By Sashi K. Kunnath,1 Associate Member, ASCE, Nader Panahshahi,2
and Andrei M. Reinhorn,3 Member, ASCE
ABSTRACT: The in-plane flexibility of floor-slab systems has been observed to influence the seismic response of many types of reinforced concrete buildings. The assumption of rigid floor diaphragms is often used to simplify engineering analyses without significant loss in the accuracy of seismic response prediction for most buildings. However, for certain classes of structures, such as long and narrow buildings (especially with dual-braced lateral load-resisting systems), and buildings with horizontal (T- or L-shaped) or vertical (setbacks or cross-walls) offsets, the effect of diaphragm flexibility cannot be disregarded. Moreover, if the floor slab panels experience cracking or yielding due to pronounced in-plane distortions, the seismic response of the entire building system may be significantly altered. This paper presents an simplified macromodeling scheme to incorporate the effect of inelastic floor flexibility in the seismic response analysis of RC buildings. The slab model includes effects of both in-plane flexure and shear. The inelastic behavior of diaphragms is emphasized through a study of narrow rectangular buildings with end walls. The study shows that the in-plane deflections of floor slabs impose a larger demand on strength and ductility of flexible frames than predicted values using the assumption of rigid or elastic slabs. These demands may in turn lead to a failure of the gravity-load supporting system. A quantitative estimate of this effect is presented in terms of the floor aspect ratios.
INTRODUCTION
Floor slabs in multistory buildings, which usually transmit gravity loads to the vertical structural system, are also required to transfer lateral inertia forces to the vertical structural system. In practical design, for simplicity of the analysis procedure and due to a lack of understanding of the in-plane behavior of floor slab systems (diaphragm action), floor slabs are frequently treated as perfectly rigid elements in their own planes.
The validity of the rigid-floor assumption was questioned as early as 1961 (Blume et al. 1961). Recent experience and research has clearly demonstrated the importance of the influence of flexible diaphragms on the seismic response of many types of buildings (Karadogan et al. 1978; Unemori et al. 1980; Nakashima et al. 1982; Button et al. 1984; Roper and Iding 1984; Jain 1984; Jain and Jennings 1985; Aktan and Nelson 1988; Liu and Yang 1988; Suto and Asayama 1988; and Lee and Moon 1989). Recently, Chen (1986) formulated a finite element model for slab behavior in the inelastic range and applied the technique to predict the cyclic behavior of independent slab components. However, such a scheme cannot be readily applied to the inelastic seismic analysis of entire building systems.
'Res. Asst. Prof., Dept. of Civ. Engrg., State Univ. of New York at Buffalo, Amherst, NY 14260.
2Asst. Prof., Dept. of Civ. Engrg., Southern Illinois Univ. at Edwardsville, Ed-wardsville, IL 62026.
3Prof., Dept. of Civ. Engrg., State Univ. of New York at Buffalo, Amherst, NY 14260.
Previous research on the influence of flexible floor diaphragms concludes, among other findings, that the influence of diaphragm flexibility is more pronounced for L- or Y-shaped plan buildings, and for long rectangular buildings, particularly, where a dual-bracing system consisting of flexible moment-resisting frames and stiff shear walls is used, and for setback building structures.
All previous analytical studies on diaphragm behavior of buildings were performed using elastic models. This overlooks a potentially serious problem if any cracking or yielding of the floor-slab system occurs. The severe damage to the Imperial County Services Building in California during the October 1979 earthquake and the collapse of the Fifth Avenue Chrysler Center, in Anchorage, Alaska, during the March 1964 earthquake have been attributed to the influence of flexible floor diaphragms (Jain and Jennings 1985). If indeed any distortion of the floor slabs did take place, the possibility that significant inelastic behavior (in-plane cracking or yielding) may have also occurred cannot be ruled out.
The purpose of this paper is, therefore, to study the behavior of flexible floor diaphragms in the inelastic range and thereby ascertain the eventual importance of diaphragm action and its influence in altering the seismic response characteristics of buildings. The flexibility of floor diaphragms (especially when cracking and yielding is expected) affects the seismic response of buildings in two predominant ways: (1) The distribution of the lateral forces to the vertical elements is altered; and (2) the dynamic characteristics of the building are influenced by local vibration modes of the floor systems. These effects are totally ignored in analyses when floor slabs are assumed to be perfectly rigid.
OBJECTIVES AND SCOPE
The primary objective of the study presented here are: (1) The development of an analytical tool that is capable of analyzing inelastic building systems in which the effects of inelastic in-plane slab flexibility have been considered without the extensive computational effort required by finite element models; and (2) the application of the model in parametric studies of the effect of diaphragm flexibility of dual-braced RC structures with a long rectangular plan. Consequently, the study attempts to highlight the importance of diaphragm action on a particular class of buildings and to establish the bounds in which diaphragm flexibility and inelasticity is critical in assessing building response.
