ACI Structural Journal/July-August 2011 453

ACI Structural Journal, V. 108, No. 4, July-August 2011.MS No. S-2009-389.R1 received May 3, 2010, and reviewed under Institute

publication policies. Copyright © 2011, American Concrete Institute. All rights reserved,including the making of copies unless permission is obtained from the copyright proprietors.Pertinent discussion including author’s closure, if any, will be published in the May-June2012 ACI Structural Journal if the discussion is received by January 1, 2012.

ACI STRUCTURAL JOURNAL TECHNICAL PAPER

Predicting the deflection serviceability of reinforced concretemembers is fraught with uncertainties, which include imperfectknowledge of the limiting serviceability criteria, the materialproperties, and the load history including construction loads andthe service load. The serviceability criteria can be immediatedeflection/curvature or incremental deflection/curvature. Mostcodes offer two methods for control of deflections. The designermay choose to calculate the deflections and check that thesecomputed deflections are less than specified allowable limits.Alternatively, the codes give specified maximum span-depth ratiosfor which serviceability can be assumed to be satisfied and deflectionsdo not need to be calculated. This paper compares the deemed-to-comply span/thickness limits of ACI 318-08, CSA A23.3-04, BS8110-97, AS 3600-2009, Eurocode 2 (2004), ACI Committee 435revisions, and the proposals of numerous other authors.

Keywords: code provisions; deflection; serviceability.

INTRODUCTIONThe object of structural design is to achieve acceptable

probabilities that structures will perform satisfactorilyduring their intended service life. For safety, the structuremust have adequate strength with a low probability ofcollapse. The required probability against collapse isachieved by increasing the specified loads by appropriateload factors and reducing the member strengths by strengthor behavior reduction factors. Although safety is the mostimportant limit state, it is not sufficient without satisfying therequirements of serviceability. Service load deflections/curvatures may be excessive, or long-term deflections/curvatures due to sustained loads may cause damage topartitions, visual discomfort, and/or perception. With theincreasing use of higher strength concretes and reinforcingsteels, as well as more efficient design procedures, there is atendency toward designing shallower section members inreinforced concrete structures with attendant reductions instiffness and, hence, larger deflections. The recent (2005)reductions in the ACI load factors have decreased membersizes, increasing the service load/design ultimate load ratioand the possibility of deflection serviceability problems.

Most codes offer two methods for control of deflections.The designer may choose to calculate the deflections andcheck that these computed deflections are less than specifiedallowable limits. Calculating the immediate deflections ofreinforced concrete members is difficult due to the concretecracking in the tension zones due to early-age constructionloads or being under service load. Calculating the additionaldeflections due to shrinkage, creep, and the consequentredistribution of stress is extremely difficult. Alternatively,the codes give specified maximum span-depth ratios forwhich serviceability can be assumed to be satisfied anddeflections do not need to be calculated.

RESEARCH SIGNIFICANCECodes give specified maximum span/thickness or span/

effective depth ratios for which serviceability can beassumed to be satisfied and deflections do not need to becalculated. The use of higher strength reinforcing steel, moreefficient calculation methods, faster construction schedules, andchanges in load factors increase the possibility of deflectionserviceability problems and warrant a review of currentcode provisions.

CODE REQUIREMENTSFOR DEFLECTION CONTROL

ACI 318-08—Building Code Requirements for Reinforced Concrete; CSA A23.3-04—Design of Concrete Structure for Buildings

The American Code, ACI 318-08,1 and the CanadianCode, CSA A23.3-04,2 are the commonly used design codesfor reinforced concrete structures in North America. Forbeams, their provisions are effectively identical to those inACI 318-71. The deflection limits are given in Table 1. Theminimum thicknesses of beams and one-way slabs notsupporting, or attached to, partitions likely to be damaged bylarge deflections required by both codes are reproduced inTable 2. No guidance is given for beams and slabssupporting or attached to partitions likely to be damaged bydeflections. Table 3 is an extended version of Table 2recommended by ACI Committee 435,3 which distinguishesbetween members that support, or are attached to, nonstructuralelements likely to be damaged by large deflections and thosethat do not. Grossman4 noted that the minimum member

Title no. 108-S43

Span/Thickness Limits for Deflection Controlby Noel J. Gardner

Table 1—Maximum permissible computed deflection (ACI 318-08 and CSA A23.3-04)

Type of memberDeflection

to be consideredDeflectionlimitation

Flat roofs not supporting or attached to nonstructural elements likely to be damaged by large deflections

Immediate deflection due to live load L

ln/180

Floors not supporting or attached to nonstructural elements likely to be damaged by large deflections

Immediate deflection due to live load L

ln/360

Roof or floor construction supporting or attached to nonstructural elements likely to be damaged by large deflections

That part of the total deflection occurring after attachment of nonstructural elements (sum of the long-term deflection due to all sustained loads and the immediate deflection due to any additional live load)

ln/480

Roof or floor construction supporting or attached to nonstructural elements not likely to be damaged by large deflections

ln/240

Note: ln = clear span.

