Funicular Quadratic Curve & Pure Compression

Parabolic Arch Calculator

Calculate quadratic vertex equations (y = (4H/W²)·(Wx − x²)), exact arc length via calculus integration, focal distance, springline tangent angles, and spandrel under-arch area (A = 2/3 W·H).

Pure Axial Compression Under UDL Parabola
e.g. 60 or 48
in / ft
e.g. 20 or 15
in / ft
For thrust calc
lb/ft
Standard Parabolic Ratios: Parabola
Parabolic Quadratic Vector & Thrust
EXACT ARC LENGTH (L) Calculus
74.72 units
+24.5% vs straight span
HORIZONTAL THRUST H = wL²/8f
22,500 lbs
22.5 kips abutment push
UNDER-ARCH AREA A = 2/3 W·H
800.0 sq units
Archimedes Quadrature
SPRINGLINE ANGLE arctan(4H/W)
53.13 °
Tangent thrust vector
QUADRATIC EQUATION y = 0.02222·x(60 − x)
FOCAL LENGTH (W²/16H) 11.25 units from apex
INTERNAL BENDING MOMENT M = 0.00 (Pure Compression)
Structural Mechanics Simulator

Interactive Parabolic Rise-to-Span & Thrust Simulator

Adjust the crown rise slider to observe how deepening the parabolic arch cuts horizontal abutment thrust force in half.

20.0 Units (Thrust = 22,500 lbs)
Horizontal Abutment Thrust 22,500 lbs H = wL² / 8f
Springline Incline Angle 53.13° Tangent Angle Thrust vector direction
Structural Efficiency 100% Compressive (M = 0) No tensile reinforcement needed
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.

Quadratic Formulas

Parabolic Arch Equations

Exact quadratic geometry and calculus formulas for parabolic curves.

1. Parabolic Curve Equations
Height Function: y(x) = (4H / W²) × (W·x − x²)
Slope / Derivative: dy/dx = (4H / W²) × (W − 2x)
Springline Tangent: θ = arctan(4H / W)
Focal Distance: a = W² / (16H)

The parabola reaches its crown vertex $y = H$ at $x = W/2$ with horizontal tangent ($dy/dx = 0$).

2. Area, Arc Length & Thrust
Under-Arch Area: A = (2 / 3) × W × H
Horizontal Thrust: H_thrust = (w × W²) / (8 × H)
Vertical Reaction: V = (w × W) / 2
Resultant Abutment: R = √(H_thrust² + V²)

Under uniform vertical loading, the resultant thrust vector $R$ points directly along the springline angle.

Step-by-Step Worked Numeric Example

Suppose you are framing a parabolic bridge rib or cathedral ceiling vault with a clear opening span of W = 48.0 inches and a vertical crown rise of H = 16.0 inches.

Step 1: Parabola Constant k = 4H / W² = 64 / 2304 k = 0.02778 in⁻¹
Step 2: Elevation @ x = 12" 16 × (1 − 4(144)/2304) y(12) = 12.00 inches
Step 3: Perimeter Arc (L) W + (8H² / 3W) − ... L = 60.19" (5' 0 3/16")
Step 4: Pure Compression Zero Bending Moment Ideal for uniform vertical dead load
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 How do you calculate the width of a parabolic arch at any height above the base (or water level)?

For a parabolic arch of span W and apex height H, the horizontal width at any elevation y above the base is W(y) = W × √(1 − y / H). For example, to find the width of an arch with span W = 40 ft and maximum height H = 20 ft at 8 ft above the water: W(8) = 40 × √(1 − 8 / 20) = 40 × √0.60 ≈ 30.98 ft (or 31 ft to the nearest foot).

Q2 What is the standard formula for a parabolic arch?

With origin at the bottom-left springline, the quadratic equation is y = (4H / W²) × (W·x − x²). With the apex centered at (0, H), the vertex form is y = H − (4H / W²) × x², where x is the horizontal distance from the centerline and y is the height above the springline.

Q3 What is a parabolic arch and why is it structurally efficient?

A parabolic arch follows a second-degree polynomial (quadratic) curve y = (4H / W²) × (W·x − x²). Under a uniformly distributed vertical gravity load (UDL such as a bridge deck or roof load), the internal bending moment is mathematically ZERO throughout the entire rib, resulting in pure axial compression with zero tension or bending stress.

Q4 How do you calculate the exact arc length of a parabolic arch?

The closed-form calculus integration for parabolic arc length is: L = 0.5 × √(W² + 16H²) + (W² / (8H)) × ln((4H + √(W² + 16H²)) / W). For example, for span W = 60 ft and height H = 20 ft: L = 0.5 × √(3600 + 6400) + (3600 / 160) × ln((80 + 100) / 60) = 50 + 22.5 × ln(3.0) = 50 + 24.72 = 74.72 ft.

Q5 What is the focal length of a parabolic arch?

The focal distance from the vertex apex is: a = W² / (16 × H). For a 60 ft span with 20 ft rise, the focal length is a = 3600 / 320 = 11.25 ft.

Q6 How do you calculate the area under a parabolic arch?

By Archimedes' quadrature of the parabola, the area under a parabolic arch is exactly two-thirds of the bounding rectangle: Area = (2 / 3) × Span (W) × Rise (H).