The present study comprises the following developments and investigation:
1. Global modeling of building structures with inelastic flexible floor diaphragms in which consideration is given primarily to in-plane flexibility.
2. Hysteretic modeling of floor-slab systems. 3. Parametric studies of building systems to identify behavior patterns arising
from the influence of diaphragm flexibility.
This study is the first phase of a comprehensive study on the influence of flexibility and inelastic behavior near collapse of slabs. The analytical model developed herein has been used for the design of experimental models and
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a shaking table testing program using 1:6 scaled specimens, and preliminary experimental studies have validated the adequacy of the model (Panahshahi et al. 1989).
MODELING OF STRUCTURAL SYSTEM
An enhanced computer code for the inelastic damage analysis of RC (IDARC) frame shear-wall buildings has been developed (Park et al. 1987). Owing to its suitability for the modeling of large frame-wall structural systems, it was decided that an inelastic flexible diaphragm element be incorporated into the framework of the modeling scheme of ID ARC. Available programs such as COMBAT (1983), with the capability to model floor diaphragm elements, do not possess the features required for the inelastic modeling of RC structures. Some of the features of ID ARC that make it particularly suitable for the modeling of RC structures include: (1) A distributed flexibility model to construct the element stiffness matrix including the effects of spread plasticity; and (2) a versatile force-deformation hysteretic model incorporating stiffness degradation, strength deterioration, and pinching behavior.
In the revised modeling scheme of ID ARC-Version 2 (Reinhorn et al. 1988), a reinforced concrete building is idealized as a series of plane frames linked together by inelastic flexible floor slabs and transverse beams. Consequently, a building is modeled using the following element types: (1) Floor slabs; (2) beam-columns; (3) shear walls; (4) edge columns; and (5) transverse beams.
A discretized section of a building using all of the presented element types is shown in Fig. 1. Only two typical frames (frame / and frame j) are shown. A building can typically be subdivided into a number of such frames. Each frame in turn can have more than two column lines or more than a single wall. However, it must be remembered that the influence of floor flexibility is predominant only for large aspect ratios. The components of the building frame as shown in Fig. 1 are further isolated to show clearly the element degrees of freedom. The structural components are modeled using end nodes in which only the significant degrees of freedom were retained for simplicity of the model. Slab elements are modeled primarily as deep beams (in which the in-plane effect of the supporting beams are also included) with two in-plane degrees of freedom per node (a lateral displacement in the y-direction, and a rotational deformation in the z-direction). Main beams are modeled with two degrees of freedom per node in the vertical plane and consist of elements in the direction of the applied load history. Columns are modeled similarly to beams but have an additional degree of freedom per node to account for axial deformations. Shear wall elements are modeled as line elements at the wall centers with rigid beams at the top and bottom and axial inelastic springs at the edges (edge-columns). Three degrees of freedom are considered per node. In addition to elements in the plane of loading, transverse beam elements are also included in the modeling. In this case, direct stiffness contributions of the transverse beams (one vertical and one torsional for each node) are added to the corresponding degrees of freedom at the respective connecting frames.
Each frame has a unique lateral degree of freedom. This corresponds to the lateral movement of the vertical elements of each frame. Each vertical
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zfc TRANSVERSE BEAM
FIG. 1. Component Modeling Scheme Showing Degrees of Freedom for Typical Building
element in a given frame at a particular floor level is constrained to move through the same lateral displacement. This ensures the compatibility of lateral displacements at each frame and floor level. The in-plane rotations of slab elements are further constrained by the torsional stiffness of individual frames, the details of which are presented in a subsequent section.
All components of the building, with the exception of transverse beams, are modeled as fully inelastic elements with concentrated plasticity at member ends, and a distributed flexibility rule to account for the spread of plasticity, as shown in Fig. 2. A linear variation of flexibility is assumed in deriving the flexibility matrix. The incremental moment-rotation relationship is established from the integration of the M/EI diagram. Two possibilities
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AM;
£ ^ AM!
AM In
(Moment Distribution)
AMI
oHTa
^ * ^ " i
f (EDo
I (El)b
Case (a) Case(b)
(Flexibility Distribution)
FIG. 2. Model of Distributed Flexibility
arise, depending upon the location of the point of contraflexure (Fig. 2). Thus
A9a\ = /AM, AeJ 8VAM„,
where
ts=L / n /i:
./21 A
(1)
(2)
where fu = coefficients of the flexibility matrix; and L = length of the member under consideration. The element stiffness matrix is constructed from the flexibility relationship expressed in Eq. 1 and the use of force equilibrium across the member. Details of the formulation may be found in Rein-horn et al. (1988).
Modeling of Inelastic Floor Slabs In-plane deformations in floor slabs caused by inertial forces (diaphragm
action) are also accompanied by out-of-plane bending due to gravity load. In modeling this behavior of the slab system, a macromodel approach is used: The slab element itself is modeled as a deep beam in the in-plane direction, while an empirical approach (based on observed experimental data) is used to alter the in-plane response to account for the effects of out-of-plane loading, the details of which are presented later in this paper.
A typical floor slab element connecting two parallel frames is shown in Fig. 3(a). Two degrees of freedom per node are assumed: an in-plane rotation and a lateral translation. In essence, this results in a deep-beam model. The similarity to shear walls comes from the fact that in-plane shear is modeled separately using an inelastic spring, which is then connected in series to the flexural spring. Consequently, a shear-type failure, otherwise not ac-
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AMb ,A6b
f30f
FIG. 3. In-Plane Modeling of Inelastic Floor Diaphragm: (a) Typical Floor Slab System; (b) Frame Distortion; (c) Modeling of Torsion for Single Frame
counted for in flexural models, is taken into account. The inclusion of the inelastic shear spring in series with the flexural spring necessitates the following modification of the flexibility matrix:
/ * = / . + 1
GA*L 1 - 1
- 1 1 (3)
where f8 = flexibility matrix of Eq. 2; G = shear modulus; and A* = effective shear area of the section.
Effects of Frame Torsion Any floor slab system that undergoes in-plane bending also experiences a
certain amount of twisting due to differential movement of the slab edges [Fig. 3(b)]. The effect of the torsional resistance of the frames on the in-plane rotation of the slabs depends on the relative stiffness of the horizontal and vertical structural systems. Generally the effect of frames in restraining the floor slab system from in-plane rotation is negligible and can be ignored; however, the influence of solid shear walls arranged in the perpendicular
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direction to the lateral loading can result in considerable rotational restraint for the floor slab, which must be included in the analysis. Modeling of the torsional restraint is achieved in the two-dimensional scheme of IDARC in the following manner.
The rotation of the slab system takes place about the center of stiffness of the frame axis, which in the example shown in Fig. 3 corresponds to the center of the frame. For a rotation 6/ about the center of stiffness, the frame moment Mf is given by
Mf=kf% (4)
The restraint provided by the columns due to the longitudinal deflection shown in Fig. 3(c) is evaluated as
P' = 3{f>J^ (5)
where EI and h = flexural rigidity and height of the vertical element, respectively.
The stiffness coefficient is then determined for a unit rotation taking into account the total moment about the center of stiffness of the frame axis
kf = -ZPilf, (6)
where .P, is obtained from Eq. 5 by setting 0/ = 1. These stiffness coefficients are determined frame by frame at each floor level and are added to the in-plane rotational stiffness contributions of the independent slab elements. This provides the necessary coupling between the longitudinal frame movements and the in-plane rotational degrees of freedom of the floor slabs. However, the inelastic biaxial interaction resulting from this bidirectional movement is not considered in the present formulation since, in the context of the floor flexibility problem being investigated in this study, such inelastic interactions are expected to be minimal.
Beam-Column Element Model Main beam-column elements form a vertical plane in the axis of loading.
They are modeled as continuous inelastic shear-flexure springs in which shear-deformation effects have been coupled by means of an equivalent spring. A typical element with rigid panel zones is shown in Fig. 4. Axial deformation effects are included in columns but ignored in beams. However, the effect of the variation of axial load on flexural capacity of the columns during the dynamic analysis is not considered.
Shear Wall Element Model The modeling of shear wall elements is similar to that for floor slabs,
enhanced by: (1) The inclusion of axial effects; and (2) the incorporation of edge columns at the ends of the wall. Walls may, however, be modeled with or without edge columns. Alternatively, the edge columns may be included only for strength computations in setting up envelope curves. The ability to treat each wall as an equivalent column with inelastic axial springs at the edges allows for the bending deformation of the wall element to be caused by the vertical movements of the boundary columns. Such a scheme has been used earlier in analytical studies by Kabeyasawa et al. (1983).
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T) SHEAR
MOMENT 4>
AXIAL
FORCE
X„=I9_ H = — rb
FIG. 4. Beam-Column Component with Rigid Zones
Transverse Beam Elements To incorporate the effects of transverse elements so as to account for their
restraining action due to the axial movements of vertical elements, especially edge columns in shear walls, and to permit flexural-torsional coupling with main elements, each transverse T-beam is modeled using elastic springs with one vertical and one rotational (torsional) degree of freedom as shown in Fig. 1. Transverse elements are basically of two types: beams that connect to shear walls and beams connected to the main beams, i.e., in the direction of loading.
HYSTERETIC MODELING OF COMPONENTS
The hysteretic model that was used for the analysis uses three parameters in conjunction with a nonsymmetric trilinear curve to establish the rules under which inelastic loading reversals take place. Details of the general meaning and effect of the parameters can be found in Kunnath et al. (1990). The three main characteristics represented in the model are: stiffness degradation a, strength deterioration p, and bond-slip or pinching y. These hysteretic parameters can be combined in various ways to achieve a range of hysteretic behavior patterns typical of reinforced concrete sections. The essential details of the hysteretic modeling parameters used to simulate inelastic behavior of the various components are described in the following sections.
Beam-Columns and Shear Walls The inelastic properties for the hysteretic behavior of beam-columns and
shear walls have been assigned based on observed behavior using past experimental data of typical components. Importance was placed on the relative strength of components, reinforcement ratios, and effective shear spans. Fig. 5 shows the behavior of typical beam-columns and walls using the selected parameters. For beam-columns, only the flexural spring is effective during analysis. For shear-walls, an additional shear spring is connected in series with the flexural spring. The coupling of shear and flexure in this manner is similar for both walls and slabs. Consequently, the hysteresis model shown in Fig. 5(b) is used for the shear spring in walls and slabs.
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Moment Shear
(a)
Curvature
a
P v
- 2.0 = 0.1 - 1.0
(b)
a = 0 .25 [J = 0.20 Y - 1.0
FIG. 5. Hysteretic Modeling of Beam-Columns and Walls Using Three-Parameter Model: (a) Flexural Spring; (b) Shear Spring
Floor Slabs Since available experimental data on inelastic behavior of diaphragms are
limited to the work done at Lehigh University, the hysteretic modeling of floor slabs has presently been established through the interpretation of these results (Nakashima et al. 1981, 1982). The test specimens used in the Lehigh tests were developed as scaled models of beam-supported floor slab systems in typical medium-rise buildings. Essential details of the test setup and specimen are shown in Fig. 6(a). The specimens were subjected to inelastic in-
L
LOAD (P)
v
iA-
39.51 J, r.
- I * 1/4
(b) -4.0 0.0 4.0
DISPLACEMENT (MM)
135.3 ^
* -* +-* 74 1562 68
dimensions in m m reinf. rat ios: r-]= 0.005
r „ = 0 . 0 0 7
f l - 4.0 kai Ec - 3000 kai €. - 0.003
f s = 6B.0 kai
E^- 28000 kai
50
0
- 5 0
-100
(c) -4.0 0.0 4.0
DISPLACEMENT (MM)
-
- ^&?w/ a - 10.5 8 - 0.33 y = 0.90
FIG. 6. Simulation of Hysteretic Behavior of Slabs: (a) Test Setup; (b) Experiment; (c) Analytical Solution (Nakashima et al. 1981)
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plane cyclic loading with and without superimposed gravity loads. It was observed that the out-of-plane bending induced by gravity loads reduced the ultimate in-plane capacity by 10-15%. Also, the deviation from elasticity occurred more gradually in the specimen with gravity load, and at an earlier stage (starting at approximately one-third of the ultimate capacity of the section).
The experimental force-deformation hysteresis obtained for the specimen with superimposed gravity load is shown in Fig. 6(b). A simple identification study was carried out to simulate this behavior using the three-parameter model. First, a fiber model analysis of the slab section is carried out to obtain the monotonic envelope, which is modified to include the effect of out-of-plane bending (as described previously). This envelope is then used in conjunction with appropriate parameters (in this case obtained by trial and error) to simulate hysteretic behavior of the test specimen. The shear spring hys-teretic model used for walls [Fig. 5(b)] is also utilized for slabs, primarily due to lack of information on the hysteretic characteristics of the shear behavior of slab panels. The analytical loops generated using the selected parameters are shown in Fig. 6(c), which is seen to adequately reproduce the slab panel behavior.
RESPONSE OF RECTANGULAR BUILDINGS WITH STIFF END WALLS
The analytical model developed herein is used to study the seismic response of low-rise multistory buildings with a rectangular plan. In particular, the effect of varying aspect ratios, the number of stories, and the use of different slab modeling schemes is investigated.
