. Theory of structures and strength of materials. Fig. 503. Then dsA=A,sece=A,^ (17) Also, since the moments of inertia of similar figures varyas the breadth and as the cube of the depth, and since thedepth in the present case is constant, dsI=Asece = I- (IB) ^ . T Hsec 0 H ^ ^ . , , , Agam, ~T- = -z 2) = X ^^^ ^^ mtensity of the thrust A A J sec A, is constant throughout. ARCHED RIB OF UNIFORM DEPTH. 789 Hence, equations (5), (15), and (16), respectively, become I — I. iijy X y -y (19) (20) (21) Equation (19) shows that the deflection at each point of therib is the same as that at correspondi
. Theory of structures and strength of materials. Fig. 503. Then dsA=A,sece=A,^ (17) Also, since the moments of inertia of similar figures varyas the breadth and as the cube of the depth, and since thedepth in the present case is constant, dsI=Asece = I- (IB) ^ . T Hsec 0 H ^ ^ . , , , Agam, ~T- = -z 2) = X ^^^ ^^ mtensity of the thrust A A J sec A, is constant throughout. ARCHED RIB OF UNIFORM DEPTH. 789 Hence, equations (5), (15), and (16), respectively, become I — I. iijy X y -y (19) (20) (21) Equation (19) shows that the deflection at each point of therib is the same as that at corresponding points of a straighthorizontal beam of a uniform section equal to that of the ribat the crown, and acted upon by the same bending moments. Ribs of uniform stiffness are not usual in practice, but theformulae deduced in the present article may be applied withoutsensible error to flat segmental ribs of uniform section. 18. Parabolic Rib of Uniform Depth and Stiffness, withRolling Load ; the Ends fixed in Direction; the Abut-ments Fig. 504. Let the axis of ;r be a tangent to the neutral curve at itssummit. Let k be the rise of the curve. Let X, y be the co-ordinates at any point C with respectto O. Then y = 4kfl_1A2 ^l (22) and dx ~ r\2 -^h di- 4k dy^ 4k d^y Sk I dx. dx 7- (23) 790 THEORY OF STRUCTURES. Let w be the dead load per horizontal unit of ;< ^1 live Let the live load cover a length D£, = rl, of the b)^ (A) formulae relating to the unloaded divisionOE, and by (B) formulae r^elating to the loaded division (7) and (8), respectively, become (A) S=S,-\-\-^-w)x; (24) (B) 5 = 5„ + (^ - ^4^- - ^^\^ - (I - r)l\. . (25)yA) Mz^M,-\-S,x^\-^-w)-- (26) (B) M=M,-^rS,x-\-\~^-w)-^-—{x-{Y-r)l\\ (27) Since the ends are fixed, ^0 =0 = ?; (28) Hence, by equations (19) and (26), (A) ^^--^\M,x-\-S--^\-^-w)-^Y (29)and by equations (19) and (27), (B) .= _J^|^.. + s| + (?^-z<,)| -^{x-ii-ryvY (30) When X ?= I, i ^^ i^ — o, and theref
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