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

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

Values for calculation

$ l $ $ \mathrm{mm} $
$ a $ $ \mathrm{mm} $
$ W $ $ \mathrm{N} $
$ E $ $ \mathrm{MPa} $
$ I $ $ \mathrm{mm^4} $
$ x $ $ \mathrm{mm} $

Calculation

Vertical end reactions $ R_A $

$$R_A=\cfrac{W}{l}\cdot\left(l-a\right)$$

Reaction end moment $ M_A $

$$M_A=0$$

Angular displacement $ θ_A $

$$θ_A=\cfrac{-W\cdot a}{6\cdot E\cdot I\cdot l}\cdot\left(2\cdot l-a\right)\cdot\left(l-a\right)$$

Deflection $ y_A $

$$y_A=0$$

Vertical end reactions $ R_B $

$$R_B=\cfrac{W\cdot a}{l}$$

Reaction end moment $ M_B $

$$M_B=0$$

Angular displacement $ θ_B $

$$θ_B=\cfrac{W\cdot a}{6\cdot E\cdot I\cdot l}\cdot\left(l^2-a^2\right)$$

Deflection $ y_B $

$$y_B=0$$

Max. moment

$$M_{max}=R_A\cdot a$$

Max. deflection

$\text{if }\ a < \cfrac{l}{2}$
$$y_{max}=\cfrac{-W\cdot a}{3\cdot E\cdot I\cdot l}\cdot\left(\cfrac{l^2-a^2}{3}\right)^{3/2}$$
$\text{else}$
$$y_{max}=\cfrac{-W\cdot \left(l-a\right)}{3\cdot E\cdot I\cdot l}\cdot\left(\cfrac{l^2-\left(l-a\right)^2}{3}\right)^{3/2}$$

Max. angular displacement

$\text{if }\ a < \cfrac{l}{2}$
$$θ_{max}=θ_A$$
$\text{else}$
$$θ_{max}=θ_B$$

Transverse shear

$\text{if }\ x\le a$
$$V=R_A$$
$\text{else}$
$$V=R_A-W$$

Bending moment

$\text{if }\ x\le a$
$$M=M_A+R_A\cdot x$$
$\text{else}$
$$M=M_A+R_A\cdot x-W\cdot\left(x-a\right)$$

Slope

$\text{if }\ x\le a$
$$θ=θ_A+\cfrac{M_A\cdot x}{E\cdot I}+\cfrac{R_A\cdot x^2}{2\cdot E\cdot I}$$
$\text{else}$
$$θ=θ_A+\cfrac{M_A\cdot x}{E\cdot I}+\cfrac{R_A\cdot x^2}{2\cdot E\cdot I}-\cfrac{W}{2\cdot E\cdot I}\cdot\left(x-a\right)^2$$

Deflection

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