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RM Bridge Professional Engineering Software for Bridges of all Types
TRAINING PRESTRESSING BASIC - RM - PART 3: DYNAMIC EARTHQUAKE ANALYSIS
RM Bridge V8i
March 2012
RM Bridge
Training Prestressing Basic - RM - Part 3: Dynamic Earthquake Analysis I
Bentley Systems Austria
Contents
1 General ................................................................................................................... 1-1
1.1 Scope .............................................................................................................. 1-1
1.2 Generating a new construction schedule ....................................................... 1-1
2 Definition of Masses .............................................................................................. 2-3
2.1 Definition of load sets for self weight and SDL ............................................ 2-3
2.1.1 Generating the load sets for SW and SDL ................................................. 2-3
2.1.2 Defining the load set for SW ..................................................................... 2-3
2.1.3 Defining the load set for SDL .................................................................... 2-4
2.2 Definition of load case for masses ................................................................. 2-4
2.2.1 Generating a load case for the masses ....................................................... 2-4
2.2.2 Assigning the load sets to the load case ..................................................... 2-5
3 Calculation of Eigenvalues .................................................................................... 3-6
3.1 Generating a stage for the calculation of the eigenvalues ............................. 3-6
3.2 Calculating the eigenvalues ........................................................................... 3-6
4 Preparation of Response Spectrum ........................................................................ 4-8
4.1 Defining a response spectrum diagram .......................................................... 4-8
4.1.1 Generating tables for horizontal and vertical response spectrum ............. 4-8
4.1.2 Defining the tables for horizontal and vertical response spectrum ............ 4-8
4.1.3 Assignment of the tables to variables (formulas) .................................... 4-13
4.2 Response Spectrum defined by formulas ..................................................... 4-13
5 Definition of the Earthquake Load ...................................................................... 5-16
5.1 Defining the earthquake events .................................................................... 5-16
6 Evaluation of the Response Spectrum ................................................................. 6-18
6.1 Generating a stage for the evaluation of the response spectrum .................. 6-18
6.2 Evaluating the response spectrum ................................................................ 6-18
6.2.1 Initializing envelopes for storing the resuts of the evaluation ................. 6-18
6.2.2 Evaluating the response spectra ............................................................... 6-18
7 Superposition of the Seismic Loads ..................................................................... 7-20
7.1 Generating a stage for the superposition of the seismic loads ..................... 7-20
7.2 Superposing the seismic loads ..................................................................... 7-20
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8 Result Presentation .............................................................................................. 8-22
8.1 Plotting the response spectra data ................................................................ 8-22
8.2 Plotting of eigenmodes ................................................................................ 8-22
8.2.1 Definition of the DoPlot action ................................................................ 8-22
8.2.2 Definition of the Plot File in the Plot Container ...................................... 8-23
9 Time History Analysis ......................................................................................... 9-25
9.1 Preparation of the tables and variables ........................................................ 9-25
9.2 Definition of the load cases and load sets .................................................... 9-27
9.2.1 Definition of the load sets for the masses ................................................ 9-27
9.2.2 Definition of the load sets for the displacements ..................................... 9-27
9.2.3 Definition of the load cases for the time history calculation ................... 9-28
9.3 Calculation of the Time History Events ...................................................... 9-30
9.4 Definition of Damping ................................................................................. 9-31
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1 General
1.1 Scope
In the following the procedure for performing a dynamic earthquake calculation using
the response spectrum analysis shall be explained. All necessary steps including load
case definitions, preparation of response spectrum, and required calculation actions and
superposition in the schedule will be given. The principle input procedure is the follow-
ing:
1.) Definition of masses of all permanent loads (i.e. self weight and superimposed dead load)
2.) Calculation of the eigenvalues
3.) Preparation of the response spectrum
4.) Definition of the earthquake load
5.) Evaluation of response spectrum for earthquake analysis
6.) Superposition of the seismic loads
7.) Result presentation
In addition in chapter 9 there is given a general overview of the necessary input proce-
dure for performing a time history analysis.
1.2 Generating a new construction schedule
To separate schedule actions and results it can be favourable to create a new schedule
variant (it is of course not obligatory), where the dynamic calculation is performed. This
is done as follows:
Create a new
schedule variant
Schedule CS will be skipped deactivate
Name earthquake
Schedule Variants Description response spectrum analysis
Sequence No 1
Top Table
Then change in the main GUI to this construction schedule (drop down menu to the
right of the Recalc button). All input of loads and stages will now refer to this (active) schedule.
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NOTE: If the final superposition file containing the results of the earthquake calculation
is needed and used in the calculation of the DEFAULT schedule (e.g. combination table), the sequence for calculating the different construction schedules has to be
changed in accordance. In the recalculation pad one can then choose Recalc all, which will use this given sequence. All envelope results (*.sup) are stored in the main project
directory of RM and are therefore available for all the construction schedules. Alterna-
tively one can just calculate a particular construction schedule by selecting it in the GUI
first or in the recalculation pad and using Recalc.
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2 Definition of Masses
All permanent loads acting on the structure, i.e. self weight and superimposed dead
loads, have to be defined as masses. These masses must be defined in one single load
case, whereas it is favourable to group them by means of load sets.
2.1 Definition of load sets for self weight and SDL
2.1.1 Generating the load sets for SW and SDL
Create new load
sets for the self
weight and super-
imposed dead load
Schedule Name LS-SW LS-SDL
Description self weight mass SDL mass
Load Definition
Load Set Defi-
nition
Top table
2.1.2 Defining the load set for SW
Define load set for
the self weight
Schedule Name LS-SW
Loading Uniform
load
Uniform load
Uniform load
Load Definition Type Self weight -
mass
Self weight -
mass
Self weight -
mass
From 101 1201 1301
Load Set Defi-
nition To 135 1204 1304
Step 1 1 1
Bottom table Rx 0 0 0
Ry -1 -1 -1
Rz 0 0 0
Gam
[kN/m3]
0 0 0
The load type for the self-weight mass can be specified as just as mass (only used for dynamic calculation) or load and mass (also used for static calculation).
These defined self weight masses act in the centre of gravity.
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2.1.3 Defining the load set for SDL
Define load set for
superimposed dead
load
Schedule Name LS-SDL
Loading masses masses masses
Load Definition Type
Element
uniform
mass + ec-
centricity
Element
uniform
mass + ec-
centricity
Element
uniform
mass + ec-
centricity
From 101 101 101
Load Set Defi-
nition To 135 135 135
Step 1 1 1
Bottom table g*mx
[kN/m] 35 6.1 6.1
g*my
[kN/m] 35 6.1 6.1
g*mz
[kN/m] 35 6.1 6.1
g*Imx
[kNm] 0 0 0
Ey [m] 0.06 0.45 0.45
Ez [m] 0 +6.3 -6.3
In RM masses are defined as forces (and moments respectively) and internally trans-
formed into masses by dividing them by the gravity acceleration value, which is set to
9.81 m/s2 by default (this can be modified in the Recalculation Pad in the menu Dy-
namic). Since mass is a scalar value the definition of the vectors has to be given for all three
force-components.
The eccentricities are not related to the center of gravity (local element coordinate
system), but to the node (node axis, i.e. connection between start and end node of the
element), that means the internal element eccentricities are added automatically!
2.2 Definition of load case for masses
2.2.1 Generating a load case for the masses
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Create new load
case for the load
sets of self weight
and superimposed
dead load
Schedule Name LC-MASS
Description dead load mass
Load Definition
Load Case Def-
inition
Top table
2.2.2 Assigning the load sets to the load case
Assign load sets to
load case
Schedule Name LC-MASS
Loading Load set input Load set input Load Set LS-SW LS-SDL
Load Definition
Load Case Def-
inition
Bottom table
This load case is needed then as reference for calculating the eigenvalues (see 3.2)
All loads not specified in this manner using the respective load types for masses will
not be considered for the calculation of the mass matrices in the dynamic analysis.
(That means on the other hand that it is possible to include load items, which should
not be taken into account as masses, but only as loads in the static or time history
analysis).
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3 Calculation of Eigenvalues
3.1 Generating a stage for the calculation of the eigenvalues
Create a stage for
the calculation of
the eigenvalues
Schedule Name EIGEN
Description calculation of eigenvalues
Stages
Activation
Top table
Activate all ele-
ments of structure
Schedule Activate From 101
Stages To 1402
Step 1
Activation
Bottom table
3.2 Calculating the eigenvalues
Calculate the ei-
genvalues
Schedule Action Calculation (Dy-
namic) List/plot actions
Type Eigen ListMod
Stages Input-1 30 eigen.mod
Input-2 LC-MASS
Schedule Actions Input-3
Output-1 eigen.mod
Bottom table Output-2 eigen.lst eigen-mod.lst
Descrip-
tion
Calculate eigen
frequencies and
eigenvectors (natural
modes) of structure
Create listfile of
binary modal file
The load case, where the effective masses are defined, has to be given as reference for
the eigenvalue analysis (see 2.2).
The number of eigenvalues (lowest natural frequencies) to be calculated has to be
given. (The tolerance value for determining the accuracy of the calculated eigenval-
ues is specified in the Recalc pad in the menu Dynamic). For receiving 90% of mass participation in vertical direction in this example approximately 30 eigenvalues
were necessary (see comments for the output of mass participation below).
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The eigenvalues and natural modes are stored in a binary modal file, which name has
to be given. A readable output list of this file can be printed using the list/plot action
ListMod The name of the output list with a protocol of all the relevant input and output data
has to be entered (or left to default *). An essential part is the output of the mass participation factors at the end of this list, which should be greater than 90% of the to-
tal mass in the relevant direction of the acceleration of the structure (see Diagram 1).
In some cases the number of calculated eigenmodes is smaller than expected. In such
cases the dimension of the iteration matrix may be increased by additional iteration
vectors (Subspace). This may lead to better iteration results and a higher number of
eigenmodes found.
Note: Eigenvalues and eigenfrequencies (natural frequencies) are calculated on the
un-damped structural system; therefore no damping parameters need to be specified
yet within this action.
After performing this action a number of n load cases is generated and can be accessed
from the load case pool (named name#n), where n is the number of the eigenmode and
the name being taken from Output-1 (e.g. eigen#3). These load cases contain normal-
ized eigenvectors as displacements and may be used for graphic presentation (see 8.2).
Diagram 1: Output of mass participation factors in eigen.lst
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4 Preparation of Response Spectrum
4.1 Defining a response spectrum diagram
The response spectrum is defined in Properties Variables in terms of tables and/or formulas according to the rules of the respective codes.
The response spectrum used in this example is in accordance with Eurocode 8 (Type 1,
Ground type A, 5% damping). It is prepared by means of tables where the particular
values have been evaluated before. Also an alternative will be shown in 4.1.3 illustrat-
ing a direct definition by formulas.
4.1.1 Generating tables for horizontal and vertical response spectrum
Create a new table
for the horizontal
and vertical re-
sponse spectrum
Properties Name resp_hor_tab resp_vert_tab
Type table table
Variables Description table for horizontal
response spectrum
table for vertical
response spectrum
Top table
Note: Do not use special characters in variable names (formulas and tables) except un-
derline _!
