[工学]材料力学课件第四版刘鸿文3.ppt

材料力学（三） 刘鸿文 弯曲变形 应力和应变分析 强度理论 §7--3 二向应力状态分析----解析法 a f e Analysis of biaxial stress states----Analytical method Normal stress is positive as it in tensile stress and shear stress is positive as its direction is up on left side of unit cube, or is down on right side of unit cube τx is positive, butτy is negative, ? is positive as N rotates counterclockwise from x axis. 讨论：1、角度为二倍。 反映二个平行平面上的应力完全相等。若斜面的夹角超过90可用小于90的处理。 Angle in trigonometry is twice the rotation angle of normal line, it reflects that stresses on two parallel planes are completely identical. If the angle of oblique exceeds to 90 degree, we can select an angle less than 90 degree. 2、正应力和剪应力为角度的函数。 （7--5）式有二个角，它们相差90。一个为最大主应力, 另一个为最小主应力。 在第二章中告诉我们，脆性材料破坏时由拉应力产生的。最大主应力方位决定了破坏的方向。 Normal stress and shear stress are functions of rotation angle. Equation 7-5 can lead two angles, their difference is 90 degree, one represents the maximum stress Another one represents the minimum stress. In Ch. 2, we have found that failure of brittle materials results from tension stress, the orientation of maximum principle stress determines the direction of failure. （3）相互垂直面的正应力之和为常数。 Sum of normal stresses on mutually perpendicular planes is a constant. Difference between ?0 and ?1 is 45 degree. The orientations of the maximum shear stresses is in 90 degree difference. The first stress invariant 例7--2 分析铸铁试件受扭时破坏面的 方位。 T T T 分析：已知D，T。由剪应力公式 可知剪应力的大小。方向以左上 右下，因内力图为负的。其方向如 图示。 破坏面如图示。方向垂直最大拉应力。 Figure out the orientation of failure plane as cast iron rod is subjected to torsion. Known diameter D and torque T, the shear stress can be calculated by shear stress equation. Failure plane is shown in the figure, it is perpendicular to the maximum tension stress. §7--4 二向应力状态分析------图解法 Analysis of biaxial stress state--method of geometric solution center radius Equation of circle d D E F A C B A1 B1 o 2 2 Build a Cartesian coordinate system using transverse and longitudinal coordinates σandτ Find the po