Worked problem

Simply Supported Beam: Reactions, Moment and Deflection

Find the reactions, maximum moment and midspan deflection of a 6 m steel beam under a 15 kN/m uniform load, and check it against L/360.

Civil & Structural · Last reviewed

Given

Simply supported steel beam, span L = 6 m, uniform service load w = 15 kN/m (including self-weight). E = 200 GPa, I = 8.0 × 10⁻⁵ m⁴ (80,000,000 mm⁴). Deflection limit L/360.

Find

Support reactions, maximum bending moment, midspan deflection, and whether the deflection limit is met.

Solution

  1. Reactions: R = wL ÷ 2 = 15 × 6 ÷ 2 = 45 kN at each support.
  2. Maximum moment at midspan: M = wL² ÷ 8 = 15 × 36 ÷ 8 = 67.5 kN·m.
  3. Midspan deflection: δ = 5wL⁴ ÷ (384 E I) = 5 × 15,000 × 6⁴ ÷ (384 × 200 × 10⁹ × 8.0 × 10⁻⁵) = 0.0158 m = 15.8 mm.
  4. Limit: L ÷ 360 = 6,000 mm ÷ 360 = 16.7 mm.
  5. 15.8 mm is below 16.7 mm, so the beam meets the limit with little margin (95% of the allowable).

Answer

R = 45 kN | Mmax = 67.5 kN·m | δ = 15.8 mm < 16.7 mm (OK)

Common mistakes

  • Mixing units. Keep E in Pa, I in m⁴, w in N/m and L in m, and the answer is in metres.
  • Using the factored (ultimate) load for a deflection check. Serviceability limits use service loads.
  • Checking only deflection. Strength (moment and shear capacity) and the other serviceability criteria still apply.

Try a variation: an 8 m span

  1. The same beam and load, now with L = 8 m: δ = 5 × 15,000 × 8⁴ ÷ (384 × 200 × 10⁹ × 8.0 × 10⁻⁵) = 0.050 m = 50 mm.
  2. The limit is L ÷ 360 = 22.2 mm, so the beam fails.
  3. The required moment of inertia for L/360 at 8 m is I = 5 w L⁴ ÷ (384 E δ) = 1.8 × 10⁻⁴ m⁴ (180 × 10⁶ mm⁴), which is 2.25 times the original.

Lengthening the span from 6 m to 8 m multiplied the deflection by (8 ÷ 6)⁴ = 3.2, because deflection grows with the fourth power of the span.

Related tools and labs

Calculators, labs and simulators are part of the All-Access Pass.

More worked problems

Questions

Why L/360?

It is a common serviceability limit for floor members that support plaster or brittle finishes. The limit that applies depends on the code, the member and the finishes.

How would a fixed-fixed beam differ?

Its midspan deflection is wL⁴ ÷ (384 E I), one fifth of the simply supported value, and the maximum moment moves to the supports.

Why does deflection grow with the fourth power of the span?

The load on the beam scales with the span, the moment scales with the span squared, and the curvature integrates twice along the length. The result is a term in L⁴.

What is the difference between live-load and total-load deflection limits?

Codes set separate limits, such as L/360 for live load and L/240 for total load, for floors and roofs, with the actual values depending on the member and the finishes it carries.

Should the beam self-weight be included?

Yes. Self-weight is part of the dead load. In this example it is included in the 15 kN/m. For deflection checks, check which loads the limit applies to.

Educational material. Examples use simplified, hypothetical inputs and are not a design, a construction document or professional engineering advice. Confirm requirements with the codes and standards adopted where the work is built and with a licensed engineer.

← All worked problems