Simply Supported Beam, Uniform Load (wL²/8) Formulas

A beam of span L on a pin and a roller carries a uniform load w over its full length.

Formulas

Reactions

R_{A} = R_{B} = \frac{w L}{2}

Shear

V (x) = w (\frac{L}{2} - x)

Bending moment

Moment

M (x) = \frac{w x}{2} (L - x)

Maximum moment

M_{ma x} = \frac{w L ^{2}}{8} at x = \frac{L}{2}

Deflection

δ (x) = \frac{w x}{24 E I} (L^{3} - 2 L x^{2} + x^{3})

Maximum deflection

δ_{ma x} = \frac{5 w L ^{4}}{384 E I} at x = \frac{L}{2}

Shear varies linearly from +wL/2 to −wL/2 and is zero at midspan, where the moment is greatest.
The moment diagram is a parabola.

Calculator

Span L (m)Load w (kN/m)E (GPa)I (×10⁶ mm⁴)

Loads act downward as drawn. Steel: E = 200 GPa. I in millions of mm⁴ (100 means 100 × 10⁶ mm⁴).

Reactions

R_A = 30 kN
R_B = 30 kN

Forces: + up. Moments: − hogging.

Maximum values

|V|_max = 30 kN at x = 0 m
M_max = 45 kN·m at x = 3 m
δ_max = 8.438 mm at x = 3 m
Span/δ = 711.1

Shear, moment and deflection diagrams

Shear force V (kN)

Bending moment M (kN·m)

Deflection δ (downward) (mm)

x is measured from the left end. Positive moment (sagging) plots above the axis. Need more loads or supports? Solve any beam with the full beam calculator.

FAQ

What are the reactions for a simply supported beam with uniform distributed load?

RA = RB = wL/2.

What is the maximum bending moment of a simply supported beam with uniform distributed load?

Mmax = wL²/8 at midspan.

What is the maximum deflection of a simply supported beam with uniform distributed load?

δmax = 5wL⁴/(384EI) at midspan.

What sign convention do these formulas use?

x is measured from the left end (from the fixed end for cantilevers and propped cantilevers). Loads act downward, reactions are positive upward, sagging moment is positive and deflection is positive downward. E is the elastic modulus and I the second moment of area of the section.

Simply Supported Beam with Uniform Distributed Load

Formulas

Reactions

Shear

Bending moment

Deflection

Calculator

Reactions

Maximum values

Shear, moment and deflection diagrams

How these formulas were checked

Other simply supported beam cases

More beam formulas

FAQ

What are the reactions for a simply supported beam with uniform distributed load?

What is the maximum bending moment of a simply supported beam with uniform distributed load?

What is the maximum deflection of a simply supported beam with uniform distributed load?

What sign convention do these formulas use?

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