15SECTION1.3Mechanical Properties of Materials1.3MECHANICALPROPERTIESOFMATERIALSThe design of machines and structures so that they will function prop-erly requiresthatwe understand themechanical behavior of thematerials being used.Ordinarily,the only wayto determinehowmaterialsbehavewhen theyare subjected to loads is toperform experiments inthe laboratory.The usual procedure is to place small specimens of thematerial in testing machines, apply the loads, and then measure theresulting deformations (such as changes in length and changes in diameter).Mostmaterials-testinglaboratoriesare equipped withmachines capableof loading specimensin avarietyof ways,includingbothstaticanddynamicloadingintensionandcompression.A typical tensile-test machine is shown in Fig.1-7.The test specimen is installedbetweenthetwo largegripsof thetestingmachineandthenloaded intension.Measuringdevices record thedeformations,andthe automatic control and data-processing systems (at the left in thephoto)tabulate andgraph theresults.Amore detailed view of a tensile-test specimen is shown in Fig.1-8on the next page.The ends of the circular specimen are enlarged wheretheyfit in the grips so that failure will not occur near the grips them-selves.Afailureattheendswouldnotproducethedesired informationabout the material, because the stress distribution near the grips is notuniform,asexplainedinSection1.2.Inaproperlydesignedspecimenfailure will occur in the prismatic portion of the specimen where thestress distribution is uniform and the bar is subjected only to puretension.This situation is shown in Fig.1-8, where the steel specimenhasjustfractured underload.Thedeviceattheleft,whichisattachedbyFIG.1-7Tensile-testmachine withautomaticdata-processingsystem(Courtesy of MTS Systems Corporation)
SECTION 1.3 Mechanical Properties of Materials 15 1.3 MECHANICAL PROPERTIES OF MATERIALS The design of machines and structures so that they will function properly requires that we understand the mechanical behavior of the materials being used. Ordinarily, the only way to determine how materials behave when they are subjected to loads is to perform experiments in the laboratory. The usual procedure is to place small specimens of the material in testing machines, apply the loads, and then measure the resulting deformations (such as changes in length and changes in diameter). Most materials-testing laboratories are equipped with machines capable of loading specimens in a variety of ways, including both static and dynamic loading in tension and compression. A typical tensile-test machine is shown in Fig. 1-7. The test specimen is installed between the two large grips of the testing machine and then loaded in tension. Measuring devices record the deformations, and the automatic control and data-processing systems (at the left in the photo) tabulate and graph the results. A more detailed view of a tensile-test specimen is shown in Fig. 1-8 on the next page. The ends of the circular specimen are enlarged where they fit in the grips so that failure will not occur near the grips themselves. A failure at the ends would not produce the desired information about the material, because the stress distribution near the grips is not uniform, as explained in Section 1.2. In a properly designed specimen, failure will occur in the prismatic portion of the specimen where the stress distribution is uniform and the bar is subjected only to pure tension. This situation is shown in Fig. 1-8, where the steel specimen has just fractured under load. The device at the left, which is attached by FIG. 1-7 Tensile-test machine with automatic data-processing system. (Courtesy of MTS Systems Corporation)
16CHAPTER1Tension,Compression,andShearFIG.1-8Typical tensile-test specimenwithextensometerattached:thespecimenhasjustfracturedintension(CourtesyofMTSSystems Corporation)two arms to the specimen,is an extensometerthatmeasures the elonga-tion during loading.In order that test results will be comparable,thedimensions of testspecimens and the methods of applying loads must be standardizedOne of the major standards organizations in the United States is theAmerican Society for Testing and Materials (ASTM), a technical societythat 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).Similarorganizations existinothercountries.TheASTM standard tension specimenhas a diameter of 0.505in.and a gage length of 2.0 in. between the gage marks, which are thepoints wherethe extensometer arms are attached to the specimen (seeFig.1-8).As the specimen is pulled, the axial load is measured andrecorded,eitherautomaticallyorbyreadingfromadial.Theelongationoverthegagelengthismeasured simultaneously,eitherbymechanical
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)
17SECTION1.3Mechanical PropertiesofMaterialsgages of the kind shown in Fig.1-8 or by electrical-resistance straingages.In a static test, the load is applied slowly and the precise rate ofloading isnot of interest becauseit does notaffectthebehavior of thespecimen.However,in a dynamic test the load is applied rapidly andsometimes in a cyclical manner.Since the nature of a dynamic loadaffects the properties of the materials,the rate of loading must also bemeasured.Compressiontests of metals arecustomarily made on small speci-mens in the shape of cubes or circular cylinders.For instance,cubesmay be 2.0 in.on a side, and cylinders may have diameters of 1 in. andlengths from1to12in.Both theload appliedbythemachineand theshorteningof the specimenmaybemeasured.The shortening should bemeasured over a gage length that is less than the total length of the spec-imeninordertoeliminateendeffects.Concreteistested in compression on importantconstructionproj-ects to ensure that the required strength has been obtained. One typeof concrete test specimen is 6 in.in diameter, 12 in.in length, and28 days old (the age of concrete is important because concretegainsstrength as it cures).Similar but somewhat smaller specimens areused whenperforming compression tests of rock (Fig.1-9,on thenextpage).Stress-Strain DiagramsTest results generally depend upon the dimensions of the specimen beingtested. Since it is unlikely that we will be designing a structure havingparts that are the same size as the test specimens, we need to expressthe test results in a form that can be applied to members of any size.A simple wayto achieve this objective is to convert the test results tostresses and strains.The axial stress in a test specimen is calculated by dividing theaxial load P by the cross-sectional area A (Eq. 1-1). When the initialarea of the specimen is used in the calculation, the stress is called thenominal stress (other names are conventional stress and engineeringstress).Amoreexactvalueoftheaxial stress,called thetruestress,canbe calculated by using the actual area of the bar at the cross sectionwhere failure occurs.Since the actual area in a tension test is always lessthan the initial area (as illustrated in Fig. 1-8), the true stress is largerthanthenominal stress.The average axial strain e in the test specimen is found by dividingthe measured elongation 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. Sincethe distance between the gage marks increases as the tensile load isapplied, we can calculate the true strain (or natural strain)at any valueof the load by using the actual distance between the gage marks.Intension, truestrainisalways smallerthan 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
18CHAPTER1Tension,Compression,andShearFIG.1-9 Rock sample being tested incompressiontoobtaincompressivestrength, elastic modulus andPoisson's ratio(Courtesyof MTSSystemsCorporation)most engineeringpurposes,nominal stressand nominal strainareadequate,asexplainedlaterinthissection.Afterperforming atensionor compressiontestand determiningthestressandstrainat various magnitudes of theload,wecanplotadiagram of stress versus strain. Such a stress-strain diagram is a char-acteristicof theparticularmaterial beingtested and conveysimportantinformation about themechanical properties and typeof behavior.'Stress-strain diagrams were originated by Jacob Bermoulli (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
19SECTION1.3MechanicalPropertiesofMaterialsThe first material we will discuss is structural steel, alsoknown asmild steel or low-carbon steel.Structural steel is one ofthemost widelyused metals and is found in buildings,bridges,cranes, ships,towers,vehicles,andmanyothertypesof construction.Astress-straindiagramforatypicalstructural steelintensionis showninFig.1-10.Strains areplotted on the horizontal axis and stresses on thevertical axis.(In orderto display all of the important features of this material, the strain axis inFig.1-10 is not drawn to scale.)The diagrambegins with a straight linefrom the origin Oto point A.which means that the relationship between stress and strain in this initialregion is not only linear butalso proportional.Beyond point A,theproportionality between stress and strain no longer exists;hence thestress at A is called the proportional limit.For low-carbon steels,thislimitisintherange30to50ksi(210to350MPa),buthigh-strengthsteels (with higher carbon content plus other alloys)can have propor-tional limitsof morethan80ksi(550MPa).Theslopeof the straightline from O to A is called the modulus of elasticity.Because the slopehas units of stress divided by strain,modulus of elasticity has the sameunits as stress.(Modulus of elasticity is discussed later in Section 1.5.)With an increase in stress beyond theproportional limit, the strainbegins to increasemorerapidly for each increment in stress.Conse-quently,the stress-strain curvehas a smaller and smaller slope,until,atpoint B, the curve becomes horizontal (see Fig.1-10).Beginning at thispoint, considerable elongation of the test specimen occurs with noEUltimatestressYield stressBFracturProportionallimitWStrainPerfectNeckingFIG.1-10 Stress-strain diagram forhardeningplasticityLineara typical structural steel in tensionor yielding(notto scale)region"Two variables are said to be proportional if their ratio remains constant. Thereforea 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 aproportional 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 butnot 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