Question

At a point on the free surface of a stressed body, the normal stresses are 20 ksi (T) on a vertical plane and 30 ksi (C) on a horizontal plane. An unknown negative shear stress exists on the vertical plane. The absolute maximum shear stress at the point has a magnitude of 32 ksi. Determine the principal stresses and the shear stress on the vertical plane at the point.

Answer 1

The principal stresses are σp1 = 27 ksi, σp2 = -37 ksi and the shear stress is zero

Step-by-step explanation:

The expression for the maximum shear stress is given:

Where

σx = stress in vertical plane = 20 ksi

σy = stress in horizontal plane = -30 ksi

τM = 32 ksi

Replacing:

Solving for τxy:

τxy = ±19.98 ksi

The principal stress is:

Where

σp1 = 20 ksi

σp2 = -30 ksi

(equation 1)

equation 2

Solving both equations:

σp1 = 27 ksi

σp2 = -37 ksi

The shear stress on the vertical plane is zero

Answer 2

The principal stresses are σp1 = 27 ksi, σp2 = -37 ksi and the shear stress is zero

Step-by-step explanation:

The expression for the maximum shear stress is given:

`\tau _(M) =\sqrt{((\sigma _(x)^(2)-\sigma _(y)^(2) )/(2))^(2)+\tau _(xy)^(2) }`

Where

σx = stress in vertical plane = 20 ksi

σy = stress in horizontal plane = -30 ksi

τM = 32 ksi

Replacing:

`32=\sqrt{((20-(-30))/(2) )^(2) +\tau _(xy)^(2) }`

Solving for τxy:

τxy = ±19.98 ksi

The principal stress is:

`\sigma _(x)+\sigma _(y) =\sigma _(p1)+\sigma _(p2)`

Where

σp1 = 20 ksi

σp2 = -30 ksi

`\sigma _(p1) +\sigma _(p2)=-10 ksi` (equation 1)

`\tau _(M) =(\sigma _(p1)-\sigma _(p2))/(2) \\\sigma _(p1)-\sigma _(p2)=2\tau _(M)\\\sigma _(p1)-\sigma _(p2)=32*2=64ksi` equation 2

Solving both equations:

σp1 = 27 ksi

σp2 = -37 ksi

The shear stress on the vertical plane is zero