Linear/Eigenvalue Buckling analysis are fairly easy to set up and post process. Typically an Eigenvalue Buckling Analysis is preceded by a static structural analysis with perturbation load leading to compressive stress field being generated in the model. The pre-stressed model from the static structural analysis is then analyzed for buckling failure modes. Results of buckling analysis are eigenvalues or load multipliers for various buckling failure modes. The product of the eigenvalue and the perturbation load gives the critical buckling load for the analyzed mode. Typically the 1st buckling mode is the most critical one with the lowest eigenvalue, however its generally recommended to find atleast 3 buckling modes to be sure that any critical buckling mode is not missed.

Here an ASME PTB-3 Validation for “Example E5.4 – Protection Against Collapse from Buckling” is carried out per ASME Sec VIII Div 2 using ANSYS.

Problem Statement:

Evaluate the following tower for compliance with respect to the Type-1 buckling criteria provided in paragraph 5.4.1.2.

Material – Shell and Heads = SA-516, Grade 70, Normalized

Design Conditions = -14.7 psig at 300^{o}F

Corrosion Allowance = 0.125 inches

ANALYSIS SUMMARY

- A mid surface shell model was constructed using corroded dimensions from the above figure.
- The model was meshed with higher order shell elements.

- Fixed Support was applied to the base of the skirt.
- External Pressure = 14.7 psi was applied as perturbation load for the static structural analysis.

- The pre-stressed model from static structural analysis was used for eigenvalue buckling analysis. 3 buckling modes were requested using analysis setup and the numerical model was solved.
- Following Eigenvalues were found from the buckling analysis

Mode | Eigenvalues |

1 | 9.0009 |

2 | 9.0016 |

3 | 15.288 |

ASME PTB-3 Eigenvalues

Mode | Eigenvalues |

1 | 7.939 |

2 | 7.940 |

3 | 14.351 |

Note the ASME PTB-3 analysis is carried out using abaqus which calculates Buckling Load as Eigenvalue + Eigenvalue*Perturbation Load. But in ANSYS Buckling Load is calculated as Eigenvalue*Perturbation Load. Hence there is a difference of about 1 between the two eigenvalues.

First Mode Shape Plot

ASME PTB-3 First Mode Shape

The plots are quite similar. Note that values of deformations have no physical meaning for buckling analysis

CRITICAL BUCKLING LOAD EVALUATION

For Type 1 buckling analysis (Elastic Stress Analysis with small deformation theory) performed here ASME SEC VIII Div 2 gives a minimum design factor of \(\Phi_B=\frac{2}{\beta_{cr}}\).

For unstiffened cylinders under external pressure \(\beta_{cr} = 0.80 \) from eq 5.14.

Therefore \(\Phi_B=\frac{2}{\beta_{cr}}=\frac{2}{0.80}=2.5\)

Critical Buckling Load (Mode 1)

= Perturbation Load * Eigenvalue/Design Factor

= 14.7*9.0009/2.5

= 52.9 psi

Critical Buckling Load (Mode 1 ASME PTB-3)

= (Perturbation Load +Perturbation Load* Eigenvalue)/Design Factor

= (14.7 + 14.7*7.939)/2.5

= 52.6 psi

The values compare fairly well.

Since external pressure = 14.7 psi < buckling pressure = 52.9 psi the structure is safe w.r.t buckling under the design conditions.

Have these results been checked against classical buckling theory? There is reason to believe that eigenvalue buckling theory is inappropriate for cylindrical shells under external pressure. Moreover for long cylinders the Code approach (which was developed in the 1930’s) is a simple calculation and proven by tests. You should check your figures against the Div 1 procedure for long cylinders. It’ll take you about 15 minutes and give you a number you can rely on. I also note that that lateral loading from wind or seismic excitation may also cause buckling near the bottom of the column since a portion of the shell will carry compression. Eigenvalue buckling should be used with great caution on thin cylindrical shells.

I was also surprised that ASME PTB-3 picked up an example which will normally be designed using code based formula. However since this post was a validation exercise, I repeated the same example given in ASME PTB-3. As suggested by you I did a code based calculation to arrive at the maximum allowable external pressure.

Here’s a quick summary:

Results for Maximum Allowable External Pressure (MAEP):

Tca, OD, SLEN, D/t, L/D, Factor A, B

1.000, 92.25, 652.00, 92.25, 7.0678, 0.0001858, 2694.06

EMAP = (4*B)/(3*(D/t)) = (4*2694.0613 )/(3*92.2500 ) = 38.9386 psig

The buckling pressure from FEA was 52.9 psi which is obviously higher than the code based calculated value. I have a couple of queries.

1. Whether code calculated Max Allowed External Pressure has some inbuilt factor of safety?

2. Why is eigenvalue buckling theory inappropriate for cylindrical shells under external pressure?

Hi Sandip.

You stated that, values of deformations have no physical meaning for buckling analysis, then what it really signifies?

For your question, Whether code calculated Max Allowed External Pressure has some inbuilt factor of safety?

Refer STP-PT-029, Where knock down factor of 5.0 is used by ASME for effect of geometrical imperfections on axial compression. In addition, a design factor DF, of 2.0 is also added to account for reduced modulus in the inelastic range and variation of material properties that cause test to deviate from theory.

You basically look at the buckling shape, not the actual values of the deformations.