《工程测试与信号处理》课程教学资源（文献资料）Strain Measurement.pdf

11ChapterStrain Measurement11.1 INTRODUCTIONThe design of load-carrying components for machines and structures requires information con-cerning the distribution offorces within the particular component.Proper design of devices such asshafts,pressurevessels,and support structures must considerload-carryingcapacityand allowabledeflections.Mechanicsofmaterials provides a basisforpredictingtheseessential characteristicsofamechanical design, and provides the fundamental understanding of the behavior of load-carryingparts.However,theoretical analysis is often not sufficient,and experimental measurements arerequired to achieve a final design.Engineering designs are based on a safe level of stress within a material. In an object that issubjecttoloads,forceswithintheobjectactto balancetheexternal loads.As a simple example, consider a slender rod that is placed in uniaxial tension, as shown inFigure 11.1.If the rod is sectioned at B-B,a force within the material at B-B is necessary tomaintain static equilibriumfor the sectioned rod. Such a force within the rod, acting per unit area, iscalled stress.Design criteria are based on stress levels within a part.In most cases stress cannot bemeasured directly.But the length of the rod in Figure 11.1 changes when the load is applied, andsuch changes in length or shape of a material can be measured. This chapter discusses themeasurement of physical displacements in engineering components. The stress is calculatedfromthesemeasured deflections.Upon completion of this chapter, the reader will be able to.define strain and delineatethe difficulty in measuring stress,? state the physical principles underlying mechanical strain gauges,.analyze strain gauge bridge circuits, and. describe methodsfor optical strain measurement.11.2STRESSANDSTRAINBeforeweproceedtodeveloptechniquesforstrainmeasurements,webrieflyreviewtherelationshipbetween deflections and stress.The experimental analysis of stress is accomplished bymeasuringthedeformation of apart underload,and inferringtheexisting stateof stressfromthemeasureddeflections.Again, consider therod in Figure 11.1.If therod has a cross-sectional area of A,and the466

E1C11 09/14/2010 13:14:1 Page 466 Chapter 11 Strain Measurement 11.1 INTRODUCTION The design of load-carrying components for machines and structures requires information concerning the distribution of forces within the particular component. Proper design of devices such as shafts, pressure vessels, and support structures must consider load-carrying capacity and allowable deflections. Mechanics of materials provides a basis for predicting these essential characteristics of a mechanical design, and provides the fundamental understanding of the behavior of load-carrying parts. However, theoretical analysis is often not sufficient, and experimental measurements are required to achieve a final design. Engineering designs are based on a safe level of stress within a material. In an object that is subject to loads, forces within the object act to balance the external loads. As a simple example, consider a slender rod that is placed in uniaxial tension, as shown in Figure 11.1. If the rod is sectioned at B–B, a force within the material at B–B is necessary to maintain static equilibrium for the sectioned rod. Such a force within the rod, acting per unit area, is called stress. Design criteria are based on stress levels within a part. In most cases stress cannot be measured directly. But the length of the rod in Figure 11.1 changes when the load is applied, and such changes in length or shape of a material can be measured. This chapter discusses the measurement of physical displacements in engineering components. The stress is calculated from these measured deflections. Upon completion of this chapter, the reader will be able to define strain and delineate the difficulty in measuring stress, state the physical principles underlying mechanical strain gauges, analyze strain gauge bridge circuits, and describe methods for optical strain measurement. 11.2 STRESS AND STRAIN Before we proceed to develop techniques for strain measurements, we briefly review the relationship between deflections and stress. The experimental analysis of stress is accomplished by measuring the deformation of a part under load, and inferring the existing state of stress from the measured deflections. Again, consider the rod in Figure 11.1. If the rod has a cross-sectional area of Ac, and the 466

11.2467Stress and StrainFNFNCross-sectionalBareaA.BFNFigure 11.1 Free-body diagramillustrating internal forcesforarod inBuniaxial tension.load is applied only along the axis of the rod, the normal stress is defined as(11.1)a=FN/Acwhere A。is the cross-sectional area and F is the tension force applied to the rod normal to the areaAc. The ratio of the change in length of the rod (which results from applying the load) to the originallengthistheaxial strain,defined as&a = 8L/L(11.2)whereis theaveragestrain overthelengthL,Lis the change in length,andListheoriginal unloadedlength.For mostengineeringmaterials, strain is a small quantity; strain is usuallyreported in units of10- in./in.or 10-m/m. These units are equivalent to a dimensionless unit called a microstrain (μ).Stress-strain diagrams are very important in understanding the behavior of a material under load.Figure 11.2 is such a diagram for mild steel (a ductile material). For loads less than that required topermanently deform the material,most engineering materials display a linear relationship betweenstress and strain.Therange of stress over whichthis linear relationship holds is called the elasticregion.The relationshipbetween uniaxial stress and strain for this elastic behavior is expressed as(11.3)Oa=Em&awhere Em is the modulus ofelasticity,or Young'smodulus, and the relationship is called Hooke'slaw.Hooke'slawapplies onlyoverthe rangeof applied stress wherethe relationshipbetween stress andstrain is linear.Different materialsrespond in a variety of waystoloadsbeyond the linear range, largelydepending on whether the material is ductile or brittle.For almost all engineering components,stresslevels aredesigned to remain well belowthe elastic limit of thematerial; thus, a direct linearrelationshipmaybeestablishedbetweenstressandstrain.Underthisassumption,Hooke'slawformsthe basis for experimental stress analysis through the measurement of strain.Lateral StrainsConsider the elongation of therod shown in Figure 11.1 that occurs as aresult of the load Fn.As therod is stretched in the axial direction,the cross-sectional area must decrease since the total mass (or

E1C11 09/14/2010 13:14:1 Page 467 load is applied only along the axis of the rod, the normal stress is defined as sa ¼ FN=Ac ð11:1Þ where Ac is the cross-sectional area and FN is the tension force applied to the rod normal to the area Ac. The ratio of the change in length of the rod (which results from applying the load) to the original length is the axial strain, defined as ea ¼ dL=L ð11:2Þ where ea is the average strain over the length L, dL is the change in length, andL is the original unloaded length. For most engineering materials, strain is a small quantity; strain is usually reported in units of 106 in./in. or 106 m/m. These units are equivalent to a dimensionless unit called a microstrain (me). Stress–strain diagrams are very important in understanding the behavior of a material under load. Figure 11.2 is such a diagram for mild steel (a ductile material). For loads less than that required to permanently deform the material, most engineering materials display a linear relationship between stress and strain. The range of stress over which this linear relationship holds is called the elastic region. The relationship between uniaxial stress and strain for this elastic behavior is expressed as sa ¼ Emea ð11:3Þ where Em is the modulus of elasticity, or Young’s modulus, and the relationship is called Hooke’s law. Hooke’s law applies only over the range of applied stress where the relationship between stress and strain is linear. Different materials respond in a variety of ways to loads beyond the linear range, largely depending on whether the material is ductile or brittle. For almost all engineering components, stress levels are designed to remain well below the elastic limit of the material; thus, a direct linear relationship may be established between stress and strain. Under this assumption, Hooke’s law forms the basis for experimental stress analysis through the measurement of strain. Lateral Strains Consider the elongation of the rod shown in Figure 11.1 that occurs as a result of the load FN. As the rod is stretched in the axial direction, the cross-sectional area must decrease since the total mass (or FN FN FN B B B B FN Ac σa = Cross-sectional area Ac Figure 11.1 Free-body diagram illustrating internal forces for a rod in uniaxial tension. 11.2 Stress and Strain 467

E1C11 09/14/2010 13:14:1 Page 468 volume for constant density) must be conserved. Similarly, if the rod were compressed in the axial direction, the cross-sectional area would increase. This change in cross-sectional area is most conveniently expressed in terms of a lateral (transverse) strain. For a circular rod, the lateral strain is defined as the change in the diameter divided by the original diameter. In the elastic range, there is a constant rate of change in the lateral strain as the axial strain increases. In the same sense that the modulus of elasticity is a property of a given material, the ratio of lateral strain to axial strain is also a material property. This property is called Poisson’s ratio, defined as yP ¼ jLateral strainj jAxial strainj ¼ eL ea ð11:4Þ Engineering components are seldom subject to one-dimensional axial loading. The relationship between stress and strain must be generalized to a multidimensional case. Consider a two-dimensional geometry, as shown in Figure 11.3, subject to tensile loads in both the x and y directions, resulting in normal stresses sx and sy. In this case, for a biaxial state of stress, the stresses and strains are ey ¼ sy Em yp sx Em ex ¼ sx Em yp sy Em sx ¼ Em ex þ ypey 1 y2 p sy ¼ Em ey þ ypex 1 y2 p txy ¼ Ggxy ð11:5Þ In this case, all of the stress and strain components lie in the same plane. The state of stress in the elastic condition for a material is similarly related to the strains in a complete three-dimensional situation (1, 2). Since stress and strain are related, it is possible to determine stress from measured strains under appropriate conditions. However, strain measurements are made at the surface of an engineering component. The measurement yields information about the state of stress on the surface 0.001 0.1 0.002 0.2 0.003 0.3 0.004 0.4 in./in. or m/m percent 0 50 100 150 200 250 300 Strain Stress (psi) Stress (MN/m2) 10,000 20,000 30,000 40,000 Figure 11.2 A typical stress– strain curve for mild steel. 468 Chapter 11 Strain Measurement

