Should we design for stiffness or strength?

I don't think it's quite correct to state that we usually design for strength first. As other have pointed out, both are important and either can take priority depending on the design. For example, when designing a structure for dynamic loads, the first concern is usually the response frequencies of major structural modes in light of the excitation to avoid resonance. This involves design for both structural stiffness and minimum mass (i.e., an efficient structure). Sometimes, the response frequency issue has non-structural implications, such as playing havoc with the operation of servo control mechanisms. The example given in the article about the tube that is stiffened when repaired is a good one as well. This situation arises in often in real world structures since most do have parallel load paths rather than a single load path; the concept of load distribution between multiple parallel load paths is always a concern, and is often different in different directions. One of the many useful features of FEA is that element forces at each grid point are all readily available, saving a lot of time. I particularly like the "free body" feature in FEMAP, which makes it easy to see such forces . As usual, the structural engineer needs to be concerned with asking the right question first. Being simply a PATRAN or FEMAP modeling whiz is not enough.

In my lecture to students today on Exercise mechanics, i gave them an example of high rate of loading and slower rate of loading. Of course stiffness is an important factor but with the higher load of rating (e.g. running/jogging) we give bone an opportunity to accumulate more loads, more stiffer and store more energy over a short time period than slower rate of loading (e.g. walking). So I think even accounting for stiffness, the rate of loading is again a significant predictor.

The example given here by Tony is one of constant displacement and not a constant load. In such cases, the stiffness is the key, more than strength. As in dynamics. Design for Dynamic loads will require stiffness optimisation more than strength optimisation.

The answer has to be "for both". In some applications, strength and stiffness are in the same direction, some other applications, they are against to each other. In the case of vibrations, stiffness determines the resonant frequency, that in turn affects the strength. When the load paths are manifold, the one with less stiffness tends to be less stressed. Usually, the concept of stiffness is related to a large portion of the structure, while strength is of a concern of the stress concentration where the bad thing start from.