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Strength of Materials: Statically Indeterminate Axial Loading

Most cases in a real word situation are statically indeterminate. Statically indeterminate means that there are more unknowns then static equations to solve for those unknowns. For a member that is axial loaded in one direction, there is only one static equation that can be used to solve for the unknown. If the situation is similar to the figure below then it would be considered statically indeterminate.

Statically Inderminate Axial Loaded Member

Superposition Method

To solve a statically indeterminate problem, additional equations would have to be derived based on what the engineer knows about the problem, this could be symmetry or something else. For the image above we could take in consideration of deflection, and use superposition method (which is breaking the problem into parts; view the image below) to derive another equation. Realize, however this is going to only derive one additional equation. View the resulting equation below.

Superposition Method Axial Loading

Using Stiffness

Another way to look at a problem is to view it as a set of springs. Basically, this would be similar to looking a circuit diagram for an electrical application. If the each spring sees the same force, the circuit of springs would be considered in parallel.

Axial Loading Stiffness Same Force

On the other hand if the circuit of springs have the same deflection they would be considered to be in series to one another.

Axial Loading Stiffness Same Deflection

Similar to the superposition method, this derives one additional equation.

If there are too many unknowns to derive equations for, then a finite element analysis can be performed to approximate the resulting forces and stresses.

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