Beam Deflection and Moment Formulas — Simply Supported, Cantilever & Fixed

This reference covers the most commonly used beam formulas for deflection (δ), slope (θ), shear (V), and bending moment (M) for standard loading conditions. All formulas assume linear elastic behavior, prismatic sections, and small deflections.

Variables

Simply Supported Beam — Midspan Point Load

P applied at midspan (x = L/2)

Max moment:   M_max = PL/4        (at midspan)
Max shear:    V_max = P/2         (at supports)
Max deflection: δ_max = PL³/48EI  (at midspan)
End slope:    θ = PL²/16EI

Example: W18x55, L=20ft, P=30 kips, EI=200×10³ kip·ft² δ_max = 30×20³ / (48×200,000) = 0.050 in

Simply Supported Beam — Uniform Distributed Load (UDL)

Total load W = wL

Max moment:   M_max = wL²/8       (at midspan)
Max shear:    V_max = wL/2        (at supports)
Max deflection: δ_max = 5wL⁴/384EI (at midspan)
End slope:    θ = wL³/24EI

This is the most common formula for floor and roof beams.

Simply Supported Beam — Off-Center Point Load

Load P at distance 'a' from left, 'b' from right (a < b)

Left reaction:  R_A = Pb/L
Right reaction: R_B = Pa/L
Max moment:    M_max = Pab/L     (at load point)

Deflection under load:
δ_P = Pa²b²/3EIL

Max deflection (if a < b, occurs at x = √((L²-b²)/3)):
δ_max = Pb(L²-b²)^(3/2) / (9√3·EI·L)

Cantilever Beam — Point Load at Free End

P at free end

Max moment:   M_max = PL         (at fixed support)
Max shear:    V_max = P          (constant throughout)
Max deflection: δ_max = PL³/3EI  (at free end)
Free end slope: θ = PL²/2EI

Cantilever Beam — Uniform Distributed Load

Total load W = wL

Max moment:   M_max = wL²/2      (at fixed support)
Max shear:    V_max = wL         (at fixed support)
Max deflection: δ_max = wL⁴/8EI  (at free end)
Free end slope: θ = wL³/6EI

Cantilever Beam — Point Load at Any Location

Load P at distance 'a' from fixed end (a < L)

Deflection at free end:
δ_tip = Pa²(3L - a) / 6EI

Deflection under load:
δ_a = Pa³/3EI

Fixed-Fixed Beam — Midspan Point Load

P at midspan, both ends fully fixed

End moment:     M_end = PL/8     (each end, hogging)
Midspan moment: M_mid = PL/8     (sagging)
Max deflection: δ_max = PL³/192EI (at midspan)

Fixed-Fixed Beam — UDL

Max end moment:     M_end = wL²/12  (each end, hogging)
Max span moment:    M_mid = wL²/24  (sagging)
Max deflection:     δ_max = wL⁴/384EI (at midspan)

Note the end moment for fixed-fixed (wL²/12) vs simply supported (wL²/8): fixity reduces midspan moment by 33% but introduces support moments.

Propped Cantilever — UDL

Fixed at left, pinned at right

Left reaction:   R_A = 5wL/8
Right reaction:  R_B = 3wL/8
Left end moment: M_A = wL²/8 (hogging)
Max + moment:    M_max = 9wL²/128 (at x = 5L/8 from fixed end)
Max deflection:  δ_max = wL⁴/185EI (at x ≈ 0.422L from pinned end)

Quick Reference — Maximum Deflection Summary

Note: Cantilever deflection is 16× greater than a simply supported beam for the same point load and span — a critical comparison for overhanging structures.

Serviceability Deflection Limits

Frequently Asked Questions

Which beam formula should I use for a uniform floor load? Use the simply supported UDL formula: max moment M_max = wL²/8 at midspan, max deflection δ_max = 5wL⁴/384EI at midspan. This is the most common formula for floor and roof framing. Convert your area load (psf) to linear load (kip/ft) by multiplying by the tributary width before applying the formula.

How does a fixed-fixed beam compare to a simply supported beam? For a fixed-fixed beam under UDL, the midspan moment is wL²/24 — one-third of the simply supported value (wL²/8). However, this comes with fixed-end moments of wL²/12 at each support that must also be designed for. For deflection, fixed-fixed gives wL⁴/384EI versus 5wL⁴/384EI for simply supported — five times stiffer. Real connections rarely achieve full fixity, so designers often use intermediate assumptions or analyze both cases.

Why is a cantilever deflection so much larger than a simply supported beam? For a point load at the free end, cantilever deflection = PL³/3EI. For a simply supported beam with the same load at midspan: PL³/48EI. The cantilever is 16× more flexible. The key reason is boundary conditions: a cantilever has only one support, so the entire beam must curve to carry the load, while a simply supported beam distributes curvature across the span with two supports restraining movement at both ends.

What is the deflection limit for a steel floor beam? AISC Design Guide and IBC typically limit live load deflection to L/360 for floor beams supporting brittle finishes (tile, plaster). For beams supporting flexible finishes, L/240 is common. Total load deflection (dead + live) is often limited to L/240 for floor beams. Always check both live load deflection (for cracking concerns) and total load deflection (for visual appearance).

How do I apply these formulas for metric units? All formulas work in any consistent unit system. For SI: use w in kN/m, L in m, E in kPa (or GPa × 10⁶), and I in m⁴. This gives M in kN·m and δ in m. In practice, engineers use E = 200 GPa = 200 × 10⁶ kN/m², convert I from mm⁴ to m⁴ (divide by 10¹²), and express results in mm by multiplying δ in m by 1000.

Run This Calculation

→ Beam Calculator — beam reactions and moment/shear diagrams for any span and loading condition.

→ Beam Deflection Calculator — compute deflections and reactions automatically for common loading cases, checking L/360 and L/240 limits.

→ Continuous Beam Calculator — reactions, moments, and deflections for multi-span beams with any loading pattern.

See Also

• W-Shape Beam Sizes — Section Properties (Ix, rx, ry)
• Beam Bending Moment Formulas — Reference Table
• Steel Beam Load Tables — W-Shape UDL Capacity
• Steel Beam Span Guide — W-Shape Span Ranges by Depth
• Lateral-Torsional Buckling — Lp, Lr, Cb
• Steel Fy & Fu Reference — Yield and Tensile Strength by Grade
• beam capacity calculator
• deflection limits reference
• Composite Beam
• Floor Vibration

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