Analysis of the Full-scale Seven-story Reinforced Concrete Test Structure

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1
This paper was published in Journal (B), The Faculty of Engineering, University of Tokyo, Vol. XXXVII,
No. 2, 1983, pp. 432 - 478.




Analysis of the Full-scale Seven-story Reinforced Concrete Test Structure


by

Toshimi KABEYASAWA*, Hitoshi SHIOHARA**, Shunsuke OTANI**
and Hiroyuki AOYAMA**



(Received May 4, 1983)





Pseudo-dynamic earthquake response tests of a full-scale seven-story
reinforced concrete wall-frame structure, conducted as a part of U.S.-Japan
Cooperative Research Program, were analyzed. A nonlinear dynamic analysis
method was used to simulate the observed behavior. The results of small-scale
sub-assemblage tests and the ful1-scale test were made available for the study,
hence, the information was reflected in the development of analytical models.
The three-dimensional structure was idealized as plane frames which
consisted of three different member models. The analysis utilized four different
hysteresis models for elements of member models. The force-deformation
relationship of member models was evaluated by approximate methods on the
basis of material properties and structural geometry.
The analytical results such as response waveforms, hysteresis relation and
local deformation were compared with the test results. A good correlation was
reported between the observed and calculated responses.





1. Introduction

This paper describes an analysis of the full-scale seven-story reinforced concrete building tested
as a part of the U.S.-Japan Cooperative Research Program Utilizing Large Scale Testing Facilities. A
general purpose computer program was developed to simulate the inelastic behavior of a structure
during an earthquake. On the basis of given structural geometry and material properties, this paper
places an emphasis to describe (a) methods to model member behavior, and (b) methods to
determine member stiffness properties. At the time of analysis, the results of small-scale
sub-assemblage tests and the ful1-scale test were made available to the authors, hence, the
information was reflected in the development of the analytical models and evaluation methods. A
good correlation of the observed and computed responses of the test structure is reported in this
paper.
* Formerly, Department of Architecture
Currently, Department of Architecture, Yokohama National University
* Department of Architecture
2
2. Outline of Test Program

A test of a full-scale reinforced concrete building was conducted, as a part of U.S.-Japan
Cooperative Research Program Utilizing Large Scale Testing Facilities, at the Large Size Structures
Laboratory of Building Research Institute, Ministry of Construction, Tsukuba, Japan (1).

A full-scale seven-story reinforced concrete building was designed and constructed in the Large
Size Structures Laboratory, Building Research Institute, in accordance with normal construction
specifications and practices (Fig. 1). The building had three bays in the longitudinal direction, and two
spans in the transverse direction. A shear wall was placed, parallel to the direction of loading, in the
middle bay of the center frame.

The structure was subjected to lateral load of an inverted triangular distribution at each level by
eight actuators; two actuators used at the roof level and one actuator each at the other six levels. The
pseudo-dynamic test method was used to control the roof-level displacement; i.e., the response
displacement under an imaginary earthquake motion was computed, in parallel with the test, for a
system having the observed restoring force characteristics. The computed response displacement
was applied to the roof level of the test structure while the load amplitude at the first to the sixth floor
levels were made proportional to the load measured in the roof level actuators; in this fashion, the
number of degrees of freedom of the test structure was reduced to one. A total of 716 channel strains,
displacements, rotations and loads were measured during the test.

The intensity of imaginary earthquake motions was varied in four test runs to yield expected
maximum roof-level displacements of approximately 1/7000, 1/400, 3/400, and 1/75 of the total height
(Table 1). The earthquake record used in the test was modified from the original record so that the
first mode response should dominate in the response of the test structure; i.e., higher frequency
components were removed from the original records. Free and forced vibration tests were carried out
between pseudo-dynamic earthquake test runs to study the change in period and damping.

After the fourth test run, epoxy resin was injected into major cracks in structural members, and
non-structural partitions were installed to the original bare structure. The repaired structure was
tested in three runs using the pseudo-dynamic test method, and finally tested statically under
reversals of uniform load distribution to a roof-level displacement of 1/50 the total height.

The second, third and fourth pseudo-dynamic earthquake tests of the bare structure were
simulated by analytical models in this paper.


3. Description of Test

The construction of the full-scale seven-story reinforced concrete test structure is described in
detail in a paper (1) by J. K. Wight and S. Nakata. The method of testing is described by S. Okamoto
et a1. (2). The information relevant to the structural analysis is summarized herein from the two
papers.

3.1 Geometry of Test Structure

A general plan view is shown in Fig. 2. The test structure consisted of three three-bay frames,
(frames A, B, and C) parallel to the loading direction, and four two-bay frames, (frames 1 , 2, 3 and 4)
perpendicular to the loading direction. The span widths were 6.0, 5.0, and 6.0 m in the longitudinal
direction, and 6.0 m each in the transverse direction. Frame B had a shear wall in the central bay
continuous from the first to the seventh story.

An elevation of Frame B is shown in Fig. 3. Floor level and story notations are defined in the figure.
The inter-story height was 3.75 m in the first story, and 3.0 m from the second to the seventh story.
Note that the girders of spans 1-2 and 3-4 were not continued through the shear wall. Frames A and
C were three-span open frames, having inter-story heights and bay widths identical to Frame B.
3

A general elevation of Frame 4 is shown in Fig. 4. Two walls were installed in the frame
perpendicular to the loading direction so as to reduce the torsional and transverse displacements of
the test structure. A 1.0-m gap was provided between the face of a column and the edge of the wall to
eliminate the contribution of the wall in the stiffness of the structure in the loading direction. Frame 1
was identical to Frame 4 except for pairs of openings in the walls for the loading beams. Frames 2
and 3 were open frames without walls.

A plan view of the foundation is given in Fig. 5, a floor plan for the second floor through the
seventh floor levels in Fig. 6, and the roof plan in Fig. 7. Notations for beams, columns and walls are
shown in the figures. The foundation was post tensioned to the test floor with 33-mm diameter
high-strength (10,000 kgf/cm
2
) rods at a stress level of 5,900 kgf/cm
2
.

The dimensions of a column section were 50 x 50 cm
2
throughout the test structure. The size of
the girders parallel to the loading direction was 30 x 50 cm
2
from the second to the roof level. The
dimensions of a transverse beam were 30 x 45 cm
2
. The wall (W
1
) parallel to the loading direction had
a thickness of 20 cm, and the transverse walls a thickness of 15 cm, both from the first to the seventh
story. The floor slab was 12 cm thick throughout the structure. Two 120 x 120 x 80 cm
3
loading points
were placed at each floor from the second to the seventh floor in the floor slab at the mid-span of
beams B
2
(Fig. 6). At the roof level, the loading point dimensions were 70 x 530 x 64 cm
3
(Fig. 7).

The as-built dimensions were reported to be very close to the nominal dimensions. In the first
story, areas of poorly compacted concrete were found near the base of the first story columns. The
voids did not penetrate into the column core even in the worst case. However, longitudinal and
transverse reinforcing bars were reported to be visible in some locations.

3.2 Reinforcement Details

Cross section reinforcement details for the foundation beams and floor beams are shown in Figs.
8 and 9. D19 and D25 deformed bars were used as flexural reinforcement, and D10 and D19
deformed bars as stirrup reinforcement. Two digits after alphabet D denote approximate bar diameter
in mm. In the foundation beams, D16 bars were used in the web to hold the stirrups in position. All of
the beam bars terminating at an exterior column or in the wall boundary columns were anchored with
a 90 degree hook. Within a region extending one-quarter of the clear span from a column face, floor
beam stirrups were spaced at an approximately one-fourth of the effective beam depth. The spacing
was increased to approximately one-half of the effective depth in the middle region.

A typical column cross section is given in Fig. 10. The columns were all 50 x 50 cm
2
, and
reinforced with 8-D22 deformed bars. All of the column bars terminated at the roof level with a 180
degree hook. Perimeter hoops were spaced at 10 cm over the total height of the columns, including
the beam-to-column joint regions. Cross ties in the first-story independent columns were provided at
a 10-cm spacing over the first 60 cm above the foundation, and at a 60-cm spacing elsewhere. Cross
ties in the boundary columns of the shear wall were provided at a 10-cm spacing over the full height
of the first three stories except in the beam-to-column joint regions, and at a 60-cm spacing
elsewhere.

The shear wall, parallel to the direction of loading, was reinforced with 2-D10 bars at a spacing of
20 cm in the horizontal and vertical directions. The horizontal wall reinforcement was anchored into
the boundary columns, and the vertical wall reinforcement into the foundation.

Reinforcement details for the floor slabs at the second through the roof levels are shown in Fig. 11.
Different spacing was used in the column strips, middle strips and in the cantilevered portion of the
floor slabs.

3.3. Materials

Deformed bars were used in the constructions of the test structure. The grade of reinforcing steel
4
was SD35. Geometrical and mechanical properties are listed in Table 2 taken from Reference 1. All
bars showed a clear yield plateau after yielding up to a strain of 0.012 to 0.022, depending on the bar
size.

The ready-mixed concrete was used in the test structure. Following the Japanese construction
practices, the concrete was placed in columns of a story and into beams and slabs immediately
above the columns in a single job. The mechanical properties of the concrete are listed in Table 3.
The values were obtained from the tests of 15 x 30-cm
2
standard cylinders cured in the field. The
sixth and the seventh story concrete strengths were found to be significantly weaker than the
specified strength of 270 kgf/cm
2
. However, the compression tests on standard cured cylinders did
not show such a change in concrete strength (1). The tensile strength was determined by the splitting
test of cylinders.

