Ultimate Stress of Unbonded Tendons in Partially Prestressed Concrete Beams Xuekang Tao Research Engineer China Academy of Building Research Beijing, People's Republic of China Gongchen Du Professor and Deputy Chief Engineer China Academy of Building Research Beijing, People's Republic of China U nbonded tendons are being used in- creasingly in partially prestressed concrete structures in China. This is be- cause unbonded tendons usually have a lower unit cost compared to bonded tendons while also offering simplicity in construction. Tests indicate that the ultimate strength of unbonded beams can be in- creased by the addition of bonded non- prestressed reinforcement. Although several research papers' focus on the presence of nonprestressed reinforce- rnent, this practice is not currently rec- ognized in many building codes 2 for computing the ultimate stress in un- bonded tendons at beam failure. That is to say, although most designs employ at least a minimum amount of nonpre- stressed steel for distributing cracks in concrete and Iimiting their width, the actual influence of this added rein- forcement on the ultimate stress of un- bonded tendons is often neglected. For example, the minimum bonded rein- forcement in AC! 318-83 is only 0.4 per- cent of the area of that part of the beam section between the flexural tension face and the neutral axis of the gross section, Since partially prestressed concrete is classified between fully prestressed concrete and ordinary reinforced con- crete, the amount of nonprestressed steel can vary widely. Therefore, re- search must he conducted to determine the effect of bonded nonprestressed steel on the ultimate stress in unbonded tendons, as well as on the ultimate strength of the beam itself. 72
Research EngineerChina Academy of Building Research
Beijing, People's Republic of China
Gongchen DuProfessor and Deputy Chief EngineerChina Academy of Building ResearchBeijing, People's Republic of China
U nbonded tendons are being used in-creasingly in partially prestressed
concrete structures in China. This is be-cause unbonded tendons usually have alower unit cost compared to bondedtendons while also offering simplicity inconstruction.
Tests indicate that the ultimatestrength of unbonded beams can be in-creased by the addition of bonded non-prestressed reinforcement. Althoughseveral research papers' focus on thepresence of nonprestressed reinforce-rnent, this practice is not currently rec-ognized in many building codes 2 forcomputing the ultimate stress in un-bonded tendons at beam failure. That isto say, although most designs employ atleast a minimum amount of nonpre-stressed steel for distributing cracks in
concrete and Iimiting their width, theactual influence of this added rein-forcement on the ultimate stress of un-bonded tendons is often neglected. Forexample, the minimum bonded rein-forcement in AC! 318-83 is only 0.4 per-cent of the area of that part of the beamsection between the flexural tensionface and the neutral axis of the grosssection,
Since partially prestressed concrete isclassified between fully prestressedconcrete and ordinary reinforced con-crete, the amount of nonprestressedsteel can vary widely. Therefore, re-search must he conducted to determinethe effect of bonded nonprestressedsteel on the ultimate stress in unbondedtendons, as well as on the ultimatestrength of the beam itself.
72
TEST PROGRAMThe main factors which may affect the
behavior of unbonded partially pre-stressed concrete beams are:
1. Amount of prestressed reinforce-ment
2. Amount of nonprestressed rein-forcement
3. Material properties4. Effective prestress in tendons im-
mediately before testing5. Span to depth ratio6. Initial tendon profile7. Form of loading8. Friction between tendon and ductIn this paper, only Items 1 through 3
were investigated, with the main vari-ables adopted in the test beams beingAp,A.andff.
All test beams (Fig. 1) were approxi-mately 160 x 280 mm (6 x 11 in.) in crosssection, 4400 mm (14 ft 6 in.) in length,and were tested with third point loadingover a 4200 mm (13 ft 9 in.) span. Thespan to depth ratio, 1/dp , of the beamswas 19.1.
Each beam has one straight tendonconsisting of two to eight high strengthwires 5 mm (0.20 in.) in diameter. Theprestressing wires were coated with athin layer of grease, approximately Ito 2mm (0.04 to 0.08 in.) thick, and thenwrapped with three layers of plasticpaper. In order to reduce the pull-inlosses, buttonhead anchorages wereadopted for the tendons. All beams weretensioned prior to testing and the effec-tive prestress of the tendons was 55 to 65percent of the yield strength of thewires. The strength of the concrete, f,was 30 to 50 MPa (4350 to 7250 psi).
In addition to the unbonded tendons,each beam also contained from two tofour additional bonded nonprestresseddeformed bars 10, 14 and 16 mm (0.4,0.55 and 0.63 in.) in diameter. This re-inforcement was selected on the basisthat the beams at failure would fall intothree categories with the nonpre-stressed steel carrying about 30, 50 or 70
SynopsisThis paper studies the effects of
varying amounts of non prestressedreinforcement on the stress in un-bonded prestressing tendons atflexural strength in partially pre-stressed concrete beams. The studywas both experimental and analyticalin scope. Altogether, twenty-two un-bonded and four bonded partially pre-stressed concrete beams were tested.
Test results show that the stress inunbonded tendons at flexural strengthis a function of the reinforcement indi-ces of both the unbonded tendons andthe bonded nonprestressed rein-forcement. The analytical data agreeclosely with the experimental results.
An empirical equation is included toestimate the ultimate stress in un-bonded tendons.
percent of the total ultimate load. Thus,it was expected that the influence of thebonded steel on the ultimate stress inunbonded tendons might also be ob-served.
At the same time, the combined rein-forcement index, q 0 , was divided intothree levels: low (q0 < 0.15), medium (qo= 0.15 to 0.25) and high (q0 > 0.25). Thisallowed the effect of steel content on theultimate stress in unbonded tendons tobe observed. The minimum A 8 ofbonded nonprestressed reinforcementused in this study was about 0.004 bdp.
The 26 test beams were divided intofour groups with details as listed inTable 1.
Group A consisted of nine beamswhich were divided into three catego-ries. Each category contained threebeams with each of the different levelsofg0 , as stated above.
Each beam in Group B was identicalto the corresponding beam in Group Aexcept that the strengths of the wire
PCI JOURNAUNovember-December 1985 73
100mmiann....-. P/2 i., nn-- P12 wnn--
SOOmm 60omm 700mm 700mm 606mm 80omm4200 mm
E
7EEEl pX =STRAW GAGES ON WIRES
f N —o =STRAiNGAGES ON TOP SURFACEOF BEAMS
16O m m
Fig. 1. Loading arrangement and instrumentation on test beams.
100mm
C9
and concrete were higher while theamount of prestressing steel in BeamsB-6 and B-9 was less than that of BeamsA-6 and A-9.
Group C consisted of four beamswhich were identical to Beams A-1, A-3,A-7 and A-9, except that cold stretchedbars with a higher strength were usedinstead of the ordinary reinforcing steel.
Finally, Group D consisted of fourheams which were bonded, Beam D-0
was an ordinary reinforced concretebeam while Beams D-1 and D-3 wereduplicates of their counterparts inGroup A except that dispersed preten-sioned wires were used instead. The lastbeam, D-10, was a fully prestressedbeam consisting of two plain bars 6.5mm (0.25 in.) in diameter incorporatedinto the pretensioned wires.
The yield strengths, f0.8 , of the highstrength wires used in the test beams of
Groups A and C, Group B and Group Dwere 1465, 1645 and 1360 MPa (212,470,238,580 and 197,240 psi), the ultimatestrength being 1790, 1840 and 1660 MPa(259,610, 266,860 and 240,750 psi), andthe elastic moduli were 205, 210 and 200GPa (about 30,000,000 psi), respec-tively.
The test load was applied by 200 kN(45 kip) Amsler hydraulic jacks (Fig. 2),and five electronic deformeters wereused for measuring displacements atmidspan, at the third points and at thesupports. Electric strain gages with agage length of 100 mm (4 in.) were fixedon the top surface of each beam over alength of 500 to 900 mm (20 to 35 in,) inthe middle of the span; Group B used
nine gages while the remaining threegroups each used five. In addition, six tonine strain gages [2 x 5 mm (0.08 x 20in.)) were placed on prestressing wiresin accordance with the number of wiresin the tendons (see Fig. 1). The bondednonprestressed bars had two straingages at midspan.
