Effectiveness of using a carbon nanotube-reinforced adhesive in steel beams strengthened with composite materials

Abstract

This work examines the effectiveness of a carbon nanotube-reinforced adhesive used in retrofitting steel beams strengthened with composite materials. The method includes applying an adhesive to the soffit of the beam, crucial for transferring tensile forces from the bonded plate to the original beam through interfacial shear and normal stresses. A key innovation is the use of the nanotube-reinforced adhesive to prevent debonding failures that can occur at the plate ends due to increased shear and normal stresses. This technique has reduced shear stresses at the ends of composite plates by approximately 50%. The enhancements not only improve beam performance but also increase their durability. Additionally, the document recommends using composite plates with a variable cross-section—specifically thinner at the ends—to effectively mitigate shear stress concentrations at the edges. The approach yields significant economic benefits, establishing it as a viable solution for beam retrofitting, combining technological innovation with cost-effectiveness, thus positioning this method favorably for engineering applications.

Keywords

composite plate interfacial stresses steel beam strengthening shear lag effect carbon nanotube

1. Introduction

The rehabilitation and enhancement of steel structures through the use of composite materials has emerged as a pivotal area of focus in engineering construction. Recent advancements in this field highlight the growing utilization of composites, particularly in the form of laminates and pultruded plates, for the reinforcement of existing steel frameworks. This shift is primarily driven by the advantageous properties of composites, which include high tensile strength, remarkable durability, corrosion and fire resistance, and a lower weight compared to traditional steel plates.^1–8 These benefits have contributed to the gradual replacement of steel plates, especially in light of steel’s notable limitations, such as susceptibility to corrosion, which complicates effective adhesion. Research in this domain has explored various methodologies and established guidelines for the implementation of composites in structural reinforcement. A significant concern arises from the potential failure modes associated with the debonding of composite plates from steel members, commonly driven by heightened interfacial stresses that occur at the ends of the plates. Therefore, it is critical to develop accurate models to predict these interfacial stresses, thereby reducing the likelihood of debonding failures.^9–16 Furthermore, numerous studies conducted worldwide have examined steel beams reinforced with composites, yielding valuable insights into effective design practices and mechanisms for preventing failures.^3,5,17–19 Collectively, this body of research underscores the importance of integrating composite materials into steel structure rehabilitation, particularly as advancements continue to enhance the reliability and performance of such reinforcements.^20–22

The design of steel beams reinforced with fiber-reinforced polymer (FRP) composites follows specific regulatory codes that outline structural performance evaluation criteria rooted in classical mechanics principles. These codes necessitate detailed assessments of key factors such as stress distribution, flexural strength, shear capacity, and serviceability limits.^23–30 A significant emphasis is placed on the bond performance between the steel and FRP, which is crucial for preventing premature debonding. The incorporation of safety factors within these codes addresses material variability and environmental influences.^28–33 Nonetheless, the existing codes have notable limitations, including the assumption of perfect bonding and uniform stress distribution, which overlook local imperfections and long-term impacts like fatigue and corrosion. As a result, while these standards provide fundamental guidance, they require further scrutiny and development to enhance their accuracy and applicability in real-world scenarios. Research on steel beams enhanced with FRP composites commonly employs a sectional analysis method based on strain compatibility. This methodology posits that all components—steel, adhesive, and FRP—experience compatible deformations under loads, maintaining the integrity of plane sections post-bending according to classical beam theory. Consequently, this leads to a linear distribution of strain across the materials involved. Stress–strain relationships are characterized by the elastic-plastic behavior of steel, while FRP exhibits linear elasticity until failure. The analysis also incorporates internal equilibrium conditions to derive resulting forces, bending moments, and identify the neutral axis location. Failure modes, including steel yielding and FRP rupture, are evaluated, yet the method simplistically assumes an ideal bond between the materials. Overall, this analytical framework proves to be effective in predicting the flexural behavior and load capacity of strengthened steel beams and is widely accepted in both theoretical and practical design contexts.^34–37

Several international seismic design standards, including ACI, AISC, and Eurocode, offer guidelines for designing structures capable of withstanding seismic loads. Nevertheless, these standards are constrained in their applicability to steel beams reinforced with fiber-reinforced polymer (FRP) composites, as they were largely conceived for conventional materials. The ductility requirements outlined in these codes clash with the inherently brittle nature of FRP, which compromises the performance of strengthened structural members. Furthermore, existing codes inadequately address critical aspects such as connection detailing and the bond behavior between steel and FRP, neglecting to consider significant factors like fatigue and debonding during seismic activities. The scarcity of experimental data further limits the codes’ applicability to complex seismic conditions, underscoring the necessity for additional research and the creation of specialized guidelines to enhance the incorporation of composite materials into seismic design practices.

The study focuses on improving the technique of externally bonding composite beams through two innovative methods emphasizing carbon nanotube reinforcement. The first method applies a low percentage (0.5% to 3%) of carbon nanotubes to an adhesive used with composite material plates. The second method employs a novel polymer-based adhesive that has a medium percentage (approximately 10% to 20%) of carbon nanotubes, which has shown significantly enhanced results compared to traditional techniques documented in existing literature. Both approaches consider the shear deformations of the substrates, which are essential for understanding stress distribution at the adhesive-substrate interface. The study suggests that shear stresses at this interface remain continuous, while the free surface should exhibit a zero shear stress, based on Tounsi’s⁸ findings concerning the delayed shear effect observed in bonded lap joints. The results obtained align closely with numerical data from previous studies, validating that the proposed methods can effectively predict maximum stress values in cut sections of the beams. Notably, the interface stresses predicted by the new model are considerably lower than those anticipated by earlier models that failed to take shear deformations into account, thus offering a more accurate depiction of the actual performance of hybrid steel beams. Insights garnered regarding interfacial stress are pivotal for the structural design and application of these composite systems, enhancing both their efficacy and reliability in engineering applications.

2. Method of solution

2.1. Research significance

The text provides a comprehensive analysis of common failure modes in composite-strengthened steel beams, particularly focusing on the debonding of composite plates that prevents these beams from achieving their ultimate flexural capacity. It emphasizes the necessity of incorporating these failure modes into design frameworks, identifying shear and normal stress concentrations within the adhesive layer as key factors leading to premature failures. To address these issues, the article advocates for the development of closed-form solutions aimed at mitigating these stress concentrations, which are vital for establishing effective design guidelines for reinforcing steel beams using composite plates.

The research presented is innovative, employing elementary steel beams to explore reinforcement through bonding with composite plates utilizing a novel carbon nanotube-reinforced adhesive applied in varied proportions (10% to 20%). This approach is proposed to enhance the stability and strength of the assemblies. Importantly, the findings suggest that increasing the percentage of the carbon nanotube-reinforced adhesive enhances bond strength and stability, aligning with the research objectives. Furthermore, the article aims to advance current engineering solutions by proposing a new analytical solution that considers the parabolic deformation effects of shear in the adhesive layer and the bonded composite plates. It aspires to deliver effective reinforcement solutions using both low and high proportions of carbon nanotube-reinforced adhesive, underscoring the method’s advocacy for a cost-efficient design strategy centered on the adhesive’s effectiveness.

2.2. Basic assumptions

The study presents a geometric model illustrated in Figures 1 and 2, detailing the steps involved in reinforcing a beam using a bonded laminate. Figure 2 specifically depicts an infinitesimal element of a composite-strengthened steel beam. The analysis incorporates the effects of transverse shear stress and strain on both the beam and the plate, while excluding the transverse normal stress. An analytical approach proposed by Hassaine Daouadji⁷ is employed to strengthen the steel beam with a bonded composite plate, serving as a basis for comparison with other analytical solutions.

Figure 1.

Steel beam strengthened by a composite plate.

Figure 2.

Forces in infinitesimal element of a steel beam bonded by composite plate.

The analytical approach outlined by Hassaine Daouadji⁷ rests on several key assumptions regarding the materials involved in the stress analysis. Firstly, it assumes an elastic stress-strain relationship exists within the steel beam, composite plate, and adhesive. This assumption lays the foundation for understanding how these materials behave under stress. Secondly, it is presumed there is a perfect bond between the composite plate and the steel beam, indicating that the two components work cohesively without any interfacial issues. Thirdly, the role of the adhesive is limited solely to the transfer of stresses from the steel beam to the composite plate reinforcement, suggesting that it does not contribute to the structural integrity in other ways. Finally, it is assumed that the stresses within the adhesive layer remain constant throughout the thickness, implying a uniform distribution of stress across the adhesive material. These assumptions are critical for accurately analyzing the performance of the composite system in structural applications. As relevant studies that demonstrate a significant improvement in performance, we cite the work of several researchers such as.^{2–4,7,8,23–25}

2.3. Mathematical formulation of the present method

A differential section, denoted as dx, can be excised from a composite-strengthened steel beam, illustrated in Figure 2. This theoretical framework enables the representation of the strains occurring within the steel beam at points close to the adhesive interface and adjacent to the external composite material reinforcement.

2.3.1. Shear stress distribution along the composite material-steel beam interface

The deformation in Steel beam in the vicinity of the adhesive layer can be expressed by Hassaine Daouadji⁷:

ε_{s} (x) = \frac{d u_{s} (x)}{d x} = \frac{y_{s}}{E_{s} I_{s}} M_{s} (x) + \frac{N_{s} (x)}{E_{s} A_{s}} + \frac{t_{s}}{4 G_{s}} \frac{d τ (x)}{d x}

(1)

Based on the theory of laminated sheets, the deformation of the composite sheet in the vicinity of the adhesive layer is given by:

ε_{c} (x) = \frac{d u_{c} (x)}{d x} = - \frac{y_{c}}{E_{c} I_{c}} M_{c} (x) + \frac{N_{c} (x)}{E_{c} I_{c}} + \frac{5 t_{c}}{12 G_{c}} \frac{d τ (x)}{d x}

(2)

Where u_s(x) and u_c(x) are the horizontal displacements of the Steel beam and the composite plate respectively. M_s(x) and M_c(x) are respectively the bending moments applied to the Steel beam and the composite plate; E_s is the Young’s modulus of steel; I_s the moment of inertia, N_s and N_c are the axial forces applied to the Steel and the composite plate respectively, b_c and t_c are the width and thickness of the reinforcement plate. In what follows, the stiffness of the reinforcement plate is significantly lower than that of the Steel beam to be reinforced. The bending moment in the composite plate can be neglected to simplify the shear stress derivation operations. The shear stress at the adhesive layer can be expressed as follows:

τ_{a} = τ (x) = K_{a} Δ u (x) = K_{a} [u_{c} (x) - u_{s} (x)]

(3)

Where K_a is the shear stiffness of the adhesive layer per unit length. From equation (3) we can deduce the expression of K_s which is given by:

K_{a} = \frac{τ (x)}{Δ u (x)} = \frac{τ (x)}{Δ u (x) / t_{a}} \frac{1}{t_{a}} = \frac{G_{a}}{t_{a}}

(4)

Δ u (x)

is the displacement relative to the adhesive interface,

G_{a}

t_{a}

are the modulus and thickness of the adhesive layer, respectively. By differentiating the equations (3), (1) and (2) with respect to x, and neglecting the bending moment of the composite plate we will have:

\frac{d τ (x)}{d x} = K_{a} [[- D_{11}^{'} \frac{y_{c}}{b_{c}} M_{c} (x) + A_{11}^{'} \frac{N_{c} (x)}{b_{c}}] - [\frac{y_{s}}{E_{s} I_{s}} M_{s} (x) + \frac{N_{s}}{E_{s} A_{s}} + \frac{t_{1}}{4 G_{s}}]]

(5)

By differentiating equation (5) we will have:

\begin{array}{l} \frac{d^{2} τ (x)}{d x^{2}} = K_{a} [(\frac{A_{s}}{b_{c}} \frac{d N_{c} (x)}{d x} - A_{c} \frac{t_{c}}{2 b_{c}} \frac{d M_{c} (x)}{d x}) - (\frac{y_{s}}{E_{s} I_{s}} \frac{d M_{s} (x)}{d x} + \frac{1}{E_{s} A_{s}} \frac{d N_{s} (x)}{d x})] \\ - K_{a} (\frac{5 t_{c}}{12 G_{c}} + \frac{1}{4 G_{s}}) \frac{d^{2} τ (x)}{d x^{2}} \end{array}

(6)

Substituting dM₁(x)/dx, dM₂(x)/dx and N(x) with their following expressions in equation (5):

N_{s} (x) = - N (x) = - N_{c} (x) = - b_{c} \int_{0}^{x} τ (x)

(7)

\frac{d M_{s} (x)}{d x} = \frac{E_{s} I_{s}}{E_{s} I_{s} + E_{c} I_{c}} [V_{T} (x) - b_{c} τ (x) (y_{s} + t_{a} + \frac{t_{c}}{2})]

(8)

\frac{d M_{c} (x)}{d x} = \frac{E_{c} I_{c}}{E_{s} I_{s} + E_{c} I_{c}} [V_{T} (x) - b_{c} τ (x) (y_{s} + t_{a} + \frac{t_{c}}{2})]

(9)

Allows us to obtain the differential equation of the shear interface stress:

\frac{d^{2} τ (x)}{d x^{2}} - \frac{b_{c} [\frac{(y_{s} + y_{c}) (y_{s} + y_{c} + t_{a})}{E_{s} I_{s} + E_{c} I_{c}} + \frac{1}{E_{s} A_{s}} + \frac{1}{E_{c} A_{c}}]}{\frac{t_{a}}{G_{a}} + \frac{t_{s}}{4 G_{s}} + \frac{5 t_{c}}{12 G_{c}}} τ (x) + \frac{[\frac{y_{s} + y_{c}}{E_{s} I_{s} + E_{c} I_{c}}]}{\frac{t_{a}}{G_{a}} + \frac{t_{s}}{4 G_{s}} + \frac{5 t_{c}}{12 G_{c}}} V_{T} (x) = 0

(10)

to simplify the equation, let’s set that:

η = {[\frac{b_{c} [\frac{(t_{s} + t_{c}) (t_{s} + t_{c} + 2 t_{a})}{4 (E_{s} I_{s} + E_{c} I_{c})} + \frac{1}{E_{s} A_{s}} + \frac{1}{E_{c} A_{c}}]}{\frac{t_{a}}{G_{a}} + \frac{t_{s}}{4 G_{s}} + \frac{5 t_{c}}{12 G_{c}}}]}^{\frac{1}{2}}

(11)

φ = \frac{[\frac{y_{s} + y_{c}}{E_{s} I_{s} + E_{c} I_{c}}]}{\frac{t_{a}}{G_{a}} + \frac{t_{s}}{4 G_{s}} + \frac{5 t_{c}}{12 G_{c}}}

(12)

the equation becomes expressed as:

\frac{d^{2} τ (x)}{d x^{2}} - η^{2} τ (x) + φ V_{T} (x) = 0

(13)

For simplicity, the general solutions presented below are limited to loading which is either concentrated or uniformly distributed over part or the whole span of the beam, or both. For such loading, d²V_T(x)/dx² = 0, and the general solution to equation (10) is given by

τ (x) = δ_{1} \cosh (η x) + δ_{2} \sinh (η x) + \frac{(t_{s} + t_{c}) {(E_{s} I_{s} + E_{c} I_{c})}^{- 1}}{2 η^{2} (\frac{t_{a}}{G_{a}} + \frac{t_{s}}{4 G_{s}} + \frac{5 t_{c}}{12 G_{c}})} V_{T} (x)

(14)

And δ₁ and δ₂ are constant coefficients determined from the boundary conditions. In the present study, a simply supported beam has been investigated which is subjected to a uniformly distributed load.

