Analysis of Distortion and Deformation Revised by Roch J. Shipley and David A Moore, Packer Engineering and William Dobson, Binary Engineering Associates. Inc Introduction THIS HANDBOOK is organized according to four general categories of failure: fracture, corrosion, wear, and the subject of this article, distortion. One reason metals are so widely used as engineering materials is that they have high strength but also generally have the capability to respond to load(stress) by deforming. In fact, much of metallurgical engineering is concerned with balancing strength and ductility. Thus, distortion often is observed in analysis of other types of failures, and consideration of the distortion can be an important part of the analysis. Energy is absorbed during deformation, and in some situations, the amount of energy absorbed may also be an important factor. Furthermore, it should be noted that not all distortion necessarily constitutes This article first considers true distortion failures that is situations in which distortion occurs when it should not have occurred and in which the distortion is associated with a functional failure. Then, a more general consideration of distortion in failure analysis is introduced. As used here, distortion will refer to a condition in which the shape of a component has changed without loss of material. Deformation will refer to the process that results in the distortion Distortion failure occurs when a structure or component is deformed so that it can no longer support the load was intended to carry, is incapable of performing its intended function, or interferes with the operation of another component. Distortion failures can be plastic or elastic and may or may not be accompanied by fracture. There are two main types of distortion: size distortion, which refers to a change in volume(growth or shrinkage), and shape distortion(bending or warping), which refers to a change in geometric form. Most of the examples in this article deal with metals, but the concepts also apply to nonmetals. Materials as diverse metals, polymers, and wood are all susceptible to distortion, although the mechanisms may differ somewhat among the different classes of material Distortion failures are ordinarily considered to be self-evident, for example, damage of a car body in a collision or bending of a nail being driven into hard wood. However, the failure analyst is often faced with more subtle situations. For example, the immediate cause of distortion(bending) of an automobile-engine valve stem is contact of the valve head with the piston, but the failure analyst must go beyond this immediate cause in order to recommend proper corrective measures. The valve may have stuck open because of faulty lubrication; the valve spring may have broken because corrosion had weakened it. The spring may have had insufficient strength and taken a set, allowing the valve to drop into the path of the piston, or the engine may have been raced beyond its revolutions per minute limit many times, causing coil clash and subsequent fatigue fracture of the spring. Without careful consideration of all the evidence, the failure analyst may overlook the true cause of a distortion failure. Several common aspects of failure by distortion are discussed in this article, and suitable examples of distortion failures are presented for illustration Analysis of Distortion and Deformation Revised by Roch ]. Shipley and David A. Moore, Packer Engineering and william Dobson, Binary Engineering Associates, Inc. Overloading Every structure has a load limit beyond which it is considered unsafe or unreliable. Applied loads that exceed this limit are known as overloads and sometimes result(depending on the factor of safety used in design)in distortion or fracture of Thefileisdownloadedfromwww.bzfxw.com
Analysis of Distortion and Deformation Revised by Roch J. Shipley and David A. Moore, Packer Engineering and William Dobson, Binary Engineering Associates, Inc. Introduction THIS HANDBOOK is organized according to four general categories of failure: fracture, corrosion, wear, and the subject of this article, distortion. One reason metals are so widely used as engineering materials is that they have high strength but also generally have the capability to respond to load (stress) by deforming. In fact, much of metallurgical engineering is concerned with balancing strength and ductility. Thus, distortion often is observed in analysis of other types of failures, and consideration of the distortion can be an important part of the analysis. Energy is absorbed during deformation, and in some situations, the amount of energy absorbed may also be an important factor. Furthermore, it should be noted that not all distortion necessarily constitutes a “failure.” This article first considers true distortion failures, that is, situations in which distortion occurs when it should not have occurred and in which the distortion is associated with a functional failure. Then, a more general consideration of distortion in failure analysis is introduced. As used here, distortion will refer to a condition in which the shape of a component has changed without loss of material. Deformation will refer to the process that results in the distortion. Distortion failure occurs when a structure or component is deformed so that it can no longer support the load it was intended to carry, is incapable of performing its intended function, or interferes with the operation of another component. Distortion failures can be plastic or elastic and may or may not be accompanied by fracture. There are two main types of distortion: size distortion, which refers to a change in volume (growth or shrinkage), and shape distortion (bending or warping), which refers to a change in geometric form. Most of the examples in this article deal with metals, but the concepts also apply to nonmetals. Materials as diverse as metals, polymers, and wood are all susceptible to distortion, although the mechanisms may differ somewhat among the different classes of material. Distortion failures are ordinarily considered to be self-evident, for example, damage of a car body in a collision or bending of a nail being driven into hard wood. However, the failure analyst is often faced with more subtle situations. For example, the immediate cause of distortion (bending) of an automobile-engine valve stem is contact of the valve head with the piston, but the failure analyst must go beyond this immediate cause in order to recommend proper corrective measures. The valve may have stuck open because of faulty lubrication; the valve spring may have broken because corrosion had weakened it. The spring may have had insufficient strength and taken a set, allowing the valve to drop into the path of the piston, or the engine may have been raced beyond its revolutions per minute limit many times, causing coil clash and subsequent fatigue fracture of the spring. Without careful consideration of all the evidence, the failure analyst may overlook the true cause of a distortion failure. Several common aspects of failure by distortion are discussed in this article, and suitable examples of distortion failures are presented for illustration. Analysis of Distortion and Deformation Revised by Roch J. Shipley and David A. Moore, Packer Engineering and William Dobson, Binary Engineering Associates, Inc. Overloading Every structure has a load limit beyond which it is considered unsafe or unreliable. Applied loads that exceed this limit are known as overloads and sometimes result (depending on the factor of safety used in design) in distortion or fracture of The file is downloaded from www.bzfxw.com
one or more structural members. Estimation of load limits is one of the most important aspects of design and is commonly computed by one of two methods--classical design or limit analysis Classical Design. The conservative, classical method of design(assuming monotonic or static loading) assumes that failure occurs whenever the stress at any point in a structure exceeds the yield strength of the material. Except for members that are loaded in pure tension, the fact that yielding occurs at some point in a structure has little influence on the ability of the structure to support the load. However, yielding has long his classical approach inherently assumes that or fracture and is therefore a reasonable basis for limiting applied loads the stress to cause fracture is greater than the stress to cause yield. As fracture mechanics analysis clearly shows, this may not be the case. Fracture may occur at loads less than that required to cause yield if a sufficiently large imperfection is present in the material Classical design keeps allowable stresses entirely within the elastic region and is used routinely in the design of parts Allowable stresses for static service are generally set at one-half the yield strength for ductile materials and one-sixth for brittle materials, although other fractions may be more suitable for specific applications. For very brittle materials, there may be little difference between the yield"and ultimate strength, and the latter is used in design computations. The reason for using such low fractions of yield (or ultimate) strength is to allow for such factors as possible errors computational assumptions, accidental overload, introduction of residual stress during processing, temperature effects, variations in material quality(including imperfections), degradation(for example, from corrosion), and inadvertent local increases in applied stress resulting from notch effects Classical design is also used for setting allowable stresses in other applications, for example, where fracture can occur by fatigue or stress rupture. In these instances, fatigue strength or stress-rupture strength is substituted for yield strength as a point of reference, typically with different factors of safety Limit Analysis. The upper limit in design is defined as the load at which a structure will break or collapse under a single application of force. This load can be calculated by a method known as limit analysis(Ref 1, 2). With limit analysis, it is unnecessary to estimate stress distributions, which makes stress analysis much simpler by this method than by classical design. However, limit analysis is based on the concept of tolerance to yielding in the most highly stressed regions of the structure and therefore cannot be used in designing for resistance to fatigue or elastic buckling or in designing flaw- tolerant structures Limit analysis assumes an idealized material-one that behaves elastically up to a certain yield strength, then does not work harden but undergoes an indefinite amount of plastic deformation with no change in stress. The inherent safety of a structure is more realistically estimated by limit analysis in those instances when the structure will tolerate some plastic deformation before it collapses. Because low-carbon steel, one of the most common materials used in structural members behaves somewhat like the idealized material, limit analysis is very useful to the designer, especially in the analysis of statically indeterminate structures Figure 1 illustrates the relative stress-strain behavior of a low-carbon steel, a strain-hardening material, and an idealized material-all with the same yield strength(the upper yield point for the low-carbon steel and the stress at 0. 2% offset for the strain-hardening material ) Load limits for parts made of materials that strain harden significantly when stressed in the plastic region can be estimated by limit analysis, as can those for parts made of other materials whose stress-strain behavior differs from that of the idealized material. In these situations the designer bases his design calculations on assumed strength that may actually lie well within the plastic region for the material Strain-hardening Low-carbon ste Idealized moterial Strain
one or more structural members. Estimation of load limits is one of the most important aspects of design and is commonly computed by one of two methods—classical design or limit analysis. Classical Design. The conservative, classical method of design (assuming monotonic or static loading) assumes that failure occurs whenever the stress at any point in a structure exceeds the yield strength of the material. Except for members that are loaded in pure tension, the fact that yielding occurs at some point in a structure has little influence on the ability of the structure to support the load. However, yielding has long been considered a prelude to structural collapse or fracture and is therefore a reasonable basis for limiting applied loads. This classical approach inherently assumes that the stress to cause fracture is greater than the stress to cause yield. As fracture mechanics analysis clearly shows, this may not be the case. Fracture may occur at loads less than that required to cause yield if a sufficiently large imperfection is present in the material. Classical design keeps allowable stresses entirely within the elastic region and is used routinely in the design of parts. Allowable stresses for static service are generally set at one-half the yield strength for ductile materials and one-sixth for brittle materials, although other fractions may be more suitable for specific applications. For very brittle materials, there may be little difference between the “yield” and ultimate strength, and the latter is used in design computations. The reason for using such low fractions of yield (or ultimate) strength is to allow for such factors as possible errors in computational assumptions, accidental overload, introduction of residual stress during processing, temperature effects, variations in material quality (including imperfections), degradation (for example, from corrosion), and inadvertent local increases in applied stress resulting from notch effects. Classical design is also used for setting allowable stresses in other applications, for example, where fracture can occur by fatigue or stress rupture. In these instances, fatigue strength or stress-rupture strength is substituted for yield strength as a point of reference, typically with different factors of safety. Limit Analysis. The upper limit in design is defined as the load at which a structure will break or collapse under a single application of force. This load can be calculated by a method known as limit analysis (Ref 1, 2). With limit analysis, it is unnecessary to estimate stress distributions, which makes stress analysis much simpler by this method than by classical design. However, limit analysis is based on the concept of tolerance to yielding in the most highly stressed regions of the structure and therefore cannot be used in designing for resistance to fatigue or elastic buckling or in designing flawtolerant structures. Limit analysis assumes an idealized material—one that behaves elastically up to a certain yield strength, then does not work harden but undergoes an indefinite amount of plastic deformation with no change in stress. The