ENGINEERING CALCULATOR
Beam Deflection Calculator
Compute how far a beam bends under load before you spend a cent on manufacturing. Pick the support condition, enter load, span, and material — get deflection, bending moment, and stress in seconds, with the formula shown so you can trace every number.
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Free Online Tool
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Deflection & Stress Results
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Allowable Deflection Check
Beam Deflection Calculator
Select beam type and load case, enter span and load to calculate deflection.
Max Deflection δ
0.45 mm
Max Moment M
25,000 N·mm
Max Stress σ
52.1 MPa
Formula
δ = F·L³ / (3·E·I)
Engineering Calculator
Beam Deflection Calculator
Select beam type, load case, material, and cross‑section, then press Calculate to see deflection, stress, and pass/fail status.
N (point) or N/mm (uniform)
Maximum Deflection δ
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Maximum Bending Moment M
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Maximum Bending Stress σ
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Span / Deflection Ratio
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Deflection Check
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Formula
Enter values and press Calculate.
HOW IT WORKS
How to Calculate Beam Deflection
There is no single deflection formula — the correct one depends on support conditions and load type. Match your structure to a load case, then apply the appropriate equation.
Cantilever End Load
δ = F·L³ / 3EI
M = F·L
Deflection scales with L³
Simply Supported Center Load
δ = F·L³ / 48EI
M = F·L / 4
Deflects ~4× more than fixed-fixed
Fixed-Fixed Center Load
δ = F·L³ / 192EI
M = F·L / 8
Stiffest common configuration
Worked Example
A steel cantilever arm: 250 mm long, 40 × 12 mm rectangular section, carrying 100 N at the free end. Steel E ≈ 200,000 MPa.
I = 5,760 mm⁴
b·h³/12
→
δ = 0.45 mm
F·L³/3EI
→
M = 25,000 N·mm
F·L
Steel Deflection
0.45 mm
Aluminum 6061
1.31 mm
Difference
2.9×
Three Levers That Control Beam Deflection
01
Span
Deflection scales with L³ for point loads and L⁴ for distributed loads. Double the span = 8–16× more deflection.
Shorten the span first
02
Moment of Inertia
Deflection is inversely proportional to I. Double the depth of a rectangle = 8× less deflection.
I = b·h³/12
03
Material Stiffness
Higher E = less deflection. Steel (200 GPa) vs aluminum (69 GPa) ≈ 3× difference — the weakest lever of the three.
E from 68.9 to 200 GPaBeam Types & Typical Deflection
Cantilever
Fixed at one end, free at the other — deflects the most. Common for brackets and sensor arms. δ = F·L³/3EI.
Simply Supported
Pivoted at both ends, free to rotate — deflects ~4× more than fixed-fixed. Standard model for beams on supports. δ = F·L³/48EI.
Fixed-Fixed
Built in at both ends, no rotation — stiffest common configuration. Weld or clamp at both ends. δ = F·L³/192EI.
MACHINE DESIGN APPLICATIONS
Why Deflection Matters in Machined Parts
Machined parts carry load more often than their shape suggests — brackets, fixture plates, sensor arms, and guide rails all deflect, and that deflection lands directly on accuracy.
A sensor bracket that moves 0.2 mm under load or a tooling arm that sags a fraction of a millimeter changes the position of whatever it holds. For vision systems, drill bushings, or probing fixtures, that difference separates a part in tolerance from a scrap part.
Deflection & Precision
Every machined part that carries load deflects. The question is whether the deflection stays within the tolerance budget. Deflection checks appear constantly in precision equipment design, and tolerance stacks always include member flexibility.
- Sensor brackets under cable pull
- Tooling arms and fixture plates
- Linear guide rails and support frames
Stiffness & Design Review
When a deflection check flags a part as too flexible, the fix is usually a design change — a deeper section, a rib, or a gusset. That is exactly the kind of improvement our engineers suggest during DFM review, before you spend money on a part that will not hold position.
