Influence Line Diagrams for Three-Hinged Arches

Influence Line Diagrams (ILDs) are essential tools in structural analysis, graphically showing how a specific reaction, shear, or moment changes as a unit load moves across a structure. For a three-hinged arch, which is statically determinate, ILDs can be readily determined using static equilibrium principles.

This explanation details the process of finding the ILD for the vertical reaction at support A ($R_A$) for a symmetrical three-hinged arch with a given span '$L$' and rise '$h$'.

Vertical Reaction ($R_A$) ILD for Three-Hinged Arch

The ordinate (the height) of an ILD at any point along the structure's span represents the value of the reaction, shear, or moment at a fixed location when a unit vertical load is positioned at that point on the span. To find the ILD for $R_A$, we analyze the arch's response to a unit load moving across it.

Applying Static Equilibrium

Consider the entire three-hinged arch structure. Let a unit vertical load ($P=1$) move across the span. We place this load at a distance '$x$' from support A.

The fundamental equations of static equilibrium for the whole structure are:

• Sum of vertical forces = 0: $R_A + R_B = 1$. This indicates that the sum of the vertical reactions at both supports equals the applied vertical load (which is 1 unit in this case).
• Sum of horizontal forces = 0: $H_A + H_B = 0$. For a symmetrical arch, the horizontal thrusts at the supports are equal in magnitude, $H_A = H_B$.

To determine the vertical reaction $R_A$, we can use the moment equilibrium equation. Taking moments about the opposite support, B, eliminates $R_B$ from the equation. It is important to consider the effect of horizontal thrusts ($H_A$ and $H_B$) as they act at a height '$h$' above the springing level (or baseline).

Applying the sum of moments about support B ($M_B = 0$):

$$ \sum M_B = 0 $$

This equation considers the moment due to $R_A$ acting at a distance $L$ from B, the moment due to the unit load acting at a distance $(L-x)$ from B, and the moments due to the horizontal thrusts $H_A$ (acting at height $h$ to the right of B) and $H_B$ (acting at height $h$ to the left of B).

$$ R_A \cdot L - 1 \cdot (L-x) + H_A \cdot h - H_B \cdot h = 0 $$

A key property of symmetrical three-hinged arches is that the horizontal thrusts at the supports are equal ($H_A = H_B$). Consequently, the terms involving the horizontal thrusts ($H_A \cdot h - H_B \cdot h$) cancel each other out.

$$ R_A \cdot L - 1 \cdot (L-x) = 0 $$

Now, we can solve for $R_A$:

$$ R_A = \frac{L-x}{L} $$

This equation can be simplified to:

$$ R_A = 1 - \frac{x}{L} $$

$R_A$ Equation Analysis for ILD

The derived equation, $R_A = 1 - \frac{x}{L}$, defines the ordinates of the ILD for the vertical reaction $R_A$ based on the load position '$x$'.

• Linear Relationship: The equation is linear with respect to '$x$', meaning the ILD is a straight line.
• Value at Support A ($x=0$): When the unit load is placed directly at support A ($x=0$), the ILD ordinate is $R_A = 1 - \frac{0}{L} = 1$.
• Value at Support B ($x=L$): When the unit load is placed at support B ($x=L$), the ILD ordinate is $R_A = 1 - \frac{L}{L} = 1 - 1 = 0$.

Plotting these values, the ILD starts at an ordinate of 1 directly above support A and linearly decreases to an ordinate of 0 directly above support B. This forms a straight line segment connecting the points $(0, 1)$ and $(L, 0)$.

ILD Shape and Maximum Ordinate

The shape described by the linear equation $R_A = 1 - \frac{x}{L}$ is a triangle.

• The base of this triangle lies along the span '$L$' of the arch on the horizontal axis.
• The maximum ordinate occurs at the point corresponding to support A ($x=0$) and has a value of 1.
• The ordinate is 0 at the other end of the span (support B, $x=L$).

It is important to note that the rise of the arch ('$h$') does not influence the shape or the maximum value of the ILD for the vertical reaction $R_A$ in a three-hinged arch. Values like $L/(8h)$ or $L/(4h)$ are typically associated with the calculation of horizontal thrusts or bending moments in arches, particularly parabolic ones, not the ILD for vertical reactions at supports.

Conclusion on $R_A$ ILD

In conclusion, the Influence Line Diagram for the vertical reaction at support A ($R_A$) of a three-hinged arch is a triangle.

The maximum ordinate of this triangular ILD is located at support A and is equal to 1.