Research on distributed strain monitoring of a bridge based on a strained optical cable with weak fiber Bragg grating array

Lina Yue; Qing Wang; Qing Wang; Fang Liu; Qiuming Nan; Guanghui He; Sheng Li

doi:10.1364/OE.518450

1. Introduction

The construction of the highway is a significant infrastructure for modern society, whose intelligence represents the crucial development direction in the transport field [1]. Intelligent highway network satisfies the requirements of highway full-time and spatial traffic information collection, which detects and traces the relevant information of vehicles by collecting the strained variation signals of bridge [2]. Intelligent system plays a meaningful role in improving the safety and maintenance efficiency of highway bridges [3].

The traditional intelligent highway network system applied video recognition [4], microwave radar [5], gyroscope [6] and inductive circle [7] technology to realize the identification of vehicle information. In contrast, distributed optical fiber sensing (DOFS) technology has the advantages of full-time, full-domain, high-density sensing and excellent reliability [8]. Meanwhile, the flexibility and linear distribution characteristics of DOFS cable are suitable for the structural form of highway bridges, which can be perfectly applied to the main girder made for concrete. Compared with the traditional discrete point monitoring technology, such as vibrating string sensor, strain gauge and other sensing elements [7,9], the most significant feature of DOFS can monitor the physical parameter along the array path, which characterizes the spatial distribution of the measured physical parameter in real time and quantitative value [10]. Although fiber Bragg grating (FBG) can be connected in series to achieve distributed measurement, the quantity of sensing nodes is limited to 30 pcs, because of the limitation of incident light bandwidth wavelength [11,12]. A few sensing nodes can't construct a high-density strain plane distributed matrix, which is hard to realize the monitoring function of intelligent highway. DOFS can be based on numerous inherent physical phenomena in optical fiber media, such as Brillouin scattering and Raman scattering. These phenomena are generated by the interaction of propagating photons with properties of the local material such as density, temperature and strain [13]. Brillouin Optical Time Domain Reflectometry (BOTDR) strained optical cable based on the Brillouin scattering principle also realize distributed measurements, with a lower level of testing accuracy. If the monitoring structure requires high accuracy, such as the strain measurement of concrete structure, BOTDR is unable to obtain accurate monitoring data. In addition, the engineering application of BOTDR faces the challenges of how to improve the spatial resolution and the temperature self-compensation [14]. Regarding Raman scattering, the disadvantage of OTDR-based strained optical fiber cable is that the distance between individual sensing nodes is related to the spatial resolution of OTDR. Within an acceptable cost range, its spatial resolution is currently limited to 6 meters [15]. If small-span bridges need to be monitored, the long distance of the sensing nodes will dramatically increase the difficulty of laying the sensing optical fiber cables. DOFS based on wFBG currently has a favorable application prospect in structural health monitoring. DOFS based on wFBG currently has a favorable application prospect in structural health monitoring. wFBG can realize the characteristics of large-capacity, high-density, high-S/N ratio with low-reflection loss, which can overcome the shortcomings of other DOFS such as FBG, BOTDR, and OTDR [16].

Currently, wFBG technology has been successfully applied in engineering scenes such as subways, airports, and tunnels [17,18]. However, the current application of wFBG is to use bare fiber or encapsulate it into simple cladding structure. Since wFBG is extremely fragile with exposed applications, the distributed serial connection of a monitoring point damage can lead to the collapse of the entire sensing network. Some researches provided insights into the encapsulated structure of optical fiber causing strain transfer reduction. Since the shear lag effect, the deformation of the optical fiber is less than the measured structure when it is deformed [19]. In monitoring applications, various types of fibers with different coatings are possible to be applied. Q. Wang calculated the shear deformation to summarize a general expression for the strain transfer coefficient k when different quantities and types of coatings are used for optical fiber [20]. F. Falcetelli developed an exponential attenuation model for optical fiber in the boundary region of cable in combination with Saint Venant principle. The relevant parameters of the fiber boundary conditions were obtained after fitting the experimental data [21]. This experimentally verified boundary condition model is also referred for the theoretical analysis in the subsequent section. F. Liu proposed a method to update the shear lag parameters in the mathematical model using finite element simulation correcting the strain transfer error of a three-layer optical fiber model [22]. This corrected model had instructive value for the sensing optical fiber which are pasted on the surface of the bridge. H. Wang analyzed a strain transfer model for optical fibers from the perspective of being embedded in an asphalt pavement [23]. The effects of the three dimensions as temperature, cracks and gluing length for strain transfer were discussed. Besides the theoretical analysis, the strain transfer coefficient between the fiber and the gluing layer has also been calibrated through heating deformation, external strain field, and simulation by scholars [24–26].

However, chirping problem might occur by encapsulating the grating array directly within the gluing layer [27]. Few researches have been investigated by considering the encapsulation of wFBG into special optical fiber cable style in order to avoid the application problems caused by the traditional encapsulated structure. In this study, a novel strained optical fiber cable based on wFBG arrays is designed. The wFBG arrays are suspended and pretensioned to be arranged between the gluing regions at both ends. The strain loss coefficient in the gluing region of optical fiber cable is accurately calculated, while the solution of high sensitivity or large-scale range of optical fiber cable structure is proposed. Theoretically, the strained sensitivity of wFBG optical fiber cable can reach to 1.55 pm/µɛ at the maximum. The wFBG optical fiber cable with large scale type is applied in the static and dynamic loading tests of Dongmiao Bridge to verify the effectiveness of wFBG optical fiber cable.

2. Design of wFBG optical fiber cable

2.1 Modulation and demodulation principle

Previously, the conventional modulation and demodulation of wFBG array generally used time division multiplexing (TDM) or wavelength division multiplexing (WDM) solution [28]. A broadband light source passing through each wFBG will create a narrow-band reflected light. TDM is achieved by optical time-domain reflectometer (OTDR) detecting the time at which narrow-band reflected light returns to the source, positioning each identical wFBG wavelength. Field programmable gate array (FPGA) is used to resolve the corresponding wFBG wavelength shift. However, the spatial resolution of the OTDR is limited, making it impossible to locate each wFBG within 2 m. When the number of wFBG in a strained optical fiber cable is excessively sparse, it is difficult to build the high-density strain plane distribution matrix for bridges. WDM means that each wFBG whose wavelength difference exceeds 1 nm is inscribed in the array, which uses various wavelengths to locate the position of the wFBG after resolving the wavelength signal of the reflected light. But the applied light source is usually in the C + L band, limiting the number of inscribed gratings within twenty. In order to overcome the drawbacks of applying WDM and TDM separately, a novel wFBG array based on periodic switching of wavelength is designed in this study with the premise of applying the existing OTDR for wavelength demodulation. As shown in Fig. 1(a), the spacing of any two adjacent wFBG in a strained optical fiber cable is L while the length of the wFBG is H. The central wavelength of wFBG varies periodically with values of 1548 nm versus 1556 nm. It has the advantage of reducing the spacing of wFBG which enhances the density of wFBG array to twice that of conventional sensing optical fiber cable. If it is necessary to have a denser wFBG scenario, the periodic wavelength can be set to more than three. For bridge health monitoring, the strain plane distribution matrix constructed by wFBGs with a spacing L of 1 m is sufficient to invert the bridge deformation as well as information for the traffic state. The fabricated process of the novel wFBG array based on periodic switching of wavelength involves the UV spectrum emitted from the laser passing through two phase masks with the periods of 1058.1 nm as well as 1063.6 nm respectively, during the manufacture of the fiber drawing tower. The fiber being produced 1 m by the drawing tower each time, the laser is controlled to emit an UV spectrum to inscribe the grating, where the phase mask is rapidly switched between different periods. The wire drawing speed of the fiber must be adapted to match the inscribing time of the wFBG, which is approximately 0.05 s, to ensure that the reflectivity of the wFBG is around 1%. The process based on wire drawing with online inscription allows for continuous multi-point inscribing of the sensing wFBG without the requirement to fuse fiber between individual gratings, significantly reducing the overall consumption of the array.

