Moment Redistribution Theoretical Studies

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CHAPTER

6 THEORETICAL STUDIES ON MOMENTREDISTRIBUTION

CONTENTS

6 THEORETICAL STUDIES ON MOMENT REDISTRIBUTION 447

6.1 Introduction 449

6.2 Literature Review 450

6.2.1 Moment Redistribution Concept 450

6.2.2 Plastic Hinge Approach 452

6.2.3 Moment Redistribution in National Standards 454

6.3 Fundamental Concept of Flexural Rigidity (EI) Approach 459

6.4 Simplified Flexural Rigidity (EI) Approach 460

6.4.1 Journal Paper: Moment Redistribution In Continuous Plated RC Flexural Members. Part 2 Flexural Rigidity

Approach461

6.4.2 Further Discussions on Simplified EI Approach 479

6.4.2.1 Derivation of Mathematical Equations for Beams with Different Hogging and Sagging Stiffnesses 479

6.4.2.1.1 One End Continuous Beam Subjected to Point Load 479

6.4.2.1.2 Both Ends Continuous Beam Subjected to Uniformly Distributed Loads 483

6.4.2.2 Comparison Between Experimental and Theoretical Results 490

6.4.2.2.1 Test Series S (Specimens With Externally Bonded Plates) 4906.4.2.2.2 Test Series NS and NB (Specimens With NSM Strips) 495

6.4.3 Parametric Studies Based on Simplified EI Approach 502

6.4.3.1 Journal Paper: Moment redistribution parametric study of CFRP, GFRP and steel surface plated RC

beams and slabs 502

6.5 Linear Flexural Rigidity (EI) Approach 521

6.5.1 Journal Paper: Moment redistribution in FRP and steel p lated reinforced concrete beams 521

6.5.2 Further Discussions on Linear EI Approach 545

6.5.2.1 Derivation of Equivalent EI 5456.5.2.1.1 One End Continuous Beam Subjected to Point Load 545

6.5.2.1.2 Both Ends Continuous Beam Subjected to Uniformly Distributed Loads 552


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6.5.2.2 Comparison Between Experimental and Theoretical Results 557

6.5.2.2.1 Test Series S (Specimens With Externally Bonded Plates) 557

6.5.2.2.2 Test Series NS and NB (Specimens With NSM Strips) 562

6.6 Summary 569

6.7 References 570

6.8 Notations 571


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Intermediate Crack Debonding of Plated RC Beams Theoretical Studies on Moment Redistribution

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6.1 INTRODUCTION

The existing design guidelines (fib 2001; Concrete Society 2000) tend to neglect any moment

redistribution occurring in plated structures. However, the experimental studies presented in Chapter 5

and tests performed by other researchers, such as Mukhopadhyaya et al. (1998), El-Refaie et al.

(2003) and Ashour et al. (2004), clearly showed that significant amounts of moment redistribution can

be obtained in both externally bonded (EB) and near surface mounted (NSM) plated beams.

Therefore, new approaches are required to analyse the moment redistribution behaviour of continuous

plated members that takes into account premature debonding failure prior to concrete crushing.

The member ductility design of reinforced concrete (RC) continuous beams or frames often uses the

plastic hingeconcept

(Darvall and Mendis 1985; Barnard 1964) and the neutral axis depth factor(ku),which is common to most national standards (i.e. AS3600), to quantify both collapse and the

associated ability to redistribute moment within a continuous beam prior to collapse. These

approaches work well in unplated reinforced concrete structures as the material ductility of the steel

tension reinforcing bars, that is their strain capacity, can be assumed to be very large which ensures

that compressive crushing of the concrete, at an often specified strain c, always controls failure of the

beam (Oehlers et al. 2004a). However for plated beams where premature debonding often occurs,

this method of determining moment redistribution is found to be unsuitable.

To date, very limited research has been carried out on moment redistribution of plated structures. A

few researchers, such as Mukhopadhyaya et al. (1998) and El-Refaie et al. (2003), have developed

different indexes to measure the ductility of beams with external reinforcement, but none can be used

to quantify moment redistribution of continuous plated structures.

Through the experimental investigations performed in Chapter 5, it has been shown that moment

redistribution is affected by the extent of cracking along the beam. That is moment redistribution is

dependent on the variation in stiffness along the beam. Therefore in this research, the flexural rigidity

(EI) approach was developed for evaluating the moment redistribution behaviour of plated and

unplated reinforced concrete members. This approach takes into account the variation in stiffness

along the beam, while assuming that there is zero rotation at the hinges. To allow for the variation in

stiffness, two methods are proposed: (1) the simplified EI approach, where the stiffness of the hogging

(EIhog) and the sagging (EIsag) are different, while within each region EI is assumed to be constant; and

(2) the linear EI approach, where the stiffness within the hogging and sagging regions of the beam is

assumed to vary linearly.


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In this chapter a literature review on the existing methods for moment redistribution analysis of

statically indeterminate beams is firstly presented, followed by a description of the fundmental concept

of the flexural rigidity approach developed in this research. The simplified EI approach is described

and verified with the experimental results of EB plated specimens (Chapter 5) in the journal paper

included in the Section 6.4.1. Further discussions on the simplified EI approach are given in Section

6.4.2, which includes the derivation of the mathematical expressions of the simplified EI approach and

the application of the approach to the NSM test specimens in Chapter 5. Parametric studies on

varying plating positions and materials were also performed on EB plated beams using the simplified

EI approach and are presented in the journal paper in Section 6.4.3. Finally, the linear EI approach is

described and verified in the journal paper in Section 6.5.1, with further discussions on the derivation

of the mathematical equations and the comparison between the experimental and test results given in

Section 6.5.2.

6.2 LITERATURE REVIEW

Moment redistribution is an important and beneficial behaviour in statically indeterminate structures as

it allows transfer of moments from the most stressed to less stressed areas, hence giving a more

economical and efficient design. The total moment redistribution in a statically indeterminate system

consist of two parts (CEB-FIP 1998): (1) related to the change of stiffness in the span and over the

support due to different cracking; (2) governed by ductility of reinforcement when passed the yielding

moments in the hinge that occurs first. In the following section, the concept of moment redistribution is

firstly revised, then the plastic hinge approach presently used to determine the moment redistribution

of reinforced concrete structures is reviewed, and finally, the neutral axis depth factor used by various

RC codes and standards to determine the amount of moment redistribution is discussed.

6.2.1 MOMENT REDISTRIBUTION CONCEPT

Consider the encastre or built in unplated reinforced concrete beam of length L in Figure 6.1c, which is

equivalent to an internal span of a continuous beam. For convenience, it is assumed that the same

longitudinal reinforcing bars are in the top and bottom of the beam. Hence, the hogging (hog) and

sagging (sag) regions have the same moment/curvature (M/) relationships as shown in Figure 6.1a,

where: the idealised perfectly elastic portion has a flexural rigidity of (EI)elasup to a moment capacity


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of Muat a curvature y; after which there is a perfectly plastic ductile plateau in which the secant

stiffness (EI)secreduces up to a curvature of uat which failure occurs when the secant stiffness is at

its minimum (EI)min. The beam in Figure 6.1c is subjected to a uniformly distributed load w, so that

whilst the flexural rigidity of the whole beam remains at EI, the moment at the supports Mhog is twicethat at mid-span Msag. Hence for this specific beam, there is no moment redistribution whilst the

maximum hogging moment Mhog is equal to twice the maximum mid-span moment Msag. Conversely,

when Mhog 2Msag, then there is moment redistribution. Therefore in this context, moment

redistribution is defined as occurring when the distribution of moment within a beam is not given by

elastic analyses that assume EI is constant within the beam.

Mu

Mu/2

M

sag2

sag1

hog1 hog2

(Mstatic)1=1.5Mu=w1L2/8

(Mstatic)2=2Mu=w2L2/8

Mhog=Mu

Msag=Mu/2

Msag=Mu

w (kN/m)

L

(c)continuousbeam

EI

elastic

non-elastic

(a) (b)

y u(EI)elas

A B

(EI)sec

(EI)min

hogging joint

ductile plateau

elastic

Figure 6.1 Moment redistribution concept

As the uniformly distributed load wis gradually applied to the beam in Figure 6.1c, the beam is initially

elastic so that Mhog = 2Msag and there is no moment redistribution. When the support moment first

reaches its moment capacity Mu as shown as the point hog1 in Figure 6.1a, then the mid-span

moment reaches a value of Mu/2 which is shown as sag1. At this stage, the static moment is (Mstatic)1=

1.5Mu = w1L2/8 as shown in Figure 6.1b and the distribution of moment is given by line A which is

labelled elastic. Up to this point, the beam behaviour remains linear elastic. As the load is increased,

the beam deflects further resulting in an increase in Msag above Mu/2 in Figure 6.1b. However, the

moment at the support remains at Mu. The only way that the increase in deflection or deformation, due

to the increased load, can be accommodated is for the curvature at the supports to be increased from

hog1 to hog2 as shown in Figure 6.1a and the hogging curvature will keep increasing until the saggingcurvature sag1 reaches sag2 in Figure 6.1a, that is the mid-span moment has reached its capacity M u


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whilst the behaviour of the hogging region is no longer elastic. The static moment has now reached

(Mstatic)2= 2Mu= w2L2/8 in Figure 6.1b, which is the maximum static moment. Hence, the maximum

load w2that can be applied as all the joints, that is the positions of maximum moments in the hogging

and the sagging regions, have reached their moment capacities and a collapse mechanism has

formed. The distribution of moment within the beam is now given by line B which has been labelled

non-elasticas shown in Figure 6.1b.

It can be seen in the example shown in Figure 6.1, that it is the hogging joints that are required to

maintain the moment whilst their curvature is increasing. Hence in this example, it is the hogging joints

that have to redistribute moment and it is their ductility that governs the amount of moment

redistribution. If for example it was necessary for hog2 in Figure 6.1a to exceed the curvature capacity

of the section u, to achieve the static moment (Mstatic)2in Figure 6.1b, then sag2 in Figure 6.1a cannot

achieve Muand the continuous beam would fail before reaching its theoretical plastic capacity. It can

be seen in this example that the sagging moment joint has only to reach its moment capacity, M u in

Figure 6.1a at point sag2, that is its curvature has only to reach y. Hence its ductility, that is its

capacity to extend along the plateau in Figure 6.1a, is of no consequence. Unless of course the beam

is required to absorb energy such as under seismic loads, in which case it may be a requirement that

point sag2 is also extended into the plastic zone to allow the beam to deflect further and absorb

energy without an increase in load.

6.2.2 PLASTIC HINGE APPROACH

To determine whether a beam is ductile enough to redistribute moment is an extremely complex

problem and there is much good ongoing research (CEB-FIP 1998; Bigaj 1999; El-Refaie et al. 2003,

2001; Mukhopadhyaya et al. 1998) to develop a comprehensive and simple solution. These

researches generally involve the development of different indexes to measure the ductility of beams,

but none of which can be used to quantify moment redistribution of continuous plated structures. The

problem is to understand how the beam can deform to accommodate the non-elastic distribution of

moment (line B in Figure 6.1b and also shown in Figure 6.2b) and then to determine whether the

deformation capacity of the beam can accommodate this required deformation. The presently used

method for analysing the redistribution of moment in statically indeterminate structures is known as

the plastic hinge approach. This method assumes that there is a discontinuity of the slope at the

supports as shown in line C in Figure 6.2e.


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Msa =Mho/2

Msag=Mhog

staticdeformations:

support momentdeformations:

overalldeformation:

dy/dx > 0: plastic hingeapproach

EI

plastichinge

(a)

(b)

(c)

(d)

A

B

C

L

EI

w

(e)

Mstatic

Mhog

plastichinge

(Mstatic)el

(dy/dx)static

(dy/dx)support

Figure 6.2 Plastic hinge approach

In the hinge approach, it is assumed that most of the beam of length L remains linear elastic at a

flexural rigidity EI as shown in Figure 6.1a, and that there are small hinge regions at the joints of

length Lhingewhere moment redistribution requires ductility. The hinge length Lhinge

Mhog/2. The discontinuity of slope can be determined from the static moment in Figure 6.2c and the

redundant moment shown in Figure 6.2d. The slope at the supports (dy/dx)staticand (dy/dx)support in

Figure 6.2c&d can be derived by integration of the curvature along the length of the beam. Hence the

discontinuity of the slope in line C in Figure 6.2e is equal to the difference between (dy/dx)static and

(dy/dx)supportwhich is accommodated by the plastic hinge in Figure 6.2a. As the length of the hinge is

very small, it is often assumed that the curvature within the hinge u is constant so that the rotationcapacity of the hinge capis:


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hingeucap L = Equation 6.1

Therefore, in order for moment redistribution to occur, the condition given in Equation 6.2 needs to be

satisfied, wherereq is the rotation required at the plastic hinge which can be determined from the

fundamental relationship given in Equation 6.3 (CEB-fib 1998). Using the concept of plastic hinge, the

available rotation capacity cap can be determined from Equation 6.1. One of the difficulties with

applying the plastic hinge theory is the determination of the plastic hinge length.

capreq Equation 6.2

= dxreq Equation 6.3

The earliest plastic hinge approaches assumed that the hinge occurs at a point, that is Lhinge0. This

posses some conceptual difficulties for beams with horizontal or falling branch moment/curvature

relationships, since the rotation of the beams requires hinges of zero length where the curvature was

increasing, whilst adjacent to the hinge the curvature was decreasing which requires a sudden step

change in the curvature at the boundary of the hinge. Acknowledging this problem, Johnson (Barnard

and Johnson 1965), Barnard (1964) and Wood (1968) proposed the concept of a finite hinge length.

This concept, however, is specifically for reinforced concrete beams only where concrete crushing

failure occurs, such that large amounts of rotation are present at the hinges to achieve the required

rotation (Equation 6.1). Therefore, the plastic hinge approach is unsuitable for plated structures where

premature debonding failure occurs prior to concrete crushing.

6.2.3 MOMENT REDISTRIBUTION IN NATIONAL STANDARDS

International standards tend to base the ability of (unplated) reinforced concrete beams and slabs to

redistribute moment on the neutral axis parameter kugiven by Equation 6.4, where dand dn are the

effective depth of the beam and the depth of the neutral axis from the compression face. This k u

factor, which measures the ductility of a structure, is based on the plastic hinge approach where the

hinge length is assumed to be equal to the depth of the beam i.e. L hinge=d. Therefore from Equation

6.1, the rotation at the hinge is given by Equation 6.5, where the curvature is equal to c/kud. For a

constant concrete crushing strain c, the rotation is, hence, directly proportional to 1/ku.


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d

dk nu = Equation 6.4

u

c

u

c

hingeucap

k

d

dk

L

=== Equation 6.5

Typical examples of kufrom five standards (DIN1045, CEB-FIP1990, BS8110, CAN-A23.2, AS3600)

are given in Figure 6.3 for the commonly used high ductility reinforcing bar steels. For these high

ductility steels, it can be assumed that the strain capacity of the steels is sufficiently large to ensure

that they never fracture prior to concrete crushing. Therefore, the ultimate failure of the RC beam is

always governed by concrete crushing at a strain cthat is often assumed to range between 0.003 to

0.004.

neutral axis parameter ku

10 %

0

20 %

30%

0.1 0.2 0.3 0.4 0.6

British

Canadian

European

German

Australian

moment

redistributionmean value

B

A

Figure 6.3 Moment redistribution dependence on neutral axis parameter ku

As shown in Figure 6.3, it can be seen that there is a general agreement for an upper bound of 30% to

the amount of moment redistribution that can occur. However, there is a fairly wide divergence

between predictions. For example, no moment redistribution is allowed when ku 0.4 for the

Australian Standard requirements but this is substantially increased to ku0.6 for both the Canadian

and British Standards; the mean value for no moment redistribution from the five approaches is ku

0.48 and is shown as point A. At the other extremity, the British Standard approach allow 30%

redistribution when ku0.3 whereas the Canadian approach uses 30% as an upper bound as ku0;


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the mean value for 30% redistribution is ku0.15 and is shown as point B. A line joining these mean

values at the extremities is shown as the mean value in Figure 6.3.

The effect of the variation A-B in Figure 6.3 of the neutral axis parameter k uon the strain profile of a

reinforced concrete (RC) section at failure is illustrated in Figure 6.4. For ease of explanation, let us

consider the case of a deep RC beam in which the effective depth d approaches the depth of the

beam h as shown in Figure 6.4a (which is the cross-section of a beam in the hogging or negative

region). It has been assumed that the concrete crushing strain cis 0.0035, as shown in Figure 6.4b.

As the strain capacity of the reinforcing bars is assumed to be very large, as previously explained,

concrete crushing always controls failure so that the strain at the compression face of c= 0.0035 is

common to all the strain profiles shown and can be considered to be a pivotal point. It can be seen in

Figure 6.4b that the neutral axis parameter kucontrols the maximum tensile strain at the tension face

tf for any depth of beam d, as tf = c(1-ku)/kuand, hence, tf is independent of d. It needs to be

pointed out that the neutral axis parameter kudoes not control the curvature for any beam depth as

this depends on the actual depth of the beam, that is = c/kud and, hence, it depends on d.

