Page:American Journal of Mathematics Vol. 2 (1879).pdf/23

The equations of condition for equilibrium in these cases, from equations 2, will be the three following:

${\tfrac {dN_{1}}{dx}}+{\tfrac {dT_{3}}{dy}}+{\tfrac {dT_{2}}{dz}}=0,$	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,}$........(3)
${\tfrac {dT_{3}}{dx}}=0,$
${\tfrac {dT_{2}}{dx}}=0,$

or

$(\lambda +\mu )\,\left({\tfrac {d^{2}u}{dx^{2}}}+{\tfrac {d^{2}v}{dxdy}}+{\tfrac {d^{2}w}{dxdz}}\right)\,+$	$\,\mu \,\left({\tfrac {d^{2}u}{dx^{2}}}+{\tfrac {d^{2}u}{dy^{2}}}+{\tfrac {d^{2}u}{dz^{2}}}\right)$	$\,=0,$	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,}$...(4)
​	$\mu \,\left({\tfrac {d^{2}u}{dxdy}}+{\tfrac {dv^{2}}{dx^{2}}}\right)$	$\,=0,$
​	$\mu \,\left({\tfrac {d^{2}w}{dx^{2}}}+{\tfrac {d^{2}u}{dxdz}}\right)$	$\,=0.$

Three other equations of condition result from the conditions that $N_{2},$ $N_{3}$ and $T_{1}$ each equal zero. These give in connection with equations (1)

$\lambda \,\left({\tfrac {du}{dx}}+{\tfrac {dv}{dy}}+{\tfrac {dw}{dz}}\right)+2\mu \,{\tfrac {dv}{dy}}$	$\,=0,$	......(5)
$\lambda \,\left({\tfrac {du}{dx}}+{\tfrac {dv}{dy}}+{\tfrac {dw}{dz}}\right)+2\mu \,{\tfrac {dw}{dz}}$	$\,=0,$	......(6)
$\mu \,\left({\tfrac {dv}{dz}}+{\tfrac {dw}{dy}}\right)$	$\,=0.$	......(7)

These equations, as it will afterwards be seen, aid in the determination of the displacements $u,$ $v$ and $w.$ The last two of equations (3) may be integrated at once, and will give

$T_{3}=f\,(y,z)$	$\,=0,$	..........(8)
$T_{2}=F\,(y,z)$	$\,=0,$	..........(9)

In which $f$ and $F$ signify any arbitrary functions of $y$ and $z$ whatever; they correspond to the "constants" of integration and must be written because the intensities of the internal stresses are, in general, each functions of $x,$ $y$ and $z.$

Denoting by $f'_{y}\,(y,z)$ and $F'_{z}(y,z)$ the partial derivatives of $T_{3}$ and $T_{2},$ respectively, in respect to the variables indicated, the first of equations (3) may be integrated, and will give

$N_{1}=-x\,[f'_{y}\,(y,z)+F'_{z}\,(y,z)]+\Psi \,(y,z)$

....(10)

The quantity $\Psi \,(y,z)$ is any arbitrary function of $y$ and $z,$ and it will now be shown that in general it is independent of $y$ and $z,$ as well as of $x,$ and that many of the cases of pure flexure it may be put equal to zero.