Influence Line Diagrams Compared

An Influence Line Diagram (ILD) shows how a structural response (like bending moment, shear force, or reaction) at a specific point changes as a unit load moves across the structure. Understanding ILDs is crucial for designing structures subjected to moving loads, such as bridges.

The question asks about the similarity between the influence line diagram for bending moment at a section in a 3-hinged stiffening girder of a suspension bridge and that of an arch structure. Let's examine the behavior of these structures.

Stiffening Girder in Suspension Bridge

A suspension bridge typically consists of cables, hangers, and a stiffening girder. The main cables are usually parabolic. The stiffening girder is connected to the cables by hangers and supports the deck. It's often designed as a continuous beam or a 3-hinged girder. The purpose of the stiffening girder is to distribute concentrated loads to the hangers and to resist bending moments and shear forces caused by live loads, which the cables alone cannot efficiently resist.

In a 3-hinged stiffening girder, the three hinges introduce flexibility. The horizontal component of the cable tension provides support to the girder. The bending moment at any section in the stiffening girder is the difference between the bending moment if the girder were a simple beam supporting the load and the moment resisted by the horizontal component of the cable tension multiplied by the vertical distance of the cable from the girder.

Mathematically, for a unit load at position $x'$ on a simple beam of span $L$, the bending moment at section $x$ is $M_{simple}(x)$. For a 3-hinged stiffening girder with cable sag $y(x)$ and horizontal tension $H$, the bending moment at section $x$ is:

$$ M_{girder}(x) = M_{simple}(x) - H \cdot y(x) $$

For a parabolic cable and girder, the term $H \cdot y(x)$ is parabolic.

3-Hinged Arch Structure

A 3-hinged arch is a statically determinate structure. For a parabolic arch subjected to a uniformly distributed load over its entire span, the internal force at any section is purely axial compression (no bending moment). However, under a concentrated live load, bending moments and shear forces are developed.

The bending moment at any section in a 3-hinged arch is the difference between the bending moment if the arch were a simple beam supporting the load and the moment resisted by the horizontal thrust $(H_{arch})$ at the support multiplied by the vertical distance of the arch centroidal axis from the springing line ($y_{arch}(x)$).

$$ M_{arch}(x) = M_{simple}(x) - H_{arch} \cdot y_{arch}(x) $$

For a parabolic arch, the term $H_{arch} \cdot y_{arch}(x)$ is parabolic.

Comparing ILDs

Comparing the expressions for $M_{girder}(x)$ and $M_{arch}(x)$, we see a structural similarity. The stiffening girder essentially carries the load that the flexible cable cannot. The cable tension provides an upward component that reduces the bending moment, similar to how the horizontal thrust in an arch provides an upward component that reduces the bending moment below the simple beam moment.

The ILD for bending moment at a section in a simple beam is triangular. The ILD for $H$ (horizontal tension in cable) in a 3-hinged stiffening girder and $H_{arch}$ (horizontal thrust) in a 3-hinged parabolic arch are similar shapes (often related to the product of load position and sag/rise). Since both structures involve subtracting a term related to the horizontal reaction multiplied by a parabolic ordinate from the simple beam bending moment, the resulting influence line diagrams for bending moment have the same characteristic shape, featuring parabolic segments. The parabolic shape of the cable in the suspension bridge and the parabolic shape of the arch profile lead to this similarity.

Therefore, the influence line diagram for bending moment at any section in a 3-hinged stiffening girder of a suspension bridge is the same in shape as that for a 3-hinged parabolic arch.

The correct option is the one stating 'Parabolic arch'.