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• Be sure we shall test you withsomething of fear and hunger, someloss in goods, lives and fruit of yourtoils, but give glad tidings to those whopatiently persevere. Who say whenafflicted with calamity “To Allah webelong and to Him is our return”. Theyare those on whom descend theBlessings from their Lord and Mercyand they are the ones that receiveguidance.
1
Design Methods
2
Design MethodsFrom the early 1900s until the early 1960s, nearly all reinforced
concrete design was performed by the working-stress design method (also
called allowable-stress design or straight-line design). In this method,
frequently referred to as WSD, the dead and live loads to be supported,
called working loads or service loads, were first estimated. Then the
members of the structure were proportioned so that stresses calculated by
a transformed area did not exceed certain permissible or allowable values.
After 1963, the ultimate-strength design method rapidly gained
popularity because:
(1) it makes use of a more rational approach than does WSD,
(2) it uses a more realistic consideration of safety, and
(3) it provides more economical designs.
3
Design MethodsWith this method (now called strength design), the working dead
and live loads are multiplied by certain load factors (equivalent to safety
factors), and the resulting values are called factored loads. The members
are then selected so they will theoretically just fail under the factored
loads.
Even though almost all of the reinforced concrete structures the
reader will encounter will be designed by the strength design method, it is
still useful to be familiar with WSD for several reasons:
1. Some designers use WSD for proportioning fluid-containing
structures (such as water tanks and various sanitary structures).
When these structures are designed by WSD, stresses are kept at
fairly low levels, with the result that there is appreciably less4
Design Methodscracking and less consequent leakage. (If the designer uses
strength design and makes use of proper crack control methods,
there should be even fewer leakage problems.)
2. The ACI method for calculating the moments of inertia to be used
for deflection calculations requires some knowledge of the
working-stress procedure.
3. The design of pre-stressed concrete members is based not only
on the strength method but also on elastic stress calculations at
service load conditions.
5
Design MethodsIt should be realized that working-stress design has several
disadvantages. When using the method, the designer has little knowledge
about the magnitudes of safety factors against collapse; no consideration
is given to the fact that different safety factors are desirable for dead and
live loads; the method does not account for variations in resistances and
loads, nor does it account for the possibility that as loads are increased,
some increase at different rates than others.
In 1956, the ACI Code for the first time included ultimate-strength
design, as an appendix, although the concrete codes of several other
countries had been based on such considerations for several decades. In
1963, the code gave ultimate-strength design equal status with working-
stress design; the 1971 code made the method the predominant method
and only briefly mentioned the working-stress method.6
Design MethodsFrom 1971 until 1999, each issue of the code permitted designers
to use working-stress design and set out certain provisions for its
application. Beginning with the 2002 code, however, permission is not
included for using the method.
Today’s design method was called ultimate-strength design for
several decades, but, the code now uses the term strength design. The
strength for a particular reinforced concrete member is a value given by
the code and is not necessarily the true ultimate strength of the member.
Therefore, the more general term strength design is used whether beam
strength, column strength, shear strength, or others are being considered.
7
Advantages of Strength Design
8
Advantages of Strength DesignAmong the several advantages of the strength design method as compared
to the no-longer permitted working-stress design method are the
following:
1. The derivation of the strength design expressions takes into
account the nonlinear shape of the stress–strain diagram. When
the resulting equations are applied, decidedly better estimates of
load-carrying ability are obtained.
2. With strength design, a more consistent theory is used
throughout the designs of reinforced concrete structures.
9
Advantages of Strength Design3. A more realistic factor of safety is used in strength design. The
designer can certainly estimate the magnitudes of the dead loads
that a structure will have to support more accurately than he or she
can estimate the live and environmental loads. With working stress
design, the same safety factor was used for dead, live, and
environmental loads. This is not the case for strength design. For
this reason, use of different load or safety factors in strength design
for the different types of loads is a definite improvement.
