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The Elements of Coplanar Force Resolution

There are many ways in which forces can be manipulated. It is often easier to work with a large, complicated system of forces by reducing it to an ever decreasing number of smaller problems. This is called the "resolution" of forces or force systems. This is one way to simplify what may otherwise seem to be an impossible system of forces acting on a body. Certain systems of forces are easier to resolve than others. Coplanar force systems have all the forces acting in in one plane. They may be concurrent, parallel, non-concurrent or non-parallel. All of these systems can be resolved by using graphic statics or algebra.

concurrent force system with all arrows meeting at one point
A concurrent coplanar force system is a system of two or more forces whose lines of action ALL intersect at a common point. However, all of the individual vectors might not acutally be in contact with the common point. These are the most simple force systems to resolve with any one of many graphical or algebraic options.
force system with all of the arrows drawn parallel
A parallel coplanar force system consists of two or more forces whose lines of action are ALL parallel. This is commonly the situation when simple beams are analyzed under gravity loads. These can be solved graphically, but are combined most easily using algebraic methods.
a force system in which none of the arrows meet at one single point and they are not parallel.  Essentially a jumble of forces.
The last illustration is of a non-concurrent and non-parallel system. This consists of a number of vectors that do not meet at a single point and none of them are parallel. These systems are essentially a jumble of forces and take considerable care to resolve.

two forces meet at the origin and point out.  A vector is drawn between them that is the resultantAlmost any system of known forces can be resolved into a single force called a resultant force or simply a Resultant. The resultant is a representative force which has the same effect on the body as the group of forces it replaces. (A couple is an exception to this) It, as one single force, can represent any number of forces and is very useful when resolving multiple groups of forces. One can progressively resolve pairs or small groups of forces into resultants. Then another resultant of the resultants can be found and so on until all of the forces have been combined into one force. This is one way to save time with the tedious "bookkeeping" involved with a large number of individual forces. Resultants can be determined both graphically and algebraically. The graphical methods that will be discussed in Lecture 8 are the Parallelogram Method and the Triangle Method. It is important to note that for any given system of forces, there is only one resultant.

three forces that have been broken down into components.  Each one has a seperate, and unique, set of two components that lie to each side of the force.
It is often convenient to decompose a single force into two distinct forces. These forces, when acting together, have the same external effect on a body as the original force. They are known as components. Finding the components of a force can be viewed as the converse of finding a resultant. There are an infinate number of components to any single force. And, the correct choice of the pair to represent a force depends upon the most convenient geometry. For simplicity, the most convenient is often the coordinate axis of a structure.

a force showing its decomposition into two forces that are orthagonally oriented to each other This diagram illustrates a pair of components that correspond with the X and Y axis. These are known as the rectangular components of a force. Rectangular components can be thought of as the two sides of a right angle which are at ninety degrees to each other. The resultant of these components is the hypotenuse of the triangle. The rectangular components for any force can be found with trigonometrical relationships: Fx = F cos theta, Fy = F sin theta (where theta is the angle). There are a few geometric relationships that seem common in general building practice in North America. These relationships relate to roof pitches, stair pitches, and common slopes or relationships between truss members. Some of these are triangles with sides of ratios of 3-4-5, 1-2-sqrt3, 1-1-sqrt2, 5-12-13 or 8-15-17. Commiting the first three to memory will simplify the determination of vector magnitudes when resolving more difficult problems.

a force showing its decomposition into two forces that are orthagonally oriented to each other

When forces are being represented as vectors, it is important to should show a clear distinction between a resultant and its components. The resultant could be shown with color or as a dashed line and the components as solid lines, or vice versa. NEVER represent the resultant in the same graphic way as its components.

three forces that have been broken down into components.  Each one has a seperate, and unique, set of two components that lie to each side of the force.
Any concurrent set of forces, not in equilibrium, can be put into a state of equilibrium by a single force. This force is called the Equilibrant. It is equal in magnitude, opposite in sense and co-linear with the resultant. When this force is added to the force system, the sum of all of the forces is equal to zero. A non-concurrent or a parallel force system can actually be in equilibrium with respect to all of the forces, but not be in equilibrium with respect to moments.

Graphic Statics and graphical methods of force resolution were developed before the turn of the century by Karl Culmann. They were the only methods of structural analysis for many years. These methods can help to develop an intuitive understanding of the action of the forces. Today, the Algebraic Method is considered to be more applicable to structural design. Despite this, graphical methods are a very easy way to get a quick answer for a structural design problem and can aid in the determination of structural form.

The following is an example of how an understanding of structural action and force resolution can aid in the interpretation of structural form. The illustration on the left is the Saltash Bridge, design and built by the Engineer Brunel in 1859. Notice the form of the structural elements. The illustration on the right is a retouched imaged to bring the bridge to a state which is akin to a pure suspension bridge. However, there are some issues with this design that need to be clearly understood.

Saltash Bridge

Since this is a railroad bridge, trains of various weights will rumble across. The weight of the train is carried by the horizontal deck which is within the space defined by the steel girders. These girders are supported by hangers which extend from the chain to the girder. Each one of these hangers is loaded in pure vertical tension. The suspension chain is loaded with a vertical load as the deck is loaded with a train. This action would tend to cause the chain to pull the towers towards the middle of the river. This can be clearly seen if one resolves the tension force into its two orthagonal components. The system is not in equilibrium without some force to resist the lateral component of the resolution.

Saltash Bridge

The system is usually put back into equilibrium by the additional tension force of a chain, or cable, continuing over the supporting pylon to an anchor on the bank of the river. This was not possible here. Thus, instead of a tensile force, Brunel resisted the lateral component with a compressive thrust which was created by the massive arch. In the illustration below this is seen as the large black arrow. The chain link suspension system works in harmony with the arch....
Saltash Bridge

An examination of the resolution of the forces at the support MUST reveal that the resultant passes through the stone pillar.....which it does.

Saltash Bridge

Components of Forces and Equilibrium


Questions for Thought

What would be other options for the Saltash bridge now that over a century has passed since it's desgin? What could/would be the most likely location for a strength or stability failure of the bridge? What might the design implication be if the chain was replaced by a wire rope?


Homework Problems


Additional Reading

Shaeffer, R. E. Elementary Structures for Architects and Builders. pp. 21-33.
Allen, Ed. Graphic Statics.

Copyright © 1995 by Chris H. Luebkeman and Donald Peting
Copyright © 1996, 1997, 1998 by Chris H. Luebkeman