Fig. 1. Structure of the strained optical fiber cable. (a) Before deformation; (b) after deformation.

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In this research, two semiconductor optical amplifier (SOA) are controlled by a pulse signal generator module to achieve the wavelength resolving and positioning of the designed wFBG array. As shown in Fig. 2, the core of the demodulator with wFBG is mainly composed of amplified spontaneous emission broadband light source (ASE-BBS), pulse signal generator module and modulation-demodulation module. The ASE-BBS connects the SOA (1) and the erbium-doped fiber amplifier (EDFA) in turn to substantially increase the power of the incident light. Furthermore, the incident light enters into the wFBG array using a three-port circulator with a 1×N optical splitter. In addition, the ultra-weak reflected light encountering the wFBG is transmitted by the circulator to another SOA (2). Finally, the real-time wavelength value of the reflected light is resolved by the field programmable gate array (FPGA) of the modulation-demodulation module. To achieve the function of TDM, the pulse signal generator module controls two periodic electrical pulse signals with different phase delays in the SOA (1) and SOA (2). The time difference of the reflected light arriving at each SOA can precisely locate the wFBG. The pulse generation module can filter out irrelevant reflected light to improve accuracy by periodically controlling the high-speed switching of the SOA. To avoid crosstalk of the signals from wFBG array, the central wavelengths of the sensing elements are designed to alternate between λ₁ and λ₂. The sensing network consists of 2N segments of wFBG sensing elements, with each wavelength partition consisting of N segments identical wFBG. Wavelength switching is based on the partitioned spectral design of the wFBG strained optical cable, rather than analyzing the reflectance spectrum of individual wFBG.

Fig. 2. The principle of wFBG demodulator.

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2.2 Structural design of the strained optical fiber cable

The exposed wFBG is extremely fragile which must be encapsulated into the structure of a strained optical fiber cable by components such as reinforcement and protective layers to enable distributed measurement in harsh engineering environment. As shown in Fig. 1(a), the distribution in the fixed end region of the strained optical fiber cable from inside to outside are the optical fiber layer, the Thermo-Plastic elastomer (TPEE) tight buffer, the Ultraviolet (UV) gluing layer, the spiral metal protective layer and the High-Density-Polyethylene (HDPE) outer protective layer. According to the cross section structural diagram of the optical fiber cable, where A-A and B-B, the distribution of components in the cross section of the wFBG position lacks the TPEE tight buffer and the UV gluing layer. The purpose is that wFBG remains in a hanging state in which chirping phenomenon of non-uniform force is avoided.

While the protective layer of the optical cable can enhance the reliability of wFBG, the influence of its strain transfer efficiency is resulted. With strain reduction, for example, the decrease in transfer efficiency is manifested in the fact that the strain measured by wFBG is smaller than the actual strain of the structure under measurement. Analyzing the fixed end region of the fiber optic cable (A-A section), multiple layers are shear deformed by external strains. This shear deformation will lag the cooperative deformation of the fiber as it is transferred to the center of the cable. As shown in Fig. 1(b), the end surface of the protective layer presents a concave arc shape. The apparent phenomenon is that the tensile deformation ΔL of the fiber in the fixed end region is smaller than the external deformation. The deformation of the two end surfaces in the fixed end region are similar. Observing the HDPE outer protective layer and optical fiber of cable from the macroscopic viewpoint, it can be found that their deformation is synchronized under the external force. According to the deformation coordination, the loss generated by the strain of the optical fiber in the fixed end region is inevitably compensated by the optical fiber in the overhanging region. The overhanging region (B-B section) of the optical cable being analyzed, the strain transfer reduction in the fixed-end region will amplify the strain coefficients in the overhanging region which is without the TPEE tight buffer and the UV gluing layer. The gluing layer at both ends will amplify the deformation of the fiber ΔH in the overhanging region. In this paper, the novel wFBG array is designed by inscribing the grating in the overhanging region. This structure of the optical cable can be more sensitive to capture the strain variation of the external structure.

2.3 Derivation of strain transfer efficiency

2.3.1 Basic assumptions

To calculate the strain transfer efficiency of the strained optical fiber cable based on wFBG, the following assumptions are proposed in this research.

1. The material of the optical fiber cable as well as the measured structure satisfy the linear elasticity and small deformation, which can be solved using elasticity theory.
2. Slippage at the contact interface can be ignored as the fiber is well bonded to each protective layer.
3. The shear deformation of the fiber is not considered since the high Young's modulus as well as only the axial deformation of the fiber causes wFBG wavelength shift.
4. The surface of the spiral metal protective layer being considered smooth, the influence of the thread dimension on the theoretical analysis is neglected. In addition, since the Young's modulus of the metal layer (SUS304 stainless steel) and the measured structure is substantially larger than the other layers, their shear deformation is neglected.

2.3.2 Analysis of the outer protective layer

The outer protective layer of the optical fiber cable consists of the metal layer and the HDPE layer. Only these two regions are continuously distributed in the whole structure of the cable. In this research, the strain transfer model based on the outer protective layer of the strained optical fiber cable is firstly developed. This model doesn't consider the influence of the internal gluing layer and the fiber region, since the gluing layer contributes poorly to the stiffness by its low modulus of elasticity and size. The purpose is that the degree of reduction of the external strain transfer to the metal layer is calculated. The measured structure, the HDPE outer protective layer and the metal protective layer from outside to inside are defined as subscript m, h and s. In this region, σ (x) and τ (r, x) represent the positive stress and shear stress of the corresponding layer, respectively. Figure 3(a) illustrates their stress components with the cylindrical coordinate system. Because of the uniform distribution of the protective layer of the optical cable with a ring, the angular variable of the cylindrical coordinate is not considered.

Fig. 3. The stress components and cylindrical coordinate system of the optical fiber cable and the measured structure. (a) Outer protective layer region; (b) gluing layer region.

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According to the cylindrical coordinate system established in Fig. 3(a), the stress state of the metal protective layer is analyzed. In the micro-element of the metal layer, the axial stress in the micro-section is equal to the shear stress exerted on its micro-surface. According to Assumption 4, the shear deformation of the metal layer is neglected when calculating the strain, thus the shear stress of the metal layer is independent of the variable r. Since the gluing layer region is not considered, the shear stress term between the metal layer and the UV layer is ignored. The equilibrium equation can be expressed as:

(1)$${A_s} \cdot ({\sigma _s}(x) + d{\sigma _s}(x)) - {A_s} \cdot {\sigma _s}(x) + 2\pi {r_s} \cdot {\tau _{s,h}}(x) \cdot dx = 0,$$

where A_s and r_s represent the circular cross-sectional area and maximum radius of the metal layer. The shear stress τ_s_,h (x) between the metal layer and the HDPE layer can be solved according to the equilibrium Eq. (1). Similarly, the shear stress τ_h_,m (r, x) between the HDPE layer and measured structure can be obtained. Their difference comes from whether it is considered shear deformation. Combined with Hooke's law, the shear stress can be obtained.

