AISC 360 / Eurocode 3
Fundamentals
Structural Mechanics
Euler-Bernoulli Beam Deflection, Shear & Moment
Calculate Shear Force Diagrams (SFD), Bending Moments (BMD), and elastic deflection profiles.
Solve differential beam equilibrium under combined concentrated point loads and uniformly distributed loads (UDL). Inspect elastic deformed shapes, maximum fiber stresses, and verify AISC L/360 deflection serviceability.
Governing Physical Law & Equations
EI \frac{d^4 w}{dx^4} = q(x), \quad \sigma_{max} = \frac{M_{max} y}{I}
Euler-Bernoulli 4th-order beam equation and flexural stress formula.
Law: Euler-Bernoulli Beam Theory & Hooke’s Law | Standard Reference: AISC 360-16 / Eurocode 3 (EN 1993-1-1) / ASTM A36 (Structural Steel Design)
Adjustable System Parameters
| Parameter |
Nominal Value |
Dynamic Range |
Physical Role |
| Span Length L (L) |
6 m |
2 to 12 m |
Clear distance between support boundaries |
| Concentrated Load P (P) |
45 kN |
0 to 150 kN |
Point load applied along beam span |
| Load Position (a) |
3 m |
0.5 to 10 m |
Distance from left support to concentrated load |
| Uniform Load q (q) |
12 kN/m |
0 to 35 kN/m |
Continuous distributed self-weight and live load |
Analytical Proof & Derivation
From Euler-Bernoulli beam theory: planar cross sections remain plane and normal to the deformed centroidal axis, yielding strain ε = -y (d²w/dx²). Combining with Hooke’s law σ = E ε and moment equilibrium d²M/dx² = -q(x) gives the 4th-order ODE: EI (d⁴w/dx⁴) = q(x). Integrating with pinned-pinned boundary conditions w(0)=w(L)=0 yields midspan deflection δ_max = (P L³)/(48 E I) + (5 q L⁴)/(384 E I).
Verification Benchmark
AISC W250×67 benchmark: Span L = 6.0 m, P = 45 kN at midspan, q = 12 kN/m (E = 200 GPa, I = 84.9×10⁻⁶ m⁴). Theoretical midspan deflection δ_mid = 11.93 mm (point) + 11.93 mm (UDL) = 23.86 mm. AISC L/360 allowable limit = 6000/360 = 16.67 mm.
Field Engineering Insights
AISC 360-16 Table L-1 specifies L/360 for floor beams carrying brittle plaster ceilings and L/240 for general structural roof members. Extreme fiber stress σ = M_max y / I must satisfy LRFD strength φ_b M_n where φ_b = 0.90.