Eng. Mazen Alshorafa

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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