# Left end simply supported 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 $R_A$

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

### Reaction end moment $M_A$

$$M_A=0$$

### Angular displacement $θ_A$

$$θ_A=\cfrac{γ}{4\cdot t\cdot l}\cdot\left(T_2-T_1\right)\cdot\left(l-a\right)\cdot\left(3\cdot a-l\right)$$

### Deflection $y_A$

$$y_A=0$$

### Vertical end reactions $R_B$

$$R_B=-R_A$$

### Reaction end moment $M_B$

$$M_B=R_A\cdot l$$

### Angular displacement $θ_B$

$$θ_B=0$$

### Deflection $y_B$

$$y_B=0$$

### Max. moment

$$M_{max}=M_B$$

### Max. deflection +

$$y_{max+}=\cfrac{γ\cdot\left(T_2-T_1\right)\cdot\left(l-a\right)}{6\cdot t\cdot\sqrt{3\cdot\left(l+a\right)}}\cdot\left(3\cdot a-l\right)^{3/2}$$

### 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

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