地盤災害システム論 前田研究室 maedalab NumericalVib 1D

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L L _L

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1 質点線形系の振動解析

Numerical Analysis Method for a Mass with Linear-System in 1-D

x y

c _k

m

m _x

y k

c

Linear Vibration System

spring-dashpot-mass system spring mass

dashpot

structure system

−１質点系の振動の数学的厳密解(Mathematical Exact Solution)

− 応答の数値解法(Numerical Solutions) : 直接積分法(Direct Integration Method)

− 地震応答スペクトル（Earthquake Response Spectrum）

運動方程式：

( ) ( ) ( ) ( ) ^{t c x t kx t f t m y} ( ) ^t

x

m & & ₊ & _{+ = = −} & &

(1.1)

（貫性力：相対加速度）+（減衰制振力：相対速度）+（復元力：相対変位）＝（外力） Inertia force viscous force recover force

釣合い式：

( ) ( )

( ⁺ ) ( ) ( ) ^{− − = 0}

− ^{m & x & t & y & t c x & t kx t}

^(1.2)

（貫性力：絶対加速度）+（減衰制振力：相対速度）+（復元力：相対変位）＝0

m

: mass [kg]

k

: spring coefficient [kg/sec²]

c

: viscous damping coefficient [kg/sec]

( ) ^t

f

: external force applied to the mass

( ) ^t

y&

&

:

f _{( )} t

is replaced with acceleration of the ceiling and the ground

₋ m & & y _{( )} t

according to D'Alembert’s law.

( ) ^t

x&

&

,

x& _{( )} t

,

x _{( )} t

: relative acceleration[m/sec²], velocity[m/sec], and displacement[m] with respect to the ceiling or the ground, respectively

( ) ( ) ^{t y t}

x ₊

: absolute displacement

( ) ^t

x

m & &

: inertia force

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L L _L

^AAA^BBB

1. Mathematical Exact Solutions: Analytical results

1.1. ^{自由振動（} Free Vibration）:

( ) ^{t ≡ 0}

f

,

& y& _{( )} t _≡ 0

(1.3)

Linear and homogeneous differential equation solution = general solution

( ) ^{t x} ( ) ^t

x ₌

_g (1.4)

Characteristic Equation using differential operator

0

2

²

2

_{+ + =}

g o g

g

^{Dx x}

x

D _{γ ω}

(1.5)

D

: differential operator with respect to time differential (

D ₌ d dt

):

x &

_g

₌ Dx

_g

and

&& x ^&

_g

₌ D

²

x

_g.

m

c

= 2

γ

^,

m

k

o²

=

ω

^(1.6)

γ

: viscosity normalized by mass[1/sec]

ω

o: referential angular frequency[1/sec] / natural circle frequency

m

k

T f

o o

o

_π

ω ^{π 2}

2

1 = =

=

^(1.7)

T

o: referential vibration period[sec] / natural period

f

o: referential vibration frequency[1/sec] / natural frequency

(

^o

)

^g

g

^x

Dx _{= γ ± γ}

²

_{− ω}

² (1.8)

Vibration mode of the system clearly depends on the relative magnitude of viscous damping:

( )

( )

( )

⎪ ⎩

⎪ ⎨

⎧

>

>

−

=

=

−

<

<

−

mk

c

mk

c

mk

c

o o o

2

0

2

0

2

0

2 2 2 2 2 2

ω

γ ^ω

γ ^ω

γ

vibration

damped

over

vibration

damped

critically

vibration

damped

normally

:

c

_cr

₌ 2 mk

(1.9)

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G G _G

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L L _L

^AAA^BBB a) Normally damped vibration

parameters

mk

c

c _≤

_cr

₌ 2

, (2.10)

( γ ⁱ ω )

D _{= − ±}

, (2.11)

2 2

2

_{ω γ}

ω =

o

−

^{, (2.12)}

period

2

2

1

1

1

2

2

h

T

h

T

^o

O

₋

− =

=

= ω ^π ω ^π

^(2.13)

motions

( ) _{[ ]}

( ) _{[ { } { } ]}

( ) _{[ { ( ) } { ( ) } ]}

⎪ ⎩

⎪ ⎨

⎧

+

−

+

−

−

=

−

−

+

+

−

=

+

=

−

−

−

t

A

B

t

B

A

e

t

x

t

A

B

t

B

A

e

t

x

t

B

t

A

e

t

x

g g

g g

t g

g g g

g t g

g g

t g

ω

γωω

ω

γ

ω

γωω

ω

γ ^{γ ω ω γ ω ω}

ω

ω

γ

γ

γ

sin

2

cos

2

sin

cos

sin

cos

2 2 2

&

2

&

&

(2.14)

initial conditions

^{( )}

⎩ ( )

⎨ ⎧

+

−

=

=

g g g

g g

B

A

x

A

x

ω

γ

0

0

&

^(2.15)

therefore,

^{( )}

( ) ( )

{ }

⎩ ⎨

⎧

+

=

=

ω

γ ⁰

0

0

g g

g

g g

x

x

B

x

A

&

^(2.16)

complex descriptions

( ) _[ { ( ) } _]

( ) _[ ( ) ( { ) } _]

( ) _[ ( ) { ( ) } _]

⎪ ⎩

⎪ ⎨

⎧

−

−

−

−

=

−

−

−

−

=

−

−

=

t

i

i

C

al

t

x

t

i

i

C

al

t

x

t

i

C

al

t

x

g g

g g

g g

ω

λ

ω

λ ^{λ ω λ ω}

ω

λ

exp

Re

exp

Re

exp

Re

&

2

&

&

(2.17)

damping

Viscous damping coefficient

c

mk

c

_cr

₌ 2

(2.18)

Viscosity normalized by mass

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L L _L

^AAA^BBB

m

c

= 2

γ

^{, (2.19)}

Logarithmic damping factor

1

ln

+

=

n n

a

D a

(2.20)

Damping factor

o

cr

mk

c

c

h c

ω ^γ

=

=

= 2

^(2.21)

( ) ( )

²

1

1

ln 2

ln

h

T h

T

t

x

t

x

a

D a

n n

= −

+ =

=

=

+

γ π

^(2.22)

0 1 2 3 4 5

-2

-1

0

1

2

t(s)

x(t)

h=0

h=0.005

h=0.01

h=0.05

h=0.1

自由減衰振動

0 1 2 3 4 5

-2 -1 0 1 2

t(s)

x(t)

x

1

1

'

x

x

2

x

3

x

4

2

'

x ^x

³

^'

4

'

x

'

T

t

1

t

₂

G G G

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G G _G

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^NNN^GGG^III^NNN^EEE^EEE^RRR^III^NNN^GGG

L L _L

^AAA^BBB 固有周期実測値(observed value of fundamental natural period)

structure ^中低層（H 45m） ^高層（H＜45m）

S

^T

1=0.061 N[階]

T

₁=0.079 N[階] SRC, RC

T

1=0.054 N[階]

T

₁=0.053 N[階]

減衰定数実測値(observed value of damping factor) 鉄骨構造(Steel structure):

h

=2％

鉄骨鉄筋コンクリート構造(Steel reinforced concrete structure):

h

=3％ 鉄筋コンクリート構造(Reinforced concrete structure)：

h

=5％ 土：

_h

_=0~25％

Here, if h<< 1, we can obtain,

<< 1

h

,

D ₌ 2 _π h

(2.23)

Consequently, we can estimate value of h by measuring D,

<< 1

h

,

π

2

h ₌ D

(2.24)

G G G

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^CCC^III^EEE^NNN^CCC^EEE^&&&

G G _G

^EEE^OOO

E E _E

^NNN^GGG^III^NNN^EEE^EEE^RRR^III^NNN^GGG

L L _L

^AAA^BBB b) Critical damped vibration

parameters

mk

c

c ₌

_cr

₌ 2

, (2.25)

γ

−

=

D

, (2.26)

2

0

2

2

_{= ω − γ =}

ω

^{o , (2.27)}

period

∞

⇒

= ω ^π

T 2

(2.28)

motions

( ) [ ]

( ) [ ( ) ( ) ]

( ) [ ( ) ]

⎪ ⎩

⎪ ⎨

⎧

+

−

=

−

+

+

−

=

+

=

−

−

−

t

B

B

A

e

t

x

t

B

B

A

e

t

x

t

B

A

e

t

x

g g

g t g

g g

g t g

g g t g

3 2

2

2

2 _{γ γ}

γ ^{γ γ γ}

γ

γ γ

γ

&

&

&

(2.29)

initial conditions

^{( )}

⎩ ( )

