Abstract
Hinge joint performance during the operation of prefabricated prestressed concrete (PC) hollow slab bridges is critical to ensure their lateral collaborative working condition and safe serviceability. Traditional performance identification of hinge joints mainly relies on manual inspection, which is inefficient and inaccurate. At the same time, existing indexes (such as acceleration and strain) can only qualitatively detect the damage to hinge joints. This study proposes a novelty detection method based on impact vibration testing to rapidly perform the quantitative assessment of the working condition of the hinge joints. The relationship between hinge joint performance and lateral load distribution (LLD) is first derived in detail by theoretical analysis. And then, the quantitative analysis of collaborative performance is converted to the identification of the LLD influence line, which is innovatively established by the lateral flexibility of the hollow slab bridge. The effectiveness of the proposed method is verified through a multibeam model using ABAQUS software, and the lateral collaborative working relationship between slabs is simulated using the connector elements. Furthermore, the LLD influence lines and hinge joints performance of a PC hollow slab Yanhu Bridge are evaluated based on the impact vibration testing with sensor lateral arrangement strategy. The detection results show that the proposed method can quickly and accurately identify the damage location and the stiffness loss of hinge joints.
1. Introduction
Prefabricated prestressed concrete (PC) hollow slab bridges are applied widely in highway networks because of their simple and efficient construction, high industrialization, and low costs, making them the most commonly used type of bridge. As environmental deterioration and traffic volume increase, the mechanical properties of PC hollow slab bridges deteriorate gradually, especially in lateral collaborative working performance, which results in the pattern abnormal of load lateral distribution (LLD) and the “single slab bearing” phenomenon [1–6]. To ensure that the structure can continue to be used normally, an efficient and accurate method of disease detection is required to evaluate different degrees of hinge joint damage quantitatively.
Hinge joints always fail before the bridge girder, significantly affecting the structure’s bearing capacity. Visual inspection is the most common and straightforward method for performance evaluation of hinge joints by detecting concrete shedding, cracks, water leakage, and other diseases. However, the pavement in most bridges is covered with highly elastic asphalt concrete, making the hidden damage of hinge joints challenging to detect. Currently, the assessment methods of the damage degree of hinge joints are mainly divided into static and dynamic processes.
The static load test is the most accurate method for evaluating the damage degree of hinge joints [7–11]. It analyzes whether the hinge joint can still transmit shear force normally through the LLD determined by the load test. Usually, the relative displacement of the slabs on both sides of the hinge joint is used as a static indicator and deformation measurement is required. This method has a more precise mechanical concept and more intuitive results. However, the measurement of relative displacement at hinge joints requires specialized equipment, and most tests need to interrupt traffic. It is difficult to achieve rapid detection of hinge joint diseases of the PC hollow slab bridges on the highway network.
The damage identification methods of hinge joints based on dynamic indicators are a process of identifying the structural response [12], such as acceleration, strain/stress, and mode shape, by applying impact excitation of traffic flow. Wen et al. [13] established a new simplified mechanical model of prefabricated girder bridges by using a sufficient sample size under statistically stable traffic flow. The proposed indicates accurately reflect prefabricated girder bridges’ lateral collaborative working performance. Dan et al. [14] developed a dynamic strain correlation coefficient index to evaluate the lateral collaborative working capability. Numerical simulation analysis and real bridge monitoring data analysis results prove that this index can characterize the function of hinge joint degradation and can successfully identify the location and degree of hinge joint damage. Dan et al. [15] established a multibeam model connected by distributed springs to analyze the modal characteristics of a fabricated girder bridge. This model proposes two types of damage detection indexes based on lateral modal shapes for hinge joint damage detection. A numerical analysis of different damage situations proves the applicability of the proposed index. But the lateral connection stiffness still cannot be identified. Zhan et al. [16] suggested a quantitative damage evaluation method, which is the shape difference index for the spectra of dynamic responses under vehicle-bridge coupling vibration analysis. Damage in single or multiple hinge joints with different extents can all be identified using the proposed method. In addition, some damage identification methods based on artificial intelligent algorithms were proposed to determine the hinged joint damage [17–19]. These methods are almost all realized by simulation calculation, which is challenging to be used in practical engineering.
The above literature shows that the dynamic indicator and intelligent algorithm have more convenient advantages than the static indicator. It is still tricky to quantitatively identify the damage degree of the hinge joints. Establishing a novel dynamic indicator to quantify joint damage is the core work of this study. Impact vibration testing with the input force measurement has merit in extracting not only dynamic (modal parameters) but also static (flexibility) characteristics of the structure [20–23]. Traditional ambient vibration testing can only identify the basic modal parameters such as frequency, damping and mode shape, which cannot directly make decisions on the structural health condition. The flexibility as the inverse of stiffness can more effectively evaluate the bridge safety performance directly. Tian et al. [24] identified the structural flexibility based on an impact vibration test and predicted the deflections of a three-span concrete box girder bridge under static loads. The deflection prediction under any static load by impact vibration testing has been verified in many kinds of literature [25–27]. It is a very effective and reliable method. However, it is not difficult to find that almost all of these literature identify the longitudinal flexibility of bridge structures. This study aims to extend this method to the lateral flexibility identification of PC hollow slab bridges.
This study proposes a rapid quantitative and qualitative identification method for the hinge joint performance evaluation of prefabricated PC hollow slab bridges based on impact vibration testing. First, the relationship between hinge joint performance and the LLD influence line is introduced. The quantitative analysis of lateral collaborative working performance is converted to the identification of the LLD influence line which is innovatively established by the lateral flexibility through the impact vibration testing. Second, the effectiveness of the proposed method is verified through a multibeam model using ABAQUS software. Third, the LLD influence line and hinge joint performance of a hollow slab Yanhu Bridge are carried out on the impact vibration test with the lateral sensor placement strategy.
2. Proposed Hinge Joint Performance Evaluation Methodology
With the increasing traffic and load levels, the PC hollow slab bridges on the highway network have many diseases, such as cracks, water leakage, and misplacement, and hinge joint damage has become the main disease form of hollow slab bridges. If it cannot be found and treated in time, the hinge joint damage may lead to the phenomenon of “single slab bearing” and seriously endanger structural safety. Therefore, it is of great significance to research hinge joint damage identification methods to ensure the safety of hollow slab bridges. This study proposed a novelty methodology for hinge joint performance rapid evaluation based on impact vibration testing as shown in Figure 1.(1)The relationship between hinge joint performance and the LLD influence line is briefly introduced based on the lateral transfer mechanical properties. The intrinsic relationship between the LLD influence line and structural lateral flexibility is deduced theoretically for the first time. The lateral flexibility identification based on the sensor lateral arrangement strategy is performed. The proposed method is verified by a multibeam finite element model (FEM) of the PC hollow slab bridge, in which the hinge joints are simulated by the connector element on the ABAQUS software. The effects of vertical and torsional modes on the lateral flexibility matrix of the structure are analyzed in detail based on the impact vibration test. Then, the identification process of the LLD influence line based on lateral flexibility is further deduced.(2)A field test method for identifying the LLD influence line based on impact vibration testing is developed for engineering applications. A strategy of sensor lateral arrangement is adopted to realize the measurement of torsion mode while reducing the placement of redundant sensors and identifying the LLD influence lines. Comparing the LLD influence line before and after hinge damage can give a comprehensive quantitative and qualitative identification of the damage location and stiffness damage.
3. Rapid Evaluation of Hinge Joint Performance Based on Impact Vibration Test
3.1. The Relationship between Hinge Joint Performance and LLD Influence Line
Most small- and medium-span bridges widely adopt economically simply supported beam structures that are composed of PC hollow slabs, as shown in Figure 2(a). Multiple hollow slabs are connected and composite a whole system through hinge joints, which achieve LLD through shear forces between hinges, as shown in Figure 2(b). The LLD influence line is the connection line of all points formed by a fixed point’s vertical amplitude (shear force and bending moment), which is produced by the vehicle load moving in the lateral direction of the bridge. Under the heavy traffic flow during the service period, the working condition of the hinge joint will gradually degenerate and reduce the condition of the lateral load transfer of the hollow slabs, further affecting the overall performance of the structure as shown in Figure 3. Therefore, accurately identifying the LLD influence line can detect the deterioration of the hinge joint.
