The prediction of the Thermo-Hydro-Mechanical (THM) behavior of large buildings with a containment role (reservoirs, dams, nuclear vessels, etc.) is a critical step towards their risk assessment. In particular, their cracking implies a considerable loss of their structural tightness that needs to be controlled, monitored and, if necessary, repaired to ensure a safe operational environment. The difficulty of performing numerical predictive analyses is related to (a) the multiphasic and multi-physical nature of concrete (b) the large number of inputs to identify at the specimen and structural scales (c) the non-negligible and intrinsic material and load related uncertainties. All these aspects strongly affect our ability to foresee the structural response of large constructions; especially in terms of cracking and tightness. In this contribution, a global finite elements based stochastic methodology is proposed to allow physically representative and efficient non-intrusive probabilistic coupling of strongly nonlinear and numerically expensive THM calculations. To this aim (a) concrete cracking is modeled using a stochastic, local and energy regularized damage model accounting for size effects (b) concrete permeability is defined using a strain-based law (c) the spatial heterogeneity of properties is modeled using discretized and FE projected Random Fields (d) uncertainties propagation is computed using adapted Surface Response based methods. For the demonstration of this strategy's efficiency and effectiveness, in terms of physical accuracy and cost optimization, a 1:3 scaled containment building named VeRCoRs is considered as an application. In particular, a complete probabilistic analysis of its dry air leakage rate (indicative of the whole structural performance) is achieved within a computational time of tens of days only. In general, such results can help during the decision-making process for the design, maintenance and risk assessment of large structures with a containment role based on a leakagerate-defined criterion under service loads.