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C ON T EN T S
Chapter
1 2-5 Flexural Strength of Rectangular
Beams 24
MATERIALS AND MECHANICS
OF BENDING, AND CONCRETE SLAB 2-6 Equivalent Stress Distribution 25
SYSTEMS 1
2-7 Balanced, Brittle, and Ductile Failure
1-1 Concrete 1 Modes 27
Chapter
3 Chapter
5
REINFORCED CONCRETE BEAMS: DEVELOPMENT, SPLICES, AND
T-BEAMS, L-BEAMS, AND DOUBLY SIMPLE-SPAN BAR CUTOFFS 104
REINFORCED BEAMS 51
5-1 Bond Stress and Development Length:
3-1 T-Beams and L-Beams: Introduction 51 Introduction 104
3-2 T-Beam and L-Beam Analysis 53 5-2 Development Length: Tension Bars 106
3-4 T-Beam and L-Beam Design (for 5-4 Development Length: Standard
Moment) 57 Hooks in Tension 112
Chapter
4 6-4 Analysis and Design of Continuous
Two-Way Slabs 145
SHEAR AND TORSION 75
References 180
4-1 Introduction 75
Problems 180
4-2 Shear Reinforcement Design
Requirements 76
7-6 Procedure for Calculating the 9-6 Summary of Procedure for Analysis
Deflection of Simply Supported and Design of Short Columns with
and Continuous Beams and Slabs 189 Small Eccentricities 241
7-7 Procedure for Calculating the 9-7 The Load-Moment Relationship 241
Deflection of Continuous Girders 190
9-8 Columns Subjected to Axial Load at
7-8 Deflection Control Measures in Large Eccentricity 242
Reinforced Concrete Structures 193
9-9 F Factor Considerations 242
7-9 Crack Control 194
9-10 Analysis of Short Columns: Large
7-10 Floor Vibrations 195 Eccentricity 243
7-11 Gross and Cracked Section Properties 9-11 Biaxial Bending 250
of Concrete Sections 197
9-12 The Slender Column 252
References 198
9-13 Concrete Column Schedule 256
Problems 198
References 257
Problems 257
Chapter
8
WALLS 200
Chapter
10
FOUNDATIONS 260
8-1 Introduction 200
10-1 Introduction 260
8-2 Lateral Forces on Retaining
Walls 201 10-2 The Geotechnical Report 261
9-5 Design of Short Columns: Small 11-2 Design Approach and Basic
Eccentricity 239 Concepts 303
x Contents
12-7 Design of Formwork for Beams 331 14-11 Cast-In Place Concrete
Specifications 379
12-8 Wall Form Design 335
14-12 Student Design Projects 381
12-9 Forms for Columns 339
Problems 342
APPENDIX A TABLES AND DIAGRAMS 386
13-3 Marking Systems and Bar Marks 345 APPENDIX C METRICATION 408
close packing is desirable because, in general, the more defined as the compressive strength of the concrete at 28
densely the aggregate can be packed, the better are the days after placement, but there are other factors to con-
strength and durability because of the inter-locking of sider in order to satisfy the concrete durability require-
the aggregates. ments of the ACI Code. For each group of structural
Aggregates are typically classified as fine or coarse. members, the exposure classes must first be assigned
Fine aggregate is generally sand and may be categorized based on ACI 318–14, Table 19.3.1.1 depending on the
as consisting of particles that will pass a No. 4 sieve degree of exposure of the concrete to freeze/thaw (F),
(four openings per linear inch). Coarse aggregate consists sulfate (S), water (W), and corrosion (C). Once the expo-
of particles that would be retained on a No. 4 sieve. The sure classes are known, ACI 318–14, Table 19.3.2.1 is
maximum size of coarse aggregate in reinforced con- used to determine the minimum design strength and
crete is governed by various ACI Code requirements. water-cementitious material ratio. This table is also used
These requirements are established primarily to ensure to determine whether or not air entrainment is needed.
that the concrete can be placed with ease into the forms If air entrainment is needed, then ACI 318–14, Table
without any danger of jam-up between adjacent bars 19.3.3.1 is used. The amount of air entrainment needed
or between bars and the sides of the forms. Section is a function of the exposure class and the aggregate
26.4.2.1 of ACI 318 states that maximum size of coarse size. Air entraining typically ranges between 4.5% and
aggregate should not exceed the least of the following: 7%. The design data for a sample mix is shown below:
(i) one-fifth the narrowest dimension between Sample Mix Design Data
sides of forms Mix Design Number: 4001-A (this number is spe-
(ii) one-third the depth of slabs cific to the concrete supplier)
(iii) three-fourths the minimum specified clear Minimum compressive strength at 28 days
spacing between individual reinforcing bars or f′c = 4000 psi
wires, bundles of bars, prestressed reinforcement, Used for: Footings, piers, and foundation walls
individual tendons, bundled tendons, or ducts Slump: 4 in.; 8 in. with HRWR1
Air content: 6.0, +>- 1.5,
Note that smaller aggregate sizes have a relatively water-to-cementitious material ratio: 0.40
larger total surface area and thus require more cement
paste to coat the surfaces of the aggregate which Weight per Cubic
results in higher drying shrinkage. In typical building Materials Yard
construction, the nominal maximum coarse aggregate Cement, Type I/II, ASTM C150 540 lbs
size of 3>4″ is commonly specified, though maximum
Fly Ash, ASTM C618 95 lbs
aggregate size of up to 1 1>2″ is also sometimes specified
Natural Sand, ASTM C33 1230 lbs
to achieve reduced shrinkage.
#1 and #2 crushed stone, ASTM C33 1780 lbs
Potable Water, ASTM C1602 255 lbs
1-5 CONCRETE MIXES Air Entrainment, ASTM C260 1.5 oz./cwt
Water Reducer, ASTM C494 Type A 3.0 oz./cwt
There are numerous factors to consider when determin-
ing all of the components of a concrete mix. The most High-Range Water Reducer, ASTM C494 7 oz. to 10 oz./cwt
Type F*
basic mix consists of water, cement, coarse aggregate,
Non-Chloride Accelerator, ASTM C494* 0 oz. to 26 oz./cwt
and fine aggregate (sand). Beyond these basic compo-
nents, one or more chemical admixtures can be used *Added at the site upon request
depending on the required performance of the concrete. The most common admixtures are outlined as follows:
Since the admixtures are a proprietary product, the dos-
age is based on the manufacturer’s recommendations. Air Entrainment
Admixtures are often dosed in units of fluid ounces per
100 pounds of cementitious material (cwt). The strength Air entrainment provides small air bubbles in the con-
of the concrete could be adversely impacted by certain crete to act as a buffer against volumetric changes during
admixtures, and there could also be compatibility issues freeze/thaw cycles and thus makes the concrete more
with multiple admixtures. To control these issues and durable. It also makes the concrete more workable. A rule
to also ensure the performance of the concrete, concrete of thumb is that the compressive strength of concrete will
suppliers will typically go through the process of creat- decrease by 5% for every 1% increase in entrained air.
ing batches of concrete and testing them. Once a certain
mix is validated, it can then be used in the future using
Water Reducers
the same components and proportions. A sample mix When it is desired to make the concrete more work-
design is shown in the below table. able, a common cost-effective solution is to simply
The most critical parameter in the design and selec- add water, but this will have a negative impact on
tion of a concrete mix is the design strength f′c, which is 1
High range water reducer (HRWR)
Materials and Mechanics of Bending, and Concrete Slab Systems 3
the strength of the concrete. Water-reducing admix- information on the temperature requirements for con-
tures will increase the slump and thus the workability crete can be found in ACI 305 - Guide to Hot Weather
of concrete while not adversely impacting the water- Concreting and ACI 306 - Guide to Cold Weather
cementitious material ratio. Concreting. Special mix considerations also apply to
When a much higher degree of workability is mass concrete pours, such as thick mat foundations,
needed, a superplasticizer (high-range water reducer) where the high temperatures generated from the
can be used which will yield slumps around 8 in. hydration process, if not adequately controlled, can
and higher. There are also products available that are lead to cracking of the concrete.
called mid-range water reducers that provide a more
moderate amount of slump increase.
1-6 CONCRETE IN
Accelerators and Retarders COMPRESSION
It is sometimes desired to decrease or increase the The theory and techniques relative to the design and
setting time of the concrete, and so an accelerator or proportioning of concrete mixes, as well as the placing,
retarder product could be used. Accelerators can finishing, and curing of concrete, are outside the scope
be used to increase the rate at which the concrete of this book and are adequately discussed in many
reaches its design strength. This may be desired due other publications [2–5]. Field testing, quality control,
to weather conditions or to meet a certain construction and inspection are also adequately covered elsewhere.
schedule. Calcium chloride is a common accelerator This is not to imply that these are of less importance in
component, but it cannot be used in reinforced con- overall concrete construction technology but only to
crete since the chloride will promote corrosion in the reiterate that the objective of this book is to deal with
steel reinforcement. the design and analysis of reinforced concrete members.
Set retarders are used to delay the chemical reac- We are concerned primarily with how a reinforced
tion with the cement that allows the concrete to set. concrete member behaves when subjected to load. It
This might be used to offset the effect of high tempera- is generally accepted that the behavior of a reinforced
tures since high temperatures can produce a faster concrete member under load depends on the stress–
setting time. They might also be used, for example, strain relationship of the materials, as well as the type
during a paving operation to allow more time between of stress to which it is subjected. With concrete used
concrete batches and reduce the possibility of a cold principally in compression, the compressive stress–
joint in the pavement. strain curve is of primary interest.
The compressive strength of concrete is denoted
as f′c and is assigned the unit of pounds per square inch
Corrosion-Inhibitor (psi). This is the unit for f′c used in the ACI Code equa-
Corrosion-inhibiting admixtures are used to deter corro- tions. For calculations, f′c is frequently used with the
sion of reinforcing steel in concrete that is exposed to unit kips per square inch (ksi).
water or salt. This will increase the durability and life A test that has been standardized by the American
span of the concrete structure. Society for Testing and Materials (ASTM C39) [6] is
Other chemical admixtures include shrinkage used to determine the compressive strength ( f′c) of
compensating and permeability reducing admix- concrete. The test involves compression loading to
tures. For more information on chemical admixtures, failure of a specimen cylinder of concrete. The com-
the reader should refer to ACI 212.3R-16-Report on pressive strength so determined is the highest com-
Chemical Admixtures for Concrete. pressive stress to which the specimen is subjected.
Concrete that is placed between 50° F and 85° F do Note in Figure 1-1 that f′c is not the stress that exists
not require any special mix design or temperature con- in the specimen at failure but that which occurs at a
siderations. When concrete is placed during extreme strain of about 0.002 (though the concrete strain at f′c
weather temperatures outside of these range, spe- may vary between approximately 0.0015 and 0.0025).
cial considerations for cold and hot weather concrete Currently, 28-day concrete strengths (f′c) range from
must be accounted for [13, 14]. 2500 to upwards of 10,000 psi, with 3000 to 4000 psi
When concrete is poured in cold temperatures being common for reinforced concrete structures
without any precautionary measures taken, the con- and 5000 to 6000 psi being common for prestressed
crete will freeze and thus destory the bond between concrete members, and higher strengths used for
the concrete and the rebar. Pouring concrete at exces- columns in high-rise buildings. High strength con-
sively high temperatures causes the concrete to set crete with 12,000 psi compressive strength was used
faster than anticipated, leading to the formation of recently for the columns, shear walls, and drilled
cold joints. High temperatures also lead to uncon- caissons for a project at the Hudson Yards in New
trolled and signigicant cracking of the concrete which York City. For normal weight concrete, the compres-
negatively impacts strength and durability. Detailed sive strength at 28 days is specified in the ACI Code
4 CHApTER ONE
f 9c = 6000 psi
6
f 9c = 5000 psi
5
Stress (psi 3 1000)
f 9c = 4000 psi
4
f 9c = 3000 psi
3
0
0 0.001 0.002 0.003 0.004 FIGURE 1-1 Typical stress–strain
Strain (in./in.) curves for concrete.
5 years
not constant. It may also be observed that the slope 4000 6 months
of the initial portion of the curve (if it approximates
a straight line) varies with concretes of different
3000 28 days
strengths. Even if we assume a straight-line portion,
the modulus of elasticity is different for concretes
of different strengths. At low and moderate stresses 2000
(up to about 0.5f′c), concrete is commonly assumed to
behave elastically. 1000
The ACI Code, Section 19.2.2, provides the
accepted empirical expression for modulus of elasticity: 0
Time
Ec = w 1.5
c 33 1f′c FIGURE 1-2 Strength–time relationship for concrete.
Materials and Mechanics of Bending, and Concrete Slab Systems 5
Concrete strength varies with time, and the speci- bending stress in a plain concrete test beam at fail-
fied concrete strength is usually that strength that ure), as a measure of tensile strength (ASTM C78) [6].
occurs 28 days after the placing of concrete. A typical The moment that produces a tensile stress just equal to
strength–time curve for normal stone concrete is shown the modulus of rupture is termed the cracking moment,
in Figure 1-2. Generally, concrete attains approximately Mcr, and may be calculated using methods discussed
70% of its 28-day strength in 7 days and approximately in Section 1-9. The ACI Code recommends that the
85% to 90% in 14 days. modulus of rupture fr be taken as 7.5l 1f′c, where f′c
Concrete, under load, exhibits a phenomenon is in psi (ACI 318 Equation 19.2.3.1). Greek lowercase
termed creep. This is the property by which concrete lambda (l) is a modification factor reflecting the lower
continues to deform (or strain) over long periods of tensile strength of lightweight concrete relative to
time while under constant load. Creep occurs at a normal-weight concrete. The values for l are as follows:
decreasing rate over a period of time and may cease
Normal-weight concrete—1.0
after several years. Generally, high-strength concretes
exhibit less creep than do lower-strength concretes. Sand-lightweight concrete—0.85
The magnitude of the creep deformations is propor- All-lightweight concrete—0.75
tional to the magnitude of the applied load as well as Interpolation between these values is permitted.
to the length of time of load application. See ACI Code Table 19.2.4.2 for details. If the aver-
age splitting tensile strength fct is specified, then
l = f ct >(6.71f cm) … 1.0, where f cm is the average mea-
1-7 CONCRETE IN TENSION sured compressive strength, in psi.
The tensile and compressive strengths of concrete
are not proportional, and an increase in compressive
strength is accompanied by an appreciably smaller 1-8 REINFORCING STEEL
percentage increase in tensile strength. According to
Concrete cannot withstand very much tensile stress
the ACI Code Commentary, the tensile strength of
without cracking; therefore, tensile reinforcement must
normal-weight concrete in flexure is about 10% to 15%
be embedded in the concrete to overcome this defi-
of the compressive strength.
ciency. In the United States, this reinforcement is in the
The true tensile strength of concrete is difficult to
form of steel reinforcing bars or welded wire reinforc-
determine. The split-cylinder test (ASTM C496) [6] has
ing composed of steel wire. In addition, reinforcing in
been used to determine the tensile strength of light-
the form of structural steel shapes, steel pipe, steel tub-
weight aggregate concrete and is generally accepted as
ing, and high-strength steel tendons is permitted by the
a good measure of the true tensile strength. The split-
ACI Code. Many other approaches have been taken in
cylinder test uses a standard 6-in.-diameter, 12-in.-
the search for an economical reinforcement for concrete.
long cylinder placed on its side in a testing machine. A
Principal among these are the fiber-reinforced concretes,
compressive line load is applied uniformly along the
where the reinforcement is obtained through the use of
length of the cylinder, with support furnished along
short fibers of steel or other materials, such as fiberglass.
the full length of the bottom of the cylinder. The com-
For the purpose of this book, our discussion will primar-
pressive load produces a transverse tensile stress, and
ily include steel reinforcing bars and welded wire rein-
the cylinder will split in half along a diameter when its
forcing. High-strength steel tendons are used mainly in
tensile strength is reached.
prestressed concrete construction (see Chapter 11).
The tensile stress at which splitting occurs is
The specifications for steel reinforcement pub-
referred to as the splitting tensile strength, fct, and may
lished by the ASTM are generally accepted for the steel
be calculated by the following expression derived
used in reinforced concrete construction in the United
from the theory of elasticity:
States and are identified in ACI 318 Section 20.2.
2P The ACI Code states that reinforcing bars should
f ct = be secure and in place prior to the placement of con-
pLD
crete, thus the practice of wet-setting of rebar is not
where permitted. While it is not uncommon in practice to see
f ct = splitting tensile strength of lightweight aggre- some contractors wanting to place rebar in wet con-
gate concrete (psi) crete, this should not be permitted because the rebar
P = applied load at splitting (lb) displaces the aggregates, and proper bond between
L = length of cylinder (in.) the concrete and rebar cannot easily be assured.
The steel bars used for reinforcing are, almost
D = diameter of cylinder (in.)
exclusively, round deformed bars with some form of
Another common approach has been to use the patterned ribbed projections rolled onto their surfaces.
modulus of rupture, fr (which is the maximum tensile The patterns vary depending on the producer, but
6 CHApTER ONE
Letter designation for mill, a number indicating the size of the bar, a symbol
producing mill or letter indicating the type of steel from which the bar
Bar size (#7 shown)
was rolled, and for grade 60 bars, either the number 60
or a single continuous longitudinal line (called a grade
Steel type: line) through at least five deformation spaces. The grade
S: Carbon steel (A615) indicates the minimum specified yield stress in ksi. For
W: Low alloy steel (A706) instance, a grade 60 steel bar has a minimum specified
SS: Stainless steel (A955)
yield stress of 60 ksi. No symbol indicating grade is rolled
B
Grade (60ksi shown) onto grade 40 or 50 steel bars (see Figure 1-3). Grade 75
bars can have either two grade lines through at least five
7
Elastic Elastic
region region
fy fy
Stress
Stress
ey ey
Strain Strain
As determined by tensile test Idealized
(a) (b)
FIGURE 1-4 Stress–strain diagram for reinforcing steel.
Materials and Mechanics of Bending, and Concrete Slab Systems 7
exhibits increasing strain with no increase in stress. wire size. WWR with wire diameters larger than about
For reinforcement without a sharply defined yield ¼ in. is usually available only in sheets.
point, ACI 318-14, similar to the ASTM standards, Both plain and deformed WWR products are avail-
defines the yield strength as the 0.2% proof stress (that able. According to ACI 318 Section 20.2.1.7, deformed
is, the offset stress at a 0.2% strain). The yield stress of wire, plain wire, welded deformed wire reinforcement,
the steel will usually be one of the known (or given) and welded plain wire reinforcement shall conform
quantities in a reinforced concrete design or analysis to ASTM A 1064 for carbon steel and ASTM A 1022
problem. See Table A-1 for the range of fy. for stainless steel. Depending on the application, both
The modulus of elasticity of carbon reinforcing materials have the maximum fy permitted for design
steel (the slope of the stress–strain curve in the elastic that varies from 60,000 psi to 100,000 psi (see ACI 318
region) varies over a very small range and has been Tables 20.2.2.4a and 20.2.2.4b). The deformed wire is usu-
adopted as 29,000,000 psi (ACI Code Section 20.2.2.2). ally more expensive, but it can be expected to have an
Unhindered corrosion of reinforcing steel will lead improved bond with the concrete.
to cracking and spalling of the concrete in which it is A rational method of designating wire sizes to
embedded. Quality concrete, under normal conditions, replace the formerly used gauge system has been
provides good protection against corrosion for steel adopted by the wire industry. Plain wires are described
embedded in the concrete with adequate cover (mini- by the letter W followed by a number equal to 100
mum requirements are discussed in Chapter 2). This times the cross-sectional area of the wire in square
protection is attributed to, among other factors, the inches. Deformed wire sizes are similarly described,
high alkalinity of the concrete. Where reinforced con- but the letter D is used. Thus a W9 wire has an area of
crete structures (or parts of structures) are subjected to 0.090 in.2 and a D8 wire has an area of 0.080 in.2 A W8
corrosive conditions, however, some type of corrosion wire has the same cross-sectional area as the D8 but is
protection system should be used to prevent deterio- plain rather than deformed. Sizes between full num-
ration. Examples of such structures are bridge decks, bers are given by decimals, such as W9.5.
parking garage decks, wastewater treatment plants, Generally, the material is indicated by the sym-
and industrial and chemical processing facilities. bol WWR, followed by spacings first of longitudi-
One method used to minimize the corrosion of the nal wires, then of transverse wires, and last by the
reinforcing steel is to coat the bars with a suitable pro- sizes of longitudinal and transverse wires. Thus
tective coating. The protective coating can be a nonme- WWR6 * 12 @ W16 * W8 indicates a plain WWR with
tallic material such as epoxy or a metallic material such 6-in. longitudinal spacing, 12-in. transverse spacing,
as zinc (galvanizing). The ACI Code requires epoxy- and a cross-sectional area equal to 0.16 in.2 for the lon-
coated reinforcing bars to comply with ASTM A775 gitudinal wires and 0.08 in.2 for the transverse wires.
or ASTM A934 and galvanized (zinc-coated) bars to Additional information about WWR, as well
comply with ASTM A767 (ACI 318 Section 20.6.2). The as tables relating size number with wire diameter,
bars to be epoxy-coated or zinc-coated (galvanized) area, and weight, may be obtained through the Wire
must meet the code requirements for uncoated bars as Reinforcement Institute [9] or the Concrete Reinforcing
tabulated in Table A-1. Steel Institute [9 and 10]. Table 1-1 contains common
Welded wire reinforcing (WWR) (commonly called WWR sizes with the area of steel in in.2/ft width.
mesh) is another type of reinforcement. It consists of Most concrete is reinforced in some way to resist
cold-drawn wire in orthogonal patterns, square or tensile forces (Figure 1-5). Some structural elements,
rectangular, resistance welded at all intersections. It particularly footings, are sometimes made of plain con-
may be supplied in either rolls or sheets, depending on crete, however. Plain concrete is defined as structural
concrete with no reinforcement or with less reinforce- foundation and basement walls), the specified con-
ment than the minimum amount specified for rein- crete cover is 30. These members are typically placed
forced concrete. Plain concrete is discussed further in without the use of forms. For concrete elements
Chapter 10. not cast against soil or ground, but exposed to the
weather or in contact with ground, the specified con-
crete cover is 20 for No. 6 through No. 18 bars, and
1-9 CONCRETE COVER 1.50 for No. 5 and smaller bars. The surfaces of these
The clear distance between the concrete surface and the members would typically be formed. The presence of
face of the rebar is called the cover. The reinforcement forms allows for greater accuracy in establishing the
in concrete needs to have this cover for several reasons: proper clear cover distance.
The fire protection requirements of the Code may
• To protect the reinforcement against corrosion sometimes necessitate a higher concrete cover than
• To provide adequate surface area for bond between the above-specified concrete covers, depending on the
the concrete and the reinforcement required fire rating. See Chapter 14 for discussions
• To protect the reinforcement against loss of on concrete cover requirements as a function of the
strength in a fire fire ratings.
The accepted expression for the maximum bend- homogeneous or nonhomogeneous beams having
ing stress in a beam is termed the flexure formula, linear (straight-line) or nonlinear stress distributions.
For reinforced concrete beams, it has the advantage of
Mc using the basic resistance pattern found in the beam.
fb =
I The following three analysis examples dealing
with plain (unreinforced) concrete beams provide
where an introduction to the internal couple method. Note
f b = calculated bending stress at the outer fiber of that the unreinforced beams are considered homo-
the cross section geneous and elastic. This is valid if the moment is
small and tensile bending stresses in the concrete
M = the applied moment
are low (less than the tensile bending strength of the
c = distance from the neutral axis to the outside
concrete) with no cracking of the concrete develop-
tension or compression fiber of the beam ing. For this condition, the entire beam cross section
I = moment of inertia of the cross section about carries bending stresses. Therefore, the analysis for
the neutral axis bending stresses in the uncracked beam can be based
The flexure formula represents the relationship on the properties of the gross cross-sectional area
between bending stress, bending moment, and the using the elastic-based flexure formula. The use of
geometric properties of the beam cross section. By rear- the flexure formula is valid as long as the maximum
ranging the flexure formula, the maximum moment tensile stress in the concrete does not exceed the
that may be applied to the beam cross section, called modulus of rupture fr. If a moment is applied that
the resisting moment, MR, may be found: causes the maximum tensile stress just to reach the
modulus of rupture, the cross section will be on the
FbI verge of cracking. This moment is called the cracking
MR = moment, Mcr.
c
These examples use both the internal couple
where Fb = the allowable bending stress. approach and the flexure formula approach so that the
This procedure is straightforward for a beam of results may be compared.
known cross section for which the moment of inertia
can easily be found. For a reinforced concrete beam, Example 1-1
however, the use of the flexure formula presents some A normal-weight plain concrete beam is 6 in. * 12 in. in
complications, because the beam is not homogeneous cross section, as shown in Figure 1-6. The beam is simply
and concrete does not behave elastically over its full supported on a span of 4 ft and is subjected to a midspan
range of strength. As a result, a somewhat different concentrated load of 4500 lb. Assume f′c = 3000 psi.
approach that uses the beam’s internal bending stress
distribution is recommended. This approach is termed a. Calculate the maximum concrete tensile stress using
the internal couple method. the internal couple method.
