Nonlinear Dynamic Finite Element Research Articles

Damage Analysis of High-Speed Railway Bridge-Track System with Unequal Pier Heights Under Near-Fault Earthquake Based on FK-FE Method

ABSTRACT Canyons have a significant influence on near-fault earthquakes, and near-fault earthquakes exhibit strong amplitude and large spatial variability. In turn, this causes the damage and deformation of the high-speed railway (HSR) bridge-track system with unequal pier heights, which affects the normal service of the unequal pier heights bridge and the safe running of the train. This study investigates the damage and deformation of HSR bridge-track systems with unequal pier height in canyon areas under near-fault seismic based on FK (frequency wavenumber)-FE (finite element) method. It aims to offer suggestions for seismic design in canyon regions of HSR unequal pier height bridges. First, a finite fault kinematic hybrid source model and a canyon site model were established, and the FK-FE hybrid method was employed to simulate the three-dimensional (3D) site motion considering the full process of the seismic source-propagation medium-complex site (SSPC). Then, the influence of near-fault earthquakes and the canyon topography on the spatial distribution of near-fault ground motions were analyzed, and the non-uniform near-fault seismic excitation at the bottom of each pier considering the canyon topography effect was obtained. Finally, a nonlinear dynamic FE model of the HSR bridge-track system was established, and its damage patterns and mechanism in canyon near-fault earthquakes were analyzed. The results indicate that near-fault earthquakes in the canyon exhibit significant vertical and sliding effects with significant spatial variability and the FK-FE method can provide more accurate ground motion input. The seismic damage to HSR bridge-track systems increases as the fault distance decreases, and topography effects further aggravates seismic damage to HSR bridges. Moreover, vulnerable areas in the track system are often found at girder ends. Therefore, for HSR bridges in canyon regions, seismic input needs to comprehensively consider the effects of near-fault earthquakes and topography, with post-earthquake focus on damage assessment at girder ends.

Seismic response of deep soft soil sites with varying shear wave velocities

The variations in seismic response between deep soft soil sites with different shear wave velocities were not fully understood. This study focuses on the seismic response of deep soft soil sites in the lower reaches of the Yangtze River, China. A nonlinear dynamic finite element model was developed for two representative deep soft soil sites with borehole profiles and the shear wave velocity tested by the single borehole method. Two nonlinear cyclic constitutive models are used and thus compared through the site seismic response. To accurately calibrate the nonlinear cyclic model parameters, resonant column tests were conducted on 21 soil samples collected from the two boreholes. The results show that the peak ground acceleration (PGA) under low-frequency (Liuan) input motion was higher for soft soil sites compared to that under medium- and high-frequency (Kobe and Nahanni) input motions. The PGA amplification factor for deep soft soil sites under different input motions can be approximated by an exponential function. The peak ground acceleration tends to be lower as the equivalent shear wave velocity (Vse) decreases. The shapes of the spectral acceleration were similar for the two sites, despite a substantial difference in the Vse between them. Additionally, a crossover point was observed in the spectral acceleration for the two sites. The period corresponding to this crossover point increased with increasing intensity of input motions, indicating that the sites became softer with higher intensity and thus generally exhibited a longer characteristic period of the spectral acceleration. This paper also highlights the significance of selecting nonlinear constitutive models and the precise calibration of model parameters in the seismic response analysis of deep soft soil sites, providing a scientific basis for future similar site analyses.

Nonlinear dynamic finite element analysis of vehicle impacts into road restraint systems

Despite that the safety of cars is continuously increasing, improving the safety of road transportation remains a key issue of modern society. The design of suitable road restraint systems plays an important role. As the tests of barriers are technically demanding and expensive, and road traffic intensity is continuously growing, it is crucial to develop advanced models that would make it possible to explore various scenarios under which road vehicles may accidently impact into the road restraint barrier. The submitted study focuses on the use of nonlinear dynamic finite element analysis (NLDFEA) in the reliability assessment of road restraint systems, with a particular focus on steel barriers subjected to impacts by heavy vehicles. The outcomes of theoretical analyses are compared with the results in a compiled unique database of full-scale road barrier impact test results. It appears that a good match between crash test results and the validated NLDFEA model can be achieved, differences being mostly within a range of 10 %. When modelling the crashes of buses and HGVs, it is particularly important to model correctly shape and geometry of the front and impacting side of the vehicle, tyre sizes, wheelbase and overall truck dimensions, and position of the centre of gravity of the vehicle including trailer and cargo. Recommendations for subsequent research studies and for practical applications are finally provided.

