1

2. Single Degree of Freedom Systems (1-DOFs)

under Harmonic Base Excitation&

Elastic Response Spectra

2. Single Degree of Freedom Systems (1-DOFs)

under Harmonic Base Excitation&

Elastic Response Spectra

October 2017

George BOUCKOVALASProfessor of N.T.U.A.

2

2.1 Equation of dynamic equilibrium

0KXXCYM

UXY

UMKXXCXM

tiO

2tiO eUUeUU

tiOeFKXXCXM

O2

O MUF με

για αρμονική διέγερση:

and, finally:

Reminder: [eiΩt = cosΩt + i sin Ωt]

X = X1+X2

where Χ1 is the «transient» component, andΧ2 is the «steady state» component of displacement

which is of greater practical importance (why?)

t/T

X

X1+X2

X2

3

NOTE: This component has frequency equal to the fundamental frequency of vibration of the 1-DOF (NOT the excitation frequency)

The “transient”component of displacement (Χ1) is adamped free vibration described by the equation:

where: , and

The picture can't be displayed.

)tsinBtcosA(eX DDt

1

/ 2C KM n

K

M 21D n n

e.g.A=1B=0

2.2 Relative displacement Χ

X = X1+X2

where Χ1 is the «transient» component, andΧ2 is the «steady state» component of displacement

which is of greater practical importance (why?)

t/T

X

X1+X2

X2

4

The “steady state” component of displacement (Χ2) is harmonic, With frequency equal to the excitation frequency (Ω) and phase difference (φ) from the excitation:

)t(iST2 e)AX(X

22O o

ST o2

F Ω ΜU ΩX U

K ωω Μ

2222 )/(4])/(1[

1A

2)/(1

)/(2tan

where

Question for the Class:

Based on the previous definitions and the figures for the Dynamic coefficient Α and the angle of phase difference φ, describeThe basic mechanisms and the physical meaning of the steady state response (X2/UO)MAX in the following benchmark cases:

I. Area Ω/ω <<<1.0II. Area Ω/ω = 1.0III. Area Ω/ω >>>1.0

Examples from engineering practice (and/or from every day life) will contribute to better understanding and are welcome.

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2.3 Total Displacement Υ (velocity & acceleration)

)t(iO

tiO eXeUXUY

tiO

tiiOO eYe)eXU(

)sinX(i)cosXU(Y OOOO

i

O

O

O

O eU

X1

U

Y

U

Y AU

X2

O

O

i

2

O

O e1U

Y

U

Y

A

U

Y

U

Y

O

O

After normalization against the base excitation,

,

and finally, or

[from a .. strange coincidence that is worth proving analytically]

Question for the Class:

Based on the previous definitions and the figures for the Dynamic coefficient Α and the angle of phase difference φ, describeThe basic mechanisms and the physical meaning of the TOTAL1-DOF response (Y/UO)MAX in the following benchmark cases:

I. Area Ω/ω <<<1.0II. Area Ω/ω = 1.0III. Area Ω/ω >>>1.0

Examples from engineering practice (and/or from every day life) will contribute to better understanding and are welcome.

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For harmonic excitation, all previous non-dimensional expressions for relative and total displacements are also valid for the corresponding velocities and accelerations. Hence, we may define the following two “dynamic factors”:

U

X

U

X

U

XAX

U

Y

U

Y

U

YAY

In this case also it is worth checking the physical meaning of the above diagrams for the characteristic cases where Ω/ω<<<1.0, Ω/ω=1 and Ω/ω>>>1.

AY

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2.4 The “Elastic Response Spectrum”Harmonic Excitation

A. For a given structure, when ω=2π/Τ=constant, it describes the maximum dynamic response for various excitations with frequency Ω=2π/Τexc.

B. For a given excitation, when Ω=constant, it describes the maximum dynamic response of various structures in terms of their fundamental vibration frequency ω=2π/Τα, i.e. it provides the SPECTRUM (possible range) of the visco-elastic structural response.

The diagram for the dynamic factor variationΑΥ – (Ω/ω) may be interpreted in two basic ways:

The second interpretation is of particular practical interest for the seismic design of structures as it provides the maximum seismic actions (displacement, velocity, acceleration) which are applied to the mass of the structure during base excitation. Hence, it was extended to non harmonic – seismic excitations, in the form of the widely used “Elastic Response Spectra”, initially by Biot (1932) and later byHousner.

Elastic response ”SPECTRUM” for harmonic excitation

Excitation with(Ü)max=0.30gandΩ=31.4 rad/sorΤexc=0.20sec

0 1 2 3Ω/ω

0

1

2

3

4

5

6

RY

0 0.1 0.2 0.3 0.4T (sec)

0

0.4

0.8

1.2

1.6

Sa

(g)

ΑΥ

8

Elastic response ”SPECTRUM” for harmonic excitation

0 0.1 0.2 0.3 0.4T (sec)

0

0.4

0.8

1.2

1.6

Sa

(g)

PGA=(Ü)max=0.30g

Τexc=0.20sec

The other way round…

i.e.

if the spectrum of a seismic motion is given to us…..

