EXPLICIT DYNAMIC ANALYSIS OF A REINFORCED CONCRETE FRAME UP TO COLLAPSE ABSTRACT: K. Kocamaz 1, K. Tuncay 2 and B. Binici 3 1 Res. Assist., Civil Eng. Department, Middle East Technical University, Ankara 2,3 Prof. Dr., Civil Eng. Department, Middle East Technical University, Ankara Email: kkocamaz@metu.edu.tr Simulating the collapse mechanism is quite important for performance based seismic design of reinforced concrete frame buildings. In this regard, classical implicit time integration methods have usually been employed where divergence of the solution is usually treated as the onset of collapse. In this study, we take a different path and employ the explicit time integration for collapse simulations by using displacement based frame elements. The results from a recently conducted pseudo dynamic test were employed for the numerical validation of the modeling approach. It was found that the numerical estimations were in reasonable agreement with the test results in terms of the global engineering demand parameters of story displacements. On the other the errors in the local parameters, such as the strain within the plastic hinge zone were also found to be acceptable. The results demonstrated that as opposed to non-convergence in implicit dynamic analysis as indicator of collapse, the developed simulation platform was able to continue the analysis without any convergence problems. KEYWORDS: explicit dynamic analysis, PID control, fiber element 1. INTRODUCTION Accurate collapse estimation of reinforced concrete structures is important for performance based seismic design. Fiber frame elements with implicit integration methods became the state of the art for the dynamic analysis of reinforced concrete structures (Mazzoni et.al. 2010). Despite their established formulation, this approach usually suffers from convergence issues especially for systems approaching collapse. A similar problem is also encountered during inelastic static analysis, (i.e. pushover analysis) of buildings at large deformation demands. A better alternative to overcome these limitations is the use of explicit time integration, which has the advantages of non-iterative formulation, ease of coding and parallelization at the cost of smaller time steps. The most important benefit of explicit integration for dynamic analysis of buildings is perhaps its ability to capture collapse without any divergence issues. In this study, an explicit dynamic analysis program is prepared for the two-dimensional dynamic analysis of reinforced concrete buildings by employing displacement based frame elements. A Proportional Integral Derivative (PID) Control algorithm was employed to extend the capability of explicit dynamic analysis methodology to static inelastic analysis of buildings. PID algorithm is usually employed for accurate control of hydraulic actuators in dynamic testing and we employ this algorithm effectively for dynamic simulations. The validation of the developed computer program was conducted by comparing the simulation results with pseudo dynamic (PsD) test results conducted at Middle East Technical University. 2. EXPLICIT DYNAMIC SIMULATION PLATFORM WITH PID CONTROL 2.1. Fiber modelling There are two approaches to capture material nonlinearity within a finite frame element framework; lumped plasticity and distributed plasticity models. In the former modelling approach, a frame element consists of one elastic frame element and two rotational springs at elements ends. Rotational springs have nonlinear moment rotation behavior and whole material nonlinearity is modelled by assigning nonlinearities to those springs. In the later modelling approach, material inelasticity is distributed over the length of the member and it is taken into account through the use of sections throughout the element. Each section consists of a fiber discretization, which
are assumed to be subjected to uniaxial loading. In this study, a displacement based element with fiber modelling approach was adopted. Figure 1 shows a discretized frame element in 2D setting. Following the Euler-Bernoulli beam theory assumptions, section axial strain (ε o ) at plastic centroid and the curvature (κ ) can be described in terms of nodal displacements (u) at the member end degrees of freedom as follows: u s = [ ε o κ ] = B u ( 1 ) where B matrix is the classical strain-displacement matrix for fiber elements as given in Equation 2. B = [ 1 0 0 L 0 6 1 + 12 x 4 1 + 6 x L 2 L 3 L L 2 1 L 0 0 0 6 1 + 12 x 2 1 + 6 x ] ( 2 ) L 2 L 3 L L 2 Strain at any fiber (ε i ) can be computed from the linear strain profile hypothesizes due to the plane section remains plane assumption: with a si = [1 y i ] Section secant force vector (r s ) at a given step is then computed from: ε i = ε y i κ = a si u s ( 3 ) n r s = i=1 a T si E si a i ( 4 ) Above E si is the secant stiffness computed from the uniaxial material model described in the next section. Figure 1. Section discretization in fiber elements 2.3. Material Models For steel fibers, an ideal elastoplastic with kinematic hardening material model is adopted and for the confined concrete fibers a simplified bilinear approach to the Mander (1988) is assumed as shown in Figure 2. The simplification for the concrete stress-strain model was found to significantly reduce the computational cost without scarifying from accuracy of results. since high computational time and effort.
Figure 2. Material Models for Analysis (a) Concrete, (b) Steel 2.2. Explicit Dynamic Analysis An explicit numerical integration is used to integrate section force to calculate element end forces. Structural force matrix is constructed from element end forces and by using the mass and damping matrices equation of motion equation (Equation 5.) is solved at each time integration time step by using the following methodology. m a i + c v i + r i = ma gi ( 5 ) where m is the mass matrix, c is the damping matrix, r i is the restoring force vector, a i, v i are the acceleration and velocity vectors, respectively, and a gi is the ground acceleration at time i. The equation of motion was solved by the explicit Chung and Lee (1994) time integration. The steps in the integration scheme whose detailed explanations could be found in [1] are as follows: 1) Measure the restoring force (r i ) by imposing displacements (u i ). (Section 2.1) 2) Compute the acceleration at time i+1. a i+1 = m 1 ( ma gi r i ) 3) Calculate the displacements at time step i+1. u i+1 = u i + tv i + β 1 a i + β 2 a i+1 4) Calculate the velocities at time step i+1. v i+1 = v i + γ 1 a n + γ 2 a i+1 5) Repeat all steps for the remaining time steps. where β 1, β 2, γ 1, γ 2 appropriate coefficients and t is time step. 2.3. PID Control for Static Analysis In order to extend the capability of the explicit dynamic analysis platform to inelastic static analysis a PID control algorithm was implemented. For the Proportional-Integral-Derivative (PID) controller derivative of force with respect to time value is chosen as the controller output variable and velocity is chosen as the desired input variable. In general, a PID controller is a loop feedback controller used in various industrial control systems and it can be stated as follows: co(t) = K p e(t) + K i e(t)dt + K d de dt ( 6 )
Above, co(t) represents the controller output, e(t) tracking error, K p is the proportional gain coefficient, K i is the integral gain coefficient and K d is the derivative gain coefficients. The error variable is defined in equation 7 with r(t) as the desired input variable and y(t) is the process measurement of the same variable. Since explicit time integration scheme is used in this study, r(t) is taken as the desired velocity at present time step and y(t) is the previous velocity value which belongs to previous time step. e(t) = r(t) y(t) ( 7 ) K p, K i, K d values are critical for the controller their estimation is difficult to calculate explicitly for complex structures. For vibration problems, whole model can be reduced to a simple model by ignoring the vibrational modes which do not contribute response of the structure (Khot et al., 2011). Since the structure is laterally loaded and the response to that force mainly consists of its first mode motion, the full model can be simplified to a first mode in order to calculate controller coefficients. This is consistent with the general idea of pushover analysis as well. Ziegler-Nichols method (Ziegler & Nicholes 1942) is employed to determine K p, K i, K d values after the calculation of the first mode s modal mass, stiffness and damping terms with eigenvalue analysis. In Figure 2, it is shown that the prescribed velocity and the response velocity are in a good agreement upon application of our control algorithm. As shown in the same figure, acceleration variable does not have a significant value therefore, it can be said that during a static analysis with explicit time integration inertial forces do not affect the overall response, validating the application of the procedure. Figure 3. Velocity and Acceleration at load applied node (a) Prescribed and Response Velocity, (b) Acceleration 2. 3 BAY-3 STORY FRAME A prototype RC frame building was designed according to the regulations of TEC (2007). 3-bay 3-story half scaled RC frame specimen was built representing the interior frame of the prototype building (Figure 3). A continuous PsD testing was employed for all specimens using synthetic ground motions compatible with the site-specific earthquake spectra developed for the city center of Duzce. In order to test the ability of the computational model in estimating the lateral deformation capacity and ductility the envelope of the measured response curve was used in the next section along the local maximum column end rotations.
(a) prototype building (b) elevation view of the test frame Figure 4. Test frame Member section details of the test specimen are given in Figure 4. Column longitudinal reinforcement ratio was about 2%. Average uniaxial compressive strength of concrete used for Specimen was 33.7 MPa. Deformed bars, used as longitudinal reinforcement, had yield strength of 480 MPa and plain bars, used as transverse reinforcement, had yield strength of 240 MPa obtained from material tests. The details of the test setup and the measurement system are shown in Figure 5. Figure 5. Section details of Specimen 2 Figure 6. Test Setup A numerical model for the test frame was prepared by using the developed computational platform. Expected plastic hinge zones in beams and columns were modelled with one element with 3 fiber sections. Beam and column regions expected to remain below yielding but to exceed cracking stage were modelled with one element with 5 fiber sections. Mass of the system was idealized with lumped mass approximation and damping was calculated by
using classical Rayleigh damping method whose coefficients are calculated by fixing first and second mode s damping as 2%. Constructing the stiffness matrix, a secant modulus approach was used. 3. COMPARISON AND RESULTS Roof displacement-base shear force results from the test and simulation are shown in Figure 7. It should be noted that the test was conducted pseudo dynamically and only the envelope curve is presented below. For the simulation, a PID control scheme was applied to obtain the envelope pushover curve. The two results seem to agree in a reasonable manner while the ultimate capacity in the positive direction was slightly overestimated. The hinging pattern obtained from the simulation is presented in Figure 8 along with the estimated damage states of the same test frame by Mutlu (2012). It can be observed that the estimated hinging sequence agrees well with estimations of Mutlu (2012) including the incipient of collapse in the first story columns. The measured maximum column base rotations are compared with those obtained from simulations in Table 1. Numerical base rotations were obtained at the maximum roof displacement demand measured during the test. It can be seen that there is an excellent agreement with the base rotation demands between the experimental and simulation results. These results provide confidence on the use of the proposed approach for future studies. 150 100 Base Shear Force (kn) 50 0-50 -100 Simulation Experimental -150-150 -100-50 0 50 100 150 200 250 Roof Displacement (mm) Figure 7. Base shear Roof Displacement Curves CP: Collapse, SD: Significant Damage, MD: Minimum Damage according to TEC (2007). Figure 8. Plastic Hinge Formation Sequence and Damage Assessment Result from Mutlu (2012)
Table 1. Comparison of bottom end rotations of 1 st story columns Column 101 Column 102 Column 103 Column 104 Experiment -0.025-0.032-0.031-0.029 Simulation -0.020-0.026-0.028-0.027 3. CONCLUDING REMARKS In this study, simulation of collapse mechanism which is quite important for performance based seismic design is done by using explicit time integration scheme with PID controlling algorithm which expands method s power to apply inelastic static analysis, (i.e. pushover analysis). Despite implicit integration methods have usually divergence problem for systems approaching collapse, it is seen that explicit time integration is capable of estimating collapse mechanisms without any convergence problems. Furthermore, program development was easier compared to the implicit schemes requiring various iterative techniques. The proposed PID algorithm to extend the explicit dynamic analysis to static pushover analysis was found to successful based on the obtained results. The developed simulation platform will be used to validate the estimations under dynamic and impact loading scenarios. REFERENCES [1] Chung J, Lee JM. A new family of explicit time integration methods for linear and non-linear structural dynamics. International Journal for Numerical Methods in Engineering 1994; 37:3961 76. [2] Khot SM, Yelve NP, Tomar R, Desai S and Vitta S (2011). Active vibration control of cantilever beam by using PID based output feedback controller. Journal of Vibration and Control 18(3), 366-372. [3] Mahin, S. A., and Shing, P. B. (1985). Pseudodynamic method for seismic testing. Journal of Structural Engineering, ASCE, Vol. 111, No. 7, pp 1482 1503. [3] Mazzoni, S., McKenna, H., Scott, M.H., and Fenves, G.L., (2010). OpenSees Manual, Pacific Earthquake Engineering Research Center, http://opensees.berkeley.edu, 01/09/2010. [4] Mutlu MB (2012). Numerical Simulations of Reinforced Concrete Frames Tested Using Pseudo-Dynamic Method, MSc Thesis, Middle East Technical University, 117 p. [5] Ziegler, J. G., & Nichols, N. B. (1942). Optimum settings for automatic controllers, Trans. ASME, vol. 64, pp. 759-768. [6] TEC, (2007). Deprem Bölgelerinde Yapılacak Binalar Hakkında Yönetmelik. Bayındırlık ve İskân Bakanlığı, Ankara, Türkiye.