Damage-based models for flexural response on RC elements
FINAL DOCTORAL EXAMINATION - May 2nd, 2012 DEPT. of CIVIL and ENVIRONMENTAL ENGINEERING at UC DAVISDEPT. of MECHANICS and MATERIALS at UNIVERSITY of REGGIO CALABRIADAMAGE-BASED MODEL FOR FLEXURAL RESPONSE OFREINFORCED CONCRETE BEAM-COLUMN ELEMENTS by Maria Grazia Santoro
1. Objective and Scope of the Research 2. Overview of Previous Studies 3. Kinematic of Planar Frames 4. Typical Progression of Damage in RC Columns 5. Methodology, Hypotheses and Degradation Scheme 6. RC Damage-Based Beam ElementTABLE OF CONTENTSTABLE OF CONTENTS 7. Damage Indices 8. Stiffness Matrix of the Damaged Member 9. Plastic functions: Yield Surface and Mixed Hardening Rule • Integration Algorithm for Path-Dependent Plasticity 10. Energy Release Rate 11. Damage Criterion 12. Damage Evolution Law • Integration Algorithm for Lumped Damage 13. Formulation of the Problem 14. Return Map Algorithm for Damage-Based Beam Element 15. Numerical Solution Strategies • Structure State Determination • Element State Determination 16. Numerical Examples 17. Remarks on Damage-Based Model Performance 18. Identification of Model Parameters dy and r
1. To implement a numerical model based on damage mechanics for predicting the nonlinear flexural response of RC members under cyclic static loadings,OBJECTIVE AND SCOPEOF THE STUDYOBJECTIVE AND SCOPEOF THE STUDY 2. To predict damage indices by combining matrix structural analysis, plastic theories and basic concepts continuum damage mechanics, 3. To introduce a different way of accounting for stiffness and strength degradation based on damage indices, 4. To provide a new damage evolution law, calibrated on experimental tests by using mechanical properties of the member, 5. The advantage of the model lies in its efficacy to evaluate the inelastic response of RC members that couples computationally inexpensive finite elements with classical moment-curvature analysis, 6. Damage indices introduced can facilitate the process of integrating structural components damage to determine the overall structural performance, 7. The model and numerical simulations are implemented on a special purpose computer program for nonlinear static and dynamic analysis of reinforced concrete developed at University of California at Berkeley (FEDEASLab, Release 3.0) and in OpenSees environment, 8. Analytical results are compared with experimental results extracted from the UW-PEER RC column performance database, which documents the performance of more than 450 columns.
LUMPED PLASTICITY MODELS the process of energy dissipation responsible for the nonlinear structural response is concentrated at the ends of the finite element, in special locations called plastic hinges Clough , Giberson  and Otani OVERVIEW OF PREVIOUS STUDIESOVERVIEW OF PREVIOUS STUDIES Filippou and Issa , Kunnath et al.  - increase of the inelastic zone length LUMPED DAMAGE-BASED MODELS interpreted as a combination between continuum damage mechanics and the concept of plastic hinge Cipollina et al.  - Flòrez–Lòpez  Marante  - Faleiro et al.  - Alva et al.  Observation: use of force parameters to control damage accumulation and tendency to overestimate damage and degradation at low to moderate damage states 1 d EA 0 0 a L 1 d i 4 d j 4 EI 4 1 d i 1 d j 2 EI K 0 4 did j L 4 did j L 4 1 d i 1 d j 2 EI 1 d j 4 d i 4 EI 0 4 did j L 4 did j L ln 1 d d= d a di dj g G Gcr q 1 d
Local and global reference systems and notation for planar frames U=[ Uf Ud] P=[Pf Pd]KINEMATIC OF PLANAR FRAMES Uf unknown displ. – free DOFsKINEMATIC OF PLANAR FRAMES Ud assigned displ. – restrained DOFs Pf nodal loads Pd support reactions Element nodal displacements Element nodal forces u=[u1 u2 .... u6]T p=[p1 p2 .... p6]T Basic element deformations Basic element forces v=[v1 v2 v3]T q=[q1 q2 q3]T
COMPATIBILITY EQUATIONS Element deformations can be expressed in terms of element end displacements using the compatibility transformation matrix ag X Y X Y KINEMATIC OF PLANAR FRAMESKINEMATIC OF PLANAR FRAMES v agu L L 0 L L 0 U f ag Y X 1 Y X 0 V AU A f Ad U L2 L 2 L2 L2 d Y X Y X STRUCTURAL COMPATIBILITY MATRIX 2 0 1 L 2 L L2 L2 EQUILIBRIUM EQUATIONS The equilibrium equations must be satisfied in the undeformed configuration, if displacements are small relative to the structure. p bg q Nodal equations of static Bf structural equilibrium P BQ Q B AT Bd bg aT applied forces P=[Pf Pd] are in equilibrium with g the resisting forces Pr, which are the sum of the P Pr 0 element contributions p(el).
CONSTITUTIVE EQUATIONS – SECTION RESPONSEKINEMATIC OF PLANAR FRAMESKINEMATIC OF PLANAR FRAMES LINEAR ELASTIC RESPONSE OF FRAME ELEMENTS L 0 0 EA 0 0 EA L L L 4 EI 2 EI q f e 1 v v 0 fe 0 ke 0 3EI 6 EI L L q ke v q0 L L 2 EI 4 EI 0 0 6 EI 3EI L L nonmechanical initial deformations v0, caused by temperature and shrinkage strains and fix–end forces q0 under nonmechanical deformations
If displacements are large, equilibrium needs to be satisfied in the deformed configuration. In this case nonlinear geometric effects must be taken into account accounting for P-∆ effectsKINEMATIC OF PLANAR FRAMESKINEMATIC OF PLANAR FRAMES using the matrix bP∆ instead of bu corotational formulation (Crisfield, 1990) p bu q NONLINEAR GEOMETRY and P-∆ GEOMETRIC STIFFNESS a g u T q k k g km q a g u a g u u v element stiffness matrix in local coordinates is composed of two contributions: the geometric stiffness kg arising from the change of the equilibrium matrix with end displacements, and the material stiffness km, which represents the transformation of the tangent basic stiffness matrix to the local coordinate system
1. Increasing of axial deformations increase strains of the concrete cover untilPROGRESSION OF DAMAGE IN RC COLUMNSPROGRESSION OF DAMAGE IN RC COLUMNS cracking and spalling. The loss of cross-sectional area imposes additional stresses on the remaining concrete core and on longitudinal reinforcement (Bresler, 1961). 2. The longitudinal reinforcement yields in tension and eventually begins to strain harden. 3. Poisson’s effect causes expansion of the concrete core, which exerts pressure on both longitudinal and transverse reinforcement. 4. Transverse reinforcement restrains the lateral deflection of longitudinal reinforcement, and it confines the expanding core. The confining pressure is not uniform: it depends on stiffness and strength of transverse reinforcement (Bresler and Gilbert, 1961). Additionally, the stiffness of the tie depends on its strain, which in turn is affected by the axial deformation of the column and by bar buckling. 5. The increased axial strain and imposed lateral deformations (due to core expansion) lead to instability of longitudinal bars (Bayrak and Sheikh, 2001). When the tie spacing is very large, longitudinal bar buckling can occur between two adjacent ties (Dhakal and Maekawa, 2002). In other situations, bar buckling can occur over several tie spacings. 6. Bar buckling is affected by the maximum tension strain and by tension strain growth (associated with cyclic inelastic deformations) in the longitudinal reinforcement (Moyer and Kowalsky, 2001). 7. The effect of cycling on the constitutive properties of the concrete and steel, is significant (Monti and Nuti, 1992). The load history and cycling affect damage progression and specifically bar buckling (Kunnath et al., 1997).
FUNDAMENTALS OF CONTINUUM DAMAGE MECHANICS DEFINITION OF DAMAGE degradation of material properties resulting from initiation, growth and coalescence of microcracks or microvoids, and it isMETHODOLOGY – HYPOTHESES represented by a scalar in the range 0-1METHODOLOGY – HYPOTHESES DEGRADATION SCHEME DEGRADATION SCHEME Sd d S EFFECTIVE STRESS the ratio between the load applied on the volume element and the effective resistance area 1 d HYPOTHESIS OF STRAIN EQUIVALENCE the strain associated with a damage state under the Cauchy stress is equivalent to the strain associated with its undamaged state under effective stress e E E 1 d
The modelling of reinforced concrete behaviour comes from the experimental observation that damage is a continuous process that initiates at very low levels of applied loads and leads to an increasing amount of damage when levels of strain increase. Conversely, the behavior of steel bars is dominated by plasticity laws and damage materializes at higherMETHODOLOGY – HYPOTHESES deformation levelsMETHODOLOGY – HYPOTHESES DEGRADATION SCHEME DEGRADATION SCHEME For reinforced concrete structures, plasticity is physically associated to the flow of reinforcement, while damage indicates cracking and crushing of the concrete
CONCENTRATED PLASTICITY-DAMAGE APPROACH simplicity, computational convenience of having stiffness matrix in a concise form FLEXURAL RESPONSE UNDER UNIAXIAL BENDING and CONSTANT AXIAL LOADMETHODOLOGY – HYPOTHESESMETHODOLOGY – HYPOTHESES hysteretic behavior of RC is governed by only few parameters DEGRADATION SCHEME DEGRADATION SCHEME SMALL DEFORMATIONS SHEAR DEFORMATIONS NEGLECTED AXIAL BEHAVIOR LINEAR ELASTIC AND UNCOUPLED FROM THE FLEXURAL BEHAVIOR simple truss element w/o second order effects UNCOUPLED PLASTIC-DAMAGE BEHAVIOR Independent constitutive eqns. (1) change of the elastic properties of the material is produced only by damage; (2) plasticity only produces incompatible strains
combined degradation of both strength and stiffness under the effect of damage indices and a mixed-hardening ruleMETHODOLOGY – HYPOTHESESMETHODOLOGY – HYPOTHESES DEGRADATION SCHEME DEGRADATION SCHEME The softening effect resulting into strength loss is caused by damage, meaning that an increment of damage at a constant plastic deformation, produces a decrease of the plastic limit. Damage also controls the degradation of the unloading stiffness.
ONE COMPONENT MODELRC DAMAGE-BASED BEAM ELEMENTRC DAMAGE-BASED BEAM ELEMENT element forces = B. MOMENTS q=[qi qj]T element deformations= ROTATIONS v=[vi vj]T q qel q pl v v el v in q kel v el kel v v in v in v pl v d
d=[di dj] Numerical quantification of flexural damage at hinges i and j Damage parameters take values in the interval [0 1] If d=0 (no damage), standard plastic hinge If d=1 (hinge totally damaged) internal hinge in elastic member d+ = d+ i d j d = d i d j DAMAGE INDICESDAMAGE INDICES for RC members subjected to cyclic loadings two different set of cracks can appear, one due to positive end moments (positive cracks) and another due to negative end moments (negative cracks). This behavior can be represented by using two sets of damage variables where the superscript + and – denote damage due to positive and negative moments
2 4 2 3 1- d i 1- d j EI 1- d j STIFFNESS MATRIX OF DAMAGED MEMBERSTIFFNESS MATRIX OF DAMAGED MEMBER -1 L 1- d i f ed ked 6 EI -1 2 4 1- d i 1- d j L 4 1- d j 2 1- d i for di=dj=0, the stiffness matrix of the damaged member is equal to the elastic stiffness matrix. If di=1 and dj=0, then ked becomes the stiffness matrix of an elastic member with an internal hinge at the left end. If di=0 and dj=1, then ked becomes the stiffness matrix of an elastic member with an internal hinge at the right end.
The yield function is obtained according to a mixed-hardening ruleYIELD SURFACE and MIXED HARDENING RULE which control the rate of transition of the yield surface.YIELD SURFACE and MIXED HARDENING RULE If isotropic and kinematic hardening are combined, the yield surface is allowed both to expand and to translate, providing a more realistic modeling of the real behavior, especially when dealing with cyclic loading. f q , qb , n q qb q pl H is 0 T PLASTIC FUNCTIONS: PLASTIC FUNCTIONS: q= q i q j q b = q bi q bj q pl = M yi M yj = i j f sign qi qbi 0 n ni n j T q 0 sign q j qbj
FLOW RULE and HARDENING RULES v pl sign q qb INTEGRATION ALGORITHM FOR PATH-INTEGRATION ALGORITHM FOR PATH- , qb H k v pl DEPENDENT PLASTICITY DEPENDENT PLASTICITY KUHN-TUCKER CONDITIONS k 0, f k 0, k f k 0, k fk 0, k i, j 0 if f 0 (no plasticity) 0 if f 0 (plastic increment) CONSISTENCY CONDITIONS f < 0 if 0 f = 0 0 if f = 0
DISCETIZATION of GOVERNING EQNS.INTEGRATION ALGORITHM FOR PATH-INTEGRATION ALGORITHM FOR PATH- DEPENDENT PLASTICITY DEPENDENT PLASTICITY f q , qb , T n kel H k n H is kel nn T kel kep ke T n kel H k n H is
g G , R , d Gt , d Rt d 0 G = ENERGY RELEASE RATE It is a function of the independent variable t, the so-called pseudo-timeDAMAGE CRITERIONDAMAGE CRITERION (for cyclic static loadings is the current loading step); It is a function of plastic curvatures; It is a function of d, the damage internal variable. R = DAMAGE THRESHOLD It is a function of d, the damage internal variable; R0 ≡ Gcr is the initial damage threshold, that is the amount of energy stored when the first crack forms Rt>R0 It is governed by a time law of evolution damage is initiated when the damage energy release rate G exceeds the initial damage threshold Gcr
G = energy stored during loading/unloading process e , p , d 1 d 0 e , p p is a set of internal plastic parameters, corresponding toENERGY RELEASE RATEENERGY RELEASE RATE isotropic and kinematic plastic hardening variables (namely, β and α) For uncoupled behavior 0 e , p el e , d pl p 1 T , d e ked d e pl pl 0 = i j 2 G : 0 e , p d ENERGY RELEASE RATE conjugated to the damage variable d
DISCETIZATION of ENERGY RELEASE RATE G Mk 1 Mk k , pl Gk L pl 1 dk 2 EI 1 d k Gi Gi G d G j G ENERGY RELEASE RATE j L pl 0.08 L 0.022 f yd d bENERGY RELEASE RATE pl y L pl at the current step the total energy release rate is equal to the energy stored at the previous step Gn, incremented by the plastic contribution at the current step ∆Gn+1, proportional to bending moment and deformation increment (curvature), and extended to the inelastic zone Lpl G at the n-th step is the summation of the initial elastic stored energy Gel and Gn+1=Gn+∆Gn+1 the plastic contribution
SOFTENING RULE The identification of an appropriate softening rule is the most important requirement for the complete g k d k k k i, j definition of the damage-based model. It is related to the mechanical properties of the RC member, Gk and accounts for cracking, yielding and ultimate bending, as well as inelastic hinge deformations.DAMAGE EVOLUTION LAWDAMAGE EVOLUTION LAW KUHN-TUCKER CONDITIONS d i d i D d k 0, g k 0, k g k 0 d j j CONSISTENCY CONDITIONS 0 if g k 0 no damage k g k 0 k 0 if g k 0 damage increment The softening rule is given in a numerical form to enable representation of the physical evolution of damage in the member. Test results as well as numerical simulations (Shah; Chung et al.; Park et al; Gupta et al.; Sadeghi et al.) show that cumulative damage increases as a function of deformations rather than forces
DISCETIZATION of DAMAGE EVOLUTION LAW R Ri Ri M y , k 1 bk R d Rk ( d k ) Gcr , k ak 1 k d R j R j 1 d k r r DAMAGE EVOLUTION LAWDAMAGE EVOLUTION LAW M u ,k M y ,k Sin ,k tg 2 du d y M y ,k S f ,k tg1 dy Sin ,k ak M y ,k bk b S f ,k 2 M cr R0 Gcr L pl 2 EI
PREDICTED RATE OF DAMAGE AT YIELDING – dy d y = 0 .0 1 d y = 0 .1 d y = 0 .2 d y = 0 .3 d y = 0 .5DAMAGE EVOLUTION LAWDAMAGE EVOLUTION LAW R 0 0 .1 0 .2 0 .3 0 .4 0 .5 0 .6 0 .7 0 .8 0 .9 1 d dY SHIFTS THE INTERSECTION POINT D OF THE TWO TANGENTS tg1 AND tg2 du = 1
EXPONENT OF THE FUNCTION – r 5 r=1.5 r=2 r=4 r=7 r=10 4 4DAMAGE EVOLUTION LAWDAMAGE EVOLUTION LAW 3 3 R 2 2 1 1 5 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 d 1 r CONTROL THE SHAPE OF THE FUNCTION R(d) r>0
INTEGRATION ALGORITHM FOR LUMPED DAMAGEINTEGRATION ALGORITHM FOR LUMPED DAMAGE g i Gi Ri 0 g i g i max 1,: g i ,trial g i Gi Ri 0 g trial g g j gj max 2,: j ,trial g G R 0 j j j ONLY ONE DAMAGE FUNCTION G IS NEEDED g G R 0 j j j FOR EACH HINGE TO CHARACTERIZE THE STATE OF DAMAGE, WHETHER THIS IS DUE TO POSITIVE OR NEGATIVE BENDING MOMENT DETERMINATION of the DAMAGE INCREMENT d G d - R d dg trial d n 1 d n d n 1 subscripts i and j and superscript + and – are omitted for simplicity Rk S f , k 1 bk Stg , k S f , k bk 1 r d k 1 d k r r
Given: (i) The geometry of the structure defined by nodes coordinates, andFORMULATION OF THE PROBLEM the connection table that identifies each member,FORMULATION OF THE PROBLEM (i) The mechanical properties of each member, (ii) The relation moment-curvature at each hinge (iii) The loading history of nodes during pseudo-time interval [Tmin Tmax], (iv) The displacement history of nodes during a pseudo-time interval [Tmin Tmax], Calculate: (i) Displacement history of nodes during at each loading step, (ii) Reactions on nodes (base shear), Calculate and update at the end of every load step (n+1): (i) The damaged stiffness matrix of the structure (ii) Basic element force vector q, basic element deformations v, plastic deformation vpl, back stresses qb, damage indices collected in the matrix D and all the remaining internal variables α and β, for each member of the structure Such that they verify: (i) Compatibility equations, (ii) Equilibrium equations, (iii) Yield functions f=[fi fj]<0 and damage functions at nodes i and j, gtrial=[gi,trial gj,trial]>0 (iv) Internal variables evolution laws.
The state determination process is made up of two nested phases: a) THE ELEMENT STATE DETERMINATION: the element resistingNUMERICAL SOLUTION STRATEGIESNUMERICAL SOLUTION STRATEGIES forces are determined for the given element end deformations b) THE STRUCTURE STATE DETERMINATION: the element resisting forces are assembled to the structure resisting force vector The resisting forces are then compared with the total applied loads and the difference, if any, yields the unbalanced forces which are then applied to the structure in an iterative solution process until external loads and internal resisting forces agree within a specified tolerance. INCREMENTATION STEP k – Advancing phase The external reference load pattern Pref is imposed as a sequence of load increments ΔPfk = ∆λPref, where Δλ is is the load factor increment. At load step k the applied load is equal to Pfk =Pfk-1+∆Pfk, with k=1,2…nstep and Pf0=0 ITERATION STEP t – Correcting phase This iteration loop yields the structural displacements Uk that correspond to applied loads Pfk
THE ELEMENT STATE DETERMINATION - s k s=0 =k t 1NUMERICAL SOLUTION STRATEGIESNUMERICAL SOLUTION STRATEGIES Final point Unbalanced deformation Starting point v s=2 =v t -v s=2 u r residual deformation v s=2 =v s=2 +v s=2 r el pl Given deformation
NUMERICAL SOLUTION STRATEGIESNUMERICAL SOLUTION STRATEGIES STATE STRUCTURE DETERMINATION
NUMERICAL EXAMPLESNUMERICAL EXAMPLES Reinforcement ratios are in the range 0.6-3.6%, while axial load ratios vary from 0 to 0.56 An average value of EI applicable to the entire length of the member was estimated by correcting the stiffness of the uncracked member through parameter γ Normal strength (NSC) and high strength (HSC) specimens For normal-strength concrete, the model proposed by Hoshikuma et al. was used for determining the confined properties, while the Muguruma et al. model was employed for high-strength concrete Moment-curvature analyses were performed using OpenSees using a zero-length element. Sections were discretized into confined and unconfined concrete regions, for which separate fiber discretizations were generated. Reinforcing steel bars were placed around the boundary of the confined and unconfined regions. Sections were discretized into ten layers inside the concrete core and two layers for the unconfined concrete on each side. For reinforcing steel, a bilinear stress-strain relationship was used, with elastic modulus Es=210 GPa and 1% strain-hardening ratio. Numerical simulations were performed under force control, imposing forces at the free end of the member in an iterative manner as to induce the desired displacement history
200 150 Moment-Curvature Analysis of the cross- M [KNm] 100 section – The section shows hardening behavior; the estimated yielding and ultimate curvatures are 50 ϕy=0.0066 and ϕu=0.07351, respectively 0 0 0.02 0.04 0.06 0.08 0.1 0.12 [1/m] 150 100 50Shear base [KN] 0 -50 -100 model test -150 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1 Top Displacement [m] Force-Displacement History – The results of the numerical simulation are depicted in blue. From the comparison with the experimental results represented with a dashed green line, it can be observed that the damage-based model is able to provide a good agreement with the test regarding the evaluation of the strength peaks during the loading/unloading process and an adequate estimation of the initial unloading stiffness
150 100 50Shear force [KN] 0 -50 -100 model test -150 0 100 200 300 400 500 600 700 iterations Shear Force Development during the Loading Cycles – It can be observed that the first cycle shows some discrepancies between analysis and experiment, because the effect of the first cycle was not included in the analysis and the difficulty of establishing the right initial conditions for the model. The likelihood of an abnormal behavior increases with the number of load cycles (iterations) because of the monotonically increasing value of the energy dissipation, which affects the degradation parameter d
200 150 100bending moment M,Qb 50 0 -50 -100 -150 M- qb-pl test -200 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 rotations , pl Bending Moment-Rotation History – The green line represents the progression of the back stress with the plastic deformation θpl. The slope of the back stress is the kinematic hardening ratio of the member which controls the increasing of plastic deformation during the cyclic loading process. The degradation of the elastic properties instead is produced only by the damage index d since the damage-based model is uncoupled
