REPORT NO.
EERC 76-11
APRIL 1976EARTHQUAKE ENGINEERING RESEARCH CENTER
INFLUENCE OF ANALYSIS AND DESIGNASSUMPTIONS ON COMPUTErn INELASTICRESPONSE OF MODERATELY TALL FRAMES
by
GRAHAM H. POWELL
and
DENNIS G. ROW
Reporf fo Sponsor:
National Science Foundation
Grant NSF-GI-36387
LULLI:Ivr: ur- I:Nl:7INttRING
UNIVERSITY OF CALIFORNIA • Berkeley, California
See back of repor'l for up to date listing ofEERC reports.
BIBLIOGRAPHIC DATA 11. Report No.SHEET EERC 76-11
4. Title and Subtitle
IlInfluence of Design and Analysis Assumptions on ComputedInelastic Response of Moderately Tall Frames ll
7. A uthor(s)Graham H. Powell and Dennis G. Row
9. Performing Organization Name and AddressEarthquake Engineering Research CenterUniversity of California, Berkeley47th Street and Hoffman Blvd.Richmond, California 94804
12 Sponsoring Organization Name and AddressNational Science Foundation1800 G StreetWashington, D. C. 20550
15. Supplementary Notes
3. Recipient's Accession No.
5. Report Date
April 19766.
8. Performing Organization Rept.No. 76-11
10. Project/Task/Work Unit No.
11. Contract/Grant No.
NSF-GI-36387
13. Type of Report & PeriodCovered
14.
16. Abstracts
The investigation described is essentially a pilot study which develops general conclusions on the inelastic behavior of buildings designed by elastic methods. The investigation is limited to single bay frames of only ten stories subjected to groundmotions of short durations.
The analysis includes effects of column overdesign, stiffness degradation and the Pdelta effect. Maximum values of story displacements, interstory drifts, beam aridcolumn hinge rotations and column axial forces are determined and compared.
17. Key Words and Document Analysis. 170. Descriptors
17b. Identifiers/Open-Ended Terms
17c. COSArI Field/Group
18. Availability Statement
Release Unlimited
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19•. Security Class (ThisReport) . .
UNCI. ASSIFIED20. Security Class (This
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21. No. of Pages
lIb22. Price fie B-c i~
mFl+of'-lSCOMM~DC 14952-P72
INFLUENCE OF DESIGN AND ANALYSIS ASSUMPTIONSON COMPUTED INELASTIC RESPONSE OF MODERATELY TALL FRAMES
by
Graham H. PowellProfessor of Civil Engineering
and
Dennis G. RowGraduate Student
Prepared Under the Sponsorship ofthe National Science Foundation
Report No. 76-11Earthquake Engineering Research Center
College of EngineeringUniversity of California
Berkeley, California
April 1976
TABLE OF CONTENTS
1. OBJECTIVE AND SCOPE .....
2. DESIGN OF EXAMPLE STRUCTURES
2.1 STRUCTURE TYPES ....
2.2 STIFFNESSES AND NATURAL PERIODS
2.3 GROUND MOTIONS ..
2.4 ELASTIC ANALYSIS
2.5 BEAM DESIGN .
2.6 COLUMN DESIGN
3. INELASTIC ANALYSIS
3.1 BASIC IDEALIZATION
3.2 PARAMETERS VARIED
3.3 RESPONSE PARAMETERS
4. RESULTS .
4.1 PRESENTATION OF RESULTS
4.2 ELASTIC RESPONSE .•..
4.3 INFLUENCE OF BEAM STRENGTH ON DISPLACEMENTS
4.4 INFLUENCE OF COLUMN STRENGTH ON DISPLACEMENTS
4.5 INFLUENCE OF P-DELTA EFFECT ON DISPLACEMENTS ..
4.6 INFLUENCE OF DEGRADING STIFFNESS ON DISPLACEMENTS
4.7 TIME HISTORIES OF DISPLACEMENT .•...
4.8 INFLUENCE OF BEAM AND COLUMN STRENGTH ON INTERSTORY
DRI FTS . . . . . . . . . . . .
4.9 INFLUENCE OF P-DELTA EFFECT ON INTERSTORY DRIFTS.
Page1
4
4
4
4
5
6
8
12
12
12
13
14
14
14
15
16
17
17
18
18
19
4.10 INFLUENCE OF DEGRADING STIFFNESS ON INTERSTORY DRIFTS 20
4.11 INFLUENCES OF BEAM AND COLUMN STRENGTH ON BEAM HINGE
ROTATIONS . . . . . .
;
20
" CI 0 • CI " • (J
4.12 INFLUENCES OF DEGRADING STIFFNESS ON BEAM HINGE
ROTATIONS . . . . . . . . . . . . . . . .
4.13 TIME HISTORIES OF BEAM HINGE ROTATIONS ....
4.14 COMPARISON OF MAXIMUM AND ACCUMULATED BEAM HINGE
ROTATIONS: STRONGER COLUMN DESIGNS .
4.15 COMPARISON OF MAXIMUM AND ACCUMULATED BEAM HINGE
ROTATIONS: WEAKER COLUMN DESIGNS .
4.16 INFLUENCE OF COLUMN STRENGTH ON COLUMN HINGE ROTATIONS
4.17 INFLUENCE OF DEGRADING BEAM STIFFNESS AND SPECIAL COLUMN
DESIGN ON COLUMN HINGE ROTATIONS .• . . . . . . . . . . .
4.18 COMPARISON OF MAXIMUM AND ACCUMULATED COLUMN HINGE
ROTATI ONS .
4.19 EXPLANATION OF HINGE REVERSAL PHENOMENA.
4.20 MAXIMUM COLUMN AXIAL FORCES .
5. CONCLUSIONS AND RECOMMENDATIONS
REFERENCES •...........
i i
21
21
22
22
23
24
25
25
27
28
32
1. OBJECTIVE AND SCOPE
Procedures for the elastic dynamic analysis of frame structures
under earthquake loading are well established, and are being used
increasingly in design. It is commonly accepted that if the beams and
columns are proportioned to resist the calculated forces, and if sound
detailing practices are followed, then the performance of the structure in
an actual earthquake will be satisfactory.
It is also commonly accepted, however, that it is economically
unrealistic to design a structure with sufficient strength to remain elastic
during a strong earthquake, and hence that inelastic behavior must occur.
It does not necessarily follow, therefore, that elastic methods of analysis
will lead to consistent designs. For example, if all beams are designed
with the same load factor, it does not necessarily follow that all will
experience the same inelastic deformations.
In order to estimate the actual inelastic response, it is
necessary to carry out inelastic analyses. A large number of analytical
techniques and computer programs have been developed for estimating
inelastic dynamic response [3-50J, several of which are applicable to
practical frame structures. The modelling of inelastic behavior is
usually rather simple, using plastic hinges with stable hysteresis loops.
However, analyses incorporating such refinements as degrading stiffness
have been reported. The results of several inelastic analyses of frame
structures have been described in research reports. However, it is
difficult to correlate the results of different investigators, and few
general conclusions have been reached. Inelastic analyses are rarely used
in building design, primarily because of cost.
The aim of the investigation described herein has been to develop
general conclusions on the inelastic behavior of buildings designed by-1-
elastic methods. The number of parameters which might be varied in a study
of this type is large, and this study has been limited in scope. Hence,
the conclusions which have been reached are only tentative. Nevertheless,
it is believed that the investigation is more consistent and comprehensive
than those which have previously been carried out. Similar procedures
could be applied in future studies, considering a wider range of structures
and design assumptions and hence drawing more definite conclusions. The
procedure was as follows.
(1) Reinforced concrete building frames of three types were
selected, namely a IItypical ll frame with slender members, a frame with stiff
spandrel beams, and a coupled-wall frame. Each frame was of only ten
stories with only a single bay, because of limited funds and the large
number of analyses required. This is the most serious limitation of the
study.
(2) A fami ly of fi ve gro und moti ons was genera ted arti fi ci ally.
The major limitation here was that the earthquakes were of short duration.
Response spectra were produced for' these motions, and a smoothed envelope
spectrum was constructed. Two design spectra were then produced by
dividing this smoothed spectrum by 3 and 6, respectively.
(3) The forces in the beams and columns were calculated for
combined gravity and earthquake loads, using the design spectra. The
members were then designed for these forces.
(4) The frames were analyzed for each of the five ground motions,
allowing plastic hinges to form at the member ends. The effects of column
overdesign, stiffness degradation, and the P-delta effect were included in
some of the analyses.
(5) The maximum values of story displacements, interstory drifts,
beam hinge rotations, column hinge rotations and column axial forces
-2-
were determined for each analysis, and plotted in graphs for
comparison.
(6) Trends were observed, and conclusions drawn.
The following chapters describe the procedures and results.
2. DESIGN OF EXAMPLE STRUCTURES
2.1 STRUCTURE TYPES
Three structures of different types were selected, namely (1) a
IItypical ll frame, with slender beams anel columns; (2) a II stiff beam ll frame,
with deep spandrel beams; and (3) a coupled-wall frame. The dimensions,
masses and stiffness are shown in Figs. 2.1 and 2.2. The properties were
selected as described in the following sections.
2.2 STIFFNESSES AND NATURAL PERIODS
First mode periods of 1.2, 0.9 and 0.6 seconds, respectively,
were selected for the three frames. Each of these periods is a typical
value for a frame of the corresponding type.
The stiffness ratios shown in Fig. 2.2 were chosen to provide
reasonable ratios between the column and beam stiffnesses, and to provide
reasonable stiffness variations over the frame heights. The values of
EIo were then selected to give the required first mode periods. In each
case the stiffness values correlate quite closely with the assumed column
and girder dimensions. The floor masses correspond to frames at
approximately 20 foot centers.
2.3 GROUND MOTIONS
A family of artificial ground motions was generated, using the
procedure described by Ruiz and Penzien [4]. This procedure uses a linear
stochastic model to generate records of filtered nonstationary shot noise,
simulating ground motion accelerograms.
From consideration of several strong motion earthquake records
obtained for similar soil conditions, epicentral distance and magnitude,
Ruiz and Penzien suggested the use of a linear filter with a natural
-4-
frequency of 2.5 cycles/sec and 60% critical damping. The resulting
accelerograms simulate strong motion earthquakes for firm soil at moderate
epicentral distances.
Five records were created for the present study, using a single
computer run of the Ruiz-Penzien program. Each record had a duration of
10 secs, made up of a 1.0 sec parabolic build-up, 6.0 sees at maximum
intensity, and a 3.0 sec exponential decay. The ground motion records
are shown in Figs. 2.3 through 2.7.
The peak ground accelerations are approximately 0.3g for each
record. The peak ground velocities are approximately 20 ft/sec for
records 1, 2 and 3, but substantially less for records 4 and 5. Records
1, 2 and 3 give comparable peak ground displacements of approximately
15 inches, with records 4 and 5 giving smaller displacements. Records 2
and 3 exhibit a tendency for the ground displacement to drift in one
direction.
The response spectra for the five records are shown in Fiq. 2.8.
In this and subsequent figures, the solid line represents record 1, and
the lines with successively shorter dashes represent records 2 through 5,
respectively. The spectra indicate that records 4 and 5 represent
generally weaker ground motions, although there is substantial overlap
among all of the spectra.
2.4 ELASTIC ANALYSIS
Elastic analyses, using the TABS [1] program, were carried out
to obtain member forces for design. The structures were idealized
assuming rigid joint zones.
For gravity load effects, weights corresponding to the masses
shown in Fig. 2.1 were applied as vertical loads, distributed over the
, -5-
clear spans of the beams. For earthquake effects, the ground motion
spectra were enveloped approximately, using engineering judgment, to give
the smooth spectrum shown in Fig. 2.9. This spectrum was then used to
define horizontal ground motion effects, vertical earthquake motions being
ignored.
Two intensities of design spectrum were used to design the frames,
name ly a "hi gher ll spectrum, obtained by divi ding the smoothed spectrum
ordinates by 3 (see Fig. 2.9), and a 1I1owerll spectrum obtained by dividing
these ordinates by 6. The TABS analysis was carried out considering 10
modes of vibration with 5% damping in each mode. The peak base shear
coefficients obtained from the analyses are shown in Table 2.1, and compared
with the design coefficients from the 1974 SEAOC recommendations [2J for
ductile moment-resisting frames. For each frame, the SEAOC recommended
base shear lies between the values obtained for the lower and higher
spectra.
Member forces were calculated for the following two load
conditions:
(1) 1.4 (D + L) ± E
(2) 0.9D ± E
where D= dead load, L = live load, and E = earthquake. The beams and
columns were designed to have ultimate strengths equal to the member
forces produced by these loads. For the calculation of gravity load
effects, the vertical load was assumed to be 80% dead load and 20% live
load.
2.5 BEAM DESIGN
(2.1)
(2.2)
At each end of each beam, the maximum positive and negative
bending moments were determined from the elastic analyses, considering the
-6-
TABLE 2.1 BASE SHEAR COEFFICIENTS
TYPICAL STIFF BEAM COUPLED
FRAME FRAME WALLS
Higher Spectrum 0.1090 0.1240 0.1720
Lower Spectrum 0.0545 0.0622 0.0860
SEAOC* 0.0856I
0.0989 0.1211I
* coefficient = 1.4 KCSwhere K = 0.67
C = 115;1
and S =soil factor, assumed = 1.5
The factor 1.4 is to convert from working load to ultimate loadvalues
-7-
two loading conditions of Eqs. 2.1 and 2.2. The maximum values are shown
in Table 2.2 as the "required" moments. Note that because of the gravity
loads, the negative moments are increased by less than 100% when the
response spectrum ordinates are doubled, whereas the positive moments are
increased by more than 100%.
The moment values for which the beams were proportioned (i .e. the
yield moments for the beams) are also shown in Table 2.2. These moments
were obtained by rounding the required values and smoothing over the
building heights. Also a positive moment capacity equal to at least 50%
of the corresponding negative moment capacity was provided at each point
to conform to usual practice.
For the subsequent inelastic analyses, plastic hinges were
allowed to form only at the ends of each beam. Hence, no consideration
was given to the moment capacities at midspan of the beams.
2.6 COLUMN DESIGN
The column design was based on an assumed interaction
relationship for column strength, as shown in Fig. 2.10. For any column
width, h, this relationship is defined completely by the single parameter
Mo. Its shape was selected to be reasonable for a typical reinforced
·concrete column.
The column sections immediately above and below each floor were
assumed to have the same strength, to be consistent with practical designs
in which the column steel is changed between floors. For the two load
conditions of Eqs. 2.1 and 2.2, combinations of gravity with positive and
negative earthquake effects were extracted from the TABS printout for both
the left and right columns above and below each floor. That combination
of axial force and bending moment which required the largest value of
-8-
oJ:x::oJ..~-
.l..
.l.....-n
TABLE 2.2 BEAM DESIGN MOMENTS
LOWER SPECTRUM DESIGN HIGHER SPECTRUM DESIGN
NEGATIVE POSITIVE NEGATIVE POSITIVE
Propor- Propor- Propor-- Required Propor-LEVEL Required tioned Requi red tioned Required tioned tioned
10 79.0 80.0 0 40.0 98.0 100.0 6.5 50.09 148.0 150.0 9.0 75.0 195.1 200.0 40.0 100.08 178.3 190.0 16.5 95.0 250.8 270.0 89.0 135.07 195.7 190.0 31.4 95.0 283.9 270.0 119.7 135.06 208.5 215.0 40.7 107.5 307.3 320.0 139.6 160.05 220.8 215.0 53.1 107.5 331.9 320.0 164.2 I 160.04 225.3 230.0 65.9 115.0 346.4 370.0 187.0 210.03 237.2 230.0 76.9 115.0 369.6 370.0 209.3 210.02 242.7 230.0 80.1 115.0 379. 1 370.0 216.5 210.01 228.2 230.0 69.9 115.0 352.9 370.0 194.5 210.0
.10 70.0 80.0 0 40.0 91.0 100.0 17.0 50.09 139.0 140.0 10.0 70.0 194.0 200.0 64.0 100.08 178.0 180.0 33.0 90.0 261.0 260.0 116.0 130.07 198.0 210.0 51.0 105.0 300.0 320.0 154.0 160.06 218.0 210.0 66.0 105.0 336.0 320.0 184.0 160.05 231.0 230.0 77.0 115.0 361.0 380.0 208.0 220.04 238.0 230.0 90.0 115.0 379.0 380.0 232.0 220.03 252.0 260.0 104.0 130.0 407.0 420.0 259.0 260.02 265.0 260.0 112.0 130.0 431 .0 420.0 278.0 260.01 250.0 260.0 100.0 130.0 402.0 420.0 252.0 260.0
10 90.0 90.0 5.0 45.0 131.0 140.0 56.0 70.09 128.0 140.0 12.0 70.0 180.0 200.0 64.0 100.08 144.0 140.0 27.0 70.0 211.0 200.0 95.0 100.07 160.0 160.0 43.0 80.0 243.0 250.0 127.0 125.06 173.0 180.0 57.0 90.0 271.0 290.0 154.0 170.05 183.0 180.0 66.0 90.0 290.0 290.0 173.0 170.04 186.0 180.0 69.0 90.0 296.0 290.0 179.0 170.03 181.0 180.0 65.0 90.0 286.0 290.0 169.0 170.02 162.0 160.0 46.0 80.0 248.0 250.0 132.0 125.01 132.0 130.0 16.0 65.0 189.0 200.0 72.0 I 100.0
II
unit of moment = kip-ft
-9-
M was then determined, to obtain the required column strength. Thearequired strengths for all columns are shown in Table 2.3.
-10-
W:E:<:0:::L1-
-J<:u>-<0>I-
~wc::o
l.1..L1--l-V)
V)
-J-J<::::s:Cll.LJ-J0..~ou
TABLE 2.3 REQUIRED COLUMN STRENGTHS
LOWER SPECTRUM HIGHER SPECTRUMDESIGN DESIGN
* *LEVEL Mo Ki p-ft Po Kips r10
Ki p-ft Po Kips
10 . 70 262 86 3249 72 270 105 3928 83 312 126 4717 87 327 127 4756 101 379 131 4905 108 404 146 5474 122 459 147 5513 128 479 167 6252 136 510 180 6751 158 592 I 219 823G 166 621 274 , 1026
10 50 187 64 2419 59 223 90 3398 72 270 109 - 4107 87 327 113 4256 93 349 122 4595 100 375 138 5194 111 418 149 5573 121 453 176 6392 129 485 187 7001 144 538 210 788G 176 660 271 1017,
10 142 133 208 1959 155 145 254 2388 193 181 269 2527 217 203 306 2876 253 237 367 3445 322 302 417 3914 401 376 482 4523 440 412 625 5862 536 502 805 7551 688 i 645 1031 967G 824 I 772 1289 I 1209
*See Fig. 2.10
-11-
3. INELASTIC ANALYSIS
3.1 BASIC IDEALIZATION
The frames were analyzed inelastically by a step-by-step method,
using the DRAIN-2D computer program [5,6]. The joint regions were
considered to be rigid, and column axial deformations were permitted but
beam axial deformations ignored. The masses were lumped at the floors,
and only horizontal inertia effects were considered. Gravity load was
first applied, followed by horizontal ground motion. These idealization
assumptions are the same as those made for the TABS analyses.
Plastic hinges in the beams were permitted to form only at the
beam and column ends. Interaction between axial force and bending moment
was taken into account for hinge formation in the columns. Strain
hardening following hinge formation was assumed to be zero (i .e. the yield
and ultimate strengths were assumed to be equal).
The beam strengths for all inelastic analyses were the values
shown in Table 2.2. The column strengths were varied, as explained in
the following section, to investigate the effects of providing I'over-strength'
columns.
3.2 PARAMETERS VARIED
In order to investigate the effects of idealization assumptions on
the computed response, the following changes were made for some analyses.
(l) The P-delta effect was included.
(2) Degrading stiffness of Takeda type was assumed for the
beam hinges, using the beam element developed by Litton [3J.
For the inelastic analyses, the case including the P-delta effect was used
as the basic case, with only a few analyses repeated ignoring the effect.
-12-
In order to investigate the influence of column strength, three
different column strength assumptions were made, as follows.
(1) Strengths equal to those shown in Table 2.3.
(2) Strengths equal to 1.4 times those shown in Table 2.3
(i.e an overstrength provision of 40%).
(3) Infinite strengths (i.e. elastic columns).
3.3 RESPONSE PARAMETERS
The parameters used to characterize the inelastic response were
as follows.
(1) Maximum horizontal displacements at floors (displacements
relative to base).
(2) Maximum interstory drifts.
(3) Maximum and accumulated plastic hinge rotations in beams.
See Fig. 3.1 for definition of these quantities.
(4) Maximum and accumulated plastic hinge rotations in columns.
(5) Maximum axial forces in columns.
For a few cases, time histories of some of these parameters have been
plotted, in order to obtain more detailed information on the response.
-13-
4. RESULTS
4.1 PRESENTATION OF RESULTS
The results of the study are presented graphically in the figures
of Appendix A. The cases studied are identified by the following notations.
(1) Frame type:
(a) Typical frame.
(b) Stiff beam frame.
(c) Coupled walls.
(2) Response spectrum used for selecting member strengths:
(a) HI = higher spectrum.
(b) LO = lower spectrum.
(3) Column strength:
(a) 1.0 = strengths as in Table 2.3.
(b) 1.4 = strengths 1.4 times those in Table 2.3.
(c) 00 = elastic columns.
In addition, some fully elastic analyses with infinite strength assumed for
both beams and columns were carried out.
For each case, results are presented for all five ground motions.
A solid line is used for ground motion 1, and dashed lines with successively
shorter dashes for motions 2 through 5.
In all cases, there are large variations in the response parameters
from one earthquake to the next. Hence, the influences of changes in the
design assumptions can be considered only in terms of qualitative trends.
4.2 ELASTIC RESPONSE
The computed story shear and floor displacement envelopes,
obtained assuming infinitely strong beams and columns, are shown in Figs.
A.l and A.2, respectively" For any frame, the roof displacements produced
-14-
by the different ground motions differ substantially, and are essentially
proportional to the spectral accelerations at the first mode period. This
reflects the fact that the first mode response dominates. The story shears
vary somewhat more erratically, indicating significant contributions from
the higher modes. The story shears obtained from a response spectrum
analysis using the smooth design spectra are also shown.
The results shown in Figs. A.l and A.2 were obtained ignoring the
P-delta effect. Results were also obtained including this effect, and were
found to be negligibly different. This indicates that the P-delta effect
produced only slight changes in the periods in the elastic range.
4.3 INFLUENCE OF BEAM STRENGTH ON DISPLACEMENTS
Displacement envelopes were computed assuming infinitely strong
columns, with beams designed for (a) infinite strength, (b) the HI spectrum,
and (c) the La spectrum. A comparison of the results indicates the
influence of beam strength on the displacement response.
Figs. A.2 through A.4 show how the displacements vary from frame
to frame, within each design. Figs. A.5 through A.7 show how the response
varies from design to design, within each frame. The following points may
be noted.
(1) As expected, the displacements decrease from the typical
frame to the stiff beam frame to the coupled walls, for all three beam
strengths (Figs. A.2 through A.4).
(2) The roof displacements for any frame remain of the same order
of magnitude as the beam strengths are decreased (Fig. A.S through A.7).
(3) For the typical and stiff beam frames, the displacements
show greater variations from ground motion to ground motion as the beam
strength is reduced (Figs. A.5 and A.6). However, there are marked
variations in response even for the elastic case.
-15-
(4) For the typical and stiff beam frames, the shapes of the
displacement envelopes change substantially as the beam strength
decreases (Figs. A.5 and A.6). As the beams are made weaker, the displace
ments in the lower stories increase substantially, whereas the roof displace
ments remaiQ generally of the same magnitude.
(5) For the coupled wall frame, there is no marked tendency for
the displacement envelopes to change shape as the beam strengths decrease
(Fig. A.7).
