# Left end simply supported, right end simply supported for concentrated intermediate moment

## Values for calculation

$l$ $\mathrm{mm}$
$a$ $\mathrm{mm}$
$M_O$ $\mathrm{Nmm}$
$E$ $\mathrm{MPa}$
$I$ $\mathrm{mm^4}$
$x$ $\mathrm{mm}$

## Calculation

### Vertical end reactions $R_A$

$$R_A=\cfrac{-M_O}{l}$$

### Reaction end moment $M_A$

$$M_A=0$$

### Angular displacement $θ_A$

$$θ_A=\cfrac{-M_O}{6\cdot E\cdot I\cdot l}\cdot\left(2\cdot l^2-6\cdot a\cdot l+3\cdot a^2\right)$$

### Deflection $y_A$

$$y_A=0$$

### Vertical end reactions $R_B$

$$R_B=\cfrac{M_O}{l}$$

### Reaction end moment $M_B$

$$M_B=0$$

### Angular displacement $θ_B$

$$θ_B=\cfrac{M_O}{6\cdot E\cdot I\cdot l}\cdot\left(l^2-3\cdot a^2\right)$$

### Deflection $y_B$

$$y_B=0$$

### Max. moment +

$$M_{max+}=\cfrac{M_O}{l}\cdot\left(l-a\right)$$

### Max. moment -

$$M_{max-}=\cfrac{-M_O\cdot a}{l}$$

### Max. deflection

$$y_{max}=\cfrac{M_O\cdot\left(6\cdot a\cdot l-3\cdot a^2-2\cdot l^2\right)^{3/2}}{9\cdot\sqrt{3}\cdot E\cdot I\cdot l}$$

### Transverse shear

$$V=R_A$$

### Bending moment

$\text{if }\ x\le a$
$$M=M_A+R_A\cdot x$$
$\text{else}$
$$M=M_A+R_A\cdot x+M_O$$

### Slope

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

### Deflection

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