On many occasions, an opening needs to be created in a structure. This could be a tunnel from a station box, a cross passage tunnel from the main tunnel, or a pipe from a fluid containing vessel. Numerical models are normally built to analyse these openings as the geometries tend to be over complicated for hand calculations. Engineers typically experience the difficulty of stress concentration around openings. You see a huge spike of stress at the opening that messes up all your contour plots. Stress concentration is normal, but how much is too much?
There are first principles that help you get a feel of space allowance for the structure, or sense-check whether the numerical model results are in the right ball park. Perfectly serving this purpose, here comes the ‘hole-in-plate’ theory, with its core principles explain in this blog post.
The stress flow of the ‘hole-in-plate’ model
On a plate ‘infinitely’ large (i.e. large enough to see the end of the ‘zone of influence’ of the opening) subject to uni-directional nominal stress, a circular opening is created. This imaginary model is called the ‘hole-in-plate’ model. From the diagram below you can see that there is stress flow in the plate that is disrupted by the existence of the opening. Tensile stress and compressive stress have the same effect but in opposite direction. In this example I will use compressive stress. The opening diverts the stress flow to its sides. As the same case with water, when some flow is squeezed to the sides, the intensity of the flow on the sides increases.
Here come the useful numbers:
Parallel to the direction of the stress flow, the stress concentration increases in an exponential manner approaching the edge of the opening, up to 3 times that of the nominal stress. The influence from the opening on the stress concentration diminishes as a given point moves from the edge of the opening, dying down to no effect at 3 times the diameter (6 times the radius) of the circular opening.
In the transverse direction to the stress flow, the diversion of the stress flow creates a ‘vacuum effect’, creating a suction (tension) force across the opening width. The tensile stress is 1 time that of the nominal stress right on the edge of the opening, and reduces down to 0 at 1x opening diameter away from the edge, and then reverses into compression further away, and then gradually reduces back to 0 transverse stress at 3xdiameter away.
Key assumptions
There are some important assumptions to this theory:
· The plate must be ‘infinitely’ large compared to the opening. If the width of the plate was finite (as usually is in reality), the narrower the plate is, the higher the nominal stress is, and the less stress concentration there is.
· The stress in the plate must be uniform
· The plate is flat. In applications to tubular structures such as tunnels, there will be additional bending effects.
· There is a single opening. Multiple openings in a row may divert the stress flow better hence resulting in less stress concentration.
· The opening is circular. If the opening is oval, the more flat the circular opening is, in the direction of the flow, the better the stress flow is diverted, and the less stress concentration there is. If the opening is rectangular, the stress concentration is usually much worse, with additional bending effect along its edges.
· The plate has the same thickness and material. Increase in thickness or stiffness in any local area is likely to attract stress hence result in stress concentration
In practical application
This is a simple, yet super useful theory for any engineers that deal with openings. It may be applicable to tunnel opening in a rectangular or circular station box, ‘child’ tunnel protruding out of ‘parent’ tunnel, or opening for pipeline within a pressure vessel. The list goes infinitely on. For tubular structures, the nominal stress in the ‘plate-in-hole’ model should be taken as the hoop stress, rather than the normal stress.
Particularly in tunnel engineering, as I am a tunnel engineer, there is a notion of ‘parent-child’ ratio, which is the ratio of the diameter of the big tunnel compared to the diameter of the small tunnel protruding out of it. The bigger the ratio is, the more it behaves akin to the ‘hole-in-plate’ analogy. As the ‘parent-child’ ratio reduces and approaches to 1, the more skewed the results will be.
The solution
Now that we know how much stress concentration exists around an opening, we need to provide a practical structural solution for it. It is often straightforward - for high compression zone, thicker concrete could be provided (if it is a concrete structure); for high tension zone, more reinforcement should be provided. Very often, for concrete structures, a ‘picture frame’ type of structure is provided around the opening.
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