InTechOpen uses cookies to offer you the best online experience. By continuing to use our site, you agree to our Privacy Policy.

Materials Science » Polymers » "Fiber Reinforced Polymers - The Technology Applied for Concrete Repair", book edited by Martin Alberto Masuelli, ISBN 978-953-51-0938-9, Published: January 23, 2013 under CC BY 3.0 license. © The Author(s).

Chapter 7

Analysis of Nonlinear Composite Members Including Bond-Slip

By Manal K. Zaki
DOI: 10.5772/51446

Article top


A typical interaction diagram of RC columns
Figure 1. A typical interaction diagram of RC columns
Figure 2. Cross-section
Column Segment
Figure 3. Column Segment
Stress-strain curves
Figure 4. Stress-strain curves
Incremental-iterative method
Figure 5. Incremental-iterative method
Verification against Shahawy
Figure 6. Verification against Shahawy
Uniaxial Moments About z and y axes
Figure 7. Uniaxial Moments About z and y axes
Contour lines (Fx=0.0)
Figure 8. Contour lines (Fx=0.0)
Contour lines (Fx=0.70)
Figure 9. Contour lines (Fx=0.70)
Force-Deflection Diagrams
Figure 10. Force-Deflection Diagrams
Linear Stiffness Matrix [Ke]
Appendix 1. Linear Stiffness Matrix [Ke]
Nonlinear Stiffness Matrix [Kg].
Appendix 2. Nonlinear Stiffness Matrix [Kg].
Bond-slip Stiffness Matrix [Kb]
Appendix 3. Bond-slip Stiffness Matrix [Kb]

Analysis of Nonlinear Composite
Members Including Bond-Slip

Manal K. Zaki1

1. Introduction

Extensive research has been carried out in recent years on the use of FRP composites in strengthening of RC structures. Concrete elements strengthened with FRP undergo significant improvement of strength, ductility and resistance to electrochemical corrosion. Moreover, strengthening concrete member with FRP has the advantages of decreased installation costs and repairs, less stiffness and weight in comparison with steel. The increase in stiffness of the structural elements is undesirable in seismic prone areas. Structural members can be strengthened with FRP jackets provided along the whole length of the member or in regions of maximum straining actions. FRP strengthening can, also, be provided on one face of the structural member as in the case of stiffening the tension fibers of a beam.

For FRP retrofitting problem, the confinement model describing the behavior of rectangular concrete columns retrofitted with externally bonded FRP material and subjected to axial stress was presented by Chaallal et al. [1]. Other researchers investigated the effect of FRP in seismic strengthening of concrete columns, Tastani and Pantazopoulou [2] and Ozcan et al. [3]. They found that FRP retrofitting remarkably increased the strength and ductility of the strengthened members. Some researchers proposed simplified equations for FRP retrofit design of difficient rectangular columns, Ozcan et al. [4].

Other researchers studied reinforced concrete members externally bonded with FRP fabric using commercial software ANSYS, Kachlakev et al. [5], Li et al. [6]. Yan et al. [7] developed an analytical stress-strain model. Purushotham et al. [8] studied piles in berthing structures under uniaxial bending. Kaba and Mahin [9] presented the concept of fiber method in their refined modeling of RC columns for seismic analysis under uniaxial bending.

Some searches were conducted to the problem of biaxial bending. Bresler [10] and Bernardo [11] studied biaxial bending for unretrofitted short rectangular columns.

At early stage of the use of layered beams, full interaction (perfect bond) was assumed in the design. It was until the mid-fifties that Newmark and his co-authors [12] pointed the influence of partial interaction on the overall elastic behavior of steel-concrete composite beams. They derived the governing equations and solved the equilibrium equations expressed in terms of the axial force. Since then, several studies have been conducted to study the problem of bond-slip, Arizumi et al. [13], Daniel and Crisinel [14], Salari et al. [15]. Gara [16] and Ranzi [17] adopted the displacement based finite element formulation to include the vertical slip. Salari et al. [18] also Valipour and Bradford [19] adopted one-dimensional element force-based element to solve the relevant problem. Other researchers [20] and [21] adopted the mixed-procedure, displacement-based together with force-based, to solve the problem. Moreover, nonlinear geometric effects were introduced to the problem by Girhammar and Gopu [22], Girhammar and Pan [23], Čas et al. [24] and Pi et al. [25]. Krawczyk and the co-authors [26,27], Battini et al. [28] developed a corotational formulation for the nonlinear analysis of composite beams with interlayer slip. Nguyen et al. [29], Sousa et al. [30] implemented a finite element model to solve a composite beam column with interlayer slip.


Figure 1.

A typical interaction diagram of RC columns

In practice, many RC columns are subjected to biaxial bending. The analysis of such problems are difficult because a trial and adjustment procedure is necessary to find the inclination and depth of the neutral axis. The problem becomes more complicated when the slenderness effect is included. A typical interaction diagram for biaxially loaded column is shown in Fig. 1. Case a and case b are the uniaxial bending about the z axis and y axis respectively. The interaction curves represent the failure envelope for different combinations of the axial load and bending moments. Case c represents the case of a RC column with biaxial bending.

The material nonlinearity is considered to account for concrete cracks and the change of the stress-strain relationship of the different materials. The material nonlinearity is thus introduced by using the FMM together with the incremental iterative solution. The geometric nonlinearity is considered in the present study to account for the deformations occurring due to excess bending moments developed by the effect of axial load. The geometric nonlinearity, thus, considers the slenderness effect of the column. The bond-slip effect is considered by introducing the bond properties of the epoxy resin applied to adhere FRP to the RC column.

The method adopted is accomplished by dividing the column into segments along the member axis to introduce the FEA for the skeletal segments. At each end of the segment, the cross-section is divided into concrete, steel and FRP fibers to introduce the FMM. The properties of a cross-section is calculated by summing up the properties of all the fibers or elemental areas of the particular section. The column segment properties are considered as the average properties of the its end cross-sections. The segment and cross-section discretization are detailed in section 2.

The load is applied incrementally until the maximum allowed strains are reached. An incremental iterative method is employed to solve the problem. After each iteration, the properties of each cross-section are computed according to the material changes occurring and governed by the stress-strain relationship for each material. The properties of each column segment is considered as the average between its end section properties. Those properties are then introduced to the tangential linear stiffness matrix. The geometric nonlinearity is accounted for through the geometric stiffness matrix. Also, the bond-slip effect is considered by the addition of the bond-slip stiffness matrix.

It is, therefore, the aim in this study to adopt the FEA to formulate the linear, geometric and the bond-slip stiffness matrices of composite members subjected to biaxial bending together with axial forces. The model is developed within an updated Lagrangian incremental formulation.

The assumptions of the present analysis are: 1)only longitudinal partial interaction is considered. Axial relative displacement occurs between different elements while the vertical displacement is the same for all elements. 2)small strains and moderate rotations are considered. This assumption represents a rigorous simplification applicable to many problems. 3) Both layers, referred to as elements in the present study, followed the Euler-Bernoulli beam theory. This considers that plane cross-sections remain plane after deformations and perpendicular to the axis of the beam. 4) Shear and torsional deformations are neglected. 5) Effect of the column weight is neglected.

2. Fiber method modelling of frp confined beam columns

The cross-section is divided into concrete, steel and FRP fibers to introduce The FMM is introduced herein to compute the properties of each fiber, thus achieving the properties of the a cross-section by summing up the properties of all its fibers or elemental areas. The meshing is given in Fig. 2(a). The column segment properties are considered as the average properties of its end cross-sections.

The same derivation in the companion paper [31] for columns under biaxial bending is adopted herein after the necessary modifications to solve the column under the effect of slip.

The strain distribution is defined by the maximum compressive strain εm, together with the depth of the neutral axis, Zn. The strains are shown in Fig. 2(b).


Figure 2.


The following section parameters are then computed


where φ y and φ z are the curvatures along the y-axis and z-axis respectively and εo is the strain at point "O".

The elemental strain is computed as:


where Z is the distance from the maximum strain to the element measured perpendicular to the N.A. After determining the strain of each fiber from eq.5, the corresponding elemental modulus of elasticity, G, is determined as detailed in section 4. The elemental properties are computed and summed up to obtain the cross-section properties as shown in the following equations:


where α is the counter of an arbitrary element. In the present study, element 1 is the RC section and element 2 is the FRP. n is the total number of elements and is equal to 2 in the present study, i is the counter of fibers, nfib is the total number of fibers of element α, ΔA i is the area of each fiber, y i , z i are distances from the center of the considered fiber to the z and y axes respectively. Those symbols are shown in Fig. 2a. It should be noted that the properties EA, ES y and ES z are given separately for each element, while the properties EI z , EI y and EI yz are the summation of the corresponding properties of both elements. The reason for this is that the axial displacement is different for each element due to the slip effect while both elements undergo the same curvatures about the z-axis and the y-axis.

3. Displacement-based fiber model with bond-slip

A one dimensional finite element analysis is adopted to solve the column segments. The segments are considered to be of unsymmetric cross-section caused by the inclination of the N.A. The finite element formulation given by Yang and McGraw [32] to solve the thin-walled, i.e. bare-steel columns, is introduced herein after applying the necessary modification to include the concrete, FRP and bond-slip.

3.1. Displacements and strain fields

The axial displacement of an arbitrary point of an element (α ) in the cross-section is given in terms of the displacements of a constant point “c” on the same element as follows


where u xcα is the axial displacement of the element α, y and z are the vertical and horizontal distances, respectively from the centroid of any follower element α to the centroid of the parent element having α=1. In the present study, the concrete section and the FRP are considered to be the parent and the follower elements respectively. And uycanduzc are the derivatives of the transverse displacements u yc and u zc .

y=yα-y1 and z=zα-z1 however, for simplicity, the reference axes are chosen such that y1=0 and z1=0, Fig.3 (a).


Figure 3.

Column Segment

It should be noted that the transverse displacements, u yc and u zc , are considered to be the same for all elements of the beam with respect to the axis of the beam. For simplicity, the symbol “c” is dropped out from the r.h.s. of the equations. The relevant linear strain field can, then, be obtained from the displacement field given by eq. 7 by differentiating the mentioned equation w.r.t. the beam coordinates as


while the nonlinear strain field is given by


and u x,x is ignored as usual practice. The slip satisfying the compatibility relation with the displacements of element 1 and element 2 is given as


3.2. Forces

In general, the the normal stress f x shown in Fig. 3 (a) is expressed as


Following the integrations at the cross-section, the stress resultant are expressed as


where the elemental properties are previously given in section 2.

3.3. Degrees of freedom and nodal forces

The local coordinates are shown in Fig. 3(b and c).

The displacement vector <u>is given by:

u=ux1A    uyA       uzA      θyA       θzA     ux2A    ux1B    uyB    uzB      θyB    θzB     ux2B

where θy and θz are the angles of rotation of the section about y and z axis respectively.

While the vector of the nodal forces <f> is given by:

   f=Fx1A   FyA   FzA  MyA   MzA   Fx2A   Fx1B   FyB  FzB  MyB   MzB  Fx2B    

3.4. Virtual work

The partial interaction problem is solved using the principle of virtual work. This is accomplished by equating the work of internal stresses to the work of external stresses in an incremental form. The principle of virtual work will be adopted to deduce:

-the linear and nonlinear stiffness matrices of a beam element which consist of an assemblage of two different

elements connected by deformable interface.

-the bond-slip stiffness matrix.

The equilibrium condition for the entire beam-column is then expressed by assembling the vectors and matrices defined for each segment according to the principle of finite elements.

The details are given below.

3.4.1. For the beam with FRP


in which 1exx and 1ηxx are the linear and non-linear incremental strain respectively, fx denotes the stress at C1, E is the modulus of elasticity of the cross-section and 2 1R and 1 1R are the external virtual work at C2 and C1 respectively but both being measured at C1 and 2 2R = the external virtual work expression

substituting equations 8 into eq.16, we get


A linear displacement field is adopted for the axial displacement, ux, and a cubic displacement field for other displacements. The incremental displacements are expressed as:



n1=1i        i≻
n3=13i2+2i3     i2i2+i3     3i22i3     i3i2

in which i is given by the value xl .

The nodal degrees of freedom ie., the ends A and B of the column segment are given by:


where l is the length of the segment. The linear part

The linear part taken from eq. 17 is


in which α is the counter of the considered elements and n is their total number.


applying the properties of the cross-section given in eqs 6, the previous expression becomes


Substituting the interpolation functions in eq. 18, the following equation applies

(24) The nonlinear part

The nonlinear part taken from eq. 17 is


when several elements participate in the nonlinear virtual work, the previous eq becomes


in which α is the counter of the considered elements and is their total number.


when n=2, as in the general case, then [0lα=1nFxαl{n3}n3di] becomes [0l(Fx1+Fx2)l{n3}n3di]. The linear and nonlinear stiffness matrices are obtained after performing the integrations in eqs 24 and 27 and are given in the appendix.

3.4.2. For bond-slip

The bond-slip expression given in eq.10 is substituted in the linear portion of the virtual work expression given in eq. 17 and the expression thus becomes


where A b is the area of the FRP per unit length of the segment.

The bond-slip matrix is obtained after performing the integrations in eq 30 and is given in the appendix.

Eqs.(24, 27 and 30) can be combined to give:


And upon simplification, the equilibrium equations (31a to 31d) are written in the form


in which [Ke] , [Kg] and [Kb] are the linear, geometric and bond-slip stiffness matrices respectively, {u} is the incremental displacement vector and {f1} and {f2} are the segment nodal forces at the beginning and the end of the incremental step.

The very simple form of the equilibrium equations is

in which


The given procedure can be applied to problems with complete bond by combining elemental properties of the elements 1 and 2 and dropping out the bond-slip stiffness matrix. In this case each [K] will be of order 10*10 instead of 12*12.

4. Stress-strain curves

The constitutive relations for concrete, steel, FRP and bond are schemetically shown in Fig.4.

4.1. Stress strain relationship for FRP

The stress-strain relationship for FRP is considered linear as shown in Fig.4.a

the incremental stress-strain relationship is given by

where G f is the elemental FRP modulus of elasticity and is expressed as

Gf=ffuεfu=Ef   when    0εfεfu

in which f f and εf are the FRP stress and strain respectively, f fu and εf u are the ultimate FRP stress and strain respectively and E f is the modulus of elasticity of FRP.


Figure 4.

Stress-strain curves

4.2. Stress strain relationship for steel

For simplicity, the stress-strain relationship for the steel bars is considered to be an elastic-perfectly plastic curve neglecting steel hardening. The relationship is shown in Fig. 4.b.

The incremental stress-strain relationship is

where Gs is the incremental steel modulus of elasticity and is expressed as

Gs=fyεy=Es                   whenεyεsεy
Gs=0                      whenεsεyorεsεy

in which f s and ε s are the steel stress and strain respectively, fy and εy are the yield stress and yield strain respectively and E s is the modulus of elasticity of steel.

4.3. Stress strain relationship for concrete

For unconfined concrete, the relationship adopted by Al-Noury and Chen [33] was chosen to express the first portion of the compressive stress-strain curve for concrete as a third-degree polynomial. The second portion is considered to be perfectly plastic as shown in Fig. 4.c. The incremental stress-strain relationships is expressed as:


Gc=fcεcγ1+2fcεc2(32γ1)εc+3fcεc3(γ12)εc2 when   0.0εcεc
Gc=0.0       when      εcεc

in which

γ1=Ecεcfc   ,    εc=0.002 and     Ec=30000fc

Ec= modulus of elasticity of concrete computed in t/m2 while fc’ and εc’ are the maximum unconfined concrete compressive strength and the corresponding strain respectively.