⎨ ⎧

+

−

=

=

g g g

g g

B

A

x

A

x

γ

γ

0

0

&

^(2.30)

therefore,

^{( )}

( ) ( )

⎩ ⎨

⎧

+

=

=

0

0

0

g g

g

g g

x

x

B

x

A

γ

&

^(2.31)

c) Over damped vibration

parameters

mk

c

c _≥

_cr

₌ 2

, (2.32)

( − γ ± ω )

=

D

, (2.33)

(

^{2 2}

)

2

_{ω γ}

ω = −

o

−

^{, (2.34)}

period

1

1

1

2

2

2

2

₋ ⁼ ₋

=

= h

T

h

T

^o

ω ^π

O

ω ^π

^(2.35)

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^CCC^III^EEE^NNN^CCC^EEE^&&&

G G _G

^EEE^OOO

E E _E

^NNN^GGG^III^NNN^EEE^EEE^RRR^III^NNN^GGG

L L _L

^AAA^BBB

motions

( ) [ ]

( ) [ { } { } ]

( ) [ { ( ) } { ( ) } ]

⎪ ⎩

⎪ ⎨

⎧

+

−

+

−

−

=

−

−

+

+

−

=

+

=

−

−

−

t

A

B

t

B

A

e

t

x

t

A

B

t

B

A

e

t

x

t

B

t

A

e

t

x

g g

g g

t g

g g g

g t g

g g

t g

ω

γωω

ω

γ

ω

γωω

ω

γ ^{γ ω ω γ ω ω}

ω

ω

γ

γ

γ

sinh

2

cosh

2

sinh

cosh

sinh

cosh

2 2 2

&

2

&

&

(2.36)

initial conditions

^{( )}

⎩ ( )

⎨ ⎧

+

−

=

=

g g g

g g

B

A

x

A

x

ω

γ

0

0

&

^(2.37)

therefore,

^{( )}

( ) ( )

{ }

⎩ ⎨

⎧

+

=

=

ω

γ ⁰

0

0

g g

g

g g

x

x

B

x

A

&

^(2.38)

- 2.5 - 2 - 1.5 - 1 - 0.5 0 0.5 1 1.5 2 2.5 3

0 2 4 6 8 10

過減衰

減衰振動

臨界減衰

An example for vibration behaviours with different h : normally damped, critical damped and over damped vibrations

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G G _G

^EEE^OOO

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^NNN^GGG^III^NNN^EEE^EEE^RRR^III^NNN^GGG

L L _L

^AAA^BBB

* Euler’s formula

( ^e

^ix

^e

^ix

)

x _{= +}

⁻

2

cos 1

,

x _{= (} e

^ix

₊ e

^−ix

₎

2

sin 1

( ^{x i x} ) ( ^{nx i nx} )

e

^inx

₌ cos ₊ sin

ⁿ

₌ cos ₊ sin

* mechanical vibration system and electric circuit system

m _x

k

c

spring

dashpot

spring-dashpot-mass system

electric circuit system

C

R L

f

∫

+

+

= ^{m v cv k vdt}

f &

x

v & ₌

∫

+

+

= ^Idt

RI C

I

L

e ^{& 1}

q e

I & ₌

* mechanical vibration system and a control system in control engineering: PID control

∫

+

+

= ^{K e dt}

e T

dt K

K de

T

S

_p

I p p

D c

1

Derivation control + Proportion control + Integration control PID

S

c: control signal

e

: error = target value – current value

G G G

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G G _G

^EEE^OOO

E E _E

^NNN^GGG^III^NNN^EEE^EEE^RRR^III^NNN^GGG

L L _L

^AAA^BBB 非線形解法 (Non-linear analysis of vibration by numerical method)

直接積分法(Direct Integration Method) 非線形運動方程式の数値積分の方法

⊂

^差分法

− 時間領域の数値積分

− _{Newmark のβ}法： 実用的に最も用いられる

( ^{t t} ) ( ) ^{x t t x} ( ) ^t ( ) ( ) ( ) ( ^{t x t t x t t} )

x _{⎟ ∆ + ∆ ⋅ + ∆}

⎠

⎜ ⎞

⎝ ⎛ −

+

⋅

∆

+

=

∆

+

^{2 2}

2

1 β ^{& &} β

&

(2.39)

