CHAPTER 1 SIMPLE STRESS AND STRAIN 1.1.Load In any engineering structure or mechanism the individual components will be subjected to external forces arising from the service conditions or environment in which the component works.If the component or member is in equilibrium,the resultant of the external forces will be zero but,nevertheless,they together place a load on the member which tends to deform that member and which must be reacted by internal forces which are set up within the material. If a cylindrical bar is subjected to a direct pull or push along its axis as shown in Fig.1.1, then it is said to be subjected to tension or compression.Typical examples of tension are the forces present in towing ropes or lifting hoists,whilst compression occurs in the legs of your chair as you sit on it or in the support pillars of buildings. Area A Stress:P/A P Tension Compression Fig.1.1.Types of direct stress. In the SI system of units load is measured in newtons,although a single newton,in engineering terms,is a very small load.In most engineering applications,therefore,loads appear in SI multiples,i.e.kilonewtons (kN)or meganewtons(MN). There are a number of different ways in which load can be applied to a member.Typical loading types are: (a)Static or dead loads,i.e.non-fluctuating loads,generally caused by gravity effects. (b)Live loads,as produced by,for example,lorries crossing a bridge. (c)Impact or shock loads caused by sudden blows. (d)Fatigue,fluctuating or alternating loads,the magnitude and sign of the load changing with time
CHAPTER 1 SIMPLE STRESS AND STRAIN 1.1. Load In any engineering structure or mechanism the individual components will be subjected to external forces arising from the service conditions or environment in which the component works. If the component or member is in equilibrium, the resultant of the external forces will be zero but, nevertheless, they together place a load on the member which tends to deform that member and which must be reacted by internal forces which are set up within the material. If a cylindrical bar is subjected to a direct pull or push along its axis as shown in Fig. 1.1, then it is said to be subjected to tension or compression. Typical examples of tension are the forces present in towing ropes or lifting hoists, whilst compression occurs in the legs of your chair as you sit on it or in the support pillars of buildings. ,Are0 A Tension Compression Fig. 1.1. Types of direct stress. In the SI system of units load is measured in newtons, although a single newton, in engineering terms, is a very small load. In most engineering applications, therefore, loads appear in SI multiples, i.e. kilonewtons (kN) or meganewtons (MN). There are a number of different ways in which load can be applied to a member. Typical loading types are: (a) Static or dead loads, i.e. non-fluctuating loads, generally caused by gravity effects. (b) Liue loads, as produced by, for example, lorries crossing a bridge. (c) Impact or shock loads caused by sudden blows. (d) Fatigue,fluctuating or alternating loads, the magnitude and sign of the load changing with time. 1
2 Mechanics of Materials §1.2 1.2.Direct or normal stress ( It has been noted above that external force applied to a body in equilibrium is reacted by internal forces set up within the material.If,therefore,a bar is subjected to a uniform tension or compression,i.e.a direct force,which is uniformly or equally applied across the cross- section,then the internal forces set up are also distributed uniformly and the bar is said to be subjected to a uniform direct or normal stress,the stress being defined as load P stress (o)= 一 area A Stress a may thus be compressive or tensile depending on the nature of the load and will be measured in units of newtons per square metre(N/m2)or multiples of this. In some cases the loading situation is such that the stress will vary across any given section, and in such cases the stress at any point is given by the limiting value of oP/6A as 6A tends to zero. 1.3.Direct strain (e) If a bar is subjected to a direct load,and hence a stress,the bar will change in length.If the bar has an original length L and changes in length by an amount oL,the strain produced is defined as follows: strain (e)-change in lengthL original length L Strain is thus a measure of the deformation of the material and is non-dimensional,i.e.it has no units;it is simply a ratio of two quantities with the same unit(Fig.1.2). Strain e=8L/L ◇ Fig.1.2. Since,in practice,the extensions of materials under load are very small,it is often convenient to measure the strains in the form of strain x 10-6,i.e.microstrain,when the symbol used becomes ue. Alternatively,strain can be expressed as a percentage strain L i.e. strain (e)= ×100% 1.4.Sign convention for direct stress and strain Tensile stresses and strains are considered POSITIVE in sense producing an increase in length.Compressive stresses and strains are considered NEGATIVE in sense producing a decrease in length
