Chapter 9 Homework A uniform rigid bar of total mass m is supported on two

1

CHAPTER 9

Problem 9.1

springs k1 and k2 at the two ends and subjected to dynamic

forces as shown in Fig. P9.1. The bar is constrained so that

it can move only vertically in the plane of the paper. (Note:

This is the system of Example 9.2.) Formulate the

Figure P9.1





1. Determine the force vector.









Wp

t



F

H

GI

K

J

2

()

(b)

R

|

U

|

p

t

2. Determine the stiffness matrix.

Apply a unit displacement u11 with u20



and

By statics,

Similarly, apply a unit displacement u21 with u10



and identify the resulting forces and the stiffness influence

By statics,

Thus the stiffness matrix is

3. Determine the mass matrix.

2

By statics,

mm

11  mmL

21 2



Similarly, impart a unit acceleration 

u21 with 

u10



,

By statics,

Thus the mass matrix is

4. Write the equations of motion.

Substituting Eqs. (c), (d), and (e) in Eq. (9.2.12), with

c0, gives

3

Problem 9.2

A uniform simply supported beam of length L, flexural

(b) Identify the DOFs to represent the inertial properties

and determine the mass matrix.

(c) Formulate the equations governing the translational

I



Figure P9.2

Solution:

translational displacements and four rotational

displacements.



u5u3u4u6

3

u



instance, to obtain the first column of the stiffness matrix,

k21

k11

k51

k31 k41

k61

By statics,

kEI

L

11 3

648

 kE

I

L

21 3

324

 k31 0



k23

k13

k53

k33 k43

k63

Other elements of the stiffness matrix are obtained

similarly. Apply a unit displacement ui1 while 0,

j

u



The complete stiffness matrix is



L

N

O

Q

LL

648 324 0 54 54 0

(a)

4

Part b

The DOF representing the inertial properties are the

two translational displacements u1 and u2 associated with

the concentrated masses.

u2

u1

To obtain the coefficients of the mass matrix for these

DOF, apply first a unit acceleration 11u

 , while 20.u







u = 1,

1

 u = 0

2

 u = 0,

1

 u = 1

2



Thus the mass matrix is

Part c

The condensed stiffness matrix for the two vertical

DOF is

The force vector is given by

mL u

u

EI

L

u

pt

3

10

01

162

5

87

78

1

23

1

2

1

2

L

N

MO

Q

PR

S

TU

V

W



L

N

MO

Q

PR

S

TU

V

WR

S

TU

V

W



()

(f)

5

Problem 9.3

Derive the equations of motion of the beam of Fig. P9.2

governing the translational displacements u1 and u2 by

p (t)

1

p (t)

2

1. Determine the stiffness matrix.

The flexibility matrix is calculated first and inverted to

obtain the stiffness matrix. The flexibility influence coef-

The deflections due to unit load at node 2 are computed to

obtain the other two influence coefficients:

Thus the flexibility matrix is

The stiffness matrix is obtained by inverting 

f:

2. Determine the mass matrix.

The mass matrix for the translational DOF is same as

in Problem 9.2:

3. Write the equations of motion.

Using Eqs. (d) and (e), the equation governing the

translational motion of the beam is the same as in Problem

9.2:

(f)

6

Problem 9.4

1. Determine the mass matrix.



2

By statics,



inertia forces.

m22

u = 1

2



By statics,

2. Determine the flexibility matrix.

Apply a unit force p11 along DOF 1 with p20



along DOF 2. The first two displacements or influence

Similarly, apply a unit force p21, with p10



. The

The stiffness matrix is computed by inverting the

flexibility matrix 

f:

7

Problem 9.5

Using the definition of stiffness and mass influence coef-

Figure P8.18

Solution:

k = – k

21

Thus, the stiffness matrix is

2. Determine the mass matrix.

Apply 

u11



, 

u20





Apply 

u21



, 

u10





m12 0



mm

22 2



Thus, the mass matrix is

3. Write the equations of motion.



8

Problem 9.6

mations in all elements.

(c) Formulate the equations governing the motion of the

frame in the DOFs in part (b).

Solution:

Part a

The elastic properties of the shear frame (neglecting

u3

u4

u1

The complete stiffness matrix is



L

O

hh

48 24 0 0 6 6

This matrix can be written in partitioned form as follows:

Part b

The DOFs representing the inertial properties are the

two translational displacements u1 and u2.

