Graphic Methods of Coplanar Force Resolution

The

The illustration shows two vectors and their resultant. The resultant
force is shown as the dashed vector. In order to resolve these forces graphically,
one must first extend the lines of action of two concurrent forces until
they intersect. This intersection is known as the **point of origin**
for the system. Both forces, as well as the resultant, must ALL act either
away from or toward the point of origin.

Animation illustrating how to resolve vectors with the Parallelogram method

The resultant can be represented graphically by the diagonal of the parallelogram
formed by using the two force vectors to determine the length of the sides
of the parallelogram. The magnitude of the resultant can be accurately measured
as the scaled length of the diagonal. __The resultant MUST go through
the point of intersection of its components!!!__

(**Remember**: graphical solutions depend upon the accuracy of the
drawing. The length of each vector should be carefully scaled to equal the
magnitude of the force).

More than two non-parallel forces can be combined by successively eliminating one of the forces. Combine any two of the forces into their resultant by the parallelogram method. Combine this resultant with any of the remaining forces (or with the resultant of any of the remaining forces) until all of the forces are included. One must remember that the vectors can only be translated (or moved) along their lines of action. Two vectors (or Forces) cannot be combined (or resolved) until both of them are meeting head-to-head or tail-to-tail!

The resolution of the this system is a single vector that has a magnitude of approximately 4k with a direction of up and to the left. Try it yourself!!!

The **Triangle of Forces Method** is another graphical method developed
to find the resultant of a coplanar force system. Since the opposite sides
of a parallelogram are equal, a force triangle may also be found instead
of using the parallelogram method. This method is quite useful because it
can be successivly applied to any number of concurrent forces.

To calculate the resultant of the force system shown above, move force A
so that it's tail meets the head of force B. Now forces A and B form a "Head-to-Tail"
arrangement. The resultant R is found by starting at the tail of B (the
point of intersection of forces A and B) and drawing a vector which terminates
at the head of the transposed A. Note that if force B had been transposed
instead of force A, the resultant would have started from the tail of A
and terminated at the head of force B. Again, this process could be repeated
for any number of force vectors.

The resultant is described by the vector's magnitude and direction. These are determined by scaling the length and angle respectively. The accuracy of these values depends upon the accuracy of the graphics.

More than two non-parallel, non-concurrent forces can be combined by
successively eliminating one of the forces. Combine any two of the forces
into their resultant by the triangle method, and then __extend that resultant
until it intersects the line of action of another force__. One continues
this process until all forces have been included. In this way, each one
of the forces is successivly combined with the resultant of the previous
triangle. One cannot simply continue to add the vectors head-to-head or
tail-to-tail because the resulting lines of action would then be incorrect!

Animation illustrating how to resolve vectors with the Triangle method

This illustration of a fixed jib crane allows one to read the forces
as they meet at the tip of the boom. The diagram indicates the forces acting
in the crane: red for tension and blue for compression.

The graphical methods of force decomposition could be used to determine
the magnitude of the forces within the crane. In this case the two components
for each of the structural elements are shown. All of the vectors are representational.
They are not drawn to scale. The actual magnitudes are simply determined
IF one would actually draw ALL of the vectors to scale and then measure
the results.

A
Thrust on a Wall

A
Comparison of the two methods

Could one determine components that are not related to the x and y axis for the crane? why might one need to do this?

For those who are interested in reading more about graphic statics:

- Digweed, E.N. Graphical Statics. The Association of Engineering and Shipbuilding Draughtsmen; Session 1929-30. The Draughtsman Publishing Co. (London) 1930.
- International Library of Technology. Graphcial Analysis of Stresses. International Texbook Company (London) 1905.

Copyright © 1995 by Chris H. Luebkeman and Donald Peting

Copyright © 1996, 1997, 1998 by Chris H. Luebkeman