Chapter. Materials. 1.1 Notations Used in This Chapter

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1 Chapter 1 Materials 1.1 Notations Used in This Chapter A Area of concrete cross-section C s Constant depending on the type of curing C t Creep coefficient (C t = ε sp /ε i ) C u Ultimate creep coefficient (on average C u = 2.35) D Diameter of cylinder for split test (Brazilian test) E c E s I L M P P sh Modulus of elasticity of concrete Modulus of elasticity of non-prestressed reinforcement Moment of inertia of section about centroidal axis Length of cylinder for split test (Brazilian test) Applied moment Applied concentrated load Correction factor for shrinkage strain

2 2 Chapter 1 Q cr T a b f c f c f r f s f sp f t f y h t T α T γ c ε ε c ε cp Correction factor for creep strain Tensile force Shear span, distance from application point of concentrated load to support Width of member Compressive stress in concrete Specified compressive strength of concrete Modulus of rupture of concrete Calculated tensile stress in reinforcement at specified loads Splitting tensile strength of concrete Concrete tensile stress due to applied loads Specified yield strength of non-prestressed reinforcement Overall thickness or height of member Time Temperature variation Coefficient of thermal expansion Density of concrete Normal strain Strain at the extreme concrete compression fibre Creep strain in concrete ε pic c Strain in concrete corresponding to f c ε cu Maximum strain at the extreme concrete compression fibre at ultimate (ε cu = ) ε i ε sh ε shu ε th λ ν σ Instantaneous elastic strain Shrinkage strain Ultimate shrinkage strain Thermal expansion strain Factor to account for low-density concrete (λ = 1 for normal-density concrete) Poisson s ratio Effective normal stress

3 Materials Concrete Concrete is a material obtained by hardening a mixture of aggregates (sand, gravel), hydraulic lime (cement), water, and additives (such as entrained air) in pre-determined proportions. Concretes are classified according to their density γ c as follows: low-density concrete with γ c 1850 kg/m 3 semi-low-density concrete with 1850 kg/m 3 < γ c 2150 kg/m 3 normal-density concrete with 2150 kg/m 3 < γ c 2500 kg/m 3 high-density concrete with 2500 kg/m 3 < γ c In addition to its density, concrete is characterized by: its mechanical properties: compressive strength f c and tensile strength f t, its elastic properties: modulus of elasticity E c, ultimate strain ε cu, and Poisson s ratio ν, its volumetric change properties: thermal expansion α T, creep strain ε cp, and shrinkage strain ε sh. Five basic types of Portland cement are produced according to their applications (Table 1.1). Table 1.1 Cement Classifications Cement Qualification Application GU MS HE LH HS General use Moderate sulphate resistant High early strength Low heat of hydration High sulphate resistant General purpose, used in ordinary construction where special properties are not required Moderate exposure of concrete to sulphate attack Used when less heat of hydration than GU cement is required Rapid achievement of a given level of strength Used when a low heat of hydration is desired Concrete exposed to severe sulphate action

4 4 Chapter 1 Compressive Strength The compressive strength of concrete, denoted by f c, is obtained from crushing tests on mm concrete cylinder samples at 28 days of aging. (If the concrete cylinder samples are mm, use 0.95 f c.) Typical stress-strain curves for concrete in compression are shown in Figure 1.1. A normal-density concrete of structural quality has a compressive strength f c ranging between 20 MPa (minimum) and 40 MPa. High-strength concrete (f c > 40 MPa) can also be used for special projects. Figure 1.1 Concrete under Compressive Load Tensile Strength The tensile strength may be obtained using three types of tests (Figure 1.2): a) direct tension, b) flexure test, c) split or Brazilian test. For guidance: f sp = 1.2 to 1.6 f t ; f r = 1.4 to 2 f t (1.1) Moreover, there is a strong relationship between λ f c and f r. Clause of the CSA A Standard provides the following relationship for f r : f r = 0. 6λ fc (1.2)

5 Materials 5 where λ = 1.0 for normal-density concrete and λ = 0.75 for low-density concrete. Figure 1.2a Figure 1.2b a) Direct tension σ = f t = T/A directly provides the tensile strength but is difficult to achieve in laboratory b) Flexure M h σ = I 2 Pa σ = f r = modulus of rupture = 6 2 bh c) Split or Brazilian test 2P σ = fsp = πld Figure 1.2c Figure 1.2 Tensile Strength of Concrete Modulus of Elasticity According to CSA A Standard (Clause 8.6.2), the modulus of elasticity, the secant modulus between σ c = 0 and σ c = 0.4f c, may be estimated by: E c γ c = 3300 fc ; 1500 γ c 2500 kg/m 3 (1.3) In addition, for concrete of normal density and compressive strength, 20 MPa f c 40 MPa, E c may be estimated using the following simplified equation: E c = 4500 f (1.4) c

