What sign convention does the calculator use?

The usual textbook convention: the shear V(x) is the sum of the upward forces to the left of the section, and a sagging moment (bottom fibre in tension) is positive. Loads are entered as positive downward and point moments are positive counter-clockwise. With this convention a simply supported beam under downward loads has a positive moment, while a cantilever has a negative (hogging) moment near the fixed end.

Why is the bending moment largest where the shear is zero?

Because dM/dx = V: the shear is the slope of the moment diagram. Where the shear changes from positive to negative, the moment stops rising and starts falling, so it peaks there. Shear can also jump straight across zero at a point load, in which case the peak moment is at that load. The tool lists both kinds of location.

How is the deflection calculated and how accurate is it?

It uses Euler–Bernoulli beam theory, EI·y″ = M(x). The moment divided by EI is integrated twice along the beam (4000 steps, trapezoid rule with a curvature correction), and the constants come from the supports: y = 0 at both supports of a simple beam, and y = 0 with zero slope at a fixed end. For the standard textbook cases the result agrees with the closed-form formulas such as 5wL⁴/384EI and PL³/3EI to about 0.01%.

Can it solve continuous or fixed-fixed beams?

No. The tool only handles statically determinate beams, where the reactions follow from equilibrium alone: simply supported, overhanging and cantilever beams. Continuous beams and beams fixed at both ends are statically indeterminate and need compatibility conditions (the force method, moment distribution and so on), which are not implemented here.

How is the second moment of area of an I-beam found?

Treat the I-beam as a b × h rectangle minus the two empty strips beside the web: I = [bh³ − (b − tw)(h − 2tf)³]/12, where b is the flange width, h the overall depth, tf the flange thickness and tw the web thickness. The maximum bending stress is σ = M·c/I with c = h/2. I/c is the section modulus, which steel section tables list directly, so you can check the value there.

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🏗️Beam Calculator - Shear Force & Bending Moment Diagrams, Deflection

An online tool for drawing shear force and bending moment diagrams. Set the beam length and supports (simple, overhanging or cantilever), then add and drag point loads, uniform loads, triangular or trapezoidal loads and point moments. You immediately get the support reactions, the shear and moment diagrams, the location of the maximum moment and the zero-shear points. Choose a material and a cross-section (rectangle, circle, tube, box or I-beam) to also get the deflected shape, the maximum deflection and the maximum bending stress.

Free · No sign-up · Runs in your browserLast updated: 2026-10-29

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How to use Beam Shear & Bending Moment Calculator

1Load a textbook example, or set the length and supports and drag the supports to make overhangs
2Add point loads, UDLs, triangular loads or couples, and drag them on the beam or type positions and values
3Read the shear, moment and deflection diagrams; move the cursor over them to read values at any section
4Pick a material and section size to get I, maximum deflection and maximum bending stress

FAQ

What sign convention does the calculator use?: The usual textbook convention: the shear V(x) is the sum of the upward forces to the left of the section, and a sagging moment (bottom fibre in tension) is positive. Loads are entered as positive downward and point moments are positive counter-clockwise. With this convention a simply supported beam under downward loads has a positive moment, while a cantilever has a negative (hogging) moment near the fixed end.
Why is the bending moment largest where the shear is zero?: Because dM/dx = V: the shear is the slope of the moment diagram. Where the shear changes from positive to negative, the moment stops rising and starts falling, so it peaks there. Shear can also jump straight across zero at a point load, in which case the peak moment is at that load. The tool lists both kinds of location.
How is the deflection calculated and how accurate is it?: It uses Euler–Bernoulli beam theory, EI·y″ = M(x). The moment divided by EI is integrated twice along the beam (4000 steps, trapezoid rule with a curvature correction), and the constants come from the supports: y = 0 at both supports of a simple beam, and y = 0 with zero slope at a fixed end. For the standard textbook cases the result agrees with the closed-form formulas such as 5wL⁴/384EI and PL³/3EI to about 0.01%.
Can it solve continuous or fixed-fixed beams?: No. The tool only handles statically determinate beams, where the reactions follow from equilibrium alone: simply supported, overhanging and cantilever beams. Continuous beams and beams fixed at both ends are statically indeterminate and need compatibility conditions (the force method, moment distribution and so on), which are not implemented here.
How is the second moment of area of an I-beam found?: Treat the I-beam as a b × h rectangle minus the two empty strips beside the web: I = [bh³ − (b − tw)(h − 2tf)³]/12, where b is the flange width, h the overall depth, tf the flange thickness and tw the web thickness. The maximum bending stress is σ = M·c/I with c = h/2. I/c is the section modulus, which steel section tables list directly, so you can check the value there.

Features

Simply supported, overhanging and left- or right-fixed cantilever beams with draggable supports
Point loads, uniform loads, linearly varying (triangular, trapezoidal) loads and point moments in any combination
Support reactions with the equilibrium equations, max sagging and hogging moment, max shear and zero-shear points
Deflection curve by numerical double integration, with the maximum deflection and span/deflection ratio
Steel, aluminium, timber and concrete with five common sections: I, section modulus and bending stress
Eight classic textbook cases to check against the standard formulas

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