Anisotropic Damage Plasticity Model for Concrete and Its Use in Plastic Hinge Relocation in RC Frames with FRP

Accepted Manuscript Anisotropic Damage Plasticity Model for Concrete and Its Use in Plastic Hinge Relocation in RC Frames with FRP

M.R. Javanmardi, Mahmoud R. Maheri PII: DOI: Reference:

S2352-0124(17)30064-4 doi:10.1016/j.istruc.2017.09.009 ISTRUC 226

To appear in:

Structures

Received date: Revised date: Accepted date:

1 August 2017 26 September 2017 27 September 2017

Please cite this article as: M.R. Javanmardi, Mahmoud R. Maheri , Anisotropic Damage Plasticity Model for Concrete and Its Use in Plastic Hinge Relocation in RC Frames with FRP. The address for the corresponding author was captured as affiliation for all authors. Please check if appropriate. Istruc(2017), doi:10.1016/j.istruc.2017.09.009

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ACCEPTED MANUSCRIPT Anisotropic damage plasticity model for concrete and its use in plastic hinge relocation in RC frames with FRP M.R. Javanmardia and Mahmoud R. Maherib a

PhD candidate, Department of Civil Engineering, Shiraz University, Shiraz Professor, Department of Civil Engineering, Shiraz University, Shiraz, Email: [email protected]

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Abstract

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In an RC frame, relocating plastic hinges in the beam away from the column face is suggested to increase ductility of the frame. This can be achieved through flange-bonded FRP retrofit of the

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joint. Relocating the plastic hinges further into the beam, shortens the effective length of the

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beam.

As the effective length of the beam is reduced, ductile flexure failure in the beam may switch to

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a brittle shear failure, resulting in a decrease in ductility of the frame. The consequences and

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limits of plastic hinge relocation are therefore important issues in need of evaluation. To achieve this, in this paper, an accurate numerical model is first developed for simulating the complex

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behaviour of concrete and its interaction with steel and FRP materials. For this purpose,

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anisotropic damage plasticity model for concrete is developed. Damage plasticity model simulates irreversible plastic deformation besides material degradation (crack). After verifying the accuracy of the developed software through comparison with experimental results, response of full-scale RC frames with different beam clear length/effective height ratios is evaluated to find advantages and drawbacks of the flange-bonded retrofitting scheme for relocating plastic hinges. The results show that for beam clear length/height ratios less than 6, the frame cannot achieve the 0.02 code-required inelastic drift and the ductile flexural failure in the beam changes

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ACCEPTED MANUSCRIPT to an undesirable brittle shear failure. Also, the results indicate that for clear length/height ratios between 6 and 9, bending moment at the face of the column may increase by up to 60% depending on the length of FRP overlay, therefore, columns should be controlled, and if necessary, be retrofitted for the increased moment.

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Keywords: seismic retrofitting; flange-bonded FRP; RC moment frame; anisotropic damage plasticity model; numerical simulation.

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ACCEPTED MANUSCRIPT 1. Introduction Fiber-reinforced polymer (FRP) laminates have been widely used for retrofitting RC structures [1-5]. FRP laminates have several advantages in comparison with other retrofitting methods, including having high tensile strength and being easy to install. In a moment resisting frame,

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beam-column joints have a crucial role in stability and seismic performance of the frame.

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Therefore, a large amount of research effort has been directed at retrofitting RC beam-column

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joints with FRP. Retrofitting RC joints with FRP may be carried out with different aims, including (i) repairing a damaged joint, (ii) increasing the moment capacity and ductility of a

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joint and (iii) relocating the plastic hinge away from the joint core. Many researchers have

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addressed the above aims with experimental investigations. The earlier experimental results reported for the effects on moment capacity and joint ductility show considerable increase in

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both parameters due to retrofitting [6-13]. Recently, in a number of experimental investigations

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FRP overlays are used to relocate the plastic hinge away from the joint core. Mahini and Ronagh [10] tested scaled beam-column RC joints retrofitted with web-bonded FRP with the

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aim of relocating the plastic hinge away from the column face. Their experiments showed that

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the FRP web-bonded scheme can upgrade its strength, stiffness and ductility, as well as, shifting the plastic hinge away from the column face further into the beam. Azarm et al. [14] used a

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flange-bonded FRP retrofitting scheme to relocate the plastic hinge. They tested a number of full-scale RC joints and showed that the flange-bonded scheme not only successfully relocates the plastic hinge, but also improves the capacity and seismic performance of the RC beamcolumn joints. Numerical simulation of FRP-retrofitted RC structures is, in itself, a challenging task, since it involves accurate simulation of the complex concrete, reinforcement and FRP materials

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ACCEPTED MANUSCRIPT behaviour and the interaction between these three materials. A number of numerical investigations have been reported on FRP-retrofitting of RC joints. One of the earliest studies was carried out by Parvin and Granata [15]. They showed that moment capacity of joints increased considerably, prior to formation of plastic hinge. Later studies by El-Amoury and

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Ghobarah [16, 17] showed similar results. Another numerical study conducted by Niroomandi et

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al. [18] also showed the effectiveness of FRP web-bonded retrofitting of joints to relocate the

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plastic hinge and as a result, to improve the seismic performance level of a retrofitted RC frame. They compared the pushover response of an RC frame retrofitted at joints by web-bonded FRP

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with the same frame retrofitted with steel X-bracing scheme [19] and pointed out that both

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methods have similar abilities to increase the behaviour factor, R, of the frame; the FRP webbonding being able to improve ductility of the frame while the X-bracing scheme being more

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suited to increase strength capacity. In another numerical study, behaviour of FRP-retrofitted RC

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beam-column joints under monotonic and cyclic loading was examined by Dalalbashi et al. [20, 21]. Their employed retrofitting scheme was directed at increasing the flexural capacity of beam-

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column connections and their results showed the effectiveness of the FRP retrofit, not only in

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increasing the moment capacity, but also in moving the plastic hinges away from the beamcolumn edge. The ability of web-bonded FRP retrofit of RC joints in relocating the plastic hinge

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was also investigated by Hadigheh et al. [22, 23]. Similar results were reported by Zarandi and Maheri [24] for flange-bonded FRP retrofitting of joints. Numerical efforts by Baji et al. [25] and Eslami and Ronagh [26] also showed that numerical analysis can reasonably accurately predict the failure mechanism and location of plastic hinges observed during the experimental tests. Relocating the plastic hinge away from the face of column and into the beam may be beneficial in avoiding undesirable failure in the joint‟s core, however, it results in a reduction in

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ACCEPTED MANUSCRIPT the effective length of the beam, leading to an increase in the beam shear force [27]. This is shown graphically in Fig. 1. As the plastic hinges are relocated further into the beam, the effective length, L', is reduced and as the plastic moments at plastic hinges (Mp) remain

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unchanged, a reduction in L' results in an increase in plastic shear (Vp).

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Fig. 1. Relocation of plastic hinge in retrofitted frame resulting in a decrease in beam clear span and an increase in plastic shear (Vp)

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If plastic shear increases noticeably, it could change the ductile flexure failure of the beam

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to undesirable brittle shear failure [27]. Furthermore, an increase in the plastic shear results in an increase in the moment at the face of the column, evident in a free-body diagram of the part of

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the beam between the plastic hinge and the face of column. An increase in the moment at the face of the column could, in turn, adversely affect the joint‟s core performance.

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In the previous numerical works listed above, researchers have mostly used classical

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plasticity models for concrete and commercial FE analysis software. The plasticity models are not able to clearly combine damage process due to microdefects (microcracks and microvoids) such as stiffness degradation with plastic deformation. The tensile cracking results in a reduction in the stiffness of concrete. Therefore, continuum damage mechanics is needed to model the degradation accurately. However, during loading of the structure, the concrete material also undergoes some irreversible deformations, hence, the continuum damage theories cannot be used

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ACCEPTED MANUSCRIPT without considering plasticity. Therefore, a combined continuum damage and plasticity theory may be used for modeling the nonlinear material behavior of concrete [28-50]. Damage plasticity models are proposed based on the thermodynamics of irreversible process. In damage plasticity model, plastic yield surface is defined to capture irreversible plastic

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deformation and damage surface is employed to involve material degradations [37-50].

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Kachanov [28] proposed the basic concept of continuum damage mechanics which is expressed

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based on equivalency of strain energy in undamaged material and damaged material. Scalar (isotropic) damage model is expressed based on a scalar damage variable, D, which represents

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the material degradation during the loading process. Due to its easy implementation in a finite

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element code, many authors have proposed isotropic damage model for both ductile and brittle materials [28-37]. However, under triaxial loading condition the significance of the direction of

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material degradation becomes more important and an analysis without considering this may

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result in loss of accuracy. For example, in many structural RC simulations, such as confined concrete in the core of beam-column connections, the initiation and propagation of cracks are not

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the same in different directions. To overcome this problem, some researchers have proposed an

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axial vector representation for damage, called vector damage variable [38-39]. Also anisotropic damage plasticity model is introduced in order to take different material degradations in different

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directions into account by attributing the second order damage tensor [40-50]. For granule and frictional materials such as concrete, two separate second order damage tensors are needed, one for tension damage and another for compression damage. Researchers showed that anisotropic damage plasticity model has great advantages over classical plasticity in numerical simulation of concrete [40-50].

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ACCEPTED MANUSCRIPT The main objectives of the current study is firstly; to investigate the effects of the shortening of beam effective length due to plastic hinge relocation on the seismic behavior of RC frame and the conditions that prevent brittle failure in the beam. Secondly, it is to study the behaviour of retrofitted joint core under complex stress conditions that arises from retrofitting of

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joints, resulting in increased shear forces and joint‟s core moment. To this end, it is necessary to

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have a powerful constitutive equation for concrete. Therefore, in the present study, the

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anisotropic damage plasticity model, proposed by Voyiadjis et al. [44] is used to develop a nonlinear finite element software in FORTRAN code. To validate the developed software, the

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results obtained from numerical simulation are compared with those from three previous

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experiments. A parametric numerical investigation is then conducted to investigate the adverse

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effects and limitations of plastic hinge relocation.

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2. Anisotropic damage model for concrete 2.1. Anisotropic damage concept

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In classical plasticity models for concrete, the evolution and coalescence of microdefects in the

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material due to increase in deformations is not considered. Anisotropic damage model, on the other hand, has the ability to consider the growth and coalescence of microdefects besides

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material plastic deformation [48]. In the available literature on anisotropic damage model, two configurations are generally considered; undamaged configuration and damaged configuration. The stress relationship between undamaged configuration and damaged configuration was first suggested by Cordebois and Sidoroff [40]. They expressed the damaged material using the constitutive laws of the undamaged material in which the Cauchy stress tensor in damaged configuration ( ) is replaced by the stress tensor in undamaged configuration ( ̅) [40-50] (In the

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ACCEPTED MANUSCRIPT following formulations, the parameters with superimposed dash represent those of the undamaged configuration and the parameters without superimposed dash signify the damaged configuration): ̅ is the fourth order damage effect tensor. Many different expressions for

have

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where,

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been proposed. In this work, the following expression, first proposed by Cordebois and Sidoroff

)

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(

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[40], is used:

where, δij is Kronecker delta and

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is defined as follows:

is the second order damage tensor. Second order damage tensor is defined as

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in which,

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follows [42-50]:

is microdamage density vector,

is the total area of the defects and

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where,

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√

is

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the total area of the surface, unit normal of which is n i . The explicit matrix representation of the fourth order damage tensor (

) is given in [43]. The linear elastic constitutive equations for

the damaged material can be written, based on the principle of elastic strain energy equivalence between the undamaged material and damaged material, as [40-50]: ̅

̅

One of the main hypotheses of the small strain theory of plasticity is the decomposition of the total strain tensor,

, into the sum of an elastic strain tensor (reversible part), 8

, and a

ACCEPTED MANUSCRIPT plastic strain tensor (irreversible part),

. Therefore, the total strain tensor,

, in the two

configurations are defined as follows:

̅

̅

̅

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The relationships between stress and elastic strain tensors in damaged and undamaged

̅

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̅

and ̅

where,

are stiffness tensor in damaged and undamaged configurations,

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̅

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configurations are:

respectively.

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Substituting Eqs. (7.a) and (7.b) in Eq. (5) and using Eq. (1) gives:

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̅

Substituting Eqs. (7.a) and (7.b) in Eq. (1) and using Eq. (8) lead to the elastic strain tensor

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relationship in damaged and undamaged configurations.

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̅

2.2. Plastic yield surface In this work, the yield surface criterion, first proposed by Lubliner et al. [36] and later modified by Lee and Fenves [37] is used. The yield surface criterion has the ability to predict plastic deformation in concrete and complies well with the experimental results. This yield surface is also implemented in the Abaqus finite element program as a concrete damage plasticity for concrete [51]. The yield surface employed in this work can be expressed in the undamaged 9

ACCEPTED MANUSCRIPT configuration as follows (the superscripts „„+” and „„-” specify tensile and compressive states, respectively): ̅

(

̅

̅̂

̅̂

)

̅

̅

̅

is deviatoric stress tensor and ̅̂

⁄

̅

maximum principle stress .The parameters

and

compressive and uniaxial tensile yield stresses, ̅ ̅

a function of accumulated plastic strain

̅ ⁄ is second invariant of the

and

are material constants and depend on ̅

is isotropic hardening function which is ̅

is Heaviside step jump function. All

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variables are in undamaged configuration.

is the

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deviatoric stress tensor,

̅

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̅ is first invariant of the stress tensor, ̅

̅

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where, ̅

√

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̅ ̅

Isotropic hardening function in tension and compression states, being a conjugate force of

̅ [

̅

where,

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̅

̅

and

̅

]

are respectively the initial uniaxial tensile and compressive yield stresses and

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̅

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the plastic part of the Helmholtz free energy function, can be expressed as [44]:

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Q, b and a are material constants.

2.3. Non-associative flow rule Flow rule relates the plastic yield surface evolution to the stress–strain curve. In numerical simulation, plastic potential surface is used in computing the evolution of the plastic strain; also a hardening law is defined for the evolution of the yield surface limit. Non-associative flow rule commonly used for granular and cohesive material, leads to antisymmetric stiffness matrix. In this work, the Drucker–Prager flow rule is employed as [44]:

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ACCEPTED MANUSCRIPT √

̅

where,

̅ is the dilation constant.

⁄ ̅

defining the direction of the plastic flow, can be

expressed as: ̅ ̅

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√

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̅

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Furthermore, the rate of plastic strain tensor, ̇ ̅ , is normal to plastic potential function (FP) and

̇̅

̇̅

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is defined in undamaged configuration as:

̅

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where, ̇ ̅ is the plastic multiplier which may be obtained by imposing the plastic consistency

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condition on plastic surface.

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2.4. Damage surface

In this article, the anisotropic damage growth function, proposed by Chow and Wang [49] is

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used. The damage surface is introduced in two separate damage surfaces (one for tension damage

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and another for compression damage). This surface can be expressed as:

√

)

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(

where,

is the tensile or compressive damage isotropic hardening function,

damage force of (

,

is conjugate

is a fourth-order symmetric tensor, expressed as follows [42]: )

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ACCEPTED MANUSCRIPT Researchers have proposed different functions for the evolution of the tensile and compressive damage in concrete. In this work, the functions proposed by Cicekli et al. [50] are used which can be expressed as: ̇

(

)+ ̇

̇

)+ ̇

is material constant showing the onset of compression damage and tension damage in is a material constant which is related to the fracture energy. Also, ̇

concrete and

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where,

(

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*

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*

Moreover,

̇

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√ ̇

is thermodynamic conjugate force of

̅

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Helmholtz free energy as [44]:

and can be obtained from damage part of

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̇

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rate of equivalent damage defined as [45]:

is the

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In damage surface, the Lagrangian damage multipliers, ̇

, is also obtained by imposing

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damage consistency conditions. For more details regarding anisotropic damage model the reader is referred to Voyiadjis et al. [44].

2.5. Numerical simulation algorithm In finite element method, Newton Raphson algorithm is often used for the solution of nonlinear incremental equations. When a reasonable initial guess of the solution is not available, this

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ACCEPTED MANUSCRIPT method cannot converge. To avoid divergence, the load must be applied in incremental form (ΔF). Within each load increment, the problem is solved by Newton Raphson algorithm. After the solution corresponding to the previous load increment has converged, the next load increment is applied.

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The tangent stiffness matrix is written for each element by using the incremental elastoplastic

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damage stress-strain law, expressed in the previous sections, and assembled into global matrix

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(kglobal), then solved for increments in nodal displacements in undamaged configuration ( ̅).

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From the definition of the strain displacement matrix, the incremental strains in undamaged configuration can then be obtained. Due to material nonlinearity, elastic predictor, plastic and

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damage corrector return mapping scheme is an appropriate approach to solve nonlinear material. The main algorithm is summarized in Box 1. As it is seen in Box 1, the trial stress in undamaged

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configuration is obtained by assuming the elastic response, termed „elastic predictor step‟.

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Therefore, the plasticity and damage consistency conditions are checked and plastic and damage corrector steps are used to restore plastic and damage consistency as stated above.

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If the trial stress is not outside the plastic surface and damage surface, the step is elastic

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and the assumption is correct. However, if the trial stress at gauss point is outside the plastic surface or damage surface, intersection point for plastic surface or damage surface must be

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calculated and if trial stress is outside both plastic surface and damage surface, two intersections should be evaluated and the lower one would be considered. The remaining portion of trial stress that does not lie within the elastic domain, must be eliminated efficiently in some way because at all times one must also make sure that the computed stresses do not drift away from the yield surface or the damage surface. The state of bond between internal reinforcement and the surrounding concrete has a

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ACCEPTED MANUSCRIPT significant effect on the behaviour of RC members. However, to include the effects of possible reinforcement slippage, a complex slip-stick rule is needed and in many cases geometric nonlinearity (large deformation) should also be considered which may notably increase computational cost. Therefore, in the present finite element simulations, the steel reinforcements

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are assumed to be fully attached to the concrete at their interface, hence ignoring possible

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slippage of reinforcement.

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In the experimental works used for verification of the models [10, 14], no debonding of FRP was reported. Therefore, in the numerical models, FRP overlay is also assumed to be fully

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attached to the concrete. To ensure that debonding does not occur, the maximum strain at the

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interface was checked against the following limiting relation suggested by ACI 440.2-2008 [51]:

is the maximum strain allowed in the FRP laminate to prevent debonding,

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where,

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√

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standard uniaxial compressive strength of concrete cylinders,

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overlay.

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thickness of FRP overlay, n is the number of FRP overlay and

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is the elastic modulus,

is the is the

is the ultimate strain of FRP

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Box 1. Elastic predictor, plastic and damage corrector algorithm

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ACCEPTED MANUSCRIPT 3. Verification of the developed software In order to test the effectiveness of the developed software and verify its accuracy, in this section three full-scale RC beam-column joint problems for which experimental results were available in the literature are presented and simulated. The simulated results are then compared with the

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experimental data as well as the results from Abaqus software [52] simulation which is based on

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classical plasticity and isotropic damage plasticity (damage plasticity model in Abaqus). The

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numerical solutions were first checked for mesh-dependency and a fine mesh was applied to the

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models for anticipating damage propagation. Nonlinear pushover analyses are conducted on the RC joints in a displacement-controlled manner in order to predict hardening and softening of the

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specimens.

