Design of Partially Prestressed Concrete Beams Based on the Cracking

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.



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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.



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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.



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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



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)


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



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.

http://-/?-http://-/?-


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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



10/15

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).



12/15

Fig. 10. Design charts for partially prestressed reinforced concrete beams with T-shaped cross-section based on the cracking control (see alsoTable 2for notation).



13/15

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.



15/15

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

[1] ACI Committee 318 (1995). Building code requirements for structural concrete.American Concrete Institute, Farmington Hills, Detroit, Michigan, USA.

[2] ACI Committee 318 (2002). Building code requirements for structural concrete.American Concrete Institute, Farmington Hills, Detroit, Michigan, USA.

[3] CEN. Eurocode 2 (1992). Design of concrete structures Part 1-1: general rulesand rules for buildings. ENV 1992-1-1:1992.

[4] CEN. Eurocode 2 (2004). Design of concrete structures Part 1-1: general rulesand rules for buildings. EN 1992-1-1: 2004:E.

[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.

[6] Naaman AE. Partially prestressed concrete: review and recommendations. PCI J1985;30(6):3071.

[7] Harajli MH, Naaman AE. Static and fatigue tests on partially prestressed beams.J Struct Eng (ASCE) 1985;111(7):160218.

[8] Au FTK, Du JS. Partially prestressed concrete. Prog Struct Eng Mater2004;6:12735.

[9] Dilger WH, Suri KM. Steel stresses in partially prestressed concrete members.PCI J 1986;31(3):88112.

[10] Naaman AE, Siriaksorn A. Serviceability based design of partially prestressedbeams. PCI J 1979;24(2):6489.

[11] Bruggeling ASG. Partially prestressed concrete structures a design challenge.PCI J 1985;30(2):14070.

[12] Chowdhury SH, Loo YC. A new formula for prediction of crack widths inreinforced, partially prestressed concrete beams. Adv Struct Eng2001;4(2):10110.

[13] Chowdhury SH. Cracking and deflection behaviour of partially prestressed

high strength concrete beams. In: Australasian structural engineeringconference (ASEC), 2627 June 2008, Melbourne Australia; 2008. Papernumber 008.

[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

members, causes, mechanism and control of cracking in concrete, SP-20.American Concrete Institute, Farmington Hills, Detroit, MI, USA; 1968. p. 87117.


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