- Email: [email protected]
C structures in the nonlinear range
Computers & Structure5 Voi. 16. No. i--4, pp. 519--529. 1983 Printed in Great Britain
0045-79,19!83/010519-11503.0010 Pergamon Preis L|d
EARTHQUAKE RESPONSE OF IRREGULAR R/C STRUCTURES IN THE NONLINEAR RANGE MEHDI SAIIDIand KENNETH E. HODSON Civil EngineeringDepartment. University of Nevada, Reno, NV 89557, U.S.A. A~tract--The application of a simple analytical model (Q-Model) for calculation of nonlinear seismic response history of irregular planar structures has been demonstrated. The model represents the structure by an equivalent SDOF oscillator. Stiffness degradation effects are accounted for through a simple hysteresis model. The Q-Model is evaluated for small-scaletest structures in addition to a full-scalehypotheticalframe. It is shown that the model exhibits satisfactory performance. The application of the Q-Model, or an approach similar to that, in seismic response computationfor three-dimensionalirregular structures is briefly discussed. INTRODUCTION Much has been learned about the nonlinear seismic behavior of structures with uniform distribution of stiffness through experimental and analytical studies as well as observation of real structures damaged during earthquakes. The findings from these studies have been incorporated in seismic codes, and simplified guidelines have been developed which appear to yield reasonable results. For irregular structures, i.e. structures with large differences in stiffness between adjacent stories and those with unsymmetric distribution of lateral force resisting elements, the amount of available information is limited. The major factor for the scarcity of information on the earthquake behavior of these structures is the fact that required investigations, whether experimental or analytical, are very involved. Irregular structures may be grouped into two categories: (I) planar structures, in which abrupt change of stiffness occurs along the height, and (2) structures in which center of mass is eccentric with respect to center of stiffness, and there may also be a drastic stiffness variation in elevation. The object of this paper is to demonstrate the application of a simple analytical model for calculation of nonlinear seismic response of irregular structures. The method is tested for small-scale physical models as well as full-scale hypothetical structures, with emphasis being placed on structures in the first category. The response for the physical models is evaluated based on the correlation between the experimental and analytical results, while the response for other structures is examined against the calculated results obtained from more elaborate analytical models.
to a model called the Q-Model[Ill, which has been found successful in nonlinear response simulation of uniform reinforced concrete structures. A new and simpler version of the Q-Model was developed and called Q-Model tL), with (L) standing for the limit analysis used in calculating the properties. The new version is described in the following sections. Information about the original version may be found in Ref. [11].
Basic principles
ANALYTICALMETHOD Representation of a multi-degree-of-freedom (MDOF) system by an equivalent single-degree-of-freedom (SDOF) oscillator has been introduced and used for elastic structures[3,4]. In order to have a successful formulation, it is necessary for the system to have a single form or displaced shape during the dynamic response. While this condition is known to be satisfied for many elastic systems, its fulfillment for nonlinear structures has been considered doubtful. Experimental investigations on the nonlinear seismic response of several small-scale reinforced concrete frames and framewalls[l, 5] have revealed that the change in the displaced shape of these structures during the motions is insignificant. These observations along with a simple representation of force-deformation variations have led
The Q-Model is an approximate SDOF model for nonlinear displacement history calculations for planar reinforced concrete structures subjected to base acceleration. Two basic assumptions have been incorporated in the model about: (1) SDOF idealization and (2) hysteretic behavior. (a) SDOF idealization. Representation of a MDOF system by an equivalent SDOF oscillator is possible because it has been known that seismic response of ordinary building structures is dominated by their fundamental mode of vibration. Such an idealization reduces computational effort especially if used for response history analysis. This reduction is due to the considerably smaller size of the SDOF system and due to the substantially smaller number of solutions steps. (b) Hysteretic behavior. Experimental studies of the cyclic behavior of reinforced concrete structures have shown that force-deformation relationships are associated with progressive stiffness degradation and "pinching" action[7]. An equivalent SDOF system would have to take these characteristics into account in order to yield satisfactory results. Several hysteresis models have been developed which include pinching or stiffness degradation, or both[10, 12]. Most of these models are relatively complex and their use in a simplified SDOF system is unjustified. A relatively simple model (called Q-hyst) was used for the original version of the QModel. Detailed information about this model may be found in Ref. [11].
519
Q-Model (L) for planar structures The Q-Model reduces the complex and lengthy nonlinear analysis of multistory structures to the analysis of a SDOF system, once the Q-Model properties are determined. However, in the process of calculating the properties, a computer program for nonlinear static analysis of reinforced concrete structures subjected to lateral loads is needed to obtain the base moment versus
%4EHD[ ~MIDI and KtsnEra E. HoDso~,
520
lateral displacement curve (primary curve). This may not be convenient for the analysts who do not have access to such program or the necessary computer facilities. An alternative approach is used in the new version utilizing the limit analysis method. This approach has been also used in Ref. [5]. The main characteristics of the new and old versions are the same. In the new version, the elastic stiffness of the spring is calculated based on a static analysis of the multistory sturcture for lateral loads applied at floor levels. These loads are proportional to floor masses and heights from base. Because beams in reinforced concrete structures are cracked under service load conditions, cracked moment of inertia is used for beams in the static analysis. The break point of the primary curve (the apparent "yield" point) represents the base moment for the collapse loads. Structural elements are assumed to behave elasto-plastic at this stage, and strain-hardening effects are taken into account at a later stage. The "yield" base moment is found from a limit analysis of the multistory structure with different hinging locations. Current guidelines for seismic design recommend an overdesign of columns to limit yielding to beams[2]. For buildings so designed, hinging takes place primarily in beams, and number of collapse mechanisms to be considered is minimal. Strain-hardening effects are accounted for by assigning a slope to the post-yielding part of the primary curve. This slope is related to increase in moment for individual sections: (1) due to increase in the moment arm as steel strain is raised and (2) due to strain hardening of reinforcement. An explicit consideration of the first factor would be involved. Therefore, the following simple relationship is used to determine the post-yielding slope of the primary curve: (1)
K2 = 2(&'IS,')K1
where K1 =elastic slope of the primary curve; K2 = post-yielding slope of the primary curve; S , ' = modulus of elasticity of reinforcement steel; &'= strain-hardening slope in stress-strain curve for reinforcement steel. The Q-hyst model is used to determine stiffness variation of the base spring. The floor displacements obtained from the static analysis are normalized with respect to the top-floor displacement. The normalized values are assumed to represent the displaced shape of the structure throughout the response. The equivalent mass and height are found from the following relationships: M,=
/
. m~¢~,2
~md,,
)
(M,)
(2)
and L, =
m,qbihi
mi¢~,
(3)
i=l
where L, = equivalent height; M, = equivalent mass; M, = total mass; N = number of floors; h~ = height from the base at floor i; m~ = mass at floor i; Oi = normalized displacement at floor i; The equation of motion is formulated as: M j + C.~ + K,x = - M,,¢~
(4)
where C = damping coefficient; K, = stiffness calculated
using the hysteresis model; .~ = base acceleratioi~ ~. and x = relative acceleration, velocity, and displacement at the equivalent mass (relative to base). The above formulation is the same as that usea m the original Q-Model[Ill. It should be noted that i~ this equation the mass on the r.h.s, of the equation is the total mass, while the mass on the l.h.s, is the equivalent mass which is smaller than the total mass. This difference should be accounted for during integration. The base moment vs displacement at equivalent height relationship forms the primary curve upon which the hysteresis model operates. The displacement at equivalent height is found from a linear interpolation of displacements at adjacent floors Equation (4) is integrated using the Newmark's /3 method[6] to determine the displacement history at the equivalent height, and the base moment history. Displacements at different floors can be obtained using the assumed displaced shape.
