Propped Cantilever, Uniform Load (wL²/8) Formulas

A beam of span L is fixed at the left end (A) and supported on a roller at the right end (B). A uniform load w acts over the full length.

Formulas

Reactions

R_{A} = \frac{5 w L}{8}, R_{B} = \frac{3 w L}{8}, M_{A} = - \frac{w L ^{2}}{8}

Shear

V (x) = \frac{5 w L}{8} - w x

Bending moment

Moment

M (x) = - \frac{w L ^{2}}{8} + \frac{5 w Lx}{8} - \frac{w x ^{2}}{2}

Maximum moments

M_{A} = - \frac{w L ^{2}}{8}, M_{ma x}^{+} = \frac{9 w L ^{2}}{128} at x = \frac{5 L}{8}

Deflection

δ (x) = \frac{w x ^{2}}{48 E I} (L - x) (3 L - 2 x)

Maximum deflection

δ_{ma x} = \frac{w L ^{4}}{185 E I} \approx 0.00542 \frac{w L ^{4}}{E I} at x = \frac{15 - 33}{16} L \approx 0.579 L

x is measured from the fixed end. The zero-moment point is at L/4 from the fixed end.
The maximum deflection is 0.4215L from the roller support.

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 = 37.5 kN
M_A = -45 kN·m
R_B = 22.5 kN

Forces: + up. Moments: − hogging.

Maximum values

|V|_max = 37.5 kN at x = 0 m
M_max = -45 kN·m at x = 0 m
δ_max = 3.51 mm at x = 3.471 m
Span/δ = 1,710

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 propped cantilever with uniform distributed load?

RA = 5wL/8 at the fixed end, RB = 3wL/8 at the roller, and a fixed-end moment of wL²/8.

What is the maximum bending moment of a propped cantilever with uniform distributed load?

wL²/8 at the fixed end, which governs, and 9wL²/128 at 3L/8 from the roller.

What is the maximum deflection of a propped cantilever with uniform distributed load?

δmax ≈ wL⁴/(185EI) at 0.4215L from the roller.

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.

Propped Cantilever with Uniform Distributed Load

Formulas

Reactions

Shear

Bending moment

Deflection

Calculator

Reactions

Maximum values

Shear, moment and deflection diagrams

How these formulas were checked

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FAQ

What are the reactions for a propped cantilever with uniform distributed load?

What is the maximum bending moment of a propped cantilever with uniform distributed load?

What is the maximum deflection of a propped cantilever with uniform distributed load?

What sign convention do these formulas use?

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