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

FORM NTIS-35 (REV. 3·72)

19•. Security Class (ThisReport) . .

UNCI. ASSIFIED20. Security Class (This

PageUNCLASSIFIED

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.

-22-

(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

-23-

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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47. Goe1, S.C. and Hanson, R.D., "Seismic Behavior of Multistory Braced SteelFrames," Proceedings, ASCE, Journal of the Structural Division, Vol. 100,No. ST 1, Jan. 1974.

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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