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)
(pp22)
END CAP CLOSURE BOLT PATTERN / TENSILE STRESSES IN BOLTS    
SUMMARY  

1) Dimensional input:
ASSUME:  A fully flooded case at pressure q.  

Enter cylinder dimensions:
Thin Wall Calcs:
Is it thin wall (1=yes)?

Number of equally spaced bolts (n):  

Inside Diameter :
Outside Diameter:
O-ring surface diameter:
Case Length: 
Cylinder Base 
Thickness:	
Wall thickness:
Meridional (axial) stress:

Change in height:

Radial displacement 
of circumference:

Circumferential (hoop) stress:

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

Internal Area of End Cap (A.in):    

Margin of Safety:

Von Mises Stress:

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

Internal Case Volume (V.in):  

Axial Stress:

Change in OD (radius):

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

Change in ID (radius):

Force on the End Cap (F.in):  

Radial Stress:

Change in Length:

2) Material properties:
Shear Stress:

Von Mises Stress:

Enter material properties:
Material:
RP-VisiJet_M3_Crystal   
Tensile Stress Area of the Bolt (A.t):  

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 (Edges Free): 
?

?

Margin:

Coeff. of Thermal Expansion:  (zero is ok) 
Surface Temp.: 
Temp at depth "q": 

Closed End Cylinder Base Calcs (Edges Fixed): 

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
Tensile Stress in Bolt (σ.n):  

B) Formulas for deformation and stresses in cylinders:
(pp23)    
END CAP Circular Plate  

1) Thick walled cylinder calculations:
Pressure Load:  

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
Conn. Line Load:  

Combined Loading:   

Cylindrical disk or shell
Calculated dimensions:

MARGIN OF SAFETY: 

END CAP tabs: 
Formulas for change in outer and inner radii, and change in length:

Outer radius:
Inner radius:
Length:  
MARGIN OF SAFETY: 

O-RING tab: 

MARGIN OF SAFETY: 

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

CASE WEIGHTS 
CASE VOLUME CHANGE (@ temp & depth)   

Longitudinal stress: 
Circumferential stress: 
Radial stress: 
Shear stress @ b: 
Case + 2ea. Caps = 

Von Mises Stress calculation: 
(pp24)       
EndCap Flange/Case Bearing Stress:    Where the allowable bearing stress =   

Bearing Area: 

End Cap Force: 

Bearing Stress: 

Corresponding margins of safety: 
Margin of safey: 

for yield strength:

for shear strength:

EndCap Flange Shear Thru Stress 
Flange Shear Area: 

Based on Von Mises stress  
Based on Max. shear stress theory 0.5*Sy 
End Cap Force: 

Diametral clearance (extrusion gap):
Pressure step:

Shear Stress: 

Margin of Safety:  

Extrusion gap (radial):

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DTMB:  

2) Thin walled cylinder calculation:
Reference: Table 28  Formulas for membrane stresses and deformations in thin-walled pressure vessels
BUCKLING OF NEAR-PERFECT THICK-WALLED CIRULAR CYLINDERS UNDER HYDROSTATIC PRESSURE - ROSS  2009  
Cylindrical shell; uniform internal or external pressure (ends capped)
Case 1c
For Thick Wall Calcs:   R.mean/T.cyl_wall < 10    

Cylindrical shell
Calculated dimensions:
Eccentricity of the case (e<.16t)

Mean radius:

Cylinder Height:

Windenburg - Trilling Thinness Ratio:
Windenburg, D.F., Trilling, F., "Collapse by Instability of Thin Cylindrical Shells under External Pressure, Trans. ASME, 11, 819-25, 1934
Thin wall stresses:
Conditions:
For the expressions in the case to be valid, the following condition must be true (i.e., condition = 1):

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

Von Mises Stress calculation:

II  Ross PKD Factor:  [Ross 2009]  
Where:  

Corresponding margins of safety:
for yield strength:

Margin of Safety:    
Where:  

Radial displacement 
of circumference:

Change in height:

Diametral clearance (extrusion gap): ----------------------------------------------------------------------------------------------------------------------------------------------
PKD step:

Pressure step:

Extrusion gap (radial):

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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 !!!

Von Mises Stress @ Critical Buckling Pressure:   

Meridional (axial) stress:
Circumferential (hoop) stress:
For Thin Wall Calcs:   R.thin/T.cyl_wall > 10    

Windenburg - Trilling Thinness Ratio:
Windenburg, D.F., Trilling, F., "Collapse by Instability of Thin Cylindrical Shells under External Pressure, Trans. ASME, 11, 819-25, 1934
Von Mises Stress calculation:

PRESS. CASE BASE (FREE):  
Table 24 Case 10a
Solid circular plate simply supported; uniformly distributed pressure from ro to a 
II  Ross PKD Factor:  [Ross pp106-109]  
Where:  

Machined Cyl.  Fig.3.4 pp106   
Solid circular plate
Uniformly distributed pressure from ro to a
(Roark Table 24, Case 10a:

Cylinder Base Thickness:	
If assumed simply supported:
Margin of Safety:    
Where:  

CLOSED END CYLINDER BASE CALCS:  

Check stress in bottom:

Plate Constant:

PKD step:

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:

2. Volume Change of Case at Depth: 
Radius change due to press: 

Length change due to press:  

Radius change due to temp:  

Length change due to temp: 

Dished End Cap Spherical Segment Vol.:  

external volume of cyl. (at depth):  

external volume of cyl.  (initial):  

Pressure Case Volume Change V:  

Assume Bulk Modulus of CH2O = 325,000 psi.  

Compressed water vol. at depth:   

Compressed water vol. due to temp:  

Wet weight at depth: