Torsion - Unified Theory of Concrete Structures

Civil Engineering Reference

In-Depth Information

ε dec =

strain in prestressing steel at decompression of concrete,

ε dec = ε pi

+ ε i ; taken as

0.005 for grade 1862 MPa (270 ksi) strands;

ε pi =

initial strain in prestressed steel after loss;

ε i =

initial strain in mild steel after loss;

E ps =

elastic modulus of prestressed steel, taken as 200 000 MPa (29 000 ksi);

E ps =

tangential modulus of Ramberg-Osgood curve at zero load, taken as 214 000 MPa

(31 040 ksi);

f pu =

ultimate strength of prestressing steel, taken as 1862 MPa (270 ksi);

shape parameter describing the curvature at knee portion, taken as 4.

Notice that the tensile stress of concrete is assumed to be zero, i.e.

σ r =

0. Thus we have a

total of 18 equations, rather than 19.

7.1.5 Method of Solution

The 18 governing equations for a torsional member (Equations [1] - [18]) are listed in Table

7.2 in three categories: the 4 equilibrium equations, the 7 compatibility equations and the

7 constitutive equations. These 18 equations contain 21 unknown variables, which are also

divided into three categories in Table 7.2. The 9 stress or force variables include

σ ,

σ t ,

τ t ,

σ d , f , f t , f p , f tp and T . The 10 strain or geometry variables include

ε ,

ε t ,

γ t ,

ε r ,

ε d ,

α r ,

and k 1 . If 3 unknown variables are given,

then the remaining 18 unknown variables can be solved using the 18 equations.

Table 7.2 also lists the 13 governing equation, 1 to 13 , used in the RA-STM analysis of

2-D elements. The similarity between this set of 13 equations for 2-D elements and the set of

, t d and

ε ds ; and the 2 material coefficients are

Table 7.2

Summary of Variables and Equations for 2-D Elements and Torsional Members

Variables

Equations

Stresses

or forces

Strains or

geometry