4.1.2 Defining the tables for horizontal and vertical response spectrum
Define the tables
for the horizontal
and vertical re-
sponse spectrum
Properties
Variables
Bottom table
The units used are period (T) [s] for the abscissa (VarA) and the elastic response spec-
trum divided by the design ground acceleration (Se/ag) [-] and (Sve/avg) [-] respectively
for the ordinate (VarB).
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Table for horizontal response spectrum
VarA T [s] VarB Se/ag [-] Interpolation
VarA T [s] VarB Se/ag [-] Interpolation
0.00 1.0000 linear
1.70 0.5882 linear
0.15 2.5000 linear
1.75 0.5714 linear
0.40 2.5000 linear
1.80 0.5556 linear
0.45 2.2222 linear
1.85 0.5405 linear
0.50 2.0000 linear
1.90 0.5263 linear
0.55 1.8182 linear
1.95 0.5128 linear
0.60 1.6667 linear
2.00 0.5000 linear
0.65 1.5385 linear
2.10 0.4535 linear
0.70 1.4286 linear
2.20 0.4132 linear
0.75 1.3333 linear
2.30 0.3781 linear
0.80 1.2500 linear
2.40 0.3472 linear
0.85 1.1765 linear
2.50 0.3200 linear
0.90 1.1111 linear
2.60 0.2959 linear
0.95 1.0526 linear
2.70 0.2743 linear
1.00 1.0000 linear
2.80 0.2551 linear
1.05 0.9524 linear
2.90 0.2378 linear
1.10 0.9091 linear
3.00 0.2222 linear
1.15 0.8696 linear
3.10 0.2081 linear
1.20 0.8333 linear
3.20 0.1953 linear
1.25 0.8000 linear
3.30 0.1837 linear
1.30 0.7692 linear
3.40 0.1730 linear
1.35 0.7407 linear
3.50 0.1633 linear
1.40 0.7143 linear
3.60 0.1543 linear
1.45 0.6897 linear
3.70 0.1461 linear
1.50 0.6667 linear
3.80 0.1385 linear
1.55 0.6452 linear
3.90 0.1315 linear
1.60 0.6250 linear
4.00 0.1250 linear
1.65 0.6061 linear
Diagram 2: Table for the horizontal response spectrum
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Table for vertical response spectrum
VarA T [s] VarB Sve/avg [-] Interpolation
VarA T [s] VarB Sve/avg [-] Interpolation
0.00 1.0000 linear
1.60 0.1758 linear
0.05 3.0000 linear
1.70 0.1557 linear
0.15 3.0000 linear
1.80 0.1389 linear
0.20 2.2500 linear
1.90 0.1247 linear
0.25 1.8000 linear
2.00 0.1125 linear
0.30 1.5000 linear
2.10 0.1020 linear
0.35 1.2857 linear
2.20 0.0930 linear
0.40 1.1250 linear
2.30 0.0851 linear
0.45 1.0000 linear
2.40 0.0781 linear
0.50 0.9000 linear
2.50 0.0720 linear
0.55 0.8182 linear
2.60 0.0666 linear
0.60 0.7500 linear
2.70 0.0617 linear
0.65 0.6923 linear
2.80 0.0574 linear
0.70 0.6429 linear
2.90 0.0535 linear
0.75 0.6000 linear
3.00 0.0500 linear
0.80 0.5625 linear
3.10 0.0468 linear
0.85 0.5294 linear
3.20 0.0439 linear
0.90 0.5000 linear
3.30 0.0413 linear
0.95 0.4737 linear
3.40 0.0389 linear
1.00 0.4500 linear
3.50 0.0367 linear
1.10 0.3719 linear
3.60 0.0347 linear
1.20 0.3125 linear
3.70 0.0329 linear
1.30 0.2663 linear
3.80 0.0312 linear
1.40 0.2296 linear
3.90 0.0296 linear
1.50 0.2000 linear 4.00 0.0281 linear
Diagram 3: Table for the vertical response spectrum
The internal variable for evaluating the response spectrum is the angular velocity () [rad/sec]. Therefore if other values are used for the abscissa in the table of the re-
sponse spectrum diagram (Frequency (Hz), Period (T), or if these are given in terms
of logarithm), a respective transformation has to be performed (see 4.1.3).
The related ordinate value of the ground motion amplitude can either be given as dis-
placement (d), velocity (v) or acceleration (a). (The used value has to be referred to
later when defining the earthquake load see 5.1). The ordinate values of the re-sponse spectrum must be given in the internal units [m] and [s]. When other units (or
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factors of the gravity constant (g)) are used for the motion amplitude of the response
spectrum, a respective transformation has to be performed.
For a detailed description for performing such transformations refer to the RM User
Guide 14.3.2.
The dimensionless ordinate value of the ground motion amplitude used in this example
shall be transformed into acceleration. This can either be done by multiplying it with the
design ground acceleration by defining a respective formula (see explanations above
and also 4.1.3) or within the definition of the earthquake load when defining the respec-
tive directions and intensities of the excitation vectors (as will be performed in this ex-
ample (see 5.1)).
For the practical application it can be helpful to prepare an Excel sheet for the genera-
tion of the values of the response spectrum tables and copy/paste them into a Tcl file,
which then can be imported (added) to the project.
Diagram 4: (Partial) Tcl file for horizontal response spectrum
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Diagram 5: Graphical presentation of horizontal and vertical response spectrum
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4.1.3 Assignment of the tables to variables (formulas)
In order to enable the correct evaluation of the data of a table the used value for its ab-
scissa (abscissa of the response spectrum) has to be defined, i.e. the table needs to be
related to the value of its abscissa. This can be done in the variable definition (as shown
in the following) or alternatively within the definition of the earthquake load when se-
lecting the table (see 5.1).
