advanced dynamics module (astar) - htw dresdenfem/docs/cosmosm/astar.pdf · cosmosm advanced...

Contents

1 Introduction

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-1Analysis Capabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-2Main Analysis Features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-2Load Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-4Base Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-5Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-5Response Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-6Stress Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-6

2 Theoretical Background

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-1Normal Mode Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-1Damping Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-3

Rayleigh Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-3Modal Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-3Concentrated Dampers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-4Composite Modal Damping . . . . . . . . . . . . . . . . . . . . . . . . . . 2-4

Solution Accuracy Considering Mode Truncation . . . . . . . . . . . 2-6Force Excitation Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-6Base Excitation Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-7

COSMOSM Advanced Modules i

Contents

ii

Modal Acceleration Method (MAM) in Time History . . . . . . 2-7Missing Mass Correction Technique in Response Spectrum . . 2-8

Excitation Due to Base Motion . . . . . . . . . . . . . . . . . . . . . . . . . 2-8Uniform Translational Base Motion . . . . . . . . . . . . . . . . . . . . 2-8Multi-Base Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-11

Time History Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-13Solution Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-13Loading Options Available . . . . . . . . . . . . . . . . . . . . . . . . . . 2-14Concentrated Dampers and Gap Elements . . . . . . . . . . . . . . 2-14

Response Spectrum Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 2-14Definition of Response Spectrum . . . . . . . . . . . . . . . . . . . . . 2-15Modal Maximum Response . . . . . . . . . . . . . . . . . . . . . . . . . 2-15Structure Maximum Response . . . . . . . . . . . . . . . . . . . . . . . 2-16Multiple Response Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . 2-18

Response Spectra Generation . . . . . . . . . . . . . . . . . . . . . . . . . 2-18Random Vibration Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 2-20

Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-21Analysis Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-22Methods of Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-25Partial (Spatial) Correlation . . . . . . . . . . . . . . . . . . . . . . . . . 2-27

Steady State Harmonic Analysis . . . . . . . . . . . . . . . . . . . . . . . 2-29Analysis Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-29Direct Spectrum Generation . . . . . . . . . . . . . . . . . . . . . . . . . 2-32Time History Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-32

3 Description of Elements

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-1

4 Brief Description of Commands

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-1

COSMOSM Advanced Modules

Part 1 ASTAR Advanced Dynamics Analysis

Main POST_DYN Menu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-1PD DAMP/GAP Submenu . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-2PD CURVES Submenu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-3PD BASE EXCITATION Submenu . . . . . . . . . . . . . . . . . . . . 4-4PD OUTPUT Submenu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-4

Postprocessing of Dynamic Results . . . . . . . . . . . . . . . . . . . . . . 4-5XY-Plot of Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-6Plot of Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-7XY-Plot of Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-7Plot of Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-9Other Postprocessing Options . . . . . . . . . . . . . . . . . . . . . . . . 4-9

Post Dynamic Input Procedure . . . . . . . . . . . . . . . . . . . . . . . . 4-10Typical Input Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-11

5 Detailed Description of Examples

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-1Dynamic Analysis of a Cantilever Beam . . . . . . . . . . . . . . . . . . 5-1

Analysis Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-1Statement of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-1Frequency Analysis of the Problem . . . . . . . . . . . . . . . . . . . . 5-3Modal Time History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-4Random Vibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-14

Dynamic Analysis of a Culvert . . . . . . . . . . . . . . . . . . . . . . . . 5-22Analysis Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-22Given . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-22GEOSTAR Commands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-23Modal Time History Analysis . . . . . . . . . . . . . . . . . . . . . . . 5-24Response Spectrum Generation . . . . . . . . . . . . . . . . . . . . . . 5-33Harmonic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-36Random Vibration Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 5-45Response Spectrum Analysis . . . . . . . . . . . . . . . . . . . . . . . . 5-54

COSMOSM Advanced Modules iii

Contents

iv

Multi-Base Motion Application with Composite Material Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-60

Dynamic Analysis of a Plate . . . . . . . . . . . . . . . . . . . . . . . . . . 5-65Analysis Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-65Statement of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-65GEOSTAR Inputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-65

6 Verification Problems

Index


First EditionCOSMOSM 2.9August 2004

Copyright

Structural Research and Analysis Corp. is a Dassault Systemes S.A. (Nasdaq: DASTY) company.This software product is copyrighted and all rights are reserved by Structural Research and Analysis Corporation. (SRAC) Copyright 1985 - 2004 Structural Research and Analysis Corporation. All Rights Reserved.The distribution and sale of this product is intended for the use of the original purchaser only and for use only on the computer system specified. The software product may be used only under the provisions of the license agreement that accompanies the product package.COSMOSM manuals may not be copied, photocopied, reproduced, translated or reduced to any elec-tronic medium or machine readable form in whole or part wit prior written consent from Structural Research and Analysis Corporation. Structural Research and Analysis Corporation makes no warranty that COSMOSM is free from errors or defects and assumes no liability for the program. Structural Research and Analysis Corporation disclaims any express warranty or fitness for any intended use or purpose. You are legally accountable for any violation of the License Agreement or of copyright or trademark. You have no rights to alter the software or printed materials.The COSMOSM program is constantly being developed, modified and checked and any known errors should be reported to Structural Research and Analysis Corporation.

Disclaimer

The authors have taken due care in preparing this manual and the examples presented herein. In no event shall SRAC assume any liability or responsibility to any person or company for direct or indirect dam-age resulting from the use of the information contained herein or any discrepancies between this infor-mation and the actual operation of the software.

Licenses & Trademarks

Use by Structural Research and Analysis Corporation of ANSYS Input Commands and Command Structure herein is licensed under agreement with Swanson Analysis Systems, Inc. All rights reserved. COSMOSM and COSMOS are registered trademarks of Structural Research and Analysis Corporation. All other COSMOSM module names are trademarks of Structural Research and Analysis Corporation.ABAQUS is the registered trademark of Hibbitt, Karlsson & Sorensen, Inc. ANSYS is a registered trademark of Swanson Analysis Systems. AutoCAD is registered in the U.S. Patent and Trademark Office by Autodesk, Inc. DXF and AutoSolid are the registered trademarks of Autodesk, Inc. CenBASE/Mil5 is the registered trademark of Information Indexing, Inc. DECStation is the registered trademark of Digital Equipment Corporation. EPSON is the registered trademark of Epson Computers. HP is the reg-istered trademark of Hewlett-Packard. IBM is the registered trademark of International Business Machines Corporation. MSC/NASTRAN is the registered trademark of MacNeal-Schwendler Corp. PATRAN is the registered trademark of PDA Engineering. PostScript, Acrobat, and Acrobat Reader are registered trademarks of Adobe Systems, Inc. SINDA/G is a registered trademark of Network Analysis Associates, Inc. Sun is the registered trademark of Sun Microsystems, Inc. SGI is the trademark of Sili-con Graphics, Inc. SoliWorks is a registered trademark of SolidWorks Corporation. Helix Design Sys-tem is a trademark of MICROCADAM Inc. SDRC I-DEAS is a trademark of Structural Dynamics Research Corporation. MicroStation Modeler is a registered trademark of Bentley Systems, Incorpo-rated. Solid/Edge is a trademark of Intergraph Corporation. Eureka is a trademark of Cad.Lab. All other trade names mentioned are trademarks or registered trademarks of their respective owners.

ii


1 Introduction

Introduction

The Advanced Dynamic module, ASTAR or POST DYNAMIC, has three principal functions. The first is to perform linear dynamic analysis of systems subject to dif-ferent categories of forcing functions; the second is to carry out stress calculations subsequent to a dynamic analysis; and finally to accommodate plot files required for graphic evaluations of the system response at specific nodes and/or solution steps.

The various dynamic capabilities of the program are based on the normal mode method. It is therefore essential that the frequency and mode shape calculations are done prior to the application of this module (except for the curve transformation problems) for the solution of the desired dynamic response problem.

Note that the GEOSTAR program controls the complete operations of the COSMOSM package and all modules are accessed from there. In the Post Dynamic analysis, after running for frequencies, the POST_DYNAMIC submenu can be accessed from the main ANALYSIS menu in order to set up the input required for any of the several different analysis options listed in the next section.

COSMOSM Advanced Modules 1-1

Chapter 1 Introduction

1-2

Analysis Capabilities

The Advanced Dynamic module is used for the solution of dynamic response prob-lems listed below:

Modal Time History Response Spectra Response Spectra Generation Random Vibration Steady State Harmonic Curve TransformationAs stated earlier, these analyses options are based on the normal mode method for which modes and frequencies must be determined in advance (except for curve transformation problems). This can be accomplished through GEOSTAR in which the model is initially created and then analyzed for its modes and frequencies based on the lumped or consistent mass options using DSTAR. Each of these analysis options can then be performed by considering any combination of the four excita-tion types noted below:

Concentrated Forces specified in any coordinate system Pressure Loads specified in any coordinate system Uniform Base Motion defined in the global or local Cartesian system Multi-Base Motion defined in the global or local Cartesian systemA brief theoretical description of the different analysis types is provided in Chapter 2.

Main Analysis Features

1. Modal Time History AnalysisResponse of structures excited by time varying forces or base excitations may be evaluated using this option. The uncoupled modal equations of motion are solved by using the step-by-step integration technique of the Wilson-Theta or the Newmark Method.



All structural elements available in the COSMOSM element library are sup-ported. Linear gap elements may also be created. Concentrated dampers at specified nodes are available for Time History analysis in addition to Modal damping, Rayleigh and composite dampings options. The evaluated response consists of displacements, velocities and accelerations as well as stresses (inter-nal forces for 1D elements). Options available for postprocessing are discussed in Chapter 4 under Postprocessing of Dynamic Results. A detailed description of solution techniques is available in Chapter 2.

2. Response Spectra AnalysisThe maximum response of a structure can be evaluated for a specified base spectrum. Several options are available for mode combination techniques including the Absolute Sum, Square Root Sum of Squares (SRSS), Complete Quadratic Combination (CQC), and the Naval Research Laboratory (NRL) methods. The evaluated response consists of maximum displacements, veloci-ties, and accelerations as well as stresses at each node. Modal, Rayleigh and composite material damping options are available for this type of analysis. Options available for postprocessing are discussed in Chapter 4 under postpro-cessing of dynamic results. A detailed description of solution techniques is available in Chapter 2.

3. Response Spectra GenerationThe response spectrum curves at a specified node are evaluated by using the already calculated modal time response of the structure at that node. The evalu-ated spectra represents the maximum response amplitude of a single degree of freedom system at various oscillating frequencies for a specified damping ratio. The response spectra can be generated for specified nodal displacements, veloc-ities and/or accelerations. Options available for postprocessing are discussed in Chapter 4 under Postprocessing of Dynamic Results. A detailed description of solution techniques is available in Chapter 2.

4. Random Vibration Analysis The response of structures to random excitations is evaluated using this option. The input may be in the form of Power Spectra Densities (PSD) of applied loads or base excitations versus frequency. The output includes response in the form of PSD of displacements, velocities, accelerations and stresses. In addition, the RMS of response (which for stationary, Gaussian excitation with mean value of zero, is equivalent to one standard deviation of the response, i.e., the actual response can be assumed to be at or below this one standard deviation value about 68% of the time) is evaluated in order to completely determine the statistical behavior of the response. Modal, Rayleigh and composite material



1-4

damping options are available for this type of analysis. Options available for postprocessing are discussed in Chapter 4 under Postprocessing of Dynamic Results. A detailed description of solution techniques is available in Chapter 2.

5. Steady State Harmonic AnalysisIn this case, the response of structures to harmonic forces or base excitations for a range of frequencies is evaluated, i.e., the amplitude of the resulting system response as well as the phase angle of the response (relative to the applied force) are calculated. The system response is evaluated for displacements, velocities, accelerations and stresses. Modal, Rayleigh and composite material damping options are available for this type of analysis. Options available for postprocess-ing are discussed in Chapter 4 under Postprocessing of Dynamic Results. A detailed description of solution techniques is available in Chapter 2.

6. Curve TransformationThe curve transformation option is used to perform the following analyses:a. Direct Spectrum Generationb. Time History GenerationFrequency analysis is not required prior to performing these analyses. The objective of Direct Spectrum Generation is to evaluate spectra curves for a given time history curve, whereas, the Time History Generation, evaluates time history records from the user defined acceleration response spectra curve.

Options available for postprocessing are discussed in the on-line Help for the PD_ATYPE (Analysis > POST_DYNAMIC > Sel PD Analysis Type) command.

Load Excitation

Load excitations may be applied as concentrated loads at specified nodes or as dis-tributed pressure applied to specified element faces. Depending on the type of anal-ysis, loads may have arbitrary patterns in the time or frequency domain. You may define up to 100 time or frequency curves and associate them with loads as desired. Force and pressure loading is applicable to all the above mentioned types of analy-ses except the Response Spectra analysis and curve transformation problems. Forces can be defined in global or local coordinates.



Base Excitation

The excitation to the structure may be specified in terms of base motion (displace-ment, velocity or acceleration) instead of or in addition to the applied mechanical loads. In this case the previously constrained nodes of the structure (during fre-quency analysis) are assumed to have motions which follow certain patterns described in the time or frequency domains. You may define up to 100 patterns by creating time or frequency curves and associating them with all the constrained nodes in a certain direction to define uniform base excitation, or associating them with a subset of the nodes constrained in a given direction to define multi-base motion. Base motions are supported in all types of analyses in ASTAR. Both uni-form and multi-base motions can be defined in local and global Cartesian coordi-nates. For a theoretical review of the subject, please refer to Chapter 2.

Damping

There are several options for consideration of structural damping in linear dynamic analysis using the ASTAR module.

1. Rayleigh damping, where the damping matrix is assumed to be proportional to the modal mass and the modal stiffness matrices (applicable to all types of anal-ysis excluding curve transformation problems).

2. Modal damping, where you directly define the modal damping ratio for each mode (applicable to all types of analysis).

3. Concentrated dampers, where you specify damping between two nodes of the structure (applicable only to Time History).

4. Composite modal damping, where you specify damping as a material property before performing frequency analysis, and the program evaluates the equivalent modal damping ratios for each mode (applicable to all types of analysis excluding curve transformation problems).

For a detailed review of this subject, please refer to the appropriate section of Chap-ter 2.



1-6

Response Calculations

For all available analysis options, the response of structures at certain nodes may be calculated for the displacement, velocity and acceleration versus time (in Modal Time History) or versus frequency (in PSD of Random Vibration, Harmonic Analysis, and Response Spectra Generation) or simply as the maximum value (in Response Spectra Analysis) or RMS (in Random Vibration). The user may request to have these results written in the output file or have them written in plot files for graphical display (see the postprocessing of dynamic results section of Chapter 4).

Stress Calculation

For all the analysis options except Response Spectra Generation and Curve Trans-formation problems, the stresses can be calculated for all element groups supported by DSTAR. Both nodal and element stresses are evaluated. Stresses versus time or frequency (similar to the response) can be requested and if so will be available in both the output file and plot file.


2 Theoretical Background

Introduction

In this chapter, a brief theoretical discussion on the various linear dynamic response analysis options supported by the Post Dynamic module (ASTAR) is presented.

In the nonlinear module (NSTAR), the equations of motion for both linear and non-linear systems are solved by direct integration. However, problems with a large number of equations or degrees of freedom (DOF) often require less computation time if the dynamic behavior of the linear structure can be approximated with suffi-cient accuracy by only the first few modes (nf), where nf

Chapter 2 Theoretical Background

2-2

(2-1)

where:

[M] = mass matrix[C] = damping matrix[K] = stiffness matrix{f(t)} = time varying load vector

and are the displacement, velocity, and acceleration vectors, respec-tively.

For linear dynamic problems, the system of equations of motion (Eq. 2-1) can be decoupled into nf single degree of freedom equations in terms of the modal dis-placement vector {x}, where:

(2-2)

and [] is the matrix of the lowest nf eigenvectors obtained from the solution of:

(2-3)

Substituting for {u} from (Eq. 2-2) into (Eq. 2-1) and premultiplying it by []T, will yield:

(2-4)

With the mode shapes satisfying the orthogonality conditions, Eq. (2-4) becomes:

(2-5)

Eq. (2-5) represents nf uncoupled single degree of freedom (second-order differ-ential) equations as shown below:

(2-6)



These equations can be evaluated using step-by-step integration or other tech-niques, and the displacements {u} and other system responses can then be deter-mined by performing the transformation shown in Eq. (2-2).

Damping Effects

The damping matrix [C] is assumed to satisfy the orthogonality conditions. It should be noted that in the majority of cases, (a) the exact damping matrix is unknown, and (b) the effect of any non-orthogonality is usually small. In the Post Dynamic module, the following damping options are available.

Rayleigh Damping

Rayleigh damping is of the form:

[C] = [M] + [K] (2-7)

This form of [C] is orthogonal with respect to the system eigenvectors, and the modal damping coefficient for the ith mode Ci may be calculated:

Ci = 2 i i = + i2 (2-8)

and in terms of the modal critical damping ratio

i = / (2 i) + i / 2 (2-9)

where and are the Rayleigh damping coefficients.

Modal Damping

Modal damping is defined as a fraction of critical damping

i = Ci / Cc (2-10)



2-4

Concentrated Dampers

Concentrated dampers can be defined between any two nodes or any one node and the ground for only the modal time history analysis option. The damping coeffi-cients are defined in terms of their components in the global X, Y, and Z-directions as shown in Figure 2-1.

Figure 2-1. Concentrated Dampers

Composite Modal Damping

Composite modal damping allows for the definition of the damping coefficient as a material property. Thus, different element groups representing different materials can be assigned different damping coefficients during a post dynamic analysis. This option provided on the basis of NRC recommendation, is defined below in the form of equivalent modal damping ratios:

(2-11)

where:

= equivalent modal damping ratio of the jth mode

= jth normalized modal eigenvector

= modified structural mass matrix constructed from element matrices formed by the product of the damping ratio for the element and its mass matrix

DY

Node_1

Node_2

Y

X

Z



Gap-Friction Element

The Gap-Friction element is defined by two nodes representing the distance between any two points in a two- or three-dimensional model. The actual gap sepa-ration is defined independent of the node locations and can assume any value including the distance between the nodes. The element resists either compression or tension depending on whether the gap distance is positive or negative. Figure 2-2 illustrates the element and its force-deflection relationship in both compressive and tensile modes.

Figure 2-2. Gap-Friction Element

V

S

Fs

Fn

Node_2

(+)

Node_1

relFs

Compressive Gap

Force

(+) Gdist

Displacement

Gstiff

1

Force

(-) Gdist

Gstiff1

Displacement

V

S

Fs

Node_2

(-)

Node_1

relFs

Tensile Gap

V

F

F

s Friction force in s-direction > G * Ffic n

rel

Comprehensive or tensile gap force in the n-direction and proportional to V

Relative velocity in the s-direction

n

=

=

=

rel

Fn

Fn

Fn



2-6

As shown in Figure 2-2, the element is assumed to be a compressive gap when the gap distance is positive, and only compressive forces result from its closure. Con-versely, it is characterized as a tensile gap element when the distance (Gdist) is neg-ative and tensile forces are produced at the closing stage.

An iterative procedure is adopted in the solution of problems involving gap ele-ments. The iteration is performed at the end of each time step on gap elements closed, to ensure convergence of the force in the element to its correct value corre-sponding to the relative displacement between the two nodes.

Frictional forces may also be considered in conjunction with gap elements if the coefficient of friction is supplied. This effect is currently accounted for in the X-Y plane only, along the direction normal to the element axis. Sliding resistance devel-ops once the gap is closed. Maximum force of friction at each time step is equal to the product of normal force Fn and the coefficient of friction. In the program, no arbitrary stiffness value is assigned to the friction element. Instead, the frictional force effects are computed on the basis of relative velocities of the two nodes defin-ing the gap element during each time step.

Solution Accuracy Considering Mode Truncation

The solution accuracy of dynamic problems based on the normal mode method depends to a large extent on the number of modes considered. Below a few possible options as well as certain remedial steps available in the Interface to improve the solution accuracy are discussed for the various analysis types and loading condi-tions.

Force Excitation Problems

In the case of systems under the influence of force excitations, it is essential that all modes which contribute to the static deformation shape of the structure are consid-ered. The following example shows that at least five modes should be included for this case.



Figure 2-3. Mode Shapes of a Simply-Supported Beam

Base Excitation Problems

In dynamic problems under the influence of base excitations, usually the number of modes considered must contribute to a total mass participation factor of at least 80% of the system mass in the direction of the base motion.

For harmonic and random vibration problems, in addition to the 80% mass partici-pation factor requirement, the range of natural frequencies considered for the analy-sis must cover the highest frequency in the excitation.

Modal Acceleration Method (MAM) in Time History

The process of mode truncation, as was explained before, introduces some error in the response. The Modal Acceleration Method (MAM), in the Time History Analy-sis, approximates the effects of the truncated modes by their equivalent static effects. This approximation can be expressed for the displacement by:

Uc = [K]-1 {Rc}

Mode 1

Mode 2

Requires Five Modes

Mode 3

Mode 4

Mode 5



2-8

where, K is the structural stiffness matrix and Rc represents the static loading. It can be shown that this static load vector can be computed in terms of the included modes, according to:

Rc = [I - M T] {P(t)}

where, M and are the mass matrix and modal matrix, respectively, {P(t)} is the applied dynamic load, and T is the mass matrix transpose. For a demonstration of the MAM on the improvement of the accuracy, please refer to Chapter 5.

Thus by considering only a few number of modes, for even very complicated geometry, you are able to evaluate the response accurately.

Missing Mass Correction Technique in Response Spectrum

Truncation of higher modes in the modal analysis always introduces some error in the results. This truncation means that some mass of the system is ignored. The dis-tribution of this missing mass is such that the inertia forces associated with it will usually produce only small displacements and stresses. However, these ignored inertia forces will often produce significant displacements (stresses) for stiff sys-tems or at the close proximity of the structural supports. The Missing Mass Correc-tion technique is incorporated in the ASTAR program for Response Spectrum analysis in order to estimate the error introduced by the ignored higher modes and to improve the results. This correction is presented as a factor in the output file immediately after the printed accelerations. The user may apply this factor to improve the accuracy of the results obtained for accelerations or stresses.

Excitation Due to Base Motion

The various analysis modes available in ASTAR may be considered in conjunction with either uniform base motion, multi-base motion, or both.

Uniform Translational Base Motion

The equations of motion for a linear dynamic system with uniform base accelera-tion, can be written as:



(2-12)

where {ur} is the structure displacement relative to the base, and {fe(t)} is an effec-tive load due to the base motion:

(2-13)

The vector {Ib} is an influence vector relating base motion to rigid body structure displacements according to:

{U} = {ur} + {Ib} ub (t) (2-14)

The equations of motion Eq. (2-12) can be transformed into uncoupled equations in terms of the modal displacements {x} where:

{ur} = [] {x} (2-15)

and the equation of motion Eq. (2-12) becomes

(2-16)

where [] is the modal participation factor for the lowest nf eigenvectors, i.e.,

(2-17)

Therefore for each mode i, the participating factor is:

(2-18)

and the participation mass for each mode i is i2. Thus:

i2 = total participating (effective) mass in the direction of motion.



2-10

Uniform Rotational Base Motion

In the presence of rotational motion, the base motion is not the same at each base point and cannot be described by a constant b as done in the case of uniform trans-lational base motion (Eq. 2-13).

Consider the absolute displacement of a base point p where position is specified by vector rp with respect to a moving frame of reference with origin o. Let the transla-tion of the origin o with respect to the absolute frame of reference be go, and let the rotation of the point o in respect to the absolute frame of reference be , then the total acceleration of base point p is specified by:

(2-19)

The acceleration term, , indicates that superposition of base rotations can-

not be used. Further, the Coriollis acceleration term, , includes the response p; making the problem nonlinear.

The assumption made here is that these two terms are negligible, and that for rota-tional base motion only the tangential acceleration term, needs to be included. Thus for rotational (and translational) base motion.

(2-20)

With this approximation, for point p can be found from Eq. 2-20 as:

(2-21)

Where xp, yp, and zp are the coordinates of the point p with respect to the rotating frame of reference with origin o.



In Eq. 2-13, the right hand term , then is replaced by , where it con-sists of the acceleration at all base points calculated by the above equation.

Multi-Base Motion

The application of different support motions to different groups of constrained nodes or supports may also be considered. In this case, the equations of motion can be written as:

(2-22)

where a denotes structure degrees of freedom and b refers to support degrees of freedom. The superscript t signifies that these displacements are total or absolute displacements differentiating them from relative displacements. In the above equa-tions, the effect of damping may have also been considered. Next, we will examine the response of the system when the support displacements ub are applied in a static manner associated with a single time function, i.e.,

(2-23)

The static application of ub means that the time variation is absent and the applied

displacement vector is equal to . Eq. (2-22) are then reduced to:

(2-24)

in which are the structure displacements and represent support forces. Eq. (2-24) gives:

(2-25)



2-12

or

(2-26)

On multiplying both sides of Eq. (2-26) by f(t), the quasi-static part of the response is obtained.

(2-27)

The total displacement uat can be expressed by the superposition of the quasi-static and the relative displacements, so that:

(2-28)

Substitution of Eq. (2-28) into the first part of Eq. (2-22) give:

(2-29)

Now because,

(2-30)

Eq. (2-29) reduces to:

(2-31)

Eq. (2-31), written in terms of the relative displacements {ua}, is equivalent to equations of motion (2-12) (ignoring damping), with the forcing function being a function of the base motion b. It can therefore be solved by the application of the mode super-position method as described before. A major simplification results when the motion is assumed to be identical for all the supports, in which case Eq. (2-31) will reduce to undamped form of Eq. (2-16) following the mode superposi-tion application.



Time History Analysis

In time history analysis problems, the equations of motion for multi degree-of-free-dom systems are solved considering different dynamic loadings and base excitation functions. The normal mode method is first used to obtain the uncoupled equations of motion as discussed earlier and shown by Eq. (2-6), i.e.,

Next, one of two step-by-step integration methods available in COSMOSM, that is either (a) the Wilson-Theta method or (b) the Newmark method (see Reference 6), is used to evaluate the response of each mode. These techniques use the results obtained in one previous step to solve for those in the next step. The integration is performed in the time domain starting from time at the last step and ending with the time at the current step (which is equal to the time-step increment). Thus, by reduc-ing the time increment between consecutive steps, accuracy of the solution can be improved. The system response is then determined at each time step using the fol-lowing transformation:

{u} = [] {x}

In Figure 2-4, some typical loading functions are shown.

Figure 2-4. Known Functions of Time

Solution Accuracy

The solution accuracy in time-history analysis problems, as in all cases, depends on:

1. Number of modes considered.

2. Now, accurately, the modes are calculated (i.e., accuracy in modeling).

Time Time Time

F(t) F(t) F(t)

Harmonic Loading Pe riodic Loading Shock Loading



2-14

3. Integration increment or time-step size, (smaller than 1/10th of the last mode period is recommended).

Loading Options Available

1. Force or pressure loadings associated with time curves.

2. Uniform base motion, i.e., the entire constraint portion of the model is subjected to the same motion specified by a time curve. (See PD_BASEFAC [Analysis > POST_DYNAMIC > PD BASE EXCITATION > Base Excitation Factor) commands.]

3. Multi-base motion application, i.e., different groups of constraint nodes are subjected to motions defined by different time curves. (See PD_SPPRT [Analysis > POST_DYNAMIC > PD BASE EXCITATION > Support Level) commands.]

4. Initial conditions can be specified in the form of initial displacements, velocities, or accelerations to a group of nodes. [See INITIAL (LoadsBC > LOAD_OPTIONS > Initial Condition) command.].

Concentrated Dampers and Gap Elements

Concentrated dampers and gap elements can be modeled only in conjunction with the time history analysis option. Gap elements are introduced in modal analysis as truss elements which can resist either tension or compression, once a certain dis-tance between two nodes is reached. Also, concentrated dampers can be defined between two nodes or between one node and the ground. Their effect is considered in the modal analysis by applying forces at the proper nodes. The forces, due to gap elements, are proportional to gap distances while the forces resulting from dampers are in proportional with differential nodal velocities. More detailed information on these two features is provided in the earlier parts of this chapter.

Response Spectrum Analysis

The capability to perform response spectrum analysis of linear elastic structures subject to base motion, is included in the ASTAR module of the COSMOSM pro-gram. The normal mode method is first used to transform the problem into uncou-



pled generalized modal coordinates. The maximum modal responses are then determined from the input response spectrum, and the structure response is found by summing the contributions from each mode.

Definition of Response Spectrum

A response spectrum is the maximum response of a single degree of freedom sys-tem subjected to a particular base motion plotted as a function of the natural fre-quency of the single DOF system. Different curves are usually plotted for different values of the modal damping ratio of the single DOF system as shown in Figure 2-5. The usual values plotted are the maximum displacement Sd(i,i), the maxi-mum pseudo-velocity Sv(i,i), and the maximum pseudo-acceleration Sa(i,i) responses.

These three responses are related by:

Sa = Sv = 2 Sd (2-32)

and all can be plotted as a single curve on appropriately scaled graph paper.

Figure 2-5. Response Spectrum Curves

Modal Maximum Response

As each mode is a single DOF system, the maximum modal response can be obtained from the input response spectrum curve as shown below.

(2-33)

Modal Damping

3

1 2

Sd



2-16

(2-34)

(2-35)

where i is the modal participation factor for each mode i (see Eq. 2-17) and i=1, 2,...nf. Then for each mode i, the structure maximum response can be found from Eq. (2-15), i.e.,

or

(2-36)

(2-37)

and

(2-38)

Structure Maximum Response

The maximum modal responses cannot be simply added to obtain the structure maximum response because the occurrences of the maximum modal response in the time domain are not known. However, several recommended approaches exist and the following mode combination techniques are available in the Interface.

Absolute Sum

This is a conservative approach in which it is assumed that all of the modes have their maximum response in the same direction at the same time, i.e.,

(2-39)



Square Root Sum of Squares (SRSS)

This is a more rational (and not necessarily conservative) approach where the modal responses are summed using the square root of sum of the squares.

(2-40)

In COSMOSM, the Absolute Sum method will only be used in conjunction with the SRSS technique if the modal frequency spacing is less than or equal to the cluster factor multiplied by the lowest frequency in the cluster. The cluster factor (clusf) is specified by the PD_ATYPE (Analysis > POST_DYNAMIC > P_Analysis Type command when defining Response Spectra Analysis options.

Complete Quadratic Combination (CQC)

The complete quadratic combination technique considers the effects of damping in combining the mode responses, i.e.,

(2-41)

where ij is the cross-mode correlation coefficient

(2-42)

and, , and i, j are modal damping coefficients for modes i and j.



2-18

Naval Research Laboratory (NRL)

The mode combination technique recommended by NRL takes the absolute value of the maximum response among all specified modes and adds it to the SRSS response of the remaining modes for each degree of freedom as noted below:

(2-43)

where {uj} represents the maximum response among responses of all nf modes.

Multiple Response Spectra

The structure response to multiple response spectra is found by the square root of the sum of the squares of the individual spectra responses. The output file lists the RMS values of relative displacements and velocities as well as the absolute acceler-ations. The accelerations at fixed nodes are not listed.

Response Spectra Generation

This dynamic analysis option allows the generation of response spectrum curves at any point of the structure for any displacement degree of freedom. The curve is generated for a single degree of freedom oscillator and represents the maximum response amplitudes as a function of oscillator frequency for a specified damping ratio.

The excitation input required for this analysis is a curve defining accelerations ver-sus time at the desired point of the structure. The program uses the results obtained from a modal time-history analysis to generate the spectrum curve at the specified node. Thus, a response spectra generation is possible only after a modal time his-tory analysis is performed.

The response spectrum generated can then be used as base excitation input for a response spectra analysis. This is particularly useful for studying the effect of struc-tures response on a secondary system attached to a point in the structure.



In order to generate a response spectra for an available time-dependent curve, use the following procedure:

1. Model a single DOF oscillator (Figure 2-6), such as a truss element with a lumped mass attached to one node and the other node fixed, and perform a frequency analysis.

2. Subject the model to the base acceleration (t) defined by the time-dependent curve (Figure 2-7) in a modal time-history analysis.

3. Use this analysis type (response spectrum generation) to generate a response spectrum curve (Figure 2-8) for the fixed node.

Figure 2-6. Single DOF Oscillator

Figure 2-7. Base Acceleration

C

k

A m

A(t)..

A(t)

t t

t

. .



2-20

Figure 2-8. Generated Response Spectrum Curve

Random Vibration Analysis

In COSMOSM, the random vibration analysis is available for linear elastic systems when subjected to a random excitation. The excitation is assumed to be stationary, Gaussian, with a mean value of zero, and one-sided (defined for positive frequencies only).

The random excitation input required for this analysis consists of curves defining values of power spectral densities (PSD) versus frequencies, which can be associ-ated with base motion, nodal forces, or element pressure.

The curves frequencies as well as the lower and upper frequency limits are input in radians per second or in cycles per second, and the units of PSD functions will be interpreted accordingly. Thus, the PSD units are either (Force)2/freq, (disp)2/freq, (vel)2/freq, or (accel)2/freq. Also, the exciting power spectral densities of nodal forces can be specified as fully correlated, partially correlated, or fully uncorre-lated. For the base motions, the power spectral density can be either fully correlated or fully uncorrelated. In the case of element pressure, the power spectral density can only be defined as fully correlated.

