Chapter 8 Homework Solve Problem 87 Asssuming That

1

CHAPTER 8

Repeat parts (a), (b), and (c) of Example 8.1 with one

change: Use the horizontal displacement at C as the gener-

alized coordinate. Show that the natural frequency, damp-

Solution:

1. Determine the shape function.

O

z

Substituting ImL

11

212 and ImL

22

2128 gives

The equation of motion (after dividing by L) is

The relation between z and the rotation



about fulcrum O

is

Substituting Eq. (d) into Eq. (a) leads to the same equation

of motion as in Example 8.1.

3. Determine natural frequency and damping ratio.

4. Solve the equation of motion.

5. Determine displacements.

2

For the rigid-body system shown in Fig. P8.2:

(a) Formulate the equation of motion governing the

δ(t), the Dirac delta function.

Figure P8.2

uxtx(,)



L8



9

2. Draw the free body diagram and write the equilibrium

equation.

pt

()

k2

()

L



m2

()

L

9

m8

()

L









Substituting ImL

1

212 and Im L

2

12 4()[()

(b)

3. Determine natural frequency and damping ratio.

(c)

4. Solve the equation of motion.

For pt t() ()





, the solution of Eq. (a) is

5. Determine displacements.

3

2. Draw the free body diagram and write the equilibrium

equation.

MO

0

where

3. Determine natural frequency and damping ratio.

(c)

4. Solve the equation of motion.

This result is identical to Eqs. (d) and (e) in Problem 8.2,

4

Problem 8.4

The rigid bar in Fig. P8.4 with a hinge at the center is

Figure P8.4

Solution:

1. Determine the shape function.

2. Draw the free body diagram and write the equilibrium

equation.

MO

0

or

The equation of motion is

and

3. Determine natural frequency and damping ratio.

5

For the rigid-body system shown in Fig. P8.5:

(c) Determine the natural vibration frequency and damping

Figure P8.5

Solution:

1. Determine the shape function.

2. Draw the free body diagram and write the equilibrium

equation.

MA

0 for bar AB 

The force in spring BC is

Substituting Eq. (b) in Eq. (c) gives

where

3. Determine natural frequency and damping ratio.

6

Problem 8.6



23

31

22

x

xLL





1. Determine the generalized properties.



2

() ()

L

kEIxxdx











2. Determine the natural period.

3. Formulate the equation of motion.

4. Determine the peak value of zt()

.

gives

A

5. Determine peak displacements of the tower.

6. Determine equivalent static forces.

7. Compute shear and bending moment at mid-height and

at the base.

L

7

Problem 8.7

A reinforced–concrete chimney 600 ft high has a hollow

circular cross section with outside diameter 50 ft at the

base and 25 ft at the top; the wall thickness is 2 ft 6 in.,

uniform over the height (Fig. P8.7). Using the approxi-

the base, and its damping ratio is estimated to be 5%. The

unit weight of concrete is 150 lb/ft3, and its elastic modulus

Ec = 3600 ksi. Assuming that the shape function is

is scaled to a peak acceleration of 0.25g.

Solution:

1. Determine properties of the chimney.

L

600 f

t

I

L

z





10 8

( ) 5.454 10 1.435 10

 

EI x x

600‘

25‘

R (x)

avg

2. Determine 

m, 

k

,





and

T

n.



22

0

() () 134.5kip–sec ft





L

mmxxdx



3. Determine the peak value of zt()

.

For 3.313 sec

n

T



and 0.05



, the design spectrum

gives

8

5. Determine equivalent static forces.

fx mx xA

o() ~() ()







6. Determine shear and bending moment at mid-height and

at the base.

2

( 2) ( ) 1426 kips

L

oo

L

Lfd







V

9

Problem 8.8

Solve Problem 8.7 asssuming that the shape function is

Solution:

1. Determine properties of the chimney.

4253

0.243 (7.102 10 ) ft

xx





2

( ) ( ) 240.27 kip – sec / ft

0





L

Lmxxdx



3. Determine the peak value of z(t).

For 3.28sec

n

T



and 0.05



, the design spectrum

5. Determine equivalent static forces.

() () ()

o

fx mx xA







6. Determine shear and bending moment at mid-height and

at base.



L

0

x

600‘

10

Problem 8.9

1. Formulate equation of motion.

()mz k z p t







2

134.5 kip-sec ftm

; 483.5 kips ftk



L

2. Solve the equation of motion.

Note that

t

T

dn

0 25 3 313 0 075.. .

. Because

t

T

3. Determine peak responses.

u

L

z

() () .





379in.

( 2) ( ) 216 kips

L

oo

Lfd





V

(2) () 36,290kip–ft

 

oo

Lfd



M

Three-story shear frames (rigid beams and flexible

columns) in structural steel (E = 29,000 ksi) are shown in

that are equal to the floor weights, determine the floor dis–

placements, story shears, and overturning moments at the

floors and base due to ground motion characterized by the

design spectrum of Fig. 6.9.5 scaled to a peak ground

Solution:

1. Determine the shape function.

z







Shape vector:

2. Determine generalized properties.

1

j

3. Determine the natural period.

where

kips100



w

4. Determine the equation of motion.

5. Determine the peak value of zt()

.

For

T



0 3405. sec and



005., the design

6. Determine floor displacements.

7. Determine equivalent static forces.

F

I

F

I

F

I

F

I

F

I

12

8. Determine story shears and overturning moments.

0.519

45. 8

41. 2 kips

160.2

1867

13

Problem 8.11

damping ratios ζn are 5% for all modes. Assuming that the

shape function is given by deflections due to lateral forces

that are equal to the floor weights, determine the floor

displacements, story shears, and overturning moments at

Solution:

1. Determine deflections to applied forces.

3. Determine generalized properties.

~().kk k

j

jjj





1

3

1

20797



4. Determine the natural period.



nkm km

~

/~./0 803

5. Determine the equation of motion.

6. Determine the peak value of z(t).

gives:

/ 2.71 0.25 0.677Ag



 ; 20.966 in.

n

A

D





7. Determine floor displacements.

k

E

8. Determine equivalent static forces.

k

9. Determine story forces.

shear

3w/2

drif

t

3w/4

k

deflectio

n

25w/12

k

w/2

w

2EI/3

14

s

0.52

0.99

15

Problem 8.12

Solve Problem 8.10 using the shape function given by

deflections due to a lateral force at the roof level.

Solution:

1. Determine the shape function.

Shape vector:

2. Determine generalized properties.

g2

3

~3

1

w

 

j

jj

mL



3. Determine the natural period.

Problem 8.9).

4. Determine the peak value of zt()

.

For 0.315 sec

n

T



and 0.05



, the design spectrum

gives

20.6577 in.

n

A

D





5. Determine peak responses.

The floor displacements using Eq. (8.4.14) are given

0.312

32.1

48.2 kips

144.5

1927

3

k

Story

p/k

Floor

3p/k

p

w

w

16

Solve Problem 8.11 using the shape function given by

1. Determine deflections due to applied forces.

2. Determine shape vector:

M

g/739.0

~3

1

2w



jjj

mm



4. Determine the natural period.

From Problem 8.11:

Substituting in above equation:



n17 58.rad/sec

5. Determine the peak value of z(t).

6. Determine the peak responses.

7. Determine equivalent static forces.

p

8. Determine story forces.

p

in Figs. (c) and (d).

p

/k

11p/6k

2p/6k

w

E

I/3

2EI/3