In this paper, we would like to consider the Cauchy problem for semi-linear σ-evolution equations with time-dependent damping for any σ≥1. Motivated strongly by the classification of damping terms in the paper34, the first main goal of the present work is to make some generalizations from σ=1 to σ>1 and simultaneously to investigate decay estimates for solutions to the corresponding linear equations in the so-called effective damping cases. For the next main goals, we are going not only to prove the global well-posedness property of small data solutions but also to indicate blow-up results for solutions to the semi-linear problem. In this concern, the novelty which should be recognized is that the application of a modified test function combined with a judicious choice of test functions gives blow-up phenomena and upper bound estimates for lifespan in both the subcritical case and the critical case, where σ is assumed to be any fractional number. Finally, lower bound estimates for lifespan in some spatial dimensions are also established to find out their sharp results.