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Building Design and Construction Handbook - Merritt - Ventech!

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5.134 SECTION FIVE<br />

where � � vertical deflection of cable due to live load<br />

M L � live-load bending moment that would be produced by the live load in<br />

a simple beam<br />

H L � increment in horizontal component of tension due to live load<br />

Subtraction of Eq. (5.223) from Eq. (5.224) yield<br />

M � Hy<br />

L L D<br />

� � (5.225)<br />

H � H<br />

D L<br />

If the cable is assumed to take a parabolic shape, a close approximation to H L may<br />

be obtained from<br />

l l<br />

HL wD 1<br />

K � � � dx � � ��� dx (5.226)<br />

AE H 0 2 0<br />

1 5 ��<br />

2 16ƒ<br />

2� D<br />

2 16ƒ<br />

2<br />

3l 4ƒ<br />

e�<br />

2 16ƒ<br />

2��<br />

� �<br />

K � l � 1 � � log � 1 � (5.227)<br />

4 2 l l 32ƒ l l<br />

where �� � d 2�/dx 2 .<br />

If elastic elongation <strong>and</strong> �� can be ignored, Eq. (5.226) simplifies to<br />

l<br />

� MLdx l<br />

0 3<br />

HL � l � � ML dx (5.228)<br />

2ƒl 0 � yDdx 0<br />

Thus, for a load uniformly distributed horizontally wL, l 3 wl L � MLdx � (5.229)<br />

0 12<br />

<strong>and</strong> the increase in the horizontal component of tension due to live load is<br />

3 3 wl L<br />

2 wl L<br />

2 wl L 8HD wL<br />

L<br />

2ƒl 12 8ƒ 8 2 wDl wD<br />

D<br />

H � � � � H (5.230)<br />

When a more accurate solution is desired, the value of H L obtained from Eq. (5.230)<br />

can be used for an initial trial in solving Eqs. (5.225) <strong>and</strong> (5.226).<br />

(S. P. Timoshenko <strong>and</strong> D. H. Young, ‘‘Theory of Structures,’’ McGraw-Hill Book<br />

Company, New York: W. T. O’Brien <strong>and</strong> A. J. Francis, ‘‘Cable Movements under<br />

Two-dimensional Loads,’’ Journal of the Structural Division, ASCE, Vol. 90, No.<br />

ST3, Proceedings Paper 3929, June 1964, pp. 89–123; W. T. O’Brien, ‘‘General<br />

Solution of Suspended Cable Problems,’’ Journal of the Structural Division, ASCE,<br />

Vol. 93, No. ST1, Proceedings Paper 5085, February, 1967, pp. 1–26; W. T.<br />

O’Brien, ‘‘Behavior of Loaded Cable Systems,’’ Journal of the Structural Division,<br />

ASCE, Vol. 94, No. ST10, Proceedings Paper 6162, October 1968, pp. 2281–2302;<br />

G. R. Buchanan, ‘‘Two-dimensional Cable Analysis,’’ Journal of the Structural<br />

Division, ASCE, Vol. 96, No. ST7, Proceedings Paper 7436, July 1970, pp. 1581–<br />

1587).

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