The stress-strain behavior of FRP-confined concrete is largely dependent on the level of FRP confinement. The bilinear stress-strain relationship suggested by Wu et al. [34] is shown in Fig. 4(c) and is adopted herein. The stress-strain curve of concrete confined with sufficient FRP displays a distinct bilinear curve with a second ascending branch as shown in Fig. 4(c). A minimum ratio of FRP confinement strength to unconfined concrete compressive strength f l /f’c of approximately 0.08 is provided to ensure an ascending second branch in the stress-strain curve. Confinement modulus (E1) and confinement strength (f l ) are considered to be the two main factors affecting the performance of FRP-confined columns. The two factors are given as:

where ρ f is the volumetric ratio of FRP to concrete, which can be determined for a rectangular section as to a circular section with an equivalent diameter taken as the length of the diagonal of the rectangular section as follows:


where h and b are the bigger and smaller dimensions of the cross-section respectively, n is the number of FRP layers and t f is the thickness of each layer.

The maximum FRP-confined concrete compressive strength and the ultimate axial strain of the FRP-confined concrete compressive stress-strain are given by Rocca et al. [35] as


where ka and kb are efficiency factors that account for the geometry of the cross-section. In the case of rectangular columns, they depend on the effectively confined area ratio Ae/Ac and the side-aspect ratio h/b. These factors are given by the following expressions:


where Ag is the total cross-sectional area, ρ g is the ratio of the longitudinal steel reinforcement to the cross-sectional area of a compression member and r is the corner radius of the cross-section.

The slope of the second branch E2 is computed from the following equation considering the intercept of the second portion with the stress axis equal to f c’ for simplicity, Rocca et al. [35].


The transition stress f t and transition strain ε t are given by the following equations


The maximum exerted confining pressure f lu is attained when the circumferential strain in the FRP reaches its ultimate strain ε fu corresponding to a tensile strength f fu [36] and Eq. (42b) becomes


where h2+b2 is the equivalent diameter for non-circular cross-sections. The following equations express the elemental modulus of elasticity for confined concrete in terms of strain.

The effective strain ε fe is computed as the product of an efficiency factor Ke and the ultimate FRP tensile strain ε fu . The factor Ke is to account for the difference between the actual rupture strain observed in FRP-confined concrete specimens and the FRP material rupture strain determined from tensile coupon testing, Wu et al. [34]. The factor ranges from 0.55 to 0.61 and is taken 0.586 in this study.

Gc=Ec   when   0εcεt 
Gc=E2   when    εtεcεccu

4.4. Stress strain relationship for FRP-Concrete Bond

The relationship is shown in Fig. 4.d.

The incremental stress-strain relationship is

where Gb is the incremental steel modulus of elasticity and is expressed as

Gb=τbub=Eb   when   0ububu

in which τ b and u b are the steel stress and strain respectively, τb and ub are the yield stress and yield strain respectively and E b is the bond elastic stiffness.

5. Steps of solution followed by the developed program

The mixed procedure is utilized to solve the nonlinear problem. This procedure utilizes a combination of the incremental and iterative (Newton-Raphson) schemes. The load is applied incrementally and after each increment successive iterations are performed. Steps of the solution are then introduced.


Figure 5.

Incremental-iterative method

The mixed procedure is utilized herein to solve the nonlinear problem. This procedure utilizes a combination of the incremental and iterative (Newton-Raphson) schemes. The load is applied incrementally and after each increment successive iterations are performed. The method is illustrated in fig.5.

The combined method is summarized in the following steps:

  1. Apply the first increment of load {f} and compute [Ko] assuming no cracks and full bond between the concrete element and the FRP element at the beginning. Compute the displacements {u1} by solving the equation [Ko]{u1}={f}

  2. Compute [K1] based on the displacement {u1} then compute the load {f} from the equation {f}=[K1]{u1}

  3. Compute {Δf} as the difference between the applied load {f} and the deduced load {f} . Then compute the corresponding displacements {u2} by solving the equation [K1]{u2}={Δf}

  4. Repeat steps 2 and 3 until {Δf} becomes very small.

  5. Repeat all steps again for the next increment.

6. Numerical examples

Two examples are given below. The first example considers a rectangular column fully confined with FRP. Complete bond is considered. The second example is a beam strengthened with FRP on the tension side. In this example the slip between the two elements is considered.

