**Yan-Hui Zhang and S J Maddox**

Structural Integrity Technology Group

TWI Limited

Granta Park

Great Abington

Cambridge

CB1 6AL, UK

Paper published in International Journal of Fatigue 2008, vol. 31, Issue 1, January 2009. pp.138-152.

http://www.sciencedirect.com/science/journal/01421123

### Abstract

The paper presents the results of the investigation on the effect of loading spectra with different mean stresses on the validity of Miner's rule, and the effect of stresses below the constant amplitude fatigue limit (CAFL) on the fatigue performance of two types of weld joint. In support of understanding the mechanism for any deficiency to Miner's rule, fracture mechanics analysis was carried out by measuring and predicting the crack growth in specimens tested under both constant amplitude and variable amplitude loading. The experimental results showed that, although Miner's rule would predict the same fatigue life for each type of specimens tested under the same spectrum, in fact the actual value of Σ(n/N) at failure strongly depended on the sequence applied. The influence of the loading sequence on Σ(n/N) was in agreement with that on crack growth rates. The deficiency in Miner's rule was attributed primarily to the stress interaction effects resulting from the type of loading sequence used. The experimental results also showed that, under certain circumstances, stress ranges well below the fatigue limit were found to be as damaging as implied by the S-N curve extrapolated beyond the CAFL without changing the slope. The value of the minimum fully damaging stress range was found to depend on the basic fatigue strength of the weld joint.

### Key words:

Variable amplitude loading, Miner's rule, welded joint, fatigue limit, crack growth.

## Nomenclature and definitions

**Block length N _{L}** : Total number of cycles in a variable amplitude loading block.

**CA**: Constant amplitude.

**Constant amplitude fatigue limit (CAFL)**: Fatigue strength under constant amplitude loading corresponding to infinite fatigue life or a number of cycles large enough to be considered infinite by a design code.

**Equivalent (constant amplitude) stress range ΔS _{eq}** : For a particular number of cycles to failure, ΔS

_{eq}is the constant amplitude stress range which, according to Miner's linear cumulative damage rule, is equivalent in terms of fatigue damage to a variable amplitude stress spectrum.

**Fully damaging stress range**: Stress range that is as damaging as implied by the CA S-N curve extrapolated beyond the CAFL without changing the slope.

**Loading block**: The stress history between successive applications of the peak stress in the spectrum.

**Miner's rule**: Fatigue failure under variable amplitude loading corresponds to the following:

Where *n _{1}* ,

*n*, etc are the numbers of cycles at applied stress ranges Δ

_{2}*S*, Δ

_{1}*S*, etc and

_{2}*N*,

_{1}*N*, etc are the corresponding numbers of cycles to failure under constant amplitude loading at those stress ranges, k is number of stress range levels.

_{2}** p _{i}** : ΔS

_{i}/ΔS

_{max}, relative stress range in a spectrum.

** R**: Stress ratio (=

*S*).

_{min}/S_{max}**Δ S**: Stress range

**Δ S _{i}** : The ith stress range in a spectrum.

**Δ S _{max}** : The maximum (peak) stress range in a spectrum.

**Δ S' _{min}** : The minimum fully damaging stress range in a spectrum, below which stresses are not as damaging as implied by the CA S-N curve extrapolated beyond the CAFL without a slope change.

**S-N curve**: Relation between applied stress range and life in cycles to failure under constant amplitude loading. For welded joints it has the general form Δ*S ^{m}N*=

*C*where

*C*and

*m*are material constants.

**Sequence A**: A loading spectrum with all stresses cycling down from a constant maximum tensile stress

**Sequence B**: A loading spectrum with all stresses cycling around a constant mean stress

**Sequence C**: A loading spectrum with all stresses cycling up from a constant minimum stress

**Spectrum irregularity factor, I**: The number of positive mean crossings divided by the total number of cycles in one block.

**VAM**: Variable amplitude.

**Wide band spectrum or loading**: This describes a loading history with an irregularity factor significantly less than 1.0.

## 1 Introduction

In service the great majority of structures and components are subjected to stresses of variable amplitude (VA). The fatigue design of welded joints in such structures is based on fatigue data obtained under constant amplitude (CA) loading, used in conjunction with a cumulative damage rule to estimate the damage introduced by cycles of various magnitudes in the service stress history. The most widely used is Miner's linear cumulative damage rule^{[1]}, which states that the following should be satisfied in fatigue design:

where *n _{1}* ,

*n*, etc are the numbers of cycles corresponding to applied stress ranges Δ

_{2}*S*, Δ

_{1}*S*, etc expected in the life of the structures and

_{2}*N*,

_{1}*N*, etc are the corresponding numbers of cycles to failure under CA loading at those stress ranges, k is number of stress range levels. Miner's rule suggests that any structure with Σ(n/N) <1.0 is safe for operation.

_{2}An implicit assumption in Miner's rule is that the fatigue damage due to the application of a particular stress cycle in a VA load sequence is exactly the same as that due to the same stress cycle under CA loading. However, there is extensive evidence^{[2-7]} to suggest that VA stress cycles could be more damaging than the same stress cycles under CA loading, with the result that Miner's rule can be unsafe (ie Σ(n/N) <1.0 at failure) under certain circumstances. Gurney has investigated the fatigue performance of welded joints under VA loading extensively, as summarised recently.^{[6]} His test results suggested that Miner's rule was generally correct or conservative when the spectrum block length was sufficiently long (N_{L} >1,000 cycles) and the mean stresses for all cycles were comparable with that used in the CA tests, which were mostly performed at R=0 or -1. However, he found that Miner's rule tended to be unsafe in the following circumstances, even when assuming that all stresses below the CAFL were fully damaging (ie as implied by the S-N curve extrapolated beyond the CAFL without changing the slope, see below):

- Short block length, typically less than 100 cycles
- High mean stresses in the spectrum.

However, the latter may have been a reflection of the type of test specimen used - plates with longitudinal fillet welded edge attachments (see *Figure 1a*). The fatigue performance under CA loading was found to exhibit mean stress dependence^{[5]} but the VA tests carried out with variable stress ratio were assessed on the basis of the CA S-N curve obtained at a constant *R* (often *R*=0).

There are also doubts^{[5, 8-12]} about the method of treating stresses below the constant amplitude fatigue limit (CAFL). The most widely used assumption is that their damaging effect can be represented by the CA S-N curve extrapolated beyond the CAFL, widely assumed to correspond to N=10^{7} cycles, at a shallower slope, typically *m* = 5 instead of 3. However, there is evidence^{[5,10]} that in fact they are more damaging than this. For example, it was reported^{[5]} that the use of a 2-slope S-N curve with the slope change at a stress range above 10.1N/mm^{2} (corresponding to about 5.5x10^{8} cycles) is potentially unsafe, particularly for loading spectra containing large numbers of small stress ranges. This was especially true for fully tensile loading.

When applying Miner's rule in the design of welded structures the CA fatigue strength is generally represented by a single S-N curve, expressed in terms of the applied stress range regardless of mean stress, eg BS 7608.^{[13]} This is to allow for the inevitable presence of high tensile residual stresses, which are assumed to produce conditions equivalent to the most severe high applied tensile mean stress conditions under either CA or VA loading. However, in view of the apparent influence of applied mean stress reported, this may be an over-simplification.

Thus, the objectives of this study were to continue Gurney's work and investigate the effects of loading spectra with different mean stresses on both the validity of Miner's rule and the damaging effect of stresses below the CAFL.

## 2 Approach

Three variable amplitude loading sequences, all based on the same spectrum (with respect to stress ranges, corresponding number of cycles at each stress range and block length) but with different mean stresses, were tested to investigate the effect of mean stress on fatigue performance under VA loading. They were:

- Sequence A, stresses cycling down from a constant high tensile stress, in which case the mean stress increased with decrease in stress range;
- Sequence B, stresses cycling about a constant mean stress;
- Sequence C, stresses cycling up from a constant minimum stress, in which case the mean stress increased with increase in stress range.

The fatigue tests were performed on two types of fillet welded joint in steel with different fatigue strengths. CA S-N curves were established for each by testing with the maximum stress held constant, as with Sequence A. In this way it was anticipated that the VA test results obtained under Sequence A would provide a unique opportunity to evaluate the accuracy of Miner's rule by eliminating any possible effect of mean stress.

A spectrum with a so called concave-up stress distribution, in which fatigue damage from small stresses was predominant, was derived to investigate the effect of small stresses. Using the same approach as Gurney^{[5]}, it was anticipated that by successively adding progressively smaller stress ranges, Miner's sum would be significantly increased when a non-damaging stress range was approached. On the other hand, if the lowest stress in a spectrum produced fatigue damage consistent with the CA S-N curve extrapolated beyond the CAFL without a slope change, Miner's sum would be expected to be almost constant.

