Inverted Hanging Chain & Hyperbolic Cosine

Catenary Arch Calculator (y = a(cosh(x/a) − 1))

Calculate hyperbolic cosine scaling parameters (a), exact catenary arc length (L = 2a sinh(W/2a)), springline thrust angles, and self-weight pure compression curves for Gaudí vaults and Gateway arches.

Pure Self-Weight Compression Catenary
e.g. 60 or 630
in / ft
e.g. 30 or 630
in / ft
Iconic Catenary Structures: Historic
Hyperbolic Cosine Catenary Vector
EXACT ARC LENGTH (L) 2a sinh(W/2a)
82.35 units
+37.2% vs straight span
CATENARY CONSTANT (a) Apex Radius
17.22 units
Scaling parameter 'a'
SPRINGLINE ANGLE arctan(sinh(W/2a))
67.3 °
Steep thrust vector
SPANDREL AREA Under-Arch
1,025 sq units
Exact hyperbolic area
CATENARY HYPERBOLIC EQUATION y = 17.22·(cosh(x / 17.22) − 1)
TENSILE / BENDING MOMENT M = 0.00 (Zero Bending Under Self-Weight)
HISTORIC PRINCIPLE Hooke's Inverted Chain (1675)
Gaudí Funicular Simulator

Interactive Inverted Hanging Chain Simulator

Adjust the rise-to-span slider to observe how Antoni Gaudí and Eero Saarinen scaled catenary arches into standing monuments of pure compression.

30.0 Units (a = 17.22)
Apex Curvature Radius (a) 17.22 units Crown scaling constant
Springline Incline Vector 67.3° Incline arctan(sinh(W/2a))
Thrust Equilibrium 100% Pure Axial Compression Funicular under self-weight
Field Layout Generator

Offset-Coordinate "Mark-Out" Table & Interactive Plotter

For large arches where the swing radius exceeds your compass or room length. Simply draw your baseline, measure out horizontal intervals from center (X), and mark vertical height (Y) to plot perfect curves directly on wood or wall.

Coordinate Stepping Points (Centerline Outward)
Based on current Span: 48" • Rise: 12" • Radius: 30.00"
Step Interval:
Pt # Dist from Center (X) Height (Y) Fraction Drop
💡 Tip: Symmetrical — use identical Y heights for both left and right sides.

Live Plywood Scribing Simulator

Point 1 of 13
Plotted Offset (X): 0.00" (Center Apex)
Height Y: 12.000" (12") Drop: 0.000"
3/4" Plywood Blank Baseline X-Axis (Springline) Y: 12"

🔨 Carpentry Technique: Tap finish brads at coordinate marks. Bend a flexible batten against the pins to scribe a smooth curve.

Hyperbolic Trigonometry

Catenary Arch Equations & Calculus

The mathematics of hyperbolic functions governing hanging chains and compression arches.

1. Hyperbolic Profile Equations
Catenary Profile: y(x) = a × (cosh(x / a) − 1)
Definition: cosh(u) = (e^u + e^(−u)) / 2
Apex Curvature Radius: R_apex = a
Springline Tangent: tan θ = sinh(W / (2a))

The scaling constant $a$ represents the ratio of horizontal tension to weight per unit length ($a = T_0 / w$).

2. Exact Arc Length & Area
Exact Arc Length: L = 2 × a × sinh(W / (2a))
Definition: sinh(u) = (e^u − e^(−u)) / 2
Spandrel Area: A = 2a² sinh(W / 2a) − W·a
Horizontal Thrust: H = w_weight × a

Unlike circular arches, catenary arches naturally steepen toward the springline to match dead load weight accumulation.

Step-by-Step Worked Numeric Example

Suppose you are building a freestanding catenary masonry arch or kiln portal with a clear base span of W = 60.0 inches and a vertical apex height of H = 30.0 inches.

Step 1: Catenary Parameter (a) Solve 30 = a(cosh(30/a)−1) a = 18.42 inches
Step 2: Elevation @ x = 15" 30 − 18.42(cosh(0.814)−1) y(15) = 23.45 inches
Step 3: Perimeter Arc (L) 2 × 18.42 × sinh(1.629) L = 90.38" (7' 6 3/8")
Step 4: Thrust Equilibrium H_thrust = w × a Pure compression (zero tension)
Structural Mechanics & Physics

Structural Thrust Dynamics & Lateral Force Engineering

Understand how gravitational loads resolve into outward lateral forces, and how to size king studs, headers, and abutments to prevent framing deflection.

Thrust Equation

The Fundamental Law of Arch Thrust

The horizontal outward thrust (H_thrust) generated at each springline abutment is inversely proportional to the rise height (H):

H_thrust = (Total Load × Span) / (8 × Rise)

Key Rule: Halving the rise height doubles the outward lateral force pushing against your king studs or masonry abutment piers!

