PRESSURE VESSEL CALCULATIONS
PROJECT:  I.S.M.S. Mass Spectrometer 
DATE:   Oct 13, 2004 
ANALYST:        J.E.  
Revisions: (last rev date 8/4/99) 

1)Added first ten buckling modes for determination of critical buckling pressure.
2) Corrected maximum bending stress equation for cap
3) Added von Mises calculation and corrected buckling calculation
4) Corrected von Mises equation
5) Added O-ring groove diameter for diametral clearance and tab bending calculations.
6) Corrected index problem with radial moment equations
The following calculations are for determining the margin of safety  for a pressure vessel at a given depth. The equations are mostly from Roark's Formulas for Stress and Strain and are referenced when used. The user must input the dimensions,  material properties,  and design depth. A factor of safety will be included in the calculations to determine the minimum margin of safety. The minimum margin of safety must be greater than zero for the design to be suitable for the depth desired.

The format for this analysis package will be as follows:

A) Input of design parameters including dimensions, material properties, and design depth 
B) Calculations for deformation and stresses in cylinders (check for thin wall or thick wall condition)
C) Buckling calculations for cylinders
D) Calculations for deformations and stresses in end plates,
E) Weight calculation
F) Summary of input and list of results including diametral gap, stresses, margins of safety and weight.

Note: The minimum margin of safety will be determined based upon the lowest margin calculated in the sections above. Thus, for a give pressure vessel, the maximum working pressure may be limited by the end plate thickness while for another it may be limited by the length of the cylindrical section. Therefor, if a desired allowable working depth is not reached, the limiting factor needs to be determined in order to effectively iterate the design to meet the requirements. Also, the diametral gap must be checked for the different pressures to assure that it falls within the maximum allowable from the o-ring vendor.

A) Design parameters:
(pp02)
(pp18)

1) Dimensional input:
Thin Wall Calcs:
Is it thin wall (1=yes)?

Enter cylinder dimensions:
Meridional (axial) stress:

Change in height:

Inside Diameter :
Outside Diameter:
O-ring surface diameter:
Case Length: 
Cylinder Base 
Thickness:	
Wall thickness:
Radial displacement 
of circumference:

Circumferential (hoop) stress:

Enter end cap and gland dimensions  (O-Ring #2-366): 

Margin of Safety:

Plug Diameter:
Groove Width:
Groove Diameter:
Groove Base Thickness:
Thick Wall Calcs

Axial Stress:

Change in OD (radius):

End Cap OD:
End Cap Thickness:
Tab Thickness:
Connector Diameter:
Hoop Stress:

Change in ID (radius):

Radial Stress:

Change in Length:

2) Material properties:
Shear Stress:

Von Mises Stress:

Enter material properties:
Material:
Titanium Alloy 6AL-4V 
Margins of Safety:
for yield strength:

for shear strength:

Modulus of elasticity:
Material density:
Poisson's ratio:
Yield Stress:
Shear stress:
Water density:

Closed End Cylinder Base Calcs: 

Margin:

3) Design depth and resulting pressure:

Number of pressure steps:

Pressure:

For the expressions in the case to be valid, the following condition must be true (i.e., condition = 1):
Buckling Calcs: 

4) Factors of safety to be used:
for buckling:

for stress:

Margin of safety on buckling:

END CAP Calcs
(pp19)
B) Formulas for deformation and stresses in cylinders:
END CAP Circular Plate  
1) Thin walled cylinder calculation:
Reference: Table 28  Formulas for membrane stresses and deformations in thin-walled pressure vessels

Cylindrical shell; uniform internal or external pressure (ends capped)
Case 1c
MARGIN OF SAFETY: 

Cylindrical shell
Calculated dimensions:
Mean radius:

Cylinder Height:

END CAP tabs: 

MARGIN OF SAFETY: 

Thin wall stresses:
Conditions:
For the expressions in the case to be valid, the following condition must be true (i.e., condition = 1):

O-RING tab: 

Meridional (axial) stress:
Circumferential (hoop) stress:

MARGIN OF SAFETY: 

Von Mises Stress calculation:

WEIGHTS 

Corresponding margins of safety:
for yield strength:

(pp19)
Radial displacement 
of circumference:
Change in height:

End Cap Tap Surface Bearing Stress: 
Diametral clearance (extrusion gap):
Pressure step:

Bearing Area: 

Extrusion gap (radial):

End Cap Force: 

Bearing Stress: 

Flange Shear Thru: 
Flange Shear Area: 

End Cap Force: 

Shear Stress: 

2) Thick walled cylinder calculations:
Table 32  Formulas for thick-walled vessels under internal and external loading
Case 1d
Cylindrical disk or shell; uniform external pressure in all directions, ends capped

Cylindrical disk or shell
Calculated dimensions:

Formulas for change in outer and inner radii, and change in length:
Outer radius:
Inner radius:
Length:  

Formulas  for normal stresses 
as a function of radial thickness, r:
radius range variable

Longitudinal stress: 
Circumferential stress: 
Radial stress: 
Shear stress @ b: 

Von Mises Stress calculation: 

Corresponding margins of safety: 
for yield strength:

for shear strength:

Based on Von Mises stress  
Based on Max. shear stress theory 0.5*Sy 

Diametral clearance (extrusion gap):
Pressure step:

Extrusion gap (radial):

C) Buckling calculation for cylinders:
Table 35 Formulas for elastic stability of plates and shells
Case 20a
Thin tube with closed ends under uniform external pressure, lateral and longitudinal; ends held circular
Thin tube with closed ends under uniform external pressure, lateral and longitudinal

Calculated dimensions:
Tube dimensions:
radius:

length:

Conditions:
For the expressions in the case to be valid, the following condition must be true (i.e., condition = 1):

Critical pressure:
The critical pressure for the initial buckling of a cylinder  is:
number of lobes:

critical pressure:

Margin of safety on buckling:

Where experimental values of pcritical are +/- 20% of pmin !!!

Table 24 Case 10a
Solid circular plate simply supported; uniformly distributed pressure from ro to a 
/??????????????
Solid circular plate
Uniformly distributed pressure from ro to a
(Roark Table 24, Case 10a:

Cylinder Base Thickness:	
If assumed simply supported:
CLOSED END CYLINDER BASE CALCS:  

Check stress in bottom:

Plate Constant:

Margin:

check small deflection assumption
(deflection < t/2)

END CAP DEFLECTION AND STRESSES:  
Table 24  Formulas for shear, moment and deflection of flat circular plates of constant thickness
This file corresponds to Cases 2a - 2d in Roark's Formulas for Stress and Strain.
Annular plate with uniformly distributed pressure q over the portion from ro to a; outer edge simply supported
Annular plate with a uniformly distributed pressure q over the portion from ro to a
Outer edge simply supported, inner edge free

Plate dimensions:
thickness:  

outer radius:

inner radius:

Shear modulus:

Plate Constant 

in*lbf / in  
in*lbf / in  

in*lbf / in

Transverse

check small deflection assumption
(deflection < t/2)

Should be < 1 ???

Margin:

Stresses in tabs:
For a conservative approach, assume the outer edge of the tabs are free and the inner edge fixed
A) Top tab: 
Table 24  Formulas for shear, moment and deflection of flat circular plates of constant thickness
Annular plate with uniform annular line load w at radius ro; outer edge free

Case 1l:
Outer edge free, inner edge fixed

Loading on tab:

Applied
Unit 
Load @
OD of
Case ??

Margin:

B) O-ring groove tab:
Table 24  Formulas for shear, moment and deflection of flat circular plates of constant thickness
Annular plate with uniform annular line load w at radius ro; outer edge free

Case 1l:
Outer edge free, inner edge fixed
For o-ring tab:

Loading on tab:

OK

Margin:

E) WEIGHT CALCULATION:
Cylinder:
cross sectional area:

internal volume:

external volume:

weight:

End Cap:
volume:
Tab:

Plug:

Cap:

Weight:

Total Weight: