Digest 2007, December 2007 1115-1127
Elasticity Theory of Concrete and Prediction of Static E-Modulus for Dam Concrete Using Composite Models†1 İlker Bekir TOPÇU* Ali UĞURLU** ABSTRACT In this study, elastic theory of concrete and estimation of static E-modulus for composite models of dam concrete is investigated. Keeping in mind the nature of concrete as a composite material, consisting of several phases, a theoretical evaluation of the elasticity modulus for concrete is attempted. Subsequently theoretical results are supported and adjusted by empirical formulas and composite modeling to attain a practical model for calculation of E-moduli for concretes. Although concrete’s elastic behavior can be correctly exemplified by composite modeling up to a certain rate of internal stresses, it is seen that such equations are not totally reliable for all cases. Especially in dam concretes, keeping in mind the difficulty of carrying out elasticity tests, the investigation focuses on the possibility of estimating elasticity moduli through composite modeling especially to attain more reliable results, also keeping in mind the uncertainties imposed by previously used “wet screening” methods, other factors that influenced elasticity moduli in bulk concretes were kept in mind to reach more pertinent results. 1. INTRODUCTION E-modulus of concrete should be determined in order to decide on rates of strain and deformation in structural designs based on elasticity. Concrete may strain under various loads due to its flexural nature. In other words, strain depends on the type and size of the structure and loading period. E-modulus of concrete is usually determined with φ150x300 mm cylindrical specimens tested under various load levels within elastic limits as to standards. It is not possible to conduct mechanical tests such as compressive strength on mass concrete owing to the larger grain size it has. Much larger test equipments and press with a much higher capacity are required for these tests. Such equipment are only available in a few test laboratories. Thus, some method have been developed to conduct mechanical tests on mass concrete. The wet-screening is the most widely used method among these methods. In this method aggregates larger than 38 (or 32) mm are removed from fresh concrete by wet sieving and the remaining part is used to prepare specimens for experiments. In practice, this method has been criticised due to reasons such as disruption of specimen homogeneity, alteration * Eskişehir Osmangazi University, Eskişehir, Turkey ** General Directorate of State Hydraulic Works, Ankara, Turkey -
[email protected] † Published in Teknik Dergi Vol. 18, No. 1 January 2007, pp: 4055-4067
Elasticity Theory of Concrete and Prediction of Static E-Modulus … of phase balances, and total area shrinkage of cement paste-aggregate interface. And today, the results of this method are still dubious for many scientists. In this study experimental results were obtained in order to minimize the negative effects of these factors, which will be discussed in detail. As it is known, concrete is a three-phased composite consists of aggregate, matrix and aggregate-cement paste interface; therefore experimental data were reviewed vis a vis composite theories throughout this work. 2. ELASTIC THEORY OF CONCRETE Concrete is defined as a three-phased anisotropic brittle material that behaves differently under various loads. Total deformation of a structural element that has an elastic property under P load is directly proportional with applied load and size of the concrete component, but is inversely proportional with cross sectional area of the element [1]. This situation can be expressed as ∆L ∝ PL/A, or ∆L/L ∝ P/A. Concrete is a brittle and a composite material consisting of various phases. However, it has elastic behavior under low stresses. Theoretically, this is equal to a value of 30-40 % of its compressive strength [1]. Therefore, concrete is considered as an elastic material in engineering calculations [2,3]. The σ−ε diagrams are used for explaining elastic behavior of concrete determined by experimental methods.
σ M K
A ο
εκ
ε
Fig. 1. Typical σ−ε diagram of concrete. A typical σ−ε behavior of concrete is shown in Figure 1. It is seen that, initially as the stress increases deformation also increases in direct proportion. Diagram is linear up to point A and permanent deformation doesn’t occur when this short term elastic load is removed. The curve deviates from linearity with the application of a greater stress on specimen at point A, and this situation continues with the increase of curvature up to point M. Micro cracks rapidly occur with increasing stress after point A and concrete yields at point M, so the failure or fracture occurs at point K. The σk and εk demonstrates the fractural stress and unit deformation at point K, respectively. Static E-modulus of concrete 1116
İlker Bekir TOPÇU, Ali UĞURLU is calculated with σ−ε curve of experimental results. For this purpose, various methods can be used.
2.1 Methods on calculation of E-modulus of concrete The experimental method mentioned above is used for calculating static E-modulus of concrete. In this method, σ−ε curve is formed by conducting tests on specimens as to TS 3502 and ASTMC 469, moreover static E-modulus is calculated with various methods applied to the curve [4,5]. The empirical equations were developed by Turkish Institution of Standards, European Committee of Concrete, American Concrete Institute for calculating E-modulus of concrete in respect to unit weight and compressive strength. In Turkey, TS 500 specifies rules for calculation and design of reinforced concrete structures [6]. Static E-modulus is calculated with equation (1). European Concrete Committee calculates E-modulus from compressive strength as defined by equation (2) [7]. American Concrete Institute also developed an empirical equation (3), which uses compressive strength together with unit weight for calculating static E-modulus of concrete [8]. E represents the E-modulus calculated with the method of secant and corresponded to 40 % of concrete compressive strength; W is the unit weight of conventional concrete, σ is the compressive strength of standard cylindrical specimens. E=14000+3250σ1/2 E=9500(σ+8)
1/3
3/2 1/2
E=0.043W σ
(1) (2) (3)
As it is seen, E-modulus of concrete can be calculated with empirical equations. These equations give similar results [1]. Apart from empirical equations developed by the mentioned institutions, E-modulus of concrete can also be calculated by sonic or ultrasonic methods. Therefore another method called dynamic E-modulus is based on the principle of transition of ultrasound through concrete. In this method transition speed of ultrasound through concrete is calculated and, then empirical equations are involved for calculating dynamic E-modulus. E-modulus calculated with dynamic method is a little larger than Emodulus calculated with the static methods. This situation is related to the fact that load isn’t applied to the specimen during experiment. Static E-modulus which is the most similar to E-modulus calculated by dynamic methods is obtained by the initial tangent method. However, in all of these approaches factors such as concrete age, E-modulus and size of aggregate, which are highly important for correct determination E-modulus, are neglected. Thus, E-modulus calculated by the mentioned equations never represents the actual Emodulus value. 2.2 Factors affecting E-modulus of concrete Microstructure of concrete that has a heterogeneous structure, exhibits different behaviors during loading due to various phases such as aggregate, cement paste matrix, various 1117
Elasticity Theory of Concrete and Prediction of Static E-Modulus … cellular systems, aggregate-cement paste interface [9]. Thus, a change in quantity or quality of one of these may result in very different consequences than expected. For example, elasticity values of concretes, which have the same compressive strength but different combination properties and components, may be different. As shown in Figure 2, stressstrain behaviors of concrete and its components are highly different [2]. The compressive strength of aggregate and hardened cement paste, which is among the phases within concrete, is higher than the over all strength of concrete and mortar. However, concrete exhibits more ductile behavior than its own components. In other words, inelastic behavior of concrete is higher than other phases and components. The deformation of concrete is higher than deformation of phase and components for an equivalent compressive strength as seen in figures. All of these facts determine material’s behavior during fracture, its stiffness, E-modulus and deformation capacity. Factors affecting E-modulus can be classified as in Figure 3. Although all of these basic elements shown in Figure 3 are important, but aggregate porosity and E-modulus of aggregate, properties of cement paste matrix, cement paste-aggregate transition area and experimental parameters are the most important factors.
σ
Coarse aggregate
Hardened cement paste
Concrete Mortar
ε Fig. 2. The σ−ε diagram of concrete and its components. Porosity and E-modulus of aggregates are important elements in determination of stiffness. Thus, E-moduli of non-porous aggregates with high density and concrete consisting of such aggregates are extremely high. E-modulus and properties of cement paste are completely determined by the water/cement ratio. E-modulus of mortar increases with the decrease of the water/cement ratio. This situation rises with the increase of mortar phase/E-modulus ratio of coarse aggregate. This occurs as a decrease on stress in the adherence area of cement paste and coarse aggregate. Cement paste porosity coming into existence due to various reasons plays an important role in influencing concrete E-modulus. Cement paste and aggregate transition area is also an important parameter for concrete E-modulus. In recent years, SEM (scanning electron microscope) micro structural analysis showed that air voids, micro cracks and diffused calcium hydroxide crystals have an effect on fracture behavior and elastic properties of concrete. Properties of aggregate and cement paste play 1118
İlker Bekir TOPÇU, Ali UĞURLU an important role in this area. Tensile and shear strengths in aggregates and cement paste adherence area increase during loading due to different E-moduli of these two different phases. This situation may result in large cracks and fractures.
