Centroids & Moment of Inertia

Centroids & Moment of Inertia EGCE201 Strength of Materials I Instructor: ดร.วรรณสิริ พันธ์อุไร (อ.ปู) ห้องทำงาน:6391 ภาควิชาวิศวกรรมโยธา E-mail: egwpr@mahidol.ac.th โทรศัพท์: 66(02) 889-2138 ต่อ 6391

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. • The centroid of any area can be found by taking moments of identifiable areas (such as rectangles or triangles) about any axis. • 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. Hollowed pipes, L shaped section have centroid located outside of the material of the section Centroidal axis or Neutral Sum MAtotal = MA1 + MA2 + MA3+ ...

y ZZ’ b h/2 centroid h/2 b area x distance | from the centroid of the area to the moment axis centroid example simple rectangular shape Sum MAtotal = MA1 + MA2 + MA3+ ... Take ZZ’ as the reference axis and take moment w.r.t ZZ’ axis

Moment of Inertia (I) • also known as the Second Moment of the Areais 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

y h/2 z dy h/2 b Moment of Inertia example simple rectangular shape Centroid or Neutral axis

h y h/2 b/2 z z b/2 h/2 y b “I” is an important value! • It is used to determine the state of stress in a section. • It is used to calculate the resistance to bending. • It can be used to determine the amount of deflection in a beam. > Stronger section

Built-up sections • It is often advantageous to combine a number of smaller members in order to create a beam or column of greater strength. • The moment of inertia of such a built-up section is found by adding the moments of inertia of the component parts

Transfer formula • There are many built-up sections in which the component parts are not symmetrically distributed about the centroidal axis. • To determine the moment of inertia of such a section is to find the moment of inertia of the component parts about their own centroidal axis and then apply the transfer formula. • The transfer formula transfers the moment of inertia of a section or area from its own centroidal axis to another parallel axis. It is known from calculus to be:

Centroids & Moment of Inertia