Rational model for calculating deflection ofreinforced concrete beams and slabs

Peter H. Bischoff

Abstract: Deflection control is an important performance criterion that needs to be satisfied to ensure serviceability ofthe structure for its intended use. The extent of cracking and amount of reinforcement affects the flexural rigidity, EI,of a reinforced concrete member and both the Canadian concrete design standard (CSA A23.3-04) and ACI BuildingCode (ACI 318-05) use an effective moment of inertia, Ie, that was originally proposed by Branson to compute beamdeflection. This is an empirically derived equation that works well within a narrow range of limits corresponding tosteel-reinforced concrete beams with a reinforcing ratio between 1% and 2%. However, the equation underestimatesdeflection for steel-reinforced concrete beams and slabs with a reinforcing ratio less than 1% and for most beams rein-forced with low-modulus, fibre-reinforced-polymer (FRP) bars. Deflection of slender tilt-up wall panels can also beunderestimated with Branson’s equation. This paper provides an explanation of why the Branson equation does not alwayswork well in predicting deflection, and presents a rational approach to develop an alternative expression for the effectivemoment of inertia that works equally well for both steel- and FRP-reinforced concrete at all reinforcing ratios. A rationalexpression is also introduced for continuous beams that uses an averaged moment of inertia, Ie,avg, to calculate beamdeflection. Changes are included in a proposed revision to deflection prediction requirements specified in clause 9.8 ofCSA A23.3-04.

Key words: reinforced concrete, deflection, effective moment of inertia, serviceability.

Résumé : Le contrôle des flèches est un critère de performance important qui doit être rencontré pour assurer un com-portement en service des structures en fonction de l’usage planifié. Le niveau de fissuration et la quantité d’armatureaffectent la rigidité en flexion EI d’un élément en béton armé. La norme canadienne de Calcul des ouvrages en béton(CSA A23.3-04) et l’« ACI Building Code » (ACI 318-05) utilisent un moment d’inertie efficace Ie , proposé initialementpar Branson, pour calculer la flèche des poutres. Il s’agit d’une équation dérivée empiriquement qui fonctionne biendans une plage étroite correspondant aux poutres de béton armé ayant un pourcentage d’armature en acier entre 1 et 2 %.Toutefois, l’équation sous-estime les flèches pour les poutres et les dalles en béton armé ayant un pourcentaged’armature en acier inférieur à 1 % ainsi que pour la plupart des poutres en béton armé avec des tiges renforcées defibres de polymères — PRF — à faible module. La flèche des panneaux muraux minces mis en place par relèvementpeut être également sous-estimée par l’équation de Branson. Le présent article expliqure pourquoi l’équation de Bransonne prédit pas toujours adéquatement la flèche et présente une approche rationnelle qui permet de développer une autreexpression pour le calcul du moment d’inertie efficace qui fonctionne aussi bien pour le béton armé d’armatures enacier ou en PRF, et ce, pour tous les pourcentages d’armature. Une expression rationnelle est également présentée pourles poutres continues pour lesquelles on utilise un moment d’inertie moyen, « Ie, avg », pour calculer la flèche des poutres.Des suggestions pour réviser les exigences du paragraphe 9.8 de la norme CSA A23.3-04 pour la prédiction des flèchessont inclus dans l’article.

Mots-clés : béton armé, flèche, moment d’inertie efficace, comportement en service.

[Traduit par la Rédaction] Bischoff 1002

Introduction

Reinforced concrete structures are typically designed forstrength using factored loads to ensure that the structure issafe from collapse. Member deflection and crack widths aresubsequently checked to ensure that the structure remains

serviceable under the specified loading conditions. Althoughdeflection was often less of a problem when design was con-servatively based on limiting stresses in the concrete andsteel at service loads, the adoption of strength design anduse of higher-strength materials has led to more slendermembers that have a greater potential for experiencing de-flection problems. Use of advanced composite materials forinternal reinforcement can also lead to more flexible mem-bers because of their lower stiffness. Hence, serviceabilityfailures related to excessive deflection are more common thanone might expect (Gilbert 2001) and can lead to costly liti-gation and repairs.

Deflection problems are avoided by either limiting thespan to depth ratio of the member (to give a minimum mem-ber thickness) or ensuring that calculated deflection values

Can. J. Civ. Eng. 34: 992–1002 (2007) doi:10.1139/L07-020 © 2007 NRC Canada

992

Received 21 June 2006. Revision accepted 4 February 2007.Published on the NRC Research Press Web site at cjce.nrc.caon 7 September 2007.

P. Bischoff. Department of Civil Engineering, University ofNew Brunswick, PO Box 4400, Fredericton, NB E3B 5A3,Canada (e-mail: bischoff@unb.ca).

Written discussion of this article is welcomed and will bereceived by the Editor until 31 December 2007.

do not exceed specified limits that are often based on maxi-mum deflection to span ratios found in most codes. Mini-mum member thickness values are intended for members notsupporting or attached to partitions or other constructionlikely to be damaged by large deflections and do not applywhen the partitions are susceptible to cracking. More oftenthan not, minimum thickness values are used regardless ofwhether damage to the partition is likely (Scanlon and Choi1999). For this reason, it has been suggested that deflectionvalues always be computed and minimum thickness valuesbe used as a guideline only for preliminary selection ofmember size. Whereas calculation of deflections may appearto be slightly onerous, it should be remembered that thisprocedure is typically included in most software packagesused for design of reinforced concrete structures, many ofwhich use an effective moment of inertia to compute deflec-tion (Scanlon et al. 2001).

