Overview
This lecture introduces two-hinged parabolic arches, explains their structural analysis, and provides essential formulas and cases to determine the horizontal thrust (H).
Introduction to Two-Hinged Parabolic Arches
- Two-hinged arches are indeterminate structures because they have more unknown reactions than equilibrium equations.
- The arch consists of two hinges (supports) and typically looks like a curved beam between points A and B with a crown at point C.
- The span (L) is the horizontal distance between hinges, and the rise (h) is the vertical height at the crown.
Structural Analysis and Forces
- Indeterminate structures cannot be analyzed by equilibrium alone; extra equations are required.
- Each hinge provides two reaction components, making four reactions in total, but only three equilibrium equations are available.
- The horizontal thrust (H) acts inward at both supports to resist opening due to applied loads.
Important Formulas
- The actual bending moment at a section X is: Mx = M - H * y (where M is the beam moment and y is the vertical coordinate).
- The key formula for horizontal thrust is:
H = ∫(M₀y dx) / ∫(y² dx),
where M₀ = moment if the arch were a simply supported beam.
- For a parabolic arch, y = 4h/L² × (Lx - x²).
- Always use the integration formula for H in exams for full marks. Only use shortcut formulas to check your answer.
Specific Loading Cases & Shortcut Formulas
- Concentrated load W at crown: H = (25/128) × (WL/h)
- Uniformly distributed load (UDL) W on left half: H = (WL²)/(16h)
- UDL W over entire span: H = (WL²)/(8h)
- UDL W over distance a from the left:
H = [Wa²/(16h³)] × [5L³ - 5L²a + 2a³]
Key Terms & Definitions
- Indeterminate Structure — A structure with more unknown reactions than equilibrium equations.
- Span (L) — Horizontal distance between the two hinges of the arch.
- Rise (h) — Vertical height from the base to the crown of the arch.
- Horizontal Thrust (H) — Inward force at the supports of an arch required to maintain equilibrium.
Action Items / Next Steps
- Practice applying H = ∫(M₀y dx) / ∫(y² dx) for different loading scenarios.
- Remember shortcut formulas only for answer checking, not for primary calculations in exams.