# On the Susceptibility of Reinforced Concrete Beam and Rigid-Frame Bridges Subjected to Spatially Varying Mining-Induced Seismic Excitation

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Characteristics of Mining-Induced Tremors and the Shock Used for Dynamic Analysis

^{5}to 10

^{7}J, although events with energy levels up to 10

^{10}J have been documented [5,6,34,35]. In the USCB region, the recorded maximum amplitudes of these tremors can reach values of up to 1.7 m/s

^{2}, and their predominant frequencies typically range from 3 to 8 Hz [36,37].

^{2}in the WE and 0.28 m/s

^{2}in the NS directions, respectively. In the vertical direction, accelerations of 0.12 m/s

^{2}were observed. Spectral analysis of the shock revealed dominant frequencies of 3.56 Hz in the WE direction, 2.8 Hz in the NS direction, and 3.4 Hz for the vertical component of the shock. For the purposes of this analysis, the mining tremor accelerations were scaled up to the maximum value of PGA = $5\xb70.35$ m/s

^{2}, resulting in 1.75 m/s

^{2}. This value corresponded to the most intense tremors experienced in the region.

#### 2.2. Theoretical Framework for the Dynamic Response of Multiple-Support Structures to Spatially Varying Ground Motion

**M**_{ss},**C**_{ss}, and**K**_{ss}—mass, damping, and stiffness matrix corresponding to non-support DOFs;**M**_{gg},**C**_{gg}, and**K**_{gg}—mass, damping, and stiffness matrix corresponding to support DOFs;**M**_{sg},**C**_{sg}, and**K**_{sg}—mass, damping, and stiffness coupling matrix;- ${\ddot{x}}_{s}$,${\dot{x}}_{s}$, and ${x}_{s}$—accelerations, velocities, and displacements vector for each of the DOFs of the structure;
- ${\ddot{x}}_{g}$, ${\dot{x}}_{g}$, and ${x}_{g}$—accelerations, velocities, and displacements vector for each of the DOFs of the ground;
- ${P}_{g}$—forces generated at the supports.

#### 2.3. Theorethical Basis of the Large Mass Method

**M**, represented by the movement of the supports acceleration ${\ddot{x}}_{g}$ into the inertia forces

_{ss}**P**. This conversion is accomplished by incorporating a significant mass

_{0}**M**to the degrees of freedom supporting the structure and the equivalent kinematic excitation ${\ddot{x}}_{0}$. Therefore, Equation (1) took the following form:

_{0}**M**_{0}—large mass added to supported DOFs;- ${P}_{0}={M}_{0}\xb7{\ddot{x}}_{0}$—inertia forces;
- ${\ddot{x}}_{0}$—vector of large mass acceleration.

**M**>>

_{0}**M**, the displacements of large mass ${x}_{0}$ were very close to the displacements ${x}_{g}$ [43]. It was estimated that the applied mass should be approximately 10

_{ss}^{6}times bigger than the total mass of the structure [42,43,44,45,46].

- n—number of modes;
- m
_{n}, k_{n}, and c_{n}—vector of modal mass, modal stiffness, and modal damping for n-th mode; - q
_{n}—generalized coordinate; - ϕ
_{n}—n-th modal vector.

**M**to the numerical model introduced rigid body modes, which are associated with extremely low frequencies (≈0 Hz) related to the movement of large mass

_{0}**M**. If all modes of a structure (including the rigid body ones) were considered when solving Equation (6), the total dynamic response of the structure was obtained. However, the quasi-static component of the overall structural response could be determined by only including the rigid body modes in Equation (6). The dynamic component is obtained by considering the modes associated with non-zero frequencies, i.e., omitting the rigid body modes.

_{0}#### 2.4. Structural Solutions for Bridges Subjected to Dynamic Analysis

#### 2.4.1. Structural Design and Numerical Model of a Beam Bridge

^{3}were adopted.

#### 2.4.2. Structural Design and Numerical Model of a Rigid-Frame Bridge

^{3}.

^{3}.

## 3. Results

#### 3.1. Numerical Evaluation of Natural Frequencies and Modes of Vibration of the Bridges

#### 3.2. Time History Analysis of Principal Stresses in the Bridges Subjected to the Mining Shock

- For point SB_1, located above the support, the maximum stress caused by the tremor is higher for the non-uniform model application (see Figure 10a) compared to the uniform model. We can observe that reducing the velocity of the shock wave results in an increase in dynamic response. It is worth emphasizing that the differences in the dynamic performance of the bridge under various excitation models are significant. The stresses obtained for two extreme excitation scenarios (uniform and non-uniform with a velocity of 250 m/s) differ 2.5-fold.
- Applying the non-uniform excitation model results in elevated stress levels compared to the uniform excitation model, as seen in element LB_2 (refer to Figure 10b). The maximum stress for non-uniform excitation with a velocity of 250 m/s is approximately 1.5 times higher than the stresses obtained for uniform excitation.

