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
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.
(pp02)
(pp21)
1) Dimensional input:
ASSUME:
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:
Internal Area of End Cap (A.in):
Von Mises Stress:
Plug Diameter:
Groove Width:
Groove Diameter:
Internal Case Volume (V.in):
End Cap OD:
End Cap Thickness:
Tab Thickness:
Connector Diameter:
Axial Stress:
Change in OD (radius):
Hoop Stress:
Change in ID (radius):
Force on the End Cap (F.in):
2) Material properties:
Radial Stress:
Change in Length:
6AL-4V Gr.5 Ti.
Tensile Stress Area of the Bolt (A.t):
Shear Stress:
Von Mises Stress:
Modulus of elasticity:
Material density:
Poisson's ratio:
Yield Stress:
Shear stress:
Water density:
Coeff. of Thermal Expansion:
Surface Temp.:
Temp at depth "q":
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):
for buckling:
for stress:
Tensile Stress in Bolt (σ.n):
(pp22)
Pressure Load:
Conn. Line Load:
Combined Loading:
MARGIN OF SAFETY:
Outer radius:
Inner radius:
Length:
MARGIN OF SAFETY:
radius range variable
CASE VOLUME CHANGE (@ temp & depth)
Case + 2ea. Caps =
(pp23)
EndCap Flange/Case Bearing Stress:
Bearing Area:
End Cap Force:
Bearing Stress:
Margin of safey:
for yield strength:
for shear strength:
EndCap Flange Shear Thru Stress
Flange Shear Area:
End Cap Force:
Pressure step:
Shear Stress:
Margin of Safety:
Extrusion gap (radial):
The pressure at which the extrusion gap (GAP thick ) is equal to the change in interior radius (Δb thick )
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DTMB:
BUCKLING OF NEAR-PERFECT
THICK-WALLED
CIRULAR CYLINDERS UNDER HYDROSTATIC PRESSURE - ROSS
For Thick Wall Calcs:
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
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:
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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Tube dimensions:
radius:
length:
For the expressions in the case to be valid, the following condition must be true (i.e., condition = 1):
The critical pressure for the initial buckling of a cylinder
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:
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:
Where:
Machined Cyl.
Solid circular plate
Uniformly distributed pressure from r o to a
(Roark Table 24, Case 10a:
Cylinder Base Thickness:
If assumed simply supported:
Margin of Safety:
Where:
Check stress in bottom:
Plate Constant:
PKD step:
Margin:
check small deflection assumption
(deflection < t/2)
Margin:
Superposition of pressure loading of the annulus and line loading of the connector/penetrator hole in the center.
Table 24
This file corresponds to Cases 2a - 2d in Roark's Formulas for Stress and Strain.
Annular plate with a uniformly distributed pressure q over the portion from r o to a
Outer edge simply supported, inner edge free
Plate dimensions:
thickness:
outer radius:
inner radius:
Shear modulus:
Margin:
check small deflection assumption
(deflection < t/2)
Should be < 1
CONNECTOR LINE LOAD:
Table 24
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 r o
Outer edge simply supported, inner edge free 1a
Area of Connector:
Force on the connector:
Line load on Conn. Hole (w):
By Superposition:
Margin:
For a conservative approach, assume the outer edge of the tabs are free and the inner edge fixed
Case 1l:
Loading on tab:
Margin:
Case 1l:
For o-ring tab:
Loading on tab:
Margin:
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.
Pressure Case Volume Change
Assume Bulk Modulus of CH2O = 325,000 psi.
Compressed water vol. at depth:
Compressed water vol. due to temp:
Wet weight at depth: