Patent Publication Number: US-2009228254-A1

Title: Method of simulating pneumatic tire based on finite element models

Description:
FIELD OF THE INVENTION 
     The present invention relates to a method of simulating a pneumatic tire based on finite element models. 
     BACKGROUND ART 
     There have been proposed in the art a variety of methods of simulating a pneumatic tire (also referred to simply as “tire”) by approximating the pneumatic tire with a finite element model which is generated by dividing the tire into a plurality of finite elements and analyzing the finite element model according to the finite element method. 
     According to one of the proposed pneumatic tire simulating methods, a composite assembly (stiffener) of a belt and a carcass, each including a plurality of cords, among other components of the tire, is converted into a simplified model, rather than a faithful model (see JP No. 11-153520 A). Specifically, the cords are modeled into a quadrilateral membrane element with defined anisotropy, and the composite assembly is calculated as a continuous body whose cross-sectional shape remains the same in the circumferential directions of the tire. 
     The reasons for the above simulation proposal are as follows: If the cords of the composite assembly are to be converted into a faithful model, then the cords have to be divided into a number of elements. Since the carcass or belt of a single tire includes more than thousand cords, the simulation needs vast computational efforts and hence is highly tedious and time-consuming to carry out. However, the simulating computational efforts and the time required for the simulating process can be greatly reduced if the cords are modeled into simply shaped elements. 
     The proposed simulating method, however, is disadvantageous in that as the cords of the composite assembly is modeled into a quadrilateral membrane element that is completely different in shape from the actual cords, the quadrilateral membrane element fails to properly analyze the behaviors of the composite assembly. The proposed simulating method thus fails to sufficiently analyze the performance of the tire with accuracy. 
     SUMMARY OF THE INVENTION 
     It is an object of the present invention to provide a method of simulating a pneumatic tire for accurately analyzing the performance of the pneumatic tire based on finite element models. 
     According to an aspect of the present invention, there is provided a method of simulating a pneumatic tire including a composite assembly comprising a rubber web and a plurality of parallel cords embedded in the rubber web at spaced intervals in the circumferential directions of the pneumatic tire, by approximating the pneumatic tire with a finite element model generated by dividing the pneumatic tire into a plurality of finite elements and analyzing the finite element model according to a finite element process, the method comprising the step of generating a composite assembly element model for the composite assembly by dividing the rubber web into rubber web elements as solid models and dividing the cords into cord elements as solid elements, according to the finite element process. 
     Since the composite assembly element model for the composite assembly is made up of the rubber web elements as solid models and the cord elements as solid elements, stresses and strains of the composite assembly can be analyzed accurately for an accurate evaluation of the performance of the pneumatic tire. 
     The above and other objects, features, and advantages of the present invention will become apparent from the following description when taken in conjunction with the accompanying drawings which illustrate a preferred embodiment of the present invention by way of example. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a transverse cross-sectional view of a pneumatic tire taken along a plane extending through the central axis thereof; 
         FIG. 2  is an enlarged fragmentary perspective view, partly in cross section, showing structural details of a composite assembly of a carcass and a belt of the pneumatic tire shown in  FIG. 1 ; 
         FIG. 3  is a plan view of the composite assembly as viewed in the direction of arrow A in  FIG. 2 ; 
         FIG. 4  is a block diagram of a computer used for carrying out a method of simulating a pneumatic tire according to an embodiment of the present invention; 
         FIG. 5  is a block diagram of a functional system provided by the computer shown in  FIG. 4 ; 
         FIG. 6(A)  is a schematic cross-sectional view of the composite assembly; 
         FIG. 6(B)  is a schematic cross-sectional view of a finite element model of the composite assembly according to the background art; 
         FIG. 6(C)  is a schematic cross-sectional view of a finite element model of the composite assembly according to the embodiment of the present invention; 
         FIG. 7  is a partial perspective view of a finite element model of the carcass of the composite assembly according to the embodiment of the present invention; 
         FIG. 8  is an enlarged perspective view of a portion of the finite element model; 
         FIG. 9  is a schematic view showing the constraint of nodes of the finite element model of the composite assembly according to the embodiment of the present invention; 
         FIG. 10  is a flowchart of the sequence of the method of simulating a pneumatic tire according to the embodiment of the present invention; 
         FIG. 11  is a view showing a distribution of stresses (strains) on a pneumatic tire simulated by a simulating method according to a comparative example; and 
         FIG. 12  is a view showing a distribution of stresses (strains) on a pneumatic tire simulated by a simulating method according to an inventive example. 