inherent safety of a structure is more realistically estimated by limit analysis in those instances when the structure will tolerate some plastic deformation before it collapses. Because low-carbon steel, one of the most common materials used in structural members, behaves somewhat like the idealized material, limit analysis is very useful to the designer, especially in the analysis of statically indeterminate structures. Figure 1 illustrates the relative stress-strain behavior of a low-carbon steel, a strain-hardening material, and an idealized material—all with the same yield strength (the upper yield point for the low-carbon steel and the stress at 0.2% offset for the strain-hardening material). Load limits for parts made of materials that strain harden significantly when stressed in the plastic region can be estimated by limit analysis, as can those for parts made of other materials whose stress-strain behavior differs from that of the idealized material. In these situations, the designer bases his design calculations on an assumed strength that may actually lie well within the plastic region for the material
Fig. 1 Comparison of the conventional stress-strain behavior of a low-carbon steel,a strain-hardening material, and the idealized material assumed in limit analysis. all have the same yield strength. Buckling Collapse due to instability under compressive stress, or buckling, may or may not be permanent deformation, depending on whether or not the yield strength was exceeded. Long, slender, straight bars, tubes, or columns under axial compressive forces will buckle when the buckling load is exceeded. Buckling failure may also be encountered on the compressive sides of tubes, I-beams, channels, and angles under bending forces. Tubes may also buckle due to torsional forces, causing waves, or folds, generally perpendicular to the direction of the compressive-stress component. Parts under bending load are also subject to buckling failures on the compressive(concave)side(Fig. 2) c Fig 2 Buckled flange(lower arrow) of an extruded aluminum section deliberately loaded with a lateral force(upper arrow). Source: Ref 3. The buckling load depends only on the dimensions of the part and the modulus of elasticity of the material. Therefore, buckling cannot be prevented by changing the strength or hardness of the metal. The modulus of elasticity of a given metal is affected only by temperature, increasing at lower temperature and decreasing at higher temperature. buckling can be prevented only by changing the size or shape of the part with respect to the load imposed on it( Ref 3) The failure analyst should be sensitive to situations in which buckling has occurred but may not be immediately apparent a beam in bending will be more susceptible to buckling on the compression side if it is relatively deep and narrow. A thin, circular shaft in torsion may buckle into a helical configuration when a critical moment is exceeded. Creep or distortion from other causes may change the dimensions of a structure so that it becomes susceptible to buckling. Further details can be found in references such as ref 4 Safety Factors. In both classical design and limit analysis, yielding is assumed to be the criterion for calculating safe loads on statically loaded structures. For a given design and applied load, the two methods differ in that the safety factor( the ratio of the theoretical capacity of a structural member to the maximum allowable load)is generally higher when calculated by limit analysis. For example, classical design limits the capacity of a rectangular beam to the bending Thefileisdownloadedfromwww.bzfxw.com
Fig. 1 Comparison of the conventional stress-strain behavior of a low-carbon steel, a strain-hardening material, and the idealized material assumed in limit analysis. All have the same yield strength. Buckling. Collapse due to instability under compressive stress, or buckling, may or may not be permanent deformation, depending on whether or not the yield strength was exceeded. Long, slender, straight bars, tubes, or columns under axial compressive forces will buckle when the buckling load is exceeded. Buckling failure may also be encountered on the compressive sides of tubes, I-beams, channels, and angles under bending forces. Tubes may also buckle due to torsional forces, causing waves, or folds, generally perpendicular to the direction of the compressive-stress component. Parts under bending load are also subject to buckling failures on the compressive (concave) side (Fig. 2). Fig. 2 Buckled flange (lower arrow) of an extruded aluminum section deliberately loaded with a lateral force (upper arrow). Source: Ref 3. The buckling load depends only on the dimensions of the part and the modulus of elasticity of the material. Therefore, buckling cannot be prevented by changing the strength or hardness of the metal. The modulus of elasticity of a given metal is affected only by temperature, increasing at lower temperature and decreasing at higher temperature. Buckling