Beam Deflection Calculator
Estimate maximum beam deflection or required section stiffness for common load cases using classic Euler–Bernoulli formulas.
Structural engineering · Serviceability checks
Beam Deflection Calculator
This guide explains the equations behind the Beam Deflection Calculator, how to choose the right load case, and how to interpret deflection limits like \(L/240\) and \(L/360\) so your beams feel solid and serviceable in real life.
Quick Start: Using the Beam Deflection Calculator Safely
The Beam Deflection Calculator above is built around classic Euler–Bernoulli closed-form solutions. It is perfect for quick sizing, sensitivity checks, and teaching examples, as long as you feed it realistic inputs and stay within its assumptions.
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Step 1
Pick what you want to solve for. Choose Maximum deflection \(\delta_{max}\) if you already know the section properties, or Required moment of inertia \(I_{req}\) if you are trying to size a beam to stay under a deflection limit.
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Step 2
Select the load case that matches your beam. Use “Simply supported – uniform load” for a typical floor/roof beam with nearly constant tributary load, “Simply supported – center point load” for a single heavy point, or “Cantilever – end point load” for a balcony or canopy bracket.
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Step 3
Enter material stiffness \(E\) and section inertia \(I\). For structural steel, \(E \approx 200 \text{ GPa}\). For concrete, \(E\) is much lower (for example 25–35 GPa). Use section tables or section-property calculators to find \(I\) in mm\(^4\), in\(^4\), or m\(^4\).
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Step 4
Set the span and load in consistent units. Span \(L\) can be in metres or feet. Uniform load \(w\) can be entered as kN/m or lb/ft. Point load \(P\) can be in kN, kips, or lb. The calculator automatically converts everything to SI internally.
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Step 5
For sizing, provide an allowable deflection \(\delta_{allow}\). If you choose “Required moment of inertia”, enter a limit such as \(L/360\) converted to mm or inches. For example, a 6 m beam with an \(L/360\) limit has \(\delta_{allow} = 6000/360 \approx 17 \text{ mm}\).
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Step 6
Check the result and span/deflection ratio. The “Quick Stats” box reports deflection in both metric and imperial, plus the effective \(L/\delta\). Compare this to project criteria (e.g. \(L/240\), \(L/360\), or stricter for vibration-sensitive spans).
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Step 7
Use “Show Steps” to audit the math. The step-by-step breakdown shows the normalized units and the exact formula used for your load case, so you can verify the calculator matches your hand calculations or course notes.
Choosing Your Method: Which Load Case and Solve-For Option?
The same span, load and material can be analysed in slightly different ways depending on what you want from the Beam Deflection Calculator. This section compares the common workflows and when each one is the best fit.
Method 1 – Check deflection for a known section
Uses the classic formulas, eg. simply supported with uniform load: \[ \delta_{max} = \frac{5 w L^4}{384 E I} \]
When it shines
- Verifying a beam size taken from manufacturer tables.
- Quickly comparing steel vs LVL for the same span.
- Confirming that an existing beam is not overly bouncy.
Limitations
- Requires you to know \(I\) accurately.
- Does not directly suggest a better size if the result fails.
Method 2 – Solve for required inertia \(I_{req}\)
Rearranges the same formula, e.g. for uniform load: \[ I_{req} = \frac{5 w L^4}{384 E\,\delta_{allow}} \]
When it shines
- Early sizing for new projects with deflection limits already specified.
- Helps you target a minimum stiffness before picking a specific section type.
- Great for side-by-side span/limit comparisons during concept design.
Limitations
- Still need section tables or software to map \(I_{req}\) to a real shape.
- Uses a single governing load case; complex patterns may need several runs.
Method 3 – Combine with code-based L/ratio limits
Translate span-to-deflection limits like \(L/240\) into numeric \(\delta_{allow}\): \[ \delta_{allow} = \frac{L}{\text{limit}} \]
When it shines
- Aligns calculator results with common building code checks.
