VIEWS: 599 PAGES: 23 CATEGORY: College POSTED ON: 2/11/2011
Lecture Notes Module 2: Centroids & Moment of Inertia Centroid • Centroid or center of gravity is the point within an object from which the force of gravity appears to act. • Centroid of 3D objects often (but not always) lies somewhere along the lines of symmetry. Hollowed pipes, L shaped section have centroid located outside of the material of the section • The Centroid of any area can be found by taking Centroidal axis or Neutral moments of identifiable areas (such as rectangles or triangles) about any axis. i.e A.y or A.x • The moment of an area about any axis is equal to the algebraic sum of the moments of its component areas. • The moment of any area is defined as the product of the area and the perpendicular distance from the Centroid of the area to the moment axis. First Moment of Area-Centroid • Composite areas – Subdivide – Select coordinate axes – Find the overall area – Apply the composite centroid formula Moment of Inertia (I) • Also known as the Second Moment of the Area is a term used to describe the capacity of a cross-section to resist bending. • It is a mathematical property of a section concerned with a surface area and how that area is distributed about the reference axis. The reference axis is usually a centroidal axis. where Cross Section of Beam Y Y Internal Forces • Naming those internal forces 2011-2-11 7 Perpendicular Axis Theorem • For flat objects the rotational moment of inertia Iy = (1/12) b3d of the axes in the plane is related to the moment of inertia perpendicular to the Ix = (1/12) bd3 d plane. Iz Ix Iy b Iz Ix Iy Iz = (1/12) bd(b2 + d2) Izz Polar Moment of Inertia Parallel-Axis Theorem for an Area • Used to calculate the moment of inertia about other axis. • The moment of inertia of an area about an axis is equal to the moment of inertia of the area about a parallel axis passing through the area’s Centroid plus the product of the area and the square of the perpendicular distance between the axes. - Ix’: Moment of Inertia about new axis - Ix: Moment of Inertia about original axis - A: Area of shape A - d: Perpendicular distance from new to original axis x x d I X ' I X Ad 2 x' x' 2011-2-11 10 Moment of Inertia - Parallel Axis Theorem Parallel axis theorem: Consider the moment of inertia Ix of an area A with respect to an axis AA’. Denote by y the distance from an element of area dA to I x y dA 2 AA’. Moment of Inertia - Parallel Axis Theorem Consider an axis BB’ parallel to AA’ through the Centroid C of the area, known as the centroidal axis. The equation of the moment inertia becomes I x y dA y d dA 2 2 y dA 2 ydA d 2 2 dA Moment of Inertia - Parallel Axis Theorem The first integral is the moment of inertia about the centroid. I x y dA 2 The second component is the first moment area about the centroid i.e if the area is located at the C.G yA ydA y 0 ydA 0 Moment of Inertia - Parallel Axis Theorem Modify the equation obtain the parallel axis theorem. Ix y 2 dA 2 ydA d 2 dA Ix d A 2 Moments of Inertia and Radius of Gyration • In general case Polar Moment of Inertia IX kX A Iy ky A JO kO A • It is a mathematical property of a section concerned with a surface area and how that area is distributed about the reference axis. Look at IX of following shapes: x x x x 15 Moment of Inertia example simple rectangular shape dA bdy Centroid or Neutral axis Rectangular Area • Moment of Inertia about centroidal axes for a rectangle To Remember 1 1 IX bh 3 (300mm )(150mm ) 3 562500 mm 4 12 12 1 IX bh 3 1 1 12 IY hb 3 hb 3 (150mm )(300mm ) 3 1125000 mm 4 1 12 12 IY hb 3 – Which one is bigger? 12 New Unit! y 1. Standard sections (C-shapes, Wide flange beams, Hollow steel sections) 150mm x x 2. Rectangular shapes: h Calculate yourself, memorize equations. y 3. Other shapes (circles, triangles, 300mm b etc..): The equation or value will be given to you. 17 Moments of Inertia for Composite Areas • If the moment of inertia of each simpler area (a part of the composite area) is known or can be determined about a common axis, then the moment of inertia of the composite area equals the algebraic sum of the moments of inertia of all its parts. • Steps to follow: – Divide the composite area into smaller and simpler parts – Find the Centroid of each part – Calculate the moment of inertia (I) for each part about its centroidal axis – Use parallel-axis theorem to calculate the moment of inertia of each part about the given axis (normally the centroidal axis of composite area, if this is the case, very likely you need to find the Centroid of that composite area first) – Take the algebraic sum of the moments of inertia of all parts to get the moment of inertia of the composite area 20 Find the M.I of the I - section Y 24 mm 6mm X3 X3 8 mm y3 h3y h2y 48 mm X X X2 X2 y2 y h1y 6mm X1 X1 48 mm y1 Y Ix = Ix1 + A1h1y2 + Ix2 + A2h2y2 + Ix3 + A3h3y2 Symmetric about Y axis hence h1x, h2x, h3x = 0 C.G x = 0 , y = (A1y1 + A2y2 + A3y3)/ (A1 + A2 + A3) Ix1 , Ix2, Ix3= (b1d13 )/12, ………. h1y = y-y1 h2y = y – y2 h3y = y – y3 Similarly find Iy. Iy1 = (d1b13 )/12 Radius of Gyration along X & Y axis