Upon subjecting concrete to moderate or high uniaxial compressive or tensile stresses, the strain response is non-linear relative to the stress applied. As the stress is removed, the resulting stress-strain curve deviates from the original path traced during loading, creating a hysteresis loop, indicative of the concrete's non-linear and non-elastic properties. Typically, a material's modulus of elasticity, which is a measure of the material's stiffness, is inferred from the linear portion of the stress-strain curve's slope. However, due to concrete's non-linear stress-strain characteristic, various methods must be employed to determine its modulus of elasticity.
The tangent modulus is calculated from the slope of a line drawn tangent to the stress-strain curve at any selected point. The secant modulus uses the slope of a line from the origin to a point on the curve, which represents 40 percent of the failure load. Conversely, the chord modulus is determined by the slope of a line connecting two points on the curve: one representing a strain of 50 microinches per inch and the other 40 percent of the ultimate load. This adjustment accounts for the initial concavity seen in the curve. The initial tangent modulus is typically derived from the slope of the tangent drawn at the curve's origin.
When a concrete specimen is subjected to moderate or high magnitude uniaxial compressive or tensile stresses, the resulting strain varies non-linearly with the applied stress.
Upon unloading the specimen, the resulting stress-strain curve traces a different path than during loading, forming a hysteresis loop. This demonstrates the material's non-linear and non-elastic behavior.
Generally, the slope of the linear portion of the stress-strain curve represents the value of the modulus of elasticity.
Since the concrete stress-strain curve is non-linear, its static modulus of elasticity is computed using different methods.
A line tangent to any point on the stress-strain curve is drawn, and its slope gives the tangent modulus specifically for loads near the considered stress value.
From the origin, a line is drawn to a point on the curve corresponding to 40 percent of the concrete's failure load, and its slope gives the secant modulus.
The point on the curve representing a strain of 50 microinches per inch is joined with the point on the curve representing 40 percent of the concrete's ultimate load, using a line, and its slope gives the chord modulus.
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