7.1.4. Efficiency in Load Bearing

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 used in long span structures to improve load efficiency. By aligning the shape of the arch closely to the expected load distribution (moment diagram), the arch can effectively carry the loads applied to it. This is important because an efficiently shaped structure will require less material, thus reducing weight and cost while maintaining strength.

Imagine holding a large, flat piece of cardboard at both ends – it tends to sag under its own weight. Now, if you create an arch by curving it, the shape helps distribute the weight more evenly and prevents excessive sagging. This principle is why bridges and large buildings utilize arches.

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.

While it is possible to use flat structures like girders for long spans, building a curved structure like an arch can often be more cost-effective, especially for really long spans (over 100 feet). This economic advantage comes not just from material savings but also from reducing labor costs and time.

Think of building a tent. If you use flat wooden beams, you will need more material and labor to support the structure, resulting in higher costs. On the other hand, using a curved fabric surface (like a dome) requires less material and is easier to set up, making it more economical and quicker to construct.

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 labor was not an issue.

Historically, arches have been a fundamental architectural feature due to their ability to effectively span distances while only needing to support compressive forces. This simplicity made arches a desirable choice, especially when materials that are strong in compression (like stone and brick) were used in construction. The durability and strength of these natural materials made arches a practical and reliable option.

Think about how ancient civilizations used stone arches in their buildings and bridges. They didn't have modern metal supports, yet the natural ability of stone to endure compression allowed them to create majestic structures like the Colosseum in Rome, which has lasted for centuries.

The basic issues of static in arch design are illustrated in Fig. 7.2 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).

In static design, the forces acting on an arch are carefully analyzed. The vertical load is distributed across the horizontal span, and thanks to the symmetrical structure of the arch, the vertical reaction force can be calculated simply as the load (w) times the length of the arch (L). Importantly, because of how the forces are transmitted, there is no shear force or bending moment acting in the middle of the arch.

Imagine a seesaw at a playground. When two kids sit evenly on either end, the seesaw doesn’t tip; it balances out. Similarly, in a symmetrical arch, forces act evenly across its shape, creating stability without extra stresses in the center.

An arch carries the vertical load across the span through a combination of axial forces and flexural ones. A well dimensioned arch will have a small to negligible moment, and relatively high normal compressive stresses. An arch is far more efficient than a beam, and possibly more economical and aesthetic than a truss in carrying loads over long spans.

Arches operate differently than beams; they use a combination of forces working along their length (axial forces) and also distribute loads through their curves (flexural forces). A properly designed arch minimizes bending moments, making it much more efficient in carrying loads than traditional beams. This efficiency not only requires less material but also allows for elegant and appealing designs.

Think of carrying a heavy backpack. If you carry it with your arms straight, it would hurt your shoulders (like a beam resisting bending). But if you attach it to a chest harness and let it rest against your back, the weight is distributed and you feel less strain (similar to how an arch distributes its weight).

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