"Centroids & Moment of Inertia"
Centroids & Moment of Inertia EGCE201 Strength of Materials I Instructor: ดร.วรรณสิ ร ิ พันธอุไร (อ.ปู) ์ ห้องทำงำน: 6391 ภำควิชำวิศวกรรมโยธำ E-mail: email@example.com โทรศัพท: 66(02) 889-2138 ์ ตอ 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. Hollowed pipes, L shaped section have centroid located outside of the material of the section Centroidal axis • The centroid of any area can be found by taking or Neutral moments of identifiable areas (such as rectangles or triangles) about any axis. Sum MAtotal = MA1 + MA2 + MA3+ ... • 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. centroid example simple rectangular shape y h/2 Sum MAtotal = MA1 + MA2 + MA3+ ... centroid area x distance | from the centroid h/2 of the area to the moment axis ZZ’ b h h h h h (bh)Y (b ) (b ) 2 2 4 2 4 Take ZZ’ as the reference axis h 3h h and take moment w.r.t ZZ’ axis (b ) 2 4 4 (b )h h 2 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 Moment of Inertia example simple rectangular shape I z y dA h 2 2 I z y bdy 2 h 2 dA bdy y3 h 2 y b 3 h 2 h/2 b h 3 h 3 z h/2 dy 3 8 8 bh 3 Centroid or Neutral axis b 12 “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. y h h/2 b/2 z z h/2 b/2 b y bh 3 > hb 3 Iz Iz 12 Stronger section 12 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: