Simpliﬁed Estimation Method for Maximum Deﬂection in Bending-Failure-Type Reinforced Concrete Beams Subjected to Collision Action and Its Application Range

: As natural disasters have become increasingly severe, many structures designed to prevent rockfalls and landslides have been constructed in various areas. The impact resistance capacity of a reinforced concrete (RC) rock shed can be evaluated using its roof deﬂection. This study establishes a method for estimating the maximum deﬂection of a bending-failure-type RC beam, subjected to collisions that is based on the energy conservation concept—in which, the transmitted energy from a collision is equivalent to the energy absorbed by the beam. However, the following assumptions have never been conﬁrmed: (1) The energy transmitted to the RC beam, due to the dropped weight, can be estimated by assuming a perfect plastic collision; and (2) the energy absorbed by the RC beam can be estimated by assuming plane conservation. In this study, these assumptions were veriﬁed using 134 previous test results of RC beams subject to weight collisions. In addition, we proposed a simple method for calculating the maximum deﬂection and its application scope. With this method, a performance-based impact-resistant design procedure for various RC structures can be established in the future. Moreover, this method will signiﬁcantly improve the maintenance and management of existing RC structures subject to collisions.


Introduction
In recent years, natural disasters have become more severe because of climate change caused by global warming. Torrential rains have occurred in various parts of the world, causing large-scale slope disasters [1][2][3]. Many rock sheds, retaining walls, and barriers are used in coastal and mountainous areas as road disaster prevention countermeasures. Figure 1 shows examples of rockfall and landslide disasters in rock sheds in coastal and mountainous regions. Instead of being designed with specification-based approaches, such as the allowable stress method, these protective structures should be designed using performance-based design methods [4,5]. In particular, the impact resistance capacity of a reinforced concrete (RC) rock shed can be evaluated using the deflection of its roof [6][7][8], which can be used to set each limit state of the shed. However, even for basic structural members, such as an RC beam, an appropriate method for estimating the maximum deflection has not yet been established. Hence, many research institutes were attempting to establish a method for estimating the maximum deflection of RC beams subjected to collision action [9][10][11][12][13][14]. Figure 2 illustrates the general impact loading method and an example of the test results. In previous studies, Kishi et al. proposed a method for estimating the maximum deflection of RC beams based on the results of their weight-falling impact tests under bending failure [9]. This estimation method was suggested based on the linear relationship between the kinetic energy of the weight and the maximum deflection of the RC beam. This method was also summarized in Structural Engineering Series 22, published by the Japan Society of Civil Engineers (JSCE) [10]. Tachibana et al. conducted impact loading tests of RC beams and proposed an estimation equation for the maximum deflection [11]. Fujikake et al. calculated the load-deflection relationship of RC beams, considering the strain rate effect of concrete and reinforcing bars and subsequently attempted to estimate the maximum deflection based on the conservation laws of momentum and energy. Their study clarified that the experimental results presented by other researchers can be evaluated conservatively [12]. Kishi et al. proposed a residual deflection estimation method for large RC beams with a clear span of 8.0 m [13]. In this estimation method, a correction formula based on the mass ratio of the weight and RC beam was empirically introduced.
Recently, Hwang et al. proposed a maximum deflection estimation method based on the conservation law of energy, considering (i) input energy (due to falling weight), (ii) energy loss at the time of collision, (iii) change in potential energy (due to the deflection of the RC beam), (iv) energy In previous studies, Kishi et al. proposed a method for estimating the maximum deflection of RC beams based on the results of their weight-falling impact tests under bending failure [9]. This estimation method was suggested based on the linear relationship between the kinetic energy of the weight and the maximum deflection of the RC beam. This method was also summarized in Structural Engineering Series 22, published by the Japan Society of Civil Engineers (JSCE) [10]. Tachibana et al. conducted impact loading tests of RC beams and proposed an estimation equation for the maximum deflection [11]. Fujikake et al. calculated the load-deflection relationship of RC beams, considering the strain rate effect of concrete and reinforcing bars and subsequently attempted to estimate the maximum deflection based on the conservation laws of momentum and energy. Their study clarified that the experimental results presented by other researchers can be evaluated conservatively [12]. Kishi et al. proposed a residual deflection estimation method for large RC beams with a clear span of 8.0 m [13]. In this estimation method, a correction formula based on the mass ratio of the weight and RC beam was empirically introduced.
Recently, Hwang et al. proposed a maximum deflection estimation method based on the conservation law of energy, considering (i) input energy (due to falling weight), (ii) energy loss at the time of collision, (iii) change in potential energy (due to the deflection of the RC beam), (iv) energy In previous studies, Kishi et al. proposed a method for estimating the maximum deflection of RC beams based on the results of their weight-falling impact tests under bending failure [9]. This estimation method was suggested based on the linear relationship between the kinetic energy of the weight and the maximum deflection of the RC beam. This method was also summarized in Structural Engineering Series 22, published by the Japan Society of Civil Engineers (JSCE) [10]. Tachibana et al. conducted impact loading tests of RC beams and proposed an estimation equation for the maximum deflection [11]. Fujikake et al. calculated the load-deflection relationship of RC beams, considering the strain rate effect of concrete and reinforcing bars and subsequently attempted to estimate the maximum deflection based on the conservation laws of momentum and energy. Their study clarified that the experimental results presented by other researchers can be evaluated conservatively [12]. Kishi et al. proposed a residual deflection estimation method for large RC beams with a clear span of 8.0 m [13]. In this estimation method, a correction formula based on the mass ratio of the weight and RC beam was empirically introduced.
Recently, Hwang et al. proposed a maximum deflection estimation method based on the conservation law of energy, considering (i) input energy (due to falling weight), (ii) energy loss at the time of collision, (iii) change in potential energy (due to the deflection of the RC beam), (iv) energy absorption (due to the bending deformation of the beam), and (v) energy loss (due to the peeling of concrete at the upper edge of the beam) [14]. Their results showed that the maximum deflection of RC beams can be estimated based on previous experimental studies. Furthermore, a numerical analysis using the finite element method was conducted [15][16][17].
However, few studies have focused on the simple estimation method for maximum deflection, and its application range for cases where a bending-failure-type RC beam is plastically deformed to absorb energy. The simple calculation method enables the appropriate maintenance based on the reliability design of existing disaster prevention structures, taking into consideration the uncertainty of action and variations in material strength [18,19].
Currently, the energy transmitted to the RC beam during a weight collision can often be calculated under the assumption of a perfect plastic collision, in which the weight and RC beam move together without repulsion after the collision [12,14]. However, no investigation exists where the validity of this assumption is verified by the impact test results of RC beams under different conditions. In addition, the energy absorbed by the RC beam is calculated to be the area surrounded by the curve by obtaining the RC beam load-deflection relationship under the assumption of plane conservation [9][10][11][12][13][14]. However, the applicable range of this assumption has not been clarified.
In this study, these assumptions were verified based on the experimental results of 134 cases conducted in previous studies, and a simple calculation method for the maximum deflection and its application range were proposed. Based on the proposed method for estimating the maximum deflection of a bending-failure-type RC beam, an impact-resistant-design procedure for various RC structures can be established in the future. Moreover, the proposed method will significantly improve the maintenance and management of existing RC structures subject to a collision action.

Energy Conservation Concept
As previously discussed, the maximum deflection is calculated as the deflection when the transmitted impact energy E t and the absorbed energy E a of the RC beam are equivalent, based on the law of conservation of energy.
In addition to the above energy, it is possible that the energy generated by the movement of the RC beams and the energy generated by the scattering of concrete pieces may have an effect. However, these effects are extremely small compared to E t and E a ; therefore, they were excluded from this study. The calculation methods for E t and E a are detailed below.