Test Structures Typical structures in which the slab panels are supported along their four
edges by monolithic concrete beam-column frames with shear walls at the ends are considered. Square slabs measuring 24 ft in each direction are used, with a 4-ft overhang on all noncontinuous sides. A single bay is used in the transverse direction, while multiple bays are chosen in the longitudinal direction.
The buildings are designed for combined gravity and seismic loads. Service gravity loads consisted of self-weight plus a live load of 80 psf. The structure is designed to satisfy the requirements of ACI Standard 318-83 (1986). The seismic design load is selected in accordance with the Zone 4 classification of the Uniform Building Code (1985). Concrete strength of 4,000 psi and Grade 40 reinforcement is used. The columns are 18 in. by 18 in., the beams are 22 in. deep and 12 in. wide, and the slab thickness is 7 in. The first story height is 18 ft, while the remaining stories are 12 ft, respectively. A plan and elevation of a typical eight-span three-story building is shown in Fig. 7.
The gravity load, dead load plus 25% of floor live load, is lumped at each transverse frame in accordance with its tributary area. The first 20 seconds of the 1940 El Centra normalized earthquake accelerogram, N-S component, with a peak acceleration of 0.5 g, is used as the input ground motion.
A total of 15 cases were analyzed. The main parameters considered were:
1. The number of bays spanning along the longitudinal direction, perpendicular to the direction of ground motion.
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(a) wall (12"X300") column (18"X18")
. ^ _ J _ 1...
—-f—
-—f-—-
.....*....
1 _
--—?-—
—-f-—
.....*.—-
_ — A . . . . 1
1
—-+—-
— - • + - - - 1
— - A -, 1 r
(W / slab (7" thick) beam (12"X22")
ihiii in 1 r
in 1
| n
U 1
1
a
II
1
1
8@24' = 192'
CM
N
h
FIG. 7. Structural Layout and Dimensions of Eight-Span Three-Story Structure: (a) Plan; (b) Elevation
2. The in-plane flexibility model used for the floor diaphragm. 3. The number of stories.
For the present study, four-, six-, and eight-span single-story structures and eight-span two- and three-story structures were analyzed. In each case, the behavior of floor diaphragms were modeled as rigid, elastic, and inelastic, respectively, while the remaining components (except for the transverse elements) were always modeled as inelastic elements. The reinforcement details of components used in the analysis for all the respective structures are given in Table 1 and Fig. 8.
TABLE 1
Building description
(D One-story
(4, 6, 8-span)
Two-story (8-span)
Three-story (8-span)
Story number
(2)
1 2 1 3 2 1
. Reinforcement Details of Components Used In Analysis
Columns
Total longitudinal steel area
(%) (3)
1.23 2.17 1.56 2.17 1.56 1.56
Effective
of transverse hoops at member ends (%)
(4)
1.7 2.7 2.7 2.7 2.7 2.7
Top steel area
(sq in.) (5)
2.37 4.53 5.06 4.53 5.06 6.00
Beams
Bottom steel area
(sq in.) (6)
1.58 2.00 2.00 2.00 2.00 2.00
Stirrups at end regions
(7)
#4 @ 4.75 in. #4 @ 4.75 in. #4 @ 4.75 in. #4 @ 4.75 in. #4 @ 4.75 in. #4 @ 4.75 in.
Walls
Horizontal reinforcement
ratio (%) (8)
0.26 0.26 0.26 0.26 0.26 0.26
Vertical reinforcement
ratio (%)
(9)
0.67 0.75 0.75 0.75 0.75 0.75
Note: Floor slab reinforcement ratios are as follows: For transverse steel, interior panels = 0.23% and exterior panels 0.20%; for longitudinal steel, n = 0.21%; r2 = 0.91%; and r3 = 0.32% (see Fig. 8 for details).
8R = rigid; FE = flexible elastic; FI = flexible inelastic. bM, My = in-plane moment and corresponding yield moment. Note: Number in parenthesis = time of occurrence.
Results and Discussion Overall Response
Peak values of top-story displacement of the middle and end frames, respectively; total base shear for the structure, end wall, and middle frame;
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TIME (SEC)
FIG. 9. Top Floor Lateral Displacement History at Middle Frame for Three-Story Eight-Span Structure (Note: 1 in. = 25.4 mm)
and in-plane slab and wall moments normalized with respect to corresponding yield moments are shown in Table 2. Also shown is the fundamental period of the structures for all cases analyzed. It is observed that the maximum wall moments occur at the base of the structure, while the peak slab moment is experienced in the interior panel near the middle frame. The overall displacement of the structure is controlled by the rigidity of the floor slab system and the end walls. The deflection of rigid diaphragms (which have a large but not infinite rigidity) is almost identical to the deflection of the end walls. However, when the diaphragm is flexible, the deformations in the center of the structure are substantially larger than at the ends (see columns 4 and 5 of Table 2). The increase is more accentuated with an increase
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o 6 -H-Z bJ 4
UJ
o ^ 2 0_ V} -f Q ,
3RD FLOOR SUB MODEL: RIGID 2ND FLOOR 1ST FLOOR
(a) WALL
1 1 1— 3 4 5
INTERIOR FRAMES WALL
1— H
Z LJ 4 -UJ . O _ ^ 2 -CL
Q n
3RD FLOOR 2ND FLOOR
_ - • -
1 1 1 1 2 3
I 4
SUB MODEL: EUSTIC
* • • _ .
