Beam Deflection Calculator

Roark's Formulas for Stress and StrainAISC Steel Construction Manual
Calculator Input
Enter beam geometry, material properties, and loading conditions to calculate deflection.

Type of beam support condition

Distribution of applied load

m

Total length of beam (1 - 100 m)

Structural material with known modulus of elasticity

Geometric shape of beam cross-section

m

Beam width in meters for rectangular section

m

Beam height/depth in meters for rectangular section

N/m

Distributed load intensity in N/m (1 - 1,000,000 N/m)

Frequently Asked Questions

Common questions about this calculator

Beam deflection is the vertical displacement of a beam under load, caused by bending. Excessive deflection causes serviceability problems: cracked finishes, visible sagging, vibration issues, and impaired functionality (doors won't close, equipment misaligns). Design limits are typically L/240 to L/360 of span length.

For simply supported beam with uniform load: δ = 5wL⁴/(384EI), where w is load per unit length, L is span, E is elastic modulus, and I is moment of inertia. Point load at center: δ = PL³/(48EI). Cantilever with end load: δ = PL³/(3EI). More complex cases require superposition or integration.

Moment of inertia measures a cross-section's resistance to bending—larger I means less deflection. For rectangle: I = bh³/12 (b=width, h=height). For I-beam: sum web and flange contributions. Steel beam tables list I values directly. I depends on axis of bending—always use I about the bending axis.

Common limits (expressed as span/deflection ratio): L/180 general structural, L/240 floors with brittle finishes, L/360 floors supporting partitions, L/480 sensitive equipment. Live load deflection is typically limited separately from total load deflection. Check applicable building code for specific requirements.

Support conditions dramatically affect deflection. Fixed-end beams deflect ~5× less than simply supported under same load. Continuous beams (multiple supports) deflect less than simple spans. Cantilevers deflect most per unit span. Overhangs can reduce deflection in adjacent spans.

To reduce deflection: increase beam depth (most effective—I ∝ h³), use higher modulus material (steel E=200 GPa vs wood E=10 GPa), reduce span (shorter span = less deflection), add intermediate supports, or use cambered beams (pre-curved upward to offset expected deflection).

Learn More

Beam deflection is the vertical displacement of structural members under applied loads, representing a critical serviceability limit state. While strength calculations prevent catastrophic failure, deflection calculations ensure serviceability by preventing cracking of finishes, misalignment of equipment, and excessive vibration. The fundamental principle relates curvature to bending moment through flexural rigidity (EI), with deflection depending on applied loads, span length, material stiffness, and cross-sectional geometry. Understanding deflection behavior enables proper structural design, member sizing, and occupant comfort while meeting code requirements and preventing long-term performance issues.

Fundamental Deflection Factors: Beam deflection depends on four primary factors: applied loads (magnitude and distribution), span length, material stiffness (elastic modulus E), and cross-sectional geometry (moment of inertia I). Deflection increases with the fourth power of span length; a beam spanning twice the length deflects 16 times as much under identical conditions. Material selection profoundly affects behavior: steel (E = 200 GPa) is eight times stiffer than wood (E = 12-14 GPa), while aluminum (E = 70 GPa) offers intermediate performance. The moment of inertia increases with depth cubed; doubling beam depth increases stiffness eight-fold while only doubling material mass.

Support Conditions: Support configuration dramatically influences deflection magnitude and distribution. Simply supported beams (pinned at both ends) exhibit maximum deflection at midspan using δ = 5wL⁴/(384EI) for uniform loads. Fixed-end beams (clamped preventing rotation) deflect significantly less due to moment restraint at supports. Cantilever beams exhibit largest deflections with maximum displacement at the free end using δ = wL⁴/(8EI). Continuous beams spanning multiple supports benefit from reduced deflections through negative moments over interior supports, enabling longer spans or shallower sections compared to simple spans.

Code Deflection Limits: Building codes specify maximum allowable deflections as fractions of span length to ensure serviceability. AISC and IBC require L/360 for floor beams supporting brittle finishes (plaster, ceramic tile), L/240 for floors with flexible finishes, and L/180 for roof members. These limits represent acceptable performance based on decades of experience correlating calculated deflections with actual behavior and occupant acceptance. Vibration considerations may impose stricter limits for long-span floors where natural frequency approaches 1.5-2.5 Hz walking pace, creating perceptible bounce requiring L/480 or deeper sections.

Material Properties: Material selection affects deflection through elastic modulus and time-dependent behavior. Steel provides consistent performance with E = 200 GPa and minimal creep. Wood at E = 10-15 GPa exhibits creep under sustained loads, particularly at high moisture content. Reinforced concrete requires analysis of cracked versus uncracked section properties as tension cracking reduces effective stiffness. Composite construction (steel beams with concrete slab) must account for construction sequence with non-composite section carrying wet concrete, then composite section carrying additional loads after curing providing significantly increased stiffness.

Deflection Control Strategies: Controlling deflection employs multiple approaches balancing performance and cost. Increasing member depth is most effective due to I∝h³ relationship. Adding intermediate supports reduces span length. Higher-stiffness materials improve performance but increase cost. Cambering provides intentional upward curvature during fabrication to offset anticipated deflections, commonly specified at 1.5 times dead load deflection for steel beams. Span-to-depth ratios of L/15-20 for simple beams and L/20-28 for continuous beams provide preliminary sizing guidance. Economic optimization balances material cost against depth constraints and building height impacts.

Standards Reference: AISC Steel Construction Manual provides comprehensive deflection calculation procedures and serviceability criteria for steel structures. ACI 318 governs concrete beam deflection including time-dependent effects. IBC Section 1604.3 establishes general deflection limits. ASCE 7 addresses dynamic considerations and vibration. NDS covers wood member deflection accounting for creep and moisture effects. These standards ensure structural performance meets safety and serviceability requirements while maintaining occupant comfort and preventing damage to building finishes and systems.

What This Calculator Computes (Scope): This tool computes elastic, small-deflection serviceability for two support conditions — simply supported and cantilever — under a uniform load or a single point load (center for simply supported, tip for cantilever). It reports the maximum deflection, the moment of inertia, and a pass/fail check against the allowable span fraction (L/360 for simply supported beams, L/180 for cantilevers). The fixed-end, continuous-beam, composite, camber, and long-term-creep formulas below are provided as reference equations for hand calculation; they are not solved by the calculator. Strength checks (maximum bending moment, section modulus, bending stress σ = Mc/I, lateral-torsional buckling) are a separate limit state — pair this serviceability check with a strength calculation such as the Steel Beam Calculator. For reinforced concrete, the tool uses the gross (uncracked) section and does not apply a cracked-section effective inertia or creep multiplier, so concrete results are non-conservative and should be treated as a first estimate only.

Residential Floor Joist - Live Load Deflection Check

Verify floor joist deflection meets code requirements for residential construction

1
Beam Type: Simply Supported
2
Load Type: Uniform Distributed
3
Beam Length: 4.0 m
4
Material Type: Wood
5
Elastic Modulus: 13.1 GPa
6
Width: 0.038 m
7
Height: 0.235 m
8
Uniform Load: 772 N/m

Result

Max Deflection:
4.8 mm (3/16") at mid-span

Calculations

  • •
    Moment of inertia: I=bh3/12=0.038×0.2353/12=4.11×10−5 m4I = bh^3/12 = 0.038 \times 0.235^3/12 = 4.11 \times 10^{-5} \text{ m}^4
  • •
    Max deflection: δ=5wL4/(384EI)=5×772×4.04/(384×13.1×109×4.11×10−5)=4.8 mm\delta = 5wL^4/(384EI) = 5 \times 772 \times 4.0^4/(384 \times 13.1 \times 10^9 \times 4.11 \times 10^{-5}) = 4.8 \text{ mm} at mid-span
  • •
    Allowable deflection (L/360): 4000/360 = 11.1 mm for live load per IBC Table 1604.3
  • •
    Deflection ratio: 4.8/11.1 = 0.43 (43% of limit)

Status

  • •
    ✅ PASS - Live load deflection is 43% of the L/360 limit
  • •
    No perceptible bounce expected from static deflection alone

Recommendations

  • •
    Static deflection passes comfortably, but residential floors are often vibration- rather than deflection-governed
  • •
    For long, lightly damped joists also verify natural frequency above ~8-15 Hz (or an L/480 deflection target) to avoid perceptible bounce
  • •
    Headroom exists to evaluate a shallower section; a 2×8 (actual 38 × 184 mm, I=1.97×10−5 m4I = 1.97 \times 10^{-5} \text{ m}^4) raises deflection to about 10.0 mm (90% of L/360)
  • •
    Acceptable for deflection but leaves little margin, so the 2×10 remains the prudent choice
  • •
    Widening joist spacing to 19.2" (0.488 m) raises the line load to about 928 N/m and deflection to about 5.7 mm (52% of limit), still within L/360
  • •
    Trades material for a small reduction in stiffness margin
Option 1: Confirm Vibration Performance Option 2: Reduce Joist Depth (Material Saving) Option 3: Increase Spacing

Additional Notes

Allowable deflection limits per IBC Table 1604.3: L/360 for floors with brittle finishes (plaster), L/240 for floors with flexible finishes, L/180 for roofs. Calculate maximum deflection: δ = (5wL⁴)/(384EI) for uniformly distributed load, δ = (PL³)/(48EI) for center point load. Verify actual deflection < allowable. Consider both dead load (permanent) and live load (temporary) deflections; this check covers live load only.

Commercial Office Building - Steel Beam Supporting Mechanical Equipment

Verify steel beam deflection for rooftop HVAC unit support meets code requirements

1
Beam Type: Simply Supported
2
Load Type: Point Center
3
Beam Length: 6.0 m
4
Material Type: Steel
5
Elastic Modulus: 200 GPa
6
Cross-Section: Custom (W12×26)
7
Point Load: 30,000 N

Result

Max Deflection:
7.9 mm (5/16") at mid-span

Calculations

  • •
    Max deflection: δ=PL3/(48EI)=30000×6.03/(48×200×109×8.49×10−5)=7.9 mm\delta = PL^3/(48EI) = 30000 \times 6.0^3/(48 \times 200 \times 10^9 \times 8.49 \times 10^{-5}) = 7.9 \text{ mm} at mid-span
  • •
    Allowable deflection (L/360): 6000/360 = 16.7 mm
  • •
    Deflection ratio: 7.9/16.7 = 0.48 (48% of limit)

Status

  • •
    ✅ PASS - 48% of the L/360 limit (also well within L/240 = 25.0 mm)

Equipment Vibration Concern

  • •
    HVAC manufacturer specifies maximum 12 mm (1/2") static deflection under operating load for proper compressor alignment and condensate drainage
  • •
    At 7.9 mm: ✅ within manufacturer limit with about 34% margin

Recommendations

  • •
    I=8.49×10−5 m4I = 8.49 \times 10^{-5} \text{ m}^4 → 7.9 mm, satisfying both the L/360 (16.7 mm) and 12 mm equipment limits
  • •
    This is the most economical section that meets every criterion
  • •
    I=9.91×10−5 m4I = 9.91 \times 10^{-5} \text{ m}^4 (238 in⁴)
  • •
    Reduces deflection to 6.8 mm if the operating mass or dynamic factor is uncertain
  • •
    I=1.21×10−4 m4I = 1.21 \times 10^{-4} \text{ m}^4 (291 in⁴)
  • •
    Gives 5.6 mm and raises the floor natural frequency, useful where compressor speed is near the beam's fundamental frequency
Option 1: Keep W12×26 (Governing Check Satisfied) Option 2: Upsize to W12×30 for Extra Vibration Margin Option 3: Upsize to W14×30 (Deeper Section)

Alternative

  • •
    Reduce span to 5 m via intermediate column
  • •
    Reduces deflection to 4.6 mm with the W12×26 (allowable L/360 = 13.9 mm, ratio 0.33)
  • •
    Adds a column in the equipment service area, so only worth it for very heavy future units

Additional Notes

Commercial building beams per IBC require deflection analysis for serviceability and occupant comfort. Long-span beams (>9m): Camber recommended to offset dead load deflection (typically L/360-L/480). Vibration considerations for floor systems: Natural frequency >3-4 Hz prevents perceptible movement. Composite beams (steel with concrete): Account for transformed section properties and long-term creep deflection.

Building Entrance Canopy - Cantilever Glass and Steel Structure

Design cantilevered steel beam for glass entrance canopy with combined dead and wind loads

1
Beam Type: Cantilever
2
Load Type: Uniform
3
Beam Length: 3.5 m
4
Material Type: Steel
5
Elastic Modulus: 200 GPa
6
Moment of Inertia: 1.68×10⁻⁵ m⁴
7
Uniform Load: 1400 N/m

Result

Max Deflection (Dead Load Only):
7.8 mm (5/16") downward at cantilever tip

Calculations

  • •
    Cantilever formula: δtip=(wL4)/(8EI)=(1,400×3.54)/(8×200×109×1.68×10−5)=0.0078 m\delta_{\text{tip}} = (wL^4)/(8EI) = (1{,}400 \times 3.5^4)/(8 \times 200 \times 10^9 \times 1.68 \times 10^{-5}) = 0.0078 \text{ m}
  • •
    Max deflection: 7.8 mm (5/16") downward at cantilever tip

Allowable Deflection

  • •
    The calculator checks cantilevers against L/180 = 3,500/180 = 19.4 mm; result is 7.8 mm (ratio 0.40), so it PASSES
  • •
    Architectural appearance criterion: L/150 = 3,500/150 = 23.3 mm; at 7.8 mm the tip is well under the visible-sag threshold (33% of L/150)
  • •
    Glass stress check: at 7.8 mm tip deflection over 3.5 m, the tip slope for a UDL cantilever is θ=wL3/(6EI)=0.30%\theta = wL^3/(6EI) = 0.30\% (0.17°)
  • •
    A 0.17° glass panel rotation is acceptable per structural glazing standards (ASTM E1300)

Wind Load Analysis

  • •
    ASCE 7-16 wind uplift on canopy: 1.5 kPa (31 psf) upward pressure
  • •
    Wind load = 1,500 N/m² × 1.2 m = 1,800 N/m upward (opposes dead load)
  • •
    Deflection from full wind uplift alone (1,800 N/m): 10.0 mm upward
  • •
    Combined service case 1.0D + 0.75W = 1,400 − 1,350 = 50 N/m net downward → 0.3 mm (negligible)
  • •
    Critical load case: Dead load without wind (7.8 mm down) controls deflection design

Recommendations

  • •
    I=1.68×10−5 m4I = 1.68 \times 10^{-5} \text{ m}^4 → 7.8 mm, within the L/180 (19.4 mm) and L/150 (23.3 mm) limits
  • •
    Most economical section that satisfies the serviceability and appearance criteria
  • •
    Upsize to HSS 250×150×10 (I=4.20×10−5 m4I = 4.20 \times 10^{-5} \text{ m}^4)
  • •
    Reduces deflection to 3.1 mm (13% of L/150) for a visually flat edge where the canopy is highly exposed
  • •
    Install an intermediate support at 2.0 m from the building, reducing the cantilever to 1.5 m
  • •
    Deflection drops to 0.3 mm with the original HSS 200×100×8
  • •
    Adds a column but may conflict with vehicle clearance
  • •
    Since the beam already passes, camber is only for appearance; a modest 8 mm upward camber offsets the full dead-load sag so the edge reads dead level under sustained load
Option 1: Keep HSS 200×100×8 (Passes Both Limits) Option 2: Upsize for a Crisper Edge Line Option 3: Intermediate Support Option 4: Pre-Camber (Optional)

Additional Notes

Industrial beams supporting heavy machinery or cranes per AISC Design Guide 7 require strict deflection limits (often L/600-L/1000) to maintain equipment alignment and prevent excessive vibration. Dynamic loading: Increase deflection calculation by impact factor (1.25-2.0× static load). Fatigue considerations for cyclic loading. Monitor critical beams with deflection gauges or strain sensors. Retrofit options: Add supplemental beams, external post-tensioning, or composite action with slab.

AISC Steel Construction Manual

AISC 360

AISCstandard

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