Ceramic -metal functiona lly graded materials (FGMs) have been extensively used in aerospace engineering where high strength and excellent heat insulation materials are desired . In this paper , the thermodynamic behavior of the Thermal Protection System (TPS) used bolted joints mad e up of porous ZrO 2/(ZrO 2+Ni) FGMs is investigated by finite -element (FE) modeling . The bolted joint is subjected to reentry heating corresponding to the Access to Space Vehicle. Thermodynamic simulations are carried out to yield the transient response of the porous ZrO 2/(ZrO 2+Ni) functionally graded bolted joint (FGBJ) . The effects of the preload on the thermomechanical behavior and service reliability of the bolted joint are numerically analyzed in detail by ABAQUS codes. It is found that the preload rela xation of the bolted joint occurs at elevated temperature, and the preload has significant influence on service reliability of the bolted joint under transient thermomechanical circumstances . With the increase of the preload , stress concentration which occ urs at the root of the first thread of the bolt increases rapidly and predominates in service reliability . Proper preload is thus defined to balance the service reliability and tightness of the bolted joint . Further studies show that the shape of the nut h as a great effect on the stress concentration of the thread, the optim ized nut is designed to reduce the stress concentration of the thread, and thus the reliability of the bolted joint is also improved. Keywords: A. Ceramic -matrix composites (CMCs); B. Thermomechanical; C. Finite element analysis (FEA); E. Joints/joining * Corresponding author. Tel.: +86 010 627 57417 Email address: sgai@pku.edu.cn and mjchen81@gmail.com . 2 1. Introduction Thermal Protection System (TPS) is usually covered on the surface of the aircraft as the thermal protecting layer . Due to the complex shape and structure of the aircraft , a unibody design and manufacture of TPS is infeasible , therefore it is unavoidable to connect lots of thermal protection components . As a result, heat short s cause d by the gap and joints between thermal protection components cannot be ignored [1]. Heat shorts may lead the interior part of the fuselage structure to an extreme high temperature , severely influencing the heat insulation property of the TPS. In the process of service, the TPS has to withstand not only thermal loads, but also mechanical loads a s well as harsh chemical environments, repeatedly without failure [2]. Therefore, in the study of the TPS, the analysis of heat short s effect and the use of high temperature resistant joints with high strength are indispensable. Many efforts have been don e on metallic thermal protection system, including the design and characterization of thermal insulation material/ structure [3-8], however, heat shorts blocking joints have been rarely studied. Also , most researches focused on metal joints [ 9-15], which had quite poor heat insulation properties. Some studies were carried out on C/SiC joints [1 6-18], yet the com plicated fabrication approaches for geometrically complex composites such as bolted joints have certain limitations in terms of size and shape, which have greatly l imited their wider applications. The studies for the thermodynamic properties of these joints which are quite necessary for joints used on TPS are less . Functionally graded materials (FGMs) are ideal candidates for the applications requiring multifunctional performance . For example, the ceramic -metal FGMs can be designed to take advantages of the mechanical strength of metals and the heat and corrosion resi stance of ceramics [ 19-20]. In our previous work, we fabricated and experimentally inve stigated the microstructure and therm omechanical properties of the porous ZrO 2/(ZrO 2+Ni) ceramics and found that they had a low thermal conductivity and high strength [21-22]. Therefore, the ceramic -metal FGMs can be a potential candidate for heat short s blocking joints. However, most of the FGMs have thermal stress due to a gradually macroscopic change in composition, which are destined to affect the mechanical property and service reliability of the FGMs [ 23- 24]. Also, the preload is an important factor t hat cannot be neglected. On the one hand, the preload can enhance the reliability and tightness of bolted joints . On the other hand, the preload may cause the fracture of bolted joints in advance under thermomechanical circumstances. So it is of great impo rtance to study the thermodynamic properties of bolted joints made up of ceramic -metal FGMs , which will guide the preload and structure design of joints. Finite element analysis (FEA) has been widely used to analyze the thermal and thermomechanical propert ies of various materials and structures [25-29], since the method through experimental 3 study to obtain the thermomechanical properties of the ceramic -metal functionally graded bolted joint (FGBJ) is accurate but time -consuming, so the theoretical simulatio n and optimization would be rather important for the successful implementation of ceramic -metal functionally graded jointing technology in a wide variety of high temperature application. In this paper, firstly a thermomechanical coupled model is constructe d to study the thermodynamic properties of the ceramic -metal functionally graded bolted joint (FGBJ) , wh ose material systems (porous ZrO 2/(ZrO 2+Ni) FGMs ) are fabricated by cold isostatic pressing and pressureless sintering (CIP -PLS) . Microstructures and mechanical properties of the FGMs are experimental ly studied in our previous work [21- 22]. Then the heat transfer and thermomechanical responses are investigated under different preload s to study the service reliability of the bolted joint . The effects of the preload on the thermomechanical responses and service reliability of the bolted joint are analyzed in detail , thus the proper preload is proposed to balance the service reliability and tightness of the bolted joint. Fi nally the optimal structural design of the bolted joint under thermomechanical circumstances is done to improve the service reliability. The findings will highlight the optimization of the preload and the structure of the bolted joint. 2. Material s and structure of the bolted joint Porous Z rO2/(ZrO 2+Ni) functionally graded samples are fabricated , with the height of the bearing load layer (layer a , 12% porosity ), transition layer (layer b , 15% porosity ) and middle layer (layer c , 30% porosity ) being la=10mm, lb=2mm, and lc=10mm in the axial direction , respectively (see Fig. 1). These samples are further manufactured into the functionally graded bolted joint s. The materials and structure of the bolted joint are shown in Fig. 2. The bolt thread , bolt head and nut are all comprised of the bearing load layer (ZrO 2+30 vol. %Ni) to ensure the mechanical strength and service reliability . The bolt shank without thread s is mainly composed of the middle layer (ZrO 2), which serve s the purpose of heat resistan ce. The transition layer ( ZrO 2+15 vol. %Ni ) on the bolt shank is designed to reduce the residual stress es which are generated inevitably due to the mismatch in coefficients of thermal expansion between different layers . The transition layer also serves the purpose of cutting the fillet. According to the standard handbook of fastening and joining [30], the diameter and height of the bolt head are s=12.8mm and k=5.5mm , respectively . The diameter of the bolt shank is D=7.8mm, which is also the major diameter of the external thread . The thread pitch is p=1.25mm . The height and diameter of the nut are m=6.8mm and s=13mm, respectively. The number of thread s on the nut is thus calculated to be five. In th is study, the number of threads is not a significant factor because stress concentration at the roots of the threads is found to vary little after the first three threads. The thread class used in this study is class 3A and 4 3B for external and internal threads respectively, since this class of thread is most used in high -precision applications . The root radius f or the threads is taken as L/6, where L is the thread height [30]. The bolted joints with the preload are called high -strength bolts, which usually need only 2~3 thread s out of the nut when the bolt and nut are fully engaged. Therefore, we suppose the numb er of thread s on bolt shank is eight, and the height of the bolt shank with threads is accordingly to be b=10mm. 3. Numerical Modeling and Finite Element Analysis 3.1. Finite element model To study the thermomechanical behaviors of the bolted joint, a two-dimensional ( 2D) axisymmetric finite element model is constructed based on ABAQUS code using the advantage of symmetry, as illustrated in Fig. 3 . The error resulting from modeling the bolt and nut axisymmetrically and neglecting the helix angle is ignor ed in this study , since t he helix angle is very small (less than 2 deg) on most bolts and thus the three -dimensional effect can be neglected . In fact, t he 2D approximation for such a problem was also adopted in the earlier experimental photoelastic studies [31]. The model consists of 48147 elements with 48976 nodes. The finite element mesh is refined using a very small element length at the crest and root of the threads. The nodes between two mutually contacted threads are coincident to improve the converge nce. The thermomechanical response of the system is obtained in two steps. First, the heat transfer problem is solved to obtain the time - dependent temperature distribution in the bolted joint due to the applied thermal loads and boundary conditions. Next, the thermal stresses caused by the temperature distributions are determined. The solutions are facilitated by using the same mesh for both the heat transfer and thermal stress problems. The DCAX 4 elements are used for the heat transfer analysis and the CAX 4 elements for the thermal stress analysis. The material parameters (mechanical properties and thermal properties) used in the numerical simulations are listed in Table 1 -2. It should be noted that, the material parameters for ZrO 2 and Ni were only experi mental ly tested at some discrete temperature points, while the parameters that have not been measured at other temperatures will be given by interpolation in the simulations. The material constants in each composition, such as elastic s modulus, thermal con ductivity, coeffic ient of thermal expansion (CTE), specific heat and Poisson’s ratio are calculated as follows [32-33]: 5 0 0 0 0 0 0 0 0 0 0 2/3 2/3 00 1/3 1/3 2/3 2/3 008 (1 )(1 )(23 8 ) 8(1 )(23 8 ) (5 )(37 8 ) ,, [(1 ) ] (1 ) [(1 ) ]g gEPEP CC PP P P P P                              (1) where P is the porosity, g is the the rmal conductivity of air , 0E, 0C, 0, 0and 0 are all dependent on the ceramic volume fraction and can be expressed from the Vegard's rule below [34]: 0 03( )[1 ]3 ( )(1 )ce ce m m m ce ce ce m ce mM M V M V V         (2) where M represents the material property parameters ( E , , C,  and  ), ceV donates the ceramic volume fraction, and subscript s ‘ce’ and ‘m’ represent ceramic and metal, respectively. The mate rial model used is based on the tension experiment of the p orous ZrO 2/(ZrO 2+Ni) FGMs . The tensile stress -strain relationship is show in Fig. 4. It should be noted that, the stress and strain obtained by experimental test are only the nominal stress and nom inal strain, which should be transformed into the true stress and true stain used in ABAQUS. The plastic strain which is input into ABAQUS is calculated as (1 )ln(1 )  nom nom pl nomE (3) where Eis the Young’s modulus , pl, nom , nom are the plastic strain, the nominal stress and nominal stain , respectively. 3.2. Simulation pro cess Firstly, the heat transfer simulations are carried out to obtain the time -dependent tem perature distribution in the bolted joint . The initial temperature of the specimen is 2 5oC (i.e., room temperature), the ambient air temperature of the top and bottom surface s of the bolted joint are set as 1200oC and 25oC, respectively. The temperature de pendent convection and radiation coefficient s can be seen in Table 1. The total loading time is 1000 s. The transient temperature dis tribution is then computed. After the heat transfer process, a coupled thermal -mechanical approach based on ABAQUS nonlinea r finite element codes is adopted . Contact is defined between all the contacted surfaces . The friction coefficient of bolted joints usually ranges from 0.1 ~0.2 [35], and thus the same friction coefficient 0.15 is adopted here between all the contacted surf aces. The axis of symmetry of the bolt is restrained in the radial direction. The upper surface of the nut contacted with the plate is restrained in the horizontal y direction, while the lower 6 surface of the nut is free. A surface pressure of F=100MPa is applied to the top surface of the bolt head . The preload is applied to the bolt through the bolt load method in ABAQUS. It should be noted that, the load s are applied gradually (little loads are applied initially) and the constraints are release d stepwise (extra constraints are need ed initially) to realize the convergence . The thermal load is applied by importing the transient temperature dis tribution calculated f rom the heat transfer process . The t hermomechanical coupling studied here is conducted by applyi ng the mechanical load and thermal load simultaneously, which is called transient thermodynamic behavior. 4. Results and Discussion 4.1. Heat transfer Heat transfer behaviors of the bolted joint in 1000s are numerical ly simulated and the temperature co ntour of the bolted joint at the end of the heat transfer is shown in Fig . 5a. The results show that the temperature of the top surface of the bolt at 1000s is approximately equal to the top surface ambient air temperature (1200oC). The temperature decrea ses along the height direction of the bolt . The bottom surface temperature of the bolt is about 150oC at 1000s , which has 125oC increase compared with the original temperature. Fig. 5b illustrates the temperature history of the bolt at different locations , i.e. the top surface of the bolt , the bottom surface of the bolt, the upper (left) surface of the middle layer (ZrO 2), the center of the middle layer (ZrO 2) and the lower (right) surface of the middle layer (ZrO 2). As shown in Fig. 5b, temperature s at different locations increase monotonously with time initially, and the n reach the steady state almost at the same time 500s. The steady temperature s of the upper (left) and lower (right) surface s of the middle layer (ZrO 2) are 1029oC and 397oC, respectively . Compared with the steady temperatures of the top surface ( 1190oC) and bottom surface (125oC) of the bolt, it is clear that the middle layer (ZrO 2) plays a domina nt role in the heat resistance. This is because that the conductivity of ZrO 2 is lower than tha t of the upper -layer material (ZrO 2+30vol. % Ni). As listed in Table 1, through Eq. (1) and (2), the conductivity of the upper -layer material (ZrO 2+30vol. %Ni) is approximately 3~4 times greater than that of ZrO 2. The equivalent heat resistance s chematic of the bolt in the height direction is shown in Fig. 6. By Fourier’ s law of heat conduction , the equivalent conductivity of the bolt in the height direction can be approximately calculated as [36]: effL AR (4) where L, A are the equivalent height and cross -sectional area of the bolt correspondingly , R is the equivalent heat resistance. The heat resistance Ri of each layer is given by 7 i i iilRA (5) where il , iA, i are the height, cross -sectional area and conductivity of each layer , respectively . The equivalent model of the whole bolt can be assumed as the series model . The equivalent heat resistance R and equivalent length L of the whole bolt can then be g iven by i i i iLl RR   (6) Combin ing Eq. ( 4)-(6), the equi valent conductivity of the bolt in the height direction is express ed as i i eff i i il l   (7) Besides the low er conductivity, the height of the ZrO 2 layer is also an important f actor influencing eff . i il  of the porous ZrO 2 layer is much greater than other layers, so the middle layer (porous ZrO 2) plays a domina nt role in the heat resistance . eff of the functional ly graded bolt is 1.54 W/(m∙K) , while eff of the corresponding pure ZrO 2 bolt (30% porosity of layer a, without the addition of metal phase to layer b and layer c) is 1.40 W/(m∙K) . The addition of metal phase to improve thermal condu ctivity of the functionally graded bolt is very limited. 4.2. Thermodynamic behavior 4.2.1 Preload relaxation of bolted joints To study the variations of the preload at elevated temperature, only t he thermal load is applied during 0 - 1000s by importing the transient temperature dis tribution calculated from the heat transfer simulations . The mechanical load (top surface pressure ) is applied after the temperature distribution in the bolt gets steady (i.e. after 1000s), and the total loading time is 2000s. Different preloads (15, 33.5 and 50MPa ) are applied to the bolted joint at the beginning of the simulation. For high strength bolts , based on the standard manual , the maximum preload is less than half of the tensile strength at room temperature. In this study , the maximum preload is set as 50MPa, which is approximately 0.3 times of the tensile strength at room temperature (168.6MPa). The axial stress -loading time curves under different preloads are illustrated in Fig. 7 , taking location A as an example. A is l ocated in the center of the middle layer ( ZrO 2). Fig. 7 shows that , as the heat transfer proceeds, the axial stress es under different preloads quickly reach the maximum value almost at the same time (t=100s ), then gradually decrease to the same value at th e end of the heat transfer process. To 8 illustrate this phenomenon, the axial stress at location A without pre load is also plotted in Fig. 7, which is also called the axial thermal stress because there is no preload in this situation. As shown in Fig. 7, the axial stress with preloads has the same variation trend as the axial thermal stress. Therefore, the maximum value of the axial stress is the result of the maximum thermal stress . For different preloads, the thermal stress gets the peak value at the same time (t=100s), so the axial stress under different preloads reaches the maximum valu e simultaneously . The thermal stress gets the maximum value when t=100s because the temperature difference between different layers at this time is the greatest. As shown in Fig. 5b, the temperature of the top layer increases much quickly than that of the middle layer (ZrO 2). The temperature difference between these layers quickly increases as the heat transfer proceeds, reaching the maximum at almost t=100s . Then it gradual ly decrease s and becomes constant as the heat transfer become steady after t=400s. The thermal stress also become constant after t=400s, which is the same trend as the temperature difference. From the simulation results it is learn ed that, preload relaxati on of the bolted joint happens at elevated temperature , the preload gradually decrease s in the heat transfer process and will decrease to a near zero value if the heat conduction process is sufficient . 4.2.2 Transient thermodynamic behavior The therm al load is applied during 0 -1000s by importing the transient temperature calculated fro m the heat transfer simulation , and is kept constant after 1000s. The mechanical load (top surface pressure of 10 0MPa) is applied simultaneously with the t hermal load . To study the effects of the preload on the thermo dynamic responses and service reliability of the bolted joint, different preloads are applied to the bolted joint at the beginning of the simulation. The most potential dangerous locations of the bolted joint are the interfac e between the bearing load layer (ZrO 2+30 vol. %Ni ) and the transition layer ( ZrO 2+15 vol. %Ni ), and the root of the first thread, which are shown in Fig. 5a and are marked as locatio n B and C. This is because the residual thermal stress at location B is the maximum tensile stress , which can be seen in our previous work [22]. Besides , stress concentration is the most severe at location C. In the present work , we only give results of three representative situations, namely, Little preload (15MPa), Critical preload (33.5MPa) and Big preload (50MPa) . For each situation, the axial stress es at some key locations, such as B and C, are particularly investigated. The a xial stress -time curves are illustrated in Fig. 8-10. Figure 8a-b plot the axial s tress-loading time curve s of location B and C under Little preload (15MP a). The corresponding temperature -time and strength -time curves are also plotted as comparison. Here, if the axial stress exceeds the material strength , the bolt is assumed to fail . In the loading process, the axial stress increases with 9 the increment of temperature, while the material strength decreases with the increment of temperature. So, the bolt will failure when the local axial stress exceeds the strength of the local material . As shown in Fig. 8a-b, material failure at location B happens when t=263s, while it occurs at location C at the time t= 284s. Therefore, when Little preload (15MPa) is applied, thermal stresses dominate the service reliability of bolted joints . The most dan gerous location of the bolt is at the interface between the bearing load layer (ZrO 2+30 vol. %Ni ) and the transition layer ( ZrO 2+15 vol. %Ni ), where the thermal stress is the maximum tensile stress . As the preload increases, the axial stress at location C increases much more rapidly than that of location B because of the existence of stress concentration at location C , so there should have a critical preload when material failure at location B and C happens at the same time. Under the Critical preload, the bolt joint will play the best performance and a key of this simulation in the present work is to find out the Critical preload based on a certain bolt. The axial stress -loading time curve s of location B and C under Critical preload (33.5MPa) are shown in F ig. 9a-b. It should be noted that this Critical preload is identified by a series of continuous simulations with different preloads increasing from Little preload (15MP a) to B ig preload (50MPa). As illustrated in Fig. 9a-b, failure almost happens simultane ously at location B and C when t= 221s under the Critical preload. Therefore, it is clear that the preload has significant influence on the service reliability of the bolted joint under transient thermomechanical circumstances . Stress concentration occurri ng at the root of the first thread of the bolt increases rapidly and predominates in the service reliability of the bolted joint with the increase of the preload. Proper preload is thus defined to be this Critical preload to balance the service reliability and tightness of bolted joints. To verify the correctness of this conclusion, the axial stress -loading time curve s of location B and C under Big preload (50MPa) are shown in Fig. 1 0a-b. As shown in Fig. 1 0a-b, failure keeps occurring at location C when th e preload exceeds the Critical preload, and with the increase of the preload , failure of the bolt happens earlier and earlier , that is , the service reliability of the bolted joint decreases. 4.3. Stress concentration Stress concentration occurring at the thread root predominates the service reliability of the bolted joint . In this section, the stress concentration of the porous ZrO 2/(ZrO 2+Ni) functionally graded bolt under transient thermomechanical circumstances is studied and compared with that of the pur e ceramic ( ZrO 2) bolt. The thermal load is applied during 0 -1000s by importing the transient temperature calculated f rom the heat transfer simulations , and is kept constant from 1000 to 2000s . The mechanical load (top surface load of 10 00N) is applied simu ltaneously with the thermal load . The preload is 33.5MPa for the functionally graded bolt, and no preload is applied to the ZrO 2 bolt. Stress concentration factor is defined as the ratio of maximum axial stress in 10 the nut-bolt system to the nominal axial stress in the shank of the bolt. Fig. 1 1 show s the axial stress contour of the functionally graded bolt and ZrO 2 ceramic bolt at the end of the thermomechanical process. The axial stress decreases progressively from the first thread to the subsequent thread s. Dividing the top surface load (1000N) by the corresponding cross -sectional area, the nominal axial stress in the shank of the bolt is calculated as 20.9MPa, so the maximum stress concentration factor occurring at the root of the first thread of the functionally graded bolt and ZrO 2 ceramic bolt is 7.22 and 8.86, respectively. It is clear that the maximum stress concentration factor of the functionally graded bolt is reduced compared with that of ZrO 2 ceramic bolt. Since stress concentration occurring at the root of the first thread of the bolt predominates the service reliability of the bolted joint, the service reliability of the functionally graded bolt is improved compared with that of ZrO 2 ceramic bolt. It is because the addition of metal phase to threads of the functionally graded bolt can release the stress due to plastic deformation. The maximum axial stress -loading time curves of the ZrO 2 ceramic bolt under different preloads are illustrated in Fig. 12. As shown in Fig. 12, even under very little p reload, the axial stress grow s rapidly and even reach es the limit strength at the initial loading process because of the stress conc entration at the first threa d. Hence , almost no preload is able to be applied on the pure ceramic bolt, which makes it vulne rable to looseness under thermodynamic circumstance. The service reliability of the pure ceramic bolt is therefore lower than that of the functionally graded bolt . Stress concentration can greatly degrade the high -temperature mechanical performance. S. Ven katesan’s research showed that grooves on the nut can significantly reduce the stress concentration factor and imp rove the mechanical performance [37 ], but whether it is effective under high temperature still needs to be studied. Inspired by S. Venkatesan’ s research, an internal groove and an external groove are constructed on the nut and studied under transient thermomechanical circumstance , as shown in Fig. 1 3-14. The one without a groove is also plotted as comparison s. Three different cases are simulated for the functionally graded bolted joint with an internal groove . (i) b/l=1/4, (ii) b/l=2/4, (iii) b/l=3/4, where b and l are the width of the nut after and before grooving. The depth of the internal groove is fixed at a=p/4, where p=1.25mm is the thread pitch. As shown in Fig. 13, the maximum stress concentration factor still occurs at the root of the first thread , and gradually decrease s as the decrease of b/l. When b/l=1/4, the maximum stress concentration factor decrease s to 5.79 for the functionally g raded bolt. Compar ed with the bolted joint without the groove , the maximum stress concentration factor of the functionally graded bolt decreas es by 19.81%. Four different cases are simulated for the functionally graded bolted joint with an external groove , (i) a/h=1/4, b/l=2/3, (ii) a/h=1/4, b/l=5/6, (iii) a/h=1/2, b/l=2/3, (iv) a/h=1/2, b/l=5/6. It is clear from Fig. 1 4 that the maximum stress concentration factors all 11 have a decrease for the four cases. When a/h=1/4, b/l=5/6, the maximum stress concentrat ion factor decrease s to 5.03 for the functionally graded bolt. Compar ed with the bolted joint without the groove , the maximum stress concentration factor for the functionally graded bolt decreas es by 30.33%. Comparing the internal groove with the external groove, the nut with an external groove ( a/h=1/4, b/l=5/6) acts be st performance to reduce the stress concentration of the thread and achieve s the best reliability of the joints . Therefore, the shape of the nut has a great effect on the stress concentratio n of the bolted joint . Both the internal and external groove can make the loading position of the nut stretches outward along the radial direction, which makes the nut bear the bending moment in an anti clockwise direction ( zM , based on the coordinate system shown in Fig. 3). Hence , the contact tightness between the first thread of the bolt and the nut decrease s, while the contact tightness between the last thread of the bolt and the nut increase s, resulting in the decrease of the maximum stress concentration factor at the first thread. 5. Conclusions Thermodynamic behavior and service reliability of the Thermal Protection System (TPS) used bolted joints made up of porous ZrO 2/(ZrO 2+Ni) FGMs are numerically analyzed in detail by ABAQUS codes. From the numerical simulations some conclusion s can be drawn: (1) The middle layer ( porous ZrO 2) plays a domina nt role in the heat resistance of the functionally graded bolt. The addition of metal phase will obviously improve the mechanical performance of the bolted joint while it to improve thermal conductivity of the bolt is very limited. (2) Preload relaxation of the bolted joint happens at elevated temperature. The preload decrea ses gradually as the heat transfer proceeds and it reaches a near zero value if the heat conduction process is sufficient . (3) Preload has significant influence on the service reliability of the bolted joint under transient thermomechanical circumstances . The critical preload 33.5MPa is recommended for the functionally graded bolt studied in this work . (4) The shape of the nut has a great effect on the stress concentration of the bolt , the nut with an external groove (a/h=1/4, b/l=5/6) can significantly reduce the stress concentration at the first thread of the bolted joint , and thus the reliability of the bolted joint is also improved. Acknowledgments The authors are grateful for the support by the National Natural Science Foundation of China (Grant nos. 11090330, 11090331, 11102003 and 11472038 ), the National Basic Research Program of China (973 Program, 12 2010CB83270 1, 2011CB610303 ), and the Foundation of the Author of National Excellent Doctoral Dissertation of China (No. 201029) .