Stiffness Of Bridge Research Articles

Study on Deflection-Span Ratio of Cable-Stayed Suspension Cooperative System with Single-Tower Space Cable

This study uses the wind–vehicle–bridge coupling vibration analysis method to investigate the bridge stiffness problem of a large-span cable-stayed-suspension cooperative system. On the basis of the particle-damping-spring vehicle model, the TMeasy surface contact tire model is introduced, and a set of universal wind–vehicle–bridge coupling analysis algorithm is built in the framework of the whole process iterative method. Based on the Latin supercube sampling principle, random traffic flow is generated and loaded onto bridge structures with different stiffness conditions to analyze the driving comfort and safety under each stiffness condition. Combining the specification requirements, engineering experience, and research results, the vertical stiffness limit applicable to the bridge of the highway cable-stayed-suspension collaborative system is proposed. Existing engineering experience shows that the vertical deflection-to-span ratio of a cable-stayed bridge under live load is distributed between 1/400 and 1/1600, and the vertical deflection span ratio under the action of lane load is recommended based on numerical analysis.

Detect the stiffness transition in beam structures by using the passive tap-scan method

Damage detection of bridges through the response of a passing vehicle has drawn much attention in recent years. Most related researches focus on the impact of the bridge stiffness on vehicle responses. However, this paper demonstrates that the vehicle response is more sensitive to the stiffness transition in bridge girders. The theoretical basis is established on a vehicle-bridge interaction model, in which the change of an effective vehicle stiffness is introduced to represent the change of beam stiffness. The solution to this problem clearly shows that the acceleration of vehicle has an abrupt change at the stiffness transition point due to the discontinuity of the stiffness gradient. This effect will be enhanced when applying a tapping force near the vehicle resonance frequency. This tapping force can be economically generated by coordinating the vehicle velocity and the geometric pattern of toothed wheels, which was called the “passive tap-scan method” in a recent paper. This theory also reveals that the amplitude of the acceleration of vehicle is a quadratic function of the stiffness change ratio. Hence, a novel damage detection method is proposed to give a quantitative evaluation of the damage. These theoretical findings are quantitatively verified by numerical simulations and experimental tests on a standard pre-stressed concrete T-beam. Then, the practical capability of this damage detection method is further demonstrated through a field test on a real bridge.

Asymmetric Arrangement of Crossed Cables in Multitower Cable-Stayed Bridges Based on Structural Stiffness

To increase the structural stiffness of multitower cable-stayed bridges, an analysis theory of crossed cable is developed, and a new-layout scheme of crossed cable for the multitower cable-stayed bridges is presented. The area ratio of the crossed cables at the same anchorage position was derived by examining their restraint stiffness to the bridge tower. Accordingly, the anchorage position of the crossed cable on the main span with a maximum restraint stiffness was examined, and an asymmetrical arrangement of the crossed cables on the main span was proposed. The finite-element models of three- and four-tower cable-stayed bridges were established. The influences of the asymmetrical arrangement of crossed cables on the deformation of the tower and girder, the stress of the bridge tower, and the dynamic characteristics of the bridge were analyzed. The results from the analysis demonstrate that the area ratio of crossed cables is equal to the ratio of the square of the cable length when their restraint stiffness is maximum. When the height–span ratio of the multitower cable-stayed bridge was between 0.2 and 0.3 and the anchorage position of the crossed cable was 0.69–0.73 times the span length from the middle tower, the restraint stiffness of the crossed cables was maximum. Under a uniform live load, the mechanical properties of asymmetrical arrangements were better than those of the symmetrical arrangement of crossed cables. When the crossed cable as adjusted to an asymmetrical arrangement for three- and four-tower cable-stayed bridges, the tower top displacement and tower bending moment were considerably reduced, and the vertical frequency of the bridge was increased. Note that the asymmetrical arrangement could considerably improve the restraint of crossed cables to the middle tower, increase the structural stiffness of the bridge, and improve the stress of the middle tower.

