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8/12/2019 Design of Partially Prestressed Concrete Beams Based on the Cracking
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8/12/2019 Design of Partially Prestressed Concrete Beams Based on the Cracking
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the other hand the analysis and design procedures for partially pre-
stressed members are not well understood or accepted. This comes
as a result from the fact that in partially prestressed members the
section may be cracked or uncracked depending on the level of
loading as well as on the level of the prestressing.
Thus the main problem is the estimation of the crack width to
be expected under full service load for a given combination of pre-
stressed and conventional (non-prestressed) steel reinforcement.Henceforth the key requirement in the design of partially pre-
stressed members is the inverse question: given the desirable or
the allowable crack width of the examined concrete member to
estimate a combination of non-prestressed and prestressed rein-
forcement that ensures it.
Dilger and Suri[9]presented a method to directly calculate the
stress in the steel. The steel stress is calculated based on the
assumptions that the prestressing steel and the non-prestressing
steel are located close to each other and that the decompression
force acts at the level of the prestressed steel. A design table has
been proposed from which the steel stress at the level of the com-
bined centroid of both steels can be obtained.
Naaman and Siriaksorn[10] proposed a rational design proce-
dure for partially prestressed concrete beams based on satisfying
ultimate strength and serviceability requirements. The use of an
estimated parameter called Partial Prestressing Ratio (PPR) was
also addressed.
In present work a proposal for the estimation of the required
partial prestressing based mainly on the crack control of concrete
is presented. Current codes (ACI 318 and Eurocode 2) do not in-
clude a method or design procedure or even scarce clues about
the design of partially prestressed concrete elements. The pro-
posed procedure merely uses design values for the allowable crack
width and design formulas for the estimation of cracks from the
codes. Thus, the procedure described in this work although is not
included in the codes it uses considerations from the codes.
According to the proposed methodology, the stress of the non-
prestressed reinforcement is first estimated based on the allowable
crack width as it is stated by ACI 318 and Eurocode 2. Then thedepth of compression zone is derived by the solution of a proposed
cubic equation which has been formed for this purpose. Further the
required effective pre-strain of the prestressing steel and hence-
forth the required prestress force are calculated. Design charts
and numerical paradigms are also presented and commented.
2. The extent of partial prestressing
As mentioned before there is a need for a numerical parameter
that can adequately describe the extent of the partial prestressing.
This index should preferably take the value of zero for convention-
ally reinforced concrete and the value of one for fully prestressed
concrete. Several indices have been proposed to characterize theextent of partial prestressing.
As the most obvious expression for the degree of prestress, j,can be considered the ratio of the applied partial prestressing force,
Ppart, to the prestressing force,Pfull, that causes full prestress under
maximum load[11], i.e. zero stress at the extreme fibre of a con-
crete member: j =Ppart/Pfull. In case that the prestressing forceshave not the same centroid the degree of prestress should be de-
fined as the ratio j =MDEC/Mmax, where MDECis the moment thatproduces zero concrete stress at the extreme fibre when added
to the action of the partial prestress andMmaxis the maximum mo-
ment caused by the total service load.
The Partial Prestressing Ratio (PPR) allows a unified treatment
of the ultimate flexural capacity for reinforced, fully prestressed
and partially prestressed concrete [6,10] and therefore is usually
used. It is defined as the ratio of the nominal moment resistance
provided by the prestressing, Mu,p, to the total nominal moment
resistance of the memberMu,p+s:
PPR Mu;pMu;ps
Apfp dp
a2
Apfp dp
a2
Asfs ds
a2
1where
Ap: cross-sectional areas of prestressed steel,
fp: tensile stress of prestressing tendons at nominal ultimate
strength,
As: cross-sectional areas of non-prestressed tensile steel,
fs: tensile stress of non-prestressed tensile steel bars at nominal
ultimate strength that usually equals to the yield stress,
a: depth of equivalent compressive stress block of concrete atultimate,
dp: distance from extreme compression fibre to centroid of pre-
stressed steel,
ds: distance from extreme compression fibre to centroid of non-
prestressed tensile steel and for the case thatds dpthe above
relationship(1) is simplified as[10]:
PPR Apfp
ApfpAsfs1-S
InFig. 1, typical behaviour curves (moment versus curvature) of
reinforced concrete (RC), fully prestressed concrete (FPC) and par-
tially prestressed concrete (PPC) flexural members with equal ulti-
mate capacity level are demonstrated (see also [10]). Fully
prestressed concrete elements do not crack under the total loading
(green area inFig. 1) and expected to exhibit brittle failure (ductil-
ity lu 1) whereas partially prestressed concrete elements are al-lowed to exhibit limited cracking under dead load (red area in
Fig. 1) and expected to exhibit a less brittle failure depending on
the partial prestressing ratio. Typical flexural reinforced concrete
elements are designed to be cracked even under the self-weightloading (solid black line in Fig. 1) and can exhibit excellent post
yielding behaviour (ductile behaviour).
3. Cracking limitation and non-prestressed reinforcement
stress
In a fully prestressed concrete beam that remains in compres-
sion under service load, cracking usually is expected only in an
overload condition. On the contrary, in a partially prestressed rein-
forced concrete beam permanent cracks of limited width may oc-
cur at service loading level. In these members, cracking initiates
when the tensile stress exceeds the modulus of rupture of concrete
and for this reason the control of cracks is necessary. Cracking con-
trol is achieved by adopting a maximum allowable crack width,such as in non-prestressed reinforced concrete members. However,
in prestressed concrete members the tendons are more sensitive to
corrosion than the ordinary steel reinforcement and therefore
smaller crack widths than those in reinforced concrete members
are recommended.
The calculation of the crack width is a complex problem since
there are many factors that cause cracking and a plethora of formu-
lae have been proposed for the calculation of crack widths in rein-
forced and prestressed concrete members. Nevertheless, loading is
the main factor that causes cracks and for this reason only the
loading is considered in design, whereas the other factors (volu-
metric change due to drying shrinkage, creep, thermal stresses
and composition of concrete; internal or external direct stresses
due to continuity; long-term deflection; environmental effects
C.G. Karayannis, C.E. Chalioris / Engineering Structures 48 (2013) 402416 403
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including differential movement in structural systems) are usually
eliminated or reduced by selecting suitable material and improving
the quality of construction[12].
Concerning the stress of the tensional reinforcement on the ba-
sis of a cracked section, it is mentioned that stress is usually esti-
mated by simple design recommendation (e.g. 60% of steel yield
for the service reinforcement stress according to ACI 318-95[1])
or by simplified expressions, such as: MEd/(As,prov .0.85d), whereasMEd is the design value of the applied bending moment, As,prov . is
the provided longitudinal steel flexural reinforcement and d is
the effective depth of the member. However, a more accurate cal-
culation of the tensile steel stress for cracked section is usually
necessary, especially in cases of partially prestressed concrete
beams where the predictions of code provisions proved to be
inconsistent with test results[13].
3.1. Eurocode
The previous version of Eurocode 2 (EC2-92 [3]) suggested the
following relations to calculate (a) the average strain of the ten-
sional steel,esmand (b) the mean final crack spacing,srm, of a rein-
forced concrete member, in order to evaluate the characteristicvalue of the design crack width,wk:
wk bsrmesm 2
where
esmrsEs
1 b1b2rsrrs
2" # 3
srm 50 0:25k1k2
qr4
b: coefficient that equals to 1.7 for load induced cracking and
for restraint cracking in sections with a minimum dimension
in excess of 800 mm, 1.3 for restraint cracking in sections with
a minimum dimension depth, breadth or thickness of 300 mm
or less (values for intermediate section sizes may be
interpolated),
b1: coefficient that takes account of the bond properties of the
bars (1.0 for high bond bars and 0.5 for plain bars),
b2: coefficient that takes account of the duration of the loading
or of repeated loading (1.0 for a single, short term loading and
0.5 for a sustained load or for many cycles of repeated loading),
k1: coefficient that takes account of the bond properties of thebars (0.8 for high bond bars and 1.6 for plain bars),
k2: coefficient that takes account of the form of the strain distri-
bution (0.5 for bending and 1.0 for pure tension),
Es: design value of modulus of elasticity of reinforcing steel,
rs: stress of the tensional reinforcement calculated on the basisof a cracked section,
rsr: stress of the tensional reinforcement calculated on the basisof a cracked section under the loading conditions causing first
cracking that equals to:fctm/qr,fctm: mean value of axial tensile strength of concrete,
qr: effective reinforcement ratio that equals to:As/Ac,eff,As: area of reinforcement contained within the effective tension
area,
Ac,eff: effective tension area that is generally the area of concretesurrounding the tension reinforcement, that equals to:
Ac;effb min
2:5hd forbeams
hx=3 for slabs
2:5c=2 for slabs or members in tension
h=2 for members intension
8>>>>>:
9>>>=>>>;
h, b: overall depth (height) and width of the member,
respectively,
d: effective depth of the member,
x: neutral axis depth,
c: clear cover to the longitudinal reinforcement and
: bar size (for mixture of bar sizes in a section, an average bar
size may be used).
Fig. 1. Typical behaviour curves (moment versus curvature) of reinforced concrete (RC), fully prestressed concrete (FPC) and partially prestressed concrete (PPC) flexural
members with equal ultimate capacity level are demonstrated (see also [10]). Fully prestressed concrete elements (green area) do not crack under the total loading and
expected to exhibit brittle failure (ductilitylu 1) whereas partially prestressed concrete elements (red area) are allowed to exhibit limited cracking under dead load andexpected to exhibit a less brittle failure depending on the partial prestressing ratio. Typical flexural reinforced concrete elements (solid black line) are designed to be cracked
even under the self-weight loading and can exhibit excellent post yielding behaviour (ductile behaviour). (For interpretation of the references to colour in this figure legend,
the reader is referred to the web version of this article.)
404 C.G. Karayannis, C.E. Chalioris/ Engineering Structures 48 (2013) 402416
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The characteristic value of the design crack width,wk, can be se-
lected based on the exposure class of the member (e.g.wk= 0.2 mm
for exposure class 2 and post-tensioned prestressing under the fre-quent load combination, as Table 4.10 of EC2-92 indicates). There-
fore, based on the previous Eqs. (2)(4) the following quadratic
equation is derived:
bq2rr2s
wkEsq2rsrm
rs bb1b2f
2ctm 0 5
From Eq.(5)given the allowable crack width, wk, the unknown va-
lue of the tensional reinforcement stress,rs, of the cracked sectioncan be evaluated. From this equation it is deduced that the maxi-
mum allowable crack width (according to EC2-92 [3] design
provisions), along with the geometrical and the mechanical charac-
teristics of the member determine the reinforcing tensile stress for
cracked section under the service load. Further, this equation shows
that the variables that influence the value of this stress are theeffective reinforcement ratio (ratio of the provided steel bars to
the cross-sectional geometrical properties), the duration of the
loading (single or sustained load), the form of the strain distribution
(bending or pure tension), the concrete strength class, the crack
width, the characteristics of the steel bar (size and bond properties)
and the minimum dimension of the cross-section.
Fig. 2ad demonstrates the influences of (a) the concrete
strength class, (b) the crack width, (c) the bar diameter and (d)
the minimum cross-sectional dimension, respectively, on the rela-
tionship of the reinforcing tensile stress for cracked section,rs, ver-sus the effective reinforcement ratio,qr. It is mentioned that thesecurves have been calculated considering the following constant
data:k1= 0.8 and b1= 1.0 for high bond bars, k2= 0.5 for bending,
b2= 0.5 for sustained load, concrete class C35 (except in Fig. 2awhere it varies), wk= 0.2 mm (except in Fig. 2b where it varies),
12 (except inFig. 2c where it varies) andb= 1.3 that corresponds
with minimum cross-sectional dimension 6300 mm (except in
Fig. 2d where it varies).The calculated curves ofFig. 2reveal that the influence of the
concrete class, the bar diameter and the minimum cross-sectional
dimension of the member to the tensile reinforcing stress for
cracked section is rather minor, whereas the crack width signifi-
cantly affects thersqr relationship, especially for high values ofqr.
The provisions of the latest version of Eurocode 2 (EC2-04[4])
modified the previous mentioned relationships for the calculation
of the design crack width, wk, based on the average strain of ten-
sional steel,esm, the average strain in the concrete between cracks,ecm, and the maximum crack spacing, sr,max, as follows:
wk sr;maxesm ecm 6
where
esm ecm rsktfct;eff=qp;eff1 aeqp;eff
EsP 0:6
rsEs
7
sr;max k3ck1k2k4
qp;eff!
k33:4 and k4 0:425
sr;max 3:4c 0:425k1k2
qp;eff8
kt is the factor dependent on the duration of the load (0.6 for short
term loading and 0.4 for long term loading),
fct;eff fctm mean value of axial tensile strength of concrete
qp;eff Asn
21Ap
Ac;eff effective reinforcement ratio
(a) (b)
(c) (d)
Fig. 2. Influence of (a) concrete strength class, (b) crack width, (c) bar diameter and (d) minimum cross-sectional dimension to the relationship of reinforcing tensile stress for
cracked section versus the effective reinforcement ratio, according to EC2-92 design provisions.
C.G. Karayannis, C.E. Chalioris / Engineering Structures 48 (2013) 402416 405
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Note: According to EC2-04, bonded tendons in the tension zone
may be assumed to contribute to crack control within a distance
6150 mm from the centre of the tendon and they may be taken
into account. In case that the contribution of the prestressing rein-
forcement is not taken into account:qp;eff AsAc;eff
qr; n1 ffiffiffiffiffiffiffiffinsp
q
(adjusted ratio of bond strength taking into account the different
diameters of prestressing and reinforcing steel), s is the largest
bar diameter of reinforcing steel, p is the equivalent diameter
of tendon (1:6ffiffiffiffiffiAp
p for bundles, 1.75wire and 1.20wire for single
7 and 3 wire strands, respectively, wire is the wire diameter)
and n is the ratio of bond strength between bonded tendons and
ribbed steel in concrete that equals to 0.3, 0.5, 0.6 and 0.7 for
smooth bars and wires, strands, indented wires and ribbed bars,
respectively for bonded, post-tensioned tendons and concrete class
6C50,
Ac;eff bmin
2:5hd
hx=3
h=2
8>:
9>=>;
ae=Es/Ecm and Ecm is the secant modulus of elasticity of concrete.Since the design crack width,wk, can be selected using the rec-
ommended values of Table 7.1N of EC2-04 (e.g. wk= 0.3 mm for
exposure class XC2 and reinforced members or/and prestressing
with unbonded tendons under the quasi-permanent load combina-
tion), the corresponding value of the tensional reinforcement
stress,rs, for cracked section and crack width,wk, can be approxi-mated directly using the following relationship:
rs wkEssr;max
ktfctmAc;effAs
EsEcm
9
Eq.(9)shows that the stress value of the tensile reinforcement for a
cracked section can be directly calculated for a given allowable
crack width. Allowable crack width values can be obtained accord-
ing to the design provisions of EC2-04 [4]. The main variable thatinfluences the value of this stress is the effective reinforcement ra-
tio (ratio of the provided steel reinforcement to the cross-sectional
geometrical properties).
Further, according to the CEB-FIP Model Code 2010[14], the cal-
culation of crack width is based on the simple case of a prismatic
reinforced concrete bar, subjected to axial tension and the concept
that under increasing deformation cracks occur in sequence. There-
fore, for the calculation of the crack width it is necessary to deter-
mine whether the crack formation stage or the stabilized cracking
stage applies. According to the simplified representation of a rein-
forced concrete member in tension with crack showed inFig. 3, the
stabilized cracking stage applies when the load is larger than the
cracking load. Hence, the crack formation stage applies when, for
imposed deformation, the stress satisfies the following condition:
Asrs Pfct;effAc;eff Asrsm !rsmecmEs rs Pfct;eff
Ac;effAs
ecmEs
!rs Pfct;effqp;eff
aeecmEcm !fct;efffctmecmEcm
rs Pfctmqp;eff
1 aeqp;eff
10a
wherersm and ecm are the steel stress and concrete strain in conti-nuity area, respectively.
The reinforcing steel tensile stress calculated with Eq.(10a)is
the maximum steel stress in a crack when stabilized cracking stage
applies. It is also mentioned that Eq.(7)defines a minimum value
of the average strain in tensional steel minus the average strain in
the concrete between cracks, (esm ecm)P 0.6rs/Es. This deter-mines the same lower limit value of the reinforcing tensile stress
for cracked section at the service load, as Eq.(10a)defines, for long
term loading (kt= 0.4):
rs P 2:5ktfctmAc;effAs
EsEcm
!kt0:4rs P
fctmqp;eff
1 aeqp;eff 10b
Since the value of the maximum steel stress in the crack formation
stage is unknown, expressions (9) and (10a)or(10b)can also yield a
lower limit of the effective reinforcement ratio. Henceforth, based
on Eqs.(9), (10a)or (10b)and (8)the following quadratic equation
can be used to evaluate the minimum value of the effective rein-
forcement ratio,qr,min = (As/Ac,eff)min:
k1k2k4 1
qr;min
!2 k3ck1k2k4ae
1
qr;min
!
k3cae wkEs1:5ktfctm
0 11
From Eq.(11)it is deduced that the value of the minimum effective
reinforcement ratio is influenced by the duration of the loading
(single or sustained load), the form of the strain distribution (bend-
ing or pure tension), the concrete strength class, the design crack
width, the characteristics of the steel longitudinal reinforcement
(size, clear cover and bond properties). Fig. 4ac demonstrates the
influences of (a) the concrete strength class, (b) the bar diameter
and (c) the clear cover of the steel bars, respectively, on the relation-
ship of the minimum effective reinforcement ratio versus the designcrack width. It is mentioned that these curves have calculated con-
sidering the following constant data: k1= 0.8 for high bond bars,
k2= 0.5 for bending, kt= 0.4 for long term loading, concrete class
C35 (except inFig. 4a where it varies), 12 (except inFig. 4b where
it varies) and clear cover of barsc= 40 mm (except inFig. 4c where
it varies). From the curves ofFig. 4it can be deduced that for the
common values of crack width (0.20.4 mm), the concrete class,
the diameter and the clear cover of the steel bars slightly influence
the value of the minimum required effective reinforcement that
ranges approximately between 1% and 2%.
Since the minimum effective reinforcement ratio can be calcu-
lated by Eq. (11) the reinforcing tensile stress for cracked section
at the service load can be estimated using Eq. (9) based on the
maximum allowable crack width according to the EC2-04[4] de-sign provisions and the characteristics of the examined member.
Fig. 3. Simplified representation of a reinforced concrete tensile member withcrack.
406 C.G. Karayannis, C.E. Chalioris/ Engineering Structures 48 (2013) 402416
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The relationship of the reinforcing tensile stress for cracked sec-
tion,rs, versus the effective reinforcement ratio,qr, is influencedby the duration of the loading (single or sustained load), the form
of the strain distribution (bending or pure tension), the concretestrength class, the crack width and the characteristics of the steel
bars (size, clear cover and bond properties).
Fig. 5ad demonstrates the influences of (a) the concrete
strength class, (b) the crack width, (c) the bar diameter and (d)
the clear cover of the longitudinal reinforcement, respectively, on
thersqrrelationship. It is noted that these curves have been cal-culated considering the following constant data:k1= 0.8 for high
bond bars, k2= 0.5 for bending, kt= 0.4 for long term loading, con-
crete class C35 (except inFig. 5a where it varies),wk= 0.2 mm (ex-
cept inFig. 5b where it varies), 12 (except in Fig. 5c where it
varies) and clear cover of bars c= 40 mm (except inFig. 5d where
it varies).
From the curves ofFig. 5it can be concluded that the influence
of the concrete class and the bar diameter to the tensile stress ofthe steel reinforcing for cracked section is rather minor. On the
contrary, the value of the crack width significantly affects thersqrrelationship, whereas the clear cover of the steel bars has a med-ial effect to this curve in cases of rather high values ofqr.
3.2. ACI 318
In ACI 318-95[1]the semi-empirical expression of Gergely and
Lutz[15]is adopted for the calculation of the crack width,w:
w 0:076bfsffiffiffiffiffiffiffiffidcA
3p
w in units of 0:001 in:
using ksi and in 12a
w 11:0225 106bfs ffiffiffiffiffiffiffiffidcA3p using MPa and mm 12bwhere
b: ratio of distances to neutral axis from extreme tension fibre
and from centroid of reinforcement, approximately equal to
1.2 for beams and 1.35 for slabs,
dc: distance between edge and centre of the lowest bar (bottomcentroid cover),
A: average effective tension concrete area surrounding each
reinforcing bar, having same centroid as reinforcement, and
fs: (=rs) stress of the tensional reinforcement for the crackedsection.
The upper design limits of the maximum crack width,wmax, that
ACI 318-95 [1] defines are 0.013 in (0.33 mm) and 0.016 in
(0.41 mm) for exterior and interior exposure, respectively. There-
fore, the stress of the tensional reinforcement for cracked section,
rs, can be calculated using the following equation (using MPaand mm):
rs wmax 106
11:0225bffiffiffiffiffiffiffiffidcA
3p 13
It is mentioned that in ACI 318-02[2]there is no longer any de-
sign expression for the evaluation of crack width, but as an indirect
control of cracking, only design limitations regarding the spacing
between the longitudinal bars are provided.
3.3. Comparisons and comments
The relationships for the calculation of the tensional reinforce-
ment stress for cracked section at service load based on the design
value of crack width according to Eurocode (EC2-92 and EC2-04)
and ACI 318 provisions exhibit differences. The major modification
between EC2-92 and EC2-04 is that the recent Eurocode introducea minimum effective reinforcement ratio for the design of the rein-
(a) (b)
(c)
Fig. 4. Influence of (a) concrete strength class, (b) bar diameter and (c) clear cover of the steel bars to the relationship of minimum effective reinforcement ratio versus the
design crack width, according to EC2-04 design provisions.
C.G. Karayannis, C.E. Chalioris / Engineering Structures 48 (2013) 402416 407
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forced concrete member. Further, the value of the clear bars cover
does not influence directly the calculations in EC2-92, whereas the
value of the minimum dimension does not influence the calcula-
tions in EC2-04. On the other hand, ACI 318-95 relationship can
be regarded as a simplified approach compared with the Eurocode
provisions.
In order to compare these three design expressions and to eval-
uate the average stress of the tensional reinforcement for cracked
section in relation with the effective reinforcement ratio, ACI
318-95 equation(13)can be transformed in the following equipol-
lent relationship:
rs wmax 10
6
11:0225bffiffiffiffiffiffiffidcAsqrn
3
q 14
where qr is the effective reinforcement ratio that equals to:qr
AsAc;eff
AsAn !A Asqrn
and n is the total number of the provided
longitudinal bars contained within the effective tensional zone of
the member, sinceA =Ac,eff/n (effective area of concrete in tension
surrounding each reinforcing bar).
Four typical applications of the design provisions of EC2-92,
EC2-04 and ACI 318-95 for the cracking control and the calculation
of therrversusqrcurves are presented inFig. 6ad. The curves ofthese figures compare the results and highlight the differences be-
tween the above mentioned design expressions.Fig. 6a and b con-
cern two typical reinforced concrete beams with different concrete
classes (C25 and C50),Fig. 6c presents the results of a typical slab
and Fig. 6d concerns the case of a beam with large dimensions
(minimum dimension = 700 mm) where the results of EC2-92,
EC2-04 and ACI 318-95 are rather identical for the common rangeof the effective reinforcement ratio.
4. Partial prestressing requirements
In order to calculate the partial prestressing requirements a
cracked cross section analysis is required with the assumption thatplane sections remain plane. The equilibrium of the forces (RF= 0)
and the moments (RM= 0) based on the strain distribution across
the depth of a T-shaped cross-section yields to the following rela-
tionships (see alsoFig. 7for annotation):
FcF0cFs2FpFs1 0 15
Fc2x
3
F0c
2 xhf
3
Fs2xds2 Fpdpx Fs1ds1x M
16
It is also essential to focus on the fact that in the case of reinforced
concrete the neutral axis of flexure of the cracked section coincides
with the point of zero stress.
Further, based on the strains of the materials, the internal forces
can be estimated as shown below:
es1 rs=Es and Fs1 As1es1Es ! Fs1 As1rs 17
ec es1x
ds1x and Fc
beffx
2 ecEc! Fc
beffx2
2aeds1xrs 18
e0c es1xhf
ds1x and F0c
beffbwxhf
2 e0cEc!
F0cbeff bwxhf
2
2aeds1x rs 19
es2 es1xds2
ds1x and Fs2 As2es2Es ! Fs2 As2
xds2
ds1xrs
20
(a) (b)
(d)(c)
Fig. 5. Influence of (a) concrete strength class, (b) crack width, (c) bar diameter and (d) clear cover of the steel bars to the relationship of reinforcing tensile stress for cracked
section versus the effective reinforcement ratio, according to EC2-04 design provisions.
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Dep es1dpx
ds1x and Fp Apept DepEs !
Fp ApeptEsApdpx
ds1xrs 21
whereMis the external imposed bending moment, ecthe maximumconcrete strain 60.002, esthe tensional steel strain 6eyd=fyd/Es(de-sign yield strain), fyd the design yield stress of the steel reinforce-
ment andept is the effective pre-strain of the prestressed tendons.Multiplying Eq.(15)with (dp x) and adding in Eq.(16), the fol-
lowing relationship is derived:
Fc dpx 2x
3
F0c dpx
2xhf
3
Fs2dpx xds2 Fs1dpx ds1x M 22
In Eq. (22), replacing the internal forces of the materials using
expressions(17)(20), the neutral axis depth, x , can be calculated
by the following cubic equation:
Ax3
23
where
A bw2
B3bwdp
2
C3hfbeff bw2dphf
2 3aeAs1ds1As2ds2 3aedpAs1
As2 3aeM
rs
D h
2fbeff bw3dp 2hf
2 3aeAs1d
2s1As2d
2s2 3aedpAs1ds1
As2ds2 3aeMds1
rs
It is mentioned that for the simplified case of a beam with rectan-gular cross-section (b/h), without compressive reinforcement
(As2= 0 and As=As1) and ds1= ds ffi dp= d, the aforementioned
parameters of the cubic Eq.(23)become:
A b
2; B
3bd
2 ; C
3aeM
rsand D
3aeMd
rs
and Eq.(23)is simplified as follows:
b
2x3
3bd
2 x2
3aeM
rsx
3aeMd
rs0 23-S
Further, based on Eq. (15) and using Eqs. (17)(21), the required
effective pre-strain of the prestressed tendons can be estimated as
follows:
(a) (b)
(c) (d)
Reinforcingstressforcrackedsection(MPa)
Effective reinforcement ratio
EC2-92
EC2-04
ACI 318-95
Data:
C25, c= 40 mm, 14, wk = 0.20 mm, highbond bars, bending, long term loading,
beam, minimum dimension 300 mm.
Coefficients:
- EC: k1 = 0.8, k2 = 0.5, kt = 0.4, = 1.3,
1 = 1.0, 2 = 0.5
- ACI: = 1.2
Reinforcingstressforcrackedsection(MPa)
Effective reinforcement ratio
EC2-92
EC2-04
ACI 318-95
Data:
C50, c= 40 mm, 14, wk = 0.20 mm, highbond bars, bending, long term loading,
beam, minimum dimension 300 mm.
Coefficients:
- EC: k1 = 0.8, k2 = 0.5, kt = 0.4, = 1.3,
1 = 1.0, 2 = 0.5
- ACI: = 1.2
Reinforcingstr
essforcrackedsection(MPa)
Effective reinforcement ratio
EC2-92
EC2-04
ACI 318-95
Data:
C25, c= 30 mm, 10, wk = 0.20 mm, high
bond bars, bending, short term loading,
slab, minimum dimension 300 mm.
Coefficients:
- EC: k1 = 0.8, k2 = 0.5, kt = 0.6, = 1.3,1 = 1.0, 2 = 1.0
- ACI: = 1.35
Reinforcingstressforcrackedsection(MPa)
Effective reinforcement ratio
EC2-92
EC2-04
ACI 318-95
Data:
C25, c= 30 mm, 16, wk = 0.30 mm, high
bond bars, bending, long term loading,
beam, minimum dimension = 700 mm.
Coefficients:
- EC: k1 = 0.8, k2 = 0.5, kt = 0.4, = 1.62,1 = 1.0,2 = 0.5
- ACI: = 1.2
Fig. 6. Typical applications of EC2-92, EC2-04 and ACI 318-95 provisions for the crack width control of reinforced concrete members.
Fig. 7. Strain distribution and internal forces across the depth of a T-shaped cross-
section.
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ept rs
ApEsds1x
beffx2 beff bwxhf
2
2ae
"
As2xds2 As1ds1x Apdpx
24
For the simplified case of a beam with rectangular cross-section (b/
h), without compressive reinforcement (As2= 0 and As=As1) and
ds1=ds ffi 1dp= d, Eq.(24)and consequently expression(21)become
as follow:
ept rsApEs
bx2
2aedx AsAp
" # 24-S
Fp rsbx
2
2aedxAs
" # 21-S
Consequently, using as data the geometrical and the mechanical
characteristics of the member along with the external imposed
bending moment, the required prestressing force,Fp, can be calcu-
lated using consecutively Eqs.(23), (24)and (21). Before that, the
average stress of the tensional reinforcement for cracked section,
rs, has to be estimated based on the properties of the member
and the maximum allowable crack width according to the designprovisions of Eurocode or ACI, as described in Section2of the pres-
ent study.
It is also mentioned that for the calculation of required effective
pre-strain using Eq. (24), an approximation of the cross-section
area,Ap, of the prestressing reinforcement has to be initially con-
sidered. However, since the value of the required effective pre-
strain should be greater than zero, an upper limit for the area of
the prestressed tendons can be obtained assuming that eptP 0.Henceforth the following expression can be derived:
Ap;max beffx
2 beffbwxhf2
2aeAs2xds2 As1ds1x
" #,dpx 25
whilex P ds2 and x 6 dp 6 ds1.
It is also noted that for the simplified case of a beam with rect-angular cross-section (b/h), without compressive reinforcement
(As2= 0 and As=As1) andds1=ds ffi dp=d, expression(25)becomes:
Ap;max bx
2
2aedxAs 25-S
The non-prestressed reinforcement of a partially prestressed
member should also fulfil the minimum amount design provisions
of Eurocode to control cracking in areas where tension is expected.
This amount is estimated from equilibrium between the tensile
force in concrete just before cracking and the tensile force in rein-
forcement at the average stress for cracked section, as follows:
As;minkckfct;effAct
rs26
where
Act: area of concrete in tension just before the formation of the
first crack,
k: coefficient for the effect of non-uniform self-equilibrating
stresses, equal to 1.0 for webs with height 6300 mm or flanges
with width6300 mm, 0.65 for webs with height P800 mm or
flanges with width 6800 mm and for intermediate values may
be interpolated,
kc: coefficient that takes account the stress distribution within
the section immediately prior to cracking and of the change of
the lever arm, equal to 1.0 for pure tension and for bending with
or without axial force: 0:4 1 rck1h=h
fct;eff
h i6 1 for rectangular
cross-sections and webs of T-shaped and box sections, where
h = min(h, 1 m) and 0:9 FcrActfct;eff
P 0:5 for flanges of T-shaped
and box sections,
rc: average concrete stress that equals to:NEd/(bh), wherebandhare the width and the height of the cross-section, respectively
andNEd is the axial force at the serviceability limit state (posi-
tive for compression),
k1: coefficient considering the effects of axial forces on the
stress distribution, that equals to 1.5 for compressive and 2h
/(3h) for tensile axial force, and
Fcr: absolute value of the tensile force within the flange imme-
diately prior to cracking due to the cracking moment calculated
withfct,eff(=fctm).
It is noted that design provisions of ACI 318 (versions of 1995
and late) also require a minimum steel reinforcement to be placed
near the tension face of prestressed flexural members in order to
control cracking under full service loads or overloads. This mini-
mum amount of steel is defined by the 0.4% of the area of that part
of cross-section between the flexural tension face and centre of
gravity of gross section.