In support of understanding the mechanisms responsible for any deficiency in Miner's rule, fracture mechanics analysis was carried out by measuring and predicting the crack growth in specimens tested under both CA and VA loading.

## 3 Test specimens

Two types of specimen were used, designated G and F, as shown in *Figure 1*. They were manufactured from two grades of carbon manganese structural steel to BS 4360, one from Grade 50D and the other Grade 50B. The chemical compositions and mechanical properties of the parent materials are given in *Tables 1 and 2*.

**Table 1 Chemical composition of the parent materials**

Element | Type G specimen (BS 4360 Grade 50D)* | Type F specimen (BS 4360 Grade 50B) | |
---|---|---|---|

Batch 1 | Batch 2 | ||

C | 0.14 | 0.16 | 0.17 |

Mn | 1.35 | 1.35 | 1.36 |

Si | 0.39 | 0.33 | 0.26 |

S | 0.012 | 0.029 | 0.002 |

P | 0.012 | 0.022 | 0.031 |

Cr | 0.022 | 0.12 | 0.017 |

Ni | 0.021 | 0.087 | 0.016 |

Al | 0.046 | 0.025 | 0.029 |

Cu | 0.019 | 0.017 | 0.011 |

Nb | 0.026 | 0.003 | <0.002 |

Ti | 0.003 | 0.002 | 0.004 |

N | 0.0056 | 0.0065 |

*: from^{[5]}.

**Table 2 Mechanical properties of parent materials**

Mechanical properties | Type G specimen* | Type F specimen | |
---|---|---|---|

Batch 1 | Batch 2 | ||

Yield stress, N/mm ^{2} |
399 | 386 | 418 |

Tensile strength, N/mm ^{2} |
541 | 509 | 554 |

Elongation | 34% | 27% | 34% |

*: from^{[5]}.

The type G specimen, *Figure 1a*, consisted of a 12mm thick plate with longitudinal attachments fillet welded to each edge, while the type F specimen involved a 12.5mm plate with longitudinal attachments fillet welded on each surface, *Figure 1b*. The former specimens were from the batch tested by Gurney^{[5]} and the latter were from the batch tested by Maddox.^{[14]} In terms of fatigue strength, they are designated as Class G and F respectively^{[13]}; hence the present designations. Both types of specimen were symmetrical and could be subjected to axial loading without any significant secondary bending.

**Fig.1. Details of test specimens and the locations where residual stresses were measured:**

**a) Longitudinal non-load-carrying attachments on plate edges (type G specimen);**

**b) Longitudinal non-load-carrying fillet welded joints (type F specimen)**

**c) Locations where residual stresses were measured**

All the specimens were fabricated so that the direction of stressing was parallel to the rolling direction. The fillet welds were made in the flat position in two runs with the stop-start positions in the middle of the attachments so as to avoid, as far as possible, the effects of end craters. The fillet welds were carried around the ends of the attachments in the type F specimens, but not in the type G specimens. All specimens were tested in the as-welded condition.

## 4 Experimental details

### 4.1 Fatigue testing

The test programme involved both CA and VA amplitude loading tests on each type of specimen. All the specimens in the testing programme were subjected to axial loading in servo-hydraulic fatigue testing machines at testing frequencies in the range 3 to 8Hz.

Apart from specimen F-14, all the CA tests were performed under a constant maximum tensile stress of 280N/mm^{2}, about 0.7 x yield strength for the type G specimen and 0.67 x yield strength for type F specimen. For specimen F-14, the maximum stress was held constant at the lower value of 135N/mm^{2}.

As noted previously, some of the VA tests also involved cycling down from a constant tensile stress, while others involved cycling about a constant tensile mean stress and others cycling up from a constant tensile minimum stress, as detailed in the next section.

Fatigue tests continued until complete failure of the specimen. Examples of fatigue failures in each type of specimen are shown in *Figure 2*. The failure modes were the same under CA and VA loading. In the type G specimens (*Figure 2a*) fatigue cracks initiated at one or more of the weld ends, propagated through the plate thickness and finally propagated across the plate width as edge cracks. In the type F specimens (*Figure 2b*) fatigue cracks initiated at one or more of the weld toes at the ends of the attachments. They then propagated through the plate thickness, adopting semi-elliptical shapes, and finally grew to failure across the width of the main plate.

**Fig.2. Examples of fatigue failures in specimen types G and F**

**a) Type G**

**b) Type F**

### 4.2 Variable amplitude loading spectrum

To investigate the effect of small stress ranges, a loading spectrum with a concave-up shape in the plot of relative stress range, *p _{i}* , the ratio of ith stress range to the maximum stress range in a spectrum, against exceedence and a reasonably long block length, N

_{L}, was derived. The block length in a given test depends on the minimum value of

*p*adopted. The length of the basic spectrum, in which the lowest value of

_{i}*p*was 0.04, was ~2x10

_{i}^{5}cycles. However, it was shorter for those cases when low

*p*values were omitted. The stress distribution was derived in such a way that small stresses made a significant contribution to fatigue damage. Details of the stress ranges and the corresponding numbers of cycles in a block are shown in

_{i}*Table 3*and the distribution is plotted in

*Figure 3*. The spectrum used by Gurney

^{[5]}is also shown for comparison. As will be seen, both spectra exhibit concave-up shapes, but the present spectrum is longer than Gurney's. The relative fatigue damage, defined as the ratio of fatigue damage at a stress level Δ

*S*against the fatigue damage at the maximum stress range

_{i}*Formula.1.*, where m was assumed to be 3, as for the BS 7608[13] Class F and G design curves, gradually increased with decreasing stress range, Figure 4. In comparison with Gurney's spectrum, included in Figure 4, this should make the results of the present VA tests more sensitive to the inclusion of small stresses in the spectrum.

**Table 3 Details of concave-up spectrum derived for the VA tests**

Relative stress range, p _{i} | Stress range, N/mm ^{2} | Cycles | Exceedence |
---|---|---|---|

1.00 | 210.0 | 1 | 1 |

0.90 | 189.0 | 3 | 4 |

0.80 | 168.0 | 6 | 10 |

0.70 | 147.0 | 12 | 21 |

0.60 | 126.0 | 23 | 44 |

0.50 | 105.0 | 48 | 92 |

0.40 | 84.0 | 109 | 202 |

0.30 | 63.0 | 296 | 498 |

0.25 | 52.5 | 544 | 1,042 |

0.20 | 42.0 | 1,125 | 2,1<67 |

0.15 | 31.5 | 2,815 | 4,982 |

0.10 | 21.0 | 9,500 | 14,482 |

0.06 | 12.6 | 43,981 | 58,463 |

0.04 | 8.4 | 148,438 | 206,901 |

### 4.2 Variable amplitude loading spectrum

To investigate the effect of small stress ranges, a loading spectrum with a concave-up shape in the plot of relative stress range, *p _{i}* , the ratio of ith stress range to the maximum stress range in a spectrum, against exceedence and a reasonably long block length, N

_{L}, was derived. The block length in a given test depends on the minimum value of

*p*adopted. The length of the basic spectrum, in which the lowest value of

_{i}*p*was 0.04, was ~2x10

_{i}^{5}cycles. However, it was shorter for those cases when low

*p*values were omitted. The stress distribution was derived in such a way that small stresses made a significant contribution to fatigue damage. Details of the stress ranges and the corresponding numbers of cycles in a block are shown in

_{i}*Table 3*and the distribution is plotted in

*Figure 3*. The spectrum used by Gurney

^{[5]}is also shown for comparison. As will be seen, both spectra exhibit concave-up shapes, but the present spectrum is longer than Gurney's. The relative fatigue damage, defined as the ratio of fatigue damage at a stress level Δ

*S*against the fatigue damage at the maximum stress range

_{i}*Formula.1.*, where m was assumed to be 3, as for the BS 7608[13] Class F and G design curves, gradually increased with decreasing stress range, Figure 4. In comparison with Gurney's spectrum, included in Figure 4, this should make the results of the present VA tests more sensitive to the inclusion of small stresses in the spectrum.

