Pipe Stress Analysis: Theories of failure

Pipe Stress Analysis: Theories of failure

Introduction:

       Piping parts fail when the stresses induced by external forces exceed their strength. The external loads cause internal stresses in the elements and the component size depend on the stresses developed. Loading may be due to:

a) Thermal loads

b) Sustain Loads

c) Inertial forces

d) Displacement loads

e) Occasional Loads

As per our last articles, the system behavior and failure are dependent on the type of loading imposed. These are mainly classified as Primary vs. Secondary or Static vs. Dynamic. We have already understood the classification of load base on Primary Loads vs. Secondary Loads.

Now Load maybe classified as:

a) Static load- Load does not change in magnitude and direction and normally  increases gradually to a steady value.

b)   Dynamic load- Load may change in magnitude, for example, Vibration and shock are types of dynamic loading.

The following figure shows load vs. time characteristics for both static and dynamic loading of machine elements.

load vs. time characteristics

Theories of failure

          In design an element is said to have failed if it ceases to perform its function. When a Piping element is subjected to a system of the complex stress system, it is important to predict the mode of failure so that the design methodology may be based on a particular failure criterion. Theories of failure are essentially a set of failure criteria developed for the ease of design. Important Theories of failure commonly used as follows;

1. Maximum principal stress theory (Rankine theory)
2. Maximum shear stress theory (Tresca theory)

Maximum principal stress theory (Rankine theory)

          According to this, if one of the principal stresses S1 (maximumprincipal stress), S2 (minimum principal stress) or S3 exceeds the yield stress, yielding would occur. In a two-dimensional loading situation for a ductile material where tensile and compressive yield stress are nearly of same magnitude

S1 = ± Sy

S2 = ±Sy

Using this, a yield surface may be drawn, as shown in the following figure. Yielding occurs when the state of stress is at the boundary of the rectangle. Consider, for example, the state of stress of a thin-walled pressurized pipe. Here S1= 2 x S2, S1 being the circumferential or hoop stress and S2 the axial stress. As the pressure in the pipe increases the stress follows the dotted line. At a point (say) the stresses are still within the elastic limit but at b, S1 reaches Sy although S2 is still less than Sy. Yielding will then begin at point b. This theory of yielding has very poor agreement with experiment. However, the theory has been used successfully for brittle materials.

Maximum principal stress theory

The Maximum Principal Stress Theory (Rankine theory) forms the basis of a piping system governed by ASME B31 and subsection NC & ND (class 2 & 3) of Section III of ASME Boiler & Pressure Vessel Code.

Maximum shear stress theory (Tresca theory)

          According to this theory, yielding would occur when the maximum shear stress just exceeds the shear stress at the tensile yield point. At the tensile yield point S2= S3 = 0 and thus maximum shear stress is Sy/2. This gives us six conditions for a three-dimensional stress situation:

S1 – S2 = ± Sy

S2 – S3 = ±Sy

S3 – S1 = ±Sy

Maximum shear stress theory

In a biaxial stress situation ( As shown in above figure) case, S3 = 0 and this gives

S1 – S2 = +Sy   if S1>0, S2<0

S1 – S2 = -Sy   if S1<0, S2>0

S2 = +Sy         if S2>S1>0

S2 = -Sy         if S2<S1<0

S1 = +Sy         if S1>S2>0

S1 = -Sy         if S1<S2<0

This criterion agrees well with the experiment.

In the case of pure shear,

 S1 = - S2 = k (say), S3 = 0 and this gives

S1- S2 = 2k= Sy

This indicates that yield stress in pure shear is half the tensile yield stress and this is also seen in the Mohr’s circle for pure shear.

The Mohr’s circle for pure shear

The maximum shear stress theory is more accurate than The Maximum Principal stress theory for predicting both Yielding and Fatigue failure in a ductile material. This theory forms the bases of Piping of subsection, NB(class 1) of ASME Section III.

Note: For Complex loading, Principal Stress S1(Maximum/Major Normalstress) and S2 (Minimum/Minor Normal Stress) & Maximum Shear Stress are calculated by the following formula; 

Complex Stresses in Piping System

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