Accurate rotation identification of flexural structures using long-gauge fiber optical sensors

Abstract

The rotation of beam and column components is a key parameter in structural health monitoring (SHM), which providing analysis of bending deformation, evaluation of structural stability, and overall structural performance. Conventional sensors directly measuring rotation are typically designed assuming linear behavior. It becomes challenging to achieve precise and rapid measurements of small deformations while accurately measuring significant large deformations. This study obtained experimental and analytical studies to identify the rotation response of flexural structures using long-gauge fiber optical sensor array. Rotation is determined through two mechanisms: the plane section assumption, utilizing strain distributions on the compression and tension sides, and the sectional fiber model (SFM)-based neutral axis identification. In order to discuss the applicability of these two mechanisms from elastic to plastic state, four identification methods are proposed: Method 1 uses strain distribution on the concrete surface to identify rotation, Method 2 uses strain on steel bars, and Methods 3 and 4 use SFM-based neutral axis identification with strain measured on the compression side concrete surface and tension side steel reinforcements, respectively. Laboratory tests of beams and columns as well as field tests were shown. First, a comparison of the rotation identification accuracy among the four methods was conducted using a reinforced concrete (RC) beam test in the elastic state. Results showed good agreement between the rotations identified by all four methods and those directly measured by the tilt meter. And then, the accuracy of rotation identification in crack state and inelastic state was discussed by using a RC column test. The results indicate that, following the occurrence of cracks in concrete surface, neither Method 1 nor Method 2 can accurately identify the rotation. This is attributed to the fact that cracks disrupt the correspondence between the strain on the tension side and the compression side. Meanwhile, Methods 3 and 4 maintain a good rotational identification accuracy even after cracks happened. Moreover, when the steel reinforcement undergoes yielding and the concrete column enters the inelastic state, the rotation results identified by Methods 3 and 4 still match with the directly measured rotations. This underscores the effectiveness of the SFM-based rotational identification under large deformation conditions. Furthermore, experimental results indicate that with the increase in deformations, slip occurred in the sensing units near the column base in the sensor array on the tension side. This shows that the sensing units installed on the steel reinforcement (Method 4) are more suitable for calculating rotations during the large deformation state compared to the sensing units positioned on the concrete surface (Method 3). At last, two case studies involving the monitoring of an actual bridge grid and a bridge column were investigated to assess the effectiveness of dynamic rotation identifications. The performance evaluation results for various rotation angle measurement sensors demonstrate that long-gauge fiber optical sensors can be used for rotation identification, ensuring the stability of dynamic rotation identification.

Keywords

structural health monitoring long-gauge fiber optical sensors rotation identification bridge monitoring structural analysis

Introduction

Rotation is one of the crucial parameters for evaluating structural performance. By measuring the rotation of reinforced concrete (RC) beams and columns, engineers can gain insights into the deformation of structures under external loads. This is essential for ensuring appropriate deformation of the structure and evaluating whether it meets design requirements. Additionally, when horizontal loads of a concrete column is excessive, larger rotations may indicate that certain parts of the structure are undergoing significant bending, which could affect the stability of the structure. Therefore, regular measurement of rotations for structures in long-term operation can be utilized in structural health monitoring (SHM). For instance, the American Association of State Highway and Transportation Officials (AASHTO) has recommended the implementation of tilt measurement in bridge monitoring (Transportation Officials, 2002). The Railway Technical Research Institute of Japan considers rotation behaviors to evaluate the damage level of the column foot and provide reinforcement recommendations (Railway Technical Research Institute, 2004). Figure 1 shows the seismic performance level specified in the revised design specifications of the Japanese code (1996). The rotation at the bottom of each bridge foot is defined as the damage index, and directly corresponds to the overall bridge damage level, which is categorized into four levels. Specifically, when the rotation is less than θc (rotation at initial cracking), the structure is categorized as Level 1, which means the bridge should behave elastically without essential structural damage. Therefore, repairment is not required. When the rotation is greater than θc but less than θy (rotation at yielding), the structure is classified as Level 2, which means cracks should occur in the concrete members of bridge, repairment is required if it is necessary. Furthermore, when the rotation is greater than θm (rotation at maximum load) the structure is classified as Level 3, which means it leads to irreversible damage, and completely repairment is required in this situation. When the rotation is greater than θn, the structure is categorized as Level 4, which means the bridge structure has experienced severe damage. The bridge not only requires extensive repairs but may also necessitate the replacement of damaged components when deemed necessary. Hence, in SHM, the evaluation of beam and column rotation stands as a pivotal parameter. This parameter assumes a critical role in the analysis of bending deformation, and the evaluation of overall structural performance.

Figure 1.

Damage assessment based on the relationship between moment and rotation.

Existing methods for measuring rotations often involve installing inclinometers on bridge piers or calculating rotations based on horizontal displacements. However, both of these methods necessitate the deployment of a large number of sensors to meet the monitoring requirements of large structures (Wang et al., 2023). Particularly, when structures experience damage or large deformations, these sensors are prone to detachment, compromising the stability of measurement accuracy. Additionally, these methods are susceptible to environmental vibrations and are more suitable for primarily static rotation measurements, making it challenging to meet the demands of dynamic rotation measurements.

The fiber Bragg gratings (FBG) sensor has the advantages of high accuracy, corrosion resistance, and anti-electromagnetic interference (Fu et al., 2021; Measures, 2002). Moreover, its capability of transferring and sensing simultaneously, and its strong multiplexing functions, in addition to its convenience in forming sensor networks, overcome the limitations of traditional electric monitoring technologies for a SHM system (Fujino and Siringoringo, 2008; Majumder et al., 2008; Rodrigues et al., 2010). Moreover, owing to their flexibility and easy multiplexing, FBG sensors can be installed at multiple locations to construct quasi-distributed sensing arrays. However, this increases the scope and cost of the measurement. Recently, a long-gauge FBG sensor technology was proposed to satisfy the requirements of SHM (Li and Wu, 2007; Ye et al., 2014). To ensure effective strain measurement, a high repeatability anchorage method for a long-gauge packaged FBG sensor has been developed (Wu et al., 2010). Moreover, based on the specific property of basalt materials, a self-sensing basalt fiber reinforced plastics (BFRP) reinforcement has been designed to provide highly accurate strain sensing (Tang et al., 2010).

Furthermore, various structural analysis and identification technologies have been proposed based on long-gauge FBG sensors. Based on the plane section assumption and the conjugate beam method, deflection distribution of flexural steel structures can be calculated from the distributed strain measurements (Li et al., 2010). A section fiber model (SFM) based deflection identification algorithm has been developed using long-gauge sensing (Li and Wu, 2010b). An improved conjugated beam method has been proposed to derive the static and dynamic deflection of beam members using long-gauge sensors (Shen et al., 2010). By utilizing long-gauge sensing technology to measure static strain distribution, it is possible to achieve the identification of internal damages such as corrosion through the use of the central axis (Fouad and Saifeldeen, 2022; Zhang et al., 2020). For minor damages, dynamic strain distribution can be measured through long-gauge sensing technology, thereby achieving the identification of small, early-stage damages such as cracks (Adewuyi and Wu, 2011). The establishment of these techniques provides a systematic theoretical foundation for utilizing long-gauge sensing technology to achieve static and dynamic rotation identification in SHM.

This study conducted analytical and experimental investigations to identify the rotation response of flexural structures using a long-gauge fiber optical sensor array. Firstly, the relationship between strain distribution of long-gauge sensors and rotation were discussed. The conditions for the plane section assumption and SFM, as well as the identification methods for neutral axis identification were systematically discussed. Consequently, algorithms for rotation identification satisfying both the elastic and large deformation states were derived. Secondly, static loading tests on a RC beam and RC column are present. In the tests, long-gauge FBG sensor arrays were installed on the compression and tension sides of the concrete surface, and the steel reinforcements. This arrangement facilitated the discussion of the effectiveness of rotation identification algorithms under various deformation state. Lastly, case studies of a bridge grid and a bridge column confirmed the effectiveness of static and dynamic rotation identification using long-gauge FBG sensors.

Theoretical background

Strain distribution of long-gauge FBG sensors

FBG itself is a sensing element sensitive to both temperature and strain. By measuring the central wavelength shift of the FBG, strain or temperature information can be obtained (Andreas et al., 2006; Bao et al., 2010). The fundamental expression is as follows:

ε = K (\frac{λ - λ_{0}}{λ_{0}} - ξ Δ T)

(1)

where

λ

is the measured central wavelength,

λ_{0}

is the initial central wavelength,

K

is a function of the Poisson’s ratio, strain-optic constants of the fiber, and refractive index,

ξ

is the thermal optic coefficient and

Δ T

is the temperature change.

