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1. correct specified lt 0 uy tol 0 cu 10 y computed with tolerance a b Figure 3 Effect of convergence tolerance on a overshoot during loading and b hysteretic response on unloading In the figure uy the yield displacement and tol the convergence tolerance The convergence tolerance is checked any time that the stiffness changes The convergence tolerance is considered to be satisfied in each case if the displacement at the end of a step during which a change in stiffness occurs does not differ by more than the user specified overshoot tolerance percentage of uy from the displacement at which the change in stiffness occurs The displacement difference is shown as 6 in Figure 3a and 3b The correct stiffness at time t Ar is then used to begin the subsequent time step If the convergence tolerance is not satisfied then the solution for the step is discarded and USEE repeats the calculation beginning at time 7 with a smaller time increment The new time step is internally set in USEE to 1 10 of the previous time step With such a large reduction in the time step subsequent time steps may not encounter a stiffness change For this reason the reduced time step is used for all subsequent steps until a change in stiffness is encountered If the convergence tolerance is satisfied for the reduced time step the program continues but reverts to the original time step for subsequent calculations I
2. o Solution for u 1 4 Discontinuity 14 2 o o 2 2 a a 1 2 1 88 0 175 0 25 0 325 0 184 0 188 0 192 Yield Strength Coefficient Cy Yield Strength Coefficient Cy a b Figure B4 The strength ductility relationship for a bilinear oscillator responding to the 1987 Whittier Narrows record for an oscillator period of a 0 20 sec and b 0 15 sec B 3 Description of the Algorithm An efficient algorithm is desirable because many nonlinear response computations may be required to compute constant ductility response spectra Figure B5 shows the strength ductility relation in the vicinity of a target ductility W for a particular oscillator and ground motion Nonlinear SDOF analyses at C and Cs determined the corresponding ductility responses indicated by dark circles in the figure Simple interpolation between these points would lead to the solution identified by C in the figure missing the higher strength solutions at A and B Greater resolution of the strength ductility relation would provide greater certainty that an unrecognized higher strength solution would not be missed but this certainty comes at the cost of a larger number of nonlinear SDOF response computations Thus a balance must be struck between the time required to obtain a solution and the possibility that an unrecognized higher strength solution may exist To address this a two phase solution procedure is employed The first phase identifies the
3. Experience indicates that C should 66 Set initial region bounded by Cyu Cy elastic and Cy i Cyu Ut Compute u C Adjust Cy Is u Cyu uulS Apoi or Switch to fast search Cyu Cy lt Cy range limit algorithm Determine C by linear interpolation between Cy and Cy and compute u C e Define new region bounded by Cy and Cyi Cye Attempt to reject the region bounded between C and C Divide this region into 4 equal segments to determine C Cy2 and C s Reject Cy and add C Tn 2 points halfwa ompute ductility A a dala the each C to compare with re all points within Yes nearest C values the expected ductility Ihe Jeglonehecked on either side Is smoothness ratio e a lt smoothness tolerance Redefine interpolation function using u C Y ISAC SAC at SS Define new region bounded by C No IS u Cye uul lt Atuort and Cyu Cye Yes Report solution as Cy Cye Figure B6 The first phase of the algorithm for determining the initial bounds on the solution 67 be selected to result in a ductility u C equal to 1 1 to 1 5 times the target ductility by To secure this result C is estimated initially as C u and then is adjusted until 1 1 u lt W C lt 1 5 nj Next linear interpolation between the current upper and lower bound strength coefficients is used to determine an expected s
4. Hysteretic Energy Weight Hysteretic Energy Weight Peak Strength Weight Graphs that plot results may be copied and pasted to other Windows applications using the right mouse key 3 3 8 Summary Data Files A summary of numerical results from the current analysis session is provided to the user Each analysis run during the session is listed At the top the date and time is provided followed by the properties specified by the user and calculated quantities and peak response values 3 4 Modeling and Response Computation 3 4 1 Load Deformation Models USEE features three commonly used load deformation models linear elastic bilinear and stiffness degrading These are described in this section 3l 3 4 1 1 Linear Elastic Model The linear elastic model Figure 6 is used in most introductory courses in structural dynamics and is applicable to the response of structures that remain elastic such as for relatively small ground shaking intensities For linear elastic response only the stiffness is needed to characterize the load deformation curve R Figure 6 Linear elastic model 3 4 1 2 Bilinear Model Bilinear models are applicable to structures that exhibit stable and full hysteretic loops and often are used for modeling steel structures The bilinear model Figure 7 is defined by three parameters yield strength initial stiffness and post yield stiffness Strength is bounded by the yield envelo
5. been determined for nonlinear systems Changes in stiffness during the response of nonlinear systems may result in equilibrium violations which ideally must be accounted for to prevent deviation from the correct solution USEE implements a variable time step algorithm to ensure accuracy and stability using methods discussed in sections 2 1 2 2 and 2 1 2 3 Additional information on numerical solution methods is available in Clough and Penzien 1993 and Chopra 1995 2 1 3 2 Time Step Selection The previous discussion indicated that the size of the time step may affect the stability and accuracy of the numerical computation and may contribute to equilibrium errors Equilibrium errors may result from large changes in stiffness within a step and therefore are reduced when smaller time steps are used Reducing the time step increases number of calculations which increases the solution time and the volume of data generated in the solution An ideal time step would be sufficiently small to maintain stability and accuracy in the numerical results while not requiring excessive solution times and not producing needlessly large quantities of data The size of the ideal time step cannot be identified a priori Instead computations are done to iteratively refine the time step either increasing it or decreasing it as conditions watrant By using smaller time steps at critical points and larger time steps elsewhere the number of calculation steps can be
6. Commands si one Ha Ie ERA SOON IND HD HIT EE 25 3 3 2 User Pretetences aus co AC e da a eu a Mesa ndm o 27 3 3 3 SDOP Analysis SI6DSo erotici Al da tite rotis to tata Met page dada s 28 3 3 4 Multistory Building Approximation Analysis Steps nenea 20 3 3 5 Response Spectra Analysis Steps ciii e soda aa abate edisieancsgecsacasanGieasoneses 29 3 3 0 Wandows Copy amp Paste oeste nd Ob Repas tie eaten eatin 30 3 3 7 Input and Output Data Files eerte kae een na ada erra oon aaa 30 3 2 0 Summary Data Piles ys de pete po cette eoe court eus Pasce anaes e iu iN 31 3 4 Modeling and Response Computation cccssecccceceeeeeeeeenneeeeeeeeeeeeeenaeees 31 3 4 1 Load Deformation Models 0 eeeecessccccccceeeeseeeenneeeeeeceeeeeeeennaeeeeeeeeeeeeees 31 3 4 2 Load Deformation Curve Properties eeeeeesssceeceeeeeeeeeeenneeeeeeeeeeeeees 34 3 4 3 Response Spectra Parameters casa cabaret cada e Ee AY NENNEN EYE ESSE oala 37 3 5 Base MobBon TBpult associe a toca eu iste te sm dci dt ee 37 3 5 1 Recorded Ground Motions ceo eee sete ia n tn e TN MR Re be sa ERR utei 37 3 5 2 Synthetic Motions iouis eb na ORO OH OR e EAR Sus dui dalta 39 Deo Se CIBIISES sc secs la e ct anal eet ura ted chc am eM lut Avea ct ette 4 JO Scale PAcCtors M T Ganei anes retinas Goes 43 3 6 Tutorials been a tu a dt M pL ae ML Es 43 3 6 T SIDOP Analysis EXamplbs s esos tac pati pa eni p PI 43 ill 3 6 2 Multistory Buil
7. Select the Entire record button to compute response for entire record duration Enter 1 0 for both the Amplitude Scale Factor and the Time Scale Factor Click the Next button to advance to Step 2 Step 2 Among the linear bilinear and stiffness degrading models select the linear model Next specify the periods by selecting the uniformly spaced radio button and entering 60 periods ranging from 0 05 to 3 seconds For the parameter to vary Viscous Damping is the only choice when linear behavior is assumed Choose 3 discrete values from the pull down menu and enter the values of 0 2 and 10 in the text boxes labeled Values Click the Compute Results command button 48 Step 3 The results of the response spectrum computations are displayed in Step 3 Spectra corresponding to each parameter value are displayed with the color indicated at the top of the window Select the Zoom to Full Screen button to enlarge the plot Lines are plotted for each parameter value with a different color Figure 15 shows spectral acceleration versus period for damping values of 0 5 and 10 Spectral Acceleration q 0 Period sec Figure 15 Response spectra example Spectral Acceleration g vs Period sec This completes the first example To modify the current analysis to begin the second analysis click on the Back button on the View Results window This takes the user back to Step 2 Ste
8. USEE makes it possible to easily master the richness and variety of response that may be developed as parameters are varied This understanding is increasingly important as greater attention is given to the seismic performance of new and existing structures in education research and practice 1 1 Program Description The USEE interface makes nonlinear analysis nearly effortless The user is guided through several data input screens A point and click interface allows the user to navigate through the menus and to select analysis options Simulated response is displayed using versatile plots that allow the user to select among a variety of response parameters to be plotted Response data is summarized on screen and may be saved as ASCII text files for subsequent processing Response plots may be copied to the Windows clipboard and then pasted into Windows applications e g Microsoft Word using the Copy and Paste functions accessed via a right mouse click An icon on screen directs the user s web browser to load the Mid America Earthquake Center home page where current USEE release information as well as research results and other products of the Mid America Earthquake Center may be obtained in addition to information about the Mid America Earthquake Center The program includes modules that provide for several kinds of analyses e The Single Degree Of Freedom SDOF analysis module determines the detailed response history of nonlinear single deg
9. and Cy The ductility at the point Cy Cy C 2 is then determined If the computed ductility is within the ductility tolerance or if the yield strength coefficient is within the tolerance on the yield strength coefficient then C is reported as the solution If neither tolerance is satisfied then the solution must lie between either Cy and C or Cy and Cyu If u C gt Iu gt U C p then C is retained and C is reset to C Otherwise u C 5 gt Iu gt U C then C is reset to Cy and C is retained The bisection procedure is then repeated using the new interval from C to C Throughout the computations a value of C is considered acceptable if the computed ductility is within a specified ductility tolerance of the specified target ductility to avoid computation that achieves unnecessary precision A tolerance on yield strength coefficient is also needed however because of the possibility that a discontinuity in the strength ductility relationship is large enough that a solution can not be obtained that satisfies the ductility tolerance The tolerances on strength and ductility are specified as percentages of the average of C and C and the target ductility so that their scales are independent of the absolute values of strength and ductility The average of the lower and upper bound strengths is used because this value becomes a good approximation of the actual solution as the strength interval is redu
10. bilinear model Next specify the periods by selecting the uniformly spaced radio button and entering 60 periods ranging from 0 05 to 3 seconds For the parameter to vary select Constant Ductility Factor Choose 3 discrete values from the pull down menu and enter the values of 2 4 and 8 in the text boxes labeled Values For Viscous Damping enter 5 and enter 0 for Post Yield Stiffness Click the Compute Results command button Step 3 The results of the response spectrum computations are displayed in Step 3 Spectra corresponding to each parameter value are displayed with the color indicated at the top of the window Select the Zoom to Full Screen button to enlarge the plot Lines are plotted for each parameter value with a different color Figure 17 plots yield strength coefficient versus period for the ductility values of 2 4 and 8 The yield strength coefficient is also plotted against yield displacement in Figure 17 50 Base Shear Weight 0 Period sec Figure 17 Traditional Constant Ductility Spectra Base Shear Weight vs Period sec Base Shear Weight 0 2 Yield displacement cm Figure18 Yield Point Spectra Base Shear Weight vs Yield Displacement cm 51 4 Program Verification 4 1 Accuracy of Computational Engine To validate the accuracy of the USEE computations response was computed for selected cases that are reported in Dynamics of Structures C
11. factor restoring force of a SDOF oscillator restoring force of a SDOF oscillator at time t geometric ratio used for the specification of non uniformly spaced periods pseudo acceleration peak spectral displacement 78 T Ti t t RD tol u u t Ug Ug t u ut Ug u t Us li 1 Uroof Uroof t Uy eg R R SANE S PP AC At Au t Au t A t AR t AP t AT u Lr initial elastic period of vibration as a function of the initial stiffness k and mass m initial elastic period of vibration of the i mode of a multistory building time pulse duration when a pulse is used for base motion input pulse record duration when a pulse is used for base motion input convergence tolerance as a of the yield displacement displacement of a system relative to the ground displacement of a system relative to the ground at time t displacement of the ground relative to a fixed datum displacement of the ground relative to a fixed datum at time t total displacement of the system total displacement of the system at time u t u t ug t velocity of the ground with respect to a fixed datum velocity of the ground with respect to a fixed datum at time t acceleration of the ground with respect to a fixed datum acceleration of the ground with respect to a fixed datum at time t roof displacement with respect to the ground roof displacement with respect to the ground at t
12. follows the algorithms used by Mahin and Lin 1983 Algorithms introduced for constant ductility iterations are described in Appendix B The graphic interface for USEE was created using Microsoft Visual Basic 6 0 Dynamic Link Libraries DLL are used to communicate between the interface and engine USEE Help was developed using VB HelpWriter Software The following versions of these programs were used for development of USEE 2001 Microsoft Developer Studio 97 for DLL files Microsoft Visual Basic 6 0 SP3 with Service Pack 3 for graphical user interface VB HelpWriter Version 4 2 11 for USEE Help USEE utilizes two Dynamic Link Library DLL files One is for computation and the other is for manual testing of the load deformation models The code for the load deformation models is identical in both DLLs 60 APPENDIX B Algorithm for Computing Isoductile Response Spectra B 1 Introduction Methods for computing the linear elastic response of a single degree of freedom SDOF oscillator to a given ground motion were established in the late 1950s e g Newmark 1959 and were subsequently extended to oscillators having nonlinear load deformation relationships e g Veletsos and Newmark 1964 Wilson et al 1973 and Petkov and Ganchev 1998 In these methods the response is computed in the time domain by a series of sequential analyses each covering a small increment of time Ar Of particular interest is the relationship between the stre
13. input files The 11 3 filename convention used in the USEE 37 distribution follows the format EQYRSTATBRG EXT where EQ 2 characters representing the earthquake YR 2 digits representing the year of the earthquake STAT 4 characters representing the name of the recording station BRG 3 digits representing the compass bearing in degrees for horizontal motions or the characters UPW or DNW for vertical components EXT 3 characters denoting the file extension set equal to MAE For example IVAOELCNI180 MAE is the NS component of ground motion recorded at El Centro in 1940 located in the Imperial Valley of California The MAE extension denotes the use of the formatting style adopted by USEE 3 5 1 2 Format of Recorded Ground Motion Files The suite of recorded ground motions provided with USEE comes from a variety of sources They have been reformatted according to the convention described in this section All files begin with a header consisting of any number of lines each line beginning with the exclamation mark For example data from the 1940 NS El Centro record is reproduced below Mid America Earthquake Center Format on June 29 1998 Corrected Recorded Ground Motion Units are cm sec GENERAL INFO Earthquake Imperial Valley Date May 19 1940 Station El Centro Site Imperial Valley Irrigation District Component N180 PGA 341 7 RECORD SOURCE Source NCEER Source Identif
14. reduced while maintaining a specified level of accuracy In USEE as well as in NONSPEC Mahin and Lin 1993 the time step Af is selected at the beginning of the time step based on the following three criteria e At does not exceed the user specified time step AT e At does not exceed the time required to reach the next point at which the input acceleration is specified in the base input motion e At is adjusted smaller or larger to satisfy the specified convergence tolerance when the stiffness changes within the time step or a previous step The first two criteria for selecting the time step are checked before the step begins the last criterion is checked at the end of the step The last criterion concerns the convergence of the results when stiffness changes during the time step Figure 3 shows instances where the computed responses overshoot the bilinear load deformation model when the stiffness changes The solid lines in Figures 3 a and 3 b represent the paths followed by the computed responses when convergence tolerances are met The dashed lines represent the correct paths the responses should have taken To prevent excessive overshoot error the user can specify the convergence tolerance as a percentage of the yield displacement uy in USEE Overshooting also modifies the shape of the hysteretic curves as seen in Figure 3 b zero velocity turning point correct specified t 2 t computed with tolerance
15. shape vector 6 may be specified in USEE If an elastic mode shape is used for then the natural period of vibration of the SDOF analogue will match the period of vibration of the multi degree of freedom system whether computed using Eq 32 or Eq 33 If the shape vector is not identical to an elastic mode shape then the period of the SDOF analogue obtained in the ATC 40 implementation Eq 33 will not match the corresponding period of vibration of the multistory system while Eq 32 assures that the period of vibration of the equivalent SDOF matches a period of the multi degree of freedom Both implementations are available in USEE 15 0 normalized to be 1 at roof level Base Shear Force Force u Roof Displacement _ Wy Displacement l a 1 a b Figure 4 Establishing the properties of an equivalent SDOF system a capacity curve determined from the nonlinear static pushover analysis of the building b load deformation curve of the SDOF analogue derived from the capacity curve 2 2 2 Implementation of the SDOF Analogue in USEE The Multistory Building Approximation analysis module within USEE provides a simple means to estimate the displacement response of a multistory building based on a SDOF analogue The user specifies the distribution of floor mass over the height of the building story heights and the deflected shape to be used in making the anal
16. specified load deformation relation of the MDOF system using the formulation given in Section 2 2 1 as K 43 3 4 2 6 Post Yield Stiffness o The post yield stiffness is the slope of the load deformation curve after yielding of the oscillator It is specified as a percentage o of the initial stiffness a 44 35 where K is the slope of the load deformation curve after yielding Figure 8 Values of o between 100 and 100 may be entered for all analyses except for constant ductility response spectra for which only non negative values of o are allowed Negative values of post yield stiffness may result in the collapse of the oscillator Figure 9 b Under static loading collapse is defined when the restoring force decreases to zero at A of Figure 9 b At larger displacements the restoring force changes sign to act in the direction of the displacement causing the displacement to grow without limit Under certain dynamic conditions it is possible for the oscillator to exceed the static collapse displacement and not collapse provided that accelerations drive the oscillator back towards the origin USEE is internally set to halt computation if displacements exceed 1 2 times the static collapse displacement When this occurs a large dot is plotted at the last completed time step in the View Results window of the SDOF and Multistory Approximation modules In the Response Spectrum module no information is plotted for
17. that might cause these differences Although the same computational time step values are used for computed results of USEE NONSPEC and NONLIN time step values used for the results reported by Chopra were not identified Also USEE and NONSPEC reduce the time step values in regions where smaller time steps are required for convergence It is not clear whether NONLIN and the code used by Chopra reduce the time steps in these regions since published information does not address this issue Figures 19 and 20 52 compare the displacement history and base shear versus displacement response for the bilinear case having a period of 0 5 sec yield strength coefficient of 0 170 and damping value of zero percent USEE NONLIN MM NONSPEC Displacement cm 0 5 10 15 20 25 30 35 Time sec Figure 19 Comparison of USEE to NONLIN and NONSPEC Displacement cm vs Time sec 53 0 20 0 15 0 10 0 05 USEE NONLIN M NONSPEC 0 00 Force Weight Displacement cm Figure 20 Comparison of USEE to NONLIN and NONSPEC Force Weight vs Displacement cm 4 2 Accuracy of Multistory Building Approximation Analysis Example Equivalent SDOF models of multistory buildings are useful for estimating the peak displacements and displacement response histories e g Saiidi and Sozen 1981 Fajfar and Fischinger 1988 Qi and Moehle 1991 M
18. the menu bar 3 3 7 Input and Output Data Files Individual analysis data input files can be saved for subsequent recall and for use in the Compare Results step The filenames are saved with the following extensions SDOF for SDOF analyses BLDG for Multistory Building Approximation analyses and RSPC for Response Spectra analyses These extensions are automatically supplied if not specified by the user The files are binary Input parameters and output summaries may be saved as ASCII text when viewing response data Complete data files will be created and saved as ASCII text by clicking the Export Output As menu item when viewing response data Output files are saved with 30 TXT extensions The quantities available for export are shown in Table 2 Table 2 Response quantities available for export Single Degree of Freedom Displacement Multistory Building Approximation Roof Displacement Response Spectra Period Absolute Velocity Absolute Roof Velocity Yield Displacement Absolute Acceleration Absolute Roof Acceleration Damping Force Weight Force Weight Post Yield Stiffness Elastic Strain Energy Weight Elastic Strain Energy Weight Ductility Input Energy Weight Input Energy Weight Yield Strength Coefficient Kinetic Energy Weight Kinetic Energy Weight Peak Displacement Damping Energy Weight Damping Energy Weight Total Acceleration
19. the potential to generate a large amount of data If the user prefers data may be output less frequently without changing the size of the time step used in the computations The output time step is specified as an integer multiple of the computation time step The user may choose the output to be reported at 1 2 5 or 10 times the user specified time step Values of 2 or more cause corresponding reductions in the size of the data files The overshoot tolerance is used to check convergence for the nonlinear response any time that the stiffness changes It is specified as percentage of the yield displacement The convergence is considered to be satisfied in each case if the displacement at the end of a step during which a change in stiffness occurs does not differ by more than the user specified overshoot tolerance percentage of uy from the displacement at which the change in stiffness occurs Section 2 1 2 2 Parameters that control the constant ductility iterations are described in Appendix B 3 3 2 3 Export Options The user can manage the size of the exported output by choosing what to report from the provided checkbox list that includes displacement absolute velocity absolute acceleration force and energy related parameters 3 3 3 SDOF Analysis Steps The wizard interface for SDOF analysis presents the user with three windows in sequence the first two provide for data input and the third displays response quantities and plots The
20. the Capacity Spectrum Method Yield Point Spectra and direct computation of the response to a ground motion using software such as USEE Of the various recommendations for determining the equivalent SDOF system USEE allows the ATC 40 formulation to be used or an alternative formulation that matches the period of the SDOF analogue to the fundamental period of the building The vertical distribution of mass a deflected shape often estimated or assumed equal to the first mode shape the lateral strength of the building and either the fundamental period of vibration of the building or the roof displacement that corresponds approximately to yielding of the system are needed to establish the properties of the SDOF analogue The lateral strength and yield displacement are those that would be observed in a nonlinear static pushover analysis of the building when lateral forces are imposed consistent with the assumed mode shape and mass distribution The USEE implementation assumes that the building may be modeled as a planar structure responding laterally with mass lumped at each floor level Second order P delta effects and multiaxial excitations transverse and vertical response are not explicitly considered 2 2 1 The Equivalent Single Degree of Freedom System The SDOF analogy relies on the assumptions that the response of the multistory 11 building is predominantly in a single mode and that the deflected shape is proportional to t
21. 1 Cy i Figure B9 Definition of smoothness ratio e a in the context of checking an interval for rejection If the smoothness ratio exceeds the smoothness tolerance at any intermediate strength the possibility that a solution may exist in the vicinity of the current yield strength is pursued further Figure B10 illustrates this case for which two additional points are added each halfway between the current C and the closest points on either side The smoothness of the strength ductility relation is now evaluated at the original points C Cy C 5 and at the added points The estimated ductilities are now based on linear interpolation making use of the ductility value that was just determined In this manner the interpolation function begins to conform more closely to the actual strength ductility relation where it previously had violated the smoothness criterion Note that points are not added if the distance between adjacent points would be less than the specified tolerance on the yield strength coefficient in this case the algorithm proceeds to the next previously established point This process is repeated for each interpolated point until all points between C and C are evaluated 70 e Computed Ductility o Interpolated Ductility Check reject region Cy Cye Cys Cy2 Cyr Cyu Figure B10 Checking for a possible solution in the check reject region If the ductility computed at an intermediate strength exceeds the targ
22. 1r01 8 0 17 4 166 4 106 2 0 90 sec cm sec 2 0 00 0 1221E 00 0 01 0 1231E 00 0 02 0 1222E 00 0 03 0 1201E 00 0 04 0 1198E 00 oop The header information beginning with the mark was inserted when the motions 40 were prepared for distribution in USEE Only the bold text lines are required for USEE to use the synthetic data in the response analyses No information about the record is displayed to the user in the Synthetic Motions tab of the base input window However the user may view the file header from this window by clicking the View File Header eqo command button The two lines following the lines beginning with originated in the RR 1 project The first of these identifies the file ID moment magnitude focal depth km epicentral distance km closest horizontal distance to the surface projection of rupture plane km and deviation from median attenuation The second line contains titles for the columns of synthetic motion data Synthetic motion data begins on the third line USEE assumes the first column is time in sec and the second column is acceleration in cm sec Selecting the View Accelerogram button in the base input step causes a window to appear where the user may view the accelerogram of the selected ground motion The View File Header button may be selected to show the ground motion file information all lines that begin with a 3 5 3 Pulses Several pulse types may be selected for the base
23. B2 Computed load deformation response to 1992 Landers earthquake at Joshua Tree Fire Station NS for a 1 second period oscillator 20 O Ductility u 0 0 2 0 4 Yield Strength Coefficient Cy Figure B3 The strength ductility relationship for a bilinear oscillator having a period of T 0 15 sec responding to the 1987 Whittier Narrows record The inverse problem with which this Appendix is concerned is to determine the strength coefficient C which causes u to be equal to a specified value Figure B3 illustrates the well known trend that u increases as C decreases Upon first inspection a reasonable solution strategy might be to compute the ductilities obtained for arbitrarily chosen strengths interpolating until a solution of adequate precision is obtained However several properties of the strength ductility relationship require that a more 63 sophisticated solution strategy be employed Multiple solutions may exist Figure B4 a shows the strength ductility relationship for a bilinear oscillator having T 0 20 sec responding to the NE component of the 1987 Whittier Narrows earthquake recorded at the Mt Wilson Caltech Seismic Station In this case a peak ductility of 1 4 is obtained for several different strengths the largest yield strength coefficient is more than 40 greater than the smallest yield strength coefficient Reporting any one of these strength coefficients as the answer would introduce a degree of a
24. Building Structures Final Report on a Study to the American Society for Engineering Education Postdoctoral Fellowship Program September 1985 ATC 40 Seismic Evaluation and Retrofit of Concrete Buildings Volumes I and 2 Applied Technology Council November 1996 Report No SSC 96 01 Black E and Aschheim M 2000 Seismic Design and Evaluation of Multistory Buildings Using Yield Point Spectra CD Release 00 04 Mid America Earthquake Center University of Illinois Urbana September Boroschek R L and Mahin S A PCNSPEC Manual A Modified Version of NONSPEC 1991 unpublished Charney F Nonlinear Dynamic Time History Analysis of Single Degree of Freedom Systems NONLIN Version 6 01 Advanced Structural Concepts Inc Golden Colorado and Schnabel Engineering Denver Colorado under a contract with Federal Emergency Management Agency 1998 Also available from http www fema gov emi nonlin htm Chopra A Dynamics of Structures Theory and Applications to Earthquake Engineering Prentice Hall Inc New Jersey 1995 Clough R W and Penzien J Dynamics of Structures 2 Edition McGraw Hill Inc New York 1993 Fajfar P and Fischinger M N2 A Method for Non Linear Seismic Analysis of Regular Structures Proceedings from the Ninth World Conference on Earthquake Engineering Tokyo Kyoto Japan 1988 FEMA 273 NEHRP Guidelines for the Seismic Rehabilitation of Buildings Report No FEMA 273 Feder
25. E SEE Help SEE Multistory SEE Response Spectra SEE SDOF SEE Waveforms SEE Waveforms Recorded SEE Waveforms Synthetic SEE Waveforms Synthetic Hard Rock SEE Waveforms Synthetic Soil C coc C CH C C The subdirectories titled Multistory Response Spectra and SDOF contain input data for the analyses that were previously saved by the user The Help folder contains files necessary for the help menus The Waveforms subdirectory contains individual files for each synthetic and recorded ground motion made available in USEE If the user wishes to use a ground motion record not supplied with USEE the file should be placed in the recorded waveforms subdirectory Formatting requirements for user supplied accelerograms are described in Section 3 5 1 2 The synthetic ground motion files are distributed into separate subdirectories based on the soil type 3 3 Using the Program 3 3 1 Description of Commands Command buttons and toolbar commands are as follows Back takes the user back to the previous step in an analysis module Compare Results takes the user to the Compare Results window This requires the current analysis to be saved raising a dialog box if needed Compute Results computes results with user specified input data and advances to the View Results window Default values are used for any steps omitted by the user Copy copies the selected plot as a bitmap image to the Windows clipboard Export Output saves the current
