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:

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

or

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

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

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

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

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

In which ${\displaystyle f}$ and ${\displaystyle F}$ signify any arbitrary functions of ${\displaystyle y}$ and ${\displaystyle z}$ 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 ${\displaystyle x,}$ ${\displaystyle y}$ and ${\displaystyle z.}$

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

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

....(10)

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