ConcretewBeta Material

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). In tension, the model has allows for either tri-linear softening or nonlinear tension stiffening based on 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 page 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 normal tensile strains, ε_n. This model uses a tri-linear relation between ε_n and β (see the Biaxial Behavior Section). Default values result in β = 1, and thus no compressive stress reduction.

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>

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$matTag	integer tag identifying material
$fpc	peak unconfined concrete compressive strength*
$ec0	compressive strain corresponding to unconfined concrete compressive strength*
$fcint, $ecint	intermediate stress-strain point for compression post-peak envelope*
$fcres, $ecres	residual stress-strain point for compression post-peak envelope*
$ftint	tensile strength of concrete
$ftint, $etint	intermediate stress-strain point for tension softening envelope
$ftres, $etres	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)
$bint $ebint	intermediate β-strain point for for biaxial effect (default 1 and 0, respectively)
$bres $ebres	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)
$Ec	initial stiffness (default 2*$fpc/$ec0; see Note 3)
$fcc $ecc	confined concrete peak compressive stress and corresponding strain* (see Eq. 1)

NOTES:

(1) *Parameters of concrete in compression should be specified as negative values.

(2) For non-zero $M, the tension stiffening behavior will govern the post-peak tension envelope. Tri-linear tension softening parameters $ftint, $etint, $ftres, $etres will have no effect, but dummy values must be specified.

(3) Value of $Ec must be between $fpc/$ec0 and 2*$fpc/$ec0 otherwise the closest value will be assigned.

Implementation

Uniaxial Behavior

Figure 1. ConcretewBeta material model behavior based on specified input parameters

Figure 1 shows the compression and tension envelopes and the input parameters. The confined concrete envelope is defined as:

Equation 1.

up to strain $ecc. The default values of $fcc and $$ecc are set equal to $fpc and $ec0, respectively, resulting in an unconfined behavior. Following this region, the compression envelope is tri-linear and passes through the points ($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:

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 tri-linear path is ignored and the nonlinear tension stiffening behavior is:

Equation 3.

It is suggested that $M = (75 mm)*ρ_l/d_b where ρ_l is the steel ratio in the direction parallel to the material direction and d_b 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

Figure 2. Relation between the concrete compressive stress reduction factor, β, and normal tensile strain, ε_n

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 the Truss2 Element page).

Figure 2 above shows the relationship between concrete compressive stress reduction factor, β, and the normal tensile strain, ε_n. For compressive stresses, the instantaneous stress value computed by the material is β*f_c where f_c is the compressive stress given by the uniaxial behavior described above. The β = 1 for positive (tensile) stress.

For compressive stress, the β-ε_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.

Example

See: Truss Model Example - Squat RC Wall

References

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.

Lu, Y., and Panagiotou, M. (2013). “Three-Dimensional Nonlinear Cyclic Beam-Truss Model for Reinforced Concrete Non-Planar Walls.” Journal of Structural Engineering, published online.

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.

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.

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Code Developed by: Yuan Lu, UC Berkeley and Marios Panagiotou, UC Berkeley