The load was applied in 10 to 15stages up to the yielding point of thenonprestressed steel, and the intervalbetween two consecutive stages wasroughly 3 minutes. All instrumentreadings were taken by a ProgrammableData Logger 7V06. The deflection of thebeam increased quickly when thebonded steel yielded so that the jack hadto he pumped up at an accelerating
PCI JOURNALJNovember-December 1985 75
1 0
12
9 02Y
080
0J0IiiJ 600
40
20
I
7
P- 8
2 p-7
1 A-4
p 1
20mm
MIDSPAN DEFLECTION (mm)
Fig. 3. Load-deflection curves for beams of Group A.
speed. Readings were taken continu-ously to the point where the load-de-flection curve descended and crushingof the concrete on the top of the beamoccurred, usually within a period of 4 to8 minutes.
TEST RESULTSLoad-Deflection Relationship
Representative load-deflection (P –A) curves for all beams in Groups A andD under short-term loading from zero toultimate load are presented in Figs. 3and 4.
It may be seen from the above figuresthat with the addition of an adequateamount of bonded nonprestressed steel,the shape of the P – 0 curve for un-
bonded beams was very similar to thatfor pretensioned beams with bondednonprestressed steel. In both cases, thecurves exhibit essentially three distinctstages, namely, uncracked elastic,cracked elastic and plastic. The wholecurve can be approximated by threestraight segments (Fig. 5).
The transition from the second stageto the third stage is the result of theyielding of bonded nonprestressedsteel, resembling an ordinary reinforcedconcrete beam, and shows an abruptchange in the curve. In the third stage,the curve is still linear up to the pointwhere the wires or concrete reaches theinelastic range of the stress-strain curve.Beams having low and medium valuesof qa exhibit all three stages of such be-havior while beams having high values
76
I,
D
z so
0
060
nw_J0a-
40
M
B
20
20mm_
MIDSPAN DEFLECTION i mm)
Fig. 4. Load-deflection curves for beams of Group D.
of q,, do not exhibit the third stage sincethe bonded reinforcement is still in theelastic range of the steel.
All beams with low values of q0 , suchas less than 0.15, are very ductile, withthe deflections at failure being 90 to 120mm (3.5 to 4.7 in.), or 1147 to 1/35. Thebeams with higher values of q0 , such asgreater than 0.25, have a larger neutralaxis depth and therefore smaller deflec-tion at failure, i.e., about 40 to 45 mm(1.6 to 1.8 in.) or 1/105 to 1/93.
It is well known that a completely un-bonded beam behaves after cracking asa shallow tied arch rather than as aflexural member. Thus, the shape of theP – 0 curve is quite different from thatof the unbonded beam having addition-al bonded nonprestressed reinforce-ment.
Stress Increase inUnbonded Tendons
The stress increase in unbonded ten-dons (A f^) was computed from the av-erage of the strain readings given by thegages placed on the wires and thestress-strain curve obtained on wire testsamples, In all beam tests, the stress in-crease obtained in the above manneragreed well with the value actuallymeasured by the load cell placed underthe anchorage of the unbonded tendon.
For the sake of brevity, only the P –f,,,, curves for the short-term tests of the
beams in Group A are presented in Fig.6. Except for the test beams with qo >0.25, the curves all consist of threestraight segments each. The stress in-crease in the unbonded tendon, similar
PCI JOURNAL/November-December 1985 77
STAGE 1
I STAGE2 1 STAGE 3
YIELDING OFI NONPRESTflES5ED5TEEL
1'CRAcKING
DEFLECTION
Fig. 5. Simplified load-deflection curve for unbondedbeam with additional bonded steel.
140
c120
Z00Y
0 80
A- 5
ZA-^
A- a
ID Op
STRESS INCREASE IN UNBONDED STEEL (Mpo)
Fig. 6. Applied load versus measured increase in stress of unbonded tendon for beamsof Group A.
