OpenSpecEngineering

Finds the maximum deflection of a simply supported beam or a cantilever under a point load or a uniform load. It can also work the other way: from a deflection limit, it gives the second moment of area I the beam needs. Pick a metal or lumber species for E and an AISC shape for I, or type your own. Designers use it to size joists, beams and brackets for stiffness.

Equations

Simple, point load at midspan: δ = P·L³/(48·E·I)     Cantilever, point load at the free end: δ = P·L³/(3·E·I)

Simple, uniform load: δ = 5·w·L⁴/(384·E·I)     Cantilever, uniform load: δ = w·L⁴/(8·E·I)

P is the point load, w the uniform load per unit length, L the span (or the cantilever’s length), E the elastic modulus and I the second moment of area about the bending axis [1]. δ is at midspan for a simple beam and at the free end for a cantilever. A listed shape fills in I_x, for bending about its strong axis [2]. A lumber species fills in its reference E [3]. To find I, the same equations are solved for I with δ set to the limit. A code limit from IBC Table 1604.3 [4] or ACI 318 Table 24.2.2 [5] allows span/n. For a cantilever, both codes take the span as twice its length.

Limits and common mistakes

• The equations are for small, elastic deflections of a straight, prismatic beam. Shear deflection is ignored.
• The point load is at midspan for a simple beam. A load off center deflects the beam less.
• Use the load the limit applies to: live load alone for L/360, dead plus live for L/240.
• The page doesn’t check strength.
• A channel bends as listed only if the load passes through its shear center or it is braced. The page says so when you pick one.

Worked example

A steel floor beam spans 6 m between supports and carries a live load of 15 kN/m (15 N/mm). The floor must meet the IBC live-load limit of L/360.

1. Choose Beam Case Simple / Uniform Load and Solve for Second Moment of Area (I).
2. Enter Uniform Load (w) 15 N/mm and Span Length (L) 6000 mm. Choose Material (E) Steel.
3. Choose Deflection Limit Live load — L/360 under Floor members, and press Solve. The Solution shows: Second Moment of Area (I): 75,937,500 mm⁴
   Allowable deflection δ = L/360: 16.6667 mm
4. That is 182 in⁴. To check a W12x26 (I_x = 204 in⁴), choose Solve for Deflection (δ), Section Type W-Shape and Section W12x26, and press Solve: Maximum Deflection (δ): 14.9053 mm

Check by hand. I_x = 204 × 25.4⁴ = 84.9 × 10⁶ mm⁴, and 5 × 15 × 6000⁴ / (384 × 200,000 × 84.9 × 10⁶) = 14.9 mm. That is under the 16.7 mm limit, as it must be with more than the 182 in⁴ required.