chapter 6 theoretical studies on moment redistribution · 6 theoretical studies on moment...

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CHAPTER

6 THEORETICAL STUDIES ON MOMENT REDISTRIBUTION

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

Approach 461

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) 490

6.4.2.2.2 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 plated reinforced concrete beams 521

6.5.2 Further Discussions on Linear EI Approach 545

6.5.2.1 Derivation of Equivalent EI 545

6.5.2.1.1 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

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 hinge concept (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 εc, 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/χ) relationships as shown in Figure 6.1a,

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


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of Mu at a curvature χy; after which there is a perfectly plastic ductile plateau in which the secant

stiffness (EI)sec reduces up to a curvature of χu at 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 twice

that 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) continuous beam

EI

elastic

non-elastic

(a) (b)

χy χχχχ χu(EI)elas

A B

(EI)sec

(EI)min

hogging joint

ductile plateauelastic

Figure 6.1 Moment redistribution concept

As the uniformly distributed load w is 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 sagging

curvature sag1 reaches sag2 in Figure 6.1a, that is the mid-span moment has reached its capacity Mu


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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 w2 that 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-elastic as 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 χu, to achieve the static moment (Mstatic)2 in Figure 6.1b, then sag2 in Figure 6.1a cannot

achieve Mu and 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, Mu in

Figure 6.1a at point sag2, that is its curvature has only to reach χy. 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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Msag=Mhog/2 Msag=Mhog

static deformations:

support moment deformations:

overall deformation:

dy/dx > 0: plastic hinge approach

EI

plastic hinge

(a)

(b)

(c)

(d)

A

B

C

L

EI

w

(e)

∆Mstatic

Mhog

plastic hinge

(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 Lhinge where moment redistribution requires ductility. The hinge length Lhinge << L, being of the

order of magnitude of the depth of the beam. It is assumed that the hinge, often referred to as the

plastic hinge, accommodates the discontinuity of slope at the supports in line C shown in Figure 6.2e.

This discontinuity is caused by the non-elastic moment distribution (line B in Figure 6.2b) where Msag >

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)static and (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)support which 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 χu is constant so that the rotation

capacity of the hinge θcap is:


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hingeucap Lχθ = Equation 6.1

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

satisfied, where θreq 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 θcap 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 θθ ≤ Equation 6.2

∫= dxreq χθ Equation 6.3

The earliest plastic hinge approaches assumed that the hinge occurs at a point, that is Lhinge→0. 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 ku given by Equation 6.4, where d and dn are the

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

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. Lhinge=d. Therefore from Equation

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

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


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d

dk n

u = Equation 6.4

u

c

u

chingeucap k

ddk

Lεεχθ === Equation 6.5

Typical examples of ku from 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 εc that 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

momentredistribution mean 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 ku ≥ 0.6 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 ku ≤ 0.3 whereas the Canadian approach uses 30% as an upper bound as ku → 0;


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the mean value for 30% redistribution is ku ≤ 0.15 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 ku on 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 εc is 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 εc = 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 ku controls the maximum tensile strain at the tension face

εtf for any depth of beam d, as εtf = εc(1-ku)/ku and, hence, εtf is independent of d. It needs to be

pointed out that the neutral axis parameter ku does not control the curvature χ for any beam depth as

this depends on the actual depth of the beam, that is χ = εc/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

χLhinge where the curvature at failure χ = εc/kud; for example, if Lhinge ≈ d then the rotation is equal to

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

tensile strain εtf.

d = h

kud= 0.48d

εc = 0.0035

(ε tf)0.15d=0.02

0%

εεεε

30% 0% to 30%

pivotal point

tension face

compression face

EB steel plates

(a) (b)

mean 30% (point B)

mean 0% (point A)

ductile reinforcing

bars

(ε tf)0.48d=0.0038

kud= 0.15d

NSM CFRP strips

EB CFRP plates

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 (εtf)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 (εtf)0.15d =

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

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

allowed, and when εtf > 0.020 then 30% redistribution is allowed. Generally speaking, ductile

reinforcing 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 ku approach is controlled by the strain at

the tension face εtf when the concrete crushes at εc. Hence, it is not the moment/curvature, M/χ,

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


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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 ( εεεεtf)

Mom

ent (

kNm

)

yield of reinforcing bars εy

A

G

D

O

C

Plate yield

HIC debonding εp.db

I plate fracture εp.fr

Jconcrete crushes at εcconcrete crushes at εc

CFRP plated

EIC debonding εp.db

Fsteel plated

unplated

concrete crushes at εc

Bconcrete crushes at εc

Figure 6.5 Typical moment/ tension face strain behaviours

The M/εtf 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 εc, the beam can only fail by concrete crushing at point B. Hence, there is

typically a very long tensile strain plateau commencing at yield of the reinforcing bars εy at point A and

terminating when the concrete crushes at a strain εc at 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/εtf relationship 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 ku factor 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, ku

cannot be used to control the moment redistribution as the ku approach 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/εtf relationship 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 ku factor 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)min of Mhog depends on the ultimate sectional curvature capacity χu as 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/2

Msag=Mhog

overall deformation:

dy/dx=0; EI approach


(a)

(b)

(c)

A

B

C D

EI approach: EIhog

L

w

(d)

EIsag

(Mstatic)el

Δ Mstatic

Mhog

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

Lhinge occurs 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 RC FLEXURAL 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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Intermediate Crack Debonding of Plated RC Beams Theoretical Studies on Moment Redistribution Moment redistribution in continuous plated RC flexural members. Part 2: Flexural Rigidity Approach

- 463 -

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. Oehlers Associate Professor School of Civil and Environmental Engineering Centre for Infrastructure Diagnosis, Assessment and Rehabilitation The University of Adelaide Adelaide SA5005 AUSTRALIA Tel. 61 8 8303 5451 Fax 61 8 8303 4359 email [email protected] **Ms. I. Liu Postgraduate student School of Civil and Environmental Engineering The University of Adelaide ***Dr. G. Ju Lecturer Department of Architectural Engineering University of Yeungnam South Korea ****Dr. R. Seracino Senior Lecturer School of Civil and Environmental Engineering The University of Adelaide Published in Engineering Structures 2004, vol. 26, pg.2209-2218


- 464 -

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. PART 2: 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 to increase both the strength and stiffness. However, because of the brittle nature of the plate debonding mechanisms, plating is often assumed to reduce the ductility to such an extent that guidelines often preclude moment redistribution. Tests on seven full-scale flexural members have shown that significant 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 moment redistribution that can occur in externally bonded steel or FRP plated members which can be used to design plated members for ductility. Keywords: Retrofitting; reinforced concrete beams; externally bonded plates; ductility; moment redistribution 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 structures depends on both the concrete crushing and the existence of a horizontal plateau in the moment/curvature relationship. Both requirements seldom occur in plated structures due to intermediate crack, IC, debonding of the plate so the ku approach is felt to be unsuitable for this new form of plated structure. Instead, an alternative approach based on flexural rigidities has been developed to quantify moment redistribution in plated members in which IC debonding controls the ultimate strength. 2. MOMENT REDISTRIBUTION CONCEPT In order to illustrate the phenomenon of moment redistribution, that is the ability of statically indeterminate beams to redistribute moment, let us consider the encastre or built in beam of length L in Fig.1(c), which can also be considered to represent an internal span of a continuous beam. For convenience, let us assume 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/χ) relationships as shown in Fig.1(a), where: the idealised perfectly elastic portion has a flexural

rigidity of (EI)elas up to a moment capacity of Mu at a curvature χy; after which there is a perfectly

plastic ductile plateau in which the secant stiffness (EI)sec reduces up to a curvature of χu at which failure occurs when the secant stiffness is at its minimum (EI)min. Let us also assume that the beam is subjected to a uniformly distributed load w, as shown in Fig.1(c), so that whilst the flexural rigidity of the whole beam remains at EI, the moment at the supports Mhog is twice that 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


- 466 -

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

EI

elastic

non-elastic

(a) (b)

χy χχχχ χu(EI)elas

A B

(EI)sec

(EI)min

hogging joint

ductile plateauelastic

Figure 1 Moment redistribution concept As the uniformly distributed load w is gradually applied to the beam in Fig.1(c), 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 Fig.1(a), 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 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 beam deflects further resulting in an increase in Msag above Mu/2 in Fig.1(b). 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 Fig.1(a) and the hogging curvature will keep increasing until the sagging curvature sag1 reaches sag2 in Fig.1(a), that is the mid-span moment has reached its capacity Mu whilst 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 collapse mechanism has formed. The distribution of moment within the beam is now given by line B which has been labelled non-elastic as 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 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 Fig.1(a) to exceed the curvature capacity of the section χu, to achieve the static moment (Mstatic)2 in Fig.1(b) then sag2 in Fig.1(a) cannot achieve Mu and 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, Mu in Fig. 1(a) at point

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


- 467 -

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. 3. MOMENT REDISTRIBUTION APPROACH FOR PLATED BEAMS To determine whether a beam is ductile enough to redistribute moment is an extremely complex problem2 and there is much good ongoing research2-6 to develop a comprehensive and simple solution. The problem is to understand how the beam can deform to accommodate the non-elastic distribution of moment (line B in Fig.1(b) and also shown in Fig. 2(a)) and then to determine whether the deformation capacity of the beam can accommodate this required deformation. Two approaches can be followed: (i) assume that there is a discontinuity of the slope at the supports as shown in line D in Fig.2(f) and this will be referred to as the hinge approach; or (ii) assume that there is no discontinuity, 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


hinge approach:

support momentdeformations:

(dy/dx)support Mhog

deformation: dy/dx=0, EI approach


EIplastichinge

Lhinge

L

(a)

(b)

(c)

(d)

(e)

(f)

A

B

CD

elastic

non-elastic

EI approach:(EI)hog (EI)hog(EI)sag

Mhog

Figure 2 Compatibility in moment redistribution


- 468 -

3.1 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 Fig.2(c), and that there are small hinge regions at the joints of length Lhinge where moment redistribution requires ductility. The hinge length Lhinge << L, being of the order of magnitude of the depth of the beam. It is assumed that the hinge, often referred to as the plastic hinge, accommodates the discontinuity of slope at the supports in line D shown in Fig.2(f). This discontinuity is caused by the non-elastic moment distribution (line B in Fig.2(a)) where Msag > Mhog/2. The discontinuity of slope can be determined from the static moment in Fig.2(d) and the redundant moment 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 of the slope in line D in Fig.2(f) is equal to the difference between (dy/dx)static and (dy/dx)support which is accommodated by the plastic hinge in Fig.2(c). As the length of the hinge is very small, it is often assumed that the curvature within the hinge is constant so that the rotation capacity of the hinge is

simply χuLhinge where χu 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 hinge approach, the flexural rigidity approach assumes that the slope at the supports is zero, as shown in line C in Fig.2(f). This can only be accommodated by allowing variations in the flexural rigidity along the length of the beam such as shown in Fig.2(b), where (EI)hog represents the flexural rigidity of the hogging region and (EI)sag that of the sagging region. It is not the magnitudes of these flexural rigidities that control the moment redistribution, but their relative values or proportions, that is (EI)hog/(EI)sag. For example when (EI)hog = (EI)sag, that is (EI)hog/(EI)sag = 1, then, in this example, the elastic distribution of moment is achieved so that Mhog = 2Msag and consequently there is no moment redistribution. Even if one were to double both flexural rigidities, (EI)hog/(EI)sag would still remain at unity and, therefore, Mhog would remain at 2Msag so there would still be no moment redistribution. However, if the secant flexural rigidity (EI)sec is taken in the hogging region, it

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

minimum flexural rigidity (EI)min of Mhog depends on the ultimate sectional curvature capacity χu as shown 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 approach or the flexural rigidity approach in Fig.2, is suitable for plated sections, let us first consider their moment/curvature responses. The theoretical non-linear full-interaction moment/curvature response for the steel plated sections in the companion paper1 is shown in Fig.3 and those for the FRP specimens1 in Fig.4. These relationships were derived from standard sectional analyses that allowed for the non-linear properties of 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 εp.max; this occurred either at debonding or just prior to debonding when there was virtually full interaction between the plate and the concrete, as discussed in the companion paper1. These strains are shown


- 469 -

for all the tests in column 4 in Table 1. Also shown are the maximum concrete strains in the compression 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/χ analyses.

Figure 3. M/χ for steel specimen

0.0001 -1

0

5

10

15

20

25

30

0 0.00002 0.00004 0.00006 0.00008 curvature (mm )

Mom

ent (

kNm

)

flexural cracking

SF4 debonds

SF3 debonds

SF2 debonds

SF1 debonds

SF4

SF1&SF2

SF3

D

E F

G

tensile reinforcing bars yield

Figure 4. M//χ for CFRP specimens

Table 1 Analysis of test results

Spec. bpxtp (mm)

plate material

εεεεp.max εεεεc.max χχχχmax/

χχχχyield

%MRtot

(εεεεp.max) αααα1 (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 0.90 SS2 112x2 steel 0.0059 0.0012 1.48 33 0.94 0.69 SS3 224x1 steel 0.0149 0.0026 3.63 48 1.25 0.29 SF1 25x2.4 CFRP 0.0020 0.0005 0.52 30 0.85 1.02 SF2 50x1.2 CFRP 0.0029 0.0007 0.75 29 0.90 1.01 SF3 80x1.2 CFRP 0.0025 0.0006 0.65 28 0.84 1.02 SF4 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 and one 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)

Mom

ent (

kNm

)

flexural cracking

plate yields

tensile reinforcing bars yield SS3 debondsSS2 debonds

SS1 debonds

(EIsec)SS2α(EIsec)SS2

A B C

(EIsec)SS3

α(EIsec)SS3


- 470 -

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 concrete compressive strains of about 0.0012 which was still well below the crushing strain of the concrete of about 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 concrete crushing strain. Because most of the tests debonded at relatively low concrete compressive strains and often whilst the sections were pseudo-elastic, it was felt that the plastic hinge approach that requires rotation to be concentrated over very small plastic regions as in Fig.2(c) would not be suitable. Hence the flexural rigidity approach in Fig.2(b) where the change in slope is accommodated over the whole hogging region, has been adopted for plated structures in the following analyses. 4. ANALYSIS OF TEST RESULTS 4.1 Flexural rigidity model In order to apply the flexural rigidity approach, the test specimens1 have been idealised as propped cantilevers about the line of symmetry, as in Fig.5, where the flexural rigidity in the hogging region EI2 and in the sagging region EI1 vary but are constant within a region. This distribution of EI is not meant to represent the general behaviour, such as would be required for determining the deflection, but it is only meant to represent moment redistribution where the differences in EI between regions affect the amount of moment redistributed.

EI1 EI2

L/2 L/2

x

P

Msag

Mhog

Figure 5. Idealised structure for moment redistribution

A stiffness analysis software package, with two elements of stiffnesses EI1 and EI2, could be used to find a solution to the beam in Fig.5; an iterative procedure is required to adjust the length of each element, by varying the length of the hogging region x, until the point of contraflexure also occurs at distance x. Alternatively, an elastic solution (for the beam in Fig.5 with two flexural rigidities and a single concentrated load at mid-span) can be derived using the force method and conjugate beam theory 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

213

2

32221

32

LxxEIEILEI

xxLLxEIEILEI

L

MP hog

−−−−−−−

= (2)

As an example of the iterative approach, for a fixed applied load P, the hogging moment Mhog could be estimated or guessed and, from Eq.1, the position of the point of contraflexure x determined. Inserting this value of x into Eq.2 would give a value for P and Mhog which would be adjusted until P from Eq.2 equalled the fixed value.


- 471 -

4.2 Calibration of flexural rigidity model In the following analyses, the flexural rigidities at the positions of the maximum hogging and sagging moments 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 EI1 was fixed at the flexural rigidity of the cracked unplated section. The flexural rigidity of the hogging region EI2 was taken as the secant flexural rigidity of the

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

equals ((EI)sec)SS2 which is the secant EI of the hogging region of the beam at εp.max in column 4 of Table 1. The secant flexural rigidities at the maximum strain (EI)sec are compared with those at yield of the reinforcing bars (EI)yield in column 9 of Table 1. It can be seen that the secant flexural rigidities are as low as 29% of that at yield of the reinforcing bars. The secant stiffness of the hogging region was

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

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

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

the average α value of all tests1 is 0.96. The α 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 α = 1 would tend to overestimate the flexural rigidity of the hogging region and, consequently, underestimate the moment redistributed except at high curvatures, such as for specimen SS3, where it is slightly

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

The use of α = 1 in predicting the moment redistribution in the test specimens is shown in Fig.8 and

gives a safe design with a mean of α = 1.18 and a standard deviation of 0.20. Using α = 0.91

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

deviation of 0.18. However for a safe design, an α of 1 is suggested as it is only slightly unconservative at high curvatures. It can be seen from Figs.6-8 that the moment redistribution estimated using the flexural rigidity approach compares well with the experimental results for a wide range 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 εεεεp.max

αα αα

SS3

SS2

SS1

SF2 SF3

SF4 SF1

Figure 6. Variation of flexural rigidity adjustment factor α with maximum plate strain


- 472 -

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

χχχχmax/χχχχyield

αα αα

SS3

SS2

SS1

SF2 SF3

SF4 SF1

Figure 7. Variation of flexural rigidity adjustment factor α with 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 flexural rigidity of the beam is the same throughout its length, that is when EI1 = EI2 in Fig.5. This definition of moment redistribution was given in the companion paper1 as

( ) ( )( )

constEIhog

testhogconstEIhog

tot M

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 hogging moment in the test when the same Mstatic was 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

χχχχmax/χχχχyield

(%M

Rto

t)ex

pt/(

%M

Rto

t)th

eo

α = 1α = 0.91


- 473 -

to consist of: an elastic moment redistribution component due to the difference between the elastic flexural rigidities, shown as (EI)elas in Fig.1(a), of the hogging and sagging regions; and a plastic moment redistribution component due to the secant stiffnesses (EI)sec that follow the horizontal or rising plateau. The plastic moment redistribution can, therefore, be expressed by the following relationship

( ) ( )( )

iedEIhog

testhogiedEIhog

plas M

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 the cracked (in this case plated) section (EIcr)hog, and the sagging region has the flexural rigidity of the cracked (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 the hogging 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 α(EIcr)hog when calculating for

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

Table 2 Moment redistribution in test specimens

Spec. bpxtp (mm)

plate material

εεεεp.max %MRtot (εεεεp.max)

%MRplas

(εεεεp.max) εεεεp.db

%MRtot (εεεεp.db)

%MRplas (εεεεp.db)

(1) (2) (3) (4) (5) (6) (7) (8) (9) SS1 75x3 steel 0.0045 22 16 0.0045 22 16 SS2 112x2 steel 0.0059 33 28 0.0059 33 28 SS3 224x1 steel 0.0149 45 43 0.0149 45 43 SF1 25x2.4 CFRP 0.0020 29 4 0.0012 32 9 SF2 50x1.2 CFRP 0.0029 29 4 0.0026 36 14 SF3 80x1.2 CFRP 0.0025 28 7 0.0025 28 7 SF4 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 that three of the CFRP specimens showed a reasonable amount of plastic moment redistribution. This

occurred when the plate strains reduced just prior to debonding (εp.db in Table 2 of the companion paper1) which was due to the slip across the plate interface and, hence, partial interaction as slip occurs at the plate/concrete interface. This slip reduces the stiffness of the beam and, hence, the rotation 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 beams conducted by several researchers2 where some moment redistribution occurred after reaching the peak moment. However, none of the models to date can accurately analyse the descending branch of the moment-curvature relationships of RC beams2.


- 474 -

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 beam with a uniformly distributed load w, the moment redistribution can be determined from the following equations which were derived using the techniques used to derive Eqs. 1 and 2:

EI1 EI2

4m

x Msag

Mhog Mhog

EI2

x

w

plate

Figure 9. Moment redistribution in an encastre beam

Lw

Mx hog

5

2= (5)

+−−

−+=

32

3212

21

6 )46)((

)2

(24

LEIxLxEIEI

xL

EIxEIMw hog (6)

The moment Mhog in Eq.5 is the hogging moment in the beam when a static moment Mstatic is applied of magnitude w5L2/8. Hence w5 in Eq.5 represents the applied static moment Mstatic. Equations 5 and 6 are applied by fixing Mhog and determining w (that is the static moment) to achieve Mhog. Hence the sequence of the analysis consists of:

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

� Estimate a value of w5 for Eq.5; as Mstatic > Mhog then w5 > 8Mhog/L2. � Derive x from Eq.5. � EI1 in Eq.6 is the elastic flexural rigidity of the cracked sagging region ((EI)cr)sag, and EI2 is 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 w6 ≠ w5, substitute w6 for w5 in 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 –

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

Mhog/Mstatic which 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 and compression reinforcing bars in both the hogging and sagging regions as shown in Fig.10 where the


- 475 -

units are in mm and the cover to the longitudinal reinforcing bars is 20 mm. For this beam, steel or FRP plates are adhesively bonded in the hogging region with the elastic axial rigidity of the plate (EA)p fixed at 3x107 N. The sectional moment/curvature and the moment/plate-strain relationships, based on non-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 sectional analyses 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 a specific debonding strain the point on the moment/curvature plot is known and, hence, the secant flexural rigidity at a specific debonding strain.