- The stress analysis of the rigid-frame bridge in the support zone leads to both qualitative and quantitative conclusions analogous to those determined in the case of the beam bridge. For point SF_1, the maximum stress is higher with the application of the non-uniform model (see Figure 11a) compared to the uniform model, and a decrease in the wave velocity corresponds to an increase in the dynamic response. The stresses obtained for the uniform and non-uniform model with a velocity of 250 m/s exhibit 3-fold differences.
- The non-uniform excitation model produces distinct performances in the span zones compared to the supports. The maximum stress at the midspan point LF_2 is higher for the uniform model than for the non-uniform model of excitation (see Figure 11b). The maximum stress values decrease as the wave velocity decreases, reaching the smallest value for the slowest wave at 250 m/s. The maximum stress for non-uniform excitation with a velocity of 250 m/s is approximately 20% lower than for the stresses obtained for uniform excitation.

#### 3.3. Dependence of the Dynamic Behavior of the Bridges on Wave Velocity

#### 3.3.1. Stress–Velocity Dependence for the Beam Bridge

#### 3.3.2. Stress–Velocity Dependence for the Rigid-Frame Bridge

#### 3.3.3. The Obtained Results in the Context of the Literature Research

## 4. Discussion on the Susceptibility of the Bridges to Quasi-Static and Dynamic Effects

#### 4.1. Rigid-Frame Bridge vs. Beam Bridge: A Comparison of Susceptibility to Quasi-Static Effect

#### 4.2. Rigid-Frame Bridge vs. Beam Bridge: A Comparison of Susceptibility to Dynamic Effects

## 5. Conclusions

- The dynamic responses of both types of bridges undergo significant changes when subjected to the model of spatially varying ground mining-induced excitation, compared to the model of uniform excitation.
- In the case of the beam bridge, the impact of non-uniform excitation is evident across the entire structure, with the most notable effects observed at the central support, where the dynamic response evidently increases with decreasing wave velocity.
- The relationship between the total dynamic response and the wave velocity is less straightforward for the rigid-frame bridge. Similarly, as observed in the case of the beam bridge, applying the spatially varying excitation model results in increased stress in the support zones. However, in the spans, the highest stresses are obtained under uniform excitation.
- The quasi-static component plays a crucial role in the overall dynamic responses for both beam and rigid-frame bridges. However, rigid-frame bridges exhibit much greater susceptibility to quasi-static effects compared to beam bridges. This is related to the higher stiffness of the rigid-frame bridge. On the other hand, beam bridges are more susceptible to the dynamic components of stresses.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) The time history of the ground acceleration registered in the USCB region in horizontal direction WE and (

**b**) its frequency spectrum.

**Figure 2.**(

**a**) The time history of the ground acceleration registered in the USCB region in horizontal direction NS and (

**b**) its frequency spectrum.

**Figure 3.**(

**a**) The time history of the ground acceleration registered in the USCB region in vertical direction and (

**b**) its frequency spectrum.

**Figure 4.**The structural details of a beam bridge (dimensions in cm): (

**a**) side view; (

**b**) cross-section of the slab in the midspan (left) and at the support zone (right); (

**c**) the system of bearings.

**Figure 5.**The numerical model of the beam bridge with representative elements selected for the dynamic analysis.

**Figure 6.**The structural details of the rigid-frame bridge (dimensions in cm): (

**a**) side view; (

**b**) cross-section of the slab at the support zone (left) and in the midspan (right); (

**c**) the system of bearings.

**Figure 7.**The numerical model of the rigid-frame bridge with representative elements selected for dynamic analysis.

**Figure 8.**Modes of vibrations for the beam bridge: (

**a**) first (vertical), (

**b**) second (vertical), and (

**c**) third (torsional). The color scale represents displacement from blue (zero) to red (maximum displacement).

**Figure 9.**Modes of vibrations for the rigid-frame bridge: (

**a**) first (longitudinal), (

**b**) second (vertical), and (

**c**) third (vertical). The color scale represents displacement from blue (zero) to red (maximum displacement).

**Figure 10.**Time histories of maximal principal stress (i.e., the maximal tensile stress in the concrete material) for the beam bridge, occurring during the intense phase of seismic shock (5 s): (

