# 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 $C_2$

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

### Plate constant $C_3$

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

### Plate constant $C_5$

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

### Plate constant $C_6$

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

### Plate constant $C_8$

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

### Plate constant $C_9$

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

### Function of the radial location $F_1$

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

### Function of the radial location $F_2$

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

### Function of the radial location $F_3$

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

### Function of the radial location $F_4$

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

### Function of the radial location $F_5$

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

### Function of the radial location $F_6$

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

### Function of the radial location $F_7$

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

### Function of the radial location $F_8$

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

### Function of the radial location $F_9$

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

### Function of the radial location $G_3$

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

### Function of the radial location $G_6$

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

### Function of the radial location $G_9$

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

### Loading constant $L_3$

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

### Loading constant $L_6$

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

### Loading constant $L_9$

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

### The plate constant

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

### Unit radial bending moment at the inner edge

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

### Unit shear force at the inner edge

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

### Vertical deflection of plate at the outer edge

$$y_a=0$$

### Unit radial bending moment at the outer edge

$$M_{ra}=0$$

### Vertical deflection of plate at the inner edge

$$y_b=0$$

### Radial slope of plate at the inner edge

$$θ_b=0$$

### Radial slope of plate at the outer edge

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

### Unit shear force at the outer edge

$$Q_a=Q_b\cdot\cfrac{b}{a}-\cfrac{w\cdot r_o}{a}$$

### Vertical deflection of plate

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

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

$$M_r=θ_b\cdot\cfrac{D}{r}\cdot F_7+M_{rb}\cdot F_8+Q_b\cdot r\cdot F_9-w\cdot r\cdot G_9$$
$$M_t=\cfrac{θ\cdot D\cdot\left(1-ν^2\right)}{r}+ν\cdot M_r$$
$\text{if }\ r> r_o$
$$Q=Q_b\cdot\cfrac{b}{r}-w\cdot\cfrac{r_o}{r}$$
$\text{else}$
$$Q=Q_b\cdot\cfrac{b}{r}$$