The tensile strength of concrete falls between 8 and 15 percent of the compressive strength.
The actual value is strongly affected by the type of test carried out to determine the tensile strength, the type of aggregate, the compressive strength of the concrete, and the presence of a compressive stress transverse to the tensile stress [3-28], [3-29], and [3-30].
Standard Tension Tests
Two types of tests are widely used. The first of these is the modulus of ruptureor flexural test
(ASTM C78), in which a plain concrete beam, generally long, is
loaded in flexure at the third points of a 24-in. span until it fails due to cracking on the tension face. The flexural tensile strength or modulus of rupture, from a modulus-of-rupture test is calculated from the following equation, assuming a linear distribution of stress and strain:
(3-9) In this equation,
moment
width of specimen overalldepth of specimen
The second common tensile test is the split cylindertest (ASTM C496), in which a standard compression test cylinder is placed on its side and loaded in compres- sion along a diameter, as shown in Fig.3-9a.
In a split-cylinder test, an element on the vertical diameter of the specimen is stressed in biaxial tension and compression, as shown in Fig. 3-9c. The stresses acting across the vertical diameter range from high transverse compressions at the top and bottom to a nearly uniform tension across the rest of the diameter, as shown in Fig. 3-9d. The split- ting tensile strength, from a split-cylinder test is computed as:
(3-10) where
maximum applied load in the test length of specimen
diameter of specimen
Various types of tension tests give different strengths. In general, the strength decreases as the volume of concrete that is highly stressed in tension is increased. A third-point-loaded modulus-of-rupture test on a 6-in.-square beam gives a modulus-of-rupture strength thatfr
d = / = P =
fct = 2P p/d fct,
6-by-12-in.
h = b = M =
fr = 6M bh2
fr,
6 in. * 6 in. * 30 in.
averages while a 6-in.-square prism tested in pure tension gives a direct tensile strength that averages about 86 percent of [3-30].
Relationship between Compressive and Tensile Strengths of Concrete
Although the tensile strength of concrete increases with an increase in the compressive strength, the ratio of the tensile strength to the compressive strength decreases as the com- pressive strength increases. Thus, the tensile strength is approximately proportional to the square root of the compressive strength. The mean split cylinder strength, from a large number of tests of concrete from various localities has been found to be [3-10]
(3-11) where and are all in psi. Values from Eq. (3-11) are compared with split-cylinder test data in Fig.3-10. It is important to note the wide scatter in the test data. The ratio of measured to computed splitting strength is essentially normally distributed.
Similarly, the mean modulus of rupture, can be expressed as [3-10]
(3-12a) Again, there is scatter in the modulus of rupture. Raphael [3-28] discusses the reasons for this, as do McNeely and Lash [3-29]. The distribution of the ratio of measured to computed modulus-of-rupture strength approaches a log-normal distribution.
ACI Code Section 9.5.2.3 defines the modulus of rupture for use in calculating deflections as
(3-12b) where for l = 1.0 normalweight concrete. Lightweight concrete is discussed in section 3-8.
fr = 7.5l2fcœ fr = 8.32fcœ
fr, 2fcœ
f-ct,fcœ,
fct = 6.42fcœ
fct, fct
1.5 times fct, Fig. 3-9
Split-cylinder test.
A lower value is used for the average splitting tensile strength (ACI Commentary Section R8.6.1.):
(3-12c) Factors Affecting the Tensile Strength of Concrete
The tensile strength of concrete is affected by the same factors that affect the compressive strength. In addition, the tensile strength of concrete made from crushed rock may be up to 20 percent greater than that from rounded gravels. The tensile strength of concrete made from lightweight aggregate tends to be less than that for normal sand-and-gravel concrete, although this varies widely, depending on the properties of the particular aggregate under consideration.
The tensile strength of concrete develops more quickly than the compressive strength.
As a result, such things as shear strength and bond strength, which are strongly affected by the tensile strength of concrete, tend to develop more quickly than the compressive strength. At the same time, however, the tensile strength increases more slowly than would be suggested by the square root of the compressive strength at the age in question. Thus, concrete having a 28-day compressive strength of 3000 psi would have a splitting tensile strength of about At 7 days this concrete would have compressive strength of about 2100 psi (0.70 times 3000 psi) and a tensile strength of about 260 psi (0.70 times 367 psi). This is less than the tensile strength of that one would compute from the 7-day compressive strength. This is of importance in choosing form-removal times for flat slab floors, which tend to be governed by the shear strength of the column–slab connections [3-31].
6.722100 = 307 psi 6.72fcœ = 367 psi.
fr = 6.7l2fcœ Fig. 3-10
Relationship between splitt- ing tensile strengths and compression strengths.
(From [3-10].)
Fig. 3-11 Biaxial stresses.
Strength under Biaxial and Triaxial Loadings
Biaxial Loading of Uncracked, Unreinforced Concrete Concrete is said to be loaded biaxiallywhen it is loaded in two mutually perpendicular directions with essentially no stress or restraint of deformation in the third direction, as shown in Fig.3-11a. A common example is shown in Fig. 3-11b.
The strength and mode of failure of concrete subjected to biaxial states of stress varies as a function of the combination of stresses as shown in Fig.3-12. The pear-shaped line in Fig. 3-12a represents the combinations of the biaxial stresses, and which cause cracking or compression failure of the concrete. This line passes through the uniaxial compressive strength, at Aand and the uniaxial tensile strength, at Band
Under biaxial tension ( and both tensile stresses) the strength is close to that in uniaxial tension, as shown by the region (zone 1) in Fig. 3-12a. Here, failure occurs by tensile fracture perpendicular to the maximum principal tensile stress, as shown in Fig. 3-12b, which corresponds to point in Fig. 3-12a.
When one principal stress is tensile and the other is compressive, as shown in Fig. 3-11a, the concrete cracks at lower stresses than it would if stressed uniaxially in ten- sion or compression [3-32]. This is shown by regions A–Band in Fig. 3-12a. In this region, zone 2 in Fig. 3-12a, failure occurs due to tensile fractures on planes perpendicular to the principal tensile stresses. The lower strengths in this region suggest that failure is governed by a limiting tensile strain rather than a limiting tensile stress.
Under uniaxial compression (points Aand and zone 3 in Fig. 3-12a), failure is initi- ated by the formation of tensile cracks on planes parallel to the direction of the compressive stresses. These planes are planes of maximum principal tensile strain.
Under biaxial compression (region and zone 4 in Fig. 3-12a), the failure pattern changes to a series of parallel fracture surfaces on planes parallel to the unloaded
A–C–A¿ A¿
A¿–B¿ B¿
B–D–B¿ s2
s1
B¿. ftœ,
A¿ fcœ,
s2, s1
sides of the member, as shown in Fig. 3-12d. Such planes are acted on by the maximum tensile strains. Biaxial and triaxial compression loads delay the formation of bond cracks and mortar cracks. As a result, the period of stable crack propagation is longer and the concrete is more ductile. As shown in Fig. 3-12, the strength of concrete under biaxial compression is greater than the uniaxial compressive strength. Under equal biaxial com- pressive stresses, the strength is about 107 percent of as shown by point C.
In the webs of beams, the principal tensile and principal compressive stresses lead to a biaxial tension–compression state of stress, as shown in Fig. 3-11b. Under such a load- ing, the tensile and compressive strengths are less than they would be under uniaxial stress, as shown by the quadrant ABor in Fig. 3-12a. A similar biaxial stress state exists in a split-cylinder test, as shown in Fig. 3-9c. This explains in part why the splitting tensile strength is less than the flexural tensile strength.
In zones 1 and 2 in Fig. 3-12, failure occurred when the concrete cracked, and in zones 3 and 4, failure occurred when the concrete crushed. In a reinforced concrete member with sufficient reinforcement parallel to the tensile stresses, cracking does not represent fail- ure of the member because the reinforcement resists the tensile forces after cracking. The biaxial load strength of cracked reinforced concrete is discussed in the next subsection.
Compressive Strength of Cracked Reinforced Concrete
If cracking occurs in reinforced concrete under a biaxial tension–compression loading and there is reinforcement across the cracks, the strength and stiffness of the concrete under com- pression parallel to the cracks is reduced. Figure3-13ashows a concrete element that has been cracked by horizontal tensile stresses. The natural irregularity of the shape of the cracks leads to variations in the width of a piece between two cracks, as shown. The compressive stress
A¿B¿
fcœ,
A
C
D
B B
fc
A s2
fc
s2
fc
s1
fc
s1
Fig. 3-12
Strength and modes of failure of unreinforced concrete sub- jected to biaxial stresses.
(From [3-32].)
Compression
Tension
(a) Compression member with cracks. (b) Free-body diagram of the shaded area in (a).
Cracks
s1
s2
t
t
Fig. 3-13
Stresses in a biaxially loaded, cracked-concrete panel with cracks parallel to the direction of the principal compression stress.
acting on the top of the shaded portion is equilibrated by compressive stresses and probably some bearing stresses on the bottom and shearing stresses along the edges, as shown in Fig. 3-13b. When the crack widths are small, the shearing stresses transfer sufficient load across the cracks that the compressive stress on the bottom of the shaded portion is not sig- nificantly larger than that on the top, and the strength is unaffected by the cracks. As the crack widths increase, the ability to transfer shear across them decreases. For equilibrium, the com- pressive stress on the bottom of the shaded portion must then increase. Failure occurs when the highest stress in the element approaches the uniaxial compressive strength of the concrete.
Tests of concrete panels loaded in in-plane shear, carried out by Vecchio and Collins [3-33], have shown a relationship between the transverse tensile strain, and the com- pressive strength parallel to the cracks,
(3-13) where the subscripts 1 and 2 refer to the major (tensile) and minor (compressive) principal stresses and strains. The average transverse strain, , is the average transverse strain measured on a gauge length that includes one or more cracks. Equation 3-13 is plotted in Fig.3-14a. An increase in the strain leads to a decrease in compressive strength. The same authors [3-34]
recommended a stress–strain relationship, for transversely cracked concrete:
(3-14) f2 = f2maxc2aP2
eob - aP2
eob2d f2–P2,
P1
P1
f2max
fcœ = 1 0.8 + 170P1
f2max:
P1,
where is given by Eq. (3-13), and is the strain at the highest point in the compres- sive stress–strain curve, which the authors took as 0.002. The term in brackets describes a parabolic stress–strain curve with apex at and a peak stress that decreases as increases.
If the parabolic stress–strain curve given by Eq. (3-14) is used, the strain for any given stress can be computed from
(3-15) If the descending branch of the curve is also assumed to be a parabola, Eq. (3-15) can be used to compute strains on the postpeak portion of the stress–strain curve if the minus sign before the radical is changed to a plus.
The stress–strain relationships given by Eqs. (3-13) and (3-14) represent stresses and strains averaged over a large area of a shear panel or beam web. The strains computed in this way include the widths of cracks in the computation of tensile strains, as shown in the inset to Fig. 3-14a. These equations are said to represent smearedproperties. Through smearing, the peaks and hollows in the strains have been attenuated by using the averaged stresses and strains. In this way, Eqs. (3-13) and (3-14) are an attempt to replace the stress analysis of a cracked beam web having finite cracks with the analysis of a continuum. This substitution was a breakthrough in the analysis of concrete structures.
Triaxial Loadings
Under triaxial compressive stresses, the mode of failure involves either tensile fracture parallel to the maximum compressive stress (and thus orthogonal to the maximum tensile strain, if such exists) or a shear mode of failure. The strength and ductility of concrete under triaxial compression exceed those under uniaxial compression, as shown in Fig.3-15. This figure presents the stress–longitudinal strain curves for cylinders each subjected to a con- stant lateral fluid pressure , while the longitudinal stress, was increased to failure. These tests suggested that the longitudinal stress at failure was
(3-16) s1 = fcœ + 4.1s3
s1, s2 = s3
P1, Pc = Pcœa1 - A
f2 fcœb
P1
eo eo f2max
0.5 1.0
0 0.005 0.010 0.015
f2 P1
P10
P10.005 P2
f2
(a)
0 0.001 0.002 0.003
(b)
Transverse strain, P1, tensile Longitudinal strain, P2, compressive fc
f2max fc
Eq. (3-13)
Eq. (3-14) fc
Fig. 3-14
Effect of transverse tensile strains on the compressive strength of cracked concrete.
Fig. 3-16
Mohr rupture envelope for concrete tests from Fig. 3-15.
Tests of lightweight and high-strength concretes in [3-8] and [3-35] suggest that their compressive strengths are less influenced by the confining pressure, with the result that the coefficient 4.1 in Eq. (3-16) drops to about 2.0.
The strength of concrete under combined stresses can also be expressed via a Mohr rupture envelope. The Mohr’s circles plotted in Fig.3-16correspond to three of the cases plotted in Fig. 3-15. The Mohr’s circles are tangent to the Mohr rupture envelope shown with the outer line.
In concrete columns or in beam–column joints, concrete in compression is sometimes enclosed by closely spaced hoops or spirals. When the width of the concrete element increases due to Poisson’s ratio and microcracking, these hoops or spirals are stressed in tension, caus- ing an offsetting compressive stress in the enclosed concrete. The resulting triaxial state of stress in the concrete enclosed or confinedby the hoops or spirals increases the ductility and strength of the confined concrete. This effect is discussed in Chapters 11 and 19.
20,000
16,000
12,000
8000
4000
00 0.01 0.02 0.03 0.04 0.05 0.06 0.07
Strain (in./in.) Axial stress, s1 (psi)
s3⫽ 0
s3⫽ 550 psi s3⫽ 1090 psi
s3⫽ 2010 psi
s3⫽ 4090 psi
s3
s1 s1
s3
Fig. 3-15
Axial stress–strain curves from triaxial compression tests on concrete cylinders;
unconfined compressive strength
(From [3-3].)
fcœ = 3600 psi.
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.