| name | arch-analysis |
| description | Arch structural analysis — three-hinged/two-hinged/fixed arches, thrust line, parabolic arch, horizontal thrust, influence lines, buckling, concrete arches, AISC/ACI arch design. |
| metadata | {"priority":7,"promptSignals":{"phrases":["arch analysis","arch structure","parabolic arch","arch thrust","three hinged arch","arch bridge"],"minScore":3}} |
Arch Structural Analysis — Complete Skill
Arch Types
Three-hinged arch: two springings + one crown hinge; statically determinate; no temperature/settlement stress
Two-hinged arch: hinges at springings only; one degree indeterminate; horizontal thrust H = reaction
Fixed arch: fully fixed at both ends; three degrees indeterminate; maximum efficiency; sensitive to temperature
Reactions and Horizontal Thrust
Three-Hinged Parabolic Arch (UDL)
Geometry: y = (4f/L²) × x × (L-x) [f = rise; L = span; x from left springing]
At crown (x = L/2): y = f
Reactions (UDL w [N/m] over full span):
V_A = V_B = wL/2 [vertical reactions, symmetric]
Horizontal thrust H (from moment about crown hinge):
Taking moments about crown from left half:
V_A × (L/2) - H × f - w(L/2) × (L/4) = 0
H = V_A × L/(2f) - w L²/(8f) = wL²/(8f) [for full UDL]
General H for arbitrary loading:
H = M₀_crown / f [M₀_crown = simply-supported beam moment at midspan position]
Two-Hinged Arch (Elastic Solution)
Compatibility equation:
H = [∫ M₀ y ds / EI] / [∫ y² ds / EI + L³ / (3EA_rib)]
M₀ = simply-supported beam moment at arch axis position
y = arch height at position s; EA_rib = axial stiffness contribution (often neglected)
Simplified (neglect axial):
H ≈ ∫₀^L M₀(x) y(x) dx / ∫₀^L y²(x) dx
For parabolic arch under UDL w:
H = 5wL²/(8f) × (1/(1 + 2f²/5L²)) ≈ wL²/(8f) for f/L < 0.3 [approx same as 3-hinged]
Internal Forces
Normal force N, shear V_shear, moment M:
For parabolic arch at angle φ (tangent to arch):
N(x) = -H cos φ - V sin φ [compression dominant; thrust arch]
V_s(x) = V cos φ - H sin φ [shear; small for ideal arch]
M(x) = M₀(x) - H × y(x) [bending moment; zero if arch shape = funicular of load]
Funicular principle:
For given load pattern, choose arch shape = bending moment diagram (scaled by H)
→ Pure compression; no bending; maximum efficiency
Parabolic arch = funicular of UDL; catenary = funicular of self-weight
Thrust Line and Kern
Thrust line: locus of resultant force (N) at each cross-section
For no tension in concrete arch: thrust line must stay within kern (middle third of cross-section)
Kern:
Rectangle: k = b/6 (each side from center axis)
Circle: k = D/8
For arch rib: e = M/N ≤ t/6 [e = eccentricity; t = rib thickness; middle-third rule]
Check: at each section, calculate e = M/N; verify e ≤ kern limit
Buckling of Arches
In-plane buckling (symmetric snap-through):
For parabolic arch (fixed at springings), critical load parameter:
λ_cr = EI / (wL⁴) × C_mode [C_mode from tables; depends on rise ratio f/L and boundary conditions]
Simplified elastic in-plane buckling:
N_cr = α × EI / L² [Euler-like; α ≈ 7.8–40 depending on arch type and mode]
For three-hinged parabolic arch: first mode symmetric; N_cr ≈ EI × (π/L)² × (4 + 4f²/L²)
Out-of-plane buckling:
W_cr = C × EI_y / L² [C depends on arch geometry, bracing]
Bracing requirement: out-of-plane bracing at 1/4, 1/2, 3/4 span for long arch bridges
Concrete Arch Design (ACI 318)
Design compressive force:
P_u = factored horizontal thrust + weight of rib
Check: P_u / (0.85 f'c A_gross) ≤ 0.8 [for tied arch]
Slenderness ratio for arch:
k = 1.0 (three-hinged); 0.7 (fixed both ends); 0.5 (fixed + symmetric)
L_eff = k × L_rib
If kL/r > 100 → use second-order analysis
Reinforcement:
Minimum longitudinal: ρ_min = 0.01 A_gross; maximum 0.08 A_gross
Ties: spacing ≤ 16 × d_bar or 48 × d_tie or least dimension of section
Steel Arch Design (AISC 360)
Compression check:
P_u ≤ φ_c P_n [φ_c = 0.90; P_n from AISC Chapter E]
Combined compression + bending (Chapter H):
P_u / P_n + 8/9 × (M_u / M_n) ≤ 1.0 [if P_u / P_n ≥ 0.2]
P_u / (2P_n) + M_u / M_n ≤ 1.0 [if P_u / P_n < 0.2]
Temperature Effects
Temperature change ΔT causes arch to change length:
Free expansion: δ_T = α × L × ΔT
For fixed arch: H_T = (α ΔT × L × EI / I) / (∫ y²/EI ds) [additional horizontal force]
α_steel = 11.7×10⁻⁶ /°C; α_concrete = 10×10⁻⁶ /°C
Three-hinged arch: statically determinate; no temperature stress (only displacements)
Output
Provide: arch type (three-hinged/two-hinged/fixed), span L [m], rise f [m] and f/L ratio, horizontal thrust H [kN] at each load case, internal forces N/V_s/M at L/4, L/2, 3L/4 [kN, kN·m], thrust line eccentricity e [mm] vs. kern limit, in-plane buckling load factor, out-of-plane bracing recommendation, temperature load contribution ΔH [kN] per ΔT [°C], rib cross-section design (area, I), and applicable standard (AISC 360 Chapter E/H or ACI 318).