**Fig.3. Concave-up spectrum used in the present VA tests. The spectrum used by Gurney ^{[5]} is also included for comparison. The dashed lines indicate the linear stress distribution for each spectrum.**

**Fig.4. Comparison of the relative fatigue damage ( Formula.2.) in the current stress distribution with that in the distribution used by Gurney. ^{[5]}**

A maximum stress range of 210N/mm^{2} was used in all VA loading tests. Initially, a VA test was conducted using a spectrum with the minimum stress range above the estimated CAFL, to ensure that all stress cycles were damaging. As in BS 7608, the CAFL was defined as the fatigue strength corresponding to 10^{7} cycles on the S-N curves generated from the CA tests on the two types of specimen. Smaller stress ranges were then gradually added in the subsequent VA tests to establish the minimum stress range, Δ*S' _{min}* , that was still contributing to fatigue damage in line with that expected according to the CA S-N curve extrapolated beyond the CAFL without a slope change. The minimum

*p*in the spectra tested ranged from 0.15 to 0.04 for type G specimen and from 0.25 to 0.1 for the type F specimen.

_{i}For the same basic stress distribution (identical number of cycles for each stress range) described above, the three types of loading sequence used to investigate the effect of mean stress were applied as follows:

Sequence A - stresses cycling down from a constant maximum stress of 280N/mm^{2};

Sequence B - stresses cycling at a constant mean stress of 175N/mm^{2};

Sequence C - stresses cycling up from a constant minimum stress of 70N/mm^{2}.

The resulting maximum and minimum stresses for each *p _{i}* value are shown schematically in

*Figure 5*. As indicated, the maximum stress and maximum stress range was the same in every case. One specimen, F-13, was tested under a variant of Spectrum A in which the maximum stress was reduced to 147N/mm

^{2}in order to investigate the effect of mean stress under this sequence.

**Fig.5. Schematic illustration showing the difference in maximum and minimum stresses for the three sequences used in the VA tests.**

The VA tests were performed in computer-controlled testing machines that were programmed to apply the stress cycles in each block in a random order. This was achieved by selecting *p _{i}* values using a random number generator. When the whole of the first block had been applied, the process started again and subsequent blocks were applied in the same random order. This process was repeated until the specimen failed. Examples of stress - time histories for each type of sequence are shown in

*Figure 6*.

**Fig.6. Examples showing the three loading sequences**

**a) Cycling-down from a constant maximum stress;**

**b) Cycling at a constant mean stress;**

**c) Cycling-up from a constant minimum stress**

### 4.3 Crack initiation monitoring and growth measurements

To support a fracture mechanics analysis of the test results, crack initiation and growth were monitored in many type F specimens under both CA and VA loading. As seen in *Figure 2b*, the cracks in these specimens propagated from a weld toe through the plate thickness with a semi-elliptical shape. The crack depth a and surface length *2c* were measured using a combination of visual inspection and the alternating current potential drop (ACPD) method.

The visual inspection was aided by the application of soap solution and a magnifier to detect fatigue-induced cracking and to monitor its propagation. When applied to specimens while they were being tested under cyclic loading, this method was able to detect a crack length of ~2mm. However, ACPD was able to detect evidence of fatigue cracking before the soap solution method. As seen in *Figure 2b*, the soap solution usually left marks on the fracture surface so that the crack depth corresponding to a measured surface length at a known endurance could be measured after failure. Additional *a*-growth data were also obtained by ACPD.

### 4.4 Residual stress measurements

Neglect of the effect of mean stress in fatigue design codes is based on the assumption that high tensile residual stresses are always present in welded structures. To confirm this for the present specimens, and to investigate any change of residual stresses during cyclic loading, residual stresses close to the plate surface were measured in both types of specimen. The hole drilling method was used, with the holes located 5mm from either the attachment end (type G specimen) or the weld toe (type F specimen), as indicated in *Figure 1*. Measurements were made at four locations in each specimen.

In addition, two more measurements were made in a type F specimen after it had been subjected to 10 blocks of cycles under VA loading, which was less than 1% of the endurance of that specimen.

## 5 Test results

### 5.1 Residual stress measurements

The results of the residual stress measurements are presented in *Table 4*. These confirmed that high tensile residual stresses acting parallel to the direction in which the specimens were to be fatigue loaded were present in both types of specimen near the areas where crack initiation was expected to occur. The residual stresses were relatively higher in the type G specimen, approaching the yield strength of the parent metal, while they were about 67% of yield in the type F specimen.

**Table 4 Results of residual stress measurements**

Specimen | Yield strength, N/mm ^{2} | Residual stress, N/mm ^{2} | ||||
---|---|---|---|---|---|---|

Location 1 | Location 2 | Location 3 | Location 4 | Average | ||

G-02, as-received | 386 - 399 | 455 | 324 | 296 | 404 | 370 |

F-02, as-received | 418 | 302 | 237 | 300 | 285 | 281 |

F-08, 10 blocks under VA loading* | 68 | 84 | - | - | 76 |

*: Residual stresses were measured after the specimen was tested under VA loading for ten blocks. The total fatigue life of this specimen was 1,147 blocks.

Under the fatigue loading, part of the residual stresses quickly relaxed. The average residual stress in the type F specimen, tested under Sequence A, was reduced by 73% after the specimen had been tested for <1% of the total life. This was in agreement with other work^{[15]} for a similar welded joint containing yield magnitude residual stress. In that case, about 80% of the residual stresses relaxed under the first application of the maximum tensile stress in the spectrum, which corresponded to about 57% of yield. Similar results were also reported by others.^{[16,17]}

Although the fatigue loading reduced the residual stresses considerably, those remaining were still significant with respect to small stress ranges in the spectrum. The average residual stress from the two measurements after 10 blocks of VA loading noted above was 76N/mm^{2}. It is expected that a similar level would remain for the rest of the life since residual stress relaxation occurs mainly in the first application of the maximum stress.^{[15,18]} A remaining residual stress of 70N/mm^{2} would result in actual stress ratios of 0.74 and 0.63 for applied stress ranges of 21 and 31.5N/mm^{2} (corresponding to *p _{i}* =0.10 and 0.15) respectively.

### 5.2 Constant amplitude tests

The results obtained from the type G specimens, together with a result obtained by Gurney from the same batch of specimens that happened to be tested with the same S _{max} ^{[5]}, are given in *Table 5* and plotted in *Figure 7*. Also shown is the best-fit S-N curve:

ΔS^{2.728} N=1.183x10^{11} [2]

**Table 5 Constant amplitude test results for the type G specimen, S _{max}=280N/mm ^{2}**

Specimen No. | Stress range, N/mm ^{2} | Stress ratio R | Cycles to failure |
---|---|---|---|

G-01 | 80 | 0.71 | 736,000 |

G-02 | 120 | 0.57 | 237,000 |

G-03 | 55 | 0.80 | 2,240,000 |

G-04 | 65 | 0.77 | 1,350,000 |

871* | 280 | 0.0 | 26,500 |

*: from^{[5]}.

**Fig.7. Constant amplitude test results for the type G specimen, tested at a constant maximum stress of 280N/mm ^{2}. The test results obtained at R=0^{[6]} were virtually on the BS 7608 Class G mean S-N curve**

It may be noted that the test results obtained at *R*=0 from this batch of specimens^{[6]} happened to lie virtually on the BS 7608 Class G mean S-N curve, as shown in *Figure 7*. The present results obtained at low stress ranges are lower than those obtained at *R*=0. This will be seen to have a significant effect on Σ(n/N) when the VA test results are evaluated.

Again five results were used to establish the CA S-N curve for the type F specimens, four from the present tests, obtained under a constant maximum stress of 280N/mm^{2}, and a fifth from tests carried out by Maddox^{[14]} at a maximum stress of 267N/mm^{2}. These results are presented in *Table 6* and plotted in *Figure 8* together with the best-fit S-N curve:

ΔS^{3.072} N=1.312x10^{12} [3]

**Table 6 Constant amplitude test results for the type F specimen**

Specimen No. | Maximum stress, N/mm ^{2} | Stress range, N/mm ^{2} | Stress ratio R | Cycles to failure |
---|---|---|---|---|

F-01 | 280 | 90 | 0.71 | 1,378,500 |

F-02 | 280 | 120 | 0.57 | 546,600 |

F-11 | 280 | 65 | 0.77 | 3,271,600 |

F12 | 280 | 140 | 0.50 | 360,400 |

SN3-31* | 267 | 240 | 0.1 | 60,100 |

F-14 | 135 | 65 | 0.52 | 3,866,500 |

**Fig.8. Constant amplitude results for the type F specimen, tested at a constant maximum stress of ~280N/mm ^{2}.**

*Figure 8* also includes the experimental data obtained from the same batch of specimens but at *R*=0^{[14]}, as well as the BS 7608 Class F mean curve, for comparison. It will be seen that the mean stress had little effect on the fatigue strength of the type F specimen investigated, as was also found for a similar type of specimen.^{[19]} This was also confirmed by comparison of the results obtained from specimens F-14 and F-11, tested at the same stress range but different maximum stresses. The fatigue life was only slightly higher for specimen F-14 tested at S_{max}=135N/mm^{2} when compared with specimen F-11 tested at S_{max}=280N/mm^{2}, a difference that was partly due to the longer crack growth length before failure under the lower maximum stress in F-14.