4. A structure designed by the strength method will have a more
uniform safety factor against collapse throughout. The strength
method takes considerable advantage of higher strength steels,
whereas working-stress design did so only partly. The result is
better economy for strength design.10
Advantages of Strength Design5. The strength method permits more flexible designs than did the
working-stress method. For instance, the percentage of steel may
be varied quite a bit. As a result, large sections may be used with
small percentages of steel, or small sections may be used with large
percentages of steel. Such variations were not the case in the
relatively fixed working stress method. If the same amount of steel
is used in strength design for a particular beam as would have been
used with WSD, a smaller section will result. If the same size
section is used as required by WSD, a smaller amount of steel will
be required.
11
Structural Safety
12
Structural SafetyThe structural safety of a reinforced concrete structure can be
calculated with two methods. The first method involves calculations of the
stresses caused by the working or service loads and their comparison with
certain allowable stresses. Usually the safety factor against collapse when
the working-stress method was used was said to equal the smaller of f’c/fc
or fy/fs .
The second approach to structural safety is the one used in
strength design in which uncertainty is considered. The working loads are
multiplied by certain load factors that are larger than 1. The resulting
larger or factored loads are used for designing the structure. The values of
the load factors vary depending on the type and combination of the loads.
13
Structural SafetyTo accurately estimate the ultimate strength of a structure, it is
necessary to take into account the uncertainties in material strengths,
dimensions, and workmanship. This is done by multiplying the theoretical
ultimate strength (called the nominal strength herein) of each member by
the strength reduction factor, φ, which is less than 1. These values
generally vary from 0.90 for bending down to 0.65 for some columns.
In summary, the strength design approach to safety is to select a
member whose computed ultimate load capacity multiplied by its strength
reduction factor will at least equal the sum of the service loads multiplied
by their respective load factors. Member capacities obtained with the
strength method are appreciably more accurate than member capacities
predicted with the working-stress method.
14
Derivation of Beam Expressions
16
Derivation of Beam ExpressionsTests of reinforced concrete beams confirm that strains vary in
proportion to distances from the neutral axis even on the tension sides
and even near ultimate loads. Compression stresses vary approximately in
a straight line until the maximum stress equals about 0.50f’c . This is not
the case, however, after stresses go higher. When the ultimate load is
reached, the strain and stress variations are approximately as shown in the
next figure.
The compressive stresses vary from zero at the neutral axis to a
maximum value at or near the extreme fiber. The actual stress variation
and the actual location of the neutral axis vary somewhat from beam to
beam, depending on such variables as the magnitude and history of past
loadings, shrinkage and creep of the concrete, size and spacing of tension
cracks, speed of loading, and so on.17
Derivation of Beam Expressions
If the concrete is assumed to crush at a strain of about 0.003
(which is a little conservative for most concretes) and the steel to yield at
fy, it is possible to make a reasonable derivation of beam formulas without
knowing the exact stress distribution. However, it is necessary to know the
value of the total compression force and its centroid.
18
Derivation of Beam ExpressionsIf the shape of the stress diagram were the same for every beam,
it would be possible to derive a single rational set of expressions for
flexural behavior. Because of these stress variations, however, it is
necessary to base the strength design on a combination of theory and test
results.
Although the actual stress distribution given in Figure 3.2(b) may
seem to be important, in practice any assumed shape (rectangular,
parabolic, trapezoidal, etc.) can be used if the resulting equations compare
favorably with test results. The most common shapes proposed are the
rectangle, parabola, and trapezoid, with the rectangular shape used in this
text as shown in Figure 3.2(c) being the most common one.