I I i 5 6 7
(b) WALL INTERIOR FRAMES WALL
c^6
u 4. Ld
o 0_
---3RD FLOOR — 2ND FLOOR — 1ST FLOOR ..
SUB MODEL: INEUSTIC
(c) WALL 3 4 5
INTERIOR FRAMES WALL
FIG. 10. Floor Deformations at Peak Top Displacement for Three-Story Eight-Span Structure (Note: 1 in. = 25.4 mm)
in the number of longitudinal spans and a decrease in the number of stories when an inelastic slab model is used. This can be attributed to the inelastic behavior of the slab panels and end walls, and to their interaction with each other. For example, in the eight-span single-story structure, the slab has just reached its yield capacity, while the end wall moment is only at 0.22 of its yield capacity, which is even below its cracking moment (0.6 My). However, as the number of stories increases, the end walls begin to crack (two-story structure) and then yield (three-story structure). This results in a progressive reduction of the ratio of middle-to-end frame displacements.
The middle frame displacement history at the top floor level for the three-story structure is plotted in Fig. 9 for the different slab models. Note that
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7.0
5.0
^ 3.0
U J - I . 0 H > <l -3 .0 UJ ^ - 5 . 0
-7 .0
(a)
SLAB MODEL - ELASTIC
W^^V//V^-^/*WNV'WJ^W^*rwrr-gywr
8 12 TIME (SEC)
16 20
TIME (SEC)
FIG. 11. Relative Displacement History between Middle Frame and End Wall at Top Floor for Three-Story Eight-Span Structure (Note: 1 in. = 25.4 mm)
the maximum displacement predicted by the inelastic floor model is 5.88 in., which is 3.24 and 2.77 times the values obtained for the rigid and elastic slab model analyses, respectively. The difference in floor deformations predicted by each of the slab models is further shown in Fig. 10 and Fig. 11. Floor displacements at each story level obtained from analyses using rigid, elastic, and inelastic slab models are plotted in Fig. 10. The relative displacement history between the middle frame and the end walls at the third floor is compared for the elastic and inelastic slab model in Fig. 11. These figures clearly show the inadequacy of a rigid or elastic slab model in the prediction of floor deformations during seismic action.
The maximum base shear of the middle frame predicted by the inelastic slab model is considerably higher than for the elastic case, and even more so than for the rigid floor assumption (see column 8 of Table 2). Again, the difference is more pronounced as the number of spans increases and the number of stories decreases.
Using the elastic slab model results in higher structure base shear than
"R = Rigid; FE = flexible elastic; FI = flexible inelastic.
rigid analysis (column 6, Table 2). This is due to the increase in the natural period of the structure (see column 11, Table 2) caused by the flexibility of floor diaphragms, and its effect in amplifying the response for the El Centro 1940 excitation used in the analysis. Furthermore, the effect of inelastic diaphragm behavior reduces the structure base shear (column 6 of Table 2) compared to the elastic slab model. This is due to dissipated energy caused by hysteretic action of the slab panels. At the same time, however, the excursion of the slab panels into the inelastic range results in increased redistribution of base shear toward the middle frames.
Base Shear Distribution The distribution of the maximum building base shear between the end
walls and the interior frames is shown in Table 3 for all cases analyzed. Although the walls carry a major portion of the base shear (column 5, Table 3), the assumption of rigid or elastic diaphragms underestimates the base shear forces of the interior frames (column 7, Table 3). Using the elastic slab model, the total base shear resisted by the interior frames are 2 .59 -7.39 times the values obtained from the rigid floor assumption. Assuming an inelastic model results in even higher ratios (3.77-11.5) of base shear compared to rigid floor assumptions for the structures considered in this study. It is seen that the lower ratios correspond to the three-story eight-span structure.