However, the ku factor controls the rotation of the plastic hinge, of length Lhinge, as this is given by

Lhingewhere the curvature at failure = c/kud; for example, if Lhingedthen the rotation is equal to

c/ku. It can, therefore, be seen that the ability to redistribute moment depends on the maximum

tensile strain tf.

d = h

kud=0.48d

c= 0.0035

( tf)0.15d=0.02

0%

30% 0% to 30%

pivotalpoint

tension face

compression face

EB steelplates

(a) (b)

mean 30%(point B)

mean 0%

(point A)

ductile

reinforcing

bars

( tf)0.48d=0.0038

kud=0.15d

NSM CFRP

strips

EB CFRPplates

Figure 6.4 Moment redistribution dependence on tension face strains

The strain profile associated with the mean 0% redistribution, that is point A in Figure 6.3, is shown in

Figure 6.4b as the line mean 0%; as the depth of the neutral axis from the compression face is 0.48d,the strain at the tensile face is (tf)0.48d= 0.0038. The strain profile for the mean 30% redistribution at


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point B in Figure 6.3 is also shown in Figure 6.4 and this requires a tensile face strain of (tf)0.15d=

0.020. Hence, when the concrete crushes and the tensile face strain tf < 0.0038 then no moment

redistribution is allowed, when 0.0038 < tf < 0.020 then between 0% and 30% redistribution is

allowed, and when tf > 0.020 then 30% redistribution is allowed. Generally speaking, ductilereinforcing bars can easily accommodate these strains. However, if a plate is adhesively bonded to

the tension face as shown in Figure 6.4a, then these strains have also to be accommodated by the

plate.

The strain capacity of FRP tension face plates depends on either its fracture strength or its IC

debonding resistance (Chapter 2). For most cases, IC debonding controls the strain capacity of

externally bonded FRP plates which commonly debond at less than half of the fracture capacity,

except for very thin plates used in the wet lay up process. Tests reported in Chapter 5 and also those

published elsewhere (Oehlers et al. 2003) conducted on 1.2 mm pultruded carbon FRP (CFRP) plated

beams, found that the IC debonding strains ranged from 0.0025 to 0.0052. This range of strains is

shown as the shaded region labelled EB CFRP plates in Figure 6.4b. The bounds of this range just

fall either side of the mean 0% profile which suggests that in general pultruded carbon FRP plated

structures have little capacity for moment redistribution. This, however, is not the case for NSM CFRP

plates, as can be seen in Figure 6.4b. Due to the strong bond that exist at the plate/concrete interface,

NSM strips are found to debond at much large strains (Chapter 5). Depending on the plating

configuration, NSM CFRP plates can accommodate around 20% redistribution (Oehlers et al. 2005),

as indicated by the unshaded region labelled NSM CFRP plates in Figure 6.4b.

Metal plates can be designed to IC debond prior to yielding in which case the behaviour is similar to

that of FRP plates. However and in contrast to FRP plates, metal plates can be designed to yield prior

to IC debonding; although it should be remembered that tests have shown that in the majority of cases

the metal plated beam will still eventually debond but at a much larger strain than if it remained elastic.

EB beam tests in Chapter 5 have shown that the IC debonding strains for 3 mm steel plates range

from 0.0045 to 0.021 which is shown as the hatched region in Figure 6.4b, and which suggests that

metal plates that have been designed to yield prior to IC debonding may have adequate capacity to

redistribute moment.

It has been shown that moment redistribution based on the kuapproach is controlled by the strain at

the tension face tfwhen the concrete crushes at c. Hence, it is not the moment/curvature, M/,

relationship that is important in moment redistribution but the moment/tension-face-strain, M/tf,


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relationship such as those shown in Figure 6.5; Figure 6.5 was derived from a standard non-linear full-

interaction sectional analysis of the externally plated beams tested in this research (Chapter 5).

0

5

10

15

20

25

30

0 0.005 0.01 0.015 0.02 0.025

tension face strain (tf)

Moment(kNm)

yield of reinforcing bars y

A

G

D

O

C

Plateyield

H

IC debondingp.db

Iplate fracturep.fr

Jconcrete crushes atcconcrete crushes atcCFRP plated

EIC debondingp.db

F

steel plated

unplated

concrete crushes

atcB

concrete crushes

atc

Figure 6.5 Typical moment/ tension face strain behaviours

The M/tf relationship for the unplated RC beam is shown as O-A-B in Figure 6.5. As the ductile

reinforcing bars are assumed to have almost unlimited strain capacity in comparison with the finite

concrete strain capacity c, the beam can only fail by concrete crushing at point B. Hence, there istypically a very long tensile strain plateau commencing at yield of the reinforcing bars yat point A and

terminating when the concrete crushes at a strain cat point B. Over this plateau, A-B, the moment

capacity remains almost constant. It may be worth noting that the national standards use of ku to

control the amount of moment redistribution implicitly applies to sections with the behaviour

represented by the curve O-A-B, that is a long tensile strain plateau that is terminated by concrete

crushing.

The M/tfrelationship O-C-D-E-F in Figure 6.5 applies to a steel plated beam in which IC debonding

has not occurred prior to concrete crushing at F. The plate yields at C and the reinforcing bars at D,

after which the moment remains fairly constant until the concrete crushes at F. This steel plated beam

behaviour, O-C-D-E-F has almost identical characteristics to that of the unplated beam O-A-B and,

hence, the kufactor used in standards can be used to control the moment redistribution.

Let us now assume that IC debonding occurs at point E in Figure 6.5 that is prior to concrete crushing

at F but after yielding of both the tension face plate and tension reinforcing bars at D. In this case, k ucannot be used to control the moment redistribution as the kuapproach implicitly requires the concrete


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to crush as illustrated in Figure 6.4b. If the steel plate debonds prior to yielding at point C, then the

behaviour is similar to that of an FRP plated section described in the following paragraph.

The M/tfrelationship for an FRP plated slab in which debonding, that is IC, PE and CDC, does not

occur prior to concrete crushing is given by O-G-H-I-J in Figure 6.5. The moment continues to

increase after the reinforcing bars yield at point G because FRP is a linear elastic material that does

not yield prior to fracturing, so that the plate keeps attracting more force, thereby, increasing the

moment. Hence, an FRP plated section does not have a near horizontal plateau, such as D-E-F or A-

B, that is ideal for accommodating moment redistribution away from the plated section. For this

reason, the FRP plated section keeps attracting more moment even though the moment is being

redistributed. Because of this rising plateau (G-I-J in Figure 6.5), FRP plated sections are less capable

of redistributing moment as compared to metal plated sections with a horizontal plateau. Furthermore,

IC debonding of FRP plates such as at point H often occurs soon after the reinforcing bars yield and

generally well before the plate fractures at point I in Figure 6.5 or the concrete crushes at point J so

that the length of the rising plateau is relatively short.

In conclusion, it is suggested that the use of the kufactor in standards to control moment redistribution

should not be applied to FRP plated structures because invariably the concrete does not crush and

there is no ductile horizontal plateau; both of which are implicitly required in national standards.

Furthermore, there is usually only a short rising plateau. Hence, it is suggested that the ku factors in

national standards for controlling moment redistribution should only be used for metal plated sections

in which the concrete crushes prior to the plate debonding.

6.3 FUNDAMENTAL CONCEPT OF FLEXURAL RIGIDITY (EI) APPROACH

Through the literature review performed in Section 6.2, it can be seen that the existing plastic hinge

approach has its limitations, especially for members retrofitted with external plates. This led to the

development of the flexural rigidity (EI) approach in this research. The EI approach assumes that the

slope at the supports is zero, as shown in line C in Figure 6.6d, and which is accommodated by

variations in the flexural rigidity along the length of the beam, such as that shown in Figure 6.6c where

EIhog represents the flexural rigidity of the hogging region and EIsag that of the sagging region. It is

worth noting that it is not the magnitudes of these flexural rigidities that control the moment

redistribution but their relative values or proportions, that is EIhog/EIsag. The minimum flexural rigidity


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(EI)minof Mhogdepends on the ultimate sectional curvature capacity uas shown in Figure 6.1a. In this

approach, the region where the flexural rigidity is reducing is referred to as the plastic hinge for ease

of comparison with the existing moment redistribution approaches although the hinge approach is not

adopted here. In the flexural rigidity approach it is assumed that the hinge is bounded by the points of

contraflexure as shown in Figure 6.6b.

Msag=Mhog/2Msa=Mho

overalldeformation:

dy/dx=0; EI approach

dy/dx > 0: plastic hinge approach

(a)

(b)

(c)

A

B

CD

EI approach:EIhog

L

w

(d)

EIsag

(Mstatic)el

Mstatic

Mho

EIhog

Figure 6.6 Fundamental concept of flexural rigidity approach

In summary, the plastic hinge approach assumes that a small plastic hinge region of unknown length

Lhingeoccurs in a statically indeterminate beam, where within the hinge the curvature is assumed to be

constant and the rotation of the hinge is greater than zero to allow for the constant stiffness EI along

the beam. In contrast, the EI approach assumes a larger plastic hinge region, where variation in

curvature is allowed for within the hinge, such that the stiffness along the beam is not constant and

that there is zero rotation at the hinge.