19
Derivation of Beam ExpressionsFor concretes with f’c > 4000 psi, β1 can be determined with the
following formula:
20
Derivation of Beam ExpressionsWhitney replaced the curved stress block [Figure 3.2(b)] with an
equivalent rectangular block of intensity 0.85f’c and depth α = β1c, as
shown in Figure 3.2(c). The area of this rectangular block should equal that
of the curved stress block, and the centroids of the two blocks should
coincide. Sufficient test results are available for concrete beams to provide
the depths of the equivalent rectangular stress blocks. The values of β1
given by the code (10.2.7.3) are intended to give this result. For f’c values
of 4000 psi or less, β1 = 0.85, and it is to be reduced continuously at a rate
of 0.05 for each 1000-psi increase in f’c above 4000 psi. Their value may
not be less than 0.65. The values of β1 are reduced for high-strength
concretes primarily because of the shapes of their stress–strain curves.
21
Derivation of Beam ExpressionsIn SI units, β1 is to be taken equal to 0.85 for concrete strengths
up to and including 30 MPa. For strengths above 30 MPa, β1 is to be
reduced continuously at a rate of 0.05 for each 7 MPa of strength in excess
of 30 MPa but shall not be taken less than 0.65.
For concretes with f’c > 30 MPa, β1 can be determined with the
following expression:
Based on these assumptions regarding the stress block, statics
equations can easily be written for the sum of the horizontal forces and for
the resisting moment produced by the internal couple. These expressions
can then be solved separately for a and for the moment, Mn.
22
Derivation of Beam ExpressionsA very clear statement should be made here regarding the term
Mn. Mn is defined as the theoretical or nominal resisting moment of a
section. In Section 3.3, it was stated that the usable strength of a member
equals its theoretical strength times the strength reduction factor, or, in
this case, φMn. The usable flexural strength of a member, φMn, must at
least be equal to the calculated factored moment, Mu, caused by the
factored loads
For writing the beam expressions, reference is made to Figure 3.3.
Equating the horizontal forces C and T and solving for a, we obtain
23
Derivation of Beam ExpressionsBecause the reinforcing steel is limited to an amount such that it
will yield well before the concrete reaches its ultimate strength, the value
of the nominal moment, Mn, can be written as
and the usable flexural strength is
If we substitute into this expression the value previously
obtained for a (it was ρfyd/0.85f’c ), replace As with ρbd, and equate φMn
to Mu, we obtain the following expression:
24
Derivation of Beam Expressions
Replacing As with ρbd and letting Rn = Mu/φbd², we can solve this
expression for ρ (the percentage of steel required for a particular beam)
with the following results:
25
Derivation of Beam ExpressionsInstead of substituting into this equation for ρ when rectangular
sections are involved, the Tables A.8 to A.13 in Appendix A will be quite
convenient. (For SI units, refer to Tables B.8 and B.9 in Appendix B.)
Another way to obtain the same information is to refer to Graph 1 in
Appendix A. This expression for ρ is also very useful for tensile reinforced
rectangular sections that do not fall into the tables. An iterative technique
for determination of reinforcing steel area is also presented later in this
chapter.
26
TABLES A-12fy = 60,000 PSI; f’c = 3000 PSI—U.S. Customary Units
TABLES A-12 (Cont’d)
GRAPH 1:Moment capacity of rectangularsections.
(Note: The upper ends of the curvesshown here for 40 ksi and 50 ksi barscorrespond to ρ values for which ϵt <0.004 in the steel.)
29
Strains in Flexural Members
30
Strains in Flexural MembersAs previously mentioned, Section 10.2.2 of the code states that
the strains in concrete members and their reinforcement are to be
assumed to vary directly with distances from their neutral axes. (This
assumption is not applicable to deep flexural members whose depths over
their clear spans are greater than 0.25.) Furthermore, in Section 10.2.3 the
code states that the maximum usable strain in the extreme compression
fibers of a flexural member is to be 0.003. Finally, Section 10.3.3 states that
for Grade 60 reinforcement and for all pre-stressed reinforcement we may
set the strain in the steel equal to 0.002 at the balanced condition.
(Theoretically, for 60,000-psi steel, it equals fy/Es = 60,000 psi/29 × 10⁶ psi =
0.00207.)
31
Strains in Flexural MembersIn Section 3.4, a value was derived for a, the depth of the
equivalent stress block of a beam. It can be related to c with the factor β1
also given in that section:
Then the distance c from the extreme concrete compression
fibers to the neutral axis is
In next example, the values of a and c are determined for the
beam previously considered in Example 2.8, and by straight-line
proportions the strain in the reinforcing ϵt is computed.
32
Strains in Flexural MembersExample 3.1
Solution
This value of strain is much greater than the yield strain of 0.002. This is an
indication of ductile behavior of the beam, because the steel is well into its yield
plateau before concrete crushes.
33
Strains in Flexural MembersExample 3.1
34
Balanced Sections, Tension-Controlled Sections,
and Compression-Controlled or Brittle Sections
35
Balanced Sections, Tension-Controlled Sections, andCompression-Controlled or Brittle Sections
A beam that has a balanced steel ratio is one for which the tensile
steel will theoretically just reach its yield point at the same time the
extreme compression concrete fibers attain a strain equal to 0.003. Should
a flexural member be so designed that it has a balanced steel ratio or be a
member whose compression side controls (i.e., if its compression strain
reaches 0.003 before the steel yields), the member can suddenly fail
without warning. As the load on such a member is increased, its
deflections will usually not be particularly noticeable, even though the
concrete is highly stressed in compression and failure will probably occur
without warning to users of the structure. These members are
compression controlled and are referred to as brittle members. Obviously,
such members must be avoided.36
Balanced Sections, Tension-Controlled Sections, andCompression-Controlled or Brittle Sections
The code, in Section 10.3.4, states that members whose
computed tensile strains are equal to or greater than 0.0050 at the same
time the concrete strain is 0.003 are to be referred to as tension-controlled
sections. For such members, the steel will yield before the compression
side crushes and deflections will be large, giving users warning of
impending failure. Furthermore, members with ϵt ≥ 0.005 are considered
to be fully ductile. The ACI chose the 0.005 value for ϵt to apply to all types
of steel permitted by the code, whether regular or pre-stressed. The code
further states that members that have net steel strains or ϵt values
between ϵy and 0.005 are in a transition region between compression-
controlled and tension-controlled sections. For Grade 60 reinforcing steel,
which is quite common, ϵy is approximated by 0.002.37
Strength Reduction or φ Factors
38
Strength Reduction or φ Factors
Strength reduction factors are used to take into account the
uncertainties of material strengths, inaccuracies in the design equations,
approximations in analysis, possible variations in dimensions of the
concrete sections and placement of reinforcement, the importance of
members in the structures of which they are part, and so on. The code
(9.3) prescribes φ values or strength reduction factors for most situations.
Among these values are the following:
0.90 for tension-controlled beams and slabs
0.75 for shear and torsion in beams
0.65 or 0.75 for columns
0.65 or 0.75 to 0.9 for columns supporting very small axial loads
0.65 for bearing on concrete
39
Strength Reduction or φ Factors
The sizes of these factors are rather good indications of our
knowledge of the subject in question. For instance, calculated nominal
moment capacities in reinforced concrete members seem to be quite
accurate, whereas computed bearing capacities are more questionable.
For ductile or tension-controlled beams and slabs where ϵt ≥
0.005, the value of φ for bending is 0.90. Should ϵt be less than 0.005, it is
still possible to use the sections if ϵt is not less than certain values. This
situation is shown in Figure 3.5, which is similar to Figure R.9.3.2 in the ACI
Commentary to the 2011 code.
40
Strength Reduction or φ Factors
41
Strength Reduction or φ Factors
Members subject to axial loads equal to or less than 0.10f’cAg may
be used only when ϵt is no lower than 0.004 (ACI Section 10.3.5). An
important implication of this limit is that reinforced concrete beams must
have a tension strain of at least 0.004. Should the members be subject to
axial loads ≥ 0.10f’cAg, then ϵt is not limited. When ϵt values fall between
0.002 and 0.005, they are said to be in the transition range between
tension-controlled and compression controlled sections. In this range, φ
values will fall between 0.65 or 0.70 and 0.90, as shown in Figure 3.5.
