The descending portion of the stress–strain curve after the maximum stress has been reached is highly variable and is strongly dependent on the testing procedure. Similarly, the maximum or limiting strain, is very strongly dependent on the type of specimen, type of loading, and rate of testing. The limiting strain tends to be higher if there is a pos- sibility of load redistribution at high loads. In flexural tests, values from 0.0025 to 0.006 have been measured.
Equations for Compressive Stress–Strain Diagrams
A common representation of the stress–strain curve for concretes with strengths up to about 6000 psi is the modified Hognestad stress–strain curve shown in Fig. 3-19a. This consists of a second-degree parabola with apex at a strain of where
followed by a downward-sloping line terminating at a stress of and a limiting strain of 0.0038 [3-38]. Equation (3-14) describes a second-order parabola with its apex at the
0.85fc– fcfl = 0.9fcœ, 1.8f–c/Ec, Pcu, Pcu, fcœ P0, Ec = 40,0002f¿c + 1.0 * 106psi Ec = 57,0002fcœ psi 145 lb/ft3, Ec Ec 0.50fœc 155 lb/ft3 lb/ft3. w Ec = 331w1.522fcœ psi
strain The reduced strength, accounts for the differences between cylinder strength and member strength. These differences result from different curing and placing, which give rise to different water-gain effects due to vertical migration of bleed water, and differences between the strengths of rapidly loaded cylinders and the strength of the same concrete loaded more slowly, as shown in Fig. 3-2.
Two other expressions for the stress–strain curve will be presented. The stress–strain curve shown in Fig. 3-19b is convenient for use in analytical studies involving concrete strengths up to about 6000 psi because the entire stress–strain curve is given by one continu- ous function. The highest point in the curve, is taken to equal to give stress-block properties similar to that of the rectangular stress block of Section 4-3 when for
up to 5000 psi. The strain corresponding to maximum stress, is taken as For any given strain The stress corresponding to that strain is
(3-20) For a compression zone of constant width, the average stress under the stress block from
to is where
(3-21) The center of gravity of the area of the stress–strain curve between and is at from the point where exists, where
(3-22) where x is in radians when computing The stress–strain curve is satisfactory for concretes with stress–strain curves that display a gradually descending stress–strain curve at strains greater than Hence, it is applicable for up to about 5000 psi for normal-weight concrete and about 4000 psi for lightweight concrete.
fœc P0. tan-1x. k2 = 1 - 21x - tan -1x2 x2b1 P k2P P P = 0 b1 = ln11 + x 22 x b1fflc, P P = 0 fc = 2f fl cx 1 + x2 P, x = P>eo. 1.71fœc>Ec. eo, fœc Pult = 0.003 0.9fcœ fflc, fcfl = 0.9fcœ, eo. Linear Pult (From [3-39].) (From [3-41].) Fig. 3-19
Expressions for the compressive stress–strain curve for concrete are reviewed by Popovics [3-40]. Thorenfeldt, Tomaszewicz, and Jensen [3-42] generalized two of these expressions to derive a stress–strain curve that applies to concrete strengths from 15 to 125 MPa. The relationship between a stress, and the corresponding strain, is
(3-23) where
stress obtained from a cylinder test (see Eq. (3-27))
curve-fitting factor equal to (see Eq. (3-24))
tangent modulus (when )
factor to control the slopes of the ascending and descending branches of the stress–strain curve, taken equal to 1.0 for less than 1.0 and taken greater than 1.0 for greater than 1.0. [See Eqs. (3-25) and (3-26).] The four constants and k can be derived directly from a stress–strain curve for the concrete if one is available. If not, they can be computed from Eqs. (3-25) to (3-27), given by Collins and Mitchell [3-43]. Equations (3-17) and (3-18) can be used to compute although they were derived for the secant modulus from the origin and through points representing 0.4 to For normal-density concrete,
(3-24) where is in psi. For less than or equal to 1.0,
(3-25) and for
(3-26) If n, and are known, the strain at peak stress can be computed from
(3-27) A family of stress–strain curves calculated from Eq. (3-23) is shown in Fig. 3-18. Equation (3-23) produces a smooth continuous descending branch. Actually, the descend- ing branch for high-strength concretes tends to drop in a series of jagged steps as the struc- ture of the concrete is destroyed. Equation 3-23 approximates this with a smooth curve, as shown in Fig. 3-18.
Traditionally, equivalent stress blocks used in design are based directly on stress–strain curves that have the peak stress equal to which is to to allow for differences between the in-place strength and the cylinder strength. For prediction of experimentally obtained behavior, the ordinates of the stress–strain curve should be computed for a strength and then multiplied by 0.90. For design based on stress–strain relationships, the stress–strain curve should be derived for a strength of and the ordinates multiplied by 0.90. As shown in Fig. 3-15, a lateral confining pressure causes an increase in the com- pressive strength of concrete and a large increase in the strains at failure. The additional
fcœ fcœ 0.9fcœ, 0.85fcœ fcfl, eo = fc¿ Eca n n - 1b Ec fcœ, k = 0.67 + a fc œ 9000b Ú 1.0 (psi) Pc/eo 7 1.0, k = 1.0 Pc/eo fc¿ n = 0.8 + a fc œ 2500b 0.5fcœ. Ec, eo, Ec, n, Pc/eo Pc/eo k = a Ecœ = fcœ/eo Pc = 0 Ec = initial Ec/1Ec - Ecœ2 n = a
eo= strain when fc reaches fcœ
fcœ = peak fc fcœ = n1Pc/eo2 n - 1 + 1Pc/eo2nk Pc, fc,
strength and ductility of confined concrete are utilized in hinging regions of structures in seismic regions. Stress–strain curves for confined concrete are described in [3-44].
When a compression specimen is loaded, unloaded, and reloaded, it has the stress–strain response shown in Fig. 3-20. The envelope to this curve is very close to the stress–strain curve for a monotonic test. This, and the large residual strains that remain after unloading, suggest that the inelastic response is due to damage to the internal struc- ture of the concrete, as is suggested by the microcracking theory presented earlier.
Stress–Strain Curve for Normal-Weight Concrete in Tension
The stress–strain response of concrete loaded in axial tension can be divided into two phases. Prior to the maximum stress, the stress–strain relationship is slightly curved. The diagram is linear to roughly 50 percent of the tensile strength. The strain at peak stress is about 0.0001 in pure tension and 0.00014 to 0.0002 in flexure. The rising part of the stress–strain curve may be approximated either as a straight line with slope and a max- imum stress equal to the tensile strength or as a parabola with a maximum strain
and a maximum stress The latter curve is illustrated in Fig. 3-21a with and based on Eqs. (3-11) and (3-18).Ec
ftœ fœt. Pœt = 1.8fœt/Ec fœt Ec 0 0.0001 0.0002 (a) T ensile stress f't 0 0.0005 T ensile stress f't (b) Crack opening, w (in.)
Tensile strain, Pt
Fig. 3-21
Stress–strain curve and stress–crack opening curves for concrete loaded in tension. Fig. 3-20
Compressive stress–strain curves for cyclic loads. (From [3-45].)
After the tensile strength is reached, microcracking occurs in a fracture process zone adjacent to the point of highest tensile stress, and the tensile capacity of this concrete drops very rapidly with increasing elongation. In this stage of behavior, elongations are concen- trated in the fracture process zone while the rest of the concrete is unloading elastically. The unloading response is best described by a stress-versus-crack-opening diagram, ideal- ized in Fig. 3-21b as two straight lines. The crack widths shown in this figure are of the right magnitude, but the actual values depend on the situation. The tensile capacity drops to zero when the crack is completely formed. This occurs at a very small crack width. A more detailed discussion is given in [3-46].
Poisson’s Ratio
At stresses below the critical stress (see Fig. 3-1), Poisson’s ratio for concrete varies from about 0.11 to 0.21 and usually falls in the range from 0.15 to 0.20. On the basis of tests of biaxially loaded concrete, Kupfer et al. [3-32] report values of 0.20 for Poisson’s ratio for concrete loaded in compression in one or two directions: 0.18 for concrete loaded in ten- sion in one or two directions and 0.18 to 0.20 for concrete loaded in tension and compres- sion. Poisson’s ratio remains approximately constant under sustained loads.
3-6
TIME-DEPENDENT VOLUME CHANGES
Concrete undergoes three main types of volume change, which may cause stresses, crack- ing, or deflections that affect the in-service behavior of reinforced concrete structures. These are shrinkage, creep, and thermal expansion or contraction.
Shrinkage
Shrinkage is the decrease in the volume of concrete during hardening and drying under con- stant temperature. The amount of shrinkage increases with time, as shown in Fig. 3-22a.
The primary type of shrinkage is called drying shrinkage or simply shrinkage and is due to the loss of a layer of adsorbed water (electrically bound water molecules) from the surface of the gel particles. This layer is roughly one water molecule thick, or about 1 percent of the size of the gel particles. The loss of free unadsorbed water has little effect on the magnitude of the shrinkage.
Shrinkage strains are dependent on the relative humidity and are largest for relative humidities of 40 percent or less. They are partially recoverable upon rewetting the con- crete, and structures exposed to seasonal changes in humidity may expand and contract slightly due to changes in shrinkage strains.
The magnitude of shrinkage strains also depends on the composition of the concrete mix and the type of cement used. The hardened cement paste shrinks, whereas the aggre- gate acts to restrain shrinkage. Thus, the larger the fraction of the total volume of the con- crete that is made up of hydrated cement paste, the greater the shrinkage. This may be particularly important with the more common use of self-consolidating concrete, which has significantly higher paste content than normally consolidated concrete of the same strength. An increase in the water/cementitious materials ratio or the total cement content reduces the volume of aggregates, thus reducing the restraint of shrinkage by the aggregate. Also, more finely ground cements have a larger surface area per unit volume, and thus, there is more adsorbed water to be lost during shrinkage. There is less shrinkage in concrete made with quartz or granite aggregates than with sandstone aggregates because quartz and granite have a higher modulus of elasticity.
Drying shrinkage occurs as the moisture diffuses out of the concrete. As a result, the exterior shrinks more rapidly than the interior. This leads to tensile stresses in the outer skin of the concrete and compressive stresses in the interior. For large members, the ratio of volume to surface area increases, resulting in less shrinkage because there is more moist concrete to restrain the shrinkage. Shrinkage also develops more slowly in large members. Autogenous shrinkage occurs without the loss of moisture due to hydration reactions inside the cement matrix. In earlier studies this was considered to be a very small portion of the total shrinkage, but with a greater use of high-performance concretes (water/cement ratio below 0.40), autogenous shrinkage may constitute a more significant percentage of the total shrinkage [3-47].
A final form of shrinkage called carbonation shrinkage occurs in carbon-dioxide rich atmospheres, such as those found in parking garages. At 50 percent relative humidity, the amount of carbonation shrinkage can equal the drying shrinkage, effectively doubling the total amount of shrinkage. At higher and lower humidities, the carbonation shrinkage decreases.
The ultimate drying shrinkage strain, for a 6-by-12-in. cylinder maintained for a very long time at a relative humidity of 40 percent ranges from 0.000400 to 0.001100
Pshu, Fig. 3-22
(400 to ), with an average of about 0.000800 [3-17]. Thus, in a 25-ft bay in a building, the average shrinkage strain would cause a shortening of about in unreinforced concrete. In a structure, however, the shrinkage strains will tend to be less for the same concrete, for the following reasons:
1. The ratio of volume to surface area will generally be larger than for the cylinder;