109-s76 - Defelection Control Slabs With HPreinforcing Steel ASTM A1035

8/22/2019 109-s76 - Defelection Control Slabs With HPreinforcing Steel ASTM A1035

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Title no. 109-S76

ACI STRUCTURAL JOURNAL TECHNICAL PAPER

ACI Structural Journal, V. 109, No. 6, November-December 2012.MS No. S-2011-017.R1 received February 16, 2011, and reviewed under Institute

publication policies. Copyright 2012, American Concrete Institute. All rightsreserved, including the making of copies unless permission is obtained from the

copyright proprietors. Pertinent discussion including authors closure, if any, will bepublished in the September-October 2013ACI Structural Journal if the discussion isreceived by May 1, 2013.

ACI Structural Journal/November-December 2012 867

Defection Control o Concrete Slabs Longitudinally

Reinorced with ASTM A1035/A1035M-07 Steel

by Admasu S. Desalegne and Adam S. Lubell

The ACI 318-08 design code or reinorced concrete construction

provides both an implicit check o slab defection control based on

minimum member thickness and a direct computation method or

defection. Similar provisions are given in the ACI ITG-6 design

guide (ACI ITG-6R-10) or members reinorced with high-peror-

mance ASTM A1035/A1035M-07 steel. This paper reports an

analytical study that compared the maximum span-depth ratios

rom the implicit defection provisions with corresponding ratios

determined rom direct defection calculations. Emphasis was

placed on defection control at the serviceability limit state (SLS)

or one-way slabs longitudinally reinorced with ASTM A1035/

A1035M-07 steel where the nominal steel stress at the ultimatelimit state (ULS) ranged rom 60 to 120 ksi (414 to 828 MPa). The

results indicate that the maximum span-depth ratio should decrease

as the span length increases, as the design load increases, as the

concrete strength decreases, or as the maximum permissible defec-

tion decreases. The maximum span-depth ratio can be increased

as the longitudinal reinorcement ratio is increased beyond that

required to satisy the fexural demand. These relationships with

the maximum span-depth ratio were all nonlinear in nature and

were o similar shape or all nominal reinorcement stress magni-

tudes considered. Furthermore, these relationships were similar

when ULS fexural design was completed using either the simplied

or general fexural design models provided in the ACI ITG-6R-10

guidelines. The study recommends that direct defection calcula-

tions should be used or the design o all slabs and proposes graph-ical design aids or use in initial thickness selection.

Keywords: cracking; defection; high-perormance reinorcement; one-way

slabs; reinorced concrete; stiness.

INTRODUCTIONReinorced concrete fexural members must have accept-

able defections at the serviceability limit state (SLS) while

providing adequate strength at the ultimate limit state (ULS).

The maximum SLS deormations o structural members,

including the eects o incremental defection rom

sustained loads, should be appropriate or their intended use

and minimize signicant damage to nonstructural elements.The longitudinal tensile reinorcement ratio r or a one-wayspanning slab is usually based on the fexural strength require-

ments at ULS. Use o higher- or lower-strength reinorce-

ment will change the required r and, hence, the reinorce-ment stresses and corresponding member curvatures at the

SLS condition. Thus, it is generally believed that minimum

slab thickness must change as a unction o the nominal

reinorcement design stress D to maintain adequate defec-tion control. The reinorcement stress at SLS is commonly

approximated as 0.67y or traditional steel reinorcement

grades (reer to ACI 318-08, Section 10.6.4).1 However, the

ratio between the stress at SLS andD used or ULS design o

ASTM A1035/A1035M-072 steel can dier rom the 0.67yapproximation, especially i the ULS design considers the

nonlinear region o the ASTM A1035/A1035M-07 steelstress-strain response.3

Defections o reinorced concrete members depend onmany actors, including the degree o cracking, the time-dependent characteristics o the concrete, the mechanicalproperties o the reinorcement, and the support and loadingconditions.4 ACI 318-081 provides two methods to satisydefection control requirements or reinorced concretemembers. The ACI ITG-6 design guide (ITG-6R-10),3 whichprovides modications to ACI 318-08 provisions or use

with ASTM A1035/A1035M-07 Grade 100 (690 MPa)steel, adopts the same two-approach method o defectioncontrol. In the rst approach, an implicit evaluation omember defection is used, whereby a member with su-ciently large overall depth h is deemed to comply (DTC)with the defection requirements. As given in ACI 318-08,Table 9.5(a) (reproduced herein as Table 1), the minimummember thickness h or spanL is based on member type(or example, slab or beam) and support coniguration(or example, simple-span or continuous). Footnote b)o Table 1 gives an adjustment coecient to increase h asthe reinorcement yield strength y increases above 60 ksi(414 MPa). In the second approach, the member defection

is directly calculated using an eective moment o inertiaIe to account or the variation in stiness along the memberlength due to cracking. The calculated defections arethen compared to established defection limits. Due to thesimplicity o the DTC approach or defection control, thismethod is usually preerred over direct defection calcula-tions or member size selection in design practice. Thus, it isimportant that the DTC approach yields members that alsosatisy the defection control criterion under the direct calcu-lation method while still promoting structural economy.

Several previous studies have proposed dierent maximumL/h relationships or defection control to replace those inTable 1. Grossman5 used computer simulations to developa simple expression or the minimum thickness o one-waymembers based on the maximum permitted defection, thelongitudinal reinorcement ratio, and the loading. Gardnerand Zhang6 used a layered, nonlinear nite element modeland approximated the required increase in the maximumL/h ratio as inversely proportional to the cube root o theservice moment-to-ultimate moment ratio Ma/Mu. Theyalso identied that the limiting L/h ratio increases as theconcrete strength c increases and as the fexural tension


2/12868 ACI Structural Journal/November-December 2012

ACI member Admasu S. Desalegne is a PhD Student in structural engineering at

the University o Alberta, Edmonton, AB, Canada. He received his BSc and MSc in

structural engineering rom Addis Ababa University, Addis Ababa, Ethiopia. His

research interests include analysis and design o concrete structures reinorced with

high-perormance materials.

ACI memberAdam S. Lubell is an Assistant Proessor o Civil Engineering at the

University o Alberta. He received his PhD rom the University o Toronto, Toronto,

ON, Canada. He is past Secretary o ACI Task Group ITG-6, High Strength Steel

Reinorcement, and is a member o ACI Committee 440, Fiber-Reinorced Polymer

Reinorcement; 544, Fiber-Reinorced Concrete; and Joint ACI-ASCE Committee 445,

Shear and Torsion. His research interests include the design and rehabilitation oreinorced and prestressed concrete structures and the development o structural

detailing guidelines to allow the use o high-perormance materials.

or compression reinorcement ratios (r and r) increase.Scanlon and Choi7 showed that the minimum slab thicknesscan be reduced asL decreases and as the live load decreases.Gardner8 compared maximum L/h relationships rom theliterature and rom several codes o practice and recom-mended maximumL/h ratios that decrease as r decreases, asc decreases, and as the ratio o maximum sustained momentto ultimate moment capacity increases. Choi et al.9 used aMonte Carlo simulation to calibrate a proposed simpli-ed expression or maximum L/h. Among the parametersincluded were span length, load intensity, support condi-tions, concrete strength, and steel strength.

Tang and Lubell10 used a holistic approach to considermember thickness and the corresponding reinorcement,which would simultaneously satisy the fexure, shear, anddefection requirements o one-way spanning members,and which could orm the basis o graphical design aids orselecting member thickness. The maximumL/h ratios romthese plots were then compared with the correspondingL/hratios derived rom the implicit defection control provisions(that is, the DTC approach rom Table 1). The study usedrequirements rom CSA A23.3-04,11 which are similar to

those in ACI 318-08. The results showed that the maximumL/h ratios should decrease as the span length L increases,as the design load w increases, or as the cracking momentMcr decreases. The study also demonstrated that the ULSdesign strength o longitudinal reinorcementD did not havea signicant eect on the minimum h required to satisy thedefection criterion in contrast to the assumed relationshiprom Footnote b) o Table 1. Bischo and Scanlon12 laterdeveloped simple expressions or the maximumL/h ratio orone-way slabs and beams that had a similar shape to thoserom Tang and Lubell10 by considering the parameters oreinorcement ratio, cracking moment, specied defectionlimits, compressive strength o concrete, and yield strength

o steel. The use o these expressions to check or adequatedefection control, however, requires prior knowledge or,

which is typically unknown until the member thickness his selected. Thus, the Bischo and Scanlon12 expressionscannot be easily used or optimized member size selection.Furthermore, the structure o the relationships prevents theirconsistent use when the ULS design considers the nonlinearstress-strain response o high-perormance reinorcement,such as ASTM A1035/A1035M-07 steel.

The study by Tang and Lubell10 used a linear elastic-perectly plastic stress-strain model or the steel reinorce-ment and only considered steel design strengths

yup to 80 ksi

(552 MPa). This paper orms an extension to the Tang andLubell10 approach by specically considering one-wayconcrete slabs longitudinally reinorced with ASTM A1035/A1035M-07 Grade 100 (690 MPa) steel. Higher nominaldesign strengths D o up to 120 ksi (828 MPa) were usedin the analytical calculations, including considerationo the nonlinear stress-strain response o ASTM A1035/A1035M-07 steel in some cases.

RESEARCH SIGNIFICANCEOne-way slabs reinorced with high-strength steel typi-

cally have low longitudinal reinorcement ratios; however,there has not been previous work to systematically establish

whether the maximumL/h ratios specied by the DTC defec-tion control method o ACI 318-08 are also appropriate orlightly reinorced members with higher-strength reinorce-ment. In the case o slabs reinorced with ASTM A1035/A1035M-07 steel, it is also important to evaluate whetherthese DTC provisions can be applied to members where thefexural design uses the nonlinear portion o the steel stress-strain response. A comparison was completed betweenmaximum L/h values determined rom a holistic designapproach considering fexure, shear, and direct defectioncalculations with L/h ratios derived rom the DTC provi-sions. The aim was to establish an appropriate method orselection o minimum member thickness with due regard or

infuences that arise both rom the overall member congu-ration and rom the dierent ULS analysis techniques givenin ACI ITG-6R-10.3

REINFORCEMENT PROPERTIESASTM A1035/A1035M-07 steel has a dierent metal-

lurgy and microstructure than conventional reinorcing steelcommonly used in most new construction.3 These changesresult in ASTM A1035/A1035M-07 steel with an eec-tive yield strength signicantly higher than conventionalASTM A615/A615M-0613 (60 or 75 ksi [414 or 518 MPa])or ASTM A706/A706M-0814 (60 ksi [414 MPa]) reinorcingsteel while also being less susceptible to corrosion.15,16 To

account or the mechanical properties o ASTM A1035/A1035M-07 Grade 100 (690 MPa) steel, ACI Innovation

Table 1Minimum thickness o non-prestressed beams or one-way slabs unless defections arecalculated (adapted rom ACI 318-08, Table 9.5(a)1)

Minimum thickness h

Simply supported One end continuous Both ends continuous Cantilever

Member Members not supporting or attached to partitions or other construction likely to be damaged by large defections

Solid one-way slabs L/20 L/24 L/28 L/10

Beams or ribbed one-way slabs L/16 L/18.5 L/21 L/8

Notes: Values given shall be used directly or members with normalweight concrete and Grade 60 (414 MPa) reinorcement. For other conditions, values shall be modied as ollows:a) For lightweight concrete having equilibrium densitywc in the range o 90 to 115 lb/t

3 (1440 to 1840 kg/m3), values shall be multiplied by (1.65 0.005wc) but not less than 1.09.

b) Fory other than 60 ksi (414 MPa), values shall be multiplied by (0.4 +y/100,000).


3/12ACI Structural Journal/November-December 2012 869

Task Group 63 developed representative analytical stress-strain curves that are suitable or use in design and wereadopted in this study

(1a)

(1b)

DESIGN PROCEDURES FOR ONE-WAY SLABSThis analytical study evaluated the defection control o

one-way simply supported slabs longitudinally reinorcedwith ASTM A1035/A1035M-07 steel in building-type struc-tures. All slabs studied were subjected to uniormly distrib-

uted foor loading. Loading consisted o the member sel-weight wDL, superimposed dead loads wSDL to account ormechanical systems and architectural nishes, and typicallive loads wLL dened in ASCE/SEI 7-05

17 or dierentbuilding occupancy conditions. Load and resistance actorsrom ACI 318-08, with specied modications romACI ITG-6R-10, were used. While all reinorcing steel consid-ered was ASTM A1035/A1035M-07 Grade 100 (690 MPa),designs corresponding to dierent nominal steel stressvaluesD o 60, 100, and 120 ksi (414, 690, and 828 MPa)at the ULS condition were developed, consistent with theACI ITG-6R-10 provisions. This allowed the nonlinearstress-strain response o the steel at ULS to be directly eval-

uated or its corresponding infuence on defection controlat the SLS. The range o parameters and member congura-tions studied are provided in Table 2.

Overview o typical design sequenceSlabs must be designed to have adequate strength in

fexure and shear at the ULS condition. The defections othese members must also be controlled to within accept-able limits or their intended use at SLS. The typical designsequence used in practice to achieve these objectives is givenby the fowchart in Fig. 1 and briefy described. Uniquedesign aspects within each step, as they pertain to dier-ences between ACI ITG-6R-10 and ACI 318-08 provisions

to account or the use o ASTM A1035/A1035M-07 steel,are provided in the ollowing sections.

Initially, a member thickness h must be selected that isexpected to satisy the ULS and SLS design criteria (Step 1).Due to its simplicity, the initial selection o h is typicallymade using the DTC defection control provisions (that is,Table 1) with the modication specied in the table oot-note or y 60 ksi (414 MPa). However, i h is directlydetermined in later steps rom the governing case o explicitdefection calculations or strength requirements, this modi-cation is not directly applicable. Next, or the member sizeselected, the longitudinal reinorcement quantityAs is deter-mined to satisy the fexural strength requirement at ULS

(Step 2). The shear capacity o the slab is evaluated at ULSand compared against the loading demand (Step 3). I the

member satises the requirements o the DTC defection

control method, including adjustment ory, no urther checko defection is required (Step 4a). Alternatively, direct

Table 2Design parameters considered

Parameter U.S. customary units Metric units

Concrete strengthc 5 and 10 ksi 34.5 and 69 MPa

Nominal steel design

strengthD60 to 120 ksi 414 to 828 MPa

Live load intensity wLL 50 and 100 lb/t2 2.4 and 4.8 kPa

Superimposed dead loadintensity wSDL

20 lb/t2 1 kPa

Span lengthL 10 to 32 t 3 to 10 m

Slab thickness h 3 to 22 in. 75 to 550 mm

Fig. 1Design procedure or optimized slab thicknessaccording to defection control.



defection calculations at SLS are completed and comparedagainst the appropriate limits (Step 4b). The defection o amember satisying Step 4a could also be evaluated at Step 4bso as to allow optimization oh. To satisy the strength ordefection criteria at Steps 2, 3, or 4, the slab thickness hat Step 1 can be adjusted. To emphasize the relationshipsbetween the various design parameters, this study reportsresults or optimized values o h that just satisy the moststringent criterion rom Steps 2, 3, or 4b, whereas industrypractice will use practical incremental thicknesses or slabs.

DESIGN AT ULSFlexural design o slabs with ASTM A1035/

A1035M-07 steelAs noted previously, the stress-strain relationship or

ASTM A1035/A1035M-07 steel given by Eq. (1) does notinclude a distinct yield point or yield plateau. Flexural designo members longitudinally reinorced with ASTM A1035/A1035M-07 steel must account or the curvilinear stress-strain response. ACI ITG-6R-10 permits the use o twodierent fexural analysis models that are based on dierentassumptions or the reinorcement stress-strain response andcorresponding moment-curvature response o the concretesection. Both fexural analysis models are used in this studyto highlight their respective infuences on the maximumL/hratios that provide adequate defection control.

