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GENERAL IMPLEMENTATION OF MODIFIED IBARRAMEDINA-KRAWINKLER MODELS FOR CONCENTRATED PLASTICITY MODELS AND FINITE LENGTH PLASTICITY MODELS Filipe L. A. Ribeiro (1) André R. Barbosa (2) Luís C. Neves
(3)
(1) Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa, Portugal (2) School of Civil and Construction Engineering, Oregon State University, Corvallis, U.S.A. (3) Nottingham Transportation Engineering Centre, University of Nottigham, U.K.
Fisciano, Italy June 2015
Scope
OpenSees Days Italy 2015
Concentrated plastic hinge (CPH) formulation Finite-length plastic-hinge (FLPH) formulation
Images adapted from: NIST GCR 10-917-5. “NEHRP Seismic Design Technical Brief No. 4”. 2010
Mod IMK Models
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The original Ibarra-Medina-Krawinkler (IMK) model is based on a backbone curve that represents the behavior for monotonic loading and defines the limits for cyclic loading and accounts for four main deterioration modes (total of six modes)
Images adapted from: Lignos (2008). “Sidesway collapse of deteriorating structural systems under seismic excitations”. PhD Thesis. Stanford University.
Mod IMK Models
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Models implemented in OpenSees (a) (b) (c) (d)
Backbone curve Bilin model Peak-Oriented mode Pinching model
Deterioration mechanisms:
Basic strength (b,c,d) Post-yielding ratio (b,c,d) Post-capping strength (b,c,d) Unloading Stiffness (b,c,d) Reloading stiffness (c,d) Pinching(d)
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Mod IMK Models Modeling the rate of deterioration
The rates of cyclic deterioration are controlled by a characteristic total hysteretic energy dissipation capacity Et and an energy based rule developed by Rahnama and Krawinkler (1993)
Ei i i Et E j j 1
c
Et Fy
Ei
In general, a parameter X, which can represent any of the six deterioration modes (e.g., basic strength) and can include a stiffness parameter or a strength parameter, can be updated through:
X i 1 i X i 1
1 i Fy
FLPH formulation
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What is it? FLPH formulation is an efficient distributed plasticity formulation with designated hinge zones at the member ends. Cross sections in the inelastic hinge zones are characterized through either nonlinear moment-curvature relationships or explicit fiber-section integrations that enforce the assumption that plane sections remain plane.
Integration scheme Scott and Fenves (2006) proposed a Modified Gauss-Radau integration scheme in order to avoid localization issues
FLPH formulation
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Advantages Avoid localization issues (occurs in distributed plasticity elements)
When compared to the concentrated plasticity formulation: Lower modeling effort (less nodes and elements) Lower computational cost Allows for clear separation between member and connection nonlinearity Disadvantages Not possible to directly use empirically calibrated moment-rotation relationships (such as the ones provided by ModIMK models) Needs a plastic hinge length to be assigned (empirical or based on moment gradient – afternoon)
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FLPH formulation How to use ModIMK models in FLPH elements?
Converting a moment-rotation relationship into moment-curvature:
M
M
K M K M Lp
Et Fy
/ Lp θ
M M / Lp
The moment-curvature relationship can then be assigned to define the nonlinear hinge sections
χ
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FLPH formulation How to use ModIMK models in FLPH elements? However, that is not enough M
Ke
6 EI Lp L
χ
Lp
L 2Lp
Lp
OpenSees Days Italy 2015
FLPH formulation How to use ModIMK models in FLPH elements? However, that is not enough
Thinking about the elastic region Original member
EI EI
EI EI
EI EI
FLPH member
6EI Lp EI L
EI EI
EI 6EI Lp L
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FLPH formulation How to use ModIMK models in FLPH elements? Proposed calibration procedure
Ribeiro, F.L.A.; Barbosa, A.R.; Scott, M.H. and Neves, L.A.C. (2014). “Deterioration Modeling of Steel Moment Resisting Frames Using FiniteLength Plastic Hinge Force-Based Beam-Column Elements. ASCE Journal of Structural Engineering.
