Beam Deflection Calculator
Calculate maximum deflection and slope for cantilever, simply supported, and fixed-fixed beams with section presets, material library, and serviceability code checks.
Simply Supported, Centre Point Load
Max deflection at mid-span
Computed I = 6.667e+3 cm⁴ = 6666.6667 cm⁴
What Is the Beam Deflection Calculator?
This calculator finds the maximum deflection and maximum slope of a structural beam under load, the two most important serviceability checks in structural and mechanical engineering. It covers the six most common configurations used in real-world design practice.
- ›Material presets, choose from Structural Steel (200 GPa), Aluminum (69 GPa), Concrete (30 GPa), Timber (12 GPa), or Titanium, with the option to enter any custom modulus.
- ›Section calculator, enter dimensions for rectangular, circular, or hollow rectangular cross-sections and I is computed automatically. Or enter I directly in cm⁴.
- ›Serviceability checks, the result is automatically compared against code deflection limits L/360, L/240, L/180, and L/150, showing pass/fail for each.
- ›Step-by-step derivation, every calculation is shown with full numeric substitution, so you can verify or reproduce the result by hand.
Formula
Cantilever Beam
Tip point load: δ = PL³ / (3EI) θ = PL² / (2EI)
Uniform load (UDL): δ = wL⁴ / (8EI) θ = wL³ / (6EI)
Simply Supported Beam
Centre point load: δ = PL³ / (48EI) θ = PL² / (16EI)
Uniform load (UDL): δ = 5wL⁴ / (384EI) θ = wL³ / (24EI)
Fixed-Fixed Beam
Centre point load: δ = PL³ / (192EI)
Uniform load (UDL): δ = wL⁴ / (384EI)
| Symbol | Name | SI Unit | Notes |
|---|---|---|---|
| δ | Maximum deflection | m (or mm) | Positive downward |
| θ | Maximum slope (rotation) | rad | At support or free end |
| P | Point load | N (or kN) | Applied at specified location |
| w | Distributed load intensity | N/m (or kN/m) | Force per unit length |
| L | Beam span / length | m | Between supports (or fixed to free) |
| E | Elastic (Young's) modulus | Pa (GPa) | Material stiffness |
| I | Second moment of area | m⁴ (cm⁴) | Cross-section bending resistance |
| EI | Flexural rigidity | N·m² | Combined stiffness of beam |
Common Cross-Section Formulas for I
| Section | Formula for I | Variables |
|---|---|---|
| Rectangle | I = bh³ / 12 | b = width, h = height in bending direction |
| Circle | I = πd⁴ / 64 | d = diameter |
| Hollow Rectangle | I = (bh³ − b_i h_i³) / 12 | b_i = b−2t, h_i = h−2t, t = wall thickness |
| I-Beam | I = (b_f d³ − (b_f−t_w)(d−2t_f)³) / 12 | b_f = flange width, d = total depth, t_w = web, t_f = flange |
How to Use
- 1Select a beam configuration: Choose from the six grid buttons at the top, cantilever, simply supported, or fixed-fixed, each with point load or uniform load.
- 2Pick a material: Click a material preset to auto-fill the elastic modulus E, or type a custom value in GPa.
- 3Define the cross-section: Choose Rectangular, Circular, or Hollow Rectangle and enter the dimensions in mm. The moment of inertia I is computed automatically. For non-standard sections, choose "Enter I directly" and type the value in cm⁴.
- 4Enter span and load: Type the beam length L in metres and the applied load in kN (point load) or kN/m (distributed load).
- 5Press Enter or click Calculate: Results appear immediately: max deflection, max slope, L/δ ratio, flexural rigidity EI, and a pass/fail serviceability check against four code limits.
- 6Review the step-by-step derivation: Expand "Show step-by-step working" to see the full numeric calculation, useful for checking work or including in a report.
Example Calculation
Steel floor beam, Simply Supported, 6 m span, 15 kN/m UDL
A W200×100 steel floor beam (I ≈ 113,000 cm⁴, enter manually) spans 6 m between columns. It carries a uniform floor load of 15 kN/m. We need to check whether deflection exceeds the L/360 limit for live loads.
Given:
E = 200 GPa = 200 × 10⁹ Pa
I = 113,000 cm⁴ = 1.13 × 10⁻³ m⁴
EI = 200 × 10⁹ × 1.13 × 10⁻³ = 2.26 × 10⁸ N·m²
Formula: simply supported, UDL
δ = 5wL⁴ / (384EI)
= 5 × 15,000 × 6⁴ / (384 × 2.26 × 10⁸)
= 5 × 15,000 × 1296 / (8.678 × 10¹⁰)
= 97,200,000 / 86,784,000,000
δ ≈ 1.12 mm
Serviceability Check
Allowable deflection (L/360) = 6,000 mm / 360 = 16.7 mm.
