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3.11

## University of Liverpool

Skip to 0 minutes and 0 seconds Hello and welcome to the mid-course review for the MOOC on Understanding Superstructures I’d like to start by recalling some of my opening comments in the introduction in week 1. Everyone needs structures Structures literally support life and without structures we would just have a random heap of parts I went on from these statements to talk about the questions that a designer and client might discuss at the concept stage of a structure Why do you need it? How long should it last? And what should it look like?

Skip to 0 minutes and 37 seconds The answers to these three questions form the beginning of the design specification by defining the purpose the material choice and form for the structure And hopefully will lead to a super structure that meets the principles of good structural design laid down by Vitruvius Utility, Durability and Beauty Utility is about providing the difference between an ordered functioning system and a random collection of non-functioning parts structure positions and protects the parts of a system but to be a super structure it should also be cheap with a tiny ecological footprint Durability is about ensuring that the structure will perform its role for as long as is required That might be for the foreseeable future if it is a bridge or longer for nuclear waste disposal system Or only the next 15 minutes if it’s a paper coffee cup Beauty is about a form that is aesthetic pleasing and fits well into its cultural historical and social contexts We can’t achieve all of these attributes at once and so there has to be compromise which is where the skill of the structural engineer comes into play So now let’s look at some of the knowledge and understanding that a structural engineer might deploy to support the design process particularly acknowledging that 100% reliability in all circumstances is not possible And hence we need to optimise structures in terms of their durability In order to achieve durability we need to consider the forces acting on and inside a structure using Newton’s laws In static equilibrium the sum of the forces in one direction should be zero So we can resolve in the x y and z directions and we can take moments to find the values of forces and moments To help us perform this type of analysis engineers like to draw free-body diagrams that are simple sketches of a structure or a part of a structure separated from its surroundings with the connections to its surroundings replaced by the corresponding forces acting at those connections I used an example of your computer sitting on a desk We can go a step further and think about structures consisting of small elements or Platonic cubes Then we can consider the forces acting on these cubes So on each face of the cube there could be direct compressive or tensile forces shear forces and moments as shown in the small inset figure The direct stress sigma is equal to force divided by the cross-section area A on which the force is acting The force causes a deflection delta and strain is defined as delta or deflection divided by original length l If we plot stress sigma as a function of strain epsilon then the gradient of the graph is the modulus of elasticity E (for a linear elastic material).

Skip to 4 minutes and 1 second When a structure deflects the work done is absorbed as strain energy U and is equal to the area under the plot of force against deflection or F times delta upon 2 The strain energy can be released and used to do work when the structure springs back to its original shape or dimensions We looked at the special case of thin-walled pressure vessels The definition of thin is that the thickness must be less than one twentieth of the radius of curvature Using a free-body diagram of a cut-section of the vessel and resolving forces we can derive expressions for the circumferential and longitudinal stresses as shown here For a cylindrical pressure vessel the circumferential stress is twice the longitudinal stress which is why sausages tend to burst open with longitudinal cracks pulled apart by the circumferential stress.

Skip to 5 minutes and 6 seconds The relationship of stress to radius of curvature implies that structures with tighter curvatures will contain lower stresses and require less substantial support to contain the same pressures and I talked about these ideas in the context of bats wings and the design of sailing ships We can use structures to transmit power and energy usually in the form of torque. Torques cause rotational deformation and shear stresses tau.

Skip to 5 minutes and 40 seconds Shear stresses and strain in elastic material are related via the bulk modulus G which is the gradient of a plot of shear stress against shear strain just like the modulus of elasticity is the gradient of the direct stress-strain graph There is a triple equality that connects shear stress tau at a radius r to the applied torque T and the polar second moment of area J For a circular shaft J is equal to pie times diameter to the power 4 upon 32 And the third part of the triplet is the bulk modulus G times the ratio of the angle of twist theta to the length of the shaft L This week we have extended our analysis skills to include beams in bending The bending stress sigma is equal to the bending moment times y bending moment times y the distance from the neutral axis divided by the second moment of area The second moment of area for a rectangular beam is b times h cubed upon 12 where b is breadth and h height When a beam bends there will be compression on the inside of the curve and tension on the outside with a transition between tension and compression where there is no change and this is the neutral axis We can draw shear force and bending moment diagrams to establish the variation of M along the length of the beam Finally there are some classes of structure which Newton’s laws and equilibrium analysis is not sufficient to allow us to determine the forces and stresses These structures are called statically indeterminate We have to use compatibility of displacements to analysis these structures This involves introducing imaginary or virtual deflections and using the requirement to maintain the continuity of structural geometry to identify a new set of relationships Initially these new relationships will be in terms of deflections delta but we can combine the definitions of stress strain and modulus of elasticity to relate deflection to forces and convert them into an additional equation describing the forces on the structure That concludes a brief overview of the material that I covered so far in this MOOC I hope it has been useful.

Skip to 8 minutes and 30 seconds You can download the summary slide to use as a reference

# Mid-course review

Please see the pdf below.