Table Of Content

Section 5 Strength of Materials BY JOHN SYMONDS Fellow Engineer (Retired), Oceanic Division, Westinghouse Electric Corporation. J. P. VIDOSIC Regents’ Professor Emeritus of Mechanical Engineering, Georgia Institute of Technology. HAROLD V. HAWKINS Late Manager, Product Standards and Services, Columbus McKinnon Corporation, Tonawanda, N.Y. DONALD D. DODGE Supervisor (Retired), Product Quality and Inspection Technology, Manufacturing Development, Ford Motor Company. 5.1 MECHANICAL PROPERTIES OF MATERIALS by John Symonds, Expanded by Staff Stress-Strain Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-2 Fracture at Low Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-7 Fatigue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-8 Creep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-10 Hardness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-12 Testing of Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-13 5.2 MECHANICS OF MATERIALS by J. P. Vidosic Simple Stresses and Strains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-15 Combined Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-18 Plastic Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-19 Design Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-20 Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-20 Torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-36 Columns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-38 Eccentric Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-40 Curved Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-41 Impact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-43 Theory of Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-44 Cylinders and Spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-45 Pressure between Bodies with Curved Surfaces . . . . . . . . . . . . . . . . . . . . . . 5-47 Flat Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-47 Theories of Failure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-48 Plasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-49 Rotating Disks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-50 Experimental Stress Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-51 5.3 PIPELINE FLEXURE STRESSES by Harold V. Hawkins Pipeline Flexure Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-55 5.4 NONDESTRUCTIVE TESTING by Donald D. Dodge Nondestructive Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-61 Magnetic Particle Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-61 Penetrant Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-61 Radiographic Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-65 Ultrasonic Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-66 Eddy Current Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-66 Microwave Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-67 Infrared Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-67 Acoustic Signature Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-67 5-1 Copyright (C) 1999 by The McGraw-Hill Companies, Inc. All rights reserved. Use of this product is subject to the terms of its License Agreement. Click here to view. 5.1 MECHANICAL PROPERTIES OF MATERIALS by John Symonds, Expanded by Staff REFERENCES: Davis et al., ‘‘Testing and Inspection of Engineering Materials,’’ McGraw-Hill, Timoshenko, ‘‘Strength of Materials,’’ pt. II, Van Nostrand. Richards, ‘‘Engineering Materials Science,’’ Wadsworth. Nadai, ‘‘Plasticity,’’ McGraw-Hill. Tetelman and McEvily, ‘‘Fracture of Structural Materials,’’ Wiley. ‘‘Fracture Mechanics,’’ ASTM STP-833. McClintock and Argon (eds.), ‘‘Me- chanical Behavior of Materials,’’ Addison-Wesley. Dieter, ‘‘Mechanical Metal- lurgy,’’ McGraw-Hill. ‘‘Creep Data,’’ ASME. ASTM Standards, ASTM. Blaz- nynski (ed.), ‘‘Plasticity and Modern Metal Forming Technology,’’ Elsevier Science. STRESS-STRAIN DIAGRAMS The Stress-Strain Curve The engineering tensile stress-strain curve is obtained by static loading of a standard specimen, that is, by applying the load slowly enough that all parts of the specimen are in equilibrium at any instant. The curve is usually obtained by controlling the loading rate in the tensile machine. ASTM Standards require a loading rate not exceeding 100,000 lb/in2 (70 kgf/mm2)/min. An alternate method of obtaining the curve is to specify the strain rate as the independent vari- able, in which case the loading rate is continuously adjusted to maintain the required strain rate. A strain rate of 0.05 in/in/(min) is commonly used. It is measured usually by an extensometer attached to the gage length of the specimen. Figure 5.1.1 shows several stress-strain curves. Fig. 5.1.1. Comparative stress-strain diagrams. (1) Soft brass; (2) low carbon steel; (3) hard bronze; (4) cold rolled steel; (5) medium carbon steel, annealed; (6) medium carbon steel, heat treated. For most engineering materials, the curve will have an initial linear elastic region (Fig. 5.1.2) in which deformation is reversible and time- independent. The slope in this region is Young’s modulus E. The propor- tional elastic limit (PEL) is the point where the curve starts to deviate from a straight line. The elastic limit (frequently indistinguishable from PEL) is the point on the curve beyond which plastic deformation is present after release of the load. If the stress is increased further, the stress-strain curve departs more and more from the straight line. Un- loading the specimen at point X (Fig. 5.1.2), the portion XX� is linear and is essentially parallel to the original line OX��. The horizontal dis- tance OX� is called the permanent set corresponding to the stress at X. This is the basis for the construction of the arbitrary yield strength. To determine the yield strength, a straight line XX� is drawn parallel to the initial elastic line OX�� but displaced from it by an arbitrary value of permanent strain. The permanent strain commonly used is 0.20 percent of the original gage length. The intersection of this line with the curve determines the stress value called the yield strength. In reporting the yield strength, the amount of permanent set should be specified. The arbitrary yield strength is used especially for those materials not ex- hibiting a natural yield point such as nonferrous metals; but it is not limited to these. Plastic behavior is somewhat time-dependent, particu- larly at high temperatures. Also at high temperatures, a small amount of time-dependent reversible strain may be detectable, indicative of anelas- tic behavior. Fig. 5.1.2. General stress-strain diagram. The ultimate tensile strength (UTS) is the maximum load sustained by the specimen divided by the original specimen cross-sectional area. The percent elongation at failure is the plastic extension of the specimen at failure expressed as (the change in original gage length � 100) divided by the original gage length. This extension is the sum of the uniform and nonuniform elongations. The uniform elongation is that which occurs prior to the UTS. It has an unequivocal significance, being associated with uniaxial stress, whereas the nonuniform elongation which occurs during localized extension (necking) is associated with triaxial stress. The nonuniform elongation will depend on geometry, particularly the ratio of specimen gage length L0 to diameter D or square root of cross- sectional area A. ASTM Standards specify test-specimen geometry for a number of specimen sizes. The ratio L0/√A is maintained at 4.5 for flat- and round-cross-section specimens. The original gage length should always be stated in reporting elongation values. The specimen percent reduction in area (RA) is the contraction in cross-sectional area at the fracture expressed as a percentage of the original area. It is obtained by measurement of the cross section of the broken specimen at the fracture location. The RA along with the load at fracture can be used to obtain the fracture stress, that is, fracture load divided by cross-sectional area at the fracture. See Table 5.1.1. The type of fracture in tension gives some indications of the quality of the material, but this is considerably affected by the testing tempera- ture, speed of testing, the shape and size of the test piece, and other conditions. Contraction is greatest in tough and ductile materials and least in brittle materials. In general, fractures are either of the shear or of the separation (loss of cohesion) type. Flat tensile specimens of ductile metals often show shear failures if the ratio of width to thickness is greater than 6:1. A completely shear-type failure may terminate in a chisel edge, for a flat specimen, or a point rupture, for a round specimen. Separation failures occur in brittle materials, such as certain cast irons. Combinations of both shear and separation failures are common on round specimens of ductile metal. Failure often starts at the axis in a necked region and produces a relatively flat area which grows until the material shears along a cone-shaped surface at the outside of the speci- 5-2 Copyright (C) 1999 by The McGraw-Hill Companies, Inc. All rights reserved. Use of this product is subject to the terms of its License Agreement. Click here to view. STRESS-STRAIN DIAGRAMS 5-3 Table 5.1.1 Typical Mechanical Properties at Room Temperature (Based on ordinary stress-strain values) Tensile Yield Ultimate strength, strength, elongation, Reduction Brinell Metal 1,000 lb/in2 1,000 lb/in2 % of area, % no. Cast iron 18–60 8–40 0 0 100–300 Wrought iron 45–55 25–35 35–25 55–30 100 Commercially pure iron, annealed 42 19 48 85 70 Hot-rolled 48 30 30 75 90 Cold-rolled 100 95 200 Structural steel, ordinary 50–65 30–40 40–30 120 Low-alloy, high-strength 65–90 40–80 30–15 70–40 150 Steel, SAE 1300, annealed 70 40 26 70 150 Quenched, drawn 1,300°F 100 80 24 65 200 Drawn 1,000°F 130 110 20 60 260 Drawn 700°F 200 180 14 45 400 Drawn 400°F 240 210 10 30 480 Steel, SAE 4340, annealed 80 45 25 70 170 Quenched, drawn 1,300°F 130 110 20 60 270 Drawn 1,000°F 190 170 14 50 395 Drawn 700°F 240 215 12 48 480 Drawn 400°F 290 260 10 44 580 Cold-rolled steel, SAE 1112 84 76 18 45 160 Stainless steel, 18-S 85–95 30–35 60–55 75–65 145–160 Steel castings, heat-treated 60–125 30–90 33–14 65–20 120–250 Aluminum, pure, rolled 13–24 5–21 35–5 23–44 Aluminum-copper alloys, cast 19–23 12–16 4–0 50–80 Wrought, heat-treated 30–60 10–50 33–15 50–120 Aluminum die castings 30 2 Aluminum alloy 17ST 56 34 26 39 100 Aluminum alloy 51ST 48 40 20 35 105 Copper, annealed 32 5 58 73 45 Copper, hard-drawn 68 60 4 55 100 Brasses, various 40–120 8–80 60–3 50–170 Phosphor bronze 40–130 55–5 50–200 Tobin bronze, rolled 63 41 40 52 120 Magnesium alloys, various 21–45 11–30 17–0.5 47–78 Monel 400, Ni-Cu alloy 79 30 48 75 125 Molybdenum, rolled 100 75 30 250 Silver, cast, annealed 18 8 54 27 Titanium 6–4 alloy, annealed 130 120 10 25 352 Ductile iron, grade 80-55-06 80 55 6 225–255 NOTE: Compressive strength of cast iron, 80,000 to 150,000 lb/in2. Compressive yield strength of all metals, except those cold-worked � tensile yield strength. Stress 1,000 lb/in2 � 6.894 � stress, MN/m2. men, resulting in what is known as the cup-and-cone fracture. Double cup-and-cone and rosette fractures sometimes occur. Several types of tensile fractures are shown in Fig. 5.1.3. Annealed or hot-rolled mild steels generally exhibit a yield point (see Fig. 5.1.4). Here, in a constant strain-rate test, a large increment of extension occurs under constant load at the elastic limit or at a stress just below the elastic limit. In the latter event the stress drops suddenly from the upper yield point to the lower yield point. Subsequent to the drop, the yield-point extension occurs at constant stress, followed by a rise to the UTS. Plastic flow during the yield-point extension is discontinuous; Fig. 5.1.3. Typical metal fractures in tension. successive zones of plastic deformation, known as Luder’s bands or stretcher strains, appear until the entire specimen gage length has been uniformly deformed at the end of the yield-point extension. This behav- ior causes a banded or stepped appearance on the metal surface. The exact form of the stress-strain curve for this class of material is sensitive to test temperature, test strain rate, and the characteristics of the tensile machine employed. The plastic behavior in a uniaxial tensile test can be represented as the true stress-strain curve. The true stress � is based on the instantaneous Fig. 5.1.4. Yielding of annealed steel. Copyright (C) 1999 by The McGraw-Hill Companies, Inc. All rights reserved. Use of this product is subject to the terms of its License Agreement. Click here to view. 5-4 MECHANICAL PROPERTIES OF MATERIALS cross section A, so that � � load/A. The instantaneous true strain incre- ment is �dA/A, or dL/L prior to necking. Total true strain � is � A A0 � dA A � ln� A0 A� or ln (L/L0) prior to necking. The true stress-strain curve or flow curve obtained has the typical form shown in Fig. 5.1.5. In the part of the test subsequent to the maximum load point (UTS), when necking occurs, the true strain of interest is that which occurs in an infinitesimal length at the region of minimum cross section. True strain for this element can still be expressed as ln (A0/A), where A refers to the minimum cross Fig. 5.1.5. True stress-strain curve for 20°C annealed mild steel. section. Methods of constructing the true stress-strain curve are de- scribed in the technical literature. In the range between initial yielding and the neighborhood of the maximum load point the relation- ship between plastic strain �p and true stress often approximates � � k�pn where k is the strength coefficient and n is the work-hardening exponent. For a material which shows a yield point the relationship applies only to the rising part of the curve beyond the lower yield. It can be shown that at the maximum load point the slope of the true stress-strain curve equals the true stress, from which it can be deduced that for a material obeying the above exponential relationship between �p and n, �p � n at the maximum load point. The exponent strongly influences the spread between YS and UTS on the engineering stress-strain curve. Values of n and k for some materials are shown in Table 5.1.2. A point on the flow curve indentifies the flow stress corresponding to a certain strain, that is, the stress required to bring about this amount of plastic deformation. The concept of true strain is useful for accurately describing large amounts of plastic deformation. The linear strain definition (L � L0)/L0 fails to correct for the continuously changing gage length, which leads to an increasing error as deformation proceeds. During extension of a specimen under tension, the change in the specimen cross-sectional area is related to the elongation by Poisson’s ratio �, which is the ratio of strain in a transverse direction to that in the longitudinal direction. Values of � for the elastic region are shown in Table 5.1.3. For plastic strain it is approximately 0.5. Table 5.1.2 Room-Temperature Plastic-Flow Constants for a Number of Metals k, 1,000 in2 Material Condition (MN/m2) n 0.40% C steel Quenched and tempered at 400°F (478K) 416 (2,860) 0.088 0.05% C steel Annealed and temper-rolled 72 (49.6) 0.235 2024 aluminum Precipitation-hardened 100 (689) 0.16 2024 aluminum Annealed 49 (338) 0.21 Copper Annealed 46.4 (319) 0.54 70–30 brass Annealed 130 (895) 0.49 SOURCE: Reproduced by permission from ‘‘Properties of Metals in Materials Engineering,’’ ASM, 1949. Table 5.1.3 Elastic Constants of Metals (Mostly from tests of R. W. Vose) E G K � Modulus of Modulus of elasticity rigidity (Young’s (shearing Bulk modulus). modulus). modulus. 1,000,000 1,000,000 1,000,000 Poisson’s Metal lb/in2 lb/in2 lb/in2 ratio Cast steel 28.5 11.3 20.2 0.265 Cold-rolled steel 29.5 11.5 23.1 0.287 Stainless steel 18–8 27.6 10.6 23.6 0.305 All other steels, including high-carbon, heat-treated 28.6–30.0 11.0–11.9 22.6–24.0 0.283–0.292 Cast iron 13.5–21.0 5.2–8.2 8.4–15.5 0.211–0.299 Malleable iron 23.6 9.3 17.2 0.271 Copper 15.6 5.8 17.9 0.355 Brass, 70–30 15.9 6.0 15.7 0.331 Cast brass 14.5 5.3 16.8 0.357 Tobin bronze 13.8 5.1 16.3 0.359 Phosphor bronze 15.9 5.9 17.8 0.350 Aluminum alloys, various 9.9–10.3 3.7–3.9 9.9–10.2 0.330–0.334 Monel metal 25.0 9.5 22.5 0.315 Inconel 31 11 0.27–0.38 Z-nickel 30 11 �0.36 Beryllium copper 17 7 �0.21 Elektron (magnesium alloy) 6.3 2.5 4.8 0.281 Titanium (99.0 Ti), annealed bar 15–16 6.5 0.34 Zirconium, crystal bar 11–14 Molybdenum, arc-cast 48–52 Copyright (C) 1999 by The McGraw-Hill Companies, Inc. All rights reserved. Use of this product is subject to the terms of its License Agreement. Click here to view. STRESS-STRAIN DIAGRAMS 5-5 The general effect of increased strain rate is to increase the resistance to plastic deformation and thus to raise the flow curve. Decreasing test temperature also raises the flow curve. The effect of strain rate is ex- pressed as strain-rate sensitivity m. Its value can be measured in the tension test if the strain rate is suddenly increased by a small increment during the plastic extension. The flow stress will then jump to a higher value. The strain-rate sensitivity is the ratio of incremental changes of log � and log �� m �� � log � � log ��� � For most engineering materials at room temperature the strain rate sen- sitivity is of the order of 0.01. The effect becomes more significant at elevated temperatures, with values ranging to 0.2 and sometimes higher. Compression Testing The compressive stress-strain curve is simi- lar to the tensile stress-strain curve up to the yield strength. Thereafter, the progressively increasing specimen cross section causes the com- pressive stress-strain curve to diverge from the tensile curve. Some ductile metals will not fail in the compression test. Complex behavior occurs when the direction of stressing is changed, because of the Baus- chinger effect, which can be described as follows: If a specimen is first plastically strained in tension, its yield stress in compression is reduced and vice versa. Combined Stresses This refers to the situation in which stresses are present on each of the faces of a cubic element of the material. For a given cube orientation the applied stresses may include shear stresses over the cube faces as well as stresses normal to them. By a suitable rotation of axes the problem can be simplified: applied stresses on the new cubic element are equivalent to three mutually orthogonal principal stresses �1, �2, �3 alone, each acting normal to a cube face. Combined stress behavior in the elastic range is described in Sec. 5.2, Mechanics of Materials. Prediction of the conditions under which plastic yielding will occur under combined stresses can be made with the help of several empirical theories. In the maximum-shear-stress theory the criterion for yielding is that yielding will occur when �1 � �3 � �ys in which �1 and �3 are the largest and smallest principal stresses, re- spectively, and �ys is the uniaxial tensile yield strength. This is the simplest theory for predicting yielding under combined stresses. A more accurate prediction can be made by the distortion-energy theory, accord- ing to which the criterion is (�1 � �2)2 � (�2 � �3)2 � (�2 � �1)2 � 2(�ys)2 Stress-strain curves in the plastic region for combined stress loading can be constructed. However, a particular stress state does not determine a unique strain value. The latter will depend on the stress-state path which is followed. Plane strain is a condition where strain is confined to two dimensions. There is generally stress in the third direction, but because of mechani- cal constraints, strain in this dimension is prevented. Plane strain occurs in certain metalworking operations. It can also occur in the neighbor- hood of a crack tip in a tensile loaded member if the member is suffi- ciently thick. The material at the crack tip is then in triaxial tension, which condition promotes brittle fracture. On the other hand, ductility is enhanced and fracture is suppressed by triaxial compression. Stress Concentration In a structure or machine part having a notch or any abrupt change in cross section, the maximum stress will occur at this location and will be greater than the stress calculated by elementary formulas based upon simplified assumptions as to the stress distribu- tion. The ratio of this maximum stress to the nominal stress (calculated by the elementary formulas) is the stress-concentration factor Kt. This is a constant for the particular geometry and is independent of the mate- rial, provided it is isotropic. The stress-concentration factor may be determined experimentally or, in some cases, theoretically from the mathematical theory of elasticity. The factors shown in Figs. 5.1.6 to 5.1.13 were determined from both photoelastic tests and the theory of elasticity. Stress concentration will cause failure of brittle materials if 3.4 3.0 2.6 2.2 1.8 1.4 1.0 Stress concentration factor, K 0.01 0.1 I I II III 0.2 r d 1.0 Note; in all cases D�d�2r D Tension or compression D 2r d/2 d d/2 II IV IV V V D d III � � Bending D d r r � � r r � � r r � � r r D d Fig. 5.1.6. Flat plate with semicircular fillets and grooves or with holes. I, II, and III are in tension or compression; IV and V are in bending. 3.4 3.0 2.6 2.2 1.8 1.4 1.0 Stress concentration factor, K 0.4 1.0 1.5 h r Sharpness of groove, 2 h d D d h h 3 4 5 6 Semi-circle grooves (h�r) Blunt grooves Sharp grooves 2 1 0.5 0.2 0.1 0.05 � 0.02 � � r r Fig. 5.1.7. Flat plate with grooves, in tension. Copyright (C) 1999 by The McGraw-Hill Companies, Inc. All rights reserved. Use of this product is subject to the terms of its License Agreement. Click here to view. 5-6 MECHANICAL PROPERTIES OF MATERIALS the concentrated stress is larger than the ultimate strength of the mate- rial. In ductile materials, concentrated stresses higher than the yield strength will generally cause local plastic deformation and redistribu- tion of stresses (rendering them more uniform). On the other hand, even with ductile materials areas of stress concentration are possible sites for fatigue if the component is cyclically loaded. 3.4 3.0 2.6 2.2 1.8 1.4 1.0 Stress concentration factor, K 0.4 1.0 1.5 h r Sharpness of fillet, 2 h h 3 4 5 6 D�d � 2h Full fillets (h�r) Blunt fillets 0.5 2.0 1.0 0.2 0.1 0.05 � 0.02 Depth of fillet � Sharp fillets h d D d � � r r Fig. 5.1.8. Flat plate with fillets, in tension. 3.4 3.0 2.6 2.2 1.8 1.4 1.0 Stress concentration factor, K 0.4 1.0 1.5 h r Sharpness of groove, 2 3 4 5 6 D�d � 2h Semi-circle grooves (h�r) 0.5 2 1 0.2 0.1 � 0.05 h d Sharp grooves D d h h � � r r Fig. 5.1.9. Flat plate with grooves, in bending. 3.4 3.0 2.6 2.2 1.8 1.4 1.0 Stress concentration factor, K 0.4 1.0 1.5 h r Sharpness of fillet, 2 3 4 5 6 D�d � 2h Full fillets (h�r) 0.05 2 1 0.2 0.5 0.1 � 0.02 h d D d h h � � r r Sharp fillets Blunt fillets Fig. 5.1.10. Flat plate with fillets, in bending. Fig. 5.1.11. Flat plate with angular notch, in tension or bending. 3.4 3.0 2.6 2.2 1.8 1.4 1.0 Stress concentration factor, K 0.5 1.0 1.5 h r Sharpness of groove, 2 3 4 5 6 Semi circ. grooves h�r 4 10 1 0.4 � 0.1 h d D d h �r 8 10 15 20 Sharp grooves Blunt grooves � 0.04 h d Fig. 5.1.12. Grooved shaft in torsion. Copyright (C) 1999 by The McGraw-Hill Companies, Inc. All rights reserved. Use of this product is subject to the terms of its License Agreement. Click here to view. FRACTURE AT LOW STRESSES 5-7 3.4 3.0 2.6 2.2 1.8 1.4 1.0 Stress concentration factor, K 0.5 1.0 h r Sharpness of fillet, 2 3 4 5 7 1 0.2 0.5 10 20 40 Blunt fillets � 0.05 0.1 h d Sharp fillets Full fillets (h�r) D�d � 2h D d h � r Fig. 5.1.13. Filleted shaft in torsion. FRACTURE AT LOW STRESSES Materials under tension sometimes fail by rapid fracture at stresses much below their strength level as determined in tests on carefully prepared specimens. These brittle, unstable, or catastrophic failures origi- nate at preexisting stress-concentrating flaws which may be inherent in a material. The transition-temperature approach is often used to ensure fracture- safe design in structural-grade steels. These materials exhibit a charac- teristic temperature, known as the ductile brittle transition (DBT) tem- perature, below which they are susceptible to brittle fracture. The tran- sition-temperature approach to fracture-safe design ensures that the transition temperature of a material selected for a particular application is suitably matched to its intended use temperature. The DBT can be detected by plotting certain measurements from tensile or impact tests against temperature. Usually the transition to brittle behavior is com- plex, being neither fully ductile nor fully brittle. The range may extend over 200°F (110 K) interval. The nil-ductility temperature (NDT), deter- mined by the drop weight test (see ASTM Standards), is an important reference point in the transition range. When NDT for a particular steel is known, temperature-stress combinations can be specified which de- fine the limiting conditions under which catastrophic fracture can occur. In the Charpy V-notch (CVN) impact test, a notched-bar specimen (Fig. 5.1.26) is used which is loaded in bending (see ASTM Standards). The energy absorbed from a swinging pendulum in fracturing the speci- men is measured. The pendulum strikes the specimen at 16 to 19 ft (4.88 to 5.80 m)/s so that the specimen deformation associated with fracture occurs at a rapid strain rate. This ensures a conservative mea- sure of toughness, since in some materials, toughness is reduced by high strain rates. A CVN impact energy vs. temperature curve is shown in Fig. 5.1.14, which also shows the transitions as given by percent brittle fracture and by percent lateral expansion. The CVN energy has no analytical significance. The test is useful mainly as a guide to the frac- ture behavior of a material for which an empirical correlation has been established between impact energy and some rigorous fracture criterion. For a particular grade of steel the CVN curve can be correlated with NDT. (See ASME Boiler and Pressure Vessel Code.) Fracture Mechanics This analytical method is used for ultra-high- strength alloys, transition-temperature materials below the DBT tem- perature, and some low-strength materials in heavy section thickness. Fracture mechanics theory deals with crack extension where plastic effects are negligible or confined to a small region around the crack tip. The present discussion is concerned with a through-thickness crack in a tension-loaded plate (Fig. 5.1.15) which is large enough so that the crack-tip stress field is not affected by the plate edges. Fracture me- chanics theory states that unstable crack extension occurs when the work required for an increment of crack extension, namely, surface energy and energy consumed in local plastic deformation, is exceeded by the elastic-strain energy released at the crack tip. The elastic-stress Fig. 5.1.14. CVN transition curves. (Data from Westinghouse R & D Lab.) Copyright (C) 1999 by The McGraw-Hill Companies, Inc. All rights reserved. Use of this product is subject to the terms of its License Agreement. Click here to view. 5-8 MECHANICAL PROPERTIES OF MATERIALS field surrounding one of the crack tips in Fig. 5.1.15 is characterized by the stress intensity KI, which has units of (lb √in) /in2 or (N√m) /m2. It is a function of applied nominal stress �, crack half-length a, and a geom- etry factor Q: K 2l � Q�2�a (5.1.1) for the situation of Fig. 5.1.15. For a particular material it is found that as KI is increased, a value Kc is reached at which unstable crack propa- Fig. 5.1.15. Through-thickness crack geometry. gation occurs. Kc depends on plate thickness B, as shown in Fig. 5.1.16. It attains a constant value when B is great enough to provide plane-strain conditions at the crack tip. The low plateau value of Kc is an important material property known as the plane-strain critical stress intensity or fracture toughness KIc. Values for a number of materials are shown in Table 5.1.4. They are influenced strongly by processing and small changes in composition, so that the values shown are not necessarily typical. KIc can be used in the critical form of Eq. (5.1.1): (KIc)2 � Q�2�acr (5.1.2) to predict failure stress when a maximum flaw size in the material is known or to determine maximum allowable flaw size when the stress is set. The predictions will be accurate so long as plate thickness B satis- fies the plane-strain criterion: B � 2.5(KIc/�ys)2. They will be conserva- tive if a plane-strain condition does not exist. A big advantage of the fracture mechanics approach is that stress intensity can be calculated by equations analogous to (5.1.1) for a wide variety of geometries, types of Fig. 5.1.16. Dependence of Kc and fracture appearance (in terms of percentage of square fracture) on thickness of plate specimens. Based on data for aluminum 7075-T6. (From Scrawly and Brown, STP-381, ASTM.) Table 5.1.4 Room-Temperature Klc Values on High-Strength Materials* 0.2% YS, 1,000 in2 Klc, 1,000 in2 Material (MN/m2) √in (MN m1/2/m2) 18% Ni maraging steel 300 (2,060) 46 (50.7) 18% Ni maraging steel 270 (1,850) 71 (78) 18% Ni maraging steel 198 (1,360) 87 (96) Titanium 6-4 alloy 152 (1,022) 39 (43) Titanium 6-4 alloy 140 (960) 75 (82.5) Aluminum alloy 7075-T6 75 (516) 26 (28.6) Aluminum alloy 7075-T6 64 (440) 30 (33) * Determined at Westinghouse Research Laboratories. crack, and loadings (Paris and Sih, ‘‘Stress Analysis of Cracks,’’ STP- 381, ASTM, 1965). Failure occurs in all cases when Kt reaches KIc. Fracture mechanics also provides a framework for predicting the occur- rence of stress-corrosion cracking by using Eq. (5.1.2) with KIc replaced by KIscc, which is the material parameter denoting resistance to stress- corrosion-crack propagation in a particular medium. Two standard test specimens for KIc determination are specified in ASTM standards, which also detail specimen preparation and test pro- cedure. Recent developments in fracture mechanics permit treatment of crack propagation in the ductile regime. (See ‘‘Elastic-Plastic Frac- ture,’’ ASTM.) FATIGUE Fatigue is generally understood as the gradual deterioration of a mate- rial which is subjected to repeated loads. In fatigue testing, a specimen is subjected to periodically varying constant-amplitude stresses by means of mechanical or magnetic devices. The applied stresses may alternate between equal positive and negative values, from zero to max- imum positive or negative values, or between unequal positive and negative values. The most common loading is alternate tension and compression of equal numerical values obtained by rotating a smooth cylindrical specimen while under a bending load. A series of fatigue tests are made on a number of specimens of the material at different stress levels. The stress endured is then plotted against the number of cycles sustained. By choosing lower and lower stresses, a value may be found which will not produce failure, regardless of the number of ap- plied cycles. This stress value is called the fatigue limit. The diagram is called the stress-cycle diagram or S-N diagram. Instead of recording the data on cartesian coordinates, either stress is plotted vs. the logarithm of the number of cycles (Fig. 5.1.17) or both stress and cycles are plotted to logarithmic scales. Both diagrams show a relatively sharp bend in the curve near the fatigue limit for ferrous metals. The fatigue limit may be established for most steels between 2 and 10 million cycles. Nonferrous metals usually show no clearly defined fatigue limit. The S-N curves in these cases indicate a continuous decrease in stress values to several hundred million cycles, and both the stress value and the number of cycles sustained should be reported. See Table 5.1.5. The mean stress (the average of the maximum and minimum stress values for a cycle) has a pronounced influence on the stress range (the algebraic difference between the maximum and minimum stress values). Several empirical formulas and graphical methods such as the ‘‘modified Goodman diagram’’ have been developed to show the influ- ence of the mean stress on the stress range for failure. A simple but conservative approach (see Soderberg, Working Stresses, Jour. Appl. Mech., 2, Sept. 1935) is to plot the variable stress Sv (one-half the stress range) as ordinate vs. the mean stress Sm as abscissa (Fig. 5.1.18). At zero mean stress, the ordinate is the fatigue limit under completely reversed stress. Yielding will occur if the mean stress exceeds the yield stress So, and this establishes the extreme right-hand point of the dia- gram. A straight line is drawn between these two points. The coordi- nates of any other point along this line are values of Sm and Sv which may produce failure. Surface defects, such as roughness or scratches, and notches or Copyright (C) 1999 by The McGraw-Hill Companies, Inc. All rights reserved. Use of this product is subject to the terms of its License Agreement. Click here to view. FATIGUE 5-9 Fig. 5.1.17. The S-N diagrams from fatigue tests. (1) 1.20% C steel, quenched and drawn at 860°F (460°C); (2) alloy structural steel; (3) SAE 1050, quenched and drawn at 1,200°F (649°C); (4) SAE 4130, normalized and annealed; (5) ordi- nary structural steel; (6) Duralumin; (7) copper, annealed; (8) cast iron (reversed bending). shoulders all reduce the fatigue strength of a part. With a notch of prescribed geometric form and known concentration factor, the reduc- tion in strength is appreciably less than would be called for by the concentration factor itself, but the various metals differ widely in their susceptibility to the effect of roughness and concentrations, or notch sensitivity. For a given material subjected to a prescribed state of stress and type of loading, notch sensitivity can be viewed as the ability of that material to resist the concentration of stress incidental to the presence of a notch. Alternately, notch sensitivity can be taken as a measure of the degree to which the geometric stress concentration factor is reduced. An attempt is made to rationalize notch sensitivity through the equation q � (Kf � 1)/(K � 1), where q is the notch sensitivity, K is the geometric stress concentration factor (from data similar to those in Figs. 5.1.5 to 5.1.13 and the like), and Kf is defined as the ratio of the strength of unnotched material to the strength of notched material. Ratio Kf is obtained from laboratory tests, and K is deduced either theoretically or from laboratory tests, but both must reflect the same state of stress and type of loading. The value of q lies between 0 and 1, so that (1) if q � 0, Kf � 1 and the material is not notch-sensitive (soft metals such as copper, aluminum, and annealed low-strength steel); (2) if q � 1, Kf � K, the material is fully notch-sensitive and the full value of the geometric stress concen- tration factor is not diminished (hard, high-strength steel). In practice, q will lie somewhere between 0 and 1, but it may be hard to quantify. Accordingly, the pragmatic approach to arrive at a solution to a design problem often takes a conservative route and sets q � 1. The exact material properties at play which are responsible for notch sensitivity are not clear. Further, notch sensitivity seems to be higher, and ordinary fatigue strength lower in large specimens, necessitating full-scale tests in many cases (see Peterson, Stress Concentration Phenomena in Fatigue of Fig. 5.1.18. Effect of mean stress on the variable stress for failure. Metals, Trans. ASME, 55, 1933, p. 157, and Buckwalter and Horger, Investigation of Fatigue Strength of Axles, Press Fits, Surface Rolling and Effect of Size, Trans. ASM, 25, Mar. 1937, p. 229). Corrosion and galling (due to rubbing of mating surfaces) cause great reduction of fatigue strengths, sometimes amounting to as much as 90 percent of the original endurance limit. Although any corroding agent will promote severe corrosion fatigue, there is so much difference between the effects of ‘‘sea water’’ or ‘‘tap water’’ from different localities that numerical values are not quoted here. Overstressing specimens above the fatigue limit for periods shorter than necessary to produce failure at that stress reduces the fatigue limit in a subsequent test. Similarly, understressing below the fatigue limit may increase it. Shot peening, nitriding, and cold work usually improve fatigue properties. No very good overall correlation exists between fatigue properties and any other mechanical property of a material. The best correlation is between the fatigue limit under completely reversed bending stress and the ordinary tensile strength. For many ferrous metals, the fatigue limit is approximately 0.40 to 0.60 times the tensile strength if the latter is below 200,000 lb/in2. Low-alloy high-yield-strength steels often show higher values than this. The fatigue limit for nonferrous metals is ap- proximately to 0.20 to 0.50 times the tensile strength. The fatigue limit in reversed shear is approximately 0.57 times that in reversed bending. In some very important engineering situations components are cycli- cally stressed into the plastic range. Examples are thermal strains result- ing from temperature oscillations and notched regions subjected to sec- ondary stresses. Fatigue life in the plastic or ‘‘low-cycle’’ fatigue range has been found to be a function of plastic strain, and low-cycle fatigue testing is done with strain as the controlled variable rather than stress. Fatigue life N and cyclic plastic strain �p tend to follow the relationship N�2p � C where C is a constant for a material when N � 105. (See Coffin, A Study Table 5.1.5 Typical Approximate Fatigue Limits for Reversed Bending Tensile Fatigue Tensile Fatigue strength, limit, strength, limit, Metal 1,000 lb/in2 1,000 lb/in2 Metal 1,000 lb/in2 1,000 lb/in2 Cast iron 20–50 6–18 Copper 32–50 12–17 Malleable iron 50 24 Monel 70–120 20–50 Cast steel 60–80 24–32 Phosphor bronze 55 12 Armco iron 44 24 Tobin bronze, hard 65 21 Plain carbon steels 60–150 25–75 Cast aluminum alloys 18–40 6–11 SAE 6150, heat-treated 200 80 Wrought aluminum alloys 25–70 8–18 Nitralloy 125 80 Magnesium alloys 20–45 7–17 Brasses, various 25–75 7–20 Molybdenum, as cast 98 45 Zirconium crystal bar 52 16–18 Titanium (Ti-75A) 91 45 NOTE: Stress, 1,000 lb/in2 � 6.894 � stress, MN/m2. Copyright (C) 1999 by The McGraw-Hill Companies, Inc. All rights reserved. Use of this product is subject to the terms of its License Agreement. Click here to view. 5-10 MECHANICAL PROPERTIES OF MATERIALS of Cyclic-Thermal Stresses in a Ductile Material, Trans. ASME, 76, 1954, p. 947.) The type of physical change occurring inside a material as it is re- peatedly loaded to failure varies as the life is consumed, and a number of stages in fatigue can be distinguished on this basis. The early stages comprise the events causing nucleation of a crack or flaw. This is most likely to appear on the surface of the material; fatigue failures generally originate at a surface. Following nucleation of the crack, it grows during the crack-propagation stage. Eventually the crack becomes large enough for some rapid terminal mode of failure to take over such as ductile rupture or brittle fracture. The rate of crack growth in the crack- propagation stage can be accurately quantified by fracture mechanics methods. Assuming an initial flaw and a loading situation as shown in Fig. 5.1.15, the rate of crack growth per cycle can generally be ex- pressed as da/dN � C0(�KI)n (5.1.3) where C0 and n are constants for a particular material and �KI is the range of stress intensity per cycle. KI is given by (5.1.1). Using (5.1.3), it is possible to predict the number of cycles for the crack to grow to a size at which some other mode of failure can take over. Values of the constants C0 and n are determined from specimens of the same type as those used for determination of KIc but are instrumented for accurate measurement of slow crack growth. Constant-amplitude fatigue-test data are relevant to many rotary- machinery situations where constant cyclic loads are encountered. There are important situations where the component undergoes vari- able loads and where it may be advisable to use random-load testing. In this method, the load spectrum which the component will experi- ence in service is determined and is applied to the test specimen artificially. CREEP Experience has shown that, for the design of equipment subjected to sustained loading at elevated temperatures, little reliance can be placed on the usual short-time tensile properties of metals at those tempera- tures. Under the application of a constant load it has been found that materials, both metallic and nonmetallic, show a gradual flow or creep even for stresses below the proportional limit at elevated temperatures. Similar effects are present in low-melting metals such as lead at room temperature. The deformation which can be permitted in the satisfactory operation of most high-temperature equipment is limited. In metals, creep is a plastic deformation caused by slip occurring along crystallographic directions in the individual crystals, together with some flow of the grain-boundary material. After complete release of load, a small fraction of this plastic deformation is recovered with time. Most of the flow is nonrecoverable for metals. Since the early creep experiments, many different types of tests have come into use. The most common are the long-time creep test under constant tensile load and the stress-rupture test. Other special forms are the stress-relaxation test and the constant-strain-rate test. The long-time creep test is conducted by applying a dead weight to one end of a lever system, the other end being attached to the specimen surrounded by a furnace and held at constant temperature. The axial deformation is read periodically throughout the test and a curve is plot- ted of the strain �0 as a function of time t (Fig. 5.1.19). This is repeated for various loads at the same testing temperature. The portion of the Fig. 5.1.19. Typical creep curve. curve OA in Fig. 5.1.19 is the region of primary creep, AB the region of secondary creep, and BC that of tertiary creep. The strain rates, or the slopes of the curve, are decreasing, constant, and increasing, respectively, in these three regions. Since the period of the creep test is usually much shorter than the duration of the part in service, various extrapolation procedures are followed (see Gittus, ‘‘Creep, Viscoelasticity and Creep Fracture in Solids,’’ Wiley, 1975). See Table 5.1.6. In practical applications the region of constant-strain rate (secondary creep) is often used to estimate the probable deformation throughout the life of the part. It is thus assumed that this rate will remain constant during periods beyond the range of the test-data. The working stress is chosen so that this total deformation will not be excessive. An arbitrary creep strength, which is defined as the stress which at a given tempera- ture will result in 1 percent deformation in 100,000 h, has received a certain amount of recognition, but it is advisable to determine the proper stress for each individual case from diagrams of stress vs. creep rate (Fig. 5.1.20) (see ‘‘Creep Data,’’ ASTM and ASME). Fig. 5.1.20. Creep rates for 0.35% C steel. Additional temperatures (°F) and stresses (in 1,000 lb/in2) for stated creep rates (percent per 1,000 h) for wrought nonferrous metals are as follows: 60-40 Brass: Rate 0.1, temp. 350 (400), stress 8 (2); rate 0.01, temp 300 (350) [400], stress 10 (3) [1]. Phosphor bronze: Rate 0.1, temp 400 (550) [700] [800], stress 15 (6) [4] [4]; rate 0.01, temp 400 (550) [700], stress 8 (4) [2]. Nickel: Rate 0.1, temp 800 (1000), stress 20 (10). 70 CU, 30 NI. Rate 0.1, temp 600 (750), stress 28 (13–18); rate 0.01, temp 600 (750), stress 14 (8–9). Aluminum alloy 17 S (Duralumin): Rate 0.1, temp 300 (500) [600], stress 22 (5) [1.5]. Lead pure (commercial) (0.03 percent Ca): At 110°F, for rate 0.1 percent the stress range, lb/in2, is 150–180 (60–140) [200–220]; for rate of 0.01 percent, 50–90 (10–50) [110–150]. Stress, 1,000 lb/in2 � 6.894 � stress, MN/m2, tk � 5⁄9(tF � 459.67). Structural changes may occur during a creep test, thus altering the metallurgical condition of the metal. In some cases, premature rupture appears at a low fracture strain in a normally ductile metal, indicating that the material has become embrittled. This is a very insidious condi- tion and difficult to predict. The stress-rupture test is well adapted to study this effect. It is conducted by applying a constant load to the specimen in the same manner as for the long-time creep test. The nomi- nal stress is then plotted vs. the time for fracture at constant temperature on a log-log scale (Fig. 5.1.21). Copyright (C) 1999 by The McGraw-Hill Companies, Inc. All rights reserved. Use of this product is subject to the terms of its License Agreement. Click here to view. CREEP 5-11 Table 5.1.6 Stresses for Given Creep Rates and Temperatures* Creep rate 0.1% per 1,000 h Creep rate 0.01% per 1,000 h Material Temp, °F 800 900 1,000 1,100 1,200 800 900 1,000 1,100 1,200 Wrought steels: SAE 1015 0.20 C, 0.50 Mo 0.10–0.25 C, 4–6 Cr � Mo SAE 4140 SAE 1030–1045 17–27 26–33 22 27–33 8–25 11–18 18–25 15–18 20–25 5–15 3–12 9–16 9–11 7–15 5 2–7 2–6 3–6 4–7 2 1 1–2 2–3 1–2 1 10–18 16–24 14–17 19–28 5–15 6–14 11–22 11–15 12–19 3–7 3–8 4–12 4–7 3–8 2–4 1 2 2–3 2–4 1 1 1–2 1 Commercially pure iron 7 4 3 5 2 0.15 C, 1–2.5 Cr, 0.50 Mo SAE 4340 SAE X3140 0.20 C, 4–6 Cr 0.25 C, 4–6 Cr � W 0.16 C, 1.2 Cu 0.20 C, 1 Mo 0.10–0.40 C, 0.2–0.5 Mo, 1–2 Mn SAE 2340 SAE 6140 SAE 7240 Cr � Va � W, various 25–35 20–40 7–10 30 30 35 30–40 7–12 30 30 20–70 18–28 15–30 10–20 10–15 18 27 12–20 5 12 21 14–30 8–20 2–12 5–4 7–10 4–10 10–15 12 4–14 2 4 6–15 5–15 6–8 1–3 1 2–8 3 2 3–4 1 20–30 8–20 3–8 25 25–28 7 30 18–50 12–18 6–11 10–18 12 8–15 6 11 8–18 3–12 1–6 1–2 3–5 2–7 7–12 6 2–8 1 3–9 2–13 2–5 1 1–2 0.5 Temp, °F 1,100 1,200 1,300 1,400 1,500 1,000 1,100 1,200 1,300 1,400 Wrought chrome-nickel steels: 18-8† 10–25 Cr, 10–30 Ni‡ 10–18 10–20 5–11 5–15 3–10 3–10 2–5 2–5 2.5 11–16 5–12 6–15 2–10 3–10 2–8 1–2 1–3 Temp, °F 800 900 1,000 1,100 1,200 800 900 1,000 1,100 1,200 Cast steels: 0.20–0.40 C 0.10–0.30 C, 0.5–1 Mo 0.15–0.30 C, 4–6 Cr � Mo 18–8§ Cast iron Cr Ni cast iron 10–20 28 25–30 20 5–10 20–30 15–25 8 3 6–12 8–15 20–25 4 9 2 8 15 10 8–15 20 20–25 10 10–15 9–15 1 2–5 2–7 20 2 3 2 15 8 * Based on 1,000-h tests. Stresses in 1,000 lb/in2. † Additional data. At creep rate 0.1 percent and 1,000 (1,600)°F the stress is 18–25 (1); at creep rate 0.01 percent at 1,500°F, the stress is 0.5. ‡ Additional data. At creep rate 0.1 percent and 1,000 (1,600)°F the stress is 10–30 (1). § Additional data. At creep rate 0.1 percent and 1,600°F the stress is 3; at creep rate 0.01 and 1,500°F, the stress is 2–3. The stress reaction is measured in the constant-strain-rate test while the specimen is deformed at a constant strain rate. In the relaxation test, the decrease of stress with time is measured while the total strain (elastic � plastic) is maintained constant. The latter test has direct application to the loosening of turbine bolts and to similar problems. Although some correlation has been indicated between the results of these various types of tests, no general correlation is yet available, and it has been found necessary to make tests under each of these special conditions to obtain satisfactory results. The interrelationship between strain rate and temperature in the form of a velocity-modified temperature (see MacGregor and Fisher, A Ve- locity-modified Temperature for the Plastic F...

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Section 5 Strength of Materials BY JOHN SYMONDS Fellow Engineer (Retired), Oceanic Division, Westinghouse Electric Corporation. J. P. VIDOSIC Regents’ Professor Emeritus of Mechanical Engineering, Georgia Institute of Technology. HAROLD V. HAWKINS Late Manager, Product Standards and Services, Columbus McKinnon Corporation, Tonawanda, N.Y. DONALD D. DODGE Supervisor (Retired), Product Quality and Inspection Technology, Manufacturing Development, Ford Motor Company. 5.1 MECHANICAL PROPERTIES OF MATERIALS by John Symonds, Expanded by Staff Stress-Strain Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-2 Fracture at Low Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-7 Fatigue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-8 Creep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-10 Hardness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-12 Testing of Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-13 5.2 MECHANICS OF MATERIALS by J. P. Vidosic Simple Stresses and Strains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-15 Combined Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-18 Plastic Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-19 Design Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-20 Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-20 Torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-36 Columns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-38 Eccentric Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-40 Curved Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-41 Impact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-43 Theory of Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-44 Cylinders and Spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-45 Pressure between Bodies with Curved Surfaces . . . . . . . . . . . . . . . . . . . . . . 5-47 Flat Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-47 Theories of Failure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-48 Plasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-49 Rotating Disks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-50 Experimental Stress Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-51 5.3 PIPELINE FLEXURE STRESSES by Harold V. Hawkins Pipeline Flexure Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-55 5.4 NONDESTRUCTIVE TESTING by Donald D. Dodge Nondestructive Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-61 Magnetic Particle Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-61 Penetrant Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-61 Radiographic Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-65 Ultrasonic Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-66 Eddy Current Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-66 Microwave Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-67 Infrared Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-67 Acoustic Signature Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-67 5-1 Copyright (C) 1999 by The McGraw-Hill Companies, Inc. All rights reserved. Use of this product is subject to the terms of its License Agreement. Click here to view. 5.1 MECHANICAL PROPERTIES OF MATERIALS by John Symonds, Expanded by Staff REFERENCES: Davis et al., ‘‘Testing and Inspection of Engineering Materials,’’ McGraw-Hill, Timoshenko, ‘‘Strength of Materials,’’ pt. II, Van Nostrand. Richards, ‘‘Engineering Materials Science,’’ Wadsworth. Nadai, ‘‘Plasticity,’’ McGraw-Hill. Tetelman and McEvily, ‘‘Fracture of Structural Materials,’’ Wiley. ‘‘Fracture Mechanics,’’ ASTM STP-833. McClintock and Argon (eds.), ‘‘Me- chanical Behavior of Materials,’’ Addison-Wesley. Dieter, ‘‘Mechanical Metal- lurgy,’’ McGraw-Hill. ‘‘Creep Data,’’ ASME. ASTM Standards, ASTM. Blaz- nynski (ed.), ‘‘Plasticity and Modern Metal Forming Technology,’’ Elsevier Science. STRESS-STRAIN DIAGRAMS The Stress-Strain Curve The engineering tensile stress-strain curve is obtained by static loading of a standard specimen, that is, by applying the load slowly enough that all parts of the specimen are in equilibrium at any instant. The curve is usually obtained by controlling the loading rate in the tensile machine. ASTM Standards require a loading rate not exceeding 100,000 lb/in2 (70 kgf/mm2)/min. An alternate method of obtaining the curve is to specify the strain rate as the independent vari- able, in which case the loading rate is continuously adjusted to maintain the required strain rate. A strain rate of 0.05 in/in/(min) is commonly used. It is measured usually by an extensometer attached to the gage length of the specimen. Figure 5.1.1 shows several stress-strain curves. Fig. 5.1.1. Comparative stress-strain diagrams. (1) Soft brass; (2) low carbon steel; (3) hard bronze; (4) cold rolled steel; (5) medium carbon steel, annealed; (6) medium carbon steel, heat treated. For most engineering materials, the curve will have an initial linear elastic region (Fig. 5.1.2) in which deformation is reversible and time- independent. The slope in this region is Young’s modulus E. The propor- tional elastic limit (PEL) is the point where the curve starts to deviate from a straight line. The elastic limit (frequently indistinguishable from PEL) is the point on the curve beyond which plastic deformation is present after release of the load. If the stress is increased further, the stress-strain curve departs more and more from the straight line. Un- loading the specimen at point X (Fig. 5.1.2), the portion XX� is linear and is essentially parallel to the original line OX��. The horizontal dis- tance OX� is called the permanent set corresponding to the stress at X. This is the basis for the construction of the arbitrary yield strength. To determine the yield strength, a straight line XX� is drawn parallel to the initial elastic line OX�� but displaced from it by an arbitrary value of permanent strain. The permanent strain commonly used is 0.20 percent of the original gage length. The intersection of this line with the curve determines the stress value called the yield strength. In reporting the yield strength, the amount of permanent set should be specified. The arbitrary yield strength is used especially for those materials not ex- hibiting a natural yield point such as nonferrous metals; but it is not limited to these. Plastic behavior is somewhat time-dependent, particu- larly at high temperatures. Also at high temperatures, a small amount of time-dependent reversible strain may be detectable, indicative of anelas- tic behavior. Fig. 5.1.2. General stress-strain diagram. The ultimate tensile strength (UTS) is the maximum load sustained by the specimen divided by the original specimen cross-sectional area. The percent elongation at failure is the plastic extension of the specimen at failure expressed as (the change in original gage length � 100) divided by the original gage length. This extension is the sum of the uniform and nonuniform elongations. The uniform elongation is that which occurs prior to the UTS. It has an unequivocal significance, being associated with uniaxial stress, whereas the nonuniform elongation which occurs during localized extension (necking) is associated with triaxial stress. The nonuniform elongation will depend on geometry, particularly the ratio of specimen gage length L0 to diameter D or square root of cross- sectional area A. ASTM Standards specify test-specimen geometry for a number of specimen sizes. The ratio L0/√A is maintained at 4.5 for flat- and round-cross-section specimens. The original gage length should always be stated in reporting elongation values. The specimen percent reduction in area (RA) is the contraction in cross-sectional area at the fracture expressed as a percentage of the original area. It is obtained by measurement of the cross section of the broken specimen at the fracture location. The RA along with the load at fracture can be used to obtain the fracture stress, that is, fracture load divided by cross-sectional area at the fracture. See Table 5.1.1. The type of fracture in tension gives some indications of the quality of the material, but this is considerably affected by the testing tempera- ture, speed of testing, the shape and size of the test piece, and other conditions. Contraction is greatest in tough and ductile materials and least in brittle materials. In general, fractures are either of the shear or of the separation (loss of cohesion) type. Flat tensile specimens of ductile metals often show shear failures if the ratio of width to thickness is greater than 6:1. A completely shear-type failure may terminate in a chisel edge, for a flat specimen, or a point rupture, for a round specimen. Separation failures occur in brittle materials, such as certain cast irons. Combinations of both shear and separation failures are common on round specimens of ductile metal. Failure often starts at the axis in a necked region and produces a relatively flat area which grows until the material shears along a cone-shaped surface at the outside of the speci- 5-2 Copyright (C) 1999 by The McGraw-Hill Companies, Inc. All rights reserved. Use of this product is subject to the terms of its License Agreement. Click here to view. STRESS-STRAIN DIAGRAMS 5-3 Table 5.1.1 Typical Mechanical Properties at Room Temperature (Based on ordinary stress-strain values) Tensile Yield Ultimate strength, strength, elongation, Reduction Brinell Metal 1,000 lb/in2 1,000 lb/in2 % of area, % no. Cast iron 18–60 8–40 0 0 100–300 Wrought iron 45–55 25–35 35–25 55–30 100 Commercially pure iron, annealed 42 19 48 85 70 Hot-rolled 48 30 30 75 90 Cold-rolled 100 95 200 Structural steel, ordinary 50–65 30–40 40–30 120 Low-alloy, high-strength 65–90 40–80 30–15 70–40 150 Steel, SAE 1300, annealed 70 40 26 70 150 Quenched, drawn 1,300°F 100 80 24 65 200 Drawn 1,000°F 130 110 20 60 260 Drawn 700°F 200 180 14 45 400 Drawn 400°F 240 210 10 30 480 Steel, SAE 4340, annealed 80 45 25 70 170 Quenched, drawn 1,300°F 130 110 20 60 270 Drawn 1,000°F 190 170 14 50 395 Drawn 700°F 240 215 12 48 480 Drawn 400°F 290 260 10 44 580 Cold-rolled steel, SAE 1112 84 76 18 45 160 Stainless steel, 18-S 85–95 30–35 60–55 75–65 145–160 Steel castings, heat-treated 60–125 30–90 33–14 65–20 120–250 Aluminum, pure, rolled 13–24 5–21 35–5 23–44 Aluminum-copper alloys, cast 19–23 12–16 4–0 50–80 Wrought, heat-treated 30–60 10–50 33–15 50–120 Aluminum die castings 30 2 Aluminum alloy 17ST 56 34 26 39 100 Aluminum alloy 51ST 48 40 20 35 105 Copper, annealed 32 5 58 73 45 Copper, hard-drawn 68 60 4 55 100 Brasses, various 40–120 8–80 60–3 50–170 Phosphor bronze 40–130 55–5 50–200 Tobin bronze, rolled 63 41 40 52 120 Magnesium alloys, various 21–45 11–30 17–0.5 47–78 Monel 400, Ni-Cu alloy 79 30 48 75 125 Molybdenum, rolled 100 75 30 250 Silver, cast, annealed 18 8 54 27 Titanium 6–4 alloy, annealed 130 120 10 25 352 Ductile iron, grade 80-55-06 80 55 6 225–255 NOTE: Compressive strength of cast iron, 80,000 to 150,000 lb/in2. Compressive yield strength of all metals, except those cold-worked � tensile yield strength. Stress 1,000 lb/in2 � 6.894 � stress, MN/m2. men, resulting in what is known as the cup-and-cone fracture. Double cup-and-cone and rosette fractures sometimes occur. Several types of tensile fractures are shown in Fig. 5.1.3. Annealed or hot-rolled mild steels generally exhibit a yield point (see Fig. 5.1.4). Here, in a constant strain-rate test, a large increment of extension occurs under constant load at the elastic limit or at a stress just below the elastic limit. In the latter event the stress drops suddenly from the upper yield point to the lower yield point. Subsequent to the drop, the yield-point extension occurs at constant stress, followed by a rise to the UTS. Plastic flow during the yield-point extension is discontinuous; Fig. 5.1.3. Typical metal fractures in tension. successive zones of plastic deformation, known as Luder’s bands or stretcher strains, appear until