can be prevented only by changing the size or shape of the part with respect to the load imposed on it (Ref 3). The failure analyst should be sensitive to situations in which buckling has occurred but may not be immediately apparent. A beam in bending will be more susceptible to buckling on the compression side if it is relatively deep and narrow. A thin, circular shaft in torsion may buckle into a helical configuration when a critical moment is exceeded. Creep or distortion from other causes may change the dimensions of a structure so that it becomes susceptible to buckling. Further details can be found in references such as Ref 4. Safety Factors. In both classical design and limit analysis, yielding is assumed to be the criterion for calculating safe loads on statically loaded structures. For a given design and applied load, the two methods differ in that the safety factor (the ratio of the theoretical capacity of a structural member to the maximum allowable load) is generally higher when calculated by limit analysis. For example, classical design limits the capacity of a rectangular beam to the bending The file is downloaded from www.bzfxw.com
moment that will produce tensile yielding in the regions farthest from the neutral axis; limit analysis predicts that It is important that the designer be abling moment 1.5 times the limiting bending moment determined by classical design complete collapse will occur at a bend le to relate the actual behavior of a structure to its assumed behavior because. for a given applied load and selected safety factor, a structure designed by limit analysis will usually have thinner sections than a structure designed by classical methods Safety factors are important design considerations because they allow for factors that cannot be computed in advance Overload failure can occur either when the applied loads increase the stress above the design value or when the material strength is degraded. If either situation is a characteristic of the fabricated structure the design must be changed to allow for these factors more realistically Example 1: Collapse of Extension Ladders by Overloading of Side Rails. Several aluminum alloy extension ladders of the same size and type collapsed in service in the same manner; the extruded aluminum alloy 6063-T6 side rails buckled, but the rungs and hardware remained firmly in place. The ladders had a maximum extended length of 6.4 m(21 ft), and the recommended maximum angle of inclination to the vertical was 15 Ivestigation. Visual examination disclosed that the side-rail extrusions, which had the I-beam shape shown in Fig. 3(a) had failed by plastic buckling, with only slight surface cracking in the most severely deformed areas. There were no isible defects in materials or workmanship, and all dimensions of the side rails were within specified tolerances 0.875 0046 3 0.046 0.6|o 0. 124(typ) 500 卫-E2E×o 400 30 00300040005000600.070 (b) Extrusion thickness in Fig 3 Aluminum alloy 6063-t6 extension- ladder side-rail extrusion that failed by plastic deformation and subsequent buckling.(a) Configuration and dimensions(given in inches) (b) Relation of maximum applied load to the section thickness of the flanges and web of the side-rail extrusion Hardness tests using a portable hardness tester, metallographic examination, and tensile tests of specimens from the buckled side rails were conducted. All results agreed with the typical properties reported for aluminum alloy 6063-T6 extrusions Stress analysis of the design of the ladder, using actual dimensions, indicated that the side-rail extrusions had been designed with a thickness that would provide a safety factor of 1. 2 at ideal loading conditions(15 maximum inclination and that use of the ladders under other conditions could subject the side rails to stresses beyond the yield strength of the material Once yielding occurred, buckling would continue until the ladder collapsed The stress analysis was extended to include an evaluation of the relation of maximum applied load to section thickness ith the ladder at its maximum extension of 6.4 m(21 ft)and at an inclination of 15. This relation is shown in Fig 3 (b) Conclusions. The side rails of the ladders buckled when subjected to loads that produced stresses beyond the yield ength of the alloy Failure was by plastic deformation, with only slight tearing in the most severely deformed regions
