In the early design stage, not much is known about the anticipated stress histories.  An exceedance spectrum based on previous experience is usually available.  However, material selection may still have to be made, and operational stress levels may still have to be selected.  Hence, it is impossible and premature to derive a detailed service life history as discussed in Section 5.4.2.  Yet, crack-growth calculations have to be made as part of the design trade-off studies.  The designer wants to know the effect of design stress, structural geometry, and material selection with respect to possible compliance with the damage-tolerance criteria, and with respect to aircraft weight and cost.  Such studies can be made only if a reasonable service stress history is assumed.  The following example shows how much a history can be derived in a simple way, if it is to be used only for comparative calculations.

 

EXAMPLE 5.4.2         Construction of a Simple Stress Sequence

Consider the exceedance spectrum for 1,000 flights shown below.  Instead of selecting stress levels for the discretization, it is much more efficient in this case to select exceedances.  Since a large number of levels is not necessary in this stage, six levels were chosen in the example.  The procedure would remain the same if more levels were to be selected.

The exceedances in the example were taken at 10 (in accordance with Section 5.4.2); 100; 1,000; 10,000; 100,000; and 500,000 (in accordance with Section 5.4.2).  Vertical lines are drawn at these numbers, and the stepped approximation is made.  This leads to the positive excursion levels, S₁-S₆, and the negative excursion levels, L₁-L₆ , as shown below.  The stress levels and exceedances are given in columns 1 and 2 of the table; subtraction gives the number of occurrences in column 3.

The highest stress level is likely to occur only once in the severest mission.  Therefore, a mission A spectrum is selected, as shown in column 4, in which S₁ occurs once, and lower levels occur more frequently in accordance with the shape of the total spectrum.  In order to use all 10 occurrences of level S₁, it is necessary to have 10 missions A in 1,000 flights.  The number of cycles used by 10 missions A is given in column 5.  The occurrences from these missions are subtracted from the total number of occurrences (column 3) to give the occurrences in the remaining 990 flights (column 6).

The next severest mission is likely to have one cycle of level S₂.  Hence, the mission B spectrum in column 7 can be constructed in the same way as the mission A spectrum.  Since 60 cycles of S₂  remain after mission A, mission B will occur 60 times in 1,000 flights.  The 60 missions B will use the cycles shown in column 8, and the cycles remaining for the remaining 930 flights are given in column 9.

 

 

Level S₃ will occur once in a mission C, which is constructed in column 10.  There remain 570 cycles S₃, so there will be 570 missions C.  These missions will use the cycles given in column 11, and the remaining cycles are given in column 12.

 

There will be 10 missions A, 60 missions B, and 570 missions C in 1,000 flights, meaning that 360 flights remain.  By dividing the remaining cycles in column 12 into 360 flights, a mission D spectrum is defined, as given in column 13.  Consequently, all cycles have been accounted for.

A mission mix has to be constructed now.  With mission A occurring 10 times per 1,000 flights, a 100-mission block could be selected.  However, a smaller block would be more efficient.  In the example, a 33-mission block can be conceived, as shown below.  After 3 repetitions of this block (99 flights) one mission A is applied.

 

The cycles in each mission are ordered in a low-high-low sequence.  The negative excursion L₁-L₆ are accounted for by combining them with the positive excursions of the same frequency of occurrence:  L₁ forms a cycle with S₁, L₂ with S₂, etc.

 

 

To arrive at the stresses an approximate procedure has to be followed also.  Given the flight duration, an acceleration spectrum (e.g., the 1,000 hours spectra given in MIL-A-8866B) can be converted approximately into a 1,000 flight spectrum.  Limit load will usually be at a known value of n_z, e.g., 7.33g for a fighter or 2.5g for a transport.  As a result, the vertical axis of the acceleration diagram can be converted into a scale that gives exceedances as a fraction of limit load.  This is done in Figure 5.4.6 for the MIL-A-8866B spectra of Figure 5.4.2.  A comparison of these figures will clarify the procedure.

Once the spectrum of the type of Figure 5.4.6 is established, design trade-off studies are easy.  Selecting different materials or different design stress levels S₁-S₆ and L₁-L₆ can be determined and the flight-by-flight spectrum is ready.  Selection of a different design stress level results in a new set of S₁-S₆., and the calculations can be re-run.

 

Figure 5.4.6.  Approximate Stress Spectrum for 1000 Flights Based on MIL-A-8866B (USAF)

This shows the versatility of the spectrum derivation shown in Example 5.4.2.  It is a result of choosing exceedances to arrive at the stepped approximation of the spectrum, which means that the cycle content is always the same.  If stress levels were selected instead, a change in spectrum shape or stress levels would always result in different cycle numbers.  In that case, the whole procedure to arrive at the spectrum in Example 5.4.2 would have to be repeated, and many more changes would have to be made to the computer program.

Example 5.4.2 shows only a few levels.  The spectrum could be approximated by more levels and more missions could be designed, but the same procedure can be used.  In view of the comparative nature of the calculations in the early design stage, many more levels or missions are not really necessary.

Note:  The stress history derived in this section is useful only for quick comparative calculations for trade-off studies.

The stress history developed in Example 5.4.2 was applied to all the s spectra from MIL-A-8866B (shown in Figure 5.4.6) to derive crack-growth curves.  These results will be discussed in Section 5.5.3.