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

Transcription

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

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. 2

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 Materials Deformed bars were used in the constructions of the test structure. The grade of reinforcing steel 3

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 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; [ m]{ x} + { f} = [ m]{1} 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, { f } = { v} f R (2) in which { v }: constant vector, each element of which represents the lateral resistance amplitude normalized to the roof-level resistance amplitude f R. 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 = {} uq (3) If the mode shape is normalized to the roof-level amplitude, the value of q represents a roof-level lateral displacement x R. 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 [ m]{ u} x + { v} f = [ m]{1} y (4) R R 4

5 Pre-multiplying {} u T to Eq. 4, Or, T T T {}[ u m]{} u x + {}{} u v f = {}[ u m]{1} y (5) R R mx + f = m( β y) (6) R T T in which m= {}[ u m]{} u : effective mass, f = {}{} u v f R : effective restoring force, and T β = {}[ u m]{1}/ m: 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 {} u T = [1.000, 0.850, 0.696, 0.540, 0.384, 0.234, 0.102] {} v T = [1.000, 0.862, 0.724, 0.586, 0.448, 0.310, 0.172] β = m = 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 ( 2 ) / q = q q + q t (7) i i+ 1 i i 1 in which, q : displacement at time step i, numerical integration. i q i : acceleration at time step i, and t : time increment for Equation 7 can be rewritten in a form, q = 2q q + t q (8) 2 i+ 1 i i 1 i 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 x (= q i + 1 ). 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 R ( = i 1 q + ), the resistance f R at the roof level was measured. The acceleration amplitude x R ( = q i + 1) was evaluated by Eq. 6 with given ground acceleration amplitude y i + 1. With a new acceleration value at time step i+1, Eq. 8 was used to calculate the displacement at further time step i+2. 5

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, q = q t m f t y (9) 2 2 i+ 1 2 i 1 ( / ) i ( β ) i or in an incremental form, q = q t m f t y (10) 2 2 i+ 1 i ( / ) i ( β ) i in which, qi = qi qi 1. 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 6

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 b in a T-shape member (one a 7

8 side) is ba = ( a/ ) a when a < 0.5 (11.a) b = 0.1 when a 0.5 (11.b) a 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 M c 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., M c cσ t Ze = (12) where c σ t : tensile strength of concrete (=20 kgf/cm 2 ), and steel. Z e : section modulus without reinforcing 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., α σb σ εb ε E = where α = σ B εb σb cε B (13) where σ, ε : compressive stress and strain, σ B : stress at compressive strength (=290 kgf/cm 2 ), strain at compressive strength (=0.0021), and E : initial tangent modulus (=2.37 x 10 5 kgf/cm 2 ). c 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 ε B : 8

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 C 1 and C 3 carried the weight of actuators and loading beams. For columns C 1 and C 1 ', or C 3 and C 3 ', 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 moment M y ; i.e., M c and yield Mc = cσ t Ze + N D/6 (14) M = 0.8 a σ D+ 0.5 N D(1 N / b D F ) (15) y t y c 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 F : compressive strength of concrete (=290 kgf/cm 2 ). c The area a t 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 ( M / θ ) at the yield point was proposed: y y M M N d 2 6EI / θ = ( n p )( ) ( ) QD bdf D y y t c (16) in which, M y : yield moment applied at two member ends, θ y : member end rotation at yielding, n: Young's modulus ratio ( = Es / Ec ), p t : tensile reinforcement ratio ( = at / bd), M/QD: shear 9

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 (= 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 (= EA / ) 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 K s was defined as GAw K = (17) s κ h in which, G: elastic shear modulus (=0.98 x 10 5 kgf/cm 2 ), A w : area of shear wall section (Fig. 18), 2 3 κ :shape factor for shear deformation (= 3(1 + u)[1 u (1 v)] / 4[1 u (1 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 Q c (in kgf), sqc = 1.4 FcAw (18) in which F c : compressive strength of concrete in kgf/cm 2 (=290 kgf/cm 2 ). s Hirosawa's empirical equation (13) was used to evaluate the ultimate shear resisting capacity Q (kgf); u ( F + ) Q p b j M / QL pt c 180 s u = σwh wh + 0.1σ0 e (19) D where p t : effective tensile reinforcement ratio (%), = 100 at / be( L ), a t : area of longitudinal 2 reinforcement in tension-side boundary column, M/QL: shear span-to-depth ratio, σ wh : yield strength 10