moment that will produce tensile yielding in the regions farthest from the neutral axis; limit analysis predicts that complete collapse will occur at a bending moment 1.5 times the limiting bending moment determined by classical design. It is important that the designer be able to relate the actual behavior of a structure to its assumed behavior because, for a given applied load and selected safety factor, a structure designed by limit analysis will usually have thinner sections than a structure designed by classical methods. Safety factors are important design considerations because they allow for factors that cannot be computed in advance. Overload failure can occur either when the applied loads increase the stress above the design value or when the material strength is degraded. If either situation is a characteristic of the fabricated structure, the design must be changed to allow for these factors more realistically. Example 1: Collapse of Extension Ladders by Overloading of Side Rails. Several aluminum alloy extension ladders of the same size and type collapsed in service in the same manner; the extruded aluminum alloy 6063-T6 side rails buckled, but the rungs and hardware remained firmly in place. The ladders had a maximum extended length of 6.4 m (21 ft), and the recommended maximum angle of inclination to the vertical was 15°. Investigation. Visual examination disclosed that the side-rail extrusions, which had the I-beam shape shown in Fig. 3(a), had failed by plastic buckling, with only slight surface cracking in the most severely deformed areas. There were no visible defects in materials or workmanship, and all dimensions of the side rails were within specified tolerances. Fig. 3 Aluminum alloy 6063-T6 extension-ladder side-rail extrusion that failed by plastic deformation and subsequent buckling. (a) Configuration and dimensions (given in inches). (b) Relation of maximum applied load to the section thickness of the flanges and web of the side-rail extrusion. Hardness tests using a portable hardness tester, metallographic examination, and tensile tests of specimens from the buckled side rails were conducted. All results agreed with the typical properties reported for aluminum alloy 6063-T6 extrusions. Stress analysis of the design of the ladder, using actual dimensions, indicated that the side-rail extrusions had been designed with a thickness that would provide a safety factor of 1.2 at ideal loading conditions (15° maximum inclination) and that use of the ladders under other conditions could subject the side rails to stresses beyond the yield strength of the material. Once yielding occurred, buckling would continue until the ladder collapsed. The stress analysis was extended to include an evaluation of the relation of maximum applied load to section thickness with the ladder at its maximum extension of 6.4 m (21 ft) and at an inclination of 15°. This relation is shown in Fig. 3(b). Conclusions. The side rails of the ladders buckled when subjected to loads that produced stresses beyond the yield strength of the alloy. Failure was by plastic deformation, with only slight tearing in the most severely deformed regions
Corrective Measure. The flange and web of the side-rail extrusion were increased in thickness from 1. 2 to 1. 4 mm(0.046 to 0.057 in. ) This increased the safety factor from 1. 2 to 1.56. After this change, no further failures were reported. Note that this example is presented for illustrative purposes only, and this safety factor may not be appropriate in other applications Amount of Distortion. When designing structures using limit analysis, the designer does not al ways consider the amount of distortion that will be encountered. a rough illustration of the distortion that resulted from overloading of small cantilever beams is given in Fig. 4. Known loads were applied to rectangular-section beams of low-carbon steel and of stainless steel, and the permanent deflection at the loading point was measured. Maximum fiber stresses were calculated from the applied load and original specimen dimensions 2.50 Type 302 stainless steel, quarter hard Tensile strength, 135, 000 pal 200 1.75 |50 lolO steel(annealed) ield strength, 26, 000 psi ensile strength, 32 000 pal 1.0o 0.75 Distortion ratio Fig. 4 Relation of distortion ratio to stress ratio for two steel cantilever beams of rectangular cross section. Distortion ratio is permanent deflection, measured at a distance from the support ten times the beam thickness, divided by beam thickness. Stress ratio is maximum stress, calculated from applied load and original beam dimensions, divided by yield strength. This type of test provides a simplistic but useful concept of distortion by showing how much distortion occurs at strains beyond the yield point. As shown in Fig. 4, the beam of low-carbon steel, which strain hardens only slightly, exhibited no distortion when the calculated maximum fiber stress was equal to the yield strength(at a stress ratio of 1.00).However this beam collapsed at a load equivalent to a fiber stress just above the tensile strength, as shown in Fig 4 where the lower curve became essentially horizontal. This collapse load agrees with the limit-analysis collapse load of 1.5 times the load at yield. The beam made of stainless steel, which strain hardens at a rather high rate, showed no distortion at fiber stresses up to 1. 47 times the yield strength. When the calculated stress equaled the tensile strength(at a stress ratio of 1.59), distortion was 0.7 times the beam thickness, and the beam supported a calculated stress of 1.5 times the tensile strength without collapse It should be noted that the preceding example is intended simply to illustrate the differences in deformation behavior between two different materials. As a practical matter, one would not substitute stainless steel for low-carbon steel to increase load capacity. One would use a heavier section, or perhaps, a higher-strength alloy When loads increase gradually, distortion is gradual, and design can be based on knowledge of the amount of distortion that can be tolerated. Thus, simple bench tests of full-size or scaled-down models can often be used in estimating the loads required to produce various amounts of distortion Effect of Impact/Very High Strain Rates. When rapid or impulse loads are applied, as in impact, shock loading, or high- frequency vibration, the amount of distortion that can occur without fracture is considerably less predictable. The crystallographic processes involved in deformation and fracture are influenced by strain rate as well as temperature. For most structural materials, measured values of strength are higher under impulse loading, and values of ductility are lower, than the values measured under static loading. Tensile and yield strengths as much as 20% higher than the slow-tension test values have been measured under very high rates of loading. Strain-rate sensitivity data have been compiled for many Thefileisdownloadedfromwww.bzfxw.com
Corrective Measure. The flange and web of the side-rail extrusion were increased in thickness from 1.2 to 1.4 mm (0.046 to 0.057 in.). This increased the safety factor from 1.2 to 1.56. After this change, no further failures were reported. Note that this example is presented for illustrative purposes only, and this safety factor may not be appropriate in other applications. Amount of Distortion. When designing structures using limit analysis, the designer does not always consider the amount of distortion that will be encountered. A rough illustration of the distortion that resulted from overloading of small cantilever beams is given in Fig. 4. Known loads were applied to rectangular-section beams of low-carbon steel and of stainless steel, and the permanent deflection at the loading point was measured. Maximum fiber stresses were calculated from the applied load and original specimen dimensions. Fig. 4 Relation of distortion ratio to stress ratio for two steel cantilever beams of rectangular cross section. Distortion ratio is permanent deflection, measured at a distance from the support ten times the beam thickness, divided by beam thickness. Stress ratio is maximum stress, calculated from applied load and original beam dimensions, divided by yield strength. This type of test provides a simplistic but useful concept of distortion by showing how much distortion occurs at strains beyond the yield point. As shown in Fig. 4, the beam of low-carbon steel, which strain hardens only slightly, exhibited no distortion when the calculated maximum fiber stress was equal to the yield strength (at a stress ratio of 1.00). However, this beam collapsed at a load equivalent to a fiber stress just above the tensile strength, as shown in Fig. 4 where the lower curve became essentially horizontal. This collapse load agrees with the limit-analysis collapse load of 1.5 times the load at yield. The beam made of stainless steel, which strain hardens at a rather high rate, showed no distortion at fiber stresses up to 1.47 times the yield strength. When the calculated stress equaled the tensile strength (at a stress ratio of 1.59), distortion was 0.7 times the beam thickness, and the beam supported a calculated stress of 1.5 times the tensile strength without collapse. It should be noted that the preceding example is intended simply to illustrate the differences in deformation behavior between two different materials. As a practical matter, one would not substitute stainless steel for low-carbon steel to increase load capacity. One would use a heavier section, or perhaps, a higher-strength alloy. When loads increase gradually, distortion is gradual, and design can be based on knowledge of the amount of distortion that can be tolerated. Thus, simple bench tests of full-size or scaled-down models can often be used in estimating the loads required to produce various amounts of distortion. Effect of Impact/Very High Strain Rates. When rapid or impulse loads are applied, as in impact, shock loading, or highfrequency vibration, the amount of distortion that can occur without fracture is considerably less predictable. The crystallographic processes involved in deformation and fracture are influenced by strain rate as well as temperature. For most structural materials, measured values of strength are higher under impulse loading, and values of ductility are lower, than the values measured under static loading. Tensile and yield strengths as much as 20% higher than the slow-tensiontest values have been measured under very high rates of loading. Strain-rate sensitivity data have been compiled for many The file is downloaded from www.bzfxw.com