7.1.1. Optimization of Dead-Load Efficiency

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.

To achieve the best performance of a long-span structure, such as an arch or a prestressed concrete beam, its shape should closely match the moment diagram, which represents the distribution of bending moments (forces that cause bending) throughout the structure. This means that structures like arches and cables are usually designed to take on a parabolic shape since this allows them to efficiently handle loads.

Think of a well-designed bridge. If the arches are shaped like a smooth, upward curve (like a parabola), they can better handle the weight of cars and trucks passing over them, distributing the load more effectively than if they were flat.

Long span structures can be built using flat construction such as girders or trusses. However, for spans in excess of 100 ft, it is often more economical to build a curved structure such as an arch, suspended cable or thin shells.

When building structures that span large distances (over 100 feet), using curved designs like arches or cables is often more cost-effective compared to using flat designs like girders or trusses. This is because curved structures can efficiently handle more weight with less material, which can lower construction costs and make these kinds of designs preferable for longer spans.

Consider the difference between a flat wooden beam and a curved archway. The flat beam might work for short distances, but for a wide hallway, a curved archway can span the area with less material while supporting the same weight, making it cheaper and lighter.

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.

Historically, arches have been a fundamental architectural feature, as they made it possible to construct large spans using materials that primarily handle compressive forces, like bricks or stone. These materials are strong when pressed together, allowing for the building of structures that can last for centuries.

Imagine ancient Roman aqueducts. These large structures used stone arches to transport water over long distances while requiring minimal labor, as the stones could be stacked in such a way to efficiently distribute weight without modern construction technology.

The basic issues of static in arch design are illustrated where the vertical load is per unit horizontal projection (such as an external load but not a self-weight). Due to symmetry, the vertical reaction is simply V = wL, and there is no shear across the midspan of the arch.

In arch design, understanding how loads affect the structure is crucial. A vertical load acting on the arch leads to a reaction force. The vertical reaction can be calculated easily because of the symmetrical nature of most arches, which helps determine how much weight can be supported without causing failure. Additionally, there’s no shear (sideways force) at the midpoint, simplifying calculations.

Think of a playground swing set. When kids swing, the force is directed downward, similar to how a load acts on an arch. The supports at each end feel the weight symmetrically, with no lateral forces complicating the support needed in the middle.

Since equilibrium requires H to remain constant across the arch, a parabolic curve would theoretically result in no moment on the arch section.

In a properly designed arch, the horizontal component of the reaction force (H) remains constant, which ideally means that there should be no bending moments acting on the arch. This condition can be achieved with a parabolic shape, allowing for efficient load distribution that minimizes stresses and potential deformation across the curve.

Consider a perfectly balanced seesaw that pivots at its center. If each side is weighted perfectly, there won't be any bending moments at the pivot point. Similarly, a parabolic arch can stay balanced and strong under a load, effectively managing the forces involved.

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