26. EBU SEHORL cope totae Poi p E SIN Meno tia a rebote ete a pa 4 2 13 3EHetey Tertiisacs uem bre po eee dete aa d a a Cate qn 4 2 1 3 Computational Aspects s ooo irae de oa pou ioa portas toe etie ad dads 6 2 2 Single Degree of Freedom Analogies of Multistory Buildings 11 2 2 1 The Equivalent Single Degree of Freedom System 11 2 2 2 Implementation of the SDOF Analogue in USEE eee 16 2 3 Computation of Response Spectra Using USEE eee 17 2 3 4 Specification of Periods sistata dea aaa ae aan to S RR REO a tates ona 18 2 3 2 Computation of Elastic Response Spectra mmm nenea nana 18 2 3 3 Computation of Inelastic Response Spectra m nenea nana 19 3 Users Manidl edo d oe qe ul inei ttd ta a al dam te 23 3 1 Installing and Maintaining the Program nenea nenea 23 Ali USEE Distribution tate etr or ur Ran rac Ra Pe ERE EC aa auda 25 3 1 2 Hardware Recommendations ccccccceeeeseeeennceeeeeeeeeeeeeennneeeeeeeeeeeseees 23 J3 Un mstellapom G de sess eee pe ate og CER RM DUE 23 3 1 4 Maintenance and Support anna nana nennen 23 3 2 Programi DCS UIs uoi ia a ai ta aa arty een gol na REIN 24 oS M d l OperatiOlic asus RE n tenes dus uadit ERU EE C a iai ra 24 3 2 2 Directory Structure ensis tetas e baie ced areas eddie 24 3 9 Using the Prost ATD za cc sets DIS na uit DANS DR a ERES IGNES ERR bd in 25 3 3 1 Description of
27. USEE 2001 Utility Software for Earthquake Engineering Report and User s Manual Mid America Earthquake Center Mehmet Inel Erich M Bretz Edgar F Black Mark A Aschheim and Daniel P Abrams Civil and Environmental Engineering University of Illinois at Urbana Champaign Urbana Illinois October 2001 ACKNOWLEDGMENTS The Mid America Earthquake Center is a National Science Foundation Center for Earthquake Engineering Synthetic ground motions distributed with USEE were developed by Y K Wen and Chiun Lin Wu as part of project RR 1 of the Mid America Earthquake Center 1999 Previous work by Mahin and Lin 1983 which included a variable time step algorithm developed by Professor R Klingner of the University of Texas at Austin Abrams 1985 and Boroschek and Mahin 1991 was used in the development of this software This work was supported primarily by the Mid America Earthquake Center under the Earthquake Engineering Research Centers Program of the National Science Foundation under Award Number EEC 9701785 Windows 95 Windows 98 Windows 2000 Windows NT 4 0 and Microsoft Word are registered trademarks of the Microsoft Corporation TERMS AND DISCLAIMER Considerable time effort and expense have gone into the development and documentation of Utility Software for Earthquake Engineering USEE The program has been thoroughly tested and used However no warranty of any kind express or implied is made with respect to
28. ad deformation response and computed response data Step 1 Specify number of stories story heights mass or weight distribution and mode shape Step 2 Select the appropriate tab to choose recorded ground motions synthetic motions or simple pulses for base input acceleration Step 3 Select load deformation model and specify parameter values to define the base shear versus roof displacement relation Step 4 View response plots and summary statistics Pull down list boxes allow various quantities to be plotted in the View Results window 3 3 5 Response Spectra Analysis Steps This module provides three windows two provide for data input and the third displays response data and plots Step 1 Select the appropriate tab to choose recorded ground motions synthetic motions or simple pulses for base input acceleration Step 2 Select a load deformation model and specify response spectra parameters Parameters that may be varied include viscous damping yield strength coefficient strength reduction factors and displacement ductilities The last case requires an iterative solution to determine oscillator strengths and is more time consuming Step 3 View summary statistics and response plots 29 Pull down list boxes allow the user to select quantities to be plotted The selection set depends on the type of response spectra that were computed Quantities may be plotted versus period the peak relative displacement or yield dis
29. al Emergency Management Agency Washington D C May 1997 FEMA 274 NEHRP Commentary on the Guidelines for the Seismic Rehabilitation of Buildings Report No FEMA 274 Federal Emergency Management Agency Washington D C October 1997 Hachem M M BISPEC Version 1 1 2 University of California Berkeley 2000 Available from http www ce berkeley edu hachem bispec index html Lawson R S Vance V and Krawinkler H Nonlinear Static Push Over Analysis Why When and How Proceedings of the Fifth U S Conference in Earthquake Engineering Earthquake Engineering Research Institute Oakland California 1994 Mahin S A and Bertero V V An Evaluation of Inelastic Design Spectra Journal of Structural Engineering ASCE Vol 107 No ST9 September 1981 83 Mahin S A and Lin J Construction of Inelastic Response Spectra for Single Degree of Freedom Systems Report No UCB EERC 83 17 Earthquake Engineering Research Center Berkeley California 1983 Miranda E Seismic Evaluation and Upgrading of Existing Buildings Ph D dissertation Department of Civil Engineering University of California Berkeley California 1991 Newmark N M A Method of Computation for Structural Dynamics Journal of the Engineering Mechanics Division ASCE Vol 85 No EM 3 July 1959 Newmark N M and Hall W J Seismic Design Criteria for Nuclear Reactor Facilities Technology Report 4 Building Practices for Disaster Mitig
30. analysis output as an ASCII text file under a user specified file name Exit SDOF Oscillator Session exits the current SDOF Oscillator analysis session closing all windows except the main USEE window Exit Multistory Building Approximation Session exits the current Multistory 25 Building Approximation analysis session closing all windows except the main USEE window Exit Response Spectra Session exits the current Response Spectra analysis session closing all windows except the main USEE window Exit USEE exits the program closing all windows Load Existing File loads an existing file of the current analysis type Single Degree of Freedom Approximate Multistory Building and Response Spectra The existing file is opened input values are loaded response is computed and results are displayed in the Compare Results window This is a shortcut to facilitate comparing responses from multiple analyses New creates a new analysis file having an extension appropriate for the current analysis type e g NEW1 SDOF NEWI BLDG OR NEWI RSPC Next takes the user to the next step in an analysis module Open locates and opens a previously saved analysis file the file name extension is appropriate to the current analysis type Refresh Plots refreshes plots in the View Results window This is needed whenever the user changes a plotting parameter such as color time interval or number of plots Return To Results Window closes the cu
31. ape i ot Desk du eb a 83 Table 1 Table 2 Table 3 Table 4 Table B1 Table B2 TABLES Constant ductility algorithm parameters eee 22 Response quantities available for export nenea nana 31 Validation of SDOF code netta tede d a a dia a tati 52 First elastic mode shape of the 12 story building frame 57 Ground motions used in the computations ceea 74 Clock time required to compute response spectra for different ductilities using different software programs 1 eene tete tortor eno Pet alta ap ena Pent a 76 Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Figure 9 Figure 10 Figure 11 Figure 12 Figure 13 Figure 14 Figure 15 Figure 16 Figure 17 Figure 18 Figure 19 Figure 20 Figure 21 Figure 22 Figure 23 Figure 24 Figure 25 Figure Bl Figure B2 Figure B3 FIGURES SDOF system subjected to ground acceleration cecene eee 4 Schematic illustration of absorbed energy for an oscillator with a bilinear load deformation relationship ceeace 6 Effect of convergence tolerance on a overshoot during loading and b hysteretic response on unloading cc i oaie za cca tectae nerd a sie 10 Establishing the properties of an equivalent SDOF system a capacity curve determined from the nonlinear static pushover analysis of the building b load deformation curve of the SDOF analogue der
32. are provided to illustrate the Single Degree of Freedom Approximate Multistory Building and Response Spectra modules To begin run USEE by selecting Utility Software for Earthquake Engineering from the Start menu This is a good time to set your preferences of units parameters and export options although they may be changed subsequently without affecting the underlying data Set the parameters by selecting Options from the Preferences menu on the menubar To specify the computation time step select the Parameters tab on the window that appears Typically a value not exceeding 1096 of the period is used The program will automatically reduce the time step if required for convergence of the solution Set the Computation Time Step to 0 01 sec To report results for every computation time step 0 01 sec select 1 for the Report Results box Enter 1 to set the overshot tolerance as 1 of the yield displacement Click on Export Options tab to choose what to export from the provided checkboxes You may set the units you prefer from either the Units tab of current window or the Units pull down list box on the toolbar 3 6 1 SDOF Analysis Example The first example computes the response of a SDOF system to the 1940 NS El Centro record The oscillator has an initial period of vibration of 0 75 sec yield strength equal to 30 of its weight viscous damping equal to 5 of critical damping and is modeled as having an elastic perfectly pla
33. ation National Bureau of Standards U S Department of Commerce 1973 Newmark N M and Riddell R Statistical Analysis of the Response of Nonlinear Systems Subjected to Earthquakes Civil Engineering Studies Structural Research Series No 468 University of Illinois Urbana August 1979 Petkov Z B Ganchev S G An algorithm for computation of inelastic response spectra Proceedings of the Eleventh European Conference on Earthquake Engineering A A Balkema Rotterdam 1998 Prakash V Powell G H and Campbell S 1993 Drain 2DX Base Program Description and User Guide Report No UCB SEMM 93 18 Structural Engineering Mechanical and Materials Berkeley California 1993 Qi X and Moehle J P Displacement Design Approach for Reinforced Concrete Structures Subjected to Earthquakes Report No EERC 91 02 Earthquake Engineering Research Center Berkeley California 1991 Satidi M and Sozen M A Simple Nonlinear Seismic Analysis of R C Structures Journal of Structural Division ASCE 1981 Vol 107 No STS Uang C M and Bertero V V The Use of Energy as a Design Criterion in Earthquake Resistant Design Report No UCB EERC 88 18 Earthquake Engineering Research Center Berkeley California Nov 1988 Vidic T Fajfar P Fischinger M Consistent Inelastic Design Spectra Strength and Displacement Earthquake Engineering and Structural Dynamics vol 23 pp507 521 1994 Wi
34. ationship If the interval between C and C is identified as smooth it may be rejected from further consideration Then the solution bounds are revised by setting Cyu Cye and the algorithm restarts with the new bounds e Computed Ductility u C o Interpolated Ductility Figure B8 The case where u C lt Lu The number of discrete values of C that is checked and the allowable deviation of the actual ductility values from the interpolated values impact the efficiency of the algorithm and determine the odds of an unrecognized higher strength solution Of particular concern is the unusual instance in which the strength ductility relation happens to coincide with the interpolated ductilities at the chosen values of C but deviates significantly from the interpolated function elsewhere Considering this possibility more reliable conclusions may be obtained when several points are checked rather than just one or two and checking several points allows the tolerances to be relaxed somewhat relative to cases in which fewer points are checked Experience indicates that a region may be discarded when the smoothness ratio is less than a smoothness tolerance of 0 20 at three successive points For this reason the interval between C C is divided into 4 segments in Figure B9 The algorithm proceeds sequentially from C to C 69 e Computed Ductility o Interpolated Ductility EN Check reject region Cy Cye Cy Cy 2 Cy
35. ator The relative input energy is ultimately dissipated through damping and hysteretic losses The dynamic portion of the response also contains kinetic energy associated with the relative velocity of the mass and potential energy associated with the elastic strain energy The relative kinetic energy Ej of the mass obtained by integrating the first term of Eq 3 Uang and Bertero 1988 is E jrm t lu E 5 The energy dissipated by viscous damping Ez given by integration of the second term of Eq 3 is E fjcu t du f clu at 6 The energy absorbed by the oscillator is composed of recoverable elastic strain energy E and irrecoverable hysteretic energy En These are obtained by integrating the third term of Eq 3 E f R u du E E 7 where rf E 2K 8 where K the initial elastic stiffness of the oscillator The recoverable strain energy Es and the dissipated hysteretic energy En are shown schematically in Figure 2 for a bilinear oscillator Force R hysteretic energy dissipated En pu E recoverable strain EN NE energy Es Relative Displacement u Figure 2 Schematic illustration of absorbed energy for an oscillator with a bilinear load deformation relationship Thus E E E E E E E E 9 USEE allows each of these quantities to be exported In the View Results step of the Single Degree of Freedom analysis and Multistory Building Approximation m
36. c and the first mode shape is shown in Figure 22 In the present case the first elastic mode shape was considered an adequate approximation of the predominant mode shape This results in the same period of vibration for the multi degree of freedom system and its equivalent system when the ATC 40 procedure is used The load deformation response of the building frame was obtained using a nonlinear static pushover analysis that was done by applying lateral forces in proportion to the mode shape amplitude and mass at each floor level Figure 23 using DRAIN 2DX A bilinear curve was fit to the capacity curve to determine the yield strength and displacement for response in the first mode The displacement of the roof at yield is 0 353 m or 0 72 of the height of the building and the base shear coefficient at yield is 0 173 The post yield stiffness is 17 5 of the initial stiffness The response of the building frame was estimated using the Multistory Building Approximation analysis module of USEE In step 1 of the USEE module a user defined mode shape was selected and values from Table 4 are used to specify the elastic first mode shape In the second step the El Centro record was selected scaled by a factor of 2 Load deformation properties established from Figure 22 were specified in the third step A bilinear model was selected specifying a base shear coefficient at yield of 0 173 a post yield stiffness of 17 5 of the initial stiffness and
37. c response spectra USEE calculates the response histories using the numerical method described in Section 2 1 and determines the peak relative displacement and spectral acceleration Sz for the specified values of period and damping Peak relative displacement or pseudo acceleration may be plotted against period for each value of damping 2 3 3 Computation of Inelastic Response Spectra Inelastic response spectra provide a convenient means to summarize the peak responses of nonlinear SDOF oscillators subjected to a specified base input motion Three types of inelastic response spectra may be computed in USEE constant strength spectra constant strength reduction factor R factor spectra and constant ductility spectra Each type of spectra may be computed using the bilinear or stiffness degrading load deformation model The excitation load deformation model damping and post yield stiffness values are kept constant in any computation The user specifies a period range and up to 5 values of the strength R factor or ductility for which the spectra are to be computed Spectral response quantities yield strength coefficient peak strength normalized by weight peak relative displacement peak ductility and absolute acceleration normalized by the acceleration of gravity g may be plotted versus period of vibration yield displacement or peak relative displacement The type of response spectrum the number of periods and the number of paramete
