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Beam Bending Calculator

Calculate deflection, stress, shear force, and bending moment for beams with various support conditions. Includes safety checks against deflection limits and allowable stress.

About This Tool

Purpose

This calculator analyzes elastic beam behavior using Euler-Bernoulli beam theory. It supports multiple load cases, cross-section types, and materials with full equation transparency.

Features

  • 10 common load/support configurations
  • 8 cross-section types including standard steel shapes
  • Material library with common engineering materials
  • Interactive diagrams showing deflected shape
  • Safety checks for deflection limits and stress utilization
  • Full derivation display with governing equations

Limitations

  • Linear elastic analysis only (small deflections)
  • Prismatic beams (constant cross-section)
  • Does not account for shear deformation (slender beams only)
  • Single-span beams only

References

  • Roark's Formulas for Stress and Strain, 8th Edition
  • AISC Steel Construction Manual, 15th Edition
  • Gere & Timoshenko, Mechanics of Materials

Quick analysis with common defaults. Switch to Advanced for full control over materials, safety factors, and deflection limits.

Inputs

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Total load on beam: 50,000 N

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E = 200 GPa, Fy = 250 MPa
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Allowable stress = Fy / SF

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Beam Configuration

Cross Section

Configure beam parameters and click Calculate to see results.

Beam Configuration

Deflected Shape (Exaggerated)

Euler-Bernoulli Beam Theory

This calculator uses classical beam theory, which assumes small deflections, linear elastic material behavior, and plane sections remaining plane after bending.

Fundamental Beam Equation
\[ EI \frac{d^4 y}{dx^4} = w(x) \]

The governing differential equation relates the distributed load \(w(x)\) to the beam deflection \(y(x)\) through the flexural rigidity \(EI\).

Moment-Curvature Relationship
\[ M(x) = EI \frac{d^2 y}{dx^2} \]

Bending moment is proportional to curvature. This is the basis for determining deflections by double integration of the moment diagram.

Shear-Moment Relationship
\[ V(x) = \frac{dM}{dx} \quad \text{and} \quad w(x) = \frac{dV}{dx} \]

Shear force is the derivative of moment; distributed load is the derivative of shear. These relationships are used to construct shear and moment diagrams.

Load Case Formulas

Specific equations for the selected load case:

Maximum Deflection
Select a load case to see equations

Maximum vertical displacement of the beam. For simply supported beams with symmetric loading, this occurs at midspan. For cantilevers, at the free end.

Maximum Bending Moment

Peak internal moment that causes bending stress. Location depends on load type and support conditions.

Maximum Shear Force

Peak internal shear force. Typically maximum at supports or at concentrated load locations.

Stress Analysis

Determining stresses from internal forces:

Bending Stress (Flexure Formula)
\[ \sigma = \frac{M \cdot y}{I} = \frac{M}{S} \]

Normal stress varies linearly through the cross-section depth. Maximum stress occurs at the extreme fibers (top and bottom). \(S = I/c\) is the section modulus.

Maximum Bending Stress
\[ \sigma_{max} = \frac{M_{max} \cdot c}{I} \]

Where \(c\) is the distance from the neutral axis to the extreme fiber. For symmetric sections, \(c = h/2\).

Shear Stress (Approximate)
\[ \tau_{avg} = \frac{V}{A_{web}} \quad \text{or} \quad \tau_{max} = \frac{VQ}{Ib} \]

Shear stress distribution is parabolic through the depth. The exact formula uses the first moment of area \(Q\). For rectangular sections, \(\tau_{max} = 1.5 V/A\).

Allowable Stress Design
\[ \sigma_{allow} = \frac{F_y}{SF} \quad \text{where } SF \geq 1.5 \]

The allowable stress is the yield strength divided by a safety factor. Common factors: 1.5-2.0 for buildings, 2.5-4.0 for critical applications.

Section Properties

Key geometric properties of cross-sections:

Moment of Inertia (Second Moment of Area)
\[ I = \int y^2 \, dA \]

Measures resistance to bending. Larger \(I\) means less deflection and lower stress for the same moment.

Common shapes:
Rectangle: \( I = \frac{bh^3}{12} \)
Circle: \( I = \frac{\pi d^4}{64} \)
Hollow circle: \( I = \frac{\pi (d_o^4 - d_i^4)}{64} \)
Section Modulus
\[ S = \frac{I}{c} \]

Convenient property for stress calculations: \(\sigma = M/S\). For symmetric sections, \(S_{top} = S_{bottom}\).

Radius of Gyration
\[ r = \sqrt{\frac{I}{A}} \]

Used in buckling calculations. Represents the distance at which the entire area could be concentrated to produce the same moment of inertia.

Deflection Limits

Serviceability requirements for acceptable deflection:

Common Deflection Limits
L/180 Roof members not supporting ceiling
L/240 Floor members, general use
L/360 Floor members supporting brittle finishes (plaster, tile)
L/480 Precision equipment, sensitive machinery
L/600 High-precision applications, optical equipment