( ^{t t} ) ( ) ( ) ( ) ^{x t t x t t x} ( ^{t t} )

x & _{+ ∆ =} & ₊ 1 _{− γ ∆ ⋅} & & _{+ γ ⋅ ∆ ⋅} & & _{+ ∆}

(2.40)

Newmark の_β^{法の特別な呼び名}

手法

_{γ β}

線形加速度法(Linear Acceleration Method) ^{0.5 1/6} 中点加速度法(Constant Average Acceleration Method) 0.5 1/4

− _{Wilson のθ}法

( ^{t t} ) ( ) ^{x t t x} ( ) ( ) ( ) ( ) ( ^{t t x t t} ( ^{x t t} ) ( ) ^{x t} )

x _{+ ∆ = + ∆ ⋅} & _{+ ∆} & & ₊ ^∆ & & _{+ θ ∆ −} & &

θ

6

2

1

2 ²

(2.41)

( ^{t t} ) ( ) ^{x t t x} ( ) ^{t t} ( ^x ( ^{t t} ) ( ) ^{x t} )

x & _{+ ∆ =} & _{+ ∆ ⋅} & & ₊ ^∆ & & _{+ ∆ −} & &

θ

2

^(2.42)

各種数値積分法の安定条件(stability condition of numerical integration)

手法 数値積分の安定条件 無条件安定条件

(unconditionally Stable) Newmark の_β^法

β

ω ^{1 4}

2

≤ −

∆

o

t

0.5

^{β ≤ 1 4}

Wilson の_θ^法

_{θ ≥} 1 . 37

中央差分法

ω

≤ 2

∆t

^なし

ω

oはモデルの最も大きい円振動数

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L L _L

^AAA^BBB

− _{直接積分法}

) 運動方程式の離散化

( ) ^{t h x} ( ) ^{t x} ( ) ^{t y} ( ) ^t

x & & & &

& ₊ 2 _{ω + ω}

²

_{= −}

(2.43)

⎭ ⎬

⎫

+

+

+

=

+

+

+

=

∆ +

∆ +

∆ +

∆ +

t t t

t t

t t

t t t

t t

t t

y

B

y

B

x

A

x

A

x

y

B

y

B

x

A

x

A

x

&

&

&

&

&

&

&

&

&

&

&

22 21

22 21

12 11

12

11 (2.44)

[ ] _⎥

⎦

⎢ ⎤

⎣

= ⎡

22 21

12 11

A

A

A

A A

,

_{[ ] ⎥}

⎦

⎢ ⎤

⎣

= ⎡

22 21

12 11

B

B

B

B B

(2.45)

[ ] [ ]

⎭ ⎬

⎫

⎩ ⎨

+ ⎧

⎭ ⎬

⎫

⎩ ⎨

= ⎧

⎭ ⎬

⎫

⎩ ⎨

⎧

∆ +

∆ +

∆ +

t t

t t

t t

t t t

y

B y

x

A x

x

x

&

&

&

&

&

&

^(2.46)

t t t t t

t t

t

^{h x x y}

x &

_+∆

₊ &

_+∆

₊

_+∆

_{= −} & &

_+∆

& 2 _{ω ω}

² (2.47)

t t t

t

^{h x x y}

x & & & &

& ₊ 2 _{ω + ω}

²

_{= −}

(2.48)

t t t t t

t t

t

^{h x x y}

x &

_+∆

₊ &

_+∆

₊

_+∆

_{= −} & &

_+∆

& 2 _{ω ω}

² (2.49)

Taylor Expansion

( )

2

2 2

x t

t

x

x

x

_t+∆t

₌

_t

₊ &

_t

_{⋅ ∆ +} & &

_t

_⋅ ^∆

(2.50)

t

x

x

x

_t+∆t

₌

_t

₊ & &

_t1

_{⋅ ∆}

(2.51)

Taylor Expansion

( ^{t t} ) ( ) ^{f t t f} ( ) ^t

¹

f _{+ ∆ = + ∆ ⋅ ′}

t _≤ t

₂

_≤ t _{+ ∆} t

(2.52)

