As all of you know, despite how many objects instantiated, only one instance of static member variables are stored in the BSS segment. Accordingly, consider having some thread which instantiates a class with a static member variable and changes static member frequently; my question is whether the static member is thread-safe or not.
Variables do not know anything about threads. Class static member variables are not inherently thread-safe. You might consider putting them in thread-local storage, or making sure the members' data structures are themselves thread-safe.
You also mentioned "static member functions" but those are immaterial to this question: it doesn't matter how you modify your static member variables, doing so is not inherently thread-safe, whether modified via static member functions, regular member functions, or non-member functions.