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CHAPTER 6: PROPERTIES OF CROSS SECTIONS 6.0 Introduction 6.1 Center of Gravity – Centroid (trọng tâm – “trọng tâm” hình học) 6.2 Moment of Inertia of an area (Moment quán tính “mặt cắt ngang”) 6.3 Moment of Inertia of composite area (Moment quán tính “mặt cắt ngang” phức tạp) 6.4 Radius of Gyration (bán kính quán tính – bán kính vặn xoắn) 6.0 INTRODUCTION Not only area but also other geometrical characteristics affecting to the resistance of the beam Called: CROSS-SECTIONAL PROPERTIES Relative resistance of beam cross-sections (with the same cross-section areas) to bending stress and deflection 6.1 CENTER OF GRAVITY - CENTROID Center of gravity or center of mass, refers to weights or masses and can be thought of a single point at which the weight could be held and be in balance in all directions Centroid usually refers to the center of lines, areas and volumes The centroid of cross-sectional areas (of beams and columns) will be used later as the reference origin for computing other section properties If the weight or object were homogeneous, the center of gravity and the centroid would coincide CENTER OF GRAVITY CENTROID 6.1 CENTER OF GRAVITY - CENTROID First moment of an area about an axis (Moment tĩnh tiết diện quanh môt trục) 6.1 CENTER OF GRAVITY - CENTROID PROPERTIES: (1) Consider a moment of an area A with respect to an u-axis: Su If Su = then u is called central axis (trục trung tâm) and the centroid lies on u-axis (2) If u-axis and v-axis are central axes which mean: Su = and Sv = then the intersection of the two axes is the centroid (3) Coordinates of the centroid C(xC, yC)can be calculated as follows: 6.1 CENTER OF GRAVITY - CENTROID DEMONSTRATION: 6.1 CENTER OF GRAVITY - CENTROID 6.1 CENTER OF GRAVITY - CENTROID 6.1 CENTER OF GRAVITY - CENTROID 6.1 CENTER OF GRAVITY - CENTROID 6.2 MOMENT OF INERTIA OF AN AREA 6.2 MOMENT OF INERTIA OF AN AREA 6.2 MOMENT OF INERTIA OF AN AREA 6.2 MOMENT OF INERTIA OF AN AREA (2) POLAR MOMENT OF INERTIA (moment quán tính độc cực) r (3) PRODUCT OF INERTIA OF AN AREA (moment quán tính ly tâm) 6.2 MOMENT OF INERTIA OF AN AREA NOTES If Ixy = then (x, y) are principle axes of inertia If (x, y) are the central axes, which mean Sx = and Sy = 0, and Ixy = then (x, y) are central principle axes of inertia 6.3 MOMENT OF INERTIA OF COMPOSITE AREAS 6.3 MOMENT OF INERTIA OF COMPOSITE AREAS Moment of inertia of a composite area can be calculated by the sum of moments of inertia of simple component areas n I x I xi i 1 n I y I yi i 1 n is the number of simple component areas 6.3 MOMENT OF INERTIA OF COMPOSITE AREAS (1) Parallel axis theorem 6.3 MOMENT OF INERTIA OF COMPOSITE AREAS (1) Parallel axis theorem 6.3 MOMENT OF INERTIA OF COMPOSITE AREAS (1) Parallel axis theorem - Example 6.3 MOMENT OF INERTIA OF COMPOSITE AREAS (1) Parallel axis theorem - Example 6.3 MOMENT OF INERTIA OF COMPOSITE AREAS (2) Rotation of axes – Transformation of inertia moment of an area 6.4 RADIUS OF GYRATION 6.4 RADIUS OF GYRATION 6.4 RADIUS OF GYRATION ... CENTROID 6. 1 CENTER OF GRAVITY - CENTROID 6. 2 MOMENT OF INERTIA OF AN AREA (1) MOMENT OF INERTIA WITH RESPECT TO AN AXIS Moment of inertia of an irregular are 6. 2 MOMENT OF INERTIA OF AN AREA 6. 2... axes of inertia 6. 3 MOMENT OF INERTIA OF COMPOSITE AREAS 6. 3 MOMENT OF INERTIA OF COMPOSITE AREAS Moment of inertia of a composite area can be calculated by the sum of moments of inertia of simple... 6. 2 MOMENT OF INERTIA OF AN AREA 6. 2 MOMENT OF INERTIA OF AN AREA 6. 2 MOMENT OF INERTIA OF AN AREA (2) POLAR MOMENT OF INERTIA (moment quán tính độc cực) r (3) PRODUCT OF INERTIA OF AN AREA (moment