park strut and tie m 2007

ACI Structural Journal/November-December 2007 657

ACI Structural Journal, V. 104, No. 6, November-December 2007.MS No. S-2006-006.R1 received April 16, 2007, and reviewed under Institute publication

policies. Copyright © 2007, American Concrete Institute. All rights reserved, includingthe making of copies unless permission is obtained from the copyright proprietors. Pertinentdiscussion including author’s closure, if any, will be published in the September-October 2008 ACI Structural Journal if the discussion is received by May 1, 2008.

ACI STRUCTURAL JOURNAL TECHNICAL PAPER

In this paper, a strut-and-tie-based method is presented for calcu-lating the strength of reinforced concrete deep beams. The proposedmethod employs constitutive laws for cracked reinforced concrete,considers strain compatibility, and uses a secant stiffness formulation.This method accounts for the failure modes due to crushing of thenodal compression zone at the top of the diagonal strut, yielding ofthe longitudinal reinforcement, as well as that of strut crushing orsplitting. This method is used to calculate the capacity of 214normal- and high-strength concrete deep beams that have beentested in laboratories. This method is illustrated to provide moreaccurate estimates of capacity than the strut-and-tie provisions ineither ACI 318-05 or the Canadian code. The comparison showsthat the proposed method consistently predicts the strengths of deepbeams with a wide range of horizontal and vertical web reinforcementratios, concrete strengths, and shear span-to-depth ratios (a/d)well. The proposed approach provides valuable insight into thedesign and behavior of deep beams.

Keywords: deep beams; failure strength; shear strength; strut-and-tie model.

INTRODUCTIONAccording to St. Venant’s principle and as illustrated by

elastic analyses, there is a complex state of strain in deepbeams. Thus, traditional sectional design approaches that arebased on the plane sections theory and use a parallel chordtruss model are not applicable for the design of deep beams.The strut-and-tie model is gaining widespread use andrespect as a rational method for the design of deep beams.Provisions for the design of deep beams using a strut-and-tiemodel have been included in several codes and guidelines forpractice, including AASHTO LRFD,1 ACI 318,2 CEB-FIP,3

and the Canadian code.4 With the addition of the procedurein Appendix A, “Strut-and-Tie Models,” of the 2005 ACIbuilding code, the deep beam provisions for evaluating Vcand Vs for deep beams as given in the 1999 version of ACI 318were deleted. In using Appendix A, designers are free tochoose the shape and dimensions of the load-resisting truss(strut-and-tie model) to carry the imposed loads through theD-region to its supports. More than one strut-and-tie modelis usually feasible and thus there is no single design solutionas there typically is with the use of sectional design procedures.Many designers are uncomfortable with the flexibilityprovided by the strut-and-tie method, which is often calledan approximate approach with undetermined accuracy. It hasbe argued that the strut-and-tie method is similar to the stripmethod for the design of two-way slabs5 in which the designerselects the load path and then provides the reinforcementaccordingly; but unlike two-way slabs, discontinuity regionsare inherently nonductile and this flexibility in design couldlead to significant cracking and local crushing under serviceload levels and inadequate strength to support factored loadswhen inappropriate strut-and-tie models are selected. Thus,it is important for the research community to further evaluate

the strengths and limitations of strut-and-tie design proceduresso that needed changes in codes of practice can be made andguidelines produced. This study focuses only on the applicationof the strut-and-tie method to deep beams.

In this study, a strut-and-tie model approach was developedfor calculating the capacity of reinforced concrete deepbeams, and the effectiveness of two strut-and-tie designmethods in current codes was evaluated. The proposedapproach is a compatibility-based strut-and-tie method thatconsiders the effects of compression softening of crackedconcrete. This concept has been previously applied to predictthe shear strengths of several discontinuity regions such asdeep beams,6 corbels,7 squat walls,8 exterior beam-columnjoints,9 and interior beam-column joints.10 The authors’approach builds upon and differs from previous methods inthat it uses a statically determinate truss for modeling theflow of forces and a secant stiffness formulation in evaluatingcapacity. The resulting model offers an effective and newapproach for the use of compatibility-based strut-and-tiemethods for predicting the strength and behavior of deep beams.

RESEARCH SIGNIFICANCEBeginning in 2002, the ACI building code stated that deep

beams should be designed using either nonlinear analysis orusing the strut-and-tie model. The strut-and-tie provisions inACI 318-02 were developed for the design of all forms ofdiscontinuity regions and not specifically deep beams. Thus,it is not surprising that this study reveals that Appendix A ofACI 318-05 provides conservative and scattered estimates ofthe strength of deep beams. The proposed compatibility-based strut-and-tie method, which considers the effects ofcompression softening, is shown to provide accurate estimatesof the measured load-carrying capacities of reinforced concretedeep beams. The findings illustrate that the inexactness ofAppendix A should not be considered an indictment on strut-and-tie design methods but rather point out areas forimprovement in provisions and the necessity of additionalguidelines for design by this method.

STRUT-AND-TIE MODELAPPROACH FOR DEEP BEAMS

Although several different types of strut-and-tie modelsare possible for describing the flow of forces in deep beams,the statically determinate model shown in Fig. 1 is used inthis study. This selection avoids the need to consider the relativestiffness of strut-and-tie members in determining member

Title no. 104-S61

Strut-and-Tie Model Analysis for Strength Prediction of Deep Beamsby Jung-woong Park and Daniel Kuchma

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ACI Structural Journal/November-December 2007658

forces. This model is used in the development of a generalapproach that considers the compression softening and websplitting phenomena as influenced by transverse tensilestraining. In this model, reinforcement for resisting thebursting of the strut is shown to be oriented normal to thedirection of applied compression. In traditional designpractice in which an orthogonal grid of reinforcement isprovided, as has been used in most tests, the contributionof this reinforcement in the direction of the ties, as shown inFig. 1, can readily be evaluated to calculate the effect ofcompression softening on the stiffness and strength ofdiagonal struts.

Experiments have shown that for the majority of simplysupported deep beams with a shear span-to-depth ratio (a/d)less than approximately 2.5, two nonflexural modes offailure are common, namely, diagonal splitting and concretecrushing. Test members, however, may often be more accuratelydescribed as having combined bending and shear failures. Theauthors’ approach considers all of these possible modes offailure. While ACI 318-05 places a limit on the angle betweenthe axes of struts and ties of 25 degrees, this limit wasdisregarded in this study to examine its relevance. The details ofthe proposed strut-and-tie approach are now presented.

Force equilibriumSoftened truss models by other researchers have typically

used statically indeterminate strut-and-tie models. Thismeans that it was necessary to assume a stiffness ratiobetween the components of these statically indeterminatetrusses to solve for member forces. As one example, inHwang’s model,6 these ratios were based on the linear elasticfinite element analyses and suggestions by Schäfer.11 It isconsidered to be a primary advantage of the authors’ modelto use a statically determinate model (refer to Fig. 1) as itavoids the necessity of such assumptions. The authors alsocontend that their selected model captures the primary flowof forces and resistance provided by reinforcement in deepbeams well. Equilibrium provides the following equations(Fig. 1)

(1)FdVθsin

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

(2)

(3)

where Fd, Fc, and Ft are the compressive forces in the diagonaland horizontal concrete struts, and the bursting tensile forcein the tie of the strut-and-tie model, as shown in Fig. 1. Whilethe traditional symbol for shear V is used to define themagnitude of the point loads and reaction forces, it should berecognized that the strut-and-tie method is a full memberdesign procedure and does not explicitly consider shear. Thecompressive force in the strut is assumed to spread at a 2:1 slopeas indicated in ACI 318-05, Section A.3.3 (Fig. RA.1.8(b)).Because the tensile force Ft is a quarter of the compressiveforce of the diagonal strut Fd, the horizontal and verticalcomponents of the tie force can be obtained from equilibriumas follows

(4)

(5)

where Fh and Fv are the horizontal and vertical componentof the tie force, respectively.

Secant stiffness formulationThe proposed compatibility-based strut-and-tie model

procedure uses an iterative secant stiffness formulation andemploys constitutive relations for concrete and steel. Withthis approach, the material stiffness properties are refined ineach iteration until convergence is achieved. The secantstiffness formulation has previously been used for predictingthe response of structural concrete, such a done by Vecchio12

in nonlinear continuum analysis applications. In the authors’approach, the strains in the horizontal concrete strut, diagonalconcrete strut, horizontal web steel tie, vertical web steel tie,and the longitudinal steel tie can be calculated using a secantstiffness approach as follows

(6)

(7)

(8)

(9)

(10)

FcVθtan

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

FtV

4 θsin--------------=

FhV4---

Fd θsin

4-----------------= =

FvV

4 θtan--------------

Fd θcos

4------------------= =

εcFc

EcAc

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

εdFd

EdAd

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

εh2Fh

EshAsh

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

εv2Fv

EsvAsv

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

εsT

EsAs

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

Jung-woong Park is a Postdoctoral Researcher in the Department of Civil andEnvironmental Engineering at the University of Illinois at Urbana-Champaign,Urbana, IL. He received his PhD from Kyungpook National University, Korea.

Daniel Kuchma, FACI, is an Assistant Professor in the Department of Civil andEnvironmental Engineering at the University of Illinois at Urbana-Champaign. He isa member of ACI Committees 318, Structural Concrete Building Code; 318E, Shearand Torsion (Structural Concrete Building Code); and Joint ACI-ASCE Committee445, Shear and Torsion. He received his PhD from the University of Toronto, Toronto,ON, Canada.

Fig. 1—Strut-and-tie model for deep beam.

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where Ac, Ad, Ash, Asv, and As are effective cross-sectionalareas for each member; Ec, Ed, Esh, Esv, and Es are secantmoduli as evaluated by

(11)

(12)

(13)

(14)

(15)

where fc, fd, fsh, fsv, and fs are stresses that are obtained fromthe constitutive relations of each member, and the initialvalues of secant moduli are the elastic moduli for concreteand reinforcing bars. Because the stiffness is calculated atthe element level, the total stiffness matrix of the systemdoes not have to be constructed in the proposed approach butrather it can be readily implemented in a spreadsheet.

Constitutive lawsCracked reinforced concrete can be treated as an orthotropic

material with its principal axes corresponding to the directionsof the principal average tensile and compressive strains.Cracked concrete subjected to high tensile strains in thedirection normal to the compression is observed to be softerthan concrete in a standard cylinder test.13-16 This phenomenonof strength and stiffness reduction is commonly referred toas compression softening. Applying this softening effect tothe strut-and-tie model, it is recognized that the tensilestraining perpendicular to the strut will reduce the capacityof the concrete strut to resist compressive stresses. Multiplecompression softening models were used in this study toinvestigate the sensitivity of the results to the selected model.All models were found to provide similarly good results aswill be illustrated later in the paper. The compression softeningmodel proposed by Hsu and Zhang13 was somewhat arbitrarilyselected for the base comparisons and is now described. Thestresses of concrete struts are determined from the followingequations proposed by Hsu and Zhang13

(16)

(17)

(18)

Ecfc

εc

----=

Edfd

εd

----=

Eshfsh

εh

-----=

Esvfsv

εv

-----=

Esfs

εs

----=

σd ζfc′ 2εd

ζε0

--------⎝ ⎠

⎛ ⎞εd

ζε0

--------⎝ ⎠

⎛ ⎞2

– for εd

ζε0

-------- 1≤=

σd ζfc′ 1εd ζεd( ) 1–⁄

2 ζ 1–( )⁄

-------------------------------⎝ ⎠

⎛ ⎞2

– for εd

ζε0

-------- 1>=

ζ5.8

fc′

--------- 1

1 400εr+---------------------------×

0.9

1 400εr+---------------------------≤=

where ε0 is a concrete cylinder strain corresponding to thecylinder strength fc′, which can be approximately defined17 as

(19)

The steel bar is assumed to be an elastic-perfectly-plasticmaterial in this approach.

Compatibility relationsThe strain compatibility relation used in this study is that

the sum of normal strain in two perpendicular directions is aninvariant, that is

εh + εv = εr + εd (20)

where εh and εv are tensile strains in the horizontal andvertical web steel ties, respectively; εd is the compressivestrain in the concrete strut; and εr is the tensile strain in thedirection perpendicular to the concrete strut. Equation (20) isderived from the strain compatibility condition as describedby Mohr’s circle of strain.

Effective depth of concrete strut and nodeThe effective depth of the top horizontal concrete strut was

taken as

wc = kd (21)

where d is the effective depth of deep beam and k was derivedfrom the classical bending theory for a single reinforced beamsection as

(22)

where n is the ratio of steel to concrete elastic moduli and ρis the longitudinal reinforcement ratio. The effective depthof the diagonal concrete strut was taken as

(23)

where a/2 should not be less than the length of the loadingplate, kd is the depth of the compression zone at the section,and the inclination angle of the diagonal strut with respect tothe horizontal axis θ can be obtained from

(24)

The notations for obtaining the effective depth wd of Eq. (23)and strut angle θ of Eq. (24) are given in Fig. 1. The proposedmethod evaluates the capacity as limited by the failure modesassociated with nodal crushing, yielding of the longitudinaltension reinforcement, and crushing or splitting of the diagonalstrut. The effective width of the top node of the horizontalconcrete strut was taken as a quarter of the overall height of

ε0 0.002 0.001fc′ 20–

80-----------------⎝ ⎠

⎛ ⎞ for 20 fc′ 100 MPa≤≤+=

0.002 0.001fc′ 2901–

11 603,

-----------------------⎝ ⎠

⎛ ⎞ for 2901 fc′ 14 504 psi ,≤≤+=

k nρ( )2 2nρ+ nρ–=

wda2--- θ kd θcos+sin=

θtan d kd 2⁄–a

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

660 ACI Structural Journal/November-December 2007

the deep beam based on the suggestion of Paulay andPriestley18 for the depth of the flexural compression zone ofa column as

(25)

where N is the axial force, Ag is the gross area, and hc is theoverall height of the column. The effective width of the topnode in the face of the diagonal concrete strut was taken as

(26)

where wp is the width of the loading plate.It is important to also account for the different behavior of

high-strength concrete deep beams with no web reinforcementor very light amounts of web reinforcements as they exhibitedmore brittle failures than normal-strength concrete beamswith similar levels of web reinforcement.17,19-21 In a recentinvestigation into the strength of high-strength concrete deepbeams19 at a/d greater than approximately 1.0, it wasobserved that web reinforcement restrained the developmentof a sudden shear failure. This indicates that if the web of adeep beam is heavily reinforced, the failure will becontrolled by strut crushing; however, without sufficientreinforcement, failure can occur suddenly due to splitting ofconcrete struts. This splitting failure becomes more evidentas concrete strength increases. To consider the brittleness ofhigh-strength concrete, area reduction factors for diagonalconcrete struts were introduced by the authors as

(27)

(28)

where ρh and ρv are the ratios (in percent) of horizontal andvertical web reinforcement, respectively. The aim of the areareduction factors was to account for the brittle failure of deepbeams for members starting with concrete strengths greaterthan 42 MPa (6092 psi) or when there was insufficient webreinforcement. This was selected to be when the reinforcementratio is less than 0.25% in each direction. It is implied thatthe strut area is reduced by horizontal web reinforcementwhen a/d is less than 0.75, and by vertical web reinforcementwhen a/d is greater than 1.0. This approach is supported byexperimental test data from which it has been generallyobserved that horizontal web reinforcement is more effectivethan vertical web reinforcement when a/d is less than 0.75,and that vertical web reinforcement is more effective thanhorizontal web reinforcement when a/d ratios are greaterthan or approximately equal to 1.0.20,22 The use of areareduction factors led to better agreement with the 214 testresults examined in this study and the effective area of thediagonal concrete strut in Eq. (7) is expressed by

