Abstract: A series of micro-loading tests on metal microbeams have revealed a significant increase in bending strength as specimen sizes decrease, demonstrating a remarkable size effect. In this study, we systematically investigate the initial yielding strength of pure metal single-crystal microbeams, utilizing a novel dislocation pile-up model that incorporates stress gradient effects. This model allows us to identify the key intrinsic length scales behind the underlying strengthening mechanisms, which are pivotal in determining the size-dependent behavior of microbeams when subjected to either pure or transverse bending. Through a synthesis of existing micro-bending experimental observations and discrete dislocation dynamic (DDD) simulation results, we firstly propose a new dislocation pile-up configuration, which is exclusively focused on dislocation sources and comprehensively accounts for the interactions between free surfaces and dislocations. Within such a pile-up configuration, we then conduct a detailed continuum analysis of dislocation pile-up behavior under linear stress gradients, which leads us to finally establish a source-controlled stress gradient yield model. Applying this yield model to micro- and submicron copper beams under bending conditions, we observe that it could effectively capture the size-dependent behavior of the initial yielding stress in microbeams within the specified size range (spanning from several micrometers to hundreds of nanometers). Notably, our model shows good agreement with corresponding experimental results. These findings suggest that, for pure metal single-crystal microbeams with specimen size of several micrometers or smaller, the dislocation pile-up strengthening mechanism plays a dominant role in the observed size-dependent behavior. Moreover, our results emphasize the importance of two material length scales — the source length and the dislocation pile-up length — in accurately quantifying the size-dependent bending strength at micro- and submicron scales. This study offers an insightful perspective in understanding the size-dependent yield stress in micro-scale crystalline materials under non-uniform loading conditions, particularly in the case of pure metal single-crystal microbeams.