Calculation Method of Transmitted Impact Energy E t
The energy E t transmitted to the RC beam during a weight collision was calculated under the assumption of a perfect plastic collision where the weight and the RC beam moved together without repulsion after the collision. The formula for calculating E t was derived, as follows.
First, the momentum conservation laws, immediately before and after the weight collides with the beam, are expressed as follows: where M be is the equivalent mass of the beam obtained by assuming that the vibration mode of the beam is equivalent to the first-order bending mode and multiplying the mass of the beam within its Appl. Sci. 2020, 10, 6941 4 of 26 clear span M b by 17/35. Furthermore, M w is the mass of the impact weight, V is the velocity of the weight immediately before collision, and V a is the velocity of the mass point, including the falling weight and the beam, immediately after the collision. Subsequently, we considered the kinetic energy before collision E k can be estimated using Equation (4), and the energy after collision E ka can be derived using Equations (5)-(7) as follows: where E ka is the kinetic energy of the combined weight and beam immediately after the collision and corresponds to the energy transmitted to the beam E t . The energy E t can be determined using Equation (8).
where ρ is the unit mass of the RC beam (=2.5), A is the sectional area of the beam, and L is the clear span of the beam.

Calculation Method for the Absorbed Energy E a of the RC Beam
E a is calculated as the area under the load-deflection curve obtained by the fiber model, considering the plane conservation of the beam section.
The calculation procedure is outlined as follows: (i) Divide the height of the section into 5 mm intervals and along the span direction into 100 mm intervals, considering the solution stability; (ii) increase the upper-edge strain by 10 µ to determine the cross-sectional neutral axis at each stage, and determine the curvature-bending moment relationship; (iii) determine the bending moment distribution along the span direction at each load step and the corresponding curvature distribution; and (iv) calculate the deflection of the span center using the elastic load method. Figure 3 shows the concept of the fiber model. The constitutive material laws for concrete and reinforcing bars were determined, as shown in Figure 4, following the JSCE Concrete Standards Design [20].  The compressive stress and strain relationships of concrete were defined as follows: = exp 0.73 1 exp 1.25 Material test results were used to establish the compressive strength of concrete and yield strength of the reinforcing steel. The tensile strength ft of concrete was estimated using the following equation, according to JSCE [20]: As the tensile fracture energy of concrete is substantially smaller than the absorbed energy owing to the bending plastic deformation of the RC beams, the tension softening of concrete was not included in the scope of this study. Moreover, perfect bonding was assumed between the rebar and concrete.
In this calculation, the strain rate cannot be considered because it changes significantly in time and space after a weight collision. It is difficult to apply it to a simple estimation formula. In addition, the applicable range of absorbed energy calculated under the assumption of plane conservation has not been clarified yet. This should be confirmed through a comparison with the numerous existing experimental results.

Experimental Results Used for This Investigation
The validity of the assumption of a perfect plastic collision was evaluated using experimental results from previous studies [9,10]. Table 1 lists the specifications of the tested RC beams and experimental conditions. The clear span of the beam, L, is the distance between the two supports, and pt is the tensile rebar ratio. The numerical values for the calculated bending capacity Pu and shear capacity Qu were obtained from the literature [9]. Table 1. List of test specimens used for analysis [9]. The compressive stress and strain relationships of concrete were defined as follows:

Size of Section
Material test results were used to establish the compressive strength of concrete and yield strength of the reinforcing steel. The tensile strength f t of concrete was estimated using the following equation, according to JSCE [20]: As the tensile fracture energy of concrete is substantially smaller than the absorbed energy owing to the bending plastic deformation of the RC beams, the tension softening of concrete was not included in the scope of this study. Moreover, perfect bonding was assumed between the rebar and concrete.
In this calculation, the strain rate cannot be considered because it changes significantly in time and space after a weight collision. It is difficult to apply it to a simple estimation formula. In addition, the applicable range of absorbed energy calculated under the assumption of plane conservation has not been clarified yet. This should be confirmed through a comparison with the numerous existing experimental results.

Experimental Results Used for This Investigation
The validity of the assumption of a perfect plastic collision was evaluated using experimental results from previous studies [9,10]. Table 1 lists the specifications of the tested RC beams and experimental conditions. The clear span of the beam, L, is the distance between the two supports, and p t is the tensile rebar ratio. The numerical values for the calculated bending capacity P u and shear capacity Q u were obtained from the literature [9]. Table 1. List of test specimens used for analysis [9].

Name
Size of Section   In all 36 experimental cases employed in this study, the steel weight was dropped only once from a predetermined height to the RC beam center. The RC beam was placed on a fulcrum with a lifting prevention jig. The boundary condition of the fulcrum was similar to that of the pinned support. The beams were all rectangular RC beams.

Size of Section
The cross-sectional width, height, and span length of the test specimens varied from 150 to 250 mm, 200 to 400 mm, and 2 to 3 m, respectively. Further, the tensile rebar ratio and mass of the falling steel weight varied from 0.8% to 3.17% and from 300 to 500 kg, respectively. Furthermore, the velocity of the falling weight before collision varied from 4 to 7.67 m/s.

Comparison Between the Estimated and Measured Maximum Deflections
The maximum deflection δ u , assuming a perfect plastic collision, was calculated as the satisfying deflection Equation (16) based on the energy conservation concept. For comparison, the maximum deflection δ uk , assuming no energy loss, was also calculated using Equation (17). Figure 5 shows a conceptual diagram of δ u and δ uk obtained with E t and E k , respectively, using the calculated P-δ curve. In all 36 experimental cases employed in this study, the steel weight was dropped only once from a predetermined height to the RC beam center. The RC beam was placed on a fulcrum with a lifting prevention jig. The boundary condition of the fulcrum was similar to that of the pinned support. The beams were all rectangular RC beams.
The cross-sectional width, height, and span length of the test specimens varied from 150 to 250 mm, 200 to 400 mm, and 2 to 3 m, respectively. Further, the tensile rebar ratio and mass of the falling steel weight varied from 0.8% to 3.17% and from 300 to 500 kg, respectively. Furthermore, the velocity of the falling weight before collision varied from 4 to 7.67 m/s.

Comparison Between the Estimated and Measured Maximum Deflections
The maximum deflection δu, assuming a perfect plastic collision, was calculated as the satisfying deflection Equation (16) based on the energy conservation concept. For comparison, the maximum deflection δuk, assuming no energy loss, was also calculated using Equation (17). Figure 5 shows a conceptual diagram of δu and δuk obtained with Et and Ek, respectively, using the calculated P-δ curve.  From Figure 5, it can be seen that δ u is smaller than δ uk . This means that in the case of considering a perfect plastic collision, energy is lost when a weight collides. Figure 6 compares the estimated maximum deflections δ uk and δ u based on E k and E t , respectively, with the experimental results of δ u.exp . The figure shows that the estimated maximum deflection δ uk based on E k significantly exceeds the experimental value and that this difference increases when the deflection is larger. In contrast, the estimated maximum deflection δ u based on the transmitted energy E t is generally larger than the experimental value; however, it is closer to the experimental value than in the case of δ uk . This indicates that the maximum deflection δ u.exp can be determined accurately and conservatively by using the transmitted energy E t . respectively, with the experimental results of δu.exp. The figure shows that the estimated maximum deflection δuk based on Ek significantly exceeds the experimental value and that this difference increases when the deflection is larger. In contrast, the estimated maximum deflection δu based on the transmitted energy Et is generally larger than the experimental value; however, it is closer to the experimental value than in the case of δuk. This indicates that the maximum deflection δu.exp can be determined accurately and conservatively by using the transmitted energy Et. Consequently, δu.exp can be predicted considering the energy loss, due to the complete plastic collision between the weight and RC beam. This is because the target RC beam is a bending-failure type and is plastically deformed; therefore, the collision with the weight behaves like a nearly perfect plastic collision. On the other hand, in some cases, δu greatly exceeds δu.exp. This is especially evident when the amount of deformation is large. It is assumed that this occurs because the RC beam has reached a region, in which the assumption of plane conservation does not hold. In the next section, based on this investigation, the scope of application of the plane-holding assumption is examined.

Examination Outline
To examine the applicable range of the plane conservation assumption of the RC beam, the experimental results of 134 cases of the bending-failure-type RC beam plastically deformed by a weight collision were collected. The specifications of the RC beams investigated in this study are shown in Appendix A (see Tables A1 and A2). The numerical values of the calculated bending capacity Pu and shear capacity Qu were obtained from the literature, shown in those tables. Table 2 shows the range of the considered specifications of the RC beams for the 134 cases. In this table, α is the shear-bending capacity ratio obtained by dividing the calculated shear capacity Qu by the calculated bending capacity Pu. RC beams with α ≥ 1.0 were selected here. In addition, when the velocity of the impactor is higher than approximately 80 m/s, the RC member is often damaged by penetration or perforation, including shear failure and backside spalling, prior to exhibiting bending deformation [14]. Moreover, it was reported that the maximum impact velocity of falling Consequently, δ u.exp can be predicted considering the energy loss, due to the complete plastic collision between the weight and RC beam. This is because the target RC beam is a bending-failure type and is plastically deformed; therefore, the collision with the weight behaves like a nearly perfect plastic collision. On the other hand, in some cases, δ u greatly exceeds δ u.exp . This is especially evident when the amount of deformation is large. It is assumed that this occurs because the RC beam has reached a region, in which the assumption of plane conservation does not hold. In the next section, based on this investigation, the scope of application of the plane-holding assumption is examined.

Examination Outline
To examine the applicable range of the plane conservation assumption of the RC beam, the experimental results of 134 cases of the bending-failure-type RC beam plastically deformed by a weight collision were collected. The specifications of the RC beams investigated in this study are shown in Appendix A (see Tables A1 and A2). The numerical values of the calculated bending capacity P u and shear capacity Q u were obtained from the literature, shown in those tables. Table 2 shows the range of the considered specifications of the RC beams for the 134 cases. In this table, α is the shear-bending capacity ratio obtained by dividing the calculated shear capacity Q u by the calculated bending capacity P u . RC beams with α ≥ 1.0 were selected here. In addition, when the velocity of the impactor is higher than approximately 80 m/s, the RC member is often damaged by penetration or perforation, including shear failure and backside spalling, prior to exhibiting bending deformation [14]. Moreover, it was reported that the maximum impact velocity of falling rocks is approximately 25 m/s [3]. Thus, those experiments with impact velocities of less than 25 m/s were considered. The diameter of the impactor is almost the same as the width of the RC beam, and the shape of the bottom surface of the weight was spherical with a small curvature. Each study confirmed that bending deformation was predominantly observed for all RC beams. Figure 7 shows the dimensions for the typical RC beams, considered here. rocks is approximately 25 m/s [3]. Thus, those experiments with impact velocities of less than 25 m/s were considered. The diameter of the impactor is almost the same as the width of the RC beam, and the shape of the bottom surface of the weight was spherical with a small curvature. Each study confirmed that bending deformation was predominantly observed for all RC beams. Figure 7 shows the dimensions for the typical RC beams, considered here.     Figure 8 shows the relationship between the estimated maximum deflection δ u and the experimental value δ u.exp . From the figure, the estimated value δ u up to approximately 40 mm corresponds well with the experimental value δ u.exp . On the other hand, when δ u becomes large, δ u may overestimate the experimental ones. It is assumed to be because the RC beams have greatly deformed and damaged, so that the assumption of plane conservation does not hold as in the case of Figure 6.
However, as the minimum and maximum clear spans of the considered RC beam are 0.9 m and 8 m, respectively, the maximum deflection is expected to be significantly different even if the degree of damage is the same. Therefore, the accuracy of the estimated values and the applicable range must be examined without the results being affected by the shape and size of the beam. Appl. Sci. 2020, 10, x FOR PEER REVIEW 9 of 18 However, as the minimum and maximum clear spans of the considered RC beam are 0.9 m and 8 m, respectively, the maximum deflection is expected to be significantly different even if the degree of damage is the same. Therefore, the accuracy of the estimated values and the applicable range must be examined without the results being affected by the shape and size of the beam.

Investigation of the estimation accuracy based on the deflection ratio RD
In this section, the accuracy of the estimation method is assessed using the deflection ratio without considering the influence of the shape and dimensions of the beams. The value obtained by dividing the maximum deflection δu by the clear span length L was defined as the deflection ratio RD. RD = δu/L (18) Figure 9 shows the relationship between the experimental and estimated deflection ratios. The figure shows that the estimated value of RD overestimates the experimental value RD.exp when 2% < RD < 9%. In contrast, when RD is larger than 9%, the estimated RD value corresponds relatively well to the experimental value RD.exp. Therefore, the accuracy and the applicable range of the estimated values are difficult to examine based on the deflection ratio RD.

