A Pioneering Integration of Structural Health Assessments and Dynamic Analyses: Bridge Pier Responses to the Impact of Floating Objects during Extreme Floods

Abstract

This study presents a transformative dynamic amplification factor for assessing the resilience of over-river bridges, informed by the real-world conditions of flood events. Through advanced finite element analysis, we unveil how the interplay between mass and velocity of floating objects significantly influences bridge pier responses, challenging conventional assessment methods. Our findings reveal potential inadequacies in current design standards, such as AASHTO and AS5100, and introduces a dynamic multiplier that enhances structural health assessment algorithms. The core contribution of this research is a data-driven analysis approach, which is critical for the proactive maintenance and risk assessment of bridge infrastructures in areas prone to flooding. By redefining the parameters for damage-level identification, our work advocates for a shift towards more resilient infrastructure systems in the face of global climate change.

1. Introduction and Literature Review

Bridges play a critical role in ensuring the functionality and safety of transportation networks. In the broader context of structural engineering, understanding the dynamic responses of bridge structures is of paramount importance. The impact of global climate change is increasingly being felt, and floods are posing a growing threat to infrastructure worldwide. Dynamic loads such as those induced by the impact of floating and FOs pose a significant threat to the structural integrity of bridge piers. Logs are one of the most significant hazards associated with various types of debris, and can cause severe damage upon impact. Given their prevalence in bodies of water and their capacity to cause structural damage, logs are particularly important. The ability to predict and comprehend these dynamic responses is essential not only for the design and assessment of individual bridge elements but also for the development of resilient and safe infrastructure systems.

Structural integrity and safety are key elements in the design and assessment of bridges, particularly when they are subjected to dynamic forces. A comprehensive understanding of the dynamic behavior of bridge structures is essential for retaining their resilience against a spectrum of loading conditions. Designing and assessing the interaction between FOs and bridge components requires a thoughtful approach that recognizes the possibility of dynamic effects triggering significant structural reactions. This also contributes to the primary objective of creating resilient, environmentally sustainable infrastructure, and aligns with the aspirations of achieving net-zero emissions [1,2]

Accurate assessments of dynamic responses play a pivotal role in crafting effective design and maintenance strategies and empowering engineers to address both conventional and unexpected scenarios. The insights derived from studying dynamic responses extend beyond the specific context of a bridge pier impacted by FO. Instead, they lay the groundwork for advancing design methodologies applicable to diverse structural elements and scenarios.

Reinforced concrete (RC) piers of over-river U-slab bridges, a prevalent bridge type in Victoria, Australia, are vulnerable to impact loads owing to their slender design, particularly under extreme floods, as highlighted in a report by the Department of Transport. These structures face unique challenges when subjected to instantaneous loading from FOs during floods. Although hydrodynamic analyses of extreme flood loading on bridges and the impacts of vehicular collisions on highways have been extensively studied [3,4,5], there is a noticeable research gap in understanding the distinctive challenges posed by FO impacts on over-river bridges, particularly on U-slab bridge piers.

Figure 1 illustrates a typical U-slab bridge, emphasizing the relevance of our study in the Victoria region, where such bridges are common. Figure 2 underscores the urgency of our research, showcasing the documented damage caused by log impacts on U-slab bridge piers in Victoria following the 2010 floods.

Structural integrity and safety are of paramount importance in the design and assessment of bridges, particularly when subjected to dynamic forces. Understanding the dynamic behavior of bridge structures is crucial for ensuring their resilience under various loading conditions. The interaction between FOs and bridge components necessitates a nuanced approach for design and assessment that considers the potential for dynamic effects to induce critical structural responses.

To address this research gap, we conducted a parametric study to analyse the dynamic response of U-slab bridge piers to determine the maximum impact forces resulting from collisions with a relatively rigid FO. The finite element model assumed a fixed connection at the soil-structure interface and a simple support at the top, simulating its connection with the bridge crosshead to represent its boundary conditions. The FO, modelled as a cylindrical shape with a rigid body with one degree of freedom, was conservatively assumed to have a constant velocity, representing the impact on the bridge pier. The interaction was analysed using the Newmark integration method in ABAQUS 6.14 [6].

