Chapter
3
ميحرلا نمحرلا الله مسب
Design of Concrete
Structure I
University of
Palestine
Instructor:
Eng. Mazen Alshorafa
Page
1
Design of Concrete
Structure I
University of
Palestine
Flexural Stress
Introduction
Instructor:
Eng. Mazen Alshorafa
The
beam
is
a
structural
member
used
to
support
the
internal
moments
and
shears
and
in
some
cases
torsion
.
C = T → M = C (jd) = T (jd)
Page
1
Design of Concrete
Structure I
University of
Palestine
Flexural Stress
Basic Assumptions in Flexure Theory:
Instructor:
Eng. Mazen Alshorafa
•
Plane
sections
remain
plane
after
bending
i
.
e
.
in
an
initially
straight
beam,
strain
varies
linearly
over
the
depth
of
the
section
.
•
The
strain
in
the
reinforcement
is
equal
to
the
strain
in
the
concrete
at
the
same
level,
i
.
e
.
ε
s
=
ε
c
at
same
level
.
•
Stress in concrete & reinforcement may be calculated from the strains
using
σ

ε
curves for concrete & steel.
•
Tensile strength of concrete is neglected in flexural strength.
•
Concrete is assumed to fail in compression, when
ε
c
=
ε
cu
=
0.003
.
•
Compressive
σ

ε
relationship for concrete may be assumed to be any
shape that results in an acceptable prediction of strength.
Page
1
Design of Concrete
Structure I
University of
Palestine
Flexural Stress
Behavior of rectangular section at ultimate limit state in flexural
Instructor:
Eng. Mazen Alshorafa
Page
1
Design of Concrete
Structure I
University of
Palestine
Flexural Stress
Behavior of rectangular section at ultimate limit state in flexural
Instructor:
Eng. Mazen Alshorafa
The compressive zone is modeled with a equivalent stress block.
In
the
Equivalent
rectangular
block,
an
average
stress
of
0
.
85
f
c
’
is
used
with
a
rectangle
of
depth
a=
β
1
c
,
where
c
is
the
distance
from
the
extreme
compression
fiber
to
the
neutral
axis
Page
1
Design of Concrete
Structure I
University of
Palestine
Flexural Stress
Behavior of rectangular section at ultimate limit state in flexural
Instructor:
Eng. Mazen Alshorafa
2
1
2
1
/
280
'
65
.
0
70
)
280
'
(
05
.
0
85
.
0
/
280
'
85
.
0
cm
kg
f
for
f
cm
kg
f
for
c
c
c
Page
1
Design of Concrete
Structure I
University of
Palestine
Flexural Stress
Requirements for analysis of reinforced concrete beams
Instructor:
Eng. Mazen Alshorafa
[
1
] Stress

Strain Compatibility
Stress at a point in member
must correspond to strain at
a point.
[
2
] Equilibrium
Internal forces balances with
external forces
Page
1
Design of Concrete
Structure I
University of
Palestine
Flexural Stress
Example of rectangular reinforced concrete beam
Instructor:
Eng. Mazen Alshorafa
s s c
x
n
0 T C
0
0
T
8
M
.5
2
A f f ab
F
a
M d
[
1
] Setup equilibrium
.
Page
1
Design of Concrete
Structure I
University of
Palestine
Flexural Stress
Example of rectangular reinforced concrete beam
Instructor:
Eng. Mazen Alshorafa
[
2
] Find flexural capacity.
d
b
A
where
f
f
b
f
f
A
a
ab
f
C
f
A
T
s
c
y
c
y
s
c
s
s
85
.
0
85
.
0
85
.
0
)
2
(
M
y
s
n
a
d
f
A
Tjd
Page
1
Design of Concrete
Structure I
University of
Palestine
Flexural Stress
Example of rectangular reinforced concrete beam
Instructor:
Eng. Mazen Alshorafa
2
(1 )
1.7
y
n y
C
f
M bd f
f
Page
1
Design of Concrete
Structure I
University of
Palestine
Flexural Stress
Example of rectangular reinforced concrete beam
Instructor:
Eng. Mazen Alshorafa
[
3
] Need to confirm
ε
s
>
ε
y
y
y
s
1
c y
t
E
a
c
d c
c
Page
1
Design of Concrete
Structure I
University of
Palestine
Flexural Stress
Types of Flexural Failure:
Instructor:
Eng. Mazen Alshorafa
Flexural failure may occur in three different ways
[
1
] Tension Failure

