## 46 Ribbed Domes

As pointed out in Art. 4.5, domes may be thin-shell, framed, or a combination. One type of framed dome consists basically of arch ribs with axes intersecting at a common point at the crown and with skewbacks, or bases, uniformly spaced along a closed horizontal curve. Often, to avoid the complexity of a joint with numerous intersecting ribs at the crown, the arch ribs are terminated along a compression ring circumscribing the crown. This construction also has the advantage of making it easy to provide a circular opening at the crown should this be desired. Stress analysis is substantially the same whether or not a compression ring is used. In the following, the ribs will be assumed to extend to and be hinged at the crown. The bases also will be assumed hinged. Thrust at the bases may be resisted by abutments or a tension ring.

Despite these simplifying assumptions, such domes are statically indeterminate because of the interaction of the ribs at the crown. Degree of indeterminacy also is affected by deformations of tension and compression rings. In the following analysis, however, these deformations will be considered negligible.

It usually is convenient to choose as unknowns the horizontal component H and vertical component V of the reaction at the bases of each rib. In addition, an unknown force acts at the crown of each rib. Determination of these forces requires solution of a system of equations based on equilibrium conditions and common displacement of all rib crowns. Resistance of the ribs to torsion and bending about the vertical axis is considered negligible in setting up these equations.

As an example of the procedure, equations will be developed for analysis of a spherical dome under unsymmetrical loading. For simplicity, Fig. 4.6 shows only two ribs of such a dome. Each rib has the shape of a circular arc. Rib 1C1' is subjected to a load with horizontal component PH and vertical component PV. Coordinates of the load relative to point 1 are (xP, yP). Rib 2C2' intersects rib 1C1' at the crown at an angle ar < W2. A typical rib rCr' intersects rib 1C1' at the crown at an angle ar < it/2. The dome contains n identical ribs.

A general coordinate system is chosen with origin at the center of the sphere which has radius R. The base of the dome is assigned a radius r. Then, from the geometry of the sphere,

FIGURE 4.6 Arch ribs in a spherical dome with hinge at crown.

And the height of the crown is

## Renewable Energy 101

Renewable energy is energy that is generated from sunlight, rain, tides, geothermal heat and wind. These sources are naturally and constantly replenished, which is why they are deemed as renewable. The usage of renewable energy sources is very important when considering the sustainability of the existing energy usage of the world. While there is currently an abundance of non-renewable energy sources, such as nuclear fuels, these energy sources are depleting. In addition to being a non-renewable supply, the non-renewable energy sources release emissions into the air, which has an adverse effect on the environment.

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