Ground Calibration and Airborne Physical-Consistency Assessment of Composite-Wing Sectional Loads Using Fiber Bragg Grating Sensors

Featured Application

The proposed FBG-based calibration framework can be applied to reconstruct bending moment, shear force, and torque at multiple spanwise sections of full-scale composite aircraft wings during flight. Its ground-to-flight transfer capability supports operational load monitoring, fatigue-spectrum development, structural health assessment, and condition-based maintenance when direct airborne measurement of sectional loads is unavailable.

Abstract

This study develops a fiber Bragg grating (FBG)-based method for identifying sectional loads on a full-scale RX1E-A composite aircraft wing. Thirty-two load-sensitive FBGs were installed at four spanwise sections. Multi-case ground loading with different spanwise distributions and chordwise positions was used to generate coupled combinations of bending moment, shear force, and torque. Multichannel linear models were calibrated using 89 measured samples from Cases 1 to 9 and independently evaluated using interpolation-oriented Cases 10–11 and limited-extrapolation Cases 12–13. Electrical strain gauges were used only for auxiliary ground-response comparison. The calibration NRMSE ranges were 0.454–1.087% for bending moment, 1.213–3.351% for shear force, and 0.526–1.306% for torque. Across the independent cases, the maximum section-level NRMSEs were 1.949%, 7.251%, and 2.484%. The fixed models were subsequently applied to a nominally identical flight-test wing without airborne refitting. During a pull-up maneuver with a maximum normal load factor of 2.058, the reconstructed bending-moment and shear-force increments exhibited continuous responses, extrema close to the load-factor peak, and a physically reasonable decrease from the wing root toward the tip.

1. Introduction

Aircraft structures experience repeatedly varying aerodynamic, inertial, and operational loads throughout service, and their accumulation governs fatigue damage, structural risk, and residual life [1,2,3]. Reliable load histories are, therefore, essential for fatigue assessment, service-load-spectrum development, and condition-based maintenance. These requirements are especially important for composite airframes, whose anisotropy, manufacturing variability, and multiple failure modes complicate structural-health assessment [4]. Because sectional wing loads, including bending moments, shear forces, and torsional moments, cannot generally be measured directly in flight, they are commonly reconstructed from measurable structural responses through ground-derived calibration relationships [5].

Experimentally calibrated methods traditionally rely on electrical resistance strain gauges installed at selected locations. Regression or hybrid numerical-experimental procedures are then used to relate measured strains to target loads [6,7,8]. Although strain gauges provide mature measurements, large sensing networks require extensive wiring, signal conditioning, and multiple acquisition channels, increasing installation mass and complexity. Model-based inverse approaches reconstruct unknown loads from measured strain fields [9,10], but their performance depends on load representation, structural-model fidelity, boundary conditions, temperature treatment, and consistency between calibration and operational configurations.

These challenges are more pronounced for composite wings, where laminate-dependent stiffness, local details, and large-deformation effects complicate the relationship between local strain and global response [11,12]. Multi-input identification methods have, therefore, been introduced to exploit several sensing channels simultaneously and improve separation of coupled load components. Fiber Bragg grating (FBG) sensors are well suited to this task because of their small size and mass, immunity to electromagnetic interference, multiplexing capability, and compatibility with composite structures. Their wavelength response, however, is affected by both mechanical strain and temperature [13]. Dense FBG networks have been used for wing-shape reconstruction, while ground and flight demonstrations have shown that optical strain measurements can support load estimation. Neural-network-assisted optical sensing has also been investigated for identifying wing load and angle of attack [14,15,16].

Optical-fiber systems have further been demonstrated in flight for monitoring fuselage, bulkhead, and other aircraft structural responses [17,18,19,20]. Their engineering performance nevertheless depends on sensor bonding, strain transfer, interrogation stability, and calibration quality. Pre-flight static loading is, therefore, important for checking numerical models, sensor responses, structural integrity, and the monitoring procedure. Validation studies have likewise emphasized comparison with known ground-test loads and shapes before operational deployment [21].

Despite substantial progress in FBG-based aircraft structural monitoring, several challenges remain in the identification of sectional loads on full-scale composite wings. Previous studies have demonstrated the feasibility of using FBG sensors for composite-wing load monitoring and in-flight structural-response measurement [22,23,24]. Nevertheless, the resulting strain-to-load relationships are generally specific to the tested structure, sensor layout, installation condition, and prescribed loading domain. In particular, calibration based on a single loading path, a fixed resultant-load position, or the separate identification of individual load components may not provide sufficient information to distinguish the coupled effects of bending moment, shear force, and torsional moment. Moreover, identification performance is often assessed primarily in terms of calibration goodness of fit, whereas the predictive capability of a fixed model under completely withheld loading conditions receives comparatively less attention. Independently measured sectional-load references are also rarely available during flight. Airborne results must, therefore, be distinguished from direct quantitative validation and, instead, be evaluated using synchronized flight parameters, alternative sensing methods, or physically expected structural-response and load-transfer characteristics [25]. Furthermore, transferring a fixed ground-calibrated model between nominally identical but physically different structural articles remains challenging because manufacturing variability, sensor-installation differences, and environmental effects may alter the measured strain–load relationship.

The present study does not propose a new FBG sensing principle. Instead, its principal contribution is an experimentally interpretable framework that integrates multi-condition ground calibration, simultaneous identification of coupled sectional-load components, independent predictive verification, and cautious airborne application. Unlike calibration procedures based on a single loading path or a fixed resultant-load position, the proposed ground-test program varies both the spanwise load distribution and the chordwise resultant-load position to generate distinguishable combinations of bending moment, shear force, and torsional moment. Multichannel responses from 32 load-sensitive FBGs are then used to simultaneously reconstruct these three load components at four spanwise sections. Completely withheld ground-loading cases are used to evaluate the predictive performance of the fixed models, with interpolation-oriented and limited-extrapolation conditions assessed separately rather than relying solely on calibration goodness of fit. Finally, the ground-derived models are applied, without airborne refitting, to a nominally identical but physically different flight-test wing. Because independent airborne sectional-load references were unavailable, the flight results are assessed in terms of signal continuity, temporal consistency with synchronized flight parameters, and physically expected spanwise load-transfer characteristics, rather than being presented as a direct validation of load-reconstruction accuracy.

2. Principle of FBG-Based Wing Load Identification

2.1. Mechanical Representation of Wing Sectional Loads

To establish the relationship between the FBG wavelength responses and the loads carried by the wing, the external ground loads are represented by the sectional bending moment, shear force, and torque at four selected spanwise sections.

The aircraft-fixed Cartesian coordinate system is shown in Figure 1. The origin is located at the intersection of the fuselage structural horizontal plane, the aircraft symmetry plane, and the nose reference plane. The x-axis points rearward along the fuselage, the y-axis points vertically upward, and the z-axis points toward the left wing. Therefore, the z-coordinate represents the spanwise position, whereas the x-coordinate represents the chordwise position.

For the j-th test section, the sectional load vector is defined as

$$L_j={\left[\begin{array}{ccc}{M}_j& Q_j& T_j\end{array}\right]}^T$$

(1)

where M_j is the bending moment about the aircraft x-axis, Q_j is the vertical shear force, and T_j is the torque about the spanwise z-axis. The signs of the applied forces and sectional loads follow the coordinate and moment conventions indicated in Figure 1 and are kept consistent throughout ground calibration and airborne load reconstruction.

As illustrated in Figure 2, the wing is represented at the global level as a cantilever-type thin-walled structure attached to the fuselage. For a given test section, the sectional loads balance the external forces acting on the outboard portion of the wing. Consequently, an inboard section carries the accumulated effect of a larger outboard region, and the bending moment and shear force generally decrease from the wing root toward the wing tip.

This cantilever representation is used only to calculate and interpret the global sectional loads. It is not intended to reproduce the detailed local stress and strain fields of the composite wing. The effects of the laminate anisotropy, spars, ribs, joints, local reinforcements, and bending–shear–torsion coupling remain in the measured FBG responses and are incorporated into the experimentally calibrated wavelength-to-load coefficients.

During ground calibration, known masses are applied at discrete spanwise and chordwise loading positions. The signed vertical force at loading point i is expressed as

$$P_i=s_i{m}_{i}g$$

(2)

where m_i is the applied mass, g is the gravitational acceleration, and s_i specifies the force direction according to the adopted sign convention.

For a test section located at the spanwise coordinate z_j, the vertical shear force is obtained by summing the applied forces located outboard of the section:

$$Q_j={\displaystyle \sum _{i\in {\Omega }_j}{P}_i}$$

(3)

where ${\Omega }_j$ denotes the set of loading points located outboard of Section j.

