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    Chapter 1Chapter 1

    Materials andMaterials and

    Mechanics of BendingMechanics of Bending

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    ConcreteConcrete

    Concrete is a mixture of cement along with fine and coarse aggregates.Concrete is a mixture of cement along with fine and coarse aggregates.

    Aggregates: sand, gravel, crushed rock, and other materials.Aggregates: sand, gravel, crushed rock, and other materials. Water is added for the chemical reaction of curing.Water is added for the chemical reaction of curing.

    Concrete strength and durabilit depend on the proportions of the mix,Concrete strength and durabilit depend on the proportions of the mix,along with the placing, finishing, and curing of the concrete.along with the placing, finishing, and curing of the concrete.

    !he compressive strength of concrete is high" the tensile strength of!he compressive strength of concrete is high" the tensile strength ofconcrete is low.concrete is low.

    Concrete is a brittle material.Concrete is a brittle material.

    #einforcing steel $in the form of bars and mesh% is used to resist#einforcing steel $in the form of bars and mesh% is used to resist

    tension.tension.

    #einforced concrete#einforced concreteis a combination of steel and concrete.is a combination of steel and concrete.

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    !he AC& 'uilding Code!he AC& 'uilding Code

    !he design and construction of reinforced concrete buildings is controlled!he design and construction of reinforced concrete buildings is controlledb theb the 'uilding Code #e(uirements for )tructural Concrete'uilding Code #e(uirements for )tructural Concrete$AC& *1+-+%$AC& *1+-+%of the American Concrete &nstitute.of the American Concrete &nstitute.

    !he term code/ generall refers to the AC& Code.!he term code/ generall refers to the AC& Code.

    !he code is revised, updated, and reissued on a *ear ccle.!he code is revised, updated, and reissued on a *ear ccle.

    !he code has no legal status, but the AC& Code is incorporated into!he code has no legal status, but the AC& Code is incorporated into

    the building codes of almost all states and municipalities throughoutthe building codes of almost all states and municipalities throughoutthe 0nited )tates.the 0nited )tates.

    When incorporated into a state or local building code, the AC& CodeWhen incorporated into a state or local building code, the AC& Codehas official sanction, becomes a legal document, and is part of the lawhas official sanction, becomes a legal document, and is part of the lawcontrolling reinforced concrete design and construction in that area.controlling reinforced concrete design and construction in that area.

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    Cement and WaterCement and Water

    )tructural concrete generall uses hdraulic cement.)tructural concrete generall uses hdraulic cement.

    Water is re(uired for the chemical reaction ofWater is re(uired for the chemical reaction of hdrationhdration.. uring hdration, the concrete hardens into a solid mass.uring hdration, the concrete hardens into a solid mass.

    2ortland cement2ortland cement, which originated in 3ngland, is the most common, which originated in 3ngland, is the most commonform of cement.form of cement.

    Cement is marketed in bulk or in 45 lb $1 cubic foot% bags.Cement is marketed in bulk or in 45 lb $1 cubic foot% bags.

    &n concrete, the ratio of the amount of water to the amount of cement, b&n concrete, the ratio of the amount of water to the amount of cement, bweight, is termedweight, is termed the water6cement ratiothe water6cement ratio..

    !his ratio is often expressed in terms of gallons of water per bag of!his ratio is often expressed in terms of gallons of water per bag ofcement.cement.

    7or complete hdration of the cement in a concrete mix, a7or complete hdration of the cement in a concrete mix, awater6cement ratio of -.*8 to -.5- $5 to 59 gal6bag% is re(uired.water6cement ratio of -.*8 to -.5- $5 to 59 gal6bag% is re(uired.

    !o increase the!o increase the workabilitworkabilitof concrete $i.e. the ease with which it canof concrete $i.e. the ease with which it canbe mixed, handled, and placed%, higher water6cement ratios arebe mixed, handled, and placed%, higher water6cement ratios arenormall used.normall used.

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    AggregatesAggregates

    &n ordinar concretes, aggregates occup approximatel -; to 8; of&n ordinar concretes, aggregates occup approximatel -; to 8; of

    the volume of the hardened mass.the volume of the hardened mass. o. 5 sieve $four

    openings per linear inch%.openings per linear inch%.

    Coarse aggregatesCoarse aggregates consist of particles $such as gravel% that areconsist of particles $such as gravel% that areretained on a >o. 5 sieve.retained on a >o. 5 sieve.

    !he maximum si=e of coarse aggregate in reinforced concrete is!he maximum si=e of coarse aggregate in reinforced concrete isgoverned b AC& Code re(uirements.governed b AC& Code re(uirements. ?aximum si=e re(uirements for aggregate are established to assure that?aximum si=e re(uirements for aggregate are established to assure that

    the concrete can be placed into forms without an danger of largethe concrete can be placed into forms without an danger of largeparticles lodging between ad@acent bars or between bars and the sides ofparticles lodging between ad@acent bars or between bars and the sides ofthe forms.the forms.

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    Concrete in Compression $continued%Concrete in Compression $continued%

    >ote the following observations from the stressstrain curves.>ote the following observations from the stressstrain curves.

    !he maximum compressive strength is generall achieved at a unit!he maximum compressive strength is generall achieved at a unitstrain of approximatel -.--B.strain of approximatel -.--B.

    Digherstrength concretes are more brittle and fracture at a lowerDigherstrength concretes are more brittle and fracture at a lowermaximum strain than the lowstrength concretes.maximum strain than the lowstrength concretes.

    !he initial slope of the curve varies $unlike that of steel% and onl!he initial slope of the curve varies $unlike that of steel% and onl

    approximates a straight line.approximates a straight line. 7or steel $that behaves elasticall up to a ield point%, the stressstrain7or steel $that behaves elasticall up to a ield point%, the stressstrain

    plot is a straight line below the ield point.plot is a straight line below the ield point. 7or concrete, the straightline portion of the plot is ver short $if it7or concrete, the straightline portion of the plot is ver short $if it

    exists at all%.exists at all%. 7or concrete, there is no constant value of modulus of elasticit for a7or concrete, there is no constant value of modulus of elasticit for a

    given concrete strength.given concrete strength. &f a straightline portion is assumed, the modulus of elasticit is different&f a straightline portion is assumed, the modulus of elasticit is different

    for concretes of different strengths.for concretes of different strengths.

    At low and moderate stresses $up to about -.8 fAt low and moderate stresses $up to about -.8 f cc%, concrete is assumed to%, concrete is assumed tobehave elasticall.behave elasticall.

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    Concrete in Compression $continued%Concrete in Compression $continued%

    !he AC& Code $)ection +.8.1% provides the accepted empirical expression for!he AC& Code $)ection +.8.1% provides the accepted empirical expression for

    thethe modulus of elasticitmodulus of elasticit..33ccE wE wcc1.81.8**Ff**Ffcc

    wherewhere

    33ccE modulus of elasticit of concrete in compression $psi%E modulus of elasticit of concrete in compression $psi%

    wwccE unit weight of concrete $lb6ftE unit weight of concrete $lb6ft**%%

    ffcc E compressive strength of concrete $psi% E compressive strength of concrete $psi%

    !he AC& Code expression for modulus of elasticit is valid for concretes!he AC& Code expression for modulus of elasticit is valid for concreteshaving whaving wccbetween 4- and 1- lb6ftbetween 4- and 1- lb6ft**..

    &f the unit weight for concrete is 155 lb6ft&f the unit weight for concrete is 155 lb6ft**

    , the expression for, the expression for

    modulus of elasticit becomesmodulus of elasticit becomes

    33ccE 8,--- FfE 8,--- Ffcc

    !able A $p. 5+ of the textbook% lists values for 3!able A $p. 5+ of the textbook% lists values for 3cc using thisusing thisexpression.expression.

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    Concrete in Compression $continued%Concrete in Compression $continued%

    Concrete strength varies with time and rates of loading.Concrete strength varies with time and rates of loading.

    !he!he specifiedspecifiedconcrete strength is usuall the strength that occurs B+concrete strength is usuall the strength that occurs B+das after the concrete is placed.das after the concrete is placed.

    Concrete attains approximatel -; of its B+da strength in das.Concrete attains approximatel -; of its B+da strength in das.

    Concrete attains approximatel +8; to 4-; of its B+da strength inConcrete attains approximatel +8; to 4-; of its B+da strength in15 das.15 das.

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    Concrete in !ensionConcrete in !ension

    !he tensile and compressive strengths of concrete are not e(ual or!he tensile and compressive strengths of concrete are not e(ual or

    proportional.proportional. !he tensile strength of normalweight concrete is approximatel 1-; to!he tensile strength of normalweight concrete is approximatel 1-; to

    18; of the compressive strength $per AC& Code Commentar%.18; of the compressive strength $per AC& Code Commentar%.

    An increase in compressive strength is accompanied b an appreciablAn increase in compressive strength is accompanied b an appreciablsmaller percentage increase in tensile strength.smaller percentage increase in tensile strength.

    !he true tensile strength of concrete is difficult to determine.!he true tensile strength of concrete is difficult to determine.

