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EFFECTS OF CREEP AND SHRINKAGE ON TIME-DEPENDENT STRAIN AND CURVATURE OF R/C MEMBERS
Abstract
The long-time deflection multiplier specified in the ACI Code is simple to use, but neglects the effects of several parameters such as age at loading and reinforcement ratio that are known to have significant effects on long-time slab deflections. Several authors including Branson (1977), Sbarounis (1984(a)), and Graham and Scanlon (1986) have made recommendations in the past to increase the long-time multiplier for application to two-way slabs, however, no systemic study has been found that would provide a sound theoretical basis to justify such recommendations.
The research documented in this thesis examines the factors that influence longtime deflections from a theoretical viewpoint at the member section level using a timestep method of analysis developed from established principles. The time-step analysis traces the effects of creep and shrinkage on the time-dependent strain and curvature of various member sections. A parametric study is performed using the developed algorithm to evaluate the limitations of the existing ACI 318-11 long-time multiplier (λΔ) used in calculating two-way slab deflections. Results of the parametric study are used to provide designers with insight as to which factors have the most influence on long-term deflections and recommend methods of dealing with design problems engineers face.
TABLE OF CONTENTS
LIST OF FIGURES ………………………………………………………………………………………..vi
LIST OF TABLES………………………………………………………………………………………….viii
LIST OF SYMBOLS………………………………………………………………………………………ix
ACKNOWLEDGEMENTS……………………………………………………………………………..xv
Chapter 1 Introduction…………………………………………………………………………………….1
1.1. Background………………………………………………………………………………………1
1.2. Problem Statement…………………………………………………………………………….2
1.3. Objectives and Scope…………………………………………………………………………3
Chapter 2 Literature Review…………………………………………………………………………….5
2.1. Time-Dependent Creep Functions……………………………………………………….5
2.1.1. ACI 209R-92 ……………………………………………………………………………6
2.1.2. CEB-FIP MC 1990……………………………………………………………………7
2.1.3. B3 Model…………………………………………………………………………………8
2.1.4. Other Functions ………………………………………………………………………..9
2.1.5. Selection of Creep Function……………………………………………………….9
2.2. Time-Dependent Shrinkage Functions …………………………………………………10
2.2.1. ACI 209R-92 ……………………………………………………………………………10
2.2.2. CEB-FIP MC 1990……………………………………………………………………11
2.2.3. B3 Model…………………………………………………………………………………13
2.2.4. Other Functions ………………………………………………………………………..14
2.1.5. Selection of Shrinkage Function …………………………………………………14
2.3. Long-Term Multiplier………………………………………………………………………..14
2.3.1. ACI 318-11 Code Provisions………………………………………………………15
2.3.2. Branson (1977)…………………………………………………………………………16
2.3.3. Sbarounis (1984(a))…………………………………………………………………..16
2.3.4. Graham and Scanlon (1986)……………………………………………………….17
2.3.5. Hossain, Vollum, and Ahmed (2011)…………………………………………..18
2.3.6. Section Analysis……………………………………………………………………….19
2.4. Time-Dependent Analysis ………………………………………………………………….19
2.4.1. Creep Superposition ………………………………………………………………….19
2.4.2. Age-adjusted Effective Modulus…………………………………………………20
2.5. Two-Way Action………………………………………………………………………………21
2.5.1. Classical Solutions…………………………………………………………………….21
2.5.2. Crossing-beam Methods…………………………………………………………….21
2.5.3. Finite Element Methods …………………………………………………………….22
2.6. Summary………………………………………………………………………………………….22 Chapter 3 Detailed Time-Step Method of Analysis……………………………………………..24
3.1. Assumptions ……………………………………………………………………………………..24
3.2. Material Properties……………………………………………………………………………..25
3.3. Axial Prism Analysis………………………………………………………………………….26
3.3.1. Development of Method…………………………………………………………….26
3.3.2. Analysis Algorithm …………………………………………………………………..34
3.4. Un-cracked Flexural Section Analysis ………………………………………………….35
3.4.1. Development of Method…………………………………………………………….35
3.4.2. Analysis Algorithm …………………………………………………………………..42
3.5. Cracked Flexural Section Analysis……………………………………………………….44
3.5.1. Development of Method…………………………………………………………….44
3.5.2. Analysis Algorithm …………………………………………………………………..45
3.6. Summary…………………………………………………………………………………………..47
Chapter 4 Parametric Studies……………………………………………………………………………49
4.1. Axial Prism……………………………………………………………………………………….49
4.2. Axial Prism Results ……………………………………………………………………………51
4.2.1. Sensitivity to the Number of Time Steps ……………………………………..51
4.2.2. Reinforcement Ratio, ρ………………………………………………………………52
4.2.3. Initial Time of Loading, t0 ………………………………………………………….53
4.2.4. Concrete Strength, f’c ………………………………………………………………..55
4.2.5. Ultimate Creep Coefficient, Cu……………………………………………………57
4.2.6. No Correction Factor vs. Correction Factor………………………………….58
4.3. Reinforced Concrete Flexural Sections …………………………………………………59
4.4. Reinforced Concrete Flexural Sections Results ……………………………………..62
4.4.1. Long-Term Effects of Shrinkage…………………………………………………62
4.4.2. Reinforcement Ratio, ρ………………………………………………………………67
4.4.3. Initial Time of Loading, t0 ………………………………………………………….70
4.4.4. Concrete Strength, f’c ………………………………………………………………..74
4.4.5. Ultimate Creep Coefficient, Cu……………………………………………………77
4.5. Summary…………………………………………………………………………………………..80
Chapter 5 Implications for Design…………………………………………………………………….82
5.1. Axial Prism……………………………………………………………………………………….82
5.2. Reinforced Concrete Flexural Sections …………………………………………………85
5.3. Summary…………………………………………………………………………………………..88
Chapter 6 Summary, Conclusions, and Recommendations…………………………………..89
6.1. Conclusions……………………………………………………………………………………….89
6.2. Recommendations………………………………………………………………………………90
References……………………………………………………………………………………………………..92
Appendix Equilibrium Verification for Flexural Sections…………………………………….96
Chapter 1 INTRODUCTION
1.1. Background
The use of two-way concrete slabs is very common in building construction, particularly for application to multistory office and apartment buildings. Excessive deflection in the two-way slab systems sometimes occurs during and after the construction phase, and causes serviceability problems such as cracked partitions, cracked door and window jambs, and uneven floors. While these problems generally do not indicate a potential life safety problem, significant economic loss can result from such damage to nonstructural elements and functionality.
