Cantilever Beam, Triangular Load (wL⁴/30EI) Formulas

A cantilever of length L is fixed at the left end (A). The load is w at the wall and decreases linearly to zero at the free end. Total load W = wL/2.

Formulas

Reactions

Fixed-end reactions

R_{A} = \frac{w L}{2}, M_{A} = - \frac{w L ^{2}}{6}

Shear

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

Bending moment

Moment

M (x) = - \frac{w}{6 L} (L - x)^{3}

Maximum moment

∣ M_{ma x} ∣ = \frac{w L ^{2}}{6} at the fixed end

Deflection

δ (x) = \frac{w x ^{2}}{120 E I L} (10 L^{3} - 10 L^{2} x + 5 L x^{2} - x^{3})

Maximum deflection

δ_{ma x} = \frac{w L ^{4}}{30 E I} at the free end

Slope at the free end

θ_{ma x} = \frac{w L ^{3}}{24 E I}

x is measured from the fixed support. If the load is reversed (zero at the wall, w at the tip), the results change: R = wL/2, M = wL²/3 and δmax = 11wL⁴/120EI.

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 = 9 kN
M_A = -9 kN·m

Forces: + up. Moments: − hogging.

Maximum values

|V|_max = 9 kN at x = 0 m
M_max = -9 kN·m at x = 0 m
δ_max = 0.81 mm at x = 3 m
L/δ = 3,704

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 cantilever beam with triangular load?

With the peak intensity w at the wall, R = wL/2 and the fixed-end moment is wL²/6.

What is the maximum bending moment of a cantilever beam with triangular load?

Mmax = wL²/6 at the fixed support.

What is the maximum deflection of a cantilever beam with triangular load?

δmax = wL⁴/(30EI) at the free end.

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.

Cantilever Beam with Triangular Load

Formulas

Reactions

Shear

Bending moment

Deflection

Calculator

Reactions

Maximum values

Shear, moment and deflection diagrams

How these formulas were checked

Other cantilever beam cases

More beam formulas

FAQ

What are the reactions for a cantilever beam with triangular load?

What is the maximum bending moment of a cantilever beam with triangular load?

What is the maximum deflection of a cantilever beam with triangular load?

What sign convention do these formulas use?

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