*3.1. Finite Element Modeling of Nonlinear Behavior of Beam-Column Joint* 3.1.1. Finite Element Method

The finite element method (FEM) is the most widely used in numerical simulation of structures [35]. Finite element models have the potential to solve a wide range of complex problems from elastic linear models for linear elements to highly plastic models for nonlinear and solid elements. FEM is one of the leading methods to simulate all types of structures (timber, steel, concrete, masonry) [36].

### 3.1.2. Abaqus Software

To perform numerical simulation of beam-column joint using RPC a FEM-based software ABAQUS/CAE was selected, which is general-purpose analysis software having the capability of solving the elastic and inelastic problems of the static and dynamic response of components [37].

ABAQUS/CAE 6.14-1 VERSION was used for modeling and analysis of beam-column joint using RPC.

### 3.1.3. Concrete Damage Plasticity Model

The concrete damage plasticity (CDP) model was selected as it has the capability and potential for modeling reinforced concrete and other quasi-brittle material for different types of structure. CDP model can define the nonlinear behavior of the RPC beam-column joint. Additionally, it takes into account the isotropic damage elasticity concepts with isotropic tensile and compressive plasticity. It also considers the degradation of elastic stiffness produced by plastic straining both in compression and tension [38]. The CDP model can show damage characteristics of a material. The main failure mechanism that this model assumes is the tensile cracking and the compressive crushing [39].

Different parameters required in the CDP model were studied and selected based on available literature both for conventional as well RPC specimens. The dilation angle for the model was taken as 36◦ . It is the angle obtained due to a change in volumetric strain produced due to plastic shearing. It depends on the angle of internal friction. Dilation angle controls the amount of plastic volumetric strain produced due to plastic shearing. Normally dilation angle is taken between 30◦ and 40◦ for concrete to avoid large variation between experimental work and numerical modeling. For the seismic design of reinforced concrete, the value of dilation angle is normally between 35◦ to 38◦ [40]. Moreover, eccentricity is the deviation from the center. The default value for eccentricity was taken, i.e., 0.1. If the value

is increased by 0.1 the curvature of flow potential is increased. If the value is decreased from the default value, the convergence problem may occur if confinement pressure is not high enough. Furthermore, the ratio of biaxial loading (*f<sup>b</sup>* ) to uniaxial loading (*fc*0) is normally taken as 1 or greater than 1. In this case default value was taken i.e., *fb*/*fc*<sup>o</sup> = 1.16. K is the shape factor and default value for K = 0.667 [41]. The viscosity parameter shows the amount of flow potential in a material. A lower viscosity parameter value is better as higher values result in a high force of reaction. Therefore, the viscosity parameter, in this case, was taken as 0.001 [42]. *εc*1 is the strain at peak stress *εcu* is the ultimate strainat which concrete fails Equations (2) and (3) are only pertinent to concrete having a cylindrical compressive strength of 50 MPa and cube compressive strength of 60 MPa at the most. On the basis of a list of the experimental results, Kmiecik and Kamiński [44] proposed the quite accurate approximating Equations (4) and (5): 1 = 0.0014 ሾ2 െ expሺെ0.024 ሻ െ expሺ0.140 ሻ (4) = 0.004 െ 0.0011ሾ1 െ expሺെ0.0215 ሻሿ (5)

The compressive behavior of concrete was calculated by using the relations of Euro-

Other values showing the position of characteristics points are strain *εc*1 at average

= 22ሺ0.1ሻ.ଷ (1)

1 = 0.7ሺሻ0.31 (2)

= 0.35% (3)

### *3.2. Compressive and Tensile Behavior Determination by Using Eurocode* Knowing the values of the output in Equations (4) and (5) one can determine the

*Crystals* **2021**, *11*, x FOR PEER REVIEW 10 of 22

3.2.1. Compressive Behavior

where:

where:

code [43] given in Equation (1).

*fcm* (MPa) is the compressive strength *Ecm* (GPa) is the modulus of elasticity

compressive strength and ultimate strain *εcu* at 0.

Compressive behavior and tensile behavior of both normal concrete and RPC were determined by using EN 1992 Eurocode 2: Design of concrete structures part 1–1 [43]. It describes different principles and requirements for the safety, serviceability, and durability of concrete structures with specific provisions of buildings. Eurocode 2 applies to the design of civil engineering works such as buildings, roads, bridges, etc. It is applied to plain, reinforced, and prestressed concretes. It complies with the specifications and requirements given in EN 1992-1-1 about safety, serviceability of the structures, the basis of their design, and verification of structures given in EN 1990; basis of structural design [38]. Compressive and tensile stress-strain curves are shown in Figures 11 and 12, respectively. The limitation of the Eurocode 2 for concrete structures is that it is concerned only with the requirements for resistance, safety, serviceability, durability, and fire resistance of the structures. Moreover, it does not consider the other requirements like thermal or sound insulation, etc. [38]. points at which the graphs intersect. Compressive stress values can be determined at any point using these relations [43]. According to Eurocode EN 1992-1-1 = ሺ െ ଶሻ/ሺ1 ሺെ2ሻ (6) where: = 1.05 ∗ ൬ 1 ൰ (7) and: = (8) *Crystals* **2021**, *11*, x FOR PEER REVIEW 11 of 22

**Figure 11. Figure 11.** Stress-strain diagra Stress-strain diagram for analysis of concrete using Eurocode 2 [ m for analysis of concrete using Eurocode 2 [43]. 43].

**Figure 12.** Tensile stress-strain curve of concrete from Eurocode [43]. **Figure 12.** Tensile stress-strain curve of concrete from Eurocode [43].

For the determination of the complete stress-strain curve for compressive behavior and the tensile behavior of normal concrete and RPC, Eurocode has been used which is capable of determining the actual response of structures closer to the experimental setup.

Simple modifications were incorporated into the Equations (1), (3), and (12). These equations were utilized in assigning material properties during numerical modeling in ABAQUS to obtain more realistic results of RPC concrete. Modified equations are shown

After the completion of assembly, a step was formed. In steps, a time period was provided for which the load is applied to the assembly. The load was then applied to the designated location according to the magnitude of the sample and boundary conditions were applied according to experimental work in which two specimens (CC\_S1 and RPC\_S1) have hinge boundary condition, i.e., (U1 = U2 = UR3 = 0) while the other two specimens have fixed boundary condition (U1 = U2 = U3 = UR1 = UR2 = UR3 = 0) at column end while roller support (U1 = UR2 = UR3 = 0) at the beam end in all specimens. "U" refers

= 22ሺ0.13ሻ.ଷ (13)

 = 0.40% (14) = 0.40 ∗ .ହ (15)

for the determination of stress-strain curves of RPC.

in Equations (13)–(15).

*3.3. Steps and Boundary Conditions* 

3.2.1. Compressive Behavior

The compressive behavior of concrete was calculated by using the relations of Eurocode [43] given in Equation (1).

$$\text{Ecm} = \text{22}(0.1fcm)^{0.3} \tag{1}$$

where:

*fcm* (MPa) is the compressive strength

*Ecm* (GPa) is the modulus of elasticity

Other values showing the position of characteristics points are strain *εc*<sup>1</sup> at average compressive strength and ultimate strain *εcu* at 0.

$$
\epsilon \varepsilon 1 = 0.7(fcm)^{0.31} \tag{2}
$$

$$
\varepsilon\omega = 0.35\% \tag{3}
$$

where:

*εc*<sup>1</sup> is the strain at peak stress

*εcu* is the ultimate strain at which concrete fails

Equations (2) and (3) are only pertinent to concrete having a cylindrical compressive strength of 50 MPa and cube compressive strength of 60 MPa at the most. On the basis of a list of the experimental results, Kmiecik and Kami´nski [44] proposed the quite accurate approximating Equations (4) and (5):

$$
\epsilon \epsilon 1 = 0.0014 \left[ 2 - \exp(-0.024 \, fcm) - \exp(0.140 \, fcm) \right] \tag{4}
$$

$$
\varepsilon u = 0.004 - 0.0011[1 - \exp(-0.0215 \, fcm)]\tag{5}
$$

Knowing the values of the output in Equations (4) and (5) one can determine the points at which the graphs intersect. Compressive stress values can be determined at any point using these relations [43].

According to Eurocode EN 1992-1-1

$$\sigma c = f c m (k \eta - \eta^2) / (1 + (k - 2)\eta \tag{6}$$

where:

$$k = 1.05 \ast Ecn \left(\frac{\varepsilon c1}{fcm}\right) \tag{7}$$

and:

$$
\eta = \frac{\varepsilon c}{\varepsilon cl} \tag{8}
$$

3.2.2. Tensile Behaviors

The tensile behavior of concrete was calculated by using the Equations (9)–(12). If *εt* ≤ *εcr*

*σt* = *Ec* ∗ *εt* (9)

and if *εt* > *εcr*

$$
\sigma t = f \text{cm}(\frac{\varepsilon \sigma}{\varepsilon t})^{0.4} \tag{10}
$$

$$ft = 0.33 \* ft^{0.5} \tag{11}$$

$$ftr = 0.30fck^{2/3} \tag{12}$$

For the determination of the complete stress-strain curve for compressive behavior and the tensile behavior of normal concrete and RPC, Eurocode has been used which is capable of determining the actual response of structures closer to the experimental setup. As RPC is a composite material and there is no official code for RPC developed yet, therefore small modifications based on literature have been made in normal concrete formulas for the determination of stress-strain curves of RPC.

Simple modifications were incorporated into the Equations (1), (3), and (12). These equations were utilized in assigning material properties during numerical modeling in ABAQUS to obtain more realistic results of RPC concrete. Modified equations are shown in Equations (13)–(15).

$$\text{Ecm} = \text{22}(0.13fcm)^{0.3} \tag{13}$$

$$
\varepsilon\upsilon = 0.40\% \tag{14}
$$

$$ft = 0.40 \* fc^{0.5} \tag{15}$$

### *3.3. Steps and Boundary Conditions*

After the completion of assembly, a step was formed. In steps, a time period was provided for which the load is applied to the assembly. The load was then applied to the designated location according to the magnitude of the sample and boundary conditions were applied according to experimental work in which two specimens (CC\_S1 and RPC\_S1) have hinge boundary condition, i.e., (U1 = U2 = UR3 = 0) while the other two specimens have fixed boundary condition (U1 = U2 = U3 = UR1 = UR2 = UR3 = 0) at column end while roller support (U1 = UR2 = UR3 = 0) at the beam end in all specimens. "U" refers to translatory motion while "UR" refers to rotation of the support. Both the boundary conditions for the column were studied and their effect on the strength and load values were observed. *Crystals* **2021**, *11*, x FOR PEER REVIEW 12 of 22 to translatory motion while "UR" refers to rotation of the support. Both the boundary conditions for the column were studied and their effect on the strength and load values were observed. *3.4. Meshing* 

### *3.4. Meshing* Meshing is the process of dividing the whole finite element model into a smaller

Meshing is the process of dividing the whole finite element model into a smaller number of chunks by the formation of different nodes at different points. Meshing is an important process as it allows us to apply load and find displacement or any other desired result at any point in the model. The greater the size of the mesh, the smaller will be the number of iterations taken to analyze the whole model and vice versa. In a greater size mesh, a lesser number of nodes are formed, hence the number of iterations and time of analysis is reduced. In our case, the size of the mesh taken was 25 mm, 40 mm, and 50 mm. Independent types of meshing for concrete and steel are selected in Figures 13–15 [42]. number of chunks by the formation of different nodes at different points. Meshing is an important process as it allows us to apply load and find displacement or any other desired result at any point in the model. The greater the size of the mesh, the smaller will be the number of iterations taken to analyze the whole model and vice versa. In a greater size mesh, a lesser number of nodes are formed, hence the number of iterations and time of analysis is reduced. In our case, the size of the mesh taken was 25 mm, 40 mm, and 50 mm. Independent types of meshing for concrete and steel are selected in Figures 13–15 [42].

**Figure 13.** Meshing of 25 mm size for concrete. **Figure 13.** Meshing of 25 mm size for concrete.

**Figure 14.** Meshing of steel embedded in concrete.

**Figure 14. Figure 14.**  Meshing of steel embedded in concrete. Meshing of steel embedded in concrete.

**Figure 13.** Meshing of 25 mm size for concrete.

**Figure 15.** Typical view of Abaqus model of reinforcement. **Figure 15.** Typical view of Abaqus model of reinforcement.

### *3.5. Load 3.5. Load*

were observed.

*3.4. Meshing* 

[42].

