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

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

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

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

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

1st moment of Area – Example (Ans)

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

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

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 3003 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 108 mm4 = 750 x 106 225 x 106 = 3.33 N/mm2