Both the Canadian concrete design standard CSA A23.3-04(CSA 2004) and ACI Building Code ACI 318-05 (ACICommittee 318 2005) use an effective moment of inertia, Ie,that was originally proposed by Branson (1965, 1977) to cal-culate beam deflection. Branson’s approach was quite inno-vative at the time; his expression for Ie (eq. [1]) represents agradual transition from the uncracked flexural stiffness, EcIg,to a fully cracked stiffness, EcIcr, as loading on the memberincreases beyond the cracking point (see Fig. 1a).

[1] I I I IM

Me cr g cr

cr

a

� � ��

���

�

�( )

3

��

���

�

� � �

�

���

�

�

�

��

�

��

�M

MI

M

MI Icr

ag

cr

acr g

3 3

1

In eq. [1], Ig represents the gross (uncracked) moment ofinertia and Icr is the cracked transformed moment of inertia.Deflection is calculated for the applied service load momentMa at the critical section and the cracking moment is givenby Mcr. The cubic term in this equation, (M Mcr a/ )3, accountsfor both tension stiffening and the variation of member stiff-ness along the beam length.

Branson’s equation is an empirically derived relationshipthat was calibrated for steel-reinforced concrete beams witha reinforcing ratio between 1% and 2%. This corresponds tobeams with an uncracked to cracked stiffness (EcIg/EcIcr =I Ig cr/ ) ratio between 2 and 3 (Bischoff 2005) and theequation has appeared to work well in the past because mostbeams fall within this narrow range of limits. However,Branson’s approach consistently underestimates short-termdeflection of lightly reinforced members (Scanlon et al.2001; Gilbert 2006) and members reinforced withlow-modulus, fibre-reinforced-polymer (FRP) bars (ACICommittee 440 2006). Problems also arise for slender tilt-upwall panels with a central layer of reinforcement (Bischoff2006). In all cases, this occurs because member stiffness isoverestimated with Branson’s equation when flexure mem-bers have an I Ig cr/ ratio greater than 3. As a result, eq. [1]begins to underpredict deflection when steel reinforcing ra-tios are less than approximately 1% for beams, correspond-ing to an I Ig cr/ ratio greater than 3 (Bischoff 2005). For verylow reinforcing ratios, the service load can be near or below

the cracking moment. The uncracked moment of inertia istypically used when the applied load is less than the crack-ing moment; however, these members may still crack overtime as additional stresses develop from shrinkage and tem-perature effects. For this reason, the Australian standardAS3600 (AS 1994) imposed an upper limit of 0.6Ig forsteel-reinforced beams and slabs with a reinforcing ratio lessthan 0.5% (Gilbert 2001).

This paper provides an explanation of why the Bransonequation does not always work well in predicting deflection,and presents a rational approach to develop an alternativeexpression for the effective moment of inertia that worksequally well for both steel- and FRP-reinforced concrete atall reinforcing ratios. The focus is on flexure members rein-forced with steel, although some consideration is given toFRP-reinforced members. Revisions to the section on controlof deflections in CSA A23.3-04 (clause 9.8) (CSA 2004) areproposed and include changes to expressions for the effectivemoment of inertia Ie and for the averaged effective moment ofinertia Ie,avg of continuous members. Direct calculation of de-flection is also considered for continuous members in lieu ofusing an averaged moment of inertia. Only immediate(short-term) deflection is considered in this study.

Modeling tension stiffening in beams

The overall stiffness and corresponding deformation of areinforced concrete member is affected by tension stiffening

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Bischoff 993

Fig. 1. (a) Flexural member response based on an effective mo-ment of inertia Ie. (b) Flexural member response incorporatingtension stiffening.

that arises from tension carried by the concrete betweencracks. The concrete contribution in beams is modeled witha tension stiffening factor � � �ts ( / )max� � � that decreaseswith increasing load once the member has cracked (Fig. 1b).The maximum tension stiffening curvature at first crackingis given by � ��max M I I E Icr cr g c cr( / ) /1 � . The change incurvature relative to the cracked member response is calledthe tension stiffening curvature � �� � �( )max� ts and is usedto define a rational expression for the effective moment ofinertia Ie. Figures 1a and 1b help show the derivation for Ie,giving

[2] IM

E

M

E M E Ie

a

c a

a

c a c cr

� ��� �( / )�

��

�I

M MIcr

ts cr ag

1 � �( / )

where � � �1 I Icr g/ and �a is the curvature at Ma. Approxi-mating the tension stiffening factor �ts with M Mcr a/ thengives

[3] II

M MIe

cr

cr ag�

��

1 2�( / )

This approximation assumes that the tension stiffeningcurvature varies inversely with bar stress at the crack loca-tions. That is, �ts b,cr b� f f/ , where f b,cr is the stress in thebar at first cracking and fb is the stress in the bar (at a crack)at Ma (Nemirovsky 1949; Rao and Subrahmanyam 1973;Rostasy et al. 1976). The same tension stiffening factor hasbeen adopted by the Comité Euro-International du Béton(CEB) model codes (CEB-FIP 1978, 1993) and CEB-basedEurocode 2 (CEN 2004). Equation [3] represents a rationaland unified approach that works well for predicting deflec-tion of both steel- and FRP-reinforced concrete beams with-out the need for correction factors (Bischoff 2005; Bischoffand Scanlon 2007). This equation is simple, easy to use, andis proposed as a suitable replacement for the Branson equa-tion.

