Structural Concrete Design

Transcription

1 50 Strutural Conrete Design Amy Grider Purdue University Julio A. Ramirez Purdue University Young Mook Yun Purdue University 50.1 Properties of Conrete and Reinforing Steel Properties of Conrete Lightweight Conrete Heavyweight Conrete High-Strength Conrete Reinforing Street 50.2 Proportioning and Mixing Conrete Proportioning Conrete Mix Admixtures Mixing 50.3 Flexural Design of Beams and One-Way Slabs Reinfored Conrete Strength Design Prestressed Conrete Strength Design 50.4 Columns under Bending and Axial Load Short Columns under Minimum Eentriity Short Columns under Axial Load and Bending Slenderness Effets Columns under Axial Load and Biaxial Bending 50.5 Shear and Torsion Reinfored Conrete Beams and One-Way Slabs Strength Design Prestressed Conrete Beams and One-Way Slabs Strength Design 50.6 Development of Reinforement Development of Bars in Tension Development of Bars in Compression Development of Hooks in Tension Splies, Bundled Bars, and Web Reinforement 50.7 Two-Way Systems Definition Design Proedures Minimum Slab Thikness and Reinforement Diret Design Method Equivalent Frame Method Detailing 50.8 Frames Analysis of Frames Design for Seismi Loading 50.9 Brakets and Corbels Footings Types of Footings Design Considerations Wall Footings Single-Column Spread Footings Combined Footings Two- Column Footings Strip, Grid, and Mat Foundations Footings on Piles Walls Panel, Curtain, and Bearing Walls Basement Walls Partition Walls Shear Walls At this point in the history of development of reinfored and prestressed onrete it is neessary to reexamine the fundamental approahes to design of these omposite materials. Strutural engineering is a worldwide industry. Designers from one nation or a ontinent are faed with designing a projet in

2 another nation or ontinent. The deades of efforts dediated to harmonizing onrete design approahes worldwide have resulted in some suesses but in large part have led to further differenes and numerous different design proedures. It is this abundane of different design approahes, tehniques, and ode regulations that justifies and alls for the need for a unifiation of design approahes throughout the entire range of strutural onrete, from plain to fully prestressed [Breen, 1991]. The effort must begin at all levels: university ourses, textbooks, handbooks, and standards of pratie. Students and pratitioners must be enouraged to think of a single ontinuum of strutural onrete. Based on this premise, this hapter on onrete design is organized to promote suh unifiation. In addition, effort will be direted at dispelling the present unjustified preoupation with omplex analysis proedures and often highly empirial and inomplete setional mehanis approahes that tend to both distrat the designers from fundamental behavior and impart a false sense of auray to beginning designers. Instead, designers will be direted to give areful onsideration to overall struture behavior, remarking the adequate flow of fores throughout the entire struture Properties of Conrete and Reinforing Steel The designer needs to be knowledgeable about the properties of onrete, reinforing steel, and prestressing steel. This part of the hapter summarizes the material properties of partiular importane to the designer. Properties of Conrete Workability is the ease with whih the ingredients an be mixed and the resulting mix handled, transported, and plaed with little loss in homogeneity. Unfortunately, workability annot be measured diretly. Engineers therefore try to measure the onsisteny of the onrete by performing a slump test. The slump test is useful in deteting variations in the uniformity of a mix. In the slump test, a mold shaped as the frustum of a one, 12 in. (305 mm) high with an 8 in. (203 mm) diameter base and 4 in. (102 mm) diameter top, is filled with onrete (ASTM Speifiation C143). Immediately after filling, the mold is removed and the hange in height of the speimen is measured. The hange in height of the speimen is taken as the slump when the test is done aording to the ASTM Speifiation. A well-proportioned workable mix settles slowly, retaining its original shape. A poor mix rumbles, segregates, and falls apart. The slump may be inreased by adding water, inreasing the perentage of fines (ement or aggregate), entraining air, or by using an admixture that redues water requirements; however, these hanges may adversely affet other properties of the onrete. In general, the slump speified should yield the desired onsisteny with the least amount of water and ement. Conrete should withstand the weathering, hemial ation, and wear to whih it will be subjeted in servie over a period of years; thus, durability is an important property of onrete. Conrete resistane to freezing and thawing damage an be improved by inreasing the watertightness, entraining 2 to 6% air, using an air-entraining agent, or applying a protetive oating to the surfae. Chemial agents damage or disintegrate onrete; therefore, onrete should be proteted with a resistant oating. Resistane to wear an be obtained by use of a high-strength, dense onrete made with hard aggregates. Exess water leaves voids and avities after evaporation, and water an penetrate or pass through the onrete if the voids are interonneted. Watertightness an be improved by entraining air or reduing water in the mix, or it an be prolonged through uring. Volume hange of onrete should be onsidered, sine expansion of the onrete may ause bukling and drying shrinkage may ause raking. Expansion due to alkali-aggregate reation an be avoided by using nonreative aggregates. If reative aggregates must be used, expansion may be redued by adding pozzolani material (e.g., fly ash) to the mix. Expansion aused by heat of hydration of the ement an be redued by keeping ement ontent as low as possible; using Type IV ement; and hilling the aggregates, water, and onrete in the forms. Expansion from temperature inreases an be redued by

3 using oarse aggregate with a lower oeffiient of thermal expansion. Drying shrinkage an be redued by using less water in the mix, using less ement, or allowing adequate moist uring. The addition of pozzolans, unless allowing a redution in water, will inrease drying shrinkage. Whether volume hange auses damage usually depends on the restraint present; onsideration should be given to eliminating restraints or resisting the stresses they may ause [MaGregor, 1992]. Strength of onrete is usually onsidered its most important property. The ompressive strength at 28 days is often used as a measure of strength beause the strength of onrete usually inreases with time. The ompressive strength of onrete is determined by testing speimens in the form of standard ylinders as speified in ASTM Speifiation C192 for researh testing or C31 for field testing. The test proedure is given in ASTM C39. If drilled ores are used, ASTM C42 should be followed. The suitability of a mix is often desired before the results of the 28-day test are available. A formula proposed by W. A. Slater estimates the 28-day ompressive strength of onrete from its 7-day strength: S28 = S7 +30 S7 (50.1) where S 28 = 28-day ompressive strength, psi, and S 7 = 7-day ompressive strength, psi. Strength an be inreased by dereasing water-ement ratio, using higher strength aggregate, using a pozzolan suh as fly ash, grading the aggregates to produe a smaller perentage of voids in the onrete, moist uring the onrete after it has set, and vibrating the onrete in the forms. The short-time strength an be inreased by using Type III portland ement, aelerating admixtures, and by inreasing the uring temperature. The stress-strain urve for onrete is a urved line. Maximum stress is reahed at a strain of in./in., after whih the urve desends. The modulus of elastiity, E, as given in ACI (Revised 92), Building Code Requirements for Reinfored Conrete [ACI Committee 318, 1992], is: E = w f 3 lb ft and psi (50.2a) E = w f kg m and MPa (50.2b) where w = unit weight of onrete, and f =ompressive strength at 28 days. Tensile strength of onrete is muh lower than the ompressive strength about 7 f for the higherstrength onretes and 10 f for the lower-strength onretes. Creep is the inrease in strain with time under a onstant load. Creep inreases with inreasing waterement ratio and dereases with an inrease in relative humidity. Creep is aounted for in design by using a redued modulus of elastiity of the onrete. Lightweight Conrete Strutural lightweight onrete is usually made from aggregates onforming to ASTM C330 that are usually produed in a kiln, suh as expanded lays and shales. Strutural lightweight onrete has a density between 90 and 120 lb/ft 3 ( kg/m 3 ). Prodution of lightweight onrete is more diffiult than normal-weight onrete beause the aggregates vary in absorption of water, speifi gravity, moisture ontent, and amount of grading of undersize. Slump and unit weight tests should be performed often to ensure uniformity of the mix. During plaing and finishing of the onrete, the aggregates may float to the surfae. Workability an be improved by inreasing the perentage of fines or by using an air-entraining admixture to inorporate 4 to 6% air.

4 Dry aggregate should not be put into the mix, beause it will ontinue to absorb moisture and ause the onrete to harden before plaement is ompleted. Continuous water uring is important with lightweight onrete. No-fines onrete is obtained by using pea gravel as the oarse aggregate and 20 to 30% entrained air instead of sand. It is used for low dead weight and insulation when strength is not important. This onrete weighs from 105 to 118 lb/ft 3 ( kg/m 3 ) and has a ompressive strength from 200 to 1000 psi (1 7 MPa). A porous onrete made by gap grading or single-size aggregate grading is used for low ondutivity or where drainage is needed. Lightweight onrete an also be made with gas-forming or foaming agents whih are used as admixtures. Foam onretes range in weight from 20 to 110 lb/ft 3 ( kg/m 3 ). The modulus of elastiity of lightweight onrete an be omputed using the same formula as normal onrete. The shrinkage of lightweight onrete is similar to or slightly greater than for normal onrete. Heavyweight Conrete Heavyweight onretes are used primarily for shielding purposes against gamma and x-radiation in nulear reators and other strutures. Barite, limonite and magnetite, steel punhings, and steel shot are typially used as aggregates. Heavyweight onretes weigh from 200 to 350 lb/ft 3 (3200 to 5600 kg/m 3 ) with strengths from 3200 to 6000 psi (22 41 MPa). Gradings and mix proportions are similar to those for normal weight onrete. Heavyweight onretes usually do not have good resistane to weathering or abrasion. High-Strength Conrete Conretes with strengths in exess of 6000 psi (41 MPa) are referred to as high-strength onretes. Strengths up to 18,000 psi (124 MPa) have been used in buildings. Admixtures suh as superplastiizers, silia fume, and supplementary ementing materials suh as fly improve the dispersion of ement in the mix and produe workable onretes with lower water-ement ratios, lower void ratios, and higher strength. Coarse aggregates should be strong, fine-grained gravel with rough surfaes. For onrete strengths in exess of 6000 psi (41 MPa), the modulus of elastiity should be taken as E = 40, 000 f (50.3) where f = ompressive strength at 28 days, psi [ACI Committee 36]. The shrinkage of high-strength onrete is about the same as that for normal onrete. Reinforing Steel Conrete an be reinfored with welded wire fabri, deformed reinforing bars, and prestressing tendons. Welded wire fabri is used in thin slabs, thin shells, and other loations where spae does not allow the plaement of deformed bars. Welded wire fabri onsists of old drawn wire in orthogonal patterns square or retangular and resistane-welded at all intersetions. The wires may be smooth (ASTM A185 and A82) or deformed ASTM A497 and A496). The wire is speified by the symbol W for smooth wires or D for deformed wires followed by a number representing the ross-setional area in hundredths of a square inh. On design drawings it is indiated by the symbol WWF followed by spaings of the wires in the two 90 diretions. Properties for welded wire fabri are given in Table The deformations on a deformed reinforing bar inhibit longitudinal movement of the bar relative to the onrete around it. Table 50.2 gives dimensions and weights of these bars. Reinforing bar steel an

