Software Verification

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1 EXAMPLE 6-6 LINK SUNY BUFFALO DAMPER WITH LINEAR VELOCITY EXPONENT PROBLEM DESCRIPTION This example comes from Section 5 of Scheller and Constantinou 1999 ( the SUNY Buffalo report ). It is a two-dimensional, three-story moment frame with diagonal fluid viscous dampers that have linear force versus velocity behavior. The model is subjected to horizontal seismic excitation using a scaled version of the SE component of the 194 El Centro record (see the section titled Earthquake Record later in this example for more information). The results for modal periods, interstory drift and interstory force-deformation are compared with experimental results obtained using shake table tests. The experimental results are documented in the SUNY Buffalo report. The model is shown in the figure on the following page. Masses representing the weight at each floor level, including the tributary weight from beams and columns, are concentrated at the beam-column joints. Those masses, 2.39 N-sec 2 /cm at each joint, act only in the X direction. In addition, small masses,.2 N-sec 2 /cm, are assigned to the damper elements. The small masses help the nonlinear time history analyses solutions converge. Diaphragm constraints are assigned at each of the three floor levels. Beams and columns are modeled as frame elements with specified end length offsets and rigid-end factors. The rigid-end factor is typically.6 and the end length offsets vary as shown in the figure. The frame elements connecting the lower end of the dampers to the Level 1 and Level 2 beams are assumed to be rigid. This is achieved in by giving those elements section properties that are several orders of magnitude larger than other elements in the model. See the section titled Frame Element Properties later in this example for additional information. The dampers are modeled using two-joint, damper-type link elements. Both linear and nonlinear properties are provided for the dampers because this example uses both linear and nonlinear analyses. See the section titled Damper Properties and the section titled Load Cases Used later in this example for additional information. EXAMPLE

2 GEOMETRY AND PROPERTIES 12.5 cm 1 cm 1 cm Joints constrained as diaphragm, typical at Levels 1, 2, and 3 7 2XST2X3 Damper 8 1 cm Level 3 Frame element end length offsets, typical. Rigid-end factor is N-sec 2 /cm mass at joints 3, 4, 5, 6, 7 and 8 acting in X direction only ST2X385 ST2X STCOL 4.25 cm Damper 8 cm 9 Stiff XST2X3 26 cm Y 2XST2X3 Stiff cm 26 cm Damper 1STCOL ST2X385 ST2X cm 15 cm 1 cm 1 cm 1.5 cm 76.2 cm 76.2 cm Level 2 Level cm X 2 2 cm Base EXAMPLE

3 FRAME ELEMENT PROPERTIES The frame elements in the model have the following material properties. E = 21,, N/cm 2 ν =.3 The frame elements in the model have the following section properties. 1STCOL A = 9.1 cm 2 I = cm 4 A v = 4.42 cm 2 ST2X385 A = 6.61 cm 2 I = 5.95 cm 4 A v = 2.2 cm 2 2XST2X3 A = cm 2 I = 11.9 cm 4 A v = 2.2 cm 2 STIFF A = 1, cm 2 I = 1, cm 4 A v = cm 2 (shear deformations not included) EXAMPLE

4 DAMPER PROPERTIES The damper elements in the model have the following properties. Linear (k is in parallel with c) k = N/cm c = 16 N-sec/cm Nonlinear (k is in series with c) k = 1,, N/cm c = 16 N-sec/cm exp = 1 The damping coefficient used for the dampers for both the linear and nonlinear analyses is c = 16 N-sec/cm. This value was determined using the average value from a series of experimental tests. As described in Scheller and Constantinou 1999, the tested values of the damping coefficient ranged from 135 to 185 N- sec/cm. The average value of 16 N-sec/cm was used for all dampers in the model LINEAR AND NONLINEAR ANALYSIS USING DAMPERS This example uses both linear and nonlinear load cases. It is important to understand that there are differences in the damper element behavior for linear and nonlinear analysis. For nonlinear analysis the damper acts as a spring in series with a dashpot and uses the specified nonlinear spring stiffness and damping coefficient for the damper. In contrast, for linear analyses the damper element acts as a spring in parallel with a dashpot and uses the specified linear spring stiffness and damping coefficient for the damper. This is illustrated in the figure to the right. c nonlinear k nonlinear Damper Properties for Nonlinear Analyses k linear c linear Damper Properties for Linear Analyses EXAMPLE

5 In this example, for the linear analysis, the linear effective stiffness, k linear, is set to zero so that pure damping behavior is achieved. For nonlinear analysis the nonlinear stiffness, k nonlinear, is set to an approximation of the stiffness of the brace with the damper. If pure damping behavior is desired from the damper element for nonlinear analysis with dampers, as is the case in this example, the effect of the spring can be made negligible by making its stiffness, k nonlinear, sufficiently stiff. The spring stiffness should be large enough so that the characteristic time of the springdashpot damper element, given by τ = c/ k nonlinear, is approximately one to two orders of magnitude smaller than the size of the load steps. Care must be taken not to make k nonlinear excessively large because numerical sensitivity may result. For this example: c 16 τ = = =.16 seconds 1,, k nonlinear Thus τ is approximately two orders of magnitude less than the.1 second load steps and the 1,, N/cm seems to be a reasonable value to obtain pure damping behavior. Important Note: In linear modal time history analysis (and response spectrum analysis) of systems with damper elements, only the diagonal terms of the damping matrix are used; the off-diagonal, cross-coupling terms are ignored. All other analyses of systems with damper elements use all terms in the damping matrix. Thus linear modal time history analysis (and response spectrum analysis) of systems with damper elements should be used with great care and should typically be considered as only an approximation of the solution. In general, nonlinear analysis should be used for final design of systems with damper elements. EXAMPLE

6 LOAD CASES USED Five different load cases are run for this example. They are described in the following table. Load Case MODAL MHIST1 NLMHIST1 DHIST1 Description Modal load case for ritz vectors. Ninety-nine modes are requested. The program will automatically determine that a maximum of ten modes are possible and thus reduce the number of modes to ten. The starting vectors are U x acceleration and all link element nonlinear degrees of freedom. Linear modal time history load case that uses the modes in the MODAL load case. This case includes modal damping in modes 1, 2 and 3. Nonlinear modal time history load case that uses the modes in the MODAL load case. This case includes modal damping in modes 1, 2 and 3. Linear direct integration time history load case. This case includes proportional damping. Nonlinear direct integration time history load case. This case includes proportional damping. The modal time history analyses use 2.71%, 1.2% and 1.4% modal damping for modes 1, 2 and 3, respectively. As described in Scheller and Constantinou 1999, those modal damping values were determined by experiment for the frame without dampers. EXAMPLE

7 The direct integration time histories use mass and stiffness proportional damping that is specified to have 2.71% damping at a the period of the first mode and 1.2% damping at the period of the second mode. The solid line in the figure to the right shows the proportional damping used in this example. Damping Ratio Mass Stiffness Rayleigh Period (sec) EARTHQUAKE RECORD The following figure shows the earthquake record used in this example. As described in Scheller and Constantinou 1999, it is the SE component of the 194 El Centro record compressed in time by a factor of two. It is compressed to satisfy the similitude requirements of the quarter length scale model used in the shake table tests. The earthquake record is provided in a file named EQ6-6.txt. This file has one acceleration value per line, in g. The acceleration values are provided at an equal spacing of.1 second..3 Acceleration (cm/sec 2 ) Time (sec) EXAMPLE

8 TECHNICAL FEATURES OF TESTED Damper links with linear velocity exponents Frame end length offsets Joint mass assignments Modal analysis for ritz vectors Linear modal time history analysis Nonlinear modal time history analysis Linear direct integration time history analysis Nonlinear direct integration time history analysis Generalized displacements RESULTS COMPARISON Independent results are experimental results from shake table testing presented in Section 5, pages 61 through 73, of Scheller and Constantinou The following table compares the modal periods obtained from and the experimental results. Modal Period Load Case Mode 1 sec Independent Percent Difference % Mode 2 sec MODAL % Mode 3 sec % The following three figures plot the analysis results and the experimental results for the story drift versus time for each of the three story levels for the load case. Similar results are obtained for the other time history load cases. The story drift for Level 3 is calculated by subtracting the displacement at joint 5 from that at joint 7 and then dividing by the Level 3 story height of 76.2 cm and multiplying by 1 to convert to percent. Similarly, the story drift for Level 2 is calculated by subtracting the displacement at joint 3 from that at joint 5 and then dividing by the Level 2 story height of 76.2 cm and multiplying by 1. The story drift for Level 1 is calculated by dividing the displacement at joint 3 by the Level 1 non-rigid story height of 81.3 cm and multiplying by 1. The interstory displacement results are obtained using generalized displacements. EXAMPLE

