General Approach to Statically Indeterminate Problems
All problems in Mechanics of Materials can be approached using the same basic four principles. These principles are:
- Free Body Diagram. If a part is separated or isolated from a body (ie: sectioned) or surroundings (ie: boundary conditions), appropriate reactions and/or internal forces must be added.
- Equilibrium of Forces. If a part or body is in equilibrium, all reactions and/or internal forces must also be in equilibrium.
- Displacement Compatibility. The nature of plausible deformation in a system imposes constraints on the geometry and/or deformation of structural elements in a given system (ie: two solid objects cannot occupy the same space, elements rigidly connected to each other must deform with each other).
- Stress-Strain (Force-Displacement) Relations. Loading and deformation in a structure are linked by material properties and structural geometry through Hooke’s Law
Once the number of equations is equal to the number of unknown reaction forces, you can solve for the reactions. Be aware of what the question actually asks for. It is common to go through the above procedure and feel like you are done. However, some questions ask about a particular stress state, deformation, or require some sort of sizing to be performed. All of these activities can only be completed once the unknown reactions are determined.
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