10.1.2. Classes of Solution

Analysis methods of statically indeterminate structures must satisfy three requirements:

1. Equilibrium
2. Force-displacement (or stress-strain) relations (linear elastic in this course).
3. Compatibility of displacements (i.e. no discontinuity).

This can be achieved through two classes of solution:

1. Force or Flexibility method;
2. Displacement or Stiffness method.

In analyzing statically indeterminate structures, we must ensure that three key requirements are met: equilibrium of forces, appropriate force-displacement or stress-strain relationships, and compatibility of displacements. Equilibrium ensures that the structure is stable under the loads applied. The force-displacement relationship considers how much the structure deforms under these forces, and compatibility ensures that all parts of the structure move together without any gaps or overlaps. To accomplish this analysis, engineers can use either the Force (or Flexibility) method, which focuses on calculating internal forces, or the Displacement (or Stiffness) method, which emphasizes displacements in finding forces.

Think of a row of dancers performing a coordinated routine. Each dancer must maintain their balance (equilibrium) while moving fluidly in time with one another (compatibility). If one dancer moves out of sync, it could disrupt the entire performance. The Force method would be like focusing on the strength of each dancer's individual movements, while the Displacement method would focus on how well they move in relationship to each other.

The flexibility method is illustrated by the following problem of a statically indeterminate cable structure in which a rigid plate is supported by two aluminum cables and a steel one. We seek to determine the force in each cable. We have three unknowns and only two independent equations of equilibrium. Hence the problem is statically indeterminate to the first degree.

In this example, we have a cable structure that is considered statically indeterminate because it has three unknowns (the forces in the cables) but only two equations of equilibrium to solve for those unknowns. Essentially, we can't determine the forces using only the typical methods applied to determinate structures. The flexibility method will help us analyze how the structure behaves under load and find the internal forces within each cable by taking into account the material properties and the way each cable will stretch or deform under those loads.

Imagine a group of gymnasts on a balance beam, trying to keep their balance while radiating out in three different directions. If two gymnasts are trying to pull in one direction, the third one's pull adds complexity to achieving balance since the team can only use a set number of supporting hands (equations of equilibrium). Just like in our cable structure example, we need a clever way to analyze how each gymnast interacts with each other to maintain their balance on the beam.

We observe that the solution of this problem, contrarily to statically determinate ones, depends on the elastic properties.

Unlike statically determinate structures, where the forces can be calculated directly using equilibrium equations, statically indeterminate structures require us to consider the material properties, such as elasticity, to solve for unknown forces. When a structure's response is determined by how materials stretch or compress, the analysis must include these material characteristics to accurately predict internal forces during loading.

Consider a rubber band stretching when you pull it. The extent to which it stretches is not just about how hard you pull (the force), but also how elastic that rubber band is (the elastic property). In our structural example, if the cables were made of different materials with varying elastic properties, their responses under the same loads would differ, similar to how different rubber bands stretch differently under the same force.

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