Flight Crew Air Transport Pilot (ATP)
Situational Awareness; Fatigue; Distraction
Deviation - Track / Heading All Types; Deviation / Discrepancy - Procedural Published Material / Policy; Inflight Event / Encounter Weather / Turbulence
Flight Crew Became Reoriented
Human Factors; Weather
During climb passing about 2;500 feet and at 220 KIAS and in a right turn to avoid weather I as Pilot Flying (PF) inadvertently allowed bank angle to reach 35-40 degrees followed by GPWS 'bank angle' aural warning. My focus was on the weather radar and thunderstorm avoidance instead of flying the plane.  Recovery was immediate and smooth with no stick shaker or physical vibration indicating a stall. Autopilot was engaged and aircraft cleaned up. No loss of PFD information and no exceedances of flight control limitations occurred.  No abnormal flight control conditions were noted and flight was continued to destination.  [Logbook] entry made.Human error - failed to aviate first. Distracted by multiple cells of weather displayed on screen. Fatigue - well rested day before event but trip revision; minimal crew rest and long duty day the day before; and this being the fourth night of hub turns had me short on sleep for the week Non- compliance - unintentional exceedance of bank angle normal limits of 30 degrees.Aviate first; use autopilot during high workload; don't fixate on one issue - keep scan going.
[Report narrative contained no additional information].
A300 flight reported bank angle limit was exceeded after takeoff when pilot flying was distracted by weather in the area.