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Left end fixed, right end fixed for intermediate externally created lateral displacement

Intermediate externally created lateral displacement a y A R A M A R B M B l θ B θ A Δ o
Intermediate externally created lateral displacement
Left end fixed, right end fixed for intermediate externally created lateral displacement Δ o a
Left end fixed, right end fixed for intermediate externally created lateral displacement

Values for calculation

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

Calculation

Vertical end reactions $ R_A $

$$R_A=\cfrac{12\cdot E\cdot I\cdot Δ_o}{l^3}$$

Reaction end moment $ M_A $

$$M_A=\cfrac{-6\cdot E\cdot I\cdot Δ_o}{l^2}$$

Angular displacement $ θ_A $

$$θ_A=0$$

Deflection $ y_A $

$$y_A=0$$

Vertical end reactions $ R_B $

$$R_B=-R_A$$

Reaction end moment $ M_B $

$$M_B=-M_A$$

Angular displacement $ θ_B $

$$θ_B=0$$

Deflection $ y_B $

$$y_B=0$$

Max. moment +

$$M_{max+}=M_B$$

Max. moment -

$$M_{max-}=M_A$$

Max. angular displacement

$$θ_{max}=\cfrac{-3\cdot Δ_o}{2\cdot l}$$

Max. deflection +

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

Max. deflection -

$$y_{max-}=\cfrac{-Δ_o\cdot a^2}{l^3}\cdot\left(3\cdot l-2\cdot a\right)$$

Transverse shear

$$V=R_A$$

Bending moment

$$M=M_A+R_A\cdot x$$

Slope

$$θ=θ_A+\cfrac{M_A\cdot x}{E\cdot I}+\cfrac{R_A\cdot x^2}{2\cdot E\cdot I}$$

Deflection

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