What is Equilibrium? To answer that question, let's turn our heads to Mechanics, the study of forces and energy among objects. An object is said to be in a state of Equilibrium if and only if all forces and moments acting on it must balance each other.
To illustrate, try balancing a pencil on the tip of your finger. Usually, it would be best to find its center of gravity first, then place your finger on that point. Your finger will act as support on said point, and the pencil stays in position; that's Equilibrium in action.
The pencil's weight and your finger's supporting force balance each other. Because of this, the pencil remains stationary and in a state of rest. If you let it go from your hand, it will fall due to its weight; When that happens, there is no force balancing the pencil's weight, and the object will NOT be in a state of Equilibrium. Eventually, when it hits the ground, it will support the pencil and be in Equilibrium again.
Equilibrium, in structural analysis, is a crucial concept if we want to analyze our structures accurately.
Types
There are two types of Equilibrium:
- Static Equilibrium deals with the Equilibrium of an object that is at rest. Meaning it doesn't move relative to another thing.
- Dynamic Equilibrium deals with the Equilibrium of an object that is moving. In this situation, the object moves at a constant velocity (zero acceleration).
Generally, we idealize structures as objects that must be at rest most of the time; hence, most analysis deals with static Equilibrium; however, there are instances when we have to consider earthquakes for buildings or wind forces for tall structures. In these situations, the frame may move considerably, and we may extend the analysis to dynamic Equilibrium.
What is Balance?
We say that an object is in balance if it satisfies the two conditions of Equilibrium regardless of whether it is static or dynamic:
- The summation of forces acting on the body must be zero: \({\sum}F=0\)
- The summation of moments acting on the body must be zero: \({\sum}M=0\)
The external forces and moments acting on any structure must balance out with the supports. In addition, when we isolate parts of the structure, the internal forces should balance with the applied external forces.
From these two conditions, we can expound it to its components depending if the structure is 2D or 3D.
Plane Equilibrium
Consider a standard 2D-Cartesian coordinate plane. For a structure to be in Equilibrium:
- The summation of all forces in the x and y-directions is zero,
- The summation of moments about the z-axis is zero;
From this, there are three Equilibrium equations.
- \({\sum}F_x=0\)
- \({\sum}F_y=0\)
- \({\sum}M_z=0\); moment about the z-axis (perpendicular to the 2D-plane)
Space Equilibrium
Consider a standard 3D-Cartesian coordinate system. For a structure to be in Equilibrium:
- The summation of forces along the x, y, and z-directions is zero,
- The summation of moments about the x, y, and z-axes is zero;
From this, there are six equations of Equilibrium.
- \({\sum}F_x=0\)
- \({\sum}F_y=0\)
- \({\sum}F_z=0\)
- \({\sum}M_x=0\)
- \({\sum}M_y=0\)
- \({\sum}M_z=0\)
Summary
Equilibrium describes the object's state and deals with balance.
There are two types of Equilibrium: static and dynamic. The former deals Equilibrium of an object at rest, while the latter deals Equilibrium of a moving object.
We say that an object is in balance if it satisfies the two conditions of Equilibrium regardless of whether it is static or dynamic: \({\sum}F=0\) and \({\sum}M=0\)
For any structure, the external forces and moments acting on the structure must balance out with the supports. In addition, when we isolate parts of the structure, the internal forces should balance with the applied external forces.
For plane equilibrium, there are at most three equilibrium equations. For space equilibrium, there are, at most, six equilibrium equations.
