材料力学06_梁的剪应力.ppt

06 Shearing Stress in Beams Introduction Calculation of Shearing Stress: Preliminary Shear on the Horizontal Face of a Beam Element Shear on the Horizontal Face: First Moment of Area Example 1 Example 1: Solution Strategy Example 1: Solution Shearing Stress in a Beam Shear Stress in a Narrow Rectangular Beam I Beam: Shear Stress txy Example 2 Example 2: Solution Longitudinal Shear on a Beam Element of Arbitrary Shape Example 3 Example 3: Solution Shearing Stresses in Thin-Walled Members Shearing Stresses in Thin-Walled Members Shearing Stresses in Thin-Walled Members Esc TOPIC School of Engineering Mechanical Engineering ENGR 323 Mechanics of Deformable Bodies ? Tulong Zhu, All rights reserved. For most beams, both the bending moment and shear force will present. When shearing stresses are exerted on the vertical faces of an element, equal stresses must be exerted on the horizontal faces. The possible stress components at any point on the cross section will be sx, txy and txz. x y z sx txz txy If V = 0 (pure bending), txy = txz =0 Now V ? 0, both txy and txz may not be zero. Distribution of shearing stresses is unknown but satisfies (? txy ?0) txz may not be zero, but the average is zero The distribution of txy can not be determined from statics alone. Analysis of deformation may not help too much. However, we know that shearing stress comes in pair, arrow to arrow and tail to tail, as shown. If we can determine tyx, then from txy = tyx, we can obtain txy. x y z sx txz txy We know that txz may not be zero, but the average is zero. However, for a narrow beam (width much smaller than height), we can assume that txz = 0 zero, since tzx ?0 on the left and right surfaces. Therefore, this topic will be focusing on txy only. txz tzx = 0 M V V+dV M+dM txy tyx s s + ds dH (Resultant of tyx) F1 F2 D′ C D dx A* y For the beam shown, consider an element CDD′C′ to calculate shear stress at y1. C′ C D A* Shear Flow First Moment of Area I is the second moment of are