3.4 Method of Testing

Test of the full-scale seven-story structure was carried out using "SDF Pseudo Dynamic
Earthquake Response Test Procedure". The theoretical background is outlined by Okamoto et al. (2).
The summary is given below.

The equation of motion of a multi-degree-of-freedom system without damping can be written in a
matrix form;


[ ]{ } { } [ ]{1}m x f m y+ = −
 
(1)

in which
[ ]m
: mass matrix,
{ }
f
: restoring force vector (resistance of structure),
{ }
x

: structural
response acceleration vector relative to the base,
{1}
: vector consisting of unit elements, and
y

:
ground acceleration.

In order to reduce the number of degrees of freedom to one, the structure was assumed to
oscillate in a single "governing" mode. The restoring force distribution pattern was assumed to remain
unchanged during an earthquake. In other words,


{ } { }
R
f
v f
=
(2)

in which
{ }v
: constant vector, each element of which represents the lateral resistance amplitude
normalized to the roof-level resistance amplitude
R
f
.

Under the specified distribution of the lateral resistance (or loads), the structure would deform in a
certain shape, reflecting the stiffness distribution of the structure. Namely, the "mode shape"
{ }u

and its "amplitude"
q
.


{ } { }
x
u q=
(3)

If the mode shape is normalized to the roof-level amplitude, the value of
q
represents a
roof-level lateral displacement
R
x
. Displacement distribution vector
{ }
u
normally varies with
stiffness deterioration associated with structural damage. However, the deflected shape pattern did
not change appreciably regardless of load amplitudes in a preliminary analysis of the test structure
under an inverted triangular distribution of lateral loads. Therefore, the structure was assumed to
respond in the fixed mode shape
{ }
u
during an earthquake, and the equation of motion was
expressed as


[ ]{ } { } [ ]{1}
R R
m u x v f m y+ = −
 
(4)

5
Pre-multiplying
{ }
T
u
to Eq. 4,


{ } [ ]{ } { } { } { } [ ]{1}
T T T
R R
u m u x u v f u m y+ = −
 
(5)

Or,


( )
R
mx f m y
β
+ = −
 
(6)

in which
{ } [ ]{ }
T
m u m u=
: effective mass,
{ } { }
T
R
f
u v f=
: effective restoring force, and
{ } [ ]{1}/
T
u m mβ =
: effective participation factor. The seven-story structure was forced to reduce to
an "equivalent" single-degree-of-freedom system in this manner.

An inverted triangular shape was used to represent the lateral resistance distribution
{ }
v
. The
corresponding deflected shape
{ }
u
was obtained as an average of deflected shapes at different
load amplitudes. Hence, the properties of the equivalent single-degree-of-freedom system are


{ }
T
u
=
[1.000, 0.850, 0.696, 0.540, 0.384, 0.234, 0.102]

{ }
T
v
=
[1.000, 0.862, 0.724, 0.586, 0.448, 0.310, 0.172]

1.442β =


m=
0.643 tonf sec
2
/cm

The pseudo-dynamic earthquake response test was carried out on the "equivalent" single
degree-of-freedom system.

The central difference method was used for a numerical integration procedure in the
pseudo-dynamic earthquake response test. The central difference method is given


(
)
2
1 1
2/
i i i i
q q q q t
+ −
= − + ∆

(7)

in which,
i
q
: displacement at time step i,
i
q

: acceleration at time step i, and
t∆
: time increment for
numerical integration.

Equation 7 can be rewritten in a form,


2
1 1
2
i i i i
q q q t q
+ −
= − + ∆

(8)

In other words, from displacements and acceleration at old time steps i-1 and i, the displacement at
new time step i+1 can be evaluated, hence the roof-level displacement
R
x
(=
1i
q
+
).

Although displacement amplitudes at other levels can also be determined by Eq. 3, only the
roof-level displacement was controlled in the test. Eight actuators, maintaining the fixed load
distribution
{ }v
, applied load to the structure until the roof-level displacement reached the specified
displacement.

When the roof-level displacement attained the calculated amplitude
R
x
( =
1
i
q
+
), the resistance
R
f
at the roof level was measured. The acceleration amplitude
R
x


1
( )
i
q
+
=

was evaluated by Eq.
6 with given ground acceleration amplitude
1
i
y
+

. With a new acceleration value at time step i+1, Eq.
8 was used to calculate the displacement at further time step i+2.

6
Repeating the procedure outlined above, the test structure was subjected to an imaginary
earthquake motion.

Equations 6 and 8 may be combined to yield a single-step procedure,


2 2
1 1
2 (/) ( )
i i i i
q q t m f t yβ
+ −
= − ∆ − ∆

(9)

or in an incremental form,


2 2
1
(/) ( )
i i i i
q q t m f t yβ
+
∆ = ∆ − ∆ − ∆

(10)

in which,
1i i i
q q q

∆ = −
. Equation 10 was used in the analysis.


4. Modeling of Structural Members

It is not feasible to analyze an entire structure using microscopic material models. Therefore, it is
necessary to develop a simple analytical model of structural members.

Nonlinear dynamic analysis of a reinforced concrete structure requires two types of mathematical
modeling: (a) modeling for the distribution of stiffness along a member; and (b) modeling for the
force-deformation relationship under stress reversals. The former models are called "member
models", and the latter "hysteresis models".

Inelastic deformation of a reinforced concrete member does not concentrate in a critical location,
but rather spreads along the member. Various member models have been proposed to represent the
distribution of stiffness within a reinforced concrete member (3, 4).

The member models used to represent the stiffness behavior of beams, columns, and walls are
presented in this part.

4.1 Beam and Column Model

Many member models have been proposed for the beam and column members; for example, (a)
One-component model (5), (b) Multi-component model (6), (c) Connected two-cantilever model (7),
(d) Distributed flexibility model (8).

The One-component model was used for beams and columns in this paper. Namely, beam or
column member was idealized as a perfectly elastic massless line element with two nonlinear
rotational springs at the two ends. The model could have two rigid zones outside the rotational
springs as shown in Fig. 12. Axial deformation is considered in the elastic element of a column
member.

The stiffness properties of a rotational spring are evaluated for an imaginary anti-symmetric
loading conditions with the inflection point at the center of the flexible portion of a member. The
rotation at a flexible end less the elastic rotation is assigned to the rotational spring. The shear
deformation within a member and the member end rotation due to bar slip within the beam-to-column
connection should be considered in the evaluation of the deformation.

The shear deformation of a beam-to-column connection panel is not considered in the analysis.

4.2 Wall Model

A shear wall is normally idealized as (a) an equivalent column taking flexural and shear
deformation into account, (b) a braced frame, in which the shear deformation is represented by the
deformation of diagonal elements, whereas the flexural deformation by the deformation of vertical
7
elements, and (c) short line segments along the height with each short segment with hysteretic
characteristics (9, 10). These models have advantages and disadvantages. In most cases, the
horizontal boundary beams (or slabs) are assumed to be rigid.

The Japanese support tests on three-story walls with connecting beams (11) indicated a large
elongation of a tension-side column due to cracking, and a small compression of the
compression-side column, with the neutral axis of wall section close to the compression-side column.
In other words, the bending deformation of a wall was caused primarily by the extension of the
tension-side boundary column. The resistance of a wall came from the resistances of the boundary
columns and that of the central wall section.

The wall member of a story was, therefore, idealized as three vertical line elements with infinitely
rigid beams at the top and bottom floor levels (Fig. 13). Two outside truss elements represented the
axial stiffness of boundary columns. The axial stiffness varied with the sign and level of axial stress,
and degraded with tensile stress history. The central vertical element was a one-component model in
which vertical, horizontal and rotational springs were concentrated at the base. A finite rigid zone
could be placed between the spring assembly and the lower rigid chord.

The model was intended to simulate the wall deformation under uniform bending, the resistance of
wall section being lumped at the locations of the outer truss elements and the central vertical spring.
The effect of strain gradient across the wall section was represented by the rotational spring in the
central element, and the shear deformation expressed by the deformation of the horizontal spring.

The stiffness matrix of a wall element was formulated as the sum of the stiffness of the three
vertical elements evaluated at the top and bottom of the two boundary columns.

4.3 Transverse Beam Model

The tensile boundary column of a wall tends to elongate extensively under bending deformation,
yielding a significant vertical displacement at a beam-to-wall-joint node, whereas the vertical
displacement of a beam-to-column-joint node of an open frame is relatively small. Consequently, the
transverse beam connecting the boundary column of a shear wall and an adjacent parallel open
frame is subjected to vertical differential displacement at the two ends, and resists the upward
movement of a wall boundary column.

Vertical spring elements, therefore, were introduced to reflect the effect of such transverse beams
to restrain the elongation of a tensile boundary column (Fig. 14). A spring was placed between the
joints of the wall and an open frame connected by a transverse beam.


5. Stiffness of Member Models

Force-deformation relationship of member models under monotonically increasing load (called
skeleton force-deformation relationship) was evaluated on the basis of idealized stress-strain
relations of the concrete and the reinforcing steel.

5.1 Force-Deformation Relation

The force-deformation relationship is described for each member model. As the analysis reported
herein was of preliminary nature, approximate methods were used in evaluating member
deformations and resistances. Nominal member dimensions and material properties obtained from
coupon tests were used.