The current SHM systems face notable challenges in effectively identifying structural damage, as they grapple with inherent limitations of either being too localized or too macroscopic in relation to structural systems. For example, the total length of a normal FBG grid is approximately 5–10 mm. Integrating multiple FBG sensors in the creation of functional sensor networks often results in increased complexity and costs. For practical adaptation to SHM, a feasible distributed long-gauge strain sensing system on the basis of FBGs is developed (Li and Wu, 2007). The long-gauge FBG sensor uses a hollow tube to package the general FBG, extends the original gauge length, and ensures the measuring stability of non-uniform strains. As shown in Figure 2, the change in the length of the long-gauge FBG sensor can be obtained by integrating the axial infinitesimal strain (hereafter referred to as macrostrain). The macrostrain $ε_{m}$ measured by the m-th long-gauge FBG sensor can be expressed as follows (Li and Wu, 2009):

ε_{m} = \frac{Δ l}{L_{m}} = \frac{y_{m}}{L_{m}} (θ_{m 1} - θ_{m 2})

(2)

where

Δ l

is the change in the length of the m-th long-gauge FBG sensor,

L_{m}

is gauge length,

y_{m}

is the neutral axis height,

θ_{m 1}

and

θ_{m 2}

are the rotational displacements at the two ends.

Figure 2.

General view of long-gauge packaged FBG sensor.

Thus, based on a reliable anchorage method, the measured macrostrain of a flexural long-gauge FBG sensor is related to the rotation change at the two ends. For a series long-gauge FBG sensor array, the rotation changes at each unit, such as the m-th long-gauge FBG sensor in equation (2), can be evaluated by considering the measured macrostrain ( $ε_{m}$ ) and neutral axis height ( $y_{m}$ ) of the unit in question.

Relationship between strain distribution and rotation

Generally, the deformation of RC columns is affected by flexure, shear, and bond deterioration. The plane section assumption is satisfied in the analysis of the columns’ flexural behavior. Unless the column is remarkably tilted, that is, the reinforcement bar slips out from the foundation section of the column and the bottom-section of the column is separated from the foundation-section, the use of moment–curvature curves to consider the nonlinear behavior of RC is the preferred procedure for framed systems, owing to the lower complexity of this procedure.

The main purpose of this study is to establish the rotation identification method which satisfies the elastic state and the large deformation state. For this purpose, the bending deformation under these two states is discussed respectively. Firstly, according to the plane section assumption during the elastic state, tensile and compressive stresses balance within the beam cross-section. Consequently, by measuring the strain values on the compression and tension sides, the position of the neutral axis within the cross-section can be directly obtained, leading to the determination of the rotation angle (Sigurdardottir and Glisic, 2013; Studer and Taras, 2022). Secondly, when concrete appears cracking or when steel reinforcement yields, the concrete in the tension zone no longer provides stress. In other words, the structure enters a nonlinear state. The entire beam is comprised of the compressed concrete on the upper side and the tensile reinforcement on the lower side (Studer and Taras, 2022). At this point, although it is not possible to directly measure the strain in the tensile zone of concrete, it is still possible to obtain the position of the neutral axis on the cross-section with the measured strain values on the compressed side or on the steel reinforcement. In addition, in SFM analysis, the flexural member is represented by unidirectional steel and concrete fibers. Because the steel and concrete fiber responses are specified in the direction of the member length, fiber analysis can be used to model any flexural member regardless of its cross-sectional shape (Feng and Xu, 2018). Hence, the change of the neutral axis can be easily identified from single or multiple measured strains. Therefore, the key issue of the rotation derivative is to identify the change of the neutral axis and stress distribution. According to the above views, the two following mechanisms were used to identify the neutral axis location in this study:

• Mechanism A: Based on the plane section assumption, directly obtain the neutral axis location by the measured compression and tension strains.

• Mechanism B: Based on the equilibrium of tensile and compressive stresses, assuming the position of the neutral axis by SFM analysis, the strain in the compressed concrete or the strain in the steel reinforcement can be utilized to calculate the stress within the cross-section. The neutral axis position is determined when the stresses within the cross-section are in equilibrium.

Furthermore, another objective of this study is to investigate the optimal arrangement of long-gauge FBG sensors to achieve effective strain for rotation identification. In accordance with the above two neutral axis identification mechanisms, three kinds of layout positions of the long-gauge FBG sensor arrays are proposed. Specifically, the long-gauge FBG sensor array installed on the compression side of the concrete surface is named as F1, the array near the steel reinforcements is named as F2, and the array on the tension side of the concrete surface is named as F3. Consequently, the four following sensor arrangement methods were compared:

• Method 1: based on Mechanism A (the plane section assumption), using the strain distribution of the compression and tension sides of concrete surface to identify the rotation.

