Torsion - Unified Theory of Concrete Structures - page 308

τ t = ( − σ d )sin α r cos α r [3]

T = τ t (2 A o t d ) [4]

Compatibility equations

ε = ε r cos 2 α r + ε d sin 2

α r [5]

= ε r sin 2 α r + ε d cos 2

ε t α r [6]

γ t
2 =

( ε r − ε d )sin α r cos α r [7]

p o
2 A o γ t

θ =

[8]

ψ = θ sin 2 α r [9]

ε ds
ψ

t d = [10]

ε ds
2

ε d = [11]

Constitutive law of concrete in compression

f c

σ d = k 1 ζ [12]

1

ε ds
ζε o

1

3 ε ds

k 1 = − ε ds /ζ ε o ≤ 1 [13 a ]

ζε o

1 1 1

2 2

ζ 1

3 ζε o

ζ

) 2 ε ds

1

3 ε ds

k 1 = − − + − ε ds /ζ ε o > 1 b ]

) 2

ε ds ζε o ζε o

(2 − ζ (2 − ζ

0 . 9

ζ = √ 1 [14]

+ 600 ε r

Constitutive law of mild steel

f = E s ε ε <ε y [15 a ]

f =

ε ≥ ε y

f y [15 b ]

f t = E s ε t ε t <ε ty [16 a ]

f t = f ty ε t ≥ ε ty [16 b ]

Constitutive law of prestressing steel

f p ≤ 0 . 7 f pu f p = E ps ( ε dec + ε s ) [17 a ][18 a ]

E ps (

ε dec + ε s )

f p > 0 . 7 f pu f p = [17 b ][18 b ]

1 E ps ( ε dec + ε s )

f pu

m

1
m

+

where
f p = stress in prestressing steel, f p becomes f p or f tp when applied to the longitudinal

and transverse steel, respectively;

ε s = strain in the mild steel, ε s becomes ε or ε t when applied to the longitudinal and

transverse steel, respectively;