4.1: Statically Equivalent Systems

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Two sets of forces are considered statically equivalent if they cause the same set of reaction forces on a body. Because this is true, the two statically equivalent sets of forces are interchangeable in statics analysis.

On the left, an adult man weighing 200 lbs stands in the middle of a 6-foot-long beam that has one foot in contact with the ground at each end. A free body diagram shows the man's weight acting downwards on the beam, balanced by a 100-lb normal force from the ground on each end of the beam. On the right, two 100-lb children stand on the same beam, each child 2 feet from one end. The free body diagram shows the two 100-lb weights acting downwards on the beam, balanced by one 100-lb contact force on each end of the beam. — Figure \(\PageIndex{1}\): A single 200-lb man standing at the center of a beam causes the same reaction forces as two 100-lb children standing evenly along the beam as shown above. The weight force of the man is therefore statically equivalent to the weight forces of the two children.

Determining if Forces are Statically Equivalent:

To determine if two sets of forces are statically equivalent, you must solve for the reaction forces in both cases. If the reaction forces are the same then the two sets of forces must be statically equivalent. For any one set of forces, there are an infinite number of sets of forces that are statically equivalent to original set of forces.

Finding a Single Equivalent Point Force:

In statics analysis, we are usually looking to simplify a problem by turning multiple forces into a single, statically equivalent force. To find a single point force that is equivalent to multiple point forces you can use the following procedure.

Solve for the reaction forces in the original scenario.
Draw a new free body diagram with these reaction forces. You will also add one force with an unknown magnitude, direction, and point of application to your diagram. This is the single point load that will be equivalent to your original set of forces.
Write out the equations of equilibrium for this scenario, including the known values for the reaction forces.
First, solve the force equations to find the \(x\) and \(y\) components of this unknown force (or \(x\), \(y\) and \(z\) components for a 3D problem). This can be used to find the magnitude and direction of the statically equivalent point force.
Next, use the moment equation (or equations, for 3D problems) to determine the location of the statically equivalent point force.

Video \(\PageIndex{1}\): Lecture video covering this section, delivered by Dr. Jacob Moore. YouTube source: https://youtu.be/r0pwzQ_Ge00.

Example \(\PageIndex{1}\)

Determine if the two sets of forces shown below are statically equivalent.

Two identical horizontal bars (10 meters long, attached to a wall at the left end) experience different forces. One experiences a downwards force of magnitude 100 N, applied 2 meters from its free end. The other experiences one downwards force of magnitude 60 N, applied at the free end, and another downwards force of magnitude 40 N applied 4 meters away from the free end. — Figure \(\PageIndex{2}\): problem diagram for Example \(\PageIndex{1}\); two identical bars experience the same total magnitude and direction of external force, applied in different locations and distributions.

Solution: Video \(\PageIndex{2}\): worked solution to example problem \(\PageIndex{1}\), provided by Dr. Jacob Moore. YouTube source: https://youtu.be/Xv5DyvQD29E.