• Method 2: based on Mechanism A (the plane section assumption), using the strain distribution of the compression and tension sides of steel reinforcements to identify the rotation.

• Method 3: based on Mechanism B (SFM), using the strain distribution of the compression sides of concrete surface to identify the rotation.

• Method 4: based on Mechanism B (SFM), using the strain distribution of the tension sides of steel reinforcements to identify the rotation.

Identification of neutral axis location

As shown in Figure 3, Mechanism A uses the strain measured at different locations to derive the neutral axis location. From the tension and compression strain of each section, the curvature can be expressed as follows:

φ = \frac{ε_{1}}{y} = \frac{ε_{2}}{y - h_{2}} = \frac{ε_{3}}{h - y}

(3)

where

φ

is the curvature of section;

y

is the neutral axis height;

ε_{1}

ε_{2}

and

ε_{3}

are the measured macrostrains of F1, F2 and F3, respectively;

h_{2}

is the distance of F2 to the compressed edge;

h

is the section width.

Figure 3.

Three kinds of layout positions (F1-F3) of the long-gauge FBG sensor array.

However, in Mechanism B, SFM uses the plane section assumption to deduce the strain distribution across each sensing cell. From the compression side of each sensing cell, the plane section is assumed to be sequentially positioned on each divided layer.

Identification of rotation

Based on SFM analysis, the strain measured on the tension side is considered as an input value in the calculation of the horizontal strain distribution under the plane section assumption. Based on the strain–stress relationship of each material, the internal force on the tension and compression sides can be expressed as follows:

\begin{array}{l} \sum N_{c} = σ_{s}^{'} A_{s}^{'} + \sum σ_{c i}^{'} A_{i}^{'} \\ \sum N_{t} = σ_{s} A_{s} + \sum σ_{c i} A_{i} \end{array}

(4)

where $N_{c}$ is the internal forces on compression side, $σ_{s}^{'}$ is the stress of steel reinforcement on compression side, $A_{s}^{'}$ is the area of steel reinforcement on compression side, $σ_{c i}^{'}$ is the stress of concrete on compression side, $A_{i}^{'}$ is the area of concrete on compression side, $N_{t}$ is the internal forces on tension side, $σ_{s}$ is the stress of steel reinforcement on tension side, $A_{s}$ is the area of steel reinforcement on tension side, $σ_{c i}$ is the stress of concrete on tension side, $A_{i}$ is the area of concrete on tension side.

Thus, with the shifts of the neutral axis toward the tension side, the curvature distribution can be calculated as follows:

φ_{i} = \frac{{ε_{1}}^{i}}{y_{i}}

(5)

where

{ε_{1}}^{i}

is the measured macrostrain.

And the definition of rotation is the following:

θ_{i} = \int_{0}^{x_{i}} φ_{i} d x

(6)

where

θ_{i}

is the rotation of the sensing unit i, and

x_{i}

is the gauge length.

Experimental verification of rotation identification for RC beams

Experimental setup

In order to investigate the effectiveness of static rotation identification in the elastic state, a RC beam specimen was tested by using long-gauge FBG sensors. The experimental setup shown in Figure 4 was used during the test. The beam specimen is 2000 mm in length, 150 mm in width, and 200 mm in height. The distance between the bottom supports of the beam is 1600 mm, and the distance between the two loading points at the top of the beam is 400 mm. Each of the upper and lower surfaces of the beam is equipped with a long-gauge FBG array to measure the distribution of compressive and tensile strains during the loading process, respectively. Each long-gauge FBG array consists of three sensing units. Specifically, one long-gauge sensing unit with a length of 400 mm was installed between the loading points at the midspan, and a long-gauge sensing unit with a length of 600 mm was positioned on each side of the loading points. The spacing between each sensing unit is 50 mm. The sampling frequency for strain measurement using the long-gauge FBG sensors was set to 10 Hz. And a displacement meter was installed in the bottom of the beam to measure the rotation value during loading.

Figure 4.

Schematic diagram of the RC beam specimen and sensor location.

Results of rotation identification under small deformation

Since this experiment is mainly to verify the accuracy of rotation identification in the elastic state, the rotation angle was calculated based on the above mention Method 1. As shown in Figure 5, during the elastic state of the beam, the deformation and rotation of the bent beam follow a geometric relationship. In the elastic state, the rotation angle can be directly calculated using tensile and compressive strains. Therefore, by using long-gauge sensors strategically placed on the compression and tension sides of the beam, the compressive and tensile deformations on both sides can be measured. Figure 6 is the comparison of calculated and measured rotation of RC beam. The results indicate that the rotation angles calculated through tensile and compressive strains align with those measured by the displacement meter. This confirms the effectiveness of calculating the rotation angles using both the tension and compression strain. Simultaneously, this also indicates that long-gauge sensing units can accurately measure strain values within a certain length range on the beam, making them suitable for identifying macroscopic responses such as rotations.

