Experimental and numerical investigations of the compressive behavior of carbon fiber-reinforced polymer-strengthened tubular steel T-joints

Abstract

A method for strengthening damaged tubular steel T-joints under axial compression by wrapping them with carbon fiber-reinforced polymer (CFRP) sheets was proposed and evaluated. The influence of the CFRP strengthening on the failure mode and load capacity of T-joints with different degrees of damage was investigated using experiments and finite element analyses. Five T-joints were physically tested: one bare joint to obtain the peak load and corresponding displacement (D_1m), two reinforced joints to provide a reference, and two pre-damaged then retrofitted joints to serve as the primary research objects. The ratio of the pre-loaded specimen chord displacement to the value of D_1m was considered to be the degree of damage of the two retrofitted joints, and was set to 0.80 and 1.20. The results demonstrate that the maximum capacity of the retrofitted specimen was increased by 0.83%–15.06% over the corresponding unreinforced specimens. However, the capacity of the retrofitted specimen was 2.51%–22.77% lesser compared with that of the directly reinforced specimens. Next, 111 numerical analysis models (0.63≤b≤0.76, 9.70≤g≤16.92) were established to parametrically evaluate the effects of different geometric and strengthening parameters on the load capacity of strengthened tubular T-joints under different degrees of damage. The numerical analysis results revealed that the development of equivalent plastic strain at the selected measuring points was moderated by strengthening with CFRP wrapping, and indicated the optimal CFRP strengthening thickness and wrapping orientation according to tubular T-joint parameters. Finally, reasonable equations for calculating the load capacity of CFRP-strengthened joints were proposed and demonstrated to provide accurate results. The findings of this study can be used to inform improved CFRP strengthening of damaged tubular steel structures.

Keywords

tubular T-joint / carbon fiber-reinforced polymer / degree of damage / numerical analysis / equivalent plastic strain

Introduction

Tubular steel structures have been rapidly developed and widely applied in stadiums, bridges, offshore platforms, and other types of structural engineering projects due to their excellent mechanical properties, aesthetic appearance, and cost effectiveness. Due to the thin wall thickness of tubular chords and the complex stress distribution in their joints, tubular chord members often fail in local buckling [¹] or yielding [¹,²], which can in turn lead to the failure of the entire structure. As a result, the joints are considered the weakest regions and most critical components of tubular structures [³]. It is important to note that tubular joint regions can be damaged by overloading, corrosion, accidental collision, wind load, or seismic action. Therefore, it is necessary to study technologies for the retrofit, strengthening, and reinforcement of damaged tubular joints.

Currently, common tubular joint reinforcement methods include external stiffeners [⁴,⁵], internal ring-stiffeners [⁶], partially thickened chord walls [⁷], double plate reinforcement [⁸–¹⁰], and collar plate reinforcement [¹¹]. However, the application of most of these methods is limited for in-service structures due to issues such as construction difficulties and associated welding damage [¹²]. Therefore, the use of carbon fiber-reinforced polymer (CFRP) wrapping, commonly implemented in concrete structures [¹³,¹⁴], has been proposed to retrofit damaged tubular structures. Compared to the more conventional reinforcement methods previously discussed, the use of CFRP strengthening has many advantages including light weight, high strength, corrosion resistance [¹⁵], and especially convenient construction.

Currently, CFRP is mainly applied in the strengthening of concrete structures [¹⁶–¹⁸], steel beams [¹⁹,²⁰], and steel columns [²¹,²²], while only some relevant research on strengthening and retrofitting tubular joints with CFRP has been reported. To investigate the effect of CFRP reinforcement on tubular joints, Lesani et al. [¹⁵,²³,²⁴] studied the failure modes and mechanical properties of T/Y-joints under axial loads using experiments and finite element analyses. The results revealed that the CFRP sheets bore 50% of the joint capacity, the strength of the joints was significantly improved compared with the unreinforced joints, and CFRP reinforcement effectively delayed the plastic failure of the chord surface. Experimental and theoretical analyses were conducted by Fu et al. [²⁵] on K-joints reinforced with CFRP to study the effects of CFRP length, layer number, and mechanical properties on the joint load capacity. The results demonstrated that the load capacity of K-joints can be effectively improved by wrapping with CFRP and that the degree of improvement is proportional to the number of CFRP layers provided, whereas the effect of the CFRP length on load capacity was not significant. Aguilera and Fam [²⁶] explored the local buckling of thin-walled rectangular hollow steel section (HSS) webs in T-joints bonded with CFRP plates under transverse compression load. Their conclusions indicated that CFRP plates were effective in improving the bearing capacity of the web, and that the reinforcement effect of CPRP increases significantly with the increase in the height-to-thickness ratio of the web. On the basis of previous experiments, Zahedi and Vatani Oskouei [²⁷] studied the reinforcement effect of multilayer CFRP on the steel members of an offshore platform structure by means of finite element analysis, and the results showed that CFRP improved the mechanical properties of steel tubular joints.

It should be noted, however, that this previous research has primarily focused on undamaged specimens reinforced directly with CFRP, and that little research into the influence of damage zones and degree of damage on the failure mode, bearing capacity, and plastic development regularity has been conducted. As a result, the previously obtained conclusions cannot provide theoretical support to guide the strengthening of in-service damaged tubular joints. At present, the influence of damage on load capacity has only been fully considered in the strengthening of concrete structures. The degree of damage considered in such research was classified as artificial damage, natural aging damage, and accidental load damage. The effects of the degree of damage have typically been analyzed by means of experiments [²⁸–³¹] and three-dimensional nonlinear finite element analyses [³²]. Gao et al. [³³] conducted experimental and theoretical analyses on the properties of concrete-filled rectangular steel tubular short columns confined by CFRP to verify the influence of degree of damage on the bearing capacity and plastic development of the joints. However, it remains necessary to establish evaluation criteria for the repair and reinforcement of in-service damaged tubular structures, and to analyze the impact of the degree of damage on the load capacity and plastic development of CFRP-strengthened joints in tubular structures.

Due to the action of live load, metallic structures often exhibit cracks in their welded joints that continuously propagate until causing failure. Research methods for such cracking problems include new experimental methods such as digital image correlation, proposed by Vanlanduit et al. [³⁴], and theoretical analysis methods such as the meshfree particle method [³⁵–³⁷] and edge-based smoothed finite element method by Nguyen-Xuan and Rabczuk [³⁸]. To investigate the ability of FRP to retrofit damaged joints, Pantelides et al. [³⁹] conducted monotonic static tests of welded aluminum connections with and without glass fiber-reinforced polymer (GFRP) strengthening, determining that the bearing capacities of the strengthened connections were 17%–25% greater than those of unstrengthen connections. Nadauld and Pantelides [⁴⁰] observed that the use of GFRP improved the strength of joints in constant amplitude fatigue tests, and proposed a fatigue reduction factor as well as bond length and ultimate static fatigue strength design criteria. Fam et al. [⁴¹] conducted static tensile tests on four welded tubular K joint specimens, finding that CFRP sheets were able to restore the full joint strength, whereas GFRP sheets were only able to restore 70% of the joint strength. Clearly, CFRP provides better retrofit performance for cracked joints. Although the use of CFRP in retrofitting fatigue cracks has been found to have considerable effect in abundant research, these results are not helpful for retrofitting joints with large plastic deformation on the surface of the chord.

Accordingly, in this study, five specimens of the same size were subjected to static loading to experimentally determine the influence of the CFRP wrapping orientation and degree of damage on the load capacity of the joint. The influences of parameters such as CFRP wrapping orientation, chord diameter, and chord thickness on the load capacity of the T-joints under axial compression were then further investigated by finite element (FE) analysis. The results demonstrate the effectiveness of CFRP in strengthening damaged tubular steel T-joints and provide reference for continued research in this field.

Experimental investigation

Overview of specimen design and experimental scheme

Five specimens were designed for the test program: one unreinforced specimen (bare specimen) to obtain the peak load and the corresponding displacement, two undamaged specimens reinforced with CFRP (reference specimens) to investigate the effects of different wrapping orientations, and two damaged CFRP-reinforced specimens (retrofitted specimens) for comparison with the undamaged specimens wrapped in the same way to determine the influence of different degrees of damage. The steel tubes were manufactured from grade Q235 steel, and all specimens were constructed to the dimensions shown in Fig. 1. Dimensions evaluated in this study were the chord outside diameter d₀, chord wall thickness t₀, chord length l₀, brace outside diameter d₁, brace wall thickness t₁, brace length l₁, length of the CFRP sheets l_c, chord diameter ratio b (d₁/d₀), and chord diameter to wall thickness ratio g (d₀/2t₀). The brace and chord were connected by full penetration welds at the joint [⁴²]. The CFRP sheets were installed in the different wrapping orientations shown in Fig. 2.

