Beam Deflection Calculator estimates how much a beam bends under a load. It computes the maximum deflection using standard beam formulas for common boundary conditions (simply supported or cantilever) and common load types.
You enter the beam length, material stiffness (Young’s modulus), cross-section moment of inertia, and load details. The calculator returns deflection in your chosen units and highlights the exact formula used.
What “beam deflection” means
Deflection is the downward (or sideways) displacement of a beam from its original shape. Engineers use deflection limits to prevent serviceability problems like cracked finishes, misaligned doors, or excessive vibration.
Deflection depends on three big factors:
- Load (how big the force is, and where it acts)
- Beam stiffness (material Young’s modulus, E, and geometry moment of inertia, I)
- Support condition (how the beam is held at the ends)
Core formula behind the calculator
Most closed-form deflection equations come from Euler–Bernoulli beam theory. They share the same stiffness term: E·I, where:
- E is Young’s modulus (Pa)
- I is the second moment of area (m4) about the bending axis
For a given load case and support condition, the deflection equation is typically a constant times a load term divided by E·I, multiplied by a power of the span L.
Variables you’ll enter
| Symbol | Meaning | Typical units (calculator default) |
|---|---|---|
| L | Beam length (span) | m |
| E | Young’s modulus | GPa |
| I | Second moment of area | cm4 |
| w | Uniformly distributed load (UDL) | kN/m |
| P | Point load | kN |
| a | Distance from the left support to the point load | m |
Support types and load cases used
The calculator covers two common boundary conditions and two common load types. It computes the maximum deflection based on standard textbook formulas.
1) Simply supported beam (pinned–roller)
The beam can rotate at the supports, and deflection is zero at each support.
- UDL w over full span: maximum deflection occurs at midspan.
- Point load P at distance a from left: maximum deflection is not always at midspan; the formula used accounts for the position.
2) Cantilever beam (fixed–free)
The fixed end prevents rotation and deflection. The free end experiences the maximum deflection.
- UDL w over full span: maximum deflection occurs at the free end.
- Point load P at distance a from the fixed end: maximum deflection occurs at the free end; the formula uses the load position.
Formulas used by the calculator (maximum deflection)
The calculator automatically converts units into SI internally, applies the selected formula, then converts the deflection into your chosen output unit.
Simply supported, UDL over full span
δmax = 5 w L4 / (384 E I)
Where w is in force per length, and L is the span.
Simply supported, point load at distance a from left
δmax = P a (L − a)(L2 − a(L − a)) / (3 E I L)
This expression accounts for the load position a and produces the maximum deflection for that location.
Cantilever, UDL over full span
δmax = w L4 / (8 E I)
Deflection is largest at the free end.
Cantilever, point load at distance a from fixed end
δmax = P a2 (3L − a)2 / (24 E I)
In this case, a is measured from the fixed end to the point load. The free end is where maximum deflection occurs.
How to use the calculator (step-by-step)
- Select support type: simply supported or cantilever.
- Select load type: UDL (uniform) or point load.
- Enter geometry: span length L.
- Enter stiffness: E and I.
- Enter load details: w (for UDL) or P and a (for point load).
- Pick output units (mm, cm, or inches).
- Click Calculate to get the maximum deflection.
Practical examples
Example 1: Simply supported steel beam under a uniform load
A simply supported beam of length L = 4 m carries a uniform load w = 6 kN/m. The steel has E = 200 GPa and the beam section has I = 200 cm4.
The calculator uses δmax = 5 w L4 / (384 E I) to estimate the peak midspan deflection. If the result is too large, you can increase stiffness (larger section, higher I) or reduce load.
Example 2: Cantilever beam with a point load
A cantilever beam of length L = 2.5 m is fixed at one end. A point load of P = 3 kN is applied at a = 1.2 m from the fixed end. Use E = 70 GPa and I = 90 cm4
The calculator applies the cantilever point-load formula and returns the maximum deflection at the free end. This helps you check serviceability without doing a full structural analysis.
Common mistakes to avoid
- Using the wrong moment of inertia: I must be about the bending axis.
- Mixing units: the calculator handles conversions, but you must still enter consistent values (e.g., L in meters).
- Forgetting load position limits: for point loads, a must be within the beam length.
- Assuming these formulas fit every scenario: they are for idealized, uniform loading and standard boundary conditions.
Frequently Asked Questions
What is the Beam Deflection Calculator used for?
The Beam Deflection Calculator estimates maximum deflection of a beam under common loads. It uses Euler–Bernoulli beam theory and standard closed-form formulas for simply supported and cantilever beams. You input span length, Young’s modulus, moment of inertia, and load details, then it returns deflection in your chosen units.
How do I find the moment of inertia (I) for my beam?
Moment of inertia depends on the beam cross-section shape and the bending axis. For common shapes (rectangles, I-beams, channels), tables or CAD/structural software provide I values. If you compute it yourself, use the correct axis and units, since deflection scales directly with 1/I.
Why does deflection increase with beam length?
For the supported beam cases used here, deflection grows with the fourth power of span (L4). That means doubling the length increases deflection by about 16 times. This strong sensitivity is why long spans require stiffer sections, higher I, or reduced loads.
Can I use these results for design approval?
These calculations provide serviceability estimates for idealized conditions. Real structures may include load combinations, supports that don’t match ideal boundary conditions, material nonlinearity, shear deformation, and deflection from multiple load cases. For final design, follow your local code and have a qualified engineer verify assumptions.
What deflection limit should I use?
Deflection limits are code- and application-specific. Common rules use a fraction of span (for example, L/360 or L/480) for floors and beams. Your project may require tighter limits for finishes, cladding, or vibration concerns. Use the limit required by your applicable standard.