Finally, this analysis outputs the root mean square (RMS) responses of displace-ments, velocities, accelerations, and stresses which for stationary, Gaussian excita-tion with mean value of zero, is equivalent to one standard deviation of response, i.e., the actual response can be assumed to be at or below this one standard devia-tion value about 68% of the time. The output of modal PSD's at selected frequen-cies is optionally available. Also, curves of power spectral density of response versus exciting frequencies at nodes (Q-plots) can be requested.

a (). .

max



Basic Definitions

In the analysis of random vibration problems, the effects of such physical phenom-ena as the intensity of an earthquake or the noise from a jet engine, where the value of the variables describing the phenomena at some future time cannot be predicted, are considered. In view of the nondeterministic or random nature of these types of phenomena, it is necessary to abandon the explicit description of the applied excita-tions and their corresponding responses in terms of time and work with quantities that are based on certain averages. A few of the relevant average quantities used in the following discussions are described below:

Autocorrelation Functions

The autocorrelation function provides information concerning the dependence of the values of a random variable f(t) at different times on one another, given by the relation:

(2-44)

where

(2-45)

is known as the mean square value of the random variable which is equivalent to the maximum value of the autocorrelation function obtained at = 0.

Power Spectral Density

The power spectral density function defines properties of a random variable, simi-lar to autocorrelation, in the frequency domain, i.e.,

(2-46)

The autocorrelation function can, therefore, be expressed by the Inverse Fourier Transform of the power spectral density function:



2-22

(2-47)

The function Sf () has properties that render the evaluation of certain averages easier.

Analysis Procedure

The equations of motion for a linear dynamic system, Eq. (2-1), are:

With certain restrictions on the form of the structure damping matrix [C], this sys-tem of equations was decoupled into nf modal equations as mentioned before.

(2-48)

for n=1, 2,..., nf, with:

(2-49)

and the vector of modal loads {r(t)} defined as:

(2-50)

Mean Square Response

Considering that in the analysis of random vibration problems, the applied excita-tions are expressed by their power spectral density functions, a frequency domain solution is used. Therefore, if {f(t)} has the known PSD matrix [Sf ()], then {r(t)}, as defined in Eq. (2-50), would have its PSD matrix defined as:

(2-51)

Now the PSD of the modal displacement response, [Sx ()], can be obtained by,



(2-52)

where [H()] is the modal transfer function matrix, and [H*()] is its complex con-jugate. For normal modes, the transfer function matrix is diagonal with diagonal elements Hn()

(2-53)

and

(2-54)

The structure displacement response PSD, [Su ()], can then be found from Eq. (2-49), i.e.,

(2-55)

Similarly, the structure velocity and acceleration PSD responses can be written as:

(2-56)

(2-57)

Now, the modal velocity PSD can be expressed in terms of the modal displacement PSD with individual matrix elements:

(2-58)

and hence, the structure velocity and acceleration response PSD's can be written as:

(2-59)



2-24

(2-60)

Next, the zero delay modal autocorrelation responses for = 0 in terms of PSD of modal response (directly from Eq. 2-52) are:

(2-61)

(2-62)

(2-63)

and from these, the mean square responses are determined as the diagonal terms of the matrices:

(2-64)

(2-65)

(2-66)

Random Base Motion

The equation of motion for a linear dynamic system with base motion vb(t) can be written:

(2-67)

(2-68)

If the base motions have p.s.d. matrix [Sb()], then the p.s.d. of the equivalent vec-tor is [Sp()] is:

(2-69)



Then the p.s.d. of the modal load is:

(2-70)

Once the p.s.d. of the modal loads Sr() has been calculated the analysis for the rel-ative displacement response proceeds as for the random nodal load problem with the following considerations.

a. In the case of Uniform Base Motion, {Ib} contains 1 in the direction(s) of motion and 0 elsewhere (at all nodes).

b. In the case of Multi-Base Motion, {Ib} corresponds to the static displacement resulting from the static application of support displacements for each one of the support levels.

Stress Mean Square Response

The element stresses {} are determined from the nodal displacements {ue} of the element,

(2-71)

or in terms of modal displacements,

(2-71b)

where [e] are the eigenvectors corresponding to the nodal displacements {ue}. The stress response correlation matrix, [R] for each element is written as:

(2-72)

Methods of Integration

Two methods of integration are available. One is the so called standard method and the second is the approximate method, as briefly discussed below:

(2-71a)



2-26

Standard Method

The standard integration method performs a classical random vibration analysis. This method proceeds in the following steps:

1. Certain frequency points are selected around each natural mode as requested. Locations of these points are based on the value of biasing parameters which is input. A biasing value of one results in points uniformly distributed between the natural frequencies; while a value greater than one helps to bias point locations toward natural frequencies. The default values for these parameters are given in Table 2-1, below, as a function of the modal damping ratio at the first mode.

2. Modal PSDs of response are evaluated at each of the selected frequency points. The cross-mode cut-off ratio (RATIO) defines a limit on the ratio of each two natural frequencies (i / j, i > j). This means for each two modes with i / j > RATIO, the cross-spectral density terms are neglected. Cross-mode effects are not considered if RATIO = l.

3. The modal PSDs are then numerically integrated over the specified frequency range to yield the mean squared values and covariances of the modal response. The integration is carried out numerically using Gauss integration of orders two or three over each frequency interval, based on a log-log relation. The mean squared response is obtained by summing the interval contributions.

4. Finally, transformation from modal to nodal yields the RMS displacements, velocities, and accelerations of the system.

Table 2-1. Default Values for Parameters num_Freq_points and Bias as a Function of

Approximate Method

The standard random vibration analysis can be computationally expensive due to the numerical integration of matrices. The approximate integration method per-forms a simplified solution by considering the following assumptions:

Modal Damping Ratio Default for num_freq_points Default for Bias

< 0.01 21 11.00.01 < < 0.1 21 - 4.34 ln ( / 0.01) 11.0 - 3.47 ln ( / 0.01)

> 0.1 11 3.0



Neglecting the cross-mode response, Sx(), that is the effect of one mode on another, i.e.,

(2-73)

Forcing PSD's are considered constant around each normal mode. Thus, each mode is assumed to be excited by white noise with power spectral density Sn, where

(2-74)

and n is the natural frequency of mode n = 1,2,...,nf. For white noise, the mean square response can be determined analytically for the modal response and they are:

(2-75)

(2-76)

(2-77)

By combining the two assumptions, the structure mean square responses can be found.

Partial (Spatial) Correlation

The excitation between two nodes could be fully correlated, fully uncorrelated or partially correlated. In COSMOSM, the partial correlation is based on the spatial distance between two nodes. If the partial (spatial) correlation is used depending on the distance between the two nodes (i) and (j) (Rij), one of the following three situ-ations will prevail:

a. If the distance Rij is smaller than a user defined RMIN, then the two nodes are considered fully correlated.

b. If the distance Rij is larger than a user defined RMAX, then the two nodes are considered fully uncorrelated.

c. If the distance Rij is between RMIN and RMAX, then the excitation is partially correlated and the degree of correlation is linearly proportional to the



2-28

distance. The above conditions can be illustrated in the following figure where nodes 1 and 2 are fully correlated, nodes 1 and 3 are partially correlated, and nodes 1 and 4 are fully uncorrelated.

Figure 2-9. Sphere of Influence for the Degree of Correlation

The effect of the distance on the cross-correlation terms (off-diagonal terms) of the part of the PSD matrix related to the two nodes (i) and (j) and a single degree of freedom k can be written as:

(2-78)

where:

Currently, partial correlation is only available for force excitations.

RMAX

Distance (R )i j

Node 1

Node 3

Node 2

Node 4

1RMIN

i j



Steady State Harmonic Analysis

The steady state harmonic analysis evaluates the maximum structural response due to harmonic excitations of varying magnitudes and varying frequencies. Maximum nodal response is evaluated at different exciting frequencies in the range specified.

The excitation input required for this analysis consists of curves defining ampli-tudes versus frequencies of a harmonic forcing function, which can be associated with nodal forces, element pressures, or base motion. Also, a phase angle can be defined for each base motion curve or nodal force. Note that element pressures are currently associated with zero phase angles. The input curves frequencies as well as the lower and upper frequency limits can be specified either in radians per sec-ond or cycles per second.

Curves of maximum nodal response magnitude versus exciting frequencies (Q-plots) may be requested for any node in the structure.

