The basics; butted tubing, stress and fatigue.
Why was the design innovation of "butting" of bicycle frame tubing developed? How does it relate to good frame design, when constructing a frame of any material?
In my opinion, THE single most important innovation in bicycle frame design is butted frame tubing and more importantly the technology and understanding connected with it. It is little understood or appreciated. It actually pre-dated the understanding in jet-aircraft (with tragic results). It's not designed to save weight. The only reference on the internet that spells out in basic terms the reason butted tubing was developed is here in the second paragraph. With this knowledge it is clear that a frame designed without some form of butting is not advanced design, because without it, no effort has been made to achieve a more uniform distribution of the levels of stress occurring in the structure. An aircraft is designed to have an optimal (even) distribution of peak stress levels throughout its structure.
I cannot achieve the structural performance that a steel frame with key design features offers for anywhere near the same price with any other material. Given that bicycle frames play on the edge of performance weight, failure is possible and good design is about optimising the choices. Should a bike frame fail in use, the principal way is by fatigue-cracks developing. So that becomes the principal design issue to manage. It is very common to make a frame stiffer to make it less prone to fatigue failure; remember stiff-riding aluminium frames? Most carbon-fibre frames aren't much better. It is the nature of carbon-fibre that weight reductions can be achieved by making the frame tubes stiffer, at the expense of the manner in which they fail. Hence in recent times the more frequent and more spectacular failures of carbon-fibre frames.
I do not sacrifice frame comfort or robustness for the sake of a few hundred grams of weight saving. At the moment nothing beats steel as a material for a custom made bike frame when you take into account real-world use. Most of us aren't team riders and get handed a new bike if we have a mechanical problem. All high-performance frames strive to achieve a low weight whilst balancing that with robustness amongst other features, but the manner in which a frame behaves when it develops fatigue crack is sometimes overlooked. Steel is the most forgiving material of all with the longest time before a dramatic failure should a fatigue crack develop. Sudden catastrophic failures are far more common with carbon-fibre frames. Forks are far simpler structure with a far more unchanging design over tiem and hence are far less prone to failure (thankfully since the consequences are far worse than for a frame).
In any structure, joints are the location where stress is concentrated. Here I am referring to Engineering Stress, the applied load divided by the original cross-sectional area of a material. Also known as Nominal stress. When the joints are at corners, stress is intensified. The joints are the point of localised stress concentration, sometimes referred to as a "stress riser'. At welded joints, there is further stress at the edge of any lumpiness (any rapid change to the wall thickness).
Heating also diminishes a material's ability to resist high levels of stress. This is why bike frames frames made of aircraft tubing break at the edge of the weld where all this converges.
To combat this inherent weak spot in frame tubes, designers can specify a heavy wall thickness. Thicker tubing walls at the joints enable thinner walls elsewhere. Known as butting. Reynolds invented it over 100 years ago. Carbon-fibre frames also have this design feature with extra layers at the joints. Besides saving weight and improving comfort, double-butted tubing principally offers improved fatigue resistance. A smooth section of tubing adjacent to a joint, with a wall that smoothly tapers down to a thin-walled center section can handle loads that would otherwise concentrate stress at the joints. This is the bit that messes most with people's heads; the ends of properly designed butted tubing can be thinner than a unbutted tube doing the same job, in resisting premature fatigue-failure. Stiffness and "strength" are a secondary issue to this. This explains how 8/5/8 tubes (.8mm at the ends and .5mm in the middle section) can dissipate stress better (and last longer) than frames built from much heavier 0.9mm wall "aircraft" tubing.