determination of preliminary cable profile in prestressed concrete girders

7/23/2019 Determination of Preliminary Cable Profile in Prestressed Concrete Girders

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Proceedings of

3rd International Conference on Recent Trends in Engineering & Technology

(ICRTET’2014)

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Determination of Preliminary Cable Profile in Prestressed Concrete Girders

Chauhan Y. J.a, *, Shah B.J. b, Patel D.D.c

aDepartment of Applied Mechanics, PG Student,L.D. Engg College,Ahmedabad,380001,India bDepartment of Applied Mechanics, Professor,L.D. Engg College,Ahmedabad,380001,India

cJoint Principal Consultant, SPECTRUM Techno Consultants Pvt. Ltd., Ahmedabad,380009,India

Abstract

The determination of cable profile in the design of Prestressed Concrete for flexure is a complex, time consuming and trial & error process. Many authors have extended the features of the original Magnel diagram[1] to determine the acceptable zone of

combinations of eccentricity and prestressing force. The purpose of the work is to present generalized design charts for uniformly

loaded simply supported girders as per Indian Standard. The paper also describes its extension for simply supported bridge girders

subjected to non-uniform loads and varying cross section which can be easily incorporated in a computer program. The computer

program can also facilitates to visualize the cable eccentricity and also to check whether the resultant of the cables fall within the

acceptable zone along the length of girder. The IRC vehicular loads Class 70R and Class A have been replaced by Conjugate

Stationary Load System (CSLS) which simplifies the bending moment calculations. Keywords: Prestress Concrete Girder; Cable profile; Acceptable zone ; Magnel diagram; Generalized design charts; IRC; CSLS.

1. Introduction

The determination of acceptable cable profile is one of the most important parameters in the design of PSC for flexure as it

involves trial and error process. The graphical method known as “Magnel Diagram” proposed by Magnel Gustave[1] is widely

used for the determination of acceptable combinations of prestressing force and eccentricity at a particular section along the span.

The method uses the linear relationship between the eccentricity (e) and the reciprocal of the prestressing force (P) and plotted on

e vs. P axis as shown in the Fig. 1.1. However the limitations of original Magnel Diagram [1] are: It does not give the visual

location of acceptable zone with respect to girder c/s and the designers have to generate a set of such diagram at various critical

sections along the span to arrive at suitable cable profile along the length of girders.

Fig.1.1 Magnel’s Graphical Method Fig. 1.2 Krishnamurthy’s Modified Magnel Diagram

* Corresponding author. Tel.:+919712845427.

E-mail address:[email protected].



Chauhan Y. J. .et.al.

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Krishnamurthy[2] has proposed a modification of the Magnel Diagram[1] by plotting (1/P) along horizontal axis and

eccentricity (e) along vertical axis with girder cross section drawn alongside with its centroidal axis located in line with the

horizontal axis as shown in Fig. 1.2. Hence the designer can visually locate the acceptable zone with respect to girder c/s and

select feasible combinations of P and e. However both the methods are limited to a particular section only and the designers have

to generate set of such diagrams at various sections along the span to arrive at suitable cable profile.

Mohammad R. Ehsani and J. Russell Blewitt[3] proposed generalized design charts which extends the idea of Magnel’s safe

zone into a relationship for the entire length of uniformly loaded simply supported girders. The limitation of these charts is thatthey are used for simply supported girders having cross sections uniform throughout the length of the girders. However, in

practice, for post tensioned girders, the cross sections need to be scaled at ends to account for large bursting force developed and

to house the anchorages. Again the method is applicable to uniformly loaded simply supported girders only which is not usually

the case and does not take into considerations the actual construction stage sequences. In actual case the girders may be subject to

any loads and specifically in bridges, the girders are subjected to vehicular loads in addition to static loads.

For a given particular girder with varying cross section, the method is proposed by the authors which can be incorporated in a

computer program. The program can also facilitate to plot the cross section on same vertical axis and suitably scaled horizontal

axis and enables the user to visualize the feasible zone. The program can also plot the CGS of the cables to check visually whether

it falls within the acceptable zone and the user can accordingly modify the eccentricity of each cable.

The paper presents:

A simplified method to determine the acceptable zone of eccentricity for both uniformly loaded as well as non-uniformly

loaded girders

Non-dimensional generalized chart for uniformly loaded girders as per IRC: 112: 2011

To extend its application for a given particular simply supported girders subjected to non-uniform loads and having varying

cross section using computer program.

Application of Conjugate Stationary Load System (CSLS) for Class 70R and Class A wheeled vehicles which simplifies

bending moment computations.

2. Basic Equations

The four fundamental conditions for stresses in a PSC section at transfer and service stages are as follows:At transfer stage:

Fig. 2.1 Stresses in PSC section during transfer stage

f t = Po/A – (Po·e)/Ztg + Msw/Ztg ≥ (f ct,all)transfer (2.1)

f b = Po/A + (Po·e)/Z bg – Msw/Z bg ≤ (f cc,all)transfer (2.2)

At service stage:

Fig. 2.2 Stresses in PSC section during service stage

At service stage,

f t = (η·Po)/A – (η·Po·e)/Ztg + (Msw + Mdeck )/Ztg + (MSIDL + MLL)/Ztgc ≤ (f cc,all)service (2.3)

f b = (η·Po)/A + (η·Po·e)/Z bg – (Msw + Mdeck )/Z bg – (MSIDL + MLL)/Z bgc ≥ (f ct,all)service (2.4)




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At transfer stage: stresses due to applied prestressing force and self weight of the girder are

accounted.

At service stage: the deck slab is cast but the concrete being green, composite action

between girder and deck slab will not occur and so the section modulus is corresponding

to girder only. When concrete of deck slab attains sufficient strength, S.I.D.L. and L.L. is

applied and the composite action between deck slab and girder will occur and so section

modulus corresponding to composite action should be accounted.

Transforming above equations in terms of e(x) and substituting, the initial prestressing

force, Po is given by,

Po = f ccg · A (2.5)

where, f ccg = (f cc,all)transfer – [(f cc,all)transfer - (f ct,all)transfer ]·Y b/h

Fig. 2.4 Initial Prestress at C.G. of the section

emax(x) ≤ Ztg/A · [ – (f ct,all)transfer /f ccg + 1 + Msw/f ccg·Ztg] or (2.6a)

emax(x) ≤ Kb · [ – (f ct,all)transfer /f ccg + 1 + Msw/f ccg·Ztg ] (2.6b)

emax(x) ≤ Zbg/A · [ (f cc,all)transfer /f ccg – 1 + Msw/f ccg·Z bg ] or (2.7a)

emax(x) ≤ Kt · [ (f cc,all)transfer /f ccg – 1 + Msw/f ccg·Z bg ] (2.7b)

emin(x) ≥ Ztg/A · [ – (f cc,all)service/(η·f ccg) + 1 + (Msw + MDeck )/(η·f ccg·Ztg) + (MSIDL + MLL)/(η·f ccg·Ztgc)] or (2.8a)

emin(x) ≥ Kb · [ – (f cc,all)service/(η·f ccg) + 1 + (Msw + MDeck )/(η·f ccg·Ztg) + (MSIDL + MLL)/(η·f ccg·Ztgc)] (2.8b)

emin(x) ≥ Zbg/A · [ (f ct,all)service/(η·f ccg) – 1 + (Msw + MDeck )/(η·f ccg·Z bg) + (MSIDL + MLL)/(η·f ccg·Z bgc)] or (2.9a)

emin(x) ≥ Kt · [ (f cc,all)service/(η·f ccg) – 1 + (Msw + MDeck )/(η·f ccg·Z bg) + (MSIDL + MLL)/(η·f ccg·Z bgc)] (2.9b)

where K t & K b are top & bottom kern distances respectively.

