On the Susceptibility of Reinforced Concrete Beam and Rigid-Frame Bridges Subjected to Spatially Varying Mining-Induced Seismic Excitation
Abstract
:1. Introduction
2. Materials and Methods
2.1. Characteristics of Mining-Induced Tremors and the Shock Used for Dynamic Analysis
2.2. Theoretical Framework for the Dynamic Response of Multiple-Support Structures to Spatially Varying Ground Motion
- Mss, Css, and Kss—mass, damping, and stiffness matrix corresponding to non-support DOFs;
- Mgg, Cgg, and Kgg—mass, damping, and stiffness matrix corresponding to support DOFs;
- Msg, Csg, and Ksg—mass, damping, and stiffness coupling matrix;
- ,, and —accelerations, velocities, and displacements vector for each of the DOFs of the structure;
- , , and —accelerations, velocities, and displacements vector for each of the DOFs of the ground;
- —forces generated at the supports.
2.3. Theorethical Basis of the Large Mass Method
- M0—large mass added to supported DOFs;
- —inertia forces;
- —vector of large mass acceleration.
- n—number of modes;
- mn, kn, and cn—vector of modal mass, modal stiffness, and modal damping for n-th mode;
- qn—generalized coordinate;
- ϕn—n-th modal vector.
2.4. Structural Solutions for Bridges Subjected to Dynamic Analysis
2.4.1. Structural Design and Numerical Model of a Beam Bridge
2.4.2. Structural Design and Numerical Model of a Rigid-Frame Bridge
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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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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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
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 StyleBoroń, 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