## 5 Test Results and Evaluation
### The Amplitude of Shape Imperfections in the Circumferential Direction  
The types of defects in the CFRP cylinder include elastic modulus deviation, layup angle deviation, thickness deviation, ply gap and overlap, manufacturing-induced defects, and shape imperfection. This study focuses on shape imperfections because the shape imperfections are significant. The authors resolved the data measured by the shape-imperfection measurement equipment into the amplitude of diameter deflection \(w_{0}\), amplitude of rigid body movement \(w_{1}\), elliptic amplitude \(w_{2}\), triangles amplitude \(w_{3}\), etc., by performing a Fourier transformation (Equation (1)). Figure 9 illustrates the meanings of \(w_{0}\)-\(w_{3}\). Because the diameter deflection \(w_{0}\) and amplitude of rigid body movement \(w_{1}\) uniformly along the axial direction are not circumferential imperfections, they are evaluated in Section 5.2 (Evaluation of Amplitude of Shape Imperfections in the Axial Direction). \[w_{j}(\theta)=w_{0,j}+w_{I,1,j}\sin\theta+w_{R,1,j}\cos\theta+\cdots+w_{I,n,j} \sin n\theta+w_{R,n,j}\cos n\theta \tag{1}\]  
with  
\(j\): index of axial direction \(1\leq j\leq 10\theta\): angle in circumferential direction  
\(n\): amplitude in wave numbers  
\(I\): coefficient for the sine part  
\(R\): coefficient for the cosine part  
The terms in Equation (1) are determined using Equations (2) and (3), where \(h_{k,j}\) are the raw data obtained by the measurement equipment at a particular circumferential position \(k\) and elevation (lengthwise) position \(j\). \[w_{R,n,j}=\sum_{k=0}^{N-1}h_{k,j}(\frac{\cos 2\pi kn}{N}) \tag{2}\]  
\[w_{I,n,j}=\sum_{k=0}^{N-1}h_{k,j}(\frac{\sin 2\pi kn}{N}) \tag{3}\]  
with \(h_{k,j}\) being the measured values at circumferential position \(k\) and elevation position \(j\). \(k\): index of circumferential direction \(0\leq k\leq 60\)  
Here, each amplitude \(w_{n}\) is determined using Equation (4). \[w_{n}=\sqrt{\sum_{j=1}^{10}\left(w_{R,n,j}\text{${}^{2}$s}\cos n\theta_{i}+w_{ I,n,j}\text{${}^{2}$}\sin n\theta_{i}\right)/10} \tag{4}\]  
with  
\(\theta_{i}\): angle in circumferential direction \(0\leq\theta_{i}\leq 360\)  
\(i\): index of circumferential direction \(0\leq i\leq 60\)  
To obtain the KDF, it is necessary to investigate the theoretical formula. It is thought that the anisotropy of the laminate and the deformation of out-plane shear have led to complex stress/multidimensional stress states. The buckling load of the cylindrical shell is calculated using a theoretical formula that takes into account anisotropy of the laminate and deformation of out-plane shear. This method is based on the deformation of out-plane shear, first-order shear theory, and anisotropic elasticity, including coupling of laminate tensile-shear, bending-torsion, and tension-torsion (bending-shear). It has the fewest approximations among the linear buckling theories and is known to agree well with experimental results [23]. However, it assumes that the shape of the cylindrical shell is ideal and that the radius and thickness are constant regardless of the measurement location. Because the theoretical formula has a huge equation, it is difficult to present in this paper. The complete formula is shown in [26]. Because the KDF on the \(y\)-axis is a dimensionless quantity, the amplitudes are divided by \(t\) to make the transformation dimensionless. It is necessary to evaluate \(w_{2}/t\)-\(w_{10}/t\) simultaneously by multiple regression analysis. The relationships between the KDF and the graphed amplitudes \(w_{2}\)-\(w_{10}\), together with linear regression lines, KDF = \(a(w_{n}/t)\) + KDF\({}_{0}\), are shown in Figure 10. The linear analysis of Figure 10a-i is simple linear regression analysis. These linear regressions are used as a reference for showing tendency.