12.1.1. Stiffness vs Flexibility

There are two classes of structural analysis methods, Table 12.1:

1. Flexibility: where the primary unknown is a force, where equations of equilibrium are the starting point, static indeterminancy occurs if there are more unknowns than equations, and displacements of the entire structure (usually from virtual work) are used to write an equation of compatibility of displacements in order to solve for the redundant forces.

2. Stiffness: method is the counterpart of the flexibility one. Primary unknowns are displacements, and we start from expressions for the forces written in terms of the displacements (at the element level) and then apply the equations of equilibrium. The structure is considered to be kinematically indeterminate to the nth degree where n is the total number of independent displacements. From the displacements, we then compute the internal forces.

This chunk introduces the two fundamental approaches to structural analysis: flexibility and stiffness. The flexibility method focuses on forces as the unknowns and begins by establishing equilibrium, shifting the focus towards displacements once enough equations are established. In contrast, the stiffness method starts with displacements as the primary unknowns, deriving forces based on those displacements and using equilibrium to solve for any unknowns. The degree of kinematic indeterminacy indicates the number of independent displacements that can occur before additional constraints need to be considered.

Imagine a swing set. If you pull on the swings (like applying force), you can think of that as the flexibility approach, where you're looking at how the swing reacts to every force you provide. On the other hand, if you observe how high the swings go when released from different heights (like measuring displacements), you're employing the stiffness approach. Understanding both methods helps us build safe structures that respond well to both forces and displacements.

This chunk provides a table comparing flexibility and stiffness methods along various dimensions. Key highlights include that the flexibility method focuses on forces, where the primary unknowns lead to static indeterminacies, while the stiffness method considers displacements as primary unknowns and leads to kinematic indeterminacies. The governing relations differ, with flexibility relying on compatibility equations and stiffness focusing on equilibrium equations. The methods of analysis also differ, with flexibility employing consistent deformation methods and stiffness utilizing techniques like slope deflection.

Think of a bridge. When testing its design using the flexibility method, engineers might examine how much weight (force) it can bear before it bends too much, whereas, in the stiffness method, the focus shifts to how far the bridge will bend at various weights, ensuring it's built to endure even while under tension. Both perspectives are critical to guarantee the structure remains safe and functional.

In the flexibility method, we started by releasing as many redundant forces as possible in order to render the structure statically determinate, and this made it quite flexible.

The flexibility method requires identifying and removing redundant forces to simplify the system, making it easier to solve. By doing so, the structure becomes statically determinate, which means that enough equations are available to solve for unknown forces without ambiguity. This process increases the flexibility of the structure, allowing it to adapt better under loads.

Imagine a large, complex puzzle where some pieces overlap unnecessarily. If you take out those overlapping pieces, it becomes easier to assemble the entire puzzle. Similarly, in structural analysis, removing redundant forces simplifies the calculations needed to understand how the structure behaves under load.

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