Properties of Area

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

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

5. 1st moment of Area - Example 30 60 30 Find centroid of the composite beam section shown 50 7 15 35 20 Dia 65

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 b d d

9. Example / Exercise • Loaded Timber beam has max BM of 5 kNm, find stress in the section. 5 kNm 100 compression 300 tension Section Stress block BMD Hence I = bd3 / 12 = 100 x 3003 mm4 12 = 102 x 33 x 1003 12 = 27 x 102 x 106 12 = 2.25 x 108 mm4 Hence f = 5 x 103 x 103Nmm x 150 mm 2.25 x 108 mm4 = 750 x 106 225 x 106 = 3.33 N/mm2