Duragrouch

V1llage, you said earlier:
I am a retired M.E., so offering a little knowledge. You may be confusing material STIFFNESS with STRENGTH.

Let's says you have two pieces of steel, identical dimensions, one is "mild" low-strength steel, the other is high-strength spring steel. The mild one bends a little bit elastically, but then YIELDS (permanent deformation). The spring steel one can be bent much further without yield, because it has a higher yield strength (as well as a higher "ultimate tensile strength", i.e, the point at which is breaks/fractures). That is strength. BUT... NOTE!: Before the mild steel part yielded, when it was functioning within the ELASTIC LIMIT, its ELASTICITY/STIFFNESS was exactly the same as the higher strength steel part, in terms of force versus movement. This is the modulus of elasticity or stiffness, also known as Young's Modulus. Note: Within a given "family" of materials (like "steel", or "aluminum"), the stiffness changes very little, even as stength changes. (Some very new metal alloys have broken this rule, but let's dismiss that for now.) Steel stiffness is about 30x10^6 PSI. Aluminum is about 10x10^6 PSI, 1/3 as stiff, for the same geometry (size). Let's look at an example:

My 1989 Cannondale crit racer is made from 6061-T6 aluminum, and has a huge (for the time) 2" diameter down tube. To make from steel at low weight, the wall thickness would be so thin, it would be very dent-prone. With aluminum, for the same weight, the wall thickness can be 3X the thickness, so durable, because generally, the flat plate stiffness of something is a function of the cube of its thickness (this neglects hoop strength of the tubing curve, but let's stay simple). So even if the aluminum is 1/3 the stiffness, being 3X the thickness will give it 3^3 times stiffness, or 27X, but the material is 1/3 as stiff, so divide by 3, and you still get 9X as dent-stiff as a steel tube of same diameter and weight. But, torsional stiffness of a tube is a function of the 4th power of the radius, so that 2" down tube, versus 1", is gonna be about 2^4 stiffer in torsion, so about 16X stiffer, and that makes for an incredible bike in sprints and climbing, at the time, it was the stiffest frame Bicycling Magazine had ever tested. And it was probably not 3X wall thickness of a steel frame, because it was also lighter, 3 lbs. But, all that stiffness has a downside; It is so stiff in longitudinal bending, it had a terrible ride. After years of punishing ride and fatiguing two sets of rims, I fit touring rims with 32mm tires, far better.

Titanium is great, because it has much better fatigue strength than aluminum, and sometimes even steel, so it can offer a bit more spring to the frame, for better ride. I forget the stiffness modulus of Ti. It's also 1/2 the density of steel, so can still have a reasonable wall thickness for durability.

Carbon fiber composite is ideal in this respect, because they can orient the fibers on the frame tube to give superb torsional stiffness, but flex more in bending, to give both good ride and desired stiffness. However it is fragile, and if you bonk it, only a few places will repair it, maybe.

So, to conclude: Usually, bike frame STRENGTH comes into play for fatigue strength (long life) before the frame yielding is a factor. (Fatigue strength is cyclic loading at stresses below the yield strength of the part.) But for riding characteristics, the stiffness/elasticity is a whole different ball game. And that is a function of the material stiffness, the tube geometry (section modulus), and other geometric factors. A frame builder with a ton of experience with a particular material and geometry, can probably tell you off the top of their head, how a frame will behave, given the knowns above. The rest of us, would have to do a complete "solid model" of the frame on computer, do a Finite Element Analysis (FEA) of the frame under various load conditions, and even then, would need "baseline" values of a known frame with known ride characteristics, to then tell you, how the prospective new frame will "feel". Fortunately, at this point in history, the above will give "high-fidelity" in predicted results versus actual tests.

What is even MORE complex, is a computer program to predict the dynamic behavior of the bike, how it "handles". Dynamics of two-wheel bicycle systems (bikes and motorcycles, et al) is an entire field unto itself, only in recent decades having the computer simulation programs to predict; It used to be all trial and error.