awb130 dynamics 04 spectrum

8/12/2019 AWB130 Dynamics 04 Spectrum

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Customer Training Material

ec ure

Res onse S ectrum

Linear and Nonlinear

L04-1ANSYS, Inc. Proprietary

2011 ANSYS, Inc. All rights reserved.Release 13.0

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ANSYS Mechanical Linear and Nonlinear Dynamics

Customer Training MaterialResponse Spectrum Analysis

Topics covered:

A. Descri tion and ur ose

Generation of the response spectrum

.

Single point response spectrum

Closely-spaced modes Ri id res onse

Missing mass

Multi point response spectrum

C. Procedure



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Customer Training MaterialDescription & Purpose

It is common to have a large models excited by transient loading.

e.g., building subjected to an earthquake

e.g., electronic component subjected to shock loading

The most accurate solution is to run a long transient analysis.

Large means many DOF. Long means many time points.

In many cases, this would take too much time and compute resources.

Instead of solvin the 1 lar e model and 2 lon transient to ether,



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it can be desirable to approximate the maximum response quickly.


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Customer Training MaterialDescription & Purpose

Idea: solve the (1) large model and (2) long transient separately and

combine the results.

Transient Anal sis Ch 6 Res onse S ectrum Anal sis

Large modelLong transient

Large modelLong transient

Large model Small model

Mode extraction

Mode shapes

Long transient

Response spectrum

Combined solution



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Fast, approximate

Slow, accurate


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Customer Training MaterialGenerating the Response Spectrum

The generation of the response spectrum will usually be done for

you, but the process can be described as follows:

1.Subject a small model to the transient loading. smallest is 1 DOF oscillator (k, c, m)

note the maximum absolute amplitude over time

m

m

k

c

e.g. for oscillator with frequency

= 30 Hz



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max a so u e va ue over me

S = 95 m/s2


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3. Repeat for oscillator with different frequency (same damping).

4. Plot the max response over time as a function of frequency.

= 30 HzS = 95 m/s2

= 50 HzS = 138 m/s2

= 70 HzS = 86 m/s2



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Note that the damping is included in the response spectrum.

Additional spectra can be generated for other damping values.



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Using many oscillators results in a more detailed curve.

The graph is typically plotted in log-log scale.

This also allows us to generate the spectrum only once and reuse it for

any model.

For efficiency, one analysis is done on an array of many oscillators.

the same damping is used for all oscillators



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We can easily convert between acceleration, velocity, and

displacement spectra by multiplying or dividing by the frequency.

remember to convert frequency units; rad/s = 2f Hz

( )2/ fSS vd = ( )fSS dv 2= ( )fSS da 2 2

=

( )22/ fSa = ( )fSa 2/= ( )fSv 2=



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Customer Training MaterialSpectral Regions

Recall: from the Harmonic Response chapter

a single DOF oscillator response to excitation frequency sweep

Note: two frequencies of interestA. Peak response amplification

Displacement Response Acceleration Response



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fBfB fAfA


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Note how phase angle relates the response to the input acceleration.

out-of-phase in-phasetransition

(uncorrelated) (correlated)



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fBfA


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These two frequencies can often be identified on a response

spectrum

This divides the spectrum into three regions

mid

frequency

high

frequency

low

frequency

1. Low frequency (belowfSP)

periodic region

modes generally uncorrelated

(periodic) unless closely spaced

. SPan ZPA

transition from periodic to rigid

modes have periodic component and

rigid component

fSP fZPA3. High Frequency (abovefZPA)

rigid region



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frequency at peak response

(spectral peak)

frequency at rigid response

(zero period acceleration) modes correlated with input frequency

and, therefore, also with themselves


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Customer Training MaterialTypes of Analyses

Types of Response Spectrum analysis:

Sin le- oint res onse s ectrum

A single response spectrum excites all specified points in the model.

-

Different response spectra excite different points in the model.



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ng e- o n

Response Spectrum




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Customer Training MaterialCommon Uses

Commonly used in the analysis of:

Nuclear power plant buildings

and components, for seismic

oa ng

Airborne Electronic equipmentfor shock loading

earthquake zones



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Customer Training MaterialAssumptions & Restrictions

The structure is linear (i.e. constant stiffness and mass).

The structure is excited b a s ectrum of known direction and

frequency components, acting uniformly on all support points.



