Equilibrium (Plasticity) Truss Model - Unified Theory of Concrete Structures - page 30

Equations (2.21) and (2.22) are in terms of stresses ( τ ty , ρ f y , ρ t f ty ). Equations (2.23) and

f c ,ω ,ω t . (2.24) are in terms of the nondimensional indices τ ty /ζ

Over-reinforced Elements
When 1, we have an over-reinforced element where the concrete crushes before the

yielding of steel in one or both directions. Since this failure mode violates the basic assumption
of the plasticity truss model, the design of such an element is unacceptable and no further
discussion is necessary.

ω + ω t >

Three Cases of the Balanced Condition
The balanced condition, ω + ω t = 1, can be divided into three cases:

Case (1) : ω = ω t

When ω = ω t , the yielding of both the longitudinal and the transverse steel occur simultaneo-

usly with the crushing of concrete. The balanced condition in this case gives ω = ω t = 0.5,

resulting in

τ ty
ζ f c =

0 . 5 (2.25)

45 ◦

α r = (2.26)

Case (2) : 0.5

In this case the transverse steel has yielded, but the concrete crushes simultaneously with the
yielding of the longitudinal steel. The longitudinal reinforcement index becomes

ω t <

ω = 1 − ω t

according to the balanced condition, and Equation (2.23) becomes

τ ty
ζ f c =

ω t (1 − ω t ) (2.27)

The corresponding angle α r from Equation (2.24) is

(1

− ω t )
ω t

45 ◦ tan α r = > 1 ,α r > (2.28)

Case (3) : 0.5

In this case the longitudinal steel has yielded, but the concrete crushes simultaneously with
the yielding of the transverse steel. The transverse steel will then be determined by the balanced
condition,

ω <

ω t = 1 − ω , and Equation (2.23) becomes

τ ty
ζ f c =

ω (1 − ω ) (2.29)

α r is

The corresponding angle

ω

45 ◦

tan α r =

− ω ) <

1 ,α r < (2.30)

(1