The "macro-morphology" of the reinforcing bars is considered in the third modelling strategy. In this case, the measurements of the minimum diameter along each corroded bar developed during the experimental tests, are exploited to discretize the bar in pieces over its corroded length. Therefore, an equivalent section is considered for each piece identified. It should be noted that the diameter measurement of bars deformed by test loads, due to yielding or distortion, is omitted from the discretization process. The longitudinal bar is then modeled through connected in series truss bond-slip reinforcement elements, to which a reduced section diameter is assigned. It is remarked that through this strategy no direct reductions are applied on the constitutive laws of the reinforcements.

3.3.5. Modelling Strategy 4: Bar Discretization and Morphology-Based Constitutive Law Reduction

Strategy 4 accounts for both the "macro-morphological" aspects, i.e., the variation of corrosion along with the reinforcement, and the "micro-morphological" aspects, i.e., pitting on the cross-section of the bar. The morphology of the bars' cross-section is considered using the model proposed by Val and Melchers [36], summarized in Figure 7, in which the maximum penetration is linked to the net sectional area of the corroded bar, considering a hemispherical form of pits.

**Figure 7.** Pit configuration, model by Val and Melchers [36].

$$\begin{cases} A\_{pit} = A\_1 + A\_2 & \text{if } p \le \frac{D\_0}{\sqrt{2}}\\\ A\_{pit} = A\_0 - A\_1 + A\_2 & \text{if } \frac{D\_0}{\sqrt{2}} \le p \le D\_0\\\ A\_{pit} = A\_0 & \text{if } p \ge D\_0 \end{cases} \tag{1}$$

$$A\_1 = 0.5 \ast \left[ \theta\_1 \left( \frac{D\_0}{2} \right)^2 - b \left| \frac{D\_0}{2} - \frac{p^2}{D\_0} \right| \right], \ A\_2 = 0.5 \ast \left[ \theta\_2 p^2 - b \frac{p^2}{D\_0} \right], \ A\_0 = \frac{\pi \ast D\_0^2}{4}, \tag{2}$$

$$b = 2 \ast p \ast \sqrt{1 - \left(\frac{p}{D\_0}\right)^2}, \; \theta\_1 = 2 \ast \arcsin\left(\frac{b}{D\_0}\right), \; \theta\_2 = 2 \ast \arcsin\left(\frac{b}{2 \ast p}\right), \tag{3}$$

Then an inverse method was used to iteratively search the "pitting factor" *R* (i.e., maximum vs. average penetration ratio) of each bar (Figure 8). Firstly an initial guess value for the "pitting factor" is assumed and, considering the measured maximum penetration of the corroded sections of the bar, the mass loss value for each piece of the corroded bar is evaluated through the Val and Melchers model [36] with Equations (1)–(3). The "calculated mass loss" of each bar is evaluated as the weighted average on the length of each bar piece. If this parameter is equal to the experimental mass loss measured after the tests, the "pitting factor" is then determined. Otherwise, a new "pitting factor" guess value is set in the iterative procedure, until the satisfaction of the check test.

Following this procedure, the influence of the pitting phenomenon has been evaluated for each bar and then the loss mass value assigned to each section of the bar has been optimized according to the pitting factor. In order to significantly compare the numerical results, the same discretization along the bars adopted in the strategy 3 was also applied in this case. Tables 5 and 6 summarize the results obtained with the methodology previously described for all corroded bars, relating to specimens B25 and B30 respectively.

**Figure 8.** Flow chart of the inverse method for morphology-based mass loss evaluation.

**Table 5.** Pitting factor and lost mass assumed for strategy 4, column B25.

Figures 9 and 10 resume the modelling assumptions made for the "macro-morphology" and "micro-morphology", respectively. Indeed, the bars' discretization is shown through the comparison between measured diameter and assumed diameter. Furthermore, the results of the morphology-based mass loss evaluation are reported through a comparison between the measured mass loss and the assumed mass loss. In these figures, the 0 cm progressive distances of the bar correspond to the column-foundation interface, while the progressive 70 cm corresponds to the top of the corroded bar piece. The longitudinal reinforcement is then modeled through connected in series truss bond-slip reinforcement elements, to which reduced constitutive laws are also set, according to [35] in the case of uniform corrosion, as a function of the calculated loss mass of each bar piece.

**Table 6.** Pitting factor and lost mass assumed for strategy 4, column B30.

(**c**) (**d**)

**Figure 9.** Assumptions for corroded bars (modelling strategies 3 and 4) compared with original measurement, Specimen B25. (**a**) Bar 1-2, (**b**) Bar 2-3, (**c**) Bar 3-4, (**d**) Bar 4-1.

**Figure 10.** Assumptions for corroded bars (modelling strategies 3 and 4) compared with original measurement, Specimen B30. (**a**) Bar 1-2, (**b**) Bar 2-3, (**c**) Bar 3-4, (**d**) Bar 4-1.

#### *3.4. Bond Modelling*

The bond between reinforcement and concrete is modeled according to [32], considering splitting failure and good bond conditions. The differences in the bond laws between the two un-corroded specimens are considered by the different confinement effects given by the stirrups. The expansion of corrosion oxides and the cracking of the concrete cover have a deterioration effect on the bond. Several experimental studies and analytical formulations are available in the literature for the evaluation of the bond between concrete and corroded bars [4,37–39]. In this paper, the law proposed by [40], based on regression analyses of experimental data is adopted for the bond degradation as a function of the corrosion penetration. The model accounts for the amount of transverse reinforcement and predicts the residual bond strength, in relationship to either corrosion penetration or surface crack width. These analyses refer to medium-low corrosion levels, i.e., corrosion penetration up to 0.5 mm. In fact, when the corrosion increases, the bond is reduced until the collaboration between steel and concrete is almost negligible. For this reason, beyond a penetration value equal to 1 mm, a perfect plastic frictional bond behaviour with a bond strength value equal to 0.1 MPa [41,42] is considered. In Figure 11 an example of the bond law reduction assumed for the bars of corroded specimens is presented. As regards the unloading/reloading behaviour, the model unloads with a linear stiffness, that is equal to the initial stiffness, until the opposite residual stress value is reached. This stress value is kept until the back-bone curve in the opposite direction can be picked up [30].

**Figure 11.** Example of bond law reduction.

#### *3.5. Expectations about Results*

Based on the visual inspection of the reinforcement (Figures 5 and 6) and of the pitting factors *R* identified through the morphological study described in the previous section (Tables 5 and 6), it is highlighted that the pitting phenomenon is more pronounced for specimen B30. Vice versa specimen B25 shows a rather uniform corrosion morphology. Based on this evidence, the following authors' expectations for the B25 specimen are made:

#### **4. Results**

*4.1. Un-Corroded Specimens*

The results of the non-linear numerical analyses, relating to the un-corroded specimens, are shown in Figure 12.

As regards column A25, the main results in terms of load capacity and drift at the maximum capacity are in perfect agreement with the experimental results, as shown in Table 7. For column A30, even if it was not possible to catch the slight asymmetry of the cyclic behaviour, the results of the numerical analyses, reported in Table 7, can be considered satisfactory. Overall, the force-drift cyclic curves obtained through the numerical model show a good fit with the experimental results. The goodness of the results in terms of capacity, displacement, and energy dissipation means that the numerical model is validated by the experimental test and that it represents a reliable base for evaluations regarding the modelling of corroded bars and their effect on the cyclic behaviour of the columns.

**Table 7.** Main results of un-corroded specimens, experimental vs. numerical.

#### *4.2. Corroded Specimen*

In this section, the numerical results of corroded specimens, with reference to each of the considered modelling strategies, are reported and compared with the reference experimental tests (Section 2). In Tables 8 and 9 the main results for the different modelling strategies are summarized respectively for columns B25 and B30, in terms of maximum load, drift at the maximum load, and percentage difference of the numerical load capacity with respect to the experimental one.

**Table 8.** Main results of corroded specimens B25, experimental vs. numerical.

**Table 9.** Main results of corroded specimens B30, experimental vs. numerical.

4.2.1. Modelling Strategy 1

It is worth reminding that in this strategy the constitutive laws of steel are reduced as a function of the loss mass, that was measured directly on the bars, following the experimental tests. The numerical results, compared with the experimental ones, are reported in Figure 13.

**Figure 13.** Comparison between experimental and numerical response, corroded specimens, MS 1. (**a**) column B25, (**b**) column B30.

The cyclic behaviour, characterized by pinching, is well-caught thanks to the bond reduction, which in the loading and unloading phase reduces the collaboration between steel and concrete. The latter aspect, however, in a modelling strategy that does not consider the variation of the section along the bar, leads to excessive slip of the bars with the consequent over-estimation of the yield drift. The crack patterns obtained through numerical simulations are in good agreement with the experimental observations, with the formation of base cracks that open and close during the cycles as shown below. This crack is precisely due to the bond losses and reinforcement sliding, which also reduces the energy dissipation. Similar comments can be drawn for column B30: a good approximation of the ultimate load, but a poor prediction of the pre-yield behaviour are found. Furthermore, given the reduction of the mechanical properties due to uniform corrosion, an overestimation of the capacity was expected for the column with pitting phenomenon, as B30. As a matter of fact, this outcome is obtained in the negative load direction only. The main results for the numerical cyclic behaviour of modelling strategy 1 are reported in Tables 8 and 9.

#### 4.2.2. Modelling Strategy 2

In this case, the constitutive law of the reinforcement rebars was reduced with the pitting corrosion degradation relationship as a function of the mass lost by the reinforcement. The numerical cyclic responses for specimens B25 and B30 are reported in Figure 14a, b respectively. Results for B25, reported in Table 8, showed an incorrect prediction of the cyclic behaviour. Neither the ultimate behaviour nor the one preceding the reinforcement yielding are accurately captured and the cyclic dissipation is underestimated. All these results were expected since the reduction of the bond by pitting is too severe for a column that has shown signs of roughly uniform corrosion. The analyses for specimen B30 are not entirely satisfactory. In fact, for this column, which is most affected by pitting, a good fit with the ultimate behaviour was expected. Instead, the numerical result slightly underestimates the experimental one in the positive load direction, while in the negative direction a good fit is obtained. This result may be acceptable since the degradation relationships for pitting used in [35] are made by data regression on several experimental campaigns, therefore a slightly lower reliability of the results is attended when the bars are heavily corroded (i.e., positive load direction). As regards the cyclic behaviour, the same considerations made for specimen B25 apply. This outcome demonstrates/confirms that in case of localization of corrosion in the bar, assuming an average value of mass loss for the whole bar can be misleading and can lead to results different from the real behaviour in the pre-yielding stage.

**Figure 14.** Comparison between experimental and numerical response, corroded specimens, MS 2. (**a**) Column B25, (**b**) Column B30.

#### 4.2.3. Modelling Strategy 3

In this strategy, every single corroded bar was modeled with connected in series truss bond-slip elements, having attributed to each one an equivalent section based on the measurements performed directly on the bar. The numerically predicted behaviours for specimens B25 and B30 are reported in Figure 15. The main results are listed in Tables 8 and 9 respectively.

**Figure 15.** Comparison between experimental and numerical response, corroded columns, MS 3. (**a**) Column B25, (**b**) Column B30.

It can be noted that for specimen B25 the results of the numerical analyses improve if compared to the two previous modelling strategies. A better fit of the post cracking behaviour is identified. Indeed, if the bar is discretized in pieces and constitutive law and bond law are assigned to each part, the stresses in the bar and at the interface with the concrete are better predicted. Furthermore also cyclic dissipation and crack patterns are better described than in previous strategies.

As expected, the result of specimen B30 excessively underestimates the observed experimental response. This outcome was predictable since the section reduction was based on the minimum measured diameter, neglecting the actual pitting morphology (Figure 7). In any case, the results of the numerical analysis, even if they are not able to validate this model for the pitting, confirm that column B30 is affected by the pitting phenomenon and what emerged from the morphological analysis reported in Section 3.

#### 4.2.4. Modelling Strategy 4

The aim of strategy 4 is the modelling of the phenomenon as detailed as possible, considering both the corrosion localization along the bar (macro-morphology) and the corrosion morphology in the cross-section of the bar (micro-morphology). The benefits in terms of fitting the numerical curves emerge from the comparison with the experimental results (Figure 16). About the B25 column, it can be noted that the behaviour in the postcracking phase has been improved. In fact, both the capacity and the drift for which the first crack and the yielding of the reinforcements occur are captured. Likewise, the ultimate behaviour of the column has been correctly identified. The cyclic dissipation is in agreement with the experimental results, as well as the cracking pattern as shown below. The latter is, also in this case, governed by the formation of the main crack at the base of the column, in which the reinforcements, given the reduction of the bond, slide exchanging minimum stresses with the concrete. This aspect is found in the pinching that characterizes the response curves. In conclusion, the results of the analyses confirm that the numerical model is very suitable for the prediction of the cyclic behaviour of reinforced concrete columns subjected to roughly uniform corrosion. The considerations made concerning column B25 are also valid for column B30. The model provides satisfactory results and is therefore validated by the experimental test. Compared to the other modelling strategies, the results considerably improve, confirming the importance of the morphological study explained in Section 3. Still, a slight underestimation of the capacity in the positive load direction is observed for column B30, but it is considered acceptable.

**Figure 16.** Comparison between experimental and numerical response, corroded specimens, MS 4. (**a**) column B25, (**b**) column B30.

#### **5. Discussion**

In the previous section, the main results relating to the numerical models for both specimens B25 and B30 and the proposed modelling strategies have been reported. In general, the authors' expectations regarding the model responses were largely met. From the comparison with the experimental results, it emerged that the modelling strategies based on the constitutive law and bond law reduction as a function of the average loss mass of the corroded bars (i.e., modelling strategies 1 and 2) are suitable for the prediction of the ultimate capacity of the structure. Furthermore, it is noted how, attributing distinct values of the bars' mass loss (instead of a single value for all the bars of the column), allows to catch the cyclic asymmetrical behaviour due to corrosion, as in the case of the column B30. As regards the post-cracking behaviour, it emerged that these strategies (MS 1 and 2) may underestimate the capacity and overestimate the yielding displacement. The authors believe that this deficiency is linked to the fact that the variation of corrosion along the bar is neglected and therefore neither the possible localized bar section reductions nor good bond areas are considered. These strategies risk being not accurate for the prediction of the Damage Control Limit State in the seismic performance assessment, especially as the corrosion degree increases, since the formulations available in the literature are affected by high results' dispersion. The authors also noted that the results for pitting corrosion could be more inaccurate than those for uniform corrosion and therefore believe that, given the danger of the phenomenon, new experimental campaigns must be conducted for the pitting degradation relationship of reinforcements, focusing on the localization of the corroded areas and the influence of the pits. The results relating to modelling strategies 3 and 4 showed that considering detailed micro and macro morphological aspects allows improving the predictivity of the model in main aspects related to the cyclic behaviour of the corroded columns. While strategy 3 proved to be suitable only for columns characterized by roughly uniform corrosion, strategy 4 proved to be very suitable for both specimens and therefore for both types of corrosion. It is also underlined that in addition to the force-drift curves, the numerical models are in agreement with the experimental results with regard to the cracking pattern. This, for corroded columns, is characterized by the opening of a main crack close to the base of the column and by the cyclic slipping of the reinforcement, with a reduction of the collaboration between steel and concrete and of the energy dissipation, as it is emphasized by the pinching in the force-drift curves. Figure 17 shows an example of comparison between the cracking pattern numerically and experimentally obtained.

**Figure 17.** Comparison between cracking pattern, numerical vs. experimental, column B25, drift = −1.5%. (**a**) Numerical, (**b**) Experimental.

The results of the four strategies of numerical analyses and the experimental response have been superimposed in terms of monotonic load-displacement curves, obtained as envelopes of the cyclic ones in Figure 18a,b relatively for B25 and B30.

**Figure 18.** Comparison between experimental and numerical response, monotonic curves. (**a**) Column B25; (**b**) column B30.

From the comparison, the coherence between the models emerges. In particular, strategy 4 stands out for being close to 1 in the case of uniform corrosion and instead tends to migrate towards strategy 2 in the case of pitting corrosion. It should be noted that numerical models tend to underestimate the energy dissipated by the specimens. As mentioned in Section 3.2, this can be due to the model adopted for the concrete (TSCM), that tends to underestimate the energy dissipation of the concrete, and also due to some asymmetric sliding shear phenomena verified during the tests, not fully captured by the numerical model. Finally, it is also noted that almost no model is able to capture the cyclic degradation of strength by a shear mechanism, which can be seen in the last cycles of the experimental tests of corroded columns. These results can be justified by the numerical

assumption of considering an average corrosion level in the stirrups, since no detailed measurements are available for the transversal reinforcement.

#### **6. Conclusions**

In this work, the cyclic behaviour of RC columns subjected to localized corrosion in the plastic hinge regions was investigated through FE modelling and non-linear analyses. The validation of the numerical models was performed through the comparison with the results of an experimental campaign carried out by some of the authors. Particular attention was paid to the modelling of the corroded rebars and their corrosion morphology. The comparison between the numerical results showed how the most common modelling strategies, which operate according to the section or constitutive law reduction as a function of the average corrosion of the bars, are suitable for the evaluation of the ultimate capacity. Nevertheless, given the uncertainties that characterize the phenomenon from the morphological point of view, they may present deficiencies in the seismic performance assessment (i.e., at the Damage Control performance stage). The strategies based on the morphological study of the corroded bars show, instead, a very good fit for all the cyclic behaviours. The results, therefore, demonstrated how the behaviour of the RC corroded columns depends both on the nature of corrosion (uniform vs. pitting) and on the localization of damage along with the structural element, both on aspects of a random nature that affect the structural response. However, it must be considered that the most refined modelling strategies are difficult to implement for the assessment of real existing structures, as they make use of detailed data and measurements that cannot be obtained with the same precision through on-site surveys. The authors, therefore, believe that to maintain the easy application approach based on the reduction of the constitutive law as a function of the mean mass loss, a new experimental campaign for the mechanical behaviour of corroded bars, which focuses on pitting and localization, is necessary. Furthermore, the assessment of corroded existing RC structures should take into account the probabilistic aspects of the morphology of the corrosion phenomenon.

**Author Contributions:** Conceptualization, F.M., F.D.C. and Z.R.; methodology, F.M., F.D.C. and Z.R.; formal analysis, F.M.; writing—original draft preparation, F.M.; writing—review and editing, F.M., F.D.C. and Z.R. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Some or all data that support the findings of this study are available from the corresponding author upon reasonable request.

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

#### **References**

## *Article* **Flexural Behavior of Reinforced Concrete Beams Retrofitted with Modularized Steel Plates**

**Min Sook Kim and Young Hak Lee \***

Department of Architectural Engineering, Kyung Hee University, Deogyeong-Daero 1732, Yongin 17104, Korea<EMAIL_ADDRESS>
**\*** Correspondence<EMAIL_ADDRESS>Tel.: +82-31-201-3815

**Abstract:** Many structural retrofitting methods tend to only focus on how to improve the strength and ductility of structural members. It is necessary for developing retrofitting strategy to consider not only upgrading the capacity but also achieving rapid and economical construction. In this paper, a new retrofitting details and technique is proposed to improve structural capacity and constructability for retrofitting reinforced concrete beams. The components of retrofitting are prefabricated, and the components are quickly assembled using bolts and chemical anchors on site. The details of modularized steel plates for retrofitting have been chosen based on the finite element analysis. To evaluate the structural performance of concrete beams retrofitted with the proposed details, five concrete beams with and without retrofitting were tested. The proposed retrofitting method significantly increased both the maximum load capacity and ductility of reinforced concrete beams. The test results showed that the flexural performance of the existing reinforced concrete beams increased by 3 times, the ductility by 2.5 times, and the energy dissipation capacity by 7 times.

**Keywords:** retrofitting; modular; finite element analysis; flexural behavior; ductility

#### **1. Introduction**

Structural members need to be strengthened through retrofitting due to revision in the current design code or change in load. The retrofitting methods using jacketing have mainly been investigated by many scholars to improve the strength and ductility of existing structural members. Steel jacketing or fiber-reinforced polymer (FRP) jacketing is a commonly employed method used to retrofit concrete beams and columns.

Steel jacketing consists of steel angles or plates with different thicknesses, widths, and spacing and they are installed by welding. Cement or epoxy mortar fills the gap between the jacket and concrete column. Several studies have been conducted evaluating the structural performance of steel jacketing. Garzon-Roca et al. [1] experimentally investigated the behavior of beam-column joint strengthened with steel caging under combined bending and axial load. The test results showed that steel caging increases both the failure load and ductility of the columns. Adam et al. [2] presented parametric study using the finite element model and carried out analyzing the behavior of reinforced columns strengthened by steel caging. Wei and Wu [3] tested concrete columns retrofitted with high strength steel wire and proposed stress-strain relationship. Tarabia and Albakry [4] conducted an axially loaded test on ten column specimens. The size of steel angles, strip spacing, and grout material were considered as parameters. They reported that most of the specimens were failed due to buckling of the steel angle by crushing of the columns. Alvarez et al. [5] presented the nonlinear analysis procedures for concrete columns retrofitted with steel jackets. Backbone curves were constructed of circular and rectangular retrofitted columns to evaluate the response of jacketing columns. Chrysanidis and Tegos [6] proposed new type of hybrid jackets composed of metal grid jacket and high strength mortar. It has been proven that the proposed systems can improve strength and ductility than conventional method.

**Citation:** Kim, M.S.; Lee, Y.H. Flexural Behavior of Reinforced Concrete Beams Retrofitted with Modularized Steel Plates. *Appl. Sci.* **2021**, *11*, 2348. https://doi.org/ 10.3390/app11052348

Academic Editor: Panagiotis G. Asteris

Received: 28 January 2021 Accepted: 3 March 2021 Published: 6 March 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

This technique is widely used in construction field due to cost-effective, developing lateral strength, and axial load carrying capacity. However, it is mainly applied to the column due to weight and welding. Additional lifting equipment is required for installation of tube type steel jacket without welding on site because the steel jacket is prefabricated considering member size.

Composite material was used in the jacketing method to improve movement and installation problems. Commonly carbon or aramid fibers with resin or epoxy resin is used for FRP jacketing. Maaddawy [7] examined the effect of eccentrically load of the concrete column retrofitted with FRP jacketing. It was concluded that the strength of columns decreased as eccentricity ratio increased. A number of studies have been evaluated the efficiency of a FRP jacket joining [8–10]. The behavior of concrete columns retrofitted with FRP jacket subjected to seismic loads was investigated based on experimental and analytical results [11,12]. Experimental investigation was performed to explore the performance of eccentrically loaded rectangular RC columns with different CFRP strengthening schemes and preloading levers by Wang et al. [13]. The results indicated that ultimate capacity can be significantly enhanced by full wrapping of CFRP. They also proposed calculation model to predict the axial load and bending moment capacity of RC columns rapped with FRP jackets.

FRP has low tensile strength and poor adhesion to concrete while FRP sheet have several advantages such as ease of installation and being light weight. Especially, structural retrofitting methods mainly focus on columns or beam-to-column connections, but only retrofitting vertical members may cause stiffness differences between the horizontal members. Few studies have dealt with retrofitting method for concrete beams than columns or beam-column joints.

In this paper, a new modularized retrofitting method for concrete beams is proposed and experimentally examined. The proposed retrofitting method is expected to resolve the problems in terms of the weight and welding of steel jacketing and adhesion and low strength of FRP jacketing. In order to reduce the self-weight of retrofitted concrete beams, modularize steel plate consisting of L- and Z-shaped steel plates are used instead of steel tube or steel jacket. The new modularized retrofitting method can offer rapid construction progress and good quality. The finite element modeling and parametric study were performed to optimize and determine the details of modularized steel plate to reduce stress concentration and exercise its full capacity. A total of four retrofitted concrete beams and one control beam specimen were fabricated and tested to prove that the proposed retrofitting details were capable of improving the load carrying capacity and ductility.

#### **2. The Retrofitting Method for Reinforced Concrete Beams**

The proposed retrofitting system aims at improving structural performance, reducing self-weight and rapid assembly based on modularization. This study proposes a new type of retrofitting details using modularized steel plates. The proposed retrofitting details consist of Z-shaped side plates, L-shaped lower plates, and bottom plates with vertical grid as shown in Figure 1a. Two L-shaped lower plates are fixed to the bottom of the concrete beam using chemical anchors. L-shaped lower plates help easy and accurate installation of the Z-shaped side plates. The Z-shaped side plate is connected to the L-shaped lower plate using bolts and then bonded to the side of the concrete beam using the chemical anchors. The vertical grid is inserted into the Z-shaped side plate and combined using bolts. The space formed under the concrete beam is filled with mortar. The details of concrete beam retrofitted with modularized steel plates are shown in Figure 1b. By assembling L-shaped lower and Z-shaped side plates with bolts, the depth of the parts being extended can be adjusted, and it is easy to design and construct regardless of the size of the members. In addition, grid is installed vertically to resist tensile stress, so no subsequent process is required to install additional flexural reinforcement on the site.

**Figure 1.** Proposed retrofitting system: (**a**) The components of proposed retrofitting system; (**b**) The retrofitted beam with the proposed system.

#### **3. Parametric Studies**

*3.1. Analytical Model*

Finite element analysis was performed to determine the details of the proposed retrofitting systems. The commercial FE analysis software ANSYS 16.0 was employed to analyze stress for beams. A three-dimensional and eight-node solid element was applied to the model. The steel plates are fixed to the concrete surface using bolts. Automatic surface to surface contact option was used to model the contact interfaces between concrete and steel plates. A surface element was placed and attached to the beam inside the steel plates and around the steel plates. The thickness, stiffness, and strength were not considered for the surface element. The surface of concrete and steel plates are designated as master and slave surface, respectively. Contact interaction was defined between the concrete and steel plates. The friction contact between the parts was defined using Coulomb friction with friction coefficient of 0.3 [14].

The steel plates and concrete were modelled as isotropic materials. Material and geometric nonlinearity are considered. The elastic-perfectly plastic uniaxial material model with bilinear isotropic hardening is used for the concrete model. The yielding stress is equal to the concrete compressive strength and tangent modulus equals zero. The steel plates and bolts are considered to be elastic-perfectly plastic. Steel are assumed to be linear elastic until the yield stress, and after that, the stress remains constant. The yield strength and ultimate strength of the steel plates were 250 MPa, and 460 MPa, respectively. The compressive strength of the concrete was 24 MPa. The material properties are listed in Table 1.

**Table 1.** Material properties used in the analytical models.

The thickness of the bottom plate (A), the thickness of the L-shaped lower plate (B), the thickness of the Z-shaped side plate (C), and the spacing of bolts (D) were considered as variables as shown in Figure 2. T-shaped beam is designed with a 1000 mm × 150 mm slab, 300 mm × 650 mm beam, and 4 m length. The concrete beam model had simply supported

condition. The displacement was applied at the node of top surface center of beam in the downward direction. The purpose of the analysis is to compare the stress distribution according to each component details, the displacement of 10 mm was applied for efficiency of analysis. The element mesh size was determined to be 1.5 mm × 1.5 mm considering that the smallest dimension among the components, L-shaped lower plate and Z-shaped side plate, was 2.5 mm. The finite element model is shown in Figure 3.

**Figure 2.** Analytical variables of the proposed retrofitting system.

**Figure 3.** Finite element model.

Four models with different mesh sizes, 20 mm, 10 mm, 5 mm, 1.5 mm, were performed to determine a reasonable mesh size that could be reasonably evaluated for the stress of each component. As a results, the stress converged from the mesh size of 5 mm as shown in Figure 4. A sufficient number of elements are needed to observe the stress distribution of each component. Therefore, the mesh size of 1.5 mm would be applicable to this study.

#### *3.2. Determinations of the Details of the Proposed Retrofitting System*

The bottom plate is located at the bottom of the beam and is the component that occurs high tensile stresses. An appropriate thickness of the bottom plate must be determined. As the thickness of the bottom plate increases, the maximum load capacity increases, which may cause compression failure of the concrete beams. Thickness of the bottom plates are 5 mm, 10 mm, 15 mm, 20 mm, and 30 mm to propose a thickness with a stable stress distribution for the analytical model. The results of the analysis are summarized in Table 2.

**Table 2.** The maximum stress of components.

The deflection of the beam decreased as the thickness of the bottom plate increased. As the thickness of the bottom plate increased, the stress of the compressive zone and bolt increased. The thickness of the bottom plate is greater than 10 mm, the stress was concentrated on the bolt hole. This may lead to the crushing of concrete before the flexural failure. The thickness of the L-shaped lower plate was alternatively set to 2.5 mm, 5 mm, 7.5 mm, and 10 mm. As the thickness of the L-shaped lower plate increased, the stress on concrete, Z-shaped side plates, and bottom plates decreased slightly, but did not significantly affect the stress distribution. As Z-shaped side plate thickness increases, the maximum stress is occurred in the compressive zone of the concrete beam. It is concluded that the thickness of the Z-shaped side plate can induce the brittle failure on the compressive zone of the concrete beam. Bolt spacing should be examined in terms of the composite behavior between the concrete and the modular steel plates. As the bolt spacing increased, the stress of the concrete decreased and the stress of the bolt hole increased. The optimal bolt spacing was determined to be 200 mm, which provided a relatively uniform stress distribution.

**Figure 4.** Meshing convergence study.

#### **4. Experimental Program**

#### *4.1. Sequence of Construction*

The specimens were manufactured in the order of steel rebar placement and formwork installation, concrete placement, combination of modularize steel plate, and mortar injection. As shown in Figure 5, L-shaped lower plates were fixed to the bottom of the concrete specimen, the process is to bolt the lower and side plates and then insert the bottom plates to tighten the bottom plate and the side plate bolts. After integrating the side plates with chemical anchors, the specimen was manufactured by injecting mortar, and the specimen was manufactured upside down to prevent filling defects in mortar.

**Figure 5.** The process of proposed retrofitting system application: (**a**) The installation of the L-shaped lower plate; (**b**) Assembly of Z-shaped side plate and vertical grid; (**c**) The process of connecting the side plate used in chemical anchors; (**d**) The process of grouting of mortar.

#### *4.2. Specimen Details and Test Parameters*

In this paper, the total of five specimens were manufactured, including one reinforced concrete beam and four reinforced concrete beams retrofitted with the modularized steel plates. The finite element analysis to optimize the dimensions of each component was performed assuming the real in-situ situations that beams are supporting the slabs. In this study, the slab (flange) parts in T-shaped section only provide the spaces to fix the steel plates by using chemical bolts. Therefore, flanges were not considered when the specimens were manufactured as shown Figure 6. The concrete beams of all specimens were designed to be 300 mm wide, 350 mm high and 4500 mm long. The thickness of Z-shaped side plate was 2.5 mm, the thickness of the L-shaped lower plates was 5 mm, and the spacing of chemical anchors and bolts was 300 mm, using the same in zigzag format. The thickness of the bottom plate, and the number of vertical grids were considered as experimental variables, which are expected to affect the flexural behavior, and represent the corresponding portion of each variable in Figure 5. The depths of the new beams were 100 and 150 mm, the thicknesses of the bottom plates were 5 and 10 mm, and the number of vertical grid were 0, 2, and 4. For the specimen with two vertical grids, the spacing of each grid was 200 mm, and for the specimen with four vertical grids, each grid was installed at spacing of 65 mm. The specimen illustration is organized in Table 3, and the details of the specimen, according to each variable, are shown in Figure 7. The 28-day compressive strength of the concrete used in this experiment was 24 MPa. For the tensile and compressive reinforcement, deformed rebar was used with a diameter of 19 mm and it's yield strength is 400 MPa. Steel plate was used with a yield strength of 275 MPa. The test results of compressive strength of concrete and steel plate are shown in Figure 8. The high-strength bolts which is F10T M16 bolts with a diameter of 16 mm was used to connect the concrete beams. The material properties are summarized in Table 4.

**Figure 6.** The test parameters: (**a**) The fabrication of proposed retrofitting system in the experiment; (**b**) test parameters.

**Table 3.** The test specimen parameters.

<sup>1</sup> A: Bottom plate thickness (A: 5 mm, B: 10 mm). <sup>2</sup> 2: The number of steel plate openings (2: 2EA, 4: 4EA, 0: 0EA).

**Figure 7.** Specimen details: (**a**) RC; (**b**) D100-A2.

#### *4.3. Test Set Up*

All specimens were tested using a Universal Testing Machine (UTM) with a 5000 kN load capacity as shown in Figure 9. The specimens were simply supported and subjected to four-point bending loading. The distance from support to loading point was 1750 mm and the distance between the two central point loads was 600 mm. Displacement was applied to each specimen a rate of 2 mm/min. During the flexural test, load and midspan deflection were measured. Deflection was measured with three linear variable displacement transducers (LVDT's) mounted onto the bottom of the specimens. For concrete, the strain gage is attached in the compression zone C1. In order to evaluate the yield by strain of the tensile reinforcing bar, the strain gage was attached in the middle of the tensile reinforcing

bar before concrete placement (R1). The strain gage (R2) was attached to the center of the vertical grid's vertical strip to evaluate the flexural behavior as tensile reinforcement. A total of 7 strain gages (S1 ~ S7) were attached to the front of the beam to evaluate the local buckling and deformation of the modularized steel plates. The installation of the LVDTs and strain gages is shown in Figure 10. All specimens were continuously observed to mark the crack pattern and note any signs of failure during the loading process. Loading, displacement and strain were recorded using a data acquisition system. The specimens were tested until the load decreased at the level of approximately 70% to 80% of the maximum load.

**Figure 8.** (**a**) Compressive strength test of concrete; (**b**) Tensile test of steel plate.

**Figure 9.** Test setup: (**a**) A schematic of the test setup; (**b**) A photograph of the test setup.

#### **5. Test Results and Discussion**

*5.1. Cracking Behavior and Modes of Failure*

Table 5 shows the crack pattern at maximum load for each specimen tested. For the D100-A2 and D100-B2 specimens with variable thicknesses of the bottom plate, it has been confirmed that the crushing of compression zone widens as the thickness of the bottom plate increases. As the number of vertical grid increased, the crushing and spalling of compression zone were more localized.

**Table 5.** Crack propagation examples.

The loading was terminated when the load was reduced 80% after the maximum load was reached. All the specimens showed flexural behavior as shown in Figure 11, cracks started appearing in the compression zone. As the load increased, the cracks of the compression zone became wider, and some spalling was observed on the mortar grouted on the bottom plate. The spalling of the mortar was observed differently according to the vertical grid. More spalling occurred in the D100-B0 specimen without the vertical grid. Finally, as the concrete crushing and spalling increased, the load gradually decreased. Local buckling of the Z-shape side plates and bolt fracture were not observed in all retrofitted specimens as shown in Figures 12 and 13. It is concluded that sliding was not observed at the end of the beam because the steel plates and concrete were perfectly bonded.

**Figure 11.** The shape of the test specimen after a flexural test.

**Figure 12.** The side plate and bottom plate after a flexural test.

**Figure 13.** Load-Deflection Relationship.

The Load-displacement curve is shown in Figure 14. Specimens retrofitted with modularized steel plates showed greater stiffness and maximum load than RC specimen. Retrofitted specimens showed similar initial stiffness. Remarkable differences were not observed in terms of the number of vertical grid between D100-B2 and D100-B4. The maximum load for D100-B2 was 602 kN and that for D100-B4 was 598 kN, but the maximum load for D100-B0 was 514 kN, which was increased by approximately 1.17 times depending on the presence of vertical grid. This is to achieve the restraining effect of reinforcing parts on the mortar injected into reinforcing areas, and the flexural strength due to reinforcement is improved depending on whether or not it is installed.

**Figure 14.** The load-deflection curve test results.

The maximum load increased as the thickness of bottom plate increased. A comparison of the load-displacement curves of the two specimens confirmed that the maximum load for the D100-A2 specimen was 508.48 kN, the maximum load for the D100-B2 specimen was 602.62 kN, and that the flexural performance was approximately 1.19 times higher for the D100-B2 specimen.

The test results are summarized in Table 6. The nominal flexural strength *Pn* is calculated from plastic stress distribution method suggested by AISC 360-10 [15]. The yield load *Py* defines a point as the yield load where a line parallel to the line connecting the origin to the maximum load meets the load-displacement curve as shown in Figure 15. The proposed retrofitting method has shown an increase in the strength, with increasing thicknesses of bottom plates and the number of vertical grids. The experiment resulted in an average increase of 3.2 times the maximum load and an average increase of 1.3 times the maximum displacement.

**Table 6.** Flexural test results.

<sup>1</sup> *Pn*: The nominal force calculated by the nominal flexural strength applied to the beam. <sup>2</sup> *Py*,*test*: The yield load of each specimen. <sup>3</sup> *Pu*,*test*: The maximum load of each specimen. <sup>4</sup> *δy*: The displacement when the yield load applied. <sup>5</sup> *δu*: The displacement when the maximum load applied.

#### *5.2. Load-Strain Relationship*

Figure 16a shows the load-strain relationship of the C1 gauge located in the upper of the concrete specimen. The strains of all the specimens were similar and after exceeding the maximum compressive strain of the concrete under the maximum load of 0.003, the gauge was found to have been damaged due to the compressions on the top of the concrete. Figure 16b shows the load-strain relationship for the R2 gauge located at the vertical grid. Although there were differences in the slopes of the initial linear depending on the specimen, the strain of the specimens increased in a similar, and the yield strain reached

0.002 under the maximum load and then yielded. It is judged that the specimens exhibited sufficient flexural behavior because vertical grid as tensile reinforcement yielded.

**Figure 15.** Determination of the yield strength.

**Figure 16.** The Load-strain relationships: (**a**) C1 Guage; (**b**) R2 Gauge.

#### *5.3. Ductility Capacity*

The ductility of flexural members, such as beams, can be calculated by the ratio of the maximum displacement and yield displacement after the maximum load is generated, without a sudden reduction of the load when the load is reduced to 70–80%. In this study, the ductility was calculated by dividing the displacement *δ*0.8*Pu* when the maximum load was reduced to 80% of the maximum load after the maximum load was applied by the displacement *δ<sup>y</sup>* when the yield load was applied. The ductility capacities of each specimen are summarized, as shown in Table 7. The ductility capacity of specimens retrofitted steel plates was shown to be at least 7.47 and up to 10.01, averaging 8.85. Compared to the RC specimen, the ductility capacity is about 2.5 times higher. The ductility of each D100-A2 and D100-B2 specimens, with variable thicknesses of the bottom plate, was 10.01 and 7.47, respectively, indicating that the ductility decreases as the thickness of the bottom plate increases. In addition, the comparison of specimens according to the number of vertical grid resulted in improved ductility as the number of vertical grid increased to 7.47 for D100- B4, 9.67 for D100-B0, and 8.26 for the D100-B0 specimen. However, due to the thickness of the bottom plate and the presence or absence of vertical grid, excessive reinforcement on the tensile zone is considered to be possible due to the crushing of concrete. The ductility capability can be improved by the application of modularized steel plates.

**Table 7.** Analysis of the ductility capacity for each specimen.

<sup>1</sup> *δ*0.8*Pu*: The displacement when the maximum load reduced to 80%. <sup>2</sup> *δy*: The displacement when the yield load applied. <sup>3</sup> *δ*0.8*Pu*/*δy*: The ductility capacity of each specimen.

#### *5.4. Energy Dissipation Capacity*

Energy dissipation is one of the structural performance evaluation elements of a member that measures the ability of the structure and member to absorb the energy generated when deformation is applied to the structure and components. An analysis of the energy dissipation capacity of specimens is shown in Figure 17.

For the energy dissipation with a thickness increase of the bottom plate, the D100-A2 specimen measured 61.6 kN-m, and the D100-B2 specimen measured 64 kN-m, and the difference with varying thickness of the bottom plate was minimal. The D100-B2 specimen, D100-B4 specimen, and D100-B0 specimen, with variable numbers of vertical grid, are shown to improve their energy dissipation capacity as the number of vertical grid increases to 64 kN-m, 71.6 kN-m, and 55 kN-m, respectively. It has been confirmed that the energy dissipation capability was approximately 7 times higher than that RC specimen.

#### *5.5. Model Validation and Discussion*

The results of the numerical model are validated and confirmed by comparing the results of the experiments and the numerical model. Analysis conditions were set the identical as for parametric study. For comparison with the experimental results, a displacement of 120 mm was applied. Figure 18 shows the comparison of the numerical predictions with the experimental results for the D100-B4 specimen. The comparison indicated that numerical results could reasonably predict the overall structural performance in terms of initial stiffness and maximum load capacity. The numerical models slightly became stiffer than the experimental results. This may be due to the numerical model not considering micro-crack or the construction error of the specimen, but it is considered to be acceptable. However, the numerical model shows a different behavior after yielding due to the limitation of the numerical models in implementing the failure mechanism. This is a topic for further investigation by the authors.

**Figure 18.** Comparison between the numerical and the experimental results of load versus midspan deflection.

#### **6. Conclusions**

The purpose of this study is to propose and examine a modularized steel plate retrofitting method resulting in improved structural performance and constructability. The details of retrofitting method were proposed based on the finite element analysis. The flexural test was conducted to evaluate the structural performance of concrete beams retrofitted with the proposed method. The main conclusions obtained from the study are as follows:

**Author Contributions:** Conceptualization, M.S.K. and Y.H.L.; methodology, M.S.K.; validation, Y.H.L.; formal analysis, M.S.K.; investigation, M.S.K. and Y.H.L.; writing—original draft preparation, M.S.K.; writing—review and editing, Y.H.L. All authors have read and agreed to the published version of the manuscript.

**Funding:** This study was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government (MSIT) (No. 2019R1A2C1090033).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** The results presented in this study are available on request from the corresponding author.

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

#### **References**

5. Alvarez, J.C.; Brena, S.F.; Arwade, S.R. Nonlinear backbone modeling of concrete columns retrofitted with fiber-reinforced polymer or steel jackets. *Aci Struct. J.* **2018**, *115*, 53–64. [CrossRef]

## *Article* **Strengthening Design of RC Columns with Direct Fastening Steel Jackets**

**Zhiwei Shan 1, Lijie Chen 2, Kun Liang 2, Ray Kai Leung Su 2,\* and Zhaodong Xu <sup>1</sup>**

**\*** Correspondence<EMAIL_ADDRESS>Tel.: +852-28592648

**Abstract:** For non-seismically designed columns with insufficient strength and flexural stiffness, intense inter-story drift can be incurred during a strong earthquake event, potentially leading to the collapse of the entire building. Existing strengthening methods mainly focus on enhancing axial or flexural strength but not the flexural stiffness of columns. In response, a novel direct fastening steel jackets that can increase both flexural strength and stiffness is introduced. This novel strengthening method features straightforward installation and swift strengthening as direct fastening is used to connect steel plates together to form a steel jacketed column. This new connection method can quickly and stably connect two steel components together by driving high strength fasteners into them. In this paper, the design procedure of RC columns strengthened with this novel strengthening method is originally proposed, which includes five steps: (1) estimating lateral load capacity of damaged RC columns; (2) determining connection spacing of steel jacket; (3) estimating the lateral load capacity of strengthened RC column; (4) evaluating the axial load ratio (ALR) of strengthened RC columns; and (5) estimating effective stiffness of strengthened RC columns. Lastly, an example is presented to illustrate the application of the proposed design procedure.

**Keywords:** RC columns; strengthening; direct fastening; steel jackets; design procedure

#### **1. Introduction**

Reinforced concrete (RC) moment-resisting frame buildings are widely used in schools, hospitals, and residential buildings. RC columns are the principal structural component in the resisting of lateral and gravity loads in frame buildings. Based on post-earthquake investigations [1–3], the stability of frame buildings is known to be critically dependent on the seismic performance of RC columns. The structural deficiencies identified in outmoded non-seismically designed RC columns [1–10] are (1) insufficient lap splice length of longitudinal reinforcement at column ends, (2) insufficient transverse reinforcement at the plastic hinge, (3) strong beam-weak column arrangement, (4) insufficient corrosion resistance, (5) insufficient strength due to new functional use of the building, (6) fireinduced damage, and (7) earthquake-induced damage. The first five above mentioned deficiencies may result in insufficient flexural strength, which can be mitigated by various available strengthening techniques such as RC jacketing [11,12], steel jacketing [13,14] and fiber reinforced polymer (FRP) jacketing [15–17]. However, in the case of fire or earthquake damaged columns, both flexural strength and stiffness can be reduced [18–21]. Figure 1 illustrates the seismic displacement demands of structurally damaged and nondamaged buildings within a demand spectrum. Any reduction in lateral strength and stiffness within a structural system can then induce higher seismic displacement demand during subsequent earthquake events. To restore the seismic capacity of columns to their undamaged state, both flexural stiffness and strength must be improved.

**Citation:** Shan, Z.; Chen, L.; Liang, K.; Su, R.K.L.; Xu, Z. Strengthening Design of RC Columns with Direct Fastening Steel Jackets. *Appl. Sci.* **2021**, *11*, 3649. https://doi.org/ 10.3390/app11083649

Academic Editor: Pier Paolo Rossi

Received: 1 April 2021 Accepted: 14 April 2021 Published: 18 April 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

**Figure 1.** Design spectrum and capacity curves for damaged and undamaged structures.

In response, a novel steel jacket that deviates from traditional steel jacketing (assembled by either welding or bolting) is developed (see Figure 2). This new strengthening method offers straightforward installation and rapid strengthening, as direct fastening is used to connect steel plates together to form a steel jacketed column. The strengthening process is briefly described herein: (1) the prefabrication of steel jackets is achieved by welding the end angles to the end of the steel plates (welding can be conducted in-shop); (2) four steel plates are fastened by anchor bolts to the top and base beams; (3) the steel plates are tightly clamped to minimize gaps between them and the RC columns, and steel angles are temporarily fixed to the adjacent steel plates; and (4) the adjacent steel plates and steel angles are tightly and quickly joined by high strength nail using a powder-actuated gun (see Figure 3).

**Figure 2.** Proposed strengthening scheme: (**a**) front view; (**b**) plane view.

**Figure 3.** Strengthening procedure.

In this paper, previous experimental and theoretical studies on shear connections joined by direct fastening, axial load strengthening and the seismic strengthening of RC columns strengthened by direct fastening steel jackets are first reviewed. Based on the findings in previous studies [22–25], a design procedure of RC columns strengthened by direct fastening steel jackets is developed. To illustrate the application of the proposed design procedure, a worked example is presented.

#### **2. Review of Previous Work**

#### *2.1. Mechanical Behavior of Shear Connectors Joined by Direct Fastening*

The connections used in this novel steel jacket resist shear loading incurred as a result of concrete expansion or lateral load. The load transfer process of these connections is similar to that of single lap joints subjected to tensile force at the ends. However, the behaviors of this type of connection have rarely been studied and relevant design specifications are thus not available. Therefore, the experimentation on the single lap joints shown in Figure 4 was carried out to study the mechanical behavior of direct fastening connections [22].

Two kinds of failure modes—bearing and shear fracture—were examined in the tests. Similar to the bearing failure seen in bolted or screwed connections, this failure occurs with enlarged nail holes and bulging of the material around the fastener holes due to large plastic deformation, leading to desirable ductile behavior (see Figure 5).

**Figure 4.** Illustration of connection test joined by direct fastening.

**Figure 5.** Mechanical behavior of connections: (**a**) bearing failure; (**b**) shear load versus displacement curve.

Although equations for predicting the maximum bearing resistance of bolted and screwed connections are available in current specifications EN-1993-1-8 [26], ANSI/AISC 360-10 [27], and AS 4100 [28], these equations cannot be applied to estimate the maximum bearing resistance (yield strength) of connections joined by direct fastening without modification. Therefore, a unified design equation is developed to evaluate the maximum bearing resistance of the connections:

$$F\_b = \psi\_{fp} \psi\_{fk} \alpha\_{br} d\_n t\_p f\_{pw} \tag{1}$$

where *αbr* is the bearing resistance factor, which is taken as 1.6; *ψfp* is the factor that considers the effect of protuberance; *ψfk* is the factor that takes the effect of knurling into account; *dn* is the nominal fastener diameter; *tp* is the thickness of the connected steel plate; and *fup* is the ultimate strength of the connected steel plate.

The two factors caused by the effects of protuberance and knurling are given as:

$$\psi\_{fp} = \begin{cases} 1.35 & \text{without pre} - \text{drilled holes on connected steel plate} \\ 1.0 & \text{with pre} - \text{drilled holes on connected steel plate} \end{cases} \tag{2}$$

$$
\psi\_{fk} = \begin{cases} 1.17 & \text{fastener with knurling} \\ 1.0 & \text{fastener without knurling} \end{cases} \tag{3}
$$

#### *2.2. Axial Strengthening of RC Columns by Direct Fastening Steel Jackets*

From the experimental study on the axial strengthening of RC columns by the direct fastening of steel jackets [23], the reliability and effectiveness of the proposed method were observed. Critical parameters (i.e., vertical spacing between adjacent connections, thickness of the steel plate and number of fasteners in each connection) affecting load bearing performance and deformation behavior were identified.

In the proposed strengthening method, steel plates can directly sustain axial load. The axial load contribution of the steel plates is determined by buckling strength as buckling occurs prior to the yield strength of steel plates. The buckling strength is presented in the following section.

Direct fastening connections behave in the manner of transverse reinforcement (i.e., stirrups). Due to the transverse dilation of the column, passive confinement stress is mobilized in stirrups and direct fastening connections, which enhances the strength of the column. To determine confined concrete strength, the equivalent passive confinement stress shown in Equation (3) should be determined.

$$f\_{est} = \frac{m\alpha\_{st} f\_{yt} A\_{st}}{s\_{st} l\_{st}} \tag{4}$$

$$f\_{cd} = \frac{2\alpha\_d n\_f F\_b}{(s\_d + d\_d)d\_c} \tag{5}$$

where *m* is the stirrup legs; *lst* is the center-to-center distance of the peripheral stirrup; *sst* is the stirrup spacing; *fyst* is the yield strength of the stirrup; *Ast* is the cross-sectional area of the stirrup; *sd* is the clear vertical spacing of the adjacent connections; *dd* is the length of the right steel angle bracket; *dc* is the column width; *αst* is the stress ratio of the stirrup; and *α<sup>d</sup>* is the shear force ratio of the direct fastening connection.

A theoretical study was carried out to examine the stress ratio of the stirrup and the shear force ratio of the direct fastening connection [24]. Based on an extensive parameter study, a lower bound value of 0.34 is used to represent the stress ratio of the stirrups with a yield strength of 400 MPa. To consider the effect of the yield strength, a yield strength factor of *γfyst* was proposed. The yield strength factors for yield strengths of 500 MPa and 600 MPa are 0.8 and 0.66, respectively. The shear force ratio of the connections is obtained using:

$$
\alpha\_d = \gamma\_{nf} (0.82 - 0.64\lambda\_d) \tag{6}
$$

where *λ<sup>d</sup>* is the normalized connection spacing (*λ<sup>d</sup>* = *sd*/*dc*); and *γnf* represents a factor of the number of fasteners on the shear force ratio of the connections. When the number of fasteners is 2 and 4, the factor is equal to 0.72 and 0.47, respectively.

The failure criterion proposed in [29] is used to derive the confined concrete strength of *fcc* . The failure criterion is given by:

$$
\tau\_{\rm oct}^\* = 6.9638 \left( \frac{0.09 - \sigma\_{\rm oct}^\*}{c - \sigma\_{\rm oct}^\*} \right)^{0.9297} \tag{7}
$$

$$x = 12.2445(\cos 1.5\theta)^{15} + 7.3319(\sin 1.5\theta)^2 \tag{8}$$

$$\cos \theta = \frac{\sqrt{3}}{2} \frac{s\_1}{\sqrt{f\_2}} \tag{9}$$

$$
\pi\_{\rm oct}^\* = \frac{\pi\_{\rm oct}}{f\_c'} \tag{10}
$$

$$
\sigma\_{\rm oct}^{\*} = \frac{\sigma\_{\rm oct}}{f\_c'} \tag{11}
$$

where *τoct* and *σoct* are the octahedral shear and normal stress, respectively; *θ* is the direction of the deviatoric stress on the deviatoric plane; *s*<sup>1</sup> is the first deviatoric stress; *J*<sup>2</sup> represents the second invariant of the deviatoric stress tensor; and *fc* is the compressive strength of the unconfined concrete.

The key parameters in the applied failure criterion are summarized as follows:

$$s\_1 = \frac{f\_{cc}^{\prime} - f^{\prime}}{3} \tag{12}$$

$$J\_2 = \frac{1}{3} \left( f\_{cc}' - f' \right)^2 \tag{13}$$

$$\cos \theta = \frac{f\_{cc}' - f'}{3\sqrt{2}\tau\_{\rm tot}} \tag{14}$$

$$\pi\_{\rm tot} = \frac{\sqrt{2(f\_{\rm cc}' - f')}}{3} \tag{15}$$

$$
\sigma\_{\rmoct} = -\frac{2f' + f'\_{\rm cc}}{3} \tag{16}
$$

where *f* cc is the axial compressive strength of the confined concrete and *f* is the equivalent passive confinement stress.

#### *2.3. Seismic Strengthening of RC Columns by Direct Fastening Steel Jackets*

To investigate the seismic performance of strengthened RC columns using direct fastening steel jackets, an experimental study was conducted [25]. Attention was given to the enhancement of flexural stiffness and strength.

On the basis of comparisons between predicted and measured effective flexural stiffness, the expressions recommended in EN 1994-1-1 [30] are advised to calculate effective flexural stiffness:

$$K\_i = (EI)\_s + 0.6(EI)\_c \tag{17}$$

where *(EI)s* is the flexural stiffness provided by the steel jacketing; and *(EI)c* is the flexural stiffness provided by the RC column.

Buckling of the steel plate on the compressive side was observed. The corresponding buckling stress of steel plate under a compressive state should thus be embedded within the theoretical model used to predict lateral capacity. Using the Rayleigh–Ritz method and assuming the initial bowing and subsequent deflection as a trigonometric function, the buckling strength of steel plate is given as:

$$
\sigma\_{p,critical} = \frac{4\pi^2 D d\_p}{s\_d^2 A\_p} (1 - \alpha\_i) \tag{18}
$$

where *D* is the bending stiffness of the steel plate per unit width; *dp* is the plate depth; *Ep* and *μ<sup>p</sup>* are elastic modulus and Poisson's ratio of steel plate; *Ap* is the cross-sectional area of steel plate; and *α<sup>i</sup>* is the imperfection factor.

The imperfection factor *α<sup>i</sup>* is introduced to consider the initial bowing effect induced by the welding and is calibrated using the experimental results:

$$
\omega\_i = 1.046 - \frac{0.73\lambda\_{sr}}{100} \tag{19}
$$

where *λsr* = *sd*/*tp* is the slenderness ratio of steel plate. As the empirical imperfection factor was calibrated with limited available data, it is useful to assume a slenderness ratio of the steel plate between 14 and 39.

It is worth noting that the steel plate detached from the RC column at the bottom of the tension fiber. This was largely due to the strain incompatibility between RC column and the steel jacket on the tension side during bending. As a result, the assumption that the plane section would remain plane after deformation throughout the entire length of the column is inaccurate. Hence, a reduction factor of *η<sup>i</sup>* = 0.6 was introduced for the tensile component of the steel plate when the lateral resistant capacity was estimated.

#### **3. Design Procedure**

In this section, the critical parameters (i.e., steel plate thickness, fastener number and connection spacing) are first roughly determined. The feasibility of these parameters are further examined by satisfying four conditions: (1) desirable flexural failure occurs prior to brittle shear failure; (2) lateral load capacity is larger than lateral load demand; (3) ALR does not exceed the limit stipulated in current specification; and (4) effective flexural stiffness should be comparable to that of the undamaged RC column.

#### *3.1. Estimating Lateral Load Capacity of Damaged RC Columns*

The deficiency of the RC column may occur for a variety of reasons (e.g., fire). It is necessary to estimate the lateral load capacity of the damaged RC column and compare this with the load demand. According to the assumption that the plane section remains plane after deformation, the strain profile and the stress of the longitudinal rebar can be defined (Figure 6).

$$\begin{cases} & -f\_{yl} \le \sigma\_{l1} = E\_s(d\_\mathsf{c} - \mathsf{c} - \phi\_{\mathsf{s}t} - \frac{\phi\_l}{2} - \mathsf{x}\_\mathsf{c})\frac{\mathsf{c}\_{\mathsf{c}l}}{\mathsf{x}\_\mathsf{c}} \le f\_{yl} \\ & -f\_{yl} \le \sigma\_{lj} = E\_s(d\_\mathsf{c} - \mathsf{c} - \phi\_{\mathsf{s}t} - \frac{\phi\_l}{2} - (j-1)\mathsf{s}\_l - \mathsf{x}\_\mathsf{c})\frac{\mathsf{c}\_{\mathsf{c}l}}{\mathsf{x}\_\mathsf{c}} \le f\_{yl} \\ & -f\_{yl} \le \sigma\_{lm} = E\_s(d\_\mathsf{c} - \mathsf{c} - \phi\_{\mathsf{s}t} - \frac{\phi\_l}{2} - (m-1)\mathsf{s}\_l - \mathsf{x}\_\mathsf{c})\frac{\mathsf{c}\_{\mathsf{c}l}}{\mathsf{x}\_\mathsf{c}} \le f\_{yl} \end{cases} \tag{20}$$

where *sl* is the longitudinal rebar spacing; subscript *lj* represents the *j*th row longitudinal rebar counting from the tensile sides (1 ≤ *j* ≤ *m*); *m* is the total rows of longitudinal rebars which is equal to the stirrup legs; *φ<sup>l</sup>* is the diameter of the longitudinal rebar; *φst* is the diameter of the stirrup; *c* is the cover thickness of the column; *xc* is the compressive depth of the column; *εcu* is the ultimate strain of concrete and is taken as 0.003 in accordance with ACI 318 [31]; *fyl* is the yield strength of the longitudinal reinforcement; and *σlj* is the stress of the *j*th row longitudinal rebar.

**Figure 6.** Vertical strain and stress profiles of RC column: (**a**) strain profile; (**b**) stress profile.

The stress profile of the concrete is quantified using the equivalent rectangular stress block. The compressive depth is then determined by the axial equilibrium:

$$N\_0 + \frac{\sum\_{j=1}^{m} k\_j A\_l \sigma\_{lj}}{\gamma\_s} - \frac{a f\_c' d\_c \beta x\_c}{\gamma\_c} = 0\tag{21}$$

where *kj* is the number of the *j*th row longitudinal rebar; *γ<sup>s</sup>* is the partial safety factor of rebar; *γ<sup>c</sup>* is the partial safety factor of concrete; *N*<sup>0</sup> is the axial load; *Al* is the cross-sectional area of longitudinal rebar; *fc* is the concrete strength; and *α* and *β* are the two factors used to determine the equivalent rectangular stress block.

By taking moment to the extreme compressive fiber of the RC column, the moment capacity is given as:

$$M\_{\rm cap} = N\_0 \frac{d\_\xi}{2} + \frac{\sum\_{j=1}^m k\_j a\_{lj} \sigma\_{lj} (d\_c - c - \phi\_{st} - \frac{\phi\_l}{2} - (j-1)\kappa\_l)}{\gamma\_s} - \frac{a f\_c' d\_c \beta x\_c \frac{\beta x\_c}{2}}{\gamma\_c} \tag{22}$$

Under the assumption that the stiffness of the upper and lower joints is comparable, the lateral load resistance is then given by:

$$V\_{cap} = \frac{M\_{cap}}{0.5L} \tag{23}$$

where *L* is the length of the column.

If lateral load capacity cannot satisfy lateral load demand, the RC column can be strengthened using the direct fastening steel jackets.

#### *3.2. Determining Connection Spacing of Steel Jacket*

Flexural failure is the desirable failure mode. To ensure that desirable flexural failure occurs, the following relationship should be satisfied in light of ASCE 41-13 [32].

$$V \le V\_{cap,stream} \le 0.6V\_{stren} \tag{24}$$

where *V* is the lateral load demand acting on the RC column; *Vcap,stren* is the lateral load corresponding to the moment capacity of the strengthened RC column; and *Vstren* is the shear strength of the strengthened RC column which is determined by the following expression:

$$\begin{split} V\_{\text{strea}} &= V\_d + V\_c + V\_s\\ &= 0.5 \frac{d\_c}{s\_d + d\_d} \frac{2n\_f F\_b}{\gamma\_s} + \frac{0.17(1 + \frac{N\_0}{144\zeta\_c})\sqrt{f\_c'}d\_c d\_w}{\gamma\_c} + \frac{mA\_{st}f\_{yst}d\_w}{\gamma\_s s\_{st}} \end{split} \tag{25}$$

where *nf* is the fastener number in a connection; *Ac* is the cross-sectional area of the column; and *dw* is the depth of the column.

The first term represents the shear resistance produced from the connections. To avoid an overestimation of the shear resistance from the steel jacket, a factor of 0.5 is introduced by following EN 1998-3:2005 [33]. As the partial safety factor for direct fastening connections is absent, the partial safety factor of steel rebar is used.

Combining the above two expressions, the connection spacing of the steel jacket can be determined.

#### *3.3. Estimating Lateral Load Capacity of Strengthened RC Column*

According to the assumption that the plane section remains plane after deformation, the strain profile and stress profile of the longitudinal rebar and steel plates of the strengthened RC column are defined, respectively (see Figure 7). It should be noted that the side plate is divided into *n* equal small parts to accurately define the stress in the side plate. Moreover, the buckling of the steel plate under a compressive state is embedded.

$$\begin{cases} \begin{aligned} \sigma\_{pt} &= E\_p(d\_c + \frac{t\_p}{2} - \mathbf{x}\_c)\frac{\varepsilon\_{01}}{\mathbf{x}\_c} \le f\_{py} \\ \sigma\_{pc} &= E\_p(\frac{t\_p}{2} + \mathbf{x}\_c)\frac{\varepsilon\_{01}}{\mathbf{x}\_c} \le \sigma\_{p,critical} \\ &- \sigma\_{p,critical} \le \sigma\_{p,idc,j} = E\_p(d\_c - \frac{d\_c - d\_p}{2} - \frac{\Delta\_i}{2} - (i-1)\Delta\_i - \mathbf{x}\_c)\frac{\varepsilon\_{01}}{\mathbf{x}\_c} \le f\_{py} \\ &- f\_{yl} \le \sigma\_{l1} = E\_s(d\_c - c - \phi\_{st} - \frac{\Phi\_l}{2} - \mathbf{x}\_c)\frac{\varepsilon\_{01}}{\mathbf{x}\_c} \le f\_{yl} \\ &- f\_{yl} \le \sigma\_{lj} = E\_s(d\_c - c - \phi\_{st} - \frac{\Phi\_l}{2} - (j-1)\mathbf{s}\_l - \mathbf{x}\_c)\frac{\varepsilon\_{01}}{\mathbf{x}\_c} \le f\_{yl} \\ &- f\_{yl} \le \sigma\_{lm} = E\_s(d\_c - c - \phi\_{st} - \frac{\Phi\_l}{2} - (m-1)\mathbf{s}\_l - \mathbf{x}\_c)\frac{\varepsilon\_{01}}{\mathbf{x}\_c} \le f\_{yl} \end{aligned} \tag{26}$$

where *i* represents the *i*th partitioned part of the steel plate parallel to the lateral load counting from the tensile side; *σpside\_i* is the stress of the *i*th partitioned part of the steel plate parallel to the lateral load; Δ*<sup>i</sup>* is the length of the equal small part and is equal to *dp/n*; and *σp,critical* is the ultimate compressive stress of the steel plate, which can be defined by Equation (8).

**Figure 7.** Vertical strain and stress profiles of strengthened RC column: (**a**) strain profile; (**b**) stress profile.

The compressive depth of the strengthened RC column is then determined by the axial equilibrium.

$$0 = N\_0 + \frac{\sum\_{j=1}^{m} k\_j A\_l \sigma\_{lj}}{\gamma\_s} + \frac{\eta\_i \sigma\_{pt} A\_p - A\_p \sigma\_{pc} + 2 \sum\_{i=1}^{n} t\_p \Delta\_i \sigma\_{pisle\underline{i}}}{\gamma\_s} - \frac{a f\_c' d\_c \beta x\_c}{\gamma\_c} \tag{27}$$

By taking moment to the extreme compressive fiber of the strengthened RC column, the moment capacity is given as:

$$\begin{split} M\_{\text{cap},\text{stren}} &= \quad N\_0 \frac{d\_c}{2} - \frac{\frac{\alpha f'}{\gamma\_c} d\_c \beta \mathbf{x}\_c \frac{\beta \mathbf{x}\_c}{2}}{\gamma\_c} + \frac{\sum\_{j=1}^{n} k\_j A\_l \sigma\_{lj} (d\_c - c - \phi\_{sl} - \frac{\phi\_l}{2} - (j-1)s\_l)}{\gamma\_s} \\ &+ \frac{\eta\_i \sigma\_{pl} A\_p (d\_c + \frac{t\_p}{2}) + A\_P \sigma\_{pl} \frac{t\_p}{2} + 2 \sum\_{i=1}^{n} t\_P \Delta\_i \sigma\_{pie} j\_c (d\_c - \frac{d\_c - d\_p}{2} - \frac{\Lambda\_i}{2} - (i-1)\Delta\_i)}{\gamma\_s} \end{split} \tag{28}$$

Under the assumption the stiffness of the upper and lower joints is comparable, the lateral load resistance is then given by:

$$V\_{cap,stream} = \frac{M\_{cap,stream}}{0.5L} \tag{29}$$

Then, the relationship between the ultimate lateral load and the lateral load resistance of the strengthened RC column can be checked.

#### *3.4. Estimating ALR of Strengthened RC Columns*

The confined concrete divisions are shown in Figure 8a. Confined concrete 1 is confined by the erected steel jacket. Confined concrete 2 is confined by the stirrup. Confined concrete 3 is confined by the stirrup and erected steel jacket. Because the area of confined concrete 1 and confined concrete 2 is relatively smaller than that of confined concrete 3, and these two areas are close, the confined concrete divisions can be simplified in the design calculation. The simplified concrete divisions are shown in Figure 8b, in which the confined effect of confined concrete 2 is imposed on confined concrete 1. Thus, the initial three varieties of confined concrete can be equivalent to one confined concrete that is confined by a stirrup and steel jacket. For the confined concrete in Figure 8b, the imposed equivalent passive confinement stress is *fl = fest + fed*, which can be determined according to (3). Using the confined concrete strength model in (5), the confined concrete strength (*fcc* ) can be determined. Thus, the axial load capacity is given as:

$$N\_{c,stream} = \frac{A\_{c\varepsilon}f\_{c\varepsilon}^{\prime} + A\_{c0}f\_{\varepsilon}^{\prime}}{\gamma\_{\varepsilon}} + \frac{(4m - 4)A\_{l}f\_{yl} + 4P\_{p,critical}}{\gamma\_{s}}\tag{30}$$

where *Acc* is the area of confined concrete in Figure 8b; and *Ac*<sup>0</sup> is the area of the unconfined concrete in Figure 8b.

The confined concrete area *Acc* and unconfined area *Ac*<sup>0</sup> are respectively given as:

$$A\_{c\varepsilon} = d\_{\varepsilon}^2 - 4\frac{d\_l^2}{6} - (4m - 4)A\_l \tag{31}$$

$$A\_{c0} = 4\frac{d\_l^2}{6} \tag{32}$$

where *dl* is the edge-to-edge horizontal distance of the connections, as shown in Figure 8b. The ALR is given as:

$$\text{ALR} = \frac{N\_0}{N\_{c,strain}}\tag{33}$$

According to the recommendation in EN 1998-1:2004 [34], the ALR must necessarily be less than 0.65.

**Figure 8.** Divisions of the confined concrete: (**a**) exact divisions; (**b**) simplified divisions.

#### *3.5. Estimating Effective Stiffness of Strengthened RC Columns*

In light of the findings relating to the seismic strengthening of RC columns by direct fastening steel jackets, the effective flexural stiffness of the strengthened RC column can be estimated by (7). The effective flexural stiffness of the strengthened RC column should be comparable to that of the undamaged RC column, with the aim of the deformation capacity and seismic load carrying capacity remaining unchanged.

#### **4. Parameter Study**

A parameter study is conducted in this section to investigate the enhancement ratio (*η<sup>M</sup>* = *Vcap,stren*/ *Vcap*) of the lateral load capacity of strengthened RC columns and roughly estimate the required thickness of steel plate. Four critical variables (i.e., thickness of steel plate, cross-sectional dimension of column, longitudinal rebar ratio *ρ<sup>l</sup>* and ALR) are investigated to plot the design curves. The range of these four variables is set out in Table 1. The yield strength and elastic modulus of the longitudinal rebar are 500 MPa and 200 GPa, respectively. The yield strength, ultimate strength and elastic modulus of the steel plate are 300 MPa, 400 MPa and 200 GPa, respectively. The cylinder strength of the concrete is 30 MPa. The compressive strength of the compressive steel plate is greatly affected by its slenderness ratio (*λsr*). For conservative consideration, *λsr* is taken as 38, which is the

upper bound value of the slenderness ratio of steel plate determined from the experimental study [23,25].

**Table 1.** Range of four variables in parametric study.

It can be determined from Figure 9 that the enhancement ratio is significantly affected by the thickness of the steel plate and longitudinal rebar ratio. The enhancement ratio increases with the thickness of the steel plate, which can be seen from Equation (17). The enhancement ratio declines by 20% when the longitudinal rebar ratio increases from 0.015 to 0.03, as RC columns comprised of more longitudinal rebar possess a higher lateral load capacity, causing the decreased enhancement ratio. Compared with the effect of the longitudinal rebar ratio, the effect of the ALR on the enhancement ratio is marginal, especially for RC columns possessing higher longitudinal rebar ratios.

**Figure 9.** Enhancement of lateral load capacity: (**a**) ALR = 0.15; (**b**) ALR = 0.3.

#### **5. Worked Example**

The prototype of the RC column is shown in Figure 10. The height of the column is 3000 mm. The cross-sectional dimension is 500 mm × 500 mm. The reinforcement cage is formed by 12T20 and R10@150. At the ultimate limit state, the factored axial load (*N*0) and factored lateral load (*V*) are 1400 kN and 300 kN, respectively. The elastic modulus of the rebar is 200 GPa. The yield strength of the stirrup and the longitudinal rebar is 400 MPa and 500 MPa, respectively. The cylinder strength (*fc* ), cube strength (*fcu*), and elastic modulus (*Ec*) of the concrete are 30 MPa, 40 MPa, and 25 GPa, respectively. The cover thickness (*c*) is 20 mm. It is postulated that the column experienced a fire event up to 570 ◦C. The strength and stiffness of the RC column can be greatly impaired after exposure to high temperature. The proposed strengthening method of directly fastening steel jackets can be used to restore fire-damaged RC columns. The elastic modulus, yield strength and ultimate strength of the steel plate are 200 GPa, 300 MPa and 400 MPa, respectively. The steel angle bracket used to connect the two adjacent steel plates measures 75 mm × 75 mm × 5 mm with a length of 50 mm. The 4 mm diameter high strength knurled fastener is used in this strengthening system. The partial safety factor of steel and concrete is taken as 1.2 and 1.5.

**Figure 10.** Reinforcement details of column: (**a**) front view; (**b**) plane view.

#### *5.1. Evaluating Lateral Load Capacity of Fire-Damaged Column*

After the concrete is exposed to fire, the concrete strength is impaired. The concrete strength post fire can be estimated according to the recommendation in EN-1992-1-2 [35]. The residual cylinder strength (*fc ,pf*) of the RC column following exposure to 570 ◦C fire is 0.5*fc* .

According to the study on the mild steel post fire in [22], the residual strength and elastic modulus of rebar can be readily determined. The residual strength of longitudinal rebar is given as:

$$\begin{array}{rcl} f\_{yl,pf} & = f\_{yl} \text{Rs}\_{y} \\ & = & 500 \times (1.0 - 0.2 \times \frac{570 - 400}{600}) \\ & = & 470 \text{ MPa} \end{array}$$

The residual strength of the stirrup is given as:

$$\begin{array}{rcl} f\_{\text{yst,pf}} & = f\_{\text{yst}} R s\_y \\ & = & 400 \times (1.0 - 0.2 \times \frac{570 - 400}{600}) \\ & = & 376 \text{ MPa} \end{array}$$

The residual elastic modulus of rebar is given as:

$$\begin{array}{rcl} E\_{s,pf} &= E\_s R s\_E\\ &=& 200 \times (1.0 - 0.07 \times \frac{570 - 350}{500})\\ &= 194 \text{ GPa} \end{array}$$

According to (20) and (21), the stresses of the four rows of longitudinal rebar and thus the compressive height of the plane section can be determined.

$$N\_0 + \frac{4A\_l\sigma\_{l1} + 2A\_l\sigma\_{l2} + 2A\_l\sigma\_{l3} + 4A\_l\sigma\_{l4}}{1.2} - \frac{\alpha f\_{c,pf}'d\_c\beta x\_c}{1.5} = 0$$

The compressive height of the plane section is solved as:

$$\mathbf{x}\_{\mathbf{c}} = 304 \text{ mm}$$

The moment capacity is derived by (22).

$$\begin{array}{ll} M\_{\text{cap}} &= N\_0 \frac{d\_c}{2} - \frac{a f'\_{c, p f} d\_c \phi x\_c \frac{p\_{\text{fix}}}{2}}{1.5} \\ &+ \frac{4 A\_l \sigma\_{l1} (d\_c - c - \phi\_{sl} - \frac{\Phi\_l}{2}) + 2 A\_l \sigma\_{l2} (d\_c - c - \phi\_{sl} - \frac{\Phi\_l}{2} - s\_l)}{1.2} \\ &+ \frac{2 A\_l \sigma\_{l3} (d\_c - c - \phi\_{sl} - \frac{\Phi\_l}{2} - 2s\_l) + 4 A\_l \sigma\_{l4} (d\_c - c - \phi\_{sl} - \frac{\Phi\_l}{2} - 3s\_l)}{1.2} \\ &= 311 \text{ kN} \cdot \text{m} \end{array}$$

The lateral load resistance of the fire-damaged RC column is given by (23).

$$V\_{cap} = \frac{M\_{cap}}{0.5L} = \frac{311 \times 10^3}{0.5 \times 3000} = 207 \text{ kN} < V = 300 \text{ kN}$$

Thus, the fire-damaged RC column is not safe, and the required enhancement ratio of the RC column is around 1.4.

#### *5.2. Evaluating Lateral Load Capacity of Strengthened RC Column*

Since the required enhancement ratio is around 1.4 and the longitudinal rebar ratio is 0.015, 4 mm steel plate is utilized to strengthen the fire-damaged RC column based on the findings in Figure 8. Additionally, four fasteners on the connection are used.

#### (a) Determining connection spacing

Flexural failure is the desirable failure mode. To realize this, Equation (24) should be satisfied.

$$300\text{ kN} = V \le V\_{cap,stream} \le 0.6 V\_{strem}$$

According to Equation (25), the shear capacity of the strengthened fire-damaged RC column is given as:

$$\begin{array}{rcl} V\_{\rm stru} &= V\_d + V\_{c,pf} + V\_{s,pf} \\ &= 0.5 \frac{d\_c}{s\_d + d\_d} \frac{2n\_f F\_d}{\gamma\_s} + \frac{0.17(1 + \frac{N\_0}{14 \lambda\_c}) \sqrt{f\_{c,pf}'} d\_c d\_w}{\gamma\_c} + \frac{mA\_{sf} f\_{gel,pf} d\_w}{\gamma\_s s\_{sl}} \\ &= 0.5 \times \frac{500}{s\_d'} \times 2 \times \frac{4 \times 1.6 \times 1.35 \times 1.17 \times 4 \times 4 \times 400}{1.2} + \frac{0.17(1 + \frac{1400 \times 1000}{13 \times 500 \times 500})}{1.5} \\ &\times \sqrt{15} \times 500 \times \left(500 - c - \phi\_{st} - \frac{\phi\_l}{2}\right) + \frac{4 \times \frac{344 \times 10^2}{4} \times 376 \times (500 - c - \phi\_{sl} - \frac{\phi\_l}{2})}{150 \times 1.2}} \end{array}$$

Solving the above two equations, the connection spacing is given as:

$$s\_d \le 160 \text{ mm}$$

Here, the connection spacing is taken as 100 mm. The corresponding shear capacity of the strengthened column is 623 kN.

(b) Determining ultimate compressive stress in compressive steel plate

The slenderness ratio of the steel plate is given as:

$$
\lambda\_{\mathcal{F}} = \frac{s\_d}{t\_P} = \frac{100}{4} = 25^\circ
$$

The initial imperfection factor is given by (19).

 $\kappa\_{l} = 1.046 - 0.0073$  $\lambda\_{sr} = 1.046 - 0.0073 \times 25 = 0.86$

The ultimate compressive stress of the steel plate is determined by (18).

$$\begin{array}{rcl} \sigma\_{p,critical} &= \frac{4\pi^2 D d\_p}{s\_d^2 A\_p} (1 - \alpha\_i) = \frac{4\pi^2 E\_p t\_p^2}{12s\_d^2 (1 - \mu\_p^2)} (1 - \alpha\_i) \\ &= \frac{4 \times 3.14^2 \times 200 \times 10^3 \times 4^2}{12 \times 100^2 \times (1 - 0.3^2)} \times (1 - 0.86) \\ &= 162 \text{ MPa} < f\_{pp} = 300 \text{ MPa} \end{array}$$

#### (c) Determining lateral load capacity of strengthened RC column

The stresses in steel plate and longitudinal rebar are given by Equation (26). The compressive depth of the strengthened fire-damaged RC column is determined by Equation (27).

$$\begin{array}{c} 0 = \quad \mathsf{N}\_{0} + \frac{4A\_{l}\sigma\_{l1} + 2A\_{l}\sigma\_{l2} + 2A\_{l}\sigma\_{l3} + 4A\_{l}\sigma\_{l4}}{1.2} \\ + \frac{0.6\sigma\_{p1}A\_{p} - A\_{p}\sigma\_{p<1} + 2\sum\_{i=1}^{n}t\_{p}\Delta\_{i}\sigma\_{p\text{side}\_{-i}}}{1.2} - \frac{af\_{c,pf}'d\_{c}\delta\chi\_{c}}{1.5} \end{array}$$

The compressive height of the plane section of the strengthened fire-damaged RC column is solved as:

$$\mathbf{x}\_c = 302 \text{ mm}$$

The moment capacity of the strengthened fire-damaged RC column is derived by Equation (28).

*Mcap*,*stren* = *N*<sup>0</sup> *dc* <sup>2</sup> <sup>−</sup> *<sup>α</sup> <sup>f</sup> <sup>c</sup>*,*p f dcβxc βxc* 2 1.5 <sup>+</sup>4*Alσl*1(*dc*−*c*−*φst*<sup>−</sup> *<sup>φ</sup><sup>l</sup>* <sup>2</sup> )+2*Alσl*2(*dc*−*c*−*φst*<sup>−</sup> *<sup>φ</sup><sup>l</sup>* <sup>2</sup> −*sl*) 1.2 <sup>+</sup>2*Alσl*3(*dc*−*c*−*φst*<sup>−</sup> *<sup>φ</sup><sup>l</sup>* <sup>2</sup> <sup>−</sup>2*sl*)+4*Alσl*4(*dc*−*c*−*φst*<sup>−</sup> *<sup>φ</sup><sup>l</sup>* <sup>2</sup> −3*sl*) 1.2 + 0.6*σpt*1*Ap*(*dc*<sup>+</sup> *tp* <sup>2</sup> )+*Apσpc*<sup>1</sup> *tp* <sup>2</sup> <sup>+</sup><sup>2</sup> *<sup>n</sup>* ∑ *i*=1 *tp*Δ*iσpside*\_*i*(*dc*<sup>−</sup> *dc*−*dp* <sup>2</sup> <sup>−</sup> <sup>Δ</sup>*<sup>i</sup>* <sup>2</sup> −(*i*−1)Δ*i*) 1.2 = 530 kN · m

The lateral load resistance of the strengthened fire-damaged RC column is given by Equation (29).

$$V\_{\text{cap},tren} = \frac{M\_{\text{cap},tren}}{0.5L} = \frac{530 \times 10^3}{0.5 \times 3000} = 353 \text{ kN} > V = 300 \text{ kN}$$

The condition in (24) is rechecked herein.

$$V\_{cap,stream} = 353 \text{ kN} < 0.6 V\_{stream} = 374 \text{ kN}$$

#### *5.3. Checking ALR*

#### (a) Estimating equivalent passive confinement stress

The equivalent passive confinement stress originated from the stirrup and the connections are respectively given by (4) and (5).

$$\begin{array}{rcl} f\_{\text{est}} &= \frac{4a\_{\text{sf}}f\_{\text{st},f}A\_{\text{tf}}}{s\_{\text{sf}}l\_{\text{st}}} & f\_{\text{cd}} &= \frac{2a\_{\text{d}}u\_{f}F\_{\text{d}}}{(s\_{\text{d}}+d\_{\text{d}})d\_{\text{c}}}\\ &= \frac{4 \times 0.34 \times 376 \times 3.14 \times \frac{10^{2}}{4}}{150 \times (500-20 \times 2-10)} & &= \frac{2 \times 0.47 \times (0.82 - 0.64 \times \frac{100}{500}) \times 4 \times 1.6 \times 1.35 \times 1.17 \times 4 \times 4400}{150 \times 500} \\ &= 0.59 \text{ MPa} & &= 0.56 \text{ MPa} \end{array}$$

#### (b) Estimating axial load capacity

For the confined concrete in Figure 7b, the imposed equivalent passive confinement stress is *fl* = *fest* + *fed* = 0.59 + 0.56 = 1.15 MPa. Using the confined concrete strength model in (7)–(16), the confined concrete strength is 22.5 MPa. Thus, the axial load capacity is given by Equations (30)–(32).

$$\begin{array}{rcl} N\_{c,strren} &=& \frac{A\_{c\uparrow}f\_{c,pf}^{\prime} + A\_{c0}f\_{c,pf}^{\prime}}{\gamma\_{c}} + \frac{12A\_{l}f\_{pl,pf} + 4A\_{l}\sigma\_{p,critical}}{\gamma\_{s}} \\ &=& \frac{(500 \times 500 - 12 \times \frac{3.14 \times 20^{2}}{4} - \frac{2}{3}(500 - 70 \times 2)^{2}) \times 22.5 + \frac{2}{3}(500 - 70 \times 2)^{2} \times 15}{1.5} \\ &+& \frac{12 \times \frac{3.14 \times 20^{2}}{4} \times 470 + 4 \times 4 \times 495 \times 162}{1.2} \\ &=& 5800 \text{ kN} \end{array}$$

The ALR is given by (33).

$$\text{ALR} = \frac{N\_0}{N\_{\varepsilon, strm}} = \frac{1400}{5800} = 0.24 < 0.65$$

#### *5.4. Checking Effective Flexural Stiffness*

Based on the study conducted by Hwang et al. [6], the residual elastic modulus of concrete retains around 0.2*Ec* after exposure to a 570 ◦C fire event. According to (17), the effective flexural stiffness of the strengthened RC column is given as:

$$\begin{array}{l} \mathcal{K}\_{i} \ = (EI)\_{s} + 0.6(EI)\_{c.pf} \\ = 2\frac{E\_{p}t\_{p}d\_{p}^{3}}{12} + E\_{P}t\_{p}d\_{p}\frac{d\_{2}^{2}}{2} + 0.6\frac{0.2E\_{c}d\_{c}^{4}}{12} \\ = 2 \times \frac{200000 \times 4 \times 495^{3}}{12} + 200000 \times 4 \times 495 \times \frac{500^{2}}{2} + 0.6 \times \frac{0.2 \times 25000 \times 500^{4}}{12} \\ = 8.1 \times 10^{13} \text{N} \cdot \text{mm}^{2} \approx 7.8 \times 10^{13} \text{N} \cdot \text{mm}^{2} = 0.6 \times \frac{25000 \times 500^{4}}{12} = 0.6(EI)\_{c} \end{array}$$

Until then, the parameters of the steel jacket (i.e., steel plate thickness, fastener number and connection spacing) are determined. The strengthening scheme is shown in Figure 11.

**Figure 11.** Strengthened RC column: (**a**) front view; (**b**) plane view.

#### **6. Conclusions**

In this paper, a novel direct fastening steel jacket is introduced. The main findings of the available experimental and theoretical studies on the proposed strengthening method are reviewed, based on which the design procedure for the proposed strengthening method is presented. The design procedure can be processed by the application of four critical steps: (1) sizing the steel plate thickness based on the required enhancement ratio and Figure 9; (2) sizing the connection spacing and fastener number by avoiding flexural failure; (3) checking lateral load demand and capacity; and (4) checking the ALR limit. If either of the final two checking conditions is not satisfied, the process should be repeated from step (1) until these two final conditions are satisfied. Subsequently, the effective flexural stiffness is examined to ensure it is similar to that of the undamaged RC column.

**Author Contributions:** Conceptualization, Z.S. and R.K.L.S.; methodology, Z.S. and R.K.L.S.; investigation, L.C. and K.L.; writing—original draft preparation, Z.S.; writing—review and editing, Z.S., R.K.L.S. and Z.X. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was supported by the Fundamental Research Funds for the Central Universities.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** The results presented in this study are available on request from the corresponding author.

**Acknowledgments:** This research was supported by the Fundamental Research Funds for the Central Universities. The authors would like to thank for their financial support.

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

#### **References**

**Jing Li 1, Lizhong Jiang 1, Hong Zheng 2, Liqiang Jiang 1,\* and Lingyu Zhou 1,\***

**Abstract:** A bolt-connected precast reinforced concrete deep beam (RDB) is proposed as a lateral resisting component that can be used in frame structures to resist seismic loads. RDB can be installed in the steel frame by connecting to the frame beam with only high-strength bolts, which is different from the commonly used cast-in-place RC walls. Two 1/3 scaled specimens with different height-tolength ratios were tested to obtain their seismic performance. The finite element method is used to model the seismic behavior of the test specimens, and parametric analyses are conducted to study the effect on the height-to-length ratio, the strength of the concrete and the height-to-thickness ratio of RDBs. The experimental and numerical results show that the RDB with a low height-to-length ratio exhibited a shear–bending failure mode, while the RDB with a high height-to-length ratio failed with a shear-dominated failure mode. By comparing the RDB with a height-to-length ratio of 2.0, the ultimate capacity, initial stiffness and ductility of the RDB with a height-to-length ratio of 0.75 increased by 277%, 429% and 141%, respectively. It was found that the seismic performance of frame structures could be effectively adjusted by changing the height-to-length ratio and length-to-thickness of the RDB. The RDB is a desirable lateral-resisting component for existing and new frame buildings.

**Keywords:** precast reinforced concrete deep beam; experimental study; seismic performance; finite element method (FEM)

#### **1. Introduction**

A reinforced concrete (RC) shear wall is commonly used as an infilled lateral loadresisting component in RC frames for high-rise buildings. The RC frames with infill RC shear walls exhibited better performance compared with the traditional bare RC frames because of the insufficient lateral stiffness and resistance of RC frames. The frame columns and beams are used to resist the vertical loads and moments due to earthquakes, while the RC shear walls (RCSW) resisted the majority of shear forces [1]. The initial researches focused on different kinds of RCSW systems and conducted experimental and theoretical investigations on their seismic performance [2–4]. Furthermore, experimental and numerical studies were performed to analyze the mechanical behavior of different types of connections between RC walls and building frames [5–7]. However, the RC shear walls may exhibit oversized lateral stiffness resulting in an insufficient lateral deformation capacity of the structure with an insufficient load-carrying capacity. Besides, the shear force carried by the RC shear walls may be transmitted as tension or compressive force to the frame columns [8]. To prevent severe damage from occurring at the end of frame columns, strong columns must be designed to comply with the design principle of "strong frame, weak wall". Furthermore, to meet the architectural requirements, the effects of different kinds of openings on the seismic performance of RCSWs also need to be investigated, which may make the seismic design of RCSWs more difficult.

Three major issues should be considered for the frames with RCSW systems. Firstly, a few RC shear walls might be placed in the first story for the stores and open lobbies due to

**Citation:** Li, J.; Jiang, L.; Zheng, H.; Jiang, L.; Zhou, L. Cyclic Tests and Numerical Analyses on Bolt-Connected Precast Reinforced Concrete Deep Beams. *Appl. Sci.* **2021**, *11*, 5356. https://doi.org/10.3390/ app11125356

Academic Editors: Pier Paolo Rossi and Melina Bosco

Received: 10 April 2021 Accepted: 3 June 2021 Published: 9 June 2021

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the architectural demands. These buildings may have a soft first story if insufficient shear walls are placed in the first story compared to the upper stories [9]. Secondly, rapid repair and retrofit methods of a severely damaged component are desired for structures after an earthquake occurs [10,11]. The RC shear walls are commonly connected with frame columns and beams cast in place; thus, the frame columns and beams may be destroyed if the retrofit work is performed on the damaged walls. Lastly, the dimensions of the infill walls are determined by the dimensions of the surrounding frame. The lateral stiffness and resistance of the structure are mainly adjusted through the thickness of the walls. However, the thickness is a strict rule for the RC walls due to stability requirements [2]. Therefore, the lateral stiffness and resistance of the RC frames with RCSWs are difficult to change on a large scale to meet the optimization design of RC structures.

In the past decades, the RC wall panel (RCWP) was introduced by Kahn and Hanson [12] as a strengthening member to enhance the performance of RC structures. It was also proposed as a retrofit/repair strategy for the RC frames [13] because the member could be connected to structures with precast construction. Engineered cementitious composite materials (ECC) were proposed for RCWP due to their good performance to dissipate energies generated from earthquakes [14], and a series of experimental and theoretical investigations were performed. In recent years, many researchers have suggested that the RC walls should be separated with frame columns, including separating the RC infill wall from the RC frame by slits [15], and slitting the separated RC infill wall [9]. Other types of steel wall panels, such as double-skin wall panels [16] and panel wall strengthened systems [17–20], were proposed and analyzed. Different from the steel wall panels, the RC wall panels were commonly connected to the boundary frames with cast-in-place construction. Thus the RC wall panels are difficult to remove or repair. The precast concrete components or structures should be developed for the rapid construction of new structures and repairing existing damaged structures under seismic hazard, and it is important to guarantee the desired structural performance for the precast structures [21,22]. The connections are the key aspects of obtaining precast structures' desired performance [23–26]. Therefore, a new type of connection could realize the rapid construction and replacement of RC wall panels and could also ensure the connecting performance between RC wall panels and boundary frames.

Based on the researches presented here, the bolt-connected precast reinforced concrete deep beam (RDB) is proposed in this paper. It connects to the frame beam with highstrength bolts. It exhibits some advantages to handle the above issues compared to the other RC walls for the following reasons. Firstly, the RDB can be flexibly placed, and thus it provides architectural space for the installation of doors and windows. Secondly, it can be easily assembled or removed from the bolted connection; thus, it can be rapidly and cost-efficiently fabricated in building repair or retrofit. Lastly, the length of the RDB could be selected in a wide range; thus, it can achieve a wide range of initial stiffness and lateral resistance. In addition, the main deformation of the deep beam is combining bending and shearing deformation; thus, RDB may exhibit better ductility by comparing it with the RC infill walls for relatively sufficient plastic development of materials. It is also a part of a project that uses a deep beam as a lateral resisting system.

In this study, experimental investigations of two precast reinforced concrete deep beam specimens that were 1/3 scaled were performed, and numerical studies of RDBs were conducted and verified by test results. The initial stiffness, as well as the mechanical behavior of the bolt connections, were also discussed and calculated. Such results provide information for the seismic design of bolt-connected precast RDB components.

#### **2. Materials and Methods**

#### *2.1. Material Properties*

In Table 1, the average cube compressive strength of concrete material, as well as the material performance of reinforcing bars, is presented. The concrete and reinforcement steel bars were designed as C30 and HPB235, respectively. Based on the Chinese standard

GB50081 [27], concrete cube specimens taken from the batches of concrete were tested. Three repeatable specimens were fabricated with 28 days of standard maintenance, and the average values were used. For the high-strength bolts used in this paper, the yield strength was 800 N/mm2, and the yield-failure ratio was 0.8. The pretension force applied to each bolt was 125 kN.

**Table 1.** Mechanical properties of concrete and reinforcing bars.

#### *2.2. Test Specimens*

In the test program, two bolt-connected precast reinforced concrete deep beams (RDB) were designed on a 1/3 scale and different height-to-length ratios (RDB-A and RDB-B were 2.0 and 0.75, respectively). The dimensions of these specimens and the test setup are shown in Figure 1. The height of RDBs was 900 mm, and the length was 450 mm for RDB-A and 1200 mm for RDB-B, respectively. The clear height of the specimens was 810 mm. The thickness of the specimens was 60 mm. The reinforcing bars were arranged at a distance of 75 mm, and the diameter was 8 mm.

**Figure 1.** Test setup and test specimens.

#### *2.3. Setup and Measurements of Test Specimens*

The construction details of the test device and the specimens are presented in Figure 2. To distribute the load in the RDBs more evenly, a distribution member was designed to connect to the actuator through M45 bolts. Two angles were placed at the two sides of RDBs, and these angles were equipped back-to-back to support the RDBs. M20 high-strength bolts

were used to connect the angles and the RDBs. The angles were connected to the base beam and distribution member by M20 bolts, and they were connected to the strong floor by anchorage bolts. To ensure that no out-of-plane deformation developed in the specimens during the tests, two rigid beams were constructed to provide lateral supports. Two pairs of sliding constraints were installed between the rigid beams and the distribution member.

**Figure 2.** Details of the test device, specimens, connection details and boundary conditions.

Three displacement-based linear transducers were used for measuring the lateral displacement of the specimen. They were placed at the two sides of the RDBs and the base beam, and they were used to measure the relative lateral deformation of the test specimens. The stress distribution of the RDBs was traced by uniaxial strain gauges, and they were placed along the reinforced bars to analyze the mechanical behavior of the RDBs, as presented in Figure 3. The loading pattern of the test specimens was designed based on the Chinese standard JGJ 101 [28], and it was controlled by displacement. The diagrams of the loading pattern are shown in Figure 4. An increment of 1 mm on displacement was used at the elastic mechanical stage, and the displacement did not repeat at the same loading level. The increment was increased at the elastic–plastic stage, and the displacement was repeated three times at the same loading level. Two methods were used to determine the yield displacement of the test specimens: some of the strain gauges exceeded the yield value, and the slope of the load–displacement curves decreased sharply.

**Figure 3.** Arrangement of uniaxial strain gauges on the reinforcing bars.

**Figure 4.** Loading patterns for the test specimens.

#### **3. Test Results**

#### *3.1. Failure Mode and Hysteretic Behavior*

The RDB-A failed to combine with the bend and shear destroy. An inclined crack was found at the bottom of the deep beam at the 2 mm loading step. At this loading step, concrete spalling was observed at the top side corner of the deep beam, but the reinforcing bars were not exposed, as shown in Figure 5a. New cracks were observed at the corner and bottom of the specimen at the story displacement of 3 mm, and a major crack had obviously developed along the direction of 45◦ in a horizontal direction. When the displacement reached 5 mm, the concrete at the bottom corners was apparently crushed, as presented in Figure 5b, which resulted in exposing the stirrups. A crisp noise was produced from the high-strength bolts at this step. After that, a relative slip deformation was found, and the cracks developed quickly. The final failure pattern of the RDB-A specimen is depicted in Figure 5c.

(**a**) Spalling of concrete (**b**) Crack at the corner (**c**) Failure of the RDB-A

**Figure 5.** Failure modes of the RDB-A.

The RDB-B failed, dominated by the shearing destroy. The first crack was found at the bottom corner nearby, with the first high-strength bolt at the story displacement of 2 mm. When the displacement had reached 3 mm, such cracks could not be closed, as shown in Figure 6a, and a crisp noise was heard from the high-strength bolts. The first diagonal crack was detected; it was formed along the direction of 45◦ in a horizontal direction when the lateral displacement was 5 mm, as shown in Figure 6b. After that, more other

is presented in Figure 6c.

(**a**) Cracks at the corner (**b**) Diagonal cracks at the center region (**c**) Failure of the RDB-B

**Figure 6.** Failure modes of the RDB-B.

The relation curves of lateral load and displacement curves of the specimens are depicted in Figure 7. The force and displacement results of some key points are presented in Table 2. When the height-to-length ratio of the RDB specimen decreased from 2.0 to 0.75, the ultimate capacity, initial lateral stiffness and ductility increased about 277%, 429% and 141%, respectively. The results indicated that the height-to-length ratio is an important factor in adjusting the strength and stiffness of the RDBs on a large scale.

inclined cracks were formed at the center region of the deep beam. The final failure mode

**Figure 7.** Loading-displacement curves of the specimens.

#### *3.2. Ductility and Energy Dissipation*

In this paper, the displacement-based ductility coefficient *μ*<sup>Δ</sup> is utilized to represent the ductility performance of RDBs. The *μ*<sup>Δ</sup> is determined by the displacements when the specimens yield and fail, and the calculation is *μ*<sup>Δ</sup> = Δmax/Δy. The equal energy method is used to obtain the value of Δy. The failure displacement is defined as 85% times the maximum value [8]. The ductility results are listed in Table 2. The RDB-B exhibited better ductility than the RDB-A, indicating that the ductility of RDBs increases if the height-tolength ratio reduces.

To evaluate the energy dissipation performance of RDB specimens, the energy dissipation coefficient *E* is used. The detailed calculation and the diagram are shown in Figure 8a [8]. The energy dissipation coefficients of the test specimens in this paper are presented in Figure 8b. The results show that the energy dissipation coefficient decreases with the increase of the lateral displacement for the most part. The energy dissipation capacity of RDB-A is close to the RDB-B, indicating that the energy dissipation capacity of RDBs changes little by decreasing the height-to-length ratio. Due to the limited number of specimens, it should be highlighted that the effects of the height-to-length ratio on ductility and energy dissipation coefficient are not regular.

**Figure 8.** Energy dissipation results of test specimens.

#### *3.3. Elastic Stiffness*

The boundary condition is an important factor to determine the stiffness as well as the ultimate strength of RC deep beams [29]. Thus, the boundary conditions of the RDBs can be simplified as a deep beam with fixed supports [8], and the idealized diagram is presented in Figure 9a. The initial stiffness of the RDB could be obtained by considering the bending deformation and shearing deformation of the RDB. The deformation diagram and force distribution diagram of the RDBs associated with lateral force *F* are shown in Figure 9b.

**Figure 9.** Deformation and force diagrams for RDBs and bolt connections.

The lateral displacement of RDBs Δ<sup>w</sup> is a combination of bending deformation ΔwM and shearing deformation ΔwV. The displacement of RDBs can be calculated based on the Graph Multiplication Method in basic theory in structural mechanics:

$$
\Delta\_{\rm W} = \Delta\_{\rm WM} + \Delta\_{\rm WV} = \frac{V\_{\rm w}h^3}{12E\_{\rm c}I\_{\rm W}} + \frac{1.2V\_{\rm w}h}{GA} \tag{1}
$$

where *I*w, *V*<sup>w</sup> and *A* are the moments of inertia, shear force and section area of RDBs, respectively; *E*<sup>c</sup> and *G* are the elastic modulus and shearing modulus of concrete. Thus, Equation (1) can be expressed as:

$$
\Delta\_{\rm w} = \frac{V\_{\rm w}}{E\_{\rm c}t} \left[ \frac{1}{\left(l/h\right)^{3}} + \frac{2.88}{\left(l/h\right)} \right] \tag{2}
$$

where *h* is the effective height and *l* is the effective length. The elastic lateral stiffness *K*e is calculated as:

$$K\_{\rm c} = \frac{E\_{\rm c}t}{\left[\left(h/l\right)^{3} + 2.88\left(h/l\right)\right]}\tag{3}$$

Table 3 presents the comparison of the elastic lateral stiffness of the test specimens obtained from Equation (3) and the tests. It was found that the error is no more than 20%, and the values derived from the equation slightly overestimate the lateral stiffness of the test specimens. It can be concluded that the imperfections of the specimens and uncertainty in test procedures were not considered in the equation. However, the overestimation can also be treated as safety reserves in the structural design of RDBs.

**Table 3.** Comparison of elastic stiffness of the theoretical and experimental test results.

#### *3.4. Mechanical Behavior of the Bolt Connection*

The frictional types of high-strength bolts were used in the tests to transfer the shear forces. Thus, it is important to ensure the effectiveness and reliability of the bolt connections. In this paper, the mechanical behavior of the bolt connections is divided into two main phases; they are: (1) phase I, the carried shear force of a bolt is less than its friction force; (2) phase II, the carried shear force of a bolt is larger than its friction force.

A mechanical diagram of RDBs at phase I is presented in Figure 9c; the shear force of bolts generated from the lateral load (*N*v) and bending moment (*N*m) are included in the diagram. The lateral force is considered as an even distribution force on the bolts; the shear force of each bolt is determined as *N*<sup>v</sup> = *V*w/*n*, where *n* is the number of the bolts in a support of RDBs. According to Figure 9c, *M*<sup>w</sup> = *V*w*h*/2 = 0.405*V*<sup>w</sup> for the RDB-A, thus the *N*<sup>m</sup> = *M*w/4*D* = 0.675*V*w, and *N*<sup>v</sup> = *V*w/3. The largest shear force is 0.75*V*w. For the mechanical diagram of RDB-B, *M*<sup>w</sup> = *V*w*h*/2 = 0.405 *V*<sup>w</sup> and *N*m4 = 0.225*V*w, and *N*v = *V*w/8. The largest shear force is 0.26 *V*w.

At phase II, due to the carried shear force of the bolts that exceeded the friction force, the bending moment is carried by the RDB, and thus the bolts carry the shear force solely. The largest shear force of the bolts is 0.33 *V*<sup>w</sup> and 0.125 *V*<sup>w</sup> for the RDB-A and RDB-B, respectively.

The shear force carried by the bolts in specimens is 10.35 kN and 39.06 kN for the RDB-A and RDB-B (Table 2), respectively. Thus the largest shear force is 7.76 kN and 10.16 kN in phase I for bolt connections in RDB-A and RDB-B, respectively, and the largest shear force is 3.42 kN and 4.88 kN in phase II for the bolt connections in RDB-A and RDB-B, respectively.

The friction force *N*vuI in phase I and shear resistance *N*vuII in phase II of the bolts used in this paper are:

$$N\_{\rm vul} = 0.9 n\_{\rm f} \mu P \tag{4}$$

$$N\_{\rm vall} = \frac{\pi d\_0^2}{4} f\_{\rm v} \tag{5}$$

where *n*<sup>f</sup> is the number of frictional faces of bolts; *μ* is the anti-sliding coefficient between bolts and angles, and it is determined as 0.3 according to suggestions [8]; *P* is the pre-tensile force, it is 125 kN in this study; *d*<sup>0</sup> is the clear diameter of bolts, and it is 20 mm; *f* <sup>v</sup> is the yield shear strength of bolts, and it is 320 MPa.

Thus*N*vuI = 67.5 kN and*N*vuII =100.5 kN can be calculated according toEquations (4) and (5), and the calculated values represent the loading resistance of a high-strength bolt at the two phases. The calculated values are larger than the maximum shear force of the bolts during tests; thus, it can be concluded that high-strength bolts are safe enough in the test specimens. According to test results recorded by the LVDTs, there was also no slipping behavior or bolt-damage failure observed in the test specimens.

#### **4. Numerical Study**

#### *4.1. Validation of Finite Element Model*

The numerical study was performed by ANSYS, as presented in Figure 10, and the "Solid 65" and "Pipe20" elements were selected out to model the mechanical behavior of reinforced concrete. A bilinear model was used for the reinforcements, and the material test results such as the elastic modulus, yield stress and ultimate stress were derived from material test results. The concrete damage plasticity model in ANSYS was used for the concrete material in RDBs, because it considers the nonlinear behavior of concrete after it cracks or crushes. The compressive strength curve was taken from the compressive test results of the concrete, and the tensile cracking strength was estimated as 5.6% of the compressive strength. The 3D 8-node solid element captured the cracking and crippling behavior of the concrete. The same mesh sizes (50 mm) were used to mesh the model. The intersectant nodes in concrete elements and reinforcing bar elements were coupled together at the same positions so that they could cause consistent deformations. The boundary conditions of the bottom and top side of the concrete panel are considered as a fixed connection (at the bottom side, fixed the X, Y, Z, Rot X, Rot Y and Rot Z; at the top side, fixed the Y, Z, Rot X, Rot Y and Rot Z; the X-direction represents the direction that applied lateral loads). The lateral load is assumed as a uniform load to apply to the top side of the concrete panel in the finite element model, and this mechanical assumption was valid by Jiang et al. [8].

**Figure 10.** Deformation and force diagrams for RDB.

The limitations of the modeling method are: (1) the effects of bolt connections, including the influence on stress concentration and the weakness on RC walls, are not considered in this model; (2) the bond-slipping deformation between reinforcements and concrete is not considered in the model.

The comparison between test and FEM results on lateral–loading curves and failure modes are presented in Figures 11 and 12, respectively. There is a relatively close agreement between the test results and FEM results in the elastic and elastic–plastic stage. For the RDB-A, the peak shear force of RDBs during the tests and FEM analysis is 10.35 kN and 12.22 kN, respectively, and it is 39.06 kN and 45.24 kN for RDB-B specimen, respectively. The difference in maximum shear force value between tests and FE models is no more than 18%. Besides, the failure modes of FEM (Figure 12) can predict the position where concrete cracking is in the test specimens at the estimated boundary conditions. Thus, it can be concluded that the numerical method could effectively simulate the nonlinear behavior of the RDBs, especially for the ultimate lateral resistance of RDBs, even though the largest error is about 18%. However, the loading, deformation and degradation changing processes are basically predicted by the numerical method, and it is assumed that the method can be used for the following parametric study to analyze the effects of different parameters on the mechanical performance of RDBs.

**Figure 11.** Comparison of the load-displacement relationship between the test and FEM results.

#### *4.2. Parametric Analysis Study*

A parametric analysis was performed to summarize the effects on geometrical and material parameters based on the verified numerical modeling method in Section 4.1. The following parameters were considered: height-to-length ratio (*α*), height-to-thickness (*β*) and material type of concrete. The height of RDBs is determined as 900 mm, which is the same as the test specimens. The thickness of RDB includes 60 mm, 80 mm and 100 mm. The height-to-length ratio of RDB is ranged from 0.75 to 2.0. The compressive strength of concrete includes three types: 25 N/mm<sup>2</sup> (C25), 30 N/mm<sup>2</sup> (C30) and 35 N/mm2 (C35). Table 4 presents the detailed information of these RDB models. The effects on lateral loading–displacement curves with parameters of height-to-length ratio, height-to-thickness and material type of concrete are shown in Figure 13.

**Figure 12.** Failure pattern comparison of the tests and FEM predictions.

**Table 4.** Parameters analyzed for RDBs.

Note: *h*, *l* and *t* are the height, length and thickness of the RDB, respectively.

As the results presented in Figure 13a show, it can be concluded that the height-tolength ratio of the RDB is showing a high contribution to the shear resistance of the RDBs. A higher height-to-length ratio of the RDB results in a higher shear resistance throughout all of the loading levels, as demonstrated by the test results. According to Figure 13b, it can be seen that the shear resistance of RDB is increasing with the increase in compressive strength of the concrete material. However, the increment is decreasing with the increase in the compressive strength. The shear resistance of RDB is obviously increasing with the decrease in the height-to-thickness ratio of RDB (see in Figure 13c), and such increment is increasing with the decrease in the height-to-thickness ratio of RDB.

Therefore, it can be concluded that the shear resistance of RDB can be efficiently strengthened by a large margin by decreasing the height-to-length ratio and increasing the height-to-thickness ratio of RDBs. However, the material type of concrete shows little effect on the seismic performance of RDBs.

**Figure 13.** Effects of parameters on seismic performance of RDBs.

#### **5. Conclusions and Discussions**

In order to understand the mechanical behavior and seismic performance of RDBs, two scaled specimens were designed and tested under cyclic loads, and the corresponding nonlinear FE models were established and verified. The conclusions are listed as follows:

ratio shows a general influence on the seismic performance of RDBs, but the effect of concrete strength on the seismic performance of RDBs is relatively small.

The bolt-connected RC deep beam (RDB) proposed in this paper is quite different from other types of RC shear walls. The bending–shear mechanical characteristic of RDBs is a benefit for the plastic development of RC materials. The length of RDBs could be an effective parameter to adjust the lateral stiffness and resistance of frame structures, and the RDBs could be flexibly arranged to achieve the architecture requirement. According to the failure patterns of RDB specimens, the bolt connection was effective to ensure the performance utilization of RDBs. Thus, the bolt connection could be treated as a reliable connection for RC deep beam and boundary frames. The authors have conducted experimental and numerical analyses on steel frames with steel panel walls, RC panel walls and composite steel panel walls [17–20]. The aim of this paper is to investigate the seismic performance of RDBs solely and to understand the collaborative working relationship between the steel frames and RDBs. We found that the seismic performance of the steel frame with RDB specimens is better than the sum of the steel frame and the RDB based on the test results. Though the numerical model developed in this paper is simplified compared to the macroscopic model [20], the simple model proposed in this paper has relatively good accuracy in predicting the lateral resistance of RDB.

**Author Contributions:** Conceptualization, H.Z. and L.J. (Liqiang Jiang); methodology, J.L. and H.Z.; software, J.L. and H.Z.; validation, J.L., H.Z. and L.Z.; investigation, J.L., H.Z. and L.Z.; writing—original draft preparation, J.L. and L.J. (Liqiang Jiang); writing—review and editing, L.J. (Lizhong Jiang) and L.Z. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the National Natural Science Foundation of China, grant number 50678025.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** The results presented in this study are available on request from the corresponding author. The data are not publicly available due to the reason that the authors are conducting further analysis on the same structural model.

**Acknowledgments:** We would like to thank the editors and reviewers for the comments on the manuscript.

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

#### **Nomenclature**

#### **References**

## *Article* **3D FEA of Infilled RC Framed Structures Protected by Seismic Joints and FRP Jackets**

**Theodoros Rousakis \*, Vachan Vanian, Theodora Fanaradelli and Evgenia Anagnostou**

Department of Civil Engineering, Democritus University of Thrace, 67100 Xanthi, Greece<EMAIL_ADDRESS>(V.V<EMAIL_ADDRESS>(T.F<EMAIL_ADDRESS>(E.A.) **\*** Correspondence<EMAIL_ADDRESS>
**Abstract:** This study focused on characteristic cases of recently tested real-scale RC framed wall infilled structures with innovative seismic protection through polyurethane joints (PUFJ) or polyurethaneimpregnated fiber grids (FRPU). The frames revealed a highly ductile response while preventing infill collapse. Herein, suitable 3D pseudo-dynamic FE models were developed in order to reproduce the experimental results. The advanced Explicit Dynamics framework may help reveal the unique features of the considered interventions. Externally applied double-sided FRPU jackets on OrthoBlock infills may maintain an adequate bond with the surrounding RC frame as well as with the brick infill substrate at up to a 3.6% drift. In a weak four-column RC structure, the OrthoBlock infills with PUFJ seismic joints may increase the initial stiffness remarkably, increase the base shear by three times (compared with the bare structure) and maintain a high horizontal drift of 3.7%. After this phase, the structure may receive FRPU retrofitting, reveal the redistribution of stress over broad infill regions, including predamaged parts, and still develop a higher initial stiffness and base shear (compared with the bare RC). The realization of a desirable ductile behavior of infilled frames through PUFJ of only 20 mm thickness, as well as through FRPU jacketing, may remarkably broaden the alternatives in seismic protection against the collapse of structures.

**Keywords:** RC frames; brick infills; prior damage; finite element analyses; seismic joint; FRP

#### **1. Introduction**

The behavior of infilled reinforced concrete (RC) frames under in-plane and out-ofplane loading is an open issue as there is a variability of existing RC frames designed and constructed according to different design codes over the years, with or without modern seismic-resistant provisions or suitable reinforcement detailing. Further, the infills may include solid or hollow clay bricks (usually with a horizontal direction of the holes of the bricks), concrete solid or hollow blocks and several other types of infill wall units. Therefore, there is a high variation in the modulus of elasticity, strength and modes of failure of the infills when interacting with the RC frames. In some cases, the effects of the infills on the RC frames may be detrimental and lead to building collapses during severe earthquakes. On the other hand, collapsed infills alone may lead to human injuries or loss. Numerous experimental campaigns have addressed the variable behavior of infilled RC frames. The high scatter in the prediction of several critical parameters—especially the ones related to displacement ductility—reveals the complexity and uncertainty involved (see the studies by [1–7], among others). Furthermore, the out-of-plane response of the infills is equally crucial to predict (see, e.g., [4,8–10], among others). The abovementioned studies suggest that, in most cases, common clay brick infills reveal an accumulation of severe damage for the lateral drift of the infilled RC frame that ranges between 0.5 and 1.5%. Hence, they lead to reduced displacement ductility levels with respect to the case of a bare RC frame, despite the fact they increase the elastic stiffness and maximum base shear capacity of the infilled frame.

**Citation:** Rousakis, T.; Vanian, V.; Fanaradelli, T.; Anagnostou, E. 3D FEA of Infilled RC Framed Structures Protected by Seismic Joints and FRP Jackets. *Appl. Sci.* **2021**, *11*, 6403. https://doi.org/10.3390/ app11146403

Academic Editor: Pier Paolo Rossi

Received: 20 June 2021 Accepted: 8 July 2021 Published: 11 July 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).