A fresh perspective on Branson’s equation

Tension stiffening is characterized by the change in curva-ture �� relative to the cracked member response and shouldnot exceed the change in curvature at first cracking ��max ifthe tensile stress in the concrete is to remain less than thecracking strength fcr (Fig. 1b). Hence, the tension stiffeningfactor � � �ts � �� �/ max 1, which gives a member responsebounded by an upper limit with constant tension stiffening��ts � 1) and a lower limit with no tension stiffening ��ts � 0).Member responses using both Branson’s expression for Ie(eq. [1]) and the unified approach (eq. [3]) are plotted inFig. 2 for beams with I Ig cr/ � 2, 3, 5, 8, and 11 (corre-sponding to steel reinforcing ratios of � = 1.7%, 1%, 0.5%,0.3%, and 0.2%, respectively). Results clearly show that ten-sion stiffening is overestimated using Branson’s approach forbeams with I Ig cr/ greater than 3 (corresponding to � < 1%for steel-reinforced concrete beams), whereas there is little dif-ference between the two approaches for steel-reinforcedmembers at higher reinforcing ratios (greater than 1%). Ten-sion stiffening is grossly overestimated for beams reinforced

with FRP bars, as these types of beams typically have anI Ig cr/ ratio between 5 and 25 (Bischoff 2005). Tilt-up wallpanels with a central layer of reinforcement also exhibit anunrealistically stiff computed response when using Branson’sequation because the I Ig cr/ ratio for this type of wall typi-cally ranges from 15 to 25 (Bischoff 2006; Bischoff andScanlon 2007; Lawson 2007).

Figure 3 plots the influence of steel reinforcing ratio onI Ig cr/ for rectangular beams and the corresponding level ofservice load relative to the cracking moment ( / )M Ma cr atwhich deflection is calculated. Service load moments( / )M Ma load� r � are approximately equal to 60% of thenominal moment capacity (Mn) provided that the beam isunderreinforced, where Mr is the factored moment resistancecalculated using material resistance factors, Mn is the nomi-nal moment capacity without material resistance factors, and� load is an average load factor equal to 1.375 for an assumeddead to live load ratio of 1:1. All calculations are based onthe requirements of CSA A23.3-04 (CSA 2004) and arecarried out for concrete with a specified compressivestrength fc � of 35 MPa and steel with a yield strength fy of400 MPa. Figure 3 shows how the I Ig cr/ ratio increases sig-nificantly at low reinforcing ratios less than about 0.6%.Corresponding service loads in this reinforcing range aretypically less than twice the cracking moment, where theinfluence of tension stiffening is most significant.

The concrete contribution in Branson’s equation is evalu-ated by calculating the corresponding tension stiffening factor�ts . This is determined by substituting Ie from eq. [1] intothe expression for the tension stiffening factor

[4] � � �ts a crcr e

cr g

� ��

�� �/ ( / )

( / )

( / )max M M

I I

I I

1

1

to get

[5] �tsa cr

cr cr g cr a

�� �

M M

M M I I M M

/

[ ( / ) ] ( / ) / ( / )1 1 3 3a

Figure 4 compares Branson’s tension stiffening factor withthe factor �ts cr a� ( / )M M incorporated into eq. [3], for beamswith I Ig cr/ equal to 2, 3, 5, 8, 25, and 75. These plots showthat the tension stiffening factor in Branson’s equation tendstowards M Ma cr/ as I Ig cr/ goes to infinity. In other words,Branson’s expression for Ie approaches Ig as the ratio I Ig cr/increases. Overestimating tension stiffening leads to a stifferresponse and less deflection than expected, which is theproblem with Branson’s equation.

Comparison with rational proposal

Branson’s equation gives a weighted average of the grossand cracked moments of inertia at any given load level. Thisis analogous to having an uncracked and cracked springplaced in parallel as shown in Fig. 5a, where the equivalentspring stiffness ke = kg + kcr approaches the uncracked stiff-ness kg of the stiffer spring as the difference between the twospring stiffness values increases. This occurs when the ratiok kg cr/ becomes larger as the cracked stiffness kcr decreasesfor smaller values of Icr. Consequently, the member responseis incorrectly “pulled” towards a weighted value of the un-

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994 Can. J. Civ. Eng. Vol. 34, 2007

cracked response E I M Mc g a( / )cr3 as the reinforcing ratio

and (or) elastic modulus of the reinforcing bar decreases.This results in an unrealistically high amount of tensionstiffening and smaller deflections than expected when Icrdrops below one-third of Ig (Bischoff 2005).

Tension stiffening is modeled properly by putting the un-cracked and cracked springs in series (Fig. 5b) to representthe variation in stiffness of the uncracked and cracked partsof the beam along the member length. Using this approach,the member stiffness is then “pulled” more realisticallytowards the response with the lower stiffness (that is, thecracked member response). The equivalent or effective springstiffness is now determined by taking a weighted average of theinverse stiffness values (1/EI), leading to a subtle change inBranson’s original eq. [1] as given below.

[6]1 1

11

I

M

M I

M

M I

m m

e

cr

a g

cr

a cr

��

���

�

� � �

�

���

�

�

�

��

�

��

�1

I g

Calibration of this equation with eq. [1] for a beam withI Ig cr/ � 2.2 (Branson calibrated his original equation usingbeams with this ratio) gives a power of m = 2 (Bischoff2005). This is equivalent to using a tension stiffening factor�ts cr a� M M/ . Equation [6] with m = 2 is a rearranged formof the proposed eq. [3] and is also the same as the curva-ture-based equation used in Eurocode 2 (CEN 2004) for de-

flection calculations. Equation [3] is thus proposed as a suit-able replacement for Branson’s equation because itrepresents a physical model that correctly accounts for ten-sion stiffening in reinforced concrete beams and slabs.

Differences between the two methods used to calculate Ie(eqs. [1] and [3]) are shown in Figs. 6a and 6b for differentlevels of applied load. A comparison of these two approachesis plotted in Fig. 7 and demonstrates that differences start tobecome evident for steel reinforcing ratios less than about1% when the ratio of applied service load moment to crack-ing moment ( / )M Ma cr equals 2. The effective moment ofinertia using the proposed approach (eq. [3]) decreasesdramatically for low reinforcing ratios less than about 0.5%,especially when the applied moment Ma is close to thecracking load, whereas Branson’s expression predicts valuesof Ie that are significantly closer to Ig. For example, a beamwith � = 0.5% that is subjected to a moment just 10% greaterthan the cracking moment (Ma = 1.1Mcr) has an effectivemoment of inertia Ie equal to about 0.8Ig (using Branson’seq. [1]) compared with a more realistic value of 0.58Ig ob-tained with the proposed approach. Computed values of Ieare typically greater than 0.6Ig at low reinforcing ratioswhen using Branson’s equation for moments less than1.2Mcr. This helps to explain the upper limit of 0.6Ig im-posed on Ie by the Australian standard AS3600 (SA 1994)when using Branson’s equation to compute deflection ofbeams and slabs with a reinforcing ratio less than 0.5% (seeIntroduction). Note how the applied service load is close tothe cracking moment at low reinforcing ratios (Fig. 3),where the influence of tension stiffening is most significant.

Evaluation and validation of deflectionprediction at service loads

Beam deflections calculated with the proposed expressionfor Ie (eq. [3]) are compared with Branson’s method (eq. [1])in Fig. 8. Results are compared at the full service loaddefined in Fig. 3, clearly showing that both the Bransonequation and proposed equation give comparable results(within a few percent of each other) for reinforcing ratiosbetween 1% and 2%. Results also compare well with existingbeam tests by Washa and Fluck (1952). However, differ-ences between the two approaches become evident once steelreinforcing ratios drop below 1%, with Branson’s expression

© 2007 NRC Canada

Bischoff 995

Fig. 2. Member response comparison for steel reinforced con-crete beams (fc� = 35 MPa).

Fig. 3. Variation of Ig/Icr and Ma/Mcr with steel reinforcing ratio �.

Fig. 4. Tension stiffening component in Branson’s equation.

underpredicting deflection by up to half of the calculatedvalue using the proposed approach for a beam with � = 0.3%.

Experimental support of this behavioural trend is providedby Gilbert (2006) for a number of steel-reinforced slabs hav-ing a reinforcing ratio between 0.18% and 0.84%. Figure 9ashows one example comparing the two approaches to Gilbert’smeasured member response for a simply supported slab witha very low reinforcing ratio of 0.18%, which results in anIg/Icr ratio of 13.3. Branson’s expression clearly produces amember response that is much too stiff at this low reinforce-

ment ratio; however, the proposed model (eq. [3]) worksquite well. Work by Bischoff and Scanlon (2007) and Law-son (2007) confirm that Branson’s expression for Ie underes-timates deflection of slender tilt-up wall panels, whereaseq. [3] gives reasonable results at service loads. Figure 9bshows that for glass FRP (GFRP) reinforced beams, Bis-choff’s expression for Ie compares well with the ACI 440

© 2007 NRC Canada

996 Can. J. Civ. Eng. Vol. 34, 2007

Fig. 5. (a) Equivalent spring model for Branson expression (eq. [1]). (b) Equivalent spring model for proposed expression (eqs. [3] and[6] with m = 2).

Fig. 6. Variation of effective moment of inertia using (a) Branson’sapproach and (b) proposed approach.

Fig. 7. Comparison of Branson (eq. [1]) and proposed (eq. [3])approach.

Fig. 8. Calculated beam deflection at service loads defined inFig. 3.

(ACI Committee 440 2006) approach using a modified formof the Branson equation. As expected, Branson’s originalexpression gives a response that is too stiff because of thehigh I Ig cr/ ratio and requires a correction factor to softenthe computed member response.

Practical examples are given in Table 1 for a rectangularconcrete section having steel reinforcement ratios of 0.4%,1.0%, and 2.0%. A GFRP-reinforced beam with a reinforc-ing ratio of 1.0% is also included for comparison and theGFRP bars are assumed to have an elastic modulus of40 GPa. While analysis is carried out for concrete with acompressive strength of 35 MPa, the proposed expressionfor Ie is expected to be equally valid for higher strengthconcretes.