6.4 SIMPLIFIED FLEXURAL RIGIDITY (EI) APPROACH

Based on the flexural rigidity concept developed, a simplified flexural rigidity approach is proposed

where it is assumed that the EI in the hogging and the sagging regions varies, while the EI within each


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of these regions remains constant as discussed in the journal paper in Section 6.4.1. Also presented

in the paper is the verification of the theoretical approach using the test results obtained in Chapter 5

on specimens with externally bonded plates. Application of the model to the NSM test specimens

reported in Chapter 5 is described in Section 6.4.2.2, along with further discussions of the comparison

between the experimental and theoretical results of the EB test specimens. In addition, the derivation

of the mathematical equations developed for the simplified flexural rigidity approach is presented in

Section 6.4.2.1 for various loading systems.

6.4.1 JOURNAL PAPER: MOMENT REDISTRIBUTION IN CONTINUOUS PLATED RCFLEXURAL MEMBERS. PART 2 FLEXURAL RIGIDITY APPROACH

In the following paper, the simplified flexural rigidity approach is presented and verified using the test

results for externally bonded plated beams presented in Chapter 5. To demonstrate the application of

the model, an encastre beam was analysed which was subjected to uniformly distributed loading and

where the beam was plated over the hogging regions with different plating materials.


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Moment redistribution in continuous plated RC flexural members. Part 2: Flexural Rigidity Approach

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Moment redistribution in continuous plated RC flexural members. Part

2: Flexural Rigidity Approach

*Oehlers, D.J., **Liu, I., ***Ju, G., and ****Seracino, R.

Corresponding author*Dr. D.J. OehlersAssociate ProfessorSchool of Civil and Environmental Engineering

Centre for Infrastructure Diagnosis, Assessment and RehabilitationThe University of AdelaideAdelaideSA5005AUSTRALIATel. 61 8 8303 5451Fax 61 8 8303 4359email [email protected]

**Ms. I. Liu

Postgraduate studentSchool of Civil and Environmental EngineeringThe University of Adelaide

***Dr. G. JuLecturerDepartment of Architectural EngineeringUniversity of YeungnamSouth Korea

****Dr. R. SeracinoSenior LecturerSchool of Civil and Environmental EngineeringThe University of Adelaide

Published in Engineering Structures 2004, vol. 26, pg.2209-2218


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Statement of Authorship

MOMENT REDISTRIBUTION IN CONTINUOUS PLATED RC FLEXURAL MEMBERS. PART

2: FLEXURAL RIGIDITY APPROACH

Published in Engineering Structures 2004, vol. 26, pg.2209-2218

LIU, I.S.T.(Candidate)

Performed all analyses, interpreted data and co-wrote manuscript.

Signed Date

OEHLERS, D.J.

Supervised development of work, co-wrote manuscript and acted as corresponding author.

Signed Date

SERACINO, R.

Supervised development of work, and manuscript review.

Signed Date


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MOMENT REDISTRIBUTION IN CONTINUOUS PLATED RC FLEXURAL MEMBERS. PART2: FLEXURAL RIGIDITY APPROACH

Oehlers, D.J., Liu, I., Ju, G., and Seracino, R.

ABSTRACT

Adhesive bonding plates to the surfaces of reinforced concrete members is now frequently used toincrease both the strength and stiffness. However, because of the brittle nature of the plate debondingmechanisms, plating is often assumed to reduce the ductility to such an extent that guidelines oftenpreclude moment redistribution. Tests on seven full-scale flexural members have shown thatsignificant amounts of moment can be redistributed from steel and carbon fibre reinforced polymer(FRP) plated regions. In this paper, a procedure is developed for quantifying the amount of momentredistribution that can occur in externally bonded steel or FRP plated members which can be used todesign plated members for ductility.

Keywords: Retrofitting; reinforced concrete beams; externally bonded plates; ductility; momentredistribution

1. INTRODUCTION

It was suggested in the companion1 paper that the neutral axis parameter (ku) approach used in

international standards for controlling the moment redistribution in reinforced concrete structuresdepends on both the concrete crushing and the existence of a horizontal plateau in themoment/curvature relationship. Both requirements seldom occur in plated structures due tointermediate crack, IC, debonding of the plate so the kuapproach is felt to be unsuitable for this newform of plated structure. Instead, an alternative approach based on flexural rigidities has beendeveloped to quantify moment redistribution in plated members in which IC debonding controls theultimate strength.

2. MOMENT REDISTRIBUTION CONCEPT

In order to illustrate the phenomenon of moment redistribution, that is the ability of staticallyindeterminate beams to redistribute moment, let us consider the encastreor built inbeam of length Lin Fig.1(c), which can also be considered to represent an internal span of a continuous beam. Forconvenience, let us assume that the same longitudinal reinforcing bars are in the top and bottom ofthe beam. Hence, the hogging (hog) and sagging (sag) regions have the same moment/curvature

(M/) relationships as shown in Fig.1(a), where: the idealised perfectly elastic portion has a flexural

rigidity of (EI)elasup to a moment capacity of Muat a curvature y; after which there is a perfectly

plastic ductile plateau in which the secant stiffness (EI)sec reduces up to a curvature of u at whichfailure occurs when the secant stiffness is at its minimum (EI)min. Let us also assume that the beam issubjected to a uniformly distributed load w, as shown in Fig.1(c), so that whilst the flexural rigidity ofthe whole beam remains at EI, the moment at the supports Mhogis twice that at mid-span Msag. Hence

for this specific beam, there is no moment redistribution whilst the maximum hogging moment M hogisequal to twice the maximum mid-span moment Msag. Conversely, when Mhog 2Msag, then there is


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moment redistribution. We will, therefore, define moment redistribution as occurring when thedistribution of moment within a beam is not given by elastic analyses that assume EI is constant withinthe beam. We will use this simple definition for convenience, as designers generally assume in theirpreliminary analyses that EI is constant within a beam in determining the initial distribution of moment

which can then be redistributed.

Mu

Mu/2

M

sag2

sag1

hog1 hog2

(Mstatic)1=1.5Mu=w1L2/8

(Mstatic)2=2Mu=w2L2/8

Mhog=Mu

Msag=Mu/2

Msag

=Mu

w (kN/m)

L

(c)continuousbeam

EI

elastic

non-elastic

(a) (b)

y u(EI)elas

A B

(EI)sec

(EI)min

hogging joint

ductile plateau

elastic

Figure 1 Moment redistribution concept

As the uniformly distributed load w is gradually applied to the beam in Fig.1(c), the beam is initiallyelastic so that Mhog = 2Msagand there is no moment redistribution. When the support moment first

reaches its moment capacity Muas shown as the point hog1in Fig.1(a), then the mid-span momentreaches a value of Mu/2 which is shown as sag1. At this stage, the static moment is (Mstatic)1= 1.5Mu=w1L2/8 as shown in Fig.1(b) and the distribution of moment is given by line A which is labelled elastic.Up to this point, the beam behaviour remains linear elastic. As the load is increased, the beamdeflects further resulting in an increase in Msagabove Mu/2 in Fig.1(b). However, the moment at thesupport remains at Mu. The only way that the increase in deflection or deformation, due to theincreased load, can be accommodated is for the curvature at the supports to be increased from hog1to hog2 as shown in Fig.1(a) and the hogging curvature will keep increasing until the saggingcurvature sag1 reaches sag2 in Fig.1(a), that is the mid-span moment has reached its capacity Muwhilst the behaviour of the hogging region is no longer elastic. The static moment has now reached(Mstatic)2= 2Mu= w2L2/8 in Fig.1(b), which is the maximum static moment and, hence, the maximum

load w2 that can be applied as all the joints have reached their moment capacities and a collapsemechanism has formed. The distribution of moment within the beam is now given by line B which hasbeen labelled non-elasticas shown in Fig. 1(b).

It can be seen in the example shown in Fig.1, that it is the hogging joints that are required to maintainthe moment whilst their curvature is increasing. Hence in this example, it is the hogging joints thathave to redistribute moment and it is their ductility that governs the amount of moment redistribution. If

for example, it was necessary for hog2in Fig.1(a) to exceed the curvature capacity of the section u,to achieve the static moment (Mstatic)2 in Fig.1(b) then sag2 in Fig.1(a) cannot achieve Mu and thecontinuous beam would fail before reaching its theoretical plastic capacity. It can be seen in thisexample that the sagging moment joint has only to reach its moment capacity, Muin Fig. 1(a) at point

sag2, that is its curvature has only to reach y. Hence its ductility, that is its capacity to extend alongthe plateau in Fig.1(a), is of no consequence. Unless of course the beam is required to absorb energy


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such as under seismic loads, in which case it may be a requirement that point sag2is also extendedinto the plastic zone to allow the beam to deflect further and absorb energy without an increase inload.

3. MOMENT REDISTRIBUTION APPROACH FOR PLATED BEAMS

To determine whether a beam is ductile enough to redistribute moment is an extremely complexproblem2 and there is much good ongoing research2-6 to develop a comprehensive and simplesolution. The problem is to understand how the beam can deform to accommodate the non-elasticdistribution of moment (line B in Fig.1(b) and also shown in Fig. 2(a)) and then to determine whetherthe deformation capacity of the beam can accommodate this required deformation. Two approachescan be followed: (i) assume that there is a discontinuity of the slope at the supports as shown in line Din Fig.2(f) and this will be referred to as the hinge approach; or (ii) assume that there is nodiscontinuity, such as at line C, and this will be referred to as the flexural rigidity (EI) approach. In

many ways, these approaches can be combined.

