Understanding the Problem: Moving Load on a 3-Hinged Parabolic Arch

The question asks us to identify the specific location, denoted by a distance '$x$' from the ends of the span, where the maximum sagging moment occurs on a symmetrical three-hinged parabolic arch. This arch has a total span of '$L$', and it is subjected to a single moving point load '$W$'.

Parabolic Arches and Sagging Moments

A parabolic shape is structurally efficient for arches, especially when carrying a uniformly distributed load, as it theoretically results in zero bending moment. However, when a concentrated load moves across the arch, bending moments are induced.

In the context of arches, a 'sagging moment' typically refers to a moment that causes tension on the *bottom* fibers of the arch structure, analogous to a simply supported beam carrying a central load. For a three-hinged arch, hinges are usually placed at the abutments and the crown. The presence of hinges means the bending moment at the hinge points is zero.

Analyzing the effect of a moving load requires understanding influence lines or performing calculations based on the load's position relative to the arch's geometry and support conditions. The exact position for the maximum sagging moment isn't immediately obvious and depends on the arch's parabolic profile and the load's position.

Determining the Location of Maximum Sagging Moment

The calculation to find the precise location '$x$' involves setting up the bending moment equation at a section of the arch as the load '$W$' moves across the span '$L$'. This equation typically incorporates the load '$W$', the position of the load, the span '$L$', the horizontal thrust '$H$' (which varies for a point load), and the rise of the arch ($y$) at the section considered. The equation for the moment '$M_x$' at a section '$x$' is generally given by:

$M_x = M_{\text{beam}} - H \cdot y$

where $M_{\text{beam}}$ is the bending moment at section '$x$' if the structure were a simply supported beam, and $y$ is the vertical rise of the arch at section '$x$'. For a parabolic arch, $y$ is related to $x$ by $y = \frac{4h}{L^2}x(L-x)$, where $h$ is the maximum rise at the crown.

To find the maximum sagging moment, we need to determine the position of the load '$W$' that maximizes this value. This usually involves calculus: finding the derivative of the moment equation with respect to the load's position and setting it to zero.

For a symmetrical 3-hinged parabolic arch subjected to a single moving point load '$W$', standard structural analysis results show that the maximum sagging moment (which is often negative in arch analysis conventions, indicating tension on the top fibres) occurs at a specific distance from the abutments.

Through detailed analysis, considering the influence line for bending moment or optimizing the load position, it has been determined that the maximum sagging moment occurs at a distance approximately equal to 0.211 L from the ends of the span.

Conclusion

Therefore, the distance '$x$' from the ends where the maximum sagging moment occurs for a single-point load '$W$' moving on a symmetrical 3-hinged parabolic arch of span '$L$' is approximately 0.211 L.