A planar harmonic mapping is a complex-valued function f : U → C of the form f (x + iy) = u(x, y) + iv(x, y), where u and v are both real harmonic. Such a function can be written as f = h + 1, where h and 1 are both analytic; the function ω = 1′/h′ is called the dilatation of f . We consider the linear combinations of planar harmonic mappings that are the vertical shears of the asymmetrical vertical strip mappingsϕ j(z) = 12i sinα j log ( 1+ze iα j 1+ze −iα j ) with various dilatations, where α j ∈ [ pi2 , pi), j = 1, 2. We prove sufficient conditions for the linear combination of this class of harmonic univalent mappings to be univalent and convex in the direction of the imaginary axis