Multi-Load Topology Optimization Design for the Structural Safety Maintenance of Low- and Intermediate-Level Radioactive Waste Packaging Containers in the Case of a Collision.

This paper presents an optimized design approach using nonlinear dynamic analysis and finite element methods to ensure the structural integrity of square-shaped containers made from ductile cast iron for intermediate- and low-level radioactive waste packaging. Ductile cast iron, with its spherical graphite structure, effectively distributes stress throughout the material, leading to a storage capacity increase of approximately 18%. Considering the critical need for containers that maintain integrity under extreme conditions like earthquakes, the design focuses on mitigating stress concentrations at the corners of square structures. Nonlinear dynamic analyses were conducted in five drop directions: three specified by ASTM-D5276 standards and two additional directions to account for different load patterns. Fractures were observed in four out of the five scenarios. For each direction where fractures occurred, equivalent loads causing similar displacement fields were applied to linear static models, which were then used for multi-load topology optimization. Three optimized models were derived, each increasing the volume by 1.4% to 1.6% compared to the original model, and the design that best met the structural integrity requirements during drop scenarios was selected. To further enhance the optimization process, weights were assigned to different load conditions based on numerical analysis results, balancing the impact of maximum stress, average stress, and plastic deformation energy. The final model, with its increased storage capacity and enhanced structural integrity, offers a practical solution for radioactive waste management, overcoming limitations in previous designs by effectively addressing complex load conditions.

Analysis of transient energy conversion performance improvement of thermoelectric generator with penetrating crack under pulsed heat source

To simulate the performance of a thermoelectric generator (TEG) in practical applications more accurately and comprehensively analyse the influence of transient heat source and crack on such devices, it is necessary to examine the transient performance of TEGs with cracks under periodic pulsed heat source. In this study, a three-dimensional nonlinear dynamic finite element model with thermal, electrical, and mechanical coupling was established, and the laws describing the macroscopic electrical energy output and thermal stress distribution were derived. The position of the maximum thermal stress changed as the crack size varied. When the crack field axis was increased from 0.2 mm to 1.8 mm, the total output energy increased by 14.4 %, whereas the thermal stress increased to 447 MPa. The total energy growth rate under pulsed heat source conditions reached 52.3 % (from 62.71 J to 95.52 J) compared with that of the steady-state heat source. However, the maximum thermal stress increased by 173.6 % (from 91 MPa to 249 MPa). The results show that the crack and pulsed heat source can effectively improve the electrical output performance of TEGs; however, their mechanical properties deteriorated. This study is useful in terms of predicting the performance of TEGs in practical applications more accurately.

Construction of multi-component finite element model to predict biomechanical behaviour of breasts during running and quantification of the stiffness impact of internal structure

This study aims to investigate the biomechanical behaviour and the stiffness impact of the breast internal components during running. To achieve this, a novel nonlinear multi-component dynamic finite element method (FEM) has been established, which uses experimental data obtained via 4D scanning technology and a motion capture system. The data are used to construct a geometric model that comprises the rigid body, layers of soft tissues, skin, pectoralis major muscle, fat, ligaments and glandular tissues. The traditional point-to-point method has a relative mean absolute error of less than 7.92% while the latest surface-to-surface method has an average Euclidean distance (d) of 7.05 mm, validating the simulated results. After simulating the motion of the different components of the breasts, the displacement analysis confirms that when the motion reaches the moment of largest displacement, the displacement of the breast components is proportional to their distance from the chest wall. A biomechanical analysis indicates that the stress sustained by the breast components in ascending order is the glandular tissues, pectoralis major muscle, adipose tissues, and ligaments. The ligaments provide the primary support during motion, followed by the pectoralis major muscle. In addition, specific stress points of the breast components are identified. The stiffness impact experiment indicates that compared with ligaments, the change of glandular tissue stiffness had a slightly more obvious effect on the breast surface. The findings serve as a valuable reference for the medical field and sports bra industry to enhance breast protection during motion.