ACI 318-11 considers deflection serviceability requirements of two-way slabs fulfilled by satisfying slab minimum thickness based on span length. The current ACI 318-11 minimum thickness requirements are essentially independent of the applied load, which can lead to inadequate serviceability performance of slabs (Bondy, 2005). The current alternative to the minimum thickness requirements is calculating the short and long-term deflection using software packages, or procedures outlined in the ACI 318-11 Code. However, the current ACI 318-11 does not require the calculation of deflections as long as the slab thickness meets the requirements of ACI 318-11 Table 9.5(c). Recommendations for improving minimum thickness requirements to extend their range of applicability have been provided by Scanlon and Lee (2006).
The general procedure for calculating long-term slab deflections is outlined in sections 9.5.2.2 – 9.5.2.5 of ACI 318-11. In the procedure, short-term instantaneous elastic deflections for a unit strip width are calculated using the equation for deflection at midspan due to a uniform distributed load (1-1).
Δi = kwln4 (1-1)
384EcIe
The long-term deflections are then calculated by applying a long-time multiplier, λΔ, to the instantaneous deflection. The long-time multiplier adopted in ACI 318-11 accounts for increased deflections due to the time-dependent effects of creep and shrinkage in the concrete.
The total slab deflection at midspan is then determined by adding the short and long-term deflection, ΔT = Δi + λΔ Δi. This method can be applied to estimate mid-panel, two-way, slab deflections by combining with the crossing-beam method presented in ACI 435.9R-91.
Several authors, including Branson (1977), Sbarounis (1984(a)), and Graham and Scanlon (1986), have found that the long-time multiplier established by ACI often substantially under-predicts long-term deflections of two-way slabs when compared to experimental data published by Heiman (1974) and Sbarounis (1984(b)). In addition, the simple code-specified multiplier does not account for the effects of varying reinforcement ratios and early age loading, which are known to affect long-time deflections (Wium, 2010).
ACI 318-11 long-time multiplier is generated based on test data from one-way slabs and beams and does not pertain to axially loaded columns and shear walls. Simbirkin and Balevicius (2004) found that the same serviceability problems also arise from differential deformations of columns and shear walls in tall reinforced concrete buildings with slender columns where small strains can cause large long-term deformations.
1.2. Problem Statement
The long-time deflection multiplier specified in the ACI Code is simple to use, but neglects the effects of several parameters such as age at loading and reinforcement ratio that are known to have significant effects on long-time slab deflections. While several recommendations have been made in the past to increase the long-time multiplier for application to two-way slabs, no systemic study has been found that would provide a sound theoretical basis to justify such recommendations.
1.3. Objectives and Scope
The objectives of the research are to evaluate the limitations of the existing ACI 318-11 long-time multiplier (λΔ) used in calculating two-way slab deflections and to investigate the factors that influence long-time deflections from a theoretical viewpoint at the member section level.
The objectives were achieved within the following scope:
- A literature review was conducted on the current state of research on longtime multipliers.
- A method of analysis based on the principle of creep superposition was developed to evaluate time-dependent curvature under sustained load.
- A parametric study was conducted to determine the sensitivity of the calculated long-time strain in an axially loaded prism, and curvature in a flexural member, both uncracked and cracked, to various parameters including reinforcement ratio and age at loading.
- Implications for design of axial members as well as flexural members were made based on the results of the parametric study to aid engineers with long-term serviceability design.
An algorithm was developed for time-stepping analysis at the sectional level of an axially loaded prism under constant load based on the principle of creep superposition, and maintaining equilibrium on the section. Creep superposition elements of the algorithm were used in an algorithm developed for time-stepping analysis of a layered beam model of a flexural section under constant moment. Equilibrium is maintained in the section assuming a linear variation of the strain with depth of the section. A compliance function was used to predict strains in each layer at the end of the time step assuming constant stress throughout the time step. Equilibrium was established in the section at each time step to determine new stress increments in the steel and concrete. A long-time multiplier value was determined by dividing the time-dependent curvature at five years by the initial curvature at loading.
The developed algorithms were used to conduct a parametric study examining the effects of various parameters on the long-time strain in an axially loaded prism section, and curvature in an uncracked and cracked flexural section. The uncracked and cracked section act as the upper and lower bounds of flexural stiffness and play a large role on the value of the long-term multiplier. Each section type was examined for a range of parameters including reinforcement ratio, ρ = 0.18 – 1.5%, age of loading, t0 = 3 – 56 days, concrete strength, f’c = 3000 – 8000 psi, and ultimate creep coefficient, Cu = 1.3 – 4.15. Design implications based on results of the study are examined, including ranges of applicability for the existing long-time multiplier and potential approaches to develop expressions for the long-time multiplier including parameters such as reinforcement ratio.
