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A computational strategy is presented for determining sensitivity coefficients for the contact-impact response of composite shell structures. The shells consist of an arbitrary number of perfectly bonded layers made of a hyperelastic, anisotropic material. The sensitivity of the response is determined with respect to various parameters (longitudinal modulus of elasticity, density and dynamic coefficient of friction). Numerical results are presented for two cylindrical structures impacting a rigid plate at an inclined angle. The local and global response quantities are obtained by the computational procedure.