5. Design procedure
The proposed design procedure of a partially prestressed rein-
forced concrete member based on the serviceability limit state pro-
visions includes the following steps:
1. Decision of the maximum allowed crack width according to
the design code and based on the exposure class specifica-
tions of the examined member.
2. Selection of the non-prestressed tensile steel reinforcement.
Check of the minimum requirements of code. Especially for
ACI 318, the assumed non-prestressed tensile steel reinforce-
ment should not be less than the minimum required amount,
as defined in paragraph 5 of the present study.
3. Calculation of the effective reinforcement ratio based on the
amount of the tensional steel reinforcement and the effective
tension area of concrete surrounding the tension
reinforcement.
4. Only applied for EC2-04: evaluation of the minimum effective
reinforcement ratio based on Eq.(11)and checking with the
calculated value of the effective reinforcement ratio of step
3. In case that the calculated effective reinforcement ratio is
less than the minimum value, an increase of the assumed
non-prestressed tensile steel reinforcement in step 2 and a
re-calculation of the increased effective reinforcement ratio
is required.
5. Calculation of the tensile steel stress for cracked section using
Eq. (5), (9) and (13) for EC2-92, EC2-04 and ACI 318-95,
respectively and based on the geometrical and mechanical
properties of the member.
6. Calculation of the neutral axis depth based on the cubic Eq.
(23).
7. Calculation of the upper limit for the prestressed tendons area
based on Eq.(25)and assumption of the amount of the pre-
stressing reinforcement.
8. Calculation of the required effective pre-strain and prestress-
ing force of the prestressed tendons based on Eqs. (24) and
(21), respectively.
9. Only applied for Eurocode: evaluation of the minimum
required amount of steel reinforcement to control cracking
based on Eq. (26)and checking with the assumed non-pre-
stressed tensile steel reinforcement (from step 2). In case that
the required minimum steel reinforcement is greater than the
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initially assumed amount of tensile steel in step 2, an increase
of this steel reinforcement and re-calculations of steps 27
are required.
A detailed flow chart of the proposed design procedure is also
presented inFig. 8.
It is noted that the proposed procedure is not included in the
codes but it uses considerations from the codes, such as the designprovisions for (a) the maximum crack, (b) the minimum steel rein-
forcement, (c) the tensile steel stress for cracked section and (d)
the minimum effective reinforcement ratio. In the above men-
tioned step-by-step design procedure, steps 15 include the design
provisions of ACI 318 or Eurocode 2 for a RC element, whereas
steps 68 constitute the base of the proposed methodology for
the design of a partially prestressed concrete element.
6. Design charts
For design purposes, the simplified design charts of eight typical
rectangular beams and eight T-beams are illustrated inFigs. 9 and
10, respectively. The specific geometrical characteristics and the
reinforcement arrangement of each examined beam are shown in
Tables 1 and 2 for the beams with rectangular and T-shaped
cross-sections, respectively. Further, the common used data of
these examined cases are as follows:
Concrete class: C30 (characteristic concrete compressive
strength equals to 30 MPa).
Exposure class: XC3 (corrosion induced by carbonation, concrete
inside buildings with moderate or high air humidity or/and
external concrete sheltered from rain).
Structural class: S4 (design working life of 50 years).
Thus, the minimum cover due to durability is: cmin,dur= 25 mm
(for reinforcement steel) and 35 mm (for prestressing steel)
and for steel bars with diameter 14 (625 mm), the minimumallowed cover is: cmin =cmin,dur, whereas the nominal cover is:
cnom = 35 mm (for reinforcement steel) and 45 mm (for pre-
stressing steel).
Therefore, the values of the effective depth for the tensional
steel reinforcement, the compressional steel reinforcement
and the prestressing tendons are:ds1=h 50 mm,ds2= 50 mm
anddp=h 100 mm, respectively.
The design charts ofFigs. 9 and 10can be used for the estima-
tion of the required effective pre-strain,ept, iny-axis, based on thevalue of the stress,rs, of the non-prestressed tensional reinforce-ment for cracked section in x-axis and for a variation of external
imposed bending moment, M, values. Consequently, the required
and the total prestressing force can be calculated using the corre-sponding value of the tendons effective pre-strain and Eq.(21).
It is mentioned that the curves inFigs. 9 and 10that correspond
to the higher values of the imposed bending moment are not con-
tinued to the end ofx-axis (for high values ofrs) because the max-imum concrete strain exceeds the upper limit of 0.002. Further, it is
emphasized that thers eptcurves that correspond to the lowervalue of the bending moment and to the beams with Ap= 2000 -
mm2 (beams Nos. 1, 2, 3 and 4) have values ofeptless than zerofor higher values ofrs. For example in Fig. 10, the r s eptcurveof the T-shaped beam No. 4 (forM= 1000 kN m) has no valid (neg-
ative) values ofept for rsP 300 MPa. This means that the maxi-mum allowed area of the prestressed tendons calculated by Eq.
(25)is less than Ap= 2000 mm2. In this case, a lower value ofAp,
such as 1000 mm2
, could be provided for design purpose, as shownin therseptcurve of the T-shaped beam No. 8 (Fig. 10).
7. Numerical examples
In order to illustrate the use of the proposed design methodol-
ogy for partially prestressed RC members three numerical exam-
ples are included herein. Geometrical and mechanical data of the
examined beams are shown in Fig. 11. The common data of the
beams are the characteristic concrete compressive strength that
equals to 30 MPa (concrete class C30 according to Eurocode and
specified concrete compressive strength equal to 4350 psi accord-
Take wkaccording to the design code and
based on the exposure class
AssumeAs(tensional steel reinforcement)
ACI 318: Select:AsAs,min
Yes
No
Calculate sfrom equations:
(5) for EC2-95, (9) for EC2-04 and
(13) for ACI 318-95
EC2-04
EC2-95 or
ACI 318-95
Calculatexfrom equation (23)
No
Yes
EC2-95 or
EC2-04
ACI 318-95
End
Calculate ptfrom equation (24)
and Fpfrom equation (21)
EC2-04: Calculate r,minfrom equation (11)
and check: r r,min
Calculate r(=As /Ac,eff )
Eurocode: CalculateAs,minfrom equation (26)
and check:AsAs,min
Data: Cross-sectional geometry, concrete class, exposure
class, type and duration of loading,bending moment
CalculateAp,maxfrom equation (25) and assumeAp
Fig. 8. Flow chart of the proposed design procedure for a partially prestressedreinforced concrete member based on the serviceability limit state provisions.
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Fig. 9. Design charts for partially prestressed reinforced concrete beams with rectangular cross-section based on the cracking control (see alsoTable 1for notation).
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Fig. 10. Design charts for partially prestressed reinforced concrete beams with T-shaped cross-section based on the cracking control (see alsoTable 2for notation).
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ing to ACI 318), the exposure class XC3 according to EC2-04 or 2
according to EC2-92 or exterior exposure according to ACI 318.
The first example is a beam with rectangular cross-sectional
dimensions (rectangular beam No. 5 ofTable 1) under an external
imposed bending moment for short term loading that equals to
M= 400 kN m. The non-prestressed steel reinforcement was se-
lected to be equal toAs1= 770 mm2 (tension bars) andAs2= 462 -
mm2 (compression bars), whereas the prestressed tendons area
was Ap= 1000 mm2. According to the provisions of Eurocode
(EC2-04) the tensile steel reinforcing stress for cracked section
was calculated to be equal to rs= 266 MPa and the requiredprestressing force wasPpt= 472 kN. Further, according to Eurocode
(EC2-92) the tensile steel reinforcing stress for cracked section was
rs= 322 MPa and the required prestressing force wasPpt= 363 kN.Moreover, according to ACI 318-95 the tensile steel reinforcing
stress for cracked section was rs= 373 MPa and the required pre-stressing force wasPpt= 266 kN.
Table 1
Geometrical characteristics and reinforcement arrangement of the examined beams with rectangular cross-section.
Beam no. b/h(mm) As1 As2 Apqs1=As1/(bds1) qs2=As2/(bds1) qp=Ap/(bdp)
1 300/600 5 14 (770 mm2) 3 14 (462 mm2) 2000 mm2
0.47 % 0.28 % 1.33 %
2 300/900 5 14 (770 mm2) 3 14 (462 mm2) 2000 mm2
0.30 % 0.18 % 0.83 %
3 400/800 8 14 (1232 mm2) 4 14 (616 mm2) 2000 mm2
0.41 % 0.21 % 0.71 %
4 400/1200 8 14 (1232 mm2) 4 14 (616 mm2) 2000 mm2
0.27 % 0.13 % 0.45 %
5 300/600 5 14 (770 mm2) 3 14 (462 mm2) 1000 mm2
0.47 % 0.28 % 0.67 %
6 300/900 5 14 (770 mm2) 3 14 (462 mm2) 1000 mm2
0.30 % 0.18 % 0.42 %
7 400/800 8 14 (1232 mm2) 4 14 (616 mm2) 1000 mm2
0.41 % 0.21 % 0.36 %
8 400/1200 8 14 (1232 mm2) 4 14 (616 mm2) 1000 mm2
0.27 % 0.13 % 0.23 %
Table 2
Geometrical characteristics and reinforcement arrangement of the examined beams with T-shaped cross-section.
Beam no. beff/bw/h/hef(mm) As1 As2 Apqs1=As1/(bwds1) qs2=As2/(bwds1) qp=Ap/(bwdp)
1 900/300/600/200 5 14 (770 mm2) 3 14 (462 mm2) 2000 mm2
0.47 % 0.28 % 1.33 %
2 900/300/900/200 5 14 (770 mm2) 3 14 (462 mm2) 2000 mm2
0.30 % 0.18 % 0.83 %
3 1200/400/800/200 8 14 (1232 mm2) 4 14 (616 mm2) 2000 mm2
0.41 % 0.21 % 0.71 %
4 1200/400/1200/200 8 14 (1232 mm2) 4 14 (616 mm2) 2000 mm2
0.27 % 0.13 % 0.45 %
5 900/300/600/200 5 14 (770 mm2) 3 14 (462 mm2) 1000 mm2
0.47 % 0.28 % 0.67 %
6 900/300/900/200 5 14 (770 mm2) 3 14 (462 mm2) 1000 mm2
0.30 % 0.18 % 0.42 %
7 1200/400/800/200 8 14 (1232 mm2) 4 14 (616 mm2) 1000 mm2
0.41 % 0.21 % 0.36 %
8 1200/400/1200/200 8 14 (1232 mm2) 4 14 (616 mm2) 1000 mm2
0.27 % 0.13 % 0.23 %
Fig. 11. Geometrical data and reinforcement arrangement of the examined cases.
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The second example is a beam with T-shaped cross-section (T-
shaped beam No. 3 ofTable 2) under an external imposed bendingmoment for short term loading that equals toM= 3200 kN m. The
non-prestressed steel reinforcement was selected to be equal to
As1= 1232 mm2 (tension bars) and As2= 616 mm
2 (compression
bars), whereas the prestressed tendons area was Ap= 2000 mm2.
According to the provisions of Eurocode (EC2-04) the tensile steel
reinforcing stress for cracked section was calculated to be equal to
rs= 267 MPa and the required prestressing force wasPpt= 4414 kN.Further, according to Eurocode (EC2-92) the tensile steel
reinforcing stress for cracked section wasrs= 315 MPa and the re-quired prestressing force was Ppt= 4,229 kN. Moreover, according
to ACI 318-95 the tensile steel reinforcing stress for cracked section
was rs= 396 MPa and the required prestressing force wasPpt= 3926 kN.
The third example is a slab under an external imposed bending
moment for long term loading that equals toM= 200 kN m/m. The
non-prestressed steel reinforcement was selected to beAs= 785 mm2
and the prestressed tendons area was Ap= 393 mm2. According to
the provisions of Eurocode (EC2-04) the tensile steel reinforcing
stress for cracked section was calculated to be equal to
rs= 280 MPa and the required prestressing force wasPpt= 1692 -
kN. Further, according to Eurocode (EC2-92) the tensile steel rein-
forcing stress for cracked section was rs= 336 MPa and therequired prestressing force wasPpt= 1577 kN. Moreover, accordingto ACI 318-95 the tensile steel reinforcing stress for cracked section
was rs= 354 MPa and the required prestressing force wasPpt= 1542 kN.
Main and intermediate results along with helpful details of the
above numerical applications are presented inTable 3.
8. Calculation of the required prestressing reinforcement based
on the PPR and the maximum allowed crack width
Applying the aforementioned design procedure and relation-
ships for the simplified case of a beam with rectangular cross-sec-
tion (b/h), without compressive reinforcement (As2= 0 andAs=As1)
and ds1=ds ffi dp=d, the required prestressing reinforcement canbe calculated for a certain level of the external imposed bending
moment that is represented by the preferred Partial Prestressing
Ratio (PPR) and for a given maximum allowed crack width.
In this direction, firstly, the tensile steel reinforcing stress,rs, iscalculated using Eq.(5)or(9)or(13)(for EC2-92 or EC2-04 or ACI
318-95, respectively), given the maximum allowed crack width,
Table 3
Results and details of the numerical applications.
1st Example (rectangular beam) 2nd Example (T-shaped beam) 3rd Example (slab)
EC2-04 EC2-92 ACI 318-95 EC2-04 EC2-92 ACI 318-95 EC2-04 EC2-92 ACI 318-95
M(kN m) 400 400 400 3200 3200 3200 200 200 200
wk (mm) 0.2 0.2 0.33 0.2 0.2 0.33 0.2 0.2 0.33
As,min (mm2) 310 210 360 451 383 800 414 297 400
As1 (mm2) 770 770 770 1232 1232 1232 785 785 785
As2 (mm2) 462 462 462 616 616 616
Ac,eff(mm2) 37500 37500 50000 50000 38667 40667
A(mm2) 6000 5000 7000
qr,min (%) 1.81 1.81 1.16 qr(%) 2.05 2.05 2.57 2.46 2.46 3.08 2.03 1.93 1.12sr,max (mm) 235 118 216 107 186 102
rs(MPa) 266 322 373 267 315 396 280 336 354x(mm) 210 193 181 306 285 256 84 78 76
Ap,max (mm2) 3079 2311 1825 20,633 17,039 13,030 12,263 9059 8,312
Ap (mm2) 1000 1000 1000 2000 2000 2000 393 393 393
ept 0.00236 0.00182 0.00133 0.01104 0.01057 0.00982 0.02154 0.02008 0.01963Ppt(kN) 472 363 266 4414 4229 3926 1692 1577 1542
Fp(kN) 699 640 287 4888 4791 4637 1748 1648 1618
Fig. 12. Relationships between crack widthwk, PPR andxp/xs. For a maximum allowed crack widthwk= 0.3 mm and an externally imposed bending moment M= 175 kN m
the value PPR = 0.29 (lightly prestressed member) and xp/xs= 0.41 are obtained. For higher value of bending moment (M= 600 kN m), higher value of Partial PrestressingRatio (PPR = 0.81) as well as higher value ofxp/xs= 4.16 are obtained.
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wk, along with the geometrical and mechanical data of the mem-
ber. Secondly, the neutral axis depth, x , is calculated using(23-S)
and the value of the bending moment,M. Thus, the required pre-
stressing force of the tendons can be calculated using Eq. (21-S).
The value of the PPR and the mechanical ratio of the required pre-
stressing reinforcement, xp, can be calculated by the followingexpressions:
PPR Apfp
ApfpAsfs F
p
FpFs!
Eq:21-SPPR 1 2A
saedxbx
2 27
xpxs
ApfpydbdfcdAsfydbdfcd
ApfpydAsfyd
ffiApfpAsfs
28
PPR Apfp
ApfpAsfs!
1
PPRApfpAsfs
Apfp1
AsfsApfp
!Eq:28
xpxs
PPR
1 PPR 29
InFig. 12an application of the above mentioned procedure is
presented. It concerns a concrete beam with rectangular cross-
section b/h= 300/600 mm, tensional reinforcement As= 770 mm
2
(5 14), ds ffi dp=d= 550 mm, without compressive reinforce-
ment, concrete class C30, exposure class XC3 and structural class
S4. First, the maximum allowed crack width is selected (for exam-
ple: wk= 0.3 mm). In the examined case the external imposed
bending moment is equal to M= 175 kN m (light green line)
slightly higher than the design bending moment at yield of the
RC member without prestressing, that equals to 165 kN m. From
this point of intersection of the vertical black dashed line with
the light green line a horizontal line is drawn that first intersects
the y-axis and yields a Partial Prestressing Ratio equal to:
PPR = 0.29 (lightly prestressed member) and then intersects the
curve on the right part of Fig. 12 and leads downwards to xp/xs= 0.41. As it can be seen from the same figure, for higher valueof bending moment (that equals toM= 600 kN m), higher value of
Partial Prestressing Ratio (PPR = 0.81) as well as higher value ofxp/xs(xp/xs= 4.16) are obtained. A fully prestressed member is pre-ferred for bending moment greater than 1200 kN m since the val-
ues of PPR and xp/xs become particularly high (approximatelygreater than 0.90 and 10, respectively).
9. Concluding remarks
The application of partial prestressing on a reinforced concrete
element exhibits in many cases certain advantages compared to
the application of full prestressing while it offers great technical
capabilities in comparison with the conventional reinforced con-
crete. In this work a design procedure for the estimation of the re-
quired partial prestressing force for a flexural reinforced concrete
element has been presented based mainly on the allowable crack
width as it stated by the contemporary major design codes (ACI
318 and Eurocode 2).
The analysis and design of the partially prestressed concrete are
mainly based on the serviceability limitations and especially the
cracking control thus the effective reinforcement ratio and the ten-
sile stress of it are critical parameters for the design procedure.
According to the design provisions of Eurocode 2 (version of
1992 and current version 2004) the influence of the concrete class,
the bar diameter and the minimum cross-sectional dimension of
the member to the tensile reinforcing stress for cracked section
is rather minor, whereas the crack width is the only parameter that
significantly affects the reinforcing tensile stress versus the effec-
tive reinforcement ratio relationship, especially for high values of
effective reinforcement ratio. Therefore first step is the choice of
the amount and the estimation of the stress of the non-prestressedreinforcement and then the calculation of the depth of the com-
pression zone using a cubic equation formed for this purpose. Fur-
ther the required effective pre-strain of the prestressed steel and
thus the required prestressed force are calculated. According to
EC2-04 design provisions and for the common values of crack
width (0.20.4 mm) the concrete class, the diameter and the clear
cover of the steel bars slightly influence the value of the minimum
required effective reinforcement that usually ranges approxi-
mately between 1% and 2%.
Helpful design charts and three numerical paradigms are also
presented and commented. In these paradigms the procedure for
the estimation of the partial prestressing has been applied on a
rectangular beam, a T-beam and a slab. Both codes (Eurocodes
and ACI-318) are used in each application for comparison pur-
poses. Through these examples can be concluded that the pre-
sented procedure is an easy-to-apply, versatile tool for the
application of partial prestressing on a flexural reinforced concrete
beam.
References
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[5] Joint ACI-ASCE Committee 423 (2000). State-of-the-art report on partiallyprestressed concrete (ACI 423.5R-99). Farmington Hills, MI, USA, AmericanConcrete Institute.
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[14] CEB-FIP. Bulletin n. 56, Model Code 2010, First complete draft, vol. 2; 2010.[15] Gergely P, Lutz LA. Maximum crack width in reinforced concrete flexural
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Recommended