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Bruhn - Analysis And Design Of Flight Vehicles Structures Episode 1 Part 8

Tham khảo tài liệu 'bruhn - analysis and design of flight vehicles structures episode 1 part 8', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | AS. 18 STATICALLY INDETERMINATE STRUCTURES SECOND In the form of eq. 14 Qi 0 .806 1.154 521 Qa Qa - Ị 0 2000 -1.564 1.00 -1.729 0 416 Q 0 _ 0 1.00_ V 4 Note that In this case consisted of only- one column inasmuch as there was only a single external load. In Art. A7.9 the strain energy was -written 2U LQdEd w 15 where matrix of member flexibility coefficients Art. A7.1Q . If now eq. 14 and its transpose are used to substitute Into IS the expression becomes note the use of 1 J r s and m n Interchangeably of external-point influence coefficients In the cut structure deflection at point m per unit load at point n. Ed Ed Ed Cd of influence coefficients relating relative displacements at the cuts to external loads displacement at cut r per unit lead at point n. Ed 5 Ed Ed Ed the drlx 18 Of Influence coefficients relating relative displacements at the outs to redundant loads at the cuts displacement at cut r per unit redundant force at cut s. with the above notation one may write 2U VdEd JEdz x Ed x EdN Ed Multiplying out LviEdEdEdEn L dEdEdEdW OjEdW------- 19 Now according to the Theorem of Least Work 5U 8qr 0 for continuity. Then differentiating eq. 19 i -EdddEdH---- í2 This last result may be verified by writing eq. 19 out in expanded form differentiating and then recombining In matrix form. Rearranged eq. 20 gives L J Ed Ed Ed N Ed id Ed Ed------------------------------------- 21 The reader may satisfy himself that the cross product term in the middle of the above result Is correct by observing that because of the symmetry of Ed L rj Ed Ed Ed pn ưdEdEd Ed Ed Eq. 21 Is a set of simultaneous equations for the redundant Internal forces q q . It may be compared with eq. 6 of Art. A8.6 to which It corresponds. Eq. 21 may be solved directly from the form there displayed or Its solution may be obtained by computing Pds3j the inverse of the matrix of coefficients giving Es -ErdEd Pn -------- The various matrix triple products occurring above are assigned the following .