Abstract The integrity of the casing is crucial for oil and gas well. Based on stress function method, a three-dimensional model of the casing-cement sheath-formation system subjected to linear crustal stress is proposed. And then an analytical solution of the model was obtained. In the process of calculation, the casing and cement sheath are simplified as the perfect cylinder. The cement sheath is closely bonded with the casing and formation. The formation is considered to be an isotropic material without the layer-block structure. And the crustal stress is assumed to be linearly increasing with the depth of the well. The analytical solution strictly meets the stress and displacement continuity condition and boundary condition, and exhibits good agreement with finite element method. The results imply that an analytical method to capture the stress and displacement field of the casing under linear crustal stress along the axis is presented. Next, a benchmark for numerical and approximate solutions is provided. In addition, a new idea about solving the casing under the non-linear loads along the axis in some special stratum (such as heterogeneity stratum, salt rock) is proposed. Finally, our understanding for the casing under complex loads will be deepened.