Non-bracket oblique traction-hoisting construction strategy for cable-truss structures

This study describes the non-bracket oblique traction-hoisting construction strategy for cable-truss structures, which is to assemble the upper and lower radial cables, hoop cables, sling cables, and compression rods without stress at a low altitude, then hoist the cable-strut system to a high altitude by oblique traction of the upper radial cables through the jack fixed on the upper radial anchorage nodes, and finally actively tension the lower radial cables to achieve the designed shape and prestress level of the entire structure. This strategy assembles at a low altitude, requires simple operations, results in high tensioning efficiency, and does not require brackets, which could guarantee both quality and quantity in terms of completing the construction of cable-truss structures. The semilune-shaped canopy of Yueqing Stadium is constructed using this strategy. The construction simulation and disturbance stability analyses of the structure in the traction-hoisting state and prestress tensioning state are conducted using a nonlinear dynamic finite element method. In the traction-hoisting stage, the deformation changes sharply, and the hoop cables and upper radial cables make up the primary bearing substructure, while the lower radial cables are in a suspended hanging state. In the forming process, the forces of the radial and hoop cables increase gradually, and the structure finally reaches the designed state. For cable-trusses with crossed upper and lower radial cables, the additional stabilizing tooling ropes should be tied at the top of the middle rods to ensure geometric stability because they are susceptible to excessive out-of-plane displacement or even overturning, which is the least desirable at the beginning of traction hoisting.

Seismic performance of shared suction caisson anchors installed in liquefiable sand for floating offshore wind turbines

Shared anchors will reduce the levelized cost of energy for floating offshore wind (FOW) arrays currently under active development in various seismically active regions. This study investigates the seismic performance of a shared suction caisson anchor installed in liquefiable sand for a FOW array using a nonlinear dynamic 3D finite element analysis. The advanced UBC3D-PLM constitutive model was calibrated against cyclic element tests then validated with centrifuge experimental data. Strong ground motions with orthogonal horizontal and vertical components were applied to the model with varying frequency content, peak ground acceleration (PGA), and orientation. The model captured the stress–strain response of the soil and evolution of excess pore pressure to the onset of liquefaction, which occurred after application of a PGA of 0.2 g and above. Despite significant liquefaction caused by a PGA of 0.8 g, displacement of the anchor remained within around 10% of anchor diameter. The results suggest some resilience to pull-out during seismic loading in liquefiable soils, even with vertical tensile loading on the anchor developed by the inverse catenary. Further investigation could be carried out under specific site conditions and serviceability criteria to optimize the size of the caisson and reduce costs while ensuring anchor performance during seismic loading.

Investigation of the displacement-based seismic performance of geogrid earth-retaining walls using three-dimensional finite element modeling

This study evaluates the earthquake-induced movement of geogrid earth-retaining (GER) walls. A thorough investigation was conducted on a GER wall model, utilizing a comprehensive finite element (FE) analysis. This research focuses on investigating and designing hollow prefabricated concrete panels and conventional gravity-type stone masonry GER walls. It also displays comparative studies such as the displacement of the wall, deflection of the wall, lateral pressure of the wall, settlement of the backfill reinforcement, vertical pressure of the backfill, lateral pressure of the backfill, vertical settlement of the foundation, and settlements of soil layers across the height and acceleration of the walls of the GER walls. The FE simulations used a three-dimensional (3D) nonlinear dynamic FE model of full-scale GER walls. The seismic performance of models has also been examined in terms of wall height. It was found that the seismic motion significantly impacts the height of the GER walls. In addition, the validity of the proposed study model was assessed by comparing it to the conventional reinforcement concrete and gravity-type GRE wall and ASSHTO guidelines using finite element (FE) simulation results. Based on the findings, the hollow prefabricated concrete panels were the most practical alternative due to their lower deflection and displacement. Based on the observation, it was also found that the hollow prefabricated GER wall is the most viable option, as the settlement and lateral pressure in the former type are high.

