I have not run across that problem myself. Perhaps you can give us more information like what brand and the name of the set that you use. Also, is the core round or hex shaped, and what is it made of.
The problem could alternately be with the bridge and saddle. So we need to know what brand guitar and the name of the model as well as the brand and style of the bridge with all of the details. Your playing style and type of pick used is also important. BTW does this guitar have a whammy bar that you use regularly?
Once you supply that information we may be able to help you with this problem.
P.S. I'm not sure if this is applicable for a 24W string but for the plain strings if one of them breaks right at the saddle I will dress it just a little piece with folded emery paper. If I do that every time the string breaks there, the problem goes away. (I can't see how the core could be damaged by a small burr on the saddle.)
One other thought: if you have a Tune-A-Matic bridge and matching tailpiece, sometimes when you lower the tailpiece for more sustain it can make the angle of the string going over the saddle too sharp.
The 24W is pretty much the smallest wound string used on an electric guitar. So the core and wrap would be the smallest of all of your wound strings and it would be more likely to break.