Summary: We present a simple low-order solid-shell element formulation – having only displacement degrees of freedom (dofs), i.e., without rotational dofs – that has an optimal number of parameters to pass the patch tests, and thus allows for efficient and accurate analyses of large deformable multilayer shell structures using elements at extremely high aspect ratio. The formulation of this element is based on the mixed Fraeijs de Veubeke-Hu-Washizu variational principle leading to a novel enhancing strain tensor method that renders the computation particularly efficient, with improved in-plane and out-of-plane bending behavior (Poisson thickness locking), especially in refined analyses of composite structures involving a large number of high aspect-ratio layers. We also review the equivalence between various choices of the enhancing strains in tensor form, and point out the relative efficiency of these choices. We discuss the enhanced assumed strain (EAS) formulations based on both the Green–Lagrange strain and the displacement gradient (the companion paper), point out the pitfalls in each approach, e.g., not passing the patch test, and the possibility and the method to remedy the problem. Shear locking and curvature thickness locking are treated using the assumed natural strain (ANS) method. The element passes the patch tests (both membrane and out-of-plane bending). We provide an optimal combination of the ANS method and the minimal number of EAS parameters required to pass the out-of-plane bending patch test. Numerical examples involving static analyses of multilayer shell structures having a large range of element aspect ratios are presented. Finally, we note that the topic in this paper is a fitting dedication to Professor Ekkehard Ramm, who has made important pioneering contributions in this research direction.