5 TABLE 50.1 Wire and Welded Wire Fabri Steels Minimum Yield Stress, a f y Minimum Tensile Strength Wire Size AST Desription Designation ksi MPa ksi MPa A82-79 (old-drawn wire) (properties apply when material is to be used for fabri) W1.2 and larger b Smaller than W1.2 A (welded wire fabri) Same as A82; this is A82 material fabriated into sheet (so-alled mesh ) by the proess of eletri welding. A (deformed steel wire) (properties apply when material is to be used for fabri) D1 D A Same as A82 or A496; this speifiation applies for fabri made from A496, or from a ombination of A496 and A82 wires. a The term yield stress refers to either yield point, the well-defined deviation from perfet elastiity, or yield strength, the value obtained by a speified offset strain for material having no well-defined yield point. b The W number represents the nominal ross-setional area in square inhes multiplied by 100, for smooth wires. The D number represents the nominal ross-setional area in square inhes multiplied by 100, for deformed wires. Soure: Wang and Salmon, TABLE 50.2 Reinforing Bar Dimensions and Weights Nominal Dimensions Bar Diameter Area Weight Number (in.) (mm) (in. 2 ) (m 2 ) (lb/ft) (kg/m) be made of billet steel of grades 40 and 60 having minimum speifi yield stresses of 40,000 and 60,000 psi, respetively (276 and 414 MPa) (ASTM A615) or low-alloy steel of grade 60, whih is intended for appliations where welding and/or bending is important (ASTM A706). Presently, grade 60 billet is the most predominately used for onstrution. Prestressing tendons are ommonly in the form of individual wires or groups of wires. Wires of different strengths and properties are available with the most prevalent being the 7-wire low-relaxation strand onforming to ASTM A416. ASTM A416 also overs a stress-relieved strand, whih is seldom used in onstrution nowadays. Properties of standard prestressing strands are given in Table Prestressing tendons ould also be bars; however, this is not very ommon. Prestressing bars meeting ASTM A722 have been used in onnetions between members. The modulus of elastiity for non-prestressed steel is 29,000,000 psi (200,000 MPa). For prestressing steel, it is lower and also variable, so it should be obtained from the manufaturer. For 7-wire strands onforming to ASTM A416, the modulus of elastiity is usually taken as 27,000,000 psi (186,000 MPa).

6 TABLE 50.3 Standard Prestressing Strands, Wires, and Bars 50.2 Proportioning and Mixing Conrete Proportioning Conrete Mix Grade Nominal Dimension f pu Diameter Area Weight Tendon Type ksi in. in. 2 plf Seven-wire strand 250 1/ / / / / Prestressing wire Deformed prestressing bars 157 5/ ¼ ⅜ Soure: Collins and Mithell, A onrete mix is speified by the weight of water, sand, oarse aggregate, and admixture to be used per 94-pound bag of ement. The type of ement (Table 50.4), modulus of the aggregates, and maximum size of the aggregates (Table 50.5) should also be given. A mix an be speified by the weight ratio of ement to sand to oarse aggregate with the minimum amount of ement per ubi yard of onrete. In proportioning a onrete mix, it is advisable to make and test trial bathes beause of the many variables involved. Several trial bathes should be made with a onstant water-ement ratio but varying TABLE 50.4 Types of Portland Cement* Type I II III IV V Usage Ordinary onstrution where speial properties are not required Ordinary onstrution when moderate sulfate resistane or moderate heat of hydration is desired When high early strength is desired When low heat of hydration is desired When high sulfate resistane is desired *Aording to ASTM C150. TABLE 50.5 Reommended Maximum Sizes of Aggregate* Maximum Size, in., of Aggregate for Minimum Dimension Reinfored-Conrete Heavily Lightly Reinfored of Setion, in. Beams, Columns, Walls Reinfored Slabs or Unreinfored Slabs 5 or less ¾ 1½ ¾ 1½ 6 11 ¾ 1½ 1½ 1½ ½ or more 1½ *Conrete Manual. U.S. Bureau of Relamation.

7 TABLE 50.6 Typial Conrete Mixes* Maximum Size of Aggregate, in. Mix Designation Bags of Cement per yd 3 of Conrete Air-Entrained Conrete Aggregate, lb per Bag of Cement Sand Conrete Without Air Gravel or Crushed Stone 1/2 A B C I A B C A B C ½ A B C A B C *Conrete Manual, U.S. Bureau of Relamation. ratios of aggregates to obtain the desired workability with the least ement. To obtain results similar to those in the field, the trial bathes should be mixed by mahine. When time or other onditions do not allow proportioning by the trial bath method, Table 50.6 may be used. Start with mix B orresponding to the appropriate maximum size of aggregate. Add just enough water for the desired workability. If the mix is undersanded, hange to mix A; if oversanded, hange to mix C. Weights are given for dry sand. For damp sand, inrease the weight of sand 10 lb, and for very wet sand, 20 lb per bag of ement. Admixtures Admixtures may be used to modify the properties of onrete. Some types of admixtures are set aelerators, water reduers, air-entraining agents, and waterproofers. Admixtures are generally helpful in improving quality of the onrete. However, if admixtures are not properly used, they ould have undesirable effets; it is therefore neessary to know the advantages and limitations of the proposed admixture. The ASTM Speifiations over many of the admixtures. Set aelerators are used (1) when it takes too long for onrete to set naturally, suh as in old weather, or (2) to aelerate the rate of strength development. Calium hloride is widely used as a set aelerator. If not used in the right quantities, it ould have harmful effets on the onrete and reinforement. Water reduers lubriate the mix and permit easier plaement of the onrete. Sine the workability of a mix an be improved by a hemial agent, less water is needed. With less water but the same ement ontent, the strength is inreased. Sine less water is needed, the ement ontent ould also be dereased, whih results in less shrinkage of the hardened onrete. Some water reduers also slow down the onrete set, whih is useful in hot weather and in integrating onseutive pours of the onrete. Air-entraining agents are probably the most widely used type of admixture. Minute bubbles of air are entrained in the onrete, whih inreases the resistane of the onrete to freeze-thaw yles and the use of ie-removal salts. Waterproofing hemials are often applied as surfae treatments, but they an be added to the onrete mix. If applied properly and uniformly, they an prevent water from penetrating the onrete surfae. Epoxies an also be used for waterproofing. They are more durable than silione oatings, but they may

8 be more ostly. Epoxies an also be used for protetion of wearing surfaes, pathing avities and raks, and glue for onneting piees of hardened onrete. Mixing Materials used in making onrete are stored in bath plants that have weighing and ontrol equipment and bins for storing the ement and aggregates. Proportions are ontrolled by automati or manually operated sales. The water is measured out either from measuring tanks or by using water meters. Mahine mixing is used whenever possible to ahieve uniform onsisteny. The revolving drum-type mixer and the ounterurrent mixer, whih has mixing blades rotating in the opposite diretion of the drum, are ommonly used. Mixing time, whih is measured from the time all ingredients are in the drum, should be at least 1.5 minutes for a 1-yd 3 mixer, plus 0.5 min for eah ubi yard of apaity over 1 yd 3 [ACI , 1973]. It also is reommended to set a maximum on mixing time sine overmixing may remove entrained air and inrease fines, thus requiring more water for workability; three times the minimum mixing time an be used as a guide. Ready-mixed onrete is made in plants and delivered to job sites in mixers mounted on truks. The onrete an be mixed en route or upon arrival at the site. Conrete an be kept plasti and workable for as long as 1.5 hours by slow revolving of the mixer. Mixing time an be better ontrolled if water is added and mixing started upon arrival at the job site, where the operation an be inspeted Flexural Design of Beams and One-Way Slabs Reinfored Conrete Strength Beams The basi assumptions made in flexural design are: 1. Setions perpendiular to the axis of bending that are plane before bending remain plane after bending. 2. A perfet bond exists between the reinforement and the onrete suh that the strain in the reinforement is equal to the strain in the onrete at the same level. 3. The strains in both the onrete and the reinforement are assumed to be diretly proportional to the distane from the neutral axis (ACI ) [ACI Committee 318, 1992]. 4. Conrete is assumed to fail when the ompressive strain reahes (ACI ). 5. The tensile strength of onrete is negleted (ACI ). 6. The stresses in the onrete and reinforement an be omputed from the strains using stressstrain urves for onrete and steel, respetively. 7. The ompressive stress-strain relationship for onrete may be assumed to be retangular, trapezoidal, paraboli, or any other shape that results in predition of strength in substantial agreement with the results of omprehensive tests (ACI ). ACI outlines the use of a retangular ompressive stress distribution whih is known as the Whitney retangular stress blok. For other stress distributions see Reinfored Conrete Mehanis and Design by James G. MaGregor [1992]. Analysis of Retangular Beams with Tension Reinforement Only Equations for M n and fm n : Tension Steel Yielding. Consider the beam shown in Fig The ompressive fore, C, in the onrete is ( ) C = 085. f ba (50.3) The tension fore, T, in the steel is T = Asfy (50.4)

9 b 0.85f a = b 1 a/2 d Neutral axis (Axis of zero strain) jd = d - a/2 f s T (a) Cross setion. (b) Atual stress distribution. () Equilvalent retangular stress distribution. FIGURE 50.1 Stresses and fores in a retangular beam. (Soure: MaGregor, 1992.) For equilibrium, C = T, so the depth of the equivalent retangular stress blok, a, is Af s y a = 085. fb (50.5) Noting that the internal fores C and T form an equivalent fore-ouple system, the internal moment is M = T ( n d - a 2) (50.6) or M = C ( n d - a 2) fm n is then ( ) fm = n ft d - a 2 (50.7) or ( ) fm = n fc d - a 2 where f = Equation for M n and fm n : Tension Steel Elasti. The internal fores and equilibrium are given by: C = T 085. fba = Af s s 085. fba = rbde e s s (50.8) From strain ompatibility (see Fig. 50.1), e s = e u Ê d - ˆ Á Ë (50.9)

10 Substituting e s into the equilibrium equation, noting that a = b 1, and simplifying gives Ê 085. f ˆ 2 Á a d a b1d Ë re e 2 + ( ) - = s u 0 (50.10) whih an be solved for a. Equations (50.6) and (50.7) an then be used to obtain M n and fm n. Reinforement Ratios. The reinforement ratio, r, is used to represent the relative amount of tension reinforement in a beam and is given by r= A s bd (50.11) At the balaned strain ondition the maximum strain, e u, at the extreme onrete ompression fiber reahes just as the tension steel reahes the strain e y = f y /E s. The reinforement ratio in the balaned strain ondition, r b, an be obtained by applying equilibrium and ompatibility onditions. From the linear strain ondition, Fig. 50.1, b eu , 000 = = = d e + e f 87, f , 000, 000 u y y y (50.12) The ompressive and tensile fores are: C = 085. f bb1 b b T = f A = r bdf b y sb b y (50.13) Equating C b to T b and solving for r b gives r b 085. f b1 Ê b ˆ = Á f Ë d y (50.14) whih on substitution of Eq. (50.12) gives 085. f b Ê ˆ 1 87, 000 rb = f Á y Ë 87, f y (50.15) ACI limits the amount of reinforement in order to prevent nondutile behavior: max r = 075. r b (50.16) ACI 10.5 requires a minimum amount of flexural reinforement: r min = 200 f y (50.17) Analysis of Beams with Tension and Compression Reinforement For the analysis of doubly reinfored beams, the ross setion will be divided into two beams. Beam 1 onsists of the ompression reinforement at the top and suffiient steel at the bottom so that T 1 = C s ; beam 2 onsists of the onrete web and the remaining tensile reinforement, as shown in Fig

11 0.003 A s d A s a = b f f s C s C d A s A s f s = f y T (a) Setion. (b) Strain distribution. () Stress distribution. (d) Internal fores. A s C s = A s f s a C (d - d ) (d - a/2) A s1 A s2 T 1 = A s1 f y (e) Beam 1. (f) Beam 2. T 2 = A s2 f y FIGURE 50.2 Strains, stresses, and fores in beam with ompression reinforement. (Soure: MaGregor, 1992.) Equation for M n : Compression Steel Yields. The area of tension steel in beam 1 is obtained by setting T 1 = C s, whih gives A s1 = A s.the nominal moment apaity of beam 1 is then ( ) M = A n1 sfy d - d (50.18) Beam 2 onsists of the onrete and the remaining steel, A s2 = A s A s1 = A s A s.the ompression fore in the onrete is C = 085. f ba (50.19) and the tension fore in the steel for beam 2 is ( ) T = A - A f s s y (50.20) The depth of the ompression stress blok is then a = ( ) A - A f s s y 085. fb (50.21) Therefore, the nominal moment apaity for beam 2 is ( ) ( - ) M = A - A f d a n2 s s y 2 (50.22) The total amount apaity for a doubly reinfored beam with ompression steel yielding is the summation of the moment apaity for beam 1 and beam 2; therefore, ( ) + ( - ) ( - ) M - A f d - d A A f d a 2 n s y s s y (50.23)