The percentage of the total base shear distributed to the interior frames (column 8, Table 3) indicates that as the number of spans increases, the share of the total base shear resisted by the interior frames increases rapidly (6.7%, 24.5%, and 37.7% of structure base shear for four-, six-, and eight-span single-story structures, respectively). This share gradually decreases as
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40
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K °
In-20
-40
,
SLAB MODEL-RIGID
'
1 , 1 ,
(a) - 2 - 1 0 1
DISPLACEMENT (IN)
(b) ~2 DISPLACEMENT (IN)
(c) DISPLACEMENT (IN)
FIG. 12. Base Shear versus Top Displacement of Middle Frame for Single Story Eight-Span Structure (Note: 1 kip = 4,450 N; 1 In. = 25.4 mm)
the number of stories increases (26.2% and 21.4% for eight-span two-story and three-story structures, respectively). However, the percentage of the total base shear distributed to the interior frames by elastic or rigid slab models is significantly less than the values obtained from inelastic analyses.
The redistribution of the base shear to the internal frames due to inelastic behavior of the slab panels imposes a larger demand on both strength and ductility of the flexible interior frames. This is clearly shown in Fig. 12 where the base shears versus the top story displacements of the middle frame
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for the single-story, eight-span structure are shown for rigid, elastic, and inelastic slab models, respectively.
It is important to emphasize that the redistribution of the base shear to the internal frames can require frame shear capacity in excess of that normally specified by seismic building codes. UBC (1985), ATC (1984), and ANSI (1982) specify that for a dual system consisting of ductile moment-resisting frames and shear walls, the frames shall have the capacity to resist at least 25% of the required lateral loads. Analytical results presented in this paper clearly indicate that for one- and two-story structures with a floor aspect ratio (ratio of building length between walls to floor slab width) of 4:1 and greater, this limit can be exceeded. Neglecting the inelastic behavior of diaphragms can result in a nonconservative design of interior frames of dual system RC buildings, leading to possible damage of interior slab panels and frames.
CONCLUSION
An enhanced modeling scheme is developed for the inelastic response evaluation of RC building structures that accounts for the inelastic in-plane deformations of floor-slab systems. The proposed model, which is essentially two-dimensional with coupling of pseudo-three-dimensional effects, can be used for regular rectangular floor layouts, as well as for nonregular buildings with horizontal and vertical offsets. The suggested model accounts for the variation of the flexibility of the slab during the inelastic response caused by changes in the plastic regions. Slabs with variable thickness and reinforcement can be used. The slab model includes influences of both in-plane shear and flexural bending. These influences are considered separately and are combined in the analysis stage to allow for the detection of both flexural and shear failures.
The influence of diaphragm flexibility on the seismic response of RC buildings with end walls is investigated using the developed model. The study indicates that the in-plane deflections of floor slabs, in the inelastic region, impose a larger strength and ductility demand on the interior frames than that predicted by the rigid or elastic slab assumption. This results in a nonconservative design of flexible frames in frame-wall structures. In the event of a strong ground motion, this can lead to severe damage of the frames and eventual loss of the vertical load-carrying capacity of the columns.
Although the effect of inelastic floor flexibility has not yet been systematically identified as the cause of structural failure in recent earthquakes, the present study clearly indicates that the distinct possibility of building collapse, as a result of diaphragm yielding, exists for the class of buildings discussed in this paper. The developed model can be utilized in extensive parametric evaluation of existing long-span reinforced concrete frame-wall construction to develop appropriate design guidelines to avoid the adverse consequences of diaphragm yielding.
ACKNOWLEDGMENTS
This study was sponsored by the National Center for Earthquake Engineering Research (NCEER), which is supported by the National Science
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Foundation (Grant No. ECE-86-07591) and the state of New York. This work is part of a larger study on inelastic floor diaphragms done in collaboration with Profs. L.-W. Lu and T. Huang from Lehigh University. Their cooperation is gratefully acknowledged. The writers also wish to thank Lee Fang for assisting in the numerical testing of some of the examples presented in this paper.
APPENDIX I. REFERENCES
Applied Technology Council. (1984). "Tentative provisions for the seismic regulations of buildings." ATC-3-06, Applied Technology Council, Palo Alto, Calif.
American Concrete Institute. (1986). "Building code requirements for reinforced concrete." ACI-318-83, American Concrete Institute, Detroit, Mich.
American National Standards Institute. (1982). "Minimum design loads for buildings and other structures." ANSI-A, ANSI A58.1, American National Standards Institute, New York, N.Y.
Aktan, A. E., and Nelson, G. E. (1988). "Problems in predicting seismic responses of RC buildings." J. Struct. Engrg., ASCE, 114(9), 2036-2056.
Blume, J. A., Sharpe, R. L., and Elsesser, E. (1961). A structural dynamic investigation of fifteen school buildings subjected to simulated earthquake motion. Division of Agriculture, Sacramento, Calif.
