INSTRUCTOR’S
SOLUTIONS MANUAL
DYNAMICS OF STRUCTURES
Theory and Applications to
Earthquake Engineering
SIXTH EDITION
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Anil K. Chopra
University of California at Berkeley
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, CHAPTER 1
Problem 1.1
Starting from the basic definition of stiffness, determine
the effective stiffness of the combined spring and write the
equation of motion for the spring–mass systems shown in
Fig. P1.1.
Figure P1.1
Solution:
If ke is the effective stiffness,
fS keu
u
k1
k1 u
fS fS
k2 u
k2
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Equilibrium of forces: fS ( k1 k2 ) u
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Effective stiffness: ke fS u k1 k2
Equation of motion: mu keu p ( t )
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, Problem 1.2
Starting from the basic definition of stiffness, determine
the effective stiffness of the combined spring and write the
equation of motion for the spring–mass systems shown in
Fig. P1.2.
Figure P1.2
Solution:
If ke is the effective stiffness,
fS keu (a)
u
k1 k2
fS
If the elongations of the two springs are u1 and u2 ,
u u1 u2 (b)
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Because the force in each spring is fS ,
fS k1u1 fS k2u2 (c)
Solving for u1 and u2 and substituting in Eq. (b) gives
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fS f f 1 1 1
S S
ke k1 k2 ke k1 k2
k1 k2
ke
k1 k2
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Equation of motion: mu keu p ( t ) .
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