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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 B Moment of Force = Fxd F d 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 x Elemental area y a y x y ax a ay a 1st moment of Area - Example 30 60 30 Find centroid of the composite beam section shown 50 7 15 35 65 20 Dia 1st moment of Area – Example (Ans) distance to centroid Area Ref breadth 1 2 3 50 35 28 hole -10 width area 30 30 60 ΣA = 1st Moments of area 1500 1050 1680 from Oy x 15 105 60 From Ox y 25 18 14 About Oy Ax 22500 110250 100800 About Ox Ay 37500 18900 23520 -314 65 15 -20420 -4712 213130 75208 Σ Ax = 3916 x= y= Σ Ax ΣA Σ Ay ΣA = 213130 = 54 = 19 3916 = 75208 3916 = Σ Ay 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 I ax 2 Standard Results for I • Using differential calculus we can formulate standard solutions, eg: b • Rectangle about its base bd 3 d I • Rectangle about its centre b d O 3 3 bd I CC 12 • 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 kNm, find stress in the section. 100 300 5 kNm Section compression tension Stress block M I BMD 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 f E y R f My Hence I 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