ACI STRUCTURAL JOURNAL TECHNICAL PAPER Title no. 95-S39

Reliability-Based Strength Reduction Factor for Bond

by David Darwin, Emmanuel K. ldun, Jun Zuo, and Michael L. Tholen

The formulation of a reliability-based strength-reduction ($)factor for developed and spliced bars is described. Conventional and high relative rib area bars, both with and without confining reinforcement, are considered. The $factor is determined using statistically-based expressions for devel­opment/splice strength and Monte Carlo simulations of a range of beams. The overall approach is applicable to the calculation of $factors for all types of loading on reinforced concrete.

A strength-reduction factor of0.9 is obtained for the design expressions for development/splice length, based on a probability of failure in bond equal to about one-fifth of the probability of failure in bending or combined bending and compression. $ = 0.9 is incorporated into two expressions for development/splice length in a manner that is transparent to the user. A major advantage of each of the final expressions is that they provide identi­cal values for development and splice length, removing the need to multiply development length by 1.3 or 1.7 to obtain the length of most splices.

Keywords: bond (concrete to reinforcement); bridge specifications; build­ing codes; deformed reinforcement; development; lap connections; rein­forcing steels; relative rib area; reliability; splicing; structural engineering; variability.

Recent work to improve the development characteristics of reinforcing bars by modifying bar deformation patterns 1-4

has included a reevaluation of existing development and splice tests and the formulation of an expression to represent the bond force of bottom-cast bars at development/splice failure. 5·6

The resulting best-fit equation for ultimate bond force Tb

can be used to determine both development and splice length. However, without modification, the expression is not usable in practice, since 50 percent of the development/ splice designs would be weaker than predicted-a situation that presents unacceptable safety risks.

The level of safety can be improved by reducing the usable bond force using a suitable strength reduction factor. A longer development length is then required to provide the desired value of Tb.

This paper describes the calculation of a reliability-based <)>-factor for developed and spliced bars with relative rib areas Rr (ratio of projected rib area normal to bar axis to the product of the nominal bar perimeter and the center-to-center rib spacing) of 0.0727 and 0.1275 (for conventional and high Rr bars, respectively). The best-fit expressions are presented first, followed by the overall approach and the details of the

434

calculation. As will be demonstrated, a major advantage of the final expressions is that they provide identical values for development and splice length. The full details of this effort are presented by Darwin, Idun, Zuo, and Tholen.7 The over­all approach is applicable to the calculation of <)>-factors for all types of loading on reinforced concrete members.

BOND FORCE EQUATIONS Based on the analysis by Darwin et al.,5•6 the best-fit

equation for ultimate bond force Tb is

where

Tb = Ab = db = fs = fc' = ld = Cm,CM =

Cs = Csi

Cso•Cb

N

= Abfs

u;)l/4

= [63licm + 0.5db) + 2130Ab]( 0.1 ~= + 0.9) (1)

NA + 2226trtd __ tr + 66

n

force in bar at development or splice failure, lb bar area, in.2

nominal bar diameter, in. steel stress at failure, psi

concrete compressive strength, psi; (fc') 114 psi development or splice length, in. minimum, maximum value of c s or cb(cM/cm:::; 3.5), in. min (csi + 0.25 in., Cs0 ), in. one-half of clear spacing between bars, in.

side cover, bottom cover of reinforcing bars, in.

number of transverse reinforcing bars (stirrup or ties) crossing ld

ACI Structural Journal, V. 95, No. 4, July-August 1998. Received January 27, 1997, and reviewed under Institute publication policies.

Copyright © 1998, American Concrete Institute. All rights reserved, including the making of copies unless permission is obtained from the copyright proprietors. Perti­nent discussion will be published in the May-June 1999 ACI Structural Journal if received by January I, 1999.

ACI Structural Journal I July-August 1998

David Darwin, FACI, is the Deane E. Ackers Professor of Civil Engineering and Director of the Structural Engineering and Materials Laboratory at the University of Kansas. He is a past member of the Board of Direction and the Technical Activities Committee and is past president of the Kansas Chapter of ACI. Darwin is also former chairman of the Publications Committee and the Concrete Research Council, and a member and former chairman of ACI Committee 224, Cracking. He chairs the TAC Technology Transfer Committee and serves on ACI Committees 408, Bond and Devel­opment of Reinforcement; 446, Fracture Mechanics; and ACI-ASCE Committees 445, Shear and Torsion, and 447, Finite Element Analysis of Reinforced Concrete Structures.

He is a recipient of the Arthur R. Anderson and ACI Structural Research Awards.

ACI member Emmanuel K. ldun is the lead structural engineer with Constructive Engineering Design, Kansas City, Missouri. He holds a BSc( Eng) in civil engineering and an MPhil in Structures from the University of Science and Technology in Ghana,

and a PhD from the University of Kansas.

ACI member Jun Zuo is a structural engineer with Constructive Engineering Design, Kansas City, Missouri. He obtained his BS in architectural engineering from Tongji University in China, and his MS and PhD in civil engineering from the University of

Kansas.

ACI member Michael L Tholen is a structural engineer with Burns and McDonnell International in Kansas City, Missouri. He holds a BS in architectural engineering and an MS and PhD in civil engineering from the University of Kansas.

A1r = area of transverse reinforcement crossing the potential plane of splitting adjacent to the reinforcement being developed, in. 2

n number of bars being developed or spliced along the plane of splitting

tr = 9.6Rr + 0.28

td = 0.72db + 0.28

The final term in Eq. (1), 66, is used only if the member has confining transverse reinforcement.

