Lab Report: Analysis of a Two-Hinged Parabolic Arch

The primary goals of this experiment are:

• To experimentally determine the influence line for the horizontal reaction (thrust) in a two-hinged parabolic arch.
• To use the constructed influence line to find the horizontal reaction for a given system of multiple point loads.
• To compare the experimental value of the horizontal reaction with the value obtained from the influence line and the value calculated from theoretical principles.

A two-hinged arch is a structural form that is supported by hinged supports at both ends. Unlike a simple beam which primarily resists loads through bending and shear, an arch carries loads mainly through axial compression. This is achieved by its curved shape, which directs vertical loads outwards.

The hinged supports prevent vertical and horizontal movement but allow rotation. Because the supports can resist horizontal forces, a key characteristic of an arch is the development of a horizontal reaction, often called horizontal thrust (H). This inward-directed force at the supports is what gives the arch its strength and allows it to span longer distances than a beam of a similar size.

An Influence Line Diagram (ILD) is a powerful tool in structural analysis. It is a graph that illustrates how a specific function (like a reaction, shear force, or bending moment) at a single point on a structure changes as a concentrated unit load moves across its entire length. For this experiment, we will construct the influence line for the horizontal thrust (H). The vertical axis of the ILD represents the value of H, and the horizontal axis represents the position of the moving unit load. Once the ILD is created, the total horizontal thrust for any combination of point loads can be found by summing the products of each load’s magnitude and the corresponding ordinate (height) of the ILD at that load’s position.

The experiment utilizes the following apparatus:

• Arch Model: A two-hinged parabolic arch with the following specifications:
  • Span (L): 1 meter
  • Rise (h): 200 mm
• Loading System: Seven hangers positioned at 125 mm intervals to apply loads.
• Measurement Device: A dial gauge installed at one of the hinged supports to measure the horizontal displacement, which is then converted to the horizontal reaction force (H).

The experiment was conducted as follows:

1. The parabolic arch model was securely set up on the test frame and carefully leveled.
2. The dial gauge was positioned at one of the hinged supports to measure the horizontal reaction.
3. To generate the influence line, a single point load of 50N was applied sequentially at each of the seven hanger locations (from A to G).
4. For each load position, the corresponding horizontal reaction (H) was recorded from the dial gauge.
5. The recorded horizontal reactions were plotted against their respective load positions to construct the influence line diagram for H.
6. Next, a specific loading system consisting of multiple loads (20N at B, 30N at D, and 15N at F) was applied to the arch.
7. The total horizontal reaction for this combined loading was measured directly using the dial gauge.
8. The theoretical horizontal reaction was also calculated using the influence line ordinates and the established theoretical formula.
9. Finally, the three values for the horizontal reaction (directly measured, from the influence line, and theoretical) were compared.

Part 1: Generating the Influence Line Diagram (ILD)

A 50N load was moved across the arch, and the horizontal reaction was recorded at each point.

Part 2: Analysis of a Combined Load System

A combined load of 20N at B, 30N at D, and 15N at F was applied.

A. Horizontal Reaction from Influence Line:

Using the ILD ordinates calculated above:

B. Directly Measured Horizontal Reaction:

For the combined load system, the horizontal reaction measured directly by the dial gauge was:

\[ H\ (measured) = 54.5\ N \]

C. Theoretical Horizontal Reaction:

The theoretical value of H is calculated using the formula:

Where \( M_{beam} \) is the bending moment on an equivalent simply supported beam and \( y \) is the height of the arch at any point \( x \).

Based on the calculations provided in the experimental notes:

• \(\int M_{beam} \cdot y \cdot dx = 1.1531\)
• \(\int y^2 \cdot dx = 0.0213\)

The three values obtained for the horizontal reaction are summarized below:

Percentage Error:

Comparing the directly measured value with the theoretical value:

The experiment successfully demonstrated the principles of a two-hinged parabolic arch. The shape of the experimentally derived influence line for the horizontal reaction was parabolic, peaking at the center, which is consistent with structural theory.

There is a very strong agreement between the three calculated values for the horizontal thrust. The value obtained from the influence line (54.45 N) is nearly identical to the directly measured value (54.5 N). Furthermore, the theoretical value (54.14 N) is also in close agreement, with a minimal percentage error of only 0.66% when compared to the measured result.

This small discrepancy can be attributed to minor experimental errors, such as:

• Friction in the hinged supports, which may resist free rotation.
• Slight inaccuracies in reading the dial gauge.
• Minor imperfections in the arch’s material or geometry.

In conclusion, the experiment successfully verified the theoretical method for calculating horizontal thrust in a two-hinged arch. It also confirmed that the influence line is an accurate and reliable tool for determining the reactions in a structure under complex loading conditions.