Span depth ratios for one-way members based on

ACI Structural Journal/September-October 2009 617

ACI Structural Journal, V. 106, No. 5, September-October 2009.MS No. S-2008-104 received March 24, 2008, and reviewed under Institute publication

policies. Copyright © 2009, American Concrete Institute. All rights reserved, including themaking of copies unless permission is obtained from the copyright proprietors. Pertinentdiscussion including author’s closure, if any, will be published in the July-August 2010ACI Structural Journal if the discussion is received by March 1, 2010.

ACI STRUCTURAL JOURNAL TECHNICAL PAPER

Span-depth ratio (l/h) expressions are developed for steel-reinforcedconcrete one-way slabs and beams to satisfy deflection control andflexural strength requirements. The expressions are presented asfunctions of reinforcement ratio and specified deflection-to-span limit.A comparative study is performed to evaluate the effects of effectivemoment of inertia, shrinkage restraint, construction loads,sustained live load, long-time deflection multiplier, supportconditions, and deflection limits on the resulting span-depth ratios.A comparison is also made with minimum thickness valuesprescribed by ACI 318 that are a function of span and supportconditions only. Effects of various input parameters are consideredin examples involving a simply supported rectangular beam and aninterior span of a continuous one-way slab. The results of the studyindicate some of the limitations associated with the ACI 318minimum thickness requirements.

Keywords: beams; concrete; deflection; slabs; span-depth ratio.

INTRODUCTIONConcrete structures are designed to satisfy strength

requirements and serviceability requirements related todeflection and crack control. Many codes and standards suchas ACI 318-08 (ACI Committee 318 2008) and CSA A23.3(2004) include provisions for deflection control by eitherrequiring a minimum member thickness based on a limitingspan-depth ratio (l/h) or ensuring that computed values ofdeflection do not exceed permissible deflection limits underspecified service loads. Satisfying appropriate span-depthratios is the simplest approach to ensure the member hassufficient stiffness to avoid deflection problems and impliesdeflection limits are satisfied.

Minimum member thickness requirements using the span-depth ratios specified in Table 9.5(a) of ACI 318 areintended for nonprestressed beams and one-way slabs notsupporting or attached to partitions or other constructionlikely to be damaged by large deflections. In other words, thespan-depth ratio limits can be used when damage to attachednonstructural elements is not a concern. Deflection must becalculated for members with a thickness less than theminimum required value or when supporting nonstructuralelements that are likely to be damaged by large deflections.Computed values of deflection, in this instance, are requiredto satisfy the deflection limits specified in Table 9.5(b) ofACI 318. These include limits for immediate deflectionunder live load (l/180 for flat roofs and l/360 for floors) andincremental deflection after installation of nonstructuralelements (l/480 when supporting nonstructural elementssusceptible to damage and l/240 when damage is not aconcern). Other deflection criteria may also need to beconsidered depending on the intended use of the structure(ACI Committee 435 1968) but are not considered in thispaper, although the proposed expressions can be easilyextended to cover other deflection control criteria.

Immediate deflection Δi is computed using a generalizedelastic equation

(1)

for a given span l. Deflection is calculated for the serviceload moment Ma at the critical section which is taken at thesupport face for a cantilever and at midspan for simple andcontinuous members. The restraint factor K is determined byequating Eq. (1) with the elastic deflection equationcorresponding to the type of loading and support conditionsbeing considered. Appropriate values are given in Table 1 fora member under uniform load w with different supportconditions. Effects of cracking and reinforcement onmember stiffness are taken into consideration with an effectivemoment of inertia Ie. The elastic modulus of normalweightconcrete Ec = 57,000 psi (4730 MPa), where fc′ isthe specified compressive strength of concrete.

Additional long-term deflection Δlt from creep andshrinkage is typically computed by taking a multiple of theimmediate deflection Δi,sus caused by the sustained load.This gives Δlt = λΔi,sus with λ = ξ /(1 + 50ρ′) and ρ′ = As′ /bdfor a given amount of compression steel As′ , width b of the

Δi K 548------

Mal2

EcIe

-----------=

fc′ fc′

Title no. 106-S57

Span-Depth Ratios for One-Way Members Based onACI 318 Deflection Limitsby Peter H. Bischoff and Andrew Scanlon

Table 1—Restraint factor K and moment Ma at critical section for different support conditions (uniformly distributed load)

Member type K* C = Ma/Mo

Cantilever (fixed end)† 2.4 4.0

Simple span 1.0 1.0

One end continuous with discontinuous end unrestrained (K = 1.20 – 0.20Mo /Mm)‡ 0.925 0.73

One end continuous with discontinuous end integral with the end support (K = 1.20 – 0.20Mo /Mm)‡ 0.8 to 0.85 0.5 to 0.57

Fixed-hinged (midspan value) 0.80 0.5

Both ends continuous (K = 1.20 – 0.20Mo /Mm)‡ 0.7 to 0.8 0.4 to 0.5

Fixed-fixed 0.60 0.33*Δ = K(5/48)Mal2/EcIe.†Deflection from rotation at support of cantilever must be included.‡Mo = wl2/8 and Mm is midspan moment for continuous member. Ma is midspan momentMm except for cantilevers where it is moment at support face.

ACI Structural Journal/September-October 2009618

member, and effective depth d of the tension steel. Values ofthe time-dependent factor ξ prescribed by ACI 318 are 1.0,1.2, 1.4, and 2.0 for 3 months, 6 months, 1 year, and 5 yearsor more, respectively. The deflection multiplier λ has amaximum value of 2.0 for the worst case (assuming nocompression steel) when loads are sustained over a period of5 years or more.

This paper uses ACI-prescribed deflection limits to evaluatemaximum span-depth ratios for one-way flexural membersand compares these to the ACI 318 minimum thicknessvalues. Requirements for incremental deflection Δincr ,which occurs after attachment of the nonstructuralelements, typically govern such that

Δincr = Δlt + Δi,L(add) ≤ l/240 or l/480 (2)

where Δlt is the long-term deflection under sustained loadsthat occurs after installation of the nonstructural elementsand Δi,L(add) equals the immediate deflection from theremaining live load that is not part of the sustained load. Inother words, Δi,L(add) = Δi,D+L – Δi,D+L(sus). When used inconjunction with Eq. (1), Eq. (2) can be rearranged to givespan-depth ratios as a function of the reinforcing ratio ρ for agiven deflection limit (l/240 or l/480). Comparison is made forrectangular and T-shaped sections as outlined later in the paper.

