| name | curved-beams |
| description | Curved beam stress analysis — neutral axis shift, Winkler-Bach formula, crane hooks, rings, chain links, correction factor K for curved beams. |
| metadata | {"priority":7,"promptSignals":{"phrases":["curved beam","crane hook","curved bar","Winkler-Bach","neutral axis shift"],"minScore":3}} |
Curved Beams — Complete Skill
Neutral Axis Location
For a curved beam, the neutral axis (NA) shifts toward the center of curvature relative to the centroidal axis.
Radius of neutral axis:
r_n = A / ∫ dA/r [integral over cross-section]
A = cross-section area, r = radial distance from center of curvature
Eccentricity: e = r̄ - r_n (always positive; r̄ = centroidal radius)
Integral ∫dA/r for common sections:
Rectangle (b×h, inner radius r_i, outer r_o): ∫dA/r = b × ln(r_o/r_i)
Circle (diameter d, centroid at r̄): ∫dA/r = π(r̄ - √(r̄² - d²/4)) (approx for r̄ >> d: ≈ πd²/(4r̄))
Trapezoidal section: sum of triangular components
Winkler-Bach Formula (Curved Beam Bending Stress)
σ = M(r_n - r) / (A × e × r)
Where:
M = bending moment [N·mm] (positive = opening the curve)
r = radial distance from center of curvature to point of interest
r_n = neutral axis radius
e = eccentricity = r̄ - r_n
Inner fiber (r = r_i):
σ_i = M(r_n - r_i) / (A × e × r_i) [tension for positive M = opening]
Outer fiber (r = r_o):
σ_o = M(r_n - r_o) / (A × e × r_o) [compression for positive M]
K-Factor Approach (Correction Factor Method)
σ = K × M × c_i / I (stress at inner fiber — always higher)
K = correction factor (K > 1; increases as r̄/c decreases)
K values (rectangular section, r̄ = centroidal radius, c = half-depth):
r̄/c = 1.2: K_i = 3.63; r̄/c = 1.4: K_i = 2.73; r̄/c = 2.0: K_i = 1.97; r̄/c = 3.0: K_i = 1.57; r̄/c = 6.0: K_i = 1.24; r̄/c → ∞: K_i → 1.0 (straight beam)
Direct Stress + Bending (Crane Hook)
For a curved member with axial load P and moment M:
σ_total,i = P/A + M(r_n - r_i)/(A × e × r_i) [at inner fiber]
σ_total,o = P/A + M(r_n - r_o)/(A × e × r_o) [at outer fiber]
Crane hook: P = hook load (tension), M = P × r̄ (bending moment about centroid)
Critical point: inner fiber of hook curve = maximum tensile stress
Thin Ring Under Diametral Load
For a thin circular ring (r >> t) under two diametrically opposite loads P:
M_max (at load point) = P × r / π = 0.318 P r
M_max (at 90°) = P × r × (1/π - 1/2) = -0.182 P r
σ = ±6M/(bt²) (thin ring approximation; for thick ring use Winkler-Bach)
Output
Provide: r_n [mm], e [mm], σ_inner and σ_outer [MPa], K correction factor, combined stress with direct load σ_total [MPa], safety factor vs. S_y.
