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:
where
F0 is the plastic strength of the structure, i.e., the limit value of the static force (
Fs);
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
VI0 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 (
MP):
where
MF is the bending moment value of the CFST section in support
F;
MN is the bending moment value of the CFST section in support
N;
MI is the bending moment value of the CFST section in impact point
I;
MP is the plastic limit value of the bending moment in the CFST section;
LF is the distance between the impact point and support
F;
LN 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].
Ms is the static bending moment of the CFST section and
DIF is its dynamic increase factor. For the different CFST sections (
I,
N,
F),
MI,s,
MN,s,
MF,s, and
MP,s are the static parameters of the bending moments.
M, including
MI,
MN,
MF, and
MP, are the dynamic parameters. These parameters are in accordance with the relationship of Equation (3), where the additional subscript “
s” indicates the static values:
where
DIF is the dynamic increase factor for the bending moment of the CFST section and
Ms 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 (
ab), 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).
F0,d is equivalent to
F0, as it combines the material resistance and the inertia effect;
F0 contains only part of the material. Additionally,
F0,d is equal to the impact action (−
maI) according to the contact relationship:
where
FIE is the inertia force of the CFST column;
ab is the acceleration value of the CFST column at the impact point; and
ρl is the mass of the CFST per unit length:
where
aI is the acceleration value of the colliding object;
m is the mass of the colliding object; and
Qd 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 (
Qd = QF + QN):
where
QN is the shear force of CFSTs at the support
N and
QF 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 F0,d through ρl ab L/2 (FIE) but also the reaction force of the supports by ρl ab LF/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 F0,d in Equation (5).
The calculation of
F0,d is based on Equations (6)–(8), where
ab is the control parameter of the inertia effect. Here, an additional condition is needed to determine the value of
ab. Following Equation (1), which is the embedding model,
aI =
ab may become the additional condition. Taking
aI =
ab into Equation (6), Equation (9) is used to obtain the required
ab value. Then,
F0,d can be obtained according to Equation (10):
In Equation (10), F0 is the dynamic material resistance of CFSTs and is the multiplication of DIF and Fs. DIF is the dynamic increase factor for the bending moment of the CFST section. Fs is the static part of the CFST resistance. Comparing Equations (10) and (2), 1 + ρl 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 + ρl 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 (
LN) whereas the distance is 700 mm between the impact point and the support
F (
LF).
In addition, some similar experiments were also applied to verify the one-step method. The details are summarized in
Table 1.
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 (
tE) is obtained by Equation (13). The calculation results are detailed in
Table 2:
where
C and
P are the strain rate parameters of the steel tube and
tE 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 (
Wmax) 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
Wmax 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. Δ
EI is the energy dissipation of the colliding object during IPS and
E0 − Δ
EI is that in SS. Then, from Equation (1), Δ
EI/F is the deflection in IPS and (
E0 − Δ
EI)
/F is the deflection in SS. The subscript “
TE” is the test value and the subscript “
Pre” is the predictive one:
where
FTE is the dynamic plastic strength of the structure obtained from the test in the stable stage;
FTE,I is the dynamic plastic strength of the structure in the initial peak stage;
WTE is the experimental deflection of CFST at the impact point after the stable stage;
FPre is the predictive value of the dynamic plastic strength;
WPre is the predictive deflection of CFST at the impact point after the stable stage; and Δ
EI is the energy dissipation of the colliding object during the initial peak stage.
As
FPre is correct in SS, (
E0 − Δ
EI)
/FTE is almost equal to (
E0 − Δ
EI)
/FPre and the deflection prediction is reasonable in this stage. Yet,
FPre is smaller than
FTE,I, which means Δ
EI/FTE,I < Δ
EI /FPre, so the predictive deflection in IPS is higher than the test one. Finally,
WTE < WPre, which is Equation (14). The deviation degree of
WTE and
WPre is related to Δ
EI/E0. Let
E0 − Δ
EI ≈
FTE ×
WTE, Δ
EI/E0 is approximately the statistics in
Table 2. Δ
EI/E0 has an average value of 21.54%, which leads the average
|1 − WPre/WTE| reaching 28.06%. When Δ
EI/E0 is 37.5%, the deflection error is 42.6%.
The relationship between Δ
EI/E0 and
WPre/WTE is summarized in
Figure 6; Equation (15) is the mathematical description of the development trend. Here, Δ
EI/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
WPre and
WTE. From Equation (15), Δ
EI/E0 ≈ 0 corresponds to
WPre/WTE ≈ 1, which indicates IPS has little importance for design accuracy. Subsequently, with the increase in Δ
EI/E0,
WPre/WTE 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 (Δ
EI/E0 ≈ 0). For more common cases of the soft impact design (Δ
EI/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):
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 (
VI = VI0), but the CFST remains stationary (
Vb = 0).
Figure 7b is the situation of IPS, where the impactor begins to contact the CFST. As CFST starts to move from
Vb = 0,
Vb needs to undergo an ascending process to reach
Vb = VI, see (
b’) in
Figure 7d,e. Finally,
Figure 7c shows SS, which always maintains
Vb = VI 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,
aI = ab = 0 before the impact contact (
a’) and there is no interaction between the impactor and CFST.
aI = ab = 0 is the sufficient condition of
Vb = 0 and
VI = VI0. When the impact process enters IPS (
b’),
aI < 0 and
ab > 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,
aI ≠ ab in IPS.
ab > 0 matches the rising process of
Vb and
aI < 0 indicates the deceleration of
VI. Last, when
aI = ab < 0 in SS, CFST and the impactor decelerate together (
Vb = VI). This phenomenon (
aI = ab< 0 and
Vb = VI in SS) is the basis of the embedding assumption in Equation (9). Reversely, the embedding assumption is false in IPS due to
aI ≠ ab, which is the reason for the calculation error in this stage.
To reduce the potential dangers from aI = ab, aI ≠ ab is needed during the IPS prediction. The necessity of aI ≠ ab 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 aI = ab 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):
where
t is the duration of the initial peak stage and
Vb is the velocity of CFST at the impact point.
The duration
t is obtained from Equation (19), where
ab is near constant in IPS [
32]:
Equation (18) is solved according to
VI = Vb. Then, Equation (20) is the estimation of the velocity (
VI and
Vb) at the end of IPS:
where
VI1 is the initial velocity of the stable stage.
ab in IPS is calculated by Equation (21), which is used to replace
aI =
ab and comes from Equation (7):
where
MP,s is the static plastic bending moment of CFST, i.e., the limit value of
MF; and
ab,I is the representative value of the CFST acceleration in IPS.
In Equations (19)–(21),
ab is a constant, which comes from the research results about the soft impact process [
32]. Equation (21) is an approximate estimation of
ab from Equation (6), which is based on the travelling hinge theory [
27,
28,
29,
30]. After taking
QF = 0 and
LF into Equation (7), the minimal value of
ab can be calculated.
QF = 0 is one mechanical characteristic of the travelling hinge [
27,
28,
29,
30], where
LF 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.
LF is the maximum location of the travelling hinge, corresponding to the minimum value of
ab,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
MP,s.
Equation (22) is the impact energy dissipation in IPS and
E0 − Δ
EI is that in SS:
where Δ
EI 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. ΔEI/E0 also describes the proportion of IPS from the perspective of energy dissipation and 1 − ΔEI/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
aI ≠
ab in IPS and
aI =
ab 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,
F0,d =
FIE +
Qd, see Equation (5). During IPS,
F0,d is predicted by Equation (23), where
ab,I is the minimal
ab from Equation (21). Here, the predictive
F0,d,I is regarded as the representative value of the CFST resistance in IPS and it is a constant:
Substituting Equation (23) into Equation (1), Equation (24) is the displacement of the impactor at the end of IPS:
where
WI,I is the displacement of the impact object during the initial peak stage.
When CFST accelerates from
Vb = 0 (static) to
VI1, its deflection is estimated to be half of the impactor:
where
Wb,I is the deflection of CFST during the initial peak stage.
Finally, Equation (26) is the limitation of the above IPS calculation:
where
θP is the rotation angle of the CFST section at the limit of elasticity, i.e.,
θP = 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 Wb,I is smaller than LFθP in IPS. LFθP become a conservative result of Wb,I, and replaces Equation (25).
Comparing Equations (21) and (23) to Equations (9) and (10), ab,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, ab < 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 ab 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 (ΔEI/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 mVI02) is subdivided into ΔEI and E0 − ΔEI 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 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, Δ
EI/E0 is the ratio of IPS from the qualitative analysis, Δ
EITE/E0 is that from the experiments; for the approximate degree of the two, see
Figure 9. On the whole, the average value of Δ
EI/E0 is 18.19% and that of Δ
EITE/E0 is 21.53%, which is relatively close. The absolute error of Δ
EI/E0 and Δ
EITE/E0 can be further evaluated by
|Δ
EI − Δ
EITE|/E0. The average value is 6.99% and the standard deviation is 0.062. The maximum error of
|Δ
EI − Δ
EITE|/E0 occurs at DZF26, which reached 21.90%. The reason for this error is mainly the position of the travelling hinge mentioned before.
LF 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.
WTE is the deflection from the experimental tests,
WPre is the prediction from the one-step method, and
WPre,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
WPre 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
LF and
MP in Equation (21), namely, the position and elastic-plastic behavior of the travelling hinge. Details on the influence of
LF and
MP 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
ab 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.