# Left end simply supported right end fixed for intermediate externally created angular deformation

## Values for calculation

$l$ $\mathrm{mm}$
$a$ $\mathrm{mm}$
$θ_o$ $\mathrm{rad}$
$E$ $\mathrm{MPa}$
$I$ $\mathrm{mm^4}$
$x$ $\mathrm{mm}$

## Calculation

### Vertical end reactions $R_A$

$$R_A=\cfrac{-3\cdot E\cdot I\cdot a\cdot θ_o}{l^3}$$

### Reaction end moment $M_A$

$$M_A=0$$

### Angular displacement $θ_A$

$$θ_A=θ_o\cdot\left(1-\cfrac{3\cdot a}{2\cdot l}\right)$$

### Deflection $y_A$

$$y_A=0$$

### Vertical end reactions $R_B$

$$R_B=-R_A$$

### Reaction end moment $M_B$

$$M_B=\cfrac{-3\cdot E\cdot I\cdot a\cdot θ_o}{l^2}$$

### Angular displacement $θ_B$

$$θ_B=0$$

### Deflection $y_B$

$$y_B=0$$

### Max. moment

$$M_{max}=M_B$$

### Max. deflection +

$$y_{max+}=θ_o\cdot a\cdot\left(1-\cfrac{2\cdot l}{3\cdot a}\right)^{3/2}$$

### Max. deflection -

$$y_{max-}=-θ_o\cdot a\cdot\left(1-\cfrac{3\cdot a}{2\cdot l}+\cfrac{a^3}{2\cdot l^3}\right)$$

### Transverse shear

$$V=R_A$$

### Bending moment

$$M=M_A+R_A\cdot x$$

### Slope

$\text{if }\ x\le a$
$$θ=θ_A+\cfrac{M_A\cdot x}{E\cdot I}+\cfrac{R_A\cdot x^2}{2\cdot E\cdot I}$$
$\text{else}$
$$θ=θ_A+\cfrac{M_A\cdot x}{E\cdot I}+\cfrac{R_A\cdot x^2}{2\cdot E\cdot I}+θ_o\cdot\left(x-a\right)$$

### Deflection

$\text{if }\ x\le a$
$$y=y_A+θ_A\cdot x+\cfrac{M_A\cdot x^2}{2\cdot E\cdot I}+\cfrac{R_A\cdot x^3}{6\cdot E\cdot I}$$
$\text{else}$
$$y=y_A+θ_A\cdot x+\cfrac{M_A\cdot x^2}{2\cdot E\cdot I}+\cfrac{R_A\cdot x^3}{6\cdot E\cdot I}+θ_o\cdot\left(x-a\right)$$