2 Mechanics of Materials $1.2 1.2. Direct or normal stress (a) It has been noted above that external force applied to a body in equilibrium is reacted by internal forces set up within the material. If, therefore, a bar is subjected to a uniform tension or compression, i.e. a direct force, which is uniformly or equally applied across the crosssection, then the internal forces set up are also distributed uniformly and the bar is said to be subjected to a uniform direct or normal stress, the stress being defined as load P stress (a) = - = - area A Stress CT may thus be compressive or tensile depending on the nature of the load and will be measured in units of newtons per square metre (N/mZ) or multiples of this. In some cases the loading situation is such that the stress will vary across any given section, and in such cases the stress at any point is given by the limiting value of 6P/6A as 6A tends to zero. 1.3. Direct strain (E) If a bar is subjected to a direct load, and hence a stress, the bar will change in length. If the bar has an original length L and changes in length by an amount 6L, the strain produced is defined as follows: change in length 6L strain (E) = =- original length L Strain is thus a measure of the deformation of the material and is non-dimensional, Le. it has no units; it is simply a ratio of two quantities with the same unit (Fig. 1.2). Strain C=GL/L Fig. 1.2. Since, in practice, the extensions of materials under load are very small, it is often convenient to measure the strains in the form of strain x i.e. microstrain, when the symbol used becomes /ALE. Alternatively, strain can be expressed as a percentage strain 6L L i.e. strain (E) = - x 100% 1.4. Sign convention for direct stress and strain Tensile stresses and strains are considered POSITIVE in sense producing an increase in length. Compressive stresses and strains are considered NEGATIVE in sense producing a decrease in length
s1.5 Simple Stress and Strain 3 1.5.Elastic materials-Hooke's law A material is said to be elastic if it returns to its original,unloaded dimensions when load is removed.A particular form of elasticity which applies to a large range of engineering materials,at least over part of their load range,produces deformations which are proportional to the loads producing them.Since loads are proportional to the stresses they produce and deformations are proportional to the strains,this also implies that,whilst materials are elastic,stress is proportional to strain.Hooke's law,in its simplest form*, therefore states that stress (o)oc strain (8) stress i.e. constant* strain It will be seen in later sections that this law is obeyed within certain limits by most ferrous alloys and it can even be assumed to apply to other engineering materials such as concrete, timber and non-ferrous alloys with reasonable accuracy.Whilst a material is elastic the deformation produced by any load will be completely recovered when the load is removed; there is no permanent deformation. Other classifications of materials with which the reader should be acquainted are as follows: A material which has a uniform structure throughout without any flaws or discontinuities is termed a homogeneous material.Non-homogeneous or inhomogeneous materials such as concrete and poor-quality cast iron will thus have a structure which varies from point to point depending on its constituents and the presence of casting flaws or impurities. If a material exhibits uniform properties throughout in all directions it is said to be isotropic;conversely one which does not exhibit this uniform behaviour is said to be non- isotropic or anisotropic. An orthotropic material is one which has different properties in different planes.A typical example of such a material is wood,although some composites which contain systematically orientated "inhomogeneities"may also be considered to fall into this category. 1.6.Modulus of elasticity-Young's modulus Within the elastic limits of materials,i.e.within the limits in which Hooke's law applies,it has been shown that stress constant strain This constant is given the symbol E and termed the modulus ofelasticity or Young's modulus. Thus stress o E= strain (1.1) P L PL FA÷D=A6L (1.2) Readers should be warned that in more complex stress cases this simple form of Hooke's law will not apply and mis-application could prove dangerous;see $14.1,page 361
$1.5 Simple Stress and Strain 3 1.5. Elastic materials - Hooke’s law A material is said to be elastic if it returns to its original, unloaded dimensions when load is removed. A particular form of elasticity which applies to a large range of engineering materials, at least over part of their load range, produces deformations which are proportional to the loads producing them. Since loads are proportional to the stresses they produce and deformations are proportional to the strains, this also implies that, whilst materials are elastic, stress is proportional to strain. Hooke’s law, in its simplest form*, therefore states that stress (a) a strain (E) i.e. stress strain -- - constant* It will be seen in later sections that this law is obeyed within certain limits by most ferrous alloys and it can even be assumed to apply to other engineering materials such as concrete, timber and non-ferrous alloys with reasonable accuracy. Whilst a material is elastic the deformation produced by any load will be completely recovered when the load is removed; there is no permanent deformation. Other classifications of materials with which the reader should be acquainted are as