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World J of Engineering and Pure and Applied Sci. 2012;2222((((3333):):):):98989898 ISSN 2249-0582

SSSSarkiarkiarkiarki et alet alet alet al., 2012. Safety of Lap Lengths Prediction in Reinforced Concrete Structures

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OriginalOriginalOriginalOriginal Article

Applied Science

Safety of Lap Lengths Prediction in Reinforced Concrete Structures

Yushau A SARKI 1, Abdulfatai A MURANA 2, Samuel O ABEJIDE 3

ABSTRACT [ENGLISH/ABSTRACT [ENGLISH/ABSTRACT [ENGLISH/ABSTRACT [ENGLISH/ANGLAISANGLAISANGLAISANGLAIS]]]] Affiliations:

1 Department of Civil Engineering, Faculty of Engineering, Ahmadu Bello University, ZARIA



Address for Correspondence/ Adresse pour la Correspondance: [email protected]

Accepted/Accepté: March, 2011

Citation: Sarki YA, Murana AA, Abejide SO. Assessing safety in predictions of lap lengths in reinforced concrete structures. World Journal of Engineering and Pure and Applied Sciences 2012;2(3):98-106.

The recommendations for lap lengths of steel bars in reinforced concrete as specified by the pseudo-

probabilistic design codes the British Standards Institution in about the past four decades and the current

relevant Eurocode have been reviewed and evaluated for their safety implications in the bar curtailment and

detailing. Results indicate that the implied safety is not uniform especially when the bar size factor in the

prevailing specified equations for lap length design is about half its value for all the codes. Also, the earlier

provision of the British codes gave the best safety indices in all the cases evaluated. However, it is suggested

herein that an adequate lap length of bars in reinforced concrete design could be about 50 times the smallest

bar size in the reinforced concrete member as opposed to the Eurocode’s provisions of 40times the smallest

bar size

Keywords: Reinforcement bars, lap lengths, concrete

RÉSUMÉ RÉSUMÉ RÉSUMÉ RÉSUMÉ [[[[FRANÇAISFRANÇAISFRANÇAISFRANÇAIS/FRENCH/FRENCH/FRENCH/FRENCH]]]]

Les recommandations pour une longueur de recouvrement des barres d'acier dans le béton armé tels que

spécifiés par les codes de conception pseudo-probabilistes de laBritish Standards Institution dans

environ les quatre dernières décennies et l'Eurocodeactuelle pertinents ont été examinés et évalués pour

leurs répercussions sur la sécuritédans la réduction bar et détaillant . Les résultats indiquent que la

sécurité implicite n'est pas uniforme en particulier lorsque le facteur de taille de la barre dans les

équations en vigueur spécifiées pour la conception longueur de recouvrement est d'environ la moitié de

sa valeur pour tous les codes. En outre, la disposition antérieure des codesbritanniques ont donné

les meilleurs indices de sécurité dans tous les cas évalués.Cependant, il est suggéré ici qu'une longueur de

recouvrement suffisante de barres debéton armé dans la conception pourrait être d'environ 50 fois la

taille plus petite barredans l'élément de béton armé, par opposition aux dispositions de

la Eurocode de40times de la taille plus petite barre

Mots-clés: Des barres d'armature, les longueurs de recouvrement, en béton

INTRODUCTIONINTRODUCTIONINTRODUCTIONINTRODUCTION In the construction of reinforced concrete, due to the

limitations in available length of bars and due to

constraints in construction, there are numerous occasions

when bars have to be joined, some of which are detailed

and hence the essence of overlapping two bars over at

least a minimum specified length called lap length [1, 2,

3]. This lap length varies depending on the bars sizes as

there are various bar sizes and where the bars are lapped

and/or which structural member or element the lapping

occurs.

As a result of load transfer, the steel bars maybe either in

axial tension or axial compression [4, 5]. Flexure, shear

and torsion may occur as effects [6], but due to

limitations of results and other factors, this paper would

focus solely on tension and compression and how they

vary with different lap lengths and bar sizes. Hence, the

distribution of tensile stresses in the concrete normal to

the axis of the bars is relevant. The overlap on the other

hand transfers or generates additional forces in the

concrete which tend to push the bars apart, so concrete

cover must be strong enough to overcome this “bursting

force” [2, 7].

This paper attempts to deal basically with the variables

of the lap splice which include lap length, the head-size

and shape, and the bar spacing.

MATERIALS AND METHODSMATERIALS AND METHODSMATERIALS AND METHODSMATERIALS AND METHODS Lap Splice Design Equations A revised equation developed resulting from

advancement in technology, submitted to building codes

ISSN 2249-0582 World J of Engineering and Pure and Applied Sci. 2012;2222((((3333):):):):99999999


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and reference documents and subsequently adopted is

given [2] in equation 1 below.

= (1)

Where

= diameter of reinforcement.

= specified yield stress of the reinforcement.

= specified compressive strength of masonry.

= required splice length of reinforcement.

= lesser of the masonry cover, clear spacing between

adjacent reinforcement or 5 times

= adherence factor (0.5, 1.0, 1.5 & 2.0).

The minimum length of lap splices for reinforcing bars in

tension or compression, , is calculated using equation 1

above but shall not be less than 300 mm (12 in)

The metric form of equation 1 above is given as:

= (2)

Equations (1) and (2) have been evaluated in a

probabilistic setting to determine the safety of their

provisions when used in reinforced concrete. The safety

checking is carried out as suggested by [8]. This

procedure is described in the next section.

Reliability analysis Reliability is defined [4] as the probability of a

performance function g(X) greater than zero i.e. P{g(X) >

0}. In other words, reliability is the probability that the

random variables Xi = (X1, …., Xn) are in the safe region

that is defined by g(X) > 0. The probability of failure is

defined as the probability P{g(X) < 0} .Or it is the

probability that the random variables Xi = (X1, ……., Xn)

are in the failure region that is defined by g(X) < 0.

In a mathematical sense, structural reliability can be

defined as the probability that a structure will attain each

specified limit state (ultimate or serviceability) during a

specified reference period and set of conditions [9]. The

idea of a `reference period’ comes into play because the

majority of structural loads vary with times in an

uncertain manner. Hence the probability that any

selected load intensity or criterion will be exceeded in a

fixed interval of time is a function of the length of that

interval. Thus, in general, structural reliability is

dependent on the time of exposure to the loading

environment.

Concept of Structural Reliability Analysis In order to incorporate structural reliability into

reinforcement lap length requirements, we shall assume

that R and S are random variables whose statistical

distributions are known very precisely as a result of a

very long series of measurements. Let R be a variable

representing the variations in strength between

nominally identical structures, whereas S represents the

maximum load effects in successive T-yr periods, then

the probability that the structure will collapse during any

reference period of duration, T years, is given by [9, 10]:

(3)

The reliability of the structure is the probability that it

will survive when the load is applied, given by [9, 10]:

(4)

Where, FR is the probability distribution function of R

and fs is the probability density function of S. Note that R

and S are statistically independent and must necessarily

have the same dimensions.

Resistant and Load Interaction In basic reliability problems, consideration is given to the

effect of a load S and the resistance R offered by the

structure [9]. Both the load and resistant S and R can be

described by a known probability density function (fS)

and (FR) respectively. S can be obtained from the applied

load through a structural analysis making sure that R

and S are expressed in the same unit [9].

Considering only safety of a structural element, it would

be said that a structural element has failed if its resistance

R, is less than stress S acting on it. The probability of

failure Pf of the structural element can be expressed in

any of the following ways [9, 10].

Pf = P(R – S)

Where R = strength (resistance) and S = loading in the

structure. The failure in this case is defined in this region

where R - S is less than zero or R is less than S, that is;

Pf = P((R – S) ≤ 0) (5)

As an alternative approach to Equation (5), the

performance function can also be given as:

(6)

where in this case, the failure is defined in the region

where Pf is less than 1, or R is less than S, that is.

Pf ≤ 1 or R ≤ S

It could also be expressed as;

Pf = P (Ln R – Ln S ≤ 1) (7)

or in general,



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Pf = P [G(R, S) ≤ 0] (8)

where G(x) is the “limit state function” and the

probability of failure is an identical with probability of

the limit state violation.

For any random variable X, the cumulative distribution

function Fx(x) is given by Equation (9) provided that x ≥

y.

(9)

It follows for the common, but special case where R and

S are independent, the expression for the probability of

failure is

Pf = P(R – S ≤ 0) = (χ)fs(χ)dx

(10)

Expression (10) is known as the “convolution integral”

and FR(x) is the probability that R ≤ x, or the probability

that the actual Resistance R of the member is less than

some value x. fs(x) represents probability that the load

effect S acting in the member has a value x and x + Δx in

the limit as Δx → 0.