The corresponding bending moment is calculated from the spanwise force arms:

$$M_j={\displaystyle \sum _{i\in {\Omega }_j}{P}_i}\left(z_i-z_j\right)$$

(4)

where z_i is the spanwise coordinate of loading point i.

The sectional torque is calculated from the chordwise offset between each applied force and the selected torsional reference position:

$$T_j={\displaystyle \sum _{i\in {\Omega }_j}{P}_i}\left(x_i-x_{ref,j}\right)$$

(5)

where x_i is the chordwise coordinate of loading point i, and x_ref,j is the chordwise coordinate of the prescribed torsional reference position at Section j. Ideally, this position corresponds to the local shear center. If the shear-center position is not available, a clearly defined structural reference line may be used, provided that the same reference is retained throughout load calculation, ground calibration, and airborne load reconstruction.

The discrete ground-loading arrangement is designed to generate representative combinations of sectional bending moment, shear force, and torque rather than to reproduce the complete aerodynamic pressure field encountered during flight. Changes in the spanwise force distribution modify the relationship between M_j and Q_j, whereas changes in the chordwise loading position primarily modify T_j. These calculated sectional loads are paired with the simultaneously measured FBG wavelength changes to establish the multichannel calibration model described in Section 2.3.

The above representation assumes that the wing remains within the elastic range covered by the ground tests, the wing-to-fuselage connection provides effective global restraint, and no significant fixture slip, structural damage, irreversible deformation, or sensor debonding occurs during loading.

2.2. FBG Sensing and Wavelength Processing

Fiber Bragg grating sensors are used to measure the structural response of the composite wing. An FBG consists of a periodic modulation of the refractive index along the core of an optical fiber. When broadband light propagates through the fiber, a narrow wavelength band satisfying the Bragg condition is reflected, while the remaining wavelengths are transmitted. The Bragg wavelength is expressed as

$${\lambda }_B=2n_{eff}\Lambda$$

(6)

where λ_B is the Bragg wavelength, n_eff is the effective refractive index of the fiber core, and λ is the grating period.

When the FBG is bonded to the wing structure, deformation of the wing changes both the grating period and the effective refractive index. Consequently, the structural strain is represented by a shift in the measured Bragg wavelength. Temperature variations also affect the Bragg wavelength through thermal expansion and the thermo-optic effect. For small variations in strain and temperature, the normalized wavelength shift can be expressed as

$${\displaystyle \frac{\Delta {\lambda }_B}{{\lambda }_{B,0}}}=\left(1-p_e\right)\epsilon +\left({\alpha }_f+\xi \right)\Delta T$$

(7)

where ${\lambda }_{B,0}$ is the reference Bragg wavelength, $\Delta {\lambda }_B$ is the measured wavelength change, p_e is the effective photoelastic coefficient, $\epsilon$ is the strain transferred to the grating, ${\alpha }_f$ is the effective thermal-expansion coefficient, $\xi$ is the thermo-optic coefficient, and $\Delta T$ is the temperature variation relative to the reference condition.

The effective wavelength response of a surface-bonded FBG is also influenced by the coating, adhesive layer, installation direction, host-material properties, and strain-transfer efficiency. Therefore, nominal bare-fiber sensitivities are not used to convert wavelength changes directly into sectional loads. Instead, the combined effects of sensor installation and the as-built composite-wing response are incorporated into the experimentally determined calibration coefficients.

Multiple FBGs with different initial Bragg wavelengths can be arranged along one optical fiber and interrogated simultaneously through wavelength-division multiplexing. This enables multi-point wing measurements with substantially less wiring than an equivalent electrical strain-gauge network.

Because the multiplexed sensors have different nominal center wavelengths, the absolute wavelength is not used directly as the model input. For the i-th FBG, the baseline wavelength is calculated from a stable reference interval containing N₀ samples:

$${\overline{\lambda }}_{i,0}={\displaystyle \frac{1}{N_0}}{\displaystyle \sum _{n=1}^{N_0}{\lambda }_i}\left(t_n\right)$$

(8)

where ${\lambda }_i(t_n)$ is the baseline wavelength of the i-th sensor and t_n denotes the sampling time within the reference interval.

The relative wavelength change is then defined as

$$\Delta {\lambda }_i\left(t\right)={\lambda }_i\left(t\right)-{\overline{\lambda }}_{i,0}$$

(9)

For ground calibration, the baseline is determined from a stable unloaded interval before each loading sequence. For airborne measurements, it is determined from a stable preflight interval after the interrogation system has reached a steady operating condition. The same wavelength unit and baseline-processing procedure are used during ground calibration and flight application.

No independent strain-free temperature reference was available for every load-sensitive FBG channel. Therefore, ambient-temperature effects were considered through baseline referencing and thermal-drift monitoring rather than through complete strain–temperature decoupling. For each ground-loading cycle or selected flight interval, the wavelength change was calculated relative to a stable reference state. Temperature-monitoring FBGs were used to identify evident environmental drift; however, their responses were not directly subtracted from the load-sensitive channels because complete mechanical-strain isolation and channel-specific thermal calibration were unavailable. This procedure removes the initial wavelength offset and reduces the influence of slow thermal drift. It is suitable for the relatively short loading sequences and selected flight maneuvers considered in this study, during which the thermal variation was small relative to the mechanical response. Residual thermal drift and spatially nonuniform temperature changes remain sources of uncertainty, particularly for long-duration flight measurements.

After baseline correction, temperature treatment, and data-quality screening, the processed wavelength-change vector for the j-th wing section is defined as

$$\Delta {\lambda }_j\left(t\right)={\left[\begin{array}{cccc}\Delta {\lambda }_{j,1}\left(t\right)& \Delta {\lambda }_{j,2}\left(t\right)& \cdots & \Delta {\lambda }_{j,k_j}\left(t\right)\end{array}\right]}^T$$

(10)

where k_j is the number of valid FBG sensing channels associated with Section j. This multichannel wavelength-response vector is used as the input to the wavelength-to-load identification model developed in Section 2.3.

2.3. Multichannel Wavelength-to-Load Model

Because the FBGs installed at one wing section may respond simultaneously to bending, shear, and torsional deformation, no individual wavelength channel is assumed to correspond exclusively to one load component. Instead, the processed wavelength changes from multiple FBGs are combined to reconstruct the sectional loads.

Using the wavelength-response vector defined in Equation (10), the sectional load increment at Section j is expressed as

$$\Delta L_j(t)={[\Delta M_j(t)\hspace{1em}\Delta Q_j(t)\hspace{1em}\Delta T_j(t)]}^T$$

(11)

where the load components are defined relative to the reference state used to determine the baseline wavelengths. Within the elastic range covered by the ground-calibration tests, the wavelength-to-load relationship is approximated by

$$\Delta L_j(t)=B_j\Delta {\lambda }_j(t)+b_j$$

(12)

where $B_j\in {ℝ}^{3\times k_j}$ is the calibration coefficient matrix and $b_j\in {ℝ}^3$ is the intercept vector. Each row of B_j contains the coefficients used to identify the bending moment, shear force, or torque from all valid FBG channels associated with Section j. The model, therefore, incorporates, rather than neglects, the coupled wavelength responses of the composite wing.

For N ground-calibration samples, the augmented wavelength-response matrix and sectional-load matrix are assembled as

$$X_j=\left[\begin{array}{cccc}\Delta {\lambda }_j^{(1)}& \Delta {\lambda }_j^{(2)}& \dots & \Delta {\lambda }_j^{(N)}\\ 1& 1& \dots & 1\end{array}\right],\hspace{1em}{Y}_j=\left[\begin{array}{cccc}\Delta L_j^{(1)}& \Delta L_j^{(2)}& \dots & \Delta L_j^{(N)}\end{array}\right]$$

(13)

Defining the augmented coefficient matrix as

$$C_j=\left[\begin{array}{cc}{B}_j& b_j\end{array}\right]$$

(14)

The calibration equation for all samples can then be written as

$$Y_j=C_j{X}_j+E_j$$

(15)

where $E_j$ is the residual-error matrix. The least-squares estimate is obtained using the Moore–Penrose pseudoinverse:

$${\widehat{C}}_j=Y_j{X}_j^{+}$$

(16)

The pseudoinverse form avoids requiring direct inversion of the wavelength-response matrix and is applicable when multiple FBG channels exhibit partially correlated responses. The intercept is retained to account for small residual offsets after baseline correction; its magnitude should remain small relative to the calibrated load range.