ACI Structural Journal/July-August 2011454

depths provided in Table 2 (ACI 318-08 Table 9-5(a)), toeliminate the need to calculate deflections, do not correlatewith the requirements of Table 1 (ACI 318-08 Table 9-5(b))of the Code. It can be noted that Table 2 does not takeaccount of several parameters that play important roles in thelong-term behavior of reinforced concrete members, that is,the effect of compression steel. Consideration should also betaken for the effect of concrete compressive strength and themagnitude of the service load relative to the ultimate load (aproxy to the extent of tension cracking of the concrete).

For slabs, the provisions of the two codes are slightlydifferent. ACI 318-08 requirements for slabs without interiorbeams or slabs with beams spanning between supports on allfour sides with αfm < 0.2, the minimum thickness is given inTable 4. For slabs with beams spanning between the supportson all sides, the minimum thickness is

for 0.2 < α < 2.0 (1)hmin ln0.8 fy/A+( )

36 5β αfm 0.2–( )+----------------------------------------------≥

for α > 2.0 (2)

where A = 1400 (SI units); A = 200,000 (U.S. customaryunits); hmin is the slab thickness; ln is the longer clear span;fy is the yield strength of tensile flexural reinforcement (MPafor SI and psi for U.S. units); α is the ratio of flexural stiffnessof beam to flexural stiffness of slab; αfm is the average valueof α; and β is the ratio of long side to short side.

At discontinuous edges, an edge beam with a stiffness αf> 0.8 must be provided or thickness of the panel with adiscontinuous edge must be increased by 10%.

For flat slabs with drop panels, meeting code-specifiedminimum thickness and dimensions, the slab thicknessbeyond the drop panel may be reduced by 10%.

CSA A23.3-042 adopted the more conservative provisionsproposed by Thompson and Scanlon5 (for flat slabs withoutedge beams, use αm = 0).

(3)

At discontinuous edges, an edge beam with a stiffnessratio αf > 0.8 must be provided or thickness of the panel witha discontinuous edge must be increased by 10%.

For slabs with drop panels, the minimum thickness isgiven by Eq. (4), where hs is the slab thickness, hd is the totaldepth of the drop panel, and xd is the distance from the faceof the column to the edge of the drop panel.

(4)

where B = 1000 (SI units); B = 145,000 (U.S. customaryunits); and fy is yield strength of tensile flexural reinforcement(MPa for SI and psi for U.S. units).

hmin ln0.8 fy/A+( )36 9β+

----------------------------≥

hmin ln0.6 fy/B+( )30 4βαm+----------------------------≥

hmin ln0.6 fy/B+( )

30----------------------------

2xd

ln

-------- hd hs–( )–≥

Noel J. Gardner, FACI, is a member of ACI Committees 209, Creep and Shrinkage ofConcrete; 231, Properties of Concrete at Early Ages; 347, Formwork for Concrete; and435, Deflection of Concrete Building Structures. His research interests are early-agemember behavior, shrinkage, creep, deflection serviceability, and formwork pressures.

Table 2—Minimum thickness of non-prestressed beams and one-way slabs unless deflections are computed (ACI 318-08 and CSA A23.3-04)

Simplysupported

One end continuous

Both ends continuous Cantilever

MemberMembers not supporting or attached to partitions

or other construction likely to be damaged by large deflection

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

Beams or ribbed one-way slabs

ln/16 ln/18.5 ln /21 ln/8

Note: For fy other than 60,000 psi (414 MPa), the values shall be multiplied by 0.4 +fy/100,000 psi units (0.4 + fy/690 SI units).

Table 3—Minimum thickness of beams and one-way slabs used in roof and floor construction(ACI Committee 435 1978)

MemberMembers not supporting, or not attached to, nonstructural

elements likely to be damaged by large deflectionsMembers supporting, or attached to, nonstructural elements

likely to be damaged by large deflections

Simplysupported

One endcontinuous

Both endscontinuous Cantilever

Simplysupported

One endcontinuous

Both endscontinuous Cantilever

Roof slab ln/22 ln/28 ln/35 ln/9 ln/14 ln/18 ln/22 ln/5.5

Floor slab and roof beam or ribbed roof slab

ln/18 ln/23 ln/28 ln/7 ln/12 ln/15 ln/19 ln/5

Floor beam or ribbed floor slab ln/14 ln/18 ln/21 ln/5.5 ln/10 ln/13 ln/16 ln/4

Table 4—Minimum thickness of slabs without interior beams unless deflections are computed(ACI 318-08)

Without drop panels With drop panels

Exterior panels Interior panels Exterior panels Interior panels

fy, MPa fy, psi Without edge beams With edge beams* Without edge beams With edge beams*

280 40,000 ln/33 ln/36 ln/36 ln/36 ln/40 ln/40

420 60,000 ln/30 ln/33 ln/33 ln/33 ln/36 ln/36

520 75,000 ln/28 ln/31 ln/31 ln/31 ln/34 ln/34

*Slabs with beams along exterior edges. The value of αf for the edge beam shall not be less than 0.8.

ACI Structural Journal/July-August 2011 455

At discontinuous edges, an edge beam with a stiffnessratio αf > 0.8 must be provided or thickness of the panel witha discontinuous edge must be increased by 10%.

The span/thickness provisions of ACI 318-08 and CSAA23.3-04 do not address the sensitivity of slab deflections toearly-age construction loads, rate of construction, orconcrete strength.

