Page 191 - Civil Engineering Formulas

CONCRETE FORMULAS 127

where M and M are bending moments about the x and y axes, and M and M 0y
0x
z
y
are the values of M for bending about these axes.
0
PROPERTIES IN THE HARDENED STATE

Strength is a property of concrete that nearly always is of concern. Usually, it is
determined by the ultimate strength of a specimen in compression, but sometimes
flexural or tensile capacity is the criterion. Because concrete usually gains
strength over a long period of time, the compressive strength at 28 days is com-
monly used as a measure of this property.
The 28-day compressive strength of concrete can be estimated from the
7-day strength by a formula proposed by W. A. Slater:
S 28 S 7 30 2 7 (5.26)
S
2
where S 28-day compressive strength, lb/in (MPa); and S 7-day strength,
28
7
2
lb/in (MPa).
Concrete may increase significantly in strength after 28 days, particularly when
cement is mixed with fly ash. Therefore, specification of strengths at 56 or 90 days
is appropriate in design.
Concrete strength is influenced chiefly by the water/cement ratio; the higher this
ratio is, the lower the strength. The relationship is approximately linear when
expressed in terms of the variable C/W, the ratio of cement to water by weight. For
a workable mix, without the use of water reducing admixtures,
C
S 28 2700 760 (5.27)
W
Tensile strength of concrete is much lower than compressive strength and,
f c
regardless of the types of test, usually has poor correlation with . As deter-
mined in flexural tests, the tensile strength (modulus of rupture—not the true
strength) is about 7 f c for the higher strength concretes and 10 f c for the
lower strength concretes.
Modulus of elasticity E , generally used in design for concrete, is a secant
c
modulus. In ACI 318, “Building Code Requirements for Reinforced Concrete,”
it is determined by
1.5
f
E c w 33 2 c (5.28)
3
3
where w weight of concrete, lb/ft (kg/m ); and f specified compressive
c
2
strength at 28 days, lb/in (MPa). For normal-weight concrete, with w 145 lb/
3
3
ft (kg/m ),
f
E c 57,000 2 c (5.29)
The modulus increases with age, as does the strength.