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Left end free, right end fixed (cantilever) for partial distributed load

Partial distributed load W a a y A R A M A R B M B l θ B θ A W l
Partial distributed load
Left end free, right end fixed (cantilever) for partial distributed load W a a W l
Left end free, right end fixed (cantilever) for partial distributed load

Values for calculation

$l$ $\mathrm{mm}$
$a$ $\mathrm{mm}$
$W_a$ $\mathrm{N/mm}$
$W_l$ $\mathrm{N/mm}$
$E$ $\mathrm{MPa}$
$I$ $\mathrm{mm^4}$
$x$ $\mathrm{mm}$

Calculation

Vertical end reactions $ R_A $

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

Reaction end moment $ M_A $

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

Angular displacement $ θ_A $

$$θ_A=\cfrac{W_a}{6\cdot E\cdot I}\cdot\left(l-a\right)^3+\cfrac{W_l-W_a}{24\cdot E\cdot I}\cdot\left(l-a\right)^3$$
$θ_A=\cfrac{W_a}{6\cdot E\cdot I}\cdot\left(l-a\right)^3+\cfrac{W_l-W_a}{24\cdot E\cdot I}\cdot\left(l-a\right)^3$
Equations in LaTeX code
θ_A=\cfrac{W_a}{6\cdot E\cdot I}\cdot\left(l-a\right)^3+\cfrac{W_l-W_a}{24\cdot E\cdot I}\cdot\left(l-a\right)^3

Deflection $ y_A $

$$y_A=\cfrac{-W_a}{24\cdot E\cdot I}\cdot\left(l-a\right)^3\cdot\left(3\cdot l+a\right)-\cfrac{W_l-W_a}{120\cdot E\cdot I}\cdot\left(l-a\right)^3\cdot\left(4\cdot l+a\right)$$
$y_A=\cfrac{-W_a}{24\cdot E\cdot I}\cdot\left(l-a\right)^3\cdot\left(3\cdot l+a\right)-\cfrac{W_l-W_a}{120\cdot E\cdot I}\cdot\left(l-a\right)^3\cdot\left(4\cdot l+a\right)$
Equations in LaTeX code
y_A=\cfrac{-W_a}{24\cdot E\cdot I}\cdot\left(l-a\right)^3\cdot\left(3\cdot l+a\right)-\cfrac{W_l-W_a}{120\cdot E\cdot I}\cdot\left(l-a\right)^3\cdot\left(4\cdot l+a\right)

Vertical end reactions $ R_B $

$$R_B=\cfrac{W_a-W_l}{2}\cdot\left(l-a\right)$$
$R_B=\cfrac{W_a-W_l}{2}\cdot\left(l-a\right)$
Equations in LaTeX code
R_B=\cfrac{W_a-W_l}{2}\cdot\left(l-a\right)

Reaction end moment $ M_B $

$$M_B=\cfrac{-W_a}{2}\cdot\left(l-a\right)^2-\cfrac{W_l-W_a}{6}\cdot\left(l-a\right)^2$$
$M_B=\cfrac{-W_a}{2}\cdot\left(l-a\right)^2-\cfrac{W_l-W_a}{6}\cdot\left(l-a\right)^2$
Equations in LaTeX code
M_B=\cfrac{-W_a}{2}\cdot\left(l-a\right)^2-\cfrac{W_l-W_a}{6}\cdot\left(l-a\right)^2

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. angular displacement

$$θ_{max}=θ_A$$
$θ_{max}=θ_A$
Equations in LaTeX code
θ_{max}=θ_A

Max. deflection

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

Transverse shear

$\text{if }\ x\le a$
$\text{if }\ x\le a$
Equations in LaTeX code
\text{if }\ x\le a
$$V=R_A$$
$V=R_A$
Equations in LaTeX code
V=R_A
$\text{else}$
$\text{else}$
Equations in LaTeX code
\text{else}
$$V=R_A-W_a\cdot\left(x-a\right)-\cfrac{W_l-W_a}{2\cdot\left(l-a\right)}\cdot\left(x-a\right)^2$$
$V=R_A-W_a\cdot\left(x-a\right)-\cfrac{W_l-W_a}{2\cdot\left(l-a\right)}\cdot\left(x-a\right)^2$
Equations in LaTeX code
V=R_A-W_a\cdot\left(x-a\right)-\cfrac{W_l-W_a}{2\cdot\left(l-a\right)}\cdot\left(x-a\right)^2

