Mathematics and Modelling
Understanding stress and why it occurs and how it can be dangerous is very important. The sort of stress we are discussing here is that which occurs in structures we use and take for granted every day. For example, the cable that holds the elevator we use, the steel frames that allow a skyscraper to stand tall, the muscles we use to lift a heavy weight will all have to withstand mechanical stresses that are induced throughout the structure. Stress in this context is due to the forces imposed on the structure.
If the stress in a structure becomes dangerously high then the structure can fail through collapse or through the formation of cracks that can subsequently lead to loss of performance and structural collapse. Fortunately mathematics, modelling and associated numerical techniques have enabled design engineers to build safe products that can withstand such stresses and operate for many years without maintenance.
One example is the design of printed circuit boards, which house semiconductor devices from companies such as AMD and Intel, found in many consumer products such as computers, mobile phones, laptops, DVD players, games consoles, etc. Surprisingly these components are constantly changing stresses. This is because when these devices are being used, their operating temperatures increase and each material making up the circuit boards wishes to expand by different amounts. This results in stress, which change every time we switch on the product. If these stresses are too high then the device that we are so fond of will cease to operate.
Fortunately mathematics helps us formulate the equations that describe the behaviour and modelling techniques allow us to understand how these stresses occur which in turn helps us design products that will not break under these stresses. Techniques such as the finite element method (FEM) and more generally Computational Fluid Dynamics (CFD) bring together mathematics, computers and software to allow design engineers to investigate how physics and the environment can lead to stress, and to then investigate and optimise the design to relieve these stresses and guarantee that the product continues to work safely for many years.
These mathematical modelling techniques, now taught in many university degree programmes, are used extensively by companies in the aerospace, automotive, medical, telecommunications, construction and many other high-technology sectors as part of the design process, providing many lucrative opportunities for mathematicians.
But it is not only the high technology sector that benefits from mathematics and its ability to model and predict stress. The massive heritage sector in the UK is now also benefiting from these technologies. An example is the world famous Cutty Sark Ship.
Built in 1869, the Cutty Sark is the world’s sole surviving clipper cargo ship and the fastest ship of her time. A symbol of Britain’s seafaring ingenuity she is placed at the gateway to the world heritage site in Greenwich and is one of the most distinctive landmarks to bee seen in the London Marathon.
When the Cutty Sark was built the ship was expected to last for 30 years. In the 1990’s it was recognised that the fabric of the Cutty Sark was in need of repair. Salt from the ocean and fifty years in a dry dock environment has taken its toll on the fabric of the ship and some of the iron in the hull of the vessel has now corroded.
An exciting plan of conservation has been put in place which requires dismantling the ship and treating and repairing the corroded structures. The ship will then be reassembled and lifted above the dry dock to create an exciting museum that will extend into the dry dock and below the ship.
To support the conservation plan the Cutty Sark employed the mathematics and modelling techniques behind finite element method to help understand how the stresses within the ships structure will change when the vessel is lifted above the dry dock.
The results from this mathematical analysis also played an important role in the subsequent lottery heritage submission. The results show that the conservation plans are feasible and these helped the ship secure £12M of lottery funding which is now allowing the conservation to take place saving this naval treasure for the nation.
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