The equations (2.6) to (2.9) gives upper and lower limits of eccentricity at any position x along the length of girder within which

the resultant of prestressing cables should lie.

The above equations can be expressed as emax(x)/ Kb; emax(x)/ Kt; emin(x)/ Kb; emin(x)/ Kt respectively.

emax(x)/Kb ≤ [{ – (fct,all)transfer/fccg + 1} + Msw/fccg·Ztg ] (2.10)

emax(x)/Kt ≤ [ {(fcc,all)transfer/fccg – 1} + Msw/fccg·Zbg ] (2.11)

emin(x)/Kb ≥ [{ – (f cc,all)service/(η·f ccg) + 1} + (Msw + MDeck )/(η·f ccg·Ztg) + (MSIDL + MLL)/(η·f ccg·Ztgc)] (2.12)

emin(x)/Kt ≥ [ {(f cc,all)service/(η·f ccg) – 1} + (Msw + MDeck )/(η·f ccg·Z bg) + (MSIDL + MLL)/(η·f ccg·Z bgc)] (2.13)

The equations (2.10) to (2.13) are non-dimensional. Introducing moment ratio, MR defined as:

MR = [Msw / {Msw + MDeck + (MSIDL + MLL) · r}] ; (2.14)

where, r = ratio of section modulus at top/bottom of girders alone to the section modulus at top/bottom of composite section

i.e. r t = Ztg/Ztgc & r b = Z bg/Z bgc.

So, the equations (2.12) and (2.13) can be written as,

emin(x)/Kb ≥ [{ – (f cc,all)service/(η·f ccg) + 1} + {Msw + MDeck + (MSIDL + MLL)·rt}/(η·f ccg·Ztg)] or (2.15a)

emin(x)/Kb ≥ [{ – (f cc,all)service/(η·f ccg) + 1} + Msw / (MR·η·f ccg·Ztg)] (2.15b)

emin(x)/Kt ≥ [ {(f ct,all)service/(η·f ccg) – 1} + {Msw + MDeck + (MSIDL + MLL)·rb}/(η·f ccg·Z bg)] or (2.16a)

emin(x)/Kt ≥ [ {(f ct,all)service/(η·f ccg) – 1} + Msw / (MR·η·f ccg·Z bg)] (2.16b)

The B.M. due to s/w at any position x along the length of girder can be expressed as:

Msw = A· γc · [ L·x – x2] / 2 (2.17)

So, the equations (3), and (4) can be written as,

emax(x)/Kb ≤ [{ – (f ct,all)transfer /f ccg + 1} + A·γc·{L·x – x2}/(2·f ccg·Ztg) ] or (2.18a)

emax(x)/Kb ≤ [{ – (f ct,all)transfer /f ccg + 1} + γc·{(x/L) – (x/L)2}·(L2/Kb)/(2·f ccg) ] (2.18b)




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emax(x)/Kt ≤ [ {(f cc,all)transfer /f ccg – 1} + A·γc·{L·x – x2}/(2·f ccg·Z bg)] or (2.19a)

emax(x)/Kt ≤ [ {(f cc,all)transfer /f ccg – 1} + γc·{(x/L) – (x/L)2}·(L2/Kt)/(2·f ccg)] (2.19b)

emin(x)/Kb ≥ [{ – (f cc,all)service/(η·f ccg) + 1} + A·γc·{L·x – x2} / (2·MR·η·f ccg·Ztg) ] or (2.20a)

emin(x)/Kb ≥ [{ – (f cc,all)service/(η·f ccg) + 1} + γc·{(x/L) – (x/L)2}·(L2/ MR·Kb)/(2·η·f ccg)] (2.20b)

emin(x)/Kt ≥ [ {(f ct,all)service/(η·f ccg) – 1} + A·γc·{L·x – x2

} / (2·MR·η·f ccg·Z bg) ] or (2.21a)emin(x)/Kt ≥ [ {(f ct,all)service/(η·f ccg) – 1} + γc·{(x/L) – (x/L)2}·(L2/ MR·Kt)/(2·η·f ccg)] (2.21b)

The equations (2.18) & (2.19) are functions of (L2/K b) & (L2/K t) whereas the equations (2.20) & (2.21) are functions of

(L2/MR ·K b) & (L2/ MR ·K t) respectively.

2.1 Uniformly Loaded Girder:

A non-dimensional generalized chart for given ratios of sectional dimensions can be obtained by taking different values of

(L2/K b), (L2/K t), (L2/MR ·K b) & (L2/ MR ·K t) lying within practical range and plotting e/K vs. x/L. Out of the four curves, only two

curve closer to the centroidal axis will govern, usually equation (2.18) and (2.21) and define the acceptable zone of eccentricity

within which the resultant of cables should lie. Charts are prepared for different grade of concrete and sectional dimension ratios

and one such chart is presented with illustrative example. In calculating the values of (L2/K b), (L2/K t), (L2/MR ·K b) & (L2/ MR ·K t),

the span is in metre and the kern distance is in millimetre.While calculating M R it should be noted that for emin(x)/Kb, r should bereplaced by rt whereas for emin(x)/Kt, rb should be used in place of r. However, the above chart is applicable for girders subjected

to uniform loads only.

2.2 Non-Uniformly Loaded Girder:

Generally, the bending moment changes at different points along the length of beam as the load is not always uniform and also

the values of Kb and Kt when the cross section is not uniform. So, to consider the non-uniformity in loading and cross section of a

particular case, the above equations can be programed in a computer program by replacing the terms as a function of position x

i.e. Kb by Kb(x), Kt by Kt(x) and MR by MR(x). For a given girder, the values of Kb(x) and Kt(x) can easily be computed. But

the calculation for MR needs concern. The method presented below simplifies the tedious calculations for MR and can be

incorporated in spreadsheet. For a given girder, the cross section can also be plotted on same e/(Kt or Kb) axis by suitably scaling

the cross section. This helps in visualizing the acceptable zone with respect to girder c/s. The user defined cable eccentricities andtheir resultant can also be plotted with suitable scale so that the user can check visually whether it falls within the acceptable zone

and can accordingly modify the eccentricity of each cables.

The bending moment can be expressed as a function of x and the same is derived for different stages as below. It is simpler to

formulate the B.M. due to D.L. and S.I.D.L. in terms of any position x along the length of girder as it is stationary loads and hence

not discussed in this paper. But the B.M. due to vehicular loads need concern as the position of loads changes at every instant. To

overcome this, the concept of Conjugate Stationary Load System or CSLS suggested by Marius B. Wechsler is utilized. The Class

70R Wheeled vehicle and Class A vehicular loads are replaced by CSLS as follows:

Fig. Class 70R Wheeled Vehicle

Fig. CSLS for Class 70R Wheeled Vehicle




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Maxi. B.M. at any position x along the span due to single train of Class 70R wheeled vehicle is given by,

Fig. Class A Vehicle

Fig. CSLS for Class A Vehicle

Maxi. B.M. at any position x along the span due to single train of Class 70R wheeled vehicle is given by,

It should be noted that the above CSLS is for spans covering one complete train of vehicular loads. For spans smaller than the

train length of vehicular loads, actual wheeled loads acting on the span should be considered for computing CSLS. For larger span

carrying consecutive trains of vehicular loads, the CSLS should be suitably modified.The B.M. calculated above should be suitable modified by multiplying with reaction factors, Impact factors and reduction

factors.

3. Illustrative Example

A post tensioned bridge girder having uniform cross section throughout

the length as shown in fig. is subjected to uniformly distributed SIDL & L.L.

of 4 kN/m and 10 kN/m respectively. The girder also supports the deck slab

of effective width 2.0 m and thickness 0.15 m. The effective span of girder is

30 m. Determine the safe acceptable range of eccentricity along the length of

girder. Take grade of concrete M40.