Equation (1) is based on the analysis of 133 development and splice tests of bottom-cast bars without confining rein­forcement and 166 tests with confining reinforcement. 3.4·8·24

Eighty-seven percent of the specimens contain Class B (ACI 318-95)25 /Class C (AASHTO Highway 1992)26 splices. The database includes specimens with concrete strengths f; between 1820 and 15,760 psi (13 and 109 MPa) and bars with relative rib areas Rr between 0.056 and 0.140; the relative rib area has been shown to significantly affect the contribution of transverse reinforcement to bond strength. 1·3 The effect of Rr is reflected in the expression for tr. Rr averages 0.0727 for conventional reinforcement and 0.1275 for newly proposed high relative rib area bars.3•5

As observed earlier by Oranjun, Jirsa and Breen,27 the analysis by Darwin et al.5·6 demonstrates that, for the same geometric and material properties, development strength and splice strength are the same. The analysis shows that Eq. (1) applies equally well to developed and spliced bars. This provides an important advantage over current design proce­dures, since the design values of ld developed in this paper and by Darwin et al.5•6 do not need to be multiplied by 1.3 (ACI) or 1. 7 (AASHTO) to obtain the length of most splices, as required by ACI 318-95 and AASHTO Highway.26 The preponderance of splice tests in the database (used for both equation calibration and <!>-factor calculations) should serve as extra assurance to those concerned with using a single value of ld for both development and splice lengths.

ACI Structural Journal I July-August 1998

Eq. (1) produces a mean test/prediction ratio of 1.00, with a coefficient of variation VTIP of 0.107 for beams in which the bars are not confined by transverse reinforcement and a mean test/prediction ratio of 1.01, with VTIP = 0.125, for beams in which the bars are confined by transverse reinforcement.

Eq. (1) can be used to calculate development/splice length ld by dropping the final term, 66, and setting N = Vs, in which s = spacing of transverse reinforcement.

(2)

( e + Ktr) 80.2--db

where e =(em+ 0.5db)(0.1eM1em + 0.9), K1r= 35.3trtdAtrlsn, and (e + K1r)ldb ~ 4.0.

Equation (2) can be simplified further by setting eM/ em = 1 and dropping 0.25 in. from the definition of es.

~-2130 ld u:)l/4

(3) db

8o.2C :~tr)

where e =(em+ 0.5db).

Converting Eq. (2) and (3) back to a form that can be used to predict Tb = Abfs gives, respectively,

Tb = Abfs

= CJ;)114{ [63/d(em + 0.5db) + 2130Ab]( 0.1 ~: + 0.9) (4)

+ 2226t t ~~tr} r d sn

(5)

+ 2226t t ..!l..__.!! l A } r d sn

where tr = 0.98 for conventional bars and 1.50 for high relative rib area bars, and es (used to determine em), is defined appropriately in the two expressions.

Equation (4) (and Eq. [2]) represents, very nearly, the best-fit equation for the full data base, Eq. (1). Equation (5) (and Eq. [3]) is, in general, more conservative than Eq. (4), but will provide the same value of ldas Eq. (4) when

em= eb = eso·

435

CALCULATION OF STRENGTH REDUCTION FACTORS

Overall approach The strength reduction factor for bond cl>b must be selected

to insure an acceptably low probability of bond failure. Con­sidering the brittle nature of bond failures, that probability should be lower than the probability of failure under a main load-carrying mechanism, such as bending or combined bending and compression. This can be achieved by using the concepts of structural reliability.

Limiting consideration to "statically" applied load for the purpose of this analysis (i.e., not seismic or shock loading), it is recognized that the bar force Tb = Abfs that appears on the left side ofEq. (4) and (5) has already been increased by a factor of 1/cp, where cp =strength reduction factor for the main loading, before development/splice design is undertaken. This point can be demonstrated using bending as an exam­ple. Following the requirements of ACI 318-95, cpMn ~ Mu. In the limiting case, ci>Mn = Mu or cpAJs(d- a/2) = Mu. Then cpAbfs = Muf(d- a/2), and Abfs = Mulcp(d- a/2) (where Mn and Mu =nominal and factored moments, respectively, cp = the strength reduction factor for bending, d =effective depth, and a = depth of stress block). Thus, the bar force Abfs is higher by a factor of 1/cp than required to meet the determin­istic value of the factored moment Mu. For development and splice strength, cl>b times the right side of Eq. (4) or (5) will also= Muf(d- a/2). Setting the design bar forces equal gives

Abfs = cpd[Right side ofEq. (4) or (5)] (6)

Since AJs is normally selected based on bending, the ef­fective strength reduction factor for use in calculating de­velopment/splice length becomes cl>d = cpb/cp, although the basic strength reduction factor against bond failure remains cl>b· Thus,

cpAbfs = cpd[Right side ofEq. (4) or (5)] (7)

Level of reliability-Determining the value of cpb (and ultimately cpd) requires the selection of the desired level of reliability, which can be represented by the reliability index ~.28 For a resistance R, and a loading Q, failure will not occur if Rl Q ~ 1. Assuming that R and Q have log normal distribu­tions (as shown in Fig. 1) and using the small-variance approximations,28 ln(RIQ) "' ln(RIQ) and crln(RIQ) "' (vi + V Q ) 112, where the overbar represents the average, cr = standard deviation, and V = coefficient of variation,

~ = ln(RIQ)"' ln(RIQ)

(Jln(R/Q) (~ + ~)112 (8)

Thus, ~ represents the number of standard deviations by which the mean value of ln(RIQ) exceeds ln(RIQ) = 0. Increases in ~ represent conditions of improved reliability and decreased probability of failure.

Under typical conditions of loading, ~"' 3.0 for reinforced concrete beams and columns. 28 A higher value of~ is needed to insure that the probability of a bond failure is lower than the probability of a failure in bending for beams or in com-

436

\I

Frequency

ln(R/0) ln(R/0)

Fig. ]-Illustration of reliability index (after Ellingwood et al. 28 ): ~ = number of standard deviations, cr In{RIQ)'

between ln(R/Q) = (ln(R/Q) and ln(R/Q) = 0

bined bending and compression for columns. Using~= 3.5 for the calculation of development/splice length produces a probability of failure, based on the assumed form of the distributions,28•29 equal to about one-fifth of that obtained with~= 3.0.

Resistance and loading random variables-Equation (8) can be used to calculate cpb, but to do so requires knowledge of R and Q, both of which are random variables. This knowl­edge can be obtained through the application of Monte Carlo analysis, used in conjunction with data obtained from field measurements and test results. The derivation that follows combines techniques used by Ellingwood et al.,28 Mirza and MacGregor,30 and Lundberg31 and is applicable to a wider range of problems than development/splice length design.

To solve Eq. (8) for cl>b requires a series of substitutions that result in the introduction of cp into the expression. Equa­tions (9) through (21) demonstrate how this is done.

The random variable for resistance R is represented as

R = X(1)RP (9)

where X(l) = test-to-predicted load capacity random variable RP = predicted capacity random variable, dependent

on material and geometric properties of member, which are also random variables

The random variable for loading Q is represented for dead load and live load as

(10)

(11)

where Qv and QL = random variables representing dead and live load effects, and Qvn = nominal dead load.

ACI Structural Journal I July-August 1998

The ratios Qvl QDn and QL/ QDn in Eq. ( 11) are expressed as

where

QD = X(2) QDn

QLn = nominal live load X(2), X(3) = actual-to-nominal dead and live load

random variables

= nominal ratio of live load to dead load

(12)

(13)

Expression for strength reduction factor-For the limiting case in design, the strength reduction factor times the nomi­nal capacity will equal the factored load:

where

<l>c strength reduction factor for the loading under consideration (for bond, <l>c = <l>b)

Rn = nominal resistance

YD· YL = load factors for dead and live loads

(14)

Factoring out QDn on the right side ofEq. (14) and setting QL)Qvn = (QdQv)n gives

(15)

Solving Eq. 15 for QDn gives

(16)

The total load Q is obtained by substituting Eq. (12), (13), and (16) into Eq. (11):

(17)

where

ACI Structural Journal I July-August 1998

Defining

gives

<l>c can now be introduced into Eq. (8),

~ = ln(R/Q) = ln(rRni<J>cqRn)

(Jln(R/Q) <Jln(rRnl<l>cqRn)

where

q =

= --------------------------X(2) + X(3)(QL)

QD n

(18)

(19)

(20)

Finally, solving Eq. (20) for <l>c (and remembering that for bond <l>c = <l>b) gives

(21)

The solution of Eq. (21) requires that the mean values, r and q, and the coefficients of variation Vr and V<l>q of the random variables, rand q, be known. This is done next.

Random variables In this section, the values associated with the resistance

random variable r, r and Vr, are obtained first, followed by the values associated with the load random variable q, q and V<iJq· While the derivation ofEq. (21) is ge~eral, most of what

437

follows applies more specifically to development/splice design.

Resistance random variable Resistance random variable r is obtained using Eq. (18):

R (18)

Test-to-predicted load random variable, X(l)-The test-to-predicted load random variable X(1) is based on a comparison of test results with Eq. (1). X(1) is treated as a normal random variable with a mean equal to the mean test/prediction ratio. X(1) = 1.00 and 1.01 for members with­out and with confining transverse reinforcement, respective­ly.5·6 The coefficient of variation Vx(l) is equal to the coefficient of variation associated with the predictive equation (or model) itself Vm as separate from uncertainties in the measured loads and differences in the actual material and geo­metric properties of the specimens from values used to calculate the predicted strength, represented by V18 • The total coeffi­cient of variation in the test/prediction ratio Vr;p is equal to ( 2 2 1/2 f 2 I' 2 2 112 vm + vts) (Re. 3 ). Theretore, Vm = ( VTIP- VIS) .

For reinforced concrete, V18 "" 0.07 (Ref. 32). For beams Without confining reinforcement, V m = ( Vi; p - V;,) 112 = (0.1072 - 0.072) 112 = 0.081. For beams with confining reinforcement, additional uncertainty occurs because the relative rib area Rr is not known for 34 of the beams used to establish Eq. (1). This is handled with VR = 0.02, giving V m = ( Vi;p- V1: - V~Y/2 = (0.1252 - 0Ji'72 - 0.022) 112 = 0.102.

Predicted capacity random variable, RP-The individual values of the predicted capacity random variable RP are obtained for hypothetical beams using the Monte Carlo method. The random variables used to calculate RP are the concrete strengthfd (adjusted for the rate of loading), the development/splice length ld, the member width b, the cover cb, the side cover c so, and the relative rib area of the developed/ spliced bar Rr- The predicted capacity RP is calculated by solving Eq. (1) for Abfs·

RP = Abfs

= (f:) 114{[63licm + 0.5db) + 2130Ab](0.1 eM+ 0.9) em (22)

NA } + 2226th7 + 66

Individual values of RP are calculated by substituting values for each of the variables that are determined based on the nominal value and statistical properties of that variable. Beams with spliced bars are used as the physical model in this study.

Concrete strength, fd [X(4)]-The random variable for concrete strength X(4) must take into consideration the strength and variability of concrete, as used in practice, and the effect of the actual load rate in the structure, as opposed

438

to the load rate used in standard tests.33 The latter point is considered first.

A relation proposed by Jones and Richart34 is used to take into account the fact that, under practical conditions, loading rates will be different than the average value of 35 psi/sec (0.24 MPa/sec) used in a standard compression test:35

fcl = 0.89fc35 ( 1 + 0.08logj) (23)

where 0.1 psi/sec:::;; J:::;; 10,000 psi/sec, and

f/J = compressive strength of concrete at stress rate J fc3s = compressive strength of concrete at J = 35 psi/sec

(0.24 MPa/sec) It is assumed that, in practice, the load rate will be such

that failure will occur in about an hour, resulting in a lower effective compressive strength than would be obtained in a standard test. The stress rate J corresponding to compressive failure in one hour is

J = fcf 3600

(24)

The values of J and f/J are obtained by iteration using Eq. (23) and (24).

The selection of the value of fc3s, which should be repre­sentative of concrete strength in the field, is affected by two considerations:

1. Splice tests are calibrated against the compressive strength of standard cylinders that are cured in the same manner as the splice test specimens, not on the actual strength of the concrete in the splice specimens. The closest thing in concrete construction is the use of field-cured specimens.

2. In practice, concrete must be proportioned to produce a higher strength than used to design the structure to insure that the strength of most of the concrete will exceed the specified value of fd.

The two considerations have opposite effects on the value of fc35 used in the analysis, since field-cured cylinders usu­ally produce a lower strength than standard laboratory-cured specimens (the basis upon whichf: is measured), while the average strength of concrete produced in the field, as measured using standard specimens, exceeds fd by a considerable amount. These opposing effects largely cancel each other out. Therefore, the specified value of fd is used as the mean value of the concrete strength for use in determining RP:

(25)

In Eq. (22),fd is replaced by the normally distributed ran­dom variable X( 4) with a mean value X( 4) = f' (Eq. [23] and [24]). Forf: = 4000 psi (28 MPa), X( 4) = fer= 3559 psi (24.54 MPa). The standard deviation <Jx(4) = Vcfc/ is based on: 1) an assumed standard deviation for standard laboratory cylinders, <Jccyl = 550 psi (3.8 MPa), representative of good jobsite quality control, and 2) an assumed variability for in-place concrete, expressed as Vc = (Vcb,z + 0.0084) 112

ACI Structural Journal I July-August 1998

(Ref. 33), where Vccyl = crccyllfc~ andfc~ =required average compressive strength of concrete= fc' + 2.33 crccyz- 500 psi (Eq. [5-2] of ACI 318-95). For fc' = 4000 psi (28 MPa), Vc = 0.147 and crx(4) = 523 psi (3.6 MPa).

Geometric properties-The balance of the random vari­ables used to calculate RP are the geometric properties of the structural member and the reinforcement. The tolerances in ACI 117-90 are used as the basis for establishing the variability of the geometric properties of concrete sections. All geometric properties are represented using normal distri­butions. The resulting values for mean and standard deviation are close to those obtained from field measurements and used in earlier reliability studies.28 In the case of splice length and side cover, however, no field data are available.

The splice length ld is represented by the random variable X(5), with a mean equal to the specified value of ld. The tolerance for the embedded length of bars and the length of bar laps in ACI 117-90 is -1 in. (25 mm) for No.3 through No. 11 (9.5 through 36 mm) bars. It is assumed that 95 percent of all bars will meet this criterion. For the normal distribution X(5), this means that 1.645crx(S) = 1 in. (25 mm), or crx(5) = 0.61 in. (16 mm). (The values of crx(i) are shown rounded to two significant figures. No rounding, however, is used in the calculation of <jl.)

Concrete cover cb is represented by random variable X(6), with a mean equal to the specified cover. The tolerance on cover in ACI 117-90 is-% in. (9.5 mm) for members less than or equal to 12 in. (305 mm) in size and - 1/ 2 in. (13 mm) for members greater than 12 in. (305 mm) in size. Again, assuming that 95 percent of all members will meet these criteria, 1.645crx(6) = 0.375 in. (9.5 mm), or crx(6) = 0.23 in. (6 mm) for members ~ 12 in. (305 mm) in size, and 1.645crx(6) = 0.5 in. (13 mm), or crx(6) = 0.30 in. (8 mm) for members> 12 in. (305 mm) in size.

Side cover c so is represented by random variable X(7), with a mean equal to the specified value of cso- In this case, the tolerances on placement of reinforcement in ACI 117-90 are ±3/ 8 in. (9.5 mm) for members between 4 and 12 in. (102 and 305 mm) in size and ±1/2 in. (13 mm) for members greater than 12 in. (305 mm) in size. Since cso is bounded on two sides, if 95 percent of all bar placements meet these criteria, the tolerances are equal to 1.96crx(7)· Using procedures sim­ilar to those used for cb and ld, crx(7) = 0.19 in. for members between 4 and 12 in. (102 and 305 mm) in size and crx(?) = 0.26 in. (7 mm) for members> 12 in. (305 mm) in size.

One-half of the clear spacing between bars c si is calculated as

(26)

where nb = number of bars. In this expression, in addition to cso• beam width b is a random variable, represented by X(8).

The tolerances on cross-sectional dimensions in ACI 117-90 are+% in. and - 1/4 in. (+9.5 mm and -6.5 mm) for members with dimensions of 12 in. (305 mm) or less and +1/2 in. and - 3/ 8 in. ( + 13 mm and -9.5 mm) for members with dimensions greater than 12 in. (305 mm), but less than 3 ft (914 mm). ACI 117-90 also provides criteria for members over 3 ft

ACI Structural Journal I July-August 1998

(914 mm) in dimension, but these are not used in the current Monte Carlo analysis. The mean value of beam width X(8) is taken as the nominal beam width plus the average of the tolerances = b + 0.0625 in. (1.6 mm) for members in both size categories. The standard deviations are selected such that 95 percent of all members have dimensions between the tolerances, giving crx(S) = 0.16 in. (4 mm) for members with b ~ 12 in. (305 mm) and crx(S) = 0.22 in. (6 mm) for members 12 < b ~ 36 in. (305 < b ~ 914 mm).

The term representing the effect of relative rib area on the effectiveness of transverse reinforcement on bond strength, tr = 9.6Rr + 0.28, depends on the random variable representing R, = X(9). Rr = X(9) = 0.0727 for conventional reinforce­ment and 0.1275 for high relative rib area reinforcement. Conservatively, the standard deviations are crx(9) = 0.0090 for conventional reinforcement and 0.0045 for high relative rib area reinforcement. 5•6

In Eq. (4) and (5), the number of stirrups crossing the splice N, (Eq. [1] and [22]) is replaced by ld!s. Of course, N must have an integer value, although ld!s is the value used in Eq. (2) and (3) to calculate development and splice length. As an example, if ld/s = 3.6, the development/splice length would be crossed by four stirrups 60 percent of the time and three stirrups 40 percent of the time, for an average of 3.6 stirrups. Thus, the average strength can be based on 3.6 stir­rups. However, using 3.6 stirrups does not account for the variability in strength that occurs because some splices are crossed by three stirrups, while others are crossed by 4. This variability is accounted for in the Monte Carlo simulation by applying the appropriate weights to the calculated strengths for the two integer values for the number of stirrups. This results in a lower <jl-factor than if N = ld/s were used to calculate RP.

Nominal strength, Rn-The nominal strength Rn is calcu­lated using Eq. (4) or (5) with the specified concrete strength J: and the nominal dimensions ofthe member.

Monte Carlo simulation-The values ofr and Vr (following Eq. [20]) are obtained using Monte Carlo simulations of a selected number of beams. For each beam and simulation, values are selected for normally distributed random variables X(1) andX(4) to X(9). To do this for each variable, a random number between 0 and 1 is used with the cumulative distri­bution function to calculate the standard normal random variable, z (-= < z <=).For variable i, X(i) = X(i) + zcrx(i)· The values of X(i) are used to calculate r (Eq. [18]) for the simulation. The results of multiple simulations are combined to obtain r and Vr-

Loading random variable

The term q (following Eq. [17]) depends on random vari­ables X(2) and X(3), representing the actual-to-nominal ra­tios for dead and live load, respectively; load factors for dead and live load, YD and YL; and the nominal live load-to-dead load ratio, (QdQv)n- YD and YL are selected based on the load factors used in design, 1.4 and 1. 7 for ACI 318-9525 and AASHTO Highway,26 and 1.2 and 1.6 for ASCE 7-95.36 Values of (QdQv)n of 0.5, 1.0, and 1.5 are normally selected for evaluating the reliability of reinforced concrete structures, with a nominal live load-to-dead load

439

ratio of 1.0 serving as the standard for calculating <)>-fac­tors or determining the reliability index ~-

For reinforced concrete structures, X(2) = Qv!Qvn = 1.03 and VQD = 0.093 (Ref. 28). X(3) = QdQLn depends on the tributary area AT, and the influence areaA1.28 For AT= 400 ft2

(37 m2) and A1 = 800 ft2 (74 m2) (representative values for a reinforced concrete flexural member),

- ( 15_] QL = Lo 0.25 + JA/ = 0.780Lo (27)

(28)

where L 0 = basic (unreduced) live load and areas are in ft2•

Thus, X(3) = QdQLn = 0.975. VQL = 0.25 (Ref. 28).

STRENGTH REDUCTION FACTORS Strength reduction (<)>) factors are calculated for Eq. (4)

and (5) using: 1) nominal live load-to-dead load ratios of 0.5, 1.0, and 1.5; 2) two combinations of dead and live load factors: (a) 1.4 and 1.7 [with<)> for bending= 0.9], and (b) 1.2 and 1.6 [with<)> for bending= 0.8]; 3) bars with relative rib areas of 0.0727 and 0.1275; and 4) members with and with­out confining transverse reinforcement. Thirty-five beams in which the bars are not confined by transverse reinforcement and 140 beams (in four groups of 35 each) in which the bars

are confined by transverse reinforcement are used in the calculations. The beams have widths of 8, 12, 18, or 24 in. (203, 305, 457, and 610 mm) and depths of 12 or 24 in. (305 and 610 mm). Concrete strengths of 3000, 4000, and 6000 psi (21, 28, and 41 MPa) are evaluated, and 2, 4, 6, or 8 bars are spliced at the same location. No. 6, No. 8, No. 10, and No. 11 bars (19, 25, 32, and 36 mm) are used. For bars with confining transverse reinforcement, No. 3 and No. 4 (9.5 and 12.5 mm) bar stirrups are spaced at values ranging from 4 to 10.8 in. (102 to 275 mm). A summary of the beams used for the analysis is presented in Ref. 7 and in Appendix A.*

For each of the 35 beams without transverse reinforce­ment, 1000 Monte Carlo simulations are carried out in which the predicted strengths are calculated using Eq. (22) and the material and geometric random variables described in this report. For each of the 140 beams with transverse reinforce­ment, 250 simulations are carried out. The programs used for the Monte Carlo simulations are presented in Ref. 7. The in­dividual predicted strengths are used to calculate r and Vr (following Eq. [20]). The selected load factors and live load-to-dead load ratios are used to calculate q and V<l>q (also following Eq. [20]). The results are combined with~= 3.5 to calculate <l>c = <i>b (Eq. [21]). The value of <i>d = <I>Y<I> is then obtained.

The results of the Monte Carlo simulations are presented in Table 1.

Load factors 1.4 and 1.7-For Eq. (4), which is based on Eq. (2) (the more accurate of the two design equations),

* The Appendix is available in xerographic or similar form from ACI headquarters, where it will be kept permanently on file, at a charge equal to the cost of reproduction plus handling at the time of request.

Table 1-Strength reduction(<!>) factors for bond Eq. (4)

YD = 1.4 YL = 1.7 (<I> bending = 0.9)

Without stirrups With stirrups

AverageRr N/A* 0.0727 0.1275

r 0.955 0.989 0.980

Vr 0.106 0.125 0.125

(QD/QL)n 0.5 1.0 1.5 0.5 1.0 1.5 0.5 1.0 1.5

q 0.675 0.647 0.631 0.675 0.647 0.631 0.675 0.647 0.631

v<l>q 0.102 0.131 0.152 0.102 0.131 0.152 0.102 0.131 0.152

<i>b 0.846 0.819 0.792 0.833 0.812 0.788 0.826 0.805 0.780

<i>d 0.940 0.910 0.880 0.926 0.902 0.875 0.917 0.894 0.867

YD = 1.2 YL = 1.6 (<!>bending= 0.8)

Without stirrups With stirrups

Average Rr N/A 0.0727 0.1275

r 0.955 0.989 0.980

vr 0.106 0.125 0.125

(QD/QL)n 0.5 1.0 1.5 0.5 1.0 1.5 0.5 1.0 1.5

q 0.759 0.716 0.693 0.759 0.716 0.693 0.759 0.716 0.693

v<l>q 0.102 0.131 0.152 0.102 0.131 0.152 0.102 0.131 0.152

<i>b 0.752 0.740 0.722 0.741 0.733 0.718 0.734 0.727 0.711

'i>d 0.940 0.925 0.902 0.926 0.917 0.897 0.917 0.908 0.889

* N/ A = not applicable

440 ACI Structural Journal I July-August 1998

<l>d equals 0.94, 0.91, and 0.88 for bars without confining transverse reinforcement at live load-to-dead load ratios of 0.5, 1.0, and 1.5, respectively; and 0.93, 0.90, and 0.88 (Rr = 0.0727) and 0.92, 0.89, and 0.87 (Rr = 0.1275) for bars with confining transverse reinforcement at the same live load-to-dead load ratios.

For Eq. (5), which is based on Eq. (3) (the more simplified of the two expressions), <l>d equals 0.89, 0.87, and 0.85 for bars without confining transverse reinforcement at live load-to-dead load ratios of 0.5, 1.0, and 1.5, respectively; and 0.99, 0.97, and 0.95 (Rr = 0.0727) and 0.97, 0.95, and 0.93 (Rr = 0.1275) for bars with confining transverse rein­forcement at the same live load-to-dead load ratios.

Loadfaetors 1.2 and 1.6-For load factors of 1.2 and 1.6, the values of <l>d increase slightly compared to those obtained for load factors of 1.4 and 1.7. Using Eq. (4) and a live load-to-dead load ratio of 1.0, <l>d equals 0.93 for bars without transverse reinforcement and 0.92 (Rr = 0.0727) and 0.91 (Rr = 0.1275) for bars with transverse reinforcement. Using Eq. (5), the respective values are 0.89, 0.99, and 0.97.

<l>d = 0.9 appears to be generally conservative and satisfactory for application with Eq. (4) and (5) for both sets of load factors. The lower values of <l>d for bars without confining reinforcement obtained for Eq. (5) compared to Eq. (4) pose no safety problems, since ld obtained with Eq. (5) is never shorter than ld obtained with Eq. (4). The lower values of <l>d

calculated for Eq. (5) are due to the greater scatter (higher Vr)

obtained with Eq. (5), as shown in Table 1.

Table 1 demonstrates that an increase in the live load­to-dead load ratio results in a reduction in the <\>-factor. This reduction is due to the increased variability, represented by Vcpq• that results from the greater uncertainty in the live load.

Design expressions-For ease in application, <l>d can be incorporated directly into the design expressions so that its value becomes transparent to the user. Multiplying the right side ofEq. (4) and (5) by <l>d = 0.9, settingfs = fy, and solving for ld/db gives, respectively,

__lt.__- 1900(0.1 eM+ 0.9) u;)l/4 em

(29)

= (30)

The development and splice lengths obtained with Eq. (29) and (30) are compared with those obtained using the provi­sions of ACI 318-95 for both conventional and high relative rib area bars by Darwin et a1.5·6

As pointed out early in this paper, the analysis described here provides an important advantage over current design procedures (ACI 318-95, AASHTO Highway 1992)25•26 in that Eq. (29) and (30) apply directly to both development and

ACI Structural Journal I July-August 1998

splice lengths, eliminating the need to multiply ld by 1.3 (ACI) or 1.7 (AASHTO) to obtain the length of most splices.

SUMMARY AND CONCLUSIONS The formulation and calculation of a reliability-based

strength-reduction(<\>) factor for developed and spliced bars is described. Conventional and high relative rib area bars, both with and without confining reinforcement, are considered. The <\>-factor is determined using statistically-based expres­sions for development/splice strength and Monte Carlo simulations of a range of beams.

A strength-reduction factor of 0.9 is obtained for the design expressions for development/splice length, based on a prob­ability of failure in bond equal to about one-fifth of the probability of failure in bending or combined bending and compression. <1> = 0.9 is incorporated into two expressions for development/splice length in a manner that is transparent to the user. A major advantage of each of the final expressions is that they provide identical values for development and splice length, removing the need to multiply development length by 1.3 or 1.7 to obtain the length of most splices.

ACKNOWLEDGMENTS Support for this research was provided by the Civil Engineering Research

Foundation under CERF Contract No. 91-N6002, the National Science Foundation under NSF Grants No. MSS-9021066 and CMS-9402563, the U.S. Department of Transportation-Federal Highway Administration, ABC Coating, Inc., AmeriSteel (formerly Florida Steel Corporation), Birmingham Steel Corporation, Chaparral Steel, Fletcher Coating Company, Herberts-O'Brien Inc., Morton Powder Coatings, Inc., North Star Steel Company, and 3M Corporation. Support was also provided by Geiger Ready-Mix, Iron Mountain Trap Rock Company, and Richmond Screw Anchor Company.

CONVERSION FACTORS I in. 25.4 mm I ft 0.3048 m I lb 4.45 N

I psi 6.895 kPa

NOTATION Ab bar area, in. 2

A1 influence area, ft2

Ar tributary area, ft2

A1r area of each stirrup or tie crossing the potential plane of splitting adjacent to the reinforcement being developed or spliced, in. 2

b beam width, in.

e (em +0.5db)(O.IeM/em + 0.9) or (em +0.5db)

eb bottom cover of reinforcing bars, in. maximum value of c, or eh, (eM/em~ 3.5), in. minimum value of cs or eb, (eM/em~ 3.5), in.

min (esi + 0.25 in., es0 ) or min (esi• es0 ), in. one-half of clear spacing between bars, in. side cover of reinforcing bars, in. nominal bar diameter, in.

stress rate, psi/sec

concrete compressive strength, in psi; Uc') 114, psi concrete compressive strength at stress rate f, psi

f/ + 2.33 accyl- 500 psi, required average concrete compressive strength, psi

concrete compressive strength at f = 35 psi/sec, psi steel stress at failure, psi yield strength of bars being spliced or developed, psi beam depth, in.

