10.1.1.2. Force-displacement (or stress-strain) relations

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)

In structural analysis, particularly for statically indeterminate structures, it's essential for the analysis methods to satisfy three conditions. The first condition is equilibrium, which ensures that the structure remains in a balanced state. The second condition is the force-displacement relations, which describes how the material deforms under applied forces – we consider these behaviors under linear elastic materials for this course. The third requirement is compatibility of displacements, which means that the course of deformation is continuous, with no abrupt changes or jumps.

Imagine a long rubber band that stretches and returns to its original shape when a weight is added and then removed. The relationship between how much you stretch it and the weight you apply is akin to the force-displacement relation. If the rubber band breaks (a discontinuity), that would violate compatibility. Likewise, a balanced seesaw fulfills equilibrium.

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

When solving problems in structural analysis, the elastic properties of materials play a pivotal role, especially for statically indeterminate structures. Unlike statically determinate structures where the solutions might depend solely on loads and geometry, statically indeterminate structures require an understanding of how materials deform under load. This dependency on elastic properties means we must consider material stiffness and yield limits as part of our analyses.

Consider constructing a bridge using steel versus rubber. The steel will only deform a little when heavy vehicles pass, while rubber would deform significantly. In a statically indeterminate structure, the way each material handles these loads will greatly affect the overall structural behavior.

Compatibility of displacements ensures the structure does not experience discontinuities in deformation.

In structural analysis, compatibility of displacements means that as loads are applied, the structure must deform in a continuous manner without any breaks or interruptions. In simpler terms, if one part stretches, adjacent parts should adjust accordingly without any gaps. This principle aids engineers in ensuring that connections between different structural parts remain intact and perform effectively under loads.

Think of a set of interconnected gears in a clock. All gears need to mesh together smoothly without any gaps. If one gear were to skip out of sync, the entire mechanism would fail to work correctly. Similarly, in structural design, every piece needs to adjust properly to maintain the structural integrity.

This can be achieved through two classes of solution: Force or Flexibility method; Displacement or Stiffness method.

To analyze statically indeterminate structures, engineers can use two main methods: the Force or Flexibility method and the Displacement or Stiffness method. The Flexibility method focuses on how forces applied at certain points affect displacements elsewhere in the structure, while the Stiffness method adopts the opposite approach, directly correlating how displacements influence the forces in the structure. Understanding these methods gives engineers options for tackling complex structural problems.

Imagine two people lifting a heavy object together. One person focuses on where to push (Force method), while the other concentrates on adjusting their grip and position according to the weight distribution (Displacement method). Both approaches can yield successful results in moving the object, much like the two methods offered for analyzing complex structures.

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