232 



THE CIVIL ENGINEER AND ARCHITECTS JOURNAL. 



[August, 



into account in the nnit of tenacity, but as the strain is supposed to be on 

 the highest section of the bar, it is therefore at this section the rupture will 

 take place, hence the length mast be taken into account in determining the 

 weight that produces fracture. 



Condition of rupture in a rectangular beam under a transrerse strain. 



Let us next consider the nature of the forces exerted by the filaments of 

 the cross section, near g, when the beam is in the state bordering upon 

 rupture. 



Let g' be any point of the curve g g', very near g ; and let the normals 

 teg and t' e' g' meet in the (point o. Let p = the radius of curvature of 

 the mean filament, e e', at the point e, that is, the filament which i.s in a 

 middle slate between 1 1' and g g' ; it is represented by h k in the cross 

 gection through t g; hk is in such a position that it is equal in length to 

 abor p q, the breadth of the beam before the weight W was applied. 



It must be observed that fig. 14 is part of fig. 13 enlarged, the same 

 Fig. 14. 



3 S 



letters represent the same parts, and that e e' is not considered neutral. 

 Let a — ee' ; i = the distance of any filament whatever as//, from the 

 mean filament, and a' the length of //' 

 of— p + s — radius of curvature of//', at the point/, then, 



.' . a p — a p -\- K a, a =: a -\- 



On account of the forces applied, the longitudinal filaments undergo 

 »mall dilations, which will destroy the equality that existed between a and 

 a' before the forces were applied. Let o = the primitive magnitude, then, 

 supposing 5 and 5' very small fractional parts of a, so that 

 <r = o (1 + 8) and ff' = a (1 4- S') 



.•.<.(l + J') = a(l+a) + a-''('+') 



. 8' = ! + 



^(1+8) 



I ar8 



= S + - + — 



Let<(' = o(l — €');g-ff' = a(l +e); n'=:a(i_5',); and let the 

 height of the fibre a.ig -i, and breadth a A or p ? := 3, that is, the breadth 

 before the force is applied, ef-x; el = y ; e g = a^; and c < =: Oj. 

 When the element or fibre at g-, becomes expanded till its length a becomes 



Q 



a (I + €), then its breadth will be ^j^T^^J.and height ! ^i .the heighti 



i« ao small, this supposes the cross section of the extreme fibre, before the 

 expansion takes place, to be similar to it after being expanded,*— mind we 



Q 



say similar, not equal, for is less than ^ ; so the breadth is di- 



minished. It will be perceived that we are here speaking of the lower 

 filaments at g. 



Tlie assumiJtion BeemB inconBisIent with theass'imed liiillening of llie beam. For it 

 IS Clear that if the proportion lietiveef the depth and width ..I the constituent elements 

 lemnn.s unnlleretl, the proportion hetwesn the depth and width oflhe whole heam must 

 «f.^ . L /,n ; I u ■; VM'" '"rf"^>^ of 'I"- I'eam bulsed out. the depth of the upper 81.,. 

 a.eiits would he , ImMnshed and the width increrijed-if we afsume the density to rem ,in 

 Hike's Law- Ed* °"'""'P''''° "'" '" contrary to our knonkiige of elastic bodies and to I 



• . the 1 



breadth of the fibre. at//- = ^^ + "-i^ 0" (TT^Ji) 

 fir ^ 



~"i i "i" f 1 — ^1 1 y^ L- The force arising from the action of 



//' is proportional to 6', and may be represented by m 8', m being a con- 

 stant depending upon the nature of the material of which the bar con- 

 sists ; therefore, the normal force which corresponds to the point /, normal 



totf, IS represented by ~^ a,— / 1 — , , , I x \ml dx, and the 

 moment of all the forces normal to eg- with respect to the asis A fc = M = 



'^/""["■-O-raV}'' 



xdx 



(1 -f 8) .r 

 but 5 _ 8 -| ; Iience, by substitution, 



Therefore M equal 

 — S rc — 4(1— _i ~)18pa? + r4 — Sri — _J_M/, . « .1 



(l-t-8)<i} I ; butp:p + a,::a(l + 8):a(l + e); 



P(l +0 . 

 P + a, 



f-^a. 



, M = 



m&(i\p 



I2(p + n,)(l+0i^ 



> (2 (1+01 + 4) ( p e «,) + ( (l-f e)J- -f 3) (I + .) a. | 

 This being established, we shall next determine the nature of the for(»» 

 between e and t, or on the area hkdc. The breadth of the extreme fibres 

 6 



at tf 



aj :y:: 



ot zz p — n„; ol = p — y, 



S - - y . 1 



(T=^i-^:^((-i=7yi-^)' 



breadth of the fibres at 1 1'. The elastic forces of compression between 

 e and t, as well as those of extension between e and g, tend to turn the 

 surface of rupture in the same direction about the axis of rupture. The 

 moment of all the forces normal to e t, with respect to the axis h k which 

 we shall call M, is equal 



— a. 



, , „, 6, p — (i;-f (1— <,'ly 



but 8 =: , 



p — a^ 



forp — o, : p — !/ :; o(l — «,) : a(l — J"). 



_a,fp-o, 



• "■ - m,i8 ^ 



.•.M, = ""^^^ X 



• latp-a,) (l-,,)i ^ 

 {(-' »(l-*.)i +4) («,p-a,)-f( -3 + 7 (l-€,)i )(!-.,)«. I 



Now the elastic forces M and M,, when th« beam is on the point of 

 breaking, together wilh the pressure applied at B or A (fig. 7), which it 

 equal to half the weight W and half the weight of the beam. 



Espresenting, therefore, by ir the weight of the beam, 

 M + M, = 4(W + t() X rB 

 Let rB = p = half the distance between the supports =:the perpendicular 

 let fall from the axis round which the section of rupture turns upon the 

 direction of the pressure at A or B. In the next number will be pointad 

 out the erroneous principle upon which Hooke's law is founded. 



(To be continued J 



