confined model abaqus

10
207 A SIMPLIFIED APPROACH FOR NONLINEAR RESPONSE ANALYSIS OF COMPOSITE STRUCTURAL MEMBERS Hao-Ze DENG 1 , Ya-Ying CHANG 1 , David T. LAU 2 , Shadi OSTOVARI 3 , and Keh-Chyuan TSAI 4 SUMMARY In recent years, extensive efforts have been directed towards the use of composite construction in bridge and building structures. Composite structural systems have the desirable characteristics of high strength and stiffness, ductility, and fire resistance, and thus are suitable for construction in seismic regions. To accurately predict the seismic performance of a composite structure, it is important to understand the cyclic behaviour of its individual components. Many of the existing analytical methods based on finite elements and fibre layer models are time consuming to use or the results are difficult to interpret. Therefore, there is still the need to develop simple but accurate methods for determining the behaviour of composite structural members. In this research, a simplified approach is developed for analyzing the nonlinear behaviour of composite structural members. The member is modeled by an equivalent member section of homogeneous nonlinear material with a specified stress-strain relationship, which is derived based on the requirement that the equivalent section has the same moment-curvature behaviour as the original section. One advantage of this simplified approach is its simplicity. It can be easily implemented in most structural analysis computer programs with nonlinear modeling capabilities. The developed model can accurately predict the nonlinear hysteretic responses of reinforced concrete structures with frame members of arbitrary shapes and reinforcing details and other structures of composite construction under severe earthquake excitations. Numerical examples of a cantilever reinforced concrete beam and a new bridge pier system of double-skinned concrete filled tubular (DSCFT) members, recently developed in Taiwan for seismic construction in environmental sensitive areas or areas with difficult terrain, are analyzed under monotonic and cyclic loadings up to failure. Results obtained from the simplified approach, fibre layer model using Open System for Earthquake Engineering Simulation framework (OpenSees), and finite element model using the computer program ABAQUS are compared. Keywords: evaluation and retrofit, structural response, nonlinear analysis, composite construction, concrete filled steel tube. INTRODUCTION The difficulty of analyzing composite structural members arises from many factors such as cracking and crushing of concrete, and yielding, strain hardening, slippage and buckling of reinforcing or composite steel. When load reversals are considered, phenomena such as pinching of the hysteresis loops due to shear, bond deterioration, Bauschinger effects, and other factors also become important. Presently, various numerical models based on fibre layer models and finite elements are available for analyzing the ultimate behaviour of composite structures under severe loading conditions. Typically, a fibre layer model with the assumption of plane sections remain plane divides the structural section into a number of discrete layers. Equilibrium at each load step is achieved by iteration based on adopting a specified convergence criterion. Similarly, finite element models generally subdivide a continuum structure or member into multiple parts modeled by finite elements. Although the existing methods of analysis can give accurate results, they often require significant amount of computation efforts or the results are difficult to interpret. Therefore, there is still 1 Graduate Research Assistant, Dept. of Civil and Environmental Engineering, Carleton University, Canada 2 Professor, Dept. of Civil and Environmental Engineering, Carleton University, Canada, email: [email protected] 3 Former Graduate Student, Dept. of Civil and Environmental Engineering, Carleton University, Canada 4 Professor, Department of Civil Engineering, National Taiwan University, Taiwan, e-mail: [email protected]

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Page 1: Confined Model Abaqus

207

A SIMPLIFIED APPROACH FOR NONLINEAR RESPONSE ANALYSIS OF COMPOSITE STRUCTURAL MEMBERS

Hao-Ze DENG1, Ya-Ying CHANG1, David T. LAU2, Shadi OSTOVARI3, and Keh-Chyuan TSAI4