Button, M. R., Kelly, T. E., and Jones, L. R. (1984). "The influence of diaphragm flexibility on the seismic response of buildings." 8th WCEE, Vol. IV, San Francisco, Calif.
Chen, S.-J. (1986). "Reinforced concrete floor slabs under in-plane monotonic and cyclic loading," thesis presented to Lehigh University, at Pennsylvania, Pa., in partial fulfillment of the requirements for the degree of Doctor of Philosophy.
International Conference of Building Officials. (1985). Uniform Building Code (UBC), Int. Conf. of Building Officials, Whittier, Calif.
Jain, S. K. (1984). "Seismic response of buildings with flexible floors." J. Engrg. Mech., ASCE, 110(1), 125-129.
Jain, S. K., and Jennings, P. (1985). "Analytical models for low-rise buildings with flexible floor diaphragms." Earthquake Engrg. Struct. Dyn., 13, 225-241.
Kabeyasawa, T., Shiohara, H., Otani, S., and Aoyama, H. (1983). "Analysis of the full scale seven story reinforced concrete test structure." J. Faculty of Engrg., Univ. of Tokyo, Tokyo, Japan, 27(2).
Karadogan, H. F., Nakashima, M., Huang, T., and Lu, L. W. (1978). "Static and dynamic analysis of buildings considering the effect of floor deformation—A state-of-the-art survey." Fritz Engineering Laboratory Report No. 422.2, Lehigh University, Pennsylvania, Pa.
Kunnath, S. K., Reinhorn, A. M., and Park, Y. J. (1990). "Analytical modeling of inelastic seismic response of RC structures." J. Struct. Engrg., ASCE, 116(4), 992-1017.
Lee, D.-G., and Moon, S.-K. (1989). "Analysis of multistory building structures with flexible floor diaphragms." Proc, 2nd East-Asia Pacific Conf. on Struct. Engrg. and Const., Chiang Mai, Thailand, Vol. 1, 799-804.
Liu, D., and Yang, C. (1988). "The effect of stiffness degradation of shear wall and floor horizontal deformation on the flame-shear wall buildings." Proc, 9th World Conf. on Earthquake Engineering, Tokyo, Japan, Vol. 6, 17-22.
Nakashima, M., Huang, T., and Lu, L.-W. (1981). "Experimental study of seismic behavior of reinforced concrete floor systems: Beam supported slab structures." Fritz Engineering Laboratory Report, No. 422.7, Lehigh Univ., Pennsylvania, Pa.
Nakashima, M., Huang, T., and Lu, L.-W. (1982). "Experimental study of beam-supported slabs under in-plane loading." ACI Struct. J., (Jan.-Feb.), 59-65.
Park, Y. J., Reinhorn, A. M., and Kunnath, S. K. (1987). "IDARC: Inelastic damage analysis of reinforced concrete frame—Shear-wall structures." Technical Re-
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port No. NCEER-87-0008, National Center for Earthquake Engineering Research, State Univ. of New York at Buffalo, Buffalo, N.Y.
Reinhorn, A. M., Kunnath, S. K., and Panahshahi, N. (1988). "Modeling of RC building structures with flexible floor diaphgrams (IDARC2)." Technical Report No. NCEER-88-0035, National Center for Earthquake Engineering Research, State Univ. of New York at Buffalo, Buffalo, N.Y.
Roper, S. C , and Iding, R. H. (1984). "Appropriateness of the rigid floor assumption for buildings with irregular features." Proc, 8th World Conf. on Earthquake Engineering, San Francisco, Calif., 751-758.
Suto, F., and Asayama, S. (1988). "Experimental considerations on earthquake behaviors of large long strip-type of actual structures." Proc, 9th World Conf. on Earthquake Engrg., Tokyo, Japan, Vol. V, 503-508.
Unemori, A. L., Roesset, J. M., and Becker, J. M. (1980). "Effect of in-plane floor slab flexibility on the response of crosswall buildings." Reinforced concrete buildings subjected to wind and earthquake forces, ACI-Special Publication No. 63, American Concrete Institute, Detroit, Mich.
APPENDIX II. NOTATION
The following symbols are used in this paper:
A* = effective shear area; EI = flexural rigidity; fy = flexibility coefficients; fs = flexibility matrix; G = shear modulus; h = height of vertical elements of frame below slab level; kf = frame stiffness; L = clear span of member; If, = distance from frame center to column position (';
Mf = frame moment; P, = lateral force on column ;';
&Mab = moment increment at ends of component; A6„it = rotation increment at ends of component;
6/ = frame rotation; and Xaii = rigid zone ratios at ends of component.