( ^{t t} ) ( ) ^{f t t f} ( ) ( )( ) ^{t f t}

²

₂ ^t

²

f _{+ ∆ = + ∆ ⋅ ′ + ′′} ^∆

t _≤ t

₂

_≤ t _{+ ∆} t

(2.53)

) 代数方程式を計算

(1) Newmark の_β^{法（内挿方式）}

( )

^{t t t}

t

^{x x}

x & _{= −} & & _{+ ⋅} & &

_+∆

&

₂

1 _{γ γ}

,

& x &

_t1

_{= (} 1 ₋ 2 _{β )} x & &

_t

₊ 2 _β & x &

_t+∆t (2.54)

( )

[ ^{1 2 x 2 x} ] ^{( )} ₂ ^t

²

t

x

x

x

_t+∆t

₌

_t

₊ &

_t

_{∆ + − β} & &

_t

_{+ β} & &

_t+∆t

^∆

(2.55)

[ ( ) ^{x x} ] ^t

x

x &

_t+∆t

₌

_t

₊ 1 _{− γ} & &

_t

_{+ γ} & &

_t+∆t

_∆

(2.56)

[ ] [ ] ^{A = ?}

[ ] [ ] ^{B = ?}

G G G

^EEE^OOO

S S _S

^CCC^III^EEE^NNN^CCC^EEE^&&&

G G _G

^EEE^OOO

E E _E

^NNN^GGG^III^NNN^EEE^EEE^RRR^III^NNN^GGG

L L _L

^AAA^BBB (2) Wilson の_θ^{法（外挿方式）}

時間

_t

∼

_{t + ∆ t}

で成立する運動方程式が、時間

_{t + θ ∆ t}

₍

_{θ ≥ 1}

_{)でも成立すると} する。

t t t t t

t t

t

^{h x x y}

x &

_+θ∆

_{+ ω} &

_+θ∆

_{+ ω}

_+θ∆

_{= −} & &

_+θ∆

& 2

² ,

_{θ ≥} 1

(2.57)

( ) ( ) ^{− + ⋅} _⎭ ^{⎬ ⎫}

=

⋅

+

−

=

∆ +

∆ +

∆ +

∆ +

t t t

t t

t t t t

t

y

y

y

x

x

x

&

&

&

&

&

&

&

&

&

&

&

&

θ

θ ^θ

θ

1

1

(2.58)

( ) ^{( )}

( ) _⎪ ^⎪

⎭

⎪⎪ ⎬

⎫

⎥⎦ ∆

⎢⎣ ⎤

⎡ _{⎟ +}

⎠

⎜ ⎞

⎝ ⎛ −

+

=

⎥⎦ ∆

⎢⎣ ⎤

⎡ _{⎟ +}

⎠

⎜ ⎞

⎝ ⎛ −

+

∆

+

=

∆ +

∆ +

∆ +

∆ +

t

x

x

x

x

x t

x

t

x

x

x

t t t

t t

t t t

t t t

θ θ

θ

θ

θ

θ θ

θ θ

θ θ

&

&

&

&

&

&

&

&

&

&

&

2

1 2

2

3

1 3

2

(2.59)

In the case of

_θ

=1,

( ) ^{( )}

( ) _⎪ ^⎪

⎭

⎪⎪ ⎬

⎫

⎥⎦ ∆

⎢⎣ ⎤

⎡ ₊

+

=

⎥⎦ ∆

⎢⎣ ⎤

⎡ ₊

+

∆

+

=

∆ +

∆ +

∆ +

∆ +

t

x

x

x

x

x t

x

t

x

x

x

t t t

t t

t t t

t t t

θ θ

θ θ

&

&

&

&

&

&

&

&

&

&

&

2

1

2

1

2

3

1

3

2

²

(2.60)

= Linear Acceleration Method (

_γ

=0.5,

_β

=1/6)