the entire specimen gage length has been uniformly deformed at the end of the yield-point extension. This behav- ior causes a banded or stepped appearance on the metal surface. The exact form of the stress-strain curve for this class of material is sensitive to test temperature, test strain rate, and the characteristics of the tensile machine employed. The plastic behavior in a uniaxial tensile test can be represented as the true stress-strain curve. The true stress � is based on the instantaneous Fig. 5.1.4. Yielding of annealed steel. Copyright (C) 1999 by The McGraw-Hill Companies, Inc. All rights reserved. Use of this product is subject to the terms of its License Agreement. Click here to view. 5-4 MECHANICAL PROPERTIES OF MATERIALS cross section A, so that � � load/A. The instantaneous true strain incre- ment is �dA/A, or dL/L prior to necking. Total true strain � is � A A0 � dA A � ln� A0 A� or ln (L/L0) prior to necking. The true stress-strain curve or flow curve obtained has the typical form shown in Fig. 5.1.5. In the part of the test subsequent to the maximum load point (UTS), when necking occurs, the true strain of interest is that which occurs in an infinitesimal length at the region of minimum cross section. True strain for this element can still be expressed as ln (A0/A), where A refers to the minimum cross Fig. 5.1.5. True stress-strain curve for 20°C annealed mild steel. section. Methods of constructing the true stress-strain curve are de- scribed in the technical literature. In the range between initial yielding and the neighborhood of the maximum load point the relation- ship between plastic strain �p and true stress often approximates � � k�pn where k is the strength coefficient and n is the work-hardening exponent. For a material which shows a yield point the relationship applies only to the rising part of the curve beyond the lower yield. It can be shown that at the maximum load point the slope of the true stress-strain curve equals the true stress, from which it can be deduced that for a material obeying the above exponential relationship between �p and n, �p � n at the maximum load point. The exponent strongly influences the spread between YS and UTS on the engineering stress-strain curve. Values of n and k for some materials are shown in Table 5.1.2. A point on the flow curve indentifies the flow stress corresponding to a certain strain, that is, the stress required to bring about this amount of plastic deformation. The concept of true strain is useful for accurately describing large amounts of plastic deformation. The linear strain definition (L � L0)/L0 fails to correct for the continuously changing gage length, which leads to an increasing error as deformation proceeds. During extension of a specimen under tension, the change in the specimen cross-sectional area is related to the elongation by Poisson’s ratio �, which is the ratio of strain in a transverse direction to that in the longitudinal direction. Values of � for the elastic region are shown in Table 5.1.3. For plastic strain it is approximately 0.5. Table 5.1.2 Room-Temperature Plastic-Flow Constants for a Number of Metals k, 1,000 in2 Material Condition (MN/m2) n 0.40% C steel Quenched and tempered at 400°F (478K) 416 (2,860) 0.088 0.05% C steel Annealed and temper-rolled 72 (49.6) 0.235 2024 aluminum Precipitation-hardened 100 (689) 0.16 2024 aluminum Annealed 49 (338) 0.21 Copper Annealed 46.4 (319) 0.54 70–30 brass Annealed 130 (895) 0.49 SOURCE: Reproduced by permission from ‘‘Properties of Metals in Materials Engineering,’’ ASM, 1949. Table 5.1.3 Elastic Constants of Metals (Mostly from tests of R. W. Vose) E G K � Modulus of Modulus of elasticity rigidity (Young’s (shearing Bulk modulus). modulus). modulus. 1,000,000 1,000,000 1,000,000 Poisson’s Metal lb/in2 lb/in2 lb/in2 ratio Cast steel 28.5 11.3 20.2 0.265 Cold-rolled steel 29.5 11.5 23.1 0.287 Stainless steel 18–8 27.6 10.6 23.6 0.305 All other steels, including high-carbon, heat-treated 28.6–30.0 11.0–11.9 22.6–24.0 0.283–0.292 Cast iron 13.5–21.0 5.2–8.2 8.4–15.5 0.211–0.299 Malleable iron 23.6 9.3 17.2 0.271 Copper 15.6 5.8 17.9 0.355 Brass, 70–30 15.9 6.0 15.7 0.331 Cast brass 14.5 5.3 16.8 0.357 Tobin bronze 13.8 5.1 16.3 0.359 Phosphor bronze 15.9 5.9 17.8 0.350 Aluminum alloys, various 9.9–10.3 3.7–3.9 9.9–10.2 0.330–0.334 Monel metal 25.0 9.5 22.5 0.315 Inconel 31 11 0.27–0.38 Z-nickel 30 11 �0.36 Beryllium copper 17 7 �0.21 Elektron (magnesium alloy) 6.3 2.5 4.8 0.281 Titanium (99.0 Ti), annealed bar 15–16 6.5 0.34 Zirconium, crystal bar 11–14 Molybdenum, arc-cast 48–52 Copyright (C) 1999 by The McGraw-Hill Companies, Inc. All rights reserved. Use of this product is subject to the terms of its License Agreement. Click here to view. STRESS-STRAIN DIAGRAMS 5-5 The general effect of increased strain rate is to increase the resistance to plastic deformation and thus to raise the flow curve. Decreasing test temperature also raises the flow curve. The effect of strain rate is ex- pressed as strain-rate sensitivity m. Its value can be measured in the tension test if the strain rate is suddenly increased by a small increment during the plastic extension. The flow stress will then jump to a higher value. The strain-rate sensitivity is the ratio of incremental changes of log � and log �� m �� � log � � log ��� � For most engineering materials at room temperature the strain rate sen- sitivity is of the order of 0.01. The effect becomes more significant at elevated temperatures, with values ranging to 0.2 and sometimes higher. Compression Testing The compressive stress-strain curve is simi- lar to the tensile stress-strain curve up to the yield strength. Thereafter, the progressively increasing specimen cross section causes the com- pressive stress-strain curve to diverge from the tensile curve. Some ductile metals will not fail in the compression test. Complex behavior occurs when the direction of stressing is changed, because of the Baus- chinger effect, which can be described as follows: If a specimen is first plastically strained in tension, its yield stress in compression is reduced and vice versa. Combined Stresses This refers to the situation in which stresses are present on each of the faces of a cubic element of the material. For a given cube orientation the applied stresses may include shear stresses over the cube faces as well as stresses normal to them. By a suitable rotation of axes the problem can be simplified: applied stresses on the new cubic element are equivalent to three mutually orthogonal principal stresses �1, �2, �3 alone, each acting normal to a cube face. Combined stress behavior in the elastic range is described in Sec. 5.2, Mechanics of Materials. Prediction of the conditions under which plastic yielding will occur under combined stresses can be made with the help of several empirical theories. In the maximum-shear-stress theory the criterion for yielding is that yielding will occur when �1 � �3 � �ys in which �1 and �3 are the largest and smallest principal stresses, re- spectively, and �ys is the uniaxial tensile yield strength. This is the simplest theory for predicting yielding under combined stresses. A more accurate prediction can be made by the distortion-energy theory, accord- ing to which the criterion is (�1 � �2)2 � (�2 � �3)2 � (�2 � �1)2 � 2(�ys)2 Stress-strain curves in the plastic region for combined stress loading can be constructed. However, a particular stress state does not determine a unique strain value. The latter will depend on the stress-state path which is followed. Plane strain is a condition where strain is confined to two dimensions. There is generally stress in the third direction, but because of mechani- cal constraints, strain in this dimension is prevented. Plane strain occurs in certain metalworking operations. It can also occur in the neighbor- hood of a crack tip in a tensile loaded member if the member is suffi- ciently thick. The material at the crack tip is then in triaxial tension, which condition promotes brittle fracture. On the other hand, ductility is enhanced and fracture is suppressed by triaxial compression. Stress Concentration In a structure or machine part having a notch or any abrupt change in cross section, the maximum stress will occur at this location and will be greater than the stress calculated by elementary formulas based upon simplified assumptions as to the stress distribu- tion. The ratio of this maximum stress to the nominal stress (calculated by the elementary formulas) is the stress-concentration factor Kt. This is a constant for the particular geometry and is independent of the mate- rial, provided it is isotropic. The stress-concentration factor may be determined experimentally or, in some cases, theoretically from the mathematical theory of elasticity. The factors shown in Figs. 5.1.6 to 5.1.13 were determined from both photoelastic tests and the theory of elasticity. Stress concentration will cause failure of brittle materials if 3.4 3.0 2.6 2.2 1.8 1.4 1.0 Stress concentration factor, K 0.01 0.1 I I II III 0.2 r d 1.0 Note; in all cases D�d�2r D Tension or compression D 2r d/2 d d/2 II IV IV V V D d III � � Bending D d r r � � r r � � r r � � r r D d Fig. 5.1.6. Flat plate with semicircular fillets and grooves or with holes. I, II, and III are in tension or compression; IV and V are in bending. 3.4 3.0 2.6 2.2 1.8 1.4 1.0 Stress concentration factor, K 0.4 1.0 1.5 h r Sharpness of groove, 2 h d D d h h 3 4 5 6 Semi-circle grooves (h�r) Blunt grooves Sharp grooves 2 1 0.5 0.2 0.1 0.05 � 0.02 � � r r Fig. 5.1.7. Flat plate with grooves, in tension. Copyright (C) 1999 by The McGraw-Hill Companies, Inc. All rights reserved. Use of this product is subject to the terms of its License Agreement. Click here to view. 5-6 MECHANICAL PROPERTIES OF MATERIALS the concentrated stress is larger than the ultimate strength of the mate- rial. In ductile materials, concentrated stresses higher than the yield strength will generally cause local plastic deformation and redistribu- tion of stresses (rendering them more uniform). On the other hand, even with ductile materials areas of stress concentration are possible sites for fatigue if the component is cyclically loaded. 3.4 3.0 2.6 2.2 1.8 1.4 1.0 Stress concentration factor, K 0.4 1.0 1.5 h r Sharpness of fillet, 2 h h 3 4 5 6 D�d � 2h Full fillets (h�r) Blunt fillets 0.5 2.0 1.0 0.2 0.1 0.05 � 0.02 Depth of fillet � Sharp fillets h d D d � � r r Fig. 5.1.8. Flat plate with fillets, in tension. 3.4 3.0 2.6 2.2 1.8 1.4 1.0 Stress concentration factor, K 0.4 1.0 1.5 h r Sharpness of groove, 2 3 4 5 6 D�d � 2h Semi-circle grooves (h�r) 0.5 2 1 0.2 0.1 � 0.05 h d Sharp grooves D d h h � � r r Fig. 5.1.9. Flat plate with grooves, in bending. 3.4 3.0 2.6 2.2 1.8 1.4 1.0 Stress concentration factor, K 0.4 1.0 1.5 h r Sharpness of fillet, 2 3 4 5 6 D�d � 2h Full fillets (h�r) 0.05 2 1 0.2 0.5 0.1 � 0.02 h d D d h h � � r r Sharp fillets Blunt fillets Fig. 5.1.10. Flat plate with fillets, in bending. Fig. 5.1.11. Flat plate with angular notch, in tension or bending. 