A characteristic smile crack at the rear of the 1929-31 Model AA Ford truck four speed transmission case is quite commonly found, and is caused by the wear and failure of the rear countershaft roller bearing rubbing against the rear thrust surface and oiling notch within the transmission case.
Pictured above is a severe crack, almost completely through the case top to bottom, side to side. Many cracks are much more subtle, but cracked none the less. Also, the crack will likely grow when the countershaft is removed.
The arrows in the pic above indicate the oiling keyholes in the front and rear countershaft thrust surfaces of the transmission case.
When a cracked transmission case is disassembled, it is typically found that a countershaft roller bearing has self-destructed, the rollers have skewed, and a part of the cage has moved into the "oiling keyhole" at the front or rear countershaft thrust surfaces.
From there, the piece of sheet metal cage gets drawn into the thrust face gap between the case and the countershaft gear, and effectively wedges and locks the transmission.
It is like "throwing a wrench in the gearbox" when the roller bearing cage fails, and engages the oiling notch in the case casting. The case cracks from the high longitudinal forces and stresses from the resulting wedge, and will also likely cause the transmission to lock up mechanically!
You may have to look at a lot of used AA trans cases to find one that is not cracked. Based on my limited experience several years ago, I had to look at about 10 as-found cases before I found an uncracked one.
In the process of rebuilding an AA tans, I also cleaned and welded three cracked cases to see what I could save. I welded one with cast iron rod and oxyacetylene, one with an arc welder and a rod for cast iron, and one just brazed with brass. I am not an expert welder and all my repair attempts were failures because the countershaft holes could not be (kept) properly aligned.
When the cracked case is disassembled, the stress will be relaxed and the front and rear countershaft holes will move out of alignment. Also, the inside of the crack is contaminated with oil and is difficult to clean, compress, and weld.
It is a lot of work to repair a cracked case correctly and I would recommend just looking for a good one, but be sure to check each case carefully. If the outside rear of the case is covered with gunk buildup the case is probably already cracked and oil has been seeping out of the crack and collecting dirt.
Here are a couple of pics of a good original uncracked four speed transmission case.
In the pic above, the Ford part number and casting date is seen, as well as "WG", which was the name of the transmission supplier, Warner Gear. Also note the imprint of the casting date tag at the top, clearly showing an original Model A era date, not a later service replacement part date.
The other main supplier of four speed Model AA transmissions was Detroit Gear & Machine, and their transmission cases and gears are usually marked "DGM".