g 140 Threshold Energy g=G-R 120 100 80R(d),G(d),g(d) 60 40 20 0 -20 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 Positive Damage index Progression of the Damage Functions with the damage index d – The damage threshold R shows an increasing exponential behavior governed by the damage index d and the exponent of the damage evolution law r=1.15. The energy function G represents the amount of energy stored during the loading process and it is a typical step function. The constant energy intervals correspond to the unloading branches since the unloading process in the damage-based model is elastic. The initial damage threshold is Gcr=0.0022 [KNm] and it corresponds to the formation of the first crack in the member. The function g represents the damage criterion adopted for the model, g=G–R>0
0.45 0.4 0.35Positive Damage index 0.3 0.25 0.2 0.15 0.1 0.05 0 0 100 200 300 400 500 600 700 No. of iterationsDamage Index Evolution – The progression of the damage index with the load cycles is captured in this graph. The damageindex is recorded only for positive bending at the bottom of the cantilever column, where the plastic hinge forms. The damageindex starts to grow after the first crack forms in the member and progresses very slowly until the first yielding of the memberitself. After that, the progression of the damage becomes steeper until failure. The maximum value of damage is dmax=0.4251
Ductility factors – The graph reports the ductility of the member for every load cycle. The cumulative ductility of themember is also estimated to μcum=41.05. The displacement at first yielding is approximated to δy=0.01032 [m]
900 800 700 Moment-Curvature Analysis of the cross-section – 600 The section shows softening behavior thus the failure is governed by the crushing of the concrete core; the M [KNm] 500 400 estimated yielding and ultimate curvatures are ϕy=0.013 300 and ϕu=0.1977, respectively 200 100 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 [1/m] 800 600 400Shear base [K ] 200 N 0 -200 -400 -600 model test -800 -0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1 Top Displacement [m] Force-Displacement History – The results of the numerical simulation are depicted in blue. From the comparison with the experimental results represented with a dashed green line, it can be observed that the damage-based model is able to provide a good agreement with the test regarding the evaluation of the strength peaks during the loading/unloading process and an adequate estimation of the initial unloading stiffness
Shear Force Development during 800 the Loading Cycles – It can be 600 observed that the first cycle shows some discrepancies between analysis 400 and experiment, because of the difficulty of establishing the right initial Shear force [KN] 200 conditions for the model. The likelihood of an abnormal behavior 0 increases with the number of load -200 cycles (iterations) because of the monotonically increasing value of the -400 energy dissipation, which affects the degradation parameter d. Furthermore, -600 due to the application of the load model test control method the nonlinear behavior -800 0 100 200 300 400 500 600 700 near the ultimate strength peaks of the iterations member is not well matched 1500 Bending Moment-Rotation History 1000 – The green line represents the progression of the back stress with thebending moment M,Qb 500 plastic deformation θpl. The slope of the back stress is the kinematic hardening ratio of the member which 0 controls the increasing of plastic deformation during the cyclic loading -500 process. The degradation of the elastic properties instead is produced only by -1000 the damage index d since the damage- M- qb-pl test based model is uncoupled -1500 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 rotations ,pl
Progression of the Damage Functions 700 with the damage index d – The damage Threshold Energy g=G-R threshold R shows an increasing exponential 600 behavior governed by the damage index d 500 and the exponent of the damage evolution law r=1.30. The energy function G 400 represents the amount of energy stored R(d),G(d),g(d) during the loading process and it is a typical 300 step function. However, in this particular 200 case, due to the relatively high number of iterations, it is not possible to distinguish the 100 constant energy intervals. The initial damage threshold is Gcr=0.0576 [KNm] and it 0 corresponds to the formation of the first -100 crack in the member. The function g 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Positive Damage index represents the damage criterion adopted for the model, g=G–R>0 0.5 Damage Index Evolution – The progression of the damage index with the 0.4 load cycles is captured in this graph. The damage index is recorded only for positivePositive Damage index 0.3 bending at the bottom of the cantilever column, where the plastic hinge forms. The damage index starts to grow after the 0.2 first crack forms in the member and progresses very slowly until the first yielding of the member itself. After that, 0.1 the progression of the damage becomes steeper until failure. The maximum value 0 of damage is dmax=0.4816 0 100 200 300 400 500 600 700 No. of iterations