SUMMARY

In recent years, extensive efforts have been directed towards the use of composite construction in bridge and building structures. Composite structural systems have the desirable characteristics of high strength and stiffness, ductility, and fire resistance, and thus are suitable for construction in seismic regions. To accurately predict the seismic performance of a composite structure, it is important to understand the cyclic behaviour of its individual components. Many of the existing analytical methods based on finite elements and fibre layer models are time consuming to use or the results are difficult to interpret. Therefore, there is still the need to develop simple but accurate methods for determining the behaviour of composite structural members. In this research, a simplified approach is developed for analyzing the nonlinear behaviour of composite structural members. The member is modeled by an equivalent member section of homogeneous nonlinear material with a specified stress-strain relationship, which is derived based on the requirement that the equivalent section has the same moment-curvature behaviour as the original section. One advantage of this simplified approach is its simplicity. It can be easily implemented in most structural analysis computer programs with nonlinear modeling capabilities. The developed model can accurately predict the nonlinear hysteretic responses of reinforced concrete structures with frame members of arbitrary shapes and reinforcing details and other structures of composite construction under severe earthquake excitations. Numerical examples of a cantilever reinforced concrete beam and a new bridge pier system of double-skinned concrete filled tubular (DSCFT) members, recently developed in Taiwan for seismic construction in environmental sensitive areas or areas with difficult terrain, are analyzed under monotonic and cyclic loadings up to failure. Results obtained from the simplified approach, fibre layer model using Open System for Earthquake Engineering Simulation framework (OpenSees), and finite element model using the computer program ABAQUS are compared. Keywords: evaluation and retrofit, structural response, nonlinear analysis, composite construction, concrete filled steel tube.

INTRODUCTION The difficulty of analyzing composite structural members arises from many factors such as cracking and crushing of concrete, and yielding, strain hardening, slippage and buckling of reinforcing or composite steel. When load reversals are considered, phenomena such as pinching of the hysteresis loops due to shear, bond deterioration, Bauschinger effects, and other factors also become important. Presently, various numerical models based on fibre layer models and finite elements are available for analyzing the ultimate behaviour of composite structures under severe loading conditions. Typically, a fibre layer model with the assumption of plane sections remain plane divides the structural section into a number of discrete layers. Equilibrium at each load step is achieved by iteration based on adopting a specified convergence criterion. Similarly, finite element models generally subdivide a continuum structure or member into multiple parts modeled by finite elements. Although the existing methods of analysis can give accurate results, they often require significant amount of computation efforts or the results are difficult to interpret. Therefore, there is still

1 Graduate Research Assistant, Dept. of Civil and Environmental Engineering, Carleton University, Canada 2 Professor, Dept. of Civil and Environmental Engineering, Carleton University, Canada, email: [email protected] 3 Former Graduate Student, Dept. of Civil and Environmental Engineering, Carleton University, Canada 4 Professor, Department of Civil Engineering, National Taiwan University, Taiwan, e-mail: [email protected]

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the need for efficient modeling and analysis tools that do not require excessive amount of time and effort in the preparation of modeling input data and are easy to use. The aim of the present study is to develop a simple and flexible modeling and analysis procedure for the evaluation of the nonlinear behaviour of composite structural members up to ultimate failure. The simplified approach developed in the present study accounts for the nonlinear material behaviour of reinforcing steel and concrete under compression. The nonlinear behaviour of concrete after cracking and the softening behaviour after peak stress are considered. The effect of confinement on the concrete behaviour is taken into account in the simplified model. The elasto-plastic behaviour of reinforcing steel with strain hardening, and the interaction between concrete and reinforcing steel materials are modeled. The developed analysis procedure can be used for nonlinear analysis of reinforced concrete frame structures with beam-column members. One advantage of the simplified method is its simplicity that it can be easily implemented in most structural analysis computer programs with nonlinear modeling capabilities. The developed model can accurately predict the nonlinear hysteretic responses of reinforced concrete and other composite structures with frame members of arbitrary shapes and reinforcing details under severe earthquake excitations.

APPROACH OF SIMPLIFIED ANALYSIS PROCEDURE The basic approach of the simplified analysis procedure is similar to the fibre layer model, except that only distinct states in the behaviour of the modeling composite member are considered in the formulation. For flexure behaviour, the following assumptions are considered in the formulation of the simplified analysis procedure: 1. Plane sections of the member remain plane when a member is subjected to deformation. Consequently, the

longitudinal strain in the steel and concrete materials at a plane section is directly proportional to the distance from the neutral axis.