Mstatic

Mhog

Msag= Mhog/2

Msag> Mhog/2

w (kN/m)

Mstatic

L

(dy/dx)static

static deformations:

hinge approach:

support moment

deformations:

(dy/dx)support Mhog

deformation: dy/dx=0, EI approach

dy/dx > 0: plastic hinge approach

EIplastic

hinge

Lhinge

L

(a)

(b)

(c)

(d)

(e)

(f)

A

B

C

D

elastic

non-elastic

EI approach:

(EI)hog (EI)hog(EI)sag

Mhog

Figure 2 Compatibility in moment redistribution


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3.1 Hinge approach

In the hinge approach, it is assumed that most of the beam of length L remains linear elastic at aflexural rigidity EI as shown in Fig.2(c), and that there are small hinge regions at the joints of length

Lhingewhere moment redistribution requires ductility. The hinge length Lhinge Mhog/2.The discontinuity of slope can be determined from the static moment in Fig.2(d) and the redundantmoment shown in Fig.2(e). The slope at the supports (dy/dx)static and (dy/dx)support in Figs.2(d) & (e)can be derived by integration of the curvature along the length of the beam. Hence the discontinuity ofthe slope in line D in Fig.2(f) is equal to the difference between (dy/dx) staticand (dy/dx)supportwhich isaccommodated by the plastic hinge in Fig.2(c). As the length of the hinge is very small, it is oftenassumed that the curvature within the hinge is constant so that the rotation capacity of the hinge is

simply uLhinge where u is the curvature capacity of the section as illustrated in Fig.1(a). The main

problem with this approach is deciding what is the length of the plastic hinge region.

3.2 Flexural rigidity approach

In contrast to the plastic hingeapproach, the flexural rigidityapproach assumes that the slope at thesupports is zero, as shown in line C in Fig.2(f). This can only be accommodated by allowing variationsin the flexural rigidity along the length of the beam such as shown in Fig.2(b), where (EI)hogrepresentsthe flexural rigidity of the hogging region and (EI)sagthat of the sagging region. It is not the magnitudesof these flexural rigidities that control the moment redistribution, but their relative values orproportions, that is (EI)hog/(EI)sag. For example when (EI)hog= (EI)sag, that is (EI)hog/(EI)sag= 1, then, inthis example, the elastic distribution of moment is achieved so that Mhog = 2Msag and consequentlythere is no moment redistribution. Even if one were to double both flexural rigidities, (EI)hog/(EI)sagwould still remain at unity and, therefore, Mhog would remain at 2Msag so there would still be nomoment redistribution. However, if the secant flexural rigidity (EI)sec is taken in the hogging region, it

reduces as increases along the plateau in Fig.1(a). Consequently (EI)hog/(EI)sag also reduces. AsMhogis constant whilst Msagis increasing, Mhog< 2Msagthat is moment redistribution is occurring. The

minimum flexural rigidity (EI)min of Mhog depends on the ultimate sectional curvature capacity u asshown in Fig.1(a).

3.3 Choice of approach for plated sections

In order to determine which of the two moment redistribution approaches, that is the hinge approachor the flexural rigidity approach in Fig.2, is suitable for plated sections, let us first consider theirmoment/curvature responses.

The theoretical non-linear full-interaction moment/curvature response for the steel plated sections inthe companion paper1 is shown in Fig.3 and those for the FRP specimens1 in Fig.4. Theserelationships were derived from standard sectional analyses that allowed for the non-linear propertiesof the materials and which assumed full interaction. The points marked A to G in Figs.3 and 4

occurred when the strains in the plate were equal to their maximum recorded strains p.max; this

occurred either at debonding or just prior to debonding when there was virtually full interactionbetween the plate and the concrete, as discussed in the companion paper1. These strains are shown


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for all the tests in column 4 in Table 1. Also shown are the maximum concrete strains in thecompression face at plate debonding in column 5, and in column 6 the curvatures at debonding in

terms of the curvature at yield of the reinforcing bars which were derived from the M/analyses.

Figure 3. M/for steel specimen

0.0001-1

0

5

10

15

20

25

30

0 0.00002 0.00004 0.00006 0.00008curvature (mm )

Moment(kNm)

flexural

cracking

SF4debonds

SF3 debonds

SF2debonds

SF1debonds

SF4

SF1&SF2

SF3

D

EF

G

tensile reinforcing barsyield

Figure 4. M//for CFRP specimens

Table 1Analysis of test results

Spec.bpxtp(mm)

platematerial p.max c.max

max/yield

%MRtot(p.max)

1 (EI)sec/(EI)yield

(1) (2) (3) (4) (5) (6) (7) (8) (9)SS1 75x3 steel 0.0045 0.0011 1.15 22 1.03 0.90SS2 112x2 steel 0.0059 0.0012 1.48 33 0.94 0.69SS3 224x1 steel 0.0149 0.0026 3.63 48 1.25 0.29SF1 25x2.4 CFRP 0.0020 0.0005 0.52 30 0.85 1.02SF2 50x1.2 CFRP 0.0029 0.0007 0.75 29 0.90 1.01SF3 80x1.2 CFRP 0.0025 0.0006 0.65 28 0.84 1.02SF4 100x2.4 CFRP 0.0041 0.0008 1.04 35 0.92 0.97

From Table 1, three of the specimens (SF1 to SF3) debonded before the reinforcing bars yielded andone specimen (SF4) debonded at about yield; for these specimens, the concrete compressive strains,measured on the compression face adjacent to the central support, were very low ranging from 0.0005

0

10

20

30

0 0.00005 0.0001 0.00015 0.0002

curvature (mm-1)

Moment(kNm)

flexuralcracking

plateyields

tensilereinforcingbars yield SS3 debondsSS2 debonds

SS1 debonds

(EIsec)SS2(EIsec)SS2

A B C

(EIsec)SS3

(EIsec)SS3


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to 0.0008, so that these sections were still pseudo-elastic and would not have formed a plastic hinge.The remaining three specimens debonded after yield; specimens SS1 and SS2 had concretecompressive strains of about 0.0012 which was still well below the crushing strain of the concrete ofabout 0.0035 so that it is felt that a plastic hinge would not have formed here either. It was only the

remaining specimen SS3 that debonded at a concrete strain of 0.0026 that approached the concretecrushing strain. Because most of the tests debonded at relatively low concrete compressive strainsand often whilst the sections were pseudo-elastic, it was felt that the plastic hinge approach thatrequires rotation to be concentrated over very small plastic regions as in Fig.2(c) would not besuitable. Hence the flexural rigidity approach in Fig.2(b) where the change in slope is accommodatedover the whole hogging region, has been adopted for plated structures in the following analyses.

4. ANALYSIS OF TEST RESULTS4.1 Flexural rigidity model

In order to apply the flexural rigidity approach, the test specimens1 have been idealised as proppedcantilevers about the line of symmetry, as in Fig.5, where the flexural rigidity in the hogging region EI2and in the sagging region EI1vary but are constant within a region. This distribution of EI is not meantto represent the general behaviour, such as would be required for determining the deflection, but it isonly meant to represent moment redistribution where the differences in EI between regions affect theamount of moment redistributed.

EI1 EI2

L/2L/2

x

P

Msag

Mhog

Figure 5. Idealised structure for moment redistribution

A stiffness analysis software package, with two elements of stiffnesses EI1and EI2, could be used tofind a solution to the beam in Fig.5; an iterative procedure is required to adjust the length of eachelement, by varying the length of the hogging region x, until the point of contraflexure also occurs atdistance x. Alternatively, an elastic solution (for the beam in Fig.5 with two flexural rigidities and asingle concentrated load at mid-span) can be derived using the force method and conjugate beamtheory as given by Eqs.1 and 2; this can be used in an iterative analysis to determine the position x.

hog

hog

MPL

LMx

2

2

+= (1)

)]128)((3[

)]33)(([1623

21

3

2

322

21

3

2

LxxEIEILEI

xxLLxEIEILEI

L

MP

hog

= (2)

As an example of the iterative approach, for a fixed applied load P, the hogging moment Mhogcould beestimated or guessed and, from Eq.1, the position of the point of contraflexure xdetermined. Insertingthis value of xinto Eq.2 would give a value for P and Mhogwhich would be adjusted until P from Eq.2equalled the fixed value.


