Structural Engineering · Beam Deflection Formula

Beam Deflection Formula – How to Calculate Beam Sag from Load, Span, and Stiffness

Learn how to use the beam deflection formula to calculate sag for common beam cases, including simply supported beams, cantilevers, point loads, and uniformly distributed loads, with practical serviceability checks and worked structural examples.

Read time \( \delta_{\max} = \dfrac{P L^3}{48 E I} \) Euler–Bernoulli beam theory Stiffness & serviceability

What the beam deflection formula means and when to use it

Core formula for a common case

\[ \delta_{\max} = \frac{P L^3}{48 E I} \]

For a simply supported beam with a single point load at midspan, maximum deflection increases with load and span, and decreases with material stiffness \(E\) and section stiffness \(I\).

Use this when you need to:

  • estimate how much a beam, joist, girder, or cantilever will sag under service load
  • check deflection limits such as \(L/240\), \(L/360\), or \(L/480\)
  • compare beam sizes or materials based on stiffness rather than strength alone
  • screen structural options before moving to more detailed analysis or FEA

Most readers want this first: the beam deflection formula depends heavily on span. In many common cases, deflection scales with \(L^3\) or \(L^4\), which means even a modest span increase can cause a dramatic increase in sag. That is why long-span serviceability often controls beam selection before bending stress does.

The phrase “beam deflection formula” usually refers to a family of standard equations from Euler–Bernoulli beam theory, not just one formula. The exact expression depends on the support condition and load pattern. A simply supported beam with a centered point load uses one equation, a simply supported beam with a full-span uniformly distributed load uses another, and a cantilever with an end load uses another again.

In real structural design, deflection is a serviceability issue. A beam may be safe in bending stress and shear but still feel too flexible, crack finishes, misalign doors, create ponding, or look visibly sagged. That is why beam deflection calculations are often just as important as strength checks in floors, roofs, balconies, and framing systems.

Editorial note: this page focuses on the standard closed-form elastic formulas most engineers and students use first. These equations are powerful for quick checks, but they assume small deflections, linear material behavior, and idealized supports and loading.

Simply supported beam with a central point load P, span L, and an exaggerated elastic curve showing maximum midspan deflection delta max
A classic beam deflection case: a simply supported beam with a point load at midspan. The largest elastic deflection occurs at midspan and is \( \delta_{\max} = \dfrac{P L^3}{48 E I} \).

Beam deflection formula variables, symbols, and units

Beam deflection formulas use the same core stiffness idea repeatedly: the beam becomes less flexible when \(E\) is larger or when \(I\) is larger. Load magnitude and span length work the other way, increasing deflection.

Common notation

SymbolMeaningTypical unitWhat it represents
\( \delta(x) \)deflection at position \(x\)m or mmThe vertical displacement of the beam along the span.
\( \delta_{\max} \)maximum deflectionm or mmThe largest magnitude of beam sag for the chosen load case.
\( P \)point loadN or kNA concentrated load applied at one location.
\( w \)uniformly distributed loadN/m or kN/mA constant load per unit length along some or all of the beam.
\( L \)span lengthm or mmThe distance between supports, or from fixed end to free end for a cantilever.
\( E \)Young’s modulusPa, MPa, or GPaThe material stiffness. Larger \(E\) means a stiffer beam response.
\( I \)second moment of aream⁴ or mm⁴The section property that controls resistance to bending deformation.
\( M(x) \)bending momentN·m or kN·mThe internal bending action that creates curvature and deflection.

Unit and usage notes

  • Use a fully consistent unit system. Deflection errors of 1,000× are common when \(E\), \(I\), and load units are mixed.
  • Most structural calculations are easiest in N and mm or in N and m, but do not mix the two carelessly.
  • \(I\) is about the bending axis, so using the wrong axis can produce a completely wrong deflection result.
  • A beam with high strength can still deflect too much if its stiffness \(EI\) is too low.
  • For quick section-property work, use the Moment of Inertia Calculator to estimate \(I\) before plugging into a beam deflection formula.

For multiple loads, custom support conditions, or beam diagrams beyond textbook formulas, use the Beam Calculator to explore deflection, reactions, and internal forces more directly.

How the beam deflection formula works in practice

Beam deflection formulas come from elastic beam theory. The governing idea is that curvature is proportional to bending moment divided by flexural stiffness \(EI\). Once the bending moment function is known, integrating it with the correct boundary conditions gives the beam’s slope and deflected shape.

Method 1: Start with the Euler–Bernoulli beam relationship

For slender elastic beams under small deflection, the curvature relationship is written as:

\[ \frac{d^2 y(x)}{dx^2} = \frac{M(x)}{E I} \]

Here \(y(x)\) is the deflection shape, \(M(x)\) is the bending moment along the beam, \(E\) is the elastic modulus, and \(I\) is the second moment of area. A larger bending moment increases curvature, while a larger \(EI\) reduces it.

Method 2: Use the standard closed-form case that matches the beam

In design work, you rarely integrate from scratch unless the loading is unusual. Instead, you match the beam and loading case to a known closed-form solution. A few of the most used formulas are:

\[ \delta_{\max} = \frac{P L^3}{48 E I} \qquad \text{simply supported beam, point load at midspan} \] \[ \delta_{\max} = \frac{5 w L^4}{384 E I} \qquad \text{simply supported beam, full-span UDL} \] \[ \delta_{\max} = \frac{P L^3}{3 E I} \qquad \text{cantilever beam, end load} \]

These formulas are popular because they let you move quickly from load and stiffness to a serviceability check. The most important lesson is that support condition changes the coefficient dramatically, so choosing the wrong case can make the result badly wrong.

