NON-LINEAR DYNAMIC ANALYSIS OF PLANE FRAME STRUCTURES UNDER SEISMIC LOAD IN FREQUENCY DOMAIN

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1 International Conference on Engineering Vibration Ljubljana, Slovenia, 7-1 September 215 NON-LINEAR DYNAMIC ANALYSIS OF PLANE FRAME STRUCTURES UNDER SEISMIC LOAD IN FREQUENCY DOMAIN Luis F. Paullo Muñoz* 1, Paulo B. Gonçalves 1, Ricardo A.M. Silveira 2, Andréa Silva 2 1 Civil Engineering Department of Pontifical Catholic University of Rio de Janeiro lfernand@tecgraf.puc-rio.br, paulo@puc-rio.br 2 Civil Engineering Department of Federal University of Ouro Preto ramsilveira@yahoo.com.br, andreadiassilva@yahoo.com.br Keywords: Nonlinear Dynamic Analysis, Analysis in Frequency Domain, Dynamic instability. Abstract. The dynamic response of large structures in the main resonance regions is a topic of great importance in the analysis of structures under time varying loads. In complex problems, the determination of the response in the frequency domain, characterized by the resonance curves is, in some cases, indirectly obtained through a series of analyses in time domain, which leads to huge computational effort when analyzing structures with a large number of degrees of freedom. In nonlinear cases, the response in the frequency domain becomes even more cumbersome because of the possibility of multiple solutions for certain forcing frequencies. Those solutions can be stable and unstable, and bifurcations can appear, in particular saddle-node bifurcation at the turning points along the resonance curves. In this work, an incremental technique for direct calculation of the nonlinear dynamic response in frequency domain of nonlinear plane frames discretized by the finite element method and subjected to base excitation is proposed. The transformation of discretized equations of motion, in the finite element method context, to the frequency domain is made here through the classical harmonic balance method in conjunction with the Galerkin method. The resulting system of nonlinear equations in terms of the modal amplitudes and forcing frequency is solved by the Newton-Raphson method together with an arc-length procedure to obtain the nonlinear resonance curves. Suitable examples are presented, and the influence of the frame geometric parameters and base motion on the nonlinear resonance curves is investigated.

2 1 INTRODUCTION The dynamic response of large structures in the main resonance regions is an important topic in the analysis of structures under time varying loads. In complex problems, the determination of the response in the frequency domain, characterized by the resonance curves is, in some cases, indirectly obtained through a series of analyses in time domain, as in Galvão [1] and Silva [2], which leads to huge computational effort when analysing structures with a large number of degrees of freedom. In nonlinear cases, the response in the frequency domain becomes even more cumbersome because of the possibility of multiple solutions for certain forcing frequencies. Those solutions can be stable and unstable, and bifurcations can appear, in particular saddle-node bifurcation at the turning points along the resonance curves. In structures subjected to seismic loads (earthquakes), there is a great interest to know the response of structure in frequency domain, since the vulnerability of structure during an earthquake is related of the relations between the natural vibration frequencies of the structure and the frequency content in the seismic load. In this work, an incremental technique for the direct calculation of the nonlinear dynamic response in frequency domain of nonlinear plane frames discretized by the finite element method and subjected to a base excitation is proposed. The transformation of discretized equations of motion, in the finite element method context, to the frequency domain is accomplished here through the classical harmonic balance method (HBM), following the formulation presented in Pasquetti [3] for the linear case. For the non-linear analysis, a particular adaptation of the HBM-Galerkin methodology presented by Cheung and Chen [4] and generalized for the use in FEM context by Chen et al [5] is here proposed. The resulting system of nonlinear equations in terms of the modal amplitudes and forcing frequency is solved by the Newton-Raphson method together with an arc-length procedure to obtain the nonlinear resonance curves. Suitable examples are presented, and the influence of the frame geometric parameters and base motion on the nonlinear resonance curves is investigated. 2 FORMULATION 2.1 Equilibrium equations and resonance curves by time domain analysis One classic way to determinate the resonance curves or bifurcation diagram of a structure is through a time domain analysis, such as the brute force method (Parker et al. [6]). For this, it is supposed that the structure is under the action of a harmonic load with a prescribed forcing frequency ω. The dynamic equilibrium is defined by: ( ) (1) where, and are the mass matrix, damping matrix and the internal forces vector, respectively, being dependent on the nodal displacement vector, and on the geometric non-linear formulation (Silveira [7] and Tokami et al. [8]), is the nodal external reference load vector, and and ω are the amplitude and frequency of the harmonic excitation, respectively. To obtain the resonance curves, it is necessary to make multiples time domain analyses by direct integration of the equations of motion and increasing step by step the excitation frequency (Parker et al. [6]). The analysis must be long enough to achieve the steady state response, as shown in Fig. 1. The resonance curve is obtained by mapping the maximum displacements in the steady state stage for each frequency value. 2

Transient state Steady state Maximum positive displacement Maximum negative displacement Figure 1. Transient and steady state response in time domain of a system under harmonic excitation.

