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Left end fixed, 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 $l$
rLeft end fixed, 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 fixed, 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 $

$$R_A=\cfrac{-6\cdot E\cdot I\cdot a\cdot γ}{t\cdot l^3}\cdot\left(T_2-T_1\right)\cdot\left(l-a\right)$$
$R_A=\cfrac{-6\cdot E\cdot I\cdot a\cdot γ}{t\cdot l^3}\cdot\left(T_2-T_1\right)\cdot\left(l-a\right)$
Equations in LaTeX code
R_A=\cfrac{-6\cdot E\cdot I\cdot a\cdot γ}{t\cdot l^3}\cdot\left(T_2-T_1\right)\cdot\left(l-a\right)

Reaction end moment $ M_A $

$$M_A=\cfrac{E\cdot I\cdot γ}{t\cdot l^2}\cdot\left(T_2-T_1\right)\cdot\left(l-a\right)\cdot\left(3\cdot a-l\right)$$
$M_A=\cfrac{E\cdot I\cdot γ}{t\cdot l^2}\cdot\left(T_2-T_1\right)\cdot\left(l-a\right)\cdot\left(3\cdot a-l\right)$
Equations in LaTeX code
M_A=\cfrac{E\cdot I\cdot γ}{t\cdot l^2}\cdot\left(T_2-T_1\right)\cdot\left(l-a\right)\cdot\left(3\cdot a-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 $

$$M_B=\cfrac{-E\cdot I\cdot γ}{t\cdot l^2}\cdot\left(T_2-T_1\right)\cdot\left(l-a\right)\cdot\left(3\cdot a+l\right)$$
$M_B=\cfrac{-E\cdot I\cdot γ}{t\cdot l^2}\cdot\left(T_2-T_1\right)\cdot\left(l-a\right)\cdot\left(3\cdot a+l\right)$
Equations in LaTeX code
M_B=\cfrac{-E\cdot I\cdot γ}{t\cdot l^2}\cdot\left(T_2-T_1\right)\cdot\left(l-a\right)\cdot\left(3\cdot a+l\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 +

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

Max. moment -

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

Max. deflection +

$$y_{max+}=\cfrac{2\cdot M_A^2}{3\cdot R_A^2\cdot E\cdot I}$$
$y_{max+}=\cfrac{2\cdot M_A^2}{3\cdot R_A^2\cdot E\cdot I}$
Equations in LaTeX code
y_{max+}=\cfrac{2\cdot M_A^2}{3\cdot R_A^2\cdot E\cdot I}

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
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