12 Equation for M n : Compression Steel Does Not Yield. Assuming that the tension steel yields, the internal fores in the beam are T = A f s y where From equilibrium, C s + C = T or C C = 085. f ba ( ) = A E e s s s s Ê b1 d ˆ e s = Á Ë a ( ) (50.24) (50.25) Ê b1d ˆ 085. fba + AE Á Ë a ( ) = Af s s s y (50.26) This an be rewritten in quadrati form as ( ) - b s s s y s s 1 ( ) + - ( ) = 085. fba AE Af a AE d 0 (50.27) where a an be alulated by means of the quadrati equation. Therefore, the nominal moment apaity in a doubly reinfored onrete beam where the ompression steel does not yield is Ê a ˆ Mn = C Ád - + Cs d - d Ë 2 ( ) (50.28) Reinforement Ratios. The reinforement ratio at the balaned strain ondition an be obtained in a similar manner as that for beams with tension steel only. For ompression steel yielding, the balaned ratio is 085. f b Ê ˆ 1 87, 000 ( r - r ) = b f Á Ë + y 87, 000 fy (50.29) For ompression steel not yielding, the balaned ratio is Ê r - r f ˆ Á b Ë = Ê ˆ s 085. f 1 87, 000 f f Á Ë + f y y 87, 000 y b (50.30) The maximum and minimum reinforement ratios as given in ACI and 10.5 are r r max min = 075. r 200 = f y b (50.31)

13 Prestressed Conrete Strength Design Elasti Flexural Analysis In developing elasti equations for prestress, the effets of prestress fore, dead load moment, and live load moment are alulated separately, and then the separate stresses are superimposed, giving F Fey f =- ± ± A I My I (50.32) where ( ) indiates ompression and (+) indiates tension. It is neessary to hek that the stresses in the extreme fibers remain within the ACI-speified limits under any ombination of loadings that may our. The stress limits for the onrete and prestressing tendons are speified in ACI 18.4 and 18.5 [ACI Committee 318, 1992]. ACI states that the loss of area due to open duts shall be onsidered when omputing setion properties. It is noted in the ommentary that setion properties may be based on total area if the effet of the open dut area is onsidered negligible. In pretensioned members and in post-tensioned members after grouting, setion properties an be based on gross setions, net setions, or effetive setions using the transformed areas of bonded tendons and nonprestressed reinforement. Flexural Strength The strength of a prestressed beam an be alulated using the methods developed for ordinary reinfored onrete beams, with modifiations to aount for the differing nature of the stress-strain relationship of prestressing steel ompared with ordinary reinforing steel. A prestressed beam will fail when the steel reahes a stress f ps, generally less than the tensile strength f pu. For retangular ross-setions the nominal flexural strength is M a = A f d - 2 n ps ps (50.33) where Aps fps a = 085. fb (50.34) The steel stress f ps an b found based on strain ompatibility or by using approximate equations suh as those given in ACI The equations in ACI are appliable only if the effetive prestress in the steel, f se, whih equals P e /A ps, is not less than 0.5 f pu. The ACI equations are as follows. (a) For members with bonded tendons: f ps Ê g È p f pu d = fpu - Í f + ( d - ) ˆ Á1 Ë b r w w 1 ÎÍ p (50.35) If any ompression reinforement is taken into aount when alulating f ps with Eq. (50.35), the following applies: È f pu d rp w w f + ( d - ) Í ÎÍ p 017. (50.36)

14 and d 015. (b) For members with unbonded tendons and with a span-to-depth ratio of 35 or less: d p f ps f f py = fse + 10 Ï, Ì 100 r f p Ó se + 60, 000 (50.37) () For members with unbonded tendons and with a span-to-depth ratio greater than 35: f ps f f Ï py = fse + 10, Ì 300 r f p Ó se + 30, 000 (50.38) The flexural strength is then alulated from Eq. (50.33). The design strength is equal to fm n, where f = 0.90 for flexure. Reinforement Ratios ACI requires that the total amount of prestressed and nonprestressed reinforement be adequate to develop a fatored load at least 1.2 times the raking load alulated on the basis of a modulus of rupture of 7.5 f. To ontrol raking in members with unbonded tendons, some bonded reinforement should be uniformly distributed over the tension zone near the extreme tension fiber. ACI speifies the minimum amount of bonded reinforement as A = s A (50.39) where A is the area of the ross setion between the flexural tension fae and the enter of gravity of the gross ross setion. ACI gives the minimum length of the bonded reinforement. To ensure adequate dutility, ACI provides the following requirement: Ï Ô Ô Ô w p Ô d wp + Ê ˆ Ô w w b Ë Á Ô Ì d ( - ) Ô p Ô Ô d wpw + Ê ˆ Ô Ô ww ww Ë Á d ( - ) Ô Ô Ó p Ô (50.40) ACI allows eah of the terms on the left side to be set equal to 0.85 a/d p in order to simplify the equation. When a reinforement ratio greater than 0.36b 1 is used, ACI states that the design moment strength shall not be greater than the moment strength based on the ompression portion of the moment ouple Columns under Bending and Axial Load Short Columns under Minimum Eentriity When a symmetrial olumn is subjeted to a onentri axial load, P, longitudinal strains develop uniformly aross the setion. Beause the steel and onrete are bonded together, the strains in the

