Design Calculation for Concrete-Filled Steel Tube under Soft Lateral Impact

Abstract

In recent decades, the rapid development of transportation construction has increased the possibility of concrete-filled steel tubes (CFSTs) subjected to soft lateral impact. Some kinds of deviation may still exist between the soft impact process and its design prediction due to frequent safety accidents. To improve the prediction accuracy, this paper contrasted the current soft impact design with the experimental impact processes of CFSTs. The current design is represented by the method and route in Eurocode-1 Actions on structures, which belongs to the one-step type. Extra parameters were added to the one-step method to describe the inertia effect of CFST. According to the experiments and predictions, the soft impact process contains two stages, which are the initial peak stage (IPS) and the stable stage (SS). The current one-step method is consistent with SS but incorrect in IPS and its accuracy is related to the proportion of IPS and SS. An empirical formula, based on the weight of IPS, is summarized to evaluate the overall error of the one-step method in the soft impact design of CFST. This error is due to the unreasonable assumption of the inertial effect in IPS. For the demands of an accurate design in IPS, a new technical route is proposed, consisting of two steps: qualitative analysis and two-stage calculation. The qualitative analysis achieved an approximate quantitative division of IPS and SS, and provided the design loads for the two-stage calculation. The two-stage calculation supplemented the prediction of the inertia effect in IPS and independently estimated the resistances and responses of CFST in the two stages. The above qualitative analysis and two-stage calculation constituted the two-step method and its accuracy and applicability are generally better than that of the one-step design.

1. Introduction

Concrete-filled steel tubes (CFSTs) have been widely used in engineering constructions, such as bridges, buildings, and municipal infrastructures [1]. The triaxial constraint of the steel tube enhances the strength of the concrete, and the filling of the concrete prevents the buckling of the steel tube. This promotes the fine bearing capacity and ductility of CFSTs [2,3]. The advantages of CFST have resulted in it becoming one conventional structural form for piers and columns.

According to the investigations of El-Tawil [4] and Grob [5], lateral impact load is one of the security threats for CFSTs when columns are located along the transportation network. The potential consequences of the impact actions include direct damage of the columns and the collapse of the structure [6,7,8,9,10,11,12,13]. Numerous objects (e.g., vehicles, rockfalls, or derailed trains) may cause lateral impact accidents. Referring to Eurocode-1 Actions on structures [14] and BS 5400-2 [15], lateral impact processes are generally classified into situations with hard impact and soft impact. Figure 1a shows hard impact, which requires the impacting body to dissipate the impact energy and the structure is rigid and immovable. Soft impact is the opposite and is shown in Figure 1b. The structure is deformable and absorbs the impact energy; the impacting body is regarded as the rigid. From the statistics of Buth [16] and Zhu [17], the response mode of CFSTs is close to the definition of soft impact when lateral impact accidents have serious consequences. Thus, the design of CFST under soft lateral impact has received attention.

The dynamic response of CFSTs is shown in the lateral impact experiments carried out by Bambach et al. [18] and Wang et al. [19]. Both CFSTs with circular or square sections exhibited beam bending behavior. Plastic deformation or flexural cracks were concentrated at the impact point and support areas. Regarding the plastic concentration areas, a full-range analysis was carried out in the research of Han et al. [20], which introduced a simplified mode to calculate the flexural capacity of CFST members under an impact load.

According to the experimental phenomena, the theory of beam bending is naturally used to predict the impact process of CFSTs. The design methods can be found in the cases of square CFSTs investigated by Bambach et al. [21] and circular CFSTs introduced by Qu et al. [22] and Deng et al. [23]. There is also the theoretical derivation based on the above principles, such as the analysis of Wang et al. [24] on ultra-lightweight cement composite (ULCC)-filled steel pipe structures. From the above calculations, the impact energy is fully dissipated by the plastic concentration areas in the experiments (plastic hinges). Outside these plastic hinges, CFSTs have little plasticity and energy absorption.

The model of the plastic hinge has been recognized by the design codes [14,15]. However, the description of the inertia effect is still one outstanding problem in this model. Although it has been proven that the inertia force has a significant effect on the dynamic response of the engineering structures [25,26], the above methods have been scarcely considered in the lateral impact design.

A description of the inertial effect is often found in the theoretical analysis of the soft impact process [27]. One classical case is response analysis of a rigid-plastic beam under lateral impact from a rigid mass [28]. Parkes [29] described the inertial effect through the acceleration distribution of a rigid-plastic beam. Then, the inertial effect can resist the lateral impact together with the material resistance of the structure. During this analysis, the material resistance of the beam was a definite parameter, but the resistance from the inertial effect was the unknown parameter. The unknown parameter makes it difficult to properly predict the dynamic response of the structural member.

To eliminate the unknowns caused by the inertia effect of the structure, a relationship is usually supplemented between the inertia effect and other given parameters. For example, one typical relationship is the embedding assumption [28,29,30,31]: the impact body is embedded into the structure and becomes an additional mass of the beam. In this situation, the acceleration of the beam is the same as that of the impact body at the impact point. Then, the inertia force of the beam can be associated with the impact force. Although the embedding assumption removes the unknowns from the inertia effect of the beam, its rationality is doubtful sometimes. Another simpler approach is to ignore the inertia effect of the beam directly, which is quite common in design calculation [14,15,16,17,18,19,20,21,22,23,24]. Because the total resistance loses part of the inertia effect and contains only the material one, this approach ensures the usability of the design calculation but sacrifices the completeness and accuracy. Logically, both methods can achieve the purpose of cancelling the unknowns from the inertia effect of the structure, but it is better to consider the influence of the inertia effect during the calculation. However, due to the difficulty in the parameter calibration of the inertia effect, the second technical route is more universal in current designs. Another reason is that an unreasonable description of the inertia effect may lead to a higher error than ignoring it.

To improve the above situations, this paper analyzes the characteristics of the current design method. Then, a new design route is established to predict the dynamic response of CFSTs under the soft impact process. The influence of the inertia effects is presented throughout the whole calculation. During the design, the new method involves two steps, which are qualitative analysis and two-stage calculation. The qualitative analysis realizes the stage division of the impact process and provides the design loads for the following dynamic response calculation. The two-stage calculation contains the predictions of the inertia effect, achieving differentiated estimation for the dynamic resistance of CFST in each response stage.

2. Current Design Method

2.1. Design with the One-Step Method

The one-step method is derived from the soft impact design in Eurocode-1: Actions on structures [14] and BS 5400-2 [15]. Figure 2 shows the basic force mechanism. Its principle is that the plastic deformation of CFST absorbs all lateral impact energy. Based on this principle, Equation (1) is recommended as the design standard [14,15], which describes the balanced relationships between the total kinetic energy of the colliding object and the impact resistance of CFSTs. Since the soft impact process is described by Equation (1) alone, it performs dynamic analysis only once during the design calculation. Thus, this kind of design mode is classified as the one-step method in this paper, which is dominant in the current soft impact design.

In Equation (1), F₀ is the interaction force between the impactor and CFSTs at the contact surface, which causes deformation of CFST and prevents the displacement of the impactor. When the plastic energy dissipation of CFST reaches the initial impact energy (E₀ = 0.5 mV_I0²), the deflection of CFST should be less than its deformation capacity (y₀) to ensure structural safety. Another perspective is that the displacement of the impactor is equal to the deflection of CFST when F₀ depletes the total energy of the impactor (E₀) completely. Therefore, Equation (1) further implies that the impact mass has the same displacement as CFST at the impact point, i.e., the colliding object is embedded in the CFST:

$$\frac{1}{2}m{V}_{I0}{}^2\le F_0{y}_0$$

(1)

where F₀ is the plastic strength of the structure, i.e., the limit value of the static force (F_s); y₀ is the deformation capacity, i.e., the displacement of the point of impact that the structure can undergo; m is the mass of the colliding object; and V_I0 is the initial velocity of the colliding object in the lateral impact direction.

Equation (2) is the equivalent value of F₀ suitable for a clamped support beam (fixed constraint). If a pin constraint replaces the clamped one, the bending moment is reduced to zero at this position. Last, the deflection (y₀) can be obtained by taking Equation (2) into (1). The bending moment in Equation (2) usually comes from the limit value related to the material resistance of CFSTs, such as the static or dynamic plastic-limit value of the bending moment in the CFST section (M_P):

$$F_0=\frac{M_I+M_F}{L_F}+\frac{M_I+M_N}{L_N}$$

(2)

where M_F is the bending moment value of the CFST section in support F; M_N is the bending moment value of the CFST section in support N; M_I is the bending moment value of the CFST section in impact point I; M_P is the plastic limit value of the bending moment in the CFST section; L_F is the distance between the impact point and support F; L_N is the distance between the impact point and support N; and L is the effective total length of the CFST member.

Equation (3) is adopted to calculate the dynamic bending moment of the CFST section [23,32]. M_s is the static bending moment of the CFST section and DIF is its dynamic increase factor. For the different CFST sections (I,N,F), M_I,_s, M_N,_s, M_F,_s, and M_P,_s are the static parameters of the bending moments. M, including M_I, M_N, M_F, and M_P, are the dynamic parameters. These parameters are in accordance with the relationship of Equation (3), where the additional subscript “s” indicates the static values:

$$M=DIF\cdot M_s$$

(3)

where DIF is the dynamic increase factor for the bending moment of the CFST section and M_s is the static bending moment of the CFST section.

2.2. Inertia Effect Description

The inertia effect is another influence on F₀, which is not mandatory in Equations (1) and (2). The acting force due to the inertia effect can be estimated based on the acceleration distributions of the CFST member, which is also called the inertia force. Figure 2 shows the linear mode of the acceleration distribution, which was used in the research of [27,28,29,30,31,32]. The features of this mode are that the acceleration has a linear distribution between the impact point and each support, the impact point has the maximum value of the acceleration (a_b), and the supports are the zero acceleration locations. Then, Equation (4) is used to describe the inertia force of CFSTs and Equation (5) is the total force balance. Here, Equation (5) is the dynamic form of Equation (2). F_0,d is equivalent to F₀, as it combines the material resistance and the inertia effect; F₀ contains only part of the material. Additionally, F_0,d is equal to the impact action (−ma_I) according to the contact relationship:

$$F_{IE}={\displaystyle \underset{0}{\overset{L_F}{\int }}{\rho }_l{a}_b}\left(1-\frac{x}{L_F}\right)dx+{\displaystyle \underset{0}{\overset{L_N}{\int }}{\rho }_l{a}_b}\left(1-\frac{x}{L_N}\right)dx=\frac{1}{2}{\rho }_{l}L{a}_b$$

(4)

where F_IE is the inertia force of the CFST column; a_b is the acceleration value of the CFST column at the impact point; and ρ_l is the mass of the CFST per unit length:

$$F_{0,d}=F_{IE}+Q_d=-m{a}_I$$

(5)

where a_I is the acceleration value of the colliding object; m is the mass of the colliding object; and Q_d is the total reaction force of the two supports, including the inertia effect.

In the following derivation, the displacement parameters are positive along the impact direction, i.e., the deflection, velocity, and acceleration. The reaction force of the support is positive along the opposite direction of the lateral impact.

Equations (6)–(8) are the extension of Equation (5). Equation (6) is the more detailed format of the force equilibrium Equation (5). Equations (7) and (8) are the equilibrium of the bending moment in the IF and IN areas, which are used to determine the reaction force of the supports (Q_d = Q_F + Q_N):

$$-m{a}_I=\frac{1}{2}{\rho }_l{a}_{b}L+\left(Q_F+Q_N\right)$$

(6)

$$Q_F=\frac{M_I+M_F}{L_F}-\frac{{\rho }_l{a}_b{L}_F}{6}$$

(7)

$$Q_N=\frac{M_I+M_N}{L_N}-\frac{{\rho }_l{a}_b{L}_N}{6}$$

(8)

where Q_N is the shear force of CFSTs at the support N and Q_F is the shear force of CFSTs at the support F.

From the above analysis, it can additionally be found that the inertia effect not only directly changes F_0,d through ρ_l a_b L/2 (F_IE) but also the reaction force of the supports by ρ_l a_b L_F/6. Therefore, it may be difficult to accurately estimate the resistance of CFSTs with little consideration of the inertia effect in the design process.

