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Outer edge simply supported, inner edge fixed

Annular plate with a uniform annular line load w at a radius ro b a Q b Q a M ra M rb w r o y b θ b θ a
Annular plate with a uniform annular line load $w$ at a radius $r_o$
Outer edge simply supported, inner edge fixed w r o
Outer edge simply supported, inner edge fixed

Values for calculation

$a$ $\mathrm{mm}$
$b$ $\mathrm{mm}$
$t$ $\mathrm{mm}$
$r_o$ $\mathrm{mm}$
$w$ $\mathrm{N/m}$
$E$ $\mathrm{MPa}$
$ν$
$r$ $\mathrm{mm}$

Calculation

Plate constant $ C_2 $

$$C_2=\cfrac{1}{4}\cdot\left[1-\left(\cfrac{b}{a}\right)^2\cdot\left(1+2\cdot\ln\left(\cfrac{a}{b}\right)\right)\right]$$
$C_2=\cfrac{1}{4}\cdot\left[1-\left(\cfrac{b}{a}\right)^2\cdot\left(1+2\cdot\ln\left(\cfrac{a}{b}\right)\right)\right]$
Equations in LaTeX code
C_2=\cfrac{1}{4}\cdot\left[1-\left(\cfrac{b}{a}\right)^2\cdot\left(1+2\cdot\ln\left(\cfrac{a}{b}\right)\right)\right]

Plate constant $ C_3 $

$$C_3=\cfrac{b}{4\cdot a}\cdot\left\{\left[\left(\cfrac{b}{a}\right)^2+1\right]\cdot\ln\left(\cfrac{a}{b}\right)+\left(\cfrac{b}{a}\right)^2-1\right\}$$
$C_3=\cfrac{b}{4\cdot a}\cdot\left\{\left[\left(\cfrac{b}{a}\right)^2+1\right]\cdot\ln\left(\cfrac{a}{b}\right)+\left(\cfrac{b}{a}\right)^2-1\right\}$
Equations in LaTeX code
C_3=\cfrac{b}{4\cdot a}\cdot\left\{\left[\left(\cfrac{b}{a}\right)^2+1\right]\cdot\ln\left(\cfrac{a}{b}\right)+\left(\cfrac{b}{a}\right)^2-1\right\}

Plate constant $ C_5 $

$$C_5=\cfrac{1}{2}\cdot\left[1-\left(\cfrac{b}{a}\right)^2\right]$$
$C_5=\cfrac{1}{2}\cdot\left[1-\left(\cfrac{b}{a}\right)^2\right]$
Equations in LaTeX code
C_5=\cfrac{1}{2}\cdot\left[1-\left(\cfrac{b}{a}\right)^2\right]

Plate constant $ C_6 $

$$C_6=\cfrac{b}{4\cdot a}\cdot\left[\left(\cfrac{b}{a}\right)^2-1+2\cdot\ln\left(\cfrac{a}{b}\right)\right]$$
$C_6=\cfrac{b}{4\cdot a}\cdot\left[\left(\cfrac{b}{a}\right)^2-1+2\cdot\ln\left(\cfrac{a}{b}\right)\right]$
Equations in LaTeX code
C_6=\cfrac{b}{4\cdot a}\cdot\left[\left(\cfrac{b}{a}\right)^2-1+2\cdot\ln\left(\cfrac{a}{b}\right)\right]

Plate constant $ C_8 $

$$C_8=\cfrac{1}{2}\cdot\left[1+ν+\left(1-ν\right)\cdot\left(\cfrac{b}{a}\right)^2\right]$$
$C_8=\cfrac{1}{2}\cdot\left[1+ν+\left(1-ν\right)\cdot\left(\cfrac{b}{a}\right)^2\right]$
Equations in LaTeX code
C_8=\cfrac{1}{2}\cdot\left[1+ν+\left(1-ν\right)\cdot\left(\cfrac{b}{a}\right)^2\right]

Plate constant $ C_9 $

$$C_9=\cfrac{b}{a}\cdot\left\{\cfrac{1+ν}{2}\cdot\ln\left(\cfrac{a}{b}\right)+\cfrac{1-ν}{4}\cdot\left[1-\left(\cfrac{b}{a}\right)^2\right]\right\}$$
$C_9=\cfrac{b}{a}\cdot\left\{\cfrac{1+ν}{2}\cdot\ln\left(\cfrac{a}{b}\right)+\cfrac{1-ν}{4}\cdot\left[1-\left(\cfrac{b}{a}\right)^2\right]\right\}$
Equations in LaTeX code
C_9=\cfrac{b}{a}\cdot\left\{\cfrac{1+ν}{2}\cdot\ln\left(\cfrac{a}{b}\right)+\cfrac{1-ν}{4}\cdot\left[1-\left(\cfrac{b}{a}\right)^2\right]\right\}

Function of the radial location $ F_1 $

$$F_1=\cfrac{1+ν}{2}\cdot\cfrac{b}{r}\cdot\ln\left(\cfrac{r}{b}\right)+\cfrac{1-ν}{4}\cdot\left(\cfrac{r}{b}-\cfrac{b}{r}\right)$$
$F_1=\cfrac{1+ν}{2}\cdot\cfrac{b}{r}\cdot\ln\left(\cfrac{r}{b}\right)+\cfrac{1-ν}{4}\cdot\left(\cfrac{r}{b}-\cfrac{b}{r}\right)$
Equations in LaTeX code
F_1=\cfrac{1+ν}{2}\cdot\cfrac{b}{r}\cdot\ln\left(\cfrac{r}{b}\right)+\cfrac{1-ν}{4}\cdot\left(\cfrac{r}{b}-\cfrac{b}{r}\right)

Function of the radial location $ F_2 $

$$F_2=\cfrac{1}{4}\cdot\left[1-\left(\cfrac{b}{r}\right)^2\cdot\left(1+2\cdot\ln\left(\cfrac{r}{b}\right)\right)\right]$$
$F_2=\cfrac{1}{4}\cdot\left[1-\left(\cfrac{b}{r}\right)^2\cdot\left(1+2\cdot\ln\left(\cfrac{r}{b}\right)\right)\right]$
Equations in LaTeX code
F_2=\cfrac{1}{4}\cdot\left[1-\left(\cfrac{b}{r}\right)^2\cdot\left(1+2\cdot\ln\left(\cfrac{r}{b}\right)\right)\right]

Function of the radial location $ F_3 $

$$F_3=\cfrac{b}{4\cdot r}\cdot\left\{\left[\left(\cfrac{b}{r}\right)^2+1\right]\cdot\ln\left(\cfrac{r}{b}\right)+\left(\cfrac{b}{r}\right)^2-1\right\}$$
$F_3=\cfrac{b}{4\cdot r}\cdot\left\{\left[\left(\cfrac{b}{r}\right)^2+1\right]\cdot\ln\left(\cfrac{r}{b}\right)+\left(\cfrac{b}{r}\right)^2-1\right\}$
Equations in LaTeX code
F_3=\cfrac{b}{4\cdot r}\cdot\left\{\left[\left(\cfrac{b}{r}\right)^2+1\right]\cdot\ln\left(\cfrac{r}{b}\right)+\left(\cfrac{b}{r}\right)^2-1\right\}

Function of the radial location $ F_4 $

$$F_4=\cfrac{1}{2}\cdot\left[\left(1+ν\right)\cdot\cfrac{b}{r}+\left(1-ν\right)\cdot\cfrac{r}{b}\right]$$
$F_4=\cfrac{1}{2}\cdot\left[\left(1+ν\right)\cdot\cfrac{b}{r}+\left(1-ν\right)\cdot\cfrac{r}{b}\right]$
Equations in LaTeX code
F_4=\cfrac{1}{2}\cdot\left[\left(1+ν\right)\cdot\cfrac{b}{r}+\left(1-ν\right)\cdot\cfrac{r}{b}\right]

Function of the radial location $ F_5 $

$$F_5=\cfrac{1}{2}\cdot\left[1-\left(\cfrac{b}{r}\right)^2\right]$$
$F_5=\cfrac{1}{2}\cdot\left[1-\left(\cfrac{b}{r}\right)^2\right]$
Equations in LaTeX code
F_5=\cfrac{1}{2}\cdot\left[1-\left(\cfrac{b}{r}\right)^2\right]

Function of the radial location $ F_6 $

$$F_6=\cfrac{b}{4\cdot r}\cdot\left[\left(\cfrac{b}{r}\right)^2-1+2\cdot\ln\left(\cfrac{r}{b}\right)\right]$$
$F_6=\cfrac{b}{4\cdot r}\cdot\left[\left(\cfrac{b}{r}\right)^2-1+2\cdot\ln\left(\cfrac{r}{b}\right)\right]$
Equations in LaTeX code
F_6=\cfrac{b}{4\cdot r}\cdot\left[\left(\cfrac{b}{r}\right)^2-1+2\cdot\ln\left(\cfrac{r}{b}\right)\right]

Function of the radial location $ F_7 $

$$F_7=\cfrac{1}{2}\cdot\left(1-ν^2\right)\cdot\left(\cfrac{r}{b}-\cfrac{b}{r}\right)$$
$F_7=\cfrac{1}{2}\cdot\left(1-ν^2\right)\cdot\left(\cfrac{r}{b}-\cfrac{b}{r}\right)$
Equations in LaTeX code
F_7=\cfrac{1}{2}\cdot\left(1-ν^2\right)\cdot\left(\cfrac{r}{b}-\cfrac{b}{r}\right)

Function of the radial location $ F_8 $

$$F_8=\cfrac{1}{2}\cdot\left[1+ν+\left(1-ν\right)\cdot\left(\cfrac{b}{r}\right)^2\right]$$
$F_8=\cfrac{1}{2}\cdot\left[1+ν+\left(1-ν\right)\cdot\left(\cfrac{b}{r}\right)^2\right]$
Equations in LaTeX code
F_8=\cfrac{1}{2}\cdot\left[1+ν+\left(1-ν\right)\cdot\left(\cfrac{b}{r}\right)^2\right]

Function of the radial location $ F_9 $

$$F_9=\cfrac{b}{r}\cdot\left\{\cfrac{1+ν}{2}\cdot\ln\left(\cfrac{r}{b}\right)+\cfrac{1-ν}{4}\cdot\left[1-\left(\cfrac{b}{r}\right)^2\right]\right\}$$
$F_9=\cfrac{b}{r}\cdot\left\{\cfrac{1+ν}{2}\cdot\ln\left(\cfrac{r}{b}\right)+\cfrac{1-ν}{4}\cdot\left[1-\left(\cfrac{b}{r}\right)^2\right]\right\}$
Equations in LaTeX code
F_9=\cfrac{b}{r}\cdot\left\{\cfrac{1+ν}{2}\cdot\ln\left(\cfrac{r}{b}\right)+\cfrac{1-ν}{4}\cdot\left[1-\left(\cfrac{b}{r}\right)^2\right]\right\}

Function of the radial location $ G_3 $

$\text{if }\ r> r_o$
$\text{if }\ r> r_o$
Equations in LaTeX code
\text{if }\ r> r_o
$$G_3=\cfrac{r_o}{4\cdot r}\cdot\left\{\left[\left(\cfrac{r_o}{r}\right)^2+1\right]\cdot\ln\left(\cfrac{r}{r_o}\right)+\left(\cfrac{r_o}{r}\right)^2-1\right\}$$
$G_3=\cfrac{r_o}{4\cdot r}\cdot\left\{\left[\left(\cfrac{r_o}{r}\right)^2+1\right]\cdot\ln\left(\cfrac{r}{r_o}\right)+\left(\cfrac{r_o}{r}\right)^2-1\right\}$
Equations in LaTeX code
G_3=\cfrac{r_o}{4\cdot r}\cdot\left\{\left[\left(\cfrac{r_o}{r}\right)^2+1\right]\cdot\ln\left(\cfrac{r}{r_o}\right)+\left(\cfrac{r_o}{r}\right)^2-1\right\}
$\text{else}$
$\text{else}$
Equations in LaTeX code
\text{else}
$$G_3=0$$
$G_3=0$
Equations in LaTeX code
G_3=0

Function of the radial location $ G_6 $

$\text{if }\ r> r_o$
$\text{if }\ r> r_o$
Equations in LaTeX code
\text{if }\ r> r_o
$$G_6=\cfrac{r_o}{4\cdot r}\cdot\left[\left(\cfrac{r_o}{r}\right)^2-1+2\cdot\ln\left(\cfrac{r}{r_o}\right)\right]$$
$G_6=\cfrac{r_o}{4\cdot r}\cdot\left[\left(\cfrac{r_o}{r}\right)^2-1+2\cdot\ln\left(\cfrac{r}{r_o}\right)\right]$
Equations in LaTeX code
G_6=\cfrac{r_o}{4\cdot r}\cdot\left[\left(\cfrac{r_o}{r}\right)^2-1+2\cdot\ln\left(\cfrac{r}{r_o}\right)\right]
$\text{else}$
$\text{else}$
Equations in LaTeX code
\text{else}
$$G_6=0$$
$G_6=0$
Equations in LaTeX code
G_6=0

Function of the radial location $ G_9 $

$\text{if }\ r> r_o$
$\text{if }\ r> r_o$
Equations in LaTeX code
\text{if }\ r> r_o
$$G_9=\cfrac{r_o}{r}\cdot\left\{\cfrac{1+ν}{2}\cdot\ln\left(\cfrac{r}{r_o}\right)+\cfrac{1-ν}{4}\cdot\left[1-\left(\cfrac{r_o}{r}\right)^2\right]\right\}$$
$G_9=\cfrac{r_o}{r}\cdot\left\{\cfrac{1+ν}{2}\cdot\ln\left(\cfrac{r}{r_o}\right)+\cfrac{1-ν}{4}\cdot\left[1-\left(\cfrac{r_o}{r}\right)^2\right]\right\}$
Equations in LaTeX code
G_9=\cfrac{r_o}{r}\cdot\left\{\cfrac{1+ν}{2}\cdot\ln\left(\cfrac{r}{r_o}\right)+\cfrac{1-ν}{4}\cdot\left[1-\left(\cfrac{r_o}{r}\right)^2\right]\right\}
$\text{else}$
$\text{else}$
Equations in LaTeX code
\text{else}
$$G_9=0$$
$G_9=0$
Equations in LaTeX code
G_9=0

Loading constant $ L_3 $

$$L_3=\cfrac{r_o}{4\cdot a}\cdot\left\{\left[\left(\cfrac{r_o}{a}\right)^2+1\right]\cdot\ln\left(\cfrac{a}{r_o}\right)+\left(\cfrac{r_o}{a}\right)^2-1\right\}$$
$L_3=\cfrac{r_o}{4\cdot a}\cdot\left\{\left[\left(\cfrac{r_o}{a}\right)^2+1\right]\cdot\ln\left(\cfrac{a}{r_o}\right)+\left(\cfrac{r_o}{a}\right)^2-1\right\}$
Equations in LaTeX code
L_3=\cfrac{r_o}{4\cdot a}\cdot\left\{\left[\left(\cfrac{r_o}{a}\right)^2+1\right]\cdot\ln\left(\cfrac{a}{r_o}\right)+\left(\cfrac{r_o}{a}\right)^2-1\right\}

Loading constant $ L_6 $

$$L_6=\cfrac{r_o}{4\cdot a}\cdot\left[\left(\cfrac{r_o}{a}\right)^2-1+2\cdot\ln\left(\cfrac{a}{r_o}\right)\right]$$
$L_6=\cfrac{r_o}{4\cdot a}\cdot\left[\left(\cfrac{r_o}{a}\right)^2-1+2\cdot\ln\left(\cfrac{a}{r_o}\right)\right]$
Equations in LaTeX code
L_6=\cfrac{r_o}{4\cdot a}\cdot\left[\left(\cfrac{r_o}{a}\right)^2-1+2\cdot\ln\left(\cfrac{a}{r_o}\right)\right]

Loading constant $ L_9 $

$$L_9=\cfrac{r_o}{a}\cdot\left\{\cfrac{1+ν}{2}\cdot\ln\left(\cfrac{a}{r_o}\right)+\cfrac{1-ν}{4}\cdot\left[1-\left(\cfrac{r_o}{a}\right)^2\right]\right\}$$
$L_9=\cfrac{r_o}{a}\cdot\left\{\cfrac{1+ν}{2}\cdot\ln\left(\cfrac{a}{r_o}\right)+\cfrac{1-ν}{4}\cdot\left[1-\left(\cfrac{r_o}{a}\right)^2\right]\right\}$
Equations in LaTeX code
L_9=\cfrac{r_o}{a}\cdot\left\{\cfrac{1+ν}{2}\cdot\ln\left(\cfrac{a}{r_o}\right)+\cfrac{1-ν}{4}\cdot\left[1-\left(\cfrac{r_o}{a}\right)^2\right]\right\}

The plate constant

$$D=E\cdot t^3/\left(12\cdot\left(1-ν^2\right)\right)$$
$D=E\cdot t^3/\left(12\cdot\left(1-ν^2\right)\right)$
Equations in LaTeX code
D=E\cdot t^3/\left(12\cdot\left(1-ν^2\right)\right)

Unit radial bending moment at the inner edge

$$M_{rb}=-w\cdot a\cdot\cfrac{C_3\cdot L_9-C_9\cdot L_3}{C_2\cdot C_9-C_3\cdot C_8}$$
$M_{rb}=-w\cdot a\cdot\cfrac{C_3\cdot L_9-C_9\cdot L_3}{C_2\cdot C_9-C_3\cdot C_8}$
Equations in LaTeX code
M_{rb}=-w\cdot a\cdot\cfrac{C_3\cdot L_9-C_9\cdot L_3}{C_2\cdot C_9-C_3\cdot C_8}

Unit shear force at the inner edge

$$Q_b=w\cdot\cfrac{C_2\cdot L_9-C_8\cdot L_3}{C_2\cdot C_9-C_3\cdot C_8}$$
$Q_b=w\cdot\cfrac{C_2\cdot L_9-C_8\cdot L_3}{C_2\cdot C_9-C_3\cdot C_8}$
Equations in LaTeX code
Q_b=w\cdot\cfrac{C_2\cdot L_9-C_8\cdot L_3}{C_2\cdot C_9-C_3\cdot C_8}

Vertical deflection of plate at the outer edge

$$y_a=0$$
$y_a=0$
Equations in LaTeX code
y_a=0

Unit radial bending moment at the outer edge

$$M_{ra}=0$$
$M_{ra}=0$
Equations in LaTeX code
M_{ra}=0

Vertical deflection of plate at the inner edge

$$y_b=0$$
$y_b=0$
Equations in LaTeX code
y_b=0

Radial slope of plate at the inner edge

$$θ_b=0$$
$θ_b=0$
Equations in LaTeX code
θ_b=0

Radial slope of plate at the outer edge

$$θ_a=M_{rb}\cdot\cfrac{a}{D}\cdot C_5+Q_b\cdot\cfrac{a^2}{D}\cdot C_6-\cfrac{w\cdot a^2}{D}\cdot L_6$$
$θ_a=M_{rb}\cdot\cfrac{a}{D}\cdot C_5+Q_b\cdot\cfrac{a^2}{D}\cdot C_6-\cfrac{w\cdot a^2}{D}\cdot L_6$
Equations in LaTeX code
θ_a=M_{rb}\cdot\cfrac{a}{D}\cdot C_5+Q_b\cdot\cfrac{a^2}{D}\cdot C_6-\cfrac{w\cdot a^2}{D}\cdot L_6