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}
$$M=M_A+R_A\cdot x-\cfrac{W_a}{2}\cdot\left(x-a\right)^2-\cfrac{W_l-W_a}{6\cdot\left(l-a\right)}\cdot\left(x-a\right)^3$$
$M=M_A+R_A\cdot x-\cfrac{W_a}{2}\cdot\left(x-a\right)^2-\cfrac{W_l-W_a}{6\cdot\left(l-a\right)}\cdot\left(x-a\right)^3$
Equations in LaTeX code
M=M_A+R_A\cdot x-\cfrac{W_a}{2}\cdot\left(x-a\right)^2-\cfrac{W_l-W_a}{6\cdot\left(l-a\right)}\cdot\left(x-a\right)^3

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}
$$θ=θ_A+\cfrac{M_A\cdot x}{E\cdot I}+\cfrac{R_A\cdot x^2}{2\cdot E\cdot I}-\cfrac{W_a}{6\cdot E\cdot I}\cdot\left(x-a\right)^3-\cfrac{W_l-W_a}{24\cdot E\cdot I\cdot\left(l-a\right)}\cdot\left(x-a\right)^4$$
$θ=θ_A+\cfrac{M_A\cdot x}{E\cdot I}+\cfrac{R_A\cdot x^2}{2\cdot E\cdot I}-\cfrac{W_a}{6\cdot E\cdot I}\cdot\left(x-a\right)^3-\cfrac{W_l-W_a}{24\cdot E\cdot I\cdot\left(l-a\right)}\cdot\left(x-a\right)^4$
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{W_a}{6\cdot E\cdot I}\cdot\left(x-a\right)^3-\cfrac{W_l-W_a}{24\cdot E\cdot I\cdot\left(l-a\right)}\cdot\left(x-a\right)^4

Deflection

$\text{if }\ x\le a$
$\text{if }\ x\le a$
Equations in LaTeX code
\text{if }\ x\le a
$$y=y_A+θ_A+θ_A\cdot x+\cfrac{M_A\cdot x^2}{2\cdot E\cdot I}+\cfrac{R_A\cdot x^3}{6\cdot E\cdot I}$$
$y=y_A+θ_A+θ_A\cdot x+\cfrac{M_A\cdot x^2}{2\cdot E\cdot I}+\cfrac{R_A\cdot x^3}{6\cdot E\cdot I}$
Equations in LaTeX code
y=y_A+θ_A+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}
$$y=y_A+θ_A+θ_A\cdot x+\cfrac{M_A\cdot x^2}{2\cdot E\cdot I}+\cfrac{R_A\cdot x^3}{6\cdot E\cdot I}-\cfrac{W_a}{24\cdot E\cdot I}\cdot\left(x-a\right)^4-\cfrac{W_l-W_a}{120\cdot E\cdot I\cdot\left(l-a\right)}\cdot\left(x-a\right)^5$$
$y=y_A+θ_A+θ_A\cdot x+\cfrac{M_A\cdot x^2}{2\cdot E\cdot I}+\cfrac{R_A\cdot x^3}{6\cdot E\cdot I}-\cfrac{W_a}{24\cdot E\cdot I}\cdot\left(x-a\right)^4-\cfrac{W_l-W_a}{120\cdot E\cdot I\cdot\left(l-a\right)}\cdot\left(x-a\right)^5$
Equations in LaTeX code
y=y_A+θ_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_a}{24\cdot E\cdot I}\cdot\left(x-a\right)^4-\cfrac{W_l-W_a}{120\cdot E\cdot I\cdot\left(l-a\right)}\cdot\left(x-a\right)^5
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