6 6 Chapter 1 Strain The strain in concrete, ε c pic, corresponding to f c increases with f c. The approximate value of ε c pic is It may also be estimated as a function of f c by: ε c pic = fc 80, (1.5) The ultimate concrete strain in compression generally varies between and However, the CSA A Standard limits the value of ε cu to: ε cu = (1.6) Poisson s Ratio For uncracked concrete, Poisson s ratio varies between 0.15 and 0.20 for a concrete compressive stress f c less than 0.7f c. Creep Creep is a phenomenon by which, under sustained loads and stresses, concrete undergoes strain. The strain increases with time, but at a progressively decreasing rate (Figure 1.3). According to the American Concrete Institute (ACI) Committee , the creep strain in concrete, ε cp, may be estimated in terms of the instantaneous elastic strain, ε i, by: ε 0. 6 t = C ε where C = 10 + t C Q cp t i t 0. 6 u cr (1.7) where ε cp = creep strain ε i = instantaneous elastic strain C t = creep coefficient = ε cp /ε i C u = ultimate creep coefficient, which varies between 1.30 and 4.15, with an average value of 2.35 Q cr = correction factor that takes into consideration the conditions of use (relative humidity, percentage of air, aggregate content, thickness of the element, type of curing) [see Table 1.2] t = time in days.

7 Materials 7 Instantaneous recovery Total strain Ultimate creep strain Unloading Progressive recovery Residual creep strain Time since the application of compressive stress Figure 1.3 Typical Strain-Time Curve for Concrete under Axial Compression Shrinkage Shrinkage is a phenomenon by which the concrete undergoes strain caused by the decrease in the volume of concrete due to drying at constant temperature. Shrinkage strain generally develops during the first two to three years after casting of concrete (Figure 1.4). Figure 1.4 Shrinkage-Time Curve for Concrete after 7 Days of Curing

8 8 Chapter 1 Table 1.2 Creep and Shrinkage Modification Factors (Adapted from Table 1.2 of CSA A Standard) Creep: Q cr = Q a Q h Q f Q r Q s Q v Q a : to account for curing Shrinkage: P sh = P c P h P f P r P s P v P c : to account for cement content Age at loading (days) Moist curing Q a Steam curing Cement content (kg/m 3 ) P c Q h : to account for humidity P h : to account for humidity Relative humidity (%) Q h Relative humidity (%) P h Q f : to account for fine aggregates P f : to account for fine aggregates Ratio of fine to total aggregates Q f Ratio of fine to total aggregates P f Q r : to account for volume/surface ratio P r : to account for volume/surface ratio Volume/Surface ratio (mm) Q r Volume/Surface ratio (mm) P r

9 Materials 9 Creep: Q cr = Q a Q h Q f Q r Q s Q v Q s : to account for slump Shrinkage: P sh = P c P h P f P r P s P v P s : to account for slump Slump (mm) Q s Slump (mm) P s Q v : to account for air content P v : to account for air content Air (%) Q v Air (%) P v According to ACI Committee , the shrinkage strain may be estimated using the following formula (Figure 1.4): ε sh t = ε C + t s shu P sh (1.8) where ε sh = shrinkage strain ε shu = ultimate shrinkage strain, ε shu In the absence of a specific value, it is recommended to use ε shu = C s = constant; C s = 35 for seven-day moist curing of concrete and C s = 55 for one- to three-day steam curing P sh = correction factor taking into account the conditions of use (relative humidity, air content, aggregate and cement contents, thickness of the element) [see Table 1.2] t = time in days. Thermal Expansion of Concrete The coefficient of thermal expansion of concrete is α T = mm/mm/ C. The thermal expansion strain, ε th, can therefore be represented as follows: where T is the temperature variation assumed. εth = αt T (1.9)

10 10 Chapter Steel Reinforcement Steel reinforcement for concrete can be achieved by using: a) deformed bars and wires, b) welded wire fabric, or c) smooth wires. Smooth wires are allowed to be used for wire fabric, spirals, stirrups, and ties with diameters of 10 mm or less. Grades The CSA G30.18 Standard defines five grades of steel reinforcement in concrete: 300R, 400R, 500R, 400W and 500W. The W grade indicates that a ductile and weldable steel is required. The number of each grade indicates the minimum guaranteed specified yield strength in MPa. Grade 400R is the most frequently used for reinforcement, with a specified yield strength f y = 400 MPa. Table 1.3 presents the geometric and physical characteristics of steel bars commonly used in practice. Stress-Strain Curves Figure 1.5 shows actual and idealized stress-strain curves for steel reinforcement. The modulus of elasticity of steel reinforcement is E s = 200,000 MPa. Bar Designation No. Table 1.3 Characteristics of Reinforcing Bars Area (mm 2 ) Nominal dimensions Diameter (mm) Perimeter (mm) Mass (kg/m) 10M M M M M M M M