0wJ£1-o 60a
40
20
78
('7 /(/(__tF
Q ^i
^lr
R
L7
Q
R^
R Q
/7/_
/ // I50 100 150 MAD $PAN DEFLECTION (mm )
Fig. 7. Measured deflection versus increase in stress of unbonded tendon for beamsof Group A.
0600a2
500OzW
0400W0Z0m2300
U-0
w200IrH
Z–100LLI
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to the increase of deflection of the beam,is insignificant before cracking. Al-though the rate of increase developsmore rapidly in the cracked elasticstage, the major part of the increase oc-curs in the third stage, that is, afteryielding of the mild steel reinforce-ment.
For all test beams with q,, < 0.15, thestress increase in the tendon at failure isgreater than 500 MPa (72,500 psi); forbeams with q,, = 0.30 to 0.47, the stressincrement A fp, remains almost constantwith a value of about 200 MPa (29,000psi).
In comparing the P – A and P – A f„acurves for each beam, it is apparent thatthese two curves are very similar inshape, indicating the close relationshipbetween the deflection and the stressincrease in unhonded tendons. Fig. 7shows such a relationship for beams inGroup A during the course of loading.For beams with low or medium levels of
the index q0 , these two curves maintaina fairly good linear relationship at thebeginning, but later this relationship isterminated as a result of the noticeabledeformation in the wires.
Pattern of Flexural CracksThe patterns of flexural cracks after
beam failure in Groups A and D areshown in Figs. 8 and 9. The main crackswere numerous for all unbonded testbeams. The average value of the mea-sured crack spacings over the constantmoment zone is as follows:
For each beam in 115 to 155 minGroups A, B and C: (4.5 to 6 in.)Average for alltest beams: 143 mm (5-6 in.)
For each beam in 136 to 155 mmGroup D: (5.3 to 6 in.)Average for alltest beams: 148 mm (5.8 in.)
PCI JOURNAUNovember-December 1985 79
The tests indicate that bonded andunbonded beams with A,lbdp > 0.004show no appreciable difference in thenumber or spacing of the main cracksover a beam length of constant moment.
The same result was obtained from testslisted in Refs. 1 and 4. However, inBeams A-1 and C-1 where two plain bars10 mm (0.4 in.) in diameter were usedfor the bonded steel, cracks of 15 and 25
Fig. 8. Crack patterns at failure for beams of Group A.
Fig. 9. Crack patterns at failure for beams of Group D.
80
0.004 0.004
AT NEAR Mu f00Muz 0.002 0.97 Mu z 0.002 0.96 Mu
0.93 Mu a 0.92 Mu0.90MU cr 0.83 Mu
MuM0.80.0 cnW CRUSHING ZONE ww F
wC i
o Z QQ U" (a) BEAM A-2 J (b} BEAM A-8J
200041- q 0.004L-
3I- I I-
I.00Muz 0.96 Mu Q 1.00 Mu
0.002 092Mu J 0.002 O.88Mu082 MU 0.73 M
0 o o 0.72Mu 0.59 Mu
(c) BEAM C-3 (d} BEAM C-9
Fig. 10. Concrete extreme fiber compressive strain distribution in constant moment zonefor Beams A2, A8, C3 and C9.
mm (0.6 and 1 in.) wide, respectively,appeared at failure of the beam due tolow bond strength.
Compressive Strain Distributionin Concrete
Representative strain distributions inthe extreme concrete fiber in compres-sion along the constant moment regionat the various loading stages up to fail-ure are presented in Figs. 10, 11 and 12.
For all test beams the distributionof concrete compressive strains turnedout to be fairly uniform, up to 80 to 90percent of the ultimate moment, M, andno significant influence of the cracks onsuch strains could be identified. Thiswas probably due to the large length ofthe strain gage in comparison with thecrack spacings. However, when the load
approached the ultimate value in beamswith a low q0 , such as in Beams A-2 andB-2, the strain distribution exhibited acomparatively wide scatter .5
The average value of the measuredcompressive concrete strains at or priorto failure is:
For each beam inGroups A, B and C 0.0022 to 0.0040Average for allbeams: 0.0028
For each beam inGroup D: 0.0023 to 0.0031
Average for allbeams: 0.0029
The corresponding maximum value is0.0025 to 0.0049 (average 0.00376) forGroups A, B and C and 0.0021 to 0.0039(average 0.00343) for Group D.
On the whole, the compressive strain
PCI JOURNAL'November-December 1985 81
0,004
AT NEAR Mu0.002 0,97Mu
aI
°0.91 MuH 0.80MuU) 0.70Muw CRUSHING ZONEw
z0U-, (a) BEAM B-2azq 0.004D
0 1eAT NEAR Muz0'j 0.002 0.98 Mu
0.95 Mu
S2 Mu0.82 Mu
CRUSHING ZONe
(b) BEAM B-7
Fig. 11. Concrete extreme fiber compressive strain distribution in constant moment zonefor Beams B2 and B7.
distribution was relatively uniform andno disparity could he seen in either thedistribution or magnitude of concretestrains between the bonded beams(Group D) and the unbonded beams(Groups A, B and C).
Ultimate Stress inUnbonded Tendons
Prior to rupture of the test beams withlow values of q0 , as further straining ofthe beam proceeded, the deflection ofthe beam and the strain in the unhondedtendon both increased rapidly with littleor no increase in load until crushing ofthe concrete occurred. Such a phenom-enon was reflected by a plateau in the P
– A and P – Af„^ curves. Therefore,strain and stress values corresponding toinitial ultimate load were taken as the
strain and stress of the tendon at beamfailure. The ultimate tendon stress forthe three beam groups at failure is listedin Table 2.
The tendon stress increment at failureAf1e , as actually measured, dependednot only on q, = pp f; I f, , but also onq, = pe fy l f, , increasing with any de-crease of q„ while A f^ was fairly con-stant for equal values of q0 . For example,Beams A-21A-7, B-2/B-7 and C-3/A-8were three pairs of beams with nearlythe same qo , yet the value of Afp, wasbasically the same regardless of the 2.5to 2.7 times difference in the area of pre-stressing steel.
Actually, the index qo reflects thedepth of the neutral axis C, on which thetendon stress increment at failure ofA f,!depends strongly. As such, when theneutral axis moves toward the extreme
82
0.0041AT NEAR Mu
0.95 Mu
Z 0.002 0.90 Mu
0.82 Mu
CRUSHING ZONEwUoZO
ia) BEAM D-1JQZEi 0.004-
Z AT NEAR MuOJ 0.002—
0.004
1.00 Mu0.002 0.96 Mu
a0.66 Mu
U) 0.76 Muw --CRUSHING ZONE
wCrUZOUJ (b) BEAM D-342
°°°°4 r-
2 ^ 1.00 Mu0 0-97 MuJ 0.002 n 0.93 Mu
-a0.97Mu 0.89 Mu0 90 Mu
CRUSHING ZONE I GRUSH1NG ZONE
{c) BEAM 0-0 (dl BEAM D -$0
Fig. 12. Concrete extreme fiber compressive strain distribution in constant moment zonefor beams of Group D.
compression fiber, the rotation ability ofbeams and the value of 0 fw increase.Therefore, factors such as any increaseof A,, and A, can induce the decrease of0f,,, since they increase the value of q,,.On the other hand, the value of A fp, in-creases with the increase of concretestrength, f,, since it makes the value ofq, decrease.
From the experimental results in Ta-bles 1 and 2, it can be seen that in thethree pairs of Beams A-6/A-8, A-21A-4and A-5/A-7, with the same amount ofABand almost the same f, for each pair,Beams A-8, A-4, and A-7 have a smallerAa and lower q, than their counterparts,and as such, have a greater A f, .
Beams A-4/A-8 and A-31A-6 have thesame amount o£A, and almost the samef, for each pair while Beams A-4 and A-3have a smaller A, and lower q„ and as a
result, they have a greater A fD, thanBeams A-8 and A-6.
The concrete strength of beams inGroup B is about 50 percent higher thanin Group A and the values of 0 f,,, inGroup B are also generall y greater thanin Group A. This proves that the index qo
roughly reflects the depth of the neutralaxis from the top of the beam cross sec-tion.
The interrelationship between theexperimental value of A fpa and q, isshown in Fig. 13. Beams A-1, C-1, A-9,and C-9 were discarded when preparingthis figure since the first two beamsfailed at a lower A fn, than expected,owing to insufficient bond of the non-prestressed steel and the higherf, valueof Beam A-1. Furthermore, the stress inthe unbonded steel of Beam A-1, hadactually already reached f,, ,2 .
PCI JOURNALJNovember-December 1985 83
The latter two over-reinforced beamsfailed at a steel stress below the yieldingstress of the bonded deformed bars, acase rarely seen in actual practice. Also,gage readings were unfortunately nottaken on Beams B-4 and B-8 at the finalstage of the testing.
The tendon stress increment and theindex qo bear a fairly good linear rela-tionship, giving rise to the following re-gression equation:
fâ, = 786– 1920q,, (1)
where t f„, is expressed in MPa and thecorrelation coefficient being r = –0.97.
The ultimate stress in the unbondedtendon is:
fW = fPe + af. (2)
where fp, = fo.aThis expression is limited to q. -_ 0.30
and f, is in the range of 0.55 to 0.65f'2.
The ultimate stress increments for thetest beams computed from the above
equation are given in Table 2. The meanvalue of the ratio (experimental ultimatestress/calculated ultimate stress) for the16 beams is 0.998, with a standard de-viation of 0.022.
Ultimate Flexural Strengthof Beam
The ultimate flexural strength of thetest beams was calculated in accordancewith the assumptions shown in Fig. 14.The assumed concrete compressivestress distribution agrees with Section10.2 of the ACI Code .2 The actual exper-imental flexural strength for all beams,together with that calculated using Eq.(2), is listed in Table 2. For all un-bonded partially prestressed concretebeams having qa < 0.30, the experi-mental results agreed well with thecalculated strengths, giving a mean ratioof 1.051 and a standard deviation of0.057.
The experimental values of some
800
700
a 600
y 500ata 400
300
200
I l l 11000.0 0.05 0.10 0.15 020 025 0.30
q o - 4 pe + qs
Fig. 13. Increase in tendon stress at failure versus combined reinforcement index.
84
Table 2. Experimental steel stress and moment details.
beams turned out to be higher; for ex-ample, Beam A-4, for which the aboveratio is 1.14. However, this can be ex-plained by the strain hardening of thebonded steel caused by excessivedeflection of the beam in the latterstage.
Comparison of Test ResultsWith Code Values
The test results of Mattock,* the au-thors and the calculated values by ACI318-83 and CP-110 are presented inTable 3. It can be seen that the A fp, val-ues of the authors and Mattock are sim-ilar for both higher and lower q0 . How-
ever, the ratios offs lftt in this study areslightly greater due to lower effectiveprestress.
On the other hand, the test values ofA f,,, and ratios of f„Iff in Table 3 aregreater than the values calculated usingACI 318-83 and CP-110 except for thosebeams with higher q. This is due to theinfluence of nonprestressed reinforce-ment on the distribution of cracks andon the ultimate tendon stress. Thisshows that for partially prestressed con-crete beams with unhonded tendons,the beneficial effect of nonprestressedreinforcement on the ultimate tendonstress should be taken into considera-tion.
PCI JOURNALJNovember-December 1985 B5
ds
0.e5 f^
a/2 C=o.85bafcpc
MU
Aptps— . fY
^ASfY
STRESSES INTERNAL FORCES
Fig. 14, Conditions at ultimate moment.