Beam Stiffness
: The beams were analyzed as a T-shaped beam, taking the contribution of slab into
account. The effective width of slab for the elastic stiffness of a beam was taken in accordance with
the Architectural Institute of Japan Standard (AIJ Standard for R/C) for Structural Calculation of
Reinforced Concrete Structure (12); i.e., the cooperating flange width
a
b
in a T-shape member (one
8
side) is


(0.5 0.6/)
a
b a a= − 
when
0.5
a
<

(11.a)

0.1
a
b = 
when
0.5
a
≥ 
(11.b)

where
a
: distance from the side of a beam to the side of the adjacent parallel T-beam (Fig. 15), and

: span length of the beam.

Equation (11.b) governed in all beams, and the total effective width B of beams parallel to the
loading direction was 150 cm in spans 1-2 and 3-4, and 130 cm in span 2-3 (Fig. 2).

The moment of inertia of a T-shaped beam section was computed about the geometrical centroid
ignoring the contribution of reinforcing steel. The elastic modulus of concrete was assumed to be 2.37
x 10
5
kgf/cm
2
, ignoring the fact that the field cured cylinders from the sixth and seventh story concrete
showed lower strength. The elastic stiffness properties were given to the perfectly elastic massless
line element of a one-component model.

Cracking moment
c
M
of a beam at the face of the supporting column was computed on the
basis of the flexural theory and an assumed concrete tensile strength of 20 kgf/cm
2
(Table 3); i.e.,


c c t e
M
Z
σ

=
(12)

where
c t
σ
: tensile strength of concrete (=20 kgf/cm
2
), and
e
Z
: section modulus without reinforcing
steel.

The value of cracking moment was different for the positive and negative bending because the
geometrical center does not locate at the mid-height of the section. The average value of positive and
negative cracking moments was used in the analysis.

Yield moment and curvature of a T-shaped beam section were calculated based on the flexural
theory. A linear strain variation across the section was assumed and the stress-strain relationships for
the longitudinal steel and concrete were considered as input factors.

Bi-linear model was used for the stress-strain relationship of steel as shown in Fig.16.a. Yield
stress (=3,650 kgf/cm
2
), and elastic modulus (=1.710 x 10
6
kgf/cm
2
), for D19 deformed bars were
determined according to the results of the material tests. The stiffness after yielding was assumed to
be zero.

The -stress-strain relation model by Aoyama (17) was used for concrete as shown in Fig. 16.b,
which defined the primary curve according to the following equation; i.e.,


c BB B
B B B
E
where
α
ε
σ σ ε ε
α
σ
ε σ
   
− −
= =
   
   
(13)

where
,
σ
ε
: compressive stress and strain,
B
σ
: stress at compressive strength (=290 kgf/cm
2
),
B
ε
:
strain at compressive strength (=0.0021), and
c
E
: initial tangent modulus (=2.37 x 10
5
kgf/cm
2
).

The slab can contribute to the resistance of a beam. The region, in which slab reinforcement
parallel to the loading direction yielded under beam negative moment, progressively spread with
increasing beam rotation. The strains measured in the slab reinforcing bars during the full-scale test
indicated that the effective slab width B (Fig. 15) was 350 cm in Frames A and C and 510 cm in Frame
B at maximum structural deformation (18). Therefore, the slab effective width B of 430 cm was used
9
in computation. Consequently, the yield moments for the positive and negative bending were
significantly different.

The inelastic beam deformation was assumed to concentrate at the locations of two nonlinear
rotational springs. The beam-end rotations at cracking and yielding were computed on the basis of
corresponding curvature distribution of the beam with an inflection point assumed to locate at the
mid-span of the flexible portion of the beam. The shear deformation was assumed to be proportional
to the flexural deformation. The calculated beam-end rotation less the elastic deformation was
assigned to the rotational spring at the end.

The skeleton moment-rotation curve was represented by a trilinear relation in each direction of
loading. The stiffness after yielding was arbitrarily assumed to be 3 % of the initial elastic stiffness.
The calculated stiffness properties of a beam model are listed in Table 4. The elastic deformation is
included in the calculated rotation.

Column Stiffness
: The dimensions of a column section and the amount of longitudinal reinforcement
were identical in all the column. The elastic stiffness properties (moment of inertia, cross sectional
area, and area effective for shear deformation) were calculated for gross concrete section, ignoring
the contribution of the steel reinforcement.

The existing axial force of a column due to the gravity loading was not the same for a column at
different story levels, and for columns of a story at different locations. The weight of slab, beams, and
girders within the tributary area of a column (Fig. 17) was used to calculate the axial load. The
calculated values (Table 3) were generally in reasonable agreement with the values obtained from
strain gauge measurement on column longitudinal bars. Columns
1
C
and
3
C
carried the weight of
actuators and loading beams. For columns
1
C
and
1
'
C
, or
3
C
and
3
'
C
, the average axial load
of the two columns was used in the analysis. The variation of axial load due to the overturning effect
of earthquake forces was not considered in evaluating flexural resisting capacity.

Simple approximate expressions (12) were used to evaluate cracking moment
c
M
and yield
moment
y
M
; i.e.,


/6
c c t e
M Z N D
σ
=
+
(14)

0.8 0.5 (1/)
y t y c
M
a D N D N b D F
σ
=
+ −
(15)

where N: axial force in column section (Table 5.2), b: width of column section (=50 cm), D: overall
depth of column section (=50 cm), and
c
F
: compressive strength of concrete (=290 kgf/cm
2
).

The area
t
a
of tensile reinforcement was 3-D22 (=11.61 cm
2
). The yield strength of D22
reinforcing bars was taken from the coupon test to be 3,530 kgf/cm
2
. The tensile strength of concrete
was assumed to be 20 kgf/cm
2
.

The rotations of a column were evaluated by a simple empirical formula by Sugano (19). The
formula was prepared for reinforced concrete beams and columns subjected to anti-symmetric
bending. The secant stiffness (
/
y y
M
θ
) at the yield point was proposed:


2
6
/(0.043 1.64 0.043 0.33 )( ) ( )
y y t
c
M
N d EI
M n p
QD bDF D
θ = + + +

(16)

in which,
y
M
: yield moment applied at two member ends,
y
θ
: member end rotation at yielding, n:
Young's modulus ratio (
/
s
c
E E=
),
t
p
: tensile reinforcement ratio
(/)
t
a bD
=
, M/QD: shear
10
span-to-depth ratio, and

: total length of member.

Ninety percent of test data studied fell within 30 per cent range of the value predicted by Eq. 16.
The Sugano's formula was used to estimate the yield rotation of a column. The column-end rotation
less the elastic deformation was assigned to the rotational spring. The skeleton moment-rotation
curve was represented by a trilinear relation with stiffness changes at cracking and yielding. The
skeleton curves were the same for positive and negative directions. The calculated stiffness
properties are listed in Table 6.

The axial rigidity (=
/EA 
) of a column in compression was defined by the gross sectional area,
elastic modulus (2.37 x 10
5
kgf/cm
2
) and height of the column (Fig. 3). When the axial force due to the
gravity effect was overcome by the overturning effect of earthquake forces, the axial rigidity was
reduced to 90% of the initial elastic stiffness. The column was assumed to yield in tension when the
net tensile load reached a tensile force equal to the sum of yield forces carried by all the column
longitudinal reinforcement (= 109.3 tonf). After tensile yielding, the stiffness was arbitrarily reduced to
0.1 % of the initial axial stiffness.

Wall Stiffness
: The boundary columns and a wall were analyzed as a unit. The wall model consists
of three sub-elements; i.e., (a) two vertical truss elements for the boundary columns, and (b) vertical
one-component element for the wall panel.

The axial rigidity (=
/
E
A 
) of a truss element (Table 7) was determined in the same way as that of
an independent column. The axial rigidity in compression remained linearly elastic. When a net axial
load changed its sign from compression to tension, the stiffness was reduced to 90% of the initial
elastic stiffness. The initial axial forces due to the gravity loads are listed in Table 5.2.b. Tensile
yielding occurred when a net tensile force reached a force level (=109.3 tonf) at which all column
longitudinal reinforcement yielded. Then the stiffness was reduced to 0.1% of the initial elastic
stiffness.

The shear resistance of a shear wall was provided by the lateral spring in the central vertical
element. The initial elastic shear rigidity
s
K
was defined as

w
s
GA
K

=
(17)
in which, G: elastic shear modulus (=0.98 x 10
5
kgf/cm
2
),
w
A
: area of shear wall section (Fig. 18),
κ
:shape factor for shear deformation (=
2 3
3(1 )[1 (1 )]/4[1 (1 )]
u u v u v
+
− − − −
, h: inter-story height,
and u, v : geometrical parameters defined in Fig. 18.

Shear cracking was assumed to occur at a shear force
s
c
Q
(in kgf),

1.4
s
c c w
Q F A=
(18)
in which
c
F
: compressive strength of concrete in kgf/cm
2
(=290 kgf/cm
2
).

Hirosawa's empirical equation (13) was used to evaluate the ultimate shear resisting capacity
s
u
Q
(kgf);


( )
0.23
0
0.0679 180
2.7 0.1
/0.12
t c
s
u wh wh e
p F
Q p b j
M QL
σ σ
 
+
= + +
 
+
 
 
(19)

where
t
p
: effective tensile reinforcement ratio (%),
100/( )
2
t e
D
a b L= −
,
t
a
: area of longitudinal
reinforcement in tension-side boundary column, M/QL: shear span-to-depth ratio,
wh
σ
: yield strength
11
of horizontal reinforcement in the wall (kgf/cm
2
),
wh
p
: effective horizontal wall reinforcement ratio =
/
wh e
a b x
,
x
: spacing of horizontal wall reinforcement (= 20 cm),
0
σ
: average axial stress over
entire wall cross sectional area (Table 5.2.a),
e
b
: average width of wall section,
7
( )
8 2
D
j L= −
, L,D:
geometrical parameters defined in Fig. 18.

Ratio
s
β
= 潦⁴桥o獥捡湴⁳瑩s普敳猠慴 = 獨敡爠祩敬搠灯楮≥⁴= ⁴桥⁥污獴楣⁳瑩= 普敳猠睡猠摥瑥牭楮敤=
敭灩物捡汬礠批e
=

0.46/0.14
s wh wh c
p F
β
σ= +
(20)

The shear stiffness reduction factor
s
β
= 睡猠慰灲wx業慴敬礠〮ㄶ⁦i r⁴桥⁳桥慲⁷慬氠慮慬祺敤==
=
周攠獴楦普敳猠ff瑥爠獨敡爠祩敬摩湧⁷a猠≥a步渠瑯≥ 扥‰⸱b┠潦⁴桥⁩湩瑩慬o 敬慳瑩挠獨敡爠物杩摩瑹⸠
䍡汣畬慴敤⁳瑩C普敳猠灲潰f牴楥 猠慲攠汩獴s搠楮dθ慢汥‸⹡⸠
=
䅸楡氠獴楦普敳猠灲潰敲瑩敳映瑨攠 捥湴牡氠癥牴楣慬⁥汥c 敮琠⡔慢汥‸⹢⤠睥牥⁤e≥敲浩湥搠楮⁴桥⁳慭e=
睡礠慳⁴桥⁴牵獳⁥汥浥湴⸠䅲敡映愠獨敡爠ea汬⁢潵湤敤n批⁴桥⁩湮敲⁦慣敳n⁴睯⁢潵湤nry⁣潬畭湳=
睡猠畳敤⁦潲⁴桥⁣牯獳⁳w捴楯湡氠慲敡c 潦⁴桥⁣敮瑲慬⁶敲瑩捡氠敬敭敮琮o
=
副Ra瑩潮慬⁳瑩≥普敳猠灲潰敲瑩 敳映瑨e⁣=湴牡氠癥牴楣慬n敬敭敮琠 ⡔慢汥‸⹣⤠睥牥a摥晩湥搠景爠睡汬⁡牥a=
扯畮摥搠dy=瑨攠楮湥爠晡捥猠潦⁴睯⁢潵湤≥特⁣潬r浮献 ⁗a汬⁲l≥a瑩潮⁷慳⁣潭灵瑥搠慳≥瑨攠灲潤畣琠潦u
瑨攠捵牶慴畲攠慴⁢慳攠慮搠瑨攠楮瑥爭e 瑯特⁨敩≥h琮⁉渠潴桥爠睯牤猬s景 爠瑨攠灵牰潳攠潦⁣r浰畴楮朠睡汬=
牯ra瑩潮Ⱐ浯浥湴⁷慳⁡獳畭敤⁴漠摩獴物扵瑥⁵湩景 rm汹⁡汯湧⁴l攠獴潲礠桥楧e琠睩瑨⁡渠慭灬楴畤攠敱≥慬a
瑯⁴桥潭敮琠慴⁷慬氠捲楴楣慬≥獥捴楯渮⁃牡捫楮朠 睡猠瑯捣畲w睨敮⁴桥⁥x瑲敭攠瑥湳楬攠晩扥爠獴牡楮≥
扥捡浥⁺敲b⁵湤敲⁴桥⁧=慶楴礠汯慤⁡nd癥牴=牮楮r潭敮琻=
=

( )
6
c
N uL
M =
(21)

Yielding moment
y
M
was taken to be the full plastic moment; moment about the centroid of wall
section caused by the yielding of all vertical wall reinforcement. The gravity load was ignored in
computing the full plastic moment. The stiffness after yielding was taken to be 0.1 % of the initial
elastic stiffness.

Transverse Beam Stiffness
: The effect of transverse beams to restrain the upward movement of a
tensile wall boundary column was represented by a vertical spring. The initial elastic stiffness
t
K

was calculated for a fixed-fixed beam as


3
12
t
EI
K =

(22)

where EI : flexural rigidity of transverse beam, and

: span length of transverse beam.

Cracking and yielding forces were determined as a shear force acting in the transverse beam
when both ends cracked and yielded simultaneously in flexure. Cracking moment, yielding moment
and curvature of T-shaped transverse beam section were evaluated based on the flexural theory in
the same way used for beam stiffness evaluation. These values calculated for positive and negative
bending moments were averaged. The effective width B (Fig. 15) of 190 cm was determined referring
to the results of the fun-scale test (18). The stiffness after yielding was reduced to 3 % of the initial
stiffness.

12
The numerical values of the stiffness properties of the vertical spring are listed in Table 9.

5.2 Hysteresis Models

A hysteresis model must be able to provide the stiffness and resistance relation under any
displacement history. Four different hysteresis models were used in the analysis; i.e., (a) Takeda
hysteresis model (14), (b) Takeda-slip hysteresis model (16), (c) Axial-stiffness hysteresis model, and
(d) Origin-oriented hysteresis model. The characteristics of each model are briefly described in this
section.

Takeda Hysteresis Model
: Based on the experimental observation on the behavior of a number of
medium-size reinforced concrete members tested under lateral load reversals with light to medium
amount of axial load, a comprehensive hysteresis model was developed by Takeda, Sozen and
Nielsen (l4). The model included (a) stiffness changes at flexural cracking and yielding, utilizing a
trilinear skeleton force-deformation relationship, (b) hysteresis rules for inner hysteresis loops inside
the outer loop ; i.e., the response point during loading moves toward a peak of the immediately outer
hysteresis loop, and (c) unloading stiffness degradation with a maximum deformation amplitude. The
unloading stiffness
r
K
is given by


| |
c y
a
m
r
c y y
F F
D
K
D D D

+
= ⋅
+
(23)

in which
(,)
c c
D F
: cracking point deformation and resistance,
(,)
y y
D F
: yielding point deformation
and resistance,
m
D
: maximum deformation amplitude greater than
y
D
,
α
㨠畮汯慤楮朠獴楦普敳f=
摥杲慤慴楯渠pa牡浥瑥爠⡮潲浡汬礠扥瑷敥渠〮〠慮搠〮㘩⸠
=
周攠来θ敲慬e桹獴敲敳楳⁲畬敳⁡牥畴汩湥搠楮⁆楧h ‱9⸠周攠摥.慩氠摥獣物灴楯渠潦⁴桥潤敬⁣慮⁢==
景畮搠楮⁒敦e牥湣敳‷⁡r搠ㄴ⸠ =
=
周攠θa步摡⁨祳瑥牥獩猠浯摥氠睡猠 畳敤⁩渠楮敬慳瑩挠牯ua瑩潮慬⁳≥ 物湧猠潦⁴桥⁩湤数敮摥湴⁣潬畭==
潮攭捯浰on敮琠浯摥氬⁡湤⁩渠愠癥牴楣慬e 獰物湧映瑨攠瑲慮獶敲s攠扥慭潤e氮l
=
周攠畮汯慤θ湧⁳瑩n普敳猠摥杲慤慴楯渠pa牡浥瑥爠
α
= 景爠慮⁩湤数e湤敮琠捯汵浮⁡湤⁡⁴牡湳癥牳n=
扥慭潤敬b睡猠慲扩瑲慲楬礠捨潳敮⁴漠扥‰⸴⸠
=
Takeda-Slip Hysteresis Model
: Half-scale beam-to-column joint assemblies with slab were tested
(15) to obtain preliminary information about possible behavior of the full-scale seven-story building.
Force-deformation relation of a beam with slab showed obvious pinching characteristics in negative
moment region (loading under which the beam top was in tension) as shown in Fig, 20. This pinching
behavior was not associated with that often observed in a member failing in shear, but rather
associated with a wide crack opening at the bottom of the beam during positive-moment loading; i.e.,
after a load reversal from positive-moment loading, the stiffness did not recover until the crack closed
at the beam bottom.

Eto and Takeda (16) introduced pinching characteristics into a hysteresis model in simulating
member-end rotation behavior due to bar slip within a beam-column connection. The Takeda and
Eto's model was modified in this paper for use in a rotational spring of a beam one-component model.

The Takeda hysteresis model was modified as follows :
(a) The pinching occurs only in one direction where the yield resistance is higher than that in the
other direction, and the pinching occurs only after the initial yielding in the direction concerned.
(b) The stiffness
s
K
during slipping is a function of the maximum response point
(,)
m m
D E
and
the point of load reversal
0 0
(,)D F
in the force-deformation plane (Fig. 21.d)

13

0 0
m m
s
m m
F D
K
D D D D
γ
 
=
 
− −
 
(24)

where
γ
㨠牥汯慤楮朠獴楦普敳猠fa牡浥瑥r.=
⡣⤠䅦瑥爠灩湣桩湧Ⱐ瑨攠牥獰潮獥≥ 灯楮琠浯癥猠瑯睡pd猠瑨攠灲敶楯畳 慸業畭⁲敳灯湳攠灯楮琠睩瑨=
獴楦普敳f
p
K
;


( )
m
p
m
F
K
D
η=
(25)

where
η
㨠牥汯慤楮:⁳瑩= 普敳猠ea牡浥瑥r.⁉渠潴桥= = 睯牤猬⁴桥⁳瑩= 普敳猠捨慮c攠潣捵牳⁡琠慮=
楮瑥牳散瑩潮i潦⁴桥⁴睯⁳o 牡楧桴楮敳⁨慶楮朠獬潰敳=
s
K
and
p
K
.

The Takeda-slip hysteresis model was used in the inelastic rotational spring of a beam
one-component model. The values of unloading stiffness degradation parameter
α
Ⱐ獬楰灩湧=
獴楦普敳猠摥杲慤慴楯渠fara浥瑥爠
γ
Ⱐ慮搠牥汯,d楮朠獴楦普敳猠fara浥瑥爠
η
= 睥we‰⸴Ⱐㄮ〬⁡湤‱⸰Ⱐ
牥獰散瑩癥汹.=
=
Axial-Stiffness Hysteresis Model
: The behavior of a column under axial load reversals is not clearly
understood. The following hysteresis model was developed and tentatively used for the axial
force-deformation relation of a column.

Referring to Fig. 22, a point Y' is defined on the elastic slope in compression at a force level equal
to the tensile yield strength
y
F
. The response point follows the regular bilinear hysteresis rules
between the two points Y and Y' (Fig. 22.a). Once tensile yielding occurs, then a response point
moves following the regular bi-linear hysteresis rules between point Y, and previous maximum tensile
response point M with a force level of
y
F
using unloading stiffness
r
K
(Fig. 22.b) :


max
a
r c
yt
D
K K
D

 
=
 
 
 
(26)

where,
yt
D
: tensile yielding point deformation,
max
D
: maximum deformation amplitude greater than
yt
D
,
α
㨠畮汯慤楮:⁳瑩=普敳猠摥杲慤慴楯渠aa牡浥瑥爠⠽ 〮㤩⸠坨敮⁴桥⁲敳灯湳攠灯楮琠牥慣桥猠瑨≥=
灲敶p潵猠浡硩浵洠瑥湳楬攠灯楮琠 M, then the response point moves on the second slope of the
skeleton curve, renewing the maximum response point M.

When the response point approaches the compressive characteristic point Y' and moves on the
elastic slope in compression, the response moves toward a point Y" from a point P of deformation
p
D
:


( )
p yc x yc
D D D D
β
= + −
(27)

where,
β
㨠:ara浥瑥爠景爠獴楦普敳猠桡牤敮楮朠灯f湴
㴰⸲⤬n
x
D
: deformation at unloading stiffness
changing point. This rule is introduced only to reduce an unbalanced force by a sudden stiffness
change at compressive characteristic point Y’. The compressive characteristic point Y' will be
maintained under any loading history.

This axial-stiffness hysteresis model was used for the axial deformation of an independent column
14
as well as a boundary column of a wall. The initial response point located in the compression zone
because a column carried gravity loads.

Origin-Oriented Hysteresis Model
: A hysteresis model which dissipates small hysteretic energy
was used for the rotational and horizontal springs at the base of the central vertical element of a wall
model.

The response point moves along a line connecting the origin and the previous maximum response
point in the direction of reloading(Fig. 23). Once the response point reaches the previous maximum
point, the response point follows the skeleton force-deformation relation renewing the maximum
response point. In this model, no residual deformation occurs, and the stiffness changes when the
sign of resistance changes. No hysteretic energy is dissipated when the response point oscillates
within a region defined by the positive and negative maximum response points. The skeleton curve of
this model can be of any shape.


6. Method of Response Analysis

The seven-story test structure was idealized as three parallel plane frames with beams, columns
and walls represented by corresponding member models. The transverse beams connecting the
shear wall boundary columns and adjacent parallel frames were idealized by vertical springs. A
routine stiffness method was used in the analysis.

Floor slab was assumed to be rigid in its own plane, causing identical horizontal displacements of
all the joints in a floor level. The mass of the structure was assumed to be concentrated at each floor
level.

Vertical displacement and rotation were two degrees of freedom at each joint. The frames and a
shear wall were assumed to be fixed at the base of the structure.

A numerical procedure was developed to simulate the "equivalent" single-degree-of-freedom
pseudo-dynamic earthquake response procedure. The mode shape
{ }u
, participation factor
β
and
resistance distribution
{ }v
were taken from the test as outlined in Section 3.4.

The test structure changed its stiffness continuously with applied displacement even in a short
time increment, whereas the analytical model assumed a constant tangent stiffness during the time
increment. Therefore, the unbalanced forces, caused by overshooting at a break point of hysteresis
rules, must be released at the next time step. The analytical procedure from time step i to i+1 is
briefly outlined below.

Step 1: Determine displacement increment ∆q
i+1
at the top floor using Eq. 10;


2 2
1
(/) ( )
i i i i
q q t m f t y
β
+
∆ = ∆ − ∆ − ∆

(28)

where
1i i i
q q q

∆ = −
: incremental displacement at roof level from time step i-1 to i,
t

: time
increment,
m
: effective mass,(
{ } [ ]{ }
T
u m u=
),
f
: effective resistance at time step i, (
{ } { }
T
R
u v f=
),
β
㨠敦晥捴楶攠fa牴楣楰慴楯渠晡捴潲Ⱘ
{ } [ ]{1}/
T
u m m=
),
i
y

: ground acceleration at time step i,
{ }u
:
assumed lateral deflection mode shape,
{ }v
: assumed lateral resistance distribution,
[ ]m
: mass
matrix, and
{1}
: vector consisting of unit element.

Step 2: Unbalance force correction. This step is necessary only when the stiffness changed
between time steps i-1 and i.
(a) Calculate displacement vector
0
{ }
x
due to unit load
{ }v


15

0 1
{ } [ ] { }
i
x
k v

=
(29)

(b) Calculate displacement vector
{'}
x
due to unbalanced force
{'}F
, at time step i,


0 1
{ } [ ] {'}
i i
x
K F

=
(30)

where
[ ]
i
K
: tangent stiffness matrix evaluated at time step i.

Step 3: Determine incremental lateral resistance
1Ri
f
+

at the roof level,


( )
0
1 1 1
'/
Ri i Ri R
f
q x x
+ + +
∆ = ∆ −
(31)

in which
'
R
x
and
0
R
x
are the values of vectors
{'}
x
and
0
{ }
x
evaluated at the roof level.

Step 4: Calculate displacement increment
{ }
i
x

due to incremental load and unbalanced forces
{'}F
,


0
1 1
{ } { } {'}
i Ri i
x x f x
+ +
∆ = ∆ +
(32)

Step 5: Calculate incremental member forces from incremental joint displacement and
tangent member stiffness. Check if member stiffness changed during time step i and i+1.

Steps 1 through 5 were repeated for each time step. Computed response was temporarily stored
on computer files, and plots of response wave forms and force-deformation curves were made if
necessary.


7. Results of Analysis

The analytical models and procedure described above was applied to the full-scale seven-story
test structure. Equivalent single-degree-of-freedom pseudo-dynamic tests PSD-2 through PSD-4
were simulated continuously so that the structural damage in the preceding test runs could be
reflected in the analysis of the following test runs. In other words, the analytical model was given the
calculated residual displacement and structural damage from the immediately preceding test run.
Initial velocity of the analytical model was set to be null at the beginning of each test run. The actual
test was conducted in the same manner.

Pseudo-dynamic test PSD-1 was not analyzed herein because the test run was carried out to
examine the reliability of the testing technique and procedure. The maximum roof-level displacement
was as small as 1/8600 of the total structural height. The structure was observed to remain in the
elastic range during the test run. Therefore, the test run was not included for study here.

The analytical response of test PSD-3 was studied in detail to examine the reliability of the
analytical method. The structure was subjected to a displacement beyond the formation of collapse
mechanism during the test run. The roof-level displacement was observed to reach 1/91 the total
story height, a displacement which may be expected from this type of a structure during a "strong"
earthquake motion. This test included a wide range of response prior to and after the yielding of
various members and the shear wall. The structure did not have non-structural elements. These were
major reasons to choose this particular test run for careful inspection. Studied are the analytical
results of (a) response waveforms at the roof level displacement and base shear, (b) hysteresis
relation between base shear and roof-level displacement, (c) base shear-local deformation relations,
and (d) force and deformation distribution at maximum computed response.
16

The calculated response waveforms and the roof-level displacement vs. base shear relations are
briefly compared with the observed for tests PSD-2 and PSD4.

7.1 Response Waveforms (Test PSD-3)

The artificial earthquake accelerogram based on EW component of the Taft record (1952) was
used in test PSD-3. The higher frequency components were removed from the original record so that
the first mode should govern the response of the test structure. The maximum ground acceleration
was 320 Gal (cm/sec
2
).

No damping was assumed in the test structure in the pseudo-dynamic response computation
during the test. Observed and calculated response waveforms are compared for the roof-level
displacement and base shear as shown in Fig, 24. The input base motion is shown in the same figure.
Note that the base motion oscillates in relatively low frequency compared with the original earthquake
record.

Response waveforms observed in test PSD-3 are shown in broken lines. Analytical responses are
in good agreement with the observed response over the entire duration of earthquake excitation.

Maximum displacement at roof-level was 238 mm from the test attained at 4.48 sec, while the
calculated maximum amplitude was slightly larger (= 248 mm) than the observed. Both maxima
occurred at the same time step. The period of oscillation elongated significantly after this time step in
the test and analysis.

At 10.16 second, the ground motion input was terminated in the test, and pseudo-dynamic
free-vibration test was started with existing residual displacement and no velocity. In the free vibration
range, the period of the analytical model appeared slightly longer than that of the test structure.

Maximum base shear of 414 tonf was attained at 4.48 sec in the test. The computed value was
425 tonf, slightly higher than the observed. Maximum base shear amplitude of an analytical model
can easily be controlled by choosing yield resistance level and post-yielding stiffness of constituent
members, especially of beam members in this analysis.

Parametric studies by varying beam yield resistance and post-yielding stiffness indicated that the
combination of the values described in Chapter 5 was most suited to the test structure; i.e., the beam
yield resistance to be computed with the contribution of slab reinforcement within an effective width of
430 cm and post-yielding stiffness to be 3 % of the initial elastic stiffness.

7.2 SDF Hysteresis Relation (Test PSD-3)

Roof-level displacement and base shear were the corresponding force and displacement in the
"equivalent" single-degree-of-freedom pseudo-dynamic test, and their relation may be called "SDF
hysteresis" as shown in Fig. 25.

As can be expected from a good correlation of observed and computed response waveforms, the
observed and computed hysteresis relations as an equivalent single-degree-of-freedom system are
in fair agreement, especially at the peaks of hysteresis loops. General shapes of the two curves are
slightly different; the stiffness of the test structure changed gradually during unloading, whereas the
stiffness of the analytical model changed when the sign of resistance changed. The latter stiffness
change was associated with that of member hysteresis models such as Takeda, Takeda-slip, and
Origin-oriented hysteresis models. The analytical model showed some pinching behavior, which was
also appreciable in the observed hysteresis relation. The pinching behavior of an analytical model
was caused by Takeda-slip hysteresis model used with beam one-component models and
axial-stiffness hysteresis model used with vertical line elements in the shear wall.


17
7.3 Local Deformations (Test PSD-3)

During the pseudo-dynamic tests, local deformations of members were measured at various
locations of the test structure; (a) flexural rotation at beam ends, (b) flexural rotation at column base,
(c) elongation of boundary columns of the wall, and (d) shear deformation of the wall panel.
Computed local deformations of typical members were compared with the observed deformations so
as to examine the reliability of the analysis method.

Beam End Rotation
: Rotations at beam ends were determined from the axial elongation and
compression measurements by two displacement gauges, one placed above the slab face and the
other placed below the beam, parallel to the beam member axis (Fig. 26.c). The gauge length was
one half the effective beam depth from the column face.

The observed base shear-beam end rotation relation of a sixth floor beam at the wall connection is
shown in Fig. 26.a. The calculated relation is shown in Fig. 26.b. The calculated and the observed
relations do not necessarily agree because the beam end rotation was measured for a given gauge
length, whereas the rotation was calculated for an entire beam under imaginary anti-symmetric
loading condition. In other words, the calculated deformation corresponds to the deformation over
one-half span length of the beam. Therefore, the measured deformations were generally smaller, and
approximately 60 to 70% of the calculated amplitudes. General shapes of the base shear-beam end
rotation relation curves of the two were similar.

The beam was subjected to larger deformation in the negative loading direction when the
connecting tension-side boundary column moved upward. The upward displacement of a boundary
column joint (node) was significantly larger than the downward displacement because the bending
deformation of a wall was mainly attributable to the elongation of a tension-side boundary column.
Both observed and calculated beam-end response show this behavior.

Negative maximum deformations were larger than the positive deformation, although positive and
negative amplitudes of overall structural displacement were comparable. Negative deformation
amplitudes at the two ends of the beam were comparable, whereas the positive deformation at the
wall end was approximately 1.3 times larger than that of the behavior observed at the further end (left
end); the behavior was observed both in the measured and calculated beam-end rotations. At the
exterior column-beam joint, beam negative moment capacity was large due to the participation of slab
reinforcement. Hence, the exterior column was subjected to higher bending moment under the
positive loading (load applied from right to left), and experienced a larger rotation at column ends.
Therefore, nodal rotation at the exterior beam-column joint under the positive loading was smaller,
resulting in a smaller beam-end rotation at the exterior column end.

Figure 27 shows the beam-end rotations at a sixth floor beam in Frame A. The observed
beam-end rotation amplitudes were smaller than the calculated amplitudes.

Column Axial Deformation
: A large vertical displacement was observed at the top (roof level) of the
tension-side boundary column of the wall during the test. Large axial elongations were measured in
the tensile region of the wall, especially at lower stories. Compressive axial deformations in the
corresponding region were small under opposite direction loading. Larger deformation was observed
in a transverse beam connected at the tensile edge (boundary column) of the wall. The boundary
columns were measured to elongate as much as 44 mm in the first story as shown in Fig. 28.a,
whereas the maximum compressive deformation reached only 5 mm.

Computed axial deformations of the boundary column, as expressed as the deformation of outer
truss elements, are shown in Fig. 28.b. General deformation amplitudes and hysteresis shapes of the
analytical model agree reasonably well with those of the test structure. The computed axial
deformation was larger.



18
7.4 Response at Maximum Displacement (Test PSD-3)

It is important from design point of view to estimate possible force amplitudes and deformation
ductility factors at various critical sections of the test structure at maximum deformation. However,
member forces could not be measured in the test. The frame analysis method may be applied to
estimate these quantities. The maximum deformation of the test structure was observed as well as
calculated to occur at 4.48 see of the earthquake time.

Member Forces
: Member forces in wall-frame B calculated at maximum structural deformation by
the analytical model are shown in Fig. 29. The wall carried smaller shear forces in the first story than
in the second story.

Vertical forces transferred by transverse beams to the wall boundary columns are also shown in
the figure. Yield force level was reached by the transverse beam connected to the boundary column
in tension.

Member Ductility
: Ductility factors are defined in the analysis as a ratio of maximum deformation
amplitude to the calculated yield amplitude. Figure 30 shows the distribution of ductility factors at the
maximum structural deformation for frames A and C and frame B.

In open frames A and C (Fig. 30.a), almost all beam ends yielded at the maximum displacement
except those at the roof level. Under this deformed configuration, the top chord was in tension at the
left end of a beam, and the bottom chord was in tension at the right end. Ductility factors, ratios of
beam end rotations to the yield rotation, ranged from 0.8 to 1.5 at the left end of the beams, and from
2.3 to 4.7 at the right end. The rotation amplitudes at the left and right ends of the beams are
comparable. The difference in ductility factors at the two ends of a beam was caused by the
difference in the yield rotations at the two ends (see Table 4); the yield: rotation amplitude under
negative moment (top chord in tension) is approximately twice as much as that under positive
moment (bottom chord in tension).

Ductility factors at the same end (left or right) of the beams varied with the level of the beam; the
ductility factor decreased with the beam level. A beam end rotation appeared to be inversely related
to the column end rotation of the joint. Ductility factors of beam ends were smaller at the upper floor
levels where the columns yielded, and they were larger at the right exterior joints where the column
rotations were smaller.

In wall-frame B (Fig. 30.b), all beams yielded. Under the deformed configuration, the top chord
was in tension at the left end of a beam, and the bottom chord in tension at right end. The distribution
of beam end ductility factors was relatively uniform along the height; 1.4 to 1.7 at left exterior beam
ends, 4.1 to 4.7 at beam ends immediately left of the wall, 3.2 to 3.4 at beam ends immediately right
of the wall, and 6.5 to 7.9 at right exterior beam ends. The beam end rotation was generally larger in
the right exterior span than that at the corresponding end in the left exterior beams, which was
caused by the large vertical displacement along the tensile boundary column.

Deflected Shape
: Observed and calculated deflected shapes at maximum structural deformation are
compared in Fig. 31. A good agreement can be noted at every floor level. The deflection mode shape
used for the equivalent single-degree-of-freedom pseudo-dynamic earthquake test slightly deviated
at lower floor levels.

7.5 Analysis of Test PSD-2

The maximum roof-level displacement during the second test run (PSD-2) reached 1/660 of the
total structural height, or 33 mm. An artificial earthquake motion, modified from the NS component of
the Tohoku University record measured during the 1978 Miyagi-Oki Earthquake, was used with the
maximum acceleration amplitude of 105 Gal (cm/sec
2
). The calculated response waveforms and
equivalent SDF hysteresis relation are examined below.

19
Response Waveforms
: Observed and calculated roof-level displacement and base shear
waveforms are compared in Fig. 32. The analysis indicated that the test structure responded
elastically up to 1.5 sec, and then started to suffer damages. The calculated response waveforms
(solid lines) are in good agreement with the observed (broken lines) in the first 2.5 sec, and then
significantly deviates from the observed.

The maximum roof-level displacement of 32.9 mm was observed at 2.03 sec, while the maximum
amplitude of 36.5 mm was calculated at 2.06 sec. The maximum base shear of 224 tonf was attained
at 2.01 sec in the test. The maximum displacement and base shear did not occur at the same time in
the test. The maximum base shear of 219 tonf, slightly smaller than the observed, was calculated at
the same time as the calculated maximum displacement.

The calculated residual displacement at the termination of the base motion was so small that the
free vibration response was not excited in the analysis.

Equivalent SDF Hysteresis Relation
: Observed and calculated roof-level displacement vs. base
shear relation is compared in Fig. 33. Note that the two curves are similar. However, a careful
inspection reveals that calculated stiffness and resistance (solid lines) were generally lower than the
observed (broken lines). The calculated stiffness in a small amplitude oscillation following a large
amplitude excursion was lower, which may be a major cause to create the discrepancy in the two
waveforms (Fig. 32) after 2.5 sec.

7.6 Analysis of Test PSD-4

After test PSD-3, the roof-level displacement during test PSD-4 reached as large as 1/64 of the
total story height, or 342 mm. The EW component of the Hachinohe Harbor record measured during
the 1968 Tokachi-Oki Earthquake was used in the test with the maximum acceleration of 350 Gal.
The analysis was carried out continuously using the PSD-2, PSD-3 , and PSD-4 input motions.
Calculated and observed response waveforms and equivalent SDF hysteresis relations are
compared below.

Response Waveforms
: Observed and calculated roof-level displacement and base shear
waveforms are compared in Fig. 34. Note the good agreement of the two waveforms over the entire
duration of the test.

Maximum roof-level displacement reached 342 mm at 4.36 sec during the test, while the maximum
amplitude of 391 mm was calculated at 4.33 sec. Observed maximum base shear of 439 tonf was
attained at 2.52 sec, much before the maximum displacement was attained. The base shear at the
maximum displacement amplitude was observed to be 433 tonf, almost of the same amplitude as the
observed maximum base shear. The maximum base shear of 463 tonf was calculated at 4.33 sec,
slightly larger than the observed. The calculated and observed waveforms oscillated in the same
phase with a common dominant period of 1 .36 sec.

Equivalent SDF Hysteresis Relations
: Observed and calculated roof-level displacement vs. base
shear relations are compared in Fig. 35. The two hysteresis curves show a pinching behavior at low
stress levels. As expected from the good agreement in the response waveforms, the two hysteresis
curves agreed well. The observed base shear in the positive direction was slightly lower than that in
the negative direction. Such degradation in resistance was not reproduced by the analytical model.


8. Concluding Remarks

A full-scale seven-story reinforced concrete structure was tested using "equivalent"
single-degree-of-freedom pseudo-dynamic earthquake response test procedure at Building Research
Institute, Tsukuba, as a part of U.S.-Japan Cooperative Research Program Utilizing Large Scale
Testing Facilities.

20
A nonlinear dynamic analysis method was used to simulate the observed behavior. The method
utilized three different member models for (a) beams and columns, (b) shear walls, and (c) transverse
beams, and four hysteresis models for elements of member models: (a) Takeda hysteresis model, (b)
Takeda-slip hysteresis model, (c) Axial-stiffness hysteresis model, and (d) Origin-oriented hysteresis
model.

A procedure was outlined as to the method to determine stiffness properties used for the analysis
on the basis of material properties and structural geometry.

The response of the test structure was computed by a numerical procedure specially developed to
simulate the "equivalent" single-degree-of-freedom pseudo-dynamic earthquake response test
procedure.

A good correlation was reported between the observed and calculated response when the
structure responded well in an inelastic range. However, it was felt more difficult to attain a good
correlation when the structural response reached barely yielding.

The method of nonlinear dynamic analysis of reinforced concrete buildings can be made
significantly reliable not only to outline the overall structural behavior, but also to describe the local
behavior.


Acknowledgement

The writers wish to express a sincere gratitude to Dr. H. Umemura, Co-Chairman, and Dr. M.
Watabe, Japanese Technical Coordinator, U.S.-Japan Cooperative Research Program Utilizing Large
Scale Testing Facilities, for providing the writers with an opportunity to actively participate in the
research project. Mr. S. Okamoto, Dr. S. Nakata and Dr. M. Yoshimura, Building Research Institute,
gave the detail description of the test structure, testing method and test results of the full-scale
seven-story structure.

The writers wish to record the assistance of Mr. H. Katsumata, formerly with Department of
Architecture, currently with Obayashi-Gumi Ltd.

The computation reported herein was carried out on the Computer System of the University of
Tokyo Large Size Computer Center.


References:

1) Wight, J.K. and S. Nakata: Construction of the Full-scale Seven-story Reinforced Concrete Test
Structure, Report Presented during The Second Joint Technical Coordinating Committee, U.S.-Japan
Cooperative Earthquake Research Program Utilizing Large-Scale Testing Facilities, Tsukuba, Japan,
1982.

2) Okamoto, S., S. Nakata, Y. Kitagawa, M. Yoshimura and T. Kaminosono: A Progress Report on
the Full-scale Seismic Experiment of a Seven-story Reinforced Concrete Building - Part of the
U.S.-Japan Cooperative Program, BRI Research Paper No. 94, Building Research Institute, Ministry
of Construction, 1982.

3) Otani, S: Nonlinear Dynamic Analysis of Reinforced Concrete Building Structures, Canadian
Journal of Civil Engineering, Vol. 7, No. 2, 1980, pp. 333 - 344.

4) Umemura, H. and H. Takizawa: A State-Of-the-Art Report On the Dynamic Response of
Reinforced Concrete Buildings, Structural Engineering Documents 2, IABSE, 1981.

5) Giberson, M. F.: The Response of Nonlinear Multi-Story Structures Subjected to Earthquake
21
Excitation, Earthquake Engineering Research Laboratory, California Institute of Technology,
Pasadena, California, EERL Report, 1967.

6) Clough, R. W., K. L. Wilson: Inelastic Earthquake Response of Tall Building, Proceedings, Third
World Conference on Earthquake Engineering, New Zealand, Vol. II Section II, 1965, pp. 68 - 89.

7) Otani, S. and M. A. Sozen: Behavior of Multistory Reinforced Concrete Frames during
Earthquake, Structural Research Series No. 392, University of Illinois, Urbana, 1972.

8) Takizawa, H.: Strong Motion Response Analysis of Reinforced Concrete Buildings (in
Japanese), Concrete Journal, Japan National Council on Concrete, Vol. 11, No. 2, 1973, pp. 10 - 21.

9) Omote, Y. and T. Takeda : Nonlinear Earthquake Response Study on the Reinforced Concrete
Chimney - Part 1 Model Tests and Analysis (in Japanese), Transactions, Architectural Institute of
Japan, No. 215, 1974, pp. 21-32.

10) Takayanagi, T. and W. C. Schnobrich: Computed Behavior of Reinforced Concrete Coupled
Shear Walls, Structural Research Series No. 434, University of Illinois, Urbana, 1976.

11) Hiraishi, H., M. Yoshimura, H. Isoishi and S. Nakata: Planer Tests on Reinforced Concrete
Shear Wall Assemblies - U.S.-Japan Cooperative Research Program -, Report submitted at Joint
Technical Coordinating Committee, U.S.-Japan Cooperative Research Program, Building Research
Institute of Japan, 1981.

12) Architectural Institute of Japan: AIJ Standard for Structural Calculation of Reinforced Concrete
Structures (Revised in 1982), 1950.

13) Hirosawa, M.: Past Experimental Results on Reinforced Concrete Shear Walls and Analysis
on them (in Japanese), Kenchiku Kenkyu Shiryo No. 6, Building Research Institute, Ministry of
Construction, 1975.

14) Takeda, T., M. A. Sozen and N. N. Nielsen: Reinforced Concrete Response to Simulated
Earthquakes, ASCE, Journal of the Structural Division, Vol. 96, No. ST12, 1970, pp. 2557 - 2573.

15) Nakata, S., S. Otani; T. Kabeyasawa, Y. Kai and S. Kimura: Tests of Reinforced Concrete
Beam-Column Assemblages, - U.S.-Japan Cooperative Research Program -, Report submitted to
Joint Technical Coordinating Committee, U.S.-Japan Cooperative Research Program, Building
Research Institute and University of Tokyo, 1980.

16) Eto. H. and T. Takeda: Elasto Plastic Earthquake Response Analysis of Reinforced Concrete
Frame Structure (in Japanese), Proceedings, Architectural Institute of Japan Annual Meeting, 1977,
pp. 1877 - l878.

17) Fujii, S, H. Aoyama and H. Umemura: Moment-Curvature Relations of Reinforced Concrete
Sections Obtained from Material Characteristics (in Japanese), Proceedings, Architectural Institute of
Japan Annual Meeting, 1973, pp. 1261 - 1262.

18) Kaminosono. T., S. Okamoto, Y. Kitagawa, S. Nakata, M. Yoshimura, S. Kurose and H.
Tsubosaki: The Full-Scale Seismic Experiment of a Seven-story Reinforced Concrete Building, - Part
1, 2 - (in Japanese), Proceedings, Sixth Japan Earthquake Engineering Symposium, 1982, pp.
865-880.

