Statics of Non-Concurrent Force Systems

In this section, we will examine various aspects of non-concurrent force systems, which principally apply to statics of rigid bodies. Besides the non-currency of forces, the main difference with the previous section is the presence of moments. For a general introduction to moments, refer to the section entitled Moment of a Force.

Loads: The word load implies force or moment. Hence, when we talk about the loads acting on an object, we are generalizing the action to mean forces or moments or the combination of the two.

Supporting a Rigid Body:

A rigid body is usually supported at one or more locations. These supports help keep the body in a stationary position. At a support location, the body exerts a load (action) on to the support and the support exerts an equal and opposite load (reaction) on the body.

In order to identify the type of reaction exerted by the support, we consider the movement that the support helps to restrain. For example, if a support prevents the translation of the body along a particular direction, then there is a reaction force in the same direction at the point of support. Similarly, if the support prevents the rotation of the body, then the reaction load is in the form of a moment. As described below, it is also possible for a support to restrain the body in more than one direction.

As a general rule, we must remeber the following fact: Movement Prevented = Reaction Created

In the following discussion, a rigid body is represented by a trapezoid. The coordinate system shown helps to identify the directions of designated reaction forces.

Support Reactions in Two-Dimensional Force Systems: Some examples of typical supports and the reactions they exert on the body are shown below.

Roller: Body is free to move in the direction parallel to the support line, and is free to rotate, but it is restrained against translation in the perpendicular or normal direction. Hence, the reaction force has a known direction and an unknown magnitude, Rn.

A roller support may also be represented in alternative forms as shown below.

Pin or Hinge: Body is free to rotate about the hinge point, but is restrained against translation in any arbitrary direction. Therefore, the direction of the reaction force is not known. As a result, it is represented in terms of its two rectangular components. Hence, the reaction force is represented by two rectangular components of assumed direction and unknown magnitude, Rx and Ry.

Weightless Link: Sometimes a body is connected to a rigid support via a weightless link. Since the force in the link is always along its axis, the direction of the reaction force is known, but its magnitude is not. Hence, the reaction force has a known direction and an unknown magnitude, R.

Fixed or Clamped: Body is restrained against translation in all directions as well as rotation about the support point. As a result, there is a reaction force (represented in terms of its rectangular components of assumed direction and unknown magnitude Rx and Ry), and a reaction moment of assumed direction and unknown magnitude, M.

Notice that the last two representations of reaction forces are equivalent because the resultant reaction force can be expressed in terms of either Rx and Ry or Rx' and Ry' components.

Free-Body Diagram of a Rigid Body: We rely on free-body diagrams to help identify the loads acting on a body prior to performing the equilibrium analysis. A very important factor in drawing a free-body diagram is the understanding of the support reactions as explained next. When drawing a free-body diagram, we must pay close attention to the following points:

1. The free-body diagram must not include forces that are internal to the body also known as internal forces. Consider the frame structure shown below. It consists of beam ABC and link BD. Two different free-body diagrams are shown for this system. In FBD #1, the beam is separated from the support at A and the link at B. In FBD #2, the beam and link are kept attached but separated from the supports at A and D. See the difference in the two free-body diagrams.

As can be seen in FBD #1 the force RB exerted on the beam is equal and opposite to that exerted on the link. Therefore, when the two parts are joined together at point B, these two forces cancel out. Thus, RB is not shown in FBD #2. So as far as FBD #2 is concerned, RB is an internal force.

2. Whenever specified, the weight of the body should be included in the equilibrium equations. The weight is typically represented by a concentrated force at the center of gravity of the body, and is always acting vertically down toward the center of earth. For a homogeneous body, the center of gravity (C.G.) coincides with its geometric center or centroid.

3. When a rigid body is under a distributed force, it is permissible to represent the distributed force by its resultant acting at the centroid of the distribution in the same direction as the distributed force.

4. Search for and identify all two-force and three-force members present in the structural system.

Two-Force Members: When a structural member is subject to forces at only two points, and there are no other loads acting on it, the member is said to be a two-force member. A two-force member is in the state of equilibrium only when the two forces acting on it are equal in magnitude, collinear, and opposite in direction as depicted in the figure.

Three-Force Members: When a structural member is subject to forces at three points, and there are no other loads acting on it, the member is said to be a three-force member. A three-force member is in the state of equilibrium when the three forces are either parallel to each other or form a concurrent force system as shown in the figure. Notice that in both cases the forces point in different directions. Otherwise, the equilibrium of forces and moments would be violated.

Example 1

Example 2

Equilibrium of a Rigid Body: A body is in the state of equilibrium if it is at rest or is moving at constant velocity. As there is no translational or rotational acceleration, the condition of equilibrium for a general three-dimensional system can be expressed as

 

(1)
(2)

Equation (1) simply states that the summation of all the forces acting on the body must be zero. Similarly, Eq. (2) shows that the summation of all the moments must be zero.

2-Dimensional Loading System: For a body under a two-dimensional loading system (e.g., in x y plane), the equations of equilibrium can be expressed as

(3)

It is also possible to use alternative sets to Eqs. (3) given as

(4)

When using the first set of equilibrium equations in (4), points A and B must be selected such that a line passing through them is not perpendicular to the x axis. Likewise, when using the second set, the line passing through points A and B must not be perpendicular to the y axis. For the last set, points A, B and C are three separate moment centers that do NOT fall on a straight line. Irrespective of which set we use, the results will be identical. Although in the last set only the equations of moment equilibrium are used, the equilibrium of forces will be met implicitly.

Example 3

Example 4

Example 5

3-Dimensional Loading System: For a body under a three-dimensional loading system, the equilibrium equations i.e., Eqs. (1) and (2) in their scalar forms are given as

 

(5)
(6)

The concept of equilibrium is without a doubt the most fundamental concept in statics. It enables us to determine the unknown forces and moments acting at joints, supports, surfaces of contact, etc. The steps in development and implementation of the equilibrium equations are described in the following examples.