The Pennsylvania State University. The Graduate School. Department of Civil Engineering TIME-DEPENDENT ANALYSIS OF PRETENSIONED

Transcription

1 The Pennsylvania State University The Graduate School Department of Civil Engineering TIME-DEPENDENT ANALYSIS OF PRETENSIONED CONCRETE BRIDGE GIRDERS A Dissertation in Civil Engineering by Brian D. Swartz 2010 Brian D. Swartz Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy May 2010

2 The dissertation of Brian D. Swartz was reviewed and approved* by the following: Andrew Scanlon Professor of Civil Engineering Dissertation Co-Advisor Co-Chair of Committee Andrea J. Schokker Professor and Head of Civil and Environmental Engineering, The University of Minnesota Duluth Adjunct Professor, The Pennsylvania State University Dissertation Co-Advisor Co-Chair of Committee Daniel G. Linzell Associate Professor of Civil and Environmental Engineering Ali M. Memari Associate Professor of Architectural Engineering William D. Burgos Professor of Civil and Environmental Engineering Professor-in-Charge of Graduate Programs in Civil and Environmental Engineering *Signatures are on file in the Graduate School

3 iii ABSTRACT The increasing use of high strength concrete in pretensioned concrete bridge girders drove the development of new prestress loss provisions that were introduced to the AASHTO LRFD Bridge Design Specifications in The provisions have led to industry concerns because of the complex implementation of the equations and seemingly unconservative results. The research documented in this thesis studies the models used historically for prestress loss analysis in bridge girders, then proposes a simplified method for design. The simplified method is derived from fundamental principles of mechanics and validated by comparison with a detailed time step analysis. Monte Carlo simulation is used to consider the inherent uncertainty in timedependent analysis of concrete girders. The simplified approach, called the Direct Method, is formatted for inclusion in the AASHTO LRFD Bridge Design Specifications.

4 iv TABLE OF CONTENTS Chapter 1 Introduction Background Problem Statement Objective and Scope Thesis Organization... 4 Chapter 2 Material Properties Shrinkage of Concrete ACI 209 (1992) AASHTO (2004) AASHTO (2005) Comparison of Methods Discussion Creep of Concrete ACI 209 (1992) AASHTO (2004) AASHTO (2005) Comparison of Methods Discussion Modulus of Elasticity of Concrete AASHTO (2004) AASHTO (2005) Discussion Relaxation of Prestressing Steel Estimating Intrinsic Relaxation Modulus of Elasticity of Prestressing Steel Summary Chapter 3 Approximate Time-Dependent Analysis AASHTO Loss due to Shrinkage Loss due to Creep Loss due to Steel Relaxation S6-06 Canadian Highway Bridge Design Code Loss due to Shrinkage Loss due to Creep Loss due to Steel Relaxation AASHTO Stages for Analysis Transformed Section Coefficient... 44

5 v Analysis Before Deck Placement Analysis After Deck Placement AASHTO 2005 Approximate Method Discussion Stages for Analysis Transformed Section Coefficient Differential Shrinkage Transformed Section Properties Chapter 4 Analysis Methods Detailed Time-Step Method Assumptions Development of the Method Algorithm Monte Carlo Simulation Summary Chapter 5 Detailed Time-Dependent Analysis Stages of Behavior Example Bridge Details PCI BDM Example FHWA Example Components of Time-Dependent Behavior Time of Deck Placement Irreversible Creep Summary Chapter 6 The Direct Method for Time-Dependent Analysis Elastic Shortening and Steel Relaxation Concrete Shrinkage Differential Shrinkage Approximate Calculation of Differential Shrinkage Strain Approximate Calculation of the Deck Creep Coefficient Approximating the Effective Differential Shrinkage Force Creep of Concrete Implementation of the Direct Method Numerical Example Differential Shrinkage Loss of Prestress Calculation of Bottom Fiber Stress at Midspan: (Tension shown Positive) Summary Chapter 7 Validating the Direct Method Uncertainty Study Monte Carlo Simulation

6 Input Parameters Uncertainty Study Results Irreversible Creep Sensitivity Study Summary Chapter 8 Conclusion Summary Future Research Recommendations References vi Appendix A Appendix B Proposed Provision for AASHTO LRFD Bridge Design Specifications..177 Numerical Example Demonstrating the Time Step Method..180

7 vii LIST OF FIGURES Figure 2-1. Comparison of shrinkage models over time for common input parameters Figure 2-2. Comparison of shrinkage models with respect to the concrete strength parameter Figure 2-3. Comparison of shrinkage models with respect to the V/S ratio parameter Figure 2-4. Comparison of shrinkage models with respect to the V/S ratio parameter Figure 2-5. Experimental results from shrinkage tests as reported in NCHRP Report 496 (Source: Tadros et. al., 2003) Figure 2-6. Creep of concrete for loads applied instantaneously Figure 2-7. Total stress-related strain as a function of the concrete age when the stress change occurs Figure 2-8. Comparison of creep models over time for common input parameters Figure 2-9. Comparison of creep models with respect to the concrete strength parameter Figure Comparison of creep models with respect to the V/S ratio parameter Figure Comparison of creep models with respect to the relative humidity parameter Figure Experimental results from creep tests in NCHRP Report 496 (Source: Tadros, 2003) Figure Summary of test data used to develop predictive models for concrete elastic modulus (Source: Tadros et. al., 2003) Figure 3-1. Timeline representing the change in prestressing force over time in a typical prestressed member (Source: Tadros, 2003) Figure 3-2. Schematic diagram demonstrating the effect of steel restraint on concrete shrinkage Figure 3-3. Generic composite cross-section to facilitate the derivation of Δf cdf Figure 3-4. Transformed cross section, shown schematically Figure 4-1. Schematic of the creep compliance relationship Figure 4-2. Diagram of the generic strain profile to facilitate development of the time step algorithm... 71

8 Figure 4-3. Schematic of the Monte Carlo simulation technique used for the uncertainty study of prestress loss methods Figure 5-1. Stage of loading for a pretensioned concrete girder - manufacturing through service Figure 5-2. Strain and stress in the girder cross section due to initial prestressing force Figure 5-3. Strain and stress in the girder cross section due to girder self-weight Figure 5-4. Strain and stress in the girder cross section due to shrinkage prior to deck placement Figure 5-5. Strain and stress in the girder cross section due to creep prior to deck placement Figure 5-6. Strain and stress in the girder cross section due to deck self-weight Figure 5-7. Strain and stress in the girder cross section due to shrinkage after deck placement Figure 5-8. Strain and stress in the girder cross section due to superimposed dead load on the composite section Figure 5-9. Strain and stress in the girder cross section due to creep after deck placement Figure Strain and stress in the girder cross section due to live load Figure Bridge section for PCI BDM Example 9.4 (Source: PCI, 1997) Figure Girder section for PCI BDM Example 9.4 (PCI, 1997) Figure Bridge section for FHWA Example (Source: FHWA, 2003) Figure Girder section for FHWA Example (Source: FHWA, 2003) Figure Effective prestress over time for PCI BDM Example 9.4 assuming deck casting at 90 days Figure Comparison between the time-step results and the AASHTO 2005 method for effective prestress in the PCI BDM Example 9.4 bridge, assuming the deck is cast at 90 days Figure Components of prestress loss for the PCI BDM Example 9.4 bridge assuming the deck is cast at 90 days Figure Components of bottom fiber stress for the PCI BDM Example 9.4 bridge assuming the deck is cast at 90 days viii

9 Figure Comparison of prestress loss components for different times of deck placement in the PCI BDM Example 9.4 bridge Figure Comparison of prestress loss components for different times of deck placement in the PCI BDM Example 9.4 bridge Figure Total effective prestress estimated by AASHTO 2005 over a range of deck placement times for the PCI BDM Example 9.4 bridge Figure Creep of concrete when loaded and unloaded (Source: Mehta and Monteiro, 2006) Figure Impact of creep recovery factor on effective prestress for the PCI BDM Example 9.4 bridge Figure Impact of creep recovery factor on bottom fiber concrete stress for the PCI BDM Example 9.4 bridge Figure 6-1. The format of the Direct Method relative to the AASHTO 2004 and AASHTO 2005 methods Figure 6-2. The effective action on the composite section due to differential shrinkage Figure 7-1. Rectangular stress block simplification used when calculating the effective width of the deck (Source: Wight and Macgregor, 2009) Figure 7-2. Conceptual depiction of the method used to consider model uncertainty in the Monte Carlo simulation Figure 7-3. Determination of the model uncertainty factor for concrete elastic modulus (Data source: Tadros et. al., 2003) Figure 7-4. Histogram of Monte Carlo simulation results for prestress loss estimates applied to PCI BDM Example Figure 7-5. Comparison of mean values from Monte Carlo simulation frequency distribution with nominal design values for prestress loss applied to PCI BDM Example Figure 7-6. Histogram of Monte Carlo simulation results for bottom fiber concrete stress estimates applied to PCI BDM Example Figure 7-7. Comparison of mean values from Monte Carlo simulation frequency distribution with nominal design values for prestress loss applied to PCI BDM Example Figure 7-8. Histogram of Monte Carlo simulation results for prestress loss estimates applied to the FHWA example ix

10 x Figure 7-9. Comparison of mean values from Monte Carlo simulation frequency distribution with nominal design values for prestress loss applied to the FHWA Example Figure Histogram of Monte Carlo simulation results for bottom fiber concrete stress estimates applied to the FHWA example Figure Comparison of mean values from Monte Carlo simulation frequency distribution with nominal design values for prestress loss applied to the FHWA example Figure Histogram of Monte Carlo simulation results for prestress loss estimates applied to PCI BDM Example 9.4, taking the creep recovery factor to be a random variable uniformly distributed between 50% and 100% Figure Comparison of mean values from Monte Carlo simulation frequency distribution with nominal design values for prestress loss applied to PCI BDM Example 9.4, taking the creep recovery factor as a random variable uniformly distributed between 50% and 100% Figure Histogram of Monte Carlo simulation results for bottom fiber stress estimates applied to PCI BDM Example 9.4, taking the creep recovery factor to be a random variable uniformly distributed between 50% and 100% Figure Comparison of mean values from Monte Carlo simulation frequency distribution with nominal design values for bottom fiber stress applied to PCI BDM Example 9.4, taking the creep recovery factor as a random variable uniformly distributed between 50% and 100% Figure Scatter plot of Monte Carlo simulation results to indicate sensitivity to the relative humidity input Figure Scatter plot of Monte Carlo simulation results to indicate sensitivity to the girder compressive strength input Figure Scatter plot of Monte Carlo simulation results to indicate sensitivity to the deck compressive strength input Figure Scatter plot of Monte Carlo simulation results to indicate sensitivity to the elastic modulus of prestressing steel input Figure Scatter plot of Monte Carlo simulation results to indicate sensitivity to the time of deck placement input Figure Sensitivity study comparing the effect on prestress loss of material property model errors with that of other common variables Figure Sensitivity study comparing the effect on prestress loss of material property model errors with that of other common variables

11 xi LIST OF TABLES Table 2-1. Summary of experimental results for creep (Source: Tadros, 2003) Table 2-2. Summary of experimental results for creep (Source: Tadros, 2003) Table 3-1. Assumptions in the AASHTO LRFD (2004) creep loss prediction Table 4-1. Stress and strain relationships for key values in the time step routine Table 5-1. Parameters for the PCI BDM Example 9.4 Bridge (Source: PCI, 1997) Table 5-2. Summary of moments at midspan (k-in) for PCI BDM Example 9.4 (Source: PCI, 1997) Table 5-3. Concrete elastic modulus for PCI BDM Example 9.4 (Source: PCI, 1997) Table 5-4. Composite section properties for PCI BDM Example 9.4 (Source: PCI, 1997) Table 5-5. Parameters for the FHWA Example Bridge (Source: FHWA, 2003) Table 5-6. Summary of moment at midspan (k-in) for the FHWA Example (Source: FHWA, 2003) Table 5-7. Concrete elastic modulus for the FHWA Example (Source: FHWA, 2003) Table 5-8. Composite section properties for the FHWA Example (Source: FHWA, 2003) Table 7-1. Probability distributions related to material properties used in Monte Carlo simulation Table 7-2. Probability distributions related to initial prestressing used in Monte Carlo simulation Table 7-3. Probability distributions related to precast girder geometry used in Monte Carlo simulation Table 7-4. Probability distributions related to cast-in-place deck geometry and behavior used in Monte Carlo simulation Table 7-5. Probability distributions related to construction schedule used in Monte Carlo simulation Table 7-6. Probability distribution related to environmental factors used in Monte Carlo simulation

12 Table 7-7. Probability distribution related to the relaxation coefficient used in Monte Carlo simulation Table 7-8. Probability distributions related to the model uncertainty factors for concrete creep, shrinkage, and elastic modulus used in Monte Carlo simulation Table 7-9. Probability distributions related to applied loads used in Monte Carlo simulation Table Summary of Monte Carlo simulation results for prestress loss estimates applied to PCI BDM Example Table Summary of Monte Carlo simulation results for prestress loss estimates applied to PCI BDM Example Table Summary of Monte Carlo simulation results for prestress loss estimates applied to the FHWA Example Table Summary of Monte Carlo simulation results for prestress loss estimates applied to the FHWA example xii

13 Chapter 1 Introduction The flexural design of pretensioned concrete bridge girders is often controlled by tension stresses at service. Limits are imposed on tension stresses in concrete to minimize cracking. In order to anticipate the stresses in a bridge girder during service, engineers must be able to estimate the loss of prestress over time. This thesis first summarizes methods available to predict the time-dependent behavior of concrete girders. Three provisions for estimating prestress losses will be examined: 1) the Old AASHTO method, last published in 2004 (AASHTO, 2004), 2) the method of the S6-06 Canadian Highway Bridge Design Code (CSA, 2006), and 3) the method adopted by AASHTO in the 2005 Interim Revisions (AASHTO, 2005), which has been modified only editorially since. Considering the past approaches to the problem, and a detailed time-step analysis of the time-dependent effects, a streamlined method is developed and proposed in this thesis. It has been termed the Direct Method to use nomenclature separate from others. The Direct Method is validated through its fundamental derivation and through an uncertainty analysis by Monte Carlo techniques Background Accurate estimates of prestress loss are vital to successful design of prestressed concrete members. The amount of force available from the prestressing strands, which is a function of prestress losses, affects the quantity of strands needed and the size of the concrete cross section.

14 2 The amount of prestressing steel and the size of the concrete section directly affect bridge efficiency and cost. In recent years, understanding of the concrete material and quality control of its production have improved such that high-strength and high-performance concrete are now common in bridge applications. Concerns have been raised (Tadros et. al., 2003) about the applicability of historical methods to the design of girders with high-strength concrete. NCHRP Report 496 (Tadros et. al, 2003) was published with an aim at extending applicability of the AASHTO LRFD Bridge Design Specifications to include time-dependent analysis of highstrength concrete girders. The recommendations of this report were adopted, almost in their entirety, into the 3 rd edition of the Specifications as part of the 2005 Interim Revisions (AASHTO, 2005). For the purposes of this thesis, AASHTO 2005 will refer broadly to the method introduced in 2005, including minor editorial revisions made since 2005 and AASHTO 2004 will refer to the method replaced by the 2005 Interim Revisions. The AASHTO 2005 method is more computationally demanding than its predecessor. This has caused designers to rely more heavily on software solutions, sometimes bringing the engineer a step farther from the fundamentals of the problem. Additionally, the AASHTO 2005 method tends to predict smaller prestress losses than the AASHTO 2004 method for the same design parameters. Smaller loss totals result in a less conservative design in service. Awareness of these concerns prompted the research documented in this thesis.

15 Problem Statement The material property model for elastic modulus, creep, and shrinkage used by the AASHTO 2004 method were developed in the mid-1970 s for a range of concrete strengths common at the time. The increasing use of high-strength concrete prompted the research documented in NCHRP Report 496 (Tadros et. al., 2003) that led to a new method for timedependent analysis in the AASHTO LRFD Bridge Design Specifications starting in The industry concern about the AASHTO 2005 method has highlighted two needs: 1) A more thorough understanding of time-dependent analysis of pretensioned girders in order to validate the AASHTO 2005 method and to understand what it represents, and 2) A simpler approach to time-dependent analysis that can be applied more efficiently at the design phase. The research documented in this thesis aims at addressing both of those needs Objective and Scope The objective of the research is to develop a simplified procedure for calculating prestress losses in bridge girders. The tasks undertaken to reach this objective are as follows: Review literature related to concrete material properties and existing prestress loss models Conduct a detailed review of the recommendations for NCHRP Report 496 that were adopted into the AASHTO LRFD Bridge Design Specifications (AASHTO, 2005)

16 4 Develop a time-step method that can be used to track prestress loss and concrete stresses through the life of a bridge girder based on assumed material property models and a specified loading history Assemble a simple, complete example problem to demonstrate the time step procedure Develop a Direct Method that can be used as an alternative to the AASHTO 2005 and detailed time step methods for time-dependent analysis Perform an uncertainty analysis through Monte Carlo simulation to compare various prestress loss methods and evaluate the proposed Direct Method Format the Direct Method into language suitable for inclusion in the AASHTO LRFD Bridge Design Specifications Prepare an example problem to demonstrate application of the Direct Method 1.4. Thesis Organization The thesis will first summarize the material property models and approximate methods for estimating time-dependent behavior common in bridge design practice in North America. A detailed time-step model is then developed and programmed. The time-step model serves as a theoretical baseline for the comparison of methods. A simplified approach, termed the Direct Method is developed from fundamental mechanics and existing material models. The Detailed method is validated through an uncertainty study using Monte Carlo simulation.

17 Chapter 2 Material Properties The behavior of a prestressed concrete member over time is dependent on the material properties. Five material characteristics are identified in this chapter as particularly relevant to the time-dependent analysis of prestressed bridge girders: 1) shrinkage of concrete, 2) creep of concrete, 3) modulus of elasticity of concrete, 4) relaxation of steel, and 5) modulus of elasticity of steel. The sections that follow detail the characteristics of each material property and present the methods often used in predicting their values Shrinkage of Concrete Shrinkage of concrete occurs at several stages during the life of a prestressed beam and is caused by different mechanisms. Not all types of shrinkage lead to loss of prestress. First, plastic shrinkage refers to a volume loss due to moisture evaporation in fresh concrete, generally at exposed surfaces (Mindess et. al., 2002). This shrinkage occurs before prestressing force is applied, and does not affect long-term prestressing forces. Drying Shrinkage is the strain due to loss of water in hardened concrete (Mindess, et. al., 2002). Since drying shrinkage occurs in hardened concrete, it affects the time-dependent behavior and loss of prestress. Drying shrinkage occurs almost entirely in the paste of the concrete matrix, with aggregate providing some restraint against volume changes. Since drying shrinkage involves moisture loss, it is largely affected by the ambient relative humidity. Drying shrinkage is also affected by the specimen s shape and size if there is a large amount of surface

18 6 area for the volume, more moisture can be drawn out of the concrete. Additionally, drying shrinkage is affected by the concrete porosity, which is a function of mixture proportions and curing conditions. Two special cases of drying shrinkage in hardened concrete are autogeneous and carbonation shrinkage. Since both occur after the concrete is hardened, they can contribute to the time-dependent behavior of concrete. Autogeneous shrinkage occurs as cement paste hydrates, because the volume of hydrated cement paste is less than the total solid volume of unhydrated cement and water (Cousins, 2005). Carbonation shrinkage results from the carbonation of the calcium-silicate-hydrate molecules in concrete, which causes a decrease in volume (Mindess, et. al., 2002). For the purposes of this thesis, shrinkage will refer to the summation of all drying shrinkage and exclude plastic shrinkage. Due to the complex and uncertain nature of shrinkage, most predictive models are empirical fits to experimental data. In most cases the models asymptotically approach an ultimate shrinkage value that was determined from the test data and is further adjusted by a series of factors which account for differences between the test conditions and the in-situ conditions. Three models are summarized and compared in the following sections: the ACI 209 (1992) method, which has long been an industry baseline, the AASHTO 2004 method, and the method adopted by AASHTO 2005, which was developed primarily for use with high-strength concrete as documented by NCHRP Report 496 (Tadros et. al., 2003) ACI 209 (1992) The ACI 209 shrinkage model recommends an ultimate shrinkage strain of in/in subject to a series of adjustment factors, γ sh, to account for non-standard conditions.

19 (2-1) The net adjustment factor is given by the product of several other factors in (2-2).,,,,,,, (2-2) The last four terms in (2-2), representing adjustments for slump,, fine aggregate content,, cement content,, and air content,, will all be taken as 1.0 as the variables cannot be easily defined by the structural designer. Also, for concrete steam-cured 1 to 3 days,, 1.0. The remaining adjustment factors are calculated by (2-3) through (2-5). Humidity correction factor: % 80%, % (2-3) Size factor:, 1.2. (2-4) Time-development factor to predict shrinkage at any time, t, for steam-cured concrete with a start of drying at time, t c :, 55 (2-5) AASHTO (2004) The AASHTO 2004 shrinkage model suggests an ultimate shrinkage strain of in/in and adjusts that value for time, specimen size, and relative humidity. The base equation, which is often expressed including the time-development term, is given in (2-6).

20 (2-6) respectively. The correction factors for size and relative humidity are determined from (2-7) and (2-8), (2-7) % % 70 (2-8) AASHTO (2005) The AASHTO 2005 material property models were developed as part of the NCHRP Report 496 study (Tadros, et. al., 2003). In developing the model, emphasis was placed on characterizing the behavior of high-strength concrete. The model suggests an ultimate shrinkage of in/in and adjusts that value for specimen size, relative humidity, concrete strength, and time development, as calculated by (2-10) through (2-13). The base equation is given in (2-9) (2-9) (2-10) (2-11)

21 9 5 1 (2-12) 61 4 (2-13) Comparison of Methods The models for shrinkage cannot be compared considering only the ultimate shrinkage strain used in the model. Each model is dependent on a set of assumptions often called the standard conditions and adjustment factors are used to account for actual conditions. If the standard conditions vary, a direct comparison of ultimate shrinkage strains is not valid. A graphical comparison is presented where a practical range of values is assigned to each variable in the models. This indicates the relative sensitivity of the model to each primary input variable. First, the time dependence of each model is investigated in Figure 2-1. The figure demonstrates that all three methods predict a similar rate in development of shrinkage strain over time. Also, each model asymptotically approaches a final maximum value. Since the development of shrinkage over time is predicted similarly by all methods, and the final timedependent analysis of a prestressed girder will depend more on the total shrinkage than on the rate of its development, the methods will be compared for the other input parameters considering only the ultimate shrinkage value predicted. Figure 2-1 also suggests that the AASHTO 2005 method predicts less shrinkage than the other methods. This conclusion, as drawn from Figure 2-1, is true

22 10 for the assumed combination of input values, and will be further validated in considering the other parameters Shrinkage Strain, ε sh Assumed Variables: f' c = 8 ksi f' ci = 6.4 ksi H = 70% V/S = 3.5 Moist Cured, 1 day AASHTO 2004 AASHTO 2005 ACI 209 (1992) Drying Time (Days) Figure 2-1. Comparison of shrinkage models over time for common input parameters Figure 2-2 compares the shrinkage models over a range of concrete strengths when other input parameters are held constant. The models are compared based only on the final shrinkage strain predicted. Figure 2-2 indicates a significant change introduced by the AASHTO 2005 method. The AASHTO 2005 model is dependent on the concrete strength input, while the other two models do not consider concrete strength.

23 Constant Values: V/S = 3.5 in H = 70% ACI 209 (1992) AASHTO 2004 AASHTO 2005 Ultimate Shrinkage Strain Concrete Compressive Strength, f' c (ksi) Figure 2-2. Comparison of shrinkage models with respect to the concrete strength parameter Figure 2-3 compares shrinkage models considering their response to the V/S input parameter. The graph indicates a slightly different treatment of the V/S ratio for the different models, although the difference over a reasonable range of values is small especially when compared with the difference in response to concrete strength (Figure 2-2). AASHTO-type prestressed concrete girders typically have a V/S ratio around 3.5; deck sections are at the higher end of the range, approximately 4.5. Figure 2-4 indicates that all three shrinkage models have a very similar trend with respect to relative humidity, decreasing the total shrinkage prediction as relative humidity increases.

24 Constant Values: f' c = 6 ksi H = 70% ACI 209 (1992) AASHTO 2004 AASHTO 2005 Ultimate Shrinkage Strain Ratio Volume:Surface Area (in) Figure 2-3. Comparison of shrinkage models with respect to the V/S ratio parameter Constant Values: f' c = 8 ksi V/S = ACI 209 (1992) AASHTO 2004 AASHTO 2005 Ultimate Shrinkage Strain Relative Humidity, % Figure 2-4. Comparison of shrinkage models with respect to the V/S ratio parameter

25 Discussion The AASHTO 2005 model for shrinkage was developed for use with high strength concrete applications. For the range of concrete strengths typical of pretensioned concrete girders (f c = 6-12 ksi), the AASHTO 2005 model predicts less shrinkage than the other two models presented here, including its predecessor in the AASHTO LRFD Bridge Design Specifications, AASHTO This implies that use of the AASHTO 2005 model will estimate smaller prestress losses and could impact the flexural design of prestressed girders. Additionally, it should be noted that the AASHTO 2005 model was developed for high strength concrete, but it is the only model currently in the specifications, implying it should be used for a broad range of concrete strengths. The scope of the specifications suggests the model is applicable up to f c = 15 ksi, with no lower limit (AASHTO, 2005). The development of AASHTO 2005 is documented in NCHRP Report 496 (Tadros et. al., 2003). Data were generated from experimentation on concrete mixes from four different states Nebraska, New Hampshire, Texas, and Washington. A summary of the experimental data is provided in Figure 2-5, which combines a number of figures from NCHRP Report 496. The labels S1, S2, and S3 indicate three different test specimens. The tests were performed at a controlled relative humidity (35-40%) and the specimens had a V/S ratio of 1.0. All specimens had a tested compression strength in the range f c = ksi. Although NCHRP Report 496 does not explicitly say so, it will be assumed that the specimens were moist-cured because a factor of 35 was used in the ACI 209 time-development term in Appendix F of NCHRP Report 496). The plots are superimposed with the shrinkage predicted by each of the three shrinkage models discussed given the test parameters. In the plots, AASHTO refers to the AASHTO 2004 model, and Proposed refers to the AASHTO 2005 model.

26 14 Figure 2-5. Experimental results from shrinkage tests as reported in NCHRP Report 496 (Source: Tadros et. al., 2003) The experimental results are further summarized in Table 4-1, which compares the observed shrinkage strain to the shrinkage strain predicted by each model. A volumetric gain (decrease in shrinkage strain) is observed between days of drying for three of the four tests. Drying shrinkage occurs when the relative humidity outside the concrete is lower than that inside the concrete and moisture evaporates. This causes a decrease in

27 15 Table 2-1. Summary of experimental results for creep (Source: Tadros, 2003) volume, and it is partially reversible, but only if the ambient humidity increases (Mindess et. al., 2002). Therefore, a gradual increase in shrinkage strain would be anticipated in a shrinkage test with constant relative humidity, and a volume gain would not be expected. Observing that three of the tests demonstrate a volumetric gain introduces skepticism in evaluating the data. It suggests an error in the experimental procedure or in the data collection. This volumetric gain, since it suggests less total shrinkage, serves to validate the new model (AASHTO 2005) that predicts smaller strains. If the experimental results are in error, an error in the proposed model follows. The parameters used in the shrinkage testing (H = 35-40%, V/S = 1.0, and moist-cured) are not indicative of typical bridge girders in the United States. Therefore, adjustment factors are needed to correlate the AASHTO 2005 model with conditions other than those used during testing. In many cases the correction factors have been drawn from other models. Factors for relative humidity, specimen size, concrete strength, and time-development are discussed here. The adjustment factor for relative humidity matched that published in the PCI Bridge Design Manual (1997) and agreed closely with that used in ACI 209 (1992). It is reproduced in (2-11). The adjustment factor for specimen size, given in (2-10) was not changed from the previous Specification (AASHTO, 2004).

28 16 The AASHTO 2005 model introduces an adjustment factor for concrete strength, shown in (2-12). Neither of the other models in this discussion considers concrete strength in calculating shrinkage. The factor introduced to AASHTO 2005 is partially validated by the fact that its response is similar to the strength correction factor used in the AASHTO 2004 creep model. The AASHTO 2004 model, however, does not apply that factor to shrinkage calculations. Furthermore, the experimental data presented in NCHRP Report 496 was collected for range of concrete strengths (f c = ksi) to narrow to justify a strength correction factor to be applied broadly for all values of f c. The time-development factor in AASHTO 2005, shown in (2-13), is similar to that used in ACI 209 (1992). However, a change to this factor has been proposed by NCHRP Report 595 (2007). Of the adjustment factors, the choice of time-development factor is of least importance to for prestress loss estimates because the shrinkage at final time is of primary importance. The rate of shrinkage strain becomes secondary. 2.2 Creep of Concrete Creep is a time-dependent volume change in concrete due to sustained load. Creep can be divided into two categories basic creep and drying creep. Both components affect prestress losses. For the purposes of this thesis, creep of concrete will indicate the sum of basic creep and drying creep. The amount of creep observed in stressed concrete over time is a function of many variables, including: mixture proportions, level of applied stress, relative humidity, maturity of concrete when load is applied, and duration of constant applied stress. Mixture proportions greatly affect concrete s ability to resist creep, including type and amount of cement, aggregate properties, and water-to-cement ratio. Different types of cement

29 17 experience different amounts of creep, and the inclusion of supplemental cementitious materials yields even more variability in predicting the creep of a concrete mixture. Creep effects are primarily a result of stress redistribution away from the paste and towards aggregate in the concrete. Stiffer aggregates resist more load and reduce creep (Cousins, 2005). Also, aggregate with a rougher surface reduces creep because load is better transferred along the paste-aggregate interface. Finally, water-to-cementitious material ratio is significant as mixes with less free water lead to smaller volume changes due to creep. As applied stress increases, greater creep can be expected. Creep is proportional to the stress level of the concrete up to a point of 40-60% of the concrete compressive strength (Cousins, 2005). Relative humidity affects drying creep, and hence total creep. In regions with lower relative humidity, more creep can be expected. Concrete that is more mature when loaded will experience less total creep (Cousins, 2005). The effects of creep are shown schematically in Figure 2-6. Concrete loaded instantaneously will undergo an elastic strain, represented by point A. If that level of stress is held constant, additional strain will result due to creep effects. The total strain of elastic and creep effects is shown by point B in Figure 2-6. Total stress-related strain (elastic and creep) is shown schematically in Figure 2-7. This assumes that the stress change is applied instantaneously, and then remains constant. Note that the same stress change applied when the concrete is older will yield less total creep strain.

30 18 Figure 2-6. Creep of concrete for loads applied instantaneously Figure 2-7. Total stress-related strain as a function of the concrete age when the stress change occurs Creep strain due to an instantaneous load is defined in terms of a creep coefficient,,, which is a factor of the elastic strain:,, (2-14)

31 19 Where:, Creep coefficient at time (t) for load applied at time (t i ) Stress change in the concrete Concrete elastic modulus at the time of the stress change Combining creep and elastic strain to express total stress-related strain:, 1, (2-15) Figure 2-6: Stress and strain can be related by an effective elastic modulus, shown graphically in, 1, (2-16) Where:, Effective elastic modulus of concrete representing elastic and creep effects Concrete elastic modulus at the time of transfer Creep effects when stress changes are introduced gradually over time can be approximately represented by use of an age-adjusted effective modulus (Bazant, 1972) and (Trost, 1967). When a stress change varies over a time period between t i and t, an age-adjusted effective modulus can be used to simplify the relationship between stress and strain:, 1, (2-17)

32 20 Where: E c,adj χ Effective elastic modulus of concrete adjusted for a slowly developing stress change Relaxation coefficient (Trost, 1967) which accounts for the reduction in creep that occurs because not all of the stress is applied at the initial time, t i (Collins, 1991). Values typically range between 0.6 and 0.9. The concept of age-adjusted effective modulus is demonstrated in Figure 2-8. For the purposes of demonstration, the same stress change shown instantaneously in Figure 2-6 is applied in three increments in Figure 2-8. Less total creep can be anticipated in cases where the stress change occurs gradually. Each of the models studied in this thesis measure creep in terms of a creep coefficient,,, which is a ratio of creep strain to elastic strain. Similar to shrinkage, creep has historically been expressed as a function of time and an ultimate creep value for time infinity. Adjustment factors are used to adjust for non-standard conditions. The models of ACI 209 (1992), AASHTO 2004 and AASHTO 2005 are summarized in the following sections ACI 209 (1992) In the method given by ACI Committee 209, the creep coefficient is expressed by (2-18) which implies an ultimate creep coefficient of 2.35., 2.35 (2-18) The correction factor,, represents the product of several adjustment factors for nonstandard conditions:,,,,,, (2-19)

33 21 The slump factor,, fine aggregate factor,, and air content factor, are often ignored and taken as 1.0 for design. An adjustment for age at loading, for steam-cured concrete, is reproduced in (2-20)., (2-20) Age of concrete at the time of the stress change, days Factors for relative humidity and specimen size (for inch-pound units) are shown in (2-21) and (2-22), respectively., (2-21), (2-22) AASHTO (2004) The AASHTO 2004 method estimates creep by (2-23)., (2-23) Where: Age of concrete at the time of interest, days Age of concrete at the time of the stress change, days The creep coefficient is adjusted for concrete strength and specimen size, as shown in (2-24) and (2-25), respectively.

34 (2-24) (2-25) AASHTO (2005) AASHTO 2005 estimates the creep coefficient by (2-26), 1.9. (2-26) The adjustment factors for specimen size, concrete strength, and time development are the same as those used in the AASHTO 2005 shrinkage model, and are shown in (2-10), (2-12), and (2-13), respectively. The factor to adjust for relative humidity differs slightly from that used in the shrinkage model. Is it shown in (2-27) (2-27) Comparison of Methods As done in the case of shrinkage, the creep models will be compared over a practical range of the input parameters. Figure 2-8 compares the three models over time for typical input values of f c, V/S, and relative humidity. The plot shows that the rate of creep in the early ages is predicted differently, where AASHTO 2004 predicts a slower gain in creep strain, but a larger

35 23 total strain. Similar to shrinkage, however, the total strain is of primary importance in timedependent analysis. Therefore, since the general trend over time is similar for all models, comparison with other inputs will be based on the total strain Creep Coefficient, ψ(t,t i ) Assumed Variables: f' c = 8 ksi f' ci = 6.4 ksi H = 70% V/S = 3.5 Moist Cured, 1 day AASHTO 2004 AASHTO 2005 ACI 209 (1992) Maturity of Concrete (days) Figure 2-8. Comparison of creep models over time for common input parameters Figure 2-9 compares the long-time creep coefficient of each model with respect to concrete strength. The AASHTO 2004 and AASHTO 2005 models demonstrate similar trends. At higher strengths, however, the AASHTO 2005 model estimates creep strain about 25% less than its predecessor, AASHTO The ACI 209 (1992) model is not sensitive to concrete strength. Figure 2-10 shows that all three models respond similarly to the V/S ratio input. In each case a small (relative to the sensitivity of the AASHTO models to concrete strength) decrease is observed as the V/S ratio increases.

36 Constant Values: V/S = 3.5 in H = 70% ACI 209 (1992) AASHTO 2004 AASHTO 2005 Ultimate Creep Coefficient Concrete Compressive Strength, f' c (ksi) Figure 2-9. Comparison of creep models with respect to the concrete strength parameter Constant Values: f' c = 8 ksi H = 70% ACI 209 (1992) AASHTO 2004 AASHTO 2005 Ultimate Creep Coefficient Ratio Volume:Surface Area (in) Figure Comparison of creep models with respect to the V/S ratio parameter

37 25 The three creep models demonstrate (Figure 2-11) sensitivity to relative humidity similar to that seen for the V/S parameter. All three models show a modest decline in estimated creep coefficient as relative humidity increases Constant Values: V/S = 3.5 in f' c = 8 ksi ACI 209 (1992) AASHTO 2004 AASHTO 2005 Ultimate Creep Coefficient Relative Humidity, % Figure Comparison of creep models with respect to the relative humidity parameter Discussion As with the shrinkage model, the AASHTO 2005 creep model was developed as part of the research in NCHRP Report 496 (Tadros et. al., 2003). It has been shown to predict smaller creep strains than the previous model, AASHTO 2004, meaning that smaller prestress losses will be predicted when using this model. A change in the prestress loss estimate affects the flexural analysis of prestressed girders.

38 26 Development of the creep model was done through the same test program that produced the AASHTO 2005 shrinkage model (refer to Sections and 2.1.5). The creep and shrinkage strains were monitored on different specimens, but the specimens were of the same concrete mixture. The shrinkage specimens, which were not loaded, were monitored for shrinkage strain over time. A set of sister specimens was maintained in the same environmental conditions, loaded, and the load was maintained. Those specimens were monitored for elastic strain when the load was applied and monitored for total strain over time. The creep strain is found by subtracting elastic strain and shrinkage strain (measured on the corresponding shrinkage specimen) from the total strain at each time increment. As such, measurements of creep strain rely on accurate elastic and shrinkage strain data. The data generated by the NCHRP Report 496 study, using concrete from four different states in the f c = ksi range, are shown in Figure ACI 209 refers to the ACI 209 (1992) creep model, AASHTO to the AASHTO 2004 model, and Proposed to the AASHTO 2005 model. The results are further summarized, considering only the final creep strain, in Table 2-2. The inconsistencies in the shrinkage data, detailed in Section 2.1.5, also contribute to inaccuracies in the creep data because the creep strain is determined by subtracting shrinkage strain from the total strain. Those inconsistencies introduce uncertainty in the AASHTO 2005 creep model. The experimental data could be supplemented to better substantiate a new model by including tests when the load is applied at various concrete ages. In the experimentation of NCHRP Report 496, all test specimens were loaded at an age of one day. However, the model proposed by the report includes an adjustment term for the age of concrete when the stress change is applied -. in (2-26). It differs from the adjustment term for age of concrete in AASHTO in (2-23) without experimental justification.

39 27 Figure Experimental results from creep tests in NCHRP Report 496 (Source: Tadros, 2003) The adjustment factors for concrete strength, specimen size, and time development are the same as those used in the AASHTO 2005 shrinkage and reproduced in (2-12), (2-10), and (2-13), respectively. The relative humidity correction factor, slightly different than that used in the shrinkage model, is shown in (2-11).

40 28 Table 2-2. Summary of experimental results for creep (Source: Tadros, 2003) 2.3. Modulus of Elasticity of Concrete The stress-strain response of concrete is non-linear because of internal micro-cracking and stress redistribution. However, for small stresses less than approximately half the ultimate strength of concrete the behavior of concrete is nearly elastic and an elastic modulus can be approximated (Wight and Macgregor, 2009). The modulus of elasticity is needed for flexural analysis of prestressed girders so that stress can be calculated from elastic strains. The elastic modulus of concrete is dependent on the stiffness of both the paste and the aggregates (Tadros et. al., 2003) and has historically been estimated as a function of concrete compressive strength and unit weight AASHTO (2004) The AASHTO LRFD Bridge Design Specifications (2004) estimates the elastic modulus of concrete by (2-28) (2-28)

41 29 Where: Specified compressive strength of concrete, ksi AASHTO (2005) The recommendations adopted in the specifications from NCHRP Report 496 (Tadros et. al., 2003) introduced an additional factor, K 1, to account for specific aggregate sources (2-29) Where: Correction factor for source of aggregate to be taken as 1.0 unless determined by physical test, and as approved by the authority of jurisdiction Discussion Use of the K 1 factor in AASHTO 2005 to adjust for aggregate source follows the recommendations of Myers and Carrasquillo (1999) who concluded that elastic modulus is a function of the course aggregate content and type. However, use of the factor is possible only if a K 1 value calibrated for the given aggregate source is available. The NCHRP Report 496 study calibrated factors for the four states in the study Nebraska, New Hampshire, Texas, and Washington but other states will be responsible for developing factors appropriate to their aggregate sources. When K 1 is taken to be one, the AASHTO 2005 and AASHTO 2004 equations are identical.

42 30 Not all of the NCHRP Report 496 recommendations were adopted into AASHTO 2005 for estimating elastic modulus. The NCHRP Report 496 model included an additional factor, K 2, to yield an upper- or lower-bound estimate of elastic modulus, as desired. Also, an equation to estimate the unit weight, as a function of f c, was proposed. Figure 2-13 is reproduced from NCHRP Report 496 to show the uncertainty involved in estimating elastic modulus. The data were combined in NCHRP Report 496 from multiple sources. Proposed refers to the method proposed in NCHRP Report 496 and partially adopted into AASHTO AASHTO-LRFD is the AASHTO 2004 method, which is identical to the AASHTO 2005 model when no information is available about the aggregate source (K 1 = 1.0). ACI 363 refers to the model proposed by ACI Committee 363 (1992). Figure Summary of test data used to develop predictive models for concrete elastic modulus (Source: Tadros et. al., 2003)

43 Relaxation of Prestressing Steel Relaxation is a loss of stress in the prestressing steel when held at a constant strain. The strands typically used in practice today are called low-relaxation strands. They undergo a strain tempering stage in production that heats them to about 660 o F and then cools them while under tension (Barker and Puckett, 2007). This process reduces relaxation losses to approximately 25% of that for stress-relieved strand. The models used by both AASHTO 2004 and AASHTO 2005 rely on the work of Magura (1964) Estimating Intrinsic Relaxation In the case of a pretensioned concrete girder, the prestressing strand is not held at constant strain because the actions of elastic shortening, shrinkage and creep of the concrete reduce the tension strain in the steel. The intrinsic relaxation of the steel assuming the strain is held constant must be considered in developing a procedure to estimate prestress loss. Magura (1964) developed the formula reproduced in (2-30), which estimates relaxation as a function of stress in the strand and the length of time the stress is maintained log 24 1 (2-30) 24 1 Where: Intrinsic relaxation loss between t 1 and t 2 (days) Stress in prestressing strands at the beginning of the period considered Yield strength of strands Age of concrete at the end of the period (days)

44 32 Age of concrete at the beginning of the period (days) 2.5. Modulus of Elasticity of Prestressing Steel The elastic response of prestressing is less uncertain than that of concrete. Both AASHTO 2004 and AASHTO 2005 recommend use of ksi for the prestressing steel elastic modulus Summary Material properties for low-relaxation prestressing steel are well-defined and their treatment in design specifications has not changed in recent years. Concrete materials properties, however, are highly variable. Recent changes to the AASHTO LRFD Bridge Design Specifications have brought about new models for the time-dependent behavior of concrete. The new models, which followed the recommendations of NCHRP Report 496, are specifically aimed at defining the behavior of high strength concrete. The material property models are fundamental to any method used for estimating time-dependent behavior and prestress loss.

45 Chapter 3 Approximate Time-Dependent Analysis The methods used by engineers in the design of prestressed concrete bridge girders to predict time-dependent effects are often based on a set of simplifications that are intended to approximate reality. Time-dependent analysis is complicated because concrete shrinkage and creep, along with steel relaxation, lead to partial loss of the initial prestressing force. As the load history of the girder is considered, there are numerous stress reversals that further complicate the analysis, especially for concrete creep. A detailed time-step analysis, discussed in Chapter 4, is often too complex for use in design. Therefore, simplified methods have been developed to estimate prestress loss. The estimate of losses is then used in predicting extreme fiber concrete stresses. This chapter summarizes the AASHTO 2004 and AASHTO 2005 (detailed and approximate) models, as well as the method of the Canadian Highway Bridge Design Code, S6-06 (CSA, 2006). These models represent common practice for bridge design in North America AASHTO 2004 The AASHTO 2004 model divides the time-dependent components leading to prestress losses into three categories: 1) Shrinkage of concrete, 2) Creep of concrete, and 3) Relaxation of steel. Barker and Puckett (1997) provide a thorough development of these provisions. A summary is provided in this section.

46 Loss due to Shrinkage Hooke s Law requires that the loss of prestress be equal to the product of the elastic modulus of prestressing steel and the change in strain at the level of the prestressing centroid. This development assumes perfect bond between the steel and concrete. Δ (3-1) Where: Δ Loss of prestress due to concrete shrinkage Elastic modulus of prestressing steel Shrinkage strain of concrete at the level of prestressing steel The AASHTO 2004 model estimates shrinkage strain by equation (2-6). The correction factor for specimen size, k s, can be taken approximately equal to 0.7 if assumptions are made for time (500 days, since most shrinkage has occurred by then) and V/S ratio (3.75, which is common for bridge girders). The humidity correction factor, k h, is reproduced in (2-8). Taking the humidity adjustment, k h, approximately equal to , a constant value of 0.7 for k s, and 28,500,000 psi for E p in (3-1) yields an expression for prestress loss due to shrinkage, shown in (3-2). Rounding leads to the equation in the Specifications (AASHTO, 2004). Δ (3-2)

47 Loss due to Creep As with shrinkage, Hooke s Law can be used to derive an expression for creep losses. Since creep is a stress-related phenomenon, concrete stress at the centroid of prestressing must be known in order to calculate creep strain. Stress changes in concrete are split into two categories for the AASHTO 2004 method: 1) Stresses introduced at prestress transfer,, and 2) Stresses introduced at deck placement or later Δ. The total concrete stress at the centroid of prestressing is the sum of those two terms, recognizing that they will have opposite directions. Δ (3-3) Where: Δ Concrete stress at center of gravity of prestressing at transfer Change in concrete stress at the centroid of prestressing due to permanent loads applied after transfer As demonstrated by (2-16), a time-dependent effective modulus for concrete can be defined as a function of the creep coefficient:,, (3-4) It follows from (3-4) that a time-dependent expression for the modular ratio between prestressing steel and concrete can be expressed:,,, (3-5) Multiplying together the modular ratio and the concrete stress at the prestressing centroid estimates the loss of prestress. A different modular ratio will apply to the two terms because the

48 stresses are applied at different times. In this approach, full creep recovery is assumed when the direction of stress reverses. 36 Δ,,,,,, Δ (3-6) Where:,,,, Creep modular ratio at transfer Age of concrete at transfer Creep modular ratio for permanent loads Age of concrete when permanent loads are applied The creep coefficient is different in the two modular ratio terms because the stress is induced at different times. AASHTO 2004 uses (2-23) to calculate the creep coefficient. The series of assumptions shown in Table 3-1 leads to reproduction of the code provision. Table 3-1. Assumptions in the AASHTO LRFD (2004) creep loss prediction T Maturity of concrete, days 365 H Relative humidity, % 70 V/S Ratio volume:surface area, in 3.75 E p Modulus of Elasticity, prestressing steel, ksi t i Concrete age at transfer, days 5 f ci Concrete strength at transfer, ksi 3.5 E ci Concrete modulus of elasticity at transfer, ksi 3400 t d Concrete age when deck is cast, days 30 f c Concrete strength when deck is cast, ksi 5 E c Concrete modulus of elasticity when deck is cast, ksi 4000

49 37 As in the assumptions leading to a shrinkage provision, the specimen size factor (k c ) can take a constant value of 0.7. Substituting the assumptions of Table 3-1 into (2-23) yields (3-7) for creep coefficient after one year when load is applied at the time of transfer , (3-7) Referencing (3-5), (3-7), and Table 3-1, the effective modular ratio at transfer is approximately 12.3., 365,5 365, (3-8) 3400 Similar to (3-7) and (3-8), the creep coefficient and effective modular ratio for stresses applied at an age of 30 days (the assumed time of deck placement) are shown in (3-9) and (3-10), respectively. 365, (3-9), 365,30 365, (3-10) 4000 for creep losses: Substituting (3-8) and (3-10) into (3-6) and rounding yields the AASHTO 2004 provision Δ Δ 12 7Δ (3-11)

50 Loss due to Steel Relaxation In AASHTO 2004, two components of relaxation are considered that occurring before transfer, and that after transfer. The relaxation losses at transfer are calculated as the intrinsic relaxation of the prestressing steel using a form of (2-30). The estimate of relaxation losses after transfer considers the interaction of prestress losses to reduce the stress in the strands and reduce the total relaxation loss. Elastic shortening and friction have a larger effect on relaxation because they occur early in the life of the girder. Since shrinkage and creep occur over time their effect is smaller. Relaxation loss after transfer for stress-relieved strands can be estimated by (3-12) Δ 0.3Δ 0.2Δ Δ (3-12) Where: Δ Δ Δ Δ Δ Loss of prestress due to relaxation after transfer Loss of prestress due to elastic shortening Loss of prestress due to friction Loss of prestress due to shrinkage Loss of prestress due to creep 30% of (3-12). In the case of low-relaxation strands, the prestress loss due to relaxation can be taken as

51 S6-06 Canadian Highway Bridge Design Code The S6-06 Canadian Highway Bridge Design Code (CSA, 2006) estimates prestress loss in a format similar to that of AASHTO Like AASHTO 2004, S6-06 separates timedependent losses into the categories of shrinkage, creep, and relaxation Loss due to Shrinkage The S6-06 estimate of shrinkage losses is identical to that of AASHTO The equation is shown in (3-2) Loss due to Creep The long-term estimate of creep loss in S6-06 is based largely on the work of Zia et. al. (1979), which proposed (3-13) Δ (3-13) Where: = 2.0 for pretensioned girder; = 1.6 for post-tensioned girder Modulus of elasticity of prestressing strands Modulus of elasticity of concrete at 28 days Net compressive stress in concrete at center of gravity of tendons immediately after the prestress has been applied to the concrete Stress in concrete at center of gravity of tendons due to all superimposed permanent dead loads that are applied to the member after it has been prestressed

52 40 S6-06 revises this formula only to include an adjustment factor for relative humidity, based on recommendations of the PCI Committee on Prestress Losses (1975). The adjustment factor, shown in (3-14), can be applied to (3-13) (3-14) Loss due to Steel Relaxation Like AASHTO 2004, S6-06 separates relaxation losses into components before and after transfer. Prior to transfer, the methods for estimating relaxation are identical to AASHTO 2004, again based on (2-30). After transfer, S6-06 considers the effect of inelastic strains in the concrete. Based on the work of Grouni (1973 and 1978), S6-06 uses (3-15) to estimate relaxation losses after transfer for low-relaxation strands, in megapascals (3-15) Where: Loss of prestress due to relaxation after transfer Stress in the prestressing steel at transfer Specified tensile strength of prestressing steel Loss of prestress due to creep Loss of prestress due to shrinkage

53 AASHTO 2005 The time-dependent analysis (prestress loss) method of AASHTO 2005 was adopted into the specification following recommendation in NCHRP Report 496 (Tadros et. al., 2003). Although the impetus of that research program was to extend applicability of the prestress loss provisions to high strength concrete, the time-dependent analysis method is independent of any material property assumptions. The AASHTO 2005 material property model is intended for use with the time-dependent analysis method for high strength concrete applications, although it could be equally implemented with any material model. The AASHTO 2005 prestress loss method is more refined that its predecessor (AASHTO, 2004) in four ways. 1) Rather than lumping all time-dependent effects into a single time increment, the AASHTO 2005 method divides time-dependent behavior into two periods before deck placement and after deck placement. 2) AASHTO 2005 explicitly represents the effect of internal restraint against creep and shrinkage of concrete by the bonded prestressing steel. A transformed section coefficient is used to model the behavior. 3) The creep response of concrete to the gradual stress changes that occur as prestress forces decrease over time is modeled using the age-adjusted effective modulus of concrete. This concept is introduced in Section ) Differential shrinkage between the precast girder and cast-in-place deck results in a theoretical prestressing gain. The AASHTO 2005 method marks the first time this behavior has been included in the specification.

54 42 Points 1) and 2) affect the AASHTO 2005 model as a whole, and are discussed before detailing each of the components considered. Application of the AASHTO 2005 method for design is presented by Al-Omaishi, et. al. (2009) Stages for Analysis The AASHTO 2005 model divides the long-term analysis of a composite girder into two phases. The model first considers the non-composite stage of behavior, prior to deck placement, and the composite phase is considered separately. Figure 3-1, from NCHRP Report 496, summarizes the sequence of steps that contribute to changes in the prestressing force over time. 1. {A-C} Loss due to prestressing bed anchorage seating, relaxation between initial tensioning and transfer, and temperature change from that of the bare strand to temperature of the strand embedded in concrete. 2. {C-D} Instantaneous prestress loss at transfer due to prestressing force and self-weight. 3. {D-E} Prestress loss between transfer and deck placement due to shrinkage and creep of girder concrete and relaxation of prestressing strands. 4. {E-F, G-H} Instantaneous prestress gain due to deck weight on the noncomposite section and superimposed dead loads on the composite section. 5. {H-K} Long-term prestress losses after deck placement due to shrinkage and creep of girder concrete, relaxation of prestressing strands, and deck shrinkage.

55 43 Figure 3-1. Timeline representing the change in prestressing force over time in a typical prestressed member (Source: Tadros, 2003) Total time-dependent losses are found by summing components, as shown in (3-16). The elastic gains due to load application are not considered. (3-16) Where: Loss due to shrinkage of girder concrete between transfer and deck placement Loss due to creep of girder concrete between transfer and deck placement Loss due to relaxation of prestressing strands between time of transfer and deck placement Loss due to relaxation of prestressing strands in composite section between time of deck placement and final time

56 44 Loss due to shrinkage of girder concrete between time of deck placement and final time Loss due to creep of girder concrete between time of deck placement and final time Prestress gain due to shrinkage of deck in composite section Sum of time-dependent prestress losses between transfer and deck placement Sum of time-dependent prestress losses after deck placement Transformed Section Coefficient AASHTO 2005 uses a transformed section coefficient to model the internal restraint that bonded prestressing imparts on the surrounding concrete against shrinkage and creep. The coefficient itself is a value less than 1.0 that represents the ratio of actual change in strain, considering the restraint provided by the prestressing steel, to the change in strain that would occur with no restraint. It is denoted by K id for the non-composite stage of behavior and K df after casting of a composite deck. The formulation of the transformed section coefficient is similar for both shrinkage and creep, before and after deck placement. For demonstration here, the term will be derived with respect to shrinkage prior to deck placement. The derivation refers to Figure 3-2. The shrinkage strain distribution across the girder section is affected by the presence of bonded prestressing steel. is the free shrinkage strain of concrete that would exist without any internal restraint. denotes the reduction in shrinkage strain, at the centroid of the prestressing, caused by the steel s

57 45 restraint. The transformed section coefficient is the ratio of the net strain to the free strain. (3-17) Figure 3-2. Schematic diagram demonstrating the effect of steel restraint on concrete shrinkage In developing an equation for K id, it will be assumed that the rate of shrinkage is uniform over the entire cross section. The concrete will undergo a free shrinkage,. Compatability requires that the same strain exist in the steel. Therefore, shrinkage of the concrete exerts a compressive force, P, on the steel equal to: (3-18) Where: Effective compression force applied to the prestressing steel by the shrinkage strain of concrete Total area of prestressing steel

58 46 Modulus of elasticity of prestressing steel Unrestrained shrinkage strain of concrete Considering equilibrium, a tension force must be applied to the cross section by the prestressing steel. The force can be represented as the sum of two components the portion applied to the gross concrete section and the portion applied to the prestressing steel. The component of that force applied to the gross concrete section can be determined by recognizing that resulting stresses must satisfy the relationship in (3-19). (3-19) Where: The portion of the restraint force effectively applied to the concrete component of the cross section Gross area of concrete Moment of inertia, based on the gross concrete section Eccentricity of the prestressing steel centroid in the section considered, usually midspan Portion of the total shrinkage strain restrained by the bonded prestressing steel, at the centroid of the prestressing Elastic modulus of concrete at the time of prestress transfer Solving (3-19) for the component of the force on the concrete: 1 (3-20) Where:

59 1 (3-21) 47 The second component of the restraint force, applied to the prestressing steel, is shown in (3-22). (3-22) Where: The portion of the restraint force effectively applied to the prestressing steel component of the cross section Summing the concrete and steel components and setting them equal to the compression force exerted by shrinkage on the steel (force equilibrium) yields (3-23). (3-23) Shrinkage is not instantaneous, but occurs gradually with time. Therefore, the stresses due to restrained shrinkage are partially relieved by concrete creep. To represent the fact that the force, P, builds gradually over time, the age-adjusted effective modulus,, will replace the concrete elastic modulus,. 1, (3-24) Where: Relaxation coefficient (Trost, 1967) which accounts for the reduction in creep that occurs because not all of the stress is applied at the initial time, t i (Collins, 1991). Values typically range between 0.6 and 0.9.

60 48, Creep coefficient at time, t, due to stresses induced at time, t i Substituting the age-adjusted effective modulus into (3-23): 1, (3-25) Solving for the strain restrained by the bonded prestressing steel, : 1 1, 1 1 1, (3-26) Where: (3-27) (3-28) Substituting (3-26) into (3-17) and simplifying leads to the form of the equation incorporated into the AASHTO 2005 model , (3-29) The AASHTO 2005 model adopts a constant value of 0.7 for the relaxation coefficient,, as recommended by Dilger (1982).

61 Analysis Before Deck Placement Time-dependent analysis of the non-composite phase is separated into three components leading to prestress loss shrinkage, creep, and relaxation Loss Due to Girder Shrinkage Prestress loss due to shrinkage is determined by Hooke s Law using the net shrinkage strain at the prestressing centroid as described in Section and depicted in Figure 3-2. The format used by AASHTO 2005 is given in (3-30). Δ (3-30) Where: Concrete shrinkage strain of girder between the time of transfer and deck placement [Eq ] Transformed section coefficient that accounts for time-dependent interaction between concrete and bonded steel in the section being considered for time period between transfer and deck placement Modulus of elasticity of prestressing steel (ksi) Loss Due to Girder Creep Again from Hooke s Law, the equation for losses due to girder creep is very similar to that for shrinkage loss. Δ (3-31)

62 50 Where: Unrestrained creep strain of girder concrete Recalling (2-14), creep strain is determined by the product of the creep coefficient and the elastic stress in the concrete. The stresses prior to deck placement are caused primarily by the initial prestress and the self-weight of the girder. Calculating the elastic stress at the centroid of the prestressing and the creep coefficient for the time of deck placement allows for a prediction of creep strain, shown in (3-32)., (3-32) 31). The creep loss equation of AASHTO 2005 is reproduced by substituting (3-32) into (3-, (3-33) Where: E p E ci f cgp t d t i K id Modulus of elasticity of prestressing steel (ksi) Modulus of elasticity of concrete at transfer (ksi) Sum of concrete stresses at the center of gravity of prestressing tendons due to the prestressing force at transfer and the self-weight of the member at the sections of maximum moment (ksi) Age at deck placement (days) Age at transfer (days) Transformed section coefficient that accounts for time-dependent interaction between concrete and bonded steel in the section being considered for time period between transfer and deck placement

63 51 ψ b (t d,t i ) Girder creep coefficient at time of deck placement due to loading introduced at transfer Loss Due to Steel Relaxation Losses due to strand relaxation from transfer to deck placement can be given as: Δ (3-34) If the ratio log 24 1 (3-35) 24 1 If 0.55 relaxation losses are assumed to be zero The reduction factor,, which accounts for the steady decrease in strand tension due to creep and shrinkage losses, is given by Tadros (1977): 1 3Δ Δ (3-36) Where: Stress in prestressing strands just after transfer Specified yield strength of strands Age at deck placement (days) Age at transfer (days) Transformed section coefficient that accounts for time-dependent interaction between concrete and bonded steel in the section being

64 52 considered for time period between transfer and deck placement = 45 for low-relaxation steel; = 10 for stress-relieved steel AASHTO 2005 allows designers to assume a total relaxation loss of 2.4 ksi, as there tends to be small variability in this term. It is recommended that half of the total loss be assigned to the time period before deck placement, and half afterwards Analysis After Deck Placement AASHTO 2005 divides the time-dependent change in prestress into four components for the composite phase after deck placement girder shrinkage, creep, relaxation, and differential shrinkage between the deck and the girder Loss Due to Girder Shrinkage Prestress loss due to girder shrinkage after deck placement is determined similar to shrinkage losses for the non-composite girder case. From Hooke s Law: (3-37) Where: Concrete shrinkage strain of girder between the time of deck placement and final time Transformed section coefficient that accounts for time-dependent interaction between concrete and bonded steel in the section being considered for time period between deck placement and final time Modulus of elasticity of prestressing steel (ksi)

65 K df is derived in the same manner as K id (refer to Section 3.3.2), except that it is relative to the full composite section , (3-38) Loss Due to Girder Creep The AASHTO 2005 equation for creep loss after deck placement is presented in (3-39).,, Δ, 0.0 (3-39) Where: Modulus of elasticity of prestressing steel Modulus of elasticity of girder concrete at transfer Modulus of elasticity of girder concrete Sum of concrete stresses at the center of gravity of prestressing tendons due to the prestressing force at transfer and the self-weight of the member at the sections of maximum moment Transformed section coefficient that accounts for time-dependent interaction between concrete and bonded steel in the section being considered for time period between deck placement and final time Δ Change in concrete stress at centroid of prestressing strands due to longterm losses between transfer and deck placement, combined with deck weight and superimposed loads

66 54,, Girder creep coefficient at final time due to loading introduced at transfer Girder creep coefficient at time of deck placement due to loading introduced at transfer, Girder creep coefficient at final time due to loading at deck placement Equation (3-39) separates the creep strain into two components: 1) Creep caused by the initial prestressing force and the girder self-weight some of which already occurred prior to deck placement, and 2) creep in the opposite direction caused by deck self-weight and superimposed dead loads. The creep coefficient difference term,,,, represents the amount of creep that remains to occur during the time from deck casting to final time, considering the elastic stresses at the centroid of prestressing due to initial conditions,. The second term represents a creep gain (assuming Δ is negative, as typical) due to the tension induced (decrease in compression) at the centroid of the prestressing strands. The tension stress increment results from prestress losses during the phase prior to deck placement and flexural stresses caused by additional permanent loads, including deck self-weight. A different creep coefficient,,, is used because the stress change occurs at the time of deck placement, t d, rather than initial time, t i. This approach, by superimposing creep strains due to both tension and compression stress increments, inherently assumes full creep recovery Loss Due to Steel Relaxation As indicated in Section , AASHTO 2005 permits an assumption of 2.4 ksi for total losses due to relaxation, with half of that amount (1.2 ksi) attributed to the time period after deck placement.

67 Gain Due to Deck Shrinkage In typical composite construction, which bonds a precast girder with a cast-in-place deck, internal stresses develop because of the differing rates of shrinkage between the two components. Since the girder is precast, and most shrinkage strain occurs during the early ages of the concrete (Section 2.1), much of the shrinkage strain occurs prior to deck casting. Therefore, only the small portion of remaining shrinkage strain occurs during the composite phase of behavior. The castin-place deck, however, experiences all of its shrinkage during the composite phase. This differential in the composite section the deck shrinks more than the girder induces an effective compression force on the composite section at the level of the deck centroid. A tension strain at the opposite face of the girder (the bottom) follows. The elongation leads to an increase in the prestress force. AASHTO 2005 estimates the prestress gain by (3-40). Δ Δ 1 0.7, (3-40) Where: Δ Modulus of elasticity of prestressing steel Modulus of elasticity of concrete Change in concrete stress at centroid of prestressing strands due to shrinkage of deck concrete Transformed section coefficient that accounts for time-dependent interaction between concrete and bonded steel in the section being considered for time period between deck placement and final time, Girder creep coefficient at final time due to loading at deck placement The age-adjusted effective modulus is used in (3-40) because the shrinkage differential builds gradually.

68 56 AASHTO 2005 provides an equation for Δ that will be derived for clarity. The derivation will be based on the generic cross section shown in Figure 3-3 where the deck is above the neutral axis of the composite section and the center of gravity of the prestressing force is below the neutral axis. Figure 3-3. Generic composite cross-section to facilitate the derivation of Δf cdf section, P. As the deck shrinks relative to the girder, it applies a compressive force on the composite (3-41) Where: Shrinkage strain of the deck concrete Effective area of the deck that behaves with the girder in composite action Modulus of elasticity of deck concrete Effective compression force on the composite section at the centroid of the deck due to differential shrinkage, as defined by AASHTO 2005

69 57 Since the force builds over time, the age-adjusted effective modulus,, will be substituted for. The change in stress due to this effective force at the level of the prestressing is a combination of axial and flexural effects. Taking strand shortening as positive since that reflects a prestress loss, the change in stress is: Δ (3-42) Where: Gross area of the composite section Moment of inertia of the gross concrete section Eccentricity of the deck, relative to the composite section Eccentricity of the prestressing centroid, relative to the composite section Substituting the expression for from (3-41) into (3-42), recalling that, and combining terms yields the AASHTO 2005 equation for Δ. Δ 1, 1 (3-43) The negative sign in this equation assumes a positive value for e d true of conventional cases where the deck is above the neutral axis of the composite section. The Δ term will be negative in most cases, indicating strand elongation a gain in prestressing force.

70 AASHTO 2005 Approximate Method AASHTO 2005 also presents an approximate method for use in preliminary design. It is a lump sum approach based on the detailed method (Section 3.3), but some simplifications and assumptions are made to arrive at an abbreviated equation. First, to summarize the detailed method, total losses are based on (3-44). Δ Δ Δ 1 (3-44) The authors of NCHRP Report 496 arranged the equation such that the first two terms relate to shrinkage of the girder, the last two terms relate to relaxation of the strands, and all the terms in between deal with creep. Differential shrinkage is not considered. The following is a summary of the assumptions made to arrive at the approximate method. A full description is provided by Tadros et. al. (2003). 1) For low-relaxation strands, the total relaxation loss is roughly 2.4 ksi 2) The total shrinkage loss can be estimated as 12 ksi assuming: a. E p = ksi b. Typical girder V/S ratios yield k s = 1.0 c. Prestressing is usually transferred at a concrete age of one day, so the loading age factor can be taken as 1.0 d. Assume K id = K df = 0.8 e. Combining coefficients yields f. The authors of NCHRP 496 used a coefficient of 12 rather than to produce an upper-bound correlation with the test results

71 59 3) The creep losses are simplified to the expression 10 through a series of steps a. The effect of girder stiffening by composite action will be ignored the girder will be assumed non-composite its entire life span b. The small prestress gain due to deck shrinkage will be ignored c. Assume K id = K df = 0.8, such that total creep losses could be given by 0.8 Δ 0.8 d. Assume modular ratios n i = 7 and n = 6 e. For a loading age of one day, load duration of infinity and V/S ratio 3in-4in, the creep coefficient can be expressed as 1.9 f. The creep coefficient for deck loads and superimposed loads is assumed to be 40% of the creep coefficient for initial loads g. The level of prestress in the girder is related to the stress at the level of the prestressing by assuming that the prestress force provided yields zero net stress in the bottom fibers at service load. It is further assumed the stress is a result of three equal components from girder self-weight, deck weight, and live load The approximate method is ultimately given by (3-45). Δ (3-45) Where: (3-46)

72 5 1 (3-47) Discussion Concerns have been expressed (Walton and Bradberry, 2004) about the complex nature of the AASHTO 2005 method, relative to the other methods. Designers have grown accustomed to the AASHTO 2004 method that separates long-term prestress losses into three components and concrete stresses are then determined from fundamental mechanics once an effective prestressing force is known. The increased complexity of the calculations in AASHTO 2005 suggests greater precision. Prestress losses are highly variable and dependent on many factors. Therefore, it may be unreasonable to expect a great deal of precision in a model Stages for Analysis The division of time-dependent behavior into two phases complicates the AASHTO 2005 model, relative to the others. It effectively doubles the computational effort, and it requires the designer to estimate the value of more variables. In particular, AASHTO 2005 requires the designer to assign an age for the variable, t d, that represents the age of the girder when the deck is cast. The sequence of construction especially the time of deck placement relative to production of the girder is highly variable and difficult for the engineer to anticipate at the time of design. The time-dependent analysis in Chapters 4 and 5 provide justification for the removal of the t d variable and for combining the two phases for design calculations.

73 Transformed Section Coefficient The transformed section coefficient, K id, for use with shrinkage and creep prior to deck placement was derived in Section and the format shown in the Specifications (AASHTO, 2005) is reproduced in (3-48) , (3-48) Two terms in (3-48) are inconsistent with its fundamental derivation. First, the K id transformed section coefficient is intended to represent the behavior of the girder concrete, when partially restrained against shrinkage and creep by bonded prestressed steel, prior to the time of deck placement. Therefore, the age-adjusted effective modulus should be determined using the creep coefficient at the time of deck placement,,, rather than that for final time,,. Secondly, the internal redistribution of stresses that occurs when the prestressed steel resists shrinkage and creep strains is partially dependent on the modular ratio between steel and concrete. Over time stresses will distribute with respect to the modular ratio of steel and final time concrete. Therefore, the modular ratio should be replaced by. The formulation for K df, the transformed section coefficient for composite section, has similar inconsistencies. The AASHTO 2005 format of the equation is given in (3-38). This coefficient is intended for use in the time period after deck placement. The inelastic strains occur during the composite phase of behavior. Therefore, the age-adjusted effective modulus used in development of K df should introduce the creep coefficient at final time for stresses induced at the time of deck placement,,, rather than,. Also, as presented in the previous paragraph, the modular ratio should be with respect to the final time concrete elastic modulus. Additionally, derivation of K df, similar to K id, assumes that shrinkage strain is constant over the

74 62 cross section. This assumption is not valid during the composite phase because differential shrinkage between the girder and deck is typical. Separate consideration of deck shrinkage partially compensates for this inconsistency. Furthermore, with respect to both K id and K df, the age-adjusted effective modulus used in the derivation represents behavior attributed to creep. Therefore, use of these coefficients to represent internal stress redistribution due to shrinkage is not entirely accurate because it partially combines actions due to creep with the shrinkage component. For the case of shrinkage, the internal stress redistribution that occurs because of the restraint of the bonded prestressing steel would be exactly the same regardless of whether shrinkage occurs instantaneously or over time, in the absence of creep. It would be better to use a transformed section coefficient very similar to K id and K df that does not include the age-adjusted effective modulus if trying to explicitly separate shrinkage and creep components Differential Shrinkage The AASHTO 2005 model introduces an estimate of prestress gain due to shrinkage differential between the girder and the deck. It was noted in Section that the effective force in the deck acting in compression on the composite section will produce elongation in the prestressing strands and an elastic gain in force. The language of the Specification (AASHTO, 2005) can create confusion, however, because differential shrinkage also induces tension stress on the bottom of the girder. If the prestress gain due to differential shrinkage is superimposed with prestress losses due to the other components, and the resulting effective prestressing force is used to calculate extreme fiber concrete stresses, an error results. The elastic gain in prestressing due to differential shrinkage does not act to further pre-compress the bottom face of the girder. This effect is similar to the elastic gain observed (refer to Figure 3-1) when

75 63 load is applied to the girder. These gains, although they are real, are not considered when calculating extreme fiber stresses. A tension increment on the bottom face of the girder accompanies the elongation of the prestressing strands in responding to applied load. Furthermore, approximating the effective force that differential shrinkage applies to the composite section should be done considering the difference in shrinkage strains between the girder and the deck after deck placement. The formulation in (3-41) suggests that is a function of total deck strain. It should, instead, be based on the difference,, where represents the shrinkage strain in the girder after deck placement. Finally, the transformed section coefficient, K df, should not be applied in considering differential shrinkage as shown in (3-40). The transformed section coefficient applies when creep or shrinkage of the concrete is partially restrained by bonded prestressing steel. In the case of differential shrinkage, however, an effective force is applied to the entire cross section. The bonded prestressing responds elastically, but does not cause an internal redistribution of stresses Transformed Section Properties As documented by Ahlborn et. al. (2000), there are various recommendations for the use of transformed section properties in calculating concrete stresses. Generally speaking, although the use of transformed section properties is more exact, gross or net section properties can be used in practice with little error (Lin and Burns, 1981). Stresses in the concrete section can then be calculated through a combined stress calculation of the general form (compression indicated negative): (3-49)

76 64 Where: Prestressing force at the stage of interest Eccentricity of the centroid of prestressing with respect to the girder centroid Location of concrete layer for which stress is being calculated, relative to the girder centroid Gross moment of inertia Gross cross-sectional area Applied moment due to external loading Transformed section properties can be used for a more accurate calculation of concrete stresses (Hennessey and Tadros, 2002), although no formal recommendation was adopted into the Specifications (AASHTO, 2005). In calculating transformed section properties, the steel area is multiplied by a factor 1 in which n is the modular ratio,. The (-1) term reflects the fact that steel is replacing an equivalent area of concrete. The transformed section can be represented schematically in Figure 3-4. Concrete is shown in light gray, while steel that has been transformed to an equivalent area of concrete is shown darker. Figure 3-4. Transformed cross section, shown schematically

77 65 Since the steel generally falls closer to the face of the beam controlled by a tension stress limit, using transformed section analysis reduces the extreme fiber tensile stress (because the theoretical neutral axis location shifts closer to the tension face) and reduces the prestressing demand (Hennessey and Tadros, 2002). By this reasoning, it is generally conservative to use gross section properties for stress analysis of pretensioned concrete members. Hennessey and Tadros (2002) state that: Prestress loss estimates by AASHTO (2004) formulas were based on the assumption that gross section properties are used in the concrete stress analysis. Unless these formulas are modified, transformed section analysis may be incorrect and misleading. If the proper loss components are accounted for, the difference in results between the approximate gross section analysis and the more accurate transformed section analysis is not expected to be large. In other words, prestress methods of the past had been calibrated to consider the fact that engineers would be using gross section properties in design because of the lack of computing power needed to make transformed section analysis efficient. When using a transformed section analysis, elastic effects such as elastic shortening due to transfer or elastic gains when external loads are applied will be automatically accounted for in the calculation of extreme fiber stress but must be explicitly calculated if the effective prestress force is needed. Example problems illustrating this concept are provided by Hennessey and Tadros (2002) and Walton and Bradberry (2004).

78 Chapter 4 Analysis Methods Two analysis methods are developed in this chapter for use later in this study. First, a time-step method is developed to facilitate a detailed time-dependent analysis of pretensioned concrete girders. The detailed analysis will serve as a baseline for comparing other methods and for developing a simplified approach. Second, the Monte Carlo simulation techniques used for the uncertainty analysis in this thesis are developed and documented Detailed Time-Step Method In determining the accuracy and variability of prestress loss prediction methods, an exact solution is needed as a baseline. Many experimental studies have been done in this area, but reliable test data are difficult to obtain because measuring prestress losses is challenging, as evidenced by the highly variable test data summarized in the literature (Tadros et. al., 2003). Even if the prestress losses can be measured correctly, the various components cannot be separated with any certainty due to the combination of elastic and inelastic strains. Therefore, a detailed time-step analysis was developed to discern the sensitivity of key variables and to validate a simplified procedure. The girder is discretized into horizontal layers representing the concrete component. One layer, at the level of the prestressing centroid, is dedicated to representing the presence of bonded steel in the cross section. At each step in the time history of the girder a strain distribution that satisfies compatability and equilibrium is calculated considering inelastic effects (i.e. creep, shrinkage, and relaxation) and the elastic response to applied loads. With the strain distribution at

79 67 each step known, the change in strain at the level of prestressing can be found, leading to an estimate of prestress loss Assumptions A time step algorithm is developed so that a theoretically precise baseline solution, for a given set of assumptions, can be obtained. The time step routine allows tracking not only of prestress losses, but also of bottom fiber concrete stresses which are often the designer s end goal in flexural design. The algorithm is based on the following assumptions: Creep effects are additive for both increasing and decreasing stress increments (creep superposition) A creep recovery factor that scales the creep function in the case of decreasing compression stress increments can be included to allow for less than full creep recovery Stresses are constant for an entire time step Strain compatability requires perfect bond between the concrete and prestressing steel (the strain in the steel matches the strain in the concrete at the same level) Plane sections remain plane Shrinkage is uniform through the cross section Material properties, as detailed in Chapter 2, are based on published models

80 Development of the Method Figure 4-1 graphically shows the relationship between increments of stress, creep, and total strain. All effects contributing to strain up to the time of interest are summed to find the total strain. Schematically, all of the stress changes are shown positive, but the stress increments may also reverse. In the case of stress reversal, full creep superposition is assumed unless a creep recovery factor is applied. The total stress-related strain elastic effects and creep is determined by superposition of the strain due to each stress increment in the time history of the concrete. (4-1) Where: Total stress-related strain at time, t Total stress-related strain at time, t, due to the i th stress increment The creep compliance function expresses the total elastic and creep strain as a function of elastic modulus and creep coefficient for a unit stress, as shown in (4-2)., 1 1, (4-2) Where:, τ Creep compliance function total stress-related strain at time, t, due to a stress increment at time,, Creep coefficient at time, t, for a stress increment at time, τ Modulus of elasticity of concrete at time,

81 69 Figure 4-1. Schematic of the creep compliance relationship The total strain at time, t, due to a series of stress increments can be written in terms of the stress of each increment and the corresponding creep compliance function as shown in (4-3), or in simplified form as in (4-4). Δ τ Ct,τ Δσ τ Ct,τ Δσ τ Ct,τ Δσ τ Ct,τ (4-3) Δ, (4-4) Where:, Creep compliance function total stress-related strain at time, t, due to a stress increment at time,

82 70 Δ Increment of stress induced at time, Not all time-dependent strain in concrete is stress-induced. Shrinkage strain must also be considered. Temperature strain will be disregarded for this study because temperature changes will have a similar effect on both concrete and steel, therefore not impacting prestress losses. (4-5) Where: Total strain at time, t Elastic strain at time, t Creep strain at time, t Shrinkage strain at time, t Creep and shrinkage effects can be lumped together and termed inelastic. (4-6) Where: Total inelastic strain at time, t Substituting (4-6) into (4-5) yields (4-7). (4-7) Rearranging (4-7) to solve for the elastic strain yields (4-8). (4-8)

83 shown in (4-9). Stress in the concrete is found as the product of elastic strain and modulus of elasticity, 71 (4-9) The effective prestressing force at any time can be found if the total strain in the prestressing steel is known. The total strain is the difference between the initial jacking strain and the compressive strain in the concrete at the prestressing center of gravity, as shown in (4-10). Refer to Figure 4-2. (4-10) Figure 4-2. Diagram of the generic strain profile to facilitate development of the time step algorithm Variables related to the strain profile in Figure 4-2 are time-dependent and are defined with respect to any given step in the time-stepping routine. Layer of interest for a given step in the routine

84 72 Area of layer k Total area of prestressing steel Total deck thickness Total girder height Vertical location of layer k, relative to the top of the deck Vertical location of the prestressing centroid, relative to the top of the deck Reference strain at time used to define the strain profile Reference curvature at time used to define the strain profile Reference strain at the time of deck placement, including the elastic response of the girder to the deck weight Reference curvature at the time of deck placement, including the elastic response of the girder to the deck weight Total strain in layer k Change in strain in the prestressing steel due to time-dependent effects Equivalent strain used to model the loss of stress due to steel relaxation Initial jacking strain in the prestressing steel Effective jacking strain in the prestressing steel, considering losses due to relaxation which are modeled as a reduction to the initial jacking strain Total effective strain in the prestressing steel The effective jacking strain in the prestressing steel is denoted, where an effective strain representing the relaxation of steel is subtracted from the initial jacking strain. The sign convention for the method is established by Figure 4-2. Tension strain in the prestressing steel is positive, while compression/shortening strain in the concrete is positive. As shortening strain in the concrete increases over time (i.e. creep or shrinkage), the strain in the

85 73 prestressing steel will become a smaller positive (tension) value to indicate loss of prestressing force. Referring to Figure 4-2, the relationships in Table 4-1 can be developed for strain and stress. Table 4-1. Stress and strain relationships for key values in the time step routine Strain Stress Concrete Layer k Mild Steel Prestressing Steel The total stress-related strain in concrete layer k at any time, t, is found by the summation of all stress changes in the time step history and the creep compliance function. 1 Δ, (4-11) (4-11) can be separated into elastic and creep components. The elastic strain is approximately equal to the elastic stress at the end of the previous time step divided by the elastic modulus of concrete. (4-12) If all time steps leading up to time, t i, are known, the total creep strain can be calculated using (4-11) and subtracting the elastic strain calculated at the end of the previous time step. 1 Δ, (4-13)

86 74 Recall the shrinkage strain is assumed constant over the cross section. Shrinkage strain will be calculated with respect to the chosen material property model. Creep and shrinkage strain can be combined as total inelastic strain. Recognizing that the cross-section must be in equilibrium, and that the applied axial force is zero, the total axial force, N, in the section must sum to zero at any time after transfer. 0 (4-14) Practically, the equilibrium expression in (4-14) will be satisfied by considering the stress in each concrete layer k and the effective stress in prestressing steel. The equilibrium expression is expanded in (4-15) using the relationships summarized in Table (4-15) The reference strain and curvature are substituted into (4-15) to reduce the total number of unknowns in the equation. Refer to Table (4-16) Grouping terms in (4-16) with respect to reference strain and reference curvature produces (4-17).

87 75 (4-17) (4-17) takes the general form of (4-18). (4-18) Where: (4-19) (4-20) (4-21) (4-22) N I and N P are effective axial forces representing the internal stresses due to creep and initial strand tension, respectively. For layers representing deck concrete (assuming the deck to be composite) additional considerations are needed. The calculations must reflect the fact that a zero strain case for the deck corresponds to an existing strain and curvature in the girder at the time of deck placement

88 76 (after deck weight has been applied to the girder, assuming unshored construction). Referring to Figure 4-2, and are the reference strain and curvature, respectively, for the girder at the time of deck placement. This line serves as the datum for calculations in deck layers. Therefore the equilibrium equation must be adjusted slightly, as shown in (4-23). (4-23) Where: Strain in layer k at the time of deck placement (after deck loading has been introduced, but before deck stiffness is considered) The form of (4-18) changes with the addition of another term. (4-24) Where: (4-25) Similar steps must be taken to ensure that flexural equilibrium is satisfied. The internal moment in the cross section must equal the external moment due to applied loads. (4-26) yields (4-27). Expanding (4-26) to include forces due to concrete and prestressing steel components

89 77 (4-27) Where: Total moment from external loads; taken negative for moments that induce compression on top of the beam Substituting the reference strain and curvature (see Table 4-1) into (4-27) and combining terms produces (4-28). (4-28) (4-28) takes the general form of (4-29). (4-29) Where: (4-30) (4-31)

90 78 (4-32) (4-33) M I and M P are effective moments due to internal stresses associated with concrete creep and initial tension in the prestressing strands, respectively. Similar to the equations for axial force equilibrium, special considerations are needed for deck layers. (4-34) is derived similar to (4-24) and can be used for deck layers when analyzing flexural equilibrium. (4-34) Where: (4-35) A strain profile that satisfies equilibrium is found by simultaneous solution of (4-18) and (4-29) before deck placement or (4-24) and (4-34) after deck placement. The solution yields the reference strain and curvature for the time step under consideration, from which the strain at any location in the section can be determined. Once the strain is known, the stress at any layer is found by Hooke s Law, shown in (4-36). (4-36)

91 Algorithm The following is an outline of the algorithm used to solve for the strain profile and stresses in each layer in any given time step. Computer code to execute the algorithm repeatedly has been developed in VBA and run through Microsoft Excel for use in this study. 1. Calculate the stress at each level k a. In the typical time step (not step 1) this is the stress found at the end of the previous step. At step 1, all layers begin with zero stress. 2. Calculate the creep strain at each level k a. The total creep strain is based on the stress increment and creep coefficient corresponding to that increment for each step leading up to the current age. This requires an assumption about creep superposition. Either full superposition can be applied, or a creep recovery factor can be defined. 3. Add shrinkage strain to creep strain to find the total inelastic strain for each level k 4. Calculate the constants for the simultaneous equations (4-18) and (4-29) or (4-24) and (4-34). 5. Solve simultaneous equations to yield reference strain and curvature for the current time step 6. Solve for the total strain at each level k, based on the reference strain and curvature 7. Find total strain in the prestressing steel from Equation Find the elastic stress at each level k by taking the difference between total strain and inelastic strain 9. Calculate the stress increment compared with the previous step to be used in future creep calculations 10. Repeat the algorithm for the next time step

92 A detailed example demonstrating the implementation of the time step routine to a simple problem is provided in Appendix B Monte Carlo Simulation Monte Carlo simulation involves repeatedly cycling different values for each uncertain input parameter through a numerical model. The values for the uncertain input parameters are determined from its probability distribution. For models with many input parameters, such as prestress loss methods, one value from each is sampled simultaneously in each repetition of the simulation. (Cullen and Frey, 1999). Monte Carlo simulation can be summarized concisely by the following steps: Identify the base input variables for the numerical model Develop a distribution to represent the uncertainty inherent in each input variable Establish the numerical model that connects the input variables to yield the desired output For each cycle in the simulation, generate a random number (between zero and one) for each independent variable Using the cumulative distribution function (CDF) for each input variable, a value can be assigned to the variable for the current simulation cycle based on the random number generated The model output, based on the randomly selected input variable values, is stored and compiled with results from all other simulation cycles The collection of the model outputs from all cycles can be used to fit a distribution representing the inherent uncertainty in the model The process is shown schematically in Figure 4-3.

93 81 Parameter 1 Parameter 2 Probability Distribution Function (PDF) Develop a distribution to represent uncertainty for each input parameter Express the distribution for each parameter in terms of cumulative probability Cumulative Distribution Function (CDF) Use a random number generator to select a value from the CDF for each parameter in each simulation cycle Input 1 Use the randomly selected Input 2 input values in the numerical model for estimating prestress losses Numerical Model for Estimating Prestress Loss Save the prestress loss estimate for all cycles in the simulation Run 1: PS 1 Run 2: PS 2 Run 3: PS 3... Run i: Ps i Run N: PS N Prestress Loss Estimate: PS i Probability Density Function Estimated Prestress Loss Use the data from all the simulation cycles to develop a distribution representing uncertainty in prestress loss estimates Figure 4-3. Schematic of the Monte Carlo simulation technique used for the uncertainty study of prestress loss methods.

94 82 For the purposes of this study, a Monte Carlo simulation routine was developed in Microsoft Excel and VBA (Visual Basic for Applications the programming language used for Excel macros). Using this technique, most of the calculations are done in the spreadsheet. A macro is needed only to drive the iterations for subsequent cycles. For each cycle, the macro generates a random number, using the Microsoft Excel random number generator, for each variable. Based on that random number, and the cumulative frequency distribution representing the uncertainty of the variable, a random value for the input variable is determined. Once a value has been determined for each variable and checked to be within the limits specified (minimum and maximum values are set by practical criteria), the spreadsheet formulas representing the numerical model for the prestress loss method calculate the output. In this case, the prestress loss and bottom fiber stress are the most important output values. Finally, the macro stores all the input and output information for each cycle in a separate table for data analysis at a later time Summary This chapter summarizes the development of the time step method used for detailed analysis of the girder s time-dependent behavior. This method will be used in subsequent chapters as a baseline for model comparison, and as the foundation for justifying a simplified method. The Monte Carlo simulation technique is used in the uncertainty analysis detailed in Chapter 7.

95 Chapter 5 Detailed Time-Dependent Analysis An approximate approach to time-dependent analysis, three of which are summarized in Chapter 3, is usually preferable for use in design due to the complexity of the problem. In validating the approximate methods, and in developing a new method, results from a detailed time-step method are valuable. The time-step method developed in Chapter 4 will be implemented in this chapter Stages of Behavior This section summarizes the construction sequence of the typical pretensioned girder and indicates the effects this sequence has on the time-dependent behavior of the girder. The major stages of loading are shown in Figure 5-1 and summarized below. A. Prestressing strands are tensioned between fixed restraints and anchored B. Concrete is cast around the tensioned strands. Once set, the concrete is bonded to the prestressed strands. C. The prestressed strands are cut. The initial force in the prestressing strands is now transferred to the concrete through bond stresses, introducing a compression force on the section. The concrete will have an elastic response to this compression load, causing the beam to shorten. When the beam shortens, some of the initial strain in the prestressing tendons is lost. With this decreasing strain, the internal force in the tendons also decreases. In the typical case where the net prestressing effect is eccentric in the girder cross section an upward deflection (camber) will result. In this

96 84 condition, the beam is supporting its own selfweight because the ends of the beam are sitting on the casting bed but the midspan has deflected upwards. D. In most cases the girder will be stored for weeks or months before being installed at a bridge site. During this time the beam is resisting a large force from the prestressing and has only its own selfweight as gravity load. The concrete is undergoing volume change due to two phenomena creep and shrinkage. Shrinkage is considered to be uniform through the cross section, causing the entire beam to shorten. The volume change due to creep is stress-dependent. Therefore the bottom of the beam (assuming eccentric prestressing and the beam in positive camber) will tend to shorten more than the top, causing an increase in camber. The combination of the creep and shrinkage reduces the strain in the prestressing strands and leads to further decrease in strand force. E. The beam is installed in its final location where additional (superimposed) dead load is applied, typically in the form of a deck slab and other bridge elements. This load causes a downward deflection and increases the strain at the level of the prestressing, thus increasing the force in the prestressing tendons. This effect is an elastic gain in prestressing. It should be noted, however, that the superimposed load contributes tension stress to the bottom face of the girder. Creep and shrinkage of concrete continue to be important factors for the in-service condition. If the girder was stored for several months before being installed, there may be very little shrinkage strain remaining to occur during the service condition. The creep effect, however, is now reversed. The region of the cross section under the highest compressive stress has now changed. During stage D most of the creep deformation occurred near the face of the girder with the highest prestressing effect generally the bottom. Now the net compressive stress is more uniform through the cross section as the stresses due to

97 85 superimposed dead load and due to prestressing eccentricity approximately negate each other. The girder will continue to shorten, resulting in a decrease in prestressing force. Also, in many cases the deck will be cast composite with the girder (meaning shear transfer is provided between the two). An effective force is created by the differential volume changes between the deck and girder. Since the deck is often cast-in-place, the fresh deck concrete is bonding to the aged concrete of the girder. The deck concrete will have more potential for shrinkage strain during their bonded lifetime because much of the girder s shrinkage strain has already occurred. This differential shrinkage effectively applies a compressive force to the composite section at the level of the deck centroid. F. When live loads (service loads) are applied, the prestressing strands experience an elastic gain while the bottom face of the girder receives additional tensile stress. This stress should be calculated based on the composite section properties if the deck is behaving compositely with the girder. The stages of behavior can be further described by the changes to the strain and stress distributions in the cross section due to each effect. The effects on prestressing force and concrete stress have been separated into eleven components for presentation here. The following figures (5-2 through 5-10) summarize changes in strain and stress due to each component. It should be recognized that the bonded prestressing steel in the cross-section provides restraint to creep and shrinkage in the concrete. This restraint causes a redistribution of stress. Any mild reinforcement will have the same restraining effect, although it is not considered for the purposes of this study. This omission is reasonable because the amount of prestressing steel is usually much greater than the amount of mild reinforcement in the primary flexural direction. Additionally, the mild reinforcement will typically be distributed across the section with

98 86 Figure 5-1. Stage of loading for a pretensioned concrete girder - manufacturing through service. little net eccentricity. In the case of a partial prestressed design, special considerations may be warranted. The sequence of figures presented here is a conceptual look at the system to aid in understanding the problem. In the case of steel relaxation, stress is lost in the strand without a change in strain. In practice, the stress loss due to relaxation is very small. Therefore the internal

99 87 stress redistribution due to relaxation is also very small. While relaxation losses will be considered, the corresponding redistribution of stresses internally will be ignored as negligibly small. An attempt has been made to indicate the relative magnitudes of the different components graphically, but in some cases scale has been sacrificed for clarity of the graphic. 1. Initial Prestressing Force Figure 5-2. Strain and stress in the girder cross section due to initial prestressing force 2. Girder Self-Weight Figure 5-3. Strain and stress in the girder cross section due to girder self-weight

100 88 3. Girder shrinkage prior to deck placement Figure 5-4. Strain and stress in the girder cross section due to shrinkage prior to deck placement 4. Girder creep prior to deck placement Figure 5-5. Strain and stress in the girder cross section due to creep prior to deck placement 5. Relaxation of steel prior to deck placement Relaxation involves a decrease of stress in the steel without corresponding change in strain. Compared to other components, relaxation losses are relatively small. The changes in stress and strain over the cross section due to relaxation are minor.

101 89 6. Deck self-weight Figure 5-6. Strain and stress in the girder cross section due to deck self-weight 7. Shrinkage after deck placement Effects due to shrinkage after deck placement are complicated by the fact that the girder and deck are shrinking at different rates. Much of the girder shrinkage has already taken place by this time, but all of the deck shrinkage will be redistributed in the composite section. It is common for the differential shrinkage to lead to a theoretical gain in prestressing force, but a corresponding tension stress at the girder bottom fiber. Figure 5-7. Strain and stress in the girder cross section due to shrinkage after deck placement

102 90 8. Super-imposed dead load on the composite section Figure 5-8. Strain and stress in the girder cross section due to superimposed dead load on the composite section 9. Creep after deck placement After the deck has been cast, the girder will continue to creep. In some cases, it may recover some of the creep from before deck placement because of the stress reversal. Creep effects in the deck concrete are very small because it is under relatively small stress. The creep gain shown in the graphic could also be a creep loss, depending on the exact nature of the system and the age of the girder concrete when the deck is cast. Creep of the deck concrete acts to soften the effect of differential shrinkage between the deck and the girder. Figure 5-9. Strain and stress in the girder cross section due to creep after deck placement

103 Relaxation of prestressing strands after deck placement As indicated in point 5, relaxation losses are small compared to the other components. 11. Live Load Figure Strain and stress in the girder cross section due to live load 5.2. Example Bridge Details The prestress loss methods are compared for a given set of numerical input, and the timestep method results are shown with respect to a particular set of input parameters. Two bridges have been identified for use in this study because they represent typical pretensioned bridge girder construction and full design calculations are readily available. One is Design Example 9.4 in the PCI Bridge Design Manual (PCI, 1997) and the other is from the Comprehensive Design Example for Prestressed Concrete Girder Superstructure Bridge with Commentary (FHWA, 2003). They will be referred to as PCI BDM Example 9.4 and FHWA Example, respectively. The PCI BDM Example 9.4 bridge will be the primary example used in this study. Since loss of prestress is determined by an analysis of the critical cross section, it is not necessary to study a broad range of bridges within a single structure type classification. The FHWA Example

104 92 is used as a comparison to validate the simplified method developed in this thesis because it has a smaller initial prestress and is therefore less affected by creep losses. The basic design parameters of both bridges are summarized in the following sections PCI BDM Example 9.4 The bridge consists of six 120-ft simple span 72-in. deep AASHTO-PCI bulb-tee girders spaced at 9 feet. An 8-in. thick composite deck is cast-in-place on the girders. Relevant design data is presented in Table 5-1. Table 5-1. Parameters for the PCI BDM Example 9.4 Bridge (Source: PCI, 1997) Average ambient relative humidity, H 70% Girder concrete strength at release, f ci 5.8 ksi Girder concrete strength at service, f c 6.5 ksi Deck concrete strength at service, f cd 4 ksi Total Area of Prestressing, A ps in 2 Prestressing Eccentricity at Midspan, e m in Prestressing Stress at Transfer, f pbt ksi Girder gross area, A g 767 in 2 Girder gross moment of inertia, I g in 4 Girder centroid, relative to girder bottom, y b 36.6 in Effective width of deck, b eff 108 in Width of haunch 42 in Height of haunch 0.5 in

105 The bridge section is shown in Figure 5-11, followed by the girder section in Figure Figure Bridge section for PCI BDM Example 9.4 (Source: PCI, 1997) Figure Girder section for PCI BDM Example 9.4 (PCI, 1997) The applied loads are summarized in terms of midspan moment in Table 5-2.

106 Table 5-2. Summary of moments at midspan (k-in) for PCI BDM Example 9.4 (Source: PCI, 1997) Dead Load Live Load plus Dynamic Load Allowance Non-composite Composite Composite Girder, M g Slab, M d M SIDL M LL The modulus of elasticity for concrete, calculated by the AASHTO 2004 method, is summarized for each component in Table 5-3. Table 5-3. Concrete elastic modulus for PCI BDM Example 9.4 (Source: PCI, 1997) Girder (Transfer) 4383 ksi Girder (Service) 4640 ksi Deck 3640 ksi The composite section properties, using an effective deck width of 108 inches, are summarized in Table 5-4. Table 5-4. Composite section properties for PCI BDM Example 9.4 (Source: PCI, 1997) Composite Area 1419 in 2 Composite Moment of Inertia in 4 Location of neutral axis, relative to girder bottom in Eccentricity of Prestress in FHWA Example The FHWA (Wassef et. al., 2003) example bridge consists of a reinforced concrete deck supported on simple span prestressed girders made continuous for live load. There are two spans of 110-feet each. Relevant design data is provided in Table 5-5.

107 95 Table 5-5. Parameters for the FHWA Example Bridge (Source: FHWA, 2003) Average ambient relative humidity, H 70% Girder concrete strength at release, f ci 4.8 ksi Girder concrete strength at service, f c 6 ksi Deck concrete strength at service, f cd 4 ksi Total Area of Prestressing, A ps in 2 Prestressing Eccentricity at Midspan, e m in Prestressing Stress at Transfer, f pbt ksi Girder gross area, A g 1085 in 2 Girder gross moment of inertia, I g in 4 Girder centroid, relative to girder bottom, y b in Effective width of deck, b eff 111 in Width of haunch 42 in Height of haunch 1 in The bridge section is shown in Figure 5-13, followed by the girder section in Figure Figure Bridge section for FHWA Example (Source: FHWA, 2003)

108 96 Figure Girder section for FHWA Example (Source: FHWA, 2003) The applied loads are summarized in terms of midspan moment in Table 5-6. Table 5-6. Summary of moment at midspan (k-in) for the FHWA Example (Source: FHWA, 2003) Dead Load Live Load plus Dynamic Load Allowance Non-composite Composite Composite Girder, M g Slab, M d M SIDL M LL The modulus of elasticity for concrete, calculated by the AASHTO 2004 method, is summarized for each component in Table 5-7.

109 97 Table 5-7. Concrete elastic modulus for the FHWA Example (Source: FHWA, 2003) Girder (Transfer) 4200 ksi Girder (Service) 4696 ksi Deck 3834 ksi The composite section properties, using an effective deck width of 111 inches, are summarized in Table 5-8. Table 5-8. Composite section properties for the FHWA Example (Source: FHWA, 2003) Composite Area 1419 in 2 Composite Moment of Inertia in 4 Location of neutral axis, relative to girder bottom in Eccentricity of Prestress in 5.3. Components of Time-Dependent Behavior The PCI BDM Example 9.4 bridge is used in this section to demonstrate the time step analysis method. The AASHTO 2005 material property model is used for each analysis. Figure 5-15 plots the effective prestress in the girder over time assuming the deck is cast at an age of 90 days. Note that Figure 5-15 matches the general behavior anticipated, shown in Figure 3-1. The effective prestress loss shown in Figure 5-15 by the time-step method is compared with the results yielded by the AASHTO 2005 method for the same design. Two cases are plotted: 1) the case where elastic gains are included in the estimate of prestress, and 2) the case, typically used in design, where elastic gains are ignored. The comparison is shown in Figure 5-16.

110 Jacking stress Loss at transfer due to elastic shortening Loss prior to deck placement due to creep, shrinkage, and relaxation Effective Prestress (ksi) Loss after deck placement due to creep, shrinkage, and relaxation coupled with an elastic gain due to differential shrinkage between the girder and deck Elastic gain due to application of superimposed dead load Elastic gain due to application of deck weight Time (days) Figure Effective prestress over time for PCI BDM Example 9.4 assuming deck casting at 90 days The prestress losses for the PCI BDM Example 9.4 (when the deck is cast at 90 days) are plotted in Figure The time step model allows explicit separation of the components by the following sequence of analyses: 1. The first analysis considers only the initial prestressing force. All timedependent effects and applied loads, including girder self-weight, are ignored. This analysis determines the effect due to initial prestressing. 2. The second analysis considers only the initial prestressing force and applied loads, ignoring time-dependent effects. The difference between the second and first analyses yield the effect due to external loads.

111 Effective Prestress (ksi) Time Step Method AASHTO 2005 Method (including elastic gains) AASHTO 2005 Method (ignoring elastic gains) Time (days) Figure Comparison between the time-step results and the AASHTO 2005 method for effective prestress in the PCI BDM Example 9.4 bridge, assuming the deck is cast at 90 days 3. The third analysis includes initial prestressing, applied loads, and girder shrinkage. Once the deck concrete is included in the analysis, after the time of deck casting, the shrinkage of the deck is artificially taken equal to the girder shrinkage. In this manner, differential shrinkage can be isolated as a separate component. Difference between the third and second analyses yields the effect due to shrinkage. 4. The fourth analysis includes initial prestressing, applied loads, girder shrinkage, and deck shrinkage. The difference between the fourth and third analyses yields the effect due to differential shrinkage.

112 The fifth analysis includes all contributors to time-dependent behavior: initial prestressing, external loads, girder shrinkage, deck shrinkage, relaxation, and creep. Relaxation and creep are both stress-dependent, so they cannot be explicitly separated. Since relaxation effects are small by comparison, they will be isolated first to minimize error. Analysis 5 represents the total effect. 6. The sixth analysis includes all effects from the fifth analysis, except creep. The calculated relaxation losses over time in the fifth analysis are artificially copied into the sixth analysis. The difference between Analysis 6 and Analysis 4 yields the effect due to relaxation. The difference between Analysis 5 and Analysis 6 yields the effect due to creep. Figure 5-17 presents the results of the six analysis steps indicated for the PCI BDM Example 9.4 bridge. For flexural design, the bottom fiber concrete stress is often the controlling factor. The time step procedure also allows tracking of the bottom fiber stress. The components have been split in the same manner as indicated above, and the results are shown in Figure One should note, in Figure 5-18, the small impact on bottom fiber stress of creep, shrinkage, relaxation, and differential shrinkage relative to the applied loads and initial prestressing. Additionally, in comparing Figures 5-17 and 5-18, note that differential shrinkage causes a prestressing gain, but a tension increment at the bottom fiber. This distinction is important in applying the provisions of AASHTO 2005 (see Section ), and in considering a simplified procedure.

113 Prestress Loss (ksi) [Positive = P/S Loss; Negative = P/S Gain] Color Key: Initial Prestressing Steel Relaxation External Loads Creep Girder Shrinkage Total Deck-Girder Differential Shrinkage Time (days) Figure Components of prestress loss for the PCI BDM Example 9.4 bridge assuming the deck is cast at 90 days

114 102 3 Bottom Fiber Stress at Midspan (ksi) [Positive Indicates Tension] Color Key: Initial Prestressing External Loads Girder Shrinkage Deck-Girder Differential Shrinkage Steel Relaxation Creep Total 5 Time (days) Figure Components of bottom fiber stress for the PCI BDM Example 9.4 bridge assuming the deck is cast at 90 days 5.4. Time of Deck Placement The time-step method is useful in determining the impact the construction schedule has on the total loss of prestress, with respect to the girder age when the deck is cast. As discussed in Section 3.5.1, the AASHTO 2005 method separates the time-dependent into two stages before and after deck placement. The analysis summarized in Figures 5-17 and 5-18 is repeated for cases when the deck is cast early in the construction sequence (girder age of 30 days) and late in the sequence (girder age of 365 days). The results are compared with the analysis for deck casting at 90 days in Figure 5-19 for prestress losses and Figure 5-20 for bottom fiber stress.

