# Failure Analysis of Locally Damaged Slender Steel Bars Strengthened with CFRP Composites: Experiments, Theory, and Computational Simulations

Authors: Bin Li; Hua Luo; Xianqiao Wang
Journal: Advances in Civil Engineering
 (2020)

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## Abstract

Carbon fiber-reinforced polymer/plastic (CFRP) composites bear attractive performance in resistance to tension, fatigue, and corrosion and, thus, have been recognized as a promising candidate for repairing and strengthening steel structures in engineering. Here, we combine experiments, theory, and numerical simulations to elucidate how the location and degree of local damages, as well as the reinforcement mode, affect the stability of slender steel bars repaired by CFRP. The deformation, failure mode, and the critical buckling load of the reinforced steel flat bars subjected to axial compressive forces are experimentally evaluated. We show that all tested specimens exhibit buckling failure, before which the damaged steel bars have entered an elastic-plastic stage. Our theoretical analysis provides an upper bound for the critical force, which is sensitive not only to the damage degree but also to the damage location. Damage locating at the middle regime of the specimens will remarkably reduce stability of the steel bars, but an optimized combination of wrapping method and number of CFRP layers can restore and even enhance the stability of the damaged structures beyond the undamaged counterparts. Finite element simulations are implemented in the same scenario as experiments, showing good agreement with our measurements. Our findings suggest that, to improve the stability of the damaged steel bars reinforced by CFRP, the load carrying capacity of the the bars, the number of CFRP layers, and the construction convenience should be taken into account.


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## Body

## 1. Introduction

Carbon fiber-reinforced polymer/plastic (CFRP) composites have attracted much attention from research and engineering communities due to their unique advantages including noncorrosive characteristics, excellent resistance to fatigue, high stiffness and strength-to-weight ratios, ease and rapidity of erection, and reduced long-term maintenance expenses [1]. With these predominant features, CFRP composites have been increasingly employed to strengthen and repair steel structures [2–4]. Furthermore, CFRP composites can be adhesively bonded to steel structures, instead of being mechanically connected to the latter through bolt holes or welds, which may significantly engender stress concentration. These advantages can put the unique material properties of each component of steel-CFRP systems into full play and, thereby, make the use of CFRP an attractive candidate for rehabilitating and retrofitting steel structures [3, 5]. For example, it was reported that, repaired with the particular CFRP plates, the elastic flexural stiffness of damaged beams can be restored up to 50% and the strength of damaged beams can be fully restored to their original, undamaged state [6].Recently, CFRP composites have been leveraged to enhance the stability of steel structures such as bars, columns, beams, plates, and tubes [2, 3, 7, 8]. For instance, it has been shown that the effect of CFRP on the critical load around the weak shaft is remarkable and is more significant than that of the steel plate with the same thickness [9]. Shaat and Fam [2] performed an experiment to examine the buckling behavior of slender hollow structural section square columns strengthened using CFRP plates with high elastic moduli. They showed that the effectiveness of the CFRP system in increasing the axial strength of the columns can be substantially enhanced by increasing the slenderness ratios. Ding [10] carried out a full-scale axial compression test on round and square steel tube specimens reinforced by CFRP, respectively. Ding’s results demonstrated that the ultimate bearing capacity of specimens depended on the direction of CFRP sticking fibers. These studies showed that the high stiffness of CFRP composites is utilized to provide bracing that resists the buckling and postbuckling of steel structures subjected to axial compression [11].Most previous studies focused on the stability of undamaged steel structures reinforced CFRP composites. However, it has been recognized that the instabilities of structures are sensitive to imperfections, especially when the structures are thin [12–14]. Both the critical loads and buckling mode can be modulated by structures’ imperfections such as local damage [15, 16], mechanical heterogeneities [17, 18], and geometrical nonuniformity [19, 20]. For steel structures, local damage usually emerges due to environmental cues and unanticipated human factors; this is also one of the main demands on CFRP composites to reinforce these damaged structures. Therefore, it is of significance to examine how local damage affects the stability of steel structures strengthened by CFRP. There are a limited number of studies related to the buckling of steel-CFRP structures available in the literature. Recently, the elastic buckling behaviors of locally damaged steel tubes and bars with fiber reinforced polymer reinforcement have been theoretically investigated under axial forces [21, 22]. Nevertheless, it remains elusive how the damage degree and position, as well as reinforcement method, jointly influence the buckling and postbuckling behaviors of steel-CFRP structures.In this paper, we use a combination of experiments and numerical simulations to address the instability of damaged slender steel bars repaired by CFRP. The failure form, deformation geometry, critical load, and postbuckling state of the steel-CFRP structure are studied. We show that the position and degree of local damage, the CFRP reinforcement method, and the CFRP paste thickness can profoundly affect the critical load and postbuckling configuration. Our experimental results are consistent with finite element simulations implemented by commercial software Abaqus.