RESEARCH SIGNIFICANCEDesign of reinforced concrete structures requires evaluation

of serviceability conditions related to deflection which iscontrolled by either choosing a member that meets minimumthickness requirements or limiting computed values ofdeflection to some fraction of the member span. Maximumspan-depth ratios (l/h) are developed using the incrementaldeflection limits given in ACI 318 and compared to the ACIminimum thickness values for one-way members. Resultsshow that, for the most part, members satisfying the ACIminimum thickness requirements do not necessarily satisfythe deflection limits prescribed by the ACI 318 Building Code.

EFFECTIVE MOMENT OF INERTIACracking is taken into consideration using an effective

moment of inertia Ie that accounts for nonlinear behaviorarising from a gradual reduction in flexural stiffness as themember cracks and loses tension stiffening under increasingload. Branson (1965) introduced the concept of an effectivemoment of inertia using the following expression to modelthe transition from a gross (uncracked) moment of inertia Igto the cracked transformed moment of inertia Icr.

(3)IeMcr

Ma

---------⎝ ⎠⎛ ⎞ 3

Ig 1Mcr

Ma

---------⎝ ⎠⎛ ⎞3

–⎝ ⎠⎛ ⎞ Icr Ig≤+=

where Ie is computed for the service load moment Ma at thecritical section, and the cracking moment Mcr (relative to theapplied moment Ma) controls the amount of tension stiff-ening in the member response. Equation (3) was adopted byACI 318 in 1971 and continues to be used for calculatingdeflection with the exception of slender (tilt-up) wall panels(ACI Committee 318 2008).

Branson’s equation works well for moderately to heavilyreinforced members with a steel reinforcing ratio greaterthan approximately 1%, but consistently overestimatesmember stiffness for lightly reinforced members and fiber-reinforced polymer (FRP) reinforced concrete members(Bischoff 2007; Bischoff and Scanlon 2007). Hence, deflectionis underestimated as a consequence. An alternative expressionproposed by Bischoff (2005) for use in ACI 318 is given byEq. (4) and used in this paper to evaluate the span-depthrelationship limits, as this expression works equally well forsteel- and FRP-reinforced concrete members over a widerange of reinforcing ratio.

(4)

SUPPORT CONDITIONSEnd restraint conditions at the supports are taken into

account with the restraint factor K used in Eq. (1) to computeelastic deflection. Values given in Table 1 are based on thedeflection equation corresponding to uniform loading for thesupport conditions indicated. For a continuous member witheither one or both ends continuous, the restraint factordepends on the assumed value of the end support momentsand does not account for redistribution of moments as themembers crack. Table 1 also gives the moment Ma at the criticalsection expressed as a ratio of the total static moment Mo = w l2/8,and is directly related to the restraint factor K = 1.2 – 0.2/C(except for cantilevers) given that C = Ma /Mo.

For a member with both ends continuous, the value of K isassumed to vary from 0.8 using the ACI moment coefficient foran interior span (based on a positive midspan moment Mm ofwl2/16 or 0.5Mo) down to 0.7 using Rangan’s (1982)assumption that the average moment at the end supports equals0.6Mo. For a member with one end continuous, the value of Kvaries from 0.85 using the ACI moment coefficient for an endspan with the discontinuous end integral with the support (basedon a positive midspan moment Mm of wl2/14 or 0.57Mo) downto 0.8 using Rangan’s assumption that Mm equals 0.5Mo. Amember with the discontinuous end unrestrained (Mm =wl2/11 = 0.73Mo) gives a value of 0.925 for K.

Equation (1) implies that deflection of continuousmembers depends on the stiffness of member at the criticalmidspan section with no consideration given to the stiffnessat the end supports (unless a weighted average is taken ofstiffness at the midspan and end supports). Using the stiffnessat midspan alone gives reasonable results and, in most practicalcases, gives a better approximation than deflection computedwith a weighted average of member stiffness (Bischoff 2007).

SHRINKAGE RESTRAINT CRACKINGAND CONSTRUCTION LOADS

Restraint to shrinkage induces tensile stresses in theconcrete that decrease the cracking moment and reduce theflexural stiffness of the member. Recent work by Scanlonand Bischoff (2008) suggests using Eq. (4) for Ie along with

IeIcr

1 η Mcr Ma⁄( )2–---------------------------------------- Ig with η 1 Icr Ig⁄–=≤=

Peter H. Bischoff, FACI, is a Professor in the Civil Engineering Department at theUniversity of New Brunswick, Fredericton, NB, Canada. He is a member of ACICommittees 224, Cracking; 360, Design of Slabs on Ground; 435, Deflection ofConcrete Building Structures; 544, Fiber Reinforced Concrete; and Joint ACI-ASCECommittee 408, Development and Splicing of Deformed Bars. His research interestsinclude serviceability behavior of concrete structures.

Andrew Scanlon, FACI, is a Professor of civil engineering at the Pennsylvania StateUniversity, University Park, PA. He is Chair of ACI Committee 435, Deflection ofConcrete Building Structures, and is a member of ACI Committees 224, Cracking; 342,Evaluation of Concrete Bridges and Bridge Elements; and 348, Structural Safety; andHis research interests include safety and serviceability of concrete structures.


a reduced cracking moment equal to two-thirds the value ofMcr computed with the rupture modulus fr = 7.5 psi(0.62 MPa) prescribed by ACI 318-08 and this corre-sponds to a reduced rupture modulus fr = 5 psi(0.42 MPa). Decreasing the cracking moment has asignificant effect on computed values of deflection forlightly reinforced members such as slabs, while deflection ofmoderately to more heavily reinforced members is notgreatly affected at full service load because the effectivemoment of inertia Ie is closely approximated by Icr at theseload levels (Scanlon and Bischoff 2008).

Long-term deflections computed with the long-term multiplierλ usually depend on calculation of the immediate deflectionunder sustained load using an effective moment of inertiaIe,sus corresponding to the sustained load moment only.Preloading from construction loads prior to installation ofthe nonstructural elements often causes additional crackingthat reduces the member stiffness and increases deflectionarising from the sustained portion of loading. Scanlon andBischoff (2008) recommend computing the immediatedeflection under sustained loads with an effective moment ofinertia (Ie,D+L) corresponding to the full (dead plus live)service load moment to account for the reduced stiffness;this simplifies the calculation procedure considerably.

SPAN-DEPTH RATIO RELATIONSHIPRelationships for the span-depth ratio developed in this

paper depend on deflection limits for incremental deflectionand allow for a portion of the live load to be sustained. Ageneral expression for the limiting span-depth ratio (l/h) isobtained by equating the incremental deflection from Eq. (2) toa deflection limit Δall.

(5)

Full details of the derivation and explanation of variables arefound in the Appendix. This expression is subsequentlysimplified for a rectangular section to give

(6)

where the factor Ω = [1 + γ(λ – 1)(Ie,D+L/Ie,sus)] accounts forlong-term effects with the deflection multiplier λ , the ratio γof sustained load to full service load, and the differencebetween the member stiffness at a sustained and full serviceload when preloading is not taken into account. SettingIe,D+L/Ie,sus = 1 takes account of preloading from constructionloads such that λMsus + ML,add = ΩMD+L. Using a reducedcracking moment 0.67Mcr to account for shrinkage restrainthas the effect of decreasing the effective moment of inertia Ie.

The strength reduction factor φ for flexure is taken equalto 0.9 for tension-controlled sections, and αD+L is an aver-aged load factor depending on the ratio of dead to live load.The flexural resistance factor Rn = Mn/bd

2 = ρfs[1 – ρfs/(2α1 fc′)] for a rectangular section with bar stress fs = fy whenthe section is under-reinforced, α1 is a rectangular stressblock factor equal to 0.85, and Ie /Ig = (Mcr /Ma)3 + [1 –(Mcr /Ma)3](Icr /Ig) using Branson’s (1965) Eq. (3) or (Icr /Ig)/[1 – η(Mcr /Ma)2] using Bischoff’s (2005) Eq. (4). The

fc′fc′

fc′fc′

l

h---

Ec d h⁄( ) Ie D L+, Ig⁄( ) Ig bd3⁄( )K 5 48⁄( )Ω φ αD L+⁄( )Rn

------------------------------------------------------------------------Δall

l--------≤

l

h---

0.8Ec Ie D L+, Ig⁄( )

KΩ φ αD L+⁄( ) d h⁄( )2Rn

-----------------------------------------------------------Δall

l--------≤

cracked-to-gross stiffness ratio Icr /Ig = 12(d/h)3[k3cr /3 + nρ(1 –

kcr )2] for a rectangular section with kcr = –nρ, ρ = As/bd, and n = Es/Ec.

The incremental deflection limit Δall/l is equal to either 1/480or 1/240 depending on whether or not the nonstructuralelements are likely to be damaged by large deflections.Equations (5) and (6) essentially express the span-depthlimit needed for deflection control as a function of thereinforcing ratio ρ needed to satisfy strength requirementsat the critical section and take implicit account of the loadingmagnitude w.

STRENGTH-CONTROLLED LIMITGiven that the moment at the critical section Ma = (φ/αD+L)Mn,

Mn = Rnbd 2, Mo = w l

2/8, and Ma = CMo (refer to Table 1)leads to the following expressions relating the l/h needed forstrength control

(7)

for a slab strip with width b subjected to a uniformly distributedload w, and

(8)

for beams with a d/b aspect ratio and distributed load-to-spanlength ratio w/l.

ALLOWABLE SPAN-DEPTH RATIOSFigure 1 shows a typical plot of the maximum span-depth

ratio (l/h) for deflection control (Eq. (6)) together with plotsof the limiting l/h controlled by strength for slabs (Eq. (7))and beams (Eq. (8)). Intersection of the deflection andstrength-controlled curves defines the l/h requirement for aflexural member, with the location of the intersection pointdepending on the loading conditions and aspect ratio of themember cross section. Loading for the slabs ranges between100 lb/ft2 (4.8 kPa) for Slab S1 (corresponding to a 4 in.[100 mm] thick slab with a live load of 50 lb/ft2 [2.4 kPa]),200 lb/ft2 (9.6 kPa) for Slab S2, and 500 lb/ft2 (24 kPa) forSlab S3 (representing a much heavier 2 ft [600 mm] thick slabwith a 200 lb/ft2 [9.6 kPa] live load). Different combinations

ηρ( )2 2ηρ+

l

h---

8 φ αD L+⁄( )Rn d h⁄( )2

C w b⁄( )-----------------------------------------------------=

l

h---

8 φ αD L+⁄( )Rn d h⁄( )3

C d b⁄( ) w l⁄( )-----------------------------------------------------3=

Fig. 1—Typical span-depth ratios (l/h) for slabs and beams.

620 ACI Structural Journal/September-October 2009

of slab thickness and live load are possible for each slabloading case that depends, of course, on the member span.Beam B1 has an assumed d/b aspect ratio of 1.0 for themember cross section and w/l equal to 75 lb/ft2 (3.6 kPa), whileBeam B2 has a higher aspect ratio of 2.0 and a larger w/lof 300 lb/ft2 (14.4 kPa).

The maximum l/h limit drops as the reinforcing ratioincreases (particularly for lightly reinforced members typicalof slabs) and is heavily dependent on the proportion ofsustained loading. The l/h limit for slabs is greater than thatfor beams as expected. The l/h limit for deflection given byEq. (6) may also not govern under certain conditions forbeams with high d/b and w/l ratios and, in this instance, themember thickness is controlled by strength requirementsonly as observed for Beam B2 in Fig. 1.

The solution to the l/h expression given by Eq. (6) is iter-ative (depending on ρ, which is unknown until the memberthickness h is chosen). While not suitable for design because ofits complexity, the developed expression is useful forunderstanding the effect of parameters such as ρ, Ie, andMcr ; preloading from construction loads; dead-to-live loadratio; fc′; and fy, on the required member thickness needed tosatisfy deflection requirements.