Example 1: The verification of the method is plotted in Fig.6 against experimental results given by Chaallal and Shahawy [1]. The column has across-section of 0.35*0.2 m2 and length 2.1m. The concrete has a compression strength 25 MPa and the column is reinforced with 4 grade 60 steel bars of diameter 19 mm each. The steel bars are of 406 MPa yield stress and 206 GPa modulus of elasticity. The specimens are confined with 1mm of carbon fiber reinforced polymer of tensile strength 530 MPa and tensile modulus of elasticity 44 GPa. This gives a confinement ratio, f l /f c =0.103. The present procedure of analysis was adopted to the same specimens and interaction diagrams were plotted. The present results show great accordance with the previous work.

A slight difference in results is observed. It is owed to the provision of corbels in the specimens of Chaallal and Shahawy. They provided large corbels at the ends of the specimens to receive a single load source applied eccentrically thus simulating the combined stress effects in columns. The corbels increased the overall stiffness of the beam column and thus the capacity of loads.

It should be noted that all wraps were characterized by a bidirectional oriented fibers (00/900) applied along the entire height of the columns. As recommended by ACI 440.2R-02 [37]. The enhancement is only of the significance in members where compression failure is the controlling mode Nanni [38]. This strength enhancement is due to the confining effect of the FRP. When the column is subjected to axial load Fx and moment Mz such that their coordinates lie below the balanced point, the column is considered to be unconfined. This is owed to the limited value of Fx which is considered insufficient to dilate the concrete in the hoop direction thus failing to activate the FRP wrapping effect to confine the concrete. In the present analysis where the wraps are of bidirectional fibers, the point of pure bending is computed accounting for the FRP in the longitudinal direction and its contribution to the flexural capacity according to ACI440.2R-02 [37]. This case was also set by Chaallal and Shahawy [1]. Fig.7shows the plots of the column subjected to uniaxial bending Mz and My.


Figure 6.

Verification against Shahawy


Figure 7.

Uniaxial Moments About z and y axes

As expected, the capacity of the about the about the y-axis is less than that about the z-axis. The same model was also subjected to biaxial bending at two axial load levels, namely: Fx=0 and Fx=0.7. The plots of the contour lines of the confined and unconfined columns are given in Figs.(8 and 9).


Figure 8.

Contour lines (Fx=0.0)

Example 2:The problem of bond-slip was verified against Aprile et al.[39]. A simply supported rectangular beam of length 2.9m loaded by two forces, each=F at the middle third of the beam. The cross-section is 0.3*0.2 m2. Top reinforcement is 226 mm2, bottom reinforcement is 339 mm2. the beam is strengthened at the bottom by carbon FRP of width 50 mm and 1.2 mm thickness. The concrete has a compression strength 25 MPa. The steel bars are of 460 MPa yield strength and 210 GPa modulus of elasticity. The carbon fiber reinforced polymer is of tensile strength 2400 MPa and tensile modulus of elasticity 150 GPa. The epoxy resin is of 100 MPa compressive strength and 12.8 GPa modulus of elasticity. The concrete element is considered supported on a roller at one end and hinged at the other end. While the FRP element is considered to be supported on rollers at both ends. Fig.10 shows the verification of the present analysis if the beam considering bond-slip against Aprile. The curves are plots of the mid-span deflection of the beam against the applied force (2F). A slight difference is observed between the two curves. Also, a plot of the reference beam, with no FRP was plotted as reference beam. Another plot of the beam with full bond between the concrete and the FRP was plotted. At the maximum deflection of the beam with bond-slip, the reference beam shows nearly 15% decrease in the load capacity while the beam with complete bond achieves nearly 20% increase in the load capacity. In addition, the later beam undergoes greater deflection and the highest capacity. The curves show two points of remarkable change in slope indicating remarkable loss of strength in the beam. The lower point indicates concrete cracking in the middle third of the beam, at the location of the applied concentrated load. The upper point indicates the start of yield of the bottom steel reinforcement.


Figure 9.

Contour lines (Fx=0.70)

7. Summary and conclusions

The FEA together with the FMM were utilized to solve the problem of RC strengthened with FRP. The structural member solved can be of any slenderness ratio, under any loading and can have any end conditions. The FRP wraps can be totally or partially bonded to the concrete member. The elastic, geometric and bond-slip stiffness matrices of the member in the three-dimension were deduced and given in an appendix.


