Structures (aeroplanes, bridges, pipelines, etc.) are made from various materials. These structures will need to withstand loads, deformations (deflections, displacements), impacts, etc.. ‘Mechanics’ is the branch of science concerned with the behaviour of these structures when they are subjected to forces or deformation. It is usually called ‘structural mechanics’. In structural mechanics we talk about ‘bodies’: this is a generic term for all structures.
This section explains some important terms, and principles we use in structural mechanics.
1. Forces and Deformations
Structural mechanics involves understanding how structures react to forces and deformation.
‘Force’ is a measure of the interaction between bodies (unit is Newton), whereas ‘load’ usually means the force exerted on a surface or body (unit of weight is kg). The force on a material is resisted by internal forces (cohesion between particles in the materials). We term this resistance, a material’s ‘strength’ (see later).
Forces act on the cross-sectional area of a structure. These forces produce deformations (or ‘displacements’, or ‘elongations’), Figure 1, and we know that when we apply a force to a structure it can deform and eventually fail; for example, we design a railway bridge to support the heaviest train that is likely to pass over it. A heavier train may damage the bridge by excessive deformations or even failure. Differing types, and differing directions, of loadings will create differing deformations.
If we apply a tensile (pulling) force to a body of original length Lo, to elongate or pull apart the body, the material will ‘elongate’ or ‘deform’ to a new length of L. Hence, there is a relationship between force and deformation, Figure 1.
In 1678, Robert Hooke (1635 – 1703) reported that, for many materials, the deformation due to the load was proportional to the force, Figure 1. The point at which a material ceased to obey his law is known as its ‘elastic limit’. In an elastic material, force is proportional to stress.
Hooke stated: ‘… the power [sic.] of any springy body is in the same proportion with the extension…’.
In simple terms… when we apply a force to a spring, it stretches. If we apply double the force, it stretches twice as much. The extension of the spring is proportional to the force. The spring will go back to its original length when the force is removed, provided we do not exceed the elastic limit, Figure 1. It behaves in an elastic manner, along this elastic line – returning to its original shape when the force is removed. The extension is ‘recovered’.
Materials will deform beyond their elastic limit, Figure 2. When the elastic limit has been reached the object can continue to deform, but it will not return to its original shape if the force is removed, and it may eventually break, Figure 2. The extension is not ‘recovered’. Hence, this elastic ‘limit’ is the greatest stress that can be applied to a material without causing permanent deformation.
Figure 2. Deformation after the Elastic Limit is not Proportional to Force.
The shaded area under the elastic line in the plots in Figure 3 is the ‘elastic stored energy’. This means the energy that will be released when we release the force. As the material is elastic, all the energy we have put into the material to obtain this deformation will be released when we unload the specimen. The specimen is behaving like a spring.
Figure 3. The Energy Created by the Force and Deformation.
Thomas Young (1773–1829) was interested in the elastic limit, and the elastic line, Figure 1. He noted that the slope of this elastic line was related to a material’s ‘stiffness’ (how easy a material bends): soft materials such as plastics are not stiff (they are ‘pliable’ or ‘flexible’), but hard materials such as steel are stiff and difficult to bend.
The’ Young’s modulus’ is the slope of the elastic region of the force-deformation curve, Figure 3. Young’s Modulus (also known as the ‘elasticity modulus’, or ‘modulus of elasticity’), E, of a material, is the ratio of the force versus the deformation within the elastic region of the curve, Figure 3. A high Young’s modulus means that the material is stiffer; for example, steel has a high E of 20.7 x 1010 N/m2 and polyethylene (plastic) has a low E of 3.45 x 108 N/m2. Hence, we would use steel for the shaft of a screwdriver (it needs to be stiff to resist twisting), and not plastic.
We now know that forces act on the cross-sectional area of a body and cause deformation. But we also know that the larger the cross-section, the smaller the deformation. Why? To understand the effect of cross-sectional area, we need to understand ‘stress’.
2. Stress and Strain
If we apply a tensile force to a body to elongate or pull it apart, the material becomes ‘stressed’. The tensile load on the body has created a ‘stress’ in the body (Figure 4):
Stress = Force (F)/Cross-sectional Area (/Ao)
where: F = applied force in Newtons (N), and Ao = cross-sectional area of the body in m2.
Figure 4. Forces on a Body create Deformation.
We see that the units of stress are N/m2 in SI units (or Ib/in2 in English units).
We have higher stresses as we increase the force, or decrease the cross-sectional area.
We want to ensure our forces do not create stresses that excessively deform or fail our structure. The material our structure is built from will have a ‘strength’ – it cannot have infinite strength. The material ‘strength’ is the limit to how much stress it can withstand.
This strength will resist the stresses; hence, we need a test to measure the strength of our material, and also quantify how the material reacts to increasing stress. We can then ensure the stresses on our structure do not exceed the material’s strength. We use the ‘tensile test’ to determine a material’s strength, and its resistance to stress.
We create a ‘force versus deformation’ plot (Figure 4) for a material by conducting a tensile test. This test plots how a material deforms under a load (force). In a tensile test a sample is gradually elongated to failure by applying a tensile force (F). The force elongates (strains) the sample.
The stress is obtained by dividing the force, F, by the original area of the cross section of the sample, A0:
The ‘strain’ in a material is defined as the fractional change in the length of the material as it is subjected to a force, Figure 5. It is obtained by dividing the elongation of the material, ΔL, by its original length, Lo:
ε = ΔLO = ΔL/LO = (L-LO)/LO
where: ΔL = change in length in m, and, Lo = original length in m.
Strain is the deformation caused by stress. Strain is thus a ‘unitless’ (has no units) quantity.
The force versus deformation plot can then be converted into a ‘stress versus strain’ plot, Figure 3 and Figure 5.
Figure 5. Strain, Yield Strength, and Ultimate Tensile Strength.
2.3 Poisson’s ratio
All engineering materials become narrower in cross section when they are stretched; for example, if you pull a rubber band, it elongates, but the width decreases. This is explained by ‘Poisson’s ratio’.
Strain (ε) is defined as the change in length divided by the original length: ε = ΔL/Lo, Figure 5 and Figure 6. Poisson’s ratio, ν, is the ratio of transverse contraction strain to longitudinal extension strain in the direction of stretching force, Figure 6. It is named after its discoverer – Siméon Poisson (1781 – 1840).
Figure 6. Poisson’s ratio.
Poisson’s (dimensionless) value in steel under axial loading is in the order of 0.28. Most metals have values between 0.25 and 0.35. Rubber is 0.5.
3. Stresses in Pipelines
Pressurising the fluid in a pipe (of diameter, D, and wall thickness, t) will create stresses in the pipe wall. The internal pressure, p, will cause two large stresses (biaxial (two) stress state) in the wall of the pipe (Figure 7):
- a ‘hoop (circumferential) stress’, h; and,
- an ‘axial (longitudinal) stress’, a.
The hoop stress is higher than the axial stress:
- h = pD/2t;
- a = pD/4t = 0.5h (if the pipe is free to expand; e.g., when the pipe is above ground with no restriction to its movement).
- a = 0.3h (if expansion of the pipe is restricted; e.g., it is buried and restrained by the surrounding soil).
where D = pipe diameter, and t = wall thickness.
There are other loads on the pipelines that can create stresses:
- temperature effects, caused by the substance temperature, pressure reduction, etc.;
- external loads, caused by ground subsidence in mining areas, vibration, etc.;
- weight effects, caused by backfill over the pipeline, weight of substance carried, etc..
Figure 7. Main Stresses caused by internal Pressure.
Properties of Materials
4. Yield Strength and Tensile Strength
As we plot our engineering stress and strain we can see a linear portion at the start of the test, and, a ‘non-linear’ portion, after this linear portion, Figure 5. The two main parameters which are used to describe the stress-strain curve of a metal are:
- yield strength or yield point; and,
- the tensile strength.
The ‘yield strength’ is the stress at which yielding (permanent, plastic deformation) occurs across the whole specimen, Figure 5. Before this stress, the deformation is purely ‘elastic’: stress is proportional to strain. If we unload in this elastic regime, all the strain is ’recovered’, and the material returns to its original shape, Figure 3 and Figure 5. After the yield strength the material is plastically deformed (it cannot return to its original shape). All the strain cannot be recovered.
As we progress from the elastic regime we approach the ultimate tensile strength. This ultimate tensile strength is the highest stress the material can withstand, before losing its load bearing capacity, and starting to fail, Figure 2.
We can view the yield strength and ultimate tensile strengths as a failure of the material:
- when a material exceeds its yield strength, it will fail to return to its original shape;
- when a material exceeds its ultimate tensile strength it fails (separates).
A material’s ductility is its ability to deform. When we load materials they deform in different ways:
- if we bend a piece of chalk it quickly breaks, with little deformation, so it is a ‘brittle’ material.
- If we bend a silver spoon, it does not break, and these is extensive deformation, so it is a ‘ductile’ material.
Structures need to be able to bend and deform; for example, line pipe needs to be able to bend (we bend pipe during onshore construction), deform (our pipeline can be dented during operation), and stretch (pipelines can be subjected to strains higher than predicted in design; for example, excessive ground movement). We need materials to be able to deform and strain.
Line pipe specifications such as API 5L  ensure the line pipe steel can strain extensively before it fails by specifying tests and values for the steel,
Ductility is also a description of how a material fails, Figure 8:
- ‘ductile’ means extensive deformation before failure, and a fracture surface that shows ductile features;
- ‘brittle’ means limited deformation before failure, and a fracture surface that shows brittle features.
‘Ductile’ and ‘brittle’ describe the amount of deformation associated with the failure mode.
Figure 8. Ductility.
‘Toughness’ is the ability of a material to withstand the presence of a defect such as a crack. Low toughness material (such as glass) cannot tolerate cracks, and can fail in a ‘brittle’ manner. Materials such as aluminium, are high toughness, and can withstand large cracks and can fail in a ‘ductile’ manner.
In pipeline standards, we measure toughness by a small notched bar test called a ‘Charpy’ test. A full size Charpy specimen is 10 mm x 10 mm in cross-section, Figure 9. This is a small ‘impact’ specimen (it is tested by hitting it with a pendulum), Figure 9.
Figure 9. The Charpy specimen and the Charpy Test Machine.
The Charpy toughness is measured in ‘Joules’ (or ft lbf). The pipeline industry uses the API 5L specification  for guidance on the required toughness for a pipeline. API 5L gives minimum Charpy requirements, based on pipe diameter and pipe strength. These requirements ensure the pipeline has adequate toughness.
Toughness is sensitive to temperature, Figure 10: at high temperatures line pipe has high toughness, but at low temperatures it can have low toughness. The ductility on the fracture faces of the toughness specimens also varies with temperature, Figure 10: at high temperatures the specimens have 100% ductility on their fracture surfaces, and at low temperatures the specimens have little/no ductility (they exhibit a brittle fracture).
The Charpy test give a qualitative measure of toughness: it is not a quantitative measure. There are quantitative measures of fracture toughness that are used is defect assessment processes (for example, Reference 2). These quantitative measures include the stress intensify factor, the J-integral, and the crack tip opening displacement.
Figure 10. Effect of Temperature on Toughness and Ductility.
‘Hardness’ generally means the resistance to indentation or deformation. Hardness is not considered a material property, as it depends on other material properties (e.g., strength and ductility), but the greater the hardness of the metal, the greater resistance it has to deformation.
Hardness is normally measured by pressing a steel ball or cone into the metal, and measuring the depth of indentation for a specified load. The indentation depth is them related to the material hardness.
There are various methods for this measurement (Brinell test, Rockwell test, Vickers test, etc.). The ‘Vickers’ hardness test uses a square-based pyramidal diamond indenter. The applied load may range from 1 to 120 kgf. ‘HV10’ indicates a Vickers Hardness (‘HV’) obtained using a 10 kgf load (‘10’).
Very few structures operate at a ‘static’ stress (Figure 11), where the stress on the structure is constant. Most structures, such as airplanes or bicycles, are subjected to repeat (‘cyclic’) and varying stresses, and this cyclic stressing can lead to cracking. These varying stresses can cause cracks, and the cracks can cause failures.
Figure 11. Cyclic Stresses cause Fatigue.
You can perform a simple experiment on fatigue. You cannot fail a paper clip by simply pulling it to failure, as it is too strong under a single (‘static’) load. But it is very weak under a small repeated (cyclic) bending load:
- take a paper clip;
- bend it repeatedly;
- it fails (separates) in a few bends (the repeated bending is your cyclic load).
You have initiated a crack in the paper clip, and it has grown to failure because of your repeated bending. The paperclip’s ‘fatigue life’ or ‘endurance’ is the number of times (‘cycles’) you have bent it before failure.
