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Category Engineering Calculators

Support Type

Load Type

Uniform Distributed Point Load (Midspan)

Beam Length, L

Distributed Load, w

kN/m

Point Load, P

Material Preset

Elastic Modulus, E

GPa

Yield Strength, σ_y

MPa

Cross-Section (Rectangular)

Width b m

Height h m

Or Override I Directly

×10⁻⁶ m⁴

Enter beam parameters and press Calculate to see results.

Reaction R_A —

Reaction R_B —

Max Shear Force V_max —

Max Bending Moment M_max —

Max Deflection δ_max —

Second Moment of Area I —

Max Bending Stress σ_max —

Safety Factor —

Deflection Ratio L/δ —

Failure Marginal Safe

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About

Incorrect beam sizing causes structural failure. A floor joist undersized by 10% in moment of inertia I can exceed allowable deflection limits under service loads, leading to cracked finishes, bouncy floors, or collapse. This calculator applies Euler-Bernoulli beam theory to compute reaction forces, maximum bending moment M, shear force V, deflection δ, and bending stress σ for simply supported and cantilever beams under point or uniformly distributed loads. Results assume linear-elastic material behavior and small deflections relative to span length.

The tool checks computed bending stress against material yield strength σ_y and reports a safety factor. A safety factor below 1.0 indicates the beam will yield under the applied load. Most building codes require a minimum factor of 1.5 to 2.0 for static loads. Note: this tool does not account for lateral-torsional buckling, dynamic loads, or connection detailing. For critical structural members, verify results with a licensed engineer.

Formulas

Bending stress at the extreme fiber of a beam cross-section under pure bending (the flexure formula):

σ = M ⋅ yI

where σ = bending stress Pa, M = bending moment N⋅m, y = distance from neutral axis to extreme fiber m, I = second moment of area m⁴.

Maximum bending moment for a simply supported beam with uniformly distributed load w:

M_max = w ⋅ L²8

Maximum deflection for a simply supported beam with uniformly distributed load:

δ_max = 5 ⋅ w ⋅ L⁴384 ⋅ E ⋅ I

Maximum bending moment for a cantilever beam with uniformly distributed load:

M_max = w ⋅ L²2

Maximum deflection for a cantilever beam with uniformly distributed load:

δ_max = w ⋅ L⁴8 ⋅ E ⋅ I

For a simply supported beam with a point load P at midspan: M_max = P ⋅ L4 and δ_max = P ⋅ L³48 ⋅ E ⋅ I. For a cantilever with point load at free end: M_max = P ⋅ L and δ_max = P ⋅ L³3 ⋅ E ⋅ I.

Safety factor: SF = σ_yσ_max, where σ_y is the material yield strength and σ_max is the maximum computed bending stress.

Reference Data

Material	Elastic Modulus E GPa	Yield Strength σ_y MPa	Density kg/m³	Typical Use
Structural Steel (A36)	200	250	7850	I-beams, columns, frames
Structural Steel (A992)	200	345	7850	Wide-flange beams (W-shapes)
Stainless Steel (304)	193	215	8000	Corrosion-resistant structures
Aluminum 6061-T6	68.9	276	2700	Lightweight frames, aerospace
Aluminum 2024-T4	73.1	324	2780	Aircraft structures
Titanium Ti-6Al-4V	113.8	880	4430	Aerospace, medical implants
Cast Iron (Gray)	100	130	7200	Machine bases, pipe fittings
Copper C11000	117	69	8940	Electrical conductors
Douglas Fir (No.1)	12.4	35	530	Joists, rafters, residential
Southern Pine (No.1)	13.1	38	570	Treated lumber, decking
Spruce-Pine-Fir (SPF)	9.5	27	420	Light framing, studs
Glulam (24F-V4)	12.4	41	500	Long-span timber beams
LVL (Laminated Veneer)	13.8	44	580	Headers, ridge beams
Concrete (f'c 25 MPa)	25	25	2400	Reinforced beams, slabs
Concrete (f'c 40 MPa)	31.6	40	2400	High-strength columns, precast
GFRP (Glass Fiber RP)	40	450	1900	Corrosive environments, bridges
CFRP (Carbon Fiber RP)	150	1500	1600	Aerospace, high-performance
Brass (C36000)	97	124	8500	Fittings, hardware
Bronze (C93200)	103	125	8800	Bearings, marine hardware
Magnesium AZ31B	45	200	1770	Ultralight structures

Frequently Asked Questions

Deflection is proportional to span length raised to the 4th power (L⁴) for distributed loads. Doubling the span increases deflection by a factor of 16 if all else remains equal. Most building codes limit deflection to L/360 for floor beams under live load and L/240 for total load. If your computed deflection exceeds these limits, increase the moment of inertia I by selecting a deeper cross-section rather than a wider one - depth contributes cubically to I for rectangular sections.

Elastic modulus E (in GPa) measures material stiffness - how much the beam resists deformation. Yield strength σ_y (in MPa) is the stress at which permanent deformation begins. A beam can have adequate stiffness (low deflection) but still fail if bending stress exceeds σ_y. Conversely, a very strong material with low E (like some polymers) may deflect excessively. Both must be checked: deflection against serviceability limits, and stress against strength limits with an appropriate safety factor.

The second moment of area I for a rectangle equals b·h³/12, where b is width and h is height (depth). This is the most common cross-section for timber beams and a useful baseline for any material. For I-beams (W-shapes), I is significantly larger per unit weight because material is concentrated at the flanges. If you are using a standard steel W-shape, input the published I value directly from the AISC Steel Manual rather than computing from b and h.

A safety factor of 1.5 assumes static, well-characterized loading on ductile material with consistent quality. Increase to 2.0-3.0 for dynamic or impact loads (machinery, vehicles), brittle materials (cast iron, concrete in tension), uncertain loading conditions, or when failure consequences are severe (public structures). Timber beams require higher factors (2.0+) due to natural variability in grain structure, moisture content, and knots.

No. The Euler-Bernoulli beam theory used here considers bending deflection only, which dominates for slender beams (span-to-depth ratio > 10). For deep beams (ratio < 5), shear deflection becomes significant and Timoshenko beam theory should be applied instead. For typical structural beams in buildings (ratio 12-24), bending deflection accounts for over 95% of total deflection, so the results here are reliable.

This calculator places the point load at midspan, which produces the maximum bending moment for a simply supported beam. For an off-center load at distance a from the left support, M_max = P·a·(L−a)/L at the load point, and the maximum deflection shifts toward the longer segment. If your load is off-center, the midspan results here are conservative (they overestimate M_max), which errs on the safe side.

No. Continuous beams over multiple supports develop negative moments at interior supports and require solving indeterminate structural equations (moment distribution, stiffness method, or finite element analysis). This tool handles single-span determinate beams only: simply supported (two supports at ends) and cantilever (one fixed end, one free end).