## 2. Experiments

### 2.1. Design of Damaged Steel Columns

Q235 rectangular section steel bars were selected for test. Before the reinforcement by using CFRP, two opposite sides of the steel bars were grinded to produce imperfections, through which we mimic the local damage of steel structures induced by corrosion and so on. In the present study, we used a grinding machine that has wheel metal grinding plates with grain fineness 240# to locally grind the steel bars. Three kinds of grinding depth (td) were produced: td = 0.5, 0.6, and 0.7 mm, which is much smaller than the bar thickness of 12 mm, as shown in Tables 1–3. A total of 24 steel bars were prepared. We divided the 24 steel bars according to the slenderness ratios. In Groups 1–3, the slenderness ratio was taken as 115.47, 144.34, and 173.21, respectively. In Group 1, we considered a single damage region with the same length but different locations; in Group 2, we considered two damage regions with the same length; and, in Group 3, two damage regions with the different length were introduced in the specimens. The specimen number, damage area, and CFRP reinforcement are shown in Tables 1–3. For reinforcement method, we considered three typical cases: (i) unreinforced, (ii) reinforced in damaged area only, and (iii) reinforced in the whole bar. In addition, different paste thicknesses (up to 3 layers) were used for bonding the CFRP plates.Table 1
Damage and reinforcement of steel specimens in Group 1.
Specimen #Geometry and damage area (cross section size 14 mm × 12 mm/slenderness ratio 115.47)Grinding depthtd (mm)Reinforcement areaPaste thicknesstn (layer)JW000-11Lossless steel bar0Unreinforced0JW152-120Whole bar2JS000-13Unit: mm0.5Unreinforced0JS142-140.5Damaged area2JS152-150.5Whole bar2JS000-16Unit: mm0.5Unreinforced0JS142-170.5Damaged area2JS152-180.5Whole bar2Table 2
Damage and reinforcement of steel specimens in Group 2.
Specimen #Geometry and damage area (cross section size 14 mm × 12 mm/slenderness ratio 144.34)Grinding depthtd (mm)Reinforcement areaPaste thicknesstn (layer)JW000-21Lossless steel bar0Unreinforced0JS000-22Unit: mm0.5Unreinforced0JS142-230.5Damaged area2JS142-240.6Damaged area2JS142-250.7Damaged area2JS152-260.5Whole bar2JS152-270.6Whole bar2JS152-280.7Whole bar2Table 3
Damage and reinforcement of steel specimens in Group 3.
Specimen #Geometry and damage area (cross section size 16  mm × 12 mm/slenderness ratio 173.21)Grinding depthtd (mm)Reinforcement areaPaste thicknesstn (layer)JW000-31Lossless steel bar0Unreinforced0JS000-32Unit: mm0.5Unreinforced0JS141-330.5Damaged area1JS142-340.5Damaged area2JS143-350.5Damaged area3JS151-360.5Whole bar1JS152-370.5Whole bar2JS153-380.5Whole bar3Note. “JW” and “JS” denote undamaged and damaged specimens, respectively. From left to right, the first digits “0” and “1” represent no reinforcement and reinforcement, respectively. Second digits “4” and “5,” respectively, indicate reinforcement only in damage area and reinforcement throughout full bar. The third digit represents the number of CFRP layers.


### 2.2. Material Parameters

We measured the mechanical properties of the steel bars by standard unidirectional tensile tests and the values were averaged over 3 sets of independent experiments, as shown in Table4. The mechanical properties of CFRP are given in Table 5. ZP-500 carbon fiber impregnated adhesive (Grade A) was adopted for the binder.Table 4
Mechanical properties of steel bars.
Yield strengthσy (MPa)Ultimate strengthσbg (MPa)Modulus of elasticityE (GPa)287.2437.0206.0Table 5
Mechanical properties of CFRP.
Wideb (mm)Densityρ(g/m2)Single layer thicknesstp (mm)Tensile strengthσbc (MPa)Modulus of elasticityE (GPa)Elongationδ (%)2003000.1673281235.01.7


### 2.3. Specimen Preparation

The preparation of the specimens included four steps. Firstly, a grinding machine with a particle size of 240# hundred-wheel metal polishing block was used to fine-polish the prescribed damage area of the steel bars (Figure1(a)). The grinding precision was carefully guaranteed by a Vernier caliper during the whole polishing process. Secondly, two strain gauges, denoted by G1 and G2, were pasted symmetrically at the midpoint of each specimen (Figure 1(b)). The insulation treatment of the terminals of the two strain gauges was performed. Thirdly, the impregnated glue was coated in the reinforcement area of the specimens and, then, CFRP plates were pasted on the adhesive surface induced by glue (Figure 1(c)). In the situation where only the damaged areas were reinforced, the length of CFRP plates was set as 110% of the length of the damage zone to avoid stress concentration at the junction between the damaged and the undamaged regions. Finally, two strain gauges, denoted by B1 and B2, were vertically and symmetrically pasted on the CFRP surface of the specimens to measure the longitudinal strain of CFRP in situ (Figure 1(d)).Figure 1
Experiment setup. (a) Grinding the damage area, (b) pastingG1 and G2 strain gauges, (c) pasting CFRP, (d) pasting B1 and B2 strain gauges, and (e) loading method.
(a)(b)(c)(d)(e)


### 2.4. Test Method

The vertical loading device is shown in Figure1(e). Two vertical displacement sensors and two horizontal displacement sensors were arranged symmetrically on both sides of the specimen to monitor the axial and lateral displacement, respectively. The vertical loading force, axial displacement, lateral displacement, and strains of the steel bars and CFRP were collected by Donghua DH3816 Net static strain tester. The position of the strain gauges was the same for all the specimens. We used the multistage, force-controlled loading mode to compress the specimens. The incremental force at each loading stage was set as 2%–10% of the predicted limit load and maintained about 1–3 min to collect data.The specimens were strictly levelled and aligned by preloading. The preload was estimated to be 10% of the ultimate load. A slow and continuous loading method was adopted to track the whole buckling process. In our experimental tests, we added teflon spacers at the connection between the specimens and the machine to reduce friction. In such a way, the specimens undergo weak rotation constraints at the two ends and the confinements can be approximately treated as hinged constraints.


## 2.1. Design of Damaged Steel Columns

Q235 rectangular section steel bars were selected for test. Before the reinforcement by using CFRP, two opposite sides of the steel bars were grinded to produce imperfections, through which we mimic the local damage of steel structures induced by corrosion and so on. In the present study, we used a grinding machine that has wheel metal grinding plates with grain fineness 240# to locally grind the steel bars. Three kinds of grinding depth (td) were produced: td = 0.5, 0.6, and 0.7 mm, which is much smaller than the bar thickness of 12 mm, as shown in Tables 1–3. A total of 24 steel bars were prepared. We divided the 24 steel bars according to the slenderness ratios. In Groups 1–3, the slenderness ratio was taken as 115.47, 144.34, and 173.21, respectively. In Group 1, we considered a single damage region with the same length but different locations; in Group 2, we considered two damage regions with the same length; and, in Group 3, two damage regions with the different length were introduced in the specimens. The specimen number, damage area, and CFRP reinforcement are shown in Tables 1–3. For reinforcement method, we considered three typical cases: (i) unreinforced, (ii) reinforced in damaged area only, and (iii) reinforced in the whole bar. In addition, different paste thicknesses (up to 3 layers) were used for bonding the CFRP plates.Table 1
Damage and reinforcement of steel specimens in Group 1.
Specimen #Geometry and damage area (cross section size 14 mm × 12 mm/slenderness ratio 115.47)Grinding depthtd (mm)Reinforcement areaPaste thicknesstn (layer)JW000-11Lossless steel bar0Unreinforced0JW152-120Whole bar2JS000-13Unit: mm0.5Unreinforced0JS142-140.5Damaged area2JS152-150.5Whole bar2JS000-16Unit: mm0.5Unreinforced0JS142-170.5Damaged area2JS152-180.5Whole bar2Table 2
Damage and reinforcement of steel specimens in Group 2.
Specimen #Geometry and damage area (cross section size 14 mm × 12 mm/slenderness ratio 144.34)Grinding depthtd (mm)Reinforcement areaPaste thicknesstn (layer)JW000-21Lossless steel bar0Unreinforced0JS000-22Unit: mm0.5Unreinforced0JS142-230.5Damaged area2JS142-240.6Damaged area2JS142-250.7Damaged area2JS152-260.5Whole bar2JS152-270.6Whole bar2JS152-280.7Whole bar2Table 3
Damage and reinforcement of steel specimens in Group 3.
Specimen #Geometry and damage area (cross section size 16  mm × 12 mm/slenderness ratio 173.21)Grinding depthtd (mm)Reinforcement areaPaste thicknesstn (layer)JW000-31Lossless steel bar0Unreinforced0JS000-32Unit: mm0.5Unreinforced0JS141-330.5Damaged area1JS142-340.5Damaged area2JS143-350.5Damaged area3JS151-360.5Whole bar1JS152-370.5Whole bar2JS153-380.5Whole bar3Note. “JW” and “JS” denote undamaged and damaged specimens, respectively. From left to right, the first digits “0” and “1” represent no reinforcement and reinforcement, respectively. Second digits “4” and “5,” respectively, indicate reinforcement only in damage area and reinforcement throughout full bar. The third digit represents the number of CFRP layers.


## 2.2. Material Parameters

We measured the mechanical properties of the steel bars by standard unidirectional tensile tests and the values were averaged over 3 sets of independent experiments, as shown in Table4. The mechanical properties of CFRP are given in Table 5. ZP-500 carbon fiber impregnated adhesive (Grade A) was adopted for the binder.Table 4
Mechanical properties of steel bars.
Yield strengthσy (MPa)Ultimate strengthσbg (MPa)Modulus of elasticityE (GPa)287.2437.0206.0Table 5
Mechanical properties of CFRP.
Wideb (mm)Densityρ(g/m2)Single layer thicknesstp (mm)Tensile strengthσbc (MPa)Modulus of elasticityE (GPa)Elongationδ (%)2003000.1673281235.01.7


