Build. Sci. Vol. 8, pp. 179-185. Pergamon Press 1973. Printed in Great Britain ] I I I I
Flexural Behaviour of Pretensioned
Concrete Beams with Limited Prestress
N. K R I S H N A RAJU*
B. S. BASAVARAJAIAH*
U. C. A H A M E D K U T T Y *
The behaviour of class-3 type prestressed concrete beams, at the limit states of cracking, deflexion and collapse are investigated by experiments on preten- sioned beams with mild steel as supplementary reinforcement. Several methods of calculating the width of cracks are examined in the light of experimental results and an empirical formula, which includes the effect of percentage of untensioned reinforcement is suggested.
The deflexions of beams, based on the method of Beeby and Taylor, were marginally conservative when compared with the experimental results. The ultimate moment capacity of concrete sections with tensioned and untensioned reinforcement, was underestimated by as much as 10, 15 and 25 per cent by the Indian, British and American Code recommendations.
I N T R O D U C T I O N
T H E P H I L O S O P H Y of design termed "Limit state approach", adopted by the Russian Code in 1954 is being adopted slowly by other National Codes. The influence of this design approach is evident in the revised American Code[l] and the British draft unified code[2]. Limit state design of prestressed concrete structural elements requires a proper knowledge of the behaviour of the members at the multiple limit states of cracking, deflexion and collapse. The use of limited or partial pre- stressing of concrete structures in which cracks of limited width are acceptable under occasional over loads or even under working loads, is embodied in the recent C E B - F I P recommendations[3]. The purpose of this investigation is to study the load- deformation, cracking and strength characteristics of pretensioned beams with different degrees of prestress and with different percentages of supple- mentary reinforcement.
Test-specimens
Flexural tests were made on beams with a rect- angular cross section, 100 mm wide by 200 mm deep, over an effective span of 2.76 m. The details of pretensioned wires and supplementary reinforce- ment used in beams are given in Table 1 and figure 1. Two beams were fully prestressed and for the remaining six beams the prestress was varied from 3.0 to 8.6 N f m m 2. Mild steel bars conforming to
* Department of Civil Engineering, Karnataka Regional Engineering College, Surathkal, (S.K.), Mysore State, India.
the Indian Standard IS :43214] with a guaranteed yield stress of 260 N / m m 2 were used as supplemen- tary reinforcement. The prestress in the concrete was obtained by using 5 mm dia high tensile wires having a guaranteed 0.2 per cent p r o o f stress o f 1500 N / m m 2 and an ultimate tensile strength of 1650 N / m m 2.
In the prestressed beams of each group the degree of prestress was varied by controlling the number of high tensile wires and the flexural strength of all the beams was maintained approximately constant by using the required amount of supplementary reinforcement in the form of mild steel.
The beams were cast in a column type preten- sioned bed fabricated in the laboratory. The con- crete for the beams was mixed in the proportions of 1:0.72:2.18 by weight with a water-cement ratio of 0.38 using ordinary portland ce~nent, together with sand and crushed granite aggregate of 20 mm, maximum size. The beams were cured by covering them with wet burlaps for 28 days before the start of tests. The average compressive strength of con- creterecorded by testing 10-cm cubes was 42 N / m m 2.
179
I N S T R U M E N T A T I O N AND E X P E R I M E N T A L PROCEDURE
The prestressed beams were tested using a load- ing frame and a 10 tonne hydraulic jack in conjunc- tion with a proving ring for loading in small incre- ments. The beams supported on rocker and roller bearings were loaded at one third points of the effective span. Batty dial gauges were used for recording the deflexions at regular load intervals.
Table I. Details o f test beamx
Nominal effective N u m b e r o f Supplementary Percentage of Approximate Beam depth (ram) 5 m m dia. reinforcement untensioned prestress at n u m b e r . . . tensioned mild steel reinforcement soffit o f beam
dr. ~ d,,, wires rods ( N / m m 2 )
F P - I A 137 180 5 I 1.0
FP IB
L P - I A 139 180 4 6 m m dia. 0.3 8.6
LP- 1 B 2 Nos.
L P - 2 A 143 180 2 10 m m dia. 1.2 5-5
LP-2B 3 Nos.
LP-3A 161 180 1 12 m m dia. 1.6 3"0
LP-3B 2 N o s . and
8 m m dia.
i No.
100 mm - - - *7
E E
0 o Tensioned Steel
!
Untensioned Steel
b r_
File. 1. Cross section o f test beams.
[
f
_ _ i _
Strain measurements were made for the central section of the beam by using a 20 cm demec gauge developed by Morice and Base[5]. Stainless steel demec targets were fixed on the surface at regular intervals to study the strain distribution across the central section of the beam. The cracks developed on the surface of the beams, and were observed through a Begg's microscope. The smallest width of cracks measurable with the instrument being 0.0015 ram. The maximum crack widths were mea- sured at loading increments of 400 kg.
DISCUSSION OF TEST RESULTS Cracking characteristics
The influence of supplementary reinforcement on the cracking characteristics of beams with varying degree of limited prestress is detailed in figure 2.
The crack width behaviour is presented for increas-
ing applied loads expressed as a function of the calculated ultimated load according to IS: 1343[6].
The crack width at design load is considerably affected by the percentage of untensioned rein- forcement used in the section. Increase of sup- plementary reinforcement significantly reduces the width of cracks at design loads. In the case of class-3 prestressed beams, Bennett and Chandra- sekhar[7] have reported increased crack widths
"~ o.~
E
~ 0.~
L P * 2 A 1 2
: ~ L P - 3 A 1 B
L P - 3B
0 2 Q 3 0 4 0 5
C ~ a c k W , a t h - ~
Fig. 2. Influence o f supplementary reinforcement on load- crack width relationship.
when smaller area of steel is used as reinforcement.
The draft unified code also recognises the import- ance of percentage of untensioned reinforcement in the cross section in limiting the allowable fictitious tensile stresses in the beam.
Prediction of crack width
In the limit state design, local damage is defined by a maximum allowable width of crack. Conse- quently the method of calculating crack width is of
Flexural Behaviour o f Pretensioned Concrete Beams with Limited Prestress 181
considerable importance. Methods suggested by Goschy[8], Borges[9], Birkenmeier[10], Baus[11] and the recent CEB-FIP report, relate crack width to the stress in the reinforcement. Abeles[12] has used the concept of fictitious tensile stress in concrete to predict the width of cracks, and the British draft unified code recommendations are based on this approach.
The reinforcement-stress method, although in- herently more accurate for crack width predictions, has the disadvantage of lengthy calculations, whereas the fictitious tensile stress approach with its relative simplicity yields reasonably accurate values for practical purposes.
For the present series of tests on class-3 type pretensioned beams, the variation of crack width with fictitious tensile stress is presented in figure 3.
24
I
~ 12 ~
'5 o
p-
Untensioned
~ ReiNorc , m e n t
4 - - rl • LP-1A. B 0.3"/, 0 0 LP-2A.B 1.2"k A • LP-3A.II 1.6"k
0 0.1 0.2 0.3 0.4 0.5
Crack Width at Soffit - mm
Fig. 3.
Relation between crack width and fictitious
tensile stress.
Based on the results of tests, it was found that the crack widths could be estimated by an expression of the type:
R
where C is the cover over reinforcement, f , is the fictitious tensile stress in concrete, Pe is the percen- tage of untensioned reinforcement, and R is a factor depending upon the type of reinforcement.
The experimental value of the constant R is 750x 10 -6 mm2/N for mild steel bars, having a guaranteed yield stress of 260 N/mm z.
In figures 4, 5 and 6, the experimental results are shown together with the recommended relations of Beeby and Taylor[13], Bennett and Chandra-
2 4 ] ~ LP-1A ---- "-- LP -1B
Beeby and Taylor
20 Bennet and
C handrasekhar Proposed Equation
~E E z i
g
p-
o
5 u.
