STRUCTURAL DESIGN I

Felix V. Garde, Jr.

January 30, 2019

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Course Objective

The main objective of this course is to develop, in the en-

gineering student, the ability to analyze and design a re-
inforced concrete member subjected to different types of
forces in a simple and logical manner using the basic prin-
ciples of mechanics and some empirical formulas based on
experimental results.

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Introduction

“Structural design is the art of using materials we do not

fully understand to create geometry we cannot accurately
analyze to withstand forces we cannot confidently predict in
such a way that the general public has no reason to suspect
our ignorance.”– (Anonymous).

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Introduction

Structural design can be defined as a mixture of art and

science, combining the engineer’s feeling for the behavior
of a structure with a sound knowledge of the principles of
statics, dynamics, mechanics of materials, and structural
analysis, to produce a safe economical structure that will
serve its intended purpose. – (Salmon and Johnson).

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Analysis versus Design

Structural Analysis:
Structural Analysis is the prediction of the performance of a given structure
under prescribed loads and/or other effects, such as support movements and
temperature change.

Structural Design:
Structural design is the art of utilizing principles of statics, dynamics and
mechanics of materials to determine the size and arrangement of structural
elements under prescribed load and/or other effects.

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Introduction

Purpose of structural design:

• Minimize Cost
• Minimize Weight of the structure
• Minimize Construction Time
• Minimize Labor Cost
• Minimize Cost of manufacture of owner’s product
• Maximize efficiency of operation to owner

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Design Procedures:
1 Functional Design – it ensures that intended results are achieved such
as
• adequate working areas and clearances;
• proper ventilation and/or air conditioning;
• adequate transportation facilities;
• adequate lighting;
• aesthetics.
2 Structural Framework Design – is the selection of the arrangement
and sizes of structural elements so that service loads may be safely
carried, and displacements are within the acceptable limits.
• Planning
• Preliminary structural configuration
• Establishment of the loads
• Preliminary member selection
• Analysis
• Evaluation
• Redesign
• Final decision
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 Planning Phase

Preliminary Structural Design

Estimation of Loads

Structural Analysis

Strength and
NO
Serviceability Revised Structural Design

Requirements

YES

Construction Phase

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Reinforced Concrete Structural System

Simply Supported beam

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Reinforced Concrete Structural System

Cantilever construction

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Reinforced Concrete Structural System

Cantilever beam system

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Reinforced Concrete Structural System

Box girder bridges

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Reinforced Concrete Structural System

Continuous arches

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Reinforced Concrete Structural System

Precast (I, T, double T)

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Reinforced Concrete Structural System

Three-hinge arch

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Reinforced Concrete Structural System

Cable stayed girder bridge

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Reinforced Concrete Structural System

Shells

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Reinforced Concrete Structural System

Free form shells

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Reinforced Concrete Structural System

Modern high-rise buildings

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Reinforced Concrete Structural System

Moment-resisting frames

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Reinforced Concrete Structural System

Moment-frame response to gravity loading.

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Reinforced Concrete Structural System

Moment-frame response to lateral loading.

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Reinforced Concrete Structural System

Frame-wall lateral force resisting frame.

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Reinforced Concrete Structural System

Joist floor system

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Reinforced Concrete Structural System

Flat plate floor system

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Reinforced Concrete Structural System

Flat slab floor system

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Reinforced Concrete Structural System

Waffle floor slab system

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Reinforced Concrete Structural System

Typical structure

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Reinforced Concrete Structural System

Typical structure

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Reinforced Concrete Structural System

Typical structure

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Reinforced Concrete Structural System

Typical structure

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Concrete is a stonelike material obtain by a mixture of cement, sand and
gravel or other aggregate, admixtures, and water to harden in forms of the
shape and dimensions of the desired structure.
Unreinforced concrete made with stone or gravel weighs 145 lb/ft3
(2320 kg/m3 ).
Reinforced concrete normally weighs 150 lb/ft3 (2400 kg/m3 ).

Classification of Concrete

• Normal-weight concrete is a reinforced concrete having a unit

weight of 145 pcf (pounds per cubic foot) or 2400 kg/m3 .
• Structural lightweight concrete having a unit weight ranging from
70 to 115 pcf or 1120 to 1840 kg/m3 .
• Lightweight concrete having a unit weight ranging down to 30 pcf
or 480 kg/m3 .

