This sounds strongly like a symbol conflict to me. I'm just surprised the linker didn't catch it, but I assume that's because you're using a static library in both your frameworks instead of simply yet another framework.
Generally speaking, I'd warn that "this is a bad idea™". What you're trying to introduce into your design is basically dependency management. Like a lot of blog articles and specifically this SO answer suggest, you should avoid packaging frameworks (and by extension libraries) into frameworks.
What most likely happens in your scenario is this (I admit I'm guessing a bit here): You linked the library into Framework A. Thus, the library becomes a fixed part of it. Its symbols are in it, even if you did not expose them to the app in any header files or the like. As long as you use only that, everything works smoothly. Then comes Framework B, of which the library is also a fixed part. Even though you can't see it from your app, the very same symbols are inside it. This, however, clashes with the symbols that were already loaded by Framework A, hence a crash (as said, this would usually be caught by the linker, but I guess you "tricked" it by building the frameworks beforehand and packaged the library in them). Maybe somebody else can explain this in more detail, but that quickly becomes besides the point as to how you would go for a solution. From how I see it, it just won't work this way.
So here's a suggestion for how you can solve your problem:
If you really, really need to split like this (two frameworks used in your app using the same dependency), I'd suggest removing the library from the frameworks (i.e. make them depend on it, but not package the actual .a file in them) and documenting that properly. Then add the library to your app project, just like you added the frameworks.
If you want to make this fancy and easily installable into other apps, I'd recommend setting up a private CocoaPods repository and turn your frameworks into private pods. You can then easily define the library (or rather "new Framework C") as a dependency for Framework A and Framework B. When you then pod install in your app, cocoapods figures out the dependency and adds Framework C automatically to your project. This scenario is exactly what cocoapods (or any dependency manager for that matter) was designed for. It automates and helps in the project setup, so that the final build (the app) doesn't have to figure out dynamically what it can and can't use. The end result is the same.
Trying to duplicate that "in code" quickly becomes messy. Frameworks trying to figure out things of the surrounding app/project that uses them (like "is my dependency so-and-so already linked? if not, can I load my own version of the library?") can lead to a lot of pain.
Okay, in response to your comment I'll try my hand at a more detailed How-To for the non-cocoapods setup. As a preface, though, let me say that that's kinda hard to do on top of my head, as I have no ready sample project to toy around with. Since this is one of those "set it up once and then forget aout it for a long time" I have to admit my recollection of these things is a bit fuzzy, so consider this as a sort of "rough direction". There might be things you need to configure differently than how I recall them. Other SO user are thus hereby invited to edit and correct my answer here, too. :)
First, I have to say I am not exactly sure whether you need to convert your static library into a framework or not for this, I think you don't so I'll go from here (I have never used a static library in that way).
That means you can leave the project that builds your library as is. On second thought, you probably have to do this to be able to link against the library without making it a part of the framework that uses it. I will still refer to this code as "library" in the below points, but assume that you're able to turn it into a dynamic framework as well.
The projects building Framework A and Framework B should be changed. I don't know how you have this set up (as one project with various targets, whether you have a "development application" as part of it to test the frameworks on themselves, etc.), but the important thing is that in the target that builds a framework, the library should be linked (i.e. in the "Link Binary With Libraries" build phase), but not copied (i.e. it must not be in the "Copy Bundle Ressources" build phase). It might be tricky to set up any development/test target you use to run, depending on how you did that so far. For example you might have to manually edit Library Search paths in your build settings to be able to properly compile the framework.
Lastly, you need to change your final app's project settings, obviously. The library that was originally part of Framework A and B now needs to be linked to from its project directly, and, obviously, it needs to be copied into the bundle as well. Note that any projects that include either of your frameworks (A or B or both) in the future must do so, otherwise they won't work, because these frameworks expect the library to be present (but no longer "bring it with them").
In spite of this long-ish text wall, it shouldn't be that complicated, I think, but you may still want to check out how cocoapods can support you with something like this, perhaps it's helpful in the future. The linked article expects basic knowledge on how to use a pod and write one, but the other guides on the site explain that pretty well. I am just saying this because in the end, when using cocoapods in a project to add multiple libraries/frameworks that introduce dependencies, it basically sets up your project in the way I described above. It uses some fancy shell scripts that ensure everything is copied to the correct folders and such, but overall speaking that's how it works: A pod's podspec tells cocoapods what additional pods to include in your app for it to work (a dependecy the pod expects to be there, but doesn't "bring in" by itself).
