Beam Deflection
Beam deflection is the displacement of a beam's neutral axis from its original position under transverse loading, measured perpendicular to the beam's length. It depends on the load magnitude, span length, cross-sectional geometry (moment of inertia), and material stiffness (Young's modulus). Limiting beam deflection is a key serviceability requirement in structural design to prevent cracking of finishes, damage to supported equipment, and discomfort to occupants.
Key Formula
delta_max = (5 × w × L^4) / (384 × E × I)
LaTeX: \delta_{max} = \frac{5wL^4}{384EI}
| Symbol | Meaning | Unit |
|---|---|---|
| δ_max | Maximum mid-span deflection (simply supported, UDL) | m |
| w | Uniformly distributed load per unit length | N/m |
| L | Span length | m |
| E | Young's modulus of beam material | Pa |
| I | Second moment of area (moment of inertia) of cross-section | m⁴ |
Worked Example
Problem
A simply supported steel I-beam of span 6 m carries a uniformly distributed load of 12 kN/m. The beam has I = 85 × 10⁻⁶ m⁴ and E = 200 GPa. Calculate the maximum deflection.
Solution
Step 1: Identify values. w = 12 000 N/m, L = 6 m, E = 200 × 10⁹ Pa, I = 85 × 10⁻⁶ m⁴ Step 2: Calculate numerator. 5 × w × L⁴ = 5 × 12 000 × (6)⁴ = 5 × 12 000 × 1 296 = 77 760 000 Step 3: Calculate denominator. 384 × E × I = 384 × 200 × 10⁹ × 85 × 10⁻⁶ = 384 × 1.7 × 10⁷ = 6.528 × 10⁹ Step 4: Compute deflection. δ_max = 77 760 000 / 6.528 × 10⁹ = 0.01191 m
Answer
Maximum deflection δ_max ≈ 11.9 mm (span/504, acceptable for most floor beams)
Maximum Deflection Formulas for Common Beam and Loading Configurations
| Configuration | Loading | δ_max Formula | Location of δ_max |
|---|---|---|---|
| Simply supported | UDL (w per unit length) | 5wL⁴ / 384EI | Mid-span |
| Simply supported | Central point load P | PL³ / 48EI | Mid-span |
| Cantilever | UDL (w per unit length) | wL⁴ / 8EI | Free end |
| Cantilever | Point load P at free end | PL³ / 3EI | Free end |
| Fixed-fixed | UDL (w per unit length) | wL⁴ / 384EI | Mid-span |
Interactive Tools
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Related Terms
Young's Modulus
Young's modulus (also called the modulus of elasticity) is the ratio of engineering stress to engineering strain in the linear-elastic region of a material's stress-strain curve. It is a fundamental mechanical property that quantifies the stiffness of a solid material — a higher value means the material resists deformation more effectively. Young's modulus is essential in structural design for calculating deflections, natural frequencies, and load-bearing capacity of components.
Torsion (engineering)
Torsion is the twisting of a structural member caused by an applied torque (twisting moment) about its longitudinal axis. In circular shafts, torsion produces a shear stress distribution that varies linearly from zero at the neutral axis to a maximum at the outer surface. Torsion analysis is fundamental for the design of drive shafts, axles, springs, and any component that transmits rotary power.
Engineering Stress
Engineering stress is defined as the applied force divided by the original cross-sectional area of a specimen, regardless of any deformation that occurs during loading. It is the conventional measure used in materials testing and structural analysis because the original dimensions are easily measured before the test begins. Engineering stress is widely used in design calculations, material data sheets, and stress-strain curves to characterise material behaviour under uniaxial loading.
From Latin "deflectere" (to bend aside), combining "de-" (away) and "flectere" (to bend). The systematic analysis of beam bending and deflection was developed by Euler (1744) and further formalised by Navier and Saint-Venant in the 19th century using the Euler–Bernoulli beam theory.