38. ced B 4 Comparison of Results with Other Programs The present algorithm is implemented in the USEE program Results obtained with this implementation are compared with those obtained using PCNSPEC Borosheck 1991 and BISPEC Hachem 2000 for several sample ground motions listed in Table 73 B1 Both the computed results and the clock time required for the computations are discussed All three programs compute response during successive time intervals using the linear acceleration method a special case of the Newmark Beta Method Clough and Penzien 1993 for which o 1 2 and B 1 6 For each program a time step of 0 01 sec was specified although each program may use different subdivisions of this interval as needed to satisfy convergence criteria All computations were performed for a bilinear load deformation model on a 300 MHz Pentium II computer with 128 MB RAM running Windows 98 Table B1 Ground motions used in the computations Record ID Earthquake Year Station Component PGA g bb92civc360 Big Bear 1992 C ee ah 300 0 545 ch85lleo010 Chile 1985 E 1 N10 0 712 iv40elcn180 Imperial Valley 1940 El Centro N180 0 312 mx85sct1270 Mexico City 1985 edes ied N270 0 171 wh87mtwl090 Whittier 1987 M N90 0 185 B 4 1 Accuracy of Constant Ductility Response Spectra Constant ductility response spectra were computed for the three programs and the five ground motions of Table B1 Figures B12 a
39. cillator 3 4 2 2 Base Shear Coefficient C The base shear coefficient C is defined as the base shear strength at yield divided by the weight of the building C 5 39 W where V base shear strength of the building at yield and W weight of the building 3 4 2 3 Period of Vibration The natural period of vibration T of the system is defined as the time required to complete one cycle of free vibration of the system if undamped This period is related to the circular frequency by T 21 0 where M 34 and K and M are the stiffness and mass of the SDOF oscillator respectively The frequency of vibration f is the inverse of the period 1 PUE 3 4 2 4 Damping Viscous damping is specified relative to critical damping with critical damping defined as c 2Mo 40 The critical damping ratio 5 is specified in the USEE input defined by c c m 2Ma 41 Values of 2 5 are typical of many common structures 3 4 2 5 Initial Stiffness The initial stiffness is the slope of elastic portion of load deformation response of oscillator It must be positive For Single Degree of Freedom analysis the stiffness may be determined as the ratio of yield strength F and yield displacement uy F K 42 u y For Multistory Building Approximation analysis the stiffness can be determined from the load deformation relation of the equivalent SDOF system This stiffness may be derived from the user
40. d prior to installing the new version 3 1 3 Un installation Guide USEE may be uninstalled using the Windows uninstall feature In Windows 95 98 2000 and NT4 0 this is accessed from the Control Panel under the Add Remove Programs icon This process will not delete files common to other installed applications 3 1 4 Maintenance and Support Support is handled electronically via the Mid America Earthquake Center web site http mae ce uiuc edu This site provides information on e The current release of the program e Comments from users and bug reports 23 e Release history information Please feel free to contact the authors to provide your comments to request new features and to report bugs inel uiuc edu ebretz Quiuc edu and aschheim uiuc edu 3 2 Program Design The program utilizes a modular design Program modules are accessed from the main window by mouse driven command menus USEE 2001 provides modules for e Single Degree Of Freedom SDOF analysis e Multistory Building Approximation analysis using SDOF analogues e Response Spectra computation Each module is implemented using a wizard that guides the user through a series of windows for data entry and viewing of results Each window is a step in the module and the user may freely navigate forwards and backwards through the data input screens Each time the Compute Results button is clicked a run number is assigned to the analysis This run nu
41. d results are displayed in this step The two plots may be used to display load deformation response and the displacement or acceleration histories as well as other quantities Peak quantities are tabulated Figure 11 and Figure 12 show load deformation response and the displacement history for the SDOF example After viewing the results the user may start a new analysis by clicking the Start New Analysis button or may click the View Summary Log button to view the input parameters for the current analysis Response quantities determined during the analysis may be saved to a text file by selecting Export Output from the File menu Plots may be copied to the Windows Clipboard using a left mouse click to select the plot and a right mouse click to copy the plot to the Clipboard The plots may be pasted into other Windows applications from the Clipboard 44 Base Shear Weight 6 Displacement cm Figure 11 SDOF example Base Shear Weight vs Displacement cm Displacement cm 0 Time sec Figure 12 SDOF example Displacement cm vs Time sec 45 3 6 2 Multistory Building Approximation Analysis Example The approximate response of a 4 story moment resisting frame structure is computed for the El Centro record In this example the drift at yield of the frame is 0 25 of the total building height The frame has a base shear coefficient at yield equal to 25 story weights of 318kN story heights of 4 m viscous damping equal
42. damping equal to 5 of the critical damping Either period or yield drift of the frame can be specified for the ATC 40 implementation Since the elastic first mode shape was used the period associated with the first mode shape was specified as 2 17 sec The response computed using USEE is compared with the response computed in the nonlinear response of the MDOF system using DRAIN 2DX in Figures 24 and 25 Figure 24 compares the roof displacement histories Figure 25 compares the base shear versus roof displacement respectively From the figures it can be observed that the roof displacement history of the equivalent SDOF model captures the essence of the roof displacement response determined for the MDOF system However base shear versus roof displacement response is poorly represented by the equivalent SDOF system While the base shear roof displacement histories are dissimilar it may be observed that the estimate based on the equivalent 56 SDOF system provides reasonable estimates of the peak quantities Although the equivalent SDOF bounds the response the details of the load deformation response of the equivalent SDOF model and MDOF model are very different The goodness of the displacement history shown for the 12 story steel moment resistant frame building demonstrates that the equivalent SDOF model based on the first mode shape can be useful for estimating peak roof displacement and roof displaceme
43. de shape with the circular frequency associated with vibration in the i mode Substituting Eqs 28 and 24 into Eq 27 results in T 2 1 M Tu _ QOL Qo 29 1 M1 2 g The yield strength of the SDOF analogue F can be expressed as o 2 k F Ku 0 Mu 30 Hence the yield strength coefficient of the SDOF oscillator is F s C229 t 31 y M g g y To cause the SDOF analogue to have a natural period of vibration that matches the i period of vibration of the MDOF system the circular frequency w should be set equal to 14 the i circular frequency Doing so results in 2 Di ast Oh 2 Ds ve l where T the natural period of vibration of the i mode This implementation Eq 32 assures that the natural period of the SDOF analogue matches a natural period of vibration of the MDOF system regardless of whether the shape vector corresponds to an elastic mode or not The ATC 40 implementation uses Eqs 29 and 32 to express Cy as Cc 33 The yield strength of the SDOF analogue is given by V C W V T representing the notion that the yield strength coefficient associated with the mass that participates in the i mode can be related to a smaller yield strength coefficient C that is associated with the total mass of the structure Eq 33 is used to determine the strength of the SDOF analogue in ATC 40 and represents one of the implementations available in USEE Any
44. ding Approximation Analysis Example 46 3 6 3 Response Spectra BXampleS ues OH ISIN NOI ERI detis Un A TRA 47 4 Propram Verification ssenari tentare a a Hoa rbi aiite a ela at ela 52 4 1 Accuracy of Computational Engime eee nenea nea 52 4 2 Accuracy of Multistory Building Approximation Analysis Example 54 APPENDIX A Software Development Tools nene nenea nenea ennnnaee 60 APPENDIX B Algorithm for Computing Isoductile Response Spectra 61 B 1 Tntrod ctio m ERROR Tc CN 61 B 2 Properties of the Strength Ductility Relationship eese 61 B 3 Description of the Algorithm soapta Ses ese rd ea acne anaes 65 B 3 1 Initial Bounding of SOIULIOH ua ee Re ph n aid eee petas teu oben 66 B 3 2 Fast Search BISEctiOni ses ei aceata oala ea io codes cc desti ed datat 71 B 4 Comparison of Results with Other Programs eee 43 B 4 1 Accuracy of Constant Ductility Response Spectra sssssssss 74 B 4 2 Computational EfBCICBCy sese iiie he ees ue tartare Mewes 75 Boo OMG i ice o ics o Sick tenis dit Sat PRI CR ICI e a Sachin a RR e te pa en ASA Ret dd B 6 Glossary Defimnitions iii cete tette eese tla eda deae da aaa ada e TI APPENDIX C NOGANOM M 78 APPENDIX D USEE Orp anzation n ip ec spi de oe inne ea a ded eie desde 8l APPENDIX E References 2 piede on beste vest eee bei ed
45. enough to require further investigation of a possible solution on either side of the current yield strength coefficient Tolerance on Target Ductility is a user specified tolerance that determines whether the current yield strength coefficient may be reported as a solution If the computed ductility is within the user specified tolerance from the target ductility the corresponding C is reported as the solution Tolerance on Yield Strength Coefficient is a user specified tolerance that halts the iteration on strength when successive values differ by less than the specified tolerance The tolerance is specified as a percentage of the average of the two adjacent values of C Target Ductility is the specified displacement ductility for which the associated yield strength coefficients are determined by iteration for each specified period Yield Strength Coefficient is the yield strength of a SDOF oscillator normalized by its weight 77 APPENDIX C Notation AANAAAS POs mS A Q i L u SSA cm On IO v v Q t Q t R t Sa Sa the distance between the target and expected ductilities pulse amplitude g when a pulse is used for base motion input yield strength coefficient of a SDOF oscillator yield strength coefficient of a SDOF analogue equivalent yield strength coefficient of a SDOF analogue yield strength coefficient obtained by bisecting the interval between C and Cy expected yield strength coef
46. et ductility the process is halted and new bounds are established by setting C equal to the current yield strength coefficient and setting C to the previous C Because the algorithm works down progressively from C if a ductility is computed that is within the user specified ductility tolerance the corresponding C is reported as solution If the check reject approach determines that the region from C to Cye can be discarded then the algorithm restarts with the upper and lower bounds set equal to C and C respectively This continues until the strength interval between C and C is smaller than a user specified value or the computed ductility is within a specified percentage of the target ductility In either of these events the algorithm switches to the fast search bisection phase B 3 2 Fast Search Bisection The fast search portion of the algorithm assumes the bounds of the solution have been narrowed sufficiently that the first solution obtained within these bounds is the correct solution that is a higher strength solution is assumed not to exist The solution bounds are those determined in the first phase of the algorithm The flowchart for the fast search interpolation is illustrated in Figure B11 Key points are described in the following 71 Divide the region bounded by Cy and Cy into 10 or less segments such that AC lt ACy to and set Cy e Cy Decrease Cy by ACy Compute u Cy e Y No Swi
47. f convergence is not obtained with the reduced time step the solution for the last step using the reduced time step is discarded and a new time step equal to 1 10 of the previous time step i e one hundredth of the original is used This process is repeated until the tolerances are satisfied However if the time step is reduced 10 5 times to 1x10 of the original time step and satisfactory convergence is not obtained the program stops and notifies the user of the failure to converge 2 2 Single Degree of Freedom Analogies of Multistory Buildings Many research studies e g Saiidi and Sozen 1981 Fajfar and Fischinger 1988 Qi and Moehle 1991 Miranda 1991 and Lawson et al 1994 have shown that the displacement response of multi degree of freedom MDOF buildings often may be approximated by a single degree of freedom SDOF system when response is predominantly in a single mode The SDOF analogue is often termed an equivalent SDOF system Various equivalent systems have been described in the literature but in some cases these systems differ from one another and hence do not represent the concept of equivalency For this reason such systems are referred to as SDOF analogues herein SDOF analogues are used to estimate displacement response in the Nonlinear Static Procedures NSPs of ATC 40 1996 and FEMA 273 274 1997 Methods for estimating the response of the nonlinear system include the Displacement Coefficient Method
48. fast search 22 3 User s Manual 3 1 Installing and Maintaining the Program 3 1 1 USEE Distribution The USEE distribution is compiled for use with the Windows 95 98 2000 and NT4 0 operating systems USEE is distributed in compact and full versions The compact version includes with the software a very limited suite of synthetic motions The full version includes the complete suite of 120 synthetic motions that were generated in the Mid America Earthquake Center Project RR 1 Both versions include a modest suite of recorded ground motions 3 1 2 Hardware Recommendations The compact version requires approximately 16 MB of disk space depending on whether files common to other applications are already present The full version requires an additional 25 MB of disk space A screen resolution of 1024 x 768 is recommended although an 800 x 600 display resolution is sufficient 3 1 2 1 Obtaining and Installing the Program The software may be obtained using any standard internet web browser from the Mid America Earthquake Center web site http mae ce uiuc edu Once downloaded installation is as simple as double clicking on the SETUP EXE file The installation routine will present a number of dialog boxes Files will be installed on the user s hard drive Existing files will not be replaced without the user s explicit consent If USEE is being installed over a pre existing installation of USEE the pre existing installation should be uninstalle
49. ficient of a SDOF oscillator obtained by interpolation between C and Cy yield strength coefficient at point i lower bound of C interval for which the corresponding peak displacement ductility LL is larger than the target ductility Li upper bound of C interval for which the corresponding peak displacement ductility u is smaller than the target ductility Li critical damping absorbed energy irrecoverable hysteretic energy relative input energy relative kinetic energy energy dissipated by viscous damping recoverable elastic strain energy deviation of the computed ductility value from the value expected based on linear interpolation yield strength of a SDOF oscillator equivalent yield strength of a SDOF analogue initial stiffness of a SDOF oscillator frequency of vibration of a SDOF oscillator acceleration due to gravity tangent stiffness of a SDOF oscillator equivalent tangent stiffness of a SDOF analogue mass of a SDOF oscillator equivalent mass of a SDOF analogue applied force to a SDOF oscillator applied force to a SDOF oscillator at time t vector of story forces at the floor levels for a multistory building vector of story forces at the floor levels for a multistory building at time t vector of equivalent story forces at the floor levels of a SDOF analogue of a multistory building vector of equivalent story forces at the floor levels of a SDOF analogue of a multistory building at time t strength reduction
50. following actions are required Step 1 Select the appropriate tab to choose base input recorded ground motions synthetic motions or simple pulses Step 2 Select a load deformation model and specify values of model parameters Step 3 View response plots and summary statistics Quantities to be displayed on the plot are selected from the pull down list boxes located on each plot axis The Zoom to Full Screen button provides greater detail With the cursor located over any plot a right click of the mouse or clicking the toolbar Copy 28 button copies the plot to the Windows clipboard The plot can now be pasted into other Windows applications such as Microsoft Word Analysis results may be exported to formatted ASCII files by clicking on the Export Output button Results may be compared to previously completed analyses in the Compare Results window accessible from this step The input files for the previous analyses must have been saved previously 3 3 4 Multistory Building Approximation Analysis Steps The displacement response of buildings that respond predominately in a single mode may be determined approximately using an analogous SDOF oscillator The oscillator characteristics may be established using the procedure described in Section 2 2 1 of this report The drift profile shape vector story heights and mass distribution are specified in the first input window of this module three subsequent windows characterize the base input lo
51. g to the 1987 Whittier Narrows record for an oscillator period of a 0 20 sec and OYONI C EE 65 Linear interpolation between Cy and C sess 66 The first phase of the algorithm for determining the initial bounds on the SO UI mL c asi 67 The case where 6 2 9 Iis iieri etie tl aei esae eon e a ean 68 The Case where di C Pt iz euer Ol a apart aa 69 Definition of smoothness ratio e a in the context of checking an interval IE Tee ct IDE en Gea iets pasa e bande 80 a a 20 Med qeu 70 Checking for a possible solution in the check reject region 71 Bisection with the fast search algorithm eeeeesesse 12 Constant ductility response spectrum for u 2 for the El Centro record 75 Constant ductility response spectrum for u 2 for the Llolleo record 75 vii 1 Introduction Utility Software for Earthquake Engineering USEE provides a Windows based user friendly graphic interface for performing simple computer simulations of the response of structures subjected to earthquake ground shaking and for accessing data and products of the Mid America Earthquake Center The visual interface allows students practicing engineers and researchers to quickly simulate nonlinear dynamic response and to understand the influence of parameter variations on response characteristics Because the results are easily obtained using a point and click interface
52. his mode shape throughout the response history The mode shape used in the analogy need not be identical to the elastic mode shape determined by traditional structural dynamics Various techniques for establishing an equivalent SDOF system have been recommended Generally a shape similar to the one that represents the displacement profile of the building at or near its peak response is adequate Calculated responses usually are not very sensitive to the precise shape selected and reasonable assumptions often lead to acceptable results The equation of motion of a multistory building may be expressed in terms of the degrees of freedom representing the lateral displacements at the floor levels relative to the ground The equation of motion for such a system is Mii r Cu r Q r Mi t 17 where terms are defined conventionally with M diagonal matrix representing lumped masses at the floors of the building C damping matrix of the building system Q t vector of story forces at the floor levels u t vector of relative displacements at the floor levels and u t vector of lateral accelerations of the floors relative to the base of the structure A shape vector i is assumed to represent the deflected shape of the MDOF system throughout its response history Displacements of the multistory building are tracked at a point known as the control node Many formulations locate the control node at the roof of the building and normalize
53. hopra 1995 Response for these cases also was computed using NONSPEC Mahin and Lin 1983 and NONLIN Charney et al 1998 Table 3 shows the properties of SDOF systems considered The SDOF systems with bilinear load deformation response were modeled as elastic perfectly plastic All cases are subjected to the 1940 NS EI Centro record that is described in Dynamics of Structures Chopra 1995 The results are summarized in the table below Table 3 Validation of SDOF code SDOF Properties Peak Displacement cm T sec 96 Fy W USEE Chopra book NONSPEC NONLIN 1 0 5 2 6 83 6 78 6 83 7 11 S 2 1 0 2 15 16 15 16 15 16 15 57 5 3 20 2 18 98 18 97 18 97 19 63 4 2 0 0 25 19 25 17 25 19 26 29 1 0 5 0 0 170 4 37 4 34 4 37 4 50 g 2 0 5 5 0 125 4 70 5 26 4 70 5 00 E 3 0 5 5 0 250 4 55 4 45 4 55 4 45 m 4 0 5 5 0 500 4 50 4 11 4 50 4 37 5 0 5 5 1 000 5 72 5 72 5 72 5 82 The computation engine of USEE is a C version of the original Fortran code used in NONSPEC The peak displacement results of USEE and NONSPEC in Table 3 are different only in the 4 digit of precision presumably due to roundoff error or other minor differences For linear elastic cases the USEE and Chopra text report similar results these results differ somewhat from those computed with NONLIN For inelastic response the results do not show the same level of agreement There are several reasons
54. ial formatting of the numerical quantities is necessary Selecting the View Accelerogram button in the base input step causes a window to appear where the user may view the accelerogram of the selected ground motion The View File Header button may be selected to show the ground motion file information all lines that begin with a Additional motions may be added by the user and these will be recognized by the program if the ground motion data files are located in the subdirectory with the other recorded ground motions Each motion must follow the format described above 3 5 2 Synthetic Motions Synthetic ground motions were developed by Professor Y K Wen and Chiun Lin Wu in Project Number RR 1 of the Mid America Earthquake Center and are included in the USEE distributions The full distribution contains the complete catalogue of synthetic motions developed in this project The motions were developed for rock and soil types at three cities Memphis TN Carbondale IL and St Louis MO and for different probabilities of exceedence Radio buttons selected by the user identify the desired city soil type and exceedence probability Specification of these parameters determines the synthetic motions listed in the window 39 3 5 2 1 Synthetic Motion Filenames The synthetic motions are stored in the USEE waveforms synthetic subdirectory The file names are identical to those used in project RR 1 Each filename contains 7 charac
55. ication EARTHQUAKE DATA Trigger Time hr min 24 hr 20436 sec 41 0 time code UTC Location latitude 32 8000 longitude 1 15 5000 depth km 0 0 Magnitude ML 6 3 MS NA MW NA STATION DATA Station No 117 Channel No Location latitude 32 79528 longitude 115 54861 elevation km 0 0 RECORD DATA Initial Velocity 4 664 Initial Displacement 2 159 Duration of Record sec 53 74 Number of Acceleration points 2688 Time Step sec 0 02 Interpolated TIME ACCELERATION 38 0 000 1 400 0 020 10 800 0 040 10 100 The actual file is simple unformatted ASCII text The selection above has additional formatting to clarify the information provided Bold text lines identify information that must be supplied for USEE to use the ground motion in response analyses Italic text lines indicate information that if supplied is extracted from the record for display to the user when the record is selected in the recorded ground motions step This information is useful but is not required No restriction is given as to the number and sequence of header lines that begin with an exclamation point must contain the text TIME and ACCELERATION Paired time acceleration The line immediately following the last oop data begin on the second line following the last line and must be in two columns Units of sec and cm sec are assumed While space must be provided between data columns no other spec
56. ile 53 Comparison of USEE to NONLIN and NONSPEC Force vs Displacement CI i iu peste i ED b b bb nie vi cb d cau 54 Multistory building approximation analysis example 55 First elastic mode shape of the 12 story building frame 58 Capacity curve obtained by applying forces proportional to the product of the elastic modal amplitude and mass at each floor in a nonlinear static pushover analysiss osc tick need spel odit ade e qeu an 58 Displacement history of a 12 story building frame subjected to 1940 El Centro record amplitude scaled by factor Of 2 esssssese 59 Base shear vs roof displacement response of the 12 story building frame subjected to 1940 El Centro record amplitude scaled by factor of 2 59 a Schematic load deformation response and b normalized load deformation response eiie e or aside UE Id as EAE ES ta aaa 62 Computed load deformation response to 1992 Landers earthquake at Joshua Tree Fire Station NS for a 1 second period oscillator 63 The strength ductility relationship for a bilinear oscillator having a period of T 0 15 sec responding to the 1987 Whittier Narrows record 63 vi Figure B4 Figure B5 Figure B6 Figure B7 Figure B8 Figure B9 Figure B10 Figure B11 Figure B12 Figure B13 The strength ductility relationship for a bilinear oscillator respondin
57. ime t yield displacement of a SDOF oscillator equivalent yield displacement of SDOF analogue base shear strength of a SDOF oscillator equivalent base shear strength of the SDOF analogue weight of a SDOF oscillator equivalent weight of SDOF analogue mass participation factor for mode i ratio of post yield stiffness to initial stiffness Newmark Beta Method parameters participation factor for mode i static collapse displacement yield displacement of a SDOF oscillator ultimate displacement of a SDOF oscillator tolerance on yield strength coefficient time interval incremental displacement at time t incremental velocity at time t incremental acceleration at time t incremental restoring force of a SDOF oscillator at time t incremental applied force to a SDOF oscillator at time t user specified time step displacement difference peak displacement ductility target displacement ductility 79 tolerance on target ductility critical damping ratio first mode displaced shape vector of a MDOF system equivalent circular frequency of SDOF analogue 80 APPENDIX D USEE Organization The organization of the Visual Basic code of USEE is described for documentation purposes The VB interface consists of Forms Modules and Help Files The Forms and Modules and their functions are described below Forms 1 2 3 4 9 10 11 12 13 14 15 16 17 Main form displays main window Copyright and agreeme
58. input acceleration These are shown in Figure 10 The motions are specified using several parameters e Pulse duration f duration in time of the pulse acceleration e Pulse amplitude as peak value of pulse acceleration e Computation time frp the duration over which the dynamic response is to be calculated The computation time must be greater than or equal to the pulse duration This creates two intervals of motion Forced vibration occurs for 0 lt t lt t and free vibration occurs for f gt 1 e Number of cycles Partial cycles may be applied by specifying non integer values For example for a half cycle of a sine wave 0 5 is specified for the number of cycles 41 Acceleration Acceleration ay a ty A tao 784 Time a Acceleration Acceleration ay a H trp Time 84 c Acceleration Acceleration ay ay H _ tmp Time a j ar e Acceleration ay t Time g Figure 10 Pulse types 42 ty _ tao Time b ty tao Time d t Time f 3 5 4 Scale Factors 3 5 4 4 Amplitude Scale Factor The amplitude scale factor scales the amplitude of the input acceleration Any non zero amplitude scale factor may be specified 3 5 4 2 Time Scale Factor The time scale factor scales the time coordinates of the acceleration Values of the time scale factor must be positive 3 6 Tutorial Annotated examples are provided in the following to introduce new users to the operation of the USEE Separate examples
59. iranda 1991 and Lawson et al 1994 The use of such equivalent systems has been formalized in ATC 40 1996 and FEMA 273 274 1997 In this section the response of a 12 story moment resistant frame building computed using Drain 2DX Prakash et al 1993 is compared to the response computed using a SDOF analogue in the Multistory Building Approximation analysis module The El Centro ground motion was applied to the building frame with amplitude scaled by a factor of 2 The 12 story steel moment resistant frame building Figure 21 was designed for uniform floor masses equal to 551 kN per floor The base shear strength was established to limit drift response the design is described in more detail as the Flexible 12 frame in 54 Black and Aschheim 2000 The frame was designed only for lateral loads in order to validate a design methodology Lateral response was computed using DRAIN 2DX Prakash et al 1993 Flexural response was modeled using beam column elements W18x46 mao wow masa uns wes uns meo unm vm a e wais Dune 8 ware aa wm Dun waha Dunn wem Dun E W14x193 8 LO 3 8 00 m Note all columns and all beams within a story are identical Figure 21 Multistory building approximation analysis example 55 Type 02 extending along beam and column centerlines the post yield stiffness was set equal to 5 of the initial stiffness The first mode of vibration has a period of 2 168 se
60. ived from the capacty CUE VS o acies ia ada a secede ERO OUR ERI AO ada a ote ei rea aaa 16 Default mode shapes available in USEE eene 17 Linear elastic model 1 se a aan a to iet t bec a i bi doi i 32 Brine ae model iiie ce a toi ataca al ala dedisse tn des essa g unica esae ala 33 Stiffness degrading model a a aa re et d i ov eee 34 Load deformation response of a bilinear oscillator with a positive post yield stiffness b negative post yield stiffness 37 Pulse DES asista tau n henes za li E at la Potes ec bs nis ina 42 SDOF example Base Shear Weight vs Displacement cm 45 SDOF example Displacement cm vs Time sec nenea 45 Multistory building approximation example Base Shear Weight vs Roof Displacement CHI as oie als ge dada a on iso dat iso Madea saute Sec 47 Multistory building approximation example Roof Displacement cm vs Title SCC Y ood eoi rac Cedex ecc Ue di a doara du at peso lut eue cold pe cbe augus 48 Response spectra example Spectral Acceleration g vs Period sec 49 Response spectra example Ductility vs Period sec 50 Yield Point Spectra Base Shear Weight vs Period sec 51 Yield Point Spectra Base Shear Weight vs Yield Displacement cm 51 Comparison of USEE to NONLIN and NONSPEC Displacement cm vs ume SCC iris met eee rena ses eee m ta are a he acea oe na na abea p
61. le B1 are reported in Table B2 It is apparent that the USEE 75 implementation is significantly faster than the other codes and that the differences are dependent on the ground motion records to some extent Many factors may contribute to differences in computation time including i the efficiency of the algorithms for the forward computation ii the efficiency of algorithms used for constant ductility iterations and ili overhead associated with graphical interfaces and iv other implementation specific details Some of the implementation specific differences are as follows e PCNSPEC requires that the lower and upper bound values of yield strength coefficient be specified along with the number of intervals within the bounds These intervals determine the discrete values of C that PCNSPEC uses PCNSPEC then determines a solution within the two adjacent values of C on either side of the target ductility This potentially may result in lower strength solutions or in no solutions at all if the solution lies outside the specified bounds In this comparison study the boundaries were defined to include the solution and the number of intervals is set to 50 e The same tolerances on strength were specified for USEE and PCNSPEC Tolerances for BISPEC are set internally and cannot be specified Table B2 Clock time required to compute response spectra for different ductilities using different software programs Com
62. lson E L Farhoomand I and Bathe K J Nonlinear Dynamic Analysis of Complex Structures Earthquake Engineering and Structural Dynamics 1973 Vol 1 pp 241 252 84 Wen Y K and Chiun Lin Wu Project RR 1of the Mid America Earthquake Center 1999 85
