A general scheme for the solution of linear elastic quasi-periodic fracture problems is presented. The simplest type of such problems is characterized by a non-periodic stress state in a domain with translational symmetry. Employing the discrete Fourier transform reduces the initial problem to a problem of a representative cell with specific boundary conditions which may be solved analytically or numerically. The procedure for solving the problem by the finite element method is developed. The suggested technique is employed for the solution of the problem of antiplane deformation of a strip weakened by a periodic array of arbitrary loaded cracks.