3.4 3.0 2.6 2.2 1.8 1.4 1.0 Stress concentration factor, K 0.5 1.0 1.5 h r Sharpness of groove, 2 3 4 5 6 Semi circ. grooves h�r 4 10 1 0.4 � 0.1 h d D d h �r 8 10 15 20 Sharp grooves Blunt grooves � 0.04 h d Fig. 5.1.12. Grooved shaft in torsion. Copyright (C) 1999 by The McGraw-Hill Companies, Inc. All rights reserved. Use of this product is subject to the terms of its License Agreement. Click here to view. FRACTURE AT LOW STRESSES 5-7 3.4 3.0 2.6 2.2 1.8 1.4 1.0 Stress concentration factor, K 0.5 1.0 h r Sharpness of fillet, 2 3 4 5 7 1 0.2 0.5 10 20 40 Blunt fillets � 0.05 0.1 h d Sharp fillets Full fillets (h�r) D�d � 2h D d h � r Fig. 5.1.13. Filleted shaft in torsion. FRACTURE AT LOW STRESSES Materials under tension sometimes fail by rapid fracture at stresses much below their strength level as determined in tests on carefully prepared specimens. These brittle, unstable, or catastrophic failures origi- nate at preexisting stress-concentrating flaws which may be inherent in a material. The transition-temperature approach is often used to ensure fracture- safe design in structural-grade steels. These materials exhibit a charac- teristic temperature, known as the ductile brittle transition (DBT) tem- perature, below which they are susceptible to brittle fracture. The tran- sition-temperature approach to fracture-safe design ensures that the transition temperature of a material selected for a particular application is suitably matched to its intended use temperature. The DBT can be detected by plotting certain measurements from tensile or impact tests against temperature. Usually the transition to brittle behavior is com- plex, being neither fully ductile nor fully brittle. The range may extend over 200°F (110 K) interval. The nil-ductility temperature (NDT), deter- mined by the drop weight test (see ASTM Standards), is an important reference point in the transition range. When NDT for a particular steel is known, temperature-stress combinations can be specified which de- fine the limiting conditions under which catastrophic fracture can occur. In the Charpy V-notch (CVN) impact test, a notched-bar specimen (Fig. 5.1.26) is used which is loaded in bending (see ASTM Standards). The energy absorbed from a swinging pendulum in fracturing the speci- men is measured. The pendulum strikes the specimen at 16 to 19 ft (4.88 to 5.80 m)/s so that the specimen deformation associated with fracture occurs at a rapid strain rate. This ensures a conservative mea- sure of toughness, since in some materials, toughness is reduced by high strain rates. A CVN impact energy vs. temperature curve is shown in Fig. 5.1.14, which also shows the transitions as given by percent brittle fracture and by percent lateral expansion. The CVN energy has no analytical significance. The test is useful mainly as a guide to the frac- ture behavior of a material for which an empirical correlation has been established between impact energy and some rigorous fracture criterion. For a particular grade of steel the CVN curve can be correlated with NDT. (See ASME Boiler and Pressure Vessel Code.) Fracture Mechanics This analytical method is used for ultra-high- strength alloys, transition-temperature materials below the DBT tem- perature, and some low-strength materials in heavy section thickness. Fracture mechanics theory deals with crack extension where plastic effects are negligible or confined to a small region around the crack tip. The present discussion is concerned with a through-thickness crack in a tension-loaded plate (Fig. 5.1.15) which is large enough so that the crack-tip stress field is not affected by the plate edges. Fracture me- chanics theory states that unstable crack extension occurs when the work required for an increment of crack extension, namely, surface energy and energy consumed in local plastic deformation, is exceeded by the elastic-strain energy released at the crack tip. The elastic-stress Fig. 5.1.14. CVN transition curves. (Data from Westinghouse R & D Lab.) Copyright (C) 1999 by The McGraw-Hill Companies, Inc. All rights reserved. Use of this product is subject to the terms of its License Agreement. Click here to view. 5-8 MECHANICAL PROPERTIES OF MATERIALS field surrounding one of the crack tips in Fig. 5.1.15 is characterized by the stress intensity KI, which has units of (lb √in) /in2 or (N√m) /m2. It is a function of applied nominal stress �, crack half-length a, and a geom- etry factor Q: K 2l � Q�2�a (5.1.1) for the situation of Fig. 5.1.15. For a particular material it is found that as KI is increased, a value Kc is reached at which unstable crack propa- Fig. 5.1.15. Through-thickness crack geometry. gation occurs. Kc depends on plate thickness B, as shown in Fig. 5.1.16. It attains a constant value when B is great enough to provide plane-strain conditions at the crack tip. The low plateau value of Kc is an important material property known as the plane-strain critical stress intensity or fracture toughness KIc. Values for a number of materials are shown in Table 5.1.4. They are influenced strongly by processing and small changes in composition, so that the values shown are not necessarily typical. KIc can be used in the critical form of Eq. (5.1.1): (KIc)2 � Q�2�acr (5.1.2) to predict failure stress when a maximum flaw size in the material is known or to determine maximum allowable flaw size when the stress is set. The predictions will be accurate so long as plate thickness B satis- fies the plane-strain criterion: B � 2.5(KIc/�ys)2. They will be conserva- tive if a plane-strain condition does not exist. A big advantage of the fracture mechanics approach is that stress intensity can be calculated by equations analogous to (5.1.1) for a wide variety of geometries, types of Fig. 5.1.16. Dependence of Kc and fracture appearance (in terms of percentage of square fracture) on thickness of plate specimens. Based on data for aluminum 7075-T6. (From Scrawly and Brown, STP-381, ASTM.) Table 5.1.4 Room-Temperature Klc Values on High-Strength Materials* 0.2% YS, 1,000 in2 Klc, 1,000 in2 Material (MN/m2) √in (MN m1/2/m2) 18% Ni maraging steel 300 (2,060) 46 (50.7) 18% Ni maraging steel 270 (1,850) 71 (78) 18% Ni maraging steel 198 (1,360) 87 (96) Titanium 6-4 alloy 152 (1,022) 39 (43) Titanium 6-4 alloy 140 (960) 75 (82.5) Aluminum alloy 7075-T6 75 (516) 26 (28.6) Aluminum alloy 7075-T6 64 (440) 30 (33) * Determined at Westinghouse Research Laboratories. crack, and loadings (Paris and Sih, ‘‘Stress Analysis of Cracks,’’ STP- 381, ASTM, 1965). Failure occurs in all cases when Kt reaches KIc. Fracture mechanics also provides a framework for predicting the occur- rence of stress-corrosion cracking by using Eq. (5.1.2) with KIc replaced by KIscc, which is the material parameter denoting resistance to stress- corrosion-crack propagation in a particular medium. Two standard test specimens for KIc determination are specified in ASTM standards, which also detail specimen preparation and test pro- cedure. Recent developments in fracture mechanics permit treatment of crack propagation in the ductile regime. (See ‘‘Elastic-Plastic Frac- ture,’’ ASTM.) FATIGUE Fatigue is generally understood as the gradual deterioration of a mate- rial which is subjected to repeated loads. In fatigue testing, a specimen is subjected to periodically varying constant-amplitude stresses by means of mechanical or magnetic devices. The applied stresses may alternate between equal positive and negative values, from zero to max- imum positive or negative values, or between unequal positive and negative values. The most common loading is alternate tension and compression of equal numerical values obtained by rotating a smooth cylindrical specimen while under a bending load. A series of fatigue tests are made on a number of specimens of the material at different stress levels. The stress endured is then plotted against the number of cycles sustained. By choosing lower and lower stresses, a value may be found which will not produce failure, regardless of the number of ap- plied cycles. This stress value is called the fatigue limit. The diagram is called the stress-cycle diagram or S-N diagram. Instead of recording the data on cartesian coordinates, either stress is plotted vs. the logarithm of the number of cycles (Fig. 5.1.17) or both stress and cycles are plotted to logarithmic scales. Both diagrams show a relatively sharp bend in the curve near the fatigue limit for ferrous metals. The fatigue limit may be established for most steels between 2 and 10 million cycles. Nonferrous metals usually show no clearly defined fatigue limit. The S-N curves in these cases indicate a continuous decrease in stress values to several hundred million cycles, and both the stress value and the number of cycles sustained should be reported. See Table 5.1.5. The mean stress (the average of the maximum and minimum stress values for a cycle) has a pronounced influence on the stress range (the algebraic difference between the maximum and minimum stress values). Several empirical formulas and graphical methods such as the ‘‘modified Goodman diagram’’ have been developed to show the influ- ence of the mean stress on the stress range for failure. A simple but conservative approach (see Soderberg, Working Stresses, Jour. Appl. Mech., 2, Sept. 1935) is to plot the variable stress Sv (one-half the stress range) as ordinate vs. the mean stress Sm as abscissa (Fig. 5.1.18). At zero mean stress, the ordinate is the fatigue limit under completely reversed stress. Yielding will occur if the mean stress exceeds the yield stress So, and this establishes the extreme right-hand point of the dia- gram. A straight line is drawn between these two points. The coordi- nates of any other point along this line are values of Sm and Sv which may produce failure. Surface defects, such as roughness or scratches, and notches or Copyright (C) 1999 by The McGraw-Hill Companies, Inc. All rights reserved. Use of this product is subject to the terms of its License Agreement. Click here to view. FATIGUE 5-9 Fig. 5.1.17. The S-N diagrams from fatigue tests. (1) 1.20% C steel, quenched and drawn at 860°F (460°C); (2) alloy structural steel; (3) SAE 1050, quenched and drawn at 1,200°F (649°C); (4) SAE 4130, normalized and annealed; (5) ordi- nary structural steel; (6) Duralumin; (7) copper, annealed; (8) cast iron (reversed bending). shoulders all reduce the fatigue strength of a part. With a notch of prescribed geometric form and known concentration factor, the reduc- tion in strength is appreciably less than would be called for by the concentration factor itself, but the various metals differ widely in their susceptibility to the effect of roughness and concentrations, or notch sensitivity. For a given material subjected to a prescribed state of stress and type of loading, notch sensitivity can be viewed as the ability of that material to resist the concentration of stress incidental to the presence of a notch. Alternately, notch sensitivity can be taken as a measure of the degree to which the geometric stress concentration factor is reduced. An attempt is made to rationalize notch sensitivity through the equation q � (Kf � 1)/(K � 1), where q is the notch sensitivity, K is the geometric stress concentration factor (from data similar to those in Figs. 5.1.5 to 5.1.13 and the like), and Kf is defined as the ratio of the strength of unnotched material to the strength of notched material. Ratio Kf is obtained from laboratory tests, and K is deduced either theoretically or from laboratory tests, but both must reflect the same state of stress and type of loading. The value of q lies between 0 and 1, so that (1) if q � 0, Kf � 1 and the material is not notch-sensitive (soft metals such as copper, aluminum, and annealed low-strength steel); (2) if q � 1, Kf � K, the material is fully notch-sensitive and the full value of the geometric stress concen- tration factor is not diminished (hard, high-strength steel). In practice, q will lie somewhere between 0 and 1, but it may be hard to quantify. Accordingly, the pragmatic approach to arrive at a solution to a design problem often takes a conservative route and sets q � 1. The exact material properties at play which are responsible for notch sensitivity are not clear. Further, notch sensitivity seems to be higher, and ordinary fatigue strength lower in large specimens, necessitating full-scale tests in many cases (see Peterson, Stress Concentration Phenomena in Fatigue of Fig. 5.1.18. Effect of mean stress on the variable stress for failure. Metals, Trans. ASME, 55, 1933, p. 157, and Buckwalter and Horger, Investigation of Fatigue Strength of Axles, Press Fits, Surface Rolling and Effect of Size, Trans. ASM, 25, Mar. 1937, p. 229). Corrosion and galling (due to rubbing of mating surfaces) cause great