## 2.3. Specimen Preparation

The preparation of the specimens included four steps. Firstly, a grinding machine with a particle size of 240# hundred-wheel metal polishing block was used to fine-polish the prescribed damage area of the steel bars (Figure1(a)). The grinding precision was carefully guaranteed by a Vernier caliper during the whole polishing process. Secondly, two strain gauges, denoted by G1 and G2, were pasted symmetrically at the midpoint of each specimen (Figure 1(b)). The insulation treatment of the terminals of the two strain gauges was performed. Thirdly, the impregnated glue was coated in the reinforcement area of the specimens and, then, CFRP plates were pasted on the adhesive surface induced by glue (Figure 1(c)). In the situation where only the damaged areas were reinforced, the length of CFRP plates was set as 110% of the length of the damage zone to avoid stress concentration at the junction between the damaged and the undamaged regions. Finally, two strain gauges, denoted by B1 and B2, were vertically and symmetrically pasted on the CFRP surface of the specimens to measure the longitudinal strain of CFRP in situ (Figure 1(d)).Figure 1
Experiment setup. (a) Grinding the damage area, (b) pastingG1 and G2 strain gauges, (c) pasting CFRP, (d) pasting B1 and B2 strain gauges, and (e) loading method.
(a)(b)(c)(d)(e)


## 2.4. Test Method

The vertical loading device is shown in Figure1(e). Two vertical displacement sensors and two horizontal displacement sensors were arranged symmetrically on both sides of the specimen to monitor the axial and lateral displacement, respectively. The vertical loading force, axial displacement, lateral displacement, and strains of the steel bars and CFRP were collected by Donghua DH3816 Net static strain tester. The position of the strain gauges was the same for all the specimens. We used the multistage, force-controlled loading mode to compress the specimens. The incremental force at each loading stage was set as 2%–10% of the predicted limit load and maintained about 1–3 min to collect data.The specimens were strictly levelled and aligned by preloading. The preload was estimated to be 10% of the ultimate load. A slow and continuous loading method was adopted to track the whole buckling process. In our experimental tests, we added teflon spacers at the connection between the specimens and the machine to reduce friction. In such a way, the specimens undergo weak rotation constraints at the two ends and the confinements can be approximately treated as hinged constraints.


## 3. Theoretical Model

We use the Bernoulli–Euler beam theory to characterize the mechanical response of both the undamaged and damaged bars. To remain consistent with experiments, we took hinged constraints at the two ends of the steel bars. The differential equation for bending of bars can be expressed as [23](1)EiIid2widx2+Fwi=0,where x is the distance along the bar from the left end and subscript i denotes the region with or without damage; wi is deflection in each region, Ei denotes Young’s modulus, and Ii is the moment of inertia in different areas. F represents the axial force applied at the end. For reinforced regions, Ei stands for the effective modulus accounting for both steel bars and CFRP. For the present theoretical model, we need only to calculate the effective bending stiffness of the beam (EiIi). In the damaged regions reinforced by CFRP, we can use the transformed section method of composite beams, as shown in Figure 2, to evaluate the effective bending stiffness (EiIicomp), which gives(2)EiIicomp=B12Esh3+2Ecδ3+6EcH+δ2δ,where Es and Ec denote Young’s moduli of steel and CFRP, respectively; B is the width of the cross section of the steel bars; h is the height of the steel part in the damaged region; and δ is the total thickness of the CFRP.Figure 2
Schematic for calculating the effective stiffness of the composite section using the transformed section method, which transforms the beam into the one made of a single material of steel.B1 denotes the width of the CFRP-reinforced zone after transformation.The general solution of equation (1) is wi=Aisinkix+Bicoskix, where ki=F/EiIi, and Ai and Bi are coefficients to be determined. By considering the boundary conditions at the two ends and the continuity conditions at the interface between damaged and undamaged regions, we can solve the critical force for buckling onset. Take Group 1 for example, where there is only one damaged region. The bar is perfect in the regions 0<x<l1 and l2<x<l and is damaged in the region l1<x<l2, with l being the total length of the bar. Therefore, we have w10=w3l=0 at the two ends. The continuity conditions read w1l1=w2l1, w1′l1=w2′l1, w2l2=w3l2, and w2′l2=w3′l2. Solving this eigenvalue problem, we obtain the transcendental equation(3)k1cosk1l1sink2l0−k2sink1l1cosk2l0k2sink1l1sink2l0−k1cosk1l1cosk2l0=k2k3tank3l−l2,where l0=l1−l2. Equation (3) has to be numerically solved to calculate the critical force. For an undamaged bar (k1=k2=k=F/EI), equation (3) reduces to the traditional solution sinkl=0, which gives the critical force(4)Fcr=π2EIl2.The analytical method provided above is complicated when the bars have many damaged regions. In this situation, we can alternatively calculate the critical force by using energy method. We takew=Asinkx to approximate the deflection curve of the whole bar under hinged constraints at the two ends. The strain energy arising from bending (U) in the bar can be written as(5)U=∑i=1,2,3...∫M22EiIidx,where M=Fw. The work (T) done by the force F is(6)T=F2∫0ldwdx2dx.The energy balance gives the critical force for buckling onset [23]. For specimens in Group 1, we obtain(7)Fcr=2π3E1E2I1I22πlE2I2l+l0−E1I1l0+l2E1I1−E2I2sin2πl1/l−sin2πl2/l.This energy method can be easily extended to calculate the critical force for specimens in Groups 2 and 3.


