There have been many attempts to manufacture and market plastic bicycles. All have been too flexible and soft. Which design-limiting property is insufficiently large?

The design-limiting property that insufficiently large is the elastic modulus (Young modulus)

Explanation:

Plastic usually have a relatively low elastic modulus, this couses the material to deform too much under stress. In the case of a bicycle, a little weight you put on it or little bumps will cause the bicycle to deform too much.