ConcretewBeta
This command is used to construct a uniaxial concrete material object that explicitly considers for the effect of normal (to the axis where the material object is used) strain to the behavior of the concrete in compression. The compressive stress-strain envelope, up to the peak compressive strength(unconfined or confined) is based on the Fujii concrete model (Hoshikuma et al. 1997). The material has two options regarding the strength degradation in tension: (a) tri-linear or (b) nonlinear [based on the tension stiffening relation of Stevens et al (1991)]. The softening behavior in compression is tri-linear.
The model accounts for the effect of normal tensile strains on the concrete compressive behavior when used with the Truss2 or CorotTruss2 elements. See the Truss2 Element for description of how the normal strain is computed. The instantaneous stress is β*f where f is the computed stress and β is the compressive stress reduction factor which depends on the normal tensile strain, ε<sub>n</sub>. The relation between ε<sub>n</sub> and β (see the Biaxial Behavior Section) is tri-linear. Default values result in β = 1.
See the Examples Section for the use of this material model in truss models for planar RC walls and a beam-truss model for a non-planar wall loaded biaxially.
uniaxialMaterial ConcretewBeta $matTag $fpc $ec0 $fcint
$ecint $fcres $ecres $ft $ftint $etint $ftres $etres
< -lambda $lambda >
< -alpha $alpha> <-beta $bint $ebint $bres $ebres >
< -M $M > < -E $Ec >
< -conf $fcc $ecc >
matTag
|
integer tag identifying material |
fpc
|
peak unconfined concrete compressive strength* |
ec0
|
compressive strain corresponding to unconfined concrete compressive strength* |
|
intermediate stress-strain point for compression post-peak envelope* |
|
residual stress-strain point for compression post-peak envelope* |
ftint
|
tensile strength of concrete |
|
intermediate stress-strain point for tension softening envelope |
|
residual stress-strain point for tension softening envelope |
Optional: |
|
lambda
|
controls the path of unloading from compression strain (default 0.5) |
alpha
|
controls the path of unloading from tensile strain (default 1) |
|
intermediate β-strain point for for biaxial effect (default 1 and 0, respectively) |
|
residual β-strain point for for biaxial effect (default 1 and 0, respectively) |
M
|
factor for Stevens et al. (1991) tension stiffening (default 0; see Note 2) |
Ec
|
initial stiffness (default 2* |
|
confined concrete peak compressive stress and corresponding strain* (see Eq. 1) |
NOTES:
- *Parameters of concrete in compression should be specified as negative values.
- For non-zero
M
, the tension stiffening behavior will govern the post-peak tension envelope. Tri-linear tension softening parametersftint, etint, ftres, etres
will have no effect, but dummy values must be specified.- Value of
Ec
must be betweenfpc
/ec0
and 2*fpc
/ec0
otherwise the closest value will be assigned.Implementation
- Value of
- For non-zero
\[ \$ \boldsymbol{E} \boldsymbol{u}=\$ \text { lambda } \cdot\left(\frac{f}{\varepsilon}\right)+(1-\$ \text { lambda }) \cdot \$ \boldsymbol{E c} \]
Uniaxial Behavior
Figure 1 shows the compression and tension envelopes and the input parameters. The confined concrete envelope is defined by Equation 1 up to strain ecc
. The default values of fcc
and ecc
are equal to fpc
and ec0
, respectively, for an an unconfined behavior. Following this region, the compression envelope is tri-linear and passes through the points (ecint
, fcint
) and (ecres
, fcres
) in that order. For compression strains larger than ecres
, the residual stress is fcres
.
For compression strain, the slope of the unloading branch is defined by Equation 2. After reaching zero stress, the material reloads linearly to the point with the largest tensile strain that occurred before.
The tension envelope is linear until it reaches ft
. If the tension stiffening parameter M
is not specified, the tension envelope after reaching ft
is tri-linear and passes through the points (etint
, ftint
) and (etres
, ftres
) in that order. For tensile strains larger than etres
, the residual stress is fcres
.
If M
is specified, the nonlinear tension stiffening behavior defined by Equation 3. It is suggested that M
= (75 mm)*ρ<sub>l</sub>/d<sub>b</sub> where ρ<sub>l</sub> is the steel ratio in the direction parallel to the material direction and db is the bar diameter in mm.
The material unloads from tension strain using a slope of Ec
. After reaching zero stress, the material targets the point (0, -alpha
ft
). Thereafter, the material loads linearly to the point where the peak compressive strain previously occurred. In the case where the slope leading to this target point is less than that for the point (0, -alpha
ft
), the material reloads directly to the point where peak compressive strain occurred.
Biaxial Behavior
The ConcretewBeta material model accounts for the biaxial strain field on the concrete compressive behavior when used in conjunction with the Truss2 element. The Truss2 element computes the strain normal to the direction of the element (see Truss2 Element).
Figure 2 shows the relationship between concrete compressive stress reduction factor, β, and the normal tensile strain, ε<sub>n</sub>. For compressive stresses, the instantaneous stress value computed by the material is \(\beta f_c\) where \(f_c\) is the compressive stress given by the uniaxial behavior described above. For positive (tensile) stress, \(\beta = 1\). For compressive stress, the \(\beta - \epsilon_n\) relationship is tri-linear and passes through the points (0,1), (ebint
, bint
), and (ebres
, bres
) in that order. For normal tensile strains larger than ebres
, β = bres
.
Examples
20-story RC core wall buildings: conventional fixed-base (video), rocking wall (video), and base isolation with rocking wall (video)
1200px |alt=20-story core walls
5-story coupled wall specimen with diagonal tension failure, see: Video of the simulation