3 Transient state Steady state Maximum positive displacement Maximum negative displacement Figure 1. Transient and steady state response in time domain of a system under harmonic excitation. A disadvantage of this method, even in the linear case is that, after each frequency increase, it is necessary to integrate the equations of motion long enough to reach the steady state response, which depends directly on the damping of the system, the frequency step and on the non-linearity, leading to a huge computational effort for systems with a large number of degrees of freedom. Another disadvantage is the difficulty of mapping all fixed points in regions where coexisting stable and unstable solutions may occur (Parker et al. [6]). Figure 2. Resonance curve of a 2-DOFs obtained through time domain analysis (25 simulations). 2.2 Classic Balance Harmonic Method (HBM) for Analysis The HBM is one the most popular methods for the nonlinear analysis of dynamical systems. In this method, the displacement field is approximated by a finite Fourier series, that is: [ ] (2) where and are the modal amplitude vectors corresponding to the n th harmonic, and N is the number of terms considered in the approximation. For the linear damped case, it is only 3

4 necessary to retain the second term in (2). Thus, the nodal displacement vector can be approximated by: Introducing Eq. (3) in Eq. (1) results in: (3) (4) Taking and, where is the linear stiffness matrix, and applying the harmonic balance to equation Eq. (4) the following system is obtained: where [ ] { } { } (5) Eq. (5) can be expressed in a more compact form as: (6) [ ], { }, { } (7) The forcing frequency ω and the modal amplitude vector are the unknowns in Eq. (7). Even in the linear case, the transformation of Eq. (1) to the frequency domain leads to a nonlinear system of algebraic equations, being necessary to use of the Newton-Raphson method to solve Eq. (7). 2.3 HBM-Galerkin for Non- Analysis For the non-linear analysis, the approach proposed by Cheung and Chen [4] and generalized for the use in a finite element formulation by Chen et al [5] is used. In this case, and considering that in steady state phase the response is periodic, it is convenient to use a periodic non-dimensional time variable defined as: Substituting Eq. (8) into Eq.(1), the equation of motion can be rewritten as: ( ) (9) and each component of the displacement vector can be approximated by: [ ] (1) For systems with quadratic and cubic non-linearity, at least the first two terms of the Fourier series must be considered (Nayfeh and Mook [9]): (11) For system with cubic nonlinearity it is necessary to consider at least the terms (Nayfeh and Mook [9]): (12) (8) Defining: (13) 4

5 With considering the following relation: where, [, (14) N = Total number of the degrees of freedom. (15) ], (16) and introducing Eq. (15) into Eq. (9), the following matrix equation is obtained: ( ) (17) Considering that the solution is periodic, multiplying both of sides of the equations Eq. (17) by and integrating the resulting expression over a period, one obtains (Bouc [1]): [ ( )] (18) In a compact form, Eq. (18) can be expressed as: ( ) (19) where:, =, ( ) ( ) (2) Equation Eq. (19) is a non-linear system of algebraic equations in which the unknowns are the frequency and the modal amplitude vector. The internal force vector in frequency domain ( ) is obtained from the transformation of ( ). In this context, it is necessary an explicit expression for ( ). The next step is the solution of the non-linear system of equation. 2.4 Solution of Non-linear System of Equations The transformation of the equations of motion from time to frequency domain results in a non-linear system of algebraic equation both for the linear and non-linear cases, as shown by Eq. (6) and Eq. (19), being the unknown variables the forcing frequency and the modal amplitudes. In this section, the technique to solve the non-linear system of equation is described, considering the possibility of frequency and amplitude limit points. In this context, the equation Eq. (6) can be defined in general form as: ( ) (21) Through an incremental analysis the dynamic equilibrium in the i th step is given by: 5

6 ( ) (22) The unknown variables in the i th step are obtained by the incremental analysis as: Introducing Eq. (22) into Eq. (21) results in: The frequency and amplitude increments, (23) ( ) (24) and can be calculated iteratively as:, (25) where and are the correctors which can be obtained through the first variation of Eq. (24) as: or, in a more compact form, as: (26) (27) It is necessary the addition of a constrain equation to solve Eq. (17). In this work, a spherical arc-length control method (Crisfield [11]; Ramm [12]) is used to solve the non-linear system defined by Eq. (27). More details about the adaptation of arc-length control methods to solve dynamic problems in frequency domain can be found in Ferreira and Serpa [13] and in Pasquetti [3]. In this work a particular adaptation of the spherical arc-length method is used. In this case the spherical arc-length constrain equation is given by: = (28) From equations Eq. (24), Eq. (25), Eq. (27) and Eq. (28), it is possible to obtain the iterative frequency corrector as: (29) where { ( ), ( ) (3) The signal of the first frequency corrector, also called frequency predictor, can be obtained using the positive work criterion as: ( ) (31) For linear systems, it can be defined by the following expressions: [ ], [ ]. (32) Analogously, for the non-linear case, it can be obtained from the following relations: ( ), = ( ) (33) 6

