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

Concentrated intermediate moment M o a y A R A M A R B M B l θ B θ A
Concentrated intermediate moment
Left end simply supported, right end simply supported for concentrated intermediate moment a M o
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

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