Ad = φc1φc2wdb (29)

Solution procedureThe solution procedure of the proposed method is

summarized as follows.

c 0.25 0.85 NAgfc′-----------+

⎝ ⎠

⎛ ⎞ hc=

wnd wp θh4--- θcos+sin=

φc1 ρh 0.75 for a d⁄ 0.75≤+=

φc2 ρv 0.75 for a d⁄ 1.0≥+=

1. For a selected value of V, the member forces are calculatedusing Eq. (1) through (5) and the strains in concrete strutsand steel ties are calculated using Eq. (6) through (10). Whenweb reinforcement yielded or was not defined, the transversestrain was conservatively assumed to be 0.0025;

2. Using Eq. (20), the softening coefficient ζ is calculatedfrom Eq. (18);

3. Using the state of strain in each member, the stresses aredetermined from the stress-strain relations of Eq. (16) through(19). The ζ factor accounts for the compression softening effect;

4. The secant moduli for each member are then determinedby Eq. (11) through (15) using the values calculated in theprevious step. If the differences between the secant moduli inthis step and those calculated by Eq. (6) through (10) are largerthan 0.1%, then the steps are repeated until convergence; and

5. The procedure is completed when the stress in either thehorizontal or diagonal concrete strut reach their capacity orwhen the stress in the longitudinal tension reinforcementreaches its yield point. Then, the nominal failure strengthdue to crushing of horizontal or diagonal concrete struts, oryielding of longitudinal steel tie can be determined. Thestrength prediction is the minimum value of the strengthcomputed in this iterative procedure.

COMPARISON WITH TEST RESULTSThe proposed method and code provisions are used to

calculate the capacity of 214 reinforced concrete deep beamsthat were tested to failure and reported in the literature of thefollowing eight investigations: Clark,23 Kong et al.,22 Smithand Vantsiotis,24 Anderson and Ramirez,25 Tan et al.,26 Ohand Shin,19 Aguilar et al.,27 and Quintero-Febres et al.21

Only those references that provided sufficiently completeinformation on the test setup and material properties wereused. This database is considered to be sufficiently largeto enable a fair critique of code provisions and validationof the proposed model. The deep beams that were consideredin this study include a/d ratios ranging from 0.27 to 2.7,concrete strengths that range from 13.8 to 73.6 MPa(2001 to 10,675 psi), and various combinations of webreinforcements. A summary of these beams is presentedin Tables 1 through 8.

Due to the small a/d, typically less than approximately 2.5for a deep beam, a large portion of the supported loads aredirectly transmitted to supports. Therefore, the shearstrength of deep beams has been experimentally observed tobe significantly greater than that of slender beams. When thea/d is less than approximately 2.5, failures are typicallyobserved to be due to crushing of concrete in the compressionzone at the head of the inclined crack and in the region adjacentto the loading plate. This type of failure is typically referredto as a shear compression failure. Other types of failuresinclude crushing of concrete in the web, splitting of theconcrete along an inclined crack, crushing of concreteunderneath the supports, and anchorage failure between theconcrete and the main reinforcements. While the developmentof cracking and the progression of failures were reported tobe similar in most specimens, a significant number ofspecimens were reported to fail by yielding of longitudinalreinforcement and combined bending and shear failures. Oneor more diagonal cracks were commonly observed to penetrateso deeply into the compression zone at the loading point thatimmediate failure occurred by crushing of the concrete inthis location. It was reported that high-strength concrete deepbeams without web reinforcement exhibited abrupt shear

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failures without any warning, regardless of the a/d ratio. At alower a/d of 0.5 and 0.85, in spite of vertical web reinforcement,most high-strength beams showed abrupt failures. At ahigher a/d of 1.25 and 2.0, it was observed that web reinforcementrestrained sudden shear failures. This brittle shear failurewas more evident for members cast with higher strength

concretes. These observations were a motivation to introducethe area reduction factors for high-strength concrete.