Given that: wSIDL = 4 kN/m2

; wLL = 10 kN/m2

; L = 30 m; f ck = 40 N/mm2

For given c/s: K b = 520.92116 mm; K t = 402.23026 mm;

r b = Z bg/Z bgc = 0.25341 / 0.3268 = 0.58021 ; w s/w = 0.63 x 25 = 15.75

kN/m

wDECK = 2.0 x 0.15 x 25 = 7.5 kN/m

MR = 15.75/[(15.75 + 7.5) + (4 + 10) x 0.58021] = 0.50202

L2/K b = 1.7277; L2/(MR ·K t) = 4.457

The values of upper and lower limits can be interpolated from the chart

The dashed line in the chart shows the acceptable zone for above values of L2/K b & L2/(MR ·K t).




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4. Conclusion

The proposed charts can be used for determining the acceptable zone of eccentricity of a uniformly loaded simply supported

girder. The modified charts take into consideration the construction stage sequence as well as some practical limitations of placing

the cable which were missing in earlier works. The simplified method presented to compute B.M. and acceptable zone of

eccentricity at any position helps in simple programing.

References

1. Krishnamurthy N. Magnel diagrams for prestressed concrete beams, ASCE Vol. 109, No. 12, Dec. 1983, pg. 2761-2769

2. Krishnamurthy N. Modified magnel diagram as design aid for prestressed concrete bridge member, ACI, SP 26-27, pg. 663-668

3 Mohammad R. Ehsani, J. Russell Blewitt. Design curves for tendon profile in prestressed concrete beams. PCI Journal / May-June 1986, pg. 114-135

4. Marius B. Wechsler, Moment Determination for Moving Load Systems. ASCE, Journal of Structural Engineering, Vol. III, No. 6, June, 1985

5. R. H. Evans & E. W. Bennett. Pre-stressed concrete theory and design (John Wiley & Sons, NY), p. 130-155

6. N Krishna Raju. PRESTRESSED CONCRETE (Fourth Edition), Tata-McGraw-Hill Company Ltd., ND

7. Prof. Devdas Menon, Prof. Amlan Kumar Sengupta. Prestressed concrete structure. NPTEL Lecture Handouts

8. Code of Practice for Concrete Road Bridges, IRC: 112: 2011, The Indian Roads Congress, New Delhi.

9. Standard Specification and Code of Practice for Road Bridge, Section – II, Loads and Stresses, IRC: 6-2010.The Indian Roads Congress, New Delhi.

Appendix-Notation

Po = initially applied prestress

η = prestress loss ratio

A = cross section area of concrete section

C.A. = centroidal axis of concrete section

CGS = centroid of prestressing cables

emax / emin = maximum eccentricity / minimum eccentricity

Ztg / Z bg = section modulus of girder alone at top / bottom

Ztgc / Z bgc = section modulus of composite section at top / bottom

Msw = Moment due to self weight

MDECK = Moment due to deck slab

MSIDL = Moment due to SIDL

MLL = Moment due to LL

Kt / Kb = Top / Bottom kern distance(f cc,all)transfer = maximum allowable compression in concrete at transfer stage

(f ct,all)transfer = maximum allowable tension in concrete at transfer stage

(f cc,all)service = maximum allowable compression in concrete at service stage

(f ct,all)service = maximum allowable tension in concrete at service stage

f ccg = centroidal stress in concrete at transfer stage.

γc = density of concrete

MR = Moment ratio

r = ratio of section modulus of girder alone to composite section

r t = ratio of section modulus of girder alone to composite section at top of the girder

r b = ratio of section modulus of girder alone to composite section at bottom of the girder

L = effective span of girder.

x = position along the length of girder.




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determination of preliminary cable profile in prestressed concrete girders

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