What units should I use in the beam deflection calculator?

You can use any consistent set of units. For example, use kN, m, GPa, and mm⁴; or lb, in, ksi, and in⁴. As long as force, length, modulus of elasticity, and moment of inertia are all in a consistent system, the deflection result will be correct.

How do I check if beam deflection is acceptable?

Compare the maximum deflection to a code or project limit such as L/240, L/360, or L/480, where L is the span length. The calculator shows both the absolute deflection and the ratio L/δ so you can quickly see whether the beam meets common serviceability criteria.

Can this calculator handle multiple loads on a beam?

Yes. You can add multiple point loads and uniformly distributed loads along the same span. The calculator uses superposition to combine their effects and reports the maximum deflection along the beam.

What assumptions does this beam deflection calculator make?

The calculator assumes a prismatic, linearly elastic beam that obeys Euler–Bernoulli beam theory: small deflections, plane sections remain plane, constant E and I along the span, and static loading. For heavily cracked concrete, deep beams, or very large deflections, more advanced analysis may be required.

Beam Deflection Calculator

Calculate maximum deflection and deflection curves for simply supported and cantilever beams with multiple point and distributed loads.

Beam setup

Beam type

Cantilever: fixed at left, free at right.

Span length L

Modulus of elasticity E

Steel ≈ 200,000 MPa, concrete ≈ 25,000–35,000 MPa, timber ≈ 8,000–14,000 MPa.

Second moment of area I

For a rectangle: I = b·h³/12 about the strong axis.

Loads

Use any consistent force and length units (e.g., kN & m, or lb & ft). The calculator converts internally.

Deflection limit (optional)

Limit type

Common limits: L/240 (roof), L/360 (floor), L/480 (sensitive finishes). Uses the same length unit as the span.

Results

Maximum deflection

–

Enter data and click Calculate.

Deflection limit check

No limit selected.

Deflection at selected positions

Position x m

How this beam deflection calculator works

This tool uses classic Euler–Bernoulli beam theory and closed-form formulas to compute the deflection curve \(w(x)\) for prismatic beams. It supports:

Simply supported and cantilever beams
Multiple point loads at arbitrary positions
Multiple uniformly distributed loads (UDL) over arbitrary segments
Custom material stiffness \(E\) and section stiffness \(I\)

For each load case, the corresponding analytical deflection function is evaluated and the results are superposed (summed) to obtain the total deflection at any position along the span.

Key beam deflection formulas

The calculator internally uses standard formulas such as the following (downward deflection taken as positive).

Simply supported beam with central point load

Span \(L\), point load \(P\) at midspan \(x = L/2\).

Maximum deflection at midspan:

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

Simply supported beam with uniform load

Span \(L\), uniformly distributed load \(w\) (force per unit length) over entire span.

Maximum deflection at midspan:

\[ \delta_\text{max} = \frac{5 w L^4}{384 E I} \]

Cantilever beam with end point load

Span \(L\), point load \(P\) at free end.

Maximum deflection at the free end:

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

Cantilever beam with uniform load

Span \(L\), uniform load \(w\) over entire span.

Maximum deflection at the free end:

\[ \delta_\text{max} = \frac{w L^4}{8 E I} \]

Deflection limit criteria

Structural design codes usually limit deflection to control cracking, vibration, and visual appearance. A common way to express limits is as a span ratio:

\(L/240\) – typical for roofs without brittle finishes
\(L/360\) – typical for floors supporting partitions or finishes
\(L/480\) – for more stringent serviceability requirements

The calculator compares the computed maximum deflection \(\delta_\text{max}\) with the selected limit \(\delta_\text{allow} = L / N\) and reports whether the beam passes or fails.

Assumptions and limitations

Euler–Bernoulli beam theory (small deflections, linear elastic behavior)
Prismatic beam: constant \(E\) and \(I\) along the span
Static loading (no dynamic or impact effects)
No shear deformation (acceptable for slender beams; deep beams may need Timoshenko theory)

For reinforced concrete beams with cracking, composite sections, or complex boundary conditions, use this tool for preliminary sizing only and verify with detailed design codes or finite element analysis.

Worked example

Consider a simply supported steel beam with:

Span \(L = 6\ \text{m}\)
Uniform load \(w = 10\ \text{kN/m}\) (including self-weight)
Modulus of elasticity \(E = 200{,}000\ \text{MPa}\)
Second moment of area \(I = 8.0 \times 10^{-6}\ \text{m}^4\)

Maximum deflection at midspan:

\[ \delta_\text{max} = \frac{5 w L^4}{384 E I} \]

Convert \(w\) to N/m: \(10\ \text{kN/m} = 10{,}000\ \text{N/m}\).

\[ \delta_\text{max} = \frac{5 \cdot 10{,}000 \cdot 6^4}{384 \cdot 200{,}000 \cdot 8.0 \times 10^{-6}} \approx 14.1\ \text{mm} \]

If the deflection limit is \(L/360\), the allowable deflection is \(L/360 = 6000 / 360 \approx 16.7\ \text{mm}\), so the beam passes the limit.

Beam deflection – frequently asked questions

What is beam deflection?

Beam deflection is the displacement of a beam from its original, unloaded position due to applied loads. It is usually measured vertically and is a serviceability criterion: even if stresses are within allowable limits, excessive deflection can cause cracking, misalignment, or discomfort.

Which units can I use in this calculator?

You can use any consistent set of units. For example:

SI: kN, m, MPa, m⁴
Metric: kN, m, GPa, cm⁴
Imperial: lb, ft or in, ksi, in⁴

The calculator converts everything internally, so just ensure that all inputs belong to the same unit system (do not mix m with in, for example).

How accurate is this beam deflection calculator?

For slender, prismatic beams with small deflections, the results match textbook solutions and standard beam tables very closely. For deep beams, composite sections, or cracked concrete, the real behavior may deviate from Euler–Bernoulli theory, so treat the results as approximate and confirm with detailed design methods if needed.

Can I model multiple spans or overhangs?

This version focuses on single-span simply supported and cantilever beams. For continuous beams with multiple spans or overhangs, you typically need more advanced analysis (e.g., slope-deflection, moment distribution, or finite element software). You can sometimes approximate an overhang as a separate cantilever if the support moment is known.