A series of analysis was performed on conventional concrete controlled specimens and RPC specimens to simulate and predict the actual response of linear and nonlinear behavior on beam-column joint and to show the behavior of RPC in improving the shear strength deformation against different structural loading. A monotonic load of 0 to 20 kN was applied at the top of the exterior joint for all specimens till the specimens reached the ultimate value. After the application of load, step was created for static analysis. Time period and increment values were given to all specimens**.** Figures 16 and 17 show the analysis of RPC samples with fixed and hinge boundary conditions, respectively. As seen in Figure 16, a small amount of buckling was observed when the column boundary condition was kept fixed, whereas no buckling was observed in the case of hinge column conditions as seen in Figure 17. A series of analysis was performed on conventional concrete controlled specimens and RPC specimens to simulate and predict the actual response of linear and nonlinear behavior on beam-column joint and to show the behavior of RPC in improving the shear strength deformation against different structural loading. A monotonic load of 0 to 20 kN was applied at the top of the exterior joint for all specimens till the specimens reached the ultimate value. After the application of load, step was created for static analysis. Time period and increment values were given to all specimens. Figures 16 and 17 show the analysis of RPC samples with fixed and hinge boundary conditions, respectively. As seen in Figure 16, a small amount of buckling was observed when the column boundary condition was kept fixed, whereas no buckling was observed in the case of hinge column conditions as seen in Figure 17.

to translatory motion while "UR" refers to rotation of the support. Both the boundary conditions for the column were studied and their effect on the strength and load values

Meshing is the process of dividing the whole finite element model into a smaller number of chunks by the formation of different nodes at different points. Meshing is an important process as it allows us to apply load and find displacement or any other desired result at any point in the model. The greater the size of the mesh, the smaller will be the number of iterations taken to analyze the whole model and vice versa. In a greater size mesh, a lesser number of nodes are formed, hence the number of iterations and time of analysis is reduced. In our case, the size of the mesh taken was 25 mm, 40 mm, and 50 mm. Independent types of meshing for concrete and steel are selected in Figures 13–15

conditions as seen in Figure 17.

**Figure 15.** Typical view of Abaqus model of reinforcement.

A series of analysis was performed on conventional concrete controlled specimens and RPC specimens to simulate and predict the actual response of linear and nonlinear behavior on beam-column joint and to show the behavior of RPC in improving the shear strength deformation against different structural loading. A monotonic load of 0 to 20 kN was applied at the top of the exterior joint for all specimens till the specimens reached the ultimate value. After the application of load, step was created for static analysis. Time period and increment values were given to all specimens**.** Figures 16 and 17 show the analysis of RPC samples with fixed and hinge boundary conditions, respectively. As seen in Figure 16, a small amount of buckling was observed when the column boundary condition was kept fixed, whereas no buckling was observed in the case of hinge column

*3.5. Load* 

**Figure 16.** Analysis of RPC with fixed column boundary condition. **Figure 16.** Analysis of RPC with fixed column boundary condition.

**Figure 17.** Analysis of concrete with hinge column boundary condition. **Figure 17.** Analysis of concrete with hinge column boundary condition.

### **4. Results and Discussion 4. Results and Discussion**

Experimental results of shear strength-deformation improvement for vulnerable beam-column connection using RPC were used to validate the developed FEM approach. Different parameters from experimental work were used in numerical modeling. This approach provided a more realistic response simulation of the actual beam-column joint. The numerical results were compared with experimental results for the verification of the model as shown in Table 3. There was a negligible deviation of numerical results from experimental results for controlled concrete samples in case of maximum loads. It was 1.13% and 0.63% for CC\_S1 and CC\_S2, respectively. For RPC samples, the divergence was comparatively higher. It was 6.05% for RPC\_S1 and 6.73% for RPC\_S2. Maximum displacement variation was 7.06%, 4.18%, 3.12%, and 6.54% for CC\_S1, CC\_S2, RPC\_S1, and RPC\_S2, respectively. The maximum variation observed was 7.06% for CC\_S1 for displacement. This shows that numerical results were in strong agreement with the experi-Experimental results of shear strength-deformation improvement for vulnerable beamcolumn connection using RPC were used to validate the developed FEM approach. Different parameters from experimental work were used in numerical modeling. This approach provided a more realistic response simulation of the actual beam-column joint. The numerical results were compared with experimental results for the verification of the model as shown in Table 3. There was a negligible deviation of numerical results from experimental results for controlled concrete samples in case of maximum loads. It was 1.13% and 0.63% for CC\_S1 and CC\_S2, respectively. For RPC samples, the divergence was comparatively higher. It was 6.05% for RPC\_S1 and 6.73% for RPC\_S2. Maximum displacement variation was 7.06%, 4.18%, 3.12%, and 6.54% for CC\_S1, CC\_S2, RPC\_S1, and RPC\_S2, respectively. The maximum variation observed was 7.06% for CC\_S1 for displacement. This shows that numerical results were in strong agreement with the experimental results.