15 onrete and steel are equal. For any given strain it is possible to ompute the stresses in the onrete and steel using the stress-strain urve for the two materials. The fores in the onrete and steel are equal to the stresses multiplied by the orresponding areas. The total load on the olumn is the sum of the fores in the onrete and steel: ( ) + P = 085. f A - A f A o g st y st (50.41) To aount for the effet of inidental moments, ACI speifies that the maximum design axial load on a olumn be, for spiral olumns, and for tied olumns, (50.42) (50.43) For high values of axial load, f values of 0.7 and 0.75 are speified for tied and spiral olumns, respetively (ACI b) [ACI Committee 318, 1992]. Short olumns are suffiiently stoky suh that slenderness effets an be ignored. Short Columns under Axial and Bending Almost all ompression members in onrete strutures are subjeted to moments in addition to axial loads. Although it is possible to derive equations to evaluate the strength of olumns subjeted to ombined bending and axial loads, the equations are tedious to use. For this reason, interation diagrams for olumns are generally omputed by assuming a series of strain distributions, eah orresponding to a partiular point on the interation diagram, and omputing the orresponding values of P and M. One enough suh points have been omputed, the results are summarized in an interation diagram. For examples on determining the interation diagram, see Reinfored Conrete Mehanis and Design by James G. MaGregor [1992] or Reinfored Conrete Design by Chu-Kia Wang and Charles G. Salmon [1985]. Figure 50.3 illustrates a series of strain distributions and the resulting points on the interation diagram. Point A represents pure axial ompression. Point B orresponds to rushing at one fae and zero tension at the other. If the tensile strength of onrete is ignored, this represents the onset of raking on the bottom fae of the setion. All points lower than this in the interation diagram represent ases in whih the setion is partially raked. Point C, the farthest right point, orresponds to the balaned strain ondition and represents the hange from ompression failures for higher loads and tension failures for lower loads. Point D represents a strain distribution where the reinforement has been strained to several times the yield strain before the onrete reahes it rushing strain. The horizontal axis of the interation diagram orresponds to pure bending where f = 0.9. A transition is required from f = 0.7 or 0.75 for high axial loads to f = 0.9 for pure bending. The hange in f begins at a apaity fp a, whih equals the smaller of the balaned load, fp b, or 0.1f Ag. Generally, fp b exeeds 0.1f Ag exept for a few nonretangular olumns. ACI Publiations SP-17A(85), A Design Handbook for Columns, ontains nondimensional interation diagrams as well as other design aids for olumn [ACI Committee 340, 1990]. Slenderness Effets [ ( ) + ] fp (. f. f A A f A max ) = n g st y st [ ( ) + ] fp (. f. f A A f A max ) = n g st y st ACI desribes an approximate slenderness-effet design proedure based on the moment magnifier onept. The moments are omputed by ordinary frame analysis and multiplied by a moment magnifier that is a funtion of the fatored axial load and the ritial bukling load of the olumn. The following gives a summary of the moment magnifier design proedure for slender olumns in frames.

16 Pure ompression e u e u A e u Axial load, P n B C Balaned failure D e y E e u Moment, M n e su > e y FIGURE 50.3 Strain distributions orresponding to points on interation diagram. 1. Length of olumn. The unsupported length, l u, is defined in ACI as the lear distane between floor slabs, beams, or other members apable of giving lateral support to the olumn. 2. Effetive length. The effetive length fators, k, used in alulating d b shall be between 0.5 and 1.0 (ACI ). The effetive length fators used to ompute d s shall be greater than 1 (ACI ). The effetive length fators an be estimated using ACI Fig. R or using ACI Equations (A) (E) given in ACI R These two proedures require that the ratio, y, of the olumns and beams be known: y =   ( EI l ) ( EI l ) (50.44) In omputing y it is aeptable to take the EI of the olumn as the unraked gross E I g of the olumns and the EI of the beam as 0.5 E I g. 3. Definition of braed and unbraed frames. The ACI Commentary suggests that a frame is braed if either of the following are satisfied: (a) If the stability index, Q, for a story is less than 0.04, where b b b  Pu u Q = D 004. Hh u s (50.45) (b) If the sum of the lateral stiffness of the braing elements in a story exeeds six times the lateral stiffness of all of the olumns in the story.

17 4. Radius of gyration. For a retangular ross setion r equals 0.3 h, and for a irular ross setion r equals 0.25 h. For other setions, r equals I A. 5. Considerations of slenderness effets. ACI allows slenderness effets to be negleted for olumns in braed frames when kl r u M1 < M b 2b (50.46) ACI allows slenderness effets to be negleted for olumns in unbraed frames when kl u < 22 r (50.47) If kl u /r exeeds 100, ACI states that design shall be based on seond-order analysis. 6. Minimum moments. For olumns in a braed frame, M 2b shall be not less than the value given in ACI In an unbraed frame ACI applies for M 2s. 7. Moment magnifier equation. ACI states that olumns shall be designed for the fatored axial load, P u, and a magnified fatored moment, M, defined by M = d M + d M b 2b s 2s (50.48) where M 2b is the larger fatored end moment ating on the olumn due to loads ausing no appreiable sidesway (lateral defletions less than l/1500) and M 2s is the larger fatored end moment due to loads that result in an appreiable sidesway. The moments are omputed from a onventional first-order elasti frame analysis. For the above equation, the following apply: d d b s Cm = P fp 1 = P f P For members braed against sidesway, ACI gives d s = 1.0. u  u  M b Cm j = M 2b (50.49) (50.50) The ratio M 1b /M 2b is taken as positive if the member is bent in single urvature and negative if the member is bent in double urvature. Equation (50.50) applies only to olumns in braed frames. In all other ases, ACI states that C m = 1.0. where or, approximately P p 2 2 EI kl = ( ) EI 5 + EI EI = 1 + b EI = u g s se EI g d b d (50.51) (50.52) (50.53)

