Flexural Analysis and Design of Beamns 5

8/14/2019 Flexural Analysis and Design of Beamns 5

1/16

Plain & Reinforced

Concrete-1CE-313

Flexural Analysis and

Design of Beams

(Ultimate Strength Design ofBeams)


2/16

a n e n orceConcrete-1

Balanced Steel Ratio, bIt is corresponding to that amount of steel which will causeyielding of steel at the same time when concrete crushes

At ultimate stage:

ys = ys ff =and0.003cu =

From the internal Force diagram

bc ab'0.85fC c =

ab = depth of equivalent rectangular stress block when balanced

steel ratio is used.

cu =0.003

Strain

Diagram

y

cb

d- cb


3/16


Balanced Steel Ratio, b (contd)ybys fd)b(fAT ==

For the longitudinal equilibrium

cCT =

bcyb ab'0.85ffd)b( =

d

a

f

'f0.85 b

y

cb =

(1)


4/16


Balanced Steel Ratio, b (contd)From the strain diagram cu =

0.003

Strain

Diagram

y

cb

d- cb

B C

A

E D

ADE&ABCs

b

y

b cd

c0.003

=

y

b0.003

0.003dc

+

=

s

s

s

yb

E

Ed

E

f0.003

0.003c

+=


5/16


Balanced Steel Ratio, b (contd)

df600

600cy

b +=

df0.003E

0.003Ec

ys

sb +=

b1b Ca =As we know

df600

600ay

1b

+

= (2)Put (2) in

(1)

y

1

y

cb

f600

600

f

'f0.85

+

=


6/16


Types of Cross Sections w.r.t. Flexure atUltimate Load Level

1. Tension Controlled SectionA section in which the net tensile strain in the extreme tension

steel is greater than or equal to 0.005 when the correspondingconcrete strain at the compression face is 0.003.

cu =0.003

StrainDiagram

s=>0.00

5

c

d- c

cd

0.005

c

0.003

d0.008

0.003c

d

8

3c d

8

3a 1and


7/16


Types of Cross Sections w.r.t. Flexure atUltimate Load Level(contd)

2. Transition Section

The section in which net tensile strain in the extreme tensionsteel is greater than y but less than 0.005 when

corresponding concrete strain is 0.003. cu =0.003

StrainDiagram

y


8/16


Types of Cross Sections w.r.t. Flexure atUltimate Load Level(contd)

2. Transition Section(contd)

cu =0.003

StrainDiagram

0.004

c

d- c

To ensure under-reinforced behavior,ACI code establishes a minimum nettensile strain of 0.004 at the ultimatestage.

cd

0.004

c

0.003

=

d7

3c = d

7

3a 1=


9/16


Types of Cross Sections w.r.t Flexure at UltimateLoad Level(contd)

Both the Tension Controlled Section and TransitionControlled Section are Under-Reinforced Section

In Under-Reinforced Sections steel starts yielding before thecrushing of concrete and:

< b

It is always desirable that the section is under-reinforcedotherwise the failure will initiate by the crushing of concrete.

As concrete is a brittle material so this type of failure will besudden which is NOT DESIREABLE.


10/16


Types of Cross Sections w.r.t Flexure at UltimateLoad Level(contd)

3. Compression Controlled Section (over-reinforcedsection)

The section in which net steel strain in the extremetension steel is lesser than y when corresponding

concrete strain is 0.003.

Capacity of steel remain unutilized.

It gives brittle failure without warning.

baa > bCC >


11/16


Strength Reduction Factor (Resistance Factor), Tension Controlled Section, = 0.9

Compression Controlled Section

Member with lateral ties, = 0.65

Members with spiral reinforcement, =0.7

Transition Section

For transition section is permitted to belinearly interpolated between 0.65 or 0.7 to 0.9.


12/16


13/16


Maximum Steel Ratio, maxFor

T = C

ab'0.85ffA cys =

ab'0.85ffbd cy =

d

a

f

'f0.85

y

c =

For tension controlled section d8

3a0.005 1s ==

So

= 1y

cmax

8

3

f

'f0.85


14/16


Maximum Steel Ratio, max (contd)

For transitionsection

d7

3

a0.004 1s ==

= 1y

cmax

7

3

f

'f0.85


15/16


Minimum Reinforcement of Flexural Members(ACI 10.5.1)

The minimum steel is always provided in structuralmembers because when concrete is cracked then all loadcomes on steel, so there should be a minimum amount of

steel to resist that load to avoid sudden failure. For slabs this formula gives a margin of 1.1 to 1.5.

This formula is not used for slabs.

yy

c

min

f

1.4

4f

'f =


16/16

Concluded

Documents

Flexural Analysis and Design of Beamns 5