reduction of fatigue strengths, sometimes amounting to as much as 90 percent of the original endurance limit. Although any corroding agent will promote severe corrosion fatigue, there is so much difference between the effects of ‘‘sea water’’ or ‘‘tap water’’ from different localities that numerical values are not quoted here. Overstressing specimens above the fatigue limit for periods shorter than necessary to produce failure at that stress reduces the fatigue limit in a subsequent test. Similarly, understressing below the fatigue limit may increase it. Shot peening, nitriding, and cold work usually improve fatigue properties. No very good overall correlation exists between fatigue properties and any other mechanical property of a material. The best correlation is between the fatigue limit under completely reversed bending stress and the ordinary tensile strength. For many ferrous metals, the fatigue limit is approximately 0.40 to 0.60 times the tensile strength if the latter is below 200,000 lb/in2. Low-alloy high-yield-strength steels often show higher values than this. The fatigue limit for nonferrous metals is ap- proximately to 0.20 to 0.50 times the tensile strength. The fatigue limit in reversed shear is approximately 0.57 times that in reversed bending. In some very important engineering situations components are cycli- cally stressed into the plastic range. Examples are thermal strains result- ing from temperature oscillations and notched regions subjected to sec- ondary stresses. Fatigue life in the plastic or ‘‘low-cycle’’ fatigue range has been found to be a function of plastic strain, and low-cycle fatigue testing is done with strain as the controlled variable rather than stress. Fatigue life N and cyclic plastic strain �p tend to follow the relationship N�2p � C where C is a constant for a material when N � 105. (See Coffin, A Study Table 5.1.5 Typical Approximate Fatigue Limits for Reversed Bending Tensile Fatigue Tensile Fatigue strength, limit, strength, limit, Metal 1,000 lb/in2 1,000 lb/in2 Metal 1,000 lb/in2 1,000 lb/in2 Cast iron 20–50 6–18 Copper 32–50 12–17 Malleable iron 50 24 Monel 70–120 20–50 Cast steel 60–80 24–32 Phosphor bronze 55 12 Armco iron 44 24 Tobin bronze, hard 65 21 Plain carbon steels 60–150 25–75 Cast aluminum alloys 18–40 6–11 SAE 6150, heat-treated 200 80 Wrought aluminum alloys 25–70 8–18 Nitralloy 125 80 Magnesium alloys 20–45 7–17 Brasses, various 25–75 7–20 Molybdenum, as cast 98 45 Zirconium crystal bar 52 16–18 Titanium (Ti-75A) 91 45 NOTE: Stress, 1,000 lb/in2 � 6.894 � stress, MN/m2. Copyright (C) 1999 by The McGraw-Hill Companies, Inc. All rights reserved. Use of this product is subject to the terms of its License Agreement. Click here to view. 5-10 MECHANICAL PROPERTIES OF MATERIALS of Cyclic-Thermal Stresses in a Ductile Material, Trans. ASME, 76, 1954, p. 947.) The type of physical change occurring inside a material as it is re- peatedly loaded to failure varies as the life is consumed, and a number of stages in fatigue can be distinguished on this basis. The early stages comprise the events causing nucleation of a crack or flaw. This is most likely to appear on the surface of the material; fatigue failures generally originate at a surface. Following nucleation of the crack, it grows during the crack-propagation stage. Eventually the crack becomes large enough for some rapid terminal mode of failure to take over such as ductile rupture or brittle fracture. The rate of crack growth in the crack- propagation stage can be accurately quantified by fracture mechanics methods. Assuming an initial flaw and a loading situation as shown in Fig. 5.1.15, the rate of crack growth per cycle can generally be ex- pressed as da/dN � C0(�KI)n (5.1.3) where C0 and n are constants for a particular material and �KI is the range of stress intensity per cycle. KI is given by (5.1.1). Using (5.1.3), it is possible to predict the number of cycles for the crack to grow to a size at which some other mode of failure can take over. Values of the constants C0 and n are determined from specimens of the same type as those used for determination of KIc but are instrumented for accurate measurement of slow crack growth. Constant-amplitude fatigue-test data are relevant to many rotary- machinery situations where constant cyclic loads are encountered. There are important situations where the component undergoes vari- able loads and where it may be advisable to use random-load testing. In this method, the load spectrum which the component will experi- ence in service is determined and is applied to the test specimen artificially. CREEP Experience has shown that, for the design of equipment subjected to sustained loading at elevated temperatures, little reliance can be placed on the usual short-time tensile properties of metals at those tempera- tures. Under the application of a constant load it has been found that materials, both metallic and nonmetallic, show a gradual flow or creep even for stresses below the proportional limit at elevated temperatures. Similar effects are present in low-melting metals such as lead at room temperature. The deformation which can be permitted in the satisfactory operation of most high-temperature equipment is limited. In metals, creep is a plastic deformation caused by slip occurring along crystallographic directions in the individual crystals, together with some flow of the grain-boundary material. After complete release of load, a small fraction of this plastic deformation is recovered with time. Most of the flow is nonrecoverable for metals. Since the early creep experiments, many different types of tests have come into use. The most common are the long-time creep test under constant tensile load and the stress-rupture test. Other special forms are the stress-relaxation test and the constant-strain-rate test. The long-time creep test is conducted by applying a dead weight to one end of a lever system, the other end being attached to the specimen surrounded by a furnace and held at constant temperature. The axial deformation is read periodically throughout the test and a curve is plot- ted of the strain �0 as a function of time t (Fig. 5.1.19). This is repeated for various loads at the same testing temperature. The portion of the Fig. 5.1.19. Typical creep curve. curve OA in Fig. 5.1.19 is the region of primary creep, AB the region of secondary creep, and BC that of tertiary creep. The strain rates, or the slopes of the curve, are decreasing, constant, and increasing, respectively, in these three regions. Since the period of the creep test is usually much shorter than the duration of the part in service, various extrapolation procedures are followed (see Gittus, ‘‘Creep, Viscoelasticity and Creep Fracture in Solids,’’ Wiley, 1975). See Table 5.1.6. In practical applications the region of constant-strain rate (secondary creep) is often used to estimate the probable deformation throughout the life of the part. It is thus assumed that this rate will remain constant during periods beyond the range of the test-data. The working stress is chosen so that this total deformation will not be excessive. An arbitrary creep strength, which is defined as the stress which at a given tempera- ture will result in 1 percent deformation in 100,000 h, has received a certain amount of recognition, but it is advisable to determine the proper stress for each individual case from diagrams of stress vs. creep rate (Fig. 5.1.20) (see ‘‘Creep Data,’’ ASTM and ASME). Fig. 5.1.20. Creep rates for 0.35% C steel. Additional temperatures (°F) and stresses (in 1,000 lb/in2) for stated creep rates (percent per 1,000 h) for wrought nonferrous metals are as follows: 60-40 Brass: Rate 0.1, temp. 350 (400), stress 8 (2); rate 0.01, temp 300 (350) [400], stress 10 (3) [1]. Phosphor bronze: Rate 0.1, temp 400 (550) [700] [800], stress 15 (6) [4] [4]; rate 0.01, temp 400 (550) [700], stress 8 (4) [2]. Nickel: Rate 0.1, temp 800 (1000), stress 20 (10). 70 CU, 30 NI. Rate 0.1, temp 600 (750), stress 28 (13–18); rate 0.01, temp 600 (750), stress 14 (8–9). Aluminum alloy 17 S (Duralumin): Rate 0.1, temp 300 (500) [600], stress 22 (5) [1.5]. Lead pure (commercial) (0.03 percent Ca): At 110°F, for rate 0.1 percent the stress range, lb/in2, is 150–180 (60–140) [200–220]; for rate of 0.01 percent, 50–90 (10–50) [110–150]. Stress, 1,000 lb/in2 � 6.894 � stress, MN/m2, tk � 5⁄9(tF � 459.67). Structural changes may occur during a creep test, thus altering the metallurgical condition of the metal. In some cases, premature rupture appears at a low fracture strain in a normally ductile metal, indicating that the material has become embrittled. This is a very insidious condi- tion and difficult to predict. The stress-rupture test is well adapted to study this effect. It is conducted by applying a constant load to the specimen in the same manner as for the long-time creep test. The nomi- nal stress is then plotted vs. the time for fracture at constant temperature on a log-log scale (Fig. 5.1.21). Copyright (C) 1999 by The McGraw-Hill Companies, Inc. All rights reserved. Use of this product is subject to the terms of its License Agreement. Click here to view. CREEP 5-11 Table 5.1.6 Stresses for Given Creep Rates and Temperatures* Creep rate 0.1% per 1,000 h Creep rate 0.01% per 1,000 h Material Temp, °F 800 900 1,000 1,100 1,200 800 900 1,000 1,100 1,200 Wrought steels: SAE 1015 0.20 C, 0.50 Mo 0.10–0.25 C, 4–6 Cr � Mo SAE 4140 SAE 1030–1045 17–27 26–33 22 27–33 8–25 11–18 18–25 15–18 20–25 5–15 3–12 9–16 9–11 7–15 5 2–7 2–6 3–6 4–7 2 1 1–2 2–3 1–2 1 10–18 16–24 14–17 19–28 5–15 6–14 11–22 11–15 12–19 3–7 3–8 4–12 4–7 3–8 2–4 1 2 2–3 2–4 1 1 1–2 1 Commercially pure iron 7 4 3 5 2 0.15 C, 1–2.5 Cr, 0.50 Mo SAE 4340 SAE X3140 0.20 C, 4–6 Cr 0.25 C, 4–6 Cr � W 0.16 C, 1.2 Cu 0.20 C, 1 Mo 0.10–0.40 C, 0.2–0.5 Mo, 1–2 Mn SAE 2340 SAE 6140 SAE 7240 Cr � Va � W, various 25–35 20–40 7–10 30 30 35 30–40 7–12 30 30 20–70 18–28 15–30 10–20 10–15 18 27 12–20 5 12 21 14–30 8–20 2–12 5–4 7–10 4–10 10–15 12 4–14 2 4 6–15 5–15 6–8 1–3 1 2–8 3 2 3–4 1 20–30 8–20 3–8 25 25–28 7 30 18–50 12–18 6–11 10–18 12 8–15 6 11 8–18 3–12 1–6 1–2 3–5 2–7 7–12 6 2–8 1 3–9 2–13 2–5 1 1–2 0.5 Temp, °F 1,100 1,200 1,300 1,400 1,500 1,000 1,100 1,200 1,300 1,400 Wrought chrome-nickel steels: 18-8† 10–25 Cr, 10–30 Ni‡ 10–18 10–20 5–11 5–15 3–10 3–10 2–5 2–5 2.5 11–16 5–12 6–15 2–10 3–10 2–8 1–2 1–3 Temp, °F 800 900 1,000 1,100 1,200 800 900 1,000 1,100 1,200 Cast steels: 0.20–0.40 C 0.10–0.30 C, 0.5–1 Mo 0.15–0.30 C, 4–6 Cr � Mo 18–8§ Cast iron Cr Ni cast iron 10–20 28 25–30 20 5–10 20–30 15–25 8 3 6–12 8–15 20–25 4 9 2 8 15 10 8–15 20 20–25 10 10–15 9–15 1 2–5 2–7 20 2 3 2 15 8 * Based on 1,000-h tests. Stresses in 1,000 lb/in2. † Additional data. At creep rate 0.1 percent and 1,000 (1,600)°F the stress is 18–25 (1); at creep rate 0.01 percent at 1,500°F, the stress is 0.5. ‡ Additional data. At creep rate 0.1 percent and 1,000 (1,600)°F the stress is 10–30 (1). § Additional data. At creep rate 0.1 percent and 1,600°F the stress is 3; at creep rate 0.01 and 1,500°F, the stress is 2–3. The stress reaction is measured in the constant-strain-rate test while the specimen is deformed at a constant strain rate. In the relaxation test, the decrease of stress with time is measured while the total strain (elastic � plastic) is maintained constant. The latter test has direct application to the loosening of turbine bolts and to similar problems. Although some correlation has been indicated between the results of these various types of tests, no general correlation is yet available, and it has been found necessary to make tests under each of these special conditions to obtain satisfactory results. The interrelationship between strain rate and temperature in the form of a velocity-modified temperature (see MacGregor and Fisher, A Ve- locity-modified Temperature for the Plastic F...

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