Vibration control of bridges under simultaneous effects of earthquake and moving loads using steel pipe dampers

Abstract

This study aims to develop a practical methodology based on the eigenfunction expansion method to assess the effects of simultaneous action of vertical earthquake excitation and moving vehicle loads in single-span and multi-span simply-supported bridges. While the effects of vertical earthquake ground motions are generally ignored in common design practice, it is shown that the influence of simultaneous vertical earthquake excitation and vehicle loads can considerably affect the structural response of the bridge, especially in near-field earthquakes with high amplitude vertical components. To address this issue, a novel vibration suppression system is proposed using steel pipe dampers and its reliability is investigated for a wide range of bridge flexural rigidity under seven different earthquake records. The results indicate that the proposed system can significantly (up to 75%) suppress the vertical vibrations generated in the bridge, especially for the systems with lower flexural rigidity. For the same maximum deflection limits, application of pipe dampers could reduce the required flexural rigidity of the bridge (up to 50%), and therefore, lead to more economic design solutions with lower structural weight.

Keywords

Bridge vibration vibration suppression vertical earthquake moving inertial loads steel pipe dampers

1. Introduction

Currently, most bridge design guidelines (e.g., American Railway Engineering and Maintenance-of-Way Association, 1997; Union Internationale des Chemins de Fer, 2002; European Standard (2003)) rely heavily on static analysis methods to account for the impact of vehicle/train loads. While such simplified approaches can be acceptable for conventional systems, previous studies have demonstrated that more sophisticated methods may be required to obtain accurate results in the case of long-span bridges traversed by high-speed loads (Nikkhoo et al., 2007; Kiani and Nikkhoo, 2012; Salcher and Adam, 2015; Yang et al., 2017; He, 2018).

The structural response of linear continuous beams under dynamic loads has been a topic of many investigations (Su and Ahmadi, 1988; Frýba, 1999; Beskou and Theodorakopoulos, 2011; Leissa and Qatu, 2011; Pi and Ouyang, 2015; Zhu et al., 2015). However, most of these studies were limited to the beam vibrations caused exclusively by a single dynamic load case (i.e., under a single moving load or a specific seismic event).

While the interaction between moving loads and bridge systems has been widely investigated (e.g., Ichikawa et al., 2000; Xia et al., 2003; Salcher and Adam, 2015; Ticona Melo et al., 2018; Xia et al., 2018; Zhang et al., 2018), there are very limited studies on the structural response of bridge systems under simultaneous effects of moving loads and earthquake excitations (Wibowo et al., 2013; Nguyen, 2015; Paraskeva et al., 2017). In one of the early studies, Yau (2009) developed a method to obtain the dynamic response of suspended beams subjected to simultaneous actions of moving vehicles and support excitations by decoupling the response into the pseudo-static and inertia-dynamic components. Frýba and Yau (2009) and Liu et al. (2011) showed that the interaction between the moving force and the vertical support excitation can considerably amplify the dynamic response of suspension bridges, especially for high vehicle speeds. Legeron and Sheikh (2009) proposed a methodology to determine the response of the bridge supports under vertical seismic loads and demonstrated that the effects of vertical seismic excitations should not be disregarded in the design process of bridges. A similar conclusion was reported by other researchers (Papazoglou and Elnashai, 1996; Shrestha, 2015; Wang et al., 2015; Chen et al., 2016).

Several research studies have investigated the dynamic behavior of single-span bridge systems under seismic ground motions by mainly focusing on the horizontal acceleration components (e.g., Su and Ahmadi, 1988; Shrestha, 2015; Wang et al., 2015; Chen et al., 2016). Zarfam et al. (2013) developed a method to assess the influence of the weight and velocity of traversing masses on the dynamic response of a beam subjected to horizontal support excitations at different frequencies. Konstantakopoulos et al. (2012) and Nguyen et al. (2017) analyzed the vertical and horizontal dynamic response of suspension bridges subjected to earthquake and moving loads acting either separately or simultaneously. They concluded that the structural responses are significantly affected by the simultaneous effects of moving and seismic loads, especially when the moving loads are relatively small. However, the convective accelerations and the inertial acts of the moving vehicles were disregarded in most of the above-mentioned studies.

Elias and Matsagar (2017) developed a modified modal analysis approach, which can deal with nonclassically damped systems, to investigate the effectiveness of tuned mass dampers on isolated reinforced concrete bridges under horizontal earthquake excitations. A similar approach was then adopted by Matin et al. (2019) to study the effectiveness of distributed multiple tuned mass dampers in seismic response control of bridges. Agrawal et al. (2009) and Tan and Agrawal (2009) presented the problem definition of a comprehensive benchmark structural control problem for the highway bridges under seismic excitations simultaneously applied in two directions. While different passive, semi-active, and active devices and algorithms can be used to study the response of the benchmark model, the effects of moving masses are not taken into account.

Since the occurrence of an earthquake during the operational time of a bridge (i.e., whilst carrying moving loads) is highly likely, the structural response of the bridge system under such loading condition has to be adequately assessed during the design process. However, existing studies on the interaction between the vertical earthquake excitations and vehicle loads are commonly limited to the investigation of suspension bridges under moving forces (Frýba and Yau, 2009; Liu et al., 2011; Konstantakopoulos et al., 2012; Nguyen et al., 2017). Therefore, more studies are required to extend the results to the general bridge systems.

In this paper, dynamic behavior of bridges under simultaneous effects of moving inertial loads/vehicles and excitation due to the vertical earthquake excitations is investigated by adopting a modal analysis approach on simply-supported beams representing typical bridge systems. In addition, it is attempted to assess the efficiency of pipe dampers, originally proposed by Maleki and Bagheri (2010), to mitigate the vibrations induced by simultaneous actions of earthquake and vehicle loads. The vertical acceleration component of seven near-field earthquake ground motions with different frequency contents is applied to a simply-supported Euler–Bernoulli beam subjected to vehicle loads (or lumped masses) traveling at uniform intervals. The pipe dampers are then employed in the locations where maximum deflections commonly occur, aiming to evaluate their efficiency for mitigation of the vertical vibrations of the bridge systems with different flexural rigidities.

2. Problem formulations

It is assumed that a uniform undamped single-span Euler–Bernoulli beam of length L undergoes a dynamic excitation $f (x, t)$ . The Cartesian system adopted for the analysis is illustrated in Figure 1. Given $D (x, t)$ as the function describing the vertical displacement along the beam at time t, the governing vibration equation is written as follows (Frýba, 1999)

ρ A \frac{\partial 2 D (x, t)}{\partial t^{2}} + EI \frac{\partial 4 D (x, t)}{\partial x^{4}} = f (x, t)

(1)

where

EI

and

ρ A

are the flexural rigidity and the mass per unit length of the beam, respectively.

f (x, t)

is obtained by combining the results of (i) a set of moving masses,

m_{k}

, spaced at uniform intervals d, and traversing the beam at a constant velocity of v, and (ii) base excitation due to the vertical acceleration of the design earthquake

\frac{d^{2} U_{g} (t)}{d t^{2}}

, resulting in the following equation (Yang et al., 1997)

\begin{matrix} f (x, t) = \sum_{k = 1}^{N} m_{k} [g - \frac{d^{2} D_{0} (t)}{d t^{2}}] \\ \cdot U_{k} - {ρ A + \sum_{k = 1}^{N} m_{k} U_{k}} . \frac{d^{2} U_{g} (t)}{d t^{2}} \end{matrix}

(2)

where, by letting

Δ H (t) = H (t - t_{k}) - H (t - t_{k} - L / v)

U_{k} = δ [x - v (t - t_{k})] . Δ H (t)

(3)

Figure 1.

Schematic view of the proposed system.

Here, δ and $H (.)$ denote the Dirac delta and the unit step function, respectively. $t_{k}$ represents the arrival time of the kth mass at the beam (i.e., $t_{k} = (k - 1) d / v$ ), and N is the total number of moving masses. The action of the kth moving mass is considered via $H (t - t_{k})$ function when the load enters the beam and is deactivated through $H (t - t_{k} - L / v)$ when the load departs from the beam. In equation (2), $D_{0} (t)$ represents the vertical displacement of a moving mass. Given the fact that the moving masses are not separated from the beam surface during the course of vibration ( $D_{0} (t) = D (x = v (t - t_{k}), t)$ ), the following equation is obtained

\begin{matrix} \frac{d^{2} D_{0} (t)}{d t^{2}} = {\frac{\partial 2 D (x, t)}{\partial t^{2}} + 2 v \frac{\partial 2 D (x, t)}{\partial x . \partial t} \\ + v^{2} \frac{\partial 2 D (x, t)}{\partial x^{2}}}_{x = v (t - t_{k})} \end{matrix}

(4)

3. Formulation of the steel pipe dampers (SPD) system

3.1. Assumptions and governing equations

While different types of dampers have been successfully utilized for vibration control of bridges, experimental and analytical studies conducted by Maleki and Bagheri (2010) indicated that SDP can provide very efficient design solutions by offering a stable hysteresis behavior and high energy absorption through metallic yielding. In this study, this type of damper is adopted to mitigate the effects of simultaneous action of vertical earthquake excitation and vehicle loads in bridges. The mathematical formulation for application of SPD is developed in the following section.

Figure 1 shows the schematic view of the proposed system. The stiffness factor of each pipe ( $K_{pipe}$ ) in elastic and plastic deformation zones varies in proportion to the selected pipe. Maleki and Bagheri (2010) showed that the damping ratio of the system is defined as $ξ = E_{D} / (4 π E_{S})$ . $E_{D}$ is the overall energy absorbed by the dampers, which is equal to the area under the hysteresis load-displacement curve for one complete cycle. $E_{S}$ represents the elastic strain energy of the pipes. The damping factor of each pipe damper can be obtained as $C_{pipe} = 2 ξ (M_{pipe} K_{pipe}) 1 / 2$ , where $M_{pipe}$ is the mass of each pipe in kilograms. At each loading phase, the pipe dampers may enter the plastic phase, while the beam itself is designed to remain in the elastic regime under the design load conditions. It should be noted that the dynamic systems solved in this study are classically damped, since the damping matrix is orthogonal and the stiffness and damping of the SPD are only applied in one direction (Veletsos and Ventura, 1986).

3.2. Configuration of the pipe damper system

As illustrated in Figure 1, the pipe damper system is a rigid box with length $L'$ located at the mid-span of the bridge. The box comprises a number of pipes with exterior diameter $d_{pipe}$ connected to the main bridge beam. Placement of the pipe damper system is based on the fact that maximum deflection of a simply-supported beam under the assumed loading condition is expected to occur at the mid-span. The rigid box is assumed to respond completely isolated from the main beam, and therefore, does not impose any constraints on the beam deformation response under external excitation. For practical applications, depending on the location of the dampers and type of the bridge, the rigid box can be connected to the ground using a structural system such as a truss, pylon, concrete pad, or pre-tensioned cables.

As shown in Figure 1, the number of pipes attached above and below the main beam can be calculated as $q = n_{pipe} / 2$ , where $n_{pipe}$ is total number of pipes used in the box. Length of the rigid box is given by $L' = q [h_{IPE} + d_{pipe}]$ , where $h_{IPE}$ is depth of the IPE section used to connect the dampers to the main beam. Based on the bi-linear model presented by Maleki and Bagheri (2010), the response of the SPD system (rigid box) is simulated using a spring-dashpot by assuming $Δ H (x) = H (x - x_{1}) - H (x - x_{q})$ , where $x_{1}$ and $x_{q}$ are the distance between the center of the first and last pipe-couple from the assumed origin, respectively

\begin{matrix} {\begin{matrix} x_{1} = \frac{L - L'}{2} + (h_{IPE} + d_{pipe}) \\ x_{q} = \frac{L + L'}{2} - (h_{IPE} + d_{pipe}) \end{matrix} \end{matrix}

(5)

The stiffness and damping of the steel pipes directly contribute to the beam vibration formula (equation (1)) as follows

\begin{matrix} ρ A \frac{\partial 2 D (x, t)}{\partial t^{2}} + EI \frac{\partial 4 D (x, t)}{\partial x^{4}} + K_{pipe}^{e} D (x, t) Δ H (x) \\ = f (x, t) - C_{pipe}^{e} \frac{\partial D (x, t)}{\partial t} Δ H (x) \end{matrix}

(6)

in which, the equivalent stiffness and damping factors of the steel pipe system are calculated by the equations below

\begin{matrix} {\begin{matrix} K_{pipe}^{e} = n_{pipe} . K_{pipe} = n_{pipe} . K_{pipe} (L' - h_{IPE}) (L' - h_{IPE}) \\ C_{pipe}^{e} = n_{pipe} . C_{pipe} = n_{pipe} . C_{pipe} (L' - h_{IPE}) (L' - h_{IPE}) \end{matrix} \end{matrix}

(7)

where

K_{pipe}^{e}

and

C_{pipe}^{e}

are the function of the stiffness and damping factors of steel pipes.

3.3. Spatial and time discretization of the governing equations

The eigenfunction expansion method (EEM) is adopted to solve the above-mentioned differential equation, as follows

D (x, t) = \sum_{i = 1}^{\infty} ϕ_{i} (x) a_{i} (t) \approx \sum_{i = 1}^{p} ϕ_{i} (x) a_{i} (t)

(8)

where

ϕ_{i} (x)

and

a_{i} (x)

respectively denote the orthogonal shape function related to the i^th beam vibration mode (in this study:

ϕ_{i} (x) = \sqrt{2 / L} sin (in π x / L);

n =

number of spans of the simply-supported beams) and the corresponding i^th time-dependent amplitude. The total number of shape functions required to reach the predefined target accuracy is p. By substituting equation (8) into equation (6), multiplying both sides of the resulting equation by

ϕ_{j} (x)

, and integrating them over the length of the beam, the following equation is attained

\begin{matrix} ρ A \sum_{i = 1}^{p} {\sum_{j = 1}^{p} \int_{0}^{L} ϕ_{i} (x) ϕ_{j} (x) d x} \frac{\partial 2 a_{i} (t)}{\partial t^{2}} \\ + EI \sum_{i = 1}^{p} {\sum_{j = 1}^{p} \int_{0}^{L} \frac{\partial 4 ϕ_{i} (x)}{\partial x^{4}} ϕ_{j} (x) d x} a_{i} (t) \\ + K_{pipe}^{e} \sum_{i = 1}^{p} {\sum_{j = 1}^{p} \int_{x_{1}}^{x_{q}} ϕ_{i} (x) ϕ_{j} (x) d x} Δ H (x) a_{i} (t) \\ + C_{pipe}^{e} \sum_{i = 1}^{p} {\sum_{j = 1}^{p} \int_{x_{1}}^{x_{q}} ϕ_{i} (x) ϕ_{j} (x) d x} Δ H (x) \frac{\partial a_{i} (t)}{\partial t} \\ = \sum_{k = 1}^{N} \sum_{i = 1}^{p} \sum_{j = 1}^{p} m_{k} \int_{0}^{L} {g - [ϕ_{i} (x) \frac{\partial 2 a_{i} (t)}{\partial t^{2}} \\ + 2 v \frac{\partial ϕ_{i} (x)}{\partial x} \frac{\partial a_{i} (t)}{\partial t} + v^{2} \frac{\partial 2 ϕ_{i} (x)}{\partial x^{2}} a_{i} (t)]} \\ \times ϕ_{j} (x) . δ [x - v (t - t_{k})] . Δ H (t) d x \\ - \frac{d^{2} U_{g} (t)}{d t^{2}} \sum_{j = 1}^{p} \int_{0}^{L} ϕ_{j} (x) \\ \times {ρ A + \sum_{k = 1}^{N} m_{k} . δ [x - v (t - t_{k})] . Δ H (t)} d x \end{matrix}

(9)

By defining $ω_{i}$ as the i^th beam's natural frequency, the beam's free vibration equation can be presented as

EI \int_{0}^{L} \frac{\partial 4 ϕ_{i} (x)}{\partial x^{4}} ϕ_{j} (x) d x = ρ A ω_{i}^{2} \int_{0}^{L} ϕ_{i} (x) ϕ_{j} (x) d x

(10)

Considering the orthogonality of the modes

\begin{matrix} á ϕ_{i} (x), ϕ_{j} (x) ñ = \int_{0}^{L} ϕ_{i} (x) ϕ_{j} (x) d x = δ_{ij} \\ = {\begin{matrix} \begin{matrix} 0 if i \neq j \\ 1 if i = j \end{matrix}; i, j = 1, 2, \dots, p \end{matrix} \end{matrix}

(11)

and by substituting equation (10) into equation (9), the matrix form of equation (9) is rewritten as

M (t) \frac{d^{2} a (t)}{d t^{2}} + C (t) \frac{d a (t)}{d t} + K (t) a (t) = F (t)

(12)

in which

a (t) = [a_{i} (t)]_{p \times 1}

(13-1)

\begin{matrix} M (t) = [ρ A δ_{ij} + \sum_{k = 1}^{N} m_{k} ϕ_{i} [v (t - t_{k})] . ϕ_{j} [v (t - t_{k})] . Δ H (t)]_{p \times p} \end{matrix}

(13-2)

\begin{matrix} C (t) = [C_{pipe}^{e} \int_{x_{1}}^{x_{q}} ϕ_{i} (x) ϕ_{j} (x) d x . Δ H (x) \\ + \sum_{k = 1}^{N} 2 m_{k} v ϕ_{i, x} [v (t - t_{k})] . ϕ_{j} [v (t - t_{k})] . Δ H (t)]_{p \times p} \end{matrix}

(13-3)

\begin{matrix} K (t) = [ρ A ω_{i}^{2} δ_{ij} + K_{pipe}^{e} \int_{x_{1}}^{x_{q}} ϕ_{i} (x) ϕ_{j} (x) d x . Δ H (x) \\ + \sum_{k = 1}^{N} m_{k} v^{2} ϕ_{i, xx} [v (t - t_{k})] . ϕ_{j} [v (t - t_{k})] . Δ H (t)]_{p \times p} \end{matrix}

(13-4)

\begin{matrix} F (t) = [[g - \frac{d^{2} U_{g} (t)}{d t^{2}}] \sum_{k = 1}^{N} m_{k} ϕ_{j} [v (t - t_{k})] . Δ H (t) \\ - ρ A \frac{d^{2} U_{g} (t)}{d t^{2}} \int_{0}^{L} ϕ_{j} (x) d x]_{p \times 1} \end{matrix}

(13-5)

By transforming equation (12) into a state-space form, the second order differential equations set in equation (12) could be reduced to a first order set of equations via the following reformulation

\frac{d}{d t} X (t) = A (t) X (t) + B (t)

(14)

where

\begin{matrix} X (t) = [\begin{matrix} a (t) \\ d a (t) / d t \end{matrix}]_{2 p \times 1}, \\ A (t) = [\begin{matrix} 0_{p \times p} & I_{p \times p} \\ - M^{- 1} K & - M^{- 1} C \end{matrix}]_{2 p \times 2 p}, \\ B (t) = [\begin{matrix} 0_{p \times 1} \\ M^{- 1} F \end{matrix}]_{2 p \times 1} \end{matrix}

(15)

A number of methods are available to solve equation (14) in the time domain. In this paper, the matrix-exponential method, proposed by Brogan (1985) and utilized by a number of other researchers (e.g., Ebrahimzadeh Hassanabadi et al., 2014; Nikkhoo, 2014; Nikkhoo et al., 2016; Niaz and Nikkhoo, 2015) is adopted for calculation of the displacement field. In the following section, the efficiency of the method is demonstrated through a benchmark numerical example.

4. Numerical example

In this benchmark example, it is assumed that a group of moving vehicles are crossing a simply-supported bridge with a span length of $L = 60 m$ (Legeron and Sheikh, 2009). The deck of the bridge is supported by a number of beams parallel to the direction of the bridge. To analyze the vibration and maximum deformation of the bridge, one of the main beams, as indicated in Figure 2, is considered. The mechanical properties of the beam section are assumed as $EI = 3.467 \times 10^{10} N m^{2}$ and $ρ A = 2956 kg / m$ . The moving vehicles consist of a group of $m = 50000 kg$ masses traveling with a constant velocity of $v = 30 m / s$ and effective distance of d. As illustrated in Figure 2, it is assumed that when a traversing mass reaches the mid-span, the next mass enters the beam (i.e., $d = L / 2$ ). This system is analyzed for the first 16 seconds implying that, in total, 15 masses traverse the beam. Additionally, it is assumed that as the first mass arrives at the beam (i.e., for $t = 0$ ), the beam undergoes a vertical excitation caused by an earthquake event.

Figure 2.

Schematic view of a simply-supported beam subjected to moving masses with uniform intervals.

To investigate the effects of input ground motion characteristics on the dynamic response of the bridge, the vertical acceleration records of seven major near-field earthquakes are considered. The characteristics of the selected earthquakes, including the soil conditions according to SEI/ASCE 7-16 classifications (American Society of Civil Engineers, 2017), are presented in Table 1 (Pacific Earthquake Engineering Research Center, n.d.). It can be seen that these earthquake records cover different soil types and fault mechanisms. Figure 3 compares the acceleration response spectrum of the selected earthquake records.

Figure 3.

Response spectrum of the selected earthquake ground motions: (a) vertical components; and (b) horizontal components.

Table 1.

Earthquake characteristics.

Number	1	2	3	4	5	6	7
Event	L'Aquila, Italy	Imperial Valley-02	Kobe, Japan	Mammoth Lakes-01	Loma Prieta	Northridge-01	Tabas, Iran
Year	2009	1940	1995	1980	1989	1994	1978
Station	Gran Sasso	El Centro Array Number 9	KJMA	Convict Creek	Corralitos	Arleta- Nordhoff	Tabas
Magnitude	6.3	6.95	6.9	6.06	6.93	6.69	7.35
Mechanism	Normal	Strike-slip	Strike-slip	Normal-oblique	Reverse-oblique	Reverse	Reverse
D5-95 (S)	8.9	24.2	9.5	9.6	7.9	13.5	16.5
Rjb (km)	6.35	6.09	0.94	1.1	0.16	3.3	1.79
Rrup (km)	6.4	6.09	0.96	6.63	3.85	8.66	2.05
Vs30 (m/s)	488	213	312	382	462	298	767
Soil type	C	D	D	C	C	D	B
Peak ground acceleration (PGA) (horizontal)	0.150g	0.281g	0.834g	0.442g	0.645g	0.345g	0.862g
PGA (vertical)	0.110g	0.178g	0.339g	0.387g	0.458g	0.552g	0.641g
Horizontal-to-vertical spectral ratio	1.36	1.58	2.46	1.14	1.41	0.63	1.34
Peak ground velocity (PGV) (horizontal) cm/s	9.71	31.31	91.06	23.75	55.94	41.09	123.34
PGV (vertical) cm/s	4.42	8.61	40.35	21.10	19.50	17.99	40.90

