Understanding Fatigue in Metal

Intergraph CAS

Ray Delaforce

What is fatigue?

Simply put, fatigue is steady crack growth. The professionals call this Incremental Crack

Growth. This raises another question: What constitutes a crack? A tear in a sheet of paper is

really a crack, but other examples are not so obvious. Consider this piece of metal that has two

thicknesses:

In fact, any abrupt change in section can be regarded as a crack. In the region of the crotch of

the crack or a change in section, the stresses are very high. These stresses are in the plastic

region that go beyond the yield stress of the material.

Let us now consider the force in the upper illustration (above). The force can be cyclic. In other

words, the force can be tensile one moment, and compressive the next. We can illustrate this

with a time curve in the form of a sine wave like this:

We are going to come back to the stress SA in a moment, but first consider what crack growth

actually is.

Crack Growth

Every time the force goes from compression to tension, if there is a little crack somewhere in the

structure, it grows progressively little by little as shown in this illustration of crack growth going

from left to right:

Ultimately the structure fails. When we design a structure, we don’t want it to fail, so we have to

understand the nature of fatigue, and how to prevent failure.

Number of Cycles and Stress Intensity

Looking again at the cycles in the above illustration, we can conclude that if we increase the

stress intensity SA, the number of cycles to failure diminishes. This intuitive reasoning turns out

to be correct. In laboratory tests where specimens were subjected to cyclic loading, a correlation

between stress intensity and cycles to failure was found. If the stress intensity is plotted against

the number of cycles on Log-Log paper, a remarkable result occurs. Here is such a plot:

If we enter the vertical axis with the stress SA, we can obtain the number of cycles N to

destruction. However, if the stress falls below the endurance limit, the structure could

experience an infinite number of cycles without failure. We really need to examine what this

stress SA really is. Below the endurance limit, the stress does not exceed the yield point of the

material.

The Stress Intensity SA

Let us visit something that is familiar to the engineer. We are referring to the Stress-Strain

curve. It is worthy of close scrutiny, because we can learn some interesting things from it. Here

it is:

The first thing to notice is that fatigue failure will not occur if the stress intensity is below the

yield point. This is very important, because in pressure vessel design, we keep all allowable

general primary membrane stresses well below this point. Look at point A on the above

diagram, which is in the Plastic Region. This is an area where fatigue (crack growth) can occur.

The actual stress is on the plastic line. However, if we extend the elastic line to point B, we have

really considered this point as a strain denoted by the point C. The point B, is the stress intensity

SA we have been discussing so far. The stress SA does not really exist, because the actual stress

follows the plastic line at A. However this ‘virtual stress’ is used in the analysis of fatigue.

Really, we are considering the strain at C rather than an actual stress. However, this is the

convention used by engineers.

The Practical Application of Fatigue Analysis

In pressure vessel design certain codes require that a fatigue analysis be carried out to ensure a

reasonable service life. We shall look at the ASME Section VIII, Division 2004 edition code to

see how this is done. Let us assume we have already determined the stress intensity SA. In that

code there is a fatigue graph that looks like this:

Suppose the stress intensity SA is 100 000 psi. An arrow is drawn horizontally from an SA value

of 100 000 psi (690 MPa) and then another arrow is drawn vertically down to find the maximum

number of allowable cycles. From the above graph, the maximum number of cycles is about 900.

It should be immediately apparent that SA exceeds the Ultimate Tensile Strength of a material

such as SA-560 Grade 70 whose UTS is only 70 000 psi (482 MPa). This is because we are

considering the virtual stress which is derived from the actual strain in the plastic region.

A Further look at the Stress Intensity SA

A lot of work has been done in the past with the investigation of cracks by Inglis and Griffith

(notably for his pioneering work laying the foundation for fracture mechanics). Inglis used the

theory of elasticity to investigate the theoretical stress that exists at the location of the crotch of

the crack. To put the analysis on a mathematical footing, he substituted an elliptical hole in a

material that is subject to a tensile force to represent the crack. His model looked like this:

Without the elliptical hole, the stress in the plate would be S which is simply Force/Area.

However, adjacent to the hole the stress would increases to SA as shown above. Now, he

considered the depth of the crack to be half the major axis of the ellipse a, and the radius at the

tip of the crack to r. The ratio SA/S is known as the Stress Concentration Factor (scf). He found

he could find the scf if the dimension a and r are known.

He developed this equation for the stress concentration factor:

In the case of a round hole, a and r are equal, so the scf = 3 according to this equation. In

ASME Section VIII, Division 2, the stress concentration factor is called a Pressure Index. Here

is an illustration of the stress distribution on a nozzle (which is essentially a hole):

This illustration shows the stresses in the shell, and the nozzle. The stresses are:

1. σ t The tangential stress

2. σ n The hoop stress

3. σ r The radial stress

These stresses have different concentration factors. The stress concentration factors (called

Pressure Indices in ASME) are shown here:

You notice that the worst concentration factor is 3.3, which is very close to what was derived

using the Inglis equation.

