1. Centroid 1st Moment of area 2nd Moment of area Section Modulus
Properties of Area
Centroid
1st Moment of area
2nd Moment of area
Section Modulus

2. CENTROID OF AREAS
Centroid of an area is the point at which the total area may be considered to be situated for calculation purposes.
Corresponds to the centre of gravity of a lamina of the same shape as the area
Often possible to deduce the centroid by SYMMETRY of the area.
Need to know position of the centroid of a section as bending occurs with compression above and tension below this axis.
Distance from centroid to axis of rotation
(x or y) is 1st moment of area /total area

3. 1st Moment of Area F F x d A x d Area A B A d Moment of Force =
Likewise; C
Area A D
G = centroid of area
First Moment of Area about the line CD =
A x d

4. CENTROID OF AREAS
Total Area A x y G
Elemental area a y x

5. 1st moment of Area - Example30 60 35 50 65
20 Dia 7 15
Find centroid of the composite beam section shown

6. 1st moment of Area – Example (Ans)

7. 2nd Moment of Area
A property of area used in many engineering calculations (e.g. stress in beams)
Elemental
Elemental area a
Second Moment of Area about the line CD = I D x C

8. Standard Results for I
Using differential calculus we can formulate standard solutions, eg:
Rectangle about its base
Rectangle about its centre
For more complicated shapes can use compound areas and parallel axes theorem
Or, easier, use tables from steel joist manufacturers b d b d

9. Example / Exercise
Loaded Timber beam has max BM of 5 kNm, find stress in the section.
5 kNm
BMD
100
300
Section
Stress block
compression
tension
Hence
I = bd3 / 12
= 100 x mm4 12
= 102 x 33 x 1003
= 27 x 102 x 106
= 2.25 x 108 mm4
Hence f = 5 x 103 x 103 Nmm x 150 mm
2.25 x mm4
= 750 x 106
225 x 106
= 3.33 N/mm2