Beam Load Calculator — reactions, shear and bending moment

The beam load calculator above works out what a loaded beam is being asked to do: how much force each support carries, how much shear the beam sees at its worst point, and how large the bending moment gets and where. These are the demands. They are the inputs to sizing a member, and they are what a structural drawing is built on.

Arb Digital publishes this as a preliminary analysis and teaching tool. It is not a structural design, it is not stamped, and it does not replace a licensed engineer or architect. It solves statically determinate cases exactly, which means it is right about the arithmetic and says nothing about whether the beam you have in mind can carry the result. Treat it as a way to understand the numbers and to check somebody else's, not as a specification.

What This Beam Load Calculator Does

It handles a simply supported beam or a cantilever carrying a uniformly distributed load plus up to two point loads placed anywhere along the length. It returns the reaction at each support, the maximum shear force and where it occurs, the maximum bending moment and where it occurs, and — if you supply an allowable bending stress — the section modulus that moment would require.

It returns forces and moments, not displacements. If you want to know how far the beam sags, that is the beam deflection calculator, and the two answer genuinely different questions: a beam can be strong enough and still too bouncy, or stiff enough and still overstressed in shear. Once you have a required section modulus from this page, the section modulus calculator converts between section geometry and that property.

How to Use It

1. Pick the support condition. Simply supported means the beam rests on two supports that can rotate freely. A cantilever is fixed at one end and free at the other, and it behaves completely differently.
2. Enter the span. Centre to centre of the supports for a simply supported beam; the projection beyond the face of support for a cantilever.
3. Enter the uniform load per unit length. This is the tributary load the beam collects, and it must include the beam's own weight.
4. Add point loads and their positions. A post landing on the beam, a girder framing into it, a piece of equipment — each is a point load at a measured distance from the left end.
5. Read the location of the maximum, not only its magnitude. Where the peak moment occurs tells you where a splice must not go and where a notch would be most damaging.

The Formulas and How They Are Calculated

Reactions come from static equilibrium. For a simply supported beam of span L carrying a uniform load w and a point load P at distance a from the left support:

R_left = wL ÷ 2 + P(L − a) ÷ L and R_right = wL ÷ 2 + Pa ÷ L

The two reactions always sum to the total applied load — that is the check worth doing every time. For a cantilever, the single reaction is simply the whole load, wL + P, and the fixed end also resists a moment of wL²/2 + Pa.

Shear and moment are then found by walking along the beam. At a distance x from the left, the shear is the left reaction minus everything applied to the left of that point, and the bending moment is the sum of moments of everything to the left about that point:

V(x) = R_left − wx − (P if x > a) and M(x) = R_leftx − wx²/2 − P(x − a) for x > a

This tool evaluates both expressions at a thousand stations along the beam and reports the largest magnitude of each. That numerical approach is deliberate: closed-form maxima exist for single load cases but get awkward as soon as two point loads and a uniform load are combined, and sampling handles every combination without special cases.

A worked example. A 16 ft simply supported beam carries 200 lb/ft plus a 2,000 lb point load 6 ft from the left. The reactions are 200 × 16 ÷ 2 + 2,000 × 10 ÷ 16 = 1,600 + 1,250 = 2,850 lb on the left, and 1,600 + 750 = 2,350 lb on the right — summing to 5,200 lb, which equals 3,200 lb of uniform load plus the 2,000 lb point load. Maximum shear is 2,850 lb at the left support. The moment peaks under the point load at 2,850 × 6 − 100 × 36 = 17,100 − 3,600 = 13,500 lb-ft. At an allowable bending stress of 1,200 psi, that requires a section modulus of 13,500 × 12 ÷ 1,200 = 135 in³.

Why the Location of the Peak Moment Matters

Under a pure uniform load, the maximum moment is at midspan and equals wL²/8. Add a point load and the peak migrates towards it, and with a single point load the peak sits directly under the load. That location is practical information, not trivia.

It tells you where not to weaken the beam. A notch, a bolt hole, a service penetration or a splice at the point of maximum moment removes material exactly where the fibres are working hardest. Conversely, the points of zero moment — near the supports for a simply supported beam — are where a splice does least harm, which is the reasoning behind where connections are located in continuous framing. It also tells you where to look when something is deflecting or cracking: a crack on the underside at midspan is a bending story, a diagonal crack near a support is a shear story.

Shear Governs More Often Than People Expect

Maximum shear in a simply supported beam is at the supports and equals the reaction. Most of the time bending is the check that sizes the beam, but shear takes over in two situations that come up regularly.

