Hindawi Mathematical Problems in Engineering Volume 2018, Article ID 8736786, 13 pages https://doi.org/10.1155/2018/8736786

Research Article Fatigue Life Characteristics of Segmental Lining Structure for the Shallow Buried Metro Tunnel under the Dynamic Load of a High-Speed Railway

Wangping Qian ,1,2 Taiyue Qi ,1,2 Qing Zhao,1,2 Bingrong Pu,1,2 Jin Zhang,1,2 and Haiyang Yi1,2

1 MOE Key Laboratory of Transportation Tunnel Engineering, Southwest Jiaotong University, Chengdu 610031, China 2School of Civil Engineering, Southwest Jiaotong University, Chengdu 610031, China

Correspondence should be addressed to Taiyue Qi; [email protected]

Received 2 May 2018; Revised 11 July 2018; Accepted 25 July 2018; Published 3 September 2018

Academic Editor: Roman Wan-Wendner

Copyright © 2018 Wangping Qian et al. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Shallow buried shield metro tunnels constructed underneath subgrade project of high-speed railways are becoming increasingly common in China, but the lower metro tunnel bears the fatigue efect of dynamic load induced by the upper high-speed railway, so the long-term durability of segmental lining is a nonnegligible problem. Te segmental lining structure of metro tunnel is in a state of static-dynamic loads for a long time, especially when a high-speed railway passes above the metro line, and the long-term durability of segmental lining needs further research. Based on theoretical analysis, the efect of diferent forms of loads on the fatigue life was analyzed, the change law of the static-dynamic loads on segmental lining was summarized, and the method was put forward to evaluate the fatigue life characteristics of segmental lining. Te research results reveal that the additional dynamic load is the fundamental reason for the fatigue failure of the structure, and the existence of static load can cause and accelerate the occurrence of structural fatigue failure simultaneously. Te results indicate that the fatigue life decreases gradually with the increase of static-dynamic load. Based on coupling analysis of static-dynamic loads of segmental lining, the fatigue life increases frst and then decreases with the increase of buried depth of metro tunnel, and it remains unchanged when the depth exceeds a certain value. According to the actual metro tunnel engineering, by using ABAQUS sofware, a three-dimensional numerical simulation was carried out to analyze the characteristics of the fatigue life and evolution rules of segmental lining. Based on the modifed fatigue life formula and metro service life, the optimization design of the buried depth was carried out to determine the most reasonable range of the buried depth. Tis study provides a valuable reference for safe operation and long-term durability of metro tunnels under high-speed railways.

1. Introduction a given depth of metro tunnels [2, 3]. During the operation of tunnels, the metro segmental lining periodically bears the Te metro segmental lining is a permanent supporting long-term metro train loads [4]. As for metro tunnels passing structure of shield tunnels. Te stability and durability of through a high-speed railway, the metro segmental lining segmental lining during the operation of tunnels directly inevitably bears cyclical dynamic loads transmitted by the infuence the service life, durability, safety, and waterproof dynamic stress of a high-speed railway subgrade [5]. Under performance of the structure of metro tunnels [1]. During the normal service life of metro tunnels, the metro segmental lin- the coupling of long-term periodic static-dynamic loads, ing must be in a state of static-dynamic loads for a long time. there must be fatigue problems in the structure of metro Te static loads acting on metro segmental lining mainly segmental lining, which directly afects the durability of include the formation pressure and additional pressure of segmental lining. Terefore, fatigue characteristics of metro buildings, and its value is constant during the operation for segmental lining deserve further investigation, especially the 2 Mathematical Problems in Engineering fatigue life of segmental lining, which directly determines the As for the engineering project of shallow shield tunnels service life of metro tunnels. passing through high-speed railways, the metro segmental Aimed at static-dynamic response and fatigue character- lining is subjected to cyclic high-speed dynamic loads trans- istics of tunnel support structures, relevant researches have mitted from the upper strata for a long period. Whether the been carried out both domestically and abroad. Caratelli [6] durability of segmental lining under the coupling of existing and Xue [7] studied the variation characteristics of segmental static loads and external dynamic loads is further reduced and lining under static and dynamic loads, respectively, and whether its fatigue life satisfes the design of service life are the analyzed diferences between two types of loads from the problems that need to be further investigated to ensure the crack development and structural displacement deforma- safety operation and long-term durability of metro tunnels. tion. Based on safety assessments of tunnels constructed by Based on above analysis, this paper studies and summa- shield driving method, under the long-term action of static- rizes the fatigue life characteristics of segmental lining under dynamic loads, there is a very diferent form of fatigue failure the upper high-speed railway dynamic loads and proposes a rather than general damage caused by short-term static loads, suitable formula for the fatigue calculation of urban shallow which causes rapid expansion and penetration of internal buried metro tunnel combined with previous researches on cracks, reduces the bearing capacity of structure, leads to concrete fatigue. Trough the theoretical analysis of fatigue excessive deformation, and directly afects the service life of formula, the fatigue life characteristics of segmental lining are tunnels [8, 9]. Yang [10] and Mussa [11] analyzed the dynamic studied and the fatigue evolution of segmental lining under response of lining structure under dynamic loads in the form the coupling of static-dynamic loads is analyzed. Aimed of blasting by taking metro tunnels in the city as research at the high-gradient area in the fatigue life curve, fatigue objects, and it is shown that the vault and inverted arch of safety coefcient is introduced to modify the fatigue curve to tunnel are more likely to be damaged. Mo [12] established better refect the actual physical state of the metro segmental a three-dimensional segmental model considering the joints lining. Concurrently, this paper analyzes the variation rule of metro tunnels during operation and analyzed the rule of of stress state of segmental lining with the change of buried dynamic responses in diferent parts of segmental linings, depth and concludes the evolution path of the fatigue life. and it was confrmed that the dynamic response value at Combining the example of Guangzhou Metro Line 9 passing the joint location of metro segmental linings was larger and through the Wuhan-Guangzhou High-Speed Railway and the problem of fatigue failures occurred more likely at the the numerical simulation of ABAQUS sofware, the fatigue joint. Luo [13] researched the initial stress feld of metro life characteristics and its evolution rule of metro segmental segmental linings under the efect of surrounding medium lining are summarized. Based on the design service life of and analyzed the infuence of initial ground stress on dynamic the metro tunnel, the optimization design of buried depth responses under blasting loads. Huang [14] investigated is carried out, and both of the critical depth without fatigue fatigue characteristics of high-speed railway tunnels under failure and the optimal depth are proposed. Tis research the efect of train dynamic loads and introduced the double provides a valuable reference for safe operation and long- scalar of tension-compressive stress to predict the fatigue life term durability of metro tunnels under high-speed railways. of the tunnel, and the research indicated that the service life of inverted arch of the tunnel was determined by tensile characteristics of concrete materials. Li [15] studied the infu- 2. Analysis of Fatigue Life Characteristics of ence law of dynamic stress of heavy-load railway tunnel bases Segmental Lining Structure by means of numerical simulation and feld measurement 2.1. Optimal Selection of S-N Formula for the Fatigue Life and investigated the lateral and vertical distribution laws of of Segmental Lining. Te segmental lining is a permanent dynamic stress of the tunnel base. Liu [16, 17] summarized structure of the shield metro tunnel, which is an integral prediction methods for the fatigue life of inverted arch structures under the conditions of sofening and voiding ring structure mainly formed by assembling concrete mate- bedrock and studied the infuence of diferent train loads on rial. Te fatigue life characteristics of the concrete material the fatigue life. Summarizing the above research focus, the determine the fatigue life of the entire segment structure. As current emphasis is mainly on analysis of dynamic responses is known to all, there is a great diference between tensile and fatigue durability under the action of metro loads, heavy- strength and compressive strength of concrete, and the tensile load railway loads, or blasting loads, and there are no more strength is only 1/10 to 1/20 of the compressive strength. reports on the study of metro tunnels passing through a A large number of scholars [11, 14, 17] verifed the tensile high-speed railway. As for the analysis of dynamic responses characteristics of the concrete as an important factor in and fatigue life under metro loads [4, 12, 16, 17], the metro the evaluation of the structural safety through numerical tunnel can meet the requirements of both normal safety simulation and laboratory tests, and the concrete structure design and long-term durability. However, there is no relevant was prone to tensile failure rather than compression failure research result to prove whether metro tunnels meet the long- in practical engineering. Tus, the tensile characteristics term durability requirement under the action of high-speed determined the service life of the tunnel to a certain extent. railway dynamic loads or not, and it remains to be further For the research of tension fatigue of concrete, many investigated. Terefore, it is necessary to study fatigue life researches have been investigated at home and abroad. characteristics and evolution rules of segmental lining under Tepfers [18] conducted a splitting test on cubic lightweight dynamic loads of high-speed railways. concrete to analyze concrete fatigue sensitivity parameters Mathematical Problems in Engineering 3

Table 1: Te S-N formula of the concrete fatigue life with diferent types and stress conditions.

No. Species S − N formula Condition Author 14.0(1 − ����/��) 1C25 lg(�) = Splitting tensile Tepfers (1 − ����/����)

24.27���� 2C25 lg(�) = 23.96 − Uniaxial Saito �� 14.52���� 2.79���� 3C50lg(�) = 14.91 − + Uniaxial, dry Cornelissen �� �� 14.52���� 2.79���� 4C50lg(�) = 13.92 − + Uniaxial, humid Cornelissen �� �� 18.518���� 5C50 lg(�) = 17.87 − Splitting tensile Zhao �� 20.0���� 6C50 lg(�) = 19.40 − Axis tensile Zhao �� 22.222���� 7C50 lg(�) = 20.933 − , Bending tensile Zhao �� 28.169���� 8C25 lg(�) = 25.408 − Splitting tensile Zhao �� 19.324���� 9C30 lg(�) = 18.968 − Uniaxial Song �� 16.67���� 5.17���� 10 C30 lg(�) = 16.67 − + ,Uniaxial Song �� �� Note. lg(�) is the fatigue life of the concrete, ���� is the maximum stress of the specimen (MPa), ���� is the minimum stress (MPa), and �� is the tensile strength of the concrete (MPa).

and obtained the equations related to those of compressive account of the environment and material composition of the stress. Saito [19] carried out the tensile fatigue test of gravel metro tunnel, the formula lg(�) = 13.92 − 14.52����/�� + regular concrete under the condition of standard humidity 2.79����/�� is selected as the basis for the subsequent fatigue and loading frequency of 5 Hz, and the ratio between the life calculation. minimum tensile stress and the specimen tensile strength is the certain value, and the fatigue S-N formula based on 2.2. Analysis of Fatigue Life Characteristics of Segments. By the probability p=0.5 is obtained. Cornelissen [20] developed classifying and summarizing the fatigue formulas in Table 1, a large number of concrete tensile fatigue tests under both a general expression of the fatigue life can be obtained as dry and humid conditions, simultaneously considered the formula (1). infuence of diferent minimum stress levels, and derived � � the fatigue S-N formula based on probability p=0.5 and (�) =�−�⋅ ��� +�⋅ ��� lg � � (1) confdence interval of 90%. Zhao [21] conducted fatigue � � tests on tensile, axial, and bending tensions of high-strength where A, B, and C are the parameters of the fatigue constant, concrete under the condition of the minimum stress level � and their values are 13.92, 14.52, and 2.79, respectively; max of 0.1. Te diferences in fatigue life between high-strength is the maximum stress (MPa), and its value is the sum of concrete and ordinary concrete in diferent stress sections � the initial static and additional dynamic stress; min is the were compared and analyzed. Song [22] took concrete of minimum stress (MPa), and that is the initial static stress. grade C30 as a sample, studied the tensile fatigue tests Considering the fatigue theory, the main formulas can be under diferent stress levels separately, and the S-N formulas got as formula (2). under diferent conditions were obtained. By comparing the concrete fatigue test of the above scholars, the fatigue formula ���� +���� �� = was obtained at diferent concrete types and stress conditions, 2�� as shown in Table 1. � � −� Comparing the formulas of Table 1, we can see that the � = � = ��� ��� � 2 2� (2) fatigue life in diferent tests decreases with the increase of � the maximum stress and increases with the increase of the ���� minimum stress. Under the same stress level conditions, it �= ���� can be found that the formula of number 4 has the smallest fatigue life and has a 90% confdence interval; that is, the S-N where �� is the average stress (MPa), �� is the stress formula life of Cornelissen in uniaxial tests of C50 concrete amplitude (MPa), �� is the stress range (MPa), and � is the under humid condition is minimal and highly reliable. Taking stress ratio. 4 Mathematical Problems in Engineering

Fatigue life Fatigue life 1.0 lg(N) 1.0 lg(N) 9.000 9.000 Relative value ＧＣＨ/＠ ＞/＠ ＠ ＠ A Medium Small

/ 8.250 / 8.250 0.8 0.8 B Large Medium

ＧＣＨ Small Medium

ＧＣＨ C

7.500  7.500  D Medium Large 0.6 6.750 0.6 6.750 B L: Fatigue life path curve 6.000 6.000 A 0.4 0.4 5.250 5.250

4.500 D 4.500 0.2 0.2 L

Minimum stress level stress Minimum C Minimum stress level stress Minimum 3.750 3.750

0.0 3.000 0.0 3.000 1.00.80.60.40.20.0 1.00.80.60.40.20.0

Maximum stress level Ｇ；Ｒ/＠ Stress range level ＞/＠ (a) (b) Fatigue life 1.0 lg(N) 9.000

8.250

＠ 0.8 /

＞ 7.500  G: High gradient region of fatigue life 0.6 6.750

6.000 0.4 5.250 G 4.500 Stress range level range Stress 0.2 3.750

0.0 3.000 1.00.80.60.40.20.0

Stress ratio Ｇin /Ｇ；Ｒ (c)

Figure 1: Te evolution of fatigue life of segmental lining under the diferent load forms.

Te transformation formula can be obtained based on the high stress of fatigue state; the local or total fatigue failure variables of �� and R, as shown in formula (4). of segmental lining will occur when the frequency of special 2 loads exceeds 10E3. � = � ��� 1−� � Based on the theoretical analysis of the fatigue formula (1), the evolution of the fatigue life of the metro segmental 2� lining is the focus of research under the coupling of static- � = � (3) ��� 1−� � dynamic loads, as follows. (1) � � 1+� Taking ��� and ��� as basic variables, the boundary � =2 � condition can be expressed as formula (4). � 1−� � 3< (�) <9, Considering the actual service life of the metro tunnel lg (4) segmental lining and the applicable range of the fatigue S-N 0<���� ≤���� <�� formula, the following assumptions are made for the fatigue life of segmental lining. When the value of ���� is equal to the value of ����, (1) Te upper limit of high-cycle fatigue: it is assumed the fatigue of segmental lining cannot be considered and that the segmental lining is under the normal train load, the the fatigue formula is not suitable to discuss the fatigue life amplitude of the dynamic stress is relatively small, and the of segmental lining. Te fatigue formula area is shown in frequency of dynamic load of the train exceeds 10E9 (the Figure 1(a) (the horizontal and vertical coordinates are both predicted years exceeds 1000 years), and there is no fatigue unitized). It can be seen that the fatigue life of segmental failure of segmental lining structure. lining gradually increases with the linear increase of the (2) Te lower limit of high-cycle fatigue: it is presumed minimum stress, and the fatigue life of segmental lining that the segmental lining is under special loads (such as gradually decreases with the maximum increase of the stress earthquakes and train impact loads) and is in a relatively level. However, the absolute value of the increment of the Mathematical Problems in Engineering 5 maximum stress is greater than the minimum stress value, regions. As for the fatigue life design of segmental lining, the which indicates that the maximum stress has an important stress state of the structure should be distributed in A and infuence on the fatigue life of segmental lining than the C regions as much as possible. Terefore, there are two stress minimum stress. conditions of the structure to ensure the long-term safety and (2) � � Taking ��� and a as basic variables, the boundary stability of segmental lining. One is that the minimum stress condition can be expressed as formula (5). is small and the additional dynamic stress is relatively large, and the other one is that the minimum stress is relatively large 3

0<���� <��, 2.3. Modifcation of Fatigue Life Based on Goodman Formula. 0<� <�, a � (5) Because of the tensile characteristics of concrete materials, there is no obvious yield strength. What is more, when the 2� <�, a � stress range is small and the stress ratio is relatively large, the fatigue life of segmental lining changes very quickly within ���� +2� <�� a this range, so it has a defect and hidden danger in the practical Formula (1) can be transformed as formula (6). application process. Combined with the research on the fatigue characteristics of commonly used railway materials � � a ��� by China Academy of Railway Sciences [23], the fatigue lg (�) =�−2�⋅ − (�−�) ⋅ (6) �� �� safety factor of concrete is introduced in this research and the Goodman curve of concrete tension-tension fatigue is In order to facilitate the analysis of its evolution, �� =2�� modifed as formula (9). is used as the analysis variable, and its fatigue area is shown in � � � � � Figure 1(b). It can be shown that the fatigue life of segmental � + � =1 0� ≤ � ≤ � lining gradually decreases with the increase of the stress �−1� �� �� �� �� range and the minimum stress, and the absolute increment (9) � � 1 �� � 1 of the stress range is greater than the minimum stress value. � + � = ≤ � ≤ � � � � � � Based on theoretical analysis, it is proved that the additional � � � � dynamic stress �� of segmental lining is the fundamental where �� is ��(�� −(�+1)�0�)/�(�� −2�0�), � is the fatigue reason for the structural fatigue failure, and the presence safety factor, �0� is the stress amplitude when stress ratio �= of the minimum tensile stress induces and accelerates the 0,and�−1� is the stress amplitude when stress ratio �=−1. fatigue of the structure. From formula (9), the modifed Goodman formula of (3) Taking � and � as basic variables, the boundary a concrete tension-tension fatigue is obtained and shown in condition can be expressed as formula (7). Figure 2. As can be seen, the curve of FHK is the modifed 3

σa separately to fnd out the fatigue life path curve in the fatigue f life surface. t � � (ℎ) =� −�⋅ � σ � � (11) max B C �� ft / n �� �� (V,ℎ,�) =�� −�⋅ (12) A ��

D where ��(ℎ) is a fatigue function related to static load, σ -1N ��(V,ℎ,�)is a fatigue function related to the dynamic load, F and ℎ, V,and�, respectively, represent the buried depth of σ σmin 0N the metro segmental lining, the train speed, and the axle load of the train. H E GKL 3.1. Evolution Rule of Static Loads of Segmental Linings. Te earth pressure acting on the metro segmental lining structure σ  f / n f σ 0N g t t m is actually the contact stress between the surrounding soil and segmental lining structure. Its size and distribution are not only related to the mechanical properties of the stratum and the stifness of segmental lining, but also related to the -σ-1N construction method, the upper building, and the geometric parameters such as the depth, the straight diameter, and the shape of the tunnel. In order to facilitate the study of its Stress envelope evolution rule, it is considered that segmental lining load Modifed Goodman fatigue formula is mainly related to the depth of the metro tunnel, and the Figure 2: Te modifed Goodman formula of concrete tension- other factors are assumed to be consistent or unchanged tension fatigue. in the calculation process. Te Terzaghi’s theory holds that the amplitude of the vertical loosening pressure of the surrounding rock tends to zero when the depth of the tunnel is increased to a certain limit, and the upper earth pressure is According to formula (1), the formula is substituted into as shown in formula (13). the static and dynamic load of segments and transformed as ∑ �ℎ, ℎ < 2.5� formula (10). { �� = { � � (13) � � { (� + � ⋅ (45∘ − )) , ℎ ≥ 2.5� � � � tan 2 lg (�) =(�� −�⋅ )+(�� −�⋅ ) (10) { �� �� where �� is the earth pressure, � and � are the unit weight where �� is the additional dynamic stress of segments(MPa), and friction angle of the overlying soil, D is the diameter of � � � segments, and � and � are the balance arch span and the i is the initial static stress (MPa), � and � are constants related to the dynamic and static stress (��+�� =A), and B sturdy coefcient. and D are fatigue parameters ( D=B-C and D>0). Considering the existence of the ground water and the It can be seen from (10) that the fatigue life of segmental upper building, the efect of the water pressure and the linings is only related to the additional dynamic stress and additional load of the building on the initial static feld of � initial static stress of segmental linings. Furthermore, there segmental lining should be analyzed [25, 26], and the � is a negative linear relationship between the fatigue life of needs to be corrected. It can be seen from formula (13) that segmental linings and the two kinds of loads, which means the stress of segmental lining increases with the increase of that the fatigue life decreases with the increase of initial static depth and then remains unchanged when the depth exceeds loads and additional dynamic loads. Terefore, in the process a certain value. According to formula (11) of the fatigue life of of design, construction, and operation of shield tunnels, it is segmental linings, the fatigue life gradually decreases with the necessary to investigate how to reduce the static and dynamic growth of the buried depth and then the fatigue life remains loads of segmental linings. By combining the research and unchanged afer a certain depth. analysis [4, 6, 9], the static loads of segmental linings are only related to the environment around. Te dynamic loads 3.2. Evolution Rule of Dynamic Load of Segmental Lin- of segmental linings, for the upper high-speed railway load, ing. Many researchers had carried out a lot of research are only related to the depth of segmental linings, the axle on the dynamic train loads from theory, simulation, and loads, and the speed of the train. Terefore, according to experiment and feld measurement. Pan [27] and Jenkins the characteristics of loads, the factors afecting the fatigue [28] summarized a classical excitation force formula taking life of segmental linings are discussed separately as shown in into consideration the train static load and track irregu- formulas (11) and (12), and the evolution of loads is analyzed larity through the empirical analysis of a large amount of Mathematical Problems in Engineering 7

N Huadu Square station

Maanshan Park Station

Guangzhou Metro Line 9 Shield Section Huaguoshan Park station

Huacheng Road Station

Huadu Station Guangzhou North Station Wuhan-Guangzhou High-speed Railway

Figure 3: Te location relation between the metro line and high-speed railway. measured and experimental data. Song [29] investigated the and also subjected to the action of the upper high-speed distribution and transfer rule of the dynamic stress of the railway load, and the stress of segmental lining is in a joint track and subgrade under diferent velocity moving loads. state of static-dynamic loads. Te fatigue failure of segmental Simultaneously, the dynamic stress of the subgrade had an lining is a mechanical phenomenon under the coupling of obvious linear relationship with the moving load speed at static-dynamic loads. According to formulas (13) and (14), the the range of 80km/h and 350km/h, and the dynamic stress trend of additional dynamic and initial static loads with depth remained unchanged when the speed was less than 80km/h or can be obtained. Terefore, the total stress of segmental lining more than the 350km/h. Jiang [30] proved that the dynamic structure decreases frst and then increases with the increase stress of the subgrade was linearly changed within the range of the buried depth, and the total stress remains unchanged of 150km/h and 300km/h through the laboratory experiment, afer reaching a certain depth. According to formulas (11) and the dynamic stress in other ranges is independent of and (12), the evolution of the fatigue life with the change of the speed of the train. Song [29] and Liu [31] had mea- buried depth can be obtained under the coupling efect of sured the dynamic stress of the subgrade of Beijing-Tianjin static-dynamic loads; that is, fatigue life path is the change and Wuhan-Guangzhou High-Speed Railway, and its value trend of area D, area C, area A, and area B in order (shown in ranged from 16.9kPa to 18.4kPa and the change rules conform Figure 1(a)). Under the dynamic load of high-speed railway, to the results of the simulation sofware and the laboratory the fatigue life of segmental lining structure appears to experiment. Based on the above analysis, the dynamic stress increase at frst and then decrease with the increase of the of the subgrade under diferent speed conditions is obtained buried depth, and the fatigue life of segmental lining remains as formulas (14) and (15). unchanged afer exceeding a certain depth. Terefore, in the � =� ⋅� actual metro design process, the impact of high-speed railway � � � (14) on the fatigue of segmental lining is further considered, and the settlement of the upper stratum or building caused by the �� metro construction does not exceed the limit value, so the {�� =�0 V ≤ 150��/h depth of the metro proposed a relatively reasonable value. { { (� −� ) (V − 150) (15) = {� = 1 0 +� 150��/ < V < 300��/ { � 0 h h { 150 {�� =�1 300��/h ≤ V 4. A Case Study of Metro Tunnel and Optimization Design of Buried Depth of � where � isthedynamicstressofthesubgradeunderacertain Segmental Lining speed condition, �� is the static stress and its value is mainly � related to the axle load of train and the type of track, � is 4.1. Project Overview. Guangzhou Metro Line 9 passes � � the dynamic stress amplifcation factor, and 0 and 1 are the through the Wuhan-Guangzhou High-Speed Railway and coefcients related to the speed of the train. the Beijing-Guangzhou Railway between Guangzhou North Te dynamic load of the train is equivalent to a stress area Railway Station and Huacheng Road Station. Te tunnel is with rectangular distribution. According to the load transfer constructed by the shield method. Te plane angle between law, the dynamic load of the train gradually attenuates during the tunnel and the railway is approximately 72 degrees, and the downward transfer process, so the dynamic response the tunnel depths range from 7m to 9m. It is the frst metro of segmental lining is gradually reduced. According to the line in the shallow section of China to pass through the high- fatigue formula (12), the fatigue life of segmental lining speed railway subgrade, and its location is shown in Figure 3. presents a nonlinear reduction state with the increase of Guangzhou North Railway Station is the halfway station of buried depth. the Wuhan-Guangzhou High-Speed Railway. Most of the trains pass through the line at a speed of about 300 km/h. 3.3. Analysis of Fatigue Life Path of Segmental Lining under Terefore, the dynamic load of the high-speed railway will the Coupling of Static-Dynamic Loads. For the actual metro inevitably have an important impact on the long-term stabil- tunnel engineering, the segmental lining is in a certain depth ity and durability of the metro segmental lining structure. 8 Mathematical Problems in Engineering

Table 2: Physical and mechanical parameter of diferent strata.

Tickness Density Elasticity modulus Friction angle Cohesive Rayleigh damping No. Poisson ratio ∘ /(m) /(kg/m3) /(MPa) /( ) force/(kPa) �� 1 2.0 1880 10.15 0.43 15 18 0.021 0.093 2 4.0 2000 57.15 0.26 18.1 17.5 0.026 0.094 3 6.2 2050 60.435 0.25 33 48 0.039 0.063 4 6.2 1910 57.15 0.32 18.1 23.5 0.054 0.045 5 4.6 2760 650 0.15 28 400 0.047 0.053

Te track plate Te base plate Note: Te crossing angle of the high-speed railway and Te support layer the tunnel centerline is 75 degrees Te stratum Dynamic load of high-speed railway

2.0m 1 180mm 4.0m 2 9.0m 240mm 300mm Two tunnels 6.2m 3

6.0m

6.2m 4 12m

4.6m 5

1. Layer of artifcial flled soil 2. Layer of plastic silty clay 3. Layer of mid-coarse sand 4. Layer of hard plastic silty clay 5. Low weathered limestone Figure 4: Geological conditions, tunnel conditions, and high-speed railway position.

4.2. Formation Conditions, Segment Conditions, and Train Te specifcation and model of the train directly afect Conditions. According to the geological survey report of the dynamic stress acting on the subgrade. According to the Guangzhou Metro Line 9, the distribution of strata is shown design plan of Wuhan-Guangzhou High-Speed Railway, the in Figure 4, and the physical parameters of each layer are model of the train is CRH380, the axle load is 15t, the length shown in Table 2. In order to reduce the impact of shield of the car is 24.175m, the distance between wheels is 17.375m, construction on the high-speed railway subgrade, the hor- and the train formation is composed of eight groups. Te izontal jet-grouting technology was used to reinforce the operating speed of the train is 300 km/h. According to the stratum under the high-speed railway subgrade before the previous research and investigation, the maximum measured shield construction. Te horizontal jet-grouting pile diameter value of dynamic stress of the subgrade is 19.8kPa. In this ∘ is 2.0m, and the reinforcement angle is 160 .Testrength calculation, the maximum stress is chosen as 20kPa and the of the solid afer hardening in the sand layer is no less than loading frequency is 3.3Hz. 3.0MPa, it is no less than 1.0MPa afer hardening in the clay layer, and the overlap thickness between the pile and the 4.3. Tree-Dimensional Model and Numerical Assumption. In pile is no less than 300 mm to form an overall load-bearing the process of numerical simulation, the dynamic calculation structure. is performed by coupling fnite element and infnite element. Te segmental lining of shield tunnel adopts a universal Te geometric dimension of the model is 60m×40m×72m wedge-shaped lining ring. Te diameter of segmental lining is (X×Y×Z). Te size of the segmental lining is based on 6.0m, the thickness is 0.3m, the width is 1.5m, and the lining Section 4.2, and the segmental lining uses homogeneous solid loop is divided into six parts, that is, three standard segments element in the model. Te three-dimensional structure is (central angle of 67.5 degrees), two adjacent segments (central adopted and the element type is CIN3D8. angle of 68 degrees), and one key segments (center angle of Te infnite element can serve as the function of absorb- 21.5 degrees). Te method of staggering of 22.5 degrees is ing the dynamic wave in the boundary generated by the assembled, two bolts of grade M30 are used between the two fnite element calculation and propagated to the far feld, parts, and sixteen bolts of grade M30 are used to connect two prevent the stress wave from propagating to the static artifcial rings. Te lining ring structure is shown in Figure 5. boundary, and reenter the fnite element calculation model, Mathematical Problems in Engineering 9

Circumferential joint A B A 21.5∘ Longitudinal B joint K 68∘ K B B B K R3000 ∘ A A 22.5 B

22.5∘ A R2700 K

B 67.5∘ A B A K: Key segment B: Stradard segment A: Adjacent segment

Figure 5: Segmental lining structure of shield tunnel.

(a) Finite element model (b) Infnite element model

B K 1

K 2 B B

2 3 A

A A 1 K: Key segment 1: Rebar mesh B: Adjacent segment 2: Circumferential joint A: Standard segment 3: Longitudinal joint (c) Sectional metro tunnel (d) Segment connection

Figure 6: Tree-dimensional model. so it can reduce the interference of the refected wave. Tus, through the soil layers, then acts on the segmental lining, and the infnite element is built at the periphery and bottom of fnally passes into the infnite element at the periphery and the fnite element model as an infnite element boundary bottom of the model. condition. Moreover, in order to simulate the actual segment, Some assumptions are proposed in the process of numer- the model of the solid element is performed for segmental ical simulation. Te Mohr-Coulomb model is used as the lining and bolt, and the contact between diferent parts constitutive model of the soil layers, and the linear elastic and bolts is established, respectively. Te three-dimensional model is selected as the constitutive model for the segmental model is shown in Figure 6. lining, the support layer, the base plate, and the track plate. In Te high-speed railway is mainly composed of the sup- the damping setting, the Rayleigh damping coefcient (con- port layer, base plate, and track plate, and the dynamic load taining � and �) is adopted and the damping ratio of analysis of high-speed railway acts directly on the track plate; the is 0.05. Its coefcient is taken by the reference frequency of location of dynamic load is shown in Figure 4. Te dynamic the frst two orders of the soil layer in the modal analysis, and load of high-speed railway propagates downward, passes the values of diferent soil layer are shown in Table 2. 10 Mathematical Problems in Engineering

Table 3: Fatigue life of segment at diferent buried depths.

Te joint between the two rings Te end of the bolt Depth Minimum stress Maximum stress Minimum stress Maximum stress Fatigue life lg(�) Fatigue life lg(�) /(MPa) /(MPa) /(MPa) /(MPa) 5m 0.7326 1.2955 7.658 0.7185 1.2104 8.108 7m 0.7911 1.2342 8.055 0.7867 1.1599 8.456 10m 0.8933 1.2192 8.135 0.8615 1.1321 8.687 13m 1.0944 1.3384 7.697 1.1554 1.2631 8.282 16m 1.2657 1.4218 7.532 1.1924 1.3198 8.011

S, Max. Principal 8.2 (Avg: 75%) +3.025e+05 +2.722e+05 +2.519e+05 +2.265e+05 +2.012e+05 8.1 A +1.759e+05 +1.505e+05 A:5m~7m +1.252e+05 +9.986e+04 +7.453e+04 +4.919e+04 8.0 +2.386e+04 B:7m~10m −1.473e+03 7.9 C:10m~13m D:13m~16m 7.8

7.7 Design life lg(Nd) Fatigue life lg(N) life Fatigue 7.6 B 7.5 A B C D A: Te end of the bolt h h B: Te joint between two rings 7.4 1 hm 2

Figure 7: Te stress of the inverted arch of the tunnel segment. 486110 1412 6 Buried depth (m) Fatigue life lg(N) 4.4. Fatigue Life Prediction of Diferent Buried Depths of Segmental Lining. By analyzing the force value in the position Figure 8: Te change trends of segment fatigue life at diferent buried depths. of the invert, it can be found that the tensile stress at the joint between two rings, as well as the end of the bolt, is relatively larger than other parts, as shown in Figure 7. Terefore, the position of the invert is the most dangerous position and the depth of segmental lining is from 10m to 16m, which further selected as the research object. Based on the fatigue means the fatigue life shows a tendency of increasing frst and formula (1), the fatigue life prediction of segmental lining is then decreasing with the increase of the depth. Terefore, this performed at diferent depths, as shown in Table 3. phenomenon is consistent with the conclusion of the fatigue From Table 3, the fatigue life at the invert of segmental life characteristics of segmental lining structure described lining is analyzed at diferent positions. Te results show that above (see Section 3.3). Meanwhile, by observing the fatigue the fatigue life at the end of the bolt is greater than the fatigue life curves at diferent depths, it can be seen that there is a life at the joints of adjacent segments, and the fatigue failure reasonable buried depth in the interval between 7m and 10m, occurs frst at the joint of adjacent ring segment. Terefore, and the fatigue life of segmental lining reaches a maximum under the dynamic load of high-speed railway, the dislocation value in that depth. When the depth of segmental lining is between adjacent segments is the main cause of the fatigue larger than 16m, it can be seen that the fatigue life tends to be failure of the concrete, and this position causes fatigue failure stable. Te main reason is that the depth exceeds 2.5D (15m), before other parts in a long-term dynamic load. the static stress remains stable, and the dynamic response When the metro segmental lining are at diferent depth, value is very small. it can be seen from the calculation results that the mini- When the buried depth of metro segment exceeds a mum stress of segmental lining gradually increases with the certain value, the fatigue characteristic value of segmental increase of the depth, and the maximum stress of segmental lining is in the region of high-gradient variation, that is, the lining decreases frst and then increases with the increase of area G in Figure 1(c). Combined with the modifed Goodman the depth. Taking the fatigue life of the joints as an example, formula, the fatigue safety factor is taken as 2, so the corrected as shown in Figure 8, the fatigue life of segmental lining is fatigue curve can be obtained as shown in Figure 9. It can increasing from lg(�) = 7.658 to lg(�) = 8.055 when be seen that, in addition to the depth of segmental lining the depth of segmental lining ranges from 5m to 7m, and of 16m, the fatigue characteristics of diferent depths are it decreased from lg(�) = 8.135 to lg(�) = 7.532 when in the safe area. Trough curve ftting and mathematical Mathematical Problems in Engineering 11

0.32 where �1 and �2 are the infuence parameters of the reinforce- ment layer above segmental lining, �1 is the proportional 0.28 parameter between the static load and depth, � is the � � �

Ｎ equivalent parameter of the upper soil layer, 1, 1,and 1

/＠ 0.24 ； are the ftting parameters of the dynamic load, and �(�) is  HK section: ；/＠Ｎ+Ｇ/＠Ｎ =1/Ｈ a function of position relative to segmental lining stress. 0.20 * F Feature fitting: ；/＠Ｎ =Ｓ0+！∗？ＲＪ(２0 Ｇ/＠Ｎ) It is assumed that the fatigue failure of segmental lining �(�) 0.16 occurs at the same position of diferent depths and is H a constant value. From the fatigue formula (1), the fatigue 0.12 Safe area Unsafe area characteristic equation of segmental lining is obtained as formula (18).

Stress amplitude level amplitude Stress 5m 0.08 7m �(� � �ℎ� (�)) E:(0.484,0.016) (�) =�− 1 1 10m lg � 0.04 � 13m 16m (18) K �(� � (� +� (ℎ + � )) � (�)) 0.2 0.3 0.4 0.60.5 0.7 0.8 0.9 1.0 − 2 � 1 1 ln 1 � Average stress level Ｇ/＠Ｎ �

Fatigue characteristics at diferent depth For a given metro designed service life ��,itcanbe Modifed Goodman fatigue formula used as a judgment standard to meet the design life. When Figure 9: Te modifed Goodman formula and fatigue characteris- the calculation life is more than the design life at diferent tic value at the diferent buried depths. depths, the segmental lining structure meets the fatigue life requirements in any case, so the segmental lining structure does not consider fatigue failure under static-dynamic loads. When the calculation life is less than the design life at � /� + interpolation, the modifed Goodman curve HK is � � diferent depths, the structural design of segmental lining � /� =1/2 � � , the fatigue characteristic curve with diferent does not meet the actual service life, so the segmental lining � /� = 0.029 + 526.31 (−24.33� /� ) depths is � � exp � � ,and design needs to be corrected. When the calculated fatigue the determination coefcient is 0.982. It can be determined life includes the design service life, the depth interval that that the coordinates of the intersection point between the meets the normal design can be determined according to the modifed Goodman curve and the fatigue characteristic designed service life (as shown in Figure 8); that is, lg(�) = curves are (0.484, 0.016), so the depth of segmental lining is lg(��). Trough the calculation of formula (18), a reasonable 14.417m. For the convenience of analysis, the depth is taken depth range can be determined to satisfy the design life of as 2.5D. Terefore, when the depth is greater than 2.5D, segmental lining. the fatigue formula (1) is not applicable to the evaluation of For the depth range that satisfes the fatigue design life, it fatigue of segmental linings and the failure rule needs to be can be seen that there is an optimal depth in the interval and evaluated according to the normal serviceability state. the fatigue life of segmental lining is the largest, so the metro has a longer safety reserve period. Te derivative of lg(�) is 4.5. Optimization Design of Buried Depth with Consideration equal to 0, and the optimal buried depth is shown in formula of Fatigue Life of Segmental Lining. Based on the evaluation (19). of fatigue life of segmental lining, it is necessary to consider �� � � ℎ = 2 1 � −� which depth can meet the design life, has the largest safety � �� � � (19) reserve life of the metro tunnel, and can ignore the efect 1 1 of fatigue life. Terefore, it is necessary to optimize the where the parameter ℎ� is mainly afected by �1,�2,and��. design of segmental lining depth. It is assumed that the Based on the ABAQUS calculation results (see Table 3), most unfavorable position of segmental lining structure is taking the results of joints as an example, the datum is the same position in diferent buried depths, and it provides processed to ft the curves of the static-dynamic load. Te 2 convenience for the optimization design and analysis of the static load curve is � = 0.0486� + 0.4557 ��� � = 0.9752 2 buried depth of segmental lining. andthedynamicloadis� = −0.334 ln(�) + 1.1024 ��� � = Trough the analysis of the additional stress and static 0.9899. Based on formula (18), the calculation life can be load on segmental lining structure, the static-dynamic stress obtained as lg(�) = 5.8338 + 1.837 ln(�) − 0.2158�. formula for diferent position of segmental lining can be According to the current survey of the high-speed railway, obtained as formulas (16) and (17) taking into account the 103 pairs of trains are arranged every day. For the 100 years of efect of strata reinforcement and the relationship between design life, the actual times of dynamic loading are �� = 100⋅ the subgrade dynamic load and depth. 365 ⋅ 103 ⋅ 8 ⋅ 2 = 60152000 and lg(��) = 7.779. Terefore, the depth interval is from 5.51� to 12.45�. According to formula � � (ℎ) =�� �ℎ� (�) (19), the optimal buried depth is calculated as lg(�) = � 1 1 (16) � (5.8338 + 1.837 ln(�) − 0.2158�) =0, and the depth is 8.51m. �� (V,ℎ,�) =�2�� (�1 +�1 ln (ℎ + �1)) � (�) (17) Combined with the modifed Goodman fatigue formula, the 12 Mathematical Problems in Engineering depth of the metro within the range of 5.51m to 15m can [2] F.Ye,C. He, and S. Wang, “Analysis of mechanical characteristic guarantee the design service life of the metro tunnel under of shield tunnel segment lining and its infuence during con- the action of a high-speed railway. struction,” Rock and Soil Mechanics,vol.32,no.6,pp.1801–1807, 2011. [3] C. Shi, M. Lei, L. Peng, and D. Zhao, “Infuenced factors of 5. Conclusion static-dynamic character about tunnel bridge structure,” Journal of Central South University, vol. 42, no. 4, pp. 1085–1091, 2011. In this paper, we analyze and summarize fatigue life char- [4]L.Li,M.-X.Zhang,andH.-M.Wu,“Infuenceofmetrotrain acteristics of metro tunnel structure based on fatigue life loading calculation methods on dynamic responses of shield formula, in terms of static load, dynamic load, and fatigue life tunnel,” Journal of Shanghai Jiaotong University,vol.49,no.7, path. Taking the Guangzhou Metro Line 9 cross underneath pp. 1030–1034, 2015. the Wu-Guang High-Speed Railway as a case, the fatigue life [5] S. Zhou, “New challenges in construction mechanics of urban of segmental lining under the action of high-speed railway rail transit,” Scientia Sinica Technologica,vol.46,no.6,pp.560– is analyzed. Te main results of the research are summarized 569, 2016. below: [6] A. Caratelli, A. Meda, Z. Rinaldi, and S. Spagnuolo, “Precast (1) Tere is a negative linear relationship between the tunnel segments with GFRP reinforcement,” Tunnelling and fatigue life and the static-dynamic load, and the fatigue life Underground Space Technology,vol.60,pp.10–20,2016. gradually decreases with the increase of static-dynamic load, [7]F.Xue,J.Ma,L.Yan,Z.Liu,andQ.Cheng,“Cyclicdynamic which can provide a feasible reference for the design of the test of water-rich loess tunnel subgrade for high-speed railway,” metro tunnel. Journal of Vibration and Shock,vol.29,no.9,pp.226–230,2010. (2) Based on the coupling analysis of the static-dynamic [8]C.He,K.Feng,andY.Fang,“Reviewandprospectsoncon- load of metro tunnels, the fatigue life of segmental lining structing technologies of metro tunnels using shield tunnelling increases at frst and then decreases with the increase of the method,” Journal of Southwest Jiaotong University,vol.50,no.1, depth. pp. 97–109, 2015. (3) Tere is a reasonable buried depth in the interval [9] Y. Yuan, Y. Bai, and J. Liu, “Assessment service state of tunnel between 7m and 10m to maximize the fatigue life of segmental structure,” Tunnelling and Underground Space Technology,vol. lining. What is more, it is concluded that the fatigue failure 27, no. 1, pp. 72–85, 2012. of segmental lining can be ignored when the depth is greater [10] Y. Yang, X. Xie, and R. Wang, “Numerical simulation of than 2.5D and the failure rule needs to be evaluated according dynamic response of operating metro tunnel induced by ground to the normal serviceability state. explosion,” Journal of Rock Mechanics and Geotechnical Engi- (4) According to the optimization design of the fatigue neering,vol.2,no.4,pp.373–384,2010. life, it is concluded that the buried depth in the range of 5.51m [11]M.H.Mussa,A.A.Mutalib,R.Hamid,S.R.Naidu,N.A. to 15m can meet the design service life of 100 years and the M. Radzi, and M. Abedini, “Assessment of damage to an optimal depth in this interval is 8.51m. underground box tunnel by a surface explosion,” Tunnelling and Underground Space Technology,vol.66,pp.64–76,2017. [12] H. Mo, F. Deng, and J. Wang, “Analysis of dynamic responses of Data Availability shield tunnel during metro operation,” Chinese Journal of Rock Mechanics and Engineering,vol.25,no.2,pp.3507–3512,2006. Te data used to support the fndings of this study are [13]K.Luo,Y.Zhao,Z.Luo,andX.Yu,“Numericalsimulation available from the corresponding author upon request. on initial stress and strain state of subway segmented tunnel,” China Civil Engineering Journal,vol.46,no.4,pp.78–84,2013. Conflicts of Interest [14] J. Huang, StudyontheVibrationResponseandFatigueLifeof High-Speed Railway Tunnels Based on Damage Teory,Central Te authors declare that there are no conficts of interest South University, 2010. regarding the publication of this paper. [15] Z. Li, M. Wang, L. Yu, and B. Li, “Dynamic pressure response of foundation base structure in heavy haul railway tunnel,” China Railway Science,vol.37,no.1,pp.71–77,2016. Acknowledgments [16] N. Liu, L. Peng, and C. Shi, “Fatigue life calculation method of tunnel base structure under the conditions of bedrock void,” Te authors would like to acknowledge the National Natural Journal of Railway Science and Engineering,vol.13,no.5,pp. Science Foundation of China (Grant No. 51278423 and 921–928, 2016. Grant No. 51478395) for their fnancial support. Equally, the [17] N. Liu, L.-M. Peng, and C.-H. Shi, “Fatigue life prediction of authors would like to thank Wenjing Lu from China for her tunnel base structure under the sofening surrounding rock help in linguistic assistance during the preparation of this conditions,” Journal of Vibration Engineering,vol.29,no.5,pp. manuscript. 936–944, 2016. [18] R. Tepfers and T. Kutti, “Fatigue strength of plain, ordinary, References and lightweight concrete,” Journal Proceedings,vol.76,no.5,pp. 635–652, 1979. [1] Q. Qihu, “Present state, problems and development trends of [19] M. Satio and S. Imai, “Direct tensile fatigue of concrete by the urban underground space in China,” Tunnelling and Under- use of friction grips,” Journal of the ACI,vol.80,no.5,pp.431– ground Space Technology, vol. 55, pp. 280–289, 2016. 438, 1983. Mathematical Problems in Engineering 13

[20] H. A. W. Cornelissen and H. W. Reinhardt, “Uniaxial tensile fatigue failure of concrete under constant-amplitude and pro- gramme loading,” Magazine of Concrete Research,vol.36,no. 129, pp. 216–226, 1984. [21]G.Zhao,P.Wu,andW.Zhan,“Tefatiguebehaviorof high-strength concrete under tension,” China Civil Engineering Journal,vol.26,no.6,pp.13–19,1993. [22] Y. Song, Fatigue Behavior and Design Principle of Concrete Structure, China Machine Press, 2006. [23] B. Xiang, J. Shi, L. Guo, X. Liu, and J. Lin, “Plotting and appli- cation of goodman fatigue limit diagram of railway common materials,” China Railway Science,vol.23,no.4,pp.72–76,2002. [24] N. Liu, L. Peng, C. Shi, and M. Lei, “Experimental and model study on dynamic behaviour and fatigue damage of tunnel invert,” Construction and Building Materials,vol.126,pp.777– 784, 2016. [25] T.-Y. Qi, “Settlement characteristics of strata and buildings caused by metro tunneling,” Chinese Journal of Geotechnical Engineering,vol.34,no.7,pp.1283–1290,2012. [26]J.Liu,T.Qi,andZ.Wu,“Analysisofgroundmovementdue to metro station driven with enlarging shield tunnels under building and its parameter sensitivity analysis,” Tunnelling and Underground Space Technology,vol.28,no.1,pp.287–296,2012. [27] C. Pan and G. N. Pande, “Preliminary deterministic fnite element study on a tunnel driven in loess subjected to train loading,” China Civil Engineering Journal,vol.17,no.4,pp.19– 28, 1984. [28] H. Jenkins, “Te efect of track and vehicle parameter on wheel- rail vertical dynamic forces,” Railway Engineering,vol.3,no.1, pp. 21–27, 1974. [29] X. Song and W. Zhai, “Dynamic stress distribution of the infrastructure of CRTS II slab ballastless track under high speed moving load,” China Railway Science,vol.33,no.4,pp.1–7,2012. [30] H. Jiang, X. Bian, Y. Chen, and J. Jiang, “Full-scale accelerated testing for simulation of train moving loads in track-subgrade system of high-speed railways,” China Civil Engineering Journal, vol.48,no.9,pp.85–95,2015. [31] X. Liu, Te Settlement Control and Dynamic Stability of the Red- Clay Subgrade of Ballastless Track of High-Speed Railway,China Railway Press, 2010. Advances in Advances in Journal of The Scientifc Journal of Operations Research Decision Sciences Applied Mathematics World Journal Probability and Statistics Hindawi Hindawi Hindawi Hindawi Publishing Corporation Hindawi www.hindawi.com Volume 2018 www.hindawi.com Volume 2018 www.hindawi.com Volume 2018 http://www.hindawi.comwww.hindawi.com Volume 20182013 www.hindawi.com Volume 2018

International Journal of Mathematics and Mathematical Sciences

Journal of Optimization Hindawi Hindawi www.hindawi.com Volume 2018 www.hindawi.com Volume 2018

Submit your manuscripts at www.hindawi.com

International Journal of Engineering International Journal of Mathematics Analysis Hindawi Hindawi www.hindawi.com Volume 2018 www.hindawi.com Volume 2018

Journal of Advances in Mathematical Problems International Journal of Discrete Dynamics in Complex Analysis Numerical Analysis in Engineering Dierential Equations Nature and Society Hindawi Hindawi Hindawi Hindawi Hindawi www.hindawi.com Volume 2018 www.hindawi.com Volume 2018 www.hindawi.com Volume 2018 www.hindawi.com Volume 2018 www.hindawi.com Volume 2018

International Journal of Journal of Journal of Abstract and Advances in Stochastic Analysis Mathematics Function Spaces Applied Analysis Mathematical Physics Hindawi Hindawi Hindawi Hindawi Hindawi www.hindawi.com Volume 2018 www.hindawi.com Volume 2018 www.hindawi.com Volume 2018 www.hindawi.com Volume 2018 www.hindawi.com Volume 2018