In the rst fexural model, herein termed the Mast model,18 asimplied elastic-plastic representation or the stress-strainresponse o ASTM A1035/A1035M-07 steel is used withan eective yield stress oy = 100 ksi (690 MPa). Flexuralcapacity at ULS is calculated with the ACI 318-08 rectan-gular stress block provisions and with an assumed concretestrain at the extreme compression ber o ecu = 0.003.The ACI ITG-6R-10 partial saety actor or fexure ff,1,according to this approach, is given by 0.65 (ff,1 = 0.45+ 50es) 0.9, where es is the reinorcement strain at ULS.Owing to its simplicity in the required calculations, the Mastmethod18 is expected to be the more widely used fexuraldesign approach in industry practice. This study used the

Mast model18 to consider members with nominal steel designstrength valuesD o 60 and 100 ksi (414 and 690 MPa).

In the second fexural model provided in ACI ITG-6R-10,herein termed the Appendix B model, the ull nonlinearstress-strain relationship or ASTM A1035/A1035M-07 steelaccording to Eq. (1) is used. The concrete strain at the

extreme compression ber is taken as ecu = 0.003 at ULSand the fexural response o the section can be solved usinga strain compatibility approach based on engineering beamtheory. Due to the nonlinear stress-strain relationship or thereinorcement, an iterative calculation approach is typically

required. A provision o adequate fexural capacity can beeasily checked when the reinorcement quantity is known.For the selection o an optimized quantity o reinorce-ment or a given fexural demand Mu, it is typically easiestto establish a target maximum nominal stress level D inthe reinorcement. ACI ITG-6R-10 gives a partial saetyactor or fexure ff,2 or members designed according to theAppendix B approach rom 0.65 (ff,2 = 0.23 + 100es) 0.9. This study used the Appendix B model to consider slabswithD values o 100 and 120 ksi (690 and 828 MPa).

To compare these two fexural design approaches,Fig. 2 plots the relationship between the nominal momentMnand curvature F or the 12 x 12 in. (305 x 305 mm) cross

section dened in the gure. Note that the ullMn-F responsein Fig. 2 was prepared using a variable ecu value19 and willhave minor variance rom Mn-F values calculated withconstant ecu at ULS or the two fexural models describedpreviously. The Mast and Appendix B models bothpredict a similar Mn-F response prior to cracking and orvalues oMn up to approximately 40 kip-t/t (178 kN-m/m). This point corresponds to the proportional limit oASTM A1035/A1035M-07 steel rom Eq. (1). Beyond thispoint, the Mast model18 gives a slightly stier response dueto the use o an eectivey larger than the proportional limitstress; however, the maximum calculated fexural capacityquickly plateaus. The Appendix B model gives increasing

fexural capacity at a decreasing rate as the curvature F (thatis, slab defection) increases. Points have been marked on theplot to correspond to nominal reinorcement stress magni-tudes D o 60, 100, and 120 ksi (414, 690, and 828 MPa).It is important to note that or each o these conditions, thecorresponding point representing the SLS condition will havesteel stresses below the proportional limit when typical loadand resistance actors are applied. Thus, or the case shown,it is possible to consider the SLS design requirements basedon the elastic methods used or traditional reinorcing steelsthat exhibit well-dened yield plateaus.

Structural slabs designed according to ACI 318-08 must havea minimum quantity o longitudinal reinorcement in the span

direction that satises the shrinkage and temperature reinorce-ment provisions o Section 7.12.2. ACI ITG-6R-10 requiresa designer to ollow these provisions and notes that or

ASTM A1035/A1035M-07 Grade 100 (690 MPa) steel, thecorresponding minimum gross reinorcement ratio is 0.14%.This requirement is easily satised or thin slabs with reason-able bar spacing, as the maximum spacing is limited to thesmaller o 18 in. (457 mm) or 5h. According to ACI ITG-6R-10, Section 4.9.4, to provide crack control at a reasonable barspacing or members with increased cover, it is necessary tolimit the steel stress at the service load to less than 67 ksi(460 MPa). Section 4.2 o ACI ITG-6R-10 also suggestslimiting the maximum strain in the reinorcement at ULS to

0.015 (that is, 144 ksi [994 MPa]) to avoid excessive crackingo members.

Fig. 2Moment-curvature relationship or fexurallycracked slab with ASTM A1035/A1035M-07 steel.



Shear design o slabs with ASTM A1035/A1035M-07 steel

The shear capacity o reinorced concrete slabs that do notcontain stirrups are infuenced by many design parameters,including the concrete strength c, the eective depth d,and the longitudinal reinorcement conguration.20-23 Notethat these same parameters will also infuence the fexuraldesign and the overall member defection. With regard tothe use o ASTM A1035/A1035M-07 steel as the longitu-dinal reinorcement in slabs, the higher nominal strengthD compared to traditional Grade 60 (414 MPa) reinorcingsteel allows slabs with a lower reinorcement ratio r andhigher steel stresss to still satisy fexural strength require-ments. These slabs, however, will exhibit larger diagonalcrack widths at the ULS condition, which will impact theshear strength.24

With additional modications to account or the possiblenonlinear stress-strain response o ASTM A1035/A1035M-07steel at ULS, Desalegne and Lubell25 showed that the shearmodel proposed by Hoult et al.26 can be used to predict theshear capacity o slabs reinorced with this steel. The Hoult etal.26 model enhances the modied compression eld theory24-based CSA A23.3-0411 shear model to better account or the

infuence on one-way shear capacity rom large longitudinalreinorcement strains, with the shear capacity at the criticalsection given as

(2a)

(2b)

where parameter ex represents the eective axial strain atmidheight and is derived rom the reinorcement stress at thecritical section25; the shear depth dv is taken as 0.9d; and theeective crack spacing parameter sze can be taken as 0.9dorthe concrete with 3/4 in. (19 mm) aggregate assumed in thisstudy. While a simplied version o this shear capacity methodthat is compatible with the simplied Mast fexural modelassumptions is included in the ACI ITG-6R-10 guide,3,25 thegeneral version (that is, Eq. (2)) is used in this study due tothe use o the Appendix B lexural method in some cases.ACI ITG-6R-10 adopts the same partial saety actor or shearfsh = 0.75, as given in ACI 318-08.

