Analytical and Numerical Models of Steel I-Section Beam-Columns Strengthened with Bonded Prestressed FRP Laminates

: Fibre Reinforced Polymer (FRP) laminate are currently used in strengthening steel structures having a significant favorable effect on the member capacity. Researches have been focused on FRP strengthened beams or columns; thus, I-section beam-columns strengthened with prestressed bonded FRP laminate are scrutinized. Steel yielding, laminate rupture and/or laminate interfacial debonding common failure modes af-fecting the strengthened beam-columns capacity: debonding commonly leads to a premature failure. Prestressing the FRP laminate may delay this premature failure. Equations are analytically developed to evaluate the interfacial shear and normal stresses between the laminate and the steel; hence, the maximum stresses at debonding are evaluated. Thus, the capacity of a beam-column with prestressed FRP laminate is analytically estimated. A nonlinear numerical model is then developed for these beam-columns. The model is verified against experimental results available from literature. Then, a parametric study is conducted using the numerical model. Based on these results, a regression analysis is performed and the capacity of beam-columns with prestressed FRP laminate is numerically estimated considering steel yielding or FRP laminate rupture. Results


INTRODUCTION
Fibre Reinforced Polymers (FRP) laminate are currently used in strengthening structural steel members (Teng el al. 2012). However, applications of these new advanced composite materials are almost limited to strengthen steel beams or columns (Liu et al. 2001, Chajes et al. 2003, Donga et al. 2013, Sundarraja et al. 2014, Deng et al. 2016. For example, Denga and Lee (2007) presented an experimental study on steel beams strengthened with bonded CFRP plates. Another two investigations (Chiew et al. 2011a(Chiew et al. , 2011b presented a model for predicting the failure load corresponding to debonding of the FRP laminate from the beams. Full-scale bending tests on FRP strengthened steel beams were carried out and then numerical analyses were conducted on these strengthened steel beams. Benachour et al. (2008) presented another theoretical investigation on the interfacial stress between the steel beam and the FRP laminate and compared their results to those developed by Al-Emrani and Kliger (2006). They argued that there is a high concentration of shear and peeling stresses at the ends of the FRP laminate. Generally, FRP laminate adopted in strengthening steel sections are not economically utilized due to the premature interfacial debonding failure (Ghareeb et al. 2013, Gharib et al. 2015, Gharib et al. 2017, Sayed-Ahmed 2004, Teng et al . 2015, Colombi and Fava 2015. In the same time, high strength adhesive will not prevent this mode of failure as the stresses generated in the adhesive layer is always significantly higher than its strength.
Herein, the ultimate capacity of steel I-section beam-columns strengthened by unprestressed or prestressed bonded FRP laminate is scrutinized where prestressing is thought to delay the premature FRP laminate debonding. The investigation starts by analytically developing equations to evaluate the interfacial shear and normal stresses associated with such interfacial debonding. Based on these equations, an expression for the ultimate capacity of beam-columns with bonded unprestressed and prestressed FRP laminate is presented: this expression considers failure by laminate debonding.
Then, a nonlinear numerical model based on the finite element method for the strengthened beam-columns is built-up and verified against previously published experimental investigation. The numerical model is used to evaluate the capacity of the steel beam-columns with bonded FRP laminate and perform a parametric study considering different parameters which are thought to affect the strength of these beam-columns. A regression analysis is performed Analytical and Numerical Models of Steel I-Section Beam-Columns Strengthened with Bonded Prestressed FRP Laminates E.Y. Sayed-Ahmed *

