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

Intermediate externally created angular deformation a y A R A M A R B M B l θ B θ A θ o
Intermediate externally created angular deformation
Left end simply supported right end fixed for intermediate externally created angular deformation a θ o
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

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