An Indeterminate Strut - Tie Model and Load Distribution Ratio for RC Deep Beams (I) - Model and Load Distribution Ratio

1. INTRODUCTIONUnlike slender beams, the ultimate strength of simplysupported reinforced concrete deep beams (hereinafter,simple deep beams) with a ratio of shear span-to-effective depth of less than 2.5 is generally governedby their shear capacities. Since the shear failures ofsimple deep beams are brittle, the shear designs must beconducted cautiously to prevent brittle failures. Thestructural behavior of simple deep beams iscomplicated and is mainly controlled by the mechanicalrelationships between the primary design variablesincluding shear span-to-effective depth ratio a/d,flexural reinforcement ratio ρ, and the compressivestrength of concrete f ′c . To closely examine the

Advances in Structural Engineering Vol. 14 No. 6 2011 1031

An Indeterminate Strut-Tie Model and Load

Distribution Ratio for RC Deep Beams - (I)

Model & Load Distribution Ratio

Byung-Hun Kim1 and Young-Mook Yun2,* 1Structural Department, Hyundai Engineering Co. LTD, Seoul 158-050, Korea

2School of Architecture & Civil Engineering, Daegu 702-701, Korea

(Received: 8 February 2010; Received revised form: 30 December 2010; Accepted: 4 January 2011)

Abstract: The ultimate strength of simply supported reinforced concrete deep beamsis governed by the capacity of the shear resistance mechanism. The structural behaviorof the deep beams is controlled mainly by the mechanical relationships between theprimary design variables including shear span-to-effective depth ratio, flexuralreinforcement ratio, and the compressive strength of concrete. In this study, a simpleindeterminate strut-tie model which reflects all characteristics of the ultimate strengthand behavior of the deep beams was presented. A load distribution ratio, defined as thefraction of load transferred by a truss mechanism, was also proposed to help structuraldesigners perform the rational design of deep beams by using the strut-tie modelapproaches of current design codes. In the determination of the load distribution ratio,a concept of balanced shear reinforcement ratio which ensures the ductile shear designof the deep beams was introduced, and the effect of the primary design variables wasreflected through numerous numerical analyses of the presented indeterminate strut-tiemodel. In the companion paper, the validity of the presented model and loaddistribution ratio was examined by applying them to the evaluation of ultimatestrength of 234 simply supported reinforced concrete deep beams tested to failure.

Key words: reinforced concrete, deep beam, indeterminate strut-tie model, load distribution ratio, ultimate strength.

complicated structural behavior of simple deep beams,analytical and experimental works have beenperformed. However, any satisfactory theories orapproaches have not yet been established.

Recently, a strut-tie model approach known as arational design method for structural concrete withdisturbed regions has been suggested for the sheardesigns of simple deep beams (Hwang et al. 2000;Foster and Malik 2002; Hwang and Lu 2002;Matamoros and Wong 2003; Quintero-Febres et al.2006; Park and Kuchma 2007; Tjhin and Kuchma 2007;Ashour and Yang 2008), and the approach has beenaccepted in the current design codes including the CSA(1984), NZS 3101 (1995), BS8110 (1997), FIB (1999),

*Corresponding author. Email address: [email protected]; Fax: +82-53-950-6564; Tel: +82-53-950-5610.

AASTHO-LRFD (2007), and ACI 318M-08 (2008).However, for all that the design codes have beenestablished on the basis of research results of simpledeep beams, an appropriate strut-tie model thatrepresents a true load transfer mechanism for simpledeep beams and reflects the effects of the primarydesign variables on shear behavior has not beenprovided. Furthermore, the fundamental concept that theload acting on top of a simple deep beam must betransferred to supports by concrete and reinforcing barshas not been satisfied.

To improve the problem, an indeterminate strut-tiemodel that includes both the arch and truss load transfermechanisms must be used for the analysis and design ofsimple deep beams. However, since the cross-sectionalforces of struts and ties in an indeterminate strut-tiemodel depend on the stiffness of struts and ties, the loadtransferred by an arch or truss mechanism defined in thisstudy as a load distribution ratio of an indeterminatestrut-tie model must be determined rationally to employthe strut-tie model approaches of the current designcodes in practice. To find a solution to the problem,much research has been conducted regarding thedevelopment of analysis and design approaches withindeterminate strut-tie models (Alshegeir 1992; Yun2000; Tjhin and Kuchma 2002; Park et al. 2005), ofindeterminate strut-tie models for simple deep beams(Hwang et al. 2000; Foster and Malik 2002; Matamorosand Wong 2003; Bakir and Boduroglu 2005; Alcocerand Uribe 2008), and of load distribution ratios ofindeterminate strut-tie models for simple deep beams(Foster and Gilbert 1998; FIB 1999). In the previousstudies, however, the analysis and design of simple deepbeams were based on the direct application of finiteelement material nonlinear analyses of indeterminatestrut-tie models, and the load distribution ratios wereproposed based on the researchers’ experience andsubjectivity.

In this study, a simple indeterminate strut-tie modelreflecting all characteristics of the ultimate strength andcomplicated structural behavior is presented for thedesign of simple deep beams. In addition, a loaddistribution ratio obtained by conducting numerousfinite element material nonlinear analyses of a singletype of indeterminate strut-tie model with thechangeable primary design variables is presented. Toensure the ductile shear failure design of reinforcedconcrete deep beams, a concept of balanced shearreinforcement ratio requiring a simultaneous failure ofinclined concrete strut and vertical steel tie is introducedin the determination of the ratio. The presented loaddistribution ratio may help structural designers performthe design of simple deep beams with the strut-tie model

approaches of the current design codes since it providesa reasonable basis to transform an indeterminate strut-tie model into a determinate one.

2. STRUT-TIE MODEL AND LOADDISTRIBUTION RATIO OF PREVIOUSSTUDIES

The strut-tie model design of simple deep beams isusually conducted by a determinate strut-tie modelrepresenting an arch mechanism, shown in Figure 1(a),in which an external concentrated load is directlytransferred to the supports by an inclined strut or a trussmechanism, shown in Figure 1(b), in which an externalconcentrated load is transferred to the supports by thecombination of inclined struts and a vertical tie. Thecross-sectional forces of struts and ties in these types ofstrut-tie models are determined regardless of thestiffness of the struts and ties.

The CSA (1984) and AASHTO-LRFD (2007) havesuggested a basic concept of a strut-tie model thatsatisfies equilibrium and constitutive relationships, and

1032 Advances in Structural Engineering Vol. 14 No. 6 2011

An Indeterminate Strut-Tie Model and Load Distribution Ratio for RC Deep Beams - (I) Model & Load Distribution Ratio

aL

d

P P

aL

d

P P

(a) Strut-tie model representing arch mechanism

(b) Strut-tie model representing truss mechanism

Figure 1. Determinate strut-tie models for simply supported

deep beams

they have allowed the design of simple deep beams withthe strut-tie model shown in Figure 1(a). This hasinfluenced the ACI 318M-08 (2008) to allow the samemodel for simple deep beams with the requirement thatthe angle between a concrete strut and a tie be greaterthan 25 degrees. When the requirement on the angle isconsidered, the strut-tie model shown in Figure 1(a)can be used for simple deep beams with a shear span-to-effective depth ratio a /d of less than 1.93 (a/z = 2.14,z = 0.9d, d = 0.9h, h = depth). In addition, according tothe design book of the ACI Subcommittee 445-1(2002), the simple deep beams with a/d ≥ 1.93 can bedesigned by using the strut-tie model shown in Figure 1(b). In the CSA, AASHTO-LRFD, and ACI318M-08, any additional provisions for the applicationof indeterminate strut-tie models to structural concreteincluding simple deep beams are not provided.

A provision for the selection of a strut-tie modelaccording to the shear span-to-moment arm length ratioa/d of simple deep beams was recommended by theFIB (1999). In the ranges of a/z ≤ 0.5 and a/z ≥ 2.0, thestrut-tie models representing an arch mechanism and atruss mechanism as shown in Figures 2(a) and 2(c),were suggested. Additionally, a strut-tie modelrepresenting a combination of arch and trussmechanisms, as shown in Figure 2(b), was suggested inthe range of 0.5 < a/z < 2.0. Since the strut-tie model inFigure 2(b) is the first-order indeterminate trussstructure, a load distribution ratio was proposed tocalculate the cross-sectional forces of struts and ties bysimply employing the force equilibrium equations atnodes. With the load distribution ratio α of Eqn 1,varying linearly as a function of a/z, the cross-sectionalforce of a vertical steel tie Pw in the truss mechanism ofFigure 2(a) is directly obtained from the followingequation:

(1)

where P is a vertically applied load and Nsd is ahorizontally applied axial load.

Similar to the FIB’s strut-tie models, three types ofmodels according to the ratio of a/z were suggested byFoster and Gilbert (1998). They were the twodeterminate strut-tie models of Figures 2(a) and 2(c) in

the ranges of a/z ≤ 1 and , and an indeterminatestrut-tie model of Figure 2(b) in the rangeof . The load distribution ratio was alsoproposed as follows:

(2)α = =−

−

P

P

a zw / 1

3 1

1 3≤ ≤a z/

a z/ ≥ 3

α = =−

−

P

P

a z

N Pw

sd

2 1

3

/

/

3. STRUT-TIE MODEL AND LOADDISTRIBUTION RATIO OF PRESENT STUDY

3.1. Indeterminate Strut-Tie Model

The ultimate behavior of simple deep beams is highlynonlinear in accordance with the design variables such asthe shear span-to-effective depth ratio, flexural and shearreinforcement ratios, load and support conditions, andmaterial properties. In the present study, one simple


Byung-Hun Kim and Young-Mook Yun

a

P

P

≤ 45°

z

θ

(a) a/z < 0.5

a

Pw

P

z

θ

(b) 0.5 ≤ a/z ≤ 2.0

a

P

z

θ

(c) a/z > 2.0

P

P

δ

δ

δ

δ

Figure 2. FIB’s strut-tie models for simply supported deep beams

indeterminate strut-tie model reflecting all characteristicsof the ultimate strength and behavior is presented toconduct the rational design of the entire range of simpledeep beams. The presented indeterminate strut-tie model,defined as a combination of arch and truss mechanisms, isshown in Figure 3. In the presented model, the role ofhorizontal shear reinforcing bars is not reflected uponbecause, according to the research by Smith and Vantsiotis(1982), Bazant (1997), Shin et al. (1999), and Zararis(2003), the effect of horizontal shear reinforcing bars onshear strength is insignificant when simple deep beams donot contain plenty of horizontal reinforcing bars.

3.2. Load Distribution Ratio

A load distribution ratio, defined as the fraction of loadtransferred by a truss mechanism, is proposed to helpstructural designers perform the design of simple deepbeams by using the presented indeterminate strut-tiemodel along with the provisions of the current designcodes. In the present study, the load distribution ratio isdetermined by conducting a finite element materialnonlinear analysis of the presented indeterminate strut-tie model. A state of simultaneous failure of the inclinedconcrete strut and vertical steel tie, defined as a state ofbalanced shear reinforcement ratio, is used as acondition for determining the load distribution ratio. Asimultaneous failure of concrete strut E and steel tie D(denoting a failure of the arch mechanism) or asimultaneous failure of concrete strut C (or F) and steeltie D (denoting a failure of the truss mechanism) isassumed to occur at the balanced shear reinforcementratio. To determine the load distribution ratio at the stateof balanced shear reinforcement ratio, the materialnonlinear finite element analysis of the indeterminatestrut-tie model is conducted by changing the magnitudeof the applied load P and the amount of the verticalshear reinforcement area Atie,D, according to theprocedure shown in Figure 4. In Figure 4, the maximum

value of P, Pmax, is determined from the flexuralstrength of the deep beams, and the maximum value ofAtie,D, AD tie,max, is the vertical shear reinforcement arearequired for Pmax. The initial values of P and Atie,D,Pinitial and AD tie,initial, respectively, are chosen as 1% and0.5% of their maximum values. With the presented loaddistribution ratio, an optimum design of a simple deepbeam may be ensured by deciding the cross-sectionalareas of reinforcing bars at a state of the simultaneousfailure. Additionally, the ductile structural behaviorcaused by the yield of the steel tie before the crushing ofthe inclined concrete strut may be assured in designpractice by using a smaller load distribution ratio thanthe one obtained at a state of the simultaneous failure.



D

Pa

P

A B

F

H

La/2

dz

bSymm.

C

E

G1 3

42 5 7

86

θθ2

1

Figure 3. Indeterminate strut-tie model of present study

Determination of initial applied load Pinitialand initial area of steel tie D AD tie, initial

(Pinitial = 0.01Pmax, Pmax = Pbending = Mn /a,AD tie, initial = 0.005AD tie, max, AD tie, max = Pmax/fy)

Determination of cross-sectional areas (Astrut, Atie) and modulus of elasticity (Ec, Et)

of struts and ties

Material nonlinear analysis of indeterminate strut-tie model(Max. incremental load step INCmax = 20)

Determination of tangentstiffness matrix Kg

Calculation of nodal displacementsand strains of struts and ties

by solving Kg−1P

Calculation of tangent modulus of elasticity (E tc, E ts) of struts and ties

for next incremental load step

INC

= IN

C +

1

No

P =

P +

0.0

1Pin

itial

INC > INCmax

Yes

AD

tie

= A

D ti

e +

0.02

AD

tie,

initi

al

Etc (of struts C, F and/or E) < 0.01 Ec

No

No

No

Yes

Yes

Yes

fs (of steel tie D) > 1.001fy

fs (of steel tie D) < 0.999fy

Determination of prime design variables a/d, f ′c,ρ

Determination of load distribution ratio (= FD tie /P) for given a/d, fc, and (FD tie:

cross-sectional force of steel tie D)

αρ

Figure 4. Algorithm for determining load distribution ratio of

indeterminate strut-tie model

Since the load distribution ratio of the present studyis determined by the nonlinear analysis of indeterminatetruss structure, the axial stiffness of struts and ties EA (E = modulus of elasticity, A = cross-sectionalareas) in accordance with the stress states of struts andties must be proffered. In the present study, the cross-sectional areas of struts and ties are decided as themaximum areas of struts and ties that they can contain,as the method of conventional strut-tie modelapproaches. As shown in Figure 5, the cross-sectionalareas of struts A and B placed at the biaxial compressionregion are decided by multiplying the width of the strutws (which is the same as the depth of the equivalentrectangular stress block a) by the thickness of the beamb, as in Eqn 3:

(3)

where, β1 is the coefficient of the equivalent rectangularstress block, c is the distance from the top of the beamto the neutral axis, As (= ξρbbd) is the cross-sectionalarea of flexural reinforcement, d is the effective depth ofthe beam, ρb is the balanced flexural reinforcementratio, and ξ is the variable of flexural reinforcement (inthe case of maximum flexural reinforcement ratio ρmax,ξ = 0.75). The cross-sectional areas of inclined struts C,E, and F placed at the shear span are decided by

A ab cbf A

f

f

fbA Bstrut

y s

c

y b

c, . .

= = =′

=′

βξρ

1 0 85 0 85dd

multiplying the thickness of the beam b by the smallerwidth of the strut and nodal zone boundary, as expressedin the following:

(4)

(5)

(6)

where, wC strut is the width of strut C, wH tie is the widthof tie, H, θ1 and θ2 are the angles between the inclinedstrut and horizontal axis, and lb,1 is the width of thebearing (or loading) plate of nodal zone 1. In the presentstudy, the width of the bearing or loading plate lb isdetermined to satisfy the ACI 318M-08’s (2008)strength requirement of nodal zone, as expressed in thefollowing:

(7)

where βn is the coefficient of the effective strength ofnodal zone. For nodal zones 1 and 4 which are classifiedas CCT and CCC nodal zones, the values of 0.8 and 1.0

lP

f bbn c

=′0 85. β

A w b w l bF strut F strut Astrut b= = +( cos sin ),θ θ2 4 2

A w b w l

w

E strut E strut G tie b,= = +min( ,cos sinθ θ1 1 1

AAstrut b,l bcos sinθ θ1 4 1+ )

A w b w l bC strut C strut G tie b= = +( cos sin ),θ θ2 1 2



wE strut = min(wG tiecos 1 + lb,1sin 1, wA strut cos 1 + lb,4sin 1)

wC strut = wG tiecos 2 + lb,1sin 2

wF strut = wA strut cos 1 + lb,4sin 2 ( 1, 2 : See fig.3)θθ θ θ

Nodalzone 1

Nodalzone 4

a

wG tie (= wG tie) wH tie

AD tie= Variable

value

= Clear cover × 2

Plb,4 =

0.85 n,4 f ́c bβ

R lb,1 =

0.85 n,1f ́c b β

C

z

T

wA strut

= β= wB strut

1c

θ

θ θ

θ θ θ

Figure 5. Maximum widths of struts and ties in indeterminate strut-tie model

are taken as the coefficients. The cross-sectional areasof ties G and H placed at the bottom of the beam aredecided as Atie = ξρbbd, the cross-sectional area offlexural reinforcing bars. The cross-sectional area of tieD is obtained by changing its area repeatedly in order toreach the state of simultaneous failure of the inclinedconcrete strut and vertical steel tie.

For the finite element material nonlinear analysis ofthe indeterminate strut-tie model, the stress-strainrelationship of concrete suggested by Pang and Hsu(1995) and expressed in Eqn 8 and Figure 6(a), wasemployed in the present study. The tangential modulusof elasticity of a concrete strut was evaluated bydifferentiating the stress-strain relationship with thestrain of a concrete strut as expressed in Eqn 9:

(8)

f fc cc c= ′

−

ζεζε

εζε

20 0

2

for εε ζε

ζε ζε

ζ

c

c ccf f

/

/

/

0

02

1

11

2 1

≤

= ′ −−

−

>for ε ζεc / 0 1

(9)

where, fc is the compressive stress of a concrete strutthat corresponds to the compressive strain of a concretestrut εc, ζ is the softening coefficient of concrete, and ε0

is the compressive strain corresponding to the peakcompressive stress of a concrete strut defined as ε0 = 2f ′c /Ec where Ec is the initial modulus of elasticityof concrete (for f ′c ≤ 30MPa, ; for

f ′c > 30MPa, ). Following theACI 318M-08’s suggestion for the effective strength ofconcrete struts, the softening coefficient of ζ = 0.85 wasemployed for concrete struts A and B located in thebiaxial compression region, and ζ = 0.638(= 0.85βs =0.85 × 0.75) was employed for inclined concrete strutsC, E, and F located in the biaxial compression-tensionregion. The tangential modulus of elasticity of a steeltie, Et

s , was evaluated by assuming a bi-linear stress-strain relationship of steel, as expressed in Eqn 10 andFigure 6(b):

(10)

where Es is the initial modulus of elasticity of steel.In Table 1, the procedure for determining a load

distribution ratio according to the algorithm of Figure 4is illustrated by using a simple deep beam with a/d =1.4, f ′c = 40 MPa, and ρ = 0.45ρb. As a result of the finiteelement material nonlinear analysis of the indeterminatestrut-tie model of the beam with the alterations of theexternal applied load P and the cross-sectional area ofthe vertical steel tie AD tie, a simultaneous failure of strutE and the vertical steel tie occurred at an applied load of145.2 kN. At this state, the cross-sectional area of thevertical steel tie and the load distribution ratio of thebeam were determined as 163.3 mm2 and 45%.

3.3. Load Distribution Ratio Associated with

Design Variables

The influence of the primary design variables ofsimple deep beams on the load distribution ratio wasscrutinized by employing the presented indeterminatestrut-tie model as a numerical analysis model withdimensions of a = 200~1200 mm, d = 400 mm, L = 600~2600 mm, b = 100 mm, and L − 2a = 200 mm.In the analysis, the values of the primary design

E E

E E

st

s s y

st

s s y

= ≤

= >

for

for

ε ε

ε ε0 001.

E fc c= ′ +3300 7700

E fc c= ′4700

E E

E E

ct

cc

c

ct

cc

= −

≤

= −

1 10

0

0

εζε

ε ζε

ε ζε

for /

/ −−

−

>

1

2 112 0

( / )/

ζε ζεfor c



f c

(a) Concrete strut

f ́c

f ́c

f ́c

0

c

f c = f ́cfc

f c

c = 0

(b) Steel tie

f s

f y Est = 0.001Es

Est = Es

�

c�0��

�

�

ζ

ζ

s�y�

Figure 6. Stress-strain relationships of concrete strut and steel tie

variables a /d, ρ /ρb, and f ′c varied in the ranges of0.5~3.0, 0.15~0.75, and 20~70 MPa, respectively.

The load distribution ratios that were determinedaccording to the algorithm of Figure 4 with differentdesign variables are shown in Figure 7. Unlike the loaddistribution ratios that were proposed by the FIB (1999)



Shear span-to-effective depth ratio (a/d)

100

90

80

70

60

50

40

30

20

10

00.5 1.0 1.5

Load

dis

trib

utio

n ra

tio

(%

)

2.0

a/d = 1.6∼1.7FIB(1999)

= 60.0%∼61.5%

Foster and Gilbert (1998)Failure of archmechanism

Failure of trussmechanism

f ́c = 30 MPaf ́c = 20 MPa

f ́c = 40 MPaf ́c = 50 MPaf ́c = 60 MPaf ́c = 70 MPa

2.5 3.0

(a) bρ ρ= 0.75


100

90

80

70

60

50

40

30

20

10

00.5 1.0 1.5 2.0

a/d = 1.8∼1.9

f ́c = 30 MPaf ́c = 20 MPa


2.5 3.0

(b) bρ ρ= 0.45


100

90

80

70

60

50

40

30

20

10

00.5 1.0 1.5 2.0

a/d = 2.0∼2.1

f ́c = 30 MPaf ́c = 20 MPa


2.5 3.0

(c) bρ ρ= 0.15

α

= 60.6%∼62.1%α

Load

dis

trib

utio

n ra

tio

(%

)α

= 61.2%∼62.6%α

αLo

ad d

istr

ibut

ion

ratio

(

%)

α

Figure 7. Load distribution ratios associated with primary

design variables

Ta

ble

1.

Illu

st

rat

ion

of

de

te

rmin

at

ion

pro

ce

du

re o

f lo

ad

dis

tri

bu

tio

n r

at

io

for,

a/

d=

1.4

, f′ c

= 4

0 M

Paa

nd

=

0.4

5b

Wid

th

s o

f lo

ad

ing

C

ros

s-s

ec

tio

na

lM

od

ulu

s o

f

& b

ea

rin

g p

lat

es

a

rea

of

st

rut

e

las

tic

ity

of

st

rut

Fa

ilu

re o

fF

ail

ure

of

(mm

)(m

m2)

(MP

a)

inc

lin

ed

st

rut

sh

ea

r re

info

rce

me

nt

P(k

N)

l b,1

l b,4

AC

str

utA

E st

rut

AF

stru

tE

t C s

trut

Et E

stru

tE

t F st

rut

St

rut

CS

tru

t E

St

rut

FA

D,t

ieF

ail

/S

afe

14.5

25.

34.

370

1789

6411

268

2856

826

232

2856

80.

0529

.04

10.7

8.5

7417

9227

1158

828

560

2359

828

560

0.21

43.5

616

.012

.878

1894

9011

909

2854

720

368

2854

70.

4858

.08

21.4

17.1

8218

9753

1222

928

530

1550

128

530

0.88

72.5

926

.721

.486

1810

016

1254

928

392

321

2839

24.

0687

.11

32.0

25.6

9018

1027

912

869

2695

526

926

955

×36

.910

1.63

37.4

29.9

9418

1054

213

189

2555

128

725

551

69.0

116.

1742

.734

.298

1910

806

1351

024

149

270

2414

9×

100.

713

0.69

48.1

38.4

1021

911

069

1383

022

720

293

2272

013

2.1

145.

2153

.442

.710

620

1133

214

150

2122

627

021

226

×16

3.3

×

Loa

d D

istr

ibut

ion

Rat

io α

= P

w/P

= F

D,t

ie/P

= (A

D,ti

e×

f y)/

P =

(163

.3 ×

400)

/145

210

=44

.9%

: Saf

e; ×

: Fai

l

ρρρρ

αα

and Foster and Gilbert (1998) in which the ratios changelinearly in proportion to the shear span-to-effectivedepth ratio, the load distribution ratios of the presentstudy change nonlinearly according to the primarydesign variables. This result implies that the presentapproach reflects not only the nonlinear structuralbehavior and ultimate strength of simple deep beams butalso the variations of the load-resistant capacities of thearch and truss mechanisms of simple deep beams due tothe primary design variables.

Figure 7 shows that the load transferred by the archmechanism is similar to that used by Foster and Gilbert(1998) when the shear span-to-effective depth ratio a /dis less than 1.0, and the load transferred by the trussmechanism increases as the ratio a /d increases.However, unlike the results of earlier studies by the FIB(1999) and Foster and Gilbert (1998) where 100% of theapplied load is transferred by the truss mechanism whenthe ratio a /d is greater than 1.80 and 1.56, respectively,the present study reveals that more than 20% of theapplied load is still carried by the arch mechanism whenthe ratio a /d is greater than 2.0. This indicates that theshear-resistant capacity by the concrete struts making upthe arch mechanism exists although the ratio a /dincreases, as proven to be true in the previous studies(Leonhardt 1965; Park and Paulay 1975; Kim et al. 2003).

Figure 7 also shows that the load distribution ratios ata state of simultaneous failures of arch and trussmechanisms are very much analogous regardless of theflexural reinforcement ratio ρ and the compressivestrength of concrete f ′c. This is inferred from the fact thatthe stiffness of both of the concrete struts and steel tiesconstituting the arch and truss mechanisms is increasedproportionally due to the increase of ρ and f ′c , althoughthe failure strength is augmented by the increases ρ ofand f ′c , as shown in Figure 8.

The ratio a /d, at which a simultaneous failure of thearch and truss mechanisms occurs, decreases when theflexural reinforcement ratio increases, as shown inFigure 7. Namely, the range of a /d where deep beamsfail due to the failure of the arch mechanism decreasesbecause the load-carrying capacity of the archmechanism improves by the increase of the flexuralreinforcement ratio. This result is similar to theprevious studies (Zsutty 1971; Okamura and Higai1980; Niwa et al. 1986; Eurocode 2 1992; Bazant 1997;ACI 318-99 1999) expressing that the load transferredby the arch mechanism in simple deep beams increasesas the flexural reinforcement ratio increases. Thisdemonstrates that the load distribution ratio used in thepresent study considers accurately the effect of theflexural reinforcement ratio on the structural behavior ofsimple deep beams.

As the strength of concrete increases, the loadtransferred by the arch mechanism increases in the rangeof a /d ≥ (1.6~1.7) and decreases in the range of, asshown in Figure 7(a). This is due to the relative increaseof the role of a /d ≤ (1.6~1.7) concrete to steel when thestrength of concrete increases, as explained in Figure 9.In addition, the load-carrying capacities of the arch andtruss mechanisms, although they are not the same,increase almost linearly as the strength of concreteincreases. This implies that the amount of shearreinforcing bars in a design must be augmented toensure the ductile shear behavior of simple deep beamsif concrete with increased strength is used.

3.4. Equation of Load Distribution Ratio

An equation of load distribution ratio is developed in thepresent study through the curve fittings of Figure 7. Theequation associated with the primary design variables




100

00.5 1.0 1.5

a/d = 1.6∼1.7 a/d = 1.8∼1.9

2.0 2.5 3.0 3.5

200

300

400

500

Fai

lure

load

(K

N)

f ́c = 70 MPa,f ́c = 30 MPa,

f ́c = 30 MPa,f ́c = 70 MPa,

bρ ρ= 0.75

bρ ρ= 0.75

bρ ρ= 0.45

bρ ρ= 0.45

Figure 8. Failure strength associated with primary design variables

450Load transferred by truss mechanism (%)Load transferred by arch mechanism (%)

Concrete strength f ́c (MPa)

Load

incr

easi

ng r

atio

p/p

f ́ c =

20(

%) 400

350

300

250

200

150

100

50

020

35.2

43.9

47.1

52.956.164.8

51.4

48.6

49.7

50.348.9

51.1

30 40 50 60 70

Figure 9. Load-carrying capacities of arch and truss mechanisms

associated with concrete strength (a/d = 1.3, ρ = 0.75ρb)



can be directly applied to the design of simple deepbeams. The developed equation is as follows:

where, α (= Pw /P,%) , defined in Eqns 1 and 2, is theload distribution ratio of the presented indeterminatestrut-tie model, and ρb is the balanced flexuralreinforcement ratio of the beam. η, expressed in termsof ρ /ρb, is the value of a /d that decides the type ofgoverning failure mechanism between the arch and trussmechanisms, and β is the parameter that considers thevariation of the load distribution ratio according toprimary design variables. The parameters are defined asfollows:

(12)

(13)

Figure 10 shows that the load distribution ratiosdetermined from Eqn 11 and the finite element

βρ ρ η

ρ ρ η=

+( ) −( ) ′ >

+( ) −(

1 40

2 3

2/ / ,

/ /

b c

b

a d f MPa

a d )) ′ ≤

<

= − ′ + +

2 40

0 07 13 1 5

,/

. .

f MPaa d

f

c

c

for η

β ρρ ρ η/ /b a d( ) ≥for

ηρρ

= −

2 1

2

3.

b

(11)

α βρ ρ

(%)/

/

/

. .