18 When omputing d b, b d = Axial load due to fatored dead load Total fatored axial load (50.54) when omputing d s, b d = Fatored sustained lateral shear in the story Total fatored lateral shear in the story (50.55) If d b or d s is found to be negative, the olumn should be enlarged. If either d b or d s exeeds 2.0, onsideration should be given to enlarging the olumn. Columns under Axial Load and Biaxial Bending The nominal ultimate strength of a setion under biaxial bending and ompression is a funtion of three variables, P n, M nx, and M ny, whih may also be expressed as P n ating at eentriities e y = M nx /P n and e x = M ny /P n with respet to the x and y axes. Three types of failure surfaes an be defined. In the first type, S 1, the three orthogonal axes are defined by P n, e x, and e y ; in the seond type, S 2, the variables defining the axes are 1/P n, e x, and e y ; and in the third type, S 3, the axes are P n, M nx, and M ny. In the presentation that follows, the Bresler reiproal load method makes use of the reiproal failure surfae S 2, and the Bresler load ontour method and the PCA load ontour method both use the failure surfae S 3. Bresler Reiproal Load Method Using a failure surfae of type S 2, Bresler proposed the following equation as a means of approximating a point of the failure surfae orresponding to prespeified eentriities e x and e y : = + - Pni Pnx Pny P 0 (50.56) where P ni =nominal axial load strength at given eentriity along both axes P nx =nominal axial load strength at given eentriity along x axis P ny =nominal axial load strength at given eentriity along y axis P 0 =nominal axial load strength for pure ompression (zero eentriity) Test results indiate that Eq. (50.46) may be inappropriate when small values of axial load are involved, suh as when P n /P 0 is in the range of 0.06 or less. For suh ases the member should be designed for flexure only. Bresler Load Contour Method The failure surfae S 3 an be thought of as a family of urves (load ontours) eah orresponding to a onstant value of P n. The general nondimensional equation for the load ontour at onstant P n may be expressed in the following form: Ê M Á Ë M nx ox a 1 ˆ Mny + Ê 2 ˆ Ë Á M = 10. oy a (50.57) where M nx = P n e y ; M ny = P n e x M ox = M nx apaity at axial load P n when M ny (or e x ) is zero M oy = M ny apaity at axial load P n when M nx (or e y ) is zero

19 The exponents a 1 and a 2 depend on the olumn dimensions, amount and arrangement of the reinforement, and material strengths. Bresler suggests taking a 1 = a 2 = a. Calulated values of a vary from 1.15 to For pratial purposes, a an be taken as 1.5 for retangular setions and between 1.5 and 2.0 for square setions. PCA (Parme Gowens) Load Contour Method This method has been developed as an extension of the Bresler load ontour method in whih the Bresler interation equation (50.57) is taken as the basi strength riterion. In this approah, a point on the load ontour is defined in suh a way that the biaxial moment strengths M nx and M ny are in the same ratio as the uniaxial moment strengths M ox and M oy, M M ny nx Moy = = b M ox (50.58) The atual value of b depends on the ratio of P n to P 0 as well as the material and ross-setional properties, with the usual range of values between 0.55 and Charts for determining b an be found in ACI Publiation SP-17A(85), A Design Handbook for Columns [ACI Committee 340, 1990]. Substituting Eq. (50.48) into Eq. (50.57), Ê bm Á Ë M ox ox a a ˆ bmoy + Ê ˆ Ë Á M = 1 oy a 2b = 1 b a = 12 (50.59) log 05. a = log b thus, Ê M Á Ë M nx ox log 05. log b log 05. log b ˆ Mny + Ê ˆ Ë Á M = 1 oy (50.60) For more information on olumns subjeted to biaxial bending, see Reinfored Conrete Design by Chu- Kia Wang and Charles G. Salmon [1985] Shear and Torsion Reinfored Conrete Beams and One-Way Slabs Strength Design The raks that form in a reinfored onrete beam an be due to flexure or a ombination of flexure and shear. Flexural raks start at the bottom of the beam, where the flexural stresses are the largest. Inlined raks, also alled shear raks or diagonal tension raks, are due to a ombination of flexure and shear. Inlined raks must exist before a shear failure an our. Inlined raks form in two different ways. In thin-walled I-beams in whih the shear stresses in the web are high while the flexural stresses are low, a web-shear rak ours. The inlined raking shear an be alulated as the shear neessary to ause a prinipal tensile stress equal to the tensile strength of the onrete at the entroid of the beam.

20 In most reinfored onrete beams, however, flexural raks our first and extend vertially in the beam. These alter the state of stress in the beam and ause a stress onentration near the tip of the rak. In time, the flexural raks extend to beome flexure-shear raks. Empirial equations have been developed to alulate the flexure-shear raking load, sine this raking annot be predited by alulating the prinipal stresses. In the ACI Code, the basi design equation for the shear apaity of onrete beams is as follows: V u fv n (50.61) where V u = the shear fore due to the fatored loads f = the strength redution fator equal to 0.85 for shear V n = the nominal shear resistane, whih is given by Vn = V + Vs (50.62) where V = the shear arried by the onrete V s = the shear arried by the shear reinforement The torsional apaity of a beam as given in ACI is as follows: T u ft n (50.63) where T u = the torsional moment due to fatored loads f = the strength redution fator equal to 0.85 for torsion T n = the nominal torsional moment strength given by Tn = T + T (50.64) where T = the torsional moment strength provided by the onrete T s = the torsional moment strength provided by the torsion reinforement Design of Beams and One-Way Slabs Without Shear Reinforement: for Shear The ritial setion for shear in reinfored onrete beams is taken at a distane d from the fae of the support. Setions loated at a distane less than d from the support are designed for the shear omputed at d. Shear Strength Provided by Conrete. Beams without web reinforement will fail when inlined raking ours or shortly afterwards. For this reason the shear apaity is taken equal to the inlined raking shear. ACI gives the following equations for alulating the shear strength provided by the onrete for beams without web reinforement subjet to shear and flexure: V = 2 f b d w (50.65) or, with a more detailed equation: V Ê Vd ˆ u = Á19. f r bd w fbd w Ë M 3. 5 w u (50.66) The quantity V u d/m u is not to be taken greater than 1.0 in omputing V where M u is the fatored moment ourring simultaneously with V u at the setion onsidered.