115 Prestress Loss (ksi) [Positive = P/S Loss; Negative = P/S Gain] Color Key: Linetype Key: Initial Prestressing External Loads Girder Shrinkage Deck-Girder Differential Shrinkage Steel Relaxation Creep Total Deck cast at 30 days Deck cast at 90 days Deck cast at 365 days Time (days) Figure Comparison of prestress loss components for different times of deck placement in the PCI BDM Example 9.4 bridge Figures 5-19 and 5-20 indicate that the time of deck placement has minimal impact on the time-dependent behavior of the girder, assuming full creep recovery. Splitting the timedependent analysis into phases before and after deck placement, as done by the AASHTO 2005 method, complicates the analysis and introduces a variable that engineers are not likely to know at the time of design. These analysis results suggest that the division between the two phases is not necessary. In examining the bottom fiber stress results in Figure 5-20, it s apparent that the small difference in bottom fiber stress due to changing the time of deck placement is entirely attributed to the differential shrinkage component. Therefore, if the two phases are combined in a

116 104 3 Bottom Fiber Stress at Midspan (ksi) [Positive Indicates Tension] Color Key: Linetype Key: Initial Prestressing Deck cast at 30 days External Loads Deck cast at 90 days Girder Shrinkage Deck cast at 365 days Deck-Girder Differential Shrinkage Steel Relaxation Creep Total 5 Time (days) Figure Comparison of prestress loss components for different times of deck placement in the PCI BDM Example 9.4 bridge simplified analysis, a conservative assumption for the time of deck placement should be made. Conservative, in this case, would be a late age for deck casting because greater shrinkage differential exists. To further justify combining the two phases in the analysis and eliminating the time-ofdeck-placement variable, a range of practical values are studied in the AASHTO 2005 method. Figure 5-21 shows the total effective prestress estimated by the AASHTO 2005 method when considering a range of deck placement times for the PCI BDM Example 9.4 bridge. These results further justify the removal of the time-of-deck-placement variable because less than a 1.0 ksi

117 difference in effective prestress is observed for a range of deck placement times from 30 days to 365 days Effective prestress after all losses estimated by AASHTO 2005 (ksi) Initial Prestress Time of Deck Placement (Days) Figure Total effective prestress estimated by AASHTO 2005 over a range of deck placement times for the PCI BDM Example 9.4 bridge 5.5. Irreversible Creep In development of the simplified methods for time-dependent analysis presented in Chapter 3, full creep recovery is assumed. The calculations are built around the premise that a compressive stress increment will cause elastic strain instantaneously, followed by additional creep strain over time. It is also assumed that for a later tension stress increment (i.e. unloading of the compressive stress) the elastic strain is fully recovered and the creep strain is recovered

118 fully according to the creep coefficient once updated for the concrete age at the time of the stress change. Figure 5-22 offers a schematic of elastic and creep strains in concrete. 106 Figure Creep of concrete when loaded and unloaded (Source: Mehta and Monteiro, 2006) Although it is beyond the scope of the current research program to quantify the effects of creep recovery, it is helpful to determine the impact of irreversible creep on prestress loss and extreme fiber stresses in order to provide guidance for future research. An approach will be used here similar to the two-function method proposed by Yue and Taerwe (1993) to predict concrete creep under decreasing stress. As a simplification, the function to represent creep recovery will be the same as the function to predict creep but multiplied by a scalar. Since creep under decreasing stress is less than creep under increasing compressive stress, the scalar will be a value less than one. Note, again, that the AASTHO 2005, AASHTO 2004, and S6-06 methods assume this scalar to be equal to one.

119 107 Since the purpose here is only to gauge the significance of creep recovery on the longterm estimate of extreme fiber stresses, a scale factor of 0.75 will be used. This value is chosen somewhat arbitrarily, although it is a realistic and practical value. Again the analysis will apply to the PCI BDM Example 9.4 bridge. Time-dependent plots are provided to compare creep recovery factors of 75% and 100% for both prestress loss (Figure 5-23) and bottom fiber stress (Figure 5-24) assuming 90 days for the time of deck placement. The analysis was done with the time step method using the AASHTO 2005 material models Effective Prestress (ksi) 100 Creep Recovery = 100% Creep Recovery = 75% Time (days) Figure Impact of creep recovery factor on effective prestress for the PCI BDM Example 9.4 bridge

120 108 1 Bottom Fiber Stress at Midspan (ksi) [Positive Indicates Tension] Creep Recovery = 100% Creep Recovery = 75% 4 Time (days) Figure Impact of creep recovery factor on bottom fiber concrete stress for the PCI BDM Example 9.4 bridge A decrease in effective prestressing of approximately 4 ksi is observed due to the 75% creep recovery factor. Also, the bottom fiber stress at midspan increased (less pre-compression) by approximately 0.2 ksi. This means that three more prestressing strands would be needed to achieve the same stress limit in design based on this analysis. While the difference in prestress loss is of concern, the difference in bottom fiber stress is even more important. It is the extreme fiber stress that will drive design decisions about the prestressing requirements for the system. Further research is needed to better characterize the creep behavior of concrete in the case of stress reversals and its impact on the flexural design of pretensioned girders.

121 Summary The time step method, developed in Chapter 4, is used to analyze the time-dependent behavior of pretensioned girders, with the PCI BDM Example 9.4 bridge used as a case study. The time step results, coupled with a sensitivity study of AASHTO 2005, suggest that separating the time-dependent behavior into phases before and after deck placement is not necessary. Also the assumption of full creep recovery impacts the estimate of prestress loss and extreme fiber concrete stress. The time step method results are needed to validate the Direct Method, which is detailed in Chapters 6 and 7.

122 Chapter 6 The Direct Method for Time-Dependent Analysis In an attempt to simplify the AASHTO 2005 method, a simplified approach coined the Direct Method to use separate nomenclature from previous AASHTO specifications is derived in the following sections. The scope of applicability for the Direct Method is the same as the AASHTO 2005 methods, currently in Articles and of the AASHTO LRFD Bridge Design Specifications (AASHTO, 2005). In order to satisfy a need for comfort and familiarity with designers, the format of AASHTO 2004 is followed as closely as possible. With this goal in mind, time-dependent losses are treated in three separate components: creep of concrete, shrinkage of concrete, and relaxation of prestressing steel. Those components will not be separated into time steps before and after deck placement, as justified by the analysis and discussion in Section The differential shrinkage component, first considered by AASHTO 2005, is also included, however, the treatment of differential shrinkage is different in the Direct Method. The AASHTO method expresses the effect of differential shrinkage in terms of a prestress gain. This approach creates the possibility of significant calculation errors in design because the prestress gain cannot be superimposed with the prestress losses and treated in the same manner. Therefore, the Direct Method will account for differential shrinkage by an effective force at the deck centroid so that its application will be more intuitive than the current format and less prone to confusion. The format of the Direct Method, relative to the AASHTO 2004 and AASHTO 2005 methods, is shown in Figure 6-1.

123 111 Figure 6-1. The format of the Direct Method relative to the AASHTO 2004 and AASHTO 2005 methods The prestress loss equations can be further simplified if a particular model for creep and shrinkage is adopted inherently. Although the creep and shrinkage models developed in NCHRP Report 496, and subsequently adopted as part of AASHTO 2005, may not be fully vetted, those models will be used in the Direct Method for the following reasons: 1) The models have already been adopted by AASHTO and are currently in the specifications 2) Although less conservative (predicting smaller creep and shrinkage strains than previous methods) in some instances, the results from this model have been developed (Tadros, 2003) considering a comparison with other creep and shrinkage predictive methods. 3) There is not a more suitable method available that considers the behavior of high-strength concrete.

124 4) The choice of a comprehensive creep and shrinkage model is not critical because creep and shrinkage are small components affecting the bottom fiber stress at final time Elastic Shortening and Steel Relaxation No changes are proposed to the AASHTO 2005 method regarding elastic shortening losses and steel relaxation losses. A constant value of 2.5 ksi, as recommended by NCHRP Report 496, should be used for relaxation of low-relaxation strands Concrete Shrinkage The effects of concrete shrinkage will be split into two categories: 1) Shrinkage of girder concrete 2) Differential shrinkage between the deck and the girder Differential shrinkage will be considered as a separate component, with shrinkage of the girder concrete treated in this section. Considering the shrinkage of the girder alone, and recognizing that the change in prestress is the product of steel elastic modulus and the change in strain at the level of the prestressing centroid (Hooke s Law), yields the general equation for shrinkage losses in (6-1). Δ (6-1) Where: Elastic modulus of prestressing steel Unrestrained shrinkage strain of girder concrete from initial to final time

125 The ratio of actual change in strain, considering the restraint provided by the prestressing steel against shrinkage, to the change in strain that would occur with no restraint. 113 This is comparable to the base equation for shrinkage loss used in AASHTO 2005, reproduced in (4-30) except that K id-sh has replaced K id. K id-sh is specified so that only the restraint effects specifically related to shrinkage are represented. The intent of the factor is the same, but a few adjustments have been made: 1. The softening effect represented by the age-adjusted effective modulus is a result of creep behavior. (refer to Section 2.2 for background on the age-adjusted effective modulus) If shrinkage and creep components are strictly separated, the results of shrinkage will be the same regardless of whether the shrinkage happens suddenly or over a long period of time. Therefore, an age-adjusted effective modulus is not applied to the case of shrinkage, and the creep term is removed from the K id equation. 2. The service-level concrete elastic modulus will be used rather than the elastic modulus at the time of transfer. Over time, stresses will be redistributed according to the final relative stiffness between concrete and steel, not the initial ratio. K id-sh can then be given by (6-2) (6-2) For typical pretensioned girders, K id-sh is approximately 0.9. Expanding the AASHTO 2005 model equation to estimate the shrinkage of concrete results in (6-3).

126 (6-3) Where: Ratio of volume to surface area for the girder Ambient relative humidity Compressive strength of girder concrete at transfer Age of the concrete (in this case the girder concrete) By adopting this model for shrinkage, the prestress loss provisions become less flexible because they cannot be adapted for use with other models. The AASHTO 2005 method maintains the flexibility to use other models, sacrificing opportunities for algebraic simplification. The following assumptions and simplifications are made: girder size factor, for common girder V/S ratios near 3.5 time-development factor, at final time 0.8 as recommended by Tadros, et. al. (2003) when t is very large, as it is for losses Incorporarting these assumptions in (6-3) and substituting into (6-1) results in a simplified equation to predict prestress loss due to girder shrinkage, shown in (6-4). Δ (6-4)

127 Differential Shrinkage Differential shrinkage between the deck and the girder, in the case of composite construction, should be considered. Furthermore, any provision related to differential shrinkage adopted into the specifications should be clear so that a non-conservative conceptual error does not follow. Such a danger exists with the AASHTO 2005 format. Examine Figure 6-2 for a clarification of these points. When differential shrinkage occurs the deck has a potential shrinkage strain greater than that in the girder concrete an effective force, P deck, builds up in the composite section. This effective force, depending on the cross-section dimensions, could cause an increase in strain at the level of prestressing and a theoretical GAIN in prestressing force. It also, in such a case, would cause an increase in tension stress at the extreme bottom fiber (presuming positive flexure). If this gain is superimposed with the prestress loss components in calculating extreme fiber stresses, suggesting that it contributes to pre-compression of the concrete, a significant error follows. Figure 6-2. The effective action on the composite section due to differential shrinkage Therefore, a non-conservative result is possible if differential shrinkage is considered just in terms of prestressing gain, as recommended by AASTHO Such

128 116 language can be applied incorrectly if the designer does not have a thorough understanding of the impact differential shrinkage has on the entire composite section. It may lead to a better conceptual understanding if, instead of considering differential shrinkage by a loss or gain of prestressing, it is considered as an effective force, P deck, applied at the centroid of the deck. The effective force, P deck, applied to the composite section can be calculated as the product of differential shrinkage, the elastic modulus of the deck, and the area of the deck that behaves compositely with the girder. The age-adjusted effective modulus of concrete should be used in this case because the strain differential builds over time and will be partially relieved by concrete creep. The effective force, P deck, can be calculated by (6-5). 1, (6-5) Where: Differential shrinkage between the deck and the girder Elastic modulus of deck concrete Effective area of the deck Relaxation coefficient (Trost, 1967) that accounts for the reduction in creep that occurs because not all of the stress is applied at the initial time, t i (Collins, 1991). Values typically range between 0.6 and 0.9. AASHTO (2005) applies a constant value of 0.7 (Tadros, 2003), Creep coefficient for deck concrete at final time due to stresses induced at the time of deck placement In (6-5), the age-adjusted effective modulus (often denoted ) is represented by the term shown in (6-6).

129 117 1, (6-6) The effects of differential shrinkage can be determined by (6-5) using any suitable creep and shrinkage model. As an alternative to calculating the creep coefficient and differential shrinkage strain in (6-5), an approximate procedure is derived in the following sections based on the AASHTO 2005 model. The following sections detail development of approximate terms for the differential shrinkage term, the creep coefficient for deck concrete, and the effective force, P deck Approximate Calculation of Differential Shrinkage Strain The differential shrinkage term is the difference between total deck shrinkage and girder shrinkage after deck placement. (6-7) Where: Shrinkage strain of girder concrete after the time of deck placement Shrinkage strain of deck concrete Using the AASHTO 2005 model for concrete shrinkage, the shrinkage of the girder after the time of deck placement can be found by (6-8).

130 (6-8) Where: Shrinkage strain of girder concrete over entire life Shrinkage strain of girder prior to deck placement The following simplifications can be made: Girder size factor, for common V/S ratios near as recommended by Tadros et. al. (2003) The age at deck placement, t d, will be assumed 150 days. An earlier age assumption would be less conservative because it would mean more girder shrinkage takes place after deck casting, reducing the differential between deck and girder shrinkage. A later age assumption would have little impact. For the assumption t d = 150 days, the product of the concrete strength factor and the time-development factors can be approximated as follows:.. 1. Considering these assumptions in (6-8), girder shrinkage after deck placement can be estimated in (6-9).

131 (6-9) 119 Total deck shrinkage will be estimated considering the following assumptions: Deck size factor, representative of a typical V/S ratio of 4.5 for decks 0.8 as recommended by Tadros et. al. (2003) The time development factor for final time, 1.0 Applying these assumptions in the AASHTO 2005 model for shrinkage, reproduced in (6-3), the total deck shrinkage is approximately given by (6-10) (6-10) Where: Shrinkage strain of deck concrete after the time of deck placement Compressive strength of the deck concrete Combining similar terms, simplifying algebraically, and rounding yields (6-11) to approximate the differential shrinkage between girder and deck (6-11)

132 Approximate Calculation of the Deck Creep Coefficient A simplified creep coefficient for the deck concrete is derived based on the AASHTO 2005 model for creep, reproduced in (6-12)., (6-12) The following assumptions and simplifications can be made: For typical deck geometry, as recommended by Tadros et. al. (2003) Time-development factor at final time, 1.0 The effective force due to differential shrinkage starts to build up as soon as the deck concrete begins gaining strength and shrinking. Therefore, the age of concrete when loading is applied, 1.0 days Applying these simplifications in (6-12) allows approximating the deck creep coefficient by (6-13)., (6-13) Approximating the Effective Differential Shrinkage Force Substituting the approximate terms given in (6-11) and (6-13) into the basic formulation of (6-5) approximates P deck, as shown in (6-14).

133 (6-14) The inputs required to calculate P deck using (6-14) will be known at the time of design. The effect of differential shrinkage can be quantified by applying the calculated effective force to the composite (deck concrete, girder concrete, and bonded prestressing steel) cross-section as indicated by Figure 6-1. Use of this approach improves the transparency of the provision because it becomes clear that, even though a theoretical prestress gain results, there will be an increase in bottom fiber tension. The tension stress increment due to P deck can be determined by methods of fundamental mechanics Creep of Concrete Loss of prestress due to creep can be determined by Hooke s Law. The change in prestress is the product of the prestressing steel elastic modulus and the creep strain in the girder at the level of the prestressing centroid. The strain is adjusted by the transformed section coefficient to represent the force redistribution caused by the restraint of bonded steel against creep. Δ (6-15) Where: Elastic modulus of prestressing steel Creep strain in the girder at the level of the prestressing steel centroid The ratio of actual change in strain, considering the restraint

134 provided by the prestressing steel against creep, to the change in strain that would occur with no restraint, approximately The formulation is identical to that shown in (4-48), except that E c is substituted for E ci. 122 Creep strain is expressed as a function of elastic strain and a creep coefficient. Δ, (6-16) Where: Δ Change of stress in the concrete (at the level of the prestressing centroid, in this case) Elastic modulus of girder concrete, Creep coefficient at time of interest, t, due to the stress change Δ applied at time, t i Substituting (6-16) into (6-15) and rearranging produces (6-17). Δ Δ, (6-17) There are three key stress changes at the level of prestressing to consider: 1) f cgp the stress at the centroid of the prestressing just after transfer 2) Δf cdp the stress change at the centroid of the prestressing due to application of deck weight and other permanent loads 3) Δf cps the stress change at the centroid of the prestressing due to shrinkage and relaxation losses, and differential shrinkage between the deck and girder

135 If the stress changes due to permanent loads and prestress losses are considered to occur at the time of deck placement, total creep losses can be found by (6-18). 123 Δ, Δ Δ, (6-18) The general equation for the creep coefficient is given in (6-12). The creep coefficient for stresses induced at transfer can be simplified by the following assumptions: 0.8 as recommended by Tadros et. al. (2003) Girder size factor, representing typical girder V/S ratios around 3.5 Time-development factor, 1.0 for losses at final time 1 to represent a typical construction cycle where transfer occurs at a concrete age of 1-day Applying these assumptions yields (6-19) to approximate the creep coefficient for stresses introduced at transfer. 195, 0.1 (6-19) 1.3 Where:, Creep coefficient at final time due to stresses applied at transfer

