Draft

Chapter 6

Case Study II: GEORGE

WASHINGTON BRIDGE

6.1 Theory.

1 Whereastheforcesin a cablecanbe determinedfromstaticsalone,itscongurationmust

be derivedfromitsdeformation. Letus considera cablewithdistributedloadp(x)per unit

horizontal projectionof thecable length(thusneglectingthe weight of thecable). An

innitesimalportionof thatcablecanbe assumedto be a straight line,Fig. 6.1andin the

absenceof any horizontalloadwe haveH=constant. Summationof theverticalforcesyields

(+?) Fy= 0)V+wdx+ (V +dV) = 0 (6.1-a)

dV+wdx = 0 (6.1-b)

whereV is theverticalcomponent of thecable tensionatx (Notethatif thecablewas

subjectedto itsownweight thenwe wouldhavewdsinsteadofwdx). Becausethecablemust

be tangent toT, we have

tan=

V

H

(6.2)

Substitutinginto Eq. 6.1-byields

d(Htan) +wdx= 0)

d

dx

(Htan) =w (6.3)

2 ButHis constant (nohorizontalloadis applied),thus,thislastequationcanbe rewrittenas

H

d

dx

(tan) =w (6.4)

3 Writtenin termsof theverticaldisplacementv, tan=

dv

dx

which whensubstitutedin Eq.

6.4yieldsthegoverningequationforcables

Hv

00

=w (6.5)

4 For a cablesubjectedto a uniformloadw, we candetermineitsshape by doubleintegration

of Eq.6.5

Hv

0

= wx+C 1 (6.6-a)