Learning topic
Constraints and Constraint Reactions
Common constraints and reactions in statics: supports, hinges, cables and contacts. Learn to construct free-body diagrams for equilibrium.
0 practice tasks · 0 subtopics
Constraints are bodies or devices that restrict the possible motion of the body being analyzed. In statics, each constraint is replaced by its reaction, after which the body is treated as free under the action of the applied forces and constraint reactions.
Principle of releasing constraints
To write equilibrium equations, conceptually remove the constraints from the body. Replace each removed constraint by a reaction force or a system of forces and moments that reproduces its mechanical action. The number and directions of unknown reaction components are determined by the motions that the constraint prevents.
Common constraints in two dimensions
| Constraint | Idealized reaction | Key feature |
|---|---|---|
| Smooth surface | Normal force $N$ perpendicular to the surface | With friction neglected, there is no tangential reaction component |
| Flexible cable or rope | Tension $T$ along the cable | The cable pulls the body away from the attachment point |
| Roller or movable pin support in 2D | One reaction, usually normal to the supporting surface | Allows motion along the permitted direction and allows rotation |
| Pin support in 2D | Two components $R_x$ and $R_y$ | Prevents two independent translations but allows rotation |
| Fixed support in 2D | $R_x$, $R_y$, and a reaction moment $M$ | Prevents translation and rotation |
| Ideal pin-connected member loaded only at its ends | Force along the member axis | The member acts as a two-force member |
Constraint reactions follow from the kinematic restrictions of the adopted model. Before writing equilibrium equations, replace each constraint by the appropriate reaction on the free-body diagram.
Smooth contact
For an ideally smooth surface, friction is neglected. The contact reaction acts along the common normal to the surfaces at the contact point. For a flat surface, the reaction direction is known in advance and only its magnitude is unknown.
Cables and two-force members
An ideal flexible cable or rope carries tension only, so the tension force acts along the cable. A straight member acted on only by forces at two pin-connected ends is a two-force member: the end forces are collinear with the member axis, equal in magnitude, and opposite in direction.
Pin and roller supports
A roller or movable support in a planar model produces one reaction in the direction in which it prevents motion. A pin support prevents two independent translations, so its reaction is usually represented by two unknown components $R_x$ and $R_y$. An ideal pin does not transmit a reaction moment.
Fixed support
A fixed support in two dimensions prevents two translations and rotation. Its action is therefore represented by two reaction components $R_x$, $R_y$ and a reaction moment $M$.
Free-body diagram
After releasing the constraints, construct a free-body diagram. Show all applied forces, applied couple moments, the body weight when relevant, and every constraint reaction. The equilibrium equations are applied to this isolated diagram.
Example
A beam supported by a pin at $A$ and a roller at $B$ has three unknown reactions in a typical planar arrangement: $A_x$, $A_y$, and $B_y$ when the roller reaction is vertical. This matches the three independent equilibrium equations available for a planar rigid body.
Common mistakes
- leaving a support on the free-body diagram while also drawing its reactions;
- adding a reaction moment at an ideal pin;
- assigning an arbitrary direction to a smooth-contact reaction instead of the normal direction;
- assuming a cable can carry compression;
- omitting one of the reaction components of a fixed support.