2.3. Dynamic Response Prediction

The description of the inertia effect is carried out to predict the relatively accurate dynamic response of CFSTs. The dynamic response is still achieved by Equation (1), but the original F₀ from Equation (2) is replaced by F_0,d in Equation (5).

The calculation of F_0,d is based on Equations (6)–(8), where a_b is the control parameter of the inertia effect. Here, an additional condition is needed to determine the value of a_b. Following Equation (1), which is the embedding model, a_I = a_b may become the additional condition. Taking a_I = a_b into Equation (6), Equation (9) is used to obtain the required a_b value. Then, F_0,d can be obtained according to Equation (10):

$$a_I=a_b=-\frac{1}{m+\frac{1}{3}{\rho }_l(L_N+L_F)}\left(\frac{M_N+M_I}{L_N}+\frac{M_F+M_I}{L_F}\right)$$

(9)

$$F_{0,d}=-m{a}_I=\frac{F_0}{1+\frac{{\rho }_{l}L}{3m}}=\frac{DIF}{1+\frac{{\rho }_{l}L}{3m}}{F}_s$$

(10)

$$F_s=\frac{M_{N,s}+M_{I,s}}{L_N}+\frac{M_{F,s}+M_{I,s}}{L_F}$$

(11)

In Equation (10), F₀ is the dynamic material resistance of CFSTs and is the multiplication of DIF and F_s. DIF is the dynamic increase factor for the bending moment of the CFST section. F_s is the static part of the CFST resistance. Comparing Equations (10) and (2), 1 + ρ_l L/3m reflects the influence of the inertia effect. It is generally believed that both Equations (10) and (2) can correctly describe the lateral impact resistance of CFSTs, but there are some differences between them regarding accuracy due to the inertia effect. Here, the inertia item (1 + ρ_l L/3m) in Equation (10) reduces the resistance of CFSTs and it is equally important with DIF. Instead, Equation (2) ignores the decrease in the resistance from the inertia effect and retains only the increase in the resistance from DIF; its calculation may tend to be dangerous.

3. Verification of the One-Step Design

3.1. Verification Experiment

Experiments were carried out to verify the one-step design [32], mainly the rationality of Equation (1) with Equation (10). The design of the experiments was based on the soft impact process of buildings near a highway or the pier column of a bridge.

According to BS EN 1991-1-7 [14], the lateral impact point is located 1.5 m above the pavement. If the column bottom has a buried depth of 0.5 m, the distance between the impact point and one support is 2.0 m. The supports of CFSTs are the clamped type (fixed support). This can include the restraining effect of both the superstructure and the ground. The lateral impact is quantified by the impact energy and its design levels are 7942, 13,233, and 18,543 kJ. These selected design values correspond to a 50 t colliding object with an initial impact velocity of 64, 83, and 98 km/h, respectively.

Another control parameter in the experiments is the ratio of the geometric similarity, which is 1:10 in the full-scale situations. The colliding object is a rigid body and the size of its impact contact surface is 30 × 80 mm.

The test equipment is shown in Figure 3. The mass of the impactor (m) is 270 kg, including the hammer and mass block. The impact loading is provided by the gravity free-fall of the mass. The free-fall height is 3.0, 5.0, and 7.0 m, the initial impact energy is 7.942, 13.233, and 18.543 kJ. This experimental impact energy satisfies the coordination of the geometric scale (1:10). The similarity ratio of the energy is 1:1000. For the other parameters, the effective total length (L) is 900 mm. The impact point is near the support N and the distance is 200 mm (L_N) whereas the distance is 700 mm between the impact point and the support F (L_F).

In addition, some similar experiments were also applied to verify the one-step method. The details are summarized in

Table 1. Design parameters.

Source	No.	D×tb(mm)	Type	m(kg)	VI0(m/s)	E0(kJ)	ρl(kg/m)	LF(mm)	LN(mm)	fy(MPa)	fcu(MPa)	MP,s(kN.m)	Fs(kN)
	YG1	114 × 2.0	Fix	270	7.67	7.94	25.93	700	200	338	55	11.93	153.39
Tests	YG4	114 × 3.5	Fix	270	11.72	18.54	28.41	700	200	323	55	18.42	236.83
	YG5	114 × 3.5	Fix	270	9.90	13.23	28.41	700	200	306	55	18.42	236.83
	CC1	180 × 3.65	Fix	465	9.21	19.72	73.26	970	970	247	75.1	49.25	203.09
Han [20]	CC2	180 × 3.65	Fix	920	6.40	18.84	73.26	970	970	247	75.1	49.25	203.09
	CC3	180 × 3.65	Fix	465	9.67	21.74	73.26	970	970	247	75.1	49.25	203.09
	DBF14	114 × 1.7	Fix	230	4.40	2.23	28.25	600	600	232	48.7	7.51	50.07
Wang [19]	DZF22	114 × 3.5	Fix	230	7.60	6.64	31.57	600	600	298	48.7	16.43	109.53
	DZF26	114 × 3.5	Fix	230	11.72	15.8	31.57	600	600	232	48.7	16.43	109.53
	SS1	180 × 3.65	Pin	465	8.05	15.07	73.26	1200	1200	247	75.1	49.25	82.08
Han [20]	SS2	180 × 3.65	Pin	920	5.69	14.89	73.26	1200	1200	247	75.1	49.25	82.08
	SS3	180 × 3.65	Pin	465	8.93	18.54	73.26	1200	1200	247	75.1	49.25	82.08
	CCFP1-1	219.1 × 10	Pin	1350	7.83	41.38	97.02	900	900	399.7	60	17.62	391.56
Wang [24]	CCFP2-1	219.1 × 6.3	Pin	1350	7.59	38.89	81.96	900	900	395.4	60	12.21	271.33
	CCFP3-1	219.1 × 5.0	Pin	1350	7.19	34.89	76.53	900	900	419.8	60	10.08	224.00

Note: Type “Fix” is the fixed-fixed constraint. Type “Pin” means the pin-pin constraint.

.

3.2. Response Mechanisms

Figure 4 shows the dynamic response of the CFSTs after lateral impact, which was used to prove the design model in Figure 2. YG2 and YG3 [32] are further destructive tests based on YG1, where the impact velocity was increased from 7.67 to 9.90 and 11.72 m/s.

The most obvious phenomena are the vertical bending cracks in YG2 or YG3, that is, the flexural deformation is the main response mode of CFSTs during the impact process. Little shear damage was observed near the bending cracks even in the fracture situation and the shear failure has a lower influence than that of the bending type. Therefore, it is suitable to apply the mechanical model of beam bending in the design.

The plastic hinges in the calculation correspond to the plastic concentration area of CFSTs, i.e., the cracked area. In YG2 or YG3, the cracks of CFSTs only appear at the impact point and the two supports, and there is no obvious strain concentration phenomenon outside the cracked ranges. Correspondingly, the plastic hinges are also set in the above three positions for the design calculation.

The validation of the acceleration mode is carried out according to the displacement distribution of CFST between the impact point and each support (N/F). The acceleration is the second-order time derivative of the displacement along the impact direction. From YG2 and YG3, the deflection of the beams has a linear distribution between points I and F and I and N. The maximum displacement is located at the impact point (I) and the minimum one is at the supports (F and N). Thus, the acceleration distribution is also linear due to the derivative relations. The inertial force from Equation (4) is reasonable.

3.3. Impact Force and Dynamic Resistance

The impact force and dynamic resistance of CFSTs are the interaction forces at the impact contact surface, and these two parameters have the same value and the opposite direction. Figure 5 shows the impact contact force versus time curves from the experiments and the predictions, which exhibits the similarities and differences between the one-step design and the experimental tests. The predicting results are the dynamic resistance or impact force versus time curves used in the design. During the calculation, DIF in Equation (10) is estimated by Equation (12) [32]. The duration of the lateral impact (t_E) is obtained by Equation (13). The calculation results are detailed in

Table 2. Design results from the one-step method.

No.	C	P	V10/LN	DIF	$1+\frac{1}{3}\frac{{\mathit{\rho }}_{\mathit{l}}\mathit{L}}{\mathit{m}}$	aI	DIF·F0	FPre	FTE	$\frac{\mathit{D}\mathit{I}\mathit{F}\cdot {\mathit{F}}_{\mathit{s}}}{{\mathit{F}}_{\mathit{T}\mathit{E}}}$	$\frac{{\mathit{F}}_{\mathit{P}\mathit{r}\mathit{e}}}{{\mathit{F}}_{\mathit{T}\mathit{E}}}$	WPre	WTE	$\frac{{\mathit{W}}_{\mathit{P}\mathit{r}\mathit{e}}}{{\mathit{W}}_{\mathit{T}\mathit{E}}}$	$\frac{{\mathit{F}}_{\mathit{P}\mathit{r}\mathit{e}}\times {\mathit{W}}_{\mathit{P}\mathit{r}\mathit{e}}}{{\mathit{E}}_0}$	$\frac{\mathbf{\Delta }{\mathit{E}}_{\mathit{I}}}{{\mathit{E}}_0}$
	s−1		rad/s			m/s2	kN					mm
YG1	6844	3.91	38.350	1.266	1.029	694.91	194.2	187.62	189.6	102.42%	99.0%	42.33	32.3	131.0%	1.000	0.229
YG4	6844	3.91	58.600	1.296	1.032	1098.23	306.9	296.52	286.8	107.02%	103.4%	62.54	49.5	126.3%	1.000	0.234
YG5	6844	3.91	49.500	1.283	1.032	1087.64	303.9	293.66	289.1	105.10%	101.6%	45.06	33.3	135.3%	1.000	0.272
CC1	6844	3.91	9.495	1.186	1.102	470.03	240.9	218.56	234.6	102.67%	93.2%	90.23	63	143.2%	1.000	0.251
CC2	6844	3.91	6.598	1.169	1.051	245.49	237.4	225.85	240.4	98.76%	93.9%	83.43	65	128.3%	1.000	0.171
CC3	6844	3.91	9.969	1.188	1.102	470.95	241.3	218.99	200.5	120.33%	109.2%	99.28	70	141.8%	1.000	0.354
DBF17	6844	3.91	7.333	1.174	1.049	243.580	58.8	56.02	60.6	97.00%	92.4%	39.74	32.7	121.5%	1.000	0.110
DZF22	6844	3.91	12.667	1.200	1.055	541.753	131.4	124.60	112.8	116.52%	110.5%	53.31	39.42	135.2%	1.000	0.331
DZF26	6844	3.91	19.533	1.223	1.055	552.33	134.0	127.04	113.2	118.33%	112.2%	124.34	87.2	142.6%	1.000	0.375
SS1	6844	3.91	6.708	1.170	1.126	183.42	96.0	85.29	84.1	114.19%	101.4%	176.65	139	127.1%	1.000	0.224
SS2	6844	3.91	4.742	1.156	1.064	96.93	94.9	89.17	75.3	126.01%	118.4%	167.01	155	107.7%	1.000	0.216
SS3	6844	3.91	7.442	1.175	1.126	184.13	96.4	85.62	78.6	122.70%	108.9%	216.54	174	124.4%	1.000	0.262
CCFP1-1	40	5	8.700	1.737	1.043	482.99	680.1	652.03	710.2	95.77%	91.8%	63.47	48.8	130.1%	1.000	0.163
CCFP2-1	40	5	8.433	1.732	1.036	335.97	469.9	453.55	513.9	91.45%	88.3%	85.73	76.3	112.4%	1.000	0.000
CCFP3-1	40	5	7.989	1.725	1.034	276.74	386.4	373.60	408.8	94.52%	91.4%	93.40	82.1	113.8%	1.000	0.038
Mean Error								|1 − FPre/FTE|		17.88%	7.71%	|1 − WPre/WTE|		28.06%	Avg. value	21.54%
Std. Dev.										0.113	0.047			0.109		0.107

:

$$DIF=1+{\left(\frac{V_{I0}/L_N}{C}\right)}^{\frac{1}{P}}$$

(12)

$$t_E=\frac{V_{I0}}{a_b}$$

(13)

where C and P are the strain rate parameters of the steel tube and t_E is total duration of the lateral impact process.

According to Figure 5, the impact process has two prominent characteristics, which are the initial peak process and the approximate horizontal stable process. The initial peak process presents a local maximum value of the CFST resistance, but its duration time is relatively short. Oppositely, the resistance is more average in the stable process and its duration is much longer. According to the above two characteristics, the soft impact process can be divided into the initial peak stage (IPS) and the stable stage (SS), respectively. Figure 5o shows the typical IPS and SS, which are indicated by OAB and BC.

Figure 5 also displays the dynamic resistance of CFSTs obtained from the one-step design. Its prediction has good accuracy for the CFST resistances in SS, the average error is 7.71%, and the standard deviation is 0.047. The main error comes from the calculation of IPS as the one-step method ignores the description of the initial peak phenomenon. As a result, the predictive results universally underestimate the initial peak resistance of CFSTs and the average error is 64.43%. From Figure 5, the maximum error of the CFST resistances is 89.21% in IPS; see the case of SS3 (l).

The overall accuracy of the one-step design is related to the weights of IPS and SS. If the weight of IPS is very low, the overall accuracy is similar to that in SS, which tends to be a “good” evaluation. Instead, the overall accuracy gradually decreases with the increase in the IPS proportion. If IPS has a high ratio, the overall accuracy may be close to the “bad” results. Therefore, the weights of IPS and SS determine the applicability of the one-step method for the soft impact design. This needs to be clarified before the calculation.

3.4. Deflection Prediction

The error of resistance prediction (F₀) is eventually transferred to the deflection prediction through Equation (1). The maximum deflection (W_max) of CFSTs can be predicted by taking the results of Equation (10) into Equation (1). The calculation results are also listed in Table 2. For the selected experiments, the average error of W_max is 28.06% and the maximum error is 43.2%, exceeding the tolerance of the design requirements.

The reason for the large error has been mentioned before, which is the underestimation of the resistance in IPS. Equation (14) is established to explain the deflection errors. In Equation (14), the deflection results are divided into the IPS part and the SS part. ΔE_I is the energy dissipation of the colliding object during IPS and E₀ − ΔE_I is that in SS. Then, from Equation (1), ΔE_I/F is the deflection in IPS and (E₀ − ΔE_I)/F is the deflection in SS. The subscript “TE” is the test value and the subscript “Pre” is the predictive one:

$$W_{TE}=\frac{E_0-\Delta E_I}{F_{TE}}+\frac{\Delta E_I}{F_{TE,I}}<W_{\mathrm{Pr}e}=\frac{E_0-\Delta E_I}{F_{\mathrm{Pr}e}}+\frac{\Delta E_I}{F_{\mathrm{Pr}e}}$$

(14)

where F_TE is the dynamic plastic strength of the structure obtained from the test in the stable stage; F_TE,I is the dynamic plastic strength of the structure in the initial peak stage; W_TE is the experimental deflection of CFST at the impact point after the stable stage; F_Pre is the predictive value of the dynamic plastic strength; W_Pre is the predictive deflection of CFST at the impact point after the stable stage; and ΔE_I is the energy dissipation of the colliding object during the initial peak stage.

As F_Pre is correct in SS, (E₀ − ΔE_I)/F_TE is almost equal to (E₀ − ΔE_I)/F_Pre and the deflection prediction is reasonable in this stage. Yet, F_Pre is smaller than F_TE,I, which means ΔE_I/F_TE,I < ΔE_I /F_Pre, so the predictive deflection in IPS is higher than the test one. Finally, W_TE < W_Pre, which is Equation (14). The deviation degree of W_TE and W_Pre is related to ΔE_I/E₀. Let E₀ − ΔE_I ≈ F_TE × W_TE, ΔE_I/E₀ is approximately the statistics in Table 2. ΔE_I/E₀ has an average value of 21.54%, which leads the average |1 − W_Pre/W_TE| reaching 28.06%. When ΔE_I/E₀ is 37.5%, the deflection error is 42.6%.

The relationship between ΔE_I/E₀ and W_Pre/W_TE is summarized in Figure 6; Equation (15) is the mathematical description of the development trend. Here, ΔE_I/E₀ is the dissipation ratio of the impact energy in IPS, which reflects the weight and importance of IPS. The results of Equation (15) provide a reference for the deviation of W_Pre and W_TE. From Equation (15), ΔE_I/E₀ ≈ 0 corresponds to W_Pre/W_TE ≈ 1, which indicates IPS has little importance for design accuracy. Subsequently, with the increase in ΔE_I/E₀, W_Pre/W_TE moves gradually away from 1.0, and the design error increases with the higher importance of IPS.

To sum up, due to the lack of a calculation for IPS, the one-step method produces an inherent limitation and its rationality depends on the low importance of IPS (ΔE_I/E₀ ≈ 0). For more common cases of the soft impact design (ΔE_I/E₀ > 0), the influence of IPS may not be negligible. So, it is better to supplement the IPS analysis in the one-step design or, at least, estimate the design error, such as according to Equation (15):

$$\frac{W_{\mathrm{Pr}e}}{W_{TE}}=1+1.204\frac{\Delta E_I}{E_0}$$

(15)

$$\frac{\Delta E_I}{E_0}=1-\frac{F_{TE}\times W_{TE}}{E_0}$$

(16)

3.5. Error Analysis of the Initial Peak Stage

The current one-step method has low accuracy in IPS, which is explained by the impact contact process in Figure 7. Figure 7a to 7c contain the interaction process of the impactor and CFSTs. Figure 7d,e are the changing trend of the velocity and acceleration at the impact contact point. Here, the acceleration is a first-order time derivative of the velocity. The acceleration of the impactor relates to the impact force, and the acceleration of CFST relates to its dynamic resistance.

Figure 7a is the image recording before the impact contact. The impactor has an initial velocity (V_I = V_I0), but the CFST remains stationary (V_b = 0). Figure 7b is the situation of IPS, where the impactor begins to contact the CFST. As CFST starts to move from V_b = 0, V_b needs to undergo an ascending process to reach V_b = V_I, see (b’) in Figure 7d,e. Finally, Figure 7c shows SS, which always maintains V_b = V_I and continues to the end of the impact process.

During the above three processes, the acceleration of the impactor and CFST continuously changes. As shown in Figure 7e, a_I = a_b = 0 before the impact contact (a’) and there is no interaction between the impactor and CFST. a_I = a_b = 0 is the sufficient condition of V_b = 0 and V_I = V_I0. When the impact process enters IPS (b’), a_I < 0 and a_b > 0. Compared (d) and (e) of Figure 7, the acceleration of CFST follows the initial impact direction and that of the impactor is in the opposite direction, a_I ≠ a_b in IPS. a_b > 0 matches the rising process of V_b and a_I < 0 indicates the deceleration of V_I. Last, when a_I = a_b < 0 in SS, CFST and the impactor decelerate together (V_b = V_I). This phenomenon (a_I = a_b< 0 and V_b = V_I in SS) is the basis of the embedding assumption in Equation (9). Reversely, the embedding assumption is false in IPS due to a_I ≠ a_b, which is the reason for the calculation error in this stage.

To reduce the potential dangers from a_I = a_b, a_I ≠ a_b is needed during the IPS prediction. The necessity of a_I ≠ a_b is related to the importance of IPS, which is evaluated by Equation (15). Unless Equation (15) shows the low importance of IPS, the one-step method based on a_I = a_b may not be applicable for the IPS calculation and, then, the design of the whole impact process.

4. Two-Step Design Method

The safety and economy of engineering structures, to a great extent, depend on the accuracy of the design calculation. If the accuracy is suspicious, such as the one-step design for some soft impact processes, it may be difficult to realize the expected design intention.

One of the ways to improve the accuracy is to supplement the calculation of the initial peak stage (IPS) in the original one-step method. Then, the one-step design is converted to a method involving both IPS and SS. However, two difficulties hinder the realization of this demand: first, the determination of the design loads for IPS and SS, respectively, which need a quantitative division of the two stages. Here, the design loads are the impact energy dissipation in IPS and SS; second, the prediction of the CFST resistance in the IPS, which lacks a description of its inertia effect. The above two steps are interrelated and indispensable.

If the impact energy dissipation has an assumed proportion of 0% in IPS and 100% in SS, it can evade the above two difficulties and become the current one-step method. The one-step design is a special case of the subsequent two-step method.

The two-step method is used to solve the above two difficulties. The first step is qualitative analysis, which provides an approximate quantitative dividing standard for IPS and SS. This achieves the prediction of the energy dissipation in the two stages. The second step is the two-stage calculation, which contains the dynamic analysis of IPS and SS. The analysis of SS retains the original one-step method, but the design load comes from the qualitative analysis. The analysis of IPS supplements several additional formulas to estimate the inertial effect of CFST in this stage and the design load is also obtained from the qualitative analysis.

4.1. Qualitative Analysis

The purpose of the qualitative analysis is to distribute the initial impact energy (E₀) into IPS and SS. Then, Equation (1) can still be used for the impact design.

The energy allocation is based on Equation (17), which is the equation of the momentum balance, obtained from the time integration of Equation (6). Equation (18) is a simplification of Equation (17), which is derived by taking Equations (2), (8), and (9) into Equation (17):

$$m{V}_{I0}=m{V}_I+0.5{\rho }_{l}L{V}_b+{\displaystyle \int (Q_N+Q_F})dt$$

(17)

$$m{V}_{I0}=m{V}_I+\frac{1}{3}{\rho }_{l}L{V}_b+F_{0}t$$

(18)

where t is the duration of the initial peak stage and V_b is the velocity of CFST at the impact point.

The duration t is obtained from Equation (19), where a_b is near constant in IPS [32]:

$$V_b={\displaystyle \underset{0}{\overset{t}{\int }}{a}_b\left(t\right)dt}=a_{b}t$$

(19)

Equation (18) is solved according to V_I = V_b. Then, Equation (20) is the estimation of the velocity (V_I and V_b) at the end of IPS:

$$V_{I1}=V_b=\frac{V_{I0}}{1+\frac{{\rho }_{l}L}{3m}+\frac{F_0}{m{a}_b}}$$

(20)

where V_I1 is the initial velocity of the stable stage.

a_b in IPS is calculated by Equation (21), which is used to replace a_I = a_b and comes from Equation (7):

$$a_{b,I}=DIF\times \frac{12M_{P,s}}{{\rho }_l{L}_F{}^2}$$

(21)

where M_P,s is the static plastic bending moment of CFST, i.e., the limit value of M_F; and a_b,I is the representative value of the CFST acceleration in IPS.

In Equations (19)–(21), a_b is a constant, which comes from the research results about the soft impact process [32]. Equation (21) is an approximate estimation of a_b from Equation (6), which is based on the travelling hinge theory [27,28,29,30]. After taking Q_F = 0 and L_F into Equation (7), the minimal value of a_b can be calculated. Q_F = 0 is one mechanical characteristic of the travelling hinge [27,28,29,30], where L_F is its reference location. The approximation of Equation (21) is reflected in the loss of the influence of the impact velocity, which is related to the reference location of the travelling hinge. L_F is the maximum location of the travelling hinge, corresponding to the minimum value of a_b,I and the conservative inertia resistance. In addition, if necessary, the elastic behaviors of the material can also be contained in Equation (21) through the modification of M_P,s.

Equation (22) is the impact energy dissipation in IPS and E₀ − ΔE_I is that in SS:

$$\Delta E_I=\frac{1}{2}m{V}_{I0}{}^2-\frac{1}{2}m{V}_{I1}{}^2$$

(22)

where ΔE_I is the energy dissipation of the colliding object during the initial peak stage.

By introducing the results of Equation (22) into Equation (15), a preliminarily estimation of the deflection error due to IPS being ignored can be obtained. ΔE_I/E₀ also describes the proportion of IPS from the perspective of energy dissipation and 1 − ΔE_I/E₀ is the proportion of SS. Therefore, the qualitative analysis provides one standard for the approximate quantitative division of the soft impact process.

4.2. Resistance in the Initial Peak Stage

As important as the energy distribution, the resistance of CFST is another factor that influences the prediction accuracy of the deflection. From Figure 5 and Figure 7, the resistance of CFST is different in IPS and SS because a_I ≠ a_b in IPS and a_I = a_b in SS. The essence of this difference is from the feature of the inertia effect in the two stages. Figure 5 also shows the rationality of the inertia effect description from the one-step method, which is reasonable in SS but incorrect in IPS. Thus, a method needs to be supplemented to additionally estimate the inertia effect of IPS and the calculation of the inertia effect in SS can follow the previous one-step method.

The total resistance of CFST in IPS is still composed of the inertia effect of CFST and the total support reaction, F_0,d = F_IE + Q_d, see Equation (5). During IPS, F_0,d is predicted by Equation (23), where a_b,I is the minimal a_b from Equation (21). Here, the predictive F_0,d,I is regarded as the representative value of the CFST resistance in IPS and it is a constant:

$$F_{0,d,I}=F_{IE}+Q_d=\frac{1}{3}{\rho }_{l}L{a}_{b,I}+F_0$$

(23)

Substituting Equation (23) into Equation (1), Equation (24) is the displacement of the impactor at the end of IPS:

$$W_{I,I}=\frac{\Delta E_I}{F_{0,d,I}}$$

(24)

where W_I,I is the displacement of the impact object during the initial peak stage.

When CFST accelerates from V_b = 0 (static) to V_I1, its deflection is estimated to be half of the impactor:

$$W_{b,I}=0.5W_{I,I}$$

(25)

where W_b,I is the deflection of CFST during the initial peak stage.

Finally, Equation (26) is the limitation of the above IPS calculation:

$$W_{b,I}\ge L_F{\theta }_P$$

(26)

where θ_P is the rotation angle of the CFST section at the limit of elasticity, i.e., θ_P = 0.002.

If Equation (26) is not satisfied, CFST may fail to reach full plasticity in IPS and its material state is mainly or partially elastic. Then, the plastic assumption of both Equations (1) and (21) is doubtful. However, the elastic state also means that the maximum of W_b,I is smaller than L_Fθ_P in IPS. L_Fθ_P become a conservative result of W_b,I, and replaces Equation (25).

Comparing Equations (21) and (23) to Equations (9) and (10), a_b,I > 0 in Equation (21) and its direction is along the initial impact. Thus, the inertia effect of Equation (23) increases the resistance of CFST in IPS. In contrast, a_b < 0 in Equation (9) and the direction is opposite to the initial impact. So, the inertia effect of Equation (10) decreases the resistance of CFST in SS. Although the basic formulas are both in Equation (5), the different directions of a_b change the function of the inertia effect in IPS and SS.

4.3. Process of Two-Step Design

The original design included only the calculation for the stable stage (SS), which is the one-step method composed of one-stage calculation. After supplementing the qualitative analysis into the one-step method, the steps are increased to two, which are the qualitative analysis and the design calculation. At the same time, the initial peak stage (IPS) is also added to the design calculation and there are two stages (IPS and SS) in the second step of the two-step method. In summary, the original method adopts the one-step design and one-stage calculation while the new one is the two-step design and two-stage calculation.

Figure 8 shows the new process of soft impact design after optimization. It is divided into two parts: qualitative analysis and design calculations. The design calculation contains the stages of IPS and SS.

Here, the first step is the qualitative analysis while the second step is the calculation. The new method does not change the original design ideas and Equation (1) is still the balance equation for the design. The qualitative analysis realizes the distribution of the total energy dissipation into IPS and SS, which provides the subdivided design loads for the two stages. In the second step, the inertia effect is different in the IPS and SS calculations, which reflects the change in the CFST resistance during the soft impact process.

The qualitative analysis mainly depends on Equations (20) and (22), which are a dynamic analysis process that is independent of the next step (two-stage calculation). The analysis result (ΔE_I/E₀) is beneficial in estimating the proportion of IPS and SS and then revealing the mechanical characteristics of CFST during lateral impact. Equation (15) is used to predict the design error caused by ignoring IPS, which shows the necessity of the IPS calculation.

The two-stage calculation is performed after the qualitative analysis. Equation (1) is the basic balance equation used to calculate in both IPS and SS. The original impact load (E₀ = 0.5 mV_I0²) is subdivided into ΔE_I and E₀ − ΔE_I and then IPS and SS are independent during the calculation. The resistance and deflection of CFST are predicted based on IPS and SS, respectively. The total deflection is the sum of the two stages and the resistance in the two stages has different representative values.

4.4. Verification of the Two-Step Method

Table 3. Design results of the two-step method.

No.	θP LF	Wb,I	Wb,s	ΔEI/E0	ΔEITE/E0	IPS,E	WPre,m	WTE	WPre/WTE	WPre,m/WTE	PI	PI,TE	PPRE/PI,TE	PI/PI,TE
	mm						mm				kN
YG1	1.40	2.32	34.70	17.8%	22.90%	5.10%	37.02	32.30	131.0%	114.6%	303.8	410.90	45.7%	73.9%
YG4	1.40	3.76	50.33	19.5%	23.40%	3.90%	54.09	49.50	126.3%	109.3%	480.9	636.30	46.6%	75.6%
YG5	1.40	2.69	36.31	19.4%	27.20%	7.80%	39.00	33.30	135.3%	117.1%	476.3	565.30	51.9%	84.3%
CC1	1.94	3.56	67.20	25.9%	25.10%	0.80%	70.76	63.00	143.2%	112.3%	718.5	870.00	25.1%	82.6%
CC2	1.94	1.92	71.39	14.5%	17.10%	2.60%	73.33	65.00	128.3%	112.8%	/	666.30	(Wb,I < θPLF)
CC3	1.94	3.92	73.93	25.9%	35.40%	9.50%	77.85	70.00	141.8%	111.2%	719.8	901.90	24.3%	79.8%
DBF17	1.20	0.88	34.30	13.9%	11.00%	2.90%	35.50	32.70	121.5%	108.6%	/	175.50	(Wb,I < θPLF)
DZF22	1.20	1.31	45.21	15.5%	33.10%	17.60%	46.51	39.42	135.2%	118.0%	393.0	319.20	39.0%	123.1%
DZF26	1.20	3.07	105.36	15.6%	37.50%	21.90%	108.43	87.20	142.6%	124.3%	400.6	539.10	23.6%	74.3%
SS1	2.40	4.11	131.44	26.0%	22.40%	3.60%	135.55	139.00	127.1%	97.5%	477.2	713.20	12.0%	66.9%
SS2	2.40	2.30	142.90	14.6%	21.60%	7.00%	145.30	155.00	107.7%	93.7%	/	588.50	(Wb,I < θPLF)
SS3	2.40	5.03	160.67	26.1%	26.20%	0.10%	165.70	174.00	124.4%	95.2%	479.9	793.00	10.8%	60.5%
CCFP1-1	1.80	0.73	54.65	14.3%	16.30%	2.00%	56.45	48.80	130.1%	115.7%	/	1020.60	(Wb,I < θPLF)
CCFP2-1	1.80	0.85	75.51	12.3%	0.00%	12.30%	77.31	76.30	112.4%	101.3%	/	829.90
CCFP3-1	1.80	0.87	82.96	11.5%	3.80%	7.70%	84.76	82.10	113.8%	103.2%	/	678.50
Mean Error		IPS,E = |ΔEI−ΔEITE|/E0				6.99%	|1 − WPre/WTE|		28.06%	10.80%	|1 − FPre/FTE|		69.00%	25.02%
Std. Dev.						0.062			0.109	0.090			0.153	0.178

shows the main calculation results of the two-step method. These results exhibit its rationality in three aspects: qualitative analysis, deflection prediction, and resistance of the initial peak stage (IPS).

In Table 3, ΔEI/E₀ is the ratio of IPS from the qualitative analysis, ΔEI_TE/E₀ is that from the experiments; for the approximate degree of the two, see Figure 9. On the whole, the average value of ΔEI/E₀ is 18.19% and that of ΔEI_TE/E₀ is 21.53%, which is relatively close. The absolute error of ΔEI/E₀ and ΔEI_TE/E₀ can be further evaluated by |ΔEI − ΔEI_TE|/E₀. The average value is 6.99% and the standard deviation is 0.062. The maximum error of |ΔEI − ΔEI_TE|/E₀ occurs at DZF26, which reached 21.90%. The reason for this error is mainly the position of the travelling hinge mentioned before. L_F in the design is the upper limit value of the position, which is used to ensure a conservative result. Even though the qualitative analysis has a relatively high error, the final design results from the two-step method are still more accurate than the one-step case, as shown by the deflection results in Table 3 and Figure 10. This means the two-step method is generally a better choice for the soft impact design.

Figure 10 shows the maximum deflection of each CFST at the impact point during the lateral impact process. W_TE is the deflection from the experimental tests, W_Pre is the prediction from the one-step method, and W_Pre,m is the result of the two-step method. The details are also shown in Table 3. Based on the deflection results, the average error of the one-step method is 28.06% and that of the two-step method is 10.80%. An improvement in the calculation accuracy is obvious. For the several cases of W_Pre that have a relatively high error, the improvement is more obvious. One example is the CC1 case. Its deflection error is decreased from 43.2% to 12.3% after the adoption of the two-step method. For DZF26, although the qualitative analysis has an error of 21.90%, the deflection error can also be reduced from 42.6% to 24.3%. Therefore, the two-stage calculation is helpful in promoting the accuracy of the deflection prediction. Moreover, the two-step method increases the stability of the calculation and the standard deviation is reduced from 0.109 to 0.090.

Table 3 also summarizes the prediction of the impact force in IPS. The average error is 25.02% when using the two-step method. Instead, the average error is increased to 69.0% by the one-step method. Although the accuracy is improved by the two-step method, it can still be optimized. According to the discussion of IPS [32], the above error is mainly due to L_F and M_P in Equation (21), namely, the position and elastic-plastic behavior of the travelling hinge. Details on the influence of L_F and M_P can be found in papers about the travelling hinge [27,28,29,30,31,32]. Here, Equations (21) and (23) provide a reference value of a_b and guarantee its conservativeness.

In summary, the two-step method sets up a more reasonable framework for the soft impact design. Its advantage is the subdivision of the design process and the description of the inertia effect. Further optimizations of the two-step method are mainly aimed at the elastic behavior of CFST and the representative location of the travelling hinges.

5. Conclusions

This paper analyzed the difference between the soft impact process and the design method from Eurocode-1: Actions on structures (one-step method). Based on the comparison results, a modified technical route was proposed to improve the accuracy of the current design calculation. The main contents are as follows:

(1)

The extra parameters (mass and acceleration of CFST) are increased in the current one-step method to describe the inertia effect of CFST. The inertia effect influences the resistance of CFST directly in addition to the reaction forces of the support points.

(2)

Experiments were applied to obtain the features of the soft impact process. From the contact force curves, the soft impact process includes at least two stages, which are the initial peak stage (IPS) and the stable stage (SS). The current one-step method has high accuracy in SS and the average error is 7.71% according to the prediction results of the CFST resistances. The accuracy of this method is low in IPS and the average error is increased to 64.43%. The total design error of the one-step method is related to the proportion of IPS and SS. An empirical formula was summarized to evaluate its overall accuracy based on the weight of IPS.

(3)

A new framework for the soft impact design was set up to improve the inertia effect prediction of the one-step method. The technique route of the new method involves the qualitative analysis and the response calculation of IPS and SS. It is defined as the two-step design method.

(4)

The qualitative analysis provides a standard for the approximate quantitative division of IPS and SS. It can also be used to determine the design loads of the two stages from the perspective of the impact energy.

(5)

The response calculation in the two-step method supplements the additional formulas to estimate the inertial effect of CFST in IPS. The resistance calculation for SS follows the previous one-step method, but the design load comes from the qualitative analysis.

(6)

The two-step method also shows that the direction of the acceleration in CFST determines the function of its inertia effect in IPS and SS. Due to the change in the acceleration direction, the inertia effect increases the resistance of CFST in IPS and decreases it in SS.

Therefore, the two-step method, as an extension of the current one-step design, has better applicability and accuracy in the soft impact design of CFSTs. It could be extended further by the introduction of empirical coefficients in both the qualitative analysis and the response calculation. The recommended correction coefficients usually come from engineering practices or full-scale simulations. The coefficients in qualitative analysis may be associated with the features of the various colliding objects, such as trucks, trains, ships, etc. The coefficients in the response calculation could be linked to complex boundary conditions, other structural types, etc.