The value of SA in the case of the nozzle (per ASME VIII Section VIII, Division 2)

In the case of ASME Section VIII, Division 2, the size of the nozzle is immaterial. If you look

again at the Inglis equation, in the case of the round hole, the scf was 3, irrespective of the size of

the hole. Let us consider an example of how the fatigue analysis is performed. We use the

example of a nozzle installed in a cylindrical shell.

The first things we need to determine are the Pressure Range, and the Number of cycles we want

the vessel to experience during its lifetime. So let us say the following:

So we have a hoop stress in the cylinder of 16120 psi. From the Pressure Index table from

ASME Section VIII, Division 2, we have stress concentration factor of 3.3. So can now compute

the stress in the critical part of the shell adjacent to the nozzle as follows:

This is the stress Range! If you look again at the cycle graph we drew, you will see the stress we

require is the amplitude (SA), which is half the stress range:

So SA is 53196 / 2 = 26598 psi. We enter the fatigue graph at the left hand side at 26598 psi and

read off the maximum allowable number of cycles as follows:

The maximum allowable cycles in out example is about 31 000 cycle. We now built a little table

like this – Supposing the vessel must sustain 25 000 cycles:

Stress Intensity N Cycle Max Allow Cycles Damage Factor

6598 psi 25 000 31 000 0.8065

The damage factor is simply the required number of cycles divided by the allowable number of

cycles: 25000 / 31000 = 0.8065. The damage factor must not exceed 1.0.

Here is a typical fatigue report from PV Elite:

Hoop stress - Unity Pressure [S] psig

As shown in example G-107 of ASME VIII Division 2:

= P * R/t + P/2

= 200.000 * 30.000 / 0.375 + 200.000 /2 = 16100.0000 psi

Adjust for Young's Modulus Ratio: = S * Emod / EmodCurve

= 16100.000 * 30000000 / 29664706 = 16281.97 psi

Stress Intensity Amplitude based on Pressure Index Sa: = Sadjust * 3.3 / 2.0 = 16281.97 * 3.3/2.0 = 26865.26 psi

Case 1 HoopStress: Adjusted below per above Index table (per Article 4.612):

------------------------------------------------------------------------------

sn 16281.975 25237.061 9769.186 8140.987 17096.072

st 16281.975 -1628.198 8140.987 -1628.198 21166.566

sr 16281.975 -101.762 0.000 -101.762 0.000

sint 16281.975 26865.258 9769.186 9769.186 21166.566

Example: for the last case: Alternating stress = max stress / 2 = 26865.258 / 2 = 13432.629 psi

Case StressIntens N cycles Nmax cycles Damage Factor

---------------------------------------------------------------

1 26865.258 25000. 0.3104E+05 0.805

-----

Total: Damage Factor: 0.805

Fatigue Analysis Passed: Damage Factor < 1.00

Per AD-160 Condition A: Fatigue analysis WAS required for this Component!

Max Stress Intensity is taken as 3.0 Sm which is 60000.000 psi

Given Number of Cycles is 25000.

Allowable Num of Cycles without a Fatigue Analysis is 2456.

Normally we might have several ranges like this:

From To Range Required cycle

100 psi 350 psi 250 psi 10 000

0 psi 150 psi 150 psi 2 000

50 psi 75 psi 25 psi 300 000

The report would look like this:

Case StressIntens N cycles Nmax cycles Damage Factor

---------------------------------------------------------------

1 33581.574 10000. 0.1523E+05 0.656

2 20148.943 2000. 0.9639E+05 0.021

3 3358.157 300000. 0.1000E+12 0.000

-----

Total: Damage Factor: 0.677

Fatigue Analysis Passed: Damage Factor < 1.00

Per AD-160 Condition A: Fatigue analysis WAS required for this Component!

Max Stress Intensity is taken as 3.0 Sm which is 60000.000 psi

Given Number of Cycles is 312000.

Allowable Num of Cycles without a Fatigue Analysis is 2456.

Note the accumulation of damage factors.

Other Pressure Vessel codes

Other codes, such as PD 5500 and EN 13445 treat fatigue slightly differently. PD 5500 for

example, classifies the type of crack to be considered. It is a little more sophisticated than

ASME Section VIII, Division 1. Here is a sample of the crack classification:

If you look at the column headed ‘Class’, you can see the crack classification, such as W, or F2

etc. The fatigue graph in PD 5500 has a curve for each classification as can be seen here:

Fatigue – Not a Catastrophic Failure

Pressure vessels suffer fatigue problems, especially vessels that have been in service for a long

time. Fatigue cracks develop at the junctions between the nozzle and shell of the vessel, where

legs are attached to the vessels shell, and other places where an abrupt change in thickness

occurs. These cracks are rarely catastrophic in nature. They are more of a nuisance value.

Usually repairs can be put in hand. The vessel can then be returned to service and operated for a

number of years.

The real difficulty is in predicting the pressure ranges and number of expected cycles the vessel

is expected to endure during it lifetime. This is usually an educated guess.

Differences in the 2007 (forward) edition of ASME Section VIII, Division 2

In the later edition of the code, the fatigue graph has been replaced by a formula to derive the

number of cycles from SA. Otherwise the principle is the same.