- Makes “bounciness” intuitive for clients and non-engineers.
- Easy to compare several deflection limits for the same span.
Limitations
- Codes may specify different limits for finishes, partitions, and total load cases.
- Dynamic vibration and human comfort are not captured by static deflection alone.
What Moves the Number: Key Drivers of Beam Deflection
Beam deflection is extremely sensitive to a handful of variables. Understanding these “levers” helps you decide whether to change span, load, material, or section size when the calculator output looks uncomfortable.
Deflection grows with \(L^3\) or \(L^4\) depending on the load case. Doubling the span can increase deflection by a factor of 8–16. Shortening spans or adding intermediate supports is often the most powerful lever.
For linear-elastic beams, deflection is proportional to the applied load. Doubling \(w\) or \(P\) doubles \(\delta_{max}\). Realistically modelling dead vs live loads, load combinations, and reduction factors is critical.
Steel is roughly 3–6 times stiffer than typical timber and significantly stiffer than cracked reinforced concrete. Switching from wood to steel may reduce deflection dramatically without changing the overall depth.
Making a beam “taller” increases \(I\) disproportionately. For a rectangular section \(I = b h^3 / 12\), so increasing depth is usually more effective than increasing width.
Cantilevers deflect far more for the same span, load, and stiffness because the fixed end must resist both moment and rotation. The calculator’s formulas explicitly account for these differences in boundary conditions.
A uniform load distributes bending along the span, whereas a single point load concentrates deflection near midspan or the free end. When your real loading is more complex, it is common to idealize the worst-case pattern into one of the supported cases.
Deflection is usually checked under serviceability combinations (often lower than ultimate strength combinations). Be sure the loads you input here match your code’s service load combination, not factored ultimate loads unless specifically required.
Real concrete and composite steel–concrete beams may crack or behave nonlinearly, reducing effective stiffness. The calculator assumes a constant, uncracked \(E I\), so you may need to use an “effective” \(E\) or \(I\) from your design code.
Worked Examples: From Inputs to Span–Deflection Ratios
The examples below mirror the Beam Deflection Calculator’s logic so you can follow the math line-by-line and confirm that the equations and units match your expectations.
Example 1 – Simply Supported Steel Beam with Uniform Load
- Span: \(L = 6 \text{ m}\)
- Material: structural steel, \(E = 200 \text{ GPa}\)
- Section inertia: \(I = 8.0 \times 10^8 \text{ mm}^4\)
- Uniform load: \(w = 20 \text{ kN/m}\) (including self-weight)
- Load case: simply supported, uniform load over full span
Normalize to SI units
Convert \(E\) and \(I\) to Pascals and m\(^4\):
\[ E = 200 \text{ GPa} = 200 \times 10^9 \text{ Pa} \] \[ I = 8.0 \times 10^8 \text{ mm}^4 = 8.0 \times 10^8 \times 10^{-12} \text{ m}^4 = 8.0 \times 10^{-4} \text{ m}^4 \]
Apply the uniform load formula
For a simply supported beam with uniform load:
\[ \delta_{max} = \frac{5 w L^4}{384 E I} \]
Here \(w = 20 \text{ kN/m} = 20\,000 \text{ N/m}\) and \(L = 6 \text{ m}\).
Compute deflection
\[ \delta_{max} = \frac{5 \times 20\,000 \times 6^4}{384 \times 200 \times 10^9 \times 8.0 \times 10^{-4}} \]
\[ 6^4 = 1296,\quad \text{numerator} \approx 5 \times 20\,000 \times 1296 = 129{,}600{,}000 \] \[ \text{denominator} = 384 \times 200 \times 10^9 \times 8.0 \times 10^{-4} \approx 6.144 \times 10^{10} \] \[ \delta_{max} \approx \frac{1.296 \times 10^8}{6.144 \times 10^{10}} \approx 2.11 \times 10^{-3} \text{ m} = 2.1 \text{ mm} \]
Check span–deflection ratio
\[ \frac{L}{\delta_{max}} = \frac{6000 \text{ mm}}{2.1 \text{ mm}} \approx 2857 \] This is significantly stiffer than an \(L/360\) limit, so deflection is unlikely to govern.
