Euler-Bernoulli Beam Theory

The Euler-Bernoulli beam equation describes the relationship between a beam's deflection and the applied load. It assumes that plane sections remain plane and perpendicular to the neutral axis after bending.

EI * d4y/dx4 = w(x)

Where E = elastic modulus, I = moment of inertia, y = deflection, and w(x) = distributed load.

• Assumptions: Linear elastic material, small deflections, no shear deformation
• Validity: Best for slender beams where length > 10x depth
• Limitations: Does not account for shear deformation (use Timoshenko theory for short beams)

Support Conditions

The boundary conditions at beam supports determine the deflection behavior:

• Simply Supported (Pinned-Roller): Zero deflection at both ends, free rotation. Most common for floor joists and bridge spans.
• Cantilever (Fixed-Free): One end rigidly fixed (zero deflection and slope), other end free. Used for balconies, brackets, and overhangs.
• Fixed-Fixed (Built-in): Both ends rigidly fixed. Results in lower deflection but higher end moments. Common in continuous beams.

Load Types

• Point Load (P): Concentrated force at a specific location. Units: N, kN, lbf
• Distributed Load (w): Load spread over length. Units: N/m, kN/m, lb/ft
• Moment (M): Applied torque/couple at a point. Units: N-m, kN-m, lb-ft

Moment of Inertia and Section Modulus

The moment of inertia (I) measures a cross-section's resistance to bending:

I = integral(y^2 dA)

• Rectangle: I = bh^3/12 (b = width, h = height)
• I-Beam: I = (BH^3 - bh^3)/12 (optimized for bending)
• Circular: I = pi*d^4/64
• Hollow Tube: I = pi*(D^4 - d^4)/64

Section Modulus (S = I/c) relates to maximum bending stress: sigma = M/S = Mc/I

Deflection Limits by Code

Building codes specify maximum allowable deflections for serviceability:

• L/360: Floor beams supporting plaster ceilings (IBC)
• L/240: Floor beams, general (most common)
• L/180: Roof beams, not supporting ceilings
• L/600: Crane runway beams (AISC)
• L/1000: Precision machinery bases

Stress Distribution in Beams