Beam Deflection Calculator

Calculate maximum deflection for simply supported and cantilever beams with common load cases.

Calculator

Deflection Formulas

Simply Supported — Center Load

δ = FL³ / (48EI)    M = FL / 4

Simply Supported — Uniform Load

δ = 5wL⁴ / (384EI)    M = wL² / 8

Cantilever — End Load

δ = FL³ / (3EI)    M = FL

Cantilever — Uniform Load

δ = wL⁴ / (8EI)    M = wL² / 2

How to Use

  1. 1
    Define Beam Geometry and Support Conditions

    Enter the beam span, cross-section dimensions (width and height for rectangular; diameter for circular), and select the support condition: simply supported, cantilever, or fixed-fixed.

  2. 2
    Specify Load Type and Magnitude

    Choose the loading: concentrated point load (with location), uniformly distributed load (UDL), or combined loading; enter the load magnitude in Newtons or pounds-force.

  3. 3
    Read Deflection, Bending Moment, and Stress Results

    The calculator outputs maximum deflection, location of maximum deflection, peak bending moment, maximum bending stress, and compares these against material yield strength and any specified deflection limits.

About

Beam deflection analysis underpins the structural design of everything from machine tool frames and conveyor systems to building floor joists and aircraft wing spars. While finite element analysis handles complex geometries, closed-form beam theory solutions from Euler-Bernoulli beam theory provide fast, accurate answers for standard geometries and loading conditions that cover the vast majority of practical engineering problems.

The AlloyFYI Beam Deflection Calculator implements the standard formulae for the most common boundary conditions and load configurations, automatically computing the section's second moment of area from user-supplied dimensions. By coupling the structural calculation directly to the material database, engineers can instantly see how material substitution — switching from structural steel to 6061-T6 aluminum, for example — affects deflection (which will increase due to lower elastic modulus) and check whether the alloy's yield strength provides adequate margin against the calculated bending stress.

FAQ

What is the difference between bending stress and shear stress in beams?
In a loaded beam, bending stress (σ = M×y/I, where M is bending moment, y is distance from neutral axis, and I is second moment of area) acts horizontally — tension on one face, compression on the other — and is maximum at the extreme fibers. Shear stress (τ = V×Q/(I×b), where V is shear force, Q is first moment of area, and b is section width) acts vertically, is maximum at the neutral axis, and is zero at the extreme fibers. For slender beams (span-to-depth ratio above 10), bending governs design; for deep beams or short cantilevers, shear stress can be critical.
How do I calculate the second moment of area for a hollow rectangular section?
The second moment of area (moment of inertia) for a hollow rectangular section about the neutral axis is I = (BH³ − bh³)/12, where B and H are the outer width and height, and b and h are the inner (void) width and height. Hollow sections are structurally efficient because material is concentrated at the extreme fibers where bending stress is highest, while the web carries shear. This is why I-beams and hollow structural sections (HSS) are preferred over solid rectangles for structural applications where weight minimization is important.
What deflection limits apply to structural beams?
Building codes typically limit live load deflection to span/360 for floor beams (to prevent visible sagging under service load) and total deflection (dead plus live load) to span/240. For crane runway girders and precision equipment support beams, more stringent limits — span/600 to span/1000 — may be specified. These limits are primarily serviceability requirements to prevent cracking of plaster ceilings, misalignment of doors, and aesthetic concerns; the structure may be structurally safe at much higher deflections but fail serviceability requirements at lower loads.
How does moment of inertia scaling affect beam stiffness?
Beam stiffness (and thus deflection resistance) scales with the fourth power of the section depth for equal-width rectangular cross sections, because I = bh³/12. Doubling the depth of a rectangular beam reduces deflection by a factor of eight (since deflection ∝ 1/I). This extremely strong dependence on depth explains why deep, slender beams and I-sections are so structurally efficient: increasing depth is far more effective per unit mass than increasing width, up to the limit where lateral-torsional buckling of the compression flange becomes the governing failure mode.
What is the Euler buckling load and when does it apply?
The Euler critical (buckling) load for a slender column is P_cr = π²EI/(KL)², where E is elastic modulus, I is the minimum second moment of area, L is the column length, and K is the effective length factor (1.0 for pinned-pinned, 0.5 for fixed-fixed, 2.0 for cantilever). Euler buckling is the relevant failure mode only for slender columns (slenderness ratio KL/r above approximately 120 for steel), where elastic instability occurs before yielding. For stocky columns with low slenderness, inelastic buckling or direct compression governs, and the Johnson parabola or Rankine-Gordon formula should be used instead.
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