Civil Engineering Reference
In-Depth Information
(2,000,000 psi to 6,000,000 psi), depending on factors such
as compressive strength and aggregate type. For normal-
density concrete with compressive strengths (¯) between
20 MPa and 35 MPa (3000 psi and 5000 psi), the modulus
of elasticity can be estimated as 5000 times the square root
of ¯ (57,000 times the square root of ¯ in psi). The mod-
ulus of elasticity for structural lightweight concrete is
between 7000 MPa and 17,000 MPa (1,000,000 psi and
2,500,000 psi).
E
for any particular concrete can be deter-
mined in accordance with ASTM C 469.
Shear Strain
Concrete, like other materials, deforms under shear forces.
The shear strain produced is important in determining the
load paths or distribution of forces in indeterminate struc-
tures—for example where shear-walls and columns both
participate in resisting horizontal forces in a concrete
building frame. The amount of movement, while not
large, is significant in short, stubby members; in larger
members it is overshadowed by flexural strains.
Calculation of the shear modulus (modulus of rigidity),
G
,
is shown in Fig. 15-25;
G
varies with the strength and
temperature of the concrete.
Deflection
Deflection of concrete beams and slabs is one of the more
common and obvious building movements. The deflec-
tions are the result of flexural strains that develop under
dead and live loads and that may result in cracking in the
tensile zone of concrete members. Reinforced concrete
structural design anticipates these tension cracks. Concrete
members are often cambered, that is, built with an upward
bow, to compensate for the expected later deflection.
Displacement
=
6wh
2
10GA
Area = A
Poisson's Ratio
w
h
When a block of concrete is loaded in uniaxial compres-
sion, as in Fig. 15-24, it will shorten and at the same time
develop a lateral strain or bulging. The ratio of lateral to
axial strain is called Poisson's ratio, µ. A common value
used is 0.20 to 0.21, but the value may vary from 0.15 to
0.25 depending upon the aggregate, moisture content, con-
crete age, and compressive strength. Poisson's ratio (ASTM
C 469) is generally of no concern to the structural designer;
it is used in advanced structural analysis of flat-plate
floors, shell roofs, arch dams, and mat foundations.
E
2(1 +
µ
)
G =
Fig. 15-25. Strain that results from shear forces on a body.
G
= shear modulus.
µ
= Poisson's ratio. Strain resulting
from flexure is not shown.
Torsional Strain
Plain rectangular concrete members can also fail in tor-
sion, that is, a twisting action caused by bending about an
axis parallel to the wider face and inclined at an angle of
about 45 degrees to the longitudinal axis of a member.
Microcracks develop at low torque; however, concrete
behaves reasonably elastic up to the maximum limit of the
elastic torque (
Hsu 1968
).
µ
=
CREEP
When concrete is loaded, the deformation caused by the
load can be divided into two parts: a deformation that
occurs immediately (elastic strain) and a time-dependent
deformation that begins immediately but continues at a
decreasing rate for as long as the concrete is loaded. This
latter deformation is called creep.
The amount of creep is dependent upon (1) the mag-
nitude of stress, (2) the age and strength of the concrete
Fig. 15-24. Ratio of lateral to axial strain is Poisson's ratio,
µ
.
