1 · Scope and conventions
Concept-stage check for two-pin parabolic arches. The workbench produces first-pass internal forces — H, V, N at crown / springing, M at crown, deck local moments, hanger and tie forces — for four arch types under three live-load patterns. It is a teaching / scoping tool; full design needs a proper FE model with second-order effects and detailed buckling checks.
Units
kN, kNm- internal force / moment outputs
m- spans, rises, deck and rib depths, hanger spacing
kN/m²- permanent and live area loads (gk, qk)
kN- concentrated point load Q
%- axial-deformation reduction proxy (creep / shrinkage / temperature)
Partial factors
γG = 1.35- permanent ULS factor
γQ = 1.50- leading variable ULS factor
Sign convention
H, V- horizontal / vertical reactions at the springing (positive inward / upward)
N- axial compression positive (arch in compression)
M_crown- sagging positive (causes outward thrust line offset)
e/d- thrust line eccentricity / rib depth — small = arch close to anti-funicular
Equations below copy the implementation verbatim. Where the workbench applies a coefficient that comes from energy minimisation (e.g. u = 25Q/16L for deck-stiffened with point load), the closed-form result is shown — the derivation lives in T4 Problem 1.
2 · Arch geometry & anti-funicular shape
The arch is a second-order parabola passing through the two springings with mid-span rise r. For uniform load g per unit horizontal length, the parabola is the anti-funicular shape — the load can be carried by axial response only, with no bending in the rib (L4 §Arches).
y(x) = 4 r · x (L − x) / L² with x measured along the horizontal projection
The 3D scene draws this profile and the deck either above (through / bowstring) or below (deck-stiffened / rigid). The thrust line — the locus where the resultant of N actually passes — is overlaid in gold and shifts away from the rib axis as axial deformation kicks in.
Anti-funicular condition (from L4): if M_simply_supported_beam(x) ∝ y(x) then the rib alignment is the anti-funicular of that load ⇒ M_arch(x) = 0 (under the matching load only)
Implemented in components/arch-bridge-workbench/scene.tsx (function archY).
3 · Permanent state — pure axial
The permanent load is taken as a uniform line load on the deck. The parabolic rib is anti-funicular for it, so the elastic horizontal reaction follows the standard formula and the rib carries no bending (until axial deformation modifies the picture in §4).
3.1 Loads on the deck
w_perm,line = g_k · b_deck [kN/m] w_perm,ULS = γG · w_perm,line (γG = 1.35)
3.2 Springing reactions (elastic, no axial deformation)
H* = w · L² / (8 r) (horizontal thrust) V = w · L / 2 (vertical reaction) Derivation: counter-balance the simply-supported BMD wL²/8 with the couple H · r at the crown.
3.3 Axial along the arch
For a parabolic anti-funicular under uniform w, the axial force varies along the rib:
N_crown = H (rib is horizontal at the crown, V = 0)
N_springings = √( H² + V² )
= (w·L / 2) · √( 1 + L² / (16 r²) )Implementation: lib/.../bridge.ts in evaluateArchBridge.
4 · Axial deformation & pre-jacking
When the arch shortens (creep, shrinkage, prestress, temperature drop) the springings cannot pull together, so the actual horizontal reaction H is smaller than the elastic H*. The difference ΔH = H* − H is reacted by the rib as bending — the arch is no longer purely axial. L4 writes this from compatibility of horizontal displacement at the cut springing using the unit-load theorem:
H = ∫ M_I(x)·M_II(x)/EI dx + ∫ N_I(x)·N_II(x)/EA dx
─────────────────────────────────────────────────
∫ M_II(x)²/EI dx + ∫ N_II(x)²/EA dx
with axial deformation neglected:
H* = ∫ M_I·M_II/EI / ∫ M_II²/EI
⇒ H ≤ H* (axial deformation always reduces thrust)
⇒ ΔH = H* − H
⇒ M(x) = (H* − H) · y(x) = ΔH · y(x)
⇒ M_crown,axial = ΔH · r (worst at mid-span)The workbench treats ΔH/H* as a single user-input proxy (Axial loss %) instead of integrating EI / EA. This keeps the inputs concept-level. The number maps to the physical effect 1-for-1: 5 % axial loss means H drops to 0.95 H* and a residual sagging M_crown of 0.05 · H* · r.
4.1 Pre-jacking at the crown
L4 describes the construction trick of jacking apart the two half-arches at the crown with a force equal to H* before stitching the closure pour. All axial shortening then happens before the rib is continuous, so when the jacks are released the rib closes at the design length with no residual moment:
Pre-jack force = ΔH = H* − H_eff With pre-jacking enabled: M_crown,axial = 0 Live-load M_crown still applies (only axial-deformation BM is cancelled).
Toggle: preJacked in the right rail.
5 · Live-load cases
Three live patterns are wired into the workbench, lifted directly from T4 Problem 1. Each produces its own H, V, arch crown M and (for deck-stiffened) deck local M.