### 5.3 Variable amplitude tests

**5.3.1 Method used to analyse VA test results**

The VA test results were analysed in terms of Miner's rule to check its validity and to examine the effect of stresses below the CAFL. The CAFL was assumed to correspond to a fatigue endurance of 10^{7} cycles, in accordance with the recommendation of BS 7608. For each type of specimen, three S-N curves based on the mean curves fitted to the CA data were used to determine N in the calculation of Σ(n/N):

- a single curve without any slope change;
- a bi-linear curve with a slope change from
*m*to (*m*+2) at the CAFL; - a single curve cut off at the CAFL so that all stresses below it were assumed to be non-damaging.

**5.3.2 Type G specimen**

The test results are summarised in *Table 7* where Σ(n/N) values at failure were calculated using Eq. (2), the CA S-N curve obtained under a constant maximum stress of 280N/mm^{2}. As will be seen:

- Miner's rule was significantly non-conservative for all tests under Sequence A, even when an S-N curve without a slope change was used in calculating Σ(n/N). It should be noted that, at the same stress range, both CA and VA tests had the same maximum and minimum stresses. This ruled out any possibility of a mean stress effect. Hence these non-conservative results must have resulted from a form of stress interaction under spectrum loading, whereby some stress cycles become more damaging than expected on the basis of their effect under CA loading.
- Small stresses below the fatigue limit were still 'fully damaging'. The effect of small stresses can be seen by comparing the Σ(n/N) values calculated using the single-slope S-N curve with those determined using either the bi-linear curve or the single curve with a cut-off at the fatigue limit. If the smallest stress range in a spectrum does not produce fatigue damage, the number of blocks to failure would be the same as that for a spectrum without this stress range. By comparing both the number of blocks to failure and the Miner's rule damage sums for the two spectra with minimum
*p*=0.06 and_{i}*p*=0.04 (corresponding to a stress range of 8.4N/mm_{i}^{2}), it can be concluded that stress ranges as low as 8.4N/mm^{2}(only 27% of the assumed fatigue limit at 10^{7}cycles) were still fully damaging. - For the specimens tested under Sequence B (constant mean stress of 175N/mm
^{2}), Miner's rule was still non-conservative, even when the S-N curve without a slope change was used. However, the Σ(n/N) value was significantly greater than those obtained under Sequence A. - For the specimens tested under Sequence C (constant minimum stress of 70N/mm
^{2}), Miner's rule was very conservative even when the S-N curve with a cut-off at the assumed fatigue limit was used, which gave Σ(n/N) =3.38.

**Table 7 VA test results for the type G specimen**

Specimen No | Spectrum | Block length, cycles | Minimum stress range, N/mm ^{2} | Cycles to failure | Number of blocks to failure | Σ(n/N) | |||
---|---|---|---|---|---|---|---|---|---|

Minimum p _{i} | Sequence | No slope change ^{a} | Bi-linear ^{b} | Cut-off at fatigue limit ^{c} | |||||

G-06 | 0.15 | A | 4,982 | 31.5 | 1.37x10 ^{6} |
275 | 0.43 | 0.43 | 0.43 |

G-07 | 0.10 | A | 14,482 | 21.0 | 3.16x10 ^{6} |
218 | 0.41 | 0.38 | 0.34 |

G-10 | C | 3.12x10 ^{7} |
2,152 | 4.08 | 3.70 | 3.38 | |||

G-11 | B | 5.71x10 ^{6} |
394 | 0.75 | 0.68 | 0.62 | |||

G-08 | 0.06 | A | 58,463 | 12.6 | 1.13x10 ^{7} |
212 | 0.48 | 0.38 | 0.33 |

G-09 | 0.04 | A | 206,901 | 8.4 | 3.66x10 ^{7} |
181 | 0.49 | 0.33 | 0.28 |

**Notes:**

a. ΔS^{2.728}N=1.183x10^{11}, the mean S-N curve obtained under a constant maximum stress of 280N/mm^{2}.

b. Slope change from m to (m+2) at 10^{7} cycles.

c. Cut-off at the fatigue limit, which was 31N/mm^{2} corresponding to an endurance of 10^{7} cycles.

These characteristics of the results are more evident in *Figure 9*, which shows them plotted in terms of the equivalent CA stress range. This is the CA stress range which, according to Miner's rule, is equivalent in terms of fatigue damage to a VA stress spectrum. It relates to the CA S-N curve for the detail under consideration as follows:

where *m* is the slope of the CA S-N curve, that is 2.728 in the present case. If an experimental result lies on or above the CA S-N curve it means that Miner's rule gave an accurate or safe estimate of the actual fatigue life. From *Figure 9*, the above observations regarding the effect of the loading sequence on Miner's rule can be readily seen. The result obtained from the test with a minimum stress range of 8.4N/mm^{2} in the spectrum agrees exactly with the line extrapolated from the results of those tests with higher minimum stresses, indicating that a stress range as low as 8.4N/mm^{2} was still fully damaging under Sequence A.

**Fig.9. Comparison of the VA test results with the CA S-N curve (ΔS2.728N =1.183x1011) obtained with Smax=280N/mm2 for the type G specimen, expressed in terms of the equivalent stress range**

The opportunity has also been taken to compare the present results with those obtained from the same batch of specimens by Gurney.^{[5]} Gurney evaluated his results in terms of the CA S-N curve obtained with R=0. As seen earlier this was less steep and slightly higher than the present curve obtained with *S _{max}* =280N/mm

^{2}. Gurney obtained consistently low Σ(n/N) values at failure, ranging from 0.49 to 0.74 for the tests with the peak stress range at R=0 and from 0.44 to 0.78 for the tests with the peak stress range at R=-1, and concluded that Miner's rule was significantly non-conservative for the conditions investigated. Clearly, re-analysis of his results using the present lower S-N curve will result in higher Σ(n/N) values. In fact, the range of Σ(n/N) increases to 0.75-0.95 for the VA tests with the peak stress range at R=0, and to 0.73-1.28 for the VA tests with the peak stress ange at R=-1, indicating that Miner's rule is more reasonable. This is also evident from

*Figure 9*. Compared with the very small amount of scatter in the CA results (

*Figure 7*), Gurney's VA data still indicate that Miner's rule is generally non-conservative but not by as much as he found using the CA S-N curve obtained at R=0.

In practice, the CA S-N curves given in design codes, for example BS 7608, would be used in fatigue design. Therefore, Σ(n/N) was also calculated using both the mean and design curves given in BS 7608 to examine the accuracy of Miner's rule. The results are presented in *Table 8*. As will be seen:

- Since the present CA results fell below the Class G mean S-N curve the non-conservatism was increased when this was used to apply Miner's rule to specimens tested under Sequences A and B. Under Sequence A the Miner's rule damage sums were as low as 0.26.
- For the specimen tested under Sequence C, it was still very conservative.
- Even when the design curve was used, the Miner's rule damage sums for all tests under Sequence A were still significantly less than unity. The average life would only be about half that predicted by the BS 7608 design curve (with a slope change at 10
^{7}cycles).

**Table 8 VA test results for the type G specimen - further analysis using the BS 7608 curves for the Class G details**

Specimen No | Spectrum | Σ(n/N) - based on the mean curve | Σ(n/N) - based on the design curve | |||||
---|---|---|---|---|---|---|---|---|

Minimum p _{i} | Sequence | No slope change ^{a} | Bi-linear ^{b} | Cut off at fatigue limit ^{c} | No slope change ^{d} | Bi-linear ^{e} | Cut-off at fatigue limit ^{f} | |

G-06 | 0.15 | A | 0.28 | 0.27 | 0.24 | 0.64 | 0.64 | 0.64 |

G-07 | 0.10 | A | 0.26 | 0.22 | 0.19 | 0.59 | 0.55 | 0.51 |

G-10 | C | 2.55 | 2.21 | 1.88 | 5.78 | 5.41 | 5.02 | |

G-11 | B | 0.47 | 0.40 | 0.35 | 1.06 | 0.99 | 0.92 | |

G-08 | 0.06 | A | 0.28 | 0.22 | 0.19 | 0.64 | 0.55 | 0.49 |

G-09 | 0.04 | A | 0.27 | 0.19 | 0.16 | 0.61 | 0.47 | 0.42 |

**Notes:**

a. ΔS^{3.0}N=5.66x10^{11}, the Class G mean curve given in BS 7608.

b. Slope change from 3.0 to 5.0 at a fatigue limit of 38N/mm^{2} (corresponding to 10^{7} cycles).

c. Cut-off at the fatigue limit of 38N/mm^{2}.

d. ΔS^{3.0}N=2.50x10^{11}, the Class G design curve given in BS 7608.

e. Slope change from 3.0 to 5.0 at a fatigue limit of 29N/mm^{2} for the design curve.

f. Cut-off at the fatigue limit, which was 29N/mm^{2} corresponding to an endurance of 10^{7} cycles.

**5.3.3 Type F specimen**

The VA test results, as well as the calculated Miner's sums using the experimentally determined mean S-N curve (Eq. (3)), are presented in *Table 9*. As with the type G specimen results, the VA results are also plotted in terms of the equivalent CA stress range in *Figure 10*. In general, the fatigue test results displayed similar characteristics to those obtained from type G specimens but there were also some significant differences, as follows:

- Miner's rule was non-conservative for the specimens tested under Sequence A, but less so than in type G specimens.
- Stresses as low as 31.5N/mm
^{2}appeared to be fully damaging. This stress is ~70% of the fatigue limit, assumed to be 45.3N/mm^{2}at 10^{7}cycles according to Eq. (3), slightly higher than the Class F design fatigue limit of 40N/mm^{2}. However, when the minimum stress range in the spectrum was reduced to 21N/mm^{2}(*p*=0.1), about 46% of the assumed CAFL, it was not fully effective, as judged by comparing the number of blocks to failure with that from the test with the slightly higher minimum_{i}*p*value of 0.15._{i} - If ΔS'
_{min}/CAFL is defined as the relative fatigue limit, below which stress ranges are no longer 'fully damaging', it appears that both the absolute value of ΔS'_{min}and ΔS'_{min}/CAFL x 100% decreased with decrease in basic fatigue performance. Thus, in the case of the type G specimen, the values were 8.4 N/mm^{2}and 27% respectively, as compared with 31.5N/mm^{2}and 70% for the higher fatigue performance type F specimen. - Miner's rule was slightly non-conservative for the specimens tested under Sequence B, but again less so than in type G specimen.
- Miner's rule was conservative for the specimens tested under Sequence C, although again the deviation from Miner's rule was not as great as that seen in type G specimens.

**Table 9 VA test results for the type F specimen**

Specimen No. | Spectrum | Block length,cycles | Minimum stress range,N/mm ^{2} | Cycles to failure | Number of blocks to failure | Σ(n/N) | ||||
---|---|---|---|---|---|---|---|---|---|---|

Minimum, p _{i} | Sequence | No slope change ^{a} | Bi-linear | Cut-off at fatigue limit ^{d} | ||||||

Note b | Note c | |||||||||

F-03 | 0.25 | A | 1,042 | 52.5 | 1.10x10 ^{6} |
1,053 | 0.46 | 0.46 | 0.46 | 0.46 |

F-15 | 1.50x10 ^{6} |
1,441 | 0.63 | 0.63 | 0.63 | 0.63 | ||||

F-09 | C | 3.81x10 ^{6} |
3,661 | 1.59 | 1.59 | 1.59 | 1.59 | |||

F-10 | B | 1.87x10 ^{6} |
1,799 | 0.78 | 0.78 | 0.78 | 0.78 | |||

F-04 | 0.20 | A | 2,167 | 42.0 | 2.21x10 ^{6} |
1,021 | 0.53 | 0.51 | 0.53 | 0.44 |

F-13 | A ^{e} |
2.51x10 ^{6} |
1,158 | 0.60 | 0.58 | 0.60 | 0.50 | |||

F-06 | C | 5.62x10 ^{6} |
2,592 | 1.34 | 1.30 | 1.30 | 1.13 | |||

F-07 | B | 3.92x10 ^{6} |
1,808 | 0.94 | 0.91 | 0.91 | 0.79 | |||

F-05 | 0.15 | A | 4,982 | 31.5 | 4.10x10 ^{6} |
822 | 0.50 | 0.45 | 0.50 | 0.36 |

F-08 | 0.10 | A | 14,482 | 21.0 | 1.66x10 ^{7} |
1,147 | 0.79 | 0.64 | 0.74 | 0.50 |

**Notes:**

a. ΔS^{3.072}N=1.312x10^{12}, the mean S-N curve obtained under a constant maximum stress of 280N/mm^{2}.

b. Slope change from 3.072 to 5.072 at N=10^{7}cycles on CA S-N curve.

c. Slope change from 3.072 to 5.072 at N=3.3x10^{7} cycles on CA S-N curve (corresponding to assumed CAFL of 31.5N/mm^{2})

d. Cut-off at the fatigue limit, which was 46.3N/mm^{2} corresponding to an endurance of 10^{7} cycles.

e. The maximum stress for this test was kept at 147N/mm^{2}, different from other tests where the maximum stress was 280N/mm^{2}.

**Fig.10. Comparison of the VA test results with the CA S-N curve (Δ S^{3.072}N =1.312x10^{12}) for the type F specimen, expressed in terms of the equivalent stress range, and the S-N curve predicted by fracture mechanics (ΔK_{th}=63N/mm^{3/2})**

To investigate the possible effect of the magnitude of the maximum stress used in Sequence A, specimen F-13 was tested under the same conditions as those for specimen F-04 except that the maximum stress was reduced from 280 to 147N/mm^{2}, so the maximum stress range cycled between 147 and -63N/mm^{2}. Compared with the result of specimen F-04, the Miner's rule damage sum was increased slightly, from 0.53 to 0.60 when the S-N curve without a slope change was used. This is still non-conservative, but it suggests a small influence of the maximum stress in Sequence A on Σ(n/N). However, it will be noted that Σ(n/N) varied from 0.46 to 0.63 for specimens F-03 and F-15, both tested under Sequence A with the same maximum stress of 280N/mm^{2}. Thus, the apparent effect of the magnitude of the maximum stress could simply reflect scatter in fatigue lives.

The test result for specimen F-13 also implies that the applied mean stress was not significant in comparison with the effect of the sequence type. For each stress range (*p _{i}* value), the mean stress in the spectrum for specimen F-13 was less than that in Sequence B, and even less than that in Sequence C for

*p*> ~0.37, see

_{i}*Figure 5*. However, the Σ(n/N) value from the test on specimen F-13 was significantly lower than those from the specimens tested under both Sequences B and C.

A possible consequence of the above observation that stress ranges below the conventional CAFL (corresponding to *N* = 10^{7} cycles on the CA S-N curve) appear to have been fully damaging is that the current method of accounting for the damaging effect of such stresses when applying Miner's rule is wrong. As noted earlier, in BS 7608 this is to assume a bi-linear S-N curve that changes slope from *m* to *m*+2 at the CAFL corresponding to *N* = 10^{7} cycles. The present results suggest that this approach may underestimate the damaging effect of stresses below the CAFL. To investigate this, Σ(n/N) values for the present VA results were calculated using two bi-linear versions of the present mean CA S-N curve for the type F specimens. The first changed slope at the conventional CAFL, corresponding to N = 10^{7} cycles, while the second changed slope at an assumed CAFL of 31.5 N/mm^{2}, corresponding to *N* = 3.3 x 10^{7} cycles. The results are included in *Table 9*. Referring particularly to the results obtained under Sequence A, it will be seen that the largest difference in Σ(n/N) between the two bi-linear curves was 0.1 (0.64 against 0.74, corresponding to the spectrum with minimum *p _{i}* =0.10), well within the data scatter range, regardless of the choice of bi-linear S-N curve. Thus, again basic scatter in the fatigue test results masks the possible influence of a variable. Nevertheless, any influence is clearly very small and it cannot be concluded from the present results that the current method of defining the bi-linear S-N curve with the slope change at 10

^{7}cycles is wrong.

Σ(n/N) values were also calculated using the BS 7608 Class F mean and design S-N curves. The results are presented in *Table 10*. Since the CA results from the present specimens fell significantly below the BS 7608 Class F mean curve, it is not surprising to see that Miner's rule was even more non-conservative in this case. Indeed, the Miner's rule damage sums were less than unity even for the tests under Sequence C. However, the rule was conservative for tests performed under Sequences B and C when Miner's rule was applied using the design curve, but still slightly non conservative for Sequence A. This suggests that the current BS 7608 design rules for this type of joint need to be reviewed.

