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Stiffness (F=Kx) is the extent to which an object resists deformation in response to an applied force. Elastic Modulus (E=Stress/Strain) is a quantity that measures an object or substance's resistance to being deformed elastically when a stress is applied to it.
Use one of the many, free, online moment of inertia calculators (like this one) to see how increasing the height of the beam will have an exponential effect on increasing the stiffness of the beam. And this site helps provide a pictorial view of the load(s) upon a beam depending upon differing configurations, such as where the supports are and ...
In simplified terms, this is because in statically determinate structures one can determine how the load is transferred through the structure to the supports without caring about the stiffness itself. In indeterminate structures, however, the stiffness directly impacts how the load is shared by the supports.
"Stiffness" quantifies the level of resistance of a structural member against deformation under loads. Two types of deformations are of particular concern - translation and rotation , in structural engineering.
Stiffness, at least when talking about the material (without introducing the geometry of the object) is I think synonymous with elastic modulus, i.e. the slope of the elastic portion of the curve. We also use stiffness when talking about objects, for instance stiffness of a spring, or stiffness of a truss.
The far end formula, $\frac{3EI}{L}$, applies if the beam is discontinuous in that point. This means that the point of reference is the end of the beam where the support is a hinge/pin.
Now, another possible definition is stiffness as the deflection a beam or structure suffers under load. This would be an analogy with a spring's stiffness (which is literally measured in force needed to move the spring a unit distance). This could be described as the whole structure's stiffness (even if that structure is a single cantilever beam).
Stiffness is the reverse of flexibility, the degree of deformation under stress, therefor a stiff material requires more stress or force to deform. eg, a cantilever beam under a normal load applied to its end will bend less if it is stiffer.
I'm an amateur at engineering, working more on musical instruments, and I'm often in confusion regarding what exactly constitutes stiffness of a beam for a load applied on it so it could resist deformation, for example on googling for a rectangular beam I get the basic definition that it is proportional to breadth × square of height.
Two 29.0mm diameter tubes weigh 1.43745 times as much as the one 40mm diameter tube (assumption: same wall thickness) and the "weight to torsional stiffness ratio" is 1.43745 accordingly. Note that "weight to stiffness ratio" is different from "stiffness to weight ratio". In my question I asked for the "stiffness to weight ratio".