A Method of Dynamic Nonlinear Analysis for RC Frames under Seismic Loadings

Transcription

1 A Method of Dynamic Nonlinear Analysis for RC Frames under Seismic Loadings M Matsuura^, I. Shimada^, H Kobayashi^ and K Sonoda^ ^Department of Civil Engineering, Osaka City University, Sugimoto, Sumiyoshi-ku, Osaka, , Japan EM ail: mickey@st. civil, eng. osaka-cu. ac.jp ^Toyo-Giken Consulting Co., Ltd., Shinkitano, Yodogawa-ku, shimada@toyogiken-ccei. co.jp Abstract The paper presents a method of seismic nonlinear response analysis for RC frames designed as a bridge pier. Rigid Body Spring Model (RBSM) is adopted as a numerical model, in which material nonlinearity is introduced into springs between adjacent rigid bodies. Moment-curvature relations of RC members are derived from numerical integration on their cross sections according to nonlinear stress-strain relations of steel and concrete, and Bernoulli's assumption. Assemblage of element matrix equations through a similar procedure to an ordinary displacement method leads a fundamental equation of motion under seismic loadings. An explicit time integration scheme is used for solving the equation of motion. 1 Introduction According to the Japanese Specifications for Highway Bridges [1] revised after the Hyogo-ken Nanbu Earthquake (Kobe, 1995), seismic nonlinear response analyses under a specified ground motion have been required to make a safety check on earthquake resistance of complicate RC structures. Though an ordinary finite element method (FEM) based on continuum mechanics may be effective for such analyses, it is not economical for a practical use, because of much time consuming due to strong material nonlinearity of RC structures. On the other hand, a simple frame analytical method is often used for saving computing time

2 134 Earthquake Resistant Engineering Structures under an assumed moment-curvature model such as Takeda model, Muto model and others. But such a method may be insufficient for RC frames, because flexural and extensional rigidities of member are coupled from each other. Therefore, the purpose of this paper is to develop an alternate simple method with a frame theory considering coupling of flexural and extensional rigidities. The proposed method is developed using the Rigid Body Spring Model [2,3]. By the proposed method, dynamic nonlinear analyses of a model RC portal frame are executed under a strong ground motion, which was recorded at the Hyogoken Nanbu Earthquake. From the numerical results obtained, discussions are made on nonlinear deformability and failure characteristics of the frame. 2 Analytical method 2.1 Rigid Body Spring Model RBSM was originally developed by Kawai [2], for statical limit analyses of RC structures. The model consists of rigid elements and springs connecting them, as shown in Fig. 1. element element j Fig. 1 Rigid Body Spring Model If elements / and j move with displacement w, v and 9 respectively, strains of springs between adjacent elements / and/ are given as follows: where 5^, y^ and K^ are axial strain, shearing strain, and curvature respectively, and ds is the distance between the gravity center of elements / and/. 2.2 RC cross section and stress-strain relations Three types of springs, namely axial, flexural and shearing ones, are set in RBSM as shown in Fig. 1. The flexural and axial springs are coupled due to nonlinear characteristics of RC cross sections. In a RC cross section shown in Fig. 2, As and As' are the cross sectional areas of the top and bottom layer reinforcements, respectively. As" is that of web reinforcements, d' and d" are concrete cover, h is height of sections and b is width of section. To obtain the

3 Earthquake Resistant Engineering Structures 135 moment-curvature relation of RC cross sections, numerical integration is executed as a multi-fiber model under Bernoulli's assumption. The strains of the concrete at the point of z, and longitudinal reinforcements are obtained as follows: 6f)K- (2) where s^ is the strain of the concrete at point of z,. s, ' and 2, are the strains of longitudinal reinforcements. T- As K -- ;,~As"" (a) Concrete Fig. 2 (b) Reinforcement RC cross section The stress-strain relations of concrete and steel prescribed in the Japanese Specifications for Highway Bridges [1] are shown in Fig. 3. a; and a; mean stress of concrete and steel respectively, a^ and s^ are strength and strain of the concrete enclosed with reinforcements respectively, ^ is ultimate strain of the concrete enclosed with reinforcements, and E^ and E, are initial modulus of concrete and steel respectively. Concrete outside reinforcements obeys a stressstrain curve excluding the strain softening part from Fig. 3(a). Elastic modulus of concrete in the unloading process is taken as the same as the initial elastic modulus in this analysis. ' E, (a) Concrete (b) Reinforcement Fig. 3 Stress-strain relations of concrete and reinforcement The stress-strain equation of concrete is expressed as follows:

4 136 Earthquake Resistant Engineering Structures (f,, < f, < 0) where a, is concrete stress, ^ is concrete strain, and E^ is decreasing gradient after the peak stress <r^. 2.3 Resultant forces on RC cross section Bending moment M, axial force N and shear force S are obtained as eqn (4) by numerical integration of the stress given in Fig. 3 as follows: M = a, /?6 (z, - where n is number of integration points, z, is distance between the top surface of cross section and each integration point, o;, is stress of concrete at an integration point, <%, is stress of a web reinforcement at an integration point, a, = 0.5(/=0 and n) and #, = 1.0(/ =1, 2,, n-l). On the other hand, the shearrigidityg, of RC cross section is assumed to remain constant during all the loading process. 2.4 Time integration scheme The equation of motion of elements at any time t is given by where M is mass matrix, A is observed seismic acceleration vector, C is damping matrix, F/ and F/ are respectively internal and external forces, IT and U ' are respectively velocity and acceleration vectors of the center of gravity on the each elements. F/ is composed of resultant forces vectors A/, N and S given by eqn (4), and is the function of displacement vector U. F/ expresses self weight of element and/or weight of superstructure on each element. Central and forward finite difference scheme with a small time increment At are respectively used in order to obtain the displacement and velocity of all elements at the next time step. Thus they are expressed at time t+at by (5) (6)

5 Earthquake Resistant Engineering Structures 137 The time increment At is taken as 113p,s in the following numerical example. The flowchart for this numerical scheme is shown in Fig. 4 c Start Determination of resultant forces Set r. At, U, U, ti Calculate $, /and K by eqn (1) in all sections Calculate s,', ^ s^ and5.,. by eqn (2) in all sections Check of loading or unloading by using nonlinear stress-strain relation of concrete and steel in Fig. 3 Determine cr,, a^ errand o;, in all sections Obtain A//, TV and S by eqn (4) in all sec lions integration scheme 1r Calculate F, and F^ about all elements, and Input seismic acceleration A Obtain U' by eqn (5) Obtain IT* and IT* by eqn (6) Fig. 4 Flowchart for analysis

6 138 Earthquake Resistant Engineering Structures 2.5 RC structure and analytical model A RC frame shown in Fig. 5 is designed according to the revised Japanese Specifications for Highway Bridges (1996) after the Hyogo-ken Nanbu Earthquake. This frame has a super-structure whose mass is looot and its gravity center is placed 2.5m above from the top of frame. The bottoms of columns are fixed to the ground, ignoring the effect of foundation. Element division for the columns and beam take 11 each the skeleton line. Superstructure mass" RC pier Ground (a) RC frame for analysis (b) RBSM model Fig. 5 RC frame and RBSM model (unitm) 3 Seismic nonlinear response analysis 3.1 Cross sectional and material properties Table 1 shows the properties of material and cross section of columns and beam. h m 2.0 Table 1 &CC N/mnf b m 2.7 Properties of material and cross section for analysis Concrete Steel EC 2 GCU /v ES N/mn N/mm^ N/mm* 145x1 0" x10^ Colunui member A, -4, mnf Beam A: mnf ^/' mm member,4," (/', J" ot, af" mm VTlyj, A VO^ mm^ mm h m 2.5 b m 2.7 A, mnf mm^ 3440 mm 150 ^L mm 1930 & mm 150 Note: /Land S, are the cross sectional area and spacing of stirrup in the beam and the column respectively.

7 3.2 Ground acceleration Earthquake Resistant Engineering Structures 139 A horizontal ground acceleration recorded at the Takatori station in the Hyogoken Nanbu Earthquake is used in this study. Its time history is shown in Fig. 6. The vertical acceleration recorded at the same place is relatively low and therefore it is neglected herein Fig. 6 Ground acceleration for analysis 3.3 Displacement response Figure 7 shows the time histories of horizontal displacements at the top of columns. The transcendent displacements appear at t=4.5 and 7.5s, but after MSs the displacements gradually decrease, and the residual displacement remains only 10 to 20mm corresponding to a drift angle of 1/1000-2/1000 rad of the columns. It can be mentioned that damage of this RC frame is still a slight though the moment-curvature relations of RC members already enter to strong nonlinear ranges as shown in Fig. 10. A deformation process is illustrated in Fig. 8. Concentration of curvature can be seen at the bottom of columns and the ends of beam Fig. 7 Displacement response at the top of columns

8 140 Earthquake Resistant Engineering Structures = 2.6s (b)f = 4.8s Fig. 8 Deformation process (c)f = 8.8s 3.4 Responses of axial forces and bending moments Figure 9 shows the axial force response at the bottom of columns. In this figure, the value at t=0 mean the static ones of axial force. The maximum response of axial force attains about twice as large as the static values. It can be, therefore, suggested that the variations of axial forces give a significant effect on the moment-curvature relations of members. The bending moment responses at the bottom of columns are shown in Fig. 10, from which a saturated value by the ultimate moment capacity can be seen. Fig. 9 Response of axial force at the bottom of columns

9 Earthquake Resistant Engineering Structures 141 t (s) Fig. 10 Response of bending moment at the bottom of columns 3.5 Hysteresis of bending moment Figures 11 and 12 show cyclic behavior on the moment-curvature relations at the bottom of columns and the ends of beam. Plastic domain becomes greater at the bottom of columns than the ends of beam though the ultimate bending strengths of cross section are the same for the columns and the beam. A plastic hinge seems to form at the bottom of both columns. The hysteresis curves for the left column is stable, but those for the right column gradually shift. Such hysteresis curves seem to be realistic from a viewpoint of the statically cyclic loading test results well-known in the past studies. Fig K (1/m) (a) Left column (b) Right column Hysteresis of bending moment at the bottom of the columns

10 142 Earthquake Resistant Engineering Structures S 0 -*i " /c(l/m) 0.01 (a) Left end (b) Right end Fig. 12 Hysteresis of bending moment at the both ends of the beam 4 Conclusion A new method within a frame theory for seismic nonlinear response analysis of RC frames was developed on the basis of the Rigid Body Spring Model. The method was characterized by considering coupling of flexural rigidity and extensional one at RC cross section. The resultant force-strain relations on the cross sections are directly derived by numerical integration of stress-strain equations for concrete and reinforcements. From a seismic response analysis for a portal RC frame, it can be known that the method gives reliable results in horizontal deflection response, axial force and bending moment responses, and hysteresis behavior of moment-curvature relations of members. References [1] Specifications for Highway Bridges, Part V: Aseismatic Design, Highway Bridge Society of Japan, Tokyo, pp , [2] T. Kawai, Some Considerations on the Finite Element Method, International Journal for Numerical Methods in Engineering, Vol.16, pp , [3] K. Sonoda, H. Kobayashi and M. Matsuura, Impact Fracture Analysis of Reinforced Concrete Rock Sheds, Structures under Shock and Impact III, Computational Mechanics Publications, Southampton, pp , 1994.