2. The stress-strain relationship for the steel is modeled by a trilinear model. 3. The tensile strength of concrete is assumed to be negligible. 4. The stress-strain relationships of confined and unconfined concrete under compression are modeled by Kent

and Park concrete model (Park et al. 1982).

In the simplified approach, the original structural member section is modeled by an equivalent member section of homogeneous nonlinear material with a specified stress-strain relationship, which is derived based on the requirement that the equivalent section has the same moment-curvature behaviour as the original section. The material models adopted for representation of the behaviour of concrete and steel are briefly described here. The constitutive model for concrete developed by Kent and Park for confined and unconfined concrete is implemented in the simplified analysis procedure. The behaviour of reinforcing steel materials is approximated by a trilinear stress-strain relationship considering the effect of strain hardening. Kent and Park Concrete Model Many studies have been conducted on the stress-strain relationship of concrete confined by transverse reinforcement under compression. Observations and laboratory tests have shown that if the compression zone of a concrete beam or column is confined by closely-spaced stirrup ties, hoops or spirals, the ductility of concrete is significantly enhanced and the member can sustain deformations of large curvature (Kent and Park 1971). In the simplified approach, the Kent and Park concrete model shown in Figure 1 is adopted for modeling the material behaviour of concrete under compression. The formulations of the stress-strain relations of confined and unconfined concrete of the model are summarized here. The constitutive model consists of an ascending branch represented by a second-degree parabolic curve and a descending linear part. The ascending parabola is expressed by Equation (1)

0

2

00

' 2 εεεε

εε k

kkfkf c

cccc ≤

−= (1)

where cε is the longitudinal concrete strain, 'cf is the compressive strength of concrete, oε is the strain of

unconfined concrete corresponding to 'cf , and k is a confinement coefficient. For unconfined concrete, the

parameter k is equal to one. A commonly accepted assumption for unconfined concrete is oε = 0.002.

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For strain greater than the corresponding strain at peak stress, the softening branch of the stress-strain relationship is approximated by a straight line presented in Equation (2) ( )[ ] 0

'0

' 2.01 εεεε kfkkZfkf cccmcc >≥−−= (2)

where mZ is the stress declining ratio for the confined concrete after peak stress. For unconfined concrete, it is assumed the stress reduces to zero at the strain of 0.0035. For concrete under confinement, a perfectly plastic residual behaviour is assumed at high strain level to account for the load carrying capacity of crushed concrete still effectively confined by the transverse steel. As shown in Figure 1, the confinement effect on the strength of concrete represented by the confinement coefficient k increases the concrete peak stress from '

cf to 'ckf . It is assumed that confined concrete can sustain a constant

stress of 0.2k 'cf at strains greater than c20ε .

Trilinear Steel Model A trilinear stress-strain relationship shown in Figure 2 is adopted to model the behaviour of reinforcing steel under tension or compression. The possibility of buckling of the compression steel is ignored at this stag of the study. Derivation of Equivalent Member Section The three critical states considered in the derivation of the equivalent member section in the formulation of the simplified approach are cracking of concrete, yielding of steel, and the ultimate resistance of the member. To obtain a more accurate and complete equivalent stress-strain relationship for the analyzed composite member, additional states in the behaviour of the composite member can be analyzed. To illustrate the procedure for the derivation of the stress-strain relationship of nonlinear material of the equivalent member section, a rectangular reinforced concrete beam member is analyzed as an example here. The procedure can be extended to the analysis of any composite structural members based on the same approach. Assuming the strain at the top surface of the rectangular member section shown in Figure 3(a) is topε , and the

neutral axis is located at a depth of c measured from the top of the member section, the curvatureφ of the section can be expressed by Equation (3)

ch

top

−=ε

φ (3)

where h is the member section height. Figure 3(b) shows the strain profile of the member section. Using the concrete constitutive model by Kent and Park (1982) and the steel material model, the neutral axis is determined by an iterative process, and then the internal forces F and moment M of the section can be obtained by integrating the stress distributions on concrete and steel over the member section area A as follows

