材料力学第五版刘鸿文主编第五章 弯曲应力ppt

(Chapter5:StressesinBeams)SCHOOLOFMECHANICALANDELECTRICALENGINEERINGMechanicsofMaterials韩光平Chapter5StressesinbeamsMechanicsofMaterials(Chapter5:StressesinBeams)SCHOOLOFMECHANICALANDELECTRICALENGINEERINGMechanicsofMaterials韩光平§5–1纯弯曲(Purebending)§5–2纯弯曲时的正应力(Normalstressesinpurebeams)§5–3横力弯曲时的正应力(Normalstressesintransversebending)§5–4梁的切应力及强度条件(Shearstressesinbeamsandstrengthproblems)第五章弯曲应力(Stressesinbeams)§5–6提高梁强度的主要措施（Measurestostrengthenthestrengthofbeams)(Chapter5:StressesinBeams)SCHOOLOFMECHANICALANDELECTRICALENGINEERINGMechanicsofMaterials韩光平mmFSM一、弯曲构件横截面上的应力(Stressesinflexuralmembers)当梁上有横向外力作用时,一般情况下,梁的横截面上既又弯矩M,又有剪力FS。§5–1纯弯曲(Purebending)(Chapter5:StressesinBeams)SCHOOLOFMECHANICALANDELECTRICALENGINEERINGMechanicsofMaterials韩光平mmFSmmM弯矩M正应力只有与正应力有关的法向内力元素dFN=dA才能合成弯矩剪力FS切应力内力只有与切应力有关的切向内力元素dFS=dA才能合成剪力所以,在梁的横截面上一般既有正应力(Normalstresses),又有切应力(Shearstresses)(Chapter5:StressesinBeams)SCHOOLOFMECHANICALANDELECTRICALENGINEERINGMechanicsofMaterials韩光平二、分析方法(Analysismethod)平面弯曲时横截面纯弯曲梁(横截面上只有M而无FS的情况)平面弯曲时横截面横力弯曲(横截面上既有FS又有M的情况)简支梁CD段任一横截面上,剪力等于零,而弯矩为常量,所以该段梁的弯曲就是纯弯曲(Purebending).若梁在某段内各横截面的弯矩为常量,剪力为零,则该段梁的弯曲就称为纯弯曲(Purebending).三、纯弯曲（Purebending)FFaaCD++FF+FaAB(Chapter5:StressesinBeams)SCHOOLOFMECHANICALANDELECTRICALENGINEERINGMechanicsofMaterials韩光平变形几何关系物理关系静力关系观察变形，提出假设§5–2纯弯曲时的正应力(Normalstressesinpurebeams)变形的分布规律应力的分布规律建立公式(Chapter5:StressesinBeams)SCHOOLOFMECHANICALANDELECTRICALENGINEERINGMechanicsofMaterials韩光平一、实验（Experiment）1、变形现象（Deformationphenomenon)纵向线且靠近顶端的纵向线缩短，靠近底端的纵向线段伸长相对转过了一个角度,仍与变形后的纵向弧线垂直各横向线仍保持为直线各纵向线段弯成弧线，横向线(Chapter5:StressesinBeams)SCHOOLOFMECHANICALANDELECTRICALENGINEERINGMechanicsofMaterials韩光平2、提出假设(Assumptions）(a)平面假设变形前为平面的横截面变形后仍保持为平面且垂直于变形后的梁轴线(b)单向受力假设纵向纤维不相互挤压,只受单向拉压(Chapter5:StressesinBeams)SCHOOLOFMECHANICALANDELECTRICALENGINEERINGMechanicsofMaterials韩光平推论：必有一层变形前后长度不变的纤维——中性层（Neutralsurface）中性轴横截面对称轴中性轴横截面对称轴中性层(Chapter5:StressesinBeams)SCHOOLOFMECHANICALANDELECTRICALENGINEERINGMechanicsofMaterials韩光平观察变形提出假设变形的分布规律变形几何关系物理关系静力关系应力的分布规律建立公式实验平面假设单向受力假设中性层、中性轴(Chapter5:StressesinBeams)SCHOOLOFMECHANICALANDELECTRICALENGINEERINGMechanicsofMaterials韩光平dx图（b）yzxo应变分布规律直梁纯弯曲时纵向纤维的应变与它到中性层的距离成正比图（a）dx二、变形几何关系（Deformationgeometricrelation）图（c）dzyxo’o’b’b’ybboo''bb()dyxbbdoo''oodyyddd)((Chapter5:StressesinBeams)SCHOOLOFMECHANICALANDELECTRICALENGINEERINGMechanicsofMaterials韩光平三、物理关系(Physicalrelationship)所以Hooke’sLawMyzOx直梁纯弯曲时横截面上任意一点的正应力，与它到中性轴的距离成正比应力分布规律?待解决问题中性轴的位置中性层的曲率半径ρ??EyE(Chapter5:StressesinBeams)SCHOOLOFMECHANICALANDELECTRICALENGINEERINGMechanicsofMaterials韩光平观察变形提出假设变形的分布规律变形几何关系物理关系静力关系应力的分布规律建立公式实验平面假设单向受力假设中性层、中性轴y(Chapter5:StressesinBeams)SCHOOLOFMECHANICALANDELECTRICALENGINEERINGMechanicsofMaterials韩光平yzxOMdAyσdAFNMzMy(Chapter5:StressesinBeams)SCHOOLOFMECHANICALANDELECTRICALENGINEERINGMechanicsofMaterials韩光平yzxOMdAyσdA四、静力关系(Staticrelationship）横截面上内力系为垂直于横截面的空间平行力系这一力系简化，得到三个内力分量中性层的曲率半径ρ中性轴的位置待解决问题FNMzMy内力与外力相平衡可得dAdAdAzyAAAFddNNFyMzMAAyAzMddAAzAyMdd0(1)0(2)M(3)NdFyMdzMd(Chapter5:StressesinBeams)SCHOOLOFMECHANICALANDELECTRICALENGINEERINGMechanicsofMaterials韩光平将应力表达式代入(1)式，得将应力表达式代入(2)式，得中性轴通过横截面形心自然满足0dNAyEFA0dAAyE0dCyAAyAzS0dAyzEMAy0dAAyzE0dAAyzyzI(Chapter5:StressesinBeams)SCHOOLOFMECHANICALANDELECTRICALENGINEERINGMechanicsofMaterials韩光平将应力表达式代入(3)式，得zIEM1MAyyEMAzdMIEzMAyEAd2(Chapter5:StressesinBeams)SCHOOLOFMECHANICALANDELECTRICALENGINEERINGMechanicsofMaterials韩光平观察变形提出假设变形的分布规律变形几何关系物理关系静力关系应力的分布规律建立公式实验平面假设单向受力假设中性层、中性轴中性轴过横截面形心EIz称为抗弯刚度(Flexuralrigidity)zEIM1yEyzIMy(Chapter5:StressesinBeams)SCHOOLOFMECHANICALANDELECTRICALENGINEERINGMechanicsofMaterials韩光平纯弯曲时横截面上正应力的计算公式:zIMyM为梁横截面上的弯矩y为梁横截面上任意一点到中性轴的距离Iz为梁横截面对中性轴的惯性矩讨论(1)应用公式时,一般将M，y以绝对值代入，根据梁变形的情况直接判断的正负号.以中性轴为界，梁变形后凸出边的应力为拉应力(为正号).凹入边的应力为压应力(为负号).(2)最大正应力发生在横截面上离中性轴最远的点处IyMzmaxmax则公式改写为WMmax引用记号——抗弯截面系数maxyIWz(Chapter5:StressesinBeams)SCHOOLOFMECHANICALANDELECTRICALENGINEERINGMechanicsofMaterials韩光平（1）当中性轴为对称轴时矩形截面实心圆截面空心圆截面bhzyzdyzDdy322/64/2/34ddddIWz62/12/2/23bhhbhhIWzDdαDW)1(3243(Chapter5:StressesinBeams)SCHOOLOFMECHANICALANDELECTRICALENGINEERINGMechanicsofMaterials韩光平zy（2）对于中性轴不是对称轴的横截面ycmaxytmaxM应分别以横截面上受拉和受压部分距中性轴最远的距离和直接代入公式求得相应的最大正应力ytmaxycmaxzIMyσmaxcσmaxtIMyσzccmaxmaxIMyσzttmaxmax(Chapter5:StressesinBeams)SCHOOLOFMECHANICALANDELECTRICALENGINEERINGMechanicsofMaterials韩光平惯性矩的两个重要公式（1）叠加公式当一个平面图形是由若干个简单的图形组成时，根据惯性矩的定义，可先算出每一个简单图形对同一轴的惯性矩，然后求其总和，即等于整个图形对于这一轴的惯性矩。niyiyII1nizizII1(Chapter5:StressesinBeams)SCHOOLOFMECHANICALANDELECTRICALENGINEERINGMechanicsofMaterials韩光平（2）平行移轴公式图形对形心轴的惯性矩、与形心轴平行直线的惯性矩之间的关系为：AaIIyCy2AbIIzCz2式中，a，b分别为图形的形心C的纵坐标和横坐标。A为图形的面积。(Chapter5:StressesinBeams)SCHOOLOFMECHANICALANDELECTRICALENGINEERINGMechanicsofMaterials韩光平当梁上有横向力作用时，横截面上既又弯矩又有剪力，梁在此种情况下的弯曲称为横力弯曲(Nonuniformbending)。§5–3横力弯曲时的正应力(Normalstressesofthebeaminnonuniformbending)横力弯曲时，梁的横截面上既有正应力又有切应力。切应力使横截面发生翘曲，横向力引起与中性层平行的纵截面的挤压应力，纯弯曲时的平面假设和单向受力假设都不成立。一、横力弯曲(Nonuniformben