Tham khảo tài liệu 'bruhn - analysis and design of flight vehicles structures episode 1 part 6', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | A7 8 D E F L ECTĨONS OF STRUCTURES A Ao l- q constant q 40 in. Ao 4 L 40 in. E 10 X 10 psi Fig. A7. 12 Solution After addition of the fictitious end load R the axial load from statics was found to tie s x R q L - x Hence since leadings other than tensile are of a secondary nature 1 fL s3dx 2 I AE J 0 1 L ft q L - Jo A s 1--J erally are easier to evaluate. Example Problem 11 Find the deflection at point s of the team of rig. . . p _ i El constant 3 3 3 Fig A7 13 Solution A fictitious load Pl was added at point 3 and the bending moment diagram was drawn in two parts. 2PL _2PX B. X 3 3 Mp - ----------- ---------- 2PL PY P L PrY Lx I Y 3- -9-3 9 3 T-2P1L __ .----------------p2 2P _ 2P. z MP1 M r 3 3- 0 X L 3 0 Y L 3 0 Z L 3 L L - x 3dx 1 2l Before evaluating the Integrals it was observed that the steps to follow In which u was to be differentiated with respect to R and the subsequent setting of R a would drop out both the first and last terms. Hence only the second term was evaluated. u X Rq 1 - m 2 X Ao 4 Then 0 dR Sr 1 - 111 2 Oil A J jit DIFFERENTIATION UNDER THE INTEGRAL SIGN An Important labor savings may be had In the calculation of deflections by Castlgllano s theorem. In the strain energy Integrals arising In this class of problems the load p with respect to which the deflection is to be found acts as an Independent parameter In the Integral. Provided certain requirements for continuity of the functions are met - and they Invariably are In these problems - the differentiation with respect to p may be carried out before the Integration is made. The resulting Integrals gen- Fig. A7 13a Then neglecting the energy of shear as being small u 2EI 1 2EI L S g 2L - 3Y 0 Zi 9 fL 3 r - 1 0 P p 2Pj 2EI 9 L 3Y dy Differentiation under the Integral sign with respect to Pi gave _ Au ỖB Ipx 1 El 1 El rL 3__ p 2P 9 3y KL 3y 2 L - 3s . -------- da dy For beams of usual high length - to - depth ratios the shear strain energy is small compared to the energy OÍ .