  In this paper, a strongly damped semilinear wave equation with a general
nonlinearity is considered. With the help of a newly constructed auxiliary
functional and the concavity argument, a general finite time blow-up criterion
is established for this problem. Furthermore, the lifespan of the weak solution
is estimated from both above and below. This partially extends some results
obtained in recent literatures and sheds some light on the similar effect of
power type nonlinearity and logarithmic nonlinearity on finite time blow-up of
solutions to such problems.
