STRESS, STRENGTH AND SAFETY
You are walking in the countryside when you come across a
creek which is too wide to jump. Problem: How can you cross to the far side without a soaking? Applying creativity, you may come up with the following possible solutions, depending upon the circumstances :-
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roll up your jeans and wade across
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dam the creek
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adapt a couple of tent-poles as stilts to walk dry-shod through the creek
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bend a sapling, secure it with a rope, climb the sapling, cut the rope and catapult across
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construct a bridge from a convenient tree branch;
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use a broken branch to pole-vault across
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position piles of boulders to use as stepping stones
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attach a rope to a tree, lassoo another tree on the far side, and swing across hand-over-hand
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build a raft and float across . . . . and so on.
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These and other solutions, apart from the first, require implements - ropes, trees and the like - whose failure may be through either
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fracture, ie. they break, or
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excessive deformation eg. they bend or stretch too much.
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We may examine an implement’s safety (absence of failure) by one of three approaches :-
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Try it with care, an approach which is not possible if failure could be hazardous or cause significant financial loss, or
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Analyse it via a mathematical model of the implement, from which the stresses and deflections may be calculated for the known loading and the existing implement’s size (dimensions) and material (strength, modulus), thus indicating whether the implement is safe or not; or
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Synthesise it, ie. analyse in reverse, 
where the material is chosen and the minimum dimensions necessary to avoid failure are calculated before the implement is later made to suit. [/li][/ul]
Analysis and synthesis thus involve four major aspects of the implement :-
An implement in mechanical engineering is more complex than those above, and requires careful design to ensure that everyone who is associated with it is satisfied with it. A well designed artefact is cheap to manufacture, and is easy and safe to use and to maintain, among other things.
Although safety is just one aspect of design, it is a fundamental necessity for all designs
Implement safety can of course be assured by building a physical model and testing it, but this is usually uneconomical and so one of the major aspects of this course is to demonstrate the formation of mathematical models of various mechanical components - bits and pieces such as shafts, brakes, welds, bearings and the like, which are assembled into machines for transforming mechanical power outside the human performance envelope. These models may be analysed to predict the prototypes’ behaviour and safety before they are built, and in conjunction with sketches enable component design to be carried out.
It must be appreciated that the techniques of (mathematical) model building which are introduced in the context of one particular component are usually applicable to many other components which cannot be considered in the course, and therefore the course emphasis is as much on how we arrive at a result as on what that result is.
In this chapter we trace the steps necessary for setting up the mathematical model (the design equation) of any component and derive the equation for one of the most common components in mechanical engineering - the round shaft. Static failure only is considered at this stage, but the same approach is used later in the course for other failure mechanisms such as fatigue and fracture. Safety factor
A component subjected to a solitary load will be considered in the first instance. This load is interpreted in the context of the component’s nature and duty - thus load usually implies a transverse force in the case of a beam component, or a longitudinal compressive force in a column, or a torque in the case of a shaft, or a pressure in a fluid containment vessel, and so on.
There are two completely different manifestations of the load, which have important consequences for the component :
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the extrinsic actual load is the load exerted on the component by its surrounds, and
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the intrinsic maximum load is the largest load that the component can withstand without failure; the maximum load is a property of the component, a function of its dimensions and material.
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Each of these two loads is expressed usually by its nominal value. ( “Nominal” means “in name only”; a nominal value of something is merely a convenient label. For example, 500mm pipes are made with an outside diameter of 508mm and various wall thicknesses - and hence come in various internal diameters - but they are all referred to by the convenient rounded “500” nominal diameter as a descriptor of size. )
There are many reasons why the actual load on a particular component at a particular time will differ from the nominal value, including -
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the inherent variability of the load (eg. in practice the mass of a “ten tonne truck” will depend on the load it’s carrying),
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static indeterminacy (when components share the load in proportion to their elastic responses - more of this anon), 
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dynamic (or shock ) effects - for example if a weight W is dropped from a height h onto an elastic component of stiffness k, then the peak force in the component is [/li]Fdyn = dynamic magnification factor (dmf ) * W
Elementary energy methods give dmf = [ 1+ ( 1 +2hk/W)1/2 ] for this type of loading, demonstrating that releasing a load W suddenly even from zero height induces a peak force of at least twice that arising from gradual release - the effective actual load is at least twice its nominal value,
. . . . and so on.
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Similarly the maximum load which a particular component can withstand at a particular moment in time will differ from the nominal maximum load due to many factors including -
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dimensions differing from their nominal or expected values - eg. we might look at our creek-bridging branch, calling it 50 mm and figuring the nominal maximum load on that basis; but the diameter may in fact be 47 mm, or 54 mm, or more likely vary along the branch;
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material strength differing from its nominal value due in turn to variations in material composition or heat treatment, to unsuspected flaws (is there an undetected split or other anomaly in our bridging branch?), [/li]. . . . and so on.