Part c

The condensed stiffness matrix is

9

Problem 9.7

Solution:

Since the beams are rigid in flexure and axial defor-



stiffness are:





EI

Apply 11u, 23

0uu

and determine ki1:

k = 0

31

Apply 21u



, 13

0uu



 and determine 2i

k:

k = – k

12



u = 1

3

k =

k

33

12 1

k













k (c)

2. Determine the mass matrix.

m/2

m31

10

m/2

m32

Apply 31u

 , 12

0uu

  

m

m13

The mass matrix is

3. Write the equations of motion.

111

1210()

uupt



 



(e)

11

Problem 9.8

Using the definition of stiffness and mass influence coef-

ficients, formulate the equations of motion for the three-

Figure P9.8

Solution:

Since the beams are rigid in flexure and axial defor-

mation is neglected in columns, the three floor displace-

ments are the three DOFs. The floor masses and story

stiffnesses are:

h

hk

1. Determine the stiffness matrix.

Apply uuu

123

10, and determine ki1:

Apply uuu

213

10



, and determine ki2:

The stiffness matrix is







025

2. Determine the mass matrix.

11 21 31 0

k = – k

32

k = – k

23

k = 0

13

m/2

m31

12

Apply  , 

uuu

213

10

mm mm

22 12 32 0

Apply  , 

uuu

312

10

The mass matrix is





001

3. Write the equations of motion.

u = 1

m/2

m32

u = 1

3



m/2

m33

m/2

13

Problem 9.9

Figure P9.9 shows a three-story frame with lumped masses

subjected to lateral forces, together with the flexural

(b) Identify the DOFs to represent the inertial properties

Figure P9.9

Solution:

u3

u8u9

The coefficients of the stiffness matrix corresponding to

these DOFs are computed following Example 9.7. The

complete stiffness matrix is







222

0020101060

hhhh

EI

k

(a)

The stiffness matrix can be written in partitioned form as

follows:

Part b

The DOFs representing the inertial properties are the

three translational displacements, u1, u2, and u3.

u3

m/2

The mass matrix is

14

The equation governing the translational motion of the

building is

(e)

15

Problem 9.10

Figure P9.10 shows a three-story frame with lumped

masses subjected to lateral forces, together with the

flexural rigidity of columns and beams.

(b) Identify the DOFs to represent the inertial properties

and determine the mass matrix. Assume the members to be

massless and neglect their rotational inertia.





Figure P9.10

Solution:

Part a

The elastic properties of the frame (neglecting axial

deformation) are represented by nine DOFs: three

horizontal displacements and six rotational displacements.



















5

4

1

u

The coefficients of the stiffness matrix corresponding to

these DOFs are computed following Example 9.7. The

complete stiffness matrix is

222

22

48 24 0 0 0 6 6 0 0

1

92000

2

90200

hh

hhh

hh

EI















(a)

The stiffness matrix can be written in partitioned form

as follows:

Part b

The DOFs representing the inertial properties are the

three translational displacements, 12 3

and,, .uu u

The mass matrix is









m/2

m

u

2

u

3





16

Part c

The condensed stiffness matrix for the three lateral

DOFs is

39.38 22.69 5.486





The equations governing the translational motion of the

building are

(e)

17

Problem 9.11

Figure P9.11 shows a three-story frame with lumped

masses subjected to lateral forces, together with the

deformation of the members.

(b) Identify the DOFs to represent the inertial properties





Figure P9.11

Solution:

Part a

The elastic properties of the frame (neglecting axial

deformation) are represented by nine DOFs: three

horizontal displacements and six rotational displacements.









5

4

u

The coefficients of the stiffness matrix corresponding to

these DOFs are computed following Example 9.7. The

complete stiffness matrix is

22 2

22

26 4 000

33

1

23

hh h

hh





















Part b

The DOFs representing the inertial properties are the

three translational displacements, 12 3

and,, .uu u





1

u

3

u8u9

m/2

m

u

2

u

3





18

The mass matrix is









001

Part c

The condensed stiffness matrix for the three lateral

DOFs is

32.67 14.58 2.172





The equation governing the translational motion of the

building is

(e)

19

Problem 9.12

(b) Identify the DOFs to represent the inertial properties

and determine the mass matrix. Assume the members to be

massless and neglect their rotational inertia.

(c) Formulate the equations governing the motion of the

frame in the DOFs in part (b).



Solution:

E





m

L

E

I/2

E

I

p

t

1

( )

p

t

1

( )

u3

u8u9

4

5

1







u

uu









The coefficients of the stiffness matrix corresponding to

these DOFs are computed following Example 9.7. The

complete stiffness matrix is

23

3

23 4

22

33

14 1 2

22 2

3

333

40 16 0 2 2 4 4 0 0

000

0

hh hh

hh

EI

hhh

h





















k

The stiffness matrix can be written in partitioned form as

follows:

Part b

20

m/2

u

3



The mass matrix is

100





Part c

The condensed stiffness matrix for the three lateral

DOFs is

The equation governing the translational motion of the

building is

(e)