Factors affecting the elasticity modulus
Loading conditions and moisture content of specimens
E-modulus of cement paste matrix
Porosity and structure of aggregatecement paste transition zone
E-modulus of Aggregate
Aggregate volumetric fraction
Porosity
Experimental parameters
Cement paste matrix
Transition zone
Porosity
Aggregate
Fig. 3. Classification of factors affecting E-modulus of concrete.
3. ESTIMATION OF E-MODULUS FOR MASS CONCRETE To calculate E-modulus in bulk concrete structures such as in concrete dams, structural design is needed for estimating strain and deformation. Technical and economical factors are considered in designing immense structures that are not encountered in building types of our daily life. Concrete specimens are usually produced with maximum grain size of 80200 mm, and the fact that there are serious concerns about mold sizes, transportation of specimens, experimental mechanisms and supply of high capacity equipments for tests. Specimens are obtained from dam or other bulk concrete structures with molds of 450x900 mm to perform on them compressive strength tests. In order to conduct tests upon specimens in such sizes, high capacity compression presses and equipments to are required for calculating deformation and strength. It is almost impossible to provide the mentioned technical conditions in many constructions and laboratories. Thus, E-modulus have been developed and tested for determination and estimation of bulk concrete considering various approaches. The common method among these is based on testing specimens obtained from remaining concrete that is obtained by sieving fresh concrete with a 38 mm screen, and is called the wet screen method. In this method the maximum grain size of the material sieved from fresh concrete is reduced and a specimen is prepared by using cylindrical molds in the size of φ150x300 mm, and tests are conducted on these specimens. However, in this 1119
Elasticity Theory of Concrete and Prediction of Static E-Modulus … process mortar/coarse aggregate ratio of the remaining concrete after sieving increase due to removal of coarse aggregates, coarse aggregate concentration in total volume decreases, water/cement ratio decreases, total area and length of coarse aggregate-cement paste adherence surface decreases. These factors considerably affect parameters of fracture mechanism and fractural strain during cracking of wet screened concrete compared to the conventional concrete. The decrease in total of coarse aggregate-cement paste adherence length in a certain volume is evaluated either as the decrease in weakness in microstructure of concrete or as strengthened micro structure against stress. Furthermore, adherence ruptures emerging due to increased forces in coarse aggregate surfaces at a certain level of loading decrease and as a result, cracking decreases. Porosity decreases as the water/cement ratio decreases. Briefly, all of these factors affect stress-strain behavior of concrete and enable to attain a more ductile behavior and a lower E-modulus. However, higher compressive strength and lower E-modulus are obtained by the wet screen method. Investigations indicate that it is more suitable to consider 85 % of compressive strength obtained through wet screened concrete [10-12]. As a result estimation of E-modulus becomes more difficult. The decision is based on the following assumption; the ratio of concrete components will change as coarse aggregates bigger than 38 mm are removed by wet screening; and hardened concrete properties will change as a result of increasing mortar/aggregate ratio. It is very complicated to estimate behavior of concrete that is anisotropic and two-phase composite during loading. This is directly related to parameters in determining concrete behavior and effects of parameters on properties of each other. For example, E-modulus of two concretes, whose compressive strengths are the same but component properties are different, may not be equal. It has been realized in recent years that aggregate-mortar interface is as important as mortar and aggregate for fracture or strain. In other words, deformation index of concrete, whose mortar strength is higher than aggregate strength, cannot be the same with concrete, whose aggregate strength is higher than mortar strength. In both situations, function of interface in concrete under load during fracture is different. In first situation, function of adherence forces is less important than the second situation [13]. As it is concluded from the explanations, there is a difference between E-modulus obtained by wet screening and E-modulus determined by conventional specimens. Thus, the results should be confirmed or tested with different approaches. Classic empirical formulae enabling estimation of E-modulus in parallel with compressive strength has been mentioned. It is considered that age of bulk concrete, coarse aggregate and specimen size which are important for E-modulus are neglected in related equations [14]. The results obtained through tests or wet screen method should be reevaluated with models and theories, where parallel, serial and both phases are used and which are based on the assumption that concrete is a two-phase composite consisting of aggregate and cement paste matrix and approaches towards E-modulus estimation for mass concrete can be tested [13]. Another important point that shouldn’t be overlooked is the fact that all of these models are based on the acceptance that concrete is a two-phase composite consisting of aggregate and mortar. Unfortunately, this acceptance should be reevaluated under today’s conditions as it doesn’t consider aggregate-mortar interface, whose existence has been understood better with SEM in recent years and that has a very important function in influencing behavior of concrete under load. 1120
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3.1 Calculation of E-modulus with composite models Simple composite models can be applied in determining E-modulus of concrete which assumes concrete roughly as a composite material consisting of cement paste and aggregate. Basic composite models are applied in parallel and serial phases to explain elastic behavior of two-phased materials as shown in Figure 4 [15,16]. In composite models, it is accepted that concrete is a three-dimensional combination of two homogeneous and isotropic phases such as matrix and coarse aggregate. The two phases behave linearly in the linear elastic zone of concrete [17]. Furthermore, concrete mixture ratio, unit weight or volume ratio of aggregate and E-modulus of each phase should also be known. The various composite models on two-phase concrete are shown in Figure 4. Eb=EchVch + EaVa
(4)
1/Eb=Vch / Ech + Va / Ea
(5)
1/Eb=(1-x) [C/Ea+1-C/Ech] + [1/C Ea+(1-C)Ech]
(6)
The most common composite models are Voight’s parallel model and Reuss’ serial model. In Voight’s parallel and Reuss’ serial models, equations (4) and (5) are used for calculating E-modulus with a constant strain and strength on composite materials, respectively. Emodulus of concrete or composite occurs as a function of cement paste, aggregate, volumetric quantity of these within concrete as shown in equations. In the Hirsch model, serial and parallel phases are equally proportioned, in assuming concrete as being a twophased material as shown in equation (6). E-modulus is based on the assumption that concrete is a two-phase material and volumetric function of aggregate and matrix phase and empirical constant have an effect upon the E-modulus [2,5].
Fig. 4. Composite models.
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4. EXAMPLES FOR E-MODULUS CALCULATIONS WITH COMPOSITE MODELS It is seen that E-modulus has only been predicted in extant studies [18]. The microstructure and behavior of concrete couldn’t be explained completely in the past, but recently the composite equations based on unit cell modeling could be rewritten by practical studies. Villardell attained various results that clarify some issues in his study as shown in Table 1 [17]. In this study, bulk concrete was produced with the maximum grain size of 120 mm, and then wet screened concrete was obtained by sieving fresh concrete with a 38 mm screen. This concrete was also sieved with screen # 4, and mortar specimens were prepared at the end of this process. According to experimental results, E-modulus increases as the age of specimen increases as shown in Table 1. Nevertheless, E-modulus of unscreened concrete known as bulk concrete is higher than wet screened concrete and mortar specimens. The increasing arrange of E-modulus is unscreened, wet screened concrete and mortar respectively. As it is seen, results are related to E-modulus of aggregate. E-modulus increases with increasing E-modulus of aggregate. This is valid for the situations where Emodulus of aggregate is higher than E-modulus of cement paste. The volumetric ratio of a phase which has a higher E-modulus is also an important parameter. Table 1. Variation of E-modulus according to cure periods Cure period
Mortar
E-Modulus, GPa Wet screened concrete
Mass concrete (unscreened)
7 28 90 180
19.6 23.8 28.2 30.7
24.8 34.5 35.1 37.2
30.3 37.3 43.0 42.2
4.1 Experimental study This study was carried out in order to evaluate and confirm theoretical synthesis in practice, and based on the comparison between actual values of elasticity modulus calculated in practice and the results obtained from model equations regarding composite materials. To this aim, experimental results and mixture parameters of Villardell were used to create data for model equations, and static E-modulus was calculated by using composite model equations [17]. Villardell obtained core specimens consisting of stone at a height/diameter ratio of 2, to use as aggregate and calculate elasticity modulus individually, and these values are determined as 35, 50, 60 GPa, respectively. Stones were categorized in 5 different grain groups of (0-1.25), (1.25-5), (5-19), (19-60) and (60-120) mm to be crushed in a rock crusher and grinded. In concrete mixtures, 130 kg Type I 45A (CEN Class I 42.5R) cement, 89 kg fly ash, 398 kg sand (0-1.25 mm), 234 kg sand (1.25-5 mm), 392 kg crushed stone (5-19 mm), 646 kg crushed stone, (19-60 mm), 558 kg crushed stone (60-120 mm), 0.55 plasticizer 0.55 liter and 45 kg mixture water were used. Aggregates mainly 1122
İlker Bekir TOPÇU, Ali UĞURLU consisting of limestone were obtained from River of Segre around the dam. In addition to this, mortars without crushed stone that has a maximum grain size of 5 mm, were also produced in experiments. Initially, concrete mixtures were produced with aggregates that have a maximum grain size of 120 mm. Fresh bulk concrete was sieved with 38 mm screen by wet screening method to prepare prismatic and cylindrical specimens in 450x450x450 mm and φ150x300 mm molds which has the maximum grain size of 38 mm, and furthermore the same process was performed before initial setting by screen # 4 in φ150x300 mm molds. Specimens were then demolded on the following day and kept in a cure bath at 23±2 oC for 90 days. Afterwards, specimens in form of saturated surface dry were then subjected to experimental tests. Strain of each cylindrical specimen was calculated with 3 strain-gages located in the middle of vertical axis of the specimen. Table 2. Volumetric fractions of mortar and aggregate in calculations Code M WS
Vmortar
Vaggregate
0.555
0.445
0.405
0.595
0.725 0.275 D M: Mortar, WS: Wet screened concrete, D: Dam(mass) concrete
Hirsch-Dougill, (x=0.3)
Hirsch-Dougill, (x=0.5)
Hirsch-Dougill, (x=0.8)
Popovics
Illston
Mehmel-Kern
Counto
Hashin-Hansen
Hobbs
Maxwell
Bache -NepperChristensen
31 32 35 37 41 39 44 50 43
Serial Model
35
Parallel Model
Experimental result 28 33 35 32 43 39 37 48 43
Ea GPa
M D WS M 90 50 D WS M 65 D WS
Cure period
Code
Table 3.E-modulus of mortars, wet screened and dam concretes calculated with composite models
30 31 35 34 38 38 37 42 40
30 31 35 35 38 38 39 44 41
31 32 35 36 39 38 40 46 41
31 32 35 37 40 39 43 48 42
15 16 17 18 20 19 22 25 21
31 32 35 37 40 39 42 47 42
31 32 35 36 39 38 40 46 41
31 32 35 36 39 38 40 46 41
31 32 35 36 39 38 40 45 41
31 32 35 36 39 38 40 45 41
31 32 35 36 39 38 41 46 41
31 32 35 36 39 38 40 46 41
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The strain-gages used in tests were 30 mm in length for mortar and 120 mm for concrete. In prismatic specimens, strain-gages demonstrating reference points were attached to both vertical surfaces and strains between adjacent discs were manually calculated with DEMEC type mechanical extensometer having 15 cm gage length. Strains in strain-gages are measured with a computer controlled data collection system. Specimens were loaded under uniaxial compression of 4.5 MN at a servo hydraulic press controlled by MTS 458 closed circuit. Loading velocity of piston was fixed at 0.004 mm/s for cylindrical specimens and 0.012 mm/s for prismatic specimens to ensure the same axial strain. It was observed that fracture of cylindrical specimens occurred approximately 4 minutes later. The specimens were loaded until fracture. E-modulus was calculated by marking two points on the curve corresponding to 30 % of fractural strength and from the slope of linear line through these two points. Stress-strain curves of the results were analyzed with mathematical equations in computer and transformed into numeric. The 14 different elasticity composite models were calculated with the equations by using volumetric ratio in Table 2, and presented together with experimental results in Table 3. In practical tests conducted after 90 days, it was realized that E-modulus obtained from mortar specimens gave the lowest result considering 3 different E-moduli of aggregates. Emodulus of concrete calculated with the wet screen method was higher than E-modulus of mortar and lower than E-modulus of bulk concrete as shown in Table 3. In other words, order is as Emass
