ConcretewBeta Material: Difference between revisions

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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.  
  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.


When used with the '''[[Truss2 Element| Truss2]]''' or '''[[CorotTruss2 Element| 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.  
&emsp;&emsp;The model accounts for the effect of normal tensile strains on the concrete compressive behavior when used with the '''[[Truss2 Element| Truss2]]''' or '''[[CorotTruss2 Element| CorotTruss2]]''' elements. See the '''[[Truss2 Element| Truss2 Element]]''' for description of how the normal strain is computed. The instantaneous stress is ''&beta;''*''f'' where ''f'' is the computed stress and &beta; is the compressive stress reduction factor which depends on the normal tensile strain, &epsilon;<sub>n</sub>. The relation between &epsilon;<sub>n</sub> and &beta; (see the '''[[#Biaxial Behavior|Biaxial Behavior Section]]''') is tri-linear. Default values result in ''&beta;'' = 1.
 
See the '''[[#Examples|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.
 
::[[File:BeyerTUB point770.jpg|thumb|upright=2.0|alt=RC C-shaped wall |Reinforced concrete wall with a C-shaped section subject to multi-axial loading, described in [[#Examples|the examples]].]]


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==Implementation==
==Implementation==
::[[File:ConcwBeta_Eq1a.png|thumb|upright=2.5|Equation 1.]]
::[[File:ConcwBeta_Eq2a.png|thumb|upright=1.8|Equation 2.]]
::[[File:ConcwBeta_Eq3.png|thumb|upright=1.8|Equation 3.]]


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


===Uniaxial Behavior===
&emsp;&emsp;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'''.
 
&emsp;&emsp;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.
 
&emsp;&emsp;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'''.
 
&emsp;&emsp;If '''$M''' is specified, the nonlinear tension stiffening behavior defined by Equation 3. It is suggested that '''$M''' = (75 mm)*''&rho;<sub>l</sub>''/''d<sub>b</sub>'' where ''&rho;<sub>l</sub>'' is the steel ratio in the direction parallel to the material direction and ''d<sub>b</sub>'' is the bar diameter in mm.
 
&emsp;&emsp;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.


::[[File:ConcwBeta_Fig1b.gif|thumb|center|upright=4.0|alt=ConcretewBeta Figure 1 |Figure 1. ConcretewBeta material model behavior based on specified input parameters]]
::[[File:ConcwBeta_Fig1b.gif|thumb|center|upright=4.0|alt=ConcretewBeta Figure 1 |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:
::[[File:ConcwBeta_Eq1.png|thumb|center|upright=3.0|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:
::[[File:ConcwBeta_Fig2b.gif|thumb|upright=2.0|alt=ConcretewBeta Figure 2 |Figure 2. Relation between the concrete compressive stress reduction factor, &beta;, and normal tensile strain, &epsilon;<sub>n</sub> ]]
::[[File:ConcwBeta_Eq2a.png|thumb|center|upright=2.0|Equation 2.]]
===Biaxial Behavior===
until reaching zero stress, which then reloads linearly to the point with the largest tensile strain that occurred before.
 
&emsp;&emsp;The ConcretewBeta material model accounts for the biaxial strain field on the concrete compressive behavior when used in conjunction with the '''[[Truss2 Element| Truss2]]''' element. The '''[[Truss2 Element| Truss2]]''' element computes the strain normal to the direction of the element (see '''[[Truss2 Element| Truss2 Element]]''').
 
&emsp;&emsp;Figure 2 shows the relationship between concrete compressive stress reduction factor, ''&beta;'', and the normal tensile strain, &epsilon;<sub>n</sub>. For compressive stresses, the instantaneous stress value computed by the material is ''&beta;''*''f<sub>c</sub>'' where ''f<sub>c</sub>'' is the compressive stress given by the uniaxial behavior described above. For positive (tensile) stress, ''&beta;'' = 1. For compressive stress, the ''&beta;''-&epsilon;<sub>n</sub> 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 '''$ebres''', ''&beta;'' = '''$bres'''.


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 '''$fcres''' is used.
If the tension stiffening parameter is specified, the concrete softens as:
::[[File:ConcwBeta_Eq3.png|thumb|center|upright=2.0|Equation 3.]]
where it is suggested that '''$M''' = (75 mm)*''&rho;<sub>l</sub>''/''d<sub>b</sub>'' where ''&rho;<sub>l</sub>'' is the steel ratio in the direction parallel to the material direction and ''d<sub>b</sub>'' 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===


::[[File:ConcwBeta_Fig2b.gif|thumb|center|upright=2.0|alt=ConcretewBeta Figure 2 |Figure 2. Relation between concrete compressive stress reduction factor, &beta;, and normal tensile strain, &epsilon;<sub>n</sub> ]]


The ConcretewBeta material model accounts for the biaxial strain field on the concrete compressive behavior when used in conjunction with the '''[[Truss2 Element| Truss2]]''' element, which gives the instantaneous normal tensile strain. Figure 2 above shows the relationship between the ''&beta;'' 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 ''&beta;''*''f<sub>c</sub>'' where ''f<sub>c</sub>'' is the compressive stress given by the uniaxial behavior described above and ''&beta;'' is determined from the instantaneous normal strain given by the '''[[Truss2 Element| Truss2]]''' element. The ''&beta;'' = 1 for positive (tensile) stress.


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




==Example==


See: [[Truss Model Example - Squat RC Wall]]
 
 
==Examples==
===20-story RC core wall buildings: conventional fixed-base ('''[http://youtu.be/r14GDOB9tgY video]'''), rocking wall ('''[http://youtu.be/DmEwyWwcRP4 video]'''), and base isolation with rocking wall ('''[http://youtu.be/FBj-mNos8gU video]''') ===
[[Image:20story_samplePic2.png|1200px |alt=20-story core walls]]
 
 
 
===5-story coupled wall specimen with diagonal tension failure, see: '''[http://youtu.be/a26aZiU5RgY Video of the simulation]'''===
[[Image:5story_samplePic.png|800px|link=http://youtu.be/a26aZiU5RgY |alt=RC coupled wall]]
 
 
 
===See: '''[[Truss Model Example - Mestyanek Squat Wall| Truss Model - Mestyanek (1986) Squat RC Wall]]''' and '''[http://youtu.be/lQpzwHF_Z94 Video of the simulation]'''===
[[Image:Mestyanek resultsPlot2.png|800px|link=Truss Model - Mestyanek Squat Wall |alt=RC squat wall, Unit 1.0 tested by Mestyanek (1986)]]
 
 
 
===See: '''[[Beam-truss Model Example - C-shaped RC Wall| Beam-truss Model - Beyer et al. (2008) RC Wall]]''' and '''[http://youtu.be/9O9Mev62Ilw Video of the simulation]'''===
[[File:BeyerWall fig2.png|800px|link=Beam-truss Model Example - C-shaped RC Wall |alt=non-planar RC wall, TUB tested by Beyer et al.]]
 
 
 
===See: '''[[Truss Model Example - Squat RC Wall| Truss Model - Massone Sanchez (2005) Squat RC Wall]]''' and '''[http://youtu.be/aq7r4HMAmvc Video of the simulation]'''===
[[File:MassoneWall displ1a.jpg|400px|link=Truss Model Example - Squat RC Wall |alt=RC squat wall, Wall WP1105-8 tested by Massone Sanchez(2005)]]  




==References==
==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.
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.
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.





Latest revision as of 03:03, 4 February 2015

  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, εn. The relation between εn 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.

RC C-shaped wall
Reinforced concrete wall with a C-shaped section subject to multi-axial loading, described in the examples.
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*
$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

Equation 1.
Equation 2.
Equation 3.

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)*ρl/db where ρl 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.

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


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

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, εn. For compressive stresses, the instantaneous stress value computed by the material is β*fc where fc is the compressive stress given by the uniaxial behavior described above. For positive (tensile) stress, β = 1. For compressive stress, the βn 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)

20-story core walls


5-story coupled wall specimen with diagonal tension failure, see: Video of the simulation

RC coupled wall


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

RC squat wall, Unit 1.0 tested by Mestyanek (1986)


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

non-planar RC wall, TUB tested by Beyer et al.


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

RC squat wall, Wall WP1105-8 tested by Massone Sanchez(2005)


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