= Solution
With zero <tension> and zero distributed load, a free interval satisfies $w''''=0$. Before contact the entire <elastic beam> is free, so $w$ is a cubic <polynomial>. The four end <boundary conditions> determine it uniquely. Reflection $x\mapsto-x$ preserves those <boundary conditions>, so the unique solution is an <even function> and its odd terms vanish. Direct substitution gives
$$
\boxed{w(x)=\frac{\alpha}{2L}(x^2-L^2).}
$$
For $\alpha\geq0$, its minimum occurs at $x=0$ and is $-\alpha L/2$. The constraint $w\geq-H$ is therefore satisfied exactly when $\alpha\leq2H/L$. The first contact is at the midpoint:
$$
\boxed{\alpha_{\mathrm{first}}=\frac{2H}{L}.}
$$
At equality the <normal contact force> is still zero. The uniqueness used here can also be checked by integrating $v''''v$ for the difference of two solutions with homogeneous <clamped boundary conditions>, which gives $\int_{-L}^L(v'')^2\,dx=0$ and hence $v=0$.
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