Strength of Materials

Transcription

1 Section 5 Strengt of Materials BY JOHN SYMONDS Fellow Engineer (Retired), Oceanic Division, Westingouse Electric Corporation. J. P. VIDOSIC Regents Professor Emeritus of Mecanical Engineering, Georgia Institute of Tecnology. HAROLD V. HAWKINS Late Manager, Product Standards and Services, Columbus McKinnon Corporation, Tonawanda, N.Y. DONALD D. DODGE Supervisor (Retired), Product Quality and Inspection Tecnology, Manufacturing Development, Ford Motor Company. 5. MECHANICAL PROPERTIES OF MATERIALS by Jon Symonds, Expanded by Staff Stress-Strain Diagrams Fracture at Low Stresses Fatigue Creep Hardness Testing of Materials MECHANICS OF MATERIALS by J. P. Vidosic Simple Stresses and Strains Combined Stresses Plastic Design Design Stresses Beams Torsion Columns Eccentric Loads Curved Beams Impact Teory of Elasticity Cylinders and Speres Pressure between Bodies wit Curved Surfaces Flat Plates Teories of Failure Plasticity Rotating Disks Experimental Stress Analysis PIPELINE FLEXURE STRESSES by Harold V. Hawkins Pipeline Flexure Stresses NONDESTRUCTIVE TESTING by Donald D. Dodge Nondestructive Testing Magnetic Particle Metods Penetrant Metods Radiograpic Metods Ultrasonic Metods Eddy Current Metods Microwave Metods Infrared Metods Acoustic Signature Analysis

2 5-5. MECHANICAL PROPERTIES OF MATERIALS by Jon Symonds, Expanded by Staff REFERENCES: Davis et al., Testing and Inspection of Engineering Materials, McGraw-Hill, Timosenko, Strengt of Materials, pt. II, Van Nostrand. Ricards, Engineering Materials Science, Wadswort. Nadai, Plasticity, McGraw-Hill. Tetelman and McEvily, Fracture of Structural Materials, Wiley. Fracture Mecanics, ASTM STP-833. McClintock and Argon (eds.), Mecanical Beavior of Materials, Addison-Wesley. Dieter, Mecanical Metallurgy, McGraw-Hill. Creep Data, ASME. ASTM Standards, ASTM. Blaznynski (ed.), Plasticity and Modern Metal Forming Tecnology, Elsevier Science. STRESS-STRAIN DIAGRAMS Te Stress-Strain Curve Te engineering tensile stress-strain curve is obtained by static loading of a standard specimen, tat is, by applying te load slowly enoug tat all parts of te specimen are in equilibrium at any instant. Te curve is usually obtained by controlling te loading rate in te tensile macine. ASTM Standards require a loading rate not exceeding 00,000 lb/in (70 kgf/mm )/min. An alternate metod of obtaining te curve is to specify te strain rate as te independent variable, in wic case te loading rate is continuously adjusted to maintain te required strain rate. A strain rate of 0.05 in/in/(min) is commonly used. It is measured usually by an extensometer attaced to te gage lengt of te specimen. Figure 5.. sows several stress-strain curves. Fig Comparative stress-strain diagrams. () Soft brass; () low carbon steel; (3) ard bronze; (4) cold rolled steel; (5) medium carbon steel, annealed; (6) medium carbon steel, eat treated. For most engineering materials, te curve will ave an initial linear elastic region (Fig. 5..) in wic deformation is reversible and timeindependent. Te slope in tis region is Young s modulus E. Te proportional elastic limit (PEL) is te point were te curve starts to deviate from a straigt line. Te elastic limit (frequently indistinguisable from PEL) is te point on te curve beyond wic plastic deformation is present after release of te load. If te stress is increased furter, te stress-strain curve departs more and more from te straigt line. Unloading te specimen at point X (Fig. 5..), te portion XX is linear and is essentially parallel to te original line OX. Te orizontal distance OX is called te permanent set corresponding to te stress at X. Tis is te basis for te construction of te arbitrary yield strengt. To determine te yield strengt, a straigt line XX is drawn parallel to te initial elastic line OX but displaced from it by an arbitrary value of permanent strain. Te permanent strain commonly used is 0.0 percent of te original gage lengt. Te intersection of tis line wit te curve determines te stress value called te yield strengt. In reporting te yield strengt, te amount of permanent set sould be specified. Te arbitrary yield strengt is used especially for tose materials not exibiting a natural yield point suc as nonferrous metals; but it is not limited to tese. Plastic beavior is somewat time-dependent, particularly at ig temperatures. Also at ig temperatures, a small amount of time-dependent reversible strain may be detectable, indicative of anelastic beavior. Fig General stress-strain diagram. Te ultimate tensile strengt (UTS) is te maximum load sustained by te specimen divided by te original specimen cross-sectional area. Te percent elongation at failure is te plastic extension of te specimen at failure expressed as (te cange in original gage lengt 00) divided by te original gage lengt. Tis extension is te sum of te uniform and nonuniform elongations. Te uniform elongation is tat wic occurs prior to te UTS. It as an unequivocal significance, being associated wit uniaxial stress, wereas te nonuniform elongation wic occurs during localized extension (necking) is associated wit triaxial stress. Te nonuniform elongation will depend on geometry, particularly te ratio of specimen gage lengt L 0 to diameter D or square root of crosssectional area A. ASTM Standards specify test-specimen geometry for a number of specimen sizes. Te ratio L 0 / A is maintained at 4.5 for flatand round-cross-section specimens. Te original gage lengt sould always be stated in reporting elongation values. Te specimen percent reduction in area (RA) is te contraction in cross-sectional area at te fracture expressed as a percentage of te original area. It is obtained by measurement of te cross section of te broken specimen at te fracture location. Te RA along wit te load at fracture can be used to obtain te fracture stress, tat is, fracture load divided by cross-sectional area at te fracture. See Table 5... Te type of fracture in tension gives some indications of te quality of te material, but tis is considerably affected by te testing temperature, speed of testing, te sape and size of te test piece, and oter conditions. Contraction is greatest in toug and ductile materials and least in brittle materials. In general, fractures are eiter of te sear or of te separation (loss of coesion) type. Flat tensile specimens of ductile metals often sow sear failures if te ratio of widt to tickness is greater tan 6 :. A completely sear-type failure may terminate in a cisel edge, for a flat specimen, or a point rupture, for a round specimen. Separation failures occur in brittle materials, suc as certain cast irons. Combinations of bot sear and separation failures are common on round specimens of ductile metal. Failure often starts at te axis in a necked region and produces a relatively flat area wic grows until te material sears along a cone-saped surface at te outside of te speci-

3 Table 5.. Typical Mecanical Properties at Room Temperature (Based on ordinary stress-strain values) Tensile Yield Ultimate strengt, strengt, elongation, Reduction Brinell Metal,000 lb/in,000 lb/in % of area, % no. STRESS-STRAIN DIAGRAMS 5-3 Cast iron Wrougt iron Commercially pure iron, annealed Hot-rolled Cold-rolled Structural steel, ordinary Low-alloy, ig-strengt Steel, SAE 300, annealed Quenced, drawn,300 F Drawn,000 F Drawn 700 F Drawn 400 F Steel, SAE 4340, annealed Quenced, drawn,300 F Drawn,000 F Drawn 700 F Drawn 400 F Cold-rolled steel, SAE Stainless steel, 8-S Steel castings, eat-treated Aluminum, pure, rolled Aluminum-copper alloys, cast Wrougt, eat-treated Aluminum die castings 30 Aluminum alloy 7ST Aluminum alloy 5ST Copper, annealed Copper, ard-drawn Brasses, various Pospor bronze Tobin bronze, rolled Magnesium alloys, various Monel 400, Ni-Cu alloy Molybdenum, rolled Silver, cast, annealed Titanium 6 4 alloy, annealed Ductile iron, grade NOTE: Compressive strengt of cast iron, 80,000 to 50,000 lb/in. Compressive yield strengt of all metals, except tose cold-worked tensile yield strengt. Stress,000 lb/in stress, MN/m. men, resulting in wat is known as te cup-and-cone fracture. Double cup-and-cone and rosette fractures sometimes occur. Several types of tensile fractures are sown in Fig Annealed or ot-rolled mild steels generally exibit a yield point (see Fig. 5..4). Here, in a constant strain-rate test, a large increment of extension occurs under constant load at te elastic limit or at a stress just below te elastic limit. In te latter event te stress drops suddenly from te upper yield point to te lower yield point. Subsequent to te drop, te yield-point extension occurs at constant stress, followed by a rise to te UTS. Plastic flow during te yield-point extension is discontinuous; to test temperature, test strain rate, and te caracteristics of te tensile macine employed. Te plastic beavior in a uniaxial tensile test can be represented as te true stress-strain curve. Te true stress is based on te instantaneous Fig Typical metal fractures in tension. successive zones of plastic deformation, known as Luder s bands or stretcer strains, appear until te entire specimen gage lengt as been uniformly deformed at te end of te yield-point extension. Tis beavior causes a banded or stepped appearance on te metal surface. Te exact form of te stress-strain curve for tis class of material is sensitive Fig Yielding of annealed steel.

4 5-4 MECHANICAL PROPERTIES OF MATERIALS cross section A, so tat load/a. Te instantaneous true strain increment is da/a, ordl/l prior to necking. Total true strain is A da A ln A 0 A0 A or ln (L/L 0 ) prior to necking. Te true stress-strain curve or flow curve obtained as te typical form sown in Fig In te part of te test subsequent to te maximum load point (UTS), wen necking occurs, te true strain of interest is tat wic occurs in an infinitesimal lengt at te region of minimum cross section. True strain for tis element can still be expressed as ln (A 0 /A), were A refers to te minimum cross section. Metods of constructing te true stress-strain curve are described in te tecnical literature. In te range between initial yielding and te neigborood of te maximum load point te relationsip between plastic strain p and true stress often approximates k p n were k is te strengt coefficient and n is te work-ardening exponent. For a material wic sows a yield point te relationsip applies only to te rising part of te curve beyond te lower yield. It can be sown tat at te maximum load point te slope of te true stress-strain curve equals te true stress, from wic it can be deduced tat for a material obeying te above exponential relationsip between p and n, p n at te maximum load point. Te exponent strongly influences te spread between YS and UTS on te engineering stress-strain curve. Values of n and k for some materials are sown in Table 5... A point on te flow curve indentifies te flow stress corresponding to a certain strain, tat is, te stress required to bring about tis amount of plastic deformation. Te concept of true strain is useful for accurately describing large amounts of plastic deformation. Te linear strain definition (L L 0 )/L 0 fails to correct for te continuously canging gage lengt, wic leads to an increasing error as deformation proceeds. During extension of a specimen under tension, te cange in te specimen cross-sectional area is related to te elongation by Poisson s ratio, wic is te ratio of strain in a transverse direction to tat in te longitudinal direction. Values of for te elastic region are sown in Table For plastic strain it is approximately 0.5. Fig True stress-strain curve for 0 C annealed mild steel. Table 5.. Room-Temperature Plastic-Flow Constants for a Number of Metals k,,000 in Material Condition (MN/m ) n 0.40% C steel Quenced and tempered at 46 (,860) F (478K) 0.05% C steel Annealed and temper-rolled 7 (49.6) aluminum Precipitation-ardened 00 (689) aluminum Annealed 49 (338) 0. Copper Annealed 46.4 (39) brass Annealed 30 (895) 0.49 SOURCE: Reproduced by permission from Properties of Metals in Materials Engineering, ASM, 949. Table 5..3 Elastic Constants of Metals (Mostly from tests of R. W. Vose) E G K Modulus of Modulus of elasticity rigidity (Young s (searing Bulk modulus). modulus). modulus.,000,000,000,000,000,000 Poisson s Metal lb/in lb/in lb/in ratio Cast steel Cold-rolled steel Stainless steel All oter steels, including ig-carbon, eat-treated Cast iron Malleable iron Copper Brass, Cast brass Tobin bronze Pospor bronze Aluminum alloys, various Monel metal Inconel Z-nickel Beryllium copper Elektron (magnesium alloy) Titanium (99.0 Ti), annealed bar Zirconium, crystal bar 4 Molybdenum, arc-cast 48 5

5 STRESS-STRAIN DIAGRAMS 5-5 Te general effect of increased strain rate is to increase te resistance to plastic deformation and tus to raise te flow curve. Decreasing test temperature also raises te flow curve. Te effect of strain rate is expressed as strain-rate sensitivity m. Its value can be measured in te tension test if te strain rate is suddenly increased by a small increment during te plastic extension. Te flow stress will ten jump to a iger value. Te strain-rate sensitivity is te ratio of incremental canges of log and log m log log For most engineering materials at room temperature te strain rate sensitivity is of te order of 0.0. Te effect becomes more significant at elevated temperatures, wit values ranging to 0. and sometimes iger. Compression Testing Te compressive stress-strain curve is similar to te tensile stress-strain curve up to te yield strengt. Tereafter, te progressively increasing specimen cross section causes te compressive stress-strain curve to diverge from te tensile curve. Some ductile metals will not fail in te compression test. Complex beavior occurs wen te direction of stressing is canged, because of te Bauscinger effect, wic 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 Tis refers to te situation in wic stresses are present on eac of te faces of a cubic element of te material. For a given cube orientation te applied stresses may include sear stresses over te cube faces as well as stresses normal to tem. By a suitable rotation of axes te problem can be simplified: applied stresses on te new cubic element are equivalent to tree mutually ortogonal principal stresses,, 3 alone, eac acting normal to a cube face. Combined stress beavior in te elastic range is described in Sec. 5., Mecanics of Materials. Prediction of te conditions under wic plastic yielding will occur under combined stresses can be made wit te elp of several empirical teories. In te maximum-sear-stress teory te criterion for yielding is tat yielding will occur wen 3 ys in wic and 3 are te largest and smallest principal stresses, respectively, and ys is te uniaxial tensile yield strengt. Tis is te simplest teory for predicting yielding under combined stresses. A more accurate prediction can be made by te distortion-energy teory, according to wic te criterion is ( ) ( 3 ) ( ) ( ys ) Stress-strain curves in te plastic region for combined stress loading can be constructed. However, a particular stress state does not determine a unique strain value. Te latter will depend on te stress-state pat wic is followed. Plane strain is a condition were strain is confined to two dimensions. Tere is generally stress in te tird direction, but because of mecanical constraints, strain in tis dimension is prevented. Plane strain occurs in certain metalworking operations. It can also occur in te neigborood of a crack tip in a tensile loaded member if te member is sufficiently tick. Te material at te crack tip is ten in triaxial tension, wic condition promotes brittle fracture. On te oter and, ductility is enanced and fracture is suppressed by triaxial compression. Stress Concentration In a structure or macine part aving a notc or any abrupt cange in cross section, te maximum stress will occur at tis location and will be greater tan te stress calculated by elementary formulas based upon simplified assumptions as to te stress distribution. Te ratio of tis maximum stress to te nominal stress (calculated by te elementary formulas) is te stress-concentration factor K t. Tis is a constant for te particular geometry and is independent of te material, provided it is isotropic. Te stress-concentration factor may be determined experimentally or, in some cases, teoretically from te matematical teory of elasticity. Te factors sown in Figs to 5..3 were determined from bot potoelastic tests and te teory of elasticity. Stress concentration will cause failure of brittle materials if Tension or compression I II III Stress concentration factor, K D D D r r r r d / r d / d d IV Bending Note; in all cases Ddr r d Fig Flat plate wit semicircular fillets and grooves or wit oles. I, II, and III are in tension or compression; IV and V are in bending. Stress concentration factor, K Semi-circle grooves (r) D Blunt grooves d Sarpness of groove, r Fig Flat plate wit grooves, in tension. IV r r V II V I D D III d 0. Sarp grooves r r r r d d

6 5-6 MECHANICAL PROPERTIES OF MATERIALS te concentrated stress is larger tan te ultimate strengt of te material. In ductile materials, concentrated stresses iger tan te yield strengt will generally cause local plastic deformation and redistribution of stresses (rendering tem more uniform). On te oter and, even wit ductile materials areas of stress concentration are possible sites for fatigue if te component is cyclically loaded. Stress concentration factor, K Dd Full fillets (r) Blunt fillets D r r d Dept of fillet 0.0 d Sarp fillets Stress concentration factor, K Dd Full fillets (r) Blunt fillets D Sarpness of fillet, r Fig Flat plate wit fillets, in bending. r r d d 0.0 Sarp fillets Sarpness of fillet, r Fig Flat plate wit fillets, in tension Fig Flat plate wit angular notc, in tension or bending. D r r d r d D Stress concentration factor, K Dd Semi-circle grooves (r) d Sarp grooves Stress concentration factor, K Semi circ. grooves r Blunt grooves d d Sarp grooves Sarpness of groove, r Fig Flat plate wit grooves, in bending Sarpness of groove, r Fig Grooved saft in torsion.

7 FRACTURE AT LOW STRESSES 5-7 Stress concentration factor, K Fig Dd Full fillets (r) Blunt fillets Sarp fillets Filleted saft in torsion. D r d d Sarpness of fillet, r FRACTURE AT LOW STRESSES Materials under tension sometimes fail by rapid fracture at stresses muc below teir strengt level as determined in tests on carefully prepared specimens. Tese brittle, unstable, or catastropic failures originate at preexisting stress-concentrating flaws wic may be inerent in a material. Te transition-temperature approac is often used to ensure fracturesafe design in structural-grade steels. Tese materials exibit a caracteristic temperature, known as te ductile brittle transition (DBT) temperature, below wic tey are susceptible to brittle fracture. Te transition-temperature approac to fracture-safe design ensures tat te transition temperature of a material selected for a particular application is suitably matced to its intended use temperature. Te DBT can be detected by plotting certain measurements from tensile or impact tests against temperature. Usually te transition to brittle beavior is complex, being neiter fully ductile nor fully brittle. Te range may extend over 00 F (0 K) interval. Te nil-ductility temperature (NDT), determined by te drop weigt test (see ASTM Standards), is an important reference point in te transition range. Wen NDT for a particular steel is known, temperature-stress combinations can be specified wic define te limiting conditions under wic catastropic fracture can occur. In te Carpy V-notc (CVN) impact test, a notced-bar specimen (Fig. 5..6) is used wic is loaded in bending (see ASTM Standards). Te energy absorbed from a swinging pendulum in fracturing te specimen is measured. Te pendulum strikes te specimen at 6 to 9 ft (4.88 to 5.80 m)/s so tat te specimen deformation associated wit fracture occurs at a rapid strain rate. Tis ensures a conservative measure of tougness, since in some materials, tougness is reduced by ig strain rates. A CVN impact energy vs. temperature curve is sown in Fig. 5..4, wic also sows te transitions as given by percent brittle fracture and by percent lateral expansion. Te CVN energy as no analytical significance. Te test is useful mainly as a guide to te fracture beavior of a material for wic an empirical correlation as been establised between impact energy and some rigorous fracture criterion. For a particular grade of steel te CVN curve can be correlated wit NDT. (See ASME Boiler and Pressure Vessel Code.) Fracture Mecanics Tis analytical metod is used for ultra-igstrengt alloys, transition-temperature materials below te DBT temperature, and some low-strengt materials in eavy section tickness. Fracture mecanics teory deals wit crack extension were plastic effects are negligible or confined to a small region around te crack tip. Te present discussion is concerned wit a troug-tickness crack in a tension-loaded plate (Fig. 5..5) wic is large enoug so tat te crack-tip stress field is not affected by te plate edges. Fracture mecanics teory states tat unstable crack extension occurs wen te work required for an increment of crack extension, namely, surface energy and energy consumed in local plastic deformation, is exceeded by te elastic-strain energy released at te crack tip. Te elastic-stress Fig CVN transition curves. (Data from Westingouse R & D Lab.)

8 5-8 MECHANICAL PROPERTIES OF MATERIALS field surrounding one of te crack tips in Fig is caracterized by te stress intensity K I, wic as units of (lb in) /in or (N m) /m.itis a function of applied nominal stress, crack alf-lengt a, and a geometry factor Q: K l Q a (5..) for te situation of Fig For a particular material it is found tat as K I is increased, a value K c is reaced at wic unstable crack propa- Table 5..4 Materials* Room-Temperature K lc Values on Hig-Strengt 0.% YS,,000 in K lc,,000 in Material (MN/m ) in (MN m / /m ) 8% Ni maraging steel 300 (,060) 46 (50.7) 8% Ni maraging steel 70 (,850) 7 (78) 8% Ni maraging steel 98 (,360) 87 (96) Titanium 6-4 alloy 5 (,0) 39 (43) Titanium 6-4 alloy 40 (960) 75 (8.5) Aluminum alloy 7075-T6 75 (56) 6 (8.6) Aluminum alloy 7075-T6 64 (440) 30 (33) Fig Troug-tickness crack geometry. * Determined at Westingouse Researc Laboratories. crack, and loadings (Paris and Si, Stress Analysis of Cracks, STP- 38, ASTM, 965). Failure occurs in all cases wen K t reaces K Ic. Fracture mecanics also provides a framework for predicting te occurrence of stress-corrosion cracking by using Eq. (5..) wit K Ic replaced by K Iscc, wic is te material parameter denoting resistance to stresscorrosion-crack propagation in a particular medium. Two standard test specimens for K Ic determination are specified in ASTM standards, wic also detail specimen preparation and test procedure. Recent developments in fracture mecanics permit treatment of crack propagation in te ductile regime. (See Elastic-Plastic Fracture, ASTM.) gation occurs. K c depends on plate tickness B, as sown in Fig It attains a constant value wen B is great enoug to provide plane-strain conditions at te crack tip. Te low plateau value of K c is an important material property known as te plane-strain critical stress intensity or fracture tougness K Ic. Values for a number of materials are sown in Table Tey are influenced strongly by processing and small canges in composition, so tat te values sown are not necessarily typical. K Ic can be used in te critical form of Eq. (5..): (K Ic ) Q a cr (5..) to predict failure stress wen a maximum flaw size in te material is known or to determine maximum allowable flaw size wen te stress is set. Te predictions will be accurate so long as plate tickness B satisfies te plane-strain criterion: B.5(K Ic / ys ). Tey will be conservative if a plane-strain condition does not exist. A big advantage of te fracture mecanics approac is tat stress intensity can be calculated by equations analogous to (5..) for a wide variety of geometries, types of Fig Dependence of K c and fracture appearance (in terms of percentage of square fracture) on tickness of plate specimens. Based on data for aluminum 7075-T6. (From Scrawly and Brown, STP-38, ASTM.) FATIGUE Fatigue is generally understood as te gradual deterioration of a material wic is subjected to repeated loads. In fatigue testing, a specimen is subjected to periodically varying constant-amplitude stresses by means of mecanical or magnetic devices. Te applied stresses may alternate between equal positive and negative values, from zero to maximum positive or negative values, or between unequal positive and negative values. Te most common loading is alternate tension and compression of equal numerical values obtained by rotating a smoot cylindrical specimen wile under a bending load. A series of fatigue tests are made on a number of specimens of te material at different stress levels. Te stress endured is ten plotted against te number of cycles sustained. By coosing lower and lower stresses, a value may be found wic will not produce failure, regardless of te number of applied cycles. Tis stress value is called te fatigue limit. Te diagram is called te stress-cycle diagram or S-N diagram. Instead of recording te data on cartesian coordinates, eiter stress is plotted vs. te logaritm of te number of cycles (Fig. 5..7) or bot stress and cycles are plotted to logaritmic scales. Bot diagrams sow a relatively sarp bend in te curve near te fatigue limit for ferrous metals. Te fatigue limit may be establised for most steels between and 0 million cycles. Nonferrous metals usually sow no clearly defined fatigue limit. Te S-N curves in tese cases indicate a continuous decrease in stress values to several undred million cycles, and bot te stress value and te number of cycles sustained sould be reported. See Table Te mean stress (te average of te maximum and minimum stress values for a cycle) as a pronounced influence on te stress range (te algebraic difference between te maximum and minimum stress values). Several empirical formulas and grapical metods suc as te modified Goodman diagram ave been developed to sow te influence of te mean stress on te stress range for failure. A simple but conservative approac (see Soderberg, Working Stresses, Jour. Appl. Mec.,, Sept. 935) is to plot te variable stress S v (one-alf te stress range) as ordinate vs. te mean stress S m as abscissa (Fig. 5..8). At zero mean stress, te ordinate is te fatigue limit under completely reversed stress. Yielding will occur if te mean stress exceeds te yield stress S o, and tis establises te extreme rigt-and point of te diagram. A straigt line is drawn between tese two points. Te coordinates of any oter point along tis line are values of S m and S v wic may produce failure. Surface defects, suc as rougness or scratces, and notces or

9 FATIGUE 5-9 Accordingly, te pragmatic approac to arrive at a solution to a design problem often takes a conservative route and sets q. Te exact material properties at play wic are responsible for notc sensitivity are not clear. Furter, notc sensitivity seems to be iger, and ordinary fatigue strengt lower in large specimens, necessitating full-scale tests in many cases (see Peterson, Stress Concentration Penomena in Fatigue of Fig Effect of mean stress on te variable stress for failure. Fig Te S-N diagrams from fatigue tests. ().0% C steel, quenced and drawn at 860 F (460 C); () alloy structural steel; (3) SAE 050, quenced and drawn at,00 F (649 C); (4) SAE 430, normalized and annealed; (5) ordinary structural steel; (6) Duralumin; (7) copper, annealed; (8) cast iron (reversed bending). soulders all reduce te fatigue strengt of a part. Wit a notc of prescribed geometric form and known concentration factor, te reduction in strengt is appreciably less tan would be called for by te concentration factor itself, but te various metals differ widely in teir susceptibility to te effect of rougness and concentrations, or notc sensitivity. For a given material subjected to a prescribed state of stress and type of loading, notc sensitivity can be viewed as te ability of tat material to resist te concentration of stress incidental to te presence of a notc. Alternately, notc sensitivity can be taken as a measure of te degree to wic te geometric stress concentration factor is reduced. An attempt is made to rationalize notc sensitivity troug te equation q (K f )/(K ), were q is te notc sensitivity, K is te geometric stress concentration factor (from data similar to tose in Figs to 5..3 and te like), and K f is defined as te ratio of te strengt of unnotced material to te strengt of notced material. Ratio K f is obtained from laboratory tests, and K is deduced eiter teoretically or from laboratory tests, but bot must reflect te same state of stress and type of loading. Te value of q lies between 0 and, so tat () if q 0, K f and te material is not notc-sensitive (soft metals suc as copper, aluminum, and annealed low-strengt steel); () if q, K f K, te material is fully notc-sensitive and te full value of te geometric stress concentration factor is not diminised (ard, ig-strengt steel). In practice, q will lie somewere between 0 and, but it may be ard to quantify. Metals, Trans. ASME, 55, 933, p. 57, and Buckwalter and Horger, Investigation of Fatigue Strengt of Axles, Press Fits, Surface Rolling and Effect of Size, Trans. ASM, 5, Mar. 937, p. 9). Corrosion and galling (due to rubbing of mating surfaces) cause great reduction of fatigue strengts, sometimes amounting to as muc as 90 percent of te original endurance limit. Altoug any corroding agent will promote severe corrosion fatigue, tere is so muc difference between te effects of sea water or tap water from different localities tat numerical values are not quoted ere. Overstressing specimens above te fatigue limit for periods sorter tan necessary to produce failure at tat stress reduces te fatigue limit in a subsequent test. Similarly, understressing below te fatigue limit may increase it. Sot peening, nitriding, and cold work usually improve fatigue properties. No very good overall correlation exists between fatigue properties and any oter mecanical property of a material. Te best correlation is between te fatigue limit under completely reversed bending stress and te ordinary tensile strengt. For many ferrous metals, te fatigue limit is approximately 0.40 to 0.60 times te tensile strengt if te latter is below 00,000 lb/in. Low-alloy ig-yield-strengt steels often sow iger values tan tis. Te fatigue limit for nonferrous metals is approximately to 0.0 to 0.50 times te tensile strengt. Te fatigue limit in reversed sear is approximately 0.57 times tat in reversed bending. In some very important engineering situations components are cyclically stressed into te plastic range. Examples are termal strains resulting from temperature oscillations and notced regions subjected to secondary stresses. Fatigue life in te plastic or low-cycle fatigue range as been found to be a function of plastic strain, and low-cycle fatigue testing is done wit strain as te controlled variable rater tan stress. Fatigue life N and cyclic plastic strain p tend to follow te relationsip N p C were C is a constant for a material wen N 0 5. (See Coffin, A Study Table 5..5 Typical Approximate Fatigue Limits for Reversed Bending Tensile Fatigue Tensile Fatigue strengt, limit, strengt, limit, Metal,000 lb/in,000 lb/in Metal,000 lb/in,000 lb/in Cast iron Copper Malleable iron 50 4 Monel Cast steel Pospor bronze 55 Armco iron 44 4 Tobin bronze, ard 65 Plain carbon steels Cast aluminum alloys SAE 650, eat-treated Wrougt aluminum alloys Nitralloy 5 80 Magnesium alloys Brasses, various Molybdenum, as cast Zirconium crystal bar Titanium (Ti-75A) 9 45 NOTE: Stress,,000 lb/in stress, MN/m.

10 5-0 MECHANICAL PROPERTIES OF MATERIALS of Cyclic-Termal Stresses in a Ductile Material, Trans. ASME, 76, 954, p. 947.) Te type of pysical cange occurring inside a material as it is repeatedly loaded to failure varies as te life is consumed, and a number of stages in fatigue can be distinguised on tis basis. Te early stages comprise te events causing nucleation of a crack or flaw. Tis is most likely to appear on te surface of te material; fatigue failures generally originate at a surface. Following nucleation of te crack, it grows during te crack-propagation stage. Eventually te crack becomes large enoug for some rapid terminal mode of failure to take over suc as ductile rupture or brittle fracture. Te rate of crack growt in te crackpropagation stage can be accurately quantified by fracture mecanics metods. Assuming an initial flaw and a loading situation as sown in Fig. 5..5, te rate of crack growt per cycle can generally be expressed as da/dn C 0 (K I ) n (5..3) were C 0 and n are constants for a particular material and K I is te range of stress intensity per cycle. K I is given by (5..). Using (5..3), it is possible to predict te number of cycles for te crack to grow to a size at wic some oter mode of failure can take over. Values of te constants C 0 and n are determined from specimens of te same type as tose used for determination of K Ic but are instrumented for accurate measurement of slow crack growt. Constant-amplitude fatigue-test data are relevant to many rotarymacinery situations were constant cyclic loads are encountered. Tere are important situations were te component undergoes variable loads and were it may be advisable to use random-load testing. In tis metod, te load spectrum wic te component will experience in service is determined and is applied to te test specimen artificially. curve OA in Fig is te region of primary creep, AB te region of secondary creep, and BC tat of tertiary creep. Te strain rates, or te slopes of te curve, are decreasing, constant, and increasing, respectively, in tese tree regions. Since te period of te creep test is usually muc sorter tan te duration of te part in service, various extrapolation procedures are followed (see Gittus, Creep, Viscoelasticity and Creep Fracture in Solids, Wiley, 975). See Table In practical applications te region of constant-strain rate (secondary creep) is often used to estimate te probable deformation trougout te life of te part. It is tus assumed tat tis rate will remain constant during periods beyond te range of te test-data. Te working stress is cosen so tat tis total deformation will not be excessive. An arbitrary creep strengt, wic is defined as te stress wic at a given temperature will result in percent deformation in 00,000, as received a certain amount of recognition, but it is advisable to determine te proper stress for eac individual case from diagrams of stress vs. creep rate (Fig. 5..0) (see Creep Data, ASTM and ASME). CREEP Experience as sown tat, for te design of equipment subjected to sustained loading at elevated temperatures, little reliance can be placed on te usual sort-time tensile properties of metals at tose temperatures. Under te application of a constant load it as been found tat materials, bot metallic and nonmetallic, sow a gradual flow or creep even for stresses below te proportional limit at elevated temperatures. Similar effects are present in low-melting metals suc as lead at room temperature. Te deformation wic can be permitted in te satisfactory operation of most ig-temperature equipment is limited. In metals, creep is a plastic deformation caused by slip occurring along crystallograpic directions in te individual crystals, togeter wit some flow of te grain-boundary material. After complete release of load, a small fraction of tis plastic deformation is recovered wit time. Most of te flow is nonrecoverable for metals. Since te early creep experiments, many different types of tests ave come into use. Te most common are te long-time creep test under constant tensile load and te stress-rupture test. Oter special forms are te stress-relaxation test and te constant-strain-rate test. Te long-time creep test is conducted by applying a dead weigt to one end of a lever system, te oter end being attaced to te specimen surrounded by a furnace and eld at constant temperature. Te axial deformation is read periodically trougout te test and a curve is plotted of te strain 0 as a function of time t (Fig. 5..9). Tis is repeated for various loads at te same testing temperature. Te portion of te Fig Typical creep curve. Fig Creep rates for 0.35% C steel. Additional temperatures ( F) and stresses (in,000 lb/in ) for stated creep rates (percent per,000 ) for wrougt nonferrous metals are as follows: Brass: Rate 0., temp. 350 (400), stress 8 (); rate 0.0, temp 300 (350) [400], stress 0 (3) []. Pospor bronze: Rate 0., temp 400 (550) [700] [800], stress 5 (6) [4] [4]; rate 0.0, temp 400 (550) [700], stress 8 (4) []. Nickel: Rate 0., temp 800 (000), stress 0 (0). 70 CU,30NI. Rate 0., temp 600 (750), stress 8 (3 8); rate 0.0, temp 600 (750), stress 4 (8 9). Aluminum alloy 7 S (Duralumin): Rate 0., temp 300 (500) [600], stress (5) [.5]. Lead pure (commercial) (0.03 percent Ca): At 0 F, for rate 0. percent te stress range, lb/in, is (60 40) [00 0]; for rate of 0.0 percent, (0 50) [0 50]. Stress,,000 lb/in stress, MN/m, t k 5 9(t F ). Structural canges may occur during a creep test, tus altering te metallurgical condition of te metal. In some cases, premature rupture appears at a low fracture strain in a normally ductile metal, indicating tat te material as become embrittled. Tis is a very insidious condition and difficult to predict. Te stress-rupture test is well adapted to study tis effect. It is conducted by applying a constant load to te specimen in te same manner as for te long-time creep test. Te nominal stress is ten plotted vs. te time for fracture at constant temperature on a log-log scale (Fig. 5..).

11 CREEP 5- Table 5..6 Stresses for Given Creep Rates and Temperatures* Creep rate 0.% per,000 Creep rate 0.0% per,000 Material Temp, F ,000,00, ,000,00,00 Wrougt steels: SAE C, 0.50 Mo C, 4 6 Cr Mo SAE 440 SAE Commercially pure iron C,.5 Cr, 0.50 Mo SAE 4340 SAE X C, 4 6 Cr 0.5 C, 4 6 Cr W 0.6 C,. Cu 0.0 C, Mo C, Mo, Mn SAE 340 SAE 640 SAE 740 Cr Va W, various Temp, F,00,00,300,400,500,000,00,00,300,400 Wrougt crome-nickel steels: Cr, 0 30 Ni Temp, F ,000,00, ,000,00,00 Cast steels: C C, 0.5 Mo C, 4 6 Cr Mo 8 8 Cast iron Cr Ni cast iron * Based on,000- tests. Stresses in,000 lb/in. Additional data. At creep rate 0. percent and,000 (,600) F te stress is 8 5 (); at creep rate 0.0 percent at,500 F, te stress is 0.5. Additional data. At creep rate 0. percent and,000 (,600) F te stress is 0 30 (). Additional data. At creep rate 0. percent and,600 F te stress is 3; at creep rate 0.0 and,500 F, te stress is Te stress reaction is measured in te constant-strain-rate test wile te specimen is deformed at a constant strain rate. In te relaxation test, te decrease of stress wit time is measured wile te total strain (elastic plastic) is maintained constant. Te latter test as direct application to te loosening of turbine bolts and to similar problems. Altoug some correlation as been indicated between te results of tese various types of tests, no general correlation is yet available, and it as been found necessary to make tests under eac of tese special conditions to obtain satisfactory results. Te interrelationsip between strain rate and temperature in te form of a velocity-modified temperature (see MacGregor and Fiser, A Velocity-modified Temperature for te Plastic Flow of Metals, Jour. Appl. Mec., Mar. 945) simplifies te creep problem in reducing te number of variables. Superplasticity Superplasticity is te property of some metals and alloys wic permits extremely large, uniform deformation at elevated temperature, in contrast to conventional metals wic neck down and subsequently fracture after relatively small amounts of plastic deformation. Superplastic beavior requires a metal wit small equiaxed grains, a slow and steady rate of deformation (strain Fig. 5.. Relation between time to failure and stress for a 3% cromium steel. () Heat treated at,740 F (950 C) and furnace cooled; () ot rolled and annealed,580 F (860 C).

12 5- MECHANICAL PROPERTIES OF MATERIALS rate), and a temperature elevated to somewat more tan alf te melting point. Wit suc metals, large plastic deformation can be brougt about wit lower external loads; ultimately, tat allows te use of ligter fabricating equipment and facilitates production of finised parts to near-net sape. In m m tan In (a) m (b) In Fig Stress and strain rate relations for superplastic alloys. (a) Log-log plot of K m ;(b) m as a function of strain rate. Stress and strain rates are related for a metal exibiting superplasticity. A factor in tis beavior stems from te relationsip between te applied stress and strain rates. Tis factor m te strain rate sensitivity index is evaluated from te equation K m, were is te applied stress, K is a constant, and is te strain rate. Figure 5..a plots a stress/strain rate curve for a superplastic alloy on log-log coordinates. Te slope of te curve defines m, wic is maximum at te point of inflection. Figure 5..b sows te variation of m versus ln. Ordinary metals exibit low values of m 0. or less; for tose beaving superplastically, m 0.6 to 0.8. Asm approaces, te beavior of te metal will be quite similar to tat of a newtonian viscous solid, wic elongates plastically witout necking down. In Fig. 5..a, in region I, te stress and strain rates are low and creep is predominantly a result of diffusion. In region III, te stress and strain rates are igest and creep is mainly te result of dislocation and slip mecanisms. In region II, were superplasticity is observed, creep is governed predominantly by grain boundary sliding. HARDNESS Hardness as been variously defined as resistance to local penetration, to scratcing, to macining, to wear or abrasion, and to yielding. Te multiplicity of definitions, and corresponding multiplicity of ardnessmeasuring instruments, togeter wit te lack of a fundamental definition, indicates tat ardness may not be a fundamental property of a material but rater a composite one including yield strengt, work ardening, true tensile strengt, modulus of elasticity, and oters. Scratc ardness is measured by Mos scale of minerals (Sec..) wic is so arranged tat eac mineral will scratc te mineral of te next lower number. In recent mineralogical work and in certain microscopic metallurgical work, jeweled scratcing points eiter wit a set load or else loaded to give a set widt of scratc ave been used. Hardness in its relation to macinability and to wear and abrasion is generally dealt wit in direct macining or wear tests, and little attempt is made to separate ardness itself, as a numerically expressed quantity, from te results of suc tests. Te resistance to localized penetration, or indentation ardness, is widely used industrially as a measure of ardness, and indirectly as an indicator of oter desired properties in a manufactured product. Te indentation tests described below are essentially nondestructive, and in most applications may be considered nonmarring, so tat tey may be applied to eac piece produced; and troug te empirical relationsips of ardness to suc properties as tensile strengt, fatigue strengt, and impact strengt, pieces likely to be deficient in te latter properties may be detected and rejected. Brinell ardness is determined by forcing a ardened spere under a m known load into te surface of a material and measuring te diameter of te indentation left after te test. Te Brinell ardness number, or simply te Brinell number, is obtained by dividing te load used, in kilograms, by te actual surface area of te indentation, in square millimeters. Te result is a pressure, but te units are rarely stated. BHN P D (D D d ) were BHN is te Brinell ardness number; P te imposed load, kg; D te diameter of te sperical indenter, mm; and d te diameter of te resulting impression, mm. Hardened-steel bearing balls may be used for ardness up to 450, but beyond tis ardness specially treated steel balls or jewels sould be used to avoid flattening te indenter. Te standard-size ball is 0 mm and te standard loads 3,000,,500, and 500 kg, wit 00, 5, and 50 kg sometimes used for softer materials. If for special reasons any oter size of ball is used, te load sould be adjusted approximately as follows: for iron and steel, P 30D ; for brass, bronze, and oter soft metals, P 5D ; for extremely soft metals, P D (see Metods of Brinell Hardness Testing, ASTM). Readings obtained wit oter tan te standard ball and loadings sould ave te load and ball size appended, as suc readings are only approximately equal to tose obtained under standard conditions. Te size of te specimen sould be sufficient to ensure tat no part of te plastic flow around te impression reaces a free surface, and in no case sould te tickness be less tan 0 times te dept of te impression. Te load sould be applied steadily and sould remain on for at least 5 s in te case of ferrous materials and 30 s in te case of most nonferrous materials. Longer periods may be necessary on certain soft materials tat exibit creep at room temperature. In testing tin materials, it is not permissible to pile up several ticknesses of material under te indenter, as te readings so obtained will invariably be lower tan te true readings. Wit suc materials, smaller indenters and loads, or different metods of ardness testing, are necessary. In te standard Brinell test, te diameter of te impression is measured wit a low-power and microscope, but for production work several testing macines are available wic automatically measure te dept of te impression and from tis give readings of ardness. Suc macines sould be calibrated frequently on test blocks of known ardness. In te Rockwell metod of ardness testing, te dept of penetration of an indenter under certain arbitrary conditions of test is determined. Te indenter may be eiter a steel ball of some specified diameter or a sperical-tipped conical diamond of 0 angle and 0.-mm tip radius, called a Brale. A minor load of 0 kg is first applied wic causes an initial penetration and olds te indenter in place. Under tis condition, te dial is set to zero and te major load applied. Te values of te latter are 60, 00, or 50 kg. Upon removal of te major load, te reading is taken wile te minor load is still on. Te ardness number may ten be read directly from te scale wic measures penetration, and tis scale is so arranged tat soft materials wit deep penetration give low ardness numbers. A variety of combinations of indenter and major load are possible; te most commonly used are R B using as indenter a 6-in ball and a major load of 00 kg and R C using a Brale as indenter and a major load of 50 kg (see Rockwell Hardness and Rockwell Superficial Hardness of Metallic Materials, ASTM). Compared wit te Brinell test, te Rockwell metod makes a smaller indentation, may be used on tinner material, and is more rapid, since ardness numbers are read directly and need not be calculated. However, te Brinell test may be made witout special apparatus and is somewat more widely recognized for laboratory use. Tere is also a Rockwell superficial ardness test similar to te regular Rockwell, except tat te indentation is muc sallower. Te Vickers metod of ardness testing is similar in principle to te Brinell in tat it expresses te result in terms of te pressure under te indenter and uses te same units, kilograms per square millimeter. Te indenter is a diamond in te form of a square pyramid wit an apical

13 TESTING OF MATERIALS 5-3 angle of 36, te loads are muc ligter, varying between and 0 kg, and te impression is measured by means of a medium-power compound microscope. V P/(0.5393d ) were V is te Vickers ardness number, sometimes called te diamondpyramid ardness (DPH); P te imposed load, kg; and d te diagonal of indentation, mm. Te Vickers metod is more flexible and is considered to be more accurate tan eiter te Brinell or te Rockwell, but te equipment is more expensive tan eiter of te oters and te Rockwell is somewat faster in production work. Among te oter ardness metods may be mentioned te Scleroscope, in wic a diamond-tipped ammer is dropped on te surface and te rebound taken as an index of ardness. Tis type of apparatus is seriously affected by te resilience as well as te ardness of te material and as largely been superseded by oter metods. In te Monotron metod, a penetrator is forced into te material to a predetermined dept and te load required is taken as te indirect measure of te ardness. Tis is te reverse of te Rockwell metod in principle, but te loads and indentations are smaller tan tose of te latter. In te Herbert pendulum, a -mm steel or jewel ball resting on te surface to be tested acts as te fulcrum for a 4-kg compound pendulum of 0-s period. Te swinging of te pendulum causes a rolling indentation in te material, and from te beavior of te pendulum several factors in ardness, suc as work ardenability, may be determined wic are not revealed by oter metods. Altoug te Herbert results are of considerable significance, te instrument is suitable for laboratory use only (see Herbert, Te Pendulum Hardness Tester, and Some Recent Developments in Hardness Testing, Engineer, 35, 93, pp. 390, 686). In te Herbert cloudburst test, a sower of steel balls, dropped from a predetermined eigt, dulls te surface of a ardened part in proportion to its softness and tus reveals defective areas. A variety of mutual indentation metods, in wic crossed cylinders or prisms of te material to be tested are forced togeter, give results comparable wit te Brinell test. Tese are particularly useful on wires and on materials at ig temperatures. Te relation among te scales of te various ardness metods is not exact, since no two measure exactly te same sort of ardness, and a relationsip determined on steels of different ardnesses will be found only approximately true wit oter materials. Te Vickers-Brinell relation is nearly linear up to at least 400, wit te Vickers approximately 5 percent iger tan te Brinell (actual values run from to percent) and nearly independent of te material. Beyond 500, te values become more widely divergent owing to te flattening of te Brinell ball. Te Brinell-Rockwell relation is fairly satisfactory and is sown in Fig Approximate relations for te Sore Scleroscope are also given on te same plot. Te ardness of wood is defined by te ASTM as te load in pounds required to force a ball in in diameter into te wood to a dept of 0. in, te speed of penetration being 4 in/min. For a summary Fig Hardness scales. of te work in ardness see Williams, Hardness and Hardness Measurements, ASM. TESTING OF MATERIALS Testing Macines Macines for te mecanical testing of materials usually contain elements () for gripping te specimen, () for deforming it, and (3) for measuring te load required in performing te deformation. Some macines (ductility testers) omit te measurement of load and substitute a measurement of deformation, wereas oter macines include te measurement of bot load and deformation troug apparatus eiter integral wit te testing macine (stress-strain recorders) or auxiliary to it (strain gages). In most general-purpose testing macines, te deformation is controlled as te independent variable and te resulting load measured, and in many special-purpose macines, particularly tose for ligt loads, te load is controlled and te resulting deformation is measured. Special features may include tose for constant rate of loading (pacing disks), for constant rate of straining, for constant load maintenance, and for cyclical load variation (fatigue). In modern testing systems, te load and deformation measurements are made wit load-and-deformation-sensitive transducers wic generate electrical outputs. Tese outputs are converted to load and deformation readings by means of appropriate electronic circuitry. Te readings are commonly displayed automatically on a recorder cart or digital meter, or tey are read into a computer. Te transducer outputs are typically used also as feedback signals to control te test mode (constant loading, constant extension, or constant strain rate). Te load transducer is usually a load cell attaced to te test macine frame, wit electrical output to a bridge circuit and amplifier. Te load cell operation depends on cange of electrical resistivity wit deformation (and load) in te transducer element. Te deformation transducer is generally an extensometer clipped on to te test specimen gage lengt, and operates on te same principle as te load cell transducer: te cange in electrical resistance in te specimen gage lengt is sensed as te specimen deforms. Optical extensometers are also available wic do not make pysical contact wit te specimen. Verification and classification of extensometers is controlled by ASTM Standards. Te application of load and deformation to te specimen is usually by means of a screw-driven mecanism, but it may also be applied by means of ydraulic and servoydraulic systems. In eac case te load application system responds to control inputs from te load and deformation transducers. Important features in test macine design are te metods used for reducing friction, wear, and backlas. In older testing macines, test loads were determined from te macine itself (e.g., a pressure reading from te macine ydraulic pressure) so tat macine friction made an important contribution to inaccuracy. Te use of macine-independent transducers in modern testing as eliminated muc of tis source of error. Grips sould not only old te test specimen against slippage but sould also apply te load in te desired manner. Centering of te load is of great importance in compression testing, and sould not be neglected in tension testing if te material is brittle. Figure 5..4 sows te teoretical errors due to off-center loading; te results are directly applicable to compression tests using swivel loading blocks. Swivel (ball-and-socket) olders or compression blocks sould be used wit all except te most ductile materials, and in compression testing of brittle materials (concrete, stone, brick), any roug faces sould be smootly capped wit plaster of paris and one-tird portland cement. Serrated grips may be used to old ductile materials or te sanks of oter olders in tension; a taper of in 6 on te wedge faces gives a self-tigtening action witout excessive jamming. Ropes are ordinarily eld by wet eye splices, but braided ropes or small cords may be given several turns over a fixed pin and ten clamped. Wire ropes sould be zinced into forged sockets (solder and lead ave insufficient strengt). Grip selection for tensile testing is described in ASTM standards. Accuracy and Calibration ASTM standards require tat commercial macines ave errors of less tan percent witin te loading range wen cecked against acceptable standards of comparison at at least five suitably spaced loads. Te loading range may be any range

14 5-4 MECHANICS OF MATERIALS troug wic te preceding requirements for accuracy are satisfied, except tat it sall not extend below 00 times te least load to wic te macine will respond or wic can be read on te indicator. Te use of calibration plots or tables to correct te results of an oterwise inaccurate macine is not permitted under any circumstances. Macines wit errors less tan 0. percent are commercially available (Tate- Emery and oters), and somewat greater accuracy is possible in te most refined researc apparatus. Two standard forms of test specimens (ASTM) are sown in Figs and In wrougt materials, and particularly in tose wic Fig Test specimen, -in (50-mm) gage lengt, -in (.5-mm) diameter. Oters available for 0.35-in (8.75-mm) and 0.5-in (6.5-mm) diameters. (ASTM). Fig Effect of centering errors on brittle test specimens. Fig Carpy V-notc impact specimens. (ASTM.) Dead loads may be used to ceck macines of low capacity; accurately calibrated proving levers may be used to extend te range of available weigts. Various elastic devices (suc as te Moreouse proving ring) made of specially treated steel, wit sensitive disortion-measuring devices, and calibrated by dead weigts at te NIST (formerly Bureau of Standards) are mong te most satisfactory means of cecking te iger loads. ave been cold-worked, different properties may be expected in different directions wit respect to te direction of te applied work, and te test specimen sould be cut out from te parent material in suc a way as to give te strengt in te desired direction. Wit te exception of fatigue specimens and specimens of extremely brittle materials, surface finis is of little practical importance, altoug extreme rougness tends to decrease te ultimate elongation. 5. MECHANICS OF MATERIALS by J. P. Vidosic REFERENCES: Timosenko and MacCulloug, Elements of Strengt of Materials, Van Nostrand. Seeley, Advanced Mecanics of Materials, Wiley. Timosenko and Goodier, Teory of Elasticity, McGraw-Hill. Pillips, Introduction to Plasticity, Ronald. Van Den Broek, Teory of Limit Design, Wiley. Hetényi, Handbook of Experimental Stress Analysis, Wiley. Dean and Douglas, Semi-Conductor and Conventional Strain Gages, Academic. Robertson and Harvey, Te Engineering Uses of Holograpy, University Printing House, London. Sellers, Basic Training Guide to te New Metrics and SI Units, National Tool, Die and Precision Macining Association. Roark and Young, Formulas for Stress and Strain, McGraw-Hill. Perry and Lissner, Te Strain Gage Primer, McGraw-Hill. Donnell, Beams, Plates, and Seets, Engineering Societies Monograps, McGraw-Hill. Griffel, Beam Formulas and Plate Formulas, Ungar. Durelli et al., Introduction to te Teoretical and Experimental Analysis of Stress and Strain, McGraw-Hill. Stress Analysis Manual, Department of Commerce, Pub. no. AD , 969. Blodgett, Welded Structures, Lincoln Arc Welding Foundation. Caracteristics and Applications of Resistance Strain Gages, Department of Commerce, NBS Circ. 58, 954. EDITOR S NOTE: Te almost universal availability and utilization of computers in engineering practice as led to te development of many forms of software individually tailored to te solution of specific design problems in te area of mecanics of materials. Teir use will permit te reader to amplify and supplement a good portion of te formulary and tabular collection in tis section, as well as utilize tose powerful computational tools in newer and more powerful tecniques to facilitate solutions to problems. Many of te approximate metods, involving laborious iterative matematical scemes, ave been supplanted by te computer. Developments along tose lines continue apace and bid fair to expand te types of problems andled, all wit greater confidence in te results obtained tereby.

15 SIMPLE STRESSES AND STRAINS 5-5 Main Symbols Unit Stress S apparent stress S v or S s pure searing T true (ideal) stress S p proportional elastic limit S y yield point S M ultimate strengt, tension S c ultimate compression S v vertical sear in beams S R modulus of rupture paraffin; 0 for cork. For concrete, varies from 0.0 to 0.0 at working stresses and can reac 0.5 at iger stresses; for ordinary glass is about 0.5. In te absence of definitive data, for most structural metals can be taken to lie between 0.5 and Extensive listings of Poisson s ratio are found in oter sections; see Tables 5..3 and Moment M bending M t torsion External Action P force G weigt of body W weigt of load V external sear Modulus of Elasticity E longitudinal G searing K bulk U p modulus of resilience U R ultimate resilience Geometric l lengt A area V volume v velocity r radius of gyration I rectangular moment of inertia I P or J polar moment of inertia Deformation e, e gross deformation, unit deformation; strain d or unit, angular s unit, lateral Poisson s ratio n reciprocal of Poisson s ratio r radius f deflection SIMPLE STRESSES AND STRAINS Deformations are canges in form produced by external forces or loads tat act on nonrigid bodies. Deformations are longitudinal, e, a lengtening () or sortening () of te body; and angular,, a cange of angle between te faces. Unit deformation (dimensionless number) is te deformation in unit distance. Unit longitudinal deformation (longitudinal strain), e/l (Fig. 5..). Unit angular-deformation tan equals approx (Fig. 5..). Te accompanying lateral deformation results in unit lateral deformation (lateral strain) e/l (Fig. 5..). For omogeneous, isotropic material operating in te elastic region, te ratio / is a constant and is a definite property of te material; tis ratio is called Poisson s ratio. A fundamental relation among te tree interdependent constants E, G, and for a given material is E G( ). Note tat cannot be larger tan 0.5; tus te searing modulus G is always smaller tan te elastic modulus E. At te extremes, for example, 0.5 for rubber and Fig. 5.. Stress is an internal distributed force, or, force per unit area; it is te internal mecanical reaction of te material accompanying deformation. Stresses always occur in pairs. Stresses are normal [tensile stress () and compressive stress ()]; and tangential, or searing. Fig. 5.. Intensity of stress, or unit stress, S, lb/in (kgf/cm ), is te amount of force per unit of area (Fig. 5..3). P is te load acting troug te center of gravity of te area. Te uniformly distributed normal stress is S P/A Wen te stress is not uniformly distributed, S dp/da. A long rod will stretc under its own weigt G and a terminal load P (see Fig. 5..4). Te total elongation e is tat due to te terminal load plus tat due to one-alf te weigt of te rod considered as acting at te end. e (Pl Gl/)/(AE) Te maximum stress is at te upper end. Wen a load is carried by several pats to a support, te different pats take portions of te load in proportion to teir stiffness, wic is controlled by material (E) and by design. EXAMPLE. Two pairs of bars rigidly connected (wit te same elongation) carry a load P 0 (Fig. 5..5). A, A and E, E and P, P and S, S are cross sections, moduli of elasticity, loads, and stresses of te bars, respectively; e elongation. e P l(e A ) P l/(e A ) P 0 P P S P /A [P 0 E /(E A E A )] S [P 0 E /(E A E A )] Temperature Stresses Wen te deformation arising from cange of temperature is prevented, temperature stresses arise tat are proportional to te amount of deformation tat is prevented. Let a coeffi-

16 5-6 MECHANICS OF MATERIALS Fig Fig Fig Fig and Fig a common form of test piece tat introduces bending stresses. Let Fig represent te symmetric section of area A wit a searing force V acting troug its centroid. If pure sear exists, S v V/A, and tis sear would be uniformly distributed over te area A. Wen tis sear is accompanied by bending (transverse sear in beams), te unit sear cient of expansion per degree of temperature, l lengt of bar at temperature t, and l lengt at temperature t. Ten l l [ a(t t )] If, subsequently, te bar is cooled to a temperature t, te proportionate deformation is s a(t t ) and te corresponding unit stress S Ea(t t ). For coefficients of expansion, see Sec. 4. In te case of steel, a cange of temperature of F (6.7 K, 6.7 C) will cause in general a unit stress of,340 lb/in (64 kgf/cm ). Fig S v increases from te extreme fiber to its maximum, wic may or may not be at te neutral axis OZ. Te unit sear parallel to OZ at a point d distant from te neutral axis (Fig. 5..9) is Fig Searing stresses (Fig. 5..) act tangentially to surface of contact and do not cange lengt of sides of elementary volume; tey cange te angle between faces and te lengt of diagonal. Two pairs of searing stresses must act togeter. Searing stress intensities are of equal magnitude on all four faces of an element. S v S v (Fig. 5..6). S v Ibe V yz dy were z te section widt at distance y; and I is te moment of inertia of te entire section about te neutral axis OZ. Note tat e d yz dy is te first moment of te area above d wit respect to axis OZ. For a rectangular cross section (Fig. 5..0a), S v 3 d V b y S v (max) 3 V b 3 V for y 0 A For a circular cross section (Fig. 5..0b), S v 4 V 3 r y r S v (max) 4 3 V r 4 V 3 A for y 0 Fig In te presence of pure sear on external faces (Fig. 5..6), te resultant stress S on one diagonal plane at 45 is pure tension and on te oter diagonal plane pure compression; S S v S v. S on diagonal plane is called diagonal tension by writers on reinforced concrete. Failure under pure sear is difficult to produce experimentally, except under torsion and in certain special cases. Figure 5..7 sows an ideal case, Fig. 5..0

17 Table 5.. Resilience per Unit of Volume U p (S longitudinal stress; S v searing stress; E tension modulus of elasticity; G searing modulus of elasticity) Tension or compression Sear Beams (free ends) Rectangular section, bent in arc of circle; no sear Ditto, circular section Concentrated center load; rectangular cross section Ditto, circular cross section Uniform load, rectangular cross section -beam section, concentrated center load S /E S v /G 6S /E 8S /E 8S /E 4S /E 5 36S /E 3 3S /E Torsion Solid circular Hollow, radii R and R Springs Carriage Flat spiral, rectangular section Helical: axial load, circular wire Helical: axial twist Helical: axial twist, rectangular section SIMPLE STRESSES AND STRAINS 5-7 4S v /G R R R 6S /E 4S /E 4S v /G 8S /E 6S /E S v 4 G For a circular ring (tickness small in comparison wit te major diameter), S v (max) V/A, for y 0. For a square cross section (diagonal vertical, Fig. 5..0c), S v V a y a a 4 y S v (max).59 V for y e A 4 For an I-saped cross section (Fig. 5..0d), S v (max) 3 V 4 a be (b a)f be 3 (b a)f 3 for y 0 Elasticity is te ability of a material to return to its original dimensions after te removal of stresses. Te elastic limit S p is te limit of stress witin wic te deformation completely disappears after te removal of stress; i.e., no set remains. Hooke s law states tat, witin te elastic limit, deformation produced is proportional to te stress. Unless modified, te deduced formulas of mecanics apply only witin te elastic limit. Beyond tis, tey are modified by experimental coefficients, as, for instance, te modulus of rupture. Te modulus of elasticity, lb/in (kgf/cm ), is te ratio of te increment of unit stress to increment of unit deformation witin te elastic limit. Te modulus of elasticity in tension, or Young s modulus, E unit stress/unit deformation Pl/(Ae) Te modulus of elasticity in compression is similarly measured. Te modulus of elasticity in sear or coefficient of rigidity, G S v / were is expressed in radians (see Fig. 5..). Te bulk modulus of elasticity K is te ratio of normal stress, applied to all six faces of a cube, to te cange of volume. Cange of volume under normal stress is so small tat it is rarely of significance. For example, given a body wit lengt l, widt b, tickness d, Poisson s ratio, and longitudinal strain, V lbd original volume. Te deformed volume ( )l ( )b( )d. Neglecting powers of, te deformed volume ( )V. Te cange in volume is ( )V; te unit volumetric strain is ( ). Tus, a steel rod ( 0.3, E lb/in ) compressed to a stress of 30,000 lb/in will experience 0.00 and a unit volumetric strain of , or part in,500. Te following relationsips exist between te modulus of elasticity in tension or compression E, modulus of elasticity in sear G, bulk modulus of elasticity K, and Poisson s ratio : E G( ) G E/[( )] (E G)/(G) K E/[3( )] (3K E)/(6K) Resilience U (inlb)[(cmkgf )] is te potential energy stored up in a deformed body. Te amount of resilience is equal to te work required to deform te body from zero stress to stress S. Wen S does not exceed te elastic limit. For normal stress, resilience work of deformation average force times deformation Pe AS Sl/E S V/E. Modulus of resilience U p (inlb/in 3 ) [(cmkgf/cm 3 )], or unit resilience, is te elastic energy stored up in a cubic inc of material at te elastic limit. For normal stress, U p S p/e Te unit resilience for any oter kind of stress, as searing, bending, torsion, is a constant times one-alf te square of te stress divided by te appropriate modulus of elasticity. For values, see Table 5... Unit rupture work U R, sometimes called ultimate resilience, is measured by te area of te stress-deformation diagram to rupture. U R 3e u (S y S M ) approx were e u is te total deformation at rupture. For structural steel, U R [35,000 ( 60,000)] 3,950 inlb/in 3 (98 cmkgf/cm 3 ). EXAMPLE. A load P 40,000 lb compresses a wooden block of cross-sectional area A 0 in and lengt 0 in, an amount e 4 00 in. Stress S 0 40,000 4,000 lb/in. Unit elongation s Modulus of elasticity E 4,000 50,000,000 lb/in. Unit resilience U p 4,000 4,000/,000,000 8inlb/in 3 (0.563 cmkgf/cm 3 ). EXAMPLE. A weigt G 5,000 lb falls troug a eigt ft;v number of cubic inces required to absorb te sock witout exceeding a stress of 4,000 lb/in. Neglect compression of block. Work done by falling weigt G 5,000 inlb (,7 6 cmkgf ) Resilience of block V 8inlb 5,000. Terefore, V 5,000 in 3 (45,850 cm 3 ). Termal Stresses A bar will cange its lengt wen its temperature is raised (or lowered) by te amount l 0 l 0 (t 3). Te linear coefficient of termal expansion is assumed constant at normal temperatures and l 0 is te lengt at 3 F (73. K, 0 C). If tis expansion (or contraction) is prevented, a termal-time stress is developed, equal to S E(t t ), as te temperature goes from t to t. In tin flat plates te stress becomes S E(t t )/( ); is Poisson s ratio. Suc stresses can occur in castings containing large and small sections. Similar stresses also occur wen eat flows troug members because of te difference in temperature between one point and anoter. Te eat flowing across a lengt b as a result of a linear drop in temperature t equals Q kat/b Btu/ (cal/). Te termal conductivity k is in Btu/()(ft )( F)/(in of tickness) [cal/()(m )(k)/(m)]. Te termal-flow stress is ten S EQb/(kA). Note, wen Q is substituted te stress becomes S E t as above, only t is now a function of distance rater tan time. EXAMPLE. A cast-iron plate 3 ft square and in tick is used as a fire wall. Te temperature is 330 F on te ot side and 60 F on te oter. Wat is te termal-flow stress developed across te plate? S E t ,360 lb/in (,00 kgf/cm ) or Q /,760 Btu/ and S ,760 /.3 9 4,360 lb/in (,00 kgf/cm )