136 124 The creep coefficient for stress changes at the time of deck placement can be simplified with the same assumptions, except that the loading age term, t i, will be taken as 150 days. This is a relatively conservative value because earlier loading ages would suggest more creep recovery when stresses are reversed. A later loading age has little effect on the equation. The approximate equation for the creep coefficient is conveniently half of the creep coefficient for loads applied at transfer. 195, 0.05 (6-20) 1.3 Where:, Creep coefficient at final time due to stresses applied at the time of deck placement Applying an approximate value of 0.85 to the K id-cr term and simplifying yields (6-21) for total losses due to concrete creep. Δ Δ Δ (6-21) 6.5. Implementation of the Direct Method The format of the Direct Method is similar to that of the AASHTO 2004 method. Use of the proposed Direct Method requires the following sequence of steps: Calculate the loss of prestress due to elastic shortening. The method for doing so has not changed as a result of the NCHRP Report 496 recommendations, nor are changes being proposed as part of the Direct Method Calculate the loss of prestress due to shrinkage using (6-4)

137 125 Calculate the loss of prestress due to steel relaxation. No changes are suggested to the recommendations of NCHRP Report 496. For low-relaxation strands, a constant value of 2.5 ksi may be assumed for total losses due to relaxation. Calculate the effective force, P deck, due to differential shrinkage using (6-5) or (6-14) Calculate the loss of prestress due to creep using (6-21). Stress at the level of prestressing due to each of the following three components must be calculated: o Initial prestressing, just after transfer (f cgp ) o Deck weight and other permanent loads (Δf cdp ) o Shrinkage and relaxation losses, and differential shrinkage between the deck and girder (Δf cps ) Having calculated each of the terms indicated above, the designer can calculate stress in the extreme concrete fiber by methods of fundamental mechanics as follows: o The stress increment due to initial prestressing and girder self-weight can be determined using the gross girder cross sectional properties and an effective prestressing force that is the difference between the initial prestressing force and that lost from elastic shortening. In the typical case where initial prestressing will cause camber, the self-weight moment of the girder should be considered. o The stress increment due to time-dependent loss of prestress can be found by considering a reduction in prestress force equal to the sum of shrinkage, creep, and relaxation losses. The extreme fiber stress change due to these losses can be calculated based on the gross section

138 126 properties of the girder. Generally speaking, most of the losses will occur before the girder becomes composite with the deck. o The stress increment due to deck self-weight (assuming unshored construction) should be calculated based on the gross section properties of the girder. o The stress increment due to super-imposed dead load will typically be calculated based on the composite girder properties this assumes that the deck and girder are behaving compositely when the super-imposed dead load is applied. o The stress increment due to differential shrinkage can be calculated by applying and effective force, P deck, at the centroid of the deck (see Figure 4). In determining stresses, the composite section properties should be used. o The stress increment due to live load should be calculated based on composite section properties Numerical Example In order to demonstrate use of the equations developed for the Direct Method, a numerical example is presented. The example problem will demonstrate calculation of extreme fiber concrete stresses at midspan for the PCI BDM Example 9.4 bridge. Details of the bridge are provided in Section

139 Differential Shrinkage The stresses induced by differential shrinkage between the girder and deck are calculated by determining an effective compressive force applied to the composite section at the centroid of the deck, P deck (6-22) (6-23) Loss of Prestress Prestress losses are computed for each of four components: elastic shortening, shrinkage, relaxation, and creep Loss Due to Elastic Shortening The calculation of prestress loss due to elastic shortening is unchanged from previous code provisions. AASHTO-LRFD Equation C a-1 (AASHTO, 2005) is applied as follows: Δ (6-24)

140 128 Δ (6-25) Loss Due to Shrinkage 140 Δ (6-26) Δ (6-27) Loss Due to Relaxation Δ 2.5 (6-28) Loss Due to Creep Δ Δ Δ (6-29) Where: (6-30) The effective prestress at transfer,, will be taken as: Δ (6-31)

141 Stress change due to application of deck weight and superimposed dead load: (6-32) Δ (6-33) Δ (6-34) The stress change due to shrinkage and relaxation losses, and differential shrinkage: Δ Δ Δ 1 (6-35) Δ (6-36) Substituting into (6-29) and solving yields: Δ (6-37)

142 Calculation of Bottom Fiber Stress at Midspan: (Tension shown Positive) midspan. The estimates of prestress loss are used to calculate the extreme fiber concrete stress at Stress at Transfer At the time of transfer, both the initial prestressing (minus elastic shortening losses) and self-weight moment are contributing to bottom fiber stress. Δ (6-38) Δ (6-39) Long-Term Losses Since the majority of the prestress loss occurs prior to deck placement, the stress is calculated based on the girder s gross section properties. Δ Δ Δ Δ 1 (6-40) Δ (6-41)

143 Deck Placement Δ (6-42) Super-Imposed Dead Load Δ (6-43) Differential Shrinkage Δ (6-44) Live Load Δ (6-45) Total Bottom Fiber Stress Δ 0.88 (6-46)

144 Summary The Direct Method is developed as a simplified approach for the time-dependent analysis of pretensioned girders. Hooke s Law is the foundation of all the equations proposed. Only the material model used to estimate the creep and shrinkage response of the concrete is empirical. The format of the method is comparable the AASHTO 2004 method, except that a provision for differential shrinkage is included. The treatment of differential shrinkage in the Direct Method is more transparent than that in the AASHTO 2005 method. A numerical example demonstrates application of the method for design. Comparison of the Direct Method results with those from other methods is provided in Chapter 7.

145 Chapter 7 Validating the Direct Method Much of the validation for the Direct Method is inherent in its derivation. Hooke s Law is the foundation of all equations proposed. Only the material model used to estimate the creep and shrinkage responses of concrete is empirical. The material model chosen for use in the Direct Method was carefully developed (Tadros et. al., 2003) and adopted into the AASHTO LRFD Bridge Design Specifications (AASHTO, 2005). This chapter documents an uncertainty study and a sensitivity study to verify the integrity of the Direct Method for time-dependent analysis. The Direct Method is compared with the AASHTO 2004 method, the AASHTO 2005 method, the AASHTO 2005 simplified method, and the time-step method developed in Section 4.1. Both the AASHTO 2004 and AASHTO 2005 material property models are considered Uncertainty Study The Direct Method can be further validated through comparison with other methods, especially with respect to the inherent uncertainty in the estimation of prestress losses and concrete extreme fiber stresses. Uncertainty in the time-dependent analysis of pretensioned girders arises from many factors: Material Properties: The material properties, especially for concrete, are highly variable. Even if a precise model existed for estimating material properties, the heterogeneous nature of the concrete material would make the response uncertain.

146 134 Model Error: The models used to estimate material properties are founded on an empirical fit to test data. Although the test data is considered to be a representative sample, the broad range of concrete materials and mixture proportions creates scenarios that are beyond the original scope of the material model. Also, the empirical nature of the model introduces uncertainty because the model is often based on a best fit since a perfect fit does not exist. Construction Tolerance: The geometry of elements, especially of cast-in-place concrete, can be variable. Quality control will ensure that manufactured elements fall within prescribed construction tolerances, but tolerances are permitted nonetheless, introducing additional uncertainty. Loads: An accurate estimate of loads is vital to time-dependent girder analysis, especially in anticipating the creep response of concrete. Since material unit weights and the geometry of elements are uncertain, the estimate of loads is also. Bridge live load is a significant factor for calculating extreme fiber stresses, and it is uncertain as well. Environmental Conditions: The most significant environmental factor affecting time-dependent behavior of pretensioned girders is relative humidity. Since relative humidity can fluctuate over a broad range through the year in many geographic areas, designers are typically left estimating an average relative humidity based on historical data. Construction Schedule: The time-dependent response is affected by the construction schedule, especially by the time of transfer and the time of deck placement. Although the timing of both events can be assumed within a reasonable range, the designer will not know either with certainty.

147 Monte Carlo Simulation The Monte Carlo simulation techniques outlined in Section 4.2 are used to quantify the uncertainty of each time-dependent analysis model studied. The base input variables are determined by the needs of the time-step method. The time-step method was developed (Section 4.1) to provide a detailed analysis with minimal assumptions. Therefore, it has the greatest demand for input. All of the other methods include some assumptions and simplifications in their development. These assumptions reduce the number of input parameters needed by the model, possibly introducing a model bias. Taken as the most precise of the methods, the time-step method will be used as a baseline for comparison. The following sections summarize the distributions used to represent the input parameters and the results of the analysis Input Parameters This section summarizes the assumed probability distributions employed in the Monte Carlo simulation for time-dependent analysis methods. In some cases, distributions have been drawn from available literature. In many cases, however, the judgment of the author was used to develop input distribution parameters. Since the primary purpose of the uncertainty analysis is to provide a relative comparison between methods, and all the methods use the same input parameters in the simulation, more rigorous development of the input distributions is not warranted and would not impact the conclusions of this study. Probability distributions have been truncated at values three standard deviations (σ) away from the mean (μ) unless a practical consideration exists that warrants truncating the distribution at another value.

148 Material Properties Probability distributions for key material properties elastic modulus of steel and compressive strength of concrete have been studied by others and the distributions used by Gilbertson and Ahlborn (2004) are used here. Since concrete strength is monitored closely using test cylinders, the distribution is truncated at a minimum value equal to the nominal (design) value for f c. If the concrete strength is tested significantly less than this target value, the girder would not be placed into service and does not need to be considered in this simulation. Although important material properties, creep, shrinkage, and elastic modulus of concrete will be considered separately, expressing their uncertainty in terms of a model uncertainty factor. Table 7-1. Probability distributions related to material properties used in Monte Carlo simulation Variable Distribution Mean, μ COV, σ/μ Min Max E p Normal 0.996*Nominal 0.02 μ-3σ μ+3σ f c Normal 1.1*Nominal Nominal μ+3σ f c(deck) Normal 1.1*Nominal Nominal μ+3σ Initial Prestressing The important variables in quantifying initial prestressing force are the area of prestressing steel and the initial jacking stress. A distribution representing the uncertainty of prestressing steel cross sectional area developed by Gilbertson and Ahlborn (2004) is used. A normal distribution with a small coefficient of variation will be used to represent the initial jacking stress. The jacking stress is closely monitored by pressure gauges on the hydraulic prestressing equipment and also by measuring observed elongation of the strands. Since the

149 137 relationship between stress and strain is consistent for steel, this is a reliable secondary check that prevents large errors in initial prestressing force. A coefficient of variation of 0.01 is selected with the mean of the distribution being the nominal value. Table 7-2. Probability distributions related to initial prestressing used in Monte Carlo simulation Variable Distribution Mean, μ COV, σ/μ Min Max A ps Normal 1.011*Nominal.0125 μ-3σ μ+3σ f pbt Normal Nominal 0.01 μ-3σ μ+3σ Precast Girder Geometry The overall geometry and placement of prestressing strands is very closely controlled in the precasting environment. In many cases girders are formed with reusable formwork that has been carefully manufactured for repeated use. Within the concrete, the location of the prestressing strands is closely controlled by the fact that they are typically located on a 2 -square grid. All prestressing hardware plates at the end of the prestressing bed, hold-down anchors, etc. are manufactured to ensure a 2 spacing between strands. Therefore, all variables related to girder geometry and prestress strand location are assigned to a normal distribution with mean equal to the nominal value and a small (0.005) coefficient of variation. Table 7-3. Probability distributions related to precast girder geometry used in Monte Carlo simulation Variable Distribution Mean, μ COV, σ/μ Min Max A g Normal Nominal μ-3σ μ+3σ I g Normal Nominal μ-3σ μ+3σ y b Normal Nominal μ-3σ μ+3σ y t Normal Nominal μ-3σ μ+3σ e m Normal Nominal μ-3σ μ+3σ V/S Normal Nominal μ-3σ μ+3σ

150 Cast-in-Place Deck Geometry and Behavior The deck is typically cast-in-place on site so there is less strict control over the geometry compared with precast construction. Additionally, the thickness of the deck and the thickness of the haunch are particularly less certain because they are partially dependent on the amount of camber in the prestressed girder a value which is difficult to predict accurately during design. Most of the impact of unpredictable camber is absorbed by the haunch, so a relatively high coefficient of variation (0.25) will be assigned for that variable. The uncertainty in the deck thickness is partially insulated from the effects of camber by the flexibility in haunch dimension, so a smaller coefficient of variation (0.05) is reasonable. This is still much larger than coefficients of variation used to represent precast elements. The width of the haunch is controlled by the width of the girder, so the same coefficient of variation (0.005) applied to the precast geometry will be used. The effective width of the deck is a variable which has more to do with deck behavior than the deck geometry. Effective width is a variable used to simplify calculations by representing an equivalent width of deck that effectively behaves with the girder in flexure, considering the effect of shear lag. Figure 7-1 depicts the concept of representing a parabolic stress distribution by an equivalent rectangular distribution with some effective width. Much of the uncertainty in the use of effective width comes from the fact that it is being used to simplify a parabolic stress distribution, not due to uncertainty in geometry. A coefficient of variation of 0.05 is assigned for this study.

151 139 Figure 7-1. Rectangular stress block simplification used when calculating the effective width of the deck (Source: Wight and Macgregor, 2009) Table 7-4. Probability distributions related to cast-in-place deck geometry and behavior used in Monte Carlo simulation Variable Distribution Mean, μ COV, σ/μ Min Max b eff-deck Normal Nominal 0.05 μ-3σ μ+3σ h deck Normal Nominal 0.05 μ-3σ μ+3σ b haunch Normal Nominal μ-3σ μ+3σ h haunch Normal Nominal 0.25 μ-3σ μ+3σ Construction Schedule Construction schedule impacts the calculation of prestress losses because of the varying times of transfer and deck placement. Force transfer (cutting the strands in the precasting facility) usually happens the day after the concrete is cast. Girders that are cast the last day of the work

152 140 week, however, may sit in the formwork over the weekend or holiday before the strands are cut. In expressing the age at transfer in a probability distribution, it s important to recognize that the nominal value is one day, but values much less than that are not feasible. As such, 18 hours is taken as a practical minimum. Values larger than one day are not unreasonable. A high coefficient of variation will be applied (0.25) but the distribution will be truncated at a minimum value of 18 hours or a maximum value of 3 standard deviations above the mean. The age of the girder when the deck is cast is typically somewhere between 30 days and one year, with no specific reason to expect typical values near either end of the range. Therefore, a uniform distribution with a range of 30 days to 365 days is used to model the time at deck placement. The age at final time is taken as a constant days for all simulation cycles. Table 7-5. Probability distributions related to construction schedule used in Monte Carlo simulation Variable Distribution Mean, μ COV, σ/μ Min Max t transfer Normal Nominal days μ+3σ t deck Uniform 30 days 365 days t final Constant days Environmental Factors The only significant environmental factor in estimating prestress losses is the ambient relative humidity. The distribution used by Gilbertson and Ahlborn (2004) is adopted here. Table 7-6. Probability distribution related to environmental factors used in Monte Carlo simulation Variable Distribution Mean, μ COV, σ/μ Min Max H Normal Nominal μ-3σ μ+3σ, 100%

153 Relaxation Coefficient The relaxation coefficient used in determining the age-adjusted effective modulus is treated as a random variable. Noting that the value typically falls in a range between (Collins and Mitchell, 1991), a mean of 0.75 will be used with a coefficient of variation equal to This yields values three standard deviations away from the mean equal to the minimum and maximum of the range, assuming a normal distribution. Table 7-7. Probability distribution related to the relaxation coefficient used in Monte Carlo simulation Variable Distribution Mean, μ COV, σ/μ Min Max χ Normal μ-3σ μ+3σ Model Uncertainty Various models are used, corresponding to the method under investigation, to estimate creep and shrinkage strains, as well as concrete elastic modulus, based on the other input parameters. These models are uncertain by their empirical nature. To account for the uncertainty of the material models, a series of uncertainty factors is used in this study. For example, the elastic modulus in the simulation is calculated as the product of elastic modulus calculated from the appropriate model and the elastic modulus uncertainty factor. The uncertainty factor is itself treated as a random variable with a mean value of and coefficient of variation determined from experimental data. If the numerical model is not inherently biased, the mean value of the uncertainty factor is 1.0. The concept of the uncertainty factor, demonstrated with respect to elastic modulus, is shown in Figure 7-2.

154 142 Elastic Modulus, Ec Figure 7-2. Conceptual depiction of the method used to consider model uncertainty in the Monte Carlo simulation The elastic modulus uncertainty factor distribution is determined based on the experimental data summarized by Tadros et. al. (2003), as shown in Figure 7-3. The range of compressive strengths from 5-12 ksi is identified as most common to North American bridge construction, so the uncertainty factor distribution will be developed with respect to that range. The approximate limits of the experimental data will be taken as two standard deviations away from the mean value. The mean value of the uncertainty factor is assumed 1.0, meaning the numerical model is not biased. The data points shown in Figure 7-3 suggest some bias in the model, however, most of the data points beyond the limits indicated by the green outline are from the same set of test specimens (represented by a triangle). This suggests there may have been something unique about the test procedure or the concrete being tested. Furthermore, the data points beyond the limits highlighted represent high elastic moduli. Ignoring these stiffer concrete mixes in considering flexural analysis at service is conservative.

155 143 4σ = 4000 ksi Figure 7-3. Determination of the model uncertainty factor for concrete elastic modulus (Data source: Tadros et. al., 2003) The model uncertainty factor distribution will be defined for the middle of the range indicated, f c = 8.5 ksi. At this point, the numerical model estimates the elastic modulus to be 5600 ksi. The limits of the observed elastic modulus at f c = 8.5 ksi are approximately 3400 ksi and 7400 ksi. Taking these limits to be two standard deviations above and below the mean, the coefficient of variation can be calculated as shown in (7-1) (7-1) 5600 Estimating model uncertainty factor distributions for creep and shrinkage is more complex because the numerical models are based on many input factors. An approach similar to that taken for elastic modulus is not reasonable because there are many dependent variables. ACI 209 (2008) compares the ACI shrinkage and creep models with the RILEM databank. In