An essential requirement of engineering design is the accurate determination of stresses and strains in components under working conditions. ‘Strength of materials’ is a subject relating to the physical nature of substances which are acted upon by external forces. No solid body is perfectly rigid, and when forces are applied to it changes in dimensions occur. Such changes are not always perceptible to the human eye since they are so small. For example, a spanner will bend slightly when tightening a nut, and the span of a bridge will sag under the weight of a car.
The change in the value of a linear dimension of a body, say x, divided by the original value of the dimension, say l, gives a great deal of information about what is happening to the material itself. This ratio is called strain, ε, and is dimensionless, i.e. ε =x/i
Stress
The force (F) acting on an area (A) of a body is called the stress, a, and is measured in pascals (Pa) or newtons per square metre (N/m2), i.e.
Young’s Modulus of Elasticity
If a solid body is subjected to a gradually increasing stress, and if both the stress and the resulting strain are measured, a graph of stress against strain may be drawn. Up to a certain value of stress the graph is a straight line. That particular value is known as the limit of proportionality and its value varies for different materials. The gradient of the straight line is a constant known as Young’s modulus of elasticity, E
Young’s modulus of elasticity is a constant for a given material. As an example, mild steel has a value of E of about 210 ð 109 Pa (i.e. 210 GPa).
If on removal of external forces a body recovers its original shape and size, the material is said to be elastic. If it does not return to its original shape, it is said to be plastic. Copper, steel and rubber are examples of elastic materials while lead and plasticine are plastic materials. However, even for elastic materials there is a limit to the amount of strain from which it can recover its original dimensions. This limit is called the elastic limit of the material. The elastic limit and the limit of proportionality for all engineering materials are virtually the same. If a body is strained beyond the elastic limit permanent deformation will occur.
In designing a structure, such as an electricity transmission tower carrying overhead power lines or support pillars and spans of new designs of bridges, the engineer is greatly concerned about the mechanical properties of the materials he is going to use. Many laboratory tests have been designed to provide important information about materials. Such tests include tensile, compression, torsion, impact, creep and fatigue tests and each attempt to provide information about the behaviour of materials under working conditions. (A typical tensile test is described in Chapter 24)
It is possible to design a structure that is strong enough to withstand the forces encountered in service, but is, nonetheless, useless because of the amount of elastic deformation. Hence, tests made on materials up to the elastic limit are of great importance. A material that has a relatively high value of Young’s modulus is said to have a high value of stiffness, stiffness being the
Thus, when the determination of Young’s modulus of elasticity, E, of a material is required, an accurate stress/strain or load/extension graph must be obtained. The actual strain is very small and this means that very small extensions must be measured with a high degree of accuracy.
The measurement of extension, and thus strain, is achieved in the laboratory with an instrument called an extensometer. Although some extensometers can be used in such practical situations as a crane under load, it is more usual to use in these situations an electrical device called a strain gauge.
A knowledge of stress and strain is the foundation of economy and safety in design.
An extensometer is an instrument used in engineering and metallurgical design to measure accurately the minute elastic extensions of materials, in order to forecast their behaviour during use. There are several different designs of extensometer including the Lindley, the Huggenburger and the Hounsfield.
This is probably the most common type of extensometer used for measuring tensile strains. This instrument consists of two arms, A and B, connected by a strip of spring steel that acts as a hinge. The unstressed specimen of the material is clamped at points C and D by pointed screws, the distance between C and D usually being 50 mm. Thus 50 mm is termed the ‘gauge length’. A dial test indicator is placed between the arms A and B as shown in the typical arrangement of the Lindley extensometer in Figure 26.1
The point D is halfway between the hinge and the indicator; hence the movement of the pointer on the test indicator will record twice the extension
of the specimen. However, the indicator is normally calibrated so that it indi- cates extension directly, each graduation representing an extension of 1 micron
(i.e. 10Ł6 m or 0.001 mm). Extensions may be measured to an accuracy of 0.0001 mm using the Lindley extensometer.
The Huggenburger extensometer
This is a simple, rugged and accurate instrument that may be used to measure tensile or compressive strains. Its construction is based on a lever multiplying system capable of obtaining magnifications in the order of 2000. Figure 26.2 shows a simplified schematic arrangement of a front view of the Huggenburger
extensometer clamped to a specimen, where Q and R are two knife-edges, usually either 0 mm or 20 mm apart. Any strain encountered by the specimen under test will alter the gauge length QR. In Figure 26.2, the specimen is shown in tension, thus QR will increase in length. This change is transmitted by pivots (labelled P) and levers S and T to the pointer, and is indicated on the scale according to the multiplication factor. The supplier who calibrates each device after manufacture supplies this factor, of approximately 2000, to the instrument user. This type of extensometer enables extensions to be recorded to an accuracy comparable with the Lindley extensometer and may be used in the laboratory or in the field.
