### Pointwise Comparison Principle for clamped Timoshenko beam

#### Abstract

We present the properties of three Green functions for:

\roster

\item"1." general complex ``clamped beam''

$$

\gather

D_{\alpha ,\beta }[y] \equiv y^{\prime \prime \prime \prime}

-(\alpha ^{2}+\beta ^{2}) y^{\prime \prime }+\alpha ^{2}\beta^{2}y=f,

\\

y(0) =y(1) =y^{\prime }(0) =y^{\prime}(1) =0.

\tag{BC}

\endgather

$$

\item"2." Timoshenko clamped beam $D_{\alpha ,\overline{\alpha }}[y] \equiv f$

with (BC).

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

with (BC).

\endroster

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 $\alpha ^{\prime }s$ 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 \cite{12}).

\roster

\item"1." general complex ``clamped beam''

$$

\gather

D_{\alpha ,\beta }[y] \equiv y^{\prime \prime \prime \prime}

-(\alpha ^{2}+\beta ^{2}) y^{\prime \prime }+\alpha ^{2}\beta^{2}y=f,

\\

y(0) =y(1) =y^{\prime }(0) =y^{\prime}(1) =0.

\tag{BC}

\endgather

$$

\item"2." Timoshenko clamped beam $D_{\alpha ,\overline{\alpha }}[y] \equiv f$

with (BC).

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

with (BC).

\endroster

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 $\alpha ^{\prime }s$ 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 \cite{12}).

#### Keywords

Timoshenko beam; Euler-Bernoulli clamped beam; Pointwise Comparison Principle; nonnegativity of Green function; fourth order ODE

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