10.1. Introduction

A statically indeterminate structure has more unknowns than equations of equilibrium (and equations of conditions if applicable).

Statically indeterminate structures are those where there are more unknown forces than equations available to solve for them. In other words, you cannot figure out the forces just by using the basic rules of equilibrium (which state that the sum of forces and moments must equal zero). This situation typically arises in situations where the structure has more supports or members than strictly necessary.

Think of trying to find the weight of several bags on a scale that can only show you the combined weight of two bags at a time. If you have more bags (unknowns) than the scale (equations) can compare, you won't be able to figure out the exact weight of each bag without more information.

The advantages of statically indeterminate structures are:

1. Lower internal forces
2. Safety in redundancy, i.e., if a support or members fails, the structure can redistribute its internal forces to accommodate the changing B.C. without resulting in a sudden failure.

Statically indeterminate structures are beneficial because they tend to have lower internal forces, which means they can safely withstand loads without excessive stress. Additionally, these structures can handle the failure of a support or member more gracefully. Instead of collapsing suddenly, they can redistribute the forces throughout the remaining structure, providing a safety advantage.

Consider a team of workers on a project. If one person is unable to perform their task (like a support failing), the rest of the team can adjust and take on some of that person's work without the project falling apart. The redundancy means that there is backup available, much like the multiple paths for forces through a statically indeterminate structure.

Only disadvantage is that it is more complicated to analyse.

While statically indeterminate structures have many advantages, their analysis is more complex than that of statically determinate structures. This complexity arises because you need to consider the relationships between forces and displacements in more detail, often requiring advanced methods of analysis to find the internal forces and reactions.

Imagine trying to solve a multi-layer puzzle instead of a simple one-layer puzzle. The multi-layer version requires you to understand how the different layers interact with each other, making it more intricate and time-consuming to complete.

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).

To properly analyze statically indeterminate structures, you need to address three main criteria: First, the structure must be in equilibrium, meaning all forces and moments must balance. Second, you must account for the material properties and how they relate force to displacement, which often involves linear elastic behavior. Lastly, the displacements within the structure must be compatible, ensuring that there are no sudden changes that disrupt its continuity.

Think of a bridge. For it to be safe and functional, all the sections must be stable (equilibrium), the materials must bend or compress predictably under load (force-displacement relations), and all parts must fit together without gaps or overlaps (compatibility of displacements).

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

When dealing with statically indeterminate structures, engineers use two primary methods to find solutions: the Flexibility Method, which focuses on how forces distribute and the reactions develop due to those forces, and the Stiffness Method, which emphasizes the displacements in the structure and how they relate to internal forces. Both methods have their own applications depending on the problem at hand.

Consider two approaches to climbing a mountain. The first approach (Flexibility Method) involves figuring out the best route based on where you can put your weight safely, while the second approach (Stiffness Method) is based on how steep the climb is and planning your climbing techniques accordingly. Both methods help you reach the summit, but they do so through different strategies.

--