Dimensional stability and durability of reinforced concrete frames under sustained loads
Abstract
This project is aimed at predicting the performance of the reinforced concrete frame, at service, accounting for different parameters that occur due to the aging of the structure. The analysis of the frame is done by the stiffness matrix method and the design of the frame is done in accordance with IS 456:2000. The concrete mix design is done by following the procedure given in IS 10262. Creep and Shrinkage are two important characteristics of concrete wich cause the timedependent dimensional changes in reinforced concrete. Corrosion of reinforcement is a prime aspect that affects the durability of the concrete. Prediction models for creep and shrinkage were adopted from ACI 209R92 and IS 456:2000. Carbonation mechanism is adopted for depassivation, during the corrosion of rebars. The combined effect of these parameters is considered to find out the deformation, durability, loss in strength and stiffness of the structure. Considering the abovementioned parameters, the performance of the frame structure is analyzed, by developing MATLAB codes. Applying loading conditions on the structure and providing the dimensions of the frame, as input to the software, maximum bending moment, maximum shear force, design aspects for reinforced concrete, total strain and total deformation, strain and deformation due to creep and shrinkage, durability of the structure can be computed. These simulated outcomes which have been predicted, based on different codal provisions(IS 456:2000, ACI 209R92) are then compared with a model developed by linear regression, of the experimental database available in the literature. All these predictions are useful in determining the age of the concrete structure and also to know which codal provision predictions are nearer to the actual experimental data.
Abbreviations
ACI  American Concrete Institute 
IS Code  Indian Standard Code 
INTRODUCTION
Background
Concrete exhibits timedependent strains due to creep and shrinkage. Creep and shrinkage have a considerable impact upon the performance of concrete structures, causing increased deflections as well as affecting stress distribution. Creep in concrete represents dimensional change in the material under the influence of sustained loading. In addition, whether subjected to load or not, concrete contracts on drying undergoes shrinkage. In the case of many actual structures, creep and shrinkage occur simultaneously and the treatment of the two together is often convenient from a practical point of view. Failure to include creep and shrinkage effects in the analysis of the structures may lead to excessive deformation and wide spread cracking.
In general, timedependent deformations of concrete may severely affect the serviceability, durability, and stability of the structure. Calculation of creep and shrinkage deformations is difficult. Various researchers have carried out extensive work to quantify and predict the effects of creep and shrinkage. The experimental studies have been conducted in specific environments and for different durations varying from 7 to 5,000 days. Therefore, the results are not in close agreement with each other, though they satisfied the specific conditions in which the tests were carried out. Nevertheless, due to this extensive research work, extensive progress have been achieved in the knowledge about creep and shrinkage of concrete.
Various prediction models have been developed to predict the creep and shrinkage in concrete^{(1, 2, 3, 4 )}. Experimental studies to measure the creep and shrinkage in concrete have also been carried out by researchers and reported in the literature. Creep and shrinkage have been predicted up to 5,000 days of observation by the ACI209R82 model, the B3 model, the CEBFIP model code 1990, and the GL2000 model for various grades of concrete (^{5}).
Objectives of the research
This project is aimed at computing the shrinkage and creep of concrete, by using different type of prediction models. Factors that influence the behaviour of the shrunk and creep affected reinforced concrete members are discussed. Shrinkage and creep influence on shortterm behaviour of reinforced concrete structures, has been investigated numerically and compared with test data reported in the literature.
LITERATURE REVIEW
Creep
The state of stresses due to sustained load, promotes strains in concrete members which progresses over time, characterizing the phenomenon known as creep. Such kind of deformations results, particularly, from the viscous behaviour associated to the adsorbed water layer at the cement grains surface, in the hardened concrete (McGREGOR, 1997).
The creep deformations are more pronounced in the first months of the structure lifetime, when develop under higher rates. It can extend for periods of time up to ten years, although in more advanced stages, it evolves over modest rates. There are cases in which the creep deformations can take magnitudes greater than three times the deformation at the instant of loading, inducing structural displacements of similar magnitude.
If a sustained load is removed after some time, the strain decreases immediately by an amount equal to the elastic strain. This strain is generally smaller than the initial elastic strain because of the increase in the modulus of elasticity with age. The instantaneous recovery is followed by a gradual decrease in strain, called creep recovery.
Shrinkage
Shrinkage is caused by loss of water by evaporation or by hydration of cement, and also by carbonation. The reduction in volume, i.e. volumetric strain, is equal to 3 times the linear contraction, and in practice we measure shrinkage simply as a linear strain. Its units are thus mm per mm usually expressed in 10^{6}.
Factors affecting shrinkage of concrete
Corrosion
Iron(Fe) does not exist in nature in isolated form, rather as oxides, sulphides (pyrites), etc., in ores. Considerable energy is expected in extracting Fe from ore thus chemical potential of Fe is higher. Hence Fe has tendency to react and attain a stable state.
Reaction of Fe with its environment causes loss of material from structural form where Fe exists. Loss of metal (deterioration) leading to impediment to functional performance is Corrosion in engineering sense. When functional performance remains unaffected by reaction, it may not be corrosion in Engineering sense.
Presence of both oxygen and moisture is necessary for corrosion reaction to progress. Completely dry or fully saturated concrete do not exhibit significant rebar corrosion. Being an electrochemical process resistivity of concrete is also important.
Depassivation is generally caused by either a) CARBONATION or b) FREE CHLORIDE. In this project carbonation mechanism is used for the depassivation process, inorder to predict the durability of the reinforced concrete structure.
Passive and Active corrosion
Passive condition refers to inert condition when corrosion reaction is slow or negligible. At high pH, oxides formed, tend to form a passive layer and protects the rebar. At lower pH or at high concentrations of chloride the passive layer is destroyed.
Carbonation Process
Carbondioxide from the atmosphere penetrates into the surface(through the pores) and in presence of moisture, the following reactions take place:
Since the alkali matter present in the concrete is being consumed in the above reactions, the pH of the concrete is reduced from 12.5 to a value < 9. When pH < 9, depassivation can occur.
Carbonation neutralisation Depth  The depth upto which all the alkali matter is consumed in the concrete, from the concrete surface.
Carbonation proceeds maximum when relative humidity(R_{H}) value is between 50% and 60%.
Carbonation Model
Initially the concentration of carbondioxide is more in the atmosphere, when compared to the concentration of carbondioxide in the concrete. Due to difference in the concentrations of carbondioxide, diffusion takes place from high concentration to low concentration. Osmotic pressure is the driving force behind diffusion. This diffusion process is governed by Fick's diffusion law.
According to Fick's diffusion law,
Where, q_{flux}  diffusion flux (kg/m^{2}sec),
D  diffusion gradient,
C_{CO}_{2}  concentration of CO_{2}.
Assume concrete has got carbonated upto x, and concentration in atmosphere as C_{0} and C_{c} in the concrete.
Applying mass balance,
a  CO_{2} consumed by alkali material in concrete per unit volume.
x  carbonation neutralisation depth.
where c_{env} = 1 or 0.5 (rain);
c_{air} = 1 or 0.7 (air entrained);
a,b  constants;
f_{m}  strength of the concrete;
f_{m} = f_{ck} + 8;
Binder  a  b 
OPC  1800  1.7 
OPC+flyash(28%)  360  1.2 
OPC+silicafume(9%)  400  1.2 
OPC+blast furnace slag(70%)  360  1.2 
ACI Model (for predicting Creep and Shrinkage)
Prediction of Creep
Prediction model, ^{3} is appropriate to normal weight of concrete subjected to a constant stress and stored under normal constant environmental conditions. It is not preferable for other loadings, storage environments.
Creep coefficient $\varnothing(t,t_0)$ is expressed as a function of time.
