Revisions: (last rev date 8/4/99)
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) 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 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.
(pp02)
(pp11)
1) Dimensional input:
Axial Stress:
Change in OD (radius):
Inside Diameter :
Outside Diameter:
Case Length:
Cylinder Base Thickness:
Wall thickness:
Hoop Stress:
Radial Stress:
Change in Length:
Shear Stress:
Von Mises Stress:
2) Material properties:
Modulus of elasticity:
Material density:
Poisson's ratio:
Yield Stress:
Shear stress:
Water density:
Is it thin wall (1=yes)?
Coeff. of Thermal Expansion:
Surface Temp.:
Temp at depth "q":
Meridional (axial) stress:
Change in height:
Radial displacement
of circumference:
Circumferential (hoop) stress:
3) Design depth and resulting pressure:
Number of pressure steps:
Pressure:
for buckling:
for stress:
(pp03)
(pp12)
CASE VOLUME CHANGE
Outer radius:
Inner radius:
radius range variable
Longitudinal stress:
Circumferential stress:
Radial stress:
Shear stress @ b:
Von Mises Stress calculation:
(pp05)
Corresponding margins of safety:
for yield strength:
for shear strength:
Cylinder Height:
Mean radius:
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:
Corresponding margins of safety:
for yield strength:
Radial displacement
of circumference:
Change in height:
(pp07)
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:
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
Solid circular plate
Uniformly distributed pressure from r o to a
(Roark Table 24, Case 10a:
Cylinder Base Thickness:
If assumed simply supported:
Check stress in bottom:
Where:
Machined Cyl.
Plate Constant:
Margin of Safety:
Where:
Margin:
check small deflection assumption
(deflection < t/2)
(pp09)
PKD step:
Cylinder:
cross sectional area:
internal volume:
external volume:
PKD.m
Lambda inverse
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:
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 density at depth:
Compressed water vol. due to temp:
Total density change:
Wet weight at depth: