Eliminating environmental temperature and humidity effects on the frequencies of bridges based on co-integration analysis

Abstract

In vibration-based structural health monitoring, frequency is widely used to judge the occurrence of damage, which is quite efficient and convenient. However, besides damage, variations of environmental factors, such as temperature and humidity can also cause the changes of frequency. To eliminate the effects of environmental factors, co-integration analysis method in the field of econometrics is adopted. First of all, the concept of co-integration analysis is presented. Then dynamic tests on a two-way curved arch bridge and three concrete simply supported model beams were introduced. The frequency, environmental temperature and humidity were recorded every one or 2 hours. After that, co-integration analysis was carried out and the co-integration equation among frequency, environmental temperature and humidity was determined. Finally, the effects of environmental temperature and humidity on the frequency were eliminated by the co-integration equation, which makes the vibration-based structural health monitoring method more accurate and reliable.

Keywords

environmental temperature humidity frequency co-integration analysis two-way curved arch bridge concrete simply supported model beam

Introduction

Bridges, as the passage for vehicles and pedestrians, play an extremely important role in transportation. With the increase of service life, the as-is conditions of bridges might be different from as-designed conditions due to accumulated damage. To make sure the safety operation of the bridge, monitoring the working condition and detecting the damage of bridge are of vital importance. In structural health monitoring (SHM), the vibration-based damage identification method is widely used (Han et al., 2021). It evaluates the bridge’s health condition by the variations of vibration features such as frequency, mode shape and damping with the advantages of being convenient, inexpensive and nondestructive (Li et al., 2023). However, besides damage, the vibration features are also affected by environmental and operational factors, such as environmental temperature, humidity, traffic loads, wind loads and so on (Entezami et al., 2023; Wang et al., 2022). In terms of the effects of environmental temperature, Yang (2020) carried out dynamic tests on a simple-supported beam bridge and a T-type rigid frame bridge. The results indicated that for the simple-supported beam bridge, the frequencies increased 0.32% when the temperature increased by one degree. For the T-type rigid frame bridge, the frequencies increased 0.21% when the temperature decreased by one degree. Siddique et al. (2007) found that the first three frequencies of a two-span integral abutment overpass structure changed 8.4% ∼ 12.8% when ambient temperatures ranging from −12°C to 40°C. The authors also simulated the effects structural damage on the frequencies by removing 0.3 to 3 m elements on the numerical model. The order of magnitude of damage induced frequency changes was limited to the fourth decimal place (i.e., 10^-4 Hz). In terms of the effects of humidity, Zhou et al. (2010) made five reinforced concrete simple supported beams and soaked them in water for 8 days. After soaking, the beams were regarded as saturated. The frequencies of saturated beams increased 9.2%. In terms of the combined effects of environmental temperature and humidity, Bolton et al. (2001) performed two modal tests on a concrete box-girder bridge. The first test was conducted on a cool and rainy day, with a temperatures range of 5°C to 10°C. The second test was conducted 9 months later on a dry day, with a temperature range of 22°C to 26°C. The frequencies tested at the second time increased by approximately 8.5%. As the structure deteriorates and becomes more flexible, the frequencies would decrease, the unexpected change was attribute to differences in environmental conditions. Xia et al. (2006) constructed a reinforced concrete slab and monitored for nearly 2 years. The vibration frequencies are measured together with the temperature and humidity during each measurement. It was found that the frequencies decreased 0.23% when the temperature increased by one degree, or decreased 0.03% when the humidity increased by one percent. Yu et al. (2008) carried out vibration tests on a steel-concrete composite beam for nearly a year. It was found that for transverse vibrations, the frequencies increased 0.2% ∼ 0.3% when the temperature increased by one degree, or increased 0.09% ∼ 0.26% when the humidity increased by one percent. For bridges that are used mostly by light vehicles, the effects of traffic loads on the dynamic properties can be neglected (Kim et al., 2003). For wind loads, only when the wind speed is high, the structures’ vibration features of cable-stayed bridge and suspension bridge will be affected (Cross et al., 2013; Deng et al., 2011; Min et al., 2009). In conclusion, environmental temperature, humidity, traffic loads and wind loads affect the structural vibration features in different degree. The order of magnitude of environmental temperature and humidity induced vibration features may be identical with or even greater than those induced by structural damage (Siddique et al., 2007; Jiao et al., 2014; Rainieri et al., 2019), which may lead to the misjudgment of the bridge’s health condition. Therefore, it is necessary to eliminate the effects of environmental factors on the vibration features of the bridge.

There are mainly two kinds of methods to eliminate the effects of environmental factors, namely input-output method and output-only method, which is classified according to whether environmental variables are required (Wang et al., 2022). The input-output method focuses on the establishment of the relationship between structural frequency and environmental conditions, such as regression analysis (RA) (Peeters and De Roeck, 2001), support vector regression (SVM) (Ni et al., 2005; Zhang et al., 2006), artificial neural network (ANN) (Bolourani et al., 2021; Hsu and Loh, 2010), etc. While the output-only method focuses exclusively on modal frequency itself without environmental measurements, such as principal component analysis (PCA) (Ding et al., 2023; Sun et al., 2024), auto-associative neural network (AANN) (Sohn et al., 2002), etc. In the SHM of arch bridges, polynomial regression models (Ding et al., 2018) was used to establish the temperature effects on the frequencies of Nanjing Dashengguan Yangtze River Bridge (six-span continuous steel truss girder arch bridge). PCA method (Borlenghi et al., 2022) was used to define a regression model to predict the evolution of natural frequencies with temperature of a three-span reinforced concrete tied arch bridge. The linear regression and PCA method (Maes et al., 2022) were compared to remove natural frequency variations of a steel bowstring railway bridge resulting from changes in the environmental conditions. Besides input-output and out-only method, finite element analysis (FEA) is also a useful method to eliminate environmental factors’ effects on the vibration features of the bridge. A hierarchical Bayesian finite element model updating framework (Luo et al., 2024) was proposed to predict the structural static and dynamic response by taking into account of temperature and traffic load and validated it on an arch bridge using 2-year monitoring data.

For input-output methods, RA methods are simple and the results can be expressed by explicit formulas, but the separation effect of non-linear and more variables is poor. SVM and ANN methods have no explicit formulas of modeling results and to ensure the accuracy of these methods the output data should be less affected by other factors except temperature. For output-only methods, PCA method can separate the structural responses under the effects of many environmental factors such as temperature and humidity, but it cannot predict the future response and the ability to deal with nonlinear environmental effects needs to be improved. AANN method can forecast the future response but it needs to select network topology structure. In FEA, it is easy to simulate the variations of elastic modulus by changing the parameters of concrete. While for the two-way curved arch bridge, the arch feet are clamped, it is difficult to simulate the internal forces caused by the changing boundary conditions.

In recent years, the co-integration analysis method from the field of econometrics was used in SHM. The basic idea of co-integration analysis is to establish relationships between nonstationary time series in order to create a stationary residual. When the established relationship cases being stationary, the structure is no longer operating under normal behavior (Dao and Staszewski, 2013). Cross and Worden (Cross and Worden, 2011) adopted nonlinear co-integration method to analyze the SHM data of Z24 Bridge. After removing the temperature trends, damage can be identified. Liu et al. (Liu et al., 2014) proposed a method to apply co-integration theory and control chart technique to identify damage from continues long-term static response. Liang et al. (Liang et al., 2018) used the first two frequencies of a steel truss bridge to construct the co-integration relationship, and the residue of this co-integration equation was calculated to eliminate the effect of the temperature and to identify the damage condition by judging if there is a sudden jump. He et al. (He et al., 2019) analyzed the monitoring data of a three-span concrete bridge and established a long-term equilibrium model about frequency-temperature-humidity based on co-integration method. Based on the mathematical model, the modified frequency model was proposed, which can eliminate the comprehensive effects of temperature and humidity. Tomé et al. (Tomé et al., 2020) employed co-integration analysis to suppress the effects of environmental and operational variations in SHM and then used the Hotelling T² control chart for damage detection. Shi et al. (Shi et al., 2018) extended the well-established co-integration method to a nonlinear context, which is to allow a breakpoint in the co-integration vector.

Even though there are increasing studies applied co-integration analysis in SHM, most studies use Engle-Granger procedure (Engle and Granger, 1987) to determine the co-integration vector, which is only suitable for two variables, temperature and frequency. By this method, only the influence of temperature was taken into account. To identify the structure damage more accurately, more variables need to be included. In this study, a multivariate co-integration analysis based on Johansen procedure (Johansen, 1988) is adopted, temperature, humidity and frequency three variables are taken into account. Dynamic tests data of a two-way curved arch bridge and three concrete simply supported model beams were used to establish the co-integration equation and verify the effectiveness of the co-integration analysis in structure damage detection.

The manuscript is organized as follows. First of all, the co-integration theory is presented in section 2. Then the dynamic tests of a two-way curved arch bridge and three concrete simply supported model beams were introduced in section 3, including the basic information of the bridges, test machine and method, as well as test result. After that, the effects of environmental temperature and humidity on the frequencies of the two-way arch bridge were analyzed by both univariate regression method and co-integration analysis method in section 4. Then the effects of environmental temperature and humidity on the frequencies of the simply supported model beams were analyzed by co-integration analysis method in section 5. Through co-integration equation, the effects of environmental temperature and humidity on the frequencies of both the two-way arch bridge and the concrete simply supported model beams can be eliminated effectively and damage caused frequency change can be identified. Finally, conclusions are made in section 6.

Co-integration theory

To solve the problem of nonstationary time series in econometrics, Engle and Granger proposed the co-integration theory in 1987 (Engle and Granger, 1987). The co-integration theory assumes that if one or more variables are nonstationary time series, but the linear combination of the variables may be a stationary time series. The stationary linear combination is co-integration equation, which means that there is a long-term stable relationship among the variables.

During the vibration-based structural health monitoring, environmental factors especially environmental temperature and humidity have a non-negligible influence on the variation of frequencies (Yang et al., 2022). Though the variations of environmental temperature and humidity are non-stationary, some linear combination of them may have a long-term stationary relationship. Existing studies often use multivariate regression analysis method to explore the effects of environmental temperature and humidity on the frequencies. This may obtain a regression equation with high regression coefficient, but spurious regression may also occur. However, co-integration regression analysis method can establish the stationary relationship among the variables and avoid the occurrence of spurious regression. What’s more, a significant advantage of the co-integration analysis is that it can determined the co-integration relationship among multiple variables. Therefore, the co-integration analysis method is suitable for eliminating the environmental effects on the frequencies in structural health monitoring.

Before the co-integration analysis, the integrated order of all the variables should be determined. For a non-stationary variable Y , if it becomes stationary after d times difference but it is still non-stationary after d-1 times difference, then the variable Y can be called integrated order of d, denoted as Y ∼ I (d) (Box et al., 2015). Therefore, for a series of d-integrated non-stationary Y ^T = ( y ₁, y ₂, …, y _n)∈Rⁿ, the variables in Y will have a long-term equilibrium relationship only when

β_{1} y_{1} + β_{2} y_{2} + \dots + β_{n} y_{n} = ɛ

(1)

where ɛ∈Rⁿ is the residual sequence, β^T = (β₁, β₂, …,β_n) ∈R is the co-integration vector.

If equation (1) can describe the long-term equilibrium relationship among non-stationary variables, then the residual sequence ɛ is a stationary variable, that is to say ɛ ∼ I (0). In co-integration analysis, all the variables are interdependent. It addresses the interdependence of all the variables by the co-integration vector, which can reflect the degree of influence among different variables. Therefore, the purpose of the co-integration analysis is to find the co-integration vector to make the residual sequence be stationary.

To obtain the co-integration vector, the stationarity and integrated order of Y should be determined. Augmented Dickey-Fuller (ADF) test (Dickey and Fuller, 1979) is a classical method to achieve this. It demonstrated that if a time series has a unite root, then it is non-stationary. In this study, EViews software (Aljandali and Tatahi, 2018) is used to do the ADF test. If the non-stationary variable Y has the same integrated order, co-integration test can be proceed.

There are two methods to carry out the co-integration test, one is Engle-Granger co-integration test (Engle and Granger, 1987) and the other is Johansen co-integration test (Johansen, 1988). Engle-Granger test is suitable for co-integration test between two variables. Johansen test is based on the vector autoregressive (VAR) model, which can transform a maximum likelihood function problem into a problem of seeking the eigenvector corresponding to the characteristic root. Johansen test extend the application of co-integration test to multiple variables.

The process of Johansen co-integration are as follows. First of all, a VAR model is established in equation (2):

y_{t} = Φ_{1} y_{t - 1} + \dots + Φ_{p} y_{t - p} + H x_{t} + ε_{t}, t = 1, 2, \dots, T

(2)

where components of y _t are I (1) variables, x _t is d-dimension exogenous vector, which represents trend term, constant term and so on, ɛ _t is k-dimension disturbance vector.

Δ y_{t} = Π y_{t - 1} + \sum_{i = 1}^{p - 1} Γ_{i} Δ y_{t - i} + H x_{t} + ε_{t}

(3)

with

Π = \sum_{i = 1}^{p} Φ_{i} - Ι, Γ_{i} = - \sum_{j = i + 1}^{p} Φ_{j}

(4)

I (0) can be obtained by the difference of I (1), that is to say Δ y _t and Δ y _t-i (i = 1, 2, …, p) in equation (3) are vectors consist of I (0). Then only if Π y _t-1 is I (0) series, namely components of y _t-1 have a co-integration relationship, Δ y _t is a stationary series. The rank of matrix Π, r, determines whether there are co-integration relationships among components of y _t-1. There are three circumstances about r:

(1) When r = k, only if components of y _t-1 is I (0), Π y _t-1 is a vector consist of I (0). While components of y _t are I (1) variables, which is in contradiction with the assumption, thus r < k.

(2) When r = 0, which means Π = 0. Then equation (3) is merely a difference process, all components are I (0) variables, there is no need to discuss whether there are co-integration relationships among components of y _t-1.

(3) When 0 < r < k, it means that there are r linearly independent linear combinations, the rest k-r relationships can be linear expressed. Then Π can be decomposed into the product of two matrices α and β, Π = αβ^T. The rank of matrices α and β is r. Substituting this into equation (3), the equilibrium error vector can be expressed as

e_{t - 1} = {(α^{T} α)}^{- 1} α^{T} (Δ y_{t} - \sum_{i = 1}^{p - 1} Γ_{i} Δ y_{t - i} + ε_{t})

(5)

According to equation (5), e _t-1 is a stationary series, namely I (0) variables. Therefore, β^T y _t-1 is a I (0) vector, β is the matrix including co-integration vector, r is the amount of co-integration vector.

The core of Johansen test is to achieve the co-integration test of y _t by analyzing matrix Π. For matrix Π, the amount of nonzero eigenroots is equal to its rank. Therefore, by testing the amount of nonzero eigenroots, the co-integration relationship and the rank of the co-integration vector can be test.

Assume that the eigenroots of matrix Π are λ₁>λ₂>…>λ_k. r maximum eigenroots can obtain r co-integration vectors. For other k-r non co-integration vectors, λ_r+1, λ_r+2, …, λ_k should be 0, thus null hypothesis and alternative hypothesis are:

H_{r 0} : λ_{r + 1} = 0

H_{r 1} : λ_{r + 1} > 0, r = 0, 1, 2, \dots, k - 1

The corresponding test statistics is

η_{r} = - T \sum_{i = r + 1}^{k} \ln (1 - λ_{i}), r = 0, 1, 2, \dots, k ‐ 1

(6)

where η_r is eigenroot trace statistics, then test the statistical significance in turn:η₀ < threshold, accept H₀₀, which indicates that there are no co-integration vectors;η₀ > threshold, reject H₀₀, which indicates that there is at least one co-integration vector;η₁ < threshold, accept H₁₀, which indicates that there is only one co-integration vector;η₁ > threshold, reject H₁₀, which indicates that there are at least two co-integration vectors;…η_r < threshold, accept H_r0, which indicates that there are only r co-integration vectors.

Introduction of bridges and dynamic tests

To validate the effectiveness of the co-integration analysis method, dynamic tests were carried on a real two-way arch bridge and three concrete simply supported model beams. The information of the bridges, test machine and test method, along with test results were introduced as follows.

Introduction of the bridges

The two-way curved arch bridge

The two-way curved arch bridge is a kind of arch bridge whose arch ring is consist of longitudinal arch rib and transverse arch wave. This kind of bridge was built in the 1960s and 1970s in China, with the characteristics of low cost, easy construction and poor integrity. Until now, the amount of in-service two-way curved arch bridge is more than 4000.

The tested two-way curved arch bridge is an empty top-bear concrete arch bridge located on the river of West most in the Xicheng District of Beijing. The clear span of which is 41 m, and the clear rise is 3.2 m. The overview of the bridge is illustrated in Figure 1. The cross section of the mid-span is shown in Figure 2.

Figure 1.

Overview of the two-way curved arch bridge.

Figure 2.

Cross section of the mid-span (mm).

The concrete simply supported model beams

Three concrete simply supported model beams were made of C30 concrete. An ordinary Portland cement P.O42.5 produced in the Lima cement plant of China was used in all compositions. Its properties are in accordance with the standard of Common Portland Cement (The People’s Republic of China National Standard, 2007). The fine aggregate is medium sand and the coarse aggregate is pebbles produced in the Zhongxing plant of Zhuozhou city. The fineness modulus, clay content, apparent density and bulk density of the sand are 2.7, 1.6%, 2.7 g/cm³ and 1559 kg/m³ respectively. The clay content, apparent density, bulk density and maximum aggregate size of the pebbles are 0.6%, 2.7 g/cm³, 1563 kg/m³ and 25 mm respectively. The grading diagram of pebbles and medium sand are shown in Figure 3. Ⅰ grade fly ash were selected as the concrete admixture and chemical admixture. The mix proportion of C30 concrete was designed according to Specification for mix proportion design of ordinary concrete (The People’s Republic of China Industry Standard, 2011) as shown in (Table 1)

Figure 3.

Grading diagram of pebbles and medium sand.

Table 1.

Mix proportions and properties of C30 concrete.

Water-cement ratio	Sand ratio (%)	Content per cubic meter (kg/m³)
Water-cement ratio	Sand ratio (%)	Cement	Water	Sand	Gravel	Fly ash
0.46	44	245	170	786	1004	73

The dimension of the concrete simply supported model beam is 2500 mm × 300 mm × 150 mm, with the layout of the reinforcement shown in Figure 4.

Figure 4.

The dimension and reinforcement layout of the concrete simply supported model beam (mm).

Three simply supported model beams named B1, B2 and B3 were made on September 13th, 2015 in an industry factory in Hebei Province of China. First of all, the rebars were tied according to the designed drawing. Then pour the mixed concrete into the timber formwork and vibrate the concrete evenly. After concrete forming, spray water on the surfaces of the concrete. 24 hours later, remove the timber formwork and water curing for 28 days. After curing, install the supports of the beam. The key processes are shown in Figure 5.

Figure 5.

Fabrication of the concrete simply supported model beam. (a) Steel bar binding, (b) Timber formwork installation, (c) Beam molding, (d) Support installation.

Test machine and method

Test machine and method of the two-way curved arch bridge

For the two-way curved arch bridge, environmental excitation method with the advantages of high efficiency and easy operation was adopted for dynamic tests. Test machine is Building Test Studio V2.4 data collection system manufactured by National Center for Quality Supervision and Test of Building Engineering. The system is made up of host machine, BETC data collection instrument, data receiving node (Figure 6(a)) and so on. Five acceleration sensors were layout on the both ends, middle, a quarter and three quarters of the arch bridge. The vibration of the bridge was triggered by a hammer, then the signals were collected by the data receiving nodes and data collection instruments. After processing the collection data, the time frequency domain data were shown on the screen in real time. The first frequency can be obtained by the crest value as shown in Figure 6(b). The environmental temperature and humidity were tested by two hygrothermographs put on the both ends of the bridge. The average values of the two hygrothermographs were regarded as the environmental temperature and humidity.

Figure 6.

Data receiving node and interface of the software. (a) Data receiving node, (b) Interface of the software.

The dynamic tests lasted for 48 hours from 22:00 March 25th to 22:00 March 27th, 2013. The frequency of the two-way curved arch bridge was tested every hour. The environmental temperature and humidity were recorded simultaneously.

Test machine and method of the concrete simply supported model beams

Hammering method, with the advantages of high precision and repeatable was used in model beams’ dynamic test. The test machine is WS-5924 hammer test system manufactured by Beijing Bopu Technology Cooperation as shown in Figure 7(a). The measurement precision is 0.1%FS (Full scale). The system is made up of charge amplifier, signal acquisition instrument, force hammer, acceleration sensors and so on. The matched software can display the frequency response curve immediately, then the frequency is calculated by modal parameter identification software. The sample frequency is 100 Hz, acquisition time is 0.5 s. The type of force hammer sensor is YFF-7-20, with the measuring range of 3000 kN, frequency range from 0 to 800 Hz, the sensitivity coefficient is 2.25 pc/N. For the acceleration sensor, the frequency range from 0.2 to 5000 Hz, the sensitivity coefficient is 433 pc/g.

Figure 7.

Test instruments and hammering process. (a) Data acquisition system, (b) Hammering process.

Multiple points excitation and single point response method is adopted in the experiment. The acceleration sensor was installed at the mid-span of the beam. For each simply supported model beam, three hammer points were chosen and each point was hammered for three times. The frequency of a hammer point is the average of three times hammering and the frequency of the beam is the average of the three points. The hammering process is shown in Figure 7(b).

The concrete simply supported model beams were made on September 13th, 2015. Continuous 29∼33 hours’ dynamic tests were carried out on November 3rd and December 25th, 2015 along with July 7th, October 17th and December 24th, 2016. The frequency of the simply supported model beam was tested every one or 2 hours. The environmental temperature and humidity were recorded simultaneously.

Test results

Test results of the two-way curved arch bridge

The variations of test frequency with time are plotted in Figure 8. Due to the failure of the equipment, frequencies at 23:00 March 25th, 0:00, 1:00, 2:00, 9:00 and 19:00 March 26th were not recorded. It can be seen that the frequency fluctuated greatly with time. The maximum frequency is 5.000 Hz at 15:00 March 27th and the minimum is 3.953 Hz at 6:00 March 26th, which changed more than 20% during 48 hours. In section 4, the variations of the frequency are analyzed.

Figure 8.

Variations of test frequency of the two-way curved arch bridge with time.

Test results of the concrete simply supported model beams

Five times 29∼33 hours’ dynamic tests were carried out in more than 1 year. More than 200 groups’ data including frequency, temperature and humidity were obtained. The variations of test frequency with time are plotted in Figure 9. The vertical dotted line is the separation of different tests.

Figure 9.

Variations of test frequency of the simply supported model beam with time.

The temperature ranges from −1.4 to 34.6°C, the humidity ranges from 26% to 100%, which covers the annual temperature change and is representative. The maximum frequency is 38.69 Hz at 8:00 December 24th, 2016 and the minimum is 28.42 Hz at 17:00 July 7th, 2016, which changed more than 26%. In section 5, the variations of the frequency are analyzed by co-integration analysis method.

Effects of environmental temperature and humidity on frequency

The structures’ vibration features are affected by environmental temperature, humidity, traffic loads, wind and so on (Entezami et al., 2023; Wang et al., 2022). According to Kim et al.’s (Kim et al., 2003) study, for bridges that are used mostly by light vehicles, the effects of traffic loads can be neglected. The two-way curved arch bridge is a municipal bridge, which is used mostly by light vehicles. Therefore, the effects of traffic loads on the structures’ vibration features can be neglected. For wind loads, when the wind speed is high, the structures’ vibration features of cable-stayed bridge and suspension bridge will be affected (Cross et al., 2013; Deng et al., 2011; Min et al., 2009). Therefore, for the two-way curved arch bridge with small flexibility, the effects of wind loads can also be neglected. In conclusion, the variations of the frequency were mainly caused by the variations of environmental temperature and humidity. First of all, the single effects of temperature or humidity were explored. Then co-integration analysis was carried out using the former 24 hours’ test data to explore the coupled effects of environmental temperature and humidity. Finally, by using the latter 24 hours’ test data, the effects of environmental temperature and humidity on frequencies were eliminated and the co-integration equation was verified.

Single effect of temperature or humidity

To show the variations of frequencies with environmental temperature or humidity visually, frequency-temperature and frequency-humidity scatter diagrams are shown in Figures 10 and 11. Then linear fitting was carried out respectively and the fitting equations are shown in equations (7) and (8).

f = 0.054 T + 3.981 R^{2} = 0.80

(7)

f = - 0.005 H + 4.777 R^{2} = 0.11

(8)

Figure 10.

Variations of frequencies with temperature.

Figure 11.

Variations of frequencies with humidity.

As shown in Figures 10 and 11, the frequency increases with temperature but decrease with humidity. However, according to equations (7) and (8), linear regression relationships are unable to describe the variation of frequency accurately, especially the variation of frequency with humidity. Therefore, co-integration analysis is adopted in the following section.

Frequency co-integration analysis

The former 24 hours’ test data was used in this section to carry out the co-integration analysis, the rest data was used to verify the co-integration equation. In this study, the effects of both environmental temperature and humidity are taken into consideration, one-variable linear regression model has low accuracy and cannot reflect the coupled effects of environmental temperature and humidity, therefore Johansen test is adopted. First of all, ADF test was carried out to verify the stationary of all the variables to prevent the occurrence of spurious regression. If two or more variables have the integrated order of two and other variables have the integrated order of one, then these time series model satisfies the assumption of Johansen co-integration test. Then the VAR model was built and the optimal lag order of the VAR model was determined. Finally, the number of co-integration equation was figured out by trace test and the co-integration equation was constructed by the vector error correction model (VECM).

ADF unite root test of the two-way arch bridge

The ADF unite root test was carried out in EViews 12, the test results of the frequency series (F), environmental temperature (T) and humidity (H) series are shown in Table 2.

Table 2.

Results of the ADF unite root test of the two-way arch bridge.

Project	Variable	Test type	t-Statistic	Probability	Critical vale 1%	Critical vale 5%	Critical vale 10%
Original series	F	(C, T)	0.7359	0.8646	−2.6998	−1.9614	−1.6066
	T	(C, 0)	−1.9977	0.2848	−3.8868	−3.0522	−2.6666
	H	(C, 0)	−3.4281	0.0256	−3.9203	−3.0656	−2.6735
Series of first difference	F	(0, 0)	−4.9681	0.0001	−2.7081	−1.9628	−1.6061
	T	(0, 0)	−2.5170	0.0153	−2.7081	−1.9628	−1.6061
	H	(0, 0)	−1.8265	0.0656	−2.7081	−1.9628	−1.6061
Series of second difference	T	(0, 0)	−5.1198	0.0000	−2.7175	−1.9644	−1.6056
Series of second difference	H	(0, 0)	−4.3754	0.0002	−2.7175	−1.9644	−1.6056

C: intercept items; T: trend items; 0 none.

According to the test results in Table 2, it is observed that the t-statistic value of the frequency series is 0.7359, which is larger than the critical value on significance level of 0.1, that is, −1.6066, so the frequency series is nonstationary. Then another ADF test is proceed for the first-order difference of the frequency series. The t-statistic value of the frequency first-order difference series is −4.9681, which is smaller than the critical value on significance level of 0.01, that is, −2.7081, so the frequency first-order difference series is stationary. Therefore, the frequency series is integrated of order one. Similarly, the temperature and humidity series is integrated of order of two, which satisfies the assumption of Johansen co-integration test.

Determination of optimal lag order of VAR model of the two-way arch bridge

After the ADF unite root test, the integrated order of all the three variables was determined, which indicates that Johansen co-integration test can be carried out. Then the vector VAR model was built in EViews 12 and the optimal lag order of the VAR model was determined. The optimal lag order is determined based on the principle of selecting the minimum values of sequential modified LR test statistic (LR), Final prediction error (FPE), Akaike information criterion (AIC), Schwarz criterion (SC) and Hannan-Quinn information criterion (HQ). The values of LR, FPE, AIC, SC and HQ of lag order 1 and 2 are shown in Table 3. Order 2 has the smaller values of LR, FPE, AIC, SC and HQ. Therefore, the optimal lag order of the VAR model is 2.

Table 3.

Chosen of optimal lag order of the two-way arch bridge.

Lag	LogL	LR	FPE	AIC	SC	HQ
1	−54.1698	NA	0.3428	7.4317	7.8729	7.4756
2	−38.3700	20.4468^a	0.1675^a	6.6318^a	7.5140^a	6.7195^a

^aIndicates lag order selected by the criterion.

Determination of the co-integration equation of the two-way arch bridge

First of all, the number of co-integration equation is determined by trace test. Then the format of the co-integration equation was established by VECM. The results of trace test are shown in Table 4. Trace test indicates that there is one co-integrating equation at the 0.05 level. According to VECM, the format of the equation can be expressed as

f = 0.010 H + 0.093 T + 3.114

(9)

where f is the frequency of the two-way curved arch bridge, T is environmental temperature and H is environmental humidity.

Table 4.

Results of Trace test of the two-way arch bridge.

Hypothesized No. of CE(s)	Eigenvalue	Trace statistic	0.05 critical value	Prob^a
None^b	0.6904	30.9622	29.7971	0.0366
At most 1	0.4204	11.0314	15.4947	0.2096
At most 2	0.0983	1.7582	3.8415	0.1849

^aMacKinnon-Haug-Michelis (1999) p-values.

^bDenotes rejection of the hypothesis at the 0.05 level.

Comparison of linear regression and co-integration analysis

To compare the accuracy of linear regression method and co-integration analysis method, coefficient of determination (R²) and mean squared error (MSE) of these two methods are compared.

R² is the proportion of the variations in the dependent variable that can be explained by the independent variable, which is an important index measuring the goodness of fit. R² is defined as

R^{2} = \sum_{i = 1}^{n} {({\hat{y}}_{i} - \bar{y})}^{2} / \sum_{i = 1}^{n} {(y_{i} - \bar{y})}^{2}

(10)

where n is the number of test data,

{\hat{y}}_{i}

is the ith predicted value,

\bar{y}

is the average of test values,y_i is the ith test value.

MSE is an index measuring the degree of deviation between test value and predicted value, which is defined as

M S E = \frac{1}{n} \sum_{i = 1}^{n} {(y_{i} - {\hat{y}}_{i})}^{2}

(11)

According to equations (10) and (11), R² and MSE of linear regression equations and co-integration equation are calculated and plotted in Figure 12. The R² of linear regression equations equations (7) and (8) are 0.800 and 0.113 respectively, while the R² of co-integration equations equation (9) is 0.888. The MSE of linear regression equations equations (7) and (8) are 0.016 and 0.071 respectively, while the MSE of co-integration equations equation (9) is 0.008. Compared with considering single effects of temperature, when considering the effects of temperature and humidity simultaneously, the R² increased by 11% and MSE decreased by 50%. This indicates that using the co-integration equation, more frequencies can be explained by the temperatures and humidity. What’s more, the deviations between the predicted frequencies and test frequencies are less. Therefore, it is necessary to account for both temperature and humidity effects simultaneously.

Figure 12.

Comparison of R² and MSE among different equations.

Verification of the co-integration equation of the two-way arch bridge

To verify the co-integration equation and eliminate the effects of environmental temperature and humidity on frequencies, the latter 24 hours’ test data was used in this section. The co-integration frequencies were obtained by substituting environmental temperature and humidity into the co-integration equation (9) proposed in section 4.2.3. Since the co-integration frequency takes the effects of environmental temperature and humidity into consideration, the difference value between test frequency and co-integration frequency, which is called frequency change, can reflect the effects of damage on the frequency. For the latter 24 hours’ test data, the variations of the test frequency and the frequency change with time are shown in Figure 13.

Figure 13.

Variations of the two-way arch bridge’s test frequency and frequency change with time.

As shown in Figure 13, the test frequency fluctuates with time fiercely while the frequency change fluctuate slightly. The maximum and minimum test frequency are 5.000 Hz and 4.177 Hz, which changed more than 15% during 24 hours. Compared with the fiercely change of test frequency, the maximum frequency change is less than 0.2 Hz and the average frequency change is merely 0.02 Hz.

As stated in the introduction, the structural vibration features are mainly affected by structural damage, environmental temperature, humidity, traffic loads and wind loads (Entezami et al., 2023; Wang et al., 2022). The effects of environmental temperature and humidity can be eliminated by the co-integration analysis. According to Kim et al.’s (Kim et al., 2003) study, for bridges that are used mostly by light vehicles, the effects of traffic loads on the dynamic properties can be neglected. The two-way curved arch bridge is a municipal bridge, which is used mostly by light vehicles. Therefore, the effects of traffic loads can be neglected. For wind loads, when the wind speed is high, the structures’ vibration features of cable-stayed bridge and suspension bridge will be affected (Cross et al., 2013; Deng et al., 2011; Min et al., 2009). Therefore, for the two-way curved arch bridge with small flexibility, the effects of wind loads can also be neglected. After excluding the effects of environmental temperature, humidity, traffic loads and wind loads, the frequency change is caused by structural damage and test noises. The frequency change fluctuates around zero, which indicates that there is no damage occurs. Therefore, through the variations of frequency change, the structural damage can be identified.

Co-integration analysis of the concrete simply supported model beams

To further validate the co-integration analysis method, Test data of B1 and B2 were used to carry out the co-integration analysis, the left test data of B3 was used to verify the co-integration equation.

ADF unite root test of the simply supported model beam

The ADF unite root test was carried out in EViews 12, the test results of the frequency series (F), environmental temperature (T) and humidity (H) series are shown in Table 5.

Table 5.

Results of the ADF unite root test of the simply supported beams.

