Assumptions for Pure Bending.Plane assumption: under pure bending, cross-sections of beams remain planar andperpendicular to beam axis and only rotate a smallangle.: Assumption of uniaxial stress state: individuallongitudinal layers are under uniaxialtension/compression along beam axis, withoutstresses acting in between.6
• Assumption of uniaxial stress state: individual longitudinal layers are under uniaxial tension/compression along beam axis, without stresses acting in between. • Plane assumption: under pure bending, crosssections of beams remain planar and perpendicular to beam axis and only rotate a small angle. Assumptions for Pure Bending 6
Neutral Surface & Neutral AxesNeutral Axis. Before:Neutral AxisAfter:NeutralSurface: Neutral Surface the longitudinal layer under neither tension norcompression: Neutral Axes: intersecting lines of the neutral surface & crosssections.7
• Neutral Axes: intersecting lines of the neutral surface & cross sections. Neutral Surface & Neutral Axes • Neutral Surface the longitudinal layer under neither tension nor compression. Neutral Axis Neutral Surface z Neutral Surface Neutral Axis z • Before: • After: 7
KinematicsNeutralaxisMRByr(p+y)do-pde12pdep12NeutralOSurfaceThe y-coordinate is02010201measured from the proposedb-aa61neutral axis228
a b 1 2 1 2 o1 o2 y ( ) y d d y d y Neutral Surface 1 2 2 ρ dθ o1 o2 a b Kinematics 8 • The y-coordinate is measured from the proposed neutral axis. y z
Hooke's Lawα(y)= Ec(y)= EyPNeutral surface: Normal stress acting on a longitudinal layer is linearly proportionalto its distance from the neutral surface, positive for layers undertension / negative for layers under compression.: Remark: the above equation can only be used for qualitative analysisof stresses in bending beams since it is difficult to measure thecurvature of radius (p) of individual longitudinal layers.9
• Normal stress acting on a longitudinal layer is linearly proportional to its distance from the neutral surface, positive for layers under tension / negative for layers under compression. • Remark: the above equation can only be used for qualitative analysis of stresses in bending beams since it is difficult to measure the curvature of radius (ρ) of individual longitudinal layers. Hooke’s Law 9 y y E y E
Static EquivalencyEE0= F =(odA=vdAA1p福6X: Neutral axis passes through the centroid: y = O.M: for an arbitrarily defined y-coordinate:1J=ZAJ./AW0= M.=/z·od1MdAM一EIC1pEM.yU0:C21pEI, : flexural rigidityVy?dA : second moment of cross-section w.r.t. z=10
y z 0 N A A E E F dA ydA Ay A A y yzdA E M z dA 0 2 1 z Z z A A z z z E E M M y dA y dA I EI E M y y I • Neutral axis passes through the centroid: EIz : flexural rigidity 2 z A I y dA : second moment of cross-section w.r.t. z. i i i y A y A • for an arbitrarily defined y-coordinate: Static Equivalency 10 y 0