63. m The total displacement of the system is given as u t u t u t By taking derivatives with respect to time the absolute acceleration ii 7 is t t t The restoring force for a linear elastic system is given as R t Ku r where K is the stiffness For a nonlinear system R t is determined as a function of the current deformation u t as represented by the load deformation relationship Figure 1 SDOF system subjected to ground acceleration 2 1 2 Energy Terms Input energy may be computed by integrating the force terms of the equation of motion with respect to the relative displacement of the oscillator Two equivalent forms of the equation of motion exist mii t cu t R r 0 Q mii t cu t R t mii t 3 Integration of the terms of these equations with respect to the relative displacement of the oscillator leads to two different energy relationships Uang and Bertero 1988 The integration of Eq 2 leads to the so called absolute energy equation while integration of Eq 3 leads to the so called relative energy equation USEE computes relative energy quantities The energy imparted to the SDOF oscillator known as the relative input energy is given by integration of the right hand term of Eq 3 E jjm t du 4 The relative input energy E represents that work done by the equivalent lateral force m moving through the relative displacements of the oscill
64. mber is unique in any analysis session 3 2 1 Module Operation Each module provides a series of windows for data input and viewing of results Base input motions and load deformation models are selected in designated windows in each analysis module The Moultistory Building Approximation analysis module has an additional input window for specifying floor masses story heights and the assumed mode shape Results for all three analysis modules are viewed in a View Results window Each step in the sequence provides guidance to the user USEE Help may be accessed by selecting Help from the menu bar The Save As Default button in each window adopts the values in the current window as default values for subsequent analyses The Compute Results button uses current input values for the computation and advances directly to the View Results window The main window provides menu choices for beginning a new analysis opening an existing file saving current analysis files and exporting the results of the current analysis to ASCII text files for subsequent processing by the user Only the input data is saved in an analysis file 3 2 2 Directory Structure The USEE program is installed to C Program Files USEE unless otherwise specified by the user during the installation Beneath the top level directory where USEE 24 is installed are four subdirectories Help Multistory Response_Spectra SDOF and Waveforms The complete USEE subdirectory structure is USE
65. namic response of a SDOF oscillator to a specified excitation is a function of its mass damping and load deformation relation The load deformation relation often is idealized as a continuous assembly of piecewise linear segments Figure B1 a shows the yield strength of the oscillator F and yield displacement A as well as a peak 61 displacement A The post yield stiffness aK is expressed as a fraction o of the initial stiffness K and the initial elastic period of vibration T is given by T 2m M K where M the mass of the SDOF oscillator Force Force Weight A A Ppa pcm Cu FU MW lesen l gt gt Ay Au Displacement Ay Au Displacement a b Figure B1 a Schematic load deformation response and b normalized load deformation response A normalized form of the load deformation relationship may be obtained by dividing the lateral force by the weight of the mass Doing so allows the load deformation relation to be expressed in Figure B1 b in terms of the yield strength coefficient C where pel B1 and W Mg where g the acceleration of gravity The dynamic response to a base excitation Figure B2 may be considerably more complex but even so the above terms define the oscillator characteristics and intensity of peak response The displacement ductility u that develops at the peak displacement is given by u u B2 62 Normalized Force Displacement Figure
66. nd B13 show the response spectra computed for the El Centro and Llolleo records respectively for u 2 The solutions obtained using the three programs were nearly identical with only a few results obtained from PCNSPEC deviating noticeably from the results obtained with BISPEC and USEE In these few cases PCNSPEC missed the highest strength solution and reported a lower strength solution that resulted in the target ductility The overall agreement of the solutions indicates that the algorithm implemented in USEE is at least as accurate as those implemented in other available codes The program NONLIN was not considered in this comparison because the constant ductility strengths are estimated by linear interpolation between C values for u 1 and u 8 rather than being computed explicitly for each value of ductility 74 0 60 BISPEC PCNSPEC 0 50 USEE 0 40 0 30 0 20 0 10 Yield Strength Coefficient C 0 00 Period sec Figure B12 Constant ductility response spectrum for u 2 for the El Centro record 1 25 BISPEC PCNSPEC USEE 1 00 0 75 0 50 0 25 Yield Strength Coefficient C 0 00 Period sec Figure B13 Constant ductility response spectrum for u 2 for the Llolleo record B 4 2 Computational Efficiency The clock times required to obtain constant ductility response spectra for u 2 and u 8 for the 5 records of Tab
67. ngth of the oscillator and the degree of nonlinear behavior that develops As noted by Newmark and Riddell 1979 the same ductility demand may result for different oscillator strengths Since the usual design objective is to ensure that ductility demands greater than the target ductility do not develop selecting the largest of the strengths that result in the target ductility demand is a useful strategy to ensure that the actual ductility responses do not exceed the target ductility considering that the actual structural properties or ground motions may differ even slightly from those assumed in the analysis An efficient algorithm is necessary because results are often sought for a large number of periods for different target ductility values and for different ground motions potentially requiring many thousands of nonlinear SDOF analyses Although algorithms for determining constant ductility strengths have been developed for research e g Newmark and Hall 1973 and Vidic et al 1994 into R u T relations strength reduction factor as a function of ductility and period for example and have been implemented in various software programs e g PCNSPEC Borosheck and Mahin 1991 and BISPEC Hachem 2000 few if any have received formal attention in the literature The present algorithm is implemented in USEE for computation of isoductile constant ductility response spectra B 2 Properties of the Strength Ductility Relationship The dy
68. nits values of parameters used in the computations and export options may be set from the Preferences menu at any time 3 3 2 1 Available Units USEE uses either in U S Customary or SI units to display and input data The units to be used may be specified from the Preferences menu or the Units pull down list box on the toolbar at the top of the screen Available force units are as follows e U S Customary pounds Ib or kips kips e International System SI Newtons N or kiloNewtons kN Available length units are as follows e U S Customary inches in or feet ft e International System SI centimeters cm or meters m The units used to display data may be changed at any time internal computations are done in KN and cm units 3 3 2 2 Parameters Parameters that the user can set are computational time step output time step and overshoot tolerance The ideal time step value cannot be identified a priori The smaller of the user specified time step and the time required to reach the next acceleration point of the base input is used at the beginning of each step Typically a value not exceeding 10 of the 27 period would be specified by the user The program will automatically reduce the time step if required for convergence of the solution Section 2 1 2 2 USEE can report response data according to the user specified output time step A large number of time steps may be needed to ensure accuracy of the solution This has
69. nt histories Table 4 First elastic mode shape of the 12 story building frame Story Normalized 1 Level Mode Amplitude 12 1 0000 11 0 9546 10 0 8868 9 0 8120 8 0 7254 7 0 6356 6 0 5409 5 0 4492 4 0 3556 3 0 2640 2 0 1704 1 0 0828 57 Height m Figure 22 First elastic mode shape of the 12 story building frame 1200 J e e O T 800 J 600r ii 400 J Base Shear Force kN 200r J D o 0 1 0 2 0 3 0 4 0 5 0 6 Roof Displacement m Figure 23 Capacity curve obtained by applying forces proportional to the product of the elastic modal amplitude and mass at each floor in a nonlinear static pushover analysis 58 Roof Displacement cm Computed USEE Multistory Approximation 0 10 20 30 40 50 Time sec Figure 24 Displacement history of a 12 story building frame subjected to 1940 El Centro record amplitude scaled by factor of 2 1500 1000 500 Base Shear kN eo 500 Computed 1000 USEE Multistory Approximation 1500 60 40 20 0 20 40 60 80 Roof Displacement cm Figure 25 Base shear vs roof displacement response of the 12 story building frame subjected to 1940 El Centro record amplitude scaled by factor of 2 59 APPENDIX A Software Development Tools The computational engine of USEE was written in C The code
70. nt of terms form About USEE form displayed in Help About USEE Base Input form contains Recorded Ground Motions Synthetic Motions and Pulse input motions for SDOF Approximate Multistory Building and Response Spectra Modules Structural Properties form for SDOF analysis and Multistory Building Approximation analysis contains structural properties input for the SDOF Multistory Building Approximation Modules Structural Properties form for Response Spectra analysis contains structural properties input for Response Spectra Modules Manual Testing for Load Deformation Models form Multistory Description form File Header View form Unit Types form Summary Log for the current session form Zoom Plots Form View SDOF Results form View Multistory Approximation Results form View Response Spectra Results form Compare SDOF and Multistory Building Approximation analysis Results form Options and user preferences form Modules 1 2 3 Input Preparation module SDOF and Multistory Building Approximation analysis module Response Spectra analysis module 81 4 File Operation module 5 Current Analysis File Handling module 6 7 8 9 Plotting module Internet Access module Help File Connectivity module Exporting SDOF and MDOF module 10 Error checking and handling module 11 Error throwing module 82 APPENDIX E References Abrams D P Nonlinear Earthquake Analysis of Concrete
71. odule the quantities are plotted with kinetic and strain energy combined 2 1 3 Computational Aspects Closed form solutions of the equation of motion of a single degree of freedom oscillator are not available for a general nonlinear system and for excitations that vary arbitrarily with time Solutions may be obtained by numerical integration of the equation of motion in a sequence of step by step analyses Each successive analysis is done over a small time interval for initial conditions determined at the conclusion of the previous time step The computation for each time interval Ar is based on an assumption of the structural characteristics that prevail during the entire time step Response during each time interval is computed using the linear acceleration method The linear acceleration method is a special case of the Newmark Beta Method with o 1 2 and B 1 6 In this method the response acceleration is assumed to vary linearly during the time step and the properties of the system are assumed to be invariant during the time step For nonlinear systems a displacement increment near a change in stiffness of the system may result in an imbalance between the dynamic equilibrium determined using the actual properties and the properties assumed during the time step The equilibrium unbalance is evaluated and if it is significant the result for that time step is discarded and a smaller time step is selected This procedure is applied recursivel
72. ogy For many buildings the distribution of mass is nearly uniform resulting in M and T being dependent only on the deflected shape The user may specify arbitrary deflected shapes or one of the three deflected shapes suggested by Abrams 1985 may be selected Figure 5 As an initial approximation the parabolic shear deflected shape may be suitable for many regular moment resistant frame buildings and the flexure beam deflected shape may be suitable for many structural shear wall buildings For many buildings the precise shape is not necessary to obtain good estimates of peak displacement response and one or two of these shapes may be used to determine approximate values or ranges of expected peak displacement response 16 The user indicates the yield strength that would be observed in a nonlinear static pushover analysis of the building via the base shear coefficient Cy The period of vibration of the building or the yield displacement is needed to establish the elastic portion of the load deformation curve Either may be specified The yield strength coefficient Cy of the SDOF analogue can be established by two alternative approaches as described in Section 2 2 2 Eq 32 assures the period of vibration of the SDOF analogue matches the period specified for the multistory building The ATC 40 implementation Eq 33 gives identical results provided that the elastic mode shape is used for the shape vector The user is cautioned to
73. olution C e The ductility corresponding to Cye is computed The computed ductility u C is compared to the target ductility The case u Cye gt W is illustrated in Figure B7 In this case the highest strength solution clearly lies between C and C Therefore the solution bounds are revised by setting C Cye and the algorithm restarts with the new bounds e Computed Ductility u C o Interpolated Ductility Figure B7 The case where u C e gt ue The case u Cye lt W is illustrated in Figure B8 This case is more complicated because undulations in the strength ductility relation might be large enough that a higher strength solution may exist between C and C The approach taken is to compare the ductilities computed at intermediate strengths with estimates based on linear interpolation between U C e and u C A smoothness ratio is defined as the ratio e a where e the difference between the interpolated and actual ductilities and a the difference between the interpolated and target ductilities as shown in Figure B9 If the smoothness ratio is less than a user specified smoothness tolerance at a sufficient number of points the strength ductility relation is considered to be smooth The possibility that an unidentified solution might exist within a region identified as smooth is considered to 68 be remote because this would require a sharp departure from the interpolated strength ductility rel
74. oscillator responses that exceed 1 2A Each instance that this occurs results in a discontinuity in the response spectrum plots 36 Force Force kK AK static collapse displacement Displacement Displacement B6 a b Figure 9 Load deformation response of a bilinear oscillator with a positive post yield stiffness b negative post yield stiffness 3 4 2 7 Yield Displacement A The yield displacement is the displacement of the structure relative to the ground at the instant that the structure reaches its yield strength 3 4 2 8 Building Drift at Yield The building drift at yield is the roof displacement relative to the base of the structure at the instant when the structure reaches its base shear strength at yield 3 4 3 Response Spectra Parameters See Section 2 3 3 5 Base Motion Input The setup program installs a suite of base motions in the subdirectory USEE Waveforms where USEE is the highest level directory for the program specified during the installation Recorded ground motions are located in the USEE Waveforms Recorded subdirectory and synthetic motions are located in the USEE Waveforms Synthetic Hard Rock and USEE Waveforms Synthetic Soil subdirectories 3 5 1 Recorded Ground Motions 3 5 1 1 Recorded Ground Motion Filenames Selecting the Recorded Ground Motions tab of the base input screen displays all files present in the USEE Waveforms Recorded subdirectory There is no restriction on naming base motion
75. p 2 Among the linear bilinear and stiffness degrading models select the bilinear model Next specify the periods by selecting the uniformly spaced radio button and enter 60 periods ranging from 0 05 to 3 seconds For the parameter to vary select the Yield Strength Coefficient Choose 3 discrete values from the pull down menu and enter the values of 0 25 0 50 and 1 00 in the text boxes labeled Values For Viscous Damping enter 5 and enter 0 for Post Yield Stiffness Click the Compute Results command button Step 3 The results of the response spectrum computations are displayed in Step 3 Spectra corresponding to each parameter value are displayed with the color indicated at the top of the window Select the Zoom to Full Screen button to enlarge the plot Lines are plotted for each parameter value with a different color Figure 16 shows ductility versus period for the yield strength coefficient values of 0 25 0 50 and 1 00 49 Ductility 0 1 2 a 6 Cy 0 25 E C 0 50 4 Cy 1 0 1 4 2 2 zl p e i 0 L 1 L l L I I 1 I I 0 0 1 2 3 Period sec Figure 16 Response spectra example Ductility vs Period sec This completes the second example To modify the current analysis to begin the third example click on the Back button on the View Results window This takes the user back to Step 2 Step 2 Among the linear bilinear and stiffness degrading models select the
