
On the additional deflexion due to shearing. 
In the problem of Saint-Venant the, additional deflexion, as we 
know, depens on the conditions at the fixed end. If the element at 
the centroid of the corresponding cross-section is vertical the addi- 
tional deflexion will ce equal to s,/ where s, is the shear at this 
element and / — the length of the beam. In the case illustrated on 
Fig. 1 the middlecross-section, in consequence of symmetry, remains 
plain, and the „local irregularity“ takes place near this cross-section. 
In case of a narrow rectangle the additivnal deflexion can be taken*) 
equal to “1 
(pra 35/_9 WI 
Fig. 1. 4 3,8 2cu 
In the following we give the approximate method for calcula- 
ting the additioual deflexion due tu shearing and to local irregularity 
in tke case of a narrow rectangle. We will use for the stress com- 
ponents theexpresions satisfying the stress equations and the cons- 
tants they contain we will choose in the manner as to make the 
potential energy to a minimum. The boundary conditions will be 
satisfied, if we take 

100 
pra | 
X =— (7°) “e aa **(5y—6p%e+ct) ( (D 
V = a, ate “x(y—e)y : 
; 
The members with a, correspond to the Saint-Venant’s solution 
andthe other terms we introduce, in order to estimate the „local 
irregularity“. The constans a, and « we choose in the manner stated 
above. Z 
The expression (1) substituted in 
SLI [ 2 2 2 kz 
V=;; jes -2A,Y,4+Y,+2 (14) X, | dx dy 
and putting 
I 
A 1 
Je X dx =— 
4 

e 
0 
| *) See N. G. FILON, London, Phil. Trans. R. Soc. (Ser. A), Vol 201 (1903). 
: je? 
| ij 
DEE ı an ; ’ m promene PV Taj u 
