# Quasistatic bending of beams

Urban and Civil

Nov 25, 2013 (4 years and 5 months ago)

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ELEMENTS OF RCC DESIGN

Q.1.

Write down the assumption of simple bending.

In

engineering mechanics
,

bending

(also known as

flexure
) charact
erizes the behavior of a
slender

structural

element subjected to an external

applied perpendicularly
to a

longitudinal axis

of the element.
The structural element is assumed to be such that at least one of its dimensions is a small fraction, typically 1/10 or
less, of the
other two.
[1]

When the length is considerably longer than the width and the thickness, the element is called
a

beam
. A

closet

rod
sagging

under the weight of clothes on

clothes hangers

is an example of a beam experiencing
bending. On the other hand, a

shell

is a structure of any geometric form where the len
gth and the width are of the
same order of magnitude but the thickness of the structure (known as the 'wall') is considerably smaller. A large
diameter, but thin
-
walled, short tube supported at its ends and loaded laterally is an example of a shell experie
ncing
bending.

In the absence of a qualifier, the term

bending

is ambiguous because bending can occur locally in all objects. To
make the usage of the term more precise, engineers refer to the

bending of rods
,
[2]

the

bending of
beams
,
[1]

the

bending of plates
,
[3]

the

bending of shells
[2]

and so on.

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]
Quasistatic bending of beams

A beam deforms and stresses develop inside it when a transverse load is applied on it. In the quasist
atic case, the
amount of bending
deflection

and the stresses that develop are assumed not to change over time. In a horizontal
beam supported at the ends and l
oaded downwards in the middle, the material at the over
-
side of the beam is
compressed while the material at the underside is stretched. There are two forms of internal stresses caused by

Shear stress

Direct

compressive stress

in the upper region of the beam, and direct

tensile stress

in the lower
region of the beam.

These last two forces form a

couple

or

moment

as they are equal in magnitude and opposite in direction.
This

bending moment

resists the sagging deformation characteristic of a beam experiencing bending. The stress
distribution in a beam can be predicted quite accurately even when some simplifying assumptions are used.
[1]

[
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Euler
-
Bernoulli bending theory

Main article:

Euler
-
Bernoulli beam equation

Element of a bent beam: the fibers form conce
ntric arcs, the top fibers are compressed and bottom fibers stretched.

Bending moments in a beam

In the

Euler
-
Bernoulli theory

of slender beams, a major assumption is that 'plane sections remain plane
'. In other
words, any deformation due to shear across the section is not accounted for (no shear deformation). Also, this linear
distribution is only applicable if the maximum stress is less than the

yield stress

of the material. For stresses that
exceed yield, refer to article

plastic bending
. At yield, the maximum stress experienced in the section (at t
he furthest
points from the neutral axis of the beam) is defined as the

flexural strength
.

The Euler
-
Bernoulli equation for the quasistatic bending of slender, isotropic, h
omogeneous beams of constant cross
-
section under an applied transverse load

q
(
x
)

is
[1]

where

E

is the

Young's modulus
,

I

is the

area moment of inertia

of the cross
-
section, and

w
(
x
)

is the deflection
of the neutral axis of the beam.

After a solution for the

displacement of the beam has been obtained, the bending moment (
M
) and shear force
(
Q
) in the beam can be calculated using the relations

Simple beam bending is often analyzed with the Euler
-
Bernoulli beam equation. The conditions for using
simple bendin
g theory are

[4]
:

1.

The beam is subject to

pure bending
. This means that the

shear force

is zero, and that
no torsional or axial loads are present.

2.

The material is

isotropic

and

homogeneous
.

3.

The material obeys

Hooke's law

(it is linearly elastic and will not deform plastically).

4.

The beam is initially straight with a cross sect
ion that is constant throughout the beam
length.

5.

The beam has an axis of symmetry in the plane of bending.

6.

The proportions of the beam are such that it would fail by bending rather than by
crushing, wrinkling or sideways

buckling
.

7.

Cross
-
sections of the beam remain plane during bending.

Deflection of a beam deflected symmetrically and principle of su
perposition

Compressive and tensile forces develop in the direction of the beam axis under bending loads. These
forces induce

stresses

on the beam. The maximum compressive st
ress is found at the uppermost edge of
the beam while the maximum tensile stress is located at the lower edge of the beam. Since the stresses
between these two opposing

maxima

vary

linearly
, there therefore exists a point on the linear path between
them where there is no bending stress. The

l
ocus

of these points is the neutral axis. Because of this area
with no stress and the adjacent areas with low stress, using uniform cross section beams in bending is not a
particularly efficient means of supporting a load as it does not use the full capac
ity of the beam until it is on
the brink of collapse. Wide
-
flange beams (
I
-
beams
) and

truss

girders

as they minimize the amount of material in this under
-
stressed region.

The classic formula for determining the bending stress in a beam under simple bending is
[5]
:

where

σ

is the bending stress

M

-

neutral axis

y

-

the perpendicular distance to the neutral axis

I
x

-

the

second moment of area

x

Q.2

Explain under reinforced section.

Reinforced concrete

is

concrete

in which reinforcement bars ("
rebars
"), reinforcement grids,

plates

or

fibers

have
been incorporated to strengthen the concrete in

tension
. It was i
nvented by

French

gardener

Joseph Monier

in 1849
and

pate
nted

in 1867.
[1]

The term Ferro Concrete refers only to concrete that is reinforced with iron or steel. Other
materials used to reinforce concrete can be organic and inorganic

fibres as well as

composites

in different forms.
Prior to the invention of reinforcement, concrete was strong in

compression
, but weak in tension. Adding
reinforcement crucially increases the strength in tension. The failure strain of concrete in tension is so low that the
reinforcement has to hold the cracked sections together
.

For a strong, ductile and durable construction the reinforcement needs to have the following properties:

High strength

High tensile strain

Good bond to the concrete

Thermal compatibility

Durability in the concrete environment

In most cases reinforced con
crete uses steel rebars that have been inserted to add strength.

Use in construction

Rebars of

's roof in construction (2009)

Concrete is reinforced to give it extra tensile strength; without reinforcement, many concrete buildings would not have
been possible.

Reinforced concrete can encompass many types of structur
es and components,
including

slabs
,

walls
,

beams
,

columns
,

foundations
,
frames

and more.

Reinforced concrete can be classified as

precast

or

cast in
-
situ concrete
.

Much of the focus on reinforcing concrete is placed on

floor
systems. Designing and implementing the most efficient
floor system is key to creating optimal building struc
tures. Small changes in the design of a floor system can have
significant impact on material costs, construction schedule, ultimate strength, operating costs, occupancy levels and
end use of a building.

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]
Behavior of reinforced concrete

[
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]
Materials

Concrete is a mixture of Coarse (stone or brick chips) and Fine (generally sand or crushed stone) aggregates with a
binder material(usually
Portland cement
)
. When mixed with a small amount of water, the cement

hydrates

to form
microscopic opaque crystal lattices encapsulating and locking the aggregate into a rigid structure. Typical concrete
mi
xes have high resistance to

compressive

stresses

psi (28

MPa)); how
ever, any
appreciable

tension

(
e.g.,

due to

bending
) will break the microscopic rigid lattice, resulting

in cracking and separation
of the concrete. For this reason, typical non
-
reinforced concrete must be well supported to prevent the development
of tension.

If a material with high strength in tension, such as

steel
, is placed in concrete, then the composite
material,

reinforced concrete
, resists not only compression but also bending and other direct tensile actions. A
reinforced concrete section where the concrete resists the compression and steel
resists the tension can be made
into almost any shape and size for the construction industry.

[
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]
Key characteristics

Three physical characteristics give reinforced concrete its special properties.

First, the

coefficient of thermal expansion

of concrete is si
milar to that of steel, eliminating large internal stresses due
to differences in
thermal

expansion or contraction.

Second, when the cement paste within the concrete hardens this conforms to the surf
ace details of the steel,
permitting any stress to be transmitted efficiently between the different materials. Usually steel bars are roughened or
corrugated to further improve the

bond

or cohesion between the concrete and steel.

Third, the

alkaline

chemical environment provided by the

alkali

reserve (KOH, NaOH) and th
e

portlandite

(
calcium
hydroxide
) contained in the hardened cement paste causes a

passivating

film to form on the surface of the steel,
making it much more resistant to

corrosion

than it would be in neutral or ac
idic conditions. When the cement paste
exposed to the air and meteoric water reacts with the atmospheric CO
2
, portlandite and the

Calcium Silicate
Hydrate

(CS
H) of the hardened cement paste become progressively carbonated and the high pH gradually decreases
from 13.5

12.5 to 8.5, the pH of water in equilibrium with

calcite

(
calcium carbonate
) and the steel is no longer
passivated.

As a rule of thumb, only to give an idea on orders of magnitude, steel is protected at pH above ~11 but starts to
corrode below ~10 depe
nding on steel characteristics and local physico
-
chemical conditions when concrete becomes
carbonated.

Carbonation

of concrete along with
chloride

ingress are amongst the chief reasons for the failure
of

reinforcement bars

in concrete.
[2]

The relative cross
-
sectional

area

of steel required for typical reinforced concrete is usually quite small and varies from
1% for most beams and slabs to 6% for some colu
mns.

Reinforcing bars

are normally round in cross
-
section and vary
in diameter. Reinforced concrete structures sometimes have provisions such as ventilated hollow cores to control
their moisture &

humidity.

Distribution of concrete (in spite of reinforcement) strength characteristics along the cross
-
section of vertical
reinforced concrete elements is inhomogeneous.
[3]

[
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]
Mechanism of composite action of reinforcement and concrete

The reinforcem
ent in a RC structure, such as a steel bar, has to undergo the same strain or deformation as the
surrounding concrete in order to prevent discontinuity, slip or separation of the two materials under load. Maintaining
composite action requires transfer of l
oad between the concrete and steel. The direct stress is transferred from the
concrete to the bar interface so as to change the tensile stress in the reinforcing bar along its length. This load
transfer is achieved by means of bond (anchorage) and is ideal
ized as a continuous stress field that develops in the
vicinity of the steel
-
concrete interface.

Q.3

What is local band? Explain.

Q.4

What do you mean by contilenes beam.

Beams are made continuous over the supports to increase structural integrity. A

continuous beam provides an alternate load path in the case of failure at a section. In

regions with high seismic risk, continuous beams and frames are preferred in buildings

and bridges. A continuous beam is a statically indeterminate structure.

The

advantages of a continuous beam as compared to a simply supported beam are as

follows.

1) For the same span and section, vertical load capacity is more.

2) Mid span deflection is less.

3) The depth at a section can be less than a simply supported beam

for the same

span. Else, for the same depth the span can be more than a simply supported

beam.

The continuous beam is economical in material.

4) There is redundancy in load path.

Possibility of formation of hinges in case of an extreme event.

5) Requires less number of anchorages of tendons.

6) For bridges, the number of deck joints and bear
ings are reduced.

Reduced maintenance

There are of course several disadvantages of a continuous beam as compared to a

simply supported beam.

1) Difficult analysis and design procedures.

2) Difficulties in construction, especially for precast member
s.

3) Increased frictional loss due to changes of curvature in the tendon profile.

4) Increased shortening of beam, leading to lateral force on the supporting columns.

5) Secondary stresses develop due to time dependent effects like creep and

shrinkage
, settlement of support and variation of temperature. Prestressed Concrete Structures
Dr. Amlan K Sengupta and Prof. Devdas Menon

6) The concurrence of maximum moment and shear near the supports needs

proper det
ailing of reinforcement.

7) Reversal of moments due to seismic force requires proper analysis and design.

Q.5

How will you design a one
-
way slab?

Q.6

What is RC column?

FOOTING AND FOOTING REINFORCE
-
MENT.

Footings

support

the

entire

structure
and

distribute

the

to

the

ground.

The

size
and

shape

of

a

footing

depend

upon

the

design
of

the

structure.