(6) For any frame, the ground motion which produces the largest
roof displacement for the elastic structure does not necessarily produce
the largest displacement for the inelastic structure. For example, for the
stiff beam frame (Fig. A.6) the ground motions can be arranged in order of
decreasing severity of effect as follows.
(a) Elastic: 3 - 4 - 5 .. 1 - 2
(b) HI/co 3 2 4 .. 1 5
(c) LO/co 1 4 3 •. 2 5
4.4 INFLUENCE OF COLUMN STRENGTH ON DISPLACEMENTS
Displacement envelopes were computed for three different column
strengths, namely with (a) elastic columns, (b) columns designed using an
overstrength factor of 1.4, and (c) columns designed using an overstrength
factor of 1.0 (i .e. no overstrength). The beams were inelastic in all cases,
and designs usinq both the HI and LO spectra were analyzed. A comparison
of the results indicates the effect of column strength on the displacement
response.
Fiqs. A.8 through A.13 show how the responses vary as the column
strenqths are changed, for each of the three frames and for the HI and LO
designs. The following points may be noted.
-16-
(1) As the column strength decreases, there is a marked tendency
for the displacements in the lower stories to increase, for all three frames.
This effect is more marked for the La designs (weaker beams and columns)
than for the HI designs.
(2) As the column strengths are reduced, the different ground
motions remain essentially the same in relative severity for each frame.
(3) The roof displacements again remain qenerally the same as the
column strengths are reduced, although the different ground motions differ
markedly in relative severity.
(4) Ground motion 3 has a much more severe effect on the coupled
wall frame than the other ground motions (Figs. A.12 and A.13). This
phenomenon is examined in a later section.
4.5 INFLUENCE OF P-DELTA EFFECT ON DISPLACEMFNTS
It was noted in Section 4.1 that the influence of the P-delta
effect was negligible for all three frames with elastic behavior. The
maximum displacements computed considering and ignoring the P-delta effect
are shown in Figs. A.14 and A.15 for the LO/l.O designs of the typical frame
and coupled wall frame. The LO/1.0 design of the typical frame is the most
flexible structure, and the P-delta effect would be expected to be most
significant for this case. Fig. A.14 indicates that the effect is
noticeable, especially for the ground motions which produce the largest
displacements, but that the changes in response are relatively small. Fig.
A.15 indicates negligible changes in displacement response for the coupled
wall frame.
4.6 INFLUENCE OF DEGRADING STIFFNESS ON DISPLACEMENTS
Analyses were carried out for the LO/1.4 designs of all three
frames, firstly assuming simple elastic-plastic behavior for the beam hinges,-17-
and secondly assuming degrading stiffness of Takeda type. The column hinges
were assumed to have simple elastic-plastic behavior in all cases. The
computed displacement envelopes are shown in Figs. A.16 through A.18. It
can be seen that substantial differences in response resulted for some
ground motions (for example, motion 2 in Fig. 2.16 and motion 3 in Fig. A.18).
Overall, however, there was no clear trend, and the displacements were
generally similar for both idealizations.
4.7 TIME HISTORIES OF DISPLACEMENT
Time histories of displacement at the roof and fifth floor levels
are shown in Figs. A.19 through A.2l, for the LO/1.4 designs of each frame
and for ground motions 1 and 3. The ground motion time histories are also
shown. Note that the floor displacements are relative to the base.
For the typical frame, in Fig. A.19, it can be seen that ground
motion 1 produces a distinctly biased displacement of the structure at about
5 seconds into the earthquake, and that the resulting displacements
oscillate about a mean value (at the roof) of approximately 3 inches.
Ground motion 1, on the other hand, produces no biased drift of the structure.
A similar effect, but less marked, is produced by ground motion 1 for the
stiff beam frame (Fig. A.20). For the coupled wall frame (Fig. A.21) a
very marked bias is again produced, but this time by ground motion 3. It
was noted earlier (see Figs. A.7, A.12 and A.13) that ground motion 3
produced more severe displacement responses for the coupled wall frame than
the other motions. Fig. A.21 clearly illustrates the different nature of
the response produced by ground motion 3 for that frame.
4.8 INFLUENCE OF BEAM AND COLUMN STRENGTH ON INTERSTORY DRIFTS
Envelopes of interstory drift were computed for the HI/1.4,
HI/1.O, LO/1.4 and LO/1.0 designs for all three frames. Interstory drift is
-18-
likely to be of more value to a designer than floor displacement, because it
gives a more direct indication of the amount of partition damage which is
likely to be produced. The results are shown in Figs. A.22 through A.27.
The following points may be noted.
(1) The interstory drifts decrease from the typical frame to the
stiff beam frame to the coupled wall frame, as would be expected from the
differing stiffnesses of the structures.
(2) As the design strengths are reduced from the HI to the LO
values for the typical and stiff beam frames, there is a distinct tendency
for the drifts to increase in the lower stories and decrease in the upper
stories (compare Fig. A.22 with A.23, and Fig. A.24 with A.25). This
tendency is not present, however, in the coupled wall frame (compare
Fig. A.26 with A.27).
(3) As the column strength is reduced from a factor of 1.4 to 1.0
for the typical and stiff beam frames, there is again a distinct tendency
for the drifts to increase in the lower stories (see Figs. A.22 through
A.25). There appears to be some tendency, although less marked, for
similar changes to occur for the coupled wall frame (see Figs. A.26 and
A.27).
4.9 INFLUENCE OF P-DELTA EFFECT ON INTERSTORY DRIFTS
For the LO/1.0 design for the typical frame (i.e. the most
flexible structure), the effect of ignoring the P-delta effect is shown in
Fig. A.28. There is a tendency for the drift to increase when the P-delta
effect is included, especially for those ground motions which produce the
largest interstory drifts. However, the increases are small.
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4.10 INFLUENCE OF DEGRADING STIFFNESS ON INTERSTORY DRIFTS
For the LO/1.4 designs of all three frames, the effects of assuming
degrading stiffness for the beams are shown in Figs. A.29 through A.31. For
the typical and stiff beam frames, the assumption of degrading stiffness
produces a significant increase in drift for ground motion-l, but generally
no significant changes for the other ground motions. For the coupled wall
frame there are significant increases for ground motions 2 and 4, but a
reduction for ground motion 3. Overall, degrading stiffness appears to have
no substantial influence.
