A treatise on concrete, plain and reinforced : materials, construction, and design of concrete and reinforced concrete; 2nd ed. . e change in total span length, the two halves of the arch being equal, is zl ^ My ^j = L (21) The change in inclination of tangent to axis at crown is 2I^Myj = 0 (22) Replacing the M of equations (21) and (22) by Mc + Hey, which is the moment at any point D,Fig. 166, in terms of moment and thrust at the crown, and making . constant, there results ^McIy + 2^jHcIf=JL mMc + He Z y = oFrom which rr I mEJL , . and Mc = -^r~^ (24) M = moment. He = crown thrust, m = number
A treatise on concrete, plain and reinforced : materials, construction, and design of concrete and reinforced concrete; 2nd ed. . e change in total span length, the two halves of the arch being equal, is zl ^ My ^j = L (21) The change in inclination of tangent to axis at crown is 2I^Myj = 0 (22) Replacing the M of equations (21) and (22) by Mc + Hey, which is the moment at any point D,Fig. 166, in terms of moment and thrust at the crown, and making . constant, there results ^McIy + 2^jHcIf=JL mMc + He Z y = oFrom which rr I mEJL , . and Mc = -^r~^ (24) M = moment. He = crown thrust, m = number divisions of half axis, s = length ofdivision of axis. I = moment inertia. L = span. E = modulus of elasticity. i<L=-change ofspan length, t = nse or fall of temperature, c = coefficient of expansion, x, y = coordinates ofa point. ARCHES 557 The moment at any point D may be found as soon as the values of H^and M^ have been determined by means of the relation M = M^ + H^ (27) or we can say that the moment at any point equals the thrust H* multi-plied by the distance from the point in question to the line 00, Fig. C. Fig. 166.—Moments and Thrusts due to Changes of Temperature. {See p. 55 6.) Above the line 00, Fig. 166, the moments are all negative, being a maxi-mum at the crown, and below 00 they are all positive, being maximum at A and B. The line 00 is below the crown a distance d = ly. At the two points where 00 intersect the arch axis the moments are zero, as is evidentfrom equations (24) and (26). Fall in Temperature. Here the thrust at crown is / c tLmE ^^^ ~^ ^ly - (lyy] ^^^^ where c is , ^^d moment at crown is H,Iy M^= - (29) and, as above, M =M, + Hj (30) In placing a numerical value for H^ in the last two equations, it shouldbe observed that it is a negative quantity. If in the equations the valuesof L and y are in feet, E is in pounds per square foot. Above 00 themoments are all positive, below they are all negative. The thrust at thecrown is real
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