11.3469ResistanceStrainGaugesoyFigure11.3Biaxial state of stress.of the part.The analysis of measured strains requires application of the relationship between stressand strain at a surface.Such analysis of strain data is described elsewhere (3),and an exampleprovided in this chapter.11.3RESISTANCESTRAINGAUGESThemeasurementof thesmalldisplacementsthatoccurinamaterial orobjectundermechanicalload canbeaccomplished bymethods as simple asobservingthe change in thedistancebetween twoscribe marks on the surface of aload-carrying member, or as advanced as optical holography.In anycase,the ideal sensorfor the measurement of strain would (1) have good spatial resolution,implyingthat the sensor would measure strain at a point; (2) be unaffected by changes in ambient conditions;and (3) have a high-frequencyresponse for dynamic (time-resolved) strain measurements. A sensorthat closely meets these characteristics is the bonded resistance strain gauge.In practical application, the bonded resistance strain gauge is secured to the surface of the testobjectbyanadhesive so that it deforms as thetest object deforms.The resistance of a strain gaugechanges when it is deformed, and this is easily related to the local strain.Both metallic andsemiconductor materials experience a change in electrical resistance when they are subjected to astrain.The amount that the resistance changes depends on how the gauge is deformed, the materialfrom which it is made, and the design of the gauge.Gauges can bemade quite small forgoodresolution and with a low mass to provide a high-frequency response.With some ingenuity,ambienteffectscanbeminimizedoreliminatedIn an1856publication in thePhilosophicalTransactions of theRoyal Society in England,LordKelvin (William Thomson)(4)laid the foundations for understanding the changes in electricalresistance that metals undergo when subjected to loads, which eventually led to the strain gaugeconcept.Twoindividuals began themodern developmentof strain measurement in thelate1930s-Edward Simmons attheCalifornia Instituteof TechnologyandArthurRugeattheMassachusettsInstitute of Technology.Their development of the bonded metallic wire strain gauge led tocommercially available strain gauges.The resistance strain gauge also forms the basis for a varietyof other transducers,such as load cells,pressure transducers,and torquemeters

E1C11 09/14/2010 13:14:1 Page 469 of the part. The analysis of measured strains requires application of the relationship between stress and strain at a surface. Such analysis of strain data is described elsewhere (3), and an example provided in this chapter. 11.3 RESISTANCE STRAIN GAUGES The measurement of the small displacements that occur in a material or object under mechanical load can be accomplished by methods as simple as observing the change in the distance between two scribe marks on the surface of a load-carrying member, or as advanced as optical holography. In any case, the ideal sensor for the measurement of strain would (1) have good spatial resolution, implying that the sensor would measure strain at a point; (2) be unaffected by changes in ambient conditions; and (3) have a high-frequency response for dynamic (time-resolved) strain measurements. A sensor that closely meets these characteristics is the bonded resistance strain gauge. In practical application, the bonded resistance strain gauge is secured to the surface of the test object by an adhesive so that it deforms as the test object deforms. The resistance of a strain gauge changes when it is deformed, and this is easily related to the local strain. Both metallic and semiconductor materials experience a change in electrical resistance when they are subjected to a strain. The amount that the resistance changes depends on how the gauge is deformed, the material from which it is made, and the design of the gauge. Gauges can be made quite small for good resolution and with a low mass to provide a high-frequency response. With some ingenuity, ambient effects can be minimized or eliminated. In an 1856 publication in the Philosophical Transactions of the Royal Society in England, Lord Kelvin (William Thomson) (4) laid the foundations for understanding the changes in electrical resistance that metals undergo when subjected to loads, which eventually led to the strain gauge concept. Two individuals began the modern development of strain measurement in the late 1930s— Edward Simmons at the California Institute of Technology and Arthur Ruge at the Massachusetts Institute of Technology. Their development of the bonded metallic wire strain gauge led to commercially available strain gauges. The resistance strain gauge also forms the basis for a variety of other transducers, such as load cells, pressure transducers, and torque meters. σx σx σy σy Figure 11.3 Biaxial state of stress. 11.3 Resistance Strain Gauges 469