1 6
12
0 0.1 0.2 0.3 0.4 0.5
Crack Width at Soffit- mrn
Fig. 4. Relation between crack width and fictitious
tensile stress.
~E E z )
o
p- ,o_
h 24
/ / 4 / / /
i I . / ' / /
0 0.1
/ / i / /
/ t ,,"
/ / "
. ~ "
,,/ /
/
//,/,"
/J O O L P - 2 A
/ ' ~ LP - 2B
/ Beeby lnd Taylor
Bennet and C handrasekhar Proposed Equation
0.2 0.3 0.4 0.5
Crack Width at Soffit-ram
Fig. 5. Relation b e t w e e n c r a c k
width and fictitious
tensile stress.
sekhar[7] and the proposed equation. The wide dis- crepancy between the different proposals is attri- buted to the different type of untensioned reinforce- ment used and the number of variables considered.
The proposed equation incorporates the percen- tage of untensioned reinforcement as an additional variable, when compared with the other recom-
~ E E z
i
L
lu
o u
2 4
2 0
1 2
i ! / / /
1
/ /
/ /
/ /
/ / A
/ ! / •
/I / Z _ .
;U
/
/t
/ / /
/
/ / /6 L P - 3 A A L P - 3 B
B e e b y and T a y l o r Bennet and
Chandrasekhar . . . . P r o p o s e d EqOation
O 0.1 0.2 0.3 0.4 0 5
C r a c k Width at Soffit-mnn
Fig. 6. Relation between crack width and fictitious tensile stress.
~E E
z
L 2 0
Max. Crack Width 0.1 m m
i e d Code
I"1 • LP-1A. B O • L P - 2 A , B A • L P - 3 A , B
I
0.5 1.0 1.5 2.0 2 . 5
U n t e n s i o n e d R e i n f o r c e m e n t - Percent Fig. 7. Relation between fictitious tensile s t r e s s a n d
untensioned reinforcement.
Table 2. Results o f maximurn crack width at working loads'
Maximum crack width Beam observed at 55 per
No. cent of ultimate load (ram)
Calculated maximum crack width at 55 per cent of calculated ultimate load
(according to IS : 1343) Beeby and Bennett and Proposed
Taylor Chandrasekhar equation
LP-1A 0.12 0'02 0.16 0.17
LP-1B 0.14 0.02 0.16 0.17
LP-2A 0-09 0'04 0"22 0 14
LP-2B 0.11 0.04 0.22 0'14
LP-3A 0"05 0"06 0.28 0.10
LP-3B 0.08 0'06 0'28 0" 10
Mean Ratio of Standard calculated deviation to observed
crack width Coefficient of variation (per cent)
0.51 2.60 1.40
0.40 1.67 0.30
78 64 21
m e n d a t i o n s . T h e i m p o r t a n c e o f the effective a r e a o f r e i n f o r c e m e n t on the p r e d i c t i o n o f c r a c k w i d t h s has been r e p o r t e d by K a a r a n d Mattock[14].
T a b l e 2 shows the results o f m e a s u r e d c r a c k w i d t h s at w o r k i n g l o a d s c o r r e s p o n d i n g to 55 p e r cent o f the t h e o r e t i c a l u l t i m a t e l o a d , c o m p a r e d with the p r e d i c t i o n s o f the v a r i o u s a u t h o r s a n d the p r o p o s e d e q u a t i o n .
T h e m e c h a n i s m o f c r a c k f o r m a t i o n is such t h a t a wide scatter is i n h e r e n t in the m e a s u r e d values o f c r a c k widths in tests a n d this is reflected in the statistics o f the r a t i o o f c a l c u l a t e d t o o b s e r v e d c r a c k w i d t h s c o m p i l e d in T a b l e 2. I n figures 7 a n d
8, the p e r m i s s i b l e fictitious tensile stresses f o r speci- fied c r a c k w i d t h s a n d percentage o f u n t e n s i o n e d r e i n f o r c e m e n t as r e c o m m e n d e d b y the 1969 D r a f t British C o d e are c o m p a r e d with the results o f p r e s e n t tests. F r o m the plot, the c o d e r e c o m m e n d a - tions a p p e a r to be c o n s e r v a t i v e for p e r c e n t a g e s o f u n t e n s i o n e d steel exceeding 1.5.
