5.3 Solutions of governing equations

We shall now apply the techniques of subsection 1.2.1 to establish the transfer matrix for a beam element subjected to an axial force $S$. The differential equation for a compressive axial force will be obtained from Eq. (5.30):
$\displaystyle \frac{d^{4}w}{dx^{4}} + {\vert}\frac{S}{EI_{y}}{\vert}\frac{d^{2}w}{dx^{2}} +
\frac{q\left(x\right)}{EI_{y}} = 0$     (5.33)

For a tensile axial force
$\displaystyle \frac{d^{4}w}{dx^{4}} - {\vert}\frac{S}{EI_{y}}{\vert}\frac{d^{2}w}{dx^{2}} +
\frac{q\left(x\right)}{EI_{y}} = 0$     (5.34)

We will be looking for the general solution of the non-homogeneous differential equation (5.33) in the form

$\displaystyle w = w_{0}w_{1} + w_{0}^{\prime}w_{2} + w_{0}^{\prime\prime}w_{3} +
w_{0}^{\prime\prime\prime}w_{4} + w_{e}\left(x\right)$     (5.35)

where $w_{0},\enspace w_{0}^{\prime},\enspace w_{0}^{\prime\prime}$ $w_{0}^{\prime\prime\prime}$ are parameter values of the searchable function at $x = x_{0}$;
$w_{1},\enspace w_{2},\enspace w_{3},\enspace w_{4}$ are a normed fundamental set to the associated homogeneous differential equation, and $w_{e}\left(x\right)$ is the particular solution of the non-homogeneous differential equation.



Subsections
andres
2014-09-09