Ductility factors – The graph reports the ductility of the member for every load cycle. The cumulative ductility of themember is also estimated to μcum=32.75. The displacement at first yielding is approximated to δy=0.01408 [m]
50 45 40 Moment-Curvature Analysis of the cross-section – 35 The section shows softening behavior thus the failure is 30 governed by the crushing of the concrete core; the M [KNm] 25 estimated yielding and ultimate curvatures are ϕy=0.023 20 and ϕu=0.2809, respectively 15 10 5 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 [1/m] 80 60 40Shear base [KN] 20 0 -2 0 m od el -4 0 te s t -6 0 -8 0 -0 .0 6 -0 .0 4 -0 .0 2 0 0 .0 2 0 .0 4 0 .0 6 T o p D is p la c e m e n t [m ] Force-Displacement History – The results of the numerical simulation are depicted in blue. From the comparison with the experimental results represented with a dashed green line, it can be observed that the damage-based model is able to provide a good agreement with the test regarding the evaluation of the strength peaks during the loading/unloading process and an adequate estimation of the initial unloading stiffness
Shear Force Development during 80 the Loading Cycles – It can be 60 observed that the first cycle shows some discrepancies between analysis 40 and experiment, because of the difficulty of establishing the right initialShear force [KN] 20 conditions for the model. The likelihood of an abnormal behavior 0 increases with the number of load -20 cycles (iterations) because of the monotonically increasing value of the -40 energy dissipation, which affects the degradation parameter d. Furthermore, -60 due to the application of the load model test control method the nonlinear behavior -80 0 200 400 600 800 1000 1200 1400 1600 1800 2000 near the ultimate strength peaks of the iterations member is not well matched 50 Bending Moment-Rotation History – The green line represents the progression of the back stress with the b bending moment M,Q plastic deformation θpl. The slope of the back stress is the kinematic 0 hardening ratio of the member which controls the increasing of plastic deformation during the cyclic loading process. The degradation of the elastic properties instead is produced only by the damage index d since the damage- M- qb-pl test based model is uncoupled -50 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1 rotations , pl
Progression of the Damage Functions with 60 the damage index d – The damage threshold Threshold Energy g=G-R R shows an increasing exponential behavior 50 governed by the damage index d and the exponent of the damage evolution law 40 r=0.90. The energy function G represents R(d),G(d),g(d) 30 the amount of energy stored during the loading process and it is a typical step 20 function. The constant energy intervals correspond to the unloading branches since 10 the unloading process in the damage-based model is elastic. The initial damage threshold 0 is Gcr=0.0019 [KNm] and it corresponds to the formation of the first crack in the 0.7 member. The function g represents the -10 0 0.1 0.2 0.3 0.4 0.5 0.6 Positive Damage index damage criterion adopted for the model, g=G–R>0 0.7 Damage Index Evolution – The 0.6 progression of the damage index with the load cycles is captured in this graph. The 0.5 damage index is recorded only for positivePositive Damage index bending at the bottom of the cantilever 0.4 column, where the plastic hinge forms. The damage index starts to grow after the 0.3 first crack forms in the member, it 0.2 progresses slowly until the first yielding of the member and after that the progression 0.1 of the damage becomes steeper until failure. The maximum value of damage is 0 dmax=0.5426 0 200 400 600 800 1000 1200 1400 1600 1800 2000 No. of iterations
Ductility factors – The graph reports the ductility of the member for every load cycle. The cumulative ductility of the member is also estimatedto μcum=38.4. The displacement at first yielding is approximated to δy=0.0113 [m]
200 model test 100 Force (KN) Force-displacement curves 0 for experimental test and numerical simulation Specimen JSCE-5 -100 -200 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 Displacement (m) 500 model test 250Force (KN) Force-displacement curves 0 for experimental test and numerical simulation Specimen 9 -250 -500 -0.12 -0.06 0 0.06 0.12 Displacement (m)
300 model test 200 100 Force-displacement curvesForce (KN) for experimental test and 0 numerical simulation Specimen OCR1 -100 -200 -300 -0.12 -0.06 0 0.06 0.12 Displacement (m) 200 model test 100 Force-displacement curves Force (KN) for experimental test and 0 numerical simulation Specimen 806040 -100 -200 -0.18 -0.12 -0.06 0 0.06 0.12 0.18 0.24 Displacement (m)