21 Combined Shear, Moment, and Axial Load. For members that are also subjet to axial ompression, ACI modifies Eq. (50.65) as follows (ACI ): V Ê N ˆ u = 2Á1+ f bwd Ë A 2000 k (50.67) where N u is positive in ompression. ACI ontains a more detailed alulation for the shear strength of members subjet to axial ompression. For members subjet to axial tension, ACI states that shear reinforement shall be designed to arry total shear. As an alternative, ACI gives the following for the shear strength of member subjet to axial tension: V Ê N ˆ u = 2Á1+ f bwd Ë 500 Ag (50.68) where N u is negative in tension. In Eq. (50.67) and (50.68) the terms units of psi. f N u /A g, 2000, and 500 all have Combined Shear, Moment, and Torsion. For members subjet to torsion, ACI gives the equation for the shear strength of the onrete as the following: where V = f b d 1+ ( 2. 5CT V ) Design of Beams and One-Way Slabs Without Shear Reinforements: for Torsion. ACI requires that torsional moments be onsidered in design if 2 (50.69) (50.70) Otherwise, torsion effets may be negleted. The ritial setion for torsion is taken at a distane d from the fae of support, and setions loated at a distane less than d are designed for the torsion at d. If a onentrated torque ours within this distane, the ritial setion is taken at the fae of the support. Torsional Strength Provided by Conrete. Torsion seldom ours by itself; bending moments and shearing fores are typially present also. In an unraked member, shear fores as well as torques produe shear stresses. Flexural shear fores and torques interat in a way that redues the strength of the member ompared with what it would be if shear or torsion were ating alone. The interation between shear and torsion is taken into aount by the use of a irular interation equation. For more information, refer to Reinfored Conrete Mehanis and Design by James G. MaGregor [1992]. The torsional moment strength provided by the onrete is given in ACI as w t u u 2 T f 05. f x y u ( ) Â 2 T f 05. f x y u ( ) Â 2 T = f x y 1+ ( 0. 4V CT ) u t u 2 (50.71)

22 Combined Torsion and Axial Load. For members subjet to signifiant axial tension, ACI states that the torsion reinforement must be designed to arry the total torsional moment, or as an alternative modify T as follows: T = f x y 1+ ( 0. 4V CT ) u t u 2 Ê N ˆ u Á1 + Ë 500 A g (50.72) where N u is negative for tension. Design of Beams and One-Way Slabs without Shear Reinforement Minimum Reinforement. ACI requires a minimum amount of web reinforement to be provided for shear and torsion if the fatored shear fore V u exeeds one half the shear strength provided by the onrete (V u 0.5fV ) exept in the following: (a) Slabs and footings (b) Conrete joist onstrution () Beams with total depth not greater than 10 inhes, 2½ times the thikness of the flange, or ½ the width of the web, whihever is greatest The minimum area of shear reinforement shall be at least bws A T f x y v ( min ) = 50 u. f < 05 2 for f y ( ) (50.73) When torsion is to be onsidered in design, the sum of the losed stirrups for shear and torsion must satisfy the following: Â A v 50bws + 2At f y (50.74) where A v = the area of two legs of a losed stirrup A t = the area of only one leg of a losed stirrup Design of Stirrup Reinforement for Shear and Torsion Shear Reinforement. Shear reinforement is to be provided when V u fv, suh that V V u s - V f (50.75) The design yield strength of the shear reinforement is not to exeed 60,000 psi. When the shear reinforement is perpendiular to the axis of the member, the shear resisted by the stirrups is Afd v y Vs = s If the shear reinforement is inlined at an angle a, the shear resisted by the stirrups is (50.76) Af v V = s y ( ) sin a + os a d (50.77) The maximum shear strength of the shear reinforement is not to exeed 8 f b w d as stated in ACI s

23 Spaing Limitations for Shear Reinforement. ACI sets the maximum spaing of vertial stirrups as the smaller of d/2 or 24 inhes. The maximum spaing of inlined stirrups is suh that a 45 line extending from midheight of the member to the tension reinforement will interept at least stirrup. If V s exeeds 4 f b w d, the maximum allowable spaings are redued to one half of those just desribed. Torsion Reinforement. Torsion reinforement is to be provided when T u ft, suh that Tu Ts - T f (50.78) The design yield strength of the torsional reinforement is not to exeed 60,000 psi. The torsional moment strength of the reinforement is omputed by where A T = s tatx 1 y 1 fy a t y t x t = + ( ) s [ ] 150. (50.79) (50.80) where A t is the area of one leg of a losed stirrup resisting torsion within a distane s. The torsional moment strength is not to exeed 4 T as given in ACI Longitudinal reinforement is to be provided to resist axial tension that develops as a result of the torsional moment (ACI ). The required area of longitudinal bars distributed around the perimeter of the losed stirrups that are provided as torsion reinforement is to be ( ) A A x + y l 2 t s 1 1 È Ê ˆ Í xs Á T u x Al Í 400 Á = A Ê 2 t f V Á Í y u Á Tu + Ë Í Ë C Î 3 t + y ˆ s 1 1 (50.81) Spaing Limitations for Torsion Reinforement. ACI gives the maximum spaing of losed stirrups as the smaller of (x 1 + y 1 )/4 or 12 inhes. The longitudinal bars are to be spaed around the irumferene of the losed stirrups at not more than 12 inhes apart. At least one longitudinal bar is to be plaed in eah orner of the losed stirrups (ACI ). Design of Deep Beams ACI 11.8 overs the shear design of deep beams. This setion applies to members with l n /d < 5 that are loaded on one fae and supported on the opposite fae so that ompression struts an develop between the loads and the supports. For more information on deep beams, see Reinfored Conrete Mehanis and Design, 2nd ed. by James G. MaGregor [1992]. The basi design equation for simple spans deep beams is V V + V f ( ) u s (50.82) where V = the shear arried by the onrete V s = the shear arried by the vertial and horizontal web reinforement