5.1 Full UDL — Case (a)
Live load q_k uniform on full span (q_k · b_deck) at γQ = 1.5 H_live = w_live · L² / (8 r) (anti-funicular — no arch BM) V_live = w_live · L / 2 (uniform at both supports) M_arch = 0 Deck local M = w_live · s² / 24 (between piers, deck-stiffened)
5.2 Half UDL — Case (c)
Half-span loading is the classical asymmetric case. T4 decomposes it into a symmetric q/2 over the full span (anti-funicular) and an antisymmetric q/2 (no net thrust, but rib bending in stiff arches):
H_live = w_live · L² / (16 r) (= half of the full-UDL H) V_left = w_live · L / 8 V_right = 3 · w_live · L / 8 Rigid / through / bowstring: M_arch,crown ≈ w_live · L² / 64 (asymmetric component) Deck local ≈ w_live · s² / 32 Deck-stiffened (T4 Problem 1c): Asymmetric u = 0 ⇒ M_arch = 0 Deck takes both halves: Deck local = w_live · s² / 24 + w_live · L² / 96
5.3 Point load at mid-span — Case (b)
A concentrated wheel / inspection load. T4 derives the deck-stiffened case in detail: the energy-minimising uniform interaction u between deck and arch is non-trivial.
Q_ULS = γQ · Q Deck-stiffened (T4 Problem 1b): u = 25 · Q / (16 · L) (energy minimisation) H_live = u · L² / (8 r) = 25 Q L / (128 r) ≈ 0.195 Q L / r V_live = u · L / 2 = 25 Q / 32 ≈ 0.781 Q Deck M_max = 7 · Q · L / 128 ≈ 0.055 Q L M_arch = 0 Rigid / through / bowstring (rib resists bending): H_live = Q · L / (4 r) (anti-funicular component from V = Q/2) V_live = Q / 2 M_arch_crown ≈ Q · L / 8 (≈ half of simply-supported QL/4)
Implementation: liveEffects() in bridge.ts.
6 · Per-arch-type behaviour
The four arch types in L4 differ in how vertical load gets from the deck to the springing. The workbench shows different "Behaviour" cards per type so the load path is visible.
6.1 Deck-stiffened arch
Arch is slender — flexural stiffness too low to take bending. The deck takes all the asymmetric live BM between piers, the piers transfer a uniform interaction u down to the rib, and the rib is in pure compression.
Pier compression f = u · s (u from energy minimisation) Deck local M: permanent = w_perm · s² / 24 live = case-dependent (see §5) Arch: N per §3.3, no bending
6.2 Rigid arch
Deeper rib resists both compression and asymmetric live-load bending. Crown M is the design driver under non-anti-funicular load.
Crown M = M_crown,axial + M_crown,live e/d = M_crown / N_crown / d_rib ← workbench check
6.3 Through arch
Deck is suspended below the arch on hangers. Arch carries axial + asymmetric live moment as in 6.2; thrust still goes to the abutments.
Hanger NEd = (w_perm,ULS + w_live,case) · s (per hanger panel)
Arch = N_crown / N_springings as in §3.3
+ crown M from asymmetric live6.4 Bowstring (tied arch)
The deck tie carries the arch thrust internally, so the abutment horizontal reaction is essentially zero. This is the only configuration that works on competent / soft ground, since the foundations only see the vertical reaction.
Tie tension = H_eff_total (= N_crown)
Hanger NEd = per panel as §6.3
Abutment H ≈ 5 % of H_total (residual from temperature differentials,
construction tolerances, etc.)7 · Proportion & detailing checks
All limits come from L4 (Arches section), supplemented by a crown eccentricity check derived from §4 and an out-of-plane stability indicator.
Rise / span PASS 0.10 – 0.25 L4
REVIEW outside but in 0.08 – 0.32
FAIL otherwise
Arch rib depth / span PASS 1/100 – 1/50 L4
REVIEW 1/140 – 1/35
FAIL otherwise
Panel / deck depth PASS 8 – 18 L4 (≈ 1/15 between piers)
REVIEW 4 – 24
FAIL otherwise
Crown e / d PASS ≤ 0.5 rib stays in compression
REVIEW 0.5 – 1.0
FAIL > 1.0 tension cracking territory
Out-of-plane (L / rib spacing)
PASS ≤ 35 with ≥ 2 ribs
REVIEW 35 – 50 or single rib
FAIL > 50
Foundation bowstring → PASS
arch on sound rock → PASS
arch on competent ground → REVIEW
arch on soft ground → FAILOut-of-plane is rough — a real check needs an out-of-plane buckling analysis with effective length, transverse bracing, and possibly torsional response. The workbench rule only catches obviously-too-slender schemes.
8 · Caveats
- Two-pin parabolic only. Three-pin, fixed-end, catenary and spline arches are not modelled.
- Axial-deformation as a proxy. The workbench uses an Axial loss % input rather than integrating EI / EA. The two are equivalent at concept stage but the % needs engineering judgement.
- First-order analysis. No second-order effects — stability and member buckling need a proper FE model.
- Half-UDL coefficient. M_arch ≈ wL²/64 is a rough energy-based estimate for stiff arches. The deck-stiffened case follows the closed-form T4 result.
- Hanger force. Static UDL only — fatigue, vortex-induced vibration and cable replacement are not addressed.
- Out-of-plane. Rule of thumb only. Spatial arches (out-of-plane geometry, lecture L4) need a separate spatial analysis.
- No detailed combinations. Wind, seismic, temperature gradient and accidental actions are not in the load model.