Structural Engineering is mainly concerned with static structures i.e. structures that do not move. In order to fulfill this condition, all loads on the structural system must be in equilibrium.
What is Equilibrium?
In the context of mechanics, equilibrium means that all the forces are in balance. In other words, the vector sum of all forces are equal to zero.
We can assess the equilibrium of a system using the three equilibrium equations shown below. V, H and M represent vertical forces, horizontal forces, and moments, respectively.
Vertical Equilibrium
For vertical forces, we can simple state that:
(The sum of all vertical forces is equal to zero)
Let's consider the simple example below.
In this case, we can write the equilibrium equation as follows:
Va + Vb - P = 0 ➞ Va + Vb = P
Since the load is at midspan, by inspection we know that the reactions at Va and Vb must be:
Va = Vb = P/2
Horizontal Equilibrium
For horizontal forces, we can simply state that:
Let's use the following simple example below:
Note that the support at B does not have any horizontal loads since it is a roller support. We can write the equilibrium equation as follows:
P2 - Ha = 0 ➞ Ha = P2
In other words, the reaction at A resists the full horizontal load P2.
Moment Equilibrium
For Bending Moments, we can state that:
For this, let's consider the slightly more complicated example shown below, with some numbers:
The members AB and CD are assumed to be rigidly connected at point C, and of the same stiffness. The Bending Moment at C due to the point load P is: 5 kN x 2 m = 10 kN-m. This moment is balanced at C by members AC and BC:
Σ M = 0;
-10 + 5 + 5 = 0
If the member BC is removed, then the full moment (-10 kN-m) will have to be resisted by member AC to maintain moment equilibrium.
Key Points
- Remember that these equilibrium equations hold true for ANY point in the structural system, including at joint locations.
- It is important to follow a consistent sign convention. Upwards forces and anti-clockwise moments are usually considered as positive.