When ϵt ≤ 0.002, the member is compression controlled, and the column φ
factors apply.
42
Strength Reduction or φ FactorsThe procedure for determining φ values in the transition range is
described later in this section. One must clearly understand that the use of
flexural members in this range is usually uneconomical, and it is probably
better, if the situation permits, to increase member depths and/or decrease
steel percentages until ϵt is equal to or larger than 0.005. If this is done, not
only will φ values equal 0.9 but also steel percentages will not be so large
as to cause crowding of reinforcing bars. The net result will be slightly
larger concrete sections, with consequent smaller deflections.
Furthermore, as you will learn in subsequent chapters, the bond of the
reinforcing to the concrete will be increased as compared to cases where
higher percentages of steel are used.
43
Strength Reduction or φ FactorsWe have computed values of steel percentages for different
grades of concrete and steel for which ϵt will exactly equal 0.005 and
present them in Appendix Tables A.7 and B.7 of this textbook. It is
desirable, under ordinary conditions, to design beams with steel
percentages that are no larger than these values, and we have shown
them as suggested maximum percentages to be used.
The horizontal axis of Figure 3.5 gives values also for c/dt ratios. If
c/dt for a particular flexural member is ≤ 0.375, the beam will be ductile,
and if it is > 0.600, it will be brittle. In between is the transition range. You
may prefer to compute c/dt for a particular beam to check its ductility
rather than computing ρ or ϵt.
44
Strength Reduction or φ FactorsIn the transition region, interpolation to determine φ using c/dt
instead of ϵt, when 0.375 < c/dt < 0.600, can be performed using the
equations
The equations for φ here and in Figure 3.5 are for the special case
where fy = 60 ksi and for pre-stressed concrete. For other cases, replace
0.002 with ϵy = fy/Es . Figure 10.25 in Chapter 10 shows Figure 3.5 for the
general case, where ϵy is not assumed to be 0.002.
45
Strength Reduction or φ FactorsThe resulting general equations in the range ϵy < ϵt < 0.005 are
The impact of the variable φ factor on moment capacity is shown
in Figure 3.6. The two curves show the moment capacity with and without
the application of the φ factor. Point A corresponds to a tensile strain, ϵt, of
0.005 and ρ = 0.0181 (Appendix A, Table A.7). This is the largest value of ρ
for φ = 0.9. Above this value of ρ, φ decreases to as low as 0.65 as shown
by point B, which corresponds to ϵt of ϵy.
46
Strength Reduction or φ Factors
ACI 10.3.5 requires ϵt not be less than 0.004 for flexural members
with compressive axial loads less than 0.10 f’m Ag. This situation
corresponds to point C in Figure 3.6. The only allowable range for ρ is
below point C. From the figure, it is clear that little moment capacity is
gained in adding steel area above point A. The variable φ factor provisions
essentially permit a constant value of φMn when ϵt is less than 0.005. It is
important for the designer to know this because often actual bar
selections result in more steel area than theoretically required. If the slope
between points A and C were negative, the designer could not use a larger
area. Knowing the slope is slightly positive, the designer can use the larger
bar area with confidence that the design capacity is not reduced.
47
Strength Reduction or φ Factors
48
Strength Reduction or φ FactorsFor values of fy of 75 ksi and higher, the slope between point A
and B in Figure 3.6 is actually negative. It is therefore especially important,
when using high-strength reinforcing steel, to verify your final design to be
sure the bars you have selected do not result in a moment capacity less
than the design value.
Continuing our consideration of Figure 3.5, you can see that when
ϵt is less than 0.005, the values of φ will vary along a straight line from
their 0.90 value for ductile sections to 0.65 at balanced conditions where ϵt
is 0.002. Later you will learn that φ can equal 0.75 rather than 0.65 at this
latter strain situation if spirally reinforced sections are being considered.
