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.
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 7.0magnitude earthquake at the container terminal of the Kobe port in 1995 [
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 largescale equipment performance under the different seismic conditions. However, it is very difficult to manufacture a complete dynamic similar model of QCC in the abovescale 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 [
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 [
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 J248 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.
The prototype of J248 is a largescale 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
Structural model of the quayside container crane.
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 [
The general relationship for dynamic is given as follows:
In the process of QCC dynamic model design, time
Further, equation (
Using the similarity constant of density
Further, the equation (
In this investigation, the manufacture material of the model is consistent with prototype material Q345; thus, the similarity constant of elastic modulus
Similarity constants of the model.
Items of similarity constants  Symbol  Value 

Similarity constant of geometry 

1/20 
Similarity constant of density 

1 
Similarity constant of Young’s modulus 

1 
Similarity constant of time 


Similarity constant of frequency 


Similarity constant of mass 

1/8000 
Similarity constant of displacement 

1/20 
Similarity constant of velocity 


Similarity constant of acceleration 

1 
Similarity constant of strain 

1 
Similarity constant of stress 

1 
Similarity constant of gravity 

1 
Most of the main components of the QCC are box girders such as the landside leg 2. From Figure
Crosssectional shape of the leg. (a) Prototype. (b) Similarity model.
Crosssectional parameters of landside leg 2.
Item  Design model  





 
1  25  27  2  192  1.74 
2  27  19  2.2  183.04  1.71 
3  30  11  2.5  180  1.69 
4  30  10  2.8  192.6  1.71 
5  32  7  3  198  1.76 
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 [
Assuming that the QCC model is similar to the prototype, there are nine main physical quantities related to the structural dynamic model:
According to the
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
Similarly, the acceleration similarity constant
Hence, the equation (
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
Generally, the value of distortion coefficient can only be solved by experimental test and simulation [
The scale models for different stiffness distortion coefficients are established based on the scale of the full similar model in Table
Finite element model and the layout of measuring points of the QCC. (a) Simulation model. (b) Acceleration monitoring point.
As shown in Figure
Acceleration prediction coefficient results under different distortion coefficients. (a) EL Centro. (b) Taft. (c) Northridge. (d) Kobe.
Figure
Acceleration prediction coefficient results under different distortion coefficients with different acceleration peaks. (a) EL Centro. (b) Taft. (c) Northridge. (d) Kobe.
Acceleration prediction coefficient results under different distortion coefficients at monitoring points A9, A13, and A26. (a) EL Centro. (b) Taft. (c) Northridge. (d) Kobe.
From Figure
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
Test models. (a) M1. (b) M2. (c) M3.
Crosssectional parameters of distortion components.
Item  Crosssectional parameters of main components of the 1/20 scale model  Distortion coefficient (  





 
M1  Landside leg 2  21  21  2.5  185  1.075 
0.63 
Girder  34  20  2.6  253.8  1.489  
Sill beam  78  21  2.6  487.76  3.627  
Waterside leg 1  22  20  2.5  185  1.160  
Sheerleg beam  36  70  2.8  562.24  11.09  


M2  Landside leg 2  20  20  2.7  186.8  0.955 
0.57 
Girder  32  20  2.5  235  1.361  
Sill beam  78  20  2.6  482  3.233  
Waterside leg 1  21  20  2.6  186.16  1.057  
Sheerleg beam  45  64  2.7  559.44  10.04  


M3  Landside leg 2  20  20.5  2.7  189.54  1.016 
0.60 
Girder  32  20  2.6  243.36  1.409  
Sill beam  88  20  2.4  495.36  3.432  
Waterside leg 1  21  20  2.8  198.24  1.105  
Sheerleg beam  44  65  2.8  579.04  10.59 
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
Experiment cases.
Number  Seismic level (g)  Seismic wave 

F7EX  0.1  EL Centro 
F7TX  0.1  Taft 
F7NX  0.1  Kobe 
F8EX  0.2  EL Centro 
F8TX  0.2  Taft 
F8NX  0.2  Kobe 
R8EX  0.4  EL Centro 
R8TX  0.4  Taft 
R8NX  0.4  Kobe 
The simulation model, unit type, material property, and boundary constraint are the same as in Section
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
Acceleration versus time data of prototype and M1model at measurement point A13.
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
Acceleration amplification factor under the different seismic wave tests at different monitoring points. (a) 0.1 g. (b) 0.2 g. (c) 0.4 g.
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.
The data used to support the findings of this study are included within the article.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
This research was funded by the China National Science Foundation (Project no. 51275369).