Similar types of formulation have expressed the span-depth ratio as a function of the distributed load w (Rangan1982; Gilbert 1985; Scanlon and Choi 1999; Scanlon andLee 2006). This approach can be expressed as

(9)l

h---

6.4Ecb Ie D L+, Ig⁄( )KCΩw

----------------------------------------------Δall

l--------3≤

with C = Ma /Mo (Table 1) and Ωw = λwsus + wL,add whenIe,D+L/Ie,sus = 1. This version of the expression also dependson the reinforcing ratio ρ unless simplifying assumptions aremade to express Ie as a fixed ratio of Ig (see, for example,Scanlon and Choi 1999). Equation (9) works well for slabsas dependence on the width b of the member is accommodatedby using a slab strip of unit width, and the equation is easilyrearranged to give an expression more appropriate for beamswith a specific d/b aspect ratio for the member cross section.

(10)

COMPARATIVE STUDYA comparative study is performed to demonstrate the

sensitivity of calculated l/h values to various parameters asdiscussed in this section. The general trend is shown in Fig. 1as discussed previously. In the results presented further in thepaper, the l/h limit is plotted against the reinforcing ratio ρfor a rectangular section using 4000 psi (27.6 MPa) concreteand Grade 60 (414 MPa) reinforcing steel. All members areassumed to have an effective depth-to-height ratio (d/h) of0.85. Unless otherwise noted, comparisons are made for asimply supported member with dead-to-live load ratioranging from 0.5 to 2.0, and are based on the l/240 incre-mental deflection requirement using a long-term deflectionmultiplier λ of 2.0. Calculations, for the most part, useBischoff’s expression for Ie (Eq. (4)), take account ofpreload from construction loads, and use two-thirds of Mcr.The effect of concrete strength, steel grade, support condi-tions, and use of T-shaped sections is also considered.

Effective moment of inertiaFigure 2 compares the l/h limit for deflection computed with

both Branson’s (Eq. (3)) and Bischoff’s (Eq. (4)) expressionsfor Ie. Results are presented for a simply supported memberwith a deflection limit of l/240. Calculation of incrementaldeflection, in this instance, is based on the full crackingmoment Mcr with no account taken of preloading. Memberthickness is heavily dependent on the proportion of sustainedloading characterized by the dead-to-live load ratio (D/L),and differences between the two approaches are evident forlightly reinforced members with a reinforcing ratio less thanapproximately 0.85%, as expected (Bischoff and Scanlon2007). At higher reinforcing ratios, the effective moment ofinertia Ie quickly approaches Icr using either approach.

Deflection requirements are satisfied for most slabs usingthe ACI minimum thickness value (with the exception ofheavily loaded slabs having a dead-to-live load ratio [D/L] of2.0) but not for beams. Span-depth ratio requirements forbeams need to be as low as 9 depending on the loadingconditions and are even more severe as the incrementaldeflection limit decreases from l/240 to l/480.

Shrinkage restraint and construction loadsFigure 3 shows that preloading from construction loads

and use of a lower cracking moment to account for shrinkagerestraint have a significant effect on computed values ofdeflection and corresponding l/h limit for reinforcing ratiosless than approximately 1%. Deflection of slabs using theACI l/h limit of 20 is only satisfied in this instance for slabswith a service load less than approximately 200 lb/ft2 (9.6 kPa)when the D/L equals 2. The l/h limit for more heavily rein-

l

h---

6.4Ec Ie D L+, Ig⁄( ) d h⁄( )KCΩ w l⁄( ) d b⁄( )

----------------------------------------------------------Δall

l--------4≤

Fig. 2—Comparison of l/h limits at full Mcr and no preload.

Fig. 3—Effect of construction loads and reduced crackingmoment.


forced beams is controlled by Icr and remains at 9 (D/L = 2)and 13 (D/L = 0.5) compared to the present requirement of 16for beams.

Sustained live loadFigure 4 shows the effect of having part of the live load

sustained. Increasing the sustained load increases incrementaldeflection and leads to a decrease in the l/h limit with asubsequent increase in the minimum member thicknesswhen deflection governs.

Long-term deflection multiplierFigure 5 shows the effect of first adding compression steel

and then installing partitions 3 months after removal of the

shoring. Adding 1% compression steel decreases the long-term deflection multiplier to give λ = ξ/(1 + 50 ρ′) = 2.0/(1 +50 × 0.01) = 1.33 when partitions are installed immediatelyupon removal of the shoring, while a 3-month delay in installationof the partitions gives λ = (2.0 – 1.0)/(1 + 50 × 0.01) = 0.67 toallow for long-term deflection that occurs during the first3 months before the partitions are installed. None of thelive load is assumed to be sustained in this example. In bothcases, the addition of compression steel and delay in installationof the partitions increases the l/h limit needed to satisfy ACIincremental deflection requirements. Even though changesto the l/h limit can be significant, beams using the existingACI minimum thickness values still do not satisfy deflectionrequirements in many cases except when the deflectionmultiplier λ = 0.67.

Concrete strength and fyFigures 6 and 7 indicate that flexural members with lower-

strength concrete and higher-strength steel require greaterminimum thickness values as expected (with a larger influencefor lower D/L ratios). While the l/h limit for the lower, 4 ksi(27.6 MPa) strength concrete, is less than the 10 ksi (69 MPa)concrete at reinforcing ratios up to approximately 2%, thelimit for the 10 ksi (69 MPa) concrete eventually drops off toa value lower than the 4 ksi (27.6 MPa) concrete at higherreinforcing ratios. Recent work by Tang and Lubell (2008)provides more detailed information on the effect of reinforcinggrade on deflection and minimum thickness requirementsfor slabs.

Fig. 4—Effect of sustained live load.

Fig. 5 Effect of compression steel (λ =1.33) and delay inpartition installation (λ = 0.67) for: (a) D/L = 2.0; and (b)D /L = 0.5.

Fig. 6—Effect of concrete strength.

Fig. 7—Effect of steel reinforcing grade.


T-shaped sectionsFigure 8 plots l/h requirements for T-shaped sections with

the reinforcing ratio expressed relative to the web width(ρw = As /bwd). Although the minimum thickness requiredto satisfy the incremental deflection limit decreases as theflange to web width (bf /bw) increases, incremental deflectionrequirements are still not satisfied using ACI 318 minimumthickness values for beams (with λ = 2.0).

l/240 versus l/480Figure 9 shows the effect of decreasing the allowable

value of incremental deflection from l/240 down to l/480when damage to attached nonstructural elements needs to beconsidered. Minimum thickness requirements can doublewith a corresponding 50% decrease in the l/h limit. For thesimply supported case shown, heavily reinforced beamswould require an l/h of 4.5 (D/L = 2) and 6 (D/L = 0.5),whereas the l/h for slabs can be as low as 12.5.

Deflection requirements, however, are not as severe asthey first appear in Fig. 9 because any required increase inmember thickness is accompanied by a decrease in reinforcingratio needed to maintain the same member capacity, and this, inturn, has the effect of increasing the limit for the l/h. Hence,the increase in member thickness will most often be less thantwice the value required to satisfy the l/240 deflection limit(particularly for lightly reinforced members such as slabswhere the l/h limit changes rapidly with reinforcing ratio)and this is demonstrated in deflection examples presentedlater in the paper. Changes in member thickness also have a

significant effect on the magnitude of service load for slabswhere the self-weight comprises a significant portion ofthe loading.

Support conditionsMinimum thickness requirements in Eq. (6) are directly

proportional to the restraint factor K taken from Table 1 toreflect support conditions of a uniformly loaded member.Once again, any changes to the minimum thickness requiredfor a member are tempered by changes in the reinforcingratio required to satisfy strength for a given span andloading condition.

Requirements for l/h are plotted in Fig. 10 for a memberwith one end continuous (K = 0.85), both ends continuous (K =0.8), and for a cantilever (K = 2.4). Figures 3 and 4 give the casefor a simply supported member (K = 1.0). In all cases, theACI minimum thickness requirement for beams does notsatisfy incremental deflection requirements when using along-term deflection multiplier λ of 2.0, whereas incremental

Fig. 8—l/h limits for T-beam sections.

Fig. 9—Effect of incremental deflection limit (l/240 versusl/480).

Fig. 10—l/h limit for: (a) one end continuous; (b) both endscontinuous; and (c) cantilever.


(l/240) deflection requirements for slabs using the ACIminimum thickness values are mostly satisfied with theexception of those that are heavily loaded with service loadsgreater than approximately 200 to 250 lb/ft2 (9.6 to 12 kPa).

Satisfying l/h requirements is important for slabs, as designoften begins by choosing a slab thickness based on minimumthickness requirements. Hence, adequate flexural stiffness ofslabs is ensured by specifying the correct l/h limits. Designof beams, on the other hand, begins by satisfying strengthrequirements where the member cross section is proportionedfor strength. Choosing a relatively low reinforcing ratio in therange of 0.25ρb to 0.40ρb will often result in a member thicknessgreater than the minimum thickness needed to satisfy deflectionrequirements (ACI Committee 435 1995), depending on thew/l ratio, aspect ratio of the member cross section, long-termdeflection multiplier, and dead-to-live load ratio.

DEFLECTION EXAMPLESDeflection examples are worked out for: 1) a simply

supported rectangular beam; and 2) an interior span of acontinuous one-way slab. Calculations for both examplesuse 4000 psi (27.6 MPa) concrete and Grade 60 (414 MPa)reinforcement to give fr = 474 psi (3.26 MPa), Ec = 3605 ksi(24.39 GPa), and n = 8.04. The d/h is assumed to remainconstant at 0.85 and λ = 2.0. Note that dimensions are notrounded off in these examples for ease of checking.

Example 1—Rectangular beamA simply supported beam with a 20 ft (6096 mm) span is

designed to support a uniformly distributed dead load of 1.4 k/ft(20.4 kN/m) and a live load of 0.7 k/ft (10.2 kN/m). This gives aw/l of 105 lb/ft2 (5 kPa), D/L of 2.0, service load moment atmidspan of Ma = 105 k-ft (142.4 kN-m), and factoredmoment Mu = 140 k-ft (189.8 kN-m). Results for thisexample are summarized in Table 2(a) and plotted in Fig. 11.

Designing the beam for strength based on an assumedreinforcing ratio ρ = 1% (0.5ρmax) and d/b of 1.5 gives abeam cross section with width b = 11.5 in. (292 mm) andheight h = 20.3 in. (515 mm). The corresponding l/h of 11.8is less than the ACI 318 limit of 16 for beams, implying thatdeflection limits should be easily satisfied. The computedvalue of incremental deflection, however, (using either Eq. (3)or (4)) equals 1.13 in. (28.8 mm), which is greater than thel/240 limit of 1 in. (25.4 mm). Hence, the member thicknessneeds to be increased to satisfy ACI 318 requirements forincremental deflection.

Using Bischoff’s expression at full Mcr and with nopreload (Fig. 11) requires a beam with an l/h limit of 11.2 ata reinforcing ratio ρ of 0.84%. The width and height of thisbeam equals 12.1 in. (307.9 mm) and 21.4 in. (543.3 mm),respectively. Accounting for preload and using 0.67Mcrdecreases the l/h limit from 11.2 to 10.6 at a lower reinforcingratio of 0.71% to account for the increased member thickness.The width and height of the beam, in this case, is 12.8 in.(324.5 mm) and 22.5 in. (572.6 mm), respectively. In bothcases, the member deflection equals the limiting value of l/240equal to 1 in. (25.4 mm).

Delaying partition installation by 3 months decreases thelong-term deflection multiplier to λ = 1. The l/h deflectioncurve no longer intersects the strength curve in this instance(Fig. 11) and the l/h limit increases to 14.0 based on strengthrequirements at a reinforcing ratio of 1.8%. The correspondingwidth and height of the beam equals 9.7 in. (246.7 mm) and17.1 in. (435.4 mm), respectively, and the midspan deflectionof 0.94 in. (24.0 mm) is less than the maximum value of 1 in.(25.4 mm) as deflection no longer governs.

Using reinforcing steel with 75 ksi (517 MPa) yieldstrength decreases the l/h to 9.7 at a reinforcing ratio of 0.425%for a beam with width of 14.0 in. (354.9 mm) and height of24.7 in. (626.2 mm). The increase in member thickness,compared to a beam with Grade 60 steel, is approximately10% in this case. A beam with 10,000 psi (69 MPa) concreteand 60 ksi (414 MPa) steel has a limiting l/h of 11.4 at areinforcing ratio of 0.84%, giving a beam with a width of11.9 in. (303.2 mm) and height of 21.1 in. (535.1 mm) that is

Table 2(a)—Simply supported beam example: 4000 psi (27.6 MPa) concrete and 60 ksi (414 MPa) steel*

l/h ρ, % Ma/Mcr Beam height h, in. (mm) Beam width b, in. (mm) Deflection, in. (mm)†

ACI 318 minimum thickness 16 — — 15 (381) — —

Strength at 0.5 ρmax‡ 11.8 1.0 3.4 20.3 (515.0) 11.5 (292.0) 1.13 (28.8)

Bischoff Ie with no preload, Mcr and λ = 2 11.2 0.84 2.9 21.4 (543.3) 12.1 (307.9) 1.0 (25.4)

Bischoff Ie with preload, 0.67 Mcr and λ = 2 10.6 0.71 3.7 22.5 (572.6) 12.8 (324.5) 1.0 (25.4)

λ = 1 14.0§ 1.8 8.4 17.1 (435.4) 9.7 (246.7) 0.945 (24.0)

fy = 75 ksi and λ = 2 9.7 0.425 2.8 24.7 (626.2) 14.0 (354.9) 1.0 (25.4)

fc′ = 10,000 psi and λ = 2 11.4 0.84 2.85 21.1 (535.1) 11.9 (303.2) 1.0 (25.4)

Δ = l/480 and λ = 2 8.2 0.31 1.7 29.4 (746.9) 16.7 (423.3) 0.5 (12.7)*l = 20 ft (6096 mm), d/h = 0.85, K = 1.0, D/L = 2.0, and w = 2.1 k/ft (30.6 kN/m).†Δ ≤ l/240 = 1 in. (25.4 mm) for control of deflection.‡Maximum reinforcing ratio ρmax ≈ 0.72ρb.§Strength-controlled.Note: 1 ksi = 1000 psi; 1 psi = 0.00690 MPa.

Fig. 11—Minimum thickness requirements for beamdeflection example.


approximately 6% less than the thickness of a beam using4000 psi (27.6 MPa) concrete.

Decreasing the allowable deflection value to 0.5 in. (12.7 mm)corresponding to the l/480 incremental deflection limitdecreases the l/h limit to 8.2 at a reinforcing ratio of 0.31%.The beam has a height of 29.4 in. (746.9 mm) that is 30%greater than the required thickness for the l/240 deflection limit.

Example 2—One-way slabDeflection requirements are considered for an interior

span of a continuous one-way slab with a 20 ft (6096 mm)span. The slab supports its own self-weight plus a live loadof 60 lb/ft2 (2.9 kPa). None of the live load is sustained andproportioning is carried out for an allowable deflection of l/240.Moments are based on the ACI design moment coefficient atmidspan giving C = 0.5 and a restraint factor K = 0.8 (Table 1).Design of the support sections is not considered in this example.

Design begins by assuming l/h = 28 based on existing ACI 318minimum thickness requirements to give h = 8.6 in. (218 mm)and corresponding dead load of 108 lb/ft2 (5.2 kPa). Areinforcing ratio of 0.2% satisfies strength requirementswith the 8.6 in. (218 mm) thick slab and this gives anuncracked slab under service loads (at full Mcr) with acomputed deflection of 0.174 in. (4.4 mm) that is considerablyless than the allowable value of l/240 = 1 in. (25.4 mm). UsingBischoff’s (2005) expression for Ie at full Mcr and with nopreload gives an l/h limit of 41.9 and a corresponding slabthickness of 5.7 in. (145.5 mm) with ρ = 0.37%. Recall that thereinforcing ratio needs to be increased as the member thicknessdecreases to maintain the same strength. The slab is nowcracked (Ma/Mcr = 1.27) and has a computed value of deflectionequal to the deflection limit of l/240 = 1 in. (25.4 mm).Branson’s (1965) expression (Eq. (3)) overestimates member

stiffness at this low reinforcement ratio and gives a muchsmaller deflection of 0.65 in. (16.4 mm). Results aresummarized in Table 2(b) and plotted in Fig. 12.

Accounting for preload and using 0.67Mcr with Bischoff’sexpression for Ie decreases the l/h to 32.2 with a correspondingreinforcing ratio of 0.24%. A slab thickness of 7.45 in.(189 mm) is needed to maintain the computed value ofdeflection at 1 in. (25.4 mm), and is used as the basis forother comparisons that include a delay in partition installation,increase in concrete strength, higher grade steel, and allowabledeflection of l/480 when damage to partitions is of concern.

Delaying partition installation by 3 months decreases thelong-term multiplier to λ = 1 and this gives a thinner 6.5 in.(165.5 mm) slab based on l/h = 36.8 at a reinforcing ratio of 0.3%.

Increasing the compressive strength of concrete to 10,000 psi(69 MPa) decreases the slab thickness considerably to 5.75 in.(145.7 mm) for l/h = 41.8 at a reinforcing ratio of 0.36%,while using Grade 75 (517 MPa) steel gives a slightly thicker,7.8 in. (198 mm) thick slab with l/h = 30.8 at a reinforcing ratioof 0.185%. The higher grade steel results in a 5% increase inslab thickness, which is less than the 15% increase computedusing the ACI 318 assumption that minimum thickness valuesfor fy other than 60,000 psi (414 MPa) shall be multiplied by0.4 + fy /100,000. Similar observations were made by Tangand Lubell (2008). Note also the change in dead load and corre-sponding D/L as the slab thickness changes.

Finally, decreasing the allowable deflection value to l/480 =0.5 in. (12.7 mm) gives a thicker, 8.3 in. (211.5 mm) slab withl/h = 28.8 at a reinforcing ratio of 0.205%. In this instance,the slab is barely cracked with Ma /Mcr = 1.12. While Eq. (6)suggests that the slab thickness doubles when the allowabledeflection drops by one half, increasing the slab thicknessalso results in a lower value of ρ and, hence, the correspondingincrease in slab thickness is only approximately 12% inthis case.

