The authors investigate the fourth order differential equation \[ u_{tt}+u_{xxxx}+bu^+=c\qquad x\in(0,\pi),\;t\in\mathbb R \] with free-end boundary conditions \(u_{xx}=u_{xxx}=0\) on \(\{0,\pi\}\) and \(2\pi\)-periodic boundary conditions in the \(t\)-variable. They also require the symmetry conditions \(u(x,-t)=u(x,t)\) and \(u(\pi-x,t)=u(x,t)\). Let \((\Lambda^-_n)_n\) be the sequence of negative eigenvalues of the linear problem \(u_{tt}+u_{xxxx}=\Lambda u\) with the same boundary and symmetry conditions. Using variational methods the authors prove the existence of two nontrivial solutions if \(b<\Lambda^-_1\). Moreover, for \(\Lambda^-_k<\Lambda^-_1\) they obtain a \(\delta_k>0\) such that there are four nontrivial solutions if \(\Lambda^-_k-\delta_k<b<\Lambda^-_k\). The associated functional is strongly indefinite. Using the limit relative category of G. Fournier, D. Lupo, M. Ramos, and M. Willem [Limit relative category and critical point theory, Dyn. Rep. 3, 1–23 (1993)] the solutions are obtained via linking arguments and suitable constraints. They are distinguished by energy estimates.