Civil Engineering Reference
In-Depth Information
1.3.1.5 Softened Truss Model
Principles:
2-D Equilibrium
condition,
Mohr's compatibility
condition and the
2-D soft-
ened constitutive law
for concrete. The constitutive law of reinforcement may
be linear or nonlinear
Applications:
Analysis and design of
V
and
T
in the main regions at both the serviceability
and the ultimate load stages
1.3.1.6 Softened Membrane Model
Principles:
2-D Equilibrium
condition,
Mohr's compatibility
condition and the
2-D soft-
ened constitutive law
for concrete. The constitutive law of reinforcement may
be linear or nonlinear. The
Poisson effect
is included in the analysis
Applications:
Analysis and design of
V
and
T
in the main regions at both the serviceability
and the ultimate load stages
1.3.2 Historical Development of Theories for Reinforced Concrete
1.3.2.1 Principles of mechanics of materials
The behavior of a beam subjected to bending was first investigated by Galileo in 1638. In his
famous topic
Dialogues on Two New Sciences
, he studied the equilibrium of a stone cantilever
beam of rectangular section and found that the beam could support twice as much load at the
center as at the free end, because a same magnitude of 'bending moment' was produced at the
fixed end. Using the rudimentary knowledge of equlibrium, he observed that, for beams 'of
equal length but unequal thickness, the resistance to fracture increases in the same ratio as the
cube of the thickness (provided the thickness-to-width ratio remains unchanged)'. Galileo's
work represented the beginning of a scientific discipline known as the '
mechanics of materials
'
.
Since Galileo's beam was considered a rigid body, the deflections of the beam could
not be evaluated, thus creating the mystery known as 'Galileo's problem'. The solution to
Galileo's problem required two additional sources of information in addition to the principle
of equilibrium. The first source came from an understanding of the mechanical properties of
materials, summarized as follows: In 1678, Hooke measured the elongations of a long, thin
metal wire suspended from a high ceiling at one end and carrying a weight at the bottom
end. By systematically varying the weight, he reported that the 'deformation is proportional to
force' for wires of various materials under light loads. In 1705, James Bernoulli, a member of
a prominent family of Swiss scholars, defined the concept of stress (force divided by area) and
strain (displacement divided by original length). This was followed by Euler's postulation in
1727 of 'stress is proportional to strain'. The proportionality constant between stress and strain
E
was measured by Young in 1804 for many materials and was known as Young's modulus. It
took 166 years to develop the well-known
Hooke's law
.
The second source of information came from the observation of deformations in beams. In
relating the radius of curvature of a beam to the bending moment, Jacob Bernoulli, James'
brother, postulated in 1705 the well-known 'Bernoulli's hypothesis', i.e. 'a plane section
remains plane'. It should be noted that Jacob Bernoulli misunderstood the neutral axis and took
it at the concave surface of the beam. As a result, his derived flexural rigidity
EI
was twice the
correct value. Nevertheless, based on Bernoulli's hypothesis and assuming the proportionality