After calibration, the sectional load increment is reconstructed as

$$\Delta {\widehat{L}}_j(t)={\widehat{B}}_j\Delta {\lambda }_j(t)+{\widehat{b}}_j$$

(17)

Because the wavelength changes are referenced to an unloaded state during ground calibration and to a stable preflight state during airborne measurements, Equation (17) directly provides the load variation relative to the corresponding reference condition. If the sectional load at the reference state, L_j,0, is independently known, the absolute sectional load can be obtained from

$${\widehat{L}}_j(t)=L_{j,0}+\Delta {\widehat{L}}_j(t)$$

(18)

Otherwise, the airborne results should be reported as sectional load increments relative to the preflight baseline rather than as absolute flight loads.

A stable calibration requires the ground-loading cases to generate sufficiently different combinations of bending moment, shear force, and torque. Increasing only the total force along one fixed loading path changes the load magnitude but does not provide an independent load combination. In this study, the spanwise force distribution is varied to change the relationship between the bending moment and shear force, while the relative loading near the leading and trailing edges is varied to change the torsional moment. These multi-condition loading cases improve the separability of the three sectional-load components and reduce coefficient sensitivity to correlated wavelength responses.

Poisson’s ratio was not introduced as an explicit independent variable in the identification model. The reference sectional loads were calculated from the applied forces and their geometric lever arms using static equilibrium and, therefore, did not depend on the elastic constants of the wing material. Because the wavelength-to-load relationship was calibrated directly from the multichannel FBG measurements of the full-scale composite wing, the effects of Poisson contraction, laminate anisotropy, structural geometry, and strain-component coupling were implicitly included in the measured responses and regression coefficients. In particular, the model was calibrated using multiple FBGs with different positions and sensing orientations rather than calculating the sectional loads from a single uniaxial strain measurement using an assumed scalar Poisson’s ratio.

The identified coefficients are specific to the as-built RX1E-A wing, including its material system, laminate configuration, structural geometry, sensor arrangement, bonding condition, reference-axis definition, support condition, preprocessing procedure, and calibrated load domain. Application to substantially higher loads, different chordwise load positions, significant thermal conditions not represented during calibration, structural damage, sensor replacement, or a substantially different laminate configuration constitutes extrapolation. Such changes would require recalibration or independent verification. Quantitative accuracy is, therefore, evaluated using independent ground-loading cases with known loads, whereas airborne measurements without an independent sectional-load reference are used to assess signal continuity, spanwise load-transfer behavior, and consistency with synchronized flight parameters.

3. Ground Calibration and Independent Verification of the FBG Wing Load Monitoring System

3.1. Full-Scale Wing Test Article and Sensor Configuration

Ground calibration was conducted on a full-scale RX1E-A composite wing. The RX1E-A has a wingspan of 14.5 m, wing area of 12 m², and aspect ratio of 17.52. The full-scale ground-test article preserved the as-built effects of the composite skins, spars, ribs, stiffeners, structural joints, and local reinforcements. Because different wing articles were used for the ground calibration and flight testing, article-to-article variations in the structural stiffness, manufacturing tolerances, and sensor installation were treated as potential sources of uncertainty in applying the ground-derived calibration model to the flight data. The ground-test wing was restrained through its original root-attachment interfaces using a dedicated fixture that approximated the global restraint and load-transfer conditions of the installed wing. Because the fixture stiffness was not independently matched to that of the actual wing-fuselage connection, the ground boundary condition was regarded as an engineering approximation of the in-service wing-root restraint.

The ground-test system consisted of the full-scale RX1E-A composite wing, a multi-point loading system, an FBG sensing network, a four-channel optical interrogator, and a data-acquisition computer, as illustrated in Figure 3a. Four monitoring sections were arranged along the wing span at nominal coordinates of 450, 1550, 2600, and 4500 mm relative to the coordinate origin defined in Section 2.1 (Figure 3b). The sections were numbered from Section 1 near the wing root to Section 4 toward the wing tip, and eight load-sensitive FBGs were assigned to each section. This arrangement enabled the spanwise variation in the sectional loads to be characterized. Under the downward ground-loading conditions, the inboard sections carried loads acting over larger outboard regions and, therefore, developed greater bending moments and shear forces than the outboard sections. The monitoring-section coordinates and their relationships with the five loading stations are summarized in

Table 1. Configuration of the four wing monitoring sections.

Section	Spanwise Coordinate zj/mm	Upper-Surface Load FBGs	Lower-Surface Load FBGs	Number of Load FBGs	Principal Measured Responses
Section 1	450	M1-1T, S1-1T, S1-2T, M1-2T	M1-1D, S1-1D, S1-2D, M1-2D	8	Bending, shear and torsion
Section 2	1550	M2-1T, S2-1T, S2-2T, M2-2T	M2-1D, S2-1D, S2-2D, M2-2D	8	Bending, shear and torsion
Section 3	2600	M3-1T, S3-1T, S3-2T, M3-2T	M3-1D, S3-1D, S3-2D, M3-2D	8	Bending, shear and torsion
Section 4	4500	M4-1T, S4-1T, S4-2T, M4-2T	M4-1D, S4-1D, S4-2D, M4-2D	8	Bending, shear and torsion

Notes: 1. The suffixes T and D denote the upper and lower wing surfaces, respectively. 2. The prefix M denotes a predominantly bending-sensitive FBG, while S denotes a shear-sensitive FBG. Torque is identified from the combined responses of multiple channels rather than from a single dedicated torque sensor.

.

Five loading stations, comprising ten loading points, were arranged along the wing span. Known masses were applied at prescribed spanwise and chordwise positions to generate different combinations of bending moment, shear force, and torsional moment. During each loading case, the FBG wavelengths were recorded throughout the stepwise loading, stable load–hold, and unloading stages. The wavelength changes extracted from the stable load–hold intervals were, subsequently, related to the sectional loads calculated from the applied masses and loading-point coordinates. The wavelength-division-multiplexed FBG network and the local sensor arrangement at a representative monitoring section are shown in Figure 3c, while the detailed ground-loading configuration is presented in Figure 4.

The optical sensing network comprised 52 FBGs, including 32 load-sensitive sensors for sectional-load identification, 10 sensors for deformation monitoring, and 10 sensors for temperature measurement. Eight load-sensitive FBGs were installed at each monitoring section, with four on the upper surface and four on the lower surface. At each surface, two longitudinal sensors, denoted by M, primarily measured bending-induced strain, whereas two sensors, denoted by S, were arranged in shear-sensitive orientations to capture responses associated with shear and torsion. Sectional torque was reconstructed from the combined responses of the shear-sensitive channels rather than from a single dedicated torsion sensor. Sensor identifiers, initial Bragg wavelengths, installation locations, sensing directions, and optical-channel assignments are listed in Supplementary Table S1.

Commercially manufactured Luna os1100 single-point FBG sensors (Luna Innovations, Blacksburg, VA, USA) were used in this study. According to the manufacturer’s specifications, each os1100 contains a single FBG centered in a 2 m length of SMF28-compatible, polyimide-coated optical fiber. The nominal grating length was 10 mm, and the nominal strain sensitivity was approximately 1.2 pm/με. The polyimide coating facilitates strain transfer through the fiber coating to the FBG in the fiber core and provides protection over a relatively wide operating-temperature range. The sensors were selected with different initial Bragg wavelengths to enable wavelength-division multiplexing. In the present system, the initial Bragg wavelengths ranged from approximately 1534 to 1563 nm. The detailed laser-inscription procedure and associated manufacturing parameters were proprietary to the manufacturer and were not available to the authors.

Before bonding, the composite-wing surfaces were lightly abraded using 600-grit abrasive paper and cleaned with isopropyl alcohol. The sensors were bonded using 3M Scotch-Weld DP460 (St. Paul, MN, USA) epoxy adhesive and cured for 24 h at 20–25 °C. The bonded regions and connecting fibers were, subsequently, protected using a thin silicone-rubber layer and polyimide tape. Strain-relief loops were incorporated near the wing-root transition and fiber-routing interfaces to limit unintended mechanical loading and excessive local bending.

The FBG sensors were arranged in wavelength-division-multiplexed optical networks. Multiple FBGs with different nominal Bragg wavelengths were connected along the optical-fiber network and interrogated simultaneously. Each sensor was uniquely identified by the combination of its initial Bragg wavelength and optical-channel assignment. A four-channel Micron Optics si155 interrogator, with a nominal operating range of 1510–1590 nm and a wavelength resolution of approximately 1 pm, was used for signal acquisition. During the ground-calibration tests, the initial Bragg wavelengths of the FBGs ranged from approximately 1534 to 1563 nm, and the data were acquired at 64 Hz. Because the multiplexed sensors had different nominal center wavelengths, the absolute wavelength was not used directly as the model input. Instead, the wavelength change of each FBG relative to its baseline value was calculated and arranged according to a fixed sensor order to form the input vector of the sectional-load identification model. The sensor identifiers, initial Bragg wavelengths, optical-channel assignments, installation locations, and sensing orientations are listed in Supplementary Table S1.

Electrical resistance strain gauges provided an independent ground-response measurement system. A total of 24 electrical channels were distributed among the four monitoring sections, with two bending-, two shear-, and two torsion-response channels at each section. The strain signals were acquired at 64 Hz using a DH5916N data-acquisition system over a measurement range of ±4000 με. The channel arrangement, measurement range, and acquisition parameters of the electrical reference system are summarized in

Table 2. Electrical strain-gauge reference measurement configuration.

Wing Section	Bending-Response Channels	Shear-Response Channels	Torsional-Response Channels	Number of Channels	Measurement Range	Sampling Rate
Section 1	Mzy1a, Mzy1b	Qzy1a, Qzy1b	Tzy1a, Tzy1b	6	−4000 to 4000 με	64 Hz
Section 2	Mzy2a, Mzy2b	Qzy2a, Qzy2b	Tzy2a, Tzy2b	6	−4000 to 4000 με	64 Hz
Section 3	Mzy3a, Mzy3b	Qzy3a, Qzy3b	Tzy3a, Tzy3b	6	−4000 to 4000 με	64 Hz
Section 4	Mzy4a, Mzy4b	Qzy4a, Qzy4b	Tzy4a, Tzy4b	6	−4000 to 4000 με	64 Hz
Total	8	8	8	24	−4000 to 4000 με	64 Hz

. Reference sectional loads were calculated from the applied masses and loading-point coordinates rather than from the electrical strain measurements. The electrical strain gauges were used only to compare the linearity, repeatability, and load-following characteristics of the two sensing systems and were excluded from both the FBG wavelength-to-load model and airborne load reconstruction.

3.2. Ground-Loading Configuration and Test Cases

A controlled multi-case ground-loading program was designed to establish the relationship between the multichannel FBG wavelength responses and the sectional loads of the full-scale wing. The objective was not to reproduce the continuous aerodynamic-pressure field encountered in flight but to generate representative combinations of bending moment, shear force, and torsional moment that could be calculated directly from the applied masses and loading coordinates. The calculated mechanical loads, rather than the electrical-strain-gauge outputs, were used as the reference quantities for calibration and independent ground verification of the FBG wavelength-to-load models.

Five loading stations, denoted by L₁–L₅, were arranged along the wing span, as illustrated in Figure 4. Each station contained one leading-edge-side loading point and one trailing-edge-side loading point, resulting in ten loading points denoted by F₁–F₁₀. Their coordinates in the aircraft coordinate system are listed in

Table 3. Coordinates of the ground-loading points.

Loading Station	Leading-Edge-Side Point	x (mm)	z (mm)	Trailing-Edge-Side Point	x (mm)	z (mm)
L1	F1	1633	1020	F2	2264	1020
L2	F3	1626	2150	F4	2269	2149
L3	F5	1627	2804	F6	2271	2804
L4	F7	1833	4782	F8	2151	4783
L5	F9	1741	6602	F10	2039	6601

. The x-coordinate defines the chordwise position, whereas the z-coordinate defines the spanwise position. The loading stations extended from approximately z = 1.020 m to z = 6.602 m, enabling both inboard- and outboard-dominated load distributions to be generated. The paired loading points at each station had approximately the same spanwise coordinate but different chordwise coordinates. The total force and effective chordwise line of action at each station were controlled through the sum and ratio of the paired loads, respectively.

The applied masses for the 13 loading cases are summarized in Supplementary Table S2. Cases 1–9 were used for model calibration and formed an approximately 3 × 3 loading design comprising three spanwise distributions and three chordwise loading patterns. Cases 1–3 combined an inboard-dominated spanwise distribution with leading-edge-side, intermediate, and trailing-edge-side loading, respectively. Cases 4–6 employed an intermediate spanwise distribution with the same three chordwise patterns, whereas Cases 7–9 employed an outboard-biased spanwise distribution. This design varied both the spanwise distribution and the effective chordwise line of action, thereby generating diverse combinations of bending moment, shear force, and torsional moment for multichannel calibration.

Cases 10–13 were withheld from model fitting and used as independent test cases. Cases 10 and 11 had a total applied mass of 210 kg and represented mixed-inboard distributions with leading-edge-side and trailing-edge-side chordwise biases, respectively. Cases 12 and 13 had a total applied mass of 310 kg and represented mixed-outboard distributions with different chordwise loading biases. These cases differed from the individual calibration cases in both load magnitude and spatial distribution and, therefore, provided an independent assessment of the predictive capability of the identification model.

The resultant-load coordinates were calculated from the force-weighted coordinates of the loading points. Because all loads were gravitational, the force weighting was equivalent to the mass weighting. The calculated coordinates and the corresponding values recorded in the original test documentation are compared in

Table 4. Resultant-load coordinates of the calibration and independent test cases.

Case	Use	Resultant xr (mm)	Resultant zr (mm)	Test-Case Character
1	Calibration	1628.9	1889.8	Calibration
2	Calibration	1948.3	1889.6	Calibration
3	Calibration	2267.6	1889.4	Calibration
4	Calibration	1677.4	3130.0	Calibration
5	Calibration	1946.1	3129.9	Calibration
6	Calibration	2214.8	3129.8	Calibration
7	Calibration	1747.0	4344.6	Calibration
8	Calibration	1955.0	4344.7	Calibration
9	Calibration	2162.9	4344.8	Calibration
10	Independent test	1810.4	2510.2	Predominantly interpolation-based
11	Independent test	2083.1	2510.0	Predominantly interpolation-based
12	Independent test	1861.6	3725.3	Limited extrapolation
13	Independent test	2047.2	3725.2	Limited extrapolation

and illustrated in Figure 5. The maximum absolute difference was less than 0.5 mm, confirming the consistency of the loading-point coordinates and applied-mass data. The calibration cases covered three principal spanwise resultant-load regions at approximately z = 1.890, 3.130, and 4.345 m. Within each spanwise region, the effective chordwise load position shifted progressively from the leading-edge side toward the trailing-edge side. The independent test cases occupied intermediate spanwise positions at approximately z = 2.510 and 3.725 m.

For each loading case, the reference sectional loads were calculated from the point forces located outboard of the corresponding monitoring section. The shear force was obtained by summing the outboard point forces, the bending moment was calculated using their spanwise lever arms, and the torsional moment was calculated from their chordwise offsets relative to the prescribed structural reference axis. The spanwise coordinates of Sections 1–4 were taken as 0.450, 1.550, 2.600, and 4.500 m, respectively. The chordwise reference-axis coordinates were determined by piecewise interpolation of the midpoint line between the corresponding leading-edge-side and trailing-edge-side loading positions. The calculated sectional bending moments, shear forces, and torsional moments are listed in

Table 5. The reference sectional loads.

Case	S1 M	S1 Q	S1 T	S2 M	S2 Q	S2 T	S3 M	S3 Q	S3 T	S4 M	S4 Q	S4 T
1	2.260	1.570	−0.502	0.845	0.981	−0.316	0.080	0.392	−0.126	0.000	0.000	0.000
2	2.260	1.570	−0.001	0.845	0.981	0.000	0.080	0.392	0.000	0.000	0.000	0.000
3	2.259	1.570	0.500	0.845	0.981	0.316	0.080	0.392	0.127	0.000	0.000	0.000
4	6.836	2.551	−0.693	4.342	1.962	−0.505	2.547	1.373	−0.316	0.935	0.785	−0.156
5	6.835	2.551	−0.007	4.342	1.962	−0.005	2.547	1.373	−0.006	0.935	0.785	−0.035
6	6.835	2.551	0.678	4.341	1.962	0.494	2.547	1.373	0.305	0.935	0.785	0.086
7	8.405	2.158	−0.436	6.135	1.962	−0.372	4.164	1.766	−0.310	1.308	1.472	−0.270
8	8.406	2.158	0.013	6.136	1.962	0.015	4.164	1.766	0.014	1.308	1.472	−0.041
9	8.406	2.158	0.462	6.136	1.962	0.402	4.164	1.766	0.338	1.308	1.472	0.188
10	4.244	2.060	−0.286	2.342	1.373	−0.129	1.253	0.589	−0.003	0.468	0.392	−0.018
11	4.244	2.060	0.276	2.342	1.373	0.123	1.253	0.589	−0.003	0.468	0.392	−0.018
12	9.960	3.041	−0.266	6.875	2.551	−0.170	4.418	2.060	−0.076	1.336	1.570	−0.040
13	9.960	3.041	0.299	6.875	2.551	0.209	4.418	2.060	0.113	1.336	1.570	−0.040

.

The calculated loads exhibited the expected spanwise transfer characteristics of a cantilever wing. Among the calibration cases, the bending-moment range decreased from 2.259–8.406 kN·m at Section 1 to 0–1.308 kN·m at Section 4. Cases 1–3 produced zero sectional loads at Section 4 because all applied forces were located inboard of its spanwise coordinate. In contrast, Cases 4–9 included loads at the two outer loading stations and, therefore, generated bending, shear, and torsional excitation at the outboard monitoring section.

Comparison with the calibration domain showed that the individual load components in Cases 10 and 11 remained within the corresponding minimum-to-maximum ranges of Cases 1–9. These cases were, therefore, treated as predominantly interpolation-based tests, although strict inclusion of the complete load vectors within the multidimensional calibration domain was not established. In Cases 12 and 13, the bending moments and shear forces exceeded the corresponding calibration maxima at several monitoring sections. These cases were consequently classified as limited-extrapolation tests and were used to assess model robustness near or slightly beyond the calibrated load range. The interpolation-based results of Cases 10–11 and the limited-extrapolation results of Cases 12–13 were evaluated separately in the subsequent verification analysis.

3.3. Ground-Test Procedure and FBG Data Processing

Before formal calibration, the FBG reflection spectra, sensor-channel correspondence, wing-root fixture, loading attachments, suspension cables, and loading-point connections were inspected. Two preliminary loading–unloading cycles were then performed, with the load increased in three steps to 30% of the final mass of Case 1, corresponding to 48 kg. After complete unloading, the wing was held for 30 s. No evident fixture slip or irreversible response was considered to have occurred when all load-sensitive FBGs returned to within ±5 pm of their preloading baselines and no visible displacement of the fixture or loading attachments was observed.

Each formal loading case began from an unloaded reference state. The final mass distribution specified in Supplementary Table S2 was applied in ten proportional levels from 10% to 100%. The leading- and trailing-edge-side loads were increased proportionally to maintain an approximately constant chordwise resultant-load position within each case. The maximum total masses were 160, 210, 220, 260, and 310 kg, depending on the loading case. Each level was maintained for at least 20 s, after which the masses were removed in reverse order. A minimum unloaded recovery period of 30 s was retained between repetitions, and each case was repeated three times.

FBG wavelengths were acquired at 64 Hz using the Micron Optics si155 interrogator, while the electrical strain-gauge responses were recorded in parallel at 64 Hz using the DH5916N system. Because the two systems did not share a hardware trigger, their records were aligned using the test log, unloaded intervals, and loading-step transitions. The electrical measurements were used only for the independent response comparison in Section 3.5 and were excluded from the FBG input vectors, sectional-load calculations, and calibration-coefficient estimation.

For each repetition, the mean wavelength over the final 10 s of the unloaded interval preceding loading was used as the baseline, corresponding to 640 samples. At each loading level, the final 10 s of the 20 s holding period was selected as the stable window, and its mean wavelength change was used as the representative response. The initial 10 s were excluded to reduce the influence of loading disturbances, local vibration, and short-term structural settling. The three repeated responses at each level were averaged to form one regression sample, and their standard deviation was retained as a repeatability indicator. Unloaded records were used only for baseline determination.

For each ground-loading cycle, the stable unloaded interval immediately preceding loading was used to establish the wavelength baseline. Cycle-specific baseline referencing removed the initial wavelength offset and reduced the influence of slow environmental drift. The temperature-monitoring FBGs were examined to identify evident temperature changes during the loading process. However, their responses were not directly subtracted from the load-sensitive channels because complete mechanical-strain isolation and channel-specific thermal calibration were unavailable. The temperature-monitoring channels were not included in the wavelength-to-load model, and residual thermal effects were retained as a source of uncertainty.

Before sample assembly, records were screened for reflection-peak loss, abnormal wavelength jumps, and unstable responses. Peak loss was defined as the absence of a valid wavelength for at least 0.5 s, an intersample change greater than 0.05 nm within a stable window was classified as an abnormal jump, and a stable-window standard deviation greater than 3 pm was treated as unacceptable. Loading-transition records were excluded, and no additional low-pass filtering was applied. If any of the eight load-sensitive FBGs at a monitoring section failed these criteria, the corresponding section-level sample was removed rather than reconstructed by interpolation.

After repetition averaging and data-quality screening, Cases 1–9 provided a maximum of 90 calibration samples per monitoring section, distributed over nine loading paths and ten load levels. Cases 10–13 were processed using the same procedure and provided a maximum of 40 independent test samples per section. Their data were withheld from preprocessing-parameter determination, coefficient estimation, and model adjustment until the calibration model had been fixed.

3.4. FBG Calibration Results and Independent Ground Verification

3.4.1. Calibration Performance

The calibration models were established separately for the four monitoring sections using the measured data from Cases 1 to 9. Because the 10% loading record of Case 1 was unavailable, this record was excluded rather than reconstructed. Case 1, therefore, contributed nine measured load levels from 20% to 100%, whereas each of Cases 2–9 contributed ten levels from 10% to 100%. Consequently, 89 measured calibration samples were retained for each monitoring section. The response at each loading level was obtained by averaging the three repeated tests described in Section 3.3.

Each sectional identification model contained an intercept and eight FBG wavelength-response variables. The resulting design matrices had a rank of nine at all four sections, indicating that the available calibration cases were sufficient to estimate the nine regression parameters. The standardized condition numbers were 187.6, 296.5, 448.1, and 620.4 for Sections 1–4, respectively. Thus, none of the models was rank deficient, although the increasing condition number toward the outboard sections indicates progressively stronger correlation among the FBG responses. This trend was associated with the smaller sectional loads and the more similar strain patterns measured near the wing tip.

The calibration results are summarized in

Table 6. Calibration performance of the sectional-load identification models.

Section	Component	Unit	Samples	Rank	Condition Number	R2	RMSE	NRMSE (%)	Maximum Error (%)
1	M	kN·m	89	9	187.6	0.9985	0.0889	1.087	2.553
1	Q	kN	89	9	187.6	0.9846	0.0802	3.351	8.707
1	T	kN·m	89	9	187.6	0.9981	0.0125	0.910	2.780
2	M	kN·m	89	9	296.5	0.9997	0.0281	0.464	2.666
2	Q	kN	89	9	296.5	0.9949	0.0391	2.098	7.678
2	T	kN·m	89	9	296.5	0.9961	0.0130	1.306	4.523
3	M	kN·m	89	9	448.1	0.9998	0.0189	0.454	3.290
3	Q	kN	89	9	448.1	0.9982	0.0209	1.213	2.981
3	T	kN·m	89	9	448.1	0.9979	0.0063	0.967	3.937
4	M	kN·m	89	9	620.4	0.9993	0.0105	0.801	3.059
4	Q	kN	89	9	620.4	0.9966	0.0250	1.700	5.491
4	T	kN·m	89	9	620.4	0.9990	0.0024	0.526	1.411

. For bending moment, the coefficients of determination ranged from 0.9985 to 0.9998, while the RMSE ranged from 0.0105 to 0.0889 kN·m, and the NRMSE ranged from 0.454% to 1.087%. For the shear force, the R² ranged from 0.9846 to 0.9982, with RMSE values of 0.0209–0.0802 kN and NRMSE values of 1.213–3.351%. For torque, the R² ranged from 0.9961 to 0.9990, the RMSE ranged from 0.0024 to 0.0130 kN·m, and the NRMSE ranged from 0.526% to 1.306%. The maximum normalized calibration errors were 3.290%, 8.707%, and 4.523% for bending moment, shear force, and torque, respectively.

The scatter among the three repeated loading sequences remained small relative to the wavelength variations caused by the applied loads. The representative sample-level wavelength scatter was approximately 1.20 pm, and no progressive offset was observed among the repeated loading–unloading sequences after baseline correction. The repeated measurements were, therefore, averaged before the coefficient estimation, while their standard deviations were retained as indicators of measurement repeatability rather than treated as additional independent regression samples.

Figure 6 compares the reference and FBG-identified loads for all 89 calibration samples. Each row corresponds to one monitoring section, while the three columns represent bending moment, shear force, and torsional moment. The bending-moment results are tightly concentrated around the equality line, with the R² values ranging from 0.9985 to 0.9998 and section-level NRMSE values ranging from 0.454% to 1.087%. The torsional-moment results show similarly close agreement, with R² values of 0.9961–0.9990 and NRMSE values of 0.526–1.306%. In contrast, the shear-force results exhibit greater scatter, with R² values of 0.9846–0.9982 and NRMSE values of 1.213–3.351%. The largest dispersion occurs at Section 1, where the shear-force NRMSE is 3.351%, and the maximum normalized error reaches 8.707%. The best shear-force agreement is obtained at Section 3, where the NRMSE is 1.213%, and the maximum normalized error is 2.981%.

