How is the radius of curvature calculated for a segmental masonry arch?

The radius of curvature (R) for a circular segmental arch is derived from the Pythagorean theorem on chord geometry: R = (S² + 4r²) / (8r), or R = [(S/2)² + r²] / (2r), where S is the clear horizontal span and r is the arch rise from the springing line to the intrados crown. For example, a 48-inch span with a 12-inch rise yields: [(24)² + (12)²] / (2 × 12) = (576 + 144) / 24 = 720 / 24 = 30.0 inches of radius. This radius defines the exact compass swing needed to cut timber centering ribs.

What is a skewback and why is the skewback angle critical in masonry arches?

A skewback is the inclined, radial masonry bearing surface at each end of the arch (the springing point) from which the arch begins its curve. The skewback angle (θ/2 from the horizontal) aligns directly with the radial line connecting the springing point to the center of curvature. Cutting the supporting jamb masonry to the precise skewback angle ensures that compressive thrust from the voussoirs transfers perpendicular to the mortar bed joint, preventing lateral shear failure, joint slippage, and arch collapse under service loads.

How much outward horizontal thrust does a segmental brick arch generate on abutments?

Per BIA Technical Notes 31B, horizontal thrust (H) under uniform vertical loading can be calculated as H = (W × S) / (8 × r), where W is the total design load spanning the opening (or w × S for distributed load), S is clear span, and r is arch rise. For an unreinforced brick arch supporting 500 lbs/ft over a 4-foot span (W = 2,000 lbs) with a 1-foot rise, horizontal thrust is: (2,000 × 4) / (8 × 1) = 1,000 lbs outward force per abutment. Flatter arches with lower rise-to-span ratios generate dramatically higher horizontal thrust.

When and how should temporary timber centering formwork be removed (struck)?

Under Brick Industry Association (BIA) guidelines, temporary timber centering must remain fully supported until the mortar achieves sufficient compressive and flexural bond strength, typically a minimum of 7 days under standard curing conditions (above 50°F / 10°C). Centering shoring should incorporate folding hardwood wedges or sealed sand boxes beneath the vertical posts. When striking, masons gently tap out the wedges in small, uniform increments across the entire span, allowing the arch ring to gradually take up its natural compressive thrust without sudden impact loading.

BIA Technical Notes 31B Engineering Engine

Masonry Arch Calculator

A 48-inch span segmental brick arch with a 12-inch rise requires 15 voussoir bricks along a 54.1-inch intrados arc, supported by a 30-inch radius of curvature and 36.9-degree skewback spring angle generating 480 lbs of horizontal thrust.

Calculate structural segmental brick arch dimensions under BIA Technical Notes 31B: radius of curvature, intrados arc length, skewback angle, voussoir unit counts, and horizontal abutment thrust.

BIA Technical Notes 31B Presets (1-Click)

Live Arch Estimate

16Bricks(55.6" Arc · 30" Rad)

Details

Clear Opening Span (inches)

Horizontal clear distance between jamb abutments.

Arch Rise (inches)

Vertical distance from springing line to intrados crown.

Brick Unit Thickness / Bed (inches)

Actual unit thickness along the arch arc, including mortar joint allowance.

Arch Ring Depth / Wall Wythe

Structural depth of the arch ring to compute extrados arc length.

Rise-to-Span Ratio: 1:4.0(Optimal Segmental Range)

Span: 4.00 ft · Rise: 1.00 ft

Arch Takeoff Results

Radius of Curvature (Intrados)

30Inches

Equivalent to: 2' - 6" (2.50 ft radius)

Intrados Arc Length55.6"Bottom surface run

Skewback Spring Angle53.1°From horizontal line

Voussoir Units (Net)16~18 with 10% waste

Horizontal Thrust250lbs lateral force

Timber Centering Formwork Template

Plywood Rib Radius Cut:29.5"(with 1/2" lagging)

Plumb Angle from Vertical:36.9°(skewback abutment bevel)

Extrados Arc: 62.4" · Tapered Mortar Joint: ~0.65" at top extrados.

Estimated Material Cost:~$77

Bricks: $23

Mortar: $9

Formwork: $45

Moderate Thrust (250 lbs): Standard brick masonry jamb piers or CMU backup walls provide sufficient lateral mass to stabilize the arch thrust.

BIA Technical Notes 31B Standard

All segmental masonry arches rely on pure compression. Never strike (remove) temporary timber centering until mortar has achieved sufficient compressive strength (minimum 7 days).

CinderCalc

Segmental Masonry Arch Takeoff & Centering Specification

cinderblockscalculator.com • BIA Technical Notes 31B Engineering Guidelines

Date: October 30, 2026

Arch Construction Proposal

Clear Opening Span: 48 inches (4.00 ft)

Arch Rise: 12 inches (1.00 ft)

Rise-to-Span Ratio: 1:4.0

Brick Bed Thickness: 3.625 inches

Arch Ring Depth: 3.625 inches (Single Wythe)

Radius of Curvature (Intrados): 30 inches (2' - 6")

Intrados Arc Length: 55.6 inches

Extrados Arc Length: 62.4 inches

Skewback Angle: 53.1° from horizontal

Horizontal Thrust: 250 lbs outward force

Centering Formwork & Material Takeoff

Specification Item	Dimension / Quantity	Notes
Timber Rib Cut Radius	29.5 inches	Accommodates 1/2" plywood lagging
Skewback Plumb Pitch	36.9°	Bevel angle from vertical jamb line
Voussoir Arch Bricks	16 units	Order ~18 units (10% waste)
Type N/S Masonry Mortar	~1 bag(s)	80-lb pre-blended mortar bags
Estimated Material Total	~$77	Planning budget excluding mason labor

BIA TN 31B Centering Guidelines: Do not strike (remove) temporary timber centering until mortar has fully set (minimum 7 days). Use folding hardwood wedges or sand boxes beneath centering vertical shores to allow gradual, vibration-free lowering.

Abutment Restraint: Outward horizontal thrust must be counterbalanced by adjacent masonry mass, buttresses, or continuous horizontal reinforcing in surrounding courses.

BIA Technical Notes 31B: Structural Brick Arch Fundamentals

Structural brick masonry arches carry transverse wall loads across window, door, and fireplace openings entirely through internal compressive stress. Unlike flat structural steel angles or reinforced concrete lintels that resist gravity loads through flexural tension, a properly engineered masonry arch resolves vertical gravity loads into a curved thrust line that terminates at supporting masonry abutments.

Pure Compression Line

The internal line of thrust must remain strictly within the middle third (kern) of the arch ring depth to eliminate tensile cracking across mortar joints.

Rise-to-Span Proportions

BIA TN 31B recommends a rise-to-span ratio between 1:4 and 1:6 for segmental arches. Ratios flatter than 1:8 generate excessive horizontal thrust that requires steel tie bars.

Abutment Stability

Supporting masonry piers must provide sufficient dead load mass or adjacent wall run so the resultant thrust vector exits inside the pier footprint into the foundation.

Mathematical Derivation of Arch Geometry

Every circular segmental arch is mathematically governed by intersecting chord geometry. By relating clear span (S) and crown rise (r), carpenters and masons can derive the exact radius of curvature, central subtended angle, and skewback inclination:

1. Radius of Curvature (R) Equation

R = [(S / 2)² + r²] / (2 × r) = (S² + 4r²) / (8r)

Where S is the horizontal clear span between jambs and r is the vertical crown rise. This radius dictates the pivot length for swinging timber centering form ribs.

2. Central Angle (θ) and Intrados Arc Length (L)

θ = 2 × arcsin[(S / 2) / R] | L = R × θ (radians)

The subtended central angle determines the curvature length along the underside (intrados). Dividing arc length L by unit brick bed thickness plus mortar allowance yields the total voussoir count.

3. Outward Horizontal Thrust (H) Formula

H = (W × S) / (8 × r) = (w × S²) / (8 × r)

Where W is total superimposed dead and live load across the span, or w is uniform load per linear foot. Flatter rises dramatically amplify horizontal spreading force on piers.

Timber Centering Formwork Construction & Striking

A masonry arch has zero structural capacity while under construction. Temporary timber centering formwork must support the entire wet weight of green brick masonry, mortar beds, and scaffolding vibration until the keystone is driven and the assembly cures.

Centering Ribs & Lagging Detailing

Cut two matching curved ribs from 3/4-inch CDX plywood using the calculated formwork radius (R - lagging thickness). Space ribs 3.5 inches apart for single-wythe or 7.5 inches for double-wythe arches, nailed with 1×2 wood lagging strips or flexible 1/4-inch plywood sheathing.

Folding Hardwood Wedges & Striking

Support vertical 2×4 or 4×4 timber shores on pairs of opposing hardwood wedges. When striking the form after the mandatory 7-day cure, gently tap out the wedges in small uniform increments. Never knock out formwork props with a sledgehammer, which induces dynamic cracking in fresh mortar joints.

Mortar Selection & Tapered Joint Detailing

Because voussoirs are laid on a radius, mortar bed joints are naturally wedge-shaped: tighter at the bottom intrados and wider at the top extrados. Proper joint proportioning prevents mortar squeeze-out and premature compression failure.

ASTM C270 Type N vs. Type S Mortar

Type N mortar (750 psi min) is ideal for residential exterior arches due to superior workability and flexural bond strength. For high-load commercial arches or seismic zones, specify Type S mortar (1,800 psi min) for increased compressive capacity.

Gauged Brick vs. Tapered Mortar Beds

Rough segmental arches utilize rectangular standard bricks with wedge-shaped mortar joints (minimum 1/4" at intrados to max 3/4" at extrados). For tight uniform 3/8" joints across the entire ring depth, bricks must be individually gauged (radial-cut).

Step-by-Step Worked Calculation: 48-Inch Fireplace Arch

Review an authoritative engineering example for a standard 48-inch residential fireplace opening with a 12-inch segmental rise using modular brick rowlocks:

Step 1: Measure Clear Span & Determine Arch Rise

Clear span S = 48.0" and arch rise r = 12.0". The rise-to-span ratio is 12 / 48 = 1:4.0, comfortably within the optimal 1:4 to 1:6 BIA segmental proportion.

Step 2: Calculate Radius of Curvature and Intrados Arc

R = [(24)² + (12)²] / (2 × 12) = (576 + 144) / 24 = 30.0" radius. Half-angle arcsin(24 / 30) = 0.9273 rad (53.1°). Total intrados arc length L = 2 × 30 × 0.9273 = 55.6".

Step 3: Compute Skewback Angle and Cut Springing Abutments

Skewback spring angle equals the central half-angle: 53.1° from horizontal (or 36.9° bevel from plumb vertical). Cut the supporting jamb masonry skewback to this radial angle so thrust acts perpendicular to the bed joint.

Step 4: Dry-Fit Voussoirs and Position Keystone

Dividing 55.6" arc by 3.625" modular brick bed dimension yields 16 voussoirs. For traditional symmetry with an odd unit count, adjust mortar beds to lay 15 voussoirs (7 on each flank + 1 central keystone), or order 18 units including 10% cutting waste.

Step 5: Check Abutment Thrust Resistance & Lower Formwork Wedges

Design load generates 250 lbs outward horizontal thrust. Verify adjacent jamb pier masonry has adequate overburden mass, bed units in Type N mortar, and strike hardwood wedges gradually after 7 days of damp curing.

Illustrative Material Cost & Budgeting Benchmarks

Planning material costs for a segmental brick masonry arch opening typically encompass facing units, pre-blended mortar bags, and temporary centering lumber:

Facing Brick (Modular)$0.95 – $1.65per brick unit

Type N / S Mortar$7.50 – $11.00per 80-lb bag

Timber Centering Plywood$35.00 – $65.00per opening form

Steel Abutment Tie Collar$25.00 – $55.00if tie rod required

Pricing figures are illustrative planning benchmarks. Always verify current prices with a dated local supplier quote before placing jobsite purchase orders.

Calculation Assumptions & References Planning Reference

Material estimates depend on the selected dimensions, product yields, and waste allowance.

Content date: October 2026

Related references: BIA Technical Notes 31B (Structural Design of Brick Masonry Arches), BIA Technical Notes 31 (Brick Masonry Arches), ASTM C216 (Standard Specification for Facing Brick), ASTM C270 (Standard Specification for Mortar for Unit Masonry), TMS 402/602 (Building Code Requirements and Specification for Masonry Structures), IRC Section R606 / IBC Chapter 21 (Masonry Construction). Confirm the applicable edition and local requirements.

A material estimate is not a structural design or code approval.

Masonry Q&A

Segmental Masonry Arch FAQ

Authoritative answers on chord geometry, radius derivations, skewback angles, and timber formwork striking under BIA TN 31B.

A segmental arch is an arch formed by a circular arc of less than 180 degrees (a segment of a circle), where the arch rise is less than half the clear opening span (typically a rise-to-span ratio of 1:4 to 1:6). In contrast, a semi-circular (Roman) arch has a rise equal to half the span (1:2 ratio) with vertical spring lines. Because a segmental arch is flatter, it produces substantial outward horizontal thrust at the springing points, requiring robust abutments, adjacent masonry mass, or tension tie rods, whereas semi-circular arches direct more force downward into supporting vertical piers.