Strength predictionsThe calculated strengths by the three methods based on the

strut-and-tie models (ACI 318-05,2 CSA,4 and the current

Table 1—Analysis results for Test Data 124

Deep beam ID

fc′, MPa

a, mm a/d

ρ,%

ρh, %

ρv , %

Vtest, kN

Vtest/Vn

Deep beam ID

fc′, MPa

a, mm a/d

ρ, %

ρh, %

ρv , %

Vtest , kN

Vtest /Vn

ACI 318-05 CSA

Current study

ACI 318-05 CSA

Current study

0A0-44 20.5 305 1.0 1.94 — — 139.5 1.90 1.19 1.14 3B3-33 19.0 368 1.21 1.94 0.45 0.77 158.4 1.86 1.87 1.69

0A0-48 20.9 305 1.0 1.94 — — 136.1 1.81 1.14 1.09 3B4-34 19.2 368 1.21 1.94 0.68 0.77 155.0 1.80 1.80 1.64

1A1-10 18.7 305 1.0 1.94 0.23 0.28 161.2 1.72 1.49 1.45 3B6-35 20.6 368 1.21 1.94 0.91 0.77 166.1 1.79 1.83 1.63

1A3-11 18.0 305 1.0 1.94 0.45 0.28 148.3 1.64 1.41 1.39 4B1-09 17.1 368 1.21 1.94 0.23 1.25 153.5 2.00 1.97 1.83

1A4-12 16.1 305 1.0 1.94 0.68 0.28 141.2 1.75 1.48 1.49 0C0-50 20.7 457 1.5 1.94 — — 115.7 2.19 1.78 1.40

1A4-51 20.5 305 1.0 1.94 0.68 0.28 170.9 1.66 1.45 1.39 1C1-14 19.2 457 1.5 1.94 0.23 0.18 119.0 2.42 1.95 1.56

1A6-37 21.1 305 1.0 1.94 0.91 0.28 184.1 1.74 1.53 1.46 1C3-02 21.9 457 1.5 1.94 0.45 0.18 123.4 1.48 1.81 1.41

2A1-38 21.7 305 1.0 1.94 0.23 0.63 174.5 1.61 1.42 1.34 1C4-15 22.7 457 1.5 1.94 0.68 0.18 131.0 1.52 1.86 1.44

2A3-39 19.8 305 1.0 1.94 0.45 0.63 170.6 1.72 1.50 1.45 1C6-16 21.8 457 1.5 1.94 0.91 0.18 122.3 1.48 1.80 1.41

2A4-40 20.3 305 1.0 1.94 0.68 0.63 171.9 1.69 1.48 1.42 2C1-17 19.9 457 1.5 1.94 0.23 0.31 124.1 1.64 1.97 1.57

2A6-41 19.1 305 1.0 1.94 0.91 0.63 161.9 1.69 1.46 1.42 2C3-03 19.2 457 1.5 1.94 0.45 0.31 103.6 1.42 1.69 1.36

3A1-42 18.4 305 1.0 1.94 0.23 1.25 161.0 1.74 1.50 1.47 2C3-27 19.3 457 1.5 1.94 0.45 0.31 115.3 1.57 1.87 1.50

3A3-43 19.2 305 1.0 1.94 0.45 1.25 172.7 1.79 1.55 1.51 2C4-18 20.4 457 1.5 1.94 0.68 0.31 124.6 1.60 1.93 1.53

3A4-45 20.8 305 1.0 1.94 0.68 1.25 178.6 1.71 1.50 1.44 2C6-19 20.8 457 1.5 1.94 0.91 0.31 124.1 1.57 1.91 1.50

3A6-46 19.9 305 1.0 1.94 0.91 1.25 168.1 1.68 1.47 1.42 3C1-20 21.0 457 1.5 1.94 0.23 0.56 140.8 1.76 2.13 1.68

0B0-49 21.7 368 1.21 1.94 — — 149.0 2.20 1.57 1.39 3C3-21 16.5 457 1.5 1.94 0.45 0.56 125.0 1.99 2.33 1.92

1B1-01 22.1 368 1.21 1.94 0.23 0.24 147.5 1.49 1.53 1.35 3C4-22 18.3 457 1.5 1.94 0.68 0.56 127.7 1.84 2.18 1.77

1B3-29 20.1 368 1.21 1.94 0.45 0.24 143.6 1.59 1.61 1.45 3C6-23 19.0 457 1.5 1.94 0.91 0.56 137.2 1.90 2.26 1.82

1B4-30 20.8 368 1.21 1.94 0.68 0.24 140.3 1.50 1.53 1.36 4C1-24 19.6 457 1.5 1.94 0.23 0.77 146.6 1.97 2.36 1.88

1B6-31 19.5 368 1.21 1.94 0.91 0.24 153.4 1.75 1.76 1.59 4C3-04 18.5 457 1.5 1.94 0.45 0.63 128.6 1.82 2.16 1.75

2B1-05 19.2 368 1.21 1.94 0.23 0.42 129.0 1.50 1.51 1.37 4C3-28 19.2 457 1.5 1.94 0.45 0.77 152.4 2.08 2.49 2.00

2B3-06 19.0 368 1.21 1.94 0.45 0.42 131.2 1.54 1.54 1.40 4C4-25 18.5 457 1.5 1.94 0.68 0.77 152.6 2.17 2.57 2.08

2B4-07 17.5 368 1.21 1.94 0.68 0.42 126.1 1.61 1.59 1.47 4C6-26 21.2 457 1.5 1.94 0.91 0.77 159.5 1.98 2.39 1.88

2B4-52 21.8 368 1.21 1.94 0.68 0.42 149.9 1.53 1.57 1.39 0D0-47 19.5 635 2.08 1.94 — — 73.4 2.16 2.16 1.32

2B6-32 19.8 368 1.21 1.94 0.91 0.42 145.2 1.64 1.65 1.49 4D1-13 16.1 635 2.08 1.94 0.23 0.42 87.4 1.99 3.04 1.92

3B1-08 16.2 368 1.21 1.94 0.23 0.63 130.8 1.79 1.76 1.65 Average 1.76 1.79 1.54

3B1-36 20.4 368 1.21 1.94 0.23 0.77 159.0 1.73 1.76 1.58 Coefficient of variation 0.12 0.21 0.14

Note: member thickness b = 102 mm (4.0 in.); overall height h = 356 mm (14.0 in.); and effective depth d = 305 mm (12.0 in.) for all specimens of Test Data 1. 1 MPa = 145.04 psi;1 mm = 0.0394 in.; and 1 kN = 2.248 kips.


Deep beam ID

fc′, MPa

h, mm

d, mm a/d

ρ, %

ρh, %

ρv , %

Vtest, kN

Vtest/Vn

Deep beam ID

fc′, MPa

h, mm

d, mm a/d

ρ, %

ρh, %

ρv , %

Vtest , kN

Vtest/Vn

ACI 318-05 CSA

Current study

ACI 318-05 CSA

Current study

1-30 21.5 762 724 0.35 0.52 — 2.45 239 3.11 2.56 1.48 3-15 21.9 381 343 0.74 1.09 2.45 — 159 2.27 1.67 1.77

1-25 24.6 635 597 0.43 0.62 — 2.45 224 2.59 2.11 1.37 3-10 22.6 254 216 1.18 1.73 2.45 — 87 1.51 1.59 1.60

1-20 21.2 508 470 0.54 0.79 — 2.45 190 2.62 2.06 1.52 4-30 22.0 762 724 0.35 0.52 0.86 — 242 3.08 2.54 1.54

1-15 21.2 381 343 0.74 1.09 — 2.45 164 2.42 1.78 1.83 4-25 21.0 635 597 0.43 0.62 0.86 — 201 2.73 2.21 1.40

1-10 21.7 254 216 1.18 1.73 — 2.45 90 1.64 1.70 1.66 4-20 20.1 508 470 0.54 0.79 0.86 — 181 2.63 2.08 1.45

2-30 19.2 762 724 0.35 0.52 — 0.86 249 4.62 2.99 1.79 4-15 22.0 381 343 0.74 1.09 0.86 — 110 1.57 1.15 1.22

2-25 18.6 635 597 0.43 0.62 — 0.86 224 3.42 2.78 1.75 4-10 22.6 254 216 1.18 1.73 0.86 — 96 1.67 1.75 1.77

2-20 19.9 508 470 0.54 0.79 — 0.86 216 3.18 2.51 1.76 5-30 18.6 762 724 0.35 0.52 0.61 0.61 240 3.62 2.99 1.68

2-15 22.8 381 343 0.74 1.09 — 0.86 140 1.93 1.42 1.54 5-25 19.2 635 597 0.43 0.62 0.61 0.61 208 3.07 2.50 1.56

2-10 20.1 254 216 1.18 1.73 — 0.86 100 1.95 2.01 1.84 5-20 20.1 508 470 0.54 0.79 0.61 0.61 173 2.51 1.98 1.46

3-30 22.6 762 724 0.35 0.52 2.45 — 276 3.43 2.83 1.66 5-15 21.9 381 343 0.74 1.09 0.61 0.61 127 1.81 1.34 1.41

3-25 21.0 635 597 0.43 0.62 2.45 — 226 3.07 2.49 1.50 5-10 22.6 254 216 1.18 1.73 0.61 0.61 78 1.36 1.43 1.45

3-20 19.2 508 470 0.54 0.79 2.45 — 208 3.16 2.50 1.68 Average 2.60 2.12 1.59

Coefficient of variation 0.31 0.25 0.10

Note: member thickness b = 76 mm (3.0 in.) and shear span a = 254 mm (10.0 in.) for all specimens of Test Data 2. Note that 1 MPa = 145.04 psi; 1 mm = 0.0394 in.; and 1 kN = 2.248 kips.



