Deflection of Statically Determinate Beams, Trusses and Frames

NCEES FE Civil spec: Structural Engineering, B. Deflection of statically determinant beams, trusses, and frames. FE Reference Handbook: Mechanics of Materials chapter (Deflection of Beams); Civil Engineering chapter, Structural Analysis section.

What the FE Civil Spec Covers

Section B covers how far determinate structures move under load. For beams you will use standard deflection formulas, superposition, and occasionally the moment-area idea that curvature equals M/EI. For trusses and frames the FE uses the unit load (virtual work) method. Expect one or two questions asking for a maximum deflection, an end slope, or the displacement of a truss joint.

Key Concepts

Flexural rigidity EI controls beam stiffness. Doubling I halves the deflection; doubling the span multiplies a uniform-load deflection by 16.
Superposition works for linear elastic, small-deflection behavior: add the deflections from each load case at the same point.
Curvature is M/EI. Integrating once gives slope, twice gives deflection, with constants set by the supports.
Unit load method: apply a fictitious unit load where and in the direction you want the displacement. For trusses only axial forces matter; for frames, bending usually dominates.

Core Formulas

Elastic curve:

E I \frac{d ^{2} y}{d x ^{2}} = M (x)

Maximum deflections for the standard cases:

Case	Maximum deflection
Simply supported, uniform load w	$\frac{5 w L ^{4}}{384 E I}$ at midspan
Simply supported, point load P at midspan	$\frac{P L ^{3}}{48 E I}$ at midspan
Cantilever, point load P at free end	$\frac{P L ^{3}}{3 E I}$ at free end
Cantilever, uniform load w	$\frac{w L ^{4}}{8 E I}$ at free end

Unit load method for a truss (f = member force from the unit load, F = member force from real loads):

Δ = \sum \frac{f F L}{A E}

For a frame where bending governs (m = moment from unit load, M = moment from real loads):

Δ = \int \frac{m M}{E I} d x

Moment-Area Shortcut

Two moment-area theorems are handy for slopes and for beams with unusual loading. The change in slope between two points equals the area of the M/EI diagram between them. The vertical distance from one point on the elastic curve to the tangent drawn at another point equals the first moment of that M/EI area about the first point. For a cantilever, the tangent at the fixed end is horizontal, so the tip deflection is simply the moment of the whole M/EI diagram about the tip.

Serviceability Limits

Deflection is a serviceability check, not a strength check. Common limits are span/360 for floor live load and span/240 for total load, but always use the limit given in the problem. Deflection is computed with unfactored (service) loads, never with LRFD factored loads.

Where It Is in the FE Reference Handbook

The beam deflection and slope table is in the Mechanics of Materials chapter (Deflection of Beams, with the Principle of Superposition). The unit load method appears in the Civil Engineering chapter, Structural Analysis section, as truss deflection and frame deflection by the unit load method.

Worked Example: Superposition on a Steel Beam

A simply supported steel beam spans L = 8 m. It carries a uniform load w = 15 kN/m and a 20 kN point load at midspan. E = 200 GPa and I = 400 × 10⁶ mm⁴. Find the midspan deflection and compare it to an L/360 limit.

Step 1: Flexural rigidity in kN·m²

E I = (200 \times 1 0^{6} kN/m^{2}) (400 \times 1 0^{- 6} m^{4}) = 80, 000 kN \cdotp m^{2}

Step 2: Deflection from the uniform load

δ_{w} = \frac{5 ( 15 ) ( 8 ) ^{4}}{384 ( 80 , 000 )} = \frac{307 , 200}{30 , 720 , 000} = 0.0100 m = 10.0 mm

Step 3: Deflection from the point load

δ_{P} = \frac{( 20 ) ( 8 ) ^{3}}{48 ( 80 , 000 )} = \frac{10 , 240}{3 , 840 , 000} = 0.00267 m = 2.67 mm

Step 4: Superpose and check

δ_{t o t a l} = 10.0 + 2.67 = 12.7 mm < \frac{L}{360} = \frac{8000}{360} = 22.2 mm

Both maximums occur at midspan, so adding them directly is valid. The beam meets the L/360 limit. If the two maximums were at different points, you would have to evaluate each case at the same location before adding.

Common Mistakes

Leaving E in GPa and I in mm⁴ and getting an answer off by a factor of a million or more.
Adding maximum deflections from load cases whose maximums occur at different points.
Using the simply supported formula for a cantilever (PL³/48EI instead of PL³/3EI is 16 times too small).
In the unit load method, forgetting that f and F must use the same sign convention (tension positive).
Counting a zero-force member as contributing to deflection: if either f or F is zero, that member adds nothing.

Deflection compatibility is also how you solve indeterminate beams; see statically indeterminate structures.

Practice Problems

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

A simply supported steel beam has span L = 6 m, uniform load w = 12 kN/m, E = 200 GPa and I = 300 × 10^6 mm^4. What is the maximum deflection? See the worked solution or practice the Beam Analysis deck
A 2 m cantilever with flexural rigidity E I = 4000 kN·m² carries a 10 kN point load at its free end. What is the deflection at the free end? See the worked solution or practice the Beam Analysis deck
A steel cantilever 3 m long (E = 200 GPa, I = 80 × 10⁶ mm⁴) carries a 10 kN point load at its free end. The tip deflection is most nearly: See the worked solution or practice the Structural Essentials deck
A 4.0 m beam segment AB has curvature κ(x) increasing linearly from 0 at A to +0.003 m⁻¹ at B, with x measured from A and y″ = κ (small deflections). Find the deviation tB/A = yB − yA − θA·L of point B from the tangent drawn at A, in millimeters. See the worked solution or practice the Structural Analysis Methods deck
A planar pin-jointed truss has joints A (0, 0), B (4, 3) and C (8, 0), coordinates in meters, with members AB, BC and AC. A is a pin and C is a roller restraining vertical displacement. A 24 kN downward load acts at B. Every member has EA = 40,000 kN. Consider axial deformation only. Find the vertical displacement of joint B, including direction. See the worked solution or practice the Structural Analysis Methods deck

Related Calculators and Examples

Frequently Asked Questions

Do I need to memorize beam deflection formulas for the FE exam?

No. The FE Reference Handbook has a table of deflections and slopes for standard simply supported and cantilever cases. You should know how to find it fast and how to combine cases with superposition.

How do I find the deflection of a truss joint on the FE?

Use the unit load method. Find the member forces from the real loads, then the member forces from a unit load applied at the joint in the direction you want. The deflection is the sum over all members of real force times unit force times length divided by AE.

What units mistakes are most common in deflection problems?

Mixing millimeters and meters in EI. Convert E and I so that EI is in kN·m² (or lb·in²) before substituting, and remember that L is raised to the third or fourth power, so a unit error is magnified.

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