Considering all possible value of x, total failure

probability is obtained as follows:

(11)

i.e. sum of all the cases of resistances for which the loads

exceed the resistances.

Limit State / Performance Function The limit state “g(x) = R - S” is a function of material

properties, loads and dimensions. The state of the

performance function g(x) of a structure or its

components at a given limit state is usually modelled in

terms of infinite uncertain basic random variable

x = (x1, x2, ……….,xn) with joint distribution function

given as;

(12)

and

(13)

where is the joint probability distribution function

of x. The region of integration of the function g(x) is

stated as follows:

g(x) > 0 : represents safety;

g(x) = 0 : represents attainment of the limit state;

g(x) < 0 : represents failure.

The probability of failure is given by P(g(x) < 0) and

therefore the reliability index β can be related to

probability of failure by the following equation

= 1 – Φ (β)

(14)

Computation of Reliability Index The basic approach to develop a structural reliability

base strength standard is to determine the relative

reliability of the design. In order to do this, reliability

assessment of existing structural components is needed

to estimate a representative value of the reliability index

β. This First Order Reliability Method is very well suited

to perform such a reliability assessment [8, 9, 10]. For the

estimation of the probability of failure, Level 2 methods

are employed, which involve approximate iterative

calculation procedures. In this method, two important

measures are used:

(a) Expectations,

(b) Covariances,

The “safety margin” is the random variable M = g(x) (also

called the `state function’). Non-normal variable are

transformed into independent standard normal

variables, by locating the most likely failure point, β-

point (called the reliability index), through an

optimization procedure. This is also done by linearizing

the limit state function in that point and by estimating

the failure probability using the standard normal

integral.

The reliability index,β, is then defined by

(15)

Where µ m = mean of M

And σ m = Standard deviation of M

If R and S are uncorrelated and with M = R - S,

Therefore,

( )βµ µ

σ σ=

−

+R s

R s2 2 1 2/

(16)

Assume that the safety margin M is linear in the basic

variable Xl,…, Xn

M C C X C Xn n= + + +0 1 1 .........

Where, C C Cn0 1, ......, are constants

µ µ µm n nC C C= + + +0 1 1 ........ and





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[ ]σ σ σ σ σ

σ σm n ni j

n

j

n

ij i

j i j

i j

xi xj

C C C CCov X X

212

12 2 2

1= + + +

= = ≠..........

,Σ Σ

(17)

β can thus be computed.

In the above, the safety margin, M, is assumed linear in

the basic variables. If on the other hand, it is non-linear in

X = {Xi,…, Xn}, then approximate values for µm and σm can

be obtained by using a linearized safety margin, M.

Let

M g x g X Xn= =( ) ( ,......., )1 (18)

Using Taylor series to expand about (Xi, ……., Xn) =

(µi,….., µn) and retaining only the linear terms, we obtain

( )M g x g Xn i i i

g

xi= = + −=( ) ( ,......., )µ µ µδ

δ1 1Σ

(19)

Where, δg/δxi is evaluated at (µI,….., µn)

( )µ µ µm ng≈ 1,......,

(20)

[ ]σδ

δδ

δ2

1 1m

i

n

j

ng

xi

gi jxj Cov x x≈

= =Σ Σ ,

(21)

From the above; the point of linearization is the so-called

mean point (µI, ….., µn). However, a more reasonable

point will be one on the failure surface. Experience shows

that an expansion based on the mean point should not be

used [8, 9, 10]. Because of this objection, the reliability

index,β, proposed [11], is particularly useful. Reliability

index, β, is defined as the shortest distance from the

origin to the failure surface in the normalized Z-

coordinate system. The point where this occurs on the

failure plane is called the design point [8, 9, 10, 11].

First of all, the set of basic variables is normalized.

ZX

i ni

i xi

xi

=−

=µ

σ,

, ,... ,1 2

(22)

where µxi and σxi are the mean and standard deviation of

the random variable, Xi, and the normalized set,

Z = (Zi, …, Zn) (23)

Since the basic variables are normally distributed,

µzi = 0 and σzi = 1, i = 1, 2,….,n

The reliability index is thus formulated as below:

β =

∈ =min

/

z Fi

i

ZΣ 2

1

1 2

(24)

where F is the failure surface in the Z-coordinate system.