Assessing bridge stiffness decline using power spectral density: A case study on Saigon Bridge

Bridges, especially civil infrastructure, serve a vital role in social activities. A damaged bridge can cause severe consequences such as traffic congestion, economic impacts, and loss of lives. Therefore, it is crucial for infrastructure management communities to detect and assess the condition of damaged bridges to ensure safety. To address this issue, in this study, the authors have put forth a novel set of parameters that can be used to evaluate the reduction in stiffness of spans over time. The stiffness is a critical factor that determines its structural integrity, and it is essential to monitor any changes in this parameter over the lifespan of the bridge. The proposed parameters take into account factors such as the age of the bridge, the material used in its construction, and environmental factors that may contribute to the degradation of the structure. They conducted experiments on a beam structure to evaluate its ability to monitor the reduction in bearing capacity of Saigon Bridge's spans using these parameters. To analyze the vibration signals from a randomized traffic load model, they utilized the power spectral density. The results of the study suggest a correlation between the frequency of harmonics and the high-frequency regions observed in the power spectral density and the decline in the stiffness of spans.

An analytical method for assessing the live load deformation of a suspension bridge with an improved central buckle

Suspension bridges lack favorable supports for global vertical stiffness, making it challenging to meet the road safety requirement due to the excessive deflection of stiffening girder when developing towards larger spans. It hinders the development of the ultimate span of suspension bridges and reduces the utilization rate of materials. This paper proposes an improved central buckle (ICB) to increase the global vertical stiffness of suspension bridges. Without changing the constraints of the stiffening girder, the central buckle and horizontal cable are placed at the mid-span and bottom of the stiffening girder, respectively, which combination is referred to as ICB. It restrains the horizontal displacement of the main cable and reduces the maximum deflection of the suspension bridge due to the asymmetric live loads. The analytical expressions of the deflection and girder-end rotation of the suspension bridge with the ICB are derived based on the deflection theory. The expression of the variable equivalent horizontal spring stiffness of the ICB is proposed and verified by a finite element model of the suspension bridge. These results prove that the ICB can effectively restrain the horizontal displacement of the main cable caused by the asymmetric live load and significantly reduce the vertical deflection of the stiffening girder. The performed parametric analysis clarified the mechanism of increasing the vertical stiffness of the suspension bridge based on the ICB force transmission path, providing a new design scheme for improving the global vertical stiffness of the suspension bridge and expanding its upper span limit.

Dead Load Allocation in Triple-Cable Suspension Bridges by Adjusting the Unstrained Lengths of Hangers: A Novel Analytical Approach

Triple-cable design of suspension bridges with an ultrawide main girder can improve their mechanical properties, making it lucrative for numerous engineering scenarios. However, its realization requires assigning an appropriate share of the dead load of the main girder supported by three main cables to avoid the middle main cable overload by the main girder’s dead load since the latter would impair the mechanical behavior of the upper transverse beams in the towers and deteriorate the whole bridge’s torsional stiffness. Therefore, this study proposes an analytical approach for dead load allocation to the suspension bridge with three main cables, which adjusts the share of the main girder’s dead load supported by three main cables via changing the unstrained lengths of the hangers. The main advantage of the proposed approach is that it can consider both uniform and nonuniform allocations. The spatial model for the suspension bridge with three cable planes is first converted into a plane model. The relationship between unstrained lengths of the three hangers in the same cross section is determined via the deformation compatibility conditions and one of the following options: (1) equality of stiffness of the two side hangers in the case of a symmetrical plane model; or (2) energy conservation conditions in case of an asymmetrical one. The proposed method was applied to an example of a triple-cable suspension bridge with a main span of 2,100 m and a width of 75 m, and its analytical solutions were compared against finite-element method calculation results. An increase in the share of the main girder’s dead load supported by the two side main cables increased the torsional stiffness of the whole bridge and reduced the vertical load exerted by the middle main cable on the top transverse beam of each tower. Meanwhile, variation of the share of the main girder’s dead load supported by the side and middle main cables only slightly influenced the whole bridge’s vertical and lateral stiffness values. The aforementioned adjustment by the proposed method could be accomplished without increasing the steel consumption of the bridge cables. The results could provide references for the design, static, and dynamic response analysis of similar triple-cable ultrawide suspension bridges.

Feasibility of Applying Mel-Frequency Cepstral Coefficients in a Drive-by Damage Detection Methodology for High-Speed Railway Bridges

In this paper, a drive-by damage detection methodology for high-speed railway (HSR) bridges is addressed, to appraise the application of Mel-frequency cepstral coefficients (MFCC) to extract the Damage Index (DI). A finite element (FEM) 2D VTBI model that incorporates the train, ballasted track and bridge behavior is presented. The formulation includes track irregularities and a damaged condition induced in a specified structure region. The feasibility of applying cepstrum analysis components to the indirect damage detection in HSR by on-board sensors is evaluated by numerical simulations, in which dynamic analyses are performed through a code implemented in MATLAB. Different damage scenarios are simulated, as well as external excitations such as measurement noises and different levels of track irregularities. The results show that MFCC-based DI are highly sensitive regarding damage detection, and robust to the noise. Bridge stiffness can be recognized satisfactorily at high speeds and under different levels of track irregularities. Moreover, the magnitude of DI extracted from MFCC is related to the relative severity of the damage. The results presented in this study should be seen as a first attempt to link cepstrum-based features in an HSR drive-by damage detection approach.

An effective prediction method for bridge long-gauge strain under moving trainloads with experimental verification

This paper presents a dynamic long-gauge strain prediction method for high-speed railway bridges when fiber Bragg grating (FBG) sensors are used. First, a refined three-dimensional numerical model of a rail-track bridge is established to calculate long-gauge strain response under moving trainloads. Then, by comparing the experiment measured and predicted acceleration histories of the bridge, the correctness of the prediction method for the dynamic response of the high-speed railway bridge is verified. A novel approach has been put forward to calculate the long-gauge strain, which can quickly obtain the dynamic long-gauge strain of high-speed railway bridges without knowing the height of the neutral axis in the section covered by the sensor. Finally, comparative field tests were conducted to verify the accuracy of the dynamic long-gauge strain prediction approach, and a health monitoring system based on distributed long-gauge strain sensing technology was installed on a high-speed railway bridge. The comparison results show that the dynamic long-gauge strain of high-speed railway bridges can be effectively predicted in the FE model. Furthermore, an envelope curve composed of long-gauge strain peak values was used for evaluating bridge stiffness degradation, and the prediction results show that the high-speed railway bridges' local and global stiffness degradation can be reflected in long-gauge strain time history. The predicted method of long-gauge dynamic strain under train load described may attract engineers who use FBG sensors to monitor the health of high-speed railway bridges.

Study on Dimensionless Mid-tower Stiffness of Three-Tower Self-Anchored Suspension Bridges Based on an Analytical Approach

Different from the earth-anchored suspension bridges, both the main cable and the stiffening girder significantly contribute to the stiffness of self-anchored suspension bridges. Meanwhile, the mid-tower effect is also non-negligible when extending the traditional self-anchored suspension bridges to the multi-tower system. This study derives a load distribution coefficient to include the coupling effect between the main cable and girder based on displacement compatibility conditions. The load distribution coefficient determines the live-load deflection of the girder and the cable forces on both sides of the mid-tower, corresponding to the lower and upper bound of the mid-tower stiffness. This study also proposed an analytical approach to determine the feasible dimensionless range of the mid-tower stiffness for three-tower self-anchored suspension bridges. The nonlinear finite element method is utilized to validate the accuracy of the analytical model, and a parametric analysis is carried out to examine the effects of different parameters, including the live-load ratio suffered by the main cable, the geometric and loading parameters, the cable-girder connections, and the tower-girder connections, on the range of dimensionless mid-tower stiffness. The results demonstrate that the feasible stiffness range of mid-tower depends on more parameters, including the live-load ratio supported by the main cable, the dead-to-live load ratio, the sag-to-span ratio, and the cable-girder connections, compared with three-tower earth-anchored suspension bridges.

In-situ testing and finite element model updating of a long-span cable-stayed bridge with ballastless track

Ballastless track is widely used in high-speed railways due to its long service life, low maintenance costs, and excellent riding comfort. However, laying a ballastless track on a long-span bridge remains a challenge because of the large deflection and uneven stiffness of the long-span bridge. Limited research has been reported on the experimental behavior of the long-span railway bridge laid with the ballastless track. This study presents the in-situ testing and the finite element model updating (FEMU) of the Ganjiang Bridge, China’s first long-span ballastless railway cable-stayed bridge with a main span that reaches up to 300 m before its opening for operation. The study’s primary purposes include evaluating the mechanical properties of the ballastless railway bridge and establishing a baseline finite element model for subsequent structural health monitoring. The general test plan, sensor placement, and measured responses are described. The static deflection, the stress of the main girder, and the rotation angle at the beam end are used to evaluate the static performance of the bridge. The natural frequencies and mode shapes are used to assess the dynamic performance of the bridge. A FEMU framework that utilizes displacement assurance criterion (DAC) and frequency to construct a single-objective function is proposed to update the initial finite element (FE) model of the bridge. The results show that the deflection to span ratio is 1/1016 (less than1/1000), and the rotation angle at the beam end under live load is 0.032 ‰ rad (<1‰), indicating that the bridge has sufficient vertical stiffness and excellent riding comfort. The stress of the main girder is smaller than the material yield strength, indicating that the bridge has sufficient load-bearing capacity. All the requirements allowing the bridge to start service have been met. The updated FE model shows that the DAC performs better than the displacement and provides more accurate updating results. The updated FE model can well reflect the static and dynamic characteristics of the actual bridge and can be used as a baseline FE model for subsequent studies.

Competing mechanism between vertical stiffness and anti-slip safety in double-cable multi-span suspension bridges

Double-cable multi-span suspension bridges (double-cable bridges) have great development prospects in long-span bridge engineering. The vertical stiffness of the bridge and the anti-slip safety between the cable and saddle are the two primary competing factors governing the preliminary design of double-cable bridges. It is essential to study the competing mechanism between these two factors to determine reasonable parameters for designing sustainable double-cable bridges. In this study, the deflection-span ratio and anti-slip safety coefficient were adopted as indices to evaluate the vertical stiffness and anti-slip safety, respectively. Finite-element analysis was performed to investigate the influence of the main design parameters on the relationship between the deflection-span ratio and anti-slip safety coefficient and to clarify the competing mechanism between these two factors. Finally, we propose some recommendations for the preliminary design of double-cable bridges. The results of our study reveal that increasing the bending stiffness of the middle tower, increasing the constant load distribution ratio, or installing vertical friction plates in the top sub-saddle are useful in expanding the application scope of double-cable bridges. This study provides new insights into the competing mechanism between the vertical stiffness and anti-slip safety of double-cable bridges.

Railway Bridge Condition Monitoring Using Numerically Calculated Responses from Batches of Trains

This study introduces a novel method to determine apparent profile of the track and detect railway bridge condition using sensors on in-service trains. The concept uses a type of Inverse Newmark-β integration scheme on data from a batch of trains. In a self-calibration process, an optimization algorithm is used to find vehicle dynamic properties and speed. For bridge health monitoring, the apparent profile of the bridge is first determined, i.e., the true profile plus components of ballast and bridge deflection under the moving train. The apparent profile is used, in turn, to calculate the moving reference deflection influence line, i.e., the deflection due to a moving (static) unit load. The moving reference influence line is shown to be a good indicator of bridge stiffness. This numerical approach is assessed using an elaborate finite element model operated by an independent research group. The results show that the moving reference influence line can be found accurately and that it constitutes an effective indicator of the condition of a bridge.

Analysis of general functional bearing model in a single-span bridge to identify structure response and suitable friction coefficient under near- and far-fault earthquakes

ABSTRACT Analysis of a single-span bridge with rubber bearing as the isolation system is performed under earthquakes. The conventional bridge seismic design requires the whole structure to be perfectly connected to avoid interrupting the transfer of earthquake energy from the ground through the bridge. A bridge with this typical design requires a high-cost construction due to the need for a huge section of the bridge to resist the earthquake force demand. Thus, many bridges in Taiwan are designed with a rubber bearing only put in between the column and girder without an anchor system. Thus, the bridge movement by rubber displacement is permissible, but the sliding displacement must be accommodated to limit the movement. The sliding displacement is the method to exploit the friction force provided by the sliding on the top and bottom interface of the rubber with the girder and column to dissipate the earthquake input energy transmitted to the structure. By involving the role of surface friction, the shear force transmitted to the structure can be reduced and the bridge performance optimized. General Functional Bearing Model (GFBM) analysis is a rubber bearing analysis which unmerges the function of friction and restoring force. In contrast with the conventional method, the rubber bearing designed with GFBM analysis may reduce the bridge stiffness and deck acceleration, and it is more convenient because only sliding displacement needs to be controlled. This research proposed GFBM analysis to simulate the rubber bearing that is reflected in the real conditions of bridges in Taiwan.

