PORTLAND CEMENT CONCRETE - Materials for Civil and Construction Engineers - page 277

8

Water-cementitious
materials ratio 0.33
0.40

50

6

40

0.50

30

4

20

0.67
1.00

2

10

FIGURE 7.29 Typical
stress-strain relations for com-
pressive Tests on 0.15 x 0.30-m
concrete cylinders at an age of
28 days.

0 0

0 1000 2000 3000 4000 5000

Strain, 10 6

the stress-strain relation to some extent. Therefore, a specific rate of loading
is required for testing concrete. It is interesting to note that the shape of the
stress-strain relationship of concrete is almost the same for both compres-
sion and tension, although the tensile strength is much smaller than the
compressive strength. In fact, the tensile strength of concrete typically is ig-
nored in the design of concrete structures.
The modulus of elasticity of concrete is commonly used in designing
concrete structures. Since the stress-strain relationship is not exactly linear,
the classic definition of modulus of elasticity (Young's modulus) is not ap-
plicable. The initial tangent modulus of concrete has little practical impor-
tance. The tangent modulus is valid only for a low stress level where the
tangent is determined. Both secant and chord moduli represent “average”
modulus values for certain stress ranges. The chord modulus (referred to as
the modulus of elasticity) in compression is more commonly used for con-
crete and is determined according to ASTM C469. The method requires
three or four loading and unloading cycles, after which the chord modulus
is determined between a point corresponding to a very small strain value
and a point corresponding to either 40% of the ultimate stress or a specific
strain value. Normal-weight concrete has a modulus of elasticity of 14 GPa
to 40 GPa (2,000 ksi to 6,000 ksi).
Poisson's ratio can also be determined using ASTM C469. Poisson's ratio
is used in advanced structural analysis of shell roofs, flat-plate roofs, and
mat foundations. Poisson's ratio of concrete varies between 0.11 and 0.21,
depending on aggregate type, moisture content, concrete age, and compressive
strength. A value of 0.15 to 0.20 is commonly used.
It is interesting to note that both aggregate and cement paste, when tested
individually, exhibit linear stress-strain behavior. However, the stress-strain
relation of concrete is nonlinear. The reason for this behavior is attributed to
the microcracking in concrete at the interface between aggregate particles and
the cement paste (Shah and Winter 1968).
The modulus of elasticity of concrete increases when the compressive
strength increases, as demonstrated in Figure 7.29. There are several empirical