E, F, and all that:
These terms refer to the elastic properties of materials, that is, how they stretch or squash under tension or compression.
Every material deforms, at least a little bit, under tension or compression.
We study this by understanding how a tiny sample unit of the given material deforms (its elastic behavior), and then thinking about what the net effect of all the individual units will be on a large piece of the material.
[[sample units: the stack of sponges / balloon / visualized points on the 2x2 demo beams / stack of illustration-board strips ]]
Poisson ratio: if one stretches the sample along one axis, it gets smaller in the perpendicular directions. One thickness gets bigger while the other s get smaller. The amount varies from material to material, and is measured by the ratio of the two changes in thickness.
Stress-strain, part I: two important terms: stress refers to the force (tension or compression) doing the deforming, and strain refers to the amount of deformation. They are NOT synonyms in this engineering use, whatever popular language does with them. A more precise definition of each is coming in a minute.
Stess-strain graphs: Different amounts of stress mean different amounts of strain. To keep track of the relationship, one plots a graph. Look at what one finds (Shigeru BanÕs engineers found) [[slide of machine and load-displacement graph]] when loading a sample of XXX material: greater stress (load) means greater strain (change in length of the test sample); part of the graph is straight; it stops (failure!)
Qualitative shapes of stress-strain graphs: brittle and ductile materials; ways things fail [[chalk under bending / laminated wood under bending / paper tube / coathanger / chalk under torsion / concrete in compression / rubber band ]]. Some things break, others yield.
=> some questions for the future, e.g. when and how to think about shear
Focus for now: the straight-line region, and the stopping point (failure or yield). Now we can define E (part I) and f (part I): the slope and failure stress, respectively.
E and f, part II: To read the literature, e.g. to understand the units of E and f, we need to think about the relation between total loads and deformations and the forces and deformations of the tiny sample units. What matters for failure is what happens down at the unit level, but what we control or impose is up at the level of totals.
Force per unit area: Side-by-side units share the load [[model with sponges]], so the force per unit is more relevant to failure than the total force. The normal to do this is to decide on a standard unit of area (sq ft or whatever) and divide total force by area to get the force per unit area acting in the given situation. This is the precise defnition of stress: force per unit area.
Designing for strength: arrange that the stress is within the allowable zone everywhere in the structure. For example: [[uniaxial tension example]]
Deformation per unit length: not so obviously, until you look at it right, the total change of length of a structural piece depends on how long it is, not just on the material it is made out of. Think of how the tiny units form a chain between one end of the material and the other. Each unit pulls or pushes on the next (in equilibrium), so each stretches or squashes. The total change in length is the sum of all the individual stretches or squashes. By knowing the material, what we know is how a given stress changes the size of a unit of that material, i.e. the deformation of a unit length (so many inches of change per inch of original length, and so forth). This is the precise definition of strain: change in length per unit of original length.
Designing for stiffness: arrange that the change of length of a structural piece is within a criterion you establish, e.g. deflection of a beam must be less that 1/360 of the beamÕs length (a very standard criterion for beams). The material property that governs this is not the failure stress. It is E, which gives the relation between stress and change in length. We saw it on the experimental graphs as the slope of the line. This is still true with our precise definitions of stress and strain: E (also called YoungÕs modulus) is the slope of the stress-strain line, i.e. delta (stress)/delta (strain)
Exercise on Ching (2nd edition) Rafter Span Table
A) If you are choosing rafters for a span of 10 feet, for a live load of 30 lbs per S.F., using wood with E = 1,200,000 psi, what is the smallest rafter cross section that will do the job? What is the next smallest?
B) Identify one variable in the table; pick a lower and a higher value for it, and calculate the ratio of the two (higher over lower); find the allowable span for each value of the variable; find the ratio of the two span values. Now write a short discussion of whether the allowable span changes in the way you would expect from the theory we have developed so far.
Do the same for two other variables in the table.
Sushi Questions: 1. What would you do if the available wood had an E value of 1,300,000 psi?
2. What would you do if you needed to deal with a rafter span of 4 feet (shorter than any allowable span in the table)?