1. The problem statement, all variables and given/known data So this isn't actually a homework problem or anything I'm just having problems understanding what it means to be statically determinate, as well as how to determine whether or not something is statically determinate or not. Any help would be greatly appreciated.
A structure is statically indeterminate if there are more unknown forces than the number of static equilibrium equations available to solve for those forces. For example, in 2D analysis of a beam, there are 3 static equilibrium equations: sum of F_x = 0, sum of F_y = 0, and sum of M_o = 0. Thus a simply supported beam with a pin at one end and a roller suport at the other, is statically determinate, because there are 3 unknown reactions (2 forces at the pin and one at the roller), and the 3 static equilibrium equations available to solve for those forces. A cantilever beam fixed at one end is also statically determinate (2 forces and a couple unknown at the fixed end, and the 3 static equilibrium equations available to solve for them). A beam on 3 simple supports, or a propped cantilever fixed at one end and simply supported at the other, would be statically indeterminate because you have more than 3 unknown forces, and you would have to resort to compatability/deformation equations to solve for the additional forces. A statically indeterminate truss is a bit different, in that while the suport reactions may sometimes be solved using the 3 static equilibrium equations, the member forces themselves framing into a joint may not be able to be determined using the standard equilibrium equations, for example, when you have multiple members framing into a single joint. There's a formula floating around somewhere that identifies to what degree a truss may be indeterminate, based on the number of members and joints, etc. , but they are too confusing for me to figure out.
A system is statically determinate provided the equations of statics are sufficient to describe it. If the system is statically indeterminate, then the displacements will have to be included in the description along with the equations of statics. For static structures, this usually means including the elastic relations of the structure in some form.