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## Theoretical and Applied Mechanics Program Northwestern University

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**Theoretical and Applied Mechanics Program**Northwestern University Ranked No. 1 by the Chronicle of Higher Education 24 Faculty: J. D. Achenbach, O. Balogun, Z.P. Bazant,T.Belytschko,C.Brinson, J.Cao, W.Chen. I.M. Daniel, S.H. Davis, G. Dvorak, H.D. Espinosa, S. Ghosal, Y.Huang, L.M. Keer, S. Keten, S.Krishnaswamy, S. Lichter, W.-K. Liu, R..Lueptow, N.A. Patankar, J.Qu, J.W. Rudnicki, J. Wang, J. Weertman.. Where to apply? http://www.tam.northwestern.edu/ 7 NAE, 3 NAS and 5 AAAS members**PERVASIVENESS OF CONCRETE**CREEP PROBLEMS IN STRUCTURES: WAKE-UP CALL FOR DESIGN CODES AND CONSEQUENCES OF NANO-POROSITY ZDENĚK P. BAŽANT COLLABORATORS: QIANG YU , MIJA HUBLER AND ROMAN WENDNER SPONSORS: NSF, DoT UNIVERSITY OF MIAMI, CORAL GABLES, 11//06/2011**Box Girder of World Record Span 241 m, Palau, 1977**Segmental Cantilever Construction Ulrich Finsterwalder 1897 - 1988 Eugène Freyssinet 1879 – 1962**Koror-Babeldaob Bridge in Palau**Built 1977, failed 1996 Babeldaob side (failed first) Delamination creep buckling of top slab. Sudden loss of prestressing force ~20,000 ton, emits wave.**1. Delamination in top slab**6. Side span slams back on end pier and fails in shear 3. Load from Babeldaob side transmitted 2. Box girder fails in compression- shear Babeldaob Koror 5. Section at pier overloaded and fails in compression 4. Bridge lifts up and hold-down bars fracture Trigger of Collapse: Delamination creep buckling of top slab 3 months after remedial prestressing Top slab subjected to longitudinal compressive forces Delamination buckling Crack Longit. force 190 MN (21400 tons) from 316 tendons in 4 layers 1 layer buckling releases about 5000 tons**18 years later: Deflection 1.61 m**(compared to design camber)**2. Release of Data on Litigated Failure Sealed in**Perpetuity**Palau: Legal litigation (1996-1998) didn’t establish**the causes of excessive deflections and collapse. All data sealed in perpetuity! Nov. 2007: Resolution, proposed by Bažant, adopted by 3rd World Congress of Structural Engineers: 1) The structural engineers gathered at their 3rd World Congress deplore the fact that the technical data on the collapses of various large structures, including the Koror-Babeldaob Bridge in Palau, have been sealed as a result of legal litigation. 2) They believe that the release of all such data would likely lead to progress in structural engineering and possibly prevent further collapses of large concrete structures. 3) In the name of engineering ethics, they call for the immediate release of all such data. 2 months later: Attorney General of Palau agreed to controlled release of the data!**Contrast:**• Commercial Aviation Withholding of technical data on any real or potential disaster is prohibited by law, as well as international agreements.**3. Why the creep deflections of KB Bridge were so**excessive?**Diaphragms**Three-Dimensional Mesh for Prestressed Box Girder 5036 eight-node isoparametric (hexahedral) elements and 6764 steel bar elements**Algorithm Utilizing ABAQUS**• From B3 (or ACI, CEB, GL) obtain axial and transverse compliance functions for each finite element, as function ofD, h T. • Use Widder’s formula to calculate discrete spectrum moduli for each finite element and each time step. • Specify for ABAQUS the (yield) strength limit of each finite element. • Loop on time steps • Loop of finite elements. • From Δt and the current strains of Kelvin units, calculate for each finite element the incremental moduliEijrs and inelastic (quasi-thermal) strain incrementsΔεrs” for each finite element. • Supply the quasi-elastic stress strain relationΔσrs = Eijrs(Δεrs – Δεrs”) with inelastic strainsΔεrs” to each finite element of ABAQUS, along with the current stress. • End of loop on finite elements. • Apply load and prescribed displacement increments, if any. • Run ABAQUS, heeding the strength limits. • End of loop on time steps: Obtain incremented nodal displacements, strains and stresses, go to 4, and start a new time step.****E 1 1 E1 1 E Replace history integrals by rate equations for internal variables (partial strains) (a concept pioneered by Biot) — use a continuous retardation spectrum J Age t‘1= 28 days Age t‘2= 365 days Compliance • Advantage: Continuous retardation spectrum D() is unique (1995), defined by Laplace transform inversion, easily obtained by Widder’s formula. • Discrete approximation Diof continuous spectrum, different in each time step, yields Kelvin chain moduli – easy step-by-step integration. log (t-t') Continuous Spectrum E Age t’1 1 1 discrete spectrum Age t’2 1 1 **Concrete creep and shrinkage models compared**• in new ACI Guide 209.2R-08 (2008): • ——————————————————————————————————————— • B3 (Bažant & Baweja 1995, 2000; • update of BP Model 1978) • 2) ACI 1971, reapproved in 2008 (sole dissenting vote: Bažant) • 3) CEB-fib (or CEB-FIP Model Code 1990, fib 1998, fib Draft 2010) • GL (Gardner & Lockman 2001) • (similar to Bažant & Panula 1978) • 5) JSCE and JRA (Japan) • — all rated in ACI Guide as approximately equivalent!**Two Questions in Comparing Various Models**1. Prediction: How do the predictions of various models differ from observations? 2. Credible Explanation: Can the observations be explained and matched with realistic parameter values?**Parameter Sets Considered for Model B3**Set 1 (prediction): q1 = 0.146, q2 = 1.04, q3 = 0.045, q4 = 0.053, q5 = 1.97 x 10-6 /psi, λ0 = 1 day, εk= 0.0013, kt = 19.2 = empirical estimates from design strengthf’c and mix composition Set 2 (explanation): q2, q5, εk,λ0, kt =same (from composition) but q1 = 0.188 based on E-modulus from truck load test q3 = 0.262, q4 = 0.140 x10-6 / psi = long-time parameters updated from Brooks (1984, 2006) 30-year creep data (Univ. of Leeds)**Does Model B3 (Set 2) match any real concrete?**It does – Brooks (1984, 2006) 30-year data 280 J(t,t’)x 10-6 /MPa 140 B3 (Set 2) w/c = 0.56 w/c = 0.67 Brooks, 1984 t’ = 14 days (University of Leeds) 1000 10 10000 100 0.1 1 t-t’ (log. scale, days)**Deflections [in m] predicted by 3D finite elements**using models ACI, CEB, GL, JSCE, and B3 (Sets 1, 2) KB BRIDGE 1D beam element model in commercial software SOFiSTiK 0 CEB (1D, SOFiSTiK) 3D element model for ABAQUS CEB (1D, SOFiSTiK) 0 JSCE (3D) JSCE (3D) -0.5 -0.4 ACI (3D) CEB (3D) CEB (3D) GL (3D) ACI (3D) -1.0 GL (3D) B3 (3D) (set1) B3 (3D) (set1) -0.8 Mean Deflections (m) -1.5 -2.0 B3 (3D) (set2) -1.2 -2.5 100 10000 100000 10 1 1000 B3 (3D) (set2) log t, time from construction end, days -1.6 2000 6000 0 8000 4000 t, time from construction end, days Fig. 6**0**CEB (1D, SOFiSTiK) JSCE (3D) -0.4 ACI (3D) CEB (3D) GL (3D) B3 (3D) (set1) -0.8 Mean Deflections (m) -1.2 B3 (3D) (set2) -1.6 2000 6000 0 8000 4000 t, time from construction end, days**Prestress loss (MN) in tendons at main pier**predicted by 3D finite elements using models ACI, CEB, GL, JSCE, and B3 (Sets 1, 2) KB BRIDGE IN PALAU Detailed report: Google “Bazant”, download “…Palau…”**New I-35W Bridge in Minneapolis**(allegedly designed using SOFiSTiK or similar)**(based on expected CoV of q1, q2, q3, q4, q5, kt,**, and humidity) Mean response (m) and 95% confidence limits of Model B3 in normal and logarithmic scales Calculated from 8 deterministic runs for random (Latin Hypercube) samples of input pararameters**Relaxation in Prestressing Steel Tendons**1. Effect of Varying Strain in Steel Ignored: Tropical sun • - The strain in the tendons decreased by 30 %. • A rate-type viscoplastic model must be used for • prestressing steel. 55ºC max. top bottom 20ºC The first layer of tendon reaches 30ºC in about 2.5 hours 2. Effect of Temperature Ignored fib Model Code Draft 2010 Temperature in top slab Relaxation (%) Temperature Location in Slab (mm) Temperature (ºC) time (s), log scale The initial strain in concrete is 0. After prestress and self-weight is applied, it is about 0.00056. After 20 years, the total strain in concrete is about 0.00103 due to creep and shrinkage. The concrete strain is almost doubled in 20 years. The initial strain in steel is about 1% ? So the strain in prestr. Steel drops by about 10% of the initial value.**Relaxation in Prestressing Steel**1. Up to now: Effect of varying strain ignored: Tropical sun • - But the strain in tendons can decrease by 30 %. • Hence, a rate-type viscoplastic model must be • used for prestressing steel. 55ºC max. top bottom 20ºC The first layer of tendons reaches 30ºC in about 2.5 hours 2. Traditionally: Effect of temperature ignored Normal relaxation grade (Shinko, Japan) 80 ºC Temperature in top slab 60 ºC Relaxation (%) 40 ºC Temperature 20 ºC Location in Slab (mm) Time (hours) time (s), log scale**+a T**Steel Relaxation Generalized to Variable Strain & Temp. Constant strain andT : • Main features: • Relaxation terminates when stress reaches certain level • No crossing for relaxation curves under different initial stress Proposed for varying strain andT : Reduced time (due to temperature): AT = Arrhenius factor Q = activation energy kB = Boltzmann constant**1320**1420 CEB (rate-type) CEB (rate-type) Stress(MPa) Stress(MPa) 1235 1270 SR9-5 and SR10-5 (Buckler & Scribner 1985) SR14-10 (Buckler & Scribner 1985) 1150 1120 1000 1000 0.1 1 10 0.1 100 1 10 100 1400 1500 Comparison with Tests: Variable