382
DOCK ENGINEERING.
of the column by 2 inches, the 1 inch annular space being grouted witli
cement. Another proposed expédient is to adapt the interiør of the
columns to the circulation and distribution of water.
Strength of Columns.—The strength of long columns or struts constitutes
a very intricate problem. Such columns, though nominally in compression,
rarely fail from a simple compressive stress; they succumb, when loaded to
collapse, to a bending stress induced by the unsymmetric application of the
load. Theoretically, the load should be applied so that the stress passes
absolutely through the centre of every transverse section of the column,
and, in that case, there would be no tendency to bind; but this ideal
condition is unattainable in practice.
Let us briefly investigate the case of a column deflected by a load, W,
acting at a distance, x, from the axis of symmetry. If 6 be the amount of
deflection produced, the bending moment at the foot will be W (æ + ô).
Assuming for the moment that the deflection curve is circular, as in the
case of a beam of uniform strength, we have by Euclid III., 35,
0 x 2R = Z2,...........................(70)
where I is the length of the column and R the (very considérable) radius
of curvature.
Now, the moment of resistance,
Equating the two moments :—
W+ 3) = 25^1
whence
w=^l- El
X+Ô l2’
and
i = ____^—.
2EI _
W12_____1
• (71)
• (72)
• (73)
From a considération of (73), if « = 0, 5 will also be zero, unless
2 E I
..........................(74)
and this is the critical value for W, when the column is in a state of neutral
equilibrium ; so that the least increase in the load will cause indefinite
bending and conséquent fracture.
If the column be of uniform transverse section, instead of uniform
strength, as assumed above, the curve will be one of sines, and the equation
becomes
ir2 El
W=T--(r.........................(75)