MOMENTS OF INERTIA, SECTION MODULI, AND RADII OF GYRATION OF PLANE SECTIONS

ISSN: 2582-4686 SJIF 2021-3.261,SJIF 20222.889, 2024-6.875 ResearchBib IF: 9.948 / 2024 VOLUME-5, ISSUE-11 308 MOMENTS OF INERTIA, SECTION MODULI, AND RADII OF GYRATION OF PLANE SECTIONS Turapov E.I. Associate Professor, PhD (Engineering), Department of Mechanical Engineering Technology, Almalyk State Technical Institute The axial moment of inertia of a plane section with respect to a given axis is defined as the sum, taken over the entire area of the section, of the products of the elementary area elements and the squares of their distances from that axis (see Fig. 1). Fig.1 Jz=∫y2 FdF, m4(sm⁴); Jy=∫z2 FdF, m⁴ (sm⁴). Axial moments of inertia are always positive and can never be equal to zero. The centrifugal moment of inertia of a plane section is defined as the sum, taken over the entire area of the section, of the products of the elementary area elements and their distances from two mutually perpendicular axes (see Fig. 1). Jzy =∫zy FdF, m4(sm⁴); The product of inertia may be positive, negative, or equal to zero. The axial and product moments of inertia of a plane section with respect to axes parallel to the centroidal axes are determined using the following formulas: Jz=Jzc+a2F; Jy=Jyc+b2F; Jzy =Jzcyc+abF, where: Jzc, Jyc — axial moments of inertia with respect to the centroidal axes; ISSN: 2582-4686 SJIF 2021-3.261,SJIF 20222.889, 2024-6.875 ResearchBib IF: 9.948 / 2024 VOLUME-5, ISSUE-11 309 Jzcyc— product of inertia with respect to the centroidal axes; a, b — distances between the chosen axes and the centroidal axes parallel to them; F — area of the section. When the axes are rotated by an angle α, the moments of inertia are determined using the following formulas: Jz1= Jzcos2α+Jzsin2α−Jzysin2α; Jy1= Jycos2α+Jzsin2α−Jzysin2α; Jz1y1=Jz−Jy 2𝑠𝑖𝑛2𝑎+Jzycos2α. When the value of the axes rotation angle reaches the magnitude determined by the following formula: tg2α= 2Jzy Jy−Jz , at that point, the axial moments of inertia reach their maximum and minimum values, while the product of inertia becomes zero. The axes occupying such positions are called the principal axes of inertia, and the corresponding axial moments of inertia are referred to as the principal moments of inertia. The principal axes of inertia that pass through the centroid of the section are called the principal centroidal axes of inertia. The magnitudes of the principal moments of inertia are determined using the following formula: Jmax min =Jz+Jy 2±1 2√(Jz+Jy)2+4Jzy2 The axial section modulus of a symmetric section is defined as the ratio of the axial moment of inertia to the distance from the neutral axis to the farthest point of the section: Wz=Jz ymax, m3(sm3); Wy=Jy zmax, m3(sm3). The radius of gyration of a section with respect to the z-axis is defined as the quantity determined by the following formula: iz=√Jz F, m,(sm) similarly, iy=√Jy F, m,(sm). The geometric properties of rolled profiles are presented in the GOST tables. In these tables, the horizontal and vertical centroidal axes are denoted as x and y. When using these tables, it is necessary to reconcile the axis designations with those adopted in the specific problem. ЛИТЕРАТУРА 1. Белявский С.М. (1967) Руководство к решению задач по сопротивлению материалов 2. Биргер И.А., Мавлютов Р.Р. (1986) Сопротивление материалов 3. Вольмир А.С. (1984) Сборник задач по сопротивлению материалов 4. Хасанов Б.Б. “РАСЧЕТ СТАТИЧЕСКИ ОПРЕДЕЛИМОЙ СТЕРЖНЕВОЙ СИСТЕМЫ”. ISSN: 2582-4686 SJIF 2021-3.261, SJIF 2022-2.889, 2023-5.384 ResearchBib IF: 8.848 / 2024. 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