Deep beam ID

fc′, MPa a, mm a/d ρ, % ρv, %

Vtest, kN

Vtest/Vn

Deep beam ID

fc′, MPa a, mm a/d ρ, % ρv, %

Vtest, kN

Vtest/Vn

ACI 318-05 CSA

Current study

ACI 318-05 CSA

Current study

A1-1 24.6 914 2.34 3.10 0.38 222.5 2.05 2.63 1.43 C2-3 24.1 610 1.56 2.07 0.69 323.7 2.01 1.94 1.36

A1-2 23.6 914 2.34 3.10 0.38 209.1 2.02 2.57 1.40 C2-4 27.0 610 1.56 2.07 0.69 288.2 1.60 1.58 1.09

A1-3 23.4 914 2.34 3.10 0.38 222.5 2.18 2.77 1.50 C3-1 14.1 610 1.56 2.07 0.34 223.7 3.61 2.13 1.66

A1-4 24.8 914 2.34 3.10 0.38 244.7 2.27 2.90 1.56 C3-2 13.8 610 1.56 2.07 0.34 200.3 3.30 1.94 1.52

B1-1 23.4 762 1.95 3.10 0.37 278.8 2.22 2.42 1.57 C3-3 13.9 610 1.56 2.07 0.34 188.1 3.07 1.81 1.41

B1-2 25.4 762 1.95 3.10 0.37 256.6 1.88 2.07 1.33 C4-1 24.5 610 1.56 3.10 0.34 309.3 2.87 1.73 1.33

B1-3 23.7 762 1.95 3.10 0.37 284.8 2.24 2.44 1.59 C6-2 45.2 610 1.56 3.10 0.34 423.8 2.13 1.41 1.10

B1-4 23.3 762 1.95 3.10 0.37 268.1 2.14 2.33 1.52 C6-3 44.7 610 1.56 3.10 0.34 434.9 2.21 1.46 1.13

B1-5 24.6 762 1.95 3.10 0.37 241.5 1.82 2.00 1.29 C6-4 47.6 610 1.56 3.10 0.34 428.6 2.05 1.37 1.12

B2-1 23.2 762 1.95 3.10 0.73 301.1 2.41 2.62 1.71 D1-1 26.2 457 1.17 1.63 0.46 301.1 1.40 1.14 1.02

B2-2 26.3 762 1.95 3.10 0.73 322.2 2.28 2.52 1.61 D1-2 26.1 457 1.17 1.63 0.46 356.7 1.67 1.36 1.21

B2-3 24.9 762 1.95 3.10 0.73 334.9 2.50 2.74 1.77 D1-3 24.5 457 1.17 1.63 0.46 256.6 1.28 1.03 0.87

B6-1 42.1 762 1.95 3.10 0.37 379.3 1.67 1.98 1.23 D2-1 24.0 457 1.17 1.63 0.61 290.0 1.47 1.18 0.99

C1-1 25.6 610 1.56 2.07 0.34 277.7 2.46 1.59 1.10 D2-2 25.9 457 1.17 1.63 0.61 312.2 1.47 1.20 1.06

C1-2 26.3 610 1.56 2.07 0.34 311.1 2.68 1.73 1.20 D2-3 24.8 457 1.17 1.63 0.61 334.4 1.65 1.33 1.15

C1-3 24.0 610 1.56 2.07 0.34 245.9 2.33 1.48 1.04 D2-4 24.5 457 1.17 1.63 0.61 334.9 1.67 1.34 1.14

C1-4 29.0 610 1.56 2.07 0.34 285.9 2.24 1.47 1.09 D3-1 28.2 457 1.17 2.44 0.92 394.9 1.71 1.31 1.08

C2-1 23.6 610 1.56 2.07 0.69 290.0 1.83 1.77 1.25 D4-1 23.1 457 1.17 1.63 1.22 312.2 1.65 1.31 1.06

C2-2 25.0 610 1.56 2.07 0.69 301.1 1.80 1.75 1.22 Average 2.10 1.85 1.29


Note: member thickness b = 203 mm (8.0 in.); overall height h = 457 mm (18.0 in.); and effective depth d = 390 mm (15.4 in.) for all specimens of Test Data 3. Note that 1 MPa =145.04 psi; 1 mm = 0.0394 in.; and 1 kN = 2.248 kips.


Deep beam ID

fc′, MPa

b, mm a/d

ρ, %

ρh, %

ρv,%

Vtest , kN

Vtest/Vn

Deep beam ID

fc′, MPa

b, mm a/d

ρ,%

ρh,%

ρv,%

Vtest , kN

Vtest /Vn

ACI 318-05 CSA

Current study

ACI 318-05 CSA

Current study

N4200 23.7 130 0.85 1.56 0.00 0.00 265.2 1.29 1.00 0.88 H43A0 50.7 120 1.25 1.29 0.00 0.13 213.6 0.90 0.88 0.99

N42A2 23.7 130 0.85 1.56 0.43 0.12 284.1 1.12 1.07 0.94 H43A1 50.7 120 1.25 1.29 0.23 0.13 260.4 1.09 1.08 1.21

N42B2 23.7 130 0.85 1.56 0.43 0.22 377.0 1.48 1.42 1.26 H43A2(2) 50.7 120 1.25 1.29 0.47 0.13 276.6 1.19 1.14 1.28

N42C2 23.7 130 0.85 1.56 0.43 0.34 357.5 1.41 1.34 1.20 H43A3 50.7 120 1.25 1.29 0.94 0.13 291.0 1.25 1.20 1.37

H4100 49.1 130 0.50 1.56 0.00 0.00 642.2 1.70 1.03 1.83 H45A2(2) 50.7 120 2.00 1.29 0.46 0.13 165.0 1.14 1.43 1.24

H41A2(1) 49.1 130 0.50 1.56 0.43 0.12 713.1 1.89 1.15 1.55 U41A0 73.6 120 0.50 1.29 0.00 0.13 438.0 1.03 0.73 1.44

H41B2 49.1 130 0.50 1.56 0.43 0.22 705.9 1.87 1.13 1.53 U41A1 73.6 120 0.50 1.29 0.23 0.13 541.8 1.27 0.90 1.54

H41C2 49.1 130 0.50 1.56 0.43 0.34 708.5 1.87 1.14 1.54 U41A2 73.6 120 0.50 1.29 0.47 0.13 548.4 1.28 0.91 1.35

H4200 49.1 130 0.85 1.56 0.00 0.00 401.1 1.06 0.87 1.04 U41A3 73.6 120 0.50 1.29 0.94 0.13 546.6 1.28 0.90 1.35

H42A2(1) 49.1 130 0.85 1.56 0.43 0.12 488.2 1.29 1.06 1.20 U42A2 73.6 120 0.85 1.29 0.47 0.13 417.6 1.20 1.19 1.31

H42B2(1) 49.1 130 0.85 1.56 0.43 0.22 456.3 1.21 0.99 1.12 U42B2 73.6 120 0.85 1.29 0.47 0.24 410.4 1.18 1.17 1.29

H42C2(1) 49.1 130 0.85 1.56 0.43 0.34 420.6 1.11 0.91 1.04 U42C2 73.6 120 0.85 1.29 0.47 0.37 408.0 1.17 1.16 1.28

H4300 49.1 130 1.25 1.56 0.00 0.00 337.4 1.08 1.18 1.22 U43A0 73.6 120 1.25 1.29 0.00 0.13 291.0 1.22 1.20 1.35

H43A2(1) 49.1 130 1.25 1.56 0.43 0.12 347.1 1.14 1.22 1.26 U43A1 73.6 120 1.25 1.29 0.23 0.13 310.2 1.30 1.28 1.43

H43B2 49.1 130 1.25 1.56 0.43 0.22 380.9 1.25 1.33 1.38 U43A2 73.6 120 1.25 1.29 0.47 0.13 338.4 1.46 1.40 1.57

H43C2 49.1 130 1.25 1.56 0.43 0.34 402.4 1.32 1.41 1.46 U43A3 73.6 120 1.25 1.29 0.94 0.13 333.0 1.43 1.38 1.55

H4500 49.1 130 2.00 1.56 0.00 0.00 112.5 0.57 0.88 0.66 U45A2 73.6 120 2.00 1.29 0.47 0.13 213.6 1.48 1.41 1.56

H45A2 49.1 130 2.00 1.56 0.43 0.12 210.6 1.08 1.65 1.23 N33A2 23.7 130 1.25 1.56 0.43 0.12 228.2 1.27 1.41 0.97

H45B2 49.1 130 2.00 1.56 0.43 0.22 237.3 1.25 1.86 1.38 N43A2 23.7 130 1.25 1.56 0.43 0.12 254.8 1.42 1.58 1.09

H45C2 49.1 130 2.00 1.56 0.43 0.34 235.3 1.24 1.85 1.37 N53A2 23.7 130 1.25 1.56 0.43 0.12 207.4 1.15 1.29 0.88

H41A0 50.7 120 0.50 1.29 0.00 0.13 347.4 0.98 0.59 1.10 H31A2 49.1 130 0.50 1.56 0.43 0.12 745.6 1.97 1.20 1.62

H41A1 50.7 120 0.50 1.29 0.23 0.13 397.8 1.12 0.67 1.07 H32A2 49.1 130 0.85 1.56 0.43 0.12 529.8 1.40 1.15 1.31

H41A2(2) 50.7 120 0.50 1.29 0.47 0.13 490.2 1.38 0.83 1.17 H33A2 49.1 130 1.25 1.56 0.43 0.12 377.7 1.24 1.33 1.37

H41A3 50.7 120 0.50 1.29 0.94 0.13 454.8 1.28 0.77 1.09 H51A2 49.1 130 0.50 1.56 0.43 0.12 702.0 1.86 1.13 1.51

H42A2(2) 50.7 120 0.85 1.29 0.47 0.13 392.4 1.13 1.12 1.25 H52A2 49.1 130 0.85 1.56 0.43 0.12 567.5 1.50 1.23 1.40

H42B2(2) 50.7 120 0.85 1.29 0.47 0.24 360.6 1.04 1.03 1.15 H53A2 49.1 130 1.25 1.56 0.43 0.12 362.7 1.19 1.27 1.30

H42C2(2) 50.7 120 0.85 1.29 0.47 0.37 373.8 1.07 1.06 1.19 Average 1.29 1.16 1.28


Note: overall height h = 560 mm (22.0 in.) and effective depth d = 500 mm (19.7 in.) for all specimens of Test Data 4. Note that 1 MPa = 145.04 psi; 1 mm = 0.0394 in.; and 1 kN = 2.248 kips.


study) are compared with the measured capacity of the 214deep beam test results. The size of this test database and theuse of these two code provisions are sufficient to obtainvaluable insight into the design and behavior of deep beamsfrom a strut-and-tie perspective. It would be useful to expandthis evaluation to include even more test data and other codeprovisions, such as the CEB-FIP Model Code,3 but this isbeyond the scope of the present work. The details of the test

specimens and strength ratios (Vtest/Vnmethod) are presented

for each group of test results in Tables 1 through 8 andcollectively in Fig. 2.

Figure 2(a) and (b) show that the predictions by the strut-and-tie model approach of ACI 318-05 and the Canadiancode are very conservative and scattered with mean values of1.77 and 1.64, and coefficients of variation (COVs) of 0.32and 0.35, respectively. One cause of this extreme conservatism


Deep beam ID

fc′, MPa d, mm a/d

ρ, %

ρh, %

ρv, %

Vtest, kN

Vtest/Vn

Deep beam ID

fc′, MPa d, mm a/d

ρ, %

ρh, %

ρv, %

Vtest, kN

Vtest/Vn

ACI 318-05 CSA

Current study

ACI 318-05 CSA

Current study

ACI-I 32 791 1.16 1.27 0.35 0.31 1357 1.41 1.36 1.48 STM-M 28 801 1.14 1.25 — 0.1 1277 1.56 1.42 1.40

STM-I 32 718 1.27 1.4 0.13 0.31 1134 1.33 1.29 1.23 Average 1.47 1.37 1.38

STM-H 28 801 1.14 1.25 0.06 0.31 1286 1.58 1.43 1.40 Coefficient of variation 0.08 0.05 0.08

Note: member thickness b = 305 mm (12.0 in.); overall height h = 915 mm (36.0 in.); and shear span a = 915 mm (36.0 in.) for all specimens of Test Data 5. Note that 1 MPa =145.04 psi; 1 mm = 0.0394 in.; and 1 kN = 2.248 kips.


Deep beam ID

fc′, MPa

b, mm

d, mm a/d

ρ,%

ρh,%

ρv,%

Vtest, kN

Vtest/Vn

Deep beam ID

fc′, MPa

b, mm

d, mm a/d

ρ, %

ρh, %

ρv,%

Vtest, kN

Vtest/Vn

ACI 318-05 CSA

Current study

ACI 318-05 CSA

Current study

A1 22.0 150 370 1.42 2.79 0.1 0.28 251 1.98 1.45 1.47 B4 32.4 150 375 0.81 2.04 — — 459 1.73 0.99 1.28

A2 22.0 150 370 1.42 2.79 0.1 0.28 237 1.87 1.37 1.39 HA1 50.3 100 380 1.57 4.08 0.15 0.38 265 1.18 1.32 1.13

A3 22.0 150 370 1.42 2.79 — — 221 1.74 1.28 1.30 HA3 50.3 100 380 1.43 4.08 — — 292 1.52 1.26 1.29

A4 22.0 150 370 1.42 2.79 — — 196 1.54 1.13 1.15 HB1 50.3 100 380 0.90 4.08 0.15 0.67 484 2.16 1.08 1.33

B1 32.4 150 375 0.89 2.04 0.1 0.23 456 1.72 1.10 1.27 HB3 50.3 100 380 0.82 4.08 — — 460 2.06 0.93 1.25

B2 32.4 150 375 0.89 2.04 0.1 0.23 426 1.60 1.02 1.19 Average 1.74 1.16 1.28

B3 32.4 150 375 0.81 2.04 — — 468 1.76 1.01 1.30 Coefficient of variation 0.15 0.14 0.08

Note: overall height h = 460 mm (18.1 in.) for all specimens of Test Data 6. Note that 1 MPa = 145.04 psi; 1 mm = 0.0394 in.; and 1 kN = 2.248 kips.


Deep beam ID

fc′, MPa

a, mm a/d

ρ,%

ρv,%

Vtest , kN

Vtest /Vn

Deepbeam ID

fc′, MPa

a, mm a/d

ρ, %

ρv,%

Vtest , kN

Vtest/Vn

ACI 318-05 CSA

Current study

ACI 318-05 CSA

Current study

A-0.27-2.15 58.8 125 0.27 1.23 0.48 675 2.08 0.93 1.89 D-1.08-2.15 48.2 500 1.08 1.23 0.48 270 1.21 1.10 1.11

A-0.27-3.23 51.6 125 0.27 1.23 0.48 630 2.07 0.99 1.88 D-1.08-3.23 44.1 500 1.08 1.23 0.48 280 1.37 1.21 1.15

A-0.27-4.30 53.9 125 0.27 1.23 0.48 640 2.06 0.96 1.87 D-1.08-4.30 46.8 500 1.08 1.23 0.48 290 1.33 1.20 1.19

A-0.27-5.38 57.3 125 0.27 1.23 0.48 630 1.97 0.89 1.80 D-1.08-5.38 48.0 500 1.08 1.23 0.48 290 1.30 1.18 1.19

B-0.54-2.15 56.0 250 0.54 1.23 0.48 468 1.48 0.87 1.39 E-1.62-3.23 50.6 750 1.62 1.23 0.48 220 1.41 1.67 1.35

B-0.54-3.23 45.7 250 0.54 1.23 0.48 445 1.56 0.93 1.46 E-1.62-4.30 44.6 750 1.62 1.23 0.48 190 1.38 1.59 1.17

B-0.54-4.30 53.9 250 0.54 1.23 0.48 500 1.61 0.93 1.52 E-1.62-5.38 45.3 750 1.62 1.23 0.48 173 1.24 1.43 1.07

B-0.54-5.38 53.0 250 0.54 1.23 0.48 480 1.56 0.89 1.45 F-2.16-4.30 41.1 1000 2.16 1.23 0.48 150 1.57 2.36 1.23

C-0.81-2.15 51.2 375 0.81 1.23 0.48 403 1.33 1.14 1.24 G-2.70-5.38 42.8 1250 2.7 1.23 0.48 105 1.31 2.61 1.08

C-0.81-3.23 44.0 375 0.81 1.23 0.48 400 1.46 1.31 1.23 Average 1.54 1.27 1.38


Note: member thickness b = 110 mm (4.3 in.) and overall height h = 500 mm (19.7 in.) for all specimens of Test Data 7. Note that 1 MPa = 145.04 psi; 1 mm = 0.0394 in.; and 1 kN = 2.248 kips.


Deep beam

IDfc′,

MPaρ, %

ρv,%

Vtest , kN

Vtest/Vn Deep beam

IDfc′,

MPaρ,%

ρv ,%

Vtest , kN

Vtest /Vn Deep beam

IDfc′,

MPaρ,%

ρv,%

Vtest , kN

Vtest/Vn

ACI 318-05 CSA

Current study

ACI 318-05 CSA

Current study

ACI 318-05 CSA

Current study

1 39.0 2.67 2.65 478.6 1.59 2.62 1.55 6 29.4 2.67 2.65 368.8 1.63 2.56 1.61 11 32.3 2.67 2.65 368.8 1.49 2.36 1.46

2 41.4 2.67 2.65 489.7 1.54 2.56 1.50 7 32.1 2.67 2.65 391 1.58 2.52 1.56 12 33.2 2.67 2.65 330.9 1.30 2.07 1.27

3 42.7 2.67 2.65 511.1 1.55 2.60 1.51 8 33.9 2.67 2.65 359.9 1.38 2.22 1.35 Average 1.60 2.56 1.57

4 27.5 2.67 2.65 439.9 2.08 3.23 2.06 9 34.4 2.67 2.65 395.4 1.49 2.40 1.46 Coefficient of variation 0.13 0.12 0.14

5 28.7 2.67 2.65 426.6 1.93 3.03 1.91 10 31.0 2.67 2.65 386.6 1.63 2.56 1.60

Note: member thickness b = 203 mm (8.0 in.); overall height h = 508 mm (20.0 in.); shear span a = 914 mm (36.0 in.); and shear span-depth ratio a/d = 2.15 for all specimens of TestData 8. Note that 1 MPa = 145.04 psi; 1 mm = 0.0394 in.; and 1 kN = 2.248 kips.


is that the effective depth of the concrete strut is very conser-vative. The other cause is that the tensile strain that is used toestimate the strength of the diagonal concrete strut in theCanadian code is too large, especially for deep beams withrelatively large a/d ratios.

It is also observed that the predictions by ACI 318-05 andthe Canadian code are quite scattered with respect to concretestrengths. The conservatism decreases as concrete strengthsincreases in both ACI 318-05 and the Canadian code. It is tobe noted that deep beams exhibit abrupt failures due to splittingof the concrete strut when web reinforcement is not sufficientand the concrete strength is higher. The examined codeprocedures, however, cannot capture these effects.

Figure 2(c) shows that the predictions by the proposedmethod are slightly conservative, but good accuracy isobtained with no significant trend with concrete strengths,amount of web reinforcement, or a/d ratio ranging from 0.27to 2.7. There was no significant decrease in conservatismwith the use of flatter struts down to 17.1 degrees thatcorrespond to a beam with an a/d ratio of 2.7. According tothe proposed approach, a limit on the angle between the axesof struts and ties of 25 degrees is not required for the designof deep beams. The mean of the ratio of the measured-to-calculated strength by the proposed method is 1.41 with a

COV of 0.18. Given the breadth of the database, these resultsare considered to be sufficiently good to suggest that theproposal method provides a reliable and safe means ofpredicting the capacity of deep beams.

As previously mentioned, the effect of using differentcompression softening relationships on the predictive capabilityof the authors’ approach was also investigated. For themodels proposed by Vecchio and Collins14-16,28 in 1982, 1986,and 1993, the mean strength ratios are 1.42, 1.39, and 1.47with COVs of 0.19, 0.20, and 0.19. This illustrates that theproposed method is applicable for a range of softeningmodels as is further examined in the following.

Softening coefficientThe compressive stress capacity in a strut is taken in most

design codes1-3 as the product of a softening coefficient andthe compressive cylinder strength of the concrete. Thesoftening coefficient is used to account for the reduction incompressive strength due to transverse tensile straining andthe shapes of the true stress fields in the real structure.Several approaches have been proposed for calculating thesoftening coefficient. A number of researchers haveconducted independent test programs and have proposed theconstitutive models to determine the degree of the softeningeffects and the parameters that influence it. The softeningcoefficients used in the authors’ proposed method arecompared with the coefficients that could be calculated if theconstitutive model proposed by Vecchio and Collins14 hadbeen used.

The constitutive model proposed by Vecchio andCollins,15 which was based on tests on reinforced concretepanels, uses the Hognestad parabola as the base curve todescribe the uniaxial compressive response of concrete. Asshown in Eq. (30), the softening parameter β in this model istaken as a function of ε1, the average principal tensile strain,and ε2, the average principal compressive strain. They alsoformulated a simplified model15 in which the softeningparameter is a function of ε1 and the strain at peak compressivestress ε0, as shown in Eq. (31).

Fig. 2—Ratio of measured-to-calculated strength by differentmethods.

Fig. 3—Evaluation and comparison of softening coefficients.


(30)

(31)

Vecchio and Collins16 found the softening effect might bemore pronounced in high-strength concrete, and proposed asomewhat more complex model for high-strength concreteas follows

(32)

where

(33)

(34)

Equation (32) was further simplified as follows16

(35)

where

(36)

The softening parameters of Eq. (30) through (32) and(35), and the softening coefficient of Eq. (18) with respect toε1 was evaluated and compared with the ACI 318-05strength reduction factor for struts as shown in Fig. 3. Thestrength of concrete struts by ACI 318-05 provides reasonableestimates for normal-strength concrete, but the softeningcoefficients of most high-strength concrete specimenscalculated by the proposed method were less than thecorresponding ACI 318-05 code coefficient, which is 0.51 inthese cases. Consequently, the ACI 318-05 coefficient isconsidered to become less conservative as the strength of theconcrete increases.

CONCLUSIONS AND IMPLICATIONS ON PRACTICEThe capacity of 214 tested deep beams were predicted by

the strut-and-tie provisions in two codes-of-practice and bythe softened strut-and-tie model proposed by the authors.From a comparison of measured and predicted strengths, thefollowing conclusions can be made:

1. The calculated capacities by the proposed method areboth accurate and conservative with little scatter or trends fordeep beams over a wide range in concrete strengths, variouscombinations and amounts of web reinforcements, andvalues of a/d that ranged from 0.27 to 2.7. The predictions bythe proposed method are sufficiently conservative and accurateto conclude that it provides a safe and reliable means ofcalculating the capacity of deep beams;

β1

0.85 0.27ε1

ε2

----–

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

β1

0.8 0.34ε1

ε0

----+

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

β1

1 KcKf+---------------------=

Kc 0.35ε1–

ε2

-------- 0.28–⎝ ⎠

⎛ ⎞0.80

1.0≥=

Kf 0.1825 fc′ 1.0 (fc′ in MPa)≥=

β1

1 Kc+---------------=

Kc 0.27ε1

ε0

---- 0.37–⎝ ⎠

⎛ ⎞=

2. The strut-and-tie model approaches of ACI 318-02 andthe Canadian code yielded generally conservative but scatteredresults. Two main causes of this extreme conservatism are:1) the difficulty in estimating the effective depth of the tophorizontal strut; and 2) the use of large tensile strains incalculating the capacity of the diagonal strut in the case ofthe CSA code. To provide for more accurate designs, it isrecommended to use Eq. (24) to estimate the location of thetop node of the direct strut-and-tie model for the design of thedeep beams using code provisions. It is also recommended thatEq. (11-23) in the Canadian code should not be used for thedeep beams when the inclination of diagonal strut θs is lessthan 25 degrees;

3. The conservatism of STM provisions in ACI 318-05 andthe Canadian code decreases as concrete strengths increases.Area reduction factors provide an effective means ofaccounting for the sudden splitting failures that can occurespecially in high-strength concrete deep beams;

4. There was no significant decrease in conservatism withthe use of flatter struts down to 17.1 degrees for the proposedapproach. And it was also observed for ACI 318-05 that theflatter struts down to 18.6 degrees have little effect on thedegree of conservatism; and

5. The strengths of concrete struts in ACI 318-05 providereasonable estimates for normal-strength concrete. However,the softening coefficients of most high-strength concretespecimens calculated by the proposed method were less thanthose by ACI 318-05, that is, 0.51. Thus, it becomes lessconservative to apply the strut strength provisions of ACI 318-05in the design of high-strength concrete deep beams.

