Goodman criteria (Goodman Line) :- Used for brittle Materials.

• A straight line joining $S_e$ on the stress amplitude axis and $S_{at}$ on the mean stress axis is called the Goodman's line,

• According to the Goodman criteria , the triangular region below this line is considered to be safe. Hence, any point laying in this triangular region will ensure the safety, of the component for infinite line. The equation for the Goodman line can be written in the form of,

$\frac{x}{a} + \frac{y}{b}= 1$ ---- (a)

Where,

a = X-axis intercept or mean stress axis intercept = $S_{vt}$

b = Y-axis intercept or stress amplitude axis intercept = $S_{e}$

consider any point $P( S_m, S_a)$ on the Goodman line. Using equation (a), the equation for the Goodman line.

$\frac{S_m}{S_{ut}} + \frac{S_a}{S_e} = 1$ --- (1)

Now, introducing the fator of safety $N_f$ as shown in fig.

$\sigma_m = \frac{S_m}{N_f} \ and \ \sigma_a = \frac{S_a}{N_f}$ --- (b)

Substituting Equation (b) in equation 1,

$\frac{\sigma_m.N_F}{S_{ut}}$ + $\frac{\sigma_o.N_F}{S_{e}}$ = 1

$\frac{\sigma_m}{S_{ut}}$ + $\frac{\sigma_o}{S_{e}}$ = $\frac{1}{N_F}$ ........(2)

Where,

$S_{ut}$ = Ultimate tensile strength of the material, $\frac{N}{mm^2}$

$S_{e}$ = endurance limit of the component, $\frac{N}{mm^2}$

$S_{m}$ = Limitting safe mean stress, $\frac{N}{mm^2}$

$S_{a}$ = Limitting safe stress amplitude, $\frac{N}{mm^2}$

$\sigma_m$ = permissible mean stress, $\frac{N}{mm^2}$

$\sigma_a$ = permissible stress amplitude, $\frac{N}{mm^2}$

$N_F$ = Factor of safety

The equations (1) and (2) represent the Goodman line.