Figure 5.

Relationship between long-gauge sensors on the tension and compression sides and rotation angle.

Figure 6.

Comparison of calculated and measured rotation of RC beam.

Experimental verification of rotation identification for RC columns

Experimental setup

In order to further validate the rotation identification from the elastic state to the appearance of cracks and into the large deformation state, a 1000-mm high RC column with a rectangular 200 × 200-mm cross-section was prepared to facilitate the above mention four methods. In accordance with the proposed three kinds of layout positions (F1-F3) of the long-gauge FBG sensor array mentioned in Figure 3, one long-gauge FBG array was installed on the compression side, tension side, and near the steel reinforcement of the concrete column specimen, respectively. Each long-gauge FBG array consists of five sensing units. Basalt FRP tubes were selected to package the FBG sensing cell, owing to the similar elongation and good bending performance between the basalt FRP and optical fiber material. The anchorage length of the tested long-gauge sensors was 150 mm. Here, FBG sensors with a gauge length of 100 mm were used, and the interval space of each unit was 30 mm. To estimate the sensing behavior of the relevant section, the sensor array was installed at 50 mm, 180 mm, 310 mm, 440 mm, and 570 mm, respectively. The sensor locations and experimental set-up are shown schematically in Figure 7.

Figure 7.

Schematic diagram of the RC column specimen and sensor location.

Displacement-controlled stepwise cyclic loading was applied. The lateral displacement was measured using a LVDT attached to the top loading point. A pressure transducer was attached at the loading point to record the applied lateral loads under the cyclic loading. The response of the column to different bending levels was measured using displacement sensors and long-gauge FBG sensors. The experimental protocol involved the application of a stepwise increasing lateral load to the column through a lateral jack. The top-lateral deformation was measured by displacement sensors, and the load-deformation. Three electronic inclinometers were installed at 50 mm, 310 mm, and 570 mm to obtain the rotation distribution during loading.

Results of tension and compression strain measurements

The measured load–displacement curve is shown in Figure 8(a). The rotation distributions measured by the three electronic inclinometers are plotted in Figure 8(b). From Figure 8(a), it can be observed that when the displacement is around 0.5 mm, there is a noticeable change in the slope of the load-displacement curve, indicating the transition of the column specimen from the elastic state to the nonlinear state. Additionally, in Figure 8(b), the measured rotations at the bottom (50 mm), middle (310 mm), and top (570 mm) of the column almost maintain a linear relationship with the displacements measured by the displacement meter. Although there is a slight change in the slope of the measured rotation angle at the bottom of the column when the displacement is at 0.5 mm, this slight variation is not sufficient to assess whether the structure has entered the nonlinear state. As mentioned earlier, the rotation angle is a macroscopic structural response and is not sensitive to localized damage in the structure. The aim of this study is to identify local rotation using the long-gauge sensing technology, which is sensitive to local damage, and utilize this rotation distribution to assess changes in both overall and localized structural performance. Next, we will discuss the effectiveness of the identification results at different states based on the four established rotation identification methods.

Figure 8.

(a) Measured load and displacement in small level deformation; and (b) measured rotation distribution in small level deformation.

In Figure 9, the measured strain distributions of F1, F2, and F3 are compared. Figure 9(a) shows the strain distribution on the concrete surface measured by F1 on the compression side, Figure 9(b) displays the strain distribution on the steel bar measured by F2 on the compression side, and Figure 9(c) illustrates the strain distribution on the concrete surface measured by F3 on the tension side. The long-gauge FBG arrays used in this experiment have five internally implanted sensing units, and each sensing unit is sequentially numbered based on its order and the center’s position. Taking F1 as an example, the numbering of its five internally implanted sensing units is as follows: F1-1@50 mm, F1-2@180 mm, F1-3@310 mm, F1-4@440 mm, F1-5@570 mm. In Figure 9(a), it can be observed that the compression sensing unit at the bottom of the column specimen (F1-1@50 mm) shows a noticeable slope deviation when the displacement reaches 0.5–1 mm, while the other four sets of sensing units maintain almost linear behavior. Similar phenomena are observed in Figure 9(b) and (c). Particularly in Figure 9(c), the tensile sensing unit at the bottom of the column specimen (F3-1@50 mm) experiences a significant increase in slope when the displacement reaches 0.5 mm, which is similar to the load-displacement curve in Figure 8(a). It can be seen that the strain distribution obtained through long-gauge FBG sensing technology is more suitable for indicating local damage and changes in the structure.

Figure 9.

Measured strain via long-gauge FBG sensors: (a) strain distribution of F-1; (b) strain distribution of F-2; and (c) strain distribution of F-3.

In addition, from equation (2), it can be observed that in the elastic state, both compressive and tensile strains can be used to calculate the neutral axis position. However, during cyclic loading, cracks or even delamination may occur on the concrete surface at the base of the column, potentially causing the installed sensors to slip, in the following sections, we will further investigate the relationship between the strain on the compression side and tension side of the concrete surface when cracks occur.

Results of neutral axis location before and after concrete cracking

In addressing the potential issue of sensor slippage due to concrete cracking mentioned earlier, we will utilize the strain measured on the tension side sensor array (F3) to estimate the strain on the compression side. This will be compared with the directly measured strain values from F1, providing further evidence to support the effectiveness of long-gauge FBG sensor technology in practical engineering applications.

Figure 10 illustrates a comparison of the compression side sensor unit at the bottom of the column (F1-1@50 mm), the sensor unit near the steel reinforcements (F2-1@50 mm), and the tension side sensor unit at the bottom of the column (F3-1@50 mm). In Figure 10(a), it can be observed that, when the displacement reaches between 0.5 and 1 mm, the strain values of the three units exhibit changes in the slope of the measured displacement. Further comparison of the absolute values of these three units is presented in Figure 10(b). In particular, the curves of F1-1@50 mm/F3-1@50 mm indicate a more pronounced change in slope, suggesting that the closure of cracks under compression results in a more significant alteration. The strain on the tension side can more effectively reflect the changes induced by cracks.

Figure 10.

(a) Comparison of measured compression strain on concrete surface (F1-1@50 mm), compression strain on steel reinforcement (F2-1@50 mm) and tension strain on concrete surface (F3-1@50 mm); (b) Comparison of absolute value of compression and tension strains.

In a normally rectangular flexural RC structure, before cracks appear in the concrete layer of the section, the neutral axis is typically located near the centroid of the flexural section. As the load increases, under the assumption of a plane section, as the concrete on the surface of the flexural RC structure reaches the design strength, the neutral axis starts to move downward with the continued increase in load. More and more concrete reaches the design strength and gradually moves down. Furthermore, when the steel reinforcement yields state, more concrete reaches its maximum design strength, causing the neutral axis to move downward once again (Studer and Taras, 2022; Xie et al., 2018). In response to the observed variations in the neutral axis, the study employed the proposed Mechanisms A and B to calculate the evolution of the neutral axis position. Mechanism A utilized both the compression side sensor unit (F1-1@50 mm) and the tension side sensor unit (F3-1@50 mm), while Mechanism B solely relied on the strain from the tension side sensor unit (F3-1@50 mm). The results are depicted in Figure 11. It is evident that before the appearance of cracks (at strains less than 200 microstrains), the neutral axis identified through strain measurements on the tension side remains at a position of 88 mm. However, after cracking occurs (between 200 and 2000 microstrains), the position of the neutral axis continues to move towards the concrete’s compression side. It is noteworthy that although Mechanisms A and B exhibit similar trends in their variations, their discrepancies gradually increase after the appearance of cracks. Consequently, it can be concluded that Mechanisms A and B, to some extent, can both fulfill the neutral axis identification from the elastic state to after concrete cracking. However, Mechanism B demonstrates relatively better identification accuracy.

Figure 11.

Changing of neutral axis location during loading.

Results of rotation identification in elastic state

Based on the observed pattern of neutral axis variation described above, this study first discusses the effectiveness of rotation identification during the elastic state. As mentioned earlier, long-gauge sensors primarily measure tension strains within the sensor’s coverage area. To accurately measure compression strains, it is necessary to incorporate a certain amount of pre-strain when fabricating long-gauge sensors. Considering that excessive pre-strain can be detrimental to the long-term stability of the sensor, this study implanted a pre-strain of approximately 300 microstrains in the long-gauge sensors used. In other words, the long-gauge FBG sensors used in this study, can measure strains ranging from −300 microstrain to 10,000 microstrain. To compare the accuracy of strain measurements on the compression side and the tension side, Figure 12 contrasts the directly measured compression side strains with the compression strains identified through the sensing units on the tension side. The results indicate that whether it is the F1 sensor installed on the compression side or the F2 sensor installed on the column side, both exhibit a linear correlation between compression and tension strains within the range of 300 microstrains. However, beyond 300 microstrains, due to insufficient pre-strain, the measured compression strain no longer maintains a linear relationship with tension strain. This is especially noticeable in the “Area 180 mm” and “Area 310 mm” located at the bottom of the column, which can no longer provide reliable measurements for high compression strains. It can be concluded that compression strain is suitable for calculations within a small strain range (before concrete cracking), but when strains exceed 300 microstrains, tension strain measurements should be used for calculations.

