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Left end fixed, right end fixed for intermediate externally created angular deformation

Intermediate externally created angular deformation a y A R A M A R B M B l θ B θ A θ o
Intermediate externally created angular deformation
Left end fixed, right end fixed for intermediate externally created angular deformation a θ o
Left end fixed, right end fixed for intermediate externally created angular deformation

Values for calculation

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

Calculation

Vertical end reactions $ R_A $

$$R_A=\cfrac{6\cdot E\cdot I\cdot θ_o}{l^3}\cdot\left(l-2\cdot a\right)$$
$R_A=\cfrac{6\cdot E\cdot I\cdot θ_o}{l^3}\cdot\left(l-2\cdot a\right)$
Equations in LaTeX code
R_A=\cfrac{6\cdot E\cdot I\cdot θ_o}{l^3}\cdot\left(l-2\cdot a\right)

Reaction end moment $ M_A $

$$M_A=\cfrac{2\cdot E\cdot I\cdot θ_o}{l^2}\cdot\left(3\cdot a-2\cdot l\right)$$
$M_A=\cfrac{2\cdot E\cdot I\cdot θ_o}{l^2}\cdot\left(3\cdot a-2\cdot l\right)$
Equations in LaTeX code
M_A=\cfrac{2\cdot E\cdot I\cdot θ_o}{l^2}\cdot\left(3\cdot a-2\cdot l\right)

Angular displacement $ θ_A $

$$θ_A=0$$
$θ_A=0$
Equations in LaTeX code
θ_A=0

Deflection $ y_A $

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

Vertical end reactions $ R_B $

$$R_B=-R_A$$
$R_B=-R_A$
Equations in LaTeX code
R_B=-R_A

Reaction end moment $ M_B $

$$M_B=\cfrac{2\cdot E\cdot I\cdot θ_o}{l^2}\cdot\left(l-3\cdot a\right)$$
$M_B=\cfrac{2\cdot E\cdot I\cdot θ_o}{l^2}\cdot\left(l-3\cdot a\right)$
Equations in LaTeX code
M_B=\cfrac{2\cdot E\cdot I\cdot θ_o}{l^2}\cdot\left(l-3\cdot a\right)

Angular displacement $ θ_B $

$$θ_B=0$$
$θ_B=0$
Equations in LaTeX code
θ_B=0

Deflection $ y_B $

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

Max. moment +

$$M_{max+}=M_B$$
$M_{max+}=M_B$
Equations in LaTeX code
M_{max+}=M_B

Max. moment -

$$M_{max-}=M_A$$
$M_{max-}=M_A$
Equations in LaTeX code
M_{max-}=M_A

Max. deflection -

$$y_{max-}=\cfrac{-2\cdot θ_o\cdot a^2}{l^3}\cdot\left(l-a\right)^2$$
$y_{max-}=\cfrac{-2\cdot θ_o\cdot a^2}{l^3}\cdot\left(l-a\right)^2$
Equations in LaTeX code
y_{max-}=\cfrac{-2\cdot θ_o\cdot a^2}{l^3}\cdot\left(l-a\right)^2

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

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

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