Problems with structures subjected to cyclic stresses were identified 150 years ago. As iron and steel structures came into widespread use, engineers were faced with failures occurring well below the tensile strength of the materials. The materials were ductile (could deform), but the failures exhibited little or no ductility (no deformation). The world became aware of ‘metal fatigue’.
The structures that were failing had been subjected to repeated stressing (cyclic stresses). The static or peak stress was not sufficient to fail the structure, but these cyclic stresses were sufficient. Hence, engineers had to understand these failures, and the effect of these cyclic stresses.
Metal fatigue was reported by railroad engineers in the 1880’s. A number of accidents involving failed train axles led engineers to describe the parts as being ‘tired’, or ‘fatigued’. Most of the early failures had developed in machine parts which were subjected to high frequency repeated loading.
The first commercial jet airplane failed in the early 1950s due to fatigue around its square-shaped windows. A combination of poor design and bad manufacturing lead to many deaths.
Today, ‘fatigue’ is defined as : ‘process of development of or enlargement of a crack as a result of repeated cycles of stress’. These repeated cycles of stress can either:
- initiate (create) and grow a crack in a structure; or,
- grow an existing crack.
These fatigue mechanisms can lead to failure.
8.1 The Three Stages of Fatigue
Fatigue is made up of three stages: initiation; propagation; and, failure. When I have no macroscopic defects present, the fatigue life consists of about 90% ‘initiation’ of a crack, and about 10% ‘propagation’ of a crack. When I have a macroscopic defect (e.g., a crack) present, most of the fatigue life is propagation.
Even small macroscopic defects (such as those in a weld) will have negligible cycles to initiation.
8.2 Fatigue Assessment using ‘S-N’ Curves
We know that fatigue is made up of three stages: initiation; propagation; and, failure. All these stages can be described by a cyclic stress (S) versus number of cycles (N) to failure curve (the ‘S-N’ curve), Figure 12, provided the existing defects in the structure are ‘insignificant’ (small); for example, they are all within quality control levels.
The S-N curve is obtained experimentally, for the specific material, environment, loading, and structure shape, and these are examples in the literature; for example, Reference 2.
We obtain a ‘fatigue life’ from the S-N curves: this is the number of cycles the material can withstand, before failure, at a specified stress range.
The S-N curves are ‘log-log’ plots. We use log-log plots to allow us to fit the wide range of cycles (N) we are interested in – typically between 1000 and 100,000 cycles. We also need a wide range for our cyclic stress range to accommodate the varying stresses experienced by various structures. We could not fit this wide range of data on a simple linear-linear graph.
Figure 12. The ‘S-N’ Curve for Fatigue Assessment.
The S-N curves are sensitive to many parameters (e.g., the mean stress in the stress cycle). The stress range has the greatest effect on fatigue life, but other parameters, such as the environment, have an effect.
The fatigue life of metals decreases when they are exposed to a corrosive environment, Figure 13. This ‘corrosion fatigue’ is usually called ‘environmentally-assisted cracking’ (EAC). It is caused by the combined actions of cyclic loading and a corrosive environment.
Figure 13. Effect of Environment on Fatigue Life.
8.3 Fatigue Assessment using Fracture Mechanics
S-N curves can be applied to ‘defect-free’ materials that have been produced to an acceptable standard. We cannot use S-N curves if we have a pre-existing defect, such as a crack. We must use fracture mechanics methods (e.g., ) to assess pre-existing defects.
8.4 Pipelines and Fatigue
Pipelines are subjected to cyclic stresses due to:
- internal pressure variations, caused by changing demands for the product;
- changes in temperatures (these changes will cause the pipeline to expand or contract, and this leads to changes in stresses);
- external loads, such as traffic loading over a buried pipeline, or movement on a seabed from sea currents).
These cyclic stresses can:
- create and grow cracks;
- cause existing cracks in the pipeline to grow; or,
- cause other defects such as gouges or dents, to crack, and grow these new cracks.
Pipelines sometimes fail due to fatigue, and these failures are usually associated with pre-existing defects in welds, or damage in the pipeline such as dents.
- Anon., ‘Pipeline Transportation Systems for Liquids and Slurries’, American Society of Mechanical Engineers, USA. ASME B31.4. 2016.
- Anon., ‘Guide to methods for assessing the acceptability of flaws in metallic structures’. BS 7910:2013. British Standards Institution. UK. 2013.
- Anon., ‘Specification for Line Pipe’, American Petroleum Institute, USA. API Spec 5L. Forty-sixth Edition. 2018.