Figure 12.

Measured and calculated strains: (a) compression strain distribution on concrete surface; (b) compression strain distribution on steel reinforcement.

Based on the conclusions drawn above, this study calculated rotations using tension strain through the four methods mentioned earlier. Figure 13 presents a comparison between the actual measured rotations and the identified rotations. Based on the above-mentioned identifications of rotation behavior, the analysis based on the propose four methods are suitable for identifying the rotational behavior of RC columns under small strains.

Figure 13.

Comparison of calculation rotation in small level deformation: (a) ration at 50 mm; (b) ration at 310 mm; (c) ration at 570 mm.

Results of rotation identification under large deformation state

Figure 14 shows the measured load-displacement and rotation-displacement relationship under large level deformation, and Figure 15 shows the strains measured on the tension side of F1 and F2, respectively. As mentioned above, the curvature of an arbitrary cross-section in a column can be calculated by Mechanism B using a single long-gauge FOS array. In this situation, the rotation analysis (Mechanism A) based on the original geometrical relationship is not used because the minimum limitation is reached and the measurement on the compression side (F3) no longer ensures the measurement accuracy.

Figure 14.

(a) Measured load and displacement in large level deformation; and (b) measured rotation in large level deformation.

Figure 15.

Measured strain via long-gauge FBG sensors: (a) strain distribution of F-1; (b) strain distribution of F-2.

The plots of the rotation calculated and measured at 50 mm, 310 mm, and 570 mm using methods 3 and 4 are presented in Figure 16 for comparison. Clearly, the calculations and measurements are in good agreement before the yield point of the reinforcement bars in the tested column. However, because the long-gauge sensors at the bottom slipped from the column, the corresponding calculated and measured rotations are not in agreement (50 mm).

Figure 16.

Comparison of calculation rotation in large level deformation: (a) Method 3; (b) Method 4.

Field verification of rotation identification for bridge grids

Bridge description

A RC bridge in Japan was considered to investigate the effectiveness of the method for identifying beam rotation using distributed long-gauge FBG sensors. The bridge has a span of 20 m. Four wireless tilt sensors were strategically placed at the bridge’s bottom, with three dedicated to measuring the vertical direction (designated as K1-3) and one positioned at the mid-span to measure torsion (designated as K4). Rotational behavior was analyzed using a total of 14 long-gauge FBG sensors, each having a gauge length of 1 m. The sensor layout is illustrated in Figure 17. Data of strain, displacement, and rotation angles were measured with the sensors and subsequently compared.

Figure 17.

Sensor location of long-gauge FBG sensor array and four tilt sensors.

Results of strain measurement

In the monitoring of an actual bridge, the vehicles passing on the bridge are the measurement target. Two measurement methods are employed as follows: (1) continuous monitoring: data are continuously obtained over a specific time period; (2) event-based monitoring: data are obtained for each passing vehicle.

Figure 18(a) shows the displacements measured by a displacement transducer, Figure 18(b) shows the strains measured by the long-gauge FBG sensors. All figures show measurements obtained from the passage of the same vehicle. Both displacements are approximately 2 mm and the measurements exceed 5 mm. This displacement difference is attributed to the passage of light and heavy vehicles. The waveform in Figure 18(a) corresponds to the strain measurements obtained by the long-gauge FBG sensors, as shown in Figure 18(b).

Figure 18.

(a) Measured results of displacement, and (b) measured results of strain (at the mid-span).

