The Variational Theory of Complex Rays applied to the shallow shell theory

•VTCR is extended to shallow shell theory.•The uniqueness property of the shallow-shell-VTCR solution is demonstrated.•p-Refinement is the most performing.•VTCR outperforms FEM at mid-frequency. In the last few decades the interest of aerospace and automotive industries towards the study of the medium-frequency response of complex shell structure frames has grown. Recently some dedicated “wave” computational approaches have been developed. Among them, a Trefftz technique called Variational Theory of Complex Rays (VTCR) is catching on as an ad hoc method to deal with such vibration problems. This work presents the development of the VTCR in the shallow shell theory to increase its effectiveness and flexibility. First, general theory is given and two key properties of the solution demonstrated. After that, two numerical examples are deeply analyzed.