Recall from strength of materials that a couple b. Repeat part (a) using the flexure formula approach.
is a pure moment composed of two equal, opposite, c. Compare the maximum concrete tensile stress
and parallel forces separated by a distance called with the value for modulus of rupture fr using the
the moment arm, which is commonly denoted Z. In ACI-recommended value based on f′c.
the internal couple method, the couple represents
an internal resisting moment and is composed of a Solution:
compressive force C above the neutral axis (assum- Calculate the weight of the beam (weight per unit length):
ing a single-span, simply supported beam that
develops compressive stress above the neutral axis) weight of beam = volume per unit length * unit weight
and a parallel internal tensile force T below the 6 in.112 in.2
= (150 lb>ft3)
neutral axis. 144 in.2 >ft2
As with all couples, and because the forces acting
= 75 lb>ft
on any cross section of the beam must be in equilib-
rium, C must equal T. The internal couple must be Calculate the maximum applied moment:
equal and opposite to the bending moment at the same
PL wL 2
location, which is computed from the external loads. It Mmax = +
4 8
represents a couple developed by the bending action
4500 lb14 ft2 75 lb>ft14 ft22
of the beam. = +
The internal couple method of determining beam 4 8
strength is more general and may be applied to = 4650 ft @ lb
10 CHApTER ONE
4500 lb
2'-0 A 6"
6"
N.A.
x x 12"
A
RA RB
4'- 0
Load Diagram Section A–A
(a) (b)
Midspan
ftop
2"
C
6"
Z = 8"
6"
T
2"
fbott
RA 2'-0
Free Body Stress Internal Couple
(a) (b) (c)
FIGURE 1-7 Stress and internal couple diagram for Example 1-1.
Materials and Mechanics of Bending, and Concrete Slab Systems 11
c. The ACI-recommended value for the modulus of rupture a. Using the internal couple method
(based on f′c) is
Z = 14 - 212.332 = 9.34 in.
fr = 7.5l 1f′c = 7.511.02 13000 C = T = 12 10.4742182172 = 13.27 kips
fr = 411 psi 13.2719.342
Mcr = CZ = TZ = = 10.33 ft. @kips
12
The calculated tensile stress (fbott) of 388 psi is about
6% below the modulus of rupture, the stress at which flex-
b. Check using the flexure formula
ural cracking would be expected.
Mc
Example 1-1 is based on elastic theory and f =
I
assumes the following: (1) a plane section before fr I
bending remains a plane section after bending (the MR = Mcr =
c
variation in strain throughout the depth of the mem- 811423
bh3
ber is linear from zero at the neutral axis), and (2) the I = = = 1829 in.4
12 12
modulus of elasticity is constant; therefore, stress
fr I 0.474118292
is proportional to strain and the stress distribution Mcr = = = 10.32 ft. @ kips
c 71122
throughout the depth of the beam is also linear from
zero at the neutral axis to a maximum at the outer
The internal couple method may also be used to
fibers.
analyze irregularly shaped cross sections, although for
The internal couple approach may also be used to
homogeneous beams it is more cumbersome than the
find the moment strength (resisting moment) of a beam.
use of the flexure formula.
Example 1-2
Example 1-3
Calculate the cracking moment Mcr for the plain concrete
Calculate the cracking moment (resisting moment) for the
beam shown in Figure 1-8. Assume normal-weight concrete
T-shaped unreinforced concrete beam shown in Figure 1-9.
and f′c = 4000 psi.
Use f′c = 4000 psi. Assume positive moment (compression
a. Use the internal couple method. in the top). Use the internal couple method and check using
b. Check using the flexure formula. the flexure formula.
Solution: Solution:
The moment that produces a tensile stress just equal to the The neutral axis must be located so that the strain and stress
modulus of rupture fr is called the cracking moment, Mcr. diagrams may be defined. The location of the neutral axis
The modulus of rupture for normal-weight concrete is calcu- with respect to the noted reference axis is calculated from
lated from ACI Equation 19.2.3.1:
Σ(Ay)
y =
fr = 7.51f′c = 7.514000 = 474 psi ΣA
412021222 + 512021102
=
For convenience, we will use force units of kips (1 kip = 41202 + 51202
1000 lb). Therefore, fr = 0.474 ksi. = 15.33 in.
2.33"
C
7"
N.A.
14" Z
T
2.33"
474 psi
Beam Bending Stress Internal
Cross Section at Max. M Couple
(a) (b) (c) FIGURE 1-8 Sketch for Example 1-2.
12 CHApTER ONE
20" ftop
4" +
C
6.67" 8.67"
4.67"
N.A.
5.33" Z
20" + 10.22"
y = 15.33"
T
R.A.
5" fbott
Beam Bending Stress Internal
Cross Section at Max. M Couple FIGURE 1-9 Sketch for
(a) (b) (c) Example 1-3.
