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

Abstract

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.

1. 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 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.

2. Outline of the Simple Estimation Method of Maximum Deflection for Bending-Failure-Type RC Beams Subject to a Weight Collision

2.1. 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.

E_t = E_a

(1)

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.

2.2. 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:

M_wV = (M_be + M_w) V_a

(2)

V_a = (M_w/(M_be + M_w)) V

(3)

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 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:

E_k = 1/2 M_w V² (before collision)

(4)

E_ka = 1/2 (M_be + M_w) V_a² (after collision)

(5)

= M_w² / (2 (M_be + M_w)) V²

(6)

= (M_w/(M_be + M_w)) E_k

(7)

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).

E_t = (M_w/(M_be + M_w)) E_k

(8)

M_be = (17/35) ρAL

(9)

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.

2.3. 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.

$$E_a={\int }_0^{\delta }P\left(\delta \right)d\delta$$

(10)

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:

$$\rho E_c=E_{0}K\left(\epsilon {\prime }_c-\epsilon {\prime }_p\right)$$

(11)

$$E_0=\frac{2·f{\prime }_c}{\epsilon {\prime }_{peak}}$$

(12)

$$K=\exp \left\{-0.73\frac{ {{\epsilon }^{\prime }}_{\max }}{{{\epsilon }^{\prime }}_{peak}}\left[1-\exp \left(-1.25 \frac{{{\epsilon }^{\prime }}_{\max }}{{{\epsilon }^{\prime }}_{peak}}\right)\right]\right\}$$

(13)

$$\epsilon {\prime }_p=\epsilon {\prime }_{max}-2.86·{{\epsilon }^{\prime }}_{peak}\left[1-exp\left(-0.35\frac{{{\epsilon }^{\prime }}_{max}}{{{\epsilon }^{\prime }}_{peak}}\right)\right]$$

(14)

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]:

$$f_t=0.23 f{\prime }_c{}^{\frac{2}{3}}$$

(15)

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.

3. Verification by Comparison with Previous Experimental Results

3.1. Validity of a Perfect Plastic Collision

3.1.1. 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. List of test specimens used for analysis [9].

Name	Size of Section	Rebar		L (m)	pt (%)	f’c (MPa)	fy (MPa)	Pu (kN)	Qu (kN)	α	Mb (t)	Mbe (t)	V (m/s)	Mw (t)	Ek (kJ)	Et (kJ)
		φ (mm)	N
G1-1	200 × 300	19	2	3.0	1.10	33.7	379	69.6	195.4	2.81	0.450	0.219	7.00	0.3	7.35	4.25
G1-1S													7.00		7.35	4.25
G2-1	150 × 250	13	2	2.0	0.80	32.2	373	38.1	139.7	3.67	0.188	0.091	4.00	0.3	2.40	1.84
G2-2													5.00		3.75	2.88
G2-3													6.00		5.40	4.14
G2L-1	150 × 250	13	2	2.0	0.80	32.2	373	38.1	139.7	3.67	0.188	0.091	4.00	0.4	3.20	2.61
G2L-2													5.00		5.00	4.07
G2L-3													6.00		7.20	5.86
G3-1	150 × 250	13	2	2.0	0.80	34.6	393	40.2	141.1	3.51	0.188	0.091	4.00	0.3	2.40	1.84
G3-2													5.00		3.75	2.88
G3-3													6.00		5.40	4.14
G4-1	150 × 250	13	2	2.0	0.80	32.3	373	38.1	139.8	3.52	0.188	0.091	4.00	0.3	2.40	1.84
G4-2													5.00		3.75	2.88
G5-1	200 × 300	19	2	3.0	1.10	39.2	379	70.4	200.4	2.85	0.450	0.219	6.00	0.4	7.20	4.66
G5-2													7.00		9.80	6.34
G6-1	250 × 250	19	2	2.0	1.09	34.7	392	87.4	191.4	2.19	0.313	0.152	5.00	0.3	3.75	2.49
G7-1	250 × 250	19	2	3.0	1.09	34.7	392	58.3	162.3	2.78	0.469	0.228	5.00	0.3	3.75	2.13
G7-2													6.00		5.40	3.07
G8-1	200 × 200	25	2	2.0	3.17	34.7	383	102.3	158.4	1.55	0.200	0.097	6.00	0.3	5.40	4.08
G9-1	200 × 200	25	2	3.0	3.17	34.7	383	68.2	136.3	2.00	0.200	0.097	5.00	0.3	3.75	2.83
G9-2													6.00		5.40	4.08
G10-1	200 × 250	19	2	3.0	1.36	23.5	404	56.6	289.3	5.11	0.375	0.182	4.00	0.3	2.40	1.49
G10-2													5.00		3.75	2.33
G10-3													6.00		5.40	3.36
G10-4													7.00		7.35	4.57
G11-1	200 × 300	22	2	2.7	1.55	23.6	401	94.4	164.8	1.66	0.405	0.197	3.13	0.5	2.45	1.76
G11-2													4.20		4.41	3.16
G11-3													5.05		6.38	4.58
G11-4													5.78		8.35	5.99
G11-5													6.42		10.30	7.39
G11-6													7.00		12.25	8.79
G12-1	200 × 300	19	3	2.7	2.72	23.6	407	103.9	168.1	1.52	0.405	0.197	7.67	0.5	14.71	10.55
G13-1	200 × 400	25	2	2.7	1.45	23.6	406	178.3	400.2	2.11	0.540	0.262	7.67	0.5	14.71	9.65
G14-1	200 × 350	25	2	2.7	1.69	23.6	406	149.8	312.2	1.96	0.473	0.230	7.67	0.5	14.71	10.08
G15-1	200 × 400	29	2	2.7	1.84	23.6	406	224	850.1	3.58	0.540	0.262	7.67	0.5	14.71	9.65
G16-1	200 × 370	25	2	2.7	1.58	23.6	406	161.2	371.7	2.16	0.500	0.243	7.67	0.5	14.71	9.90

φ: Nominal diameter of the tensile rebar, N: Number of rebars, L: Clear span length of the beam, p_t: Tensile rebar ratio, f’_c: Compressive strength of concrete, f_y: Yield strength of the tensile rebar, P_u: Calculated bending capacity, Q_u: Calculated shear capacity, α: Shear-bending capacity ratio (= Q_u/P_u), M_b: Mass of beam, M_be: Equivalent mass of beam, M_w: Mass of weight V: Impact velocity of weight, E_k: Kinetic energy of weight (input energy), E_t: Transmitted energy.

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].

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.

3.1.2. 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.

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.

$$E_{\mathrm{t}} = (M_{\mathrm{w}}/(M_{\mathrm{be}} + M_{\mathrm{w}})) E_{\mathrm{k}} = {\int }_0^{{\delta }_u}P(\delta )d\delta$$

(16)

$$E_{\mathrm{k}} = {\int }_0^{{\delta }_{uk}}P\left(\delta \right)d\delta$$

(17)

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.

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.

3.2. Scope of the Application of the RC Beam Plane Conservation Assumption

3.2.1. 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

Table A1. Test specimens were used for examination.

Name	Size of Section	Rebar		L (m)	pt (%)	f’c (MPa)	fy (MPa)	Pu (kN)	Qu (kN)	α	Mb (t)	Mbe (t)	Mw (t)	V (m/s)	Ek (kJ)	Et (kJ)
		ϕ (mm)	N
Tachibana et al. [11]
1-A2	150 × 250	13	2	2.0	0.8	31.0	383	33.3	91.1	2.74	0.188	0.091	0.15	3.50	0.92	0.57
2-A2													0.3	2.40	0.86	0.66
3-A2													0.45	2.00	0.90	0.75
4-A2													0.15	4.90	1.80	1.12
5-A2													0.3	3.50	1.84	1.41
6-A2													0.45	2.80	1.76	1.47
7-A2													0.15	6.00	2.70	1.68
8-A2													0.3	4.20	2.65	2.03
9-A2													0.45	3.50	2.76	2.29
10-A2													0.3	1.00	0.15	0.12
11-A2-1													0.3	2.00	0.60	0.46
11-A2-2
12-A2-1													0.3	3.00	1.35	1.04
12-A2-2
13-A2-1													0.3	4.00	2.40	1.84
13-A2-2
14-A2-1													0.3	5.00	3.75	2.88
14-A2-2
14-A2-3
15-A1-1				1.0				66.7	91.1	1.37	0.094	0.046	0.3	5.00	3.75	3.26
15-A1-2
15-A1-3
15-A1-4
16-A4-1				4.0				16.7	91.1	5.46	0.375	0.182	0.3	5.00	3.75	2.33
16-A4-2
16-A4-3
17-B-1	300 × 150	13	4	2.0	1.53	27.3	378	31.8	65.9	2.07	0.225	0.109	0.3	5.00	3.75	2.75
17-B-2
18-C-1	150 × 250	16	2	2.0	1.26		402	50.3	94.8	1.88	0.188	0.091	0.3	5.00	3.75	2.88
18-C-2
19-D-1	150 × 250	10	2	2.0	0.45		393	20.2	87.1	4.31	0.188	0.091	0.3	5.00	3.75	2.88
19-D-2
20-E-1	150 × 400	13	2	2.0	0.47		378	59.5	145.6	2.45	0.3	0.146	0.3	5.00	3.75	2.52
20-E-2
21-F-1	150 × 400	10	2	2.0	0.26		393	34.9	140.6	4.03	0.3	0.146	0.3	5.00	3.75	2.52
21-F-2
Fujikake et al. [12]
S1616-1	150 × 250	16	2	1.4	1.26	42	426	91.1	232	2.55	0.131	0.064	0.4	1.72	0.59	0.51
S1616-2														2.43	1.18	1.02
S1616-3														3.43	2.35	2.03
S1616-4														4.85	4.70	4.06
S1322-1		22	2	1.4	2.46		418	162	245.4	1.51	0.131	0.064	0.4	2.43	1.18	1.02
S1322-2														3.43	2.35	2.03
S1322-3														4.85	4.70	4.06
S1322-4														6.86	9.41	8.12
S2222-1														2.43	1.18	1.02
S2222-2														3.43	2.35	2.03
S2222-3														4.85	4.70	4.06
S2222-4														6.86	9.41	8.12
Kishi et al. [13]
PB-880	1000 × 1000	32	7	8.0	0.65	33.3	382	881	2882	3.27	20.00	9.71	2.0	9.90	98.07	16.74
PB-880														14.00	196.1	33.49
PB-880														17.15	294.2	50.23
PB-880														19.81	392.2	66.97
PC-620	1000 × 850	29	7	8.0	0.64	31.2	400	621	1794	2.89	17.00	8.26	2.0	9.90	98.07	19.12
PC-620														14.00	196.1	38.24
Pham et al. [16]
Beam 1	150 × 250	10	2	1.9	0.46	46	500	54.5	306.3	5.62	0.178	0.087	0.2035	6.26	3.99	2.80
Beam 2						52		54.2	476.1	8.78				6.26	3.99	2.80
Konno et al. [21]
W2H10	1000 × 1000	25	7	8.0	0.42	29.2	382	613	2002	3.27	20.00	9.714	2.0	14.00	196.1	33.49
W5H4													5.0	8.86	196.1	66.65
W10H2													10.0	6.26	196.1	99.49
W2H5													2.0	9.90	98.07	16.74
W10H1													10.0	4.43	98.07	49.74
Sakai et al. [22]
N-H0.1	450 × 150	13	4	2.0	1.02	25.2	383	39.9	87.8	2.20	0.338	0.164	0.3	1.24	0.23	0.15
N-H0.25														2.06	0.64	0.41
N-H0.5														2.98	1.33	0.86
N-H1.0														4.20	2.65	1.71
N-H1.5														5.13	3.95	2.55
Kurihashi et al. [23,24]
N-H300	60 × 100	6	1	0.9	0.75	31.6	380	4.1	7.18	1.75	0.014	0.007	0.02	2.30	0.05	0.04
N-H600														3.20	0.10	0.08
N-H900														3.90	0.15	0.11
Zhan et al. [25]
6-C27-3	120 × 120	6	6	1.2	0.45	27	235	13.9	47.4	3.41	0.043	0.021	0.0336	7.67	0.99	0.61
6-C27-4														8.85	1.32	0.81
8-C27-3		8		1.2	0.81			15.6	51.1	3.28	0.043	0.021	0.0336	7.67	0.99	0.61
8-C27-4														8.85	1.32	0.81
10-C27-2.5		10		1.2	1.26			27.2	54.4	2.00	0.043	0.021	0.0336	7.00	0.82	0.51
10-C27-3														7.67	0.99	0.61
10-C27-4														8.85	1.32	0.81
6-C40-5		6		1.2	0.45	40		15.8	49.8	3.15	0.043	0.021	0.0336	9.90	1.65	1.01
6-C40-6														10.84	1.97	1.22
6-C40-7														11.71	2.30	1.42
6-C40-8														12.52	2.63	1.62
8-C40-2		8		1.2	0.81			19.4	54	2.79	0.043	0.021	0.0336	6.26	0.66	0.41
8-C40-3														7.67	0.99	0.61
8-C40-4														8.85	1.32	0.81
8-C40-5														9.90	1.65	1.01
10-C40-5		10		1.2	1.26			29.9	57.8	1.93	0.043	0.021	0.0336	9.90	1.65	1.01
10-C40-6														10.84	1.97	1.22
10-C40-7														11.71	2.30	1.42
Adhikary et al. [26]
DR3.8_0.8 _0.11_H0.6	160 × 240	13	2	1.6	0.79	38.5	520	67.8	93.3	1.38	0.154	0.075	0.3	3.43	1.76	1.41
DR3.8_0.8 _0.11_H0.9														4.20	2.65	2.12
DR3.8_0.8 _0.11_H1.2														4.85	3.53	2.83
DR3.8_0.8 _0.15_H0.6								67.8	102.3	1.51	0.154	0.075		3.43	1.76	1.41
DR3.8_0.8 _0.15_H0.9														4.20	2.65	2.12
DR3.8_0.8 _0.15_H1.2														4.85	3.53	2.83
DR5.7_1.6 _0.15_H0.3	120 × 170	13	2	1.6	0.79	38.5	520	42.4	50.5	1.19	0.082	0.040	0.3	2.43	0.89	0.78
DR5.7_1.6 _0.15_H0.45														2.97	1.32	1.17
DR5.7_1.6 _0.15_H0.6														3.43	1.76	1.56
DR5.7_1.6 _0.20_H0.3								42.4	56.4	1.33	0.082	0.040		2.43	0.89	0.78
DR5.7_1.6 _0.20_H0.45														2.97	1.32	1.17
DR5.7_1.6 _0.20_H0.6														3.43	1.76	1.56

φ: Nominal diameter of tensile rebar, N: Amount of the rebar, L: Clear span length of the beam, p_t: Tensile rebar ratio, f’_c: Compressive strength of concrete, f_y: Yield strength of the tensile rebar, P_u: Calculated bending capacity, Q_u: Calculated shear capacity, α: Shear-bending capacity ratio (= Q_u/P_u), M_b: Mass of beam, M_be: Equivalent mass of beam, M_w: Mass of weight, V: Impact velocity of weight, E_k: Kinetic energy of weight (Input energy), E_t: Transmitted energy.

and

Table A2. Estimated maximum deflection δ_u.

Ref. ID	Name	δu.exp (mm)	Ek (kJ)	Et (kJ)	δuk (mm)	δu (mm)	RD (%)	RD.exp (%)	Rp	Rp.exp	Py (kN)	δy (mm)	δud (mm)	AEI
9	G1-1	64.3	7.35	4.25	114.0	67.3	2.24	2.14	7.42	7.09	67.1	9.07	67.9	1.06
	G1-1S	58.0	7.35	4.25	114.0	67.3	2.24	1.93	7.42	6.39	67.1	9.07	67.9	1.17
	G2-1	28.3	2.40	1.84	71.2	54.0	2.70	1.42	10.09	5.29	35.8	5.35	54.1	1.91
	G2-2	44.0	3.75	2.88	112.9	86.0	4.30	2.20	16.07	8.22	35.8	5.35	83.0	1.89
	G2-3	57.0	5.40	4.14	164.1	124.9	6.25	2.85	23.35	10.65	35.8	5.35	118.4	2.08
	G2L-1	44.2	3.20	2.61	95.9	77.7	3.89	2.21	14.52	8.26	35.8	5.35	75.5	1.71
	G2L-2	66.8	5.00	4.07	151.7	122.9	6.15	3.34	22.97	12.49	35.8	5.35	116.4	1.74
	G2L-3	89.7	7.20	5.86	220.2	178.5	8.93	4.49	33.36	16.77	35.8	5.35	166.5	1.86
	G3-1	36.7	2.40	1.84	67.4	51.2	2.56	1.84	9.46	6.78	37.7	5.41	51.5	1.40
	G3-2	52.0	3.75	2.88	107.0	81.4	4.07	2.60	15.05	9.61	37.7	5.41	79.0	1.52
	G3-3	70.6	5.40	4.14	155.6	118.5	5.93	3.53	21.90	13.05	37.7	5.41	112.6	1.59
	G4-1	39.7	2.40	1.84	71.2	54.0	2.70	1.99	10.09	7.42	35.8	5.35	54.1	1.36
	G4-2	56.1	3.75	2.88	112.9	86.0	4.30	2.81	16.07	10.49	35.8	5.35	83.0	1.48
	G5-1	63.5	7.20	4.66	113.9	74.2	2.47	2.12	7.99	6.84	66.8	9.29	74.3	1.17
	G5-2	83.4	9.80	6.34	154.6	100.4	3.35	2.78	10.81	8.98	66.8	9.29	99.5	1.19
	G6-1	26.4	3.75	2.49	47.7	30.9	1.55	1.32	5.46	4.66	83.8	5.66	32.5	1.23
	G7-1	45.8	3.75	2.13	72.9	42.5	1.42	1.53	3.65	3.93	55.9	11.65	44.0	0.96
	G7-2	60.9	5.40	3.07	104.8	59.9	2.00	2.03	5.14	5.23	55.9	11.65	60.7	1.00
	G8-1	36.5	5.40	4.08	59.4	45.5	2.28	1.83	5.10	4.09	101.5	8.93	44.7	1.22
	G9-1	43.2	3.75	2.83	65.2	50.8	1.69	1.44	2.77	2.35	67.6	18.37	51.1	1.18
	G9-2	57.9	5.40	4.08	91.4	70.5	2.35	1.93	3.84	3.15	67.6	18.37	69.5	1.20
	G10-1	33.7	2.40	1.49	49.3	32.7	1.09	1.12	2.48	2.56	55.5	13.18	33.5	0.99
	G10-2	49.5	3.75	2.33	74.6	48.0	1.60	1.65	3.64	3.76	55.5	13.18	48.6	0.98
	G10-3	67.8	5.40	3.36	105.8	67.2	2.24	2.26	5.10	5.14	55.5	13.18	67.1	0.99
	G10-4	83.9	7.35	4.57	142.9	90.1	3.00	2.80	6.84	6.37	55.5	13.18	89.0	1.06
	G11-1	20.5	2.45	1.76	29.5	22.6	0.84	0.76	2.24	2.03	97.7	10.08	23.0	1.12
	G11-2	33.2	4.41	3.16	50.1	36.9	1.37	1.23	3.66	3.29	97.7	10.08	37.4	1.13
	G11-3	43.1	6.38	4.58	71.2	51.9	1.92	1.60	5.15	4.28	97.7	10.08	51.9	1.20
	G11-4	55.5	8.35	5.99	92.5	67.0	2.48	2.06	6.65	5.51	97.7	10.08	66.4	1.20
	G11-5	67.2	10.3	7.39	113.6	82.0	3.04	2.49	8.13	6.67	97.7	10.08	80.7	1.20
	G11-6	83.4	12.2	8.79	134.7	97.3	3.60	3.09	9.65	8.27	97.7	10.08	95.0	1.14
	G12-1	85.4	14.7	10.55	143.9	104.0	3.85	3.16	10.09	8.28	109.2	10.31	101.8	1.19
	G13-1	60.6	14.7	9.65	82.4	54.9	2.03	2.24	7.87	8.68	185.6	6.98	55.5	0.92
	G14-1	63.7	14.7	10.08	98.9	68.6	2.54	2.36	8.22	7.63	156.3	8.35	68.7	1.08
	G15-1	40.5	14.7	9.65	65.9	44.3	1.64	1.50	6.06	5.54	233.7	7.31	44.9	1.11
	G16-1	52.9	14.7	9.90	91.5	62.5	2.31	1.96	7.89	6.68	168.5	7.92	62.7	1.19
11	1-A2	13.6	0.92	0.57	26.6	17.4	0.87	0.68	3.33	2.60	36.2	5.23	18.4	1.35
	2-A2	16.3	0.86	0.66	25.0	19.8	0.99	0.82	3.79	3.12	36.2	5.23	20.9	1.28
	3-A2	17.9	0.90	0.75	26.0	22.0	1.10	0.90	4.21	3.42	36.2	5.23	23.3	1.30
	4-A2	25.4	1.80	1.12	51.9	32.1	1.61	1.27	6.14	4.86	36.2	5.23	33.6	1.32
	5-A2	31.6	1.84	1.41	53.1	40.5	2.03	1.58	7.74	6.04	36.2	5.23	41.6	1.31
	6-A2	33.3	1.76	1.47	50.7	42.1	2.11	1.67	8.05	6.37	36.2	5.23	43.1	1.30
	7-A2	37.0	2.70	1.68	78.8	48.4	2.42	1.85	9.25	7.07	36.2	5.23	49.0	1.32
	8-A2	43.7	2.65	2.03	77.3	58.7	2.94	2.19	11.22	8.36	36.2	5.23	58.7	1.34
	9-A2	48.4	2.76	2.29	80.7	66.6	3.33	2.42	12.73	9.25	36.2	5.23	65.9	1.36
	10-A2	4.50	0.15	0.12	6.60	5.70	0.29	0.23	1.09	0.86	36.2	5.23	5.79	1.29
	11-A2-1	12.8	0.60	0.46	18.2	14.7	0.74	0.64	2.81	2.45	36.2	5.23	15.3	1.20
	11-A2-2	12.3	0.60	0.46	18.2	14.7	0.74	0.62	2.81	2.35	36.2	5.23	15.3	1.25
	12-A2-1	25.8	1.35	1.04	38.7	29.8	1.49	1.29	5.70	4.93	36.2	5.23	31.2	1.21
	12-A2-2	28.0	1.35	1.04	38.7	29.8	1.49	1.40	5.70	5.35	36.2	5.23	31.2	1.12
	13-A2-1	42.2	2.40	1.84	69.8	53.1	2.66	2.11	10.15	8.07	36.2	5.23	53.5	1.27
	13-A2-2	40.6	2.40	1.84	69.8	53.1	2.66	2.03	10.15	7.76	36.2	5.23	53.5	1.32
	14-A2-1	60.1	3.75	2.88	110.5	84.2	4.21	3.01	16.10	11.49	36.2	5.23	82.1	1.37
	14-A2-2	57.2	3.75	2.88	110.5	84.2	4.21	2.86	16.10	10.94	36.2	5.23	82.1	1.44
	14-A2-3	57.6	3.75	2.88	110.5	84.2	4.21	2.88	16.10	11.01	36.2	5.23	82.1	1.43
	15-A1-1	24.1	3.75	3.26	56.0	48.5	4.85	2.41	37.02	18.40	72.4	1.31	45.6	1.89
	15-A1-2	24.2	3.75	3.26	56.0	48.5	4.85	2.42	37.02	18.47	72.4	1.31	45.6	1.89
	15-A1-3	22.8	3.75	3.26	56.0	48.5	4.85	2.28	37.02	17.40	72.4	1.31	45.6	2.00
	15-A1-4	25.2	3.75	3.26	56.0	48.5	4.85	2.52	37.02	19.24	72.4	1.31	45.6	1.81
	16-A4-1	119.9	3.75	2.33	216.7	133.8	3.35	3.00	6.26	5.61	18.3	21.36	138.2	1.15
	16-A4-2	116.8	3.75	2.33	216.7	133.8	3.35	2.92	6.26	5.47	18.3	21.36	138.2	1.18
	16-A4-3	107.9	3.75	2.33	216.7	133.8	3.35	2.70	6.26	5.05	18.3	21.36	138.2	1.28
	17-B-1	78.0	3.75	2.75	132.2	95.3	4.77	3.90	8.04	6.58	35.7	11.85	82.9	1.06
	17-B-2	75.9	3.75	2.75	132.2	95.3	4.77	3.80	8.04	6.41	35.7	11.85	82.9	1.09
	18-C-1	41.1	3.75	2.88	69.2	53.3	2.67	2.06	8.57	6.61	58.3	6.22	52.5	1.28