where D, = displacement at equivalent height; D~ =displacement at floor i; 4', = normalized displacement at equivalent height; ¢~;= normalized displacement at floor i, Q-Model [or space [rames Many buildings are composed of unsymmetric structural systems that may not be idealized as planar systems. The earthquake response of such buildings may contain significant contribution from torsional mode of vibration even if only one translational component of the earthquake is taken into account. The contribution from torsion may lead to a deflected shape with considerable variation during the earthquake. Such variation is expected to result in an unsuccessful response calculation based on the Q-Model or a similar model. For structures with close torsional and translational periods, variations in the deflected shape are relatively small and an approach similar to that used in the Q-Model may be successful. The measured response in the north-south direction of the Imperial County Services Building (ICSB) supports this observation. The building was a six-story reinforced concrete structure with unsymmetric distribution of stiffness on the ground floor (Fig. 1). The deflected shapes at times of some of larger peaks are shown in Fig. 2. It can he seen that the variations in the shape are minor, blethods are being developed to predict the shape. These methods along with a modified version of the Q-Hyst model will be presented elsewhere. STRUCTURES To check the performance of the model, three classes of irregular reinforced concrete structures have been studied: (1) small-scale planar structures subjected to simulated earthquakes, (2) full-scale planar structures, and (3) full-scale three-dimensional structure. This paper presents the results for structures in the first category and for one structure in the second category. Test structures This group consisted of four small-scale reinforced concrete structures tested at the Newmark Civil Engineering Laboratory, University of Illinois at Urbana-Champaign by Moehle and Sozen[5].
Earthquake response of irregular RIC structures in the nonlinear range
-!
521
Element dimensions were chosen such that a "weakgirder" set of structures was accomplished. Floor masses were simulated by concrete blocks, weighing approx. 465 kg, attached to the frames at the intersection of beams and columns through hinged connections. As a result, the beams were not subjected to any dead loads other than their own weight. Yielding in beams, therefore, was limited to beam ends. The reinforcement in all elements was symmetrical. A considerable amount of shear reinforcement was provided to avoid shear failure. The longitudinal steel ratios are shown in Table 3. The values for beams and walls represent the steel area per face divided by width and effective depth, and the values for columns are the total steel area over column gross section. The concrete cover on shear reinforcement was 5 mm for all cases except for the vertical legs in beams where the cover was 8 ram. (b) Dynamic testings. The structures were subjected to extensive dynamic testings in the direction of their planes. Of relevance to this study were the first earthquake runs which are described here. The north-south component of the E1 Centro 1940 earthquake acceleration was simulated. The maximum acceleration was normalized to approx. 40% of gravity acceleration. This value had been used as the maximum acceleration of the design earthquake. Because the structures were of small scale, the earthquake time coordinate was lapsed by a factor of 2.5. The test data collected included floor displacement histories relative to
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(a)
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0
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171
D
en
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Co) Fig. 1. The Imperial County Services building. (a) Typical floor plan view. CO) Ground floor plan view.
../:
DEFLECTED ~
~RZG~NnL
P~SZT:ON~' ~ . - ' ~
POS:T;ON
(o) No Wall FNW
(b) Full-Height Wall °
FEW 3x300
r;
nml
/-/j
II I
Fig. 2. North-south deflection of the County Building near the peak responses.
(a) Structural properties. Each structure was composed of two identical frames placed parallel to each other on the shake table platform. Three of the structures had a central shear wail which was connected to "floor" levels by hinged links. Different wall heights were used for different structures (Fig. 3). The shear walls provided for approx. 75% of lateral stiffness during the elastic stage. Beam, column, and wall dimensions were 38 x 38, 38 x 51, and 38 x 203 ram, respectively. The measured material properties are shown in Table 1.
I I I I I I •I
i
p
(c) Four-Story Wall
(d) One-Story Wall
FHW
Fig. 3. Elevation of test structures.
FSW
522
MEHDI SAIIDIand KENNETHE. FIODSON
Table I. Material properties for test structures Concrete
Structure
Compressive Tensile Strength, f~ Strength (MPa) (MPa)
FNW FSW FHW FFW
39.9 37.l 35.9 34.5
3.5 3.1 3.6 3.0
Strain at ¢~ 0.003 0.003 0.003 0.003
Ultimate Strain 0.004 0.004 0.004 0.004
Mod. of Elast. (MPa)
Descend. Slope (MPa)
20,300 18,000 19,000 18,700
6,000 5,570 5,390 5,180
Steel Element
Yield Stress (MPa)
Modulus of Elasticity (MPa}
Wall
339
200,000
0.005
1680
399
200,000
0.0045
I060
Beams and Columns
Strain at Strain-Hardening Start of Slope Strain-Hardening (MPa)
Note: Tensile Strength is based on s p l i t cylinder test
Table 2. MateriLI properties for EFS Concrete
Compressive Strength (f~) Tensile Strength Strain at f ' c Strain at Ultimate Point Modulus of E l a s t i c i t y
4.0 ksi 0.474 ksi 0.002 0.003 3,600 ksi 300 ksi
Descending Slope Yield stress
Steel
Modulus of Elasticity Strain ~t Start of Strain Hardening Strain-Hardenlng Slope
60 ksi 29,000 ksi 0.00207 2,900 ksi
1 ksi : 6.895 MPa
Table 3. geinforcemont ratios ( x I00) in test structures FrN) ~ t ~ t U ~ Frme-~ll Level
Exterior Column
Interior CoI~
Beams
Columns
9
0.88
0.88
0.74
0.88
8
"
"
"
"
7
"
"
"
"
6
"
,f
.
i,
4
"
Oo74
"
3
"
0~88
I.I0
"
2
0.88
1.75
1 .I0
"
1
1.75
1 ~75
l.lO
0.88
St~ru Beams
Wall
O. 74
O. 90
O. 74
0.90
Earthquake response of irregular R/C structures in the nonlinear range base. floor absolute acceleration histories, and b~se acceleration history [5].
Structure with setbacks An eight-story reinforced concrete frame with sixty seven percent setbacks was analyzed to determine the reliability of the analytical models in response prediction for this class of structures. The frame was called EFS for Eight-Story Frame with Setbacks. The dimensions of the frame and cross sections are shown in Fig. 4 and the assumed material properties are shown in Table 2. Floor weights were taken equal to 200 kips (890 KN) at the lower three floors and 80 kips (356 KN) at the upper floors. The structural dimensions, the size of the members, and reinforcement were taken within the customary ranges. The period of the structure with gross section properties was found equal to 1.48 sec. More information about the design of the structure may be found in Ref. [9]. EVALUATION OF RESPONSES
Response of test structures (a) Q-Model. The earthquake response of the test structures was determined using the Q-Model. The properties of the equivalent SDOF systems are listed in Table 4. The displacement responses at the top level of each structure for the first earthquake runs were found using program LARZAK[8], and were superimposed on the measured response histories (Fig. 5). Comparison of the calculated and measured top-level responses is adequate to evaluate the model because both the measured and calculated responses for other levels are similar to those at the top levels. The displacements at the time of maximum top-level displacements and story drifts are presented in Fig. 6. It can be seen that for all four structures the calculated curves were in close agreement with the measured data. The correlation for the first half of the response was better than the correlation for the second half. During the low-amplitude responses (between the third and fifth sec), the analytical results deviated from the experimental responses for structure FSW. However, the Q-Model was able to predict the same type of waveform and amplitude as the measured data. Analytical results showed lower amplitudes than the experimental values after the fifth sec for all structures. This was particularly pronounced for FNW, the structure with no wall. In terms of maximum displacements and story drifts, the correlation between the calculated and measured data was reasonably good (Fig. 6). (b) Q-Model (L). The test structures were also analyzed using the Q-Model (L). To carry out the limit analyses, the yield moments were used as the plastic moments. Several yield mechanisms were considered for each structure. Because the columns had higher yield points than the beams, plastic hinging in columns was limited only to the points needed to satisfy compatibility of deformations. Each structure was analyzed for the collapse load. and the floor displacements were found and normalized with respect to the top-level displacement to obtain the deflected shapes and SDOF system properties (Table 5). It can be seen that the SDOF system properties are very close to those of the Q-Model. Each structure was analyzed for the first 6 sec of the measured first earthquake acceleration data. The response histories and maximum responses designated by
523
Q-Model (L), are shown in Figs. 5 and 6. Close agreement with the experimental data during the first five seconds is eviclent. Between the fifth and sixth second, the amplitudes of the calculated curve were smaller than those of the measured data except for FSW. In comparing the Q-Model and Q-Model (L) responses, it can be seen that the results from the Q-Model (L) showed better or equal agreement with the experimental data. It can be concluded that the overall performance of the Q-Model (L) was at least as satisfactory as that of the Q-Model.