This may be used in conjunction with a Hounsfield Tensometer (which is a universal portable testing machine capable of applying tensile or compressive forces to metals, plastics, textiles, timber, paper and so on), or with any other testing machine. The extensometer is a precision instrument for measuring the extension of a test specimen over a 50 mm gauge length, while the test specimen is loaded in the testing machine. The instrument can be attached to round specimens of material of up to 25 mm in diameter or rectangular sections of material of up to 25 mm square at precisely 50 mm gauge length without prior marking of the specimen. Figure 26.3 shows a typical Hounsfield extensometer viewed from two different elevations.
The gauge length rod is screwed into position, making the fixed centres exactly 50 mm apart. The extensometer is then clamped to the test piece before the gauge length rod is unscrewed. With the test piece still unloaded the micrometer is wound in until the platinum contacts meet, thus completing the circuit (shown by the lamp lighting up). The micrometer reading is then taken and the micrometer head unwound. After the load is placed on the
specimen the micrometer head is again wound in and a new reading taken when the lamp lights. The difference between the two micrometer readings is an indication of the extension of the test piece for the particular load applied. Each division of the micrometer wheel is equal to 0.002 mm. The accuracy of the Hounsfield extensometer compares favourably with other extensometers, and an advantage, in certain circumstances, of this instrument is its small overall size.
A strain gauge is an electrical device used for measuring mechanical strain, i.e. the change in length accompanying the application of a stress. The strain gauge consists essentially of a very fine piece of wire that is cemented, or glued strongly, to the part where the strain is to be measured. When the length of a piece of wire is changed, a change in its electrical resistance occurs, this change in resistance being proportional to the change in length of the wire. Thus, when the wire is securely cemented to the part that is being strained, a change of electrical resistance of the wire occurs due to the change in length. By measuring this change of resistance the strain can be determined. The strain gauge was first introduced in the USA in 1939 and since that time it has come into widespread use, particularly in the aircraft industry, and is now the basis of one of the most useful of stress analysis techniques. A typical simple strain gauge is shown in Figure 26.4.
Rolling out a thin foil of the resistive material, and then cutting away parts of the foil by a photo-etching process to create the required grid pattern form a modern strain gauge. Such a device is called a foil strain gauge and a typical arrangement is shown in Figure 26.5. A foil-strain gauge has many advantages over the earlier method and these include:
(i) better adhesion between conductor and backing material,
(ii) better heat dissipation,
(iii) a more robust construction,
(iv) easier to attach leads to,
(v) accurate reproducibility of readings, and
(vi) smaller sizes are possible.
In order to obtain a deflection on a galvanometer, G, proportional to the strain occurring in the gauge, it must be connected into one arm of a Wheatstone
bridge, as shown in Figure 26.6. A Wheatstone bridge circuit having four equal resistances in the arms has zero deflection on the galvanometer, but when the resistance of one or more of the arms changes, then the bridge galvanometer deflects from zero, the amount of deflection being a measure of the change in resistance. If the resistance change occurs in a strain gauge as a result of applied strain, then the bridge galvanometer deflection is a measure of the amount of strain. For very accurate measurements of strain there are a number of possible sophistications. These are not described in detail in this chapter but include:
(i) the use of a temperature-compensating dummy gauge to make the bridge output independent of temperature, since the resistance of a gauge varies with temperature and such a resistance change may be misread as strain in the material,
(ii) an additional bridge balancing circuit to obtain zero galvanometer deflection for zero strain, and
(iii) the addition of an amplifier to amplify the signal output from the bridge in applications where the level of strain is such that the bridge deflection is too small to readily detect on a galvanometer.
A typical selection of practical situations where strain gauges are used include:
(i) the airframe and skin of an aircraft in flight,
(ii) electricity pylons, cranes and support pillars and spans of new designs of bridges, where strain must be tested to validate the design, and
(iii) applications in harsh environments and remote positions, such as inside
nuclear boilers, on turbine blades, in vehicle engines, on helicopter blades in flight and under water on oil rig platforms, where a knowledge of strain is required.
