5. LA ESPERANZA COMO ACTITUD VITAL: CUATRO CUARTETOS
5.4. Little Gidding
What we would like to have is some method of quantifying the combined loads into a single value to compare with some simple strength or stress value for the material of the tube. For example, if Y is the yield stress determined from a uniaxial test and Ψ is the combined load, we might compare them thusly:
no yield yield Y
Y
> Ψ →
≤ Ψ → (6.1)
The only problem we face is how to convert the combined loads on the casing into a single value like Ψ. Fortunately this is what a yield criterion allows us to do, and for this course we will use a yield criterion called the von Mises yield criterion which works very well with metals such as steel. When we plot the von Mises criterion in principal stress space (a three dimensional plot where the coordinate axes are the principal stress components rather than a direction measure) it plots as a cylinder. The surface of the cylinder is called a yield surface. The central axis of the cylinder is along a line where all three principal stress components are equal, in other words a hydrostatic pressure or spherical stress. The radius of the cylinder is the value yield stress of the metal. Any value of Ψ that plots within the cylinder will not cause yield, but any value that plots on the surface or outside the surface of the cylinder will cause the metal to yield.
Figure 6 - 3. A von Mises yield surface in principal stress space.
The von Mises yield surface is a mathematical cylinder; it is not a tube or portion of a casing joint. This is not a picture of a tube. Do not confuse the von Mises yield surface with a tube.
That is straight forward enough. Now the only remaining question is how do we calculate Ψ ? Here is the formula in terms of principal stress components.
( ) ( ) ( )
1
2 2 2
1 2 2 3 3 1
1
2 σ σ σ σ σ σ
Ψ = − + − + −
2 (6.2)
Now what is Ψ called? Here we get into a technicality that deserves a little explanation.
I use the symbol Ψ and call it the yield indicator because that is as good a name and description for it as I know. In petroleum engineering applications it is often and called the “von Mises stress”. This is a somewhat unfortunate misnomer, because it is not a stress and calling it a stress can lead to a lot of confusion because the actual stresses can never cause the actual von Mises quantity to exceed the yield stress. We will save those technicalities for later, but for now we will just call it a yield indicator, but if you want to call it the von Mises stress then you are welcome to do so, but that term will not be
used in this manual and it never appears in any text on plasticity or continuum mechanics.
Here is another picture of a von Mises yield surface with the three principal stress components for a particular combined load plotted as point a .
Figure 6 - 4. A load that falls outside the yield surface.
Notice point a , which represents the combined load, is outside the yield surface. The yield indicator Ψ then is the scalar magnitude of a radius vector from the central axis of the yield surface to the load point. Remember that Y is the yield stress of the material and is the magnitude of the radius of the yield cylinder. So all we have to do is compare the magnitudes of Y and Ψ, and in this case it is easy to see that
Y < Ψ
so this load condition will cause yield of the material. It plots outside the yield surface.
If you can visualize Y as the radius of the yield surface cylinder and Ψ as the distance to the load point, then the whole idea of a yield surface becomes a simple picture. All you have to do is compare the magnitudes of Y to Ψ and decide which is larger.
One more important point concerns sign convention.
Tensile stress components are positive, and compressive stress components are negative.
While that is not absolutely necessary it is the standard in most disciplines of mechanics, and you can sometimes get strange results if you use something else. (Rock mechanics people for instance quite often use the opposite sign convention since most of the stresses in that field are compressive. But even the rock mechanics community often has to use the normal sign convention when doing complicated calculations because the typical rock mechanics convention can lead to negative displacement vectors in positive directions.)
The next most obvious step in this process is to answer the question, how do we calculate the principal stress components? After all our loads on casing are in terms of tension/compression, pressure, and possibly even torque if we are trying to rotate it.
When we work with tubes it is convenient (and often essential) that we work with a coordinate system that is suited to shape of a cylinder. We will use a right-hand cylindrical polar coordinate system which is standard for this type of work.
Figure 6 - 5. A cylindrical polar coordinate system shown with a tube.
Unlike the picture of the yield surface this picture is a tube or section of casing. The origin of the coordinate system lies on the longitudinal central axis of the tube. The radial coordinate extends from the origin of the coordinate system outward, the axial coordinate extends from the origin upward along the longitudinal central axis of the tube, and the tangential coordinate is measured as an angle counterclockwise from
some predetermined reference point. The coordinates are labeled r, θ , z . Now we need to define the stress components in this coordinate system, and we do so in the next picture.
Figure 6 - 6. Cylindrical coordinate stress components shown on the wall of a tube.
In this figure we have an axial stress component, σz , in the direction of the longitudinal axis of the tube, the tangential stress component, σ θ, tangent to the perimeter of the pipe wall, and the radial stress component, σ r , acting normal to the pipe wall. In the case where there is torsion in the tube there will also be a shear stress component, σ θr . There is a possibility of other shear stresses but we will not be considering those here as they are usually minor compared to the other stress components.
The axial stress is the load divided by the cross sectional area of the tube body.