The bare specimen and the reference specimens were subjected to one-step loading, whereas the retrofitted specimens were loaded in two steps: in the first loading step, a pre-designed degree of damage was induced, then the specimen was unloaded and strengthened with CFRP prior being loaded to failure in the second loading step. In this paper, the peak load and corresponding displacement of the bare specimen are represented by P_1m and D_1m, respectively, and the ratio of the pre-damaged chord surface deformation of the retrofitted specimens to D_1m was defined as the degree of damage. The degree of damage of the retrofitted specimens was accordingly designed to be 0.80D_1m and 1.20D_1m. The peak load and corresponding displacement of the retrofitted specimens were defined as P_2m and D_2m, respectively.

Detailed information describing the physical parameters of all specimens is provided in Table 1 using the following specimen nomenclature: the first letter ‘T’ denotes a T-joint, the number ‘1’ represents the specimen group (groups 2–18 are used later in the numerical analyses), the number following the letter ‘D’ indicates the degree of pre-damage (for example, ‘D120’ refers to a concave displacement of the chord of 1.20D_1m), the presence (or absence) of the letter ‘C’ at the end of the label represents the presence (or absence) of CFRP reinforcement, the letters ‘F’ or ‘V’ indicate that the wrapping orientation was 45°/135° or 0°/90°, respectively, and the final number, as in ‘6’, refers to a 6-mm chord wall thickness (which is varied later in the numerical analyses).

Material properties

Uniaxial tensile tests were conducted according to GB/T 228.1-2010 [⁴³] to obtain the mechanical properties of the brace and chord with the results shown in Table 2. The CFRP sheets had unidirectional fibers and were obtained from Toray Industries with the manufacturer-reported properties shown in Table 3. The adhesive was composed of two components obtained from the Sinopec Baling petrochemical company, E-51 epoxy resin and 593 curing agent, with the manufacturer-reported properties provided in Tables 4 and 5, respectively.

Test setup and procedures

The loading experiment was conducted at Shandong University of Science and Technology, and the test setup is pictured in Fig. 3. The specimens were axially compressed by a 500-kN hydraulic jack monitored by a high-sensitivity pressure sensor located between the jack and the specimen. Vertical compression was applied to the brace end through a 12-mm thick steel plate welded to it in order to avoid stress concentration. To simulate the boundary conditions, the chord was placed on the support through end plates on both sides. A displacement-controlled loading mode was adopted that applied 2 mm per stage at a loading rate of 2 mm/min. Loading was stopped when the vertical displacement of the brace reached 60 mm, or the component was otherwise judged to be damaged. Seven linear variable displacement transducers (LVDTs) were mounted on the specimens to measure the displacement at the selected points shown in Fig. 4. The concave chord displacement was obtained from these measurements according to (D3+ D4)/2-D9. Three, three-element rosette strains gauges were also positioned around the joint region to investigate the stress distribution in the tubular joint under load.

Experimental results and discussion

Failure modes

First step loading

The final failure modes of single-step loaded specimens T1-6 and T1-CV6 (also representative of T1-CF6) are shown in Fig. 5. As can be seen from Fig. 5(a), concave deformation at the intersection region and bulging deflection of the chord wall occurred in T1-6 and no cracks were observed in the weld seam. Similar failure phenomena can be observed in T1-CV6, as shown in Fig. 5(b), which indicates that the CFRP sheets had little effect on the joint failure mode. For the two-step loaded retrofitted specimens T1-D80-CV6 and T1-D120-CF6, the first step loading induced the intended concave chord displacement. To ensure that the plasticity of the joints continually developed, this load was held for 30 min while maintaining the intended displacement. The damaged joints were then wrapped with CFRP after unloading and allowed to cure for 10 d. Because of the limitations of the experimental conditions, such as the need for curing time, the joints could not be retrofitted in their loaded state.

Second step loading

Specimens T1-D80-CV6 and T1-D120-CF6 were retrofitted by CFRP after being loaded to damage, and then tested under the second step loading. The failure modes of the two retrofitted specimens are exhibited in Figs. 5(c) and 5(d). Clearly, the deformation characteristics of these specimens were similar to those of the three previously tested specimens. Specifically, there was no obvious peeling of the CFRP sheets in the intersection region, though different degrees of bulging and cracking were observed, clearly indicating matrix breakage, especially in the saddle areas.

