Calculation of a stepped rod under longitudinal-transverse bending with discrete axis loading

S. V. Bakushev

Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 77 (2022), pp. 38-53

Résumé

This paper aims at obtaining formulas for bending moments and shear forces in a rectilinear elastic stepped rod under plane longitudinal-transverse bending. Each step of the rod (segment) can consist of different materials and have its own shape and cross- sectional dimensions. The rod can be loaded by an axial longitudinal force at the beginning of each step. The eccentricity of longitudinal forces at the beginning of each step (segment) is taken into account, which occurs due to the mismatch of longitudinal axes at the current and previous steps. Each segment of the rod can be exposed to a transverse action represented as concentrated bending moments, concentrated forces, and uniformly distributed loading. The resulting algebraic equations of the bending moments and shear forces are obtained for the stepped rod under longitudinal-transverse bending. The numerical model has been considered. The study results show that allowance for longitudinal action on the stepped rod bending with discrete axial loading leads to an increase in the ordinates of epures and bending moments, as well as in shear forces as compared to transverse bending caused by transverse loading only. Moreover, the internal transverse forces do not remain constant on the rod segments which are free from uniformly-distributed transverse loading. The obtained formulas for bending moments and transverse forces can be applied in calculations of elastic stepped rods under longitudinal-transverse bending.

Détail

Keywords: elastic step rod, longitudinal-transverse bending, bending moments
Mots-clés : transverse forces.

Bibliographie

[1] Akhmetzyanova D.R., Ivanov S.P., “Prodolno-poperechnyi izgib fizicheski nelineinykh balok”, Nauchnomu progressu - tvorchestvo molodykh, 2018, no. 1, 88–90

[2] Semenov V.V., Ulambayar Kh., “Raschet gibkikh sterzhnei na prodolno-poperechnyi izgib”, Izvestiya vuzov. Investitsii. Stroitelstvo. Nedvizhimost, 8:2 (25) (2018), 148–158

[3] Zhubrin A.N., Kulikov Yu.A., Chernyshova O.I., “Raschetno-eksperimentalnoe issledovanie uprugogo deformirovaniya pryamogo gibkogo sterzhnya pri prodolno-poperechnom izgibe”, Nauchnomu progressu - tvorchestvo molodykh, 2018, no. 1, 70–73

[4] Lykina N.A., “Ustoichivost i prodolno-poperechnyi izgib fermennoi machty sotovoi svyazi”, Molodezhnyi nauchno-tekhnicheskii vestnik, 2016, no. 6, 13

[5] Ulitin G.M., Tsarenko S.N., “Prodolno-poperechnyi izgib i ustoichivost vesomoi sterzhnevoi sistemy”, Stroitelnaya mekhanika i raschet sooruzhenii, 2017, no. 2 (271), 18–23

[6] Tsarenko S.N., “Prodolno-poperechnyi izgib sterzhnei peremennoi zhestkosti”, Izvestiya vysshikh uchebnykh zavedenii. Stroitelstvo, 2016, no. 1 (685), 5–13

[7] Proskuryakov N.E., Lopa I.V., “Prodolno-poperechnyi izgib vintov zapornoi armatury”, Izvestiya Tulskogo gosudarstvennogo universiteta. Tekhnicheskie nauki, 2017, no. 12-2, 277–282

[8] Kagan-Rozentsveig L.M., “Razvitie prikladnogo metoda rascheta szhato-izognutykh sterzhnei”, Vestnik grazhdanskikh inzhenerov, 2017, no. 4 (63), 130–134 | DOI

[9] Chupeev G.V., “Prodolno-poperechnyi izgib sterzhnei peremennogo poperechnogo secheniya”, Molodoi uchenyi, 2015, no. 8 (88), 340–344

[10] Glazkov A.S., Klimov V.P., Gumerov K.M., “Prodolno-poperechnyi izgib truboprovoda na uchastkakh gruntovykh izmenenii”, Problemy sbora, podgotovki i transporta nefti i nefteproduktov, 2012, no. 1 (87), 63–70

[11] Ovsyanko V.M., “Modelirovanie differentsialnykh uravnenii pri raschete na poperechnyi i prodolno-poperechnyi izgib sterzhnevykh sistem bez ucheta i s uchetom vyazkouprugikh svoistv materialov”, Nauka i tekhnika, 2012, no. 4, 35–43

[12] Gumerov A.K., Shadrin V.S., Valekzhanin D.Yu., Idrisov R.Kh., Khazipov R.Kh., “Uravneniya prodolno-poperechnogo izgiba i sdviga truboprovoda s uchetom iskhodnoi krivizny uchastkov”, Problemy sbora, podgotovki i transporta nefti i nefteproduktov, 2013, no. 4 (94), 77–82

[13] Kaurov P.V., Timofeev A.A., “Novyi sposob opredeleniya peremeschenii sterzhnya maloi zhestkosti pri prodolno-poperechnom izgibe”, Izvestiya Peterburgskogo universiteta putei soobscheniya, 2011, no. 1 (26), 163–171

[14] Akhtyamov A.M., Zakharova M.A., “Obratnaya zadacha dlya prodolno-poperechnogo izgiba sterzhnya”, Stroitelnaya mekhanika inzhenernykh konstruktsii i sooruzhenii, 2011, no. 4, 31–33

[15] Nemirovskii Yu.V., Tikhonov S.V., “Prodolno-poperechnyi izgib mnogosloinogo sterzhnya iz fizicheski nelineinogo materiala”, Vestnik Chuvashskogo gosudarstvennogo pedagogicheskogo universiteta im. I.Ya. Yakovleva. Ser. Mekhanika predelnogo sostoyaniya, 2020, no. 3 (45), 247–253 | DOI

[16] Nemirovskii Yu.V., Tikhonov S.V., “Prodolno-poperechnyi izgib mnogosloinykh sterzhnei iz betonov i stalefibrobetonov”, Izvestiya Altaiskogo gosudarstvennogo universiteta, 2021, no. 1 (117), 40–46 | DOI

[17] Nemirovskii Yu.V., Tikhonov S.V., “Predelnoe sostoyanie betonnykh i zhelezobetonnykh sterzhnei pri slozhnom i prodolno-poperechnom izgibe”, Vestnik Permskogo natsionalnogo issledovatelskogo politekhnicheskogo universiteta. Mekhanika, 2020, no. 1, 60–73

[18] Nemirovskii Y., Tikhonov S., “Complex bend of multilayered concrete rods”, EPJ Web of Conferences, 221 (2019), 01048 | DOI

[19] Cho C.-G., Lee S.-J., “Inelastic Responses and Finite Element Predictions of Fiber Cementitious Composite and Concrete Columns”, Materials, 14 (2021), 2180 | DOI

[20] Milanovic M., Cvetkovska M., Knezevic P., “Load-bearing capacity of fire exposed composite columns”, Gradevinar, 67:12 (2015), 1187–1197 | DOI | MR

[21] Agapov V.P., Vasil'ev A.V., “Account for geometrical nonlinearity in the analysis of reinforced concrete columns of rectangular section by finite element method”, Vestnik MGSU, 2014, no. 4, 37–43 | DOI

[22] Rogovskii I.L., Titova L.L., Davydenko O.O., Trokhaniak V.I., Trokhaniak O.M., “Technology of producing reinforced concrete columns of circular cross-sectional and investigation of their strain-stress state at transverse-longitudinal bending”, Acta Polytechnica, 59:5 (2019), 510–517 | DOI

[23] Trapko T., Musial M., “Effect of PBO-FRCM Reinforcement on Stiffness of Eccentrically Compressed Reinforced Concrete Columns”, Materials, 13:5 (2020), 1221 | DOI

[24] Tamrazyan A., Avetisyan L., “Comparative analysis of analytical and experimental results of the strength of compressed reinforced concrete columns under special combinations of loads”, MATEC Web of Conferences, 86 (2016), 01029 | DOI

[25] Yin S.P., Hu X.Q., Hua Y.T., “Compression performance and bearing capacity calculation model of small-eccentricity columns strengthened with textile-reinforced mortar (TRM)”, Materiales de Construccion, 69:335 (2019), e195 | DOI

[26] Sarafraz E.M., “Flexural Strengthening of RC Columns with Low Longitudinal Steel Ratio using GFRP Bars”, International Journal of Concrete Structures and Materials, 13:1 (2019), 1–11 | DOI

[27] Arslan G., Hacisalihoglu M., “Nonlinear analysis of RC columns using the Drucker-Prager model”, Journal of Civil Engineering and Management, 19:1 (2013), 69–77 | DOI

[28] Zong-Cai D., Daud J.R., Hui L., “Seismic Behavior of Short Concrete Columns with Prestressing Steel Wires”, Advances in Materials Science and Engineering, 2014 (2014), 180193 | DOI

[29] Li Q., Kuang Y., Guo W., Zhang Y., “Experimental Research on Mechanical Performance of SSRC Columns under Eccentric Compression”, Applied Sciences, 10:16 (2020), 5629 | DOI

[30] Rubinin M.V., Soprotivlenie materialov. Teoriya, Mashgiz, M., 1961, 467 pp.

[31] Rubinin M.V., Rukovodstvo k prakticheskim zanyatiyam po soprotivleniyu materialov, 3-e izd., Mashgiz, M., 1957, 603 pp.