Construction Engineering Dept., the American University in Cairo, Egypt
A.T. Abdelrazik Civil, Arch. and Env. Engineering Dept., Missouri Univ. of Science & Technology, USA ABSTRACT: Fibre Reinforced Polymer (FRP) laminate are currently used in strengthening steel structures having a significant favorable effect on the member capacity. Researches have been focused on FRP strengthened beams or columns; thus, I-section beam-columns strengthened with prestressed bonded FRP laminate are scrutinized. Steel yielding, laminate rupture and/or laminate interfacial debonding common failure modes affecting the strengthened beam-columns capacity: debonding commonly leads to a premature failure. Prestressing the FRP laminate may delay this premature failure. Equations are analytically developed to evaluate the interfacial shear and normal stresses between the laminate and the steel; hence, the maximum stresses at debonding are evaluated. Thus, the capacity of a beam-column with prestressed FRP laminate is analytically estimated. A nonlinear numerical model is then developed for these beam-columns. The model is verified against experimental results available from literature. Then, a parametric study is conducted using the numerical model. Based on these results, a regression analysis is performed and the capacity of beam-columns with prestressed FRP laminate is numerically estimated considering steel yielding or FRP laminate rupture. Results of the numerical and analytical models lead to a proposal for evaluating the FRP strengthened beam-column capacity and the need for prestressing the FRP laminate. using the numerical model data to introduce an equation for the ultimate strength of steel beam-columns with bonded prestressed FRP laminate: the equation considers both the steel yielding and the FRP rupture.
The analytically and numerically developed equations are adopted to introduce a methodology for estimating the ultimate strength and the failure mode of steel beam-columns with bonded FRP laminate. The need for prestressing to delay the FRP laminate debonding is arbitrated using this methodology.
2 THE ANALYTICAL MODEL Benachour et al. (2008) assumptions are utilized to develop a model for steel beam-columns with bonded FRP laminate. Figure 1a shows a schematic sketch of a beam-column strengthened using a bonded prestressed laminate. The prestressed laminate is attached to the tension flange and has the same length of the beam-column. The beam-column is subjected to constant bending moment and normal force along its span. No end anchorages are applied to the prestressed laminate: prestressing is transferred to the beam-column via bond.

Interfacial Shear Stress
Referring to Figure1, the interfacial shear stress τ(x) was introduced as (Abdelrazik 2013)  where Al, bl, and El are the FRP laminate's cross section area, width and elastic modulus, respectively. As, Is, h, and Es are the steel beam-column's cross sectional area, second moment of area, section's height and modulus of elasticity, respectively. ta and Ga are the adhesive layer's thickness and shear modulus, respectively.
The solution of the above second order differential equation is given by where B1 and B2 are arbitrary constants and λ is a properties of the steel section with the bonded FRP laminates. Two boundary conditions may be written and utilized to find the constants of integration of Eq. (2). The first one requires the interfacial shear stresses at the middle of the FRP laminate to be zero because this point lies on the axis of symmetry. Thus, The second boundary condition implies that at the starting point of the FRP laminate, the compression force in the steel beam-column due to the prestressing force is equal to zero and the FRP laminate moment contribution is also equal to zero so the steel beamcolumn's moment contribution has a value equals to the externally acting bending moment. Thus, where P and M are the external normal force and bending moment, respectively and P0 is the laminate prestressing force. Based on Equations 2 and 3, the interfacial shear stress τ(x) may given by

Interfacial Normal Stress
where εa is the vertical strain in the adhesive material, ta is the adhesive material's thickness, νs is the steel beam's lower flange vertical displacement, νl is the vertical displacement of the bounded prestressed laminate and Ea is the modulus of elasticity of adhesive material.
The solution of this 4 th order differential equation is given by where C1 and C2 are arbitrary constants; D11 is the flexural flexibility of the externally bonded prestressed laminate; ta and Ea are the thickness and the modulus of elasticity of the adhesive layer; and tl, bl and El are the laminate thickness, width and modulus of elasticity, respectively. In Equation 6, the shear stress τ(x) is given by Equation 4 while, D11, n1 and β are properties of the steel section with the bonded FRP laminate. Two boundary conditions may be written and utilized to find the constants of integration of Equation 6. The first one requires moment contribution at the starting point of the FRP laminate to be zero, so the steel beam-column's moment contribution has a value equals to the externally acting bending moment. The second boundary condition requires both the FRP laminate and steel beam-column's shear contribution to be to zero; this is because the externally acting shear force is equal to zero. Thus, the two boundary conditions may be written as Using on Equation 7, the two constants C1 and C2 are given by Based on Equations 6 and 8, the interfacial normal stresses σ developed at x = 0 is given by where C1, n1, λ and B2 are given by Equations 8, 6, 2 and 4, respectively.