In the following the defined tables of the horizontal and vertical response spectrum are
assigned to formulas to enable the evaluation of their data for the response spectrum
analysis.
Create two varia-
bles of the type
formula for assign-
ing the two tables
of the response
spectrum
Properties Name resp_hor resp_vert
Type formula formula
Variables Expression resp_hor_tab(2*pi/om
ega)
resp_vert_tab(2*pi/om
ega)
Description formula for horizontal
response spectrum
formula for vertical
response spectrum
Top table
resp_hor and resp_vert are the variables (formulas) that describe the response spec-
trum using the values of the tables resp_hor_tab and resp_vert_tab, which have been
defined as functions of the period T. Since the internal variable for the evaluation of
the response spectrum is omega, a respective transformation has to be performed
within the assignment in terms of F = f(T) = f(2*pi/omega):
resp_hor = resp_hor_tab(2*pi/omega) and
resp_vert = resp_vert_tab(2*pi/omega).
Referring to 4.1.2 it would be possible to perform the transformation into acceleration
here by including a factor for the design ground acceleration (ag) and (avg) respectively
in the expressions above (e.g. resp_hor = ag*resp_hor_tab(2*pi/omega)). In this case
the normalized vectors have to be used within the definition of the earthquake load in-
stead of referring to the design ground acceleration there (see 5.1).
4.2 Response Spectrum defined by formulas
An alternative to setting up a table with a series of particular evaluated values is to di-
rectly define the formulas that describe the response spectrum. The definition for the
horizontal response spectrum is illustrated below. The particular parameters are defined
by variables and thus easily can be adjusted by the user. In that way also other types of
the response spectrum can be defined by changing the respective values.
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Name
EC
Denota-
tion
Description Definition (default values)
S S soil factor 1 [-]
xi viscous damping ratio in percent 5 [%]
eta damping correction factor max(SQR(10/(5+xi)),0.55) [-]
TB TB limit of the constant spectral acceleration branch (1) 0.15 [s]
TC TC limit of the constant spectral acceleration branch (2) 0.4 [s]
TD TD value of beginning of constant displacement response range 2.0 [s]
Se Se/ag function for elastic response spectrum in terms of Se/ag (T) S*Se_tab(2*pi/omega) [-]
Se_tab table of resp. spectr. in terms of Se/ag/S (T) VarA VarB
0 1+TabA/TB*(eta*2.5-1)
TB eta*2.5
TC eta*2.5*(TC/TabA)
TD eta*2.5*(TC*TD/TabA^2)
4 eta*2.5*(TC*TD/4^2)
Diagram 6: Definition of the horizontal response spectrum by formulas
The definition of the vertical response spectrum is performed analogously. Both sets of
definitions can be viewed in the corresponding example (see Properties Variables; Group EC8_RESP_form).
The advantage of this method is the general definition that can easily be adapted for all
types of response spectra and the exact evaluation of the particular values of the re-
sponse spectrum (while the accuracy of the values derived from a table relies on the
(linear) interpolation of the defined values).
REMARK:
The formulas of the elastic response spectra and design spectra according to Euro-
code 8 can be generated automatically within RM Bridge by using the respective
options in the menu Extras > Loading and Stages > Response Spectra:
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Diagram 7: Generating the design spectrum in RM Bridge
See online-help (F1) within the input window for more details.
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5 Definition of the Earthquake Load
The earthquake load is defined in Schedule Load Definition Earthquake Load.
5.1 Defining the earthquake events
The upper table contains the basic parameters of the particular earthquake events.
Define a seismic
event for each of
the three directions
Schedule Number 1 2 3
Modal-File eigen.mod eigen.mod eigen.mod
Load Definition Rule CQC CQC CQC
Duration [s] 60 60 60
Earthquake
Load Description
earthquake in
x-dir (longi-
tudinal)
earthquake in
y-dir (verti-
cal)
earthquake in
z-dir (trans-
versal)
Top table
Number of the earthquake event for storage in the database
Name of the modal file containing the results of the eigenvalue analysis (see 3)
Combination rule for superposing the particular contributions of the different natural
modes (see RM User Guide 14.3.1 for the different available rules in RM)
Duration [s] of the seismic event (influencing the results only in combinations with
rules using duration dependent correlation factors (DSC, CQC, CQCX))
The lower table contains the related ground motion parameters and the assigned re-
sponse spectrum.
Define a seismic
event for each of
the three directions
Schedule
Type of re-
sponse spec-
trum graph
a a a
Vec-Vx 1.5
Load Definition Vec-Vy 1.35
Vec-Vz 1.5
Earthquake
Load Damp-Fact 0.05 0.05 0.05
Var-
Name(Graph) resp_hor resp_vert resp_hor
Bottom table
Type of the specified ground motion in the response spectrum (displacement (d), ve-
locity (v) or acceleration (a)) (see 4.1.2)
Vector of the ground motion, multiplied by the respective design ground acceleration
to transform the dimensionless ground motion amplitude of the defined response
spectrum into an acceleration value (see 4.1.2). The design ground accelerations in
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this example are assumed with ag = 1.5 m/s2 and avg = 0.9 ag = 1.35 m/s
2 respec-
tively.
Referring to the comments given in 4.1.3 the normalized components (value of 1) of
the vectors have to be given here, if the design ground acceleration is already in-
cluded in the values of the ground motion amplitude of the response spectrum.
Damping value only used for the calculation of the correlation factors with combina-
tion rules DSC, CQC and CQCX (note that the damping ratio for the response spec-
trum is already incorporated into the table!).