35.3trtdArrlsn

441

L0 basic (unreduced) live load

ld development or splice length, in.

N number of transverse reinforcing bars (stirrups or ties)

crossing ld

n number of bars being developed or spliced

along the plane of splitting

nb number of bars

Q totalload

Qv random variable representing dead load effects

Qvn nominal dead load

QL random variable representing live load effects

QLn nominal live load

(QL!Qvln = nominal ratio of live to dead load

q random loading

R random variable for resistance

Rn nominal resistance

RP predicted capacity random variable

R, ratio of projected rib area normal to bar axis to the product of

the nominal bar perimeter and the center-to-center rib spacing

r R1Rn=X(1)Rp1Rn

spacing of transverse reinforcement, in.

Tb total force in a bar at development or splice failure, lb

Ts contribution of confining steel to total bar force at bond failure

td 0.72db + 0.28, term representing the effect of bar size on Ts

t, 9.6R, + 0.28, term representing the effect

of relative rib area on Ts

V coefficient of variation

V R coefficient of variation for random variable for resistance

V Q coefficient of variation for random variable for total load

Vc ( Vc~yl + 0.0084)112, assumed coefficient of variation

for in-place concrete

Vccyl (jccyzlfc~ V m coefficient of variation associated with the predictive equation

(or model) itself

V QD coefficient of variation of random variable representing dead

load effects

V QL coefficient of variation of random variable representing live

load effects

V R coefficient of variation of relative rib area

V, r coefficient of variation of resistance random variable r

V TIP coefficient of variation of test/prediction ratio

V1s coefficient of variation of the predictive equation caused by uncertainties in the measured loads and differences in the actual

material and geometric properties of the specimens from values

used to calculate the predicted strength

V X(i) coefficient of variation of random variable X(i)

V<l>q coefficient of variation of loading random variable q

X (I) test-to-predicted load capacity random variable

X(2) actual-to-nominal dead load random variable

X(3) actual-to-nominallive load random variable

X(4) concrete strength!; random variable

X(5) splice length ld random variable

X(6) concrete cover cb random variable

X(7) side cover c so random variable

X(8) beam width b random variable

X(9) relative rib area R, random variable

~ reliability index

<!> strength reduction factor for the main loading

<i>b overall strength reduction factor against bond failure

<l>c strength reduction factor for the loading under consideration

<i>d <i>b/<1>, effective strength reduction factor for use in calculating development/splice length

YD load factor for dead loads

YL load factor for live loads

cr standard deviation

crccyl standard deviation for standard laboratory cylinders

Overbar represents average value of the variable

442

REFERENCES l. Darwin, David and Graham, Ebenezer K., "Effect of Deformation

Height and Spacing on Bond Strength of Reinforcing Bars," SL Report 93-l, University of Kansas Center for Research, Lawrence, Jan. 1993, 68 pp.

2. Darwin, David, and Graham, Ebenezer K., "Effect of Deformation Height and Spacing on Bond Strength of Reinforcing Bars," ACI Structural Journal, V. 90, No.6, Nov.-Dec. 1993, pp. 646-657.

3. Darwin, David; Tholen, Michael L.; Idun, Emmanuel K.; and Zuo, Jun, "Splice Strength of High Relative Rib Area Reinforcing Bars," SL Report 95-3, University of Kansas Center for Research, Lawrence, May 1995, 58 pp.

4. Darwin, David; Tholen, Michael L.; Idun, Emmanuel K.; and Zuo, Jun, "Splice Strength of High Relative Rib Area Reinforcing Bars," ACI Structural Journal, V. 93, No. 1, Jan.-Feb. 1996, pp. 95-107.

5. Darwin, David; Zuo, Jun; Tholen, Michael L.; and Idun, Emmanuel K., "Development Length Criteria for Conventional and High Relative Rib Area Reinforcing Bars," SL Report 95-4, University of Kansas Center for Research, Lawrence, May 1995,70 pp.

6. Darwin, David; Zuo, Jun; Tholen, Michael L.; and Idun, Emmanuel K., "Development Length Criteria for Conventional and High Relative Rib Area Reinforcing Bars," ACI Structural Journal, V. 93, No. 3, May-June 1996,pp.347-359.

7. Darwin, David; Idun, Emmanuel K.; Zuo, Jun; and Tholen, Michael L., "Reliability-Based Strength Reduction Factor for Bond," SL Report 95-5, University of Kansas Center for Research, Lawrence, May 1995,47 pp.

8. Chinn, James; Ferguson, Phil M.; and Thompson, J. Neils, "Lapped Splices in Reinforced Concrete Beams," ACI JOURNAL, Proceedings, V. 52, No.2, Oct. 1955, pp. 201-214.

9. Chamberlin, S. J., "Spacing of Reinforcement in Beams," ACI JOUR­NAL, Proceedings, V. 53, No. I, July 1956, pp. 113-134.

10. Chamberlin, S. J., "Spacing of Spliced Bars in Beams," ACI JouR­NAL, Proceedings, V. 54, No. 8, Feb. 1958, pp. 689-698.

11. Mathey, Robert, and Watstein, David, "Investigation of Bond in Beam and Pull-Out Specimens with High-Yield-Strength Deformed Bars," ACI JOURNAL, Proceedings, V. 32, No.9, Mar. 1961, pp. 1071-1090.

12. Ferguson, Phil M., and Thompson, J. Neils, "Development Length of High Strength Rein- forcing Bars," ACI JOURNAL, Proceedings, V. 62, No. 1, Jan. 1965, pp. 71-94.

13. Ferguson, Phil M., and Breen, John E., "Lapped Splices for High Strength Reinforcing Bars," ACI JOURNAL, Proceedings, V. 62, No. 9, Sept. 1965,pp. 1063-1078.

