Stress Concentration Crack

Simple riveted lap joint

Failure of rivet in single shear

Butt joint witti two cover plates l-ailjre of ri yet in douole shear

1.16. Allowable working stress-factor of safety

The most suitable strength or stiffness criterion for any structural element or component is normally some maximum stress or deformation which must not be exceeded. In the case of stresses the value is generally known as the maximum allowable working stress.

Because of uncertainties of loading conditions, design procedures, production methods, etc., designers generally introduce a factor of safety into their designs, defined as follows:

maximum stress factor of safety = —--—----(1.9)

allowable working stress

However, in view of the fact that plastic deformations are seldom accepted this definition is sometimes modified to

. _ yield stress (or proof stress)

allowable working stress

In the absence of any information as to which definition has been used for any quoted value of safety factor the former definition must be assumed. In this case a factor of safety of 3 implies that the design is capable of carrying three times the maximum stress to which it is expected the structure will be subjected in any normal loading condition. There is seldom any realistic basis for the selection of a particular safety factor and values vary significantly from one branch of engineering to another. Values are normally selected on the basis of a consideration of the social, human safety and economic consequences of failure. Typical values range from 2.5 (for relatively low consequence, static load cases) to 10 (for shock load and high safety risk applications)—see §15.12.

1.17. Load factor

In some loading cases, e.g. buckling of struts, neither the yield stress nor the ultimate strength is a realistic criterion for failure of components. In such cases it is convenient to replace the safety factor, based on stresses, with a different factor based on loads. The load factor is therefore defined as:

. ,, load at failure load factor = —-r-.--—¡—r (1.10

allowable working load

This is particularly useful in applications of the so-called plastic limit design procedures.

1.18. Temperature stresses

When the temperature of a component is increased or decreased the material respectively expands or contracts. If this expansion or contraction is not resisted in any way then the processes take place free of stress. If, however, the changes in dimensions are restricted then stresses termed temperature stresses will be set up within the material.

Consider a bar of material with a linear coefficient of expansion a. Let the original length of the bar be L and let the temperature increase be t. If the bar is free to expand the change in length would be given by

and the new length

If this extension were totally prevented, then a compressive stress would be set up equal to that produced when a bar of length L (1 + at) is compressed through a distance of Lai. In this case the bar experiences a compressive strain

In most cases at is very small compared with unity so that

This is the stress set up owing to total restraint on expansions or contractions caused by a temperature rise, or fall, t. In the former case the stress is compressive, in the latter case the stress is tensile.

If the expansion or contraction of the bar is partially prevented then the stress set up will be less than that given by eqn. (1.10). Its value will be found in a similar way to that described above except that instead of being compressed through the total free expansion distance of Lat it will be compressed through some proportion of this distance depending on the amount of restraint.

Assuming some fraction n of Lai is allowed, then the extension which is prevented is (1 -n)Lxt. This will produce a compressive strain, as described previously, of magnitude

or, approximately, e = (1 - n)Lat/L = (1 - n)at. The stress set up will then be E times e.

Thus, for example, if one-third of the free expansion is prevented the stress set up will be two-thirds of that given by eqn. (1.12).

1.19. Stress concentrations-stress concentration factor

If a bar of uniform cross-section is subjected to an axial tensile or compressive load the stress is assumed to be uniform across the section. However, in the presence of any sudden change of section, hole, sharp corner, notch, keyway, material flaw, etc., the local stress will rise significantly. The ratio of this stress to the nominal stress at the section in the absence of any of these so-called stress concentrations is termed the stress concentration factor.

1.20. Toughness

Toughness is defined as the ability of a material to withstand cracks, i.e. to prevent the transfer or propagation of cracks across its section hence causing failure. Two distinct types of toughness mechanism exist and in each case it is appropriate to consider the crack as a very high local stress concentration.

