Torsion - Unified Theory of Concrete Structures - page 310

18 equations for torsion members stems from the fact that a 2-D element in the shear flow zone
is subjected to the in-plane truss action of a membrane element. The additional 5 equations for
a torsional member and the modifications of the required constitutive relationships are caused
by the warping of the 2-D element in the shear flow zone.
For a member subjected to pure torsion the normal stresses

σ and σ t actingona2-D

element in the shear flow zone are equal to zero, i.e. ε d is selected as the

third variable because it varies monotonically from zero to maximum, then the remaining 18
unknown variables can be solved (Hsu, 1991b, 1993). The series of solutions for various

σ = σ t = 0. If

ε d

values allows us to trace the loading history of the torque-twist curve.

7.1.5.1 Characteristics of Equations

An efficient algorithm for solving the 18 equations was derived based on a careful observation
of the six characteristics of the equations:

−

1. The three equilibrium equations [1]- [3], and the three compatibility equations [5] [7],are

− σ , σ t ,

transformation-type equations. In other words, the stresses in the t coordinate (

τ t ) are expressed in terms of stresses in the r − d direction ( σ d , σ r = 0), and the strains in the

ε d ).
2. Equations [12], [13] and [14] for the constitutive laws of concrete in compression involve
only four unknown variables in the r

− t coordinate ( ε , ε t , γ t ) are expressed in terms of strains in the r − d direction ( ε r ,

− d coordinate ( σ d , ε r , ε d and ε ds ). If the strains ε r ,

σ d can be calculated from these three equations.
3. Equations [3] and [4] are independent from all other equations, because they contain two
variables,

ε d and ε ds , are given, then the stresses

τ t and T , which are not involved in any other equations. In other words, these
two equations need not be involved in the iteration process of the solution algorithm.
4. The longitudinal steel stresses f and f p in equilibrium equation [1] are coupled to the

longitudinal steel strain ε in compatibility equation [5] through the longitudinal steel

stress-strain relationship of Equations [15] and [17]. Similarly, the transverse steel stresses
f t and f tp in equilibrium equation [2] are coupled to the transverse steel strain

ε t in com-
patibility equation [6] through the transverse steel stress-strain relationships of Equations
[16] and [18].
5. Equations [7]-[10] sequentially relate the four variables, γ t , θ , ψ and ε ds . Hence, these

four equations can easily be combined into one equation.
6. The variable t d is involved in Equations [1], [2], [8] and [10] through the terms A o , p o ,

ρ

and ρ t . Since these four equations are coupled, it is necessary to first assume the variable t d

and then check it later. The flow chart for solution algorithm will require a nested DO-loop
to determine the variable t d .

7.1.5.2 Thickness t d as a Function of strains
The thickness of shear flow zone, t d , can be expressed in terms of strains using the compatibility
equations [5]-[10]. To do this we will first combine the four compatibility equations [7]-[10]
into one equation according to characteristic 5 of Section 7.1.5.1. Inserting

γ t from Equation

[ 7 ] into [ 8 ] gives

p o
A o

θ = ( ε r − ε d )sin α r cos α r (7.50)