Example 2 – Cantilever with End Point Load, Sizing for \(I_{req}\)
- Cantilever span: \(L = 2.0 \text{ m}\)
- End load: \(P = 8 \text{ kN}\)
- Material: structural steel, \(E = 200 \text{ GPa}\)
- Allowable tip deflection: \(\delta_{allow} = 10 \text{ mm}\)
- Load case: cantilever, end point load
Convert inputs to SI
\[ E = 200 \text{ GPa} = 200 \times 10^9 \text{ Pa} \] \[ L = 2.0 \text{ m}, \quad P = 8 \text{ kN} = 8\,000 \text{ N} \] \[ \delta_{allow} = 10 \text{ mm} = 0.010 \text{ m} \]
Use the end-loaded cantilever formula
Maximum deflection at the free end of a cantilever with end load:
\[ \delta_{max} = \frac{P L^3}{3 E I} \]
Rearranging to solve for \(I_{req}\):
\[ I_{req} = \frac{P L^3}{3 E\,\delta_{allow}} \]
Calculate required inertia
\[ I_{req} = \frac{8\,000 \times 2.0^3}{3 \times 200 \times 10^9 \times 0.010} = \frac{8\,000 \times 8}{6 \times 10^9} = \frac{64\,000}{6 \times 10^9} \approx 1.07 \times 10^{-5} \text{ m}^4 \]
In mm\(^4\): \[ I_{req} = 1.07 \times 10^{-5} \times 10^{12} \approx 1.07 \times 10^{7} \text{ mm}^4 \]
Choose a real section
You would now pick a steel section with \(I \ge 1.07 \times 10^{7} \text{ mm}^4\) about the relevant axis. If you select a section with larger \(I\), the calculator can be run in “Maximum deflection” mode to see the resulting, usually smaller, tip deflection.
Common Layouts & Variations
Real projects rarely match textbook examples perfectly, but many everyday cases can be idealized into the calculator’s “simply supported” or “cantilever” models. The table below summarizes typical layouts and how to approximate them.
| Use case | Approximate model | Typical material & \(E\) | Service deflection limit | Notes |
|---|---|---|---|---|
| Interior floor beam supporting joists | Simply supported, uniform load | Steel, \(E \approx 200 \text{ GPa}\) | \(L/360\) for finishes, often stricter for sensitive partitions | Include self-weight and superimposed dead load in \(w\); live load from code tables. |
| Roof purlin under metal sheeting | Simply supported, uniform load | Cold-formed steel, \(E \approx 200 \text{ GPa}\) | \(L/180\)–\(L/240\) depending on cladding requirements | Wind uplift can be modelled with negative \(w\) if needed; check both gravity and uplift. |
| Timber joist supporting light residential floor | Simply supported, uniform load | Softwood, \(E \approx 8\text{–}12 \text{ GPa}\) | \(L/360\) or stricter to avoid “bouncy” perception | Use duration-adjusted \(E\) if required by timber design codes. |
| Balcony or canopy bracket | Cantilever, end point load | Steel or aluminium | \(L/180\)–\(L/240\) at the tip | Occupant comfort and ponding sensitivity may require tighter limits. |
| Overhanging beam with small backspan | Cantilever segment idealized as separate cantilever | Steel or concrete | Project-specific; often align with floor limits | Use reaction from backspan analysis as the effective end load \(P\). |
| Simple pipe support or cable tray support | Simply supported, uniform load | Light-gauge steel or aluminium | \(L/200\)–\(L/300\) | Check manufacturer span tables; use calculator to verify intermediate spans or unusual loads. |
Specs, Logistics & Sanity Checks
Once the Beam Deflection Calculator suggests a span/size combination that looks reasonable, there are still practical checks to run before you lock in a member size or send drawings out the door.
