tailieunhanh - Lecture Strength of Materials I: Chapter 5 - PhD. Tran Minh Tu

Chapter 5 - Geometric properties of an area. The following will be discussed in this chapter: First moment of area, moment of inertia for an area, moment of inertia for some simple areas, parallel - axis theorem. | Lecture Strength of Materials I: Chapter 5 - PhD. Tran Minh Tu STRENGTH OF MATERIALS 1/10/2013 TRAN MINH TU - University of Civil Engineering, 1 Giai Phong Str. 55, Hai Ba Trung Dist. Hanoi, Vietnam CHAPTER 5 Geometric Properties of an Area 1/10/2013 Contents . Introduction . First moment of area . Moment of inertia for an area . Moment of inertia for some simple areas . Parallel - axis theorem . Examples 1/10/2013 3 . Introduction Dimension, shape? 1/10/2013 4 . First Moment of Area . Definition • The first moment of a plane A about the x- and y-axes are defined as Sx ( A) ydA Sy xdA ( A) • Value: positive, negative or zero • Dimension: [L3]; Unit: m3, cm3,. • Centroidal axes: are axes, which first moment of a plane A about them is zero . The centroid of an area • The centroid C of the area is defined as the point in the xy-plane that has the coordinates 1/10/2013 5 . First Moment of Area Sy Sx xC yC A A yC C • If the origin of the xy-coordinate system xC is the centroid of the area then Sx=Sy=0 • Whenever the area has an axis of symmetry, the centroid of the area will lie on that axis • If the area can be subdivided in to simple geometric shapes (rectangles, circles, etc., then n n Sx S i x S y S yi i 1 i 1 1/10/2013 6 . First Moment of Area y . The centroid of composite area n yC1 C1 Sy x Ci Ai C2 xC i 1 n A A i i 1 C3 xC1 x n Sx y Ci Ai yC i 1 n A A i i 1 1/10/2013 7 . Moment of Inertia for an Area . Moment of inertia Ix ( A) y 2dA Iy ( A) x 2dA . Polar moment of inertia Ip ( A) 2 dA I x I y . Product of inertia • The value of moment of inertia and polar moment of inertia always positive, but the I xy xydA ( A) product of inertia can be positive, negative, or zero • Dimension: [L4]; Unit: .

• Kiến trúc - Xây dựng
• Strength of Materials
• Sức bền vật liệu
• Bài giảng Sức bền vật liệu
• Geometric properties of an area
• Moment of inertia
• Parallel axis theorem
• Finite element modelling
• Rectangular concrete filled steel tube stub columns incorporating high strength
• Ultra high strength materials
• Concentric axial compression
• Axial compression
• Theory and problems of strength of materials
• Statically indeterminate force systems tension
• Direct shear stresses
• Bending moment
• Moments of inertia
• Lecture Strength of materials
• Modulus of elasticity
• Statically indeterminate systems
• Statically indeterminate
• Stress transformation
• Plane stress state
• Transformation of plane stress
• Failure theories
• Permanently distorted
• Ductile material deforms
• Properties of cross sections
• Center of gravity – centroid
• Moment of Inertia of an area
• Radius of gyration
• Beam deflection
• Beam under transverse loading
• Statically indeterminate beams
• Torsional loads on circular shafts
• Torsion of noncircular members
• Helical spring under axial load
• Combined loading
• Eccentric axial loadings
• General combined loadings
• The secant formula
• Stability of structures
• Eccentric loadings
• Cementitious materials
• high strength concrete
• Strength required
• Placing procedures
• Splitting tensile strength
• Concentrically loaded columns
• Selected topics on strength
• The strength of materials
• Basic hypothesis
• Internal forces
• Three dimensional state
• Dimensional state
• Deflection of beams
• Bending stress
• Shearing stress
• Strength condition
• Torsional loads
• Circular shafts
• Engineering materials
• Their properties
• Price and availability
• The elastic moduli
• Yield strength
• Tensile strength
• Fast fracture
• Toughness and fatigue
• Mechanical engineering principles
• Forces acting at a point
• Forces in structures
• Analysis of stress
• The elastic limit
• Deflection of transversally leaded beams
• Statically indeterminate problems
• Energy of strain
• Combines bending
• Helical spring
• Close coiled
• Critical load
• Applied strength of materials
• Engineering stress
• Thermal expansion
• Properties of areas
• Torsion in round shafts
• Beam reactions
• Shear diagrams
• Moment diagrams
• Stresses in beams
• Combined stresses
• Visualizing stress
• Elastic deflection
• Double integration method
• Plastic deformations
• Mechanics and strength of materials
• The stress tensor
• The strain tensor
• Constitutive law
• Fundamental concepts
• Shear force
• Bending deflections
• Structural stability
• Energy theorems
• The total potential energy