DEFLECTION OF REINFORCEDCONCRETE SLABS

ACI 318-08 DTC with defection limitsTable 9.5(a) in ACI 318-08 (Table 1 in this paper)

provides minimum thickness values or members thatare DTC with defection requirements or members notsupporting or attached to partitions or other constructionlikely to be damaged by large defections. By this defectioncontrol specication, members sized using this techniquewould be expected to limit the total incremental long-termdefection ater installation o nonstructural items to DincL/240.1 Values or minimum thickness h are providedas unctions o the span length L, based on the member

type and the support condition. For members conorming toTable 1 and the defection limit specication noted previ-

ously, the fexural stiness does not need to be directly

determined because ACI 318-08 and ACI ITG-6R-10 do

not require direct checks o the predicted defection or

these members. In the case o lightly reinorced members,

however, ACI ITG-6R-10 recommends making direct

defection calculations.

Direct defection calculations or slabsDeormations o slender, one-way spanning reinorced

concrete slabs without shear reinorcement are assumed tobe consistent with the well-known hypothesis that plane

sections beore bending remain plane ater bending. The

defection is determined by considering the corresponding

curvatures along the member length. Thus, the instantaneous

defection o a member subjected to uniorm transverse

loading can be computed with the well-known relationship

(3)

where Di is the instantaneous defection; w is the uniormtransverse loading considered;L is the span length;Ec is the

secant modulus o elasticity o concrete taken as 57,000cpsi (4735c MPa);Ie is the eective moment o inertia othe transormed cross section; and K is a coecient based

on the boundary conditions (1.0 or simple span; 0.416 or

xed-pin; 0.2 or xed-xed). Time-dependent infuences

on defection must also be considered. According to the

ACI 318-08 provisions, the incremental long-term defection

Dinc resulting rom creep and shrinkage o fexural memberscan be determined by multiplying the immediate defection

caused by the sustained load by the actor lD

(4)

where r is the compression reinorcement ratio taken atmidspan or simple and continuous spans, and at the support

location or cantilevers; x is the time-dependent actor or thesustained loads taken equal to 1.0, 1.2, 1.4, and 2.0 or loads

sustained or 3, 6, 12, or more than 60 months, respectively.

ACI 318-08 defection provisions limit the imme-

diate defection Di rom live loads to Dmax,imm = L/180 orL/360 or roos or foors, respectively, when not supporting

or attached to nonstructural items likely to be damaged by

large delections. Defection limits o Dmax,inc = L/240 andL/480 are used or the portion o defection that occurs ater

attachment o nonstructural elements (sum o the incremental

long-term defection due to all sustained loads Dinc and theimmediate defection Di due to any additional transient liveload) i they are not likely, or are likely, to be damaged by

large defections, respectively. According to Gardner,8 there

is general agreement that this total long-term defection ater

installation o nonstructural items (that is, Dmax,inc) is typicallythe more critical case compared to the immediate transient

live load defection limit Dmax,imm. While both criteria werechecked in this study, the Dmax,inc criterion was conrmed

to be the Dmax governing case or all maximum L/h ratiospresented in this study.



Calculating fexural stinessAs part o the defection calculation or Eq. (3), an evalua-

tion o an appropriate moment o inertia or the cross sectionis required to address the variable cracked nature along themember length. Bischo27 developed a ormulation oreective moment o inertiaIe that gives estimates o memberdefection that are in better agreement with test results thanthose using theIe ormulation in ACI 318-08 developed byBranson.28 The Bischo27Ie model given by Eq. (5) was

adopted by ACI ITG-6R-10 and was used in this study

(5)

whereMa is the maximum characteristic moment under theload being considered, taken herein as the maximum servicemoment (ull dead load + ull live load) as a simplied tech-nique to consider infuences rom early-age loading duringconstruction29;Mcris the cracking moment;Ig is the moment

o inertia o the gross section about the centroidal axis,neglecting the reinorcement; andIcr is the cracked momento inertia o a singly reinorced section, given by

(6)

where the modular ratio n = Es/Ec; bw is the member width; d

is the eective depth o the reinorcement rom the compres-

sion ace;

22 ( )k n n n= r + r - r ;

andAs is the area o fexural tension reinorcement, with the

reinorcement ratio evaluated as r = (As/(bwd)).According to ACI 318-08, the cracking moment or

normalweight concrete, Mcr, is related to the modulus o

rupturer= 7.5c psi (0.623c MPa) and the gross sectionproperties through the expression

(7)

where gcr is a coecient adopted in this study to accountor a reduced cracking moment due to restrained shrinkageand is taken as 0.67, as per the recommendation o Bischoand Scanlon.30