Deflexions o f class-3 type prestressed b e a m s
A k n o w l e d g e o f the m o m e n t - c u r v a t u r e r e l a t i o n - ship is essential to c a l c u l a t e the deflexions o f p a r - tially p r e s t r e s s e d m e m b e r s in which c r a c k s a r e
Flexural Behaviour of Pretensioned Concrete Beams with Limited Prestress 183 permitted under service loads. Investigations of
moment-curvature relationships are reported by Beeby[15] in which a number of possible bilinear relationships are proposed and tested against experimental data. The moment-curvature relation- ship of each of the beams with limited prestress, was established by analyzing the cracked prestressed section on the lines suggested by Beeby and
E z i
9- u.
2 4
20
Max. Crack Width 0.2 mm
I Z~
I
16
13
J
/
J D r a f t Unified Cote
[] • L~- 1A, B 0 • LP- 2A,I]
A • LP- 3A, B
0 0.5 1.0 1.5 2.0 2,5
U n t e n s i o n e d R e i n f o r c e m e n t - P e r c e n t
Fig. 8. Relation between fictitious tensile stress and untensioned reinforcement.
4800
.4 o 3200
2400
sssss
~1~" ~ _ Design._ L o a d /
--
/
11 [] n LP - 1A
/ H LP -IB
BeellV and T a y l o r
O 8 16 2 4 3 2 4 0
Deflexion - m m Fig. 9. Load-deflexion characteristics.
4 8 0 0
3200
i
J
1600
/
0
,' I
// I
C ~ O LP -2A
] 0 " - ' - - ' 0 LP -2B
i
J Beeby and Taylor
8 16 2 4 3 2 4 0
D e f l e x i o n - m m
Fig. 10. Load-deflexion characteristics.
==
i
.J , " " Design L o a d
1~0C
f/
iJ
" ~ ~ LP - 3A
,' t • • LP -3B
Beeby and Taylor
0 8 16 2 4 32 40
D e f l e x i o n - mm Fig. 11. Load-deflexion characteristics.
Taylor[ 13]. Theoretical load-deflexion curves shown in figures 9, 10 and 11 were derived by using an approximate method developed by Beeby and Miles[16], in which the maximum deflexion of the beam is expressed as a function of the maximum curvature, length and a constant which depends on the shape of the curvature diagram.
The theoretical predictions of deflexions of beams when tested against the experimental results in the figures has indicated that the theoretical estimates are consistently larger for all the three groups of beams and the difference being larger for loads over and above the design load. In Table 3, the theo- retical and observed deflexions of beams at design load, corresponding to 55 per cent of the calculated ultimate load, are compiled. The ratio of the ob- served to the calculated defexion varied from 0.56 to 0.82, indicating the conservative estimates of the theoretical procedure used in the computations.
Table 3. Deflexions of prestressed beams Deflexion at 55 per cent of
ultimate load (mm)
Beam Ratio of observed
number Calculated to calculated
Observed (Beeby and deflexion Taylor)
FP-IA 5-0 ....
FP-IB 4.0 . . . .
LP-IA 5.0 9.0 0.56
LP-IB 6.0 9.0 0.67
LP-2A 5.0 8-5 0.59
LP-2B 6.5 8.5 0.77
LP-3A 5.5 8.0 0.69
LP-3B 6.5 8.0 0.82
The observed deflexions at design load are well within the limit of span/350 suggested by Bate[17]
in the summary of basic requirements for limit state design and within the limit of span/360 proposed by the new A C I building code for floors not support- ing or attached to non-structural elements likely to be damaged by large deflexions.
Ultimate m o m e n t capacity
The ultimate moment capacity of beams with full and limited prestress is compiled in Table 4, in which the observed moments are compared with the theo- retical predictions based on the American, British[18]
and the Indian Standard codes. The ratio of ob- served to calculated ultimate moment was never
Table 4. Moment capacity of prestressed beams
Beam number
Observed ultimate moment (t.cm)
Ratio of observed to calculated ultimate moment using IS:1343 CP:II5 ACI-318 FP-1 A 180 1 "06 1 "08 1 '28 FP-1 B 172 1 "01 1-04 1-22 LP-IA 184 1"06 1'12 1"28
LP-I B 172 1-00 1 "05 1-20
LP-2A 195 1-11 1-17 1-20
LP-2B 196 1" 12 1.18 1-21
LP-3A 216 1"20 1'25 1"32
LP-3B 225 1-24 1"34 1-36
less than 1.0. The Indian, British and American code recommendations under-estimate the ultimate strength of prestressed beams used in this investiga- tion by as much as 10, 15 and 25 per cent respec- tively, it is well established that the maximum steel stress at failure in prestressed beams is a function of the effective proportion of steel. A comparative study by Ramakrishnan[19] indicates that the ACI recommendations grossly underestimate the stress in steel at failure, compared to the British and Indian Codes which is reflected in the low values of the ultimate moment capacity of the cross section.
However a more accurate method such as that pre- sented by Warwaruk, Sozen and Siess[20] for calcu- lation of stress in steel at failure would result in moments comparable to that of the observed values.
C O N C L U S I O N S
The following conclusions have been drawn by the study of the limit state behaviour of concrete beams with mild steel as supplementary reinforce- ment and limited prestress.
1. The percentage of untensioned reinforcement was found to have an important influence on the width of cracks.
2. The proposed expression for predicting the width of cracks accommodating the variables like the type and percentage of untensioned reinforce- ment, cover and fictitious tensile stress was found to give good correlation with the experimental results.
3. The formula proposed by Beeby and Taylor, underestimates the width of cracks in beams with mild steel as supplementary reinforcement.
4. The British draft code recommendations of permissible fictitious tensile stress for maximum crack width of 0-1 and 0.2 mm appear to be con- servative for percentages of untensioned reinforce- ment exceeding 1.5.
5. The deflexions of beams, based on the method recommended by Beeby and Taylor were always on the safe side when compared with the experimental results.
6. The ultimate moment capacity of concrete sections with tensioned and untensioned reinforce- ment was underestimated by as much as 10, 15 and 25 per cent by the Indian, British and American Code recommendations respectively.
Acknowledgement--The experimental investigations were carried out in the laboratory of the Department of Civil Engineering, Karnataka Regional Engineering College, Surathkal, India. The Junior Author (Mr. U. C. Ahamed Kutty) is indebted to the Ministry of Education, Government of India for the financial help provided during the project work.
Flexural Behaviour o f Pretensioned Concrete Beams with Limited Prestress 185 REFERENCES
I. A.C.I. Building Code requirements for reinforced concrete--ACI.318-71. Detroit, Michigan (1971).
2. British Standards Institution. Draft British Standard Code of Practice for the structural use of concrete, BSI, London (I 969).
3. CEB-FIP Joint Committee. International Recommendations for the design and con- struction of concrete structures. Cement and Concrete Association, London, June (1970).
4. Indian Standard Specification for mild steel and medium tensile steel bars and hard drawn steel wire for concrete reinforcement. IS :432 [Part I], 8 (1966).
5. P. B. Mo~cE and G. D. BASE, The design and use of a demountable mechanical strain gauge for concrete structures. Mag. Concr. Res., London 5, 37 (1953).
6. Indian Standard Code of Practice for Prestressed Concrete, IS:1343, Indian Standards Institution, New Delhi (1960).
7. E. W. BENNETT and C. S. CHANDRASEKHAR, Calculation of the width of cracks in class-3 prestressed beams. Proc. Inst. Civ. Engrs, London 49, 333 (1971).
