Free calculator

Shear & Moment — Simply Supported Beam

Enter a span, a full-span uniform load, and up to two point loads. The calculator returns the reactions, the shear just left and right of every load point, the moment at every key station, and the maximum sagging and hogging moments with their locations.

R_A = wL/2 + Σ P_i(L − a_i)/L R_B = wL/2 + Σ P_ia_i/L

V(x) = R_A − wx − Σ P_i⟨x − a_i⟩⁰

M(x) = R_Ax − wx²/2 − Σ P_i⟨x − a_i⟩¹

w	uniform load over the full span, positive downward
P_i	point load i, positive downward
a_i	distance of point load i from the left support
⟨x − a⟩	Macaulay bracket: zero when x ≤ a, otherwise (x − a)

Sign convention: shear is positive when it acts upward on the left face of a segment; moment is positive when it causes sagging. A moment extreme occurs wherever the shear crosses zero inside a segment, or under a point load where the shear jumps across zero.

Unit system Span L (m) Uniform load w (kN/m, full span) Point load P1 (kN, 0 = none) P1 position a1 from left support (m) Point load P2 (kN, 0 = none) P2 position a2 from left support (m)

For study and checking only. These calculators run entirely in your browser — nothing is sent anywhere. They are meant to verify your hand calculations while you study; on the exam you will work by hand with the NCEES reference handbook.

Worked example: beam reactions, shear and moment

A simply supported beam spans L = 8.0 m with uniform load w = 6.0 kN/m over the full span, plus a point load P = 20 kN at a = 3.0 m from the left support.

Reactions: R_A = wL/2 + P(L − a)/L = 24.0 + 12.5 = 36.5 kN; R_B = 24.0 + 20(3.0)/8.0 = 31.5 kN (check: 36.5 + 31.5 = 68 = 6.0(8.0) + 20 ✓)
Moment under the point load: M(3) = 36.5(3.0) − 6.0(3.0)²/2 = 109.5 − 27.0 = 82.5 kN·m
Shear is positive everywhere left of the load and negative everywhere right of it — it never crosses zero inside a segment — so the moment peaks under the point load: M_max = 82.5 kN·m at 3.0 m from A

Exam tip: a point load jumps the shear diagram and only kinks the moment diagram; a point couple jumps the moment diagram, not the shear.

Go deeper

Preparing for exam day? The 110-question FE Civil practice exam is a full 5-hour-20-minute rehearsal with detailed solutions.