solutions manual Essentials of Structural Dynamics Hector Estrada Luke S. Lee 1st Edition

Essentials of Structural Dynamics


Chapter 1 Introduction

1.1 Consider a rigid column having mass m, length L, and constrained to move horizontally at
mid-height by a spring having a stiffness k as shown below. Draw a free-body diagram
(FBD) of the column after a time-dependent vertical force is applied and formulate the
equation of motion using D’Alembert’s Principle.
p(t)



m L
k




Answer
i) Draw the FBD including inertial force and moment
 = L
p(t)

L/2
W
I G k
G müx


L/2
Ax

Ay
ii) Formulate the equation of motion by applying dynamic equilibrium and D’Alembert’s
principle
𝛥
∑ 𝐹𝑥 = 0; 𝐴𝑥 − 𝑚𝑢̈ 𝑥 + 𝑝(𝑡) − 𝑘 = 0
2
𝑚𝜃̈𝐿 𝑘θ𝐿
𝐴𝑥 = + − 𝑝( 𝑡 )
2 2
𝛴𝐹𝑦 = 0: 𝐴𝑦 − 𝑊 = 0
𝐴𝑦 = 𝑊
𝐿 Δ 𝐿 θ𝐿
∑ 𝑀𝐴 = 0; 𝑚𝑢̈ 𝑥 ( ) − 𝑊 ( ) + 𝐼𝐺 𝛼 − 𝐿𝑝(𝑡) + (𝑘 ) = 0
2 2 2 2
𝑚𝜃̈ 𝐿2 𝑊𝜃𝐿 𝑚𝐿2 𝜃̈ 𝜃𝐿2 𝑘
⇒ − + + = 𝐿𝑝(𝑡)
4 2 12 4
Equation of Motion in terms of rotation:



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, Essentials of Structural Dynamics

𝑚𝐿 𝐿𝑘 𝑊
𝜃̈ + ( − ) 𝜃 = 𝑝(𝑡)
3 4 2

1.2 Consider a rigid column having mass m, length L, and constrained to move horizontally at
the base by a rotational spring having a stiffness k as shown below. Draw a free-body
diagram (FBD) of the column after a time-dependent vertical force is applied and formulate
the equation of motion using D’Alembert’s Principle.
p(t)



m L




k

Answer
i) Draw the FBD including inertial force and moment
 = L
p(t)

L/2
W
I G
G müx


𝑀𝐴 = 𝑘𝜃 𝜃 L/2
Ax

Ay
ii) Write the equations of motion by applying dynamic equilibrium and D’Alembert’s Principle
∑ 𝐹𝑥 = 0; 𝐴𝑥 − 𝑚𝑢̈ 𝑥 + 𝑝(𝑡) = 0
𝑚𝜃̈𝐿
𝐴𝑥 = − 𝑝( 𝑡 )
2
𝛴𝐹𝑦 = 0: 𝐴𝑦 − 𝑊 = 0
𝐴𝑦 = 𝑊
𝐿 Δ
∑ 𝑀𝐴 = 0; 𝑚𝑢̈ 𝑥 ( ) − 𝑊 ( ) + 𝐼𝐺 𝛼 − 𝐿𝑝(𝑡) + 𝑘𝜃 𝜃 = 0
2 2
𝑚𝜃̈ 𝐿2 𝑊𝜃𝐿 𝑚𝐿2 𝜃̈
⇒ − + + 𝑘𝜃 𝜃 = 𝐿𝑝(𝑡)
4 2 12
Equation of Motion in terms of rotation:



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, Essentials of Structural Dynamics

𝑚𝐿 𝑘𝜃 𝑊
𝜃̈ + ( − ) 𝜃 = 𝑝(𝑡)
3 𝐿 2

1.3 Consider a rigid beam having mass m, span L, and constrained to move vertically at the right
support by a linear spring having a stiffness k and constrained at the left support by a
rotational spring having a stiffness k as shown below. Draw a free-body diagram (FBD) of
the beam after a time-dependent vertical force is applied and formulate the equation of
motion in terms of the rotation of the beam using D’Alembert’s Principle.
p(t)

k m
k

50ft
Answer
i) Draw the FBD including inertial force and moment
p(t)
W I G
𝑀𝐴 = 𝑘𝜃 𝜃
G

Ax
Ay  = L
L/2 L/2 k
müy

ii) Formulate the equation of motion by applying dynamic equilibrium and D’Alembert’s
principle
∑ 𝐹𝑦 = 0; 𝐴𝑦 − 𝑚𝑢̈ 𝑦 + 𝑊 + 𝑝(𝑡) − 𝑘𝛥 = 0
𝑚𝐿𝜃̈
𝐴𝑦 = + 𝑘𝐿θ − 𝑊 − 𝑝(𝑡)
2
𝛴𝐹𝑥 = 0: 𝐴𝑥 = 0
𝐿 L
∑ 𝑀𝐴 = 0; 𝑚𝑢̈ 𝑦 ( ) − 𝑊 ( ) + 𝐼𝐺 𝛼 − 𝐿𝑝(𝑡) + 𝑘∆𝐿 + 𝑘𝜃 𝜃 = 0
2 2
𝑚𝐿2 𝜃̈ 𝑊𝐿 𝑚𝐿2 𝜃̈
⇒ − + + 𝑘𝐿2 𝜃 + 𝑘𝜃 𝜃 = 𝐿𝑝(𝑡)
4 2 12
Equation of Motion in terms of rotation:
𝑚𝐿 𝑘𝜃 𝑊
𝜃̈ + (𝑘𝐿 + ) 𝜃 = 𝑝(𝑡) +
3 𝐿 2


1.4 Determine the effective weight at the roof and floor levels for a 60ft by 150ft, three-story
office building with equal story heights of 10ft and the following loading:




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, Essentials of Structural Dynamics


Roof DL = 20psf,
Roof LL = 20psf,
Floor DL = 20psf,
Floor LL = 60psf,
Wall DL = 10psf,




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