Framing Reinforcement Rules
✓Span ≤ 48": Double 2x4 king studs each side
✓Span 48"–72": Triple king studs + 2x8 structural header
✓Span > 72": Engineered LVL header with structural tie straps
✓Masonry Arches: Skewback width must be ≥ 1.5× wall thickness

General residential partition guidelines. For load-bearing walls, second-story floor loads, or heavy masonry, consult a structural engineer and local building codes (IRC R602).

Low Lateral Thrust

Semicircular (Roman) Arch

Lateral Thrust Ratio20% – 30%

Because the curve meets the springline at a true vertical 90° tangent, the majority of dead load is directed vertically down into the supporting jambs.

Vector: Primarily Vertical (Downwards)
Moderate Lateral Thrust

Gothic (Pointed) Arch

Lateral Thrust Ratio35% – 50%

The pointed apex divides the curve into steeper arcs, reducing horizontal outward vector forces compared to flatter round arches.

Vector: Steep Diagonal Vector
High Lateral Thrust

Segmental (Shallow) Arch

Lateral Thrust Ratio75% – 90%

As the rise decreases relative to the span, the thrust vector angles sharply outward, exerting immense lateral force on the supporting side walls.

Vector: Strong Outward Diagonal Kick
Severe Lateral Thrust

Jack (Flat) Lintel Arch

Lateral Thrust Ratio150% – 200%

Flat arches rely solely on wedge friction and severe horizontal thrust to hold the voussoirs aloft. Requires massive abutments or structural tie rods.

Vector: Extreme Horizontal Force
Trade Gotchas & Quality Control

Common Arch Thrust & Abutment Engineering Mistakes

Ensure proper equilibrium, verify line of thrust within middle third, and prevent lateral abutment displacement.

⚠️ THRUST EQUILIBRIUM

Treating Arch Thrust Like Vertical Beam Reactions

Common Mistake:

Assuming abutment piers only support downward vertical gravity loads without providing sufficient mass or tie rods to resist outward lateral thrust kick.

Pro Trade Solution:

Compute horizontal thrust $H_t approx (w imes L^2) / (8 imes r)$ and design buttresses, tie beams, or micropiles to absorb lateral shearing forces.

⚠️ LINE OF THRUST

Thrust Line Escaping the Middle Third of the Arch Ring

Common Mistake:

Allowing the compressive funicular line of thrust under asymmetric live loads to pass outside the middle third of masonry joints, inducing tensile joint opening.

Pro Trade Solution:

Verify thrust lines with graphic statics or finite element analysis, adjusting arch ring thickness or adding surcharge fill to maintain pure compression.

⚠️ DECENTERING SEQUENCE

Sudden or Asymmetric Centering Formwork Release

Common Mistake:

Knocking out timber falsework wedges from one side first, transferring eccentric shock loads to green masonry or concrete arch ribs.

Pro Trade Solution:

Gradually ease sand jacks or screw wedges symmetrically from crown to springline simultaneously across the entire span.

Field Master Pro-Tips for Clean Execution

★ Conduct graphic statics equilibrium polygon analysis for unbalanced half-span snow or vehicle live loads.
★ Ensure abutment footings rest on unyielding bedrock or driven pile caps capable of resisting outward sliding.
★ Monitor crown elevation with dial indicators or precision optical levels during temporary centering decentering.
★ Use high-early-strength non-shrink grout under bearing shoes to prevent localized stress concentrations.
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Expert Knowledge Base

Frequently Asked Questions

Common carpentry, masonry, and geometric questions regarding arch calculations and construction layout.

Q1 What is a catenary arch and how does it work?

A catenary arch is the mathematical inversion of a freely hanging flexible chain supported at both ends under its own uniform self-weight dead load (described by Robert Hooke in 1675: 'As hangs the flexible chain, so but inverted will stand the rigid arch'). Because a hanging chain is in pure tension with zero bending, the inverted arch is in 100% pure axial compression with zero bending moments or tensile stress.

Q2 What is the mathematical equation of a catenary arch?

The vertical profile is governed by the hyperbolic cosine function: y(x) = a × (cosh(x / a) − 1), where 'a' is the catenary scaling parameter (the radius of curvature at the apex vertex), x is the horizontal distance from the centerline (−W/2 ≤ x ≤ W/2), and cosh(u) = (e^u + e^(−u)) / 2.

Q3 How do you calculate the exact arc length of a catenary arch?

Using hyperbolic calculus, the exact perimeter length is: Arc Length L = 2 × a × sinh(W / (2 × a)), where sinh(u) = (e^u − e^(−u)) / 2.

Q4 What famous monuments use a true catenary arch?

The most famous catenary arch in the world is the Gateway Arch in St. Louis, Missouri (designed by Eero Saarinen and Hannskarl Bandel, spanning 630 ft with a 630 ft rise). Antoni Gaudí famously used hanging weighted string models to design the catenary vaults of the Sagrada Família and Colònia Güell in Barcelona.