Properties of Area Centroid 1 st Moment of

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

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 1 st moment of area /total area

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

CENTROID OF AREAS y Total Area A x G x Elemental area y x

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

1 st moment of Area – Example (Ans)

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

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

Example / Exercise • Loaded Timber beam has max BM of 5 k. Nm, find stress in the section. 100 300 5 k. Nm BMD bd 3 I= / 12 = 100 x 3003 mm 4 12 = 102 x 33 x 1003 12 = 27 x 102 x 106 12 = 2. 25 x 108 mm 4 Section compression tension Stress block Hence f = 5 x 103 Nmm x 150 mm 2. 25 x 108 mm 4 = 750 x 106 225 x 106 = 3. 33 N/mm 2