Simply Supported Beam Deflection Calculator

Calculate maximum deflection for simply supported beams under various loading conditions.

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Beam Deflection Depends Strongly on Span and Bending Stiffness

A simply supported beam resists transverse loading through bending, and its deflection is governed by the product EI called flexural rigidity. Young’s modulus E describes material stiffness while the second moment of area I describes how the cross-section distributes material about the neutral axis. Because common deflection formulas contain high powers of span L, increasing beam length can increase deflection dramatically even when load and section remain unchanged.

The exact coefficient depends on load placement and distribution. A center point load gives δmax=PL3/(48EI), while a uniform load over the full span gives δmax=5wL4/(384EI). These Euler-Bernoulli formulas assume small deflections, linear elasticity, and slender beams where shear deformation is negligible.

Center load: δmax=PL3/(48EI);   UDL: δmax=5wL4/(384EI)
SymbolMeaningWhy it appears / units
PPoint loadN; applied at the specified location.
wDistributed loadN/m; load per unit beam length.
EYoung’s modulusPa; material stiffness.
ISecond moment of aream⁴; geometric bending stiffness.

Deflection and bending stress are related but not interchangeable design checks. A beam can remain below yield stress yet deflect too much for serviceability, alignment, or vibration requirements.

Support conditions provide two immediate deflection checks. An ideal simply supported beam has zero vertical deflection at both supports, while its slope there need not be zero. For symmetric loading, the maximum deflection should occur at the center and the slope should vanish there.

Worked Examples

Example 1: Floor joist: P=3000N at centre, L=5m, E=200GPa, I=500cm⁴
δ=3000×125/(48×200e9×5e-6)
Result: 7.8 mm
L/5000=640 — acceptable for structural floor
Example 2: UDL w=2000N/m, L=4m, E=70GPa, I=200cm⁴
δ=5×2000×256/(384×70e9×2e-6)
Result: 23.8 mm
⚠️ L/168 — check serviceability limit
Example 3: Center point load
P=1000N, L=2m, E=200GPa, I=8×10−6m⁴
Result: δmax≈0.104mm
The cubic span dependence makes length a major influence.
Example 4: Effect of doubling span
Same center load, E, and I; L doubles
Result: deflection increases by 8×
For a center point load, δ is proportional to L³.

Common Mistakes

⚠️
Using the wrong loading formula

Point load, uniform load, end moment, and off-center load cases have different deflection expressions.

⚠️
Mixing millimeters and meters in I or L

Because I uses length to the fourth power and deflection uses L³ or L⁴, unit errors can be enormous.

⚠️
Assuming the formula remains accurate for deep beams or large deflection

Euler-Bernoulli small-deflection theory may need shear-deformation or geometric-nonlinearity corrections outside its assumptions.

Frequently Asked Questions

Deflection vs strength?
A beam can be strong (doesn't break) but not stiff (deflects too much). Structural design checks BOTH. Strength: σ=Mc/I < allowable. Stiffness: δ < L/360 (or whatever code requires).
Ways to reduce deflection?
Increase I (add depth — most effective since I∝h³). Increase E (choose stiffer material). Reduce span (most effective — δ∝L³ or L⁴). Add intermediate supports.
Why does increasing beam depth reduce deflection so much?
For a rectangular section, I=bh³/12. Increasing depth h therefore raises I with the cube of depth, strongly increasing flexural rigidity EI.
Where is maximum deflection for a center-loaded simply supported beam?
By symmetry it occurs at midspan, the same location as the point load. Other loading patterns can place the maximum elsewhere.
Does a stiffer material always solve deflection problems?
Higher E reduces deflection, but changing cross-section can be even more effective because I is highly sensitive to geometry. Weight, stress, buckling, and cost also matter.
What is the difference between deflection and slope?
Deflection is transverse displacement of the beam centerline. Slope is the local rotation of that centerline and is the spatial derivative of deflection in small-deflection beam theory.
Why does the load position matter for a simply supported beam?
Beam reactions, bending moment, slope, and deflection depend on where the load is applied. The familiar center-load maximum-deflection formula does not apply unchanged to an off-center point load or a distributed load. Select the matching load case before substituting E, I, span, and load.

Formula Explorer connections

Interpretation: This engineering-physics relationship connects load, material property, geometry, deformation or system response. Assumption: Confirm material linearity, geometry, support conditions and safety convention. Small deformation, elastic behavior and ideal loading are common assumptions.

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