9 Level 3 Story Drift (%) Time (sec) Level 2 Story Drift (%) Time (sec) Level 1 Story Drift (%) Time (sec) EXAMPLE

10 The following table compares the maximum and minimum values of story drift obtained from and the experimental results at each story level for each of the four time history load cases. Output Parameter Load Case Story Level Maximum Story Drift Minimum Story Drift MHIST1 NLMHIST1 DHIST1 MHIST1 NLMHIST1 DHIST1 Independent Percent Difference Level % Level % Level % Level % Level % Level % Level % Level % Level % Level % Level % Level % Level % Level % Level % Level % Level % Level % Level % Level % Level % Level % Level % Level % The three figures on the following page plot the analysis results and the experimental results for the story drift versus normalized story shear for each of the three story levels for the load case. Similar results are obtained for the other time history load cases. The story shears are normalized by dividing them by 14,7 N. EXAMPLE

11 Level 3 Story Shear / Weight.4 Structure Weight for Shear Normalization = 14,7 N.3 Story Height for Drift = 76.2 cm Level 3 Story Drift (%) Level 1 Story Shear / Weight Level 2 Story Shear / Weight.4 Structure Weight for Shear Normalization = 14,7 N.3 Story Height for Drift = 76.2 cm Level 2 Story Drift (%).4 Structure Weight for Shear Normalization = 14,7 N.3 Story Height for Drift = 81.3 cm Level 1 Story Drift (%) EXAMPLE

12 The following table compares the maximum and minimum values of normalized story shear obtained from and the experimental results at each story level for each of the four time history load cases. Output Parameter Load Case Story Level Maximum Normalized Story Shear Minimum Normalized Story Shear MHIST1 NLMHIST1 DHIST1 MHIST1 NLMHIST1 DHIST1 Independent Percent Difference Level % Level % Level % Level % Level % Level % Level % Level % Level % Level % Level % Level % Level % Level % Level % Level % Level % Level % Level % Level % Level % Level % Level % Level % The inaccuracies associated with ignoring the off-diagonal, cross-coupling terms in the damping matrix for the linear modal time history load case MHIST1 are more significant in the story shear results than they have been in other results displayed in this example. EXAMPLE

13 The Level 1 story shear results shown for the MHIST1 load case do not include the force in the Level 1 damper. This damper force is not reported because, for the linear modal time history, the damping associated with the dampers is converted to modal damping and added to any other modal damping that may be specified. If a stiff frame element was included in the model below the Level 1 damper, similar to the stiff element at the other levels, the Level 1 story shear could be cut through three frame elements and all of the shear would be accounted for. However, the inaccuracies caused by ignoring the off-diagonal terms in the damping matrix would still be present. COMPUTER FILE: Example 6-6 CONCLUSION The results show an acceptable comparison with the independent results. The clearest comparison of results is evident in the graphical comparisons. The results using linear modal time history analysis (load case MHIST1) are slightly different from the other analyses because the linear modal time history analysis uses only the diagonal terms in the damping matrix, ignoring any offdiagonal, cross-coupling terms. The other analyses use all terms in the damping matrix. For this example load case MHIST1 shows a good approximation of the other solutions. The error introduced by ignoring the cross-coupling terms tends to improve the comparison with experimental results in some items and to make it worse in other items. In general we recommend that linear modal time history analysis of models with damper elements only be used for quick, preliminary checks, and that another type of analysis be used for final analysis. EXAMPLE

Software Verification

Software Verification EXAMPLE 6-003 LINK GAP ELEMENT PROBLEM DESCRIPTION This example uses a single-bay, single-story rigid frame to test the gap link element. This link element carries compression loads only; it has zero stiffness