	18-C-2	43.7	3.75	2.88	69.2	53.3	2.67	2.19	8.57	7.03	58.3	6.22	52.5	1.20
	19-D-1	95.5	3.75	2.88	189.3	143.7	7.19	4.78	27.63	18.37	21.7	5.20	135.2	1.42
	19-D-2	92.5	3.75	2.88	189.3	143.7	7.19	4.63	27.63	17.79	21.7	5.20	135.2	1.46
	20-E-1	29.6	3.75	2.52	60.1	40.3	2.02	1.48	13.39	9.83	63.7	3.01	41.1	1.39
	20-E-2	28.6	3.75	2.52	60.1	40.3	2.02	1.43	13.39	9.50	63.7	3.01	41.1	1.44
	21-F-1	43.0	3.75	2.52	100.7	66.9	3.35	2.15	23.72	15.25	38.1	2.82	67.7	1.57
	21-F-2	44.7	3.75	2.52	100.7	66.9	3.35	2.24	23.72	15.85	38.1	2.82	67.7	1.51
12	S1616-1	5.80	0.59	0.51	7.80	6.90	0.49	0.41	2.37	1.99	89.2	2.91	7.18	1.24
	S1616-2	10.2	1.18	1.02	14.2	12.4	0.89	0.73	4.26	3.51	89.2	2.91	12.9	1.26
	S1616-3	18.8	2.35	2.03	28.0	24.2	1.73	1.34	8.32	6.46	89.2	2.91	24.2	1.29
	S1616-4	36.0	4.70	4.06	56.5	48.7	3.48	2.57	16.74	12.37	89.2	2.91	46.9	1.30
	S1322-1	6.30	1.18	1.02	8.70	7.70	0.55	0.45	2.24	1.83	165.7	3.44	7.87	1.25
	S1322-2	11.3	2.35	2.03	16.0	13.9	0.99	0.81	4.04	3.28	165.7	3.44	14.0	1.24
	S1322-3	21.3	4.70	4.06	30.9	26.8	1.91	1.52	7.79	6.19	165.7	3.44	26.2	1.23
	S1322-4	41.2	9.41	8.12	60.9	52.7	3.76	2.94	15.32	11.98	165.7	3.44	50.7	1.23
	S2222-1	6.30	1.18	1.02	8.60	7.60	0.54	0.45	2.32	1.93	166.2	3.27	7.76	1.23
	S2222-2	11.2	2.35	2.03	15.7	13.7	0.98	0.80	4.19	3.43	166.2	3.27	13.8	1.24
	S2222-3	20.7	4.70	4.06	30.5	26.4	1.89	1.48	8.07	6.33	166.2	3.27	26.1	1.26
	S2222-4	39.6	9.41	8.12	60.4	52.2	3.73	2.83	15.96	12.11	166.2	3.27	50.5	1.27
13	PB-880	29.8	98.07	16.74	123.2	29.5	0.37	0.37	1.48	1.49	823.6	20.00	30.3	1.02
	PB-880	56.0	196.1	33.49	248.1	49.2	0.62	0.70	2.46	2.80	823.6	20.00	50.7	0.90
	PB-880	82.6	294.2	50.23	378.6	67.8	0.85	1.03	3.39	4.13	823.6	20.00	71.0	0.86
	PB-880	106.6	392.3	66.97	510.1	89.9	1.12	1.33	4.50	5.33	823.6	20.00	91.3	0.86
	PC-620	44.8	98.1	19.12	177.7	44.6	0.56	0.56	1.73	1.74	572.8	25.80	46.3	1.03
	PC-620	87.7	196.1	38.24	370.7	76.1	0.95	1.10	2.95	3.40	572.8	25.80	79.7	0.91
16	Beam 1	52.3	3.99	2.80	113.5	79.5	4.18	2.75	15.90	10.46	35.7	5.00	80.9	1.55
	Beam 2	48.0	3.99	2.80	113.1	79.0	4.16	2.53	15.58	9.47	35.9	5.07	80.5	1.68
21	W2H10	66.4	196.1	33.49	377.5	69.5	0.87	0.83	3.45	3.30	531.2	20.12	73.1	1.10
	W5H4	129.3	196.1	66.65	377.5	125.9	1.57	1.62	6.26	6.43	531.2	20.12	135.5	1.05
	W10H2	166.7	196.1	99.49	377.5	183.5	2.29	2.08	9.12	8.29	531.2	20.12	197.3	1.18
	W2H5	40.2	98.0	16.74	180.7	40.3	0.50	0.50	2.00	2.00	531.2	20.12	41.6	1.03
	W10H1	89.8	98.0	49.74	180.7	97.4	1.22	1.12	4.84	4.46	531.2	20.12	103.7	1.15
22	N-H0.1	10.5	0.23	0.15	11.3	9.15	0.46	0.53	0.80	0.91	37.6	11.48	9.71	0.92
	N-H0.25	17.6	0.64	0.41	21.6	16.0	0.80	0.88	1.39	1.53	37.6	11.48	16.7	0.95
	N-H0.5	24.2	1.33	0.86	39.9	26.9	1.35	1.21	2.34	2.11	37.6	11.48	28.6	1.18
	N-H1.0	43.5	2.65	1.71	85.7	52.4	2.62	2.18	4.56	3.79	37.6	11.48	51.2	1.18
	N-H1.5	59.9	3.95	2.55	133.0	82.1	4.11	3.00	7.15	5.22	37.6	11.48	73.6	1.23
23, 24	N-H300	10.0	0.05	0.04	13.8	10.8	1.20	1.11	3.45	3.19	3.50	3.13	12.9	1.29
	N-H600	15.1	0.10	0.08	35.1	24.9	2.77	1.68	7.96	4.82	3.50	3.13	23.6	1.56
	N-H900	21.9	0.15	0.11	58.1	41.7	4.63	2.43	13.33	7.00	3.50	3.13	34.3	1.57
25	6-C27-3	47.0	0.99	0.61	94.8	58.2	4.85	3.91	8.29	6.69	11.5	7.02	56.4	1.20
	6-C27-4	63.7	1.32	0.81	126.3	77.5	6.46	5.31	11.04	9.07	11.5	7.02	73.9	1.16
	8-C27-3	30.6	0.99	0.61	55.1	34.7	2.89	2.55	4.26	3.75	19.6	8.15	35.1	1.15
	8-C27-4	39.4	1.32	0.81	72.9	45.5	3.79	3.29	5.58	4.84	19.6	8.15	45.4	1.15
	10-C27-2.5	18.5	0.82	0.51	31.2	20.4	1.70	1.54	2.21	2.00	29.7	9.25	21.7	1.17
	10-C27-3	20.0	0.99	0.61	37.0	23.9	1.99	1.67	2.58	2.16	29.7	9.25	25.1	1.26
	10-C27-4	28.7	1.32	0.81	48.3	30.8	2.57	2.39	3.33	3.10	29.7	9.25	31.9	1.11
	6-C40-5	62.7	1.65	1.01	156.1	95.3	7.94	5.22	14.59	9.59	11.8	6.53	89.2	1.42
	6-C40-6	87.7	1.97	1.22	186.7	114.5	9.54	7.31	17.53	13.44	11.8	6.53	106.2	1.21
	6-C40-7	119.1	2.30	1.42	218.2	133.9	11.16	9.92	20.51	18.23	11.8	6.53	123.4	1.04
	6-C40-8	119.1	2.63	1.62	249.8	153.3	12.78	9.93	23.48	18.24	11.8	6.53	140.6	1.18
	8-C40-2	22.9	0.66	0.41	36.4	23.0	1.92	1.91	3.17	3.16	20.2	7.26	23.7	1.03
	8-C40-3	35.7	0.99	0.61	54.0	33.6	2.80	2.97	4.63	4.91	20.2	7.26	33.7	0.95
	8-C40-4	35.1	1.32	0.81	71.8	44.4	3.70	2.93	6.12	4.84	20.2	7.26	43.7	1.24
	8-C40-5	38.8	1.65	1.01	89.5	55.3	4.61	3.24	7.62	5.35	20.2	7.26	53.8	1.39
	10-C40-5	33.7	1.65	1.01	58.8	37.0	3.08	2.81	4.50	4.09	30.6	8.23	37.2	1.11
	10-C40-6	45.2	1.97	1.22	69.9	43.9	3.66	3.77	5.33	5.49	30.6	8.23	43.8	0.97
	10-C40-7	62.5	2.30	1.42	82.3	50.9	4.24	5.21	6.18	7.59	30.6	8.23	50.5	0.81
26	DR3.8_0.8 _0.11_H0.6	18.6	1.76	1.41	28.6	23.1	1.44	1.16	5.03	4.05	65.9	4.59	23.7	1.28
	DR3.8_0.8 _0.11_H0.9	34.5	2.65	2.12	42.7	34.3	2.14	2.16	7.47	7.52	65.9	4.59	34.5	1.00
	DR3.8_0.8 _0.11_H1.2	41.0	3.53	2.83	56.8	45.6	2.85	2.56	9.93	8.93	65.9	4.59	45.2	1.10
	DR3.8_0.8 _0.15_H0.6	19.6	1.76	1.41	28.6	23.1	1.44	1.23	5.03	4.27	65.9	4.59	23.7	1.21
	DR3.8_0.8 _0.15_H0.9	28.8	2.65	2.12	42.7	34.3	2.14	1.80	7.47	6.27	65.9	4.59	34.5	1.20
	DR3.8_0.8 _0.15_H1.2	39.2	3.53	2.83	56.8	45.6	2.85	2.45	9.93	8.54	65.9	4.59	45.2	1.15
	DR5.7_1.6 _0.15_H0.3	20.0	0.89	0.78	24.9	22.1	1.38	1.25	2.90	2.63	41.7	7.61	22.6	1.13
	DR5.7_1.6 _0.15_H0.45	30.0	1.32	1.17	35.9	32.0	2.00	1.88	4.20	3.94	41.7	7.61	31.8	1.06
	DR5.7_1.6 _0.15_H0.6	39.1	1.76	1.56	47.3	42.1	2.63	2.44	5.53	5.14	41.7	7.61	41.2	1.05
	DR5.7_1.6 _0.20_H0.3	19.1	0.89	0.78	24.9	22.1	1.38	1.19	2.90	2.51	41.7	7.61	22.6	1.18
	DR5.7_1.6 _0.20_H0.45	28.8	1.32	1.17	35.9	32.0	2.00	1.80	4.20	3.78	41.7	7.61	31.8	1.11
	DR5.7_1.6 _0.20_H0.6	37.9	1.76	1.56	47.3	42.1	2.63	2.37	5.53	4.98	41.7	7.61	41.2	1.09

δ_u.exp: Experimental result of maximum deflection, E_k: Kinetic energy of weight (Input energy), E_t: Transmitted energy, δ_uk: Estimated maximum deflection based on the E_k, δ_u: Estimated maximum deflection, R_D: Deflection ratio (= δ_u /L), R_D.exp: Experimental result of the deflection ratio (= δ_u.exp/L), R_p: Plasticity ratio (= δ_u/δ_y), R_p.exp: Experimental result of the plasticity ratio (= δ_u.exp /δ_y), P_y: Calculated yield bending capacity of the beam, δ_y: Calculated yield deflection of the beam, δ_ud: Estimated maximum deflection for design, AEI: Accuracy evaluation indicator (= δ_u /δ_u.exp).

). 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. Range of RC beam specifications (total of 134 cases).

Item of Specification	Symbol	Unit	Range of Value
			Min.	Max.
Size of section		mm	60 × 100	1000 × 1,000
Clear span length	L	m	0.9	8.0
Tensile rebar ratio	pt	%	0.26	1.26
Compressive strength of concrete	f’c	MPa	25.2	52.0
Yield strength of the tensile rebar	fy	Mpa	235	520
Calculated bending capacity	Pu	kN	4.10	881
Calculated shear capacity	Qu	kN	7.18	2,882
Shear-bending capacity ratio	α		1.19	8.78
Mass of beam	Mb	t	0.014	20.0
Mass of weight	Mw	t	0.020	10.0
Impact velocity of weight	V	m/s	1.00	19.8
The kinetic energy of weight (input energy)	Ek	kJ	0.05	392.2

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.

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.

3.2.2. 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.

R_D = δ_u/L

(18)

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.

3.2.3. 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.

R_p = δ_u/δ_y

(19)

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.

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.

3.3. 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:

AEI = δ_u/δ_u.exp.

(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.

4. 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).

E_a = P_y δ_ud − P_y δ_y/2

(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:

E_t = P_y δ_ud − P_y δ_y/2

(22)

δ_ud = E_t/P_y + δ_y/2

(23)

$${\mathsf{\delta }}_{\mathrm{ud}}=\frac{M_{\mathrm{w}}^2{V}^2}{{2P}_{\mathrm{y}}\left(\frac{17}{35}{\mathsf{\rho }AL+M}_{\mathrm{w}}\right)}+\frac{{\mathsf{\delta }}_{\mathrm{y}}}{2}$$

(24)

1 < R_p ≤ 10

(25)

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.

5. 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.