Investigation of the estimation accuracy based on the plasticity ratio Rp
The value obtained by dividing the maximum deflection δu by the tensile rebar yield deflection, δy, is defined as the plasticity ratio Rp. It is an index generally used to evaluate ductility in the seismic design of RC piers.

Investigation of the Estimation Accuracy Based on the Deflection Ratio R D
In this section, the accuracy of the estimation method is assessed using the deflection ratio without considering the influence of the shape and dimensions of the beams. The value obtained by dividing the maximum deflection δ u by the clear span length L was defined as the deflection ratio R D . Figure 9 shows the relationship between the experimental and estimated deflection ratios. The figure shows that the estimated value of R D overestimates the experimental value R D.exp when 2% < R D < 9%. In contrast, when R D is larger than 9%, the estimated R D value corresponds relatively well to the experimental value R D.exp . Therefore, the accuracy and the applicable range of the estimated values are difficult to examine based on the deflection ratio R D .
Appl. Sci. 2020, 10, x FOR PEER REVIEW 9 of 18 However, as the minimum and maximum clear spans of the considered RC beam are 0.9 m and 8 m, respectively, the maximum deflection is expected to be significantly different even if the degree of damage is the same. Therefore, the accuracy of the estimated values and the applicable range must be examined without the results being affected by the shape and size of the beam.

Investigation of the estimation accuracy based on the deflection ratio RD
In this section, the accuracy of the estimation method is assessed using the deflection ratio without considering the influence of the shape and dimensions of the beams. The value obtained by dividing the maximum deflection δu by the clear span length L was defined as the deflection ratio RD. RD = δu/L (18) Figure 9 shows the relationship between the experimental and estimated deflection ratios. The figure shows that the estimated value of RD overestimates the experimental value RD.exp when 2% < RD < 9%. In contrast, when RD is larger than 9%, the estimated RD value corresponds relatively well to the experimental value RD.exp. Therefore, the accuracy and the applicable range of the estimated values are difficult to examine based on the deflection ratio RD.

Investigation of the estimation accuracy based on the plasticity ratio Rp
The value obtained by dividing the maximum deflection δu by the tensile rebar yield deflection, δy, is defined as the plasticity ratio Rp. It is an index generally used to evaluate ductility in the seismic design of RC piers.

Investigation of the Estimation Accuracy Based on the Plasticity Ratio R p
The value obtained by dividing the maximum deflection δ u by the tensile rebar yield deflection, δ y , is defined as the plasticity ratio R p . It is an index generally used to evaluate ductility in the seismic design of RC piers. Figure 10 shows the relationship between the estimated and experimental values of the plasticity ratio. The figure shows that when the plasticity ratio R p is large, the estimated ratio is higher than the experimental result R p.exp . It is observed that the maximum deflection δ u can be estimated with relatively high accuracy when the plasticity ratio R p ≤ 10. In addition, if R p > 10, the estimation accuracy is low. A high plasticity ratio indicates that the RC beam has a high degree of damage.
Appl. Sci. 2020, 10, x FOR PEER REVIEW 10 of 18 Figure 10 shows the relationship between the estimated and experimental values of the plasticity ratio. The figure shows that when the plasticity ratio Rp is large, the estimated ratio is higher than the experimental result Rp.exp. It is observed that the maximum deflection δu can be estimated with relatively high accuracy when the plasticity ratio Rp ≤ 10. In addition, if Rp > 10, the estimation accuracy is low. A high plasticity ratio indicates that the RC beam has a high degree of damage. The estimated value of Rp is calculated by assuming a bending deformation based on the plane conservation of the section of the RC beam in the calculation. Therefore, it is assumed that all transmitted energy is consumed by the bending deformation of the beam. In contrast, in the experiment, energy is usually consumed by not only the bending deformation, but also damage to the concrete at the collision point, pull-out of the reinforcing steel, and opening of shear cracks. Therefore, the estimated Rp tended to be higher than Rp.exp when the plasticity ratio Rp was high.

Investigation of the Accuracy and Applicable Range of the Deflection Estimation Formula
The accuracy of the maximum deflection, δu, can be calculated using Equation (20), and its application range is discussed below. The accuracy evaluation indicator (AEI) is defined as follows: AEI = δu/δu.exp.
(20) Figure 11 shows the relationship between the AEI and plasticity ratio for the design Rp (= δu/δy) in the same manner as that in Reference [14]. From the figure, many plots can be obtained when the plasticity ratio Rp ≤ 10, and the AEI is densely distributed between 0.8 and 1.6. Conversely, when the plasticity ratio Rp > 10, the AEI is widely distributed between 1.0 and 2.0. The estimated value of R p is calculated by assuming a bending deformation based on the plane conservation of the section of the RC beam in the calculation. Therefore, it is assumed that all transmitted energy is consumed by the bending deformation of the beam. In contrast, in the experiment, energy is usually consumed by not only the bending deformation, but also damage to the concrete at the collision point, pull-out of the reinforcing steel, and opening of shear cracks. Therefore, the estimated R p tended to be higher than R p.exp when the plasticity ratio R p was high.

Investigation of the Accuracy and Applicable Range of the Deflection Estimation Formula
The accuracy of the maximum deflection, δ u , can be calculated using Equation (20), and its application range is discussed below. The accuracy evaluation indicator (AEI) is defined as follows: (20) Figure 11 shows the relationship between the AEI and plasticity ratio for the design R p (= δ u /δ y ) in the same manner as that in Reference [14]. From the figure, many plots can be obtained when the plasticity ratio R p ≤ 10, and the AEI is densely distributed between 0.8 and 1.6. Conversely, when the plasticity ratio R p > 10, the AEI is widely distributed between 1.0 and 2.0. Therefore, the application range of the proposed formula can be determined as 1 < R p ≤ 10. When R p ≤ 1, the deflection of the RC beam is within the elastic range, and it does not exhibit a complete plastic collision. Therefore, R p < 1 was excluded from the above application range.
To examine the estimation accuracy, Figure 12 shows the AEI when 1 < R p ≤ 10. As shown in the figure, the AEI was distributed between 0.81 and 1.56, with an average value of 1.15, and a coefficient of variation of 0.113. Appl. Sci. 2020, 10, x FOR PEER REVIEW 11 of 18 Figure 11. Accuracy of the estimated maximum deflection for the design δud (Rp < 40).
Therefore, the application range of the proposed formula can be determined as 1 < Rp ≤ 10. When Rp ≤ 1, the deflection of the RC beam is within the elastic range, and it does not exhibit a complete plastic collision. Therefore, Rp < 1 was excluded from the above application range.
To examine the estimation accuracy, Figure 12 shows the AEI when 1 < Rp ≤ 10. As shown in the figure, the AEI was distributed between 0.81 and 1.56, with an average value of 1.15, and a coefficient of variation of 0.113.

Simplified Estimation Method for the Maximum Deflection of the RC Beam Subjected to an Impact Load
Thus far, we have proposed a method for estimating the maximum deflection based on the assumption that the energy Et transmitted to the beam by a weight collision is equivalent to the absorption energy Ea, due to the bending deformation of the RC beam. Here, if the load-deflection curve of the RC beam can be simplified to a bilinear model, as shown in Figure 13, Ea can be easily calculated, as expressed in Equation (21). Ea = Py δud − Py δy/2 (21) Figure 11. Accuracy of the estimated maximum deflection for the design δ ud (R p < 40). Figure 11. Accuracy of the estimated maximum deflection for the design δud (Rp < 40).
Therefore, the application range of the proposed formula can be determined as 1 < Rp ≤ 10. When Rp ≤ 1, the deflection of the RC beam is within the elastic range, and it does not exhibit a complete plastic collision. Therefore, Rp < 1 was excluded from the above application range.
To examine the estimation accuracy, Figure 12 shows the AEI when 1 < Rp ≤ 10. As shown in the figure, the AEI was distributed between 0.81 and 1.56, with an average value of 1.15, and a coefficient of variation of 0.113.

Simplified Estimation Method for the Maximum Deflection of the RC Beam Subjected to an Impact Load
Thus far, we have proposed a method for estimating the maximum deflection based on the assumption that the energy Et transmitted to the beam by a weight collision is equivalent to the absorption energy Ea, due to the bending deformation of the RC beam. Here, if the load-deflection curve of the RC beam can be simplified to a bilinear model, as shown in Figure 13, Ea can be easily calculated, as expressed in Equation (21). Ea = Py δud − Py δy/2 (21) Figure 12. Accuracy of the estimated maximum deflection for the design δ ud (1 < R p ≤ 10).

Simplified Estimation Method for the Maximum Deflection of the RC Beam Subjected to an Impact Load
Thus far, we have proposed a method for estimating the maximum deflection based on the assumption that the energy E t transmitted to the beam by a weight collision is equivalent to the absorption energy E a , due to the bending deformation of the RC beam. Here, if the load-deflection curve of the RC beam can be simplified to a bilinear model, as shown in Figure 13, E a can be easily calculated, as expressed in Equation (21).
The deflection, when E a corresponds to E t , is the maximum deflection for the design of δ ud . Therefore, δ ud can be estimated as follows: To confirm the difference between δ ud and δ u , Figure 14 illustrates the relationship between these values. The figure shows that δ ud is nearly equivalent to δ u . Therefore, the maximum deflection δ ud can be calculated using the transmitted energy E t , yield load P y of the RC beam, and yield deflection δ y . Such an evaluation is possible when the relationship between the load and deflection is nearly bilinear, as in the case of a single bar RC beam. However, if the load-deflection relationship is composed of a curved line or multi-line, such as a prestressed concrete (PC) beam and other composite structural members, further investigations are required.
To confirm the difference between δud and δu, Figure 14 illustrates the relationship between these values. The figure shows that δud is nearly equivalent to δu. Therefore, the maximum deflection δud can be calculated using the transmitted energy Et, yield load Py of the RC beam, and yield deflection δy. Such an evaluation is possible when the relationship between the load and deflection is nearly bilinear, as in the case of a single bar RC beam. However, if the load-deflection relationship is composed of a curved line or multi-line, such as a prestressed concrete (PC) beam and other composite structural members, further investigations are required.  The deflection, when Ea corresponds to Et, is the maximum deflection for the design of δud. Therefore, δud can be estimated as follows: To confirm the difference between δud and δu, Figure 14 illustrates the relationship between these values. The figure shows that δud is nearly equivalent to δu. Therefore, the maximum deflection δud can be calculated using the transmitted energy Et, yield load Py of the RC beam, and yield deflection δy. Such an evaluation is possible when the relationship between the load and deflection is nearly bilinear, as in the case of a single bar RC beam. However, if the load-deflection relationship is composed of a curved line or multi-line, such as a prestressed concrete (PC) beam and other composite structural members, further investigations are required.

Conclusions
A method to estimate the maximum deflection of a bending-failure-type RC beam subjected to a collision action was established in this study. This method is based on the energy conservation concept where the transmitted energy acting on the beam by the weight collision is equivalent to the absorbed energy of the beam. The method was proposed based on reliable experimental results, and the validity and applicable range of this method were subsequently examined using the previous test results of 134 cases. The findings of this study are as follows: (1) The maximum deflection can be estimated with relatively high accuracy by using the transmitted impact energy obtained by assuming a vibration mode equal to the primary bending mode of the beam and a perfect plastic collision; (2) However, if the deflection is large, the estimated value overestimates the experimental value. This is thought to be because the assumption of the plane conservation of the cross-section of the RC beam does not hold; (3) Regardless of the shape and dimensions of the RC beam, if the estimated value of the plasticity ratio exceeds approximately 10, the assumption of plane conservation tends to fail; (4) A simplified estimation method for the maximum deflection was proposed by modeling the load-deflection relationship of the RC beams in a bilinear form. Assuming that the range of the plasticity ratio R p is from 1 to 10, the estimated value is approximately 15% larger than the experimental value. The coefficient of variation was approximately 0.11.

Appendix A
The specifications of the RC beams investigated in this study are listed in Table A1. Maximum deflections δ u estimated by the proposed method are listed in Table A2. Table A1. Test specimens were used for examination.

Name
Size of Section  Table A1. Cont.

Size of Section
Rebar  Table A1. Cont.

Size of Section
Rebar