# Buckling of Composite Structures with
## 6 Three Parametrical Shell Theories
### Axi-Symmetric Spherical Shells under External Pressure  
The buckling of delaminated spherical shells and segments has also been studied in the literature [17; 62; 63; 64; 65; 66; 67; 68]. The investigations deal with the analysis of shallow shells [17; 62; 63; 64; 65; 66; 67] or spherical shell segments [68].  
The cross-section of spherical shells is presented in Figure 11. The shell geometry is characterized by the shallowness parameter \(\rho\) defined below.  
\[\rho=\sqrt[4]{12(1-\
u_{12}\
u_{21})}\frac{a}{\sqrt{Rh}}. \tag{25}\]  
For isotropic (quasi-isotropic) material properties, the dimensionless value of the external buckling pressure is described by the following relation:  
\[p_{\mathrm{dim}}=\frac{2E_{quasi}}{\sqrt{3(1-\
u^{2})}}\bigg{(}\frac{h}{R} \bigg{)}^{2} \tag{26}\]  
Figure 11: Cross-section and geometry of spherical shells.  
Figure 10: Effects of symmetrical delaminations on buckling loads (\(\mathrm{h}_{(3)}/\mathrm{h}_{(1)}=0.5\), the thickness parameter \(\mathrm{h}/\mathrm{L}_{\mathrm{x}}=0.02\)).  
The forms of the spherical shells with delaminations are plotted in Figure 12.  
For shallow shells, the fundamental relations are presented and discussed in Refs. [50,59,69].  
\[\begin{array}{l}M\Delta^{2}u_{3}-\frac{1}{R}\Delta f-\frac{1}{\xi_{1}}\Big{(} \frac{df}{d\xi_{1}}-\frac{n^{2}}{\xi_{1}}f\Big{)}\frac{d^{2}u_{3}}{d\xi_{1}^{2} }-\frac{1}{\xi_{1}}\Big{(}\frac{du_{3}}{d\xi_{1}}-\frac{n^{2}}{\xi_{1}}u_{3} \Big{)}\frac{d^{2}f}{d\xi_{1}^{2}}\\ +2\frac{n^{2}}{\xi_{1}^{2}}\Big{(}\frac{df}{d\xi_{1}}-\frac{n^{2}}{\xi_{1}^{2} }f\Big{)}\Big{(}\frac{du_{3}}{d\xi_{1}}-\frac{n^{2}}{\xi_{1}}u_{3}\Big{)}-p_{2n }=0,M=\frac{\bar{D}-\bar{B}^{2}}{(1-v^{2})\bar{A}}\end{array} \tag{27}\]  
\[\frac{1}{\bar{A}}\Delta^{2}f+\frac{1}{R}\Delta u_{3}-\frac{n^{2}}{\xi_{1}^{2} }\bigg{(}\frac{du_{3}}{d\xi_{1}}-\frac{1}{\xi_{1}}u_{3}\bigg{)}^{2}+\bigg{(} \frac{du_{3}}{d\xi_{1}}-\frac{n^{2}}{\xi_{1}}u_{3}\bigg{)}\frac{d^{2}u_{3}}{d \xi_{1}^{2}}=0 \tag{28}\]  
\(\Delta\) is the Laplace operator, and \(f\) denotes the Airy stress function. The above relations are reduced to the eight-order differential equation with respect to the normal displacement \(u_{3}(\xi_{1})\).  
The main goal of the present section is to characterize the influence of the form of delaminations on the reduction of buckling pressures occurring for ideal/perfect spherical shells. The analysis was carried out via imperfection sensitivity analysis to determine the behaviour analogous to the curve plotted in Figure 7, i.e., to find the distributions of buckling pressures with respect to the values of parameters \(\mathrm{q_{k}}\) describing the shapes of geometrical imperfections plotted in Figure 13:  
\[\mathrm{Min\ p_{cr}(q_{k})\ k=1,2,\ldots,K} \tag{29}\]  
The buckling pressures of spherical shells were derived with the use of the Bubnov-Galerkin method. The detailed form of the series expansions is discussed in Refs. [59,69]. Buckling pressures are much more sensitive to increased radius  
Figure 12: Forms of the loss of stability of delaminations in spherical shells.  
Figure 13: Geometrical parameters illustrating the imperfect axi-symmetric spherical shallow shells: (**a**) the convex imperfection in the form of an increased radius polar radius, (**b**) the concave imperfection characterized by the cosine function.