**a**) over the central support (element SB_1); (

**b**) in the middle of the span (element LB_2).

**Figure 11.**Time histories of maximal principal stress (i.e., the maximal tensile stress in the concrete material) for the rigid-frame bridge, occurring during the intense phase of seismic shock (5 s): (

**a**) over the central support (element SB_1); (

**b**) in the middle of the span (element LB_2).

**Figure 12.**The maximum principal stress envelope plot along the beam bridge axis: (

**a**) the total response; (

**b**) the quasi-static component; (

**c**) the dynamic component.

**Figure 13.**Dependence of the absolute maximum of the total principal stresses, as well as their quasi-static and dynamic components, on wave velocity for the beam bridge: (

**a**) in the middle of the span (element LB_2); (

**b**) over the central support (element SB_1).

**Figure 14.**The maximum principal stress envelope for the rigid-frame bridge: (

**a**) the total response; (

**b**) the quasi-static component; (

**c**) the dynamic component.

**Figure 15.**Dependence of the absolute maximum of the total principal stresses and their quasi-static and dynamic components on wave velocity for the rigid-frame bridge: (

**a**) in the middle of the span (element LF_2); (

**b**) over the central support (element SF_1).

**Figure 16.**Envelope of ratio of ${\sigma}_{q-s}/{\sigma}_{tot}$ for both bridge types for non-uniform excitation with velocities of (

**a**) 250 m/s, (

**b**) 500 m/s, and (

**c**) 1000 m/s.

**Figure 17.**Envelope of ratio of ${\sigma}_{dyn}/{\sigma}_{tot}$ for both bridge types for non-uniform excitation with velocities of (

**a**) 250 m/s, (

**b**) 500 m/s, and (

**c**) 1000 m/s.

Basic Characteristics | Natural Seismic Shocks | Minig-Induced Seismic Shocks |
---|---|---|

Source of the shock | Natural tectonic processes | Human activities related to mining, such as the extraction of minerals |

Intensity and occurrence frequency | High magnitudes, frequency varying due to geological factors | Lower magnitudes but frequent in areas with extensive mining operations |

Influence range | Widespread; significant environmental impacts, including infrastructure damage and life loss | Up to 10 km; primarily affect the mining area, potentially causing ground instability and damage to mine infrastructure |

Shock duration | Lasts in minutes | Lasts up to 15 s |

Intense phase time | Minutes | 0.5–5 s |

Dominant frequency | Low range: 0.5–2 Hz | Higher range: 2–7 Hz |

Seismic wave arrival sequence | Different types of seismic waves (primary, shear, and Raleigh) reach the receiver sequentially | Due to the proximity of the source, all types of body waves, along with surface waves, arrive at the receiver almost simultaneously |

Magnitude of shock spatial components | The greatest amplitudes occur in the horizontal direction, parallel to the Raileigh wave propagation | Amplitudes in three directions are comparable; vertical amplitudes may even exceed those of horizontal vibrations |

Decay rate of the shock | Decrease in vibration amplitudes depends on the geological site conditions | Impulse-like nature of amplitudes; the decay in amplitudes with increasing distance from the source occurs much more rapidly |

Acceleration and frequency content | Acceleration and frequency range predictable, typical for given energies and epicentral distances | Unpredictable, wide acceleration and frequency range for given energies and epicentral distances |

Mode | Natural Frequency [Hz] | |
---|---|---|

Beam Bridge | Rigid-Frame Bridge | |

I | 4.12 | 4.04 |

II | 6.18 | 7.97 |

III | 6.97 | 8.48 |

IV | 7.38 | 10.96 |

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**MDPI and ACS Style**

Boroń, P.; Drygała, I.; Dulińska, J.M.; Burdak, S.
On the Susceptibility of Reinforced Concrete Beam and Rigid-Frame Bridges Subjected to Spatially Varying Mining-Induced Seismic Excitation. *Materials* **2024**, *17*, 512.
https://doi.org/10.3390/ma17020512

**AMA Style**

Boroń P, Drygała I, Dulińska JM, Burdak S.
On the Susceptibility of Reinforced Concrete Beam and Rigid-Frame Bridges Subjected to Spatially Varying Mining-Induced Seismic Excitation. *Materials*. 2024; 17(2):512.
https://doi.org/10.3390/ma17020512

**Chicago/Turabian Style**

Boroń, Paweł, Izabela Drygała, Joanna Maria Dulińska, and Szymon Burdak.
2024. "On the Susceptibility of Reinforced Concrete Beam and Rigid-Frame Bridges Subjected to Spatially Varying Mining-Induced Seismic Excitation" *Materials* 17, no. 2: 512.
https://doi.org/10.3390/ma17020512