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Bradford (In Balance) Sculpture
DESCRIPTION AND LOCATION
The sculpture is made of two opposite eighth segments of a sphere joined at the centre and with open sections. It’s made of stainless steel and bronze, is 4.5m in diameter and can be found outside the main entrance to the University of Bradford.
APPLIED LOADS
There’s not too many loads to consider with this structure, as it isn’t really meant to have people interact with it. It’s also in a pedestrianised zone so cars or other vehicles shouldn’t be much of an issue.
To begin with you have the self-weight of the structure itself. As we’ve mentioned it has a 4.5m diameter, a large structure then, with quite a significant self-weight.
There will also be variable loads on this sculpture, most notably wind, but also maybe people trying to climb on it, or other antics you would occasionally expect for a sculpture in a public square.
Of course as a piece of art, it will also have to bear the weight of the hopes and dreams of our current generation, and the judgement of future ones. However, these loads can be considered outside the scope of our analysis for now!
CONCEPTUAL/QUALITATIVE BEHAVIOUR
Supports:
The sculpture is only connected to the ground in one place, but there are actually three legs at the connection. Why is this?
Well, in terms of vertical loads it effectively widens the base, similar to how a 3-legged stool works. As long as you’re sitting on the stool so that your centre of mass lies somewhere inside the triangle created by the 3 legs then it won’t topple over.
The same principle is at works in our sculpture. Not toppling over in this case is the same as saying that all the reaction forces will be vertical, and as long as the load stays within that triangle of supports we won’t have any eccentric loads. The larger the triangle, the more likely this is.
In terms of horizontal loads it means that whichever plane the load acts (whichever way the wind blows if you will) there will be two legs offset by some distance which can take the load in tension and compression.
This is why you need a minimum of 3 legs, or there will be bending moments at the support.
For our structure, we’ll assume that everywhere is equally dense, and that calculating the centroid is equivalent to calculating the centre of mass.
As it’s a 3-dimensional object, we can do this by finding 3 lines where the moments of area balance in 3 non-parallel planes. They should meet at a unique point: our centroid.
This sounds fairly complicated, so let’s do it to get a feel for the process.
When we look at the sculpture from above, we get two quarter circles arranged as below. We can draw a line where moments of area will balance like this. Notice that because of the symmetry between the two sections this also happens to be a line of rotational symmetry, but in more uneven objects this isn’t necessarily the case.
We can think about the bird again to show this. You could draw a section through the bird separating the wing tips from the body and head which would be unsymmetrical but would nevertheless pass through the centre of mass. This is possible because the wings include more material than the rest of the model which compensates for the fact that they are closer to the pivot point.
Centre of mass is not necessarily the same thing as the centroid (or centre of area) of an object. Some parts of the object may be denser than others and contribute more mass relative to the area they occupy.
Dead Loads (Permanent Actions):
The structure itself looks pretty unstable, but when we look at the self-weight we can see that some geometrical trickery is keeping it standing.
The key is to look at the centre of mass. The centre of mass of an object is basically the point at which the moments generated by the object’s mass will balance each other in all directions. So if you put your finger there it should balance perfectly (provided you’re strong enough).
For the child’s toy on the right the centre of mass is at the tip of the beak, however unlikely that might seem.
We can take a side elevation and get a view of the structure looking like this:
Again, the line of equal moments is also a line of symmetry. You can explore these views on the 3D model to get a feel for what you’re seeing.
Looking at a final angle directly down into one of the sphere sections we get yet another view entirely.
Again, it’s important to note that the line of equal moments is not guaranteed to be a line of symmetry.
All that remains is for us to combine all those lines on our structure to find the centre of mass where all 3 lines intersect.
It may not come as much of a surprise that they all meet in the centre of the sphere, but the important result of this is that this centre is within the triangle of supports.
The centre of mass can also be thought of as the point where the total weight of the structure can be thought of as acting. This means that as this resultant force is directly over the supports, the loading isn’t eccentric, and the structure easily stands.
OTHER THINGS TO THINK ABOUT
In this example we haven’t really looked at variable loads like wind on the structure. This would be a bit complex and we’ll cover it in later tutorials but it’s worth starting to think about how lateral loads from the wind could be carried down to the base of the sculpture, trying to trace a load path.
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