Menu

Left end free, right end fixed for uniform temperature variation from top to bottom from $a$ to $l$

Uniform temperature variation from top to bottom from (and36)a(and36) to (and36)l(and36) a y A R A M A θ B M B l R B θ A T 1 T 2
Uniform temperature variation from top to bottom from $a$ to $l$
Left end free, right end fixed for uniform temperature variation from top to bottom from (and36)a(and36) to (and36)l(and36) a T 1 T 2
Left end free, right end fixed for uniform temperature variation from top to bottom from $a$ to $l$

Values for calculation

$l$ $\mathrm{mm}$
$a$ $\mathrm{mm}$
$t$ $\mathrm{mm}$
$T_1$ $\mathrm{°C}$
$T_2$ $\mathrm{°C}$
$γ$ $\mathrm{mm/mm/°C}$
$E$ $\mathrm{MPa}$
$I$ $\mathrm{mm^4}$
$x$ $\mathrm{mm}$

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 $

$$θ_A=\cfrac{-γ}{t}\cdot\left(T_2-T_1\right)\cdot\left(l-a\right)$$
$θ_A=\cfrac{-γ}{t}\cdot\left(T_2-T_1\right)\cdot\left(l-a\right)$
Equations in LaTeX code
θ_A=\cfrac{-γ}{t}\cdot\left(T_2-T_1\right)\cdot\left(l-a\right)

Deflection $ y_A $

$$y_A=\cfrac{γ}{2\cdot t}\cdot\left(T_2-T_1\right)\cdot\left(l^2-a^2\right)$$
$y_A=\cfrac{γ}{2\cdot t}\cdot\left(T_2-T_1\right)\cdot\left(l^2-a^2\right)$
Equations in LaTeX code
y_A=\cfrac{γ}{2\cdot t}\cdot\left(T_2-T_1\right)\cdot\left(l^2-a^2\right)

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 $

$$θ_B=0$$
$θ_B=0$
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

$$M_{max}=0$$
$M_{max}=0$
Equations in LaTeX code
M_{max}=0

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

$$V=R_A$$
$V=R_A$
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}
$$θ=θ_A+\cfrac{M_A\cdot x}{E\cdot I}+\cfrac{R_A\cdot x^2}{2\cdot E\cdot I}+\cfrac{γ}{t}\cdot\left(T_2-T_1\right)\cdot\left(x-a\right)$$
$θ=θ_A+\cfrac{M_A\cdot x}{E\cdot I}+\cfrac{R_A\cdot x^2}{2\cdot E\cdot I}+\cfrac{γ}{t}\cdot\left(T_2-T_1\right)\cdot\left(x-a\right)$
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{γ}{t}\cdot\left(T_2-T_1\right)\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}
$$y=y_A+θ_A\cdot x+\cfrac{M_A\cdot x^2}{2\cdot E\cdot I}+\cfrac{R_A\cdot x^3}{6\cdot E\cdot I}+\cfrac{γ}{2\cdot t}\cdot\left(T_2-T_1\right)\cdot\left(x-a\right)^2$$
$y=y_A+θ_A\cdot x+\cfrac{M_A\cdot x^2}{2\cdot E\cdot I}+\cfrac{R_A\cdot x^3}{6\cdot E\cdot I}+\cfrac{γ}{2\cdot t}\cdot\left(T_2-T_1\right)\cdot\left(x-a\right)^2$
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{γ}{2\cdot t}\cdot\left(T_2-T_1\right)\cdot\left(x-a\right)^2
On this page