### Below are the following stress acting in piping system

A. Axial or Longitudinal direction.

B. Circumferential or Hoop’s direction.

C. Radial direction.

The stresses in the pipe wall are expressed as axial(SL), Hoope’s (SH) and Radial (SR).These stresses which stretch or compress a grain/ crystal are called normal stress because they are normal to the surface of the crystal.

### Hoop Stresses calculation

Stresses which are generated circumstantially due to the action of Internal pressure of pipe are called Hoop Stress. It is calculated by; –

Hoop Stress (Sh) = Pdo / 2t

Where P = Force Acting from Inside.

do = OD of Pipe.

t = Pipe Thickness.

Affect Hoop Stress in the system

As per membrane theory for pressure design of cylinders, as long as hoop stress is less than yield stress of Moc, the design is safe. Hoop stress induced by thermal pressure is twice the axial stress (SL). This is widely used for pressure thickness calculation for pressure vessel.

### Other stresses in piping system

A. Principal stress.

B. Shear stress.

The above stresses (Axial, Circumferential and Radial stress) have stress component in direction normal to faces of randomly oriented crystal. Each crystal thus faces normal stresses. One of these orientations must be such that it maximizes one of the normal stresses. Normal stresses for such orientation (maximum normal stress orientation) are called principal stresses and are designated as S1 (maximum), S2 and S3 (minimum).

Principal stresses are way of defining the worst case scenario as far as the normal stresses are concerned.

In addition to the normal stresses, a grain can be subjected to shear stresses as well. These stress act parallel to the crystal surface. The shear stresses occur if the pipe is subjected to torsion, bending etc. Just as there is an orientation for which normal stresses are maximum, there is an orientation which maximizes shear stress. The maximum shear stress in a 3-D state of stress can be shown to be as:-

τ max = (S1 – S3)/2

i.e. half of the difference between the maximum and minimum principal stresses.

Calculation of principal stresses and maximum shear stresses

Use of Mohr’s circle allows calculating the two principal stresses and maximum shear stress as:-

S1 = (SL + SH)/2 + [{(SL + SH)/2}2 + τ2 ] 0.5

S2 = (SL + SH)/2 – [{(SL + SH)/2}2 + τ2 ] 0.5

τ max = 0.5 [(SL – SH) 2+ 4 τ2] 0.5

The third principal stress (minimum i.e. S3) is zero.