A method of fabricating a mechanical member for aircraft, including a plurality of operations of braiding and depositing layers of braided reinforcing fibers on a mandrel (11) by using braiding machine. Each operation comprises braiding a braided layer and depositing it by moving the mandrel (11) along a central axis of the braiding machine. Each of the various superposed braided layers comprises both longitudinal fibers (12, 12G) that are parallel to a main direction of the mandrel (11), and interlacing fibers that are inclined. At least one operation is configured to form and deposit a braided layer having, in at least one cross-section of the member, a density of longitudinal fibers that differs depending on whether consideration is given to one angular region (S1) or another angular region (S2) of the same extent around the center of gravity (G) of the mandrel (11) in the section under consideration.