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Left end simply supported, right end simply supported for concentrated intermediate load

Concentrated intermediate load W a y A R A M A R B M B l θ B θ A
Concentrated intermediate load
Left end simply supported, right end simply supported for concentrated intermediate load W a
Left end simply supported, right end simply supported for concentrated intermediate load

Values for calculation

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

Calculation

Vertical end reactions $ R_A $

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

Reaction end moment $ M_A $

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

Angular displacement $ θ_A $

$$θ_A=\cfrac{-W\cdot a}{6\cdot E\cdot I\cdot l}\cdot\left(2\cdot l-a\right)\cdot\left(l-a\right)$$
$θ_A=\cfrac{-W\cdot a}{6\cdot E\cdot I\cdot l}\cdot\left(2\cdot l-a\right)\cdot\left(l-a\right)$
Equations in LaTeX code
θ_A=\cfrac{-W\cdot a}{6\cdot E\cdot I\cdot l}\cdot\left(2\cdot l-a\right)\cdot\left(l-a\right)

Deflection $ y_A $

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

Vertical end reactions $ R_B $

$$R_B=\cfrac{W\cdot a}{l}$$
$R_B=\cfrac{W\cdot a}{l}$
Equations in LaTeX code
R_B=\cfrac{W\cdot a}{l}

Reaction end moment $ M_B $

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

Angular displacement $ θ_B $

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

Deflection $ y_B $

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

Max. moment

$$M_{max}=R_A\cdot a$$
$M_{max}=R_A\cdot a$
Equations in LaTeX code
M_{max}=R_A\cdot a

Max. deflection

$\text{if }\ a < \cfrac{l}{2}$
$\text{if }\ a < \cfrac{l}{2}$
Equations in LaTeX code
\text{if }\ a < \cfrac{l}{2}
$$y_{max}=\cfrac{-W\cdot a}{3\cdot E\cdot I\cdot l}\cdot\left(\cfrac{l^2-a^2}{3}\right)^{3/2}$$
$y_{max}=\cfrac{-W\cdot a}{3\cdot E\cdot I\cdot l}\cdot\left(\cfrac{l^2-a^2}{3}\right)^{3/2}$
Equations in LaTeX code
y_{max}=\cfrac{-W\cdot a}{3\cdot E\cdot I\cdot l}\cdot\left(\cfrac{l^2-a^2}{3}\right)^{3/2}
$\text{else}$
$\text{else}$
Equations in LaTeX code
\text{else}
$$y_{max}=\cfrac{-W\cdot \left(l-a\right)}{3\cdot E\cdot I\cdot l}\cdot\left(\cfrac{l^2-\left(l-a\right)^2}{3}\right)^{3/2}$$
$y_{max}=\cfrac{-W\cdot \left(l-a\right)}{3\cdot E\cdot I\cdot l}\cdot\left(\cfrac{l^2-\left(l-a\right)^2}{3}\right)^{3/2}$
Equations in LaTeX code
y_{max}=\cfrac{-W\cdot \left(l-a\right)}{3\cdot E\cdot I\cdot l}\cdot\left(\cfrac{l^2-\left(l-a\right)^2}{3}\right)^{3/2}

Max. angular displacement

$\text{if }\ a < \cfrac{l}{2}$
$\text{if }\ a < \cfrac{l}{2}$
Equations in LaTeX code
\text{if }\ a < \cfrac{l}{2}
$$θ_{max}=θ_A$$
$θ_{max}=θ_A$
Equations in LaTeX code
θ_{max}=θ_A
$\text{else}$
$\text{else}$
Equations in LaTeX code
\text{else}
$$θ_{max}=θ_B$$
$θ_{max}=θ_B$
Equations in LaTeX code
θ_{max}=θ_B

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}
$$V=R_A-W$$
$V=R_A-W$
Equations in LaTeX code
V=R_A-W

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}
$$M=M_A+R_A\cdot x-W\cdot\left(x-a\right)$$
$M=M_A+R_A\cdot x-W\cdot\left(x-a\right)$
Equations in LaTeX code
M=M_A+R_A\cdot x-W\cdot\left(x-a\right)

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}
$$θ=θ_A+\cfrac{M_A\cdot x}{E\cdot I}+\cfrac{R_A\cdot x^2}{2\cdot E\cdot I}-\cfrac{W}{2\cdot E\cdot I}\cdot\left(x-a\right)^2$$
$θ=θ_A+\cfrac{M_A\cdot x}{E\cdot I}+\cfrac{R_A\cdot x^2}{2\cdot E\cdot I}-\cfrac{W}{2\cdot E\cdot I}\cdot\left(x-a\right)^2$
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}{2\cdot E\cdot I}\cdot\left(x-a\right)^2

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