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 Classification of Concrete

• All-lightweight concrete when lightweight materials are used for

both coarse and fine aggregates in structural concrete.
• Sand-lightweight concrete when only the coarse aggregate is of
lightweight material but normal weight sand is used for the fine
aggregate.
• Heavy weight concrete a high-density concrete is used for shielding
against gamma and x radiation in nuclear reactor containers and
other structures. It typically having a unit weight ranging from 200 to
350 pcf or 3200 to 5600 kg/m3 .

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Materials

A. Cementitious materials
Cementitious materials shall conform to the relevant specifications as
follows:
• Portland Cement; ASTM C150
• Blended hydraulic cements: ASTM C595;
• Expansive hydraulic cement: ASTM C845;
• Fly ash and natural pozzolan: ASTM C618;
• Ground-granulated blast-furnace slag: ASTM C989;
• Silica fume: ASTM C1240
B. Aggregates
Concrete aggregates shall conform to one of the following specifications:
• Normalweight: ASTM C33;
• Lightweight: ASTM C330.
C. Water
Water used in mixing concrete shall conform to ASTM C1602M.

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Materials
Nominal maximum size of coarse aggregate shall be not larger than:
a. 1/5 the narrowest dimension between sides of forms, nor
b. 1/3 the depth of slabs, nor
c. 3/4 the minimum clear spacing between individual reinforcing bars or wires,
bundles of bars, individual tendons, bundled tendons, or ducts.

D. Steel Reinforcement
Reinforcement shall be deformed reinforcement, except that plain reinforce-
ment shall be permitted for spirals or prestressing steel;
Deformed reinforcement
a) Carbon steel: ASTM A615M;
b) Low-alloy steel: ASTM A706M;
c) Stainless steel: ASTM A955M;
d) Rail steel and axle steel: ASTM A996M. Bars from rail steel shall be
Type R.

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Materials

Plain Reinforcement

Plain bars for spiral reinforcement shall conform to

a) Carbon steel: ASTM A615M;
b) Low-alloy steel: ASTM A706M;
Plain wire for spiral reinforcement shall conform to ASTM A82M, except
that for wire with fy exceeding 420 MPa, the yield strength shall be taken
as the stress corresponding to a strain of 0.35 percent.

F. Admixtures
Admixtures for water reduction and setting time modification shall
conform to ASTM C494M. Admixtures for use in producing flowing
concrete shall conform to ASTM C1017M.
Air-entraining admixtures shall conform to ASTM C260.

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Mixing, Placing, Formwork, Embedments and Construction
Joints
All equipment for mixing and transporting concrete shall be clean;
All debris shall be removed from spaces to be occupied by concrete;
Reinforcement shall be thoroughly clean of deleterious coatings;
All concrete shall be mixed until there is a uniform distribution of materials
and shall be discharged completely before mixer is recharged.
Mixer shall be rotated at a speed recommended by the manufacturer;
Mixing shall be continued for at least 1-1/2 minutes after all materials are in
the drum, unless a shorter time is shown to be satisfactory by the mixing
uniformity tests of ASTM C94M;
Concrete shall be conveyed from mixer to place of final deposit by methods
that will prevent separation or loss of materials.
Concrete that has partially hardened or been contaminated by foreign
materials shall not be deposited in the structure.
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Mixing, Placing, Formwork, Embedments and Construction
Joints

Concrete (other than high-early-strength) shall be maintained above 10◦ C

and in a moist condition for at least the first 7 days after placement
High-early-strength concrete shall be maintained above 10◦ C and in a moist
condition for at least the first 3 days
Forms shall be substantial and sufficiently tight to prevent leakage of mortar.
Forms and their supports shall be designed so as not to damage previously
placed structure (ACI Committee 347).
Any aluminum embedments in structural concrete shall be coated or covered
to prevent aluminum-concrete reaction or electrolytic action between
aluminum and steel.
Conduits and pipes, with their fittings, embedded within a column shall not
displace more than 4 percent of the area of cross section on which strength
is calculated or which is required for fire protection.

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Mixing, Placing, Formwork, Embedments and Construction
Joints

Immediately before new concrete is placed, all construction joints shall be

wetted and standing water removed.
Construction joints shall be so made and located as not to impair the
strength of the structure. Provision shall be made for transfer of shear and
other forces through construction joints.
Construction joints in floors shall be located within the middle third of spans
of slabs, beams, and girders.
Construction joints in girders shall be offset a minimum distance of two
times the width of intersecting beams.
Beams, girders, or slabs supported by columns or walls shall not be cast or
erected until concrete in the vertical support members is no longer plastic.

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 Properties of Concrete

A. Compressive strength
Standard acceptance test for measuring the strength of concrete
involves short-time compression tests.
• Cylinder 600 × 1200 (ASTM Standards C31 and C39)
• Cylinder 400 × 800 (ASTM Standards)

Factors affecting concrete compressive strength

1. Water/cement ratio: A lower water/cement ratio reduces the
porosity of the hardened concrete and thus increases the number of
interlocking solids.

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Effect of water-cement ratio on 28-day compressive and flexural tensile strength.

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Factors affecting concrete compressive strength
2. Type of cement:
1. Type I, Normal: used in ordinary construction, where special properties
are not required.
2. Type II, Modified: lower heat of hydration than Type I; used where
moderate exposure to sulfate attack exists or where moderate heat of
hydration is desirable.
3. Type III, High early strength: used when high early strength is desired;
has considerably higher heat of hydration than Type I.
4. Type IV, Low heat: developed for use in mass concrete dams and other
structures where heat of hydration is dissipated slowly. In recent years,
very little Type IV cement has been produced. It has been replaced
with a combination of Types I and II cement with fly ash.
5. Type V, Sulfate resisting: used in footings, basement walls, sewers, and
so on that are exposed to soils containing sulfates.

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Effect of type of cement on strength gain of concrete (moist cured, water/cement
ratio = 0.49).

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 Properties of Concrete

Factors affecting concrete compressive strength

3. Aggregate:
The strength of concrete is affected by
1. strength of aggregate
2. surface texture
3. grading
4. size of aggregates
Strong aggregates: Weak aggregates:
felsite sandstone
traprock marble
quartzite metamorphic rocks

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 Typical stress-strain curves in compression.

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Typical compressive stress-strain curves (a) normal-density concrete, wc = 145
pcf (b) lightweight concrete, wc = 100 pcf

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Factors affecting concrete compressive strength
4. Moisture conditions during curing.

Effect of moist-curing conditions at 70◦ F (21◦ C) and moisture content of

concrete at time of test on compressive strength of concrete.

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 Factors affecting concrete compressive strength
5. Temperature conditions during curing.

Observations:
The 7- and 28-day strengths are re-
duced by cold curing temperatures
High temperatures during the first
month increase the 1- and 3-day
strengths but tend to reduce the 1-
year strength
Concrete placed and allowed to set
at temperatures greater than 80◦ F
(27◦ C) will never reach the 28-day
strength of concrete placed at lower
Effect of temperature during the first 28 days on the strength
temperatures.
of concrete (w/c ratio = 0.41, Type I cement, specimens cast
and moist-cured at indicated temperature for first 28 days).

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 Factors affecting concrete compressive strength
6. Age of concrete.

Rate of strength gain for concrete made

from Type I cement and moist-cured at
70◦ F (21◦ C) [ACI Committee 209]
 
0 0 t
fc(t) = fc(28)
4 + 0.85t

where
0
fc(t) = compressive strength at age t
Effect of temperature during the first 28 days on the strength
of concrete (w/c ratio = 0.41, Type I cement, specimens cast
and moist-cured at indicated temperature for first 28 days).

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 Factors affecting concrete compressive strength
7. Rate of Loading
The standard cylinder test is carried out
at a loading rate of roughly 35 psi per
second.
The maximum load is reached in 1.5 to
2.0 minutes, corresponding to a strain
rate of 10 microstain/sec
Under very slow rates of loading, the axial
compressive strength is reduced to about
75 percent of the standard test strength
At high rates of loading, the strength in-
creases, reaching 115 percent of the stan-
dard test strength when tested at a rate
Effect of temperature during the first 28 days on the strength of 30,000 psi/sec, approximate the rate
of concrete (w/c ratio = 0.41, Type I cement, specimens cast of loading experienced in a severe earth-
and moist-cured at indicated temperature for first 28 days). quake.

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B. Tensile strength
The tensile strength of concrete falls between 8 and 15 percent of the
compressive strength.
Standard Tension Tests:
1. Modulus of Rupture or Flexural Test (ASTM C78)
The 600 × 600 × 3000 plain concrete beam is loaded in flexure.
It assumed that the concrete is elastic material

Schematic diagram of flexure test.

