Seismic Assessment of Asymmetric Buildings

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direction of seismic input at each nonlinear stage is proposed. ... Keywords: seismic assessment, asymmetric building, direction of seismic input, pushover ...
Seismic Assessment of Asymmetric Buildings Considering the Critical Direction of Seismic Input K. Fujii Chiba Institute of Technology, Japan

ABSTRACT: In this paper, a simplified seismic assessment procedure for asymmetric frame building considering the critical direction of seismic input at each nonlinear stage is proposed. In this procedure, the critical direction of seismic input which produces the maximum response at flexible side frame is assumed to coincide with the principal direction of the first modal response, which is evaluated based on pushover analyses results at each nonlinear stage, and the peak response of flexible side frame at each seismic intensity is estimated based on equivalent linearization technique. In the numerical example, the nonlinear time-history analyses of four-story reinforced concrete asymmetric frame buildings under various directions of seismic inputs. The results show that the upper bound of peak responses at flexible-side frame by given seismic intensity and arbitrary direction of seismic input can be predicted by the results obtained from proposed procedure. Keywords: seismic assessment, asymmetric building, direction of seismic input, pushover analysis

1. INTRODUCTION It is well accepted that asymmetric buildings are vulnerable to earthquakes. This is because the excessive deformation may occur at the flexible and/or weak side frame due to the unfavorable torsional effect. That may lead to the premature failure of brittle members and finally collapse of whole buildings. For the seismic assessment of asymmetric building, it is essential to carry out 3-dimensional analysis considering the all possible direction of seismic input. However it is very time-consuming task to evaluate seismic response under all possible seismic intensity by incremental dynamic analysis (IDA) (Vanvasikos and Cornell, 2002) for all possible directions of seismic input, because the horizontal ground motion may act in any horizontal direction. Another approach for the seismic assessment of asymmetric building would be the evaluation of seismic response for critical direction of seismic input, which would produce the maximum response. Dolšek and Fajfar proposed a simplified performance assessment procedure for asymmetric buildings (Dolšek and Fajfar, 2007). In this procedure, seismic performance of asymmetric building is assessed based on pushover analysis in each of main orthogonal axis of building. However, since the critical direction of seismic input may not coincide with the main orthogonal axis of buildings; it may depends on the nonlinear structural characteristics and also characteristics of seismic inputs. In this paper, a simplified seismic assessment procedure for asymmetric frame building considering the critical direction of seismic input at each nonlinear stage is proposed. In this procedure, the critical direction of seismic input which produces the maximum response at flexible side frame is assumed to coincide with the principal direction of the first modal response, which is evaluated based on pushover analyses results at each nonlinear stage, and the peak response of flexible side frame at each seismic intensity is estimated based on equivalent linearization technique. In the numerical example, the nonlinear time-history analyses of four-story reinforced concrete asymmetric frame buildings under various directions of seismic inputs. The relationship of seismic intensity and upper bound of peak response at flexible-side frame, which is often obtained by IDA, is also discussed.

2. THEORETICAL BACKGROUND 2.1. Equation of Motion of Equivalent SDOF Model Considering ξ-axis in X-Y plane and angle ψ as shown in Fig. 2.1(a), the equation of motion for N-story asymmetric building model can be written as follows:

 + Cd + f = −Mα a Md R ξ gξ

(2.1)

⎡M 0 M = ⎢⎢ 0 ⎢⎣ 0

(2.2)

0 M0 0

0⎤ 0 ⎤ 0⎤ ⎡ m1 ⎡ I1 ⎥ ⎢ ⎥ ⎢ ⎥ 0 ⎥ , M0 = ⎢ % % ⎥ , I0 = ⎢ ⎥ ⎢⎣ 0 ⎢⎣ 0 mN ⎥⎦ I N ⎥⎦ I 0 ⎥⎦

⎧ d = { x1 " xN y1 " y N θ1 "θ N }T ⎪⎪ T ⎨ f R = { f RX 1 " f RXN f RY 1 " f RYN f MZ 1 " f MZN } ⎪ T ⎪⎩α ξ = {cosψ " cosψ − sinψ " − sinψ 0" 0}

(2.3)

where M is the mass matrix, C is the damping matrix and is assumed to be proportional to the stiffness matrix, d and fR are displacement and restoring force vector at the center of mass, mi and Ii are the mass and moment of inertia, respectively of the i-th floor, and agξ is the ground acceleration in ξ-direction. The principal direction of the first modal response of N-story asymmetric building model T is determined based on the first mode vector φ1 = {φX 11 "φXN1 φY 11 "φYN1 φΘ11 "φΘN1} and its tangent is given as Eq.(2.4), which is discussed by the author (Fujii, 2010). tanψ 1 = −∑ m jφYj1

∑m φ

(2.4)

j Xj1

j

j

In this paper, U-axis is taken as the principal axis of the first modal response. The equivalent first modal mass of U-axis, M1U*, and the first modal participation factor Γ1U is determined by Eqs. (2.5) and (2.6). 2

2

⎛ ⎞ ⎛ ⎞ ⎜ ∑ m jφ Xj1 ⎟ + ⎜ ∑ m jφYj1 ⎟ j ⎠ ⎝ j ⎠ M 1U * = Γ1U φ1T Mα U = ⎝ 2 2 + + m φ m φ I ∑ j Xj1 ∑ j Yj1 ∑ jφΘj12 j

j

j

2

Γ1U

2

⎛ ⎞ ⎛ ⎞ m jφ Xj1 ⎟ + ⎜ ∑ m jφYj1 ⎟ ⎜ ∑ φ T Mα U ⎝ j ⎠ ⎝ j ⎠ = 1T = 2 2 φ1 Mφ1 ∑ m jφ Xj1 + ∑ m jφYj1 + ∑ I jφΘj12 j

j

(2.5)

(2.6)

j

α U = {cosψ 1 "cosψ 1 − sinψ 1 " − sinψ 1 0"0}

T

(2.7)

And equivalent displacement of the first modal response D1U* and equivalent acceleration of the first modal response A1U* is defined as Eq.(2.8).

Figure 2.1 Plan of asymmetric building model and corresponding equivalent SDOF model

D1U * =

Γ1U φ1T Md Γ1U φ1TfR * , A = 1U M 1U * M 1U *

(2.8)

It is assumed that the asymmetric building oscillates predominantly by the first mode and d and fR can be written in Eq.(2.9), even if the building responses beyond the elastic range.

d = Γ1U φ1 D1U * , fR = M ( Γ1U φ1 ) A1U *

(2.9)

By substituting Eq.(2.9) into Eq.(2.1), Eq.(2.10) is obtained.

 * ) + C ( Γ φ D * ) + M ( Γ φ ) A * = −Mα a M ( Γ1U φ1 D 1U 1U 1 1U 1U 1 1U ξ gξ

(2.10)

By multiplying Γ1U φ1T of Eq.(2.10) from the left side and considering Eqs. (2.11) through (2.13), the equation of motion of the equivalent SDOF model considering the principal direction of the first modal response is obtained as Eq.(2.14).

C1U * = Γ1U 2 ( φ1TCφ1 )

(2.11)

∑m φ

j Xj1

cosψ 1 =

φ1T Mα ξ φ1T Mα U

2

⎛ ⎞ ⎛ ⎞ ⎜ ∑ m jφ Xj1 ⎟ + ⎜ ∑ m jφYj1 ⎟ ⎝ j ⎠ ⎝ j ⎠

∑m φ = ∑m φ

j Xj1

j

j

j

∑m φ

j Yj1

j

2

,sinψ 1 = −

cosψ − ∑ m jφYj1 sinψ j

Xj1 cosψ 1 − ∑ m jφYj1 sinψ 1

j

2

⎛ ⎞ ⎛ ⎞ ⎜ ∑ m jφ Xj1 ⎟ + ⎜ ∑ m jφYj1 ⎟ ⎝ j ⎠ ⎝ j ⎠

2

= cosψ cosψ 1 + sinψ sinψ 1 = cos (ψ − ψ 1 )

(2.12)

(2.13)

j

= cos Δψ

 * + C * D * + M * A * = − M 1U * D 1U 1U 1U 1U 1U

φ1T Mα ξ T

φ1 Mα U

M1U *agξ = − M 1U *agξ cos Δψ

(2.14)

Fig. 2.1(b) shows the equivalent SDOF model representing the first modal response. It can be seen that the largest first modal response may occur in case of the direction of seismic input coincides with the principal direction of the first modal response (Δψ = 0).

2.2. Determination of the Capacity Curve of Building Considering the Principal Direction of the First Modal Response To estimate the upper bound of the peak response at flexible side frame, let we consider the most severe case for the first modal response; the largest first modal response occurs. It is assumed that the direction of seismic input coincides with the principal direction of the first modal response at each nonlinear stage (Δψ = 0). The equivalent acceleration A1U* − equivalent displacement D1U*, which is referred to as the capacity curve, is determined based on the pushover analysis considering the change of the first mode shape at each nonlinear stage. In the presented pushover analysis, which is referred to as “displacement-based mode-adaptive pushover analysis”, the following assumptions are made. 1)

2) 3) 4)

All beams, columns and structural wall are modelled as as one-component models with two nonlinear flexural springs and rigid zones at both ends and one nonlinear shear spring at the middle of line element. The envelope curve of each nonlinear spring of members is symmetric in positive and negative range. The equivalent stiffness of each nonlinear spring can be defined by their secant stiffness at peak deformation previously experienced in the calculation. The first mode shape at each loading stage n φ1 can be determined based on the equivalent stiffness. The deformation shape imposed on a model is same as the first mode shape obtained in 2) and 3).

Fig. (2.2) show the flow of displacement-based mode-adaptive pushover analysis applied in this paper. The equivalent acceleration and equivalent displacement at each loading stage, nA1U* and nD1U*, are determined by substituting displacement and restoring force vector, nd and nfR, respectively into Eq.(2.15).

D1U = *

n

∑(m j

j n

x j 2 + m j n y j 2 + I j nθ j 2 )

, n A1U = *

2

⎛ ⎞ ⎛ ⎞ ⎜ ∑ mj n xj ⎟ + ⎜ ∑ mj n yj ⎟ ⎝ j ⎠ ⎝ j ⎠

2

∑( j

n

f RXj n x j + n f RYj n y j + n f MZj nθ j ) 2

⎛ ⎞ ⎛ ⎞ ⎜ ∑ mj n xj ⎟ + ⎜ ∑ mj n yj ⎟ ⎝ j ⎠ ⎝ j ⎠

Figure 2.2 Flow of displacement-based mode-adaptive pushover analysis 

2

(2.15)

3. APPLICATION EXAMPLES 3.1. Building data The buildings investigated in the present study are two four-story asymmetric buildings, as shown in Fig. 3.1. The story height is 4.05 m for the first story and 3.60 m for the upper stories. The floor mass mi and moment of inertia Ii are assumed 630 t and 6.01 × 104 tm2, respectively. The columns are assumed to be supported as fixed-ends by the foundation. The compressive strength of the concrete σB is assumed to be 24 N/mm2. In addition, SD345 (yield strength: σy = 345 N/mm2) is used for the longitudinal reinforcement, and SD295 (σy = 295 N/mm2) is used for the shear reinforcement. Each frame structure is designed according to weak-beam strong-column concept; the longitudinal reinforcements of concrete sections are determined so that the potential hinges are located at all beam-ends and bottoms of columns and structural wall in the first story. Sufficient shear reinforcement assumed to be provided to prevent premature shear failure. Table 3.1 shows the longitudinal reinforcement of each member. The crack moment Mc and yield moment My of each concrete member are calculated according to the AIJ Design Guidelines (AIJ, 1999). The base shear coefficients obtained from planar pushover analysis in both the X- and Y-directions, which are the values when the roof displacement reaches 1% of the total height HN, are 0.536 (X-direction) and 0.597 (Y-direction). The building structure is modelled as pseudo-three dimensional frame models, in which the floor diaphragms are assumed to be rigid in their own planes, there is assumed to be no out-of-plane

Figure 3.1 Model building Table 3.1 Longitudinal reinforcement of member Member Location Boundary Beam 2 to R floor 4 to R floor Beam 2 to 3 floor 2 to 4 Story Column 1st Story Structural Wall All Story

Reinforcement 6-D25 (Top and Bottom) : SD345 3-D25 (Top and Bottom) : SD345 4-D25 (Top and Bottom) : SD345 20-D29 (Top and Bottom) : SD345 8-D29 (Bottom), 20-D29 (Top) : SD345 D10@200 Double : SD295