8. B. GOSCHY, Partially stressed reinforced concrete. Melyepitestud, Szle 14, 81 (1964).
9. J. F. BORGES, CEB Commission IVa--Cracking. Commission meeting held on 18th April, 1968 in Lausanne, preliminary report. Laboratorio de Engeharia Civil, Lisbon, March (1968).
10. M. BIRKEN~rER, Design of partially prestressed structures in accordance with the [1966]
standards of the Swiss Association of Engineers and Architects. Ann. Inst. Tech. Batim.
21, 57 (1968).
11. R. BAgs et al., Fissuration des Poutres Armees Precontraintes; Calcul de l'ouverture des Fissures, University of Liege, Provisional Report (1968).
12. P. W. ABELES, Design of partially prestressed concrete beams. J. Am. Concr. Inst. 65, 669 (1967).
13. A.W. BEEBY and H. P. J. TAYLOR, Cracking in partially prestressed members, preliminary publication, 6th F.LP. Congress, Prague, Concrete Society, London, June (1970).
14. H. KAAR and A. H. MATTOCK, High strength bars as concrete reinforcement, control of cracking. J. Portland Cement Assoc. Res. Devel. Lab., Part 4, 15 (1963).
15. A.W. BEEBY, Short term deformations of reinforced concrete members, Technical Report TRA-408, Cement and Concrete Association, London, March (1968).
16. A. W. BEEBY and J. R. MILES, Proposals for the control of deflection in the new unified code. Concrete 3, 101 (1969).
17. S.C.C. BATE, Why limit state design? Concrete 2, 103 (1968).
18. British Standard Code of Practice, CP.115:1959. The structural use of prestressed concrete in buildings. The British Standards Institution, London (1959).
19. V. RAMAKRISHNAN and P. D. ARTHUR, Ultimate Strength Design for Structural Concrete, p. 46, Pitman, London (1969).
20. J. WARWARtJK, A. M. SOZEN and C. P. Smss. Investigation of prestressed reinforced concrete for highway bridges. Part 3--Strength and behaviour in flexure of prestressed concrete beams. Bulletin No. 464, Engineering Experiment Station, University of Illinois, Urbana (1962).
Le comportement de poutres en b6ton pr6-tension6 de la classe 3 aux 6tats limites de cassure, de d6flection et d'effondrement sont 6tudi6s sur des poutres pr6-tension6es avec de l'acier doux comme renforcement suppl6mentaire. Plusieurs m6thodes de calcul sugg6r6es pour la largeur des fentes sont examin6es en vue des r6sultats exp6rimentaux et des formules empiriques, qui comprennent l'effet du pourcentage de renforcement non-tension&
Les fl6chissements de poutres, fond6es sur la m6thode de Beeby at Taylor sont sensiblement conservateurs lorsqu'on les compare aux r6sultats exp6rimentaux. La capacit6 de moment ultime de sections en b6ton avec renforcement tension6 et non- tension6, a 6t6 sous-estim6e par 10, 15 et 25 pourcent m~me par les recommandations des codes Indien, Anglais et Am6ricains.
Das Verhalten von Spannbetontr/igern der Klasse 3 wird durch Versuche an Span- nungstr/igern mit Flusseisen als zus/itzliche Bewehrung in den Grenzzust/inden der Rissbildung, der Durchbiegung und des Zusammenbruchs untersucht. Mehrere Verfahren zur Berechnung der Rissbreite werden auf Grund experimenteller Ergeb- nisse untersucht, und es wird eine empirische Formel vorgeschlagen, welche die Wirkung prozentualer, nicht vorgespannter Bewehrung einschliesst.
Die Durchbiegungen von Tr/igern auf Grund des Verfahrens von Beeby and Taylor waren ziemlich m~issig im Vergleich mit den experimentellen Ergebnissen. Die /iusserste Momentenleistung yon Betonabschnitten mit vorgespannter und nicht vorgespannter Bewehrung wurde bei den Indischen, Britischen und Amerikanischen empfohlenen Vorschriften bis auf 10, 15 und 25 Prozent untersch/itzt.
