Beam Deflection Calculator
Choose a beam and load case, enter inputs with units, and calculate the maximum deflection.
- Euler–Bernoulli beam theory for small deflections: deflection varies inversely with E·I.
- Standard closed-form max-deflection cases are used for simply supported and cantilever beams.
- The off-center simply supported point-load case evaluates the valid stationary point to get the true maximum deflection.
This beam deflection calculator estimates elastic displacement for common cantilever and simply supported beam cases. Choose the support and load pattern carefully, then enter span, load, elastic modulus, and second moment of area in a consistent unit system.
What Beam Deflection Means
Deflection is the distance a loaded beam moves from its original position. Engineers evaluate it because a member can remain below its material strength limit yet still sag enough to damage finishes, disturb equipment, change drainage, or feel uncomfortable.
The result here is a serviceability estimate for idealized conditions. It is not a complete structural design or proof of safety.
Inputs the Calculator Needs
- Span, L: the modeled distance between support points or from the fixed end to the free end.
- Point load, P: a concentrated force applied at the location shown.
- Distributed load, w: force per unit length, not the total force over the span.
- Elastic modulus, E: the material’s stiffness in the modeled direction.
- Second moment of area, I: a cross-section property about the bending axis.
Choose the Correct Support Condition
A cantilever is fixed against translation and rotation at one end and free at the other. A simply supported beam is modeled with supports that allow end rotation and do not transfer fixing moments. Real connections may behave somewhere between these ideals, so selecting the wrong case can dominate the error.
Also match the load position. A point load at midspan, a point load at the cantilever tip, and a uniformly distributed load use different equations.
Common Beam Deflection Formulas
For a cantilever with a point load at the free end: δmax = PL3 / (3EI)
For a cantilever with a uniform load over the full length: δmax = wL4 / (8EI)
For a simply supported beam with a point load at midspan: δmax = PL3 / (48EI)
For a simply supported beam with a uniform load over the full span: δmax = 5wL4 / (384EI)
These standard small-deflection expressions are summarized in university engineering references, including Iowa State beam-deflection formulae.
Why Span Has Such a Large Effect
Span is raised to the third power for many point-load cases and the fourth power for uniform-load cases. Doubling the span therefore increases the calculated deflection dramatically if everything else stays constant. Increasing E or I reduces deflection in direct proportion.
This sensitivity is why a small error in effective span can matter more than rounding the final answer.
Finding the Second Moment of Area
The second moment of area depends on cross-section shape and bending orientation. For a rectangle bending about its centroidal axis, I = bh3/12, where h is the dimension parallel to the deflection direction. Rotating a rectangular member can therefore change stiffness substantially.
For built-up, hollow, or manufacturer-specific sections, use verified section properties. Do not confuse the second moment of area with mass moment of inertia.
Keep Units Consistent
If force is in newtons, length is in millimeters, E can be entered in N/mm² and I in mm⁴ to obtain deflection in millimeters. In a US customary calculation, a consistent combination might use pounds, inches, psi, and in⁴. Mixing metres with millimetres or total distributed load with load per length produces misleading results.
Assumptions and Limits
Classic Euler–Bernoulli formulas assume a slender, prismatic beam, linear-elastic material, small deflection and rotation, and the ideal support/load arrangement shown. They commonly omit shear deformation, connection slip, cracking, creep, residual stress, vibration, local buckling, and second-order effects.
Deep beams, composites, tapered members, reinforced concrete, temperature effects, moving loads, or large deflections may require a different model. For structural decisions, have a qualified engineer check strength, stability, serviceability, load combinations, and applicable codes.
Beam Deflection Calculator FAQs
Is maximum deflection always at midspan?
No. It is at midspan for several symmetric simply supported cases, but at the free end for common cantilever cases. An off-center load can move the maximum location.
Does a negative deflection mean the calculation failed?
Usually not. The sign reflects the chosen coordinate convention and direction. Compare the magnitude and direction with the model’s diagram.
Can this calculator size a structural beam?
No. Deflection is only one design check. A complete design also considers stresses, shear, buckling, connections, bearing, vibration, durability, load factors, and governing code limits.
