200 



THE CIVIL ENGINEER AND ARCHITECTS JOURNAL. 



[September, 



But, area N L F F E = area N S U \' K 

 Hence we liave the equation, 



p(.r-d"^ pd"b" / .r \_q(d-d'- 

 2 ^ + -2^^ V+ ^^-d"))-^'2-(^ 



qd' bdd—lx — d') 



d' — 'Y 



2 6' (X 

 qd' b(2d—2T 



■d") 



dy-+t^x 



■d') 



{x — d")/ 2(x — d") 

 And multiplying by 2 (j- — d"), 



+ 



(.r— i" + j) =g(d—d' — xr- + 



p b' {x—d"y +p d"b" (2 x—d") — 



q b' {d—d' — xY + 5 (/' 6 (2 d — 2 x — d'). 



Ex|)anding tlie squares we have 

 ;, b' (j>^ — 2d"x-\-d"^)^p b" d" (2x — d") = qb' (d^ + d'- + x'+ 

 2d' x — 2dd' —2d x) + qd' b(2d—2 x—d'), or 



p b' X' ~2 p b' d" X -i- p b' d"^ + 2 p b" d" x—p b" d"^ = 



V 6' (d'- + d" —2 d d')-\-q h' a^ + 2 q V d' x — 2q U d x + 



V d' b (2 d—d') — 2qd' b x. 

 And by transposition, 



(pb' — >i U) .r2 + (2 p b" d" — 2p b' d" -\--2i b' d — 2 'j b' d' + 

 2 V d' b) x=p b" d"^—p b' d"'' + q b' {d'-\-d"' — 2 d d') + 

 1/ d' h (2d—d'), 



lience .r' + 



2(pd" {!/'- 



■b')+g!>' (.d — d') + qd'i) ^_ 



p d'—qb' 

 p d'" (b" — b') + q [V (d—d'Y + d' b (2 d—d')-\ 



p //—qd' 



Ami comi)leting the square, extracting the root, &c., we have 



•»■ =A / l pd"Hi'" — i^) + 9li''(d — d'y+d'b(2d-d') ], 



'V \ pb' — q^ ^ 



( 



■pd^ (i" — yj + qb' ( d—d') + qd'6Y \ 



pb' — qb' 



) 



J 



(^.) 



Fig- 

 Q 



a. 



p d" (b" — b') + q V {d— d') +qd ' b 

 p b' — qV 



And thus we obtain tlie situation of the neutral axis. It must be 

 admitted that the equation is rather forbidding in appearance, but the 

 reductiiiu of the value of x, will not be found so tedious as may at 

 first be imagined, since the quantities are simple, and the same com- 

 binations often repeated. 



5. Assuming that N L, the dis- 

 tance from the centre of the neutral 

 axis, to the bottom of the girder, 

 has been found by equation 2 ; ue 

 shall at once be able to determine 

 the dimensions of a rectangular 

 beam, whose strength shall be equal 

 to that of the girfler. 



The figure being constructed as 

 before, produce N L to O, and N E 

 to P, and let O P be drawn at right 

 angles to N O, the distance L O 

 being supposed to be such, that 

 tlie area of the figure M E P ( ), may 

 be eqmd to that of the figure 

 I . '■" M L G F, and consequentlv, that 



;': ; \ the triangle N O F, and the figure 



I ; i \ K L G F E, may have equal areas. 



■: i I \ ^ince therefore the area XLGFE, 



"o P which represents the resistance to 



fracture of that portion of the gir- 

 der below the neutral axis, is equal 

 to the triangle N O P, which will 

 indicate the resistance of the middle rib, produced to an imafinarv 

 point O : by finding the distance L O, we shall obtain the dimension's 

 of a simple rectangular rib, having a strength equivalent to that of the 

 portion of the girder b^:l(Av the neutral axis N. 



i 



Let the known distance N L, be represented by (aj, and L O by (j), 

 and the other dimensions remain as before. 



h" a i" 



Tliena-i": ^ : : a : LGorLG = j^^— ^^. 



and d" | ^ ^ ;,, J = area of figure M L G F. 



\ 2 2 (a — d )/ 



2~ 



A' b' (a + .r) 



Again.a-d" : - •. : a+x : O P or O P = ^^£^J. 



b' , b' (a-j-x) 

 And {x + d")Xf^2'^ 2 (a — d") )= ="■«=! of ^S^^^ M E P O, 

 V 2 ^ 



henc^, by the construction we have the equation, 



^'(t+4^^"))=(^ + '^'^x {l + U^=d^) 



d" b' 



"*Vij__?_\ f' C' + d") /, , a + ^ \ 



4-V +^"^^7" 4— (.l + ^TTj^J 



Multiplying by 4 (o — d") we have, 



d" b" (a^d" + a)z=z b- (xJf-d") ■/. (a — d" + a + x) 

 or, d"b"(2a—d")=^b'(x+d")X (2a + '^ — d"), 

 and, d" b" (2 a — d") = V (x' + 2 ax -^2 ad" — d'--) 

 d"b"(2a — d") 



bence, 



b' 



= j^ + 2 a x + 2 a d" — d" 



by transposition, — ■ +(f"- — 2 ad" z:z x' + 2 a j, 



completing the square, ^^- +d"' — 2ad" + a'h:^i' + 2ax + a' 



extracting the root, x + a-= ^ / L " •* ^ (^11 (^y 



or. No^^ '^-^-cy-g ^T^^ri^. ,3.) 



The points Q and R, having been assumed in the same manner as O 

 and P, we shall have the proportion, O P : Q R ; ; ^ ; y, 

 and as the triangles X Q R, Is' O P, are equal, 



O K ; N Q inversely as pio ^ 

 _p>i ON 



that is, N Q : 



and the whole depth, O Q = N O + 



/' X K O 



(4.) 



It will be seen therefore, that by applying the equation No. 2, to 

 ascertain the position of the neutral axis, and subsequently Nos. 3 and 

 4, we obtain the depth of an imaginary rectangular beam, havii»g the 

 same thickness as the middle rib, whose strength shall be equal to that 

 of the girder, thus bringing us within the reach of the usual formuki. 



It may be as well to repeat the remark made at the commeneement, 

 that this system is not proposed for (iractical application in ccmro< n 

 cases; the essay being merely intended, as an exposition of another 

 mode of viewing the action of the transverse strain, and as affording a 

 means, should the principles be found correct, of testing the accuracy 

 of the common approximate methods. 



Derby, Augmt 11, 1S41. 



Pacific Stcnm Nafieaiion. — Extract of a letter rcceiVcil by the Directors of 

 tlictompany from Mr. W'heelw ligh;, ilated Lima, April 28. 1841 .— '• Cap;aiu 

 Peacock arrived here en Saturday, the 24ih. liaving consumed nothing I'Ut 

 Cliili coal diirinL' the voyage :— his calculations have Icon most beautifully 

 carried out, for he li.is not Lean 15 moments out of his time, on arriving at 

 and sailing from each port in the voyage, from Talcahuano to this place, a 

 distance of nearly l.TCO miles: and it affords me p'casurc to remark, that his 

 zeal in the cause of the Company merits the highest praise. His ship is. 

 am happy to state, well regulated with a due regard to economy, and tlie 

 several departments are most judiciously arranged.'" 