4.11 INFLUENCES OF BEAM AND COLUMN STRENGTH ON BEAM HINGE ROTATIONS
Maximum beam hinge rotations were computed for all three frames,
for the HI and La designs, and for three different column strengths (elastic
and factors of 1.4 and 1.0). Hinge rotations give a direct indication of
local structural damage. The results are shown in Figs. A.32 through A.37.
The following points may be noted.
(1) For the strong column designs (factor = 1.4), the maximum
hinqe rotations correlate very closely with the maximum interstory drifts,
the rotations being somewhat less than the drift divided by the story height
(compare, for example, Figs. A.22 and A.32). This is because there is
little column yielding for the strong column designs (as shown subsequently),
so that there is a simple geometrical relationship between the inelastic
part of the drift and the hinge rotation.
(2) For the weaker column designs (factor = 1.0) there is an
approximate correlation between drift and hinge rotation, but it is less
direct (compare, for example, Figs. A.22 and A.32 again). The correlation
is less close in these cases because of column hinging.
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(3) As the design strenqths are reduced from the HI to the LO
values for the typical and stiff beam frames, the beam hinge rotations
increase substantially in the lower stories and decrease in the higher
stories (compare Fig. A.32 with A.33 and Fig. A.34 with A.35). This
tendency is not present to such a marked extent in the coupled wall frame
(compare Fig. A.36 with A.37).
(4) As the column strength is reduced from a factor of 1.4 to 1.0
for the typical and stiff beam frames, the beam hinge rotations aqain
increase in the lower stories and decrease in the higher stories (see
Figs. A.32 through A.35). There is a similar tendency, although less
marked, for the coupled wall frame (see Figs. A.36 and A.37).
4.12 INFLUENCES OF DEGRADING STIFFNESS ON BEAM HINGE ROTATIONS
The effects on hinge rotation of assuming degrading stiffnesses
for the beams are shown for the LO/1.4 designs in Figs. A.38 through A.40.
Because the hinge rotations correlate closely with the interstory drifts
for these designs, the same point as in Section 4.9 may be noted.
4.13 TIME HISTORIES OF BEAM HINGE ROTATIONS
Time histories of bending moment and beam hinge rotation at the
left ends of the fifth floor beams are shown in Fios. A.41 through A.43,
for the LO/1.4 designs of each frame and ground motions 1 and 3. These
time histories show the same tendencies for bias as was observed previously
for the displacement time histories, especially for the coupled wall f~ame
in Fig. A.43. The hinge rotations reverse several times for each frame,
with a substantially larger number of reversals for the coupled wall frame.
This reflects the higher frequency of vibration for this frame.
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4.14 COMPARISON OF MAXIMUM AND ACCUMULATED BEAM HINGE ROTATIONS:STRONGER COLUMN DESIGNS
If a plastic hinge rotation undergoes reversal, then the maximum
and accumulated rotations, as defined in Fig. 3.1, will differ. The
difference provides some indication of the amount of reversal.
Maximum and accumulated beam hinge rotations are shown in Figs. A.44
through A.46, for the LO/1.4 designs of the three frames. The following
points may be noted.
(1) There are substantial differences between the accumulated and
maximum rotations for all three frames, indicating reversal of rotation in
all cases.
(2) The ratio between accumulated and maximum rotation increases
from the typical frame to the stiff beam frame to the coupled wall frame,
being roughly 1.5, 3 and 6, respectively. This reflects the higher
frequencies of vibration of the stiffer frames.
(3) The ratios between accumulated and maximum rotation decrease
at the base of each frame, especially for the coupled wall frame. This is
a consequence of column hinge formation, as explained subsequently.
(4) The products of yield moment and accumulated hinge rotation,
summed over all beams, are of the same order of magnitude for all three
frames. These products represent the amounts of inelastic energy absorbed
by the beams.
4.15 COMPARISON OF MAXIMUM AND ACCUMULATED BEAM HINGE ROTATIONS: WEAKERCOLUMN DESIGNS
Maximum and accumulated beam hinge rotations are shown in Figs.
A.47 and A.48 for the LO/l.G desiqns of the typical and coupled wall frames
only. The following points may be noted.
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(1) For the typical frame the ratio between the accumulated and
maximum rotations is substantially less (approaching 1.0) than for the
LO/1.4 design (compare Figs. A.44 and A.47). That is, the beam hinge
rotations reversed less often for the weaker column design. The reason for
this is presented in a later section.
(2) For the coupled wall frame there is still substantial
reversal of beam hinge rotation, although the ratio of accumulnted to
maximum rotation is generally less than for the LO/1.4 design (compare
Fiqs. A.46 and A.48).
(3) For the typical frame the maximum rotations increase
significantly as the column strengths are decreased, particularly in the
lower stories (compare Figs. A.44 and A.47). This result was unexpected,
because it was felt that reduction in the column strength would allow
hinging of the columns and hence require less hinging in the beams. The
reason for this result is presented in a later section.
(4) For the coupled wall frame the maximum hinge rotations tend
to increase in the lower stories and decrease in the upper stories as the
column strength is reduced (compare'Figs. A.46 and A.48). However, the increases
are generally small.
(5) For the coupled wall frame there is a marked tendency for the
accumulated hinge rotations, and the amount of hinge reversal, to decrease
in the upper stories as the column strength is reduced. Again, this is
explained in a later section.
4.16 INFLUENCE OF COLUMN STRENGTH ON COLUMN HINGE ROTATIONS
Maximum column hinge rotations were computed for all three frames,
for the HI and LO designs, and for column strength factors of 1.4 and 1.0.
The results are shown in Figs. A.49 through A.54. In these figures, the
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hinge rotation shown in any story is the larger of the rotations at the top
and bottom of the column in that story. For story 1, the larger rotation
is invariably at the foundation level. The following points may be noted
(1) The column hinge rotations are small for all designs with a
column strength factor of 1.4. There is, however, a tendency for the
rotations to be significant at the ground floor levels for these designs.
