The magnitude and distribution of stresses around suddenly punched holes in initially stressed plates and shells is of interest to insure that cracks will not precipitate from stress concentration. This problem is of practical interest to pressure vessel designers to preclude catastrophic failure when holes are punched in vessels to release gas. This paper presents a finite element analysis of several problems investigating static and dynamic stress fields around suddenly punched circular holes. The first problem deals with the investigation of the radial and tangential stress fields in the vicinity of a suddenly punched hole in a stretched, elastic, isotropic plate subjected to an initial hydrostatic stress field. The wave propagation from a punched hole in the plate under a hydrostatic state of stress was solved analytically, using transform techniques, by Miklowitz; the finite element analysis of this problem presented in this paper confirms the analytical solution. Two grid meshes were investigated and results are presented to show the effect of grid mesh on solution accuracy and the power of finite element techniques for solving stress unloading problems. A formula for determining integration step size is found to be a function of the minimum element length and the wave propagation velocity. A similar investigation into the stress effects around a suddenly punched hole in the plate subjected to an initial uniaxial state of stress was also carried out as a prerequisite for the final problem studied. The last problem is an anisotropic composite shell of varying thickness under an initial stress field due to internal pressure. The static and dynamic stress fields are computed from an unloading wave that radiates outward from a reinforced circular hole that is cut in the shell in 20 μs. A finite-element model of the shell is developed using quadrilateral and triangular plate elements and both in-plane and bending stiffness is included in the analysis as is nonlinear differential stiffening incorporated into the analysis as a single step approximation. Both bending and in-plane waves radiate outward from the cut hole and the dynamic stresses around the hole edge are computed for both unloading waves. The effects of the unloading waves are temporally spaced due to different wave velocities. The paper demonstrates that fast response stress problems are readily amenable to finite-element analysis. For holes other than circular, the power of finite-element methods is apparent since these shapes lead to mathematically intractable problems if closed form solutions are attempted.