This study delves into the intricate relationships between the mass, velocity, and structural responses of bridge piers, providing fundamental insights that extend to broader applications in structural health assessment. By emphasizing the importance of dynamic effects and challenging conventional design paradigms, this study advocates for a paradigm shift in the assessment approach. The findings presented herein not only contribute to the specific domain of bridge engineering, but also pave the way for refining design codes and fostering the development of safer and more resilient infrastructure against dynamic challenges in the broader field of structural engineering.

While existing studies on structural health monitoring using cutting-edge Internet of tools (IoT) technologies provide valuable insights into the initial stages of structural condition focusing on real-time data collection and undeveloped assessment of structural functionality by introducing groundbreaking innovative methods for data analytics [7,8,9], a comprehensive analysis that considers the intricacies of algorithmic interpretation remains scarce. Our research fills this void, offering a novel perspective by assessing the dynamic impact forces on bridge structures through the lens of advanced computational models. This distinction in focus and methodology underpins the unique contributions of our work to the field of structural engineering.

Our study aimed to provide valuable insights into the dynamic responses and vulnerabilities of U-slab bridge piers under the impact of FO during flooding events, emphasizing the importance of identifying damage levels. Recognizing and assessing these damage levels is crucial for ensuring the resilience of over-river bridges, although the specific identification of these levels is beyond the scope of the current study. Additionally, our research aligns with the growing importance of real-time monitoring and risk assessment in infrastructure management [10]. By focusing on the unique challenges faced by U-Slab bridge piers during flood events, our study provides insights that can complement risk-monitoring ecosystems, aiding in the remote assessment of infrastructure conditions and the establishment of threshold levels for risk mitigation. The significance of damage-level identification has been discussed more broadly in the assessment of structural resilience [11]. Advancements in algorithmic analytics are pivotal to this endeavor, providing a more profound and accurate understanding of structural health, something that is indispensable for the resilience of our infrastructure.

2. Quantifying Impact Load in Structural Design

Impact load, an elusive yet critical parameter in structural design, has the potential to significantly influence structural performance and diminish the overall capacity. Empirical evidence from experimental studies [12] underscores the tangible impacts of loading on structures. The results showed a substantial reduction of 11.4–15.4% in the reaction forces throughout the energy dissipation in reinforced concrete plates subjected to impact loading.

In the realm of structural analysis, nonlinear finite element methods (FEM) are invaluable tools for comprehending dynamic interactions between FOs and structures. Notable contributions in this area include the work of [13], which offers a sophisticated analysis of impact scenarios and sheds light on the complex energy dissipation mechanisms involved. For example, studies examining logs as FOs have revealed their substantial momentum due to their mass and the acceleration provided by floodwaters. Based on empirical evidence, studies on the dynamic behavior of logs as floating objects in relation to hydraulic structures have highlighted their considerable kinetic energy, resulting from their substantial mass and accelerated movement within floodwater currents. This presents a notable risk to the structural stability of bridge piers, particularly during flood events and log impact [1,14,15,16].

2.1. Approaches to Estimate Maximum Impact Force

As we navigate the diverse landscape of impact force estimation, three foundational approaches, outlined in various standards, guide structural engineers.

• Contact Stiffness (AASHTO): Delving into the details of the effective contact stiffness, this approach requires a thorough understanding of the structural response during FO impact events [17].
• Impulse-Momentum (FEMA, June 2014): The concept of stopping time takes center-stage, providing insights into momentum transfer dynamics during impact, a facet elucidated by Stolle et al. [18] in their comprehensive review [19].
• Work Energy (NAASRA, 1990): Focusing on the stopping distance, this approach considers the intricate balance of energy expenditure during an impact. Notably, ref. [20] provides in-depth insights into the interdependence of the stopping distance with the effective contact stiffness, mass, and velocity of the debris.

In a one degree-of-freedom system, the maximum impact force can be defined using the following equation:

$$F_{i,max}=v\sqrt{\hat{k}{m}_l}$$

(1)

where $F_{i,max}$ represents the maximum impact force, v denotes the object velocity, $\hat{k}$ is the effective contact stiffness, and $m_l$ stands for the mass of the object. This equation encapsulates the intricate interplay of velocity, stiffness, and mass, and offers a comprehensive understanding of the dynamic forces involved in impact scenarios. The effectiveness of this formulation lies in its ability to quantify the impact force based on key parameters, thereby providing valuable insights for structural analysis and design in the context of one-DOF systems.

This equation is rooted in foundational works such as the pioneering research by Jones [21], laying the groundwork for understanding the quantitative aspects of the impact forces in structural dynamics.

