SRESSESNBENDING 85-2 NORMAL STRESS ON THE CROSS SECTION OF THE BEAM IN PLANAR BENDING 1、 NORMAL STRESS ON THE CROSS SECTION OF THE BEAM IN Longitudinal PLANAR BENDING plane of (1 Geometric law of the symmetry eutral 接多多多多 deformation: Neutral 1. Experiment of pure bending of the beam Lateral lines(a b, cd) keep I straight lines and rotate through some angles after deformation, longitudinal straight lines change into curves with M M upper fibers constructed and lower fibers elongated, lateral lines are still normal to longitudinals lines after deformation
§5－2 NORMAL STRESS ON THE CROSS SECTION OF THE BEAM IN PLANAR BENDING 1.Experiment of pure bending of the beam Lateral lines(a b、c d）keep l straight lines and rotate through some angles after deformation, longitudinal straight lines change into curves with upper fibers constructed and lower fibers elongated, lateral lines are still normal to longitudinals lines after deformation. （1）Geometric law of the deformation： 1、 NORMAL STRESS ON THE CROSS SECTION OF THE BEAM IN PLANAR BENDING Neutral layer Longitudinal plane of symmetry b d a c a b c d M M
意画应 §5-2平面弯曲时梁横截面上的正应力 纯弯曲时梁横截面 修纵向对称面 上的正应力 中性层性轴 (一)变形几何规律 1.梁的纯弯曲实验 横向线(ab、cd)变 b 形后仍为直线,但有转动; M M纵向线变为曲线,且上缩 下伸;横向线与纵向线变 形后仍正交
§5－2 平面弯曲时梁横截面上的正应力 1.梁的纯弯曲实验 横向线(a b、c d）变 形后仍为直线，但有转动； 纵向线变为曲线，且上缩 下伸；横向线与纵向线变 形后仍正交。 （一）变形几何规律： 一、 纯弯曲时梁横截面 上的正应力 中性层 纵向对称面 中性轴 b d a c a b c d M M
STRESSESINBENDING 2. Two concepts (Neutral layer: A layer at a certain height inside the beam in which the ongitudinal fibers are neither to be elongated nor to be shortened and they are neither subject to tension nor compression. This layer is called the neutral layer P@Neutral axis: The intersection of the neutral layer with any cross section 3. hYpothesis of plane section: The cross sections remain still planes and on/y Deduction rotate through some angles around their neutral axes after deformation 2 There are only normal stresses on cross sections May be proved by symmetry and the method of infinite division
There are only normal stresses on cross sections. Hypothesis of plane section：The cross sections remain still planes and only rotate through some angles around their neutral axes after deformation. （May be proved by symmetry and the method of infinite division） Neutral layer：A layer at a certain height inside the beam in which the longitudinal fibers are neither to be elongated nor to be shortened and they are neither subject to tension nor compression.This layer is called the neutral layer Neutral axis：The intersection of the neutral layer with any cross section. 2. Two concepts: 3. Deduction:
意画应 2.两个概念 ①中性层:梁内一层纤维既不伸长也不缩短,因而纤维不 受拉应力和压应力,此层纤维称中性层。 ②中性轴:中性层与横截面的交线。 3.推 论 ①平面假设:横截面变形后仍为平面,只是绕中性轴发生转动, 距中性轴等高处,变形相等。 ②横截面上只有正应力。 (可由对称性及无限分割法证明)
横截面上只有正应力。 平面假设：横截面变形后仍为平面，只是绕中性轴发生转动， 距中性轴等高处，变形相等。 （可由对称性及无限分割法证明） 3.推论 2.两个概念 中性层：梁内一层纤维既不伸长也不缩短，因而纤维不 受拉应力和压应力，此层纤维称中性层。 中性轴：中性层与横截面的交线
STRESSESINBENDING 4. Geometric equation: A,B,-ABA,B,-00, de b AB OO1 B (p+ y)d0-pd8 y B od e E.= y Z Neutral axis o(M) Y
A1 B1 O O1 4. Geometric equation： ...... (1) y x = a b c d A B dq x y 1 1 A1 B1 OO1 AB A B AB x − = − = ) ) ) OO1 ) q y q q y = + − = d ( )d d Neutral axis Y x z