5 Rigid-Body Equilibrium and Free-Body Diagrams

Learn how to isolate rigid bodies, model support reactions, apply planar equilibrium equations, and check solutions for support-reaction problems.

Constructing a free-body diagram

A represents one selected body separated from its surroundings. Each removed support or contact is replaced by the force and, where appropriate, the it exerts on the body. The diagram should include all applied loads, the body's weight when relevant, useful dimensions, and a clear coordinate system.

Equilibrium equations can account only for forces and moments shown on the FBD. Show external forces acting on the isolated body. Forces between parts inside a selected connected assembly are internal to its overall FBD; if only one part is isolated, forces exerted on it by the other parts become external.

Drawing the diagram

Select the body or connected assembly to analyze.
Sketch it separately from its surroundings.
Add all known external forces and couples, including weight when relevant.
Replace each removed support or contact with its possible reaction components.
Label unknowns, dimensions, angles, and coordinate directions.

Draw possible reactions based on what the support can exert. If an assumed reaction direction is wrong, solving the equations will generally give a negative value, indicating that the actual direction is opposite the assumed arrow.

Modeling support reactions

A acts in a direction that the support prevents motion. The common two-dimensional idealized models are:

Cable or ideal rope: one tension force along the cable, pulling away from the body.
Smooth surface or roller: one force normal to the surface. A roller on a horizontal surface can exert a vertical force but not a horizontal one.
Pin or hinge: two force components, usually horizontal and vertical, and no reaction couple.
Fixed support: two force components and a reaction couple; it prevents both translation and rotation.

These descriptions are ideal support models. Use the reaction components each model permits when replacing supports on the FBD.

Applying planar equilibrium

For a rigid body in , both the net force and the net must be zero. The three independent scalar equations are

\sum F_x=0,\qquad \sum F_y=0,\qquad \sum M_O=0.

Here, $O$ is any convenient point, and $F_x$ and $F_y$ are the force components along the coordinate axes. In two dimensions, counterclockwise moments are often taken as positive and clockwise moments as negative. A force's about a point depends on its perpendicular distance from that point; if its line of action passes through the point, its there is zero.

These three equations are the available independent equilibrium equations for a planar rigid body. Choosing a center through the lines of action of unknown forces can eliminate those forces from the equation.

Solving a beam support-reaction example

Consider a horizontal, weightless beam of length $4\,\text{m}$ , pinned at end $A$ and resting on a roller at end $B$ . A downward point load of $10\,\text{kN}$ acts $1\,\text{m}$ from $A$ . The unknown pin components are $A_x$ and $A_y$ ; because the roller rests on a horizontal surface, its reaction is vertical, $B_y$ . Take positive $x$ to the right and positive $y$ upward.

Taking moments about $A$ eliminates both pin reactions because their lines of action pass through $A$ :

\sum M_A=0:\quad 4B_y-(10)(1)=0, \qquad B_y=2.5\,\text{kN}.

Vertical force equilibrium gives

\sum F_y=0:\quad A_y+B_y-10=0, \qquad A_y=7.5\,\text{kN}.

Horizontal force equilibrium gives

\sum F_x=0:\quad A_x=0.

Thus, the roller pushes upward with $2.5\,\text{kN}$ , the pin pushes upward with $7.5\,\text{kN}$ , and the horizontal pin reaction is zero. Check that the vertical reactions sum to the applied load and that their moments balance the load.

Checking the setup and solution

A reliable planar support-reaction procedure is:

Choose a body whose external loads and supports are clear.
Draw and label its complete FBD before writing equations.
Count the unknown reaction components and compare that count with the three planar equilibrium equations.
Choose a center that removes as many unknowns as possible.
Solve the equations while preserving the assumed directions and signs.
Substitute the results into the remaining equilibrium equations as a check.

If there are more unknown reactions than independent equilibrium equations, statics alone cannot generally determine them. Additional information, such as material deformation or compatibility conditions, is needed. A body may also be unstable if its supports do not adequately prevent motion, even when the number of unknowns appears suitable.