    !he!he split,clinder testsplit,clinder test $A)!? C54% is used to determine the tensile$A)!? C54% is used to determine the tensilestrength of lightweight aggregate concrete.strength of lightweight aggregate concrete.

    )plitting tensile strength)plitting tensile strength, f, fctct, ma be calculated b the following, ma be calculated b the following

    expression $derived from the theor of elasticit%:expression $derived from the theor of elasticit%:ffctctE B26GHE B26GH wherewhere

    ffctctEE splitting tensile strengthsplitting tensile strengthof lightweight aggregate concrete $psi%of lightweight aggregate concrete $psi%2 E applied load at splitting $lb%2 E applied load at splitting $lb%H E length of clinder $inch%H E length of clinder $inch%

    E diameter of clinder $inch% E diameter of clinder $inch%

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    Concrete in !ension $continued%Concrete in !ension $continued%

    Another common approach to measure tensile strength is to use theAnother common approach to measure tensile strength is to use the modulusmodulusof ruptureof rupture, f, frr..

    !he!he modulus of rupturemodulus of ruptureis the maximum tensile bending stress in a plainis the maximum tensile bending stress in a plainconcrete test beam at failure $A)!? C+%.concrete test beam at failure $A)!? C+%.

    !he moment that produces a tensile stress @ust e(ual to the modulus of!he moment that produces a tensile stress @ust e(ual to the modulus ofrupture is termed therupture is termed the cracking momentcracking moment, ?, ?crcr..

    !he AC& Code recommends that the modulus of rupture f!he AC& Code recommends that the modulus of rupture f rr $psi% be$psi% betaken astaken as

    ffrrE .8 I FfE .8 I Ffcc

    wherewhere

    I $

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    #einforcing )teel#einforcing )teel

    ?ost concrete is reinforced in some wa to resist tensile forces.?ost concrete is reinforced in some wa to resist tensile forces.

    !ensile reinforcement is embedded in concrete to withstand the tensile!ensile reinforcement is embedded in concrete to withstand the tensilestress.stress.

    #einforcement is in the form of steel reinforcing bars or welded wire#einforcement is in the form of steel reinforcing bars or welded wirereinforcing $often called mesh/% composed of steel wire.reinforcing $often called mesh/% composed of steel wire.

    #einforcing in the form of structural steel shapes, steel pipe, steel#einforcing in the form of structural steel shapes, steel pipe, steeltubing, and highstrength steel tendons is also permitted b the AC&tubing, and highstrength steel tendons is also permitted b the AC&Code.Code.

    Jther economical reinforcement includes fiberreinforced concreteJther economical reinforcement includes fiberreinforced concrete$using short fibers of steel or fiberglass%.$using short fibers of steel or fiberglass%.

    )pecifications for steel reinforcement are published b A)!? $American)pecifications for steel reinforcement are published b A)!? $American)ociet for !esting and ?aterials%.)ociet for !esting and ?aterials%.

    !he specifications are generall accepted for the steel used in!he specifications are generall accepted for the steel used inreinforced concrete construction in the 0nited )tates.reinforced concrete construction in the 0nited )tates.

    !he A)!? specifications are identified in the AC& Code $)ection *.8%.!he A)!? specifications are identified in the AC& Code $)ection *.8%.

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    #einforcing )teel $continued%#einforcing )teel $continued%

    )teel bars used for reinforcing are generall round deformed bars.)teel bars used for reinforcing are generall round deformed bars.

    #einforcing bars feature patterned ribbed pro@ections rolled onto#einforcing bars feature patterned ribbed pro@ections rolled ontotheir surfaces conforming to A)!? specifications.their surfaces conforming to A)!? specifications.

    )teel reinforcing bars are available in straight lengths of - feet.)teel reinforcing bars are available in straight lengths of - feet.

    )maller bar si=es are available in coil stock for use in automatic)maller bar si=es are available in coil stock for use in automaticbending machines.bending machines.

    'ars designations var from >o. * to >o. 11, along with >o. 15 and >o.'ars designations var from >o. * to >o. 11, along with >o. 15 and >o.1+.1+. !he designation for the >o. * to the >o. + bars represents the!he designation for the >o. * to the >o. + bars represents the

    bar diameter in eighths of an inch $e.g. >o. 8 bar is 86+/ inbar diameter in eighths of an inch $e.g. >o. 8 bar is 86+/ indiameter%.diameter%.

    !he >o. 4, >o. 1-, and >o. 11 bars have diameters that provide!he >o. 4, >o. 1-, and >o. 11 bars have diameters that provideareas e(ual to 1/ s(uare bars, 1.1B8/ s(uare bars, and 1.B8/ s(uareareas e(ual to 1/ s(uare bars, 1.1B8/ s(uare bars, and 1.B8/ s(uarebars, respectivel.bars, respectivel.