In

a

small

footing

(fig.

7
-
2),
“steel

mats”

o
r

reinforcements

are

generally
preassembled

and

placed

after

the

forms

have
been set. In
large or continuous footings, such as
those found under bearing walls,
steel mats are
constructed

in

place.
COLUMN

AND

COLUMN

REINFORCE
-
MENT.

A

column

i
s

a

slender,

vertical

member
that

carries

a

superimposed

Concrete
columns,

especially

those

subjected

to

bending
stresses, must always be reinforced with steel.
A
PIER

or

PEDESTAL

is

a

compressive

member
that is short (usually
the

height is less than three
times the least lateral dimension) in relation
to its
cross
-
sectional

area

and

carries

no

bending

stress.
In concrete
columns, vertical reinforcement is
the

principal

reinforcement.

However,

a

column shortens vertica
lly and expands
laterally;
hence, lateral reinforcements in the form of lateral
ties are used
to restrain the expansion. Columns
reinforced in this manner are called
tied columns
(fig.
7
-
3,

view A). If the restraining reinforcement
is a
continuous winding spiral that encircles the
core and longitudinal steel,
the column is called
a

spiral

column

(fig,

7
-
3,

view

B).
BEAM

AND

BEAM

REINFORCE

-
MENT.

-
carrying
horizontal members. They take the load directly
from

the

floor

and

carry

it

to

the

columns.
Concrete

beams

can

either

be

cast

in

place

or
precast and transported to

the jobsite.

Figure 7
-
4
shows several
common types of beam reinforcing
steel shapes. Both straight and bent
-
up principal

Q.7

Compare prestressed concrete beam and reinforced concrete

beam.

Prestressed concrete

is a method for overcoming

concrete
's natural weakness in

tension
. It can be u
sed to
produce

beams
,

floors

or

bridges

with a longe
r

span

than is practical with ordinary
reinforced concrete
. Prestressing
tendons
(generally of high

tensile

steel

cable

or

rods
) are used to provide a clamping load which produces
a

compressive stress

that balances the

tensile stress
that the concrete

compression member

would otherwise
al

reinforced concrete

is based on the use of

steel

reinforcement
bars,

rebars
, inside poured

concrete
.

Prestressing can be accomplished in three ways: pre
-
tensioned concrete, and bonded or unbonded post
-
tensioned
concrete.

Pre
-
tensione
d concrete is cast around already tensioned tendons. This method produces a good bond
between the tendon and concrete, which both protects the tendon from corrosion and allows for
direct transfer of tension. The cured concrete adheres and bonds to the bars

and when the
tension is released it is transferred to the concrete as compression by static

friction
. However, it
requires stout anchoring points between which the tendon is to be stretched

and the tendons are
usually in a straight line. Thus, most pretensioned concrete elements are

prefabricated

in a
factory and must be transported to the construction site, which
limits their size. Pre
-
tensioned
elements may be

balcony

elements,

lintels
, floor slabs, beams or foundation

piles
. An
innovative

bridge
construction method using pre
-
stressing is the

stressed ribbon bridge

design.

Reinforced concrete

is

concrete

in which reinforcement bars ("
r
ebars
"), reinforcement grids,

plates

or

fibers

have
bee
n incorporated to strengthen the concrete in

tension
. It was invented by

French

gardener

Joseph Monier

in 1849
and

patented

in 1867.
[1]

The term Fer
ro Concrete refers only to concrete that is reinforced with iron or steel. Other
materials used to reinforce concrete can be organic and inorganic fibres as well as

comp
osites

in different forms.
Prior to the invention of reinforcement, concrete was strong in

compression
, but weak in tension. Adding
reinforcement crucially increases the strength in tension. The failure strain of concrete in tension is so low that the
reinforcement has to hold the cracked sections together.

For a strong, ductile and durable construction th
e reinforcement needs to have the following properties:

High strength

High tensile strain

Good bond to the concrete

Thermal compatibility

Durability in the concrete environment

In most cases reinforced concrete uses steel rebars that have been inserted to