There are growing concerns for the remaining service life of concrete infrastructure under normal service conditions and the structural resilience under extreme climate events. Therefore, advanced and reliable computational tools are required for the assessment of existing structures' condition, and the estimation of their serviceability. Traditionally, advanced structural simulations are conducted using nonlinear Finite Element Analysis (FEA) that exhibits major drawbacks hindering its application for large-scale simulations, particularly in real-time or nearly real-time. Those drawbacks include high computational time/power, convergence problems, and limitations in modelling the actual (than ideal or theoretical) condition of the structure and, more importantly, model updating as the structure deteriorates or undergoes changes. This paper proposes a closed-loop and computationally affordable cyber-physical solution for comprehensive structural health monitoring. The proposed approach is based on real-time prediction of the structural response for a concrete structure by creating, updating, validating, and maintaining a Structural Digital Twin founded on the framework of Physics-Informed Neural Networks (PINNs). PINN-powered structural digital twins present a novel simulation scheme that combines the physics-based models (represented by differential equations governing the structural behavior) with data-driven models (trained on the response data collected through sensors) into a robust computational model. The proposed method, implemented in a lab-scale case study, is presented in detail, and future areas of research will be discussed.