11 of horizontal reinforcement in the wall (kgf/cm 2 ), p wh : effective horizontal wall reinforcement ratio = a / b x, x : spacing of horizontal wall reinforcement (= 20 cm), σ 0 : average axial stress over wh e entire wall cross sectional area (Table 5.2.a), geometrical parameters defined in Fig. 18. b e : average width of wall section, 7 D j = ( L ), L,D: 8 2 Ratio β s of the secant stiffness at shear yield point to the elastic stiffness was determined empirically by β = 0.46 p σ / F (20) s wh wh c The shear stiffness reduction factor β s was approximately 0.16 for the shear wall analyzed. The stiffness after shear yielding was taken to be 0.1 % of the initial elastic shear rigidity. Calculated stiffness properties are listed in Table 8.a. Axial stiffness properties of the central vertical element (Table 8.b) were determined in the same way as the truss element. Area of a shear wall bounded by the inner faces of two boundary columns was used for the cross sectional area of the central vertical element. Rotational stiffness properties of the central vertical element (Table 8.c) were defined for wall area bounded by the inner faces of two boundary columns. Wall rotation was computed as the product of the curvature at base and the inter-story height. In other words, for the purpose of computing wall rotation, moment was assumed to distribute uniformly along the story height with an amplitude equal to the moment at wall critical section. Cracking was to occur when the extreme tensile fiber strain became zero under the gravity load and overturning moment; ( ) N ul M c = (21) 6 Yielding moment M y 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 K t was calculated for a fixed-fixed beam as 12EI K = (22) t 3 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. 11

12 The numerical values of the stiffness properties of the vertical spring are listed in Table 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 K is given by r K r F + F Dm = D + D D c y a c y y (23) in which ( Dc, F c) : cracking point deformation and resistance, ( Dy, F y) : yielding point deformation and resistance, D m : maximum deformation amplitude greater than D y, α : unloading stiffness degradation parameter (normally between 0.0 and 0.6). The general hysteresis rules are outlined in Fig. 19. The detail description of the model can be found in References 7 and 14. The Takeda hysteresis model was used in inelastic rotational springs of the independent column one-component model, and in a vertical spring of the transverse beam model. The unloading stiffness degradation parameter α for an independent column and a transverse beam model was arbitrarily chosen to be 0.4. 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 K s during slipping is a function of the maximum response point ( Dm, E m) and the point of load reversal ( D0, F 0) in the force-deformation plane (Fig. 21.d) 12

13 K s F D m m = Dm D0 Dm D0 γ (24) whereγ : reloading stiffness parameter. (c) After pinching, the response point moves towards the previous maximum response point with stiffness K ; p K p Fm = η( ) (25) D m where η : reloading stiffness parameter. In other words, the stiffness change occurs at an intersection of the two straight lines having slopes Ks and K p. 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, slipping stiffness degradation parameter γ, and reloading stiffness parameter η were 0.4, 1.0, and 1.0, respectively. 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 F y. 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 F using unloading stiffness K (Fig. 22.b) : y r K r D K max = c D yt a (26) where, D yt : tensile yielding point deformation, D max 13 : maximum deformation amplitude greater than D yt, α : unloading stiffness degradation parameter (=0.9). When the response point reaches the previous maximum tensile point 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 D : p D = D + β ( D D ) (27) p yc x yc where, β : parameter for stiffness hardening point (=0.2), D x : 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; q = q t m f t y (28) 2 2 i+ 1 i ( / ) i ( β ) i where qi = qi qi 1 : incremental displacement at roof level from time step i-1 to i, t : time T T increment, m : effective mass,( = {}[ u m]{} u ), f : effective resistance at time step i, ( = {}{} u v f R ), T β : effective participation factor,( = {}[ u m]{1}/ m), y i : 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. 0 (a) Calculate displacement vector { x } due to unit load {} v 14

15 0 1 { x } [ k] i { v} = (29) (b) Calculate displacement vector { x '} due to unbalanced force { F '}, at time step i, 0 1 { x } [ K] i { F'} i = (30) where[ K : tangent stiffness matrix evaluated at time step i. ] i Step 3: Determine incremental lateral resistance f Ri+ 1 at the roof level, 0 ( ' )/ f = q x x (31) Ri+ 1 i+ 1 Ri+ 1 R in which x ' and R 0 x R are the values of vectors { x '} and 0 { x } evaluated at the roof level. Step 4: Calculate displacement increment { x} i due to incremental load and unbalanced forces { F '}, 0 { x} i { x } fri+ 1 { x'} i+ 1 = + (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. 15

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 PSD 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 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. 16

