Rotating Angle Shear Theories - Unified Theory of Concrete Structures - page 178

Figure 5.15 Stress-strain relationship of prestressing strands

It is interesting to observe the character of the Ramberg-Osgood curve in two ways.
When the first term, 1, in the denominator of Equation 11 b or 12 b is omitted, the curve
degenerates into the horizontal, asymptotic, straight line f p = f pu . When the second term,

E ps (

f pu m , in the denominator of Equation 11 b or 12 b is omitted, the curve
degenerates into the asymptotic straight line f p =

ε dec + ε s ) /

E ps ( ε dec + ε s ), with a slope of E ps .

5.4.3 Solution Algorithm
The twelve governing equations for a PC 2-D element, Equations 1 to 12 , contain fifteen
unknown variables. These fifteen unknown variables include eight stresses ( σ , σ t , τ t , σ d , f ,

f t , f p , f tp ) and five strains ( ε , ε t , γ t , ε r , ε d ), as well as the angle α r and the material coefficient

ζ . If three unknown variables are given, then the remaining twelve unknown variables can be

solved using the twelve equations.
If the stresses and strains are required throughout the post-cracking loading history, then
two variables must be given, and a third variable selected. In general,

ε d is selected as the third
variable because it varies monotonically from zero to a maximum. For each given value of

ε d ,
the remaining twelve unknown variables can be solved. The series of solutions for a sequence
of

ε d values allows us to trace the loading history.
In most of the structural applications, the two given variables come from the three externally
applied stresses σ , σ t and τ t . They are generally given in two ways. First, the applied normal

stresses τ t is a variable to be solved.

This type of problem occurs in nuclear containment structures. The stresses

σ and σ t , are given as constants, while the shear stress

σ t in a wall
element are induced by the internal pressure and are given as constants, while the shear stress
τ t is caused by earthquake motion and is considered as an unknown variable. This first type
of problem is treated in Sections 5.4.3 and 5.4.4.
In the second type of relationship, the three applied stresses

σ and

σ , σ t and τ t increase pro-

portionally. They are related to the applied principal tensile stress σ 1 by given constants (see

Section 5.4.5). In this way, the three variables are reduced to one variable. This type of prob-
lem occurs in all elastic structures where the three stresses on a wall element are produced