**Table 10 VA test results for the type F specimen - further analysis by using the BS 7608 curves for the Class F details**

Specimen No | Spectrum | Σ(n/N) - based on the mean curve | Σ(n/N) - based on the design curve | |||||
---|---|---|---|---|---|---|---|---|

Minimum, p _{i} | Sequence | No slope change ^{a} | Bi-linear ^{b} | Cut off at fatigue limit ^{c} | No slope change ^{d} | Bi-linear ^{e} | Cut-off at fatigue limit ^{f} | |

F-03 | 0.25 | A | 0.25 | 0.25 | 0.20 | 0.69 | 0.69 | 0.69 |

F-15 | 0.34 | 0.34 | 0.28 | 0.94 | 0.94 | 0.94 | ||

F-09 | C | 0.87 | 0.85 | 0.71 | 2.39 | 2.39 | 2.39 | |

F-10 | B | 0.43 | 0.42 | 0.35 | 1.18 | 1.18 | 1.18 | |

F-04 | 0.20 | A | 0.29 | 0.27 | 0.20 | 0.80 | 0.80 | 0.80 |

F-13 | A ^{g} |
0.33 | 0.30 | 0.22 | 0.91 | 0.91 | 0.91 | |

F-06 | C | 0.74 | 0.68 | 0.50 | 2.04 | 2.04 | 2.04 | |

F-07 | B | 0.52 | 0.47 | 0.35 | 1.42 | 1.42 | 1.42 | |

F-05 | 0.15 | A | 0.28 | 0.23 | 0.16 | 0.76 | 0.72 | 0.65 |

F-08 | 0.10 | A | 0.45 | 0.33 | 0.22 | 1.22 | 1.05 | 0.90 |

**Notes:**

a. ΔS^{3.0}N=1.726x10^{12}, the Class F mean curve given in BS 7608.

b. Slope change from 3.0 to 5.0 at 10^{7} cycles for the above mean curve.

c. Cut-off at the fatigue limit, which was 56N/mm^{2} corresponding to an endurance of 10^{7} cycles.

d. ΔS^{3.0}N=6.30x10^{11}, the Class F design curve given in BS 7608.

e. Slope change from 3.0 to 5.0 at 10^{7} cycles for the above design curve.

f. Cut-off at the fatigue limit, which was 40N/mm^{2} corresponding to an endurance of 10^{7} cycles.

g. The maximum stress for this test was kept at 147N/mm^{2}, different from other tests where the maximum stress was 280N/mm^{2}.

The various observations about the validity of Miner's rule and the damaging effect of stresses below the CAFL are also evident from *Figure 10*, which shows the VA test results expressed in terms of Δ*S _{eq}* in comparison with the CA results. The point made earlier about scatter in the Sequence A results masking any difference between the results obtained with

*S*= 280 or 147N/mm

_{max}^{2}is also evident. Similarly, the graph highlights the fact that the specimen tested under Sequence A with the lowest equivalent stress range performed proportionately better than those tested at higher stresses, suggesting that the lowest stress range (21N/mm

^{2}) was not fully damaging.

It is interesting to note that the only two specimens tested under Sequence B (with Δ*S _{min}* = 52.5 or 42N/mm

^{2}) also showed a similar trend, but occurring at a higher stress range. Although the data were very limited, it is speculated that the minimum fully effective stress range Δ

*S'*would be higher under Sequence B than that under Sequence A. This would mean that Δ

_{min}*S'*is dependent on the loading-sequence as well as the basic fatigue performance of the weld detail. Further work is required to confirm this speculation.

_{min}## 6 Fracture mechanics analysis

### 6.1 Approach

The crack growth measurements made on type F specimens tested under both CA and VA loading were used to develop a fracture mechanics model for calculating fatigue lives. This entailed first defining the fracture mechanics parameter Δ*K*, the stress intensity factor range, for the observed cracks. Then, the relationship between observed rate of crack growth (da/dN) measured under CA loading and Δ*K* was integrated between an initial crack size and the crack size at failure along the lines detailed in BS 7910^{[20]} to calculate the progress of fatigue cracks and the fatigue lives of specimens tested under VA loading. That relationship was assumed to adopt the usual form:

**da/dN = A(ΔK) ^{m}**

[5]

where *A* and *m* are material constants.

Most measurements were of the surface crack lengths and it is this measure of crack size that is compared with that calculated using fracture mechanics. In all the fracture mechanics calculations, the crack size first recorded for each specimen was used as the starting point in the integration and the calculated life was compared with the actual one remaining after that initial crack detection.

### 6.2 Stress intensity factor

As illustrated in *Figure 2b)*, the case under consideration is a semi-elliptical crack at the toe of a fillet weld. For such cases:

where *M _{k}* is a function of the stress concentration effect of the weld detail,

*Y*is a function of the crack depth to plate thickness ratio

*a/B*and the crack aspect ratio

*a/2c*, where

*2c*is the surface crack length.

The solution for Y for semi-elliptical surface cracks in BS 7910 was used in the present analysis. Specific values of the crack aspect ratio *a/2c* were based on examinations of the fracture surfaces of the tested specimens. These showed that, in the early stages of crack growth, the crack aspect ratio was ~0.25. This was in agreement with other observations in a similar type of specimen.^{[3]} It gradually increased with increasing crack size and was about 0.3 when a crack just became a through-thickness crack.

The magnification factor, *M _{k}* , due to the stress concentration effect of the joint geometry is defined as

^{[26]}:

**M _{k} = K_{in plate with weld}/K _{in plate without weld}**

[7]

*M _{k}* quantifies the change in stress intensity factor as a result of the surface discontinuity at the weld toe.

*M*decreases sharply with increasing distance from the weld toe in the thickness direction and usually reaches unity at crack depths of typically 30% of plate thickness.

_{k}The *M _{k}* solution

^{[3]}derived on the basis of 3D finite element analysis of the same type welded joint, using a model with a weld toe radius of 0.16mm, was adopted. This gave:

**M _{k}=0.845(a/B)^{-0.316} for a/B ≤ 0.1**

M_{k}=0.853(a/B)^{-0.312} for a/B > 0.1

[8]

These expressions provide *M _{k}* only at the deepest point of the crack. Crack growth in this direction also depends on the crack aspect ratio

*a/2c*. In the present calculations of crack growth, initially the

*M*value at the tips of the crack at the plate surface,

_{k}*M*, was assumed to be identical to the

_{kc}*M*value in depth,

_{k}*M*, at

_{ka}*a*= 0.1mm since this represents a good approximation

^{[21]}and the predicted crack aspect ratio also agreed well with that observed experimentally. However, the stress concentration effect of the welded joint decreases as the crack grows across the plate width away from the end of the stiffener. To reflect this observation, the

*M*values were varied, decreasing from an initial value of about 3.9 (corresponding to

_{kc}*M*at

_{ka}*a*= 0.1mm) to

*M*= 1.0 when the crack had just grown beyond the weld (about 32mm long) in front of the attachment. At and beyond this crack length it was assumed that the effect of the weld on surface crack growth could be neglected.

_{kc}### 6.3 Determination of fatigue crack growth relationship

Crack growth was monitored in CA specimens F-11 and F-12, tested under a maximum stress of 280N/mm^{2} but with different stress ranges, and specimen F-14, tested at the same stress range as specimen F-11 but with the lower maximum stress of 135N/mm^{2}. The observed crack growth behaviour was in good agreement with fracture mechanics calculations when they were based on the following fatigue crack growth relationships, independently of the mean stress:

*da/dN* = 0, when Δ*K*≤ Δ*K _{th}* = 63N/mm

^{3/2}

*da/dN* = 2.1x10^{-13} Δ*K*^{3}, when Δ*K* > 63N/mm^{3/2}

[9]

The assumed threshold stress intensity factor range, Δ*K _{th}* , is the lower bound value in BS 7910. The crack growth rate was slightly lower than that corresponding to the simplified mean growth rate given in BS 7910 for R>0.5. An example of the good agreement between actual and calculated surface crack growth is shown in

*Figure 11*. It will be seen that the crack growth data did not exhibit smooth curves. This was not surprising because the crack growth was along the curved weld toe initially and then grew away from the weld at the attachment end. Another relevant factor could be the decreasing magnitude of the residual stresses with increasing distance from the centreline of the attachment plate.