These conclusions are applicable to the design of deepbeams by the strut-and-tie method and should be takeninto consideration in any future adjustments to codeprovisions and in the development of design guidelines.Additional analytical and experimental investigations arerequired for furthering the understanding of the applica-bility and limitations of the strut-and-tie method for thebroad range of the discontinuity regions that can bedesigned by this method.

NOTATIONEc, Ed = secant moduli of horizontal and diagonal concrete strutsEs, Esh, Esv = secant moduli of longitudinal, horizontal, and vertical steel

tiesFc, Fd = compressive force in horizontal and diagonal strutsFh, Fv = tensile force in horizontal and vertical ties in webfc, fd = compressive stress in horizontal and diagonal strutsfs, fsh, fsv = tensile stress of longitudinal, horizontal, and vertical tiesT = tensile force of longitudinal steel tieVtest, Vn = experimental failure load and nominal failure strength wc, wd = effective widths of horizontal and diagonal concrete strutswp = width of loading plateβ = softening parameter of compression concreteε1, ε2 = principal tensile and compressive strains εc, εd = compressive strains in horizontal and diagonal strutsεr = tensile strain of direction perpendicular to diagonal strut εs, εh, εv = tensile strains in longitudinal, horizontal, and vertical ties φc1, φc2 = area reduction factors for diagonal concrete strutθ = angle of inclination of diagonal strut with respect to

horizontal axisρh, ρv = ratio of horizontal and vertical web reinforcementζ = softening coefficient of concrete strut

REFERENCES1. AASHTO, “AASHTO LRFD Bridge Design Specifications,” American

Association of State Highway and Transportation Officials, 2004.2. ACI Committee 318, “Building Code Requirements for Structural

Concrete (ACI 318-05) and Commentary (318R-05),” American ConcreteInstitute, Farmington Hills, MI, 2005, 430 pp.

ACI Structural Journal/November-December 2007666

3. CEB-FIP, “CEB-FIP Model Code for Concrete Structures,” Comite Euro-International du Beton, Federation International de la Precontrainte, 1993.

4. CSA Committee A23.3, “Design of Concrete Structures: Structures(Design)—A National Standard of Canada,” Canadian Standards Association,1994.

5. Hillerborg, A., Strip Method of Design, E&FN Spon, London, UK,1975.

6. Hwang, S.-J.; Lu, W.-Y.; and Lee, H.-J., “Shear Strength Predictionfor Deep Beams,” ACI Structural Journal, V. 97, No. 3, May-June 2000,pp. 367-376.

7. Hwang, S.-J.; Lu, W.-Y.; and Lee, H.-J., “Shear Strength Predictionfor Reinforced Concrete Corbels,” ACI Structural Journal, V. 97. No. 4,July-Aug. 2000, pp. 543-552.

8. Hwang, S.-J.; Fang, W.-H.; Lee, H.-J.; and Yu, H.-W., “AnalyticalModel for Predicting Shear Strength of Squat Walls,” Journal of StructuralEngineering, ASCE, V. 127, No. 1, 2001, pp. 43-50.

9. Hwang, S.-J., and Lee, H.-J., “Analytical Model for Predicting ShearStrengths of Exterior Reinforced Concrete Beam-Column Joints for SeismicResistance,” ACI Structural Journal, V. 96, No. 5, Sept.-Oct. 1999, pp. 846-858.

10. Hwang, S.-J., and Lee, H.-J., “Analytical Model for Predicting ShearStrengths of Interior Reinforced Concrete Beam-Column Joints for SeismicResistance,” ACI Structural Journal, V. 97, No. 1, Jan.-Feb. 2000, pp. 34-44.

11. Schäfer, K., “Strut-and-Tie Models for the Design of StructuralConcrete,” Notes from Workshop, Department of Civil Engineering,National Cheng Kung University, Tainan, Taiwan, 140 pp.

12. Vecchio, F. J., “Nonlinear Finite Element Analysis of ReinforcedConcrete Membranes,” ACI Structural Journal, V. 86, No. 1, Jan.-Feb.1989, pp. 26-35.

13. Hsu, T. T. C., and Zhang, L. X. B., “Nonlinear Analysis of MembraneElements by Fixed-Angle Softened-Truss Model,” ACI Structural Journal,V. 94, No. 5, Sept.-Oct. 1997, pp. 483-492.

14. Vecchio, F. J., and Collins, M. P., “Response of Reinforced Concreteto In-Plane Shear and Normal Stresses,” Report No. 82-03, University ofToronto, Toronto, ON, Canada, 1982.

15. Vecchio, F. J., and Collins, M. P., “Modified Compression Field Theoryfor Reinforced Concrete Elements Subjected to Shear,” ACI JOURNAL,Proceedings V. 83, No. 2, Mar.-Apr. 1986, pp. 219-231.

16. Vecchio, F. J., and Collins, M. P., “Compression Response ofCracked Reinforced Concrete,” Journal of Structural Engineering, ASCE,V. 119, No. 12, 1993, pp. 3590-3610.

17. Foster, S. J., and Gilbert, R. I., “The Design of Nonflexural Memberswith Normal and High-Strength Concretes,” ACI Structural Journal, V. 93,No. 1, Jan.-Feb. 1996, pp. 3-10.

18. Paulay, T., and Priestley, M. J. N., Seismic Design of ReinforcedConcrete and Masonry Buildings, John Wiley and Sons, 1992, 744 pp.

19. Oh, J. K., and Shin, S. W., “Shear Strength of Reinforced High-Strength Concrete Deep Beams,” ACI Structural Journal, V. 98, No. 2,Mar.-Apr. 2001, pp. 164-173.

20. Rogowsky, D. M.; MacGregor, J. G.; and Ong, S. Y., “Tests ofReinforced Concrete Deep Beams,” ACI JOURNAL, Proceedings V. 83, No. 4,July-Aug. 1986, pp. 614-623.

21. Quintero-Febres, C. G.; Parra-Montesinos, G.; and Wight, J. K.,“Strength of Struts in Deep Concrete Members Designed Using Strut-and-Tie Method,” ACI Structural Journal, V. 103, No. 4, July-Aug. 2006,pp. 577-586.

22. Kong, F. K.; Robins, P. J.; and Cole, D. F., “Web ReinforcementEffects on Deep Beams,” ACI JOURNAL, Proceedings V. 67, No. 12, Dec.1970, pp. 1010-1017.

23. Clark, A. P., “Diagonal Tension in Reinforced Concrete Beams,” ACIJOURNAL, Proceedings V. 48, No. 10, Oct. 1951, pp. 145-156.

24. Smith, K. N., and Vantsiotis, A. S., “Shear Strength of Deep Beams,”ACI JOURNAL, Proceedings V. 79, No. 3, May-June 1982, pp. 201-213.

25. Anderson, N. S., and Ramirez, J. A., “Detailing of Stirrup Reinforcement,”ACI Structural Journal, V. 86, No. 5, Sept.-Oct. 1989, pp. 507-515.

26. Tan, K. H.; Kong, F. K.; Teng, S.; and Guan, L., “High-StrengthConcrete Deep Beams with Effective Span and Shear Span Variations,”ACI Structural Journal, V. 92, No. 4, July-Aug. 1995, pp. 395-405.

27. Aguilar, G.; Matamoros, A. B.; Parra-Montesinos, G.; Ramirez, J. A.;and Wight, J. K., “Experimental Evaluation of Design Procedures for ShearStrength of Deep Reinforced Concrete Beams,” ACI Structural Journal,V. 99. No. 4, July-Aug. 2002, pp. 539-548.

28. Vecchio, F. J., and Aspiotis, J., “Response of High-Strength ConcreteElements Subjected to Shear,” ACI Structural Journal, V. 91, No. 4, July-Aug. 1994, pp. 423-433.

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