Reinforced
Concrete
Design
Strength Design
Engr. Alexander D. Co
Engr. Alexander D. Co 2
Reinforced Concrete Design
REFERENCES
Jack C. McCormac and James K. Nelson, Design of Reinforced Concrete 7th Edition, John Wiley & Sons
Inc. 2006
Arthur H. Nilson, David Darwin and Charles W. Dolan, Design of Concrete Structures, Fourteenth Edition
in SI units, Mc Graw Hill, 2010
Association of Structural Engineers of the Philippines Inc. (ASEP), National Structural Code of the
Philippines, 6th Edition, 2010
Not for Sale
Engr. Alexander D. Co 3
Reinforced Concrete Design
Introduction
Not for Sale
Engr. Alexander D. Co 4
Reinforced Concrete Design
PROPERTIES OF REINFORCED CONCRETE
Compressive Strength
The compressive strength of concrete is determined by testing to failure 28 day old 6-
inches by 12-inches concrete cylinders at a specified rate of loading. For 28 day period the
cylinders are usually kept under water or in a room with constant temperature and 100%
humidity. Although concretes are available with 28-day ultimate strengths from 2500 psi (17.50
MPa) up to as high as 10,000 to 20,000 psi (70 to 140 MPa), most of the concretes used fall into
the 3000 to 7000 psi (21 to 49 MPa) range.
The stress-strain curves of Fig. 1.1 represent the results obtained from compression tests of sets
of 28-day-old standard cylinders of varying strengths. Careful study of these curves will bring
out several significant points:
a) The curves are roughly straight while the load is increased from zero to about one-third
to one-half the concrete’s ultimate strength.
b) Of particular importance is the fact that regardless of strengths, all concretes reach their
ultimate strengths at strain of about 0.002.
c) Concrete does not have definite yield strength; rather, the curves run smoothly on to
the point of rupture at strain of from 0.003 to 0.004. It will be assumed for the purpose
of future calculations in this text that concrete fails at 0.003. The reader should not that
Not for Sale
Engr. Alexander D. Co 5
Reinforced Concrete Design
this value, which is conservative for normal-strength concretes, may not be conservative
for higher-strength concretes in the 8000-psi and above range.
d) Many tests have clearly shown that stress-strain curves of concrete cylinders are almost
identical to those for the compression sides of beam.
e) It should be further noticed that the weaker grades of concrete are less brittle than the
stronger ones.
Static Modulus of Elasticity (NSCP 6th ed, page 4-38)
Concrete has no clear-cut modulus of elasticity. It value varies with different concrete strengths,
concrete age, type of loading, and the characteristics and proportions of the cement and
aggregates.
= varying from 1500 to 2500
= Compressive strength of concrete in
Should normal crushed stone or gravel concrete with mass approximately 2320 kg/m3 be used
Concrete with strength above 6000 psi (42 MPa) are referred to as high strength concretes.
Tests have indicated that the usual ACI equations for when applied to high strength concretes
result in value that are too large. Based on studies at Cornell University, the expression to
follow has been recommended for normal-weight concretes with values greater than 42
MPa and up to 84 MPa and for light weight concretes with greater than 42 MPa and up to
63 MPa.
Tensile Strength
The tensile strength of concrete varies from about 8 to 15% of its compressive strength. A major
reason for this small strength is the fact that concrete is filled with fine cracks.
The tensile strength of concrete doesn’t vary in direct proportion to the square root of . This
strength is quite difficult to measure with direct axial tension loads because problems in
gripping test specimens so as to avoid stress concentrations and because of difficulties
aligning the loads. As a result of these problems, two rather indirect tests have been
developed to measure concrete’s tensile strength. These are the modulus of rupture and
the split-cylinder tests.
Not for Sale
Engr. Alexander D. Co 6
Reinforced Concrete Design
The tensile strength of concrete is flexure is quite important when considering beam
cracks and deflections. For these considerations the tensile strength obtained with the
modulus of rupture test have long been used. The modulus of rupture is usually
measured by loading a 6-in x 6-in x 30-in plain rectangular beam with simple supports
placed 24 in. on center) to failure with equal concentrated loads at its one-third points
as per ASTM C496-96.
The stress determined in this manner is not very accurate because in using the flexure formula
we are assuming the concrete is perfectly elastic, with stresses varying in direct proportion to
distance from the neutral axis. These assumptions are not very good.
Based on hundred of tests, the Code provides a modulus of rupture equal to 0.7 where
is in MPa.
The tensile strength of concrete may also be measured with the split-cylinder test.
Not for Sale
Engr. Alexander D. Co 7
Reinforced Concrete Design
The cylinder will split in half from end to end when its tensile strength is reached. The tensile
strength at which splitting occurs is referred to as the split-cylinder strength and can be
calculated with the following expression,
In which is the maximum compressive force, is the length, and is the diameter of the
cylinder. The tensile strengths obtained are not very accurate.
SI BAR SIZE
Not for Sale
Engr. Alexander D. Co 8
Reinforced Concrete Design
IDENTIFYING MARKS ON REINFORCING BARS
Not for Sale
Engr. Alexander D. Co 9
Reinforced Concrete Design
Flexure Analysis of Beam
Not for Sale
Engr. Alexander D. Co 10
Reinforced Concrete Design
In this section it is assumed that a small transverse load is placed on a concrete beam with tensile
reinforcing and that the load is gradually increased in magnitude until the beam fails. As this takes place
we will find that the beam will go through three distinct stages before collapse occurs. These are: (1)
the uncracked concrete stage, (2) the concrete cracked-elastic stress stage, and (3) the ultimate-strength
stage. A relatively long beam is considered for this discussion so that shear will not have a large effect
on its behaviour.
Uncracked Concrete Stage
At small load when the tensile stresses are less than the modulus of rupture (the bending tensile stress
at which the concrete begins to crack), the entire cross section of the beam resist bending with
compression on one side and tension on the other.
Concrete Cracked-Elastic Stresses Stage
As the load is increased after the modulus of rupture of the concrete is exceeded, cracks begin to
develop in the bottom of the beam. The moment at which these cracks begin to form-that is, when the
tensile stress in the bottom of the beam equals the modulus of rupture-is referred to as the cracking
moment, As the load is further increased, these cracks quickly spread up to the vicinity of the
neutral axis, and then the neutral axis begins to move upward. The cracks occur at those places along
the beam where the actual moment is greater than the cracking moment.
Now that the bottom has cracked, another stage is present because the concrete in the cracked zone
obviously cannot resist tensile stresses – the steel must do it.
Not for Sale
Engr. Alexander D. Co 11
Reinforced Concrete Design
Beam Failure – Ultimate-Strength Stage
As the load is increased further so that the compressive stresses are greater than one-half of , the
tensile cracks move further upward, as does the neutral axis ad the concrete compression stresses begin
to change appreciably from a straight line. For this initial discussion it is assumed that the reinforcing
bars have yielded.
Not for Sale
Engr. Alexander D. Co 12
Reinforced Concrete Design
CRACKING MOMENT
The area of reinforcing as a percentage of the total cross-sectional area of a beam is quite small (usually
2% or less), and its effect on the beam properties is almost negligible as long as the beam is uncracked.
Where:
Is the bending Cracking Moment; = gross moment of inertia of the cross section; is the
modulus of rupture; is the distance from the centroidal axis of the section to its extreme fiber in
tension
SAMPLE ILLUSTRATION
1. Assuming the concrete is uncracked, compute the flexure stresses in the extreme fibers of the
beam as shown in the figure for a bending moment of 34 kN-m. The concrete has an of
27.60 MPa; determine also the cracking moment of the section
380 455
75
300
Solution:
a) Flexure stress
Not for Sale
Engr. Alexander D. Co 13
Reinforced Concrete Design
Since the flexural stress is less than the tensile strength or Modulus of rupture of the concrete, the
section is assumed not to have cracked.
b) Cracking Moment
ELASTIC STRESSES – CONCRETE CRACKED
When the bending moment is sufficiently large to cause the tensile stress in the extreme fibers to be
greater than the modulus of rupture, it is assumed that all of the concrete on the tensile side of the beam
is cracked and must be neglected in the flexure calculations.
On the tensile side of the beam an assumption of perfect bond is made between the reinforcing bars
and the concrete. Thus the strain in the concrete and in the steel will be equal at equal distances from
the neutral axis. But if the strains in the two materials at a particular point are the same, their stresses
cannot be the same since they have different moduli of elasticity. Thus their stresses are in proportion
to the ratio of their moduli of elasticity. The ratio of the steel modulus to the concrete is called the
modular ration n:
For the figure shown, the steel area is replaced with an equivalent area of fictitious cement , which
supposedly can resist tension. This area is referred to as the transformed area. The resulting revised
cross section or transformed section is handled by the usual methods for elastic homogeneous beams.
Not for Sale
Engr. Alexander D. Co 14
Reinforced Concrete Design
SAMPLE ILLUSTRATION
1. Calculate the bending stresses in the beam shown below by using transformed area method: n =
10 and M = 100 kN-m
SOLUTION
Taking Moments about Neutral Axis
Moment of Inertia
Bending Stresses
a) Concrete
b) Tension steel
Discussion
For a given beam, the concrete and steel will not usually reach their maximum allowable stresses at
exactly the same bending moments. Such is the case for this example beam.
Not for Sale
Engr. Alexander D. Co 15
Reinforced Concrete Design
CE BOARD – August 1971
2. A beam having an 8 meter simple span has a cross-section shown in the figure. Determine the
maximum concentrated load that this beam can carry besides its own weight. Use fc = 7MPa,
fs = 124 MPa, n = 10
Taking Moments about Neutral Axis (assuming x>100mm)
Moment of Inertia
Bending Stresses
a) Concrete
Not for Sale
Engr. Alexander D. Co 16
Reinforced Concrete Design
b) Tension steel
Use
Weight of the beam
Discussion
The beam could carry a concentrated load amounting to a force of 44.34 KN and the beam is
approximately balance in tension and compression.
CE BOARD – February 1957
3. A concrete beam is carrying a uniform load on a simple span of 10 meters. If the allowable
stresses are fc = 6MPa, fs = 124 MPa and n = 15, what uniform load can the beam carry besides
its own weight?
Not for Sale
Engr. Alexander D. Co 17
Reinforced Concrete Design
Taking Moments about Neutral Axis
Moment of Inertia
Bending Stresses
a) concrete
b) Tension steel
Use
Not for Sale
Engr. Alexander D. Co 18
Reinforced Concrete Design
Uniform load that the beam can carry
Example no 4 illustrates the analysis of a double reinforced concrete beam. Should the
compression side of a beam be reinforced, the long-term stresses in that reinforcing will be greatly
affected by the creep in the concrete. As time goes by, the compression concrete will compact more
tightly, leaving the reinforcing bars to carry more and more of the load.
As a consequence of this creep in the concrete the stresses in the compression bars computed
by the transformed-area method are assumed to equal times their area .
On the subject of “hair splitting,” it will be noted in the example that the compression steel area
is really multiplied by . The transformed area of the compression side equals the gross
compression area of the concrete plus minus the area of the holes in the concrete , which
theoretically should not have been included in the concrete part. This equals the compression concrete
area plus . Similarly, is used in the moment of inertia calculations.
4. The beam has a simple span of 4 meters and carries a super-imposed load of 25 kN/m, what
additional load may be placed at the center of the span if the deflection due to this load is not to
exceed 1/800 of the span. Use fc = 9MPa, fs = 124 MPa, n = 10, fv = 2MPa, and Ec = 20700 MPa
Taking Moments about Neutral Axis
Not for Sale
Engr. Alexander D. Co 19
Reinforced Concrete Design
Moment of Inertia
Bending Stresses
a) concrete
b) compression steel
c) Tension steel
Use
Weight of the beam
Shear Stress
Deflection
Not for Sale
Engr. Alexander D. Co 20
Reinforced Concrete Design
Use P = 12.48 KN
Not for Sale
