S/N curves define the fatigue life at any given stress range S. However, in most situations in practice the material or structural component will not be sub-jected to cycles of a single stress range, but to dif-fering numbers of cycles of different stress ranges.
For example wind action will result in a few cycles at a high stress range from severe storms and a large number of cycles at lower stresses from lesser strength winds. To estimate the effect of this cumu-lative fatigue damage Miner’s rule is used. This accounts for the partial effect of the number of cycles at each particular stress range by considering that, if the material is stressed for n1 cycles at a stress range that will cause failure in a total of N1 cycles, then a fraction n1/N1 of the fatigue life is used up;
failure occurs when the sum of all the fractions, Σ ni/N1, reaches 1, irrespective of the sequence of application of the various cycles of loading.
Figure 2.20 illustrates the case of three stress ranges, S1, S2 and S3, being applied for n1, n2 and n3 cycles respectively, and for which the total fatigue lives are N1, N2 and N3. The n1 cycles at the stress range S1 use up n1/N1 of the total fatigue life. The same applies for the n2 cycles at S2 and the n3 cycles at S3. Therefore the proportion of the total fatigue life used up by all three stress ranges is:
n1/N1 + n2/N2 + n3/N3
Fatigue endurance limit
Non-ferrous metal e.g. copper alloy 200
300 400 500
100
Cycles to failure – N
Stress range – S (MPa)
103 104 105 106 107 108
Mild steel
Fig. 2.19 Fatigue life data (S/N curves) for mild steel and a copper alloy.
N1 N2 N3
n2 n1 n3
S1
S2
S3
Stress range (S) S/N curve
Number of cycles (N) Applied
loads
Fig. 2.20 Example of the application of Miner’s rule of cumulative fatigue damage.
Failure will occur when this sum reaches 1. This will be achieved with continued cyclic loading, in which one or all of n1, n2 or n3 may increase, depending on the nature of the loading. The general expression of Miner’s rule is that for failure:
∑ n1/N1 = 1 (2.22)
2.9 Impact loading
Structures and components of structures can be subjected to very rapid rates of application of stress and strain in a number of circumstances, such as explosions, missile or vehicle impact, and wave slam. Materials can respond to such impact loading by:
• an apparent increase in elastic modulus, but this is a third or fourth order effect only – a 104 times increase in loading rate gives only a 10% increase in elastic modulus
• an increase in brittle behaviour, leading to fast brittle fracture in normally ductile materials. This can be very dangerous – we think we are using a ductile material that has a high work to fracture and gives warning of failure, but this reacts to impact loading like a brittle material. The effect is enhanced if the material contains a pre-existing defect such as a crack.
The latter effect cannot be predicted by extrapolat-ing the results of laboratory tensile or compression tests such as those described earlier, and impact test procedures have been developed to assess the be-haviour of specimens containing a machined notch, which acts a local stress raiser. The Charpy test for metals is a good example of such a test. In this, a heavy pendulum is released and strikes the standard specimen at the bottom of its swing (Fig. 2.21). The specimen breaks and the energy needed for the frac-ture is determined from the difference between the starting and follow-through positions of the pendu-lum. The energy absorbed in the fracture is called the Charpy impact value. As we have discussed earlier in the chapter, brittle materials require less energy for failure than ductile materials, and an impact value of 15J is normally used as a somewhat arbitrary division between the two, i.e. brittle mater-ials have a value below this, and ductile matermater-ials above.
An example of the use of the test is in determining the effect of temperature on ductile/brittle behaviour.
Many materials that are ductile at normal tempera-tures have a tendency to brittleness at reducing
temperatures. This effect for a particular steel is shown in Fig. 2.22. The decrease in ductility with falling temperature is rapid, with the 15J division occurring at about -20°C, which is called the transi-tion temperature. It would, for example, mean that this steel should not be used in such structures as oil production installations in Arctic conditions.
Impact behaviour and fast fracture are an import-ant part of the subject called fracture mechanics, which seeks to describe and predict how and why cracking and fracture occur. We will consider this in more detail in Chapter 4.
10 mm 8 mm 55 mm
−75 −50 −25 0 25 50 75
Temperature (deg C)
Charpy impact energy (J)
15
Transition temperature Brittle Ductile
0 50 100 150
Fig. 2.21 Charpy impact test specimen (from dimensions specified in BS EN ISO 148-3:2008).
Fig. 2.22 Variation of the Charpy impact energy of a steel with temperature (after Rollason, 1961).
Mechanical properties of solids
2.10 Variability, characteristic strength and the Weibull distribution
Engineers are continually faced with uncertainty.
