Semi-Analytical Approach to the Determination of Free-Edge Stress Fields in Cylindrically Curved Composite Laminated Shells

Abstract

This paper aims at contributing to the field of highly efficient analysis methods by presenting a semi-analytical approach which enables an adequate prediction of the three-dimensional stress fields in cylindrically curved orthotropic laminated composite shells subjected to bending load. The approach incorporates a layerwise plane-strain analysis in the innermost regions of the shell wherein the stresses are specified in terms of Airy’s stress function. In order to realize an accurate determination of the expected three-dimensional stress field in the vicinity of the free edge, the plane-strain formulation is upgraded by a displacement-based approach wherein each laminate layer is discretized into a number of mathematical layers with respect to the thickness direction. The underlying differential equations and boundary conditions that govern the free-edge effect in cylindrically curved composite shells are derived by virtue of the principle of minimum elastic potential. The results of the developed analysis method are verified by comparison with detailed finite element computations, and it is found that the semi-analytical approach works with comparable accuracy, however, only at a fraction of the required computational effort.

This contribution is an abridged version of [1].

A. Abbreviations

$$\begin{aligned}&C_{opij \gamma }^{\left( k \right) } = \int \limits _{r_{\left( k-1 \right) }}^{r_{\left( k \right) }} C_{op}^{\left( k \right) } \varepsilon _{ij,0}^{\left( k \right) } \frac{\partial \psi _{\gamma }^{\left( k \right) }}{\partial r} r \,dr \textit{,} \quad D_{opij \gamma }^{\left( k \right) } = \int \limits _{r_{\left( k-1 \right) }}^{r_{\left( k \right) }} C_{op}^{\left( k \right) } \varepsilon _{ij,0}^{\left( k \right) } \frac{\psi _{\gamma }^{\left( k \right) }}{r} r \,dr \textit{,} \\&E_{op \gamma \delta }^{\left( k \right) } = \int \limits _{r_{\left( k-1 \right) }}^{r_{\left( k \right) }} C_{op}^{\left( k \right) } \psi _{\gamma }^{\left( k \right) } \psi _{\delta }^{\left( k \right) } r \,dr \textit{,} \quad F_{op \gamma \delta }^{\left( k \right) } = \int \limits _{r_{\left( k-1 \right) }}^{r_{\left( k \right) }} C_{op}^{\left( k \right) } \psi _{\gamma }^{\left( k \right) } \frac{\partial \psi _{\delta }^{\left( k \right) }}{\partial r} r \,dr \textit{,} \\&G_{op \gamma \delta }^{\left( k \right) } = \int \limits _{r_{\left( k-1 \right) }}^{r_{\left( k \right) }} C_{op}^{\left( k \right) } \frac{\partial \psi _{\gamma }^{\left( k \right) }}{\partial r} \frac{\partial \psi _{\delta }^{\left( k \right) }}{\partial r} r \,dr \textit{,} \quad H_{op \gamma \delta }^{\left( k \right) } = \int \limits _{r_{\left( k-1 \right) }}^{r_{\left( k \right) }} C_{op}^{\left( k \right) } \frac{\psi _{\gamma }^{\left( k \right) } \psi _{\delta }^{\left( k \right) }}{r} r \,dr \textit{,} \\&I_{op \gamma \delta }^{\left( k \right) } = \int \limits _{r_{\left( k-1 \right) }}^{r_{\left( k \right) }} C_{op}^{\left( k \right) } \frac{\psi _{\gamma }^{\left( k \right) }}{r} \frac{\partial \psi _{\delta }^{\left( k \right) }}{\partial r} r \,dr \textit{,} \quad J_{op \gamma \delta }^{\left( k \right) } = \int \limits _{r_{\left( k-1 \right) }}^{r_{\left( k \right) }} C_{op}^{\left( k \right) } \frac{\psi _{\gamma }^{\left( k \right) } \psi _{\delta }^{\left( k \right) }}{r^2} r \,dr \textit{,} \\&\varDelta ^{\left( k \right) }_{ij} = \int \limits _{r_{\left( k-1 \right) }}^{r_{\left( k \right) }} \psi ^{\left( k \right) }_i \psi ^{\left( k \right) }_j \,dr \end{aligned}$$