Analyses a parabolic or circular arch, three-hinged, two-hinged, fixed at both springings or tied, with a constant or varying section: the thrust, bending moment, normal and shear force and displacements under own weight, uniform and point loads, a moving force, temperature and movements of the springings. Every step shows its formula.

What it checks

The axis: the ratio f/l, the axis y = 4f·x(l − x)/l² for a parabola or the arc through both springings and the crown, the slope at the springings and the length s of the axis. The axis is modelled by at least 48 straight members with their nodes on the curve.
Redundants: the reactions r against 3 equations. A three-hinged arch is statically determinate; a two-hinged arch has one redundant, the thrust; a fixed arch three.
Loads: the vertical load as entered ΣF, ΣF_var for the worst-case loads all present, the moving force P_m, and the movements Δ_x and Δ_y of springing B.
Worst arrangement: a uniform load in the worst case is split into 48 parts, each present or absent on its own; the temperature changes act one at a time; the moving force stands at its worst node. By superposition this covers every arrangement exactly.
Support reactions: the thrust H, inward, and the vertical reaction R_y, largest and smallest, with the equilibrium check ΣR_y = ΣF.
Along the arch: the results at the divisions of the span you choose, with x, y and the slope φ of the axis at each.
Deflection, optional: the largest vertical displacement of the axis against l/n_lim.

What you enter and what you get

You enter the span l between the springings, the rise f of the axis at the crown, a parabola or a circular segment, and the supports: three-hinged, two-hinged, fixed at both springings, or tied. Then E, the section along the arch (constant, thicker towards the springings as 1/cos φ, or given at the crown and at the springings), I_c and A_c at the crown, and whether the rib shortens under its compression.

For the loads: the own weight g_a per metre of arch, up to four uniform loads downward in plan, downward along the arch or horizontal, over the whole or part of the span, up to six vertical forces, a moving force P_m, up to two temperature changes with a uniform part and a difference between underside and top, the spreading Δ_x of the springings and the settlement Δ_y of B. Then the deflection limit and the number of divisions for the table.

You get the shape, the reactions, the forces and displacements along the arch with their envelope, and a pass or fail for the deflection.

Assumptions and limits

An arch carries its loads mainly in compression. The moment at a section is the simple-beam moment reduced by H times the height of the axis there; an axis that follows the line of its loads, the funicular, has no bending at all.
A three-hinged arch is determinate: settlements and temperature produce no forces in it.
A tied arch has a pin at one springing and a roller at the other, and a tie takes the thrust: the supports see vertical forces only.
With rib shortening included, the rib shortens under N, which lowers the thrust of an indeterminate arch. Neglected, the rib is axially rigid, as in many hand formulas.
Two temperature changes in the worst case are alternatives, a warm and a cold day, never taken together.
At a force, N and V are given just right of it.
An analysis only: no section is verified.

Worked example

A two-hinged parabolic arch of 60 m span and 10 m rise, thicker towards the springings as 1/cos φ with I_c = 85,000·10⁶ mm⁴ and A_c = 1,200,000 mm² at the crown, E = 34,000 N/mm², rib shortening included; own weight 30 kN/m, 60 kN/m in plan as entered and 25 kN/m in the worst arrangement, a moving force of 300 kN, ±20 °C as alternatives, a spreading of 5 mm, and a limit of l/500. The deflection check passes. These are the module's own results for its worked example, computed when this page was built.

Inputs
Input	Value	Ref.
Span between the springings l	60 m
Rise of the axis at the crown f	10 m
Shape of the axis	Parabola
Supports and hinges	Two-hinged
Modulus of elasticity of the arch E	34000 N/mm²
Cross-section along the arch	Thicker towards the springings (1/cos φ)
Second moment of area at the crown I_c	85000 10⁶ mm⁴
Cross-section area at the crown A_c	1200000 mm²
Shortening of the rib under its compression	Included
Own weight per metre of arch g_a	30 kN/m
Number of uniform loads	2
Direction	Downward, per metre in plan
Over	Whole span
Intensity q₁	60 kN/m
Arrangement	As entered
Direction	Downward, per metre in plan
Over	Whole span
Intensity q₂	25 kN/m
Arrangement	Worst case
Number of vertical forces	0
A force moving along the span P_m	300 kN
Number of temperature changes	2
Coefficient of thermal expansion α_T	10 10⁻⁶/°C
Uniform change of the rib ΔT₁	20 °C
Difference: underside minus top ΔT_d,1	0 °C
Arrangement	Worst case
Uniform change of the rib ΔT₂	-20 °C
Difference: underside minus top ΔT_d,2	0 °C
Arrangement	Worst case
Spreading of the springings Δ_x	5 mm
Settlement of B Δ_y	0 mm
Check the deflection against a limit	Yes
Limit: the span divided by n_lim	500
Divisions of the span for the table	12