(2)$$\left\{ \begin{array}{l} {\tau_{s,h}}(x) ={-} \frac{{{A_s}}}{{2\pi {r_s}}}\frac{{{E_s} \cdot d{\varepsilon_s}(x)}}{{dx}}\\ {\tau_{h,m}}(r,x) ={-} \frac{{{A_s}}}{{2\pi r}}\frac{{{E_s} \cdot d{\varepsilon_s}(x)}}{{dx}} - \frac{{{r^2} - r_s^2}}{{2r}}\frac{{{E_h} \cdot d{\varepsilon_h}(x)}}{{dx}} \end{array} \right.,$$

where E_s, ɛ_s(x) and dɛ_s(x)/dx represent the Young's modulus, strain and strain gradient of the metal layer, respectively. In addition, E_h, ɛ_h(x) and dɛ_h(x)/dx represent the Young's modulus, strain and strain gradient of the HDPE layer, respectively.

Since the measured structure, the HDPE layer and the metal layer satisfy the condition of synergistic deformation, the strain gradients can be considered equal. G_h represents the shear modulus of the HDPE layer is defined. Moreover, only the axial deformation of the optical cable is considered, ignoring the radial deformation caused by the Poisson effect. With the influence of the shear deformation of the HDPE layer, the difference between the axial deformation of the measured structure u_m and the metal layer u_s can be expressed as:

(3)$${u_m} - {u_s}(x) = \int_{{r_s}}^{{r_h}} {\frac{{{\tau _{h,m}}(r,x)}}{{{G_h}}}dr} ,$$

where r_h represents the maximum radius of the HDPE layer, which is the radius of the entire optical cable. The optical cable is an extremely small component with respect to the entire measured structure. It can be assumed that the local deformation of the measured structure is uniform, where u_m is constant. Substituting Eq. (2) into Eq. (3) with integration obtains a detailed expression for the difference. The difference between the axial strain ɛ_m and ɛ_s(x) of the measured structure can be obtained by further derivation of the variable x.

(4)$${\varepsilon _m} - {\varepsilon _s}(x) ={-} \lambda _{sm}^2\frac{{{d^2}{\varepsilon _s}(x)}}{{d{x^2}}} ={-} \frac{1}{{{G_h}}}[\frac{{{E_s}{A_s} - {E_h}\pi r_s^2}}{{2\pi }}\ln \frac{{{r_h}}}{{{r_s}}} + {E_h}\frac{{r_h^2 - r_s^2}}{4}]\frac{{{d^2}{\varepsilon _s}(x)}}{{d{x^2}}},$$

where λ_sm is constant as a positive constant. The structure of Eq. (4) is second-order nonhomogeneous linear differential equation. Its special solution is the constant ɛ_m. At the same time, the discriminant Δ is greater than 0 as well as the solution has a pair of real roots with opposite values. Its general solution can be expressed as:

(5)$${\varepsilon _s}(x) = {C_1}{e^{\frac{x}{{{\lambda _{sm}}}}}} + {C_2}{e^{ - \frac{x}{{{\lambda _{sm}}}}}} + {\varepsilon _m},$$

where C₁ and C₂ are coefficients to be determined. In the structure of strained optical fiber cable, both end boundaries of the metal layer are not subjected to the constraint effect, so their strain is 0. Define the whole length of the optical fiber cable as B, the boundary condition of the metal layer is:

(6)$${\varepsilon _s}(x)|{_{x ={\pm} {\raise0.7ex\hbox{$B$} \!\mathord{/ {\vphantom {B 2}}}\!\lower0.7ex\hbox{$2$}}}} = 0.$$

Taking the boundary condition (6) into the generalized solution (5), the expression for ɛ_s(x) can be solved as:

(7)$${\varepsilon _s}(x) = {\varepsilon _m} \cdot {\delta _{sm}} = {\varepsilon _m} \cdot [1 - \frac{{\cosh ({x / {{\lambda _{sm}}}})}}{{\cosh ({B / {2{\lambda _{sm}}}})}}].$$

In the formula, δ_sm represents the strain transfer coefficient of ɛ_m transferred to the metal layer. Equation (7) is the basis for the analysis of the sensitivity of strained optical fiber cable. The metal layer is a significant medial layer at the interface between the continuous and discontinuous layers in the structure of the optical cable. It can be used to solve the strain transfer efficiency in the gluing layer region as well as the strain value and boundary condition of the fiber in the overhanging region can be obtained.

2.3.3 Strain transfer efficiency in the gluing layer region

Differently from the metal layer, the gluing layer shows a discrete distribution in the internal region of the optical cable. The boundary condition of the UV gluing layer cannot simply be considered as 0, which needs to be discussed in conjunction with the outer protective layer. In this section, the strain transfer model is developed for the gluing layer region based on the strained optical fiber cable. The optical fiber, TPEE tight buffer and UV gluing layer are defined from the inside to the outside by the subscript f, t and g. The metal layer is still denoted by the subscript s. In this region, σ’ (x) and τ’ (r, x) represent the positive stress and shear stress of the corresponding layer, respectively. The stress state of the optical fiber within the gluing layer is analyzed according to the cylindrical coordinate system established in Fig. 3(b). On the micro-element of the fiber, the axial stress in the micro-section is equal to the shear force exerted on its micro-surface. The equilibrium equation can be expressed as:

(8)$${A_f} \cdot ({\sigma ^{\prime}_f}(x) + d{\sigma ^{\prime}_f}(x)) - {A_f} \cdot {\sigma ^{\prime}_f}(x) + 2\pi {r_f} \cdot {\tau ^{\prime}_{f,t}}(x) \cdot dx = 0.$$

Consistent with the solving method of Eq. (2), combining with Hooke's law, the shear stresses between each layer in the gluing layer region can be expressed as:

(9)$$\left\{ \begin{array}{l} {{\tau^{\prime}}_{f,t}}(x) ={-} \frac{{{A_f}}}{{2\pi {r_f}}}\frac{{{E_f} \cdot d{{\varepsilon^{\prime}}_f}(x)}}{{dx}}\\ {{\tau^{\prime}}_{t,g}}(r,x) ={-} \frac{{{A_f}}}{{2\pi r}}\frac{{{E_f} \cdot d{{\varepsilon^{\prime}}_f}(x)}}{{dx}} - \frac{{{r^2} - r_f^2}}{{2r}}\frac{{{E_t} \cdot d{{\varepsilon^{\prime}}_t}(x)}}{{dx}}\\ {{\tau^{\prime}}_{g,s}}(r,x) ={-} \frac{{{A_f}}}{{2\pi r}}\frac{{{E_f} \cdot d{{\varepsilon^{\prime}}_f}(x)}}{{dx}} - \frac{{{A_t} - {A_f}}}{{2\pi r}}\frac{{{E_t} \cdot d{{\varepsilon^{\prime}}_t}(x)}}{{dx}} - \frac{{{r^2} - r_t^2}}{{2r}}\frac{{{E_g} \cdot d{{\varepsilon^{\prime}}_g}(x)}}{{dx}} \end{array} \right.,$$

where A_i, E_i, ɛ_i(x) and dɛ_i(x)/dx represent the area, Young's modulus, strain and strain gradient of the layer i, respectively. Moreover, u_i and G_i are defined as the axial deformation and shear modulus of the layer i of the gluing layer, respectively. Since the fiber, the TPEE tight buffer, the UV glue and the metal layer satisfy the condition of synergistic deformation, their strain gradients can be considered similar.

(10)$$\frac{{d{{\varepsilon ^{\prime}}_f}(x)}}{{dx}} \approx \frac{{d{{\varepsilon ^{\prime}}_t}(x)}}{{dx}} \approx \frac{{d{{\varepsilon ^{\prime}}_g}(x)}}{{dx}} \approx \frac{{d{\varepsilon _s}(x)}}{{dx}}.$$

With the influence of the shear deformation of the TPEE tight buffer as well as the UV glue layer, the differential value between u_f and u_g can be expressed as:

(11)$${u_g}(x) - {u_f}(x) = \int_{{r_f}}^{{r_t}} {\frac{{{{\tau ^{\prime}}_{t,g}}(r,x)}}{{{G_t}}}dr} + \int_{{r_t}}^{{r_g}} {\frac{{{{\tau ^{\prime}}_{g,s}}(r,x)}}{{{G_g}}}dr} .$$

According to the solving method of Eq. (4), the differential value between ɛ'_f (x) and ɛ'_g (x) is:

(12)$${\varepsilon ^{\prime}_g}(x) - {\varepsilon ^{\prime}_f}(x) ={-} \lambda _{fg}^2\frac{{{d^2}{{\varepsilon ^{\prime}}_f}(x)}}{{d{x^2}}} ={-} \left\{ \begin{array}{l} \frac{{{E_f}{A_f} - {E_t}{A_f}}}{{2\pi {G_t}}}\ln \frac{{{r_t}}}{{{r_f}}} + {E_t}\frac{{r_t^2 - r_f^2}}{{4{G_t}}} + \\ \frac{{({E_t} - {E_g}){A_t} + ({E_f} - {E_t}){A_f}}}{{2\pi {G_g}}}\ln \frac{{{r_g}}}{{{r_t}}} + {E_g}\frac{{r_g^2 - r_t^2}}{{4{G_g}}} \end{array} \right\}\frac{{{d^2}{{\varepsilon ^{\prime}}_f}(x)}}{{d{x^2}}}.$$

λ_fg is a constant. Equation (12) is a typical second-order nonhomogeneous linear differential equation. Based on the assumption that the shear deformation of the metal layer is ignored, its existence necessarily causes a little strain reduction. Therefore, the strain in the external and internal ring of the metal layer is approximately equal.

(13)$${\varepsilon ^{\prime}_g}(x) \approx {\varepsilon _s}(x) = {\varepsilon _m} \cdot [1 - \frac{{\cosh ({x / {{\lambda _{sm}}}})}}{{\cosh (B/2{\lambda _{sm}})}}].$$

Combining the structural form of Eq. (13), the special solution ɛ'_f,p(x) can be assumed:

(14)$${\varepsilon ^{\prime}_{f,p}}(x) = {C_3} + {C_4} \cdot \cosh ({x / {{\lambda _{sm}}}}),$$

where C₃ and C₄ are the unknown coefficients. Substituting the special solution (14) and the nonhomogeneous term (13) into the differential Eq. (12), the unknown coefficients can be found to be:

(15)$${C_3} = {\varepsilon _m},{C_4} = \frac{{{\varepsilon _m}}}{{({{\lambda _{fg}^2} / {\lambda _{sm}^2}} - 1) \cdot \cosh ({B / {2{\lambda _{sm}}}})}}.$$

The differential Eq. (12) has a discriminant Δ greater than 0, hence it has a pair of real roots with opposite values. Its generalized solution can be expressed as the summation of the homogeneous solution and the special solution. Combining equations (12) to (15), the strain general solution for the fiber layer along the length direction x is:

(16)$${\varepsilon ^{\prime}_f}(x) = {C_5}{e^{\frac{x}{{{\lambda _{fg}}}}}} + {C_6}{e^{ - \frac{x}{{{\lambda _{fg}}}}}} + {\varepsilon _m} + \frac{{{\varepsilon _m} \cdot \cosh ({x / {{\lambda _{sm}}}})}}{{({{\lambda _{fg}^2} / {\lambda _{sm}^2}} - 1) \cdot \cosh ({B / {2{\lambda _{sm}}}})}},$$

where C₅ and C₆ are unknown coefficients that need to be solved by combining the boundary conditions of the fiber layer. According to Eq. (6), the boundary condition at both ends of the metal layer is 0. However, the boundary condition of the fiber in the gluing region cannot continue to be followed. Because the designed strained optical fiber cable exists the hanging region where the wFBG is arranged. The optical fiber is a continuous material throughout the cable. There is an external load at the end of the fiber in the glue region. It is impossible for the strain of the fiber to suddenly turn to 0. Since the fiber optic cable is a symmetric structure, only the deformation of its right side needs to be analyzed in this section. The subscript k is defined as the serial number of the gluing region on the right side of the cable. The boundary conditions at both ends of the fiber layer can be established:

(17)$${\varepsilon ^{\prime}_{f,k}}(x)|{_{x = k \cdot (H + L) \pm {\raise0.7ex\hbox{$L$} \!\mathord{/ {\vphantom {L 2}}}\!\lower0.7ex\hbox{$2$}}}} = {\delta _{sm}}{\eta _k}{\varepsilon _m},$$

where H and L are the length of the overhanging region and gluing layer of the optical fiber cable, respectively. The δ_sm coefficient is added to the above equation, which means that the reduction of the external strain transfer to the fiber in the overhanging region is considered. Meanwhile, η_k represents the percentage of residual strain of the fiber within half of the gluing layer, whose value satisfies the interval [0,1]. Dr. Falcetelli gave an answer for the percentage of residual strain combining principle of Saint Venant [21]. This article follows his conclusions.

(18)$${\eta _k} = \frac{{\tanh \frac{{k(H + L) \pm {\raise0.7ex\hbox{$L$} \!\mathord{/ {\vphantom {L 2}}}\!\lower0.7ex\hbox{$2$}}}}{{{\lambda _{fg}}}}}}{{\frac{{{\lambda _{fg}}}}{{{\lambda _g}}} + \tanh (\frac{{k(H + L) \pm {\raise0.7ex\hbox{$L$} \!\mathord{/ {\vphantom {L 2}}}\!\lower0.7ex\hbox{$2$}}}}{{{\lambda _{fg}}}})}},$$

where λ_g is a constant which can be expressed as:

(19)$${\lambda _g} = \sqrt {\frac{1}{{{G_t}}}(\frac{{{E_f}{A_f} - {E_t}{A_f}}}{{2\pi }}\ln \frac{{{r_t}}}{{{r_f}}} + {E_t}\frac{{r_t^2 - r_f^2}}{4})} .$$

Substituting the boundary conditions (17) into Eq. (16), the coefficients C₅ and C₆ can be resolved. Finally, the expression for the strain generic solution ɛ'_f,k(x) in the gluing layer region of the strained optical fiber cable is obtained as:

(20)$${\varepsilon ^{\prime}_{f,k}}(x) = {\varepsilon _m} \cdot [\frac{{{\delta _{sm}}{\eta _k} - 1 - \frac{{\cosh (\frac{{k(H + L) + {\raise0.7ex\hbox{$L$} \!\mathord{/ {\vphantom {L 2}}}\!\lower0.7ex\hbox{$2$}}}}{{{\lambda _{sm}}}})}}{{({{\lambda _{fg}^2} / {\lambda _{sm}^2}} - 1)\cosh ({B / {2{\lambda _{sm}}}})}}}}{{\cosh (\frac{{k(H + L) + {\raise0.7ex\hbox{$L$} \!\mathord{/ {\vphantom {L 2}}}\!\lower0.7ex\hbox{$2$}}}}{{{\lambda _{fg}}}})}}\cosh (\frac{x}{{{\lambda _{fg}}}}) + 1 + \frac{{\cosh ({x / {{\lambda _{sm}}}})}}{{(\frac{{\lambda _{fg}^2}}{{\lambda _{sm}^2}} - 1)\cosh (\frac{B}{{2{\lambda _{sm}}}})}}].$$

The equation in the middle parenthesis is defined as the symbol δ_fm,k, which represents the strain transfer coefficient of ɛ_m transferring to the fiber in the k-th gluing layer region.

