5.2: Two-Force Members

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A two-force member is a body that has forces (and only forces, no moments) acting on it in only two locations. In order to have a two-force member in static equilibrium, the net force at each location must be equal, opposite, and collinear. This will result in all two-force members being in either tension or compression, as shown in the diagram below.

Two identical horizontal rods. The one above has tension forces of equal magnitude acting on each end, each force vector pointing away from the midpoint of the rod. The one below has compression forces of equal magnitude acting on each end, each force vector pointing towards the midpoint of the rod. — Figure \(\PageIndex{1}\): The forces acting on two-force members need to be equal, opposite, and collinear for the body to be in equilibrium.

Why the Forces Must Be Equal, Opposite and Collinear:

Imagine a beam where forces are only exerted at each end of the beam (a two-force member). The body has some non-zero force acting at one end of the beam, which we can draw as a force vector. If this body is in equilibrium, then we know two things:

the sum of the forces must be equal to zero, and
the sum of the moments must be equal to zero.

In order to have the sum of the forces equal to zero, the force vector on the other side of the beam must be equal in magnitude and opposite in direction. This is the only way to ensure that the sum of the forces is equal to zero with only two forces.

In order to have the sum of the moments equal to zero, the forces must be collinear. If the forces were not collinear, then the two equal and opposite forces would form a couple. This couple would exert a moment on the beam when there are no other moments to counteract the couple. Because the moment exerted by the two forces must be equal to zero, the perpendicular distance between the forces \((d)\) must be equal to zero. The only way to achieve this is to have the forces be collinear.

Two identical horizontal rods. In the one above, two diagonal forces of equal magnitude but opposite direction act on the ends; the line of action for each force is extended, and the perpendicular distance d between these lines is shown. In the one below, two horizontal forces of equal magnitude and opposite direction are applied at the ends; the distance d between their lines of action is given as 0. The moment equation M = F*d is provided. — Figure \(\PageIndex{2}\): In order to have the sum of the moments be equal to zero, the forces acting on two-force members must always be collinear, acting along the line connecting the two points where forces are applied.

Why Two-Force Members Are Important:

By identifying two-force members, we greatly reduce the number of unknowns in our problem. In two-force members, we know that the forces must act along the line between the two connection points on the body. This means that the direction of the force vectors is known on either side of the body. Additionally, we know the forces are equal and opposite, so if we determine the magnitude and direction of the force acting on one side of the body, we automatically know the magnitude and direction of the force acting on the other side of the body.

A vertical rod experiences a pair of forces of equal magnitude and opposite direction on its ends, each force vector pointing away from the midpoint. A rectangular body experiences forces of equal magnitude and opposite direction, one pointing up and to the left and the other pointing down and to the right, on its upper left and lower right corners repsectively. An L-shaped beam experiences forces of equal magnitude and opposite direction, one pointing up and to the left and the other pointing down and to the right, on its top left and bottom right corner respectively with the two forces sharing a line of action. — Figure \(\PageIndex{3}\): The forces in two-force members will always act along the line connecting the two points where forces are applied.

Two-force members are also important in distinguishing between trusses, and frames and machines. When we analyze trusses using either the method of joints or the method of sections, we will assume everything is a two-force member. If this assumption is incorrect, this will cause serious problems in the analysis. By making this assumption, though, we can use some shortcuts that will make truss analysis easier and faster than the analysis of frames and machines.

Video lecture covering this section, delivered by Dr. Jacob Moore. YouTube source: https://youtu.be/m3GhMaZwHJw.