Beam Formulas

Reactions, shear, bending moment and deflection for 15 standard beam cases. Pick a case for the full equations and a calculator.

Simply supported beams

PABL

Center point load

Mmax = PL/4, under the load at midspan.

PABab

Point load at any position

Mmax = Pab/L, directly under the load.

wABL

Uniform load

Mmax = wL²/8 at midspan.

wABabc

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.

wABL

Triangular load

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

M₀ABL

Moment at one end

Mmax = M₀ at the loaded support, falling linearly to zero at the other support.

Cantilever beams

PAL

Point load at free end

Mmax = PL at the fixed support.

wAL

Uniform load

Mmax = wL²/2 at the fixed support.

wAL

Triangular load

Mmax = wL²/6 at the fixed support.

M₀AL

Moment at free end

The bending moment is M₀ at every section (pure bending).

Fixed-fixed beams

PABL

Center point load

PL/8 hogging at both supports and PL/8 sagging under the load.

wABL

Uniform load

wL²/12 hogging at the supports, which governs, and wL²/24 sagging at midspan.

Propped cantilever beams

PABL

Center point load

3PL/16 at the fixed end, which governs, and 5PL/32 under the load.

wABL

Uniform load

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

Overhanging beams

wABLa

Uniform load

The positive moment in the span is w(L + a)²(L − a)²/(8L²) and the negative moment over the support next to the overhang is wa²/2. Design for whichever is larger.

Sign convention

On every page x is measured from the left end, or 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 is the second moment of area about the bending axis. Each formula was cross-checked numerically against the stiffness-method solver behind the beam calculator.

FAQ

Which beam formulas are most used?

Simply supported with a uniform load (M = wL²/8, δ = 5wL⁴/384EI), simply supported with a center point load (M = PL/4, δ = PL³/48EI), and cantilevers with an end load (M = PL, δ = PL³/3EI) or a uniform load (M = wL²/2, δ = wL⁴/8EI).

Can I add the results of several load cases?

Yes. For linear elastic beams, reactions, shear, moment and deflection from separate loads add together (superposition). Add the functions along the beam rather than the maxima, since the maxima can occur at different points.

Are these the same formulas as the AISC beam diagrams?

They are the standard elastic beam results tabulated in the AISC Manual (Table 3-23), the FE Reference Handbook and mechanics of materials textbooks. Each page states where x is measured from, since tables differ.

What if my beam is not one of these cases?

Use the beam calculator, which solves any combination of supports, point loads, distributed loads and moments with the stiffness method.

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