Is it desirable to use concrete of very high strength i.e exceed 60MPa? What are the potential problems associated with such high strength concrete?

It is not a question of being desirable, the question is do you need it. If I see columns1400x1400 under a tower C30/37, yes it is desirable. Use it when needed, not as a design feature to use high strength concrete.

The problems with this type of concrete is pouring it and then compact it correctly. Its better to use it as self compacting concrete, then you take out the human factor of failure during compacting. Also a problem is the long term strength of the concrete, not much is known about that. The are doing tests with it and it looks promising, but promising results aren't enough for an engineer!

To increase the strength of concrete, say from40 MPa to80 MPa, it definitely helps in improving the structural performance of the structure by producing a denser, more durable and higher load capacity concrete. The size of concrete members can be significantly reduced resulting in substantial cost savings. However, an increase of concrete strength is also accompanied by the occurrence of thermal cracking.  With an increase in concrete strength, the cement content is increased and this leads to higher thermal strains.Consequently, additional reinforcement has to be introduced to control these additional cracks caused by the increase in concrete strength. Moreover, the ductility of concrete decreases with an increase in concrete strength. Attention should be paid during the design of high strength concrete to increase the ductility of concrete. In addition, fire resistance of high strength concrete is found to be less than normal strength concrete Though the tensile strength of high strength concrete is higher than that of normal concrete, the rate of increase of tensile strength is not proportional strength concrete, creep and shrinkage increases. nal to the increase of compressive strength. For normal concrete, tensile strength is about one-tenth of compressive strength. However, for high strength concrete, it may only drop to5% of compressive strength. Moreover, owing to a low aggregate content of high strength concrete, creep and shrinkage increases.