SS - Determinacy

In statics, when the static equilibrium equations are sufficient to determine the internal forces and reactions of the structure, that structure is said to be statically determinate.

In solving for determinacy, three summations of forces (which are obtained by Newton's laws of motion) are used to determine whether a structure is determinate. If for three equations, three variables can be solved, then that structure is called statically determinate.

The three equations for equilibrium state the following:

Σ N = 0: the sum of the horizontal components of the forces equals zero;

Σ V = 0: the sum of the vertical components of forces equals zero;

Σ M = 0: the sum of the horizontal components of the forces equals zero;

Where N is considered the normal force, V the shear force and M the moment force. The units for N and V are either pounds (lb), kips (thousands of pounds) (k) or Newtons (N) / kiloNewtons (kN).

The Free Body Diagram

In beams, columns or other composite structural members, it is best to break the key components of bar forces into separate reactions acting against or along those members. This is called a free body diagram, and it is the fundamental building block of all statics in determining force vectors.

The classical scenario typically used for physics is the block on a slope. The block has some weight, w, which acts downward toward the surface of the earth. Against the slope, however, a partial fraction of that weight, called the normal force (-N) acts perpendicularly against the slope. An equal, but opposite reaction (N) is the result of the sum of vertical forces equaling zero.

In structural beams, N is a horizontal force, acting linearly throughout the member. A change of connotation in how one determines force is all structural analysis suggests. Instead of negative forces acting downward and positive forces acting upward, structural analysis is more interested in what members are in tension, and which are in compression. If a member is in tension, that force is outward and its value is positive. Conversely, if it is in compression, the force is inward and its value is negative.

Or put it more simply, when your muscles are in tension, you build bigger biceps, and that typically is a positive outcome. Structurally, you're better with bigger, stronger muscles. Alternatively, when your muscles are compressed, there is a force acting against you, a negative force (which may include a bench press, dumbbells or some other voluntary load). You must prove elastic against that force in order to withstand it (which is the definition of compression).

A standard free body diagram may look something like this:

Where there may be a point load (or concentrated load), distributed load (in kN/m), reaction forces (at A in this diagram), distances for the loading forces and reaction forces, and subsequent shear vectors (V) and moment vectors (M) at cut cross-sections in the member. As a result of where the cut is taken, V and M with vary either linearly or parabolically. Or not at all. It all depends on the degree of loading (whether it is one point, continuously linear or curvilinear). The way it affects V and M, however, is another blog, however, and will not be included here.

All in all, though, there are two main considerations when analyzing a structure. One is stability, which is also another post. And the other one (which tells you something about stability) is determinacy. Without determinacy, it is very hard to predict what happens for a structure. Therefore, it is the preeminent consideration for all of structural analysis.

Now take a look at the top diagram. Why are the members and trusses either determinate or indeterminate (not solvable by equilibrium equations)? Recall that roller reactions have only one force they exhibit (vertical to loading), pins have two (vertical and horizontal) and fixed connections have three (vertical, horizontal and rotational to loading). Consider this a bit of a quiz. But feel free to take your time and look closely.