The Cauchy problem for a fourth-order Boussinesq-type quasilinear wave equation (QWE-4) of the form \begin{document}$u_{tt} = -(|u|^n u)_{xxxx}\;\;\; \mbox{in}\;\;\; \mathbb{R} × \mathbb{R}_+, \;\;\;\mbox{with a fixed exponent} \, \, \, n>0, $ \end{document} and bounded smooth initial data, is considered. Self-similar single-point gradient blow-up solutions are studied. It is shown that such singular solutions exist and satisfy the case of the so-called self-similarity of the second type. Together with an essential and, often, key use of numerical methods to describe possible types of gradient blow-up, a homotopy approach is applied that traces out the behaviour of such singularity patterns as \begin{document}$n \to 0^+$\end{document} , when the classic linear beam equation occurs \begin{document}$u_{tt} = -u_{xxxx}, $ \end{document} with simple, better-known and understandable evolution properties.