A
B
Secant modulus at stress B Initial tangent modulus Tangent modulus
at stress A
O
Stress
Strain Fig. 3-17
Tangent and secant moduli of elasticity.
0.001
Stress, psi
0 2000 4000 6000 8000 10,000 12,000 14,000 16,000 18,000
0.002 Strain in/in.
0.003 0.004
fc 4500 psi
fc 8800 psi fc 13,500 psi fc 17,500 psi
Fig. 3-18
Typical concrete stress–strain curves in compression.
[Plotted using Eqs. (3-20) to (3-26).]
The length of the descending branch of the curve is strongly affected by the test con- ditions. Frequently, an axially loaded concrete test cylinder will fail explosively at the point of maximum stress. This will occur in axially flexible testing machines if the strain energy released by the testing machine as the load drops exceeds the energy that the spec- imen can absorb. If a member is loaded in compression due to bending (or bending plus axial load), the descending branch may exist because, as the stress drops in the most highly strained fibers, other less highly strained fibers can resist the load, thus delaying the failure of the highly strained fibers.
The stress–strain curves in Fig. 3-18show five properties used in establishing math- ematical models for the stress–strain curve of concrete in compression [3-36]:
1. The initial slope of the curves (initial tangent modulus of elasticity) increases with an increase in compressive strength.
The modulus of elasticity of the concrete, is affected by the modulus of elasticity of the cement paste and by that of the aggregate. An increase in the water/cement ratio increases the porosity of the paste, reducing its modulus of elasticity and strength. This is accounted for in design by expressing as a function of
Of equal importance is the modulus of elasticity of the aggregate. Normal-weight aggregates have modulus-of-elasticity values ranging from 1.5 to 5 times that of the cement paste. Because of this, the fraction of the total mix that is aggregate also affects Lightweight aggregates have modulus-of-elasticity values comparable to that of the paste;
hence, the aggregate fraction has little effect on Ecfor lightweight concrete.
Ec. fcœ.
Ec Ec,
The modulus of elasticity of concrete is frequently taken as given in ACI Code Section 8.5.1, namely,
(3-17) where is the weight of the concrete in This equation was derived from short-time tests on concretes with densities ranging from 90 to and corresponds to the secant modulus of elasticity at approximately [3-37]. The initial tangent modulus is about 10 percent greater. Because this equation ignores the type of aggregate, the scatter of data is very wide. Equation (3-17) systematically overestimates in regions where low- modulusaggregates are prevalent. If deflections or vibration characteristics are critical in a design, should be measured for the concrete to be used.
For normal-weight concrete with a density of ACI Code Section 8.5.1 gives the modulus of elasticity as
(3-18) ACI Committee 363 [3-8] proposed the following equation for high-strength concretes:
(3-19) 2. The rising portion of the stress–strain curve resembles a parabola with its vertex at the maximum stress.
For computational purposes the rising portion of the curves is frequently approximated by a parabola [3-36], [3-38], and [3-39]. This curve tends to become straighter as the con- crete strength increases [3-40].
3. The strain, at maximum stress increases as the concrete strength increases.
4. As explained in Section 3-2, the slope of the descending branch of the stress–strain curve results from the destruction of the structure of the concrete, caused by the spread of microcracking and overall cracking. For concrete strengths up to about 6000 psi, the slope of the descending branch of the stress–strain curve tends to be flat- ter than that of the ascending branch. The slope of the descending branch increases with an increase in the concrete strength, as shown in Fig. 3-18. For concretes with greater than about 10,000 psi, the descending branch is a nearly vertical, discontinuous
“curve.” This is because the structure of the concrete is destroyed by major longitudinal cracking.
5. The maximum strain reached, decreases with an increase in concrete strength.
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 = 40,0002fc¿+ 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 4-3when 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) wherexis 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 + x22
x b1fflc,
P P = 0
fc = 2fflcx 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
Analytical approximations to the compressive stress–strain curve for concrete.
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 kcan 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) Ifn, 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 fcœ
9000b Ú 1.0 (psi) Pc/eo 7 1.0,
k = 1.0 Pc/eo
fc¿
n = 0.8 + a fcœ 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)
Tensile stress
f't
0 0.0005
Tensile 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.