What is bending moment?

Bending moment is the internal moment at any cross-section of a beam that resists the external loads trying to bend it. It is measured in Newton-meters (N·m) and determines how much stress the beam material experiences.

What causes beam deflection?

Deflection occurs when external loads create bending moments that cause the beam to curve. The amount of deflection depends on the load magnitude, beam length, material stiffness (E), and cross-section shape (I).

What is moment of inertia?

Moment of inertia (I) describes the cross-sectional shape's resistance to bending. A rectangular beam has I = bh^3/12. An I-beam shape is efficient because it places material far from the neutral axis, maximizing I for minimal weight.

When is a cantilever better than a simply supported beam?

Cantilevers are ideal when you need an unsupported overhang, like a balcony, diving board, or aircraft wing. They trade higher stress at the support for the ability to extend freely into space.

What is the factor of safety for beams?

Factor of safety is the ratio of the beam's ultimate strength to the actual working stress. Typical values range from 1.5 to 3.0 for structural steel, depending on the application. Building codes specify minimum factors.

Beam Calculator

Calculate beam deflection, bending stress, and reactions for common load cases.

Beam Calculator

Simply supported beam with center load

Beam Calculator

Max deflection for center point load

Load (N)

Span Length (m)

Elastic Modulus (Pa)

Moment of Inertia (m4)

Formula

delta_max = PL^3 / (48EI), M_max = PL/4

What Is a Beam Calculator?

A beam calculator solves the structural mechanics of beams under various loading conditions. Beams are horizontal structural members that carry loads perpendicular to their length, transferring forces to supports. Engineers use beam calculations to ensure structures are safe — predicting how much a beam will deflect under load and whether the bending stress stays within the material's allowable limits.

This calculator handles common beam configurations: simply supported beams with point loads or uniform distributed loads, and cantilever beams. It computes maximum deflection (δ), maximum bending moment (M), and reaction forces at the supports. These values are essential for selecting the right beam cross-section and material in construction and mechanical design.

How to Use This Calculator

Select the beam type: simply supported or cantilever.
Select the load type: point load (concentrated) or uniform distributed load (UDL).
Enter the beam span (L), load magnitude (F or w), and the material's elastic modulus (E) and moment of inertia (I).
Click Calculate to see maximum deflection, bending moment, and support reactions.

Formula & Explanation

Simply supported, center point load:
  δmax = FL³ / (48EI)
  Mmax = FL / 4

Simply supported, UDL:
  δmax = 5wL⁴ / (384EI)
  Mmax = wL² / 8

Cantilever, end point load:
  δmax = FL³ / (3EI)
  Mmax = FL

F = point load (N), w = distributed load (N/m), L = span (m), E = elastic modulus (Pa), I = second moment of area (m⁴). Deflection δ is downward; moment M causes bending stress σ = M×c/I.

Worked Examples

Example 1 — Simply Supported with Center Load

Steel beam: L = 4 m, F = 10,000 N, E = 200 GPa, I = 8.33×10⁻⁶ m⁴. δmax = (10000×64)/(48×200×10⁹×8.33×10⁻⁶) = 8 mm. Mmax = 10000×4/4 = 10,000 N·m.

Example 2 — Simply Supported with UDL

Timber beam: L = 3 m, w = 2,000 N/m, E = 12 GPa, I = 1×10⁻⁵ m⁴. δmax = 5×2000×81/(384×12×10⁹×10⁻⁵) ≈ 8.8 mm. Mmax = 2000×9/8 = 2,250 N·m.

Example 3 — Cantilever with End Load

Steel cantilever: L = 2 m, F = 5,000 N, E = 200 GPa, I = 4×10⁻⁶ m⁴. δmax = (5000×8)/(3×200×10⁹×4×10⁻⁶) ≈ 1.67 mm. Mmax = 5000×2 = 10,000 N·m.

Frequently Asked Questions

What is deflection in a beam?▾

Deflection is the displacement of the beam from its original horizontal position under load. Excessive deflection can cause cracking in finishes, misalignment of machinery, or structural failure. Building codes typically limit deflection to L/360 for floors and L/240 for roofs.

What is the second moment of area (moment of inertia)?▾

The second moment of area (I) measures a cross-section's resistance to bending. A larger I means less deflection and lower bending stress for the same load. It depends on the cross-sectional shape — an I-beam has a much higher I than a solid rectangle of the same area.

What is elastic modulus?▾

Elastic modulus (E, or Young's modulus) measures a material's stiffness. Steel: ~200 GPa, aluminum: ~70 GPa, timber: ~10–15 GPa, concrete: ~30 GPa. A stiffer material (higher E) deflects less under the same load.

What is a simply supported beam?▾

A simply supported beam rests on two supports — one pinned (fixed against translation but free to rotate) and one roller (free to translate horizontally and rotate). This is the most common beam type in buildings and bridges.

What is a cantilever beam?▾

A cantilever beam is fixed at one end and free at the other. The fixed end resists both vertical forces and bending moments, while the free end deflects under load. Balconies, diving boards, and aircraft wings are examples of cantilever structures.