BEAM OP BECTANOULAR CROSS-SECTTOX UNDER ANY ,SY8TE>f OF LOAD. 129 
modem engineering structure, such as railway bridges, &c., cases of this kind are of 
constant occurrence, and the strength of the structure depends, to a very great 
extent, upon the strength of the individual rivets. It becomes tlierefore a problem 
of the very greatest practical importance to know how the distribution of shear 
inside such a rivet varies with the dimensions of the rivet and Avith the thickness of 
the plates. At present our knowledge of the subject is purely empirical; and 
although the results of the present paper apply only to a rivet of rectangular 
section, and even then are only an approximation, yet they should furnish some 
indications AAduch may he of value in other cases. 
Anothei point which is illustrated by these results is the manner in which 
DE SaintA'enant’s solutions are modified, when we gradually bring the terminal 
systems of load closer together. We see tliat tlie modifications introduced are 
practically insensible at distances from the section where tlie load is applied Avhich 
are greater tlian the height of the beam. This is of inpiortance, as it tells us witliiu 
wliich limits, in any experiment, Ave may assume tlie state of a beam to be given by 
one of tlie “uniform” solutions AAfiilch only depend ipion the total terminal conditions 
and Avhich are transmitted AAuthout change of type. 
PAET IV. 
Solution foe a Beam who>se upper and lower Boundaries are acted upon 
EY Sheaeinu Stress only. 
§ 33. Expressions Jor the Dispdacements and Stresses in Series and Integrals. 
Let us noAA^ consider a beam acted upon liy sliearing stress alone, oAnr tlie 
boundaries y = A h. Then, in the general solution of I 7, = y,, = = 0. 
If fuither AA"e suppose the shear to reduce to a single concentrated force L at one 
point (0, h) we have ^ L = k„ = 0 = 0, = v,. 
Putting in these values into (44), (45), (46), (47), (48), (54), and (55) aau obtain 
IT — _ . 3V + 2;^ LV . 1 L// , . 
' ' + VlLy X m m: + t: 
aJj p ah n, 8a 
_j_ V A “t /i 
,1 = 1 ici'ni 
4—\ CO,si I nth — i rnh sinh mh 
P'/_ n 
.siiih 2ml) + 2)nh 
cosh my cos mx 
-T 1 / H - ) sinh ml) — — mh eosli mh 
4 _ W lA + _ /^/_ /^ ■ . 1 
^ .A 2am ■ ^ siihi 2))ih - 2)))h my cos mx 
L 1 //co.shmUsiuh5«?/ - 1 v sinh mA cosli m?/ 
cos mx + N--^- ■’ pos mx. 
i,=i 2a fjb .sinh 2mh — 2))ih ’’ ’ 
+ V - 
n=i 2a jx sinh2/»i + 2mh 
YOL. CCI,—A, 
(103) 
