ConcretewBeta Material

This command is used to construct a uniaxial concrete material object with a compressive stress-strain envelope based on the Fujii concrete model. The model has options for tri-linear softening behavior in tension and compression as well as nonlinear tension stiffening.

When used with the Truss2 or CorotTruss2 elements, the model accounts for the effect of normal tensile strains on the concrete compressive behavior. This model uses a tri-linear relation between the normal tensile strain and the associated compressive stress reduction factor.

See this section for an example of how to use this material with the Truss2 element.

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

The instantaneous stress computed by the material model is defined to be β*f where f is the uniaxial stress calculated for the given strain history, and β is the compressive stress reduction factor defined in the Biaxial Behavior Section. For tensile stress, β = 1 while for compressive stress, the beta varies depending on inputs of the -beta option and the computed normal strain. Default values result in β = 1, and thus no compressive stress reduction.

Uniaxial Behavior

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

The above figure shows the shape of the compression and tension envelopes, based on the specified input parameters. If the confined concrete option is given, the compression loading envelope is defined as:

Equation 1.

up until strain $ecc. If the confined concrete option is not specified, the above equation for compression strains less than $ec0 is used. Following this region, the compression envelope goes linear to the points ($ecint, $fcint) and ($ecres, $fcres) in that order. For compression strains larger than $ecres, a residual stress value of $fcres is used.

Unloading from compression strain, the following slope is used:

Equation 2.

until reaching zero stress, which then reloads linearly to the point with the largest tensile strain that occurred before.

The tension envelope is linear until it reaches the specified tension strength $ft. If the tension stiffening parameter $M is not specified, the tension envelope after reaching $ft will go linearly to the specified points of ($etint, $ftint) and ($etres, $ftres) in that order. For tensile strains larger than $etres, a residual stress value of $fcres is used. If the tension stiffening parameter is specified, the concrete softens as:

Equation 3.

where 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.

Unloading from tension strain, a slope of $Ec, the initial material tangent, is used. After reaching zero stress, the material targets a compression stress equal to -$alpha*$ft at zero strain. Thereafter, the material loads linearly to the point where the peak compressive strain occurred. In the case where the slope leading to this target point is less than that for the point with stress -$alpha*$ft at zero strain, the material reloads directly to the point where peak compressive strain occurred without passing through the point with stress -$alpha*$ft at zero strain.

Biaxial Behavior

Figure 2. Relation between 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, which gives the instantaneous normal tensile strain. Figure 2 above shows the relationship between the β factor and the normal tensile strain, based on the specified input parameters. For compressive stresses, the instantaneous stress value computed by the material is given to be β*f_c where f_c is the compressive stress given by the uniaxial behavior described above and β is determined from the instantaneous normal strain given by the Truss2 element. The β = 1 for positive (tensile) stress.

At zero normal tensile strain, β = 1, resulting in no reduction of compression strength. With increasing normal tensile strain, the β factor goes linearly to the specified points of ($ebint, $bint) and ($ebres, $bres) in that order. For normal tensile strains larger than $ebres, a residual β value of $bres is used.

Example

Figure E1. (a) Concrete panel under pure shear deformation; (b) Truss representation of the panel under shear.

This example illustrates how to use the biaxial concrete material ConcretewBeta with the Truss2 to compute the effects of normal tensile strain on the concrete compressive strength. For an application of the element and material in truss models for reinforced concrete walls, please see the References.

Download the relevant tcl files here: File:ConcwBetaExample1.zip.

Model Description

In this example, we are testing a 10” x 10” x 1” thick concrete panel in pure shear deformation, as shown in Figure E1 (a). Figure E1 (b) shows two perpendicular diagonal trusses, which represent the two principle stresses for the panel. The area of each diagonal is 5√2 ". The specified input parameters for the material are listed below.

$fpc	-4. ksi
$ec0	-0.002
$fcint, $ecint	-2. ksi, -0.004
$fcres, $ecres	-0.1 ksi, -0.01
$ftint	0.3 ksi
$ftint, $etint	0.01 ksi, 0.001
$ftres, $etres	0. ksi, 1.0
$Ec	3600. ksi
$alpha	0.5
$bint $ebint	0.3, 0.01
$bres $ebres	0.1, 0.025

Results

Figure E2(a) shows the shear force-displacement computed for the prescribed cyclic loading history. In this simple example, the strain in diagonal A is equal to 0.5*Θ while the strain in diagonal B is negative of that. Θ is the displacement ratio and is equal to Δ/10” where Δ is the imposed displacement as shown in Figure E1(a). Figure E2(b) shows the stress-strain history and Figure E2(c) shows the computed β factor vs. strain for diagonal A [as defined in Figure E1(b)].

Figure E2. (a) Shear force-displacement computed for the panel; (b) Concrete stress-strain history in the diagonal A; (c) Computed beta factor vs. strain for diagonal A.

References

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.

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