2Y12

6Y12 140

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)

Mom

ent (

kNm

)

Steel plated

FRP plated

unplated (sagging)

EIdb

tensile reinforcing bars yield

steel plate yield

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 εεεεp

Mom

ent (

kNm

)

FRP plated

Steel plated

tensile reinforcing barsyield

Steel plateyielded

(EIcr)plated

A

BC

D

Figure 12. Moment/plate-strain relationship


- 476 -

Due to the addition of the external plates, the flexural rigidity of the hogging region is always greater than that of the unplated sagging region at the start of loading as illustrated in Fig.11. As the plate elastic axial rigidity is assumed constant, the moment/curvature and moment/plate-strain relationships for 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 in curvature results in yielding of the reinforcing bars at point C where the ultimate moment of 34kNm is reached and the moment remains constant upon further loading. For the FRP plated beam, a bilinear moment-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 until yielding of the reinforcing bars at point D at which the stiffness of the beam is reduced. The slope of the lines that emanate from the origin of the graph in Fig.11 are the secant stiffnesses EIdb at various IC debonding strains shown in Fig.12 for the same moment considered. Also shown in Figs. 11 and 12 are lines marked (EIcr)plated that represent the cracked plated section up to plate yield, and also shown is that of the cracked unplated section (EIcr)unplated. Note that for the same curvature, the FRP plated beam 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 the structure for different debonding strains8-10. To determine the amount of moment redistribution, the moment was first determined for a specific debonding strain from Fig.12 and the curvature and moment for that debonding strain from Fig.11. Knowing the moment and curvature for the specific debonding strain, the relative secant stiffness EIdb was then derived from Fig.11. Based on this secant stiffness, the flexural rigidity approach given by Eqs.5 and 6 was used to determine the moment in the sagging region and, hence, the amount of moment redistributed from the hogging to sagging was then evaluated. The total moment redistribution and that due to the plastic component is shown in Fig.13 for 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

omen

t red

istri

butio

n

steel plate andbar yielded

bar yielded (FRP plated beams)

steel plateyieldedcracked

Steel 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 steel plated beams, upon initial loading i.e. region A-B for the steel plated beam and region A-D for the FRP plated beam in Fig. 11, the flexural rigidity in the plated hogging region (EIcr)plated is 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


- 477 -

twice the sagging moment results in negative moment redistribution such as shown in Fig.13, between points A and B for the steel plated beam and between point A and D for the FRP plated beam. This means that debonding occurs whilst the hogging moment is greater than that anticipated by an elastic analysis 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 sagging design 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), the secant stiffness EIdb of the hogging region gradually reduces i.e. EIdb<(EIcr)plated. As a result, the moment redistributed from the hogging to the sagging region is increased in relation to the increase in debonding strain as shown in Fig.13 beyond point B and point D for steel and FRP plated beams respectively. When the secant stiffness reduces to (EIcr)unplated in Fig.11, there is zero moment redistribution at points E after which there is positive moment redistribution as shown in Fig.13. In evaluating the amount of moment redistribution due to the plastic component using Eq.4, the influence of the variation in EI due to the flexural rigidity in the plated hogging region (EIcr)plated being higher than that in the unplated sagging region (EIcr)unplated is accounted for in the calculations i.e. the analysis allows for the fact that the hogging region attracts more moment than that anticipated by the elastic analysis where EI is constant. Therefore when the plate debonds before yielding of the steel plate such as in region AB in Fig.13 or before yielding of the reinforcing bars for FRP plated beams such as in region AD, there is zero plastic moment redistribution. When debonding occurs after the steel plate yields, or after the bar yields for CFRP plated beams, the plastic moment redistribution increases gradually for increase in debonding strain. It should be noted that since the plastic moment redistribution takes into account the effect of more moment being attracted to the hogging region, the amount of moment redistributed due to the plastic component is actually greater than that found for total moment redistribution. From tests1 it was found that steel plated beams debond at strains ranging from 0.004 to 0.02 for a maximum plate thickness of 3mm, which from Fig.13 shows that a large amount of moment redistribution is possible. From Fig.13, one can see that a greater debonding strain is required for moment redistribution to occur in FRP plated beams compared with steel plated beams, however previous research1,8,9 have shown that CFRP plates usually debond much earlier than steel plates. Therefore, although some moment redistribution can occur with CFRP plated beams, the scope is much less than that available for steel plated specimens. 7. SUMMARY A mathematical model has been developed for quantifying the amount of moment redistribution that can occur in steel or FRP externally bonded plated beams that debond prior to concrete crushing. The mathematical model is based on the maximum strain at debonding and a parametric study suggests that substantial amounts of moment redistribution can occur in steel plated sections if designed with care. However, CFRP plated sections show a limited ability to redistribute moment at their maximum strain. However tests have shown that unlike steel plated beams that tend to debond at their maximum strain, the strains in CFRP plates tend to reduce prior to debonding due to partial interaction which allows some moment redistribution.


- 478 -

REFERENCES

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

2. CEB-FIP. (1998). “Ductility of Reinforced Concrete Structures-synthesis report and individual contributions”. Bulletin 242, Comité Euro-International du Béton, Switzerland

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

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

5. El-Refaie, S.A., Ashour, A.F, and Garrity, S.W.(2001), ‘Strengthening of reinforced concrete continuous beams with CFRP composites’, The International Conference on Structural Engineering, 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, New York. 2002.

8. Oehlers D.J., Park S.M. and Mohamed Ali, M.S. (2003) “A Structural Engineering Approach to Adhesive 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 debonding in RC beams and slabs.” Construction and Building Materials, Vol.17, No.6-7, pp 447-462.


- 479 -

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 AND

SAGGING 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 Castigliao’s 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 Mhog is

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 P at

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

where Msag and Mhog are the maximum moments in the sagging and hogging regions respectively, with

the point of contraflexure, poc, occurring at a distance x from the fixed end H. Based on the simplified

flexural rigidity approach, the stiffnesses in the hogging EIhog and sagging regions EIsag are assumed

to be constant at EI2 and EI1 respectively, as illustrated in Figure 6.7, where EI1≠EI2.

EI1 EI2

a L-a

x

P

Msag

Mhog

Line A poc R

E S 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,


- 480 -

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 R as given by Equation 6.8,

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

Figure 6.7.

RaM sag = 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 θhog and θ’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; αhog is the angular flexibility coefficient i.e. it

measures the angular displacement per unit couple moment. To determine the terms θhog and αhog in

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

hoghoghoghog Mhog

αθθθ −=−= '0 Equation 6.11


- 481 -

xL-x

EI1 EI2

Pa Mhog

E HEI1 EI2

Pa

θhog

E H

(a) (b) primary structure

EI1

EI2

θ'hog=Mhogαhog

E H

(c) redundant

Mhog

E'H'

(d) conjugate beamof primary structure

a L-a-x x

M'EEI1

=0

V'H=θhog

Pa(L-a)L.EI1

Pa(x)L.EI2

Pa(x)L.EI1

+=

E' H'

(e) conjugate beam of

L-x x

V'H=αhog

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 θhog in Equation 6.11, consider the primary structure in Figure 6.8b. From

conjugate beam theory, the conjugate beam loaded with the real beam’s 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 V’hog is the shear that is developed in the conjugate support H, which is equal to the slope θhog

at the real support (Figure 6.8b).

Therefore, rearranging Equation 6.12 gives:

Now consider the redundant in Figure 6.8c. To determine αhog in 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

Ehog

Equation 6.12

+ ( )

−++−−== xLEIL

xPaxLxLaL

EIL

PaV hoghog 3

2

2

.23

6'

22

23223

12

θ Equation 6.13


- 482 -

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

at the real support (Figure 6.8c).

Hence, αhog can be evaluated by rearranging Equation 6.14:

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

The load P can 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 x is evaluated using Equation 6.10 for Mhog

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

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

corresponding EI2 is computed based on the secant stiffness EIsec determined from the full interaction

M/χ relationship. For beams with a ductile M/χ 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

22

12

3 xLxxL

EILEIL

xLV hoghogα Equation 6.15

( )

( )

+−+−−

−++−−=

3

1

3

3

2

2

.23

60

322

22

12

3

22

23223

12

xLxxL

EILEIL

xLM

xLEIL

xPaxLxLaL

EIL

Pa

hog

Equation 6.16

)23()23(

)33()(2

123223

2

221

32

xLEIxxLxLaLEI

xxLLxEIxLEI

a

MP

hog

−++−−+−+−

= Equation 6.17


- 483 -

6.4.2.1.2 BOTH ENDS CONTINUOUS BEAM SUBJECTED TO UNIFORMLY DISTRIBUTED LOADS

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

method and Castigliano’s 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 x gives Equation 6.20.

x

Mhog poc

S A

L/2

x

poc

L/2

B

w

EI2 EI2 EI1

Mhog

Msag

moment distribution

Figure 6.9 Simplified EI approach: both ends continuous beam

hogsag MwL

M −=8

2

Equation 6.18

022

2

=−+ wLxwxM hog 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 primary

structure 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 αAA and αAB 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 θA, αAA and αAB, for the beam under

uniformly distributed loading Castigliano’s theorem was used, as illustrated in Figure 6.11, and will be

explained in the following.


- 484 -

EI1

xL-2x

EI1 EI2

MB=Mhog

A BθB

(a) (b) primary structure

EI2

θ'AA=MAαAA

(c) redundant MA

+=EI2

x

MA=Mhog

EI2EI2θA

A BEI1EI2

A B

MA=Mhog

EI2

θ'AB=MBαAB

(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: Castigliano’s theorem

To determine the term θA in Equation 6.22, consider the primary structure in Figure 6.10b, which is

simply supported and subjected to an UDL w. Castigliano’s 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 RA at A, given by Equation 6.23, is determined by taking moments about point B, where from

rotation equilibrium MB=0.

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

ABAA

AhogM

ααθ+

= Equation 6.22

L

MwLRA

'

2−= Equation 6.23


- 485 -

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 y3 ranges from A to C, C to D, and D to B respectively.

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

each y co-ordinate as explained in the following.

For y1, the internal moment M1 in 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 M2 in region C-D (Figure 6.11a) and its partial derivative is given by:

For y3, the internal moment M3 in region D-B (Figure 6.11a) and its partial derivative is given by:

+

2'

21

11

wyyRMM A −+= Equation 6.24

+

2

'

2'

21

111

wyy

L

My

wLMM −−+= Equation 6.25

L

y

M

M 11 1'

−=∂∂

Equation 6.26

+

2

)()(

'

2'

2

)()('

22

2

22

22

yxwyx

L

MwLM

yxwyxRMM A

+−+

−+=

+−++= Equation 6.27

L

yx

M

M 22 1'

+−=∂∂

Equation 6.28


- 486 -

From Castigliano’s theorem, substituting Equation 6.25-Equation 6.30 into Equation 6.31 and setting

M’ = 0 gives:

Consider the redundant MA in Figure 6.10c. To determine the term αAA in Equation 6.22, let MA=1. The

same procedures as above apply, where a moment M’ is placed at A as shown in Figure 6.11b. The

reaction force RA at A, given by Equation 6.34, is determined by taking moments about point B, where

from rotation equilibrium MB=0.

+

2

)()(

'

2'

2

)()('

23

3

23

33

yxLwyxL

L

MwLM

yxLwyxLRMM A

+−−+−

−+=

+−−+−+= Equation 6.29

( )33 1

1'

yxLLM

M +−−=∂∂

Equation 6.30

+

∫

∫∫∫

∂∂+

∂∂+

∂∂=

∂∂=

−

x

xLxL

A

EI

dy

M

MM

EI

dy

M

MM

EI

dy

M

MM

EI

dy

M

MM

02

333

2

01

2220

2

1110

'

'''θ

Equation 6.31

( )

−+

++−+

+−=

+−−

+−−+−+

+−

+−++

−

−=

∫

∫

∫

−

L

xx

EI

wLxLx

EI

w

L

xxLx

EI

w

dyL

yxLyxLwyxL

wL

EI

dyL

yxyxwyx

wL

EI

dyL

ywyy

wL

EI

x

xL

x

A

43212322834

12

)()(

2

1

12

)()(

2

1

122

1

43

2

332

1

432

2

0 33

23

32

2

0 22

22

21

0 11

21

12

θ

Equation 6.32

−+

++−=⇒

6412322

32

2

332

1

xLx

EI

wLxLx

EI

wAθ Equation 6.33

L

MRA

'1+= Equation 6.34


- 487 -

To analyse the redundant system in Figure 6.11b, the same co-ordinates as those used in analysing

the primary structure (Figure 6.11a) apply. Using the method of sections, the internal moments M and

its partial derivatives are computed for each y co-ordinate as follows.

For y1, the internal moment M1 in region A-C (Figure 6.11b) and its partial derivative is given by:

For y2, the internal moment M2 in region C-D (Figure 6.11b) and its partial derivative is given by:

For y3, the internal moment M3 in region D-B (Figure 6.11b) and its partial derivative is given by:

From Castigliano’s theorem:

Substituting Equation 6.35-Equation 6.40 into Equation 6.41 and setting M’ = 0 gives:

+ '1

'1'1 111 My

L

MMyRM A −−

+=−−= Equation 6.35

1'

11 −=∂∂

L

y

M

M Equation 6.36

+ ( ) '1'1

'1)( 222 MyxL

MMyxRM A −−+

+=−−+= Equation 6.37

1'

22 −+

=∂∂

L

yx

M

M Equation 6.38

+'1)(

'1'1)( 333 MyxL

L

MMyxLRM A −−+−

+=−−+−= Equation 6.39

1'

33 −+−=∂∂

L

yxL

M

M Equation 6.40

+

2

33

0 31

222

0 22

11

0 1 ''' EI

dy

M

MM

EI

dy

M

MM

EI

dy

M

MM

xxLx

AA

∂∂

+

∂∂

+

∂∂

= ∫∫∫−

α Equation 6.41


- 488 -

Finally, to determine the term αAB in Equation 6.22, consider the redundant MB in Figure 6.10d and let

MB=1. A moment M’ is placed at A as shown in Figure 6.11b. The reaction force RA at A, given by

Equation 6.44, is determined by taking moments about point B, where from rotation equilibrium:

To analyse the redundant system in Figure 6.11c, the same co-ordinates as those used in analysing

the primary structure (Figure 6.11a) applies. Using the method of sections, the internal moments M

and its partial derivatives are computed for each y co-ordinate as follows.

For y1, the internal moment M1 in region A-C (Figure 6.11c) and its partial derivative is given by:

For y2, the internal moment M2 in region C-D (Figure 6.11c) and its partial derivative is given by:

+

+−−+

+−=

−+−+

−++

−=

∫

∫∫−

2

3

22

32

1

2

2

3

2

30

2

3

2

2

22

0

2

11

2

0

1

2

3

1

33

21

3

1

11

11

11

L

x

EI

Lx

L

x

L

x

EIx

L

x

L

x

EI

dyL

yxL

EI

dyL

yx

EIdy

L

y

EI

x

xLx

AAα

Equation 6.42

+−+

+−−=⇒ x

L

x

L

x

EI

Lx

L

x

L

x

EIAA

2

2

3

22

32

1 3

21

33

21α

+ Equation 6.43

L

MRA

1'−= Equation 6.44

+ '

1'' 111 My

L

MMyRM A −

−=−= Equation 6.45

1'

11 −=∂∂

L

y

M

M Equation 6.46

+ ( ) '1'

')( 222 MyxL

MMyxRM A −+

−=−+= Equation 6.47

1'

22 −+

=∂∂

L

yx

M

M Equation 6.48


- 489 -

For y3, the internal moment M3 in region D-B (Figure 6.11c) and its partial derivative is given by:

Based on Castigliano’s theorem (Equation 6.51), substituting Equation 6.45-Equation 6.50 into

Equation 6.51 and setting M’ = 0 gives:

Therefore, by substituting Equation 6.33, Equation 6.43 and Equation 6.53 into Equation 6.22, Mhog is

given by:

The uniformly distributed load w can hence be evaluated by Rearranging Equation 6.54:

+')(

1'')( 333 MyxL

L

MMyxLRM A −+−

−=−+−= Equation 6.49

1'

33 −+−=∂∂

L

yxL

M

M Equation 6.50

+

2

33

0 31

222

0 22

11

0 1 ''' EI

dy

M

MM

EI

dy

M

MM

EI

dy

M

MM

xxLx

AB

∂∂

+

∂∂

+

∂∂

= ∫∫∫−

α Equation 6.51

−+

++−+

−=

−+−

+−−+

−+

+−+

−

−=

∫

∫∫−

2

32

22

32

12

32

2

30

33

2

2

2

0

22

110

11

2

32

1

63

21

32

1

11

11

11

L

x

L

x

EI

L

L

x

L

x

EIL

x

L

x

EI

dyL

yxL

L

yxL

EI

dyL

yx

L

yx

EIdy

L

y

L

y

EI

x

xLx

ABα

Equation 6.52

+

−+

++−=⇒

2

32

22

32

1 3

21

63

21

L

x

L

x

EI

L

L

x

L

x

EIABα Equation 6.53

[ ])

2(

)46()46(24

21

321

3322

xL

EIxEI

xLxEILxLxEIw

M hog

−+

−+++−= Equation 6.54


- 490 -

An iterative procedure, as described in Section 6.4.2.1.1, is required to solve Equation 6.55.

6.4.2.2 COMPARISON BETWEEN EXPERIMENTAL AND THEORETICAL RESULTS

In the following Section, the simplified EI approach is adopted to analyse the statically indeterminate

test specimens reported in Chapter 5.

6.4.2.2.1 TEST SERIES ‘S’ (SPECIMENS WITH EXTERNALLY BONDED PLATES)

Unlike the plastic hinge approach (Section 6.2.2), the proposed flexural rigidity approach does not rely

on concrete crushing failure to occur, therefore it can be applied to determine the moment

redistribution of beams at any loading stage. This is important in plated structures as it has been

shown experimentally in Chapter 5 that maximum plate strain and maximum moment redistribution

may be obtained prior to failure. In the journal paper presented in Section 6.4.1, the test specimens

were analysed only at the load when the maximum plate strain was achieved. Further details on the

analyses of the seven test specimens with EB steel or FRP plates (refer to Chapter 5 for beam details)

for various loading stages are shown in Figure 6.12, Figure 6.13 and Figure 6.14, along with the test

results from Chapter 5. The analyses were performed based on α =1; EI1 = flexural rigidity of the

cracked unplated section; and EI2 = secant flexural rigidity determined from the full-interaction M/χ

relationship for the Mhog guessed.

The abscissas of Figure 6.12 and Figure 6.13 are the applied static moment, Mstatic given by Equation

5.2, as a proportion of the ultimate maximum static moment, (Mstatic)u = (Msag)u + (Mhog)u/2, based on

nonlinear full interaction analysis of the ultimate capacity of the hogging and sagging sections, (Mhog)u

and (Msag)u, and ignoring IC debonding in the case of the hogging region. Figure 6.12 illustrates the

variation of total percentage of moment redistribution %MRtot calculated using Equation 5.1 for the

seven EB test specimens. Figure 6.13 shows the variation of the maximum hogging moment in the

beam Mhog as a proportion of the maximum sagging moment in the beam Msag, where line A

represents the elastic analysis for constant EI. Figure 6.14 shows the variation of the maximum

hogging Mhog moments as the applied loads P increased, where the straight line represents the elastic

analysis for constant EI.

+−−

−+= 3

232

12

21

)46)((

)2

(24

LEIxLxEIEI

xL

EIxEIMw hog Equation 6.55


- 491 -

-20

0

20

40

60

80

100

0 0.2 0.4 0.6 0.8 1

Mstatic/(Mstatic)u

%M

Rto

t

testtheo.

-10

0

10

20

30

40

50

0 0.2 0.4 0.6 0.8 1

Mstatic/(Mstatic)u

%M

Rto

t

testtheo.

-10

0

10

20

30

40

50

60

70

0 0.2 0.4 0.6 0.8 1Mstatic/(Mstatic)u

%M

Rto

t

testtheo.

Beam SS1

plate debond

Beam SS2

plate debond

Beam SS3

shear failure

-10

0

10

20

30

40

50

60

70

0 0.2 0.4 0.6 0.8 1

Mstatic/(Mstatic)u

%M

Rto

t

test.theo.

Beam SF1plate debond

-10

0

10

20

30

40

50

60

70

0 0.2 0.4 0.6 0.8 1

Mstatic/(Mstatic)u

%M

Rto

t

test.theo.

Beam SF2

plate debond

-10

0

10

20

30

40

50

60

0 0.2 0.4 0.6 0.8 1

Mstatic/(Mstatic)u

%M

Rto

t

test.theo.

Beam SF3plate debond

-20

-10

0

10

20

30

40

50

60

70

0 0.2 0.4 0.6 0.8 1

Mstatic/(Mstatic)u

%M

Rto

t

testtheo.

Beam SF4

plate debond

Figure 6.12 Simplified EI approach: Percentage moment redistribution of EB test specimens


- 492 -

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.2 0.4 0.6 0.8 1

Mstatic/(Mstatic)u

Mho

g/M

sag

test.theo.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.2 0.4 0.6 0.8 1

Mstatic/(Mstatic)u

Mho

g/M

sag

testtheo.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.2 0.4 0.6 0.8 1

Mstatic/(Mstatic)u

Mho

g/M

sag

testtheo.

Beam SS1

A

plate debondBeam SS2

A

plate debond

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.2 0.4 0.6 0.8 1

Mstatic/(Mstatic)u

Mho

g/M

sag

testtheo.

Beam SS3

shear failure

A

Beam SF1

A

plate debond

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.2 0.4 0.6 0.8 1

Mstatic/(Mstatic)u

Mho

g/M

sag

test.theo.

Beam SF2

A

plate debond

0

0.2

0.4

0.6

0.8

1

1.2

1.4


Mho

g/M

sag

test.theo.

Beam SF3

A

plate debond

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6


Mho

g/M

sag

testtheo.

Beam SF4

A

plate debond

Figure 6.13 Simplified EI approach: Mhog/Msag of EB test specimens


- 493 -

0

5

10

15

20

25

30

35

40

0 20 40 60 80applied load (kN)

Mho

g (k

Nm

)

testtheo.

0

5

10

15

20

25

30

0 10 20 30 40 50 60 70applied load (kN)

Mho

g (k

Nm

)

testtheo.

0

5

10

15

20

25

30

0 20 40 60 80applied load P (kN)

Mho

g (k

Nm

)

testtheo

Beam SS1

elastic

plate debondBeam SS2

elastic

plate debond

Beam SS3

shear failure

elastic

0

5

10

15

20

25

0 20 40 60 80load (kN)

Mho

g (k

Nm

)

testtheo.

Beam SF1

elastic

plate debond

0

5

10

15

20

25

0 20 40 60 80load (kN)

Mho

g (k

Nm

)

testtheo.

Beam SF2

elastic

plate debond

0

5

10

15

20

25

30

35

40

0 20 40 60 80load (kN)

Mho

g (k

Nm

)

testtheo.

Beam SF3

elastic

plate debond

0

5

10

15

20

25

30

0 20 40 60 80 100load (kN)

Mho

g (k

Nm

)

testtheo.

Beam SF4

elastic

plate debond

Figure 6.14 Simplified EI approach: Mhog of EB test specimens


- 494 -

In general, good correlations between experimental and theoretical results are observed throughout

the loading of the beams, especially for the steel plated beams i.e. ‘SS’ series in Figure 6.12 to Figure

6.14. The simplified EI approach tends to underestimate the percentage moment redistribution upon

debonding failure as shown in Figure 6.12, that is it overestimates the hogging moment (Figure 6.14),

hence providing a safe design. At initial loading, the beams were still settling which resulted in greater

differences between test and theoretical results as can be seen from Figure 6.12 to Figure 6.14.

However upon further loading, better correlations were observed. For specimens where significant

debonding occurred, such as the ‘SF’ series beams with FRP plates, greater differences between test

and theoretical results were observed. This is due to the use of the full interaction moment/curvature

relationship when determining the secant stiffness EIsec in the hogging region, where in reality partial

interaction occurred at the interface because of the slip between the plate and the concrete.

It is worth noting that although the errors in the percentage moment redistribution predicted in Figure

6.12 appeared to be quite significant, this is due to the magnitudes and the measure of the %MR of

the beams which enhanced the error. Much better accuracies are observed when considering

Mhog/Msag (Figure 6.13) and Mhog (Figure 6.14). This can also be seen from the standard deviations SD

and means for the %MR, Mhog/Msag, and Mhog of the specimens presented in Table 6.1, where the

ratios between the experimental and the theoretical results of the beams after settling were

considered i.e. the large discrepancies that occurred at low loads, as shown in Figure 6.12 to Figure

6.14, are neglected.


- 495 -

Table 6.1 Simplified EI approach: experimental vs test results for EB beams

Test %MRexp/%MRtheo (Mhog)exp/(Mhog)theo (Mhog/Msag)exp/( Mhog/Msag)theo

SD 0.155 0.016 0.023

mean 1.388 0.959 0.940 SS1

SD/mean 0.112 0.017 0.024

SD 0.234 0.022 0.032

mean 1.319 0.948 0.927 SS2

SD/mean 0.178 0.023 0.035

SD 0.300 0.026 0.035

mean 1.435 0.917 0.886 SS3

SD/mean 0.209 0.028 0.040

SD 0.147 0.052 0.071

mean 1.112 0.961 0.948 SF1

SD/mean 0.132 0.054 0.075

SD 0.058 0.017 0.023

mean 1.126 0.963 0.949 SF2

SD/mean 0.052 0.018 0.025

SD 0.191 0.068 0.092

mean 1.056 0.979 0.974 SF3

SD/mean 0.180 0.069 0.095

SD 0.093 0.042 0.054

mean 1.100 0.959 0.946 SF4

SD/mean 0.085 0.043 0.057

SD 0.168 0.035 0.047

mean 1.220 0.955 0.938 Average

of all tests

SD/mean 0.135 0.036 0.050

6.4.2.2.2 TEST SERIES ‘NS’ AND ‘NB’ (SPECIMENS WITH NSM STRIPS)

In the paper presented in Section 6.4.1 and also from the previous section, it has been shown that the

simplified EI approach correlates well with the experimental test results for the test beams externally

bonded with steel or FRP plates (Chapter 5). In this section, the model was applied to the nine NSM

test specimens, ‘NS’ and ‘NB’ series, reported in Chapter 5. Analyses were performed at each load

increment, where for each guess of Mhog, the secant stiffness EIsec was determined based on full

interaction moment/curvature relationship. The beams were analysed using α =1; EI1 = flexural rigidity

of the cracked unplated section; and EI2 = EIsec.


- 496 -

The analysis results are shown in the following figures, where Figure 6.15 and Figure 6.18 illustrates

the variation of total percentage of moment redistribution %MRtot calculated using Equation 5.1 for the

‘NS’ and ‘NB’ series respectively. Figure 6.16 and Figure 6.19 shows the variation of the maximum

hogging moment in the beam Mhog as a proportion of the maximum sagging moment in the beam Msag

for the ‘NS’ and ‘NB’ series respectively, where line A represents the elastic analysis for constant EI.

Figure 6.17 and Figure 6.20 shows the variation of the maximum hogging Mhog moments as the

applied loads P increased for the ‘NS’ and ‘NB’ series respectively, where the straight line represents

the elastic analysis for constant EI.

-10

0

10

20

30

40

50

0 0.2 0.4 0.6 0.8 1

Mstatic/(Mstatic)u

%M

Rto

t

testtheo

-10

0

10

20

30

40

50

60

0 0.2 0.4 0.6 0.8 1

Mstatic/(Mstatic)u

%M

Rto

t

testtheo

-505

1015202530354045


%M

Rto

t

testtheo

-5

0

5

10

15

20

25

30

35

40


%M

Rto

t

testtheo.

-5

0

5

10

15

20

25

0 0.2 0.4 0.6 0.8 1

Mstatic/(Mstatic)u

%M

Rto

t

testtheo.

Beam NS_F4

plate debond

Beam NS_F1

shear failurein sag

Beam NS_F2

conc crushingfailure in sag


Beam NS_F3


Beam NS_S1

-5

0

5

10

15

20

25

30

35

0 0.2 0.4 0.6 0.8 1

Mstatic/(Mstatic)u

%M

Rto

t

testtheo

shearfailure in sag

Beam NS_S2

Figure 6.15 Simplified EI approach: Percentage moment redistribution of ‘NS’ test series


- 497 -

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.2 0.4 0.6 0.8 1

Mstatic/(Mstatic)u

Mho

g/M

sag

testtheo

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.2 0.4 0.6 0.8 1

Mstatic/(Mstatic)u

Mho

g/M

sag

testtheo

0

0.2

0.4

0.6

0.8

1

1.2

1.4


Mho

g/M

sag

testtheo

0

0.2

0.4

0.6

0.8

1

1.2

1.4


Mho

g/M

sag

testtheo

0

0.2

0.4

0.6

0.8

1

1.2

1.4


Mho

g/M

sag

testtheo.

0

0.2

0.4

0.6

0.8

1

1.2

1.4


Mho

g/M

sag

testtheo.

Beam NS_F1

A

shear failurein sag

A

A

Beam NS_F4

A

plate debond

A

Beam NS_F2 conc crushingfailure in sag


Beam NS_F3

Beam NS_S1


A

Beam NS_S2shear

failure in sag

Figure 6.16 Simplified EI approach: Mhog/Msag of ‘NS’ test series


- 498 -

0

5

10

15

20

25

30

0 10 20 30 40 50 60 70load (kN)

Mho

g (k

Nm

)

testtheo

0

5

10

15

20

25

30

0 10 20 30 40 50 60 70load (kN)

Mho

g (k

Nm

)

testtheo

0

5

10

15

20

25

30

0 10 20 30 40 50 60 70load (kN)

Mho

g (k

Nm

)

testtheo

0

5

10

15

20

25

30

0 10 20 30 40 50 60 70load (kN)

Mho

g(kN

m)

testtheo.

0

5

10

15

20

25

30

0 20 40 60 80load (kN)

Mho

g (k

Nm

)

testtheo.

elastic

Beam NS_F4

elastic

plate debond

Beam NS_S1

elastic

Beam NS_F1shear failure

in sag

elastic

Beam NS_F2


elastic


Beam NS_F3


0

5

10

15

20

25

30

0 20 40 60load (kN)

Mho

g (k

Nm

)

testtheo

Beam NS_S2

elastic

shearfailure in sag

Figure 6.17 Simplified EI approach: Mhog of ‘NS’ test series


- 499 -

-10

0

10

20

30

40

50

60

0 0.2 0.4 0.6 0.8 1 1.2

Mstatic/(Mstatic)u

%M

Rto

t

testtheo

-10

0

10

20

30

40

50

60

0 0.2 0.4 0.6 0.8 1 1.2

Mstatic/(Mstatic)u

%M

Rto

t

testtheo

-10

0

10

20

30

40

50

60

0 0.2 0.4 0.6 0.8 1 1.2

Mstatic/(Mstatic)u

%M

Rto

t

testtheo

Beam NB_F1 Beam NB_F2

debonding failure

Beam NB_F3


debonding failure

Figure 6.18 Simplified EI approach: Percentage moment redistribution of ‘NB’ test series

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.2 0.4 0.6 0.8 1 1.2

Mstatic/(Mstatic)u

Mho

g/M

sag

testtheo

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.2 0.4 0.6 0.8 1 1.2

Mstatic/(Mstatic)u

Mho

g/M

sag

testtheo

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.2 0.4 0.6 0.8 1 1.2

Mstatic/(Mstatic)u

Mho

g/M

sag

testtheo Beam NB_F1

A

debonding failure

A

A

Beam NB_F2

Beam NB_F3


debonding failure

Figure 6.19 Simplified EI approach: Mhog/Msag of ‘NB’ test series


- 500 -

0

20

40

60

80

100

120

0 50 100 150 200 250load (kN)

Mho

g (k

Nm

)

testtheo

0

20

40

60

80

100

120

0 50 100 150 200 250load (kN)

Mho

g (k

Nm

)

testtheo

0

20

40

60

80

100

0 50 100 150 200 250load (kN)

Mho

g (k

Nm

)

testtheo


debonding failure

Beam NB_F3


elastic elastic

debonding failure

elastic

Figure 6.20 Simplified EI approach: Mhog of ‘NB’ test series

From the comparison between experimental and theoretical results shown in Figure 6.15 to Figure

6.20, it can be seen that the simplified EI approach gives reasonable predictions of the moment

distribution of the beams. Compared to the EB test series (Figure 6.12-Figure 6.14), the approach was

found to be slightly less accurate when used to analyse NSM beams, as can be seen from comparing

the means and standard deviations in Table 6.1 and Table 6.2. Table 6.2 shows the standard

deviations SD and means for the %MR, Mhog/Msag, and Mhog of the specimens, where the ratios

between the experimental and the theoretical results of the beams after they settled were considered

i.e. the large discrepancies that occurred at low loads, as shown in Figure 6.15 to Figure 6.20, are

neglected.

It is worth noting that from the simplified EI analyses of all the beams, negative %MR occurs at initial

loading while the beam is still elastic. This is due to the additional plates which caused a larger

stiffness in the hogging region. Hence theoretically, when the beam is elastic, moment is redistributed

from the sagging to the hogging region, as denoted by the negative %MR and where Mhog/Msag is

greater 1.2 in Figure 6.15and Figure 6.16 respectively for the ‘NS’ series; and in Figure 6.18 and

Figure 6.19 for the ‘NB’ series.


- 501 -

Table 6.2 Simplified EI approach: experimental vs test results for NSM beams


SD 0.157 0.047 0.051

mean 0.913 0.984 1.029 NS_F1

SD/mean 0.172 0.047 0.049

SD 0.074 0.032 0.044

mean 0.963 1.020 1.028 NS_F2

SD/mean 0.077 0.031 0.042

SD 0.068 0.051 0.068

mean 0.844 1.083 1.113 NS_F3

SD/mean 0.081 0.047 0.061

SD 0.085 0.021 0.029

mean 1.152 0.954 0.937 NS_F4

SD/mean 0.074 0.022 0.031

SD 0.080 0.019 0.022

mean 1.452 0.875 0.832 NS_S1

SD/mean 0.055 0.022 0.027

SD 0.062 0.016 0.022

mean 1.030 0.994 0.991 NS_S2

SD/mean 0.060 0.016 0.022

SD 0.090 0.108 0.140

mean 0.875 1.123 1.159 NB_F1

SD/mean 0.102 0.097 0.121

SD 0.039 0.060 0.077

mean 0.824 1.160 1.207 NB_F2

SD/mean 0.047 0.052 0.063

SD 0.066 0.069 0.091

mean 0.847 1.119 1.157 NB_F3

SD/mean 0.078 0.062 0.079

SD 0.080 0.047 0.060

mean 0.989 1.035 1.050 Average

of all tests

SD/mean 0.083 0.044 0.055

In general, at initial loading the beams were still settling which resulted in larger differences between

test and theoretical results, as can be seen from Figure 6.15 to Figure 6.20. Upon further loading, after

the beam has settled, better correlations were observed. However as the beam was further loaded,


- 502 -

greater errors between the experimental and the theoretical results were found at large loads near

beam failure (except for NS_F4, NS_S1 and NS_S2), with the theoretical Mhog/Msag less than the

experimental Mhog/Msag as can be seen from Figure 6.16 and Figure 6.19. This indicates that the EIsec

determined at the hogging region is underestimated. Hence the Mhog from theoretical analyses was

less than that measured from tests causing overestimations of the %MR that occurred in the beams.

This suggests that the simplified EI approach can be unconservative when used to analyse NSM

beam.

6.4.3 PARAMETRIC STUDIES BASED ON SIMPLIFIED EI APPROACH

For a continuous beam with externally bonded plates, the positions, material and geometry of the

plates can largely affect the moment redistribution behaviour of the member. The purpose of this

parametric study is to investigate the ability of plated beams to redistributed moment using different

plating materials and geometric properties. The results and detailed discussions of the parametric

studies are presented in the following journal paper.

6.4.3.1 JOURNAL PAPER: MOMENT REDISTRIBUTION PARAMETRIC STUDY OF CFRP, GFRP AND STEEL SURFACE PLATED RC BEAMS AND SLABS

Intermediate Crack Debonding of Plated RC Beams Theoretical Studies on Moment Redistribution Moment redistribution parametric study of CFRP, GFRP and steel surface plated RC beams and slabs

- 503 -

Moment redistribution parametric study of CFRP, GFRP and steel

surface plated RC beams and slabs

Irene Liu*, Deric John Oehlers**, Rudolf Seracino***and Gisu Ju**** *Ms. I.S.T. Liu Postgraduate student School of Civil and Environmental Engineering The University of Adelaide Corresponding author **Dr. D.J. Oehlers Associate Professor School of Civil and Environmental Engineering The University of Adelaide Adelaide SA5005 AUSTRALIA Tel. 61 8 8303 5451 Fax 61 8 8303 4359 email [email protected] ***Dr. R. Seracino Senior Lecturer School of Civil and Environmental Engineering The University of Adelaide ****Dr. G. Ju Lecturer Department of Architectural Engineering University of Yeungnam South Korea Published in International Journal of Construction and Building Materials 2005, vol. 20, pp59-70


- 504 -


MOMENT REDISTRIBUTION PARAMETRIC STUDY OF CFRP, GFRP AND STEEL

SURFACE PLATED RC BEAMS AND SLABS

Published in International Journal of Construction and Building Materials 2005, vol. 20, pp59-70


Performed all analyses, interpreted data and wrote manuscript.

Signed Date

OEHLERS, D.J.

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

Signed Date

SERACINO, R.


Signed Date


- 505 -

MOMENT REDISTRIBUTION PARAMETRIC STUDY OF CFRP, GFRP AND STEEL SURFACE PLATED RC BEAMS AND SLABS Irene Liu*, Deric John Oehlers**, Rudolf Seracino***and Gisu Ju**** ABSTRACT It is common practice these days to retrofit reinforced concrete (RC) beams and slabs by adhesive bonding FRP or steel plates to their surfaces. However research has shown that these external plates can debond prematurely at relatively low strains such that the ability of the plated section to redistribute moment to other sections is severely limited and to such an extent that guidelines often preclude moment redistribution. This restriction may severely limit the use of plating in buildings where ductility is a requirement. Tests on steel and FRP plated continuous beams have shown that moment redistribution can occur and a design procedure has been developed to determine the amount of moment redistribution for any type of adhesively bonded plated section, and for any type of plate material, such as steel or FRP plates. In this paper, an analysis approach to quantify the amount of moment redistribution is described and the concepts of positive and negative moment redistribution introduced. A parametric study is then used to illustrate how the plate material and geometric properties affect moment redistribution; in particular, the study looks at the effect of using carbon FRP plates and glass FRP plates, as well as steel plates that have been designed to either debond prior to yielding or yield prior to IC debonding. In summary, the paper will show that plated sections can redistribute moment and, hence, the present restriction can be removed which should extend the use of retrofitting by plating. Furthermore, the moment redistribution analysis procedure allows the engineer the freedom to choose the properties of the plate to design for moment redistribution. Keywords: Retrofitting; reinforced concrete beams; externally bonded plates; ductility; moment redistribution 1. INTRODUCTION

Structures retrofitted by adhesively bonding plates to the external surfaces are prone to premature debonding failures at low plate strains. There are in general three types of debonding mechanisms: (1) plate end (PE) debonding that is induced by the curvature in the beam; (2) critical diagonal crack (CDC) debonding, which is caused by the shear deformation in the beam; and (3) intermediate crack (IC) debonding, which is initiated by the formation of flexural or flexural-shear cracks intercepting the plates. Both PE and CDC debonding can be easily prevented using existing design rules1,2, however IC debonding is much more difficult to prevent and is considered as the dominant mode of debonding3-

6 as it affects both the flexural strength and the ductility of the beam. Due to premature debonding failures at low plate strains, the ductility of plated members and their ability to redistribute moment is less than that of unplated RC beams. It is suggested by both fib7 and Concrete Society8 that, moment redistribution should not be allowed for plated RC beams, however Mukhopadhyaya et al.9 showed that the ductility of a plated beam could be higher than that of an unplated beam if designed properly. Also from experimental studies performed by Oehlers et al.10, it is found that moment redistribution does occur for both steel and FRP plated beams.