Expanding upon these foundational studies not only enhances our understanding of impact load quantification but also underscores the interdisciplinary nature of structural engineering, drawing on insights from materials science, physics, and mathematical modelling.

2.1.1. AASHTO (2017)

The AASHTO standard employs the contact stiffness approach to express the impact force on a bridge pier resulting from an FO collision [17]. The impact force (Fi) is determined as the maximum collision force, and is computed based on the deadweight tonnage of the vessel (DWT) in long tons and the vessel velocity (v) in ft/s. The formula for calculating the impact force is as follows.

$$F_i\left(Kips\right)=8.15v\sqrt{DWT} (\mathrm{U}.\mathrm{S}. \mathrm{u}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{s}),$$

(2)

This approach simplifies the impact force calculation and requires only the effective contact stiffness, mass, and velocity of FO. In addition, AASHTO extends the analysis to consider the kinetic impact energy of a floating barge, which is empirically related to the static collision load applied to a bridge in a static analysis procedure [17,22].

2.1.2. NAASRA (1990)

NAASRA (1990) adopted the work energy concept to evaluate the impact force, assuming that the velocity of an object is reduced to zero when impacting a structure. The maximum impact force ($F_{i,max}$) is expressed as follows:

$$F_{i,max}=\frac{m{v}^2}{S}$$

(3)

where m is the mass of the object, v is the velocity of the object, and S is the stopping distance of the object (the distance the object travels from the point of contact with the target until it is completely stopped (u = 0)) [23].

2.1.3. AS5100

Following NAASRA (1990), AS5100 suggests a practical function for determining the total impact force as a work equation, where the force is multiplied by distance. Specifically, for log impact, the force ($F_{Lu}^{\ast }$) exerted by a log is calculated using the kinetic energy formula.

$$F_{Lu}^{\ast }=\frac{0.5m{v}_u^2}{s}$$

(4)

where m is the mass of the log, $v_u$ is the mean flood velocity, and s is the stopping distance. This standard stipulates the design considerations for log impacts on piers or superstructures, assuming a minimum log mass of 2 t (2000 kg) and stopping distances of 150 mm and 75 mm for hollow concrete piers and solid concrete piers, respectively.

In scenarios where fender piles or sheathing are strategically positioned upstream from the pier to absorb impact energy, the standard mandates adjustment of the stopping distance accordingly. The design forces were calculated using the mean water flow velocity at the flood level v_s in serviceability limit state (SLS) or v_u in the ultimate limit state (ULS). It is noteworthy that the forces resulting from the log impact should not be concurrently applied with other water flow forces but should be appropriately considered in conjunction. The impact force of log ($F_{Lu}^{\ast }$) is calculated based on the kinetic energy formula mentioned above, considering mass m and stopping distance s [24].

To reinforce the design against debris forces, the designer is advised to consider a force equivalent to that exerted by a 2-tonne log traveling at stream velocity, arrested within distances of 150 mm for hollow-column-type piers and 75 mm for solid-type concrete piers. Notably, international instances of 3-tonne logs traveling at 10 mph have been reported, and the force exerted when stopped within 75 mm may be estimated from kinetic energy considerations [25,26,27].

3. Research Significance

The objective of this study is to delve into the understudied area of design considerations for bridge piers subjected to impact loading, particularly in the context of flooding. Conventional engineering practices typically employ an equivalent static force to simulate the influence of impact forces, which may not fully capture the dynamic complexities of flooding events. Recent observations have revealed notable changes in the mass and velocity of floating objects (FO) during flooding, underscoring the importance of a more nuanced understanding and consideration of these evolving characteristics. The model can be applied to simulate the impact of logs on piers to reflect common flood conditions.

This study addresses this critical gap by conducting a comprehensive parametric investigation of the forces exerted by FO on bridge piers, with a specific focus on the influential parameters of mass and velocity. The findings of this research not only contribute valuable insights into the dynamic behavior of structures under FO impact during flooding but also offer a comparative analysis with design loads specified in selected standards. By juxtaposing numerical analysis outputs with existing provisions, this study critically evaluated the limitations and potential risks associated with current design practices.

The synthesis of this research with existing knowledge in the field marks a significant advancement, shedding light on the inadequacies of relying solely on static forces and advocating for a more refined and adaptive approach to impact design loads. As the climate-induced intensification of flooding events becomes more pronounced, this study provides a forward-looking framework for engineers and practitioners to enhance the resilience of bridge piers, thereby mitigating risks and ensuring the longevity and safety of the critical infrastructure in flood-prone regions.

