Coplanar Force Systems
 

Vectors and their properties

In mathematics, a vector is a directed quantity represented by an arrow.
A vector is therefore suitable for representing directed physical quantities, such as forces.
A vector has three properties: For geometric vectors, the magnitude is their length.
In two-dimensional space, the direction is the angle between the line along which the vector is aligned and the positive side of the x-axis.
The orientation determines on which side of the arrow the tip lies (endpoint) and where the other end is (footpoint).

In two-dimensional space, a vector can be represented by its two coordinates.
The two coordinates of a vector can be read from the coordinate system as the coordinates of the arrowhead if the footpoint is placed at the origin.

Vectors can be added graphically relatively easily.
The individual vectors are simply concatenated (in any order).
The sum vector is then the connecting vector from the beginning of the vector chain to its end.
In mechanics, this is called the sum vector of force vectors is called the resultant force, or simply the resultant.

Concurrent Force System

A concurrent force system exists when the forces under consideration act at a common point of application.
For rigid bodies, each force can be moved along its line of action without changing its effect on the rigid body.
Then a concurrent force system also exists if the lines of action of all the forces under consideration intersect at a single point.
The resultant of a concurrent force system can be generated by vectorial addition of the forces under consideration.
The line of action of the resultant passes through the common point of application of the group of forces.
The resultant generated in this way is then equivalent to the original force system. It can therefore be used instead of the force system.

Graphically, the resultant is the connecting vector from the beginning to the end of the force polygon of all forces.

Analytically, the resultant can be calculated from the coordinates of the forces involved:
FRx = F1x + F2x + ... + Fnx
FRy = F1y + F2y + ... + Fny

Non-Concurrent Force System

Even in a general planar force system, the resultant force can be generated by vectorial addition of the forces.
However, the line of action of this resultant force must still be determined by requiring that the moment of the force group on the one hand
and of the resultant force on the other hand be equal with respect to an arbitrary reference point (e.g., the origin).

The moment of a force with coordinates Fx and Fy and point of application A(x/y)
with respect to point B(xB/yB) is calculated as follows:
M(B) = Fy · (x-xB) - Fx · (y-yB).
The moment of a force with coordinates Fx and Fy and point of application A(x/y)
with respect to the origin is then calculated as:
M(0) = Fy · x - Fx · y.
Note that moments have a sign depending on the direction of action. They are negative if they act clockwise in the plane of the drawing.
The correct sign is obtained by using the coordinates of the force and the point of application in the formula, taking their signs into account.

The moments of all forces in a force group can then be added (taking their signs into account) to obtain the total moment.

If the magnitude of the resultant of a general force system is not equal to 0, the force system can always be replaced by a single force.
However, determining the position of the line of action is also necessary.

If the resultant has a magnitude of 0, there are two possibilities:
- either the force system corresponds to a couple (moment)
- or the force system is free of forces and moments (state of equilibrium).

Force diagram for a concurrent force system

Equilibrium in a concurrent force system exists precisely when the polygon of all force vectors closes.
The force diagram consists of the vector chain of all participating force vectors.
These are:
- given load vectors
- reaction forces such that a closed polygon results.

Using the force diagram, the reaction forces required for equilibrium can thus be determined.

In the case of a general coplanar force system, an additional requirement is needed to investigate equilibrium:
The sum of all moments must be zero.
See Statics of Rigid Body Systems