,
are determined with a failure
function which gives the compressive stress which causes failure as a function
of the confining stresses in the lateral directions.
If the material is cracked in the lateral direction, the parameters are
reduced with the factor
for the peak strain, and with the factor
for the peak stress.
A possible relationship is given in
§18.2.7.2.
It is tacitly assumed that the base curve in compression is
determined by the peak stress value
f_{p} = f_{cf}
, and the
corresponding peak strain value
=
.
In summary,
f_{p} 
= ^{ . }f_{cf} 
(18.93) 

= ^{ . } 
(18.94) 
The base function in compression, with the parameters
f_{p}
and
,
is modeled with a number of different predefined and userdefined curves.
The predefined curves are the constant curve and the brittle curve.
Also available are the linear hardening curve and
the saturation hardening curve.
Figure 6.6 shows
the available hardeningsoftening curves in compression.
For the curve according to Thorenfeldt and
the parabolic curve we give some background theory.
Figure 18.10:
Thorenfeldt compression curve

This curve [Fig.18.10] is described by
with
n = 0.80 + ; k = 
(18.96) 
The parabolic curve in DIANA is a formulation based on fracture energy,
according to Feenstra [27].
The parabolic curve is described by three characteristic values
[Fig.18.11].
Figure 18.11:
Parabolic compression curve

The strain
,
at which
onethird of the maximum compressive strength
f_{c}
is reached,
is
The strain
,
at which the maximum compressive strength is reached, is
Note that
and
are determined irrespective of of the element size or
compressive fracture energy.
Finally, the ultimate strain
,
at which the material is completely softened in compression, is
The parabolic compression curve in DIANA is now described by
f = 
(18.100) 
It could now easily be verified that
the fracture energy
G_{c}
and
the characteristic element length h
govern the softening part of the curve only:
The total compressive fracture energy which has been found in experiments
ranges from 10
to 25
[Nmm/mm^{2}]
which is about 50
to 100
times the
tensile fracture energy [27].
The increase of the strength with increasing isotropic stress is modeled with
the fourparameter HsiehTingChen failure surface which is defined as
f = 2.0108 +0.9714 +9.1412 +0.2312  1 = 0 
(18.102) 
with the invariants J_{2}
and I_{1}
defined in terms of the stress in the
concrete
according to
J_{2} 
= (  )^{2} + (  )^{2} + (  )^{2} 
(18.103) 
I_{1} 
= + + 
(18.104) 
and
f_{c1}
the maximum concrete stress
f_{c1} = max(,,) 
(18.105) 
which is not the maximum tensile stress but the maximum principal
stress [12].
The parameters in (18.102) are determined by fitting of the
uniaxial tensile and compressive strength, the biaxial compressive strength,
and experimental data of triaxial tests on concrete specimen.
The stress
f_{c3}
is assumed to result in failure and is determined
by scaling the linear elastic stress
vector
_{c} = s E _{nst}
such that the equation
(18.102) holds.
The compressive failure stress in multiaxial stress situation is then given by
f_{c3} = s^{ . }min(,,) 
(18.106) 
If the scaling factor s
is negative, thus resulting in a positive failure
stress
f_{c3}
, the stress vector is scaled to the tensile side of the
failure surface and the failure strength is set equal to a large negative value
(
30 f_{cc}
).
The failure strength
f_{cf}
is given by
f_{cf} =  f_{c3} 
(18.107) 
The peak stress factor
K_{}
is given by
Selby [96, Eq.(2.7)]
K_{} = 1 
(18.108) 
and the peak strain factor is assumed to be given by
In unconfined compression, the values at the peak are given by the uniaxial
values compressive strength, and the peak stress factor is equal to one.
The parameters of the compressive stressstrain function now become
f_{cf} 
= K_{}f_{cc} 
(18.110) 

= K_{} 
(18.111) 
with the value of the initial strain
is given by the relationship
The equations given above result in a gradual increase of the maximum
strength in confined compression, with an initial slope of the stressstrain
diagram given by the Young's modulus.
In a full triaxial stress situation
the failure surface cannot be reached and a linear stressstrain relation
is obtained [Fig.18.12].
Figure 18.12:
Influence of lateral confinement on compressive stressstrain curve

The increased ductility of confined concrete is modeled by a linear adaption
of the descending branch of the Thorenfeldt curve according to
f_{j} =  f_{p}1  (1  r)  rf_{p} 
(18.113) 
with r
the factor which models the residual strength of the material
[Fig.18.13].
Figure 18.13:
Compressive behavior under lateral confinement

The ultimate strain in compression is assumed to be
determined by the ratio between the peak strength and the compression strength
and the strain at peak according to
with the scalar
to be determined, currently
= 3
is assumed.
The residual strength
r f_{p}
also depends on the ratio between the
peak strength and the compressive strength according to
with r_{0}
an initial value, assume
r_{0} = 0.1
.
The linear compressionsoftening relationship is only applied for the
Thorenfeldt curve if the peak value
f_{p}
is considerably larger than the compressive strength
f_{cc}
,
assume
f_{p}/f_{cc} > 1.05
.
In case of lateral compression and lateral cracking resulting in
f_{p}/f_{cc} < 1.05
, will not increase the
ductility of the material.
18.2.7.2 Compressive Behavior with Lateral Cracking
In cracked concrete, large tensile strains perpendicular to the principal
compressive direction reduce the concrete compressive strength.
The
compressive strength
f_{p}
is consequently not only a function of the
internal variable
, but also a function of the internal variables
governing the tensile damage in the lateral directions,
and
.
The reduction factors due to lateral cracking are denoted as
= ()
and
= ()
,
which are functions of the average lateral damage variable given by
=
.
The relationship for reduction due to lateral cracking is
the model according to
Vecchio & Collins [109, model B] [Fig.18.14]
in which
K_{c} = 0.27   0.37 
(18.117) 
The factor
is equal to one.
Figure 18.14:
Reduction factor due to lateral cracking

Next: 18.2.8 Combination with Thermal
Up: 18.2 Total Strain Crack
Previous: 18.2.6 Shear Behavior
Contents
Index
DIANA9.4.2 User's Manual  Material Library
First ed.
Copyright (c) 2010 by TNO DIANA BV.