Part 6: Dynamic design analysis

Transcription

1 Part 6: Dynamic design analysis

2 BuildSoft nv All rights reserved. No part of this document may be reproduced or transmitted in any form or by any means, electronic or manual, for any purpose, without written consent by BuildSoft. The programs described in this manual are subject to copyright by BuildSoft. They may only be used by the licensee and may only be copied for the purpose of creating a security copy. It is prohibited by law to copy them for any other purpose than the licensee s own use. Although BuildSoft has tested the programs described in this manual and has reviewed this manual, they are delivered As Is, without any warranty as to their quality, performance, merchantability or fitness for any particular purpose. The entire risk as to the results and performance of the programs, and as to the information contained in the manual, is assumed by the enduser. 2 PowerFrame Manual Part 6: Dynamic Design Analysis

3 1 Table of Contents 1 TABLE OF CONTENTS DYNAMIC ANALYSIS INTRODUCTION MODAL ANALYSIS DESIGN GRAVITY LOADS Eurocode P MODAL SUPERPOSITION DYNAMIC DESIGN ANALYSIS USING POWERFRAME INTRODUCTION DYNAMIC FUNCTIONS IN THE LOADS -WINDOW Load groups for dynamic analysis Definition of design gravity loads Definition of dynamic loads DYNAMIC FUNCTIONS DURING ELASTIC ANALYSIS DYNAMIC FUNCTIONS IN THE PLOT -WINDOW PowerFrame Manual Part 6: Dynamic Design Analysis 3

4 2 Dynamic analysis 2.1 Introduction Let us consider the example of a column, clamped at its base where it is subjected to a uniformly distributed horizontal load. It is assumed that this horizontal load varies as a function of time such that vibrations are induced in the column. It can be shown that those column vibrations can be described as a linear superposition of the column s eigenmodes, in which the eigenmodes react completely independent of each other to the applied dynamic action: = * + * + γ * + * + The combination factors,, γ,, are a priori unknown, but can be calculated as a function of the applied dynamic loading and as a function of the structure s damping properties. The eigenmodes, on the other hand, are independent of the applied dynamic loading and are mostly also independent of the damping properties of the structure. They can be calculated from the structure s stiffness and mass properties. This calculation is usually referred to as modal analysis. In theory, an infinite number of eigenmodes can be calculated for any building structure. In practice however, it will be largely sufficient to consider only the N lowest eigenmodes for further use during a dynamic analysis. This explains the advantages of such an approach as compared to a direct dynamic analysis, during which the structure s response to the imposed excitation is obtained through a direct integration of the equations of movement as a function of time: using modal analysis techniques, the N lowest eigenfrequencies and eigenmodes are calculated, independent of the applied loading next, the structural response to the applied dynamic loading is calculated, using a modal superposition techniques, as a combination (or superposition) of the N lowest eigenmodes, having 4 PowerFrame Manual Part 6: Dynamic Design Analysis

5 the benefit that only a limited number ( N ) of equations must be solved. Furthermore, the previuously calculated eigenmodes can be re-used when other dynamic loads must be considered. 2.2 Modal analysis The objective of a modal analysis is to calculate the N lowest eigenfrequencies f i (expressed in Hertz) of a structure, along with the corresponding eigenmodes i. A number of remarks: very often, different terminology is used for eigenfrequency: o eigenperiod T i, being the inverse of the eigenfrequency f i (expressed in seconds). o eigenpulsation i = 2* f i eigenmodes i cannot be interpreted in absolute terms: only the eigenmode shape can be interpreted, not its amplitude at least not without additional information the additional information needed for absolute interpretation is the socalled modal mass m i corresponding to an eigenmode i. Modal mass can most easily be explained as the fraction of the total mass of the structure which effectively participates in the displacements described by the eigenmode i. For example, it can be derived from the figure below that for the first eigenmode 1 all nodes of the column always move in-phase and that consequently the distributed mass of the structure will globally move in-phase. Only the amplitude of movement of the distributed mass will be variable (zero at the column basis, maximum at top of column). With the second eigenmode 2, not all nodes will move in-phase any more. As a consequence, part of the distributed mass will move in one direction while the remaining part of the distributed mass will move in the opposite direction. As a whole, less mass is effectively mobilized by this mode shape, resulting in a lower modal mass for the second eigenmode (assuming the maximum displacement of both eigenmodes 1 and 2 to be equal). In general, it can indeed be stated that with increasing eigenfrequency f i, the wave length of the eigenmodes i will decrease and modal mass m i will also decrease (again assuming that maximum displacement of all eigenmodes i is equal). PowerFrame Manual Part 6: Dynamic Design Analysis 5

