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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 $

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

Reaction end moment $ M_A $

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

Angular displacement $ θ_A $

$$θ_A=0$$

Deflection $ y_A $

$$y_A=0$$

Vertical end reactions $ R_B $

$$R_B=-R_A$$

Reaction end moment $ M_B $

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

Angular displacement $ θ_B $

$$θ_B=0$$

Deflection $ y_B $

$$y_B=0$$

Max. moment +

$$M_{max+}=M_A$$

Max. moment -

$$M_{max-}=M_B$$

Max. deflection +

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

Transverse shear

$$V=R_A$$

Bending moment

$$M=M_A+R_A\cdot x$$

Slope

$\text{if }\ x\le a$
$$θ=θ_A+\cfrac{M_A\cdot x}{E\cdot I}+\cfrac{R_A\cdot x^2}{2\cdot E\cdot I}$$
$\text{else}$
$$θ=θ_A+\cfrac{M_A\cdot x}{E\cdot I}+\cfrac{R_A\cdot x^2}{2\cdot E\cdot I}+\cfrac{γ}{t}\cdot\left(T_2-T_1\right)\cdot\left(x-a\right)$$

Deflection

$\text{if }\ x\le a$
$$y=y_A+θ_A\cdot x+\cfrac{M_A\cdot x^2}{2\cdot E\cdot I}+\cfrac{R_A\cdot x^3}{6\cdot E\cdot I}$$
$\text{else}$
$$y=y_A+θ_A\cdot x+\cfrac{M_A\cdot x^2}{2\cdot E\cdot I}+\cfrac{R_A\cdot x^3}{6\cdot E\cdot I}+\cfrac{γ}{2\cdot t}\cdot\left(T_2-T_1\right)\cdot\left(x-a\right)^2$$