As recently discovered, disregard of the size effect is a major source of uncertainty in the work-of-fracture (or Hillerborg's) test of the fracture energy and in the calibration of the softening law underlying the cohesive (or fictitious) crack models or the nonlocal softening damage models for concrete and other quasibrittle materials. The classical work-of-fracture test of a notched beam of one size, pioneered for concrete by Hillerborgal, can deliver acceptable results only for the total fracture energy GF. It is shown that the same complete load-deflection curve of a notched specimen can be closely approximated with stress-separation curves in which the values of tensile strength f't differ by 77% and the values of initial fracture energy Gf by 68%. This ambiguity means that that the one-size work-of-fracture test alone cannot provide a sufficient basis for quasibrittle fracture analysis, especially not for predicting the load capacity of quasibrittle structures. It is found, however, that if this test is supplemented by size effect tests with a sufficient size range, the cohesive crack model or nonlocal softening damage model can be identified unambiguously and render the fracture analysis of quasibrittle structures unambiguously realistic. Vice versa, the size effect fracture tests alone do not suffice for determining the total fracture energy GF. They provide a sufficient basis only for maximum load predictions of normal structures, but not for computing the post-peak softening with the energy absorption capability, nor the peak loads of extremely large quasibrittle structures. However, if the size effect tests of sufficient range are supplemented by one complete softening loaddeflection curve of notched specimens, an unambiguous identification of the peak loads and postpeak responses of structures becomes possible. To this end, the notched specimen tests must be conducted within a certain optimum size range in relation to the material inhomogeneity size. This range is established by extending the previous analysis of Cusatis and Schauffert. A further important general implication of the demonstrated non-uniqueness is that numerous recent studies in which multiscale or other softening damage models are verified by a few tests of one size only are virtually worthless.