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4.2 Calibration of flexural rigidity model

In the following analyses, the flexural rigidities at the positions of the maximum hogging and saggingmoments were used to represent the flexural rigidities in the idealised beam shown in Fig.5. As in the

test beams1, moment was redistributed from the hogging plated region to the sagging unplated region,the flexural rigidity of the sagging region EI1was fixed at the flexural rigidity of the cracked unplatedsection. The flexural rigidity of the hogging region EI2was taken as the secant flexural rigidity of the

specimen at maximum plate strain prior to debonding p.maxat the points A to C in Fig. 3 and points Dto G in Fig. 4. As an example for specimen SS2 in Fig.3, the stiffness of the hogging region (EI)2

equals ((EI)sec)SS2which is the secant EI of the hogging region of the beam at p.max in column 4 ofTable 1. The secant flexural rigidities at the maximum strain (EI)secare compared with those at yield ofthe reinforcing bars (EI)yieldin column 9 of Table 1. It can be seen that the secant flexural rigidities areas low as 29% of that at yield of the reinforcing bars. The secant stiffness of the hogging region was

adjusted to a value of (EI)sec (as shown in Fig. 3 for specimens SS2 and SS3 where the arrowsindicate the magnitude and direction of the adjustment) so that the theoretical moment redistributionobtained from Eqs.1 and 2 was equal to the experiment moment redistribution1 listed in column 7 of

Table 1; the derived factors are given in column 8.

The variation of the factor with the maximum plate strain prior to debonding is shown in Fig.6 where

the average value of all tests1is 0.96. The factor remains fairly constant over debonding strains

up to about 0.006 and within this range the mean value is 0.91. It can be seen that a value of = 1would tend to overestimate the flexural rigidity of the hogging region and, consequently, underestimatethe moment redistributed except at high curvatures, such as for specimen SS3, where it is slightly

unconservative. The variation of the factor with curvature is shown in Fig.7; it is fairly constant overa wide range of curvatures from about 50% to 150% of the curvature at yield of the reinforcing bars.

The use of = 1 in predicting the moment redistribution in the test specimens is shown in Fig.8 andgives a safe design with a mean of = 1.18 and a standard deviation of 0.20. Using = 0.91

improves the redistribution prediction by reducing the mean value of to 1.07 with a standard

deviation of 0.18. However for a safe design, an of 1 is suggested as it is only slightlyunconservative at high curvatures. It can be seen from Figs.6-8 that the moment redistributionestimated using the flexural rigidity approach compares well with the experimental results for a widerange of debonding strains and curvatures.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016

maximum plate strain p.max

SS3

SS2

SS1

SF2SF3

SF4SF1

Figure 6. Variation of flexural rigidity adjustment factorwith maximum plate strain


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0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.5 1 1.5 2 2.5 3 3.5 4

max/yield

SS3

SS2

SS1

SF2SF3

SF4SF1

Figure 7. Variation of flexural rigidity adjustment factorwith curvature

Figure 8. Prediction of moment redistribution in test specimens

5. ELASTIC AND PLASTIC COMPONENTS OF MOMENT REDISTRIBUTION

Moment redistribution has been defined1 as the change in the moment from that when the flexuralrigidity of the beam is the same throughout its length, that is when EI1= EI2in Fig.5. This definition ofmoment redistribution was given in the companion paper1as

( )constEIhog

testhogconstEIhog

totM

MMMR

.

.

= (3)

where for a given applied load and hence applied static moment Mstatic, the hogging moment

(Mhog)EI.const is derived from an elastic analysis where EI is constant and (Mhog)test is the hoggingmoment in the test when the same Mstaticwas applied. This moment redistribution can be considered

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00

max/yield

(%MRtot)expt/(%MRtot)theo

= 1

= 0.91


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to consist of: an elastic moment redistribution component due to the difference between the elasticflexural rigidities, shown as (EI)elas in Fig.1(a), of the hogging and sagging regions; and a plasticmoment redistribution component due to the secant stiffnesses (EI)sec that follow the horizontal orrising plateau.

The plastic moment redistribution can, therefore, be expressed by the following relationship

( )iedEIhog

testhogiedEIhog

plasM

MMMR

var.

var.

= (4)

where for a given applied load and hence applied static moment Mstatic, the hogging moment(Mhog)EI.varied is the hogging moment when the hogging region has the elastic flexural rigidity of thecracked (in this case plated) section (EIcr)hog, and the sagging region has the flexural rigidity of thecracked (in this case unplated) section (EIcr)sag. Hence, the plastic moment redistribution is due to the

ductile plateau in Fig.1(a); it is due to the secant stiffness of the hogging region reducing as thehogging curvature moves along the ductile plateau.

Equation 4 was applied to the analysis of the test specimens and in order to minimise the scatter of

results, the flexural rigidities of the hogging region was taken as (EIcr)hog when calculating for

(Mhog)EI.varied, where is given in column 8 in Table 1. The results at the maximum plate strain areshown in column 6 in Table 2. As specimens SF1 to SF4 debonded at or prior to yielding of thereinforcing bar, the plastic moment redistribution is approximately zero. In contrast, much of the totalredistribution of the steel specimens is due to the plastic moment redistribution.

Table 2Moment redistribution in test specimens

Spec.bpxtp(mm)

platematerial

p.max%MRtot(p.max)

%MRplas

(p.max)p.db

%MRtot(p.db)

%MRplas(p.db)

(1) (2) (3) (4) (5) (6) (7) (8) (9)SS1 75x3 steel 0.0045 22 16 0.0045 22 16SS2 112x2 steel 0.0059 33 28 0.0059 33 28SS3 224x1 steel 0.0149 45 43 0.0149 45 43SF1 25x2.4 CFRP 0.0020 29 4 0.0012 32 9SF2 50x1.2 CFRP 0.0029 29 4 0.0026 36 14SF3 80x1.2 CFRP 0.0025 28 7 0.0025 28 7SF4 100x2.4 CFRP 0.0041 35 14 0.0017 44 25

Column 8 in Table 2 is the total moment redistribution at debonding. It can be seen in column 9 thatthree of the CFRP specimens showed a reasonable amount of plastic moment redistribution. This

occurred when the plate strains reduced just prior to debonding (p.db in Table 2 of the companionpaper1) which was due to the slip across the plate interface and, hence, partial interaction as slipoccurs at the plate/concrete interface. This slip reduces the stiffness of the beam and, hence, therotation capacity of the member is increased allowing a greater amount of moment to be redistributed.This is in agreement with the studies on moment redistribution of unplated reinforced concrete beamsconducted by several researchers2where some moment redistribution occurred after reaching thepeak moment. However, none of the models to date can accurately analyse the descending branch ofthe moment-curvature relationships of RC beams2.


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6. APPLICATION

Let us consider the amount of moment redistribution that can occur in an internal span which is

represented by the encastre beam in Fig. 9 and which is plated in the hogging region. For this beamwith a uniformly distributed load w, the moment redistribution can be determined from the followingequations which were derived using the techniques used to derive Eqs. 1 and 2:

EI1 EI2

4m

xMsag

MhogMhog

EI2

x

w

plate

Figure 9. Moment redistribution in an encastre beam

Lw

Mx

hog

5

2= (5)

+

+

= 32

32

12

21

6)46)((

)

2

(

24LEIxLxEIEI

xL

EIxEI

Mw hog (6)

The moment Mhogin Eq.5 is the hogging moment in the beam when a static moment Mstaticis appliedof magnitude w5L2/8. Hence w5in Eq.5 represents the applied static moment Mstatic. Equations 5 and 6are applied by fixing Mhogand determining w (that is the static moment) to achieve Mhog. Hence thesequence of the analysis consists of:

Fix Mhog; any value can be chosen as Eqs. 5 and 6 represent an elastic analysis and are usedto determine not the magnitude of the moments but the ratio of the moments Mstatic:Mhog:Msag.

Estimate a value of w5for Eq.5; as Mstatic> Mhogthen w5> 8Mhog/L2. Derive x from Eq.5. EI1in Eq.6 is the elastic flexural rigidity of the cracked sagging region ((EI)cr)sag, and EI2is the

secant flexural rigidity of the hogging region at plate debonding. Insert x from Eq.5 into Eq.6 to determine w say w6. If w6w5, substitute w6for w5in Eq.5 to get x and iterate until w6= w5. Hence Mstatic for the chosen value of Mhog has been determined from which Msag = Mstatic

Mhogcan be determined if required. As this is an elastic analysis, what has been achieved is the ratio of moments such as

Mhog/Mstaticwhich can be used in Eqs. 3 and 4 for a specific applied static moment Mstatic.

The beam shown in Fig.9 spans 4 m between the fixed end supports and has the same tension andcompression reinforcing bars in both the hogging and sagging regions as shown in Fig.10 where the


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units are in mm and the cover to the longitudinal reinforcing bars is 20 mm. For this beam, steel orFRP plates are adhesively bonded in the hogging region with the elastic axial rigidity of the plate (EA)pfixed at 3x107N. The sectional moment/curvature and the moment/plate-strain relationships, based onnon-linear full interaction sectional analyses, are shown in Figs.11 and 12 respectively. Figure 12 is

not required directly in the ensuing analyses but it is included to show how the same sectionalanalyses can be used to determine both the moment/curvature and moment/plate-strain variations.That is for a specific point on the moment/curvature plot, the plate strain is known, or conversely for aspecific debonding strain the point on the moment/curvature plot is known and, hence, the secantflexural rigidity at a specific debonding strain.