The applications of the model and the comparisons of analytical with experimental results demonstrate the ability of the proposed beam element to describe several complexities of the hysteretic behavior of structural members, such as softening and stiffness degradation.REMARKS ON MODEL PERFORMANCEREMARKS ON MODEL PERFORMANCE Unlike material response, in which the possibility to reproduce cyclic degradation and predict failure is directly related to the ability of the material models to characterize post-yield softening of concrete, buckling of reinforcing bars, fracture of confining reinforcement, etc., in this damage-based element model the simulation of the softening behavior as well as the stiffness degradation of the structural member, are a direct result of the adopted yield surface and the ability of the flexibility matrix of the damaged member implemented in the one component model, to trace this behavior. The agreement with the experimental results is good, especially, if the envelope curve is compared. The first cycles shows some discrepancies between analysis and experiment, because the effect of the first cycle was not included in the analysis and the difficulty of establishing the right initial conditions for the model. All subsequent analytical cycles show good agreement with the experimental results. The main difficulty in the use of the model consists in the selection of the calibration parameters, and, even if all of them bear a direct relation to the physical properties of the structural member, it is possible to arrive at a physically unreasonable hysteretic behavior by injudicious selection. The selection of the load step size was, governed by the desire to obtain a smooth load-displacement relation for the presentation of the results.
The calibration parameters selection not only affects the accuracy of the constitutive relation of the model, but has an impact on the numerical convergence characteristics as well. In this example convergence was always achieved veryREMARKS ON MODEL PERFORMANCEREMARKS ON MODEL PERFORMANCE rapidly at each load step never requiring more than 10-15 iterations. The damage-based model works as a one component element model in which the response is path-dependent. This implies that at the end of every incrementation step, the final force depends on how the strain increment is partitioned between the number of iterations. Therefore, small changes in the iteration strategy, that is the choice of the load step size, can sensitively alter the final value of the force, even if the final displacement solution remains the same. The likelihood of an abnormal behavior increases with the number of load cycles, because of the monotonically increasing value of energy dissipation, which affects the degradation parameter d. Load control method applied provides good results when the member stiffness is high, but fail to trace the nonlinear behavior near the ultimate strength of the member and the post-peak response (Filippou and Issa, 1988). The model becomes a standard elastoplastic model with linear kinematic and isotropic hardening if damage remains constant. On the contrary, the damage produces a softening effect. Thus, the achievement of an accurate numerical prediction, results from the competition between the hardening due to plastic deformations and the softening that is the consequence of damage.
REMARKS ON MODEL PERFORMANCEREMARKS ON MODEL PERFORMANCE The stiffness reduction of the model during unloading is not as pronounced as in the experimental results: while the model exhibits a practically straight unloading branch, the specimens display a gradual loss of stiffness during unloading. This can be attributed to the following factors: (a) even though the model accounts for stiffness loss between cycles, it cannot accommodate graduate stiffness reduction during the unloading phase; thus, the unloading branch of the model is straight until reaching the yield strength. This is also due to the fact that, since we are dealing with hysteretic nonlinear behavior modeled in lumped plasticity, the modification of the extension of the inelastic zone during reversals cannot be taken into consideration. (b) the use of the same degradation parameter d under, both positive and negative loading, fails to reproduce properly opening and closure of the cracks.
Regression-based identification to provide guidelines for estimatingIDENTIFICATION OF MODEL PARAMETERS dyIDENTIFICATION OF MODEL PARAMETERS dy the required model parameters Regression analyses of the data set were performed to understand which properties of the member affect the accumulation of damage during the loading process, and with which intensity as well as the weighted choice of the calibration parameters dy and r which do not bear a directed relation to properties of the structural member. L s d y 0.1ln 0.17 ln 0.44 ln ln l 0.3 h db L NSC r 19.24 1.62 0.24 f c 0.06 f y 22.17 h AND r AND r s d y 0.12 ln 2.64 ln 1.65 ln sh 2.53 db HSC s r 0.18 54.53l 0.002 f y 2.28 db