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Deviations may be highly significant or they may be insignificant - it’s important that we have an idea of their magnitudes. Sometimes we haven’t much of a clue and therefore have to fear the worst and make suitable allowance for our ignorance.
Clearly a component is safe only if the actual load applied to the component does not exceed the component’s inherent maximum sustainable load. The degree of safety is usually expressed by the safety factor, n :-
COLOR=blue n = maximum load / actual load = Fmax / F [/color]
. . . . and it follows that :
if n = 1 then the component is on the point of failure
if n < 1 then the component is in a failed state
if n > 1 then the component is safe.
The safety factor is usually expressed as a ratio of nominal loads. A higher value of the safety factor seems to indicate a safer component - however this is not necessarily the case as the inevitable variations must be kept in mind.
In Fig.A, n = 1.25 based on nominal values, but because of the relatively large variations in both actual and maximum loads, there is a significant probability of the actual load exceeding the maximum, and hence of failure.
For a negligible failure probability with these levels of variations, the nominal safety factor must be increased by eg. reducing the actual load applied to the component, as indicated in Fig.B - alternatively the maximum load could be increased, by increasing the material’s strength or the component’s dimensions.
Fig.C illustrates how the probability of failure may be decreased also by reducing the variations of the actual and/or maximum loads. This can be accomplished by a better understanding of what’s going on - we’ll amplify this later.
When an assemblage of components is subjected to a single load, the assembly’s safety factor is the smallest of the component safety factors - ‘a chain is only as strong as its weakest link’. If a component or an assemblage is subjected to a number of different simultaneous loads, then the concept of a single safety factor may be inappropriate - but nevertheless all potential failure mechanisms must be investigated when deciding whether an implement is safe to use or not.
IF - and only IF - the stress in a component is proportional to the actual load on the component, then the safety factor may be interpreted also as a stress ratio :-
COLOR=blue n = S / σ [/color]
where S is the strength of the component’s material, and
σ is the stress in the component due to the actual load, F.
Such an interpretation is evidently inapplicable to assemblages. Stresses are proportional to load for the majority of practical elements, so we usually assume that the stress ratio interpretation ( 1a) applies. But this is NOT the case with some not uncommon failure mechanisms (examined later) such as buckling and Hertzian contact - in these cases we must fall back on the fundamental load interpretation ( 1).
The strength of a material is the maximum stress it can withstand without failure; it is obtained from a tensile test on a specimen of the material. Common metals follow stress-strain relations of the forms illustrated :
The stress in a brittle material, Fig.D, cannot exceed the ultimate strength, Su.
Ductiles also display such an upper limit, Fig.E, but the yield strength, Sy, is often more relevant as it forms a bound above which plasticity and excessive deformation may occur. Most modern ductiles do not possess a distinct yield so in this case an artificial value is usually defined - the offset yield, Fig.F - based upon some maximum acceptable permanent deformation (eg 0.2%) remaining after load release. Yields and ultimates are material properties; representative values for many materials are cited in the literature.
Let us derive a design equation for the simplest of all components 
- the tensile bar.
The bar’s cross-sectional area is A, it is subjected to a tensile force P and the strength of its material is S - which may be the ultimate if fracture is important, or the yield if the material is ductile and excessive deformation is relevant.
Assuming uniform stress across the cross-section, the maximum load that the component can sustain occurs when the stress σ reaches the material strength, and is Pmax = A.S - this expression when combined with ( 1) leads to the design equation for direct normal stress, namely :-
COLOR=blue n P = A S [/color]
The design equation embodies conveniently and directly the four aspects mentioned previously - safety, load, dimensions and material (strength). The equation may be used either :
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for analysis to determine the degree of safety n = A S /P for given S, A, P, or
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for synthesis (design) to ascertain the dimensions required A ≥ n P/S to withstand a given P with material of strength S and specified degree of safety n.
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Codes often cite a material’s allowable or design stress, σd = S/n as a characteristic in its own right, rather than stipulating both the strength and the necessary safety factor. The required area would then be A ≥ P/σd.
Equation ( 1b) applies also to compressive loading, however components in compression are usually more likely to fail by buckling than by any strength limitation being exceeded. Buckling (ie. geometric instability) is dictated as much by the component’s overall geometry as by the inherent strength of the component’s material. A cardboard tube can withstand a larger compressive load than a steel wire. The design equation for buckling is derived in a later chapter.