Ultimate Strength of Beam-Columns with Bonded FRP Laminate
The interfacial shear stress causes premature debonding of the FRP laminate from the steel section. Debonding occurs when the maximum interfacial shear stress given by Equation 4 reaches the ultimate shear strength of the adhesive material; i.e.
where τmax is the interfacial shear strength of the adhesive materials.
External acting bending moment M can be written as P·e where P is the external acting normal force and e is the eccentricity of this external acting normal force from the beam-column cross section centroid.
At debonding, with no laminate prestressing, the external load reaches Pmax1 where Thus, the maximum failure load Pmax1 that can be supported by a beam-column with bonded FRP unprestressed laminate with no premature debonding failure may be given by When e < emin, the FRP laminate is will be subjected to compression, so Pmax1 has a value that tends to infinity; i.e. no need for strengthening using the FRP laminate.
For prestressing the FRP laminate, the maximum initial prestressing force in the laminate preventing the premature debonding failure (i.e. at zero values for P and M) is using Equation 10 as When the laminate is prestressed, the external load reaches Pmax2 at debonding of the FRP laminate where If the load increases beyond Pmax1 or Pmax2, premature debonding failure is encountered as the shear stress in the laminate will exceed its ultimate strength.
In Figure 2 and based on the previous equations for Pmax1 and Pmax2, three curves are schematically plotted. The first curve presents the beam-column eccentric axial load capacity Pu in case of no premature debonding failure; in other words, this beam-column fails by steel yielding or FRP laminate rupture; Pu will be defined for steel beam-columns with bonded prestressed and unprestressed FRP laminate in the next sections. The second curve presents the load causing premature debonding failure Pmax1 for steel beam-column with bonded unprestressed FRP laminate. The third curve presents the load causing premature debonding failure Pmax2 for steel beam-column with bonded prestressed FRP laminates, which is prestressed to the maximum value of the prestressing force. As shown in Figure 2, Pu curve is divided at the intersection points with Pmax1 and Pmax2 curves and with the vertical line presenting emin into four zone as follow: • Zone L1: no need for strengthening using FRP laminate.
• Zone L2: no premature failure is encountered and no need for prestressing the FRP laminate. • Zone L3: no premature failure is encountered if the laminate is prestressed below P0max. • Zone L4: premature failure will take place even if the FRP laminate is prestressed with P0max. P0max is the maximum initial prestressing force that can be applied to the FRP laminate without debonding from the steel beam-column.

THE NUMERICAL MODEL
Steel I-section beam-columns with bonded prestressed or unprestressed FRP are numerically simulated using nonlinear finite element models ( Figure  3): 8-node shell elements are adopted to model the flanges and web of the I-section as well as the FRP laminates, while 20-node solid elements are used to model the adhesive layer. After testing multiple models, fine mesh of 10 mm element size is used in modeling the FRP laminate, the adhesive, the lower flange of the steel beam and the lower part of the steel web, while a 25 mm element size is used in modeling of the rest of web and the upper flange of the steel I-section.
Full contact is assumed between the adhesive material and the steel flange or the FRP laminate. Thus, debonding failure is encountered when stresses in the interfacial material reaches its strength: strength of the interface between the adhesive and the steel or the FRP is almost equal to the adhesive material strength. The adopted load model is shown schematically in Figure 4. The model accuracy is validated by comparing its results to previously obtained experimental results (Denga andLee 2007, Chiew et al. 2011a,b). Full details of the model and its verification are given elsewhere (Abdelrazik 2013) with a summary of the verification analysis presented herein.
Three beams (Beams 1, 2 and 3) were tested in 4points flexural loading by Chiew et al. (2011a,b). These beams are simulated using the proposed numerical model via ANSYS software. In order to avoid web crippling rigid stiffener are assigned below the loading point. The concentrated load is distributed into five concentrated loads around this stiffener as shown in Figure 3. The stiffeners have a width equal to the flange width and a depth equal to its width; a thick stiffener is chosen in order to uniformly distribute the concentrated loads. In order to avoid the flange local failure the hinged support is assigned as a line support along the flange nodes, a stiffener is also assigned above this line support as shown in Figure 3. Beam 1 was fully modeled, then a quarter beam is modeled and the appropriate restrains are assigned along the planes symmetry: a difference of only 3.8% in the failure loads between the two models is recorded. Thus, Beam 2 and Beam 3 are modeled using a quarter-beam model. Table 1 summarizes the details of geometric and material properties of the modelled beam. The table also compares the failure load and failure mode recorded experimentally and obtained numerically. The model-predicted failure loads well agree with the experimental investigation's outcomes.

