Essentials in the theory of framed structures . red. Hence Ml = 45X. The origin for h is at B, hence M2 = 15a;, and EIJo EI A / I 6 ,, ,. 9,720 9,720 A = h-\ {h — k) = ^^ = -—^^ = in. 24 EI 1,500,000 The expression / M\xdx represents the area-moment of PQV about P, hence ..-f/.,oX3X4) = 5f? • Likewise, the expression j MiXdx represents the area-moment of SQV about S, hence I / ,, .,1 s 29,160 = £7*^^70 X 9 X 12) = -^^ Thus it is clear that, when the M-diagram can be convenientlydivided into portions whose areas and centroids are easilyfound, a semigraphic or geometric solution can be qui
Essentials in the theory of framed structures . red. Hence Ml = 45X. The origin for h is at B, hence M2 = 15a;, and EIJo EI A / I 6 ,, ,. 9,720 9,720 A = h-\ {h — k) = ^^ = -—^^ = in. 24 EI 1,500,000 The expression / M\xdx represents the area-moment of PQV about P, hence ..-f/.,oX3X4) = 5f? • Likewise, the expression j MiXdx represents the area-moment of SQV about S, hence I / ,, .,1 s 29,160 = £7*^^70 X 9 X 12) = -^^ Thus it is clear that, when the M-diagram can be convenientlydivided into portions whose areas and centroids are easilyfound, a semigraphic or geometric solution can be quicklymade. The area of the M-diagram to be considered in eachcase is included between two ordinates. One ordinate passesthrough th« point of tangency, on the other ordinate the tan-gential deviation is found; and the moment of this area is takenabout the latter ordinate. In Fig. 1296 the tangent is drawn through A. The area- 214 THEORY OF FRAMED STRUCTURES Chap. V moment for ti is PQS about 5; and for ti, the area-moment isPQV about . _ ^r 270 X 9 X12 = 29,1601 * EI1270 X 3(18 + 2) = i6,26oJ J I / ^y ^y \ 1,620 h = ^(270 X 3 X 2) = -^ A + «3 = ^ = ii^^^ 4 £/ ^ 11,340 — 1,620 9,720 , , 45»36oEI Sec. II DEFLECTION OF BEAMS 215 In the algebraic solution, the origin for ti is at 5. M = 15Xfor values of x between o and 18, and M = i^x — 60 {x — 18) =1,080 — 4^x, for values of x between 18 and 24, hence ^4 = -£77 I Mxdx = -=^ I iSjxHx + -=r^ f (1,080a; — /i^i,x^)dx-ti-Jo Eijo -b-Ijis _ 29,160 + 16,200 _ 45,360 ~ EI EI The origin for /s is at V, hence Jlf = 45 (6 — a;), and 1,620 h = ^ Cidx - x^)dxEl Jo EI The geometric solution is considerably shorter when M is not acontinuous function of x as in the case of Fig. 129c the tangent is drawn through B. EI t = ^[270 X 3 X 4 = 3,240] ^° E/L270 X 9(6 + 6) = 29,i6oJ .. = J^(27oX9X6)=iM8? 4 EI 24,300 — 14,580 9,720 , J. A = —^^ ^^^— = -^^-^— as before. EI EI ? 141. Maxunum Deflection.—^Let
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Keywords: ., bookcentury1900, bookdecade1920, booksubjectstructu, bookyear1922
