A new method for solving the biharmonic equation in an arbitrary convex polygon with arbitrary linear boundary conditions is applied to a class of mixed boundary-value problems involving a semi-infinite strip. Emphasis is placed on boundary-value problems for which an explicit solution can be constructed. A variety of mixed boundary-value problems are shown to admit explicit solutions. This class includes, but is not limited to, the canonical problems of the elastostatic semi-strip. New integral representations of generalizations of these canonical problems are derived where the sidewall boundary conditions are inhomogeneous.