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\title{Behavior of I-Section GFRP Beam Including Retrofitting for Damage Effects}
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             \author[1]{Mamadou  Konate}

             \author[2]{Mamadou  Konate}

             \author[3]{Zia  Razzaq}

             \affil[1]{  Old Dominion University}

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\date{\small \em Received: 12 December 2013 Accepted: 2 January 2014 Published: 15 January 2014}

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\begin{abstract}
        


This paper presents the outcome of a study of an I-section Glass Fiber Reinforced Polymer (GFRP) beam including retrofitting for damage effects. A total of three beam tests were conducted in the following sequence: GFRP beam with no retrofitting and a single mid-span web brace; the partially damaged (cracked) beam with GFRP plates used for retrofitting; and the retrofitted beam re-tested with the lateral brace at the top flange level. Both cracking and lateral-torsional buckling behavior is studied and experimental load-deflection relationships recorded. Using the mechanical properties of GFRP based on the experimental data, theoretical predictions are made for the buckling load values. The results show that retrofitted damaged beam provided about half of the original strength of the undamaged beam. The study also shows that the mid-span brace played a significant role in beam behavior and strength.

\end{abstract}


\keywords{I-section GFRP, retrofitting, lateral bracing.}

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\let\tabcellsep& 	 	 		 
\section[{Introduction}]{Introduction}\par
lass fiber reinforced polymer (GFRP) composites are increasingly been used for civil and mechanical structures. Under real-life use, situations can arise where a damaged GFRP structure needs to be retrofitted to restore all or a significant portion of its original strength. The damage could be a result of accidental overloading, misuse, or environmental conditions. A number of papers have previously been published about reinforced concrete structures retrofitted with GFRP composites \hyperref[b0]{[1,}\hyperref[b3]{4,}\hyperref[b4]{5,}\hyperref[b10]{11,}\hyperref[b13]{14]}. This paper presents the outcome of a study of retrofitting a GFRP beam with GFRP plates. 
\section[{a) Problem Statement}]{a) Problem Statement}\par
This investigation details an experimental and theoretical study of bending and lateral-torsional buckling of an I-section GFRP beam first loaded to its maximum load capacity, and then retrofitted with GFRP and re-tested. The beam has shear-type boundary conditions and mid-span lateral bracing. In each case, the beam is subjected to a gradually increasing midspan load P until it reaches its maximum load-carrying capacity. The small moment resistance of the shear type steel end connections is considered to be negligible. The main objectives of this paper are to both experimentally and analytically investigate the cracking loads for a GFRP beam with and without GFRP retrofitting including lateral torsional buckling effects. Figure  {\ref 1} shows the schematic of the GFRP beam of length L studied herein. 
\section[{Fig. 1 : Schematic of GFRP beam}]{Fig. 1 : Schematic of GFRP beam}\par
A three-fold problem has been studied in the present paper. First, the behavior of a GFRP beam with no retrofitting and a single mid-span web brace is studied. Next, the partially damaged beam with midspan web brace is retrofitted with GFRP plates and its behavior observed. Lastly, the GFRP beam is re-tested, however with the mid-span brace provided at the top flange level. A comparison of the experimental peak loads is also made to those obtained with approximate analysis. Figure \hyperref[fig_0]{2} shows the cross section with two alternative mid-span brace locations. Often one or more lateral braces are provided in order to increase the loadcarrying capacity of a GFRP beam. The ultimate load is also influenced by whether a lateral brace is provided on the web or top flange. A comparison of the experimental peak loads is also made to those obtained with approximate analysis.  
