【文章內(nèi)容簡(jiǎn)介】 ? Summations of principal stresses obtained from the previous steps were calculated ( 1? and 3? ) for all the elements. Note that the principal stresses (1? and 3? ) for the calculation of resilient modulus in the first step were found by considering only the weight of the ballast and subgrade layers. ? By importing 1? and 3? into Eqs.(13) and (14), the nonrecoverable strain ( p? ) and the resilient modulus ( r? ) for the i th loading step were calculated for each element. ? Substituting the resilient modulus obtained for each element for E in Eqs.(1)–(10), the elements’ linear strains e? due to the i th loading increment were calculated. ? Resilient modulus obtained for each element was modified by multiplying it by the ratio of elastic strain to the summation of nonrecoverable and elastic strain obtained for the element at the i th loading step. ? By taking the modified resilient modulus into Eqs.(1)–（ 10） , the final stresses and strains due to the i th loading increment were calculated. ? Total stresses and strains at the end of the i th loading step were obtained in a matrix format from 8 where []? = stress matrix at the end of the i th loading step。 []i? = stress matrix obtained due to thei th loading increment。 []? = strain matrix at the end of the i th loading step。 and []i? = strain matrix due to the i th loading increment. The final stress and strain matrices were obtained from the summation of stresses and strains obtained in all 100 loading steps. Model Validation For the validation of the proposed model, field investigations were conducted into the structural behavior of a railway track system and the results were pared with those from the proposed model. Field Measurements Rail deflection, loads transferred from the rail to the sleeper rail seat load and the pressure distribution between sleepers and the ballast were measured in a railway track system. Tests were con ducted on an Iranian railway main line near the Mobareke steel factory in a suburb of Isfahen （ the second largest city of Iran） . The properties of the track were as follows: rail is UIC60, sleepers were prestressed type B70 with 600 mm sleeper spacing, and the fastening system was Vossloh. The ballast thickness was 450 mm. Ballast aggregates were granite sized 40–60 mm. The subgrade was well pacted soil ranked A 2–7in the AASHTO soil classification. A theodoliteT16 was used to measure the deflection of the rail at certain points. Fourteen load cells were installed between the rail and 14 sleepers to measure rail seat loads. Four load cells were installed between sleeper number 9 and the ballast to record the ballast pressure distribution under the sleeper. Load cells were selected and fitted considering the possible loads and workingconditions. Load cells of 100 and 25 kN with an accuracy of were fitted under the rail seat and beneath the sleeper at the sleeperballast interface in Sleeper No. 9, respectively. It was assumed that the sleeper loading pattern is symmetrical so load cells were only installed under onehalf of the sleeper. The load cells were 100 mm in diameter and 11 mm thick. They were installed inside the sleeper during the casting process. The load cells between the sleeper and the rail seat were installed in such a way that the rail seat loads were totally transferred to the sleeper via the load cells. The installation ensured that no geometrical deficiencies were allowed. Since the ballast consists of granular materials, four separate plates were installed beneath the load cells to provide footing on the ballast. A data logger linked to a personal puter was used to trigger the data and transfer it to the software. The 9 software converted the electrical signals from the load cells in terms of millivolts to the data in terms of force kN. A freight train with the axle load of 185 kN 18,500 kgwas used for the load. The freight details are presented in Fig. 4. The distance between axles of the bogie was m and the center to center distance of the bogies was m. The loads under the sleeper, rail deflections and rail seat loads obtained from the field measurement are pre sented in Tables 1 and 2. Fig. 4. Schematic diagram and picture of the freight Theoretical Analyses Due to the symmetric loading of the track in the longitudinal and transverse directions, only half of the system was modeled. The loads were the point loads and the deflections were considered as point deflections. In the longitudinal and transverse models, the nodes along the vertical boundaries were restrained from horizontal movement. The nodes along the bottom boundary were restrained both from horizontal and vertical movements to simulate a rigid boundary. For the longitudinal direction, a length of m was considered, based on the dimensions of the freight used in the field. In the transverse direction, a length of m from the center of the sleeper was considered. The loading conditions in the model were set by the freight axle loads. Based on the number of cars that passed the track since the latest track tamping and stabilization, 2 million load cycles （ N） were considered in the analysis. As suggested in Sadeghi （ 2021） , the angle of distribution used for the modified planestrain was 15176。. It was assumed that the springs representing the sleepers in the longitudinal analysis cannot take tensile stresses. Tables 3 and 4 indicate the model inputs. MATLAB software was used to solve the equations in matrix formats. K1 to K5 were obtained from the results of the laboratory and field tests on various ballast and subgrade materials con ducted previously by these writers （ Sadeghi 2021） . The theoretical analyses were conducted under two different conditions in order to investigate the importance of the consideration of the substructure ma