The comparison of the load-displacement curve obtained from experimental and ABAQUS simulations are shown in Figures 18–21. The shape of the ABAQUS simulations curves is quite close to the experimental curves. The maximum average discrepancy between modeling and experimental results of conventional concrete was 3–7%. Almost linear behavior was obtained using ABAQUS modeling for CC\_S1 whereas in experimental work the pattern of the graph showed nonlinearity which might be due to non-uniform

mental results. **Table 3.** Comparison of load and displacement between experimental and modeling.


CC\_S1 21.03 27,497.88 15,500 15,675.21 35.98 33.44 2.54

RPC\_S2 45.24 37,433.67 16,020 17,097.80 43.33 40.50 2.83

4.1.1. Conventional Concrete Controlled Specimens

increment of load in the experimental setup.

*4.1. Load Displacement Curve* 

### *4.1. Load Displacement Curve*

## 4.1.1. Conventional Concrete Controlled Specimens

The comparison of the load-displacement curve obtained from experimental and ABAQUS simulations are shown in Figures 18–21. The shape of the ABAQUS simulations curves is quite close to the experimental curves. The maximum average discrepancy between modeling and experimental results of conventional concrete was 3–7%. Almost linear behavior was obtained using ABAQUS modeling for CC\_S1 whereas in experimental work the pattern of the graph showed nonlinearity which might be due to non-uniform increment of load in the experimental setup. *Crystals* **2021**, *11*, x FOR PEER REVIEW 15 of 22

**Figure 18.** Displacement at beam end (roller support) with a hinge boundary condition at column **Figure 18.** Displacement at beam end (roller support) with a hinge boundary condition at column end for CC\_S1 (mesh size 25 mm). 5

### 4.1.2. RPC Specimens 0

end for CC\_S1 (mesh size 25 mm).

15 20 CC S2 Modelling CC S2 Experimental The maximum discrepancy between modeling and experimental results of RPC in the case of RPC\_S1 was 6.05% while that of RPC\_S2 was 6.7%. The deviation of experimental results from modeling in RPC\_S1 was due to non-uniform increment of load and time period in the experimental setup while RPC\_S2 showed quite accurate results. Mesh size effect was studied for RPC specimens and compared with the experimental results Figures 22 and 23. Mesh size 25 was considered for RPC\_S1 and mesh size 40 for RPC\_S2 for comparison with the experimental values. **Figure 18.** Displacement at beam end (roller support) with a hinge boundary condition at column end for CC\_S1 (mesh size 25 mm). 0 10 20 30 40 Dispacement (mm)

the case of RPC\_S1 was 6.05% while that of RPC\_S2 was 6.7%. The deviation of experimental results from modeling in RPC\_S1 was due to non-uniform increment of load and time period in the experimental setup while RPC\_S2 showed quite accurate results. Mesh **Figure 19.** Displacement at beam end (roller support) with a fixed boundary condition at column end for CC\_S2 (mesh size 25 mm). **Figure 19.** Displacement at beam end (roller support) with a fixed boundary condition at column end for CC\_S2 (mesh size 25 mm).

size effect was studied for RPC specimens and compared with the experimental results

The maximum discrepancy between modeling and experimental results of RPC in the case of RPC\_S1 was 6.05% while that of RPC\_S2 was 6.7%. The deviation of experimental results from modeling in RPC\_S1 was due to non-uniform increment of load and time period in the experimental setup while RPC\_S2 showed quite accurate results. Mesh size effect was studied for RPC specimens and compared with the experimental results Figures 22 and 23. Mesh size 25 was considered for RPC\_S1 and mesh size 40 for RPC\_S2

for comparison with the experimental values.

for comparison with the experimental values.

4.1.2. RPC Specimens

**Figure 20.** Displacement at beam end (roller support) with hinge boundary condition at column end for RPC\_S1. **Figure 20.** Displacement at beam end (roller support) with hinge boundary condition at column end for RPC\_S1. 5

### *4.2. Comparison between Conventional Concrete and RPC Specimens* 0 10 20 30 40 50

20

0

25 RPC S1 EXP RPC mesh 25 mm RPC mesh 40 mm RPC mesh 45 mm Comparison between conventional concrete and RPC specimen is shown in Figures 22 and 23. RPC specimens took 10–15% more load as compared to conventional concrete-controlled specimens. Delayed peaks were obtained for RPC specimens which shows delayed damaging effect in the samples. **Figure 20.** Displacement at beam end (roller support) with hinge boundary condition at column end for RPC\_S1. Displacement (mm)

22 and 23. RPC specimens took 10–15% more load as compared to conventional concretecontrolled specimens. Delayed peaks were obtained for RPC specimens which shows delayed damaging effect in the samples. **Figure 21.** Displacement at beam end (roller support) with fixed boundary condition at column end for RPC\_S2. **Figure 21.** Displacement at beam end (roller support) with fixed boundary condition at column end for RPC\_S2.

Comparison between conventional concrete and RPC specimen is shown in Figures 22 and 23. RPC specimens took 10–15% more load as compared to conventional concretecontrolled specimens. Delayed peaks were obtained for RPC specimens which shows de-

layed damaging effect in the samples.

*4.2. Comparison between Conventional Concrete and RPC Specimens* 

**Figure 22.** Comparison between concrete and RPC specimens for S1 (column hinge condition). **Figure 22.** Comparison between concrete and RPC specimens for S1 (column hinge condition).

### CC S2 Modelling CC S2 Experimental *4.3. Stiffness of Concrete and RPC Specimens* 10 Load (KN)

10 15 20 Load (KN) RPC S2 Modelling RPC S2 Experimental For the validation of the model, the stiffness of all specimens was calculated. It can be seen from Tables 4 and 5 that the initial stiffness in the RPC specimens is low in comparison to conventional concrete. Moreover, it can be observed that as the load increased the structure lost its rigidity and stiffness (the ability of a structure to resist deformation when subjected to the applied force). However, after 20% of loading RPC was still taking more load in comparison to conventional concrete as shown in Tables 4 and 5. It can also be observed from Figures 24–26 that the initial stiffness of RPC specimens is low compared to controlled concrete specimens. Figures 25 and 26 depict that as the load was increased to 25% and 50% of the ultimate load, RPC showed to have high stiffness comparatively. **Figure 22.** Comparison between concrete and RPC specimens for S1 (column hinge condition). 0 5 0 10 20 30 40 Displacement (mm)

pared to controlled concrete specimens. Figures 25 and 26 depict that as the load was increased to 25% and 50% of the ultimate load, RPC showed to have high stiffness compar-**Figure 23.** Comparison between concrete and RPC specimens for S2 (column fixed condition). **Figure 23.** Comparison between concrete and RPC specimens for S2 (column fixed condition).

be observed from Figures 24–26 that the initial stiffness of RPC specimens is low compared to controlled concrete specimens. Figures 25 and 26 depict that as the load was increased to 25% and 50% of the ultimate load, RPC showed to have high stiffness compar-

For the validation of the model, the stiffness of all specimens was calculated. It can be seen from Tables 4 and 5 that the initial stiffness in the RPC specimens is low in comparison to conventional concrete. Moreover, it can be observed that as the load increased the structure lost its rigidity and stiffness (the ability of a structure to resist deformation when subjected to the applied force). However, after 20% of loading RPC was still taking

*4.3. Stiffness of Concrete and RPC Specimens* 

atively.

atively.


**Table 4.** Stiffness of all specimens from experimental work.

*Crystals* **2021**, *11*, x FOR PEER REVIEW 18 of 22

**Table 5.** Stiffness of all specimens at different loading rates (numerically). 50% 1.48 1.96 2.53 2.43


CC\_S1\_EXP CC\_S1\_MOD CC\_S2\_EXP CC\_S2\_MOD

**Stiffness (KN/mm)**

**Figure 24.** Initial stiffness of all specimens at 5% of ultimate loading. **Figure 24.** Initial stiffness of all specimens at 5% of ultimate loading.

### 223

**Stiffness (KN/mm)**

RPC\_S1\_EXP RPC\_S1\_MOD RPC\_S2\_EXP RPC\_S2\_MOD

**Figure 25.** Stiffness of all specimens at 25% of ultimate loading. **Figure 25.** Stiffness of all specimens at 25% of ultimate loading. CC\_S1\_EXP CC\_S1\_MOD CC\_S2\_EXP CC\_S2\_MOD

### *4.4. Ductility of Concrete and RPC Specimens* 5

1.28

1.48

0

(17), respectively.

1

1

1.96 2.53 1.83 2.43 1.94 2 3 CC\_S1\_EXP CC\_S1\_MOD CC\_S2\_EXP CC\_S2\_MOD RPC\_S1\_EXP RPC\_S1\_MOD RPC\_S2\_EXP RPC\_S2\_MOD The ductility displacement factor (R), as depicted in Figure 27, according to the Committee Euro International Du Beton, 1996, is defined as the ratio between failure displacement and yield displacement. The yield displacement is the lateral displacement at 80% of the ultimate load at ascending part of the curve while the failure displacement is the lateral displacement at 80% of the ultimate load at descending part of the curve [26]. Ductility factor R and ductility displacement (DD) can be obtained from Equations (16) and (17), respectively. 2.24 2.49 3.23 3.06 4.59 4.11 4.38 4.04 2 3 4

$$R = \frac{\Delta f}{\Delta y} \tag{16}$$

where ∆*f* = failure displacement ∆*y* = yield displacement 0

**Figure 25.** Stiffness of all specimens at 25% of ultimate loading.

$$DD = \frac{\Delta i}{\Delta y} \tag{17}$$

∆ (16)

**Stiffness (KN/mm)** where ∆*i* = maximum displacement in any cycle I ∆*y* = yield displacement

1

1.47

**Figure 26.** Stiffness of all specimens at 50% of ultimate loading. *4.4. Ductility of Concrete and RPC Specimens*  The ductility displacement factor (R), as depicted in Figure 27, according to the Committee Euro International Du Beton, 1996, is defined as the ratio between failure displacement and yield displacement. The yield displacement is the lateral displacement at 80% of the ultimate load at ascending part of the curve while the failure displacement is the lateral displacement at 80% of the ultimate load at descending part of the curve [26]. Ductility factor R and ductility displacement (DD) can be obtained from Equations (16) and (17), respectively. = ∆ ∆ (16) 1.48 1.28 1.96 1.47 2.53 1.83 2.43 1.94 0 1 2 3 1 CC\_S1\_EXP CC\_S1\_MOD CC\_S2\_EXP CC\_S2\_MOD RPC\_S1\_EXP RPC\_S1\_MOD RPC\_S2\_EXP RPC\_S2\_MOD

where ∆*f* = failure displacement ∆*y* = yield displacement **Stiffness (KN/mm)**

The ductility displacement factor (R), as depicted in Figure 27, according to the Committee Euro International Du Beton, 1996, is defined as the ratio between failure displacement and yield displacement. The yield displacement is the lateral displacement at 80% of the ultimate load at ascending part of the curve while the failure displacement is the lateral displacement at 80% of the ultimate load at descending part of the curve [26]. Ductility factor R and ductility displacement (DD) can be obtained from Equations (16) and

> = ∆

**Figure 26.** Stiffness of all specimens at 50% of ultimate loading. **Figure 26.** Stiffness of all specimens at 50% of ultimate loading.

where ∆*f* = failure displacement ∆*y* = yield displacement

**Figure 27.** Ductility displacement. **Figure 27.** Ductility displacement.

Table 6 displays ductility factor R and displacement ductility (DD) for all the specimens both for experimental and numerical modeling. Experimental results showed an increase of 27% and 11% in R for S1 and S2, respectively. Similarly, DD for RPC was enhanced by 29% and 12% for S1 and S2, respectively. The same trend was shown by numerical modeling results. Table 6 displays ductility factor R and displacement ductility (DD) for all the specimens both for experimental and numerical modeling. Experimental results showed an increase of 27% and 11% in R for S1 and S2, respectively. Similarly, DD for RPC was enhanced by 29% and 12% for S1 and S2, respectively. The same trend was shown by numerical modeling results.

=

where ∆*i* = maximum displacement in any cycle I ∆*y* = yield displacement

∆

∆ (17)

### **Table 6.** Ductility factor and ductility displacement. **Table 6.** Ductility factor and ductility displacement.


### A series of analysis were performed on conventional concrete and RPC beam-column **5. Conclusions**

**5. Conclusions** 

joint specimens. Following conclusions were made based on experimental and numerical testing. 1. The use of RPC only in the joint region increased the overall strength of the structure A series of analysis were performed on conventional concrete and RPC beam-column joint specimens. Following conclusions were made based on experimental and numerical testing.


6. To obtain actual results, displacement control analysis should be used rather than load control analysis. With displacement control analysis it is easier to obtain the converged solutions in ABAQUS in case of highly nonlinear problems.

### **6. Recommendations**

Based on this research following recommendations can be used for future research work.

In the case of RPC, a decrease in the initial stiffness of the specimen was observed in the joint region as coarse aggregates were not used. Therefore, the use of suitable size of coarse aggregate will not only increase the initial stiffness but also increase the strength of the structure.

Steel fibers of a longer length should be used so that they can help in controlling the crack from widening. In the case of RPC, shear cracking was observed in the joint region. Combining the RPC technique with some other technique will convert the failure mechanism from the joint to the beam through their combined effect.

Besides the CDP model smeared crack modeling and brittle concrete modeling of the RPC can be used to determine the complex behavior of RPC in structure.

As there is no official Eurocode for RPC. Therefore, the development of Eurocode for RPC with and without steel fibers will enable us to clearly understand the complicated behavior of the material.

**Author Contributions:** Supervision, review, and editing, A.N.; data curation and methodology, M.A.; investigation and review, M.F.J.; conceptualization, data analysis, writing original draft preparation, M.F.J.; formal analysis and modeling, F.A.; validation, proofreading, review, M.A.M. and N.I.V. All authors have read and agreed to the published version of the manuscript.

**Funding:** The research is partially funded by the Ministry of Science and Higher Education of the Russian Federation under the strategic academic leadership program 'Priority 2030' (Agreement 075-15-2021-1333 dated 30 September 2021).

**Data Availability Statement:** The data used in this research was collected from published literature.

**Conflicts of Interest:** The authors declare no conflict of interest.

### **References**


**Ruihe Zhou \* , Longhui Guo and Rongbao Hong**

School of Civil Engineering and Architecture, Anhui University of Science and Technology, Huainan 232001, China; guolonghui7864@163.com (L.G.); cherishrb2020@163.com (R.H.) **\*** Correspondence: zrhaust@163.com; Tel.: +86-187-5602-3671

**Abstract:** In order to study the energy evolution characteristics and damage constitutive relationship of siltstone, the conventional triaxial compression tests of siltstone under different confining pressures are performed, and the evolution laws of input energy, elastic strain energy and dissipative energy of siltstone with axial strain and confining pressure are analyzed. According to the test results, the judgment criterion of the rock damage threshold is improved, and an improved three-shear energy yield criterion is proposed., The damage constitutive equation of siltstone is established based on the damage mechanics theory through the principle of minimum energy consumption and by considering the residual strength of rock, and lastly, the rationality of the model is verified by experimental data. The results reveal that (1) both the input energy and dissipative energy gradually increase with the increase of axial strain, and the elastic strain energy first increases and then decreases with the increase of axial strain, and reaches its maximum at the peak. (2) The input energy and dissipation energy increase exponentially with the increase of the confining pressure, and the elastic strain energy increases linearly with the increase of confining pressure. (3) According to the linear relationship between the sum of shear strain energy and hydrostatic pressure, an improved three-shear energy yield criterion is established. (4) The model curve can better describe the strain softening stage and the residual strength characteristics of siltstone. The relative standard deviation between the model results and the test results is only 4.35%, which verifies the rationality and feasibility of the statistical damage constitutive model that is established in this paper.

**Keywords:** energy evolution; minimum energy dissipation principle; three-shear energy yield criterion; damage variable; constitutive model
