Overview of Topics. Stress-Strain Behavior in Concrete. Elastic Behavior. Non-Linear Inelastic Behavior. Stress Distribution.

Transcription

1 Stress-Strain Behavior in Concrete Overview of Topics EARLY AGE CONCRETE Plastic shrinkage shrinkage strain associated with early moisture loss Thermal shrinkage shrinkage strain associated with cooling Dr. Kimberly Kurtis School of Civil Engineering Georgia Institute of Technology Atlanta, Georgia LATER AGE CONCRETE Drying shrinkage -shrinkage strain associated with moisture loss in the hardened material Deformations occur under loading - Elastic - Viscoelastic Elastic Behavior Non-Linear Inelastic Behavior Under loading, concrete deforms in a non-linear, inelastic manner However, an estimate of E is useful for determining stresses induced when strain is produced is very small (e.g., by environmental effects) Initial Tangent Modulus/Dynamic Modulus - slope of the tangent to the curve at the origin (D) Tangent Modulus slope of a line drawn tangent to curve at any point (T) Secant Modulus slope of line drawn from the origin to a point on the curve, usually corresponding to 0.40 ultimate stress (S) Chord modulus slope of the line drawn between 2 points, one of which corresponds to 50 microstrain and the other generally occurs at 0.40 ultimate stress (C) Elastic Modulus Stress Distribution Concrete is highly heterogeneous Localized stress/strain can be quite different from nominal applied stress/strain Largest strains often occur at the interface -> microcracking

2 For E agg >E paste, (1) Tensile bond failure (2) Shear bond failure (3) Tensile matrix failure (4) Occasional aggregate failure Stress Distribution Paste shows more ductility Paste shows broader high stress region Due to stress concentrations and heterogeneity of concrete Stress Distribution Elastic Modulus: Two Phase Models E c =E concrete E p =E cement paste E a =E agg V c =vol concrete V p =vol paste V a =vol agg K=bulk modulus G=shear modulus Elastic Modulus: Two Phase Models Hirsch, Counto, and H-S models give fairly good representations of E in most concrete Deviations from actual behavior are believed to be due to ITZ effects Elastic Modulus: Three Phase Models Elastic Modulus: Three Phase Models E c =E concrete E p =E cement paste E a =E agg E i =E ITZ V c =vol concrete V p =vol paste V a =vol agg V i =vol ITZ K=bulk modulus G=shear modulus Need K,G for agg, paste, and ITZ t r = ratio of ITZ thickness to the equivalent radius of the nominally spherical inclusions

3 Estimations of E Estimations of E: High Strength Concrete ACI 318 gives equations to estimate E from compressive strength and unit weight: E c = 33 w c 1.5 f c E c = elastic modulus of concrete, psi W = unit weight, pcf f c =28d compressive strength of standard cylinders Valid to at least 6000 psi (perhaps to as high as 9000 psi) The unit weight is used to account for the presence and density of the aggregate E agg is rarely known and this is a useful way to include its effect in E For normal weight concrete (145pcf), this reduces to E c = f c Estimations of E: High Strength E c = 33 w c 1.5 f c may underestimate E for high strength concrete Estimations of E: High Strength Best fit, E c in psi: E c =w 2.55 f c ACI 363 also gives an equation for high strength concrete: E c = 40000f c + (1x10 6 ) For f c psi With E c expressed in psi ASTM C469 Measures E and Poisson s ratio Measure f c Load to 0.40f c Measured by compression loading of 6x12 cylinders psi/sec Measure longitudinal and lateral strains Take average of chord moduli of 2 nd and 3 rd curves Measurements of E Measurement of E

4 Poisson s Ratio Poisson s Ratio - ratio of lateral strain to axial strain Not typically required for design Measured by compression loading of 6x12 cylinders psi/sec typical no consistent relationship with mixture design or material properties however, it is generally higher in high strength concrete, saturated concrete ( is typical), and dynamically loaded concrete Dynamic E Dynamic Modulus of Elasticity (initial tangent modulus) initial slope of stress-strain curve; corresponds to very small instantaneous strain; important for earthquake loading A measure of the vibration of a specimen in response to a small applied load No microcracking; No creep -> it is usually, 20-40% higher than chord modulus Can be estimated by: E c = 0.83E d Dynamic E Can be measured by sonic techniques, (e.g., resonant frequency method, ACI 215) Because E d is very sensitive to microcracking, measurements of changes in E d can be used to monitor damage by physical and chemical attack For a long prism or rod composed of an isotropic, homogeneous, perfectly elastic material, the resonant frequency of vibration, N, may be calculated from N = m 2 k(e d) where m is the mass of the rod 2πL 2 k is the radius of gyration of the section about an axis perpendicular to the plane of bending (k=t/(12) for a rectangular cross section of thickness, t), E is the dynamic modulus of elasticity d is the density of the material L is the length of the specimen The equipment essentially is composed of a generator of mechanical vibrations and a sensor of mechanical vibrations. The generator consists of an oscillator which produces electrical audio-frequency voltages and an amplifier. A driver converts the amplified voltages into mechanical vibrations that are applied to the specimen. The pick up circuit consists of a sensor, amplifier, and indicator, but may also include an oscilloscope. A piezoelectric transducer acts as the sensor and converts the mechanical vibrations to an electrical AC voltage of the same frequency. These voltages are amplified and are read by a needle on an indicator. The resonant frequency is determined by measuring the frequency at which the needle experiences maximum deflection. An oscilloscope provides additional information and can be used to verify that the fundamental modes of the specimen has been reached. variable frequency o s c illa t o r oscilloscope a m plifie r t=3.5" driver specimen pick-up w=4.5" The relationship can be extended to heterogeneous materials when the sample dimensions are considered to be large in relation to the size of the variations in the material. indicator a m plifie r L=1 6.0" Resonant Frequency Performed in the longitudinal and transverse directions From the fundamental transverse frequency n (Hz), the dynamic modulus of elasticity can be obtained from: Dynamic E = CWn 2 where W is the weight of the specimen (lb.) and C= (L 3 T/bt 3 ) (s 2 /in 2 ) for a prism, where L is the specimen length (in) b and t are the specimen cross sectional measurements, t being in the direction in which the specimen is driven T is a correction factor related to K/L where K is the radius of gyration of the prism. From the fundamental longitudinal frequency n (Hz), the dynamic modulus of elasticity can be obtained from: Dynamic E =DW(n ) 2 Where, for a prism, D= (L/bt) (s 2 /in 2 ) For 4.5x3.5x16 specimens, C= s 2 /in 2 D= s 2 /in 2 g ASTM C597 Measures time for an ultrasonic wave to pass through concrete Related to age, moisture condition, agg/cement ratio, type of aggregate, location of reinforcement, degree of cracking, presence of voids, homogeneity Can be used to assess changes in properties due to damage (such as fire) Estimation of strength and elastic modulus are more difficult. In a homogeneous, elastic and isotropic material the square of the wave velocity (V 2 )is related to E, but concrete does not fit those criteria Ultrasonic Pulse Velocity Direct Semidirect Indirect

5 E in Flexure Factors Affecting E Flexural modulus of elasticity measured by a center-point flexure test E = PL 3 /(48Iy) P = load L = span length I = moment of inertia y = midspan deflection Important property for rigid pavements Aggregate Aggregate acts to restrain matrix strains; E agg is important dense agg, higher E More porous agg, lower E granite, basalt 10-20x10 6 psi sandstone, limestone 3-7x10 6 psi Lightweight aggregate 1-4 x10 6 psi Aggregate As MSA, shape, surface texture, gradation, and mineralogical composition all influence microcracking in the ITZ, these will each influence the slope of the stress-strain curve. In lightweight concrete, E is more likely to be affected by aggregate choice than f c HCP E paste is related to its porosity Values of 1-4 x 10 6 psi have been reported E paste is related to w/c, air content, use of mineral admixtures, and degree of cement hydration, degree of curing The properties of the ITZ can have a strong influence on E microcracking, increased porosity, and aligned, large hydration products Testing Parameters When tested wet E is ~15% higher than when dry (remember that the f c was 15% lower when tested wet) Why? Drying of concrete has different effects on paste and ITZ On drying, paste will gain strength because of strengthening of VDW bonds in the C-S-H ITZ microcracking will progress on drying (one explanation)

6 Testing Parameters Load Rate At faster load rates, less time for deformation, higher E For very slow load rates, also get creep effects, and E will be lower.