Revisions: (last rev date 1/23/08)
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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 deformations, stresses and elastic buckling failure in hemispherical end caps,
C) Weight & Displacement calculation
D) 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)
(pp14)
1) Dimensional input:
Hemisphere Cap OD:
Hemisphere Cap ID:
End Dome Thickness:
Thin Wall Condition (i.e., condition = 1):
Meridional stress:
Circumferential stress:
Tab OD:
Tab Thickness:
Radial displacement of circumference in R2 direction:
Von Mises Stress:
2) Material properties:
MARGIN OF SAFETY:
for yield strength:
Titanium Alloy 6AL-4V
Modulus of elasticity:
Material density:
Poisson's ratio:
Yield Stress:
Shear stress:
Water density:
Critical Buckling pressure:
MARGIN OF SAFETY:
Hemisphere CAP Thick Wall Calcs:
Coeff. of Thermal Expansion:
Surface Temp.:
Temp at depth "q":
Meridional stress:
Circumferential stress:
Radial stress:
3) Design depth and resulting pressure:
Maximum shear stress (at inner radius, r=b):
Number of pressure steps:
Pressure:
for buckling:
for stress:
(pp20)
Spherical shell
Uniform internal or external pressure, q (force per unit area), tangential edge support
for yield strength:
for shear strength:
DOME VOLUME CHANGE
DOME Disp CHANGE (in LB.)
Shell thickness:
Angle from centerline to edge:
Mean radius:
Note
:
Distance from centerline to edge measured perpendicular to centerline:
Height of section:
Meridional stress:
Circumferential stress:
Radial displacement of circumference in R direction:
Radial displacement of circumference in R2 direction:
Change in height:
Von Mises Stress calculation:
Corresponding margins of safety:
for yield strength:
Table 35 Formulas for elastic stability of plates and shells
Thin sphere under uniform external pressure
MACHINED NEAR-PERFECT HEMISPHERICAL SHELLS (in Elastic Buckling Range)
DTMB Report #1854 Dadley.
Assumptions :
Thin wall Shell:
Sphere dimensions:
radius:
Elastic Yield
< >
thickness:
Definition of Elastic Buckling Range !!!!!
:
Empirical Elastic Buckling Pressure for Machined Spherical Shells:
Critical pressure:
For ideal case,
However, the probable actual minimum value of q' is
Equilibrium Yield Pressure:
(Presure at which the material Yield stress is reached) ??
Margin of safety on buckling:
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IN-ELASTIC BUCKLING RANGE:
Table 3
2
Spherical
Outer radius:
Inner radius:
Uniform external pressure:
Formulas for change in outer and inner radii, and change in length:
Outer radius:
Inner radius:
(See Ref. 3)
radius range variable
Secant Modulus:
point of interest along range
Tangent Modulus:
Meridional stress:
Empirical In-elastic Buckling pressure:
Circumferential stress:
Radial stress:
Maximum shear stress (at inner radius, r=b):
for yield strength:
for shear strength:
1. Case Weight Calc.:
Spherical End Caps (2 ea.forming a sphere:
internal volume:
external volume:
weight sphere:
Tab (Flange) Vol:
Weight (dome):
Total Weight:
2. Volume Change of Case at Depth:
Spherical Vol. change due to Pressure:
Hemisphere radius change due to press.:
NOTE:
external volume of Dome (at depth):
Spherical volume change due to Temp change from depth zero to Depth:
Sphere radius Condition (Thick or Thin):
Sphere Volume Change
Sphere Displacement Change at depth:
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Sea Water Displacement CHANGE due to Pressure (in LB.)
Assume Bulk Modulus of CH2O = 325,000 psi.
Compressed water vol. at depth:
Compreseed water weight change at depth:
Sea Water Displacement CHANGE due to Temp. (in LB.)
TOTAL Sea Water affected Displacement change:
?--------------------------------------------------------ROSS. KAO--------------------------------------------------------------?
(R.2 Mean Radius)
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