Deflection is evaluated at full service load for thesteel-reinforced concrete sections based on a service loadmoment Ma = Mr /1.375 (as explained in an earlier section)that is equal to 60% of the nominal moment capacity Mn,which gives a strain in the reinforcement of about 1200microstrains. The service load for the GFRP section is basedon a limiting bar strain of 3000 microstrains that allows forlarger crack widths (Bischoff 2005), which gives an appliedservice load equal to about 25% of the nominal momentcapacity for this section (compared with 60% forsteel-reinforced concrete).

Little difference is seen between Branson’s approach (eq. [1])and the proposed approach (eq. [3]) for the sections withsteel reinforcing ratios of 1% and 2%. Note how theeffective moment of inertia Ie is within a few percent of Icrfor these two reinforcing ratios. This occurs because of therelatively high ratio of Ma/Mcr at full service load when thereinforcing ratio is greater than 1% (see p. 120 of Branson1977 where Ie � Icr when Ma/Mcr > 3). Branson’s approachunderestimates deflection by 34% (compared with Bischoff’sapproach) when the steel reinforcing ratio � drops to 0.4%.This occurs because the lower � value increases the I Ig cr/ratio beyond 3 to a value of about 5, whereas the serviceload has dropped down to 1.3Mcr as indicated by Fig. 3.Recall that the influence of tension stiffening is more signif-icant when service loads are less than twice the crackingload. The GFRP beam with a reinforcing ratio of 1.0% hasan I Ig cr/ ratio close to 10 and Ma/Mcr ratio of 1.6. In thiscase, Branson’s equation gives a deflection that is only halfthe expected value. The solution to dealing with this problemin the past has been to apply a correction factor to Branson’sequation for computing deflection of FRP members (ACICommittee 440 2006). For steel-reinforced concrete, CSAA23.3-04 (CSA 2004) uses one-half the cracking momentto compute Ie for two-way slabs that are typically lightly re-inforced. The reduced cracking moment compensates for thestiffened bulge evident with Branson’s computed member re-sponse at low �’s (Fig. 2) and also accounts for a decreasein member stiffness resulting from shrinkage restraintcracking.

Continuous members

For continuous members, CSA A23.3-04 (CSA 2004)provides an averaged moment of inertia Ie,avg based on aweighted average of the effective moment of inertia taken atmidspan and the end supports. This is given by Ie,avg = 0.7Iem +

0.15(Ie1 + Ie2) for two continuous ends and Ie,avg = 0.85Iem +0.15Iec for one continuous end. Iem is the value of Ie atmidspan, Ie1 and Ie2 are the values of Ie at the two end sup-ports, and Iec is the value of Ie at the one continuous end.ACI 318-05 (ACI 2005) suggests using a simple averagedvalue of Ie. Both approximations were proposed by Branson(1977) and are based on work carried out by ACI 435 (ACICommittee 435 1973).

The averaged value for Ie is used in conjunction with ageneralized elastic equation for midspan deflection, such asthat given by

[7] � ��

���

�

�K

M L

E I

5

48

2m

c e,avg

where the beam deflection coefficient K M M� � �1 01. ( ) /a1 a2M m for a beam with uniformly distributed load. Mm, Ma1,and Ma2 are the moments at midspan and the two end sup-ports, respectively. The value of K can also be expressedwith the equation 1.2 – 0.2Mo/Mm, where Mo is the totalstatic moment wL2/8. Note that algebraic (negative) valuesare used for the two end moments.

This simplification uses a single averaged value of flexuralrigidity to approximate deflection and is evaluated by com-paring computed values of deflection to a beam having threesegments of constant flexural rigidity corresponding to the

© 2007 NRC Canada

Bischoff 997

Fig. 9. (a) Slab response comparison for � = 0.18% and Ig/Icr =13.3 (after Gilbert 2006). (b) Beam response comparison forGFRP-reinforced concrete.

two negative end span regions and one positive midspan regionwith each segment defined by the location of the inflectionpoints (ACI Committee 435 1973). Both methods (single valueand three-segment solution) are approximate since the variationof Ie within each segment and stiffening from the uncrackedparts of the beam span are not considered. Comparisons aremade for steel-reinforced concrete sections only; extensionto FRP-reinforced concrete requires further evaluation.

The approximate method using a single value of Ie,avggives the same computed value of deflection as the threesegment solution only when the rigidity at the end supportsis equal to the rigidity of the midspan region; that is, whenIe1/Iem = Ie2/Iem = 1 (except when the beam is simply sup-ported with end support moments equal to zero). Down-wards deflection of the beam increases as rigidity of the endsupports increases (relative to the midspan rigidity), whichimplies that the average moment of inertia Ie,avg shoulddecrease and drop below the midspan value Iem when0 5 1. ( ) /I I Ie1 e2 em� � . Comparison is made for a given setof support moments and redistribution of moments fromchanges in rigidity at the end supports is not considered.Hence, deflection is actually underestimated with either theCSA A23.3-04 (CSA 2004) or ACI 318 (ACI Committee318 2005) approach whenever the stiffness at the end sup-ports is greater than the stiffness at midspan, as the com-puted value of Ie,avg is greater than Iem using either of thesetwo approaches. Using the midspan value Iem alone in thiscase (that is, substituting Iem for Ie,avg) would give a betterapproximation, although deflection will still be underesti-mated, but not to the same extent.

