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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


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


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 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


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


41


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


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


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


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


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


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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