Load-displacement behavior

The load-displacement curves of the five specimens during the first step loading are shown in Fig. 6, in which the horizontal axis represents the concave chord deflection and the vertical axis indicates the axial load applied to the brace end. The peak loads and corresponding displacements of the specimens are provided in Table 6, where D_pre represents the pre-damage displacement applied during the first loading step. The D_1m value of bare specimen T1-6 was 5.82 mm, so according to the design scheme, the pre-damage displacements of retrofitted specimens T1-D80-CV6 (0.80D_1m) and T1-D120-CF6 (1.20D_1m), were 4.65 and 6.98 mm, respectively. As shown in Fig. 6, the load-displacement curves of the retrofitted specimens basically coincide with that of T1-6, indicating that all three specimens possessed largely consistent material properties and welding quality. Moreover, the load-deflection curve of bare specimen T1-6 substantially coincides with those of specimens T1-CV6 and T1-CF6 until the concave chord deformation reaches 2.75 mm, indicating that the addition of CFRP sheets had virtually no effect on the joint bearing capacity at this stage. However, after a concave chord deformation of 2.75 mm, the load-displacement curves of the three specimens gradually separated, and reference specimens T1-CV6 and T1-CF6 exhibited peak loads 7.1% and 2.98% higher, respectively, than that of bare specimen T1-6. This indicates that though the added CFRP indeed provided an increase in joint strength, it was only engaged once the chord had sufficiently deformed. Comparing CFRP installation configurations, T1-CV6 provided a peak load 3.9% higher than that of T1-CF6, indicating that the 0°/90° wrapping orientation provides a superior strengthening effect compared to the 45°/135° configuration.

The load-displacement responses of the retrofitted specimens during the second step loading are shown in Fig. 7, and the test results are presented in Table 6. The load-displacement curves of the other specimens are included for comparison. Generally, the stiffness of T1-D80-CV6 was better than that of T1-6 in the elastic stage. For example, when the concave chord displacement reached 2.75 mm, the stiffness of T1-D80-CV6 was 8.21% higher than that of T1-6, considered to be the result of material hardening, whereas the stiffness of T1-D120-CV6 was slightly smaller than that of T1-6 because of excessive concave chord deformation. At higher loads, the two reference specimens remained in the elastic stage whereas the two retrofitted specimens had already entered the plastic stage. From the data in Table 6, the peak loads of retrofitted specimens T1-D80-CV6 and T1-D120-CV6 can be observed to be slightly greater than that of bare specimen T1-6, but 4.6% and 2.0% less than those of their corresponding reference specimens. This indicates that the CFRP indeed exerted a certain repair effect on the damaged specimens, though the existing damage ultimately weakened its potential strengthening effect. Notably, the curves of all five specimens can be observed to exhibit a tendency to decrease nonlinearly after peak load, and the rates of decrease in the sustained load of the retrofitted specimens is clearly faster than those of the reference specimens but slower than that of the bare specimen. It can thus be inferred that CFRP played an important role in improving the ductility of the damaged specimens, but the presence of damage reduced the overall ductility of the specimens compared to the undamaged specimens.

Finite element analysis

Numerical analysis models

The FE software Abaqus was utilized to systematically analyze the CFRP-retrofitted joints due to its excellent nonlinear analysis functionality. The tubular members were modeled using solid element C3D8R with reduced integration. This element in particular has the ability to avoid shear self-locking and achieving convergence despite serious element distortions. The CFRP sheets were modeled using membrane-type element M3D4R because this element only has in-plane stiffness, similar to the mechanical characteristics of CFRP. To simplify the FE analysis, it was assumed that relative slipping did not occur between tubular members and the CFRP sheet; therefore, they were connected using the *TIE constraint. To obtain accurate results with reasonable calculation costs, the regions where the CFRP and the chord were in contact with each other and the portion of the brace near the joint were fine-meshed, while the mesh was coarser farther away from these interaction regions. A mesh sensitivity analysis was performed using specimen T1-CV6 to determine the optimal element size and verify the FE results, as shown in Fig. 8. Because the simulated maximum capacity error was less than 2% when compared with the test result, an approximate global element size of 15 mm and an approximate local element size of 8 mm were selected. Note that as the element size decreased further, no significant change was observed, although the time required to evaluate the model continued to increase. Typical FE meshes of a T-joint and CFRP sheet are shown in Fig. 9. The boundary conditions and loading method of the FE model are shown in Fig. 10. Both ends of the chord were considered to be hinged supports such that one end of the chord constrained the degrees of freedom in the U1, U2, and U3 directions, and the other end constrained the degrees of freedom in the U1 and U2 directions. The end of the brace released all the constraints and applied displacement in U2 direction. The wrapping orientation of the CFRP sheets was defined by setting the direction of the material in the FE model. Finally, please note that the numerical analysis model ignored the effects of factors such as welding and geometric defects.