4. Time Sensitivity Analysis of Impact on ABAQUS Explicit/Dynamic

4.1. Numerical Simulation of Impact Loading on FE Model of RC Pier

This study employs the finite element method (FEM) to conduct a time sensitivity analysis of the impact on an ABAQUS explicit/dynamic finite element model. Emphasizing the pivotal role of time integration algorithms in the FEM, this study distinguishes between implicit and explicit algorithms, highlighting their treatment of inertia, damping, and stiffness terms in the governing differential equation of motion during impact and subsequent free vibration response [28,29]. Explicit formulations, as exemplified by ABAQUS/Explicit, are ideal for impact analysis, whereas implicit formulations are conventionally used for static and vibrational analyses. This model can be employed to simulate the impact of logs on piers in flood conditions. It is designed to reflect the effects of common flood scenarios.

4.2. Numerical Simulation of Impact Loading on FE Model of RC Pier

In this study, a representative reinforced concrete (RC) U-slab bridge substructure, a pier of the common bridge type in Victoria, served as the prototype. The critical features of the simulated U-slab bridge pier and finite element (FE) model for the impact analysis are shown in Figure 3. The pier was 3 m high with a cross-sectional area of 0.355 m × 0.355 m. The structural composition includes four longitudinal rebars, each with a diameter of 28 mm, complemented by stirrups with a 6-mm diameter and spaced at intervals of 100 mm (refer to Figure 3).

Strategic boundary conditions were set to present more realistic conditions for the model. The bottom of the pier is assumed to be fixed and supported by piles embedded within the soil. By contrast, the top is considered simply supported and connected to the crosshead. The partial axial load originating from the dead load of the superstructure was factored into the numerical simulations. This load was applied to the top of the pier according to the methodology outlined in [17]. These design choices and parameters aim to accurately capture the structural behavior of the U-slab bridge pier under impact conditions, laying the foundation for a comprehensive and insightful analysis. Figure 4 illustrates the FE model of the pier exposed to the impact loading simulated in ABAQUS.

4.3. Concrete Damage Plasticity (CDP) Model

The material model employed in this study was based on the concrete damage plasticity (CDP) model developed by Lubliner et al. [30]. This model incorporates two primary failure modes—compression crushing and tensile cracking—to capture the nonlinear elastic-plastic cracking behavior of concrete [31]. The modulus of elasticity ($E_c$) of concrete, based on a 28-day compressive strength ranging from 21 to 83 MPa, was calculated according to NZS 3106:2006 [32], as outlined in Equation (5).

$$E_c=3320\sqrt{f_c}+6900$$

(5)

In this context, the assumed compressive strength (f_c) was 25 MPa, sourced from the Victorian Road Authority. Additional material properties, including density, Young’s modulus, and Poisson’s ratio, were evaluated, with values of 2.4 × 10⁻⁹ ton/mm³, 23,500 MPa, and 0.15, respectively. Figure 5 and Figure 6 provide insights into the constitutive input and damage parameters of the compressive and tensile strengths of concrete (assuming a compressive strength of 25 MPa). These values were derived from the CDP definitions in ABAQUS 6.14 [6]. The adopted damage parameters, including the delamination angle, eccentricity, $\frac{f_{b0}}{f_{c0}}$, K, and viscosity, were set as 35, 0.1, 1.16, 0.667, and 0.01, respectively. The solid concrete was modelled using linear hexahedron element type C3D8R, with reinforcements fully embedded to prevent sliding within the concrete, facilitated by the *CONSTRAINT-EMBEDE option.

4.4. Impact Modeling Methodology

The constant-velocity approach was adopted to simulate the impact of FO using the *FIELD_VELOCITY and isotropic *POINT_MASS commands in ABAQUS. An FO diameter of 100 mm was used in the simulation. Given that damage to the impactor is not the primary focus of this research, the FO is treated as a rigid body nodal mass. This ensures that the total impact energy of the system is transferred to the structure, resulting in a more conservative structural response. Although a generic FO is modeled, the methodology can be applied to logs, which are often encountered as debris during flooding events. This approach is particularly relevant because of the uncertainties associated with impact loading procedures. A time-dependent impulse function was utilized to model the interaction between the impactor and concrete according to Abbate’s recommendation [33].

