Vertical axis wind turbines (VAWT) are subjected to blade lift forces which vary continuously in both magnitude and direction. These blade lift forces are transmitted via the blade support arms to the tower. The resulting tower force vector is a composite of: a downwind and a crosswind average force component, rotating force vectors, and force vectors oscillating in the crosswind direction. The frequency of the rotating and oscillating forces are multiples of the product of Bω, where B is the number of blades used and ω is the rotor angular velocity. The magnitude of the largest tower shake force vector is of the same order as the average downwind force component, and may represent a serious design constraint in the calculation of the required tower stiffness. A closed-form solution for the tower force vectors has been derived, by introducing a suitable wind interference model. It shows that the magnitude of the largest tower shake force vector, using a three-bladed rotor, is four times smaller than with a two-bladed rotor. The Betz limit and the optimum tip speed ratio as a function of solidity has been derived by comparison with two semicylindrical actuators in series.