6 The above explanations allow to understand that the contribution of the higher eigenmodes will decrease quickly in the context of a dynamic analysis, as an increasingly smaller mass will be mobilized by the eigenmodes with increasing eigenfrequency. Taking into account all those considerations, the introduction of this section should be modified as follows: the objective of a modal analysis is to calculate the N lowest eigenfrequencies f i (expressed in Hertz) of a structure, along with the corresponding eigenmodes i and modal masses m i The knowledge of the eigenmodes i and corresponding modal masses m i allows for an unambiguous and absolute interpretation of the response of a building structure subjected to a dynamic type of excitation. It can be shown that such an eigenmode i actually behaves as an equivalent mass-spring-system with (modal) mass m i and (modal) stiffness k i. The eigenfrequency f i of such a mass-spring-system is given by f i = ωi = 2π 1 2π The knowledge of the eigenmodes i and corresponding modale masses m i allows for an unambiguous calculation of the response levels for a structure subject to dynamic loads. Structures which have a high degree of symmetry can have 2 (or more) eigenmodes at the same eigenfrequency. Such modes are referred to as double modes (or triple modes, ). According to modal analysis theory, the corresponding mode shapes are mutually orthogonal. In practice (definitely in view of a correct multi-modal response analysis) it is crucial that eigenvalue solver of an analysis program is capable to calculate such double modes while respecting their orthogonality properties. The PowerFrame 6 PowerFrame Manual Part 6: Dynamic Design Analysis ki m i

7 analysis core is perfectly capable to deal with this situation and thus guarantees at any time a correct basis for a modal superposition procedure. 2.3 Design gravity loads A modal analysis of a building structure delivers its N lowest eigenfrequencies and eigenmodes. The results of such an analysis strongly depend on the gravity loads that are considered during the calculations. Such gravity loads are of course not only related to the structure s selfweight, but also to the permanent loads and (to a lesser extent) to the variable loads. Then the question arises how those permanent loads and variable loads are to be considered to derive appropriate design gravity loads for modal analysis (and any subsequent response analysis through modal superposition). This question is addressed in both seismic design standards Eurocode 8 & PS92 (France). In this reference manual, we present a short overview of the main principles Eurocode 8 To derive design gravity loads to be used for modal analysis of a building structure, gravity loads related to following combination of loads must be considered: Σ j>=1 G k,j + Σ i>=1 ψ E,i Q k,i in which k : characteristic value G: permanent action Q: variable action ψ E :. ψ 2 : correlation coefficient accounting for the degree to which different stories are occupied simultaneously. Values for need to be taken from EC8 ψ 2 : combination coefficient for quasi-permanent value of variable action PowerFrame Manual Part 6: Dynamic Design Analysis 7

8 2.3.2 P92 To derive design gravity loads to be used for modal analysis of a building structure, gravity loads related to following combination of loads must be considered: Σ j>=1 G k,j + ψ E1,1 Q k,1 + Σ i>=2 ψ E2,i Q k,i in which k : characteristic value G: permanent action Q: variable action Q 1 : most unfavourable variable action ψ Ex :. ψ x (x = 1, 2) : correlation coefficient accounting for the degree to which different stories are occupied simultaneously. Values for need to be taken from PS92 ψ 1 : ψ 2 : combination coefficient for frequent value of variable action combination coefficient for quasi-permanent value of variable action 2.4 Modal superposition The objective of a dynamic response analysis through modal superposition is to calculate the response of a structure through its eigenmodes i (with corresponding eigenfrequencies f i and modal masses m i ) that are effectively being excited by the applied dynamic loading. In case a structure is subject to a dynamic load P(t), then the structure s displacement responce can be calculated by solving the complete system of equations of motion as a function of time: [ M ]{ u } + [ C]{ u } + [ K]{ u} = { P(t) } Now suppose that fo such a structure a total of N eigenmodes i has been calculated and assembled into the eigenmode matrix [], then this structure s displacement response can theoretically be described as a linear combination of the N eigenmodes { u } = [ φ]{ q} 8 PowerFrame Manual Part 6: Dynamic Design Analysis