It is important to acknowledge certain simplifications that were implemented to balance computational efficiency with the complexity of real-world behaviors. The cohesive cracked plasticity damage model for concrete and the assumption of a fixed foundation are standard in the industry, chosen to efficiently capture the behavior of reinforced concrete under impact loads. To ensure these simplifications did not unduly affect result accuracy, a rigorous mesh sensitivity analysis was performed. The results of this analysis, which are detailed in Section 4.5, underscore our confidence in the model’s ability to provide reliable insights into the dynamic responses of bridge piers to impact loading.

4.5. Calibration and Validation

The effectiveness of the impact modelling methodology was calibrated using an experimental study conducted by Fujikake et al. [34]. This study involved a simply supported specimen, similar to piers in size and geometry, that was subjected to dropped hammer impact loading. A mesh convergence analysis of the FE model was performed, revealing that the simulation results were in good agreement with the experimental results when an element size of 12.5 mm was adopted. Further refinement of the elements reduces computational efficiency without improving the accuracy of the results.

Figure 7 presents a comparison of the three-dimensional FE model and the crack pattern of the RC beam in the current study with the Fujikake K. et al. test. The observed crack propagations in both the methods demonstrated compatibility, thereby supporting the accuracy of the simulated model.

4.6. Validation through Experimental Study

Figure 8 offers insights into the time history of mid-span deflection and induced impact force when the RC beam is exposed to the impact loading of a freely dropped hammer with a drop height of 0.3 m and a mass of 400 kg. The damage is illustrated through damage parameter contours ranging from zero to one. Notably, the analysis revealed a difference of approximately 2.2% in the peak impact force and approximately 2.3% in the maximum deflections between the simulated model and laboratory test outcomes. This analysis serves to validate the simulation technique, material modelling (concrete damage plasticity), and contact modelling of impact adopted in the current FE model, demonstrating a very acceptable correlation with laboratory test outcomes [21].

5. Results and Discussion

This study employs numerical three-dimensional simulations using ABAQUS/Explicit to investigate the structural response of a RC pier subjected to horizontal impact loading from an FO. The explicit solution method was chosen because of the dynamic nature of high-speed impact events, in which inertia is the dominant factor, making it a time-efficient approach (ABAQUS 6.14, 2013) [6]. These results have important implications for understanding the impact of logs on bridge piers.

Parametric studies were conducted to analyze the responses of a bridge pier under varying masses and velocities of the impacting FO. The objective was to understand the influence of FO mass and velocity on the structural response, with a focus on comparing the results with a designated design standard. Conservatively, water flow effects were neglected to reduce the complexity of the analysis, as suggested by [11,35]. The impact velocities ranged from 1 to 7 m/s in 1 m/s increments, and the FO masses were 0.2, 0.4, 0.6, 0.8, 1, 1.2, 1.5, and 2 t.

Under higher FO masses and velocities, the impact intensity is expected to drive the structure beyond the elastic region, promoting a more ductile behavior and resulting in increased damage. A structural performance analysis under impact velocities exceeding 6 m/s revealed pier instability with substantial deformation. Flow velocities above 6 m/s were disregarded, considering that Australia’s maximum water flow velocity rarely reaches this threshold [36]. Crack propagation and concrete crushing were observed at FO speeds below 6 m/s, which influenced structural characteristics such as ductility, stiffness, and stability.

5.1. Crack Propagation and Concrete Crushing; Damage Development

The implications of these findings for structural integrity are significant. The observed crack propagation and concrete crushing phenomena provided valuable insights into the dynamic responses of U-slab bridge piers. Figure 9 shows an in-depth view of the concrete damage contours. This represents the damage development in the simulated structures impacted by FO at different velocities, maintaining a constant FO mass of one ton. High-velocity impact loading induces structural damage and dissipates energy over a small area near the impact point. This observation aligns with findings by Cantwell and Morton’s findings [36]. Tensile cracks were prevalent at lower velocities (up to 3 m/s), whereas at higher velocities, concrete crushing was the primary failure mechanism.

This comprehensive simulation aimed to enhance our understanding of the intricate dynamics associated with impact loading on bridge piers, thereby providing valuable insights for future structural design considerations and improvements.