The first type of mechanism relates particularly to ductile materials which are generally regarded as tough. This arises because the very high stresses at the end of the crack produce local yielding of the material and local plastic flow at the crack tip. This has the action of blunting the sharp tip of the crack and hence reduces its stress-concentration effect considerably (Fig. 1.15).

area of local yielding of material reducing the stress-concentration effect

Fig. 1.15. Toughness mechanism-type

The second mechanism refers to fibrous, reinforced or resin-based materials which have weak interfaces. Typical examples are glass-fibre reinforced materials and wood. It can be shown that a region of local tensile stress always exists at the front of a propagating crack and provided that the adhesive strength of the fibre/resin interface is relatively low (one-fifth the cohesive strength of the complete material) this tensile stress opens up the interface and produces a crack sink, i.e. it blunts the crack by effectively increasing the radius at the crack tip, thereby reducing the stress-concentration effect (Fig. 1.16).

This principle is used on occasions to stop, or at least delay, crack propagation in engineering components when a temporary "repair" is carried out by drilling a hole at the end of a crack, again reducing its stress-concentration effect.

Weak interfaces

Weak interfaces

Stress Concentration Crack

Region of local tensile stress in Interface opens up front of crack tip +o reduce s.c.f. at tip of notch Fig. 1.16. Toughness mechanism-type 2.

Region of local tensile stress in Interface opens up front of crack tip +o reduce s.c.f. at tip of notch Fig. 1.16. Toughness mechanism-type 2.

1.21. Creep and fatigue

In the preceding paragraphs it has been suggested that failure of materials occurs when the ultimate strengths have been exceeded. Reference has also been made in §1.15 to cases where excessive deformation, as caused by plastic deformation beyond the yield point, can be considered as a criterion for effective failure of components. This chapter would not be complete, therefore, without reference to certain loading conditions under which materials can fail at stresses much less than the yield stress, namely creep and fatigue.

Creep is the gradual increase of plastic strain in a material with time at constant load. Particularly at elevated temperatures some materials are susceptible to this phenomenon and even under the constant load mentioned strains can increase continually until fracture. This form of fracture is particularly relevant to turbine blades, nuclear reactors, furnaces, rocket motors, etc.

area of local yielding of material reducing the stress-concentration effect

Fig. 1.15. Toughness mechanism-type

Fracture

Fracture

Primary creep

Secondary creep

Tertiary creep

Primary creep

Secondary creep

Tertiary creep

Initial strain

Time

Fig. 1.17. Typical creep curve.

The general form of the strain versus time graph or creep curve is shown in Fig. 1.17 for two typical operating conditions. In each case the curve can be considered to exhibit four principal features.

(a) An initial strain, due to the initial application of load. In most cases this would be an elastic strain.

(b) A primary creep region, during which the creep rate (slope of the graph) diminishes.

(c) A secondary creep region, when the creep rate is sensibly constant.

(d) A tertiary creep region, during which the creep rate accelerates to final fracture.

It is clearly imperative that a material which is susceptible to creep effects should only be subjected to stresses which keep it in the secondary (straight line) region throughout its service life. This enables the amount of creep extension to be estimated and allowed for in design.

Fatigue is the failure of a material under fluctuating stresses each of which is believed to produce minute amounts of plastic strain. Fatigue is particularly important in components subjected to repeated and often rapid load fluctuations, e.g. aircraft components, turbine blades, vehicle suspensions, etc. Fatigue behaviour of materials is usually described by a fatigue life or S-N curve in which the number of stress cycles N to produce failure with a stress peak of S is plotted against S. A typical S-N curve for mild steel is shown in Fig. 1.18. The particularly relevant feature of this curve is the limiting stress S„ since it is assumed that stresses below this value will not produce fatigue failure however many cycles are applied, i.e. there is infinite life. In the simplest design cases, therefore, there is an aim to keep all stresses below this limiting level. However, this often implies an over-design in terms of physical size and material usage, particularly in cases where the stress may only occasionally exceed the limiting value noted above. This is, of course, particularly important in applications such as aerospace structures where component weight is a premium. Additionally the situation is complicated by the many materials which do not show a defined limit, and modern design procedures therefore rationalise the situation by aiming at a prescribed, long, but finite life, and accept that service stresses will occasionally exceed the value Sn. It is clear that the number of occasions on which the stress exceeds S„, and by how

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