PRESSURE VESSEL END CAP CALCULATIONS
PROJECT:  BattTest Tube
DATE:   Oct 24, 2013 
ANALYST:        J.E.  
Revisions: (last rev date 10/30/12) 

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)Modified to End Cap Calcs only.
8)Added Connector Line Load to End Cap Deflection/Stress calculation (Table 24 Case 1a).
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 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)
(PP 13)
INPUT VALUES: 

1) Dimensional input:
Enter end cap and gland dimensions  (O-Ring #2-xxx):
Enter end cap and gland dimensions  (O-Ring #2-xxx): 
O-ring surface diameter ? (ASSUMED !!!):
Plug Diameter:
Groove Width?:
Groove Diameter?:
Groove Base Thickness?:
O-ring surface diameter:
Plug Diameter:
Groove Width:
Groove Diameter:
Groove Base Thickness:

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

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

ASSUMED: 

2) Material properties:

Enter material properties:
Material:
Steel (A36)
Modulus of elasticity:
Material density:
Poisson's ratio:
Yield Stress:
Shear stress:
Water density:
Summary: 
END CAP Calcs  

CLOSED END CYLINDER BASE (FREE):  
Coeff. of Thermal Expansion:  (zero is ok) 
Surface Temp.: 
Temp at depth "q": 

MARGIN OF SAFETY: 

3) Design depth and resulting pressure:
CLOSED END CYLINDER BASE (FIXED):  

Number of pressure steps:

Pressure:

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

MARGIN OF SAFETY: 

PRESS. CASE BASE (FREE):  
Table 24 Case 10a
Solid circular plate simply supported; uniformly distributed pressure from ro to a 
(pp03)
END CAP ANNULAR PLATE:  
(PP14)
Pressure Load:  

Solid circular plate
Uniformly distributed pressure from ro to a
(Roark Table 24, Case 10a: 
Simply Supported)

Conn. Line Load:  

Combined Loading:   

MARGIN OF SAFETY: 

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

Check stress in bottom:

MARGIN OF SAFETY: 

O-RING tab: 
Plate Constant:

MARGIN OF SAFETY: 

WEIGHTS 

Margin:

check small deflection assumption
(deflection < t/2)

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

PRESS. CASE BASE (FIXED):  
(pp15)
Bearing Area: 

Table 24 Case 10b
Solid circular plate Fixed support; uniformly distributed pressure from ro to a 
(Roark Table 24, Case 10b: 
Fixed Edge)
End Cap Force: 

Bearing Stress: 

Margin of safey: 

EndCap Flange Shear Thru Stress 

Flange Shear Area: 

End Cap Force: 

Margin:

Shear Stress: 

Margin of Safety:  

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 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:

Boundry Values:  

in*lbf / in  
in*lbf / in  

Transverse Bending moment / stress

in*lbf / in
check small deflection assumption
(deflection < t/2)

Should be < |1| 

Margin:

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

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:

OK

Margin:

E) WEIGHT CALCULATION:
End Cap:
volume:
Tab:

Plug:

Cap:

Weight:

Total Weight: