PRESSURE VESSEL CALCULATIONS
PROJECT:  CAMTAG Press. Case 
DATE:   Jan 26, 2014 
ANALYST:        J.E.  
Revisions: (last rev date 1/24/08) 

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
7) Added end cap tab bearing stress and tab shear thru.
8)Added Connector Line Load to End Cap Deflection/Stress calculation (Table 24 Case 1a).
9) Added Case Vol. change and CH2O Bulk Modulus
10) Added Thermal Dimension Chg (Radial & Length) @ depth.
11)Added Von Mises stress @ Critical Buckling Pressure.
12)Added Fixed case End Cap (Fixed Base) calc.
13)For Thin Wall calc's: Added λ from Windenburg & Trilling, Added PKD of p.min, from Ross 2008.
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 (Sixth Edition) 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 & displacement 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)
1) Dimensional input:
Enter cylinder dimensions:
Inside Diameter :
Outside Diameter:
O-ring surface diameter:
Case Length: 
Cylinder Base 
Thickness:	
Wall thickness:

Enter end cap and gland dimensions  (O-Ring #2-118)  

Plug Diameter:
Groove Width:
Groove Diameter:
Groove Base Thickness:

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

2) Material properties:
Enter material properties:
Material:
RP-VisiJet_M3_Crystal   
Modulus of elasticity:
Material density:
Poisson's ratio:
Yield Stress:
Water density:

3) Design depth and resulting pressure:

Number of pressure steps:

Pressure:

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

for stress:

B) Formulas for deformation and stresses in cylinders:

1) 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: 

Diametral clearance (extrusion gap):
Pressure step:

Extrusion gap (radial):

The pressure at which the extrusion gap (GAPthick) is equal to the change in interior radius (Δbthick)

PRESS. CASE BASE (FREE):  
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)

PRESS. CASE BASE (FIXED):  
Table 24 Case 10b
Solid circular plate Fixed support; uniformly distributed pressure from ro to a 
(Roark Table 24, Case 10b: 
Fixed Edge)

Margin:

END CAP DEFLECTION AND STRESSES:  
Superposition of pressure loading of the annulus and line loading of the connector/penetrator hole in the center.
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  2a

Plate dimensions:
thickness:  

outer radius:

inner radius:

Shear modulus:

Plate Constant 

in*lbf / in  
in*lbf / in  

in*lbf / in

Transverse Bending moment / stress

Margin:

check small deflection assumption
(deflection < t/2)
Should be < 1 

CONNECTOR LINE LOAD:  
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 simply supported
This file corresponds to Cases 1a - 1d in Roark's Formulas for Stress and Strain.
Annular plate with a uniform annular line load w at a radius ro
Outer edge simply supported, inner edge free 1a

Area of Connector:  

Force on the connector:  

Line load on Conn. Hole (w):  

Transverse Bending moment / stress

in*lbf / in
By Superposition:   

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:

Margin:

E) WEIGHT & DISPLACEMENT CHANGE CALCULATION:
1. Case Weight Calc.:  
Cylinder:
cross sectional area:

internal volume:

external volume:

weight:

End Cap:
volume:
Tab:

Plug:

Cap:

Weight:

Total Weight:

(pp17)
SUMMARY  

Thick Wall Calcs
Axial Stress:

Change in OD (radius):

Hoop Stress:

Change in ID (radius):

Radial Stress:

Change in Length:

Shear Stress:

Closed End Cylinder Base Calcs (Edges Free): 

Margin:

Closed End Cylinder Base Calcs (Edges Fixed): 

END CAP Calcs
(pp18)    
END CAP Circular Plate  
Pressure Load:  

Conn. Line Load:  

Combined Loading:   

MARGIN OF SAFETY: 

END CAP tabs: 

MARGIN OF SAFETY: 

O-RING tab: 

MARGIN OF SAFETY: 

CASE WEIGHTS 

Case + 2ea. Caps = 

(pp19)       
EndCap Flange/Case Bearing Stress:    Where the allowable bearing stress =   

Bearing Area: 

End Cap Force: 

Bearing Stress: 

Margin of safey: 

EndCap Flange Shear Thru Stress 
Flange Shear Area: 

End Cap Force: 

Shear Stress: 

Margin of Safety: