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

21 BEAMS 5- f deflection i slope l distance between supports r radius of gyration r c radius of curvature w distributed load per longitudinal unit A simple beam rests on supports at its ends wic permit rotation. A cantilever beam is fixed (no rotation) at one end. Wen computing reactions and moments, distributed loads may be replaced by teir resultants acting at te center of gravity of te distributed-load area. Reactions are te forces and/or couples acting at te supports and olding te beam in place. In general, te weigt of te beam sould be accounted for. Te bending moment (pound-feet or pound-inces) (kgf m) at any section is te algebraic sum of te external forces and moments acting on te beam on one side of te section. It is also equal to te moment of te internal-stress forces at te section, M sda/y. A bending moment tat bends a beam convex downward (tensile stress on bottom fiber) is considered positive, wile convex upward (compression on bottom) is negative. Te vertical sear V (lb) (kgf ) effective on a section is te algebraic sum of all te forces acting parallel to and on one side of te section, V F. It is also equal to te sum of te transverse sear stresses acting on te section, V S s da. Moment and sear diagram may be constructed by plotting to scale te particular entity as te ordinate for eac section of te beam. Suc diagrams sow in continuous form te variation along te lengt of te beam. Moment-Sear Relation Te sear V is te first derivative of moment wit respect to distance along te beam, V dm/dx. Tis relationsip does not, owever, account for any sudden canges in moment. x x x 3 lx 6 x3,if is in pounds per foot and weigt of beam is 6l neglected. Te vertical sear V R x x l l 6 x. Note again tat V l d dx lx 6 x3 6l l 6 x l. Table 5.. gives te reactions, bending-moment equations, vertical sear equations, and te deflection of some of te more common types of beams. Maximum Safe Load on Steel Beams See Table 5..3 To obtain maximum safe load (or maximum deflection under maximum safe load) for any of te conditions of loading given in Table 5..5, multiply te corresponding coefficient in tat table by te greatest safe load (or deflection) for distributed load for te particular section under consideration as given in Table Te following approximate factors for reducing te load sould be used wen beams are long in comparison wit teir breadt: Ratio of unsupported (lateral) lengt to flange widt or breadt Ratio of greatest safe load to calculated load Teory of Flexure A bent beam is sown in Fig Te concave side is in compression and te convex side in tension. Tese are divided by te neutral plane of zero stress ABBA. Te intersection of te neutral plane wit te face of te beam is in te neutral line or elastic curve AB. Te intersection of te neutral plane wit te cross section is te neutral axis NN. Fig Fig Fig EXAMPLES. Figure 5..9 illustrates a simple beam subjected to a uniform load. M R x wx x w/x wx and V R wx wl wx. Note also tat V dx d wlx wx wl wx. Figure is a simple beam carrying a uniformly varying load; M R x It is assumed tat a beam is prismatic, of a lengt at least 0 times its dept, and tat te external forces are all at rigt angles to te axis of te beam and in a plane of symmetry, and tat flexure is sligt. Oter assumptions are: () Tat te material is omogeneous, and obeys Hooke s law. () Tat stresses are witin te elastic limit. (3) Tat every layer of material is free to expand and contract longitudinally and laterally under stress as if separate from oter layers. (4) Tat te tensile and compressive moduli of elasticity are equal. (5) Tat te cross section remains a plane surface. (Te assumption of plane cross sections is strictly true only wen te sear is constant or zero over te cross section, and wen te sear is constant trougout te lengt of te beam.) It follows ten tat: () Te internal forces are in orizontal balance. () Te neutral axis contains te center of gravity of te cross section, were tere is no resultant axial stress. (3) Te stress intensity varies directly wit te distance from te neutral axis. Te moment of te elastic forces about te neutral axis, i.e., te stress moment or moment of resistance, is M SI/c, were S is an elastic unit stress at outer fiber wose distance from te neutral axis is c; and I is te rectangular moment of inertia about te neutral axis. I/c is te section modulus. Tis formula is for te strengt of beams. For rectangular beams, M 6Sb, were b breadt and dept; i.e., te elastic strengt of beam sections varies as follows: () for equal widt, as te square of te dept; () for equal dept, directly as te widt; (3) for equal dept and widt, directly as te strengt of te material; (4) if span varies, ten for equal dept, widt, and material, inversely as te span.

22 5- MECHANICS OF MATERIALS Table 5.. Beams of Uniform Cross Section, Loaded Transversely R W M x Wx M max Wl, (x ) Q x W f Wl 3 3EI (max) R W, R W M x Wx M max Wl 4, x l Q x W f W EI l 3 48 (max) R Wc, R l Wc l M x Wc x, Mx Wcx l l M max Wcc,(x c or x c) l Q x Wc, Q l x Wc l f Wc 3EIl c(l c ) 3/ (max) 3 Max f occurs at x c(l c )/3 R 5 6 W, R 6 W M x 5 6 Wx M x Wl 5 3 x 6 l M max 3 6 Wl, x l Q x 5 6 W, Q x 6 W Q max 6 W, x l to x l f W EI 7l R W, R W M x Wl M x Wl x l 4 x l 3 4 M max Wl 8, x l Q x W, Q x W f W EI l 3 9 (max) R W R W M x W c const Q W to R W Q R to 0 R Q R to W W f Wcl EI8 (max) f Wc EI3c 3l (max) If a beam is cut in alves vertically, te two alves laid side by side eac will carry only one-alf as muc as te original beam. Tables 5..6 to 5..8 give te properties of various beam cross sections. For properties of structural-steel sapes, see Sec... Oblique Loading It sould be noted tat Table 5..6 includes certain cases for wic te orizontal axis is not a neutral axis, assuming te common case of vertical loading. Te rectangular section wit te diagonal as a orizontal axis (Table 5..6) is suc a case. Tese cases must be andled by te principles of oblique loading. Every section of a beam as two principal axes passing troug te center of gravity, and tese two axes are always at rigt angles to eac oter. Te principal axes are axes wit respect to wic te moment of inertia is, respectively, a maximum and a minimum, and for wic te product of inertia is zero. For symmetrical sections, axes of symmetry are always principal axes. For unsymmetrical sections, like a rolled angle section (Fig. 5..3), te inclination of te principal axis wit te X axis may be found from te formula tan I xy /(I y I x ), in wic angle of inclination of te principal axis to te X axis, I xy te product of inertia of te section wit respect to te X and Y axes, I y moment of inertia of te section wit respect to te Y axis, I x moment of inertia of

23 BEAMS 5-3 Table 5.. Beams of Uniform Cross Section, Loaded Transversely (Continued ) R W wl M x wx M max wl,(x l) Q x wx Q max wl, (x l) f W EI l 3 8 (max) R W wl R W wl M x wx (l x) M max wl 8,(x l) Q x wl wx Q max wl,(x 0) f W EI 5l (max) R 3 8 W 3 8 wl R 5 8 W 5 8 wl M x wx 3 4 l x M max 9 8 wl,x 3 8 l M max wl,(x l) 8 Q x 3 wl wx 8 Q max 5 8 wl f W EI l 3 85 (max) R W wl, R W wl M x wl 6 x l x l M max wl,(x 0, or x l) Q x wl wx Q max wl f W EI l (max) R W total load M x W 3 x 3 l M max Wl 3 Q x Wx l Q max W f W EI l 3 5 (max) R 3 W, R 3 W M x Wx x 3 l M max 9 3 Wl, x 3 Q x W 3 x l Q max W,(x l) 3 f Wl 3 EI (max)

24 5-4 MECHANICS OF MATERIALS Table 5.. Beams of Uniform Cross Section, Loaded Transversely (Continued ) R W, R W M x Wx x l x 3l M max Wl, x l Q x W x x l Q max W,(x 0) f W EI 3l 3 30 (max) l R W, R W x l M x Wx 3 M max Wl 6, x l Q x W x l Q max W,(x 0) f W EI l 3 60 (max) R W 5, R 4W 5 M x Wx 5 x 3l M max Wl at support 5 Q x W 5 x l Q max 4W 5 f 6Wl 3,500 5EI Wl 3 (max) EI R W wl, R W wl M x Wx c x x (x c) l, M x Wx,(x c) l M max Wl 4 c l, c l Q x W wx (x c) Q x wx (x c) Concentrated load W Uniformly dist. load W wl R W c (3c c ) 3 l 3 8 W R W (c 6cc 3c )c 5 l 3 8 W M W cc (c c ) W l 8 l M W W cc (3c c ) W (3c c)c l 3 8l W (a) W l 5c 3c 4c 3c c M c max R W l, x R l (b) W W W l (3c 5c) 4c(c 6cc 3c) M c max Wc (R W) W Deflection under W f W EI l,x R W W c c(4c 3 3c ) W cc(3c c ) l 3 EI 48l l

25 BEAMS 5-5 Table 5.. Beams of Uniform Cross Section, Loaded Transversely (Continued ) Concentrated load W Uniformly dist. load W wl; c c (a) (b) R W c W l R W c l W W W c c c x M max R R l W, x R l W W c c c W M max W W cc,(x c ) l Deflection of beam under W: c c R W (3c c )c W l 3 R W (c 3c )c W l 3 M max M W cc l Wl Deflection under W 4l f EIW c3 c 3 3l 3 W c c f W l cc 8cc W c c 3EIl Table 5..3 Uniformally Distributed Loads on Simply Supported Rectangular Beams -in Wide* (Laterally Supported Sufficiently to Prevent Buckling) [Calculated for unit fiber stress at,000 lb/in (70 kgf/cm ): nominal size] Total load in pounds (kgf ) including te weigt of beam Span, ft Dept of beam, in (cm) (m) ,090,40,800,0,690 3,00 3,750 4,350 5,000 5, ,80,500,850,40,670 3,30 3,630 4,70 4, ,00,90,590,90,80,680 3,0 3,570 4, ,0,390,680,000,350,70 3,30 3, ,000,30,490,780,090,40,780 3, ,0,340,600,880,80,500, ,00,0,450,70,980,70, ,0,330,560,80,080, ,030,30,440,680,90, ,40,340,560,790, ,070,50,450,670, ,000,70,360,560, ,00,80,470, ,040,0,390, ,50,30, ,090,50, ,40, ,040, , , * Tis table is convenient for wooden beams. For any oter fiber stress S, multiply te values in table by S/,000. See Sec.. for properties of wooden beams of commercial sizes. To cange to kgf, multiply by To cange to m, multiply by To cange to cm, multiply by.54. te section wit respect to te X axis. Wen tis principal axis as been found, te oter principal axis is at rigt angles to it. Calling te moments of inertia wit respect to te principal axes I x and I y, te unit stress existing anywere in te section at a point wose coordinates are x and y (Fig ) is S My cos /I x Mx sin /I y, in wic M bending moment wit respect to te section in question, te angle wic te plane of bending moment or te plane of te loads makes wit te y axis, M cos te component of bending moment causing bending about te principal axis wic as been designated as te X axis, M sin te component of bending moment causing bending about te principal axis wic as been designated as te Y axis. Te sign of te two terms for unit stress may be determined by inspection in te usual way, and te result will be tension or compression as determined by te algebraic sum of te two terms.

26 5-6 MECHANICS OF MATERIALS Table 5..4 Approximate Safe Loads in Pounds (kgf) on Steel Beams,* Simply Supported, Single Span Allowable fiber stress for steel, 6,000 lb/in (,7 kgf/cm ) (basis of table) Beams simply supported at bot ends. L distance between supports, ft (m) a interior area, in (cm ) A sectional area of beam, in (cm ) d interior dept, in (cm) D dept of beam, in (cm) w total working load, net tons (kgf ) Greatest safe load, lb Deflection, in Sape of section Load in middle Load distributed Load in middle Load distributed Solid rectangle Hollow rectangle Solid cylinder Hollow cylinder I beam 890AD L 890(AD ad) L 667AD L 667(AD ad) L,795AD L,780AD L,780(AD ad) L,333AD L,333(AD ad) L 3,390AD L wl 3 3 AD wl 3 5 AD wl 3 3(AD ad ) wl 3 4AD wl 3 38AD wl 3 4(AD ad ) wl 3 58AD wl 3 93AD wl 3 5(AD ad ) wl 3 38(AD ad ) In general, it may be stated tat wen te plane of te bending moment coincides wit one of te principal axes, te oter principal axis is te neutral axis. Tis is te ordinary case, in wic te ordinary formula for unit stress may be applied. Wen te plane of te bending moment does not coincide wit one of te principal axes, te above formula for oblique loading may be applied. Fig Fig Internal Moment Beyond te Elastic Limit Ordinarily, te expression M SI/c is used for stresses above te elastic limit, in wic case S becomes an experimental coefficient S R, te modulus of rupture, and te formula is empirical. Te true relation is obtained by applying to te cross section a stress-strain diagram from a tension and compression test, as in Fig Figure sows te side of a beam of dept d under flexure beyond its elastic limit; line sows te distorted cross section; line 3 3, te usual rectilinear Table 5..5 Coefficients for Correcting Values in Table 5..4 for Various Metods of Support and of Loading, Single Span Max relative deflection under Max relative max relative safe Conditions of loading safe load load Beam supported at ends: Load uniformly distributed over span.0.0 Load concentrated at center of span 0.80 Two equal loads symmetrically concentrated l/4c Load increasing uniformly to one end Load increasing uniformly to center Load decreasing uniformly to center 3.08 Beam fixed at one end, cantilever: Load uniformly distributed over span 4.40 Load concentrated at end Load increasing uniformly to fixed end Beam continuous over two supports equidistant from ends: Load uniformly distributed over span. If distance a 0.07l l /(4a ). If distance a 0.07l l l 4a 3. If distance a 0.07l 5.83 Two equal loads concentrated at ends l/(4a) NOTE: l lengt of beam; c distance from support to nearest concentrated load; a distance from support to end of beam.

27 BEAMS 5-7 Table 5..6 Properties of Various Cross Sections* (I moment of inertia; I/c section modulus; r I/A radius of gyration) N.A. I b3 I c b 6 r 0.89 b 3 3 b b 3 3 6(b ) b 6 b b 6(b ) b ( cos a b sin a) b cos a b sin a 6 cos a b sin a cos a b sin a I b (H 3 3 ) I c b H H H 4 4 H H r H 3 3 (H ) H H 4 4 H 4 4 H H b 3 36 ; c 3 b 4 8 N.A. I b3 I c b R R R R 3 R 4 r R 0.475R NOTE: Square, axis same as first rectangle, side ; I 4 /; I/c 3 /6; r Square, diagonal taken as axis: I 4 /; I/c ; r Fig relation of stress to strain; and line, an actual stress-strain diagram, applied to te cross section of te beam, compression above and tension below. Te neutral axis is ten below te gravity axis. Te outer material may be expected to develop greater ultimate strengt tan in simple stress, because of te reinforcing action of material nearer te neutral axis tat is not yet overstrained. Tis leads to an equalization of stress over te cross section. S R exceeds te ultimate strengt S M in tension as follows: for cast iron, S R S M ; for sandstone, S R 3S M ; for concrete, S R.S M ; for wood (green), S R.3S M. In te case of steel I beams, failure begins practically wen te elastic limit in te compression flange is reaced. Because of te support of adjoining material, te elastic limit in flexure S p is also greater tan in tension, depending upon te relation of breadt to dept of section. For te same breadt, te difference decreases wit

28 5-8 MECHANICS OF MATERIALS Table 5..6 Properties of Various Cross Sections* (Continued ) Equilateral Polygon A area R rad circumscribed circle r rad inscribed circle n no. sides a lengt of side Axis as in preceding section of octagon I A 4 (6R a ) A 48 (r a ) AR 4 (approx) I c I r I R cos 80 n AR 4 (approx) I A 6R a R 4 r a 48 I 6b 6bb b 36(b b ) 3 I c 6b 6bb b (3b b ) r b bb b 6(b b ) c 3 3b b b b I BH 3 b 3 I c BH 3 b 3 6H r BH 3 b 3 (BH b) I BH 3 b 3 I c BH 3 b 3 6H r BH 3 b 3 (BH b) I 3(Bc 3 B 3 bc 3 3 b 3 ) c ah B d b d (H d ) ah B d b d r I Bd bd a( ) I 3(Bc 3 b 3 ac) 3 c ah bd ah bd c H c r I Bd a(h d) I d4 64 r 4 4 A 4 r 0.05d 4 (approx) I c d3 3 r 3 4 A 4 r 0.d 3 (approx) I A r d 4 increase of eigt. No difference will occur in te case of an I beam, or wit ard materials. Wide plates will not expand and contract freely, and te value of E will be increased on account of side constraint. As a consequence of lateral contraction of te fibers of te tension side of a beam and lateral swelling of fibers at te compression side, te cross section becomes distorted to a trapezoidal sape, and te neutral axis is at te center of gravity of te trapezoid. Strictly, tis sape is one wit a curved perimeter, te radius being r c /, were r c is te radius of curvature of te neutral line of te beam, and is Poisson s ratio. Deflection of Beams Wen a beam is subjected to bending, te fibers on one side elongate, wile te fibers on te oter side sorten (Fig ). Tese canges in lengt cause te beam to deflect. All points on te beam except tose directly over te support fall below teir original position, as sown in Figs and Te elastic curve is te curve taken by te neutral axis. Te radius of curvature at any point is r c EI/M

29 BEAMS 5-9 Table 5..6 Properties of Various Cross Sections* (Continued ) d m (D d) s (D d) I 64 (D4 d 4 ) 4 (R 4 r 4 ) 4A(R r ) 0.05(D 4 d 4 ) (approx) 4 I r r 4 I 0.098(R 4 r 4 ) 0.83R r (R r) R r 0.3tr 3 (approx) wen t is very small r I a3 b a 4 3 b I c D 4 d 4 3 D R 4 r 4 4 R 0.8dms (approx) wen s is very small d m I c 0.908r 3 I c 0.587r c 0.444r c 4 R Rr r 3 R r c R c I A R r D d 4 I A 9 64 r 0.64r 6 I I A (R r ) 0.3r I c a b a 4 b r a (approx) I 4 (a3 b a 3 b 4 a (a 3b)t (approx) I c a(a 3b)t 4 (approx) r I (ab a b ) a a 3b a b (approx) I 3 6 d 4 b( 3 d 3 ) b 3 ( d) I c d 4 b( 3 d 3 ) b 3 ( d) I t 4 B3 I c I H t 3 6 B B 3 H B r r I d b( d) 4 (approx) I B 4 t Fig A beam bent to a circular curve of constant radius as a constant bending moment. Replacing r c in te equation by its approximate geometrical value, /r c d y/dx, te fundamental equation from wic te elastic curve of a bent beam can be developed and te deflection of any beam obtained is, M EI d y/dx (approx) Substituting te value of M, in terms of x, and integrating once, gives te slope of te tangent to te elastic curve of te beam at point x; tan i dy/dx x Mdx/(EI). Since i is usually small, tan i i, 0

30 5-30 MECHANICS OF MATERIALS Table 5..6 Properties of Various Cross Sections* (Continued ) I (b b 3 ), were r 3I t(b 5.H) (H t) b 4(B.6t) (H t) b 4(B.6t) Corrugated seet iron, parabolically curved I c I H t Approximate values of least radius of gyration r r D 0.95D D/4.58 D/3.54 D/6 r D/4.74 D/5 BD/[.6(B D)] D/4.74 * Some of te cross sections depicted in tis table will be encountered most often in macinery as castings, forgings, or individual sections assembled and joined mecanically (or welded). A number of te sections sown are obsolete and will be encountered mainly in older equipment and/or building structures. expressed in radians. A second integration gives te vertical deflection of any point of te elastic curve from its original position. EXAMPLE. In te cantilever beam sown in Fig , te bending moment at any section P(l x) EI d y/(dx). Integrate and determine constant by te condition tat wen x 0, dy/dx 0. Ten EI dy/dx P/x Px. Integrate again; and determine constant by te condition tat wen x 0, y 0. Ten EIy Plx Px 3 /6. Tis is te equation of te elastic curve. Wen x l, y f Pl 3 /(3EI). In general, te two constants of integration must be determined simultaneously. Deflection in general, f, may be expressed by te equation f Pl 3 / (mei ), were m is a coefficient. See Tables 5.. and 5..4 for values of f for beams of various sections and loadings. For coefficients of deflection of wooden beams and structural steel sapes, see Sec... Since I varies as te cube of te dept, te stiffness, or inverse deflection, of various beams varies, oter factors remaining constant, inversely as te load, inversely as te cube of te span, and directly as te cube of te dept. Tis deflection is due to bending moment only. In general, owever, te bending of beams involves transverse searing stresses wic cause searing strains and tus add to te total deflection. Te contribution of searing strain to overall deflection becomes significant only wen te beam span is very sort. Tese strains may affect substantially te strengt as well as te deflection of beams. Wen deflection due to transverse sear is to be accounted for, te differential equation of te elastic curve takes te form EI d y dx EI d y b dx d y s M kei dx AG d M dx were k is a factor dependent upon te beam cross section. Sergius Sergev, in Te Effect of Searing Forces on te Deflection and Strengt of Beams (Univ. Was. Eng. Exp. Stn. Bull. 4) gives k. for rectangular sections, 0/9 for circular sections, and.4 for I beams. He also points out tat in te case of a deep, rectangular-section cantilever, carrying a concentrated load at te free end, te deflection due to sear may be up to 3. percent of tat due to bending moment; if tis beam supports a uniformly distributed load, it may be up to 4. percent. A deep, simple beam deflection may increase up to 5.6 percent wen carrying a uniformly distributed load and up to.5 percent wen te load is concentrated at midspan. Design of beams may be based on strengt (stress) or on stiffness if deflection must be limited. Deflection rater tan stress becomes te criterion for design, e.g., of macine tools, for wic te relative positions of tool and workpiece must be maintained wile te cutting loads are applied during operation. Similarly, large steam-turbine safts supported on two end bearings must maintain alignment and tigt critical clearances between te rotating blade assemblies and te stationary stator blades during operation. Wen more tan one beam sares a load, eac beam will assume a portion of te load tat is proportional to its stiffness. Superposition may be used in connection wit bot stresses and deflections. EXAMPLE. (Fig ). Two wooden stringers one (A)8 6 in in cross section and 0 ft in span, te oter (B) 8in 8in 6 ft carrying te center load P 0,000 lb are required, te load carried by eac stringer. Te deflections f of te two stringers must be equal. Load on A P, and on B P. f P l 3 /(48EI ) P l /(48EI 3 ). Ten P /P l 3 I /(li 3 ) 4. P 0 P P 4P P, wence P,000/5 4,400 lb (,998 kgf ) and P 4 4,400 7,600 lb (7,990 kgf ). Fig Relation between Deflection and Stress Combine te formula M SI/c Pl/n, were n is a constant, P load, and l span, wit formula f Pl 3 /(mei), were m is a constant. Ten f CSl /(Ec) were C is a new constant n/m. Oter factors remaining te same, te deflection varies directly as te stress and inversely as E. If te span is

31 BEAMS 5-3 Table 5..7 Beam Load n m C Cantilever Concentrated at end 3 3 Cantilever Uniform 8 4 Simple Concentrated at center 4 48 Simple Uniform 8 384/ Fixed ends Concentrated at center Fixed ends Uniform One end fixed, one end supported Concentrated at center 6/3 768/ One end fixed, one end supported Uniform 8/ Simple Uniformly varying, maximum at center constant, a sallow beam will submit to greater deformations tan a deeper beam witout exceeding a safe stress. If dept is constant, a beam of double span will attain a given deflection wit only one-quarter te stress. Values of n, m, and C are given in Table 5..7 (for oter values, see Table 5..). Grapical Relations Referring to Fig , te sear V acting at any section is equal to te total load on te rigt of te section, or V w dx Since wdxis te product of w, a loading intensity (wic is expressed as a vertical eigt in te load diagram), and dx, an elementary lengt along te orizontal, evidently wdxis te area of a small vertical strip of te load diagram. Ten wdxis te summation of all suc vertical strips between two indefinite points. Tus, to obtain te sear in any moment diagram Mdxup to tat point; and a slope diagram may be derived from te moment diagram in te same manner as te moment diagram was derived from te sear diagram. If I is not constant, draw a new curve wose ordinates are M/I and use tese M/I ordinates just as te M ordinates were used in te case were I was constant; tat is, (M/I)dx E(i C). Te ordinate at any point of te slope curve is tus proportional to te area of te M/I curve to te rigt of tat point. Again, since ie Edf/dx. ie dx E df E( f C) and tus te ordinate f to te elastic curve at any point is proportional to te area of te slope diagram idxup to tat point. Te equilibrium polygon may be used in drawing te deflection curve directly from te M/I diagram. Tus, te five curves of load, sear, moment, slope, and deflection are so related tat eac curve is derived from te previous one by a process of grapical integration, and wit proper regard to scales te deflection is tereby obtained. Te vertical distance from any point A (Fig ) on te elastic curve of a beam to te tangent at any oter point B equals te moment of te area of te M/(EI) diagram from A to B about A. Tis distance, te tangential deviation t AB, may be used wit te slope-area relation and te geometry of te elastic curve to obtain deflections. Tese teorems, togeter wit te equilibrium equations, can be used to compute reactions in te case of statically indeterminate beams. Fig section mn, find te area of te load diagram up to tat section, and draw a second diagram called te sear diagram, any ordinate of wic is proportional to te sear, or to te area in te load diagram to te rigt of mn. Since V dm/dx, Vdx M By similar reasoning, a moment diagram may be drawn, suc tat te ordinate at any point is proportional to te area of te sear diagram to te rigt of tat point. Since M EI d f/dx, Mdx EI (df/dx C) EI(i C) if I is constant. Here C is a constant of integration. Tus i, te slope or grade of te elastic curve at any point, is proportional to te area of te Fig EXAMPLE. Te deflections of points B and D (Fig ) are y B t AB moment area EIB M A A EI Pl 4 l 4 l Pl EI C C area M B EIC B EI Pl 4 l 4 Pl 6EI y D C l 4 DC t Pl 6EI l 4 l EI Pl 8 l 8 l Pl 3 768EI

32 5-3 MECHANICS OF MATERIALS Resilience of Beams Te external work of a load gradually applied to a beam, and wic increases from zero to P, is Pf and equals te resilience U. But, from te formulas P nsi/(cl) and f nsl /(mce), were n and m are constants tat depend upon loading and supports, S fiber stress, c distance from neutral axis to outer fiber, and l lengt of span. Substitute for P and f, and m k c U n S V E were k is te radius of gyration and V te volume of te beam. For values of U, see Table 5... Te resilience of beams of similar cross section at a given stress is proportional to teir volumes. Te internal resilience, or te elastic deformation energy in te material of a beam in a lengt x is du, and If te two moving loads are of unequal weigt, te condition for maximum moment is tat te maximum moment will occur under te eavy weel wen te center of te beam bisects te distance between te resultant of te loads and te eavy weel. Figure 5..4 sows tis position and te sear and moment diagrams. Wen several weel loads constituting a system occur, te several suspected weels must be examined in turn to determine wic will cause te greatest moment. Te position for te greatest moment tat can occur under a given weel is, as stated, wen te center of te span bisects te distance between te weel in question and te resultant of all te loads ten on te span. Te position for maximum sear at te support will be wen one weel is passing off te span. U M dx/(ei ) M di were M is te moment at any point x, and di is te angle between te tangents to te elastic curve at te ends of dx. Te values of resilience and deflection in special cases are easily developed from tis equation. Rolling Loads Rolling or moving loads are tose loads wic may cange teir position on a beam. Figure represents a beam wit two equal concentrated moving loads, suc as two weels on a crane girder, or te weels of a Fig Fig truck on a bridge. Since te maximum moment occurs were te sear is zero, it is evident from te sear diagram tat te maximum moment will occur under a weel. x a/: R P x a l l M Pl a l x a l l 4x R P x a l l l M Pl a l a x 3a 4x l l l M max wen x 4a M max wen x 3 4a M max Pl l a P ll a l EXAMPLE. Two weel loads of 3,000 lb eac, spaced on 5-ft centers, move on a span of l 5 ft, x.5 ft, and R,500 lb. M max M, (,35.90) 5,600 lb ft (,59 kgf m). Figure sows te condition wen two equal loads are equally distant on opposite sides of te center. Te moment is equal under te two loads. Fig Constrained Beams Constrained beams are tose so eld or built in at one or bot ends tat te tangent to te elastic curve remains fixed in direction. Tese beams are eld at te ends in suc a manner as to allow free orizontal motion, as illustrated by Fig A constrained beam is stiffer tan a simple beam of te same material, because of te modification of te moment by an end resisting moment. Figure sows te two most common cases of constrained beams. See also Table 5... Fig Fig Continuous Beams A continuous beam is one resting upon several supports wic may or may not be in te same orizontal plane. Te general discussion for

33 BEAMS 5-33 beams olds for continuous beams. S v A V, SI/c M, and d f/dx M/(EI). Te sear at any section is equal to te algebraic sum of te components parallel to te section of all external forces on eiter side of te section. Te bending moment at any section is equal to te moment of all external forces on eiter side of te section. Te relations stated above between sear and moment diagrams old true for continuous beams. Te bending moment at any section is equal to te bending moment at any oter section, plus te sear at tat section times its arm, plus te product of all te intervening external forces times teir respective arms. To illustrate (Fig ): V x R R R 3 P P P 3 M x R (l l x) R (l x) R 3 x P (l c x) P (b x) P 3 a M x M 3 V 3 x P 3 a Table 5..8 gives te value of te moment at te various supports of a uniformly loaded continuous beam over equal spans, and it also gives te values of te sears on eac side of te supports. Note tat te sear is of opposite sign on eiter side of te supports and tat te sum of te two sears is equal to te reaction. Figure sows te values of te functions for a uniformly loaded continuous beam resting on tree equal spans wit four supports. Continuous beams are stronger and muc stiffer tan simple beams. However, a small, unequal subsidence of piers will cause serious Fig Fig Figure sows te relation between te moment and sear diagrams for a uniformly loaded continuous beam of four equal spans (see Table 5..8). Table 5..8 also gives te maximum bending moment wic will occur between supports, and in addition te position of tis moment and te points of inflection (see Fig ). Fig Table 5..8 Uniformly Loaded Continuous Beams over Equal Spans (Uniform load per unit lengt w; lengt of equal span l) Sear on eac side of support. L left, R rigt. Reaction at any Distance to point Distance to point Notation of support is L R Moment Max of max moment, of inflection, Number of support over eac moment in measured to rigt measured to rigt supports of span L R support eac span from support from support or None , , , , , , , , , , , Values apply to: wl wl wl wl l l NOTE: Te numerical values given are coefficients of te expressions at te foot of eac column.

34 5-34 MECHANICS OF MATERIALS Table 5..9 Beams of Uniform Strengt (in Bending) Elevation Beam Cross section and plan Formulas. Fixed at one end, load P concentrated at oter end Rectangle: widt (b) constant, dept (g) variable Elevation:, top, straigt line; bottom, parabola., complete parabola Plan: rectangle y 6P x bs s 6Pl bs s Deflection at A: f 8P be l 3 Rectangle: widt (y) variable, dept () constant Rectangle: widt (z) variable, dept (y) variable z y k(const.) Elevation: rectangle Plan: triangle Elevation: cubic parabola Plan: cubic parabola y 6P x S s b 6Pl S s Deflection at A: f 6P be l 3 y 3 6P x ks s z ky 3 6Pl ks s b k Circle: diam (y) variable Elevation: cubic parabola Plan: cubic parabola y 3 3P x S s d 3 3Pl S s. Fixed at one end, load of total magnitude P uniformly distributed Rectangle widt (b) constant, dept (y) variable Elevation: triangle Plan: rectangle y x 3P bls 3Pl bs s f 6 P be l 3 Rectangle: widt (y) variable, dept () constant Elevation: rectangle Plan: two parabolic curves wit vertices at free end y 3Px ls s b 3Pl S s Deflection at A: f 3P be l 3

35 BEAMS 5-35 Table 5..9 Beams of Uniform Strengt (in Bending) (Continued ) Elevation Beam Cross section and plan Formulas. Fixed at one end, load of total magnitude P uniformly distributed Rectangle: widt (z) variable, dept (y) variable, z y k Elevation: semicubic parabola Plan: semicubic parabola y 3 3Px ks s l z ky 3 3Pl ks s b k Circle: diam (y) variable Elevation: semicubic parabola Plan: semicubic parabola y 3 6P ls s x d 3 6Pl S s 3. Supported at bot ends, load P concentrated at point C Rectangle: widt (b) constant, dept (y) variable Elevation: two parabolas, vertices at points of support Plan: rectangle y 3P S s b x 3Pl bs s f Eb P l 3 Rectangle: widt (y) variable, dept () constant Elevation: rectangle Plan: two triangles, vertices at points of support y 3P S s x b 3Pl S s f 3Pl3 8Eb 3 Rectangle: widt (b) constant, dept (y or y ) variable Elevation: two parabolas, vertices at points of support Plan: rectangle 6P(l p) y bls s y 6P p x bls s x 6P(l p)p bls s Load P moving across span Rectangle: widt (b) constant, dept (y) variable Elevation: ellipse Major axis l Minor axis Plan: rectangle x l 3Pl bs s y 3Pl bs s

36 5-36 MECHANICS OF MATERIALS Table 5..9 Beams of Uniform Strengt (in Bending) (Continued ) Elevation Beam Cross section and plan Formulas 4. Supported at bot ends, load of total magnitude P uniformly distributed Rectangle: widt (b) constant, dept (y) variable Elevation: ellipse x l 3Pl 4bS s y 3Pl 4bS s Rectangle: widt (y) variable, dept () constant Plan: rectangle Elevation: rectangle Plan: two parabolas wit vertices at center of span Deflection at O: f Pl 3 64 EI 3 P 6 be l y 3P S s x x b 3Pl 4S s 3 l canges in sign and magnitude of te bending stresses. reactions, and sears. Maxwell s Teorem Wen a number of loads rest upon a beam, te deflection at any point is equal to te sum of te deflections at tis point due to eac of te loads taken separately. Maxwell s teorem states tat if unit loads rest upon a beam at two points A and B, te deflection at A due to te unit load at B equals te deflection at B due to te unit load at A. Castigliano s teorem states tat te deflection of te point of application of an external force acting on a beam is equal to te partial derivative of te work of deformation wit respect to tis force. Tus, if P is te force, f te deflection, and U te work of deformation, wic equals te resilience, du/dp f According to te principle of least work, te deformation of any structure takes place in suc a manner tat te work of deformation is a minimum. Beams of Uniform Strengt Beams of uniform strengt vary in section so tat te unit stress S remains constant, te I/c varies as M. For rectangular beams, of breadt b and dept d, I/c bd /6; and M Sbd /6. Tus, for a cantilever beam of rectangular cross section, under a load P, Px Sbd /6. If b is constant, d varies wit x, and te profile of te sape of te beam will be a parabola, as Fig If d is constant, b will vary as x and te beam will be triangular in plan, as sown in Fig Sear at te end of a beam necessitates a modification of te forms determined above. Te area required to resist sear will be P/S v in a cantilever and R/S v in a simple beam. Te dotted extensions in Figs and sow te canges necessary to enable tese cantilevers to resist sear. Te extra material and cost of fabrication, owever, make many of te forms impractical. Table 5..9 sows some of te simple sections of uniform strengt. In none of tese, owever, is sear taken into account. Fig Fig TORSION Under torsion, a bar (Fig ) is twisted by a couple of magnitude Pp. Elements of te surface becomes elices of angle d, and a radius rotates troug an angle in a lengt l, bot d and being expressed in radians. S v searing unit stress at distance r from center; I p polar moment of inertia; G searing modulus of elasticity. It is assumed tat te cross sections remain plane surfaces. Te strain on te cross section is wolly tangential, and is zero at te center of te section. Note tat ld r. In te case of a circular cross section, te stress S v increases directly as te distance of te strained element from te center. Te polar moment of inertia I p for any section may be obtained from I p I I, were I and I are te rectangular moments of inertia of te section about any two lines at rigt angles to eac oter, troug te center of gravity.

37 TORSION 5-37 Te external twisting moment M t is balanced by te internal resisting moment. For strengt, M t S v I p /r. For stiffness, M t agi p /l. Te torsional resilience U Ppa S vi p l/(r G) a GI p /(l). Fig Te state of stress on an element taken from te surface of te saft, as in Fig , is pure sear. Pure tension exists at rigt angles to one 45 elix and pure compression at rigt angles to te opposite elix. Reduced formulas for safts of various sections are given in Table 5... Wen te ratio of saft lengt to te largest lateral dimension in te cross section is less tan approximately, te end effects may drastically affect te torsional stresses calculated. Failure under torsion in brittle materials is a tensile failure at rigt angles to a elical element on te surface. Plastic materials twist off squarely. Fibrous materials separate in long strips. Torsion of Noncircular Sections Wen a section is not circular, te unit stress no longer varies directly as te distance from te center. Cross sections become warped, and te greatest unit stress usually occurs at a point on te perimeter of te cross section nearest te axis of twist; tus, tere is no stress at te corners of square and rectangular sections. Te analyses become complex for noncircular sections, and te metods for solution of design problems using tem most often admit only of approximations. wic as te same sape as te bar and ten inflated. Te resulting tree-dimensional surface provides te following: () Te torque transmitted is proportional to twice te volume under te inflated membrane, and () te sear stress at any point is proportional to te slope of te curve measured perpendicular to tat slope. In recent years, several oter matematical tecniques ave become widely used, especially wit te aid of faster computational metods available from electronic computers. By using finite-difference metods, te differential operators of te governing equations are replaced wit difference operators wic are related to te desired unknown values at a gridwork of points in te outline of te cross section being investigated. Te finite-element metod, commonly referred to as FEM, deals wit a spatial approximation of a complex sape wic is ten analyzed to determine deformations, stresses, etc. By using FEM, te exact structure is replaced wit a set of simple structural elements interconnected at a finite number of nodes. Te governing equations for te approximate structure can be solved exactly. Note tat inasmuc as tere is an exact solution for an approximate structure, te end result must be viewed, and te results tereof used, as approximate solutions to te real structure. Using a finite-difference approac to Poisson s partial differential equation, wic defines te stress functions for solid and ollow safts wit generalized contours, along wit Prandl s membrane analogy, Isakower as developed a series of practical design carts (ARRADCOM- MISD Manual UN 80-5, January 98, Department of te Army). Dimensionless carts and tables for transmitted torque and maximum searing stress ave been generated. Information for circular safts wit rectangular and circular keyways, external splines and milled flats, as well as rectangular and X-saped torsion bars, is presented. Assuming te stress distribution from te point of maximum stress to te corner to be parabolic, Bac derived te approximate expression, S s M 9M t /(b ) for a rectangular section, b by, were b. For closer results, te searing stresses for a rectangular section (Fig. 5..5) may be expressed S A M t /( A b ) and S B M t /( B b ). Te angle of twist for tese safts is M t l/(gb 3 ). Te factors A, B, and are functions of te ratio /b and are given in Table In te case of composite sections, suc as a tee or angle, te torque tat can be resisted is M t Gb 3 ; te summation applies to eac of te rectangles into wic te section can be divided. Te maximum stress occurs on te component rectangle aving te largest b value. It is computed from S A M t A b A /( A b 3 ) Torque, deflection, and work relations for some additional sections are given in Table 5... Fig Torsion problems ave been solved for many different noncircular cross sections by utilizing te membrane analogy, due to Prandtl, wic makes use of te fact tat te matematical treatment of a twisted bar is governed by te same equations as for a membrane stretced over a ole Fig Table 5..0 Factors for Torsion of Rectangular Safts (Fig. 5..5) b/b A B

38 5-38 MECHANICS OF MATERIALS Table 5.. Torsion of Safts of Various Cross Sections (For strengt and stiffness of safts, see Sec. 8.) Angular twist, Torsional (lengt in, radius in) resisting moment In terms of tor- In terms of max Work of torsion Cross section M t sional moment sear (V volume) 6 6 d M 3 S t v 3 M t GI P d 4 G D 4 d 4 D 3 M S t v (D 4 d 4 ) G S vmax G S vmax G d D S v max 4 G V (Note ) S v D d max 4 G D (Note ) V 6 b S v ( b) 6 b M t b 3 3 G S vmax G b S v b max b 8 G (Note 3) V 9b S v ( b) 3.6* b b 3 3 M t G 0.8* S vmax G b 4 S v b max b 45 G (Note 4) V 9 3 S v 7. 4 M t G.6 S vmax G 8 45 S v max G V (Note 5) b 0 S v 46. M t b 4 G.3 S vmax G b b 3.09 S v M t b 4 G 0.9 S vmax G b *Wen /b 4 8 Coefficient 3.6 becomes Coefficient 0.8 becomes NOTES: () S vmax at circumference. () S vmax at outer circumference. (3) S vmax at A; S vb 6M t /b. (4) S vmax at middle of side ; in middle of b, S v 9M t /b. (5) S vmax at middle of side. COLUMNS Members subjected to direct compression can be grouped into tree classes. Compression blocks are so sort (slenderness ratios below 30) tat bending of member is unlikely. At te oter limit, columns so slender tat bending is primary, are te long columns defined by Euler s teory. Te intermediate columns, quite common in practice, are called sort columns. Long columns and te more slender sort columns usually fail by buckling wen te critical load is reaced. Tis is a matter of instability; tat is, te column may continue to yield and deflect even toug te load is not increased above critical. Te slenderness ratio is te unsupported lengt divided by te least radius of gyration, parallel to wic it can bend. Long columns are andled by Euler s column formula, P cr n EI/l n EA/(l/r) Te coefficient n accounts for end conditions. Wen te column is pivoted at bot ends, n ; wen one end is fixed and oter rounded, n ; wen bot are fixed, n 4; and wen one end is fixed wit te oter free, n 4. Te slenderness ratio tat separates long columns from sort ones depends upon te modulus of elasticity and te yield strengt of te column material. Wen Euler s formula results in (P cr /A) S y, strengt rater tan buckling causes failure, and te column ceases to be long. In round numbers, tis critical slenderness ratio falls between 0 and 50. Table 5.. gives additional facts concerning long columns. Sort Columns Te stress in a sort column may be considered partly due to compression and partly due to bending. A teoretical equation as not been derived. Empirical, toug rational, expressions are, in general, based on te assumption tat te permissible stress must be reduced below tat wic could be permitted were it due to compression only. Te manner in wic tis reduction is made determines te type of equation as well as te slenderness ratio beyond wic te equation does not apply. Figure 5..5 illustrates te situation. Some typical formulas are given in Table EXAMPLE. A macine member unsupported for a lengt of 5 in as a square cross section 0.5 in on a side. It is to be subjected to compression. Wat maximum safe load can be applied centrally, according to te AISC formula? At te com-

39 Table 5.. Strengt of Round-ended Columns according to Euler s Formula* Low- Medium- Wrougt carbon carbon Material Cast iron iron steel steel Ultimate compressive strengt, lb/in 07,000 53,400 6,600 89,000 Allowable compressive stress, lb/in 7,00 5,400 7,000 0,000 (maximum) Modulus of elasticity 4,00,000 8,400,000 30,600,000 3,300,000 Factor of safety Smallest I allowable at worst section, in 4 Pl Pl Pl Pl 7,500,000 56,000,000 60,300,000 6,700,000 Limit of ratio, l/r Rectangle (r b ), l/b Circle (r 4d), l/d Circular ring of small tickness (r d 8), l/d * P allowable load, lb; l lengt of column, in; b smallest dimension of a rectangular section, in; d diameter of a circular section, in; r least radius of gyration of section. Tis material is no longer manufactured but may be encountered in existing structures and macinery. COLUMNS 5-39 Table 5..3 Typical Sort-Column Formulas Formula Material Code Slenderness ratio S w 7, r l Carbon steels AISC l/r 0 S w 6, (l/r) Carbon steels Cicago l/r 0 S w 5,000 50r l Carbon steels AREA l/r 50 S w 9, (l/r) Carbon steels Am. Br. Co. 60 l r 0 S cr * 35, c l r Alloy-steel tubing ANC S w 9, l r Cast iron NYC S cr * 34, S cr * 5, S cr * S y S cr * c l r 07ST Aluminum ANC l c r Spruce ANC S y 4n Er l S y ec r sec l r P 4AE Steels Jonson Steels * S cr teoretical maximum, c end fixity coefficient, c, bot ends pivoted; c.86, one pivoted, oter fixed; c 4, bot ends fixed; c, one end fixed, one end free. e is initial eccentricity at wic load is applied to center of column cross section. Secant cr 65 l r 70 cr 94 l cr 7 l r n E S y l r critical puted load, wat size section (also square) would be needed, if it were to be designed according to te AREA formula? l/r 5/0.5/ 04 sort column P/A 7, (04),730 or P 0.5,730,940 lb (,335 kgf ),940 5,000 and a 50 5la 5,000,600 a tus a 0.73a or a in (.36 cm) Combined Flexure and Longitudinal Force Figure sows a bar under flexure due to transverse and longitudinal loads. Te maximum fiber stress S is made up of S 0, due to te direct action of load P, and S b, due to te entire bending moment M. M is te algebraic sum of two bending moments, M due to longitudinal load ( for compression and for tension), and M due to transverse load. M M M. Here M Pf and f CS b / /(Ec). Fig. 5..5

40 5-40 MECHANICS OF MATERIALS FOR THE CASE OF LONGITUDINAL COMPRESSION. S b I/c M CPS b l /(Eo), or S b M c(i CPl /E). Te maximum stress is S S b S 0 compression. Te constant C for te case of Fig is derived from te equations Pl/4 S b I/c and f Pl 3 /(48EI ). Solving for f; f S b l /(Ec), or C. For a beam supported at te ends and uniformly loaded, C Oter cases can be similarly calculated. FOR THE CASE OF LONGITUDINAL TENSION. M M Pf, and S b M c/(i CPl /E). Te maximum stress is S S b S 0, tension. Fig ECCENTRIC LOADS Wen sort blocks are loaded eccentrically in compression or in tension, i.e., not troug te center of gravity (cg), a combination of axial and bending stress results. Te maximum unit stress S M is te algebraic sum of tese two unit stresses. In Fig a load P acts in a line of symmetry at te distance e from cg; r radius of gyration. Te unit stresses are () S c, due to P,as if it acted troug cg, and () S b, due to te bending moment of P acting wit a leverage of e about cg. Tus unit stress S at any point y is S S c S b P/A Pey/I S c ( ey/r ) y is positive for points on te same side of cg as P, and negative on te opposite side. For a rectangular cross section of widt b, te maximum stress S M S c ( 6e/b). Wen P is outside te middle tird of widt b and is a compressive load, tensile stresses occur. For a circular cross section of diameter d, S M S c ( 8e/d). Te stress due to te weigt of te solid will modify tese relations. NOTE. In tese formulas e is measured from te gravity axis, and gives tension wen e is greater tan one-sixt te widt (measured in te same direction as e), for rectangular sections; and wen greater tan one-eigt te diameter for solid circular sections. If, as in certain classes of masonry construction, te material cannot witstand tensile stress and tus no tension can occur, te center of moments (Fig ) is taken at te center of stress. For a rectangular section, P acts at distance k from te nearest edge. Lengt under compression 3k, and S M 3P/(k). For a circular section, S M [ (k/r)]P/k rk, were r radius and k distance of P from circumference. For a circular ring, S average compressive stress on cross section produced by P; e eccentricity of P; z lengt of diameter under compression (Fig ). Values of z/r and of te ratio of S max to average S are given in Tables 5..4 and Fig CHIMNEY PROBLEM. Weigt of cimney 563,000 lb; e.56 ft; OD of cimney 0 ft 8 in; ID 6ft6 in. Overturning moment Pe 878,000 ft lb, r /r 0.6. e/r 0.9. Tis gives z/r. Terefore, te entire area of te base is under compression. Area under compression 55.8 ft ; I 546; S 563,000/55.8 (878, )/546 8,700 (max) and,500 (min) lb compression per ft. From Table 5..5, by interpolation, S max /S av.85. S max (563,000/55.8).85 8,685 lb/ft (9,33 kgf/m ). Te kern is te area around te center of gravity of a cross section witin wic any load applied will produce stress of only one sign trougout te entire cross section. Outside te kern, a load produces stresses of different sign. Figure sows kerns (saded) for various sections. For a circular ring, te radius of te kern r D[ (d/d) ]/8. Fig Fig Fig

41 CURVED BEAMS 5-4 Table 5..4 Values of te Ratio z/r (Fig ) r e r r e r Table 5..5 Values of te Ratio S max /S avg (In determining S average, use load P divided by total area of cross section) r r e r e r For a ollow square (H and lengts of outer and inner sides), te kern is a square similar to Fig a, were r min H H H H For a ollow octagon R a and R i radii of circles circumscribing te outer and inner sides; tickness of wall 0.939(R a R i ), te kern is an octagon similar to Fig c, were 0.56R becomes 0.56R a [ (R i /R a ) ]. CURVED BEAMS Te application of te flexure formula for a straigt beam to te case of a curved beam results in error. Wen all fibers of a member ave te same center of curvature, te concentric or common type of curved beam exists (see Fig ). Suc a beam is defined by te Winkler-Bac teory. Te stress at a point y units from te centroidal axis is S AR M y Z(R y)

42 5-4 MECHANICS OF MATERIALS M is te bending moment, positive wen it increases curvature; Y is positive wen measured toward te convex side; A is te cross-sectional area; R is te radius of te centroidal axis; Z is a cross-section property defined by Z A y R y da Analytical expressions for Z of certain sections are given in Table Also Z can be found by grapical integration metods (see any advanced Fig strengt book). Te neutral surface sifts toward te center of curvature, or inside fiber, an amount equal to e ZR/(Z ). Te Winkler-Bac teory, toug practically satisfactory, disregards radial stresses as well as lateral deformations and assumes pure bending. Te maximum stress occurring on te inside fiber is S M i /(AeR i ), wile tat on te outside fiber is S M 0 /(AeR 0 ). EXAMPLE. A split steel ring of rectangular cross section is subjected to a diametral force of,000 lb as sown in Fig a. Compute te stress at te point 0.5 in from te outside fiber on plane mm. Also compute te maximum stress. Z R ln R C R C 0 0 ln S.5 AR M y Z (R y) F A, (0.5),000 8,50 5,5 lb/in (compr.)(79 kgf/cm ) S M, (0 ),000 8,30 5,355 lb/in (66 kgf/cm ) or e ZR Z and S M M i AeR i F A, , ,355 lb/in (66 kgf/cm ) Te deflection in curved beams can be computed by means of te moment-area teory. If te origin of axes is taken at te point wose deflection is wanted, it can be sown tat te component displacements in te x and y directions are x s 0 My ds EI and y s 0 Mx ds EI Te resultant deflection is ten equal to 0 x y in te direction defined by tan y / x. Deflections can also be found conveniently by use of Castigliano s teorem. It states tat in an elastic system te displacement in te direction of a force (or couple) and due to tat force (or couple) is te partial derivative of te strain energy wit respect to te force (or couple). Stated matematically, z U/F z. If a force does not exist at te point and/or in te direction desired, a dummy force may be applied. Tis force must ten be eliminated by equating it to zero at te end. EXAMPLE. A quadrant of radius R is fixed at one end as sown in Fig b. Te force F is applied in te radial direction at te free end B. Find te deflection of B. By moment area: B y FR 3 y R sin x R( cos ) ds Rd M FR sin B x FR 3 / EI 0 / EI 0 sin d FR 3 4EI sin ( cos ) d FR 3 and B FR 3 EI 4 EI FR 3 at x tan EI 4EI 3 tan FR 3.5 By Castigliano: B x U F B y U F y F y 0 FR 3 EI F x / 0 / F R 3 EI sin d FR 3 4EI [FR sin F y R ( cos )] Rd EI Te F y, assumed downward, is equated to zero, after te integration and differentiation are performed to find B y. Te remainder of te computation is exactly as in te moment-area metod. Fig Eccentrically Curved Beams Tese beams (Fig ) are bounded by arcs aving different centers of curvature. In addition, it is possible for eiter radius to be te larger one. Te one in wic te section dept sortens as te central section is approaced may be called te arc beam. Wen te central section is te largest, te beam is of te crescent type. Crescent I denotes te beam of larger outside radius and crescent II of larger inside radius. Te stress at te central section of suc beams may be found from S KMC/I. In te case of rectangular cross section, te equation becomes S 6KM/(b ) were M is te bending moment, b is te widt of te beam section, and its eigt. Te stress factors K for te inner boundary, establised from potoelastic data, are given in Table Te outside radius is denoted by R o and te inside by R i. Te geometry of crescent beams is suc tat te stress can be larger in off-center sections. Te stress at te central section determined above must ten be multiplied by te position factor k, given in Table As in te concentric beam, te neutral surface sifts sligtly toward te inner boundary (see Vidosic, Curved Beams wit Eccentric Boundaries, Trans. ASME, 79, pp., 37 3). Fig

43 IMPACT 5-43 Table 5..6 Section Analytical Expressions for Z Expression Z R ln R C R C Z R r R r R r Z R At ln(r C ) (b t) ln(r C 3 ) b ln(r C 3 ) A tc (b t)c 3 bc Z R Ab ln R C R C (t b) ln R C R C A [(t b)c bc ] Table 5..7 Stress Factors for Inner Boundary at Central Section (See Fig ). For te arc-type beams (a) K if R o R i 5. R o R i (b) K if 5 R o R i 0. R o R i (c) In te case of larger section ratios use te equivalent beam solution.. For te crescent I-type beams (a) K if R o R i. R o R i (b) K if R o R i 0. R o R i (c) K R o R i R o R i if For te crescent II-type beams (a) K if R o R i 8. R o R i (b) K.9 (c) K R o R i R o R i if R o R i if R o R i Table 5..8 Crescent-Beam Position Stress Factors (See Fig ) Angle k, deg Inner Outer H/ 0.03H/ H/ 0.0H/ H/ 0.5H/ H/ 0.467H/ 50.5 ( H/) H/ ( H/) H/ ( H/) H/ ( H/) H/ H/ NOTE: All formulas are valid for 0 H/ Formulas for te inner boundary, except for 40 deg, may be used to H/ H distance between centers. IMPACT A force or stress is considered suddenly applied wen te duration of load application is less tan one-alf te fundamental natural period of vibration of te member upon wic te force acts. Under impact, a

44 5-44 MECHANICS OF MATERIALS compression wave propagates troug te member at a velocity c E/, were is te mass density. As tis compression wave travels back and fort by reflection from one end of te bar to te oter, a maximum stress is produced wic is many times larger tan wat it would be statically. An exact determination of tis stress is most difficult. However, if conservation of kinetic and strain energies is applied, te impact stress is found to be S S W W b 3W 3W W b Te weigt of te striking mass is ere denoted by W, tat of te struck bar by W b, wile S is te static stress, W/A (A is te cross-sectional area of te bar). Above is te case of sudden impact. Wen te ratio W/W b is small, te stress computed by te above equation may be erroneous. A better solution of tis problem may result from S S S W W b 3 If a weigt W falls a distance before striking a bar of mass W b, energy conservation will yield te relation S S e 3W 3W W b Te elongation e l SI/E. Wen te striking mass W is assumed rigid, te elasticity factor is taken equal to. Tus te equation becomes S S ( /e) If, in addition, is taken equal to zero (sudden impact), te radical equals, and so te stress becomes S S. Since Hooke s law is applicable, te relations e e ( /e) and e e are also true for te same conditions. Te expression may be converted, by using v g, to S S [ v /(eg)] Tis migt be called te energy impact form. If te natural frequency f n of te bar is used, te stress equation is S S ( 0.04f n) In general, te maximum impact stress in a beam and a saft can be approximated from te simplified falling-weigt equation. It is necessary, toug, to substitute te maximum deflection y for e, in te case of beams, and for te angle of twist in te case of safts. Of course S Mc/I and M t c/j, respectively. Tus S S y (or ) For a more exact solution, elastic yield in eac member must be considered. Te teory ten yields S S y 35W 35W 7W b for a simply supported beam struck in te middle by a weigt W. THEORY OF ELASTICITY Loaded members in wic te stress distribution cannot be estimated fail of solution by elementary strengt-of-material metods. To suc cases, te more advanced matematical principles of te teory of elasticity must be applied. Wen tis is not possible, experimental stress analysis as to be used. Because of te complexity of solution, only some of te more practical problems ave been solved by te teory of elasticity. Te more general concepts and metods are presented. Two kinds of forces may act on a body. Surface forces are distributed over te surface as te result of, for instance, te pressure of one body on anoter. Forces due to gravity, inertia, magnetism, etc., wic act over te entire volume of a body, are called body forces. Bot surface and body forces can be best andled if resolved into tree ortogonal components. Surface forces are tus designated X, Y, and Z, wile body forces are labeled X, Y, and Z. In general, tere exists a normal stress and a searing stress at eac point of a loaded member. It is convenient to deal wit components of eac of tese stresses on eac of six ortogonal planes tat bound te point element. Tus tere are at eac point six stress components, x, y, z, yx xy, xz zx, and yz zy. Similarly, if te normal unit strain is designated by te letter and searing unit strain by, te six components of strain are defined by x u/x y v/y z w/z xy u/y v/x yz v/z w/y and xz u/z w/x Te elastic displacements of particles on te body in te x, y, and z directions are identified as te u, v, and w components, respectively. Since metals ave te usually assumed elastic as well as isotropic properties, Hooke s law olds. Terefore, te interrelationsips between stress and strain can easily be obtained. x E [ x ( y z )] y E [ y ( x z )] z E [ z ( x y )] xy xy /G xz xz /G and yz yz /G Te general case of strain can be obtained by superposing te elongation strains upon te searing strains. Problems depending upon teories of elasticity are considerably simplified if te stresses are all parallel to one plane or if all deformations occur in planes perpendicular to te lengt of te member. Te first case is one of plane stress, as wen a tin plate of uniform tickness is subjected to central, boundary forces parallel to te plane of te plate. Te second is a case of plane strain, suc as a gate subjected to ydrostatic pressure, te intensity of wic does not vary along te gate s lengt. All particles terefore displace at rigt angles to te lengt, and so cross sections remain plane. In plane-stress problems, tree of te six stress components vanis, tus leaving only x, y, and xy. Similarly, in plane strain, only x, y, and xy will not equal zero; tus te same tree stresses x, y, and xy remain to be considered. Plane problems can tus be represented by te element sown in Fig Equilibrium considerations applied to tis particle result in te differential equations of equilibrium wic reduce to x x xy y X 0 and y y xy x Y 0 Since te two differential equations of equilibrium are insufficient to find te tree stresses, a tird equation must be used. Tis is te compatibility equation relating te tree strain components. It is x y y x xy x y If strains are expressed in terms of te stresses, te compatibility equation becomes x y ( x y ) 0 Now, in any two-dimensional problem, te compatibility equation along wit te differential equilibrium equations must be simultaneously

45 CYLINDERS AND SPHERES 5-45 solved for te tree unknown stresses. Tis is accomplised using stress functions, wic permit te integration and satisfy boundary conditions in eac particular situation. CYLINDERS AND SPHERES A tin-wall cylinder as a wall tickness suc tat te assumption of constant stress across te wall results in negligible error. Cylinders aving internal-diameter-to-tickness (D/t) ratios greater tan 0 are usually considered tin-walled. Boilers, drums, tanks, and pipes are often treated as suc. Equilibrium equations reveal te circumferential, or oop, stress to be S pr/t under an internal pressure p (see Fig. 5..6). If te cylinder is closed at te ends, a longitudinal stress of pr/(t) is developed. Te tensile stress developed in a tin ollow spere subjected to internal pressure is also pr/(t). Fig In tree-dimensional problems, te tird dimension must be considered. Tis results in tree differential equations of equilibrium, as well as tree compatibility equations. Te six stress components can tus be found. Te complexity involved in te solution of tese equations is suc, owever, tat only a few special cases ave been solved. In certain problems, suc as rotating circular disks, polar coordinates become more convenient. In suc cases, te stress components in a two-dimensional field are te radial stress r, te tangential stress, and te searing stress r. In terms of tese stresses te polar differential equations become r r r r r R 0 r and r r r r 0 r Te body force per unit volume is represented by R. Te compatibility equation in polar coordinates is r r r r r r r 0 r is again a stress function of r and tat will provide a solution of te differential equations and satisfy boundary conditions. As an example, te exact solution of a simply supported beam carrying a uniformly distributed load w yields x w I (I x ) y I w y3 3 c y 5 Te origin of coordinates is at te center of te beam, c is te beam dept, and l is te span lengt. Tus te maximum stress at x 0 and y c is x wl c wc 3. Te first term represents te stress as I 5 I obtained by te elementary flexure teory; te second is a correction. Te second term becomes negligible wen c is small compared to l. Te important case of a flat plate of unit widt wit a circular ole of diameter a at its center, subjected to a uniform tensile load, as been solved using polar coordinates. If S is te uniform stress at some distance from te ole, r is measured from te center of te ole, and is te angle of r wit respect to te longitudinal axis of te member, te stresses are r S a S r 3a4 r 4a 4 r cos S a r S 3a4 r 4 cos r S 3a4 r 4 a r sin Fig Wen tin-walled cylinders, suc as vacuum tanks and submarines, are subjected to external pressure, collapse becomes te mode of failure. Te sell is assumed perfectly round and of uniform tickness, te material obeys Hooke s law, te radial stress is negligible, and te normal stress distribution is linear. Oter, lesser assumptions are also made. Using te teory of elasticity, R. G. Sturm (Univ. Ill. Eng. Exp. Stn. Bull., no, Nov., 94) derived te collapsing pressure as W c KED3 t lb/in Te factor K, a numerical coefficient, depends upon te L/R and D/t ratios (D is outside-sell diameter), te kind of end support, and weter pressure is applied radially only, or at te ends as well. Figures to 5..66, reproduced from te bulletin, supply te K values. N on tese carts indicates te number of lobes into wic te sell collapses. Tese values are for materials aving Poisson s ratio 0.3. It may also be pointed out tat in te case of long cylinders (infinitely long, teoretically) te value of K approaces /( ). Fig Radial external pressure wit simply supported edges. Wen te cylinder is stiffened wit rings, te sell may be assumed to be divided into a series of sorter sells, equal in lengt to te ring spacing. Te previous equation can ten be applied to a ring-to-ring lengt of cylinder. However, te flexural rigidity of te combined

46 5-46 MECHANICS OF MATERIALS stiffener and sell EI c necessary to witstand te pressure is EI c W s D 3 L s /4. W s is te pressure, L s te lengt between rings, and I c te combined moment of inertia of te ring and tat portion of te sell assumed acting wit te ring. sown in Fig b and te equation is integrated, te general tangential and radial stress relations, called te Lamé equations, are derived. S t r p r p ( p p )r r /r r r and S r r p r p (p p )r r /r r r Wen te external pressure p 0, te equations reduce to S t r p r r r r r and S r r p r r r Fig Radial external pressure wit fixed edges. In some instances, cylinders collapse only after a stress in excess of te elastic limit as been reaced; tat is, plastic range stresses are present. In suc cases te same equation applies, but te modulus of elasticity must be modified. Wen te average stress S a is less tan te proportional limit S p, and te maximum stress (direct, plus bending) is S, te modified modulus S u S a E E 4 S S l S u is te modulus of rupture. Wen te average stress is larger tan te proportional limit, te modified modulus is taken as te tangent at te average stress. Fig Radial and end external pressure wit simply supported edges. In tick-walled cylinders (Fig a) te circumferential, oop, or tangential stress S t is not uniform. In addition a radial stress S r is present. Wen equilibrium is applied to te annulus taken out of Fig a and Fig Radial and end external pressure wit fixed edges. At te inner boundary te tangential elongation t is equal to t (S t S r )/E Te increase in te bore radius r resulting terefrom is r r p E r /r r /r Similarly a solid saft of r radius under external pressure p will ave its radius decreased by te amount r rp E s ( ) In te case of a press or srink fit, p p p. Te sum of r and r absolute is te radial interference; twice tis sum is te diametral interference or r p E r /r r /r E s If te ub and saft materials are te same, E E s E, and 4r r p E (r r ) If te equation is solved for p and tis value is substituted in Lamé s equation, te maximum tangential stress on te inner surface of te ub is found to be S t E (r 4r r r )

47 FLAT PLATES 5-47 PRESSURE BETWEEN BODIES WITH CURVED SURFACES Fig EXAMPLE. Te barrel of a field gun as an outside diameter of 9 in and a bore of 4.7 in. An internal pressure of 6,000 lb/in is developed during firing. Wat maximum stress occurs in te barrel? An investigation of Lamé s equations for internal pressure reveals te maximum stress to be te tangential one on te inner surface. Tus, r r S t p 6,000 ( ) r r ,000 lb/in (,97 kgf/cm ) Oval Hollow Cylinders In Fig , let a and b be te semiminor and semimajor axes. Te bending moments at A and C will ten be M 0 pa / pi x /(S) pi y /(S) M M 0 p(a b )/ were I x and I y are te moments of inertia of te arc AC about te x and y axes, respectively. Te bending moment at any point will be M M 0 pa / px / py / Tick Hollow Speres Wit an internal pressure p, were p T/0.65. r r [(T 0.4p)/(T 0.65p)] /3 Te maximum tensile stress is on te inner surface, in te direction of te circumference. Wit an external pressure p, were p T/.05, r r [T/(T.05p)] /3 In bot cases T is te true stress. Fig (See Hertz, Gesammelte Werke, vol., pp. 59 et seq., Bart.) Two Speres Te radius A of te compressed area is obtained from te formula A P(c c )/(/r /r ), in wic P is te compressing force, c and c ( /E and /E ) are reciprocals of te respective moduli of elasticity, and r and r are te radii. (Reciprocal of Poisson s ratio is assumed to be n 0/3.) Te greatest contact pressure in te middle of te compressed surface will be S max.5(p/a ), and S 3 max 0.35P(/r /r ) /(c c ) Te total deformation of te two speres will be Y, wic is obtained from Y P (c c ) (/r /r ) For c c /E, i.e., two speres wit te same modulus of elasticity, it follows tat A 3.36 P/E(/r /r ), S 3 max 0.059PE (/r /r ), and Y 3.84P (/r /r )/E. If te radii of tese speres are also equal, A Pr/E 0.34Pd/E; S 3 max 0.35PE /r 0.94PE /d ; and Y P /(E r) 7.36P /(E d). Spere and Flat Plate In tis case r r and r, and te above formulas become A Pr(c c ).36Pr/E, and S 3 max 0.35P/[r (c c ) ] 0.059PE /r Y P (c c ) /r.84p /(E r) Two Cylinders Te widt b of te rectangular pressure surface is obtained from (b/4) 0.9P(c c )/l[(/r ) (/r )], were r and r are te radii, and l te lengt S max [4P/(bl)] 0.35P(/r /r )/l(c c )] For cylinders wit te same moduli of elasticity, c c /E, and (b/4) 0.58P/El[(/r ) (/r )]; and S max 0.75PE(/r /r )/l. Wen r r r, (b/4) 0.9Pr/(El), and S max 0.35PE/(lr). Cylinder and Flat Plate Here r r, r, and te above formulas reduce to (b/4) 0.9Pr(c c )/l 0.58Pr/(El), and S max 0.35P/[lr(c c )] 0.75PE/(lr) For application to ball and roller bearings and to gear teet, see Sec. 8. FLAT PLATES Te analysis of flat plates subjected to lateral loads is very involved because plates bend in all vertical planes. Strict matematical derivations ave terefore been accomplised only in some special cases. Most of te available formulas contain some amount of rational empiricism. Plates may be classified as () tick plates, in wic transverse sear is important; () average-tickness plates, in wic flexure stress predominates; (3) tin plates, wic depend in part upon direct tension; and (4) membranes, wic are subject to direct tension only. However, exact lines of demarcation do not exist. Te flat-plate formulas given apply primarily to symmetrically loaded average-tickness plates of constant tickness. Tey are valid only if te maximum deflection is small relative to te plate tickness; usually, y 0.4t. In te matematical analyses, allowance for stress redistribution, because of sligt local yielding, is usually not made. Since tis yielding, especially in ductile materials, is beneficial, te formulas generally err on te side of safety. Certain cases of symmetrically loaded circular and rectangular plates are presented in Figs and Te maximum stresses are calculated from S M k wr S t M k P or S t M k C t Te first equation is for a uniformly distributed load w, lb/in ; te second supports a concentrated load P, lb; and te tird a couple C, per unit lengt, uniformly distributed along te edge. Combinations of tese loadings may be treated by superposition. Te factors k and k are given

48 5-48 MECHANICS OF MATERIALS Table 5..9 ( 0.3) Coefficients k and k for Circular Plates Case k k R/r Case k k k k k k k k k k k k in Tables 5..9 and 5..0; R is te radius of circular plates or one side of rectangular plates, and t is te plate tickness. [In Figs and 5..70, r R for circular plates and r smaller side rectangular plates.] Te maximum deflection for te same cases is given by wr y M k 4 PR y Et 3 M k CR and y Et 3 M k Et 3 Te factors k are also given in te tables. For additional information, including sells, refer to ASME Handbook, Metals Engineering: Design, McGraw-Hill. Fig Rectangular and elliptical plates. [R is te longer dimension except in cases () and (3).] Fig Circular plates. Cases (4), (5), (6), (7), (8), and (3) ave central ole of radius r; cases (9), (0), (), (), (4), and (5) ave a central piston of radius r to wic te plate is fixed. THEORIES OF FAILURE Material properties are usually determined from tests in wic specimens are subjected to simple stresses under static or fluctuating loads. Te attempt to apply tese data to bi- or triaxial stress fields as resulted in te proposal of various teories of failure. Figure 5..7 sows te principal stresses on a triaxially stressed element. It is assumed, for simplicity, tat S S S 3. Compressive stresses are negative.. Maximum-stress teory (Rankine) assumes failure occurs wen te largest principal stress reaces te yield stress in a tension (or compression) specimen. Tat is, S S y.. Maximum-sear teory (Coulomb) assumes yielding (failure) occurs wen te maximum searing stress equals tat in a simple tension (or compression) specimen at yield. Matematically, S S 3 S y. 3. Maximum-strain-energy teory (Beltrami) assumes failure occurs wen te energy absorbed per unit volume equals te strain energy per

49 PLASTICITY 5-49 Table 5..0 ( 0.3) Coefficients k and k for Rectangular and Elliptical Plates R/r Case k k k k k k k k k k * * * Lengt ratio is r/r in cases and 3. unit volume in a tension (or compression) specimen at yield. Matematically, S S S 3 (S S S S 3 S 3 S ) S y. 4. Maximum-distortion-energy teory (Huber, von Mises, Hencky) assumes yielding occurs wen te distortion energy equals tat in simple tension at yield. Te distortion energy, tat portion of te total energy wic causes distortion rater tan volume cange, is U d 3E (S S S 3 S S S S 3 S 3 S ) Tus failure is defined by S S S 3 (S S S S 3 S 3 S ) S y 5. Maximum-strain teory (Saint-Venant) claims failure occurs wen te maximum strain equals te strain in simple tension at yield or S (S S 3 ) S y. 6. Internal-friction teory (Mor). Wen te ultimate strengts in tension and compression are te same, tis teory reduces to tat of maximum sear. For principal stresses of opposite sign, failure is defined by S (S uc /S u ) S S uc ; if te signs are te same S S u or S uc, were S uc is te ultimate strengt in compression. If te principal stresses are bot eiter tension or compression, ten te larger one, say S, must equal S u wen S is tension or S uc wen S is compression. A grapical representation of te first four teories applied to a biaxial stress field is presented in Fig Stresses outside te bounding lines in te case of eac teory mean failure (yield or fracture). A comparison wit experimental data proves te distortion-energy teory (4) best for ductile materials of equal tension-compression properties. Wen tese properties are unequal, te internal friction teory (6) appears best. In practice, judging by some accepted codes, te maximum- Fig Fig sear teory () is generally used for ductile materials, and te maximum-stress teory () for brittle materials. Fatigue failures cannot be related, teoretically, to elastic strengt and tus to te teories described. However, experimental results justify tis, at least to a limited extent. Terefore, te teory evaluation given above olds for fluctuating stresses, provided tat principal stresses at te maximum load are used and te endurance strengt in simple bending is substituted for te yield strengt. EXAMPLE. A steel saft, 4 in in diameter, is subjected to a bending moment of 0,000 in lb, as well as a torque. If te yield strengt in tension is 40,000 lb/in, wat maximum torque can be applied under te () maximum-sear teory and () te distortion-energy teory? S x M c 0,000 I.55 9,00 lb/in S xy TC J T T and S M,m S x S x S xy S M S m S y or 9,00 (0.0798T) () (40,000) or T,000 inlb (54,50 cm kgf) S M Sm S M S m S y () substituting and simplifying, (9,550) 3 9,00 (0.0798T) (40,000) or T 55,000 in lb (93,50 cm kgf ) PLASTICITY Te reaction of materials to stress and strain in te plastic range is not fully defined. However, some concepts and teories ave been proposed. Ideally, a purely elastic material is one complying explicitly wit Hooke s law. In a viscous material, te searing stress is proportional to te searing strain. Te purely plastic material yields indefinitely, but only after reacing a certain stress. Combinations of tese are te elastoviscous and te elastoplastic materials. Engineering materials are not ideal, but usually contain some of te elastoplastic caracteristics. Te total strain t is te sum of te elastic strain o plus te plastic strain p, as sown in Fig , were te stress-strain curve is approximated by two straigt lines. Te natural strain, wic is at te same time te total strain, is l dl/l lo ln (l/l o ). In tis equation, l is te instantaneous lengt, wile l o is te original lengt. In terms of te normal strain, te natural strain becomes ln( o ). Since it is assumed tat te volume remains constant, l/l o A o /A, and so te natural stress becomes S P/A (P/A o )( o ). A o is te original cross-sectional area. If te natural stress is plotted against strain on log-log paper, te grap is very nearly a straigt line. Te plastic-range relation is tus approximated by S K n, were te proportionality factor K and te strain-ardening coefficient n are deter-

50 5-50 MECHANICS OF MATERIALS mined from best fits to experimental data. Values of K and n determined by Low and Garofalo (Proc. Soc. Exp. Stress Anal., vol. IV, no., 947) are given in Table 5... Fig Te geometry of Fig can be used to arrive at a second approximate relation S S o ( p o ) tan S o H E p H were H tan is a kind of plastic modulus. Te deformation teory of plastic flow for te general case of combined stress is developed using te above concepts. Certain additional assumptions involved include: principal plastic-strain directions are te same as principal stress directions; te elastic strain is negligible compared to plastic strain; and te ratios of te tree principal searing strains ( ), ( 3 ), ( 3 ) to te principal searing stresses (S S )/, (S S 3 )/, (S 3 S )/ are equal. Te relations between te principal strains and stresses in terms of te simple tension quantities become /S[S (S S 3 )/] /S[S (S 3 S )/] 3 /S[S 3 (S S )/] If tese equations are added, te plastic-flow teory is expressed: S [(S S ) (S S 3 ) (S 3 S ) ]/ ( 3)/3 In te above equation and [(S S ) (S S 3 ) (S 3 S ) ]/ S e ( 3)/3 e are te effective, or significant, stress and strain, respectively, EXAMPLE. An annealed, stainless-steel type 430 tank as a 4-in inside diameter and as a wall in tick. Te ultimate strengt of te stainless steel is 85,000 lb/in. Compute te maximum strain as well as te pressure at fracture. Te tank constitutes a biaxial stress field were S pd/(t), S pd/(4t), and S 3 0. Taking te power stress-strain relation S e K n e or /S S(n)/n e /K /n tus S e (n)/n 3 K /n 4 S 0, and 3 S e (n)/n K /n 3 4 S Te maximum-sear teory, wic is applicable to a ductile material under combined stress, is acceptable ere. Tus rupture will occur at S S 3 S u, and S e S S S 3 4 u S 3 4 [(3/4)/ S u ] (n)/n K /n Since in/in ( cm/cm) or p ROTATING DISKS S u S pd t, ten p ts u d 3 4 S 3 / 4 0.9/ ,000 43, ,000,550 lb/in 4 (09 kgf/cm ) S u /0.9 Rotating circular disks may be of various profiles, of constant or variable tickness, wit or witout centrally and noncentrally located oles, and wit radial, tangential, and searing stresses. Solution starts wit te differential equations of equilibrium and compatibility and te subsequent application of appropriate boundary conditions for te derivation of working-stress equations. If te disk tickness is small compared wit te diameter, te variation of stress wit tickness can be assumed to be negligible, and symmetry eliminates te searing stress. In te rotating case, te disk weigt is neglected, but its inertia force becomes te body-force term in te equilibrium equations. Tus solved, te stress components in a solid disk become r 3 8 (R r ) 3 8 R 3 8 r were Poisson s ratio; mass density, lb s /in 4 ; angular speed, rad/s; R outside disk radius; and r radius to point in question. Te largest stresses occur at te center of te solid disk and are r 3 8 R A disk wit a central ole of radius r (no external forces) is subjected to te following stresses: r 3 8 R r R r r r 3 8 R r R r 3 r 3 r Te maximum radial stress r M occurs at r Rr, and r M 3 8 (R r ) Table 5.. Constants K and n for Seet Materials Material Treatment K, lb/in n 0.05%C rimmed steel Annealed 77, %C killed steel Annealed and tempered 73, Decarburized 0.05%C steel Annealed in wet H 75, /0.07% pos. low C Annealed 93, SAE 430 Annealed 69, SAE 430 Normalized and tempered 54, Type 430 stainless Annealed 43, Alcoa 4-S Annealed 55, Reynolds R-30 Annealed 48,450 0.

51 EXPERIMENTAL STRESS ANALYSIS 5-5 Te largest tangential stress M exists at te inner boundary, and M 3 4 R 3 r As te ole radius r approaces zero, te tangential stress assumes a value twice tat at te center of a rotating solid disk, given above. Stresses in Turbine Disks Explicit solutions for cases oter tan tose cited are not available; so approximate solutions, suc as tose proposed by Stodola, Tomson, Hetényi, and Robinson, are necessary. Manson uses te calculus of finite differences. See commentary under previous discussion of torsion for alternate metods of approximate solution. Te problem illustrated below is a prime example of te elegance of te combination of approximate metods and electronic computers, wic allow a rapid solution to be obtained. Te speed wit wic te repetitive calculations are done allows equally rapid solutions wit canges in design variables. Te customary, simplifying assumptions of axial symmetry no variation of stress in te tickness direction and a completely elastic stress situation are made. Te differential equations of equilibrium and comparibility are rewritten in finite-difference form. Solution of te finite-difference equations, appreciation of teir linear nature, and successive application of tem yield te stresses at any station in terms of tose at a boundary station suc as r 0. Te equations tus derived are r,n A r,n t,r0 B r,n (5..) t,n A t,n t,r0 B t,n Te finite-difference expressions yield Eqs. (5..), wic permit te coefficients at station n to be computed from tose at station n. A r,n K n A r,n L n A t,n A t,n K n A r,n L n A t,n B (5..) r,n K n B r,n L n B t,n M n B t,n K n B r,n L n B t,n M n Te coefficients at te first station can be establised by inspection. For a solid disk, for instance, were bot stresses are equal to te tangential stress at te center, te coefficients in Eqs. (5..) are A r,n A t,n and B r,n B t,n 0. In te case of te disk wit a central ole, were r,r 0, A r,r B r,r B t,r 0 and A t,rl. Knowing tese, all oters can be found from Eqs. (5..). At te outer boundary, r,r A r,r t,r0 B r,r and t,r0 ( r,r B r,r )/A r,r. Te radial and tangential stresses at eac station are successively obtained, knowing t,r0 and all te coefficients, using Eqs. (5..). Te remaining coefficients in Eqs. (5..), extracted from te finitedifference equations, are defined below, were E is Young s modulus at te temperature of te point in question, is te profile tickness, is te termal coefficient of expansion, T is te temperature increment above tat at wic te termal stress is zero, is Poisson s ratio, is angular velocity of disk, and is te mass density of disk material. C n r n / n C n n /E n ( n )(r n r n )/(E n r n ) D n (r n r n ) n D n /E n ( n )(r n r n )/(E n r n ) F n r n n F n ( n /E n ) ( n )(r n r n )/(E n r n ) G n (r n r n ) n G n (/E n ) ( n )(r n r n )/(E n r n ) H n (r n r n )( n n r n n n r n) H n n T n n T n K n (F n D n F n D n )/(C n D n C n D n ) K n (C n F n C n F n )/(C n D n C n D n ) L n (G n D n G n D n )/(C n D n C n D n ) L n (C n G n C n G n )/(C n D n C n D n ) M n (H n D n H n D n )/(C n D n C n D n ) M n (C n H n C n H n )/(C n D n C n D n ) Situations need not be equally spaced between te two boundaries. It is best to space tem more closely were te profile, temperature, or oter property is canging rapidly. In cases of sudden or abrupt section canges, it is best to fair in across te cange; te material density sould, owever, be adjusted to give a total mass equal to te actual. Six to ten stations are often sufficient. Te modulus of elasticity as a significant effect, and its exact value at te temperature of eac station sould be used. Te coefficients of termal expansion are usually averaged for te temperature between te station and at wic no termal stress occurs. Te first two Eqs. (5..) and te last two must be worked simultaneously. At te outer boundary, loads external to te disk may be imposed, e.g., te radial stress r,r from te centrifuged pull of a bucket. At te center, te disk may be srunk on a saft wit te fit pressures causing a radial external pus at tis boundary. Numerical solutions are most expeditiously accomplised by use of a table wit column-to-column procedures. Tis tecnique lends itself readily to programmable computers or calculators. Disks wit Noncentral Holes Tis case as not been solved explicitly, but approximations are useful (e.g., Armstrong, Stresses in Rotating Tapered Disks wit Noncentral Holes, P.D. dissertation, Iowa State University, 960). Te area between te oles is considered removed and replaced by uniform spokes, eac one wit a cross-sectional area equal to te original minimum spoke area and wit a lengt equal to te diameter of te noncentral oles. Te iger stress in suc a spoke results in an additional extension, wic is ten applied to te outer annulus according to tin-ring teory and based on te average radius of te ring. Te additional stress is considered constant and is added to te tangential stress wic would be present in a disk of te same dimensions but filled (tat is, no noncentral oles). Te stress in te substitute spoke is computed by adjusting te stress at te ole-center radius in te solid or filled disk in proportion to te areas, or S sp r, (A g /A sp ), were r, is te radial stress in te filled disk at te radius of te ole circle, A g is te gross circumferential area at te same radius of te filled disk, and A sp is te area of te substitute spoke. Te increase in total strain is r, /[E(A g /A sp )l sp ], were l sp is te lengt of te substitute spoke. Te spoke-effect correction to be applied to te tangential stress is terefore c E/r, were r is te average outer-rim or annulus radius. Tis is added to te tangential stress found at te corresponding radius in te filled disk. Te final step is to adjust te tangential and radial stresses as determined for stress concentrations caused by te oles in te actual disk. Te factors for tis adjustment are tose in an infinite plate of uniform tickness aving te same size ole. Te metod is claimed to yield stresses witin 5 percent of tose measured potoelastically at points of igest stress. EXPERIMENTAL STRESS ANALYSIS Analytical metods of stress analysis can reac limits of applicability. Many experimental tecniques ave been suggested and tried; several ave been developed to a state of great usefulness, e.g., potoelasticity, strain-gage measurement, brittle coating, birefringent coating, and olograpy. Potoelasticity Most transparent materials exibit temporary double refraction, or birefringence, wen stressed. Ligt is resolved into components along te two principal plane directions. Te effect is temporary as long as te elastic stress is not exceeded and is in direct proportion to te applied load. Te stress magnitude can be establised by te amount of component wave retardation, as given in te wite and black band field (fringe pattern) obtained wen a monocromatic ligt source is used. Te polariscope, consisting of te ligt source, te polarizer, te model in a loading frame, an analyzer (same as polarizer), and a screen or camera, is used to produce and evaluate te fringe effect. Quarter-wave plates may be placed on eiter side of te model, making te ligt components troug te model independent of te absolute orientation of polarizer and analyzer. Te polarizer is a plane polariscope and yields te direc-

52 5-5 MECHANICS OF MATERIALS tions of principal stresses (te isoclinics); te analyzer is a circular polariscope yielding te fringes (isocromatics) as well. Figure sows te fringe pattern and te 0 isoclinics of a disk loaded radially at four places. Te isocromatics in te fringe pattern depict te difference between principal stresses. At free boundaries were te normal stress is zero, te difference automatically becomes te tangential stress. Starting at suc a boundary and proceeding into te interior, te stresses can be separated by numerical calculation. Fig Te Stress-Optic Law In a transparent, isotropic plate subjected to a biaxial stress field witin te elastic limit, te relative retardation R t between te two components produced by temporary double refraction is R t Ct( p q) n, were C is te stress-optic coefficient, t is te plate tickness, p and q are te principal stresses, n is te fringe order (te number of fringes wic ave passed te point during application of load), and is te wavelengt of monocromatic ligt used. Tus, ( p q)/ M n/(ct) nf/t If te material-fringe value f is determined wit te same ligt source (generally a mercury-vapor lamp emitting ligt aving a wavelengt of 5,46 Å) as used in te model study, te maximum searing stress, or one-alf te difference between te principal stresses, is directly determined. Te calibration is a matter of obtaining te material-fringe value in lb/in per fringe per inc (kgf/cm per fringe per cm). Isoclinics, or te direction of te principal planes, can be obtained wit a plane polariscope. A new isoclinic parameter is observed eac time te polarizer and analyzer are rotated simultaneously into a new position. A wite-ligt source reveals a more distinct isoclinic, as te black curve is more distinguisable against a colored background. Isostatics, or stress trajectories, are curves te tangents to wic represent te progressive cange in principal-plane directions. Tey are constructed grapically using te isoclinics. Since tere are two principal planes at eac point, two families of ortogonal curves are drawn. Care must be exercised in te drawing of trajectories for practical accuracy. Stress Separation If knowledge of eac principal stress is required, te potoelastic data must be treated to separate te stresses from te difference given by te data. If te sum of te two stresses is also obtained someow, a simultaneous solution of te sum and difference values will yield eac principal stress. One can also start at a boundary were te normal stress value is zero. Tere, te potoelastic reading gives te principal stress parallel to te boundary. Starting wit te single value, metods ave been developed wic can be used to proceed wit te separation. Typical of te former are lateral-extensometer, iteration, and membrane-analogy tecniques; typical of te latter are te slope-equilibrium, sear-difference, grapical-integration, alternatingsummation metods, and oblique incidence. Often, owever, te surface stresses are te maximum valued ones. (See Froct, Potoeleasticity, McGraw-Hill.) EXAMPLE. Te fringe pattern of a Homalite disk.3 in in diam, 0.8 in tick, and carrying four radial loads of 55 lb eac is sown in Fig A closed solution is not known. However, by counting te fringe order at any point, te stress can be determined potoelastically. For instance, te dark spot at te center marks a fringe of zero order, as do te disk edges except in te immediate vicinity of te concentrated loads. Te point at te center, wic remained dark trougout te loading, is an isotropic point (zero stress difference and normal stresses are equal in all directions). Counting out from te center toward te load, te first circular fringe is of order 3. Terefore, anywere along it ( p q)/ M nf/t 3 65/ lb/in (49 kgf/cm ). Carefully inspected, fringe can be counted at te point of load application. Terefore, r M 65/0.8,770 lb/in (95 kgf/cm ). THREE-DIMENSIONAL PHOTOELASTICITY Stress freezing and slicing, werein a plastic model is brougt up to its critical temperature, loaded as desired, and wile loaded, slowly brougt back to room temperature, are tecniques wic freeze te fringe pattern into te model. Te model can be cut into slices witout disturbing te frozen strains. Two-dimensional models are usually macined from plate stock, and tree-dimensional models are cast. Te frozen stress model is sliced so tat te desired information can be obtained by normal incidence using te previous formulations. Wen normal incidence is not possible, oblique incidence becomes necessary. Oblique-incidence patterns are usable in two-dimensional as well as tree-dimensional stress separation. Te measurement of fractional fringes is often required wen using oblique incidence. Wit a crossed, circular, monocromatic polariscope, oriented to te principal stresses at a point, te analyzer is rotated troug some angle until extinction occurs. Te fringe value n is n n n /80, were n n is te order of te last visible fringe. Weter te fractional term is added or subtracted depends upon te direction in wic te analyzer is rotated (establised by inspection). Fig Oblique-incidence calculations are based on te stress-optic law: n n R t( p q)/f tp/f tg/f n p n q. Also, wen polarized ligt is directed troug te slice at an angle x to a principal plane, eiter by rotating te slice away from normal to te ligt ray or by cutting it at te angle x (see Fig ), te fringe order becomes n x t t ( p q) ( p q cos f f cos x ) x (n p n q cos x )/cos x

