Three-hinged parabolic arch — horizontal thrust for partial uniformly distributed load A three-hinged parabolic arch (hinged at both springings and the crown) has a horizontal span L = 4.8 m and a central rise f = 1 m. It carries a uniformly distributed load of 0.75 tonne per metre over the left half of the span only. Determine the horizontal thrust H at the supports.

Introduction / Context:Three-hinged arches are statically determinate. For a parabolic arch, the bending moment at the crown hinge must be zero. This condition allows direct calculation of the horizontal thrust H by comparing the bending moment at midspan for the equivalent simply supported beam with the arch rise at the crown. This is especially handy for unsymmetrical loading (such as a uniform load over half the span).

Given Data / Assumptions:

• Span L = 4.8 m; rise at crown f = 1 m.
• UDL w = 0.75 t/m applied over left half: 0 ≤ x ≤ L/2 = 2.4 m.
• Arch is three-hinged (hinges at supports and crown), parabolic in profile.

Concept / Approach:At the crown hinge of a three-hinged arch, internal bending moment is zero. Therefore: H * y_crown = M_beam at crown (with the same loading on a simply supported beam). Thus, H = M_beam(crown) / f. Compute the simply supported beam's reaction forces for the given partial UDL, then find M at midspan, and divide by f to obtain H.

Step-by-Step Solution:

Verification / Alternative check:Because only half the span is loaded, there is a nonzero vertical shear at midspan in the beam analogy, but the crown-hinge moment condition remains valid. Units check: t·m divided by m gives t, appropriate for thrust.

Why Other Options Are Wrong:

• 1.8 t and 0.8 t: incorrect magnitudes relative to the computed beam moment and rise.
• 10.8 t: off by a factor of 10; likely from unit or decimal misplacement.
• None of these: unnecessary since 1.08 t is correct.

Common Pitfalls:Forgetting to use the beam moment at the crown location, misplacing the resultant load of the partial UDL, or confusing rise f with total depth. Also, mixing tonne and kiloNewton units can cause errors if conversions are not tracked.

Final Answer:1.08 tonnes