19) Sugano, S.: Experimental Study on Restoring Force Characteristics of Reinforced Concrete
Members (in Japanese), Doctor of Engineering Thesis, University of Tokyo, 1970.
22
Table 1: Test Program

Test No. Brief Description
VT-1 Free and forced vibration tests
PSD-1 Pseudo-dynamic earthquake test
Modified Miyagi-ken Oki Earthquake (1978)
Tohoku University Record (NS), a
max
*= 23.5 Gal
R
max
**= 2.52 mm, S
max
***= 31.5 tonf
PSD-2 Pseudo-dynamic earthquake test
Modified Miyagi-ken Oki Earthquake (1978)
Tohoku University Record (NS), a
max
*= 105 Gal
R
max
**= 32.5 mm, S
max
***= 226 tonf
PSD-3 Pseudo-dynamic earthquake test
Modified Tehachapi Shock (1952)
Taft Record (EW), a
max
*= 320 Gal
R
max
**= 238 mm, S
max
***= 411 tonf
PSD-4 Pseudo-dynamic earthquake test
Modified Tokachi-oki Earthquake (1968)
Hachinohe Harbor Record (EW), a
max
*= 350 Gal
R
max
**= 342 mm, S
max
***= 439 tonf
VT-2 Free and forced vibration tests
Repair of test structure by epoxy injection
VT-3 Free and forced vibration tests
Placement of non-structural elements in test structure
PSD-5 Pseudo-dynamic earthquake test
Modified Miyagi-ken Oki Earthquake (1978)
Tohoku University Record (NS), a
max
*= 23.5 Gal
R
max
**= 3.03 mm, S
max
***= 26.7 tonf
PSD-6 Pseudo-dynamic earthquake test
Modified Miyagi-ken Oki Earthquake (1978)
Tohoku University Record (NS), a
max
*= 105 Gal
R
max
**= 65.3 mm, S
max
***= 234 tonf
PSD-7 Pseudo-dynamic earthquake test
Modified Tehachapi Shock (1952)
Taft Record (EW), a
max
*= 320 Gal
R
max
**= 244 mm, S
max
***= 452 tonf
SL Static test under uniform load distribution
R
max
**= 326 mm, S
max
***= 597 tonf
*a
max
: maximum acceleration of input ground motion
**R
max
: maximum roof level displacement
***S
max
: maximum base shear
23
Table 2: Properties of Reinforcing Bars


Bar
size
Nominal
Diameter
mm
Nominal
Perimeter
mm
Nominal
Area
mm
2

Yield
Strength
Kgf/cm
2

Strain
Hardening
Strain
Tensile
Strength
Kgf/cm
2


Fracture
Strain
D10 9.5 30.0 71 3870 0.018 5670 0.17
D16 15.9 50.0 199 3850 0.019 5720 0.18
D19 19.1 60.0 287 3650 0.017 5730 0.20
D22 22.2 70.0 387 3530 0.012 5750 0.21
D25 25.4 80.0 507 3780 0.022 5660 0.20




Table 3: Properties of Filed Cured Concrete

Story Test
Age
(days)
Compressive
Strength
(kgf/cm
2
)
Strain at
Compressive
Strength
Tensile
Strength
(kgf/cm
2
)
7 67 189 0.0019 13.2
6 87 144 0.0019 13.3
5 98 295 0.0019 23.6
4 111 290 0.0023 23.3
3 119 274 0.0023 22.8
2 132 292 0.0024 24.6
1 145 289 0.0022 24.2




Table 4: Skeleton Moment-rotation Relations of Beams

Stiffness Properties Top in Tension Bottom in Tension
Cracking Moment (tonf-m) 9.1 4.2 (6.6)*
Cracking Rotation (x10
-3
x

rad)
1.08 0.50 (0.79)*
Yield Moment (tonf-m) 43.9 9.8
Yield Rotation (x10
-2
x

rad)
1.08 0.52
Note: Elastic deformation included in rotation.


: span length of beam
*: average values used in the analysis

24
Table 5: Initial Axial Loads in Vertical Members
(a) Independent Columns, tonf
Story C
1
* C
1
’ C
2
C
3
* C
3

1 87.9 70.1 96.8 127.9 92.3
2 74.2 59.4 82.8 107.7 78.1
3 61.1 49.2 69.1 88.3 64.6
4 47.9 39.0 55.3 68.9 51.2
5 34.8 28.8 41.6 49.5 37.7
6 21.6 18.6 27.8 30.2 24.2
7 8.4 8.4 14.0 10.8 10.8
Note: Column notation given in Fig. 3.5
* Loading-side column carried additional weight of actuators and loading beam

(b) Shear Wall and Boundary Columns, tonf
Story Boundary
Column C
4
Wall
Panel W
1

7 97.5 87.2
6 83.6 73.8
5 69.9 61.2
4 56.2 48.6
3 42.5 36.0
2 28.8 23.4
1 15.1 10.8


Table 6: Calculated Stiffness Properties of Columns


Type

C
1


C
2


C
3

Story N,
tf
Mc,
tf-m
My
tf-m
y
θ

N
tf
Mc,
tf-m
My
tf-m
y
θ

N
tf
Mc,
tf-m
My
tf-m
y
θ

7 8.4 4.9 18.5 2.50 14.0 5.4 19.8 2.67 10.8 5.1 19.0 2.57
6 20.1 5.9 21.3 2.88 27.8 6.5 23.1 3.12 27.2 6.4 22.9 3.09
5 31.8 6.8 24.0 3.24 41.6 7.7 26.2 3.34 43.6 7.8 26.6 3.59
4 43.5 7.8 26.6 3.59 55.3 8.8 29.1 3.93 60.6 9.2 30.0 4.05
3 55.2 8.8 29.1 3.93 69.1 9.9 31.9 4.31 76.5 10.5 33.0 4.46
2 66.8 9.8 31.4 4.24 82.8 11.1 34.5 4.66 92.9 11.9 36.3 4.90
1 79.0 10.8 33.8 4.56 96.8 12.3 37.0 5.00 110.1 13.3 39.3 5.31
Yield rotation
y
θ
in 10
-3
rad for a unit length column.


Table 7: Axial Stiffness Properties for Shear Wall (Outside Truss Element)

Elastic Stiffness
Story
Compression (tonf/m) Tension (tonf/m)
Tension Yield
Load (tonf)
First Story 158,000 142,000 109.3
Second through
Seventh stories
198,000 178,000 109.3

25
Table 8: Stiffness Properties of Shear Wall (Central Element)

(a) Shear Stiffness Properties


Story
Elastic Shear
Rigidity Ks
(tonf/cm)
Cracking
Shear
(tonf)
Ultimate
Shear
(tonf)
Yield Story
Displacement
(mm)
First story 2,770 238 381 4.17
Second story
and above
3,400 238 381 3.33

(b) Axial Stiffness Properties
Elastic Stiffness


Story
Compression
(tonf/cm)
Tension
(tonf/cm)
Tensile Yield
Load
(tonf)
First story 5,690 5,120 121.4
Second story
and above
7,110 6,400 121.4

(c) Rotational Stiffness Properties
Story Elastic Rotation
Stiffness
(tonf-m/rad)
Cracking
Moment
(tonf-m)
Yielding
Moment,
(tonf-m)
Yielding
Rotation
X10
-5


rad
1 1920000 8.1 144.7 2.75
2 17.6 154.2 2.84
3 27.0 163.6 2.93
4 36.5 173.1 3.01
5 45.9 182.5 3.10
6 55.4 192.0 3.19
7


2400000
65.4 202.0 3.23


: span length




Table 9: Stiffness Properties of Vertical Spring for Transverse Beams

Elastic Spring
(tonf/cm)
Cracking Force
(tonf)
Yield Force
(tonf)
Yield Displacement
(cm)
14.88 2.0 5.6 1.16

26

(a) Typical Floor Plan (b) Typical Elevation

Fig. 1: Test Structure






Fig. 2: General Plan View and Frame Notations

27


Fig. 3: Elevation of Frame B



Fig. 4: Elevation of Frame 4
28


Fig. 5: Plan View of Foundation




Fig. 6: Plan View of Second through Seventh Floor Levels
29


Fig. 7: Plan View at Roof Level






Fig. 8: Reinforcement in Foundation Beams

30


Fig. 9: Reinforcement in Beams at Second through Roof Levels





Fig. 10: Typical Column Cross Section
31


Fig. 11: Slab Reinforcement




Fig. 12: One-component Model for Beams and Columns Fig. 13: Wall Model
32



Fig. 14: Transverse Beam Model




Fig. 15: Notation for Effective Width Evaluation




Fig. 16: Stress-strain Relationships assumed in the Flexural Theory
33


Fig. 17: Tributary Area for Gravity Axial Load Computation of Columns and Wall



Fig. 18: Notation for Shear Wall Section



Fig. 19: Takeda Hysteresis Model
34


Fig. 20: Observed Behavior of Beam-to-column Connection with Slab




Fig. 21: Takeda-slip Hysteresis Model
35

Fig. 22: Axial Stiffness Hysteresis Model





Fig. 23: Origin-oriented Hysteresis Model
36


Fig. 24: Observed and Calculated Response Waveforms of Test PSD-3




Fig. 25: Observed and Calculated SDF Hysteresis Relation of Test PSD-3
37


Fig. 26: Observed and Calculated Beam-end Rotation at Wall Connection (Test PSDD-3)
38


Fig. 27: Observed and Calculated Beam End Rotation in Frame A (Test PSD-3)
39


Fig. 28: Observed and Calculated Axial Deformation of Wall Boundary Columns (Test PSD-3)
40


Fig. 29: Calculated Member Forces at Maximum Structural Deformation in Test PSD-3






(a) Frames A and C (b) Frame B

Fig. 30: Calculated Ductility Factors at Maximum Structural Deformation in Test PSD-3
41



Fig. 31: Deflected Shape at Maximum Structural Deformation in Test PSD-3




Fig. 32 Observed and Calculated Response Waveforms in Test PSD-2
42


Fig. 33 Observed and Calculated SDF Hysteresis Relations in Test PSD-2


Fig. 34: Observed and Calculated Response Waveforms in Test PSD-4
43


Fig. 35: Observed and Calculated SDF Hysteresis Relations in Test PSD-4