Actual deflection = 1.12 mm, well within limits (L/δ ≈ 5,357). ✓
For a lighter W200×46 section (I ≈ 45,800 cm⁴), deflection would be ≈ 2.76 mm, still passing. This illustrates how doubling the section's I roughly halves the deflection.
Understanding Beam Deflection
Why Beam Deflection Matters
Every structural beam deflects under load, that is simply how elastic materials work. The question is not whether deflection occurs, but whether it is small enough to be acceptable. Excessive deflection causes plaster ceilings to crack, doors and windows to jam, floor finishes to fracture, and occupants to feel an uncomfortable springiness underfoot.
Building standards such as AISC 360 (steel), ACI 318 (concrete), and Eurocode 3 and 5 set serviceability limits on deflection expressed as fractions of the span, typically L/360 for floors carrying live load, L/240 for total load, and L/180 for roofs. The calculator checks all four common limits simultaneously.
The Four Key Variables
Deflection is governed by four quantities, and the span L is by far the most powerful:
- ›Span L, the dominant factor. Deflection scales with L³ for point loads and L⁴ for distributed loads. Doubling the span increases deflection by 8× or 16×. This is why long-span beams are the hardest to control.
- ›Applied load P or w. Deflection is directly proportional to load. Double the load, double the deflection, simple linear scaling.
- ›Elastic modulus E. Stiffer materials deflect less. Steel (200 GPa) deflects about 3× less than aluminum (69 GPa) under the same conditions, and about 17× less than timber (12 GPa).
- ›Moment of inertia I. This is the engineer's main design lever. A deeper section, or an I-beam profile, increases I dramatically for the same cross-sectional area. An I-beam is not just lighter; it is far more efficient in bending.
Boundary Conditions and Their Effect
The same beam and load give very different deflections depending on how the ends are restrained:
| Configuration | UDL Formula | Relative Deflection | Typical Use |
|---|---|---|---|
| Fixed-Fixed (encastré) | wL⁴ / 384EI | 1× (least deflection) | Embedded beams, moment frames |
| Simply Supported | 5wL⁴ / 384EI | 5× more | Most floor and roof beams |
| Cantilever | wL⁴ / 8EI | 48× more | Balconies, overhanging eaves |
The cantilever deflects 48 times more than an equivalent fixed-fixed beam under a UDL, which explains why balconies are so deflection-sensitive and why long cantilevers demand heavy, stiff sections or pre-cambering.
Superposition: Combining Multiple Loads
For linearly elastic beams, deflections from different loads can be added together, this is the principle of superposition. A simply supported beam with both a central point load and a UDL has a total deflection equal to the sum of the individual deflections calculated separately. This calculator handles one load case at a time; run it separately for each load and add the results for complex setups.
Pre-Cambering
Long steel beams are sometimes fabricated with an intentional upward curve (camber) equal to the calculated dead-load deflection. When the dead load is applied, the beam becomes flat. This eliminates the visual sag that would otherwise be visible in long-span structures, without requiring a stiffer section. Camber values are typically set at 75–80% of the calculated dead-load deflection to allow for construction tolerance.
Common Material Elastic Moduli
| Material | E (GPa) | Typical Applications |
|---|---|---|
| Structural Steel (A36, S275) | 200 | Beams, columns, industrial frames |
| Stainless Steel (304) | 193 | Architectural, food processing |
| Aluminum (6061-T6) | 69 | Aerospace, facades, light structures |
| Concrete (normal weight) | 25–35 | Slabs, beams (use cracked I for service) |
| Timber (structural grade) | 9–14 | Joists, rafters, glulam |
| Titanium (Ti-6Al-4V) | 114 | Aerospace, medical implants |
| Carbon Fibre Reinforced (CFRP) | 70–200+ | Aerospace, sports, high-tech |
Limitations of This Calculator
- ›Assumes linear elastic, homogeneous material, valid for steel and aluminum within their working range.
- ›Does not account for shear deformation, significant only for very deep, short-span beams (span-to-depth ratio < 5).
- ›Concrete: use the effective cracked moment of inertia (I_eff from ACI 318 or Eurocode 2), not the gross section.
- ›Dynamic loads and vibration are not covered, for machinery or pedestrian bridges, check natural frequency separately.
- ›Tapered or non-prismatic beams, and beams with holes or notches, require FEA or specialised design software.
Frequently Asked Questions
What is beam deflection and why is it important?
Beam deflection is the vertical displacement a beam experiences under applied loads. Every beam deflects, the question is whether that deflection is small enough to be safe and comfortable.