CAD-CAM resin composites: Effective components for further development

ObjectiveThis paper summarizes the effective components of computer-aided design and computer-aided manufacturing (CAD-CAM) resin composites that contribute to achieving greater mechanical properties and further development. MethodsIn silico multi-scale analysis, in silico nonlinear dynamic finite element analysis (FEA), and artificial intelligence (AI) were used to explore the effective components of CAD-CAM resin composites. The effects of the filler diameter and silane coupling ratio on the mechanical properties of CAD-CAM resin composites have been clarified through multi-scale analysis. The effects of the filler contents, and filler and monomer compositions have been investigated by AI algorithms. The fracture behavior of CAD-CAM composite crown was analyzed using in silico non-linear dynamic FEA. The longevity of CAD-CAM composite crown was assessed through step-stress accelerating life testing (SSALT). ResultsAs the filler diameter decreases, there is an increase in elastic moduli and compressive strengths at the macroscale. At the nanoscale, a decrease in the filler diameter results in a decrease in the maximum value of the maximum principal strain. When the silane coupling ratio decreases, there is a decrease in the elastic modulus and compressive strength. According to the exhaustive search and feature importance analysis based on the AI algorithm, the combination of certain components was narrowed down to achieve a flexural strength of 269.5 MPa. The in silico non-linear FEA successfully detected the sign of the initial crack of the CAD-CAM composite molar crown. The SSALT revealed that CAD-CAM resin composite molar crowns containing nanofillers with a high fraction of resin matrix demonstrated great longevity. SignificanceThis paper summarized the effective components of CAD-CAM resin composites for their further development. The integration of in vitro and in silico approaches will expedite the advancement of CAD-CAM resin composites, offering benefits such as time efficiency and reduction of material waste for researchers and manufacturers.

Seismic Response Compression of Various MSE Walls Based on 3D Modeling

This study evaluates the earthquake-induced movement of mechanically stabilized earth (MSE) walls. A thorough investigation was conducted on an MSE wall model, utilizing a comprehensive finite element (FE) analysis. This research focuses on investigating and designing MSE walls made of reinforcement concrete and hollow precast concrete panels. It also involves comparative studies such as on the vertical pressure of the wall, horizontal pressure of the wall, lateral pressure of the wall, settlement of the wall, settlement of the backfill reinforcement, vertical pressure of the backfill, horizontal pressure of the backfill, lateral pressure of the backfill, vertical settlement of the foundation, and settlements of soil layers across the height of the MSE walls. The FE simulations used a three-dimensional (3D) nonlinear dynamic FE model of full-scale MSE walls. The seismic performance of MSE walls has also been examined in terms of wall height. It was found that the seismic motion significantly impacts the height of the walls. In addition, the validity of the proposed study model was assessed by comparing it to the reinforcement concrete wall and ASSHTO guidelines using finite element (FE) simulation results. Based on the findings, the hollow prefabricated MSE wall was the most practical alternative due to its lower displacement and settlement. The specifics of the modeling approach used in this study and the lessons learned serve as benchmarks for future comparable lines of inquiry and practitioners, especially as the computational power of desktop computers continues to rise.

Experimental and numerical investigation of cold-formed steel telescopic studs under compression

This paper presents a thorough experimental and numerical investigation into the structural behaviour of a novel cold-formed steel (CFS) telescopic stud system under axial compression. A total of 40 specimens were tested, considering variations in screw configurations, steel grades, and section thicknesses. The test results revealed that tilting and bearing led to failure in all screw connections. In comparison to the single-screw connection with a gap at the middle of the web, the telescopic studs with 2 screws in the web, 2 screws in the flanges, and 4 screws in both the web and flanges showed approximately 35–37% higher strengths. A nonlinear explicit dynamic finite element model was then developed and validated against the test results in terms of its connection strengths and failure modes. The experimental results were compared with the design strengths determined by the current design standards (AISI S100 (2016), AS/NZS 4600 (2018), Eurocode 3 (2006), BS5950–5 (1998) and GB50018 (2002).). The comparison results revealed that the AISI S100 (2016) and AS/NZS 4600 (2018) are conservative, underestimating the connection strengths of telescopic studs with different screw layouts by 1–13%.

Comparative study of shell element formulations as NLFE parameters to forecast structural crashworthiness

Abstract This work made a comparison of the effects of selected element formulations (EFs) through nonlinear finite element analysis (NLFEA) and physical configurations in scenario design, particularly target locations. The combined results help in quantifying structural performance, focusing on crashworthiness criteria. The analysis involves nonlinear dynamic finite element methods, using an explicit approach applied to an idealized system. This system models ship-to-ship collisions, specifically the interaction between Ro and Ro and cargo reefer vessels, with one striking the other. Summarizing initial NLFEA results reveals that the chosen EF significantly influences the crashworthiness criteria. Notably, differences in formulations lead to different calculation times. The Belytschko–Tsay (BT) EF is the quickest, followed by the Belytschko–Leviathan (BL), with around a 36% difference. Conversely, formulations such as the Hughes–Liu involve much longer processing times, more than twice that of BT. To address the potential impact of shear locking and hourglassing on calculation accuracy during impact, the fully integrated (FI) version of the EF is used. It mitigates these undesired events. For formulations with the same approach, the FI BT formulation suppresses hourglassing effectively, unlike others that show orthogonal hourglassing increments. To ensure reliability, rules were set to assess hourglassing. The criterion is that the ratio of hourglass energy to internal energy should be ≤10%. All formulations meet this criterion and are suitable as geometric models in NLFEA. Regarding reliability and processing time, analyzing the computation time offers insights. Based on calculations, BL is the fastest, followed by Belytschko–Wong–Chiang, while the FI BT formulation takes more time for the same collision case.

Fire‐Induced Collapse of Automated Rack‐Supported Warehouses

AbstractAutomated self‐supporting rack warehouses are becoming increasingly common in the international logistics sector. Their static and seismic design has been studied in the literature; however, the fire design still needs to be improved. Some Standards, such as the Eurocodes or Italian Fire Code, accept the collapse of rack warehouses during fires but require avoiding a progressive collapse or obtaining an implosive collapse regardless of any other active or passive fire protection systems. A hierarchy that guides the kinematics of collapse needs to be defined appropriately but without any Code going into further details. Moreover, the scientific literature focuses more on optimizing fire protection systems for racks, such as sprinklers, rather than preventing fire‐induced progressive collapse. Here, a case study is presented, which, using finite element modelling, provide a first robustness evaluation and highlights some possible aspects to be considered in the structural design to avoid a progressive collapse in the event of a fire. The analyses are performed using the LOCAFI method for localized fires and nonlinear dynamic finite element simulations.

Three-dimensional Non-linear Finite Element Analysis of Impact Response and Damage in Laminated Composite Shells

The impact response and the impact-induced damage in curved composite laminates subjected to transverse impact by a metallic impactor are studied using three-dimensional non-linear transient dynamic finite element formulation. A layered version of isoparametric eight-noded hexahedral element with incompatible modes is developed which incorporates geometrical non-linearity based on total Langragian approach. The non-linearity of both strain displacement relation and contact loading are simultaneously solved using Newton-Raphson incrementaliterative method. Impact-induced damages (matrix cracking and delamination) are predicted using appropriate three-dimensional stress-based failure criteria. Some example problems of graphite/epoxy laminated cylindrical shells with variation of important parameters such as impactor velocity, shell curvature, laminate dimension and fibre orientation of plies are solved and the influence of geometrical non-linear effect on both impact response and resulting damages is demonstrated.

Development and Validation of a Dynamic Abdominal Pressure Twin Sensor Finite Element Model

<div>Some anthropomorphic test devices (ATDs) currently being developed are equipped with abdominal pressure twin sensors (APTS) for the assessment of abdominal injuries and as an indicator of the occurrence of the submarining of an occupant during a crash event. The APTS is comprised of a fluid-filled polyurethane elastomeric bladder which is sealed by an aluminum cap with an implanted pressure transducer. It is integrated into ATD abdomens, and fluid pressure is increased due to the abdomen/bladder compression due to interactions with the seatbelt or other structures. In this article, a nonlinear dynamic finite element (FE) model is constructed of an APTS using LS-PrePost and converted to the LS-Dyna solver input format. The polyurethane bladder and the internal fluid are represented with viscoelastic and isotropic hypoelastic material models, respectively. The aluminum cap was considered a rigid part since it is significantly stiffer than the bladder and the fluid. To characterize the APTS, dynamic compression tests were conducted on a servo-hydraulic load frame under displacement control and held at the peak compression to allow for stress relaxation prior to slowly releasing the compression amount. The initial peak pressures and loads were 15–17% above the level observed at a 10-second hold period with 50% of the decay occurring within 300 ms. The material properties are identified using an inverse method that minimizes the difference between measured and predicted load and pressure time histories. Further, the bio-fidelity static specifications of the APTS manufacturer are used as a basis to identify the quasi-static material parameters. This approach resulted in a reasonable match between physical test data and model-simulated data for dynamic compressions of 10 mm and 15 mm (~50% compression). Additional compression tests are conducted at two compression levels (5 and 10 mm) and at four load offset configurations for use in the model validation. The FE model was used to predict peak pressure responses within approximately 10% error at full-load capacity and achieved CORA ratings >0.99 for the pressure time history. The proposed inverse method is expected to be generally applicable to the component characterization of other models and sizes of APT sensors.</div>