    !he >o. 15 and >o. 1+ bars have diameters that provide areas!he >o. 15 and >o. 1+ bars have diameters that provide arease(ual to 1.8/ and B/ s(uare bars, respectivel, and are commonle(ual to 1.8/ and B/ s(uare bars, respectivel, and are commonlavailable b special order.available b special order.

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    #einforcing )teel $continued%#einforcing )teel $continued%

    !he most useful phsical properties of reinforcing steel for reinforced!he most useful phsical properties of reinforcing steel for reinforced

    concrete design calculations are the ield stress $fconcrete design calculations are the ield stress $f% and the modulus of% and the modulus ofelasticit $3elasticit $3ss%.%.

    !able A1 $p. 5+* of the textbook% lists the available steel grades and!able A1 $p. 5+* of the textbook% lists the available steel grades andthe associated ield stress.the associated ield stress.

    !he modulus of elasticit is taken as B4,---,--- psi $per AC& Code,!he modulus of elasticit is taken as B4,---,--- psi $per AC& Code,

    )ection +.8.B%.)ection +.8.B%.

    Corrosion of reinforcing steel leads to cracking and spalling of theCorrosion of reinforcing steel leads to cracking and spalling of theconcrete in which it is embedded.concrete in which it is embedded.

    Kualit concrete, with ade(uate cover, provides good protectionKualit concrete, with ade(uate cover, provides good protection

    against corrosion.against corrosion. 2rotective coatings ma be used to minimi=e the corrosion of the2rotective coatings ma be used to minimi=e the corrosion of thereinforcing steel.reinforcing steel. >onmetallic materials: epox coated, compling with A)!? A8 or>onmetallic materials: epox coated, compling with A)!? A8 or

    A)!? A4*5 $per AC& Code%.A)!? A4*5 $per AC& Code%. ?etallic materials: =inc $galvani=ing%, compling with A)!? A $per AC&?etallic materials: =inc $galvani=ing%, compling with A)!? A $per AC&

    Code%.Code%.

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    #einforcing )teel $continued%#einforcing )teel $continued%

    Welded wire reinforcing $WW#% $mesh% is another tpe of reinforcing.Welded wire reinforcing $WW#% $mesh% is another tpe of reinforcing.

    Welded wire reinforcing consists of colddrawn wire in s(uare orWelded wire reinforcing consists of colddrawn wire in s(uare orrectangular patterns, welded at all intersections.rectangular patterns, welded at all intersections.

    Welded wire reinforcing ma be supplied in rolls or sheets dependingWelded wire reinforcing ma be supplied in rolls or sheets dependingon the wire si=e.on the wire si=e.

    Wire si=es are designated b the smbol WW#, followed b theWire si=es are designated b the smbol WW#, followed b the

    spacing of the longitudinal wires, then the spacing of the transversespacing of the longitudinal wires, then the spacing of the transversewires, b the si=e of the longitudinal wire, and b the si=e of thewires, b the si=e of the longitudinal wire, and b the si=e of thetransverse wires.transverse wires.

    7or example: WW# x 1B L W1 x W+7or example: WW# x 1B L W1 x W+

    !his is the designation for a plain WW# with / longitudinal spacing,!his is the designation for a plain WW# with / longitudinal spacing,1B/ transverse spacing, and a cross sectional area e(ual to -.1 in1B/ transverse spacing, and a cross sectional area e(ual to -.1 in BBforforthe longitudinal wires and -.-+ inthe longitudinal wires and -.-+ inBBfor the transverse wires.for the transverse wires.

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    'eams: ?echanics of 'ending #eview'eams: ?echanics of 'ending #eview

    !he expression for the maximum bending stress in a beam $i.e. the!he expression for the maximum bending stress in a beam $i.e. the flexureflexureformulaformula% is% is

    ffbbE ?c6&E ?c6&

    wherewhere ffbbE the calculated bending stress at the outer fiber of the cross sectionE the calculated bending stress at the outer fiber of the cross section

    ? E the applied maximum moment? E the applied maximum moment

    c E the distance from the neutral axis to the outside tension or compressionc E the distance from the neutral axis to the outside tension or compressionfiber of the beamfiber of the beam

    & E the moment of inertia of the cross section about the neutral axis& E the moment of inertia of the cross section about the neutral axis

    !he!he flexure formulaflexure formularepresents the relationship between bending stress,represents the relationship between bending stress,bending moment, and the geometric properties of the beam cross section.bending moment, and the geometric properties of the beam cross section.