THEORETICAL ANALYSISOF ULTIMATE STRESS
IN UNBONDED TENDONSIt can be seen from the foregoing
analysis that the use of additionalbonded reinforcement can overcome theundesired shortcomings of unbondedbeams, such as the formation of sparselyspaced wide cracks and the concentra-tion of compressive strain which canlead to premature failure of the beam.Hence, it is possible to calculate the ul-timate stress in the unhonded tendonand the deflection at ultimate on atheoretical basis by means of the mo-ment-curvature method,s,r,a
However, in order to determine theunknown stress increment in the un-bonded tendons (apart from the funda-mental assumptions for the analysis ofbonded beams), the requirement forstrain compatibility must be resorted to.The increase in length of the unbondedtendon should be equal to the totalchange in length of the adjacent con-crete beam.
The distribution of curvature alongthe span of the beam at ultimate load isshown in Fig. 15, When computing theultimate tendon stress of an unbondedbeam, an appropriate value of the ten-don stress has to be assumed first and
later checked with the calculated valueobtained from considering the straincompatibility. Such a trial and errormethod usually requires two or three it-erations before a comparatively accurateresult can be obtained. Hence, a suit-able computer program is necessary.
Table 4 lists the stress increment inunbonded tendons, flexural strength,and deflection at midspan under ulti-mate load as calculated by means of themoment-curvature method. In the cal-culations, the experimental stress-straincurve for the prestressing steel, as wellas the following data, were used:
{()2jf fc
= 0.003
E° = 0.002It can be seen from Table 4 that the
ratio of the measured ultimate stress inthe unbonded tendons of the 20 beamsto the calculated value has a mean of0.981 and a standard deviation of 0.028.In the case of ultimate strength, themean is 1.031 and the standard devi-ation is 0.053. Both values have a rela-tively high degree of precision; how-ever, the results for deflections are notas satisfactory. The two correspondingstatistical indices are 1.111 and 0.171,respectively.
86
nC-0zz0
CD
aCD
CD
23mcnmcn
Table 3. Comparison between experimental results and code values.
Lid,
Experimental AC! 318-83 CP110
q qo ppfp fw a f^ f'. I^Beam MPa MPa MPa (2) MPa (4) MPa (5) x 10
The close agreement between the ex-perimentaI results and the calculatedvalues as stated above clearly indicatesthe reliability of the experimental re-sults in regard to ultimate tendon stressand the feasibility of the moment-cur-vature method of analysis.
The theoretical calculations for thetest beams also indicate:
1. The ultimate flexural strength oftheunbonded beam can be increased by theaddition of bonded nonprestressed re-inforcement. Such an increase is due tothe resistance of the bonded steel itself,as well as to its influence in distributingcracks.
2. The ultimate tendon stress can besubstantially enhanced by adding anadequate amount of bonded nonpre-stressed steel for distributing cracks.However, when the bonded steel ex-ceeds the needed amount, even thoughthe ultimate flexural strength of thebeam as a whole will be increased ac-cordingly, the ultimate tendon stresstends to decrease due to the lowering ofthe position of the neutral axis.
3. The ratio ofA, to A. has some influ-
ence on the values of f,,, for the testbeams having the same value of q0 , butthis influence is rather small and canbe neglected.
CONCLUDING REMARKSThe ultimate stress in unbonded ten-
dons of partially prestressed concretebeams which have bonded nonpre-stressed steel carrying 30 percent ormore of the total ultimate load (A,Ibdp>
0.004 in this study), is closely related tothe combined reinforcement index q0.
For beams with a span to depth ratio of20, under the action of third point oruniform loading, the following relation-ship exists between f, and qo
fg=fp +(786– 1920g0 )
expressed in MPa, where f,,„ -_ fo.zThis expression is limited to qo -_ 0.30
and toff in the range of 0.55 to 0.65J. .Unbonded beams which have an ade-
quate amount of bonded nonprestressedreinforcement in the form of medium
Table 4. Theoretical steel stress and moment details.
Note: 1 ]!Pa - 145 psi; I kN-rn = 738 ft-16; 1 trim = 0.394 i .co
grade deformed bars distribute cracksand compressive concrete strains almostthe same as bonded prestressed con-crete beams. In addition, the ultimatetendon stress can be calculated satis-factorily on a theoretical basis by meansof the moment-curvature method sup-plemented with the strain compatibilitycondition for unhonded tendons.
Finally, in view of the effects ofspan-depth ratio, tendon profile andform of loading, the values of f,,, shouldbe properly reduced in the design ofprestressed concrete structures. Al-though the analytical research showsthat the effect of span-depth ratio couldbe neglected in partially prestressedconcrete beams under third-point load-ing (Fig. 15), further experimental in-
vestigation should be carried out to ver-ify this research.
ACKNOWLEDGMENTS
The tests were conducted at thestructural laboratory of the Institute ofBuilding Structures, China Academy ofBuilding Research. The authors are in-debted to Messrs, Pan, Li, Tang, andMingyan for their participation duringthe testing.
The authors also wish to thank Prof.Alan H. Mattock for his encouragementto make the research work being carriedout in China known to other countries.Valuable suggestions were also offeredby the PCI JOURNAL Review Com-mittee.
REFERENCES
1. Tam, A., and Pannell, F. N., "The Ulti-mate Moment of Resistance of UnbondedPartially Prestressed Reinforced ConcreteBeams," Magazine of Concrete Research,December 1976.
2. ACI Committee 318, "Building Code Re-quirements for Reinforced Concrete (ACI.318-8.3)," American Concrete Institute,Detroit, Michigan, 1983.
3. Code of Practice for the Structural Use ofConcrete, Part 1, Design, Materials, andWorkmanship, CP 110, British StandardInstitute, 1972.
4. Mattock, A. H., Yamasaki, J., and Kattula,B. T., "Comparative Study of PrestressedConcrete Beams With and Without Bond,"ACI journal, Proceedings V. 68, No. 2,February 1971, pp. 116-125.
5. Cooke, N., Park, R., and Yong, P.,
"Flexural Strength of Prestressed Con-crete Members With Unbonded Ten-dons," PCI JOURNAL, November-De-cember 1981, pp. 52-80.
6. Warwaruk, J., Sozen, M., and Siess, C. P.,"Strength and Behavior in Flexure of Pre-stressed Concrete Beams," University ofIllinois, Experiment Station, Bulletin No.464, University of Illinois, Urbana, Illi-nois, 1962.
7. Lin, T. Y., and Burns, N. H., Design ofPrestressed Concrete Structures, JohnWiley and Sons, New York, Third Edition,1981.
8. Zhao, Jida, and Mo, Lu, "ComputationalStudy of Partially Prestressed ConcreteBeams With Unbonded Tendons," Tech-nical Report, China Academy of BuildingResearch, 1984-1985.
90
APPENDIX -- NOTATION
a = depth of equivalent stress blockAp = area of unbonded prestressed re-
inforcementA, = area of bonded nonprestressed
reinforcementb = width of beamC = depth of neutral axis4 = effective depth of beam (to cen-
troid of prestressing steel)= effective depth to centroid of
nonprestressed reinforcement= compressive strength of concrete
(150 x 300 mm cylinder)fe = effective prestress in unhondedtendon prior to loading
= ultimate stress in unbonded ten-don at failure of beam
fv = yield stress of nonprestressedreinforcement.f,.2 = 0.2 percent proof stress of pre-stressing steel
= span length of member general-ly center-to-center of supports
= bending moment to producefirst crack
M. = ultimate flexural momentM„ = bending moment to produce
yielding of nonprestressed rein-forcement
P = concentrated external loadqo = qne + q'1
= prestressing steel index =Apff1
bd f'= nonprestressing steel index =
A,.f /bd.^ffA = deflection due to loadingA fm = stress increment in unbonded
tendons at intermediate stagesof loading.
= stress increment in mph onnh•dtendons at failure of beam
A e^ = increase in strain in unbondedtendons
E„ = limiting strain at which concretein beam crushes = 0.003
p, = Aplbrl,p, = A,/bd,,
f ^ ^
NOTE: Discussion of this paper is invited. Please submityour comments to PCI Headquarters by July 1, 1986.