These component-dependent differences are consistent with the corresponding sensing mechanisms. Bending produces relatively large and opposite-sign longitudinal responses on the upper and lower wing surfaces, providing a strong and distinguishable multichannel strain pattern. Torsional moment is reconstructed from the combined responses of multiple shear-sensitive FBGs, and the chordwise variation included in the calibration cases improves its separation from bending. By comparison, shear force produces smaller wavelength variations and strain patterns that are more strongly correlated with the bending- and torsion-related responses. Its identification is, therefore, more sensitive to wavelength noise, baseline drift, local strain-transfer differences, fixture compliance, loading-coordinate uncertainty, and correlations among the FBG channels. The section-to-section variation is not monotonic, indicating that identification accuracy depends on the combined effects of response amplitude and input-channel correlation rather than on spanwise location alone. It should also be emphasized that Figure 6 describes agreement within the calibration dataset; independent predictive performance is evaluated separately using the completely withheld Cases 10–13 in Section 3.4.

3.4.2. Interpolation-Oriented Ground Verification

After the calibration coefficients had been fixed, Cases 10 and 11 were used for independent ground verification. These cases employed asymmetric chordwise mass distributions that differed from those used in the principal symmetric calibration cases. However, the bending moments, shear forces, and torques at all four monitoring sections remained within the load ranges covered by Cases 1–9. Cases 10 and 11 were, therefore, treated as interpolation-oriented verification cases rather than additional calibration cases. None of their measurements were used for sensor selection, coefficient estimation, or model adjustment.

As summarized in

Table 7. Independent ground-verification performance.

Section	Component	Cases 10–11 RMSE	Cases 10–11 NRMSE (%)	Cases 10–11 Maximum Error (%)	Cases 12–13 RMSE	Cases 12–13 NRMSE (%)	Cases 12–13 Maximum Error (%)
1	M	0.1463 kN·m	1.788	2.741	0.1717 kN·m	2.098	3.492
1	Q	0.1754 kN	7.327	13.494	0.1717 kN	7.174	13.222
1	T	0.0213 kN·m	1.556	2.796	0.0229 kN·m	1.669	2.961
2	M	0.0313 kN·m	0.518	1.434	0.0754 kN·m	1.247	2.485
2	Q	0.0645 kN	3.459	8.891	0.0563 kN	3.023	6.994
2	T	0.0162 kN·m	1.626	3.652	0.0093 kN·m	0.932	1.785
3	M	0.0229 kN·m	0.550	1.047	0.0175 kN·m	0.422	0.997
3	Q	0.0731 kN	4.234	8.680	0.0427 kN	2.471	4.422
3	T	0.0126 kN·m	1.926	4.368	0.0192 kN·m	2.938	5.321
4	M	0.0141 kN·m	1.075	2.030	0.0167 kN·m	1.276	2.370
4	Q	0.0206 kN	1.398	3.769	0.0329 kN	2.232	6.065
4	T	0.0020 kN·m	0.439	1.023	0.0070 kN·m	1.538	3.588

, the bending-moment RMSE values for Cases 10–11 ranged from 0.0141 to 0.1463 kN·m across the four sections, corresponding to NRMSE values of 0.518–1.788%. The shear-force RMSE ranged from 0.0206 to 0.1754 kN, with NRMSE values of 1.398–7.327%. For torque, the RMSE ranged from 0.0020 to 0.0213 kN·m and the NRMSE ranged from 0.439% to 1.926%. The results show that the models retained approximately their calibration-level performance for bending moment and torque when applied to previously unseen load combinations within the calibrated domain.

The largest interpolation-oriented verification error occurred in the Section 1 shear-force result. Its maximum absolute error was approximately 0.323 kN, corresponding to 13.494% of the Section 1 shear-force calibration range. In comparison, the maximum normalized errors for bending moment and torque were 2.741% and 4.368%, respectively. The relatively large shear-force error was concentrated at individual loading levels and did not appear as a persistent systematic offset throughout the complete loading sequences.

The RMSE values in Table 7 are expressed in physical units. The NRMSE and maximum normalized errors were calculated using the corresponding load ranges covered by calibration Cases 1–9 rather than the smaller ranges of the individual verification cases. This common normalization allows for the interpolation-oriented and limited-extrapolation results to be compared directly.

Figure 7 compares the reference and FBG-identified loads for all independent verification samples. The results from Cases 10 to 11 and Cases 12 to 13 are displayed using different symbols. For Cases 10–11, most bending-moment and torque points remained close to the equality line, whereas greater dispersion was observed in the shear-force results, particularly at Section 1.

3.4.3. Limited-Extrapolation Assessment

Cases 12 and 13 were used to examine the behavior of the fixed identification models near and moderately beyond the calibrated load domain. Compared with the maximum loads in Cases 1–9, the full-load bending moments in Cases 12–13 exceeded the corresponding calibration maxima by approximately 18.5%, 12.0%, 6.1%, and 2.1% at Sections 1–4, respectively. The shear forces exceeded the corresponding calibration maxima by approximately 19.2%, 30.0%, 16.6%, and 6.7%. In contrast, the positive and negative torques of Cases 12–13 remained within the torque ranges represented in the calibration dataset. Consequently, Cases 12–13 constituted limited extrapolation for bending moment and shear force but remained predominantly interpolative for torque.

The bending-moment RMSE values for Cases 12–13 ranged from 0.0167 to 0.1717 kN·m, with NRMSE values of 0.422–2.098%. The shear-force RMSE ranged from 0.0329 to 0.1717 kN, corresponding to NRMSE values of 2.232–7.174%. The torque RMSE ranged from 0.0070 to 0.0229 kN·m, with NRMSE values of 0.932–2.938%. The largest normalized error was again associated with the Section 1 shear force and reached 13.222%, equivalent to approximately 0.317 kN when expressed in physical units.

No monotonic increase in the identification error was observed from the interpolation-oriented Cases 10–11 to the limited-extrapolation Cases 12–13. For example, the bending-moment NRMSE increased at Sections 1, 2, and 4 but decreased from 0.550% to 0.422% at Section 3. The shear-force NRMSE decreased from 7.327% to 7.174% at Section 1, from 3.459% to 3.023% at Section 2, and from 4.234% to 2.471% at Section 3 but increased from 1.398% to 2.232% at Section 4. The torsional-moment NRMSE decreased at Section 2 but increased at Sections 1, 3, and 4. These component- and section-dependent trends show that exceeding the calibration range of an individual load component did not, by itself, determine the identification error.

The observed differences can be explained by the combined effects of the loading distribution and the multichannel sensing mechanism. Changing the chordwise mass distribution shifts the chordwise position of the resultant load and, therefore, changes the relative combination of bending moment, shear force, and torsional moment. It also produces a different multichannel wavelength-response pattern. Because an individual FBG may respond to more than one sectional-load component, the three loads are reconstructed from correlated multichannel inputs rather than from mutually independent sensor responses. Thus, even when an individual load component remains within its calibration range, the complete combination of load components and FBG responses may be less well represented by the calibration cases. Bending produces comparatively large and opposite-sign longitudinal responses on the upper and lower wing surfaces, whereas shear produces smaller wavelength variations and responses that are more strongly correlated with bending- and torsion-related patterns. Shear-force identification is, therefore, more sensitive to measurement noise, baseline drift, local strain-transfer differences, and correlations among the FBG inputs. This interpretation is consistent with the comparatively large shear-force errors and with the standardized condition numbers, which increase from 187.6 at Section 1 to 620.4 at Section 4. However, because the signal-to-noise ratio was not independently quantified for each FBG channel, the present results do not establish a separate quantitative relationship between channel-level signal-to-noise ratio and load-identification error.

Considering all four independent verification cases, the maximum section-level NRMSE values were 1.949% for bending moment, 7.251% for shear force, and 2.484% for torque. The corresponding comparison between Cases 10–11 and Cases 12–13 is presented in Figure 8. The limited extrapolation did not produce a consistent increase in the bending-moment or shear-force errors at all sections. The relatively large verification errors remained concentrated in the shear-force results, whereas the bending-moment errors remained below 2.10% in both verification groups.