= ′ −( ) +− ( )

−

fa d

a d

cb40

200 40

1 1 0 25ln

ρρ ρη

α β η

/

(%) .

b

a

d

a

d

( )

<

= −

+

for

61 5−−

≥2

ρρ

ηb

a

dfor

material nonlinear analyses of the presentedindeterminate strut-tie model agree well, thus allowingstructural designers to employ them in the strut-tiemodel design of simple deep beams subject to variousdesign conditions. Figure 11 illustrates a designprocedure that utilizes the load distribution ratio of theindeterminate strut-tie model.

By numerical analysis


100

10

90

80

70

60

50

40

30

20

01.00.5 1.5 2.0 2.5 3.0

By proposed eqn. (11)

(b) f ́c = 65 MPa and = 0.55ρ ρb

By numerical analysis


100

10

90

80

70

60

50

40

30

20

01.00.5 1.5 2.0 2.5 3.0

By proposed eqn. (11)

(a) f ́c = 28 MPa and = 0.55ρ ρb

Load

dis

trib

utio

n ra

tio

(%

)α

Load

dis

trib

utio

n ra

tio

(%

)α

Determination of initial design conditions includinga/d, fc, , loading and support conditions, etc.

Selection of indeterminate strut-tie modelfor given initial design conditions

Determination of load distribution ratioby using eqn. (11)

Determination of cross-sectional forces of everystrut and tie by using load distribution ratio

and force equilibrium conditions

Modification of initialdesign conditions

Increase of eff. strenghof concrete strut by

confining concrete strutusing reinforcement

Increase of eff. strengthof nodal zone by modifying

design conditionson loading and/or

bearing plates

Determination of required area of reinforcement(= areas of steel ties). regulations of current design

codes for minimum ratio, anchorage, details ofreinforcement, etc. are effective.

Yes

Yes

No

Nofstrut< «s fc(fstrut: compressive stress of concrete strut;

s: coefficient of eff. strength of concrete strut)

fnode< «nfc(fnode: compressive stress of Nodal Zone Face; n:

coefficient of eff. strength of Nodal Zone Face)

ρ

β

β

β

β

Figure 11. Design procedure utilizing load distribution ratio

Figure 10. Comparison of load distribution ratio obtained from numerical analysis and proposed equation

4. SUMMARY AND CONCLUSIONThe structural behavior of simply supported reinforcedconcrete deep beams is very complicated by themechanical relationships between the shear span-to-effective depth ratio, flexural reinforcement ratio, loadand support conditions, and material properties. Toestablish a strut-tie model approach as a rational designmethod, a proper strut-tie model reflecting true loadtransfer mechanisms of the deep beams must bepresented, and the primary design variables influencingthe ultimate strength and behavior of the deep beamsmust be deliberated in the design process as well.

In this study, a simple indeterminate strut-tie modelthat reflects the characteristics of the ultimate strengthand behavior is presented for the design of simplysupported reinforced concrete deep beams. In addition,a load distribution ratio of the indeterminate strut-tiemodel is proposed to help structural engineers designthe deep beams by using the strut-tie modelapproaches of the current design codes. In thedetermination of the load distribution ratio, a conceptof a balanced shear reinforcement ratio requiring asimultaneous failure of inclined concrete strut andvertical steel tie is introduced to ensure the ductileshear design of the deep beams. The effect of theprimary design variables including shear span-to-effective depth ratio, flexural reinforcement ratio, andcompressive strength of concrete are also reflectedthrough the numerous finite element material nonlinearanalyses of the indeterminate strut-tie model withdifferent primary design variables. With the proposedload distribution ratio, the changes of stiffness of allelements constituting the load transfer mechanisms ofthe deep beams may be appropriately reflected indesign. An opportunity to help structural designersconduct the practical strut-tie model design of the deepbeams may also be provided from the present study,which provides an equation of load distribution ratioembracing various design conditions.

In the companion paper, the validity of the presentedmodel and load distribution ratio is examined byevaluating the ultimate strength of various simplysupported reinforced concrete deep beams tested tofailure.

ACKNOWLEDGEMENTThis work was supported by the Korea ResearchFoundation Grant funded by the Korean Government(MOEHRD, Basic Research Promotion Fund) (KRF-2006-214-D00157).

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An Indeterminate Strut - Tie Model and Load Distribution Ratio for RC Deep Beams (I) - Model and Load Distribution Ratio