49
Minimum Percentage of Steel
50
Minimum Percentage of SteelA brief discussion of the modes of failure that occur for various
reinforced beams was presented in Section 3.6. Sometimes, because of
architectural or functional requirements, beam dimensions are selected
that are much larger than are required for bending alone. Such members
theoretically require very small amounts of reinforcing.
Actually, another mode of failure can occur in very lightly
reinforced beams. If the ultimate resisting moment of the section is less
than its cracking moment, the section will fail immediately when a crack
occurs. This type of failure may occur without warning. To prevent such a
possibility, the ACI (10.5.1) specifies a certain minimum amount of
reinforcing that must be used at every section of flexural members where
tensile reinforcing is required by analysis, whether for positive or negative
moments.51
Minimum Percentage of SteelIn the following equations, bw represents the web width of beams.
The (200bwd)/ fy value was obtained by calculating the cracking moment of a
plain concrete section and equating it to the strength of a reinforced
concrete section of the same size, applying a safety factor of 2.5 and solving
for the steel required. It has been found, however, that when fc is higher
than about 5000 psi, this value may not be sufficient. Thus, the
value is also required to be met, and it will actually control when f’c is
greater than 4440 psi.52
Minimum Percentage of SteelThis ACI equation (10-3) for the minimum amount of flexural
reinforcing can be written as a percentage, as follows:
Values of ρmin for flexure have been calculated by the authors and are shown
for several grades of concrete and steel in Appendix A, Table A.7 of this text.
They are also included in Tables A.8 to A.13. (For SI units, the appropriate
tables are in Appendix B, Tables B.7 to B.9.). Section 10.5.3 of the code states
that the preceding minimums do not have to be met if the area of the tensile
reinforcing furnished at every section is at least one-third greater than the
area required by moment.
53
Minimum Percentage of SteelFurthermore, ACI Section 10.5.4 states that for slabs and footings of
uniform thickness, the minimum area of tensile reinforcing in the direction
of the span is that specified in ACI Section 7.12 for shrinkage and
temperature steel which is much lower. When slabs are overloaded in
certain areas, there is a tendency for those loads to be distributed laterally
to other parts of the slab, thus substantially reducing the chances of sudden
failure. This explains why a reduction of the minimum reinforcing percentage
is permitted in slabs of uniform thickness. Supported slabs, such as slabs on
grade, are not considered to be structural slabs in this section unless they
transmit vertical loads from other parts of the structure to the underlying
soil.
54
Balanced Steel Percentage
55
Balanced Steel PercentageIn this section, an expression is derived for ρb, the percentage of
steel required for a balanced design. At ultimate load for such a beam, the
concrete will theoretically fail (at a strain of0.00300), and the steel will
simultaneously yield (see Figure 3.7).
The neutral axis is located by the triangular strain relationships
that follow, noting that Es = 29 × 10⁶ psi for the reinforcing bars:
This expression is rearranged and simplified, giving
56
Balanced Steel Percentage
In Section 3.4 of this chapter, an expression was derived for depth
of the compression stress block, a, by equating the values of C and T. This
value can be converted to the neutral axis depth, c, by dividing it by β1:
57
Balanced Steel Percentage
Two expressions are now available for c, and they are equated to
each other and solved for the percentage of steel. This is the balanced
percentage, ρb:
Values of ρb can easily be calculated for different values of f’c and fy and
tabulated for U.S. customary units as shown in Appendix A, Table A.7. For SI
units, it’s Appendix B, Table B.7.
58
Balanced Steel Percentage
Previous codes (1963–1999) limited flexural members to 75% of
the balanced steel ratio, ρb. However, this approach was changed in the
2002 code to the new philosophy explained in Section 3.7, whereby the
member capacity is penalized by reducing the φ factor when the strain in
the reinforcing steel at ultimate is less than 0.005.
59
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