tailieunhanh - mechanical design an integrated approach Part 4

Tham khảo tài liệu 'mechanical design an integrated approach part 4', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 160 PART Fundamentals v a 0 0 ứ 0 0 d 0 M a đ Ố Figure Boundary conditions a fixed end b simply supported end c free end d guided or sliding support. V a - 0 M a 0 c V 0 The deflection V of a beam can be found by solving any one of the foregoing equations by successive integrations. The choice of equation depends on the ease with which an expression of load shear or moment can be formulated and individual preference. The approach to solving the deflection problem beginning With Eq. or is known as the multiple-integration method. When Eq. is used because two integrations are required to obtain the V this is called the double-integration method. The constants of the integration are evaluated using the specified conditions on the ends of the beam that is the boundary conditions. Frequently encountered conditions that may apply at the ends ụ a of a beam are shown in Figure . We see from the figure that the force static variables M V and the geometric kinematic variables V Ỡ are 0 for common situations. If the beam has a cross-sectional width b that is large compared to the depth h . b h the beam is stiffer and the deflection is less than that determined by Eqs. for narrow beams. The large cross-sectional width prevents the lateral expansion and contraction of the material and the deflection is thereby reduced as shown in Section . An improved value for the deflection V of wide beams is obtained by multiplying the result given by the equation for a narrow beam by 1 v2 where V is Poisson s ratio. EXAMPLE I Finding Beam Deflections by the Double-Integration Method A simply supported beam is subjected to a concentrated load at a distance a from the left end as shown in Figure . Develop a The expressions for the elastic curve. b The deflection at point c for the case in which a b LỊ2. Figure Example . 161 CHAPTER 4 Deflection AND Impact IỆỆỆỆT Solution The reactions are noted in the figure. a The moments for the segments AC