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Stress-Strain Behavior of Concrete
Typical Stress-Strain Plot of Concrete
(1) At stress below 30% of ultimate strength, the transition zone cracks remain stable. The stress-strain plot remains linear. (2) At stress between 30% and 50% of ultimate strength, the transition zone microcracks begin to increase in length, width and numbers. The stress-strain plot becomes non-linear.
(3) At 50 to 60% of the ultimate stress, cracks begin to form in the matrix. With further increase to about 75% of the ultimate stress, the cracks in the transition become unstable, and crack propagation in the matrix will increase. The stress-strain curve bends towards the horizontal.
(4) At 75 to 80% of the ultimate stress, the stress reaches a critical stress level for spontaneous crack growth under a sustained stress. Cracks propagate rapidly in both the matrix and the transition zone. Failure occurs when the cracks join together and become continuous.
Elastic Moduli of Concrete
- Static Modulus of Elasticity - The slope of the stress-strain curve for concrete under uniaxial tension or compression loading.
- Secant Modulus - The slope of a line drawn from the origin to the point on the stress-strain curve corresponding to 40% of the failure stress.
- Tangent Modulus - The slope of a line drawn tangent to the stress-strain curve at any point on the curve.
- Chord Modulus - The slope of a line drawn between two points on the stress-strain curve.
- Dynamic Modulus - The modulus of elasticity corresponding to a small instantaneous strain. It can be approximated by the tangent modulus drawn at the origin. It is generally 20, 30 and 40% higher than the secant modulus for high-, medium- and low-strength concretes, respectively.
Secant Modulus: Slope of SO (Note: S is at 40% failure stress.)
Chord Modulus: Slope of SC
Tangent Modulus: Slope of TT’ Dynamic Modulus: Slope of OD
Test for Static Modulus of Elasticity and Poisson’s
Ratio of Concrete in Compression (ASTM C 469)
- The test uses a 6” X 12” cylindrical specimen, which is loaded in compression. A compressometer is used to measure the longitudinal strains, and an extensometer is used to measure the transverse strains on the specimen.
- The chord modulus (E) is calculated as: E = (S 2 - S 1 ) / (ε 2 - 0.00005) where S 2 = stress corresponding to 40% of ultimate strength S 1 = stress corresponding to a strain of 50 X 10^ - ε 2 = longitudinal strain produced by stress S (^2)
- The Poisson’s ratio (μ) is calculated as: μ = (ε (^) t2 - ε (^) t1 ) / (ε 2 - 0.00005) where ε (^) t2 , ε (^) t1 = transverse strains produced by S 2 & S 2 , respectively.
Drying Shrinkage and Creep
- Phenomenon of drying shrinkage
- When a fresh concrete is initially placed down, the cement paste is usually saturated. As the saturated cement paste is exposed to ambient humidities that are below 100% saturation, moisture will be lost to the environment. The loss of physically adsorbed water from C-S-H results in shrinkage of the cement paste and thus shrinkage of the concrete.
- Induced stresses from drying shrinkage
- If a concrete is fully constrained from movement, the drying shrinkage will result in a tensile stress (σ), which is usually estimated by: σ = E εsh where E = elastic modulus of the concrete εsh = shrinkage strain of the concrete
- Induced stresses from drying shrinkage (continued)
- The use of the above equation usually results in an overestimation of actual induced stress, since the concrete also creeps when under a sustained stress. This creep strain under a tensile stress is opposite in direction to the shrinkage strain, and helps to relieve the total induced tensile stress in the concrete.
- A more realistic estimation of the induced stress is: σ = E (εsh - εcr ) where εcr = creep strain
- Cracking due to drying shrinkage
- When the induced shrinkage tensile stress exceeds the tensile strength of the concrete, cracking will occur.
- Cracking due to drying shrinkage usually occurs at the early age of concrete when the strength of the concrete is relatively low.
Influence of Shrinkage and Creep on Concrete Cracking
Similarities between Drying Shrinkage and Creep
- Both originate from the the same source, the hydrated cement paste.
- Their strain-time curves are similar.
- The factors that influence drying shrinkage also influence the creep in generally the same manner.
- Both are partially reversible.
- Strains from both drying shrinkage and creep can not be ignored in structural design.
