Structural Concrete - Hassoun

16.12 Rotation of Plastic Hinges 591
where f y is the yield strength of steel bars and E s is the modulus of elasticity of steel = 29 × 10 6 psi.
Therefore,
θ = 0.0035 − ε c 1
λ λ = 0.0035 f y
−
(16.18)
λ E s (1 − λ)
For grade 40 steel, f y = 40 ksi, and using a maximum value of λ of 0.50, then
θ min = 0.0035
0.50 − 40
= 0.00424 rad
29,000 ×(1 − 0.50)
For grade 60 steel, f y = 60 ksi and λ max = 0.44;
θ min = 0.0035
0.44 − 60
= 0.00426 rad
29,000(1 − 0.44)
The θ min calculated here is from one side only, and the total permissible rotation at the plastic
hinge equals 2θ or 2θ min . The actual λ can be calculated as follows, given α = β 1 c and β 1 = 0.85
for f c ′ ≤ 4ksi:
c =
a
0.85 = A s f y
(0.85) 2 f ′ cb
λ = c d =
A s f y
0.72 f cbd ′ = ρ f y
0.72 f c
′ ≤ 0.5 (16.19)
where ρ = A s /bd. (λ max is obtained when ρ max is used.)
If the rotation provided is not adequate, one can increase the section dimensions or reduce
the percentage of steel reinforcement to obtain a smaller c, a smaller λ, and greater θ. Baker [3]
indicated that if special binding or spirals are used, the ultimate crushing strain in bound concrete
may be as high as 0.012.
For a compression plastic hinge (as in columns),
θ = ε pl p
(16.20)
h
where h is the overall depth of the section and l p is the length over which yielding occurs. In
compression hinges, l p varies between 0.5h and h.
At a concrete ultimate stress of f c,ε ′ c = 0.002; thus, ε p = ε ′ c − 0.002 = 0.0035 − 0.002 =
0.0015 is the minimum angle of rotation on one side. Therefore,
0.0015 × 0.5h
θ min = = 0.00075 rad
h
With special binding or spirals, θ may be increased to
θ max =(0.012 − 0.002)× 0.5h = 0.005 rad
h
The extreme value of ε ′ c = 0.012 is quite high, and a smaller value may be used with proper spirals;
otherwise a different section must be adopted.
In reinforced concrete continuous beams containing steel fibers, the plastic rotation may be
estimated as follows [14]:
( )
0.0035 f y
θ p = λβ −
(16.21)
λ E s (1 − λ)
where
λ =(4.3 + 2.24ρ s − 0.043f y + 4.17ρρ s ) (16.22)
β = 0.56 − 0.16ρ s (16.14)