Beams

A beam is a structural member loaded transversely to its longitudinal axis, carrying load primarily by bending and shear. Internal forces are bending moment M(x) and shear force V(x), related by V = dM/dx and load w = -dV/dx. For linear elastic bending, normal stress varies through depth: σ = -M·y / I, maximum at outer fiber σ_max = M / S where S = I / c is elastic section modulus. For compact steel sections that can form plastic hinge, nominal flexural strength is plastic moment Mp = Fy·Z, where Z is plastic section modulus (Z > S). Design condition φb·Mn ≥ Mu, φb=0.90. Limit states for steel beams: 1) Yielding: Mn = Mp = Fy·Z for compact laterally braced sections (Lb ≤ Lp). 2) Lateral-Torsional Buckling (LTB): beam buckles sideways when compression flange unbraced. If Lp < Lb ≤ Lr, Mn = Cb·[Mp - (Mp-0.7Fy·Sx)(Lb-Lp)/(Lr-Lp)] ≤ Mp. If Lb > Lr, Mn = Cb·π²·E·Iy/(Lb²)·... = Fcr·Sx ≤ Mp. Cb accounts for moment gradient (Cb>1 for non-uniform moment). Lp = 1.1·ry·E/Fy (approx), Lr depends on rts and Fy. 3) Local buckling: flange/web slenderness λ = b/t must be ≤ λp for compact to reach Mp. Shear strength: Vn = 0.6·Fy·Aw·Cv1 where Aw = d·tw for rolled shapes, Cv1 shear buckling coefficient ≈1 for most rolled beams with h/tw ≤ 2.24√(E/Fy). φv=0.90. Deflection: δ = ∫ M·m/EI dx. For uniform load: δ_max = 5·w·L⁴/(384·E·I). Serviceability limits typically L/240 for total, L/360 for live load. Beam theory assumes prismatic, small deflections, plane sections remain plane, material homogeneous, and lateral supports as assumed.