Equilibrium of Rigid Bodies

NCEES FE Civil spec: Statics, C. Equilibrium of rigid bodies. FE Reference Handbook: Statics chapter, systems of forces and equilibrium.

A rigid body is in equilibrium when the resultant force and the resultant moment on it are both zero. On the FE Civil exam, "equilibrium of rigid bodies" mostly means finding support reactions for beams, frames, brackets and cable systems, and it feeds directly into the shear, moment and truss questions later in the exam.

What the NCEES spec covers

Statics section C covers free body diagrams, support types and their reactions, and the equations of equilibrium in 2D (with simple 3D cases). Typical problems: a simply supported or overhanging beam, a cantilever, a two-cable support, or a bracket with a pin and a link.

Key concepts

Free body diagram (FBD). Isolate the body, show every applied load, and replace each support with the reactions it can provide.
Support reactions. A roller gives one force perpendicular to its surface. A pin gives two force components. A fixed support gives two forces and a moment. A cable or link gives one force along its axis.
Three equations in 2D, so at most three unknown reactions can be found from statics alone. More than that means the structure is statically indeterminate.
Two-force and three-force members. A member loaded only at two points carries force along the line joining them. Three non-parallel forces in equilibrium must meet at one point.

Core formulas

\sum F_{x} = 0 \sum F_{y} = 0 \sum M_{any point} = 0

In 3D there are six equations:

\sum F_{x} = \sum F_{y} = \sum F_{z} = 0 \sum M_{x} = \sum M_{y} = \sum M_{z} = 0

Take moments about a point where two unknowns intersect so only one unknown remains in that equation.

Where it is in the FE Reference Handbook

The Statics chapter states the equilibrium requirements under systems of forces. Beam reaction formulas for standard load cases are in the Mechanics of Materials chapter beam tables, and they are worth recognizing so you can check a hand calculation in seconds.

Worked example

An 8 m beam has a pin at A (x = 0) and a roller at B (x = 6 m), so it overhangs 2 m past B. It carries a uniform load of 4 kN/m from A to B and a 10 kN point load at the free end (x = 8 m). Find the reactions.

Step 1: replace the distributed load. $4 \times 6 = 24 kN$ acting at x = 3 m.

Step 2: moments about A (counterclockwise positive):

\sum M_{A} = B_{y} (6) - 24 (3) - 10 (8) = 0 \Rightarrow B_{y} = \frac{72 + 80}{6} = 25.33 kN

Step 3: vertical equilibrium.

A_{y} + B_{y} - 24 - 10 = 0 \Rightarrow A_{y} = 34 - 25.33 = 8.67 kN

Step 4: check with moments about B.

\sum M_{B} = - A_{y} (6) + 24 (3) - 10 (2) = - 52.0 + 72 - 20 = 0 ✓

With no horizontal loads, $A_{x} = 0$ . The roller takes most of the load because the tip load levers about B and relieves A.

Common mistakes

Skipping the FBD, then missing a reaction component (often the horizontal pin reaction or a fixed-end moment).
Using the full distributed load at the wrong location, or forgetting to include it in the moment equation.
Assuming reactions are equal on a beam that is not symmetric.
Getting a negative reaction and dropping the sign. A negative answer means the assumed direction was wrong.
Not checking with a second moment equation, which catches most arithmetic slips.

Practice Problems

Try these from the free PolyCalc FE Civil problem bank. Each one has a full worked solution.

A simply supported beam spans 10 m between supports A (left) and B (right). A single 20 kN downward point load acts 3 m from A. What is the vertical reaction at B? See the worked solution or practice the Statics & Equilibrium deck
A 4 m cantilever is fixed at its left end and carries a 5 kN downward point load at the free end. What is the moment reaction at the fixed support? See the worked solution or practice the Statics & Equilibrium deck
A 100 N weight hangs from a ring supported by two cables. Each cable makes 30° with the horizontal, symmetric about the vertical. What is the tension in each cable? See the worked solution or practice the Statics & Equilibrium deck
A beam rests on a pin at A and a roller at B, 6 m apart, and overhangs 2 m beyond B. A 30 kN load acts at the tip of the overhang. The reaction at A is: See the worked solution or practice the Statics & Equilibrium deck
An 8 m simply supported beam carries a 12 kN downward point load 2 m from the left support. Neglect self-weight. What is the left vertical reaction? See the worked solution or practice the Statics & Equilibrium deck

Related Calculators and Examples

Frequently Asked Questions

How many unknowns can be solved with equilibrium equations?

In 2D there are three independent equations (sum of x forces, sum of y forces and sum of moments), so three unknown reactions can be found. In 3D there are six.

What reactions does each support type provide?

A roller provides one force perpendicular to its surface, a pin provides two force components, and a fixed support provides two force components and a moment. A cable or link provides one force along its axis.

What does a negative reaction mean?

It means the reaction acts in the opposite direction to the one you assumed on the free body diagram. Keep the sign when you use it in later equations.

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