(a)
(b)
3.2. The Relationship between LLD and Hinge Joint Stiffness
In general, the hinge joint as a lateral connecting member is considered to transmit only lateral shear force, and the loss of shear stiffness is an intuitive index of damage to the hinged joint. This subsection will optimize the method for detecting hinge joint damage by establishing the relationship between the LLD and the hinge joint shear stiffness. In the transversely hinged-connected slab method, the midspan section of each slab is analyzed by the independent body (#) as shown in Figure 4. The shear force () generated by the hinge joint acts on both sides of the slabs under the external load ().
If the hinge joints between the slabs can transmit the shear force well, according to the calculation assumption of the hinged slab method, the relative displacement at the joints between the slabs should be zeros. However, the hinge joints will crack and fail during long-term operation. At this time, the relative displacement between hinge joints is no longer zeros. The isolation body is selected from Figure 4 for deformation analysis, as shown in Figure 5. The relative displacement of the slabs ( and ) at the hinge joint is composed of the vertical displacement and as the following equation:where and are the relative displacements generated by the hinge force and external load at hinge joint, respectively; is the stiffness of the hinge joint; and is the shear force generated at the hinge joint.
Based on the deformation relation of hinge joints, and can be written as the following equations:where is the relative displacement generated by the unit hinge force in the hinge joint at the hinge joint; is the shear force generated at the hinge joint; is the external load at slab; and represents the displacement at the i hinge joint generated by the unit load applied to the hinge joint.
According to the geometric relationship in Figure 5, it can be obtained,where is the width of the slab; is rotational; and is the vertical deflection under the action of the unit moment or force of the slab.
Substituting equations (2a)–(3b) into equation (1), we obtain equation (4a). Dividing both sides of the equation by , we obtain equation (4b).where .
According to the above equations, the LLD influence line can be expressed in terms of load and shear force in equation (5a), and the recursive expression of shear force can be obtained after sorting (equation (5b)).where the is the vertical value of the LLD influence line at slab; is the shear force generated at the hinge joint; and is the external load at slab.
By substituting into equation (4b), the stiffness can be calculated, and written as shown in the following equation:
It can be seen that the LLD influence line of each slab can independently calculate a set of independent hinge joint stiffness. The average result obtained by taking the LLD influence line of the two slabs adjacent is the average stiffness of the hinge joint.
When a bridge has n slabs, it can calculate n sets of lateral distribution influence lines. Every set of influence lines can independently calculate a set of independent hinge joint stiffness. Therefore, can be used to further describe the stiffness of the i hinge joint depending on the j set (in the order of slab number) lateral distribution influence lines. The average result obtained by taking the LLD influence lines of the two slabs adjacent to the hinge is the average stiffness of the hinge joint. The logarithm of the average stiffness of each hinge joint is taken aswhere is the stiffness index of the i hinge joint; is the i hinge joint stiffness calculated by the i set LLD influence line; and is the i hinge joint stiffness calculated by the i + 1 set LLD influence line.
3.3. LLD Influence Line Identification Based on Impact Vibration Theory
3.3.1. Estimation of FRF with Theory
The vibration equation of the multidegree-of-freedom structure can be expressed in the modal coordinate system as follows:where is the displacement mode shape, is the first-order damping ratio, is the modal mass, is the rfirst-order natural frequency, and is the input force. The response of the system o point can be expressed by Duhamel integration and mode shape superposition method as follows:where and are the values of the rfirst-order displacement mode shape at nodes o and i; ; and is the damped natural frequency.
When is impulse function, the response of the multi-degree-of-freedom structure can be calculated as follows:where .
The impulse response function (IRF) is Fourier transformed to obtain the displacement frequency response function (FRF),
3.3.2. Estimation of FRF Based on Impact and Ambient Vibration
By performing the impact vibration test on the structure and measuring the input force and structural displacement at the same time, the H1 algorithm can be used to estimate the FRF of the structure.where represents the self-power spectral density function; represents the cross-power spectral density function; and are the Fourier transform of and , respectively; is the average order; is the complex conjugate value of .