Where the failure surface is non-linear, calculation for β

is done using iterative procedures.

A relationship can be drawn between the probability of

failure, Pf, and the reliability index, β.

All methods are approximates and the problems become

more difficult as the number of random variables and the

complexity of the limit state functions increases; and

when statistical dependence between random variables is

present. Reliability analysis consists of the calculation

that resistance will effectively exceed the applied load of

a structure during its design life time. With this reason,

safety becomes a measure of the probability of survival.

A limit state function g(x), thus defines the resistance and

load effect and various regions can be defined as failure

of safety which, is dependent on whether the g(x) < 0 or

not.

Method of Reliability Safety Checking All methods are approximate and the problems becomes

more difficult as the number of random variables and the

complexity of the limit state function increases and when

statistical dependence between random variable is

present.

Of the two broad classes of methods of structural

reliability analysis (level 2 and level 3 method of safety

checking), level 2 method shall be employed. Level 2 is

known as second moment, First Order Reliability

Method (FORM). The random variables are defined as

terms of means and variance and are considered to be

normally distributed. The measure of reliability is based

on the reliability index. It involves use of certain iteration

correlation procedure to obtain an approximation to the

probability of failure of a structure or structural system.

This generally requires an idealization of failure domain

and it is often associated with a simplified representation

of the joint probability distribution of a variable.

The necessity to have a method of reliability analysis

which is computationally fast and efficient and which

produces result with degree of accuracy prompted the

use of this level 2 method.

RESULTSRESULTSRESULTSRESULTS The stochastic models generated are analysed using the

First Order Reliability Method as proposed [8], to give

values for safety index, β, for the various diameter of

reinforced concrete bars. Algorithms developed into

FORTRAN modules were designed for failure modes in

relation to some building codes, namely, the British [12,

13] and the current European [14] codes. The diameter

and compressive strength of masonry were varied for all



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yield stresses in the algorithms to get the various safety

levels. These safety criteria are plotted against the

various respective diameters of reinforced concrete bars.

Some of the results of the safety values against their

respective diameters are shown in figures 1 to 18.

DISCUSSIONDISCUSSIONDISCUSSIONDISCUSSION RECOMMENDATIONRECOMMENDATIONRECOMMENDATIONRECOMMENDATION With regards to the safety values in the figures 1 - 18 as

shown, the lap lengths in bars can be said to be safest at

the lowest possible values of the variables (that is, bar

diameter, yield stress and adherence factor). It would be

observed that at a constant yield strength and adherence

factor, the optimum safety (i.e. the maximum β value)

occurs at small diameters of reinforced concrete bars

when the specified compressive strength of concrete is

high. But the safety of bar lap lengths reduces as the bar

diameter increases and the compressive strength of

concrete decreases simultaneously.

Therefore it could be seen that the safety of lap lengths of

reinforced concrete bars is inversely proportional to the

diameter of reinforced concrete bars and directly

proportional to the specified compressive strength of

concrete at constant yield stress of reinforcement bars

and adherence factor.

The results also indicate that the British [12, 13] and

European [14] codes for the design of reinforced concrete

structures all have similar results. Also the provision of

the American Concrete Institute [15] is similar to the

three codes used as examples.

Results show that among the building codes

investigated, the provisions for bar lap lengths in

concrete of the British code [12] gave the highest safety.

This implies that the provisions, which stipulates lap

lengths of reinforced concrete bars to be equal to

(25 , gives the best provisions for lap lengths of

steel in tension. But the provision for lap length of bars in

compression is smaller than that for tension. Also,

separate formulations will have to be remembered in

design or construction of reinforced concrete structures.

However, it is necessary to prescribe a format as in the

current European code [13] which will be very easy to

comprehend while still achieving both safety and

economy when considering laps of bars in reinforced

concrete structures.

CONCLUSIONCONCLUSIONCONCLUSIONCONCLUSION There are separate formulations for bar lap lengths in

compression and tension of reinforced concrete members

in the British codes, while only one provision is made in

the current Eurocode [14] for design and construction of

reinforced concrete members. One of the British codes

[13] has attempted a single value provision. However,

the safety intrinsic in the single value provisions in the

Eurocode and British code is not commensurate with the

economy in bar curtailment achieved. Thus, it is

necessary to prescribe a single value provision that will

be safe while at the same time achieving the required

economy, which will be applicable to all codes for the

design and construction of reinforced concrete structures.

