By Dr. Attaullah Shah Swedish College of Engineering and Technology Wah Cantt.

By

Dr. Attaullah Shah

Swedish College of Engineering and Technology Wah Cantt.

Reinforced Concrete Design-4

Design of doubly reinforced beams

− Due to size limitations compression reinforcement may be required in addition to anchor bars to support stirrups.

− If the applied ultimate moment is more than the factored nominal capacity allowed by maximum steel ratio, the additional steel may be required in compression and tension to support the excess moment.

Analysis of Doubly Reinforced SectionsAnalysis of Doubly Reinforced Sections

Effect of Compression Reinforcement on the Strength and BehaviorLess concrete is needed to resist the T and thereby moving the neutral axis (NA) up.

TC

fAT

ys

Analysis of Doubly Reinforced Analysis of Doubly Reinforced SectionsSections

Effect of Compression Reinforcement on the Strength and Behavior

12

2sc

1c

and2

; C

ReinforcedDoubly 2

;

ReinforcedSingly

aa

adfAMCC

adfAMCC

ysn

ysn

Reasons for Providing Compression ReinforcementReasons for Providing Compression Reinforcement

− Reduced sustained load deflections.− Creep of concrete in compression zone− transfer load to compression steel− reduced stress in concrete− less creep− less sustained load deflection

Doubly Reinforced BeamsDoubly Reinforced Beams

− Under reinforced Failure− ( Case 1 ) Compression and tension steel yields− ( Case 2 ) Only tension steel yields

− Over reinforced Failure− ( Case 3 ) Only compression steel yields− ( Case 4 ) No yielding Concrete crushes

Four Possible Modes of Failure

Analysis of Doubly Reinforced Analysis of Doubly Reinforced Rectangular SectionsRectangular Sections

Strain Compatibility Check Assume s’ using similar triangles

s

s

0.003'

'*0.003

c d c

c dc

Analysis of Doubly Reinforced Analysis of Doubly Reinforced Rectangular SectionsRectangular SectionsStrain Compatibility

Using equilibrium and find a

s s yc s

c

s s y y

1 1 c 1 c

0.85

' 0.85 0.85

A A fT C C a

f b

A A f d facf b f


Strain Compatibility The strain in the compression steel is

s cu

1 c

y

1

0.851 0.003

'

dc

f dd f


Strain Compatibility Confirm

;E ys

s

yys

f

y y1 cs 3

y s

0.851 0.003

' E 29 x 10 ksif ff d

d f


Strain Compatibility Confirm

y1 c

y

1 c

y y

870.85' 87

0.85 87'

87

ff dd f

f dd f f


Find c

confirm that the tension steel has yielded

s y c s y

ss s y1

c 1

0.85

0.85

A f f ba A f

A A fc a c

f b

ys cu y

sEfd c

c


If the statement is true than

else the strain in the compression steel

n s s y s y2aM A A f d A f d d

s sf E


Return to the original equilibrium equation

s y s s c

s s s c 1

s s cu c 1

0.85

0.85

1 0.85

A f A f f ba

A E f b c

dA E f b cc


Rearrange the equation and find a quadratic equation

Solve the quadratic and find c.

s y s s cu c 1

2c 1 s s cu s y s s cu

1 0.85

0.85 0

dA f A E f b cc

f b c A E A f c A E d


Find the fs’

Check the tension steel.

s s cu1 1 87 ksid df Ec c

ys cu y

sEfd c

c

Analysis of Doubly Reinforced Analysis of Doubly Reinforced Rectangular SectionsRectangular SectionsAnother option is to compute the stress in the compression steel using an iterative method.

1 c3s

y

0.8529 x 10 1 0.003

' f d

fd f

Analysis of Doubly Reinforced Analysis of Doubly Reinforced Rectangular SectionsRectangular SectionsGo back and calculate the equilibrium with fs’

s y s sc s

c

1

s

0.85

1 87 ksi

A f A fT C C a

f ba

c

dfc

Iterate until the c value is adjusted for the fs’ until the stress converges.


Compute the moment capacity of the beam

n s y s s s s2aM A f A f d A f d d

Limitations on Reinforcement Ratio Limitations on Reinforcement Ratio for Doubly Reinforced beamsfor Doubly Reinforced beams

Lower limit on

same as for single reinforce beams.

yy

cmin

200 3

fff

(ACI 10.5)

Example: Doubly Reinforced Example: Doubly Reinforced SectionSection

Given:

f’c= 4000 psi fy = 60 ksi

A’s = 2 #5 As = 4 #7

d’= 2.5 in. d = 15.5 in

h=18 in. b =12 in.

Calculate Mn for the section for the given compression steel.


Compute the reinforcement coefficients, the area of the bars #7 (0.6 in2) and #5 (0.31 in2)

2 2s

2 2s

2s

2s

4 0.6 in 2.4 in

2 0.31 in 0.62 in

2.4 in 0.012912 in. 15.5 in.

0.62 in 0.003312 in. 15.5 in.

A

A

Abd

Abd


Compute the effective reinforcement ratio and minimum

y

c

y

min

0.0129 0.0033 0.00957

200 200 0.00333 60000

3 3 4000or 0.0031660000

0.0129 0.00333 OK!

eff

f

ff

Example: Doubly Reinforced Example: Doubly Reinforced SectionSectionCompute the effective reinforcement ratio and minimum

1 c

y y

0.85 87' 87

0.85 0.85 4 ksi 2.5 in. 87 0.039860 ksi 15.5 in. 87 60

f dd f f

0.00957 0.0398 Compression steel has not yielded.


Instead of iterating the equation use the quadratic method

2c 1 s s cu s y s s cu

2

2 2

2

2

2

0.85 0

0.85 4 ksi 12 in. 0.85

0.62 in 29000 ksi 0.003 2.4 in 60 ksi

0.62 in 29000 ksi 0.003 2.5 in. 0

34.68 90.06 134.85 0

2.5969 3.8884 0

f b c A E A f c A E d

c

c

c c

c c


Solve using the quadratic formula

2

2

2.5969 3.8884 0

2.5969 2.5969 4 3.88842

3.6595 in.

c c

c

c


Find the fs’

Check the tension steel.

s s cu2.5 in.1 1 87 ksi

3.659 in.27.565 ksi

df Ec

s15.5 in. 3.659 in. 0.003 0.00971 0.00207

3.659 in.

Example: Doubly Reinforced Example: Doubly Reinforced SectionSectionCheck to see if c works

2 2s y s s

c 1

2.4 in 60 ksi 0.62 in 27.565 ksi

0.85 0.85 4 ksi 0.85 12 in.

3.659 in.

A f A fc

f b

c

The problem worked

Example: Doubly Reinforced Example: Doubly Reinforced SectionSectionCompute the moment capacity of the beam

s y s s s s

2

2

2

2

2.4 in 60 ksi 0.85 3.659 in.15.5 in.

2 0.62 in 27.565 ksi

0.62 in 27.565 ksi 15.5 in. 2.5 in.

1991.9 k - in. 166 k - ft

na

M A f A f d A f d d

Example: Doubly Reinforced Example: Doubly Reinforced SectionSectionIf you want to find the Mu for the problem

u u

3.66 in. 0.23615.5 in.

0.375 0.9

0.9 166 k - ft

149.4 k - ft

cd

cd

M M

From ACI (figure R9.3.2)or figure (pg 100 in your text)

The resulting ultimate moment is

By Dr. Attaullah Shah Swedish College of Engineering and Technology Wah Cantt.

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