#### Abstract

The scale model is an effective method to research the performance of quayside container crane (QCC) under the seismic condition, but the model distortion usually exists in the similar design process which leads to the incomplete similarity between the scale model and prototype. In this investigation, the distortion theory and the prediction coefficient correction method are used to upgrade the quality of 1/20 QCC scale model and, then, the seismic response of the QCC prototype is obtained from the shake table scale model test. In the first step, the similarity ratio of the 1/20 QCC scale model is calculated by the similitude law and the size of scale model is obtained from the similarity constants. In the second step, the bending stiffness is selected and determined as the distortion term and, then, the relationship between the distortion coefficient and the prediction coefficient is obtained by the finite element prediction coefficient method. Furthermore, the three different scale models are manufactured and tested in the shake table experiment under different seismic conditions. It is found that the experimental test results are consistent with the numerical simulation results of the QCC prototype. It can be concluded that the QCC scale model can be used to predict the performance of the prototype under the different seismic conditions after corrected by distortion theory, and the distortion theory is an effective method to solve the incomplete similarity between the scale model and prototype.

#### 1. Introduction

Quayside container crane (QCC) is an important loading and unloading equipment in the port, which has been usually damaged by the earthquake. For example, 52 numbers of QCC were destroyed by the 7.0-magnitude earthquake at the container terminal of the Kobe port in 1995 [1]. The 8.8-magnitude earthquake occurred in Kone Rolle, which led to collapse of some loading and unloading equipment in 2010. Thus, it is necessary to investigate the structure and the seismic performance of QCC through effective experimental and numerical methods.

In the past 20–30 years, the scale models of QCC were manufactured from 1 : 15 to 1 : 50 for the different carrying capacities of the shake table, which were fully used to investigate the large-scale equipment performance under the different seismic conditions. However, it is very difficult to manufacture a complete dynamic similar model of QCC in the above-scale ratio range as the geometric size of each component in QCC cannot be magnified or reduced in equal proportions. Although the method of adding mass [2, 3], amplifying input acceleration peak [4] and adjusting the section moment of inertia [5, 6], was used to satisfy the similarity design, these methods have their limitations; for example, the method of adding mass has a negative influence on the QCC stability of overall structure.

At present, distortion theory has been used to solve the problem of incomplete similarity between the model and prototype in the dynamic investigation process of mechanical manufacture [7, 8] and architecture design [9]; in addition, the design process and confirmation method of the scale model were also analyzed. Wang et al. [10] confirmed the dynamic similarity relationship between the scale model and prototype in the design process of the circular plate; the model of thin wall part was designed to investigate the dynamic characteristics of the prototype [11] to solve the distortion problem of the thickness of the beam that cannot be adjusted by the same scale as the width in the QSCC scale model design process; the numerical correction method was presented to compensate the gap of experiment and simulation [12, 13]. However, the research results are mostly focused on the theoretical and numerical analysis, and there is lack of a corresponding physical model test which leads to the low reliability of the conclusion.

In this investigation, the scale model is obtained to carry out seismic tests of QCC and the finite element prediction coefficient method is used to correct the test results. The bending stiffness is selected and determined as the distortion term by calculation of similar relation and analysis of distortion reasons of QCC, and then, the relationship between the distortion coefficient and the prediction coefficient is obtained by the finite element prediction coefficient method. Moreover, three 1/20 scaled structural models of QCC are designed by the dynamic similarity theory and scale model design method with the object of J-248 QCC. Then, shake table tests and simulation analysis are carried out to prove the feasibility of the scale model and the finite element prediction coefficient method.

#### 2. Similar Model Design

##### 2.1. Subject of Investigation

The prototype of J-248 is a large-scale QCC with a rail distance of 3.5 × 10^{4} mm, a base distance of 2 × 10^{4} mm, and the total weight of 992*t*. For details, the maximum horizontal distance of the front girder is 1.4 × 10^{5} mm and the maximum vertical height is 8 × 10^{4} mm. Its main structure is composed of legs, columns, beams, braces, beams, and rods, as shown in Figure 1. Obviously, it is difficult to directly investigate the QCC performance with the actual dimensions. In order to quickly and efficiently obtain the required performance of the QCC under the seismic condition, the experimental model should be designed by an appropriate method.

