Chapter 7 Homework T0 is The Tension The Lowest Point Solution

PROBLEM 7.116

Cable ACB supports a load uniformly distributed along the

horizontal as shown. The lowest Point C is located 9 m to the

right of A. Determine (a) the vertical distance a, (b) the length

of the cable, (c) the components of the reaction at A.

SOLUTION

Free body: Portion AC

0: 9 0

yy

F AwΣ= − =

9

y

w=A

PROBLEM 7.116 (Continued)

(b) Length =AC+CB

Portion AC:

PROBLEM 7.117

Each cable of the side spans of the Golden Gate Bridge

supports a load w= 10.2 kips/ft along the horizontal.

Knowing that for the side spans the maximum vertical

distance h from each cable to the chord AB is 30 ft and

occurs at midspan, determine (a) the maximum tension in

each cable, (b) the slope at B.

SOLUTION

FBD AB:

0: (1100 ft) (496 ft) (550 ft) 0

A By Bx

M TTWΣ= − − =

11 4.96 5.5

By Bx

T TW−=

(1)

PROBLEM 7.118

A steam pipe weighting 45 lb/ft that passes between two

buildings 40 ft apart is supported by a system of cables as

shown. Assuming that the weight of the cable system is

equivalent to a uniformly distributed loading of 5 lb/ft,

determine (a) the location of the lowest Point C of the

cable, (b) the maximum tension in the cable.

SOLUTION

Note:

40 ft

BA

xx−=

or

40 ft

AB

xx= −

(a) Use Eq. 7.8

PROBLEM 7.119*

A cable AB of span L and a simple beam A′B′of the same span are

subjected to identical vertical loadings as shown. Show that the

magnitude of the bending moment at a point C′ in the beam is equal

to the product T0h, where T0 is the magnitude of the horizontal

component of the tension force in the cable and h is the vertical

distance between Point C and the chord joining the points of support

A and B.

SOLUTION

0 loads

0: 0

B Cy B

M LA aT MΣ = + −Σ =

(1)

FBD Cable:

(Where

loadsB

MΣ

includes all applied loads)

PROBLEM 7.120

Making use of the property established in Problem 7.119,

solve the problem indicated by first solving the

corresponding beam problem.

PROBLEM 7.94 Knowing that the maximum tension in

cable ABCDE is 25 kN, determine the distance dC.

SOLUTION

Free body: beam AE

0: (20) 2(15) 4(10) 6(5) 0

E

MAΣ= − + + + =

5 kN=A

0:5246 0

y

FEΣ = −−−+ =

7 kN=E

PROBLEM 7.121

Making use of the property established in Problem 7.119, solve the

problem indicated by first solving the corresponding beam problem.

PROBLEM 7.97 (a) Knowing that

3 m,

C

d=

determine the distances

dBand

D

d

.

SOLUTION

PROBLEM 7.122

Making use of the property established in Problem

7.119, solve the problem indicated by first solving the

corresponding beam problem.

PROBLEM 7.99 (a) If dC= 15 ft, determine the

distances dB and dD.

SOLUTION

Free body: Beam AE

0: (30) 2(24) 2(15) 2(9) 0

E

MAΣ= − + + + =

3.2 kips=A

PROBLEM 7.123

Making use of the property established in Problem

7.119, solve the problem indicated by first solving the

corresponding beam problem.

PROBLEM 7.100 (a) Determine the distance dC for

which portion BC of the cable is horizontal.

SOLUTION

Free body: Beam AE

0: (30) 2(24) 2(15) 2(9) 0

E

MAΣ= − + + + =

3.2 kips=A

0: 3.2 3(2) 0

y

FBΣ = − +=

PROBLEM 7.124*

Show that the curve assumed by a cable that carries a distributed load w(x) is defined by the differential

equation d2y/dx2=w(x)/T0, where T0is the tension at the lowest point.

SOLUTION

FBD Elemental segment:

0: ( ) () () 0

yy y

F Tx x Tx wx xΣ = +∆ − − ∆ =

PROBLEM 7.125*

Using the property indicated in Problem 7.124, determine the curve assumed by a cable of span L and sag

h carrying a distributed load w=w0 cos (

π

x/L), where x is measured from mid–span. Also determine the

maximum and minimum values of the tension in the cable.

PROBLEM 7.124 Show that the curve assumed by a cable that carries a distributed load w(x) is defined

by the differential equation d2y/dx2=w(x)/T0, where T0is the tension at the lowest point.

SOLUTION

PROBLEM 7.126*

If the weight per unit length of the cable AB is w0/cos2

θ

, prove that the

curve formed by the cable is a circular arc. (Hint: Use the property

indicated in Problem 7.124.)

PROBLEM 7.124 Show that the curve assumed by a cable that carries a

distributed load w(x

) is defined by the differential equation

d2y/dx2=w(x)/T0, where T0is the tension at the lowest point.

SOLUTION

Elemental Segment:

Load on segment*

0

2

() cos

w

w x dx ds

θ

=

PROBLEM 7.127

A 25–ft chain with a weight of 30 lb is suspended between two points at the same elevation. Knowing that

the sag is 10 ft, determine (a) the distance between the supports, (b) the maximum tension in the chain.

SOLUTION

(30 lb)/(25 ft) 1.2 lb/ftw= =

PROBLEM 7.128

A 500–ft–long aerial tramway cable having a weight per unit length of 2.8 lb/ft is suspended between two

points at the same elevation. Knowing that the sag is 125 ft, find (a) the horizontal distance between the

supports, (b) the maximum tension in the cable.

SOLUTION

PROBLEM 7.129

A 40-m cable is strung as shown between two buildings. The

maximum tension is found to be 350 N, and the lowest point

of the cable is observed to be 6 m above the ground.

Determine (a) the horizontal distance between the buildings,

(b) the total mass of the cable.

SOLUTION

PROBLEM 7.130

A 50-m steel surveying tape has a mass of 1.6 kg. If the tape is stretched between two points at the same

elevation and pulled until the tension at each end is 60 N, determine the horizontal distance between the

ends of the tape. Neglect the elongation of the tape due to the tension.

SOLUTION

PROBLEM 7.131

A 20-m length of wire having a mass per unit length of 0.2 kg/m is

attached to a fixed support at A and to a collar at B. Neglecting the

effect of friction, determine (a) the force P for which h= 8 m, (b) the

corresponding span L.

SOLUTION

FBD Cable:

PROBLEM 7.132

A 20-m length of wire having a mass per unit length of 0.2 kg/m is

attached to a fixed support at A and to a collar at B. Knowing that

the magnitude of the horizontal force applied to the collar is

P = 20 N, determine (a) the sag h, (b) the span L.

SOLUTION

FBD Cable:

PROBLEM 7.133

A 20-m length of wire having a mass per unit length of 0.2 kg/m is

attached to a fixed support at A and to a collar at B. Neglecting the

effect of friction, determine (a) the sag h for which L= 15 m, (b) the

corresponding force P.

SOLUTION

FBD Cable:

PROBLEM 7.134

Determine the sag of a 30–ft chain that is attached to two points at the same elevation that are 20 ft apart.

SOLUTION