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Left end simply supported, right end simply supported 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 simply supported, right end simply supported for intermediate externally created lateral displacement Δ o a
Left end simply supported, right end simply supported for intermediate externally created lateral displacement

Values for calculation

$ \{l\in\mathbb{R}\mid l>0\} $
This set includes all real numbers that are greater than zero. Therefore, it contains only positive numbers. Zero and negative numbers are not included in this set.
$ l $ $ \mathrm{mm} $
$ \{a\in\mathbb{R}\mid a\geq0\} $
This set includes all real numbers that are greater than or equal to zero. Therefore, it contains all positive numbers and zero, but no negative numbers.
$ a $ $ \mathrm{mm} $
$ \{Δ_o\in\mathbb{R}\mid Δ_o>0\} $
This set includes all real numbers that are greater than zero. Therefore, it contains only positive numbers. Zero and negative numbers are not included in this set.
$ Δ_o $ $ \mathrm{mm} $
$ \{E\in\mathbb{R}\mid E>0\} $
This set includes all real numbers that are greater than zero. Therefore, it contains only positive numbers. Zero and negative numbers are not included in this set.
$ E $ $ \mathrm{MPa} $
$ \{I\in\mathbb{R}\mid I>0\} $
This set includes all real numbers that are greater than zero. Therefore, it contains only positive numbers. Zero and negative numbers are not included in this set.
$ I $ $ \mathrm{mm^4} $
$ \{x\in\mathbb{R}\mid x\geq0\} $
This set includes all real numbers that are greater than or equal to zero. Therefore, it contains all positive numbers and zero, but no negative numbers.
$ x $ $ \mathrm{mm} $

Calculation

Vertical end reactions $ R_A $

$$R_A=0$$
$R_A=0$
Equations in LaTeX code
R_A=0

Reaction end moment $ M_A $

$$M_A=0$$
$M_A=0$
Equations in LaTeX code
M_A=0

Angular displacement $ θ_A $

$$θ_A=\cfrac{-Δ_o}{l}$$
$θ_A=\cfrac{-Δ_o}{l}$
Equations in LaTeX code
θ_A=\cfrac{-Δ_o}{l}

Deflection $ y_A $

$$y_A=0$$
$y_A=0$
Equations in LaTeX code
y_A=0

Vertical end reactions $ R_B $

$$R_B=0$$
$R_B=0$
Equations in LaTeX code
R_B=0

Reaction end moment $ M_B $

$$M_B=0$$
$M_B=0$
Equations in LaTeX code
M_B=0

Angular displacement $ θ_B $

$$θ_B=\cfrac{-Δ_o}{l}$$
$θ_B=\cfrac{-Δ_o}{l}$
Equations in LaTeX code
θ_B=\cfrac{-Δ_o}{l}

Deflection $ y_B $

$$y_B=0$$
$y_B=0$
Equations in LaTeX code
y_B=0

Max. deflection +

$$y_{max+}=\cfrac{Δ_o}{l}\cdot\left(l-a\right)$$
$y_{max+}=\cfrac{Δ_o}{l}\cdot\left(l-a\right)$
Equations in LaTeX code
y_{max+}=\cfrac{Δ_o}{l}\cdot\left(l-a\right)

Max. deflection -

$$y_{max-}=\cfrac{-Δ_o\cdot a}{l}$$
$y_{max-}=\cfrac{-Δ_o\cdot a}{l}$
Equations in LaTeX code
y_{max-}=\cfrac{-Δ_o\cdot a}{l}

Transverse shear

$$V=R_A$$
$V=R_A$
Equations in LaTeX code
V=R_A

Bending moment

$$M=M_A+R_A\cdot x$$
$M=M_A+R_A\cdot x$
Equations in LaTeX code
M=M_A+R_A\cdot x

Slope

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

Deflection

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