Response of EFS (a) Multi-degree-of.freedom model results. Structure EFS was analyzed for the first 15 sec of the El Centro 1940 NS acceleration record normalized to a maximum acceleration of 0.5 g (Fig. 7). The maximum roof displacement was equal to 17.7 in or approx. 1.5% of the total heighL The frame developed significant nonlinear deformations. The maximum rotational ductility demand (defined as the ratio of maximum rotation and yield rotation) was 2.5 for beams and 1.5 for columns. Comparison of the responses at the top five floors shows that the displacements were generally in phase. The same was true for the second, third, and fourth floor displacements. Comparison of the roof and fourth floor responses reveals that, although "zero-crossings" were close for most parts, they were not quite simultaneous. This suggests that the motion in the part with setbacks was slightly out-of-phase with respect to the lower part. Further indication of an out-of-phase motion can be observed in the peak points in the top five floor responses. (b) Q-Model results. The MDOF analysis revealed that the displacement response had visible contribution of an apparent second mode, and that the displacement shape was variable especially near the peaks. This could be an indication that an equivalent SDOF model, utilizing a single deflected shape, might not be successful in response prediction. To explore the problem, EFS was analyzed using both the old and new versions of the Q-Model. The Q-Model properties are listed in Table 6. The displacement histories obtained from different models were superimposed (Fig. 7). Because the fourth floor response shape was somewhat different from the roof response, the response at both floors was considered. It can be observed that the roof displacement histories found using different models were in general agreement. The SDOF model responses had a slightly shorter effective period. The waveforms and amplitudes were close. The absolute maximum displacement obtained from the Q-Model was seven percent larger than the MDOF result. The Q-Model (L) overestimated the response by nine percent. The correlation at the fourth floor was not as satisfactory as that at the roof. The response based on the Q-Model (L) appeared to be slightly closer to the MDOF model result. The correlation between the SDOF model results and that of the MDOF model was poor after the twelfth second, indicating that perhaps the simple hysteresis rules incorporated in the SDOF models led to an overall structural stiffness which was not close to what was considered by the more complex hysteresis rules used in the MDOF model. Nevertheless. in terms of the overall response, the SDOF model results were reasonably close to the response from the MDOF model.
2
3
4
5
6
7
B
ROOF
FLOOR
2
3
4
5
6
7
8
STORY
7;,
I--
I [
2,4'
24' --
l
I
7
x
-
3 2
3
l
4
5 4
2
5
6
7 6
7
B
story
8
Roof
Floor
I 3"
I 0"
lO"
";3"
Beams
5#9
"
5#8
5#7
B#9
Columns
Exterior
REINFORCEMENT SCHEDULE
.r |
(per face)
COLUMN SECTION
' 'g •
26"
Fig. 4. The eight-story frame with setbacks. (a) Elevation. (b) Reinforcement detail.
~ I _
][ 1I _
l _{
~j _
1 ....i
l !~
24 '
[ 1 [-- ,,,,,I I ,1
l
D
~
:~-
14"
" 3"
L
8#11
---
Columns
Interior
BE~4 SECTION
16"
z
¥
-m
m
~K
=
Earthquake response of irregular RIC structures in the nonlinear range cSW
~OF 30r-
I
/~ q
i
FNI
'~, °
~ i|
It
A
.
_ fllt
;11 [1 I~t "1
Ill
,,.,
,.
/f,
^
FNW
BASE
DISPLACEMENT, MM ..... --r.
A ,~
ME,CSu~ED M.,DEL
Ir - -
A
II
/'~
,JI,f, 2;
? ,
FSW
~o,-
DISPLACEMENT, MM ~ MEASURED i~ Q-MODEL tL.
II~1
~
/I
* /I
1
I
- MEASUNEL
-- - -- MEASU~E£ - r,2- MroDEL
j~
~SL 50,.-
DISPLACEMENT, MM
I
FNW DISPLACEMENT, MM
i ~5F-
525
ACCELERATION,
451I
G
FSW BASE
'
I
ACCELERATION,
G
'
,r ' I
I 50 i-
30 F
/
-
~
FNW
|
i-
~ .....
3
DISPLACEMENT,
/I
~
~
TIME, SEC
6
MM
,~L--._-L
__i_
i
3oF
....
MEASURED
...
O-M~E~
FFW
.
2
.._L. 5
.
.
.
.
.
.
.
.
*;ME, SEE
DISPLACEMENT,
6
MM __I
:~ MODE~.
0 t !
15 t
~g,,-
FHW
50 r.
3_,r-
~
¢
HW
BASE
--
FFW
DISPLACEMENT, MM
Ii
- -- - MEASURED
~ l]i/ ~i]~I~
Itl
i
DISPLACEMENT, MM
i 15~_1
rj
~'
--(;'MODEL ILl
....
!5 ~
A A
t14
MEASURED O-MODEL h_~
45 r-
ACCELERATION, G
F~W BASE ACCELERATION, G
L-
/
i
L 50b
t - ~ I
; 2
__
I
5
1
...L_
TIME, SEC
.
t
6
45 . . . . . . .
i
L. . . . . 2
~
J .I ~
"IME, $EC
i
.i E~
Fig. 5. Top-level response of test structures.
DISCUSSIONOF RESULTSAND RESEARCHNEEDS In the previous section it was seen that the Q-Model (L) provided a simpler alternative to the original form of the Q-Model in terms of construction of the primary curves, while the results were very similar to those found
from the Q-Model. The method used (limit analysis) in the Q-Model (L) to form the moment-deflection curve is known to many structural engineers and requires a relatively small amount of computation. The only element strength property used in Q-Model
.EVEL
8 ! ]
2
3
u DRIFTS,
i MM
II I0
'Ii i II I
II!
I
2
3
.............
/) S P L A C E M E N
L ...........
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MM
3~_ ...........
- - - MEASURED
//
....... o-MOOE
//,;
f'//'
~L~
I
I
L.-i.-_]
DRIFTS, MM
'
,| 1 j l
4
I,
I
I
i0
6
7
8
9
I
2
¸
o
/
..6"
Q- MODEL
DISPI.ACEMENT, MM
.....
/,7 ..... O-M~
FFW
.I
MEASURED
/
Q-MODEt_
- --
Q-MODEL (L)
......