(2) For the typical and stiff beam frames, reduction of the
column strength factor to 1.0 results in substantial increases in the
column hinge rotations (see Figs. A.49 through A.52). The rotations are
larger for the LO design, especially in the lower stories, and reach values
which would'indicate severe column damage.
(3) For the coupled wall frame, reduction of the column strength
factor again increases the column hinge rotations (see Figs. A.53 and A.54),
especially for the LO design. The rotation magnitudes are much less than
for the typical and stiff beam frames, and exhibit interesting peaks in the
middle stories.
4.17 INFLUENCE OF DEGRADING BEAM STIFFNESS AND SPECIAL COLUMN DESIGNON COLUMN HINGE ROTATIONS
The influence of degrading beam stiffness on the column hinge
rotations is shown in Figs. A.55 through A.57 for the L0/1.4 designs of all
three frames. The column stiffnesses were assumed not to degrade in all
cases. Again, the influence of degrading stiffness is not significant.
Results for ('special" designs are also shown in Figs. A.55 through
A.57. In these designs, the columns strengths were based on an overstrength
factor of 1.4 at all points except at the foundation, where a factor of 1.0
was assumed. It was thought that allowing the column to hinge more freely
at the base might eliminate the column hinging at other levels of the
~tructure. The results indicate that this was not the case, the only
-24-
significant change being an increase in the column hinge rotations at
the base.
4.18 COMPARISON OF MAXIMUM AND ACCUMULATED COLUMN HINGE ROTATIONS
For the typical frame only, and for the LO/l.4 and LO/l.O designs,
the maximum and accumulated column rotations are compared in Figs. A.58
and A.59. It can be seen that except at the base of the structure the
maximum and accumulated rotations are identical, indication that there is
no reversal of the column hinges except at the foundation level. This is
explained in the following section.
4.19 EXPLANATION OF HINGE REVERSAL PHENOMENA
It has been noted in earlier sections that (1) the beam hinges
exhibit larger maximum rotations but reduced reversal as the column
strengths are decreased, and (2) the column hinges do not reverse, except
at the base of the structure. The reasons for this are as follows.
For a simple single storY frame, subjected to both gravity load
and earthquake, the behavior is illustrated qualitatively in Fig. 4.1.
Fig. 4.la shows the elastic moment diagram for gravity load only, and
Figs. 4.1b and 4.1c show these diagrams for gravity load plus lateral loads
acting to the right and left, respectively. For an inelastic frame with
strong columns, the moments at both the left and right ends may exceed the
corresponding yield values. Hence, plas~ic hinges may form at both ends,
as shown in Figs. 4.1d and 4.1e, and under earthquake loading there will be
hinge reversal. Note that the required positive moment capacities at the
beam ends are always less than the required negative moment capacities,
because of the gravity loads. Note also that for the La spectrum designs,
the required positive moments were all less than one half of the required
negative moments (see Table 2.2), and hence additional capacities beyond
-25-
the required values were assumed for the design.
If the column strengths are reduced, there is an increasing
tendency for hinges to form in the columns. This tendency is particularly
marked on the compression side of the frame, firstly because the column
moments are substantially larger on this side (the earthquake moments
reinforce the gravity load moments) and secondly because the column
compression reduces the moment capacity. On the tension side, however, the
moments are smaller and the decreased compression tends to increase the
moment capacity. Hence, column hinges tend to form on one side of the
frame only, and hinges will form as shown in Figs. 4.1f and 4.1g. It is
clear that for a single bay frame the column hinges will rotate in one
direction only, without reversal. It can also be seen from FiQs. 4.1d and
4.lf that the beam hinge rotation for the positive moment hinges will be
related to the interstory drifts. As noted previously, because the
interstory drifts occurring before yield are comparatively small, the beam
hinge rotations are essentially equal to the story drifts divided by the
story heights for both positive and negative beam hinges. For the strong
column case the analyses generally showed comparable maximum hinge
rotations for both positive and negative rotation (for example, see
Fig. A.4l). For the weaker column case, however, the negative hinge
rotations were greatly reduced. Hence, the effect of reducing the column
strength is to suppress negative hinge formation in the beams, without
suppressing positive hinge formation. Further, because the column strength
reduction leads to increased interstory drifts, the positive beam hinge
rotations are generally increased above the values for a strong column
design. Note, however, that this is not likely to be critical for the beam,
because the positive IIhinge ll is less localized in an actual frame than the
negative hinge.
-26-
4.20 MAXIMUM COLUMN AXIAL FORCES
An interesting question to explore is whether, at any time during
the earthquake, it is possible for positive and negative hinges to form in
all beams simultaneously. If this can happen, then the maximum possible
column axial force in any story can be determined by statics from the
gravity loads and the beam strengths. For selected designs of the typical
and coupled wall frames, the maximum computed column compression forces are
shown in Figs. A.60 through A.62, for four designs of the typical frame and
two designs of the coupled wall frame.
Virtually identical results are produced by all ground motions for
each frame. The lines of theoretical maximum compression (assuming hinges
in all beams) are not shown in the figures, because they correspond very
closely to the computed maximum compressions. Some computed values exceed
the theoretical maximum by small amounts, but this results from overshoot
of nominal yield values within the computer program. Clearly, for these
example frames the situation with all beams hinging simultaneously is
produced by all ground motions.
The maximum computed compressions for the HI/l.O and LO/l.O designs
of the typical frames may be seen to be slightly less than for the HI/l.4
and LO/l.4 designs. This is because the hinges tend to form in the columns on
the compression side of the frame for the weaker column designs, as explained in
Section 4.19. The maximum compressions are reduced only slightly because
the beam moments are close to their yield values when the column hinges
form. This reduction is negligible for the coupled wall frame.
-27-
5. CONCLUSIONS AND RECOMMENDATIONS
The investigation described in this report has been limited in
scope, and is essentially only a pilot study. In particular, it has been
limited to single bay frames of modest height subjected to ground motions of
short duration. The procedures used to conduct the study are believed to
be sound, particularly with regard to the design of the example structures
and the selection of the ground motions.
