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How Does Internal Pressure affects Pipes-Basic study

Imagine a pipe Element, of which's both ends are closed. An Internal Pressure “P” applies on the internal walls of the pipe. The pressure causes the following Stresses on each element of the pipe.

a) Circumferential (Hoop) Stress (σH)

b) Longitudinal Stress (σL)

c) Radial Stress(σR) 


Graphical Representation of Stress on pipe wall



Imagine a condition in which the Pressure "P" Exceeds the Resisting capacity of the wall. The pipe may split as shown below.


Splitting of Pipe along the Longitudinal Plane




The failure may possible across any line containing diameter and axis of the pipe.Here it's the horizontal plane.Elements which resists this failure experience stress and direction of the same will be along the circumference. Hence it’s called as Circumferential or “Hoop Stress”


Let's try to quantify this 
here,
                             D=Diamter of the Pipe
                             t=Thikness of the Pipe
                             L=Legth of the pipe Element

Magnitude of force causing the bursting =Pressure X Area
                                                                 =P x D x L
Resisting Forces (Stress exerted
                               on the Pipe Material)= Area of material Resisting  x Stress
                                                                  =2 t x L x σH
                                                    
 Lets Equate Bursting Force with the Stress Generated,


                                                       P x D x L=2 t X L X σH


Solving This will give, 

Hoop  Stress= 


For a typical System where Pressure (P), Diameter (D) and Material circumferential Stress (σH ) is Known, Thickness (t) is the unknown variable and can be tabulated by rearranging the above equation.

Now consider another type of failure 


Failure along Circumference
When a failure, as depicted in the above figure, happens, Elements resisting this will experience stress along the Longitudinal Plane. Therefore this stress is called as Longitudinal stress 

                          Bursting Force =Pressure X Area
          
                

                                  Resisting Force=Metal Area x Stress (σL ) 
                                                                               =π X Dt X σL

                                                     
Solving This will Give

Longitudinal Stress= 


It is Clear From the above equations of Longitudinal and Circumferential Stress that, For a given diameter D, thickness t and which is subjected to a pressure P -

The Induced Circumferential Stress is double the induced longitudinal Stress. 

Now consider the nature of Radial stress. Each Element of the Pipe is subjected to Radial Stress, Which acts in Radial direction.

                                              σR=P





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