Name of the variable or table representing the respective response spectrum (see
4.1.2). The particular response spectra must be given in terms of the same type.
The selected variable must already be defined as a function of the value of the abscis-
sa of the response spectrum in the variable definition (as done in this example). If a
table is selected it must be defined as a function of the value of the abscissa of the re-
sponse spectrum within the input here.
If the response spectra are given by the formulas as illustrated in 4.2, the respective
input for Var-Name(Graph) would be the variables Se and Sve respectively. The input
is performed in the schedule variant formula in the corresponding example.
So the possible inputs for the horizontal loads would be:
Var-Name(Graph) Remark
resp_hor resp_hor = resp_hor_tab(2*pi/omega) as defined in the Variable menu
resp_hor_tab(2*pi/omega) The reference value of the defined table has to be given here
Se Se = S*Se_tab(2*pi/omega) as defined in the Variable menu
S*Se_tab(2*pi/omega) The reference value of the table is given and the multipl. with soil factor is performed
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6 Evaluation of the Response Spectrum
6.1 Generating a stage for the evaluation of the response spectrum
Create a stage for
the evaluation of
the response spec-
trum
Schedule Name RESP
Description response spectrum evaluation
Stages
Schedule Ac-
tions
Top table
6.2 Evaluating the response spectrum
6.2.1 Initializing envelopes for storing the resuts of the evaluation
First a superposition file (envelope) has to be created for the storage of the results of the
evaluation of the response spectrum for all three directions.
Initialize envelopes
for the storage of
the results of the
response spectrum
evaluation
Schedule Action
LC/Envelope
actions
LC/Envelope
actions
LC/Envelope
actions
Type SupInit SupInit SupInit
Stages Input-1
Input-2
Schedule Actions Input-3 - - -
Output-1 resp-x.sup resp-y.sup resp-z.sup
Bottom table Output-2 - - -
Descrip-
tion
envelope for
storing results
of x-dir (lon-
gitudinal)
envelope for
storing results
of y-dir (ver-
tical)
envelope for
storing results
of z-dir
(transversal)
6.2.2 Evaluating the response spectra
The evaluation of a response spectrum is performed with the dynamic calculation action
RespS.
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Evaluate the re-
sponse spectra for
the three directions
Schedule Action
Calcula-tion (Dynam-
ic)
Calcula-tion (Dynam-
ic)
Calcula-tion (Dynam-
ic)
Type RespS RespS RespS
Stages Input-1 1 2 3
Input-2 ALL ALL ALL
Schedule Actions Input-3
Output-1 resp-x.sup resp-y.sup resp-z.sup
Bottom table Output-2 * * *
Descrip-
tion
evaluate
response
spectrum in
x-dir (longi-
tudinal)
evaluate
response
spectrum in
x-dir (verti-
cal)
evaluate
response
spectrum in
x-dir (trans-
versal)
Number of the seismic event for evaluating the structural response referring to the
given number in the earthquake load definition
Selection of elements to be considered for the calculation (ALL or ACTIVE)
Name of the envelope where the results shall be stored
Name of the output list for the data of the response calculation (default resp0001.lst
for seismic event 1) NOTE: In this example constant damping is assumed (no input in optional Input-3).
For how to apply modal damping with sinlge response spectrum, tables of response
spectra and weighted element damping see separately available example.
The results of this action are stored in the given superposition file and are extreme
forces and displacements. As the superposition rules are statistic, only leading values
may be obtained. With the use of a special algorithm, called TDV-Superposition method (set in the Recalc-option), it is possible to obtain affiliated results in the superposition file.
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7 Superposition of the Seismic Loads
7.1 Generating a stage for the superposition of the seismic loads
Create a stage for
the superposition
of the seismic
loads
Schedule Name SUP
Description superposition of the seismic loads
Stages
Schedule Ac-
tions
Top table
7.2 Superposing the seismic loads
The particular results of the three basic directions are now combined according to the
30%-rule. An envelope is initialized for the maximum results of each direction (resp-
max-x.sup, resp-max-y.sup, resp-max-z.sup), where the particular results of the respec-
tive directions are added up with 100% (factor 1.0) and the other two with 30% (factor
0.3). Then one final envelope (resp-max.sup) is initialized for evaluating the maximum
results out of these three envelopes.
Initialize envelopes
for the superposi-
tion of the maxi-
mum results for
each direction and
superpose results
according to 30%-
rule
Schedule Action
LC/Envelope
actions
LC/Envelope
actions
LC/Envelope
actions
Type SupInit SupAddSup SupAddSup
Stages Input-1 resp-x.sup resp-max-
x.sup
resp-max-
x.sup
Input-2 1.0 resp-y.sup resp-z.sup
Schedule Actions Input-3 - 0.3,0.3 0.3,0.3
Output-1 resp-max-
x.sup
Bottom table Output-2 - - -
Descrip-
tion
Initialize
envelope for
max results in
x-dir and add
up 100% of
x-direction
Add up 30%
of y-direction
Add up 30%
of z-direction
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Action LC/Envelope actions LC/Envelope actions LC/Envelope actions Type SupInit SupAddSup SupAddSup
Input-1 resp-y.sup resp-max-y.sup resp-max-y.sup
Input-2 1.0 resp-x.sup resp-z.sup
Input-3 - 0.3,0.3 0.3,0.3
Output-1 resp-max-y.sup
Output-2 - - -
Descrip-
tion
Initialize envelope for
max results in y-dir and
add up 100% of y-
direction
Add up 30% of x-
direction
Add up 30% of z-
direction
Action LC/Envelope actions LC/Envelope actions LC/Envelope actions Type SupInit SupAddSup SupAddSup
Input-1 resp-z.sup resp-max-z.sup resp-max-z.sup
Input-2 1.0 resp-x.sup resp-y.sup
Input-3 - 0.3,0.3 0.3,0.3
Output-1 resp-max-z.sup
Output-2 - - -
Descrip-
tion
Initialize envelope for
max results in z-dir and
add up 100% of z-
direction
Add up 30% of x-
direction
Add up 30% of y-
direction
Initialize envelope
for the maximum
results of the three
envelopes and
superpose
Schedule Action
LC/Envelope
actions
LC/Envelope
actions
LC/Envelope
actions
Type SupInit SupOrSup SupOrSup
Stages Input-1 resp-max-
x.sup resp-max.sup resp-max.sup
Input-2 resp-max-
y.sup
resp-max-
z.sup
Schedule Actions Input-3 -
Output-1 resp-max.sup
Bottom table Output-2 - - -
Descrip-
tion
Initialize
envelope for
max results
and add up
first envelope
Superpose
second enve-
lope with OR
Superpose
third enve-
lope with OR
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8 Result Presentation
8.1 Plotting the response spectra data
The tables and diagrams of the response spectra defined in Properties Variables can be plotted either directly in this menu by clicking on the blue info button in the top table
or within the schedule actions using the List/plot action PlVar and selecting the respec-
tive table.
8.2 Plotting of eigenmodes
The load cases deriving from the eigenvalue calculation (e.g. eigen#1, eigen#2, eigen#3
etc., see 3.2) containing the normalized eigenvectors as displacements can be used for
graphic presentation.
The corresponding RM training to this document contains a sample using Plot Contain-
er with variables and special settings within the DoPlot action to allow for plotting mul-
tiple load cases (i.e. all eigenmodes) with just one command. Also special internal vari-
ables can be used for printing the values of the particular eigenmodes.
A brief principle description shall be given in the following:
8.2.1 Definition of the DoPlot action
Diagram 8: Definition of the DoPlot action
The names of the prepared plot container and plot file have to be given as usual.
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The plot file contains a variable EigLC for the load case to be plotted (see 8.2.2). The
load cases to be plotted are eigen#1, eigen#2 up to eigen#30. Therefore the variable is
defined to be the load case eigen#, i.e. the consecutive number is again de-
fined by another variable (given in angle brackets), which is defined in a second in-
put: num = {1 30} to define the range of values that it should take.
Note that also the output file name needs to be changed for each generated plot file,
therefore also here a variable is used (e.g. names of the load cases, i.e. eigen#1, eig-
en#2 etc.).
See also the input help when pressing F1 within the input window of the action.
8.2.2 Definition of the Plot File in the Plot Container
A plot file can be set up from the scratch or easier can automatically be generated by
making use of the Macro function (option Eigenform) and be adapted afterwards.
The specific input shall be explained below:
Diagram 9: Definition of the load case to be plotted
The load case is not given directly, but referred to by the variable EigLC.
To print the particular values of the eigenmodes a text field can be added and defined by
variables:
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Diagram 10: Definition of the text field for plotting the values of the eigenmodes
E.g. to plot the values of the angular velocity or frequency of the particular eigenmode
load cases, the internal variables _OMEGA and _FREQU can be used.
In order to retrieve the values of each load case it has to be referred to by its variable
name using the following syntax:
or
.
Also press F1 for a description and a list of all available internal variables within the
general window of the plot definition.
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9 Time History Analysis
The following input procedures can be followed when changing to the schedule variant
time_history in the corresponding example.
9.1 Preparation of the tables and variables
The data sets describing the time history events are set up in terms of tables and varia-
bles as described in chapter 4 of this document.
Three different real events with a probability of exceedance of 2% in 50 years are exam-
ined in this example. The respective tables are stored in the group Time_Hist with the
names T1_acc, T2_acc and T3_acc.
The amplitude (ordinate) is given in terms of acceleration values (m/s2). RM Bridge
provides an option for automatically converting a table using acceleration or velocity
values into a table using displacements values (see diagram below).
Diagram 11: Conversion of tables from acceleration to displacements
Using this option the three tables are converted into (additional) tables with the names
T1_dis, T2_dis and T3_dis (the names have to be given within the input window).
In the same way as shown in chapter 4.1.3 the tables have to be assigned to respective
variables in dependency of the time (t).
In regard to this one has to take care that the internal variable for the time (t) possibly is
not equal to zero at the time the time history calculation is performed (e.g. in case creep
calculation has been performed in advance). Therefore a new variable for the time his-
tory calculation tint is set up in terms of
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tint = t tstart,
where
t is the internal variable for the time (global time axis in the schedule) and
tstart the internal variable for the time at the point of the global time axis where the ac-
tion is started.
That means in that way at the begin of the action of the time history calculation tint = 0
(since t = tstart) and at the end of the time interval t of the action tint = t (since t = tstart + t).
Diagram 12: Definition of the variable tint
The definitions of the tables and variables are summarized below:
Name Type Expression Description
T1_acc Table - time history set1 in terms of acceleration
T2_acc Table - time history set2 in terms of acceleration
T3_acc Table - time history set3 in terms of acceleration
T1_dis Table - time history set1 in terms of displacements
T2_dis Table - time history set2 in terms of displacements
T3_dis Table - time history set3 in terms of displacements
var_T1_dis Variable T1_dis(tint) variable for the time history set T1
var_T2_dis Variable T2_dis(tint) variable for the time history set T2
var_T3_dis Variable T3_dis(tint) variable for the time history set T3
tint Variable t-tstart time to be used for time history analysis
t
tint = t = 40 tint = 0
tint = t tstart
t = tstart + t = 120 t = tstart = 80
t = 40
80
t2 = 30 t1 = 50
STG 1 STG 2 STG 3 STG 4
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9.2 Definition of the load cases and load sets
A load case to be used for the time history analysis needs to contain the masses of the
structure and the displacements of the time history event described by the variable. In
this example for each of the three particular events two load cases are set up in regard to
the two horizontal directions (x and z); each load case consists of one load set for the
self-weight, one for the superimposed dead loads and one for the displacements in the
particular direction.
