Hoop Stress in Thin Cylinders

In our daily life, either technical or non-technical field, we come across object with the dominant shape of a cylinder. Looking around our-self, there are innumerous cylindrical objects, so to study the behavior of this shapes under stress has become an important aspect.

Expression

Cylinder subjected to internal pressure

Consider a cylinder of thickness t filled with pressurized gas exerting a force on the inner walls of the cylinder.

Due to pressure exertion the block undergoes volumetric expansion and stresses are being develop.

Specification of cylinder:

Radius r, Diameter D

Thickness t

Length L

Small element taken for further integration

We consider a small element of the cylinder, analyze the forces acting on this element and integrating the resultant to get the total force.

Length of arc subtended from the center: r*𝛅θ

Integral element under pressure

Area of this element: (D/2)*𝛅θ*𝛅L

Small force acting on the inner surface due to pressurized gas =dp= (D/2)*𝛅θ*𝛅L*p

This force can be further resolved into components along x-coordinate and y-coordinate.

Components:

Vertical component:  ((D/2)*𝛅θ*𝛅L*p)*sinθ

Horizontal component:  ((D/2)*𝛅θ*𝛅L*p)*cosθ

The horizontal component reduces to zero upon integration over the semicircular portion.

In the limit 𝛅 changes to d.

The integral of vertical component from 0 to π:    ∫((D/2)*dθ*dL*p)*sinθ

                                                                                      =p*(D/2)*(dL)*(-cosθ)     ...0 to π

                                                                                      =p*D*dL

In response to the vertical force developed due to pressurized gas, stress will be developed to oppose this force. The stress developed will be acting in the opposite direction with respect to the generated force and the area under stress is shown shaded.

Therefore, area opposing the force=2*dL*t

The Circumferential or hoop stress, σ_{c =Force/Resisting Area}

                                                                                        ₌₍p*D*dL)/(2*dL*t)

                                                                  =(p*D)/(2*t)