For many users, beam deflection is really about understanding what drives sag the most. Span is often the dominant variable. In common formulas, deflection grows with \(L^3\) or \(L^4\), which is why extending a beam span is often far more damaging to stiffness than a modest load increase. This is also why adding an intermediate support can reduce deflection so effectively.

Worked examples for beam deflection formulas

These examples cover the three most common search intents: a simply supported beam with a point load, a simply supported beam with a uniformly distributed load, and a cantilever with a tip load.

1

Example 1: Simply supported beam with a midspan point load

Scenario: A simply supported steel beam spans \(6.0\ \text{m}\) and carries a concentrated load of \(30\ \text{kN}\) at midspan. Use \(E=210\ \text{GPa}\) and \(I=8.5\times10^{-6}\ \text{m}^4\). Estimate maximum deflection.

\[ \delta_{\max} = \frac{P L^3}{48 E I} \]
\[ \delta_{\max} = \frac{30{,}000(6.0)^3}{48(210\times10^9)(8.5\times10^{-6})} \approx 0.0076\ \text{m} \]
\[ \delta_{\max} \approx 7.6\ \text{mm} \]

Steps:

  • Use the simply supported midspan point-load formula.
  • Convert load and modulus into consistent SI units.
  • Evaluate the result and convert metres to millimetres for interpretation.

Result: the beam deflects about 7.6 mm at midspan.

Interpretation: this is a serviceability-style result. The next step is usually comparing it against an allowable deflection limit such as \(L/360\).

2

Example 2: Simply supported beam with a uniform load

Scenario: A beam spans \(4.5\ \text{m}\) and carries a uniformly distributed load of \(10\ \text{kN/m}\). Use \(E=200\ \text{GPa}\) and \(I=1.2\times10^{-5}\ \text{m}^4\). Estimate maximum deflection.

\[ \delta_{\max} = \frac{5 w L^4}{384 E I} \]
\[ \delta_{\max} = \frac{5(10{,}000)(4.5)^4}{384(200\times10^9)(1.2\times10^{-5})} \approx 0.0045\ \text{m} \]
\[ \delta_{\max} \approx 4.5\ \text{mm} \]

Steps:

  • Choose the full-span UDL formula, not the point-load formula.
  • Keep load in N/m and stiffness in SI units.
  • Convert the final answer to mm for design review.

Result: the beam deflects about 4.5 mm.

Interpretation: this is a common floor-beam type calculation and shows how distributed loading is handled differently from a concentrated load case.

3

Example 3: Cantilever beam with an end load

Scenario: A cantilever balcony beam projects \(2.0\ \text{m}\) from a fixed support and carries \(12\ \text{kN}\) at the free end. Use \(E=210\ \text{GPa}\) and \(I=3.0\times10^{-6}\ \text{m}^4\). Estimate free-end deflection.

\[ \delta_{\max} = \frac{P L^3}{3 E I} \]
\[ \delta_{\max} = \frac{12{,}000(2.0)^3}{3(210\times10^9)(3.0\times10^{-6})} \approx 0.0051\ \text{m} \]
\[ \delta_{\max} \approx 5.1\ \text{mm} \]

Steps:

  • Use the cantilever tip-load equation, which has a much larger coefficient than the simply supported case.
  • Keep all terms in one consistent unit system.
  • Interpret the deflection at the free end, where serviceability is often most visible to users.

Result: the free end deflects about 5.1 mm.

Interpretation: cantilever stiffness matters strongly for comfort and appearance, so even modest millimetre-scale movement can matter in balcony or canopy design.

Common mistakes, assumptions, and engineering checks

Closed-form beam deflection equations are powerful because they are quick, but they only stay reliable when the right case and the right stiffness assumptions are used. Most mistakes happen before the math, not during it.

Choose the exact beam and load case

A beam deflection formula is only correct for the support condition and loading pattern it was derived for.

  • Do not use a simply supported formula for a cantilever or fixed beam.
  • Do not use a point-load equation when the beam is carrying a distributed load.
  • Be careful with off-center loads, partial-span UDLs, and multiple loads, since the standard one-line formulas may no longer apply.
Check stiffness assumptions before trusting the number

The elastic formulas assume constant \(E\) and \(I\), but many real sections or materials do not behave that ideally.

  • Composite beams may need transformed section properties.
  • Concrete and timber may need long-term or effective stiffness adjustments.
  • Cracking, creep, and connection flexibility can all increase real deflection beyond the short-term elastic estimate.
Compare against serviceability limits, not just strength

Many users search “beam deflection formula” because the design issue is sag or bounce, not failure. That means deflection ratio checks matter directly.

  • Compare \(\delta_{\max}\) against limits such as \(L/240\), \(L/360\), or \(L/480\) as appropriate.
  • Remember that a beam can pass bending stress but still fail serviceability.
  • If deflection is close to the limit, consider reducing span, increasing \(I\), or revising the structural layout rather than only increasing material strength.

Beam deflection formula FAQ

What is the beam deflection formula in simple terms?

It is an equation that estimates how much a beam will sag under load. The exact formula depends on the support condition and load pattern, but all standard forms relate deflection to load, span, material stiffness, and section stiffness.

What does \(EI\) mean in the beam deflection formula?

\(EI\) is the beam’s flexural stiffness. \(E\) is the material’s Young’s modulus, and \(I\) is the second moment of area about the bending axis. A larger \(EI\) means a stiffer beam and less deflection.

How do I know which beam deflection formula to use?

First identify the support condition, then identify the loading pattern. A simply supported beam, a cantilever, and a fixed-end beam all use different formulas, even if the span and load magnitude are the same.

Is beam deflection the same as beam stress?

No. Beam stress is a strength issue, while beam deflection is a stiffness and serviceability issue. A beam can be strong enough but still deflect too much for comfort, appearance, or code limits.

References and further reading

Scroll to Top