## 4. Results

### 4.1. Buckling and Postbuckling Configuration

Our experiments showed that, at the initial stage of loading, the steel bars reduced their length and maintained a straight configuration. However, when the vertical forces were increased up to a threshold magnitude, the specimens lost stability and gave rise to buckling.Further loading beyond the threshold increased the lateral displacement of the bars, leading to a bent, postbuckling configuration [23]. During this process, fine sound of CFRP peeling or breaking was heard in individual specimens. After the test was completed, the loading force was removed. The bent postbuckling deformation was recovered partly, while a bent configuration was still retained, as shown in Figure 3. It suggests that, under the experimental loading, the specimens entered the elastic-plastic state already.Figure 3
Buckling of steel bars with or without CRFP reinforcement. (a) Group 1; (b) Group 2; (c) Group 3.


### 4.2. Critical Forces and Displacements

The average of readouts of two vertical displacement sensors are taken as the axial displacement, as shown in Figure4(a). It reveals that the axial displacement increases as axial load rises. At the initial stage of loading, the axial displacements of all specimens in each group almost coincide, indicating that the specimens bear the same axial stiffness. In other words, the effect of local damage or the reinforcement of CFRP does not emerge at this stage.Figure 4
The relation between loading forces and displacements of the specimens. (a) Axial displacement; (b) lateral displacement at the midpoint.
(a)(b)When the loading force exceeds the buckling threshold, the influence of both the local damage and CFRP appears. Take Group 1 as an example. We first compare our experimental results with theoretical prediction by equation (7) (Table 6), which shows that the experimentally measured critical forces for all specimens are lower than the theoretical prediction. This is because of the following: (i) The energy method always gives the upper bound of the critical force. (ii) In the theoretical model, all bars are assumed to be ideal, while there exist inevitable imperfections such as defects and predeformation in experimental specimens, which reduce the real critical force.Table 6
Comparison between experimental and theoretical results.
Group 1Reinforcement areaCritical forceFcr (kN)Specimen #ExperimentTheoryJW000-11Unreinforced22.2925.62JW152-12Whole bar27.9930.50JS000-13Unreinforced20.3322.83JS142-14Damaged area22.1424.86JS152-15Whole bar25.6827.37JS000-16Unreinforced19.8522.44JS142-17Damaged area20.9624.74JS152-18Whole bar24.3726.92For a damaged bar, the minimum critical force is 19.85 kN, emerging in specimen JS000-16 without reinforcement, and the maximum one is 27.99 kN, corresponding to specimen JS152-15 reinforced by CFRP covering the whole bar surface. This suggests that the CFRP can remarkably enhance the stability of the steel structures, especially when the bar is wrapped over the whole surface. Besides, it can be observed that CFRP can also enhance the critical axial displacement at the buckling onset (see specimen JW152-12). In addition, our experiment shows that the damage at the midpoint region can lower the stability of the bar more than that outside the midpoint region.Figure4(b) shows the relation between the loading force and the lateral displacement at the middle point of the specimens in the three groups. Before buckling onset, the lateral displacement is quite small and can be ignored. Beyond the buckling threshold, an increasing axial force induces the larger lateral displacement. Since the reinforcement of CFRP is able to enhance both the critical force and the axial displacement, a steel-CFRP structure may exhibit larger deflection during the initial postbuckling stage, compared to those unreinforced structures. This is because most of the axial displacement has been converted to the bending deformation.


### 4.3. Strains of Specimens

Strains of tested specimens during the whole loading process can be recorded by strain gauges. Here, we representatively show strains of Group 1 in Figure5. Strains of Groups 2 and 3 are provided in Figures S1 and S2 in Supplementary Materials, respectively. G1 and G2 represent the longitudinal strains of the steel bars, and B1 and B2 denote the strains of the CFRP plates at the concave and the convex surface of the structure midpoint, respectively.Figure 5
Strains of the specimens in Group 1. (a) JS000-11, (b) JW152-12, (c) JS000-13, (d) JS142-14, (e) JS152-15, (f) JS000-16, (g) JS142-17, and (h) JS152-18.
(a)(b)(c)(d)(e)(f)(g)(h)At the initial stage, the longitudinal compressive strains increase linearly with the increase of loading force. However, the strains on both sides are not equal, indicating that the specimens have some initial deformation or other defects. As the load continues to increase, the compressive strainG1 and the compressive CFRP strain B1 on the concave side continuously increase, while the compressive strain G2 and the CFRP strain B2 gradually transition from a compressive strain to a tensile one, suggesting that bending emerges. During this process, the longitudinal strains of the CFRP on both sides of the surface are basically consistent with the strains on the surface of the steel. It is indicated that the CFRP and the steel surface remain bonded by impregnating glue. Our experiment showed that CFRP plates are well bonded to the steel bars before axial force reaches the threshold. Beyond the threshold, strains markedly rise, while the loading forces decrease. The features of steel strains are consistent with the axial and lateral displacements shown in Figure 4. However, the CFRP strains strikingly deviate from the steel strains during postbuckling stage, which may arise from the integrated effects including the nonlinear constitutive law and surface roughness of the CFRP plates, as well as the mechanical property of the glue.


## 4.1. Buckling and Postbuckling Configuration

Our experiments showed that, at the initial stage of loading, the steel bars reduced their length and maintained a straight configuration. However, when the vertical forces were increased up to a threshold magnitude, the specimens lost stability and gave rise to buckling.Further loading beyond the threshold increased the lateral displacement of the bars, leading to a bent, postbuckling configuration [23]. During this process, fine sound of CFRP peeling or breaking was heard in individual specimens. After the test was completed, the loading force was removed. The bent postbuckling deformation was recovered partly, while a bent configuration was still retained, as shown in Figure 3. It suggests that, under the experimental loading, the specimens entered the elastic-plastic state already.Figure 3
Buckling of steel bars with or without CRFP reinforcement. (a) Group 1; (b) Group 2; (c) Group 3.


## 4.2. Critical Forces and Displacements

The average of readouts of two vertical displacement sensors are taken as the axial displacement, as shown in Figure4(a). It reveals that the axial displacement increases as axial load rises. At the initial stage of loading, the axial displacements of all specimens in each group almost coincide, indicating that the specimens bear the same axial stiffness. In other words, the effect of local damage or the reinforcement of CFRP does not emerge at this stage.Figure 4
The relation between loading forces and displacements of the specimens. (a) Axial displacement; (b) lateral displacement at the midpoint.
(a)(b)When the loading force exceeds the buckling threshold, the influence of both the local damage and CFRP appears. Take Group 1 as an example. We first compare our experimental results with theoretical prediction by equation (7) (Table 6), which shows that the experimentally measured critical forces for all specimens are lower than the theoretical prediction. This is because of the following: (i) The energy method always gives the upper bound of the critical force. (ii) In the theoretical model, all bars are assumed to be ideal, while there exist inevitable imperfections such as defects and predeformation in experimental specimens, which reduce the real critical force.Table 6
Comparison between experimental and theoretical results.
Group 1Reinforcement areaCritical forceFcr (kN)Specimen #ExperimentTheoryJW000-11Unreinforced22.2925.62JW152-12Whole bar27.9930.50JS000-13Unreinforced20.3322.83JS142-14Damaged area22.1424.86JS152-15Whole bar25.6827.37JS000-16Unreinforced19.8522.44JS142-17Damaged area20.9624.74JS152-18Whole bar24.3726.92For a damaged bar, the minimum critical force is 19.85 kN, emerging in specimen JS000-16 without reinforcement, and the maximum one is 27.99 kN, corresponding to specimen JS152-15 reinforced by CFRP covering the whole bar surface. This suggests that the CFRP can remarkably enhance the stability of the steel structures, especially when the bar is wrapped over the whole surface. Besides, it can be observed that CFRP can also enhance the critical axial displacement at the buckling onset (see specimen JW152-12). In addition, our experiment shows that the damage at the midpoint region can lower the stability of the bar more than that outside the midpoint region.Figure4(b) shows the relation between the loading force and the lateral displacement at the middle point of the specimens in the three groups. Before buckling onset, the lateral displacement is quite small and can be ignored. Beyond the buckling threshold, an increasing axial force induces the larger lateral displacement. Since the reinforcement of CFRP is able to enhance both the critical force and the axial displacement, a steel-CFRP structure may exhibit larger deflection during the initial postbuckling stage, compared to those unreinforced structures. This is because most of the axial displacement has been converted to the bending deformation.


## 4.3. Strains of Specimens

Strains of tested specimens during the whole loading process can be recorded by strain gauges. Here, we representatively show strains of Group 1 in Figure5. Strains of Groups 2 and 3 are provided in Figures S1 and S2 in Supplementary Materials, respectively. G1 and G2 represent the longitudinal strains of the steel bars, and B1 and B2 denote the strains of the CFRP plates at the concave and the convex surface of the structure midpoint, respectively.Figure 5
Strains of the specimens in Group 1. (a) JS000-11, (b) JW152-12, (c) JS000-13, (d) JS142-14, (e) JS152-15, (f) JS000-16, (g) JS142-17, and (h) JS152-18.
(a)(b)(c)(d)(e)(f)(g)(h)At the initial stage, the longitudinal compressive strains increase linearly with the increase of loading force. However, the strains on both sides are not equal, indicating that the specimens have some initial deformation or other defects. As the load continues to increase, the compressive strainG1 and the compressive CFRP strain B1 on the concave side continuously increase, while the compressive strain G2 and the CFRP strain B2 gradually transition from a compressive strain to a tensile one, suggesting that bending emerges. During this process, the longitudinal strains of the CFRP on both sides of the surface are basically consistent with the strains on the surface of the steel. It is indicated that the CFRP and the steel surface remain bonded by impregnating glue. Our experiment showed that CFRP plates are well bonded to the steel bars before axial force reaches the threshold. Beyond the threshold, strains markedly rise, while the loading forces decrease. The features of steel strains are consistent with the axial and lateral displacements shown in Figure 4. However, the CFRP strains strikingly deviate from the steel strains during postbuckling stage, which may arise from the integrated effects including the nonlinear constitutive law and surface roughness of the CFRP plates, as well as the mechanical property of the glue.


## 5. Factors Affecting the Critical Force

### 5.1. Effect of Damage Location

We first examine the influence of damage position on the buckling of the steel-CFRP structures. Our experimental data show that the critical force for specimens with damaged regions located in the center, that is, JS000-16, JS142-17, and JS152-18, is lower by 2.2%, 5.3%, and 5.9%, respectively, than that of JS000-13, JS142-14, and JS152-15 with damage outside the middle region, regardless of reinforcement method (Figure6 and Table S1 in Supplementary Materials). It is suggested that the damage at the middle region can more significantly affect the loading capacity of the bars than the damage at other regions. This is because a bar with end constraints in present experiment tends to undergo Euler buckling, during which the largest deflection will emerge at the middle region. Therefore, the system is quite sensitive to imperfections there.Figure 6
Influence of damage location on the critical force in Group 1.


### 5.2. Effect of Damage Degree

The effect of damage degree on the critical forces in Group 2 is shown in Figure7 and Table S2 in the Supplementary Materials. In this group, different grinding thicknesses were introduced to mimic damage. Two layers of CFRP were used to reinforce the damage zones or the whole steel bars. Our experimental data show that, with local reinforcement, the critical force for damage depth td = 0.5 mm, 0.6 mm, and 0.7 mm is reduced, respectively, by 0.6%, 3.1%, and 5.3%, compared to the undamaged specimen (see specimens JS142-23, JS142-24, and JS142-25). When the whole surface of these damaged bars was wrapped with CFRP, the critical force can be enhanced by 11.3%, 7.9%, and 2.1% (see specimens JS142-26, JS142-27, and JS142-28), respectively. However, with either local or whole reinforcement, the damage depth tends to weaken the stability of the steel bars.Figure 7
Influence of damage degree on the critical force in Group 2.


### 5.3. Effect of the CFRP Layers

The influence of the number of CFRP layers on the critical force in Group 3 is shown in Figure8 and Table S3 in the Supplementary Materials. In this group, the steel bars are reinforced by using 1–3 layers of CFRP. It can be seen that the critical force for the damaged steel bars can be remarkably enhanced by increasing CFRP layers. When three layers of CFRP were used, the critical force for the damaged bars can be higher than that of the undamaged ones.Figure 8
Influence of paste thickness of CFRP on the critical force in Group 3.


### 5.4. Effect of CFRP Reinforcement Method

In the present study, two methods, local reinforcement and whole wrapping, were employed to strengthen the damaged steel bars using CFRP. Take Group 3 as an example, as shown in Figure8. The critical force of the specimen was completely restored and exceeded the lossless system of 7.0% when the damage zone was strengthened by using 3 layers of CFRP plate. The total length of CFRP used is 60 cm. When the whole bar was wrapped by one-layer CFRP, the critical force of the damaged specimen will completely recover and exceed the lossless component of 1.2%. In this situation, the total length of CFRP is also 60 cm. It can be seen that when the same amount of CFRP is allowed, the two reinforcement methods may give rise to similar strengthening effect on enhancing stability and recovering loading capacity of the damaged structures. Therefore, the amount of CFRP, the simplicity of the reinforcement, and the strengthening effect should be synthetically considered in engineering. As for the case in Group 3, the whole wrapping method using one layer of CFRP could be more reasonable, because it can not only simplify the construction technology but also attain the on-demand reinforcement effect.


## 5.1. Effect of Damage Location

We first examine the influence of damage position on the buckling of the steel-CFRP structures. Our experimental data show that the critical force for specimens with damaged regions located in the center, that is, JS000-16, JS142-17, and JS152-18, is lower by 2.2%, 5.3%, and 5.9%, respectively, than that of JS000-13, JS142-14, and JS152-15 with damage outside the middle region, regardless of reinforcement method (Figure6 and Table S1 in Supplementary Materials). It is suggested that the damage at the middle region can more significantly affect the loading capacity of the bars than the damage at other regions. This is because a bar with end constraints in present experiment tends to undergo Euler buckling, during which the largest deflection will emerge at the middle region. Therefore, the system is quite sensitive to imperfections there.Figure 6
Influence of damage location on the critical force in Group 1.


