Design Principles For Thick Cylinders (Part - 1) | Additional Study Material for Mechanical Engineering PDF Download

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 σ₁ = σ₂ = ± 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 σ₁ and σ₂ are given by

Here σ₁ is tensile and σ₂ 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 σ₂ = σ₃ = 0 and σ₁ = σ_yp 
E_d at the tensile yield point

We consider σ₁ = σ_θ , σ₂ = σ_r and σ₃ = 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.