TAILIEUCHUNG - 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. | 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 m4 cm4 . 1 10 2013 8 . Moment of Inertia for an Area - The product of inertia Ixy for an area will be zero if either the x or the y axis is an axis of symmetry for the area - The area with hole then the hole s area is given by minus sign. - The composite areas n Sx S n S y S yi i x i 1 i 1 n n Ix I i x I y I yi i 1 i 1 1 10 2013 9 . .

• Cơ khí - Chế tạo máy
• 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