See: Truss Model - Mestyanek (1986) Squat RC Wall and Video of the simulation

See: Beam-truss Model - Beyer et al. (2008) RC Wall and Video of the simulation

See: Truss Model - Massone Sanchez (2005) Squat RC Wall and Video of the simulation

References
Lu, Y., Panagiotou, M, and Koutromanos, I. (2014). “Three-dimensional beam-truss model for reinforced concrete walls and slabs subjected to cyclic static or dynamic loading.” Report PEER 2014/18, Pacific Earthquake Engineering Research Center, University of California, Berkeley, Berkeley, CA.
Lu, Y. and Panagiotou, M. (2014). “Earthquake Damage Resistant Multistory Buildings at Near Fault Regions using Base Isolation and Rocking Core Walls.” 1st Huixian International Forum on Earthquake Engineering for Young Researchers, August 16-19, Harbin, China.
Lu, Y., and Panagiotou, M. (2014). “Three-Dimensional Nonlinear Cyclic Beam-Truss Model for Reinforced Concrete Non-Planar Walls.” Journal of Structural Engineering, 140 (3), DOI: 10.1061/(ASCE)ST.1943-541X.0000852.
Panagiotou, M., Restrepo, J.I., Schoettler, M., and Kim G. (2012). “Nonlinear cyclic truss model for reinforced concrete walls.” ACI Structural Journal, 109(2), 205-214.
Beyer, K., Dazio, A., and Priestley, M. J. N.(2008). “Quasi-Static Cyclic Tests of Two U-Shaped Reinforced Concrete Walls.” Journal of Earthquake Engineering, 12:7, 1023-1053.
Hoshikuma, J., Kawashima, K., Nagaya, K., and Taylor, A. W. (1997). “Stress-strain model for confined reinforced concrete in bridge piers.” Journal of Structural Engineering, 123(5), 624-633.
Massone Sanchez, L. M. (2006). “RC Wall Shear—Flexure Interaction: Analytical and Experimental Responses.” PhD thesis, University of California, Los Angeles, Los Angeles, CA, 398 pp.
Mestyanek, J. M. (1986). “The earthquake resistance of reinforced concrete structural walls of limited ductility.” ME thesis. University of Canterbury.
Stevens, N. J., Uzumeri, S. M., Collins, M. P., and Will, T. G. (1991). “Constitutive model for reinforced concrete finite element analysis.” ACI Structural Journal, 88(1), 49-59.
Code Developed by: Yuan Lu, UC Berkeley and Marios Panagiotou, UC Berkeley