Beam Calculator

Calculate beam deflection, bending stress, and shear forces for common loading scenarios. Analyze simply supported and cantilever beams.

Unit System

Metric

Imperial

Beam Type

Beam Length (m)

Load Type

Load Magnitude (N)

Load Position from Left (m) — center assumed for calculations

Cross-Section

Width (mm)

Height (mm)

Material

About Beam Calculator

Beam Analysis Fundamentals

Beam analysis is a cornerstone of structural engineering, used to predict how beams respond to applied loads. Engineers calculate bending moments, shear forces, and deflections to ensure structures are safe and serviceable. The Euler–Bernoulli beam theory provides the mathematical foundation for these calculations, assuming that plane sections remain plane after bending and that deformations are small relative to the beam length.

Beam Types and Support Conditions

Type	Supports	Characteristics
Simply Supported	Pin + Roller	Free rotation at ends, max moment at mid-span
Cantilever	Fixed end	One free end, max moment at fixed support
Fixed-Fixed	Both ends fixed	No rotation at ends, reduced mid-span deflection

Loading Scenarios

Point Load: F applied at a single location
Uniform Distributed Load (UDL): w applied per unit length

Point loads — concentrated forces from columns, machinery, or equipment. Creates discontinuities in the shear force diagram.
Uniform loads — evenly distributed weight such as floor loads, snow loads, or self-weight of the beam. Produces parabolic bending moment diagrams.
Combined loads — real structures often experience multiple load types simultaneously, analyzed using superposition.

Stress and Deflection Concepts

Bending stress (σ): Varies linearly from zero at the neutral axis to maximum at the extreme fibers. Calculated as σ = M·y/I, where M is bending moment, y is distance from neutral axis, and I is moment of inertia.
Shear stress (τ): Maximum at the neutral axis and zero at the extreme fibers. For rectangular sections, τ_max = 3V/(2A), where V is shear force and A is cross-sectional area.
Deflection (δ): The vertical displacement of the beam under load. Depends on load magnitude, span length, material stiffness (E), and cross-section geometry (I).
Moment of inertia (I): For a rectangular section, I = bh³/12. Increasing height is far more effective than increasing width for reducing deflection.

Safety Factors and Limitations

Safety factor: The ratio of yield strength to maximum working stress. Values above 1.0 indicate the beam will not yield under the given load. Typical design requires factors of 1.5–3.0.
Elastic range only: These calculations assume the beam remains in the elastic range (stress below yield strength). Plastic deformation, buckling, and fatigue are not considered.
Simplified models: Real beams may have complex cross-sections, varying loads, connections, and dynamic effects not captured by simple beam theory.
Professional review: Always have structural calculations reviewed by a licensed professional engineer before use in actual construction projects.

Frequently Asked Questions

What is the difference between simply supported, cantilever, and fixed-fixed beams?

A simply supported beam rests on two supports at its ends and is free to rotate. A cantilever beam is fixed at one end and free at the other, like a diving board. A fixed-fixed (or fixed-end) beam is rigidly attached at both ends, preventing rotation. Each type distributes loads differently: cantilevers experience maximum stress at the fixed end, simply supported beams at mid-span, and fixed-fixed beams have reduced mid-span moments but develop moments at both supports.

What are typical allowable deflection limits for beams?

Building codes typically limit beam deflection to L/240 or L/360 of the span length, depending on the application. For floor beams supporting plaster ceilings, L/360 is common; for roof beams, L/240 may be acceptable. For example, a 6-meter floor beam should deflect no more than about 16.7 mm (6000/360). These limits prevent cracking of finishes, ensure occupant comfort, and maintain structural integrity.

What is the moment of inertia and why does it matter?

The moment of inertia (I) measures a cross-section's resistance to bending. A larger moment of inertia means less deflection and lower bending stress for the same load. For a rectangular section, I = bh³/12, where b is the width and h is the height. This is why beams are oriented with their taller dimension vertical — doubling the height increases I by a factor of 8, while doubling the width only doubles it.

How is the safety factor determined for beam design?

The safety factor is the ratio of the material's yield strength to the maximum stress in the beam. A safety factor greater than 1 means the beam can handle the applied load without yielding. Typical engineering practice requires safety factors of 1.5 to 3.0 depending on the application, material variability, and consequences of failure. Building codes specify minimum safety factors through load and resistance factor design (LRFD) or allowable stress design (ASD) methods.

Additional Resources

Wikipedia: Euler–Bernoulli Beam Theory Wikipedia: Structural Engineering Wikipedia: Second Moment of Area (Moment of Inertia)

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