11 Materials 11 Figure 1.5 Actual and Idealized Stress-Strain Curves for Steel Reinforcement Thermal Expansion of Steel The coefficient of thermal expansion of steel is α T = mm/mm/ C. 1.4 Examples Example 1.1 Stress, Creep, and Shrinkage Problem Statement Consider a 3-m-high reinforced concrete column with a cross-section of 400 mm 400 mm. It is reinforced with 4 No. 30M steel bars. The column is subjected to an axial compression load of 1600 kn after one week of moist curing. a) Calculate the instantaneous compressive and tensile stresses in concrete and steel and the corresponding instantaneous strain. b) What is the shortening of the column after 180 days of loading? Use: f c (at 7 days) = 20 MPa; Type GU cement (300 kg/m 3 ); relative humidity = 60%; air content = 5%; slump of fresh concrete = 125 mm; sand = 670 kg/m 3 ; coarse aggregate = 1000 kg/m 3.

12 12 Chapter 1 Solution a) Instantaneous Stresses and Strain Stress in concrete, f ci E c = 4500 f E c = = 20, 120 MPa c Es 200, 000 n = n = = 9. 9 E 20, 120 c A c = net concrete area = A g A s A c = 160, = 157, 200mm A ce = equivalent concrete area = A c + na s A ce = 157, = 184, 920mm 2 2 f ci P = f ci = A 184, 920 ce 3 = MPa Stress in steel reinforcement, f si f si = nf f si = = MPa ci Instantaneous strain, ε i ε i fci = ε i = = mm/mm E 20, 120 c The instantaneous reduction is: l i = ε l 6 l i = = mm i b) Shortening of the Column at t = 180 Days Shortening due to creep C u = (average value) Q = Q Q Q Q Q Q (see Table 1.3) cr a h f r s v Q cr = = Note: Ratio (volume/surface) = ( ) ( ) ( ) = 100

13 Materials 13 C t 180 = C Q C 0. 6 t = 10 + t t u cr = ε = C ε 6 ε cp = = mm/mm cp t i l cp = ε l 6 l cp = = 1. 6 mm cp Shortening due to shrinkage C s = 35 ε shu = mm/mm (suggested average value in the absence of a specific value) P = P P P P P P P sh = = sh c h f r s v ε sh t = ε C + t s shu P sh 187 ε sh = = mm/mm l sh = ε l 6 l sh = = mm sh Total Shortening l = l + l l = = mm cp sh 1.5 Problems Problem 1.1 By analyzing the creep and shrinkage strain equations (Equations 1.7 and 1.8) and the modification factors Q cr and P sh (Table 1.2), determine the three factors that have the most influence on creep and shrinkage. Problem 1.2 Consider a rectangular section of a prestressed concrete column with dimensions 700 mm 700 mm 4 m. The section is subjected to a prestressed force of 2500 kn acting at the centroid of the section. The force is applied after seven days of moist curing. a) Calculate the instantaneous stress and the instantaneous strain in concrete. b) Determine the shortening of the column one year after the prestressed force was applied.

14 14 Chapter 1 Use: f c (at seven days) = 25 MPa; Type GU cement (300 kg/m 3 ); relative humidity = 70%; air content = 5%; slump of fresh concrete = 120 mm; sand = 660 kg/m 3 ; coarse aggregate = 1050 kg/m 3. Problem 1.3 Consider a 4-m-high concrete column having a 500 mm 500 mm square section. The longitudinal steel reinforcement consists of 4 No. 25M bars, that is, one No. 25M bar in each corner. The beam is subjected to a specified dead load of 1000 kn (unfactored) and a specified live load of 900 kn (unfactored). The dead load is applied 14 days after concrete casting. a) What are the stresses in concrete and steel reinforcement, assuming an elastic behaviour and perfect compatibility between the concrete and steel strains, for the following load cases: specified dead load (unfactored)? total factored load? b) What is the total strain experienced by the column due to creep and shrinkage, 365 days after concrete casting? Use: f c (at 14 days) = 25 MPa; seven-day moist curing; Type GU cement: 300 kg/m 3 ; sand: 700 kg/m 3 ; coarse aggregate: 1000 kg/m 3 ; slump: 100 mm; air content: 6%; relative humidity: 60%; unit weight of concrete = 24 kn/m 3 ; C u = 2.35.

Flexure: Behavior and Nominal Strength of Beam Sections

4 5000 4000 (increased d ) (increased f (increased A s or f y ) c or b) Flexure: Behavior and Nominal Strength of Beam Sections Moment (kip-in.) 3000 2000 1000 0 0 (basic) (A s 0.5A s ) 0.0005 0.001 0.0015