follows: A material which has a uniform structure throughout without any flaws or discontinuities is termed a homogeneous material. Non-homogeneous or inhomogeneous materials such as concrete and poor-quality cast iron will thus have a structure which varies from point to point depending on its constituents and the presence of casting flaws or impurities. If a material exhibits uniform properties throughout in all directions it is said to be isotropic; conversely one which does not exhibit this uniform behaviour is said to be nonisotropic or anisotropic. An orthotropic material is one which has different properties in different planes. A typical example of such a material is wood, although some composites which contain systematically orientated “inhomogeneities” may also be considered to fall into this category. 1.6. Modulus of elasticity - Young’s modulus Within the elastic limits of materials, i.e. within the limits in which Hooke’s law applies, it has been shown that stress strain -- - constant This constant is given the symbol E and termed the modulus of elasticity or Young’s modulus. Thus stress 0 strain E E=-- _- P 6L PL A’ L ASL =--I-- -- * Readers should be warned that in more complex stress cases this simple form of Hooke’s law will not apply and misapplication could prove dangerous; see 814.1, page 361
4 Mechanics of Materials s1.7 Young's modulus E is generally assumed to be the same in tension or compression and for most engineering materials has a high numerical value.Typically,E=200 x 109 N/m2 for steel,so that it will be observed from (1.1)that strains are normally very small since 0 (1.3) In most common engineering applications strains do not often exceed 0.003 or 0.3%so that the assumption used later in the text that deformations are small in relation to original dimensions is generally well founded. The actual value of Young's modulus for any material is normally determined by carrying out a standard tensile test on a specimen of the material as described below. 1.7.Tensile test In order to compare the strengths of various materials it is necessary to carry out some standard form of test to establish their relative properties.One such test is the standard tensile test in which a circular bar of uniform cross-section is subjected to a gradually increasing tensile load until failure occurs.Measurements of the change in length of a selected gauge length of the bar are recorded throughout the loading operation by means of extensometers and a graph of load against extension or stress against strain is produced as shown in Fig.1.3; this shows a typical result for a test on a mild (low carbon)steel bar;other materials will exhibit different graphs but of a similar general form see Figs 1.5 to 1.7. Elastic Partially piastic Test E specimen Circular cross-section 6 Gauge length Extension or strain Fig.1.3.Typical tensile test curve for mild steel. For the first part of the test it will be observed that Hooke's law is obeyed,i.e.the material behaves elastically and stress is proportional to strain,giving the straight-line graph indicated.Some point A is eventually reached,however,when the linear nature of the graph ceases and this point is termed the limit of proportionality. For a short period beyond this point the material may still be elastic in the sense that deformations are completely recovered when load is removed (i.e.strain returns to zero)but
4 Mechanics of Materials $1.7 Young’s modulus E is generally assumed to be the same in tension or compression and for most engineering materials has a high numerical value. Typically, E = 200 x lo9 N/m2 for steel, so that it will be observed from (1.1) that strains are normally very small since 0 E=- E In most common engineering applications strains do not often exceed 0.003 or 0.3 % so that the assumption used later in the text that deformations are small in relation to original dimensions is generally well founded. The actual value of Young’s modulus for any material is normally determined by carrying out a standard tensile test on a specimen of the material as described below. 1.7. Tensile test In order to compare the strengths of various materials it is necessary to carry out some standard form of test to establish their relative properties. One such test is the standard tensile test in which a circular bar of uniform cross-section is subjected to a gradually increasing tensile load until failure occurs. Measurements of the change in length of a selected gauge length of the bar are recorded throughout the loading operation by means of extensometers and a graph of load against extension or stress against strain is produced as shown in Fig. 1.3; this shows a typical result for a test on a mild (low carbon) steel bar; other materials will exhibit different graphs but of a similar general form see Figs 1.5 to 1.7. Elastic Partially plastic tP Extension or strain Fig. 1.3. Typical tensile test curve for mild steel. For the first part of the test it will be observed that Hooke’s law is obeyed, Le. the material behaves elastically and stress is proportional to strain, giving the straight-line graph indicated. Some point A is eventually reached, however, when the linear nature of the graph ceases and this point is termed the limit of proportionality. For a short period beyond this point the material may still be elastic in the sense that deformations are completely recovered when load is removed (i.e. strain returns to zero) but
§1.7 Simple Stress and Strain 5 Hooke's law does not apply.The limiting point B for this condition is termed the elastic limit. For most practical purposes it can often be assumed that points A and B are coincident. Beyond the elastic limit plastic deformation occurs and strains are not totally recoverable. There will thus be some permanent deformation or permanent set when load is removed. After the points C,termed the upper yield point,and D,the lower yield point,relatively rapid increases in strain occur without correspondingly high increases in load or stress.The graph thus becomes much more shallow and covers a much greater portion of the strain axis than does the elastic range of the material.The capacity of a material to allow these large plastic deformations is a measure of the so-called ductility of the material,and this will be discussed in greater detail below. For certain materials,for example,high carbon steels and non-ferrous metals,it is not possible to detect any difference between the upper and lower yield points and in some cases no yield point exists at all.In such cases a proof stress is used to indicate the onset of plastic strain or as a comparison of the relative properties with another similar material.This involves a measure of the permanent deformation produced by a loading cycle;the 0.1% proof stress,for example,is that stress which,when removed,produces a permanent strain or "set"of 0.1%of the original gauge length-see Fig.1.4(a). 6 1b) Proof stress 6 Stron e Stra1n∈ 01% Permanent 'set' Fig.1.4.(a)Determination of 0.1%proof stress. Fig.1.4.(b)Permanent deformation or "set"after straining beyond the yield point. The 0.1proof stress value may be determined from the tensile test curve for the material in question as follows: Mark the point P on the strain axis which is equivalent to 0.1%strain.From P draw a line parallel with the initial straight line portion of the tensile test curve to cut the curve in N.The stress corresponding to Nis then the 0.1%proofstress.A material is considered to satisfy its specification if the permanent set is no more than 0.1%after the proof stress has been applied for 15 seconds and removed. Beyond the yield point some increase in load is required to take the strain to point E on the graph.Between D and E the material is said to be in the elastic-plastic state,some of the section remaining elastic and hence contributing to recovery of the original dimensions if load is removed,the remainder being plastic.Beyond E the cross-sectional area of the bar
51.7 Simple Stress and Strain 5 Hooke’s law does not apply. The limiting point B for this condition is termed the elastic limit. For most practical purposes it can often be assumed that points A and B are coincident. Beyond the elastic limit plastic deformation occurs and strains are not totally recoverable. There will thus be some permanent deformation or permanent set when load is removed. After the points C, termed the upper yield point, and D, the lower yield point, relatively rapid increases in strain occur without correspondingly high increases in load or stress. The graph thus becomes much more shallow and covers a much greater portion of the strain axis than does the elastic range of the material. The capacity of a material to allow these large plastic deformations is a measure of the so-called ductility of the material, and this will be discussed in greater detail below. For certain materials, for example, high carbon steels and non-ferrous metals, it is not possible to detect any difference between the upper and lower yield points and in some cases no yield point exists at all. In such cases a proof stress is used to indicate the onset of plastic strain or as a comparison of the relative properties with another similar material. This involves a measure of the permanent deformation produced by a loading cycle; the 0.1 % proof stress, for example, is that stress which, when removed, produces a permanent strain or “set” of 0.1 % of the original gauge length-see Fig. 1.4(a). b ”7 E i5 c r-=F I’ \ I 0 I % 1 Permanent ‘set’ Fig. 1.4. (a) Determination of 0.1 % proof stress. Fig. 1.4. (b) Permanent deformation or “set” after straining beyond the yield point. The 0.1 % proof stress value may be determined from the tensile test curve for the material in question as follows: Mark the point P on the strain axis which is equivalent to 0.1 % strain. From P draw a line parallel with the initial straight line portion of the tensile test curve to cut the curve in N. The stress corresponding to Nis then the 0.1 %proof stress. A material is considered to satisfy its specification if the permanent set is no more than 0.1 %after the proof stress has been applied for 15 seconds and removed. Beyond the yield point some increase in load is required to take the strain to point E on the graph. Between D and E the material is said to be in the elastic-plastic state, some of the section remaining elastic and hence contributing to recovery of the original dimensions if load is removed, the remainder being plastic. Beyond E the cross-sectional area of the bar