The flexural tensile strength or modulus of rupture, fr is computed as

6M
fr =
bh2
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B. Tensile strength
Standard Tension Tests:
2. Split Cylinder Test (ASTM C496)

Schematic diagram of split cylinder test.

The split tensile strength, fct is computed as

2P
fct =
πLD

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Compressive Stress Behavior

Curves are almost linear up to about

one-half the compressive strength
The peak of the curve for high-
strength concrete is relatively sharp
The strain at the maximum stress is
approx. 0.002

Typical stress-strain curves for concrete

cylinders loaded in uniaxial compression.
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Compressive Stress Behavior

A widely used approximation for the

shape of the stress-strain curve be-
fore maximum stress is a second-degree
parabola.

Idealized stress-strain curves for concrete

in uniaxial compression.

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Compressive Stress Behavior

Stress-strain curves for concrete cylinder with high-intensity repeated axial

compressive cyclic loading.

Repeated high-intensity compressive loading produces a pronounced hysteresis

effect in the stress-strain curve.
Tests, indicated that the enveloped curve was almost identical to the curve
obtained from a single continuous load application.
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Compressive Stress Behavior
Poisson’s ratio, v:
The ratio between the transverse strain and the strain in the direction of applied
uniaxial loading found to be in the range
v = 0.15 − 0.20

Strains in concrete uniaxially loaded in compression.

The failure of a specimen loaded uniaxially in compression is generally

accompanied by splitting in the direction parallel to the load and volume increase.
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Compressive Stress Behavior
Strength under Combined Loadings:
Biaxial Loading
Concrete is said to be loaded biaxially when it is loaded in two mutually perpen-
dicular directions with essentially no stress or restraint of deformation in the third
direction

Biaxial stresses.

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Compressive Stress Behavior

Kupfer, Hilsdori, and Rusch con-

cluded that the strength of con-
crete subjected to biaxial compres-
sion may be as much as 27% higher
that the uniaxial.
For equal biaxial compressive stress,
the strength increase is approxi-
mately 16%.
The strength under biaxial tension is
approximately equal to the uniaxial
Biaxial strength of concrete, fu =uniaxial tensile strength.
strength.

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Compressive Stress Behavior

Triaxial Loading
The strength of concrete cylinders loaded
axially to failure while subjected to confin-
ing fluid pressure is (Richart, Brandtzaeg
and Brown)
0
fcc = fc0 + 4.1fl

where,
0
fcc = axial compressive strength
of confined specimen
fc0 = uniaxial compressive strength
Axial stress-strain curves from triaxial
compressions tests. of unconfined specimen
fl = lateral confining pressure

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Compressive Stress Behavior

Concrete Confinement by Reinforcement

Stress-strain curves for concrete Axial load-strain curves for 4.5” square
cylinders confined by circular spirals. concrete prism with various contents of
square ties.

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Compressive Stress Behavior

Time-Dependent Volume Changes

1 Shrinkage is the decrease in the volume of concrete during hardening and

drying under constant temperature.
2 Creep is the change in volume of concrete due to sustained load.
3 Thermal expansion or contraction is a change in volume due to change in
temperature. The coefficient of thermal expansion or contraction, α is

α = 5 − 7 × 10−6 /◦ F (normal-weight concrete)

α = 3.6 − 6.2 × 10−6 /◦ F (lightweight concrete)
α = 6 × 10−6 /◦ F (reinforcing steel)

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Compressive Stress Behavior
Behavior of Concrete Exposed to High and Low Temperatures

Compressive strength of concretes at high temperatures.

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Reinforcement

Hot-Rolled Deformed Bars

1. ASTM A 615: Standard Specification for Deformed and Plain Carbon-Steel

Bars for Concrete Reinforcement. The most commonly used reinforcing bars.
Defines three grades of metric reinforcing bars: Grades 300, 420, and 520,
having specified yield strengths of 300, 420, and 520 MPa, respectively.
2. ASTM A 706: Standard Specification for Low-Alloy Steel Deformed and
Plain Bars for Concrete Reinforcement. This specification covers bars
intended for special applications such as weldability, bendability, or ductility.
3. ASTM A 996: Standard Specification for Rail-Steel and Axle-Steel
Deformed Bars for Concrete Reinforcement. The specification covers bars
from discarded railroad or from train car axles. It is less ductile and less
bendable that A 615 steel.