53 EXPERIMENTAL STRESS ANALYSIS 5-53 Solving algebraically, n p (n x cos x n n cos x )/sin x and n q (n x cos x n n )/sin x If orders n n and n x are tus measured at a point, n p and n q can be computed. Te principal stresses are ten determined from p fn p /t and q fn q /t. Te material-fringe value f in tese equations is at te freezing temperature (critical temperature). Te angle of incidence, as well as te fringe orders, must be accurately measured if errors are to be minimized. Bonded Metallic Gages Strain measurements down to one-milliont inc per inc (one-milliont cm/cm) are possible wit electrical-resistance wire gages. Suc gages can be used to measure surface strains (stress by Hooke s law) on any sape or size of object. Figure illustrates scematically te gage construction wit a grid of fine alloy wire or tin foil, bonded to paper and covered for protection wit a felt pad. In use, te gage is cemented rigidly to te surface of te member to be analyzed. Te strain relation is (R/R)(/G f ) in/in (cm/cm). Tus, if te resistance R and gage factor G f (given by te gage manufacturer) are known and te cange in resistance R is measured, te strain wic caused te resistance cange can be determined and Hooke s law can be applied to determine te stress. containing te active gage, te electric-resistance temperature effect is canceled out. Tus te active gage reports only tat wic is taking place in te stressed plate. Te power supply can be eiter ac or dc. It is sometimes useful to make bot gages active e.g., mounted on opposite sides of a beam, wit one gage subjected to tension and te oter to compression. Temperature effects are still compensated, but te bridge output is doubled. In oter instances, it may be desirable to make all four bridge arms active gages. Te experimenter must determine te most practical arrangement for te problem at and and must bear in mind tat te bridge unbalances in proportion to te difference in te strains of gages located in adjacent legs and to te sum of strain in gages located in opposite legs. Fig Fig Gages must be properly selected in accordance wit manufacturer s recommendations. Te surface to wic te gage is applied must be clean, te proper cement must be used, and te gage assembly must be coated for protection against environmental conditions (e.g., moisture). A gaging unit, usually a Weatstone bridge or a ballast circuit (see Fig and Sec. 5), is needed to detect te signal resulting from te cange in resistance of te strain gage. Te strain and, terefore, te signal are often too small for direct andling, so tat amplification is needed, wit a metering discriminator for magnitude evaluation. Te signal is read or recorded by a galvanometer, oscilloscope, or oter device. Equipment specifically constructed for strain measurement is available to indicate or record te signal directly in strain units. Static strains are best gaged on a Weatstone bridge, wit strain gages wired to it as indicated in Fig a. Wit te bridge set so tat te only unbalance is te cange of resistance in te active-strain gage, te potential difference between te output terminals becomes a measurement of strain. Since te gage is sensitive to temperature as well as strain, it will measure te combined effect. However, if a dummy gage, cemented to an unstressed piece of te same metal subjected to te same climatic conditions, is wired into te bridge leg adjacent to te one Dynamic strains can be detected using circuits suc as te ballast type sown in Fig b. Te capacitor coupling passes only rapidly varying or dynamic strains. Te capacitor s infinite impedance to a steady voltage filters out any static effects or strains. Te circuit is dc powered. Transverse Sensitivity Grid-type gages possess some strain sensitivity in te direction perpendicular to te gage axis. In a uniaxial stress field, tis transverse sensitivity is of no concern because te gage factor was obtained in suc a field. However, in a biaxial stress field, neglect of transverse sensitivity will give sligtly erroneous strains. Wen accounted for, te true strains in te axial direction of gage,, and at rigt angles to it,, are ( k)( a k a )/( k ) and ( k)( a k a )/( k ), were te apparent strains are a R /(RG f ) and a R /(RG f,), measured by cementing a gage in eac direction and. Te factor is Poisson s ratio of te material to wic gages are cemented, and k (usually provided by te gage manufacturer) is te coefficient of transverse sensitivity of te gage. Te gage is cemented to te test piece, a uniaxial stress is applied in its axial direction, and te resistance cange and strain are measured. Te gage factor G R /(R ) is computed. A uniaxial stress is next applied transversely to te gage. Again te resistance cange and strain are measured and G computed. Ten k (G G )/(G G ). Strain Rosettes In a general biaxial stress field, te principal plane directions, as well as te stresses, are unknown. Tus, tree gages mounted in tree differing directions are needed if te tree unknowns are to be determined. Tree standard gage combinations, called strain rosettes, are commercially available and are best for te purpose. Tese are te rectangular strain rosette (Fig a), wic covers a minimum of area and is terefore best were te strain gradient is ig; te equiangular strain rosette (Fig b), were te gages do not overlap and wic can be used were te strain gradient is low; te T-delta strain rosette (Fig c), wic occupies no more area tan te equiangular rosette and wic provides an extra ceck, or insurance

54 5-54 MECHANICS OF MATERIALS Fig gage. Te wiring and instrumentation of gages in rosettes do not differ from tose of individual gages. Te true strains along te gage-lengt directions are found according to te following equations, in wic R n R n /[RF ( k )] and b /k. RECTANGULAR ROSETTE (SEE FIG a) R R 3 /b R ( /b) (/b)(r R 3 ) 3 R 3 R /b EQUIANGULAR ROSETTE (SEE FIG b) R (/b)(r R 3 ) R (/b)(r R 3 ) 3 R 3 (/b)(r R 3 ) T-DELTA ROSETTE (SEE FIG c) R ( /b) (/b)(r 3 R 4 ) R ( /b) (/b)(r 3 R 4 ) 3 R 3 (/b)r 4 4 R 4 (/b)r 3 Foil Gages Foil gages are produced from tin foil by potoetcing tecniques and are applied, instrumented, read, and evaluated just like te wire-grid type. Foil gages, being muc tinner, may be applied easily to curved surfaces, ave lower transverse sensitivity, exibit negligible ysteresis under cycling loads, creep little under sustained loads, and can be stacked on top of eac oter. Brittle-Coating Analysis Brittle coatings wic adere to te surface well can reveal te strain in te underlying material. Probably te first suc coating used was mill scale, a tin iron oxide wic forms on ot-rolled steel stock. Many coatings suc as witewas, portland cement, and sellac ave been tried. Te most popular of presently available strain-indicating brittle coatings are te wood-rosin lacquers supplied by te Magnaflux Corporation under te trade name Stresscoat. Several Stresscoat compositions are available; te suitability of a particular lacquer depends upon te prevailing temperature and umidity. Te lacquer is usually sprayed to a tickness of to in (0.0 to 0.0 cm) upon te surface, wic must be clean and free of grease and loose particles. Calibration bars are sprayed at te same time. Bot must be dried at an even temperature for up to 4. To facilitate observation of cracks, an undercoating of brigt aluminum is often applied. Wen te cured test piece is subjected to loads, te lacquer will first begin to crack at its tresold sensitivity in te area of te largest principal stress, wit te parallel cracks perpendicular to te principal stress. Tis information is often sufficient, as it reveals te critical area and te direction of normal stress. Te tresold sensitivity of Stresscoat lacquers is 600 to 800 microinces per inc (600 to 800 microcentimeters per centimeter) in a uniaxial stress field. Exact control of lacquer selection, tickness, curing, and testing temperatures may reduce te tresold to 400 microinces per inc (400 microcentimeters per centimeter). If desired, te approximate strain (probably witin 0 percent) may be establised using te calibration strip sprayed wit te test part. Te strip is placed in a loading device and bent as a cantilever beam by means of a cam at te free end, causing te coating to crack on te tension surface. Crack spacing varies wit te strain, being close at te fixed end and diminising toward te free end down to tresold sensitivity values. Te strip is placed in a older containing strain graduations. A visual comparison of cracks on te testpart surface wit tose on te strip reveals te strain magnitude wic caused te cracks. Birefringent Coatings A birefringent coating is one wic becomes double refractive wen strained. Te principle is quite old, but plastics, wic adere to all kinds of materials, wic ave stable optical-strain constants, and wic are sufficiently sensitive to be practical, are of recent development. Te trade name applied to tis tecnique is Potostress. Potostress plastics can be obtained eiter as tin seets (0.040, 0.080, and 0.0 in) or in liquid form. Te seet material can be bonded to a surface wit a special adesive. Te liquid can be brused or sprayed on, or te part can be dipped in te liquid. Te layer sould be at least in (0.00 cm) tick. It is often necessary to apply several successive coatings, wit eat curing of eac layer in turn. Two seet types and two liquids are available; tese differ in stretcing ability and in magnitude of te strain-optical constant. Eac of te seet materials is available metallized on one face, to reflect polarized ligt even wen cemented to a dull surface. Te principles involved are te same as tose for conventional potoelasticity. One frequent advantage is te fact tat te plastic (seet or liquid) can be applied directly to te part, wic can ten be subjected to actual operating loads. A special reflecting polariscope must be used. It contains only one polarizer and quarter-wave disk because te ligt passes back troug te same pair after reflection by te stressed surface-plastic interface. Te only limitation rests in te geometry of te structural component to be examined; not only must it be possible to apply te plastic to te surface, but te surface must be accessible to ligt. Te strain-optic law, since te ligt passes te plastic tickness twice, becomes p q n E t K( ) were n is fringe order, E is modulus, is Poisson s ratio of workpiece material, and K (supplied by te manufacturer) is te strain-optic coefficient of te plastic. As in conventional potoelasticity, isoclinics are present as well. Holograpy A more recently developed tecnique applicable to stress, or rater, strain analysis as well as to many oter purposes is tat of olograpy. It is made possible by te laser, an instrument wic produces a igly concentrated, tin beam of ligt of single wavelengt. Te elium-neon (He-Ne) laser, emitting at te red end of te visible spectrum at a wave-

55 PIPELINE FLEXURE STRESSES 5-55 lengt of 633 nanometers, as found muc favor. Te output of a elium-cadmium (He-Cd) laser is at alf te wavelengt of te He-Ne laser; accordingly, te He-Ne laser is twice as sensitive to displacements. Te laser beam is split into two components, one of wic is directed upon te object (or specimen) and ten onto te potograpic plate. It is identified as te object beam. Te oter component, referred to as te reference beam, propagates directly to te plate. Interference between te beams resulting from retardations caused by displacements or strains forms fringes wic in turn provide a measure of te disturbance. Spacing of suc fringes depends upon Bragg s law: d sin (/) were d is te distance between fringes, is te wavelengt of te ligt source, and is te angle between te object and reference ray at te plate. A simple olograpic setup consists of te laser source, beam splitter, reflecting surfaces, filters, and te recording plate. A possible arrangement is depicted in Fig Some arrangement for loading te specimen must also be provided. Additional auxiliary and refining ardware becomes necessary as te analysis assumes greater complexity. Tus te system layout is limited only by test requirements and te experimenter s imagination. However, only a toroug understanding of te laws of optics and interferometry will make possible a reliable investigation and interpretation of results. Stability of setup must be assured via a rigid optical benc and supporting brackets. Component instruments must be spaced upon te benc so tat beam coerence is assured, required coerence dept satisfied, and te object/reference angle consistent wit te fringe spacing desired. Te film must also possess adequate sensitivity in te spectral range of te laser beam used. It is important to recognize te inerent azards of te ig-intensity radiation in laser beams and to practice every precaution in te use of lasers. Fig Simple olograpic setup. () Laser source; () beam splitter; (3) reflecting surfaces; (4) circular polarizers; (5) loaded specimen (birefringent); (6) potograpic plate. Holograpy, using pulsed lasers, can be used to measure transient disturbances. Tus vibration studies are possible. Fatigue detection using olograpic tecniques as also been undertaken. Holograpy as been used in acoustical studies and in automatic gaging as well. It is a versatile engineering tool. 5.3 PIPELINE FLEXURE STRESSES by Harold V. Hawkins EDITOR S NOTE: Te almost universal availability and utilization of personal computers in engineering practice as led to te development of many competing and complementary forms of piping stress analysis software. Teir use is widespread, and individual packaged software allows analysis and design to take into account static and dynamic conditions, restraint conditions, aboveground and buried configurations, etc. Te reader is referred to te tecnical literature for te most suitable and current software available for use in solving te immediate problems at and. Te brief discussion in tis section addresses te fundamental concepts entailed and sets fort te solution of simple systems as an exercise in application of te principles. REFERENCES: Sipman, Design of Steam Piping to Care for Expansion, Trans. ASME, 99, Wal, Stresses and Reactions in Expansion Pipe Bends, Trans. ASME, 97. Hovgaard, Te Elastic Deformation of Pipe Bends, Jour. Mat. Pys., Nov. 96, Oct. 98, and Dec. 99. M. W. Kellog Co., Te Design of Piping Systems, Wiley. For details of pipe and pipe fittings see Sec Nomenclature (see Figs and 5.3.) M 0 end moment at origin, inlb (Nm) M max moment, inlb (Nm) F x end reaction at origin in x direction, lb (N) F y end reaction at origin in y direction, lb (N) S l (Mr/I) max unit longitudinal flexure stress, lb/in (N/m ) S t (Mr/I) max unit transverse flexure stress, lb/ in (N/m ) S s (Mr/I) max unit searing stress, lb/in (N/m ) x relative deflection of ends of pipe parallel to x direction caused by eiter temperature cange or support movement, or bot, in (m) y same as x but parallel to y direction, in. Note tat x and y are positive if under te cange in temperature te end opposite te origin tends to move in a positive x or y direction, respectively. t wall tickness of pipe, in (m) r mean radius of pipe cross section, in (m) constant tr/r I moment of inertia of pipe cross section about pipe centerline, in 4 (m 4 ) E modulus of elasticity of pipe at actual working temperature, lb/in (N/m ) K flexibility index of pipe. K for all straigt pipe sections, K (0 )/( ) for all curved pipe sections were (see Fig ),, ratios of actual max longitudinal flexure, transverse flexure, and searing stresses to Mr/I for curved sections of pipe (see Fig ) Fig Fig. 5.3.

56 5-56 PIPELINE FLEXURE STRESSES A, B, C, F, G, H constants given by Table 5.3. angle of intersection between tangents to direction of pipe at reactions cange in caused by movements of supports, or by temperature cange, or bot, rad ds an infinitesimal element of lengt of pipe s lengt of a particular curved section of pipe, in (m) R radius of curvature of pipe centerline, in (m) Fig Flexure constants of initially curved pipes. General Discussion Under te effect of canges in temperature of te pipeline, or of movement of support reactions (eiter translation or rotation), or bot, te determination of stress distribution in a pipe becomes a statically indeterminate problem. In general te problem may be solved by a sligt modification of te standard arc teory: x KMy ds(ei), y KMx ds/(ei), and KM ds/(ei) were te constant K is introduced to correct for te increased flexibility of a curved pipe, and were te integration is over te entire lengt of pipe between supports. In Table 5.3. are given equations derived by tis metod for moment and trust at one reaction point for pipes in one plane tat are fully fixed, inged at bot ends, inged at one end and fixed at te oter, or partly fixed. If te reactions at one end of te pipe are known, te moment distribution in te entire pipe ten can be obtained by simple statics. Since an initially curved pipe is more flexible tan indicated by its moment of inertia, te constant K is introduced. Its value may be taken from Fig , or computed from te equation given below. K for all straigt pipe sections, since tey act according to te simple flexure teory. In Fig are given te flexure constants K,,, and for initially curved pipes as functions of te quantity tr/r. Te flexure constants are derived from te equations. K (0 )/( ) wen K (5 6 )/8.47 K(6 )/(6 5).47 8/ ( ) [ (3 0/3) (4/3) 5/8] ( ) wen 0.58 ( 8 ) / ( ) wen 0.58 Table 5.3. General Equations for Pipelines in One Plane (See Figs and 5.3.) Bot ends fully fixed Symmetric Type of supports Unsymmetric about y-axis M 0 EI x(cf AB) EI y(bf AG) ABF CGH B H A G CF F x EI x(ch A ) EI y(bh AF) ABF CGH B H A G CF F y EI x(bh AF) EI y(gh F ) ABF CGH B H A G CF 0 Bot ends inged M 0 0 Origin end only inged, oter end fully fixed In general for any specific rotation and movement x and y... F x F y M 0 0 F x F y EI x C EI y B CG B EI x B EIy G CG B x(ab CF) y(ag BF) CG B EI x C EI y B CG B EI x B EI y G CG B x(ab CF) y(ag BF) CG B M 0 M 0 EI x(cf AB) EI y(bf AG) EI (CG B ) ABF CGH A 3 G CF B H F x EI x(ch A ) EI y(bh AF) EI (CF AB) ABF CGH A G CF B H F y EI x(bh AF) EI y(gh F ) EI (BF AG) ABF CGH A G CF B H F x F y 0 0 M 0 0 EI x F GH F EI x H GH F EI x F x G F y 0 xf G

57 PIPELINE FLEXURE STRESSES 5-57 Te increased flexibility of te curved pipe is brougt about by te tendency of its cross section to flatten. Tis flattening causes a transverse flexure stress wose maximum is S t. Because te maximum longitudinal and maximum transverse stresses do not occur at te same point in te pipe s cross section, te resulting maximum sear is not one-alf te difference of S l and S t ;itiss s. In te straigt sections of te pipe,, te transverse stress disappears, and. Tis discussion of S s does not include te uniform transverse or longitudinal tension stresses induced by te internal pressure in te pipe; teir effects sould be added if appreciable. Table 5.3. gives values of te constants A, B, C, F, G, and H for use in equations listed in Table Te values may be used () for te solution of any pipeline or () for te derivation of equations for standard sapes composed of straigt sections and arcs of circles as of Fig Equations for sapes not given may be obtained by algebraic addition of tose given. All measurements are from te left-and end of te pipeline. Reactions and stresses are greatly influenced by end conditions. Formulas are given to cover te extreme conditions. Te following suggestions and comments sould be considered wen laying out a pipeline: Avoid expansion bends, and design te entire pipeline to take care of its own expansion. Te movement of te equipment to wic te ends of te pipeline are attaced must be included in te x and y of te equations. Maximum flexibility is obtained by placing supports and ancors so tat tey will not interfere wit te natural movement of te pipe. Tat sape is most efficient in wic te maximum lengt of pipe is working at te maximum safe stress. Excessive bending moment at joints is more likely to cause trouble tan excessive stresses in pipe walls. Hence, keep pipe joints away from points of ig moment. Reactions and stresses are greatly influenced by flattening of te cross section of te curved portions of te pipeline. It is recommended tat cold springing allowances be discounted in stress calculations. Application to Two- and Tree-Plane Pipelines Pipelines in more tan one plane may be solved by te successive application of te preceding data, dividing te pipeline into two or more one-plane lines. EXAMPLE. Te unsymmetric pipeline of Fig as fully fixed ends. From Table 5.3. use K for all sections, since only straigt segments are involved. Upon introduction of a 0 in (3.05 m), b 60 in (.5 m), and c 80 in (4.57 m), into te preceding relations (Table 5.3.3) for A, B, C, F, G, H, te equations for te reactions at 0 from Table 5.3. become M 0 EI x ( ) EI y ( ) F x EI x ( ) EI y ( ) F y EI x ( ) EI y ( ) Also it follows tat M M 0 F y a EI x ( ) EI y ( ) M M F x b EI x( ) EI y ( ) M 3 M F y c EI x ( ) EI y ( ) Tus te maximum moment M occurs at 3. Te total maximum longitudinal fiber stress (a for straigt pipe) S l F x rt M 3r I Tere is no transverse flexure stress since all sections are straigt. Te maximum searing stress is eiter () one-alf of te maximum longitudinal fiber stress as given above, () one-alf of te oop-tension stress caused by an internal radial pressure tat migt exist in te pipe, or (3) one-alf te difference of te maximum longitudinal fiber stress and oop-tension stress, wicever of tese tree possibilities is numerically greatest. Fig EXAMPLE. Te equations of Table 5.3. may be employed to develop te solution of generalized types of pipe configurations for wic Fig is a typical example. If only temperature canges are considered, te reactions for te rigt-angle pipeline (Fig ) may be determined from te following equations: M 0 C EI x/r F x C EI x/r 3 F y C 3 EI x/r 3 In tese equations, x is te x component of te deflection between reaction points caused by temperature cange only. Te values of C, C, and C 3 are given in Fig for K and K. For oter values of K, interpolation may be employed. Fig Rigt angle pipeline. EXAMPLE 3. Wit a/r 0 and b/r 3, te value of C is 0.85 for K and 0.65 for K. If K.75, te interpolated value of C is Elimination of Flexure Stresses Pipeline flexure stresses tat normally would result from movement of supports or from te tendency of te pipes to expand under temperature cange often may be avoided entirely troug te use of expansion joints (Sec. 8.7). Teir use may simplify bot te design of te pipeline and te support structure. Wen using expansion joints, te following suggestions sould be considered: () select expansion joint carefully for maximum temperature range (and deflection) expected so as to prevent damage to expansion fitting; () provide guides to limit movement at expansion joint to direction permitted by joint; (3) provide adequate ancors at one end of eac straigt section or along teir midlengt, forcing movement to occur at expansion joint yet providing adequate support for pipeline; (4) mount expansion joints adjacent to an ancor point to prevent sagging of te pipeline under its own weigt and do not depend upon te expansion joint for stiffness it is intended to be flexible; (5) give consideration to effects of corrosion, since corrugated caracter of expansion joints makes cleaning difficult.

