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**Application of theories of failure for thick walled pressure vessels.**

Having discussed the stresses in thick walled cylinders it is important to consider their failure criterion. The five failure theories will be considered in this regard and the variation of wall thickness to internal radius ratio t/ri or radius ratio r_{o}/r_{i} with p/σ_{yp} for different failure theories would be discussed. A number of cases such as p_{o} =0, p_{i} =0 or both non-zero p_{o} and p_{i} are possible but here only the cylinders with closed ends and subjected to an internal pressure only will be considered, for an example.

**9.3.1.1 Maximum Principal Stress **

theory According to this theory failure occurs when maximum principal stress exceeds the stress at the tensile yield point. The failure envelope according to this failure mode is shown in **figure-9.3.1.1.1** and the failure criteria are given by σ_{1} = σ_{2 }= ± σ_{yp}. If po =0 the maximum values of circumferential and radial stresses are given by

Here both σ_{θ} and σ_{r} are the principal stresses and σ_{θ} is larger. Thus the condition for failure is based on σ_{θ} and we have

where σ_{yp} is the yield stress.

(2)

**9.3.1.1.1F- Failure envelope according to Maximum Principal Stress Theory.**

**9.3.1.2 Maximum Shear Stress**

theory According to this theory failure occurs when maximum shear stress exceeds the maximum shear stress at the tensile yield point. The failure envelope according to this criterion is shown in figure- 9.3.1.2.1 and the maximum shear stress is given by

where the principal stresses σ_{1} and σ_{2} are given by

Here σ_{1} is tensile and σ_{2} is compressive in nature. τmax may therefore be given by

(3) and since the failure criterion is τ_{max} = σ_{yp }/ 2 we may write

**9.3.1.2.1F- Failure envelope according to Maximum Shear Stress theory. **

**9.3.1.3 Maximum Principal Strain theory**

According to this theory failure occurs when the maximum principal strain exceeds the strain at the tensile yield point.

and this gives where ε_{yp} and σ_{yp} are the yield strain and stress respectively. Following this the failure envelope is as shown in figure-9.3.1.3.1. Here the three principle stresses can be given as follows according to the standard 3D solutions:

(5)

The failure criterion may now be written as

(6)

**9.3.1.3.1F- Failure envelope according to Maximum Principal Strain theory**

**9.3.1.4 Maximum Distortion Energy**

Theory According to this theory if the maximum distortion energy exceeds the distortion energy at the tensile yield point failure occurs. The failure envelope is shown in** figure-9.3.1.4.1** and the distortion energy Ed is given by

Since at the uniaxial tensile yield point σ_{2 }= σ_{3 }= 0 and σ_{1} = σ_{yp }

E_{d} at the tensile yield point

We consider σ_{1} = σ_{θ} , σ_{2} = σ_{r} and σ_{3 }= σ_{z }and therefore

(7)

The failure criterion therefore reduces to

(8)

**9.3.1.4.1F- Failure envelope according to Maximum Distortion Energy Theory**

Plots of pi/σ_{yp} and t/r_{i} for different failure criteria are shown in **figure9.3.1.4.2.**

**9.3.1.4.2F- Comparison of variation of against t/r _{i} for different failure criterion. **

The criteria developed and the plots apply to thick walled cylinders with internal pressure only but similar criteria for cylinders with external

pressure only or in case where both internal and external pressures exist may be developed. However, on the basis of these results we note that the rate of increase in p_{i}/σ_{yp} is small at large values of t/r_{i} for all the failure modes considered. This means that at higher values of p_{i} small increase in pressure requires large increase in wall thickness. But since the stresses near the outer radius are small, material at the outer radius for very thick wall cylinders are ineffectively used. It is therefore necessary to select materials so that pi/σ_{yp} is reasonably small. When this is not possible prestressed cylinders may be used.

All the above theories of failure are based on the prediction of the beginning of inelastic deformation and these are strictly applicable for ductile materials under static loading. Maximum principal stress theory is widely used for brittle materials which normally fail by brittle fracture.

In some applications of thick cylinders such as, gun barrels no inelastic deformation can be permitted for proper functioning and there design based on maximum shear stress theory or maximum distortion energy theory are acceptable. For some pressure vessels a satisfactory function is maintained until inelastic deformation that starts from the inner radius and spreads completely through the wall of the cylinder. Under such circumstances none of the failure theories would work satisfactorily and the procedure discussed in section lesson 9.2 is to be used.

**9.3.1.5 Failure criteria of pre-stressed thick cylinders**

Failure criteria based on the three methods of pre-stressing would now be discussed. The radial and circumferential stresses developed during shrinking a hollow cylinder over the main cylinder are shown in** figure9.3.1.5.1.**

**9.3.1.5.1F- Distribution of radial and circumferential stresses in a composite thick walled cylinder subjected to an internal pressure. **

Following the analysis in section 9.2 the maximum initial (residual) circumferential stress at the inner radius of the cylinder due to the contact pressure p_{s} is

and the maximum initial (residual) circumferential stress at the inner radius of the jacket due to contact pressure p_{s} is

Superposing the circumferential stresses due to p_{i} (considering the composite cylinder as one) the total circumferential stresses at the inner radius of the cylinder and inner radius of the jacket are respectively

These maximum stresses should not exceed the yield stress and therefore we may write

It was shown in section-9.2 that the contact pressure p_{s} is given by

From (9), (10) and (11) it is possible to eliminate ps and express t/r_{i }in terms of p_{i}/σ_{yp} and this is shown graphically in **figure-9.3.1.5.2.**

**9.3.1.5.2F- Plot of p _{i}/σ_{yp} vs t/r_{i} for laminated multilayered, single jacket and solid wall cylinders.**

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