Stress Transformation CHAPTER 14 STRESS TRANSFORMATION 14.1 Plane Stress As shown in Figure 14.1, the general state of stress at a point is characterized by six independent normal and shear stress components, which act on the faces of an element of material located at the point. Figure 14.1 This state of stress, however, is not often encountered in engineering practice. Instead, engineers frequently make approximations or simplifications of the loadings on a body in order that the stress produced in a structural member or mechanical element can be analyzed in a single plane. If there is no load on the surface of a body, then the normal and shear stress components will be zero on the face of an element that lies on the surface. Consequently, the corresponding stress components on the opposite face will also be zero, and so the material at the point will be subjected to plane stress. As shown in Figure 14.2, the general state of plane stress at a point is represented by a combination of two normal-stress components, x, y, and one shear-stress component, xy, which act on four faces of the element. 14.1 平面应力状态 如图 14.1 所示， 一点处的应力状态由作用在该点处的单元体的面上的六个独立的正应力和切应力来表示。 zxyxyyxyzzyxzzx工程实践中并不经常遇到这种应力状态。相反的，工程师们经常会对加载做一些简化，以便在一个平面内分析结构件或机械零件中的应力。若物体的表面没有受力，则在表面的单元体的面上应力为零。相应的，与该面相对的面上的应力也为零，此时，这点的材料处于平面应力状态。 如图 14.示，一点平面应力状态是由作用在单元体的四个面上的两个正应力x，y和一个切应力xy来表示的。 57Stress Transformation xyxyFigure 14.1 14.2 General Equations of Plane Stress Transformation Sign Convention. Before the transformation equations are derived, a sign convention for the stress components must be established first. A normal stress component is positive provided it is a tensile stress, and a shear stress component is positive when it causes clockwise rotation of the member on which it acts. Normal and Shear Stress Components. Using the established sign convention, the element in Figure 14.2(a) is sectioned along the inclined plane and the segment shown in Figure 14.2(b) is isolated. Figure 14.2 (a) (b) The angle is measured from the positive x to the positive x axis. It is positive provided it rotates counterclockwise as shown in Figure 14.2(b). Assuming the sectioned area is A, the horizontal and vertical faces of the segment have an area of Asin and Acos, respectively. The resulting free-body diagram of the segment is shown in Figure 14.3. Applying the equations of force equilibrium, the following equations are obtained, 14.2 平面应力变换的 一般公式 符号规定：符号规定：推导变换公式之前应先对应力做符号规定。正应力是拉应力时为正，使 单元体产生顺时钟转动的切应力为正。 正应力和切应力分量：正应力和切应力分量：应用符号规定， 图 14.2(a)中的单元由一斜截面分开，并分离出来如图 14.2(b)所示。 xyx y sin cos xyxy 如图 14.2(b)所示，角是自正 x 方向到正 x方向，逆时钟为正。 假设斜面面积为A， 则水平和垂直面的面积分别为Asin 和Acos。合力的受力图如图14.3所示。应用力的平衡方程，可得如下方程： 58Stress Transformation xx y x y sin x cos x y xy sin xy cos Figure 14.3 0sin)sin(cos)sin(cos)cos(sin)cos(0=+=AAAAAFyxyxxyxx0cos)sin(sin)sin(sin)cos(cos)cos(0=+=AAAAAFyxyxxyyxyThere two equations may be simplified by using the trigonometric identities sin2 = 2sin cos, sin2 = (1-cos2)/2, and cos2 = (1+cos2)/2, in which case, 2sin2cos22xyyxyx x+=(14.1) 2cos2sin2xyyx yx+=(14.2) 14.3 Principal Stresses and Maximum In-Plane Shear Stress From equations 14.1 and 14.2, it can be seen that x and xy depend on the angle of inclination of the planes on which these stresses act. In engineering practice it is often important to determine the orientation of the planes that causes the normal stress to be a maximum and a miminum and the orientation of the planes that causes the shear stress to be a maximum. 14.3 主应力和面内最 大切应力 由式 14.1 和 14.2 可知x 和xy 取决于 的值。 通常工程实践中确定最大和最小正应力以及最大切应力的作用面的方位角是重要的。 59Stress Transformation In-Plane Principal Stresses. To determine the maximum and minimum normal stress, equation 14.1 must be differentiated with respect to and set the result equal to zero. Solving this equation the orientation = p of the planes of maximum and minimum normal stresses can be obtained as, yxxy p=22tan (14.3) The solution has two roots p1 and p2. These values must be substituted into equation 14.1 to obtain the required normal stresses. Therefore, the maximum and minimum in-plane normal stresses are, 22minmax 22xyyxyx+ += (14.4) It can be seen that there is no shear stress acts on the planes on which maximum and minimum normal stresses act. The plane that has no shear stress acts on is called the principal plane. The normal stresses that act on the principal planes are called the in-plane principal stresses, namely 1 and 2, according to the magnitude. For a general three-dimensional case, there are three principal stresses, 123. Maximum In-Plane Shear Stress. The orientation of an element that is subjected to maximum shear stress on its faces can be determined by taking the derivative of equation 14.