 

Figure 3.2 Hysteresis of nonlinear spring

stiffness, and the frames oriented in the X- and Y-directions are modelled independently. All beams, columns, and structural walls are modelled as one-component models with rigid zones to express the depth of intersecting members. To determine the flexibility of the nonlinear flexural springs, an anti-symmetric curvature distribution is assumed for beams and columns, whereas a uniform curvature distribution is assumed for structural walls. Fig. 3.2(a) shows the envelope curve of the force-deformation relationship of each nonlinear spring. The envelopes are assumed to be symmetric in both the positive and negative loading directions. In Fig 3.2(a-1), the secant stiffness degradation ratio of flexural spring at yielding point, αy, is assumed 0.25 for all beams and columns, while is it assumed 0.12 for the structural walls at the bottom of first story and 0.19 for others. The tangent stiffness degradation ratio after yielding α2 is assumed 0.01 for all beams and 0.001 for all columns and structural walls. In Fig 3.2(a-2), the secant stiffness degradation ratio at “yielding” point, βy’, is assumed 0.16. The axial stiffness of the columns and walls are assumed to remain elastic, and the effects of both biaxial bending and axial-flexural interaction are ignored. The torsional stiffness of members is also neglected. No second-order effect (e.g., P-Δ effect) is considered. The Muto hysteretic model (Muto et al., 1973) with one modification is used to model the flexural springs, as shown in Fig. 3.2(b-1). Specifically, the unloading stiffness after yielding decreases in proportion to μ−0.5 (μ: ductility ratio of the flexural spring) to represent the degradation of unloading stiffness after yielding of R/C members, as per the model employed by Otani (1981). The origin-oriented model (Fig. 3.2(b-2)) is used to model the shear spring of the structural wall. The shear springs of beams and columns are assumed to be elastic. The damping matrix is assumed to be proportional to the instant stiffness matrix and 3% of the critical damping for the first mode. Fig. 3.3 shows the mode shapes and natural periods in elastic range Tie (i = 1, 2, 3). In this figure, the angle of incidence of principal direction of modal response in elastic range ψie is also shown. As shown in this figure, the response of frame-Y6 is the largest in the first mode. Therefore, frame-Y6 may be considered as “flexible-side frame” in elastic range. This figure also shows that the angle between principal axes of the first and second modal responses is close to 90 degree. 3.2. Ground motion data In the present paper, the seismic excitation is considered unidirectional in X-Y plane, and three artificial ground motions are used. The target spectrum with 5% of critical damping SA0(T), which is prescribed in Japanese Building Code, is determined by Eq. (3.1): ⎧ 4.8 + 45T m s 2 ⎪ S A0 (T ) = ⎨12 ⎪12 ( 0.576 T ) ⎩

T < 0.16 s 0.16 s ≤ T < 0.576 s T ≥ 0.576 s

(3.1)

where T is the natural period of the SDOF models. The three of following records (first 60 seconds of major horizontal components) are used to determine the phase angle of artificial ground motions; El Centro 1940 (referred to as ELC), Hachinohe 1968 (HAC) and JMA Kobe 1995 (JKB). Fig. 3.4 shows the elastic acceleration spectra and demand curve (SA0 – SD0 relationship, where SD0 is displacement spectrum) of artificial ground motions with 5% of critical damping, and Table 3.1 lists the artificial

Figure 3.3 Mode shape in elastic range

Table 3.1 List of Artificial ground motion Max. Ground Phase Acc. Motion Angle (m/s2) ID Art-ELC ELC 4.494 Art-HAC HAC 4.499 Art-JKB JKB 5.753