Results of identified rotation

Due to the experimental setup, sensors were exclusively installed at the bottom of the bridge (tension side). In the process of rotation identification, the rotational values at the mid-span were iteratively obtained using the 14 sets of long-gauge FBG sensor arrays installed at the bottom of the bridge through Method 4. Figure 19(a) shows the deflection distribution when the vehicle was at sensor K1, Figure 19(b) shows the deflection distribution when the vehicle was at sensor K2, and Figure 19(c) shows the deflection distribution when the vehicle was at sensor K3. The monitoring was conducted on the bridge, focusing on vehicles in the traffic flow. Hence, the position of the load imposed by the vehicle weight changed over time, and the data at the moment when each vehicle passed over the respective inclinometer sensor were predicted based on the strain data obtained through the distribution measurement made by the long-gauge FBG sensors. The deflection distributions obtained by the inclinometer sensors and long-gauge FBG sensors are presented. Obviously, the inclinometer sensors used in this study can measure the dynamic inclination changes occurring during vehicle passage. Furthermore, it is demonstrated that the distribution measurement of the long-gauge FBG sensors enables displacement measurement in response to traffic loads. These measurement results confirm the effectiveness of the distribution measurement made by the long-gauge FBG sensors for SHM.

Figure 19.

Identified deflection distribution results: (a) measured and identified deflection on K1, (a) measured and identified deflection on K2, (c) measured and identified deflection on K3.

Field verification of rotation identification for bridge columns

Bridge description

In 2012, in the aftermath of the Great East Japan Earthquake, the author’s research team was invited to design and implement a continuous monitoring system for a high-speed railway line. To achieve the above-mentioned objectives of structural health management, a monitoring system implementing the area-distributed sensor-based SHM strategy was designed. The monitoring object was a concrete railway viaduct with a length of 7075 mm and height of 4500 mm. Based on the above-mentioned considerations, the monitoring object was divided into four distinct sensing areas, namely, two beam areas and two column areas. Each sensing area contained eleven long-gauge FBG sensor units, among which eight sensors were used for strain measurement and three sensors were used for temperature compensation. The long-gauge FBG sensors were prepared with a gauge length of 500 mm, and pre-tension of 300–500 με was given to each sensor to ensure negative (compression) strain measurement. The installed locations are shown in detail in Figure 20.

Figure 20.

Sensors arrangement of the bridge column: (a) size of bridge column and sensor location; (b) long-gauge sensor on the bridge column.

Results of strain measurement

The topical strain distributions of different areas subjected to excitations produced by a moving train are plotted in Figure 21. As can be seen, during the excitations produced by the moving train, the strain responses in the beam areas were larger than the responses in the column areas. Therefore, the strain response in the beam areas is more suitable for evaluating moving train excitations. The strain distribution measured on the bridge column is plotted in Figure 22. As can be seen, when a train passed over the bridge, the strain distribution on the bridge column exhibited alternating tension and compression. Therefore, for each excitation, the neutral axis location at different locations of the structure can be calculated to reduce the effect of variable excitation and environmental conditions.

Figure 21.

Strain response under train excitations of the column.

Figure 22.

Strain distribution of the column.

Results of rotation identification

The rotation under train excitation was investigated. Figure 23 shows the calculated rotation distribution. As can be seen, the peak rotation value occurred at both ends of the column, because the monitored object, that is, the continuous beam bridge, is a rigid structure, and the upper and lower ends of the column are fixed. The obtained data help in understanding the bending state of the columns and monitoring the daily operational status of bridge piers.

Figure 23.

Identified rotation distribution results.

Conclusions

This paper presents algorithms for identifying the rotational behavior of RC columns based on long-gauge optical fiber strain sensing, analysis based on the plane section assumption and the geometrical relationship between the detection target and multiple long-gauge FBG sensors (Mechanism A), and SFM-based analysis using a single long-gauge FBG sensor on the tension side (Mechanism B). Based on the experimental investigation of a RC column, which models the behavior of a bridge column under axial and lateral loading, the bending behavior was successfully identified using three long-gauge FBG sensor arrays installed at different locations. Under small deformation, both Mechanism A and Mechanism B are suitable for identifying the neutral axis location. Mechanism A does not require the experimental calibration of the parameters, and can directly calculate the neutral axis location from the measured long-gauge macrostrain on the tension and compression sides. However, this requires that the long-gauge FBG sensors on both the tension and compression side are within the measuring range. Because the minimum measurement limitation of long-gauge FBG sensors is determined by the pre-tension level, analysis based on Mechanism A is only useful under small deformation. In contrast, the analysis of the test results obtained using Mechanism B reveal that this mechanism is effective under both small and large deformations. Finally, field verifications considering a bridge grid and a bridge column were presented. In conclusion, long-gauge-sensor-based rotation identification is essential for effectively calculating the rotation distribution for both the elastic and large deformation states, and can help administrators in determining the change of the operation status of flexural structures.

Footnotes

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.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China (grant number 52178115) and National Key Research and Development Program of China (2019YFC1511103).

ORCID iDs

Huang Huang

Xin Wang

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