Suitable nominal factors of safety for use in elementary design work are as follows :
[RIGHT]SUGGESTED SAFETY (DESIGN) FACTORS FOR ELEMENTARY WORK based on yield strength - according to Juvinall & Marshek op cit. 1 1.25 - 1.5 for exceptionally reliable materials used under controllable conditions and subjected to loads and stresses that can be determined with certainty - used almost invariably where low weight is a particularly important consideration 2 1.5 - 2 for well-known materials under reasonably constant environmental conditions, subjected to loads and stresses that can be determined readily. 3 2 - 2.5 for average materials operated in ordinary environments and subjected to loads and stresses that can be determined. 4 2.5 - 3 for less tried materials or for brittle materials under average conditions of environment, load and stress. 5 3 - 4 for untried materials used under average conditions of environment, load and stress. 6 3 - 4 should also be used with better-known materials that are to be used in uncertain environments or subject to uncertain stresses. 7 Repeated loads : the factors established in items 1 to 6 are acceptable but must be applied to the endurance limit (ie. a fatigue strength ) rather than to the yield strength of the material. 8 Impact forces : the factors given in items 3 to 6 are acceptable, but an impact factor (the above dynamic magnification factor ) should be included. 9 Brittle materials : where the ultimate strength is used as the theoretical maximum, the factors presented in items 1 to 6 should be approximately doubled. 10 Where higher factors might appear desirable, a more thorough analysis of the problem should be undertaken before deciding on their use.
The table illustrates clearly the need to increase safety factors when designing with loads either actual or maximum which are not known with certainty. In order to avoid waste of material to cater for ignorance, we try to forecast loads as accurately as possible. For example, rather than use a large design factor to allow for unknown shock effects, we may obtain a realistic dynamic magnification factor from the literature citing other folks’ experience with similar components, and then increase the nominal actual load by this dmf. Statically determinate assemblies are generally preferred to indeterminate because component loads are predictable from simple statics and do not rely on a complex interaction between the components.
For the same reason we employ mathematical models which describe real behaviour as accurately as possible - rather than use simplistic models which are known to be poor predictors. Don’t confuse a model’s descriptive accuracy with its numerical accuracy - a mathematical model might give an answer to 53 decimal places, but it may describe a component’s actual behaviour so simplistically that only one decimal place is significant. [/right]
Design factors are increased also when the consequences of failure - economic, social, environmental or political - are serious. For example, a local manufacturer of processing machinery destined for the headwaters of the Yangtze River, doubled the size of every motor predicted by the design process because of the delay in providing replacements if failure should occur due to overload. This is another example of the designer having to foresee the whole future life of an artefact.
The tensile component is the simplest of all machine members. Investigation into the safety of a more complex component involves searching over all elements of the component (ie. over all locations in the component) to find the weakest link - the element with the smallest safety factor. In practice for a given component only a few elements are potentially critical, so an intelligent search is not necessarily protracted. At each element, safety is assessed by means of the following general approach :-
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Find the loads on the component, given the loads on the machine or structure (assemblage). If the assemblage is statically determinate then free bodies will give the components’ loads immediately, but if it is statically indeterminate then a three-pronged attack based on equilibrium, compatibility and the constitutive laws of the assemblage’s individual members must be carried out.
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Ascertain the stress components (often Cartesian) at the element in question again by the use of free bodies. This is essentially a superposition of load building blocks such as tension (or compression), shear, bending and torsion - though others will be encountered. If the member is itself indeterminate then the three-pronged attack mentioned above must be applied to elements within the member - it will be recalled that the elementary equations for bending and torsional stresses were deduced in this manner based on an assumed deformation geometry.
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Resolution of the stress components into principal stresses, using either analytic techniques or Mohr’s circles. We shall not consider tensor resolution of the general 3-dimensional state - it is somewhat complex and seldom encountered in routine design.
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Implementation of an appropriate failure theory whereby the biaxial or triaxial stresses at the element may be correlated with material strengths, which are derived from uniaxial tests, to evaluate the safety factor.
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These four steps will be examined individually in the following sections, and will be examplified in setting up the design equation for a very common component - a shaft of circular cross-section subjected to simultaneous bending and torsion.
But before embarking on this and writing down stress equations such as σ = P/A, we’ve got to appreciate the limitations of such mathematical models - we must be able to visualise qualitatively how stresses vary throughout a body and to appreciate the concept of 
stress concentration.
For example consider equation ( 1b) applied to the assembly comprising two short coaxial bars, 1 and 2, of cross-sectional areas 100 & 200 sq.mm and strengths 600 & 400 MPa respectively ( recall that 1 MPa is equivalent to 1 N/sq.mm).
The assembly is compressed by 50 kN distributed uniformly over each end.
Two distinct force paths are suggested in (g) - that of P from the eye directly into the support and byepassing the bolt shank and sleeve, and that of the self-equilibrating initial load Fo in the bolt shank and sleeve. These make physical sense as no external support is necessary to cater for a temperature rise of the sleeve or for a tightening of the nut - these effects, embodied in Fo, are resisted internally in the assemblage. Note that the force paths have been greatly simplified here to avoid confusion in the sketch. 
superposition of load building blocks as in the stress example of the previous section. The two- and one-dimensional cases illustrated here are particularisations of what is essentially three-dimensional. 
Fortunately the great majority of practical cases are two -dimensional since the relatively simple loading in conjunction with the natural choice of axes leads to one of the axes being principal automatically. For example there can be no stress on the free surface of a body, so the surface normal is an obvious choice for one of the three Cartesian axes - as there can be no shear on this surface, the axis is automatically a principal axis.
