A Simplified Damage-following Model for Reinforced Concrete Columns

OpenSees Days Portugal 2014
A Simplified Damage-following Model for Reinforced Concrete Columns:
Nuno Pereira
Xavier Romão
Pereira and Romão
Simplified Damage-following Model for RC Columns
Porto,4 July2014

Overview
1.Distributed inelasticity models: Objectivity
2.Damage-following Adaptive Force-based elements
3.Implementation using updateparametercommands
4.Application –Comparison with experimental RC column
5.Final Comments
Distributed inelasticity models
Damage-following Adaptive Force-based
Implementation usingupdateparameter
Application
Conclusions

Inelastic Beam-Column elements
oConcentrated Inelasticity Model –Plastic Spring (Moment-Rotation Spring, Interfaces)
oDistributed Inelasticity Model with Moment –Curvature relations for Sections
oDistributed Inelasticity Model with Fiber or Layer Sections
Rodrigues, 2012
Application
Conclusions

Concentrated Inelasticity Model
Concentrated Inelasticity Model
•Rotational springs: represents a given length where the curvatures are integrated. In OpenSees, zero-length elements.
•Proposed strategies available for different problems*). Calibration Issues. Dependent on the test configurations.
•Consistency: Condensation of the internal dofsto avoid global degrees of freedom and damping effects.
a) equaldofcommand to ensure rotational continuity
b) adjust damping/stiffness(Zareian& Medina, 2010)
•Suitable to include global element failure criteria and interface mechanisms**
Application
Conclusions
* Haseltonet al. 2007; Lignos& Krawinkler2009, 2010
** Elwood, 2004; LeBorgneand Ghannoum, 2009; Lodhiand Sezen, 2012; Shorakaand Elwood, 2013; LeBorgneand Ghannoum, 2014

Distributed Inelasticity Model
Distributed inelasticity model: Definition
•Model defined by any formulation that consistently integrates sectional deformations to define the element deformations.
Main Principles: the Fiber-based methodology
•Define uniaxial material modelling constitutive laws for the materials
•Discretize the sectional configuration in a grid of uniaxial cells (fibers) and assign to each fiber a material constitutive law
•In RC sections it encloses the uniaxial representation of Confined Concrete, Unconfined Concrete and Steel.
Application
Conclusions

Distributed Inelasticity Model –Strain Localization
Strain localization –physical viewpoint
•Experimental failures of concrete samples show a localization of deformations, both in tension (crack) and in compression (damage zone).
•Size dependency.
Application
Conclusions
Adapted from Borges et al.2004

Strain localization -Objectivity in Softening Response
Objectivity: account for numerical localization issues
SofteningSectional Behaviour
HardeningSectionalBehaviour
•The selection of the number of IPs along with adequate integration schemes needs to ensure objectivity in the response
The strains concentrate at the
integration point location subjected to the highest bending moment. The damage zone is related to the correspondent integration weight
Increased accuracy with higher number of IPs *
Application
Conclusions
* See Calabrese et al.2010
Damage/Characteristic length (Physical -Ldamage)
=
Localization length
(Numerical –ω1)

Ensure objectivity with Force-based elements: Limitations of general approaches
퐿푝=0.08∙퐿+0.022∙푓푦∙휙푏
퐿푝=0.08∙퐿+0.022∙휆푠푝
Objectivity in Softening Response
Gauss-Lobatto integration
Gauss-Legendre integration
Comparison of the different numerical options against an empirical length for the localization (Lp given by Paulay and Priestley, 1992)
퐿푝=0.08∙퐿+0.022∙푓푦∙휙푏
퐿푝=0.08∙퐿+휆푠푝
Application
Conclusions

Solution: Use Regularized Force-based methods
In OpenSees:
•A robust model defined according to Scott and Fenves, 2006.
•Modified Gauss-Radauintegration
•Advantages
•Ensures that the model localizes over a length equal to the inelastic zone length (defined a priori).
•Disadvantages:
•Relies on empirical expressions to define plastic hinge length.
•Inaccurate response for hardening or partially hardening models
Regularized methods in Softening Response
Application
Conclusions

How to have a consistent Damage-following beam formulation?
Application
Conclusions
Summary:
•Commute between objective hardening and softening phases in RC sectional response –adaptive.
•Most correlate real flexural damage and computed damage

Adaptive Force Based elements
•Proposed By Almeidaet al., 2012 –Computers and Structures
•Essentially 4 steps:
1 –Define a Gauss -Lobattointegration method with 7 IPs
2 –Check at each analysis step if the extreme sections have hardening or softening properties.
3 –If hardening is observed, next step. If softening is found, use a regularized model.
4 –Set the regularized model by re-computing the integration weights using the characteristic length as the reference for the extreme IPs.
Application
Conclusions
Commute between objective hardening and softening -Adaptive Force Based elements

•How to re-compute the integration weights?
•Define a characteristic length value (Lp)
•Use expressions proposed by Almeida et al relying on the solution of the Vandermondematrix, i.e. setting the error function to zero for f(x)=1,x,x2,xN-2, N=Number of IPs.
Application
Conclusions
Commute between objective hardening and softening -Adaptive Force Based elements
Adapted from Almeida et al., 2011
•The update of the integration weights can be done
•Inside N-R
•Inside Nested Element-Section verification
•At the end of Each time step -Simplified Approach adopted herein

Correlate real and computed damage -Damage-following adaptive elements
Application
Conclusions
•Basic Idea: Associate a damage measure to describe the characteristic length, i.e. the length of the inelastic zone along the length of the structural member.
1.ChecktheMomentdiagramtolookfortheMy(LeeandFilippou,2006;MergosandKappos,2008).
2.Check the conditional material states within the fiber section. Check for the length of Plastification of the longitudinal steel bars (i.e.휀>휀푦)
Rationale:Use 7IPs and construct the profile of longitudinal strains of exterior bars.
•BUT still…relying on the definition of a priori characteristic length

•Simplified approach considered:
1) Get the sectional Moment, Curvature and strains at the current step.
2) Check for softening. If softening responses are observed, use updating routine. Else skip it and continue. If Softening is already in place, check for higher curvature.
3) With the strains, use Piecewise Hermetian Cubic Interpolation to find the characteristic length (λpl)according to Ly: 휀>휀푦).
4) Compute thenewintegration weights (previous slide) | w1→λpl.
5) UpdatetheIntegrationweights.
6)Storethenewmaximumcurvatureas thereferencevalue.
7)Goto nextanalysisstep.
Simplified Damage-following adaptive elements
Application
Conclusions

•Tools: Make use of OpenSees framework flexibility
•All done in tcl.
•Starts with a low order integration schemebut defining the weights and positions corresponding to Gauss Lobatto integration.
•Sets the integration weights as parameters (can be changed during the analysis)
•Uses eleResponse recorders to get the flexure response at the extreme sections at the current step –Moment, Curvature, strains.
•Verify if softening has started or after that if the current curvature is higher than the previous maximum value. If so, compute the λpl and the new ω**.
•Use updateparameterto redefine the integration weights.
Implementation of the simplified modelling strategy
Application
Conclusions

Application
Conclusions
Application –RC column with moderate axial load
Column S24-5T tested by Bae (2005).
Aprox. 3.00 m height
Section 0.610 x 0.610 m2
Moderate axial load level (20%)
Damage Length = 0.47H=0.287m
•Case Study
Adapted from Bae (2005)

Application – Effect of Empirical definition of Lp in BWH
Distributed inelasticity models Damage-following Adaptive Force-based Implementation using updateparameter Application Conclusions
Characteristic Length:
Priestley et al. ~ 0.434 m Experimental ~ 0.287m
-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2
-1500
-1000
-500
0
500
1000
1500
Top Displacement
Lateral Load
Experimental
BWH LP,Priestley et al
-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2
-400
-300
-200
-100
0
100
200
300
400
Top Displacement
Lateral Load
Experimental
BHW 0.47H
-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2
-400
-300
-200
-100
0
100
200
300
400
Top Displacement
Lateral Load
Experimental
BWH Lp,Priestley et al
-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2
-1500
-1000
-500
0
500
1000
1500
Top Displacement
Lateral Load
Experimental
BWH
Curvature
Curvature
Moment
Moment
Element issues

Application – Damage-following 7 IPs Adaptive Force-based element
0 200 400 600 800 1000 1200 1400 1600
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0.22
0.24
0.26
Step
Characteristic Length
-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2
-400
-300
-200
-100
0
100
200
300
400
Top Displacement
Lateral Load
Experimental
Adaptive strain 7IP (Curvature trigger approach)
-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25
-1500
-1000
-500
0
500
1000
1500
Top Displacement
Lateral Load
Experimental
Adaptive stran 7IPs (curvature trigger approach)
Remarks:
Experimental ~ 0.287 m
Priestley et al. ~ 0.434 m
Damage-following ~ 0.242 m
Material Dependent
Stable element Response
Element’s Response starts to be Material Dependent
Curvature
Moment

Application – Damage-following 7 IPs Adaptive Force-based element
-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2
-400
-300
-200
-100
0
100
200
300
400
Top Displacement
Lateral Load
Experimental
Adaptive strain 7IP (Curvature trigger approach)
-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2
-400
-300
-200
-100
0
100
200
300
400
Top Displacement
Lateral Load
Experimental
BWH Lp,Priestley et al
• Better fitting between the experimental and the damage-following model
• BWH model starts softening at the predefined rate associated to the empirical
definition of Lp. Adaptive model “softens” this transition
• Are the errors extremely important? Uncertainty quantifications needed for general
use, particularly if Damage indexes based on curvatures are used (f.i. Park & Ang)

Conclusions
Application
Conclusions
Main Remarks
•An Adaptive Force based element was implemented in a tcl environment using basic output and updating strategies available in OpenSees.
•A damage-following adaptive strategy to update the inelastic zone of RC columns presented with basis on the computation of the yielding length of longitudinal steel bars.
•An experimental test involving moderate axial loads was used to compare with robust finite length (force-based) element available in OpenSees.
•The results have shown that different types of results can be obtained when different assumptions are used for the inelastic zone length.
•Thesimplifieddamage-followingadaptivestrategyshowsasatisfactoryperformanceagainstexperimentalresults,particularlyingettingthefinaldamagelength.

Conclusions
Application
Conclusions
Challenges:
•Only one experimental test as been used. Shows the possible implications of modelling options of one analyst.
•More tests needed to find the real robustness of the model.
•Cost of making a full verification of all the structural elements.
•Check the errors associated to other models regarding local and global EDPs. Compute them and provide them towards the definition of guidelines. Disaggregate modelling uncertainties.

“Remember that all models are wrong;
the practical question is how wrong do they have to be to not be useful.”
George E. Box
“…it is nevertheless a big mistake to equate
“low” with zero in probability -land.”
T. Hanks and A. Cornell
Nuno Pereira
www.seismosafety.weebly.com
nmsp@fe.up.pt
Faculty of Engineering –University of Porto, Portugal

Acknowledgements
Dr. Sungjin Bae-Bechtel Power Corporation, USA
Dr. João Pacheco de Almeida –EPFL | École Polytechnique Fédérale de Lausanne, Switzerland
Nuno Pereira
www.seismosafety.weebly.com
nmsp@fe.up.pt
Faculty of Engineering –University of Porto, Portugal

A Simplified Damage-following Model for Reinforced Concrete Columns

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