| name | cable-structures |
| description | Cable structure analysis — catenary vs. parabolic approximation, sag-tension, cable stiffness, pretensioning, suspension bridge, cable-stayed structures, wind/ice loading, cable fatigue, AISC Design Guide 9. |
| metadata | {"priority":7,"promptSignals":{"phrases":["cable structure","cable analysis","cable tension","suspension cable","catenary","cable-stayed","sag tension"],"minScore":3}} |
Cable Structures — Complete Skill
Catenary vs. Parabolic Cable
Catenary (Exact — Self-Weight Only)
Shape (uniform distributed weight w per unit arc length):
y(x) = a × cosh(x/a) - a [a = H/w; H = horizontal tension component]
Arc length: s = a × sinh(L/(2a)) × 2 [total length; L = span]
Tension at any point:
T(x) = H × cosh(x/a) = w × s [maximum tension at anchor (end); minimum at midspan]
H = horizontal tension (constant along cable)
Parabolic Approximation (Uniform Load per Horizontal Span)
Valid when: sag d < 10% of span L (sag ratio d/L < 0.1)
Used in: most bridge and structural engineering
Parabolic equation:
y(x) = (4d / L²) × x × (L - x) [origin at left support; d = midspan sag]
Horizontal tension H (from moment at midspan):
H = wL²/(8d) [N; w = load per unit horizontal length [N/m]; d = sag [m]; L = span [m]]
Maximum tension at supports:
T_max = √(H² + (wL/2)²) = H × √(1 + (4d/L)²) ≈ H(1 + 8(d/L)²) [for small sag]
Sag-to-span ratio effect:
d/L = 0.1: T_max/H = 1.077 (7.7% above H)
d/L = 0.05: T_max/H = 1.020 (2% above H)
Elastic Cable (Stiffness)
Cable stiffness under vertical load at midspan:
k = H³ / (w L³/8) × 1/(1 + H/(AE)) [N/m; extremely nonlinear due to geometry change]
k_geometric = 8 T_p / L [N/m; T_p = pretension; L = span; approximate]
k_elastic = AE/L [N/m; elastic stretching component]
Combined (Ernst formula for inclined cable in bridges):
k_equiv = AE / L_chord × 1 / (1 + (w L_chord)² AE / (12 T³))
Second term: sag reduction; large sag → low equivalent stiffness
Cable area for given sag:
From: H = wL²/(8d); T = AE×ε → A = T_max / (ε_allow × E_cable)
d_allow → design sag for acceptable deflection
Pretensioning
Required pretension T_p:
T_p ≥ (W_traffic + W_dead) × L / (8 d_max) + T_wind + T_temp [N]
Goal: keep cable taut under all service loads (no compression in cable allowed)
Temperature effects:
ΔL = α × L × ΔT → changes pretension significantly
ΔT = +50°C: ΔT_tension = -AE α ΔT (tension drops; cable relaxes; sag increases)
Design for worst cold case (maximum tension) and worst hot case (minimum tension, maximum sag)
Cable Properties
Wire rope (structural cable):
Ordinary (6×19 class): ductile; flexible; E ≈ 80–100 GPa (lower than solid due to wire twist)
Locked-coil strand: E ≈ 150–165 GPa; smooth outer surface; less air space
Parallel wire strand: E ≈ 195–205 GPa; maximum stiffness; non-bending application
Ultimate strength:
Wire rope (1770 MPa ultimate): typically f_u = 1570–1860 MPa (individual wire)
Working tension ≤ f_u/2.0 to f_u/3.5 (safety factor 2.0–3.5)
Cable area:
From: required tension T and allowable stress
A = T / (f_u / FS) [mm²]
Standard cable sizes: from 10 mm to 150+ mm diameter
Suspension Bridge (Main Cable)
Main cable parabolic approximation:
H = (w_dead + w_live/2) × L² / (8 × d)
For asymmetric loading: use influence line for maximum H
Side span (anchor):
Anchor force: A = H / cos θ_anchor [θ_anchor = inclination angle of back cable]
Anchor must resist A in bearing against soil or rock
Hanger tension:
T_hanger = P_deck × hanger_spacing / (number of hangers sharing load)
Deck stiffness interaction:
Deck + cable system: stiffened cable (cable + deck girder share load)
Rigidity ratio: EI_deck / (H × L²); higher → more uniform load distribution
Cable-Stayed Structures
Cable force components:
Inclined cable at angle θ from horizontal:
Vertical component: V = T × sin θ [kN; compresses pylon]
Horizontal component: H = T × cos θ [kN; applies prestress to deck]
Deck design:
Deck acts as prestressed beam; cables apply vertical support + horizontal compression
Required anchor zone capacity: T_max (cable tension at deck attachment)
Typical cable inclination: θ = 25–65° (steeper → more efficient for vertical; less horizontal compression)
Wind and Ice Loading
Wind drag on cable (ASCE 7-22):
P_wind = q × D × C_D [N/m; q = wind pressure [Pa]; D = cable diameter [m]; C_D = drag coefficient ≈ 1.2]
q = 0.5 × ρ × V² × K_z × K_zt [Pa; ρ = 1.25 kg/m³; V = design wind speed; K factors from ASCE 7]
Galloping (aerodynamic instability):
Occurs for ice-coated or non-circular cross-sections at critical reduced velocity
V_gallop = k_f × f × D / (ρ D C_D_slope)
Prevention: cable detuners, dampers, spiral strakes, helical wires
Ice load:
W_ice = π t_ice (D + t_ice) × ρ_ice × g [N/m; t_ice = ice thickness; ρ_ice = 900 kg/m³]
Ice adds weight → increased sag and tension; design check critical
Cable Fatigue
Fatigue of wire rope:
Alternating bending at saddles and guide points; fretting fatigue between wires
S-N curve: ΔT/T_break vs. N (cycles); for T/T_break = 0.4: Δσ_allow ≈ 30–50 MPa (net wire stress range)
Minimum bending radius:
r_min = 10–15 × d_rope [to limit bending fatigue; d_rope = rope diameter]
Saddle radius > r_min always
Standards
| Standard | Scope |
|---|
| AISC Design Guide 9 | Cable stays and suspension structures |
| EN 1337 | Structural bearings (for cable anchors) |
| ASCE 7-22 | Wind loading for cables |
| PTI DC80.3 | Recommendations for stay cable design |
| ASTM A586 / A603 | Structural strand and wire rope |
Output
Provide: cable geometry (span L [m], sag d [m], d/L ratio), load per unit length w [kN/m], horizontal tension H [kN], maximum cable tension T_max [kN], cable size/area A [mm²] and type (strand/rope/locked coil), pretension T_p [kN], temperature sensitivity ΔT_tension per ΔT [kN/°C], wind and ice load contribution [kN/m], equivalent cable stiffness k [kN/m], fatigue check (ΔT/T_break vs. allowable), safety factor FS = T_break/T_max, and applicable standard (AISC DG9, PTI DC80.3, ASTM A586).
