Finite element modelling of T-shaped RC beams with locally reduced cross sections

Abstract

A novel seismic retrofit technique, known as the section reduction (SR) technique, was proposed for reinforced concrete (RC) frames to realize the strong column and weak beam hierarchy. The SR technique involves the removing a portion of concrete at the bottom zone of the beam near the joint, along with shear strengthening of the weakened area using fiber-reinforced polymer (FRP). Preliminary experimental studies on RC beams with locally reduced cross sections (i.e., a gap) have demonstrated that the inclusion of a gap can effectively weaken the negative bending capacity of a T-shaped RC beam. However, there was no finite element (FE) approach for predicting the behaviour of such RC beams. Therefore, this paper aims at establishing a reliable FE approach for T-shaped RC beams with a gap. A total of five gap sizes were examined, including three small gap sizes and two large gap sizes. The applied FRP-strengthening system consisted of the CFRP U-jackets below and on both sides of the gap. In the proposed FE approach, two concrete cracking models available in ABAQUS/Explicit, including the concrete damaged plasticity (CDP) model and the brittle cracking (BC) model, were examined. Moreover, two tensile damage models were considered. The FE results showed that the CDP model with a power law tensile damage model is suitable for specimens that failed in a flexural mode, while the BC model with secant modulus of concrete adopted is suitable for specimens failing in a shear mode. The FE results also verified that increasing the gap height reduces the negative bending capacity of the beam more effectively than increasing the gap length, and the application of the FRP-strengthening system can enhance the ductility of the weakened beam. The proposed FE approach can be used for the subsequent proposal of strength model and design method of such beams.

Keywords

Finite element (FE) model reinforced concrete (RC)beam section reduction FRP strengthening dynamic analysis approach

Introduction

In order to ensure the adequate energy dissipation capacity of reinforced concrete (RC) frames, the RC frames are normally designed based on the strong column and weak beam (SCWB) design philosophy (Nie et al., 2020c). The SCWB design philosophy aims at promoting the formation of a beam-sway mechanism in RC frames, which enhances ductility and reduces the likelihood of a collapse mechanism of RC frames, as opposed to a story-sway mechanism that can lead to continuous collapse of RC frames. However, post-earthquake investigations on RC frames have revealed that the beam-sway mechanism was rarely observed (e.g., ATC-40, 1996; CABR, 2008; Chen et al., 2016). The main reason of such a phenomenon is that the design codes (mainly previous codes) did not implement the SCWB requirement adequately. Notably, some of the previous versions of design codes (e.g., GB-50011, 2008; ACI-318, 1983) did not take the contribution of the cast-in-place floor slabs to the negative bending capacities of the supporting beams into consideration (Nie et al., 2020c). Numerous existing studies have revealed that the bending capacities of RC beams under negative loading can be greatly improved by the presence of the cast-in-place floor slabs, as the cast-in-place floor slabs (especially the steel reinforcements in the slabs) participate in bearing the tension force (e.g., LaFave and Wight, 1999; Pantazopoulou and French, 2001; Shin and LaFave, 2004a, 2004b; Canbolat and Wight, 2008; Ning et al., 2016; Ghomi and EI-Salakawy, 2019). Consequently, it is not unreasonable to deduce that certain existing RC frames are considered non-compliant with the SCWB requirements and these frames need to be seismically retrofitted immediately.

To retrofit RC frames that violate the SCWB requirements, researchers have suggested that strengthening the columns may be an option. The strengthening techniques of RC columns include concrete jacketing (e.g., Deng et al., 2019; Sakr et al., 2020; Yuan et al., 2022), steel jacketing (e.g., He et al., 2017; Shan et al., 2020; Wang et al., 2017), FRP jacketing (e.g., Bai et al., 2023; Ferracuti et al., 2020; Liu et al., 2022; Wang et al., 2022), and a combination of the above (e.g., Ke et al., 2024; Wang et al., 2021). However, concrete jacketing and steel jacketing can increase the mass and stiffness of columns, resulting in greater seismic forces; although FRP jacketing has been widely used in recent years, the strengthening effect may be limited when FRP is used for non-circular columns. In addition, even if the strength of column is sufficiently enhanced after strengthening, the failure location of RC frames may be shifted from the column ends to the beam-column joints or foundation, which are quite difficult to strengthen. Therefore, column strengthening alone is often not sufficient to achieve the SCWB hierarchy, and joint strengthening is always required. For the retrofitting of RC joints, researchers have also proposed many methods, such as epoxy repair (e.g., Karayannis et al., 1998), concrete jacketing (e.g., Karayannis et al., 2008; Palomo et al., 2024), steel jacketing (e.g., Ebanesar et al., 2022; Huang et al., 2023; Li et al., 2017) and FRP jacketing (e.g., Akhlaghi and Mostofinejad, 2020; Alkhawaldeh and Alrousan, 2023; Mostofinejad and Hajrasouliha, 2019; Saad et al., 2023).

Against the above background, Teng et al. (2013) proposed a novel seismic retrofit method for the RC frames violating the SCWB hierarchy, which was called the beam-end weakening and fiber-reinforced polymer (FRP) strengthening (BWFS) method. Three promising techniques have been suggested for the implementation of the BWFS method (Teng et al., 2013), namely: (1) the beam section reduction (SR) technique; (2) the beam opening (BO) technique; and (3) the slab slit (SS) technique. For detailed information on these techniques, readers are referred to Teng et al. (2013). In the case of the BO technique, which weakens the ability of the beam to resist bending loads by creating openings in the beam web, the authors (Nie, 2018; Nie et al., 2018, 2020a, 2020b, 2020c, 2021a, 2021b, 2021c) have carried out a comprehensive experimental, numerical and theoretical investigation on the mechanical properties of T-shaped RC beams with FRP-strengthened web openings and the effectiveness of the BO technique has been proved. In the present study, however, attention is directed towards the SR technique, as illustrated in Figure 1, which involves removing a portion of concrete at the bottom zone of the beam near the joint and applying an FRP-strengthening system to the weakened area with locally reduced cross-sections (referred to as a gap hereafter for simplicity, as depicted in Figure 1). This technique weakens the negative bending capacity of the beam by reducing the cross-sectional effective height of the region with a gap and increases the shear capacity of the weakened area by applying an FRP shear strengthening system. Through implementing the SR technique, the contribution of the cast-in-place floor slab to the negative bending capacity of the supporting beam can be counteracted and the reduced shear capacity of the weakened area can be also offset. For an RC beam-column joint with a strong beam and weak column (SBWC) hierarchy, the flexural strength ratio (i.e., the ratio of the sum of the bending capacities of the columns at a joint to that of the beams framing into the joint) is less than 1. When the SR technique is applied to this joint, the negative bending capacity of the beam end can be effectively reduced by creating a gap at the bottom region of the beam end close to the joint, while its positive bending capacity almost keeps unchanged. If the weakening effect of the gap on the negative bending capacity of the beam end is sufficient, the flexural strength ratio of the joint can be converted from a value less than 1 to a value greater than 1, and thus the SCWB hierarchy can be realized.

Figure 1.

The beam section reduction (SR) technique.

In a recent study by the authors (Zhang et al., 2024b), an experimental investigation was carried out on T-shaped RC beams with an un-strengthened or FRP-strengthened gap, providing preliminary verification of the effectiveness of the SR technique. While experimental study is fundamental for the understanding of the structural behaviour of RC beams with an un-strengthened/FRP-strengthened gap, it is generally laborious and time-consuming. As an economical and powerful alternative to experimental study, finite element (FE) modelling can be used to more effectively and comprehensively investigate the behaviour of such beams. The focus of the present study is to establish a reliable FE approach specifically for T-shaped RC beams with an un-strengthened/FRP-strengthened gap, as tested by the authors (Zhang et al., 2024b). Once an FE approach for such beams is established and verified using the test results, parametric studies can be conducted using the FE approach to generate substantial data, which is essential for the establishment of the strength model and design method of such beams.

Introduction of the test specimens

To validate the effectiveness of the proposed SR technique, a total of nine full-size RC beams, including one rectangular beam (without a gap) and eight T-shaped beams (one beam without a gap and seven beams with a gap), were fabricated and tested under three-point negative bending by the authors (Zhang et al., 2024b). Table 1 summarized the details of the nine test specimens. The rectangular beam (Specimen CR, used to represent the situation where the effect of the cast-in-place slab is not taken into account in the design of the RC beam) and the T-shaped beam without a gap (Specimen CT, used to represent the actual situation where enhancement of cast-in-place slab in the negative bending capacity of the RC beam exists) were served as control beams. All T-shaped RC beams were dimensionally consistent. The examined parameters in the tests covered the dimensions of the gap (i.e., height × width) and whether an FRP-strengthening system was employed. A total of five gap sizes were examined, including three small gap sizes (with the gap height × width respectively being 150 mm × 100 mm, 260 mm × 100 mm and 260 mm × 150 mm) and two large gap sizes (with the gap height × width respectively being 260 mm × 200 mm and 300 mm × 100 mm). A single un-strengthened beam (i.e., Specimens G-150 × 100, G-260 × 100 and G-260 × 150) was tested for each of the three small gap sizes, while two specimens were tested for each of the two large gap sizes, including an un-strengthened beam (i.e., Specimens G-260 × 200 and G-300 × 100) and an FRP-strengthened beam (i.e., Specimens G-260 × 200-F and G-300 × 100-F). All specimens were placed upside down and subjected to a three-point negative bending test using a 100 t hydraulic jack to apply a downward point load at the beam mid-span. A load cell was installed to measure the precise load at the loading point. A preload of 30 kN (less than 10% of the estimated ultimate load) was first applied to ensure the accuracy of the measuring instruments. After completion of the preloading, the load was released and then the test was started. The loading rate was quasi-static with the typical time taken to complete each test being about 4 hours. A data acquisition system capable of digitally recording and storing load, deflection and strain data at a sampling frequency of 1 Hz was used in the tests.

Table 1.

Details of test specimens.

Specimen	Gap size		FRP strengthening	Cubic compressive strength of concrete f_cu (MPa)	Steel reinforcements in the web			Steel reinforcements in the flange
Specimen	Height (mm)	Width (mm)	FRP strengthening	Cubic compressive strength of concrete f_cu (MPa)	Tension reinforcements	Compression reinforcements	Stirrups	Steel reinforcements in the flange
CR	NA	NA	No	46.4				-
CT	NA	NA	No	47.4
G-150 × 100	150	100	No	46.4
G-260 × 100	260	100	No	43.3	4ϕ20	3ϕ20	ϕ8@100	12ϕ8 (plain, placed in two rows)
G-260 × 150	260	150	No	45.9
G-260 × 200	260	200	No	46.5	(deformed)	(deformed)	(plain)
G-300 × 100	300	100	No	41.1
G-260 × 200-F	260	200	Yes	49.9
G-300 × 100-F	300	100	Yes	50.3

The details of the representative test specimens, including Specimens CR, CT and G-150 × 100, are depicted in Figure 2. The dimensions of the beams, the detailed arrangement of the steel reinforcements and the location of the gap are clearly shown in Figure 2. Figure 3 shows the adopted FRP strengthening system for the two test specimens with an FRP-strengthened gap. As shown in Figure 3, two-layer vertical unidirectional carbon FRP (CFRP) U-jackets were installed onto both sides of the gap and below the gap, and the nominal thickness of each CFRP layer was 0.167 mm. The FRP-strengthening system was applied by the wet-layup process as follows: (1) rounding the corners of the strengthening area to a radius of 25 mm to prevent stress concentration; (2) roughening the concrete surface using a needle gun; (3) cleaning the concrete surface and then applying a well-mixed primer (Sikadur 330) to the surface; (4) applying two layers of CFRP sheets which were impregnated with a well-mixed epoxy resin (Sikadur 300); and (5) rolling the CFRP sheets slowly to distribute the resin evenly and release the air bubble. Table 1 shows the cubic compressive strength of the concrete for each specimen and Table 2 shows the material properties of the steel reinforcements and CFRP sheets used in the tests.

Figure 2.

Details of test specimens (dimensions in mm). (a) CR, (b) CT, (c) G-150 × 100.

Figure 3.

FRP-strengthening system (dimensions in mm). (a) G-300 × 100-F, (b) G-260 × 200-F.

Table 2.

Material properties of the used steel bars and FRP.

Properties	Steel bar of 20 mm in diameter	Steel bar of 8 mm in diameter	CFRP sheet
Yield stress (MPa)	456.7	338.7	-
Ultimate stress (MPa)	577.7	509.2	3212.6
Elastic modulus (GPa)	202.6	198.4	277.3

After the tests, it was clearly seen that the two control specimens (i.e., Specimens CR and CT) failed in a typical flexural mode, which occurred when the compressive concrete crushed after the yielding of the tension steel bars; the three T-shaped RC beams with a small un-strengthened gap (i.e., Specimens G-150 × 100, G-260 × 100 and G-260 × 150) exhibited a mixed flexural-shear failure mode due to the formation and progression of shear cracks and the crushing of concrete below the gap; the two T-shaped RC beams with a large un-strengthened gap (i.e., Specimens G-300 × 100 and G-260 × 200) failed in an apparent shear mode due to the formation and development of shear cracks below and on the right side of the gap; and the two T-shaped RC beams with an FRP-strengthened gap (i.e., Specimens G-300 × 100-F and G-260 × 200-F) respectively exhibited a mixed flexural-shear failure mode and a shear failure mode due to the debonding of the applied CFRP U-jackets.

Proposed FE approach

General

A two-dimensional (2D) FE model was established using the general purpose FE software ABAQUS (2014) to conduct FE modelling of T-shaped RC beams with a gap. In the proposed FE model, the concrete was simulated using four-node plane stress element CPS4R; the tension steel bars and steel stirrups were simulated using the two-node truss element T2D2, while the compression steel bars were simulated using the two-node beam element B21, as the remaining compression steel bars in the gap need to transfer the bending moment; the externally-bonded FRP was simulated using the two-node truss element T2D2; in addition, the bond-slip behaviour between the concrete and both the FRP and the steel bars was modelled using the four-node interfacial element COH2D4. The two-node truss element representing the FRP was arranged along the fiber direction of the applied FRP U-jackets; moreover, the nodes representing the corners of the applied FRP U-jackets were restrained with the corresponding concrete nodes (i.e., the surface below the gap and the top surface of the beam within the strengthened region). As a 2D FE model was established, when calculating the cross-sectional area of the truss elements T2D2 for steel bars, it is necessary to take into account the total area of a row of the steel bars; and for the beam element B21, the total area of the compression steel bars must be converted to the area of a single steel bar. All the element size was set to be 10 mm on the basis of the results of a convergence study. The boundary conditions and applied load for the test beams are shown in Figure 4, in which the red lines represent the applied FRP U-jackets. To solve the possible stress concentrations at the two supports and the loading point, the pads with the same elastic modulus as the steel bars were placed at the supports and the loading point respectively, which had a length of 60 mm. The right support allowed rotation and horizontal displacement, while the left support allowed rotation only. It should be noted that for both the rectangular beam and the T-shaped beam, the thickness of the concrete element CPS4R was equal to its actual width, which was 250 mm for the rectangular beam and the web of the T-shaped beam, and 1450 mm for the flange of the T-shaped beam; the thickness of the pads was equal to the thickness of the corresponding concrete elements.

Figure 4.

Typical FE mesh.

Constitutive modelling of concrete

The behaviour of T-shaped RC beams with a gap is dominated by the behaviour of concrete. In the present study, the concrete was simulated using two different concrete cracking models available in ABAQUS/Explicit, that is, the concrete damage plasticity (CDP) model and the brittle cracking (BC) model, for comparison purpose. For the CDP model, the concrete is assumed to be an isotropic material, the damage factor is introduced to simulate the concrete stiffness degradation, and the tensile/compressive stiffness recovery coefficient is introduced to account for the tensile/compressive stiffness recovery under the elastic-plastic state. As a result, the CDP model is suitable for simulating the plastic behaviour of concrete. For the BC model, the concrete tensile behaviour is considered as a non-linear behaviour, which is more suitable for simulating the mechanical properties of concrete in tension and shear state. The two models both have their own advantages in modelling concrete properties and applicable scenarios. Chen et al. (2011) have successfully implemented the CDP model to simulate RC beams and found that it can accurately predict the behaviour of RC beams failing in a flexural mode, while the authors (Nie et al., 2020b, 2021b) have successfully implemented the BC model to simulate RC beams with a web opening failing in a shear mode. Therefore, this study aims at finding out which of the two models is more suitable for simulating T-shaped RC beams with a gap.

Concrete damaged plasticity (CDP) model

The CDP model is based on the assumption that the uniaxial compressive and tensile responses of concrete are characterized by plastic damage, as shown in Figure 5. It should be noted that only the model under monotonic loading is shown in Figure 5, as the focus of this paper is the performance of RC beams under monotonic loading. The uniaxial stress-strain constitutive relationships of concrete under compression and tension are expressed as follows:

σ = (1 - d_{c}) E_{0} (ε - ε^{p l})

(1)

σ_{t} = (1 - d_{t}) E_{0} (ε_{t t} - ε_{t}^{p l})

(2)

where

σ

σ_{t}

and

E_{0}

are the compressive stress, tensile stress and initial elastic modulus of concrete, respectively;

σ_{0}

is the value of the initial yield point, before which the uniaxial compressive response is linear;

σ_{p}

is the maximum compressive stress of concrete;

f_{t}

is the tensile strength of concrete;

ε

and

ε_{t t}

are the compressive strain and tensile strain of concrete, respectively;

ε^{p l}

and

ε_{t}^{p l}

are the compressive plastic strain and tensile plastic strain of concrete, respectively;

ε^{e l}

and

ε_{t}^{e l}

are the compressive elastic strain and tensile elastic strain of concrete, respectively;

d_{c}

and

d_{t}

are respectively the compressive damage and tensile damage, which calibrate the deterioration of the elastic modulus and range from zero (representing the undamaged material) to one (representing total loss of strength).

Figure 5.

Uniaxial stress-strain constitutive relationships of concrete under compression and tension (ABAQUS, 2014). (a) Compression, (b) Tension.

In this study, the compressive stress-strain behaviour of concrete was simulated using the Saenz’s (1964) stress-strain relationship:

σ = \frac{α ε}{1 + [(α ε_{p} / σ_{p}) - 2] (ε / ε_{p}) + {(ε / ε_{p})}^{2}}

(3)

where

ε_{p}

is strain at ultimate uniaxial concrete compressive stress; and

α

is the elastic modulus of concrete. In addition, if no test data has been provided,

σ_{p}

is cylinder compressive strength of concrete

f_{c}

and

ε_{p}

can be set as 0.002;

α

is set to be equal to 4730

\sqrt{f_{c}}

according to ACI-318 (2022).

For the tensile behaviour of concrete, the tension-softening relationship developed by Hordijk (1991) was adopted:

\frac{σ_{t}}{f_{t}} = [1 + {(c_{1} \frac{w_{t}}{w_{c r}})}^{3}] e^{(- c_{2} \frac{w_{t}}{w_{c r}})} - {(1 + c_{1}^{3})}^{3} \frac{w_{t}}{w_{c r}} e^{(- c_{2})}

(4)

w_{c r} = 5.14 \frac{G_{f}}{f_{t}}

(5)

f_{t} = 1.4 {(\frac{f_{c} - 8}{10})}^{\frac{2}{3}}

(6)

G_{f} = (0.0469 D_{a}^{2} - 0.5 D_{a} + 26) {(\frac{f_{c}}{10})}^{0.7}

(7)

where

w_{t}

and

w_{c r}

are respectively the crack opening displacement and the crack opening displacement at the fracture energy complete release;

c_{1}

and

c_{2}

are determined from concrete tensile experiments, which are 3.0 and 6.93, respectively;

G_{f}

is the tensile fracture energy;

f_{t}

and

G_{f}

are calculated using the equations from CEB-FIP (1993), given by equations (6) and (7), respectively, if no experimental data is available;

D_{a}

is the size of the largest aggregate in the concrete, which is equal to be 20 mm in this study.

For the concrete tensile damage, the following two damage models adopted in different FE studies (Chen et al., 2011; Rots, 1988) were employed to evaluate the effect of tensile damage on the simulating results:

d_{t} = 1 - {(1 - \frac{ε_{t}}{ε_{c r}})}^{n}

(8)

d_{t} = \frac{w_{t}}{[w_{t} + (h_{c} σ_{t}) / E_{0}]}

(9)

where

ε_{t}

and

ε_{c r}

are respectively the concrete cracking strain and the concrete cracking strain when the fracture energy completely releases; n influences the degradation rate of the shear retention factor, which was set as five in this study, following Nie (2018); and

h_{c}

is the normal crack band width. For ease of subsequent discussions, the concrete tensile damage defined using equations (8) and (9) was simplified as PL (representing power law) model and ELA (representing elastic) model, respectively. A comparison between the PL model and the ELA model is shown in Figure 6. It can be concluded from Figure 6 that concrete tensile damage defined using the ELA model develops more rapidly than the PL model.

Figure 6.

Comparison between the PL model and the ELA model.

Brittle cracking (BC) model

The BC model simplifies the compressive property of concrete by assuming that the material is linearly elastic under compression. Therefore, only the tensile behaviour of concrete needs to be defined when using the BC model. For the definition of the tensile behaviour of concrete and the shear retention factor, the same models employed in the CDP model were also adopted. In particular, the BC model has the option of a power law expression in defining the tensile damage evolution, and thus only the PL model was adopted to simulate the shear retention factor $β_{s}$ :

β_{s} = {(1 - \frac{ε_{t}}{ε_{c r}})}^{5}

(10)

Furthermore, the tensile behaviour of concrete is treated as linearly elastic before reaching its tensile strength in the BC model, while the actual tensile behaviour of concrete is nonlinear before reaching its tensile strength (Ye 2005), as shown in Figure 7. Therefore, the initial elastic modulus of concrete $E_{0}$ , and the secant modulus of concrete $E_{\sec}$ , which is equal to $0.5 E_{0}$ referred to Ye (2005) and Pimanmas (2010), were both employed in the modelling for comparison.

Figure 7.

Actual tensile stress-strain curve of concrete.

Modelling of steel bars and bond behaviour between steel bars and concrete

The steel bars were assumed as a perfect elastic-plastic material, and the CEB-FIP’s (1993) bond-slip model was used to describe the bond-slip behaviour between the steel bars and the concrete:

τ^{s} = {\begin{cases} τ_{\max}^{s} {(\frac{s}{s_{1}})}^{a} & f o r s \leq s_{1} \\ τ_{\max}^{s} & f o r s_{1} < s \leq s_{2} \\ τ_{\max}^{s} - (τ_{\max}^{s} - τ_{f}^{s}) \frac{s - s_{2}}{s_{3} - s_{2}} & f o r s_{2} < s \leq s_{3} \\ τ_{f}^{s} & f o r s > s_{3} \end{cases}

(11)

where

τ^{s}

and s are respectively the shear stress between steel bars and concrete and the corresponding shear slip;

a

is the parameter for defining interface between steel bars and concrete;

s_{i}

is the parameters for defining the interface between steel bars and concrete (i = 1, 2, 3);

τ_{\max}^{s}

is the ultimate shear strength at the interface between steel bars and concrete;

τ_{f}^{s}

is the residual shear strength at the interface between steel bars and concrete; for deformed steel bars,

a

= 0.4,

s_{1}

s_{2}

= 0.6 mm and

s_{3}

= 1 mm,

τ_{\max}^{s} = 2.0 f_{c}^{0.5}

and

τ_{f}^{s} = 0.15 τ_{\max}^{s}

; for plain steel bars,

a

= 0.5,

s_{1}

s_{2}

s_{3}

= 0.1 mm, and

τ_{\max}^{s} = τ_{f}^{s} = 0.3 \sqrt{f_{c}}

Modelling of FRP and bond behaviour between FRP and concrete

The FRP sheets were modelled as a linear-elastic-brittle material, and the model proposed by Lu et al. (2005) was used to describe the bond behaviour between the concrete and FRP:

τ = {\begin{cases} τ_{\max} \sqrt{s_{f} / s_{0}} (0 \leq s_{f} \leq s_{0}) \\ τ_{\max} e^{- α_{0} (s_{f} / s_{0} - 1)} (s_{0} \leq s_{f}) \end{cases}

(12)

s_{0} = 0.0195 β_{w} f_{t}

(13)

τ_{\max} = {α_{1} β}_{w} f_{t}

(14)

β_{w} = \sqrt{\frac{2 - b_{f} / b_{c}}{1 + b_{f} / b_{c}}}

(15)

f_{t} = 0.395 {(f_{c u})}^{0.55}

(16)

α_{0} = \frac{1}{G_{F} / τ_{\max} s_{0} - 2 / 3}

(17)

G_{F} = 0.308 β_{w}^{2} \sqrt{f_{t}}

(18)

where

τ

and

τ_{\max}

are respectively the shear stress and the bond strength between FRP and concrete;

s_{f}

and

s_{0}

are the slip between FRP and concrete and the slip when the shear stress reaches

τ_{\max}

, respectively;

b_{f}

and

b_{c}

are respectively the width of FRP and the width of the beam;

α_{0}

is the parameter for defining interface between FRP and concrete;

β_{w}

and

G_{F}

are respectively the width scaling factor and the interface fracture energy;

f_{c u}

is the cubic compressive strength of concrete; and

α_{1}

= 1.5, which is the constant for defining interface between FRP and concrete.

Dynamic analysis approach

This study adopted a dynamic analysis approach available in ABAQUS (i.e., ABAQUS/Explicit) to solve the numerical convergence of the FE model. According to the previous studies of the authors (Nie et al., 2020b; Nie et al., 2021b), the loading time was set to be 50T₁, where T₁ is the period of the first mode of the simulated beam; the loading scheme was set as a ramp loading scheme to stabilize the kinetic energy generated during the whole loading process; and the damping factor was set as 1 × 10⁻⁵.

Examined schemes

As explained above, the CDP model and the BC model both have potential to accurately predict the behaviour of T-shaped RC beams with a gap, and thus both models were employed in this study. A total of four numerical schemes were considered for comparison: (1) the behaviour of concrete was simulated with the CDP model and the concrete tensile damage was simulated with the PL factor (simplified as CDP-PL model); (2) the behaviour of concrete was simulated with the CDP model and the concrete tensile damage was simulated with the ELA model (simplified as CDP-ELA model); (3) the behaviour of concrete was simulated with the BC model and the initial elastic modulus of concrete was adopted (simplified as BC-IM model); and (4) the behaviour of concrete was simulated with the BC model and the secant modulus of concrete was adopted (simplified as BC-SM model).

For the control beams, only the two CDP models (i.e., CDP-PL model and CDP-ELA model) were investigated; for the T-shaped RC beams with an un-strengthened/FRP-strengthened gap, the CDP model and the BC model (i.e., the above four numerical schemes) were both investigated.

FE results and comparison

Load-deflection curves

Control beams

The failure mode of the two control beams (Specimens CR and CT) was a typical flexural failure mode, and therefore only the CDP model which has been proved proper in modelling RC beams failing in a flexural mode (Chen et al., 2011; Nie et al., 2021b) to simulate the behaviour of the two control beams.

Figure 8 presents the predicted load-deflection curves of the two control beam (Specimens CR and CT) from the two DP models (i.e., CDP-PL model and CDP-ELA model), in which the test results of the control beams are also shown for comparison. Figure 8 reveals that for both two control beams, the predictions of the load-deflection curves obtained from the CDP-PL model are quite close to the experimental results, except that for Specimen CT, the slop (i.e., stiffness) of the second stage of the predicted load-deflection curve is higher than the experimental result. The possible reasons for such a result are as follows: (1) at the beginning of the loading process of the T-shaped beam, the flange experienced the shear lag effect, resulting in uneven stress distribution in the concrete and steel bars in the flange. However, the 2D model was unable to accurately simulate the shear lag effect, which led to an overestimation of the stress in the concrete and steel bars in the flange, resulting in an overestimation of the beam stiffness; and (2) when the T-shaped RC beam reached the yielding state, the steel bars in the flange all reached the yielding stress and the tensile stress was uniformly distributed, and therefore the 2D model was able to accurately simulate the ultimate load. Moreover, it can be seen from Figure 8 that the predicted stiffness and load-carrying capacity from the CDP-ELA model are both higher than those from the CDP-PL model, which implies that for the simulation of RC beams failing in a typical flexural mode, the CDP-PL model is more suitable than the CDP-ELA model in defining the concrete tensile damage. This conclusion can be supported by the results of the authors’ previous FE studies (Nie et al., 2020b; Nie et al., 2021b).

Figure 8.

Comparison of load-deflection curves between FE predictions and tests: control beams. (a) CR, (b) CT.

T-shaped RC beams with an un-strengthened gap

A total of five T-shaped RC beams with an un-strengthened gap were tested (i.e., Specimens G-150 × 100, G-260 × 100, G-300 × 100, G-260 × 150 and G-260 × 200). The three T-shaped RC beams with a small un-strengthened gap (i.e., Specimens G-150 × 100, G-260 × 100 and G-260 × 150) exhibited a mixed flexural-shear failure mode, and the two T-shaped RC beams with a large un-strengthened gap (i.e., Specimens G-300 × 100 and G-260 × 200) exhibited an apparent shear failure mode.

Figure 9 presents the predicted load-deflection curves from the four numerical schemes (i.e., CDP-PL model, CDP-ELA model, BC-IM model and BC-SM model) and the test results of these beams. Figure 9 reveals that for Specimens G-150 × 100, G-260 × 100, G-260 × 150 with a small un-strengthened gap, the predicted load-deflection curves from the two CDP models experience a descending branch at the third stage of the curves; the BC-IM model overestimates the ultimate loads and the stiffness; while the BC-SM model gives similar predictions of the ultimate loads but overestimates the stiffness of the second stage of the curve, due to the drawback of the 2D FE model, as explained earlier. As can be seen from Figure 9, for Specimens G-300 × 100 and G-260 × 200 with a large un-strengthened gap, the two CDP models underestimate the ultimate loads but overestimate the stiffness, the BC-IM model overestimates the ultimate loads and the stiffness, while the predictions obtained from the BC-SM model show the greatest agreement with the test results. This illustrates that the BC-SM model is able to simulate the RC beams exhibiting a shear failure mode well. The better performance of the BC-SM model over the CDP model is attributed to the difference between the scopes of application of the BC model and the CDP model. The CDP model is more suitable for RC specimens failing in a flexural mode, while the BC model is more suitable for RC specimens failing in a shear mode. The failure of T-shaped RC beams with an un-strengthened gap is mainly governed by the tensile and shear behaviour of cracked concrete but not the compressive behaviour of concrete, and therefore the BC-SM model performs better than the CDP model. Furthermore, the stiffness of the predicted curves from the BC-IM model are higher than that from the BC-SM model, because the former adopts the initial elastic modulus of concrete while the latter adopts the secant modulus of concrete. As can be seen from Figure 7, when the secant modulus of concrete is used, the adopted stress-strain curve of concrete and the actual stress-strain curve of concrete is quite similar; however, when the initial elastic modulus of concrete is used, the adopted stress-strain curve of concrete obviously differs from the actual stress-strain curve of concrete. This explains the better performance of the BC-SM model over the BC-IM model.

Figure 9.

Comparison of load-deflection curves between FE predictions and tests: T-shaped RC beams with an un-strengthened gap. (a) G-150 × 100, (b) G-260 × 100, (c) G-260 × 150, (d) G-300 × 100, (e) G-260 × 200.

T-shaped RC beams with an FRP-strengthened gap

Two T-shaped RC beams with an FRP-strengthened gap were tested, including Specimens G-300 × 100-F and G-260 × 200-F, which respectively exhibited a mixed flexural-shear failure mode and a shear failure mode after the debonding of the applied CFRP U-jackets.

Figure 10 presents the predicted load-deflection curves from the four numerical schemes and the test results of these beams. It can be seen from Figure 10 that for Specimen G-300-100-F, all four models give similar prediction of the load-deflection curve and slightly underestimate the ultimate load. For Specimen G-260 × 200-F, all four models give similar prediction of the ultimate load, which matches well with the test result; the two BC models (i.e., BC-IM model and BC-SM model) provide better predictions of the load-deflection curve than the two DP models (CDP-PL model and CDP-ELA model), as the predicted load-deflection curves from the two DP models experience sharper decline after reaching the ultimate loads than the test curve. The reason why the BC-SM model performs better than the CDP model has been explained earlier.

Figure 10.

Comparison of load-deflection curves between FE predictions and tests: T-shaped RC beams with an FRP-strengthened gap. (a) G-300 × 100-F, (b) G-260 × 200-F.

Comparison of ultimate loads

Table 3 provides a comparison between the test and FE predicted ultimate loads of all the test RC beams. Table 3 reveals that for the two control specimens (i.e., Specimens CR and CT), the CDP-PL model provides the best predictions of the ultimate loads, with the average prediction-to-test ratio, standard deviation (STD) and coefficient of variation (CoV) being 0.989, 0.024, and 0.024, respectively; the CDP-ELA model gives an average prediction-to-test ratio of 1.039, a STD of 0.024 and a CoV of 0.023. For the RC beams with an un-strengthened/FRP-strengthened gap (i.e., Specimens G-150 × 100, G-260 × 100, G-260 × 150, G-300 × 100, G-260 × 200, G-300 × 100-F and G-260 × 200-F), the CDP-PL model and the CDP-ELA model exhibit average prediction-to-test ratios of 0.973 and 0.956, STDs of 0.027 and 0.028, and CoVs of 0.027 and 0.029, respectively; the BC-IM model and the BC-SM model give average prediction-to-test ratios of 1.025 and 0.994, STDs of 0.054 and 0.030, and CoVs of 0.053 and 0.030, respectively. The test ultimate loads of the RC beams with a gap are compared with the results obtained from different FE models, as shown in Figure 11. As can be seen from Figure 11, the ultimate loads obtained from the CDP model are significantly lower than those obtained from the BC model. For the CDP model, the choice of tensile damage model has a significant influence on the FE results, with the model with a faster evolution of the damage factor (i.e., the ELA model) providing lower FE results than the model with a slower evolution of the damage factor (i.e., the PL model), which is consistent with the results of previous FE studies (Nie, 2018). For the BC model, the results obtained from the BC-IM model are generally higher and have a larger scatter than those from the BC-IM model, and the ultimate loads predicted from the BC-SM model match very well with the test results. It can be seen from the comparisons that the predicted results from the BC-SM model match best with the test results of the specimens exhibiting a shear failure mode (i.e., specimens with an un-strengthened/FRP-strengthened gap), while the predicted results from the CDP-PL model match best with the test results of the specimens exhibiting a flexural failure mode (i.e., the two control specimens). This finding coincides with the findings from FE studies on RC beams with an un-strengthened/FRP-strengthened web opening conducted by the authors (Nie et al., 2020b; Nie et al., 2021b). The FE results also verified that increasing the gap height reduces the negative bending capacity of the beam more effectively than increasing the gap length, and the application of the FRP-strengthening system can enhance the ductility of the weakened beam.

Table 3.

Test and predicted ultimate loads.

Specimen	Test result (kN)	CDP-PL		CDP-ELA		BC-IM		BC-SM
		Prediction	Prediction	Prediction	Prediction	Prediction	Prediction	Prediction	Prediction
		(kN)	/Test	(kN)	/Test	(kN)	/Test	(kN)	/Test
CR	327.6	316.0	0.965	332.4	1.015	-	-	-	-
CT	467.1	473.1	1.013	496.6	1.063	-	-	-	-
Average	-	-	0.989	-	1.039	-	-	-	-
STD	-	-	0.024	-	0.024	-	-	-	-
CoV	-	-	0.024	-	0.023	-	-	-	-
G-150-100	428	408.4	0.954	413.2	0.965	435.6	1.018	421.9	0.986
G-260-100	368.3	349.3	0.948	339.5	0.922	368.6	1.001	365.6	0.993
G-260-150	343.4	340.4	0.991	328.3	0.956	360.8	1.051	354.2	1.031
G-300-100	310.0	292.6	0.944	288.6	0.931	318.9	1.029	319.4	1.030
G-260-200	297.6	297.9	1.001	300.4	1.009	337.7	1.135	298.9	1.004
G-300-100-F	336.0	320.7	0.954	314.1	0.935	318.2	0.947	316.4	0.942
G-260-200-F	332.6	337.4	1.014	324.6	0.976	330.9	0.995	322.8	0.971
Average	-	-	0.973		0.956	-	1.025		0.994
STD	-	-	0.027		0.028	-	0.054		0.030
CoV	-	-	0.027		0.029	-	0.053		0.030

Figure 11.

Comparison of ultimate loads obtained from the FE models and tests.

Failure process and failure mode

Figure 12 presents the predicted crack patterns at the failure of the test specimens, in which the predicted crack patterns of the control beams (i.e., Specimens CR and CT) were obtained using the CDP-PL model, and the predicted crack patterns of the beams with an un-strengthened/FRP-strengthened gap (i.e., Specimens G-150 × 100, G-260 × 100, G-260 × 150, G-300 × 100, G-260 × 200, G-300 × 100-F and G-260 × 200-F) were obtained using the BC-SM model; the crack patterns of all test specimens obtained from the tests are also shown in the same figure for comparison. It can be seen from Figure 12 that for all test specimens, the predicted crack patterns match well with the test observations. For the two control specimens, it can be seen from Figure 12(a) and (b) that the flexural cracks fully develop, and strain analyses of the two models showed that the tension steel bars all yield and the compressive strain of the concrete near the loading point exceeds the ultimate compressive strain of concrete, which imply that the two specimens fail in a flexural mode due to the compressive crushing concrete after yielding of tension steel bars. For the three T-shaped RC beams with a small un-strengthened gap, it can be seen from Figure 12(c)–(e) that the diagonal cracks fully develop below the gap, and strain analyses of the three models showed that the tension steel bars all yield and the compressive strain of the concrete near the bottom of the gap exceeds the ultimate compressive strain of concrete, while the compressive strain of the concrete near the loading point does not exceed the ultimate compressive strain of concrete, which imply that the three specimens fail in a mixed flexural-shear mode due to the formation and development of shear cracks and the crushing of concrete below the gap. For the two T-shaped RC beams with a large un-strengthened gap, it can be seen from Figure 12(f) and (g) that the diagonal cracks fully develop below and on the right side of the gap, and the strain analyses of the two models showed that the tension steel bars do not yield and the compressive strain of the concrete near the bottom of the gap and the loading point does not exceed the ultimate compressive strain of concrete, which imply that the two specimens fail in an apparent shear mode due to the formation and development of shear cracks below and on the right side of the gap. For the two T-shaped RC beams with an FRP-strengthened gap, it can be seen from Figure 12(h) and (i) that the applied CFRP U-jackets effectively limit the progression of shear cracks near the gap; the strain analyses of the model of Specimen G-300 × 100-F showed that the tension steel bars all yield and the compressive strain of the concrete near the bottom of the gap exceeds the ultimate compressive strain of concrete, while the compressive strain of the concrete near the loading point does not exceed the ultimate compressive strain of concrete, which imply that this specimen fails in a mixed flexural-shear mode; the strain analyses of the model of Specimen G-260 × 200-F showed that the tension steel bars all yield, but the compressive strain of the concrete near the bottom of the gap and the loading point does not exceed the ultimate compressive strain of concrete, which imply that this specimen fails in a shear mode. In order to further verify the accuracies of the FE models in predicting the crack development of test specimens, three typical specimens (i.e., Specimens CR, G-260 × 100, and G-260 × 200-F) were selected as the example to compare the predicted results on the crack development of different types of RC beams (i.e., control beam, beam with an un-strengthened gap and beam with an FRP-strengthened gap).

Figure 12.

Comparison of crack patterns at failure of test specimens between FE predictions and tests. (a) CR, (b) CT, (c) G-150 × 100, (d) G-260 × 100, (e) G-260 × 150, (f) G-300 × 100, (g) G-260 × 200, (h) G-300 × 100-F, (i) G-260 × 200-F.

Figure 13 shows the predicted crack distributions of Specimen CR at different load levels. The observations from Figure 13 are as follows: a flexural crack first appeared in the mid-span of the beam when the applied load reached 63.3 kN, and the actual cracking load measured in the test was 50 kN; at the load of 100.3 kN, several flexural cracks appeared near the mid-span of the beam; as the load increased to higher levels, the cracks had a tendency to develop towards the loading point, and at the same time, new shear cracks appeared; when the load reached 313.5 kN, the bottom tention steel bars reached the yield strain, and the cracks developed more fully; when the load further increased to 316.0 kN, the failure of the beam occured due to the concrete in the compression zone reached the ultimate compressive strain (0.0033), and the cracks were nearly symmetrically distributed according to the loading point. Meanwhile, it can be seen by comparing Figures 13(f), (g) and 12(a) that the crack distribution and failure mode obtained from the FE model are consistent with the experimental observations.

Figure 13.

Predicted failure process of Specimen CR. (a) 63.3 kN, (b) 100.3 kN, (c) 140.3 kN, (d) 252.4 kN, (e) 313.5 kN, (f) 316.0 kN, (g) Local failure mode.

Figure 14 presents the predicted crack distributions of Specimen G-260 × 100 at different load levels. The observations from Figure 14 are as follows: a flexural crack first appeared near the left edge of the gap when the applied load reached 135.4 kN, and the cracking load recorded in the test was 110 kN, which shows that the 2D FE model slightly overestimates the cracking load of the T-shaped RC beam; as the load increased to 228.1 kN, several flexural cracks appeared in the left shear span of the beam and tended to develop towards the loading point; when the load reached 286.8 kN, diagonal shear cracks continued to appear and develop around the gap; when the load reached 354. 9 kN, the bottom tensile steel bars reached the yield strain, and the cracks developed more fully; when the load further increased to the ultimate load of 368.6 kN, the cracks developed sufficiently in the area directly below and on the right side of the gap. It can be seen from Figure 14(f) that the concrete at the bottom of the gap was crushed, the concrete cover near the two ends of the remaining compression steel bars in the gap peeled, and the remaining compression steel bars in the gap bent, which are consistent with the test observations (see Figure 12(d)).

Figure 14.

Predicted failure process of Specimen G-260 × 100. (a) 135.4 kN, (b) 228.1 kN, (c) 286.8 kN, (d) 354.9 kN, (e) 368.6 kN, (f) Local failure mode.

The predicted crack distributions of Specimen G-260 × 200-F at different load levels are shown in Figure 15. The observations from Figure 15 are as follows: a flexural crack first appeared directly below the left edge of the gap when the applied load reached 112.0 kN, and the cracking load measured in the test was 105 kN; as the load increased to 197.2 kN, a flexural crack appeared in the mid-span of the beam; when the load reached 263.5 kN, the flexural cracks in the left shear span of the beam had a tendency to develop towards the loading point, and at the same time, a shear crack began to develop on the right side of the gap; when the load reached 310.9 kN, yielding of the specimen occurred, and flexural and shear cracks continued to develop around the gap; when the load further increased to 322.8 kN, the beam reached the failure load, and the compression steel bars at the gap were bent, which are quite consistent with the phenomenon observed in the tests (Figure 12(i)). It can be seen by comparing Figure 15(e) and 12(g) that the application of the FRP-strengthening system effectively limited the development of the cracks around the gap.

Figure 15.

Predicted failure process of Specimen G-260 × 200-F. (a) 112.0 kN, (b) 197.2 kN, (c) 263.5 kN, (d) 310.9 kN, (e) 322.8 kN, (f) Local failure mode.

Concluding remarks

In the present study, a numerical study has been carried out to simulate the structural behaviour of T-shaped RC beams with an un-strengthened/FRP-strengthened gap. The established FE models were developed using ABAQUS, and a total of four FE schemes (i.e., the concrete damaged plasticity model with a power law tensile damaged model (CDP-PL model), the concrete damaged plasticity model with a elastic tensile damaged model (CDP-ELA model), the brittle cracking model with initial modulus of concrete adopted (BC-IM model) and the brittle cracking model with secant modulus of concrete adopted (BC-SM model)) were examined to find out the most proper model for simulating the behaviour of T-shaped RC beams with a gap. Based on the results of the FE modelling, the following conclusions can be made:

(1) For the control specimens (i.e., RC rectangular beam and T-shaped beam without a gap) which exhibited a flexural failure mode, the CDP-PL model provides the best predictions in terms of load-deflection curves, ultimate loads, failure modes and crack distributions, while the CDP-ELA model overestimates the ultimate strength and stiffness of beams;

(2) For the T-shaped RC beams with an un-strengthened/FRP-strengthened gap which exhibited a shear-dominated failure mode, the BC-SM model provides the best predictions, while the CDP model normally underestimates the ultimate strength and the BC-IM model usually overestimates the ultimate strength and stiffness of beams;

(3) It can be deduced from the FE results of the present study that the selection of an appropriate FE approach for RC beams should depend on the failure modes of beams. Specifically, the CDP-PL model is recommended to be adopted for the RC beams with a flexural failure mode, while the BC-SM model is recommended to be adopted for the RC beams with a shear failure mode;

(4) The established 2D FE models in this study overestimate the stiffness of the T-shaped RC beams, and thus a more advanced 3D FE model needs to be proposed in future work. To develop an accurate 3D model, the greatest challenge is the building of the cohesive element between steel bars and concrete. In a 3D model, the steel bars are usually simulated using the two-node element, while the concrete is usually simulated using the eight-node element, so the cohesive element between steel bar and concrete cannot be directly built as in a 2D model. Furthermore, the proposed FE approach can be used for the subsequent proposal of strength model and design method of RC beams with a gap, in order for the engineering application of the section reduction technique; and

(5) It is worth noting that the authors (Zhang et al., 2024a) have conducted another FE study on RC beam-to-column joints with beam web openings, and the FE models of the joint specimens used the same constitutive models of materials as those used in the FE models in this study. The FE predictions matched well with the test results, demonstrating the effectiveness of the FE approach in simulating the behaviour of RC beam-to-column joints with beam gaps in the future study.

Footnotes

Declaration of conflicting interests

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

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The authors are grateful for the financial support received from the National Natural Science Foundation of China (Project No. 52008183).

ORCID iDs

Shishun Zhang

Xuefei Nie

Appendix

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