Following the observations of crack propagation and concrete crushing, it is important to consider the initial modeling parameters that inform these results. The simplification to cylindrical FO shapes and the assumption of rigid material properties are conservative measures to provide a baseline for analyzing impact forces. This approach ensures broad coverage of the forces potentially exerted by various FOs during floods, with an emphasis on the predominant forms such as logs. By parameterizing mass and impact area, our study maintains relevance to practical engineering applications and stays within the conservative bounds of the AS5100 design standard.

5.2. Dynamic Analysis of Impact Forces on Bridge Pier: Comprehensive Insights and Standards Comparison

Impact Force Analysis

Figure 10 presents a detailed time history of the impact forces resulting from the kinetic energy transferred to a bridge pier owing to the collision of an FO. This section explores the responses of the pier to different FO velocities with constant mass. While the masses remained constant, the graph indicated the effectiveness of the FO speed on the maximum impact forces transferred to the structure. A notable difference is observed in the post-peak responses, particularly at higher velocities (3–6 m/s). The dynamic effect of the maximum impact force should be prioritized, and the post-peak behavior in the design procedure can be neglected. The increased impact forces at higher velocities are particularly concerning for logs given their potential for significant kinetic energy.

5.3. Time History of Impact Loads

Figure 11 compares the time history of the impact loads resulting from hitting the FO onto the bridge pier for a series of simulations with a constant velocity and varying masses. Clear differences were observed in the maximum forces and the rate of change was explored in this study. The bouncing-back behavior of FO became more significant with increasing initial hitting velocity, and heavier masses exhibited less post-vibration behavior. Figure 12 illustrates the variations in the maximum impact forces. A significant response in the peak impact forces and large deformations was observed.

5.4. Variation of Maximum Impact Forces

Figure 12 illustrates the variations in the maximum impact forces. A significant response in the peak impact forces and large deformations was observed when the velocity of the object or the impact loading was increased to 7 m/s (Figure 9). The structure collapses at velocities exceeding 6 m/s, which emphasizes a significant impact on structural stability. Considering the impracticality of water flow velocities exceeding 6 or 7 m/s, this discussion focuses on speeds between 1–6 m/s. The observed increase in the impact forces with velocity is critical for evaluating the potential damage from logs impacting bridge piers at high speeds.

5.5. Effect of Squared Velocity

Figure 13 shows the changes in the maximum impact forces versus the squared values of FO velocity for a constant mass. The graphs show a relatively steady pattern of change as the impact velocity increased. A critical role of the 3 m/s velocity was identified in increasing the peak impact forces for masses greater than 400 kg. The rate of increase in the peak impact forces was faster for velocities below 3 m/s and slower at higher speeds (>3 m/s).

5.6. Three-dimensional Analysis of Peak Impact Force

Figure 14 illustrates the variation in the peak impact force with an increase in the velocity and mass of FO in a three-dimensional graph. The peak values were normalized using the mass and square values of the velocity to obtain a dimensionless factor for the peak impact forces (${\phi }_{pi}$) (Figure 15). Higher masses and velocities exhibit a lower coefficient of ${\phi }_{pi}$, suggesting a reverse relationship between the velocity, mass, and design impact force. The introduced equation for the peak impact force is:

$$F_{ip}={\phi }_{pi}m{v}^2$$

(6)

5.7. Comparison with Design Standards: AASHTO vs. AS5100

AS5100 and AASHTO standards are widely acknowledged and implemented in bridge design and assessment. AS5100, an Australian standard, delivers guidelines for the construction and design of pedestrians and road bridges. Conversely, AASHTO, an American standard, establishes design specifications for highway bridges. The choice of these standards is predicated on their pertinence to the geographical context of the study (Victoria, Australia) and extensive coverage of a wide array of bridge design codes and loading conditions.

This section presents a comprehensive comparison of the analytical dynamic peak impact forces obtained in this study with the design loads recommended by Australian (AS5100) and AASHTO standards. The analysis revealed a correlation between the qualitative variables using three approaches. It was observed that the forces based on the Australian standard AS5100 were significantly underestimated. The use of a simplified assumption, treating the FO as a rigid body in the FEM analysis, could explain the comparatively higher values obtained from the analysis, in contrast to the values given by AS5100 and AASHTO. However, further research is necessary to fully understand the influence of the mass and velocity of the impactor under different modelling conditions.

