16 CHAPTER 1 Tension,Compression,and Shear FIG.1-8 Typical tensile-test specimen with extensometer attached:the specimen has just fractured in tension. (Courtesy of MTS Systems Corporation) two arms to the specimen,is an extensometer that measures the elonga- tion during loading. In order that test results will be comparable,the dimensions of test specimens and the methods of applying loads must be standardized. One of the major standards organizations in the United States is the American Society for Testing and Materials (ASTM),a technical society that publishes specifications and standards for materials and testing. Other standardizing organizations are the American Standards Associa- tion (ASA)and the National Institute of Standards and Technology (NIST).Similar organizations exist in other countries. The ASTM standard tension specimen has a diameter of 0.505 in. and a gage length of 2.0 in.between the gage marks,which are the points where the extensometer arms are attached to the specimen (see Fig.1-8).As the specimen is pulled,the axial load is measured and recorded,either automatically or by reading from a dial.The elongation over the gage length is measured simultaneously,either by mechanical
16 CHAPTER 1 Tension, Compression, and Shear two arms to the specimen, is an extensometer that measures the elongation during loading. In order that test results will be comparable, the dimensions of test specimens and the methods of applying loads must be standardized. One of the major standards organizations in the United States is the American Society for Testing and Materials (ASTM), a technical society that publishes specifications and standards for materials and testing. Other standardizing organizations are the American Standards Association (ASA) and the National Institute of Standards and Technology (NIST). Similar organizations exist in other countries. The ASTM standard tension specimen has a diameter of 0.505 in. and a gage length of 2.0 in. between the gage marks, which are the points where the extensometer arms are attached to the specimen (see Fig. 1-8). As the specimen is pulled, the axial load is measured and recorded, either automatically or by reading from a dial. The elongation over the gage length is measured simultaneously, either by mechanical FIG. 1-8 Typical tensile-test specimen with extensometer attached; the specimen has just fractured in tension. (Courtesy of MTS Systems Corporation)
SECTION 1.3 Mechanical Properties of Materials 17 gages of the kind shown in Fig.1-8 or by electrical-resistance strain gages. In a static test,the load is applied slowly and the precise rate of loading is not of interest because it does not affect the behavior of the specimen.However,in a dynamic test the load is applied rapidly and sometimes in a cyclical manner.Since the nature of a dynamic load affects the properties of the materials,the rate of loading must also be measured. Compression tests of metals are customarily made on small speci- mens in the shape of cubes or circular cylinders.For instance,cubes may be 2.0 in.on a side,and cylinders may have diameters of 1 in.and lengths from I to 12 in.Both the load applied by the machine and the shortening of the specimen may be measured.The shortening should be measured over a gage length that is less than the total length of the spec- imen in order to eliminate end effects. Concrete is tested in compression on important construction proj- ects to ensure that the required strength has been obtained.One type of concrete test specimen is 6 in.in diameter,12 in.in length,and 28 days old (the age of concrete is important because concrete gains strength as it cures).Similar but somewhat smaller specimens are used when performing compression tests of rock (Fig.1-9,on the next page). Stress-Strain Diagrams Test results generally depend upon the dimensions of the specimen being tested.Since it is unlikely that we will be designing a structure having parts that are the same size as the test specimens,we need to express the test results in a form that can be applied to members of any size. A simple way to achieve this objective is to convert the test results to stresses and strains. The axial stress o in a test specimen is calculated by dividing the axial load P by the cross-sectional area A (Eq.1-1).When the initial area of the specimen is used in the calculation,the stress is called the nominal stress (other names are conventional stress and engineering stress).A more exact value of the axial stress,called the true stress,can be calculated by using the actual area of the bar at the cross section where failure occurs.Since the actual area in a tension test is always less than the initial area (as illustrated in Fig.1-8),the true stress is larger than the nominal stress. The average axial strain e in the test specimen is found by dividing the measured elongation 6 between the gage marks by the gage length L (see Fig.1-8 and Eq.1-2).If the initial gage length is used in the calcula- tion (for instance,2.0 in.),then the nominal strain is obtained.Since the distance between the gage marks increases as the tensile load is applied,we can calculate the true strain (or natural strain)at any value of the load by using the actual distance between the gage marks.In tension,true strain is always smaller than nominal strain.However,for
gages of the kind shown in Fig. 1-8 or by electrical-resistance strain gages. In a static test, the load is applied slowly and the precise rate of loading is not of interest because it does not affect the behavior of the specimen. However, in a dynamic test the load is applied rapidly and sometimes in a cyclical manner. Since the nature of a dynamic load affects the properties of the materials, the rate of loading must also be measured. Compression tests of metals are customarily made on small specimens in the shape of cubes or circular cylinders. For instance, cubes may be 2.0 in. on a side, and cylinders may have diameters of 1 in. and lengths from 1 to 12 in. Both the load applied by the machine and the shortening of the specimen may be measured. The shortening should be measured over a gage length that is less than the total length of the specimen in order to eliminate end effects. Concrete is tested in compression on important construction projects to ensure that the required strength has been obtained. One type of concrete test specimen is 6 in. in diameter, 12 in. in length, and 28 days old (the age of concrete is important because concrete gains strength as it cures). Similar but somewhat smaller specimens are used when performing compression tests of rock (Fig. 1-9, on the next page). Stress-Strain Diagrams Test results generally depend upon the dimensions of the specimen being tested. Since it is unlikely that we will be designing a structure having parts that are the same size as the test specimens, we need to express the test results in a form that can be applied to members of any size. A simple way to achieve this objective is to convert the test results to stresses and strains. The axial stress s in a test specimen is calculated by dividing the axial load P by the cross-sectional area A (Eq. 1-1). When the initial area of the specimen is used in the calculation, the stress is called the nominal stress (other names are conventional stress and engineering stress). A more exact value of the axial stress, called the true stress, can be calculated by using the actual area of the bar at the cross section where failure occurs. Since the actual area in a tension test is always less than the initial area (as illustrated in Fig. 1-8), the true stress is larger than the nominal stress. The average axial strain e in the test specimen is found by dividing the measured elongation d between the gage marks by the gage length L (see Fig. 1-8 and Eq. 1-2). If the initial gage length is used in the calculation (for instance, 2.0 in.), then the nominal strain is obtained. Since the distance between the gage marks increases as the tensile load is applied, we can calculate the true strain (or natural strain) at any value of the load by using the actual distance between the gage marks. In tension, true strain is always smaller than nominal strain. However, for SECTION 1.3 Mechanical Properties of Materials 17
18 CHAPTER 1 Tension,Compression,and Shear FIG.1-9 Rock sample being tested in compression to obtain compressive strength,elastic modulus and Poisson's ratio(Courtesy of MTS Systems Corporation) most engineering purposes,nominal stress and nominal strain are adequate,as explained later in this section. After performing a tension or compression test and determining the stress and strain at various magnitudes of the load,we can plot a diagram of stress versus strain.Such a stress-strain diagram is a char- acteristic of the particular material being tested and conveys important information about the mechanical properties and type of behavior. "Stress-strain diagrams were originated by Jacob Bemoulli(1654-1705)and J.V.Poncelet (1788-1867):see Ref.1-4
18 CHAPTER 1 Tension, Compression, and Shear most engineering purposes, nominal stress and nominal strain are adequate, as explained later in this section. After performing a tension or compression test and determining the stress and strain at various magnitudes of the load, we can plot a diagram of stress versus strain. Such a stress-strain diagram is a characteristic of the particular material being tested and conveys important information about the mechanical properties and type of behavior.* FIG. 1-9 Rock sample being tested in compression to obtain compressive strength, elastic modulus and Poisson’s ratio (Courtesy of MTS Systems Corporation) * Stress-strain diagrams were originated by Jacob Bernoulli (1654–1705) and J. V. Poncelet (1788–1867); see Ref. 1-4
SECTION 1.3 Mechanical Properties of Materials 19 The first material we will discuss is structural steel,also known as mild steel or low-carbon steel.Structural steel is one of the most widely used metals and is found in buildings,bridges,cranes,ships,towers, vehicles,and many other types of construction.A stress-strain diagram for a typical structural steel in tension is shown in Fig.1-10.Strains are plotted on the horizontal axis and stresses on the vertical axis.(In order to display all of the important features of this material,the strain axis in Fig.1-10 is not drawn to scale.) The diagram begins with a straight line from the origin O to point A, which means that the relationship between stress and strain in this initial region is not only linear but also proportional."Beyond point A,the proportionality between stress and strain no longer exists;hence the stress at A is called the proportional limit.For low-carbon steels,this limit is in the range 30 to 50 ksi (210 to 350 MPa),but high-strength steels (with higher carbon content plus other alloys)can have propor- tional limits of more than 80 ksi(550 MPa).The slope of the straight line from O to A is called the modulus of elasticity.Because the slope has units of stress divided by strain,modulus of elasticity has the same units as stress.(Modulus of elasticity is discussed later in Section 1.5.) With an increase in stress beyond the proportional limit,the strain begins to increase more rapidly for each increment in stress.Conse- quently,the stress-strain curve has a smaller and smaller slope,until,at point B,the curve becomes horizontal (see Fig.1-10).Beginning at this point,considerable elongation of the test specimen occurs with no Ultimate- stress Yield stress B Fracture Proportional limit FIG.1-10 Stress-strain diagram for Perfect Strain Necking a typical structural steel in tension Linear plasticity hardening (not to scale) region or yielding "Two variables are said to be proportional if their ratio remains constant.Therefore. a proportional relationship may be represented by a straight line through the origin. However,a proportional relationship is not the same as a linear relationship.Although a proportional relationship is linear,the converse is not necessarily true,because a rela- tionship represented by a straight line that does not pass through the origin is linear but not proportional.The often-used expression"directly proportional"is synonymous with "proportional"(Ref.1-5)
SECTION 1.3 Mechanical Properties of Materials 19 The first material we will discuss is structural steel, also known as mild steel or low-carbon steel. Structural steel is one of the most widely used metals and is found in buildings, bridges, cranes, ships, towers, vehicles, and many other types of construction. A stress-strain diagram for a typical structural steel in tension is shown in Fig. 1-10. Strains are plotted on the horizontal axis and stresses on the vertical axis. (In order to display all of the important features of this material, the strain axis in Fig. 1-10 is not drawn to scale.) The diagram begins with a straight line from the origin O to point A, which means that the relationship between stress and strain in this initial region is not only linear but also proportional. * Beyond point A, the proportionality between stress and strain no longer exists; hence the stress at A is called the proportional limit. For low-carbon steels, this limit is in the range 30 to 50 ksi (210 to 350 MPa), but high-strength steels (with higher carbon content plus other alloys) can have proportional limits of more than 80 ksi (550 MPa). The slope of the straight line from O to A is called the modulus of elasticity. Because the slope has units of stress divided by strain, modulus of elasticity has the same units as stress. (Modulus of elasticity is discussed later in Section 1.5.) With an increase in stress beyond the proportional limit, the strain begins to increase more rapidly for each increment in stress. Consequently, the stress-strain curve has a smaller and smaller slope, until, at point B, the curve becomes horizontal (see Fig. 1-10). Beginning at this point, considerable elongation of the test specimen occurs with no * Two variables are said to be proportional if their ratio remains constant. Therefore, a proportional relationship may be represented by a straight line through the origin. However, a proportional relationship is not the same as a linear relationship. Although a proportional relationship is linear, the converse is not necessarily true, because a relationship represented by a straight line that does not pass through the origin is linear but not proportional. The often-used expression “directly proportional” is synonymous with “proportional” (Ref. 1-5). FIG. 1-10 Stress-strain diagram for a typical structural steel in tension (not to scale) Fracture Linear region Perfect plasticity or yielding Strain hardening Necking A O B C D E E' Proportional limit Yield stress Ultimate stress e s
20 CHAPTER 1 Tension,Compression,and Shear noticeable increase in the tensile force (from B to C).This phenomenon is known as yielding of the material,and point B is called the yield point.The corresponding stress is known as the yield stress of the steel. In the region from B to C(Fig.1-10),the material becomes perfectly Load plastic,which means that it deforms without an increase in the applied load.The elongation of a mild-steel specimen in the perfectly plastic region is typically 10 to 15 times the elongation that occurs in the linear region (between the onset of loading and the proportional limit).The presence of very large strains in the plastic region (and beyond)is the reason for not plotting this diagram to scale. After undergoing the large strains that occur during yielding in the region BC,the steel begins to strain harden.During strain hardening,the material undergoes changes in its crystalline structure,resulting in increased resistance of the material to further deformation.Elongation of the test specimen in this region requires an increase in the tensile load, and therefore the stress-strain diagram has a positive slope from C to D. The load eventually reaches its maximum value,and the corresponding stress (at point D)is called the ultimate stress.Further stretching of the bar is actually accompanied by a reduction in the load,and fracture finally occurs at a point such as E in Fig.1-10. The yield stress and ultimate stress of a material are also called the yield strength and ultimate strength,respectively.Strength is a general term that refers to the capacity of a structure to resist loads.For instance, Region the yield strength of a beam is the magnitude of the load required to cause of yielding in the beam,and the ultimate strength of a truss is the maximum necking load it can support,that is,the failure load.However,when conducting a Region tension test of a particular material,we define load-carrying capacity by the of stresses in the specimen rather than by the total loads acting on the speci- fracture men.As a result,the strength of a material is usually stated as a stress. When a test specimen is stretched,lateral contraction occurs,as previously mentioned.The resulting decrease in cross-sectional area is too small to have a noticeable effect on the calculated values of the stresses up to about point C in Fig.1-10,but beyond that point the reduction in area begins to alter the shape of the curve.In the vicinity of the ultimate stress,the reduction in area of the bar becomes clearly visi- ble and a pronounced necking of the bar occurs(see Figs.1-8 and 1-11). If the actual cross-sectional area at the narrow part of the neck is used to calculate the stress,the true stress-strain curve (the dashed line CE'in Fig.1-10)is obtained.The total load the bar can carry does indeed diminish after the ultimate stress is reached (as shown by curve DE),but this reduction is due to the decrease in area of the bar and not to a loss in strength of the material itself.In reality,the material withstands an increase in true stress up to failure (point E').Because most structures are expected to function at stresses below the proportional limit,the conven- Load tional stress-strain curve OABCDE,which is based upon the original FIG.1-11 Necking of a mild-steel bar in cross-sectional area of the specimen and is easy to determine,provides tension satisfactory information for use in engineering design
20 CHAPTER 1 Tension, Compression, and Shear noticeable increase in the tensile force (from B to C). This phenomenon is known as yielding of the material, and point B is called the yield point. The corresponding stress is known as the yield stress of the steel. In the region from B to C (Fig. 1-10), the material becomes perfectly plastic, which means that it deforms without an increase in the applied load. The elongation of a mild-steel specimen in the perfectly plastic region is typically 10 to 15 times the elongation that occurs in the linear region (between the onset of loading and the proportional limit). The presence of very large strains in the plastic region (and beyond) is the reason for not plotting this diagram to scale. After undergoing the large strains that occur during yielding in the region BC, the steel begins to strain harden. During strain hardening, the material undergoes changes in its crystalline structure, resulting in increased resistance of the material to further deformation. Elongation of the test specimen in this region requires an increase in the tensile load, and therefore the stress-strain diagram has a positive slope from C to D. The load eventually reaches its maximum value, and the corresponding stress (at point D) is called the ultimate stress. Further stretching of the bar is actually accompanied by a reduction in the load, and fracture finally occurs at a point such as E in Fig. 1-10. The yield stress and ultimate stress of a material are also called the yield strength and ultimate strength, respectively. Strength is a general term that refers to the capacity of a structure to resist loads. For instance, the yield strength of a beam is the magnitude of the load required to cause yielding in the beam, and the ultimate strength of a truss is the maximum load it can support, that is, the failure load. However, when conducting a tension test of a particular material, we define load-carrying capacity by the stresses in the specimen rather than by the total loads acting on the specimen. As a result, the strength of a material is usually stated as a stress. When a test specimen is stretched, lateral contraction occurs, as previously mentioned. The resulting decrease in cross-sectional area is too small to have a noticeable effect on the calculated values of the stresses up to about point C in Fig. 1-10, but beyond that point the reduction in area begins to alter the shape of the curve. In the vicinity of the ultimate stress, the reduction in area of the bar becomes clearly visible and a pronounced necking of the bar occurs (see Figs. 1-8 and 1-11). If the actual cross-sectional area at the narrow part of the neck is used to calculate the stress, the true stress-strain curve (the dashed line CE in Fig. 1-10) is obtained. The total load the bar can carry does indeed diminish after the ultimate stress is reached (as shown by curve DE), but this reduction is due to the decrease in area of the bar and not to a loss in strength of the material itself. In reality, the material withstands an increase in true stress up to failure (point E). Because most structures are expected to function at stresses below the proportional limit, the conventional stress-strain curve OABCDE, which is based upon the original cross-sectional area of the specimen and is easy to determine, provides satisfactory information for use in engineering design. Load Load Region of necking Region of fracture FIG. 1-11 Necking of a mild-steel bar in tension