Assumptions made regarding tension stiffening have aconsiderable effect on computed values of deflection whenservice loads are close to the cracking value, as is the case forthe slab in this example, with Ma/Mcr ranging between 1 and1.5. For beams with a reinforcing ratio greater than 1%, theMa/Mcr ratio is typically greater than 3 and the effectivemoment of inertia is then closely approximated with Icr.

In closing, it should be noted that deflection requirementsare satisfied in all cases for the slab in this example whenusing the ACI 318 minimum thickness value, whereas this isnot the case for the beam in Example 1. The slab wassubjected to a service load of approximately 150 lb/ft2

(7.2 kPa), and a slab supporting a heavier live load of 125 lb/ft2

(6 kPa) or more corresponding to a full service load of at least230 lb/ft2 (11 kPa) would not have satisfied deflection

Table 2(b)—Continuous slab example: 4000 psi (27.6 MPa) concrete and 60 ksi (414 MPa) steel*

l/h ρ, % Ma /Mcr D/L h, in. (mm) Deflection, in. (mm)

ACI 318 minimum thickness 28 0.2 0.72 (uncracked) 1.8 8.6 (218) 0.174 (4.4)

Bischoff Ie with no preload, Mcr and λ = 2 41.9 0.367 1.27 1.19 5.73 (145.5) 1.0 (25.4)

Bischoff Ie with preload, 0.67 Mcr and λ = 2 32.2 0.243 1.31 1.55 7.45 (189) 1.0 (25.4)

λ = 1 36.8 0.301 1.58 1.35 6.5 (165.5) 1.0 (25.4)

fy = 75 ksi and λ = 2 30.8 0.185 1.23 1.62 7.8 (198) 1.0 (25.4)

fc′ = 10,000 psi and λ = 2 41.8 0.361 1.20 1.20 5.74 (145.7) 1.0 (25.4)

Δ = l/480 and λ = 2 28.8 0.205 1.12 1.74 8.32 (211.5) 0.5 (12.7)*l = 20 ft (6096 mm), d/h = 0.85, K = 0.8, C = 0.5; and live load = 60 lb/ft2 (2.9 kPa).Note: 1 ksi = 1000 psi; 1 psi = 0.00690 MPa.

Fig. 12—Minimum thickness requirements for slabdeflection example.


requirements with a slab thickness based on the ACI (l/28)minimum value.

CONCLUSIONSSpan-depth ratio expressions for one-way members have

been developed based on deflection calculation equations interms of the reinforcement ratio required to meet flexuralstrength requirements and deflection control for specifieddeflection limits.

Results of a comparative study show that the span-depthratio is highly sensitive to reinforcement ratio (and hence,loading) for lightly reinforced members typically associatedwith one-way slabs (less than approximately 0.5% reinforce-ment). Slabs designed using the ACI minimum thicknessrequirement do not always satisfy the l/240 incrementaldeflection requirement for heavier loaded slabs.

Values obtained for moderate to high reinforcement ratiostypical for beams indicate that the ACI 318 minimum thicknessvalues for these members often do not satisfy the span/240deflection limit for additional deflection that occurs afterinstallation of nonstructural elements. Despite the fact thatbeams satisfying the ACI minimum thickness requirementmay not satisfy the l/240 requirement for incremental deflec-tion, strength will often govern in many cases (particularlyfor beams with high w/l and d/b) to give a beam with adeflection less than l/240.

Although the developed expressions are not intended forhand calculation as a design tool, they can be used to checkdesign equations for span-depth limits intended for deflectioncontrol because no simplifying assumptions have been madeother than those implicit in ACI 318 calculation procedures.

Results of the study indicate limitations inherent in thecurrent ACI 318 span-depth ratios for one-way members,particularly for beams. The results suggest that changes arerequired to account for these limitations so that the minimumthicknesses specified are consistent with the span/240 limitationfor members not supporting or attached to nonstructuralelements likely to be damaged by large deflections.

ACKNOWLEDGMENTSFinancial support was provided by the Natural Sciences and Engineering

Research Council of Canada and is gratefully appreciated.

NOTATIONAs = tension steel areaAs′ = compression steel areab = width of memberbf = flange widthbw = web widthC = Ma/Mod = effective depth of tension steelEc = elastic modulus of concreteEs = elastic modulus of reinforcing steelFD/L = dead-to-live load (D/L) ratio (MD /ML)FS = sustained live load ratio (ML,sus/ML)fc′ = specified compressive strength of concretefr = rupture modulus of concretefs = bar stressfy = yield strength of reinforcing steelh = height or depth of memberIcr = cracked transformed moment of inertiaIe = effective moment of inertiaIe,D+L = effective moment of inertia corresponding to full (dead +

live) service loadIe,sus = effective moment of inertia corresponding to sustained load

momentIg = gross (uncracked) moment of inertiaK = end restraint factor

kcr = – nρ

l = span lengthMa = service load moment (at critical section)Mcr = cracking momentMD = dead load momentMD+L = full (dead + live) service load momentML = live load momentML,add = part of live load moment that is not sustainedML,sus = sustained part of live load momentMm = midspan momentMn = nominal moment capacityMo = static moment capacity (wl 2/8)Msus = sustained (dead + sustained live) momentMu = factored momentn = modular ratio (Es/Ec)Rn = nominal flexural resistance factor (Mn/bd2)w = uniformly distributed loadwL,add = portion of live load not sustainedwsus = sustained loadα1 = rectangular stress block factor (0.85)αD+L = average load factorΔall = permissible (allowable) deflection (l/240 or l/480)Δi = immediate deflectionΔi,D+L = immediate deflection from full (dead + live) service loadΔi,D+L(sus)= immediate deflection from sustained (dead + sustained

live) loadΔi,L(add) = immediate deflection from remaining part of live load not

sustainedΔincr = incremental deflection (λΔi,sus + Δi,L(add))Δi,sus = immediate deflection from sustained loadΔlt = long term deflection (λΔi,sus)φ = strength reduction factorγ = ratio of sustained to full service load (Msus/MD+L)η = 1 – Icr /Igλ = long term (deflection) multiplier ξ/(1 + 50ρ′)ρ = reinforcing ratio (As/bd)ρ′ = compression reinforcing ratio (As′ /bd)ρb = balanced reinforcing ratio (bar strain of fy /Es at nominal

strength)ρmax = maximum reinforcing ratio (bar strain of 0.004 at nominal

strength)ρw = web reinforcing ratio (As/bwd)ξ = time-dependent factorΩ = 1 + γ(λ – 1)(Ie,D+L/Ie,sus)

REFERENCESACI Committee 318, 2008, “Building Code Requirements for Structural

Concrete (ACI 318-08) and Commentary,” American Concrete Institute,Farmington Hills, MI, 465 pp.