     
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     A method of simulating a pneumatic tire according to an embodiment of the present invention will be described in detail below with reference to the drawings. 
     First, a composite assembly (stiffener) of a tire and a process of modeling the composite assembly will be described below. 
       FIG. 1  is a transverse cross-sectional view of a pneumatic tire  10  taken along a plane extending through the central axis thereof.  FIG. 2  is an enlarged fragmentary perspective view, partly in cross section, showing structural details of a composite assembly of a carcass and a belt of the pneumatic tire shown in  FIG. 1 .  FIG. 3  is a plan view of the composite assembly as viewed in the direction of arrow A in  FIG. 2 . 
     As shown in  FIG. 1 , the tire  10  comprises a pair of laterally spaced beads  12 , a pair of laterally spaced sidewalls  14 , a tread  16 , a carcass  18 , and a belt  20 . The carcass  18  and the belt  20  jointly serve as a composite assembly as described later. 
     The beads  12 , which serve to mount the tire  10  on the rim of a wheel, support the opposite ends of the carcass  18 . The beads  12  include bead wires  12 A and bead fillers. 
     The sidewalls  14  connect the beads  12  and the tread  16  to each other. 
     The tread  16 , which serve to contact the ground, has an outer circumferential surface  16  with grooves  16 A defined therein that provide a tread pattern. 
     The carcass  18  extends in and along the inner surfaces of the sidewalls  14  and the tread  16  between the beads  12 . The carcass  18  has its opposite ends folded back on themselves around the bead wires  12 A and the bead fillers from the inner to outer sides of the beads  12 . The carcass  18  extends fully in the circumferential directions of the tire  10 , keeping the shape of the tire  10 . 
     In the tread  16 , the carcass  18  extends circumferentially in an inner portion of the tread  16 . 
     As shown in  FIGS. 2 and 3 , the carcass  18  comprises a rubber web  18 A and a plurality of parallel cords  18 B embedded in the rubber web  18 A at spaced intervals in the circumferential directions of the tire  10 . The cords  18 B extend transversely across the tire  10 . 
     The rubber web  18 A may be made of any of various known rubbers. The cords  18 B may be made of any of various known steels or plastics. 
     The cords  18 B may be spaced at equal intervals. Alternatively, the cords  18 B may be spaced at different intervals which may be constant or may vary in their longitudinal direction. The cords  18 B may be straight throughout their entire length or may be curved or tortuous partly or fully along their entire length. 
     The belt  20  is disposed in an inner portion of the tread  16  and extends in the circumferential directions of the tire  10 . 
     Specifically, the belt  20  is disposed between the outer circumferential surface of the carcass  18  and the inner circumferential surface of the portion of the tread  16  which has the tread pattern. The belt  20  is tightly held against the carcass  18  radially inwardly of the tire  10 , stiffening the tire  10  to prevent the tire  10  from being unduly expanded when it is inflated. 
     As shown in  FIGS. 2 and 3 , the belt  20  comprises a first belt member  22  disposed on the outer circumferential surface of the carcass  18  and a second belt member  24  disposed on the outer circumferential surface of the first belt member  22 . 
     The first belt member  22  comprises a rubber web  22 A and a plurality of parallel cords  22 B embedded in the rubber web  22 A and spaced from each other. 
     The second belt member  24  comprises a rubber web  24 A and a plurality of parallel cords  24 B embedded in the rubber web  24 A and spaced from each other. 
     The rubber webs  22 A,  24 A may be made of any of various known rubbers. The cords  22 B,  24 B may be made of any of various known steels or plastics. 
     As shown in  FIGS. 2 and 3 , the cords  22 B of the first belt member  22  extend obliquely across the cords  18 B of the carcass  18 , and the cords  24 B of the second belt member  24  extend obliquely across both the cords  22 B of the first belt member  22  and the cords  18 B of the carcass  18 , as viewed in the radial directions of the tire  10 . 
     The cords  22 B,  24 B may be spaced at equal intervals. Alternatively, the cords  22 B,  24 B may be spaced at different intervals which may be constant or may vary in their longitudinal direction. The cords  22 B,  24 B may be straight throughout their entire length or may be curved or tortuous partly or fully along their entire length. 
     In the present embodiment, the rubber web  18 A of the carcass  18  and the rubber webs  22 A,  24 A of the belt  20  are made of a polymer such as natural rubber or synthetic rubber and a filler such as carbon black or silica. Therefore, the rubber webs  18 A,  22 A,  24 A are made of a viscoelastic material which is referred to as a compound. 
     The beads  12 , the sidewalls  14 , the tread  16 , the carcass  18 , and the first and second belt members  22 ,  24  are integrally joined together when their rubber materials are vulcanized. 
     As shown in  FIG. 1 , the inner surface of the tire  10 , i.e., the inner circumferential surface of the carcass  18 , is covered with an inner liner  26  of rubber which serves to prevent air from leaking out of the tire  10 . 
       FIG. 4  shows in block form a computer  30  that is used for carrying out the method of simulating a pneumatic tire according to the embodiment of the present invention. 
     As shown in  FIG. 4 , the computer  30  comprises a CPU  32 , a ROM  34 , a RAM  36 , a hard disk drive  38 , a disk drive  40 , a keyboard  42 , a mouse  44 , a display  46 , a printer  48 , and an input/output interface  50  which are interconnected by interface circuits, not shown, and bus lines. 
     The ROM  34  stores a control program, and the RAM  36  provides a memory space referred to as a working area. 
     The hard disk drive  38  stores programs for performing the method of simulating a pneumatic tire according to the embodiment of the present invention. 
     The disk drive  40  serves to record data in and/or read data from a recording medium such as a CD or a DVD. 
     The keyboard  42  and the mouse  44  serve to enter input signals from the operator into the computer  30 . 
     The display  46  serves to display data. The printer  48  serves to print data. Therefore, the display  46  and the printer  48  serve to output data from the computer  30 . 
     The input/output interface  50  sends data to and receives data from an external device that is connected to the computer  30 . 
       FIG. 5  shows in block form a functional system provided by the computer  30  shown in  FIG. 4 . 
     As shown in  FIG. 5 , the computer  30  functionally comprises an input means  30 A, a processing means  30 B, and an output means  30 C. 
     As shown in  FIGS. 4 and 5 , the CPU  32 , the keyboard  42 , the mouse  44 , the disk drive  40 , and the input/output interface  50  jointly make up the input means  30 A. The CPU  32  makes up the processing means  30 B. The CPU  32 , the display  46 , the printer  48 , the disk drive  40 , and the input/output interface  50  jointly make up the output means  30 C. 
     The input means  30 A serves to enter data required to determine stresses or strains on the tire  10  including a composite assembly to be described below according to a finite element method. The data entered through the input means  30 A will be described later. 
     The processing means  30 B functions to produce stresses or strains on the tire  10  based on the data entered through the input means  30 A according to the finite element method. The processing means  30 B with such a function is implemented when a corresponding program stored in the hard disk drive  38  is loaded into the RAM  36  and run by the CPU  32 . 
     The processing means  30 B also functions to receive various data entered through the input means  30 A for setting a finite element model. The processing means  30 B with such a function is also implemented when a corresponding program stored in the hard disk drive  38  is loaded into the RAM  36  and run by the CPU  32 . 
     The output means  30 C serves to output data that are calculated by the processing means  30 B. 
       FIG. 6(A)  is a schematic cross-sectional view of the composite assembly,  FIG. 6(B)  is a schematic cross-sectional view of a finite element model of the composite assembly according to the background art, and  FIG. 6(C)  is a schematic cross-sectional view of a finite element model of the composite assembly according to the embodiment of the present invention. 
       FIG. 7  is a partial perspective view of a finite element model of the carcass  18  of the composite assembly according to the embodiment of the present invention.  FIG. 8  is an enlarged perspective view of a portion of the finite element model. 
       FIG. 9  is a schematic view showing the constraint of nodes of the finite element model of the composite assembly according to the embodiment of the present invention. 
       FIG. 10  is a flowchart of the sequence of the method of simulating a pneumatic tire according to the embodiment of the present invention. 
     The method of simulating a pneumatic tire according to the embodiment of the present invention will be described in detail below with reference to  FIGS. 6(A) through 10 . 
     First, the tire  10  is divided into a plurality of finite elements according to a finite element method, and finite element models are generated using the finite elements. Specifically, finite element models are generated respectively of a portion of the tire  10  which is exclusive of the composite assembly, i.e., the beads  12 , the sidewalls  14 , and the tread  16 , and the composite assembly, i.e., the carcass  18  and the belt  20 . 
     The portion of the tire  10  which is exclusive of the composite assembly is divided into a number of finite elements according to the finite element method, generating a first finite element model (step S 10  in  FIG. 10 ). 
     Specifically, step S 10  is carried out as follows: The processing means  30 B displays on the display  46  an input screen for prompting the operator to enter various data required to generate a finite element model. The operator enters the required data through the keyboard  42  and the mouse  44 , and the processing means  30 B receives the entered data. 
     The first finite element model of the portion of the tire  10  which is exclusive of the composite assembly may be any of various known solid element models. 
     Then, the composite assembly is divided into a number of finite elements according to the finite element method, generating a second finite element model (step S 12  in  FIG. 10 ). 
     Step S 12  is carried out in the same manner as with step S 10 . Specifically, the processing means  30 B displays on the display  46  an input screen for prompting the operator to enter various data required to generate a finite element model. The operator enters the required data through the keyboard  42  and the mouse  44 , and the processing means  30 B receives the entered data. 
     As shown in  FIG. 6(A) , the carcass  18  comprises a rubber web  18 A and a plurality of parallel cords  18 B embedded in the rubber web  18 A. 
     Likewise, the first belt member  22  comprises a rubber web  22 A and a plurality of parallel cords  22 B embedded in the rubber web  22 A, and the second belt member  24  comprises a rubber web  24 A and a plurality of parallel cords  24 B embedded in the rubber web  24 A. 
     According to the background art, as shown in  FIG. 6(B) , for modeling the carcass  18  as an element model  60 , the rubber web  18 A is modeled into two quadrilateral membrane elements  60 A,  60 B, and the cord  18 B is modeled into a quadrilateral membrane element  60 C sandwiched between the quadrilateral membrane elements  60 A,  60 B, thus defining the anisotropy of the quadrilateral membrane element  60 C. 
     According to the embodiment of the present invention, as shown in  FIG. 6(C) , the carcass  18  (the first and second belt members  22 ,  24 ) is modeled as a composite assembly element model  70  (second finite element model) by dividing the rubber web  18 A ( 22 A,  24 A) into rubber web elements  70 A as solid models, and dividing the cords  18 B ( 22 B,  24 B) into cord elements  70 B as solid elements. 
     Specifically, as shown in  FIGS. 7 and 8 , the composite assembly element model  70  of the carcass  18  is made up of the rubber web elements  70 A and the cord elements  70 B. 
     Each of the first and second belt members  22 ,  24  is modeled as a composite assembly element model  70  (second finite element model) by dividing the rubber web  22 A,  24 A into rubber web elements  70 A as solid models, and dividing the cords  22 B,  24 B into cord elements  70 B as solid elements. 
     Specifically, as shown in  FIGS. 7 and 8 , the composite assembly element model  70  of each of the first and second belt members  22 ,  24  is made up of the rubber web elements  70 A and the cord elements  70 B. 
     The belt  20  comprises a plurality of belt members, e.g., the first and second belt members  22 ,  24 , and the belt members are arranged in at least two layers such that their cords extend across each other. In this case, each of the belt members or layers is individually modeled. 
     The cord elements  70 B of the composite assembly element models  70  of the carcass  18  and the first and second belt members  22 ,  24  have a polygonal cross-sectional shape, i.e., at least a quadrangular cross-sectional shape, which remains constant throughout the full length of the cord elements  70 B. 
     The number of cord elements  70 B is smaller when their cross-sectional shape is polygonal than when it is circular. Accordingly, the computational efforts required to simulate the tire  10  are reduced. 
     Actually, the cords  18 B,  22 B,  24 B have a substantially circular cross-sectional shape. If they are modeled as nearly circular cross-sectional shapes using a number of elements, then the modeling process requires a very long computational time which makes the modeling process practically infeasible. 
     In particular, each of the cords  22 B,  24 B of the belt  20  is often made of a plurality of twisted metal wires. If the cords  22 B,  24 B are modeled as nearly circular cross-sectional shapes, they need to be divided into many small elements. Therefore, the modeling process requires a very long computational time which makes the modeling process practically infeasible. 
     According to the embodiment of the present invention, cords having a circular cross-sectional shape or cords made up of a plurality of metal wires are modeled as a finite element model made up of finite elements having identical polygonal cross-sectional shapes, i.e., at least quadrangular cross-sectional shapes. The number of finite elements required is thus reduced, and the modeling process requires reduced computational efforts and a shortened computational time. The modeling process is highly advantageous in easily simulating motion of the cords while the tire  10  is rotating. 
     The polygonal cross-sectional shape of the cord elements  70 B for the purpose of reducing the number of elements should preferably be a quadrangular cross-sectional shape, a hexagonal cross-sectional shape, or an octagonal cross-sectional shape. 
     As described above, the computational efforts for simulating the tire  10  can be reduced by reducing the number of elements with the polygonal cross-sectional shape of the cord elements  70 B. However, since the carcass  18  or each of the first and second belt members  22 ,  24  of the single tire  10  has 1000 to 1500 cords, there is a certain limitation on attempts to reduce the computational efforts if the number of cord elements  70 B is equal to the number of actual cords. 
     Consequently, it is more preferable to make the number of cord elements  70 B smaller than the number of actual cords for reducing the computational efforts. 
     According to the embodiment of the present invention, the number of cord elements  70 B per unit length in a direction perpendicular to the directions in which the cord elements  70 B extend is made smaller than the number of cords  18 ,  22 B, or  24 B per unit length in a direction perpendicular to the directions in which the cords  18 ,  22 B, or  24 B extend, thereby reducing the computational efforts. 
     In a specific example, actual 1500 cords are modeled as 500 cord elements  70 B. If the number of cord elements  70 B is reduced too much, then the accuracy of the simulation is unduly lowered. For keeping the desired simulation accuracy while reducing the computational efforts, it is preferable to reduce the number of cord elements  70 B to about one-fifth of the actual number of cords. 
     If the product of the number of cord elements  70 B per unit length in the direction perpendicular to the directions in which the cord elements  70 B extend and the cross-sectional area of the cord elements  70 B is equal to the product of the number of cords  18 B,  22 B, or  24 B per unit length in the direction perpendicular to the directions in which the cords  18 B,  22 B, or  24 B extend and the cross-sectional area of the cords  18 B,  22 B, or  24 B, then the number of cord elements  70 B is reduced and hence the computational efforts are reduced while modeling the cords  18 B,  22 B, or  24 B without sacrificing the simulation accuracy. 
     Alternatively, the product of the number of cord elements  70 B per unit length in the direction perpendicular to the directions in which the cord elements  70 B extend, the cross-sectional area of the cord elements  70 B, and the modulus of the cord elements  70 B may be equal to the product of the number of cords  18 B,  22 B, or  24 B per unit length in the direction perpendicular to the directions in which the cords  18 B,  22 B, or  24 B extend, the cross-sectional area of the cords  18 B,  22 B, or  24 B, and the modulus of the cords  18 B,  22 B, or  24 B. 
     Since the modulus of the cords  18 B,  22 B, or  24 B is reflected in generating the cord elements  70 B, the above alternative is more effective to reduce the number of cord elements  70 B and hence the computational efforts while modeling the cords  18 B,  22 B, or  24 B to keep the simulation accuracy at a higher level. 
     As shown in  FIG. 9 , the composite assembly element model  70  and a tire portion element model  80  (first finite element model generated in step S 10  shown in  FIG. 10 ), which is generated from the portion of the tire  10  which is exclusive of the composite assembly, are joined to each other by an interfacial boundary  72 . The tire portion element model  80  is made up of a plurality of finite elements  80 A. The finite elements  80 A of the tire portion element model  80  which faces the interfacial boundary are greater in size than the rubber web elements  70 A and the cord elements  70 B of the composite assembly element model  70  which faces the interfacial boundary  72 . 
     This is because since the portion of the tire  10  which is exclusive of the composite assembly is relatively simple in structure, the number of the finite elements  80 A per unit area of the interfacial boundary  72  may be smaller than the number of the rubber web elements  70 A and the cord elements  70 B per unit area of the interfacial boundary  72 . 
     Usually, the diameter of the cords of the composite assembly is different from the distance between adjacent ones of the cords of the composite assembly. Therefore, the rubber web elements  70 A and the cord elements  70 B are different in size from each other. 
     Since the rubber web elements  70 A, the cord elements  70 B, and the finite elements  80 A are different in size from each other, some nodes of the composite assembly element model  70  on the interfacial boundary and some nodes of the tire portion element model  80  on the interfacial boundary are not in conformity with each other. 
     According to the embodiment of the present invention, when the composite assembly element model  70  is generated, the number of the rubber web elements  70 A and the cord elements  70 B per unit area of the interfacial boundary  72  is greater than the number of the finite elements  80 A per unit area of the interfacial boundary  72 , and nodes  7002  of the composite assembly element model  70  on the interfacial boundary are constrained within the plane of the tire portion element model  80 . Stated otherwise, a boundary condition is established for the element models  70 ,  80  whose finite elements have different sizes such that the nodes  7002  of the composite assembly element model  70  on the interfacial boundary are constrained within the plane of the tire portion element model  80  according to boundary conditions. 
     The above boundary condition allows the tire  10  including the composite assembly to be modeled more faithfully for more accurate simulation than simply when some nodes of the composite assembly element model  70  on the interfacial boundary and some nodes of the tire portion element model  80  on the interfacial boundary are not in conformity with each other. 
     In the illustrated embodiment, the composite assembly element model  70  corresponds to the carcass  18 , and the tire portion element model  80  to an element model of the beads  12  and the sidewalls  14  lined with the carcass  18 . The composite assembly element model  70  also corresponds to the belt  20 , and the tire portion element model  80  to an element model of the tread  16  lined with the belt  20 . 
     Then, the processing means  30 B performs a simulating process for determining, by way of a finite element analysis, stresses which the tire  10  receives from the ground while running thereon and strains which the tire  10  undergoes due to the stresses, based on an analytic data that are entered through the input means  30 A (step S 14  in  FIG. 10 ). 
     Specifically, as shown in  FIG. 5 , the operator enters shape data D 1  of the composite assembly element model  70  and the tire portion element model  80 , material data D 2 , boundary data D 3 , and load data D 4  as the an analytic data through the input means  30 A. The processing means  30 B converts the analytic data into stresses and strains at local coordinates, thereby determining stresses and strains at one point of the composite assembly element model  70  and the tire portion element model  80 . 
     The processing means  30 B successively determines stresses and strains at all points of the composite assembly element model  70  and the tire portion element model  80  until the entire tire  10  is covered, and generates data of the determined stresses and strains. 
     According to the above simulating process, the tire  10  is dynamically simulated on the assumption that the tire  10  is running at a certain speed. Though the running speed of the tire  10  is optional, the tire  10  is dynamically simulated more effectively if it is assumed that the tire  10  is running at a speed of 60 km/h or higher. 
     The tire  10  may be statically simulated on the assumption that the tire  10  is held at rest. However, since the static simulation produces simulated data that are not greatly different from simulated data produced by the background art, the dynamic simulation according to the embodiment of the present invention is more effective to analyze the performance of the tire  10  accurately. 
     The processing means  30 B supplies the generated data to the output means  30 C. The output means  30 C outputs the data as simulated data D 10  (step S 16  in  FIG. 10 ). 
     The simulated data D 10  are not limited to the stresses and strains of the tire  10 , but may include heat data, for example, and may also include various known evaluative data for evaluating the durability and performance of the tire  10 . 
     According to the embodiment of the present invention, as described above, a composite assembly of a pneumatic tire comprising a rubber web, such as the carcass  18  or the belt  20 , and a plurality of cords embedded in the rubber web at spaced intervals is modeled as a composite assembly element model  70  by dividing the rubber web into rubber web elements  70 A as solid models, and dividing the cords into cord elements  70 B as solid elements. Consequently, stresses and strains of the composite assembly can accurately be analyzed, and the performance of the pneumatic tire  10  can accurately be evaluated. 
     For accurately judging the durability of the tire  10 , it is necessary to determine shearing forces of the rubber web between the cords and to faithfully express flexing of the cords. 
     According to the embodiment of the present invention, since both the rubber web elements  70 A and the cord elements  70 B are modeled as solid elements, the deformation of the rubber web between the cords, the shearing forces of the rubber web between the cords, and the flexing of the cords can faithfully be analyzed, and the effects that they have on deformation, stresses, and strains of the tire  10  can be predicted with high accuracy. 
     The rubber webs  22 A,  24 A and the cords  22 B,  24 B of the belt  20  are separately modeled as the rubber web elements  70 A and the cord elements  70 B. The rubber web elements  70 A and the cord elements  70 B which are separate from each other make it possible to analyze the propagation of a belt edge separation which has heretofore been difficult to grasp according to an analyzing process of the background art. It is also made possible to analyze which path the belt edge separation tends to follow in an initial phase thereof. Consequently, the belt edge separation can be analyzed in detail. 
     The term “belt edge separation” refers to cracking in the rubber webs at the ends of the cords in the belt members or layers. Specifically, since the ends of the cords are not constrained, the ends of the cords tend to be greatly deformed while the tire is rotating, causing the rubber webs to crack due to shearing forces generated between the belt layers. 
     According to the embodiment of the present invention, furthermore, the rubber webs  22 A,  24 A and the cords  22 B,  24 B of each of the first and second belt members  22 ,  24  are separately modeled as the rubber web elements  70 A and the cord elements  70 B. Therefore, it is possible to analyze the propagation of a belt edge separation between the first and second belt members  22 ,  24 , i.e., the belt layers. 
     The propagation of a belt edge separation between the belt layers leads to the growth of a crack produced in edges of the belt members due to a stress concentration on the crack. Particularly, a crack between the belt layers tends to spread because the cords in the belt layers are inclined at different angles and hence large shearing forces tend to be applied between the belt layers. 
     According to the embodiment of the present invention, moreover, the number of cord elements  70 B per unit length in a direction perpendicular to the directions in which the cord elements  70 B extend is smaller than the number of cords  18 ,  22 B, or  24 B per unit length in a direction perpendicular to the directions in which the cords  18 ,  22 B, or  24 B extend. As the number of divided finite elements of the composite assembly element model  70  is reduced, the computational efforts required to simulate the tire  10  are reduced, allowing the performance of the tire  10  to be evaluated with accuracy. 
     Simulated data obtained by simulating methods according to comparative and inventive examples will be described below. 
       FIG. 11  is a view showing a distribution of stresses (strains) on a pneumatic tire  10  simulated by a simulating method according to a comparative example, and  FIG. 12  is a view showing a distribution of stresses (strains) on a pneumatic tire  10  simulated by a simulating method according to an inventive example. 
     The simulating method according to the comparative example is a method according to the background art. In the simulating method according to the comparative example, a carcass was modeled as a quadrilateral membrane element with defined anisotropy, and the inclination of the carcass was defined by the angle of an anisotropic material. 
     In the simulating methods, the tires  10  were simulated under centrifugal forces corresponding to a running speed of 200 km/h. 
       FIGS. 11 and 12  show a simulated distribution of stresses (strains) f (kgf/mm 2 ) of the tires  10  when the angle of carcass cords positioned at 7 to 8 o&#39;clock indicated by a clock&#39;s short hand as viewed in plan was different from the angle of other carcass cords. 
     Specifically, the angle of carcass cords positioned at 7:30 o&#39;clock indicated by the clock&#39;s short hand was 88 degrees with respect to the circumferential direction of the tire, and the angle of other carcass cords was 90 degrees with respect to the circumferential direction of the tire, i.e., the other carcass cords extended radially of the tire. 
     In the simulating method according to the comparative example, as shown in  FIG. 11 , the strains f are indicated as being distributed substantially uniformly in the circumferential direction of the tire  10 . 
     In the simulating method according to the inventive example, as shown in  FIG. 12 , the stresses f are indicated as being distributed differently at the different angles of carcass cords. The simulating method according to the inventive example is thus capable of an analyzing the strains and stresses of the carcass (composite assembly) and evaluating the performance of the tire  10  more accurately than the simulating method according to the comparative example. 
     In the illustrated embodiment of the present invention, the carcass  18  is of a single-layer structure, and the cords  18 B of the carcass  18  extends radially of the tire  10 . The belt  20  is of a double-layer structure comprising the first and second belt members  22 ,  24 , and the cords  22 B,  24 B of the first and second belt members  22 ,  24  extend across each other. 
     However, the simulating method according to the present invention is also applicable to different pneumatic tires having various composite assemblies (stiffeners), i.e., various structural details of the carcass  18  and the belt  20 . 
     Although a certain preferred embodiment of the present invention has been shown and described in detail, it should be understood that various changes and modifications may be made therein without departing from the scope of the appended claims.