PARAMETRIC INFLUENCES ON DEFLECTIONFor the DTC defection control technique (Table 1), the

correspondingL/h ratios are constant or each member typeand support condition. Only the infuence o reinorce-ment yield strength y is given additional considerationthrough Footnote b) o Table 1. However, the ULS designmethods or fexure and shear (Steps 2 and 3 in Fig. 1) andor detailed defection computations at SLS (Step 4b in

Fig. 1) will be infuenced by various parameters that areapplicable to each particular design case. In general, these

can be classied as: 1) parameters infuencing the servicemoment magnitude and its raction relative to the ulti-mate moment; and 2) parameters infuencing the propor-tion o the member that will be cracked in fexure. For thisstudy using ASTM A1035/A1035M-07 steel reinorce-ment, the nominal design strength D and correspondingfexural design method represents a third classication. Asystematic evaluation o the infuence on defection romthe main parameters in these three primary classicationswas completed. The limiting L/h ratios were developed oreach case by considering the total defection D as the incre-mental defection Dinc rom 28 days (assumed time o appli-cation o sustained live load and superimposed dead load)until 60 months (that is, x = 2) combined with the immediatedefection Di o the transient live load raction. The limitingL/h ratios represent the case where D = Dmax. A sustained liveload raction ogLL = 70% was used or all analysis reportedin this study, but the infuence o this parameter was oundto be relatively minor within the typical range o 40 to 70%applicable or many structures.10

Factors aecting SLS and ULS momentsThe defection o a one-way slab is a unction o the magni-

tude o the cracking momentMcr, the service momentMa, andthe corresponding ultimate momentMu, as shown in Eq. (3)and (5). The moments Ma and Mu are related to the spanlengthL, the support conguration, and the applied loadingw. BecauseMu is used or determining the amount o longi-tudinal reinorcement, a higherMu will result in an increasedr or a constant slab thickness h, which thereby increasesIcr. I the slab thickness is allowed to adjust, however, theratio o dead load to live load will change. Because bothACI 318-08 and ASCE/SEI 7-05 use basic load actors o1.2 and 1.6 applied to dead load and live load, respectively,the ratioMa/Mu will change as h changes. In addition, as thesuperimposed dead load wSDL or items such as architectural

nishes increases as a raction o the total load, the Ma/Muratio will also change. In both cases, this change in ratio willaectIe and, hence, the defections at SLS.

The relationship between the maximum L/h ratio andmember span L according to the direct defection calcula-tion method was determined or dierent live load inten-sities wLL and concrete strengths using the D values andcorresponding fexural design methods identied previously(reer to Fig. 3 and 4). Figure 5 illustrates the variation inmaximumL/h as the superimposed dead load wSDL changesor a slab with a span oL = 20 t (6.1 m). Superimposeddead load wSDL reers to dead load other than the sel-weighto the member. It is observed that the maximum L/h ratio

or adequate defection control will decrease as L increasesor all values oD. Furthermore, the maximum L/h ratiodecreases as the applied live load wLL increases or as wSDLincreases, while the other parameters are kept constant. Bycomparing Fig. 3 and 4, it is also observed that increasingthe concrete strength c increases the maximum allowableL/h ratio or given values owLL and D. It is observed thatdefection control o lightly loaded slabs is more sensitive tothe span length and the superimposed dead load because theslopes o the maximum L/h-to-L and L/h-to-wSDL relation-ships decrease as the live load intensity increases. For thecases considered, thinner slabs can be used or shorter spansor or lighter loading conditions than the corresponding

minimum thickness determined rom the DTC defectionprovisions o ACI 318-08.



Fig. 3Infuence o span length on maximum span-depth ratio or normal-strength concrete. (Note: 1 ksi =6.89 MPa; 1 lb/t2 = 0.048 kPa.)

Fig. 4Infuence o span length on maximum span-depth ratio or high-strength concrete. (Note: 1 ksi =6.89 MPa; 1 lb/t2 = 0.048 kPa.)

Factors aecting degree o concrete crackingAccording to Eq. (7), the cracking momentMcris directly

related to the modulus o rupture r and, hence, c. Thus,as the concrete strength increases, Mcr will also increase,thereby increasingIe (reer to Eq. (5) and (7)) and allowingthinner sections or a given span lengthL. This is observedby comparing Fig. 3(a) and 4(a) or Fig. 3(b) and 4(b). Therelationship betweenc and maximumL/h can be observedrom Fig. 6 or typical residential foor loading and a spanoL = 20 t (6.1 m). It is observed that slabs satisying theDTC approach will typically also satisy the defectionrequirements rom direct defection calculations or prac-

tical concrete strengths when designed using higher-strengthsteel. However, or L = 20 t (6.1 m) slabs designed using

Grade 60 (414 MPa) steel and concrete strengths lower thanapproximately 5 ksi (35 MPa), the DTC approach underesti-mates the required member thickness in comparison to directdefection calculations.

Providing excess reinorcementIt is common practice that theAs provided in a slab exceeds

that required by the ULS criteria due to practical consider-ations, including the use o convenient bar spacing. Further-more, criterion or minimum longitudinal reinorcementquantities may exceed that required by the fexural demands.From Eq. (6), it is observed that the providedAs will impact

the cracked moment o inertia Icr and the correspondingdefection. Figure 7 depicts the relationship between the



maximumL/h ratio and the area o steel provided normalizedby the area o steel required or the fexural demand (Asp/Asr).As Asp increases beyond the fexural capacity requirementAsr, the maximum L/h ratio increases almost linearly orthe case oL = 20 t (6.1 m) and wLL = 50 lb/t

2 (2.4 kPa).Similar relationships occur iwLL is increased, except that, asexpected, the maximumL/h ratio is lower or higher valuesowLL. In general, providing excess longitudinal reinorce-ment within practical limits will increase the member sti-

ness and decrease the SLS defection o one-way slabs,allowing a minor reduction in the required thickness h.

Infuence o defection limitFigure 8 shows the variation o the maximum L/h ratio

or dierent limits o maximum midspan defection Dmax ora span oL = 20 t (6.1 m). To acilitate comparisons totypical design code defection requirements, the commonDmax limits oL/240, L/360, and L/480 are also indicated.The gure shows that the maximum L/h ratio increasesas the permitted Dmax increases. The maximum L/h valuesdiverge or dierent values oD at large values o Dmax,resulting in the need or thicker slabs or higher D (that is,Grade 100 and 120 [690 and 828 MPa] steel). However,or the typical design code limits o Dmax smaller thanL/240, the dierence in required h or allD values consid-ered was small. It is also noted rom Fig. 8 that slabs withGrade 60 (414 MPa) steel sized according to the DTCmethod may not satisy common defection control require-ments compared to those sized using direct defection calcu-lations, especially or the case o higher live load intensi-ties. According to Ramsay et al.,31 delection predictionscan have an error o 20% or common ratios oMcr/Ma.I a target maximum delection 20 or 30% smaller thana typical design code limit was desired to accommodatethis error range, Fig. 8 suggests that the required changein the maximumL/h ratio or a slab would be minimal incomparison to the discrepancy between DTC and direct

delection calculations.

Infuence rom fexural design methodAs discussed previously, two fexural analysis models

rom ACI ITG-6R-10 were used in this study: the Mastmethod and the Appendix B method. Due to the dierentreinorcement stress-strain models in these methods, therequired r to satisy the ULS fexural strength requirementscan dier or the same member geometry and applied load.Changes in r will have corresponding impacts on the defec-tion calculations and could alter the maximumL/h ratios oradequate defection control.

As observed in Fig. 3 through 8, the shapes o the respec-

tiveL/h curves are nearly the same, regardless o theD valueor fexural design method. Some divergence among the

Fig. 5Infuence o superimposed dead load intensity onmaximum span-depth ratio. (Note: 1 ksi = 6.89 MPa; 1 lb/t2= 0.048 kPa.)

Fig. 6Infuence o concrete strength on maximum span-depth ratio. (Note: 1 ksi = 6.89 MPa; 1 lb/t2 = 0.048 kPa.)

Fig. 7Infuence o providing excess reinorcementcompared to ULS requirement on maximum span-depthratio. (Note: 1 ksi = 6.89 MPa; 1 lb/t2 = 0.048 kPa.)



curves within each plot occurs when other design parametersare varied, but in general, the choices oD resulted in osetsto the curves with higher D values, resulting in smaller L/hlimits or a given member conguration. Furthermore, theL/h limits or higherD (100 and 120 ksi [690 and 828 MPa])had negligible sensitivity to the fexural design method, asthe reinorcement conguration was typically controlled byminimum reinorcement requirements or the cases studied.This demonstrates that member design at ULS using thenonlinear response o the ASTM A1035/A1035M-07 steelshould not have defection control provisions that result in adisproportionate impact on the maximum permittedL/h ratio.

A case study was used to urther examine the infuenceo the selected fexural design method on the holistic design

o slabs with minimum thickness h. An L/240 incrementaldefection limit was used and both direct defection calcula-tions and the DTC approach were considered. Results arepresented or a typical residential-type foor with a live loado wLL = 50 lb/t

2 (2.4 kPa) and superimposed dead loadwSDL = 20 lb/t

2 (1.0 kPa), but similar trends in the resultsare ound or other loading cases, such as oce occupancyloads, where the live load intensity is larger. Slab widthswere taken as 39.4 in. (1.0 m) and the ASTM A1035/A1035M-07 longitudinal reinorcement quantities corre-sponded toAs,min and 2As,min. The Mast fexural method wasused orD = 100 ksi (690 MPa), and the Appendix B method

was used or D = 120 and 144 ksi (828 and 994 MPa). Asshown in Table 3, the same minimum h results, regard-

Fig. 8Infuence o permissible defection limit on maximum span-depth ratio. (Note: 1 ksi = 6.89 MPa;1 lb/t2 = 0.048 kPa.)

Table 3Design example data and results summary

Flexural design method

Direct defection calculations DTC

Mast Appendix B Appendix B Mast

D, ksi (MPa) 100 (690) 120 (828)* 144 (994) 100 (690)

c= 5 ksi (35 MPa)L= 20 t (6 m)

As=As,min

h, in. (mm) 12.3 (313) 12.3 (313) 12.3 (313) 16.5 (420)

Mr, kip-t (kN-m) 56 (76) 67 (91) 79 (108) 77 (105)

As, in.2 (mm2) 0.68 (438) 0.68 (438) 0.68 (438) 0.91 (588)

Dh, % 0 0 34

DMr, % 20 42 38DAs, % 0 0 34

c=5 ksi (35 MPa)L= 20 t (6 m)

As=2As,min

h, in. (mm) 11.3 (286) 11.3 (286) 11.3 (286) 16.5 (420)

Mr, kip-t (kN-m) 90 (123) 108 (147) 128 (174) 140 (190)

