# Left end guided, 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 $R_A$

$$R_A=0$$

### Reaction end moment $M_A$

$$M_A=-M_O$$

### Angular displacement $θ_A$

$$θ_A=0$$

### Deflection $y_A$

$$y_A=\cfrac{M_O\cdot a}{2\cdot E\cdot I}\cdot\left(2\cdot l-a\right)$$

### Vertical end reactions $R_B$

$$R_B=0$$

### Reaction end moment $M_B$

$$M_B=0$$

### Angular displacement $θ_B$

$$θ_B=\cfrac{-M_O\cdot a}{E\cdot I}$$

### Deflection $y_B$

$$y_B=0$$

### Max. moment

$$M_{max}=-M_O$$

### Max. angular displacement

$$θ_{max}=θ_B$$

### Max. deflection

$$y_{max}=y_A$$

### 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

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