G G G

^EEE^OOO

S S _S

^CCC^III^EEE^NNN^CCC^EEE^&&&

G G _G

^EEE^OOO

E E _E

^NNN^GGG^III^NNN^EEE^EEE^RRR^III^NNN^GGG

L L _L

^AAA^BBB - 地震応答スペクトル（Earthquake Response Spectrum）

（相対）変位応答スペクトル(Displacement Response Spectrum):

x

（相対）速度応答スペクトル(Velocity Response Spectrum) :

x&

（絶対）加速度応答スペクトル(Acceleration Response Spectrum) :

x & & ₊ y & &

( ) ^{h T}

S

_d

,

: 最大相対変位応答値(Displacement Response Spectrum)

( ) ^{h T}

S

_v

,

: 最大相対速度応答値(Velocity Response Spectrum)

( ) ^{h T}

S

_a

,

: 最大絶対加速度応答値(Acceleration Response Spectrum)

Plots of maximum response values again selected parameters of the system or of forcing function (earthquake considered) are called ‘response spectra’.

For one-degree system, the natural period (or frequency) is the characteristic that determined its response to a given forcing function

ratio of maximum dynamic stress in a structure to the corresponding static stress

Calculation results of response for each of systems

one-degree systems with different natural periods (frequency)

input force (earthquake)

Maximum response values of systems

G G G

^EEE^OOO

S S _S

^CCC^III^EEE^NNN^CCC^EEE^&&&

G G _G

^EEE^OOO

E E _E

^NNN^GGG^III^NNN^EEE^EEE^RRR^III^NNN^GGG

L L _L

^AAA^BBB

v v

d

^S

S T

S ω ² π

1 =

≒

: 最大相対変位応答値(Displacement Response Spectrum)

v

v

^S

S ₌

: 最大相対速度応答値(Velocity Response Spectrum)

v v

a

^S

S T

S ^≒ _{ω =} ^{2 π}

: 最大絶対加速度応答値(Acceleration Response Spectrum)

(2.61)

Sa vs period

Sv vs period

SD vs period

G G G

^EEE^OOO

S S _S

^CCC^III^EEE^NNN^CCC^EEE^&&&

G G _G

^EEE^OOO

E E _E

^NNN^GGG^III^NNN^EEE^EEE^RRR^III^NNN^GGG

L L _L

^AAA^BBB

− 3 重応答スペクトル（Tripartite response spectrum）

( ) ( ) ⁻ _⎭ ^{⎬ ⎫}

+

=

+

−

=

T

S

S

T

S

S

d v

a v

log

2

log

log

log

log

2

log

log

log

π π

(2.62)

G G G

^EEE^OOO

S S _S

^CCC^III^EEE^NNN^CCC^EEE^&&&

G G _G

^EEE^OOO

E E _E

^NNN^GGG^III^NNN^EEE^EEE^RRR^III^NNN^GGG

L L _L

^AAA^BBB

− 応答スペクトル（response spectrum）について

・フーリエ・スペクトルは地震波そのものの周波数特性。応答スペクトルは構造物（_{1 質点} 減衰系）を含んだ地震動の全体像

・モード解析（modal analysis）

加速度応答スペクトル：地震力、ベース・シア係数(base shear coefficient)、動的震度 地震力

( )

max

max

^{m x y}

Q ₌ & & ₊ & &

(2.63)

ベース・シア係数(base shear coefficient)、動的震度

( ) ( )

g

T

h

S

g

y

x

W

C ₌ Q

^max

₌ ^{& & + & &}

^max

₌

^a

^,

(2.64)

静的震度_{: 静的耐震設計}

k

h

W

k

Q

_max

₌

_h

_⋅

(2.65)

速度応答スペクトル：地震動が構造物に与える最大のエネルギー 最大ひずみエネルギー(strain energy):

_{( )}

_max ²

2

1 k x

(2.66)

単位質量あたりの最大エネルギー_:

_{( ) ( )}

2 2 max 2

max

2

1

2

1

2

1

S

v

x

m x

k = =

⋅ ω

^(2.67)

(maximum energy per unit mass)

スペクトル強度(Spectral intensity):

I

_h

_{= ∫}

^2.5

S

_v

h T dT

1 .

0

^{( , )}

^(2.68)

変位応答スペクトル：ひずみの大きさ∼応力の大きさ 最大せん断力_:

max

max

^{k x}

Q _{= ⋅}

(2.69)

( ^{x y} )

^max

^kx

^max

m _⋅ & & ₊ & & ₌

(2.70)