As, in.2 (mm2) 1.24 (802) 1.24 (802) 1.24 (802) 1.82 (1176)

Dh, % 0 0 47

DMr, % 20 41 54

DAs, % 0 0 47*Yield stress assumed as 0.2% oset value.Yield stress assumed corresponding to strain o 0.015.



less o the fexural design method when direct defectioncalculations are considered. In contrast, the DTC approachto defection control with D = 100 ksi (690 MPa) requiresslabs 34 and 47% greater in thickness or the two Asconditions studied. The total reinorcement quantitiesor the slabs designed using the Appendix B approach arelower, but similar fexural capacities at the ULS conditionare achieved compared to the DTC-designed slabs. It shouldbe noted that the thinner slabs resulting rom the use o the

Appendix B method can be essential in structures that areheadroom-constrained. Furthermore, the larger sel-weighto thicker slabs rom the use o the DTC approach can resultin signicant increases in the overall structural cost due tothe corresponding infuences on the design o members inthe gravity- and seismic-orce-resisting systems. Hence,the application o the Appendix B fexural method in combi-nation with direct defection calculations can be advanta-geous over the more common use o simplied fexuralmethods and the DTC approach to defection control.

RECOMMENDED METHOD TO SELECTSLAB THICKNESS

As discussed previously, the DTC defection provisions in

ACI 318-08 require the use o a minimum thickness h ordierent member types, static conguration, and reinorce-ment yield strength. On the other hand, the parametric studiespresented in this study based on direct defection calcula-tions have shown that the maximum L/h ratio to achievesatisactory defection control is sensitive to span length,loading, concrete strength, and the longitudinal reinorce-ment ratio. To compare these approaches, the relationshipsor the maximum L/h ratio derived rom the DTC provi-sions or dierent D values are provided in Fig. 3 through8. The DTC defection provisions result in smaller estimateso the maximum permissible L/h ratio or practical rangeso the design parameters considered compared toL/h ratios

obtained rom the direct defection calculation method. Thedierence in results is especially pronounced or the caseso slabs with small spans, small live load intensities, and/or those that are designed with higher values oD. On theother hand, the maximum L/h ratio rom direct defectioncalculations is typically smaller than the DTC estimate orsmaller D values combined with longer span lengths (reerto Fig. 3 and 4) and/or smaller concrete strength (reer toFig. 3, 4, and 6).

Another important observation rom this study is thatthe use o ASTM A1035/A1035M-07 steel with a nominaldesign strength oD = 120 ksi (828 MPa) in place o thecommon Grade 60 (414 MPa) bars can result in uneco-

nomical estimates o the minimum h by the DTC methodcompared with direct defection calculations. For example,in the case o a one-way slab with c = 10 ksi (69 MPa)and typical residential loading (Fig. 4(a)), the required hto satisy defection requirements should be increasedby 38% according to the DTC method as D changesrom 60 to 120 ksi (414 to 828 MPa). The direct defectioncalculations indicate that an increase in h o only 11% isrequired. Similar results are observed or dierent loadingconditions and concrete strength.

From Fig. 3 to 8, it can be observed that the discrepancybetween the L/h ratios rom the DTC and direct defectionapproaches varied as common design parameters changed.

This was especially pronounced as the D value changed.This suggests that the DTC approach will not provide a

consistent level o reliability or defection control o slabsvarying by common design parameters and thus should notbe used or detailed design.

It is recommended that the direct defection calculationtechnique be used to conrm adequate defection controlo all slabs, rather than the DTC relationships given inTable 1. This recommendation applies regardless o thereinorcement strength. To acilitate rapid initial selectiono a member thickness that is expected to satisy the defec-tion criteria and ULS requirements, graphical design aidssimilar to Fig. 3 can be prepared by ollowing the proceduresdescribed in Fig. 1. Each design aid should directly includethe main target design parameters o live load intensity, spanlength, concrete strength, and nominal design strength or thereinorcement. Similar design charts can also be producedor the other common support conditions given in Table 1.

CONCLUSIONSThis analytical study used a holistic approach or ULS

and SLS design to identiy the infuence o common designparameters on the defection response o one-way slabsreinorced with ASTM A1035/A1035M-07 steel. Based onthe results, the ollowing conclusions can be drawn:

1. The maximumL/h ratio or lightly reinorced slabs wasdemonstrated to be sensitive to the span length, the appliedloading, and the concrete strength. Failure to account or allo these infuences in ACI 318-08 DTC defection controlprovisions gives results that are in poor agreement with themaximum L/h ratios determined rom ACI 318-08 directdefection calculations.

2. For slabs reinorced with ASTM A1035/A1035M-07steel, the ACI 318-08 DTC defection provisions adopted byACI ITG-6R-10 can result in excessively thick slabs comparedto direct defection calculations or the practical ranges o thedierent design parameters considered. On the other hand,excessively fexible slabs can result or slabs reinorced with

lower-strength steel by using the DTC approach.3. The use o the Appendix B method or fexural design

o slabs reinorced with ASTM A1035/A1035M-07 steelcan be advantageous compared to the simplied fexuralmethod (that is, the Mast model) because a higher nominalmoment resistance is considered, whereas the sameL/h andreinorcement ratios will be required to satisy the defectioncontrol provisions.

ACKNOWLEDGMENTSThe authors grateully acknowledge unding or this ongoing research

program provided by the Natural Sciences and Engineering Research

Council o Canada.

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109-s76 - Defelection Control Slabs With HPreinforcing Steel ASTM A1035