9.2.1 Definition of the load sets for the masses
The definition of the load sets for the masses of the self-weight and superimposed dead
loads are defined as shown in chapter 2.1.
The load sets for the masses just needs to be defined once and can be used for all the six
load cases.
9.2.2 Definition of the load sets for the displacements
For each direction of each time history event one load set defining the element end dis-
placements of the soil springs in the particular direction is generated as shown below.
Create new load
sets for the time
history events
Schedule Name LS-T1_dis_x LS-T1_dis_z
Description displacements
event T1 for x-dir
displacements
event T1 for z-dir
Load Definition
Load Set Defi-
nition
Top table
Name LS-T2_dis_x LS-T2_dis_z LS-T3_dis_x LS-T3_dis_z
Description displacements
event T2 for x-dir
displacements
event T2 for z-dir
displacements
event T3 for x-dir
displacements
event T3 for z-dir
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Define load sets
for the for the time
history events for
each direction
Schedule Name LS-T1_dis_x LS-T1_dis_z
Loading Actions on Element End
Actions on Element End
Load Definition Type Element end
displacements
Element end
displacements
From 1100 1100
Load Set Defi-
nition To 1400 1400
Step 100 100
Bottom table Vx -1 0
Vy 0 0
Vz 0 -1
Rx 0 0
Ry 0 0
Rz 0 0
Global / Local Global Global Begin / End End End
Name LS-T2_dis_x LS-T2_dis_z LS-T3_dis_x LS-T3_dis_z
Loading Actions on Element End
Actions on Element End
Actions on Element End
Actions on Element End
Type Element end
displacements
Element end
displacements
Element end
displacements
Element end
displacements
From 1100 1100 1100 1100
To 1400 1400 1400 1400
Step 100 100 100 100
Vx -1 0 -1 0
Vy 0 0 0 0
Vz 0 -1 0 -1
Rx 0 0 0 0
Ry 0 0 0 0
Rz 0 0 0 0
Global / Local Global Global Global Global Begin / End End End End End
9.2.3 Definition of the load cases for the time history calculation
The prepared load sets are now assigned to the respective load cases to be used for the
time history calculation; each load case consists of the load set for the self-weight, the
load set of the SDL and the respective load set of the displacements. As can be seen in
the input below the load set for the displacements gets its constant factor set to zero, but
the variable factor is defined by the variable describing the time dependent loading of
the time history event which has to be entered in the respective input field.
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Create new load
cases for the time
history calculation
for each direction
Schedule Name LC-T1-x LC-T1-z
Description load case for event
T1 for x-dir
load case for event
T1 for z-dir
Load Definition
Load Case Def-
inition
Top table
Name LC-T2-x LC-T2-z LC-T3-x LC-T3-z
Description load case for event
T2 for x-dir
load case for event
T2 for z-dir
load case for event
T3 for x-dir
load case for event
T3 for z-dir
Assign load sets to
the load cases
Schedule Name LC-T1-x
Loading Load set
input
Load set input
Load set input
Load Definition Load Set LS-SW LS-SDL LS-
T1_dis_x
Const-Fac 1 1 0
Load Case Def-
inition Var-Fac 0 0 var_T1_dis
Bottom table
Name LC-T1-z LC-T2-x
Loading Load set
input
Load set input
Load set input
Load set input
Load set input
Load set input
Load Set LS-SW LS-SDL LS-
T1_dis_z LS-SW LS-SDL
LS-
T2_dis_x
Const-Fac 1 1 0 1 1 0
Var-Fac 0 0 var_T1_dis 0 0 var_T2_dis
Name LC-T2-z LC-T3-x
Loading Load set
input
Load set input
Load set input
Load set input
Load set input
Load set input
Load Set LS-SW LS-SDL LS-
T2_dis_z LS-SW LS-SDL
LS-
T3_dis_x
Const-Fac 1 1 0 1 1 0
Var-Fac 0 0 var_T2_dis 0 0 var_T3_dis
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Name LC-T3-z
Loading Load set
input
Load set input
Load set input
Load Set LS-SW LS-SDL LS-
T3_dis_z
Const-Fac 1 1 0
Var-Fac 0 0 var_T3_dis
9.3 Calculation of the Time History Events
The defined load cases describing the time dependent loadings are now evaluated using
the schedule action Tint. The maximum and minimum results are stored in envelopes,
which have to be initialized (SupInit) in advance.