Figure 16 presents a detailed visual comparison of the peak impact forces obtained from this study with the impact load definitions of AS5100 and AASHTO. The results indicate that the impact forces align more closely with AASHTO’s suggested impact load as the speed of impact increases, depending on the mass of the FO. However, there was a significant difference when compared with the AS5100 design impact load. Despite the simplified assumptions made in the analysis, it was evident that AS5100 underestimated the impact forces. Therefore, it is recommended that the standard design considerations related to the impact of FO on bridge structures be re-evaluated and revised.

The disparity evident between AS5100, the numerical analysis, and AASHTO underscores the potential necessity of incorporating a correction factor to reconcile the disparity in standard design parameters. A comparative analysis provides a foundation for identifying substantial disparities between the outcomes of parametric studies and design standards. Any recurring patterns of underestimation or overestimation by standards were meticulously recorded. This step is critical in pinpointing areas where prevailing design codes may be inadequate to accurately reflect the dynamic consequences of FO impacts on bridge piers.

5.8. Dynamic Multiplier of Impact Forces (DMIF)

This study introduces a novel dynamic multiplier of impact forces (DMIF) to address the discrepancies between the outcomes of parametric studies and prevailing design standards. DMIF functions as a corrective factor, accounting for nuance dynamic effects that are inadequately addressed by existing standards. The formulation and application of DMIF aims to refine the design codes and enhance the accuracy of the representation of the structural response to the impact of FO on bridge piers.

This study emphasizes the importance of understanding the dynamic effects resulting from FO loading, particularly vertical movement, followed by flooding. The quasi-static definition of the maximum FO loading provided by AS5100 sets the stage for introducing DMIF. The DMIF is defined as the ratio of the suggested impact load to the peak dynamic load for identical mass and velocity parameters, and it is instrumental in gauging the dynamic effects in bridge design procedures. It is particularly useful for refining the impact force calculations for logs, allowing for a more accurate and representative design response.

The analysis conducted in this study explains the variation in the dynamic impact factor based on AS5100 and AASHTO (Figure 17 and Figure 18). The DMIF values relative to the Australian code (AS5100) consistently remained under 70% of the numerically estimated impact forces for varying masses and velocities of FO. In contrast, the DMIF values for the AASHTO design loads exhibited significant variations, ranging from 50% to >200%. These dynamic multipliers serve as essential tools for considering the dynamic impacts in bridge design procedures and offer insights into FO dynamics under different standard considerations. The application of DMIF serves to enhance the accuracy of the representation of the structural response to the impact of FO on bridge piers, as illustrated in Figure 17 and Figure 18.

The DMIF introduced in this research was validated against experimental data to ensure an accurate prediction of structural responses to impact loading. The DMIF was designed to enhance both the precision and computational efficiency of impact force assessment. Comparative analysis with the standards from both Australian and US codes indicates that the method developed herein accurately models real-world scenarios while requiring less computational effort. This efficiency is a significant advancement over traditional methods, particularly in engineering practices where accuracy and computational speed are crucial.

5.9. DMIF Formulation

The DMIF was formulated as a corrective factor to be applied to the impact forces specified in the design standards. This factor was derived using the observed relationships between FO mass, velocity, and impact forces obtained through a parametric study. This formulation ensures that the DMIF accounts for dynamic nuances overlooked by existing standards, thus providing a more accurate representation of the structural response.

The DMIF is defined as:

DMIF = f(m,v)

where DMIF is the dynamic multiplier of impact forces, m is the mass of the FO, v is the velocity of the FO, and f(m,v) represents the dynamic relationship between the FO mass, velocity, and impact forces. To derive the equations, a regression analysis was conducted on the given datasets, each of which contained pairs of velocity values and their corresponding linear coefficients that relate the mass to DMIF. It was observed that the coefficients varied with velocity, suggesting a polynomial relationship. By fitting a quadratic polynomial to these pairs, two distinct formulas were obtained for AS1500 and AASHTO.

The formulas for the DMIF for AS1500 and AASHTO are as follows:

DMIF_AS1500 (m,v) = (0.0081v2 + 0.004v + 0.0054)m

(7)

DMIF_AASHTO (m,v) = (0.002v2 + 0.084v + 0.018)m

(8)

These models enable the prediction of DMIF for any mass and velocity within the observed data range.

5.10. Application of DMIF

The integration of the DMIF into existing design standards such as AS5100 and AASHTO represents a significant advancement in bridge engineering, specifically addressing the dynamic forces on U-slab bridge piers. By factoring in the characteristics of FO and their interaction with the structural response, this study aimed to refine the design codes for enhanced accuracy.

This research is particularly critical for bridges in flood-prone areas, where climate change increases the likelihood of severe weather events, potentially exacerbating their impact on structures. This study underscores the urgency of updating current standards to fortify bridge resilience under these dynamic conditions—a vital consideration in areas where bridges are essential to the infrastructure and face frequent flood-related challenges.

This study underscores the necessity of incorporating dynamic impacts into bridge design and serves as a foundation for further inquiry into structural behavior under dynamic loads. The DMIF’s potential to improve bridge safety, longevity, and adaptability is clear, and future research should extend these efforts to a wider array of bridge designs and materials to enhance overall resilience. The methodology established here, while focusing on U-slab piers, promises broader applicability to diverse structural forms, enriching our understanding of structural dynamics.

6. Conclusions

This paper presents a comprehensive parametric study that investigates the effects of mass and velocity on the impact dynamics of an FO colliding with a U-slab reinforced concrete (RC) bridge pier. Employing the advanced capabilities of the ABAQUS 6.14-1 software, using CPD modelling approach for simulation of the reinforced concrete pier with fixed-bed boundary conditions at the base and a simply supported configuration at the top. A comprehensive mesh sensitivity analysis was conducted to counterbalance potential inaccuracies. An explicit dynamic analysis was carried out to characterize the response of the concrete pier to FO impacts, and the modelling approach was validated against experimental data from the existing literature. While the research focused on FOs in general, the findings are particularly relevant for logs, encouraging further investigation of their impact on bridge pier design.

The core objective of this study was to shed light on the effects of the mass and velocity of an FO on its collision with concrete structures. By examining the outcomes of this study along with the Australian AS5100 and American AASHTO design criteria, several key insights can be gleaned. Specifically, it was revealed that higher impact velocities tended to lead to considerable damage, with the likelihood of structural failure increasing beyond 6 m/s, regardless of the mass of the impactor. Moreover, a nonlinear association was identified between the mass and velocity of the FO and the peak impact forces, thereby calling into question the conventional design approaches.

Our findings suggest that the Australian AS5100 standard significantly underestimates the impact forces experienced by bridge piers, whereas the AASHTO standard provides more accurate predictions at higher FO velocities and masses. These disparities indicate that the current design standards may not adequately account for the nonlinear and differential behaviours of bridge piers subjected to the impact of floating objects.

To bridge the gap between theoretical and practical knowledge, this study introduces the DMIF for both Australian and American standards and proposes an enhancement to the design impact loads. These formulas were developed to adjust the standard-prescribed impact forces to better reflect the dynamic responses observed in real-world scenarios, particularly at higher velocities and FO masses. While this research is focused on a slender U-slab bridge pier, its findings lay the groundwork for broader applications and highlight the need for further investigation to develop a more generalized method.

According to the implications of our findings on the dynamic behavior of bridge piers under impact loading, it is pertinent to consider the practical application of the proposed method. To effectively utilize the dynamic amplification factor in real-world scenarios, detailed information on bridge pier geometry, material properties, and geotechnical conditions is necessary. Engineers would also require data on hydrodynamic conditions during flood events, including flow velocities, water levels, and debris characteristics, as well as historical flood data and records of past bridge damage. These data are invaluable for calibrating the model to reflect actual conditions and for refining the predictive accuracy of the proposed approach.

7. Future Work

Although this study provides a foundational framework for future research, it is imperative to acknowledge certain limitations. It is conceived for simplicity of adoption by professionals, including engineers and bridge inspection teams. The DIF’s user-friendly and robust design anticipates the diverse requirements of modern bridge assessment, streamlining the integration into existing structural health evaluation processes. The study did not delve into the influence of critical environmental factors, such as wind and seismic activity, on bridge piers, limiting the comprehensiveness of the findings. Moreover, the study overlooks the effects of various bridge pier geometries on the dynamic responses to FO impact, hindering a nuanced understanding of the design implications. Future studies should explore the behavior of specific FOs, such as logs, under various environmental conditions. The lack of an in-depth analysis of the dynamic behavior under diverse loading conditions and the absence of exploration of material effects further constrain the scope. Addressing these gaps in future research is vital for refining design methodologies and ensuring the resilience of structures facing dynamic challenges, ultimately advancing the field of structural safety and condition assessment.