Stress

1. KSI ENGINEERING
PROPERTIES
OF AREAS
Toe Myint Naing
Curtin Malaysia
Intern

2. 2Normal Stress Distribution
• When external axial loads are applied,
they are resisted by internal normal
stresses acting over the cross sectional
area of the section.
• When external bending moments are
applied, they need to be resisted by an
internal resisting couple. It therefore
follows that we need a first moment of area
to help determine the stresses in a beam
subjected to a moment.
(lbf/𝑖𝑛2)
(lbf/𝑖𝑛2)
©Curtin,CEM, Lec 2

3. 3Properties of Areas
Need to have an understanding of Properties of Areas to be able to determine the stresses
due to bending
• Centre of Gravity,
Centroid (Neutral Axes)
• Second Moment of Area (I value)
• Elastic Section Modulus;
First Moment of Area (Z value)
The Centroid of a body is the point where
the entire weight of the body appears to be
concentrated.
Centroidal axis ≠ Equal area
axis
©Curtin,CEM, Lec 2

4. Centroid, Neutral Axis
General equation about the x-axis
4
©Curtin,CEM, Lec 2

5. Centroid, Neutral Axis
General equation about the y-axis
5
©Curtin,CEM, Lec 2

6. I-value, Second Moment of Area
I-value is a measure of the geometric stiffness of a shape
I-value allows us to quantify deflections and stresses depending upon the orientation of
the beam.
6
©Curtin,CEM, Lec 2
Ixx (second moment of area about the x-axis) in
order to quantify deflection and stresses when
bending about the x-axis.
Iyy (second moment of area about the y-axis) in
order to quantify deflection and stresses when
bending about the y-axis.

7. I-value, Second Moment of Area 7
©Curtin,CEM, Lec 2
𝐼(rectangle) =(
𝑏𝑑3
12
) + 𝐴ℎ2
=(
𝑏𝑑3
12
)
0
Equation for rectangular section
𝐼 = (
𝑏𝑑3
12
+ 𝐴ℎ2
)

8. I-value, Second Moment of Area 8
©Curtin,CEM, Lec 2
𝐼 = (
𝑏𝑑3
12
+ 𝐴ℎ2
)
General equation about the x-axis
Between centroid of whole section &
centroid of rectangle being considered

9. I-value, Second Moment of Area 9
©Curtin,CEM, Lec 2
General equation about the y-axis
Between centroid of whole section &
centroid of rectangle being considered
𝐼 = (
𝑏𝑑3
12
+ 𝐴ℎ2
)

10. I-value, Second Moment of Area 10
©Curtin,CEM, Lec 2
General equation about the x-axis
(Symmetric)
𝐼 = (
𝑏𝑑3
12
+ 𝐴ℎ2
)

11. I-value, Second Moment of Area 11
©Curtin,CEM, Lec 2
General equation about the y-axis
(Symmetric)
𝐼 = (
𝑏𝑑3
12
+ 𝐴ℎ2
)

12. First Moment of Area, Elastic Section Modulus (Z) 12
©Curtin,CEM, Lec 2
Equation for rectangular section
𝑍 𝑥 =
𝐼 𝑥
𝑦 𝑚𝑎𝑥

13. Normal Bending Stress 13
©Curtin,CEM, Lec 2
𝜎 𝑚𝑎𝑥 =
𝑀𝑦 𝑚𝑎𝑥
𝐼
𝑍 =
𝐼
𝑦 𝑚𝑎𝑥
∴ 𝜎 𝑚𝑎𝑥 =
𝑀
𝑍
∴ Stresses are related to the Z value
1
𝑅
=
𝑀
𝐸𝐼
∴ ∆=
𝑀
𝐸𝐼
(constant)
∴ Deflections are related to the I value
(and E)

14. Normal Bending Stress
Stresses when bending moments are applied about the x-axis
𝜎 𝑥 =
𝑀 𝑥 𝑦
𝐼 𝑥
𝑍𝑡𝑜𝑝 =
𝐼 𝑥
𝑦𝑡𝑜𝑝 𝑓𝑖𝑏𝑟𝑒
𝜎max 𝑡𝑜𝑝 =
𝑀 𝑥
𝑍𝑡𝑜𝑝
𝑍 𝑏𝑜𝑡 =
𝐼 𝑥
𝑦 𝑏𝑜𝑡 𝑓𝑖𝑏𝑟𝑒
𝜎max 𝑏𝑜𝑡 =
𝑀 𝑥
𝑍 𝑏𝑜𝑡
©Curtin,CEM, Lec 2
14

15. Normal Bending Stress
Stresses when bending moments are applied about the y-axis
𝜎 𝑦 =
𝑀 𝑦 𝑥
𝐼 𝑦
𝑍 𝐿𝐻𝑆 =
𝐼 𝑦
𝑦 𝐿𝐻𝑆 𝑓𝑖𝑏𝑟𝑒
𝜎max 𝐿𝐻𝑆 =
𝑀 𝑦
𝑍 𝐿𝐻𝑆
𝑍 𝑅𝐻𝑆 =
𝐼 𝑦
𝑦 𝑅𝐻𝑆 𝑓𝑖𝑏𝑟𝑒
𝜎max 𝑅𝐻𝑆 =
𝑀 𝑦
𝑍 𝑅𝐻𝑆
©Curtin,CEM, Lec 2
15

16. Thank
sFor your attention

17. Any
Question?