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Left end guided, right end simply supported 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_guided,_right_end_simply_supported_for_partial_distributed_load a W a W l
Left_end_guided,_right_end_simply_supported_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 $

$$R_A=0$$

Reaction end moment $ M_A $

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

Angular displacement $ θ_A $

$$θ_A=0$$

Deflection $ y_A $

$$y_A=\cfrac{-W_a}{24\cdot E\cdot I}\cdot\left(l-a\right)^2\cdot\left(5\cdot l^2+2\cdot a\cdot l-a^2\right)-\cfrac{W_l-W_a}{120\cdot E\cdot I}\cdot\left(l-a\right)^2\cdot\left(9\cdot l^2+2\cdot a\cdot l-a^2\right)$$

Vertical end reactions $ R_B $

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

Reaction end moment $ M_B $

$$M_B=0$$

Angular displacement $ θ_B $

$$θ_B=\cfrac{W_a}{6\cdot E\cdot I}\cdot\left(l-a\right)^2\cdot\left(2\cdot l+a\right)+\cfrac{W_l-W_a}{24\cdot E\cdot I}\cdot\left(l-a\right)^2\cdot\left(3\cdot l+a\right)$$

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

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