# Left end simply supported, right end simply supported for partial distributed load

## Values for calculation

$l$ $\mathrm{mm}$
$a$ $\mathrm{mm}$
$W_a$ $\mathrm{N/mm}$
$W_l$ $\mathrm{N/mm}$
$E$ $\mathrm{MPa}$
$I$ $\mathrm{mm^4}$
$x$ $\mathrm{mm}$

## Calculation

### Vertical end reactions $R_A$

$$R_A=\cfrac{W_a}{2\cdot l}\cdot\left(l-a\right)^2+\cfrac{W_l-W_a}{6\cdot l}\cdot\left(l-a\right)^2$$

### Reaction end moment $M_A$

$$M_A=0$$

### Angular displacement $θ_A$

$$θ_A=\cfrac{-W_a}{24\cdot E\cdot I\cdot l}\cdot\left(l-a\right)^2\cdot\left(l^2+2\cdot a\cdot l-a^2\right)-\cfrac{W_l-W_a}{360\cdot E\cdot I\cdot l}\cdot\left(l-a\right)^2\cdot\left(7\cdot l^2+6\cdot a\cdot l-3\cdot a^3\right)$$

### Deflection $y_A$

$$y_A=0$$

### Vertical end reactions $R_B$

$$R_B=\cfrac{W_a+W_l}{2}\cdot\left(l-a\right)-R_A$$

### Reaction end moment $M_B$

$$M_B=0$$

### Angular displacement $θ_B$

$$θ_B=\cfrac{W_a}{24\cdot E\cdot I\cdot l}\cdot\left(l^2-a^2\right)^2+\cfrac{W_l-W_a}{360\cdot E\cdot I\cdot l}\cdot\left(l-a\right)^2\cdot\left(8\cdot l^2+9\cdot a\cdot l+3\cdot a^2\right)$$

### Deflection $y_B$

$$y_B=0$$

### Transverse shear

$\text{if }\ x\le a$
$$V=R_A$$
$\text{else}$
$$V=R_A-W_a\cdot\left(x-a\right)-\cfrac{W_l-W_a}{2\cdot\left(l-a\right)}\cdot\left(x-a\right)^2$$

### Bending moment

$\text{if }\ x\le a$
$$M=M_A+R_A\cdot x$$
$\text{else}$
$$M=M_A+R_A\cdot x-\cfrac{W_a}{2}\cdot\left(x-a\right)^2-\cfrac{W_l-W_a}{6\cdot\left(l-a\right)}\cdot\left(x-a\right)^3$$

### Slope

$\text{if }\ x\le a$
$$θ=θ_A+\cfrac{M_A\cdot x}{E\cdot I}+\cfrac{R_A\cdot x^2}{2\cdot E\cdot I}$$
$\text{else}$
$$θ=θ_A+\cfrac{M_A\cdot x}{E\cdot I}+\cfrac{R_A\cdot x^2}{2\cdot E\cdot I}-\cfrac{W_a}{6\cdot E\cdot I}\cdot\left(x-a\right)^3-\cfrac{W_l-W_a}{24\cdot E\cdot I\cdot\left(l-a\right)}\cdot\left(x-a\right)^4$$

### Deflection

$\text{if }\ x\le a$
$$y=y_A+θ_A+θ_A\cdot x+\cfrac{M_A\cdot x^2}{2\cdot E\cdot I}+\cfrac{R_A\cdot x^3}{6\cdot E\cdot I}$$
$\text{else}$
$$y=y_A+θ_A+θ_A\cdot x+\cfrac{M_A\cdot x^2}{2\cdot E\cdot I}+\cfrac{R_A\cdot x^3}{6\cdot E\cdot I}-\cfrac{W_a}{24\cdot E\cdot I}\cdot\left(x-a\right)^4-\cfrac{W_l-W_a}{120\cdot E\cdot I\cdot\left(l-a\right)}\cdot\left(x-a\right)^5$$