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Left end guided, right end fixed for concentrated intermediate moment

Concentrated intermediate moment M o a y A R A M A R B M B l θ B θ A
Concentrated intermediate moment
Left end guided, right end fixed for concentrated intermediate moment M o a
Left end guided, right end fixed for concentrated intermediate moment

Values for calculation

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

Calculation

Vertical end reactions $ R_A $

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

Reaction end moment $ M_A $

$$M_A=\cfrac{-M_O\cdot\left(l-a\right)}{l}$$
$M_A=\cfrac{-M_O\cdot\left(l-a\right)}{l}$
Equations in LaTeX code
M_A=\cfrac{-M_O\cdot\left(l-a\right)}{l}

Angular displacement $ θ_A $

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

Deflection $ y_A $

$$y_A=\cfrac{M_O\cdot a\cdot\left(l-a\right)}{2\cdot E\cdot I}$$
$y_A=\cfrac{M_O\cdot a\cdot\left(l-a\right)}{2\cdot E\cdot I}$
Equations in LaTeX code
y_A=\cfrac{M_O\cdot a\cdot\left(l-a\right)}{2\cdot E\cdot I}

Vertical end reactions $ R_B $

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

Reaction end moment $ M_B $

$$M_B=\cfrac{M_O\cdot a}{l}$$
$M_B=\cfrac{M_O\cdot a}{l}$
Equations in LaTeX code
M_B=\cfrac{M_O\cdot a}{l}

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

$$y_{max}=y_A$$
$y_{max}=y_A$
Equations in LaTeX code
y_{max}=y_A

Transverse shear

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

Bending moment

$\text{if }\ x\le a$
$\text{if }\ x\le a$
Equations in LaTeX code
\text{if }\ x\le a
$$M=M_A+R_A\cdot x$$
$M=M_A+R_A\cdot x$
Equations in LaTeX code
M=M_A+R_A\cdot x
$\text{else}$
$\text{else}$
Equations in LaTeX code
\text{else}
$$M=M_A+R_A\cdot x+M_O$$
$M=M_A+R_A\cdot x+M_O$
Equations in LaTeX code
M=M_A+R_A\cdot x+M_O

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