58 Table 5.3. Values of A, BC, F, G, and H for Various Piping Elements A Kx ds B Kxy ds C Kx ds F Ky ds G Ky ds H K ds s (x x ) Ay s s 3 x x sy Fy s sx A (y y ) Ax s (y y ) s s 3 y y s s (x x ) A 3 (y y ) s 3 (x x x x s (y y ) s 3 (y y y y ) s s 6 (x y x y ) KRx Ay R Ax x R (y R)KR Fy y R KR KR Ay R (y R)KR Fy y R KR x R KR Ay x R KR Ax R 4 x KR y R KR Fy R 4 y KR KR

59 x R KR Ay x R KR Ay x R KR Ax R 4 x KR x R KR Ay x R KR Ax R 4 x KR y R KR Fy R 4 y KR y R KR Fy R 4 y KR y R KR Fy R 4 y KR KR x 4 R KR x 4 R KR Ay x R KR 4 Ay x R KR 4 Ay x R KR 4 Ay x R KR 4 Ax 8 4 R Ax 8 4 R x KR x KR y 4 R KR y 4 KR R y 4 KR R y 4 KR R Fy y 8 4 R KR Fy y 8 4 R KR Fy y 8 4 R KR Fy y 8 4 R KR KR 4 [r( R(sin sin )]KR Ay x(cos cos ) R (sin sin ) KR Ax x(sin sin ) R 4 (sin sin ) R ( ) KR [y( ) R(cos cos )]KR Fy y(cos cos ) R 4 (sin sin ) R ( ) KR ( )KR

60 5-60 PIPELINE FLEXURE STRESSES Fig Reactions for rigt-angle pipelines.

61 PENETRANT METHODS 5-6 Table Example Sowing Determination of Integrals Part of Values of integrals pipe A B C F G H 0 a ab 3 Total ab a 3 3 a b 0 0 a c (a c) bc (a c) c 3 3 ac(a c) bc b c c a ab c (a c) ab bc (a c) a 3 3 a b c3 3 ac(a c) b b bc b 3 b b c a b c b 5.4 NONDESTRUCTIVE TESTING by Donald D. Dodge REFERENCES: Various autors, Nondestructive Testing Handbook, 8 vols., American Society for Nondestructive Testing. Boyer, Metals Handbook, vol., American Society for Metals. Heuter and Bolt, Sonics, Wiley. Krautkramer, Ultrasonic Testing of Materials, Springer-Verlag. Spanner, Acoustic Emission: Tecniques and Applications, Intex, American Society for Nondestructive Testing. Crowter, Handbook of Industrial Radiograpy, Arnold. Wiltsire, A Furter Handbook of Industrial Radiograpy, Arnold. Standards, vol , ASTM, Boiler and Pressure Vessel Code, Secs. III, V, XI, ASME. ASME Handbook, Metals Engineering Design, McGraw-Hill. SAE Handbook, Secs. J358, J359, J40, J45 J48, J4, J67, SAE. Materials Evaluation, Jour. Am. Soc. Nondestructive Testing. Nondestructive tests are tose tests tat determine te usefulness, serviceability, or quality of a part or material witout limiting its usefulness. Nondestructive tests are used in macinery maintenance to avoid costly unsceduled loss of service due to fatigue or wear; tey are used in manufacturing to ensure product quality and minimize costs. Consideration of test requirements early in te design of a product may facilitate testing and minimize testing cost. Nearly every form of energy is used in nondestructive tests, including all wavelengts of te electromagnetic spectrum as well as vibrational mecanical energy. Pysical properties, composition, and structure are determined; flaws are detected; and tickness is measured. Tese tests are ere divided into te following basic metods: magnetic particle, penetrant, radiograpic, ultrasonic, eddy current, acoustic emission, microwave, and infrared. Numerous tecniques are utilized in te application of eac test metod. Table 5.4. gives a summary of many nondestructive test metods. MAGNETIC PARTICLE METHODS Magnetic particle testing is a nondestructive metod for detecting discontinuities at or near te surface in ferromagnetic materials. After te test object is properly magnetized, finely divided magnetic particles are applied to its surface. Wen te object is properly oriented to te induced magnetic field, a discontinuity creates a leakage flux wic attracts and olds te particles, forming a visible indication. Magneticfield direction and caracter are dependent upon ow te magnetizing force is applied and upon te type of current used. For best sensitivity, te magnetizing current must flow in a direction parallel to te principal direction of te expected defect. Circular fields, produced by passing current troug te object, are almost completely contained witin te test object. Longitudinal fields, produced by coils or yokes, create external poles and a general-leakage field. Alternating, direct, or alfwave direct current may be used for te location of surface defects. Half-wave direct current is most effective for locating subsurface defects. Magnetic particles may be applied dry or as a wet suspension in a liquid suc as kerosene or water. Colored dry powders are advantageous wen testing for subsurface defects and wen testing objects tat ave roug surfaces, suc as castings, forgings, and weldments. Wet particles are preferred for detection of very fine cracks, suc as fatigue, stress corrosion, or grinding cracks. Fluorescent wet particles are used to inspect objects wit te aid of ultraviolet ligt. Fluorescent inspection is widely used because of its greater sensitivity. Application of particles wile magnetizing current is on (continuous metod) produces stronger indications tan tose obtained if te particles are applied after te current is sut off (residual metod). Interpretation of subsurface-defect indications requires experience. Demagnetization of te test object after inspection is advisable. Magnetic flux leakage is a variation wereby leakage flux due to flaws is detected electronically via a Hall-effect sensor. Computerized signal interpretation and data imaging tecniques are employed. Electrified particle testing indicates minute cracks in nonconducting materials. Particles of calcium carbonate are positively carged as tey are blown troug a spray gun at te test object. If te object is metalbacked, suc as porcelain enamel, no preparation oter tan cleaning is necessary. Wen it is not metal-backed, te object must be dipped in an aqueous penetrant solution and dried. Te penetrant remaining in cracks provides a mobile electron supply for te test. A readily visible powder indication forms at a crack owing to te attraction of te positively carged particles. PENETRANT METHODS Liquid penetrant testing is used to locate flaws open to te surface of nonporous materials. Te test object must be torougly cleaned before testing. Penetrating liquid is applied to te surface of a test object by a brus, spray, flow, or dip metod. A time allowance ( to 30 min) is required for liquid penetration of surface flaws. Excess penetrant is ten carefully removed from te surface, and an absorptive coating, known as developer, is applied to te object to draw penetrant out of flaws, tus sowing teir location, sape, and approximate size. Te developer is typically a fine powder, suc as talc usually in suspension in a liquid. Penetrating-liquid types are () for test in visible ligt, and () for test under ultraviolet ligt (3,650 Å). Sensitivity of penetrant testing is greatest wen a fluorescent penetrant is used and te object is observed

62 5-6 NONDESTRUCTIVE TESTING Table 5.4. Acoustic emission Nondestructive Test Metods* Metod Measures or detects Applications Advantages Limitations Acoustic-impact (tapping) D-Sigt (Diffracto) Eddy current Magneto-optic eddycurrent imager Eddy-sonic Electric current Electrified particle Crack initiation and growt rate Internal cracking in welds during cooling Boiling or cavitation Friction or wear Plastic deformation Pase transformations Debonded areas or delaminations in metal or nonmetal composites or laminates Cracks under bolt or fastener eads Cracks in turbine weels or turbine blades Loose rivets or fastener eads Crused core Enances visual inspection for surface abnormalities suc as dents protrusions, or waviness Crused core Lap joint corrosion Cold-worked oles Cracks Surface and subsurface cracks and seams Alloy content Heat-treatment variations Wall tickness, coating tickness Crack dept Conductivity Permeability Cracks Corrosion tinning in aluminum Debonded areas in metalcore or metal-faced oneycomb structures Delaminations in metal laminates or composites Crused core Cracks Crack dept Resistivity Wall tickness Corrosion-induced wall tinning Surface flaws in nonconducting material Troug-to-metal pinoles on metal-backed material Tension, compression, cyclic cracks Brittle-coating stress cracks Pressure vessels Stressed structures Turbine or gearboxes Fracture mecanics researc Weldments Sonic-signature analysis Brazed or adesive-bonded structures Bolted or riveted assemblies Turbine blades Turbine weels Composite structures Honeycomb assemblies Detect impact damage to composites or oneycomb corrosion in aircraft lap joints Automotive bodies for waviness Tubing Wire Ball bearings Spot cecks on all types of surfaces Proximity gage Metal detector Metal sorting Measure conductivity in % IACS Aluminum aircraft Structure Metal-core oneycomb Metal-faced oneycomb Conductive laminates suc as boron or grapitefiber composites Bonded-metal panels Metallic materials Electrically conductive materials Train rails Nuclear fuel elements Bars, plates oter sapes Glass Porcelain enamel Nonomogeneous materials suc as plastic or aspalt coatings Glass-to-metal seals Remote and continuous surveillance Permanent record Dynamic (rater tan static) detection of cracks Portable Triangulation tecniques to locate flaws Portable Easy to operate May be automated Permanent record or positive meter readout No couplant required Portable Fast, flexible Noncontact Easy to use Documentable No special operator skills required Hig speed, low cost Automation possible for symmetric parts Permanent-record capability for symmetric parts No couplant or probe contact required Real-time imaging Approximately 4-in area coverage Portable Simple to operate No couplant required Locates far-side debonded areas Access to only one surface required May be automated Access to only one surface required Battery or dc source Portable Portable Useful on materials not practical for penetrant inspection Transducers must be placed on part surface Higly ductile materials yield lowamplitude emissions Part must be stressed or operating Interfering noise needs to be filtered out Part geometry and mass influences test results Impactor and probe must be repositioned to fit geometry of part Reference standards required Pulser impact rate is critical for repeatability Part surface must reflect ligt or be wetted wit a fluid Conductive materials Sallow dept of penetration (tin walls only) Masked or false indications caused by sensitivity to variations suc as part geometry Reference standards required Permeability variations Frequency range of.6 to 00 khz Surface contour Temperature range of 3 to 90 F Directional sensitivity to cracks Specimen or part must contain conductive materials to establis eddy-current field Reference standards required Part geometry Edge effect Surface contamination Good surface contact required Difficult to automate Electrode spacing Reference standards required Poor resolution on tin coatings False indications from moisture streaks or lint Atmosperic conditions Hig-voltage discarge

63 PENETRANT METHODS 5-63 Table 5.4. Nondestructive Test Metods* (Continued) Filtered particle Metod Measures or detects Applications Advantages Limitations Infrared (radiometry) (termograpy) Leak testing Magnetic particle Magnetic field (also magnetic flux leakage) Microwave (300 MHz 300 GHz) Liquid penetrants (dye or fluorescent) Fluoroscopy (cinefluorograpy) (kinefluorograpy) Cracks Porosity Differential absorption Hot spots Lack of bond Heat transfer Isoterms Temperature ranges Leaks: Helium Ammonia Smoke Water Air bubbles Radioactive gas Halogens Surface and sligtly subsurface flaws; cracks, seams, porosity, inclusions Permeability variations Extremely sensitive for locating small tigt cracks Cracks Wall tickness Hardness Coercive force Magnetic anisotropy Magnetic field Nonmagnetic coating tickness on steel Cracks, oles, debonded areas, etc., in nonmetallic parts Canges in composition, degree of cure, moisture content Tickness measurement Dielectric constant Loss tangent Flaws open to surface of parts; cracks, porosity, seams, laps, etc. Troug-wall leaks Level of fill in containers Foreign objects Internal components Density variations Voids, tickness Spacing or position Porous materials suc as clay, carbon, powdered metals, concrete Grinding weels Hig-tension insulators Sanitary ware Brazed joints Adesive-bonded joints Metallic platings or coatings; debonded areas or tickness Electrical assemblies Temperature monitoring Joints: Welded Brazed Adesive-bonded Sealed assemblies Pressure or vacuum cambers Fuel or gas tanks Ferromagnetic materials; bar, plate, forgings, weldments, extrusions, etc. Ferromagnetic materials Sip degaussing Liquid-level control Treasure unting Wall tickness of nonmetallic materials Material sorting Reinforced plastics Cemical products Ceramics Resins Rubber Wood Liquids Polyuretane foam Radomes All parts wit nonabsorbing surfaces (forgings, weldments, castings, etc.). Note: Bleed-out from porous surfaces can mask indications of flaws Flow of liquids Presence of cavitation Operation of valves and switces Burning in small solid-propellant rocket motors Colored or fluorescent particles Leaves no residue after baking part over 400 F Quickly and easily applied Portable Sensitive to 0. F temperature variation Permanent record or termal picture Quantitative Remote sensing; need not contact part Portable Hig sensitivity to extremely small, ligt separations not detectable by oter NDT metods Sensitivity related to metod selected Advantage over penetrant is tat it indicates subsurface flaws, particularly inclusions Relatively fast and low-cost May be portable Measurement of magnetic material properties May be automated Easily detects magnetic objects in nonmagnetic material Portable Between radio waves and infrared in electromagnetic spectrum Portable Contact wit part surface not normally required Can be automated Low cost Portable Indications may be furter examined visually Results easily interpreted Hig-brigtness images Real-time viewing Image magnification Permanent record Moving subject can be observed Size and sape of particles must be selected before use Penetrating power of suspension medium is critical Particle concentration must be controlled Skin irritation Emissivity Liquid-nitrogen-cooled detector Critical time-temperature relationsip Poor resolution for tick specimens Reference standards required Accessibility to bot surfaces of part required Smeared metal or contaminants may prevent detection Cost related to sensitivity Alignment of magnetic field is critical Demagnetization of parts required after tests Parts must be cleaned before and after inspection Masking by surface coatings Permeability Reference standards required Edge effect Probe lift-off Will not penetrate metals Reference standards required Horn-to-part spacing critical Part geometry Wave interference Vibration Surface films suc as coatings, scale, and smeared metal may prevent detection of flaws Parts must be cleaned before and after inspection Flaws must be open to surface Costly equipment Geometric unsarpness Tick specimens Speed of event to be studied Viewing area Radiation azard

64 5-64 NONDESTRUCTIVE TESTING Table 5.4. Nondestructive Test Metods* (Continued) Metod Measures or detects Applications Advantages Limitations Neutron radiology (termal neutrons from reactor, accelerator, or Californium 5) Gamma radiology (cobalt 60, iridium 9) X-ray radiology Radiometry X-ray, gamma ray, beta ray) (transmission or backscatter) Reverse-geometry digital X-ray X-ray computed tomograpy (CT) Searograpy electronic Termal (termocromic paint, liquid crystals) Sonic (less tan 0. MHz) Hydrogen contamination of titanium or zirconium alloys Defective or improperly loaded pyrotecnic devices Improper assembly of metal, nonmetal parts Corrosion products Internal flaws and variations, porosity, inclusions, cracks, lack of fusion, geometry variations, corrosion tinning Density variations Tickness, gap, and position Internal flaws and variations; porosity, inclusions, cracks, lack of fusion, geometry variations, corrosion Density variations Tickness, gap, and position Misassembly Misalignment Wall tickness Plating tickness Variations in density or composition Fill level in cans or containers Inclusions or voids Cracks Corrosion Water in oneycomb Carbon epoxy oneycomb Foreign objects Small density canges Cracks Voids Foreign objects Lack of bond Delaminations Plastic deformation Strain Crused core Impact damage Corrosion in Al oneycomb Lack of bond Hot spots Heat transfer Isoterms Temperature ranges Blockage in coolant passages Debonded areas or delaminations in metal or nonmetal composites or laminates Coesive bond strengt under controlled conditions Crused or fractured core Bond integrity of metal insert fasteners Pyrotecnic devices Metallic, nonmetallic assemblies Biological specimens Nuclear reactor fuel elements and control rods Adesive-bonded structures Usually were X-ray macines are not suitable because source cannot be placed in part wit small openings and/or power source not available Panoramic imaging Castings Electrical assemblies Weldments Small, tin, complex wrougt products Nonmetallics Solid-propellant rocket motors Composites Container contents Seet, plate, strip, tubing Nuclear reactor fuel rods Cans or containers Plated parts Composites Aircraft structure Solid-propellant rocket motors Rocket nozzles Jet-engine parts Turbine blades Composite-metal oneycomb Bonded structures Composite structures Brazed joints Adesive-bonded joints Metallic platings or coatings Electrical assemblies Temperature monitoring Metal or nonmetal composite or laminates brazed or adesivebonded Plywood Rocket-motor nozzles Honeycomb Hig neutron absorption by ydrogen, boron, litium, cadmium, uranium, plutonium Low neutron absorption by most metals Complement to X-ray or gamma-ray radiograpy Low initial cost Permanent records; film Small sources can be placed in parts wit small openings Portable Low contrast Permanent records; film Adjustable energy levels (5 kv 5 mev) Hig sensitivity to density canges No couplant required Geometry variations do not affect direction of X-ray beam Fully automatic Fast Extremely accurate In-line process control Portable Hig-resolution 0 6 pixel image wit ig contrast Measures X-ray opacity of object along many pats Large area coverage Rapid setup and operation Noncontacting Video image easy to store Very low initial cost Can be readily applied to surfaces wic may be difficult to inspect by oter metods No special operator skills Portable Easy to operate Locates far-side debonded areas May be automated Access to only one surface required Very costly equipment Nuclear reactor or accelerator required Trained pysicists required Radiation azard Nonportable Indium or gadolinium screens required One energy level per source Source decay Radiation azard Trained operators needed Lower image resolution Cost related to source size Hig initial costs Orientation of linear flaws in part may not be favorable Radiation azard Dept of flaw not indicated Sensitivity decreases wit increase in scattered radiation Radiation azard Beta ray useful for ultratin coatings only Source decay Reference standards required Access to bot sides of object Radiation azard Very expensive Trained operator Radiation azard Requires vacuum termal, ultrasonic, or microwave stressing of structure to cause surface strain Tin-walled surfaces only Critical time-temperature relationsip Image retentivity affected by umidity Reference standards required Surface geometry influences test results Reference standards required Adesive or core-tickness variations influence results