Deactivation and activation of CFRP elements

The damage and strengthening of the T-joint under sustained loading was replicated by a two-step implicit dynamic analysis. In the first step, the CFRP elements were deactivated in the simulation, at which time the elements were displayed in gray. In the second step, the CFRP elements were activated, at which time the elements were displayed in blue and coordinated with the chord deformation. By deactivating and activating the CFRP elements in this manner, the pre-damage, strengthening, and continued loading of the specimens under sustained load, which was not possible in the experiments, was realized in the numerical analysis, as would be consistent with the true state of a strengthened tubular structure.

Constitutive model of materials

The mechanical properties of the chord, brace, and CFRP sheet materials were defined according to Tables 2 and 3. The Von-Mises stress criterion and double-linear constitutive model were applied in the FE analysis. The nominal stresses and strains were converted into true stresses and strains. In previous research, Msekh et al. [⁴⁴] used the phase field method to study the Young’s modulus, tensile strength, fracture toughness, and dissipation energy generated by fracture in polymeric nanocomposites. Hamdia et al. [⁴⁵] evaluated the uncertainty of the phase field method and several other popular models. Bao and Wierzbicki [⁴⁶] performed numerical simulations on a series of tests and chose the isotropic plastic model from the material input data in Abaqus in order to obtain the individual components of stress and strain tensors at fracture locations. As the observation of microscopic voids and fine cracks in the model was outside the scope of the present study, the B-W criterion was selected to study the damage instead. Based on the B-W criterion, the material damage model curve [⁴⁷] was proposed and the damage fracture parameters [⁴⁸,⁴⁹] were defined as:

(1)

Damage was defined for ductile metals in the mechanical properties of the Abaqus material definition dialog, and ductile damage and shear damage were selected. The material damage constitutive model curve was defined according to the relationship between the stress triaxiality h and the equivalent plastic strain ε₀^−pl. The model invalidated the element by defining the displacement at failure of the element.

Model verification

The experimental and numerically simulated failure modes of T1-6 and T1-CV6 are shown in Fig. 11, in which it can be seen that the deformation characteristics of the numerical models were similar to those of the experimental results, particularly in terms of concave deformation and chord bulging.

The numerically and experimentally determined load–displacement curves of the three undamaged specimens are shown in Fig. 12(a). The subscripts ‘test’ and ‘num’ denote the experimental specimen and numerical specimen, respectively. In the elastic stage, the experimentally and numerically determined curves basically coincide. Once the chord deformation reaches about 2 mm, the experimentally determined curves gradually separate from the numerically determined curves. The peak load and corresponding displacement of T1-6_num were 227.67 kN and 6.94 mm, respectively, representing 94.98% and 119.2% of the T1-6_test values. The numerically determined peak loads of T1-CV6_num and T1-CF6_num were 245.41 kN and 240.35 kN, respectively, representing 95.60% and 97.70% of their corresponding experimentally determined values. After achieving peak load, the numerically and experimentally determined curves of T1-6 exhibited considerably different trends, with the descending stage of the experimentally determined curve being relatively gentler. However, for T1-CV6 and T1-CF6, the numerically determined curves exhibited no considerable difference compared to the experimentally determined curves after reaching peak load.

A comparison of the numerically and experimentally determined load-displacement curves of the retrofitted specimens is shown in Fig. 12(b), indicating the same general behaviors as those in Fig. 12(a). However, it should be noted that due to the material hardening caused by unloading and reloading during the experiments, the slopes of the experimentally determined curves are slightly steeper than those of the numerically determined curves by 17.23%–25.44% in the elastic stage. Specimens T1-D80-CV6_num and T1-D120-CV6_num exhibited peak loads of 235.35 and 228.21 kN, respectively, that were 3.74% and 5.40% lower than their corresponding experimentally determined loads. The displacements of T1-D80-CV6_num and T1-D120-CV6_num at their peaks loads were 9.49 and 7.43 mm, respectively, representing 121.68% and 111.56% of their corresponding experimental results. Thus, the numerically determined load-displacement curves can be considered to be basically consistent with the experimental results prior to their peak loads. As a result, it was considered valid to use the FE models in the following parametric analysis. It should be pointed out, however, that there was a more significant difference between the characteristics of the experimentally and the numerically determined curves following peak load, so the use of these FE models in evaluations under post-peak load conditions was approached with caution.

