Optimization of dynamic engineering systems requires an integrated approach that accounts for the coupling between embodiment design and control system design, simultaneously. Generally known as combined design and control optimization (co-design), these methods offer superior system performance and reduced costs. Despite the widespread use of co-design techniques in the literature, extremely limited research has been done to address the issue of uncertainty in co-design problem formulations. This is problematic as all engineering models contain some level of uncertainty that might negatively affect the system’s performance, if overlooked. Accounting for these uncertainties transforms the deterministic problem into a stochastic one, requiring the use of appropriate stochastic optimization approaches. Therefore, this dissertation serves as the starting point for research on stochastic co-design problems when the uncertainty is propagated into the system from random design decision variables and/or problem parameters. Specifically, a simultaneous co-design formulation within multidisciplinary dynamic system design optimization (MDSDO), along with a special class of direct methods, known as direct transcription (DT), are consistently used throughout this research as the basis for uncertainty considerations.