4.1. Vertical seismic loads and their interaction with moving inertial loads

The mid-span beam deflection response of the case study bridge (Figure 2) due to the moving masses (dash line) and the simultaneous effects of moving masses and earthquake excitations (solid line) are depicted in Figure 4. As observed, in most cases, the seismic loads amplified the effects of moving vehicles and resulted in deflections up to twice the case excluding seismic actions. This effect was especially evident in Kobe and Tabas near-field earthquakes with large vertical components (see Figure 3). For better comparison, Table 2 presents the maximum deflection at the beam's mid-span locations under the two above loading conditions. As it is seen, the discrepancies in responses vary for different earthquakes. The increase in the beam vertical deflection response due to the vertical component of the input earthquake is represented by $M_{n}$ factor. It can be noticed that, on average, the vertical deflections increased by over 40% under the simultaneous effect of moving mass and earthquake vertical excitations. These results clearly demonstrate the importance of considering the interaction of the vertical components of earthquakes and moving masses in the design of beam-like structures.

Figure 4.

Time-history deflection of the beam due to external excitations: (a) L'Aquila; (b) Imperial-Valley; (c) Kobe; (d) Mammoth-Lakes; (e) Loma-Prieta; (f) Northridge; and (g) Tabas.

Table 2.

Impact of vertical earthquake excitation on maximum deflections at beam's mid-span.

Number	Earthquake event	Maximum mid-span beam deflection (m)		Impact of vertical earthquake excitation on mid-span deflection M_n (%)
Number	Earthquake event	Moving mass	Moving mass + earthquake
1	L'Aquila, Italy	0.133	0.138	3.8
2	Imperial-Valley-02	0.133	0.141	6.0
3	Kobe, Japan	0.133	0.270	103.0
4	Mammoth-Lakes-01	0.133	0.177	33.1
5	Loma-Prieta	0.133	0.165	24.1
6	Northridge-01	0.133	0.180	35.3
7	Tabas, Iran	0.133	0.264	98.5

4.2. Efficiency of SPD in vibration suppression

This sub-section investigates the efficiency of the SPD in mitigating the bridge vibrations induced by simultaneous actions of moving masses and vertical seismic excitations. As in the study conducted by Maleki and Bagheri (2010), the exterior diameter of the pipe dampers was set to be $d_{pipe} = 119 mm$ , while $E_{D} = 173 kNmm$ and $E_{S} = 34 kNmm$ . The stiffness factor of each pipe ( $K_{pipe}$ ) in elastic and plastic deformation zones were given by $0.0034 l (kN / mm)$ and $0.00017 l (kN / mm)$ , respectively. The yield strength of the pipe dampers was estimated as $F_{y} = 0.0088 l (kN)$ , where l is the length of each pipe in millimeters (mm). Based on these assumptions, the deformation corresponding to $F_{y}$ is calculated as 2.6 mm. An IPE270 section with the depth and flange width of $h_{IPE} = 270 mm$ and $b_{IPE} = 135 mm$ (in accordance with DIN 1025-5 (2014)) was used to connect the pipes to the main beam. The length of the pipe box $L'$ was selected to be equal to the sitting length of the beam (for a beam length of 60 m, $L'$ was considered to be 3 m). Thus, the maximum number of pipe dampers for this case study example was set to be 16 (should be a multiple of four as shown in Figure 1).

4.2.1. Single span bridge

Figure 5 demonstrates the variations in the deflection of the beam's mid-span as a function of the initial flexural rigidity factor ( $K_{n} = {0.5, 0.6, \dots, 1} EI$ ) for the single span bridge excluding dampers ( $n_{pipe} = 0$ ) and the one including 4, 8, 12, and 16 SPD ( $n_{pipe} = 4, 8, 12, 16$ ) under the seven different earthquake scenarios. The results in Figure 5 indicate that, as expected, in the bridge systems without dampers, the maximum deflection of the beam under all seven selected earthquakes decreases as the flexural rigidity increases. In general, this effect is more prominent for the initial flexural rigidity factors, $K_{n}$ , smaller than 0.8. However, it is shown that the maximum deflection of the bridge systems with SPD can be considered to be practically independent of the initial flexural rigidity factor of the bridge. It can be noted by using 16 pipe dampers (only across 5% of the total beam length), the maximum response amplitude of the bridge was significantly (up to 75%) reduced, particularly for the systems with low initial flexural rigidity factors.

Figure 5.

Effect of initial flexural rigidity factor $K_{n}$ on the variations of the single span beam deflection with and without dampers: (a) L'Aquila; (b) Imperial-Valley; (c) Kobe; (d) Mammoth-Lakes; (e) Loma-Prieta; (f) Northridge; and (g) Tabas.

The base line in Figure 5 shows the maximum deflection at the beam's mid-span without pipe dampers (i.e., $K_{n} = 1$ and $n_{pipe} = 0$ ). The aim is to find the initial flexural rigidity factor $K_{n}$ , at which the maximum deflection of the beam with pipe dampers reaches the base line level. The reduction of the $K_{n}$ is mainly achieved by reducing the size of the beam sections (slenderness of the base beam is assumed to be constant). This implies that, for the same performance target, using pipe dampers can considerably reduce the required structural weight. It is worth mentioning that the weight of these dampers is generally less than 1% of the total beam's weight.

Table 3 shows the required initial flexural rigidity factor

K_{n}

for the bridge systems with 16 pipe dampers to exhibit the same maximum mid-span deflection as the initial beam (

K_{n}

= 1,

n_{pipe} = 0

) under the seven earthquake records. It is shown that by using 16 pipe dampers, the flexural rigidity of the main beams can be reduced by up to 27% (on average 16%). According to Table 3, the performance of the pipe dampers seems to be mainly dependent on

M_{n}

and

V_{s} 30 (m / s)

. In general, as expected, the efficiency of the dampers increased by increasing the effect of vertical earthquake excitation (

M_{n}

). Similarly, in the case of stiff soils with high values of

V_{s} 30 (m / s)

, the efficiency of pipe dampers became more profound. Based on the results, these two parameters can affect the response of the system independently. For instance, in the case of the L'Aquila earthquake, although

M_{n}

is negligible, due to the soil type (high value of

V_{s} 30 (m / s)

), pipe dampers could reduce the required flexural rigidity by 14%. In the opposite situation, under the Kobe earthquake with small value of

V_{s} 30 (m / s)

and high value of

M_{n}

, utilizing the pipe dampers resulted in 17% reduction in the

K_{n}

. In the case of the Imperial-Valley, where both

V_{s} 30 (m / s)

and

M_{n}

were small, the efficiency of dampers was considerably less than the other earthquake scenarios. On the contrary, the pipe dampers exhibited their maximum efficiency (27% reduction in

K_{n}

) under Tabas, where both

V_{s} 30 (m / s)

and

M_{n}

were maximum.

Table 3.

Effects of using steel pipe dampers and different initial flexural rigidity $K_{n}$ on the maximum single span bridge deflection.

Number	Event	V_s30 (m/s)	Effect of vertical earthquake M_n (%)	K_n	Number of pipe dampers	Maximum mid-span deflection (m)
1	L'Aquila, Italy	488	3.8	1	0	0.138
1	L'Aquila, Italy	488	3.8	0.86	16	0.138
2	Imperial-Valley	213	6.0	1	0	0.141
2	Imperial-Valley	213	6.0	0.95	16	0.141
3	Kobe, Japan	312	103.0	1	0	0.270
3	Kobe, Japan	312	103.0	0.83	16	0.270
4	Mammoth-Lakes	382	33.1	1	0	0.177
4	Mammoth-Lakes	382	33.1	0.83	16	0.177
5	Loma-Prieta	462	24.1	1	0	0.165
5	Loma-Prieta	462	24.1	0.9	16	0.165
6	Northridge	298	35.3	1	0	0.180
6	Northridge	298	35.3	0.8	16	0.180
7	Tabas, Iran	766	98.5	1	0	0.264
7	Tabas, Iran	766	98.5	0.73	16	0.264

The Fourier amplitude spectra curves illustrated in Figure 6 indicate that the dominant fundamental frequencies affecting the structural response of the beam system under different earthquake events do not change significantly after using pipe dampers.

Figure 6.

Effect of frequency on the Fourier amplitude of beam deflection with and without dampers: (a) L'Aquila $K_{n}$ = 0.86; (b) Imperial-Valley $K_{n}$ = 0.95; (c) Kobe $K_{n}$ = 0.83; (d) Mammoth-Lakes $K_{n}$ = 0.83; (e) Loma-Prieta $K_{n}$ = 0.90; (f) Northridge $K_{n}$ = 0.80; and (g) Tabas $K_{n}$ = 0.73.

4.2.2. Multiple span bridge

As indicated in Figure 7, a three-span bridge with simple ends and total length of $L = 180 m$ (each span is $60 m$ ) and $d = L / 6$ is assumed. The pipe damper system is installed at the mid-span of each span. The rest of the properties are the same as the single span example.

Figure 7.

Schematic view of a three-span beam equipped with pipe damper system.

Figure 8 illustrates the spectrum of the maximum deflection at the mid-span as a function of the initial flexural rigidity factor $K_{n}$ . In addition, Table 4 shows the effects of using SPD and different initial flexural rigidity on the maximum deflection of the three-span bridge example under the seven selected earthquakes.

Figure 8.

Effect of initial flexural rigidity factor $K_{n}$ on the variations of three-span beam deflection with and without dampers: (a) L'Aquila; (b) Imperial-Valley; (c) Kobe; (d) Mammoth-Lakes; (e) Loma-Prieta; (f) Northridge; and (g) Tabas.

Table 4.

Effects of using steel pipe dampers and different initial flexural rigidity $K_{n}$ on the maximum multi span bridge deflection.

Number	Event	V_s30 (m/s)	Effect of vertical earthquake M_n (%)	K _n	Number of pipe dampers	Maximum mid-span deflection (m)
1	L'Aquila, Italy	488	5.0	1	0	0.045
1	L'Aquila, Italy	488	5.0	0.72	8	0.045
2	Imperial-Valley	213	8.8	1	0	0.049
2	Imperial-Valley	213	8.8	0.83	8	0.049
3	Kobe, Japan	312	113.5	1	0	0.097
3	Kobe, Japan	312	113.5	0.54	8	0.097
4	Mammoth-Lakes	382	22.4	1	0	0.055
4	Mammoth-Lakes	382	22.4	0.64	8	0.055
5	Loma-Prieta	462	35.5	1	0	0.061
5	Loma-Prieta	462	35.5	0.77	8	0.061
6	Northridge	298	28.1	1	0	0.058
6	Northridge	298	28.1	0.56	8	0.058
7	Tabas, Iran	766	140.4	1	0	0.109
7	Tabas, Iran	766	140.4	0.5	8	0.109

Similar to the single span bridge example, utilizing the pipe damper system could considerably reduce the dynamic response of the three-span bridge system. However, the efficiency of the pipe dampers was more prominent in the three-span bridge. This can be due to the higher values of $M_{n}$ in the three-span bridge compared to the single-span example in most of the selected earthquakes, and also utilizing three packs of dampers for the three-span bridge system. The results in Table 4 indicate that by employing eight pipe dampers, for the same mid-span deflection, the flexural rigidity of the main beam can be reduced up to 50% (on average 35%).

5. Conclusions

This study aimed to assess the vibration behavior and develop a mitigation technique for simply-supported bridge systems subjected to simultaneous effects of moving vehicles and vertical earthquakes. Through an extensive analytical study using EEM, it was shown that if a near-field earthquake with a strong vertical component strikes while vehicles are traveling across the bridge, depending on the frequency content of the earthquake, displacement response of the bridge could be severely intensified (by 100%). While the effects of vertical earthquake loads are usually ignored in the current design guidelines, this highlights the importance of considering the simultaneous effects of vertical earthquake excitation and vehicle loads in the design process of bridges in seismic regions. Subsequently, the efficiency of using SDP was examined to mitigate the induced vibrations in single-span and multi-span bridges. It was shown that the proposed system can significantly (up to 75%) reduce the maximum response amplitude of the bridge, especially for the systems with low initial flexural rigidity factors. Regardless of the input earthquake characteristics, the maximum deflection of the bridges appeared to be less sensitive to the variations of flexural rigidity when pipe dampers are employed. However, the efficiency of pipe dampers generally improved by increasing the $M_{n}$ and $V_{s} 30 (m s)$ of the input earthquake. For the same maximum mid-span deflection limit, application of pipe dampers in bridges could reduce the required flexural rigidity by up to 50%. This implies that the proposed vibration mitigation method can potentially reduce the constructional costs by reducing the structural weight of the bridge system.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

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