\section[{Experimental Investigation}]{Experimental Investigation}\par
Three experiments are conducted using a GFRP beam with a clear length of 93 inches. The damaged GFRP is retrofitted with GFRP in the last two experiments. The load-deflection curves and the peak loads are recorded. Figure  {\ref 3} shows the experimental test setup. In this figure, a dial gage (DG4) is also shown which is used to record the mid-span vertical deflection. A total of seven dial gages were mounted to record both vertical and lateral deflections. A hydraulic jack of 50-kip capacity with load cell and a loading device are also shown in Figure  {\ref 3}. 
\section[{Fig. 3 : Experimental test setup}]{Fig. 3 : Experimental test setup}\par
The hydraulic jack is controlled by the system console. This arrangement gradually transmits load from the hydraulic jack to the GFRP beam. The test procedure involved applying the load, P, in small increments and recording the resulting deflections. The loading process is continued until the member's loadcarrying capacity is reached. 
\section[{a) Beam with Mid-span Lateral Brace on Web}]{a) Beam with Mid-span Lateral Brace on Web}\par
The mid-span web brace is provided on both sides of the web at 0.81 in below the bottom surface of the top flange. When approaching failure, the GFRP beam first buckled and then cracked. Figure  {\ref 4} shows the view showing the top flange cracks and length of the GFRP beam. The buckling mode observed in the horizontal plane was S-shaped. The beam developed lateral-torsional buckling at a load of 8,426 lbs, and subsequently cracked at a load of 8,542 lbs. The beam exhibited elastic behavior up to the attainment of buckling load. Figure  {\ref 5} shows the beam load-deflection curves of the GFRP beam for the lateral deflection (DG2) and the vertical deflection (DG3) both at the beam quarter length from the left support, and for the midspan deflection (DG4). The GFRP damaged beam from Experiment 1 is first retrofitted with CFRP plates on both sides of the web and top flange and then re-tested. Figure  {\ref 6} shows the GFRP plates used to retrofit the mid-span top flange and two sides of the web, respectively. The plates were 0.5-inch thick and mounted to the beams using 0.875inch diameter steel bolts. Fig.  {\ref 6} : GFRP retrofitting plates used at mid-span Figure \hyperref[fig_3]{7} shows a part of the retrofitted beam for Experiment 2. For Experiment 2, the arrangement for the dial gages, mid-span web brace Location 1 indicated in Figure \hyperref[fig_0]{2}, the applied load location, and the beam end connection remain the same as for Experiment 1. The resulting load-deflection curves for Experiment 2 are shown in Figure  {\ref 8} showing a buckling load of 3,910 lbs. The damaged GFRP beam with GFRP retrofitting tested in Experiment 2 is tested again in Experiment 3 in which the mid-span braces are located at the top flange indicated as Location 2 in Figure \hyperref[fig_0]{2}. In this experiment, the beam buckled at a load of 4,372 lbs. The load in Experiment 3 is approximately 12 percent greater than that found in Experiment 2 indicating a greater effectiveness of the brace at the top flange as compared with the one on the web. Figure  {\ref 9} shows the load-deflection relations obtained for Experiment 3.  
\section[{Analysis and Comparison of Results}]{Analysis and Comparison of Results}\par
The following deflection equation from Reference 13 is used to calculate the longitudinal modulus of elasticity, ?? 11 :Î?" = ???? 3 48?? 11 ?? (1)\par
In this equation, P and Î?" are obtained from the load-deflection curves for each experiment in the linear range. These values are also used to calculate the relative stiffness values, K. The value of the shear modulus is estimated using the following ratio \hyperref[b7]{[8]}:?? 12 ?? 11 = 1 8\par
