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Civil
Bending moment at a three-hinged arch section
Bending moment at a section of a three-hinged arch, equal to the moment M0 in the equivalent simply supported beam minus the moment of the inward thrust H about the section, where y is the height of the section above the support chord.
- Ref
- CIV-39
- Rev
- v1
- Field
- Civil
- Result
- M [N·m]
At any section of a three-hinged arch with vertical loads and supports at the same level, the bending moment is the beam moment minus the moment of the thrust: M = M0 − Hy. The thrust lowers the moment by an amount that grows with the height y of the arch axis. If the arch is shaped so that M = 0 everywhere, it carries load by axial force alone.
Bending moment at a section of a three-hinged arch, equal to the moment M0 in the equivalent simply supported beam minus the moment of the inward thrust H about the section, where y is the height of the section above the support chord. Equal-level supports and vertical loads only; sagging is positive.
Calculator
- Moment relief from the thrust, H·y80,000 N·m
- Fraction of the beam moment removed0.8421
- Ordinate giving zero moment, M₀/H2.375 m
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- The thrust removes 84.21 % of the beam moment, leaving 15 kN·m at this section.
- The moment would vanish if the arch axis were at y = 2.375 m here, which is the funicular shape for this load.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (N·m) | Local step | Change in result |
|---|---|---|---|---|---|---|
| M0 Beam moment at the section | 0 to 5,000,000 N·m | −80,000 | 4,920,000 | 5,000,000 | +10% | +63.3 % |
| H Horizontal thrust | 100 to 1,000,000 N | −1,905,000 | 94,800 | 2,000,000 | +10% | −53.3 % |
| y Arch ordinate above the chord | 0 to 20 m | −705,000 | 95,000 | 800,000 | +10% | −53.3 % |
Sweep of Arch ordinate above the chord
| Arch ordinate above the chord (m) | Arch bending moment (N·m) |
|---|---|
| 0 | 95,000 |
| 2.5 | −5,000 |
| 5 | −105,000 |
| 7.5 | −205,000 |
| 10 | −305,000 |
| 12.5 | −405,000 |
| 15 | −505,000 |
| 17.5 | −605,000 |
| 20 | −705,000 |
Sweep one variable
Variables and how each one contributes
Arch bending moment
What the equation gives you. Everything else feeds it.
Beam moment at the section
The bending moment at the section in the equivalent simply supported beam, in N·m. Raises the arch moment in direct proportion. Raising it by 10 % (to 104,500) moves the result by +63.3 %, from 15,000 to 24,500 N·m.
Horizontal thrust
The horizontal thrust at the supports, in N. Reduces the moment in direct proportion to H. Raising it by 10 % (to 44,000) moves the result by −53.3 %, from 15,000 to 7,000 N·m.
Arch ordinate above the chord
The height of the arch axis above the support chord at the section, in m. A higher axis gives the thrust a larger lever arm, reducing the moment. Raising it by 10 % (to 2.2) moves the result by −53.3 %, from 15,000 to 7,000 N·m.
Worked example with the values above
- Beam moment: M0 = 95,000 N·m.
- Thrust moment: H·y = 40,000 × 2 = 80,000 N·m.
- M = 95,000 − 80,000 = 15,000 N·m.
Assumptions and limits
- Supports at the same level and vertical loads only, for arches and for cables.
- A flexible cable with negligible bending stiffness and a load uniform along the horizontal span (parabola).
Common mistakes
- Using the cable formulas for a cable under its own weight, which hangs as a catenary.
- Using sag measured from the wrong reference line.
- Forgetting the vertical component of tension at the supports.
Statistics and uncertainty
Cable tension is inversely proportional to sag, so the formulas are very sensitive to the measured sag: a 5 % error in sag gives a 5 % error in tension. Real cables also stretch and have some bending stiffness, which shifts results by a few per cent.
Where it is used
- Sizing cables, anchors, abutments and ties.
- Finding thrust and moments in three-hinged arches.
Maths you need
- Funicular shape: the shape that carries load by axial force only
- Three-hinge condition: the moment at the crown hinge is zero
- Horizontal tension: the constant horizontal component along a loaded cable