Resultants of Force Systems

NCEES FE Civil spec: Statics, A. Resultants of force systems. FE Reference Handbook: Statics chapter, resultants and systems of forces.

A resultant is the single force (and, for non-concurrent systems, the single couple) that has the same external effect on a rigid body as a whole group of forces. Almost every statics problem on the FE Civil exam starts with this skill: you break forces into components, add them, and put them back together.

What the NCEES spec covers

Statics section A, "Resultants of force systems", tests whether you can combine forces in two dimensions (and occasionally three). Expect concurrent forces acting at a point, parallel forces on a beam, and general coplanar systems where you also need the location of the resultant. Questions are short and numeric, so speed with components and sign conventions matters more than theory.

Key concepts

Components. A force $F$ at angle $θ$ from the +x axis has components $F_{x} = F cos θ$ and $F_{y} = F sin θ$ . Measure every angle from the same axis and let the signs take care of direction.
Concurrent forces all pass through one point, so their resultant also passes through that point and there is no moment to track.
Non-concurrent forces produce a resultant force plus a moment about any chosen point. The resultant's line of action is where a single force would produce that same moment.
Parallel forces (for example, loads on a beam) reduce to one force equal to their sum, located at the moment-weighted average position.

Core formulas

R_{x} = \sum F_{x} R_{y} = \sum F_{y} R_{z} = \sum F_{z}

R = R_{x}^{2} + R_{y}^{2} + R_{z}^{2} θ = tan^{- 1} (\frac{R _{y}}{R _{x}})

For a coplanar system, the location of the resultant along a beam (x measured from a reference point O):

\overset{x}{ˉ} = \frac{\sum M _{O}}{R} = \frac{\sum F _{i} x _{i}}{\sum F _{i}}

In 3D, the direction cosines are $cos θ_{x} = R_{x} / R$ , $cos θ_{y} = R_{y} / R$ and $cos θ_{z} = R_{z} / R$ .

Where it is in the FE Reference Handbook

Look in the Statics chapter, under the headings for force, resultants in two dimensions, resolution of a force and systems of forces. The Mathematics chapter (vectors and trigonometry) covers the vector algebra if you need a refresher. Use the search function in the PDF for "resultant" to jump there quickly.

Worked example

Three concurrent forces act at point O: $F_{1} = 500 N$ at 30° above the +x axis, $F_{2} = 300 N$ at 120° from the +x axis, and $F_{3} = 200 N$ straight down. Find the magnitude and direction of the resultant.

Step 1: resolve each force.

F_{1} : 500 cos 3 0^{\circ} = 433.0, 500 sin 3 0^{\circ} = 250.0

F_{2} : 300 cos 12 0^{\circ} = - 150.0, 300 sin 12 0^{\circ} = 259.8

F_{3} : 0, - 200.0

Step 2: add components.

R_{x} = 433.0 - 150.0 + 0 = 283.0 N R_{y} = 250.0 + 259.8 - 200.0 = 309.8 N

Step 3: combine.

R = 283. 0^{2} + 309. 8^{2} = 419.6 N

θ = tan^{- 1} (\frac{309.8}{283.0}) = 47. 6^{\circ} above the +x axis

Both components are positive, so the resultant points into the first quadrant and no angle correction is needed. The answer is about 420 N at 47.6°.

Common mistakes

Mixing angle references (some angles from horizontal, some from vertical) and swapping sin and cos.
Trusting the calculator's arctan without checking the quadrant from the signs of $R_{x}$ and $R_{y}$ .
Adding magnitudes instead of components. Forces only add directly when they are collinear.
Forgetting that a non-concurrent system also has a moment, so the resultant has a specific line of action.
Leaving the calculator in radians when the problem gives degrees.

For more practice, see the resultants of force systems example and the vector operations lesson.

Practice Problems

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

Two concurrent forces act at a point: 30 N along +x and 40 N along +y. What is the magnitude of the resultant? See the worked solution or practice the Statics & Equilibrium deck
A 500 N force acts at 30° above the horizontal. What is its horizontal component? See the worked solution or practice the Statics & Equilibrium deck
Concurrent forces of 3 kN east and 4 kN north act at a point. What is the resultant magnitude? See the worked solution or practice the Statics & Equilibrium deck
Two parallel downward forces act along a horizontal beam: 10 kN at x = 0 and 20 kN at x = 3 m. Where does their resultant act, measured from x = 0? See the worked solution or practice the Statics & Equilibrium deck
Two forces act at a point: F₁ = 3i + 4j kN and F₂ = −1i + 2j kN. The magnitude of their resultant is most nearly: See the worked solution or practice the Math & Statistics Refresher deck

Related Calculators and Examples

Frequently Asked Questions

How do I find the resultant of several forces on the FE exam?

Resolve every force into x and y components using angles measured from the same axis, add the components, then combine them with R = sqrt(Rx^2 + Ry^2) and find the angle with arctan(Ry/Rx), checking the quadrant from the signs.

What is the difference between concurrent and non-concurrent force systems?

Concurrent forces all act through one point, so the resultant is just a force through that point. Non-concurrent forces also create a moment, so the resultant has a specific line of action found by matching the moment of the original system.

Where are resultant formulas in the FE Reference Handbook?

In the Statics chapter, under force, resultants in two dimensions, resolution of a force and systems of forces. Vector algebra is in the Mathematics chapter.

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