(
2 2) (
42 2)
z
o i o i
P P P
A r r d d
σ = =π − =π − (6.3)
The axial stress component is the same at any point within the wall of the tube.
The radial and tangential stress components require a little more effort. The formulas for radial and tangential stress components in an elastic tube were derived by Lamé in 1833. Here are the full formulas which we will simplify later.
( )
Notice that both of these two formulas contain a term, r . That is the radius at the point of interest in the wall of the tube. Fortunately we do not care about the stress at different points within the tube wall; we only care about it where it is a maximum. As so happens, it is a maximum at one of the tube walls. Which one? Look at this picture of internally applied pressure.
Figure 6 - 7. A tube with internal pressure.
Assume we apply increasing pressure internally until the tube first begins to yield.
Does the yield occur first at the inner wall or the outer wall? Look at the next picture.
Figure 6 - 8. A tube with external pressure.
Now we apply increasing pressure only to the outer wall of the tube until the onset of yield. We again ask the question does the tube yield first at the inner wall or at the outer wall? The answer is not intuitively obvious – most people who do not already know the answer guess wrong.
Yielding due to pressure on a tube always occurs first at the inner wall regardless of whether the pressure is applied from the outside or the inside.
Let us look at two charts for a 7.00” OD x 6.633” ID tube where we have applied the Lamé equations for both internal and external pressures.
Internal Pressure
0 2000 4000 6000 8000 10000 12000
0 20000 40000 60000 80000 100000 120000 140000
Combined Stress (psi)
Internal Presure (psi)
Inner Wall Outer Wall
80000 psi yield
Figure 6 - 9. Internal pressure effects at the inner and outer wall of 7" OD casing.
Notice that when we apply internal pressure we reach yield at the inner wall at about 7200 psi where yield at the outer wall does not take place until the internal pressure reaches about 8300 psi. In the next figure observe the effects of external pressure.
External Pressure
0 20000 40000 60000 80000 100000 120000 140000
Combined Stress (psi)
External Pressure (psi)
Inner Wall Outer Wall
80000 psi yield
Figure 6 - 10. External pressure effects at the inner and outer wall of a 7" OD tube.
Notice in this figure that the externally applied pressure causes yield at the inner wall when it reaches a magnitude of 7000 psi, but it does not cause yield at the outer wall until it reaches 8000 psi.1
Like we said, it is not intuitive. But at least it simplifies our calculations. We substitute the inner radius, ri , in the above equations in place of r , do a little algebra, and get the following equations for the radial and tangential stress components at the inner wall of the tube.
r pi
σ = − (6.6)
and
1 In these examples the tubes are assumed to be fully elastic until yield occurs at the inner or outer wall to illustrate which occurs first. When an actual tube yields at the inner wall then the yielding will propagate from the inner wall to the outer wall internally as the pressure continues to increase and the pressure when the yielding occurs at the outer wall will be less that shown in the charts. In some cases with thick-walled cylinders the propagation will become unstable and the pressure required to yield the outer surface will be less than that required to yield the inner surface, however in those cases the initial yield will still always begin at the inner wall.
(
2 2)
2Why is the radial stress equal to the negative value of the internal pressure? That is because it is a compressive stress.
Remember, tensile stress components are positive and compressive stress components are negative.
The stress components of radial, tangential, and axial stress may or may not be principal stress components. If there is torsion in the pipe caused by rotation of the pipe in the presence of borehole friction the equation for calculating the maximum torsional shear stress is:
There are no conversion units in that equation. For typical oilfield units the radii or diameters would be in inches and the torque would be in lb-ft. That means you must multiply the torque by a conversion factor of 12 in/ft to get it into lb-in.
Another thing you may notice is that the shear stress due to torque is a maximum at the outer wall of the tube rather than the inner wall. This can lead to an inconsistency in our calculations if we combine the stresses at different points in the tube. We will address this again later and show all of the equations at both the inner wall and at the outer wall.
In the absence of torsion the radial, tangential, and axial stress components are the principal stress components and can be substituted directly into Equation (6.2).
If there is a torsional shear component present then the radial, tangential, and axial stress components are not the principal stress components and we must determine the principal stress components before testing for yield. Here are the formulas for the principal stress components when torsion is present.
2
Before we go further let us look at a simple example.
Suppose we have a point in a casing string where the internal pressure is 6000 psi, the external pressure is 4000 psi, and the tension in the pipe at this point is 100,000 lb. Our casing is 7” OD, 23 lb/ft, K55. It has an internal diameter of 6.366 in. Will the pipe yield under this load?
Since there is no torsional stress these three stress components are also the principal stress components, and we can substitute them directly into Equation (6.2) to calculate the yield indicator.
( ) ( ) ( )
1 15025 17129 17129 6000 6000 15025 2
Therefore we see that the yield indicator is well below the yield stress of the pipe which is 55,000 psi.
?
55000 22152 no yield Y> Ψ
> →
This is a fairly simple example. In a later example we will look at more complex loading.