A treatise on the strength of materials : with rules for by Barlow, Peter William; Barlow, Peter; Barlow, W. H.; Humber,

By Barlow, Peter William; Barlow, Peter; Barlow, W. H.; Humber, William

This complete paintings from the nineteenth century covers the energy of fabrics with reference to development of structures, bridges and railways, and so on. and comprises an appendix at the energy of locomotive engines and the impact of susceptible planes and gradients.

summary: This accomplished paintings from the nineteenth century covers the energy of fabrics with reference to development of constructions, bridges and railways, and so on. and comprises an appendix at the energy of locomotive engines and the influence of prone planes and gradients

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Extra info for A treatise on the strength of materials : with rules for application in architecture, the construction of suspension bridges, railways, etc., and an appendix

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F In speaking of similar beams, it must be understood that each beam projects the same distance from the wall, that is, the leverage must be the same in all cases. 2 2 TRANSVERSE STRAIN. 25 ments," Art. ) It is obvious, indeed, that the ultimate strength of a beam does not depend upon its original position, but upon that which it has attained immediately before the fracture takes place. It may be proper to observe, that in the preceding expression, f= IW cos I, that force only is included which has a tendency to turn the beam about the point C : there is, however, also another exciting force, but which does not act at any mechanical advantage, that is, the force represented by K W, which in the declining posi­ tion of the beam A F C I acts by tension, and in the ascending position of A ' F' C I' by pressure : the entire expression, therefore, for the exciting force, is / = J W c o s I + &W sin I, the value of k depending upon the proportion between the areas of compression and of tension.

6 ad 2 2 6 2 4. When the beam fixed as in either of the last two cases is loaded at any other point than the centre, We shall have in the former case, by denoting the two unequal lengths by m and n, 2 m nW «. „~ m n W sec A - — - — sec A = a d S, or : = S: I lad 7 2 J 2 and in the second, 2 m n W 2 _ 2 m n W sec A sec A = a d? S,' or —:— 3 I a d* = S, 2 3 I still the same constant quantity. TRANSVERSE 27 STRAIN. The first formula will also apply to a beam fixed at any given angle of inclination; observing only, that the angle A , in this case, will represent the angle of the beam's inclination, increased or diminished by the angle of its deflection, according as its first position is ascending or descending; or rather, it will denote the angle of the beam's inclination at the moment of fracture.

M ~ m 2 ) 2 d, &c. Whence the whole deflection m D will be expressed by the series 2 2 2 m D = ~ - { m + (m - I ) + (m - 2 ) + &c. I 2 ]• . . (1), or by the summation of the series, 3 2 _ c£ [ m m D = — j — m C 3 m m ) + — • + — \ . ; but when m is infinite, then the two latter terms vanish, as being inconsiderable with regard to the first; and we have m D = - ^ - ^ . o In the same manner, if V were the length of any other beam of which the number of parts were m', but the parts individually in length equal to the former, and the element of deflection d\ we should have ™' , TV 2 D ' whence 2 2 r m D : m' D' : : m d : m' d ; but m : m' : : I : V; therefore -01 D vanes as — — ;* o that is, the deflection varies as the square of the length, and the element of deflection; but the element d obviously varies as the * We have used the above process for the convenience of those who may not be acquainted with the Huxional or differential calculus: those who are will see immedi­ ately that the summation, expressed in equation (1), is equal to ~ times the integral of mi 2 x d x ; that is, z d /% dx — Ix a x — = mj o 7ri z T 2 d m .

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