Alternatively, deflection can be computed by the integra-tion of curvature along the beam length. Assuming a para-bolic distribution of curvature along the span of a continuousbeam subjected to a uniformly distributed load, midspandeflection � is approximated by

[8] � ��

���

�

� � �

Lm

2

1 296

10( )� � �

where L is the span length, �� and �� are the curvaturesM E Ia1 c e1/ and M E Ia2 c e2/ at the two end supports, respec-tively, and �m is the curvature M E Im c em/ at midspan. Adetailed explanation of this approximation is given by Ghali(1993). When compared with the three-segment solution,this approach overestimates deflection when the end sup-

ports are stiffer than the midspan region and underesti-mates deflection when the end supports are more flexible.The accuracy of this method is also tempered by the factthat the uncracked parts of the beam are neglected in this ap-proximation.

In lieu of using this type of direct approach for calculatingdeflection, the deflection value from eq. [8] can be equatedwith the generalized elastic equation for midspan deflection(eq. [7]) to give a rational expression for the averaged momentof inertia.

[9] IKI

IM I M I

e avgem

1 1 2 2g

01,

. .�

� ��

1 01� � � �

with �M M M1 2 a1 2 m, , /� and �I I I1 2 em e1 2, ,/� .This gives Ie,avg = Iem for a simply supported beam and

I I I Ie,avg em em e1� �0 8 1 0 2. / ( . / ) for a typical interior spanwith a positive midspan moment equal to wL2/16 and havingthe same effective moment of inertia at the two end supports(Ie1 = Ie2). Therefore, the new expression for Ie,avg gives amore flexible member than using the midspan value of Iemwhenever the end supports are stiffer than the midspanregion.

The CSA A23.2-04 approach (CSA 2004) is comparedwith eq. [9] in Fig. 10 for the interior span conditions justdescribed. Comparison is also made with deflection calcu-lated using a single moment of inertia equal to the midspanvalue and with the three-segment solution. Redistribution ofmoments after cracking is not considered. Results show thatthe existing code approach stiffens the member responsewhen Ie is greater at the ends (Ie1,2 > Iem), as explained previ-ously, whereas the new proposal using eq. [9] gives a moreflexible response that results in greater deformation as ex-pected (recall that midspan deflection should increase as theend supports become stiffer). Thus, when member stiffnessis greater at the end supports than at midspan, deformationof continuous beams can be underpredicted by up to 50% ormore using the present code approach for the average mo-ment of inertia Ie,avg. The comparison in this case is beingmade with the three-segment approximation. Once again, de-flection is more appropriately reflected by taking a weightedaverage of the inverse stiffness values (1/EcIe) as opposed tothe averaged stiffness (EcIe) values stipulated in eitherA23.3-04 (CSA 2004) or ACI 318-05 (ACI Committee318 2005). A word of caution, as the proposed approach

© 2007 NRC Canada

998 Can. J. Civ. Eng. Vol. 34, 2007

Steel, reinforcementratio (%)

GFRP, reinforcementratio (%)

2.0 1.0 0.4 1.0

Ig /bd3 (10–3)a 114.3 114.3 114.3 114.3

Icr /bd3 (10–3) 75.2 45.7 21.8 12.0Ig /Icr 1.5 2.5 5.2 9.6

Ma/Mcr 5.5 3.1 1.3 1.6Ie,Branson/bd3 (10–3) 75.5 48.0 63.6 39.1Ie,Bischoff /bd3 (10–3) 76.1 48.8 41.7 19.0�Branson/�Bischoff 1.02 1.01 0.66 0.49

ad/h = 0.90.

Table 1. Steel- and GFRP-reinforced beam deflection comparison using Ie.

underpredicts deflection significantly when the stiffness atthe end supports drops below one-half of the midspan stiff-ness for this example (Fig. 10). Note that eq. [9] does notapply to fixed-end cantilevers.

Deflection example of continuous beam

To help evaluate the different approaches proposed forcomputing deflection of continuous members, calculated mid-span deflection values under full dead plus live load arecompared for both the exterior and interior span of a contin-uous T beam in a building frame. Only short-term deflec-tions are computed. Details of this example are given inFig. 11 and Table 2. Effective moments of inertia are basedon moment envelopes using the CSA (CSA 2004) momentcoefficients, whereas calculated deflections use the bendingmoment corresponding to the loading case for maximummoment at midspan (typically based on the moment coeffi-cient at midspan). In this example, the average moment atthe supports ranges from 0.75 to 1 times the midspan mo-ment, and the effective moment of inertia at the end supportsis between 1.5 to 2.25 times the value at midspan.

Results are summarized in Table 3 and show that in bothspans the use of Ie from midspan alone (Iem) gives the clos-est approximation to the three-segment solution. Deflectionis underestimated by less than 2% even though the averagesupport stiffness is more than twice the midspan stiffness inthe end span (see Fig. 10). Deflection using the weightedaverage from eq. [9] overestimates midspan deflection byabout 8%, whereas the CSA (CSA 2004) weighted averageunderestimates deflection by up to 30%. However, thesecomparisons need to be put into perspective by realizing thatcomputed deflections using the three-segment solution areoverestimated by about 15% in this example, as the variationin Ie for each segment and the uncracked parts of the beamare neglected. Despite this, the weighted average recom-mended by CSA A23.3-04 (CSA 2004) still underestimatesdeflection when considering the uncracked parts of the beam,whereas the other methods overestimate deflection. Notehow Branson’s equation overestimates Ie by about 25% atthe supports for the interior span, where the service load mo-ment is only 30% greater than the cracking value.

