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Static equilibrium equations

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Equilibrium equations are used to describe the state of a body that is in equilibrium. Such a body does not change its position, i.e. it is at rest or not accelerates.

The sum of all forces and moments acting on a body at rest must balance, we describe it mathematically as follows:


Plane system (2D) - equilibrium equations


The above equations apply in a planar structure. We use them to determine reactions for beams, frames and trusses. These types of elements are fundamental issues in statics. In such tasks, we want the elements to remain at rest. If bodies move, we are talking about other branches of mechanics such as kinematics or dynamics.



3 degrees of freedom of the body in flat movement, on a plane
Fig.1. Three degrees of freedom of the body in planar movement.

In a plane system, the body can move in the x and y directions and rotate around the z axis directed towards us. In a flat system, the body has 3 degrees of freedom. And for the body to remain at rest, we must balance these three degrees of freedom. And that's why we use equilibrium equations. If the balance of x and y forces and the sum of moments are equal to zero, it means that the body does not move or rotate.


A specific type of plane system is the plane system of convergent forces (also called the middle system). It is a system in which the forces intersect at one point. They converge on this very point. For such a system, the equilibrium condition is only two equations:

Spatial system (3D) - equilibrium equations


We have a little more degrees of freedom in the spatial arrangement. There are as many as six of them. For each degree of freedom, we have an equation describing the equilibrium equation. So we get six equations. In tasks where we determine support reactions in spatial systems, finding a solution is more difficult because we have as many as six equations.


Remember how many equations there are as many unknowns as possible. 6 equations = maximum 6 unknowns.

This means that a body in space can move in three directions and rotate around three axes.


Degrees of freedom - spatial system - solverEdu
Fig.2. Degrees of freedom - spatial arrangement

That's all from me on the topic of equilibrium equations in statics.


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