Simply Supported Beam, Triangular Load Formulas

A beam of span L on a pin and a roller carries a load that increases linearly from zero at the left support to w at the right support. Total load W = wL/2.

Formulas

Reactions

R_{A} = \frac{w L}{6}, R_{B} = \frac{w L}{3}

Shear

V (x) = \frac{w L}{6} - \frac{w x ^{2}}{2 L}

Bending moment

Moment

M (x) = \frac{w x}{6 L} (L^{2} - x^{2})

Maximum moment

M_{ma x} = \frac{w L ^{2}}{9 3} \approx 0.0642 w L^{2} at x = \frac{L}{3} \approx 0.577 L

Deflection

δ (x) = \frac{w x}{360 E I L} (7 L^{4} - 10 L^{2} x^{2} + 3 x^{4})

Maximum deflection

δ_{ma x} \approx 0.00652 \frac{w L ^{4}}{E I} at x = L 1 - 8/15 \approx 0.519 L

Two thirds of the total load goes to the support under the heavy end.
The maximum moment and the maximum deflection are both slightly toward the heavy end of the span, at different points.

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 = 10 kN
R_B = 20 kN

Forces: + up. Moments: − hogging.

Maximum values

|V|_max = 20 kN at x = 6 m
M_max = 23.09 kN·m at x = 3.464 m
δ_max = 4.226 mm at x = 3.116 m
Span/δ = 1,420

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 triangular load?

RA = wL/6 at the light end and RB = wL/3 at the heavy end, where w is the peak intensity.

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

Mmax = wL²/(9√3) ≈ 0.0642wL² at x = L/√3 ≈ 0.577L from the light end.

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

δmax ≈ 0.00652wL⁴/EI at x ≈ 0.519L from the light 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.

Simply Supported 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 simply supported beam cases

More beam formulas

FAQ

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

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

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

What sign convention do these formulas use?

Related tools and topics