5)Study of Flexural Behavior (Derivation)  

Design Principles of Concrete Structures

Lecture 5

Study of Flexural Behavior (Derivation)

Balanced Steel Ratio, ρb

𝐴𝑆

𝜌 =𝑏𝑑

It is corresponding to that amount of steel which will cause yielding of steel at the same time when concrete crushes.

At ultimate stage:

Strain Diagram

ab = depth of equivalent rectangular stress block when balanced steel ratio is used.

Balanced Steel Ratio, Pb , ontd...

For -tl1 1 e lo]1gih.td:i.Jtal equilibrium

T = C

(pb x bx d) x f y = 0. 85cf

‘xb x a

P b == 0. . 8. 5 f ‘ ab

— (1)

 Ratio, ρb (contd…)

From the strain diagram εcu= 0.00

Types of Cross Sections w.r.t.

Flexure at Ultimate Load Level

1. Tension Controlled Section
   
2. Transition Section
   
3. Compression Controlled Section
   

Types of Cross Sections w.r.t. Flexure at Ultimate Load Level

1. Tension Controlled Section
   

   A section in which the net tensile strain in the extreme tension steel is greater than or equal to 0.005 when the corresponding concrete strain at the compression face is 0.003.
   

   
   

2. Transition Section
   

The section in which net tensile strain in the extreme tension steel is greater than εy but less than 0.005 when corresponding concrete strain is 0.003.

εcu= 0.003

Strain Diagram

εy<εs<0.005

1. Transition Section (contd…)
   

   
   

   c
   

   To ensure under-reinforced behavior, ACI code establishes a minimum net tensile strain of 0.004 at the ultimate stage.
   

   εcu= 0.003
   

    0.004
   

   Strain Diagram
   

   Both the “Tension Controlled Section” and “Transition Section” are “Under-
   

   Reinforced Section“
   

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

   ρ < ρ b
   

   It is always desirable that the section is under-reinforced otherwise the failure will initiate by the crushing of concrete. As concrete is a brittle material so this type of failure will be sudden which is NOT DESIREABLE.
   

2. Compression Controlled Section (over-reinforced section)
   

• The section in which net steel strain in the extreme tension steel is lesser than εy when corresponding concrete strain is 0.003.
  
  • Capacity of steel remain unutilized.
    
  • It gives brittle failure without warning.
    

 

Strength Reduction Factor (Resistance Factor), Φ

• Tension Controlled Section, Φ = 0.9
  
• Compression Controlled Section
  
  • Member with lateral ties, Φ = 0.65
    
  • Members with spiral reinforcement, Φ = 0.75
    
• Transition Section
  
• For transition section Φ is permitted to be linearly interpolated between 0.65 or 0.75 to 0.9.
  

  Strength Reduction Factor (Resistance Factor), Φ
  

  
  

  0.25
  

  Transition
  

  Tension Controlled
  

  Compression Controlled
  

  • Transition Section (contd…) Φ
    

0.9

Φ

0.65

ε

εy εt 0.005

εt = strain in extreme tension steel when concrete crushes.

Maxitnutn S, teel Ra tio, Pnax

:For

1

C

T = C

A x fy

== 0. 8.

5f 1‘xb x a.

For tension. conhu fille ,d

So

e chon

Maximum Steel Ratio, ρmax (contd…)

For transition section

Minimum Reinforcement of Flexural Members

(ACI – 318)

• The minimum steel is always provided in structural members because when concrete is cracked then all load comes on steel, so there should be a minimum amount of steel to resist that load to avoid sudden failure.
  
• This formula is not used for slabs.
  

Concluded