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 July 2014

Overview

1. Distributed inelasticity models: Objectivity

2. Damage-following Adaptive Force-based elements

3. Implementation using updateparameter commands

4. Application – Comparison with experimental RC column

5. Final Comments

Distributed inelasticity models Damage-following Adaptive Force-based Implementation using updateparameter Application Conclusions

Inelastic Beam-Column elements

o Concentrated Inelasticity Model – Plastic Spring (Moment- Rotation Spring, Interfaces)

o Distributed Inelasticity Model with Moment – Curvature relations for Sections

o Distributed Inelasticity Model with Fiber or Layer Sections

Rodrigues, 2012


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 dofs to avoid global degrees of freedom and damping effects.

a) equaldof command to ensure rotational continuityb) adjust damping/stiffness (Zareian & Medina, 2010)

• Suitable to include global element failure criteria and interface mechanisms**


* Haselton et al. 2007; Lignos & Krawinkler 2009, 2010** Elwood, 2004; LeBorgne and Ghannoum, 2009; Lodhi and Sezen, 2012; Shoraka and Elwood, 2013; LeBorgne and 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.


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.


Adapted from Borges et al. 2004

Strain localization - Objectivity in Softening Response

Objectivity: account for numerical localization issues

Softening Sectional Behaviour

Hardening Sectional Behaviour

• 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 *


* See Calabrese et al. 2010

Damage/Characteristic length (Physical - Ldamage)

= Localization length

(Numerical – ω1)

Ensure objectivity with Force-based elements: Limitations of general approaches

𝐿𝑝=0.08∙𝐿

+0.022∙𝑓 𝑦∙𝜙𝑏

𝐿𝑝=0.08∙𝐿

+0.022∙𝜆

𝑠𝑝Objectivity in Softening Response

Gauss-Lobatto integrationGauss-Legendre integration

Comparison of the different numerical options against an empirical length for the localization (Lp given by Paulay and Priestley, 1992)

𝐿𝑝=0.08 ∙ 𝐿 + 0.022 ∙ 𝑓𝑦 ∙ 𝜙𝑏

𝐿𝑝=0.08 ∙ 𝐿 + 𝜆𝑠𝑝


Solution: Use Regularized Force-based methods

In OpenSees:

• A robust model defined according to Scott and Fenves, 2006.

• Modified Gauss-Radau integration

• 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


How to have a consistent Damage-following beam formulation?


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 Almeida et al., 2012 – Computers and Structures

• Essentially 4 steps:

1 – Define a Gauss - Lobatto integration 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.


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 Vandermonde matrix, i.e. setting the error function to zero for f(x)=1,x,x2,xN-2, N=Number of IPs.


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


• 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. Check the Moment diagram to look for the My (Lee and Filippou, 2006; Mergosand Kappos, 2008).

2. Check the conditional material states within the fiber section. Check for the length of Plastification of the longitudinal steel bars (i.e. 𝜀 > 𝜀𝑦)

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: 𝜀 > 𝜀𝑦).

4) Compute the new integration weights (previous slide) | w1→ λpl.

5) Update the Integration weights.

6) Store the new maximum curvature as the reference value.

7) Go to next analysis step.

Simplified Damage-following adaptive elements


• Tools: Make use of OpenSees framework flexibility

• All done in tcl.

• Starts with a low order integration scheme but 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 updateparameter to redefine the integration weights.

Implementation of the simplified modelling strategy



Application – RC column with moderate axial load

Column S24-5T tested by Bae (2005). Aprox. 3.00 m heightSection 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


Ch

arac

teri

stic

Le

ngt

h:

Pri

estl

ey e

t a

l. ~

0.4

34

m

Ex

per

imen

tal ~

0.2

87

m

-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

Late

ral 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

Late

ral 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

Late

ral 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

Late

ral Load

Experimental

BWH

Curvature

Curvature

Mo

men

tM

om

ent

Element issues

Application – Damage-following 7 IPs Adaptive Force-based element


0 200 400 600 800 1000 1200 1400 16000.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22

0.24

0.26

Step

Chara

cte

ristic L

ength

-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

Late

ral 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

Late

ral 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 ResponseElement’s Response starts to be Material Dependent

Curvature

Mo

men

t

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

Late

ral 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

Late

ral 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


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.

• The simplified damage-following adaptive strategy shows a satisfactoryperformance against experimental results, particularly in getting the finaldamage length.

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

[email protected] 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

[email protected] Faculty of Engineering – University of Porto, Portugal