Beam Formula Calculator

Shear force and bending moment diagrams

This beam calculator draws the shear force diagram (SFD), the bending moment diagram (BMD) and the deflected shape for every case, and you can read a value off any of them by hovering at a position along the span. The three diagrams are not independent: shear is the slope of the moment diagram, and the moment at any point equals the area under the shear diagram up to it. That relationship is the fastest way to check a diagram by eye.

A worked check on the default case below: the shear diagram runs from +30 kN at the left support down to -30 kN at the right, crossing zero at midspan. The triangle it forms over the left half has area 0.5 x 30 x 3 = 45 kNm, which is exactly the peak bending moment the tool reports. If those two numbers do not agree in your own case, one of the inputs is wrong.

Where the maximum moment occurs

The bending moment reaches a local maximum where the shear force passes through zero, because shear is the derivative of moment. For a symmetric load that is midspan; for an unsymmetric or point-load case it is wherever the shear diagram crosses the axis, which is generally not the middle of the beam.

The tool reports the location of each extreme value, not just its magnitude, so you get the position of maximum moment, maximum shear and maximum deflection directly. That matters for detailing: the peak moment position is where you need the largest section or the most reinforcement, and it moves as the load pattern changes.

Sign conventions in the output

Sagging bending moment is positive and hogging is negative, so the moment diagram of a simply supported beam sits entirely on the positive side while a cantilever or a continuous beam over an interior support goes negative. Shear is positive where the left-hand portion of the beam is pushed upward relative to the right.

This matters when reading a continuous or fixed-end case: the support moment is negative and usually larger in magnitude than the span moment, so the governing design moment is often the hogging one at the support rather than the sagging one at midspan. The diagram makes the sign explicit rather than reporting an unsigned peak.

Reactions

Reactions are solved before the diagrams are drawn, and reported per support. For a determinate case (simply supported, cantilever, overhanging) they follow from statics alone. For an indeterminate case (fixed-end, propped cantilever, continuous) the tool solves the compatibility condition as well, so the reactions already account for the redundancy rather than assuming a determinate distribution.

On the default case the two reactions come out at 30 kN each, which is the total load w x L = 60 kN split equally, as symmetry requires. Checking that the reactions sum to the applied load is the first thing worth doing on any beam result.

Worked example - simply supported beam, 6 m span, 10 kN/m

A simply supported steel beam spanning 6 m carries a uniformly distributed load of 10 kN/m. The section has I = 8500 cm4 and E = 210 GPa. These are the values the calculator opens with, so every figure below is reproducible without entering anything.

Reactions R_A = R_B = wL/2

10 x 6 / 2 = 30.0 kN

Maximum shear, at the supports

30.0 kN

Maximum moment, M = wL^2/8

10 x 6^2 / 8 = 45.0 kNm at midspan

Maximum deflection, 5wL^4/384EI

9.45 mm at midspan

Span-to-deflection ratio

L / 635

Diagram values along the span, as the tool plots them:

Diagram values along the span

x = 0 (left support)

V = +30.0 kN, M = 0 kNm, deflection 0 mm

x = 1.5 m (quarter point)

V = +15.0 kN, M = 33.75 kNm, deflection 6.74 mm

x = 3.0 m (midspan)

V = 0 kN, M = 45.0 kNm, deflection 9.45 mm

x = 4.5 m (quarter point)

V = -15.0 kN, M = 33.75 kNm, deflection 6.74 mm

x = 6.0 m (right support)

V = -30.0 kN, M = 0 kNm, deflection 0 mm

Three things to notice, all of which are general and not specific to this case. Shear is zero exactly where the moment peaks. The moment is zero at both pinned ends, because a pin cannot carry moment. And the moment at the quarter point is 33.75 kNm, which is 75 per cent of the peak even though it is only a quarter of the way along - the parabolic moment diagram is flat near midspan, which is why a small shift in load position changes the peak moment very little.

The deflection of 9.45 mm is span over 635. Against a common serviceability limit of span over 360 for a floor beam this passes comfortably; against span over 1000 for a beam supporting brittle finishes it would not. The tool reports the ratio so the comparison is direct.

Formula reference

Simply supported, UDL - moment

$M_{\max} = \dfrac{w\,L^{2}}{8}$

Simply supported, UDL - deflection

$\delta_{\max} = \dfrac{5\,w\,L^{4}}{384\,EI}$

Simply supported, central point load

$M_{\max} = \dfrac{P\,L}{4}$

Simply supported, point load - deflection

$\delta_{\max} = \dfrac{P\,L^{3}}{48\,EI}$

Cantilever, UDL - moment

$M_{\max} = \dfrac{w\,L^{2}}{2}$

Cantilever, end point load

$M_{\max} = P\,L$

Cantilever, end load - deflection

$\delta_{\max} = \dfrac{P\,L^{3}}{3\,EI}$

Fixed both ends, UDL - support moment

$M_{support} = \dfrac{w\,L^{2}}{12}$

Shear-moment relationship

$V = \dfrac{dM}{dx}$

These are the closed-form results for the standard cases; the tool evaluates them (and the indeterminate cases, which have no single-line form) and reports the position of each extreme value as well as its magnitude. The theory page derives each expression.

Assumptions and limits

All results are linear-elastic Euler-Bernoulli beam theory: constant EI along the span, small deflections, plane sections remaining plane, and material within the elastic range. Shear deformation, warping and second-order (P-delta) effects are not included.

The beam is assumed laterally restrained, so no lateral-torsional buckling check is implied by these numbers - a bending moment the beam can carry in theory may still require a slenderness check under EN 1993-1-1. Torsion, axial load and combined-action effects are outside the scope: this tool solves in-plane bending only.

Deflections are elastic and instantaneous. For concrete they exclude creep and shrinkage, which typically multiply the long-term value substantially, and for timber they exclude the load-duration and moisture effects that EN 1995 requires.

Frequently asked questions (FAQ)