(under

reinforced beam)
[
2
] Compression Failure

(over

reinforced beam)
[
3
] Balanced Failure

(balanced reinforcement)
Page
1
Design of Concrete
Structure I
University of
Palestine
Flexural Stress
Balanced Failure
Instructor:
Eng. Mazen Alshorafa
ε
cu
=
0.003
ε
cu
=
0.003
c
b
b
h
d
ε
t
=
ε
y
The
concrete
crushes
and
the
steel
yields
simultaneously
.
Such
a
beam
has
a
balanced
reinforcement,
its
failure
mode
is
brittle
,
thus
sudden,
and
is
not
allowed
by
the
ACI
Strength
Design
Method
.
Page
1
Design of Concrete
Structure I
University of
Palestine
Flexural Stress
Tension Failure
Instructor:
Eng. Mazen Alshorafa
ε
cu
=
0.003
ε
cu
=
0.003
c<c
b
b
h
d
The
reinforcement
yields
before
the
concrete
crushes
.
Such
a
beam
is
called
under

reinforced
beam
,
and
its
failure
mode
is
ductile
,
thus
giving
a
sufficient
amount
of
warning
time,
and
is
adopted
by
the
ACI
Strength
Design
Method
Page
1
Design of Concrete
Structure I
University of
Palestine
Flexural Stress
Compression Failure
Instructor:
Eng. Mazen Alshorafa
ε
cu
=
0.003
ε
cu
=
0.003
c>c
b
b
h
d
ε
t
<
ε
y
The
concrete
will
crush
before
the
steel
yields
.
Such
a
beam
is
called
over

reinforced
beam
,
and
its
failure
mode
is
brittle
,
thus
sudden,
and
is
not
allowed
by
the
ACI
Strength
Design
Method
.
Page
1
Design of Concrete
Structure I
University of
Palestine
Flexural Stress
Types of sections:
Instructor:
Eng. Mazen Alshorafa
[
1
] Tension

controlled section
When
the
net
tensile
strain
in
the
extreme
tension
steel
is
equal
to
or
greater
than
0
.
005
when
the
concrete
in
compression
reaches
its
crushing
strain
of
0
.
003
.
[
2
] Compression

controlled section
When
the
net
tensile
strain
in
the
extreme
tension
steel
is
equal
to
or
less
than
ε
y
(
ε
y
=
0
.
002
for
fy=
4200
kg/cm
2
)
when
the
concrete
in
compression
reaches
its
crushing
strain
of
0
.
003
.
[
3
] Transition section
When
the
net
tensile
strain
in
the
extreme
tension
steel
is
between
0
.
005
and
ε
y
(
ε
y
=
0
.
002
for
fy=
4200
kg/cm
2
)
when
the
concrete
in
compression
reaches
its
crushing
strain
of
0
.
003
.
Page
1
Design of Concrete
Structure I
University of
Palestine
Flexural Stress
Strength reduction factors
Φ
Instructor:
Eng. Mazen Alshorafa
y
y
s
1
c y
t
E
a
c
d c
c
ε
cu
=
0.003
ε
cu
=
0.003
c
d
ε
t
Page
1
Design of Concrete
Structure I
University of
Palestine
Flexural Stress
Minimum percentage of steel
Instructor:
Eng. Mazen Alshorafa
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