  Figure 3.4 Elastic response spectra

  ground motions. In this paper, 3 x 24 = 72 cases are considered for each ground motion; the angle of incidence of ξ-axis (axis of seismic input) varies with interval of 15 degrees from ψ1e = −19.3 degrees, and artificial ground motions are scaled so that SA1(T1e) = λ SA0(T1e) is equal to 2.4m/s2, 7.2m/s2, 12.0m/s2, and 16.8m/s2 where λ is the scale factor and is 0.2, 0.6, 1.0, and 1.4, respectively. 3.3. Analysis results and discussions Fig. 3.5 shows the results of pushover analysis described in section 2.2. In Fig. 3.5(c), normalized displacement is defined as the absolute value of roof displacement at each frame divided by the equivalent displacement. In Fig. 3.5(a), the capacity curve (A1U* − D1U* relationship) of model building and demand curve (SA1(T1e) = 12m/s2) are shown. As shown in this figure, the equivalent linearization technique (Otani, 2000) is applied to predict the peak response of building in this paper. It can be seen from Fig. 3.5 (b) that the principal axis of the first modal response varies depend on D1U*; ψ1 varies significantly from −19.3 to −51.0 degrees. And from Fig. 3.5(c), it can be seen that the largest displacement occurs at frame Y6. Therefore in the following discussions, attention will be paid to the response of frame Y6, which is considered the critical one for the seismic assessment of the building considered. Fig. 3.6 shows the peak of roof displacement and drift at 2nd story of frame Y6 in comparisons with time-history analyses results (SA1(T1e) = 12m/s2, 24 cases of various direction of seismic input) and the predicted result by equivalent linearization technique. As shown in this figure, the predicted result is well agreed with the upper bound of peak drift obtained from time-history analyses results. It should be pointed out that the largest peak response occurs when ψ = 145.7 and 325.7 degrees, which is different from the main orthogonal axes of building (X- and Y-axis); it is between the principal axis of the first modal response correspond to elastic stage and predicted peak response. Similar observations can be made in case of SA1(T1e) = 2.4, 7.2 and 16.8m/s2. Fig. 3.7 compares the relationship of the seismic intensity SA1(T1e) and the peak of roof displacement and drift at 2nd story of frame Y6, obtained from time-history analysis and the predicted results. In this figure, the plots of time-history analyses results are the largest of those obtained from 24 cases of various direction of seismic input. As shown in this figure, the predicted result is approximated the upper bound of time-history analyses results.

Figure 3.5 Pushover analysis results

  Figure 3.6 Prediction of upper bound of peak roof displacement and drift at frame Y6(SA1(T1e) = 12m/s2)

  Figure 3.7 Relationship of seismic intensity and peak roof displacement and drift at frame Y6 

 

4. CONCLUSIONS In this paper, a simplified seismic assessment procedure for asymmetric frame building considering the critical direction of seismic input at each nonlinear stage is proposed. The results show that the upper bound of peak responses at flexible-side frame by given seismic intensity and arbitrary direction of seismic input can be predicted by the results obtained from proposed procedure. AKCNOWLEDGEMENT The author acknowledges support by Grant-in-Aid for Young Scientist (Category (B), No. 21760443), Japan Ministry of Education, Culture, Sport, Science and Technology (MEXT). REFERENCES Vanvasikos, D., Cornell, C. A. (2001). Incremental dynamic analyisis. Earthquake Engineering and Structural Dynamics. 31: 491-514. Dolšek, M., Fajfar P. (2007). Simplified probabilistic seismic performance assessment of plan-asymmetric buildings. Eartquake Engineering and Structural Dynamics. 36:13, 2021-2041. Fujii, K. (2010). Nonlinear static procedure for multi-story asymmetric frame buildings considering bi-directional excitation. Journal of Earthquake Engineering. (Accepted at Feburary 14, 2010 and in press). Architectural Institute of Japan (AIJ). (1999). Design Guidelines for Earthquake Resistant Reinforced Concrete Buildings Based on Inelastic Displacement Concept. Architectural Institute of Japan, Tokyo (in Japanese). Muto, K., Hisada, T., Tsugawa, T., and Bessho, S. (1973), Earthquake Resistant Design of a 20 Story Reinforced Concrete Buildings. Proceedings of the 5th World Conference on Earthquake Engineering. 1960-1969. Otani, S. (1981), Hysteresis Models of Reinforced Concrete for Earthquake Response Analysis. Journal, Faculty of Engineering, University of Tokyo. 36:2, 125-156. Otani, S. (2000) New seismic design provision in Japan. The Second U.S.-Japan Work-shop on Performance-Based Earthquake Engineering Methodology for Reinforced Concrete Structures, PEER Report 2000/10, 3-14.