This may be in the estimation of the loading on a structure (e.g. what is the design load due to a hur-ricane that has a small but finite chance of occurring sometime in the next 100 years?), analysis (e.g. what assumptions have been made in the computer model-ling and are they valid?) or with the construction materials themselves. When dealing with uncertainty in materials, with natural materials such as timber we have to cope with nature’s own variations, which can be large. With manufactured materials, no matter how well and carefully the production process is controlled, they all have some inherent variability and are therefore not uniform. Furthermore, when carrying out tests on a set of samples to assess this variability there will also be some unavoidable variation in the testing procedure itself, no matter how carefully the test is carried out or how skilful the operative. Clearly there must be procedures to deal with this uncertainty and to ensure a satisfac-tory balance between safety and economy. Structural failure can lead to loss of life, but the construction costs must be acceptable.
In this section, after some basic statistical consid-erations for describing variability, we will discuss two approaches to coping with variations of strength – characteristic strength and the Weibull distribu-tion. We will take strength as being the ultimate or failure stress of a material as measured in, say, a tension, compression or bending test (although the arguments apply equally to other properties such as the yield or proof stress of a material).
2.10.1 desCriPTions oF variabiliTy
A series of tests on nominally identical specimens from either the same or successive batches of mater-ial usually gives values of strength that are equally spread about the mean value with a normal or Gaussian distribution, as shown in Fig. 2.23.
The mean value, sm, is defined as the arithmetic average of all the results, i.e.:
sm = (∑ s)/n (2.23) where n is the number of results.
The degree of spread or variation about the mean is given by the standard deviation, s, where
s2 = ∑(s - sm)2/(n - 1) (2.24) s2 is called the variance, and s has the same units as s.
Materials can have any combination of mean strength and variability (or standard deviation) (Fig. 2.24).
For comparison between materials, the coefficient of variation, c, is used, where
c = s/sm (2.25)
c is non-dimensional, and is normally expressed as a percentage. Typical values of c are 2% for steel, which is produced under carefully controlled condi-tions, 10–15% for concrete, which is a combination of different components of different particle sizes, and 20–30% for timber, which has nature’s own variations. Steel and timber are at the two ends of the variability scale of construction materials.
For structural use of a material, we need a ‘safe’
stress that takes into account of both the mean failure stress and the variability. This is done by considering the normal distribution curve (Fig. 2.23) in more detail. The equation of the curve is
y= s - -s
Probability density or Frequency of occurrence
Failure stress (σ)
Fig. 2.24 Two combinations of mean strength and variability.
Fig. 2.23 Typical normal distribution of failure stresses from successive tests on samples of a construction material.
Some important properties of this equation are:
• The curve encloses the whole population of data, and therefore not surprisingly, integrating the above equation between the limits of -∞ and +∞ gives an answer of 1, or 100 if the probability density is expressed as a percentage.
• 50% of the results fall below the mean and 50%
above, but also, as shown in Fig. 2.25:
� 68.1% of results lie within one standard deviation of the mean
� 95.5% of results lie within two standard deviations of the mean
� 99.8% of results lie within three standard deviations of the mean.
2.10.2 CharaCTerisTiC sTrengTh
A guaranteed minimum value of stress below which no sample will ever fail is impossible to define – the nature of the normal distribution curve means that there will always be a chance, albeit very small, of a failure below any stress value. A value of stress called the characteristic strength is therefore used, which is defined as the stress below which an ac-ceptably small number of results will fall. Engineer-ing judgement is used to define ‘acceptably small’.
If this is very small, then there is a very low risk of failure, but the low stress will lead to increased cross-sectional area and hence greater cost. If it is higher, then the structure may be cheaper but there is an increased risk of failure.
Clearly a balance is therefore required between safety and economy. For many materials a stress below which 1 in 20 of the results occurs is con-sidered acceptable, i.e. there is a 5% failure rate.
Analysis of the normal distribution curve shows that this stress is 1.64 standard deviations below
the mean. This distance is called the margin and so, as shown in Fig. 2.26:
characteristic strength = mean strength – margin
schar = sm - ks (2.27)
where k, the standard deviation multiplication factor, is 1.64 in this case.
The value of k varies according to the chosen failure rate (Table 2.1), and, as we said above, judge-ment and consensus are used to arrive at an acceptable failure rate. In practice, this is not always the same in all circumstances; for example, 5% is typical for concrete (i.e. k = 1.64), and 2% for timber (i.e.
k = 1.96).
There is a further step in determining an allow-able stress for design purposes. The strength data used to determine the mean and standard deviation for the above analysis will normally have been obtained from laboratory tests on small specimens, which generally will have no apparent defects or damage. They therefore represent the best that can be expected from the material in ideal or near ideal circumstances. In practice, structural elements and members contain a large volume of material, which
Probability density
Failure stress (σ) 34.1% 34.1%
13.6% 13.6%
2.1% 2.1%
0.1% 0.1%
% of results
−3s −2s −s σm s 2s 3s
Fig. 2.25 Proportion of results in the regions of the normal distribution curve.
Probability density
Failure stress (σ) σm
5% of results
σchar
Margin 1.64s
Fig. 2.26 Definition of characteristic strength (schar) and margin for a 1 in 20 (5%) failure rate criterion.
Table 2.1 Values of k, the standard deviation multiplication factor, for various failure rates
Failure rate (%) k
50 0
16 1
10 1.28
5 1.64
2 1.96
1 2.33
Mechanical properties of solids has a greater chance of containing manufacturing
and handling defects. This size effect is taken into account by reducing the characteristic strength by a partial materials’ safety factor, gm.
It is normal practice for gm to be given as a value greater than one, so the characteristic strength has to be divided by gm to give the allowable stress.
Hence:
allowable design stress
= characteristic strength/gm
= (mean strength - margin)/gm (2.28) As with the failure rate, the value of gm is based on knowledge and experience of the performance of the material in practice. For example, typical values recommended in the European standard for struc-tural concrete design (Eurocode 2, BS EN 1992) are 1.15 for reinforcing steel and 1.6 for concrete.
2.10.3 The weibull disTribuTion
An alternative statistical approach to the distribu-tion of strength, particularly for brittle materials, was developed by the Swedish engineer Waloddi Weibull. As we have discussed in section 2.8 (and will consider further in Chapter 4) brittle fracture is initiated at flaws or defects, which are present to a greater or lesser extent in all materials. Therefore the variations of strength can be attributed to vari-ations in the number and, more particularly, the maximum size of defect in a test specimen. Larger specimens have a higher probability of containing larger defects and therefore can be expected to have a lower mean strength (as just discussed in relation to the partial materials safety factor, gm).
Weibull defined the survival probability, Ps(V0), as the fraction of identical samples of volume V0
that survive after application of a stress s. He then proposed that:
Ps(V0) = exp{-(s/s0)m} (2.29) where s0 and m are constants. Plots of this equation for three values of m are shown in Fig. 2.27. In each case, when the stress is low, no specimens fail and so the survival probability is 1, but at increasing stress more and more samples fail until eventually, when they have all failed, the survival probability is zero. Putting s = s0 in equation (2.29) gives Ps(V0) = 1/e = 0.37, so s0 is the stress at which 37% of the samples survive. The value of m, which is called the Weibull modulus, is a measure of the behaviour on either side of s0, and therefore
indi-cates the degree of variability of the results (and in this sense has a similar role to the coefficient of variation as defined in equation 2.25). Lower values indicate greater variability; m for concrete and bricks is typically about 10, whereas for steel it is about 100.
We can extend this analysis to give an estimate of the volume dependence of survival probability.
Ps(V0) is the probability that one specimen of volume V0 will survive a stress s. If we test a batch of n such specimens, then the probability that they will all survive this stress is {Ps(V0)}n. If we then test a volume V = nV0 of the material, which is the equiva-lent of combining all the smaller specimens into a single large specimen, then the survival probability, Ps(V), is still {Ps(V0)}n. value for Ps(V), the design stress for structural element of volume V can be calculated.
References
Case J, Chilver H and Ross C (1999). Strength of Mater-ials and Structures, Elsevier Science & Technology, London, p. 720.
Rollason EC (1961). Metallurgy for Engineers, Edward Arnold, London.
Fig. 2.27 The Weibull distribution for three values of the Weibull modulus, m.
In Chapter 1 we discussed the various ways in which atoms bond together to form solids, liquids and gases, and some of the principles involved in the changes between these states. In Chapter 2 we described the behaviour of solids when subjected to load or stress and the various rules and con
stants used to characterise and quantify this. We now go on to consider the structure of solids in more detail, which will provide an explanation for much of the behaviour described in Chapter 2.
Although the type of bonding between atoms goes some way towards explaining the properties of the resulting elements or compounds, it is equally important to understand the ways in which the atoms are arranged or packed together. We start by con sidering the relatively ordered structure of crystalline solids, and then discuss some aspects of the less ordered structures of ceramics and polymers.
3.1 Crystal structure
Many construction materials, particularly metals and some ceramics, consist of small crystals or grains within which the atoms are packed in regular, repeating, threedimensional patterns giving a long-range order. The grains are ‘glued’ together at the grain boundaries; we will consider the importance of these later, but first we will discuss the possible arrangements of atoms within the grains. For this, we will assume that atoms are hard spheres – a considerable but convenient simplification.
It is also convenient to start with the atomic structure of elements (which of course consist of singlesized atoms) that have nondirectional bond
ing (e.g. pure metals with metallic bonds). The simplest structure is one in which the atoms adopt a cubic pattern i.e. each atom is held at the corner of a cube. For obvious reasons, this is called the simple cubic (SC) structure. The atoms touch at
the centre of each edge of the cube (Fig. 3.1a). The structure is sometimes more conveniently shown as in Fig. 3.1b. We can use this figure to define some properties of crystalline structures:
• the unit cell: the smallest repeating unit within the structure, in this case a cube (Fig. 3.1)
• the coordination number: the number of atoms touching a particular atom or the numbers of its nearest neighbours, in this case 6 (Fig. 3.2)
• the closed-packed direction: the direction along which atoms are in continuous contact, in this case any of the sides on the unit cell