    Assumptions used in developing the flexure formula include:Assumptions used in developing the flexure formula include: 'eams $such as steel and wood% are composed of homogeneous material.'eams $such as steel and wood% are composed of homogeneous material.

    'eams $such as steel and wood% exhibit elastic behavior up to a certain limit.'eams $such as steel and wood% exhibit elastic behavior up to a certain limit.

    !he stress distribution developed at an cross section is linear, varing from!he stress distribution developed at an cross section is linear, varing from=ero at the neutral axis to a maximum at the outer fibers.=ero at the neutral axis to a maximum at the outer fibers.

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    'eams: ?echanics of 'ending #eview $continued%'eams: ?echanics of 'ending #eview $continued%

    !he maximum moment that ma be applied to the beam cross section,!he maximum moment that ma be applied to the beam cross section,

    called thecalled the resisting momentresisting moment$?$?##%, ma be found b rearranging the terms%, ma be found b rearranging the termsof the flexure formula.of the flexure formula.

    ??##E 7E 7bb&6c&6c

    wherewhere

    77bbE the allowable bending stressE the allowable bending stress

    !he use of the!he use of the flexure formulaflexure formulapresents some complications when appliedpresents some complications when appliedto concrete.to concrete.

    #einforced concrete is not a homogeneous material.#einforced concrete is not a homogeneous material.

    Concrete does not behave elasticall over its full range of strengthConcrete does not behave elasticall over its full range of strength$stress%.$stress%.

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    'eams: ?echanics of 'ending #eview $continued%'eams: ?echanics of 'ending #eview $continued%

    A different approach, called theA different approach, called the internal couple methodinternal couple method, is used for the, is used for the

    design and analsis of concrete beams.design and analsis of concrete beams. !he internal couple $moment% is composed of a compressive force C/!he internal couple $moment% is composed of a compressive force C/

    above the neutral axis and a parallel tensile force !/ below theabove the neutral axis and a parallel tensile force !/ below theneutral axis forming a couple.neutral axis forming a couple. A single span, simpl supported beam with a positive bending moment isA single span, simpl supported beam with a positive bending moment is

    assumed.assumed.

    7orces acting at the cross section must be e(ual and opposite in7orces acting at the cross section must be e(ual and opposite indirection to satisf e(uilibrium re(uirements $i.e. MD E -%, thusdirection to satisf e(uilibrium re(uirements $i.e. MD E -%, thus

    C E !C E !

    !he couple formed b C and ! must be e(ual and opposite to the!he couple formed b C and ! must be e(ual and opposite to thebending moment at the same location produced b the external loads.bending moment at the same location produced b the external loads.

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    'eams: ?echanics of 'ending #eview $continued%'eams: ?echanics of 'ending #eview $continued%

    !he first example that follows demonstrates the concept of!he first example that follows demonstrates the concept of crackingcracking

    momentmoment ??crcr and the modulus of rupture $fand the modulus of rupture $frr% for a plain $nonreinforced%% for a plain $nonreinforced%concrete beam.concrete beam.

    !he!he cracking momentcracking moment??crcr causes the maximum tensile stress @ust tocauses the maximum tensile stress @ust toreach the modulus of rupture $freach the modulus of rupture $frr%.%.

    !he!he cracking momentcracking moment??crcrcauses the cross sectional area to be on thecauses the cross sectional area to be on the

    verge of cracking.verge of cracking. !he first example compares the modulus of rupture developed from!he first example compares the modulus of rupture developed from

    the cracking moment with the AC&recommended modulus of rupture.the cracking moment with the AC&recommended modulus of rupture.

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    'eams: ?echanics of 'ending #eview $continued%'eams: ?echanics of 'ending #eview $continued%

    !he second example of a plain $nonreinforced% concrete beam!he second example of a plain $nonreinforced% concrete beam

    demonstrates thedemonstrates the internal couple methodinternal couple method.. >onreinforced concrete beams are considered homogeneous and>onreinforced concrete beams are considered homogeneous and

    elastic $valid for small tensile stresses%.elastic $valid for small tensile stresses%.

    !he analsis for bending stresses in the uncracked/ beam is based on!he analsis for bending stresses in the uncracked/ beam is based onthe properties of the gross cross sectional area using the elasticthe properties of the gross cross sectional area using the elastic

    based flexure formula.based flexure formula. !he use of the flexure formula is valid as long as the maximum tensile!he use of the flexure formula is valid as long as the maximum tensile

    stress does not exceed the modulus of rupture $fstress does not exceed the modulus of rupture $f rr%.%.

    !he second example compares the internal couple method and the!he second example compares the internal couple method and theflexure formula approach.flexure formula approach.