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Unbonded Tendon Stress Increases in Multi-Span
Members
Presenter: Marc Maguire William Collins, Kedar Halbe and
Carin Roberts-Wollmann
Unbonded Tendons
• Strain compatibility cannot predict Δfps
• Many research programs and design codes have empirical or semi-empirical design predictions
Δfps Calibration/Validation
• All design equation predictions were calibrated or validated using mostly simple span test results and very limited multi-span tests
• The largest known database of multi-span tests (Harajili 2006) contains 15 individual tests from only three research programs.
Δfps Calibration/Validation
• Literature suggests the multi-span tests used for design code calibration may not be ideal candidates: • Burns, Charney, and Vines (1978)
• 6 Tests – Brittle Bond Failure • Scordelis et al. (1959), Brotchie and Beresford (1967),
Burns and Hemakon (1977), and more • Punching Shear Failure
• Many programs performed collapse load tests on the same specimen multiple times
• Odd test setups
Prediction Equations
• Three prediction equations were selected for comparison • ACI 318 – 08 –100% Empirical • AASHTO LRFD – Not Empirical (Mechanical Model) • Naaman and Alkhairi (1991) – Partially Empirical
Current ACI 318 Equation
'10,000 cps se
p
ff fψρ
= + +
Span-to-depth ratio ≤ 35: Span-to-depth ratio ≥ 35:
100300
ψψ
==
psp
ps
Ab d
ρ =×
not greater than lesser of fpy or (fse + 60,000)
Current ACI 318 Equation
'10,000 cps se
p
ff fψρ
= + +
Entirely Empirical (Mojtahedti and Gamble 1978)
Current AASHTO Equation
( )2
2eLN
=+
900 psps pe
e
d cf f
− = +
where N equals number of support hinges crossed by tendon
zp
δθ
Lp
∆
L/2L/2
cdps N=0
θ/2
θ
θ
θ/2 θ/2
N=1
N=2
Current AASHTO Equation
N=1
N=2
( )2
2eLN
=+
900 psps pe
e
d cf f
− = +
Naaman and Alkahairi Equation
• Bonded Stress Reduced to Unbonded Stress Ωu = Bond Reduction Coefficient • Simple Span Converted to Continuous L1 = Length of Loaded Span L2 = Total Tendon Length
1
2
1psps pe u ps cu
d Lf f Ec L
ε
= + Ω −
Database
• Previous equation calibration combined simple and multi-span data points (Naaman, Mojtahedi, Mattock, Harajili etc.) • Note that AASHTO equation IS NOT CALIBRATED
• Should we mix simple and multi-span beams? • Mechanisms are similar, but there are significant
differences in behavior at ultimate • Pattern loadings • Moment redistribution • Strand elongation is over longer distance
• Database was created using same criteria as other test programs
Simple Span Database
• Du and Tao (1985) • Cooke, Park and Yong (1981) • Mattock, Yamazaki and Kattula (1971) • Tam and Pannell (1969) • Pannell • Harajli and Kanj (1991) • Campbell and Chouinard (1991) • Chakrabarti et al. (1994)
• Total 146
Multi - Span Database
• Burns et al. (1978) • Mattock et al. (1971) • Scordelies et al. (1959) • Burns and Hemakom (1977) • Lim et al. (2003) • Hemakom (1970) • Chen (1971) • Kosut et al. (1985) • Burns et al. (1991) • Macgregor (1989) • Brotchie and Beresford (1967) • Halbe (2007)
• Total 58
ACI 318-08 Comparison
0
20
40
60
80
100
0 20 40 60 80 100
Cal
cula
ted
delta
fps,
ksi
Measured delta fps, ksi
ACI 318-08 Single Span
0
20
40
60
80
100
0 20 40 60 80 100C
alcu
late
d de
lta fp
s, k
si
Measured delta fps, ksi
ACI 318-08 - Multi-Span
AASHTO LRFD Comparison
0
20
40
60
80
100
0 20 40 60 80 100
Cal
cula
ted
Del
ta fp
s, k
si
Measured Delta fps, ksi
AASHTO LRFD – Simple Span
0
20
40
60
80
100
0 20 40 60 80 100C
alcu
late
d D
elta
fps,
ksi
Measured Delta fps, ksi
AASHTO LRFD – Multi-Span
Naaman and Alkhairi Comparison
0
20
40
60
80
100
0 20 40 60 80 100
Cal
cula
ted
Del
ta fp
s, k
si
Measured Delta fps, ksi
Naaman – Single Span
0
20
40
60
80
100
0 20 40 60 80 100C
alcu
late
d D
elta
, fps
, ksi
Measured Delta fps, ksi
Naaman – Multi-Span
Multi-Span Database Shortcomings
• Many of these tests have been used for equation calibration by various researchers! • 3 tests ended in shear failure (more were “borderline”) • 15 tests ended in bond failure • 8 test used improper test setups • The majority of tests did not indicate how Δfps was
measured
Multi-Span Database Shortcomings
• Remove non-flexural failures: 41 tests remain • Remove improper test setups only 33 remain • Of the remainder, alternate span loading is
not very well represented • Adjacent Span Loading – 22
• Primary hinge forms at negative moment • Alternate Span Loading – 11
• Primary hinge forms in positive moment
0
20
40
60
80
100
0 20 40 60 80 100
ACI P
redi
cted
∆fp
s, k
si
Measured delta fps, ksi
ACI 318-08 - Multi-Span
0
20
40
60
80
100
0 20 40 60 80 100
Cal
cula
ted
Del
ta fp
s, k
si
Measured Delta fps, ksi
AASHTO LRFD – Multi-Span
0
20
40
60
80
100
0 20 40 60 80 100
Cal
cula
ted
Del
ta, f
ps, k
si
Measured Delta fps, ksi
Naaman – Multi-Span
Trimmed Database
0
20
40
60
80
100
0 20 40 60 80 100
ACI P
redi
cted
∆fp
s, k
si
Measured delta fps, ksi
ACI 318-08 - Multi-Span
0
20
40
60
80
100
0 20 40 60 80 100
Cal
cula
ted
Del
ta fp
s, k
si
Measured Delta fps, ksi
AASHTO LRFD – Multi-Span
0
20
40
60
80
100
0 20 40 60 80 100
Cal
cula
ted
Del
ta, f
ps, k
si
Measured Delta fps, ksi
Naaman – Multi-Span
Adjacent Span
Loading
0
20
40
60
80
100
0 20 40 60 80 100
ACI P
redi
cted
∆fp
s, k
si
Measured delta fps, ksi
ACI 318-08 - Multi-Span
0
20
40
60
80
100
0 20 40 60 80 100
Cal
cula
ted
Del
ta fp
s, k
si
Measured Delta fps, ksi
AASHTO LRFD – Multi-Span
0
20
40
60
80
100
0 20 40 60 80 100
Cal
cula
ted
Del
ta, f
ps, k
si
Measured Delta fps, ksi
Naaman – Multi-Span
Alternate Span
Loading
Multi-Span Beams Simple Statistics
• R2 values are too low to be of statistical significance! (< 0.1!!!)
• Bias (Calculated)-(Measured) • A measure similar to accuracy of prediction • Negative Value indicates “conservative”
• Mean Square Error Bias2
• A measure similar to precision of prediction • Similar to R2 value • Smaller value indicates better fit
• Percent Error 100*|Bias|/(Measured)
Multi-Span Beams Simple Statistics
Whole Multi-span Database “Trimmed Database”
Average Bias MSE Average % Error AASHTO -18.01 1164 85%
ACI -9.75 869 134% Naaman -5.13 1661 224%
Average Bias MSE Average % Error
AASHTO -27.40 1801 54%
ACI -18.68 1406 68%
Naaman -9.78 2734 208%
Multi-Span Beams Simple Statistics
Whole Multi-span Database “Trimmed Database”
Average Bias MSE Average % Error AASHTO -18.01 1164 85%
ACI -9.75 869 134% Naaman -5.13 1661 224%
Average Bias MSE Average % Error
AASHTO -27.40 1801 54%
ACI -18.68 1406 68%
Naaman -9.78 2734 208%
Multi-Span Beams Simple Statistics
Whole Multi-span Database “Trimmed Database”
Average Bias MSE Average % Error AASHTO -18.01 1164 85%
ACI -9.75 869 134% Naaman -5.13 1661 224%
Average Bias MSE Average % Error
AASHTO -27.40 1801 54%
ACI -18.68 1406 68%
Naaman -9.78 2734 208%
Statistical Analysis What does this mean?
• There are huge amounts of scatter in the available data from many factors
• It is evident that the current equations do not adequately reflect the behavior of the data set
• All methods are conservative, but the AASHTO and ACI methods seem to be the most conservative
Addition to the Database
• Four representative slabs were fabricated at the Thomas M. Murray Structural Engineering Laboratory at Virginia Tech.
• Goal was to add high quality ductile and design relevant, failures to the database.
Specimen Design
Four Representative Slabs • Two Continuous 20 ft spans • Parabolic 0.5” unbonded tendon • Minimum mild reinforcement
Testing Scheme
• Moment Redistribution and Pattern Loading • Single Span Loading – Maximum Positive Moment
• Two Span Loading – Maximum Negative Moment
Instrumentation
• Tendon Force at Anchorages • Deflection at Expected Hinge Locations
Test Setup
• Four point loading on loaded spans to simulate distributed load
• Concrete blocks for dead load on unloaded span
Test Results – Load vs. Deflection
• Four
Test Results – Unloaded Span Failure
• Failure on unloaded span
• Over-strength at second hinge
Test Results – Unloaded Span Failure
• Recommend minimum mild steel throughout member to prevent brittle failure
Test Results
• Ultimate Load Comparison:
• All methods result in very conservative ultimate strength
Load Case Max Distributed Load
(lb/ft) (Measured)/(Predicted) Ratio
AASHTO ACI Naaman Specimen 1* South Span 467 1.03* 1.00* 0.99* Specimen 2 North Span 582 1.29 1.24 1.23 Specimen 3 Both Spans 607 1.34 1.28 1.16 Specimen 4 Both Spans 698 1.54 1.47 1.34
Average 1.39 1.33 1.25
Test Results
• Δfps Comparison:
• Very conservative estimation, even for the premature failure
Load Case Tendon Force Increase
(lb) Estimated
Δfps (psi)
Δfps (Measured)/(Predicted) Ratio
Live End Dead End AASHTO ACI Naaman
Specimen 1 South Span 5,373 2,647 26,209* 1.74 1.17 1.10
Specimen 2 North Span 6,087 6,852 42,284 2.81 1.89 1.77
Specimen 3 Both Spans 8,578 7,419 52,278 3.48 2.34 1.18
Specimen 4 Both Spans 9,110 9,270 60,065 3.99 2.68 1.36
Summary
• Issues can be identified with many multi-span beam testing programs (including this one) • Failure Modes • Test Setups
• Issues have been identified with the current pool of data, resulting in only 33 tests considered “acceptable”
• This testing program added three data points to the pool
Recommendations and Conclusions
• Section over-strength should be considered when specifying reinforcement cut-off lengths • OR specify minimum reinforcement as continuous
throughout beam
• AASHTO, ACI and Naaman prediction equations resulted in conservative Δfps predictions for the specimens tested and for the database.
• We need more tests
References
Brotchie, J. F., and Beresford, F. D., “Experimental Study of a Prestressed Concrete Flat Plate Structure,” Civil Engineering Transactions, V. 9, No. 2, 1967, pp. 276-282.
Burns, N. H.; Charney, F. A.; and Vines, W. R., “Tests of One-Way Post-Tensioned Slabs With Unbonded Tendons,” PCI Journal, V. 33, No. 5, Sep.-Oct. 1978, pp. 52-80.
Burns, N. H., and Hemakom, R. "Test of Scale Model Post Tension Flat Plate," Journal of the Structural Division, ASCE, V. 103, No. 6, Jun. 1977, pp. 1237-1255.
AASHTO, “LRFD Bridge Design Specifications,” American Association of State Highway and Transportation Officials, Washington, DC, 2010, 1632 pp.
ACI Committee 318, “Building Code Requirements for Structural Concrete (ACI 318-08) and Commentary,” American Concrete Institute, Farmington Hills, MI, 2008, 473 pp.
Naaman, A. E., and Alkhairi, F. M., “Stress at Ultimate in Unbonded Post-Tensioning Tendons: Part 2- Proposed Methodology,” ACI Structural Journal, V. 88, No. 6, Nov.-Dec. 1991, pp. 683-692.
Scordelis, A. C.; Lin, T. Y.; and Itaya, R., “Behavior of a Continuous Slab Prestressed in Two Directions,” ACI Journal, V.
31, No. 6, 1959, pp. 441-459. Harajili, M., H.; “On the stress in Unbonded Tendons at Ultimate: Critical Assessment and Proposed Chagnes” ACI
Structural Journal, V. 103, No. 6, 2006, pp. 803-812