ACI Committee 435, 1995, “Control of Deflection in Concrete Structures(ACI 435R-95),” American Concrete Institute, Farmington Hills, MI, 88 pp.

ACI Committee 435 (Subcommittee 1), 1968, “Allowable Deflections,”ACI JOURNAL, Proceedings V. 65, No. 8, Aug., pp. 433-444.

Bischoff, P. H., 2005, “Re-Evaluation of Deflection Prediction forConcrete Beams Reinforced with Steel and Fiber-Reinforced PolymerBars,” Journal of Structural Engineering, ASCE, V. 131, No. 5, pp. 752-767.

Bischoff, P. H., 2007, “Rational Model for Calculating Deflection ofReinforced Concrete Beams and Slabs,” Canadian Journal of CivilEngineering, V. 34, No. 8, pp. 992-1002.

Bischoff, P. H., and Scanlon, A., 2007, “Effective Moment of Inertia forCalculating Deflections of Concrete Members Containing Steel Reinforcementand Fiber-Reinforced Polymer Reinforcement,” ACI Structural Journal,V. 104, No. 1, Jan.-Feb., pp. 68-75.

Branson, D. E., 1965, “Instantaneous and Time-Dependent Deflectionsof Simple and Continuous Reinforced Concrete Beams,” HPR Report No. 7,Part 1, Alabama Highway Department, Bureau of Public Roads, Alabama,78 pp.

CSA A23.3, 2004, “Design of Concrete Structures,” CSA StandardA23.3-04, Canadian Standards Association (CSA), Rexdale (Toronto), ON,Canada, 214 pp.

Gilbert, R. I., 1985, “Deflection Control of Slabs Using Allowable Span toDepth Ratios,” ACI JOURNAL, Proceedings V. 82, No. 1, Jan.-Feb., pp. 67-72.

Rangan, B.V., 1982, “Control of Beam Deflections by AllowableSpan-Depth Ratios,” ACI JOURNAL, Proceedings V. 79, No. 5, Sept.-Oct., pp. 372-377.

Scanlon, A., and Bischoff, P. H., 2008, “Shrinkage Restraint andLoading History Effects on Deflection of Flexural Members,” ACI Structural

nρ( )2 2nρ+

ACI Structural Journal/September-October 2009626

Journal, V. 105, No. 4, July-Aug., pp. 498-506.Scanlon, A., and Choi, B.-S., 1999, “Evaluation of ACI 318 Minimum

Thickness Requirements for One-Way Slabs,” ACI Structural Journal,V. 96, No. 4, July-Aug., pp. 616-621.

Scanlon, A., and Lee, Y. H., 2006, “Unified Span-to-Depth Ratio Equationfor Nonprestressed Concrete Beams and Slabs,” ACI Structural Journal, V. 103,No. 1, Jan.-Feb., pp. 142-148.

Tang, J., and Lubell, A. S., 2008, “Influence of Longitudinal ReinforcementStrength on One-Way Slab Deflection,” Canadian Journal of CivilEngineering, V. 35, No. 10, pp. 1076-1087.

APPENDIX: DERIVATION OF SPAN-DEPTH LIMITCombining Eq. (1) and (2) gives a general expression for

incremental deflection.

(A-1)

(A-2)

(A-3)

with Δi,L(add) = Δi,D+L – Δi,D+L(sus) and Δall = l/240 or l/480.Equation (A-3) accounts for the case where there is nopreloading from construction loads (Ie,sus > Ie,D+L).

A dead-to-live load (D/L) ratio FD/L = MD/ML andsustained live load ratio FS = ML,sus/ML are used to definethe sustained moment Msus (arising from the dead plussustained part of the live load) in terms of the full serviceload moment MD+L.

Msus = MD + ML,sus = γMD+L (A-4)

with γ = Msus/MD+L = (FD/L + FS)/(1 + FD/L).Substitution of Eq. (A-2) to (A-4) into (A-1) then gives

(A-5)

with Ω = [1 + γ(λ – 1)(Ie,D+L/Ie,sus)]. ΩMD+L = λMsus +ML,add when Ie,sus = Ie,D+L. Equation (A-5) also gives theimmediate deflection from live load for the case where λ = 0and the sustained load equals the dead load alone (Fs = 0giving γ = FD/L/(1 + FD/L)).

Rearranging Eq. (A-5) in terms of Ie,D+L/Ig, withMD+L = (φ/αD+L)Mn and Mn = Rnbd2 leads to

(A-6)

in which φ is the strength reduction factor for flexure andαD+L is an averaged load factor equal to (1.2FD/L + 1.6)/(FD/L + 1) for a dead load factor of 1.2 and live load factorof 1.6. Rearranging Eq. (A-6) in terms of the span-depthratio (l/h) gives

(A-7)

Δincr λΔi sus, Δi L add( ), Δall≤+=

Δi sus, K 548------

Msusl2

EcIe sus,

------------------=

Δi L add( ), K 548------

MD L+ l2

EcIe D L+,

---------------------- K 548------

Msusl2

EcIe sus,

------------------–=

Δincr K 548------

ΩMD L+

Ie D L+,

-------------------- l2

Ec

----- Δall≤=

Δincr K 548------

Ω φ αD L+⁄( )Rnl2

Ecd Ie D L+, Ig⁄( ) Ig bd3⁄( )------------------------------------------------------------ Δall≤=

l

h---

Ec d h⁄( ) Ie D L+, Ig⁄( ) Ig bd3⁄( )K 5 48⁄( )Ω φ αD L+⁄( )Rn

------------------------------------------------------------------------Δall

l--------≤

Technology

Span depth ratios for one-way members based on