Reinforcement ratio of the beam section (ρ) 1. Over reinforce, is a state where max. concrete strain (εu =

0.003) on the top of compression zone of beam is achieved first of the strain yield (εy = fy / Es) of reinforcing steel. Struktur Brittle

2. Strain in balanced condition (ρb) , is a state where max. concrete strain (εu = 0.003) on the top of compression zone of beam is achieved simultaneously with strain yield (εy = fy/Es) of reinforcing steel.

3. Under reinforce, is a state where strain yield (εy = fy/Es) of reinforcing steel is achieved first of the max. concrete strain (εu = 0.003) on the top of compression zone of beam .

Struktur Ductail

STAIN IN BALANCED CONDITION

From strain diagram : xb/d = 0.003/(0.003 + εy) xb = 0.003 d/(0.003 + εy) . . . . . . . . . . . . . (1)

ρ = As/bd ρb= Asb/bd Asb = ρb bd . . . . . . . . . . (2)

Hor. Equilibrium : C = T 0.85 f’c xb b = Asb fy = ρb b d fy

ρb = 0.85 f’c xb / d fy = (0.85 f’c /fy)*( xb /d) . . . . . . . . (3)

d

xb

0.003

εs = εy

xb

0.85 f’c

xb/2

l = (d - xb/2)

C

TAsb

section strain stress forces

b

0.85 f’c 0.003 d

(1) + (3) ρb = fy (0.003 + εy) d

0.85 f’c 0.003 ρb =

fy (0.003 + εy)

With the value of E = 200 x 103 MPa.

To guarantee the collapse pattern ductile, ACI limited:

0.85 f’c 600 ρb =

fy 600 + fy

ρmax = 0.75 ρb

ρmin :

Requirements of collapse ductile require a minimum reinforcement is used that produces the same strength with beams without reinforcement.

Relationship desired strength : Mn ≥ Mcr . . . . . . . . . . . . . . . . . . . . (1)

Mcr achieved when the concrete tension fibers reach modulus of collapsed (Modulus of Repture) fr

fr = 7.5 √f’c . . . . . . . . . . . . . . . . . . . . . . (2)ACI : Pure concrete Mcr = fr Ig/yt . . . . . . . . . . . . . . . . . . . . . . . . (3)

Where : Ig : Momen of Inertia of gross section

yt : distance of neutral axis to extreme tensile fiber.

For a square cross-section :

bh3/12 7.5 √f’c bh2

Mcr = 7.5 √f’c = . . . . . . . . . . (4) h/2 6

Reinforced square section : Mn = As fy (d - x/2 ) . . . . (5)

(5) ≥ (4) As fy (d - x/2 ) ≥ 7.5 √f’c bh2/6 . . . . . .(6)

As = ρ bd and x/2 = 0. 95 d

For small value of ρ ρ bd fy(0.95 d) ≥ 1.25 √f’c bh2

1.25 √f’c or : ρ ≥ ---------- (h/d)2 . . . . . . . . . . . (7)

0.95 fy

if d ≈ 0.9 h : 1.62 √f’c

ρmin = . . . . . . . . . . . . . . . . . (8) fy

ACI 10.5.1 presuppose : ρmin ≥ 1,4/fy . . . . . . . (9)

SK SNI T-15 – 1991 – 03 : ρmin ≥ 1,4/fy . . . . . . . (10)

RSNI – 2002, presuppose : ρmin ≥ √fc’/4fy or

should not be smaller than 1,4/fy . . . . . . . . . . . . (11)