Parametric Study
The numerical model is now adopted to investigate the effects of five parameters on the ultimate capacity of beam-column with bonded FRP laminate are investigated. These parameters are the prestressing force P0, the axial load eccentricity e, the FRP laminate axial stiffness ELAL, the steel beam-column depth h, and the steel beam-column flexure stiffness EsIs.
Each parameter is changed separately in the numerical analysis while the rest of the parameters are kept at the constant values shown in Table 2. The investigated range for the said five parameters are as follow: • The prestressing force P0 is considered as 0.0, 0.3P0max, 0.5P0max, and 0.7P0max where P0max is the maximum prestressing force that can be applied to the FRP laminate without debonding from the steel beam-column. • The axial load eccentricity-to-depth ratio e/h is taken as 0.5, 1.0, 2.0, 3.0 and 5.0. • The FRP laminate axial stiffness ELAL was based on the number of laminate layers where 1, 2 and 3 layers are considered with laminate thickness of 1.4 mm, 2.8 mm and 4.2 mm. These values correspond to axial stiffness of 3.8×10 4 kN, 7.6×10 4 kN and 11.4×10 4 kN, respectively. • The steel beam-column depth h is varied between 200 mm and 400 mm keeping a constant flexural stiffness EsIs of 15×10 9 kN·mm 2 by adjusting the beam-column flange thickness. • The steel beam-column flexure stiffness EsIs is varied between 8.4×10 9 kN·mm 2 to 17.7×10 9 kN·mm 2 keeping a constant beamcolumn height of 300 mm and changing the flange thickness to reflect this variation. As shown in Table 3, a total of 126 models is needed to perform the parametric study covering the above mentioned five parameters. The beam models previously developed for the verification of the numerical analysis results is adjusted herein to the beam-column loading, which is an eccentric normal force, and the beam-column's boundary conditions. The eccentric normal force is divided into two concentrated loads (P1 and P2) one at each flange ( Figure  4) where  Full contact is assumed between the steel beamcolumn flange and the adhesive material and also between FRP and the adhesive material; i.e. no contact element is applied as the strength of the interface between the steel and the adhesive material and/or between the adhesive and the FRP laminate is almost equal to the strength of the adhesive itself.
The parametric study outcomes are presented in the form of graphs, and then the 126 models results are used to form an equation to estimate the load capacity of the beam-column as a function of these parameters considering only the steel yielding failure criteria; the capacity is shown as the Pu curve of Figure 2. Figure 5 shows the effect of changing the prestressing force on the ultimate load of the beam-column for different eccentricity-to-depth ratio, laminate stiffness, beam-column stiffness and beam-column depth. It is evident from Figure 5a that changing the prestressing force for the same e/h ratio has no significant effect on the value of the ultimate load, while changing the e/h for any value of the prestressing force has a significant effect on the beam-column's ultimate load. Figure 5b also reveals that changing the prestressing force has no significant effect on the beam-column's ultimate load for low values of the laminate stiffness, while it has a higher effect for high values of the laminate stiffness. Figure 5c indicates that changing the prestressing force for the same beam-column stiffness has no significant effect on the beam-column's ultimate load, while the change in the beam-column stiffness caused significant change in the beam-column ultimate load for any value of the prestressing force. Finally, Figure 5d shows that changing the prestressing force for the same beam-column depths has no significant effect in the changing the value of the ultimate load.

Results of the Numerical analysis
Thus, with the exception of very thick FRP laminate, the change in the level of laminate prestressing has no significant effect in changing the value of the beam-column's ultimate load. In other words, laminate prestressing level has no significant effect on the beam-column ultimate load when considering only the yield failure criteria of the beam-column. The effect of changing e/h on the beam-column ultimate load is shown in Figure 6 for different laminate stiffness, beam-column stiffness and beam-column depth. The figure reveals that laminate stiffness has a no effect on the beam-column ultimate load for e/h ratios less than 2; while it has a small effect on this load e/h ratios greater than 2. On the other hand, the figure shows that e/h ratio has a pronounced effect on the beam-column's ultimate load for any value for the laminate stiffness. The figure shows the logical effect of both the beam-column stiffness and depth on the ultimate beam-column load.  Figure 7 shows the effect of changing the laminates stiffness for different beam-column stiffness and beam-column depth on the ultimate beam-column's load. Figure 7a indicates that changing the laminate stiffness is more noticeable for beam-columns with small flanges (lower beam-column stiffness) compared to beam-columns with heavier flanges (larger beam-column stiffness). Figure 7b also indicates that the effect of laminate stiffness on the beam-column ultimate capacity is more noticeable for beam-columns with small depths compared to its effect on deeper beam-columns. It is evident from Figure 7 that laminates stiffness has no significant effect on the beam-column ultimate load when considering only the yield failure criteria of the beam-column.  Figure 8 shows the effect of the beam-column stiffness on the ultimate load. It indicates that variation of any of the depth of the beam-column or the stiffness of the beam-column have a significant effect on the beam-column's ultimate load.
where, e is the load eccentricity from the steel beamcolumn section centroid, h is the steel section's depth, Es is the modulus of elasticity of the steel, Is is the second moment of area of the beam-column cross section, As is the area and of the beam-column's cross section, AL is the cross sectional area of the FRP laminate, and EL is the modulus of elasticity of the FRP laminate material. Equation 17 has a 0.965 correlation coefficient with the numerically obtained ultimate loads. A sample of the comparisons between the ultimate loads numerically obtained using the finite element model and those calculated using Equation 17 is shown in Table  4 Recalling that Equations 12 and 15 results Pmax1 and Pmax2, which are the ultimate load for beam-columns with unprestressed or prestressed FRP laminates without debonding. Thus, the need for FRP laminate prestressing is defined by the relation between Pu, Pmax1 and Pmax2 (Figure 2) as discussed earlier.

