Question

Listed below is a combination of stresses acting at a point and referred to axes x and y in an elastic material. Using Mohr’s circle of stress determine the principal stresses at the point and their directions for each combination.

i) sigma x=-60 N/mm^2, sigma y=-36N/mm^2, shear stress (xy) =5N/mm^2

ii) sigma x=30 N/mm^2, sigma y=-50N/mm^2, shear stress (xy) =30N/mm^2

Answer 1

Solution;

(i)

Firs draw the Mohr Circle;

With the following points;

(-60MPa,-5MPa)

And;

(-36Mpa,5MPa)

The centre of the Mohr Circle is;

$C=-48MPa$

The Radius is;

$R=\sqrt{(60-48)^2+5^2}=-13MPa$

Hence the principal stresss;

$\sigma_{1,2}=C_-^+R$

$\sigma_1=-48+13=-35MPa$

$\sigma_2=-48-13=-61MPa$

(ii)

Points;

(30MPa,-30MPa)

(-50MPa,30MPa)

Center of the Circle;

$C=-10MPa$

Radius of the circle;

$R=\sqrt{30^2+40^2}=50MPa$

Therefore,the principal stressses are;

$\sigma_{1,2}=C_-^+R$

$\sigma_1=-10+50=40MPa$

$\sigma_2=-10-50=-60MPa$