Parametric analysis

Parametric design

A series of FE models with different dimensionless geometric parameters (0.63≤b≤0.76, 9.70≤g≤16.92) selected from a range of practical applications were analyzed to study the influence of five variables on the performance of strengthened tubular T-joints: CFRP installation method (q_f), chord wall thickness (t₀), chord diameter (d₀), CFRP thickness (t_c), and CFRP elastic modulus (E_c). Each variable was evaluated using 2–4 values, resulting in the 17 groups shown in Table 7. The specimens in each group were evaluated under five different degrees of damage: of 0.50D_1m, 0.80D_1m, 1.20D_1m, 1.50D_1m, and 2.00D_1m.

Arrangement of equivalent plastic strain measurement points

To analyze the strengthening mechanism of the damaged members, ten measurement points were selected on the T-joint FE models to observe the variation in equivalent plastic strain. The locations of these measurement points are shown in Fig. 13: all measurement points were located near the intersection between the chord and brace, with Point 2 at the saddle point and Point 7 at the crown point.

Chord damage development process

By comparing Figs. 14(a) and 14(b), it can be observed that the equivalent plastic strain at each measuring point was zero until the concave chord displacement reached 5 mm, indicating that effect of CFRP sheets on the joint was negligible under small chord surface deformations. With the continued increase in chord surface deformation, the equivalent plastic strain increased by different degrees, such that the value at Point 2 was soon obviously larger than that at Point 7. Under the same concave chord displacement, the equivalent strains at Point 2 and Point 7 in Fig. 14(b), with CFRP, were lower than those in Fig. 14(a), with no CFRP; for example, at a concave chord displacement of 10 mm, the equivalent plastic strain at measuring Point 2 on T2-6 was 0.076, or 23.03 times that at the same point on T2-CV6. This indicates that the CFRP sheets play a significant role in reducing the stress concentration in the joint area. However, once the concave chord displacement reached 15 mm, the plastic strain at Point 2 on T2-6 was 2.16 times of that at the same point on T2-CV6, indicating that sufficient increase in chord displacement results in a remarkable decrease in the strengthening effect of the CFRP sheets. Furthermore, when the concave chord displacement reached 25 mm, the equivalent plastic strain in T2-6 was 1.94 times that in T2-CV6, indicating that the CFRP had been destroyed, basically invalidating any strengthening effect.

Influence of parameters on load capacity

Effect of CFRP wrapping orientation

Because the characteristics of the load-displacement curves for groups T2–T4 were similar, the load-displacement curves of the different damage states in Group T2 were taken as representative, and are shown in Fig. 15. The pre-damage displacement of the retrofitted specimens during the first loading step is indicated by the “Step1” dimension in the figure. It can be observed that after the CFRP sheets were activated during the second step, the load-displacement curves of the retrofitted specimens, especially those with pre-damage displacements of 1.20D_1m, 1.50D_1m, and 2.00D_1m, changed from exhibiting a downward trend to rising again, indicating that the CFRP sheets began to play a role in strengthening the damaged joints.

The FE simulation data for specimens using different CFRP wrapping orientations is presented in Table 8: for a pre-damage displacement of 0.5D_1m, a maximum load capacity of 237.28 kN can be observed for the specimen with a wrapping orientation of 0°/90°. This value is 1.03 and 1.01 times higher than that for specimens with wrapping orientations of 45°/135° and 60°/120°, respectively. To further analyze the influence of different wrapping orientation on the CFRP reinforcement effect, three specimens with the same degree of damage, T2-D120-CV6, T3-D120-CF6, and T4-D120-CS6, were selected for comparison of their equivalent plastic strains at the measurement points when the concave chord displacement reached 20 mm, as shown in Fig. 16. It can be observed that the maximum equivalent plastic strains in T2-D120-CV6, T3-D120-CF6, and T4-D120-CS6 were 0.193, 0.240, and 0.221, respectively. Further comparisons under the other consistent conditions also indicated that specimens wrapped with CFRP in the 0°/90° orientation exhibited the minimum plastic strain and the best reinforcement effect, followed by the 60°/120° orientation, then by the 45°/135° orientation.

Effect of chord wall thickness and diameter

To explore the influence of t₀ and d₀ on the bearing capacity of CRFP-strengthened tubular joints, specimen groups T5–T8 and T9–T12, respectively, were evaluated with the results shown in Fig. 17(a) and Fig. 18(a). It is clear from Fig. 17(a) that g values reported in Table 7 had a clear effect on the maximum joint capacity. For the same diameter, the thicker the chord wall (and thus the smaller the value of g), the greater the load capacity of the specimen. It should be pointed out that when the chord concave displacements of T7-D120-CV8 and T8-D120-CV10 reached 16.67 and 10.23 mm, respectively, their capacities dropped suddenly due to the occurrence of buckling failure at their brace members, as shown in Fig. 17(b).

For specimen groups T9–T12, it can be seen in Fig. 18(a) that the decrease in g and the increase in b reported in Table 7 led to an increase in the bearing capacity. It should also be noted that when the chord concave displacement of T12-D120-CV6 reached 30.29 mm, the load capacity suddenly dropped without further appreciable change in the chord displacement because the punching shear of the chord was observed as the failure mode, as shown in Fig. 18(b).

Effect of CFRP thickness and elastic modulus

The load-displacement curves of groups T13 and T14, in which the CFRP thickness t_c was varied_, are shown in Fig. 19, and the detailed results are summarized in Table 9. Through comparison of the two curve groups, it can be observed that thicker CFRP sheets made a greater contribution to the joint capacity. For example, Table 9 indicates that the peak loads of T13-CV6 and T14-CV6 were 255.82 and 262.14 kN, respectively, for CFRP sheet thicknesses of 0.111 and 0.167 mm. The equivalent plastic strains in T13-D120-CV6 and T14-D120-CV6 were measured when the concave chord displacement reached 20 mm, and are shown in Fig. 20. By comparing these maximum strains, it can be observed that the strains in the specimens were effectively decreased by the increase in the CFRP thickness t_c.

The load-displacement curves of the four specimens with a pre-damage displacement of 1.20D_1m in groups T15–T18, in which the CFRP elastic modulus E_c was varied, are shown in Fig. 21. The curve characteristics of the other pre-damage displacement specimens were similar to those in Fig. 21, and thus are not provided. It can be observed from these curves that E_c had a critical impact on joint performance. Specifically, the curves of T17-D120-CV6 and T18-D120-CV6 exhibit a sudden drop in the post-peak stage because the large E_c improved the load capacity of the specimens, but also made the CFRP stiffer, causing the sudden destruction of the component and the occurrence of brace bulging.

Calculation of load capacity of tubular T-joints strengthened with CFRP

Calculation of the maximum capacity of undamaged T-joints strengthened with CFRP

Based on the recommendations of Fu et al. [²⁵], this study considered four non-dimensional variables to quantify the maximum capacity of tubular T-joints strengthened with CFRP. According to the observed load-deflection curve shapes, the following expression is proposed:

(2){\mathrm{\Delta }}_{\mathrm{f}\mathrm{r}\mathrm{p}}=a{\left({{\displaystyle E}}_{\mathrm{f}\mathrm{r}\mathrm{p}}/{{\displaystyle E}}_{\mathrm{s}}\right)}^b{\left({{\displaystyle f}}_{\mathrm{f}\mathrm{r}\mathrm{p}}/{{\displaystyle f}}_{\mathrm{s}}\right)}^c{\left({{\displaystyle D}}_{\mathrm{s}}/{{\displaystyle T}}_{\mathrm{s}}\right)}^d{\left({{\displaystyle T}}_{\mathrm{f}\mathrm{r}\mathrm{p}}/{{\displaystyle T}}_{\mathrm{s}}\right)}^e,

where D_frp represents the portion of the ultimate capacity of the joint due to the CFRP strengthening; E_frp and f_frp are the CFRP elastic modulus and tensile strength, respectively; E_s, f_s, D_s, and T_s represent the elastic modulus, tensile strength, diameter, and thickness, respectively, of the chord member; and a, b, c, d, and e are constants. The results of the FE analyses were used to conduct multiple linear regression analyses, and the values of the five constants were obtained as a = 15.6, b = 1.12, c = 2.55, d = -1.66, e = 0.18.

The maximum capacity of the undamaged specimens strengthened with CFRP can then be calculated as follows:

(3){{\displaystyle F}}_{\mathrm{f}\mathrm{r}\mathrm{p}}={{\displaystyle F}}_{\mathrm{0}}+{\mathrm{\Delta }}_{\mathrm{f}\mathrm{r}\mathrm{p}},

where D_frp is as defined in Eq. (2) and F₀ is the maximum capacity of the joint without CFRP strengthening, which can be calculated according to Shen [⁵⁰] as:

(4){{\displaystyle F}}_{\mathrm{0}}=\mathrm{14.59}{\left({{\displaystyle D}}_{\mathrm{s}}/{{\displaystyle T}}_{\mathrm{s}}\right)}^{\mathrm{0.2}}{{\displaystyle K}}_{\mathrm{t}}{{\displaystyle K}}_{\mathrm{n}}{{{\displaystyle T}}_{\mathrm{s}}}^{\mathrm{2}}{{\displaystyle f}}_{\mathrm{y}},

where

and

K_{\mathrm{t}}=\{\begin{array}{cc}\hfill \mathrm{0.069}+\mathrm{0.93}\beta ,\hfill & \hfill \beta \le \mathrm{0.7},\hfill \\ \hfill \mathrm{2}\beta -\mathrm{0.68},\hfill & \hfill \beta >\mathrm{0.7}.\hfill \end{array}

Calculation of the ultimate capacity of damaged T-joints strengthened with CFRP

The modification coefficient z is proposed to account for the damage to the T-joint for application in Eq. (3), resulting in the following expression for the ultimate capacity of damaged CFRP-strengthened T-joints:

(5){{\displaystyle F}}_{\mathrm{d}\mathrm{f}\mathrm{r}\mathrm{p}}={{\displaystyle F}}_{\mathrm{f}\mathrm{r}\mathrm{p}}\cdot \zeta ,

where the modification coefficient z applies the effects of the degree of damage, calculated, for ease of expression, based on the degree of damage coefficient m. The degree of damage m and its corresponding modification coefficient were fitted according to the range of damaged CFRP-strengthened T-joint capacities determined by the FE analyses detailed in Section 4, as shown in Fig. 22. The linear equation z = a₀+ b₀·m was thus obtained, where a₀ = 0.8892±0.0737 and b₀ = -0.0366±0.0417.

The ultimate capacities obtained from Eq. (5) were compared with those determined by the FE analyses to evaluate the accuracy of the proposed equation, with the results shown in Table 10, where F_num is the load capacity of the CFRP-retrofitted specimens obtained using the numerical simulation. Note that only a demonstrative selection of the overall data set is provided in Table 10 due to space limitations. Clearly, the results from the proposed equation and from the numerical analyses are in good agreement.

Conclusions

The load capacity of CFRP-strengthened tubular steel T-joints was studied using experiments and FE analyses, with the conclusions summarized as follows.

1) Failure modes of obvious concave chord deflection, chord punching shear, and brace buckling were all observed in the experiments and FE analyses. Strengthening with CFRP delayed the development of plasticity in the joint area and decreased the stress concentrations near the saddle point.

2) The experiments and FE analyses showed that CFRP wrapping was able to enhance the maximum capacity of the retrofitted specimens over that of the bare joints when the degree of damage was less than 0.80∆_1m. However, the strengthening effect decreased under greater degrees of damage (1.20∆_1m–2.00∆_1m). The results demonstrated that the maximum capacity of the retrofitted specimen was 0.83%–15.06% higher compared with that of the corresponding unreinforced specimens, while being 2.51%–22.77% lesser compared with that of the directly reinforced specimens.

3) A CFRP wrapping orientation of 0°/90° exhibited the best repair effect on the capacity of the joint. Furthermore, it was found that the load capacity of the retrofitted joints increased with increasing values of b, E_c, and t_c, and decreased with increasing values of g.

4) The failure characteristics and stress state of tubular steel T-joints retrofitted with CFRP wrapping under sustained loading were obtained by deactivating and activating the CFRP elements and establishing the appropriate constitutive model. The ultimate load capacity equations proposed using a regression analysis of the simulation results using this approach were determined to provide reasonable values for CFRP-repaired damaged tubular steel T-joints.

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