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3.1. Non-retrofitted RC joint

The first test example is a full-scale RC beam-column joint tested by Mahini and Ronagh [10].

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Fig. 2 shows the geometry and reinforcement details of this joint. In the numerical simulation,

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the column was first subjected to a P2=305 kN constant axial load as in the experimental test to simulate gravity load in the column. The P1 load was then gradually applied as an incremental

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displacement at the tip of the beam. The numerical model consisted of 8-noded, 3D,

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isoparametric elements to represent solid concrete and 2-noded isoparametric line elements for rebars. Due to symmetry, only one half of the joint is considered in the finite element simulation, therefore, the nodal degrees of the cut surface are restrained normal to the plane of symmetry.

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Fig. 2. Details of the non-retrofitted RC joint [10] (all dimensions are in mm)

The material parameters used in the proposed model, as well as those used for simulation

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with Abaqus isotropic damage plasticity model are presented in Table 1 for concrete and Table 2 for steel rebars. For definition of the Abaqus damage plasticity model parameters the reader is

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referred to the Abaqus manual theory [52]. are adapted from Mahini and Ronagh [10]. Equivalent

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Parameters such as E, 𝜐 and

plastic strain evolution obtained using the proposed simulation is illustrated in Fig. 3 and equivalent damage growth is illustrated in Fig. 4. Failure of the joint was typically attributed to the formation of plastic hinge in the beam, close to the face of the column. Location of plastic hinge formation obtained from the simulation (Fig. 4) is compatible with the observed experimental cracks in the beam reported in [10].

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ACCEPTED MANUSCRIPT Moreover, another numerical simulation based on isotropic damage and classical plasticity model proposed by Lee and Fenves [37] was carried out using Abaqus software. The forcedisplacement curve of the joint obtained using the proposed simulation is compared with those obtained from Abaqus simulation and experimental test in Fig. 5. Good correlation is noted

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between the proposed simulation and experimental load-displacement curves. Fig. 5 also

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indicates that the proposed anisotropic damage plasticity model predicts the response more

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accurately compared with the classical plasticity and isotropic damage plasticity model (Abaqus) by about 7%.

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Table 1. Mechanical properties of concrete in the non-retrofitted RC joint model

̅ (GPa)

Value

27.6

0.2

(MP a)

(MPa)

40

3.4

0.12

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Concrete property

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Parameters used in simulation with anisotropic damage plasticity model

0.2

(Pa)

(Pa)

Q (MPa)

191

6310

50

0.54

0.12

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Additional parameters used in simulation with Abaqus (isotopic damage plasticity model) Dilatation angle

Eccentricity

Value

25°

0.1

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Concrete property

1.16

K

Viscosity parameter

0.67

0.001

Table 2. Mechanical properties of steel in the non-retrofitted RC joint model

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Steel property Value

(GPa)

longitudinal (MPa)

200

500

18

stirrup (MPa)

382

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Fig. 3. Evolution of plastic strain in the proposed simulation of non-retrofitted RC joint model at load a) 8.26kN, b) 14.22kN, c) 19.51kN and d) 21.64kN

Fig. 4. Evolution of equivalent damage variable (EDV) in the proposed simulation of non-retrofitted RC joint model at load a) 8.26kN, b) 14.22kN, c) 19.51kN and d) 21.64kN 19

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3.2. FRP web-bonded retrofitted RC joint

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Fig. 5. Comparison of load-displacement response curves of non-retrofitted RC joint

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The second example is another full-scale, 2D, RC exterior joint similar to the first test model (Fig. 2). However, the joint is retrofitted at its web by FRP laminates [10]. Three layers of FRP

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laminates; each 0.165 mm thick are used. The FRP laminates cover the entire joint web and

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extend 200 mm into the beam. In numerical simulation of this model, 4-noded isoparametric shell elements are used to represent the FRP laminates. Mechanical properties of FRP laminates

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used for numerical simulation are presented in Table 3. Since during the experimental test no

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failure in FRP laminates was reported, linear behaviour is assumed for FRP laminates in the analysis. Mechanical properties of concrete and steel rebars are similar to the non-retroffitted joint model (Tables 1 and 2). Table 3. Mechanical properties of FRP laminates FRP property Value

x (GPa) In fibers direction

240

y=Ez (GPa) Perpendicular to fiber direction

18.58

20

(MPa) in fibers direction

3900

(MPa) Perpendicular to fiber direction

53.7

ACCEPTED MANUSCRIPT The evolution of the equivalent damage variable obtained using the proposed simulation is illustrated as contour plots in Fig. 6. It can be seen that during the early stages of loading, slight damage is formed close to the beam-column interface. As the load is gradually increased, the maximum damage zone moves gradually into the beam towards the end of FRP laminates

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and localizes there. At final stages of loading, damage is highly localized around the end of FRP

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laminates indicating that the plastic hinge has been relocated away from the face of the column

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to the end of FRP laminates.

Fig. 6. Evolution of equivalent damage variable (EDV) in the web-bonded retrofitted joint at load a) 10.58kN, b) 17.46kN, c) 19.22kN and d) 21.6kN

Abaqus FE software was again used to simulate this test problem. The forcedisplacement response curves of the retrofitted RC joint obtained using the proposed simulation is compared with those from Abaqus simulation and experiment in Fig. 7. Again, close agreement is noted between the load-displacement curves of the proposed simulation and the 21

ACCEPTED MANUSCRIPT experiment. Fig. 7 also indicates that the proposed anisotropic damage plasticity model predicts the response much more accurately than the classical plasticity and isotropic damage plasticity

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model (Abaqus).

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Fig. 7. Comparison of load-displacement response curves of FRP web-bonded retrofitted joint

3.3. FRP flange-bonded retrofitted RC joint

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The third example is a flange-bonded FRP retrofitted RC joint tested by Azarm et.al. [14]. Fig. 8

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shows the geometry and steel reinforcement and FRP retrofitting details of the retrofitted joint. The locations of strain gauges placed on the longitudinal reinforcement bars of the beam in the experiment are also shown in Fig. 8. The flange-bonded retrofitting scheme was carried out using 5 layers of FRP laminates. Similar to the experimental setup, in the numerical simulations, the column was first subjected to a P2=225 kN constant axial load. The P1 load was then gradually applied as an incremental displacement at the tip of the beam. Mechanical properties of concrete, steel and FRP sheets used for numerical simulations are presented in Table 4, 5 and 6,

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ACCEPTED MANUSCRIPT respectively. Additional parameters used for simulation with Abaqus isotropic damage plasticity model are also presented in Table 4. The same types of element as in the previous test problem were used to model concrete, steel reinforcement and FRP sheets. Due to symmetry, only one

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half of the joint is considered in the numerical simulations.

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Fig. 8. Details of the flange-bonded FRP retrofitted joint [14] (All dimensions are in mm)

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Table 4. Mechanical properties of concrete at the flange bonded retrofitted joint test Parameters used in simulation with anisotropic damage plasticity model

Concrete property Value

̅ (GPa)

25.1

0.2

(MP a)

(MPa)

25.3

3.02

0.12

0.2

(Pa)

(Pa)

Q (MPa)

191

6310

50

0.54

0.12

Additional parameters used in simulation with Abaqus (isotopic damage plasticity model) Concrete property

Dilatation angle

Eccentricity

Value

28°

0.1

1.16

23

K

Viscosity parameter

0.67

0.001

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Table 5. Mechanical properties of steel at the flange bonded retrofitted joint test Steel property

longitudinal (MPa)

(GPa)

200

Value

stirrup (MPa)

400

340

Table 6. Mechanical properties of FRP laminates at the flange bonded retrofitted joint test y=Ez (GPa) Perpendicular to fiber direction

240

18.58

3900

(MPa) Perpendicular to fiber direction

53.7

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Value

(MPa) In fibers direction

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x (GPa) In fibers direction

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FRP property

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Azarm et al. [14] reported that; although application of flange-bonded FRP sheets increased the joint capacity, it failed to relocate the plastic hinge away from the column face

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towards the end of FRP overlay. Fig. 9 shows the evolution of the equivalent damage variable obtained using the current simulation. It can be noted that at earlier loading stages, some damage

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is detected near the beam-column interface and as the load is gradually increased, the maximum

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damage zone remains close to the face of column.

Fig. 9. Propagation of equivalent damage variable (EDV) in the FRP flange-bonded retrofitted RC joint at load a) 22.71kN, b) 33.93kN, c) 55.8kN and d) 62.88kN 24

ACCEPTED MANUSCRIPT To compare the accuracy of the proposed simulation with the more conventional isotropic damage and classical plasticity model [37] used in Abaqus software, the latter was also used to carry out a nonlinear static pushover analysis of the retrofitted joint. The force-displacement response curves obtained from the two numerical simulations are compared with experimental

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results in Fig. 10. Good correlation may be noted in the load-displacement curves measured in

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the experiment and evaluated in the proposed numerical simulation. Fig. 10 also indicates that

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anisotropic damage plasticity model predicts the concrete behaviour more accurately than the classical plasticity and isotropic damage plasticity by about 11%. To further verify the accuracy

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of the proposed simulation, strain variation in the longitudinal reinforcement at the end of FRP

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laminates over the beam is compared with experimental test results reported by Azarm et al. [14] in Fig. 11 (The location of strain gauge is illustrated in Fig.8). A close agreement can also be

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variations in concrete with accuracy.

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seen between the two curves, indicating the ability of the proposed method to predict strain

Fig. 10. Comparison of the load-displacement capacity curves of the flange-bonded retrofitted joint

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Fig. 11. Load-strain curves of the flange-bonded retrofitted joint

Comparative results of the three test problems indicate that the anisotropic damage

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plasticty model can be used as a powerful constituative equation for concrete and the developed

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software can predict the behaviour of retrofitted RC joints with improved accuracy.

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4. Numerical simulation of FRP-retrofitted RC moment frame

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Despite the relatively large amount of research conducted on FRP-retrofitted RC joints, little work has been carried out on full RC moment frames. In the following, a full frame, consisting

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of a beam and two columns (half columns from two adjacent stories) is considered for evaluating the possible side-effects of relocating plastic hinges in an intermediate ductility RC beam. As it was mentioned earlier, in an RC frame, relocating plastic hinges in the beam, away from the column face, may be achieved through flange-bonded FRP retrofitting of the joint. Relocating the plastic hinges further into the beam, however, shortens the effective length of the beam. As the effective length of the beam is reduced, ductile flexure failure in the beam may

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ACCEPTED MANUSCRIPT change to a brittle shear failure due to the increase in plastic shear force. In intermediate-ductility RC moment frames, the clear span should be at least 4 times greater than the effective depth of the beam to prevent undesirable brittle shear failure [27]. Figs. 12.a and 12.b show the location of plastic hinges in a non-retrofitted RC frame and as it is illustrated, plastic hinges have only

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rotation. In a retrofitted moment frame, beam may be modelled as consisting of three parts; two

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rigid parts at the two ends of the beam (retrofitted sections) and one flexible part in the middle of

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the beam (Figs. 12.c and 12.d). The flexible part of the beam must struggle for more deformation because plastic hinges at the ends of this section have rotation and translation, simultaneously.

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Therefore, it is logical to impose a limit on the effective length/effective depth (L'/d) of the beam

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for preventing brittle shear failure in the beam.

Fig. 12. Deformed shape and plastic hinge locations in (a, b) non-retrofitted moment frame and (c, d) FRP-retrofitted moment frame

The geometry, boundary conditions and the retrofitting details of the frame selected for the parametric investigation are illustrated in Fig. 13. In this figure, L is the total length of beam,

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ACCEPTED MANUSCRIPT is the effective (clear) length of beam, h is beam total depth, a is the length of FRP overlay, typically chosen from 0.5d to 2d, where d is the effective depth of beam, H is the column height, δ is storey displacement and Pu is column axial load, used in the analyses to consider the effects of gravity load. The length of the beam and the length of flange-bonded FRP overlay are the

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main variables in this investigation. Other parameters, including column height, beam and

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column dimensions are kept constant. Frames having different beam length and FRP overlay

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lengths are subjected to inelastic story drift to find the minimum L'/d ratio which prevents shear

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failure in the beam. A ratio of FRP lengths to effective depth is defined here as a/d. Dimensions and reinforcement details of the frame with beam length measuring 3000 mm and FRP overlay

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length equal to 2d are shown in Fig. 14. Gravity load is not applied to the beam in order to avoid any change in the beam end moments arising from inelastic design displacement only.

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retrofitted joint discussed earlier.

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Mechanical properties of concrete, steel and FRP overlay are similar to the FRP flange-bonded

Fig. 13. Typical retrofitted frame geometry and boundary conditions 28

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Fig. 14. Dimensions and reinforcement details of the frame having a beam with total length of 3000 mm and 2d length of FRP overlay (All dimensions are in mm)

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The nonlinear static pushover analyses of the frames were carried out using the developed software. The analyses were conducted in a displacement-controlled form by

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gradually increasing storey drift (δ in Fig. 13) and evaluating the corresponding forces. The limit

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for storey drift is taken as inelastic design storey displacement, δ = 0.02H, specified in a number of building codes, including ASCE- 07 [53] and Iranian Seismic Building Code (Standard 2800)

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[54]. The evolution of the equivalent damage variable obtained for the RC frame during the pushover analysis is illustrated in the contour plots of Fig. 15. It can be seen that as the drift is gradually increased, damage zones form and localize in the beam at the end of FRP laminates, signifying the formation of plastic hinges at these locations. At final stages of loading, damage is highly localized around the end of FRP laminates.

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Fig. 15. Evolution of equivalent damage variable (EDV) in retrofitted concrete frame of Fig. 14 at inelastic drift a) 0.0088%, b) 0.0129%, c) .0176% and d) 0.02%

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Fig. 16 shows inelastic drift versus L'/d for FRP overlay lengths to beam effective depth

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ratios (a/d) of 0.5, 1.0, 1.5 and 2.0. It shows that if L'/d ratio is less than 6, the frame cannot achieve the 0.02 code-specified inelastic drift. Therefore, in retrofitted frames with L'/d less than 6, the plastic shear force dominates the response, changing it from a ductile flexure failure to a brittle shear failure. This, in turn, decreases the ductile plastic hinges which would adversely affect the seismic performance of the frame. Therefore, it is recommended that plastic hinges to be relocated to an extent that L'/d ratios remain greater than 6.

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Fig. 16. Inelastic drift versus L'/d for FRP overlay lengths 0.5d, d, 1.5d and 2d

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On the other hand, an increase in the plastic shear force in the beam results in an increase

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in the moment at the face of the column which may affect the strong column-weak beam design principle. In numerical simulations, to be able to obtain the moment at the face of the column

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using free body diagram, the column is initially assumed to be elastic. In this example, moment

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at the face of the unretrofitted frame‟s column (i.e. the beam plastic moment) is about 62 kN.m. Fig. 17 shows the ratio between column face moment in the retrofitted frame and column face

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moment in the unretrofitted frame (termed Mret/M0) versus L'/d. As it is illustrated, for beams

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with L'/d more than 6 the required drift (0.02) is achieved; therefore, the moment at the face of column is acceptable, whereas, for beams with L'/d less than 6, the required drift is not achieved; therefore, the moment value is not acceptable. Fig. 17 also shows that in beams with L'/d greater than 6, moment at the face of column could increase by up to 61%, depending on FRP overlay length. This moment increase is considerable and it may result in formation of plastic hinge in the column which is not desirable.

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Fig. 17. Moment of retrofitted frame/moment of unretrofitted frame at the face of column versus L'/d

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Fig. 18 shows the force-displacement pushover curves for beams with a/d = 2 and four different effective lengths of Lˊ=1520, 1220, 920 and 620 mm. It is noted that for Lˊ=1520 mm,

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the required storey drift is satisfied and ductile flexural failure occurs in the beam. However, for

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Lˊ=1220, 920 and 620 mm, the storey drift is evaluated as 0.019, 0.0176 and 0.0112, respectively, indicating that brittle shear failure has occurred in the beam. To gain a quantitative

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insight into the frames seismic performance, ductility ratio, µ, and toughness (as a measure of

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energy dissipation capacity) are evaluated and compared in Table 7. The ductility ratio is calculated as the ratio between the ultimate displacement, δu, and the equivalent yield displacement, δy. To determine the equivalent yield displacement, different procedures have been proposed [55-57]. In this work, the method based on reduced stiffness equivalence elastic-plastic yield proposed by Park [56] is used. This method computes a secant stiffness at 75% of the ultimate load to obtain an ideal bilinear representation of the response curve as is illustrated in Fig. 19.

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Fig. 18. Force displacement curves for a/d = 2 and four beam clear length (Lˊ)

Fig. 19. Ideal bi-linearization of force-displacement curve [56]

Table 7 shows that as the plastic hinges are relocated further into the beam (Lˊ is reduced) both ductility ratio and toughness are decreased. The decreases may be as high as 46% for ductility ratio and 68% for toughness. 33

ACCEPTED MANUSCRIPT Table 7. Ductility ratio and toughness for a/d = 2

δy (mm)

δu (mm)

µ

Toughness (kN.mm)

2600

24.9

79.77

3.2

11664

2300

25.1

52.1

2.07

6395

2000

21.03

45.95

2.18

6914

1700

18.03

29.37

1.63

4986

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Lˊ (mm)

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5. Conclusions

A numerical software is developed for modelling RC, based on anisotropic damage plasticity

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model for concrete, which is suitable for simulating concrete under triaxial loading. The software is then used to study the effects of relocating plastic hinges with the aid of flange-bonded FRP

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laminates on the performance of RC beams. The conclusions drawn from the results of these

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investigations may be summarized as follows:

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1- The comparison between the numerical and experimental results of FRP-retrofitted joints showed that the constitutive damage model presented here, is able to more accurately simulate

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the damage behaviour of concrete, compared to the classical plasticity and isotropic damage

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plasticity model.

2- The numerical parametric investigation shows that for beam effective length-to-effective

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depth ratios (L'/d) less than 6, the RC frame cannot achieve the minimum code-required 0.02 inelastic drift and the failure is an undesirable brittle shear failure. Therefore, plastic hinge relocation should be carried out to an extent that L'/d is more than 6. 3- The results also indicate that for L'/d ratios between 6 and 9, the moment at the face of columns may increase by about 60%. This increased demand on the columns should be addressed when using FRP overlays to relocate the plastic hinges further into the beams.

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ACCEPTED MANUSCRIPT 4- As the plastic hinges are relocated further into the beam (Lˊ is reduced), both ductility ratio and energy dissipation capacity (toughness) are decreased. The decreases may be as high as 46%

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for ductility ratio and 68% for toughness.

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ACCEPTED MANUSCRIPT References

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[1] Oehlers DJ, Seracino R. Design of FRP and steel plated RC structures: retrofitting beams and slabs for strength, stiffness and ductility. UK: Elsevier; 2004. [2] Bank LC. Composites for constructions: structural design with FRP materials. Hoboken, NJ: Wiley; 2006. [3] Zhao LY. Static and fatigue behaviors of RC beams strengthened with carbon fiber sheet. Ph.D. dissertation, Univ. of Alabama, Huntsville, 2005. [4] Rabinovitch O, Frostig O. Experiments and analytical comparison of RC beams strengthened with CFRP composites. J Compos Part B. 2003;34:663-77. [5] Van Den Eindea L, Zhaob L, Seible F. Use of FRP composites in civil structural applications. Constr Build Mater. 2003;17:389-403. [6] Granata P, Parvin & A, An experimental study on kevlar strengthening of beam-column connections. Composite Structures. 2001;53(2):163-171. [7] Mukherjee A, Joshi M. FRPC reinforced concrete beam-column joints under cyclic excitation. Compos Struct. 2005;70: 185-99. [8] Mosallam AS, Banerjee S, Shear enhancement of reinforced concrete beams strengthened with FRP composite laminates. J Compos Part B. 2007;38-7:81-93. [9] Le-Trung K, Lee K, Lee J, Lee D, Woo S. Experimental study of RC beam-column joints strengthened using CFRP composites. Composites, Part B: Engineering, Vol. 41, No. 1. 2010:7685. [10] Mahini SS, Ronagh HR. A new method for improving ductility in existing RC ordinary moment resisting frames using FRPS. Asian J Civ Eng (Build Housing). 2007;8(6)-5: 81–95. [11] Zhu JT, Li X, Wang Z, Xu D, Weng CH. Experimental study on seismic behavior of RC frames strengthened with CFRP sheets, Composite Structures NO. 2011;93:1595-1603. [12] Eslami A, Ronagh HR. Experimental investigation of an appropriate anchorage system for flange-bonded carbon fiber–reinforced polymers in retrofitted RC beam–column joints. Compos. for Construct, ASCE, 04013056, 2014. [13] Mahini SS, Ronagh HR. Web-bonded FRPs for relocation of plastic hinges away from the column face in exterior RC joints. Compos Struct. 2011;93-24:60-72. [14] Azarm R, Maheri MR, Torabi A. Retrofitting RC Joints Using Flange-Bonded FRP Sheets, Iran J Sci Technol, Trans Civ Eng, 2017;41(1): 27-35. [15] Parvin A, Granata P, Investigation on the effects of fiber composites at concrete joints. J Compos, Part B. 2000;31:499-509. [16] El-Amoury T, Ghobarah A. Seismic rehabilitation of beam-column joint using GFRP sheets. Eng Struct. 2002;24-1:397-407. [17] Ghobarah A, El-Amoury T. Seismic rehabilitation of deficient exterior concrete frame joints. J Compos Constr. 2005;9(5):408-16. [18] Niroomandi A, Maheri A, Maheri MR, Mahini SS. Seismic performance of ordinary RC frames retrofitted at joints by FRP sheets. Eng Struct, 2010;32-23:26-36. [19] Maheri MR, Akbari R. Seismic behaviour factor, R, for steel X-braced and knee-braced RC buildings. Eng Struct, 2003;25:1505-1513. [20] Dalalbashi A, Eslami A, Ronagh HR. Plastic hinge relocation in RC joints as an alternative method of retrofitting using FRP. Comp Struct. 2012;94-243:3-9. [21] Dalalbashi A, Eslami A, Ronagh HR. Numerical investigation on the hysteretic behavior of RC joints retrofitted with different CFRP configurations. Compos for Const. 2013;17-3:71-82. 36

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[22] Hadigheh SA, Maheri MR, Mahini SS. Performance of weak-beam, strong-column RC frames strengthened at the joints by FRP, Iranian J. of Sci and Tech, Trans Civil Eng, 2013;37(C1):33-51. [23] Hadigheh SA, Mahini SS, Maheri MR. Seismic behaviour of FRP-retrofitted reinforced concrete frames. Earthq Eng, 2014;18:1171-1197. [24] Zarandi S, Maheri MR. Seismic performance of RC frames retrofitted by FRP at joints using a flange-bonded scheme. Iranian J. of Sci and Tech, Trans Civil Eng, 2015;39(C1):103-123. [25] Baji H, Eslami A, Ronagh HR. Development of a nonlinear FE modelling approach for FRP-strengthened RC beam-column connections. Structures. 2014: 272–281. [26] Eslami A, Ronagh HR, Numerical Investigation on the Seismic Retrofitting of RC Beam– Column Connections Using Flange-Bonded CFRP Composites, Compos for Const, ASCE, 2015; 04015032,. [27] ACI Committee 318. Building code requirements for structural concrete (ACI 318-14) and commentary (ACI 318R-14). Farmington Hills, Mich.: American Concrete Institute, 2014. [28] Kachanov LM, On rupture time under condition of creep. IzvestiaAkademiNauk USSR, Otd,Techn. Nauk, Moskwa. 1958;8:26-31. [29] Rabotnov IN, On the equations of state for creep. In: Progress in Appl. Mech.,The Prager Anniversary Volume, MacMillan, New York, 1963: 307-315. [30] Murakami S. Damage mechanics and its recent development, JSME. 1985;A51: 1651-1659. [31] Grassl P, Jira´sek M, Damage-plastic model for concrete failure, International Journal of Solids and Structures. 2006;43:7166-7196. [32] Kachanov LM, Introduction to Continuum Damage Mechanics, Martinus Nijhoff, Dordrecht, The Netherlands, 1986. [33] Lemaitre J, Chaboche JL, Mechanics of Solid Materials, Cambridge University Press, London, 1990. [34] de Souza Neto EA, Perić D, Owen DRJ. Computational methods for plasticity, theory and applications, John Wiley & Sons, United Kingdom, 2008. [35] Koh CG, Teng MQ, Wee TH, A Plastic-Damage Model for Lightweight Concrete and Normal Weight Concrete, International Journal of Concrete Structures and Materials, Vol.2, No.2. 2008: 123-136. [36] Lubliner J, Oliver J, Oller S, Onate E. “A Plastic damage Model for Concrete,” Int J Solids and Struct, 1989;25(3):299-326. [37] Lee J, Fenves GL. A plastic–damage model for cyclic loading of concrete structures. J. Eng. Mech. ASCE. 1998;24:892-900. [38] Krajcinovic D. Continuum damage mechanics, J. APPL. MECH.-T ASME. 1983;a-37: 1–6. [39] Krajcinovic D, Fonseka GU. The continuous damage theory of brittle materials, Part 1: General theory, J. APPL. MECH.-T ASME. 1981;a-48: 809-815. [40] Cordebois JP, Sidoroff F. “Anisotropic damage in elasticity and plasticity”, Journal de Mecanique Theorique et Applilquee (Numero Special). 1979:45-60. [41] Hayakawa K, Murakami S, Y Liu. “An irreversible thermodynamics theory for elastic– plastic-damage materials”, Eur. J. Mech. A/Solids. 1998;17(1):13-32. [42] Voyiadjis GZ, Kattan PI. Advances in Damage Mechanics: Metals and Metal Matrix Composites, with an Introduction to Fabric Tensors, second ed., Elsevier, Oxford, 2006. [43] Voyiadjis GZ, Dorgan RJ, Framework using functional forms of hardening internal state variables in modeling elasto-plastic-damage behavior, INT. J. PLASTICITY. 2007;23:18261859. 37

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[44] Voyiadjis GZ, Taqieddin ZN, Kattan PI. Anisotropic damage–plasticity model for concrete, INT. J. PLASTICITY. 2008; 24:1946-1965. [45] Voyiadjis GZ, Mozaffari N, Nonlocal damage model using the phase field method: Theory and applications, International Journal of Solids and Structures No. 2013;50:3136-3151. [46] Mozaffari N, Voyiadjis GZ, Phase field based nonlocal anisotropic damage mechanics model, Physica D. 2015;308:11-25. [47] Azizsoltani H, Kazemi MT, Javanmardi MR. An Anisotropic damage model for metals based on irreversible thermodynamics framework, IJST, Transactions of Civil Engineering, Vol. 38, No. C1+, 2014:157-173. [48] Khaloo AR, Javanmardi MR, Azizsoltani H. Numerical characterization of anisotropic damage evolution in iron based materials,Scientia Iranica A, , 2014; 21(1): 53-66. [49] Chow CL, Wang J. An anisotropic theory of elasticity for continuum damage mechanics. Int. J. Fracture, 1987;33:2-16. [50] Cicekli U, Voyiadjis GZ. Abu Al-Rub, R.K., A plasticity and anisotropic damage model for plain concrete. Int. J. Plasticity 2007;23:1874-1900. [51] ACI 440.2R-08. ACI Committee 440-02. Guide for the design and construction of externally bonded FRP system for strengthening concrete structures. USA, 2008. [52] ABAQUS Theory Manual, version 6.14, SIMULIA, 2016. [53] ASCE. Minimum Design Loads for Buildings and Other Structures. ASCE/SEI Standard 716. ASTM International, 2016. [54] Iranian code of practice for seismic resistance design of buildings. Standard, No. 2800. 4nd. ed. 2015. [55] Paulay T, Priestley MJN. Seismic design of reinforced concrete and masonry buildings. Wiley, New York, 1992. [56] Park R. Evaluation of ductility of structures and structural assemblages from laboratory testing. Bul.l N. Z. Soc. Earthq. Eng. 1989;22:155–166,. [57] Elnashai AS, Di Sarno L, Fundamentals of Earthquake Engineering: From Source to Fragility, 2nd Edition, willy, 2015.

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ACCEPTED MANUSCRIPT List of Tables

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Table 1. Mechanical properties of concrete in the non-retrofitted RC joint model Table 2. Mechanical properties of steel in the non-retrofitted RC joint model Table 3. Mechanical properties of FRP laminates Table 4. Mechanical properties of concrete at the flange bonded retrofitted joint test Table 5. Mechanical properties of steel at the flange bonded retrofitted joint test Table 6. Mechanical properties of FRP laminates at the flange bonded retrofitted joint test Table 7. Ductility ratio and toughness for a/d = 2.

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ACCEPTED MANUSCRIPT List of Figures

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Fig. 1. Relocation of plastic hinge in retrofitted frame resulting in a decrease in beam clear span and an increase in plastic shear (Vp) Fig. 2. Details of the non-retrofitted RC joint [10] (all dimensions are in mm) Fig. 3. Evolution of plastic strain in the proposed simulation of non-retrofitted RC joint model at load a) 8.26kN, b) 14.22kN, c) 19.51kN and d) 21.64kN Fig. 4. Evolution of equivalent damage variable (EDV) in the proposed simulation of nonretrofitted RC joint model at load a) 8.26kN, b) 14.22kN, c) 19.51kN and d) 21.64kN Fig. 5. Comparison of load-displacement response curves of non-retrofitted RC joint Fig. 6. Evolution of equivalent damage variable (EDV) in the web-bonded retrofitted joint at load a) 10.58kN, b) 17.46kN, c) 19.22kN and d) 21.6kN Fig. 7. Comparison of load-displacement response curves of FRP web-bonded retrofitted joint Fig. 8. Details of the flange-bonded FRP retrofitted joint [14] (All dimensions are in mm) Fig. 9. Propagation of equivalent damage variable (EDV) in the FRP flange-bonded retrofitted RC joint at load a) 22.71kN, b) 33.93kN, c) 55.8kN and d) 62.88kN Fig. 10. Comparison of the load-displacement capacity curves of the flange-bonded retrofitted joint Fig. 11. Load-strain curves of the flange-bonded retrofitted joint Fig. 12. Deformed shape and plastic hinge locations in (a, b) non-retrofitted moment frame and (c, d) FRP-retrofitted moment frame Fig. 13. Typical retrofitted frame geometry and boundary conditions. Fig. 14. Dimensions and reinforcement details of the frame having a beam with total length of 3000 mm and 2d length of FRP overlay (All dimensions are in mm) Fig. 15. Evolution of equivalent damage variable (EDV) in retrofitted concrete frame of Fig. 14 at inelastic drift a) 0.0088%, b) 0.0129%, c) .0176% and d) 0.02% Fig. 16. Inelastic drift versus L'/d for FRP overlay lengths 0.5d, d, 1.5d and 2d Fig. 17. Moment of retrofitted frame/moment of unretrofitted frame at the face of column versus L'/d Fig. 18. Force displacement curves for a/d = 2 and four beam clear length (Lˊ) Fig. 19. Ideal bi-linearization of force-displacement curve [56]

40