FAILURE MODES OF FRP STRENGTHENED BEAM-COLUMN
The beam-column's ultimate load is calculated using Equations 12, 15 and 17 for different load eccentricities and adhesive properties. The load causing yielding failure criteria is not affected by the adhesive properties, while the load causing FRP laminate debonding is affected by both the load eccentricityto-depth ratio and the adhesive layer properties. The ultimate loads and moments for different eccentricities are determined for adhesive layer thicknesses of 1.0 mm, 0.5 mm and 0.25 mm with the parameters shown in Table 5 and plotted in Figures 9-11.   For adhesive layer thicknesses of 0.5 mm and 0.25 mm, Figures 10 and 11 show similar behaviour to that mentioned above for the 1.0 mm adhesive thickness except that the e/h limits defining the three zones are 1.35 and 1.95 for 0.5 mm adhesive layer thickness and 1.08 and 1.68 for 0.25 mm thickness. Figures 9 to 11 indicate that the adhesive thickness has minor effect on the ultimate load causing steel yielding Pu, while it has a significant effect on the ultimate load causing debonding for both cases of the prestressed and unprestressed FRP laminate; Pmax1 and Pmax2, respectively.
The three zones of behaviour defined earlier are also plotted in Figure 12 against the e/h ratio and the adhesive layer thickness ta. The two curves of Figure   12 define Zone 1 in which no FRP prestressing is required where failure will be dominated by steel yielding, Zone 2 in which adequate prestressing of the FRP laminate will serve to avoid FRP laminate debonding and thus failure can occur by steel yielding and Zone 3 in which failure will always occur by FRP laminate debonding even if the FRP laminate is prestressed.

CONCLUSIONS
The ultimate strength of a beam-column with bonded prestressed FRP laminate has been scrutinized. The investigation took two parallel paths: • First, a closed form solution was developed to estimate the actual interfacial shear and normal stresses in the adhesive layer. Using these stresses, an equation was analytically developed to estimate the ultimate load causing debonding of the FRP laminate from the steel beam-column. Figure 11. Ultimate load and moment for different e/h ratios and adhesive layer thickness of 0.25 mm. Figure 13. Relation between the eccentricity, the adhesive layer thickness and the three zones defining the failure mode of beamcolumns with bonded FRP laminate.
• Then, a numerical model was built-up in order to estimate the ultimate load causing steel yielding of the beam-column. The developed finite element model results were verified using previously published experimental data. Using the finite element model data and performing a regression analysis, another equation is numerically developed to estimate the ultimate load causing steel yielding of the beam-column. Based on the two equations, a methodology is proposed to determine ultimate capacity of beam-columns strengthened with FRP prestressed laminate where the minimum load resulting from the two equations control the ultimate capacity of the FRP strengthened beam-column. This methodology also defines the failure mode and the need for FRP prestressing.
The analyses shows that strengthening small eccentricity-to-depth ratio beam-columns using FRP laminate has minor effect in raising their ultimate load; this technique is significant in delaying the laminate debonding for larger eccentricity-to-depth ratio and, thus, increasing the beam-column's capacity. Furthermore, beam-columns with small eccentricity-todepth ratio reach steel yielding before experiencing debonding of the FRP laminate. On the other hand, failure of beam-columns with very large eccentricityto-depth ratio is always controlled by premature debonding failure of the FRP laminate even when they are prestressed.
The investigation also confirms that both the FRP laminate prestressing and the mechanical properties of the adhesive layer does not affect the ultimate load causing yielding but it has a significant effect on delaying the premature debonding of the laminate from the steel section.