There are two main methods the natural excitation technique and the random reduction technique for estimating the structural displacement FRF by processing the ambient vibration data (only the output data). When the external force acts on the structure point , the displacement responses of the nodes and are, respectively, and , and the corresponding cross-correlation function of the two node displacements can be expressed as
Loads are random under ambient excitation. The above formula is further written aswhere is the white noise excitation force. When multiple white noise input forces are applied to the structure, the cross-correlation function is the sum of all input positions.where , indicates that only relevant to output data; and , .
The above formula is subjected to Fourier transform to obtain the displacement frequency response function (unscaled FRF) calculated only with the output data under ambient vibration.
3.3.3. The Relation between FRFs
After the displacement frequency response function is obtained, the relationship between the amplitude of the displacement frequency response function under the impact vibration test and the ambient vibration test is further studied.
The amplitude ratio of the displacement FRF is [21]
The FRFs estimated from the ambient vibrations and analytical solutions, respectively, have similar shapes but different magnitudes as shown in Figure 6. Both input forces and output responses were measured, and the FRFs estimated from the impact testing data are as exact as the analytical calculation, from which structural modal flexibility can be accurately identified [22, 27]. So, the ambient vibration test can only identify the basic parameters, which cannot directly to make a decision on the health condition (such as load carrying capacity). The impact vibration test can provide completed structure characteristics, more realistic and detailed information on structural characteristics, and a rapid assessment of bridges.
3.3.4. LLD Influence Line Identification Based on Lateral Flexibility
Usually, structural displacement flexibility identification has been carried out for slender structures (such as beam bridges), but the hollow slab bridge studied in this research has a width-to-span ratio of about 0.5. This means that structural torsion effects cannot be ignored, and it has a great contribution to the flexibility matrix integration. In order to predict the deflection of the bridge under the lateral action of the vehicle, the concept of lateral displacement flexibility is proposed in this study. The complex mode indicator function method was employed to identify structural flexibility by using the impact vibration data.[22] The lateral displacement flexibility matrix formula can be expressed aswhere is the lateral displacement flexibility matrix; is the lateral mode shape vector with mass normalization from sensor lateral arrangement, the is the i-order frequency, and n is the modal order.
The lateral deflection of any slab can be calculated by the lateral displacement flexibility matrix under lateral vehicle load arrangement. The formula is expressed aswhere is the lateral deflection of the i slab when the load vector is applied to the slab.
For the elastic slab-beam structure, the load is proportional to the deflection. So, the estimation of LLD can be transferred from calculating the load distribution to calculating the displacement distribution. According to the definition of the LLD influence line, the deflection produced by the midspan of the i slab when the unit load () acts on the midspan of the slab be . Then, the vertical value of the LLD influence line of the i slab can be expressed as follows:where is the LLD influence line of the i slab and is the flexibility coefficient. Therefore, the relationship of LLD influence line and lateral displacement flexibility is shown in Figure 7.
4. Numerical Verifications
4.1. Numerical Model Establishing
For the simulation work of hollow slab bridges, the most important part is to effectively simulate the lateral transmission mechanism and restraint effect of hinge joints. The virtual rigid arm is generally used for the simulation of hinged joints, and the connector element based on ABAQUS [28] software is used for the simulation of hinged joints in this study. A PC hollow slab bridge consists of 7 hollow slabs with a span of 12 meters as shown in Figure 8(a). Cross-section parameters of slabs are = 0.2065 and = 0.377 . B31 element is adopted to simulate the hollow slab, and the hinge joint is simulated with seven sets of connector elements. The slab number is represented by numbers 1∼7, and the hinge number is represented by letters a∼f.