BS 8110-19976—Code of Practice for Designand Construction of Concrete Structures

The provisions of British Standard BS 8110-97,6 thecurrent evolution of the British Code of Practice CP 110-72and BS 8110-85, were based on the work of Beeby.7

Between the 1985 standard and the 1997 standard, the steelmaterial partial safety factor γm changed from 1.15 to 1.05.The span/effective depth requirements for rectangular orflanged beams are based on limiting the total deflection tospan/250. These span/effective depth ratios should normallyensure that the part of the deflection occurring afterconstruction of finishes and partitions will be limited tospan/350 or 20 mm (0.8 in.), whichever is less, for spans upto 10 m (34 ft). The basic ratios are given in Table 5.

The basic ratios are modified according to the ratios oftension and compression reinforcement provided and theservice load steel stress at the center of the span (or at thesupport in the case of a cantilever). These factors are listedin Tables 6 and 7. The span/effective depth ratios takeaccount of normal shrinkage (< 750 × 10–6) and normalcreep (creep coefficient < 3).

Tables 5 and 6 can also be used for slabs using the reinforce-ment ratio at midspan. The reinforcement ratio for a two-wayslab supported by walls or stiff beams should be based on theshorter span and the reinforcement ratio in that direction andthe longer span for flat slabs.

Eurocode 2-048

Eurocode 28 requires the calculated deflection of a beamor slab subjected to quasi-permanent loads should not exceedspan/250. A deflection (incremental) limit after constructionof span/500 is normally considered an appropriate limit toavoid deflections that could damage adjacent parts of thestructure. The limiting span/depth may be estimated usingEq. (5a) and (5b) modified by factors for boundary conditionsand type of reinforcement.

if ρ < ρ0 (5a)

if ρ > ρ0 (5b)

where A = 1 MPa units (145 psi units), fck28 is the 28-daycharacteristic concrete strength, l/d is the limiting span/

ld--- K 11 1.5 fck/A

ρ0

ρ----- 3.2 fck/A

ρ0

ρ----- 1–⎝ ⎠⎛ ⎞

3/2

+ +≤

ld--- K 11 1.5 fck/A

ρ0

ρ ρ′–-------------- 1

12------ fck/A ρ′

ρ0

-----+ +≤

effective depth, K is the structural system factor (Table 8), ρ isthe midspan tensile steel ratio, ρ′ is the midspan compressionsteel ratio, and ρo is the reference reinforcement ratio =0.001(fck)

1/2 (MPa units) [0.001(fck × 145)1/2 (psi units)].Equations (5a) and (5b) were derived assuming the midspan

steel stress at the serviceability limit state is 310 MPa (44,000 psi).For flanged sections where the ratio of flange breadth to webbreadth exceeds 3, the values should be multiplied by 0.8. Forbeams and slabs, other than flat slabs, with spans exceeding 7 m(23 ft), which support partitions liable to be damaged byexcessive deflections, the l/d values should be multiplied by 7/l(l in meters) or 23/l (l in feet). For flat slabs, with spansexceeding 8.5 m (28 ft), which support partitions liable to bedamaged by excessive deflections, the l/d values should bemultiplied by 8.5/l (l in meters) or 28/l (l in feet).

Table 8 gives the limiting span/effective depth ratios forbeams spanning up to 7 m (23 ft) and flat slabs spanning upto 8.5 m (28 ft) derived on the assumption that the steel stressat midspan is 310 N/mm2 (44 ksi) and the concretecharacteristic strength is 30 MPa (4.4 ksi). For two-wayslabs, the calculation should be based on the shorter span andon the longer span for flat slabs. The limits for flat slabscorrespond to a less severe limitation than a midspandeflection of span/250 relative to the columns.

AS 3600-2009 Australian Standard9—concrete structures

The serviceability requirements of the Australian StandardAS 3600-20099 limit the total deflection to span/250 and the

Table 5—Basic span/effective depth ratios for beams (Table 3.9, BS 8110-97)

Basic span/effective depth ratio

Support conditions Rectangular sectionsFlanged beams

bw/b < 0.3

Cantilever 7 5.6

Simply supported 20 16.0

Continuous 26 20.8

Table 6—Modification factor for tension reinforcement (Table 3.10, BS 8110-1997)Steel service

stress Nondimensional moment Mu /bd2, MPa (psi)

MPa ksi0.50 (72)

0.75 (109)

1.00 (145)

1.50 (218)

2.00 (290)

3.00 (435)

4.00 (580)

5.00 (725)

6.00 (870)

100 14.5 2.00 2.00 2.00 1.86 1.63 1.36 1.19 1.08 1.01

150 21.8 2.00 2.00 1.98 1.69 1.49 1.26 1.11 1.01 0.94

200 29.0 2.00 1.95 1.76 1.51 1.35 1.14 1.02 0.94 0.88

250 36.3 1.90 1.70 1.55 1.34 1.20 1.04 0.94 0.87 0.82

300 41.8 1.60 1.44 1.33 1.16 1.06 0.93 0.85 0.80 0.76

307 44.5 1.56 1.41 1.30 1.14 1.04 0.93 0.85 0.80 0.76

Table 7—Modification factor for compression reinforcement (Table 3.11, BS 8110-1997)Reinforcement ratio of compression reinforcement100As′ /bd

