A feasibility study was conducted to develop a novel method to determine the temporal changes of tensile forces in bridge suspender cables using time-frequency analysis of ambient vibration measurements. An analytical model of the suspender cables was developed to evaluate the power spectral density (PSD) function of a cable with consideration of cable flexural stiffness. Discrete-time, short-time Fourier transform (STFT) was utilized to analyze the recorded acceleration histories in both time and frequency domains. A mathematical convolution of the analytical PSD function and time-frequency data was completed to evaluate changes in cable tension force over time. The method was implemented using acceleration measurements collected from an in-service steel arch bridge with a suspended deck to calculate the temporal variation in cable forces from the vibration measurements. The observations served as proof of concept that the proposed method may be used for cable fatigue life calculations and bridge weigh-in-motion studies.
The long-term viability of critical structural members is a major concern as our nation’s infrastructure ages. Tension members of bridge systems such as suspender cables are susceptible to fatigue under vehicular traffic. Cyclical tensile loading of these members may lead to premature fatigue damage or failure [
Example of a Haigh diagram for fatigue life estimation.
Since the mean stress has a substantial impact on the fatigue life of the member, it is critical to obtain an accurate history of the tension forces in fatigue-prone members such as cables. Suh and Chang conducted an experimental study on the fatigue behavior of wire ropes for hanger cables in suspension bridges and found that the mean stress on the cables had a significant effect on the fatigue performance of the cables [
Determining the mean tensile force in cables through vibration methods has been widely studied [
However, this method may only be used to calculate mean tension force in the cables. Temporal changes in suspender cables under live load have a significant effect on the fatigue life of the member. According to NCHRP Report 538, there is no recognized standard to inspect or evaluate the in situ condition and strength of bridge cables [
Several research projects have been conducted to evaluate the performance of cables used on suspension and cable-stayed bridges. Nakamura and Hosokawa [
The results presented in this paper are the results of a new methodology which was developed as a complementary part of an extensive research project aimed to evaluate the differences of cable tension forces in groups of cables through a series of stationary frequency analyses of cable acceleration data. In this complementary work, the feasibility of implementing a nonconventional, nonstationary frequency analysis was investigated. This paper presents the development and validation of this feasibility study and the formation of a new method to determine the fluctuation of tension forces in suspender cables over time with both adequate time and frequency resolution through acceleration measurements of the cable vibration. This study focuses on the application of nonstationary analysis on cable vibration data in time-frequency domain. The results of the stationary frequency analysis are presented in a detailed report by Stromquist-LeVoir et al. [
The authors have developed a methodology which demonstrates the ability to relate temporal changes of the cable frequency to variations in axial tension force of the cable. The results show that the amplitude of the oscillatory tension force and corresponding fatigue stress in a suspender cable may be determined by relating vehicle traffic passing over the bridge to the vibration response of the cable. Developing a relationship between the in situ live load on a bridge and the tension in suspender cables has the potential to improve the current understanding of their short- and long-term performance under daily traffic. If the suspender cable forces are known under typical service conditions, then variation of the forces due to different environmental loads such as wind and thermal forces may be accounted for under extreme conditions. This allows the tensile force history in one or more cables to be directly related to the vibration frequency of the cable caused by the passing vehicles. This approach may be used to determine the remaining fatigue life of the member based on the long-term variation and cyclical loading of the cable tension forces.
Time-frequency analysis may be an effective and practical approach for engineers to estimate the remaining fatigue life in suspender cables. The unique load carrying mechanisms of bridges which use suspender cables prevent the use of traditional bridge monitoring techniques. A time history of cable acceleration may be used to measure the change in cable tension force induced by crossing vehicles. Therefore, a methodology may be developed to facilitate fatigue life analysis using a collection of cable acceleration time history data. The simplicity and accuracy of this proposed method would significantly benefit the engineering community.
Normally when accelerations are used in the majority of health monitoring applications, they only require knowing the frequency of the structure. However, to find the variance of tensile forces, the temporal variation of the vibration frequency is needed. Power spectrograms depict the change in frequency over time, but power spectral density (PSD) functions cannot show the time-varying nature of the signal. Thus, power spectrogram analysis was used to relate the acceleration time history data of the vibrating cable to the traffic load applied to the structure. The power spectrograms were determined using discrete-time short-time Fourier transform (STFT).
The discrete-time STFT described was used to analyze the acceleration data in time-frequency domain to identify the cable tension over time. The methodology does not require elaborate or intrusive instrumentation as does strain-based methods and direct measurement techniques. It also did not involve the development of an extensive model. An effective and efficient analytical dynamic model of the cable was developed and validated with experimental data. The novelty of this research lies in the development of averaging variations of the first few natural frequencies to achieve adequate resolution and accuracy in temporal analysis of suspender cables.
This method was used in a case study to evaluate variations in cable tension forces caused by vehicles passing over a bridge. Acceleration data were collected from suspender cables of a steel arch bridge with a suspended deck located in Connecticut to evaluate the applicability of the method for in-service bridges. The methodology and results presented in this study may be adapted and used to document tension force data for all applicable bridges in the National Bridge Index [
The following sections outline the procedure for the proposed methodology. The time variation in cable tension forces was determined from acceleration time histories as follows: (1) the components needed to develop a representative analytical dynamic model of the cable system were identified; (2) each experimental acceleration data was transformed into a PSD function; and (3) the experimental PSD function was compared to the analytical model by varying the tension forces in the analytical model. The equations presented were used to estimate the cable tension force for a user-defined time period which could range from short term to long term depending on the frequency of data collection.
An analytical model was developed to simulate the dynamics of a suspender cable. The system is composed of a single suspender cable with length,
Analytic dynamic model of a suspender cable.
From the free body diagram, an infinitesimal section of cable at equilibrium has the following partial differential equation (PDE) of motion:
Modal frequencies may be derived from the PDE of motion using modal coordinates. The modal mass and modal stiffness of the system for the
The natural frequency of the
The modal damping for the system,
The mass, stiffness, and damping of the analytical model are used to develop state-space equations. The state-space equations are used to determine the frequency response function. The modal coordinate requires the deformed shape function,
The state-space form may be rewritten to determine the frequency response function of the analytical model:
Equations (
Bode plots were used to compare the magnitude of the system’s response in dB to the frequencies of the cable vibration under loading. A bode plot provides a powerful visualization for system responses in the frequency domain. The bode plot includes
The following definition allows a Fourier transform to be modified into a PSD function by the subsequent process. The PSD function is defined as the square of the Fourier transform series divided by its length:
To understand the effect of certain significant cable parameters on the PSD function of the analytical model, a parametric study was performed. The parameters considered were cable length, tension force, cable bending stiffness, and damping ratio. Only one parameter was varied for each mode to show the effect of that parameter on the first four fundamental frequencies of the analytical system. Figures
Analytic PSD function length comparison.
Analytic PSD function tension force comparison.
Analytic PSD function cable stiffness comparison.
Analytic PSD function damping ratio comparison.
This section outlines the method for generating a PSD function from field data. On a suspension-type bridge, the tension forces in cables change due to varying excitation from traffic loading. Because the tension force in a particular cable changes as the vehicle passes, a frequency domain analysis will show the dominant frequencies of vibration. STFTs were used to obtain valuable information from a smaller window of data to show the time-varying vibration characteristics. These STFTs showed changes in the natural frequencies of the cable over time and consequently the corresponding variations in cable tension.
For a given acceleration time history,
Hamming window.
Equation (
The tension force time history was determined by evaluating the temporal changes of frequency content of a cable because the frequency response of a dynamic cable system varies with respect to the tension force in the cable. The time resolution of the PSD function from ( The PSD function of the collected acceleration data was obtained, and the fundamental frequency was found. Equation ( A range for cable tension forces was assumed based on loads imposed by the weight and axle configuration for a crossing vehicle with a central value of Using the analytical model, the analytical PSD function was calculated for each tension force at the location where the experimental measurements were collected. The time-frequency domain transformation of the measured data was obtained using STFT. For each analytical PSD function obtained in Step 4, the convolution with the experimental PSD function of the For each time window, the tensile force that corresponds with value A plot for the tensile force of Step 7 versus the time representing the
In practice, Steps 1–4 may be precalculated and Steps 5–8 may be computed as the data is collected. The tensile force history may be used to gain a better understanding of the in situ fatigue life of the cables by accurately determining the true number and magnitude of the stress range in the cables. These potential applications will be further discussed in a case study of the Arrigoni Bridge in the following section.
A case study of the Arrigoni Bridge was used to demonstrate the proposed methodology for obtaining time variations of the tension force in suspender cables. The Arrigoni Bridge was selected because it is a steel arch bridge with suspender cables attached to the bridge deck and a predictable force distribution in the suspender cables. The experimental results collected from this bridge were compared with the analytical model to verify the proposed methodology.
The Arrigoni Bridge is a through arch bridge with a cable suspended deck connecting the towns of Middletown and Portland, Connecticut. The bridge crosses the Connecticut River and was opened to traffic in 1938. The two main spans are each 183 m (600 ft). The bridge deck is a composite girder system supported by sets of four vertical, helical suspender cables which connect the deck to the trusses. The cables vary in length from 3.35 m to 22.86 m (11 ft to 75 ft). The full breakdown of all the cable sets is provided in Table
Cable lengths.
Cable set | L2/L18 | L3/L17 | L4/L16 | L5/L15 | L6/L14 | L7/L13 | L8/L13 | L9/L11 | L10 |
---|---|---|---|---|---|---|---|---|---|
Length (m) | 3.4 | 7.9 | 11.9 | 15.2 | 18.0 | 20.1 | 21.6 | 22.6 | 22.9 |
Arrigoni Bridge.
The two spans of the bridge are similar. Therefore, data were only collected from cables on one of the spans and because of symmetry these data may be accurately extended to represent the other span.
The numbering system used to identify the cables was aligned with the vertical members of the steel truss, labeled L2 through L18. Within each set, the individual cables were labeled 1–4, starting in the SW corner and continuing clockwise. The numbering convention is shown in Figure
Cable sets are spaced at 9.14 m (30 ft). The length of each cable was determined from design plans provided by the Connecticut Department of Transportation and verified with field measurements. The length of the four cables in each cable set was assumed to be equal.
The Arrigoni Bridge suspender cables were examined using magnetic flux, nondestructive testing to determine their overall condition and identify abnormalities such as broken or loose strands [
Suspender cable layout.
The mass per unit length of the cables was determined by multiplying the cable material density by the cross-sectional area. The cables were assumed to have been made of standard structural steel with a density of 77 kN/m3 (490 lb/ft3). This resulted in a weight per unit length of 79 N/m (5.41 lb/ft) and a mass per unit length of 8.04 g/m (0.168 lb-s2/ft2). As stated in ASTM A586, the minimum Young’s modulus of the cables is 164.5 kN/mm2 (24,000 ksi).
A Bridge Diagnostics Incorporated (BDI) Structural Testing System with wireless data acquisition (STS-WiFi) was used to collect the cable acceleration histories. The system was comprised a base station, three nodes, twelve BDI 50 g accelerometers where g refers to the acceleration due to gravity, and a laptop PC. Four accelerometers and one of the nodes are shown in Figure
Cross section of the individual suspender cables.
Data were collected at a sampling rate of 100 Hz under normal ambient vibration induced by traffic and typical wind. Data collection did not depend on the type of traffic, excitation, and so on The method proved to be capable of generating reliable results even in cases with high noise-to-signal ratios because of the type of sampling used and the nonstationary windowing implemented. A subset of the data for cables L5, L6, and L7 on the south side of the Arrigoni Bridge was considered in the present analysis and discussion. The acceleration data for these cables were collected simultaneously. These cables represented a large variety of distinct vibration modes in a spectrogram analysis of the time history data. Figure
Four accelerometers of the STS-WiFi system used on Arrigoni Bridge.
Effective transformation of the acceleration data into the time-frequency domain greatly influences the efficiency of the procedure. MATLAB v7.10.0 [
Time and force resolution for different window sizes.
Window size (number of points) | Number of Fourier transform points | Time resolution (sec) | Force resolution (kN) |
---|---|---|---|
64 | 64 | 0.06 | 6.98 |
128 | 128 | 0.13 | 3.47 |
|
|
|
|
512 | 512 | 0.51 | 0.85 |
Only the first four modes of vibration were considered in the analytical model for this case study. It was observed that higher modes of vibrations were not clearly depicted on the FFT curves due to the signal-to-noise ratio of the acceleration data collected at higher frequencies. A 3% damping was used for each mode. This damping value was chosen for the analytical model to ensure that the frequency of each mode of cable vibration was captured since adjusting the damping for the cable shifts the frequency of each mode. A damping ratio of 3% was chosen because of its strong agreement with the experimental results of the modal frequency of the cables. The modal frequency of the models was found to be the critical value for determining the force in the cables. Other damping ratios were investigated, but did not capture the modal frequencies found from the field measurements.
The force time history results for all of the cables except for cables L5-1 and L5-2 are presented in this section. None of the modes of cables L5-1 and L5-2 had strong magnitude values to accurately use for the PSD analysis. The magnitude values were low for the PSD analysis when the energy between the natural frequencies was too great or the excitation was too low. If the amount of noise between the modal frequencies was too large, it made it impossible to identify distinct peaks in the data. The natural frequency of the cable must be identified to accurately determine the tension force of the cable. Therefore, the tension force changes in cables L5-1 and L5-2 could not be measured.
Initially, the forcing function participation factors were determined by comparing the experimental and analytical PSD functions. Table
Alpha values for each cable set.
Cable set |
|
|
|
|
---|---|---|---|---|
L5 | 0.05 | 0.25 | 0.50 | 1.00 |
L6 | 0.10 | 0.25 | 0.30 | 0.50 |
L7 | 0.05 | 0.25 | 0.50 | 0.75 |
Acceleration time histories.
The range of tensile forces was selected based on the approximate
The cable tension force histories obtained using the proposed algorithm are presented in Figure
Comparison between analytical and experimental PSD functions.
Cable tension force histories.
A close-up view of the cable tension force histories is shown in Figure
Time history of variation of tensile forces of individual cables in each set.
Peak force (kN) change results.
Cable set | Cable number | Average of the peak change (kN) | Maximum total change (kN) | |||
---|---|---|---|---|---|---|
1 | 2 | 3 | 4 | |||
L5 | — | — | 9.061 | 7.442 | 8.251 | 33.006 |
L6 | 7.024 | 5.978 | 9.524 | 9.759 | 8.074 | 32.294 |
L7 | 9.141 | 8.963 | 9.648 | 8.020 | 8.945 | 35.781 |
The consistent values presented in Table
The speed of the vehicle may also be estimated from the results presented by calculating the time lag between the peak moving load impacts on each of the suspender cable sets since the cables are spaced at equal spacing. The spacing between the cables is 9.14 m (30 ft) which corresponds to the distance which the vehicle traveled. By dividing the spacing between the cables by the time between the peak forces in each cable, the speed of the vehicle may be calculated. Since the peaks in the cable sets in Figure
Time history of variation of the total forces in each cable set.
This method not only gives the amplitude of tension forces in the individual cables, but also provides the mean tension force that was present from the permanent loads on the structure. The mean tension force coupled with actual temporal variation in cable forces is invaluable for evaluating fatigue life. Using these methods would allow authorities to capture a more accurate picture of the cable stress history and inform bridge owners of the remaining fatigue life of the suspender cables.
In this study, a novel method is presented to estimate temporal variation of cable tension forces. An analytic model subjected to an arbitrary, time-varying, transverse forcing function was used to develop a system of state-space equations. The model accounted for both cable tension force and bending stiffness. These equations were used to evaluate the power spectral density (PSD) functions of the analytical model. Likewise, a PSD function was developed based on a short-time Fourier transform (STFT) of experimentally measured accelerations of a bridge suspender cable. The cable tension force was determined by identifying the analytical force that maximized the result of the convolution of the two PSD functions found using the beta algorithm equation. This process identified not only the temporal variation of cable tension, but also the baseline cable tension using the recorded suspender accelerations. Including the baseline tension force was critical when examining the fatigue stress of the cable.
To determine the robustness and applicability of the proposed method, a case study was performed. Experimental data were collected from the Arrigoni Bridge located between Middletown and Portland, Connecticut. Results from this case study showed the method had the ability to measure the fluctuation of tension forces caused by a passing vehicle. The measured change in suspender tension force due to an unknown passing truck was estimated as 33.4 kN (7.5 kips). The results of this case study are evidence that the presented method may be used to determine the existing baseline cable tension as well as measuring variations in cable tension due to traffic.
Further research is required in order improve upon the methodology presented in this paper. Experimental tests should be conducted using direct for measuring techniques in order to quantify the accuracy of the proposed method. A full probabilistic analysis should also be completed to determine the effects of each parameter on the results, so that the uncertainties in the results may be accurately identified. The results presented in this paper demonstrate that time-frequency analysis may be used as an effective and practical approach to measure the time variation of suspender cable forces. Adapting this methodology may allow engineers to better estimate the remaining fatigue life of suspender cables by calculating the variation in tensile force based on its relationship to the frequency of vibration. This method may also adopted for use for bridge weight-in-motion (BWIM) studies if a complete analytical model of the structure is developed and calibrated by loading the bridge with a truck of known weight. The procedure presented may be applicable to any bridge type which includes cables due to the basic theory of the relationship between force and frequency of vibration. However, attention should be taken when applying the method to tension rods as opposed to cables due to the added flexural stiffness.
The authors declare that they have no conflicts of interest.
The authors gratefully acknowledge the support of this research by the Connecticut Department of Transportation and the U.S. Federal Highway Administration. The authors would like to thank the Connecticut Transportation Institute and the in Connecticut Department of Transportation for their help in collecting the data used in this paper. The student support of Dr. Kevin Zmetra, Dr. Alicia Echevarria, and Alexandra Hain is also thanked.