Analysis Procedure

The frequency response analysis of the structure is determined by using the normal mode method. Let the nodal force vector {P} be harmonic and defined as:

(2-79)

or,

(2-80)

where Pk is the magnitude of the force in the kth degree of freedom direction, is the exciting frequency, and k is the phase angle of the force. Considering that for linear systems, the structure equations of motion are decoupled into nf modal equations,

(2-81)

the substitution of the force vector {P} into Eq. (2-81) will give:



2-30

(2-82)

where:

The steady state solution to Eq. (2-82) is:

(2-83)

Taking the real part of Eq. (2-83) will result in,

(2-84)

where



Now the structure displacement, u, is given by:

or

(2-85)

The magnitude of the structure displacement and its corresponding phase angle for the kth degree of freedom are:

(2-86)

(2-87)

The structure velocity and acceleration response can be found by taking the deriva-tives of Eq. (2-85). The amplitude of the structure velocity and acceleration are:

(2-88)

(2-89)

where the velocity and the acceleration are 90 and 180 advanced from the dis-placement in terms of their phase angles, respectively.



2-32

Direct Spectrum Generation

This analysis evaluates different types of spectra curves for a given frequency range when a time history curve is used as input. For this analysis there is no need to per-form frequency analysis. The following spectrums are calculated and can be viewed both in output and in terms of Q-plots.

Pseudo acceleration Pseudo velocity Relative displacement Relative velocity Absolute accelerationThe direct spectrum generation analysis is based on the central difference time inte-gration procedure where the maximum response of a Single Degree of Freedom system for different frequencies is calculated. This analysis is more accurate, more time consuming and has shorter input than the Response Spectra Generation analy-sis noted earlier. Finally, this direct generation approach is highly recommended in cases where random time history excitation records (i.e., earthquake) are used as input.

The input time history must be defined as an acceleration base excitation curve.

The results are written also in a form which can be used as input curves to other problems (see PD_ATYPE command notes in the Command Reference Manual).

Time History Generation

The objective of this analysis is to generate time history records from the user defined acceleration response spectra curve. For this analysis there is no need to perform frequency analysis. The method is based on the use of summation of a series of sine terms with selected amplitudes and frequencies. The spacing of the frequencies is chosen in such a manner that the half power points of response spec-tra due to a single frequency input will overlap.

In order to obtain time histories that resemble the proper excitation record (i.e., earthquake at a given site), a modulating profile is introduced which has the shape of a given time dependent excitation record (i.e., trapezoidal, box or El-Centro N-S earthquake record). Also, the peak value of acceleration in the time history record can be limited to a specified amount defined for a particular site.



A maximum of two time histories with relatively low correlation between them can be generated. To develop the second time history, a method based on shifting the frequencies of the first record to the adjoining midfrequencies is used. Also, the Base Line Correction method is used in order to modify the generated time histories such that the final velocity and displacement will be equal to zero.

The method used in the Time History Generation analysis is an iterative procedure where the values of the spectra and generated time history curves can be printed at the user specified print intervals, however, plot is available only for the last step). This iterative algorithm will be terminated either at the time where the calculated spectrum converges to the target spectrum, within the user specified tolerance, or when the iteration number is equal to the maximum permitted number of iterations. In case of convergence, the results will be given both in the output and also in terms of Q-plots (spectrum and corresponding time-history curves).

It should be noted that once the iterative procedure is used to develop the first time history, the second record and corrected curves will be generated at the final itera-tion number.

The input target spectrum curve must be defined as frequency dependent accelera-tion base excitation curve, where the unit of frequency must be in Hertz. The profile or envelope curve should also be defined as a time dependent acceleration base excitation curve. The program will assume the El-Centro earthquake envelope if the default value of zero is taken as the profile curve label.

The results are written also in a form which can be used as input to other problems (see PD_ATYPE command notes in the Command Reference Manual.)

References

1. Clough, R. W. and Penzien, J., Dynamics of Structures, McGraw-Hill Book Company, New York, 1975.

2. Cook, R. D., Malkus, D. S. and Plesha, M. E., Concepts and Applications of Finite Element Analysis, Third Edition, John Wiley & Sons, Inc., 1989.

3. Fung, Y. C., Foundations of Solid Mechanics, Prentice-Hall, Inc., Englewood Cliffs, 1965.

4. Zienkiewicz, O. C., The Finite Element Method, McGraw-Hill Book Company, New York, 1977.



2-34

5. Meirovitch, L., Elements of Vibration Analysis, McGraw-Hill Book Company, New York, 1986.

6. Bathe, K.J., Wilson, E. L., Numerical Methods in Finite Element Analysis, Prentice Hall, Englewood Cliffs, 1976.

7. Numar, J. L., Dynamics of Structures, Prentice-Hall, Inc., Englewood Cliffs, 1990.


3 Description of Elements

Introduction

The entire set of structural elements furnished in the COSMOSM element library can be used in conjunction with the Advanced Dynamic Analysis module. The table below provides a complete list of the available elements. For detailed descrip-tions of each element, you are referred to

Chapter 4 of the COSMOSM User Guide Manual.


Chapter 3 Description of Elements

3-2

Table 3-1. Elements for Linear Structural Dynamics Analysis (STAR, DSTAR and ASTAR)

Element Type Element Name

2D Spar/Truss TRUSS2D

2D Elastic Beam BEAM2D

3D Elastic Beam BEAM3D

3D Spar/Truss TRUSS3D

2D 4- to 8-node Plane Stress, Strain, Body of Revolution PLANE2D

3D 3- to 6-node Plane Stress, Strain, Body of Revolution TRIANG

Triangular Thin Shell SHELL3

6-Node Triangular Thin Shell SHELL6

Quadrilateral Thin Shell SHELL4

Triangular Thick Shell SHELL3T

Quadrilateral Thick Shell SHELL4T

6-Node Triangular Thick Shell SHELL6T

Triangular Composite Shell SHELL3L

Quadrilateral Composite Shell SH3LL4L

8 or 9-node Isoparametric Shell Element SHELL9

8 or 9-node Isoparametric Composite Shell SHELL9L

Axisymmetric Shell SHELLAX

3D 8- to 20-node Continuum Brick SOLID

8-node Composite Solid SOLIDL

3D 4-node Tetrahedron Solid TETRA4

3D 4-node Tetrahedron Solid with Rotation TETRA4R

3D 10-node Tetrahedron Solid TETRA10

Spring Element SPRING

General Stiffness GENSTIF

2-node Rigid Bar RBAR

Elastic Straight Pipe PIPE

Boundary Element BOUND

General Mass Element MASS

Elastic Curved Pipe ELBOW

3D 8- to 20-Node Isoparametric Piezoelectric Solid SOLIDPZ

2-Node Gap with Friction GAP


4 Brief Description of Commands

Introduction

The commands grouped together in the menu for Post Dynamic analysis provide the options required to set up and run Post Dynamic analysis problems following frequency and mode shape calculations. The steps required to generate various results and system response quantities are also briefly discussed. Detailed descrip-tion of all commands associated with Post Dynamic analysis is given in the COSMOSM Command Reference Manual.

Main POST_DYN Menu

The commands of this menu provide general guidelines on the various analyses options available according to the following:

POST_ DYN

PD_ATYPE

PD_ALIST

R_DYNAIMC

PD_PREPARE

PD_DAMP/GAP

PD_CURVES

PD_BEXCIT

PD_OUTPUT


Chapter 4 Brief Description of Commands

4-2

A more detailed description of each submenu is provided in the following:

PD DAMP/GAP Submenu

The Rayleigh, modal and composite damping coefficients specified by the first three commands can be used in conjunc-tion with all the post-dynamic analysis options available. The concentrated damper and gap elements, however, are applica-ble only in the case of modal time history problems.

Command (Path) Intended Use

PD_ATYPE(Analysis > POST_DYNAMIC > Sel PD Analysis Type)

Specifies the analysis type and related inputs.

PD_ALIST(Analysis > POST_DYNAMIC > List PD Analysis Options)

Lists the assignments made by PD_ATYPE (Analysis > POST_DYNAMIC > Sel PD Analysis Type).


R_DYNAMIC(Analysis > POST_DYNAMIC > Run Post Dynamic)

Runs dynamic analysis.

PD_PREPARE(Analysis > POST_DYNAMIC > Prepare PD Plot)

Makes preparation for extreme/relative response calculations.

PD DAMP/GAP(Analysis > POST_DYNAMIC >)

Submenu related to management of damping and gap elements.

PD CURVES(Analysis > POST_DYNAMIC >)

Submenu related to the excitation curves.