76. pe curves Unloading from the curves occurs with stiffness equal to the initial elastic stiffness The elastic perfectly plastic model is a special case obtained by specifying post yield stiffness to be zero 32 Figure 7 Bilinear model 3 4 1 3 Stiffness Degrading Model Various stiffness degrading models have been used to represent reinforced concrete structures The stiffness degrading model implemented in USEE is suitable for structures that do not exhibit substantial degradation due to shear or bond deterioration which can cause severe strength degradation and or pinching of the hysteretic curves This model uses a bilinear envelope curve defined by three parameters the yield strength the initial stiffness and post yield stiffness Figure 8 shows the stiffness degrading characteristics of this model during load reversals Unloading begins with the initial elastic stiffness when the load changes sign crossing the displacement axis the stiffness changes and the model loads toward the previous peak in the direction of motion If prior yielding has not occurred in the direction of motion the model loads toward the yield point 33 Figure 8 Stiffness degrading model 3 4 2 Load Deformation Curve Properties 3 4 2 1 Yield Strength Coefficient C The yield strength coefficient C is defined as yield strength of the oscillator divided by its weight C 2 38 where F yield strength of oscillator and W weight of os
77. placement When constant ductility or constant strength reduction factor R factor spectra are computed Yield Point Spectra may be displayed by plotting yield strength coefficient versus yield displacement Alternatively constant ductility or constant strength reduction factor spectra may be displayed in a Peak Capacity Spectrum Method format by plotting peak strength coefficient versus peak displacement When Yield Point Spectra or peak spectra are plotted the cursor can be positioned on screen to provide a schematic illustration of the yield and peak points for any computed period While in any of the analysis modules the user may move backward and forward to different windows steps modify input parameters and then advance to last step to view the results Advancing to the last step in the Response Spectra module causes the spectra to be recomputed using the modified values This may be time consuming if numerous calculations are required particularly in the case of constant ductility spectra 3 3 6 Windows Copy amp Paste Response plots may be copied to Windows applications such as Microsoft Word using the Copy and Paste functions These may be accessed via the toolbar or a right mouse click To copy a response plot first left click on the plot to select it and then select Copy from the toolbar or simply right click on the plot and select Copy on the submenu Then switch to another application such as Microsoft Word and select Paste from
78. putation Time sec BISPEC PCNSPEC USEE Record ID u 2 u 8 u 2 u 8 u 2 u 8 bb92civc360 15 21 17 74 6 7 ch85lleo010 52 111 57 71 24 31 iv40elcn180 5 12 4 36 2 3 mx85sct1270 53 87 21 26 10 12 wh87mtwl090 7 11 95 153 3 4 For the foregoing reasons one can not conclude from Table B2 that the constant ductility algorithm implemented in USEE is necessarily more efficient that those implemented in other software programs It is clear however that the combination of the constant ductility algorithm the efficiency of the forward computation and other 76 implementation specific details work together to result in relatively fast computations using USEE B 5 Conclusion An algorithm consisting of an initial bounding of the solution phase and a fast search bisection phase was described This algorithm was implemented in the USEE program Comparisons with other programs indicates 1 the USEE program is at least as accurate as PCNSPEC and BISPEC and ii the USEE computation is relatively fast B 6 Glossary Definitions Smoothness Ratio e a is the ratio of deviation e of the computed ductility value from the value expected based on linear interpolation to the distance a between the target and expected ductilities See Figure B9 Smoothness Tolerance is a user specified non dimensional value that is compared to the Smoothness Ratio to determine whether the departure from the interpolated ductility is large
79. rbitrariness to the solution and would lead to inconsistencies in the results computed using different codes As a matter of engineering practice to ensure that ductilities no larger than the specified value develop the largest strength corresponding to the target ductility should be identified by the algorithm indicated by Point A in the figure An efficient algorithm must strike a balance between the computational cost of obtaining better resolution of the strength ductility relationship and the possibility of not identifying a higher strength solution An exact solution may not exist Figure B4 b shows a close up view of the strength ductility relationship in the vicinity of u 2 for an oscillator having T 0 15 seconds responding to the same record of the 1987 Whittier Narrows earthquake Several discontinuities in the ductility response are apparent upon close inspection If the target ductility lies on a discontinuity then an exact solution may not be available For example Figure 5b indicates that no oscillators exist that respond to this earthquake record with a peak ductility response of exactly 2 for the damping and load deformation model considered Given this finding instead of requiring an exact solution the algorithm should identify the strength coefficient for which the ductility is nearly equal to but does not exceed the specified target value Such an algorithm would identify Point A in Figure B4 b as the solution 64
80. red to determine oscillator properties so that a given response characteristic is obtained Constant ductility spectra are computed by iterating on strength to identify the strength required to obtain a ductility response equal to the specified ductility value for each oscillator Up to five ductility values may be specified The excitation load deformation model damping and post yield stiffness are kept constant throughout the computation The yield strength coefficients required to limit ductility demands to the specified values may be displayed in the View Results window along with the response parameters stated above The iterative nature of the computation requires significantly more computational time than is required for the other response spectra The algorithm is described in detail in Appendix B The user may change the parameters that control the accuracy and efficiency of the constant ductility computation These parameters are shown in Table 1 with their corresponding limits and default values Terms are defined and discussed in more detail in Appendix B 21 Table 1 Constant ductility algorithm parameters nies Lower Bound Upper Bound Advisor Advisor Tolerance on target ductility as of target ductility Tolerance on yield strength coefficient Smoothness tolerance Ductility as of target ductility Strength interval as of isoductile Criteria to switch to bounding
81. ree of freedom structures e The Multistory Building Approximation analysis module uses an equivalent SDOF representation of the building to estimate the displacement response history of multistory buildings e The Response Spectra module computes linear and nonlinear response spectra for a range of parameter values These capabilities are described further in Section 3 3 The analyses may be conducted with any of the following load deformations models e linear e bilinear e stiffness degrading Properties of the oscillator such as period of vibration viscous damping yield strength and post yield stiffness may be specified by the user Further details are provided in Section 3 4 The analyses may be done using base input accelerations selected from the following categories e recorded ground motions e synthetic motions e pulse waveforms The program is distributed with a basic complement of motions in these categories as described in Section 3 5 Users may add additional accelerograms of their choosing to the recorded ground motions category Accelerogram formatting requirements are described in Section 3 5 1 2 Document Overview This report describes e capabilities of the USEE program Chapter 1 e the theoretical basis of the program Chapter 2 e use of the program its organization and base motion file formats Chapter 3 e validation of the accuracy of the USEE program using several test cases Chapter 4 It i
82. region in which a solution is to be obtained This is done by applying a check reject test to determine if a higher strength region might contain a solution If the test determines that an unrecognized higher strength solution is unlikely the higher strength region is rejected This process is applied to narrow the bounds on the solution Once the initial bounds are narrowed sufficiently the second 65 phase is begun In the second phase a bisection approach is applied to determine a solution as rapidly as possible within the bounds determined by the first phase The two phases of the algorithm are described next e Computed Ductility u C o Interpolated Ductility Figure B5 Linear interpolation between C and Cy B 3 1 Initial Bounding of Solution The first phase of the algorithm narrows the interval in which the solution lies the objective is to discard regions of C in which it is determined that a solution is unlikely to be found The algorithm is described in detail in Figure B6 Key concepts are described below To begin the upper bound of the interval C is set equal to the strength coefficient required for elastic response This value is determined by computing the response of a linear elastic oscillator having the same period of vibration and viscous damping The lower bound of the interval C must result in a ductility greater than the target ductility to ensure that a solution lies between C and C
83. rrent window and returns to View Results window Save saves the current analysis file Save As saves the current analysis file under a user provided file name Save As Default saves the current window data as default values for use in subsequent analysis sessions Show Constant Period Line draws a constant period line on the plot in the View Results step a Response Spectra analysis if the capacity spectra peak strength weight vs peak relative displacement or yield point spectra yield strength coefficient vs yield displacement is active Start New Analysis creates a new analysis having an extension appropriate for the current analysis type SDOF BLDG RSPC Test Model allows the selected load deformation model to be exercised manually by incrementing displacements step by step 26 View Accelerogram plots the accelerogram of the selected input motion whether recorded or synthetic View File Header displays header information from the selected input motion whether recorded or synthetic View Summary Log views summary log file for the current analysis session Visit Mid America Earthquake Center Homepage loads the Mid America Earthquake Center Homepage using a previously installed web browser Zoom To Full Screen zooms in on results plot F1 function key brings main window for USEE Help The user can go to the main help window from any step of the USEE program by pressing the F1 key function 3 3 2 User Preferences U
84. rs for which the response spectra are to be computed affects the time required for computation In particular computation of constant ductility spectra is an iterative process that requires substantially more time to compute To compute inelastic response spectra the following actions are required of the user 1 The user selects a base input acceleration 19 2 The user selects a load deformation model bilinear or stiffness degrading 3 The user specifies a period range and the number of periods as well as the distribution of periods 4 The user specifies a viscous damping ratio as a percentage of critical damping 5 The user specifies a post yield stiffness as a percentage of initial stiffness 6 The user specifies the parameter to be varied in the inelastic response spectra computation as well as specific values of this parameter USEE calculates the response histories for the specified periods and parameter values using the numerical method described in Section 2 3 1 Appendix B describes the iterative algorithm used for computing isoductile spectra Peak response quantities are retained for each case yield strength coefficient peak ductility and R factor These quantities may then be plotted as a function of period yield displacement or peak relative displacement 2 3 3 1 Constant Strength Spectra Constant strength spectra refer to the response of oscillators having constant yield strength The excitation load deforma
85. rs linear behavior for three viscous damping values equal to 0 2 and 10 of critical damping In the second example elastic perfectly plastic behavior is considered with yield strengths equal to 25 50 and 100 of the oscillator weight Viscous damping is assumed to be 5 of critical damping In the third example elastic perfectly plastic behavior is considered and ductility is held constant equal to 2 4 and 8 Viscous damping of 5 of critical damping is assumed The spectra are computed for 60 uniformly spaced periods ranging from 0 05 to 3 seconds Base Shear Weight 3 T 4 ve T a a a a 0 2 0 1 0 1 02 A 02 Roof Displacement in Figure 13 Multistory building approximation example Base Shear Weight vs Roof Displacement cm 47 Roof Displacement in Figure 14 Multistory building approximation example Roof Displacement cm vs Time sec To begin from the main USEE window select New Response Analysis and then select Response Spectra This brings forth a series of windows titled Step 1 through Step 3 that pertain to the current analysis Data for the first example are entered as follows Step 1 This window displays three tabs Recorded Ground Motions Synthetic Motions and Pulses Since in this example the base input is a recorded ground motion select the Recorded Ground Motion tab A list of recorded ground motions is presented Select IV40ELCN180 MAE record from the list
86. s recommended that the reader install the USEE software and use it in conjunction with the reading of this report 1 3 Typographical Conventions Throughout this manual the following typographical conventions are used Roman type is used throughout this report unless otherwise noted Commands and command buttons are in bold type e g Export Output References to USEE modules and features are in italic type e g Approximate Multistory Building Computer directory names are in bold type e g USEE Waveforms Recorded Individual file names such as ground motion records are in capital letters e g CO2_01S MAE Arial type is used for ground motion file header data Variables in equations are in italics e g Sa matrices and vectors are in bold type e g Q and variables that are represented using Greek symbols are shown in regular text e g I 2 Theoretical Basis 2 1 Response of SDOF Systems 2 1 1 Theoretical Formulation The equation of motion for a viscously damped single degree of freedom system subjected to ground acceleration ii t see Figure 1 is given as a function of time t by Mut Cu t R t Mi t 1 where Mz mass of the system C viscous damping coefficient R t restoring force and ii 1 ground acceleration The term u t is the displacement of the system relative to the ground and represents the deformation of the structure while u t is the displacement of the ground relative to a fixed datu
87. sed to obtain a solution to Eq 14 over successive time steps Ar Given the structural properties and motion at time f and the acceleration applied at the base of structure during the time increment Af the incremental acceleration Aii t the incremental velocity Zu t and the incremental displacement Au t are computed The displacement and velocity values at time t Af are u t At u t Ault 15a u t At u t Au t 15b The acceleration at t At is calculated with a correction for any unbalance in equilibrium P t At Cu t At R t Ar M t At 16 The accuracy and stability of the integration method are important considerations The linear acceleration method is known to be stable for linear elastic systems only if the time step is less than the period of the system multiplied by 0 551 e g Chopra 1995 This is described as conditionally stable in the literature because the stability of the solution is assumed only under the condition that a small enough time step is used However the stability limit is not restrictive in practice because the time step must be considerably smaller than this limit to ensure adequate accuracy in the numerical solutions For linear elastic systems a time step not exceeding 1 10 of the structural period is a good rule of thumb to ensure reasonably accurate numerical results Chopra 1995 Theoretical limits on the time step required for stability of the solution have not
88. sented Select IVAOELCN180 MAE 1940 NS El Centro record from list Select the Entire Record option to compute response for the entire record duration Enter 1 0 for the Amplitude Scale Factor and the Time Scale Factor Click the Next button Step 3 The user must select one of the linear elastic bilinear and stiffness degrading load deformation models For this example select the bilinear model Enter 5 for Viscous Damping and 0 for Post Yield Stiffness In this example eng will be computed as per ATC 40 so select the corresponding radio button The user must specify the drift at yielding so enter 0 25 in the corresponding text box and enter 0 25 for C Click the Compute Results command button to advance to Step 4 Step 4 The results are displayed in Step 4 Figure 13 and Figure 14 show load deformation and displacement response history of the building The base shear coefficient and roof displacement are point of interest After viewing the results the user may start a new analysis by clicking the Start New 46 Analysis button Clicking the View Summary Log button allows the user to view the input parameters for the current analysis Response quantities of the building may be saved to a text file by selecting Export Output from the File menu 3 6 3 Response Spectra Examples The computations of three types of response spectra are illustrated for the 1940 NS El Centro in this example The first example conside
89. stic load deformation response From the main USEE window select New Response Analysis and then select SDOF Oscillator This brings forth a series of windows titled Step 1 through Step 3 that are 43 used for analysis of SDOF systems Data for this example are entered as follows Step 1 This window displays three tabs Recorded Ground Motions Synthetic Motions and Pulses Since El Centro is a recorded ground motion select the Recorded Ground Motion tab A list of ground motions is displayed Select and verify that IV40ELCN180 MAE is indeed the record you seek by looking at the information presented on the right side of the screen Select the Entire Record option to analyze response over the entire duration of the record Set the Amplitude Scale Factor and the Time Scale Factor to 1 0 To advance to Step 2 click the Next button Step 2 A linear bilinear or stiffness degrading model must be selected For elastic perfectly plastic response select the bilinear model Enter 5 to set Viscous Damping to 5 of critical damping enter 0 for the Post Yield Stiffness to obtain elastic perfectly plastic response and enter 0 3 for the yield strength coefficient Either the period of vibration or the yield displacement must be specified Since the period is 0 75 sec enter 0 75 USEE reports the corresponding yield displacement Click the Compute Results command button to advance to Step 3 Step 3 The compute
90. tch to bisection with the region bounded by C and Cy 2 Cye Yes No IS Cy e bilS Ahoi Set Cy 0 5 Cy u Cy and compute u Cy Report solution as C Cye Report solution Yes Is u Cy o u S Ato or Cy Cyb lt ACy tol Define new region Define new region bounded by Cy bounded by Cy and Cyu Cy Cy and Cyu Figure B11 Bisection with the fast search algorithm The fast search algorithm begins by dividing the previously identified solution bounds into 10 segments if these segments are larger than the specified tolerance on C If not fewer segments are used such that each segment is larger than the specified tolerance on C The ductilities corresponding to each strength coefficient are computed beginning with the largest strength coefficient and stopping when the computed ductility 72 exceeds the target ductility The yield strength coefficient corresponding to the larger ductility is assigned to C and the next larger yield strength coefficient is assigned to C The division into as many as 10 segments is not strictly necessary but was implemented in case the solution bounds determined in the first phase are relatively broad such as might happen if a user should specify relatively large tolerances A bisection procedure is then applied recursively to the two adjacent yield strength coefficients in which a solution lies The procedure begins with the points C
91. ters according to the following format LPR_SQS MAE where L 1 character representing city location M for Memphis C for Carbondale and L for St Louis P 2 digits representing the probability of exceedence in a 50 year interval SQ 2 digits representing a sequential number in each earthquake set S 1 character representing the soil type R for hard rock S for soil MAE 3 character extension to denote the use of the Mid America Earthquake Center format For Example C02 OIS MAE is a synthetic motion for a soil site in Carbondale having 2 of probability of exceedence in 50 years 3 5 2 2 Format of Synthetic Motion Files The synthetic motions conform to a consistent file format The file header is illustrated below for the C02 O1S MAE file Mid America Earthquake Center Format on December 16 1999 Synthetic Motion Units are cm sec Source Mid America Earthquake Center Created by Prof Wen Y K in the Project Number RR1 at the MAE Center City Location Carbondale Soil Type Soil Exceedence Probability level in 50 yrs 0296 in 50 yrs Focal Depth km 17 4 Epicentral Distance km 166 4 Closest Horizontal Dist to the Surface Projection of Rupture Plane 106 2 Deviation from Median Attenuation 0 90 Duration 149 99 sec Peak Ground Acceleration 513 400 cm sec 2 at time 9 84 sec Peak Ground Velocity 52 500 cm sec at time 10 57 sec Peak Ground Displacement 18 390 cm at time 16 92 sec c061
92. the USEE software product and specifically no warranty is made that USEE is merchantable or fit for any particular purpose Any description of USEE shall not be deemed to create an express warranty that USEE conforms to this description Receiver assumes all risk and liability for loss damage claims or expense resulting from use possession or distribution of any software products furnished by the developer Receiver agrees to indemnify defend and hold harmless the developer its officers agents and employees from and against any and all claims liability loss damage or expense including reasonable attorney s fees arising from or by reason of receivers use possession or distribution with respect to any of the software products furnished by the developer and such obligation shall survive acceptance of said products therefore by receiver Receiver agrees that it will not resell the software products furnished hereunder although free distribution to others is permitted CONTENTS T BtrodBcHosc cada dt ove dtt t C c E 1 1 1 Prosram DeSCPpUODSs s sens ata a iei tete io bo petet ated ea tetas bunts etatis 1 1 2 Document OYervIew iuis ite ei Ud dee ERE Eae eO d HI BEN due 2 1 3 Typographical Conventions i sasea at c i sl aa ada ea ai ae alaiul 2 Z Theoretical Basts joace sa ga te ire ante Sa auda a aia tel dd aaa na dle Sa eee 4 2 1 Response of SDOF Systems eee te a vana eoe ay at FANS SEE ERU e a 4 21 1 Theoretical FO
93. the shape vector i to have unit amplitude at the roof Following this approach the relative displacement vector may be expressed as the product of the shape vector and the roof displacement uroo as u t Qiu ut 18 Substituting Eq 18 in Eq 17 gives M oof T Chit oof Q Miu 19 The displacement of the SDOF analogue u_ is defined as a OTM u t o MI U roof lt 20 12 Pre multiplying Eq 19 by H and substituting for uroo t using Eq 20 results in the following differential equation for the equivalent SDOF system M t C 0 0 t M ii t 21 where M o M1 22a C 0 CoS 22b O RA 22c T A 22d 9 M The term T is also known as the modal participation factor for the i mode The value of Ij calculated using Eq 22d depends on how the shape vector is normalized in this presentation Q is normalized to have unit amplitude at the roof level The quantity TM is the mass that participates in the response associated with u t Giuroo 1 The mass participation factor oj is the ratio of the participating mass TM to the total mass T T 4 MMI M1 23 o M I MI The value of the mass participation factor ai is independent of the manner in which the shape vector is normalized The load deformation relation of the SDOF analogue usually is determined from the capacity curve obtained from a nonlinear static pushover analysis of the struct
94. tion The geometric distribution provides a denser spacing of periods at the lower end of the period range where response often has greater variation The geometric ratio r of a set of N periods is given by r apj Ch EE 34 where qp 35 and T lower period Ty upper period T an intermediate period and N number of periods 2 3 2 Computation of Elastic Response Spectra The peak response of linear elastic SDOF oscillators subjected to a specified input motion is conveniently described by the elastic response spectrum For each oscillator the peak displacement of the mass relative to the base the peak relative displacement often called the peak displacement or peak deformation Sa is computed for the user specified periods of vibration and viscous damping The pseudo acceleration Sa is computed as m S 36 where circular frequency of vibration 27 T 18 The response spectrum module in USEE guides the user through three data input screens The following actions are required to compute the elastic response spectra 1 The user selects the input excitation 2 The user selects the linear elastic load deformation model 3 The user specifies the range of periods number of periods and whether a uniform or geometric distribution of periods is desired 4 The user specifies the viscous damping ratio as a percentage of critical damping Up to 5 damping ratios may be specified in each computation of elasti
95. tion model damping and post yield stiffness are kept constant over a range of periods Up to 5 values of the yield strength coefficient may be specified The peak relative displacement and the peak ductility responses are often of interest However other response quantities may also be plotted in the View Results window 2 3 3 2 Constant Strength Reduction Factor R Factor Spectra Constant R factor spectra may be of interest when constant strength reduction factors are used for determining the strength of SDOF oscillators Inelastic response spectra are computed for user specified R factors for the specified excitation load deformation model damping and post yield stiffness To determine the strengths of the oscillators USEE first computes the elastic response spectrum over the specified vibration periods Yield strength coefficients are calculated for each period and R factor as Sa c Sut 37 where C yield strength coefficient S pseudo acceleration associated with linear elastic response g acceleration of gravity and R strength reduction factor 20 Peak displacement ductility demands may be viewed in the View Results window along with other parameters including absolute acceleration yield strength coefficient and peak relative displacement 2 3 3 3 Constant Ductility Spectra For the preceding types of spectra the response for specified oscillator properties is computed for a specified excitation In some cases it is desi
96. to 5 of critical damping and an idealized elastic perfectly plastic response The ATC 40 formulation for the base shear strength will be used Story weights are uniform To begin from the main USEE window select New Response Analysis and then select Multistory Building Approximation This brings forth a series of windows titled Step 1 through Step 4 that pertain to the current analysis Data for this example are entered as follows Step 1 This window provides for the input of parameters that describe the building The number of stories weight of each story story heights and deflected shapes are specified Since a four story building will be analyzed select 4 from the pull down menu labeled Number of Stories Set the story weights equal by selecting the appropriate button in the box labeled Are Story Weights Equal The weight of each story is 318 kN Next enter 4 in the text box labeled Story Height m and make every story this height by selecting the appropriate button Select the prescribed deflected shape Shear Beam Parabolic for this analysis to approximate the response this moment frame Click the Next command button to advance to Step 2 Step 2 This window displays three tabs for specifying the base input acceleration Recorded Ground Motions Synthetic Motions and Pulses Since in this example the base input is a recorded ground motion select the Recorded Ground Motion tab A list of the recorded ground motions is pre
97. ure The capacity curve plots the base shear force versus roof displacement of the structure Figure 4 shows an idealized capacity curve that was obtained by applying lateral forces proportional to the product of amplitude of the shape vector and mass at each floor level A bilinear curve was fit to the capacity curve for use in determining the load displacement relation of the SDOF analogue Eq 20 may be restated to more concisely express the relation between the yield displacement of the SDOF analogue and the yield displacement of the multistory system as 13 u roof y 24 Different approaches have been recommended for relating the base shear strength of the multistory system to the yield strength of the SDOF analogue USEE allows the user to select from two implementations that are described generally by Figure 4 b These implementations are defined as follows The bilinear curve fit to the capacity curve represents a case when yielding occurs at a sharply defined point The vector of lateral forces at the instant of yielding Fy can be expressed as F Ku Ko u 25 roof y The yield strength of the multistory building observed in the pushover analysis also known as the base shear strength at yield is the sum of the story forces V 1TF 26 The base shear coefficient at yield is given by T Orthogonality relations Clough and Penzien 1993 Eq 11 39 provide that 1 K 2 o 11 M Q8 if 6 is an elastic mo
98. validate results by other means where assumptions may be in question or when the consequences are significant 100 80 Q 60 Triangular Shape o 5 Parabolic Flexure Beam 5 40 5 l Parabolic Shear Beam a 2096 096 0 0 5 1 Deformed Shape Amplitude Figure 5 Default mode shapes available in USEE 2 3 Computation of Response Spectra Using USEE USEE provides robust tools for computing various types of response spectra for both linear elastic and nonlinear response The spectra plot the peak response values that occur 17 over the duration of shaking for a specified range of vibration periods Computed results may be plotted as a function of period or the yield or peak displacement of the oscillator providing various representations of the underlying data Elastic spectra constant strength spectra constant strength reduction factor spectra and constant ductility spectra may be computed using the linear bilinear and stiffness degrading load deformation models for any of the base input accelerations available for the SDOF analysis Details of the computation of response spectra in USEE and the required parameters are described in the following sections 2 3 1 Specification of Periods The range of periods used in the computation of response spectra is specified by the user Either of two distributions of periods within this range may be selected a uniform distribution or a geometric distribu
99. y until the desired level of convergence is achieved Any remaining unbalance is added to the response acceleration at the end of this time step to enforce dynamic equilibrium If a reduced time step is used then after a successful solution is obtained larger time steps are attempted in subsequent time steps and the larger time steps are retained if the desired level of convergence is achieved 2 1 3 1 Incremental Equation of Motion The equation of motion is presented in terms of time t in Eq 1 In this section the incremental equation of motion is developed First the equation of motion at time t At is Mii t At Cu t At 9 R t At P t Ar 10 This assumes that the time step Ar is small enough such that the system properties remain constant during the time step Subtracting Eq 1 from Eq 10 yields M lt At ii t Clil Ac u R t At R 0 P Ac P 0 UD Denoting A t ii t At ii t 12a Au t u t At u r 12b Au t u t At u t 12c AR t R t At R t 12d AP t P t At P t 12e allows Eq 11 to be restated as MAii t CAu t AR t AP t 13 By denoting AR t as KAu T Eq 13 may be restated as MAii t CAu t KAu t AP t 14 where K tangent stiffness of the structure at time f Eq 14 is the incremental equation of motion representing conditions required to maintain dynamic equilibrium during a time step Ar The linear acceleration method is u

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