Certain conclusions can be drawn from the study, but a sub
stantially broader range of structures will need to be investigated before
definite conclusions and recommendations for design can be formulated. The
conclusions which can be drawn from the study, and the areas in which
further study is needed, are as follows.
(1) The computed inelastic response varied greatly from one
ground motion to the next, even for motions having apparently similar
characteristics. Hence, if inelastic analyses are to be carried out to
estimate inelastic deformations, the computations should be repeated for
several ground motions. It may be noted, however, that the earthquakes
considered in this study were of short duration, with only a small number
of cycles of ground displacement. It is possible that more consistent
results would be obtained if earthquakes of longer duration were used.
This aspect should be investigated.
(2) Although the computed elastic responses of the frames could
be correlated with the elastic spectral accelerations for the first mode
periods, there was no correlation between these spectral accelerations and
the computed inelastic responses. Also, there was no obvious correlation
between the maximum ground accelerations, velocities and displacements on
the one hand and the computed inelastic responses on the other. It is
possible that the computed response might correlate more closely with
-28-
inelastic response spectra. The possibility of such a correlation has not
been considered in this study, but should definitely be investigated in
future studies.
(3) The computed roof displacements were of essentially the same
magnitude regardless of whether the structure yielded or remained elastic.
This is a well known result. However, the pattern of story drift changed
substantially as the frames were weakened, with large drifts in the lower
stories and smaller drifts in the upper stories. The computed beam hinge
rotations similarly were largest in the lower stories. This indicates that
a more uniform distribution of inelastic deformation, and hence presumably
a better structural design, would be produced if the members in the lower
stories were proportioned using more conservative load factors than those
in the upper stories. This aspect of design warrants further
investigation.
(4) When the columns were designed using more conservative
load factors than the beams, then inelastic behavior in the columns was
reduced or eliminated. However, when similar load factors were used, the
columns yielded substantially. The collapse of a structure during an
earthquake will usually occur because the earthquake weakens the structure
to such an extent that it can no longer support gravity loads, and it is
generally accepted that the structure will be weakened more if the columns
are damaged than if the beams are damaged. It would be sound practice,
therefore, to design the columns using more conservative load factors than
those used for the beams. An overstrength factor of 1.4 appears to be
reasonable, although additional investigation of this point is desirable.
(5) If the yielding is limited to the beams, and if hinges form
only at the beam ends, then the beam hinge rotations can be calculated
from the story drifts. This indicates that it might be feasible to use a
-29-
simplified structural model (for example, an inelastic shear beam) for
inelastic dynamic analyses. Provided the story drifts computed for the
simplified model corresponded to those computed for a more elaborate model,
the simplified analyses could be used to predict beam hinge rotations. A
shear beam analysis using load-deflection relationships calculated for single
story subassemblages of the actual frame is a possibility which warrants
further study.
(6) For strong column designs in which hinges form only in the
beams, reversal of hinge rotation can be expected to occur. For weaker
column designs, however, a hinge may form, say, on the left side of the
frame with a column hinge on the right side. When the direction of sway
reverses then a beam hinge forms on the right and a column hinge on the
left. That is, there is no reversal of rotation at the hinges. Because
of the column hinging, however, this is not necessarily a desirable
situation.
(7) For a column which is fixed at the foundation level,
significant inelastic deformation is likely to occur at this level even
though the column may be below yield at all other levels. Hence, the
column should be detailed to provide substantial ductility at this point.
(8) The computed hinge rotations were smaller for the stiffer
frames. The members of such frames might be inherently less ductile, so
that smaller hinge rotations do not necessarily indicate less damage.
Nevertheless~ the smaller computed deflections, interstory drifts and
hinge rotations for the stiffer frames appear to imply superior performance.
(9) For the frames considered in this study, the P-delta effect
exerted only a small influence. This effect would be larger, however, for
taller structures.
(10) The computed hinge rotations were not markedly increased if
-30-
beams with degrading stiffnesses were assumed. This confirms a
conclusion reached by Chopra and Kan [ 7 ], and indicates that elaborate
inelastic models of the structural members may not be needed for practical
inelastic analysis. This aspect also warrants further investigation.
(11) For the structures studied, the maximum computed
compression forces in the columns were equal to the theoretical maxima for
all earthquakes. This might not be the case for taller frames or for
multibay frames. However, if this were generally true, then it could be an
important consideration for column design. Further investigation,
particularly of tall multibay frames, is needed.
(12) Ground motion 3 exerted a markedly more severe effect on the
coupled wall frame, producing a distinctly biased drift of the frame. There
is no obvious feature of this ground motion which might indicate such a
response. It might be of value to investigate this phenomenon in greater
detail, to determine what combination of circumstances can produce this
type of behavior.
-31-
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
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48. Otani, S. and Sozen, M.A., "Simu1ated Earthquake Tests of RIC Frames,"Proceedings, ASCE, Journal of the Structural Division, Vol. 100, No. 5T 3,March 1974.
49. Mahin, S.A. and Bertero, V.V., "Non1inear Seismic Response Evaluation Charaima Building," Proceedings, ASCE, Journal of the Structural Division,Vo1. 100, No. ST 6, June 1974.
50. Otani t S., IIIne1astic Analysis of RIC Frame Structures, II Proceedings,ASCE, Journal of the Structural Division, Vol. 100, No. 5T 7, July 1974.
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FIG. 2.10 ASSUMED INTERACTION RELATIONSHIP FORCOLUMNS.
44
MAXIMUMNEGATIVEROTATION
MOMENT ACCUMULATED POSITIVE ROTATION=SUM OF POSITIVE YIELD EXCURSiONS
MAXIMUMPOSITIVEROTATION
HINGE ROTATION
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FIG. 3.1 DEFINITION OF HiNGE ROTATIONQUANTITIES
45
(0) GRAVITY LOAD MOMENTS
(b), (c): MOMENTS FOR GRAVITY PLUS LATERAL LOAD
(d), (e): HINGES WITH STRONG COLUMNS
( f), (g): HINGES WITH WEAKER COLUMNS
FIG. 4.1 HINGE REVERAL UNDER CYCLIC LOAD
46
APPENDIX A
GRAPHICAL PRESENTATIONSOF RESULTS
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