7. INTERNAL FORCES IN STRUCTURES

This chapter will concentrate on the analysis of arches. The concepts used are identical to the ones previously seen; however, the major (and only) difference is that equations will be written in polar coordinates.

This section introduces the chapter focused on arches. It clarifies that while the concepts of analysis may be familiar, the equations will switch to polar coordinates, which is a different mathematical approach than typically used.

Think of polar coordinates like describing a location on a map using angles and distances instead of using a grid system. This shift may initially seem challenging, but it provides a different perspective on the same concepts.

Like cables, arches can be used to reduce the bending moment in long span structures. Essentially, an arch can be considered as an inverted cable, and it transmits the load primarily through axial compression, but can also resist flexure through its flexural rigidity.

This chunk discusses how arches operate similarly to cables by helping to minimize bending moments in long structures. Arches primarily handle loads through axial compression rather than bending, which can enhance their structural efficiency.

Imagine bending a stiff straw versus pushing down on a bent arch. The straw bends under pressure, while the arch distributes the load without bending, making it a stronger and more stable structure.

A parabolic arch uniformly loaded will be loaded in compression only. A semi-circular arch uniformly loaded will have some flexural stresses in addition to the compressive ones.

This section elaborates on how different shapes of arches respond to loads. A parabolic arch supports loads without bending, while a semi-circular arch experiences both compression and some bending, which could affect its performance.

Consider a playground swing set. The curved swing seat can handle the weight of kids without bending too much. However, if it were a flat board, it might bend or sag under the same weight.

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

This chunk explains that the design of long-span structures, like arches, should reflect load requirements. Ideal shapes closely align with the distribution of forces (moment diagrams), improving overall stability and efficiency.

Think of how a well-designed bicycle frame distributes the weight of the rider efficiently. Just like the arch must be shaped properly to handle loads, the bicycle frame must be structured to uphold its strength during rides.

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.

This chunk highlights the historical significance of arches. They have been a fundamental method in building durable structures over time since materials available had strengths primarily in compression, which suits arch designs perfectly.

Consider ancient Roman aqueducts, which effectively used arches to transport water over long distances, showcasing how early builders capitalized on the properties of compression to create strong, lasting structures.

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).

This section introduces basic static principles relevant to arch construction focused on load distribution and symmetry. In a well-structured arch, forces are balanced, resulting in no shear forces or moments at the midspan.

Think of balancing a seesaw. If both sides are equal, there is no additional pressure or force in the middle. Similarly, the arch balances loads perfectly, resulting in a stable structure.

Since H varies inversely to the rise h, it is obvious that one should use as high a rise as possible. For a combination of aesthetic and practical considerations, a span/rise ratio ranging from 5 to 8 or perhaps as much as 12 is frequently used.

This piece addresses the relationship between the height of an arch and its efficiency. A higher arch reduces horizontal forces, making it more stable. However, practical ratios must be followed to avoid complications like buckling.

Consider a tall bridge arch. The higher it rises, the less it has to bend sideways, making it more robust. However, if it gets too extreme, problems arise just like how a very high tower can sway too much in the wind.

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