**Fig.11. Comparison between the measured and predicted fatigue crack growths in specimen F-14, tested under a constant amplitude stress range of 65N/mm ^{2} and at a maximum stress of 135N/mm ^{2}**

### 6.4 Fracture mechanics fatigue life calculations

The above fracture mechanics model (including *K* solution, *M _{k}* factor and the crack growth rate) was also used to calculate the total fatigue endurance of the type F specimen. By assuming an average initial flaw depth of 0.15mm

^{[22]}a crack length

*2c*=0.6mm (

*a/2c*=0.25), and that fatigue endurance was controlled by fatigue crack propagation only, the calculated fatigue endurance S-N curve for the type F specimen is included in

*Figure 10*. It will be seen that it agreed very well with the experimental data. This supported the fracture mechanics model and also implied that the fatigue endurance of this type of specimen was predominantly controlled by a crack propagation process, not crack initiation.

Turning to the specimens tested under VA loading, fatigue crack growth was measured in five specimens under Sequence A (F-04, F-05, F-08, F-13 and F-15), two under Sequence B (F-07 and F-10) and two under Sequence C (F-06 and F-09). It was found that crack growth under spectrum loading could not be predicted accurately using the crack growth rate relationship obtained under CA loading. This significantly and modestly under-estimated the crack growth rate observed under Sequences A, and B respectively and over-estimated that under Sequence C. Examples are shown in *Figure 12*. To predict the crack growth accurately, the crack growth rate parameter A in Eq. 5 would need to be in the ranges 2.5x10^{-13} to 5.5x10^{-13} for Sequence A, 1.9x10^{-13} to 2.6x10^{-13} for Sequence B and 1.4x10^{-13} to 1.8x10^{-13} for Sequence C. The ratio of the A value for CA to the average A value for each loading sequence varied, from 0.52 for Sequence A, to 0.93 for Sequence B and to 1.31 for Sequence C. These agree well with the Miner's rule damage sums obtained under the three Sequences. In other words, the use of fracture mechanics crack growth analysis to calculate the fatigue lives of the type F specimens under VA loading would produce essentially the same errors as calculations based on Miner's rule. This implies that the factors responsible for the deviations between the actual lives and those calculated by Miner's rule were mainly related to crack growth, rather than to crack initiation as suggested by others. ^{[23,24]}

**Fig.12. Comparison between the measured and predicted fatigue crack growths in specimens F-15, F-10 and F-09. They were tested under VA loading with the same minimum p _{i} of 0.25, and under Sequence A, Sequence B and Sequence C, respectively**

## 7 Discussion

### 7.1 Validity of Miner's rule

The present work is relevant with respect to two basic assumptions made in the fatigue design of welded joints. Firstly, the fatigue strength of welded joints is dependent only on stress range and the effect of mean stress can be ignored.^{[25,26]} This is based on the assumption that high tensile residual stresses, up to the yield strength of the base metal, will be present in welded joints. Because of this, an applied stress is considered to be superimposed on such residual stress to give an effective stress of the same range but cycling down from tensile. Secondly, there is no interaction between applied stress cycles, so the fatigue damage due to the application of a particular stress cycle in a VA load sequence is exactly the same as that due to the same stress cycle under CA loading, which is the basis of Miner's linear cumulative damage rule.

According to the above two assumptions, it is expected that:

- The CA S-N curves obtained at R=-1, R=0 and at a constant maximum stress would be the same for the type G specimen.
- The fatigue endurance of specimens F-04 and F-13, which were tested under the same spectrum but with different maximum stress, would be similar.
- The specimens tested under Sequences A, B and C would have similar fatigue endurance when the minimum
*p*value in each sequence was the same._{i} - The average value of Σ(n/N) for each sequence would be about 1.0.

However, the results in the present work were different from the above expectations, suggesting the breakdown of the two basic assumptions under certain circumstances. It is acknowledged that a larger CA database, even from the same batch of specimens, could well exhibit more scatter and so lend support to the conclusion that Miner's rule was reasonably accurate. However, there is less doubt about the accuracy of the rule when considering just the present results.

It should be noted that, for each stress range, the minimum and maximum stresses were the same between the CA and the VA under Sequence A. This ruled out any possible effect of mean stress between the CA and VA loading on fatigue endurance. The test results suggested that some form of interaction between the applied stresses was the major factor contributing to the difference in Miner's rule damage sums obtained for the three sequences.

### 7.2 Stress interaction - underloading effect

Although stress interaction under VA loading has been well recognised, much of the work in this area has focused on the tensile overloading effect whereby the effective stress intensity factor range, Δ*K _{eff}* , for the following lower stress ranges is smaller than the applied due to the crack closure effect, resulting in reduced crack growth rates when compared to those under CA loading. The results of the specimens tested under Sequence C in the current investigation could be explained by this mechanism. However, work on the effect of underloading, where the absolute magnitude of unloading is significantly greater than that in the subsequent cycle, that causes an increase in the rate of crack growth under the following cycles is comparatively limited.

^{[27-29]}

Fleck^{[27]} conducted CA and VA fatigue crack growth tests on specimens of two plain materials: steel and aluminium alloy. These involved a VA spectrum with two stress ranges but the same maximum stress: one major cycle followed by different numbers of smaller stress range cycles, *n*. The ratio of the major to the minor stress range, Φ, varied. All tests were carried out at high stress ratios (*R*=0.5 and 0.75) in a Δ*K* range between 14-30MPa√m. A parameter called the acceleration factor, Γ, defined as the ratio of the measured growth rate per block to that predicted by a linear summation of the CA crack growth response, was used to indicate the stress interaction effect. The maximum Γ value was found to be 1.79 which corresponded to *n* =~10 and Φ=~0.5. In the present investigation, the ratio of the maximum crack growth rate under Sequence A to the crack growth rate under CA was 2.6, greater than the maximum value obtained by Fleck.

Enhanced crack propagation rates due to underloading in two aluminium alloys were also reported recently .^{[28]} The spectrum involved two stress ranges with identical maximum stress. The acceleration in crack growth rate was found to be about 30% in one Al alloy but up to 1200% in another, depending on the number of small cycles between underloads, as well as other loading conditions. However, the very high effect seen in the second alloy was due to a distinct change in crack growth mechanism associated with the particular microstructure of the alloy, which is unlikely to be relevant to common weldable metals.

The detrimental effect on fatigue performance of underloading has also been observed in welded specimens. In an early study, Gurney^{[6]} investigated the fatigue performance of the type G specimen under three simple spectra, all containing a major cycle followed by two smaller cycles. The characteristics of the smaller stress cycles varied, one having the same maximum stress as the major cycle (referred to by Gurney as type C), one having the same minimum stress as the major cycle (type B) and the third having the same mean stress and the major cycle (type A). It was found that the specimens tested under type C spectrum always produced the lowest fatigue endurance. Later, Gurney^{[6]} investigated the same type of specimen in a high strength steel under two spectra with identical stress distributions (number of cycles at each stress range was the same) and block length (10^{4} cycles), one under a constant maximum stress of 500N/mm^{2} and the other at a constant minimum stress of zero (R=0). He found that the fatigue endurance of the specimens tested under the former conditions were all significantly less than those tested under the latter conditions.

Unlike overloading, the mechanisms for enhanced crack growth due to underloading are not well established. One possible mechanism could be related to the introduction of tensile residual stresses around the crack tip.^{[30]} This mechanism may be applicable for plain materials but questionable for welded joints since welding-induced tensile residual stresses are present in welds. Other possible mechanisms proposed include: reduced ductility due to strain hardening of material ahead of the crack tip from the major cycles;^{[27]} minor stresses following an under-load experiencing a higher tensile mean stress at the crack tip than that occurring under CA loading;^{[27]} increased ΔK_{eff} due to reduced crack opening stress intensity factor, K_{op}, [28,29]. It may be claimed that the higher Σ(n/N) value of specimen F-13, when compared with specimen F-04, was associated with this mechanism. However, with reduced maximum stress applied to F-13, more of the residual tensile stress induced by welding would be expected to remain after loading. As a result, the actual stress range could still be effectively fully tensile even under an applied compressive stress of 63N/mm^{2} for this specimen.

By relating to the results of crack growth measurements in both CA and VA tests, it seems certain that for specimens with the same crack size and loaded with the same stress range, the effective stress intensity factor ranges for CA and VA loading are different. Furthermore, the same is true for the three VA loading sequences, and this led to different crack growth rates for the different loading conditions.

### 7.3 Constant amplitude S-N curve used with Miner's rule

A possible reason for Σ(n/N) values below unity is that crack closure is inhibited under some VA loading regardless of the applied stress ratio or mean stress, such that cyclic stresses in such VA loading spectra produce fatigue damage associated with high tensile mean stress CA loading conditions. Support for this comes from the re-analysis of VA test results obtained from type G specimens by Gurney^{[5]} at *R*=0 and *R*=-1 using the present CA data obtained with *S _{max}* = 280 N/mm

^{2}. In spite of the presence of high tensile residual stresses, this was lower than that obtained by Gurney at R=0, which he had used when applying Miner's rule. The accuracy of Miner's rule was improved considerably. It may be noted that a dependence on mean stress of fatigue endurance for some weld details has also been reported elsewhere

^{[5,17,31-33]}, even when high tensile residual stresses were known or expected to be present in the welded joints.

