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

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