tailieunhanh - Introduction to Elasticity Part 7

Tham khảo tài liệu 'introduction to elasticity part 7', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | The quantity y2 dA is the rectangular moment of inertia with respect to the centroidal axis denoted I. For a rectangular cross section of height h and width b as shown in Fig. 3 this is I r y2 bdy J-h 2 12 Solving Eqn. 4 for v xx the beam curvature is M v xx EI 5 6 5. An explicit formula for the stress can be obtained by using this in Eqn. 3 V M yEEĨ EI -My I 7 The final expression for stress Eqn. 7 is similar to Tez Tr J for twisted circular shafts the stress varies linearly from zero at the neutral axis to a maximum at the outer surface it varies inversely with the moment of inertia of the cross section and it is independent of the material s properties. Just as a designer will favor annular drive shafts to maximize the polar moment of inertia J beams are often made with wide flanges at the upper and lower surfaces to increase I. Example 1 Figure 4 A cantilevered T-beam. Consider a cantilevered T-beam with dimensions as shown in Fig. 4 carrying a uniform loading of w N m. The maximum bending moment occurs at the wall and is easily found to be Mmax wL L 2 . The stress is then given by Eqn. 7 which requires that we know the location of the neutral axis since y and I are measured from there . The distance y from the bottom of the beam to the centroidal neutral axis can be found using the composite area theorem see Prob. 1 . This theorem states that the distance from an arbitrary axis to the centroid of an area made up of several subareas is the sum of the subareas times the distance to their individual centroids divided by the sum of the subareas . the total area _ P A y y Ei Ai For our example this is 4 _ d 2 cd d b 2 ab y cd ab The moments of inertia of the individual parts of the compound area with respect to their own centroids are just ab3 12 and cd3 12. These moments can be referenced to the horizontal axis through the centroid of the compound area using the parallel axis theorem see Prob. 3 . This theorem states that the moment of inertia Iz0 of an area A