## 2 Aircraft Wing Load Tests
### Equation for Strain Load Relationship  
In numerous instances, the effects of external loads on structural components can be effectively represented by a load vector system comprising concentrated forces and moments. When distributed loads are applied to a structure, their representation should be confined to a minimal set of parameters to simplify the description. Moments, torques, and shear forces on certain sections may be selected as parameters or constants included in expressions describing distributed load functions. During calibration tests, when a system of forces acts on the structure, each signal from the strain measurement bridge corresponds to the cumulative effect of the forces applied individually within the system. Consequently, the following vectors can be directly utilized in the calibration process of the load measurement test:Figure 1: Wing strain gauge mounting location plan. Figure 3: Wing leading edge actual patch location. Figure 2: Diagram of the strain gauge arrangement. \[\epsilon= \begin{bmatrix}\epsilon_{1}\\ \epsilon_{2}\\ \vdots\\ \epsilon_{k}\end{bmatrix} ,\ \boldsymbol{P}= \begin{bmatrix}P_{1}\\ P_{2}\\ \vdots\\ P_{s}\end{bmatrix} ,\ \boldsymbol{L}= \begin{bmatrix}L_{1}\\ L_{2}\\ \vdots\\ L_{n}\end{bmatrix} , \tag{1}\]  
where \(\epsilon\) represents the strain signal vector (dimensions \(k\times 1\)); P denotes the force vector applied for calibration (dimensions \(s\times 1\)); and L is the load parameter vector (dimensions \(n\times 1\); e.g., bending, shear, and torsion can be used as a \(3\times 1\) vector of load parameters). The relationships between these vectors can be calculated as follows:  
\[\boldsymbol{L}=\boldsymbol{T_{R}}\boldsymbol{P},\ \epsilon=\boldsymbol{K_{I}} \boldsymbol{L},\ \boldsymbol{L}=\boldsymbol{K_{D}}\epsilon, \tag{2}\]  
where \(\boldsymbol{T_{R}}\) is a transformation matrix (dimensions \(n\times s\)); \(\boldsymbol{K_{I}}\) is the inverse calibration coefficient matrix (dimensions \(k\times n\)); and \(\boldsymbol{K_{D}}\) is the coefficient matrix, which is directly calibrated (dimensions \(n\times k\)). The bending moment \(M\) and torque \(T\) can be calculated by controlling the magnitude of the shear force \(Q\) and the location of its point of application during the aircraft load calibration test, thus obtaining the corresponding relationship of the output response of the bending moment \(M\), shear force \(Q\), and torque \(T\) strain bridge on the tested section. According to the magnitude of the applied load \(Q\), the position of the center of pressure, the coordinates of the rigid center of the measured section, and the sweep angle of the rigid shaft, the relationships between the bending moment \(M\), the torque applied to the measured section, and the applied load can be derived, as shown in Figure 5.  
In the diagram, \(x\) and \(z\) are the pressure center coordinates of the calibration load \(Q\), denoted by \(A\); \(x_{0}\) and \(z_{0}\) are the coordinates of the rigid center of the measured section, denoted by \(E\); \(\alpha\) is the rigid axis sweep acute angle of the measured section, with the left wing, left tailplane, and vertical specified as positive in the given coordinate system, such that the right wing and right tailplane are negative and, so, \(\alpha\) has a negative value; \(B\) is the point of intersection between the rigid axis and the line perpendicular from \(A\) to the rigid axis; \(D\) is the point of intersection between the line from \(A\) in the direction of the \(z\)-axis point and the perpendicular straight line passing through \(E\); and \(C\) is the point of intersection between \(AD\) and the rigid axis. From the above, one can obtain the following:  
\[M=Q[(z-z_{0})\text{cos}\alpha+(x-x_{0})\text{sin}\alpha] \tag{3}\]  
\[T=Q[(z-z_{0})\text{sin}\alpha-(x-x_{0})\text{cos}\alpha] \tag{4}\]  
Figure 4: Data collector: image and mounting diagram. The aforementioned relation utilizes the previously established matrix equation \(\mathbf{L}=\mathbf{T_{R}}\mathbf{P}\).