7 Amplitude norm Luis F. Paullo Muñoz, Paulo B. Gonçalves, Ricardo A. M. Silveira, Andréa R. D. Silva 3 NUMERICAL EXAMPLES As first example, a simple tower model with a concentrated mass at the top and an elastic rotational support at the base is studied to validate de linear formulation. The finite element model is composed by 1 beam-column Euler-Bernoulli elements. The geometrical and material properties are shown in Figure 3. The tower is under the action of horizontal harmonic base-displacement (Paullo et al. [14]). Figure 3: Tower model with concentrate mass at top and elasto-plastic support. Figure 7 shows the resonance curves of the displacement at top of the tower, considering two values of the elastic support stiffness. The results obtained with the present formulation agree with the results obtained from a time domain analysis. This shows the coherence of the result obtained with the proposed method. It can be also observed that the present method is able to obtain the resonance curve with more accuracy in the vicinity of first resonant frequency. H.B.M. k = infinite H.B.M. k = 1 11 knm/rad Time dom. int. k = infinite Time dom. int. k = 1 11 knm/rad ω(rad/s) Figure 4: Maximum amplitude norm at top vs. frequency. A=.8g. As second example, a simple mass with a non-linear elastic spring under harmonic load is studied to validate the non-linear formulation. The schematic representation of the system and relevant properties are show in Figure 5. For this example, damping is not considered. m=1 kg m=1 kg m f(t)=.4m.cos(ωt) Figure 5: 1-DOFs model with cubic non-linear stiffness under harmonic excitation. 7

8 6. Amplitude norm. 1. Luis F. Paullo Muñoz, Paulo B. Gonçalves, Ricardo A. M. Silveira, Andréa R. D. Silva Figure 6 shows frequency-amplitude relation obtained through HBM-Galerkin and time domain simulations. The results obtained with the present HBM-Galerkin scheme and with time domain simulations are very close, validating the non-linear formulation. The effect of positive cubic non-linearity on the response is clear; the resonance curve exhibits a strong hardening behavior with three coexisting solutions in a large frequency range. It can be also observed that the algorithm is able to bypass the limit point associated with a saddle-node bifurcation. Time Integração domain no simulation tempo Proposed Método method proposto ω(rad/s) Figure 6: Maximum amplitude vs. frequency As a last example, a frame with a pitched-roof fixed at the base and with a constant cross section is studied to assess the behavior of this structure under horizontal and vertical base excitation. The vertical vibration is directly proportional to the horizontal vibration, following the criteria of some seismic code, such as Peruvian seismic-code RNE-E3 [15]. In this work it is considered that the intensity of the vertical base motion is.67 times the intensity of the horizontal base motion. The geometry and material properties of frame are shown in Figure 7, as well as the configuration of the first four natural vibration modes, being the first and third antisymmetric and the second and fourth symmetric. It is also studied the influence of the roof height on the response. The structure is modeled with twenty bean-column elements; 8 elements of the same size elements for the columns and 12 elements for the roof. 1. H Cross Seção estrutural section: W14x132 E =21GPa 1 st mode 2 nd mode 3 th mode 4 th mode Figure 7 Pitched-roof frame model and four first natural mode of vibration. H 1 st Mode Natural frequency of vibration (rad/s) 2 nd Mode 3 th Mode 4 th Mode Table 1: Four first natural frequency of pitched-roof frame 8

