Container Quay Crane Structural Response under Trolley Traveling

Structural response of quay crane should take more attention with the growth of worldwide logistics demand and quay crane size. is paper analyzes the responsemechanism of the metallic structure of the quay crane induced by trolley traveling. An on-site test has been conducted. e time domain of crane girder acceleration signal shows that the main inuence of the trolley traveling on crane structure is the high-frequency impact passing through the hinge point. e spectrum shows the main frequency of structure vertical vibration and impact vibration. Single girder model and theoretical derivation reveal that the vertical displacement response is quasistatic. A rened numerical model of the whole crane structure is established. e interaction between the trolley and the girder is realized by S-M contact method. e calculated results are basically consistent with the measured signal. e acceleration result indicates the same information as on-site test such that the structural acceleration response of inservice crane is aected by trolley in high-frequency band. e displacement results in three directions of 10 points on the girder show that the trolley traveling has signicant eect on the vertical displacement. e lateral displacement shows the eccentric loading condition of girder. e displacement spectrum indicates that the trolley has little dynamic eect on displacement response of crane structure. e response can be treated as quasistatic.


Introduction
With the continuous growth of international trade, the logistics industry is developing rapidly. e quayside container crane, referred to as the quay crane, as the frontline logistics equipment of the port, is an important part of maintaining the container throughput of the port. With the growth of port logistics throughput, the quay crane structure continues to increase in size. Trolley with hoisting load is the only moving load in crane structure. e Port Equipment Manufacturers Association pointed out that the alternating stress on the bridge crane structure mainly comes from the high-frequency moving load in the service state. e e ect of trolley traveling on safety of quay crane structure must be studied [1].
Studies at home and abroad have been done on beam structure response caused by car running. Huang Guoping [2] analyzed the longitudinal dynamic response of the bridge under the tra c ow based on the Euler beam theory and found that the bridge displacement caused by the tra c ow would cause fatigue wear on the bridge parts. Chen et al. [3] used beam elements to simulate bridges, dealt with the wheel-rail contact relationship with the close-tting hypothesis and creep theory, and established a train-bridgevehicle coupling model to achieve the coupled vibration of the vehicle and the bridge. Multiworking condition calculations are carried out for vehicle velocity, route, track spectrum, damping ratio, and other factors.
Based on the plate and shell theory, Prakash and Barik [4] analyzed the dynamic response of the bridge deck when the vehicle moved at a constant velocity and obtained the center de ection of the bridge deck before and after the reinforcement with sti ening and found that the I-shaped reinforcement contributed more to the strength of the bridge deck (Cheng et al. [5]). e three models of moving load, moving mass, and spring damping mass were compared when analyzing the vehicle-bridge system, and it was believed that the moving load model was the most stable. e improved moving load method is used to numerically model the vehicle-bridge system and the response of the bridge midspan under different mass and spring stiffness is analyzed. e simplified model has better calculation results except for the first resonance velocity.
At home and abroad, the research on the dynamic response of the trolley beam system mainly takes the bridge structure and high-velocity railway as the research object, and the high-velocity railway is the majority. Few take port crane trolley and the girder as the research object, and most of them take the single beam structure as the research object, lacking the structural dynamics analysis of the whole machine.
At present, there are three main research methods for the car beam system [6,7]. e car beam system is simplified to the dynamic equation of the loaded beam [8][9][10][11]. Modeling and analysis of the actual car beam 3D model are based on multibody dynamics [6,[12][13][14][15] and transient dynamic analysis of vehicle beam using finite element modeling [16][17][18]. e analysis methods based on dynamic equations are mostly used to analyze the dynamic characteristics of vehicles and simple track structures such as high-velocity trains, but they have certain limitations in the noncentral load analysis of complex beam structures [19][20][21]. Multibody dynamics is suitable for dynamic analysis of complex structures, and it can establish refined models of beams and trolleys and realize vehicle-rail coupling. Based on the transient dynamic analysis of finite element vehicle-beam coupling, complex metallic structures and vehicle-beam coupling models can be established after simplification. e main methods of vehicle-beam coupling model based on finite element theory include coupling method, birth-death element method, and contact method. All three simplify the vehicle load into a mass element. e difference is that the coupling method couples the node where the mass element is located with the node on the beam element and couples the mass element with the next node of the beam element at each time step to realize vehicle movement. e birth-and-death element method arranges a mass element on each node of the beam element to simulate the vehicle running on the beam in the form of a birth-and-death element; the contact method establishes a contact element between the mass element and the beam element to realize their interaction [22,23]. Due to the refined modeling of the finite element method being relatively complex, the beam elements are used mostly as the main analysis tool for vehicle-beam coupling modeling domestically and abroad. e above three finite element analysis methods of vehicle beam are limited to the coupling modeling between mass element and beam element. e research object of this paper is the quay crane, which is a kind of port logistics equipment with a relatively large structure. e girder structure of the large quay crane is a thin-walled box girder shaped like an irregular trapezoid. e front and rear girders are connected by hinge points, and there is a gap between the trolley tracks at the hinge points. e trolley runs on the track beam in side of the girder, so the girder is under no-central force during the trolley running. e complexity of the structure at the girders and hinge points of the quay crane should be considered when analyzing the interaction between the vehicle and the girder. According to the particularity and load characteristics of the quay crane structure, the finite element plate and shell element are used in this paper to establish the refined structure model of the quay crane, and the mass-shell contact method is used for the first time to realize the modeling analysis of the trolley running on the refined girder. e trolley is simplified as mass elements, the girder structure is refined using shell elements, and the interaction between them is simulated by contact elements.
It should be pointed out that because of the chain rule, the amplitude will be reduced in acceleration of the frequency below 1/(2π) Hz. While the amplitude of acceleration of the frequency above 1/(2π) Hz will be enlarged, this makes the signal above 1/(2π) Hz more significant in acceleration and the signal below 1/(2π) Hz more significant in displacement. e content of this paper is arranged as follows. Firstly, on-site test data were used to study the acceleration signal characteristics of the quay crane under normal working conditions. Secondly, based on the Euler beam theory, the single girder model with moving load is established. e displacement result under different working conditions is analyzed. Dynamic effect of trolley is derived. en, a refined model of the quay crane was established. e S-M contact method is used to realize the interaction of the trolley mass and girder flange. e acceleration response is used to analyze the high-frequency component induced by trolley traveling. e displacement responses of ten measuring points on the girder of the quay crane model are analyzed. Time domain signal is used for displacement characteristic analysis. Frequency domain signal is used for low-frequency component analysis and proves the theoretical derivation of quasistatic status of response. e main influence of the trolley on the displacement of the quay crane is summarized.

Metallic Structure of Quay Crane
As shown in Figure 1, the main structure of the quay crane is the water and land side gantry. e water and land side girder are suspended from the gantry. e tension rod connects the ladder frame on the top of the gantry and the front and rear girder to form a balanced structure. e struts maintain stability of the sea side and land side structures. When loading and unloading goods on the quay crane, the trolley runs along the track on the girder, and the container spreader is lowered to the ship deck or the ground for loading and unloading through the pulley and wire rope winding system on the trolley. e metallic structure of the quay crane consists of slender thin-walled girders. Among them, the girder is divided into multisection beams by suspension and tie rod support to increase the overall stability. erefore, the quay crane girder is composed of several simply supported beams in the mechanical model.

On-Site Test
For long-term monitoring, a structural health monitoring (SHM) system has been established on the investigated quay crane. e vertical acceleration of girder hinge region for the last five years is shown in Figure 2. e acceleration signals of critical sites are converted to effective value. e energy of vertical vibration increases yearly. is is because of longtime structure abrasion. e signal in red circle is significantly lower than before, which is the time after the girder was maintained.
e effective value can reduce the data size for SHM purpose. However, it can only represent the tendency of the quantity and remove the extreme value. To obtain the real time signal of the structure response, an on-site test of quay crane girder under normal operating conditions is carried out. e total length of the quay crane girder is 127 m, the nominal lifting capacity is 60 t, and the full load including the trolley and spreader is 80 t. e nominal velocity of the trolley is 4 m/s. e acceleration sensor is installed in the hinge point area of the front and rear girder, and the model of the sensor is IMI 608A11. e test acceleration direction of the sensor is perpendicular to the ground, and the installation position and direction are shown in Figure 3. e sampling frequency is 2500 Hz, and the sampling time is 60 s. Figure 4 shows the measured acceleration signal of the hinge points. When the trolley is running, the acceleration value of crane structure gradually increases and reaches maximum value in the middle. is is because the position of the measured site happens to be in the middle of the entire girder structure. e trolley rail gap which induces impact load takes place in the hinge region. When the trolley is getting closer to the hinge point, the vibration of girder system has a trend of increasing gradually. e trolley track has unwelded gaps at the hinge point, and the vibration signals surge at the hinge point compared with adjacent incremental signals, indicating that the vibration of the   trolley is sharply enhanced here. e impact caused by the vibration will accelerate the wear of the parts of track system and influence the safety and efficiency of the quay crane. e spectrum of the measured vertical acceleration data is shown in Figure 5. e signal of 50 Hz and frequency doubling signal come from the electromagnetic interference of the sensor power line. e main frequency of vertical acceleration takes place within the frequency range of 0-200 Hz. e frequency of maximum amplitude is 1.25 Hz in the low-frequency band and about 100 Hz in high-frequency band. e quay crane is a large metallic structure. e main frequency of the free vibration acceleration is in low-frequency band located in 0-5 Hz. e high frequency of the measured results is mainly due to the dynamic load impact of the trolley passing through rail gap.
For the purpose of distinguishing analysis on normal signal and impact signal, the 20 s-30 s signal of time domain is extracted from the vertical acceleration test data of crane girder structure. During this time segment, the trolley was passing through hinge point. e 0-20 s and 30 s-60 s signals are extracted for the analysis of normal operation of the quay crane without hinge point impact induced by trolley. e spectrum of the extracted signal is shown in Figure 6. When the trolley is running on the landside and waterside girder, the main frequency band of the vertical acceleration of the quay crane structure is 0-5 Hz. But from landside to waterside, the amplitude increases in high-frequency band of 10-200 Hz significantly. e main reason for this phenomenon is that the waterside girder is supported by hinge and tie rods, which form simple support beam, while the landside girder is supported by gantry and diagonal brace, which is more rigid. e section inertia of waterside girder is smaller than that of the landside girder. ere is no large static load such as machine room on landside girder. So the overall stiffness is smaller. e impact induced by trolley passing through hinge point increases the amplitude of highfrequency band following the acceleration response.
When the trolley is passing through the hinge point, the frequency of acceleration is mainly concentrated around 100 Hz. e duration of each impact is short, but the accumulated energy in the frequency domain is high. e ultralow frequency under 1/(2π) Hz is not obvious, while the high-frequency signal is evident in the acceleration signal, which is the frequency band that the trolley mainly influences on crane structural response. e high-frequency dynamic impact caused by trolley passing through rail gap results in the wear of the rail pad as well as some bolt failures as shown in Figure 7. e failure mechanism analysis is not the purpose of this paper, but SHM on logistics equipment including structural response is necessary to avoid the productivity reduction and safety of the quay crane. e front and rear girder hinge points of the quay crane and the girder segment between the front girder and middle tie rod hinge points are taken as the research objects. e girder segment is simplified as a simply supported beam, and the trolley is a moving mass acting on the simply supported beam.

Model of Quay Crane
After simplification, the dynamic balance equation of the beam segment can be expressed as where EI is the stiffness of the girder, m and m t are the mass per unit length of the girder and the mass of the trolley respectively, c is the damping of the girder structure, x is the longitudinal coordinate value along the girder, and y is the vertical displacement value of the girder. y( is the Dirac function. When the value in the parentheses is 0, the function value is 1. e right side of the equal sign is the external load, and the second-order partial derivatives in square brackets represent the vertical acceleration of the girder vibration at the location of the trolley, the vertical acceleration caused by the vertical velocity change of the girder caused by the trolley moving, and the centrifugal acceleration caused by the trolley moving on the vertical curve due to the curvature of the girder in the process of the vibration. e matrix expression of the dynamic balance equation of the girder is  Advances in Materials Science and Engineering Equation (2) is a second-order differential equation with variable coefficients. e mass matrix and external force vector are only related to the mass of the trolley, the mass of the girder, and the order of the mode shape. e centrifugal acceleration term in the dynamic balance equation (1) is omitted from the calculation in this section, and its influence on the results is calculated and compared in Section 4.3.

Result.
e actual working conditions of the quay crane and the load of the girder include four parts: the trolley, the spreader, the spreader hanger, and the hoisting weight. In this section, vertical response analysis at midspan of the single girder under simple working condition is conducted. e parameters of the model are shown in Table 1.
In this section, the Newmark method [7] is used to solve the problem, and the midspan vertical displacement of the girder segment calculated from the parameters in Table 1 is shown in Figure 8.
It can be seen from Figure 8(a) that the deflection of the vertical displacement time-history curve at midspan increases with the growth of the moving load. e maximum dynamic deflection of the trolley is less than 5 mm when it is

Advances in Materials Science and Engineering
not loaded and about 15 mm when it is fully loaded. When the trolley moves at a constant velocity of 240 m/min, there is a small fluctuation in the displacement time-history curve, which is related to the inertial impact on the structure during the movement of the load. As the dynamic load value of the girder increases, the fluctuation value caused by the vertical inertia impact of the load increases. e time for the trolley to pass through the single beam is inversely proportional to the velocity, which means total time span shortens as the trolley velocity increases. Under the same moving mass, the maximum displacement value is proportional to the velocity. e static load conditions of the trolley in the span of the girder under no-load and full-load conditions are calculated. As shown in Table 2, the displacement

Influence of Centrifugal Acceleration on Midspan Vertical
Displacement. Considering the influence of centrifugal acceleration, the midspan displacement response girder under each working condition in 4.2 is calculated. e results are shown in Figure 9. e comparison of the maximum displacement values in the midspan is shown in Table 3. e centrifugal acceleration increases the displacement amplitude. When the overloaded trolley is running at a velocity exceeding the nominal velocity, the effect of centrifugal items on the displacement result is not more than 2%. e response of single girder under trolley traveling is easy to realize with Euler beam theory, while the actual condition of quay crane girder is more complex because of the multihanging structure of girder.
With the conclusion of 4.2, the trolley mass in (1) is treated as constant load. e equation becomes e dynamic displacement of the girder y can be expressed as where the nth mode of vibration is x is the arbitrary longitudinal position of the girder, and t is time. If the damping term is ignored, the single girder model with moving constant load can be solved as where Ω n � (nπv\L) is the driving frequency of the trolley.
When the velocity of the trolley is small, the driving frequency of the trolley is correspondingly small, which is much less than ω n . e response of the quay crane structure is quasistatic under these circumstances.
When n � 1, the vertical displacement of the girder is

Overall Structure Model of Quay Crane
e refined modeling analysis of the whole metallic structure is carried out for the measured quay crane. e model of the overall crane structure and the details of the hinge points between landside and water side girder are shown in Figure 10. e quay crane studied is double-girder type. e main structure is a box-section girder. e material of main structure is Q235 (equal to ASTM A306GR55). e trolley is traction type, and the total weight of the trolley and the hoisting load is about 80 t when fully loaded. In this section, the finite element commercial software ANSYS is used to carry out the structural modeling and response analysis of the whole machine. e SHELL181 element is used to model the girders to better simulate the dynamic response caused by the girder structure and the load of the trolley. e metallic structures such as gantry and diagonal braces are built with BEAM44 and PIPE16 elements. LINK10 element is used to model the backstay and forestay to avoid multidegree-of-freedom analysis caused by hinge point coupling. e MASS21 element is used for trolley load modeling. Traditional key-point-line contact cannot realize the interaction between mass and shell elements. In this paper, the S-M contact method is applied to the dynamic analysis of the quay crane for the first time. By establishing the target element on the flange of trolley girder and the contact element on the trolley mass element, the inertial force of the trolley motion and its mass are applied to the thin-walled shell element of the trolley girder. e DOF of trolley mass has  been constrained in other directions except the motion direction along the track beam. Displacement commands were applied to mass element in the moving direction to realize the interaction between mass and shell surface.

Modal Analysis.
Modal analysis is used to obtain the nature frequency of the quay crane structure. e first seven modal shapes of the whole machine are shown in Table 4. e first three orders are lateral vibrations, the fourth and fifth orders are longitudinal vibrations, and the rest are vertical vibrations. e first seven orders are located in 0-2 Hz band. Most of them are lower than 1 Hz, which is difficult to be detected for the available equipment. e integration time step is 0.0004 s. Figure 11 shows the results of vertical dynamic load at the same hinge point as the measured position. e spectrum of the calculated acceleration signal is shown in Figure 12. Similar to the measured acceleration signal spectrum, the main energy of the hinge point acceleration signal is concentrated within 200 Hz. Since the fitting of the rail and the beam has been       simplified in FE model, the rail, the rail pad, the rail bolt, and the fitting of track and the track beam in the actual quay crane structure have been ignored. So the energy of the calculated impact signal is smaller than the measured one, especially in high-frequency band caused by the omitted component mentioned above. e impact segment of calculated acceleration signal at the rail discontinuity points is extracted and compared with the impact segment of the measured signal. e comparison of time domain signal is shown in Figure 13. e shape of the signal and the peak size of the shock are similar. e error analysis is shown in Table 5. In this section, three indexes related to shock signals are selected for calculation and comparison. Among them, the amplitude refers to the maximum value of the absolute value of the difference between adjacent peaks and valleys. e relative error is the quotient of the absolute error and the test value. e main frequency of the impact signal obtained by calculation and test is 80.65 Hz, and the amplitude is about 290 m/s 2 . It can be seen that the simulation result is consistent with the test one. It can be used as a reference for dynamic analysis of quay crane structures.

Displacement of Crane Structure.
e structural response of the quay crane caused by the trolley is further analyzed. Figure 14 shows the positions of the ten measuring points of the quay crane girders and the specific descriptions of vertical, longitudinal, and transverse. e vertical direction is perpendicular to the horizontal plane. e longitudinal direction is parallel to the girder, that is, the running direction of the trolley. e lateral direction is the horizontal direction perpendicular to the girder. e results of displacement are shown in Figures 15-17. With the movement of the trolley, the measuring point at the rear end of the girder firstly produces displacement. e whole machine's girder span, the front girder midspan, and the front end of the front girder produce displacement responses one after another. As the quay crane girder structure is suspended by four gantry points and supported by six front and rear tie rods to form a multisection simply supported beam, therefore the displacement response of the previous section of the beam will produce reverse fluctuate when the trolley moves to the next section of the beam. e magnitude and fluctuation direction of the same measuring point on the left and right sides of the vertical displacement are almost the same. Among all the measuring points, the largest displacement fluctuation is located on girder end and boom head, followed by the four measuring points of the waterside girder. e displacement of girderleft, girder-right, mid-girder-left, and mid-girder-right is smaller than those of other measuring points. e measuring points at both ends of the girder are supported by tie rods with low rigidity. e four measuring points named boomleft-1, boom-left-2, boom-right-1, and boom-right-2 are far from the rigid support of the gantry. So the vertical displacement results are larger than those closer to the gantry, such as the measuring points of mid-girder and girder-left and grider-right. When the trolley moves to the next beam segment, the reverse fluctuation of the vertical displacement response is more significant than the displacement responses in the other two directions. e longitudinal displacement is caused by the inertial load of the trolley moving forward, and the longitudinal displacement curves of all measuring points show a roughly upward trend with time. e five measuring points on the land side first generate longitudinal displacement, and the amplitude is greater than that of the five measuring points on the water side, which is also related to the working condition of the trolley moving from the land side to the water side. If the trolley moves in the opposite direction, the above results will also have the opposite trend. e lateral displacement of the same measuring point on right and left girders has the opposite direction, which is related to the eccentric loading condition of the rail girder. When the trolley was passing through the measuring point, the left girder twists to the right and produces the right lateral displacement and vice versa. In the lateral displacement results, the displacement fluctuations of measuring points on the water side and land side are smaller than those of other side points, as shown in Figure 17, and the trend is gentle. According to the above analysis, the lateral displacement of girder under trolley traveling is mainly due to the torsional lateral deformation caused by the trolley load bias relative to the girder, as in Figure 10. e measure points at both ends of the girders are almost unaffected by the torsion of the girder, so the results are the least obvious. In the displacement results of the three directions, the maximum vertical displacement is 120.6 mm, and the minimum longitudinal displacement is 11.47 mm, all of which are at the measurement point of the last end of the girder. e vertical displacement spectrum of each measuring point is shown in Figure 18. e result shows that the lowfrequency vibration information is retained more completely in the displacement signal. e first frequency of the displacement of each measuring point is v/L � 4/88 � 0.454 Hz, which is the disturbance frequency of the structure under trolley traveling proposed in 4.3. is means the nominal velocity of the crane trolley is slow enough and has little dynamic effect on quay crane structure response. e velocity of the trolley exceeding resonant velocity will induce large dynamic effect, which is the length of the girder times the natural frequency of the quay crane. But this condition can never happen. Because the maximum trolley velocity for quay crane is less than 10 m/s, while the minimum resonant velocity of the studied crane is 27 m/s. is condition can be used in bridge with large span or the structure with low velocity moving load compared to the length of the structure.

Conclusion
(1) e acceleration response at the girder hinge point under the working condition of the quay crane in service is measured on site. e trolley passed from landside to waterside girder. e time series of acceleration shows the high-frequency impact induced by trolley passing through hinge point. e spectrum shows that the main frequency band of the structure acceleration is 0-200 Hz. e separation analysis shows that the impact induced by trolley passing through will affect the following acceleration response on the high-frequency band of about 100 Hz. e acceleration shows weakness in ultralow frequency analysis. response is mainly in high frequency, which is shown in acceleration response. e displacement response of crane structure reveals that the trolley has little dynamic effect on it. e displacement response can be treated as quasistatic status. e trolley velocity should exceed the first resonant velocity, which is 27 m/s in this case far beyond the nominal velocity of the trolley, to have significant dynamic effect on the crane structure response. is deduction can be used in bridges with large span or structures with low velocity moving load compared to the length of the structure.

Data Availability
e data used to support the findings of this study can be obtained from the corresponding author upon request.

Advances in Materials Science and Engineering 13
Conflicts of Interest e authors declare no conflicts of interest in this article.