65 RADIOGRAPHIC METHODS 5-65 Table 5.4. Nondestructive Test Metods* (Continued) Metod Measures or detects Applications Advantages Limitations Ultrasonic (0. 5 MHz) Termoelectric probe Internal flaws and variations; cracks, lack of fusion, porosity, inclusions, delaminations, lack of bond, texturing Tickness or velocity Poisson s ratio, elastic modulus Termoelectric potential Coating tickness Pysical properties Tompson effect P-N junctions in semiconductors Metals Welds Brazed joints Adesive-bonded joints Nonmetallics In-service parts Metal sorting Ceramic coating tickness on metals Semiconductors * From Donald J. Hagemaier, Metal Progress Databook, Douglas Aircraft Co., McDonnell-Douglas Corp., Long Beac, CA. Most sensitive to cracks Test results known immediately Automating and permanent-record capability Portable Hig penetration capability Portable Simple to operate Access to only one surface required Couplant required Small, tin, or complex parts may be difficult to inspect Reference standards required Trained operators for manual inspection Special probes Hot probe Difficult to automate Reference standards required Surface contaminants Conductive coatings in a semidarkened location. After testing, te penetrant and developer are removed by wasing wit water, sometimes aided by an emulsifier, or wit a solvent. In filtered particle testing, cracks in porous objects (00 mes or smaller) are indicated by te difference in absorption between a cracked and a flaw-free surface. A liquid containing suspended particles is sprayed on a test object. If a crack exists, particles are filtered out and concentrate at te surface as liquid flows into te additional absorbent area created by te crack. Fluorescent or colored particles are used to locate flaws in unfired dried clay, certain fired ceramics, concrete, some powdered metals, carbon, and partially sintered tungsten and titanium carbides. RADIOGRAPHIC METHODS Radiograpic test metods employ X-rays, gamma rays, or similar penetrating radiation to reveal flaws, voids, inclusions, tickness, or structure of objects. Electromagnetic energy wavelengts in te range of 0.0 to 0 Å(Å 0 8 cm) are used to examine te interior of opaque materials. Penetrating radiation proceeds from its source in straigt lines to te test object. Rays are differentially absorbed by te object, depending upon te energy of te radiation and te nature and tickness of te material. X-rays of a variety of wavelengts result wen ig-speed electrons in a vacuum tube are suddenly stopped. An X-ray tube contains a eated filament (catode) and a target (anode); radiation intensity is almost directly proportional to filament current (ma); tube voltage (kv) determines te penetration capability of te rays. As tube voltage increases, sorter wavelengts and more intense X-rays are produced. Wen te energy of penetrating radiation increases, sorter wavelengts and more intense X-rays are produced. Also, wen te energy of penetrating radiation increases, te difference in attenuation between materials decreases. Consequently, more film-image contrast is obtained at lower voltage, and a greater range of tickness can be radiograped at one time at iger voltage. Gamma rays of a specific wavelengt are emitted from te disintegrating nuclei of natural radioactive elements, suc as radium, and from a variety of artificial radioactive isotopes produced in nuclear reactors. Cobalt 60 and iridium 9 are commonly used for industrial radiograpy. Te alf-life of an isotope is te time required for alf of te radioactive material to decay. Tis time ranges from a few ours to many years. Radiograps are potograpic records produced by te passage of penetrating radiation onto a film. A void or reduced mass appears as a darker image on te film because of te lesser absorption of energy and te resulting additional exposure of te film. Te quantity of X-rays absorbed by a material generally increases as te atomic number increases. A radiograp is a sadow picture, since X-rays and gamma rays follow te laws of ligt in sadow formation. Four factors determine te best geometric sarpness of a picture: () Te effective focal-spot size of te radiation source sould be as small as possible. () Te source-to-object distance sould be adequate for proper definition of te area of te object fartest from te film. (3) Te film sould be as close as possible to te object. (4) Te area of interest sould be in te center of and perpendicular to te X-ray beams and parallel to te X-ray film. Radiograpic films vary in speed, contrast, and grain size. Slow films generally ave smaller grain size and produce more contrast. Slow films are used were optimum sarpness and maximum contrast are desired. Fast films are used were objects wit large differences in tickness are to be radiograped or were sarpness and contrast can be sacrificed to sorten exposure time. Exposure of a radiograpic film comes from direct radiation and scattered radiation. Direct radiation is desirable, image-forming radiation; scattered radiation, wic occurs in te object being X-rayed or in neigboring objects, produces undesirable images on te film and loss of contrast. Intensifying screens made of or 0.00-in- (0.3-mm or 0.5-mm) tick lead are often used for radiograpy at voltages above 00 kv. Te lead filters out muc of te low-energy scatter radiation. Under action of X-rays or gamma rays above 88 kv, a lead screen also emits electrons wic, wen in intimate contact wit te film, produce additional coerent darkening of te film. Exposure time can be materially reduced by use of intensifying screens above and below te film. Penetrameters are used to indicate te contrast and definition wic exist in a radiograp. Te type generally used in te United States is a small rectangular plate of te same material as te object being X-rayed. It is uniform in tickness (usually percent of te object tickness) and as oles drilled troug it. ASTM specifies ole diameters,, and 4 times te tickness of te penetrameter. Step, wire, and bead penetrameters are also used. (See ASTM Materials Specification E94.) Because of te variety of factors tat affect te production and measurements of an X-ray image, operating factors are generally selected from reference tables or graps wic ave been prepared from test data obtained for a range of operating conditions. All materials may be inspected by radiograpic means, but tere are limitations to te configurations of materials. Wit optimum tecniques, wires in (0.003 mm) in diameter can be resolved in small electrical components. At te oter extreme, welded steel pressure vessels wit 0-in (500-mm) wall tickness can be routinely inspected by use of ig-energy accelerators as a source of radiation. Neutron radiation penetrates extremely dense materials suc as lead more readily tan X-rays or gamma rays but is attenuated by ligter-atomic-weigt materials suc as plastics, usually because of teir ydrogen content. Radiograpic standards are publised by ASTM, ASME, AWS, and API, primarily for detecting lack of penetration or lack of fusion in welded objects. Cast-metal objects are radiograped to detect condi-

66 5-66 NONDESTRUCTIVE TESTING tions suc as srink, porosity, ot tears, cold suts, inclusions, coarse structure, and cracks. Te usual metod of utilizing penetrating radiation employs film. However, Geiger counters, semiconductors, pospors (fluoroscopy), potoconductors (xeroradiograpy), scintillation crystals, and vidicon tubes (image intensifiers) are also used. Computerized digital radiograpy is an expanding tecnology. Te dangers connected wit exposure of te uman body to X-rays and gamma rays sould be fully understood by any person responsible for te use of radiation equipment. NIST is a prime source of information concerning radiation safety. NRC specifies maximum permissible exposure to be a.5 R/ 4 year. ULTRASONIC METHODS Ultrasonic nondestructive test metods employ ig-frequency mecanical vibrational energy to detect and locate structural discontinuities or differences and to measure tickness of a variety of materials. An electric pulse is generated in a test instrument and transmitted to a transducer, wic converts te electric pulse into mecanical vibrations. Tese low-energy-level vibrations are transmitted troug a coupling liquid into te test object, were te ultrasonic energy is attenuated, scattered, reflected, or resonated to indicate conditions witin material. Reflected, transmitted, or resonant sound energy is reconverted to electrical energy by a transducer and returned to te test instrument, were it is amplified. Te received energy is ten usually displayed on a catode-ray tube. Te presence, position, and amplitude of ecoes indicate conditions of te test-object material. Materials capable of being tested by ultrasonic energy are tose wic transmit vibrational energy. Metals are tested in dimensions of up to 30 ft (9.4 m). Noncellular plastics, ceramics, glass, new concrete, organic materials, and rubber can be tested. Eac material as a caracteristic sound velocity, wic is a function of its density and modulus (elastic or sear). Material caracteristics determinable troug ultrasonics include structural discontinuities, suc as flaws and unbonds, pysical constants and metallurgical differences, and tickness (measured from one side). A common application of ultrasonics is te inspection of welds for inclusions, porosity, lack of penetration, and lack of fusion. Oter applications include location of unbond in laminated materials, location of fatigue cracks in macinery, and medical applications. Automatic testing is frequently performed in manufacturing applications. Ultrasonic systems are classified as eiter pulse-eco, in wic a single transducer is used, or troug-transmission, in wic separate sending and receiving transducers are used. Pulse-eco systems are more common. In eiter system, ultrasonic energy must be transmitted into, and received from, te test object troug a coupling medium, since air will not efficiently transmit ultrasound of tese frequencies. Water, oil, grease, and glycerin are commonly used couplants. Two types of testing are used: contact and immersion. In contact testing, te transducer is placed directly on te test object. In immersion testing, te transducer and test object are separated from one anoter in a tank filled wit water or by a column of water or by a liquid-filled weel. Immersion testing eliminates transducer wear and facilitates scanning of te test object. Scanning systems ave paper-printing or computerized video equipment for readout of test information. Ultrasonic transducers are piezoelectric units wic convert electric energy into acoustic energy and convert acoustic energy into electric energy of te same frequency. Quartz, barium titanate, litium sulfate, lead metaniobate, and lead zirconate titanate are commonly used transducer crystals, wic are generally mounted wit a damping backing in a ousing. Transducers range in size from 6 to 5 in (0.5 to.7 cm) and are circular or rectangular. Ultrasonic beams can be focused to improve resolution and definition. Transducer caracteristics and beam patterns are dependent upon frequency, size, crystal material, and construction. Test frequencies used range from 40 khz to 00 MHz. Flaw-detection and tickness-measurement applications use frequencies between 500 khz and 5 MHz, wit.5 and 5 MHz being most commonly employed for flaw detection. Low frequencies (40 khz to.0 MHz) are used on materials of low elastic modulus or large grain size. Hig frequencies (.5 to 5 MHz) provide better resolution of smaller defects and are used on fine-grain materials and tin sections. Frequencies above 5 MHz are employed for investigation and measurement of pysical properties related to acoustic attenuation. Wave-vibrational modes oter tan longitudinal are effective in detecting flaws tat do not present a reflecting surface to te ultrasonic beam, or oter caracteristics not detectable by te longitudinal mode. Tey are useful also wen large areas of plates must be examined. Wedges of plastic, water, or oter material are inserted between te transducer face and te test object to convert, by refraction, to sear, transverse, surface, or Lamb vibrational modes. As in optics, Snell s law expresses te relationsip between incident and refracted beam angles; i.e., te ratio of te sines of te angle from te normal, of te incident and refracted beams in two mediums, is equal to te ratio of te mode acoustic velocities in te two mediums. Limiting conditions for ultrasonic testing may be te test-object sape, surface rougness, grain size, material structure, flaw orientation, selectivity of discontinuities, and te skill of te operator. Test sensitivity is less for cast metals tan for wrougt metals because of grain size and surface differences. Standards for acceptance are publised in many government, national society, and company specifications (see references above). Evaluation is made by comparing (visually or by automated electronic means) received signals wit signals obtained from reference blocks containing flat bottom oles between 64 and 8 64 in (0.40 and 0.35 cm) in diameter, or from parts containing known flaws, drilled oles, or macined notces. EDDY CURRENT METHODS Eddy current nondestructive tests are based upon correlation between electromagnetic properties and pysical or structural properties of a test object. Eddy currents are induced in metals wenever tey are brougt into an ac magnetic field. Tese eddy currents create a secondary magnetic field, wic opposes te inducing magnetic field. Te presence of discontinuities or material variations alters eddy currents, tus canging te apparent impedance of te inducing coil or of a detection coil. Coil impedance indicates te magnitude and pase relationsip of te eddy currents to teir inducing magnetic-field current. Tis relationsip is dependent upon te mass, conductivity, permeability, and structure of te metal and upon te frequency, intensity, and distribution of te alternating magnetic field. Conditions suc as eat treatment, composition, ardness, pase transformation, case dept, cold working, strengt, size, tickness, cracks, seams, and inomogeneities are indicated by eddy current tests. Correlation data must usually be obtained to determine weter test conditions for desired caracteristics of a particular test object can be establised. Because of te many factors wic cause variation in electromagnetic properties of metals, care must be taken tat te instrument response to te condition of interest is not nullified or duplicated by variations due to oter conditions. Alternating-current frequencies between and 5,000,000 Hz are used for eddy current testing. Test frequency determines te dept of current penetration into te test object, owing to te ac penomenon of skin effect. One standard dept of penetration is te dept at wic te eddy currents are equal to 37 percent of teir value at te surface. In a plane conductor, dept of penetration varies inversely as te square root of te product of conductivity, permeability, and frequency. Hig-frequency eddy currents are more sensitive to surface flaws or conditions wile low-frequency eddy currents are sensitive also to deeper internal flaws or conditions. Test coils are of tree general types: te circular coil, wic surrounds an object; te bobbin coil, wic is inserted witin an object; and te probe coil, wic is placed on te surface of an object. Coils are furter classified as absolute, wen testing is conducted witout direct comparison wit a reference object in anoter coil; or differential, wen compar-

67 ACOUSTIC SIGNATURE ANALYSIS 5-67 ison is made troug use of two coils connected in series opposition. Many variations of tese coil types are utilized. Axial lengt of a circular test coil sould not be more tan 4 in (0. cm), and its sape sould correspond closely to te sape of te test object for best results. Coil diameter sould be only sligtly larger tan te test-object diameter for consistent and useful results. Coils may be of te air-core or magneticcore type. Instrumentation for te analysis and presentation of electric signals resulting from eddy current testing includes a variety of means, ranging from meters to oscilloscopes to computers. Instrument meter or alarm circuits are adjusted to be sensitive only to signals of a certain electrical pase or amplitude, so tat selected conditions are indicated wile oters are ignored. Automatic and automated testing is one of te principal advantages of te metod. Tickness measurement of metallic and nonmetallic coatings on metals is performed using eddy current principles. Coating ticknesses measured typically range from to 0.00 in ( to 0.5 cm). For measurement to be possible, coating conductivity must differ from tat of te base metal. MICROWAVE METHODS Microwave test metods utilize electromagnetic energy to determine caracteristics of nonmetallic substances, eiter solid or liquid. Frequencies used range from to 3,000 GHz. Microwaves generated in a test instrument are transmitted by a waveguide troug air to te test object. Analysis of reflected or transmitted energy indicates certain material caracteristics, suc as moisture content, composition, structure, density, degree of cure, aging, and presence of flaws. Oter applications include tickness and displacement measurement in te range of 0.00 in (0.005 cm) to more tan in (30.4 cm). Materials tat can be tested include most solid and liquid nonmetals, suc as cemicals, minerals, plastics, wood, ceramics, glass, and rubber. INFRARED METHODS Infrared nondestructive tests involve te detection of infrared electromagnetic energy emitted by a test object. Infrared radiation is produced naturally by all matter at all temperatures above absolute zero. Materials emit radiation at varying intensities, depending upon teir temperature and surface caracteristics. A passive infrared system detects te natural radiation of an uneated test object, wile an active system employs a source to eat te test object, wic ten radiates infrared energy to a detector. Sensitive indication of temperature or temperature distribution troug infrared detection is useful in locating irregularities in materials, in processing, or in te functioning of parts. Emission in te infrared range of 0.8 to 5 m is collected optically, filtered, detected, and amplified by a test instrument wic is designed around te caracteristics of te detector material. Temperature variations on te order of 0. F can be indicated by meter or grapic means. Infrared teory and instrumentation are based upon radiation from a blackbody; terefore, emissivity correction must be made electrically in te test instrument or aritmetically from instrument readings. ACOUSTIC SIGNATURE ANALYSIS Acoustic signature analysis involves te analysis of sound energy emitted from an object to determine caracteristics of te object. Te object may be a simple casting or a complex manufacturing system. A passive test is one in wic sonic energy is transmitted into te object. In tis case, a mode of resonance is usually detected to correlate wit cracks or structure variations, wic cause a cange in effective modulus of te object, suc as a nodular iron casting. An active test is one in wic te object emits sound as a result of being struck or as a result of being in operation. In tis case, caracteristics of te object may be correlated to damping time of te sound energy or to te presence or absence of a certain frequency of sound energy. Bearing wear in rotating macinery can often be detected prior to actual failure, for example. More complex analytical systems can monitor and control manufacturing processes, based upon analysis of emitted sound energy. Acoustic emission is a tecnology distinctly separate from acoustic signature analysis and is one in wic strain produces bursts of energy in an object. Tese are detected by ultrasonic transducers coupled to te object. Growt of microcracks, and oter flaws, as well as incipient failure, is monitored by counting te pulses of energy from te object or recording te time rate of te pulses of energy in te ultrasonic range (usually a discrete frequency between khz and MHz).