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The ability of a flexural member to redistribute moment is often dependent on the type of plating material and technique used1. In some cases if moment redistribution is not allowed or cannot be allowed then the increase in strength due to plating may be minimal and the structure not worth plating. Therefore the purpose of this parametric study is to investigate the ability of plated beams to redistributed moment using different plating materials and geometric properties. The commonly used neutral axis parameter (ku) approach for determining the moment redistribution of RC beams requires that concrete crushing occurs and that the beams have a large ductile/horizontal plateau in the moment/curvature relationship. However, due to premature plate debonding, both requirements usually cannot be satisfied in plated structures. An alternative approach based on flexural rigidities has been developed by Oehlers et al.11 which can quantify moment redistribution in plated structures where IC debonding controls the ultimate strength. In the following context, the different concepts of moment redistribution are first described. It is shown that the simple flexural rigidity (EI) approach11, for quantifying the amount of moment redistribution that can occur, can be used for adhesively bonded plated structures, because adhesively bonded plates tend to debond at relatively low strains. An internal continuous beam with the same hogging and sagging reinforcing bars and which is subjected to uniformly distributed loads is used to illustrate the ability of plated beams to redistribute moment. In the first parametric study, the same steel plates are used in both the hogging and sagging regions and this example is used to show how the IC debonding strain1 affects moment redistribution. In the second parametric study, the hogging and sagging regions are plated with either glass FRP (GFRP) or carbon FRP (CFRP) plates of the same axial stiffness EA and this example clearly shows the benefit of glass over carbon FRP as far as moment redistribution is concerned. In the third parametric study, only the hogging region is plated to illustrate the concept of negative moment redistribution, where it is shown how the hogging moment at IC debonding, in comparison with the sagging moment, is larger than that anticipated from an elastic analysis. In the final parametric study, the gradual formation of hinges is illustrated as well as the importance of moment redistribution in allowing large increases in flexural strengthening. Through this study, it will be shown that plated sections can redistribute moment and, hence, the present restriction can be removed which should extend the use of retrofitting by plating. Using the flexural rigidity approach developed, engineers can quantify the amount of moment redistribution in RC beams with externally bonded plates for beams. By allowing for moment redistribution, designers can now determine the required moment capacities at the positions of maximum moment to resist the applied load, and in particular the type of plate material or bonding technique that can be used. 2. MOMENT REDISTRIBUTION The concept of moment redistribution1 is illustrated in Fig.1 for the case of an encastre or built-in beam of span L that is subjected to a uniformly distributed load w. When the flexural rigidity EI of the beam is constant throughout its length as in Fig.1(a), then the distribution of moment is line A in Fig.1(b) where the maximum hogging moment Mhog is twice the maximum sagging moment Msag. Any divergence from this 2:1 ratio, such as line B, will be assumed to signify some degree of moment redistribution. The static deformation of the beam is shown in Fig.1(e) and that due to the support moments in Fig.1(f). The resultant of these deformations is shown in Fig.1(g) where the slope at the supports maybe finite, as in line D, or zero as in line C. The plastic hinge approach shown in Fig.1(d) assumes that the support slope is finite, as in line D in Fig.1(g), and that this finite slope at the supports is allowed for by rotation within the plastic hinges in Fig.1(d). In contrast, the flexural rigidity (EI)


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approach shown in Fig.1(c) assumes that the support slope is zero as in line C in Fig.1(g), that is any divergence from the elastic 2:1 ratio of line A in Fig.1(b) is accounted for by variations in the flexural rigidity along the length of the beam as in Fig.1(c).

Msag=Mhog/2

Msag=Mhog


plastic hinge approach:

support momentdeformations:

overalldeformation:

dy/dx=0; EI approach


EI

plastichinge

(a)

(b)

(c)

(d)

(e)

(f)

A

B

CD

EI approach:EI2

LEI

w

(g)

EI2EI1

(Mstatic)elastic

ΔMstatic

Mhog

Figure 1 Moment redistribution

Tests1,10-11 on two span continuous beams plated with steel and carbon FRP plates have shown that moment redistribution in plated beams can be accounted for by the flexural rigidity approach depicted in Fig.1(c). This is probably because IC debonding of externally bonded plates usually occurs when the maximum compressive strains in the concrete are well below their crushing strain and, hence, before the sectional curvature is sufficient to cause a plastic hinge. Figure 1(b) also illustrates the importance of moment redistribution, that is beam ductility, on the ability to strengthen a structure. Let us assume that the sectional ductility, that is the moment/curvature (M/ψ) relationship, of both the hogging cross-section and sagging cross-section of the beam are as shown in Fig.2. The hogging region has an initial elastic flexural rigidity EI along O-F up to the hogging capacity Mhog followed by a horizontal plastic plateau F-H, and failure occurs at H which could be due to concrete crushing or plate debonding. The sagging region also has an initial elastic flexural rigidity EI up to an increased sagging capacity of Msag followed by a ductile plateau D-E.


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EI (EI)sec

((EI)sec)H

A

B C

D E

H F

G

sag

hog

Plastic plateaus

ψH

Curvature ψ

Mo

men

t

Mhog/2

Mhog

Msag

O

Figure 2 Sectional ductility

Let us first consider the case where the cross-sections are brittle, so that the M/ψ relationships are given by O-F and O-D, that is the cross-sections do not have a plastic plateau. For this case of an encastre beam with a uniformly distributed load as shown in Fig.1(a), failure will occur when the hogging capacity Mhog is reached and the sagging moment is Mhog/2, point A in Fig.2. The moment distribution is shown as line A in Fig.1(b) where the static moment is (Mstatic)elastic. Let us now consider the case where the hogging region is ductile so that the plastic plateau F-H in Fig.2 exists. In this case, the hogging curvature can keep increasing along F-H whilst the sagging moment increases above that at point A. Let us assume that the sagging moment equals the Mhog (point B) when the curvature in the hogging region is at point G. This is shown as line B in Fig.1(b). It can now be seen that the increase in the static moment ∆Mstatic is due to ductility in the hogging region where the curvature has increased from that at point F to that at point G and the flexural rigidity reduced from EI to (EI)sec. Hence, if the hogging section had been strengthened using a brittle plating system this increase in the static moment could not have been achieved. It can also be seen that the increase in the sagging moment, that is the limit to strengthening the sagging region, is controlled by the ductility of the hogging region. For example, if the hogging plate debonds at the curvature at point H when the sagging moment is at point C, then there is no point in strengthening the sagging region above the moment at point C. It is also worth pointing out that because in this example moment is being redistributed from the hogging to sagging regions, the hogging region needs to be ductile whilst the sagging region needs only to be brittle. Hence, a ductile plating system is required in the hogging region, however, a brittle plating system can be used in the sagging region. Ductility is only required in the sagging region, that is path D-E in Fig.2, if the structure is required to absorb substantial energy as might be required in seismic design. In such a case, the plating system will have to be designed to follow both ductile plateaus. 3. FLEXURAL RIGIDITY APPROACH The analysis for the flexural rigidity approach1,10-11 is depicted in Fig.3. The flexural rigidity of the beam within the hogging region is assumed to be constant at EI2 and that within the sagging region is also constant at EI1. It may be worth noting that the amount of moment redistribution does not depend on the magnitudes of the flexural rigidities, EI1 and EI2, but on their relative difference that is the ratio EI1:EI2. Hence, these flexural rigidities are chosen to represent the proportional change in stiffness between the hogging and sagging regions. For the following parametric studies, the continuous beam


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has been designed so that plate IC debonding will always occur in the hogging region. Therefore, the hogging flexural rigidity EI2 is the secant flexural rigidity at the position of maximum hogging moment and at plate IC debonding such as ((EI)sec)H in Fig.2. The sagging flexural rigidity EI1 is that at the position of maximum sagging moment such as EI in Fig.2.

EI1 EI2

L/2

L/2

Msag=kMhog

w

EI2

Mstatic

x

Point of contra flexure

Mhog

M

db

Mhog

Mdb

Point of contra flexure

Figure 3 Flexural rigidity analysis

For this case of an encastre beam with a uniformly distributed load, the percentage moment redistribution, %MR, is defined as the change in the hogging moment from that for a beam with a constant EI as in Fig.1(a). It is given by the following equation

100

3

23

2

% ×−

=static

hogstatic

M

MMMR (1)

where Mhog is the hogging moment when a static moment Mstatic is being applied as shown in Fig.3. Furthermore, for the case of plate debonding in the hogging region, Mhog would equal the moment capacity at plate debonding Mdb. Full details of the flexural rigidity analysis procedure are given elsewhere1,10-11. The following governing equations apply for the case of an encastre beam with a uniformly distributed load:

+−=

k

kLx

11

2 (2)

( )( ) 146

23

32

3212

212 −

+−−

−+=

LEIxLxEIEI

xL

EIxEI

Lk (3)

where x is the distance of the point of contraflexure from the nearest support as shown in Fig.3, L is the span of the beam, k is given by the ratio Msag/Mhog, EI1 is the flexural rigidity of the sagging region


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at the position of maximum sagging moment and EI2 is the flexural rigidity of the hogging region at the position of maximum hogging moment. The steps in the analysis are:

1. Guess k, that is the ratio between Msag and Mhog. This fixes the position of the point of contraflexure x, in Fig.3 and given by Eq.2.

2. Insert x from Eq.2 into Eq.3 to derive k. If k from Eq.3 does not agree with that used in Eq.2, iterate until the two values converge to determine k.

3. Having determined k, we have the ratio Msag/Mhog for the moment distribution in Fig.3 and hence, Mhog/Mstatic or Msag/Mstatic if required.

It needs to be emphasised that as this is an elastic analysis, the analysis determines the ratio of moments and not the magnitude of the moments directly. Hence, if Mhog is equated to the moment at which the plate debonds in the hogging region, Mdb, then k will give the static moment at which this occurs and therefore, the applied load to cause debonding. To illustrate the application of the flexural rigidity approach, let us assume the beam in Fig.3 is strengthened with FRP plates in both the hogging and sagging regions and that the M/ψ relationships of the FRP plated sections are as shown in Fig.4. The M/ψ relationships have been idealised to a typical bi-linear relationship for an FRP plated section where the initial flexural rigidity is that of the cracked plated section followed by another linear portion after the reinforcing bars have yielded. This is probably the most difficult problem that can occur in the flexural rigidity analysis as we have to deal with two rising plateaus.

[(EI)sag]db [(EI)hog]db

[(EI)hog]cr

[(EI)sag]cr

curvature

(Msag)yield

(Mhog)yield

(Msag)max

(Mhog)max

Moment

D

B

C

A hog

sag

Figure 4 Sectional ductility of FRP plated sections

The simplest solution is to assume that both plates debond at the same time so that EI2 = [(EI)hog]db and EI1 = [(EI)sag]db. Inserting these values into the flexural rigidity approach will give Msag/Mhog. This can be compared with the ratio of the plating capacities (Msag)max /(Mhog)max to determine which region debonds first. Let us assume that the analysis shows that the hogging region debonds first in which case Msag < (Msag)max. This is depicted as point B in Fig.4 which does not lie on the M/ψ relationship for the sagging region. It is, therefore, a matter of increasing the flexural rigidity of the sagging region until the moment in the sagging region lies on the M/ψ relationship, such as at point D.


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4. INTERNAL CONTINUOUS SLAB USED IN PARAMETRIC STUDY In the flexural rigidity analysis described above, and in the following parametric study, no distinction is made between slabs and beams as the same fundamental laws of structural mechanics applies to both slabs and beams, that is flexural members. The RC slab of 4 m span in Fig.5 was used in the parametric study. The slab is always plated over the hogging region and sometimes plated in the sagging region. The cross-sectional details of 1 m width of slab in the sagging region are shown in Fig.6. The hogging region slab is exactly the same as the sagging region but inverted so that the flexural rigidity of the cracked hogging and sagging regions were identical.

L = 4m

Hogging region Hogging region Sagging region

EI2 EI1 EI2

plates

Figure 5 Plated slab in parametric study

2Y12

5Y16 150

1000

bp

Bars: Y12 fsy = 540MPa Y16 fsy = 600MPa Concrete: fc = 35MPa

40

110

Figure 6 RC slab used in parametric study

4.1 Study 1: Steel plates in hogging and sagging region In this study, the slab was plated in both the hogging and sagging regions in Fig.5 and with identical steel plates of width per meter slab of bp=225 mm, thickness tp=2 mm, Young’s modulus Es=200 GPa and yield capacity of fy=300 MPa. The M/ψ relationship for both the hogging region and sagging region are identical and given in Fig.7 as the curve labelled full interaction which was derived from a non-linear sectional analysis. The slope of the lines that emanate from the origin of the graph in Fig.7 are the secant stiffnesses at various IC debonding strains shown in the legend. Also shown are lines that represent the flexural rigidity of the uncracked section (EI)elastic, that of the cracked plated section up to plate yield (EIcr)plated at B, that at yield of the tension reinforcing bars (EIbar)yield at C, and that of the cracked unplated


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section (EIcr)unplated. The flexural rigidity of the sagging region was held at (EIcr)plated and that of the hogging region followed the secant stiffness associated with the whole M/ψ relationship. Hence, if IC debonding occurred in the hogging region prior to the plate yielding (that is the flexural rigidity of the hogging region is the same as that of the sagging region at (EIcr)plated) then the percentage moment redistribution would be zero.

0

20

40

60

80

100

0.0E+00 5.0E-05 1.0E-04 1.5E-04 2.0E-04Curvature (mm -1)

Mom

ent (

kNm

)

debond strain=0.015

debond strain=0.01

debond strain=0.007

debond strain=0.005

debond strain=0.003

debond strain=0.02

debond strain=0.001

debond strain=0.00158

Example of secant stiffness

C

B

(EI)elastic

(EIcr)plated

(EIcr)unplated

Tensile reinforcing bars yield

Plate yield

Full interaction M/ψ

Idealised M/ψ

(EIbar)yield

Figure 7 Moment-curvature relationship for steel plated cross-sections

The results of the parametric study are shown in Fig.8. The real M/ψ curve uses the non-linear full interaction moment-curvature curve in Fig.7. In the region A-B in Fig.8, the hogging region plate debonded prior to the plate yielding so that there is no moment redistribution. In the region B-C, which starts at an IC debonding strain of about 0.002, the plate debonded after yielding but prior to the reinforcing bars yielding. Finally in the region beyond C, debonding occurred after the reinforcing bars yielded, that is in the almost horizontal plateau in Fig.7. Tests on steel plated beams1 have shown that steel plates of up to about 3 mm thickness can debond at strains that range from 0.004 to 0.020 which suggests from Fig.8 that between 10% to 50% moment redistribution can occur. Hence, substantial moment redistribution can occur in steel plated beams if they are designed properly.

0

10

20

30

40

50

60

0 0.005 0.01 0.015 0.02 0.025

debonding strain

% m

omen

t red

istr

ibut

ion

0

10

20

30

40

50

60

0 0.005 0.01 0.015 0.02 0.025

debonding strain

% m

omen

t red

istr

ibut

ion

Cracked

A B C

Plate yield

Plate & bars yield

Real M/ψ

Idealised M/ψ

Figure 8 Moment redistribution: steel plated hogging and sagging regions


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It is felt that most practical design will be based on the conservative bi-linear idealised M/ψ curve shown as the broken line in Fig.7. When this idealised bi-linear plot is used, moment redistribution will only occur if IC debonding occurs after the tension reinforcing bars have yielded. The use of this idealised plot gives the results of the idealised M/ψ curve in Fig.8 in which for the range of IC debonding strains of 0.005 to 0.020, the moment redistribution varies from 0% to 45%. Hence, some steel plated beams will not be able to redistribute moment but there is a potential for reasonable amounts of redistribution if the plates are designed to debond at large strains. 4.2 Study 2: FRP plates in hogging and sagging region In this study, both the hogging and sagging regions in Fig.5 have been plated with either carbon or glass FRP plates. The plate axial stiffness EpAp was maintained constant at 1.0x107 N so that the same moment-curvature relationship in Fig.9 applied to all cross-sections. The plate thicknesses tp were varied from 0.2 mm to 1.2 mm and are shown in the legend in Fig.9 and the Young’s modulus Ep was assumed to be either 160 or 125 GPa for carbon FRP, and 50 or 25 GPa for glass FRP. The strains at which IC debonding occurred were derived from the research of Teng et al4 and the curvatures at which this occurs are shown in the M/ψ plot in Fig.9.

Figure 9 Moment-curvature relationship for CFRP and GFRP plated cross-sections

Compared with the moment-curvature relationship for steel plates in Fig.7 which is basically tri-linear with a horizontal plateau, the moment-curvature relationship for FRP plated flexural members as in Fig.9, has the typical bi-linear relationship with the gradually rising plateau. The results of the FRP parametric study are shown as the lower curve in Fig.10 which also shows the results of the steel plated analyses taken from Fig.8 for comparison. It can be seen from the lower FRP curve that redistribution only occurs when the IC debonding strain exceeds about 0.0046 which for FRP plates is a relatively high strain4. Furthermore, the moment redistribution varies from 0% to 20%. Hence, some moment redistribution can occur with FRP plated beams but the scope is much less than that available for steel plated specimens. The results also show that glass FRP is a better option for moment redistribution than carbon FRP and that moment redistribution only occurs if debonding occurs after the bars yield, which is obvious, but worth stating.

0

10

20

30

40

50

60

70

80

90

0 0.00002 0.00004 0.00006 0.00008

Curvature (mm -1)

Mom

ent (

kNm

)

tp=1.2, Ep=160 tp=0.6;Ep=160tp=0.2;Ep=160 tp=1.2;Ep=125tp=0.6;Ep=125 tp=0.4;Ep=125tp=0.2;Ep=125 tp=1.2;Ep=50tp=0.6;Ep=50 tp=0.6;Ep=25tp=0.4;Ep=25

E (EI)elastic

(EIcr)plated (EIcr)unplated

Flexural

cracking

Tensile reinforcing

bars yielded

Full interaction


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-20

-10

0

10

20

30

40

50

60

0 0.004 0.008 0.012 0.016 0.02

Debonding strain

% M

omen

t red

istr

ibut

ion

CFRP Ep=160CFRP Ep=125GFRP Ep=50GFRP Ep=25

B

A

C

Cracked Bar yield

Cracked

Plate yield

Plate & bar yield

Steel plate

FRP plate

Figure 10 Moment redistribution: FRP plated hogging and sagging regions

4.3 Study 3: Slab plated over hogging region only In this study only the hogging regions of the slab in Figs. 5 and 6 were plated with steel or FRP plates. The moment/curvature relationships in Figs. 7 and 9 still apply but in this case the flexural rigidity of the plated hogging region is always greater than that of the unplated sagging region at the start of loading. The results of the study are shown in Fig.11. For the FRP plated slabs, the plate axial stiffness EpAp was maintained at 1.0x107 N whilst for the steel plated beams it was maintained at 9.0x107 N to reflect what occurs in practice. Let us first consider the upper curve in Fig.11, which are the results for the FRP plated slabs. When IC debonding occurred in the cracked region D-E in Fig.11, the flexural rigidity of the hogging region was (EIcr)plated in Fig.9. The flexural rigidity of the hogging region (EIcr)plated is greater than that of the unplated sagging section (EIcr)unplated in Fig.9. Therefore, the hogging region attracts more moment than if EI is constant along the beam so that the hogging moment is more than twice the sagging moment. Hence, there is now a negative moment redistribution. That is debonding occurs whilst the hogging moment is greater than that anticipated by an elastic analysis in which EI is constant. Therefore, 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 sagging design capacity was reached. After the tensile reinforcing bars have yielded, that is beyond point E in Fig. 11, the secant stiffness of the hogging region gradually reduces from the point marked E in Fig.9 as it follows the gradually rising branch. When the secant stiffness reduces to (EIcr)unplated in Fig.9, there is zero moment redistribution after which there is positive moment redistribution as shown in Fig.11. The FRP curve in Fig.11 lies below that in Fig.10 which means that the capacity for moment redistribution is less than if both regions were plated. It can be seen that there are circumstances when even assuming that there is no moment redistribution can be unsafe. The results for the steel plated beams in Fig.11 show that at least theoretically negative moment redistribution can be greater in steel plated beams than in FRP plated beams. This is because the axial stiffness of steel plates used in practice tends to be greater than that of FRP plates used in practice. However, steel plates when designed to yield prior to IC debonding tend to have a larger IC debonding strain than FRP plates so that the results to the right of point C in Fig.11 are more likely to occur in practice, that is, steel plates are more likely to have positive moment redistribution capacities.


- 515 -

-50

-25

0

25

50

0 0.005 0.01 0.015 0.02

debonding strain

% m

omen

t red

istr

ibut

ion

CFRP Ep=160

CFRP Ep=125

GFRP Ep=50

GFRP Ep=25

Cracked Bar yield

E

D

Plate & bar yield

Plate yield Cracked

C

B

A

Steel plate

FRP plate

Figure 11 Moment redistribution: slabs plated in hogging regions only

4.4 Study 4: FRP and steel plated slabs To illustrate the gradual formation of hinges, and in particular the effect of moment redistribution on strengthening, let us consider a continuous beam that is retrofitted with steel plates in the hogging region and with FRP plates in the sagging region. This is the first case, case 1, of two that will be considered. The M/ψ relationships for the two cross-sections are given in Fig.12 which are taken from Figs.7 and 9. The steel plate used in this study is identical to that used in Study 1, and the FRP plate used has a thickness tp of 1.2mm, width bp of 67mm and Ep of 125GPa.

0

20

40

60

80

100

120

0 0.00005 0.0001 0.00015 0.0002 0.00025 0.0003 0.00035

Curvature (mm -1)

Mom

ent (

kNm

)

steel platedFRP platedunplated

A

B C

Bar yield

Bar yield

(EIcr)steel (EIcr)FRP

Plate yield

Figure 12 Idealised moment-curvature relationship for CFRP and steel plated cross-sections

The amount of moment redistribution at various applied static moments Mstatic is given by the lower curve in Fig.13. Note that in Fig.13 it is assumed that debonding of the plates does not occur. Initially, before the steel plate yields, the cracked elastic stiffness (EIcr)steel at the hogging region is much greater than that at the sagging region (EIcr)FRP. This results in a negative moment redistribution, that is Mhog is greater than that given by an elastic analysis with a constant EI, as the stiffer section attracts more moment. After the plate yields at point A in Fig. 12, the secant stiffness of the hogging region EIhog reduces while the stiffness of the sagging region remains at (EIcr)FRP and so, moment redistribution increases as shown in Fig.13 at point A. As the tensile bars in the hogging region yield at


- 516 -

point B in Fig.12, EIhog now becomes less than that in the sagging region (EIsag =(EIcr)FRP), causing positive moment redistribution beyond point B in Fig.13. That is, as the ratio of EIhog/EIsag is greater than 1, there is negative moment redistribution whereby the moment is being redistributed from the sagging region to the hogging region, and as EIhog/EIsag reduces Mhog/Msag decreases, such that when EIhog/EIsag becomes less than 1, the percentage of moment redistribution is positive due to moment being redistributed from the hogging to the sagging region. The large ductile plateau of the steel plated section indicates that there is a large reduction in stiffness to accommodate the increase in curvature in the hogging region, hence allowing much moment to be redistributed. Therefore, it is the ductility of the steel plated hogging region which allowed the amount of moment redistribution to continue to increase as Mstatic increases.

-40

-30

-20

-10

0

10

20

30

40

0 50 100 150 200

Mstatic (kNm)

% m

omen

t red

istr

ibut

ion

Steel plate in hog, CFRP plate in sagCFRP plate in hog, Steel plate in sag

cracked Plate yield

in sag Bar yield

in hog Bar yield

in sag

cracked

Plate yield in hog

Bar yield in hog

Bar yield in sag

Case 2

Case 1 A

B

X

C D Y E

F

Figure 13 Moment redistribution vs Mstatic

In the above analysis, the hogging region was plated with steel plates as these can be designed to be more ductile than FRP plates as steel plates can be designed to yield prior to debonding. FRP plates were used in the sagging region, as in this example moment is being shed from the hogging region to the sagging region so that the plated sagging region can be brittle. To illustrate the importance of choosing a plating technique to suite the situation, let us consider reversing the situation by applying FRP plates to the hogging region and steel plates to the sagging region, this is case 2. The amount of moment redistribution for case 2 is given by the upper curve in Fig. 13. It can be seen that owing to the large cracked elastic stiffness in the sagging region (EIcr)steel compared to that in the hogging region (EIcr)FRP, there is much moment redistribution even prior to either the plate or reinforcing bars yielding. That is, at the initial stage of loading, as the sagging region is stiffer than the hogging, the moment in the sagging region is greater than that calculated using the elastic analysis of constant EI. Hence, this causes the resultant moment in the hogging region to be less than that given by the elastic analysis with constant EI. Upon further loading, the steel plate in the sagging region will yield (point D in Fig. 13), resulting in a reduction in EIsag, however the stiffness in the hogging region remains at (EIcr)FRP. As the ratio of EIhog/EIsag increases, the amount of moment redistribution reduces. When the tensile bars in the hogging region yield at point E, EIhog decreases while the sagging region remains relatively stiff (that is EIhog/EIsag reduces). Hence, more moment is redistributed up to a maximum at point F where the bars in the sagging region also yield and the response is now on the plateau of the steel plated M/ψ curve in Fig.12. Consequently, there is a large reduction in EIsag to allow for the rapid increase in curvature with increasing Mstatic causing EIhog/EIsag to increase and the amount of moment redistribution to reduce.


- 517 -

It appears as though there is a larger amount of moment redistribution in Case 2, with steel and FRP plates in the sagging and hogging regions respectively, compared to Case 1 with the reverse plating arrangement. However, this percentage of moment redistribution is due to the variation in stiffness along the member, which is not accounted for in the elastic analysis with constant EI, and not because of the ductility of the member. Therefore, if thinner steel plate was used in the sagging region, such that the elastic stiffness in the sagging region is less than or equal to that of the hogging region, or if we continue to increase Mstatic, provided that failure does not occur, the system with steel and FRP plates in hogging and sagging regions respectively will allow more moment redistribution. For the two cases considered in this study, the FRP plates are expected to debond at a plate strain of 0.0033 based on Teng et al.’s IC debonding model4, while from previous studies1,2 it was found that these steel plates, designed to yield prior to debonding, would achieve much higher strains at debonding. Therefore, it will be assumed in this study that the slabs will fail due to IC debonding of the FRP plates. Figures 14 and 15 show the FRP and steel plate strains required to achieve different amounts of moment redistribution respectively. It can be seen in Fig.14 that the amount of moment redistribution achieved at FRP debonding for Case 1 with FRP plates in the sagging region (point X) is less than that for Case 2 with FRP plates in the hogging region (point Y). However, the strain required in the steel plates at failure in Fig.15 is much greater in Case 1 at X than in Case 2 at Y. Furthermore, despite the fact that a higher moment redistribution is achieved in Case 2, it is found from Fig.16 that Mstatic at debonding failure in Case 2 was only 87kNm, whereas it was 123kNm for Case 1.

-40

-30

-20

-10

0

10

20

30

40

0 0.005 0.01 0.015 0.02 0.025 0.03

FRP plate strain

% m

omen

t red

istr

ibut

ion

FRP plate in sag (case 1) FRP plate in hog (case 2)

FRP plate debonds

Y

X

Case 2

Case 1

Figure 14 Moment redistribution vs FRP plate strain

-40

-30

-20

-10

0

10

20

30

40

0 0.005 0.01 0.015 0.02 0.025 0.03

steel plate strain

% m

omen

t red

istr

ibut

ion

steel plate in hog (case 1)

steel plate in sag (case 2)

X

Y

FRP plate debonds

Case 2 Case 1

Figure 15 Moment redistribution vs steel plate strain


- 518 -

0

20

40

60

80

100

120

140

160

180

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035

plate strain

Mst

atic

(kN

m)

Steel plate in hog (case 1)FRP plate in sag (case 1)FRP plate in hog (case 2)Steel plate in sag (case 2)

X

Y

Case 1: Steel plate in hog, FRP plate in sag

Case 2: FRP plate in hog, Steel plate in sag

FRP plate debonds

Figure 16 Mstatic vs plate strain

Figure 17 shows the maximum moments in the hogging and sagging regions for the two plating arrangements considered. It can be seen that for Case 1, the moment capacity of the steel plated hogging region was reached at point B. As Mstatic continues to increase, the moment in the sagging region Msag increases more rapidly while the hogging moment Mhog remains constant along B-C until the FRP plate debonds at point D at a static moment X. Whereas for Case 2, the FRP plate debonded in the hogging region at E with the corresponding moment in the sagging region given by point G, at static moment Y soon after the steel plate yielded in the sagging region at F. The maximum moment achieved in the FRP plated regions in Cases 1 and 2 (points D and E respectively in Fig.17) was considerably less than the 60kNm moment capacity of the unplated section (Fig.12). This demonstrates the ineffectiveness of applying FRP plates, which are prone to premature debonding at relatively low strains, in regions where ductility is required.

0

10

20

30

40

50

60

70

80

90

0 20 40 60 80 100 120 140 160 180

Mstatic (kNm)

Mom

ent (

kNm

)

Mhog (case 1)

Msag (case 1)

Mhog (case 2)

Msag (case 2)

A

F

C B

Plate yields

Bars yield in hog

Bars yield in hog

Bars yield in sag

Y X

D

E

G

FRP debonds

Figure 17 Moment vs Mstatic


- 519 -

Case 1 allowed greater strains in the steel plates (Fig.15) and enabled a larger load at failure (Fig.16). In Case 1, the moment capacity of the steel plated section was reached when the FRP debonded in the sagging region. In Case 2 the maximum moments obtained in both the hogging and sagging regions at failure were lower than the moment capacity of the unplated section, indicating that applying a brittle FRP plate and ductile steel plate in the hogging and sagging regions respectively is an ineffective strengthening approach. It can therefore be concluded that using steel plates to ensure ductility in the hogging region where hinges form and FRP plates in the sagging region where ductility is not required is a more effective way of plating, and permitting moment redistribution. 5. SUMMARY

� Tests have shown that the ability to redistribute moment in a plated beam can be determined from the variation in the flexural rigidity, probably because externally bonded plates tend to debond when the strains in the concrete are well below the crushing strain.

� A procedure has been developed for quantifying the ability of a plated beam to redistribute moment which depends on the IC debonding strain and which can be applied to any plate material and for any plate position and shape.

� Steel plated beams can be designed to yield prior to IC debonding, although IC debonding does eventually occur. A parametric study of this form of plating showed that the percentage moment redistribution can vary from 0% to 45%. Therefore, there is the potential for substantial moment redistribution. However, if designed poorly, there may be no moment redistribution.

� A parametric study of FRP plated beams showed that the percentage moment redistribution can vary from 0% to 20%. Hence, there is the potential for a reasonable amount of moment redistribution but this is mainly restricted to glass FRP plates.

� Furthermore, moment redistribution in FRP plated beams only occurs if IC debonding occurs after the tension reinforcing bars have yielded.

� The concept of negative moment redistribution has been introduced where it was shown that poorly designed beams can debond prematurely in a region, that is before the other region has achieved its design moment capacity.

� Also illustrated is the importance of carefully choosing the plating technique to optimise the increase in strength

� It is suggested that moment redistribution should be considered in design and that simply allowing for no moment redistribution may not always be a safe assumption.

6. REFERENCES

1. Oehlers, D.J., and Seracino, R. Design of FRP and steel plated RC structures: retrofitting beams and slabs for strength, stiffness and ductility. Elsevier, Oxford, September, 2004.

2. Teng, J.G, Chen, J.F., Smith, S.T. and Lam, L. FRP strengthened RC structures. Wiley, New York, 2002

3. Niu, H., Wu, Z. Strengthening effects of RC flexural members with FRP sheets affected by adhesive layers.. Journal of Applied mechanics, JSCE, 2002, vol.5, 887-897

4. Niu, H., Wu, Z. Interfacial Debonding Mechanism Influenced by Flexural Cracks in FRP-strengthened Beams. Journal of Structural Engineering, 2001, 47A: 1277-1288.

5. Niu, H., Wu, Z. Prediction of debonding failure load due to flexural cracks of concrete for FRP-strengthened structures., FRPRCS-5: Bond, 2001, p.361-370


- 520 -

6. Sebastian, W.M. Significance of midspan debonding failure in FRP-plated concrete beams. Journal of Structural engineering, ASCE, 2001, vol.127, no.7, p.792-798

7. fib Task Group 9.3. Externally bonded FRP reinforcement for RC structures, Technical report. International Federation for Structural Concrete, Lausanne, 2001

8. Concrete Society Committee. Design guidance for strengthening concrete structures using fibre composite materials, Concrete Society Technical Report no. 55. The Concrete Society, UK, 2000

9. Mukhopadhyaya, P., Swamy, N. and Lynsdale, C. Optimizing structural response of beams strengthened with GFRP plates. Journal of Composites for Construction, ASCE, 1998, vol. 2, no. 2, p87-95

10. Oehlers, D.J. Ju, G., Liu, I and Seracino, R. Moment redistribution in continuous plated RC flexural members. Part 1: neutral axis depth approach and tests. Engineering structures, 2004, vol.26, p.2197-2207

11. Oehlers, D.J. Liu, I., Ju, G. and Seracino, R. Moment redistribution in continuous plated RC flexural members. Part 2: flexural rigidity approach. Engineering structures, 2004, vol.26, p.2209-2218


- 521 -

6.5 LINEAR FLEXURAL RIGIDITY (EI) APPROACH

The simplified flexural rigidity approach proposed in the previous Section allows for the variations in

the flexural rigidity EI along the beam by assuming that EI of the hogging region EIhog is different to

that of the sagging region EIsag, but with the EI being constant within each of the hogging and sagging

regions. Although this approach was found to give good and conservative estimations of the amount

of moment redistribution for EB plated beams, the accuracy of the model can vary when applied to

NSM beams where the plate strains reached are found to be much higher than that achieved by EB

plated beams. Due to the assumption of constant flexural rigidities within the sagging and hogging

regions, this approach tends to underestimate the actual flexural rigidity along the region of the beam

redistributing moment, which led to unconservative design when used for NSM beams. In the following

journal paper, the linear EI approach is developed where the flexural rigidity approach has been

modified to allow for the variation in EI along the beam. This new approach gives a better

representation of the moment redistribution behaviour of plated beams, and also, can be applied to

beams with different failure modes such as premature plate debonding or concrete crushing failure.

6.5.1 JOURNAL PAPER: MOMENT REDISTRIBUTION IN FRP AND STEEL PLATED REINFORCED CONCRETE BEAMS

In the following journal paper, the linear flexural rigidity approach is presented, and is verified using

the test results of the EB and NSM specimens reported in Chapter 5.


- 522 -

Intermediate Crack Debonding of Plated RC Beams Theoretical Studies on Moment Redistribution Moment redistribution in FRP and steel plated reinforced concrete beams

- 523 -

Moment redistribution in FRP and steel plated reinforced concrete

beams

*Liu, I.S.T., **Oehlers, D.J. and ***Seracino, R.

*Ms. I.S.T. Liu Postgraduate student School of Civil and Environmental Engineering The University of Adelaide Corresponding author **Dr. D.J. Oehlers Associate Professor School of Civil and Environmental Engineering The University of Adelaide Adelaide SA5005 AUSTRALIA Tel. 61 8 8303 5451 Fax 61 8 8303 4359 Email: [email protected] ***Dr. R. Seracino Senior Lecturer School of Civil and Environmental Engineering The University of Adelaide Accepted for publication in ASCE Journal of Composites for Construction August 2005

Intermediate Crack Debonding of Plated RC Beams Theoretical Studies on Moment Redistribution Moment redistribution in FRP and steel plated reinforced concrete beams

- 524 -


MOMENT REDISTRIBUTION IN FRP AND STEEL PLATED REINFORCED CONCRETE BEAMS

Accepted for publication in ASCE Journal of Composites for Construction August 2005


Performed all analyses, interpreted data and wrote manuscript.

Signed Date

OEHLERS, D.J.

Supervised development of work, edited manuscript and acted as corresponding author.

Signed Date

SERACINO, R.


Signed Date

Liu, I.S.T., Oehlers, D.J., and Seracino, R. (2005) Moment redistribution in FRP and steel plated reinforced concrete beams ASCE Journal of Composites for Construction, accepted for publication, August 2005

NOTE: This publication is included on pages 523 - 544 in the print copy of the thesis held in the University of Adelaide Library.


- 545 -

6.5.2 FURTHER DISCUSSIONS ON LINEAR EI APPROACH

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

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

6.5.2.1 DERIVATION OF EQUIVALENT EI

In the following Section, the equivalent flexural rigidities EIeq for the linear EI approach are derived for

two plating systems: (1) one end continuous beam under point loading; and (2) both ends continuous

beam under uniformly distributed loading. These constant EIeq determined for the hogging (EIeq)hog

and sagging (EIeq)sag regions can then be substituted into the simplified EI approach (Section 6.4) to

analyse the moment redistribution behaviour of statically indeterminate structures.

6.5.2.1.1 ONE END CONTINUOUS BEAM SUBJECTED TO POINT LOAD

Consider the one end continuous beam in Figure 6.21a of span L, which is subjected to a point load P

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

where Msag and Mhog are the maximum moments in the sagging and hogging regions respectively, with

the point of contraflexure, C, occurring at a distance x from the fixed end H. Based on the linear

flexural rigidity approach, the stiffnesses in the hogging and sagging regions are assumed to vary

linearly as illustrated in Figure 6.21b. Due to the applied load, there is a discontinuity in the variation of

EI at point S. Therefore the three regions, H-C, C-S and S-B, are treated separately, each bounded by

a point of contraflexure (poc), or the support B for region S-B, where the moment is zero i.e.

MB=MC=0, and the position of maximum moment, Msag or Mhog. The length of H-C and C-S depends

on the hogging and sagging moment ratio.

The flexural rigidity varies linearly along the beam, with a maximum at the point of contraflexure EIC for

H-C and C-S and at the exterior support EIB for S-B, and a minimum at the position of maximum

moment EIH or EIS as shown in Figure 6.21b. The values of EIH, EIC, EIS, and EIB to be used in the

analyses depend on the moments in the beam. If the maximum moment in the region analysed is

greater than the moment to cause cracking, then EIC (or EIB for region S-B) is taken as the flexural

rigidity of the cracked section EIcr, else the flexural rigidity of the uncracked section EIg is adopted. EIH

and EIS are given by the secant stiffness EIsec determined from the full interaction M/χ relationship for

the Mhog and Msag considered, such as illustrated in Figure 6.1a.


- 546 -

EIB

a L-a

x

P

Msag

B S

C H

EIS

(EIC)hog

Mhog/EIH

MC/EIC=0 Msag/EIS

MB/EIB=0 θS/B

θS/C

θH/C

Msag

(EIeq)sag

Mhog/(EIeq)hog

Msag/(EIeq)sag

(EIeq)hog

EI1 EI2

(a)

(b)

(c)

(d)

(e)

(f)

poc

Mhog

line A

(EIC)sag EIH

χ

0

0

EI

z1 z3 z2

EIeq

χeq

0

0

θS/C

θH/C θS/B

a L-a

x

P

B S C H

Mhog

line A

Figure 6.21 Linear EI approach: one end continuous beam

One method of analysing the beam in Figure 6.21 is to express the varying EIs in each of the hogging

and sagging regions as an equivalent flexural rigidity EIeq that is constant over the region, as in Figure

6.21e, but which provides the region with the same rotation capacity. This is done by dividing the

linear distribution of moment M in Figure 6.21a by the linear variation in flexural rigidity EI in Figure

6.21b, which gives the distribution of curvature χ in Figure 6.21c. The rotation θ of the region

considered is obtained by integrating the curvature over that region, i.e. the area of the curvature

distribution. By dividing the linear moment distribution in Figure 6.21d with the constant flexural rigidity

in Figure 6.21e gives the variation in curvature in Figure 6.21f, and by equating the rotations in Figure

6.21c with that in Figure 6.21f, the equivalent constant EIs for each region can be obtained. The

evaluation of equivalent EIs in the hogging, (EIeq)hog, and the sagging, (EIeq)sag, regions shown in

Figure 6.21e is explained in the following.

Consider the region H-C in Figure 6.21a, where the co-ordinate z1 is the distance from the support H

with 0 ≤ z1 ≤ x. From similar triangles, the moment Mz1 at distance z1 from H is given by:

11

1

)(z

x

MM

x

zxMM

hoghog

hogz −=

−= Equation 6.56

Based on the linear varying flexural rigidity in Figure 6.21b, the flexural rigidity EIz1 at distance z1 from

H is:


- 547 -

11 zx

EIEIEIEI HC

Hz

−+= Equation 6.57

The curvature distribution χz1 in region H-C can, hence, be evaluated by dividing Equation 6.56 by

Equation 6.57 in Equation 6.58, which is illustrated in Figure 6.21c.

1

1

1

11

zx

EIEIEI

zx

MM

EI

M

HCH

hoghog

z

zz

−+

−==χ Equation 6.58

The rotation capacity of region H-C, θH/C, is obtained by integrating the curvature (Equation 6.58) over

the region of length x:

( )

( )

( )

−+

−=⇒

−−

−+

−=

−+

−== ∫∫

1ln

lnln

2/

2

2

0

1

1

1

0

11/

C

H

H

C

HC

ChogCH

HC

hog

H

C

HC

Hhog

H

C

HC

hog

x

HCH

hoghogx

zCH

EI

EI

EI

EI

EIEI

xEIM

xEIEI

x

x

M

EI

EI

EIEI

xEI

x

M

EI

EI

EIEI

xM

dzz

x

EIEIEI

zx

MM

dz

θ

χθ

Equation 6.59

Now consider an equivalent constant EI over the region H-C, (EIeq)H-C = (EIeq)hog in Figure 6.21e. For

this equivalent system with a linear varying moment distribution, the equivalent curvature χeq

distribution, given by Equation 6.60, varies linearly as shown in Figure 6.21f. The rotation capacity of

the region θH/C is obtained by integrating Equation 6.60 over the region H-C in Equation 6.61, which is

equal to the area under the χeq distribution. By rearranging Equation 6.61, (EIeq)hog is given in

Equation 6.62.

( ) ( )hogeq

z

zeq EI

M 1

1=χ

Equation 6.60


- 548 -

( ) ( ) ( ) 21

0

11

01/

x

EI

Mdz

EI

Mdz

hogeq

hogx

hogeq

zx

zeqCH === ∫∫ χθ Equation 6.61

( )2/

xMEI

CH

hog

hogeq θ= Equation 6.62

As the rotation capacity in region H-C is the same for the two systems, Figure 6.21a and Figure 6.21d,

substituting Equation 6.59 into Equation 6.62 gives:

( )( )

( ) ( )

−+

−=⇒

−+

−

=

1ln2

21ln

2

2

C

H

H

CC

HC

hogeq

C

H

H

C

HC

Chog

hog

hogeq

EI

EI

EI

EIEI

EIEIEI

x

EI

EI

EI

EI

EIEI

xEIM

MEI

for EIC≠EIH Equation 6.63

Consider the region C-S in Figure 6.21a, where the co-ordinate z2 is the distance from the point S with

0 ≤ z2 ≤ L-a-x. From similar triangles, the moment Mz2 and flexural rigidity EIz2 at distance z2 from S

are given by:

( )[ ]2

22 z

axL

MM

axL

zaxLMM sag

sagsag

z

−−−=

−−−−−

= Equation 6.64

22 zaxL

EIEIEIEI SC

Sz

−−−

+= Equation 6.65

Dividing Equation 6.64 by Equation 6.65 gives the curvature distribution χz2 in region C-S, which is

illustrated in Figure 6.21c; and by integrating χz2 (Equation 6.66) over the region C-S of length L-x-a,

the rotation capacity of region C-S, θS/C, is evaluated in Equation 6.67.

2

2

2

22

zaxL

EIEIEI

zaxL

MM

EI

M

SCS

sagsag

z

zz

−−−

+

−−−

==χ Equation 6.66


- 549 -

( )( )

( )( )

( )( )

( ) ( )

( )( )

−+

−

−−=⇒

−−−

−−−−

−

−−−

−−+

−−−

=

−−−

+

−−−

== ∫∫−−−−

1ln

lnln

2/

2

2

0

2

2

2

0

22/

C

S

S

C

SC

CsagCS

SC

sag

S

C

SC

Ssag

S

C

SC

sag

axL

SCS

sagsagaxL

zCS

EI

EI

EI

EI

EIEI

EIaxLM

axLEIEI

axL

axL

M

EI

EI

EIEI

axLEI

axL

M

EI

EI

EIEI

axLM

dz

zaxL

EIEIEI

zaxL

MM

dz

θ

χθ

Equation 6.67

Now consider an equivalent constant EI over the region S-C, (EIeq)S-C, with the equivalent curvature

χeq distribution given by Equation 6.68, which varies linearly with zero curvature at C and a maximum

at S. The rotation capacity of the region θS/C is obtained by integrating Equation 6.68 over the region

S-C in Equation 6.69, which is equal to the area under the χeq distribution.

( ) ( )CSeq

z

zeq EI

M

−

= 2

2χ

Equation 6.68

( ) ( ) ( )

−−===−

−−

−

−−

∫∫ 22

0

22

02/

xaL

EI

Mdz

EI

Mdz

CSeq

sagxaL

CSeq

zxaL

zeqCS χθ Equation 6.69

(EIeq)S-C is obtained by rearranging Equation 6.69, and substituting Equation 6.67 into Equation 6.70:

( )

−−=− 2/

xaLMEI

CS

sag

CSeq θ Equation 6.70

( ) ( )( )

( ) ( )

−+

−=⇒

−−

−+

−

−−=

−

−

1ln2

21ln

2

2

C

S

S

CC

SC

CSeq

C

S

S

C

SC

Csag

sag

CSeq

EI

EI

EI

EIEI

EIEIEI

xaL

EI

EI

EI

EI

EIEI

EIaxLM

MEI

for EIC≠EIS

Equation 6.71


- 550 -

For the region S-B in Figure 6.21a, the co-ordinate z3 was used, which is the distance from the point S

with 0 ≤ z3 ≤ a. From similar triangles, the moment Mz3 and flexural rigidity EIz3 at distance z3 from S

are given by:

( )3

33 z

a

MM

a

zaMM sag

sagsag

z

−=

−= Equation 6.72

33 za

EIEIEIEI SB

Sz

−+= Equation 6.73

Dividing Equation 6.72 by Equation 6.73 gives the curvature distribution χz3 in region S-B, which is

illustrated in Figure 6.21c; and by integrating χz3 (Equation 6.74) over the region S-B of length a, the

rotation capacity of region S-B, θS/B, is evaluated in Equation 6.75.

3

3

3

33

za

EIEIEI

za

MM

EI

M

SBS

sagsag

z

zz

−+

−

==χ Equation 6.74

( )

( )

( )

−+

−=⇒

−−

−+

−=

−+

−

== ∫∫

1ln

lnln

2/

2

2

0

3

3

3

0

33/

B

S

S

B

SB

Bsag

BS

SB

sag

S

B

SB

S

sag

S

B

SB

sag

a

SBS

sag

saga

zBS

EI

EI

EI

EI

EIEI

aEIM

aEIEI

a

a

M

EI

EI

EIEI

aEI

a

M

EI

EI

EIEI

aM

dz

za

EIEIEI

za

MM

dz

θ

χθ

Equation 6.75

Now consider an equivalent constant EI over the region S-B, (EIeq)S-B, with the equivalent curvature

χeq distribution given by Equation 6.76, which varies linearly with zero curvature at B and a maximum

at S. The rotation capacity of the region θS/B is obtained by integrating Equation 6.77 over the region

S-B in Equation 6.78, which is equal to the area under the χeq distribution.


- 551 -

( ) ( )BSeq

z

zeq EI

M

−

= 3

3χ

Equation 6.76

( ) ( ) ( )

===−−

∫∫ 23

0

33

03/

a

EI

Mdz

EI

Mdz

BSeq

saga

BSeq

za

zeqBS χθ Equation 6.77

(EIeq)S-B is obtained by rearranging Equation 6.77, and substituting Equation 6.75 into Equation 6.78:

( )

=− 2/

aMEI

BS

sag

BSeq θ Equation 6.78

( )( )

( ) ( )

−+

−=⇒

−+

−

=

−

−

1ln2

21ln

2

2

B

S

S

BB

SB

BSeq

B

S

S

B

SB

Bsag

sag

BSeq

EI

EI

EI

EIEI

EIEIEI

a

EI

EI

EI

EI

EIEI

aEIM

MEI

for EIB≠EIS Equation 6.79

For an equivalent constant EI for over the sagging region (EIeq)sag as illustrated in Figure 6.21e, the

rotation capacity of the sagging region C-B, θC/B, is given by the area under the equivalent curvature

distribution in Figure 6.21f. That is:

( )

−=2/

xL

EI

M

sageq

sagBCθ

Equation 6.80

θC/B is also given by the sum of Equation 6.69 and Equation 6.77:

( ) ( )

+

−−=+=−−

22///

a

EI

MxaL

EI

M

BSeq

sag

SCeq

sagBSSCBC θθθ Equation 6.81

Therefore, by equating Equation 6.80 and Equation 6.81, the equivalent EI over the sagging region,

(EIeq)sag, is given by:


- 552 -

( ) ( ) ( )

+

−−=

−

−−222

a

EI

MxaL

EI

MxL

EI

M

BSeq

sag

SCeq

sag

sageq

sag

Equation 6.82

[ ][ ] [ ]xLEIEIEIa

xLEIEIEI

BSeqBSeqSCeq

BSeqSCeqsageq −+−

−=

−−−

−−

)()()(

)()()( Equation 6.83

After determining the equivalent rigidities (EIeq)hog and (EIeq)sag, the beam can now be analysed using

the simplified EI approach discussed in Section 6.4, which assumes a constant within the hogging

(EI2) and the sagging (EI1) regions as shown in Figure 6.21d, where EI1 = (EIeq)sag and EI2 = α(EIeq)hog.

A stiffness analysis software package, with two elements of stiffness EI1 and EI2, can be used to find a

solution to the beam in Figure 6.21. Alternatively, the elastic solution shown in Section 6.4.2.1.1 can

be used.

6.5.2.1.2 BOTH ENDS CONTINUOUS BEAM SUBJECTED TO UNIFORMLY DISTRIBUTED LOADS

Consider the both ends continuous beam in Figure 6.22a of span L, which is subjected to an uniformly

distributed loading of w. The moment distribution of the beam is denoted by line A, where Msag and

Mhog are the maximum moments in the sagging and hogging regions respectively, with the point of

contraflexure, C, occurring at a distance x from the fixed end H. As the loading is symmetrical,

therefore only half of the beam needs to be analysed. Linear variation in stiffness EI is assumed as

illustrated in Figure 6.22b, with a maximum EI at the point of contraflexure EIC, and a minimum EI at

the positions of maximum moment EIH and EIS. If the maximum moment in the region analysed is

greater than the moment to cause cracking, then EIC is taken as the flexural rigidity of the cracked

section EIcr, else the flexural rigidity of the uncracked section EIg is adopted. EIH and EIS are the

secant stiffness EIsec determined from the full interaction M/χ relationship for Mhog and Msag

considered.

The evaluation of equivalent EIs in the hogging, (EIeq)hog, and the sagging, (EIeq)sag, regions shown in

Figure 6.22e is explained in the following. For the loading system considered, the static moment

Mstatic=wL2/8, and the sagging moment can be expressed in terms of Mhog by Equation 6.84. As the

system is symmetrical, therefore the reaction force at each support R is given by Equation 6.85.

hogsag MwL

M −=8

2

Equation 6.84


- 553 -

2

wLR = Equation 6.85

EIS

EIH

x

w

Msag

L/2

x

w

S C H

poc

Mhog line A

L/2 (a)

(b)

(c)

(d)

(e)

(f)

χ

EI

EIeq

χeq

Mhog/EIH

MC/EIC=0

Msag/EIS θS/C

θH/C

L/2 L/2

Mhog/(EIeq)hog

Msag/(EIeq)sag θS/C

θH/C

(EIeq)sag (EIeq)hog

EI1 EI2 S C H

line A

EI2 C

(EIeq)hog

C H

(EIC)hog

(EIC)sag

z1 z2

Figure 6.22 Linear EI approach: both ends continuous beam

Consider the hogging region H-C in Figure 6.22a, where the co-ordinate z1 is the distance from the

support H with 0 ≤ z1 ≤ x. Using the method of sections and Equation 6.85, the moment Mz1 at

distance z1 from H is given by:

221

21

1

wLzwzMM hogz −+= Equation 6.86

Based on the linear varying flexural rigidity in Figure 6.22b, the flexural rigidity EIz1 at distance z1 from

H is:

11 zx

EIEIEIEI HC

Hz

−+= Equation 6.87

The curvature distribution χz1 in region H-C is evaluated by dividing Equation 6.86 by Equation 6.87,

as illustrated in Figure 6.22c.


- 554 -

1

121

1

11

22

zx

EIEIEI

wLzwzM

EI

M

HCH

hog

z

zz

−+

−+==χ

Equation 6.88

The rotation capacity of region H-C, θH/C, is obtained by integrating the curvature (Equation 6.88) over

the region of length x:

( ) ( )

( )( ) ( )

−

−+

−+

−−

−+

−=

−+

+

−+

−=

−+

−+==

∫∫

∫∫

HCHC

H

CH

HC

HCH

C

HC

H

H

C

HC

hog

x

HCH

x

HCH

hog

x

HCH

hogx

zCH

EIEIEIEI

EI

EIEI

EIEI

xw

EIEI

xwL

EI

EI

EIEI

xEIwL

EI

EI

EIEI

xM

dz

zx

EIEIEI

wz

dz

zx

EIEIEI

wLzM

dz

zx

EIEIEI

wLzwzM

dz

32

ln2

2ln

2ln

22

22

2

3

3

2

2

2

0

1

1

21

0

1

1

1

0

1

1

121

0

11/ χθ

( ) ( )

( ) ( )( )HC

HCHC

HC

H

HC

H

HC

hog

H

CCH

EIEIEIEI

xw

EIEI

xwL

EIEI

EIxw

EIEI

xEIwL

EIEI

xM

EI

EI

342

22ln

2

32

3

23

2

2

/

−−

+−

−

−+

−+

−=⇒θ

for EIC≠EIH

Equation 6.89

Now consider an equivalent constant EI over the region H-C, (EIeq)hog in Figure 6.22e with the

equivalent curvature χeq distribution calculated by Equation 6.90 and illustrated in Figure 6.22f. The

rotation capacity of the region θH/C can be obtained by substituting Equation 6.86 into Equation 6.90

and integrating Equation 6.90 over the region H-C, as in Equation 6.91.

( ) ( )hogeq

z

zeq EI

M 1

1=χ

Equation 6.90


- 555 -

( ) ( ) ( )

( )

−+=

−+=== ∫∫∫

46

1

22

1

23

0

11

21

0

11

0

11/

LwxwxxM

EI

dzwLzwz

MEI

dzEI

Mdz

hog

hogeq

x

hog

hogeq

x

hogeq

zx

zeqCH χθ

Equation 6.91

Therefore, the (EIeq)hog can be expressed in terms of θH/C by rearranging Equation 6.91 as given in

Equation 6.92. Since the rotation capacity in region H-C is identical for the two systems, Figure 6.22a

and Figure 6.22d, (EIeq)hog can, hence, be evaluated by substituting Equation 6.89 into Equation 6.92.

( )

−+=

−+=

4646

1 2

/

23

/

wxLwxM

xLwxwxxMEI hog

CHhog

CHhogeq θθ

Equation 6.92

Consider the sagging region C-S in Figure 6.22a, where the co-ordinate z2 is the distance from the

point S with 0 ≤ z2 ≤ L/2-x. Using method of sections and Equation 6.85, the moment Mz2 at distance

z2 from S are given by Equation 6.93. From the linear EI distribution in Figure 6.22b, the flexural

rigidity EIz2 at z2 can be calculated using Equation 6.94.

2

221

2

wzMM sagz −= Equation 6.93

22 2/z

xL

EIEIEIEI SC

Sz

−−+= Equation 6.94

Dividing Equation 6.93 by Equation 6.94 gives the curvature distribution χz2 in region C-S, which is

illustrated in Figure 6.21c; and by integrating Equation 6.95 over the region C-S of length L/2-x, the

rotation capacity of region C-S, θS/C, is evaluated in Equation 6.96.

2

22

2

22

2/

2

zxL

EIEIEI

wzM

EI

M

SCS

sag

z

zz

−−+

−==χ Equation 6.95


- 556 -

( )

( )( )

( )( )

−−+

−−−

−−

=

−−+

−

−−+

=

−−+

−==

∫∫

∫∫

−−

−−

SCSC

S

CS

SC

S

C

SC

sag

xL

SCS

xL

SCS

sag

xL

SCS

sagxL

zCS

EIEIEIEI

EI

EIEI

EIEI

xLw

EI

EI

EIEI

xLM

dz

zxL

EIEIEI

wz

dz

zxL

EIEIEI

M

dz

zxL

EIEIEI

wzM

dz

32

ln2/

2

ln2/

2/

2

2/

2/

2

2

3

3

2/

0

2

2

22

2/

0

2

2

2/

0

2

2

22

2/

0

22/ χθ

( )( )2

3

3

2/

3)2/(

4

2/

2

)2/(ln

SC

SC

SCS

SC

sag

S

CCS

EIEI

EIEIxLw

EIEI

xLEI

w

EIEI

xLM

EI

EI

−−−−

−−−

−−

=⇒θ for EIC≠EIS

Equation 6.96

Now consider an equivalent constant EI over the region S-C, (EIeq)sag, with the equivalent curvature

χeq distribution given by Equation 6.97, which varies linearly with zero curvature at C and a maximum

at S. The rotation capacity of the region θS/C is obtained by integrating Equation 6.97 over the region

S-C in Equation 6.98 and substituting Equation 6.93.

( ) ( )CSeq

z

zeq EI

M

−

= 2

2χ

Equation 6.97

( ) ( ) ( )

( )

−−

−=

−=== ∫∫∫−−−

6

2

2

1

2

1

3

2/

0

2

22

2/

0

22

2/

0

22/

xL

w

xL

MEI

dzwz

MEI

dzEI

Mdz

sag

sageq

xL

sag

sageq

xL

sageq

zxL

zeqCS χθ

Equation 6.98

(EIeq)sag can be expressed in terms of θS/C by rearranging Equation 6.98 as given in Equation 6.99.

Since the rotation capacity in region S-C is identical for the two systems, Figure 6.22a and Figure


- 557 -

6.22d, (EIeq)sag can, hence, be evaluated by substituting Equation 6.96 and Equation 6.84 into

Equation 6.99.

( )

( ) ( )

−−−−=

−−

−=

24

2

82

2

6

2

2

1

22

/

3

/

xLwM

wLxL

xL

w

xL

MEI

hogCS

sagCS

sageq

θ

θ Equation 6.99

After determining the equivalent rigidities (EIeq)hog and (EIeq)sag, the beam can now be analysed using

the simplified EI approach discussed in Section 6.4, where EI1 = (EIeq)sag and EI2 = α(EIeq)hog. A

stiffness analysis software package, with two elements of stiffness EI1 and EI2, can be used to find a

solution to the beam in Figure 6.22. Alternatively, the elastic solution shown in Section 6.4.2.1.2 can

be used.

6.5.2.2 COMPARISON BETWEEN EXPERIMENTAL AND THEORETICAL RESULTS

In the following Section, the linear EI approach is adopted to analyse the statically indeterminate test

specimens reported in Chapter 5.

6.5.2.2.1 TEST SERIES ‘S’ (SPECIMENS WITH EXTERNALLY BONDED PLATES)

In the journal paper presented in Section 6.5.1, the linear EI model was verified by analysing the

seven test specimens with EB steel or FRP plates (refer to Chapter 5 for beam details) for various

loading stages, along with the test results from Chapter 5. Further details of the analyses performed

are shown in Figure 6.23, Figure 6.24 and Figure 6.25, where the abscissae of Figure 6.23 and Figure

6.24 are the applied static moment, Mstatic given by Equation 5.2, as a proportion of the ultimate

maximum static moment, (Mstatic)u = (Msag)u + (Mhog)u/2, based on nonlinear full interaction analysis of

the ultimate capacity of the hogging and sagging sections, (Mhog)u and (Msag)u. Figure 6.23 illustrates

the variation of total percentage of moment redistribution %MRtot calculated using Equation 5.1 for the

seven EB test specimens. Figure 6.24 shows the variation of the maximum hogging moment in the

beam Mhog as a proportion of the maximum sagging moment in the beam Msag, where line A

represents the elastic analysis for constant EI. Figure 6.25 shows the variation of the maximum

hogging Mhog moments as the applied loads P increased, where the straight line represents the elastic

analysis for constant EI.


- 558 -

-20

-10

0

10

20

30

40

50

60

70

0 0.2 0.4 0.6 0.8 1

Mstatic/(Mstatic)u

%M

Rto

t

testtheo(plated)theo(unplated)

-10

0

10

20

30

40

50

60

70

0 0.2 0.4 0.6 0.8 1

Mstatic/(Mstatic)u

%M

Rto

t

test.theo(plated)theo(unplated)

-10

0

10

20

30

40

50

60

70

0 0.2 0.4 0.6 0.8 1

Mstatic/(Mstatic)u

%M

Rto

t


-10

0

10

20

30

40

50

60

70

0 0.2 0.4 0.6 0.8 1

Mstatic/(Mstatic)u

%M

Rto

t


-20

0

20

40

60

80

100

0 0.2 0.4 0.6 0.8 1

Mstatic/(Mstatic)u

%M

Rto

t

testtheo.(plated)theo.(unplated)

-10

0

10

20

30

40

50

60

0 0.2 0.4 0.6 0.8 1

Mstatic/(Mstatic)u

%M

Rto

t


-10

0

10

20

30

40

50

60

70


%M

Rto

t


Beam SS1

plate debond

Beam SS2

plate debond

Beam SS3

shear failure

Beam SF1

plate debond

Beam SF2

plate debond

Beam SF3

plate debond

Beam SF4

plate debond

Figure 6.23 Linear EI approach: Percentage moment redistribution of EB test specimens


- 559 -

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6


Mh

og/

Msa

g


0

0.2

0.4

0.6

0.8

1

1.2

1.4


Mh

og/

Msa

g


0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.2 0.4 0.6 0.8 1

Mstatic/(Mstatic)u

Mho

g/M

sag


0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.2 0.4 0.6 0.8 1

Mstatic/(Mstatic)u

Mho

g/M

sag

test.theo(plated)theo(unplated)0

0.2

0.4

0.6

0.8

1

1.2

1.4


Mho

g/M

sag


0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.2 0.4 0.6 0.8 1

Mstatic/(Mstatic)u

Mho

g/M

sag


0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.2 0.4 0.6 0.8 1

Mstatic/(Mstatic)u

Mho

g/M

sag

testtheo.(plated)theo.(unplated) Beam SS1

A

plate debond

Beam SS2

A

plate debond

Beam SS3

shear failure

A

Beam SF1

A

plate debond

Beam SF2

A

plate debond

Beam SF3

A

plate debond

Beam SF4

A

plate debond

Figure 6.24 Linear EI approach: Mhog/Msag of EB test specimens


- 560 -

0

5

10

15

20

25

30

0 10 20 30 40 50 60 70 80 90load (kN)

Mho

g (k

Nm

)

testtheo(plated)theo(plated)

0

5

10

15

20

25

30

35

40

0 20 40 60 80load (kN)

Mh

og

(kN

m)


0

5

10

15

20

25

0 20 40 60 80load (kN)

Mho

g (k

Nm

)


0

5

10

15

20

25

0 20 40 60 80load (kN)

Mho

g (k

Nm

)

testtheo(plated)theo(plated)

0

5

10

15

20

25

30

35

40

0 20 40 60 80applied load (kN)

Mho

g (k

Nm

)


0

5

10

15

20

25

30

0 10 20 30 40 50 60 70

applied load (kN)

Mho

g (k

Nm

)


0

5

10

15

20

25

30

0 20 40 60 80

applied load P (kN)

Mho

g (k

Nm

)


Beam SS1

elastic

plate debond Beam SS2

elastic

plate debond

Beam SS3shear failure

elastic

Beam SF1

elastic

plate debond

Beam SF2 elastic

plate debondBeam SF3

elastic

plate debond

Beam SF4

elastic

plate debond

Figure 6.25 Linear EI approach: Mhog of EB test specimens


- 561 -

Also included in Figure 6.23, Figure 6.24 and Figure 6.25 are the analyses of the beams without EB

plates. In general, good correlations between experimental and theoretical results were observed

throughout the loading of the beams, especially for the steel plated beams i.e. ‘SS’ test series. Note

that in the journal paper presented in Section 6.5.1, the steel plated beams were analysed using the

ultimate plate stress fp.ult, while the analyses shown in Figure 6.23-Figure 6.25 were carried out based

on the proof yield stress, which were found to give even better correlations with the test results.

At initial loading, the beams were still settling which resulted in greater differences between test and

theoretical results as can be seen from Figure 6.23-Figure 6.25. However upon further loading, better

correlations were observed. For specimens where significant debonding occurred, such as the ‘SF’

series beams with FRP plates, large differences between test and theoretical results were observed

due to the assumption of full interaction M/χ when evaluating for the secant stiffness EIsec in the

analyses. In fact, it can be seen from Figure 6.23-Figure 6.25 that as debonding becomes severe, the

moment distribution behaviour of the beam tends towards that of the unplated beam. After the plates

debonded, good correlation was observed between test results and the theoretical results obtained for

the same unplated beam. This suggests that the flexural rigidity approach can also be applied to

unplated RC beams.

The standard deviations SD and means for the %MR, Mhog/Msag, and Mhog of the specimens are given

in Table 6.3, where the ratios between the experimental and the theoretical results of the beams after

settling were considered. Compared to the simplified EI approach (Table 6.1), the linear EI approach

is tends to be more accurate.


- 562 -

Table 6.3 Linear EI approach: experimental vs test results for EB beams


SD 0.205 0.028 0.040

mean 1.509 0.948 0.925 SS1

SD/mean 0.136 0.030 0.044

SD 0.139 0.174 0.041

mean 1.001 0.921 0.965 SS2

SD/mean 0.139 0.189 0.042

SD 0.047 0.185 0.061

mean 1.257 0.694 0.954 SS3

SD/mean 0.037 0.267 0.064

SD 0.153 0.053 0.073

mean 1.122 0.958 0.944 SF1

SD/mean 0.136 0.055 0.077

SD 0.202 0.070 0.096

mean 1.072 0.975 0.968 SF2

SD/mean 0.188 0.072 0.099

SD 0.072 0.021 0.028

mean 1.122 0.965 0.952 SF3

SD/mean 0.064 0.021 0.030

SD 0.119 0.050 0.065

mean 1.137 0.947 0.931 SF4

SD/mean 0.105 0.053 0.069

SD 0.134 0.083 0.058

mean 1.174 0.915 0.948 Average

of all tests

SD/mean 0.115 0.098 0.061

6.5.2.2.2 TEST SERIES ‘NS’ AND ‘NB’ (SPECIMENS WITH NSM STRIPS)

In this section, the linear EI approach was applied to the nine NSM test specimens, ‘NS’ and ‘NB’

series, reported in Chapter 5. Analyses were performed at each load increment, where for each guess

of Mhog, the secant stiffness EIsec was determined based on full interaction moment/curvature

relationship. The analysis results are shown in the following figures, where Figure 6.26 and Figure

6.29 illustrate the variation of total percentage of moment redistribution %MRtot calculated using

Equation 5.1 for the ‘NS’ and ‘NB’ series respectively. Figure 6.27 and Figure 6.30 shows the variation

of the maximum hogging moment in the beam Mhog as a proportion of the maximum sagging moment


- 563 -

in the beam Msag for the ‘NS’ and ‘NB’ series respectively, where line A represents the elastic analysis

for constant EI. Figure 6.28 and Figure 6.31 shows the variation of the maximum hogging Mhog

moments as the applied loads P increased for the ‘NS’ and ‘NB’ series respectively, where the straight

line represents the elastic analysis for constant EI. Also included in Figure 6.26-Figure 6.31 are the

analyses of the beams without NSM plates.

-10

0

10

20

30

40

50

60

0 0.2 0.4 0.6 0.8 1

Mstatic/(Mstatic)u

%M

Rto

t


-505

1015202530354045


%M

Rto

t


-10

0

10

20

30

40

50

0 0.2 0.4 0.6 0.8 1

Mstatic/(Mstatic)u%

MR

tot


-5

0

5

10

15

20

25

30

35

40

45


%M

Rto

t


-10

0

10

20

30

40

50

0 0.2 0.4 0.6 0.8 1

Mstatic/(Mstatic)u

%M

Rto

t


-10

0

10

20

30

40

50

0 0.2 0.4 0.6 0.8 1

Mstatic/(Mstatic)u

%M

Rto

t


Beam NS_F4

plate debond

Beam NS_F1

shear failurein sag

Beam NS_F2



Beam NS_F3


Beam NS_S1

shearfailure in sag

Beam NS_S2

Figure 6.26 Linear EI approach: Percentage moment redistribution of ‘NS’ test series


- 564 -

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.2 0.4 0.6 0.8 1

Mstatic/(Mstatic)u

Mho

g/M

sag


0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.2 0.4 0.6 0.8 1

Mstatic/(Mstatic)u

Mho

g/M

sag


0

0.2

0.4

0.6

0.8

1

1.2

1.4


Mho

g/M

sag


0

0.2

0.4

0.6

0.8

1

1.2

1.4


Mho

g/M

sag


0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.2 0.4 0.6 0.8 1

Mstatic/(Mstatic)u

Mho

g/M

sag


0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.2 0.4 0.6 0.8 1

Mstatic/(Mstatic)u

Mho

g/M

sag


Beam NS_F1

A

shear failurein sag

A

A

Beam NS_F4

A

plate debond

A

Beam NS_F2



Beam NS_F3

Beam NS_S1


A

Beam NS_S2shear

failure in sag

Figure 6.27 Linear EI approach: Mhog/Msag of ‘NS’ test series


- 565 -

0

5

10

15

20

25

30

0 10 20 30 40 50 60 70load (kN)

Mho

g (k

Nm

)


0

5

10

15

20

25

30

0 10 20 30 40 50 60 70load (kN)

Mho

g (k

Nm

)


0

5

10

15

20

25

30

0 10 20 30 40 50 60 70load (kN)

Mho

g(kN

m)


0

5

10

15

20

25

30

0 10 20 30 40 50 60 70load (kN)

Mho

g (k

Nm

)


0

5

10

15

20

25

30

0 10 20 30 40 50 60 70load (kN)

Mho

g (k

Nm

)


0

5

10

15

20

25

30

0 20 40 60load (kN)

Mho

g (k

Nm

)


elastic

Beam NS_F4

elastic

plate debond

Beam NS_S1

elastic

Beam NS_F1

shear failurein sag

elastic

Beam NS_F2


elastic


Beam NS_F3


Beam NS_S2

elastic

shearfailure in sag

Figure 6.28 Linear EI approach: Mhog of ‘NS’ test series


- 566 -

-10

0

10

20

30

40

50

60

0 0.2 0.4 0.6 0.8 1 1.2Mstatic/(Mstatic)u

%M

Rto

t


-10

0

10

20

30

40

50

60

0 0.2 0.4 0.6 0.8 1 1.2

Mstatic/(Mstatic)u

%M

Rto

t


-10

0

10

20

30

40

50

60

0 0.2 0.4 0.6 0.8 1 1.2

Mstatic/(Mstatic)u

%M

Rto

t



debonding failure

Beam NB_F3


debonding failure

Figure 6.29 Linear EI approach: Percentage moment redistribution of ‘NB’ test series

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.2 0.4 0.6 0.8 1 1.2

Mstatic/(Mstatic)u

Mho

g/M

sag


0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.2 0.4 0.6 0.8 1 1.2

Mstatic/(Mstatic)u

Mho

g/M

sag

testtheo(plated)theo(unplated)0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.2 0.4 0.6 0.8 1 1.2

Mstatic/(Mstatic)u

Mho

g/M

sag


Beam NB_F1

A

debonding failure

A

A

Beam NB_F2

Beam NB_F3


debonding failure

Figure 6.30 Linear EI approach: Mhog/Msag of ‘NB’ test series


- 567 -

0

20

40

60

80

100

120

0 50 100 150 200 250load (kN)

Mho

g (k

Nm

)


0

20

40

60

80

100

120

0 50 100 150 200 250load (kN)

Mho

g (k

Nm

)


0

20

40

60

80

100

0 50 100 150 200 250load (kN)

Mho

g (k

Nm

)



debonding failure

Beam NB_F3


elastic elastic

debonding failure

elastic

Figure 6.31 Linear EI approach: Mhog of ‘NB’ test series

Good correlation was found between the experimental and theoretical results as shown in Figure 6.26

to Figure 6.31. The discrepancies between test and theoretical results at initial loading are due to the

beams still settling. Upon further loading, after the beam has settled, better correlations were

observed. Table 6.4 shows the standard deviations SD and means for the %MR, Mhog/Msag, and Mhog

of the specimens, where the ratios between the experimental and the theoretical results of the beams

after it has settled were considered i.e. the large discrepancies that occurred at low loads, as shown in

Figure 6.26 to Figure 6.31, are neglected. Compared with the simplified EI approach (Table 6.2), the

linear EI approach is tends to be more accurate.


- 568 -

Table 6.4 Linear EI approach: experimental vs test results for NSM beams


SD 0.124 0.031 0.027

mean 1.051 0.960 0.992 NS_F1

SD/mean 0.118 0.033 0.027

SD 0.061 0.016 0.023

mean 1.057 0.982 0.976 NS_F2

SD/mean 0.058 0.017 0.023

SD 0.066 0.033 0.045

mean 0.910 1.043 1.059 NS_F3

SD/mean 0.073 0.032 0.043

SD 0.052 0.011 0.014

mean 1.243 0.927 0.901 NS_F4

SD/mean 0.042 0.011 0.016

SD 0.106 0.023 0.027

mean 1.573 0.857 0.808 NS_S1

SD/mean 0.067 0.026 0.033

SD 0.089 0.019 0.026

mean 1.102 0.975 0.965 NS_S2

SD/mean 0.081 0.019 0.027

SD 0.058 0.052 0.067

mean 0.915 1.069 1.089 NB_F1

SD/mean 0.063 0.049 0.062

SD 0.024 0.015 0.020

mean 0.904 1.069 1.090 NB_F2

SD/mean 0.026 0.014 0.018

SD 0.024 0.051 0.023

mean 0.911 1.015 1.074 NB_F3

SD/mean 0.027 0.050 0.021

SD 0.067 0.028 0.030

mean 1.074 0.989 0.995 Average

of all tests

SD/mean 0.062 0.028 0.030


- 569 -

6.6 SUMMARY

Through the literature review performed, it has been shown that the commonly used plastic hinge

approach for modelling the moment redistribution behaviour of RC beams cannot be applied to plated

structures due to the brittle behaviour of these retrofitted structures as a result of premature

debonding. Hence two approaches, the simplified flexural rigidity approach and the linear flexural

rigidity approach, were developed in this research to determine the moment redistribution of RC

beams with externally bonded and near surface mounted plates as well as for unplated RC beams.

Both methods have been validated with seven continuous beams with externally bonded CFRP or

steel plates and nine continuous beams with near surface mounted CFRP or steel strips, and all

showed good correlations with the test results. Unlike the classical hinge approach which requires

concrete crushing to occur, the proposed model can be applied to beams with any failure mode, and,

hence, allows for the premature debonding failure that often occurs in externally plated structures.

Using the simplified EI approach, parametric studies were performed to study the effects of different

plating materials and positions on the moment redistribution behaviour of plated beams. It was found

through the studies that:

• Substantial amounts of moment redistribution can occur in steel plated sections if designed with

care. However, CFRP plated sections show a limited ability to redistribute moment at their

maximum strain.

• Steel plated beams can be designed to yield prior to IC debonding, although IC debonding does

eventually occur. A parametric study of this form of plating showed that the percentage moment

redistribution can vary from 0% to 45%. Therefore, there is the potential for substantial moment

redistribution. However, if designed poorly, there may be no moment redistribution.

• A parametric study of FRP plated beams showed that the percentage moment redistribution can

vary from 0% to 20%. Hence, there is the potential for a reasonable amount of moment

redistribution but this is mainly restricted to glass FRP plates.

• Furthermore, moment redistribution in FRP plated beams only occurs if IC debonding occurs after

the tension reinforcing bars have yielded.

• The concept of negative moment redistribution has been introduced where it was shown that

poorly designed beams can debond prematurely in a region, that is before the other region has

achieved its design moment capacity.

• It is suggested that moment redistribution should be considered in design and that simply allowing

for no moment redistribution may not always be a safe assumption.


- 570 -

6.7 REFERENCES

AS 3600. (1994). Australian Concrete Structures Standard. Standards Australia, Sydney, Australia.

Ashour, A.F., El-Refaie, S.A., and Garrity, S.W. (2004). “Flexural strengthening of RC continuous beams using CFRP laminates”. Cement & Concrete Composites, 26, 765-775.

Barnard, P. R. (1964). “The Collapse of Reinforced Concrete Beams”. Proceedings of the International Symposium on the Flexural mechanics of Reinforced Concrete, Miami, Fla.

Barnard, P. R., and Johnson, R.P. (1965). “Plastic behaviour of continuous composite beams”. Proceedings of the International Symposium on the Flexural mechanics of Reinforced Concrete, Miami, Fla, 501-511.

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

BS 8110. (1995). Structural use of concrete – Part 1. British Standards Institution.

CAN-A23.2. (1994). Code for the design of concrete structures for buildings. Canadian Standards Association.

CEB-FIP (1998). Ductility of Reinforced Concrete Structures-synthesis report and individual contributions. Comité Euro-International du Béton, Switzerland.

CEB-FIP. (1990). Model code for concrete structures – Europe. International System of Unified Standard Codes of Practise for Structures.

Concrete Society Committee (2000). Design guidance for strengthening concrete structures using fibre composite materials, Concrete Society Technical Report no. 55, The Concrete Society, UK.

Darvall, P.L., and Mendis, P.A. (1985). “Elastic-plastic-softening analysis of plane frames”. Journal of Structural Engineering, ASCE, 111(4), 871-888

DIN1045. (1997). Deutsches Institut fur Normung. German Intstitute of Standards. Germany

El-Refaie, S. A., Ashour, A.F., and Garrity, S.W. (2003). “Sagging and Hogging Strengthening of Continuous Reinforced Concrete Beams Using Carbon Fiber-Reinforced Polymer Sheets.” ACI Structural Journal, 100(4): 446-453.

fib Task Group 9.3 (2001). Externally bonded FRP reinforcement for RC structures, Technical report, International Federation for Structural Concrete, Lausanne.

Hibbeler, R.C. (1999). Structural Analysis. 4th edition. Prentice-Hall, USA.

Mukhopadhyaya, P., Swamy, N. and Lynsdale, C., (1998). “Optimizing structural response of beams strengthened with GFRP plates”, Journal of Composites for Construction, ASCE, 2(2) 87-95.

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

Oehlers, D.J., Ju, G., Liu, I., and R. Seracino. (2004). “Moment redistribution in continuous plated RC flexural members. Part 1: neutral axis depth approach and tests.” Engineering structures, 26, 2197-2207.

Oehlers, D.J., Liu, I.S.T., and Seracino, R. (2005). “A generic design approach for EB and NSM longitudinally plated RC beams.” International Journal of Construction and Building Materials, accepted for publication

Wood, R.H. (1968). “Some controversial and curious developments in the plastic theory of structures”. Engineering Plasticity, Cambridge University Press, 665-691.


- 571 -

6.8 NOTATIONS

The following symbols are used in this section: %MRexp experimental percentage total moment redistribution %MRtheo theoretical percentage total moment redistribution %MRtot percentage of total moment redistribution (dy/dx)static slope due to static deformation (dy/dx)support slope due to support moment deformation (EI)elas flexural rigidity based on elastic analysis (EI)min minimum secant flexural rigidity (Mhog)u ultimate maximum hogging moment (Msag)u ultimate maximum sagging moment (Mstatic)el static moment based on elastic analysis a shear span d beam effective depth dn neutral axis depth dy/dx slope EI flexural rigidity EIcr flexural rigidity of fully cracked section EIeq equivalent flexural rigidity EIg flexural rigidity of uncracked section EIhog flexural rigidity of the hogging region EIsag flexural rigidity of the sagging region EIsec secant flexural rigidity fp.ult ultimate plate stress h beam depth ku neutral axis depth factor L span Lhinge hinge length M moment Mhog moment in hogging region at maximum moment position Msag moment in sagging region at maximum moment position Mstatic static moment Mu moment capacity P load Papplied applied load R reaction force V’hog shear at the conjugate support w uniformly distributed load x distance between the fixed end to the point of contraflexure y1,y2,y3 co-ordinates z1,z2,z3 co-ordinates ∆Mstatic change in static moment α angular flexibility coefficient; calibration coefficient αhog angular flexibility coefficient of the hogging region

χ curvature

χeq equivalent curvature

χy curvature at yield

χu ultimate curvature ε strain εc concrete strain εtfp maximum tensile strain at the tension face εy yielding strain εp.db plate debonding strain


- 572 -

εp.fr plate fracture strain θ rotation θcap rotation capacity θhog rotation in the hogging region θreq required rotation capacity The following acronyms are used in this section: CFRP carbon fibre reinforced polymer conc concrete EB externally bonded FRP fibre reinforced polymer hog hogging region IC Intermediate crack MR moment redistribution NSM near surface mounted poc point of contraflexure RC reinforced concrete sag sagging region SD standard deviation The following subscripts are used in this section: el elastic exp experimental hog hogging region sag sagging region sec secant theo theoretical u ultimate

- 573 -

CHAPTER

7 CONCLUSIONS AND RECOMMENDATIONS

CONTENTS

7 CONCLUSIONS AND RECOMMENDATIONS 573

7.1 Introduction 574

7.2 Investigations on Intermediate Crack Debonding 574

7.3 Investigations on Critical Diagonal Crack Debonding 575

7.4 Investigations on Moment Redistribution 576

7.5 Summary 579

7.6 Recommendations for Future Research 579

7.7 Reference 580

7.8 Notations 580

Intermediate Crack Debonding of Plated RC Beams Conclusions and Recommendations

- 574 -

7.1 INTRODUCTION

Through the literature review presented in this thesis, it has clearly been shown that there is a lack of

existing research on the behaviour of continuous reinforced concrete (RC) beams retrofitted with

externally bonded (EB), or near surface mounted (NSM) plates. Extensive studies were hence

performed in this research to allow a better understanding of this retrofitting technique. These include

numerical simulations of intermediate crack (IC) debonding using the proposed partial interaction

model; development of new design approaches and guidelines for predicting critical diagonal crack

(CDC) debonding; and experimental investigations on the moment redistribution behaviour of beams

retrofitted with EB and NSM plates. This has led to the development of new design approaches. This

chapter summarises the findings of the studies, listing the main conclusions and suggestions for future

research.