(2) Table \hyperref[tab_0]{1} presents the elastic limit load and deflection for each experiment, the relative stiffness values, the calculated modulus of elasticity, and the shown in this table reveal that GFRP beam with midspan lateral brace on web appeared to be much stiffer with K = 11,354 lbs/in. compared to both retrofitted damaged GFRP beam with mid-span web brace (K = 7,915 lbs/in.) and re-tested retrofitted damaged GFRP beam with flange brace (K = 8,922 lbs/in.). The K values also indicate that the retrofitted damaged GFRP beam with mid-span top web brace has a smaller stiffness than that of re-tested retrofitted damaged GFRP beam with mid-span flange brace. which is found to be 0.6 using the GFRP material properties given in Reference 8. The beam lateral-torsional buckling moment, M cr , is calculated using the following equation \hyperref[b7]{[8]}:?? ???? = ?? ?? ?? ???? ?? ?( ???? ?? ?? ?? ?? ) 2 ?? ?? ?? ?? + ?? ?? ?? ?? ?? 12 ??\textbf{(3)}\par
In this expression, C w = warping constant (in 6 ); J = torsional constant (in 4 ); C b = moment gradient multiplier; L b = unbraced length (in); k = effective length coefficient; and I y = moment of inertia about the minor axis. Table \hyperref[tab_1]{2} presents the predicted lower, upper bound, and interpolated approximated buckling loads designated as P L , P U , P IB, and P IU respectively, for the three cases both with and without mid-span web brace. The lower bounds loads were found by neglecting the GFRP retrofitting plates in the cross-section properties calculations.\par
The upper bound loads were calculated as if the retrofitting plates existed for the entire length of the beam. Also, for the theoretical buckling load calculations corresponding to Experiments 2 and 3, it was assumed that the beam is un-cracked.\par
Presented in Table \hyperref[tab_1]{2} are also the interpolated approximate theoretical buckling loads P IB and P IU calculated by using the upper and lower bound estimates for the buckling loads. The interpolation is done by using a weighted average involving the retrofitted and non-retrofitted portions of the beam length, namely, 15.5 in., and 77.5 in., respectively. For example, P IB for the beam in Experiment 2 is calculated as follows: P IB = [15.5(4995) + 77.5(2670)]/93.0 = 3058 lbs The beam in Experiment 1 was not retrofitted, however, the upper bound and interpolated buckling loads are still included in Table \hyperref[tab_1]{2} to determine the theoretical effect of retrofitting. The results in Table \hyperref[tab_1]{2} also clearly show that adding a brace at the beam midspan results in a dramatic increase in the buckling load capacity.\par
Table \hyperref[tab_2]{3} summarizes a comparison between theoretical estimates (P t ) for the buckling loads and those determined experimentally (P e ). Since no retrofitting was used for Experiment 1, the P L value from Table \hyperref[tab_1]{2} is taken as its P t value in Table \hyperref[tab_2]{3}. The P IB values corresponding to Experiments 2 and 3 from Table \hyperref[tab_1]{2} are taken as their respective P t values in Table \hyperref[tab_2]{3}. Both theoretical and experimental investigation revealed a reasonable agreement between the theoretically estimated and experimental buckling load values for Experiment 1. However, for Experiments 2 and 3, there was a difference of about 20 percent between the predicted loads and the experimental ones. This may be attributable to the complex nature of the retrofitted beam behavior with pre-existing cracking that Experiment 1 created. All of the predicted load values, however, are found to be on the conservative side.  
\section[{Conclusions}]{Conclusions}\par
