7.1.7. Analysis of Loads on Arches

In order to optimize dead-load efficiency, long span structures should have their shapes approximate the corresponding moment diagram, hence an arch, suspended cable, or tendon configuration in a prestressed concrete beam all are nearly parabolic.

Arches are structures designed to effectively distribute loads while minimizing material usage. This section suggests that for long spans, the shape of these structures should align with the forces acting on them, similar to the shapes illustrated in moment diagrams for beams. This is important because by mimicking the moment distribution, arches can be built with materials that only resist compressive forces efficiently, ensuring maximum strength and stability.

Think of it like a bow in archery. The string and the shape of the bow are designed to handle the tension and compression from the draw. Similarly, an arch's design allows it to manage loads across its span efficiently, just like a bow's design allows it to shoot arrows effectively.

Since the dawn of history, mankind has tried to span distances using arch construction. Essentially this was because an arch required materials to resist compression only (such as stone, masonry, bricks), and labour was not an issue.

Arches have been a part of architectural history for centuries, primarily used due to their ability to support heavy loads with minimal material. The ability of materials like stone and bricks to withstand compression made arches a favorable choice in construction. This historical perspective emphasizes the efficiency and effectiveness of arches in early engineering, where materials were limited and labor was relied upon.

Consider ancient Roman aqueducts. They utilized arches to transport water over long distances. The strength of the arch allowed them to build impressive structures without the need for excessive materials, demonstrating both beauty and functionality.

The basic issues of static in arch design are illustrated where the vertical load is per unit horizontal projection. Due to symmetry, the vertical reaction is simply V = wL, and there is no shear across the midspan of the arch (nor a moment).

Understanding static forces is crucial for arch design. When an arch supports a load, the vertical reaction force can be determined by multiplying the uniformly distributed load (w) by the length of the arch (L). Importantly, there are no shear forces or bending moments at the center (midspan) of the arch if it is symmetrical, which simplifies the analysis and design process.

Imagine a seesaw balanced evenly on a fulcrum. If you place an equal weight on both sides, there is no tipping or bending in the middle. Similarly, in a symmetrical arch, as long as the loads are balanced, it maintains its shape and integrity without additional complexity.

Solving for H, we find that H = (wL^2) / (8h). We recall that a similar equation was derived for arches, and H is analogous to the CT forces in a beam, and h is the overall height of the arch.

This equation for horizontal thrust (H) helps to calculate the forces acting horizontally within the arch. A higher arch (h) reduces the horizontal force, which is desirable as it minimizes the risk of structural instability. The relationship depicted here helps to ensure that the design remains within safe limits while maintaining necessary load-bearing capacity.

Think of a tall tree swaying in the wind. A tall and sturdy tree will withstand the wind better than a short, flimsy one. Similarly, taller arches experience less horizontal stress due to their design, which allows them to maintain stability under loads.

Three-hinged arches are statically determinate structures whose shape can accommodate support settlements and thermal expansion without secondary internal stresses.

Three-hinged arches are unique in that they allow for slight movements or changes (like settlements of the ground or thermal expansion due to temperature changes) without inducing internal stress that could lead to failure. This characteristic is a significant advantage for structures under varying conditions, enhancing their durability and longevity.

Consider a bridge designed to expand and contract with temperature changes. Just like the bridge may have sections that can move slightly to adjust to changes, three-hinged arches can flex slightly without harming the structure, preventing cracks and other issues.

If the arch has only two hinges, or if it has no hinges, then bending moments may exist either at the crown or at the supports or at both places.

Arches with fewer hinges can experience moments, or bending forces, which can lead to stress and potential failure. Depending on how many hinges are included in the design, moments can be concentrated at certain points in the arch, meaning additional calculations and design work are needed to ensure stability and safety.

Imagine bending a flexible straw at its ends. If you bend it too much in the middle, it will crumple or kink. In an arch, fewer hinges can lead to similar 'kinks' in terms of stress distribution, showing the importance of careful design for load handling.

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