----
I~
DISPLACEMENT,MM
FSW
Fig, 6. Rcs.pons¢ of lcsl slruclurcs al lime of maximun| lop-level displaccmcnl.
j,
L_
F "F'--I
--O-MODEL (L) ---Q-MODEL ..... MEASURED
FHW STORY DRIFTS
4
i_ i
I --r-i I
4
7
1
1 2
I
iI
4
I
50
II
MEASURED
------
(L)
Q-MODEL
----
--Q-MODEL
FNW S T O R Y DRIFTS
5
FHW
DISPLACEMENI,MM
- Q-MO~,
Q_MODEL (L)
/4'
I#
/I;
I/
5
6
7
8
9
C
,///S~/~'/
2,
--
///
3
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FNW
4
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,
8
LEVEL 9
#,{~
50
i
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-'~1
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. . . . . . . Q-MODEL (L) --- - - Q-MODEL . . . . . MEASURED
FFW STORY DRIFTS
..11 2 3 4, DRIFTS, MM
LI
I:
......... Q-MODEL (L) - - - - Q- MODEL ------ MEASURED
FSW STORY DRIFTS
iO
~r
g
z
T:
Earthquake response of irregular RIC structures in the nonlinear range
5~
Table 4(a) Normalized displaced shapes for Q-Model analysis of test structures Level
9
8
7
6
5
FNW
1.00
0.96
0.95
0.89
0.81
4 0.7]
3
2
1
0.60
0.48
0.34
FSW
1.00
0.98
0.93
0.85
0.74
0.60
0°32
0°25
O.ll
FHW
1.00
0.98
0.94
0.87
036
0.62
0.38
0.23
0.20
FFW
l.O0
0.94
0.67
0.78
0.68
0.57
0.44
0.32
0.19
Table 4(b) Q-Model properties for the test structures Structure
Eq. Mass
Eq. Height
Mom. at Break Pt. (KN-mm)
Dis° at BreakPt. (mm)
Post Yield.
(Kg)
(mm)
FNW
3540.
1544.
14500.
7.0
108.
FSW
3342.
1646.
14500.
6°6
214.
FHW
3340.
1606.
16920.
8.3
160.
FFW
3152.
1620.
18!30.
8.2
166.
Slope (KN-mm/mm)
Table 5(a) Normalized displaced shapes for Q-Model (L) analysis of test structures Level
9
8
7
6
5
4
3
2
l
FNW
1.00
0.96
0.91
0.83
0.74
0.64
0.53
0.42
0.30
FSW
l.O0
0.95
0.88
0.78
0.66
0.52
0.37
0.22
0.I0
FHW
1.00
0.95
0.87
0.78
0.66
0.52
0.39
0.25
0.14
FFW
t.00
0.92
0.83
0.73
0.61
0,49
0.37
0.24
0.13
Table 5Co) Q-MOdel (L) properties for the test structures Structure EQ. Mass (K9 )
Eq. Height . (mm)
Mom. at Break Pt. (Kn-~n)
Disp. at Break Pt. (nwn)
Post-Yieldo Slope (KN-mmlmm)
FNW
3260.
1570.
12100.
5.2
FSW
3190.
1668.
13630.
6.2
144.
174.
FHW FFW
3170. 3190.
1655. 1661
13650. 16090.
4.7 5.7
192. 192.
Table 6(a) Normalized displaced shapes for EFS Level
Roof
8
7
6
5
4
3
2
Q-Model Q-Model(L)
1.0 l.O
0.91 0.89
0.80 0.75
0.65 0.58
0.48 0.40
0.33 0.25
0.22 0.15
0.11 0.06
Table 6(b) Q-Model properties for EFS
Parameter
Eq. Mass (kip-mass)
Q-Model Q-Model (L) l kip = 4.448 kN l inch = 25.4 nln
l .74 l . 74
Eq. Height (in) 776 823
Mom. at Break Pt. (k-in) 150,000 142,000
Disp. at Break Pt. (in) 6.26 5.19
Post-Yield. Slope (k-in/in) 6340 5490
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(L) is the yield point which can be found using routine methods or tables and charts available in most structural engineering offices. On the other hand, for a Q-Modei analysis, the complete moment-curvature relationship i~ necessary for each element. To determine laterai displacements due to static loads, the original Q-Model requires a computer program for inelastic analysis of structures. Very few design offices have access to such a program. In contrast, the Q-Model (L) requires an elastic analysis, for which relatively simple and short com0uter programs are widely available. Once the properties of the equivalent SDOF system are determined, both Q-Model and Q-Model (L) use the same dynamic analysis. The computer program for the dynamic analysis, LARZAK[8], is relatively short, and it can be implemented on small computers. With respect to the response for EFS, the reasonably close agreement between the SDOF and MDOF results was contrary to the expectation that, because of the presence of an apparent second mode, an equivalent SDOF model would not work. The assumed displaced shape used in the SDOF models affects all of the main properties of the equivalent systems. A closer look at the responses found for the MDOF model (Fig. 7) shows that the drastic changes happening in the deflected shape are only for a few tenths of a second near the peaks. The success of the SDOF models can be attributed to the closeness of the assumed displaced shapes with an average shape predicted by the MDOF model, and it can be concluded that such an agreement has led to a reasonable estimate of the properties of the equivalent SDOF systems. With respect to the maximum response, the correlation at time of maximum response predicted by the MDOF model was not consistent along the height. Excellent agreement was seen at the fourth floor, while at the roof, the MDOF model result was 34% larger than the result from the Q-Model (L) and 48% larger than that from the Q-Model. The performance of the SDOF models for EFS may be viewed satisfactory in terms of overall waveforms, amplitudes, and frequency contents, although significant differences were observed in a few instances. It is not certain whether the models would exhibit an acceptable performance if the frame had even more drastic stiffness interruption. The planar structures discussed m the previous secuons represent only a few cases. A considerably larger number of structures, preferably with experimental data, need to be analyzed before the reliability of the models is established. Inconsistent correlation between the maximum displacements obtained from the MDOF model and the Q-Model calls for improvement in the Q-Model to estimate the response of structures with setbacks. The development of a simple model comparable to the Q-Model for unsymmetric structures is at its preliminary stage. This is a very important research area. and if it leads to a reasonably simple model, it will be a step forward in the direction of more realistic design and it may bring about a drastic change in the customary methods of seismic design.
I
15
(b)
Fig. 7. Response of EFS. (a) MDOF results. (b) SDOF results,
CONCLUSIONS
The satisfactory performance of both the original and new version of Q-Model for the test structures and the eight story frame along with its successful response
Earthquake response of irregular R/C structures in the nonlinear range estimation for uniform structures[l l] suggest that the model is likely to produce acceptable results for other similar structures, with or without stiffness interruption, subjected to earthquake loads in their strong direction. The limitations on the use of the Q-Model deserve attention. The structures considered in this paper were planar systems subjected to one horizontal component of earthquakes. Actual earthquakes have three translation components. The Q-Model in its present form is unable to take the effects of the vertical and transverse components into account, and the Q-Model result is not to be regarded as the "true" response. However, compared to the conventional method of utilizing an equivalent static load and compared to the elastic or inelastic response spectra method, the Q-Model provides a more realistic alternative in that it provides information regarding the behavior (variation in frequency, number of moderate-tolarge amplitudes, etc.) of the structure. Because the model is simple and economical, it provides the opportunity to determine the response for a variety of earthquakes without any concern about the computation cost. Acknowledgement--The study presented in this paper was part of an investigation supported by the U.S. National Science Foundation under Grant PFR-80-06423.The writers are thankful to Dr. Michael Gaus, the program manager of the project, for his advice and support. REFERENCES
1. D. P. Abrams and M. A. Sozen, Experimental study of
529
frame-wall interaction in reinforced concrete structures subjected to strong earthquake motions. Civil Engineering Studies SRS 460, University of Illinois, Urbana (1979). 2. ACI-ASCE Committee 352. Recommendations for design of beam-column joints in monolithic reinforced concrete structures. ACIJ. 73, 375-393 (1976). 3. J. M. Biggs, Introduction to Structural Dynamics. McGrawHill, New York (1964). 4. R. W. Clough and J. Penzien, Dynamics of Structures. McGraw-Hill, New York (1975). 5. J. P. Moehle and M. A. Sozen, Experiments to study earthquake response of R/C structures with stiffness interruptions. Civil Engineering Studies, SRS 482, University of Illinois, Urbana (1980). 6. N. M. Newmark, A Method of computation for structural dynamics. ASCE J. Engr. Mech. 85, 69-86 (1959). 7. E. P. Popov, Seismic behavior of structural subassemblages. A$CE J. Stract. Div. 106, 1451-1474(1980). 8. M. Saiidi, User's Manual for the LARZ Family. Civil Engineering Studies, SRS 466, University of Illinois, Urbana (1979). 9. M. Saiidi and K. E. Hodson, Analytical seismic study of irregular plane structures using simple nonlinear models. Engineering Report, Number 59, University of Nevada, Reno (1982). 10. M. Saiidi and M. A. Sozen, Simple and complex models for nonlinear seismic analysis of reinforced concrete structures. Civil Engineering Studies, SRS 465, University of Illinois, Urbana (1979). 11. M. Saiidi and M. A. Sozen, Simple nonlinear seismic analysis of RC structures. ASCE J. Struct. Div. 107, 937-952 (1981). 12. T. M. Takeda, M. A. Snzen and N. N. Nielsen, Reinforced concrete response to simulated earthquake. ASCE J. Struct. Division 96, 2557-2573 (1970).
0045-79,19!83/010519-11503.0010 Pergamon Preis L|d
EARTHQUAKE RESPONSE OF IRREGULAR R/C STRUCTURES IN THE NONLINEAR RANGE MEHDI SAIIDIand KENNETH E. HODSON Civil EngineeringDepartment. University of Nevada, Reno, NV 89557, U.S.A. A~tract--The application of a simple analytical model (Q-Model) for calculation of nonlinear seismic response history of irregular planar structures has been demonstrated. The model represents the structure by an equivalent SDOF oscillator. Stiffness degradation effects are accounted for through a simple hysteresis model. The Q-Model is evaluated for small-scaletest structures in addition to a full-scalehypotheticalframe. It is shown that the model exhibits satisfactory performance. The application of the Q-Model, or an approach similar to that, in seismic response computationfor three-dimensionalirregular structures is briefly discussed. INTRODUCTION Much has been learned about the nonlinear seismic behavior of structures with uniform distribution of stiffness through experimental and analytical studies as well as observation of real structures damaged during earthquakes. The findings from these studies have been incorporated in seismic codes, and simplified guidelines have been developed which appear to yield reasonable results. For irregular structures, i.e. structures with large differences in stiffness between adjacent stories and those with unsymmetric distribution of lateral force resisting elements, the amount of available information is limited. The major factor for the scarcity of information on the earthquake behavior of these structures is the fact that required investigations, whether experimental or analytical, are very involved. Irregular structures may be grouped into two categories: (I) planar structures, in which abrupt change of stiffness occurs along the height, and (2) structures in which center of mass is eccentric with respect to center of stiffness, and there may also be a drastic stiffness variation in elevation. The object of this paper is to demonstrate the application of a simple analytical model for calculation of nonlinear seismic response of irregular structures. The method is tested for small-scale physical models as well as full-scale hypothetical structures, with emphasis being placed on structures in the first category. The response for the physical models is evaluated based on the correlation between the experimental and analytical results, while the response for other structures is examined against the calculated results obtained from more elaborate analytical models.
to a model called the Q-Model[Ill, which has been found successful in nonlinear response simulation of uniform reinforced concrete structures. A new and simpler version of the Q-Model was developed and called Q-Model tL), with (L) standing for the limit analysis used in calculating the properties. The new version is described in the following sections. Information about the original version may be found in Ref. [11].
Basic principles
ANALYTICALMETHOD Representation of a multi-degree-of-freedom (MDOF) system by an equivalent single-degree-of-freedom (SDOF) oscillator has been introduced and used for elastic structures[3,4]. In order to have a successful formulation, it is necessary for the system to have a single form or displaced shape during the dynamic response. While this condition is known to be satisfied for many elastic systems, its fulfillment for nonlinear structures has been considered doubtful. Experimental investigations on the nonlinear seismic response of several small-scale reinforced concrete frames and framewalls[l, 5] have revealed that the change in the displaced shape of these structures during the motions is insignificant. These observations along with a simple representation of force-deformation variations have led
The Q-Model is an approximate SDOF model for nonlinear displacement history calculations for planar reinforced concrete structures subjected to base acceleration. Two basic assumptions have been incorporated in the model about: (1) SDOF idealization and (2) hysteretic behavior. (a) SDOF idealization. Representation of a MDOF system by an equivalent SDOF oscillator is possible because it has been known that seismic response of ordinary building structures is dominated by their fundamental mode of vibration. Such an idealization reduces computational effort especially if used for response history analysis. This reduction is due to the considerably smaller size of the SDOF system and due to the substantially smaller number of solutions steps. (b) Hysteretic behavior. Experimental studies of the cyclic behavior of reinforced concrete structures have shown that force-deformation relationships are associated with progressive stiffness degradation and "pinching" action[7]. An equivalent SDOF system would have to take these characteristics into account in order to yield satisfactory results. Several hysteresis models have been developed which include pinching or stiffness degradation, or both[10, 12]. Most of these models are relatively complex and their use in a simplified SDOF system is unjustified. A relatively simple model (called Q-hyst) was used for the original version of the QModel. Detailed information about this model may be found in Ref. [11].
519
Q-Model (L) for planar structures The Q-Model reduces the complex and lengthy nonlinear analysis of multistory structures to the analysis of a SDOF system, once the Q-Model properties are determined. However, in the process of calculating the properties, a computer program for nonlinear static analysis of reinforced concrete structures subjected to lateral loads is needed to obtain the base moment versus
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520
lateral displacement curve (primary curve). This may not be convenient for the analysts who do not have access to such program or the necessary computer facilities. An alternative approach is used in the new version utilizing the limit analysis method. This approach has been also used in Ref. [5]. The main characteristics of the new and old versions are the same. In the new version, the elastic stiffness of the spring is calculated based on a static analysis of the multistory sturcture for lateral loads applied at floor levels. These loads are proportional to floor masses and heights from base. Because beams in reinforced concrete structures are cracked under service load conditions, cracked moment of inertia is used for beams in the static analysis. The break point of the primary curve (the apparent "yield" point) represents the base moment for the collapse loads. Structural elements are assumed to behave elasto-plastic at this stage, and strain-hardening effects are taken into account at a later stage. The "yield" base moment is found from a limit analysis of the multistory structure with different hinging locations. Current guidelines for seismic design recommend an overdesign of columns to limit yielding to beams[2]. For buildings so designed, hinging takes place primarily in beams, and number of collapse mechanisms to be considered is minimal. Strain-hardening effects are accounted for by assigning a slope to the post-yielding part of the primary curve. This slope is related to increase in moment for individual sections: (1) due to increase in the moment arm as steel strain is raised and (2) due to strain hardening of reinforcement. An explicit consideration of the first factor would be involved. Therefore, the following simple relationship is used to determine the post-yielding slope of the primary curve: (1)
K2 = 2(&'IS,')K1
where K1 =elastic slope of the primary curve; K2 = post-yielding slope of the primary curve; S , ' = modulus of elasticity of reinforcement steel; &'= strain-hardening slope in stress-strain curve for reinforcement steel. The Q-hyst model is used to determine stiffness variation of the base spring. The floor displacements obtained from the static analysis are normalized with respect to the top-floor displacement. The normalized values are assumed to represent the displaced shape of the structure throughout the response. The equivalent mass and height are found from the following relationships: M,=
/
. m~¢~,2
~md,,
)
(M,)
(2)
and L, =
m,qbihi
mi¢~,
(3)
i=l
where L, = equivalent height; M, = equivalent mass; M, = total mass; N = number of floors; h~ = height from the base at floor i; m~ = mass at floor i; Oi = normalized displacement at floor i; The equation of motion is formulated as: M j + C.~ + K,x = - M,,¢~
(4)
where C = damping coefficient; K, = stiffness calculated
using the hysteresis model; .~ = base acceleratioi~ ~. and x = relative acceleration, velocity, and displacement at the equivalent mass (relative to base). The above formulation is the same as that usea m the original Q-Model[Ill. It should be noted that i~ this equation the mass on the r.h.s, of the equation is the total mass, while the mass on the l.h.s, is the equivalent mass which is smaller than the total mass. This difference should be accounted for during integration. The base moment vs displacement at equivalent height relationship forms the primary curve upon which the hysteresis model operates. The displacement at equivalent height is found from a linear interpolation of displacements at adjacent floors Equation (4) is integrated using the Newmark's /3 method[6] to determine the displacement history at the equivalent height, and the base moment history. Displacements at different floors can be obtained using the assumed displaced shape.
where D, = displacement at equivalent height; D~ =displacement at floor i; 4', = normalized displacement at equivalent height; ¢~;= normalized displacement at floor i, Q-Model [or space [rames Many buildings are composed of unsymmetric structural systems that may not be idealized as planar systems. The earthquake response of such buildings may contain significant contribution from torsional mode of vibration even if only one translational component of the earthquake is taken into account. The contribution from torsion may lead to a deflected shape with considerable variation during the earthquake. Such variation is expected to result in an unsuccessful response calculation based on the Q-Model or a similar model. For structures with close torsional and translational periods, variations in the deflected shape are relatively small and an approach similar to that used in the Q-Model may be successful. The measured response in the north-south direction of the Imperial County Services Building (ICSB) supports this observation. The building was a six-story reinforced concrete structure with unsymmetric distribution of stiffness on the ground floor (Fig. 1). The deflected shapes at times of some of larger peaks are shown in Fig. 2. It can he seen that the variations in the shape are minor, blethods are being developed to predict the shape. These methods along with a modified version of the Q-Hyst model will be presented elsewhere. STRUCTURES To check the performance of the model, three classes of irregular reinforced concrete structures have been studied: (1) small-scale planar structures subjected to simulated earthquakes, (2) full-scale planar structures, and (3) full-scale three-dimensional structure. This paper presents the results for structures in the first category and for one structure in the second category. Test structures This group consisted of four small-scale reinforced concrete structures tested at the Newmark Civil Engineering Laboratory, University of Illinois at Urbana-Champaign by Moehle and Sozen[5].
Earthquake response of irregular RIC structures in the nonlinear range
-!
521
Element dimensions were chosen such that a "weakgirder" set of structures was accomplished. Floor masses were simulated by concrete blocks, weighing approx. 465 kg, attached to the frames at the intersection of beams and columns through hinged connections. As a result, the beams were not subjected to any dead loads other than their own weight. Yielding in beams, therefore, was limited to beam ends. The reinforcement in all elements was symmetrical. A considerable amount of shear reinforcement was provided to avoid shear failure. The longitudinal steel ratios are shown in Table 3. The values for beams and walls represent the steel area per face divided by width and effective depth, and the values for columns are the total steel area over column gross section. The concrete cover on shear reinforcement was 5 mm for all cases except for the vertical legs in beams where the cover was 8 ram. (b) Dynamic testings. The structures were subjected to extensive dynamic testings in the direction of their planes. Of relevance to this study were the first earthquake runs which are described here. The north-south component of the E1 Centro 1940 earthquake acceleration was simulated. The maximum acceleration was normalized to approx. 40% of gravity acceleration. This value had been used as the maximum acceleration of the design earthquake. Because the structures were of small scale, the earthquake time coordinate was lapsed by a factor of 2.5. The test data collected included floor displacement histories relative to
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(a) Structural properties. Each structure was composed of two identical frames placed parallel to each other on the shake table platform. Three of the structures had a central shear wail which was connected to "floor" levels by hinged links. Different wall heights were used for different structures (Fig. 3). The shear walls provided for approx. 75% of lateral stiffness during the elastic stage. Beam, column, and wall dimensions were 38 x 38, 38 x 51, and 38 x 203 ram, respectively. The measured material properties are shown in Table 1.
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(d) One-Story Wall
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Fig. 3. Elevation of test structures.
FSW
522
MEHDI SAIIDIand KENNETHE. FIODSON
Table I. Material properties for test structures Concrete
Structure
Compressive Tensile Strength, f~ Strength (MPa) (MPa)
FNW FSW FHW FFW
39.9 37.l 35.9 34.5
3.5 3.1 3.6 3.0
Strain at ¢~ 0.003 0.003 0.003 0.003
Ultimate Strain 0.004 0.004 0.004 0.004
Mod. of Elast. (MPa)
Descend. Slope (MPa)
20,300 18,000 19,000 18,700
6,000 5,570 5,390 5,180
Steel Element
Yield Stress (MPa)
Modulus of Elasticity (MPa}
Wall
339
200,000
0.005
1680
399
200,000
0.0045
I060
Beams and Columns
Strain at Strain-Hardening Start of Slope Strain-Hardening (MPa)
Note: Tensile Strength is based on s p l i t cylinder test
Table 2. MateriLI properties for EFS Concrete
Compressive Strength (f~) Tensile Strength Strain at f ' c Strain at Ultimate Point Modulus of E l a s t i c i t y
4.0 ksi 0.474 ksi 0.002 0.003 3,600 ksi 300 ksi
Descending Slope Yield stress
Steel
Modulus of Elasticity Strain ~t Start of Strain Hardening Strain-Hardenlng Slope
60 ksi 29,000 ksi 0.00207 2,900 ksi
1 ksi : 6.895 MPa
Table 3. geinforcemont ratios ( x I00) in test structures FrN) ~ t ~ t U ~ Frme-~ll Level
Exterior Column
Interior CoI~
Beams
Columns
9
0.88
0.88
0.74
0.88
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"
"
"
7
"
"
"
"
6
"
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.
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4
"
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3
"
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2
0.88
1.75
1 .I0
"
1
1.75
1 ~75
l.lO
0.88
St~ru Beams
Wall
O. 74
O. 90
O. 74
0.90
Earthquake response of irregular R/C structures in the nonlinear range base. floor absolute acceleration histories, and b~se acceleration history [5].