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Reinforcement
Strength at High Temperatures

Strength of reinforcing steels at high temperatures.

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 Axial stress, σ is calculated by divid-
ing the axial load P by the cross-
sectional area, A.
Nominal stress or engineering stress
when initial area of the specimen is
used in the calculation.
True stress when actual area of the
bar at the cross section is used in the
calculation where failure occurs.
Stress-strain diagram for a typical structural
steel in tension.

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 Modulus of Elasticity, Es the slope
of the straight line from O to A
Yielding is a phenomenon where
considerable elongation of the test
specimen occurs with no noticeable
increase in the tensile force (from B
to C).
Strain hardening the material un-
dergoes changes in its crystalline
structure, resulting in increased resis-
tance of the material to further de-
formation (C to D).
Stress-strain diagram for a typical structural Ultimate stress the maximum value
steel in tension. (D).
Fracture or failure (point E).

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 Necking (reduction of area) is clearly
visible at the vicinity of the ultimate
stress.

Necking of a mild-steel bar in tension.

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Reinforcement
Stress–strain Curves for Reinforcement

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Stress–strain Curves for Reinforcement

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Stress–strain Curves for Reinforcement

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Stress–strain Curves for Reinforcement

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Reinforcement

Stress–strain Curve for Alloy

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Reinforcement

Stress–strain Curve for Rubber

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Reinforcement
Stress–strain Curve for Brittle Material

Typical stress-strain diagram for a brittle material showing the proportional limit
(point A) and fracture stress (point B).

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Reinforcement

Stress–strain Curve for Arbitrary Material

Arbitrary yield stress determined by the offset method.

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 1 If the strain in the bar returns to zero as
the load goes to zero, the material has
remained perfectly elastic.
2 The maximum stress for which the ma-
terial remains perfectly elastic is referred
to as elastic limit, σEL .
3 The stress at the limit of linear elasticity
is referred to as proportional limit, σP L
(point A).
4 When the stress, σ (point J) exceeds the
elastic limit, the strain does not disap-
pear upon unloading, this strain is re-
ferred to as permanent strain, P . The
recovered stain when the load is removed
Stress-strain curve of alloy steel. is called the elastic strain, e .

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 • Upper yield point is ignored in
design.
• Modulus of resilience is a mea-
sure of energy per unit volume
(energy density) absorbed by a
material up to the time it yields
under load.
• Modulus of toughness, UF is a
measure of the ability of a ma-
terial to absorb energy prior to
fracture. It represents the strain
energy per unit volume (strain-
energy density) in the material
at fracture. The strain-energy
Engineering stress-strain diagram for tension specimen of structural density is equal to the area un-
steel. (a) Stress-strain diagram (b) Diagram for small strain der the stress- strain diagram to
( < 0.007) (c) Idealized diagram for small strain ( < 0.007). fracture.

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 • Modulus of rupture is the max-
imum tensile or compressive
stress in the extreme fiber of a
beam loaded to failure in bend-
ing.
• Poisson’s ratio is a dimension-
less measure of the lateral strain
that occurs in a member owing
to strain in its loaded direction.
It is found by measuring both
the axial strain a and the lateral
strain l in a uniaxial tension
test and is given by the value,
Engineering stress-strain diagram for tension specimen of structural l
v=−
steel. (a) Stress-strain diagram (b) Diagram for small strain a
( < 0.007) (c) Idealized diagram for small strain ( < 0.007).

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Flexural Analysis and Design

• Modulus of Rupture (ACI 9.5.2.3)

• Modulus of Elasticity of Concrete
p
fr = 0.62λ fc0 MPa (ACI 8.5)
p
Ec = wc1.5 0.043 fc0 MPa
• Cracking Moment (ACI 9.5.2.3)

fr Ig for wc = 1440 − 2560 kg/m3

Mcr =
yt for normalweight concrete,
p
where, Ec = 4700 fc0 MPa

λ = modification factor (ACI 8.6.1) • Modulus of Elasticity of Steel (ACI

= 1.0 normalweight concrete 8.5)
= 0.85 sand-lightweight concrete
Es = 200, 000 MPa
= 0.75 all-lightweight concrete

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Flexural Analysis and Design

Location of Reinforcements:
Concrete cracks due to tension, and a result, reinforcement is required where
flexure, axial loads, or shrinkage effects cause tensile stresses.