Comparison of results from the example described abovesuggests that the best and most simple estimate of deflection

for a continuous beam in a frame is obtained by approximat-ing the average moment of inertia with the midspan valueIem. This approach gives a reasonable approximation of mid-span deflection provided that the support moments do notexceed twice the midspan moment (corresponding to a beamwith fixed end conditions) and the average stiffness at thesupports is within 0.5 to 3 times the midspan stiffness. Simi-lar recommendations were made by ACI 435 (ACI Commit-tee 435 1973). Within this range, use of the midspan valuefor Ie,avg underestimates deflection by less than 12% (when thesupport stiffness is greater than midspan stiffness) and over-estimates deflection by 30% at most when the end supportsare more flexible than the midspan stiffness. More accuratemethods should be used outside this range.

When assessing methods to compute deflection for de-sign, it is important to realize that deflection is a highly vari-able process even under controlled laboratory conditions.Assumed values of the support and midspan moments incontinuous members are also not exact and, to a large extent,depend on the loading pattern and flexural rigidities devel-oped along the beam and in other members framing into thesupports. Moments are often based on the gross (uncracked)flexural rigidity and do not account for redistribution offorces when the members crack. Long-term deflection onlycomplicates matters further. Therefore, computed values ofdeflection are at best an approximation of what is likely tooccur and as such should be used to provide only an indica-tion of whether problems with deflection are likely to be en-countered. With this in mind, almost any of the expressionsdiscussed in this section would be adequate, but use of theeffective moment of inertia at midspan is the simplest ap-proach and appears to provide the most reasonable result forpractical applications.

Proposed revisions to clause 9.8 in CSAA23.3-04

This part of the paper presents proposed revisions to clause9.8 (entitled “Control of Deflections”) in the CSA concretedesign standard CSA A23.3-04 (CSA 2004). Changes aremade to the definitions of Ie and Ie,avg in clauses 9.8.2.3 and9.8.2.4, respectively, and a more direct approach for calcu-lating deflection is included. Additional notation and sug-gested commentary explaining reasons for the changes arealso provided. Other parts of the clause remain unchanged.

9.8. Control of deflections

9.8.2.3. E and I

Unless deflections are determined by a more comprehen-sive analysis, immediate deflection shall be computedusing elastic deflection equations, a modulus of elastic-ity, Ec, for concrete as specified in Clause 8.6.2, and theeffective moment of inertia as follows:

I I M M Ie cr cr a g� � � �/ [ ( / ) ] ( )1 9 12�

where

� �1 cr g� �I I/ ( )9 2

M f I ycr r g t� �/ ( )9 3

and fr is as given in Clause 8.6.4.

Note: The moment Ma may be due to construction loads.

© 2007 NRC Canada

Bischoff 999

Fig. 10. Comparison of average moment of inertia Ie,avg usingcode equation and proposed approaches.

Commentary: Branson’s equation used for Ie in earliereditions of A23.3 for computing deflection underestimatesdeflection of steel-reinforced concrete beams with a rein-forcing percentage less than 1%. Service load deflectionof slender (tilt-up) walls is also underestimated. Equation(9-1) computes reasonable values of deflection for both

steel- and fibre-reinforced-polymer (FRP)- reinforced con-crete members at all reinforcing ratios.

9.8.2.4. Moment of inertia for continuous spans

For continuous prismatic members, the effective momentof inertia may be based on Ie values obtained from equa-

© 2007 NRC Canada

1000 Can. J. Civ. Eng. Vol. 34, 2007

Fig. 11. Continuous T-beam example used for deflection comparison: (a) beam elevation and (b) beam cross-section. All dimensions inmillimetres.

End spana Interior spanb

Support 1 Midspan Support 2 Support 1 Midspan Support 2

Ig /107 (mm4) 1157.7 1157.7 1157.7 1157.7 1157.7 1157.7

Icr /107(mm4) 245.2 332.5 344.6 329.4 332.5 329.4Ig /Icr 4.72 3.48 3.36 3.51 3.48 3.51

Mcr (kN�m) 243.9 99.9 243.9 243.9 99.9 243.9

Moment envelope using coefficients wLn2/16 wLn

2/14 wLn2/10 wLn

2/11 wLn2/16 wLn

2/11Ma (kN�m) 215.9 246.7 345.4 314.0 215.9 314.0Ma/Mcr 0.89 2.47 1.42 1.29 2.16 1.29Ie/107 (Bischoff) 1157.7 376.5 530.3 579.6 392.4 579.6Ie/107 (Branson) 1157.7 387.3 630.9 717.6 414.3 717.6

Bending moment wLn2/16 wLn

2/14 wLn2/22.4 wLn

2/16 wLn2/16 wLn

2/16Ma (kN·m) 215.9 246.7 154.2 215.9 215.9 215.9aK = 0.85, (Ma1 + Ma2)/2Mm = 0.75, (Ie1 + Ie2)/2Iem = 2.24.bK = 0.80, (Ma1 + Ma2)/2Mm = 1.0, (Ie1 + Ie2)/2Iem = 1.48.

Table 2. Details for deflection comparison of continuous members (wa = 46.7 kN/m; fc� = 35 MPa; fy = 400 MPa).

End span Interior span

Averaging approach using Ie

proposed by Bischoff (eq. [3])Ie,avg /107

(mm4)Deflection(mm)

Ie,avg /107

(mm4)Deflection(mm)

Integrationa — 14.9 — 11.6Three segmentsb — 16.3 — 12.9Equations [8] or [9] 345.1 17.6 363.1 13.8CSA A23.3-04 (CSA 2004) 516.7 11.7 448.6 11.1Iem (at midspan) 376.5 16.1 392.4 12.7

aIntegration of moment curvature includes variation of Ie and uncracked parts of beam.bIntegration of three segments with constant Ie (Ie1, Iem, and Ie2).

Table 3. Results of deflection comparison for continuous members.

tion (9-1) for the critical positive and negative momentsections. An average value for the effective moment ofinertia can be approximated with the midspan value asgiven below

I Ie,avg e,m� �( )9 4

provided that

(a) ( )/M M Ma1 a2 m� �2 2 and

(b) 0 5 2 3. ( )/I I I Iem e1 e2 em� � �

In lieu of using equation (9-4), deflection may be approx-imated directly as follows:

� ��

���

�

� � � �

Lm

2

1 29610 9 5( ) ( )� � �

for continuous members with a uniformly distributed load,where

�1 = Ma1/EcIe1

�2 = Ma2/EcIe2

�m = Mm/EcIem

Note: Ma1 and Ma2 are taken as algebraic values (–vevalue for negative moments).

Commentary: Deflection is affected by the effectivemoment of inertia at the end supports and at midspan,and may be approximated by using an appropriate elasticdeflection equation and assuming that the average effec-tive moment of inertia is equal to the value at midspan.An alternative approach for calculating deflection directlyis based on integration of an assumed parabolic distribu-tion of curvature.

Additional notation:

� = cracked stiffness factor (1 – Icr /Ig)

Ma1,2 = support moments at end 1 (Ma1) and end 2 (Ma2)

Conclusions

The tension stiffening component in Branson’s equationincreases unrealistically for beams with an Ig/Icr ratio greaterthan 3 and because of this flexural stiffness is overpredictedfor most FRP beams and for steel-reinforced concrete beamsand slabs with a reinforcing ratio less than about 1%. Com-puted deflections are then underestimated. Similar problemsare encountered for slender walls with a central layer ofreinforcement. A rational alternative to this equation is pro-posed that correctly accounts for tension stiffening, givingan effective moment of inertia I I M Me cr cr a� �/ [ ( / ) ]1 � 2

with � �� � I Icr g/ . This equation predicts deflection equallywell for both steel- and FRP-reinforced concrete members.A rational expression giving the averaged moment of inertiaIe,avg for continuous beams is developed based on the integra-tion of curvature. The existing approach in CSA A23.3-04(CSA 2004) averages stiffness at the critical sections and un-derestimates deflection when the flexural stiffness at the endsupports is greater than the stiffness at midspan.

Proposed revisions to deflection requirements contained inCSA A23.3-04 (CSA 2004) provide more realistic values ofdeflection for flexural members with low reinforcementratios. Changes to the average effective moment of inertia incontinuous members reflect the unconservative nature of theexisting equation under certain conditions, namely when the

stiffness at the end supports is greater than the stiffness atmidspan.

Acknowledgements

Support for this work was provided by the Natural Sci-ences and Engineering Research Council of Canada and theUniversity of New Brunswick.

References

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List of symbols

b beam widthd effective depth of reinforcing steelE elastic modulus

Ec elastic modulus of concretefb bar stress (at a crack)

fb,cr bar stress at first crackingfcr concrete cracking strengthfc� compressive strength of concretefy yield stress of reinforcing steelh beam heightI moment of inertia

Icr cracked transformed moment of inertiaIe effective moment of inertia

Ie,avg averaged effective moment of inertia

Ie1 effective moment of inertia at first supportIe2 effective moment of inertia at second supportIec effective moment of inertia at continuous end

Iem effective moment of inertia at midspanIg gross moment of inertiaK beam deflection coefficient

kcr cracked spring stiffnesske equivalent spring stiffnesskg uncracked spring stiffnessL beam or slab span length

Ln clear spanm power coefficient (=2 in eq. [6])

Ma applied service load momentMa1 support moment at first supportMa2 support moment at second supportMm midspan momentMcr cracking momentMn nominal moment capacityMo total static moment (wL2/8)Mr factored moment resistancew uniformly distributed load

wa uniformly distributed service load�load average load factor for dead plus live load�M1,2 ratio of end support moments (Ma1 and Ma2) to the

midspan moment (Mm)�I1,2 ratio of the effective moment of inertia at midspan (Iem) to

the effective moment of inertia at the supports (Ie1 and Ie2)� deflection

�a service load deflection�cr deflection at first cracking�� tension stiffening curvature

��max tension stiffening curvature at first cracking� stiffness reduction coefficient (1 – Icr /Ig)� reinforcing ratio� curvature

�1 curvature at first support using Ie1�2 curvature at second support using Ie2�a curvature at Ma

�cr curvature at first cracking (Mcr /EcIg)�m curvature at midspan using Iem�ts tension stiffening factor (= � ��� �max)

© 2007 NRC Canada

1002 Can. J. Civ. Eng. Vol. 34, 2007