<p>We present a reliability-based topology optimization method for structural assemblies represented by geometric primitives with uncertain dimensions and position using the geometry projection method. Compared with the density-based and level-set representations widely used in topology optimization, the primitive-based representation reduces the number of random variables, making the computational expense of incorporating geometric uncertainties via surrogate models practical. Moreover, the random variables are directly related to the dimensions and positions of the primitives, which is a natural description of dimensional and positional variability for assemblies made of stock material. The proposed method adopts a decoupled quantile-based scheme, whereby at each outer iteration, a deterministic topology optimization with a constraint on the limit-state function is solved; a surrogate of the limit-state function at this design is subsequently built via the univariate dimension reduction method to estimate the quantile corresponding to the target reliability, which in turn is used to update the constraint limit for the next outer iteration. The proposed approach is demonstrated through several examples and the results are compared with those of a single-loop formulation based on the first-order reliability method, showing closer satisfaction of the prescribed performance reliability targets at practical computational cost.</p>