Perform a time
history calculation
for the respective
time dependent
load cases
Schedule Action Calculation
(Dynamic)
Calculation (Dynamic)
Type TInt TInt
Stages Input-1
(Load Case) LC-T1-x LC-T1-z
Input-2
(Delta-t) 40.94 40.94
Schedule Actions Input-3
(RM-Set) T-x-Mz T-z-My
Output-1 T1-x.sup T1-z.sup
Bottom table Output-2 tint_T1-x-Mz tint_T1-z-My
Description
Perform time
history calcula-
tion for load
case LC-T1-x
Perform time
history calcula-
tion for load
case LC-T1-z
Action Calculation
(Dynamic)
Calculation (Dynamic)
Calculation (Dynamic)
Calculation (Dynamic)
Type TInt TInt TInt TInt
Input-1
(Load Case) LC-T2-x LC-T2-z LC-T3-x LC-T3-z
Input-2
(Delta-t) 40.94 40.94 40.94 40.94
Input-3
(RM-Set) T-x-Mz T-z-My T-x-Mz T-z-My
Output-1 T2-x.sup T2-z.sup T3-x.sup T3-z.sup
Output-2 tint_T2-x-Mz tint_T2-z-My tint_T3-x-Mz tint_T3-z-My
Description
Perform time
history calcula-
tion for load
case LC-T2-x
Perform time
history calcula-
tion for load
case LC-T2-z
Perform time
history calcula-
tion for load
case LC-T3-x
Perform time
history calcula-
tion for load
case LC-T3-z
The load case to be evaluated (Input-1) and the time period to be considered (Input-2)
have to be given. The results are stored in the given envelope (Output-1). Optionally an
RM-Set of the type Time integration (TINT) can be given (Input-3), within which
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specific result components at particular element positions for graphical output can be
defined. These results are as well printed in a list file (Output-2).
The particular envelopes of the evaluated load cases are then superposed exclusively
(SupOr) in one final envelope T-final.sup which thus contains the maximum and mini-
mum results deriving from the three events.
9.4 Definition of Damping
Differently from the response spectrum analysis where the damping is incorporated by a
constant value, in the time history analysis the Rayleigh damping is dependent on the
frequency. Design codes usually define a damping as a percentage of the critical damp-
ing not dependent on frequency and the Raleigh coefficients are usually not known.
Therefore RM Bridge provides a function to approximate this presumption with Ray-
leigh damping as accurate as possible by recalculating the coefficients from damping
ratios given for two relevant frequencies. In order to do that the user has to determine
the natural frequencies and to assign the given damping ratio to the two most important
ones. The assignment of these values is done in the Recalc pad in the submenu Dy-namic. When clicking on one of the buttons next to the Rayleigh coefficients Alpha or Beta a menu opens where two pairs of values (w1, xi1 and w2, xi2) describing the first
and second frequency value (w1, w2) with the corresponding predefined damping ratio
(xi1, xi2) have to be given. After confirming with Ok the Rayleigh coefficients Alpha
and Beta are evaluated (note that the input of the factors w1, w2, xi, xi2 is not stored).
In this example the natural modes 1 and 6 have been assumed to be the two relevant
frequencies derived from the list eigen.lst:
MASS PARTICIPATION FACTORS [%]
MODE phi*M*phi X Y Z SUM-X SUM-Y SUM-Z HERTZ
-------------------------------------------------------------------------
1 0.4550E+04 89.10 0.00 3.21 89.10 0.00 3.21 0.849
2 0.2055E+04 2.40 0.00 72.05 91.50 0.00 75.26 1.581
3 0.1016E+04 0.01 5.21 0.04 91.50 5.21 75.30 2.929
4 0.2175E+04 1.14 0.01 0.05 92.65 5.22 75.36 3.568
5 0.1288E+04 0.24 0.01 0.01 92.89 5.23 75.36 5.112
6 0.1349E+04 0.00 58.91 0.01 92.89 64.14 75.37 5.921
7 0.2050E+04 0.44 0.02 7.41 93.33 64.16 82.79 6.683
8 0.1079E+04 0.10 0.00 0.05 93.43 64.16 82.83 9.120
9 0.6110E+04 0.11 0.04 3.61 93.54 64.20 86.44 9.449
10 0.1840E+04 0.17 0.00 0.01 93.72 64.21 86.46 10.958
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Therefore the factors for evaluating the coefficients of the Rayleigh damping for a pre-
defined damping of 5% are approximated as follows:
Relevant Modes Frequency f [Hz] w (in terms of angular velocity [rad/s]) xi (damping)
1 0.85 w1 = 5 xi1 = 0.05
6 5.9 w2 = 35 xi2 = 0.05
Thus the evaluated Rayleigh coefficients used for the calculation are:
Alpha = 0.4375
Beta = 0.0025
1 General1.1 Scope1.2 Generating a new construction schedule
2 Definition of Masses2.1 Definition of load sets for self weight and SDL2.1.1 Generating the load sets for SW and SDL2.1.2 Defining the load set for SW2.1.3 Defining the load set for SDL
2.2 Definition of load case for masses2.2.1 Generating a load case for the masses2.2.2 Assigning the load sets to the load case
3 Calculation of Eigenvalues3.1 Generating a stage for the calculation of the eigenvalues3.2 Calculating the eigenvalues
4 Preparation of Response Spectrum4.1 Defining a response spectrum diagram4.1.1 Generating tables for horizontal and vertical response spectrum4.1.2 Defining the tables for horizontal and vertical response spectrum4.1.3 Assignment of the tables to variables (formulas)
4.2 Response Spectrum defined by formulas
5 Definition of the Earthquake Load5.1 Defining the earthquake events
6 Evaluation of the Response Spectrum6.1 Generating a stage for the evaluation of the response spectrum6.2 Evaluating the response spectrum6.2.1 Initializing envelopes for storing the resuts of the evaluation6.2.2 Evaluating the response spectra
7 Superposition of the Seismic Loads7.1 Generating a stage for the superposition of the seismic loads7.2 Superposing the seismic loads
8 Result Presentation8.1 Plotting the response spectra data8.2 Plotting of eigenmodes8.2.1 Definition of the DoPlot action8.2.2 Definition of the Plot File in the Plot Container
9 Time History Analysis9.1 Preparation of the tables and variables9.2 Definition of the load cases and load sets9.2.1 Definition of the load sets for the masses9.2.2 Definition of the load sets for the displacements9.2.3 Definition of the load cases for the time history calculation
9.3 Calculation of the Time History Events9.4 Definition of Damping
Recommended