Excessive deflection causes real problems:
- ›Plaster ceilings and tile floors crack from the movement below them
- ›Doors and windows bind in their frames when the structure moves
- ›Occupants feel uncomfortable springiness in floors
- ›Water ponds on flat roofs that sag between supports
Building codes control this by setting deflection limits as fractions of the span, for example, L/360 for floors under live load in most steel design standards.
What is the moment of inertia (I) and how does it affect deflection?
The moment of inertia (formally the second moment of area, I) measures how effectively a cross-section resists bending. Deflection is inversely proportional to I, double I, halve the deflection.
Key insight: height enters the formula cubically. For a rectangle, I = bh³/12:
- ›A 100×200 mm beam (h=200): I = 100×200³/12 = 66,700,000 mm⁴
- ›A 100×400 mm beam (h=400): I = 100×400³/12 = 533,000,000 mm⁴, 8× larger, 8× less deflection
- ›An I-beam achieves a large h efficiently by concentrating material in the flanges, far from the neutral axis
This is why structural steel uses I-sections rather than solid rectangles, they maximize I for minimum material weight.
What is the difference between a cantilever and a simply supported beam?
The boundary conditions, how the beam ends are restrained, have a dramatic effect on deflection:
- ›Cantilever: one end is fully fixed (no rotation, no displacement); the other end is completely free. The entire beam rotates from the fixed end, creating large tip deflections.
- ›Simply supported: both ends rest on pin/roller supports. They cannot move vertically but can rotate freely. Most floor and roof beams are simply supported.
- ›Fixed-fixed (encastré): both ends are fully built-in. Rotation is prevented at both supports, which dramatically reduces mid-span deflection, to 1/5 of simply supported for UDL.
Under a UDL, the simply supported beam deflects 5× more than fixed-fixed, and the cantilever deflects 48× more than fixed-fixed for the same span and load.
How do I read the L/360, L/240, L/180 deflection limits?
Deflection limits are expressed as fractions of the span (L) to make them proportional for any beam length:
- ›L/360, typical limit for floors under live load only (AISC 360, most steel codes). For a 6 m span: 6000/360 = 16.7 mm maximum.
- ›L/240, limit for floors under total load (live + dead). More generous than L/360.
- ›L/180, typical limit for roof members not supporting a ceiling. Even more generous.
- ›L/150, secondary structural members and elements where appearance is less critical.
The appropriate limit depends on what finishes are attached (brittle plaster is more sensitive than a bare slab), what the beam supports, and which design standard applies in your jurisdiction.
What elastic modulus (E) should I use for concrete?
Concrete is trickier than steel or timber because it cracks in tension under service loads.
- ›Elastic modulus E_c: typically 25–35 GPa for normal-weight concrete (≈ 4,700√f'c in MPa for ACI, or 22(f_cm/10)^0.3 for Eurocode 2).
- ›Gross I vs cracked I: the gross section I over-estimates stiffness. ACI 318 uses an effective I (I_e) from the Branson equation that accounts for cracking.
- ›Long-term creep: concrete deflects further over time. ACI 318 applies a multiplier λ to long-term deflection (typically 1.2–2.0 depending on compression steel).
For preliminary estimates this calculator gives a useful starting point; use the cracked I for final concrete design.
Can I use superposition to combine multiple loads?
Yes, superposition is exact for linearly elastic beams. If your beam has both a UDL and a central point load:
- ›Run the calculator with the UDL only → get δ₁
- ›Run the calculator with the point load only → get δ₂
- ›Total deflection = δ₁ + δ₂
Superposition works because the governing differential equation (EI·d²y/dx² = M) is linear, load effects scale proportionally and add independently.
It does NOT apply when: material plasticity is involved, large displacements change the geometry significantly, or there are contact/buckling non-linearities.
What is flexural rigidity (EI)?
Flexural rigidity (EI) combines the two material/geometry properties that resist bending into a single number:
- ›E, how stiff the material is (steel deflects less than timber under the same stress)
- ›I, how effectively the cross-section uses that material (deep beams resist bending better than shallow ones)
Every deflection formula in this calculator has EI in the denominator, meaning deflection is inversely proportional to EI. A beam twice as stiff (2EI) deflects half as much.
Units: N·m² (or kN·m², GN·m²). A typical IPE 300 steel beam has EI ≈ 200 × 10⁹ × 8.356 × 10⁻⁵ ≈ 16,700 kN·m².
Does the calculator save my inputs?
Yes, your inputs are automatically saved to your browser's localStorage as you work. This means:
- ›Close the tab accidentally? Your beam configuration and dimensions are restored on the next visit.
- ›The beam type, material selection, section dimensions, span, and load are all remembered.
- ›Nothing is sent to any server, all data stays in your browser.
Click Reset All to clear both the form fields and the saved localStorage data, returning the calculator to its defaults.