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Left end fixed, right end fixed 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 fixed, right end fixed for concentrated intermediate load W a
Left end fixed, right end fixed 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 $

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

Reaction end moment $ M_A $

$$M_A=\cfrac{-W\cdot a}{l^2}\cdot\left(l-a\right)^2$$

Angular displacement $ θ_A $

$$θ_A=0$$

Deflection $ y_A $

$$y_A=0$$

Vertical end reactions $ R_B $

$$R_B=\cfrac{W\cdot a^2}{l^3}\cdot\left(3\cdot l-2\cdot a\right)$$

Reaction end moment $ M_B $

$$M_B=\cfrac{-W\cdot a^2}{l^2}\cdot\left(l-a\right)$$

Angular displacement $ θ_B $

$$θ_B=0$$

Deflection $ y_B $

$$y_B=0$$

Max. moment +

$$M_{max+}=\cfrac{2\cdot W\cdot a^2}{l^3}\cdot\left(l-a\right)^2$$

Max. moment -

$$M_{max-}=\min\left\{M_A, M_B\right\}$$

Max. deflection

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

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

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