Philippines is one the countries in recent years, experiencing construction
boom due to economic growth and increasing population. At every corner,
residential buildings, low-rise, and high-rise structures are built to meet the
increasing demands of the populace such as dwellings and workplaces. Billions
of pesos are allotted for construction of roads, highways, and briges to
connect remote areas to urban areas.  Most
these structures are made of reinforced concrete (RC). Reinforced concrete, as
a structural material have the advantages against other materials such as low
cost, availability, adaptability and environmental considerations. Due to the
rapid use of RC in building constructions, this designed buildings should
behave in the most desirable manner in response to the applied loads.

of the previous studies, the desired behavior of RC members especially on the
verge of collapse should be ductile in manner.  Ductility is defined as the ability to undergo
deformations without a substantial reduction in the flexural capacity of the
member (Park & Ruitong 1988). Ductility of reinforced structures is a
desirable property where resistance to brittle failure during flexure is
required to ensure structural integrity (Olivia& Mandal 2005).

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the actual practice, structural designs if not all may have disregarded the
importance of the ductility in the designed structures which result to collapse.
One reason is that structural designers need to meet the requirements given to
them by the architects. A common scenario, structural designers are forced to
reduce dimensions, the capacity may not be sacrificed but the ductility is
affected. Meeting ductile behaviour in the designs may be too complex for some
structural engineers. Contractors on the other hand can be blamed because they
focus more on the practicality than safety. A learned structural designer should
consider the benefits of having structural elements behaving in a ductile

The first part the study is focused
on the theoretical-analytical calculation of the quantities in an RC beam using
stress-strain models for concrete and steel.  Parametric study is being carried out by varying
the amount tensile reinforcement, varying the yield strength of steel, varying
the compressive strength of concrete and varying the dimensions as to how they affect
the overall flexural behavior. The flexural behavior refers to the overall
evaluation of the flexural capacity or strength and the flexural ductility.

second part of the study is the plotting of the moment-curvature relationship
on a beam section, based on the control specimen. The analysis of the beam
section follows the Models for unconfined concrete and Models of different
grades of steel. The flexural behavior of the beam is evaluated using the shape
of the moment-curvature diagram.

Ductility/Curvature Ductility

way of measuring the ductility of a beam is the determining of its curvature. In
RC flexural members, ductile failure is attained when steel has yielded before
concrete crushes. The yielding of steel occurs if it reaches and exceeds its specified
yield strength. It is desired to have a ductile mode of failure in designing
structures for concrete is a brittle material. Brittle materials do not exhibit
substantial deformations before failure.

The effect of
curvature in the ductility of the beam is shown in Fig. 1a. The presented figure is based on the typical strain
diagram of the RC beam at crack initiation, flexural yielding, and ultimate
state. The strain diagrams are all linear in nature to conform to the basic
assumption in flexure analysis that plane sections before bending remain plane
after bending. The figure shows that the depth of compression block increases
with slight increase in the strain of concrete in the compression side as the
analysis progresses from initial to the ultimate stage. Also, the intensity of
the curvature ? increases as the stage progresses.












Figure 1a. Curvature in
Reinforced Concrete Beam








?s < ?y   ?cc   ?ct ?s' ?s AFTER CRACKING ?cc   ?s' ?s = ?y   YIELDING ?cu  =0.003   ?s' ?s > ?y






The calculation of
curvature ? is given by the formula,

           in rad/mm

?cc is the strain of concrete at the extreme compression fiber and c is the depth of the compression block.
As ductile behavior is followed, the rate of increase in the strain of steel
must be larger compared to that of the concrete. Thus, as the strain of steel
increases the depth of compression decreases which causes the increase in the
curvature of the beam. 

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