Structure with setbacks An eight-story reinforced concrete frame with sixty seven percent setbacks was analyzed to determine the reliability of the analytical models in response prediction for this class of structures. The frame was called EFS for Eight-Story Frame with Setbacks. The dimensions of the frame and cross sections are shown in Fig. 4 and the assumed material properties are shown in Table 2. Floor weights were taken equal to 200 kips (890 KN) at the lower three floors and 80 kips (356 KN) at the upper floors. The structural dimensions, the size of the members, and reinforcement were taken within the customary ranges. The period of the structure with gross section properties was found equal to 1.48 sec. More information about the design of the structure may be found in Ref. [9]. EVALUATION OF RESPONSES
Response of test structures (a) Q-Model. The earthquake response of the test structures was determined using the Q-Model. The properties of the equivalent SDOF systems are listed in Table 4. The displacement responses at the top level of each structure for the first earthquake runs were found using program LARZAK[8], and were superimposed on the measured response histories (Fig. 5). Comparison of the calculated and measured top-level responses is adequate to evaluate the model because both the measured and calculated responses for other levels are similar to those at the top levels. The displacements at the time of maximum top-level displacements and story drifts are presented in Fig. 6. It can be seen that for all four structures the calculated curves were in close agreement with the measured data. The correlation for the first half of the response was better than the correlation for the second half. During the low-amplitude responses (between the third and fifth sec), the analytical results deviated from the experimental responses for structure FSW. However, the Q-Model was able to predict the same type of waveform and amplitude as the measured data. Analytical results showed lower amplitudes than the experimental values after the fifth sec for all structures. This was particularly pronounced for FNW, the structure with no wall. In terms of maximum displacements and story drifts, the correlation between the calculated and measured data was reasonably good (Fig. 6). (b) Q-Model (L). The test structures were also analyzed using the Q-Model (L). To carry out the limit analyses, the yield moments were used as the plastic moments. Several yield mechanisms were considered for each structure. Because the columns had higher yield points than the beams, plastic hinging in columns was limited only to the points needed to satisfy compatibility of deformations. Each structure was analyzed for the collapse load. and the floor displacements were found and normalized with respect to the top-level displacement to obtain the deflected shapes and SDOF system properties (Table 5). It can be seen that the SDOF system properties are very close to those of the Q-Model. Each structure was analyzed for the first 6 sec of the measured first earthquake acceleration data. The response histories and maximum responses designated by
523
Q-Model (L), are shown in Figs. 5 and 6. Close agreement with the experimental data during the first five seconds is eviclent. Between the fifth and sixth second, the amplitudes of the calculated curve were smaller than those of the measured data except for FSW. In comparing the Q-Model and Q-Model (L) responses, it can be seen that the results from the Q-Model (L) showed better or equal agreement with the experimental data. It can be concluded that the overall performance of the Q-Model (L) was at least as satisfactory as that of the Q-Model.
Response of EFS (a) Multi-degree-of.freedom model results. Structure EFS was analyzed for the first 15 sec of the El Centro 1940 NS acceleration record normalized to a maximum acceleration of 0.5 g (Fig. 7). The maximum roof displacement was equal to 17.7 in or approx. 1.5% of the total heighL The frame developed significant nonlinear deformations. The maximum rotational ductility demand (defined as the ratio of maximum rotation and yield rotation) was 2.5 for beams and 1.5 for columns. Comparison of the responses at the top five floors shows that the displacements were generally in phase. The same was true for the second, third, and fourth floor displacements. Comparison of the roof and fourth floor responses reveals that, although "zero-crossings" were close for most parts, they were not quite simultaneous. This suggests that the motion in the part with setbacks was slightly out-of-phase with respect to the lower part. Further indication of an out-of-phase motion can be observed in the peak points in the top five floor responses. (b) Q-Model results. The MDOF analysis revealed that the displacement response had visible contribution of an apparent second mode, and that the displacement shape was variable especially near the peaks. This could be an indication that an equivalent SDOF model, utilizing a single deflected shape, might not be successful in response prediction. To explore the problem, EFS was analyzed using both the old and new versions of the Q-Model. The Q-Model properties are listed in Table 6. The displacement histories obtained from different models were superimposed (Fig. 7). Because the fourth floor response shape was somewhat different from the roof response, the response at both floors was considered. It can be observed that the roof displacement histories found using different models were in general agreement. The SDOF model responses had a slightly shorter effective period. The waveforms and amplitudes were close. The absolute maximum displacement obtained from the Q-Model was seven percent larger than the MDOF result. The Q-Model (L) overestimated the response by nine percent. The correlation at the fourth floor was not as satisfactory as that at the roof. The response based on the Q-Model (L) appeared to be slightly closer to the MDOF model result. The correlation between the SDOF model results and that of the MDOF model was poor after the twelfth second, indicating that perhaps the simple hysteresis rules incorporated in the SDOF models led to an overall structural stiffness which was not close to what was considered by the more complex hysteresis rules used in the MDOF model. Nevertheless. in terms of the overall response, the SDOF model results were reasonably close to the response from the MDOF model.
2
3
4
5
6
7
B
ROOF
FLOOR
2
3
4
5
6
7
8
STORY
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story
8
Roof
Floor
I 3"
I 0"
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Beams
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"
5#8
5#7
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Columns
Exterior
REINFORCEMENT SCHEDULE
.r |
(per face)
COLUMN SECTION
' 'g •
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Fig. 4. The eight-story frame with setbacks. (a) Elevation. (b) Reinforcement detail.
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--(;'MODEL ILl
....
!5 ~
A A
t14
MEASURED O-MODEL h_~
45 r-
ACCELERATION, G
F~W BASE ACCELERATION, G
L-
/
i
L 50b
t - ~ I
; 2
__
I
5
1
...L_
TIME, SEC
.
t
6
45 . . . . . . .
i
L. . . . . 2
~
J .I ~
"IME, $EC
i
.i E~
Fig. 5. Top-level response of test structures.
DISCUSSIONOF RESULTSAND RESEARCHNEEDS In the previous section it was seen that the Q-Model (L) provided a simpler alternative to the original form of the Q-Model in terms of construction of the primary curves, while the results were very similar to those found
from the Q-Model. The method used (limit analysis) in the Q-Model (L) to form the moment-deflection curve is known to many structural engineers and requires a relatively small amount of computation. The only element strength property used in Q-Model
.EVEL
8 ! ]
2
3
u DRIFTS,
i MM
II I0
'Ii i II I
II!
I
2
3
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3~_ ...........
- - - MEASURED
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....... o-MOOE
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DRIFTS, MM
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,| 1 j l
4
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6
7
8
9
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2
¸
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Q- MODEL
DISPI.ACEMENT, MM
.....
/,7 ..... O-M~
FFW
.I
MEASURED
/
Q-MODEt_
- --
Q-MODEL (L)
......
----
I~
DISPLACEMENT,MM
FSW
Fig, 6. Rcs.pons¢ of lcsl slruclurcs al lime of maximun| lop-level displaccmcnl.
j,
L_
F "F'--I
--O-MODEL (L) ---Q-MODEL ..... MEASURED
FHW STORY DRIFTS
4
i_ i
I --r-i I
4
7
1
1 2
I
iI
4
I
50
II
MEASURED
------
(L)
Q-MODEL
----
--Q-MODEL
FNW S T O R Y DRIFTS
5
FHW
DISPLACEMENI,MM
- Q-MO~,
Q_MODEL (L)
/4'
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/I;
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5
6
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LEVEL 9
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. . . . . . . Q-MODEL (L) --- - - Q-MODEL . . . . . MEASURED
FFW STORY DRIFTS
..11 2 3 4, DRIFTS, MM
LI
I:
......... Q-MODEL (L) - - - - Q- MODEL ------ MEASURED
FSW STORY DRIFTS
iO
~r
g
z
T:
Earthquake response of irregular RIC structures in the nonlinear range
5~
Table 4(a) Normalized displaced shapes for Q-Model analysis of test structures Level
9
8
7
6
5
FNW
1.00
0.96
0.95
0.89
0.81
4 0.7]
3
2
1
0.60
0.48
0.34
FSW
1.00
0.98
0.93
0.85
0.74
0.60
0°32
0°25
O.ll
FHW
1.00
0.98
0.94
0.87
036
0.62
0.38
0.23
0.20
FFW
l.O0
0.94
0.67
0.78
0.68
0.57
0.44
0.32
0.19
Table 4(b) Q-Model properties for the test structures Structure
Eq. Mass
Eq. Height
Mom. at Break Pt. (KN-mm)
Dis° at BreakPt. (mm)
Post Yield.
(Kg)
(mm)
FNW
3540.
1544.
14500.
7.0
108.
FSW
3342.
1646.
14500.
6°6
214.
FHW
3340.
1606.
16920.
8.3
160.
FFW
3152.
1620.
18!30.
8.2
166.
Slope (KN-mm/mm)
Table 5(a) Normalized displaced shapes for Q-Model (L) analysis of test structures Level
9
8
7
6
5
4
3
2
l
FNW
1.00
0.96
0.91
0.83
0.74
0.64
0.53
0.42
0.30
FSW
l.O0
0.95
0.88
0.78
0.66
0.52
0.37
0.22
0.I0
FHW
1.00
0.95
0.87
0.78
0.66
0.52
0.39
0.25
0.14
FFW
t.00
0.92
0.83
0.73
0.61
0,49
0.37
0.24
0.13
Table 5Co) Q-MOdel (L) properties for the test structures Structure EQ. Mass (K9 )
Eq. Height . (mm)
Mom. at Break Pt. (Kn-~n)
Disp. at Break Pt. (nwn)
Post-Yieldo Slope (KN-mmlmm)
FNW
3260.
1570.
12100.
5.2
FSW
3190.
1668.
13630.
6.2
144.
174.
FHW FFW
3170. 3190.
1655. 1661
13650. 16090.
4.7 5.7
192. 192.
Table 6(a) Normalized displaced shapes for EFS Level
Roof
8
7
6
5
4
3
2
Q-Model Q-Model(L)
1.0 l.O
0.91 0.89
0.80 0.75
0.65 0.58
0.48 0.40
0.33 0.25
0.22 0.15
0.11 0.06
Table 6(b) Q-Model properties for EFS
Parameter
Eq. Mass (kip-mass)
Q-Model Q-Model (L) l kip = 4.448 kN l inch = 25.4 nln
l .74 l . 74
Eq. Height (in) 776 823
Mom. at Break Pt. (k-in) 150,000 142,000
Disp. at Break Pt. (in) 6.26 5.19
Post-Yield. Slope (k-in/in) 6340 5490
528
gEHD[ SAIlDI and KENNETH E. HODSO~ 3
6
9
TIME,
20
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ROOF
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I
(L) is the yield point which can be found using routine methods or tables and charts available in most structural engineering offices. On the other hand, for a Q-Modei analysis, the complete moment-curvature relationship i~ necessary for each element. To determine laterai displacements due to static loads, the original Q-Model requires a computer program for inelastic analysis of structures. Very few design offices have access to such a program. In contrast, the Q-Model (L) requires an elastic analysis, for which relatively simple and short com0uter programs are widely available. Once the properties of the equivalent SDOF system are determined, both Q-Model and Q-Model (L) use the same dynamic analysis. The computer program for the dynamic analysis, LARZAK[8], is relatively short, and it can be implemented on small computers. With respect to the response for EFS, the reasonably close agreement between the SDOF and MDOF results was contrary to the expectation that, because of the presence of an apparent second mode, an equivalent SDOF model would not work. The assumed displaced shape used in the SDOF models affects all of the main properties of the equivalent systems. A closer look at the responses found for the MDOF model (Fig. 7) shows that the drastic changes happening in the deflected shape are only for a few tenths of a second near the peaks. The success of the SDOF models can be attributed to the closeness of the assumed displaced shapes with an average shape predicted by the MDOF model, and it can be concluded that such an agreement has led to a reasonable estimate of the properties of the equivalent SDOF systems. With respect to the maximum response, the correlation at time of maximum response predicted by the MDOF model was not consistent along the height. Excellent agreement was seen at the fourth floor, while at the roof, the MDOF model result was 34% larger than the result from the Q-Model (L) and 48% larger than that from the Q-Model. The performance of the SDOF models for EFS may be viewed satisfactory in terms of overall waveforms, amplitudes, and frequency contents, although significant differences were observed in a few instances. It is not certain whether the models would exhibit an acceptable performance if the frame had even more drastic stiffness interruption. The planar structures discussed m the previous secuons represent only a few cases. A considerably larger number of structures, preferably with experimental data, need to be analyzed before the reliability of the models is established. Inconsistent correlation between the maximum displacements obtained from the MDOF model and the Q-Model calls for improvement in the Q-Model to estimate the response of structures with setbacks. The development of a simple model comparable to the Q-Model for unsymmetric structures is at its preliminary stage. This is a very important research area. and if it leads to a reasonably simple model, it will be a step forward in the direction of more realistic design and it may bring about a drastic change in the customary methods of seismic design.
I
15
(b)
Fig. 7. Response of EFS. (a) MDOF results. (b) SDOF results,
CONCLUSIONS
The satisfactory performance of both the original and new version of Q-Model for the test structures and the eight story frame along with its successful response
Earthquake response of irregular R/C structures in the nonlinear range estimation for uniform structures[l l] suggest that the model is likely to produce acceptable results for other similar structures, with or without stiffness interruption, subjected to earthquake loads in their strong direction. The limitations on the use of the Q-Model deserve attention. The structures considered in this paper were planar systems subjected to one horizontal component of earthquakes. Actual earthquakes have three translation components. The Q-Model in its present form is unable to take the effects of the vertical and transverse components into account, and the Q-Model result is not to be regarded as the "true" response. However, compared to the conventional method of utilizing an equivalent static load and compared to the elastic or inelastic response spectra method, the Q-Model provides a more realistic alternative in that it provides information regarding the behavior (variation in frequency, number of moderate-tolarge amplitudes, etc.) of the structure. Because the model is simple and economical, it provides the opportunity to determine the response for a variety of earthquakes without any concern about the computation cost. Acknowledgement--The study presented in this paper was part of an investigation supported by the U.S. National Science Foundation under Grant PFR-80-06423.The writers are thankful to Dr. Michael Gaus, the program manager of the project, for his advice and support. REFERENCES
1. D. P. Abrams and M. A. Sozen, Experimental study of
529
frame-wall interaction in reinforced concrete structures subjected to strong earthquake motions. Civil Engineering Studies SRS 460, University of Illinois, Urbana (1979). 2. ACI-ASCE Committee 352. Recommendations for design of beam-column joints in monolithic reinforced concrete structures. ACIJ. 73, 375-393 (1976). 3. J. M. Biggs, Introduction to Structural Dynamics. McGrawHill, New York (1964). 4. R. W. Clough and J. Penzien, Dynamics of Structures. McGraw-Hill, New York (1975). 5. J. P. Moehle and M. A. Sozen, Experiments to study earthquake response of R/C structures with stiffness interruptions. Civil Engineering Studies, SRS 482, University of Illinois, Urbana (1980). 6. N. M. Newmark, A Method of computation for structural dynamics. ASCE J. Engr. Mech. 85, 69-86 (1959). 7. E. P. Popov, Seismic behavior of structural subassemblages. A$CE J. Stract. Div. 106, 1451-1474(1980). 8. M. Saiidi, User's Manual for the LARZ Family. Civil Engineering Studies, SRS 466, University of Illinois, Urbana (1979). 9. M. Saiidi and K. E. Hodson, Analytical seismic study of irregular plane structures using simple nonlinear models. Engineering Report, Number 59, University of Nevada, Reno (1982). 10. M. Saiidi and M. A. Sozen, Simple and complex models for nonlinear seismic analysis of reinforced concrete structures. Civil Engineering Studies, SRS 465, University of Illinois, Urbana (1979). 11. M. Saiidi and M. A. Sozen, Simple nonlinear seismic analysis of RC structures. ASCE J. Struct. Div. 107, 937-952 (1981). 12. T. M. Takeda, M. A. Snzen and N. N. Nielsen, Reinforced concrete response to simulated earthquake. ASCE J. Struct. Division 96, 2557-2573 (1970).