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Customer Training MaterialParticipation Factors

A modal analysis must first be completed to determine the natural

frequencies, mode shapes, and participation factors for each mode.

This procedure was covered in Chapter 2: Modal Analysis.

K =+ 02

{ }[ ]{ }DMTi

i =

mode frequency mode shapespectrum

value

participation

factor

mode

coefficientresponse

1

1 1

1 1 1

2 2 {}2 S2 2 A2 {R}2



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Customer Training MaterialSpectrum Values

Log-log interpolation is done for

For each natural frequency, the spectrum value can be determined by a

simple look-up from the response-spectrum table.

modal frequencies between

spectrum points. No extrapolation is done for

fre uencies outside of the s ectrum

range; i.e. the value at the closest

point is used.


value

participation

factor

mode

coefficientresponse

1

1 1

1 1 1

2 2 {}2 S2 2 A2 {R}2



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Customer Training MaterialMode Coefficients

The mode coefficients can be determined from the participation factors,

depending on the type of spectrum input.

2

onacce erave oc ynsp aceme

iii

iiiiii

SASASA

===

Recall: participation factors measure the amount of mass moving in each direction

for a unit displacement.


value

participation

factor

mode

coefficientresponse

1

1 1

1 1 1

2 2 {}2 S2 2 A2 {R}2



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Customer Training MaterialResponse

The response (displacement, velocity or acceleration) for each mode can

then be computed from the frequency, mode coefficient, and mode shape.

{ } { } responsetyfor veloci

2

iiii

iii

AR

==

If there is more than one significant mode, the response for each mode must

be combined using some method.

iiii


value

participation

factor

mode

coefficientresponse

1

1 1 1

1 1

2 2 {}2 S2 2 A2 {R}2



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Customer Training MaterialMode Combination

One straightforward method for calculating the combined response is

to find the square root of sum of squares (SRSS).

{ } { } { } { } { }==+++=N

i

iN RRRRR1

222

2

2

1 K

For modes with sufficiently separated frequencies, the SRSS is

approximately the probable maximum value of response

SRSS is also known as the Goodman-Rosenblueth-Newmark rule


value

participation

factor

mode

coefficientresponse

1 1 1 1 1 12 2 {}2 S2 2 A2 {R}2



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There are circumstances in which this SRSS rule must be modified:

1. Accounting for modes with closely-spaced frequencies.

2. Adjusting for modes with partially- or fully-rigid responses.

3. Including high-frequency mode effects without extracting all modes.



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Closel -S aced Modes




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If modes are sufficiently spaced, the modes are uncorrelated.

1 2 3 4 5 6 7 8

It is valid to combine the res onses usin the SRSS method.

=N

2



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=i 1


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If modes with closely spaced frequencies exist, the SRSS method is

not applicable.

Modes with closely spaced frequencies become correlated.

0.0 1.0

= 0: uncorrelated

= 1: fully correlated1

2 3

4 5 6

7 8

. .

partially correlated

Idea: come up with a coefficient () to determine the amount ofcorrelation between modes

Complete Quadratic Combination (CQC) method



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Rosenblueth (ROSE) method


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Whereas the SRSS method takes the following form,

2

1N

1

==i

iRR

correlation coefficient.

ROSECQC

{ } { } { } { } { } { }2

1

1 1

2

1

1

=

= = == =

N

i

N

j

jiij

N

i

N

ij

jiij RRRRRkR

Each method has a formula for the correlation coefficient, , which



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is designed to vary between 1 (fully correlated) and 0 (uncorrelated)


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The definition of modes with closely spaced frequencies is a function

of the critical damping ratio:

For critical damping ratios 2%, modes are considered closely spaced if

the frequencies are within 10% of each other forfi 2%, modes are considered closely spaced if

the frequencies are within five times the critical damping ratio of each

forfi


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Ri id Res onse




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In the low-frequency range, the periodic responses predominate.

low

frequency

The responses are uncorrelated (unless closely spaced) and can be

combined using the SRSS, CQC, or ROSE methods.

{ } { }{ } 2

1

=N N

i RRR



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1 1= =i j


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In the high-frequency range, the rigid responses predominate.high

frequency

The responses are fully correlated (with the input frequency and

consequently with themselves) and can therefore be combined

algebraically, as follows:

{ } { }=N

rr RR



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=i 1


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Customer Training MaterialRigid Response

Lindley-Yow method, =(Sa)

i

ZPA==1ZPA

ia=0

The ri id res onse coefficient

attains a minimum value atSa,max

increases for decreasingSa

attains a maximum value of 1 atS = ZPA



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is set to zero whereSa< ZPA

S S


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Gupta method, =(f)

= max,Sa

=

( )+= 12

max,

3

2

2

fff

S

ZPA

v

ZPA

( )

= 21

1

1

for

/ln

for0

fff

ff

ff

i

i

i

i

=0

The ri id res onse coefficient

f1 fZPAf2 2for1 ffi

attains a minimum value of 0 atf f1

increases linearly in log-log space betweenf1 andf2

attains a maximum value of 1 at



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ANSYS Mechanical Linear and Nonlinear D namics


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The individual modal responses are calculated as before.

With rigid response turned on, these modes are not combined directly.

The responses will be split into periodic and rigid components.

mode frequencyspectrum

valueresponse

rigid response

coefficient

periodic

component

rigid

component

1 1 1 1 p 1 r 12 2 S2 {R}2 2 {Rp}2 {Rr}2



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The rigid response coefficient is calculated according to the method

selected.

( )/ln 1== i ffZPA

0.10.0

n 12

i

ai


valueresponse

rigid response

coefficient

periodic

component

rigid

component

1 1 1 1 p 1 r 12 2 S2 {R}2 2 {Rp}2 {Rr}2



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The periodic and rigid components are calculated from the rigid

response coefficient.

{ } { } componentrigid

componentperiodic1

iiir

iiip

RR

=

=

As before, the response for each mode must be combined using

some method.

Now, the eriodic modes and ri id modes will be treated se aratel .


valueresponse

rigid response

coefficient

periodic

component

rigid

component

1 1 1 1 p 1 r 12 2 S2 {R}2 2 {Rp}2 {Rr}2



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Example: rigid response components using Gupta method.

1 2 3 4 5 6

mode frequency response

rigid response

coefficient

periodic

component

rigid

component

1 5 {R}1 0.0 {R}1 {0}2 1 4 {R}2 0.0 {R}2 {0}

periodic

3 3 6 {R}3 0.18 0.98 {R}3 0.18 {R}34 6 4 {R}4 0.61 0.79 {R}4 0.61 {R}4

12

transition



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12 .6 25 {R}6 1.0 {0} {R}6

rigid



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The periodic modes are combined according to the desired

combination rule (SRSS, CQC, or ROSE).

Again, CQC or ROSE are used if closely spaced modes are present.

ROSECQCSRSS

{ } { } { } { }{ } { } { }{ } 2

1 1

2

1 1

2

1

1

2

=

=

=

= == ==

N

i

N

jjpipijp

N

i

N

jjpipijp

N

iipp

RRRRRkRRR


valueresponse

rigid response

coefficient

periodic

component

rigid

component

1 1 1 1 p 1 r 12 2 S2 {R}2 2 {Rp}2 {Rr}2



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The rigid modes are summed algebraically.

{ } { }== iirr RR

1


valueresponse

rigid response

coefficient

periodic

component

rigid

component

1 1 1 1 p 1 r 12 2 S2 {R}2 2 {Rp}2 {Rr}2



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y


The combined periodic and combined rigid modes are finally

combined to give the total response.

{ } { } { }22

prt RRR +=


valueresponse

rigid response

coefficient

periodic

component

rigid

component

1 1 1 1 p 1 r 12 2 S2 {R}2 2 {Rp}2 {Rr}2



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Missin Mass




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Customer Training MaterialRecall: Effective Mass

Recall: printed in the modal output file is the effective mass.

Generall we cannot racticall extract all ossible modes to

account for 100% of the mass of the structure.

Many structures may have important natural vibration modes at

frequencies higher than the ZPA frequency,fZPA.

{ }[ ]{ } 2, iieff

T

ii MDM ==



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Customer Training MaterialMissing Mass

So far, we have addressed the low-, mid-, and high-frequency rangesof the spectrum.

We could have many modes extend far beyondfZPA.

mid

frequency

high

frequency

low

frequency

ZPA

R1 R2 R3 R4 R5 R6 R7 R8 R9 R10 R11

Idea: if we can figure out how much mass is missing from the modal

analysis, then we dont have to extract all modes abovef .