dAF ∫= σ (4)

dAyM ∫= σ (5)

where y is the moment arm of the stress in the member section. As discussed earlier, the equivalent member section is assumed to be composed of a uniform homogeneous material. The characteristic of this material is that the stress-strain relationship of the equivalent section should have the same moment-curvature behaviour as the original member section. The equivalent section is assumed to have the same total area of the original member section. The behaviour of the equivalent section is formulated to have the same curvature as the original member section. The homogeneous material is assumed to have the same stress-strain behaviour in both tension and compression as shown in Figures 3-5 of the derivation of the equivalent section for the crushing, yielding and ultimate states. From the linear strain variation of the equivalent member section with the same curvature as the original member, the maximum equivalent strain at the extreme fiber of the equivalent member section is given by Equation (6)

=

2h

eqeq φε (6)

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210

where φφ =eq and h is the height of the composite member section. Considering the first critical state of initial cracking of concrete, the equivalent cracking stress of the equivalent member section required to give the same moment-curvature behaviour as the original analyzed composite section is determined as follows

eq

eqcreqcr

IcM

=)(σ (7)

where crM is the cracking moment of the composite structural member section, the equivalent neutral axis

depth 2hceq = , and eqI is the moment of inertial of the equivalent member section. The stress and strain

profiles at the cracking state of concrete are shown in Figure 3. The derivation procedure can be extended to analyze the yielding and other nonlinear states of the original reinforced concrete member. For the yielding state as shown in Figure 4, the yielding moment is expressed by Equation (8)

eqeqcreqcry bdhddhdddM )}2()32

()21()()2()

22()()2()

32(

21{ 2

22

2)(11)( −∆+−+= σσσ (8)

After obtaining the equivalent yielding strain from the yield curvature of the original member section, the stress increment σ∆ in Equation (8) from the cracking stress can be solved to give the equivalent yielding stress as follows σσσ ∆+= )()( eqcreqy (9) For the ultimate state, the same procedure employed in determining the equivalent yield stress is repeated to determine the equivalent ultimate stress and strain for the equivalent section. Corresponding to the equivalent strain and stress distributions shown in Figure 5, the ultimate moment is expressed by Equation (10). The equivalent ultimate stress )(equσ is determined by Equation (11) after solving Equation (10) for σ∆ as follows

eqeqy

eqcreqy

eqcreqcru

bdhhdhddhdh

ddddd

dddddddM

)}2()]2

(31

2[)

2(

21)2(])

2(

21[)

2(

)2(])(32)[())(

21(

)2(])(21[)()2()

32(

21{

22222)(

11212)()(

11212)(11)(

−−−∆++−−+

+−−−+

+−−+=

σσ

σσ

σσ

(10)

σσσ ∆+= )()( eqyequ (11) To more accurately define the stress-strain relationship of the equivalent nonlinear material, the above procedures can be repeated to analyze additional states of the behaviour of the composite member. For each state, the equivalent stress can be evaluated based on the approach of Equations (8) and (9), as illustrated in Figure 6. The derivation of the equivalent stress-stain curve can be generalized as follows

iiiiiiiii dAdAdAM σσ ∆++∆= −−−−−∑ 2

1)]21([

22

)1(2

)1(1

)1(1

)1()1( (12)

iieqi σσσ ∆+= − )1()( (13) where )(eqiσ is the equivalent stress at the ith state, iσ∆ is the change of stress from the (i-1)th to the ith state,

iA is the sectional area at the ith state, and id is the moment arm. A linear interpolation is assumed between two consecutive states in the stress-strain relationship of the equivalent material.

COMPUTER IMPLIMENTATION In the present study, computer modules based on the computer-aided toolbox Mathcad have been developed for the implementation of the simplified equivalent analysis model and procedures to generate the required

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211

stress-strain relationship of the equivalent nonlinear material. The developed computer modules can be used in conjunction with other computer structural analysis programs with user specified nonlinear material modeling capacities for detailed nonlinear composite structural members of arbitrary shape. Open System for Earthquake Engineering Simulation framework (OpenSees) is a software framework in development at the University of California, Berkeley for simulating the seismic response of structural and geotechnical systems subjected to earthquakes (Mazzoni et al. 2000). It has a library of different material and structural element modeling modules, and computational platforms which can be selected and assembled into specific analysis tools for advanced analysis of complex structural systems. In the present study, ongoing development is being carried out to implement modules of the simplified approach in the OpenSees.

