Load Capacity Evaluation of ECC and GFRP Strengthened RC Beams Under Combined Bending and Shear

Abstract

This study presents a mechanics based analytical framework for predicting the flexural–shear capacity of reinforced concrete (RC) beams strengthened with Engineered Cementitious Composites (ECCs) and a hybrid ECC–GFRP near surface mounted (NSM) system. Building upon previously reported experimental observations, the present work aims to establish rational prediction models capable of capturing the interaction between flexural and shear mechanisms in strengthened beams. The analytical approach integrates sectional analysis for flexural capacity with a modified truss analogy for shear resistance, explicitly incorporating the strain hardening tensile contribution of ECC and the tensile and confinement effects of GFRP reinforcement. An interaction based failure criterion is subsequently employed to identify the governing failure mode under combined flexural shear actions. The proposed model is validated against experimental results obtained from twenty seven beam specimens with varying flexural and shear reinforcement ratios and strengthening configurations. The predicted ultimate loads show good agreement with experimental values, with an average deviation within ±10%. The analytical framework accurately captures the transition between flexural dominated, combined flexural–shear, and diagonal tension failures observed experimentally. Results demonstrate that ECC significantly enhances ductility and shear crack control, while the hybrid ECC–GFRP system provides substantial strength enhancement with a controlled shift in failure mode. Overall, the developed analytical models offer a reliable and computationally efficient tool for predicting the flexural–shear capacity and failure behavior of ECC and hybrid ECC–GFRP-strengthened RC beams, supporting performance based design and practical strengthening applications.

1. Introduction

Reinforced concrete (RC) beams in existing infrastructure frequently exhibit inadequate flexural and shear capacity due to aging, increased load demands, or design deficiencies. Strengthening interventions must therefore enhance not only strength but also deformation capacity and crack control, particularly under combined flexural–shear actions where brittle failure may govern. While several strengthening techniques have been proposed, the reliable prediction of ultimate capacity and failure mode remains a key challenge, limiting their broader adoption in performance based design. Engineered Cementitious Composites (ECCs) have demonstrated considerable potential for strengthening RC members due to their strain hardening behavior and ability to sustain tensile stresses beyond cracking through multiple microcracking [1,2,3]. Recent studies on FRP-strengthened RC systems have demonstrated that the long term durability and performance of FRP (fiber reinforced polymer) composites are strongly influenced by environmental exposure conditions, adhesive systems, and degradation mechanisms, highlighting the need for reliable predictive approaches including machine learning based durability assessment [4,5]. Also, recent studies have shown that FRP-based composite systems integrated with UHPC (ultra high performance concrete) and UHP-ECC significantly improve the flexural strength, axial capacity, confinement efficiency, and ductility of structural members. The use of FRP grids, FRP micro-bars, and FRP confinement provides lightweight, corrosion resistant, and durable alternatives to conventional steel reinforcement, particularly for aggressive environmental conditions. Furthermore, these studies highlighted the effectiveness of FRP-UHPC composite systems in strengthening and rehabilitation applications, while also emphasizing the need for improved analytical and stress–strain models for accurate prediction of structural behavior [6,7,8]. When applied as an overlay or as a partial replacement of concrete in tension and shear regions, ECC improves ductility and delays diagonal crack localization [9]. To further enhance flexural capacity, ECC has been combined with Near Surface Mounted (NSM) Fiber Reinforced Polymer (FRP) reinforcement, particularly glass FRP (GFRP), offering a cost effective and corrosion resistant solution [10]. The use of ECC as an epoxy free bonding medium for NSM GFRP bars has recently emerged as a promising hybrid strengthening technique [11].

In the authors’ earlier experimental study, RC beams strengthened with ECC overlays and hybrid ECC–GFRP NSM systems were tested under combined flexural and shear actions. The results revealed significant improvements in load carrying capacity, ductility, and crack control compared to control beams. ECC strengthened beams predominantly exhibited flexural or combined flexural–shear failure, whereas hybrid ECC–GFRP beams achieved higher strength but, in some cases, shifted toward shear dominated failure due to increased shear demand [12]. These observations highlight the importance of flexure–shear interaction in governing the ultimate response of strengthened beams [13]. Despite these experimental advances, existing analytical models are largely inadequate for predicting the behavior of ECC and hybrid ECC–GFRP-strengthened RC beams. Most available approaches treat flexure and shear independently and neglect the post-cracking tensile contribution of ECC and its influence on shear transfer mechanisms. Furthermore, the interaction between enhanced flexural capacity and shear resistance in hybrid systems is not explicitly addressed in current design-oriented models, resulting in limited ability to predict governing failure modes.

To address these limitations, the present study develops a mechanics-based analytical framework for predicting the flexural–shear capacity and failure behavior of RC beams strengthened with ECC and hybrid ECC–GFRP systems. The approach combines sectional analysis for flexural capacity with a modified truss analogy for shear resistance, explicitly incorporating the strain hardening tensile contribution of ECC and the elastic response of GFRP reinforcement. The studied models are validated against experimental results from twenty seven beam specimens covering a wide range of reinforcement ratios and strengthening configurations.

The present study introduces the following key advancements:

•

Explicit incorporation of ECC’s tensile strength contribution within both sectional analysis and truss analogy method with suitable material models and stress block parameters.

•

Upgrading of models to include ECC–GFRP composite action, accounting for multi-material interaction in the load-carrying mechanism

•

Capability to predict governing failure modes (flexural, shear or combined) within a single analytical framework in the case of the truss analogy method.

Existing models generally treat flexure and shear independently and do not explicitly account for ECC tensile behavior or hybrid strengthening effects.

The proposed load evaluation method demonstrates good agreement with experimental capacities and accurately captures the transition between flexural, combined flexural–shear, and shear dominated failure modes. Owing to its simplicity, transparency, and predictive capability, the model is well suited for performance based assessment and preliminary design of ECC and hybrid ECC–GFRP strengthened RC beams, consistent with the analytical and structural mechanics focus of Engineering Structures.

2. Overview of Experimental Study

The present analytical study is based on an extensive experimental investigation previously conducted to evaluate the performance of RC beams strengthened using ECC and hybrid ECC–GFRP systems [14]. A total of twenty seven beam specimens were tested under three point bending, comprising nine control RC beams (CB), nine ECC strengthened beams (ECB), and nine hybrid GFRP + ECC strengthened beams (GECB). All beams were designed to fail in flexure, with identical span length, cross-sectional geometry, and internal steel reinforcement to ensure consistent comparison across strengthening schemes. The details and the geometry of the beam specimen are presented in

Table 1. Details of beam specimens.

Sl. No.	Series Id	Beam Id	Flexural Reinforcement (%)	Shear Reinforcement (%)	Cross-Section of Beams	Strengthening Material
1.	Series 1	CB1	0.40	0.15		-
2.		ECB1				ECC
3.		GECB1				ECC + GFRP
4.	Series 2	CB2	0.61			-
5.		ECB2				ECC
6.		GECB2				ECC + GFRP
7.	Series 3	CB3	0.88			-
8.		ECB3				ECC
9.		GECB3				ECC + GFRP
10.	Series 4	CB4	0.40	0.19		-
11.		ECB4				ECC
12.		GECB4				ECC + GFRP
13.	Series 5	CB5	0.61			-
14.		ECB5				ECC
15.		GECB5				ECC + GFRP
16.	Series 6	CB6	0.88			-
17.		ECB6				ECC
18.		GECB6				ECC + GFRP
19.	Series 7	CB7	0.40	0.25		-
20.		ECB7				ECC
21.		GECB7				ECC + GFRP
22.	Series 8	CB8	0.61			-
23.		ECB8				ECC
24.		GECB8				ECC + GFRP
25.	Series 9	CB9	0.88			-
26.		ECB9				ECC
27.		GECB9				ECC + GFRP

and Figure 1, respectively.

The tension and compression properties of concrete, ECC, steel and GFRP rod are presented in

Table 2. Details of material properties.

Materials		Density (kg/m3)	Compressive Strength (MPa)	Yield Strength (MPa)	Tensile Strength (MPa)	Modulus of Elasticity (GPa)
Concrete		2400	15.9	-	-	25
Steel rebar	6 mm	7850	-	467	467	200
	10 mm	7850	-	598	567
	12 mm	7850	-	610	558
GFRP rebar	6 mm	2022	-	-	1113.56	52.114
	8 mm	1988	-	-	1052.54	54.823
ECC		2400	39.1	-	6.3	25

. The tensile stress–strain response of ECC and the three-point loading test setup of all beam specimens are shown in Figure 2.

Table 3. Summary of experimental results.

Sl. No.	Beam Id	Load (kN)	Increase in Load (%)	Deflection (mm)	Ductility Index	Failure Mode
1.	CB1	55.8	-	21.83	2.04	Flexural tension
2.	ECB1	56.8	1.76	56.58	3.33	Flexural tension + diagonal tension
3.	GECB1	142	60.70	36.6	1.98	Diagonal tension
4.	CB2	73.1	-	38.59	2.00	Flexural tension
5.	ECB2	74.1	1.35	64.88	2.15	Flexural tension + diagonal tension
6.	GECB2	115.7	36.82	35.42	1.57	Diagonal tension
7.	CB3	95.1	-	20	1.71	Flexural tension
8.	ECB3	105.2	9.60	67.16	2.08	Flexural tension + diagonal tension
9.	GECB3	125.1	23.98	25.5	1.82	Diagonal tension
10.	CB4	53.8	-	20.88	2.33	Flexure tension
11.	ECB4	56	3.93	36.53	3.31	Flexural tension
12.	GECB4	99	45.66	30.85	6.64	Flexural tension
13.	CB5	81.1	-	24.05	3.63	Flexural tension
14.	ECB5	85.4	5.04	36.57	3.65	Flexural tension
15.	GECB5	114.1	28.92	30.98	3.04	Flexural tension
16.	CB6	94.4	-	18	3.11	Flexural tension
17.	ECB6	104.2	9.40	34.46	4.01	Flexural tension
18.	GECB6	125.9	25.02	29.39	3.38	Flexural tension
19.	CB7	95.01	-	18	3.49	Flexural compression
20.	ECB7	103	7.76	35.26	3.53	Flexural compression
21.	GECB7	96.56	42.9	18.87	2.33	Flexural compression + diagonal tension
22.	CB8	78	-	29.19	2.86	Flexural compression + diagonal tension
23.	ECB8	84	7.14	41.39	2.81	Flexural compression + diagonal tension
24.	GECB8	121	35.54	38.56	2.76	Flexural compression + diagonal tension
25.	CB9	100.6	-	32.45	1.85	Flexural compression + diagonal tension
26.	ECB9	110.7	9.12	38.37	3.39	Diagonal tension
27.	GECB9	128.5	21.71	30.94	2.47	Diagonal tension

presents the summary of the experimental results from the authors’ previous study [14]. During testing, the load deflection behavior, crack initiation and propagation, stiffness degradation, ductility indices, and failure modes were recorded and analyzed. The experimental results demonstrated that the ECC strengthening significantly enhanced crack control, ductility, and energy absorption capacity, while the hybrid ECC–GFRP system provided substantial improvements in flexural–shear capacity with controlled failure mechanisms. These experimentally observed responses form the primary database for the development, calibration, and validation of the analytical models proposed in the present study.

3. Analytical Modeling and Section Analysis for Load Prediction

This section presents the analytical framework adopted to predict the flexural–shear capacity of all twenty seven beams, which includes RC beams strengthened with ECC and hybrid ECC + GFRP systems. The proposed framework is mechanics based and combines the sectional analysis for flexural resistance with a modified truss analogy for shear resistance, followed by a flexure–shear interaction criterion to identify the governing failure mode. The modeling approach is intentionally kept simple and transparent to facilitate the practical application while retaining the essential material and structural behavior observed experimentally.

3.1. General Modeling Assumptions

The following assumptions are adopted in the analytical development, consistent with experimental observations and established RC theory:

•

Plane sections remain plane after bending, and perfect composite action exists between concrete, ECC overlay, steel reinforcement, and GFRP bars [13]. This assumption is justified by the absence of debonding observed (during experimentation) in the strengthened specimens due to the use of ECC as a compatible bonding medium. The model is applicable primarily to cases with ECC as the bonding material, and future work will consider interface slip and debonding effects with more experimental evidence.

•

Tensile resistance of conventional concrete after cracking is neglected, whereas ECC is assumed to sustain tensile stress beyond first cracking due to its strain-hardening and multiple microcracking behavior [16,17].

•

Steel reinforcement is modeled as an elastic–perfectly plastic material, while GFRP reinforcement is assumed to behave linearly elastic up to failure [18,19].

•

Shear transfer is governed by the combined action of the concrete/ECC matrix, transverse steel reinforcement, and GFRP contribution, with crack bridging and confinement effects provided by ECC explicitly considered [20].

•

Failure is assumed to occur when either the flexural capacity or the shear capacity is reached, whichever governs, and the corresponding failure mode is identified through a flexure–shear interaction criterion.

3.2. Section Analysis for Flexural Capacity

The sectional analysis procedure adopted to predict the flexural capacity of RC beams strengthened with ECC and hybrid ECC–GFRP systems. The analysis is based on strain compatibility, internal force equilibrium, and material constitutive relationships, explicitly accounting for the tensile contribution of ECC and GFRP reinforcement. The procedure applies only to the beams (CB, ECB, GECB) that failed in flexure tension and compression; i.e., from twenty seven beams only fourteen beams are selected.

3.3. Assumed Strain Distribution at Ultimate Limit State

Stress–strain distributions of CB, ECB and GECB beam specimens are shown in Figure 3, Figure 4 and Figure 5, respectively.

At the ultimate limit state, a linear strain distribution is assumed across the depth of the strengthened beam section, in accordance with the plane section hypothesis. The maximum compressive strain at the extreme compression fiber of concrete is taken as given in Equation (1):

$${\mathit{\epsilon }}_{\mathit{c}}={\mathit{\epsilon }}_{\mathit{c}\mathit{u}}$$

(1)

The tensile strains in steel reinforcement, ECC overlay, and GFRP bars are computed based on their respective distances from the neutral axis using Equation (2):

$${\mathit{\epsilon }}_{\mathit{i}}={\mathit{\epsilon }}_{\mathit{c}\mathit{u}}\left({\displaystyle \raisebox{1ex}{$\left({\mathit{d}}_{\mathit{i}}-\mathit{c}\right)$}\!\left/ \!\raisebox{-1ex}{$\mathit{c}$}\right.}\right)$$

(2)

where ${\epsilon }_{cu}$ = ultimate compressive strain of concrete; ${\epsilon }_i$ = strain in the i^th tensile component; $d_i$ = effective depth in the i^th component; and $c$ = depth of the neutral axis from the extreme compression fiber.

3.4. Stress Resultants of Constituent Materials

The compressive stress distribution in concrete is represented using an equivalent rectangular stress block as shown in Figure 3. The resultant compressive force ($C_c$) is given by Equation (3) [21]:

$$C_c=0.85f_c^{\prime }ba$$

(3)

where $C_c$ = the line of action located at a distance a/2 from the extreme compression fiber; $f_c^{\prime }$ = compressive strength of concrete; $b$ = width of the beam; and $a={\beta }_{1}c$; ${\beta }_1$ = stress block parameter as per standard RC theory. Steel is assumed to behave as an elastic–perfectly plastic material. The tensile force in the steel reinforcement ($T_s$) is calculated using Equation (4) [22]:

$$T_s=A_s{f}_s$$

(4)

where $A_s$ = area of steel reinforcement; $f_s$ = stress in steel (${\epsilon }_s\le f_y$); ${\epsilon }_s$ = strain in steel; and $f_y$ = yield strength of steel. Unlike conventional concrete, ECC is assumed to sustain tensile stress beyond cracking due to its strainhardening behavior. The tensile force contributed by ECC ($T_{ECC}$) is calculated using Equation (5) [23]:

$$T_{ECC}=A_{ECC}{\sigma }_{ECC}$$

(5)

where $A_{ECC}$ = effective tensile area of the ECC overlay; ${\sigma }_{ECC}$ = ECC tensile stress corresponding to ${\epsilon }_{ECC}$, obtained from the experimentally derived ECC tensile stress–strain relationship; and ${\epsilon }_{ECC}$ = strain in ECC. For analytical simplicity, the ECC tensile stress may be idealized as a constant equivalent stress over the effective ECC tensile zone. The constant stress assumption is an analytical idealization for simplicity and practical applicability. It represents an equivalent average tensile stress over the effective ECC tensile zone.

For hybrid ECC–GFRP-strengthened beams, the tensile contribution of GFRP bars ($T_{gfrp}$) is calculated assuming linear elastic behavior using Equation (6) [24]:

$$T_{gfrp}=A_{gfrp}{E}_{gfrp}{\epsilon }_{gfrp}$$

(6)

where $A_{gfrp}$ = area of GFRP reinforcement; $E_{gfrp}$ = elastic modulus of GFRP; and ${\epsilon }_{gfrp}$ = tensile strain in GFRP bars.

3.5. Force Equilibrium and Neutral Axis Depth

At flexural equilibrium, the sum of tensile forces must balance the compressive force, which is calculated using Equation (7) [25]:

$$C_c=T_s+T_{ECC}+T_{gfrp}$$

(7)

For ECC only strengthened beams, the term $T_{gfrp}$ is omitted. The neutral axis depth $c$ is obtained iteratively by satisfying the above equilibrium condition using strain compatibility.

3.6. Flexural Moment Capacity

Once the neutral axis depth is determined, the ultimate flexural moment capacity ($M_u$) is calculated using Equation (8):

$$M_u=\sum \left(T_i{z}_i\right)$$

(8)

where $T_i$ = tensile force contributed by steel, ECC, or GFRP; $z_i$ = corresponding lever arm measured from the line of action of the compressive force. The lever arm for each tensile component is defined in Equation (9):

$$z_i=d_i-\left({\displaystyle \frac{a}{2}}\right)$$

(9)

The total flexural moment carrying capacity of all beam specimens is given by Equation (10) [26]:

$$\begin{gathered} M_u=\left(A_s{f}_y\left\{d_s-\left[{\displaystyle \frac{{\beta }_{1}c}{2}}\right]\right\}\right)+\left(A_{ECC}{\sigma }_{tECC}\left\{d_{ECC}-\left[{\displaystyle \frac{{\beta }_{1}c}{2}}\right]\right\}\right) \\ +\left(A_{gfrp}{E}_{gfrp}{\epsilon }_{gfrp}\left\{d_{gfrp}-\left[{\displaystyle \frac{{\beta }_{1}c}{2}}\right]\right\}\right) \end{gathered}$$

(10)

Based on Equation (10), the moment capacities of all beam specimens are calculated and presented in

Table 4. Summary of moment capacity results.

Sl. No.	Beam ID	Depth of Neutral Axis (mm)	Exp. Moment (Mu,Exp.) (kN.m)	Predicted Moment (Mu,Pre.) (kN.m)	Mu,Exp./Mu,Pre.
1.	CB1	59	27.90	23.3	1.20
2.	CB2	91	36.55	34.8	1.05
3.	CB3	137	47.55	51.6	0.92
4.	CB4	63	26.90	24.3	1.11
5.	ECB4	102	28.00	28.1	1.00
6.	GECB4	105	49.50	32.9	1.50
7.	CB5	95	40.55	37.0	1.10
8.	ECB5	126	42.70	40.4	1.06
9.	GECB5	129	57.05	44.6	1.28
10.	CB6	132	47.20	49.8	0.95
11.	ECB6	164	52.10	56.1	0.93
12.	GECB6	167	62.95	61.4	1.03
13.	CB7	58	47.51	26.9	1.77
14.	ECB7	98	51.50	35.2	1.46

.

From Table 4, the experimental and predicted ultimate moment capacities show generally good agreement, with Mu, _Exp./Mu, _Pre. ratios mostly close to unity. Control and ECC-strengthened beams exhibit consistent predictions, typically within ±10% of experimental values. Larger discrepancies are observed in some GFRP + ECC-strengthened beams, where the model tends to underestimate the experimental capacity, indicating a stronger synergistic contribution than predicted. Overall, the prediction through sectional analysis is reasonably accurate, though refinement is needed for hybrid strengthening systems where combined failure mechanisms are witnessed. Hence, the following section proposes a truss analogy-based procedure for the analysis of strengthened beams.

4. Capacity Prediction Using Truss Analogy Method

This section presents a truss based analytical model for predicting the capacity of RC beams strengthened with ECC and hybrid ECC–GFRP systems. The truss analogy serves as a prevalent foundation for the majority of shear design methodologies applied to reinforced concrete (RC) beams [27,28,29,30] and can be used for predicting the capacity of RC beams strengthened with ECC and hybrid ECC–GFRP systems. The proposed approach is consistent with the observed diagonal cracking patterns and failure mechanisms from the experimental investigation.

4.1. General Modeling Assumptions

In formulating the current analysis procedure, the following assumptions were considered:

• A truss structure in two dimensions (2D), with the ability of withstanding experiencing solely axial tension and compression, is employed to model the rectangular beam during analysis. The two dimensional model can effectively capture the primary stress state, particularly in shear of a beam, reducing computational complexity without significant loss of accuracy, especially for large-scale simulations [31].
• Compression members (struts) symbolize concrete elements, while tension members (ties) represent steel reinforcement and ECC in the case of RC beams strengthened with ECC and steel reinforcement, or ECC and GFRP in the case of RC beams strengthened with hybrid ECC–GFRP systems. The ties are the elements in the lower portion and vertical portion of the truss that directly or indirectly represent the combined effect of longitudinal reinforcement, shear reinforcement, ECC and GFRP [32].
• The compression members and node’s dimension adhere to the guidelines outlined by the American Concrete Institute (ACI) for specifying struts and nodes [33].

4.2. Analytical Approach

The analytical procedure incorporates the treatment of nonlinear strain distribution profiles in reinforced concrete beams while incorporating the influence of first-order shear deformation. Additionally, the formulation evaluates the structural response at two distinct loading stages, namely ultimate and yielding. For the purpose of estimating the depth of the neutral axis, the assumption is made that the mid-span section is fully cracked in flexure. Initially, application of conventional beam theory is employed to determine the compression zone depth. This provides a preliminary estimate of the neutral axis position. Subsequently, the strain in the extreme compression fiber due to bending (Ԑ_cf) is determined using the yield strain of the primary steel (Ԑ_sy) and neutral axis depth (y) [34]. The strain compatibility condition is used to relate the concrete and steel strains across the section depth. The nonlinear nature of the strain profile is incorporated to improve the accuracy of the response prediction. This approach enables a more realistic representation of the behavior of RC beams near yielding. The resulting strain distribution at the yielding stage is illustrated in Figure 6, highlighting the relationship between the neutral axis depth, steel yield strain, and extreme fiber compression strain.

4.3. Truss Model Considered for Control Beam and Strengthened Beam

The reinforced concrete (RC) beam is modeled using a statically determinate parallel truss analogy to represent its internal force transfer mechanism. Within this framework, compressive stresses in the concrete are idealized as inclined compression struts, while tensile forces are resisted by tie elements. The longitudinal reinforcement functions as the primary horizontal tie, whereas the contribution of shear reinforcement may be represented either discretely or through an equivalent vertical-tie formulation. The truss model is established based on equilibrium and compatibility conditions, ensuring force transfer between compression and tension components. In the control beam, the resisting mechanism consists of the concrete compression zone, longitudinal tensile reinforcement, and transverse steel stirrups. These elements collectively form the load-carrying skeleton of the beam. For beams strengthened with Engineered Cementitious Composite (ECC), the ECC layer is incorporated into the truss system as an additional tensile- and shear-resisting component, thereby enhancing stress redistribution capacity. In hybrid-ECC–GFRP-strengthened beams, the glass fiber-reinforced polymer (GFRP) layer further contributes to tensile force resistance and confinement effects. Accordingly, the complete truss assembly for each configuration includes the concrete core, longitudinal reinforcement, transverse reinforcement, and strengthening materials where applicable. The idealized half-span truss model, including geometric parameters of the struts and ties, is shown in Figure 7.

4.4. Stage I—Yield Load

At this stage of loading, the horizontal compression strut located in the upper region of the truss reaches a maximum compressive strain equal to Ԑ_cf. This strain corresponds, through compatibility requirements, to the yield strain of the bottom tie element. Strain compatibility between the compression and tension components governs the internal deformation profile of the system. The inclined compression struts are assumed to behave in a linearly elastic manner up to the yield condition. At the onset of yielding, the strain in the diagonal strut is considered to correspond to a compressive stress level of approximately 0.7f_c, representing the limit of the linear elastic range of concrete in compression. Beyond this stress level, nonlinear effects are expected to become significant. Simultaneously, the lower tie element, primarily composed of longitudinal reinforcement (and strengthening material where applicable), is assumed to have just reached its yield strain. The vertical tie elements, representing shear reinforcement, are considered to remain within the linear elastic range. The surrounding concrete within the core is also assumed to behave elastically at this stage. Based on these assumptions, equilibrium equations are formulated to determine the yield load (P) of the truss model. The analytical expressions account for different configurations, including the control beam (concrete core with steel ties), ECC strengthened beam, and hybrid ECC–GFRP-strengthened beam.

$$f_{c}by=f_{sy}{A}_{st}$$

(11)

In the formulation, f_c denotes the compressive strength of the reinforced concrete core, b represents the width of the beam section, and y corresponds to the neutral axis depth measured from the extreme compression fiber. The term A_st refers to the area of tensile reinforcement provided at the tension face of the section, while f_sy indicates the corresponding stress developed in the steel reinforcement. By solving Equation (11), the neutral axis depth y, which defines the effective compression zone, can be determined. This depth governs the portion of the cross-section actively resisting compressive stresses. The effective depth associated with the tension zone is evaluated in accordance with the provisions of ACI 318-19 [33] and the corresponding expression is presented in Equation (12).

$$y_t=2\left[clear cover+diameter of stirrups+\left({\displaystyle \frac{diameter of longitudinal bar}{2}}\right)\right]$$

(12)

The vertical distance between the total compressive and tensile forces in Figure 7 can be determined using Equation (13).

$$j_d=h-0.5y-0.5y_t$$

(13)

$$Angle of inclination,\mathsf{\theta }={tan}^{-1}\left({\displaystyle \frac{jd}{width of panel}}\right)$$

(14)

The tie reinforcement plays a crucial role in enhancing the shear capacity of strengthened beams by increasing the load-carrying capability and limiting crack propagation. In beams strengthened with ECC, the fibers dispersed within the matrix work synergistically with the tie reinforcement to establish an efficient crack bridging mechanism. This interaction improves shear resistance, delays the onset of critical shear cracking, and promotes a more ductile response, often shifting the failure mode toward flexural behavior rather than brittle shear failure. The inclination angle of the tie reinforcement significantly affects its contribution to shear resistance. Reinforcement placed at steeper angles is generally more effective because it aligns more closely with the principal tensile stress trajectories within the beam. Such orientation enhances the ability of the ties to intercept and control diagonal cracks. Moreover, steeper inclinations facilitate distributed cracking and gradual stiffness degradation, thereby supporting a more ductile and progressive failure mechanism. In the present analysis, the beams are designed with tie reinforcement inclined at angles greater than 25°, as reported in previous studies [32,33,34,35]. The force in the tie member of the control beam can be determined using Equation (15).

$$F_{tie}={\sigma }_{st}.A_{st}=\left(E_{st}\times {\epsilon }_{st}\right)A_{st}$$

(15)

The force in the tie member of the ECC-strengthened beam can be determined using Equation (16).

$$F_{tie}={\sigma }_{st}.A_{st}+{\sigma }_{ECC}.A_{ECC}=\left(E_{st}\times {\epsilon }_{st}\right)A_{st}+{\sigma }_{ECC}.A_{ECC}$$

(16)

The force in the tie member of the ECC-GFRP-strengthened beam can be determined using Equation (17).

$$F_{tie}={\sigma }_{st}.A_{st}+{\sigma }_{ECC}.A_{ECC}+{\sigma }_{gfrp}.A_{gfrp}$$

(17)

The force in the inclined strut member can be determined using Equation (18).

$$F_{istrut}={\sigma }_{ce}.A_{istrut}=\left(E_c\times {\epsilon }_{ce}\right).\left(w_s\times b_{core}\right)$$

(18)

In this context, Ԑ_ce denotes the compressive strain associated with a stress level of 0.7f_c, as illustrated in Figure 8. The parameter w_s represents the effective depth of the inclined compression strut shown in Figure 7. The depth of the diagonal strut is determined by considering the geometric constraints of the section. Specifically, it is taken as the smaller value between the inclined strut width at the top and that at the bottom. This ensures a conservative and realistic representation of the compression zone.

The force in the horizontal strut member can be determined using Equation (19).

$$F_{hstrut}={\sigma }_c^{~}.A_{bstrut}={\sigma }_c^{~}.\left(y\times b_{core}\right)$$

(19)

The term ${\sigma }_c^{~}$ represents the compressive stress at the centroid of the horizontal strut. It is assumed to correspond to the mean stress across the compression zone. Accordingly, ${\sigma }_c^{~}$ is equivalent to the average of 0.7f_c and the stress at the extreme fiber (σ_c), where stress at extreme fiber (σ_c) is determined using the nonlinear stress-strain relationship proposed by Hognestad. This approach provides a realistic estimation of the compressive stress distribution within the strut.

$${\sigma }_c=f_c\left[2.{\displaystyle \frac{{\epsilon }_{cf}}{{\epsilon }_c}}-{\left({\displaystyle \frac{{\epsilon }_{cf}}{{\epsilon }_c}}\right)}^2\right]$$

(20)

Since Ԑ_cf represents the extreme compressive strain at the top fiber of the beam section, and the centroid of the horizontal compression strut is located at mid-depth of the compression zone, the internal force resultants within the truss analogy are established based on strain compatibility and sectional equilibrium. The forces in the horizontal strut (F_hstrut), inclined strut (F_istrut), and tie member (F_tie) are expressed as functions of the applied load P using classical truss relationships. The truss model represents the transformation of the beam’s internal stress field into discrete compression and tension trajectories aligned with principal stress directions. The diagonal compression struts correspond to principal compressive stress paths within the concrete core, while the tie members represent principal tensile stress trajectories intercepted by reinforcement and strengthening materials. The nodal zones formed at the intersection of struts and ties play a critical role in force transfer, as stress concentrations develop in these regions. Proper stress distribution within nodal zones ensures compatibility between compressive and tensile force components.

At the yield stage (Stage I), the tensile tie in the control RC beam consists solely of longitudinal steel reinforcement. Yielding of this steel defines the primary strain-based criterion for transition into Stage II. Once the tensile strain reaches the steel yield strain, plastic redistribution begins within the tie, while the compression struts continue to resist load through nonlinear stress–strain behavior. As a secondary strain-based indicator, the peak compressive strain in the horizontal strut may be considered, particularly when strain localization or measurement uncertainty exists. For beams strengthened with ECC, the tensile tie mechanism is fundamentally altered. The ECC layer provided at the bottom region contributes to tensile resistance through its strain-hardening behavior and multiple micro-cracking capacity. Unlike conventional concrete, ECC sustains increasing tensile stress beyond first cracking, allowing redistribution of principal tensile stresses and delaying strain localization. This modifies the internal force equilibrium of the truss and enhances ductility. In the hybrid ECC–GFRP configuration, the GFRP further contributes to the tie mechanism. Due to its high tensile strength and linear elastic behavior up to rupture, GFRP significantly increases tensile stiffness and ultimate tie capacity. The combined action of steel, ECC, and GFRP results in a composite tie system with improved crack control, reduced crack widths, and enhanced energy dissipation capacity.

4.5. Stage II—Ultimate Load

During Stage II, the beam continues to be represented by the same determinate truss framework; however, material nonlinearities become dominant. The incremental load between yield and ultimate stages, denoted as ΔP_u, reflects additional capacity arising from strain-hardening of ECC, redistribution of stresses in steel after yielding, confinement effects within the compression zone, and progressive engagement of strengthening materials. In the compression region, confinement provided by transverse reinforcement enhances the effective compressive strength and strain capacity of the concrete core. This confinement delays crushing and increases the usable compressive strain in the horizontal and inclined struts. The stress state within the nodal zone transitions from predominantly elastic to nonlinear, and crushing may initiate if compressive strains exceed permissible limits. For the control RC beam, ultimate failure is typically governed by either crushing of the concrete core or rupture of the tensile steel. In ECC-strengthened beams, the strain-hardening and crack-bridging characteristics of ECC delay crack coalescence, thereby increasing the load corresponding to flexural or shear dominated failure. In hybrid ECC–GFRP-strengthened beams, rupture of GFRP or crushing of the confined concrete core may govern ultimate capacity, depending on the relative stiffness and strength of the components. The ultimate load may be determined using load-based criteria by identifying the governing failure mechanism through equilibrium of truss member forces. The incremental force in the tie member during Phase II is calculated using Equations (21) and (22), accounting for composite action of steel only (control beam), steel and ECC (ECC-strengthened beam), steel, or ECC and GFRP (hybrid strengthened beam). This enhanced two phase analytical framework integrates principal stress behavior, nodal zone mechanics, confinement effects, and material nonlinearities, thereby providing a comprehensive representation of load transfer and failure evolution in control and strengthened reinforced concrete beams. The force increment in the tie member of the ECC strengthened beam can be determined using Equation (21).

$$∆F_{tie}=∆{\sigma }_s.A_{st}+∆{\sigma }_{ECC}.A_{ECC}=E_s^{-}\times {\epsilon }_{st}.A_{st}+∆{\sigma }_{ECC}.A_{ECC}$$

(21)

The force increment in the tie member of the ECC-strengthened beam can be determined using Equation (22).

$$∆F_{tie}=\left(E_s^{-}\times {\epsilon }_{st}\right).A_{st}+∆{\sigma }_{ECC}.A_{ECC}+∆{\sigma }_{gfrp}.A_{gfrp}$$

(22)

${\mathrm{E}}_{\mathrm{s}}^{-}$ is the hardening modulus of steel after yielding, assumed to be a percentage of the original steel modulus of elasticity E_st [29]. The incremental force in the inclined strut can be computed using the following Equation (23).

$$∆F_{istrut}=∆{\sigma }_{ce}.A_{istrut}=\left({\sigma }_c-{\sigma }_{ce}\right).\left(w_s.b_{core}\right)$$

(23)

The incremental force in the horizontal strut can be computed using Equation (24).

$$∆F_{hstrut}=∆{\sigma }_c^{~}.A_{hstrut}=∆{\sigma }_c^{~}.\left(y\times b_{core}\right)$$

(24)

The three force increments mentioned (ΔF_tie, ΔF_istrut, and ΔF_hstrut) are equated to their corresponding force increments obtained from conventional truss equilibrium analysis, expressed in terms of the load increment ΔP_u. For each truss member, the relationship between internal-force increment and external load increment is established through compatibility and equilibrium conditions. The smallest computed value of ΔP_u is adopted as the governing load increment corresponding to the transition from yield to ultimate stage. This ensures that the most critical member response controls the capacity of the truss system. The model considers different configurations, including the concrete core with a steel tie (control beam); steel and ECC tie (ECC-strengthened beam); and steel, ECC, and GFRP composite tie (hybrid-strengthened beam). The ultimate load is subsequently determined using Equation (25), which incorporates the evaluated load increment and the previously calculated yield load.

P_u = P + ΔP_u

(25)

Based on Equation (25) the total shear capacity is calculated and presented in

Table 5. Summary of predicted capacity.

Sl. No.	Beam ID	${\mathit{F}}_{\mathit{T}\mathit{i}\mathit{e}}$ (kN)	${\mathit{F}}_{\mathit{i}\mathit{s}\mathit{t}\mathit{r}\mathit{u}\mathit{t}}$ (kN)	${\mathit{F}}_{\mathit{h}\mathit{s}\mathit{t}\mathit{r}\mathit{u}\mathit{t}}$ $\left(\mathbf{k}\mathbf{N}\right)$	${\mathit{P}}_{\mathit{M}\mathit{o}\mathit{d}\mathit{e}\mathit{l}}$ (kN)	Exp. Capacity (Pu,Exp.) (kN)	Predicted Capacity (Pu,Pre.) (kN)	Pu,Exp./Pu,Pre.
1.	CB1	55.29	75.55	62.14	55.29	55.80	55.29	0.99
2.	ECB1	60.48	76.01	61.56	60.48	56.80	60.48	1.06
3.	GECB1	181.44	154.78	168.11	154.78	142.00	154.78	1.09
4.	CB2	69.86	77.75	87.76	69.86	73.10	69.86	0.96
5.	ECB2	77.81	71.29	85.46	71.29	74.10	71.29	0.96
6.	GECB2	139.7	123.78	155.62	123.78	115.70	123.78	1.07
7.	CB3	100.69	106.49	113.20	100.69	95.10	100.69	1.06
8.	ECB3	112.12	100.28	119.38	100.28	105.20	100.28	0.95
9.	GECB3	147.52	141.36	168.18	141.36	125.10	141.36	1.13
10.	CB4	52.82	73.77	68.96	52.82	53.80	52.82	0.98
11.	ECB4	58.88	79.27	61.01	58.88	56.00	58.88	1.05
12.	GECB4	108.9	121.87	115.33	108.90	99.00	108.90	1.10
13.	CB5	79.86	83.84	90.37	79.86	81.10	79.86	0.98
14.	ECB5	88.81	90.22	95.03	88.81	85.40	88.81	1.04
15.	GECB5	100.13	114.42	120.86	100.13	114.10	100.13	0.88
16.	CB6	101.21	105.22	116.72	101.21	94.40	101.21	1.07
17.	ECB6	117.03	124.09	130.14	117.03	104.20	117.03	1.12
18.	GECB6	140.17	147.53	160.28	140.17	125.90	140.17	1.11
19.	CB7	97.42	106.76	90.89	90.89	95.01	90.89	0.96
20.	ECB7	109.22	110.37	94.05	94.05	103.00	94.05	0.91
21.	GECB7	124.402	110.31	112.46	110.31	96.56	110.31	1.14
22.	CB8	92.05	79.39	77.25	77.25	78.00	77.25	0.99
23.	ECB8	100.53	81.01	74.76	74.76	84.00	74.76	0.89
24.	GECB8	137.25	109.31	117.1	109.31	121.00	109.31	0.90
25.	CB9	103.34	101.12	95.82	95.82	100.60	95.82	0.95
26.	ECB9	120.28	103.07	121.63	103.07	110.70	103.07	0.93
27.	GECB9	133.2	125.20	135.49	125.20	128.50	125.20	0.97

, which shows the comparison of experimental and predicted shear capacities of all beam specimens. Figure 9 illustrates the idealized sample truss model used for the analysis for the control specimen (CB1), ECC strengthened specimen (ECB1) and hybrid ECC-GFRP strengthened specimen (GECB1).

All beam categories exhibit mean Exp/Pred ratios very close to unity, demonstrating the absence of significant systematic bias in the proposed analytical model. The coefficient of variation for each group remains below 10%, indicating statistically acceptable accuracy and consistency for structural prediction purposes. The lowest variability is observed in the control RC beams, confirming that the model is highly reliable for conventional reinforced concrete behavior. In comparison, the ECC strengthened and hybrid ECC–GFRP strengthened beams show slightly higher dispersion, reflecting the increased analytical complexity associated with material nonlinearity, strain compatibility effects, and interaction between multiple strengthening components. Overall, while the model maintains good predictive capability across all configurations, the level of scatter increases with the complexity of the strengthening system.

5. Failure Prediction

Table 6. Comparison of failure mode.

Sl. No.	Beam ID	Exp. Capacity (Pu, Exp.) (kN)	Predicted Capacity (Pu, Pre.) (kN)	Predicted Failure Mode	Exp. Failure Mode
1.	CB1	55.80	55.29	Flexure tension	Flexure tension
2.	ECB1	56.80	60.48	Flexure tension	Flexural tension + diagonal tension
3.	GECB1	142.00	154.78	Diagonal tension	Diagonal tension
4.	CB2	73.10	69.86	Flexural tension	Flexural tension
5.	ECB2	74.10	71.29	Diagonal tension	Flexural tension + diagonal tension
6.	GECB2	115.70	123.78	Diagonal tension	Diagonal tension
7.	CB3	95.10	100.69	Flexural tension	Flexural tension
8.	ECB3	105.20	100.28	Diagonal tension	Flexural tension + diagonal tension
9.	GECB3	125.10	141.36	Diagonal tension	Diagonal tension
10.	CB4	53.80	52.82	Flexural tension	Flexure tension
11.	ECB4	56.00	58.88	Flexural tension	Flexural tension
12.	GECB4	99.00	108.90	Flexural tension	Flexural tension
13.	CB5	81.10	79.86	Flexural tension	Flexural tension
14.	ECB5	85.40	88.81	Flexural tension	Flexural tension
15.	GECB5	114.10	100.13	Flexural tension	Flexural tension
16.	CB6	94.40	101.21	Flexural tension	Flexural tension
17.	ECB6	104.20	117.03	Flexural tension	Flexural tension
18.	GECB6	125.90	140.17	Flexural tension	Flexural tension
19.	CB7	95.01	90.89	Flexural compression	Flexural compression
20.	ECB7	103.00	94.05	Flexural compression	Flexural compression
21.	GECB7	96.56	110.31	Diagonal tension	Flexural compression + diagonal tension
22.	CB8	78.00	77.25	Flexural compression	Flexural compression + diagonal tension
23.	ECB8	84.00	74.76	Flexural compression	Flexural compression + diagonal tension
24.	GECB8	121.00	109.31	Diagonal tension	Flexural compression + diagonal tension
25.	CB9	100.60	95.82	Flexural compression	Flexural compression + diagonal tension
26.	ECB9	110.70	103.07	Diagonal tension	Diagonal tension
27.	GECB9	128.50	125.20	Diagonal tension	Diagonal tension

presents the comparison of experimental and predicted failure modes of all beam specimens. The predicted failure modes obtained using the proposed model show good agreement with experimental observations. Control beams predominantly exhibited flexural or brittle flexural–shear failure. ECC strengthened beams were accurately predicted to fail in flexural or combined flexural–shear modes, consistent with their multiple cracking patterns and high ductility. Hybrid ECC–GFRP strengthened beams were correctly identified as shear dominated or combined failure cases depending on reinforcement ratios and shear span. Minor discrepancies occurred only in specimens lying within the transition zone, where experimental behavior naturally exhibited mixed characteristics.

6. Validation of the Proposed Analytical Model

This section evaluates the accuracy and reliability of the proposed analytical framework by comparing the predicted flexural capacity, shear capacity, and governing failure modes with experimental results obtained from ECC and hybrid ECC–GFRP strengthened RC beams. The objective is to assess the predictive capability of the sectional analysis and truss analogy under combined flexural–shear actions.

6.1. Comparison of Predicted and Experimental Capacities

The analytically predicted ultimate flexural moment ($M_{u, p}$) and shear capacity ($P_{u, p}$) are compared with the experimentally measured values ($M_{u, e}$ and $P_{u, e}$). The accuracy of the model is quantified using the prediction ratios (Equation (26)):

$${\displaystyle \frac{M_{u, p}}{M_{u, e}}} and {\displaystyle \frac{P_{u, p}}{P_{u, e}}}$$

(26)

A prediction ratio close to unity indicates good agreement between analytical and experimental results. The results show that the proposed sectional analysis predicts flexural capacity with reasonable accuracy for both ECC and hybrid ECC–GFRP strengthened beams. Minor deviations are attributed to idealizations adopted for ECC tensile behavior and bond assumptions between ECC and the existing concrete substrate.

6.2. Statistical Assessment of Model Accuracy

To further quantify the model’s reliability, statistical indicators including the mean prediction ratio ($\mu$) [36] and coefficient of variation (COV) [37] are evaluated as given by Equation (27) and Equation (23), respectively:

$$\mu ={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n}{\left({\displaystyle \frac{X_p}{X_e}}\right)}_i$$

(27)

$$COV={\displaystyle \frac{\sigma }{\mu }}$$

(28)

where $X_p$ and $X_e$, represent predicted and experimental capacities, respectively. The proposed model demonstrates low scatter and acceptable variability, indicating stable predictive performance across different strengthening configurations and reinforcement ratios.

Table 7. Statistical assessment of model accuracy.

Beam ID	No. of Beams	Capacity Type	Mean Predicted Ration ($\mathit{\mu }$)	Standard Deviation ($\mathit{\sigma }$)	Coefficient of Variation ($\mathit{C}\mathit{O}\mathit{V}$)
CB	9	Flexural capacity (Mu)	1.05	0.08	0.076
		Shear capacity (Pu)	0.99	0.04	0.043
ECB	9	Flexural capacity (Mu)	1.04	0.06	0.058
		Shear capacity (Pu)	0.99	0.07	0.080
GECB	9	Flexural capacity (Mu)	1.05	0.05	0.048
		Shear capacity (Pu)	1.04	0.10	0.095
All beams	27	Flexural capacity (Mu)	1.05	0.06	0.057
		Shear capacity (Pu)	1.01	0.08	0.078

presents the calculated model accuracy using statistics.

The statistical evaluation of the prediction accuracy confirms that the proposed analytical framework is capable of reliably estimating both flexural and shear capacities. The mean ratios of predicted to experimental values for all beam categories are close to unity, indicating strong agreement with experimental observations. In the case of prediction of flexural capacity, beams strengthened with ECC and hybrid ECC–GFRP systems exhibit lower coefficients of variation than control specimens, highlighting the enhanced consistency resulting from ECC-induced crack control and the stable elastic response of GFRP reinforcement. Overall, the coefficients of variation remain within approximately 5–9% for both flexural and shear capacity predictions, demonstrating limited dispersion and robust predictive capability across a wide range of reinforcement ratios and strengthening configurations.

6.3. Validation of Failure Mode Prediction and Limitation of the Model

The predicted failure modes obtained from the flexural–shear interaction framework (Section 5) are compared with experimentally observed failure patterns. The analytical model successfully captures:

•

Flexural failure in ECC strengthened beams;

•

Combined flexural–shear failure in transition cases;

•

Shear-dominated failure in hybrid ECC–GFRP strengthened beams with high flexural strengthening and limited shear reinforcement.

Discrepancies are primarily observed in specimens close to the interaction boundary, where experimental failure naturally exhibits mixed characteristics. The proposed analytical framework is applicable to RC beams strengthened with ECC and hybrid ECC–GFRP systems subjected to combined flexural and shear actions within the following bounds:

•

Moderate ECC overlay thickness;

•

Proper bond between ECC and concrete substrate;

•

Linear elastic behavior of GFRP reinforcement.

The model does not explicitly account for long term effects, debonding failure, or cyclic loading conditions, which should be addressed in future studies.

7. Conclusions

•

An analytical framework was developed to predict the flexural and shear capacities of the RC beams strengthened with ECC and hybrid ECC–GFRP systems, explicitly accounting for the contributions of concrete, transverse steel reinforcement, ECC, and GFRP rebar.

•

The proposed models showed good agreement with the experimental results from 27 beam specimens, with mean predicted-to-experimental ratios close to unity and limited scatter for both flexural and shear capacity predictions.

•

ECC and ECC–GFRP strengthened beams exhibit lower coefficients of variation compared to control beams, indicating improved consistency and reliability of the proposed design methods.

•

The use of truss analogy successfully captured flexure controlled, shear controlled, and flexure–shear interaction governed failure mechanisms, achieving approximately 90% agreement with experimentally observed failure modes.

•

The analytical results confirmed that ECC and hybrid ECC–GFRP strengthening significantly enhance shear resistance and delay diagonal cracking, leading to a more ductile and predictable structural response.

•

Overall, the proposed analytical framework provides a reliable and rational tool for evaluating flexural–shear interaction and failure behavior in strengthened RC beams and offers practical applicability for the design and assessment of advanced cementitious and composite strengthening systems.