Figure 10.

Force-Deflection Diagrams


Appendix 1.

Linear Stiffness Matrix [Ke]


All the elements of the geometric stiffness matrix are multiplied by the factor α=1nFxα which in the general case when α=2, the previous term is ( Fx1+Fx2 ).

Appendix 2.

Nonlinear Stiffness Matrix [Kg].


Appendix 3.

Bond-slip Stiffness Matrix [Kb]

Two examples were studied. The first example considers a rectangular column fully confined with FRP. Complete bond was considered. Contour lines can be plotted at any load level. The second example is a beam strengthened with FRP on the tension side. In this example the slip between the two elements was considered. Load-deflection diagrams show that there exist two points of drop in stiffness, the first is due to concrete cracking under the concentrated loads and the second is due to the yield of steel. Extensive research is required to study the effect of the aspect ratio of the concrete cross-section, the strength of the concrete, the strength of FRP, the thickness of FRP and the properties of the epoxy resin used.


1 - O. Challal, M. Shahawy, M. Hassan, Performance of axially loaded short rectangular columns strengthened with carbon fiber-reinforced polymer wrapping. J Comp Const, ASCE 2003 7 3 200 208
2 - S. P. Tastani, S. J. Pantazopoulou, Detailing procedures for seismic rehabiliation of reinforced concrete members wiyh fiber reinforced polymers. Engineering Structures 2008 2 450 461
3 - O. Ozcan, B. Binici, G. Ozceke, Improving seismic performance of dificient reinforced concrete columns using carbon fiber reinfored polymers. Engineering Structures 2008 30 6 1632 1646
4 - O. Ozcan, B. Binici, G. Ozceke, Seismic strengthening of rectangular reinforced concrete columns using fiber reinfored polymers. Engineering Structures 2010 32 4 964 973
5 - D. Kachlakev, M. Thomas, S. Yim, Finite element modeling of reinforced concrete structures strengthened with frp laminates. Report for Oregon Department of Ransportation Salem 2001
6 - G. K. Li, P. S. Su-Seng, J. E. Helms, M. A. Stukklefield, Investigation into frp repaired RC columns. J Comp. Struct 2003 62 83 80
7 - Z. Yan, C. P. Pantelides, L. D. Reaveley, Fiber reinforced polymer jacketed and shape-modified compression members: I-experimental behavior. Struct J, ACI 2006 103 6 885 893
8 - B. R. Purushotham, P. Alagusundaramoorthy, R. Sundaravalivelu, 2009 Retrofitting of RC piles using GFRP composites. Journal of Civil Engineering, KSCE 13 1 39 47
9 - S. A. Kaba, S. A. Mahin, Refined modeling of reinforced concrete columns for seismic analysis. Report No. UBC/EERC-84/3. Ca: University of California, Berkeley 1984
10 - B. Bresler, Design criteria for reinforced columns under axial load and biaxial bending. ACI J. 1960 32 5 481 490
11 - A. L. Bernardo, Investigation of biaxial bending of reinforced concrete columns through fiber method modeling. Journal of Research in Science, Computing and Engineering 2007 4 3 61 73
12 - M. N. Newmark, C. P. Siess, I. M. Viest, 1951 Tests and analysis of composite beams with incomplete interaction. Proceedings of the Society for Experimental Stress Analysis 9 175 92
13 - Y. Arizumi, S. Hamada, T. Kajita, Elastic-plastic analysis of composite beams with incomplete interaction by finite element method. Comp. Struct. 1981
14 - B. J. Daniel, M. Crisinel, Composite slab behavior and strength analysis. Part I: Calculation procedure. J. Struct. Engrg. ASCE 119 1 16 35 1993
15 - M. R. Salari, E. Spacone, P. B. Shing, D. M. Frangopol, Behavior of composite structures under cyclic loading. Build. To Last, Proc., ASCE Struct.Congr. VX, Kempner Jr. L and Brown CB, eds. ASCE New York 1 1997
16 - F. Gara, G. Ranzi, G. Leoni, Displacement-based formulations for composite beams with longitudinal slip and vertical uplift. Internat J Numer Methods Engrg. 2006 65 8 1197 220
17 - G. Ranzi, F. Gara, P. Ansourian, General method of analysis for composite beams with longitudinal and transverse partial interaction. Computers and Structures 2006
18 - M. R. Salari, E. Spacone, Finite element formulations of one-dimensional elements with bond-slip. Eng Struct. 2001 23 815 26
19 - Goudarzi. H. Valipour, M. Bradford, M.A.2012 A new shape function for tapered three-dimensional beams with flexible connections. Journal Of Constructional Steel Research 70 43 50
20 - A. Dall’Asta, A. Zona, Three-field mixed formulation for the nonlinear analysis of composite beams with deformable shear connection. Finite Elem Anal Design 2004 40 425 48
21 - A. Ayoub, F. C. Filippou, Mixed formulation of nonlinear steel-concrete composite beam element. J Struct Eng 2002 126 3 371 81
22 - U. A. Grihammar, V. K. A. Gopu, Composite beam-columns with interlayer slip-Exact analysis. J Struct Eng. ASCE 1993
23 - U.A.P. Grihammar, Exact static analysis of partially composite beams and beam-columns. Int J mech Sci 2007 49 139 55
24 - B. Čas, M. Sage, I. Planinc, Non-linear finite element analysis of composite planar frames with interlayer slip. Comput Struct. 2004 82 1901 12
25 - Y. L. Pi, M. A. Bradford, B. Uy, 2006 second order nonlinear inelastic analysis of composite steel-concrete members. I: Theory. J Struct Eng. ASCE 132 5 751 61
26 - P. Krawczyk, F. Frey, A. P. Zielinsky, Large deflections of laminated beams with interlayer slips Part 1: Model development. Eng comput. 2007 24 1 17 32
27 - P. Krawczyk, B. Rebora, Large deflections of laminated beams with interlayer slips Part 2: finite element development. Eng comput. 2007 24 1 33 51
28 - J. M. Battini, Q. H. Nguyen, M. Hjiaj, Non-linear finite element analysis of composite beams with interlayer slip. Comput Struct. 2009 87 904 12
29 - Q. H. Nguyen, M. Hjiaj, Exact finite element model for shear-deformable two-layer beams with discrete shear connection. Finite Elements in Analysis and Design 2011 47 718 727
30 - J. B. M. Sousa Jr., C. E. M. Oliveira, A. R. da Silva, Displacement-based non-linear finite element analysis of composite beams with partial interaction. Journal of constructional Steel Research. 2010 66 772 779
31 - M.K. Zaki, Investigation of FRP strengthened circular columns under biaxial bending. Engineering Structures 2011 33 5 1666 1679
32 - Y. B. Yang, W. Mc Graw, Stiffness Matrix for Geometric Nonlinear Analysis. Journal of Structural Engineering ASCE 1986 112 853 877
33 - S. I. Al-Noury, W. F. Chen, Behavior and design of reinforced and composite concrete sections. Journal of Structural Division ASCE 1982 17169 1266 1284
34 - G. Wu, Z. T. Lu, Z. S. Wu, Strength and ductility of concrete cylinders confined with FRP composites. Construction and Building Materials 2006 20 134 148
35 - S. Rocca, N. Galati, A. Nanni, Interaction diagram methodology for desgin of FRP-confined reinforced concrete columns. Construction and Building Materials 2009 23 1508 1520
36 - G. Wu, Z. T. Lu, Z. S. Wu, Stress-strain relationship for FRP-confined concrete cylinders. Proceedings of the 6th international symposium on FRP reinforcement for concrete structures (FRPRCS), Singapor 2003 552 560
37 - American Concrete Institute ACI440.2R, Guide for the design and construction of externally bonded FRP systems for strengthening of concrete structures. Farmington Hills, MI, USA: American Concrete Institute 2002
38 - A. Nanni, N.M. Bradford, FRP jacketed concrete under uniaxial compression. Constr. and Build. Mat. 1995 9 2 115 124
39 - A. Aprile, E. Spacone, S. Limkatanyu, Role of bond in RC beams strengthened with steel and FRP plates. Journal of Structural Division ASCE 2001 22694 1445 11452