A number of conclusions are drawn based on the results presented in this paper. The damaged or cracked GFRP beam retrofitted with GFRP plates carried nearly 46 percent of the load capacity of the originally undamaged GFRP beam without retrofitting but with the same mid-span web brace location. The re-tested and retrofitted GFRP beam with a mid-span brace at the top flange carried nearly 52 percent of the load carried by the originally undamaged GFRP beam. The mid-span lateral bracing played a significant role in the beam behavior and strength. Placing a lateral mid-span brace at the compression flange location results in a higher buckling capacity compared to that obtained using web bracing. Lastly, the results show that the use of lateral bracing dramatically increases the buckling capacity of the beam in comparison with that without the bracing.\begin{figure}[htbp]
\noindent\textbf{2}\includegraphics[]{image-2.png}
\caption{\label{fig_0}Fig. 2 :}\end{figure}
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\noindent\textbf{415}\includegraphics[]{image-3.png}
\caption{\label{fig_1}Fig. 4 : 1 Fig. 5 :}\end{figure}
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\noindent\textbf{7}\includegraphics[]{image-4.png}
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\noindent\textbf{829}\includegraphics[]{image-5.png}
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\noindent\textbf{}\includegraphics[]{image-6.png}
\caption{\label{fig_5}}\end{figure}
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\noindent\textbf{1} \par 
\begin{longtable}{P{0.07836879432624112\textwidth}P{0.18687943262411347\textwidth}P{0.042198581560283686\textwidth}P{0.22907801418439716\textwidth}P{0.15673758865248225\textwidth}P{0.15673758865248225\textwidth}}
Experiment\tabcellsep P lbs\tabcellsep Î?" in.\tabcellsep K = lb/in ?? ??\tabcellsep E 11 x10 6 psi\tabcellsep G 12 x10 6 psi\\
1\tabcellsep \multicolumn{2}{l}{7948 0.7}\tabcellsep 11354\tabcellsep 2.33\tabcellsep 0.29\\
2\tabcellsep \multicolumn{2}{l}{3641 0.46}\tabcellsep 7915\tabcellsep 1.88\tabcellsep 0.23\\
3\tabcellsep \multicolumn{2}{l}{4372 0.49}\tabcellsep 8922\tabcellsep 1.94\tabcellsep 0.24\\
\tabcellsep \tabcellsep \tabcellsep ?? ??\tabcellsep \tabcellsep \\
\tabcellsep \tabcellsep \tabcellsep ?? ??\tabcellsep \tabcellsep \end{longtable} \par
  {\small\itshape [Note: If ?? 11 = ?? ?? , the modified ?? ?? can be calculated based on an averaged ratio ? =]} 
\caption{\label{tab_0}Table 1 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{2} \par 
\begin{longtable}{P{0.0255\textwidth}P{0.272\textwidth}P{0.12324999999999998\textwidth}P{0.102\textwidth}P{0.187\textwidth}P{0.068\textwidth}P{0.07225000000000001\textwidth}}
\tabcellsep \multicolumn{4}{l}{Interpolated Buckling Loads}\tabcellsep \tabcellsep \\
\tabcellsep \multicolumn{3}{l}{Buckling Load (lbs)}\tabcellsep \multicolumn{3}{l}{Buckling Load (lbs)}\\
\tabcellsep \tabcellsep with brace\tabcellsep \tabcellsep \multicolumn{3}{l}{without brace}\\
Exp\tabcellsep P L\tabcellsep P U\tabcellsep P IB\tabcellsep P L\tabcellsep P U\tabcellsep P IU\\
\tabcellsep lbs\tabcellsep lbs\tabcellsep lbs\tabcellsep lbs\tabcellsep lbs\tabcellsep lbs\\
1\tabcellsep 8148\tabcellsep 12702\tabcellsep \multicolumn{2}{l}{8907 1040}\tabcellsep 1545\tabcellsep 1124\\
2\tabcellsep 2670\tabcellsep 4995\tabcellsep 3058\tabcellsep 409\tabcellsep 608\tabcellsep 442\\
3\tabcellsep 3330\tabcellsep 5191\tabcellsep 3640\tabcellsep 425\tabcellsep 631\tabcellsep 459\end{longtable} \par
 
\caption{\label{tab_1}Table 2 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{3} \par 
\begin{longtable}{P{0.18133333333333335\textwidth}P{0.204\textwidth}P{0.204\textwidth}P{0.26066666666666666\textwidth}}
Experiment\tabcellsep P t\tabcellsep P e\tabcellsep P t /P e\\
\tabcellsep lbs\tabcellsep lbs\tabcellsep lbs\\
1\tabcellsep 8148\tabcellsep 8426\tabcellsep 0.97\\
2\tabcellsep 3058\tabcellsep 3910\tabcellsep 0.78\\
3\tabcellsep 3640\tabcellsep 4372\tabcellsep 0.83\\
IV.\tabcellsep \tabcellsep \tabcellsep \end{longtable} \par
 
\caption{\label{tab_2}Table 3 :}\end{figure}
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