Numerical and Experimental Studies of the Use of Fiber-Reinforced Polymers in Long-Span Suspension Bridges

For the construction of transport infrastructure (including pipeline bridges for oil and gas transportation) in the conditions of the Far North, it is necessary to improve modern regulatory and technological base for using the fiber-reinforcing polymers. It is necessary to conduct searching research to determine the conditions and shapes of application of the fiber-reinforced polymer (FRP) in the load-bearing structures of bridges and pipelines through barriers. One such searching research is the study of the use of a suspension hybrid bridge with a superstructure of FRP. For this purpose, the calculations of finite-element models of pedestrian suspension bridges were performed and their aerodynamic stability was investigated on the section models in a wind tunnel. The novelty of the study consists in the proposed additions to the structure of the bridge, and the permissible geometric of the cross-sections of the superstructure were established for ensuring aerodynamic stability. Finally, this was the first time that it was directly established that the strength, stiffness and aerodynamic stability of a suspension hybrid bridge were provided.

Response Surface‐Based Finite Element Model Updating of Steel Box‐Girder Bridges with Concrete Composite Decks

The orthotropic steel deck (OSD) using the steel‐steel fiber reinforced concrete (SFRC) deck can be used in small and medium‐span bridges for its advantages of construction convenience and lower construction cost. The seismic performance of THESE steel box girder bridges in strong earthquake zones needs to be evaluated, and a reasonable finite element model is the important basis for an accurate evaluation of these bridges. Finite element analysis (FEA) and field measurements of the dynamic characteristics of steel box girder bridges were carried out in this study, and the finite element model was updated based on the response surface method. The studies show that the bridge modal parameters can be reasonably identified based on the measurements of the bridge under ambient excitation without artificial excitation; the first six orders of mode shapes in the FEA results are consistent with the measured ones, but there is a difference between the natural frequencies; the model updating found that the elasticity modulus, density, and thickness of the SFRC have some influence on the first six orders of natural frequency of the bridge, but the SFRC thickness has the most significant influence on the bridge frequency, and the influence of epoxy asphalt pavement on bridge frequency is small. The study concluded that the SFRC layer of the bridge deck is no longer the traditional bridge deck pavement without considering the stiffness, but it integrates with the OSD as a steel‐concrete composite structure. Therefore, the structure formed by the SFRC layer through the combination of shear studs and OSD shall be reasonably simulated in the finite element model so as to reasonably reflect the contribution of the SFRC layer in the composite structure of the bridge stiffness. The obtained updated finite element model gives results closer to the measured modalities, which provides a more reasonable bridge model for seismic analysis of bridges located in strong seismic zones and also demonstrates the effectiveness of the finite element analysis and the model updating method based on the response surface method in this study.

Fundamental mode shape estimation and element stiffness evaluation of girder bridges by using passing tractor-trailers

A comprehensive indirect method is proposed in this study to estimate the dynamic characteristics of the fundamental mode and evaluate the element stiffness of a girder bridge using a passing tractor with two identical trailers. The novelty of this study is the use of the dynamic interaction between the two identical trailers and the bridge rather than that between the truck and the bridge. One of the advantages of this is that the dynamic characteristics of trailers are easier to model and analyse. Under the proposed method, the residual acceleration of the two trailers is first used to eliminate the adverse effects of road surface roughness. The fundamental natural frequency of the bridge is obtained from the Fourier transform of the resulting signal. Next, the component corresponding to this fundamental frequency was extracted using bandpass filtering. The corresponding damping ratio and mode shape can be reconstructed through a short-time Fourier transform (STFT) under the assumption that the fundamental mode shape amplitude peak is located at the midspan. Finally, the element stiffness of the bridge was evaluated using the reconstructed fundamental mode shape. In this study, the proposed method is investigated via numerical examples and performs well in different scenarios, for example, in the presence of high-level measurement noise and low-level road surface roughness. Furthermore, the method was validated through field testing on the Li-Zi-Wan bridge in Chongqing. The extracted fundamental mode shape agrees well with that obtained using the stochastic subspace identification (SSI) method. The evaluated element stiffness matches very well with both the original design and the static deflection test results.