14. Thompson, M.A.; Jirsa, J. 0.; Breen, J. E.; and Meinheit, D. F., "The Behavior of Multiple Lap Splices in Wide Sections," Research Report No. 154-1, Center for Highway Research, The University of Texas at Austin, Feb. 1975, 75 pp.

15. Zekany, A. J.; Neumann, S.; and Jirsa, J. 0., "The Influence of Shear on Lapped Splices in Reinforced Concrete," Research Report No. 242-2, Center for Transportation Research, Bureau of Engineering Research, Uni­versity of Texas at Austin, July 1981, 88 pp.

16. Choi, Oan Chul; Hadje-Ghaffari, Hossain; Darwin, David; and McCabe Steven L., "Bond of Epoxy-Coated Reinforcement to Concrete: Bar Parameters," SL Report No. 90-1, University of Kansas Center for Research, Lawrence, Jan. 1990,43 pp.

17. Choi, Oan Chul; Hadje-Ghaffari, Hossain; Darwin, David; and McCabe Steven L., "Bond of Epoxy-Coated Reinforcement: Bar Parame­ters," ACI Materials Journal, V. 88, No.2, Mar.-Apr. 1991, pp. 207-217.

18. DeVries, R. A.; Moehle, J.P.; and Hester, W., "Lap Splice Strength of Plain and Epoxy-Coated Reinforcement," Report No. UCB/SEMM-9 1/02, University of California, Berkeley, California, Jan. 1991,86 pp.

19. Hester, Cynthia J.; Salamizavaregh, Shahin; Darwin, David; and McCabe, Steven L., "Bond of Epoxy-Coated Reinforcement to Concrete: Splices," SL Report 91-1, University of Kansas Center for Research, Lawrence, May 1991, 66 pp.

20. Hester, Cynthia J.; Salamizavaregh, Shahin; Darwin, David; and McCabe, Steven L., "Bond of Epoxy-Coated Reinforcement: Splices," ACI Structural Journal, V. 90, No. 1, Jan.-Feb. 1993, pp. 89-102.

21. Rezansoff, T.; Konkankar, U.S.; and Fu, Y. C., "Confinement Limits for Tension Lap Splices under Static Loading," Report, University of Saskatchewan, Saskatoon, Aug. 1991, 24 pp.

22. Rezansoff, T.; Akanni, A; and Sparling, B., "Tensile Lap Splices under Static Loading: A Review of The Proposed ACI 318 Code Provi­sions," ACI Structural Journal, V. 90, No.4, July-Aug. 1993, pp. 374-384.

23. Azizinamini, A; Stark, M.; Roller, John J.; and Ghosh, S. K., "Bond Performance of Reinforcing Bars Embedded in lligh-Strength Concrete," ACI Structural Journal, V. 95, No.5, Sept.-Oct. 1993, pp. 554-561.

ACI Structural Journal I July-August 1998

24. Azizinamini. A; Chisala. M.; and Ghosh, S. K., "Tension Develop­ment Length of Reinforcing Bars Embedded in High-Strength Concrete," Engineering Structures, V. 17, No.7, 1995, pp. 512-522.

25. ACI Committee 318, Building Code Requirements for Reinforced Concrete (ACI 318-95) and Commentary-ACI 318R-95, American Con­crete Institute, Farmington Hills, MI, 1995, 369 pp.

26. AASHTO Highway Sub-Committee on Bridges and Structures, Standard Specifications for Highway Bridges, 15th edition, American Association of State Highway and Transportation Officials, Washington, D.C., 1992, 686 pp.

27. Orangun, C. 0.; Jirsa, J. 0.; and Breen, J. E., "Reevaluation of Test Data on Development Length and Splices," ACI JOURNAL, Proceedings, V. 74, No.3, Mar. 1977, pp. 114-122.

28. Ellingwood, Bruce; Galambos, Theodore V.; MacGregor, James G.; and Cornell, C. Allin, "Development of a Probability Based Criterion for American National Standard AS 8," NBS Special Publication 577, U.S. Dept. of Commerce, Washington, D.C., June 1980, 222 pp.

29. Hogg, Robert V., and Ledolter, Johannes, Applied Statistics for Engi­neers and Physical Scientists, Macmillan Co., New York, 1992,472 pp.

30. Mirza, S. Ali, and MacGregor, James G., "Strength Variability of Bond of Reinforcing Bars in Concrete Beams," Civil Engineering Report Series No. CE-86-1, Lakehead University, Thunder Bay, Ontario, Jan. 1986, 35 pp.

ACI Structural Journal I July-August 1998

31. Lundberg, Jane E., "The Reliability of Composite Columns and Beam Columns," Structural Engineering Report No. 93-2, University of Minnesota, Minneapolis, June 1993, 233 pp.

32. Grant, Leon H.; Mirza, S. Ali; and MacGregor, James G., "Monte Carlo Study of Strength of Concrete Columns," ACI JOURNAL, Proceed­ings, V. 75, No.8, Aug. 1978, pp. 348-358.

33. Mirza, S. Ali; Hatzinikolas, Michael; and MacGregor, James G., "Statistical Descriptions of Strength of Concrete," Journal of the Structural Division, ASCE, V. 105, No. ST6, June 1979, pp. 1027-1037.

34. Jones, P. G., and Richart, F. E., "The Effect of Testing Speed on Strength and Elastic Properties of Concrete," Proceedings, ASTM, V. 36, Part II, 1936, pp. 380-391.

35. "Standard Test Method for Compressive Strength of Cylindrical Concrete Specimens (ASTM C 39-94)," 1995 Annual Book of ASTM Stan­dards, V. 04.02, American Society of Testing and Materials, Philadelphia, 1995, pp. 17-21.

36. Minimum Design Loads for Buildings and Other Structures, ASCE 7-95, American Society of Civil Engineers, New York, 1995.

37. ACI Committee 117, "Standard Specifications for Tolerances for Concrete Construction (ACI 117 -90)," American Concrete Institute, Farm­ington Hills, 1990, 12 pp.

443