(inertia) acceleration applied to themass of a structure with period T

Period, Τ

excitation

Elastic response ”SPECTRUM” for SEISMIC excitation

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Max. acceleration of seismic motion

Range of predominant excitation periods

0 0.4 0.8 1.2 1.6 2Tstr (s)

0

0.4

0.8

1.2

1.6

2

Sa

(g)

LONG

TRANS

Lefkas earthquake(14/08/2003Μ=6.4

Elastic response ”SPECTRUM” for SEISMIC excitation

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Max. relative displacement ΧΟ.[Max. base shear force FΟ=K XO]

Max. PSEUDO-velocityVΟ=ω ΧΟ=(2π/Τ)ΧΟ

[Max. stored elastic energyΕΟ= ½ (Κ ΧΟ

2) = ½ (ΜVO2)]

Max. PSEUDO-accelerationΑΟ=ω2 ΧΟ=(2π/Τ)2ΧΟ

[Max. base shear force FΟ=K XO = ΜΑΟ]

Alternative types of Elastic Response Spectra

El Centro earthquake, ξ=2%

Comparison:

Pseudo-velocity

vs.

max. relative velocity

Harmonic base excitation

o o

o

o

o ο

V ω XV ω 1

Ω (Ω / ω)XX Ω Χ

Ω/ω

o

o

VX

1.0

1.0

good agreement at resonance . .

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Comparison:

Pseudo-velocity

vs.

max. relative velocity

Relative velocity

Pseudo velocity

Actual seismic excitation

good agreement at the range of predominant excitation

periods

Harmonic base excitation2

o o

o ο2

oo2 2

o ο o

Α ω XA Χ1 1

Α U(Ω / ω)ΥΥ Ω Υ Ω Α U

2o

o

o

o

X Ωbut ΑU ω

Aso that ....... 1.0 !

Y

good overall agreement ….

Comparison:

Pseudo-acceleration

vs.

max. absolute acceleration

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Comparison:

Pseudo-acceleration

vs.

max. absolute acceleration

Actual seismic excitation

good overall agreement ….

ΧΟ

VΟ=ω ΧΟ

ΑΟ=ω2 ΧΟ

Combined THREE-LOGARITHMICpresentation

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Combined THREE-LOGARITHMICpresentation

14

Problem solving for Chapter 2

HWK 2.1:For harmonic ground excitation, with displacement amplitude UΟ=0.15m and period Τexc..=0.40sec, compute:

1) The maximum absolute acceleration, velocity and displacement at the floor level of the frame.

2) The maximum shear force Τ and moment Μ developing at the top of the two columns.

Assume that the floor is rigid with a total weight of 500 kN, while both columns are

flexible without mass.

ΕΙ=2x104 kNm2

H=5m, ξ=5%ΕΙ=2x104 kNm2

H=5m, ξ=5%

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HWK 2.2:For harmonic seismic excitation, with displacement amplitude UΟ=0.15m and period Τδιεγ.=0.40sec, compute the following diagrams:

1) The elastic response spectrum for the relative displacement, forξ=2%.

2) The elastic response spectra for the Pseudo-velocity and the max. relative velocity, for ξ=2% (in the same diagram)

3) The elastic response spectra for the Pseudo-acceleration and the max. absolute acceleration, for ξ=2% (in the same diagram)

4) The ratio of the spectra of question 2 and the ratio of the spectra of question 3 (in the same figure)

Compare with the respective diagrams – spectra for El Centro earthquake. Discuss the similarities and the differences.

HWK 2.2:For harmonic seismic excitation, with displacement amplitude UΟ=0.15m and period Τδιεγ.=0.40sec, compute the following diagrams:

1) The elastic response spectrum for the relative displacement, forξ=2%.

2) The elastic response spectra for the Pseudo-velocity and the max. relative velocity, for ξ=2% (in the same diagram)

3) The elastic response spectra for the Pseudo-acceleration and the max. absolute acceleration, for ξ=2% (in the same diagram)

4) The ratio of the spectra of question 2 and the ratio of the spectra of question 3 (in the same figure)

Compare with the respective diagrams – spectra for El Centro earthquake. Discuss the similarities and the differences.

HWK 2.3:Repeat HWK 2.1 for Lefkada Earthquake

(any one of the two components)

ΕΙ=2x104 kNm2

H=5m, ξ=5%ΕΙ=2x104 kNm2

H=5m, ξ=5%

0 0.4 0.8 1.2 1.6 2Tstr (s)

0

0.4

0.8

1.2

1.6

2

Sa

(g)

LONG

TRANS