2Y12

6Y12140

1000

Plate (in hogging region only)

compression face

tension face

Figure 10.Specimen cross-sectional details

0

10

20

30

40

50

60

70

0 0.00004 0.00008 0.00012 0.00016 0.0002

curvature (mm-1

)

Moment(kNm)

Steel plated

FRP plated

unplated (sagging)

EIdb

tensile reinforcing bars yield

steelplateyield

B

A

D

C

(EIcr)plated

(EIcr)unplated

Figure 11.Moment/curvature relationship

0

10

20

30

40

50

60

70

0 0.005 0.01 0.015 0.02 0.025

plate strain p

Moment(kNm)

FRP plated

Steel plated

tensile reinforcing barsyield

Steel plateyielded

(EIcr)plated

A

B

C

D

Figure 12.Moment/plate-strain relationship


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Due to the addition of the external plates, the flexural rigidity of the hogging region is always greaterthan that of the unplated sagging region at the start of loading as illustrated in Fig.11. As the plateelastic axial rigidity is assumed constant, the moment/curvature and moment/plate-strain relationshipsfor the FRP and steel plated beams are identical before yielding of the steel plate i.e. region A-B in

Figs. 11 and 12. For the steel plated beam, beyond yielding of the plate at point B, further increase incurvature results in yielding of the reinforcing bars at point C where the ultimate moment of 34kNm isreached and the moment remains constant upon further loading. For the FRP plated beam, a bilinearmoment-curvature graph is obtained (as opposed to the trilinear relationship of the steel plated beam)with the moment increasing linearly with curvature at a cracked flexural stiffness of (EIcr)plate untilyielding of the reinforcing bars at point D at which the stiffness of the beam is reduced. The slope ofthe lines that emanate from the origin of the graph in Fig.11 are the secant stiffnesses EI dbat variousIC debonding strains shown in Fig.12 for the same moment considered. Also shown in Figs. 11 and 12are lines marked (EIcr)platedthat represent the cracked plated section up to plate yield, and also shownis that of the cracked unplated section (EIcr)unplated.Note that for the same curvature, the FRP platedbeam allows greater moment than the steel plated beam of the same elastic axial rigidity provided that

debonding does not occur prematurely.

Analyses were carried out to investigate the variation in the amount of moment redistribution in thestructure for different debonding strains8-10. To determine the amount of moment redistribution, themoment was first determined for a specific debonding strain from Fig.12 and the curvature andmoment for that debonding strain from Fig.11. Knowing the moment and curvature for the specificdebonding strain, the relative secant stiffness EIdbwas then derived from Fig.11. Based on this secantstiffness, the flexural rigidity approach given by Eqs.5 and 6 was used to determine the moment in thesagging region and, hence, the amount of moment redistributed from the hogging to sagging was thenevaluated.The total moment redistribution and that due to the plastic component is shown in Fig.13for both steel and FRP plated beams.

-30

-20

-10

0

10

20

30

40

50

0 0.002 0.004 0.006 0.008 0.01

debonding strain

%m

omentredistribution

steel plate and

bar yielded

bar yielded (FRP plated beams)

steel plate

yieldedcrackedSteel plated (plastic %MR)

Steel plated (tot %MR)

FRP plated (plastic %MR)

FRP plated (tot %MR)

cracked

A

A

B

D

C

E E

Figure 13. % moment redistribution vs debonding strain

Let us first consider the total moment redistribution of the encastre beams. For both the FRP and steelplated beams, upon initial loading i.e. region A-B for the steel plated beam and region A-D for the FRPplated beam in Fig. 11, the flexural rigidity in the plated hogging region (EIcr)platedis higher than that in

the unplated sagging region (EIcr)unplated. When calculating the total moment redistribution from Eq.3,which is based on assuming EI is constant along the beam, the fact that hogging moment is more than


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twice the sagging moment results in negative moment redistribution such as shown in Fig.13, betweenpoints A and B for the steel plated beam and between point A and D for the FRP plated beam. Thismeans that debonding occurs whilst the hogging moment is greater than that anticipated by an elasticanalysis in which EI is constant. Hence, if the beam was designed for the elastic distribution of

moment with a constant EI, the beam would fail prematurely due to IC debonding before the saggingdesign capacity was reached. When debonding occurs after the plate yields for steel plated beams(point B in Fig. 11) or after the reinforcing bars yields for CFRP plated beams (point D in Fig.11), thesecant stiffness EIdb of the hogging region gradually reduces i.e. EIdb


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REFERENCES

1. Oehlers, D.J., Ju, G., Liu, I, and Seracino, R. Moment redistribution in continuous plated RCflexural members. Part 1: neutral axis depth approach and tests, submitted for publication.

2. CEB-FIP. (1998). Ductility of Reinforced Concrete Structures-synthesis report and individualcontributions. Bulletin 242, Comit Euro-International du Bton, Switzerland

3. Bigaj, A. (1999) Structural Dependence of Rotation Capacity of Plastic Hinges in RC Beamsand Slabs,Delft University of Technology.

4. El-Refaie, S.A., Ashour, A.F, and Garrity, S.W.(2003) Sagging and hogging strengthening ofcontinuous reinforced concrete beams using carbon fiber-reinforced polymer sheets, ACIStructural Journal, vol.100, No.4, pp.446-453

5. El-Refaie, S.A., Ashour, A.F, and Garrity, S.W.(2001), Strengthening of reinforced concretecontinuous beams with CFRP composites, The International Conference on StructuralEngineering, Mechanics and Computation, Cape Town, South Africa, Apr. 2-4, pp.1591-1598

6. Mukhopadhyaya, P., Swamy, R.N., and Lynsdale, C. (1998), Optimizing structural response

of beams strengthened with GFRP plates, Journal of Composites for Construction, May,pp.87-95

7. Teng, J.G, Chen, J.F., Smith, S.T. and Lam, L. FRP strengthened RC structures. Wiley, NewYork. 2002.

8. Oehlers D.J., Park S.M. and Mohamed Ali, M.S. (2003) A Structural Engineering Approach toAdhesive Bonding Longitudinal Plates to RC Beams and Slabs. Composites Part A,Vol. 34,pp 887-897.

9. Oehlers, D.J. and Seracino, R. (2004) Design of FRP and Steel Plated RC Structures:retrofitting beams and slabs for strength, stiffness and ductility. Elsevier,September.

10. Teng, J.G., Smith, S.T., Yao, J. and Chen, J.F. (2003) Intermediate crack-induced debondingin RC beams and slabs. Construction and Building Materials,Vol.17, No.6-7, pp 447-462.


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6.4.2 FURTHER DISCUSSIONS ON SIMPLIFIED EI APPROACH

In the following Section, the full derivation of the mathematical equations developed for the simplified

EI approach and additional details on the verification of the approach are presented.

6.4.2.1 DERIVATION OF MATHEMATICAL EQUATIONS FOR BEAMS WITH DIFFERENT HOGGING ANDSAGGING STIFFNESSES

To illustrate how elastic solutions for the simplified EI approach are derived, two plating systems are

considered in the following: (1) one end continuous beam under point loading using the force method

and conjugate beam theory (Hibbeler 1999); and (2) both ends continuous beam under uniformly

distributed loading using the force method and Castigliaos theorem (Hibbeler 1999). These equations

can be used in an iterative analysis to determine the moment distribution along a statically

indeterminate beam, where for an applied load considered, the moment in the hogging region Mhogis

iterated until the applied load calculated from the mathematical equations derived equals to the load

considered.

6.4.2.1.1 ONE END CONTINUOUS BEAM SUBJECTED TO POINT LOAD

Consider the one end continuous beam in Figure 6.7 of span L, which is subjected to a point load Pat

a distance a from the exterior support. The moment distribution of the beam is denoted by line A,

where Msagand Mhogare the maximum moments in the sagging and hogging regions respectively, withthe point of contraflexure, poc, occurring at a distance xfrom the fixed end H. Based on the simplified

flexural rigidity approach, the stiffnesses in the hogging EIhogand sagging regions EIsagare assumed

to be constant at EI2and EI1respectively, as illustrated in Figure 6.7, where EI1EI2.

EI1 EI2

a L-a

x

P

Msag

Mhog

Line A pocR

ES H

Figure 6.7 Simplified EI approach: one end continuous beam

To solve this statically indeterminate system with two different EIs, the length of EI2, that is distance x,

is evaluated as follows:

Consider the moment distribution of the beam. Based on similar triangles,


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saghog M

xaL

M

x =

Equation 6.6

Rearranging Equation 6.6 gives

)( aLMM

Mx

saghog

hog

+= Equation 6.7

The sagging moment can be expressed in terms of the reaction force Ras given by Equation 6.8,

where Rcan be calculated using Equation 6.9, which is derived by taking moments about point H in

Figure 6.7.

RaMsag = Equation 6.8

L

MaLPR

hog=

)( Equation 6.9

Substituting Equation 6.8 and Equation 6.9 into Equation 6.7 gives:

)1)((

)(

L

aPaM

aLMx

hog

hog

+

=

Equation 6.10

To determine the moment distribution of the statically indeterminate system in Figure 6.7 (also shown

in Figure 6.8a), the force method (Hibbeler 1999) is used, where the unknown moment in the hogging

region Mhog is taken as the redundant as shown by Figure 6.8c, and the primary structure is

illustrated in Figure 6.8b. Therefore from rotational compatibility, the rotation at H is given by Equation

6.11, where hogand hog is the rotation at H caused by the applied load P in Figure 6.8b and the

redundant moment Mhog in Figure 6.8c respectively; hog is the angular flexibility coefficient i.e. it

measures the angular displacement per unit couple moment. To determine the terms hogand hogin

Equation 6.11, the conjugate beam theory (Hibbeler 1999) is used.

hoghoghoghog Mhog == '0 Equation 6.11


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xL-x

EI1 EI2

Pa Mhog

E HEI1 EI2

Pa

hogE H

(a) (b) primary structure

EI1

EI2

'hog=Mhoghog

E H

(c) redundant

Mhog

E'H'

(d) conjugate beam

of primary structure

a L-a-x x

M'EEI1

=0

V'H=hog

Pa(L-a)L.EI1

Pa(x)L.EI2

Pa(x)L.EI1

+=

E' H'

(e) conjugate beam of

L-x x

V'H=hog

redundant with Mhog=1

1EI2

M'EEI1

=0(L-x)L.EI1

(L-x)L.EI2

Figure 6.8 One end continuous beam: force method and conjugate beam theory

To determine the term hog in Equation 6.11, consider the primary structure in Figure 6.8b. From

conjugate beam theory, the conjugate beam loaded with the real beams M/EI diagram (i.e. M/EI of

Figure 6.8b) is shown in Figure 6.8d, where the supports E and H correspond to supports E and H on

the real beam. From rotation equilibrium and taking moments about point E:

where Vhogis the shear that is developed in the conjugate support H, which is equal to the slope hog

at the real support (Figure 6.8b).

Therefore, rearranging Equation 6.12 gives:

Now consider the redundant in Figure 6.8c. To determine hogin Equation 6.11, a moment of Mhog=1

is considered, and the corresponding conjugate beam is illustrated in Figure 6.8e. Therefore, taking

moments point E and from rotation equilibrium:

0'

'3

2

2

.

2

)(.

3

2

2

)(

3

2

2

)(

12

2

1

1

2

1

2

==

+

++

++

EI

MLVxL

LEI

xPaxaL

LEI

xaLxPa

xaL

LEI

xaLPaa

LEI

aLPa

E

hog

Equation 6.12

+ ( )

++== xL

EIL

xPaxLxLaL

EIL

PaV hoghog

3

2

2

.23

6'

2

2

23223

1

2 Equation 6.13


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- 482 -

where Vhogis the shear that is developed in the conjugate support H, which is equal to the slope hog

at the real support (Figure 6.8c).

Hence, hogcan be evaluated by rearranging Equation 6.14:

By substituting Equation 6.13 and Equation 6.15 into Equation 6.11 gives:

The load Pcan hence be evaluated by rearranging Equation 6.16:

Therefore, to determine the moment distribution along the beam an iterative analysis is required,

where for an applied load Papplied considered, the distance xis evaluated using Equation 6.10 for Mhog

guessed. The value of x is then substituted into Equation 6.17 to calculate P. If Pcalculated is not

equal to Papplied, then Mhog is iterated until a solution is found, where for each guess of Mhog the

corresponding EI2is computed based on the secant stiffness EIsecdetermined from the full interaction

M/ relationship. For beams with a ductile M/ behaviour, such as Fig.3 of the journal paper in

Section 6.4.1, where the moment is constant after yielding occurs, the curvature is iterated instead to

determine the new EIsec. Note that the test beams presented in Chapter 5 is equivalent to Figure 6.8

with a concentrated load applied at midspan i.e. a = L/2.

( ) 0'

'322

)(

3

2

2

)(

12

2

21

2

==

+

EI

MLV

xL

LEI

xxL

LEI

xLxxL

LEI

xL Ehog Equation 6.14

+( )

++

==

3

1

3'

322

2

2

1

2

3x

LxxLEILEIL

xLV hoghog Equation 6.15

( )

( )

++

++=

3

1

3

3

2

2

.23

60

322

2

2

1

2

3

2

2

23223

1

2

xLxxL

EILEIL

xLM

xLEIL

xPaxLxLaL

EIL

Pa

hog

Equation 6.16

)23()23(

)33()(2

1

23223

2

22

1

3

2

xLEIxxLxLaLEI

xxLLxEIxLEI

a

MP

hog

++

++= Equation 6.17


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6.4.2.1.2 BOTH ENDS CONTINUOUS BEAM SUBJECTED TO UNIFORMLY DISTRIBUTED LOADS

For both ends continuous beams under uniformly distributed loads wsuch as in Figure 6.9, the force

method and Castiglianos theorem (Hibbeler 1999) were used to derive an elastic solution. For the

loading system considered, the static moment Mstatic = wL2/8 = Mhog+ Msag. Therefore, the sagging

moment can be expressed by Equation 6.18. Consider the free body between support A and the point

of contraflexure poc, by taking moments about poc, Equation 6.19 is obtained, and solving for the

length of the hogging region xgives Equation 6.20.

x

Mhogpoc

SA

L/2

x

poc

L/2

B

w

EI2 EI2EI1

Mhog

Msag

moment distribution

Figure 6.9 Simplified EI approach: both ends continuous beam

hogsag M

wL

M = 8

2

Equation 6.18

022

2

=+ wLxwx

Mhog Equation 6.19

w

MLLx

hog2

42

2

= Equation 6.20

Based on the force method, the beam can be represented by the superposition of the primarystructure and the redundants MA and MB in Figure 6.10b,c&d respectively, where from symmetry

MA=MB=Mhog. At support A, from compatibility and principle of superposition, Equation 6.21 is

obtained, where AA and AB are the angular displacement at A caused by a unit couple moment

applied to supports A and B respectively, as defined in Figure 6.10b,c&d. Rearranging Equation 6.21

gives Mhog in terms of rotations (Equation 6.22). To solve A, AA and AB, for the beam under

uniformly distributed loading Castiglianos theorem was used, as illustrated in Figure 6.11, and will be

explained in the following.


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EI1

xL-2x

EI1 EI2

MB=Mhog

A BB

(a) (b) primary structure

EI2

'AA=MAAA

(c) redundant MA

+=EI2

x

MA=Mhog

EI2EI2A

A BEI1EI2

A B

MA=Mhog

EI2

'AB=MBAB

(d) redundant MB

+ EI1EI2A B

MB=Mhog

w w

Figure 6.10 Both ends continuous beam: force method

EI1

(a) primary structure

EI2EI2A B

x x

y1

C D

w

y2 y3

MB=0

RA

M'

EI1

(b) redundant MA

EI2EI2A B

x x

y1

C D

y2 y3

MB=0

RA

M'

MA=1

EI1

(c) redundant MB

EI2EI2A B

x x

y1

C D

y2 y3

MB=1

RA

M'

Figure 6.11 Both ends continuous beam: Castiglianos theorem

To determine the term Ain Equation 6.22, consider the primary structure in Figure 6.10b, which issimply supported and subjected to an UDL w. Castiglianos theorem is carried out by placing the

moment M at A in order to determine the rotation at that point as shown in Figure 6.11a. The reaction

force RAat A, given by Equation 6.23, is determined by taking moments about point B, where from

rotation equilibrium MB=0.

0)('' =+== ABAAhogAABBAAAAABAAA MMM Equation 6.21

ABAA

A

hogM

+= Equation 6.22

L

MwLRA

'

2= Equation 6.23


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Three co-ordinates, y1, y2, and y3, are used to determine the internal moments within the beam , since

there are discontinuities at C and D due to the different EIs in the hogging and the sagging regions.

This is shown in Figure 6.11a where y1, y2, and y3ranges from A to C, C to D, and D to B respectively.

Using the method of sections, the internal moments Mand its partial derivatives are computed for

each y co-ordinate as explained in the following.

For y1, the internal moment M1in region A-C (Figure 6.11a) is given by:

Substituting Equation 6.23 into Equation 6.24 gives:

Hence, the partial derivative of Equation 6.25 is:

For y2, the internal moment M2in region C-D (Figure 6.11a) and its partial derivative is given by:

For y3, the internal moment M3in region D-B (Figure 6.11a) and its partial derivative is given by:

+

2'

2

1

11

wyyRMM A += Equation 6.24

+

2

'

2'

2

1

111

wyy

L

My

wLMM += Equation 6.25

L

y

M

M 11 1'

=

Equation 6.26

+

2

)()(

'

2'

2

)()('

2

22

2

222

yxwyx

L

MwLM

yxwyxRMM A

++

+=

+++=

Equation 6.27

L

yx

M

M 22 1'

+=

Equation

Moment Redistribution Theoretical Studies

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