(a)
(b)
(c)
Slide plane was used to simulate the stiffness of the joint in translation, which consists of two directions to simulate the stiffness of the actual joint in the normal and tangential directions, while still maintaining the degree-of-freedom in rotation and using align. The detailed mechanical diagram is shown in Figure 8(b). The combination of Universal + join is used at the end points of the two connector elements to release only the rotational degrees-of-freedom in the axial and normal directions. The stiffness of the damaged hinge joint (No.: b) is set to , and the stiffness of the intact hinge is . The schematic diagram of the lateral mechanics of the model is shown in Figure 8(c).
The following work is to (1) verify the accuracy of the hinge joint simulation based on the connector element; (2) verify the correctness of the sensor lateral arrangement method proposed in this study; and (3) identify the LLD influence line based on the impact vibration theory.
In order to verify the accuracy of this model, the LLD influence line was calculated by the transversely hinged-connected slab (THSM) method in the theoretical calculation, and the LLD influence line was obtained by the FEM. And then, the results are drawn for comparison as shown in Table 1. The relative errors of obtained results between the simulation and the theoretical are not more than 5%. This model can accurately simulate the lateral collaborative working mechanical properties of a PC hollow slab bridge.
4.2. Identification of Lateral Flexibility
Bridge deflection usually analyzes the vertical deflection and sensors are generally arranged in the longitudinal direction from the bridge. However, the vehicle load will not only produce longitudinal deflection deformation but also transverse deflection deformation for the hollow slab bridges. Then, the flexibility matrix calculated by the traditional sensor layout cannot predict the lateral deformation of the structure under the action of nonaxial load. Therefore, it is necessary to use the lateral arrangement of sensors to calculate the lateral flexibility matrix. The feasibility and effectiveness of lateral arrangement are verified as follows:
Based on the previous theoretical derivation, the first five-order modal identification results of the PC hollow slab bridge model are in Figure 9. The first order is the bending mode shape, and the remaining four are the torsional mode shapes. This phenomenon shows that the torsional mode shape will significantly influence the flexibility identification, because the contribution of lower-order modes is more considerable. It should be pointed out that Figure 9 shows the mode shapes plotted for all the measuring points with the blue lines. Mode shapes identified with only sensor lateral arrangement strategy are drawn with red lines. It can be seen that the lateral mode shapes obtained by the two sensor arrangement strategies are consistent. Then, the lateral mode shape is extracted at the midspan to identify the lateral flexibility. According to the theory in Section 3.3, the lateral flexibility matrix was identified as shown in Figure 10. In field testing, the lateral mode shapes will be obtained by placing the sensors laterally on the midspan of the structure.
4.3. Identification of LLD Influence Line and Damage
According to the relationship between the lateral flexibility matrix and the LLD influence line, the LLD influence lines of each slab are identified based on the impact vibration test. At the same time, the results of numerical and theoretical are also calculated and plotted in Figure 11. The results obtained by the three methods are in good agreement. The results show a significant change in the slope of the LLD influence line between the 2# and 3# slabs, which indicates that the hinged joint between them is damaged. This result can also be seen from the perspective of symmetry, the change of influence line between 2# and 3# is different from that between 5# and 6#.
The stiffness of each hinge joint can be identified according to the proposed damage index (equation (7)) in Section 3.2, and the results are shown in Figure 12. The damage that significantly decreased at the b# hinge joint is clearly detected. The obtained stiffness is basically consistent with other good hinge joints. In addition, the general measuring points are arranged horizontally and vertically to realize the identification of the model LLD, but this method uses too many measuring points and sensors. It is difficult to directly adopt this method in bridge testing. For this problem, the sensor lateral arrangement strategy at the midspan is proposed and verified by numerical simulation. It will be further verified in Section 5 in the real bridge test.
5. Field Test on Yanhu Bridge
5.1. Bridge Description and Field Test
The Yanhu Bridge is 20 m long, and the lane width is 14 m, as shown in Figure 13(a). 14 one-meter-wide hollow slabs are connected with hinged joints. Both ends of the hollow slab bridge are simply supported on the piers. The cross-sectional view of the bridge is shown in Figure 13(b). It should be noted that the entire bridge includes the middle lane and the sidewalks on both sides, and there is no connection between them. When testing the LLD of the middle lanes, it can be ignored due to the sidewalks.
