Understanding ILD for Thrust in 2-Hinge Parabolic Arches

An Influence Line Diagram (ILD) is a graph that shows how a specific structural response (like reaction, shear force, bending moment, or thrust) at a particular point in a structure changes as a unit load moves across the span of the structure. Thrust in an arch is primarily the horizontal force component developed to resist the outward pushing effect of the applied loads. In a 2-hinged arch, this horizontal thrust is a significant internal force.

ILD for Horizontal Thrust (H) in 2-Hinged Arches

For a statically indeterminate structure like a 2-hinged arch, the horizontal thrust (H) is a primary indeterminate force. The value of H can be determined using compatibility equations. For a 2-hinged arch with supports at the same level, the horizontal displacement at one support relative to the other is zero. This principle is used to find H.

The formula for the horizontal thrust H in a 2-hinged arch under a unit vertical load placed at a distance 'a' from the left support is given by:

\begin{equation*} H = \frac{\int_0^L M_0(x) y(x) dx}{\int_0^L y(x)^2 dx} \end{equation*}

Where:

• \(M_0(x)\) is the bending moment at section x due to the unit load in a simply supported beam of the same span L.
• \(y(x)\) is the vertical ordinate of the arch axis at section x, measured from the horizontal line connecting the hinges.
• L is the span of the arch.

The denominator \(\int_0^L y(x)^2 dx\) is a constant for a given arch shape and span.

According to Muller-Breslau's Principle, the ILD for a reaction component or an internal force component (like horizontal thrust H) is proportional to the deflected shape of the structure when a unit displacement corresponding to that component is applied, with the structure made determinate and released in that direction.

For the horizontal thrust H in a 2-hinged arch, imagine releasing the horizontal constraint at one hinge and applying a unit horizontal displacement. The resulting deflected shape of the arch axis, under certain assumptions (like neglecting axial shortening and shear deformation, and assuming EI is proportional to the moment of inertia of the arch cross-section), is proportional to the original shape of the arch axis itself.

Shape of the ILD for Parabolic Arches

A parabolic arch has its axis shaped according to a parabolic curve. The equation for a parabolic arch with supports at the same level and crown rise 'h' at the center is typically given by:

\begin{equation*} y(x) = \frac{4hx(L-x)}{L^2} \end{equation*}

This equation describes a parabola that is zero at \(x=0\) and \(x=L\) and has a maximum value of h at \(x=L/2\).

Since the ILD for the horizontal thrust H is proportional to the shape of the arch axis, and the arch axis of a parabolic arch is parabolic, the ILD for H will also be parabolic.

The maximum ordinate of the ILD for H occurs at the point where the arch rise y(x) is maximum, which is typically the crown for a symmetric parabolic arch. The shape of the ILD will be a parabola that is zero at the hinges (where y=0) and has its maximum value at the crown (where y is maximum).

Analyzing the Options

Let's evaluate the given options for the ILD of thrust in a 2-hinge parabolic arch:

1. Parabolic: As discussed, the ILD for the horizontal thrust H in a 2-hinged parabolic arch is proportional to the parabolic shape of the arch axis. This matches our derivation based on structural principles.
2. Circular: This would be the shape of the ILD if the arch itself were circular, not parabolic.
3. Triangular: Triangular ILDs are common for shear force or bending moment in simply supported beams under a unit load, but not for the thrust in an arch.
4. Funicular Polygon: A funicular polygon represents the shape taken by a cable under a given set of concentrated loads or the ideal shape of an arch under those loads if it carries only axial compression. While related to load paths, the ILD for horizontal thrust for a single moving unit load across the span is not generally a funicular polygon in this context; it follows the principle related to the arch's inherent shape.

Based on structural analysis principles and the application of Muller-Breslau's principle, the ILD for the horizontal thrust (commonly referred to as thrust in this context) in a 2-hinged parabolic arch is parabolic.

Conclusion

The Influence Line Diagram of thrust (specifically, the horizontal thrust H) in a 2-hinge parabolic arch is parabolic. This shape is directly related to the parabolic geometry of the arch axis itself.

Additional Information on Arch Analysis and ILDs

Understanding ILDs is crucial in structural analysis for determining the maximum effect of moving loads (like vehicles on bridges) on different parts of a structure. For arches, the horizontal thrust H is a critical force as it determines the magnitude of axial compression in the arch rib and influences bending moments if they exist.

In a 3-hinged parabolic arch, which is statically determinate, the horizontal thrust H is easier to calculate. For a unit load at 'a' from the left support, H is given by \(H = \frac{M_0(c)}{y(c)}\), where \(M_0(c)\) is the moment at the crown (c) in a simply supported beam, and \(y(c)\) is the rise of the arch at the crown. Since \(M_0(c)\) varies linearly with the load position 'a' on either side of the crown and \(y(c)\) is constant, the ILD for H in a 3-hinged parabolic arch is triangular, peaking at the crown.

However, the question specifically asks about a 2-hinge parabolic arch, where the structure is indeterminate, and the method involving the integral of \(M_0 y\) is applied, leading to the parabolic ILD for H.

The concept that the ILD for H is proportional to the arch shape applies generally to 2-hinged arches of any shape, not just parabolic. If the arch were circular, the ILD for H would be proportional to the circular shape.