We present work on the application of the asymptotic numerical method (ANM) [1] in a solid mechanics context for a homogeneous elastic material subjected to homogeneous pressure. We propose a bifurcation analysis of thin structures under conservative and non-conservative loads. ANM is used to perform continuation, detection and branch-switching at simple bifurcation. It relies on the approximation of the unknowns of the non linear problem by truncated power series. Thus, the non linear problem is transformed into several linear problem to solve using the same tangent operator. The path following of the branches uses a pseudo arc-length method. The detection of simple bifurcations uses ANM series analysis. Continuation and branching are specifically adapted to the pressure follower loading case. Recall that the ANM continuation for pressure follower was recently proposed in [6]. In this work, we propose to detect singularities more effectively by ANM series analysis [4] and also to calculate post-bifurcation solution branches using a branching method at a single bifurcation point. This involves applying the branching method developed in [2], and adapted in the works [3, 5], for the case of non-conservative pressure follower. For the proposed branch-switching approach, we obtain a system of equations analogous to the works [2, 3, 5]. As a preliminary study, we examine the case of a cylindrical panel subjected to homogeneous pressure on its upper surface. A comparison with the commercial software DS Abaqus is proposed. We note very good agreements between ANM and Abaqus continuation in the case of pressure follower. We are able to detect accurately the singularities. In addition, every post-bifurcated branches are found with the proposed method. Bifurcation diagrams, critical pressure values and visualizations of singular solutions and associated bifurcation modes will be presented.