Simply Supported Beam, Partial Uniform Load Formulas

A beam of span L on a pin and a roller carries a uniform load w over a length b, starting at distance a from the left support. The unloaded length on the right is c = L − a − b.

Formulas

Reactions

Left reaction

R_{A} = \frac{w b}{2 L} (2 c + b)

Right reaction

R_{B} = \frac{w b}{2 L} (2 a + b)

Shear

V = R_{A} (x < a), V = R_{A} - w (x - a) (a \leq x \leq a + b), V = - R_{B} (x > a + b)

Bending moment

Moment (a ≤ x ≤ a + b)

M (x) = R_{A} x - \frac{w}{2} (x - a)^{2}

Maximum moment

M_{ma x} = R_{A} (a + \frac{R _{A}}{2 w}) at x = a + \frac{R _{A}}{w}

Deflection

Deflection (Macaulay brackets)

E I δ (x) = - [\frac{R _{A} x ^{3}}{6} - \frac{w ⟨ x - a ⟩ ^{4}}{24} + \frac{w ⟨ x - a - b ⟩ ^{4}}{24} + C_{1} x]

Integration constant

C_{1} = - \frac{1}{L} [\frac{R _{A} L ^{3}}{6} - \frac{w ( L - a ) ^{4}}{24} + \frac{w c ^{4}}{24}]

The maximum moment is inside the loaded length, where the shear crosses zero.
There is no short closed form for the maximum deflection: it occurs where the slope is zero, which the calculator finds numerically from the exact deflection equation. For a load centered on the span it is at midspan.
⟨x − a⟩ is a Macaulay bracket: it equals x − a when x > a and zero otherwise.

Calculator

Span L (m)Distance a (m)Length b (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 = 20 kN
R_B = 20 kN

Forces: + up. Moments: − hogging.

Maximum values

|V|_max = 20 kN at x = 0 m
M_max = 60 kN·m at x = 4 m
δ_max = 19 mm at x = 4 m
Span/δ = 421.1

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 partial uniform load?

RA = wb(2c + b)/(2L) and RB = wb(2a + b)/(2L), where b is the loaded length and a and c are the unloaded lengths at the left and right ends.

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

Mmax = RA(a + RA/2w), at x = a + RA/w from the left support, inside the loaded length where the shear is zero.

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

It occurs where the slope is zero and has no short closed form. Use the Macaulay deflection equation on this page, or the calculator, which evaluates it exactly.

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 Partial Uniform 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 partial uniform load?

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

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

What sign convention do these formulas use?

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