18 5-8 MECHANICS OF MATERIALS COMBINED STRESSES In te discussion tat follows, te element is subjected to stresses lying in one plane; tis is te case of plane stress, or two-dimensional stress. Simple stresses, defined as suc by te flexure and torsion teories, lie in planes normal or parallel to te line of action of te forces. Normal, as well as searing, stresses may, owever, exist in oter directions. A particle out of a loaded member will contain normal and searing stresses as sown in Fig Note tat te four searing stresses must be of te same magnitude, if equilibrium is to be satisfied. If te particle is cut along te plane AA, equilibrium will reveal tat, in general, normal as well as searing stresses act upon te plane AC (Fig. 5..). Te normal stress on plane AC is labeled S n, and searing S s. Te application of equilibrium yields and S n S x S y S s S x S y S x S y sin S xy cos cos S xy sin 4,000 8, S n 7,000 lb/in 60 S s S M,m at tan 4,000 8,000 4,000 8,000 4,000 8,000 ( ) 0 (0.8660) 0,73 lb/in 4,000 8, ,000,000 8,000 and 4,000 lb/in (564 and 8 kgf/cm ) 0 0 or 90 and 0 4,000 8,000 S sm,m 4,000 8,000 0,000 lb/in ( 4 kgf/cm ) Mor s Stress Circle Te biaxial stress field wit its combined stresses can be represented grapically by te Mor stress circle. For instance, for te particle given in Fig. 5.., Mor s circle is as sown in Fig Te stress sign convention previously defined must be adered to. Furtermore, in order to locate te point (on Mor s circle) tat yields te stresses on a plane from te vertical side of te particle (suc as plane AA in Fig. 5..), must be laid off in te same Fig. 5.. Fig. 5.. A sign convention must be used. A tensile stress is positive wile compression is negative. A searing stress is positive wen directed as on plane AB of Fig. 5..; i.e., wen te searing stresses on te vertical planes form a clockwise couple, te stress is positive. Te planes defined by tan S xy /S x S y, te principal planes, contain te principal stresses te maximum and minimum normal stresses. Tese stresses are S M, S m S x S y S x S y S xy Te maximum and minimum searing stresses are represented by te quantity Fig direction from te radius to (S x, S xy ). For te previous example, Mor s circle becomes Fig Eigt special stress fields are sown in Figs to 5..3, along wit Mor s circle for eac. S sm,m S x S y S xy and tey act on te planes defined by tan S x S y S xy EXAMPLE. Te steam in a boiler subjects a paticular particle on te outer surface of te boiler sell to a circumferential stress of 8,000 lb/in and a longitudinal stress of 4,000 lb/in as sown in Fig Find te stresses acting on te plane XX, making an angle of 60 wit te direction of te 8,000 lb/in stress. Find te principal stresses and locate te principal planes. Also find te maximum and minimum searing stresses. Fig Fig Combined Loading Combined flexure and torsion arise, for instance, wen a saft twisted by a torque M t is bent by forces produced by belts or gears. An element on te surface, suc as ABCD on te saft of Fig. 5..4, is subjected to a flexure stress S x Mc/I 8Fl(d 3 ) and a

19 PLASTIC DESIGN 5-9 compression) occurring at a point in two rigt-angle directions, and te cange of te angle between tem is xy. Te strain e at te point in any direction a at an angle wit te x direction derives as Fig Fig Fig Fig Fig Fig. 5.. e a e x e y e x e y cos xy sin Similarly, te searing strain ab (cange in te original rigt angle between directions a and b) is defined by ab (e x e y ) sin xy cos Inspection easily reveals tat te above equations for e a and ab are matematically identical to tose for S n and S s. Tus, once a sign convention is establised, a Mor circle for strain can be constructed and used as te stress circle is used. Te strain e is positive wen an extension and negative wen a contraction. If te direction associated wit te first subscript a rotates counterclockwise during straining wit respect to te direction indicated by te second subscript b, te searing strain is positive; if clockwise, it is negative. In constructing te circle, positive extensional strains will be plotted to te rigt as abscissas and positive alf-searing strains will be plotted upward as ordinates. For te strains sown in Fig. 5..6a, Mor s strain circle becomes tat sown in Fig. 5..6b. Te extensional strain in te direction a, making an angle of a wit te x direction, is e a, and te searing strain is ab counterclockwise. Te strain 90 away is e b. Te maximum principal strain is e M at an angle M clockwise from te x direction. Te oter principal or minimum strain is e m 90 away. Fig. 5.. Fig torsional searing stress S xy M t c/j 6M t (d 3 ). Tese stresses will induce combined stresses. Te maximum combined stresses will be S n (S x S x 4S xy) and S s S x 4S xy Te above situation applies to any case of normal stress wit sear, as wen a bolt is under bot tension and sear. A beam particle subjected to bot flexure and transverse sear is anoter case. Fig Fig Combined torsion and longitudinal loads exist on a propeller saft. A particle on tis saft will contain a tensile stress computed using S F/A and a torsion searing stress equal to S s M t c/j. Te free body of a particle on te surface of a vertical turbine saft is subjected to direct compression and torsion. Fig Wen combined loading results in stresses of te same type and direction, te addition is algebraic. Suc a situation exists on an offset link like tat of Fig Mor s Strain Circle Strain equations can also be derived for planestrain fields. Strains e x and e y are te extensional strains (tension or PLASTIC DESIGN Early efforts in stress analysis were based on limit loads, tat is, loads wic stress a member wolly to te yield strengt. Euler s famous paper on column action ( Sur la Force des Colonnes, Academie des Sciences de Berlin, 757) deals wit te column problem tis way. More recently, te concept of limit loads, referred to as limit, or plastic, design, as found strong application in te design of certain structures. Te teory presupposes a ductile material, absence of stress raisers, and fabrication free of embrittlement. Local load overstress is allowed, provided te structure does not deform appreciably. To visualize te limit-load approac, consider a simple beam of uniform section subjected to a concentrated load of midspan, as depicted in Fig. 5..7a. According to elastic teory, te outermost fiber on eac side and at midspan te section of maximum bending moment will first reac te yield-strengt value. Across te dept of te beam, te stress distribution will, of course, follow te triangular pattern, becoming zero at te neutral axis. If te material is ductile, te stress in te outermost fibers will remain at te yield value until every oter fiber reaces te same value as te load increases. Tus te stress distribution assumes te rectangular pattern before te plastic inge forms and failure ensues.

20 5-0 MECHANICS OF MATERIALS Te problem is tat of finding te final limit load. Elastic-flexure teory gives te maximum load triangular distribution as F y S yb 3l For te rectangular stress distribution, te limit load becomes F L S yb l Te ratio F L /F y.50 an increase of 50 percent in load capability. Te ratio F L /F y as been named sape factor (Jenssen, Plastic Design in Welded Structures Promises New Economy and Safety, Welding Jour., Mar. 959). See Fig. 5..7b for sape factors for some oter sections. Te sape factor may also be determined by dividing te first moment of area about te neutral axis by te section modulus. Fig A constant-section beam wit bot ends fixed, supporting a uniformly distributed load, illustrates anoter application of te plasticload approac. Te bending-moment diagram based on te elastic teory drawn in Fig (broken line) sows a moment at te center equal to one-alf te moment at eiter end. A preferable situation, it migt be argued, is one in wic te moments are te same at te tree stations solid line. Tus, applying equilibrium to, say, te left alf of te beam yields a bending moment at eac of te tree plastic inges of M L wl 6 DESIGN STRESSES If a macine part is to safely transmit loads acting upon it, a permissible maximum stress must be establised and used in te design. Tis is te allowable stress, te working stress, or preferably, te design stress. Te design stress sould not waste material, yet sould be large enoug to prevent failure in case loads exceed expected values, or oter uncertainties react unfavorably. Te design stress is determined by dividing te applicable material property yield strengt, ultimate strengt, fatigue strengt by a factor of safety. Te factor sould be selected only after all uncertainties ave been torougly considered. Among tese are te uncertainty wit respect to te magnitude and kind of operating load, te reliability of te material from wic te component is made, te assumptions involved in te teories used, te environment in wic te equipment migt operate, te extent to wic localized and fabrication stresses migt develop, te uncertainty concerning causes of possible failure, and te endangering of uman life in case of failure. Factors of safety vary from industry to industry, being te result of accumulated experience wit a class of macines or a kind of environment. Many codes, suc as te ASME code for power safting, recommend design stresses found safe in practice. In general, te ductility of te material determines te property upon wic te factor sould be based. Materials aving an elongation of over 5 percent are considered ductile. In suc cases, te factor of safety is based upon te yield strengt or te endurance limit. For materials wit an elongation under 5 percent, te ultimate strengt must be used because tese materials are brittle and so fracture witout yielding. Factors of safety based on yield are often taken between.5 and 4.0. For more reliable materials or well-defined design and operating conditions, te lower factors are appropriate. In te case of untried materials or oterwise uncertain conditions, te larger factors are safer. Te same values can be used wen loads vary, but in suc cases tey are applied to te fatigue or endurance strengt. Wen te ultimate strengt determines te design stress (in te case of brittle materials), te factors of safety can be doubled. Tus, under static loading, te design stress for, say, SAE 00, wic as a yield strengt of 45,000 lb/in (3,70 kgf/cm ) may be taken at 45,000/, or,500 lb/in (,585 kgf/cm ), if a reasonably certain design condition exists. A Class 30 cast-iron part migt be designed at 30,000/5 or 6,000 lb/in (43 kgf/cm ). A 07S-0 aluminumalloy component (3,000 lb/in endurance strengt) could be computed at a design stress of 3,000/.5 or 5,00 lb/in (366 kgf/cm )inte usual fatigue-load application. BEAMS For properties of structural steel and wooden beams, see Sec... Fig Notation I rectangular moment of inertia I p polar moment of inertia I/c section modulus M bending moment P, W concentrated load Q or V total vertical sear R reaction S unit normal stress S s or S v transverse searing stress W total distributed load