0.15 0.25 0.50 0.75 1.0 1.5 2.0 3.0 or greater

Factor to be applied 1.05 1.08 1.14 1.20 1.25 1.33 1.40 1.50

Table 8—Basic ratios of span/effective depthfor reinforced concrete members(Table 7.4N, Eurocode 2-04)

Structural system KSteel ratio =

1.5%Steel ratio =

0.5%

Simply supported beam or two-waysimply supported slab 1.0 14 20

End span of continuous beam or one-way continuous slab or two-way slab continuous over one long side

1.3 18 26

Interior span of continuous beam ortwo-way slab 1.5 20 30

Flat slab (based on longer span) 1.2 17 24

Cantilever 0.4 6 8

456 ACI Structural Journal/July-August 2011

incremental deflection to span/500 where a provision is madeto minimize the effect of movement; otherwise, span/1000.

Limiting, deemed-to-comply, span-depth ratios for beamscan be calculated from the following equation

(6)

where Δ/leff is the total or incremental deflection limit, beffis the effective width, D is the dead load, Ec is the modulusof elasticity of concrete, and Fd.ef is the effective designload/unit length.

a) (1.0 + kcs)D + (ψs + kcsψl)L for total deflectionb) kcsD + (ψs + kcsψl)L for incremental deflection

k1 = Ief /bef d3;k2 = deflection constant 5/384, 2.4/384, and 1.5/384 for

simply supported, one end continuous, and interiorspan, respectively;

kcs = [2 – 1.2As′ /As] > 0.8;leff = effective span;L = live load;ψl = 0.25 for offices and domestic occupancy (0.5 to 0.8

for storage); andψs = 0.5 for offices (1.0 for storage).

A similar equation is given for deem-to-comply span-depth ratios for one-way flat slabs and slabs supported onfour sides by walls or stiff beams.

(7)

where D is the dead load, Ec is the modulus of elasticity ofconcrete, and Fd.ef is the effective design load/unit area.

a) (1.0 + kcs)D + (ψs + kcsψl)L for total deflectionb) kcsD + (ψs + kcsψl)L for incremental deflection

kcs = [2 – 1.2As′ /As] > 0.8;k3 = 1.0 for a one-way slab;

= 0.95 for a two-way flat slab without drop panels;= 1.05 for a two-way flat slab with drop panels;

k4 = deflection constant 1.4 for simply supported slabs,1.75 for an end span, or 2.1 for an interior span;

L = live load; andleff = effective span.

For two-way slabs supported by walls or stiff beams, k3 =1.0 and k4 is given in a table as a function of boundary conditionand panel aspect ratio.

Gardner and Zhang10—beamsUsing a layered, nonlinear finite element program,

Gardner and Zhang10 determined the span thicknessrequirements to satisfy a specified deflection limit interms of specified, or characteristic, concrete strength;tension and compression steel ratios; and the ratio of thesustained moment to the moment capacity of the beam. Ahybrid method was used to calculate the long-term behaviorusing a reduced modulus to account for creep; a conventionaltime-dependent load vector was used for shrinkage. Thepositive reinforcement was reduced at the theoretical 50%cutoff point. Characteristic concrete strengths of 20, 30,and 40 MPa (2900, 4400, and 5800 psi) were considered. Totake advantage of the mean concrete strengths being largerthan the characteristic concrete strengths, the mean concrete

leff

d------

k1 Δ/Leff( )bef Ec

k2Fd .ef

-------------------------------------1/3

≤

leff

d------ k3k4

Δ/Leff( )1000Ec

Fd .ef

-------------------------------------1/3

≤

strengths were determined using fcm′ = fck′ + 8 MPa (fcm′ = fck′ +1160 psi), implying shrinkage strains and creep coefficients of700, 660, and 590 × 10–6; and 2.72, 2.51, and 2.37, respectively.

The span/thickness ratio requirements for simplysupported beams, satisfying the span/500 deflection criterionunder a service load/ultimate load ratio of 50%, are given inTable 9. For the same service moment/design ultimatemoment ratio, the span-depth ratios for deflection limitsother than span/500, the limiting span-depth ratio is simplymultiplied by 500/required span deflection ratio, that is,span-depth ratios for a span/250 deflection criterion can beobtained by doubling the values for the span/500 deflectioncriterion. The immediate deflection limit of span/375 wasnot found to be critical.

Span-depth ratios for continuous beams may be obtainedby multiplying the values for simply supported beams, usingthe positive moment steel ratio, by the following factors:

Support condition FactorSimply supported: 1.0One end continuous discontinuous end unrestrained: 1.2 discontinuous end integral with support: 1.3Both ends continuous: 1.4Cantilever: 0.35

The modifying factors were determined assuming curvatureis proportional to the moment coefficients given in ACI 318-08,Section 8.3.3, and CSA A23.3-04, Section 9.3.3.

The required span/thickness ratio for a specified deflectionlimit decreases with an increase in tensile steel ratio, anincrease in service moment/ultimate moment, and decreaseswith an increase in compression reinforcement and anincrease in concrete strength. Increasing the service moment,as a fraction of the beam section design ultimate moment,reduces the limiting span-depth ratio. As a first approximation,the limiting span-depth ratio is inversely proportional to thecube root of the ratio of the moment levels. Similarly, it canbe deduced that using a higher yield strength steel, whichwill increase the concrete stress for a given service moment/design ultimate moment ratio, will also result in smallerpermissible span-depth ratios.

Scanlon and Choi11—one-way slabsScanlon and Choi11 proposed the following equation

based on an incremental deflection limit.

(8)

where b is the width of beam; Ec is the modulus of elasticityfor concrete; WL(var) is the variable portion of live load; Wsis the sustained load; α is the ratio of Ieffective to Igross; λ isthe ACI 318 long-term deflection multiplier; and K is thedeflection coefficient = 5, 2, and 1.4 for simply supported,one end continuous, and both ends continuous, respectively.

This expression requires an iterative procedure to determinethe minimum thickness.

Bischoff and Scanlon12—beams and one-way slabsBischoff and Scanlon12 derived expressions to determine

limiting span/thickness ratios including the effects ofreinforcement ratio, shrinkage restraint, construction loads,sustained live load, support conditions, and deflection limits.

ln

h---

Δinc

ln

---------⎝ ⎠⎛ ⎞ 32Ecαb

K λWs WL var( )+( )-------------------------------------------

1/3≤

ACI Structural Journal/July-August 2011 457

For rectangular section members, the following expressionwas given. Bischoff13 has proposed an alternative formulationto determine Ie that can be used in Eq. (9).

(9)

where Ec is the modulus of elasticity for concrete; Ie,D+L isthe effective moment of inertia under full service load; Ie,susis the effective moment of inertia under sustained load; Ig isthe gross (uncracked) second moment of area; K is the endrestraint factor = 1, 0.85, and 0.8 for simply supported, oneend continuous, and both ends continuous, respectively; Mnis the nominal moment capacity; Rn is the nominal flexuralresistance factor Mn/bd2; αD+L is the average load factor;Δall is the permissible (allowable) deflection; γ is the ratio ofsustained load to full service load; λ is the ACI 318 long-term deflection multiplier; Ω = {1 + γ(λ – 1)(Ie,D+L/Ie,sus)];and φ = 0.9 strength reduction factor.

Results from a comparative study showed that lightlyreinforced slabs or beams satisfying the ACI minimumthickness requirement may not satisfy the l/240 incrementaldeflection requirement.

Thompson and Scanlon5—flat slabsThompson and Scanlon5 reported the results of a parametric

study of the effects of restraint cracking, concrete strength,design live load, construction load, and panel aspect ratio onthe deflections of flat slabs. Deflections were calculatedusing a plate-bending finite-element program with an effectivesecond moment of area to account for the reduced stiffnessdue to cracking. It was observed that the calculated deflectionswere sensitive to the assumed value of the modulus ofrupture. Thompson and Scanlon5 used serviceability criteriaof incremental deflection less than span/480 and total deflectionless than span/240. From their parametric study, Thompsonand Scanlon5 proposed a more conservative minimumthickness requirement, for both interior and edge panels, forthe control of deflections of two-way slabs. The live loaddeflection limit of span/360 was not found to be critical.

(10)

where k(β) = (1.20 – 0.20β) > 0.9 and β is the ratio of longerclear span to shorter clear span.

Thompson and Scanlon5 also recommended that theminimum slab thickness could be reduced by 10% for flatslabs with drop panels whose thickness is greater than orequal to 1.25 times the slab thickness, or by 20% if thedrop panel thickness is greater than 1.50 times the slabthickness. Thompson and Scanlon5 did not investigate theeffect of age at which the construction load was appliedon the calculated deflections.

Ofosu-Asamoah and Gardner14—flat slabsThe deflection serviceability of flat slabs is determined by

the loads imposed during the construction process (methodand rate of construction), taking account of the concretestrength available when the construction loads are imposed,and the expected sustained service load. The shore-reshoreprocedure used to construct many reinforced concrete flat slab

l

h---

0.8Ec Ie D L+, /Ig( )

KΩ φ/αD L+( ) d/h( )2Rn

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

l--------≤

hminln

30------k β( )≥

structures leads to the imposition of large early-age constructionloads,15 typically of the same order of magnitude as the serviceloads, on the partially cured supporting slabs. Consequently, itis necessary that both the construction load and design load betaken into account during the design phase of reinforcedconcrete floor slab construction. The appropriate serviceabilitycriterion will depend on the location of the critical location,midpanel, or column line, and can be immediate or incrementaldeflection, slope, or curvature.

Assuming one level of forms and three levels of reshores(1 + 3), the construction load is 1.25D (D is the self-weightof the slab). Assuming the formwork weighs 0.1D and noconstruction live load, the total unfactored construction loadis 1.35D.15 Slabs were loaded at 3, 4, 7, 14, and 28 days aftercasting. At 28 days, the construction load was removed,reducing the slab load to self-weight. Assuming the slab isput to service at 28 days, it is subjected to its own self-weightplus some fraction of the live load. The sustained load waschosen to be self-weight plus 50% of the live load plus asuperimposed dead load of 0.1D. A layered finite elementprogram was used to study the effects of age of imposition ofconstruction loading (age of supporting slab when successiveslab is cast), span, panel aspect ratio, live load to dead loadratio, and concrete strength on the deflection serviceabilityof flat slab systems.15 It was determined that the age ofloading (age superimposed slab cast-construction cycle) andspan have significant effects on the slab thickness required tosatisfy serviceability.

The following equation summarizes the slab thicknessesrequired to satisfy an exterior panel, interior column lineincremental (28 to 5000 days of sustained load) deflection ofclear span/240.

(11)

where fck28 is the 28-day characteristic concrete strength,MPa; fcm28 = fck28 + 8 = 28-day mean concrete strength,MPa; h is the slab thickness, m; and ln is the longer clearspan, m.

(A11)

where fc28′ is the 28-day specified concrete strength, psi;fcm28 = fc28′ + 1160 = 28-day mean concrete strength, psi; his the slab thickness, ft; ln is the longer clear span, ft; β is theratio of long clear span to short clear span; k1 = 0.9 for aninterior panel and 1.0 for edge and corner panels; k2 = 0.9 forslabs with drop panels; to is the age at which the constructionload is applied to the slab; L is live load; and D is dead load.

The use of the code-recommended minimum drop panelthickness of 1.25 times the slab thickness reduces the slabthickness required to satisfy the serviceability criterion byapproximately 18%; hence, the ACI 318-08 recommendationof a 10% reduction in slab thickness is conservative. Forinterior panels, the thickness given by Eq. (11) can bereduced by 10% as recommended by ACI 318-08.

DISCUSSIONThe methods of determining limiting span-depth ratios fall

into two main categories: modified elastic beamanalysis9,11,12 and parametric studies using finite element

hk1k2

53.4----------

ln1.5

to0.2

------ 38fcm28

----------0.6

1.4 1.7L/D+2.25

--------------------------------0.25

1.15 0.15β–( )≥

hk1k2

96.7----------

ln1.5

to0.2

------ 5500fcm28

------------0.6 1.4 1.7L/D+

2.25--------------------------------

0.25

1.15 0.15β–( )≥

458 ACI Structural Journal/July-August 2011

analysis.5,6,10,14 All beam analyses use an effective momentof inertia to approximate the extent of cracking. Finiteelement calculations can be done using an effective momentof inertia or using several layers through the thickness of themember. Long-term deflections can be done using a simple,combined shrinkage and creep multiplier on the calculatedimmediate deflection(s) or summing separate calculationsfor the deflections caused by load effects with the deflectiondue to shrinkage.

A single long-term deflection multiplier9,11,12 to takeaccount of both shrinkage and creep should include loadmagnitude and reinforcement ratio.

All results are consequences of the input data assumptions,namely, magnitude of service loads (assumed applied at 28 days),concrete strength, reinforcement ratio, and, for flat slabs, the ageof imposition and magnitude of the construction loads.

Obviously the deflections calculated for membersdesigned using thicknesses given by deemed-to-complyprovisions should satisfy the code-specified deflectionlimitations. While the characteristic live load—live load notexceeded 95% of the time—should be used for ultimate limitstate calculations, the experimental work of Choi16 indicatedthat an average live load of 50% of the extreme (assumedspecified/characteristic) live load would be reasonable forserviceability limit state calculations. The percentage isdependent on the load factors and the material/member under-strength factors used in the ultimate limit state calculations. BS8110-976 states that when calculating deflections, theportion of the live load to be considered permanent should be25 to 30% for office use but at least 75% for storage. AS3600-20099 suggests that for deflection calculations, thecharacteristic live load can be multiplied by 0.6 for offices(1.0 for storage) for immediate deflections and 0.25 for long-term deflections (0.5 to 0.8 for storage). The expected valueof the concrete strength, not the lower-bound characteristicconcrete strength, can be used in deflection calculations.

There appears to be agreement that incremental deflectionafter construction of partitions and finishes is more criticalthan immediate deflection.5,9,10 There is also general agree-ment that the limiting incremental deflections are span/500for brittle partitions; otherwise, span/250.

Table 9 illustrates the dependence of the limiting span/thickness ratio on sustained moment/moment capacity ratio,concrete strength, and flexural reinforcement.

Table 10 compares the deem-to-comply span-thicknessratios for simply-supported rectangular section beams for adeflection criterion of span/250. It is reassuring that all theratios are circa span/20. It must be noted that ACI, CSA, andGardner and Zhang10 use span/thickness but BS 8110, AS3600, and Eurocode 2 use span to effective depth, which arecorrected to span/thickness using h = 1.18d in the table. Spanto effective depth is appropriate for section strength calcula-tions but span to thickness is more appropriate for deflectionserviceability. The proposals of AS 3600-09, Gardner andZhang,10 and Bischoff and Scanlon12 formally accommo-date incremental deflection limits other than span/250. Theprovisions of ACI and CSA do not accommodate the effectof compression reinforcement. The modifying factors forboundary conditions other than simply supported, given inTable 11, are similar for all proposals.

Table 12 compares the limiting span/thickness ratios forthe interior panels of flat slabs. Only the provisions of Ofuso-Asamoah and Gardner14 take account of the constructioncycle–age of first/construction loading. Ofuso-Asamoah andGardner14 assumed a form-plus-three reshores construction

Table 9—Proposed span/thickness requirements to satisfy span/500 incremental deflection limit*

ρ, % ρ′, %

M = 30% Mu M = 50% Mu M = 70% Mu

fck = 30 MPa(4400 psi)

20 MPa(2900 psi)

30 MPa(4400 psi)

40 MPa(5800 psi)

20 MPa(2900 psi)

30 MPa(4400 psi)

40 MPa(5800 psi)

< 0.5 0 12.7 8.2 9.8 10.9 7.9 9.8 10.3

1.0 0 11.1 8.4 9.9 10.8 6.8 8.5 9.9

1.5 0 10.8 7.7 8.8 10.1 6.4 8.1 9.2

2.0 0 9.7 7.1 8.3 9.3 5.9 7.4 8.3

1.5 0.5 13.9 11.0 11.6 12.7 8.8 10.7 11.5

2.0 0.5 12.6 9.9 10.8 11.6 8.0 9.7 10.3

2.5 0.5 11.8 9.2 10.2 10.8 7.7 8.7 9.5

2.0 1.0 15.0 12.6 13.3 14.2 10.2 11.9 12.7

2.5 1.0 14.2 11.5 12.4 12.6 9.9 10.8 11.3

3.0 1.0 13.7 11.0 11.4 12.3 9.2 10.1 10.6

2.5 1.5 16.6 14.0 15.0 14.6 12.3 13.2 13.5

3.0 1.5 14.7 13.3 13.7 14.0 11.2 12.2 13.0*For deflection limit of span/250, multiply values by 2.

Table 10—Comparison of simple span beam span/thickness ratios: incremental deflection < span/250

ρ, % ρ′, %ACI

318-08*CSA

A23.3†BS

8110-97‡§Eurocode

2||Gardner

and Zhang#

< 0.5 0 16 16 21.2 17 19.6

1 0 16 16 16.9 13.2 19.8

1.5 0 16 16 14.8 11.9 17.6

2 0 16 16 13.5 11.3 16.6

1.5 0.5 16 16 16.9 13.5 23.2

2.5 0.5 16 16 14.3 11.6 20.4

2.5 1 16 16 15.6 12.4 24.8*Steel yield stress 60,000 psi (414 MPa).†Steel yield stress 400 MPa (58,000 psi).‡Calculated assuming steel service load stress 250 MPa (36 ksi).§Code provision written as span/effective depth-span/thickness calculated using deff =0.85h.||Calculated assuming steel service load stress 310 MPa (45 ksi).#For Msustained = 50% moment capacity and fck = 30 MPa (4350 psi) (from Table 9).

ACI Structural Journal/July-August 2011 459

sequence. The provisions of AS 3600-099 and Ofuso-Asamoah and Gardner14 (3-day construction cycle) are moreconservative than the other proposals.

All the provisions except ACI 318 and CSA A23.3 requireiteration to determine the limiting deem-to-satisfy span-depth ratio either by calculating/assuming steel ratio or themember self-weights. Tables 8, 9, 10, and 12, however, canbe used as design aids.

RECOMMENDATIONSThe live loads for which deflections should be calculated

should be clearly specified in the codes, taking note of thedifference between expected live load and extreme orcharacteristic live load. For purposes of calculating theincremental deflections, it is suggested that the service loadbe calculated from the equation that follows, which is acompromise between the provisions of BS 8110-97 andAS 3600-2009.

service load = D + αL

where α = 0.4 for offices, apartments, etc.; and 0.8 for storage.

For beams and one-way slabs, the deemed-to-complyminimum thicknesses given in Table 9 for incrementaldeflection limit of span/500 can be adopted. For incrementaldeflection limits other than span/500, at the same serviceload moment/nominal section design ultimate moment ratio,the limiting span-depth ratio is simply multiplied by 500/required span-deflection ratio. As a first approximation, thelimiting span-depth ratio is inversely proportional to thecube root of the ratio of the moment levels. Similarly it canbe deduced that using a higher-yield-strength steel, whichwill increase the concrete stress for a given service moment/design ultimate moment ratio, will also result in smallerpermissible span-depth ratios. For other than simple spans,the modification factors suggested by Gardner and Zhang10

(Table 11) should be used. For flanged sections where theratio of flange breadth to web breadth exceeds 3, the valuesshould be multiplied by 0.8.

The deflection serviceability of flat slabs is determined bythe loads imposed during the construction process (methodand rate of construction), taking account of the concretestrength available when the construction loads are imposed,and the expected sustained service load. The appropriateserviceability criterion will depend on the location of the critical

Table 11—Span/thickness factors for other than simple beams*

Maximum positive moment ACI-CSA BS 8110 AS 3600 Eurocode Gardner and Zhang

One end continuous—discontinuous end unrestrained 1.2 — 1.3 1.3 1.2

One end continuous—other end integral with support 1.2 — 1.3 1.3 1.3

Both ends continuous 1.4 1.3 1.5 1.5 1.4

Cantilever 0.5 0.35 0.4 0.4 0.35

*Use midspan positive moment steel ratio from Table 9.

wln2

11--------

wln2

14--------

wln2

16--------

wln2

2--------

Table 12—Comparison flat slab interior panel span/thickness ratios without drop panels

Span, m Span, ft ACI 318-08 CSA A23.3-04 BS 8110-97 AS 3600-09 Eurocode* O-A&G† 7-day O-A&G† 3-day

Live load: 50 lb/ft2 (2.4 kPa)

6.00 20 33 30 34‡ 26.7‡ 37‡ 36.0 31.3

7.00 23 33 30 34§ 25.6§ 37§ 33.9 28.9

8.50 28 33 30 34|| 24.4|| 37|| 31.5 26.8

Live load: 100 lb/ft2 (4.8 kPa)

6.00 20 33 30 33.8‡ 25.3‡ 37‡ 33.8 28.9

7.00 23 33 30 33.5§ 24.4§ 37§ 31.9 27.3

8.50 28 33 30 33.1|| 23.7|| 37|| 29.7 25.4

Edge panel thickness/interior panel thickness

10% +10% +20% +10% +10%

Drop panels—reduce slab thickness by 10%*ρ assumed to be ρ0/2 (minimum positive moment steel).†Ofosu-Asamoah and Gardner14: fc′ = 4350 psi (fck = 30 MPa).‡h assumed to be clear cover 20 mm + 10 mm bar diameter.§h assumed to be clear cover 20 mm + 15 mm bar diameter.||h assumed to be clear cover 20 mm + 20 mm bar diameter.

ACI Structural Journal/July-August 2011460

location, midpanel, or column line, and can be immediate orincremental deflection, slope, or curvature. The slab thick-nesses required to satisfy an exterior panel, interior columnline incremental (28 to 5000 days sustained load) deflectionof clear span/240 can be calculated using Eq. (11). The interiorpanels’ slab thicknesses should be calculated as 90% of theexterior panel thicknesses.

REFERENCES1. ACI Committee 318, “Building Code Requirements for Structural

Concrete (ACI 318-08) and Commentary,” American Concrete Institute,Farmington Hills, MI, 2008, 473 pp.

2. CSA A23.3-04, “Design of Concrete Structures for Buildings,” CanadianStandards Association, Rexdale, ON, Canada, 2004, 358 pp.

3. ACI Committee 435, “Proposed Revisions by Committee 435 to ACIBuilding Code and Commentary Provisions on Deflections,” ACI JOURNAL,Proceedings V. 75, No. 6, June 1978, pp. 229-238.

4. Grossman, J.S. “Simplified Computations for Effective Moment ofInertia Ie and Minimum Thickness to Avoid Deflection Computations,”

ACI JOURNAL, Proceedings V. 78, No. 6, Nov.-Dec. 1981, pp. 423-439.5. Thompson, D. P., and Scanlon, A., “Minimum Thickness Requirements

for Control of Two-Way Slab Deflections,” ACI Structural Journal, V. 85,No. 1, Jan.-Feb. 1988, pp. 13-22.

6. BS 8110-1997, “Structural Use of Concrete, Part 1: Code of Practicefor Design and Construction,” British Standards Institute, London, UK,1997, 117 pp.

7. Beeby, A. W., “Modified Proposals for Controlling Deflections byMeans of Ratios of Span to Effective Depth,” Technical Report 456(Publication 42.456), Cement and Concrete Association, UK, 1971, 19 pp.

8. EC 2-1-1 (2004), “Eurocode 2: Design of Concrete Structures–Part 1.1: General Rules and Rules for Buildings,” Management Centre,rue de Stassart, 36 B-1050, Brussels, EN 1992-1-1, 2004, 225 pp.

9. Australian Standard AS 3600-2009, “Concrete Structures,” StandardsAssociation of Australia, North Sydney, Dec. 2009, 208 pp.

10. Gardner, N. J., and Zhang, J., “Controlling Deflection Serviceabilityby Span/Depth Limits and Long-Term Deflection Multipliers for ReinforcedConcrete Beams,” Recent Developments in Deflection Evaluation ofConcrete, SP-161, E. G. Nawy, ed., American Concrete Institute, FarmingtonHills, MI, 1996, pp. 165-195.

11. Scanlon, A., and Choi, B.-S., “Evaluation of ACI 318 MinimumThickness Requirements for One-Way Slabs,” ACI Structural Journal,V. 96, No. 4, July-Aug. 1999, pp. 616-621.

12. Bischoff, P. H., and Scanlon, A., “Span-Depth Ratios for One-WayMembers Based on ACI 318 Deflection Limits,” ACI StructuralJournal, V. 106, No. 5, Sept.-Oct. 2009, pp. 617-626.

13. Bischoff P. H., “Re-evaluation of Deflection Prediction for ConcreteBeams Reinforced with Steel and Fiber-Reinforced Polymer Bars,” Journalof Structural Engineering, ASCE, V. 131, No. 5, pp. 752-767.

14. Ofosu-Asamoah, K., and Gardner, N. J., “Flat Slab ThicknessRequired to Satisfy Serviceability Including Early-Age ConstructionLoads,” ACI Structural Journal, V. 94, No. 6, Nov.-Dec. 1997, pp. 700-707.

15. Agarwal, R. K., and Gardner, N. J., “Form and Shore Requirementsfor Multi-Floor Flat Slab Buildings,” ACI JOURNAL, Proceedings V. 71, No. 11,Nov. 1974, pp. 559-569.

16. Choi, E. C. C., “Live Load in Office Buildings—Lifetime Maximumand Influence of Room Use,” Proceedings of the Institution of CivilEngineers, V. 94, Issue 3, Aug. 1992, pp. 307-314.