7.2 INVESTIGATIONS ON INTERMEDIATE CRACK DEBONDING

Intermediate crack debonding is one of the most commonly occurring debonding modes as it is very

difficult to prevent. It is caused by the formation of flexural or flexural shear cracks intercepting the

plate which induce slip at the plate/concrete interface. Therefore, IC debonding is likely to occur in the

hinge of a continuous beam, affecting the rotation behaviour or rotation capacity of the beam, and

hence the ability to redistribute moment. Through this research, it was discovered that the IC

debonding mechanism of plated members is similar to the sliding action at the interface of

steel/concrete composite beams and at the reinforcement/concrete interface in unplated RC beams.

Therefore, it was recognised that the partial interaction approach originally developed for composite

beams can also be applied to both plated and unplated structures. This led to the following findings

from the investigations on IC debonding:

• A numerical partial-interaction model has been developed that simulates the sliding action at the

reinforcement/concrete interfaces in both plated and unplated reinforced concrete beams, as well

as in both plated and unplated tensile specimens. The model is based on the following three

boundary conditions: the slip-strain and slip tend to zero at the full-interaction boundary of the

hinge; the boundary stress resultants at flexural cracks; and the rigid body rotation of the crack

faces of flexural cracks that induce a linear variation in crack width and slip difference. Through

the use of these boundary conditions, the model: can cope with any number of reinforcing layers,


- 575 -

as well as plated beams with and without reinforcing bars; automatically predicts the occurrence

of flexural cracks; distinguishes between disturbed and undisturbed regions; and allows for the

interaction between all of the discrete blocks along the length of the beam.

• From the numerical simulations performed on unplated reinforced concrete beams and based on

the proposed model, the following have been illustrated: the different mechanics associated with

the behaviours of ductile and brittle hinges; the sequence of the formation of primary and

secondary flexural cracks as well as their closure; the sliding of the reinforcement through the

concrete blocks as well as the movement of the zero slip position; how the curvature in the

concrete block is not at its greatest at a flexural crack; the large slip-strains at flexural cracks; the

gradual widening of the disturbed regions; the gradual spread of the partial-interaction hinge; and

between flexural cracks the position of zero slip section occurs at a point with finite slip-strain and

is hence not at full interaction.

• Numerical simulations on plated RC beams using the proposed model have illustrated: the

different debonding behaviours associated with the behaviours of ductile and brittle hinges; the

sequence of the formation of primary and secondary flexural cracks as well as their closure; and

the gradual widening of the disturbed regions.

• The model developed has been used to illustrate the complex interaction between the debonding

strain and the flexural crack distribution through a series of parametric studies. It is shown that a

lower bound to the debonding strain is a beam with single flexural crack (which is equivalent to a

pull test) and that the debonding strain can be significantly increased when there are increasing

numbers of flexural cracks. Substantial increase in debonding strain can also occur as the crack

spacing reduces, i.e. as secondary cracks form in between existing flexural cracks, which shows

the importance of locating cracks and the problem with existing models that require predefined

crack positions. Furthermore the rate of change of moment, that is the vertical shear force, can

significantly affect the debonding strain.

7.3 INVESTIGATIONS ON CRITICAL DIAGONAL CRACK DEBONDING

When a critical diagonal crack (CDC) forms, the sliding and rotation action that occurs across the

crack causes the plate to separate from the beam by a debonding crack which propagates from the

root of the diagonal crack to the plate end. As this critical diagonal crack intercepts the plate, it

induces an axial force in the plate which is dependent on the IC debonding resistance at the

plate/concrete interface. Through investigations on CDC debonding carried out in this research, it was


- 576 -

found that in cases of CDC debonding, externally bonded plates can be idealized as passive

prestressing tendons. That is the IC debonding force is analogous to the force in prestressing

tendons. Hence, a passive prestress approach was developed in this research to quantify the CDC

debonding of plated RC beams, with the following outcomes:

• A generic design procedure for CDC debonding has been developed that can be used for any

type of plate material and for plates bonded to any surface of an RC beam.

• A data base of seventy-seven plated beam tests has been brought together in which CDC

debonding was the primary mode of debonding. This has been used to validate the mathematical

models developed in this paper and which can be used in future research on CDC debonding.

• It has been shown that the concrete shear capacity of a plated beam, and hence its resistance to

CDC debonding, is analogous to the shear capacity of a prestressed beam.

• A mathematical model has been developed based on the passive prestress induced by the plate

and has been shown to correlate well with a very wide range of tests in which the plates were

bonded to any surface of the beam. The mathematical model is shown to give a lower bound and,

hence, can be used with safety.

• The implication of this research is that the shear resistance of prestressed beams in national

standards can be used to predict the CDC debonding resistance.

• It has also been shown that there is no interaction between CDC debonding and PE debonding so

that a plated beam can be designed for each debonding mechanism independently.

• The research has shown that the axial force in an FRP plate can be considered as an additional

prestress, so that the concrete shear capacity of prestressed beams can be used to design

against CDC debonding. The implication of this research is that design rules in national standards

for predicting the concrete component of the shear capacity of prestressed beams can be used to

predict CDC debonding. However, it is recommended that only prestress code models for the

concrete shear capacity that can be applied to both prestressed and unprestressed beams, such

as the Eurocode model, should be used.

7.4 INVESTIGATIONS ON MOMENT REDISTRIBUTION

From a consideration of the moment redistribution allowances in international standards that are

based on the neutral axis parameter ku, it is suggested that the ku approach should only be used for

metal plated beams in which concrete crushing precedes debonding. However, through the


- 577 -

experimental investigation performed in this research on seven beams plated with externally bonded

steel and FRP plates, it has been shown that substantial amounts of moment redistribution can occur.

For carbon FRP EB plated beams this ranged from 28% to 35% and for steel EB plated beams from

22% to 48%.

In this research, moment redistribution (MR) is considered to consist of two components: (1) the

elastic MR due to the difference between the elastic flexural rigidities of the hogging and sagging

regions; and (2) the plastic MR due to the secant stiffness EIsec that follow the horizontal or rising

moment-curvature plateau after the yielding of the reinforcing bars and/or plates. Although, the EB

plated test specimens showed significant amounts of moment redistribution, the majority of it was due

to elastic MR. This led to further experimental investigations on continuous NSM beams, which were

expected to achieve greater moment redistributions than externally plated beams, as they are less

susceptible to premature debonding failure and, hence, result in a more ductile system. Nine almost

full scale two-span continuous RC beams were retrofitted with NSM steel or CFRP plates on either the

tension face or sides. Through these tests, the following observations were made:

• The formation of the herringbone cracks clearly showed the directions of shear flow as the beam

is loaded, which has been studied through numerical simulation.

• The tests also showed the opposing shear flows between flexural cracks when the beam is first

loaded and then the reversal in direction of parts of the shear flow to accommodate the

longitudinal pulling out of the plate.

• The NSM steel plates achieved strains of up to 0.042 which allowed 39% moment redistribution

and the NSM CFRP plates achieved strains of up to 0.014 which allowed 32% moment

redistribution.

• Unlike externally bonded plated beams, these tests showed that beams retrofitted with NSM

plates have substantial ductility so that NSM plated beams can be used to retrofit RC structures

that require ductility, which should significantly expand the use of retrofitting by plating.

From the experimental studies performed, it has clearly been shown that significant amounts of

moment redistribution can occur in plated structures, and by precluding moment redistribution as

suggested by the current design guidelines, this severely restricts the use of this retrofitting technique,

as efficient design of RC structures relies on member ductility. Therefore theoretical investigations

were carried out in this research, aimed to quantify the moment redistribution of plated structures.

Through the literature review performed, it has been shown that the hinge approach and the neutral


- 578 -

axis parameter used in the national codes for RC beams relies on a horizontal moment-curvature

plateau and concrete crushing failure to occur, which is unlikely to occur in retrofitted structures. This

led to the development of two alternative moment redistribution approaches based on the variation in

flexural rigidities (EI) along the beams. The two EI approaches proposed are:

1. Simplified EI approach, which assumes that the flexural rigidity between the hogging (EIhog) and

the sagging (EIsag) regions varies, while the stiffness remains constant within each region.

2. Linear EI approach, which is based on the relative variation in stiffness between the hogging and

sagging regions, where it is assumed that the flexural rigidity varies linearly within each region.

Both the simplified and linear EI approaches have been verified with seven continuous beams with

externally bonded CFRP or steel plates and nine continuous beams with near surface mounted CFRP

or steel strips, and all showed good correlations with the test results. The mathematical models

developed can determine the moment redistribution at any stage of loading and for any failure

mechanism. Unlike the classical hinge approach which requires concrete crushing to occur, the

proposed models can be applied to beams with any failure mode, and, hence, allows for the

premature debonding failure that often occurs in externally plated structures. It was also found from

the analyses of experimental results that, NSM plating systems with steel or CFRP strips are much

more ductile than the EB plating systems, allowing significant amounts of moment redistribution to be

achieved which should now allow structures to be plated for ductility as well as strength

Through the parametric studies performed on continuous beams with EB plates using the simplified EI

approach, it was found that:

• Steel plated beams can be designed to yield prior to IC debonding, although IC debonding does

eventually occur. A parametric study of this form of plating showed that the percentage moment

redistribution can vary from 0% to 45%. Therefore, there is the potential for substantial moment

redistribution. However, if designed poorly, there may be no moment redistribution.

• A parametric study of FRP plated beams showed that the percentage moment redistribution can

vary from 0% to 20%. Hence, there is the potential for a reasonable amount of moment

redistribution but this is mainly restricted to glass FRP plates. Furthermore, moment redistribution

in FRP plated beams only occurs if IC debonding occurs after the tension reinforcing bars have

yielded.


- 579 -

• The concept of negative moment redistribution has been introduced where it was shown that

poorly designed beams can debond prematurely in a region, that is before the other region has

achieved its design moment capacity.

• Also illustrated is the importance of carefully choosing the plating technique to optimise the

increase in strength. It is suggested that moment redistribution should be considered in design

and that simply allowing for no moment redistribution may not always be a safe assumption.

7.5 SUMMARY

Through the investigations of the failure mechanisms of plated structures from this research, the

strength of the retrofitted section can now be determined with confidence as all the major debonding

mechanisms can be modelled. The proposed moment redistribution design approach will also allow

engineers to design ductile beams such that the current restriction on the use of retrofitted structures,

that is the exclusion of moment redistribution can be eliminated.

As most of the fundamental and unique failure modes of plated structures have now been identified

and quantified, this will eventually enable the development of generic design rules for all types of

adhesive plating. From the outcomes of this research, along with other existing studies on debonding

mechanisms, a generic design procedure has been proposed by Oehlers et al. 2005 for adhesively

bonding longitudinal plates to reinforced concrete beams and slabs. This procedure applies to both EB

and NSM plates which are bonded to prestressed or unprestressed beams, with all types of plating

materials, configurations and adhesion to any surface. It covers all major debonding mechanisms at

serviceability or ultimate limit states and quantifies the flexural and shear strengths as well as the

ductility associated with moment redistribution.

7.6 RECOMMENDATIONS FOR FUTURE RESEARCH

• The ductility of a reinforced concrete beam is dependent on its rotational behaviour, which

controls the ability of the member to absorb energy and redistribute moment and which in turn

controls the collapse of reinforced concrete frames. This rotational behaviour at the hinges is

affected by cracking, and hence, by the debonding behaviour of plated beams. Therefore, by

using the partial interaction model proposed in this research, which simulates the local


- 580 -

deformation within the cracked region, one can perform further studies on the rotational behaviour

of plated hinges and its effects on the moment redistribution behaviour of continuous RC beams.

• In the theoretical studies carried out in this research on the moment redistribution of plated

members, the classical full interaction M/χ sectional analysis of a beam was assumed. Further

research can be carried out to incorporate partial interaction moment-curvature relationship into

the flexural rigidity approach when determining the moment redistribution of continuous members.

• The development of generic design rules for all types of adhesive plating.

• The development of a specific ductile FRP plating system for a hinge in an RC beam that can be

used in conjunction with the current brittle plating systems so that ductile plated structures can be

designed.

• The development of a generic beam partial interaction ductility mathematical model that quantifies

the plated member’s ductility, such as the ability to redistribute moment and to absorb energy, that

can be applied to plating systems of all ductilities.

• Development of a finite element formulation that incorporates the partial interaction model and the

flexural rigidity approach, so that the finite element analysis program, it can predict the ability of

plated frames to resist static and earthquake loads for the development of frame design

guidelines.

7.7 REFERENCE

Oehlers, D.J., Liu, I.S.T., and Seracino, R. (2005). “A generic design approach for EB and NSM longitudinally plated RC beams.” International Journal of Construction and Building Materials, accepted for publication

7.8 NOTATIONS

The following symbols are used in this chapter: EI flexural rigidity EIcracked stiffness of a full cracked section EIg elastic stiffness EIsec secant stiffness M moment

χ curvature The following acronyms are used in this chapter: CDC critical diagonal crack EB externally bonded


- 581 -

FRP fibre reinforced polymer IC Intermediate crack MR moment redistribution NSM near surface mounted RC reinforced concrete


- 582 -

- 583 -

APPENDIX A.1.

PARTIAL INTERACTION ANALYSIS OF

PLATED TENSILE SPECIMENS

The fortran code developed in this research for the partial interaction analysis of plated tensile

specimens with any number of reinforcing layers (Chapter 2) is summarized in the following context,

where a flow chart explaining the steps involved in modelling is presented.

Intermediate Crack Debonding of Plated RC Beams Appendix A.1.

- 584 -

A.1.1. ANALYSIS PROCEDURES

A summary of the analysis procedures is presented in the flow chart in Figure A.1. A detailed

explanation of the analysis procedures can be found in Chapter 2. It is worth noting that Figure A.1 is

based on a fortran code written for analysing specimens under load control, where the tensile force

applied is fixed. For analyses under slip or crack width control, the code can be modified easily by

fixing the plate slip at crack 1 and iterating the applied load to obtain a solution. In Figure A.1, ‘Loop’

refers to ‘Do’ or ‘Do while’ functions used in the fortran code where automatic iterations are made to

find a solution.

A.1.2. NOTATIONS

c crack c ds/dxb slip-strain at bar/concrete interface ds/dxp slip-strain at plate/concrete interface e increment size used in segmental analysis P load sb slip at bar/concrete interface sp slip at plate/concrete interface Xcrack(c) the distance of crack c from crack 1 εct concrete cracking strain


- 585 -

Figure A.1. Flow chart for tensile specimen program code

Fix applied load P

Guess bar force at crack 1

Input known crack positions Xcrack(c)

Guess plate slip sp at crack c-1

Segmental analysis at fixed increments e to crack c

Sectional analysis at crack c-1

Check εct=0 at crack c

Proceed to next crack c=c+1

Repeat analysis for adjacent uncracked segment between cracks

Next crack position known Xcrack(c)>0

Yes εct=0

Analysis between cracks c-1 & c

If εct≠0Loop 3

Loop 2

Next crack position unknown i.e. Xcrack(c)=0

Guess plate slip sp at crack c-1

Next crack position

unknown(exit Loop 2)

Segmental analysis at increments e until slip=0 at each interface

sectional analysis at crack c-1 ds/dxp≠0at sp=0Loop 5

Check at plate slip-strain ds/dxp=0 at sp=0 plate/concrete interface

Check ds/dxb=sb=0 at bar/concrete interface

sp=0 at ds/dxp=0

ds/dxb=0 at sb=0

Check for cracking

If no new cracks

Solution

If new cracks formed

ds/dxb≠0 at sb=0, input new guess of bar force at crack 1

& reanalyse specimen

Loop 1


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APPENDIX A.2.

CONFERENCE PAPER ON INITIAL PARTIAL

INTERACTION MODEL

In the following appendix, a conference paper is presented where the initial partial interaction model

(Chapter 2) developed in this research was used to perform parametric studies to observe the effects

of variations in crack spacing and rate of change of moment on the intermediate crack debonding

behaviour of plated flexural members.

Conference Paper

Liu, I.S.T., Oehlers, D.J., and Seracino, R. (2004). “Parametric study of intermediate crack (IC)

debonding on adhesively plated beams.” Proc., the 2nd International Conference on FRP Composites

in Civil Engineering, CICE, Adelaide, Australia, 515-522

Liu, I.S.T., Oehlers, D.J., and Seracino, R.(2004) Parametric study of intermediate crack (IC) debonding on adhesively plated beams Proc., the 2nd International Conference on FRP Composites in Civil Engineering, CICE, Adelaide, Australia, pp. 515-527


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APPENDIX A.3.

PARTIAL INTERACTION ANALYSIS OF

PLATED AND UNPLATED RC BEAM

The fortran code developed in this research for the partial interaction analysis of plated and unplated

reinforced concrete beams with any number of reinforcing layers (Chapter 2) is summarized in the

following context, where a flow chart explaining the steps involved in modelling is presented.


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A.3.1. ANALYSIS PROCEDURES

A summary of the analysis procedures is presented in the flow chart in Figure A.2. A detailed

explanation of the analysis procedures can be found in Chapter 2. It is worth noting that Figure A.2 is

based on a fortran code written for analysing specimens under load control, where a concentrated

load P applied at crack 1 is fixed. For analyses under slip or crack width control, the code can be

modified easily by fixing the plate slip at crack 1 and iterating the applied load to obtain a solution. In

Figure A.2 ‘Loop’ refers to ‘Do’ or ‘Do while’ functions are used in the fortran code where automatic

iterations are made to find a solution.

A.3.2. NOTATIONS

(sp)r right crack face slip at plate/concrete interface c crack c ds/dxb slip-strain at bar/concrete interface ds/dxp slip-strain at plate/concrete interface e increment size used in segmental analysis Mcrack(c) moment at crack C P load sb slip at bar/concrete interface sp slip at plate/concrete interface Xcrack(c) the distance of crack c from crack 1 Xdisturb(c) length of disturbed region εb bar strain εct concrete cracking strain εp plate strain


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Figure A.2. Flow chart for plated and unplated RC beams program code

Input disturb region length Xdisturb if >0

Input known crack positions Xcrack(c)

Disturbed analysis at crack c-1

Undisturbed sectional analysis from Xdisurb(c-1) to crack c at increments e

Check Mcrack(c)=M next to crack c

Proceed to next crack c=c+1

Repeat analysis for adjacent uncracked segment between cracks

Next crack position known Xcrack(c)>0

M≠Mcrack

Analysis between cracks c-1 & c

M≠Mcrack

Loop B Loop A

Next crack position unknown

Next crack position

unknown (exit Loop A)

Fix crack height

Fix plate strain εp at crack 1

Fix bar strain εb at crack 1

ds/dxb≠sb

Loop X

Xdisturb increased

guess right crack face plate slip (sp)r at crack c-1

Disturbed region analysis Xdisurb(c-1)>0

Disturbed analysis at crack c Mcrack(c)

guess right crack face plate slip (sp)r at crack c-1

Disturbed analysis at crack c-1

Undisturbed sectional analysis from Xdisurb(c-1) at increments e until reach

boundary point where ds/dx=0 or s=0 at each interface

ds/dxp≠sp

Loop G Disturbed region analysis

Xdisurb(c-1)>0

Check at plate interface ds/dxp=sp=0

Xdisurb(c-1)=0

Xdisurb(c-1)=0

Check at bar interface ds/dxb=sb=0

ds/dxp=sp=0

Check if Xdisturb has extended

ds/dxb=sb=0

Check for new cracks No

Solution No new cracks

Yes

Yes


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- 601 -

APPENDIX A.4.

CONFERENCE PAPER ON MODIFIED PARTIAL

INTERACTION MODEL

In the following appendix, a conference paper is presented, where the modified partial interaction

model (Chapter 2) developed in this research was used to study the behaviour of intermediate crack

debonding.

Conference Paper

Liu, I.S.T., Oehlers, D.J., and Seracino, R. (2005). “FRP plated reinforced concrete hinges: Partial

interaction numerical simulation.” Proc., 3rd International Conference on Composites in Construction,

CCC, Lyon, France

Liu, I.S.T., Oehlers, D.J., and Seracino, R.(2005) FRP plated reinforced concrete hinges: partial interaction numerical simulation Proc., the 3rd International Conference on Composites in Construction, CCC, Lyon, France


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