Beam Deflection Calculator | Simply Supported & Cantilever

Beam Deflection Calculator | ProEngCalc

🏗 Structural Engineering

Beam Deflection Calculator

Calculate maximum deflection and slope for 4 common beam and loading configurations

Reference: AISC Steel Construction Manual | Roark’s Formulas for Stress and Strain | ASCE 7

Beam & Loading Configuration

Maximum Deflection

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📐 Solution Breakdown

Variable Definitions

Symbol	Variable	SI Unit	Imperial Unit
δ	Maximum Deflection	mm	inches
P	Point Load	kN	kips
w	Uniform Distributed Load	kN/m	kips/ft
L	Beam Span / Length	m	ft
E	Modulus of Elasticity	GPa	ksi
I	Moment of Inertia	mm⁴ × 10⁶	in⁴

⚠ Assumptions & Limits — Structural Use

Assumes linear elastic behavior — valid only when stresses remain below the yield strength of the material
Assumes prismatic beam (constant cross-section and uniform E and I throughout the span)
Small deflection theory applies — deflections must be small relative to beam length (typically δ < L/100)
Does not account for shear deflection — significant for deep beams (L/d < 10)
Does not include long-term creep deflection — multiply by creep factor per ACI 318 or AISC for sustained loads
Serviceability limits: L/360 for live loads on floors, L/240 for combined loads per AISC and IBC
All structural designs must be verified by a licensed structural engineer per applicable building codes

What Is Beam Deflection?

Beam deflection is the vertical displacement of a structural beam under applied loads. Deflection control is a serviceability requirement — separate from strength. A beam can be strong enough to safely carry a load but still deflect excessively, cracking plaster ceilings, jamming doors, damaging glass facades, or creating an uncomfortable bouncy floor. For many modern long-span structures and lightweight steel construction, deflection governs the design before strength does.

⚠ Engineering Notice: All structural calculations must be reviewed and sealed by a licensed structural engineer. This calculator provides preliminary values only and does not constitute a structural analysis or engineering approval for construction.

Deflection Formulas

Simply Supported — Center Point

δ = PL³ / 48EI

Simply Supported — Uniform Load

δ = 5wL⁴ / 384EI

Cantilever — End Point Load

δ = PL³ / 3EI

Cantilever — Uniform Load

δ = wL⁴ / 8EI

Serviceability Limits

Application	Limit	Reference
Floor beams — live load	L/360	AISC / IBC 1604.3
Floor beams — total load	L/240	AISC / IBC 1604.3
Roof — live load	L/240	AISC / IBC 1604.3
Roof — total load	L/180	AISC / IBC 1604.3
Beams supporting plaster	L/360	IBC 1604.3
Beams supporting brittle finishes	L/480	Engineer judgment
Cantilever beams	L/180	AISC
Beams supporting masonry	L/600	TMS 402

Worked Examples

Example 1 — Office Floor Beam Under Live Load

A W12×26 steel beam spans 6m simply supported, carrying a uniform live load of 8 kN/m. Does it meet L/360?

Given

w = 8 kN/m | L = 6,000 mm | E = 200 GPa | I = 48.7×10⁶ mm⁴ (W12×26)

Deflection

δ = 5×8×6000⁴ / (384×200,000×48.7×10⁶) = 6.97 mm

L/360 Limit

6000/360 = 16.7 mm

Solution

δ = 6.97 mm < 16.7 mm ✓ PASSES L/360

W12×26: I_x = 204 in⁴ = 84.9×10⁶ mm⁴ strong axis. This beam has significant reserve — could carry nearly double this load before failing deflection limits.

Example 2 — Cantilever Balcony Beam

A cantilever balcony beam extends 2.4m with a uniform load of 5 kN/m. Using a W8×18 steel beam, check the L/180 deflection limit.

Given

w = 5 kN/m | L = 2,400 mm | E = 200 GPa | I = 26.0×10⁶ mm⁴ (W8×18)

Deflection

δ = 5×2400⁴ / (8×200,000×26.0×10⁶) = 7.97 mm

L/180 Limit

2400/180 = 13.3 mm

Solution

δ = 7.97 mm < 13.3 mm ✓ PASSES L/180

Cantilevers deflect much more than simply supported beams — compare PL³/3EI vs PL³/48EI, a factor of 16 difference for point loads. Always use appropriate cantilever formulas for cantilevered elements.

Example 3 — Long Span Timber Beam

A Douglas Fir timber beam (4×12 actual, I = 415 in⁴) spans 20 feet simply supported with a 1,200 lb point load at midspan. Check deflection.

Given

P = 1,200 lb | L = 240 in | E = 1,800,000 psi | I = 415 in⁴

Deflection

δ = 1200×240³ / (48×1,800,000×415) = 0.486 in

L/360 Limit

240/360 = 0.667 in

Solution

δ = 0.486 in < 0.667 in ✓ PASSES L/360

For timber, also check long-term creep deflection. Multiply elastic deflection by 1.5–2.0 for sustained dead loads per NDS. Total deflection including creep = 0.486 × 1.5 = 0.73 in — may exceed limits for sustained loads.

Real World Applications

🏢

Office Floor Systems

Checking L/360 live load deflection limits to prevent cracking of non-structural partitions and ensure floor stiffness for occupant comfort.

🌉

Bridge Girders

Highway bridge deflection checked per AASHTO — typically L/800 for vehicular live load to prevent damage to wearing surfaces.

🏗

Crane Runways

Crane runway girders checked to very tight limits (L/600–L/1000) to prevent misalignment of crane rails and wheel wear.

🏠

Residential Framing

Floor joist and beam sizing to meet L/360 (live) and L/240 (total) limits per IRC, avoiding bouncy floors and cracked finishes.

Common Mistakes Engineers Make

❌ Mistake 1 — Using Wrong Moment of Inertia Axis

W-shapes have two moment of inertia values: strong axis (I_x, larger) and weak axis (I_y, much smaller). Always use I_x for beams bending about their strong axis. Using I_y by mistake results in massively underestimated deflection.

❌ Mistake 2 — Not Checking Both Live and Total Load Deflection

Building codes require checking deflection under live load alone (L/360) AND under total load (L/240). A beam that passes L/240 for total load may still fail L/360 for live load if the live-to-dead load ratio is high.

❌ Mistake 3 — Ignoring Long-Term Creep in Wood and Concrete

Steel deflection is essentially instantaneous and stable. Wood and concrete creep significantly under sustained loads. ACI 318 requires multiplying initial deflection by factors of 1.0–2.0 for long-term sustained loads. Ignoring creep in timber and concrete beams leads to excessive long-term deflection.

❌ Mistake 4 — Applying Simply Supported Formulas to Partially Fixed Beams

Real beam connections are rarely perfectly pinned or perfectly fixed. A beam with semi-rigid connections deflects between the simply supported and fixed-end cases. Using simply supported formulas for beams with partial fixity is conservative but can lead to oversizing. Consult AISC for moment connections with partial restraint.

Frequently Asked Questions

What elastic modulus should I use for different materials?

Standard values: Structural steel = 200 GPa (29,000 ksi). Stainless steel = 193 GPa. Aluminum alloys = 69 GPa (10,000 ksi). Concrete = 4,730√f’c MPa per ACI 318 (approximately 25–32 GPa for normal weight concrete with f’c = 28–45 MPa). Douglas Fir = 12.4 GPa (1,800 ksi). Southern Yellow Pine = 11.0 GPa. Always use the actual E for your specific material and grade.

How do I find the moment of inertia for a steel W-shape?

Moment of inertia for standard steel sections is tabulated in the AISC Steel Construction Manual (Table 1-1 for W-shapes). Values are given in in⁴ for Imperial and cm⁴ for metric. Common values: W8×18: I_x = 61.9 in⁴. W12×26: I_x = 204 in⁴. W16×40: I_x = 518 in⁴. W24×55: I_x = 1,350 in⁴. For rectangular sections: I = bh³/12.

Why is cantilever deflection so much larger than simply supported deflection?

For a point load at the free end: cantilever δ = PL³/3EI vs simply supported δ = PL³/48EI — a factor of 16 difference. For uniform load: cantilever δ = wL⁴/8EI vs simply supported δ = 5wL⁴/384EI — a factor of 4.8 difference. The fixed end prevents rotation, concentrating all curvature in the span rather than distributing it between two supports. This is why cantilever beams require much stiffer sections than simply supported spans of equal length.

Does this calculator account for shear deflection?

No — this calculator computes bending deflection only. Shear deflection is negligible for most beams (typically less than 5% of bending deflection) but becomes significant for short, deep beams where the span-to-depth ratio (L/d) is less than 10. For deep beams and short transfer structures, add shear deflection using δ_shear = VL/(GA_v), where A_v is the shear area of the web.