9 provided that the number N (number of available eigenmodes) is sufficiently high to include all eigenmodes that are effectively excited by the applied dynamic loading. Introducing the above linear combination of N eigenmodes into the afore mentioned system of equations of motion, allows to rewrite this system of equations as [ m ]{ q } + [ c]{ q } + [ k]{ q} = { p(t) } in which T T [ m ] = [ φ ] [ M][ φ] [ c ] = [ φ ] [ C][ φ] [ k ] = [ φ ] T [ K][ φ] met [m] modal mass matrix ; [c] modal damping matrix ; [k] modal stiffness matrix ; [p(t)] load vector (projected on the available modal basis). [ p(t) ] = [ φ] T [ P(t) ] [m] and [k] can easily be calculated from the structure s known stiffness and mass distribution, where-as [c] is mostly defined through a set of modal damping values, expressed as a percentage of critical damping (for most steel and R.C. building structures, modal damping values vary between 1% and 4% of critical damping). The above transformation offers the substantial benefit that the solution of an extensive set of equations with hundreds or thousands of unknowns [ M ]{ u } + [ C]{ u } + [ K]{ u} = { P(t) } is replaced by a very compact set of equations [ m ]{ q } + [ c]{ q } + [ k]{ q} = { p(t) } containing as little as N unknowns. This set of equations can be solved very fast for the unknowns {q}, from which the actual displacmenents {u} can be derived through modal superposition: { u } = [ φ]{ q} The above modal superposition approach has several advantages: * dynamic response can be evaluated extremely fast PowerFrame Manual Part 6: Dynamic Design Analysis 9

10 * as the structure s eigenmodes eigenmodes are independent of the applied dynamic loading and are mostly also independent of the damping properties of the structure (at least for lightly damped systems), a new dynamic analysis for different loadings or modified damping properties does not require a time-consuming modal analysis. It is sufficient to repeat only the superpositie phase, and in no time dynamic response can be re-evaluated. Of course, it should be borne in mind that a number of restrictions are applicable to the modal superposition approach. In general, it can be stated that the method is suited for linear structures which are lightly damped (and for which damping is mostly due to global - material damping). With the above approach, the question remains how many eigenmodes N effectively need to be calculated to allow for sufficient accuracy with modal superposition techniques. To be able to answer this question, it is essential to understand how dynamic loading varies as a function of time or what the frequency content is of the applied dynamic loads. It is generally accepted that a correct application of modal superposition techniques requires the structure s eigenmodes to be calculated up to a frequency which is at least 2 times the maximum frequency for which the dynamic loading is defined. 10 PowerFrame Manual Part 6: Dynamic Design Analysis

11 3 Dynamic design analysis using PowerFrame 3.1 Introduction This section of the manual on dynamic design analysis describes in more detail the practical use of PowerFrame s capabilities. As a starting point, it can be said that the dynamic design analysis is an integral part of the entire structural design analysis. In other words, dynamic design analysis does not represent an extra step to be taken at the end of the structural design analysis, but is completely embedded in the full design analysis process. Therefore, structural analysis for dynamic actions is part of the standard elastic analysis procedure within PowerFrame. From a user point of view, there will be little difference between elastic analyses with and without dynamic actions. The major differences related to the introduction of dynamic actions, are: in the Loads -window: o the explicit definition of a loads group in which the design gravity loads are managed (as a function of the permanent loads and the variable loads), as they are to be used for any type of dynamic analysis o the explicit definition of one or more dynamic loads group. The spatial distribution of dynamic loads and the partial safety factors & combination factors are defined in exactly the same way as is done for static type of loads. For dynamic type of loads, those data need of course to be completed with more information on how loading varies as a function of time during the elastic analysis: o when the elastic analysis is launched, a modal superposition analysis will automatically be performed. The results of this analysis will then be combined with the effects of the static loads, consistent with the definition of all loads combinations in the Plot -window o of course, you now have access to all analysis results for all loads combinations containing static and/or dynamic actions. Apart from this, no major differences will be observed in comparion with a traditional static type of analysis. Code checks (for steel, PowerFrame Manual Part 6: Dynamic Design Analysis 11

12 concrete or timber) will of course take into account all available combinationes: ULS FC (fundamental combinations) ULS SC (seismic combinations) SLS QP (quasi-permanent combinations) SLS FC (frequent combinations) SLS RC (rare combinations) 3.2 Dynamic functions in the Loads -window Load groups for dynamic analysis Through the Load factor icon in the icon toolbar of the Loads -window, 2 types of load groups can be defined that are specific to a modal and/or dynamic analysis: Gravity loads for vibration analysis: this group includes all gravity loads which must be taken into account during any type of dynamic analysis. Those design gravity loads can quickly be derived from the permanent loads and the variable loads that have defined in several load groups, through the use of the icon. At any time, it is also possible to add discrete masses (or gravity loads) manually at nodes by simply selecting those nodes and then use the icon. Dynamic: this type of group contains the definition of a dynamic action. The definition of the time-variation of the dynamic action can be done through the icon. The spatial distribution of dynamic loads is defined in exactly the same way as for static type of loads. The load groups described above can be configured using the dialogue window shown below. It should be noted that the icon shown in the column at the right, will automatically adapt itself appropriately in case of gravity loads for vibration analysis. In case of dynamic loads, it is necessary to change this icon manually into. 12 PowerFrame Manual Part 6: Dynamic Design Analysis

13 3.2.2 Definition of design gravity loads Through the icon of the icon toolbar, the dialogue window as shown below is launched. This dialogue window enables the definition of the correlation coefficients to be used for the calculation of design gravity loads (refer to 2.3 for more information on correlation coefficients). For each load group, a value for can be defined manually, based on the specifications provided by the selected design standard. Based on the correlation coefficients, PowerFrame will automatically calculate design gravity loads to be used during any type of dynamic analysis (modal, harmonic response, seismic, ) based on the static type of loads that are part of the selected load groups. Those design gravity loads are visualized in the Loads -window on the geometry of the analysis model, in case Gravity loads for vibration analysis is selected as the active load group. PowerFrame Manual Part 6: Dynamic Design Analysis 13

14 3.2.3 Definition of dynamic loads The icon provides access to the dialogue window shown below, which enables you to define all parameters that describe a dynamic action. In the dialogue window below, you can (depending on your PowerFrame license) define 2 types of dynamic actions: * periodic actions (available to all PowerFrame users): a choice can be made between a number of pre-defined signal types and an arbitrary spectrum. Such a spectrum is defined by means of following parameters: o number of frequencies o fundamental frequency o amplitude & phase at each frequency Furthermore, the dialogue below enables you to save a defined spectrum to an external TXT-file ( ) or to import a spectrum definition from an external TXT-file ( ). * non-periodic actions (only available to those users who have access to the optional license Advanced dynamic analysis ). The variation of the dynamic loading as a function of time is defined through the definition of signal amplitude for a well-defined number of discrete time steps. 14 PowerFrame Manual Part 6: Dynamic Design Analysis

15 3.3 Dynamic functions during elastic analysis A design analysis in which dynamic actions should be accounted for, is launched in the same way as a design analysis related to static loads only. If however, you are only interested in the eigenfrequencies and eigenmodes of a structure, then you should use the tab-page Modal analysis rather than Elastic analysis in the dialogue window below. PowerFrame Manual Part 6: Dynamic Design Analysis 15

16 When an elastic analysis is launched after combinations have been generated which include both static and dynamic loads, PowerFrame will by itself automatically take the necessary extra steps that are required in this situation. Indeed, as a first step PowerFrame will perform a modal superposition analysis in order to calculate the response of the structure corresponding to the dynamic load groups. Next, this response will be included in the appropriate way in all ULS SC combinations. If the user decides to perform a second-order analysis or takes into account the effects of global structural imperfections, then those options will only be applicable to the static type of loads and to their contribution in the load combinations. The response to dynamic actions will always be evaluated according to a linear first-order theory without consideration of structural imperfections, and will be combined with the afore-mentioned results for static type of loads. For the calculation of response due to dynamic loads, PowerFrame will ask you to define values for modal damping (see dialogue window below, which is presented to the user when dynamic response is calculated for the first time after a modal analysis has been performed). 16 PowerFrame Manual Part 6: Dynamic Design Analysis

17 Modal damping is defined as a percentage of critical damping, and is initially assumed to be identical for all available eigenmodes. This definition can be refined, as will be shown later on in this reference manual. If no eigenmodes are available in memory, the modal superposition analysis will start with a calculation of the N lowest eigenfrequencies of the building structure using the subspace iteration method. As a user, you will specify the number N along with the maximum number of iterations to be used for the calculation of eigenfrequencies and corresponding eigenmodes. Note that the number of eigenfrequencies N is limited to an absolute maximum of 40. Depending on the effective aantal number of DOFs (degrees of freedom) in the analysis model, a stricter limit may be applicable: in case the number of DOFs (#dof) is lower than 16, the maximum number of eigenfrequencies is limited (#dof)/2. in case the number of DOFs (#dof) is larger than or equal to 16, the maximum number of eigenfrequencies is limited (#dof 8), with an absolute maximum of 40. Up front however, it is not known how many eigenfrequencies and eigenmodes are really needed for a high-quality dynamic response analysis. Therefore, it is recommended to start the analyses with a relatively small number of eigenmodes and then evaluate the highest eigenfrequency to the frequency content of the dynamic loads. It is recommended to calculate eigenmodes of the structure up to a frequency which is at least twice the maximum frequency for which dynamic loads are defined. Decoupling the modal analysis and the dynamic response analysis offers a number of distinct advantages: * it becomes possible to verify up-front if the modal analysis has delivered a set of eigenmodes that complies with the above recommendations, prior to launching the actual response analysis. * furthermore, the calculated eigenmodes are kept in memory after calculation and are also saved to your PowerFrame project when you exit PowerFrame. As a consequence, it remains possible an any time to launch a modal superposition analysis from a known set of eigenmodes. The importance of this feature can not be overstressed, as the calculation PowerFrame Manual Part 6: Dynamic Design Analysis 17

18 cost of a modal superposition is quite limited as compared tthe calculation cost of a modal analysis. As long as the stiffness characteristics and the mass distribution of the PowerFrame model have not been changes, the same set of eigenmodes will remain available to the user. Once such a set of eigenmodes is available, any changes to definition of dynamic loads or modal damping can be made without having an impact on the structure s eigenfrequencies or eigenmodes. Damping values that have been assigned to calculated eigenmodes can be modified at any time through the icon from PowerFrame s main icon toolbar. This icon becomes available as soon as a first modal superposition has been performed (using constant modal damping values). The following dialogue window is presented: In this dialogue window, modal damping can be changed for each individual eigenmode. Just double-click on the value to be changed, and enter the new value. 3.4 Dynamic functions in the Plot - window The icon toolbar of the Plot -window enables you to visualize all familiar types of analysis results (displacements, internal forces, stresses, reaction forces) for all types of load groups and load combinations, including the contributions from static, dynamic & seismic loads. 18 PowerFrame Manual Part 6: Dynamic Design Analysis

19 During results interpretation, it should not be overlooked that results for dynamic load groups are always presented as symmetric envelopes, as they are a result of symmetric vibrations with respect to the undeformed geometry. At any time, you can also post-process the full set of calculated mode shapes (with animation capability for all results types). PowerFrame Manual Part 6: Dynamic Design Analysis 19