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Flexural Analysis and Design

Location of Reinforcements:

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Flexural Analysis and Design

Location of Reinforcements:

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Flexural Analysis and Design

Location of Reinforcements:

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Flexural Analysis and Design
Methods of Analysis and Design:

A. Elastic Design
• is considered valid for the homogenous plain concrete beam as long as
the tensile stress does not exceed the modulus of rupture, fr
B. Working Stress Design or Alternate Strength Design
• Working (service) loads are used and a member is designed based on
an allowable compressive bending stress normally 0.45fc0
C. Strength Design
• A more rational approach where service loads are multiplied by
appropriate load factors. A member is designed so that its strength is
reduced by a reduction safety factor. The strength at failure is
commonly called the Ultimate Strength.
Service Loads or working loads refer to loads encountered in the everyday
use of the structure. Specified loads without load factors.

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Flexural Analysis and Design

Working Stress Design or Alternate Strength Design

Behavior of beam under load.

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Flexural Analysis and Design

Assumption in WSD method

• A plain section before bending
remains plane after bending.
• Stress is proportional strain
• Tension in concrete is
neglected and reinforcing steel
carries all the tension.
• The bond between the concrete
and steel is perfect, no
concrete-steel slip.
Behavior of beam under load.

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 Comparison between WSD and SDM

• Service loads are amplified using partial safety

• Working loads are used and a member is design based factors
on an allowable compressive bending stress, normally
• A member is designed so that its strength is
0.45f 0 )c.
reduced by a reduction safety factor.
• Compressive stress pattern is assumed to vary linearly
• The strength at failure is commonly called the
from zero at the neutral axis.
ultimate strength.
• Formula
• Formula
m
Rn X m
≥ Li X
FS φRn ≥ γi Li
i=1
i=1

Advantage of SDM over WSD

1. Consider mode of failure
2. Nonlinear behavior of concrete
3. More realistic factor of safety
4. Ultimate load prediction ≈ 5%

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Strength Design Method (SDM)

Design Strength ≥ Required Strength (U)

where
Design Strength = Strength Reduction Factor (φ) × Nominal Strength (N)
Strength Reduction Factor (φ) accounts for
a) variations in material strength and dimensions
b) inaccuracies in the design equations
c) ductility and reliability of the members
d) importance of the member in the structure

Nominal Strength = Strength calculated by using SDM

Required Strength = Load Factors × Service Load

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Strength Design Method (SDM)
Required Strength, (U)

where
D = dead loads
E = load effects of earthquake
F = loads due to weight and pressures of fluid
H = loads due to weight and pressure of soil, water in soil
L = live loads
Lr = roof live load
R = rain load
W = wind load
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Flexural Analysis and Design

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Flexural Analysis and Design

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Basic Assumptions in Flexure Theory:

(a) (b)
Assumed stress–strain relationship for (a) reinforcing steel (b) concrete.

The strain that corresponds to peak concrete stress, εo = 0.002

"     #
2
0 εc εc
The equation for parabola is, fc = fc 2 −
εo εo
  
Z εc − εo
Beyond the strain εo , fc = fc0 1 −
1000 εo
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Behavior of Beam Under Load

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Behavior of Beam Under Load

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Behavior of Beam Under Load

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Behavior of Beam Under Load

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Behavior of Beam Under Load

First beam fails in shear while second beam fails in bending.

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Behavior of Beam Under Load

Types of flexural failure of a structure member

1. Tension Failure: Steel may reach its yield strength before the concrete
reaches its maximum εcu = 0.003. (Under-reinforced section).
2. Balanced Failure: Steel reaches yield at same time as concrete reaches
ultimate strength. (Balanced section).
3. Compression Failure: Concrete may fail before the the yield of steel due to
the presence of a high percentage of steel in the section. (Over-reinforced
section).

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Behavior of Beam Under Load

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Behavior of Beam Under Load

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Behavior of Beam Under Load

Steps in analysis of moment and curvature for a singly reinforced section.

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Effect of Major Section Variables on Strength and Ductility

Cracking Point
Flexural tension cracking will occur in the section when the stress in the
extreme tension fiber equals the modulus of rupture, fr .
Cracking moment, Mcr is defined as the moment that causes the stress in
the extreme tension fiber to reach the modulus of rupture
fr Ig
Mcr =
yt

Failure of plain concrete beam section in tension is sudden, to prevent this brittle
failure the code (ACI 10.5) specifies minimum area of longitudinal reinforcement,
As,min in positive bending as,
p
0.25 fc0
As,min = bw d
fy

1.4bw d
and not less than
fy
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Effect of Major Section Variables on Strength and Ductility

Moment–curvature relationship for the section using fc0 = 4000 psi and
fy = 60, 000 psi

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Effect of Major Section Variables on Strength and Ductility

Effect of increasing tension steel area, As .

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Effect of Major Section Variables on Strength and Ductility

Effect of increasing d, b, and A0s .

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Simplifications in Flexure Theory for Design
1. The tensile strength of concrete is neglected in flexural-strength calculations.
(ACI 10.2.5)
2. The section is assumed to have reached its nominal flexural strength when
the strain in the extreme concrete compression fiber reaches the maximum
useable compression strain, εcu = 0.003 (ACI 10.2.3)

(a) (b)
Ultimate strain from tests of (a) reinforced members (b) plain concrete
specimen.

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Simplifications in Flexure Theory for Design
3. The compressive stress–strain relationship for concrete may be based on mea-
sured stress–strain curves or may be assumed to be rectangular, trapezoidal,
parabolic, or any other shape that results in prediction of flexural strength in
substantial agreement with the results of comprehensive tests (ACI 10.2.6).

Mathematical description of compression stress block.

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Simplifications in Flexure Theory for Design
3. The compressive stress–strain relationship for concrete may be based on mea-
sured stress–strain curves or may be assumed to be rectangular, trapezoidal,
parabolic, or any other shape that results in prediction of flexural strength in
substantial agreement with the results of comprehensive tests (ACI 10.2.6).

Whitney equivalent rectangular stress block.

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Simplifications in Flexure Theory for Design

The factor β1 shall be taken as follows:

(a) For fc0 up to and including 28 MPa

β1 = 0.85

(b) For 28 MPa < fc0 ≤ 56 MPa

fc0 − 28 MPa
β1 = 0.85 − 0.05
7 MPa

(c) For fc0 > 56 MPa

Values of β1 from tests of concrete β1 = 0.65

prisms.

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Analysis of Nominal Moment Strength for Singly
Reinforced Beams Sections

Nominal Strength, Mn is the strength of a particular structural member calculated

using the current established procedures, i.e., using the equations of equilibrium
and the properties of concrete and steel reinforcements.

Steps in analysis of Mn for singly reinforced rectangular sections.

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 Internal forces on rectangular section.

Code notations:
As = area on nonprestressed tension reinforcement
Es = modulus of elasticity of steel (200,000 MPa)
a = depth of equivalent rectangular stress block
b = width of compression face of member
c = distance from extreme compression fiber to neutral axis

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 Internal forces on rectangular section.

Code notations:
d = distance from extreme compression fiber to centroid of
longitudinal tension reinforcement
fc0 = specified compressive strength of concrete
fs = calculated tensile stress in tension reinforcement
fy = specified yield strength of reinforcement

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 Internal forces on rectangular section.

Code notations:
d = distance from extreme compression fiber to centroid of
longitudinal tension reinforcement
fc0 = specified compressive strength of concrete
fs = calculated tensile stress in tension reinforcement
fy = specified yield strength of reinforcement

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Example

1. Compute the nominal moment strength, Mn of a beam shown below with

fc0 = 20 MPa, fy = 420 MPa, b = 250 mm, d = 500 mm, and three No. 25
bars giving As = 3 × 510 = 1530 mm2 . Note that the difference between the
total section depth, h, and the effect depth, d, is 65 mm, which is a typical
value for beam sections designed with metric dimensions

Beam section.

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Strength Design Method:

In the strength design procedure, the margin of safety is provided by multiplying

the service load by a load factor and the nominal strength by a strength reduction
factor.

The general requirement in Strength Design Method is [ACI 9.1],

Design Strength ≥ Required Strength

φ (Nominal Strength) ≥ U

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Strength Design Method:

Required Strength, U, shall be at least equal to the effects of factored loads. The
effect of one or more loads not acting simultaneously shall be investigated [ACI
9.2 or NSCP 409.3.1].