FLPH formulation How to use ModIMK models in FLPH elements? Proposed calibration procedure
If LpI LpJ Lp :
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OpenSees Days Italy 2015
CPH formulation What is it? Linear elastic element
Rigid-plastic zero-length springs
How to overcome the additional flexibility? Increase the elastic stiffness of the springs
Consequently, change the post-yielding ratios
K e, s n
'
6EI L
KT , s K e,s 1 n (1 )
Change the stiffness of the element interior
EI mod EI
n 1 n
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CPH formulation Advantages:
“Directly” assign moment-rotation relationships to the zero-length springs. Disadvantages:
The ideal n value is not trivial, as low values lead to erroneous results and high values result in numerical instability. The use of the parameter n, which modifies the elastic stiffness and the post-yielding ratios does not allow for direct use of expression proposed by Rahnama and Krawinkler (1993) for update of model parameters due to deterioration
X i 1 i X i 1
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CPH formulation Problems due to the parameter n
Take the unloading stiffness update (due to deterioration) as an example
Elastic stiffness:
Ke
Energy dissipated: Es 100kNm Deterioration: i
Ei i
Et E j
100 0.1 1000 0
j 1
Updated unloading stifffness:
Ku 1 i K e 0.9 K e
In the case of a CPH formulation: Ke n Ke Thus: Ku 1 i n K e 0.9 n K e If n=100: Ku 90 K e K e
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CPH formulation Problems due to the parameter n
Take the unloading stiffness update (due to deterioration) as an example
Deterioration parameter 1 i should be seen as a post-yielding ratio thus being computed through:
KT , s ' K e,s 1 n (1 ) '
Ku (1 i ) K e 1 n (1 (1 i ))
CPH formulation
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Problems due to the parameter n Deterioration mechanisms:
Basic strength Post-yielding ratio Post-capping strength Unloading Stiffness Reloading stiffness Pinching
Ribeiro, Neves, Barbosa (2015). Implementation and calibration of nite-length plastic hinge elements for use in seismic structural collapse analysis. Journal of Earthquake Engineering (in press)
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CPH formulation Proposed implementation
Implemented ModIMK models should be prepared to be used with any formulation
Add an additional input parameter n (default value is 0) Internal amplification of the elastic stiffness Ke (n 1) Ke Modification of the post-yielding ratios due to parameter n
'
KT K e 1 n (1 )
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CPH formulation Proposed implementation Unloading stiffness
(1i(1i ) K u ,i 1 KKuu,i,i j j ) K0 In case of CPH formulation:
ij (1 j ) K u ,i 1 n 1 ij (1 j )
K0
CPH formulation Proposed implementation Post-yielding stiffness
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Implementation
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How to use ModIMK models in OpenSees? Bilin02 model: uniaxialMaterial Bilin02 $matTag $Ke0 $as_Plus $as_Neg $My_Plus $My_Neg $Lamda_S $Lamda_C $Lamda_A $Lamda_K $c_S $c_C $c_A $c_K $theta_p_Plus $theta_p_Neg $theta_pc_Plus $theta_pc_Neg $Res_Pos $Res_Neg $theta_u_Plus $theta_u_Neg $D_Plus $D_Neg
nFactor = 0 (by default)
No Changes introduced (backwords
ModIMKPeakOriented02 model: and $My_Plus FLPH implementation) uniaxialMaterial ModIMKPeakOriented02 $matTag $Ke0compatibility $as_Plus $as_Neg $My_Neg $Lamda_S $Lamda_C $Lamda_A $Lamda_K $c_S $c_C $c_A $c_K $theta_p_Plus $theta_p_Neg nFactor >$Res_Pos 0 Change model properties (for use in $theta_pc_Plus $theta_pc_Neg $Res_Neg $theta_u_Plus $theta_u_Neg $D_Plus $D_Neg CPH models) ModIMKPinching02 model: uniaxialMaterial ModIMKPinching02 $matTag $Ke0 $as_Plus $as_Neg $My_Plus $My_Neg $FprPos $FprNeg $A_pinch $Lamda_S $Lamda_C $Lamda_A $Lamda_K $c_S $c_C $c_A $c_K $theta_p_Plus $theta_p_Neg $theta_pc_Plus $theta_pc_Neg $Res_Pos $Res_Neg $theta_u_Plus $theta_u_Neg $D_Plus $D_Neg
Validation Example Simply supported beam
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OpenSees Days Italy 2015
Validation Example 3-Story steel frame
Beam B1
Conclusions
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Accurate results can be achieved either by using FLPH models or CPH models, since the proposed algorithm is used if the CPH models are employed A significant reduction in model complexity is obtained when FLPH models are employed The implementation procedure for FLPH models is significantly simpler than that required for the CPH models and the use of ad-hoc parameters simulating rigid plastic behavior can be avoided Current implementation in OpenSees (Bilin02, ModIMKPeakOriented02, ModIMKPinching02) accurately reproduce Modified Ibarra-Medina-Krawinkler model laws for both CPH and FLPH models Further work is needed in order to quantify errors associated with nonconsideration of nFactor in the update of model parameters
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Thank you Filipe L. A. Ribeiro
[email protected]
Fisciano, Italy June 2015