9 Table 1 shows the first four natural frequencies for the three values of H considered in the analysis. Figure 8 shows the variation of the horizontal (A x ) and vertical displacements (A y ) at the top of frame, considering both a linear (blue) and non-linear (red) formulation. All vibration modes are excited. The influence of geometric non-linearity is evident in the two first resonance regions, where a hardening behavior leading to possible dynamic jumps is observed. As the roof height increases, the magnitude of A x corresponding to the first resonant peak increases, while the magnitude of second peak decreases. On the other hand, the magnitude of vertical displacement A y is reduced in the two first resonance regions with the increase of the roof height. It is important to point out that the non-linear resonance curves exhibit amplitude and frequency limit points, which can only be mapped by the use of continuation techniques H =.m H =.m Não Non-linear Não.6 Non-linear H =3.m.8.6 Não Non-linear H =3.m.3 Não Non-linear Figure 8: Horizontal and vertical amplitude norm vs. frequency. Agx =.8g, Agy =.667Agx. 4 CONCLUSIONS Não Non-linear H =6.m 5 In this work, an incremental technique for the direct calculation of the nonlinear resonance curves of plane frames discretized by the finite element method and subjected to a base excitation is proposed. The transformation of discretized equations of motion, in the finite element method context, to the frequency domain is accomplished here through the classical harmonic balance method together with the Galerkin method. The resulting system of nonlinear equations in terms of the modal amplitudes and forcing frequency is solved by the Newton-Raphson method together with an arc-length procedure to obtain the nonlinear resonance curves. The formulation of the proposed method is validated comparing the present results with time domain simulations showing coherence and precision. The algorithm is able to map regions with coexisting stable and unstable solutions. A pitched-roof frame under horizontal Não Non-linear H =6. m 9

10 and vertical base excitation is studied. The results show a nonlinear hardening behavior with more influence in the two first resonance regions. The roof height has a strong influence on the resonant peaks. The proposed method is shown to be a useful method for the analysis of slender frame structures in frequency domain, since it is able to obtain the resonance curves with precision and small computational effort. REFERENCES [1] A. S. Galvão, Instabilidade Estática e Dinâmica de Pórticos Planos com Ligações Semirígidas. DSc. thesis, Civil Engineering Department-PUC-Rio/Rio de Janeiro, Brazil, 24. [2] A.R.D. Silva, Sistema Computacional para Análise Avançada Estática e Dinâmica de Estruturas Metálicas. DSc thesis, Departamento de Engenharia Civil Escola de Minas UFOP/Ouro Preto, Brazil, 29. [3] E. Pasquetti, Métodos aproximados de Solução de Sistemas Dinâmicos Não-es. DSc thesis, Civil Engineering Department-PUC-Rio/Rio de Janeiro. Brazil, 28. [4] Y.K. Cheun and S.H. Chen, Application of the incremental harmonic balance method to cubic non-linearity systems. Journal of Sound and Vibration, 14 (2), , 199. [5] S.H. Chen, Y.K. Cheun and H.X. Xing, Nonlinear vibration of plane structures by finite element and incremental harmonic balance method. Nonlinear Dynamics, 26, 87-14, 21. [6] T. S. Parker and L.O. Chua, Practical numerical algorithms for chaotic systems. New York: Springer, [7] R.A.M. Silveira, Análise de Elementos Estruturais Esbeltos com Restrições Unilaterais de Contato. Dsc thesis, Civil Engineering Department-PUC-Rio/Rio de Janeiro. Brazil, [8] M.A.M. Torkamani, M. Sonmez and J.E. Cao, Second-order elastic plane-frame analysis using finite-element method. Journal of Structural Engineering, 12 (9), , [9] A. H. Nayfeh and D.T. Mook, Nonlinear oscillations. John Wiley & Sons, 28. [1] R. Bouc, Sur la méthode de Galerkin-Urabe pour les systèmes différentiels périodiques. International Journal of Non- Mechanics, 7 (2), , [11] M.A. Crisfield, A fast incremental/iterative solution procedure that handles snapthrough. Computers & Structures, 13, 52-62, [12] E. Ramm, Strategies for Tracing the Non- Response Near Limit-Points. W. Wunderlich, Springer-Verlag eds. Nonlinear Finite Element Analysis in Structural Mechanics, Berlin, Germany, 63-89, [13] J.V. Ferreira and A.L. Serpa, Aplication of the arc-length method in nonlinear frequency response. Jornal of Sound and Vibration, 284, , 25. [14] L.F. Paullo, P.B. Gonçalves, R.A.M. Silveira and A.R.D. Silva, Nonlinear Dynamic Analysis of Frame Structures under Seismic Excitation Considering Soil-Structure Interaction and Elasto-Plastic Soil Behavior. Z. Dimitrovová et al. eds. 11 th International Conference on Vibration Problems, Lisbon, Portugal, 213. [15] V.C. Ministério, RNE E3: Norma de diseño sirmorresistente. Reglamento Nacional De construcciones. Lima, Perú, 27. 1

NONLINEAR DYNAMIC ANALYSIS OF FRAME STRUCTURES UNDER SEISMIC EXCITATION CONSIDERING SOIL-STRUCTURE INTERACTION AND ELASTO-PLASTIC SOIL BEHAVIOR

11 th International Conference on Vibration Problems Z. Dimitrovová et al. (eds.) Lisbon, Portugal, 9-12 September 2013 NONLINEAR DYNAMIC ANALYSIS OF FRAME STRUCTURES UNDER SEISMIC EXCITATION CONSIDERING