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Left end fixed, right end fixed for partial distributed load

Partial distributed load W a a y A R A M A R B M B l θ B θ A W l
Partial distributed load
Left end fixed, right end fixed for partial distributed load a W a W l
Left end fixed, right end fixed for partial distributed load

Values for calculation

$ l $ $ \mathrm{mm} $
$ a $ $ \mathrm{mm} $
$ W_a $ $ \mathrm{N/mm} $
$ W_l $ $ \mathrm{N/mm} $
$ E $ $ \mathrm{MPa} $
$ I $ $ \mathrm{mm^4} $
$ x $ $ \mathrm{mm} $

Calculation

Vertical end reactions $ R_A $

$$R_A=\cfrac{W_a}{2\cdot l^3}\cdot\left(l-a\right)^3\cdot\left(l+a\right)+\cfrac{W_l-W_a}{20\cdot l^3}\cdot\left(l-a\right)^3\cdot\left(3\cdot l+2\cdot a\right)$$

Reaction end moment $ M_A $

$$M_A=\cfrac{-W_a}{12\cdot l^2}\cdot\left(l-a\right)^3\cdot\left(l+3\cdot a\right)-\cfrac{W_l-W_a}{60\cdot l^2}\cdot\left(l-a\right)^3\cdot\left(2\cdot l+3\cdot a\right)$$

Angular displacement $ θ_A $

$$θ_A=0$$

Deflection $ y_A $

$$y_A=0$$

Vertical end reactions $ R_B $

$$R_B=\cfrac{W_a+W_l}{2}\cdot\left(l-a\right)-R_A$$

Reaction end moment $ M_B $

$$M_B=R_A\cdot l+M_A-\cfrac{W_a}{2}\cdot\left(l-a\right)^2-\cfrac{W_l-W_a}{6}\cdot\left(l-a\right)^2$$

Angular displacement $ θ_B $

$$θ_B=0$$

Deflection $ y_B $

$$y_B=0$$

Transverse shear

$\text{if }\ x\le a$
$$V=R_A$$
$\text{else}$
$$V=R_A-W_a\cdot\left(x-a\right)-\cfrac{W_l-W_a}{2\cdot\left(l-a\right)}\cdot\left(x-a\right)^2$$

Bending moment

$\text{if }\ x\le a$
$$M=M_A+R_A\cdot x$$
$\text{else}$
$$M=M_A+R_A\cdot x-\cfrac{W_a}{2}\cdot\left(x-a\right)^2-\cfrac{W_l-W_a}{6\cdot\left(l-a\right)}\cdot\left(x-a\right)^3$$

Slope

$\text{if }\ x\le a$
$$θ=θ_A+\cfrac{M_A\cdot x}{E\cdot I}+\cfrac{R_A\cdot x^2}{2\cdot E\cdot I}$$
$\text{else}$
$$θ=θ_A+\cfrac{M_A\cdot x}{E\cdot I}+\cfrac{R_A\cdot x^2}{2\cdot E\cdot I}-\cfrac{W_a}{6\cdot E\cdot I}\cdot\left(x-a\right)^3-\cfrac{W_l-W_a}{24\cdot E\cdot I\cdot\left(l-a\right)}\cdot\left(x-a\right)^4$$

Deflection

$\text{if }\ x\le a$
$$y=y_A+θ_A+θ_A\cdot x+\cfrac{M_A\cdot x^2}{2\cdot E\cdot I}+\cfrac{R_A\cdot x^3}{6\cdot E\cdot I}$$
$\text{else}$
$$y=y_A+θ_A+θ_A\cdot x+\cfrac{M_A\cdot x^2}{2\cdot E\cdot I}+\cfrac{R_A\cdot x^3}{6\cdot E\cdot I}-\cfrac{W_a}{24\cdot E\cdot I}\cdot\left(x-a\right)^4-\cfrac{W_l-W_a}{120\cdot E\cdot I\cdot\left(l-a\right)}\cdot\left(x-a\right)^5$$