Calculation: The arch
Step	Value	Ref.
Rise to span1 : 6.0	f/l = 0.167
Axisslope at the springings 33.7°	y(x) = y = 4f·x(l − x)/l²
Length of the axisthe axis is modelled by 48 straight members with their nodes on the curve (at least 48, plus the load and table points)	s = 64.185 m
Redundantsr = 4 reactions against 3 equationsstatically indeterminate: the thrust depends on the stiffness of the rib; temperature and movements produce forces	n = 1

Calculation: The loads
Step	Value	Ref.
Vertical load as enteredown weight, uniform loads and forces	ΣF = 5,525.56 kN
Worst-case loads, all present	ΣF_var = 1,500.00 kN
Moving forceat the worst node, or absent	P_m = 300.0 kN
Movements of the springing B	Δ_x; Δ_y = 5.0; 0.0 mm

Calculation: The worst arrangement
Step	Value	Ref.
Loads that may each be present or absentuniform load 2 (48 parts), temperature change 1, temperature change 2; the moving force at any of 47 interior nodes	k = 50
How the envelope is foundeach load part analysed once; at every section the parts that increase a value are added for the largest result, those that decrease it for the smallest — exactly every arrangement; the temperature changes one at a time; the moving force at its worst node	superposition

Calculation: Support reactions
Step	Value	Ref.
Equilibrium: sum of the vertical reactions, loads as enteredequals the vertical load as entered, 5,525.56 kN	ΣR_y = 5,525.56 kN

Step	1Ax = 0	2Bx = 60.00 m	Ref.
Horizontal reaction (thrust), inward: largest / smallest H kN	5,582.3 / 4,085.5	5,582.3 / 4,085.5
Vertical reaction, upward: largest / smallest R_y kN	3,806.5 / 2,762.8	3,806.5 / 2,762.8

Calculation: Along the arch
Step	M largestkN·m	M smallestkN·m	N smallestkN	\|V\| largestkN
x = 0.00 my = 0.000 m, φ = 33.1°, springing A	0.0	0.0	-6,681.4	454.2
x = 5.00 my = 3.056 m, φ = 28.4°	2,142.5	-1,271.8	-6,370.3	329.7
x = 10.00 my = 5.556 m, φ = 23.3°	3,087.6	-2,013.2	-6,106.4	253.2
x = 15.00 my = 7.500 m, φ = 17.7°	3,162.7	-2,246.3	-5,892.6	267.7
x = 20.00 my = 8.889 m, φ = 11.8°	2,706.1	-2,025.4	-5,732.6	316.4
x = 25.00 my = 9.722 m, φ = 5.6°	2,045.4	-1,459.9	-5,628.3	343.2
x = 30.00 my = 10.000 m, φ = -0.8°, crown	1,769.4	-1,022.1	-5,582.3	337.5
x = 35.00 my = 9.722 m, φ = -7.1°	2,045.4	-1,459.9	-5,628.3	336.3
x = 40.00 my = 8.889 m, φ = -13.3°	2,706.1	-2,025.4	-5,732.6	307.7
x = 45.00 my = 7.500 m, φ = -19.1°	3,162.7	-2,246.3	-5,892.6	256.5
x = 50.00 my = 5.556 m, φ = -24.6°	3,087.6	-2,013.2	-6,106.4	266.9
x = 55.00 my = 3.056 m, φ = -29.7°	2,142.5	-1,271.8	-6,370.3	346.0
x = 60.00 my = 0.000 m, φ = -33.1°, springing B	0.0	0.0	-6,681.4	454.2

Calculation: Check — deflection
Step	Value	Ref.
Largest vertical displacement, at x = 15.00 m\|w\|_max ≤ l/n_lim	η = 0.834 · Pass83 % of the capacity used

How this module is verified

Verified against independent calculations used on real projects, before release.

Questions

Which loading bends an arch most?

Half-span loading. Set the variable load to the worst case: every part of it may then be present or absent, and the envelope finds the arrangement for each section.

What does a yielding abutment do?

The spreading of the springings, B moving away from A, lowers the thrust of a two-hinged or fixed arch. It is always present in the results.

Can I print the calculation?

The full calculation is shown on screen, free and without an account. A print-ready A4 calculation note is part of the Basic plan.

Page updated: 14 November 2026

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