Shear and Moment Diagram Calculator

Every shear force diagram (SFD) and bending moment diagram (BMD) is built with the same four-move routine. The calculator above automates it — and its step-by-step mode replays each move on the real numbers of the case you selected, so you can learn the method on the exact beam you care about.

1. Find the support reactions. Draw the free body of the whole beam and apply global equilibrium: ΣFy = 0 and ΣM = 0. For a simply supported beam, take moments about one support to get the other reaction directly. (For a fixed-fixed beam this is not enough — see the FAQ on indeterminate beams.)
2. Cut the beam region by region. A new region starts at every support, point load, and start/end of a distributed load. Inside one region nothing changes, so ONE pair of equations covers it.
3. Write V(x) and M(x) for each region. Keep the left free body and sum: V(x) equals the upward forces to the left of the cut; M(x) equals their moments about the cut. That gives the exact polynomials this page prints under "Segment equations".
4. Plot and close. Point loads and reactions create vertical jumps in V equal to their magnitude; distributed load makes V slope down at rate w. The moment curve is one degree higher than the shear in every region. Past the last support, both diagrams must return to zero — if they don't, a reaction is wrong. This "closure check" catches most exam mistakes.

Where the shear crosses zero, the bending moment is at a local peak — that is usually the section that governs the flexural design of the member.

Two differential identities generate every rule of thumb used to sketch diagrams. With w(x) the downward load intensity:

dV/dx = −w(x) — the slope of the shear diagram equals minus the load intensity.

dM/dx = V(x) — the slope of the moment diagram equals the shear ordinate.

From them, the shape rules:

Two consequences worth memorizing: M is extreme where V = 0 (because dM/dx = 0 there), and the change in M between two sections equals the area under the V diagram between them. You can verify both on any preset above — switch to step-by-step mode and watch the equations confirm it region by region.

This calculator uses the classic strength-of-materials convention, the one used by Hibbeler, Beer & Johnston, and most design codes:

• Positive shear V: the resultant of the forces on the left free body acts upward (the pair tends to rotate the element clockwise).
• Positive moment M: sagging — the beam smiles, compression on the top fibre, tension on the bottom. Hogging moments (cantilever over a wall, moment over the middle support of an overhanging beam) are negative.
• Applied loads P and w are entered as positive-down magnitudes; the solver handles the signs internally.

Some European and Brazilian schools plot the BMD on the tension side — positive moments drawn downward. The numbers are identical; only the plotting direction flips. If your textbook draws sagging moments below the axis, mentally mirror the BMD above (or compare the signed values at the labeled critical points, which never change).

On the diagrams above, positive regions are shaded green and negative regions red, so the convention is visible at a glance — a detail that matters when you check moment transfer at the supports of continuous or overhanging beams.

Every case in the preset bar has a textbook closed form. The calculator reproduces all of them exactly (that is part of its automated verification), and stays exact when you change L, P, w or the positions — which the formulas below cannot do for you once the loading gets mixed.

The overhanging beam is the one students get wrong most often: the moment is negative over the interior support, positive in the span, and crosses zero at the point of contraflexure — the calculator marks that point explicitly on the BMD, since it is where you would splice reinforcement or check lateral bracing.

Real beams rarely match a textbook card, so the Custom case builder accepts any combination the classic tables cannot: up to 6 point loads anywhere on the span, up to 4 distributed loads that may be partial (covering only part of the span) and trapezoidal (different intensities at each end), and up to 4 concentrated applied moments — on simply supported, cantilever, fixed-fixed or overhanging support layouts.

Two behaviors worth knowing:

• Applied moments follow the clockwise-positive convention: a clockwise moment M₀ makes the BMD jump UP by +M₀ as you traverse it left to right, while the SFD does not change at all (a pure couple carries no net vertical force). The printed segment equations treat the moment exactly; the reactions come from the same stiffness-method engine as everything else.
• Every result stays exact and auditable. The per-segment polynomials, the critical points, the step-by-step equilibrium narration and the CSV export all work for custom cases exactly as they do for the presets — including statically indeterminate layouts like fixed-fixed, which statics-only tools cannot solve at all.

When you are done, the Share link button encodes the whole configuration in the URL, so a colleague (or your students) can open the identical beam with one click — no account, no file.

A bending moment diagram is a means, not an end — the number you actually need is a steel section. Below the diagrams, this page closes that loop:

1. Required section modulus. From the governing moment Md = max(|Mmax|, |Mmin|), the elastic requirement is Wx,req = Md / (fy/γa1) with γa1 = 1.10 per NBR 8800 (the AISC 360 LRFD factor φb = 0.90 lands within a few percent). Pick fy for your steel — A36, A572 Gr.50 or S355.
2. Catalog sweep. The ~1,309-profile CalcSteel catalog (W/HP, channels, tubes — ASTM, EN and NBR series) is swept for the three lightest sections with Wx ≥ Wx,req, each shown with its weight and bending utilization, and each linking to its profile page where the full NBR 8800 / AISC 360 checks (LTB, shear, deflection) run on the real section properties.
3. One click to a real model. "Open in 3D editor" recreates the exact beam you configured — span, supports, every point load, every distributed load — in the CalcSteel 3D structural editor, with the suggested profile already assigned, ready for load combinations, code checks and IFC/DXF export. The diagram you were studying becomes a design you can finish.

This is the difference between a formula card and a tool that sits inside a real structural product: the numbers keep living after the diagram.

• Diagram does not close. If V or M is not zero after the last support, a reaction is wrong. Recompute ΣM about a support. The step-by-step mode's final step performs this check for you.
• Missing the jump at a point load. V must drop by exactly P at the load. If your jump differs, you mixed up the left/right free body.
• Peak moment at the wrong place. The maximum of M is NOT always at midspan — for the off-center point load it sits under the load; for the triangular load at x = L/√3 ≈ 0.577L. Find where V = 0 first.
• Sign error on the overhang. Hogging over the support is negative. If your BMD is all positive on an overhanging beam, the sign convention slipped.
• Treating a fixed-fixed beam as determinate. Two clamped ends give 4 unknowns and only 2 equilibrium equations — statics alone cannot solve it. This page runs a real stiffness-method solve, which is why its fixed-fixed preset shows the correct −wL²/12 end moments instead of a wrong "wL²/8".
• Forgetting units. V carries force units (kN, kip); M carries force×length (kN·m, kip·ft). The SI/imperial toggle above converts the equations' coefficients too, so you can hand equations to a colleague in either system without redoing algebra.