The recently developed LuGre model describes the three-dimensional tyre friction dynamics given in the brush representation. The model has a three-state lumped-parameter form and an analytical solution for tyre static curves. It is simplified using the assumption of a constant slip speed along the contact patch length. This assumption is abandoned in this paper in order to extend the model with several secondary but potentially important effects such as camber, carcass compliance, conicity, ply steer and rolling resistance. Two approaches to the introduction of a variable-slip-speed effect are considered. The first is based on stepwise approximation of the slip speed along the contact patch length, and the second includes a specific, spatially distributed additive deflection term. Both approaches preserve the three-state lumped-parameter model form and an analytical steady-state solution. Qualitative and quantitative validation results point to consistency and benefits of the proposed model extensions.