## 5.2. Effect of Damage Degree

The effect of damage degree on the critical forces in Group 2 is shown in Figure7 and Table S2 in the Supplementary Materials. In this group, different grinding thicknesses were introduced to mimic damage. Two layers of CFRP were used to reinforce the damage zones or the whole steel bars. Our experimental data show that, with local reinforcement, the critical force for damage depth td = 0.5 mm, 0.6 mm, and 0.7 mm is reduced, respectively, by 0.6%, 3.1%, and 5.3%, compared to the undamaged specimen (see specimens JS142-23, JS142-24, and JS142-25). When the whole surface of these damaged bars was wrapped with CFRP, the critical force can be enhanced by 11.3%, 7.9%, and 2.1% (see specimens JS142-26, JS142-27, and JS142-28), respectively. However, with either local or whole reinforcement, the damage depth tends to weaken the stability of the steel bars.Figure 7
Influence of damage degree on the critical force in Group 2.


## 5.3. Effect of the CFRP Layers

The influence of the number of CFRP layers on the critical force in Group 3 is shown in Figure8 and Table S3 in the Supplementary Materials. In this group, the steel bars are reinforced by using 1–3 layers of CFRP. It can be seen that the critical force for the damaged steel bars can be remarkably enhanced by increasing CFRP layers. When three layers of CFRP were used, the critical force for the damaged bars can be higher than that of the undamaged ones.Figure 8
Influence of paste thickness of CFRP on the critical force in Group 3.


## 5.4. Effect of CFRP Reinforcement Method

In the present study, two methods, local reinforcement and whole wrapping, were employed to strengthen the damaged steel bars using CFRP. Take Group 3 as an example, as shown in Figure8. The critical force of the specimen was completely restored and exceeded the lossless system of 7.0% when the damage zone was strengthened by using 3 layers of CFRP plate. The total length of CFRP used is 60 cm. When the whole bar was wrapped by one-layer CFRP, the critical force of the damaged specimen will completely recover and exceed the lossless component of 1.2%. In this situation, the total length of CFRP is also 60 cm. It can be seen that when the same amount of CFRP is allowed, the two reinforcement methods may give rise to similar strengthening effect on enhancing stability and recovering loading capacity of the damaged structures. Therefore, the amount of CFRP, the simplicity of the reinforcement, and the strengthening effect should be synthetically considered in engineering. As for the case in Group 3, the whole wrapping method using one layer of CFRP could be more reasonable, because it can not only simplify the construction technology but also attain the on-demand reinforcement effect.


## 6. Finite Element Simulation

We use the finite element method (FEM) to simulate our experiments using Abaqus software. The constitutive laws for the steel bars and CFRP are extracted from our experimental measurement, as shown in Figure9.Figure 9
Constitutive laws. (a) Steel; (b) CFRP.
(a)(b)Solid elements and shell elements are used to mesh the steel bars and CFRP, respectively. TIE is adopted to bond the steel and CFRP at their interface, which is basically consistent with our experimental tests. Hinged supports are prescribed at the two ends of the bars to remain consistent with our experiments. In the simulations, we reduced the cross section height according to the grinding depth in our experiments to mimic the local damage. The nonlinear buckling analysis is implemented by RIKS algorithm to consider the effects of both material and geometric nonlinearity. Simulated configurations are representatively shown in Figure10, in consistence with experimental observation. The numerical results of the axial displacement, the lateral displacement, and the critical force are compared with experimental measurement, as shown in Figure 11 and Tables S4–S6 in the Supplementary Materials.Figure 10
Buckling configuration of representative specimens. (a) JS000-11, (b) JS000-13, (c) JS142-23, and (d) JS153-38.
(a)(b)(c)(d)Figure 11
Comparison between finite element simulation and experimental results. (a) Axial displacement; (b) lateral displacement.
(a)(b)Although the critical force obtained by FEM simulation is slightly higher than the experimental value, the maximum difference between them is below 5.3%, confirming our experimental tests. It reveals that our numerical method can well capture the buckling and postbuckling features of damaged steel bars reinforced by CFRP. This provides a robust numerical approach to explore other mechanical responses of steel-CFRP structures with imperfections under complicated environments that are inaccessible experimentally.


## 7. Conclusions

In the present study, we have examined the buckling and postbuckling behaviors of damaged steel bars reinforced with CFRP using experimental measurement, theoretical analysis, and numerical simulation. The critical force at buckling onset and postbuckling characteristics are probed in three groups of 24 specimens subjected to axial compression. We demonstrate that the damage location and degree, CFRP layers, and reinforcement method play a significant role in the stability of steel bars. The theoretical prediction provides an upper bound for the critical force, in consistency with our experimental measurement. We show that an optimized combination of these influence factors can remarkably enhance the stability of the damaged structures even beyond the undamaged counterparts. Our FEM simulations can not only predict the buckling mode and configuration but also quantitatively confirm the experimentally measured critical force and displacements. These findings provide important guides for designing steel-CFRP systems and strengthening damaged steel structures during later maintenance.Finally, the bonding between steel and CRFP is perfectly retained during our experiment, theory, and simulation. It should be pointed out that CFRP delamination may happen in some other cases, which will weaken the reinforcement effect of CFRP. This issue merits further study in the future.



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*Source: 8831701-2020-10-20.xml*