Spherical pressure vessel
Spherical pressure vessel
(OP)
what is the formula for calculating stress-intensity factor for thin-walled spherical pressure vessel with axial or circumferential through crack
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Spherical pressure vessel
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RE: Spherical pressure vessel
Tada-Paris also has one, for through crack length '2a' in a sphere: KI=sigma*sqrt(pi*a)*F(lambda)
where KI==Mode I Stress Intensity Factor
sigma==0.5*p*R/t (p=internal pressure, R=radius, t=
shell thickness)
F(lambda)=sqrt(1+1.41*lambda^2+0.04*lambda^3)
lambda==a/sqrt(R*t)
Restrictions? 0<lambda<=3
Hope you understand the clumsy way I have written this.
How is an axial crack defined in a sphere?
RE: Spherical pressure vessel
RE: Spherical pressure vessel
RE: Spherical pressure vessel
also, for bulging effects in a fuselage i use rooke and cartwright, fig 197, for which my curve fit is ...
?fig197 = Gm+Gb
= [0.725+0.85*(a/(Rt)^0.5)] + [0.4-0.184*(abs(a/(Rt)^0.5-1.55))^1.77]
This curve fit is accurate for 0.5 < a/(Rt)^0.5 < 3.1.
For a/(Rt)^0.5 < 0.5, assume ?fig197 = 1.0
For a/(Rt)^0.5 > 3.1, ?fig197 = Gm-Gb
= [0.725+0.85*(a/(Rt)^0.5)] - [0.4-0.184*(abs(a/(Rt)^0.5-1.55))^1.77]
RE: Spherical pressure vessel
By my equilibrium calcs, the stress in the shell is pR/(2t), and this is the only stress. My mechanics book calls this the 'circumferential' or 'hoop' stress.
RE: Spherical pressure vessel
even tho' the OP started talking about a spherical pressure vessel, he's changed that to a cyclindrical one. the solution i gave was for a longitudinal crack in a cyclindrical pressue vessel. Is there a different solution in Tada-Paris for this ?
RE: Spherical pressure vessel
For a single axial through crack (that is, crack oriented so it propagates in the axial direction):
a=half length of crack
sigma=p*R/t
lambda=a/sqrt(R*t)
KI=sigma*sqrt(pi*a)*F(lambda)
F(lambda)=sqrt(1+1.25*lambda^2) if 0<lambda<=1
or F(lambda)=0.6+0.9*lambda if 1<=lambda<=5
For a single circumferential through crack:
a=half length of crack
sigma=0.5*p*R/t 'Note difference in sigma definition!
lambda=a/sqrt(R*t)
KI=sigma*sqrt(pi*a)*F(lambda)
F(lambda)=sqrt(1+0.3225*lambda^2) if 0<lambda<=1
or F(lambda)=0.9+0.25*lambda if 1<=lambda<=5
Note: I have validated quite a few FE solutions against these Tada-Paris solutions for comparisons, but NOT for these particular sphere and cylinder solutions, so I cannot tell you how accurate these are. Tada Paris claims 1% on the KI
RE: Spherical pressure vessel
Crack is 2c (surface) x a (into thickness), located on internal or external surface of a pressurized pipe. Crack grows axially, so that the hoop stress is the stress that opens it.
beta(a/c,a/t,theta)=0.97*(M1+M2*(a/t)^2+M3*(a/t)^4)*g*fphi*fc*fi
M1=1.13-0.09*a/c
M2=-0.54+0.89/(0.2+a/c)
M3=0.5-(1/(0.65+a/c))+14*(1-a/c)^24
g=1+(0.1+0.35*(a/t)^2)*(1-sin(theta))^2
fphi=((a/c)^2*cos(theta)^2+sin(theta)^2)^(1/4)
((cos(theta)^2 is cos(theta)*cos(theta))
theta here I think is still the elliptical angle
typically called phi, not theta.
fi=1 (internal crack) fi=1.1 (external crack)
fc=((1+k^2)/(1-k^2)+1-0.5*sqrt(a/t))*(t/(D/2-t))
where k=1-2t/D
t==thickness of cylinder, D=outside diameter of cylinder.
sigma=p*(D-2t)/2t
K1=beta*sigma*sqrt(pi*a)/PHI, PHI is equal to:
if (a/c<=1), PHI=1+1.464*(a/c)^1.65
if (a/c>1) PHI=1.464*(c/a)^1.65
(Love those functions, M3, with the 24th power!)
RE: Spherical pressure vessel
RE: Spherical pressure vessel
Since R and t are the two parameters used to define the shell (be it spherical or cylindrical), what difference does it make? As far as I can tell, my version of Tada-Paris has only through crack solutions. And, almost all the stresses in the Stress Intensity Factor solutions are defined pR/t, which you may recall from mechanics of materials is the thin shell approximation for the hoop stress in a pressurized cylinder. Unfortunately, I do not know the history of these solutions, but if I had to guess, I would guess that these Tada Paris solutions are also thin shell approximations. which if true, should mean that any FEA computation you use to compare to the Tada Paris solution should be for a 'thin' shell.
If I had the time, I would just go get some of the references, such as Folias 1965 "An Axial Crack in a pressurized Cylindrical Shell," Int. Journal of Fracture Mechanics, Vol. 1, pp. 104-113, 1965
RE: Spherical pressure vessel
RE: Spherical pressure vessel
RE: Spherical pressure vessel
other question mark sec{(?/2)....I take it those are PI=3.1415...?
Admittedly, not an expert on LBB (leak before burst) concepts. However, I have a very nice book by Farahmand, chapter 3.7.5, that discusses LBB. You know the through crack solution, you know the part through crack solution, you know the material. Farahmand then says the LBB criteria are:
K(part through)=K(through)--that is, the part through crack grows until the 'a' (depth) crack length breaks through to the other side. Further criteria:
K(part through)<K1e
and
K(through)<Kc or K1c
K1e is the Part through fracture toughness (ASTM standard E740), Kc and K1c are the usual plane stress and plane strain fracture toughness. These last two conditions are stability conditions--is the K low enough that that crack growth is stable?
RE: Spherical pressure vessel
RE: Spherical pressure vessel
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Chapters: 1-Intro to Fatigue and Fracture Mechanics
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5-Fracture Control Program and NDI
6-The Fracture Mechanics of Ductile Metals Theory
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