##### 2.2. Theory of Dynamic Similarity

At present, there is no appropriate shake table platform to test the QCC performance under the different seismic conditions and it is difficult to arrange sensors in the testing process. The similar theory has been widely used in the mechanical engineering and industrial test [14]; thus, it is imaginable that the characteristic of QCC can be effectively obtained using the scale model in the shake table testing process. In this investigation, the dynamic similar method is used to design the scale model of QCC. Generally, the scale model is designed with a certain similarity relation based on the prototype; thus, most of the similarity conditions between the scale model and the prototype should be satisfied such as the geometric similarity, material similarity, load similarity, mass similarity, and time similarity. It should be noted that not only the length size and force but also the time and the inertial force need to be considered in the structure dynamic similarity model design process.

The general relationship for dynamic is given as follows:where *m*, *c*, and *k* are the mass, damping coefficient, and stiffness coefficient of the system, respectively, *a* is the acceleration, is the gravity acceleration, is the velocity, and *u* is the displacement.

In the process of QCC dynamic model design, time *t* is the basic physical quantity. The performance of model and prototype should also be same, and then, the mathematical and physical equations of QCC dynamic phenomena (equation (1)) should be similar. Equation (1) shows that the inertial force is the main load acting on the structure, so it is necessary to consider the similarity of the density of material between the model and the prototype structure, and the related physical quantities are mass *m*, geometry *l*, gravity acceleration , density *p*, and acceleration *a*. At the same time, damping force and restoring force are important indicators and the related physical quantities are damping coefficient *c*, stiffness coefficient *k*, displacement *u*, and velocity . This article mainly focuses on the stress state and dynamic characteristics of QCC at the elastic stage; thus, the material properties should be satisfied for the basic assumption of elastic theory. The relationship between material stress *σ* and material strain *ε* is linear which can be approximately expressed by Hooke’s law *σ* = *Eε*. The physical quantities elastic modulus *E* and stress *σ* are selected. Thus, the similar physical quantities can be selected as mass *m*, elastic modulus *E*, geometry *l*, density *ρ*, time *t*, stress *σ*, damping coefficient *c*, stiffness coefficient *k*, displacement *u*, velocity , acceleration *a*, and gravity acceleration in the vibration experiment. The similar constant between the prototype and model is defined as *S*; for example, *S*_{l} means the similarity constant of geometry, *S*_{l} = *l*_{m}/*l*, where subscript *m* is used to represent the QCC model.

Further, equation (1) can be written as follows:

Using the similarity constant of density *S*_{ρ}, the similarity constant of acceleration *S*_{a}, the similarity constant of elastic modulus *S*_{E}, and the similarity constant of geometry *S*_{l}, equation (2) can be expressed as follows [2]:

Further, the equation (3) can be simplified as follows:

In this investigation, the manufacture material of the model is consistent with prototype material Q345; thus, the similarity constant of elastic modulus *S*_{E} and stress *S*_{σ} can be confirmed as 1. According to the scale of the shake table platform and the parameters of the J-248 prototype, the similarity constant of length *S*_{l} can be defined as 1 : 20. The QCC is a long cantilever structure, and the gravity effect cannot be ignored in the seismic test; thus, the similarity constant of gravity is set as 1. In order to simplify the shake table test, the similarity constant of acceleration *S*_{a} for QCC is set as 1. The other similarity constant can be deduced as shown in Table 1.

##### 2.3. Size Design of the Similar Model

Most of the main components of the QCC are box girders such as the landside leg 2. From Figure 2(a), the section of the leg with the length of the *L* 2210 mm, the length of the *H* 1290 mm, the length of the *t*_{1} 12 mm, and the length of the *t*_{2} 10 mm. According to the similarity design principle, the similarity model of leg should be designed with the length of the *L*_{m} 110.5 mm, the length of the *H*_{m} 64.5 mm, the length of the *t*_{1m} 0.6 mm, and the length of the *t*_{2m} 0.5 mm, as shown in Figure 2(b). The main response form of QCC is the bending vibration of the leg around *z*-axis under seismic condition; thus, the content of Section 2.2 can be referred to design the cross section size of the leg. The sectional dimensions of landside leg 2 and the *z*-axis moment of inertia are 186.7 mm^{2} and 3.42*e*5 mm^{4}, respectively. However, it should be noted that the size of the similarity leg is difficult to achieve in practice. Therefore, the 5 groups of cross-sectional parameters of leg are designed based on the principle of same mass and bending stiffness, as shown in Table 2.

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#### 3. Correction of Similar Model

##### 3.1. Distortion Theory

Beams of QCC (steel) are built up by arc welding. In order to ensure welding quality, the dimension of steel plate is larger than 20 mm × 20 mm × 2.5 mm. Therefore, the parameters of the leg should be adjusted based on the constraint condition.

It is well known that the similarity theory points out that the performance of the similar model must satisfy all the design conditions and, then, the similar model can be called a complete similarity model; otherwise, it should be called distorted model [15, 16]. Thus, two auxiliary coefficients are introduced: prediction coefficient *δ* and distortion coefficient *β* to correct the distorted model.

Assuming that the QCC model is similar to the prototype, there are nine main physical quantities related to the structural dynamic model: *m*, *k*, *c*, *l*, , *a*, *t*, *E*, and *p*; here, subscript *m* is used to represent the QCC model. These physical quantities can be expressed as functional form:

According to the *π* theorem, equation (5) can be written as *π* functional equation [17]:where *π* can be described aswhere superscripts *a*_{i} and *b*_{i} are power exponent.

If the model is similar to the prototype, the following equation must be satisfied:

Taking the stiffness of QCC as the example, the stiffness similarity constant *S*_{k} = *k*_{m}/*k* in the similar conditions. Assuming that the stiffness of the model is distorted as , the expression of distortion coefficient *β* can be deduced as follows:

Similarly, the acceleration similarity constant *S*_{a} = *a*_{m}/*a*. Assuming that the acceleration of the model is distorted as , thus the expression of acceleration coefficient *δ* can be deduced as follows:

Hence, the equation (7) can be written as

In order to make the acceleration of the model similar to the prototype after stiffness distortion, the similarity relationship should satisfy the following requirement:

Thus, it can be seen that the prediction coefficient can be expressed by the distortion coefficient, where *r* = −*b*_{2}/*b*_{6} is an unknown number which can be obtained by the experimental test or simulation.

Generally, the value of distortion coefficient can only be solved by experimental test and simulation [18]. In this investigation, in order to predict the relationship between the distortion coefficient and the prediction coefficient by the finite element method, the bending stiffness of legs around *z*-axis is set as the distortion term.

##### 3.2. Relationship between the Distortion Coefficient and Prediction Coefficient

The scale models for different stiffness distortion coefficients are established based on the scale of the full similar model in Table 2. From Figure 3, the simulation model and the monitoring points are shown. In finite element simulation software (Abaqus), the Beam 23 element is selected to simulate bending beams such as landside legs; the material is Q345 with the yield limit of 345 MP, modulus of elasticity 2.06 × 10^{11} N/m^{2}, Poisson’s ratio of 0.3, and density of 7850 kg/m^{3}. The wheel rail contact model is used to define the simulation of boundary condition. The different seismic waves of the EL Centro, the Taft, the Northridge, and the Kobe are adopted and considered in the ABAQUS software.

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As shown in Figure 4, the relationship between the different distortion coefficients and the acceleration prediction coefficient is obtained by extracting the corresponding acceleration of each monitoring point. It is clear that the no linear relationship exists between the acceleration prediction coefficient and the distortion coefficient.

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Figure 5 shows the relationship between the prediction coefficient of acceleration and the different distortion coefficients with different acceleration peaks (0.1 g, 0.2 g, and 0.4 g). These results show that the acceleration peak has a slight influence on the relationship between the prediction coefficient of acceleration and the different distortion coefficients. Figure 6 shows the relationship between acceleration prediction coefficient and the different distortion coefficients at monitoring points A9, A13, and A26. These results show that the relationship results of monitoring points A9, A13, and A26 are very similar.

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From Figure 6, the trend of the relationship between the prediction coefficient of acceleration and the distortion coefficient is consistent with the results of Figure 5, which evidences that the prediction coefficient of acceleration is only related to the distortion coefficient, but not to the acceleration peaks and the position of monitoring point. Specially, when the distortion coefficient is changed from 0 to 1, the prediction coefficient of acceleration has remarkably decreased under the seismic wave condition of EL Centro, but the prediction coefficient of acceleration increased first and then decreased under the three other seismic wave conditions. When the distortion coefficient is changed from 1 to 5, the acceleration prediction coefficient shows a steady upward trend under the seismic wave condition of EL Centro, but the acceleration prediction coefficient shows a slightly change.

#### 4. Experiment and Simulation Validation

##### 4.1. Experimental Model

In order to verify the correctness of the similar model which is corrected by the distortion theory, the three sets of QCC models with different magnification distortions are fabricated, which are represented by M1 (red), M2 (blue), and M3 (yellow), respectively, as shown in Figure 7. The section sizes of the main beams in the three models are shown in Table 3.

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##### 4.2. Experimental Condition

The most commonly seismic waves of EL Centro, Taft, and Kobe are selected as the experiment input condition, and the peak acceleration of these seismic waves are adjusted according to the different fortification intensities. Each seismic wave is set in three different conditions: 0.1 g (level 7), 0.2 g (level 8), and 0.4 g (level 8 rare), respectively. The detailed experiment input conditions are shown in Table 4.

##### 4.3. Simulation Model and Condition

The simulation model, unit type, material property, and boundary constraint are the same as in Section 3.2. In addition, the seismic wave is inputted as the same as in Table 4, and the length of the simulation step is set as 0.02 seconds.

##### 4.4. Results and Discussion

The acceleration versus time data is a very important index which is usually used to describe the performance of QCC under the seismic wave test. Figure 8 shows the results of acceleration versus time data of prototype and M1 similar model at measuring point A13. It is clear that the response acceleration of M1 is consistent with the result of prototype simulation, which evidences that the distortion theory is an effective approach to improve the quality of the similar model, and it can be used to predict the dynamic response of QCC after corrected by distortion theory.

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There are so many monitoring point data which lead to large document; thus, the results of monitoring points A5, A9, A13, A17, A26, and A2 are selected. Figure 9 shows the acceleration amplification factor under different seismic wave tests at the different monitoring points. The acceleration amplification factor is so sensitive to the different seismic wave tests, and it can be seen that the results of prototype are consistent with those of the similar model. The maximum error between the experiment results and simulation values at the monitoring point of A26 is only 8.52% under the condition of 0.4 g EL Centro; this is due to the difference of sensor arrangement at the monitoring point and the manufacturing quality of the similar model. But, in general, it can be concluded that the similar model of QCC can be used to predict the performance under the condition of seismic after corrected by distortion theory.

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#### 5. Conclusion

In this investigation, it can be concluded that the scale model can be used to investigate the QCC performance which is corrected by distortion theory. Further, the distortion theory is an effective method to solve the incomplete similarity between the scale model and prototype. Depending on the specified reasons, it can be said that using the simple and effective scale model causes realistic results and prevents financial consuming.

With the help of the distortion theory, taking structural seismic tests for QCC will be simpler. The scale model can also be used as a verification model in numerical simulation studies so that it will be available for researchers in numerical model studies to improve existing models. This article can lead to more developed model test studies in the future.

#### Data Availability

The data used to support the findings of this study are included within the article.

#### Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

#### Acknowledgments

This research was funded by the China National Science Foundation (Project no. 51275369).