Beam Deflection Calculator

What this calculator does

Find out roughly how far a beam will sag under a load. Every beam bends a little, and that bend is called deflection. How much depends on four things. There's the load and the span (the distance between supports). There's the stiffness of the material, called its Young's modulus, E. And there's the shape of the beam's cross-section, measured by its second moment of area, I.

Deep beams resist bending far better than shallow ones. That's because I grows with the cube of the depth: double the depth and I goes up 8 times. It's why joists stand on edge.

The calculator uses the classic formulas for four common cases. A beam can be simply supported, resting on a support at each end. Or it can be a cantilever, fixed at one end and free at the other. Each can carry a single point load or a load spread evenly along the span.

You'll get the maximum deflection, the maximum bending moment (the bending effect that drives stress in the beam) and the support reactions. You'll also get the span-to-deflection ratio. It's for learning and rough estimates. Sizing real joists, beams, headers or brackets is a job for a licensed engineer or the span tables your building code adopts. The local code and inspector govern any structural work.

Where it's used

Why this math matters outside the classroom:

• Engineering homework. A student checks that a 6 m steel beam with I = 8,000 cm⁴ under 10 kN/m sags 5wL⁴ ÷ 384EI = 10.55 mm.
• Shelf design. A woodworker estimates how far a 1 m shelf sags under books before choosing a thicker board.
• Comparing sections. A designer sees a 2×8 on edge has 23 times the I of the same board laid flat, so it bends far less.
• Cantilever brackets. A fabricator estimates the tip deflection of a 2 m steel arm with a 1,000 N end load as PL³ ÷ 3EI = 1.33 mm.
• Lab experiments. Physics students hang weights at the center of a ruler and compare the sag with P·L³ ÷ 48·E·I.

How to use it

1. Choose a simply supported beam or a cantilever, and a point load or a uniform load.
2. Enter the span, then the load. That's either a point load and where it sits, or a load per unit length.
3. Pick a material to use its typical Young's modulus, or enter E yourself.
4. Describe the cross-section as a rectangle or round bar, or enter I directly. Then read the deflection and moment.

What the inputs mean

Load position

For a simply supported beam, 50% puts the load at midspan (the middle), the worst place for deflection. For a cantilever, 100% puts it at the free tip.

Uniform load (w)

Load per unit length, spread evenly over the whole span. Include the beam's own weight if it matters.

Material (Young's modulus E)

The preset stiffness values are typical textbook figures: steel 200 GPa, aluminum 70 GPa and wood 13 GPa along the grain. Real lumber varies a lot with species, grade and moisture. Graded lumber has a published E for its grade. Steel is commonly taken as about 29,000 ksi.

Depth (h), in the direction of the load

The depth is the measurement in the direction the load pushes. A 2×8 joist on edge has b = 1.5 in and h = 7.25 in. Laid flat, those swap, and it becomes about 23 times more flexible.

Second moment of area (I)

Use I about the line the beam bends around. Steel shape tables list it as Ix for the strong direction. Make sure the unit matches: in⁴, cm⁴, mm⁴ or m⁴.

Formula and method

Simply supported, point load P at a (b = L − a):  M = P·a·b ÷ L,  δmax = P·c·(L² − c²)^1.5 ÷ (9√3·E·I·L), c = smaller of a and b
Simply supported, centered point load:  M = P·L ÷ 4,  δmax = P·L³ ÷ (48·E·I)
Simply supported, uniform load w:  M = w·L² ÷ 8,  δmax = 5·w·L⁴ ÷ (384·E·I)
Cantilever, point load P at a from the fixed end:  M = P·a,  δtip = P·a²·(3L − a) ÷ (6·E·I)
Cantilever, uniform load w:  M = w·L² ÷ 2,  δtip = w·L⁴ ÷ (8·E·I)
Rectangle: I = b·h³ ÷ 12     Solid circle: I = π·d⁴ ÷ 64

δ (delta) is the deflection and M is the bending moment. P is a point load and w is an even load per unit length. L is the span, and a is how far the point load sits from the left support (or the fixed end). b is the rest of the span, and c is the smaller of a and b. E is the material's stiffness and I is the cross-section's second moment of area. For the shapes, b is the width, h the depth and d the diameter.

These are the standard results of Euler–Bernoulli beam theory. It assumes small deflections and a straight beam of one material with the same cross-section all along. The calculator converts everything to metric (SI) units, meters, newtons and pascals, before using the formulas. Then it converts the answer back.

Deflection grows with the cube of the span for a point load, and the fourth power for an even load. So doubling a span makes a beam 8 to 16 times bendier. Deflection shrinks in proportion to E and I. And I for a rectangle grows with the cube of its depth.

Worked example

A 1.5 × 7.25 in wood beam (a 2×8 on edge) spans 10 ft between two supports and carries 1,000 lbf at midspan, with E taken as 13 GPa.

1. I = 1.5 × 7.25³ ÷ 12 = 47.63 in⁴.
2. E = 13 GPa, which is about 1,885,000 psi.
3. Maximum moment: P × L ÷ 4 = 1,000 × 10 ÷ 4 = 2,500 lbf·ft.
4. Maximum deflection: P × L³ ÷ (48 × E × I) = 1,000 × 120³ ÷ (48 × 1,885,491 × 47.63) = 0.4008 in.

Result: The beam sags about 0.4008 in at midspan, roughly L/299, with a 2,500 lbf·ft maximum bending moment. That's only a deflection estimate. It says nothing about whether the beam is strong enough.

Understanding the result

Span-to-deflection ratios are a common way to judge stiffness. L/360 means the sag is 1/360 of the span. Building codes set limits such as L/360 or L/240, depending on the part and the load. Compare against the limit your code and engineer specify, not a number from here.

Deflection is only half the story. The bending moment drives the stress in the beam: stress = M × distance from the neutral axis ÷ I. (The neutral axis is the line through the beam that neither stretches nor squeezes.) A stiff beam can still be too weak. It can also fail in shear (sliding apart), bearing (crushing at the supports) or buckling.

Assumptions and limitations

• The material springs back fully once the load is removed, and stretches in proportion to the load. Deflections are small. The beam is straight with the same cross-section all along. Loads push in the same direction the beam bends.
• Supports are ideal. Simple supports are free to rotate, and a cantilever's fixed end is perfectly rigid. The beam's own weight isn't included unless you add it to the uniform load.
• Not for designing structural members. Joists, rafters, headers, beams and connections must be sized by a licensed engineer or from code span tables. Local codes and the inspector govern.
• It leaves out shear deformation, lateral-torsional buckling (the beam twisting sideways), creep in wood (slow sagging over time) and long-term deflection. It also leaves out combined load cases and continuous beams over several supports.
• Preset E values are typical, not design values. Lumber grades and alloys differ.

Frequently asked questions

Sources

• HyperPhysics: Elasticity, Elastic Properties (Georgia State University) (checked 2026-10-19)

How this calculator is checked

Method: Standard Euler–Bernoulli beam formulas for simply supported and cantilever beams under point and uniform loads; I = bh³/12 and πd⁴/64; typical E values from HyperPhysics (Georgia State University). The calculation is covered by automated tests with known reference values and edge cases. Last updated 2026-10-18. Found a problem? Tell us.

Embed this calculator

Put this Beam Deflection Calculator on your own website or blog for free. Paste this code where you want it to appear: