Supports and reactions
Supports hold structures in place by pushing or pulling back. Each type of support restrains certain motions, and each restraint produces a reaction.
Rule of thumb
A support produces a reaction force in every direction it prevents translation, and a reaction couple about every axis it prevents rotation. Identifying exactly what each support can and cannot do is the key to drawing correct free-body diagrams of structures.
Rollers and smooth surfaces
A roller, rocker, or smooth surface prevents motion only perpendicular to the surface, so it gives one reaction force perpendicular to that surface. Bridges often rest on rollers at one end so they can expand and contract with temperature without stressing their supports.
Pins and hinges
A smooth pin or hinge prevents translation in any direction in the plane but allows rotation, giving two unknown force components, usually drawn as horizontal and vertical. Door hinges and the pinned ends of many beams work this way.
Fixed supports
A fixed support, such as a beam built into a wall or a post set in concrete, prevents translation and rotation. In 2D it gives two force components and a couple: three unknowns. A diving board clamped at one end is a fixed support, or cantilever.
Cables and links
A cable gives one tension force along the cable, away from the body. A short link with pins at both ends gives one force along the link, which may be tension or compression.
Supports in 3D
In three dimensions, a ball-and-socket joint gives three force components; a hinge gives forces and possibly couples depending on its alignment; and a fixed support gives three forces and three couples, six unknowns in all.
Counting unknowns
A 2D rigid body has three independent equilibrium equations. If the supports introduce exactly three unknowns, properly arranged, the body is statically determinate. A beam pinned at one end and on a roller at the other has exactly three unknowns: two at the pin and one at the roller.
Worked example
A 4 m diving board is pinned at its back end A and rests on a roller support B, 1.2 m from A. The free-body diagram shows Ax and Ay at the pin, a vertical reaction By at the roller, the board's weight at its centre, and the diver's weight at the tip. That is three unknowns, so the three equilibrium equations will give all the reactions.
Key takeaways
Supports produce a reaction force for each prevented translation and a reaction couple for each prevented rotation. Rollers and smooth surfaces give one perpendicular force; pins and hinges give two force components; fixed supports give two forces and a couple in 2D, and six reactions in 3D. Cables give one tension, and two-force links one force along the link. A 2D rigid body with exactly three properly arranged unknown reactions is statically determinate.
Flashcard
Force or couple preventing motion
Check your understanding
How many unknown reactions does a 2D fixed support provide?