Unit shear force at the outer edge

$$Q_a=Q_b\cdot\cfrac{b}{a}-\cfrac{w\cdot r_o}{a}$$
$Q_a=Q_b\cdot\cfrac{b}{a}-\cfrac{w\cdot r_o}{a}$
Equations in LaTeX code
Q_a=Q_b\cdot\cfrac{b}{a}-\cfrac{w\cdot r_o}{a}

Vertical deflection of plate

$$y=y_b+θ_b\cdot r\cdot F_1+M_{rb}\cdot\cfrac{r^2}{D}\cdot F_2+Q_b\cdot\cfrac{r^3}{D}\cdot F_3-w\cdot\cfrac{r^3}{D}\cdot G_3$$
$y=y_b+θ_b\cdot r\cdot F_1+M_{rb}\cdot\cfrac{r^2}{D}\cdot F_2+Q_b\cdot\cfrac{r^3}{D}\cdot F_3-w\cdot\cfrac{r^3}{D}\cdot G_3$
Equations in LaTeX code
y=y_b+θ_b\cdot r\cdot F_1+M_{rb}\cdot\cfrac{r^2}{D}\cdot F_2+Q_b\cdot\cfrac{r^3}{D}\cdot F_3-w\cdot\cfrac{r^3}{D}\cdot G_3

Radial slope of plate

$$θ=θ_b\cdot F_4+M_{rb}\cdot\cfrac{r}{D}\cdot F_5+Q_b\cdot\cfrac{r^2}{D}\cdot F_6-w\cdot\cfrac{r^2}{D}\cdot G_6$$
$θ=θ_b\cdot F_4+M_{rb}\cdot\cfrac{r}{D}\cdot F_5+Q_b\cdot\cfrac{r^2}{D}\cdot F_6-w\cdot\cfrac{r^2}{D}\cdot G_6$
Equations in LaTeX code
θ=θ_b\cdot F_4+M_{rb}\cdot\cfrac{r}{D}\cdot F_5+Q_b\cdot\cfrac{r^2}{D}\cdot F_6-w\cdot\cfrac{r^2}{D}\cdot G_6

Unit radial bending moment

$$M_r=θ_b\cdot\cfrac{D}{r}\cdot F_7+M_{rb}\cdot F_8+Q_b\cdot r\cdot F_9-w\cdot r\cdot G_9$$
$M_r=θ_b\cdot\cfrac{D}{r}\cdot F_7+M_{rb}\cdot F_8+Q_b\cdot r\cdot F_9-w\cdot r\cdot G_9$
Equations in LaTeX code
M_r=θ_b\cdot\cfrac{D}{r}\cdot F_7+M_{rb}\cdot F_8+Q_b\cdot r\cdot F_9-w\cdot r\cdot G_9

Unit tangential bending moment

$$M_t=\cfrac{θ\cdot D\cdot\left(1-ν^2\right)}{r}+ν\cdot M_r$$
$M_t=\cfrac{θ\cdot D\cdot\left(1-ν^2\right)}{r}+ν\cdot M_r$
Equations in LaTeX code
M_t=\cfrac{θ\cdot D\cdot\left(1-ν^2\right)}{r}+ν\cdot M_r

Unit shear force

$\text{if }\ r> r_o$
$\text{if }\ r> r_o$
Equations in LaTeX code
\text{if }\ r> r_o
$$Q=Q_b\cdot\cfrac{b}{r}-w\cdot\cfrac{r_o}{r}$$
$Q=Q_b\cdot\cfrac{b}{r}-w\cdot\cfrac{r_o}{r}$
Equations in LaTeX code
Q=Q_b\cdot\cfrac{b}{r}-w\cdot\cfrac{r_o}{r}
$\text{else}$
$\text{else}$
Equations in LaTeX code
\text{else}
$$Q=Q_b\cdot\cfrac{b}{r}$$
$Q=Q_b\cdot\cfrac{b}{r}$
Equations in LaTeX code
Q=Q_b\cdot\cfrac{b}{r}
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