U = 1.4(D + F )
U = 1.2(D + F + T ) + 1.6(L + H) + 0.5(Lr or R)
U = 1.2D + 1.6(Lr or R) + (1.0L or 0.80W )
..
.
U = 0.9D + 1.0E + 1.6H

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Strength Design Method:
Design Strength, φSn , provided by a member shall be taken as the nominal strength
multiplied by strength-reduction factor φ [ACI 9.3.2 or NSCP 409.4.1].

Strength Reduction Factors:

• Tension controlled section

φ = 0.90

• Compression controlled section

1. Members with spiral reinforcement

φ = 0.75

2. other reinforced members

φ = 0.65

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Strength Design Method:

Strength Reduction Factors:

• Shear and Torsion

φ = 0.75

• Bearing on concrete

φ = 0.65

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Definition of Balanced Conditions
Balanced condition is a condition when the steel strain, εs corresponding to section
equilibrium is equal to the yield strain, εy and the strain in the extreme concrete
fiber is equal to the maximum useable compression strain, εcu = 0.003.

The expression for balanced reinforcement ratio, ρbal is

As(bal)
ρbal =
bd
0.85β1 fc0
 
600
=
fy 600 + fy
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Code Definitions of Tension-Controlled and
Compression-Controlled Sections:

If a beam section with good ductile behavior was overloaded accidentally, it would
soften and experience some plastic rotations that would permit loads to be redis-
tributed to other portions of the continuous floor system. Thus, it is desirable to
have a ductile section.

[ACI 10.3.3] Sections are compression-controlled if the net tensile strain in the
extreme tension steel, εt , is equal to or less than the compression-controlled strain
limit when the concrete in compression reaches its assumed strain limit of 0.003.
The compression- controlled strain limit is the net tensile strain in the reinforcement
at balanced strain conditions. For Grade 420 reinforcement, and for all prestressed
reinforcement, it shall be permitted to set the compression-controlled strain limit
equal to 0.002.

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Code Definitions of Tension-Controlled and
Compression-Controlled Sections:

Strain distribution and net tensile strain.

[ACI 10.3.4] Sections are tension-controlled if the net tensile strain in the extreme
tension steel, εt , is equal to or greater than 0.005 when the concrete in compression
reaches its assumed strain limit of 0.003. Sections with εt between the compression-
controlled strain limit and 0.005 constitute a transition region between compression-
controlled and tension-controlled sections.
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Code Definitions of Tension-Controlled and
Compression-Controlled Sections:

Strain distribution and net tensile strain.

[ACI 10.3.4] A Transition-zone section has a tension-reinforcement area such

that when the beam reaches its nominal flexural strength, the net tensile strain in
the extreme layer of tensile,εt is between 0.002 and 0.005.

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[ACI 10.3.5] For nonprestressed flexural members and nonprestressed members
with factored axial compressive load less than 0.10fc0 Ag , εt at nominal strength
shall not be less than 0.004.

Strain distributions at tension-controlled and compression-controlled limits.

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Variation of the strength-reduction factor, φ

Variation of φ with tensile strain in extreme tension steel, εt and c/dt for Grade
420 reinforcement and for prestressing steel.

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 • Section with stirrup-tie (or
hoop) transverse reinforcement
250
φ = 0.65 + (εt − 0.002) ×
 3
1 5
φ = 0.65 + 0.25 −
c/dt 3

• section with spiral transverse

reinforcement (column section)

φ = 0.75 + (εt − 0.002) × 50

 
1 5
φ = 0.75 + 0.15 −
c/dt 3

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Upper Limit on Beam Reinforcement

Typical beam section with as a variable.

Typical beam section with as a variable.

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Upper Limit on Beam Reinforcement

Relationship between ρ and values for Mn and φMn .

For nonprestressed flexural members and nonprestressed members with factored

axial compressive load less than 0.10fc0 Ag , εt at nominal strength shall not be
less than 0.004 (ACI 10.3.5).
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Example

2. Compute the nominal moment strengths, and the strength reduction factor,
for three singly reinforced rectangular beams, each with a width b = 300 mm
and a total height h = 550 mm as shown.

Beam section.

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Example

3. Figure below shows a simply supported beam and the cross section at midspan.
The beam supports a uniform service (unfactored) dead load consisting of its
own weight plus 20.4 kN/m and a uniform service (unfactored) live load of
2 kN/m. The concrete strength is 24 MPa, and the yield strength of the
reinforcement is 420 MPa. The concrete is normal-weight concrete. For the
midspan section shown below, compute φMn and show that it exceeds Mu .

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 END OF PRESENTATION

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