The first is short, heavily loaded spans. Moment scales roughly with the square of span while shear scales linearly, so as the span shortens the moment falls away faster than the shear does. A stubby, heavily loaded beam can be comfortable in bending and critical in shear.

The second is timber with a notch at a support. Notching the underside of a beam at the bearing removes material precisely where the shear stress is highest and creates a re-entrant corner that concentrates stress at the notch root. Design standards treat notched bearings as a special case for exactly this reason, and the American Wood Council's National Design Specification for Wood Construction is where the provisions live for timber in the United States. Steel members have their own analogous problems — web crippling and local yielding under concentrated loads — covered by the standards published by the American Institute of Steel Construction.

Determinate Versus Indeterminate: What This Tool Cannot Do

Both cases here are statically determinate — the reactions can be found from equilibrium alone, without knowing anything about the beam's stiffness. Add a third support, fix both ends, or make the beam continuous over several spans, and that stops being true. The load distribution then depends on the relative stiffness of the members, and equilibrium is no longer enough to solve it.

The difference is not academic. A two-span continuous beam develops a negative moment over the interior support that can exceed the positive moment in the spans, and the interior support carries considerably more than half the total load — often around five-eighths of it for equal spans under uniform load. Modelling that beam as two separate simple spans understates the interior reaction and misses the negative moment entirely, which is exactly the moment that would tell you the top of the beam is in tension over the support. If your beam is continuous, this page is the wrong tool. The theory behind indeterminate analysis is set out in university mechanics courses such as MIT's Mechanics of Materials module series.

Tributary Width: Where the Uniform Load Comes From

The uniform load field asks for force per unit length, and getting that number is often harder than the analysis that follows. A beam supporting a floor collects load from half the distance to the framing on each side of it — its tributary width. A beam mid-way between two supports with joists spanning 12 ft on either side collects a 12 ft tributary width, not 24, because half of each joist's load goes the other way.

Multiply the floor's load per unit area by that tributary width to get the load per unit length, then add the beam's self-weight. Load per unit area itself splits into dead load, which is the permanent weight of the construction, and live load, which is the occupancy the code assigns to the space. They are combined according to load combination rules that differ between allowable stress design and strength design, and getting the combination wrong is a bigger error than anything in the arithmetic here. Roof beams add snow, which the snow load calculator deals with, and the members carrying the reaction down to the foundation are handled by the column buckling calculator and, for concrete, the concrete column calculator.

Required Section Modulus, and the Factor You Must Supply Yourself

The last output converts a bending moment into the section property that would carry it: S_required = M ÷ F_b, where F_b is the allowable bending stress. This is allowable stress design — the safety margin lives entirely inside the value of F_b that you enter.

This page applies no factor of its own and publishes no stress values, which is deliberate. Allowable bending stress for timber is a base design value adjusted by a chain of factors for load duration, moisture, temperature, size, repetitive member action and lateral stability, and every one of those depends on your situation. For steel the equivalent is derived from the yield strength and a resistance or safety factor set by the design method you are using. If you type a number into that field without knowing where it came from, the section modulus you get out is decoration. The factor of safety calculator covers the general relationship between capacity, demand and margin, and the wood beam span calculator and floor joist calculator take the prescriptive route for common residential members.

Common Mistakes to Avoid

• Leaving out the beam's self-weight — it is part of the uniform load, and on long-span steel or glulam members it is not a small part.
• Treating a continuous beam as two simple spans — that misses the negative moment over the interior support entirely and understates the interior reaction.
• Using the full joist span as tributary width — a beam collects half the span from each side, and doubling that is the most common load-takedown error.
• Checking bending and stopping — shear governs short heavily loaded spans and notched timber bearings, and bearing stress governs at narrow supports.
• Entering an allowable stress you cannot source — the entire safety margin in this calculation sits inside that one number.

Related Free Tools From Arb Digital

Pair this with the beam deflection calculator for the serviceability side, the section modulus calculator for section properties, the column buckling calculator for the supports, the snow load calculator for roof loading and the factor of safety calculator for margins. The floor joist calculator and wood beam span calculator cover prescriptive residential framing. The full free online tools hub lists every calculator we publish.

Frequently Asked Questions

This tool produces preliminary calculations for teaching and checking only. It is not a structural design, it is not stamped, and it does not replace a licensed engineer or architect. Load combinations, allowable stresses and member capacities are governed by the building code and design standard adopted in your jurisdiction, and local amendments differ by state, province and country.