where $\varnothing(t,t_0)$is ratio of creep c(t,t_{0}_{}) at age t due to a unit stress applied at the age t_{0} to the initial elastic strain under a unit stress applied at t_{0}; age is measured in days. Since the initial elastic strain under a unit stress is equal to the reciprocal of the modulus of elasticity E_{c}(t_{0}), we can write
In Eq. (1), (tt_{0}) is the time since application of load and Φ_{∞}(t,t_{0}) is the ultimate creep coefficient, which is given by
For ages at application of load greater than 7 days for moist curing, or greater than 1 to 3 days for steam curing, the coefficient k_{1} is estimated from:
The coefficient k_{2} is dependent upon the relative humidity h (per cent):
The coefficient k_{3} allows for member size in terms of the volume/surface ratio, V/S, which is defined as the ratio of the crosssectional area to the perimeter exposed to drying. For values of V/S smaller than 37.5mm, k3 is given in Table 1. When V/S is between 37.5 and 95mm, k_{3} is given by:
When V/S ≥ 95mm
Period of moist curing(days)  Shrinkage Coefficient k_{1}' 
1  1.2 
3  1.1 
7  1.0 
14  0.93 
28  0.86 
90  0.75 
The coefficients to allow for the composition of the concrete are k_{4}, k_{5} and k_{6}. Coefficient k_{4} is given by:
where s = slump (mm) of the fresh concrete.
Coefficient k_{5} depends on the fine aggregate/total aggregate ratio, A_{f}/A, in per cent and is given by
Coefficient k_{6} depends upon the air content 'a' (per cent)
Prediction of Shrinkage
Shrinkage s_{h}(t,τ_{0}) at time t (days), measured from the start of drying at τ_{0 }(days), is expressed as follows:
for moist curing:
where s_{h∞} = ultimate shrinkage, and
For curing time different from 7 days for moistcured concrete, the age coefficient k_{1}' is given im Table 2; for steam curing with a period of 1 to 3 days, k_{1}' = 1.
Volume/Surface ratio, V/S mm  Coefficient k_{3}^{'} 
12.5  1.35 
19  1.25 
25  1.17 
31  1.08 
37.5  1.00 
The humidity coefficient k_{2}' is
where h = relative humidity (per cent)
Since k_{2}^{'} = 0 when h = 100 per cent, the ACI method does not predict swelling.
Coefficient k_{3}^{'} allows for the size of the member in terms of the volume/surface ratio V/S. For values of V/S < 37.5 mm k_{3}^{'} is given in Table 3. When V/S is between 37.5 and 95 mm :
for (tτ_{0}) ≤ 1 year
for (tτ_{0}) ≥ 1 year
When V/S ≥ 95 mm
The coefficients which allow for the composition are :
Where s = slump of fresh concrete (mm) and
where A_{f}/A = fine aggregate/total aggregate ratio by mass (per cent). Also,
Where Ɣ = cement content (kg/m^{3})
where A = air content (per cent)
IS Code creep and shrinkage Model(IS 456:2000 Annex C)
The total deflection shall be taken as the sum of the shortterm deflection and the longterm deflection. The shortterm deflection may be calculated by the usual methods for elastic deflections using the shortterm modulus of elasticity of concrete, E_{c} and an effective moment of inertia I_{eff}. The long term deflection is calculated by considering the effects of creep and shrinkage.^{4}
Deflection due to Shrinkage
The deflection due to shrinkage a_{cs} is calculated by using the fiollowing formula :
where
k_{3} is a constant depending upon the support conditions,
0.5 for cantilever,
0.125 for simply supported beam,
0.086 for members continuous at one end,
0.063 for fully continuous members.
Ψ_{cs} is shrinkage curvature equal to k_{4}ε_{cs}/D.
where, ε_{cs} is the ultimate shrinkage strain of concrete.
where P_{t} = 100A_{st}/bd and P_{c} = 100A_{sc}/bd
and D is the total depth of the section, and l is the length of the span.
Deflection due to creep
The creep deflection due to permanent loads a_{cc(perm.)} may be obtained from the following equation:
a_{i,cc(perm.)} = initial plus creep deflection due to permanent loads obtained using an elastic analysis with an effective modulus of elastcity,
E_{ce}_{ } = E_{c}/(1+θ) ; θ being the creep coefficient, and
a_{i(perm.)} = shortterm deflection due to permanent load using E_{c}.
Age of Loading  Creep Coefficient 
7 days  2.2 
28 days  1.6 
1 year  1.1 
METHODOLOGY
Stiffness matrix method
It is one of the two matrix methods, used for analysis of a member. Stiffness matrix method is also known as displacement matrix method. The other matrix method for analysis is Flexibility matrix method. Stiffness matrix and Flexibility matrix are converse of each other. Flexibility is defined as the displacement caused by a unit force, and the stiffness is defined as the force required for a unit displacement.
Of the two methods, the matrix displacement method of analysis is commonly preferred particularly where the degree of static indeterminacy is high. The stiffness method aims at solving for unknown joint equations at the joints. The steps involved in the displacement method of analysis are presented in the following section.
Steps to be Followed^{5}
 As a first step the degree of freedom or kinematic indeterminacy of the structure is determined. The coordinates for the structure are established identifying the location and direction of joint displacements. Restraining forces are applied at the coordinates to prevent joint displacements. In this method there is no choice exercised in the section of joint displacement.
 The restraining forces are determined as a sum of fixed end forces for the members meeting at the joint.
 Then the forces required to hold the restrained structure with a unit displacement at one of the coordinates only, and with zero displacements at all other coordinates, are determined. This is done for all the coordinates one by one and the forces required are determined. These forces from the elements of the stiffness matrix [K].
 The values of the displacements {D} necessary to ensure the equilibrium of the joints are determined using the relation
in which
{P} = restraining forces at joints,
{K} = stiffness matrix corresponding to the coordinates,
and {D} = unknown displacements at the coordinates.
Displacements are obtained by solving Equation 22.
 The forces in the given structure are obtained by adding the forces on the restrained structure and the forces caused by the joint displacements found above.
 The end moments are determined by using the joint displacements found above. Finally bending moment profile and shear force profile on the structural member are obtained.
In the proposed analysis, the following assumptions have been taken into account:
Assumption :
 All members in the frame have uniform crosssection, and there is no initial imperfection in other geometric properties like straightness.
 Connection dimension are assumed to be negligible compared to the length of the other structural parts. It means rotational spring is considered as a point at the end of each element where required.
 The effect of eccentricity at joints is neglected.
 Stiffness matrix is modified considering only the behavior of the momentrotation curve. Shear and axial deformation in connection are ignored.
 It has been assumed that member displays linear elastic behavior and connections display linear or nonlinear forcedeformation behavior.
 Only static loading is considered with sway behavior of the frame structure.
Frame Structure : The structure having the combination of beam and column, to resist the lateral and gravity loads. The beam carries loads and transfers it to the column. Then column transfers it to the base of the structure.
Design of Reinforced Frames
Different parameters like maximum bending moment, maximum shear force and support reactions are obtained from the analysis of the frame structure, by using stiffness matrix method, mentioned above. These parameters are adopted for designing flexural and compressive members of a structure.
Assumptions :
 Maximum strain in bending compression at concrete is taken as 0.0035 or less than 0.0035, after which crushing takes place
 Strain in tension at ultimate is equal to or less than (f_{y}/1.15)/E_{s}+0.002 or 0.87f_{y}/E_{s}+0.002, where E_{s} is Young's modulus of concrete.^{10}
The design of the beams and columns is done by considering the above mentioned assumptions.
Prediction of creep and shrinkage
Creep and Shrikage are predicted by using IS 456:2000 and ACI 209R92, as prescribed in the literature review above and matlab codes are developed in order to obtain the creep and shrinkage deflections of the structural members at different time periods.
Prediction of durability by carbonation corrosion model
Carbonation mechanism is adopted for the depassivation of the concrete layer. When the carbonation neutralisation depth becomes equal to the clear cover thickness of the structural members, then the service life of the structural member is ended. So, by computing the relationship between the time and carbonation neutralisation depth, durability of concrete can be determined. This relationship between time and carbonation depth is obtained by using Fick's Law as metioned in the literature review.
RESULTS AND DISCUSSION
All the predictions in this project are obtained by developing the MATLAB codes. The following set of data is taken as the input, (i) The dimensions of the frame, (ii) loading conditions, (iii) climatic conditions, (iv) concrete characteristics, (v) curing conditions, (vi) curing time, etc. and the set of data obtained as an output is as follows:
 End moments,
 Bending moment diagram,
 Shear force diagram,
 Design considerations of reinforced concrete members like area of crosssection and area of steel,
 Creep strain and Shrinkage strain derived from the emperical relations given in IS code and ACI code,
 Total strain in a structural member taking into account the deflection due to loading, creep and shrinkage,
Example : The portal frame in fig 6, is considered as an example for computing the results from the MATLAB codes.
Assumed data for this example, Relative humidity = 65%; moist curing; no. of days of curing = 14 days, fck = 30MPa; fy = 415MPa.
The following results were obtained for the above mentioned example
Position  End Moment 
M_{12}  248.5882 
M_{21}  85.7647 
M_{23}  85.7647 
M_{32}  295.4118 
M_{34}  295.4118 
M_{43}  223.1765 
Member  Max. BM  Min. BM  Max. SF  Min. SF 
1  89.1059  248.5882  56.2824  43.7176 
2  191.7371  295.4118  193.7371  170. 9647 
3  223.1765  295.4118  30  20 
Members  Crosssection  Area of steel (mm^{2}) 
1  300 x 300  900 
2  558 x 392  2941.3 
3  300 x 300  900 
Time(days)  Creep Strain(ACI; per MPa)  Shrinkage(ACI; mm) 
28  3.65 x 10^{8}  5.84 x 10^{5} 
56  4.83 x 10^{8}  8.44 x 10^{5} 
112  5.29 x 10^{8}  1.06 x 10^{4} 
224  5.61 x 10^{8}  1.21 x 10^{4} 
365  5.78 x 10^{8}  1.27 x 10^{4} 
730  5.98 x 10^{8}  1.34 x 10^{4} 
1460  6.12 x 10^{8}  1.37 x 10^{4} 
2000  6.17 x 10^{8}  1.38 x 10^{4} 
5000  6.17 x 10^{8}  1.39 x 10^{4} 
From the creep strain and shrinkage data in table 8, the increase in creep strain and shrinkage is significant in the initial days and decreases drastically later and eventually reaches saturation at the time period of 2000 days.
CONCLUSION
This study is an attempt to use the programming aid in finding the dimensional stability of reinforced concrete. The strain of the structure accounting to different parameters like creep, shrinkage, elastic deformations and loading is computed.
FUTURE SCOPE
 The program aided predictions of creep and shrinkage can be done for other codal provisions, which are not included in this project, like the B3 model, the CEBFIP model code 1990, and the GL2000 model.
 All the predictions can be compared with each other, can be computed how close they are to each other.
 Predictions for creep and shrinkage in compressive members and on reinforced concrete walls(Shariff, et al ,2018).
 By using the creep and shrinkage database^{5} which is available in the literature, a linear regression model can be developed. And this model obtained from the experimental data can be then compared with the predictions of different codal provisions, which are obtained from the software simulations^{10}.
 The above mentioned comparisions allow us to know the accuracy of the creep and shrinkage predcitions of different codal provisions.
ACKNOWLEDGEMENTS
I would like to express my deep sense of gratitude and indebtedness to Prof. Ananth Ramaswamy for his sincere guidance, constant encouragement and all the necessary support despite his extremely busy schedule. I would also like to thank the Indian Academy of Sciences for providing me this opportunity to work in such a great environment in the Indian Institute of Science. Also I would like to thank Biswajith Pal for extending his support and encouragement throughout the course of this project.
I am also thankful to seniors working in the structural laboratory in the Civil Engineering Department for providing a solution to my problems. I especially thank my fellow interns, Ankit and Saswata Chattopadhyay for constant support throughout my project.
References

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Shariff, Najeeb & Menon, Devdas. (2018). Creep and shrinkage effects on reinforced concrete walls: Experimental study

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ACI 209R92 Prediction of Creep, Shrinkage, and Temperature Effects in Concrete Structures Reported by ACI Committee 209

IS 456:2000 PLAIN AND REINFORCED CONCRETE CODE OF PRACTICE

Mija H. Hubler, Roman Wendner, Zdeněk P. Bažant, 2015, Comprehensive Database for Concrete Creep and Shrinkage: Analysis and Recommendations for Testing and Recording, ACI Materials Journal, vol. 112, no. 4

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Source

Fig 1: A.M.Neville, Concret Technology 2nd Edition, J J Books

Fig 2: A.M.Neville, Concrete Technology 2nd Edition, J J Books

Fig 3: A.M.Niville, Concrete Technology 2nd edition, J J Books

Fig 4: A.M.Niville, Concrete Technology 2nd Edition, J J Books
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