Constraint and Reaction Forces and Moments (Part - 1) - Civil Engineering (CE) PDF Download

Machines and structures are made up of large numbers of separate components. For example, a building consists of a steel frame that is responsible for carrying most of the weight of the building and its contents. The frame is made up of many separate beams and girders, connected to one another in some way. Similarly, an automobile’s engine and transmission system contain hundreds of parts, all designed to transmit forces exerted on the engine’s cylinder heads to the ground.

To analyze systems like this, we need to know how to think about the forces exerted by one part of a machine or structure on another.

We do this by developing a set of rules that specify the forces associated with various types of joints and connections.

Forces associated with joints and connections are unlike the forces described in the preceding section. For all our preceding examples, (e.g. gravity, lift and drag forces, and so on) we always knew everything about the forces – magnitude, direction, and where the force acts.

In contrast, the rules for forces and moments acting at joints and contacts don’t specify the forces completely. Usually (but not always), they will specify where the forces act; and they will specify that the forces and moments can only act along certain directions. The magnitude of the force is always unknown.

2.4.1 Constraint forces: overview of general nature of constraint forces

The general nature of a contact force is nicely illustrated by a familiar example – a person, standing on a floor (a Sumo wrestler was selected as a model, since they are particularly interested in making sure they remain in contact with a floor!). You know the floor exerts a force on you (and you must exert an equal force on the floor). If the floor is slippery, you know that the force on you acts perpendicular to the floor, but you can’t make any measurements on the properties of the floor or your feet to determine what the force will be.

In fact, the floor will always exert on your feet whatever force is necessary to stop them sinking through the floor. (This is generally considered to be a good thing, although there are occasions when it would be helpful to be able to break this law).

We can of course deduce the magnitude of the force, by noting that since you don’t sink through the floor, you are in equilibrium (according to Newton’s definition anyway – you may be far from equilibrium mentally). Let’s say you weight 300lb (if you don’t, a visit to Dunkin Donuts will help you reach this weight). Since the only forces acting on you are gravity and the contact force, the resultant of the contact force must be equal and opposite to the force of gravity to ensure that the forces on you sum to zero. The magnitude of the total contact force is therefore 300lb. In addition, the resultant of the contact force must act along a line passing through your center of gravity, to ensure that the moments on you sum to zero.

From this specific example, we can draw the following general rules regarding contact and joint forces

(1) All contacts and joints impose constraints on the relative motion of the touching or connected components – that is to say, they allow only certain types of relative motion at the joint. (e.g. the floor imposes the constraint that your feet don’t sink into it)
(2) Equal and opposite forces and moments act on the two connected or contacting objects. This means that for all intents and purposes, a constraint force acts in more than one direction at the same time. This is perhaps the most confusing feature of constraint forces.
(3) The direction of the forces and moments acting on the connected objects must be consistent with the allowable relative motion at the joint (detailed explanation below)
(4) The magnitude of the forces acting at a joint or contact is always unknown. It can sometimes be calculated by considering equilibrium (or for dynamic problems, the motion) of the two contacting parts (detailed explanation later).

Because forces acting at joints impose constraints on motion, they are often called constraint forces. They are also called reaction forces, because the joints react to impose restrictions on the relative motion of the two contacting parts.

2.4.2 How to determine directions of reaction forces and moments at a joint

Let’s explore the meaning of statement (3) above in more detail, with some specific examples.

In our discussion of your interaction with a slippery floor, we stated that the force exerted on you by the floor had to be perpendicular to the floor. How do we know this?

Because, according to (3) above, forces at the contact have to be consistent with the nature of relative motion at the contact or joint. If you stand on a slippery floor, we know

(1) You can slide freely in any direction parallel to the floor. That means there can’t be a force acting parallel to the floor.
(2) If someone were to grab hold of your head and try to spin you around, you’d rotate freely; if someone were to try to tip you over, you’d topple. Consequently, there can’t be any moment acting on you.
(3) You are prevented from sinking vertically into the floor. A force must act to prevent this.
(4) You can remove your feet from the floor without any resistance. Consequently, the floor can only exert a repulsive force on you, it can’t attract you.

You can use similar arguments to deduce the forces associated with any kind of joint. Each time you meet a new kind of joint, you should ask

(1) Does the connection allow the two connected solids move relative to each other? If so, what is the direction of motion? There can be no component of reaction force along the direction of relative motion.
(2) Does the connection allow the two connected solids rotate relative to each other? If so, what is the axis of relative rotation? There can be no component of reaction moment parallel to the axis of relative rotation.
(3) For certain types of joint, a more appropriate question may be ‘Is it really healthy/legal for me to smoke this?’

2.4.3 Drawing free body diagrams with constraint forces

When we solve problems with constraints, we are nearly always interested in analyzing forces in a structure containing many parts, or the motion of a machine with a number of separate moving components. Solving this kind of problem is not difficult – but it is very complicated because of the large number of forces involved and the large number of equations that must be solved to determine them. To avoid making mistakes, it is critical to use a systematic, and logical, procedure for drawing free body diagrams and labeling forces.

The procedure is best illustrated by means of some simple Mickey Mouse examples. When drawing free body diagrams yourself, you will find it helpful to consult Section 4.3.4 for the nature of reaction forces associated with various constraints.

2D Mickey-mouse problem 1. The figure shows Mickey Mouse standing on a beam supported by a pin joint at one end and a slider joint at the other.

We consider Mickey and the floor together as the system of interest. We draw a picture of the system, isolated from its surroundings (disconnect all the joints, remove contacts, etc). In the picture, all the joints and connections are replaced by forces, following the rules outlined in the preceding section.

