle vibration is decisive. The initial angular amplitude of the vehicle's sprung mass vibration can be neglected in the analysis.

vibration. An eigenvector is arbitrary to the extent that a scalar multiple of it is also a solution of Eq. (35). It is convenient to choose this multiplier in such a way that ϕ has some desirable

Bridges Subjected to Dynamic Loading http://dx.doi.org/10.5772/intechopen.73193 125

To avoid creating complicated and sophisticated numerical models involving extensive assumptions in modeling (boundary and initial conditions, mechanisms of bridge flexibility and energy dissipation, inertia, etc.), it is useful to develop an appropriate model with realistic prediction of their dynamic response upon the comparison of the experimental results and theoretical predictions. This enables also the realistic and optimal economical designs. Nowadays, very popular and useful numerical method for engineering analysis is finite element method. FEM is a numerical procedure for obtaining solutions to many of the problems encountered in civil and structure engineering. Numerical solutions of the bridge dynamic analysis problems in many cases need experimental verification in situ, e.g., [20, 27, 29, 30, 31]. To create relevant analytical models with real dynamic bridge structure with input parameters, it is useful to apply experimental modal analysis (EMA) which provides mainly structure natural modes engine frequencies and damping parameters of the tested bridge structure [31]. For such type of bridge dynamic tests performance in most cases, the real bridge service conditions are too restrictive for performance such type bridge tests. In these cases, operational modal analysis (OMA) procedure is applicable, which enables to perform bridge dynamic testing and also bridge health monitoring measurements without interrupting bridge service. A wellpresented review of bridge testing methods explaining their conditions, advantages and limi-

The bridge dynamic analysis programs are commonly available and computational problems are not complicated to solve. A lot of FEM software packages are used in this field mainly for structures modal analysis and dynamic response of bridges (ANSYS Civil FEM Bridge, BRASS, BRIDGES, BridgeSoft, BRIDGADES (ABAQUS), ADINA, DYNSOLV, LUSAS, etc.).

In situ dynamic testing of bridges gives very useful information for numerical modeling and assessment of real bridge dynamic parameters and service conditions. In many countries, the requirement of putting the bridges into operation is the execution of bridge static and dynamic loading tests, which aim is to prove and confirm the projected parameters (standards criteria, serviceability, safety limit states, etc.) of tested bridge structures according to technical standards, e.g., in Slovakia by standard STN—Slovak Technical Standards [28]. Results from static or dynamic test enable to calibrate a bridge analytical model and can be utilized as basic data for a bridge health monitoring program and for other sophisticated calculations of the bridge dynamic response

property. Such eigen vectors are called normalized eigenvectors.

tations was presented by Salawu and Williams [27].

(seismic, fatigue, etc.).

4. Dynamic loading tests of bridges and monitoring

3.2. Numerical procedure application in bridge structure dynamic analysis

4. The character of the road irregularities (joints, potholes, inserted hinges, frozen snow, etc.).

The effects of the damping of the vehicles and bridges, as well as the ratio of the sprung vehicle mass to the bridge mass, are not significant for long-span bridges. However, they are a significant influence on short-span bridges vibration. It was confirmed by the theoretical analysis and the experimental tests that the curve expressing the dependence of the dynamic coefficients on the vehicle speed is not a smooth curve but has many local projections and branching points [26].