Deployment analysis of composite thin-walled lenticular tubes with effect of storage time and temperature

Composite Thin-walled Lenticular Tube (CTLT) is increasingly utilized in small satellites missions as a lightweight, foldable, and rollable structural material that facilitates the construction of large deployable systems. The CTLT is initially flattened and coiled around a central hub for storage before launch, during which elastic energy is stored as deformation energy, allowing it to be self-deployed on demand for use in orbit. This work presents a comprehensive investigation into the coiling, storage and deployment behaviors of CTLT that wraps around a central hub. A nonlinear explicit dynamic finite element model was developed with both deformable CTLT and rigid-bodies mechanisms including the central hub and guide rollers, as well as the complex interactions among them. The coiling mechanics characteristics such as stored strain energy and rotational moment were presented and validated against experimental data in the literature. Then, the dynamic deployment behaviors were analyzed in terms of two different deployment methods, namely, controlled deployment and free deployment. The effect of material property change during storage was also discussed through numerical experiments.

Numerical Simulation of Motion Process of a Projectile with Sliding Driving Band in Bore

For the information acquisition of the whole motion of a projectile with sliding driving band in gun bore, the nonlinear dynamic explicit finite element software LS-DYNA was used. The pressure-time curve in bore is equivalent to the boundary input condition of the gun, and the large deformation process of projectile driving band is described by ALE algorithm. The simulation results show that a number of calculated data are in good agreement with the experimental values, providing a reference basis for the design and simulation of similar guns.

Energy Absorption Capacity of Empty and Foam- Filled Concentric Cylindrical Tubes

Axial folding of metal tubes has been renowned over ages as a superb energy-absorbing method. High-volume industrial items like automobiles, trains, and other sectors where energy must be absorbed in a controlled manner during a crush situation are using components based on this concept. From the perspective of passenger automobile safety design, it is crucial and anticipated to investigate crushing energy absorption.
 These are thoroughly investigated experimentally and computationally how aluminium foam-filled sections behave when compressed axially. To represent quasi-static test circumstances, nonlinear dynamic finite element studies are conducted. It is discovered that the predicted crushing force and fold formation are rather similar to the experimental facts. The mean crushing force of the foam-filled sections is calculated using straightforward closed-form methods based on the computational models. It is shown that the increase in the mean crushing force of a full column increases linearly with the cross-sectional area and foam compressive resistance. The proposal, for a variety of column designs, materials, and foam strengths, has been within 8% of the experimental data.
 Ultimately, the results highlight the advantages of using concentric cylindrical tubes to absorb impact energy in circumstances with axial loads. It only understands well how control the absorbed energy by using the geometrical features of such structures. The purpose of this study is to suggest design solutions on how to use concentric cylindrical tubes in energy absorption applications like crash-worthiness.

Stochastic Dynamic Response and Long-Term Settlement Performance of Superstructure–Underground Tunnel–Soil Systems Subjected to Subway-Traffic Excitation

The vibration impact force from the long-term operation of a subway affects the comfort of the residents living above the superstructure and the long-term settlement deformation of the tunnel foundation. A method for evaluating the dynamic responses of superstructure–tunnel systems is important, especially because of the randomness of vibration impact force. The coupling effect of the randomness of train-vibration excitation and the nonlinearity of the geotechnical properties that are subjected to dynamic action leads to challenges in the evaluation of the performance of superstructure–underground tunnel–soil systems under train vibration. In this study, a stochastic dynamic model of subway vibration-load excitation was established; then, the time histories of samples with rich probability characteristics in the same set system were generated. According to the nonlinear dynamic finite element analysis, several nonlinear dynamic responses of the deterministic superstructure–tunnel soil-foundation system were obtained. Finally, the probabilistic performance evolution of the superstructure–tunnel soil-foundation system was obtained by integrating the first-passage and probability density evolution theories, and the long-term deformation performance of the tunnel foundation was evaluated using time-varying reliability. This study presents a novel probabilistic method and a more objective performance index for the dynamic performance assessment of superstructure–underground tunnel–soil systems that are subjected to subway-traffic excitation.