Before you trust the number
- Confirm that the load case (uniform vs point) is a good representation of reality.
- Verify that the loads correspond to serviceability combinations, not ultimate factored loads.
- Check that you used the correct modulus \(E\) for the material and code provisions (cracked vs uncracked).
- Compare the resulting span/deflection ratio with at least two limits (e.g. \(L/240\) and \(L/360\)).
Coordinating with suppliers
- Match the required \(I_{req}\) with actual sections available from local suppliers.
- Check stock lengths, splice locations, and camber options if deflection is tight.
- Confirm that the beam depth is compatible with architectural clearances and services.
- For proprietary systems, cross-check your deflection with the manufacturer’s span tables.
Field sanity checks
- Be cautious when spans or loads are far outside “normal” ranges from your practice.
- Watch for obvious red flags: unusually shallow beams, extreme live loads, or missing load paths.
- After construction, visible sag or vibration complaints may indicate that stiffness, not strength, is governing.
- Use the calculator as a diagnostic tool when troubleshooting serviceability issues in existing structures.
Frequently Asked Questions
What is beam deflection and why does it matter?
Beam deflection is the vertical displacement of a beam under load, typically measured at midspan or at the free end of a cantilever. Even when strength checks pass, excessive deflection can cause cracking of finishes, misaligned doors, ponding of water, and a “bouncy” feel that occupants dislike. Codes therefore set serviceability limits, often expressed as a span-to-deflection ratio like \(L/360\), which the Beam Deflection Calculator helps you evaluate.
Which deflection limit (L/240 vs L/360, etc.) should I use?
The correct deflection limit depends on your design code, occupancy and the type of element. Floor beams supporting brittle finishes or partitions often use \(L/360\) or stricter, while roof members without ceilings may allow \(L/240\) or \(L/180\). The calculator itself does not enforce a limit—it reports the span and computed deflection so you can compare to the code-specified ratio for your situation.
Can the Beam Deflection Calculator handle multiple loads or complex patterns?
The Beam Deflection Calculator focuses on the most common textbook cases: uniform load on a simple span, a point load at midspan, and an end-loaded cantilever. For multiple loads, you can often approximate the worst case by combining them into an equivalent uniform load or a single governing point load. For more complex or continuous systems, you should use frame-analysis or finite-element software and treat the calculator as a quick sense-check rather than the final word.
How accurate is this Beam Deflection Calculator compared with FEA?
For prismatic beams that follow the underlying assumptions—constant \(E I\), small deflections, linear elastic behaviour—the closed-form solutions used here typically match well-set-up finite-element models to within a few percent. Larger differences usually come from modelling choices, such as partial fixity at supports, composite action, or cracked sections, rather than from the calculator itself.
What value of modulus of elasticity E should I enter?
Use the design value of \(E\) specified in your material code or manufacturer data. For steel, a common value is \(E = 200 \text{ GPa}\) (or \(29{,}000 \text{ ksi}\)). For timber and concrete, codes may require adjustment for duration of load, moisture, or cracking. If you are checking an existing cracked concrete beam, consider using an effective \(E\) or \(I\) based on code guidance rather than the uncracked value.
Does the calculator account for composite action or cracked concrete?
No. The Beam Deflection Calculator assumes a constant, linear-elastic stiffness \(E I\) along the span. Composite steel–concrete beams, post-tensioned members, and cracked reinforced concrete sections often have different stiffness in tension and compression or change stiffness as loads increase. In those cases you should obtain an effective stiffness from your design standard or detailed analysis and enter that as an equivalent \(E\) or \(I\) for approximate checks only.
Can I use the Beam Deflection Calculator to sign off a final design?
The calculator is intended as an educational and preliminary design tool. It can highlight trends, compare options, and help you sanity-check spans and deflection limits, but it does not replace a full design in accordance with the relevant structural code. Always perform detailed strength, stability, and connection design before issuing drawings or accepting responsibility for a structure.