The results indicate that the multichannel FBG models were comparatively robust for bending-moment and torsional-moment reconstruction and remained usable under the limited bending- and shear-force extrapolation represented by Cases 12 and 13. Nevertheless, these results should not be interpreted as evidence of unrestricted extrapolation capability. Additional calibration cases and independent verification cases serve different purposes. An additional calibration case is included in the model-fitting dataset and, therefore, contributes to the re-estimation of the regression coefficients and expansion of the calibrated load domain. By contrast, an independent verification case is completely excluded from sensor selection and coefficient estimation through to model adjustment and is used only to evaluate the predictive performance of the fixed model. If the fixed model is to be assessed outside the present calibration domain without coefficient adjustment, new out-of-domain loading cases should be reserved exclusively for independent verification. If the intended application domain is to be expanded to higher load levels, different chordwise loading positions, or new load combinations, those conditions should first be incorporated as additional calibration cases, after which the refitted model should be evaluated using a separate set of independent verification cases.

The differences among the three load components can be explained by their sensing mechanisms and signal amplitudes. Bending produced comparatively large and opposite-sign longitudinal responses on the upper and lower wing surfaces, providing the regression model with a strong and distinguishable response pattern. Torque was reconstructed from the combined responses of several shear-sensitive channels, and the asymmetric loading cases improved its separation from bending. Shear force produced smaller wavelength variations and was more strongly correlated with bending- and torsion-related responses. Its identification was, therefore, more sensitive to wavelength noise, baseline drift, sensor-bonding differences, fixture compliance, uncertainties in loading-point coordinates, and correlations among the FBG channels.

The increasing condition numbers toward the outboard sections further indicate stronger correlations among the wavelength inputs. However, because the absolute loads at the outboard sections were smaller, relatively small physical errors could appear as comparatively large normalized errors. The present ground-verification results, therefore, support quantitative use within the calibrated load domain and cautious use under the limited extrapolation covered by Cases 12 and 13, rather than unrestricted application outside the tested domain.

3.5. Auxiliary Comparison with Electrical Strain-Gauge Measurements

Electrical resistance strain gauges were used as an auxiliary ground-response measurement system to examine whether the FBG and electrical measurements followed the same structural loading process. The electrical measurements were not used to calculate the reference sectional loads, select the FBG channels, estimate the FBG calibration coefficients, or reconstruct the flight loads. The reference sectional loads remained those calculated from the applied masses, loading-point coordinates, and monitoring-section positions. Moreover, because the si155 interrogator and the DH5916N electrical acquisition system did not share a common hardware trigger, the comparison was restricted to the averaged responses obtained from the stable load–hold intervals. Transient peaks, response phases, and exact load-step onset times were not compared.

For both sensing systems, the final 10 s of each stable load–hold interval was extracted according to the procedure described in Section 3.3. The mean response over the 640 samples in each interval was used as the representative measurement. The values from the three repeated tests were subsequently averaged, while their standard deviations were retained to evaluate repeatability. Because the FBG wavelength changes are expressed in pm and the electrical strain responses are expressed in με, their absolute amplitudes could not be directly compared. The response of each selected channel was, therefore, normalized by its maximum absolute response range:

$$r_i={\displaystyle \frac{s_i-s_0}{{\max }_j\left|s_j-s_0\right|}}$$

(19)

where s_i is the stable-window mean at the i-th loading level, s₀ is the corresponding unloaded baseline, and r_i is the normalized response. This normalization retained the response sign and enabled the load-following trends of the two sensing systems to be compared without implying equivalence between their physical units or sensitivities.

Cases 5 and 6 were selected as representative loading cases. Case 5 represented a chordwise-balanced loading configuration, whereas Case 6 introduced an aft-biased asymmetric loading pattern and a more pronounced torsional response. Representative bending- and shear-sensitive channels at Section 2 were selected because this section provided measurable response amplitudes without the strong root-fixture influence associated with Section 1 or the relatively small response levels near the wing tip. As shown in Figure 9, both sensing systems exhibited monotonic stepwise responses during loading and followed the reverse trend during unloading. The blue curves with circular markers represent the normalized FBG responses, whereas the orange curves with square markers represent the normalized electrical-strain-gauge responses.

The two curves are very close for the bending-sensitive response under Case 5, while more visible differences occur for the shear-sensitive response under Case 5 and the bending- and torsion-sensitive responses under Case 6. The largest differences between the normalized FBG and electrical responses at the stable loading levels were approximately 0.034 for the bending-sensitive channels, 0.061 for the shear-sensitive channels, and 0.052 for the torsion-sensitive channels. These differences can be attributed to the fact that the two sensing systems were installed nearby but not exactly at coincident positions and had different sensing directions, effective sensing lengths, and strain-transfer mechanisms. Bending generated a comparatively uniform and high-amplitude longitudinal strain field, resulting in close agreement between the two normalized responses. By contrast, shear- and torsion-related local strain fields were smaller, more spatially nonuniform, and more sensitive to sensor position and orientation, resulting in larger differences between the corresponding curves.

The FBG and electrical responses were independently normalized by their respective maximum absolute response ranges. Consequently, both curves approached zero in the unloaded state and unity at the maximum loading level by construction. Their differences at the intermediate loading levels, therefore, represent differences in normalized load-following behavior rather than differences in absolute strain sensitivity. Moreover, the two sensing systems used different acquisition hardware and did not share a common hardware trigger. The comparison was, therefore, restricted to stable-window mean responses and was not used to assess transient timing or phase differences. No persistent response reversal, unexplained discontinuity, or nonrecoverable zero shift was observed.

The linearity, repeatability, zero-return behavior, and loading–unloading hysteresis of the two measurement systems were further evaluated using representative channels from the four monitoring sections. For each channel, R² was calculated between the normalized applied load and the stable sensor response during loading. The repeatability error was defined as the maximum standard deviation among the three repetitions divided by the full response span. The zero-return error was calculated from the residual response after complete unloading, and hysteresis was evaluated from the maximum difference between the loading and unloading responses at the same nominal load level. The summarized results are presented in

Table 8. Response-quality comparison between the FBG and electrical strain-gauge systems.

Response Type	Sensor System	Number of Channels	Median R2	Maximum Repeatability Error (% FS)	Maximum Zero-Return Error (% FS)	Maximum Hysteresis (% FS)
Bending-sensitive	FBG	16	0.9987	1.42	0.74	1.38
Bending-sensitive	Electrical	8	0.9979	1.66	0.89	1.62
Shear-sensitive	FBG	16	0.9928	1.95	1.12	2.16
Shear-sensitive	Electrical	8	0.9911	2.31	1.43	2.48
Torsion-response	FBG	16	0.9956	1.78	0.96	1.84
Torsion-response	Electrical	8	0.9938	2.08	1.27	2.21

.

Both sensing systems showed the highest linearity for bending-sensitive responses. The shear-sensitive responses exhibited moderately larger repeatability and hysteresis errors because their signal amplitudes were smaller and were more strongly influenced by the coupling among bending, shear, and torsion. The FBG measurements showed slightly smaller median response-quality errors, but these differences should not be interpreted as demonstrating an inherent accuracy advantage over the electrical strain gauges. The two systems differed in sensor number, sensing position, sensing direction, strain-transfer mechanism, and acquisition hardware and, therefore, did not form a strictly matched sensor comparison.

The electrical measurements, consequently, provide only auxiliary evidence that the FBG wavelength variations followed the structural responses produced by the prescribed stepwise application, stable holding, and removal of known masses at predefined loading points. They do not independently validate the accuracy of the reconstructed sectional loads.

4. Airborne Application and Physical-Consistency Assessment

4.1. Airborne FBG Measurement and Load Reconstruction

Airborne measurements were conducted using RX1E-A aircraft No. 01005, registration B-01AU, at Faku Caihu Airport, Shenyang, China. Four flight tests were performed on 19 July, 20 July, 25 August, and 6 September 2023. The measured test mass, including the aircraft, battery pack, pilot, and onboard instrumentation, was 565.0 kg. The flight program included level flight, steady turns, and pull-up maneuvers. Two representative intervals from the 25 August flight were selected: a 40 s steady-turn interval with a maximum bank angle of 47.73° and a maximum normal load factor of 1.433, and a 12 s pull-up interval with a maximum normal load factor of 2.058. The airborne measurement arrangement is shown in Figure 10, and the detailed configuration is provided in Supplementary Table S3.

The flight-test wing and the ground-calibration wing were nominally identical RX1E-A composite-wing articles but were not the same physical article. Four monitoring sections were located at nominal spanwise coordinates of 450, 1550, 2600, and 4500 mm. Eight load-sensitive FBGs were assigned to each section, giving 32 model-input channels. Their nominal surface locations, sensing orientations, identification rules, and input order corresponded to those used during ground calibration. The archived airborne network contained 47 FBG channels, including the 32 load-sensitive channels and auxiliary deformation- and temperature-monitoring channels. Before reconstruction, each load-sensitive flight channel was matched to its ground-calibration counterpart using the sensor identifier, initial Bragg wavelength, installation location, sensing orientation, and optical-channel assignment. All 32 required channels remained valid during the selected intervals; intervals containing sustained peak loss or invalid output in any required channel were excluded without interpolation.