The bottom of the cross section is stressed in tension. The total tensile force can be evaluated as follows:
Note that the stress at the bottom will be numerically larger T = average stress * area
than at the top because of the relative distances from the
= 12 10.4742115.332152 = 18.17 kips
N.A. The stress at the bottom of the cross section will be set
equal to the modulus of rupture (l = 1.0 for normal-weight and its location below the N.A. is calculated from
2
concrete): 3 115.332 = 10.22 in. 1below the N.A.2
f bott = fr = 7.5l1f′c = 7.511.02 14000 = 474 psi = 0.474 ksi The compressive force will be broken up into compo-
nents because of the irregular area, as shown in Figure 1-10.
Using similar triangles in Figure 1-9b, the stress at the Referring to both Figures 1-9 and 1-10, the component
top of the flange is internal compressive forces, component internal couples, and
MR may now be evaluated. The component forces are first
8.67
ftop = 10.4742 = 0.268 ksi calculated:
15.33
C1 = 0.14441202142 = 11.55 kips
The modulus of elasticity of the concrete, C2 = 1
10.123621202142 = 4.94 kips
2
1
Ec = 57,0001fc′ = 57,00014000 psi = 3,605,000 psi C3 = 2 10.1444215214.672 = 1.686 kips
total C = C1 + C2 + C3 = 18.18 kips
The concrete strain at the top of the beam is
C ≈ T 1O.K.2
ftop 268 psi
etop = = = 0.000074 in.>in. Next we calculate the moment arm distance from each
Ec 3,605,000 psi component compressive force to the tensile force T:
1
Similarly, the stress at the bottom of the flange is Z1 = 10.22 + 4.67 + 2 14.002 = 16.89 in.
2
4.67 Z2 = 10.22 + 4.67 + 3 14.002 = 17.56 in.
fbott of flange = 10.4742 = 0.1444 ksi 2
15.33 Z3 = 10.22 + 3 14.672 = 13.33 in.
C3 C1
C2
N.A.
0.1444 ksi
0.1444 ksi
0.1236 ksi
FIGURE 1-10 Component compression forces for Example 1-3.
Materials and Mechanics of Bending, and Concrete Slab Systems 13
18"
MR1 = 11.55116.892 = 195.1 in. @ kips
A
MR2 = 4.94117.562 = 86.7 in. @ kips
MR3 = 1.686113.332 = 22.5 in. @ kips
l1 = 33'-0"
Mcr = MR = MR1 + MR2 + MR3 = 304 in. @ kips
36'-0"
Check using the flexure formula. The moment of inertia
is calculated using the transfer formula for moment of inertia
from statics: B
I = Σ Io + Σ Ad 2
18"
18" l2 = 13'-6" 18"
1 1
I = (20)(43) + 12 (5)(203) + 4(20)(6.672) + 5(20)(5.332)
12
= 9840 in.4 FIGURE 1-11 Load distribution in one-way slab
systems.
fr I 0.474198402
Mcr = MR = = = 304 in. @ kips
c 15.33
1Checks O.K.2 Two-Way Slab Systems
As mentioned previously, the three examples are Two-way concrete slab systems are supported on col-
for plain, unreinforced, and uncracked concrete beams umns and span in two orthogonal directions, or the
that are considered homogeneous and elastic within slab is supported on all four sides by beams or walls
the bending stress limit of the modulus of rupture. The and the curvature of the slab, and the load transfer
internal couple method is also applicable to nonho- occurs in both orthogonal directions (see Figure 1-12).
mogeneous beams with nonlinear stress distributions A two-way slab system occurs when the clear span
of any shape, however. Because reinforced concrete of the longer side of the rectangular slab panel (/1)
beams are nonhomogeneous, the flexure formula is is less than twice the clear span of the shorter side
not directly applicable. Therefore, the basic approach of the slab panel (/2); that is, /1 >/2 6 2. Examples of
used for reinforced concrete beams is the internal cou- two-way slabs include flat plates, flat slabs, flat slabs
ple method (see Chapters 2 and 3). with beams, and slabs supported on stiff beams or
walls on all four sides of a rectangular slab panel.
Note that /1 is the larger dimension of the rectangu-
1-11 CONCRETE SLAB lar slab panel bounded on all four sides by columns
SYSTEMS or beams or walls and /2 is the smaller dimension of
the rectangular slab panel. Further treatment of two-
The two types of floor systems used in reinforced and way slab systems is covered in Chapter 6.
prestressed concrete structures are one way and two-
way slab systems.
1 2
One-Way Slab Systems 30'-0"
18"
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