PD BASE EXCITATION(Analysis > POST_DYNAMIC >)

Submenu related exclusively to base excitation.

PD OUTPUT(Analysis > POST_DYNAMIC >)

Submenu, specifying options generating output and plot files.

PD_ DAMP/GAP

PD_RDAMP

PD_MDAMP

PD_DAMPREAD

PD_DAMPLIST

PD_CDAMP

PD_CDDEL

PD_CDLIST

PD_GAP

PD_GAPDEL



Concentrated Damper Element

Damper elements can be considered between any two nodes or any one node and the ground as shown in Figure 4-1. The damping coefficient is defined in terms of its components along the global X-, Y-, and Z-directions. The commands in this submenu define, list and delete concentrated dampers (damper elements).

Figure 4-1. Concentrated Damper Element Between Two Nodes

Gap-Friction Element

The Gap-Friction element is defined by two nodes representing the distance between any two points in a two- or three-dimensional model. The actual gap sepa-ration is defined independent of the node locations and can assume any value including the distance between the nodes. The element resists either compression or tension depending on whether the gap distance is positive or negative (see Chapter 2 for more details).

PD CURVES Submenu

The commands assembled in this menu are used exclusively for defining, modifying, deleting, or listing different curve types used in conjunction with the analysis options of the Post Dynamic module. Each curve can be defined by at least two and at most 5,000 points. In addition, if a curve is harmonic time dependent, it can be specified by its amplitude and fre-quency instead of curve points.

Note that at least one curve is usually required for any of the dynamic analyses types, and up to 100 different curves can be defined.

DY

Node_1

Node_2

Y

X

Z

PD_ CURVES

PD_CURTYPE

PD_CURDEF

PD_CURDEL

PD_CURLIST



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The time or frequency curves should be activated prior to defining nodal forces, element pressures, or base excitations. The dynamic loads imposed to the system are then defined by the product of the applied loads and the corresponding curve values at each time or frequency step. The command used for the activation of both time and frequency curves is the ACTSET (Control > ACTIVATE > Set Entity) command.

PD BASE EXCITATION Submenu

The commands of this submenu define parameters related to uniform base motion and multiple support excitations.

In the case of multiple support excitations, the following con-ditions can be considered:

May define up to 100 support levels (a support level consists of nodes which move in the same direction with the same amplitude).

May consider up to 100 excitation curves. May use both multi-base motion and uniform motion at the same time. Is available for all Post Dynamic analysis options.

PD OUTPUT Submenu

The PD_OUTPUT (Analysis > POST_DYNAMIC) submenu provides an extensive array of commands for the automatic generation of plot files representing time-history response of various result quantities, deformed shape and stress contour plots, and other related postprocessing information. These files can be formed for any desired time or frequency step within the defined range and for any element and direction in the speci-fied model. Printing of results in the ASCII output file can also be controlled. All plot units will correspond to units used in defining the in-put excitations. The response plots can be obtained in terms of either absolute or relative displacements and velocities, and absolute accelerations only.

Commands PD_PRINT (Analysis > POST_DYNAMIC > PD OUTPUT > Set Print Options), PD_PLOT (Analysis > POST_DYNAMIC > PD OUTPUT > Set Plot Options), and PD_NRESP (Analysis > POST_DYNAMIC > PD OUTPUT > Set Response Options) must be given prior to the R_DYNAMIC (Analysis > POST_

PD_ BEXCIT

PD_BASEFAC

PD_BSLIST

PD_SPPRT

PD_SPPRTLIST

PD_SPPRTDEL

PD_ OUTPUT

PD_PRINT

PD_PLOT

PD_NRESP

PD_PLTINT

PD_RELRESP

PD_SXYSET

PD_PLTLIST

PD_MAXMIN

PD_MAXLIST



DYNAMIC > Run Post Dynamic) command for the program to write the requested print and plot information. These commands are in general required if detailed postprocessing results are to be generated for graphical evaluations. The PD_SXYSET (Analysis > POST_DYNAMIC > PD OUTPUT > Stress Graph) command should be issued before executing the stress module to generate a file that stores stress components for specified elements as a function of time or frequency. The PD_PLTINT (Analysis > POST_DYNAMIC > PD OUTPUT > Specify Inter-val) command can be used both before or after R_DYNAMIC (Analysis > POST_ DYNAMIC > Run Post Dynamic) and R_STRESS (Analysis > STATIC > Run Stress Analysis) commands.

To obtain information on the relative response between any two nodes (or a node and the base), first you are required to submit the PD_RELRESP (Analysis > POST_DYNAMIC > PD OUTPUT > Relative Response) command and then the PD_PREPARE (Analysis > POST_DYNAMIC > Prepare PD Plot) which will result in the computation of the relative response values and their writing in the out-put file. Finally the combination of PD_MAXMIN (Analysis > POST_DYNAMIC > PD OUTPUT > List Peak Response) and PD_PREPARE (Analysis > POST_DYNAMIC > Prepare PD Plot) commands will initiate the search for nodes with highest response values among a set of nodes and within a given time (or fre-quency) domain and will list the peak responses for the specified number of nodes.

Other GEOSTAR commands such as PRINT_NDSET (Analysis > OUTPUT OPTIONS > Set Nodal Range) and PRINT_ELSET (Analysis > OUTPUT OPTIONS > Set Element Range) can be used if the printout of displacements, velocities, accelerations [using PRINT_NDSET (Analysis > OUTPUT OPTIONS > Set Nodal Range)] and stresses [using PRINT_ELSET (Analysis > OUTPUT OPTIONS > Set Element Range)] is desired for limited number of nodes and/or elements.

Postprocessing of Dynamic Results

Postprocessing of the system response to the different dynamic loading conditions of this module can be performed with commands provided in the various submenus of the RESULT menu.

Graphical depiction or numerical listing of the time or frequency dependent responses can readily be obtained for any part of the structure or for any time or fre-quency step.



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Several detailed examples involving the thorough coverage of the pre- and postpro-cessing features available for Post Dynamic analysis are presented in the next chap-ter.

XY-Plot of Response

Graphs of system response (displacement, velocity, and acceleration) versus time or frequency may be requested at specified nodes. The XY-plots available for various types of analyses options are listed below:

Time History Plots of system response versus time. Harmonic Analysis Plots of amplitude and phase of system

response versus frequency. Random Vibration Plots of PSD of response versus frequency. Response Spectra Generation Plot of the spectra versus frequency. Response Spectra Analysis Not applicable.

For all the above analyses except for Spectra Generation, you need to specify the nodes at which XY-plots are required using the following commands:

The first command must be used prior to running the analysis. However, the second command can be used both before and after the R_DYNAMIC (Analysis > POST_DYNAMIC > Run Post Dynamic) command. If it is used before, it limits the information in data base for plot to those steps requested, whereas, using it after, limits the display only to the requested steps and range. For Response Spectra Gen-eration the node and range assignments are made by using the PD_ATYPE (Analy-sis > POST_DYNAMIC > Sel PD Analysis Type) command. To plot the response use the commands in the XY_PLOTS submenu available under the DISPLAY menu. Use the PD_PLTLIST (Analysis > POST_DYNAMIC > PD OUTPUT > List


PD_NRESP(Analysis > POST_ DYNAMIC > PD OUTPUT > Set Response Options)

To select nodes.

PD_PLTINT(Analysis > POST_DYNAMIC > PD OUTPUT > Specify Interval)

To specify the range and step interval (otherwise (optional) the entire range will be considered)



Plot Info) command to list all requested outputs including the XY-plots. For the proper application of these options, please refer to the examples in Chapter 5.

Plot of Response

Contour plots representing the response of the structure (displacement, velocity, and acceleration) at specified time or frequency steps are available for:

Time History Plot of response at specified time steps. Harmonic Analysis Plot of the amplitude and phase at specified

frequency steps. Random Vibration Plot of the PSD of response at specified

frequency steps and the RMS of response. Response Spectra Analysis Plot of the maximum response.

For random vibration the N requested PSD steps are saved at the first N steps of plot file and at the step N+1 the RMS of response is saved as default. For response spectra analysis, the maximum response is always saved at step number one. Use the PD_PLOT (Analysis > POST_DYNAMIC > PD OUTPUT > Set Plot Options) command to specify the desired steps prior to executing the dynamic pro-gram. To display the results choose the ACTDIS (Results > PLOT > Displacement) command to activate a solution step and then choose DISPLOT (Results > PLOT > Displacement) command to display the response, or use DEFPLOT (Results > PLOT > Deformed Shape) of the same submenu to plot the deformed shape. Use the PD_PLTLIST (Analysis > POST_DYNAMIC > PD OUTPUT > List Plot Info) command to list all requested outputs including the requested plots. The RESULTS (Results > Available Result) command also lists available steps in the plot file. For the proper application of these options, please refer to the examples in Chapter 5.

XY-Plot of Stresses

Stresses can be plotted versus time or frequency for the specified elements. This option is available for:

Time History Plot of the stress components versus time. Harmonic Analysis Plot of the stress component amplitude

versus frequency. Random Vibration Plot of the PSD stress components versus

frequency.



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Response Spectra Analysis Not applicable. Response Spectra Generation Not available.

For all the above analyses options you need to use the following two commands prior to running the stress module:

For one-dimensional elements such as beams, trusses, pipes, elbows, and springs, the internal forces are plotted instead of stresses. To display XY-plots use XY_PLOT submenu of DISPLAY menu to activate postprocessing for Post Dynamic then the following set of commands may be issued:

Use the PD_PLTLIST (Analysis > POST_DYNAMIC > PD OUTPUT > List Plot Info) command to list all requested outputs including the XY-plots.

For the proper application of these options, please refer to the detailed examples described in Chapter 5.


PD_SXYSET(Analysis > POST_DYNAMIC > PD OUTPUT > Stress Graph)

Specify element number and stress component.

PD_PLTINT (optional)(Analysis > POST_DYNAMIC > PD OUTPUT > Specify Interval)

Specify the range and step interval (otherwise the entire range will be considered).


INITXYPLOTDisplay > XY PLOTS > Initialize)

To initialize the XY-plot buffer.

ACTXYPOST(Display > XY PLOTS > Activate Post-Proc)

To activate the desired components.

XYPLOT(Display > XY PLOTS > Plot Curves)

To generate the plot.

Other commands(Analysis > POST_DYNAMIC > PD OUTPUT > List Plot Info)

To setup the graph, specify the range, etc., are also available.



Plot of Stresses

Contour plot of stresses are available at specified time or frequency steps. This option is available for:

Time History Plot of the stress components at specified time steps.

Harmonic Analysis Plot of stress component amplitude at specified frequency steps.

Random Vibration Plot of the PSD of stress components at specified frequency steps and the RMS of stresses.

Response Spectra Analysis Plot of the maximum stress components.

For random vibration the N requested PSD of the stress components are saved at the first N steps of the stress file (provided that the appropriate option in the PD_ATYPE (Analysis > POST_DYNAMIC > Sel PD Analysis Type) command is activated) and at the step N+1 the RMS value of the stress component is saved by default. For the Response Spectra Analysis, the maximum response is saved at step one (provided that the appropriate options in PD_ATYPE (Analysis > POST_ DYNAMIC > Sel PD Analysis Type) command is activated). Use PD_PLOT (Analysis > POST_DYNAMIC > PD OUTPUT > Set Plot Options) command to specify the desired steps prior to executing the stress program. To display the stresses use the ACTSTR (Results > PLOT > Stress) command to specify the step number and the desired stress component, then use the STRPLOT (Results > PLOT > Stress) command to display the stress. Use the PD_PLTLIST (Analysis > POST_ DYNAMIC > PD OUTPUT > List Plot Info) command to list all requested outputs including the stress plots. For the proper application of these options, please refer to the detailed example in Chapter 5.

Other Postprocessing Options

GEOSTAR provides additional postprocessing tools for the better comprehension of results, such as:

1. Plot Animation: The response (displacement, velocity, and acceleration) as well as stresses can be animated in time or frequency domain using this option. After you plot for certain step, use ANIMATE (Results > PLOT > Animate) command to animate within a desired range.



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2. Section Animation: You may use LSECPLOT (Results > PLOT > Path Graph) and ANIMATE (Results > PLOT > Animate) to animate the results along a specified section.

3. Extreme Values of Results: You may use the EXTREMES menu to find the maximum response (or the maximum stress) at a desired solution step or across all solution steps.

4. Sorting Nodes with Maximum Response: Using the commands PD_MAXMIN (Analysis > POST_DYNAMIC > PD OUTPUT > List Peak Response), PD_PREPARE (Analysis > POST_DYNAMIC > Prepare PD Plot) and PD_MAXLIS (Analysis > POST_DYNAMIC > PD OUTPUT > List Peak Value), the user can request a list of nodes with highest responses within a range of time or frequency.

5. Response of a Node Relative to Another Node: Using commands PD_RELRES (Analysis > POST_DYNAMIC > PD OUTPUT > Relative Response) and PD_PREPARE (Analysis > POST_DYNAMIC > Prepare PD Plot) the user may request the calculation of relative responses between two nodes or one node relative to the base. The results will be written in the output file. Since most of these analyses generate RMS or ABS types of responses, therefore the relative response calculations are carried at the mode level where the modal displacements are first calculated.

6. Combining Responses: Combining the static response (displacement and stresses) with dynamic response of a specified time or frequency step.

Options 4 and 5 are only available for responses, (displacement, veloci-ties, and acceleration, they are not available for stresses). For the proper appli-cation of these options, please refer to the example in Chapter 5.

Post Dynamic Input Procedure

The input commands for Post Dynamic module may either be given directly from the terminal (interactively) or read from a file (in a batch mode) with the use of FILE (File > Load...) command. In the following a typical input procedure is dis-cussed.



Typical Input Procedure

In general, a specific command sequence is not required for setting up a dynamic response problem for the ASTAR module. However, some logic should be

observed in the order in which commands are issued to prepare the problem input. The following instructions are applicable to all solution options available through GEOSTAR. Note that this is only a suggested order which does not have to be strictly followed.

Before using the ASTAR module, the finite element model must have been devel-oped in GEOSTAR and processed for frequency analysis to store modes and fre-quencies essential to the normal mode method (except for curve transformation problems).

1. From the ANALYSIS-POST_DYNAMIC menu tree, select the PD_ATYPE (Analysis > POST_DYNAMIC > Sel PD Analysis Type) command to specify the type of Post Dynamic analysis option, the number of modes to be used, and other relevant solution parameters.

2. The program provides complete freedom in the selection of the response information for printing or plotting. Commands like PD_PRINT (Analysis > POST_DYNAMIC > PD OUTPUT > Set Print Options), PD_PLOT (Analysis > POST_DYNAMIC > PD OUTPUT > Set Plot Options) and others from the PD OUTPUT submenu that control printing and plotting of output results, can be issued for this purpose.

3. The time or frequency-dependent load functions associated with the analysis type under consideration are subsequently defined in conjunction with base motion commands, nodal force or element pressure multipliers to define excitation for the analysis using the PD_CURTYP (Analysis > POST_DYNAMIC > PD CURVES > Curve Type) and PD_CURDEF (Analysis > POST_DYNAMIC > PD CURVES > Define) commands. The data can be specified interactively or may be read from an external file.

4. Next, using the PD_RDAMP (Analysis > POST_DYNAMIC > PD DAMP/GAP > Rayleigh Damp) or PD_MDAMP (Analysis > POST_DYNAMIC > PD DAMP/GAP > Modal Damp) commands, Rayleigh or Modal damping coefficients may be defined before issuing the R_DYNAMIC (Analysis > POST_DYNAMIC > Run Post Dynamic) command to perform the post-dynamic analysis. Also, damping coefficients due to material (composite)



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damping can be read into modal damping locations using the PD_DAMPREAD (Analysis > POST_DYNAMIC > PD DAMP/GAP > Read Material Damp) command if damping is specified as a material property prior to frequency calculation.

5. With ASTAR analysis performed, the requested output results can be graphically animated. The deformed shapes or nodal response graphs can be plotted for the specified solution steps and nodes.

6. The corresponding dynamic stresses can then be computed by issuing the R_STRESS (Analysis > STATIC > Run Stress Analysis) command. Prior to that PD_PLOT (Analysis > POST_DYNAMIC > PD OUTPUT > Set Plot Options) and PD_SXYSET (Analysis > POST_DYNAMIC > PD OUTPUT > Stress Graph) commands should be used to specify the time steps for stress contour plots and the elements for which stresses are to be stored as a function of time or frequency for stress plots.

7. Finally the PD_PREPARE (Analysis > POST_DYNAMIC > Prepare PD Plot) command will provide the option to obtain relative response information as well as extreme nodal response. This command must be used in conjunction with the PD_RELRESP (Analysis > POST_DYNAMIC > PD OUT

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