Balanced Design and Section Ductility

8/10/2019 Balanced Design and Section Ductility

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BALANCED DESIGN AND SECTION DUCTILITY

Fig.1.Balanced design for singly reinforced rectangular section

Balanced design

At failure it is recommended that concrete crushes in the same time with yielding

of the reinforcement under ultimate applied load. So, the strain in the reinforcement

reaches its yield value at the same time as the strain in the extreme concrete fiber in

compression reaches ult , as is presented in Fig. 1. For computation the material factors

of safety are initially omitted. Than by similar triangles on the strain diagram

xd

Ef

x

syult

=

! where sys Ef != "1#

Assuming fy$ %&' (!mm), *ult$ '.''+ and -s$)'' '''(!mm), euation "1#becomes

&'+.'!

'')+.'''+,.'

)'''''!%&'''+,.'

=

=

=

dx

xdx

xdx

")#

This is true for all sections regardless of their shape. The precise amount of

reinforcement which gives a balanced design can be calculated by euilibrium of axialforces acting on the section.

The area of reinforcementfor which a design is balanced depends in the cross/

section being considered.

0n the case of euivalent rectangular stress bloc for rectangular section of Fig. 1euilibrium gives


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[ ]

#"11))

%&'#&'+.'"2.'#2,.'"

#2.'#""

)mmbdf

A

Abdf

fAxbf

cks

sck

ysck

=

=

=

"+#

0f the area of reinforcement provided is greater than that given by euation "+#,

the beam is over/reinforced, that is the reinforcement does not yield before the concretecrushes. Similarly, if the area of reinforcement provided is less than that given by

euation "+#, the beam is under/reinforced and the reinforcement will yield before the

concrete crushes. For failure only under reinforcedbeams are acceptable. To ensure that

-3) recommends that the ratio x!d does not exceed the values given inTable..............4lastic moment redistribution reuires ductility which is affected by the

degree to which the beam is under/reinforced. -ven without such redistribution, beams

should always be under/reinforced. "from the smallest values for x!d from the Table 1 itcan see that a small x implies large strain in the steel#

Table 1.

3oncrete cylinder

strength, fc"(!mm)#

5plastic moment redistribution

' 1' 1 )' ) +'

1) to + '.% '.%1 '.+6 '.++ '.)7 '.) '.)1

%' and over '.+ '.+1 '.)6 '.)+ '.17 '.1 '.11

The rectangular section from Fig. 1 has tension reinforcement only and isconsidered as singly reinforced section. 0f is considered x!d8'.&'+, the tension

reinforcement yields under ultimate loads. Assuming the use of simplified rectangular

stress bloc, the forces acting on the section are as in Fig.1. 9eintroducing materialfactors of safety, the moment capacity of the section is given by the compressive force "or

tensile force# multiplied by the lever arm, :

#%.'"%,+.'

#%.'"#2.'"

xdxbfM

xdbxf

zFM

ckult

c

ckcult

=

==

for ;$'.2 and


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For ensuring ductile failure, the allowable values for are calculated using the

limits on x!d, given in -3), "Table 1# Substituting for x!d in euation "#, results in the

maximum values for , given in Table 6.%Thus, the safe upper limit for ultimate moment capacity, regardless of

reinforcement provided, for singly reinforced rectangular section is given by)

bdfM ckult =here the appropriate value for is taen from Table 6.%.

DUCTILITY

Beams and slabs of reinforced concrete are designed to have certain ductility

under ultimate loads. This ductility ensures that the member is capable of undergoing a

certain amount of rotation after yielding of the tension steel reinforcement and beforecrushing of the concrete in compression. The ideali:ed moment curvature relationship

assumed for a member in bending can be seen in the graph between yand ultrepresents

the ductility.

For a beam with homogeneous cross/section the linear elastic relationship can be

expressed asEIkM = "%#

here $1!9 is nown as the curvature "9 is the radius of curvature#

The reinforced concrete beams are not homogeneous. 3oncrete and steel haveconsiderably different values for the elastic modulus. The section can be transformed in

an euivalent homogeneous concrete section and to calculate an euivalent second

moment of area "using the modular ratio#. hen the internal moment, C is very small,

the concrete is uncraced and the euivalent second moment of area is denoted 0u.3onsidering the elastic modulus for concrete as -c, euation "%# becomes

kIEM uc= "#

For a small moment, the section cracs, the euivalent second moment of area

drops to a much lower value, and euation "# becomeskIEM cc= "

here 0cis the euivalent second moment of area of the craced section. This

relationship is represented in Fig.).


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Fig. ).Coment!curvature relationship for reinforced concrete

For a properly designed reinforced concrete section, the steel yields before the

concrete crushes. This happens at an applied moment of C y, as illustrated in Fig. ).Assteel is a ductile material, the section too is ductile, and beyond the yield point the

curvature increases greatly for a relatively small increase in the applied moment.

3omplete failure of the section occurs when the concrete at the extreme fiber incompression crushes. The curvature at this stage is ult.

The moment curvature relationship of Fig. ) can be ideali:ed by the simplified

relationship from Fig. +.

Fig. +.0deali:ed moment!curvature relation ship for reinforced concrete

As it was shown a limitation on the neutral axis depth is the mechanism by whichductility is guaranteed. hen members in bending have this ductility, they have the

potential to continue to resist load beyond the time of initial yield.

4lastic moment redistribution is an approximate method by which the elasticbending moment diagram is adDusted to account for the ductile behavior of reinforced and

prestressed members in bending. The amount of redistribution allowed is dependent on


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the grade of concrete and the ductility characteristics of the reinforcement as well as the

neutral axis depth.

For example, the following limits are imposed on the ratio of the redistributedmoment to the moment before redistribution, E. For concrete grades with cylinder

compressive test strength of less than or eual to + (!mm)

d

x),.1%%.' + "6#

For concrete with stregths greater than + (!mm)

d

x),.1,&.' + "2#

here high/ductility steel reinforcement is used, E must be greater than or eualto '.6 "i.e. maximum of +' per cent redistribution#. here normal ductility steel is used, E

must be greater than or eual to '.2.

Also, the optimum amount of moment redistribution which should be carried outdepends greatly on the geometry and loading of the member.

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Balanced Design and Section Ductility