Triangle Meshes are static objects. They are designed for complex rapid collision detection purposes for stuff like buildings, terrains and caves. The polygons are stored in RAM for the PhysX engine to detect the collision quickly, without any dynamic transformation.
The first solution should be to deal with the separate parts of the mesh, then combine into a compound shape using DYN MAKE COMPOUND FROM SHAPES (int objectID, float density) . You should be able perform a dyn make shape command on all parts of the object to then produce the compound shape. But I have failed to get this to work because the feature is either broken in the PhysX API DBPro uses or the dynamix plugin has errors. Dyn Add Shapes command has a parameter missing error, and Dyn make shape commands seem to only produce 1 shape in the shape list.
The next best solution is to use Convex Meshes. If the mesh is not in convex form the engine will try its best to calculate a convex shape out of a concave mesh (according the the documentation).
You can also use the rigid joint example if the object is similar to a table or hull and the convex shape method will not suffice. (any other joint seems to be nightmare to work with but the ragdoll example is pretty helpful)