What is beam bending?

Beam bending occurs when a transverse load is applied to a structural member, causing it to deflect. The beam develops internal bending moments and shear forces to resist the load. The bending stress varies linearly from zero at the neutral axis to maximum at the extreme fibers.

What is the difference between a simply supported beam and a cantilever?

A simply supported beam rests on two supports at its ends and is free to rotate. A cantilever beam is rigidly fixed at one end and free at the other. Cantilevers generally experience higher deflections and moments at the fixed support compared to simply supported beams of the same span and load.

How do I calculate beam deflection?

Beam deflection depends on the load type, span, elastic modulus (E), and moment of inertia (I). For a simply supported beam with central point load P: δ_max = PL³/(48EI). For UDL w: δ_max = 5wL⁴/(384EI). For cantilever with tip load P: δ_max = PL³/(3EI).

What is the moment of inertia in beam calculations?

The moment of inertia (I) is a geometric property of the beam cross-section that measures its resistance to bending. A larger I means less deflection and lower stress for the same load. For a rectangle: I = bh³/12. For a circle: I = πd⁴/64.

What is the maximum allowable deflection for a beam?

Common deflection limits are L/360 for floor beams under live load, L/240 for total load, and L/180 for roofs. Industrial applications may use L/500 or stricter. Always check the applicable building code or design standard for your specific application.

Free Engineering Tool

Beam Bending Calculator

Calculate maximum bending moment, deflection, and stress for simply supported and cantilever beams under various load types.

Point Load UDL Cantilever #164

Load Case

Beam Length L (m)

Point Load P (kN)

Elastic Modulus E (GPa)

Moment of Inertia I (cm⁴)

Distance to Extreme Fiber c (mm)

Quick presets

Results

Maximum Bending Moment

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Maximum Deflection

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Maximum Bending Stress

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Span/Deflection Ratio

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Load Case

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Simply Supported — Point Load at Center

Simply Supported — UDL

Cantilever — Point Load at Tip

Bending Stress

Where σ is bending stress, M is bending moment, c is distance to extreme fiber, and I is moment of inertia.

Load Case	M_max	δ_max
SS — Center Point	PL/4	PL³/(48EI)
SS — Point at a	Pa(L−a)/L	complex formula
SS — UDL w	wL²/8	5wL⁴/(384EI)
Cantilever — Tip Point	PL	PL³/(3EI)
Cantilever — UDL w	wL²/2	wL⁴/(8EI)

⚠️ Note: These formulas assume linear-elastic behavior, small deflections, and prismatic (constant cross-section) beams. For complex loading, use superposition.

Beam bending occurs when a transverse load is applied to a structural member, causing it to deflect. The beam develops internal bending moments and shear forces. Bending stress varies linearly across the section — zero at the neutral axis and maximum at the extreme fibers.

A simply supported beam rests on two supports at its ends and is free to rotate at those points. A cantilever beam is rigidly fixed at one end and free at the other.

Cantilevers have larger deflections and moments at the fixed support than equivalently loaded simply supported beams.

Deflection depends on load type, span length (L), elastic modulus (E), and moment of inertia (I):

SS center point load: δ = PL³/(48EI)
SS UDL: δ = 5wL⁴/(384EI)
Cantilever tip load: δ = PL³/(3EI)

The second moment of area (I) is a geometric property that measures the cross-section’s resistance to bending. Larger I means less deflection and stress for the same load. For a rectangle: I = bh³/12. For a circle: I = πd⁴/64.

Common limits per building codes:

L/360 — Floor beams (live load)
L/240 — Total load
L/180 — Roof beams
L/500 — Precision machinery supports

Always check applicable design standards.

The procedures described on this page were developed and field-tested by the Vibromera team using the Balanset-1A, a portable dual-channel balancer and vibration analyzer for on-site rotor balancing.