Strain Fit by proposed formula Stress(MPa) Stress(MPa) 1200 1250 SR14-10 (Buckler & Scribner 1985) SR9-5 and SR10-5 (Buckler & Scribner 1985) logt (hours) logt (hours) 1000 1000 1000 1000 0.1 1 10 0.1 100 1 10 100**Comparison with Tests: Step-Wise T History**Proposed formula (constants calibrated by relaxation test at 20 ºC) 130 ºC 130 ºC T (ºC) T (ºC) 70 ºC 70 ºC 20 ºC 45 ºC 20 ºC Time (hours) Time (hours) Stress (MPa) Stress (MPa) Rostasy & Thienel, 1991 Rostasy & Thienel, 1991 Time (hours) Time (hours)**Programs for Structural Creep Analysis**Programs now in use (e.g. SOFiSTiK) are based on the state of the art 40 years ago – history integrals of linear aging viscoelasticity, computed in time steps — obsolete, misleading. Reasons: 1) Poor material model. 2) Beam-type analysis instead of 3D 3) Variation of drying effects due to thickness variation 4) Cracking (which is not hereditary), nonlinear deformations 5) Gradual prestress relaxation (nonlinear viscoplasticity) 6) Variable environment Required:Rate-type creep law in which the history is taken into account by current values of internal variables (partial strains).**Causes of Error**Prestress Dead load • Incorrect standard recommendations on concrete compliance and shrinkage functions (esp. for > 3 years, and on prestressing steel relaxation. • Differential shrinkage and drying creep compliance due to: a) plate thickness, b) temperature (both CEB 1972, 1990 and ACI 1972, 2008 are incorrect). • Beam elements instead of 3D FEM • No updating by short-time tests. • No statistical estimate (95% confidence limit).**4. Aren’t Other Presstressed Segmentally Errected**Bridges Behaving Similarly?**Tsukiyono Bridge**Completion 1982 Age 26years Max. span 84.5m After Y. Watanabe, Shimizu Constr. Co., Concreep 2008**CONCREEP8**Time dependent deflection –Tsukiyono Bridge 1 10 100 1000 10,000 0 JSCE 50 100 150 Data courtesy Y. Watanabe, Chief Engineer, Shimizu Constr. Co., 2008**Koshirazu Bridge**Completion 1987 Connected 1997 Age 10 years Max. span 59.5m After Y. Watanabe, Chief Engineer, Shimizu Constr. Co., Concreep 2008**CONCREEP8**Time dependent deflection – Koshirazu Bridge 1 10 100 1000 10,000 0 20 JSCE 40 60 80 100 Data courtesy Y. Watanabe, Chief Engineer, Shimizu Constr. Co., 2008**Elapsed days**Konaru Bridge 4000 6000 8000 10000 2000 0 JSCE 200 Deflection (mm) 400 100 1000 10000 10 0 Deflection (mm) 200 400 Data courtesy Y. Watanabe, Chief Engineer, Shimizu Constr. Co., Concreep 2008**Děčín Bridge, Continuous Box Girder No Hinge at Midspan**over Elbe River, Northern Bohemia**Mid-Span Deflection of DěčínBridge**2000 4000 6000 0 Days, linear scale Prediction Deflection (cm) 15 Measurements 30 100000 100 1000 10000 Days, log scale 0 Prediction Deflection (cm) 15 Measurements Data courtesy Dr. Vrablik, CTU Prague 30**5. Wake-Up Call:Excessive deflections are not isolated**but endemic to this kind of bridges**56 Bridges: Excessive Deflections. Any Bound? -1**102 103 104 102 103 104 102 103 104 102 103 104 102 103 104 102 103 104 102 103 104 0 0 0 0 0 0 0 Parrots Ferry Bridge 195 m U.S.A., 1978 Nordsund Bru 142 m Norway, 1971 Urado Bridge 230 m Japan, 1972 0.12 0.5 0.2 0.3 0.2 0.13 0.4 KB Bridge 241 m Palau, 1977 Konaru Bridge 101.5 m Japan, 1987 Pelotas River Bridge 189 m Brazil, 1966 Tunstabron 107 m Sweden, 1955 1 0.24 0.4 0.4 0.26 0.8 0.6 102 103 104 102 103 104 102 103 104 102 103 104 102 103 104 102 103 104 102 103 104 0 0 0 0 0 0 0 SavinesBridge Span l 77 m France, 1960 Zvíkov-Vltava Bridge Hinge 4 84 m Czech Rep., 1963 Zvíkov-Otava Bridge Hinge 1 84 m Czech Rep., 1962 Zvíkov-Otava Bridge Hinge 2 84 m Czech Rep., 1962 Savines Bridge Span k 77 m France, 1960 Zvíkov- Vltava Bridge Hinge 3 84 m Czech Rep. 1963 Maastricht Bridge 112 m Netherlands, 1968 0.1 0.1 0.1 0.15 0.1 0.1 0.1 0.2 0.2 0.3 0.2 0.2 0.2 0.2 102 103 104 102 103 104 102 103 104 102 103 104 102 103 104 102 103 104 102 103 104 0 0 0 0 0 0 0 Savines Bridge Span j 77 m France, 1960 Savines Bridge Span h 77 m France, 1960 Savines Bridge Span b77 m France, 1960 Savines Bridge Span g 77 m France, 1960 0.1 Savines Bridge Span f 77 m France, 1960 Savines Bridge Span c77 m France, 1960 0.08 0.1 0.1 0.1 0.1 0.1 Alnöbron Hinge 1 134 m Sweden, 1964 0.2 0.16 0.2 0.2 0.2 0.2 0.2 102 103 104 102 103 104 102 103 104 102 103 104 102 103 104 102 103 104 102 103 104 0 0 0 0 0 0 0 Grubben- vorst Bridge 121 m Netherlands, 1971 Källösundsbron Hinge 2 107 m Sweden, 1958 Källösundsbron Hinge 1 107 m Sweden, 1958 Alnöbron Bridge Hinge 5 134 m Sweden, 1964 Alnöbron Hinge 4 134 m Sweden, 1964 0.08 Alnöbron Hinge 3 134 m Sweden, 1964 Alnöbron Hinge 2 134 m Sweden, 1964 0.05 0.05 0.05 0.08 0.06 0.07 0.1 0.16 0.12 0.1 0.16 0.1 0.14 [Deflection/Span (%) ] vs. Time Elapsed (days, log-scale)**56 Bridges: Excessive Deflections. Any Bound? -2**102 103 104 102 103 104 102 103 104 102 103 104 102 103 104 102 103 104 102 103 104 0 0 0 0 0 0 0 Ravenstein Bridge 139 m Netherlands, 1975 Gladesville Arch 300 m Australia, 1962 Empel Bridge 120 m Netherlands, 1971 Narrows Bridge 97.5 m Australia, 1960 Wessem Bridge 100 m Netherlands, 1966 Koshirazu Bridge 59.5 m Japan, 1987 Captain Cook Bridge 76.2 m Australia, 1966 0.035 0.06 0.06 0.1 0.04 0.07 0.05 0.08 0.07 0.12 0.12 0.2 0.1 0.14 104 102 103 104 102 103 102 103 104 102 103 104 102 103 104 102 103 104 102 103 104 0 0 0 0 0 0 0 Macquarie Bridge Australia 1969 Želivka Bridge, Hinge 1 102 m Czech Rep., 1968 Victoria Bridge Australia, 1966 Želivka Bridge, Hinge 2 102 m Czech Rep., 1968 Art Gallery Bridge Australia, 1961 Děčίn Bridge 104 m Czech Rep. 1985 0.2 .008 0.01 .03 0.06 0.025 .008 Konaru Bridge 101.5 m Japan, 1987 0.4 .016 0.02 0.12 .016 .06 0.05 102 103 104 102 103 104 102 103 104 102 103 104 102 103 104 102 103 104 102 103 104 0 0 0 0 0 0 0 Zuari Bridge Span C 120 m Goa, 1986 Zuari Bridge Span E 120 m Goa, 1986 La Lutrive Bridge Hinge 3 131 m Switzerland, 1973 La Lutrive Bridge Hinge 2 131 m Switzerland, 1973 Heteren Bridge 121 m Netherlands, 1972 Stenungs- undsbron 94 m Sweden Tsukiyono Bridge 84.5 m Japan, 1982 0.1 0.04 0.07 0.04 0.06 0.2 0.05 0.2 0.08 0.08 0.1 0.4 0.14 0.12 104 102 103 104 102 103 102 103 102 103 104 104 102 103 104 102 103 104 102 103 104 0 0 0 0 0 0 0 Zuari Bridge Span L 120 m Goa, 1986 Zuari Bridge Span P 120 m Goa, 1986 Zuari Bridge Span J 120 m Goa, 1986 Zuari Bridge Span O 120 m Goa, 1986 Zuari Bridge Span M 120 m Goa, 1986 Zuari Bridge Span F 120 m Goa, 1986 Zuari Bridge Span H 120 m Goa, 1986 0.07 0.05 0.05 0.05 0.04 0.04 0.04 0.14 0.1 0.1 0.08 0.1 0.08 0.08 [Deflection/Span (%) ] vs. Time Elapsed (days, log-scale)**6. Why have the standard**recommendations for concrete creep been so incorrect, for 40 years?