This is a bid to reduce failure of structures which have

occurred in recent times in many parts of the world;

hence the review of the current international codes of

practices.

Figure 1:Figure 1:Figure 1:Figure 1:: This figure shows effect of Bar Size on Safety

of Lap Lengths at fy = 250 and α = 0.5 (BS8110)


of Lap Lengths at fy = 250 and α = 2 (BS8110)





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of Lap Lengths at fy = 250 and α = 0.5 (CP110)


of Lap Lengths at fy = 250 and α = 2 (CP110)



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of Lap Lengths at fy = 250 and α = 0.5 (EC2)


of Lap Lengths at fy = 250 and α = 2 (EC2)





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RECOMMENDATIONRECOMMENDATIONRECOMMENDATIONRECOMMENDATION

It is therefore suggested that for every diameter of

reinforced concrete bar, the lap length could be 50 times

the diameter of the smallest reinforcement bar

irrespective of the prevailing compressive or tensile

strength and yield stress; that is;

Lap Length 50

where = diameter of reinforcement bar.

Also, this suggestion is applicable to lap lengths of bars

in reinforced concrete designs and construction

notwithstanding the design code used.

REFERENCESREFERENCESREFERENCESREFERENCES

[1] Alling ES. Some Comment On Flexural and

Anchorage Bond Stresses, U.S. Department of

Agriculture Soil Service; Engineering Division,

Design Branch, Design Note;1968;No.5,pp.1–4.

[2] Thompson MK, Ledesma AL, Jersa JO, Breen JE,

Klinger RE, Anchorage Behavior of Headed

Reinforcement, Part A: Lap Splices andPart B:

Design Provisions and Summary; 2002;pp.1-45.

[3] Erico and Cagley Associates. Mechanical vs Lap

Splices in Reinforced Concrete. Rockville

2005;1:1–6.

[4] Canbay, E, Frosch, JR., Design of Lap Spliced

Bars: Is Simplification Possible? 2006;pp. 1-7.

[5] Mosley WH, Bungey JC. Reinforced Concrete

Design. 5th Edition. Macmillan Press Ltd,

London. 1990.

[6] Bilal SH, Ahmad AR, Khaled AS. Bond Strength

of Tension Lap Splices in High Strength

Concrete Beams Strengthened with Glass Fiber

Reinforcement. J. Compos. For Constr. 2004;

8:14-21.



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[7] American Concrete Institute, Building Code of

Requirements for Structural Concrete and

Commentary. ACI, Michigan. 2002. p.32 – 36.

[8] Gollwittzer S, Abdo T, Rackwitz R. First Order

Reliability Method,(FORM) Manual, RCP

Gmbh, Nymphenburger Str. 134, MÜNCHEN;

Germany. 1988.

[9] Ditlevsen O, Madsen HO. Structural Reliability

Method. First edition. John Wiley & Sons Ltd;

Chichester. 1996. Retrieved from:

http://www.mek.dtu.dk/staff/od/books.htm.

[10] Ayyub B, McCuen RH. Probability, Statistics,

and Reliability for Engineers CRC Press LLC;

New York. 1997.

[11] Hasofer, AM, Lind, NC. Exact and Invariant

Second – moment Code Format. Journal of

Engineering Mechanics 1974;1:111–21.

[12] British Standards Institution. The Structural Use

of Concrete, Part 1: Code of Practice for Design

and Construction. British Standards

Institution; London. 1972.

[13] British Standards Institution. The Structural Use

of Concrete, Part 1: Code of Practice for

Design and Construction. British Standards

Institution; London. 1997.

[14] European Committee, Design of Concrete

Structures, Part 1.1. European Committee for

Standardization; Brussels. 2008.

[15] American Concrete Institute. Building Code

Requirement for Structural Concrete and

Reinforced Concrete. American Concrete

Institute; Michigan. 2005.

ACKNOWLEDGEMENT ACKNOWLEDGEMENT ACKNOWLEDGEMENT ACKNOWLEDGEMENT / / / / SOURCE OF SUPPORTSOURCE OF SUPPORTSOURCE OF SUPPORTSOURCE OF SUPPORT Nil

CONFLICT OF INTERESTCONFLICT OF INTERESTCONFLICT OF INTERESTCONFLICT OF INTEREST No conflicts of interests were declared by authors.

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