The finite element method based on first-order shear deformation theory is utilized to obtain the nonlinear transient response of delaminated composite plates and shell panels under hygrothermal conditions. The von-Karman nonlinear strain-displacement relations are used in the kinematics to include geometric nonlinearity in the formulation. The nonlinear transient responses are obtained by implementing the Newmark constant average acceleration method along with the Newton-Raphson iteration scheme. The point continuity method is implemented to satisfy the continuity condition at the delaminated interfaces. First, the results are validated for constant stiffness composite plates and shell panels having single and multiple delaminations. After that, the results are obtained for variable stiffness composite plates and shell panels having delamination. The effect of a change in the center and edge angles of the curvilinear fiber path is studied along with delamination in the structure. Additionally, the impact of changes in delamination size and delamination position is examined in variable stiffness composite plates and shells under hygrothermal conditions.