**Criteria for Selecting Material Creep Model**a) Ability to fit individual compliance and shrinkage curves of one concrete — essential validation b) Statistical comparison with laboratory database for all kinds of concretes— code committees’ way c) Correct prediction of multi-decade deflection of structures — additional essential validation d) Theoretical foundation— essential — only this is discussed today**11821 creep data points**Histograms of Data Points in New NU-ITI Database Age at Loading log t’ Effective Size Loading Duration log(t-t’) DAYS 8326 shrinkage data points Drying Duration log(t-t0) Effective Size**(1/MPa)**Examples of Almost Useless Statistical Comparisons for Creep, NU-ITI Database, Unweighted Data Why so little difference among models? — Because all data points got equal weights. Calculated Measured MPa-1**shrinkage**Almost Useless Comparisons of Residuals (Errors) creep Time [days] After Al-Manaseer & Lam, ACI Mat.J. 2005**How to optimize and compare models?—By multivariate**regression statistics, countering bias by proper weights 1D Boxes (Intervals) 2D Boxes 5 y=J(t,t') mi =number of all points in intervali log (t-t') 4 5 10 y=J(t,t') 4 3 3 9 n 2 2 10 8 9 i=1 i=1 8 Problem with multi- dimensional boxes: Some contain only 0 or 1 point. 7 7 t'=t'1 6 6 15 14 12 13 t'2 12 11 11 t'3 mi =number of all points in boxi points j=1,2,…mi log t' (age of loading) 20 (or thickness,or humidity h) 19 18 17 i=16 n Standard Error of Regression, s: Yij 22 yij 21 x=log (t-t') Weights: Number of all data points Weighted mean of all data points**Comparisons by Weighted Least-Square Regression**Coefficients of Variation (C.o.V.) of Errors a) Compliance b) Shrinkage 50 2D boxes of log(t-t’) and H 28 2D boxes of log(t-t0) and 47.4 31.0 30.2 41.9 27.3 42.6 31.0 44.4 28.5 42.3 The differences are as marked as those for Palau**Essential Criterion: Capability to Fit All Test Curves for**One and the Same Concrete 0.5 0.8 Kommendant et al., (b),1976 sealed Optimum fit Wittmann et al.,1987 • des∞tsh • 10-3 (days) • 83 0.891 120.7 • 0.893 264.3 • 300 0.812 699.7 d = 83 mm d = 160 mm J(t, t')(×10-6/psi) Strain (×10-3) 0.3 0.4 t' = 90 days t' = 28 days d = 300 mm t' = 270 days Shrinkage 0.0 0.1 Creep 102 103 10-1 100 101 10-2 102 103 10-2 10-1 100 101 1.0 0.9 Hansen and Mattock,1966 Canyon Ferry Dam,1958 Optimum fit RH = 50% t0= 8 Days T = 21°C 11.5 × 4.25 in. 0.7 t' = 2 days J(t, t') (×10-6/psi) 23 × 8.5 in. t' = 7 days Strain (×10-3) 0.5 0.5 t' = 28 days 46 × 17 in. t' = 90 days 0.3 t' = 365 days I-section 0.1 0.0 102 103 100 101 102 103 101 t-t' (days) t-t0 (days)**7. Idea:Exploit box girder deflections for inverse analysis**to calibrate multi-decade material model for creep(helped by new RILEM TC-MDC)