^{[5,17]}

Under VA loading, the actual mean stresses (a combination of residual and applied stresses) for small stress cycles are expected to be lower than those under CA loading because of the relaxation of residual stresses under the peak stress in the spectrum. This would have yielded Σ(n/N)>1.0, but all tests under Sequences A and B gave values less than 1.0. This suggests that stress interaction effects associated with the type of loading sequence are more significant than the mean stress effect.

### 7.4 Small stresses in spectrum

Under Sequence A loading, the lowest stress range that was found to be 'fully damaging', defined as ΔS'_{min}, was found to be as low as 8.4N/mm^{2} for the type G specimen. This was lower than the value of 10.1N/mm^{2} found by Gurney^{[5]} for the same type of specimen under a spectrum with the peak stress range applied at R=0. The stress range of 8.4N/mm^{2} corresponds to a fatigue endurance of 3.5x10^{8} cycles on the present type G specimen S-N curve, significantly greater than the endurance of 10^{7} cycles at which a slope change from *m* to (*m*+2) is recommended in BS 7608 to allow for fatigue damage due to stresses below the CAFL under VA loading. A consequence was that the conventional bi-linear S-N curve with the slope change at *N*=10^{7} cycles under-estimated the damaging effect of stresses below the CAFL and was therefore non-conservative.

As noted previously, the relative fatigue limit, ΔS'_{min}/CAFL, increased with increase in the basic fatigue strength of the weld detail. Thus, for the type F specimen, ΔS'_{min} appeared to be ~31.5N/mm^{2}, which corresponded to a fatigue endurance of 3.3x10^{7} cycles. In spite of this, allowing for basic scatter in the fatigue test results, it was found that the damaging effect of stress ranges between 31.5 N/mm^{2} and the conventional CAFL, corresponding to *N* = 10^{7} cycles, of 45.3 N/mm^{2} was still accounted for satisfactorily when applying Miner's rule by the mean CA S-N curve (slope *m*) extrapolated beyond 10^{7} cycles at the shallower slope of *m*+2, as widely recommended. In other words, in contrast to the results for type G specimens, the present type F specimen results obtained under Sequence A support the current design method for accounting for the damage due to stresses below the CAFL. However, this conclusion is based on very limited data and clearly there is the need for further test results, particularly for loading spectra with a greater proportion of low stresses than those used here. It will also be recalled that published VA data, some obtained from type F specimens [e.g., 2,3] have shown that the bi-linear S-N curve with the slope change at 10^{7} cycles can seriously under-estimate the damaging effect of stresses below the CAFL.

None of the tests performed under Sequences B and C included sufficient low stresses to establish the best approach for assessing their damaging effect when applying Miner's rule.

### 7.5 Implication of the current work to fatigue design

The present work suggests that Miner's rule could be non-conservative under spectrum loading where cycling down from a constant maximum stress is predominant. On the other hand, Miner's rule can be unduly conservative in a spectrum where most cycles involve cycling-up from a constant minimum stress. Although loading spectra like Sequence A are primarily chosen to simulate the severe conditions that are thought to exist in welded joints containing high tensile residual stresses, they do represent the actual loading sequence for some engineering structures. As summarised by Fleck,^{[27]} a number of engineering components can be subjected to periodic underloading, such as gas storage vessels, gas turbine blades, railway lines and aircraft wings. A simple example for producing the Sequence A spectrum could be a combination of the following two loadings: a constant high load to keep the maximum stress constant and cyclic loads, similar to Sequence C spectrum, loading in a direction opposite to the constant load. Furthermore, some loading spectra that do not involve cycling down from a constant high tensile stress may also exhibit under-loading behaviour. One such example is the wide band spectrum^{[5]} but the present Sequence B is another. Therefore, due consideration of the type of variable amplitude loading spectrum involved is required when Miner's rule is used in fatigue design.

With regard to current design guidance, if a service loading spectrum is similar to Sequence A or predominantly involves stresses cycling down from a fixed tensile level, the present and other test results suggest that the Miner's rule damage sum should not take a value greater than 0.4, i.e. Σ(n/N) ≤ 0.4. Furthermore, the CA S-N curve, to be used in the calculation of Σ(n/N), should be established using the same maximum stress as that in the spectrum. In this respect the evidence from the present study, particularly in the case of the type F weld detail, suggests that the current design S-N curves in BS 7608 may not be low enough to allow for cycling down from a high fixed tensile stress. Clearly, this needs further investigation, including consideration of simply very high mean stress or stress ratio conditions. Finally, with regard to the assumption to be made about the damaging effect of stresses below the CAFL, it may be necessary to draw a distinction between weld details depending on their basic fatigue strengths. On the basis of the present test results for the relatively low fatigue strength type G specimen, it is recommended that Class G details are assessed on the basis of the CA S-N curve extrapolated beyond the CAFL without a slope change. In contrast, the current approach of extrapolating the CA S-N curve beyond the CAFL, assumed to correspond to *N* = 10^{7} cycles, at a shallower slope (*m* changing to *m*+2) could be suitable for Class F and higher details. However, since evidence for this from the present project and published work is very limited, for critical cases the CA S-N curve extrapolated beyond the CAFL without a slope change is still recommended.

Such stringent measures should not be required for spectra of the types that do not produce significant crack growth acceleration (such as Sequence B) or encourage crack growth retardation (such as Sequence C), or perhaps for higher fatigue strength weld details. However, insufficient relevant experimental evidence is available at this stage to make specific recommendations.

## 8 Conclusions

On the basis of fatigue tests performed on two types of welded specimen, one corresponding to BS 7608 design Class G and the other to Class F, under both constant and variable amplitude loading with three different sequences, the following conclusions can be drawn:

- The CA fatigue performance of the type G specimen decreased with increase in applied mean stress, but this was not the case for the type F specimen.
- Residual stress measurements confirmed the presence of high tensile residual stresses in regions near crack initiation sites for both types of specimen. They were reduced significantly under fatigue loading for a small proportion of the fatigue life.
- For the same basic VA load spectrum, Σ(n/N) at failure depended on the sequence applied. Miner's rule was substantially non-conservative (Σ(n/N) < 1 at failure) for all tests under Sequence A, with values down to around 0.4 for both specimen types. It was modestly non-conservative (Σ(n/N ~ 0.8) for tests under Sequence B but conservative (Σ(n/N > 1.3) for Sequence C.
- The above findings agreed very well with the results from crack growth measurements. Crack growth under spectrum loading could not be predicted accurately using the crack growth rates obtained under constant amplitude loading. They significantly underestimated the crack growth in specimens tested under Sequence A, marginally underestimated that under Sequence B and overestimated that under Sequence C, in all cases in similar proportions to the Σ(n/N) values at failure.
- There were strong indications that the most significant effect on fatigue behaviour of the type of loading was stress interaction, whereby high stresses caused significant crack growth acceleration from subsequent lower stresses under Sequence A, moderate crack growth acceleration under Sequence B and crack growth retardation under Sequence C. Variations in the applied mean or maximum stress levels were of secondary importance.
- From tests performed under Sequence A, stress ranges well below the fatigue limit were found to be as damaging as implied by the CA S-N curve extrapolated beyond the CAFL without changing the slope. The value of the minimum fully damaging stress range, referred to as ΔS'
_{min}, depended on the basic fatigue strength of the welded joint, being lower for the Class G detail (≤ 8.4N/mm^{2}) than the Class F (≤ 31.5 N/mm^{2}). - For loading conditions similar to Sequence A, or other spectra that are expected to cause crack growth acceleration from stress interaction, the present results indicate that Miner's rule should be applied assuming Σ(n/N) ≤ 0.4 at the end of the required life and in conjunction with the CA S-N curve extrapolated beyond the CAFL without a slope change. Limited evidence suggested that the less stringent bi-linear S-N curve with the slope change from
*m*to*m*+2 at 10^{7}cycles may be suitable for assessing Class F and higher details, but further work is required to confirm this.

It should be noted that the conclusions drawn in the present work were based on a limited number of tests. Verification is required by conducting tests covering different maximum stress range and maximum peak stress in a spectrum, welded joint types and types of spectra (wide band).

## 9 Acknowledgements

This work was supported by the Industrial Members of TWI. In addition, the authors would like to thank the staff of the Fatigue Laboratory in carrying out the experimental work.

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