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its effect can be lumped into an additional response



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Recall, abovefZPA, the acceleration response is rigid (in-phase), so itcan be fairly accurately represented by a static acceleration analysis.

We can calculate what the total inertia force should be abovefZPA.

=

We can also calculate the inertia force contributed by each mode.

ZPAT

[ ]{ } ZPAjjj SMF =

The sum of inertia force contributed by all modes is simply

NN



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== jZPAj

j

j

11



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The missing inertia force must simply be the difference betweenthe total inertia force and the sum of modal inertia forces.

{ } { } { } [ ] { } { } ZPAN

jj

N

jTM SDMFFF == == 11

The missing mass response is the static shape due to the missing

.

1=



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If rigid response is included, then the missing mass is added to therigid response.

{ } { } { }= +=N

i

Mirr RRR1

The total response is then calculated by combining the periodic and

rigid components.

22= rp



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Recommended Solution

Procedure




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Customer Training MaterialRecommended Solution Procedure

The recommended solution method is generally specified by yourdesign code.

combination method

rigid response method

missing mass effects

Alternatively, the best solution method can be determined by

extracting the modes to be used for combination and

com arin them to the res onse s ectrum



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1 2 3 4 5 6 7 8



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Customer Training MaterialRecommended Solution Procedure

Modes only in low-frequency region

1 2 3 4 5 6 7 8

.

No rigid response effects. No missing mass effects.

Modes only in mid- to high-frequency region

or or c ose y space mo es .

Rigid response by Lindley or Gupta method. Missing mass on.

Modes in all frequency regions



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SRSS (or CQC/ROSE for closely spaced modes).

Rigid response by Gupta method. Missing mass on.


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u - o n

Response Spectrum




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Customer Training MaterialMulti-Point Response Spectrum

In single-point response spectrum (SPRS), it was assumed that allconstrained points were being subjected to the same spectrum.



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M lti P i t R S t


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In multi-point response spectrum (MPRS), different constrainedpoints can be being subjected to different spectra (up to 100 different

excitations).



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M lti P i t R S t


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Each spectrum is treated individually first, so all of the previousinformation on single-point response spectrum applies

closely spaced modes (SRSS, CQC, ROSE)

rigid response (Lindley-Yow, Gupta)

missing mass

Finally, the response of the structure is obtained by combining the

responses to each spectrum using the SRSS method.

{ } { } { } K++= 22

2

1 SPRSSPRSMPRS RRR

{RMPRS} is the total response of the MPRS analysis

{RSPRS}1 is the total response of the SPRS analysis for spectrum 1

R is the total res onse of the SPRS anal sis for s ectrum 2



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etc


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Procedure




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Procedure


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Customer Training MaterialProcedure

Drop a Modal (ANSYS) system into the project schematic.



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Procedure


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Customer Training MaterialProcedure

Drop a Response Spectrum system onto the Solution cell of theModal system.



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Preprocessing


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Customer Training MaterialPreprocessing

Verify materials, connections, and mesh settings. This was covered in Workbench Mechanical Intro.



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Preprocessing


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Add supports to the model. Displacement constrains must have a magnitude of zero.



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Solution Settings


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Customer Training MaterialSolution Settings

Choose the number of modes toextract.

If needed, upper and lower bounds

on frequency may be specified to

extract the modes within a specified

range.



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C t T i i M t i lPostprocessing


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Customer Training MaterialPostprocessing

Review the modal results beforeproceeding.



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Insert an Acceleration, Velocity, or Direction response spectrum. Set the Boundary Condition, Spectrum (Tabular) Data, and Direction.



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Customer Training Materialep ocess g

Turn on rigid response effect if needed. Lindley-Yow method requires setting theZPA.

Gupta method requires settingf1 andf2.



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Customer Training Materialp g

Turn on missing mass effect if needed This requires setting the Zero Period Acceleration (ZPA).



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Customer Training MaterialPostprocessing


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gg

Stress (normal, shear, equivalent) and Strain (normal, shear) resultscan also be reviewed.



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Customer Training MaterialResponse Spectrum Analysis


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In this workshop, you will determine the response of a

prestressed suspension bridge subjected to a seismic load.

See your Dynamics Workshop supplement for details

WS4: Response Spectrum Analysis - Suspension Bridge



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awb130 dynamics 04 spectrum

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