Determine the maximum deflection of a circular plate on an elastic foundation subjected to a concentrated load F in the center of the plate.

Question

Accepted Answer

From Table 7.1, it is seen that as r approaches infinity, [latex] Z_{1}[/latex] and [latex] Z_{2}[/latex] also approach infinity. Therefore, [latex] C_{1}[/latex] and [latex] C_{2}[/latex] must be set to zero. Thus,

[latex] w=C_{3} Z_{3}(\alpha r)+C_{4} Z_{4}(\alpha r)[/latex]

and

[latex] heta=\frac{ d w}{ d r}=C_{3} \alpha Z_{3}^{\prime}(\alpha r)+C_{4} \alpha Z_{4}^{\prime}(\alpha r)[/latex]

As r approaches zero, 𝜃 must be zero due to symmetry. But from Table 7.1, [latex] Z_{4}^{\prime}[/latex] approaches infinity as r approaches zero. Hence, [latex] C_{4}[/latex] must be set to zero.

Thus,

[latex] w=C_{3} Z_{3}(\alpha r)[/latex]

[latex] \frac{ d w}{ d r}=C_{3} \alpha Z_{3}^{\prime}(\alpha r)[/latex]

[latex] \frac{ d ^{2} w}{ d r^{2}}=C_{3} \alpha^{2}\left[Z_{4}(\alpha r)-\frac{1}{\alpha r} Z_{3}^{\prime}(\alpha r) ight][/latex]

[latex] \frac{ d ^{3} w}{ d r^{3}}=C_{3} \alpha^{2}\left[\alpha Z_{4}^{\prime}(\alpha r)-\frac{1}{r} Z_{3}^{\prime \prime}(\alpha r) ight][/latex]

Substituting these derivatives into Eq. (7.10) and equating this to F gives

[latex] Q=D\left(\frac{ d ^{3} w}{ d r^{3}}+\frac{1}{r} \frac{ d ^{2} w}{ d r^{2}}-\frac{1}{r^{2}} \frac{ d w}{ d r} ight)[/latex] (7.10)

[latex] C_{3}=\frac{F}{4 \alpha^{2} D}[/latex]

and

[latex] w=\frac{F}{4 \alpha^{2} D} Z_{3}(\alpha r)[/latex]

and

[latex] w_{\max }=\frac{F}{8 \alpha^{2} D}[/latex]

Table 7.1 Limits of Z functions

Function	Limit as x → 0		Limit as x → ∞
[latex] Z_{1}(x)[/latex]	1.0		[latex] \zeta \cos \kappa[/latex]
[latex] Z_{2}( x )[/latex]	[latex] \frac{-x^{2}}{4}[/latex]		[latex] -\zeta \sin \kappa[/latex]
[latex] Z_{3}(x)[/latex]	0.5		[latex] v \sin \psi[/latex]
[latex] Z_{4}(x)[/latex]	[latex] \frac{2}{\pi} \ln \frac{\gamma x}{2}[/latex]		[latex] -v \cos \psi[/latex]
[latex] \frac{ d Z_{1}(x)}{ d x}[/latex]	[latex] \frac{-x^{3}}{16}[/latex]		[latex] \frac{\zeta}{\sqrt{2}}(\cos \kappa-\sin \kappa)[/latex]
[latex] \frac{ d Z_{2}(x)}{ d x}[/latex]	[latex] \frac{-x}{2}[/latex]		[latex] \frac{-\zeta}{\sqrt{2}}(\cos \kappa+\sin \kappa)[/latex]
[latex] \frac{ d Z_{3}(x)}{ d x}[/latex]	[latex] \frac{x}{\pi} \ln \frac{\gamma x}{2}[/latex]		[latex] \frac{v}{\sqrt{2}}(\cos \psi-\sin \psi)[/latex]
[latex] \frac{ d Z_{4}(x)}{ d x}[/latex]	[latex] \frac{2}{\pi x}[/latex]		[latex] \frac{v}{\sqrt{2}}(\cos \psi+\sin \psi)[/latex]
[latex] \zeta=\frac{1}{\sqrt{2 \pi x}} \exp \frac{x}{\sqrt{2}}[/latex]			[latex] v=\sqrt{\frac{2}{\pi x}} \exp \frac{-x}{\sqrt{2}}[/latex]
[latex] \kappa=\frac{x}{\sqrt{8}}-\frac{\pi}{8}[/latex]			[latex] \psi=\frac{x}{\sqrt{2}} +\frac{\pi}{8}[/latex]
[latex] \zeta=\frac{1}{\sqrt{2 \pi x}} \exp \frac{x}{\sqrt{2}}[/latex]		[latex] v=\sqrt{\frac{2}{\pi x}} \exp \frac{-x}{\sqrt{2}}[/latex]
[latex] \kappa=\frac{x}{\sqrt{8}}-\frac{\pi}{8}[/latex]		[latex] \psi=\frac{x}{\sqrt{2}}+\frac{\pi}{8}[/latex]

Function	Limit as x → 0		Limit as x → ∞
$Z_{1}(x)$	1.0		$\zeta \cos \kappa$
$Z_{2}( x )$	$\frac{-x^{2}}{4}$		$-\zeta \sin \kappa$
$Z_{3}(x)$	0.5		$v \sin \psi$
$Z_{4}(x)$	$\frac{2}{\pi} \ln \frac{\gamma x}{2}$		$-v \cos \psi$
$\frac{ d Z_{1}(x)}{ d x}$	$\frac{-x^{3}}{16}$		$\frac{\zeta}{\sqrt{2}}(\cos \kappa-\sin \kappa)$
$\frac{ d Z_{2}(x)}{ d x}$	$\frac{-x}{2}$		$\frac{-\zeta}{\sqrt{2}}(\cos \kappa+\sin \kappa)$
$\frac{ d Z_{3}(x)}{ d x}$	$\frac{x}{\pi} \ln \frac{\gamma x}{2}$		$\frac{v}{\sqrt{2}}(\cos \psi-\sin \psi)$
$\frac{ d Z_{4}(x)}{ d x}$	$\frac{2}{\pi x}$		$\frac{v}{\sqrt{2}}(\cos \psi+\sin \psi)$
$\zeta=\frac{1}{\sqrt{2 \pi x}} \exp \frac{x}{\sqrt{2}}$			$v=\sqrt{\frac{2}{\pi x}} \exp \frac{-x}{\sqrt{2}}$
$\kappa=\frac{x}{\sqrt{8}}-\frac{\pi}{8}$			$\psi=\frac{x}{\sqrt{2}} +\frac{\pi}{8}$
$\zeta=\frac{1}{\sqrt{2 \pi x}} \exp \frac{x}{\sqrt{2}}$		$v=\sqrt{\frac{2}{\pi x}} \exp \frac{-x}{\sqrt{2}}$
$\kappa=\frac{x}{\sqrt{8}}-\frac{\pi}{8}$		$\psi=\frac{x}{\sqrt{2}}+\frac{\pi}{8}$

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