Saint-Venant Principal

[dutchguy]

Dear all,

I am looking for a good reference to a document, where a description can be found what the allowed minimum distance is between two damages (eg two holes).

To determine the distance I have used the Saint-Venant principal.

This principal describes following:

The applied distance should be twice the distance where the peak stress on the edge of the hole (a stress factor times the nominal stress) is decreased to a level equal to the nominal stress (EtE = edge to edge).

I have assumed that the distance can be taken 3D minimal (where D = the diameter of the damage).

I hope to see a quick response,

Yours

Andre

 

[RPstress]

If it's more background on Saint Venant's principal that you're after, then I think there's a reasonable description in "Theory of Elasticity", by Timoshenko and Goodier.

If it's a more general statement of permissible hole separations (be they deliberate penetrations or damage), then that tends to be manufacturer and product specific.

One general way is to take the elastic stress contration distribution, also available in Timoshenko and Goodier or Peterson, and demonstrate that the stress concentration has dropped to some acceptably low value at some distance away. For a round hole in an infinite isotropic plate, the peak tangential stress drops off with stress/gross_stress = 0.5*[2 + a^2/r^2 - 3*a^4/r^4], where 'r' is the distance away from the hole centre that 'stress' is being calculated, and 'a' is hole radius. If the plate is composite then Barbero ("Introduction to Composite Materials Design") has good coverage of stress concentration distribution.

 

[edbgtr]

RPstress

I think you need a (+) instead of a (-) in front of the third term of the function, otherwise you do not get an SCF of 3 next to the hole. Using this function, for all intents and purposes, the SCF dies away to unity at 2.86 radii from the centre of the hole.

Any comments from the FEA community on the accuracy of this function?

Ed

 

[prost]

For an infinite plate with a centered open hole, (2D) plane stress or strain, it appears from the derivation that the function in Timoshenko solves the linear elasticity differential equations and boundary conditions exactly. If you are doing this centered hole problem with FEA, the biggest difference between your results and the Timoshenko equation is the finite boundary you impose in your FEA--your FEA model is always 'finite' in extent, unless you have some kind of element that uses basis functions that assume the displacements go to zero at one of the edges, therefore your FEA model will always sense the finite boundary. The trick is to determine how far away the boundaries have to be in order for your stresses near the hole not be influenced by the boundaries. By moving the boundaries farther and farther away, and tracking a relevant engineering quantity such as max. stress at the hole (the hole's stress concentration factor, that is), you can determine from the FEA model how far away 'infinity' is. Normally that's about 3 times hole diameter.

If you are using h-element software such as ANSYS, etc. in which you don't bother to check numerical convergence by solving progressively denser meshes, then that could be a source of error too, error being defined as [Kt(FEA)-Kt(Timoshenko)]/Kt(Timoshenko). From the equation, you know that the Kt(Timoshenko) is 3.0 of course.

 

[RPstress]

edbgtr: re +/-, quite correct. The original formula I used was

tangential_stress_at_theta/gross_stress = 1/2*(1+a^2/r^2) - 1/2*(1+3*a^4/r^4)*cos(2*theta)

where theta is the angle around from in line with the stress, and so at 90 degrees, where the stress is max, cos(2*theta) is -1. I missed the varibility of theta off for simplicity's sake and thereby cocked it up...

If anyone's interested the radial and shear stress components also have nice simple formulas:

radial_stress_at_theta/gross_stress = 1/2*(1-a^2/r^2) + 1/2*(1-4*a^4/r^2+3*a^4/r^4)*cos(2*theta)
shear_stress_at_theta/gross_stress = -1/2*(1+2*a^4/r^2-3*a^4/r^4)*sin(2*theta)

-RP.

 

[dutchguy]

All,

Thank you for your response on this topic.

I assume these formulas are only applicable to holes and dents.

Any suggestions in case of a scratch ?

Andre

 

[prost]

Couldn't you idealize the scratch as an edge notch? From Peterson's, Ktn==s(max)/s(nom), s(nom) is the average tensile stress at the notch section. If the notch of radius 'r' is semicircular, then
Ktn=C1+C2*(r/H)+C3*(r/H)^2+C4*(r/H)^3
H is the gross section width.
C1=3.055, C2=-8.871, C3=14.036, C4=-7.219.

if the notch is 'u-shaped', then C1, etc. are defined relative to (t/r), where 't' is the depth of the notch, all the way to the bottom of the notch from the edge surface.
(Also, Ktn replaces 'r' with 't' for u-shaped notch).

 

[dutchguy]

All,

Sorry not having mentioned this before, but does the mentioned methods above also apply to composite structures ?

Andre

 

[prost]

My posting was for isotropic, linear elastic material. Composites is an entirely different matter; only a few analytic solutions for special geometry/loads have been derived, to the best of my knowledge.

 

[RPstress]

For holes in composite see Barbero (op cit), Hoskin and Baker's "Composite Materials for Aircraft Structures" or (I think) Keith Armstrong and Richard Barrett's "Care and Repair of Advanced Composite Structures".

The analysis methods usually center around finding a stress concentration from
Kt = 1 + sqroot( [2*sqroot(Ex/Ey) - nuxy] + Ex/Gxy )
where x is the bypass load direction (alternatively the laminate's stiffness matrix terms may be used in a slightly more comlicated formula). For a hole in material of limited width the isotropic finite strip width correction factor is applied, and considerable complication ensues.

Usually a dent can be conservatively considered as a hole, provided NDT can come up a damage size (internal delaminations spread out from the dent). However, for important structures, compression after impact strength is required.

For scratches, usually the affected plies are simply considered ineffective over the length of the scratch. If it's a bad scratch affecting more than one or maybe two plies (more properly called a gouge) then it may be necessary to consider it as a large hole. In metal this sort of thing often requires a crack growth and fracture analysis. In composite it depends on the nature of the structure, but generally speaking fracture mechanics approaches are not used. Scratches affecting only one ply are often considered acceptable without repair. Where in metal the scratch would mandatorily be blended out, in composite that might do more harm than good. A bad scratch or gouge in composite will be repaired. The repair method depends on the nature of the structure and the regulatory environment.

The whole area of damage tolerance of composites is quite complicated.

More specific advice probably depends on more specific details of your situation.

Acquisition of one of the above references is recommended, though there may well be others as good as or better.

-RP.