Project	Variable	Test type	t-Statistic	Probability	Critical vale 1%	Critical vale 5%	Critical vale 10%
Original series	F	(C, 0)	−3.3422	0.0146	−3.4707	−2.8792	−2.5762
	T	(C, 0)	−2.8447	0.0544	−3.4707	−2.8792	−2.5762
	H	(C, 0)	−6.8566	0.0000	−3.4709	−2.8793	−2.5763
Series of first difference	F	(0, 0)	−14.0075	0.0000	−2.5793	−1.9428	−1.6154
Series of first difference	T	(0, 0)	−13.52739	0.0000	−2.5793	−1.9428	−1.6154

C: intercept items; T: trend items; 0 none.

According to the ADF unite root test results in Table 5, the humidity series is integrated of order zero, the temperature and frequency series is integrated of order one, which satisfies the assumption of Johansen co-integration test.

Determination of optimal lag order of VAR model of the simply supported model beam

The vector VAR model was built in EViews 12, the values of LR, FPE, AIC, SC and HQ of lag order 1 and 2 are shown in Table 6. Order 2 has the smaller values of LR, FPE, AIC, SC and HQ. Therefore, the optimal lag order of the VAR model is 2.

Table 6.

Chosen of optimal lag order of the simply supported beams.

Lag	LogL	LR	FPE	AIC	SC	HQ
1	−1257.5900	NA	1240.5310	15.6369	15.8085	15.7066
2	−1215.586	80.8969^a	825.4471^a	15.2295^a	15.5725^a	15.3687^a

^aIndicates lag order selected by the criterion.

Determination of the co-integration equation of the simply supported model beam

The number of co-integration equation is determined by trace test. Then the format of the co-integration equation was established by VECM. The results of trace test are shown in Table 7. Trace test indicates that there is one co-integrating equation at the 0.05 level. According to VECM, the format of the equation can be expressed as

f = ‐ 0.001 H ‐ 0.202 T + 36.707

(12)

where f is the frequency of the two-way curved arch bridge, T is environmental temperature and H is environmental humidity.

Table 7.

Results of Trace test of the simply supported beams.

Hypothesized No. of CE(s)	Eigenvalue	Trace statistic	0.05 critical value	Prob^a
None^b	0.2700	70.7752	35.1928	0.0000
At most 1	0.0812	19.7975	20.2618	0.0578
At most 2	0.0368	6.0764	9.1645	0.1849

^aMacKinnon-Haug-Michelis (1999) p-values.

^benotes rejection of the hypothesis at the 0.05 level.

Verification of the co-integration equation of the simply supported model beam

To verify the co-integration equation and eliminate the effects of environmental temperature and humidity on frequencies, the test data of B3 were used in this section. The co-integration frequencies were obtained by substituting environmental temperature and humidity into the co-integration equation (12) proposed in section 5.3. The variations of the test frequency and the frequency change with time are shown in Figure 14.

Figure 14.

Variations of the simply-supported model beams’ test frequency and frequency change.

As shown in Figure 14, the test frequency fluctuates with time fiercely while the frequency change fluctuate slightly. The maximum and minimum test frequency are 38.69 Hz and 29.81 Hz, which changed more than 22%. Compared with the fiercely change of test frequency, the maximum frequency change is less than 1.6 Hz and the average frequency change is merely 0.02 Hz. The frequency change caused by structural damage and test noises fluctuates around zero, which indicates that there is no damage occurs. Therefore, through the variations of frequency change, the structural damage can be identified.

Conclusions

This study eliminates the effects of environmental temperature and humidity on the frequency by co-integration analysis method. Dynamic tests of a two-way curved arch bridge and three concrete simply supported model beams were used to construct and verify the co-integration equation. Based on this investigation, the following findings and conclusions are drawn:

(1) For the two-way curved arch bridge tested in this study, the frequency increases with the increase of environmental temperature. For the simply supported model beam, the frequency decreases with the increase of environmental temperature.

(2) The co-integration theory can be used to deal with the non-stationary time series data. This study adopted Johansen co-integration analysis method to effectively eliminate the effects of both environmental temperature and humidity on the frequencies and identify the occurrence of structural damage, which can offer more accurate judgement for the structural health monitoring.

There is a limitation of this study. The fitted co-integration equation is only specific to the two-way curved arch bridge studied in this manuscript. For each individual bridge, the co-integration vectors need to be recalibrated using the co-integration analysis method introduced in section 4.2.

Footnotes

Acknowledgements

The authors would like to acknowledge their colleagues and collaborators as part of this Foundation program. The statements made herein are solely the responsibility of the authors. The authors would like to acknowledge members of the Disaster Prevention and Reduction research group at University of Science and Technology Beijing for their endless support in creation of this work.

ORCID iDs

Shuzhen Yang

Zhongliang Zou

Baodong Liu

Bo Song

Author contributions

Shuzhen Yang: Conceptualization, Methodology, Experiment, Writing. Zhongliang Zou: Writing. Baodong Liu: Supervision. Bo Song: Supervision

Funding

This work was supported by the National Natural Science Foundation of China (Grant No. 51278031).

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Data Availability Statement

The data are available from the corresponding author on reasonable request.*

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