Terminology used in Drying Shrinkage and Creep
- Basic creep - Creep (the increase in strain over time under a sustained stress) of a concrete specimen under 100% relative humidity.
- Free drying shrinkage - drying shrinkage of a concrete when it is unloaded.
- When concrete is under load and simultaneously exposed to low relative humidity environments, the total strain is higher than the sum of elastic strain, free shrinkage strain and basic creep strain. The additional creep is called drying creep.
- Specific creep - creep strain per unit of applied stress.
- Creep coefficient - ratio of creep strain to elastic strain.
Strain-time Plot of Concrete under a Constant
Stress and exposed to Low Relative Humidity
Effects of Aggregate Volume Fraction on Shrinkage of Concrete
Sc/Sp
Effects of Aggregate Volume Fraction on Creep of Concrete
Effects of Cement Content and Water-Cement Ratio on Drying Shrinkage of Concrete
- Higher drying shrinkage is caused by a higher cement paste volume fraction.
Effects of Cement Type on Creep of Concrete
- A lower creep is due to a higher strength of the concrete.
(2) Effects of Concrete Strength - An increase in concrete strength reduces shrinkage and creep.
Effects of Cement Content on Drying Shrinkage and Creep
- An increase in cement content increases the cement paste volume, which increases drying shrinkage.
- The effect on creep is reversed. The increase in strength due to a higher cement content has a dominating effect in reducing the creep.
(3) Effects of Time - Shrinkage and creep increase with time.
Drying Shrinkage versus Time
(5) Effects of Effective Thickness - Shrinkage and Creep decrease as the length of path traveled by water to the atmosphere increases.
Influence of Effective Concrete Thickness on Creep
Influences of Exposure Time and Specimen Size on Shrinkage
(6) Effects of Temperature - Exposure to higher temperature during time of loading increases creep of concrete.
Effects of Concrete Temperature on Creep
(7) Effects of Magnitude of Applied Stress - At stress- strength ratio of less than 0.4, creep is linearly proportional to stress. At stress-strength ratio of greater than 0.4, a creep correction factor must be applied.
Effects of Magnitude of Stress on Creep
Correction Factor for Computing Creep Coefficient at High Stress-Strength Levels
Thermal Shrinkage
- Concrete expands on heating and contract on cooling.
- Thermal expansion or contraction strain (εT) is linearly related to coefficient of thermal expansion (μ) and the
change in temperature (εT)
εT = μ ΔT
- If the concrete is fully restrained, the induced stress due to the temperature change (ΔT) will be equal to:
σT = E (μ ΔT - εcr )
- At the early ages of concrete, concrete usually rises in temperature as the cement hydrates. This results in compressive stresses. However, stress relaxation is high and E is low at early ages. Therefore, the resulting compressive stress will be small, and usually does not cause any problem.
Effects of Placing Temperature on Temperature of Mass Concrete
Effects of Volume to Surface Ratio on Temperature Rise in Mass Concrete
Effects of Cement and Pozzolan Content on Temperature Rise in Mass Concrete
Thermal Properties of Concrete
- Coefficient of thermal expansion ( α )
- The length change per unit length per degree of
temperature change.
- It can be estimated from the weighted average
of the coefficients of thermal expansion of its
components, i.e. the aggregate and the cement
mortar.
- Typical value of α for concrete varies from 6 to
12 X 10 -6^ /°C
* α of steel = 11 X 10 -6^ /°C
Influence of Aggregate Type on Coefficient of
Thermal Expansion of Concrete
Specific heat (c) The quantity of heat required to raise the temperature of a unit mass of a material by one degree. Typical value varies from 0.22 to 0.25 Btu/lb. °F (or cal/g °C)
- Specific heat of water = appr. 1 Btu/lb °F (or 1 cal/g °C) Specific heat of steel = appr. 0.1 Btu/lb °F
Thermal conductivity (K) The heat flux transmitted through a unit area of a material under a unit temperature gradient. Typical value varies from 13 to 24 Btu in./h. ft 2 °F (number of Btu’s transmitted per hour per square ft of material 1 in. thick per degree F temperature difference between the two faces)
- Thermal conductivity of steel = appr. 300 Btu in./h. ft 2 °F Thermal conductivity of pine wood = appr. 1 Btu in./h. ft^2 °F