NUMERICAL EXAMPLES

To demonstrate the validity of the equivalent model and analysis procedure, two examples of composite structural members are analyzed using the simplified approach. The two examples are a reinforced concrete cantilever beam and a double skinned concrete filled tubular (DSCFT) member, which is an innovative system recently developed in Taiwan for seismic construction. Results from the simplified approach are compared with fibre layer model using OpenSees, and finite element model using the computer program ABAQUS. ABAQUS is a finite element numerical analysis program with extensive linear and nonlinear material modeling and structural analysis capabilities. The simplified approach results on the DSCFT columns are also compared to test measurements obtained in experimental research by Tsai et al (2002) under monotonic loading. Cantilever Beam Example The cantilever beam analyzed in this study is a doubly reinforced concrete structural member with only longitudinal reinforcement subjected to an end applied moment. The rectangular section is reinforced by compression steel (As’ = 6300 mm2) at an effective depth of 76.2mm and tension steel (As = 6300 mm2) at an effective depth of 732.8mm. For the example here, the concrete material is assumed to be unconfined due to the absence of transverse reinforcement, and the steel is modeled by a bilinear model with perfectly plastic behaviour after yielding. Figure 7 and Table 1 show the dimensions and material properties, respectively, of the example cantilever beam. For the cantilever beam, the moment-curvature behaviour at the tip of the cantilever beam obtained by OpenSees, ABAQUS and the simplified approach are shown in Figure 8. Comparison of the results indicates that the simplified approach can accurately predict the behaviour of the cantilever beam up to the ultimate capacity. It is noted that since the concrete material in the cantilever beam is unconfined, therefore, large portion of the concrete material does not contribute to the resistance of the load when the strain of the concrete fibre exceeds 0.002. Consequently, this leads to a softening behaviour in its moment capacity, as shown in Figure 8. The derived equivalent stress-strain relationship for the reinforced cantilever beam is presented in Figure 9. Double Skinned Concrete Filled Tube Example Figure 10 shows the DSCFT member consisting of two concentric circular thin steel tubes with concrete filled between them. Research carried out in Taiwan (Lin 2002; Tsai et al. 2002; Tsai and Lin 2002; Tsai and Wei 2002) has demonstrated that the DSCFT members possess light weight, moderated axial capacity and high flexural strength characteristics. An important advantage of the DSCFT column system is that the reduced weight of the column can lead to a significant reduction of seismic force on the member. Therefore, DSCFT is an innovative structural system particularly suitable for construction of tall bridge priers in seismic regions. Since the DSCFT is a complex composite structural system, it is difficult to analyze using general analysis methods. The simplified approach is applied here to analyze the behaviour of the DSCFT. The experimental specimen analyzed using the simplified approach has an effective length of 1100mm. The external and internal tube diameters are 300mm and 180mm, respectively. The thickness is 2mm for both the external and internal steel tubes. Table 2 lists the material properties used in the analysis. For the case study here, the concrete filler is modeled as confined concrete with k=1.231 and a bilinear model is adopted for representing the behaviour of the steel tubes. Figure 11 shows the comparison of the analytical results obtained by OpenSees and the simplified approach, and test results from the experiments conducted by Tsai et al. (2002). The derived equivalent stress-strain curve is

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shown in Figure 12. It is evident from Figures 8 and 11 that the simplified approach gives consistent results with OpenSees. When comparing with the experimental results, the simplified approach gives a conservative lower moment capacity. One possible reason for the lower moment capacity is due to the inadequate modeling of strength and stiffness of the confined concrete core. To more accurately account for the total confinement of concrete by the steel tubes, the confinement coefficient k is increased to 2.5 which is twice as that determined by the Kent and Park model based on open spiral transverse reinforcement. The moment-curvature of the DSCFT under cyclic loading is presented in Figure 13. The effect of strength degradation, stiffness degradation, and pinching effects are not considered in the proposed model for cyclic loadings at this stage.

CONCLUSION This paper presents the derivation of a simplified modeling and analysis approach for the nonlinear analysis of composite structural members. In the simplified approach, the original structural member section is modeled by an equivalent member section of homogeneous nonlinear material with a derived stress-strain relationship such that the equivalent section has the same moment-curvature behaviour as the original composite section. Two numerical examples are presented to illustrate the procedures of the simplified analysis method. A rectangular reinforced concrete cantilever beam subjected to an end moment and a double skinned concrete filled tubular (DSCFT) member subject to a lateral load are analyzed. Results obtained from the simplified approach are in good agreement with the analysis results obtained from the structural analysis software framework OpenSees and finite element program ABAQUS. The results also correlate well with test data. The research shows that the nonlinear response of composite members can be accurately predicted by the simplified method. The simplified method is an efficient tool for evaluating the nonlinear behaviour of composite structural members up to ultimate failure. The advantage of the simplified method is that it can be easily implemented with most structural analysis programs with the capabilities of user specified nonlinear material models. It can analyze structural members of arbitrary shapes and reinforcing details.

REFERENCES Kent, D.C. and Park, R. (1971), “Flexural Members with Confined Concrete,” Journal of Structural Division,

Proceedings of the American Society of Civil Engineers, Vol. 97, No. ST7, 1969-1990. Lin, M.L. (2002), “A study of double-skin concrete filled steel tubes under the combined axial and bending

loads,” Ph.D. Thesis, Department of Civil Engineering, National Taiwan University, Taipei, Taiwan, July. Mander, J.B., Priestley, M.J.N., and Park, R., Fellow (1988), “Theoretical stress-strain model for confined

concrete,” Journal of Structural Engineering, ASCE, Vol. 114, No.8, 1804-1826. Mazzoni, S., Mckenna, F., Scott, M.H., and Fenves, G.L. (2000), “OpenSees User Manual,” Version 2, Pacific

Earthquake Engineering Research Centre, University of California, Berkeley. Ostovari, S. (2002), “Structural performance analysis of reinforced concrete arch bridge,” M.A.Sc. Thesis,

Department of Civil and Environmental Engineering, Carleton University, Canada, June. Ozcebe, G. and Saatcioglu, M. (1987). “Confinement of concrete columns for seismic loading,” ACI Structural

Journal, Technical Paper, Vol. 84, No. 4, 308-315. Park, T. and Paulay, T. (1975), Reinforced Concrete Structures, John Wiley & Sons, Inc., New York, USA. Park, R., Priestley, M.J.N., and Gill, W.D. (1982), “Ductility of square-confined concrete columns,” Journal of

Structural Engineering, ASCE, Vol. 108, No.4, 929-950. Tsai, K.C. and Chang, L.C. (2001), “The platform and visualization of inelastic structural analysis of 2D systems

PISA2D and VISA2D,” CEER Report, National Center for Research on Earthquake Engineering, College of Engineering, National Taiwan University, Taipei, Taiwan (in Chinese).

Tsai, K.C., Lin, M.L., Lin, Y.S., and Wang, T.F. (2002), “Double skinned concrete filled tubes for bridge piers,” Research Paper, National Taiwan University and Sinotech Engineering Consultant, Taiwan.

Tsai, K.C. and Lin, Y.H. (2002), “Cyclic response of double-skin concrete filled steel tube column-to-foundation connections,” CEER Report, National Center for Research on Earthquake Engineering, College of Engineering, National Taiwan University, Taipei, Taiwan (in Chinese).

Tsai, K.C. and Wei, H.H. (2002), “Cyclic response of embedded double-skin concrete filled steel tube column-to-foundation connections,” CEER Report, National Center for Research on Earthquake Engineering, College of Engineering, National Taiwan University, Taipei, Taiwan (in Chinese).

Wang, C.K. and Salmon, C.G. (1998), Reinforced Concrete Design, Sixth Edition, Addison Wesley Educational Publishers Inc., New York, U.S.A.

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Table 1 Material properties of cantilever beam

Concrete Steel Parameter Value Parameter Value

Ec 24648 MPa Es 200000 MPa '

Cf 30 MPa σy 400 MPa σcr 3.286 MPa σu 400 MPa εo 0.002 εu 0.159

Table 2 Material properties of DSCFT

Concrete Steel Parameter Value Parameter Value

Ec 25045 MPa Es 201000 MPa '

Cf 28 MPa σy 305.2 MPa σcr 3.175 MPa σu 416.1 MPa εo 0.002 εu 0.159

Figure 1 Concrete confined or unconfined – Kent and Park Model

Figure 2 Steel trilinear model

ε

σ(MPa)

TENSION

COMPRESSION

εu εy ES

σu

σy

εsh

ε

(MPa) σ

ε

fc'

0 0εκ

κfc'

0.2κfc'

Confined

Unconfined

ε20C

Page 8: Confined Model Abaqus

214

Original Section Equivalent Section

beqb

h

crφ

hεcr

Mcr

c

crφ

cr(eq)ε

d

cr(eq)ε σcr(eq)

cr(eq)σ

Strain

Stre

ss

(a) (b) (c) (d) (e) (f)

(a) Reinforced concrete section under cracking moment (b) Cracking strain profile (c) Equivalent section (d) Equivalent cracking strain profile (e) Equivalent cracking stress profile (f) Equivalent stress-strain relationship for cracking state

Figure 3 Equivalent section at cracking moment

Strain

Stre

ss

(a) (b) (c) (d)

(a) Equivalent section (b) Equivalent yielding strain profile (c) Equivalent yielding stress profile (d) Equivalent stress-strain relationship up to yielding state

Figure 4 Equivalent section at yielding moment

Strain

Stre

ss

(a) (b) (c) (d)

(a) Equivalent section (b) Equivalent strain profile (c) Equivalent stress profile (d) Equivalent stress-strain relationship up to ultimate state

Figure 5 Equivalent section at ultimate moment

Page 9: Confined Model Abaqus

215

hMi A(i-1)

1

2A(i-1) A(i)

d(i-1

)

σ(i-2) σ(i-1) σ(i)

d(i-1

)2

1

d(i)

(i)(i-1)

φ(i)

(i-2)beq

(a) (b) (c) (d) (e)

(a) Equivalent section (b) Equivalent strain at ith state (c) Equivalent stress at ith state and corresponding moment arm (d) (i-1)th state equivalent sectional area (e) ith state equivalent sectional area

Figure 6 Simplified equivalent model for composite structural member section

Figure 7 Cantilever beam dimension

0

600

1200

1800

0 3 6 9 12Curvature (rad*10-2/m)

Mom

ent (

kN-m

)

ABQUSOpenSeesSimplied Equivalent Approach

0

4

8

12

16

20

0 0.02 0.04 0.06

Strain

Stre

ss (M

Pa)

Figure 8 Analysis results of moment-curvature for

cantilever beam Figure 9 Derived equivalent stress-strain

relationship for cantilever beam

1100 mm

300 mm

180 mm

Concrete

De

Di

ti

to

Figure 10 Double-Skinned Concrete Filled Tube Details, Tsai et al. (2002)

Page 10: Confined Model Abaqus

216

-150

-100

-50

0

50

100

150

-10 -5 0 5 10

Curvature (10-2/m)

Mom

ent (

kNm

)

OpenSees - cyclic

Experimental - monotonic

Figure 13 Analysis result of cyclic response for DSCFT

0

20

40

60

80

100

120

140

0 10 20 30Curvature (rad*10-2/m)

Mom

ent (

kN.m

)

Experimental resultsSimplified approach, k=2.5Simplified approach, k=1.231OpenSees 0

10

20

30

40

50

60

0 0.005 0.01 0.015 0.02Strain

Stre

ss (M

Pa)

Simplified approach, k=1.231

Figure 11 Analysis results of moment-curvature for DSCFT

Figure 12 Derived equivalent stress-strain relationship for DSCFT