- Deepen the section for 8× stiffness gain
- Add ribs and webs away from neutral axis
- Convert cantilevers with gussets or supports
Machining & Stiffness
Baetro machines on 5‑axis centers to ±0.001″ and produces the complex brackets and frames where stiffness matters most. Upload your CAD file and see your price and lead time in about 60 seconds, with free manufacturability feedback.
- 5‑axis CNC machining to ±0.001″
- 50+ metals & plastics, ISO 9001 & AS9100
- Free DFM feedback on every quote
Beam Deflection Questions
Beam Deflection FAQs
Beam deflection is the distance a beam bends from its straight, unloaded position under load. It depends on the load, the span, the material’s modulus of elasticity (E), and the cross-section’s moment of inertia (I). Excessive deflection is a serviceability problem, sagging, binding, or misalignment, even when the beam is far from failing in strength.
Use the closed-form formula that matches your support condition and load case. For a cantilever with an end load, δ = FL³/(3EI). For a simply supported beam with a center load, δ = FL³/(48EI). The general method, used when no closed-form formula fits, is to double-integrate the bending moment equation M(x)/EI and apply the boundary conditions.
For a point load at the free end, δ = FL³/(3EI), with a maximum bending moment of FL at the fixed end. For a full-span uniform load, δ = wL⁴/(8EI), with a maximum moment of wL²/2.
For a point load at mid-span, δ = FL³/(48EI), with a maximum moment of FL/4. For a uniform load, δ = 5wL⁴/(384EI), with a maximum moment of wL²/8.
Common serviceability limits are L/360 for floor beams under live load, L/240 for total load, L/180 for cantilevers and roof members, and tighter limits of L/480 to L/1000 for machine frames and precision equipment. On a 3,000 mm span, L/360 allows 8.3 mm and L/1000 allows just 3.0 mm.
Deflection scales with the cube of the span for point loads and the fourth power for distributed loads. Doubling the span increases deflection by 8 to 16 times, which is why unsupported length is the first thing to attack in a stiffness problem.
Shorten the unsupported span, deepen or widen the cross-section to raise the moment of inertia, add ribs or gussets, add a support to change the load path, and only then switch to a stiffer material. For a rectangle, doubling the depth cuts deflection eightfold.
Aluminum 6061 has about one third the stiffness of steel (E ≈ 69 versus 200 GPa), so an identical beam in aluminum deflects roughly three times as far. Titanium sits between them at about 57% of steel’s stiffness. For most machined parts, section geometry affects deflection more than a material change.
Use the alloy’s published modulus: steel ≈ 200 GPa, 304/316 stainless ≈ 193 GPa, Ti-6Al-4V ≈ 113.8 GPa, aluminum 6061 ≈ 68.9 GPa, 7075 ≈ 71.7 GPa, brass ≈ 100 GPa, PEEK ≈ 3.6 GPa, nylon ≈ 2.4–3.5 GPa. The material’s datasheet is authoritative.
Most often it is a long unsupported span, a thin or narrow section with a low moment of inertia, or a lighter-than-necessary material. The fix is usually geometry, a deeper section, a rib, or a gusset, not a material change. Our engineers flag exactly these issues in the free DFM feedback you get with every quote.
Use finite element analysis for non-prismatic parts, complex supports, combined or dynamic loads, stress concentrations, or when the deflection lands near your application’s tolerance. The beam deflection calculator is a fast first-pass stiffness check; FEA is the confirmation for a critical design.
PRECISION MANUFACTURING SUPPORT
From Deflection Check to Machined Part
You have a deflection number. Now you need the part that holds it. Whether this beam deflection calculator confirmed your design or flagged an arm that flexes too far, the next step is the same: turn the geometry into a machined part that meets your stiffness requirement.
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5‑axis CNC machining to ±0.001″
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50+ metals & plastics, ISO 9001 & AS9100
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No minimum order, standard parts in 3–7 days