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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.

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By dr attaullah shah swedish college of engineering and technology wah cantt
ByDr. Attaullah ShahSwedish College of Engineering and Technology Wah Cantt.

Reinforced Concrete Design-4

Design of doubly reinforced beams


By dr attaullah shah swedish college of engineering and technology wah cantt

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 sections
Analysis of Doubly Reinforced Sections required in addition to anchor bars to support stirrups.

Effect of Compression Reinforcement on the Strength and Behavior

Less concrete is needed to resist the T and thereby moving the neutral axis (NA) up.


Analysis of doubly reinforced sections1
Analysis of Doubly Reinforced Sections required in addition to anchor bars to support stirrups.

Effect of Compression Reinforcement on the Strength and Behavior


Reasons for providing compression reinforcement
Reasons for Providing Compression Reinforcement required in addition to anchor bars to support stirrups.

  • 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 beams
Doubly Reinforced Beams required in addition to anchor bars to support stirrups.

Four Possible Modes of Failure

  • 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


Analysis of doubly reinforced rectangular sections
Analysis of Doubly Reinforced Rectangular Sections required in addition to anchor bars to support stirrups.

Strain Compatibility Check Assume es’ using similar triangles


Analysis of doubly reinforced rectangular sections1
Analysis of Doubly Reinforced Rectangular Sections required in addition to anchor bars to support stirrups.

Strain Compatibility

Using equilibrium and find a


Analysis of doubly reinforced rectangular sections2
Analysis of Doubly Reinforced Rectangular Sections required in addition to anchor bars to support stirrups.

Strain Compatibility The strain in the compression steel is


Analysis of doubly reinforced rectangular sections3
Analysis of Doubly Reinforced Rectangular Sections required in addition to anchor bars to support stirrups.

Strain Compatibility Confirm


Analysis of doubly reinforced rectangular sections4
Analysis of Doubly Reinforced Rectangular Sections required in addition to anchor bars to support stirrups.

Strain Compatibility Confirm


Analysis of doubly reinforced rectangular sections5
Analysis of Doubly Reinforced Rectangular Sections required in addition to anchor bars to support stirrups.

Find c

confirm that the tension steel has yielded


Analysis of doubly reinforced rectangular sections6
Analysis of Doubly Reinforced Rectangular Sections required in addition to anchor bars to support stirrups.

If the statement is true than

else the strain in the compression steel


Analysis of doubly reinforced rectangular sections7
Analysis of Doubly Reinforced Rectangular Sections required in addition to anchor bars to support stirrups.

Return to the original equilibrium equation


Analysis of doubly reinforced rectangular sections8
Analysis of Doubly Reinforced Rectangular Sections required in addition to anchor bars to support stirrups.

Rearrange the equation and find a quadratic equation

Solve the quadratic and find c.


Analysis of doubly reinforced rectangular sections9
Analysis of Doubly Reinforced Rectangular Sections required in addition to anchor bars to support stirrups.

Find the fs’

Check the tension steel.


Analysis of doubly reinforced rectangular sections10
Analysis of Doubly Reinforced Rectangular Sections required in addition to anchor bars to support stirrups.

Another option is to compute the stress in the compression steel using an iterative method.


Analysis of doubly reinforced rectangular sections11
Analysis of Doubly Reinforced Rectangular Sections required in addition to anchor bars to support stirrups.

Go back and calculate the equilibrium with fs’

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


Analysis of doubly reinforced rectangular sections12
Analysis of Doubly Reinforced Rectangular Sections required in addition to anchor bars to support stirrups.

Compute the moment capacity of the beam


Limitations on reinforcement ratio for doubly reinforced beams
Limitations on Reinforcement Ratio for Doubly Reinforced beams

Lower limit on r

same as for single reinforce beams.

(ACI 10.5)


Example doubly reinforced section
Example: Doubly Reinforced Section beams

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.


Example doubly reinforced section1
Example: Doubly Reinforced Section beams

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


Example doubly reinforced section2
Example: Doubly Reinforced Section beams

Compute the effective reinforcement ratio and minimum r


Example doubly reinforced section3
Example: Doubly Reinforced Section beams

Compute the effective reinforcement ratio and minimum r

Compression steel has not yielded.


Example doubly reinforced section4
Example: Doubly Reinforced Section beams

Instead of iterating the equation use the quadratic method


Example doubly reinforced section5
Example: Doubly Reinforced Section beams

Solve using the quadratic formula


Example doubly reinforced section6
Example: Doubly Reinforced Section beams

Find the fs’

Check the tension steel.


Example doubly reinforced section7
Example: Doubly Reinforced Section beams

Check to see if c works

The problem worked


Example doubly reinforced section8
Example: Doubly Reinforced Section beams

Compute the moment capacity of the beam


Example doubly reinforced section9
Example: Doubly Reinforced Section beams

If you want to find the Mu for the problem

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

The resulting ultimate moment is