Dynamic Characteristics of a Double-Pylon Cable-Stayed Bridge with Steel Truss Girder and Single-Cable Plane

The dynamic characteristics are closely linked to the seismic stability and wind-resistant of the bridge. But different bridge types have different dynamic characteristics. In order to study the dynamic characteristics of a double-pylon cable-stayed bridge with a single-cable plane and steel truss girder whose main span is the longest in the world, the dynamic load test was done, and the finite element and the subspace iteration methods were used to analyze the vibration mode of the bridge. The influence of different structural parameters on the dynamic characteristics of the bridge was analyzed. The changed structural parameters are cable layout, stiffness of steel truss girder, stiffness of stayed cables, stiffness of pylons, the concentration of dead load, number and location of auxiliary piers, and structural system. The results show that the bending and torsion resistance of the double-pylon cable-stayed bridge with a single-cable plane and steel truss girder is weak. The torsional stiffness of the cable-stayed bridge with a double-cable plane is stronger than that of the cable-stayed bridge with a single-cable plane. The seismic stability and wind-resistant of the bridge can be improved by using light dead load, improving the stiffness of pylon and girder, and adding auxiliary piers scientifically. However, the change of cable stiffness has a complex influence on the dynamic characteristics of the bridge. The conclusion can offer references for the construction, maintenance, and design of the same type of bridges.

Theoretical study on the seismic response of a continuous beam bridge with safe-belt devices

To study the seismic responses of a continuous beam bridge with safe-belt devices, theoretical formulas were derived (i.e., natural period, beam-end displacement, shear force at the bottom of the fixed pier). The effectiveness of these formulas was then verified by comparing the theoretical results with finite element results. Finally, the influence of the continuous beam bridge's total lateral stiffness and of the safe-belt device's installation position on the aseismic rates was analyzed. The results showed that the simplified model of a continuous beam bridge with safe-belt devices is effective, and the proposed theoretical formulas can be used for the seismic design of this type of bridges. The total lateral stiffness of continuous beam bridges and the installation position of the safe-belt device have both a great influence on the aseismic rates, which should be optimized to obtain a good aseismic effect. When the stiffness ratio is greater than 7.0 and safe-belt devices are installed on the top of all sliding piers in continuous bridges, safe-belt devices have a good aseismic effect. Furthermore, the natural period of continuous beam bridges with safe-belt devices should not be similar to the characteristic period of the response spectrum after the activation of the safe-belts.

Analysis and Comparison on Dynamic Characteristics of CFST Arch Bridge

In order to obtain the natural frequency and vibration mode of concrete-filled steel tube (CFST) arch bridge with different parameters, the modal analysis of a double-span arch bridge with a total length of 257.4 meters and a full width of 31.8 meters is carried out. By controlling the three variables of concrete strength, steel tube thickness and the number of transverse braces in arch ribs, the regularity of the natural frequency and vibration mode of CFST arch bridge with different variables is discussed. The results show that with the increasing of the strength of concrete-filled steel tube in arch ribs, the fundamental frequency increases gradually. The vertical stiffness of the arch bridge is positively related to the thickness of the arch rib steel pipe. The fundamental frequency of the arch bridge can be improved by setting transverse brace, and the change of the number of transverse braces has a significant effect on the sixth-order natural frequency. It is limited for the arch bridge to improve the stiffness by changing the number of transverse braces. These can lay the foundation for the dynamic response analysis of the CFST arch bridge.

Stiffness Estimation of Girder Bridges Using Influence Lines Identified from Vehicle-Induced Structural Responses

The influence line is an important static property in bridges. The shape and magnitude of the girder bridge influence line contain stiffness information for the bridge supports and beam. This research sought to directly identify the rotational stiffness of bridge supports and the flexural stiffness of beams from extracted influence lines, which provides a quantification method for bridge damage and performance degradation. The paper first reviews the influence line identification method from vehicle-induced bridge responses. Then, the complete theoretical formulations of the deflection, rotation, and strain influence line for a general girder bridge are derived. Based on these derivations, the rotational stiffness at two supports and the equivalent flexural stiffness value of the beam are identified. Next, a certain span of a continuous girder bridge is converted to an equivalent simply supported system, and the flexural stiffness distribution of the beam is estimated based on the bending moment-curvature relationship and Betti’s law. The paper then presents a three-span continuous girder bridge verification example. The stiffness estimation results under different test conditions combined with the bridge influence line identification algorithm are presented to verify the reliability of the proposed method. This stiffness estimation method can be applied to baseline-free bridge performance evaluations based on vehicle-induced bridge responses.