Pointwise Comparison Principle for clamped Timoshenko beam

Abstract

We present the properties of three Green functions for:

1. general complex "clamped beam" \begin{equation}\tag{${\textrm{BC}}$} \begin{split} D_{\alpha ,\beta }[y] &\equiv y'''' -(\alpha ^{2}+\beta ^{2}) y''+\alpha ^{2}\beta^{2}y=f, \\ y(0)& =y(1) =y'(0) =y'(1) =0. \end{split} \end{equation}

2. Timoshenko clamped beam $D_{\alpha ,\overline{\alpha }}[y] \equiv f$ with (BC).

3. Euler-Bernoulli clamped beam $D_{k(1+i) ,k(1-i)} [ y] \equiv f$ with (BC).

In case 1. we represent solution via a Green operator expressed in terms of Kourensky type system of fundamental solutions for homogeneous case. This condense form is, up-to our knowledge, new even for the Euler-Bernoulli clamped beam and it allows to recognize the set of $\alpha's$ for which the Pointwise Comparison Principle for the Timoshenko beam holds. The presented approach to positivity of the Green function is much straightforward then ones known in the literature for the case 3 (see [J. Schröder, Zusammenhängende mengen inverspositiver differentialoperatoren vierter ordnung, Math. Z. 96 (1967) 89-110]).