Notice how we’ve introduced variables to denote the unknown force components. It is sensible to use a convention that allows you to quickly identify both the position and direction associated with each variable. It is a good idea to use double subscripts – the first subscript shows where the force acts, the second shows its direction. Forces are always taken to be positive if they act along the positive x, y and z directions.

We’ve used the fact that A is a pin joint, and therefore exerts both vertical and horizontal forces; while B is a roller joint, and exerts only a vertical force. Note that we always, always draw all admissible forces on the FBD, even if we suspect that some components may turn out later to be zero. For example, it’s fairly clear that R_Ax = 0 in this example, but it would be incorrect to leave off this force. This is especially important in dynamics problems where your intuition regarding forces is very often incorrect.

2D Mickey Mouse problem 2 Mickey mouse of weight WM stands on a balcony of weight W_B as shown. The weight of strut CB may be neglected.

This time we need to deal with a structure that has two parts connected by a joint (the strut BC is connected to the floor AB through a pin joint). In cases like this you have a choice of (a) treating the two parts together as a single system; or (b) considering the strut and floor as two separate systems. As an exercise, we will draw free body diagrams for both here.

A free body diagram for the balcony and strut together is shown on the right. Note again the convention used to denote the reactions: the first label denotes the location of the force, the second denotes the direction. Both A and C are pin joints, and therefore exert both horizontal and vertical forces.

The picture shows free body diagrams for both components. Note the convention we’ve introduced to deal with the reaction force acting at B – it’s important to use a systematic way to deal with forces exerted by one component in a system on another, or you can get hopelessly confused. The recommended procedure is

1. Label the components with numbers – here the balcony is (1) and the strut is (2)
2. Denote reaction forces acting between components with the following convention. In the symbol   the superscript (1/2) denotes that the variable signifies the force exerted by component (1) on component (2) (it’s easy to remember that (1/2) is 1 on 2). The subscript Bx denotes that the force acts at B, and it acts in the positive x direction

3. The forces   exerted by component (1) on component (2) are drawn in the positive x and y directions on the free body diagram for component (2).

4. The forces exerted by component (2) on component (1) are equal and opposite to . They are therefore drawn in the negative x and y directions on the free body diagram for component (1). You need to think of the reaction force components as acting in two directions at the same time. This is confusing, but that’s the way life is.

2.4.4 Reaction Forces and Moments associated with various types of joint

Clamped, or welded joints

No relative motion or rotation is possible.

Reaction forces: No relative motion is possible in any direction. Three components of reaction force must be present to prevent motion in all three directions.

Reaction moments: No relative rotation is possible about any axis. Three components of moment must be present to prevent relative rotation.

The figure shows reaction forces acting on the two connected components. The forces and moments are labeled according to the conventions described in the preceding section.

2D versions of the clamped joint are shown below

Pinned joint.

A pinned joint is like a door hinge, or the joint of your elbow. It allows rotation about one axis, but prevents all other relative motion.

Reaction forces: No relative motion is possible at the joint. There must be 3 components of reaction force acting to prevent motion.

Reaction moments: Relative rotation is possible about one axis (perpendicular to the hinge) but is prevented about axes perpendicular to the hinge. There must be two components of moment acting at the joint.

2D pinned joints are often represented as shown in the picture below

Roller and journal bearings

Bearings are used to support rotating shafts. You can buy many different kinds of bearing, which constrain the shaft in different ways. We’ll look at a couple of different ones.

Example 1: The bearing shown below is like a pin joint: it allows rotation about one axis, but prevent rotation about the other two, and prevents all relative displacement of the shaft.

Reaction forces: No relative motion is possible at this kind of bearing. There must be 3 components of reaction force.

Reaction moments: Relative rotation is allowed about one axis (parallel to the shaft), but prevented about the other two. There must be two components of reaction moment.

Example2: Some types of bearing allow the shaft both to rotate, and to slide through the bearing as shown below

Reaction forces: No relative motion is possible transverse to the shaft, but the shaft can slide freely through the bearing. There must be 2 components of reaction force.

Reaction moments: Relative rotation is allowed about one axis (parallel to the shaft), but prevented about the other two. There must be two components of reaction moment.

Roller bearings don’t often appear in 2D problems. When they do, they look just like a pinned joint.

Swivel joint: Like a pinned joint, but allows rotation about two axes. There must be 3 components of reaction force, and 1 component of reaction moment.

Reaction forces: All relative motion is prevented by the joint. There must be three components of reaction force.

Reaction moments: rotation is permitted about two axes, but prevented about the third. There must be one component of reaction force present.

Swivel joints don’t often appear in 2D problems. When they do, they look just like a pinned joint.

Ball and socket joint Your hip joint is a good example of a ball and socket joint. The joint prevents motion, but allows your thigh to rotate freely relative to the rest of your body.

Reaction forces: Prevents any relative motion. There must be three components of reaction force.

Reaction moments. Allows free rotation about all 3 axes. No reaction moments can be present.

Ball joints don’t often appear in 2D problems. When they do, they look just like a pinned joint.

Slider with pin joint Allows relative motion in one direction, and allows relative rotation about one axis

Reaction forces: Motion is prevented in two directions, but allowed in the third. There must be two components of reaction force, acting along directions of constrained motion.

Reaction moments: Relative rotation is prevented about two axes, but allowed about a third. There must be two components of reaction moment.

2D slider joints are often represented as shown in the picture below

Slider with swivel joint: Similar to a swivel joint, but allows motion in one direction in addition to rotation about two axes.

Reaction forces: Relative motion is prevented in two directions, but allowed in the third. There must be two components of reaction force acting to prevent motion.

Reaction moments: Rotation is permitted around two axes, but prevented around the third. There must be one component of reaction moment.

In 2D, a slider with swivel looks identical to a slider with a pin joint.