2.3.4 Sensitivity of strained optical fiber cables based on wFBG

As shown in Fig. 1(b), the sensing cable utilizes the external deformation transfer to the fiber in the gluing layer region will have a strain transfer reduction to amplify the deformation of the fiber in the overhanging region. Thus, the design of the sensitized structure is realized. The strain amplification factor for each wFBG is determined by calculating the strain transfer reduction in the gluing layer. The strained optical fiber cable consists of multiple wFBGs, tensioned and arranged in suspension between protective layers. When subjected to an external strain of ɛ_m, the deformation u_f,k of the fiber in half of the gluing layer can be calculated by integrating along its half-length L/2.

(21)$${u_{f,k}} = \int_{k(L + H)}^{k(L + H) + {\raise0.7ex\hbox{$L$} \!\mathord{/ {\vphantom {L 2}}}\!\lower0.7ex\hbox{$2$}}} {{\varepsilon _m} \cdot {\delta _{fm,k}}dx} .$$

By combining Eq. (7) with Eq. (21), the differential deformation ΔH_k_{, k}_+ 1 between the optical fiber and the external structure in the overhang region between the k-th and k + 1-th adhesive layers is:

(22)$$\Delta {H_{k,k + 1}} = \int_{k(L + H) + {\raise0.7ex\hbox{$L$} \!\mathord{/ {\vphantom {L 2}}}\!\lower0.7ex\hbox{$2$}}}^{(k + 1)(L + H) - {\raise0.7ex\hbox{$L$} \!\mathord{/ {\vphantom {L 2}}}\!\lower0.7ex\hbox{$2$}}} {{\varepsilon _m} \cdot {\delta _{sm}}} dx + {\varepsilon _m}L - ({u_{f,k}} + {u_{f,k + 1}}).$$

The strain variation Δɛ acting on the wFBG induces linear offset of λ_B, which the relationship between the changes can be expressed as:

(23)$$\Delta \lambda \textrm{/}{\lambda _B}\textrm{ = }(1 - {P_e})\Delta \varepsilon,$$

where P_e and λ_B are the equivalent elastic optical coefficient and Bragg center wavelength of the fiber, respectively. The value of P_e for the optical fiber used in this research is 0.22 [29]. Since the grating wFBG is inscribed on the fiber in the overhang region, the wavelength shift of the wFBG in the overhang region can be obtained:

(24)$$\Delta {\lambda _k} = (1 - {P_e}){\lambda _B} \cdot \frac{{\Delta {H_{k,k + 1}}}}{H}.$$

According to the principle of modulation and demodulation in Section 2.1, the Bragg center wavelength values are set to alternately 1548 nm and 1556 nm. Finally, the theoretical sensitivity S_k of the designed strained optical fiber cable based on wFBG is:

(25)$${S_k} = \frac{{\Delta {\lambda _k}}}{{{\varepsilon _m}}} = (1 - {P_e}){\lambda _B} \cdot \frac{{\int_{k(L + H) + {\raise0.7ex\hbox{$L$} \!\mathord{/ {\vphantom {L 2}}}\!\lower0.7ex\hbox{$2$}}}^{(k + 1)(L + H) - {\raise0.7ex\hbox{$L$} \!\mathord{/ {\vphantom {L 2}}}\!\lower0.7ex\hbox{$2$}}} {{\delta _{sm}}} dx + L - \int_{k(L + H)}^{k(L + H) + {\raise0.7ex\hbox{$L$} \!\mathord{/ {\vphantom {L 2}}}\!\lower0.7ex\hbox{$2$}}} {{\delta _{fm,k}}dx} - \int_{(k + 1)(L + H)}^{(k + 1)(L + H) + {\raise0.7ex\hbox{$L$} \!\mathord{/ {\vphantom {L 2}}}\!\lower0.7ex\hbox{$2$}}} {{\delta _{fm,k}}dx} }}{H}.$$

2.4 Structural analysis of strained optical fiber cable

2.4.1 Finite element model

To illustrate the validity of the sensing optical fiber cable model based on the strain transfer theory, the finite element software ABAQUS was used for verification. In the simulation analysis, the slender optical fiber and the metal layer are analyzed using secondary beam element B32 and secondary shell element S8R, respectively. Beside these, others are meshed using the secondary hexahedral solid element C3D20R. As with the theoretical model, the different material layers are processed with a hard contact model without slip between them. The boundary condition for the whole model is the fixed end constraint on the right side of the concrete structure. The face loading of 6 N/mm² was applied on the left side. The measured structure strain ɛ_m is 1000 µɛ as the modulus of elasticity Em of C60 concrete is 60 GPa. The finite element model of the assembly is shown in Fig. 4. The specific structural parameters are adjusted accordingly based on the following analysis.

Fig. 4. Finite element model of strained optical fiber cable.

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2.4.2 Material parameters

Normally, sensing cable with higher strain sensitivity helps to capture micro strain variation in the external structure. In this section, the structure is established by analyzing the pivotal factors affecting the sensitivity through the appropriate dimensional parameters of the optical cable. According to Eq. (20), it can be found that the sensitivity is mainly related to the material parameters, gluing layer length L, and section size. The material selection and distribution of the strained optical fiber cable is shown in Fig. 1(a), whose relevant material parameters are shown in Table 1.

Table 1. Relevant material parameters

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2.4.3 Structural parameter analysis

Based on the requirements of the molding equipment and material technology, the following parameters of the structure can be determined. The cladding diameter r_f of the fiber, the diameter r_h of the cable, and the thickness t of the metal layer are limited to 125 µm, 3 mm, and 0.3 mm, respectively. To enhance the density of sensing deployment points for monitoring, the interval of the wFBG in the cable is required to be 1 m. Therefore, the summation of H and L is 1 m. To ensure the intensity of reflected light for WDM in demodulation, the wFBG length must be set to 20 mm. Consequently, length H must be more than 20 mm, while L must also be more than 20 mm for secure adhesion of the adhesive glue.

Firstly, the influence of the length B from the optical cable is discussed. Since the gluing layer is discontinuously distributed in the cable, the sensitivity equation is not convenient to discuss the effect of B. The optimal solution for B can be obtained by calculating the strain transfer efficiency δ_sm of the external structure. By integrating the material parameters from Table 1 with the assumed radius r_s of 1.5 mm, λ_sm can be calculated. The black line represents the theoretical analysis as well as the red line represents the simulation. The theoretical and simulation results are shown in Fig. 5. The shape of the distribution δ_sm are similar for different lengths of optical fiber cables. The smooth region of δ_sm expands as B grows. In addition, δ_sm decreases sharply close to the boundary region. Normally, the application of a sensing optical cable requires to abandon some of the measured points which are close to the boundary. Therefore, the designed strain-sensing optical cables are processed by connecting both ends to ordinary communication fibers of the same outside diameter. The communication fiber, lacking wFBG sensing points, serves to connect to the modulation and demodulation instrument for optical signal transmission. For the theoretical analysis, the sensor portion of the entire cable is away from the boundary condition region. δ_sm is all smooth and infinitely convergent to 100%. In the subsequent theoretical analysis, B can be assumed to be infinite.

Fig. 5. Relationship between strain transfer coefficient δ_sm and cable length B.