(a)
(b)
A total of 14 high-sensitivity unidirectional accelerometers (PCB 393B04) with a maximum measurement range of ±5 g are arranged on the center of each slab in the midspan, as shown in Figure 13(a). In this test, a bigger hammer (PCB086D50) with a maximum measurement range of 5 tons was used to impact the 14 measuring points, respectively. An NI PXIe-1082 data acquisition instrument was used to collect the acceleration data and impact force simultaneously. The sampling frequency for the entire field test was 1000 Hz.
5.2. Identification of Hinge Joint Performance
The dynamic test was carried out on the Yanhu Bridge, each measuring point was impacted three times, and the acquisition time was 30 seconds. The data with a better excitation effect were selected for parameter identification, and the results of the first eight modes of the bridge as shown in Figure 14.
The first-order frequency is 8.3 Hz, corresponding to a relatively flat mode shape. Because the basis of this mode is the combination of the vertical bending modes in the midspan of each hollow slab. The second-order frequency is 10.74 Hz, which corresponds to the first-order torsion of the entire bridge. The remaining identified modes, 3 to 8, show they are symmetric or asymmetric. In other words, they correspond to the torsional mode shapes of the structure, because the Yanhu Bridge has a sizeable length-width ratio. In addition, the identified modal shapes are smooth and it can be preliminarily judged that the hinge joints are not damaged. Then, the identified mode shapes are used to perform the lateral flexibility identification of the structure. The calculated lateral flexibility matrix is plotted in Figure 15.
According to the relationship between the lateral flexibility matrix and the LLD influence line, the LLD influence line of each hollow slab in the Yanhu Bridge is calculated. For the convenience of comparison, two groups of the identified results are drawn in the same graph, as shown in Figure 16. The symmetrical plates (such as 1# and 14#) reflect the symmetrical LLD influence lines. The identified results indicate that the lateral collaborative working condition of the bridge is normal. If the “single slab bearing” phenomenon is in the structure, these identification results are enough to judge the damage of the hinge joints. However, the possibility of symmetrical damage cannot be ruled out in practical structures. Therefore, the stiffness of the hinge joint requires further rigorous identification to judge its lateral working performance.
According to the identified LLD influence line, the stiffness of each hinge joint can be identified according to Section 3.2. The estimated results based on the proposed damage index are shown in Figure 17. It can be concluded that the hinge joints of Yanhu Bridge have no obvious damage.
6. Conclusions
During the operation of the PC hollow slab bridges, the transverse collaborative working performance of hinge joints is critical to their safety and serviceability. Traditional performance identification of hinge joints mainly relies on manual inspection, which is inefficient and inaccurate. Therefore, this study proposes a detection method based on impact vibration testing to rapidly perform the quantitative assessment of the working condition of the hinge joint. The main contributions of this study are summarized as follows:(1)This study presents the lateral flexibility of a PC hollow slab bridge for the first time. The theoretical relationship between LLD influence line and lateral flexibility is established. The identification of the LLD influence line based on traditional static load tests can be converted to lateral flexibility identification based on impact vibration tests. Furthermore, in order to qualitatively detect the hinge joint’s damage degree, the stiffness of the hinge joint is derived from the lateral flexibility matrix. The proposed novelty detection method can overcome and improve the efficiency of traditional testing.(2)The effectiveness of the proposed method is verified through a multibeam model using ABAQUS software, and the connection between the beams is simulated using the connector elements. The identified results are in good agreement with the theoretical and numerical methods. Finally, a PC hollow slab Yanhu Bridge is the test structure for on-site impact vibration testing with the sensor lateral arrangement strategy. The detection results show that the proposed method can quickly and accurately identify the damage location and the stiffness loss of hinge joints.
Data Availability
The data used for supporting this study are available from the corresponding author upon reasonable request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The authors wish to express appreciation for the funding support of this research from the Jiangsu Provincial Double-Innovation Doctoral Program (Grant no. JSSCBS20210129); the Natural Science Foundation of Jiangsu Province (Grant no. BK20220852); and National Natural Science Foundation of China (Grant no: 52208305).