The FBG wavelengths were recorded at 100 Hz, whereas the independently acquired flight parameters—including airspeed, altitude, normal load factor, roll angle, and pitch angle—were recorded at 50 Hz. Because the two systems did not share a verified hardware trigger, the records were aligned using common flight events and corrected for clock drift before the flight parameters were resampled onto the FBG time base. For each channel, the mean wavelength over a stable 30 s preflight interval was used as the flight baseline, and the wavelength change was calculated according to the baseline-referencing procedure described in Section 2.2. After data-quality screening, a fourth-order zero-phase Butterworth low-pass filter with a cutoff frequency of 5 Hz was applied. Temperature-monitoring FBGs were used to identify slow thermal trends during the selected flight intervals. The maximum variation in the temperature-monitoring channels was less than 15 pm within each selected interval. Because complete mechanical-strain isolation and channel-specific thermal calibration were unavailable, the temperature-channel responses were not directly subtracted from the load-sensitive channels. The selected maneuvers were of short duration, and residual thermal effects were retained as an uncertainty rather than interpreted as mechanical-load variations.

For Section j, the eight processed wavelength changes were arranged according to the ground-calibration channel order and introduced into the fixed model:

$${\widehat{L}}_j(t)=B_j\Delta {\lambda }_j(t)+b_j$$

(20)

where B_j and b_j are the coefficient matrix and intercept vector obtained from ground calibration.

At the preflight reference time t₀, the baseline-corrected wavelength input was zero and the model output was b_j. The reconstructed sectional load increment was, therefore,

$$\Delta {\widehat{L}}_j(t)={\widehat{L}}_j(t)-{\widehat{L}}_j(t_0)=B_j\Delta {\lambda }_j(t)=\left[\begin{array}{c}\Delta {\widehat{M}}_j(t)\\ \Delta {\widehat{Q}}_j(t)\\ \Delta {\widehat{T}}_j(t)\end{array}\right]$$

(21)

The intercept represents the regression output at zero baseline-corrected wavelength input and was not interpreted as the physical load acting on the stationary aircraft. The ground-derived coefficients were applied without airborne refitting or adjustment. Absolute sectional loads were not reported because the loads corresponding to the preflight reference state were unavailable. Accordingly, the results represent a direct article-to-article transfer of the ground-calibrated model, with structural, manufacturing, and sensor-installation differences retained as uncertainty sources.

4.2. Reconstructed Load Histories and Physical-Consistency Assessment

Figure 11 presents the synchronized flight parameters and reconstructed sectional load increments during the selected 12 s pull-up maneuver. All 32 load-sensitive FBG channels remained continuous, with no sustained reflection-peak loss or abrupt wavelength discontinuity. The reconstructed bending-moment and shear-force increments increased during the pull-up and reached their extrema near the normal-load-factor peak, whereas the torsional-moment increments showed a weaker temporal correspondence. Signal recovery was evaluated using 1 s stable windows immediately before and after the maneuver. The difference between the mean increments in these windows, normalized by the corresponding maneuver-induced peak-to-peak change, was used as a recovery indicator. The maximum residual among the 12 reconstructed load histories was 3.6%. The corresponding differences in normal load factor and airspeed were less than 0.03 and 4 km/h, respectively. Residual baseline changes were interpreted cautiously because mechanical and thermal contributions could not be completely separated.

The peak magnitudes of the reconstructed bending-moment increments at Sections 1–4 were 6.58, 4.80, 3.26, and 1.08 kN·m, respectively, while the corresponding shear-force increments were 1.90, 1.72, 1.55, and 1.29 kN. When normalized by the Section 1 values, the bending-moment peaks were 1.00, 0.73, 0.50, and 0.16, whereas the shear-force peaks were 1.00, 0.91, 0.82, and 0.68. The peak relationships, therefore, satisfied:

$$\left|\Delta {\widehat{M}}_{1,pk}\right|>\left|\Delta {\widehat{M}}_{2,pk}\right|>\left|\Delta {\widehat{M}}_{3,pk}\right|>\left|\Delta {\widehat{M}}_{4,pk}\right|$$

(22)

$$\left|\Delta {\widehat{Q}}_{1,pk}\right|>\left|\Delta {\widehat{Q}}_{2,pk}\right|>\left|\Delta {\widehat{Q}}_{3,pk}\right|>\left|\Delta {\widehat{Q}}_{4,pk}\right|$$

(23)

During most of the main pull-up phase, the bending-moment and shear-force increments also decreased generally from the inboard to the outboard sections. The more rapid spanwise decrease in bending moment is mechanically reasonable because an inboard section accumulates both a larger outboard resultant force and a longer effective moment arm, whereas shear force primarily represents the resultant transverse load acting outboard of the section.

The Pearson correlation coefficients with the normal load factor ranged from 0.918 to 0.966 for bending moment and from 0.894 to 0.951 for shear force. The corresponding peak-time differences were 0.02–0.08 s, which were of the same order as the estimated synchronization uncertainty. These indicators demonstrate temporal consistency but do not establish the numerical accuracy of the reconstructed load magnitudes. The peak torsional-moment increments were 0.56, 0.41, 0.29, and 0.14 kN·m from Section 1 to Section 4, with correlations of 0.48–0.71 with the normal load factor. Because torsion is strongly affected by the chordwise aerodynamic-load position, control-surface activity, and structural coupling, it was evaluated from signal continuity, smooth temporal evolution, and its component-wise magnitude relative to the ground-calibration range rather than from a strict proportional or monotonic relationship.

The peak values of all three reconstructed load components remained within the corresponding component-wise ranges represented by Cases 1–9, although inclusion of the complete (M, Q, T) vectors within the multidimensional calibration domain was not independently established. The steady-turn segment provided an additional lower-amplitude condition in which the bending-moment and shear-force increments varied consistently with the bank-angle magnitude and normal load factor and exhibited the same overall inboard-to-outboard decrease.

Overall, the results demonstrate the principal advantage of the proposed method: a lightweight multiplexed FBG network can simultaneously reconstruct bending moment, shear force, and torsional moment at multiple wing sections using a fixed, physically interpretable ground-calibration model, without airborne refitting or extensive electrical wiring. The consistent temporal and spanwise responses further indicate the engineering plausibility of transferring the model between nominally identical composite-wing articles under the selected flight conditions. Nevertheless, because independently measured airborne sectional-load truth was unavailable, the flight results provide a physical-consistency assessment rather than a direct accuracy validation, and the ground-test errors were not used as uncertainty bounds for the reconstructed flight loads.

5. Conclusions

This study developed an FBG-based framework for simultaneously identifying bending moment, shear force, and torsional moment at four sections of a full-scale RX1E-A composite wing. Multichannel linear models were established using ground-loading cases with different spanwise distributions and chordwise resultant-load positions. Electrical strain gauges were used only for auxiliary ground-response comparison.

The methodological contribution of this study is fourfold. First, the calibration program represents variations in both spanwise load distribution and chordwise resultant-load position, rather than relying on a single prescribed loading path. Second, the multichannel FBG framework simultaneously identifies bending moment, shear force, and torsional moment at multiple wing sections. Third, the predictive capability of the fixed models is evaluated using completely withheld ground-loading cases, with interpolation-oriented and limited-extrapolation conditions distinguished explicitly. Fourth, the ground-derived models are applied without refitting to a different but nominally identical flight-test wing, while the absence of independent airborne sectional-load references is explicitly reflected in the interpretation of the results.

For the 89 calibration samples from Cases 1 to 9, the section-level NRMSE values were 0.454–1.087% for bending moment, 1.213–3.351% for shear force, and 0.526–1.306% for torsional moment. Across the independent Cases 10–13, the corresponding maximum NRMSE values were 1.949%, 7.251%, and 2.484%. Bending-moment and torsional-moment identification remained comparatively stable, whereas shear-force identification showed greater dispersion because of its smaller response amplitudes and stronger coupling with the other load components. The limited-extrapolation results do not support unrestricted application beyond the tested load domain.

During a representative pull-up maneuver with a maximum normal load factor of 2.058, the reconstructed bending-moment and shear-force increments decreased from the inboard to the outboard sections and showed correlations of 0.918–0.966 and 0.894–0.951 with normal load factor, respectively. These results support the physical consistency of the reconstructed loads but do not constitute direct airborne accuracy validation. The principal limitations are the use of different ground- and flight-test wing articles, incomplete strain–temperature decoupling, the finite calibration domain, and the absence of independent airborne sectional-load references. Future work should focus on same-article ground-to-flight calibration, improved thermal compensation, expanded calibration-domain coverage, and independent airborne load measurements.