For our case of a uniformly distributed load, the formula of the shear stress is given by the following equation:

\begin{array}{l} τ (x) = [- \frac{y_{s} M_{t} (0)}{η E {}_{s}I_{s}} (\frac{t_{a}}{G_{a}} + \frac{t_{s}}{4 G_{s}} + \frac{5 t_{c}}{12 G_{c}})] e^{- η x} + \frac{(t_{s} + t_{c}) {(E_{s} I_{s} + E_{c} I_{c})}^{- 1}}{2 η^{2} (\frac{t_{a}}{G_{a}} + \frac{t_{s}}{4 G_{s}} + \frac{5 t_{c}}{12 G_{c}})} (a q + x - \frac{e^{- η x}}{η} q) \\ 0 \leq x \leq L_{P} \end{array}

(15)

2.3.2. Normal stress distribution along the composite material -steel beam interface

The following governing differential equation for the interfacial normal stress⁷:

\frac{d^{4} σ_{n} (x)}{d x^{4}} + \frac{E_{a}}{t_{a}} (\frac{b_{c}}{E_{s} I_{s}} + \frac{b_{c}}{E_{c} I_{c}}) σ_{n} (x) + \frac{E_{a}}{t_{a}} (\frac{b_{c} y_{s}}{E_{s} I_{s}} - \frac{b_{c} y_{c}}{E_{c} I_{c}}) \frac{d τ (x)}{d x} + \frac{q E_{a}}{t_{a} E_{s} I_{s}} = 0

(16)

The general solution to this fourth–order differential equation is

\begin{array}{l} σ_{n} (x) = e^{- Δ x} [δ_{3} \cos (Δ x) + δ_{4} \sin (Δ x)] + e^{Δ x} [δ_{5} \cos (Δ x) + δ_{6} \sin (Δ x)] \\ - [\frac{y_{s} E_{c} I_{c} - y_{c} E_{s} I_{s}}{E_{s} I_{s} + E_{c} I_{c}}] \frac{d τ (x)}{d x} - [\frac{E_{c} I_{c}}{b_{c} (E_{s} I_{s} + E_{c} I_{c})}] q \end{array}

(17)

For large values of x it is assumed that the normal stress approaches zero, as a result, δ₅= δ₆ = 0.

to simplify the equation, let’s set that:

Δ = {[\frac{E_{a}}{4 t_{a}} (\frac{b_{c}}{E_{s} I_{s}} + \frac{b_{c}}{E_{c} I_{c}})]}^{\frac{1}{4}}

(18)

The general solution therefore becomes

σ_{n} (x) = e^{- Δ x} [δ_{3} \cos (Δ x) + δ_{4} \sin (Δ x)] - \frac{y_{s} E_{c} I_{c} - y_{c} E_{s} I_{s}}{E_{s} I_{s} + E_{c} I_{c}} \frac{d τ (x)}{d x} - \frac{E_{c} I_{c}}{b_{c} (E_{s} I_{s} + E_{c} I_{c})} q

(19)

As is described by Tounsi,⁸ the constants δ₃ and δ₄ in equation (17) are determined using the appropriate boundary conditions and they are written as follows:

\begin{array}{l} δ_{3} = 0, 5 Δ^{- 3} [\frac{Δ E_{a} [V_{T} (0) + Δ M_{T} (0)]}{E_{s} I_{s}} - \frac{E_{a}}{t_{a}} (\frac{y_{s} b_{c}}{E_{s} I_{s}} - \frac{y_{c} b_{c}}{E_{c} I_{c}}) τ (0) + \\ + \frac{(y_{s} E_{c} I_{c} - y_{c} E_{s} I_{s})}{(E_{s} I_{s} + E_{c} I_{c})} (\frac{d^{4} τ (0)}{d x^{4}} + Δ \frac{d^{3} τ (0)}{d x^{3}})] \end{array}

(20)

δ_{4} = 0, 5 Δ^{- 2} [\frac{E_{a} M_{T} (0)}{t_{a} E_{s} I_{s}} + \frac{y_{s} E_{c} I_{c} - y_{c} E_{s} I_{s}}{(E_{s} I_{s} + E_{c} I_{c})} \frac{d^{3} τ (0)}{d x^{3}}]

(21)

The above expressions for the constants η₃ and η₄ has been left in terms of the bending moment M_T (0) and shear force V_T(0) at the end of the soffit plate. With the constants δ₃ and δ₄ determined, the interfacial normal stress can then be found using equation (17).

3. Numerical results and discussions

3.1. Material used

The present studies investigate a steel beam that is reinforced with a composite plate, primarily focusing on its structural response when subjected to a uniformly distributed load. The geometric and material properties of the beam are summarized in Table 1, along with associated visuals in Figures 3 and 4. Key specifications include a span (Lp) of 2400 mm and a distance (a) from the support to the end of the composite plate measuring 200 mm. The beam is designed to withstand a uniformly distributed load specified as 55 kN/ml. The mechanical characteristics of the carbon nanotubes illustrated in the table are given from the work of Wattanasakulpong.¹⁸ These parameters are crucial for understanding the performance and capacity of the bonded composite system in practical applications, We also cite other researchers who are working on carbon nanotubes.^38,39

Table 1.

Geometric and mechanical properties of the materials used.

Component	Width (mm)	Depth(mm)	Young’s modulus (MPa)	Poisson’s ratio
Ordinaire Adhesive layer	b_a = 100	t_a= 2	E_a=3000	0.30
Reinforced Adhesive with 0,5 % CNT	b_a = 100	t_a= 2	E_a=6323.59	0.3384
Reinforced Adhesive with 1 % CNT	b_a = 100	t_a= 2	E_a=10548.9	0.3368
Reinforced Adhesive with 2 % CNT	b_a = 100	t_a= 2	E_a=18997.8	0.3384
Reinforced Adhesive with 3 % CNT	b_a = 100	t_a= 2	E_a=27446.7	0.3306
Reinforced Adhesive with 10 % CNT as reinforcing materials plate	b_c=b_a=100	t_c=t_a= 4	E_c=E_a=86589	0.3386
Reinforced Adhesive with 15 % CNT as reinforcing materials plate	b_c=b_a=100	t_c=t_a= 4	E_c=E_a=128833.5	0.3386
Reinforced Adhesive with 20 % CNT as reinforcing materials plate	b_c=b_a=100	t_c=t_a= 4	E_c=E_a=171078	0.3386
Sika Carbodur composite plate	b_c = 100	t_c = 1.2	E_c=165000	0.28
Sikadur 30 Adhesive layer	b_a = 100	t_a= 2	E_a=12800	0.32
Steel beam IPE 200	b_s = 100	T_s = 200	E_s=210000	0.30

^*CNT: Carbon NanoTube.

Figure 3.

Geometric characteristic of a steel beam bonded by composite plate.

Figure 4.

Geometric characteristic of a steel beam bonded by adhesive reinforced with CNT as reinforcing materials plate.

3.2. Comparison with analytical solutions

As a verification of the present solution, comparisons of the present model with typical existing analytical solutions are made Hassaine Daouadji Model.⁷ For example, for the span strengthening of the steel beam is compared to the Hassaine Daouadji Model, as shown in Figure 5(a) and (b).

Figure 5.

(a) Comparison of interfacial shear stress for the steel beams reinforced with cardodur composite plate; (b) Comparison of interfacial normal stress for the steel beams reinforced with cardodur composite plate.

Figure 5(a) and (b) illustrate the interfacial shear and normal stresses near the end of a steel beam that is reinforced with a Sika Carbodur plate, employing Sikadur 30 as an adhesive along with two enhanced adhesives containing varying percentages of carbon nanotubes (0.5% and 2%) under a uniformly distributed load. The results indicate that the different analytical solutions show strong agreement with one another. Notably, the interfacial normal stress exhibits a sign change at a short distance from the plate end. Comparing these findings with predictions made by Hassaine Daouadji,⁷ it is evident that incorporating the adherend shear deformation effect into both the beam and the soffit plate results in lower maximum interfacial shearτ_max and normal stresses σ_max .This analysis demonstrates that adherent shear deformation serves to diminish the concentration of interfacial stresses, leading to a more uniform adhesive shear distribution. Consequently, the primary objective of mitigating interfacial stress concentration has been effectively achieved.

3.3. Interfacial stresses for different parameters

After verifying the accuracy of the present analytical model, a parameter study is carried out to better understand the effects of various parameters on the interfacial stresses under the unevenly distributed load.

3.3.1. Influence of adhesive improved reinforced by CNT on interface stresses compared to ordinary adhesive

Figure 6(a) and (b) demonstrate the comparative effect of carbon nanotube (CNT)-reinforced adhesives on interface shear and normal stresses against conventional adhesives. Notable improvements in interface stresses were observed in steel beams joined with Cardodur composite plates when evaluated at varying CNT concentrations of 0.5%, 1%, 2%, and 3% versus a standard adhesive with a base strength of 300 MPa. The findings indicate that as the Young’s modulus increases, so does the stress concentration along the edge of the reinforcing plate. Specifically, ordinary adhesives with a low Young’s modulus of 3000 MPa are outperformed by CNT-reinforced adhesives, which exhibit higher Young’s moduli of Ea = 6323.59 MPa, Ea = 10548.9 MPa, Ea = 18997.8 MPa, and Ea = 27446.7 MPa, respectively. This variation in moduli is clearly presented in Figure 6(a) and (b). The study emphasizes the significant beneficial impact of the CNT-reinforced adhesive on the stability of reinforced steel beams, aligning with the primary goals of the research. As a result, the study advocates for manufacturers to adopt this advanced CNT-reinforced improved adhesive to leverage its advantageous properties.

Figure 6.

(a) Influence of adhesive improved reinforced by CNT on interface shear stresses compared to ordinary adhesive; (b) Influence of adhesive improved reinforced by CNT on interface normal stresses compared to ordinary adhesive.

3.3.2. Influence of adhesive improved reinforced by CNT on interface stresses and unreinforced length

Figure 7(a) and (b) illustrate the impact of the composite-reinforced beam zone length on edge shear stresses when adhesive reinforcement is enhanced with varying concentrations of carbon nanotubes (CNTs), specifically 0.5%, 1%, 2%, and 3%. The findings indicate that shear stresses at the interface increase significantly as the termination point of the reinforcement plate is positioned further away from the supports. This suggests that in any reinforcement context, particularly renovation work, it is crucial to extend the reinforcement strip as close as possible to the supports to mitigate stress concentration at the edges. The research indicates that such extension of the composite reinforcement plate towards the supports can effectively reduce the maximum shear stress to nearly zero. This enhancement ensures the stability of the reinforced beam and aligns with the study’s objectives of preventing the detachment of the composite plate while maintaining the structural integrity of the CNT adhesive-reinforced beam in relation to interfacial shear stresses and the unreinforced length.

Figure 7.

(a) Influence of adhesive improved reinforced by CNT on interface shear stresses and unreinforced length; (b) Influence of adhesive improved reinforced by CNT on interface normal stresses and unreinforced length.

3.3.3. Adhesive influence improve reinforced by CNT on interface stresses as reinforcing materials compared with other materials GFRP, CFRP and steel plate

Figure 8 presents the benefits of implementing an enhanced adhesive augmented by carbon nanotube (CNT) reinforcement at different concentrations of 10%, 15%, and 20% on interface shear stresses. This innovative method is juxtaposed against conventional reinforcement materials noted in existing literature, including Glass Fiber Reinforced Polymer (GFRP), Carbon Fiber Reinforced Polymer (CFRP), and steel plates. The results demonstrate a high level of satisfaction regarding the performance of the CNT-reinforced materials, positioning them as a promising alternative to mitigate the issue of plate detachment, a challenge that has been extensively addressed by various researchers in the discipline.

Figure 8.

Adhesive influence improve reinforced by CNT on interface shear stresses as reinforcing materials compared with other materials GFRP, CFRP and Steel plate.

3.3.6. Geometric edge shape based optimization for interfacial shear stress reduction

In the study of taper end plates, a comparison is made between a composite plate with constant thickness (Figure 9(a)) and a plate featuring a tangent parabolic variation section (Figure 9(b)). The findings reveal a substantial reduction in interfacial shear stresses, approximately by 50% when employing the tangent parabolic variation section, as illustrated in Figure 10. This reduction can be achieved relatively easily in situ while reinforcing a steel beam in service, highlighting its significance from a design standpoint. In the case of a tangent parabolic variation section, a specific geometric configuration of the composite plate is examined. The analysis reveals that the end section of the composite plate, characterized by this tangent parabolic variation, demonstrates several notable outcomes. According to results derived from an analytical approach, interfacial shear stresses are notably diminished when compared to cases involving a uniform plate end cross section. Specifically, as illustrated in Figure 10, the shear stress evolution resulting from the tangent parabolic variation section presents a significant reduction. The findings indicate that shear stresses in this scenario are lowered by approximately 50% when juxtaposed with the standard uniform cross section plate end. This reduction emphasizes the effectiveness of using a tangent parabolic variation section in design applications aimed at enhancing structural performance.

Figure 9.

Geometric shape of end section of plate.

Figure 10.

Geometric edge shape based optimization for interfacial shear stress reduction.

Thus, the proposed solution, intended to reduce shear stresses at the ends of composite plates, demonstrates a notable reduction of roughly 50% in those stresses. It is recommended to employ composite plates with a variable cross-section, particularly thinner at the ends, as this design aids in diminishing shear stress concentrations at the edges. Furthermore, this method offers substantial economic advantages, thereby increasing its practicality and applicability in relevant contexts.

4. Conclusions

The study explores interfacial stresses in composite-steel beams under uniformly distributed bending loads, employing a refined theoretical method that integrates adherend shear deformations via a parabolic shear stress distribution across both the steel beam and the bonded composite plate. Understanding these interfacial stresses is vital for addressing debonding failures and aiding the creation of effective design principles. The methodology highlights the utilization of carbon nanotube (CNT)-reinforced adhesive in steel beams augmented with composite materials, offering a novel approach to enhance structural stability. Numerical comparisons between previous solutions and the newly introduced framework reveal detailed insights into interfacial shear and normal stresses at the termini of steel beams reinforced with Sika Carbodur plates, which made use of Sikadur 30 adhesive. Additionally, the research evaluated two enhanced adhesives containing varying CNT percentages (0.5% and 2%), demonstrating results that closely matched theoretical expectations.

The analysis also investigates the benefits of adhesives with different CNT ratios (10%, 15%, and 20%) on interface shear stresses relative to conventional reinforcement materials such as Glass Fiber Reinforced Polymer (GFRP), Carbon Fiber Reinforced Polymer (CFRP), and steel plates. The findings indicate that CNT-reinforced materials excel in performance, significantly mitigating prior concerns about plate detachment. Moreover, the study illustrates that plates with a tangent parabolic variable cross-section can greatly diminish stress concentrations at the reinforcement ends, achieving nearly a 50% reduction in interfacial shear stresses. This emphasizes the efficacy of implementing a tangent parabolic variable cross-section in design projects aimed at bolstering structural performance. The broad applicability of this innovative solution suggests its relevance across different material types.

Strengthening steel beams with fiber-reinforced polymer (FRP) composites has proven effective for enhancing structural performance, increasing load-carrying capacity, and extending the lifespan of existing structures. However, significant research gaps persist. Notably, the bond behavior between steel and composites raises concerns regarding the long-term durability of the adhesive interface under environmental influences such as temperature changes, humidity, and corrosion. Additionally, the fatigue and cyclic loading performance of FRP-strengthened steel beams is underexplored, which is vital for the safety of structures like bridges that experience repeated loads. Moreover, there is a shortage of comprehensive analytical and numerical models that accurately characterize the behavior and potential failure modes of these hybrid systems; the impact of design parameters—including fiber orientation, composite thickness, adhesive type, and strengthening configuration—also requires further research. Present design standards and guidelines for composite-strengthened steel structures are insufficient, hindering their broader application in engineering. Thus, further experimental and theoretical studies are essential to enhance the understanding of steel-composite interactions and to establish reliable design methods for these reinforced systems.

In perspective, future work will primarily focus on developing a more comprehensive experimental investigation to better characterize the actual behavior of the studied structures.^40,41 This stage will involve conducting tests on beams and plates under various boundary conditions, geometric configurations, and loading types (both static and dynamic). Appropriate instrumentation will be employed to accurately measure displacements, strains, and stresses, thereby providing a reliable experimental database for model validation. Subsequently, the experimental results will be used to improve and calibrate the existing analytical model. In parallel, a numerical analysis based on the finite element method will be carried out to simulate the structural response under different scenarios. This combined approach will enable a rigorous comparison between analytical, experimental, and numerical results. The study will also investigate the influence of key parameters, including boundary conditions, beam and plate geometries, and the nature of mechanical loading. Particular attention will be given to dynamic effects and seismic design considerations. Finally, this methodology may be extended to various materials such as concrete, wood, and aluminum, provided that their mechanical properties are accurately characterized, ensuring the reliability and relevance of the obtained results.⁴²

Footnotes

Acknowledgement

The authors warmly thank the reviewers for agreeing to examine this present work. Thank you.

ORCID iD

Tahar Hassaine Daouadji

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Appendix

References

Suárez-Suárez

Domínguez-Ramírez

Susarrey-Huerta

. “Numerical formulation solid-layer finite element to simulate reinforced concrete structures strengthened by over-coating”. Coupled Systems Mechanics 2023; 12(6): 481–501. https://doi.org/10.12989/csm.2023.12.6.481

Benachour

Benyoucef

Tounsi

, et al. Interfacial stress analysis of steel beams reinforced with bonded prestressed FRP plate. Eng Struct 2008; 30: 3305–3315. https://doi.org/10.1016/j.engstruct.2008.05.007

Smith

Teng

. Interfacial stresses in plated beams. Engineering Structures 2002; 23(7): 857–871. https://doi.org/10.1016/S0141-0296(00)00090-0

Ameur

Tounsi

Benyoucef

, et al. Stress analysis of steel beams strengthened with a bonded hygrothermal aged composite plate. International Journal of Mechanics and Materials in Design 2008; 5(2): 143–156. https://doi.org/10.1007/s10999-008-9090-2

Ljukovac

Ibrahimbegovic

Cohodar-Husic

. “Time varying LQR-based optimal control of geometrically exact Reissner’s beam model”. Coupled Systems Mechanics 2024; 13(1): 73–93. https://doi.org/10.12989/csm.2024.13.1.073

Habibi

Mollazadeh

Bazrafkan

, et al. “Comparative analysis of multiple mathematical models for prediction of consistency and compressive strength of ultrahigh performance concrete”. Coupled Systems Mechanics 2023; 12(6): 539–555. https://doi.org/10.12989/csm.2023.12.6.539

Hassaine

Rabahi

Benferhat

, et al. Flexural behaviour of steel beams reinforced by carbon fibre reinforced polymer: Experimental and numerical study. Structural Engineering and Mechanics 2019; 72(4): 409–419. https://doi.org/10.12989/sem.2019.72.4.409

Tounsi

. “Improved theoretical solution for interfacial stresses in concrete beams strengthened with FRP plate”. International Journal of Solids and Structures 2006; 43(14–15): 4154–4174. https://doi.org/10.1016/j.ijsolstr.2005.03.074

AbouEl-Hamd

Sweedan

AMI

El-Ariss

, et al. (2023) “Experimental and Numerical Investigation of the Behavior of Steel Beams Strengthened by Bolted Hybrid FRP Composites”. Buildings; 13: 824. https://doi.org/10.3390/buildings13030824

10.

Nguyen

, et al. The plate on the nonlinear dynamic foundation under moving load. Coupled Systems Mechanics 2023; 12(1): 083–102. https://doi.org/10.12989/csm.2023.12.1.083

11.

Lluis

Pérez-Aparicio

Maimí

, et al. Analytical crack growth in unidirectional composite flywheel. Coupled Systems Mechanics 2023; 12(2): 183–197. https://doi.org/10.12989/csm.2023.12.2.183

12.

Aliyev

Gafarbayli

Mammadov

, et al. Unsteady gas dynamics modeling for leakage detection in parallel pipelines. Coupled Systems Mechanics 2025; 14(4): 371–393. https://doi.org/10.12989/csm.2025.14.4.371

13.

Hung

Huandi

Nguyễn

, et al. Smart design of steel fiber effects on soft computing of steel fiber reinforced self-compacting concrete. Coupled Systems Mechanics 2025; 14(3): 247–267. https://doi.org/10.12989/csm.2025.14.3.247

14.

Kožar

. “Analysis of 3D pendulum sliding along a rope”. Coupled Systems Mechanics 2024; 13(5): 459–470. https://doi.org/10.12989/csm.2024.13.5.459

15.

Saini

Lalthazuala

Singh

. “Flexural behaviour of hybrid high-strength steel I-beams”. Steel and Composite Structures 2025; 55(4): 319–332. https://doi.org/10.12989/scs.2025.55.4.319

16.

Nguyen

Dao

, et al. Column buckling under stochastic three-dimensional material properties: A stochastic finite element analysis. Coupled Systems Mechanics 2025; 14(4): 347–369. https://doi.org/10.12989/csm.2025.14.4.347

17.

Han

S-J

Kim

M-S

Heo

, et al. Performances of precast concrete composite double wall systems with discontinuous steel members as shear connector. Steel and Composite Structures 2025; 55(4): 283–295. https://doi.org/10.12989/scs.2025.55.4.283

18.

Nuttawit

Chaikittiratana

. “Exact solutions for static and dynamic analyses of carbon nanotube-reinforced composite plates with Pasternak elastic foundation”. Applied Mathematical Modelling 2025; 39(18): 5459–5472. https://doi.org/10.1016/j.apm.2014.12.058

19.

X-J

Zhou

C-Y

Wang

. Interfacial stresses in reinforced concrete cantilever members strengthened with fibre-reinforced polymer laminates. Advances in Structural Engineering 2019; 23: 1–12. https://doi.org/10.1177/1369433219868933

20.

Seo

Lee

. “Flexural strength of reinforced concrete members strengthened by fabric reinforced cementitious matrix considering bond characteristics”. Structural Engineering and Mechanics 2025; 94(2): 71–83. https://doi.org/10.12989/sem.2025.94.2.071

21.

Wang

Y-H

Liu

J-P

, et al. Experimental study on assembled monolithic steel-prestressed concrete composite beam in negative moment. Journal of Constructional Steel Research 2020; 167: 105667. https://doi.org/10.1016/j.jcsr.2019.06.004

22.

Zhang

Hou

, et al. “Post-buckling reliability analysis of stiffened composite panels based on adaptive iterative sampling”. Engineering with Computers 2022; 38(Suppl 4): 2651–2661. https://doi.org/10.1007/s00366-021-01424-5

23.

Daouadji Tahar

Abderezak

Rabia

. “Analysis of mechanical performance of continuous steel beams with variable section bonded by a prestressed composite plate”. Steel and Composite Structures 2024; 50(2): 183–199. https://doi.org/10.12989/scs.2024.50.2.183

24.

Daouadji Tahar

Abbes

Tayeb

, et al. “Analytical and Numerical Investigation of Adhesive-Bonded T-Shaped Steel–Concrete Composite Beams for Enhanced Interfacial Performance in Civil Engineering Structures”. Inventions 2025; 10(4): 61. https://doi.org/10.3390/inventions10040061

25.

Daouadji Tahar

Abbes

Tayeb

, et al. “Analysis of Interface Sliding in a Composite I-Steel–Concrete Beam Reinforced by a Composite Material Plate: The Effect of Concrete–Steel Connection Modes”. Journal of Composite Science 2025; 9(6): 273. https://doi.org/10.3390/jcs9060273

26.

Jongvivatsakul

Thongchom

Mathuros

, et al. Enhancing bonding behavior between carbon fiber-reinforced polymer plates and concrete using carbon nanotube reinforced epoxy composites. Case Studies in Construction Materials 2022; 17: e01407. https://doi.org/10.1016/j.cscm.2022.e01407

27.

Majeed

Allawi

Chai

, et al. “Behavior of CFRP strengthened RC multicell box girders under torsion”. Structural Engineering and Mechanics 2017; 61(3): 397–406. https://doi.org/10.12989/sem.2017.61.3.397

28.

AbdulMajeed Allawi

. Behavior of Strengthened Composite Prestressed Concrete Girders under Static and Repeated Loading. advance in civil engineering 2017; 2017: 1–13. https://doi.org/10.1155/2017/3619545

29.

Allawi

Chai

Majeed

, et al. “Experimental and finite element analysis of reinforced concrete multi-cell box girders retrofitted with carbon fiber reinforced polymer strips under torsion” advance in structural engineering. Adv Struct Eng 2023; 26(Issue 14): 2636–2656. https://doi.org/10.1177/13694332231196511

30.

Elamir

Alqhtani

Alhadad

, et al. Active control procedure for minimizing the dynamic response of carbon nanotube reinforced plates. Structural Engineering and Mechanics 2026; 97(3): 311–337. https://doi.org/10.12989/sem.2026.97.3.311

31.

Soares

Campos Filho

Lazzari

, et al.

“Finite element analysis of reinforced concrete beams shear-strengthened with CFRP”

Structural Engineering and Mechanics 2025; 96(2): 155–166. https://doi.org/10.12989/sem.2025.96.2.155

32.

Saini

Lalthazuala

Singh

. Shear performance of hybrid high-strength steel girders. Steel and Composite Structures 2026; 58(3): 387–414. https://doi.org/10.12989/scs.2026.58.3.387

33.

Allawi

Nadhom Shubber

Gharawi

, et al. Enhancement of RC T-beams toughness using laced stirrups reinforcement for blast response predictions. structural concrete of the fib 2022. https://doi.org/10.1002/suco.202200894

34.

Gong

, et al. Steel-concrete composite beams strengthened with NSM CFRP systems at the hogging-moment regions. Engineering Structures 2023; 292: 116576. https://doi.org/10.1016/j.engstruct.2023.116576

35.

Yusuf

Ahmed

Elgholmy

, et al. “Strengthening Techniques for Steel–Concrete Composite Beams. A Comprehensive Review”. Eng 2025; 6(11): 307. https://doi.org/10.3390/eng6110307

36.

Tayeb

Daouadji

Abbes

, et al. Analysis behavior of composite cellular steel-concrete beams reinforced by composite plate. Earthquakes and Structures 2025; 30(2): 131–153. https://doi.org/10.12989/eas.2026.30.2.131

37.

Rabia

Daouadji

Abbes

, et al. Analysis of interfacial stresses of composite bridge box steel-concrete beam. Coupled Systems Mechanics 2025; 14(5): 431–450. https://doi.org/10.12989/csm.2025.14.5.431

38.

Shi

Jin

Wang

, et al. “Enhancing Flexural Behavior of Reinforced Concrete Beams Strengthened with Basalt Fiber-Reinforced Polymer Sheets Using Carbon Nanotube-Modified Epoxy”, Materials (Basel) 2024; 17(13): 3250. https://doi.org/10.3390/ma17133250

39.

Amany

Ahmed

. “Effects of steel fibers and carbon nanotubes on the flexural behavior of hybrid GFRP/steel reinforced concrete beams”, Salman et al. Beni-Suef Univ J Basic Appl Sci 2024; 13: 127. https://doi.org/10.1186/s43088-024-00584-9

40.

Shen

Jiang

Qiang *

, et al. “Experimental Study on Steel‒Concrete Composite Beams Strengthened by Externally Prestressed CFRP Strip”, Prestress Technology 2023; 1: 39–52. https://doi.org/10.59238/j.pt.2023.01.004

41.

Szewczyk

. Experimental and Numerical Study of Steel–Concrete Composite Beams Strengthened under Load. Materials 2024; 17(18): 4510. https://doi.org/10.3390/ma17184510

42.

Hassan

Asce

Riveros

, et al. Experimental Investigation of Steel Beams with Mechanically Reduced Sections Strengthened with Basalt Fiber–Reinforced Polymers. Journal of Structural Engineering 2024; 151(4): 04025014. https://doi.org/10.1061/JSENDH.STENG-13853