Giáo trình Strength of Materials - Chapter 6: Properties of cross sections

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 của một “mặt cắt ngang”) 
6.3 Moment of Inertia of composite area 
(Moment quán tính của một “mặt cắt ngang” phức tạp) 
6.4 Radius of Gyration 
(bán kính quán tính – bán kính của sự vặn xoắn) 
6.0 INTRODUCTION 
Not only area but also other geometrical 
characteristics affecting to the resistance 
of the beam 
Relative resistance of 4 beam cross-sections (with the 
same cross-section areas) to bending stress and 
deflection 
Called: 
CROSS-SECTIONAL PROPERTIES 
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. 
CENTER OF GRAVITY 
CENTROID 
If the weight or object were homogeneous, the center of gravity and the 
centroid would coincide. 
The centroid of cross-sectional areas (of beams and columns) will be 
used later as the reference origin for computing other section 
properties. 
6.1 CENTER OF GRAVITY - CENTROID 
First moment of an area about an axis (Moment tĩnh của một tiết 
diện quanh môt trục) 
6.1 CENTER OF GRAVITY - CENTROID 
(1) Consider a moment of an area A with respect to an u-axis: Su 
If Su = 0 then u is called central axis (trục trung tâm) and the centroid 
lies on u-axis 
PROPERTIES: 
(2) If u-axis and v-axis are central axes which mean: Su = 0 and Sv = 0 
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 
6.1 CENTER OF GRAVITY - CENTROID 
6.1 CENTER OF GRAVITY - CENTROID 
EXAMPLE 
6.1 CENTER OF GRAVITY - 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 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 = 0 then (x, y) are principle axes of inertia 
If (x, y) are the central axes, which mean Sx = 0 and Sy = 0, and Ixy = 0 
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. 
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n
i
i
xx II
1
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n
i
i
yy II
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