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. 17

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. 18

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. 19

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, ) 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, ) Otani, S: Nonlinear Dynamic Analysis of Reinforced Concrete Building Structures, Canadian Journal of Civil Engineering, Vol. 7, No. 2, 1980, pp ) Umemura, H. and H. Takizawa: A State-Of-the-Art Report On the Dynamic Response of Reinforced Concrete Buildings, Structural Engineering Documents 2, IABSE, ) Giberson, M. F.: The Response of Nonlinear Multi-Story Structures Subjected to Earthquake 20

21 Excitation, Earthquake Engineering Research Laboratory, California Institute of Technology, Pasadena, California, EERL Report, ) 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 ) Otani, S. and M. A. Sozen: Behavior of Multistory Reinforced Concrete Frames during Earthquake, Structural Research Series No. 392, University of Illinois, Urbana, ) 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 ) 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 ) Takayanagi, T. and W. C. Schnobrich: Computed Behavior of Reinforced Concrete Coupled Shear Walls, Structural Research Series No. 434, University of Illinois, Urbana, ) 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, ) Architectural Institute of Japan: AIJ Standard for Structural Calculation of Reinforced Concrete Structures (Revised in 1982), ) 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, ) 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 ) 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, ) 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 l ) 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 ) 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 ) Sugano, S.: Experimental Study on Restoring Force Characteristics of Reinforced Concrete Members (in Japanese), Doctor of Engineering Thesis, University of Tokyo,

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 22

23 Nominal Diameter mm Nominal Perimeter mm Table 2: Properties of Reinforcing Bars Nominal Area mm 2 Yield Strength Kgf/cm 2 Strain Hardening Strain Tensile Strength Kgf/cm 2 Bar size Fracture Strain D D D D D Story Table 3: Properties of Filed Cured Concrete Test Age (days) Compressive Strength (kgf/cm 2 ) Strain at Compressive Strength Tensile Strength (kgf/cm 2 ) Table 4: Skeleton Moment-rotation Relations of Beams Stiffness Properties Top in Tension Bottom in Tension Cracking Moment (tonf-m) (6.6)* Cracking Rotation (x10-3 x rad) (0.79)* Yield Moment (tonf-m) Yield Rotation (x10-2 x rad) Note: Elastic deformation included in rotation. : span length of beam *: average values used in the analysis 23

24 Table 5: Initial Axial Loads in Vertical Members (a) Independent Columns, tonf Story C 1 * C 1 C 2 C 3 * C 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 Table 6: Calculated Stiffness Properties of Columns Type C 1 C 2 C 3 Story N, Mc, My θ N Mc, My y θ N Mc, My y tf tf-m tf-m tf tf-m tf-m tf tf-m tf-m θ y Yield rotation θ in 10-3 rad for a unit length column. y Table 7: Axial Stiffness Properties for Shear Wall (Outside Truss Element) Story Elastic Stiffness Tension Yield Compression (tonf/m) Tension (tonf/m) Load (tonf) First Story 158, , Second through Seventh stories 198, ,

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, Second story 3, and above (b) Axial Stiffness Properties Elastic Stiffness Tensile Yield Story Compression (tonf/cm) Tension (tonf/cm) Load (tonf) First story 5,690 5, Second story and above 7,110 6, Story (c) Rotational Stiffness Properties Cracking Moment (tonf-m) Elastic Rotation Stiffness (tonf-m/rad) Yielding Moment, (tonf-m) Yielding Rotation X10-5 rad : span length Table 9: Stiffness Properties of Vertical Spring for Transverse Beams Elastic Spring Cracking Force Yield Force Yield Displacement (tonf/cm) (tonf) (tonf) (cm)

26 (a) Typical Floor Plan (b) Typical Elevation Fig. 1: Test Structure Fig. 2: General Plan View and Frame Notations 26

27 Fig. 3: Elevation of Frame B Fig. 4: Elevation of Frame 4 27

28 Fig. 5: Plan View of Foundation Fig. 6: Plan View of Second through Seventh Floor Levels 28

29 Fig. 7: Plan View at Roof Level Fig. 8: Reinforcement in Foundation Beams 29

30 Fig. 9: Reinforcement in Beams at Second through Roof Levels Fig. 10: Typical Column Cross Section 30

31 Fig. 11: Slab Reinforcement Fig. 12: One-component Model for Beams and Columns Fig. 13: Wall Model 31

32 Fig. 14: Transverse Beam Model Fig. 15: Notation for Effective Width Evaluation Fig. 16: Stress-strain Relationships assumed in the Flexural Theory 32

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 33