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3.3.6 Concrete modelling using modified DRUCKER-PRAGER model
The yield condition consists of two yield criteria (equations (3-19), (3-20)), whereby the concrete strength
can be described closed to the reality as well in the compressive as in the tensile domain.
F σ + β σ − σ~
Ω
1
= (3-19)
S
t =
R
3
β
2
S
t
m
( R − R )
d
d
+ R
F σ + β σ − σ~
Ω
β =
c
where:
c
z
m
z
yt
yc
1
2
~ σ
yt
=
2 R
d
3 ( R
d
R
z
+ R
z
)
= (3-20)
3
2R
( R − R )
u
u
− R
d
d
~ σ
yc
=
R
3 ( 2
u
R
R
u
d
− R
d
)
σm hydrostatic stress
I2 second invariant of the deviatoric main stresses
Rz uniaxial tensile strength
Rd uniaxial compression strength
Ru biaxial compression strength
Ω hardening and softening function (in the pressure domain Ω1 = Ω2 = Ωc, in the tensile
domain Ω1 = Ωt).
The plasticity potentials are:
Q = σ + δ β σ
1
2
S
S
t
c
t
c
m
Q = σ + δ β σ
m
where: δt, δc are dilatancy factors
16
(3-21)
The yield condition is shown in Fig. 3-9 and Fig. 3-10 in different coordinate systems. The comparison
with the concrete model made by Ottosen [6-15] is shown in Fig. 3-9 and illustrates the advantages of the
Drucker-Prager model consisting of two yield criteria. While there is a very good correspondence in the
compressive domain, the chosen Drucker-Prager model can be well adjusted to realistic tensile strength.
In opposite to that, the Ottosen model overestimates these areas significantly! A further advantage lies
within the description of the yield condition using the three easily estimable and generally known
parameters Rz, Rd and Ru.
Fig. 3-9 Singular Drucker-Prager flow conditions – Illustrated in the octaeder system
USER’S MANUAL, January, 2013