LENGTH OF CABLE SUBJECTED TO UNIFORMLY DISTRIBUTED LOAD

LENGTH OF CABLE SUBJECTED TO UNIFORMLY DISTRIBUTED LOAD

Consider the figure in which cable of span l is subjected to uniformly distributed load of w/m throughout the entire span. Since cable is a flexible structure, it deflects in a parabolic way when subjected to udl. Let h is the central dip of the cable. The equation of cable is considered by taking point C as origin.

We know that the equation for dip for a parabola is given by

Y = 4hx (l -x)/ (l)² ..........(1)      

Considering a section X- X of span x, for which the deflected length of the cable has to be calculated. Let s be the length of the arc for span x

Therefore EQ (1) becomes

Y = 4hx (x-0)/ (l)²

Y = 4hx ²/ (l) ²

We know that slope is given by tanϴ

tanϴ = dY / dX = 8hx/(l) ²

Deflected length of cable of X-X section is given by

 Sec ϴ = √ (1+ tan²ϴ)

ds/ dx = √ (1+ (dy/dx)²)

ds/ dx = √ (1+ (8hx/(l) ²)²)

ds/ dx = √ (1+ (64 h²x²/(l) ⁴)

Neglecting the smaller terms, length of deflected arc for section x- x is given by

ds = [1+ (32 h²x²/(l) ⁴]dx

Total deflected length of cable

L = 2 {  ^{l /2} ∫₀ [1+ (32 h²x²/(l) ⁴]dx }        (limits 0 to (l/2))

Solving the above equation

L = l + 8/3(h²/l)

Where l = length of cable

L = deflected length of cable

h = central dip