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Furthermore, the influence of the length L and the serial number k of the gluing layer for the sensitivity S_k is discussed. The serial number k takes values from 0 to 20, meaning that the sensitivity of 20 pcs wFBG are analyzed. The parameters r_t and r_g were set to be 0.5 mm and 1.2 mm, respectively. When r_g is determined, the outer structural parameters of the optical cable can be obtained since the thickness of the metal layer is limited to 0.3 mm. L is taken to be between 20 mm and 980 mm. The calculation of Eq. (25) using the mathematical software Origin is shown in Fig. 6. The sensitivity S_k is independent of the wFBG distributed location due to the infinite length of the optical cable, making B infinitely large. The cable with 30 m length B and 900 mm length L is simulated. The strain data along the optical fiber path is output without considering the effect of the access communication fiber during the numerical simulation. Based on Fig. 7, a conclusion similar to the theoretical analysis is reached. Excluding sensing points near the boundary condition region, strain values in other regions are consistent. After ɛ_m transfers to the gluing layers, each layer experiences the same external stress. Thus, the theoretical sensitivity S_k analysis need not consider the wFBG distributed location, making it independent of k.

Fig. 6. Relationship between sensitivity S_k and gluing layer length L.

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Fig. 7. Simulation result along the fiber path and strain cloud.

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The strain sensitivity S_FEA, which is defined to be obtained from the simulation result, is calculated according to the following equation:

(26)$${S_{\textrm{FEA}}} = (1 - {P_e}){\lambda _B} \cdot \frac{{{\varepsilon _{\textrm{wFBG}}}}}{{{\varepsilon _m}}},$$

where ɛ_wFBG is the fiber strain in the hanging region during simulation. The strain cloud in the hanging region of the strained optical fiber cable is shown in Fig. 7. The result show that its simulated strain value ɛ_wFBG is 1029 µɛ. Substituting into Eq. (26), the simulated sensitivity of the strained optical fiber cable can be obtained as 1.25 pm/µɛ. This numerical result is consistent with the theoretical analysis verifying the validity of the strain transfer theory. Continuing to analyze Fig. 6, the sensitivity increases exponentially as the length L gets longer. When the length L is set to 980 mm and 20 mm, the sensitivity S_k attains its maximum and minimum values of 1.434 pm/µɛ and 1.213 pm/µɛ, respectively. Additionally, the correlation between S_k variation and the length L remains notably low for L less than 800 mm.

The influence of the UV gluing layer radius r_g and the TPEE tight buffer radius r_t are theoretically analyzed with the addition of the variable of the gluing layer length L. The reason is that L has a significant effect on the sensitivity S_k, whose value might impact the relevant analysis of these parameters. L is taken between 20 mm and 980 mm. The effectiveness of r_g on S_k is discussed. Let the parameter r_t be 0.5 mm. r_g takes values between 1 mm and 2.5 mm. On the other hand, the effectiveness of r_t on S_k is discussed. Let the parameter r_g be 1.2 mm. r_t takes values between 0.2 mm and 1 mm. The calculated results are shown in Fig. 8. When L turns to be low, both r_g and r_t have insignificant effect on the sensitivity S_k, which can't even be analyzed by the demodulator. However, when L exceeds 900 mm, the values of r_g and r_t have a dramatic effect on the S_k. Increasing r_g or decreasing r_t can effectively increase S_k. As a conclusion, it is necessary to determine L before discussing how to set r_g and r_t. When the UV layer is short, their values can be set based on the convenience required by the production.

Fig. 8. Sensitivity S_k analysis. (a) UV gluing layer radius r_g effect; (b) TPEE tight buffer radius r_t effect.

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By designing the optical fiber structure based on the above information, if extremely high sensitivity needs to be pursued, it can be achieved by increasing the L. However, from formula (22), the strained optical fiber cable measures the average deformation within the scale range, which is the length H. When the gluing layer becomes too long it reduces the optical cable scale range. Sensing cable with low scale range has the following defects.

1. As a scale range of a sensor is reduced, the region which it can effectively monitor will be reduced. Additional sensors will be required to cover the same region, increasing the cost and complexity of the installation.
2. The low-scale sensor may detect worse signal quality in its particular region, which lead to inaccurate or misinterpreted data. For instance, in concrete engineering monitoring applications, even tiny cracks in the structure can cause distortion in the strained optical fiber cable with a long gluing layer.
3. The flexibility of the monitoring system is reduced. The arrangement of sensing points is a very critical task, where lower scale range will limit the flexibility of the monitoring system.

Therefore, in designing new strain optical cables, balancing the increased sensitivity with reduced scale range is crucial for optimal performance and cost-effectiveness. Combining actual field application needs is essential to find the optimal balance between these factors and determine the appropriate L.

In application scenarios where the above three defects are acceptable, increasing L of the gluing layer within the optical cable to 900 mm or more can effectively improve sensitivity. At the same time, increasing the r_g or decreasing the r_t, the sensitivity will be significantly improved. According to Fig. 8, the sensitivity can reach up to 1.55 pm/µɛ, which can realize 30% strain amplification compared to the ordinary strain sensors. It means that sensing optical cable, which transfers external deformation to the gluing layer, will incur strain transfer reduction magnifying the deformation of the fiber in the hanging region.

The application of this research is bridge structure monitoring. For concrete bridge, long gauge of sensing optical fiber cables should be chosen for building a high-density strain plane distributed matrix to invert the bridge strain as well as traffic information. Therefore, the minimum value of 20 mm is required for L. The values of r_g and r_t can be combined with the convenient requirements of the production. After the above analysis, the relevant structural parameters of the strained optical fiber cable based on the weak fiber Bragg grating array are shown in Table 2. According to the table data, it can be observed that the number of sensing points of the cable is 260 pcs, which means that 260 m of the cable is the sensing portion, while the remaining 40 m of the cable is for communication. Finally, the theoretical sensitivity S_k of the strained optical fiber cable is 1.21 pm/µɛ.

Table 2. Relevant structural parameters

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3. Testing and discussion of strained optical fiber cable

3.1 Test solution

The address for the performance test of strained optical fiber cable was chosen to be Dongmiao Bridge which is located at Ezhou Airport Highway in Hubei Province, China. The structure is a steel-concrete composite bridge with a span length of 65 meters. The scene photo of the Dongmiao Bridge is shown in Fig. 9(a). The solution of bridge dynamic and static loading experiments was used to test the performance of the designed strained optical fiber cable. Each wFBG wavelength value from the test on the strained optical fiber cable can be processed using formula (27) to obtain the strain value ɛ_k at the respective monitoring point.

(27)$${\varepsilon _k} = \frac{{{{\lambda ^{\prime}}_k} - {\lambda _k}}}{{{S_k}}},$$

where λ_k′ and λ_k are the real-time and initial values of the center wavelength of wFBG at the monitoring point, respectively. Further, the monitoring value ɛ_k is compared with the bridge calculated result ɛ_FEA of formula (26) to verify the effectiveness of the strained optical fiber cable in the monitoring progress. In this section, the bridge finite element software Midas Civil is used to simulate the dynamic and static loading experiments of Dongmiao Bridge. Initially, the cross-section characteristics of the concrete components in the upper part are equivalently modeled based on beam grid theory [30]. Following these, the steel girders and concrete components of the bridge are simulated using beam elements. The steel girders are rigidly connected to the concrete deck to simulate shear spike members. The optical cables are not simulated during the bridge modeling process since their presence essentially has insignificant effect on the structural stresses. The boundary conditions are set with the bridge having a hinged spherical bearing at one end and a vertically restrained bearing at the other. Different from static loading, the simulation of dynamic travel process of bridge is based on time history analysis method. For the mesh elements touched by the vehicle travelling through, the time history function of the triangular wave excitation is entered for each one according to the arrival time. Finally, the concrete strain data are extracted which consistent with the distributed location of optical fiber cable. The finite element model of Dongmiao Bridge is shown in Fig. 9(b).

Fig. 9. Dongmiao Bridge. (a) Engineering scene Photo; (b) finite element model.

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The optical cable was embedded within the concrete structure of the upper part in the bridge. Subsequently, the created recesses were filled with cured in C60 concrete. A burial depth of 10 cm was maintained to prevent any slippage or cracking of the concrete that may affect the optical cable. The arrangement of the strained optical fiber cable was shown in Fig. 10(a), which were integrally installed along the entire length of the four lanes in the bridge. The communication fiber was partially connected to the wFBG modulation and demodulation instrument at both ends. The instrument employed utilized a hybrid solution combining WDM with TDM, whose operational principle is illustrated in Fig. 2. It was developed independently by Wuhan University of Technology with a wavelength resolution of 0.1 pm and an accuracy of 1 pm. The sampling frequency of these tests was set to 10 Hz.

Fig. 10. Arrangement and installation of strained optical fiber cable. (a) Plane arrangement and numbering; (b) procedure for installing optical cable.

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The loading period was intentionally scheduled during night to obviate the need for temperature compensation in the data. At this time, the temperature variation was smooth without direct sunlight.

For the static and dynamic loading test, gravel trucks weighing 18 tons and 48 tons were selected as the loading method. During the static loading research, upon reaching the central lane of the bridge, λ_k and λ_k′ of each wFBG in the strained optical fiber cable were recorded respectively using modulation and demodulation instrument. In the dynamic loading research, vehicles drove back and forth along the second lane at a low velocity. Real-time data λ_k′ being recorded with instrument, curve smoothing was achieved through Fast Fourier Transformation (FFT). By substituting into Eq. (27), the strain measured by the wFBG was obtained. The vehicle loading conditions were detailed in Table 3. The velocity was defined as the symbol v.

Table 3. Loading test operation of Dongmiao Bridge

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3.2 Vehicle static loading test

To visualize the strain of the bridge being loaded, a strain matrix E was created to observe the global strain result. The elements of the matrix consist of strain data of the Eq. (27) after FFT smoothing, which comes from the wFBG. Serial number k is associated with Fig. 10 of the strained optical fiber cable arrangement. The row of the matrix represented the distribution along the span, while the column represented the four lanes of the bridge. The strain matrix E can be represented as:

(28)$${\textbf {E}} = \left[ {\begin{array}{{ccccc}} {{\varepsilon_{64}}}&{{\varepsilon_{63}}}& \ldots &{{\varepsilon_2}}&{{\varepsilon_1}}\\ {{\varepsilon_{67}}}&{{\varepsilon_{68}}}& \ldots &{{\varepsilon_{128}}}&{{\varepsilon_{129}}}\\ {{\varepsilon_{194}}}&{{\varepsilon_{193}}}& \ldots &{{\varepsilon_{131}}}&{{\varepsilon_{132}}}\\ {{\varepsilon_{197}}}&{{\varepsilon_{198}}}& \ldots &{{\varepsilon_{259}}}&{{\varepsilon_{260}}} \end{array}} \right].$$

The strain results obtained by wFBG testing, along with the simulated strain responses under various loading conditions from Table 3 were presented in Fig. 11, as we show in Dataset 1, (Ref. [31]). Comparative analyses of test, simulation, and smoothing were displayed on the left side of the figure, while the distribution of the strain matrix E (28) was presented on the right side.

Fig. 11. Static loading tests and FEA simulations for various level. (a) Static item 1; (b) Static item 2; (c) Static item 3; (d) Static item 4.

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Analyzing the resultant curves of the simulation in Fig. 11, the strained optical fiber cable arranged above the neutral axis of its cross-section exhibited compressive strain. In addition, the strain at the two end supports is close to 0. This phenomenon is because of the structural type of the bridge which is a simply supported beam. The simulation of static strain is related to the structure and loading location of the bridge as well as the span. Firstly, the experimental curves basically coincided with the simulation results based on beam grid theory as well as the maximum compressive strain ɛ_max were similar, which preliminarily verified the effectiveness of strained optical fiber cable. The error between ɛ_max and the finite element analysis for each item was within 10%. ɛ_max was associated with the lane containing the heaviest vehicle. For item 1 to item 3, the maximum strain ɛ_max was observed in the second lane, whereas for item 4, it occurred in the third lane. The reason for which is that the vehicle weight in the third lane significantly exceeds that in the third lane. Additionally, for the four loading conditions, ɛ_max was observed at the gravel truck placement location. In item 2, the corresponding location was at the optical fiber cable with k = 119, exhibiting a compressive strain value of -13.0 µɛ. This numbered location corresponded to the quarter-span position of the Dongmiao Bridge. While in the remaining conditions, the numbered positions can be approximately matched to the position of 1/2 span. The maximum compressive strain values were -10.5 µɛ, -25.4 µɛ and -32.2 µɛ, respectively. The above analysis demonstrated that the strained optical fiber cable based on wFBG can effectively perform traffic lane identification. As indicated by the strain matrix E, the traffic lane with the heavy vehicle exhibited higher strain compared to the other lanes.

Secondly, the identification of heavy vehicle tonnage was examined. An analysis of the loading conditions at the middle span revealed a linear amplification tendency in ɛ_max. Due to engineering environment constraints, arranging more vehicles for the experiment was not feasible. The functionality for recognizing multiple heavy vehicle tonnage was considered from the perspective of materials mechanics theory. When the strain matrix E of a simply supported bridge had been successfully captured, as well as the stiffness matrix K was established in conjunction with the geometric characteristics, material properties, and boundary conditions, the load matrix F can be solved by applying the linear elasticity theory equations [32].

To further demonstrate the effectiveness of strained optical fiber cables in functions like lane identification, heavy vehicle tonnage recognition, and velocity discrimination during health monitoring, dynamic vehicle testing was essential.

3.3 Vehicle dynamic loading test

The loaded vehicle traveled back and forth across the second lane of the bridge at a low rate, with item I and item II occurring sequentially. The vehicle loading rates were shown in Table 3. Moreover, the 18 tons and 48 tons of gravel trucks were traveling in opposite directions. The wavelength values of wFBG during loading were recorded in real time using a modulation and demodulation instrument. These values were substituted into Eq. (27) to determine the strain measured by the wFBG. The time history data for the sensing points at the middle span and quarter span of the bridge sections were output in Fig. 12. Analyzing the measured data in Fig. 12, the real time strain of the dynamic loading process was basically in accordance with the simulated values. In addition, the strain curve of the bridge section conformed to the distribution shape of the influence line. When the vehicle traveled to the vicinity of the measuring point, its strain reached the maximum value. The dynamic loading test showed that the strained optical fiber cable was able to monitor the effective strain data during the traveling process.

Fig. 12. Dynamic loading tests and FEA simulations. (a) 1/2 span; (b) 1/4 span; (c) 3/4 span.

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An interesting phenomenon could be found that the FEA result of the second lane didn't overlap properly with the experimental data. Sudden strain variation was observed in the second lane being loaded. The reason was that the external forces acting on the wFBG when the wheels moved into normal contact with the optical cable beneath the bridge deck, which were not solely come from bridge deformation. Crushing of the wheels likewise caused the wavelength shift of the wFBG, resulting in strain distortion in the test. This phenomenon was hard to simulate. However, it was possible to determine the lane in which the vehicle was located. The lanes without direct vehicle contact exhibited remaining wFBG that accurately reflected the overall strain in the bridge. Figure 12 was illustrated that the strained optical fiber cable based on wFBG can effectively realize the function of lane recognition, as we show in Dataset 1, (Ref. [31]). Furthermore, the heavy vehicle tonnage identification for the static loading process utilized the strain matrix E of Eq. (28). However, direct computation of the load matrix F using E inaccurately amplified the tonnage due to wFBG contact distortion. The solution was that FFT smoothing eliminated the sudden strain. The load matrix F from all frames within a one-second interval was selected for averaging, ultimately determining the correct tonnage of the heavy vehicle.

The trough in the real time strain curve indicated the moment when vehicle traveled over the corresponding wFBG. There was a time difference observed between the i-th and j-th wFBG strains in reaching the trough along the same lane. The vehicle rate v was determined by dividing the distance Δl_i_,j between the both wFBG segments by their time difference Δt_i_,_j. According to the static loading analysis and Fig. 11, it was known that the strain variation at the beginning and end of the bridge was not obvious. Therefore, the quarter span of the bridge is selected as the location of the speed measurement. Subsequently, the 4 pcs rate values on the same section were averaged. The vehicle rate $\nu$ could be obtained.

(29)$$\bar{v} = \frac{1}{4} \cdot \sum {\frac{{\Delta {l_{i,j}}}}{{\Delta {t_{i,j}}}}} .$$

To prove that the strained optical fiber cable had a velocity discrimination function, the real time data at the quarter span in the first lane were shown in Fig. 13, as we show in Dataset 1, (Ref. [31]). Meanwhile, data at crucial moments were magnified to more effectively illustrate the differences in the troughs. The remaining lanes of the curve were similar in shape. Comparing the actual loading velocity in Table 3, the error for all vehicles was within 10%. The designed velocity discrimination function, in conjunction with a standard camera, was utilized to develop a vehicle speeding monitoring system for the intelligent highway.

Fig. 13. Dynamic loading test at the quarter span in the first lane.

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Finally, the installed strained optical fiber cable based on wFBG enabled the establishment of the health monitoring system for the Dongmiao Bridge. The maximum strain value at each monitoring point by the designed load was calculated using FEA software to serve as the monitoring threshold. The strained optical fiber cable was applied to monitor the bridge in real time to obtain the strain matrix E. When most of the elements in the matrix E exceeded the monitoring threshold, the lane identification and tonnage calculated algorithm was executed. In turn, various known bridge damage models were employed to assess the safety status of the bridge [33].

4. Conclusions

In this research, a novel wFBG strained optical fiber cable based on wavelength periodical switching was designed. The sensitivity model of the wFBG which located in the hanging region was established by calculating the strain transfer reduction and deformation coordination in the gluing region. Compared to conventional encapsulated cable, this structural style exhibited high sensing density, tunable sensitivity, and no chirp. Meanwhile, the optimal dimensions of the optical fiber cable for various application scenarios were obtained by the parametric analysis method, which presented the limiting application of the optical cable model with high sensitivity. Extensive engineering experiments showed that the monitoring data of the optical cable approached the simulated result. The proposed theory and method could be effectively used in the design of the wFBG strained optical fiber cable. In addition, a health monitoring system enabling lane identification, heavy vehicle tonnage, as well as velocity determination was established for the Dongmiao Bridge. The solution was provided for the innovation of special optical fiber cable and the construction of intelligent highway.

Funding

National Key Research and Development Program of China (2021YFB3202901); Major Program (JD) of Hubei Province (2023BAA017); National Natural Science Foundation of China (61875155).

Acknowledgments

L. N. Yue representing Wuhan University of Technology thanks FENGLI Optoelectronics Technology Company Limited for helping to produce the optical fiber cable and testing analysis.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are available in Dataset 1, Ref. [31].

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33. M. Artus and C. Koch, “State of the art in damage information modeling for RC bridges - A literature review,” Adv. Eng. Inform. 46, 101171 (2020). [CrossRef]

Symbol	Denotation	Value	Unit
E_m	Elastic modulus of C60 concrete	60	GPa
E_f	Elastic modulus of fiber	72	GPa
E_t	Elastic modulus of TPEE tight buffer	1000	MPa
G_t	Shear modulus of TPEE tight buffer	384.6	MPa
E_g	Elastic modulus of UV glue	290	MPa
G_g	Shear modulus of UV glue	111.5	MPa
E_s	Elastic modulus of metal	190	GPa
E_h	Elastic modulus of HDPE	680	MPa
G_h	Shear modulus of HDPE	261.5	MPa

Static item	Loading vehicle	Loading position	Dynamic item	Loading vehicle	Loading velocity
0	#N/A	No loading	I	18 tons (Second lane)	Forward v: 6 km/h. Reverse v: 2.2 km/h
1	18 tons (Second lane)	Middle Span
2	48 tons (Second lane)	Quarter Span
3	48 tons (Second lane)	Middle Span	II	48 tons (Second lane)	Reverse v: 5 km/h Forward v: 10 km/h.
4	18 tons (Second lane) 48 tons (Third lane)	Middle Span	II	48 tons (Second lane)	Reverse v: 5 km/h Forward v: 10 km/h.

Symbol	Denotation	Value	Unit
E_m	Elastic modulus of C60 concrete	60	GPa
E_f	Elastic modulus of fiber	72	GPa
E_t	Elastic modulus of TPEE tight buffer	1000	MPa
G_t	Shear modulus of TPEE tight buffer	384.6	MPa
E_g	Elastic modulus of UV glue	290	MPa
G_g	Shear modulus of UV glue	111.5	MPa
E_s	Elastic modulus of metal	190	GPa
E_h	Elastic modulus of HDPE	680	MPa
G_h	Shear modulus of HDPE	261.5	MPa

Static item	Loading vehicle	Loading position	Dynamic item	Loading vehicle	Loading velocity
0	#N/A	No loading	I	18 tons (Second lane)	Forward v: 6 km/h. Reverse v: 2.2 km/h
1	18 tons (Second lane)	Middle Span
2	48 tons (Second lane)	Quarter Span
3	48 tons (Second lane)	Middle Span	II	48 tons (Second lane)	Reverse v: 5 km/h Forward v: 10 km/h.
4	18 tons (Second lane) 48 tons (Third lane)	Middle Span	II	48 tons (Second lane)	Reverse v: 5 km/h Forward v: 10 km/h.

Research on distributed strain monitoring of a bridge based on a strained optical cable with weak fiber Bragg grating array

Abstract

1. Introduction

2. Design of wFBG optical fiber cable

2.1 Modulation and demodulation principle

2.2 Structural design of the strained optical fiber cable

2.3 Derivation of strain transfer efficiency

2.3.1 Basic assumptions

2.3.2 Analysis of the outer protective layer

2.3.3 Strain transfer efficiency in the gluing layer region

2.3.4 Sensitivity of strained optical fiber cables based on wFBG

2.4 Structural analysis of strained optical fiber cable

2.4.1 Finite element model

2.4.2 Material parameters

2.4.3 Structural parameter analysis

3. Testing and discussion of strained optical fiber cable

3.1 Test solution

3.2 Vehicle static loading test

3.3 Vehicle dynamic loading test

4. Conclusions

Funding

Acknowledgments

Disclosures

Data availability

References

Supplementary Material (1)

Data availability

Cited By

Figures (13)

Tables (3)

Equations (29)

Optics Express

Symbol	Denotation	Value	Unit
B	Cable length	300	m
k	Maximum serial number of gluing layer	260	Pcs
H	Length of hanging region	980	mm
L	Length of gluing layer region	20	mm
r_f	Diameter of optical fiber	125	µm
r_t	Radius of TPEE tight buffer	0.5	mm
r_g	Radius of gluing layer	1.2	mm
r_s	Radius of metal layer	1.5	mm
r_h	Radius of HDPE layer	3	mm
S_k	Strain sensitivity	1.21	pm/µɛ