Architects hate columns. Columns are ugly, block views, and are the biggest enemies of architectural elegance. When columns appear in buildings, they are almost always designed to be as few as possible and as slender as possible. Dome is a well-known type of structure with a long track record of creating large-span structures without columns. The Pantheon, Hagia Sofia, and the Grand Pagoda are all famous dome structures. This blog post explains why these structures chose to adopt domes and how domes work.
Before we delve in, we must lay down some foundations for understanding. I need to introduce a new coordinate system: the cylindrical coordinate system. What we normally use is called the ‘Cartesian’ coordinate system, where x,y,z are all straight lines and are perpendicular to each other. In a cylindrical coordinate system, the ‘x’ develops in the ‘radial’ direction, telling you that how big a circle you can draw based on the centre point and a radius. ‘y’ develops in the ‘hoop’, or ‘circumferential’ direction. It starts from an initial point on the given circle, and tells you how far away along the circle it is from the initial point. ‘z’ is the same as ‘Cartesian’ coordinate system, developing in the vertical direction, basically telling you how high/low the circle should be drawn. The cylindrical coordinate system is extremely useful for analysis of tubular/cylindrical structures such as tunnels and shafts. The conventional Cartesian coordinate system is more suitable for buildings/bridges where most things are rectangular in shape.
Now that we know the cylindrical coordinate system, consider the comparison between a flat slab and a dome slab for the same span. A flat slab generates a lot of bending at the centre. It creates tension on the bottom side and compression on the top side. In addition, it creates a large amount of deflection. This is usually unacceptable either from aesthetic point of view or compromises the function of the structure. For large span structure, it is un-economical to limit deflection as it takes a lot of ‘extra’ material to stiffen up the structure, and sometimes downright unfeasible.
What domes do is that they convert flexural stresses into axial stresses. In a dome, tension and compression also co-exist, but they happen at completely different locations of the structure. Compression is located at the centre of the span, whereas tension is located around the perimeter of the span; both are across the full depth of the slab.
A dome is an upside-down dish. If you imagine you put a dish upside-down onto a table and try to crush it with your hand, the dish will deform and tend to expand outwards. Cracks will tend to appear first around the perimeter, and radiates out widening gradually. This means in the hoop direction, the dome is in tension around its perimeter and in compression at its centre. You also will never see cracks developing in a circular manner, which means that in the radial direction, it is all compression across the entire dome.
Also bear in mind that, even after radial cracks occur, the dome remains stable as the originally intact dome disintegrate into a series of arches. Effectively the dome automatically resort to its secondary self-protection mechanism, which utilises radial bending rather than hoop tension. Since this mechanism uses bending rather than tension, it results in larger deformation, but is nonetheless stable. In design code language, it may fail the serviceability limit state by excessive deflection, but remain a pass under the ultimate limit state.
You may ask – some domes are masonry structures and they are weak in carrying tension because of mortar joints. How do they deal with hoop tension? The answer is that they can use either metal tension rings, such as metal chains, that wraps around the perimeter, or ‘tongue-and-grooves’ that interlocks the bricks. For modern reinforced concrete structures, the tension in dome can be negated by pre-stressing, in order to eliminate cracks. Nuclear containment buildings that have very stringent requirements on cracks usually adopts this approach.
Kommentare