Beam Formulas & Structural Calculations
Beam Formulas
Deflection: δ = (W * L³) / (48 * E * I)
Bending Stress: σ = (M * y) / I
Continuous Beams
Moment Distribution Method and Clapeyron's Theorem are applied.
Fixed-End Moment Formula: FEM = (w × L²)/12
Ultimate Strength of Continuous Beams
Mu = φ × Mp (Plastic moment capacity method)
Beams of Uniform Strength
σ = (M * y)/I = constant → I varies with M(x)
Safe Loads for Beams
Safe Load = (Allowable Stress × Section Modulus)/Factor of Safety
Rolling and Moving Loads
Use Influence Line Diagrams (ILDs) for bending and shear calculations.
Curved Beams
σ = (M * (R - y)) / (A * y * (R - e))
Elastic Lateral Buckling
Critical Load: Pcr = (π² × E × I) / (L²)
Combined Axial & Bending Loads
σ = (P/A) ± (M*y/I)
Unsymmetrical Bending
σ = (M * y) / I about inclined neutral axis.
Eccentric Loading
σ = (P/A) ± (P*e*y/I)
Natural Circular Frequencies
ωn = √(k/m), T = 2π/ωn
Torsion in Structural Members
Torsional Stress: τ = (T*r)/J
Strain Energy in Members
U = ∫(M² / 2EI) dx
Fixed-End Moments in Beams
FEM for UDL: M = (w × L²)/12, for Point Load: M = (P × L)/8
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