A COMPARISON OF TWO HYSTERETIC MODELS IN PREDICTING THE RESPONSE OF A CONNECTOR UNDER CYCLIC LOADING by Yvonne P.Y. Lo B.Sc , University of Alberta, Canada, 2001 A THESIS SUBMITTED TN PARTIAL FULFILMENT OF THE REQUIREMENT FOR THE DEGREE OF M A S T E R OF APPLIED SCIENCE in THE F A C U L T Y OF G R A D U A T E STUDIES DEPARTMENT OF CIVIL ENGINEERING We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH C O L U M B I A September 2002 © Yvonne P.Y. Lo, 2002 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department o f The University of British Columbia Vancouver, Canada Date Abstract The effect of seismic loading on structures has become a main concern due to the number of collapses or incidents of substantial damage during recent earthquakes. The inelastic behavior of a fastener under seismic loading is a determinant factor of the response of the whole structure; hence, the dynamic behaviour of the fasteners is a main issue for study. The objective of this research is to compare the response of a fastener predicted by a finite-element model (HYST) and the empirical Bouc-Wen-Baber-Noori model (BNW). HYST is a mechanics-based model that uses material properties of the fastener and the characteristics of its surrounding medium to predict the response. B N W is a mathematical model involving 13 parameters. The hysteretic results calculated by this model are obtained by integration of a first order differential equation, given a displacement history A(t). The B N W parameters are calibrated using the response to a particular cyclic displacement history, and then the responses are evaluated for other different histories or seismic excitations. Results are compared to the corresponding HYST outputs. The results show that B N W gives comparable values for relative maximum displacements with respect to the medium, and restoring forces when the response of the structure remains elastic. However, the discrepancies increase with nonlinearities, and the model cannot accurately predict, for example, the residual deformations. i i Table O f Contents Abstract i i Table O f Contents i i i L is t O f Abbreviat ions A n d Symbols v i L is t O f Tables ix L is t O f Figures x Acknowledgement x i i i Chapter 1 In t roduct ion 1 1.1 Preliminary Remarks 1 1.2 Objective And Scope 1 1.3 Thesis Organization 2 1.4 Model Descriptions 2 1.4.1 Finite-element Model (HYST) 3 1.4.2 Empirical Mathematical Model (BNW) 6 1.4.2.1 Parameters Constraints 8 Chapter 2 Parameter Estimation A n d B N W Model F i t t ing To A n Experimental Record 9 2.1 Introduction 9 2.2 Parameter Estimation 9 2.3 OPT Routine 10 2.3.1 Runge-Kutta Method 11 2.4 Model Fitting To A n Experimental Record 12 Chapter 3 Evaluation O f The Cal ibrated B N W Model Accuracy For Dif ferent Cyclic Records 18 3.1 Introduction 18 ii i 3.2 Results 19 3.2.1 Test 1: Cyclic History With Increasing Peaks 19 3.2.2 Test 2: Cyclic History With Decreasing Peaks 22 3.2.3 Test 3: Earthquake Displacement 25 3.2.3.1 Test 3a: Displacement With A Strong Motion Segment Of Average Duration 25 3.2.3.2 Test 3b: Displacements Correspond To A Shock Of Short Duration 28 3.2.3.3 Test 3c: Displacements With Strong Motion Of Long Duration 31 3.3 Discussion 34 Chapter 4 Comparison For SDOF Seismic Response 35 4.1 Introduction 35 4.2 Procedure 36 4.3 Results 37 4.3.1 Relative Maximum Displacement With Respect To The Medium 37 4.3.2 Residual Displacement 40 4.4 Comparison Of The B N W Results And The HYST Results 43 4.4.1 Maximum Displacement 43 4.4.2 Residual Displacement 45 4.4.3 Hysteresis Loops 47 4.5 Discussion 57 Chapter 5 Concluding Remarks 58 5.1 Research Contributions 58 5.2 Conclusions 58 iv 5.3 Recommendation For Future Investigations 58 References 59 Appendix I Newmark's Method For Nonlinear Systems 60 A l . l Theory 60 A1.2 Determination Of The Root Of x i + i 61 Appendix II Ground Acceleration Records 63 A2.1 Duration Of Seismic Excitation: Short 63 A2.2 Duration Of Seismic Excitation: Intermediate 67 A2.3 Duration Of Seismic Excitation: Long 71 v List Of Abbreviations And Symbols Abbreviations B N W Bouc-Wen-Baber-Noori Method mm millimeter kN kilo Newton P G A peak ground acceleration SDOF single-degree-of-freedom sec seconds Symbols ajj = the coefficients in Runge-Kutta method bj = the coefficients in Runge-Kutta method c = linear viscous damping coefficient D = total discrepancy F = hysteretic forcing function F(t) = forcing function F e x p = actual restoring force Ftheory = theoretical restoring force calculated from the B W B N model F u = the ultimate value of F fs = lateral force h = step size h(F) = pinching function k = stiffness X = maximum search radius m = lumped mass vi n = no. of data points p = control the rate of initial drop in slope q = a fraction of z u at the pinching level R = random number generator s = number of stages in Runge-Kutta method sgn( ) = signum function T n = natural period of vibration in seconds u = relative displacement of the mass, m, with respect to the ground motion u = relative velocity of the mass, m. with respect to the ground motion ii - relative acceleration of the mass, m, with respect to the ground motion X = parameter set X B = lower bound of the parameter range X U = upper bound of the parameter range XNEW = new parameter set XOLD = previous parameter set A = hysteretic displacement a = rigidity ratio y,(3,n = hysteresis shape parameters v = strength degradation parameter r\ = stiffness degradation parameter C, = damping ratio C,\ = the severity of pinching or magnitude of initial drop in dz/du 2 = controls the spread of the pinching region s = hysteretic function vii £ l 0 = measure of total slip y/0 = controls to the amount of pinching Sv = constant specified for the desired rate of pinching spread A = parameter that controls the rate of change of £2as£i changes At = time step viii List Of Tables Table 2.1 The calibration result 15 Table 3.1 The optimum parameters 18 Table 4.1 Maximum displacement (in mm) for a 20.0kg mass 37 Table 4.2 Maximum displacement (in mm) for a 35.0kg mass 38 Table 4.3 Maximum displacement (in mm) for a 50.0kg mass 39 Table 4.4 Residual displacement (in mm) for a 20.0kg mass 40 Table 4.5 Residual displacement (in mm) for a 35.0kg mass 41 Table 4.6 Residual displacement (in mm) for a 50.0kg mass 42 Table 4.7 Correlation p under different types of motion phase 43 Table 4.8 rms under different types of motion phase 43 Table 4.9 Correlation p with all different types of motion phases 44 Table 4.10 rms with all different types of motion phases 44 Table 4.11 Correlation p under different types of motion phase 45 Table 4.12 rms under different types of motion phase 45 Table 4.13 Correlation p with all different types of motion phases 46 Table 4.14 rms with all different types of motion phases 46 ix List Of Figures Figure 1.1 Load-displacement relationship of a fastener under seismic events 3 Figure 1.2 Fastener subjected to lateral loads; degrees of freedom u(x) and w(x) 4 Figure 1.3 Finite element between nodes i and j 5 Figure 1.4 Embedment response 5 Figure 2.1 The calibration specimen 13 Figure 2.2 Calibration displacement history 14 Figure 2.3 The HYST hysteresis (Calibration data) 14 Figure 2.4 The hysteretic model B N W (Calibration data) 16 Figure 2.5 Comparison of the experimental and computed restoring forces with the same input displacement 16 Figure 3.1 Displacement record for Test 1 19 Figure 3.2 The hysteretic model B N W for Test 1 20 Figure 3.3 The experimental hysteresis HYST for Test 1 20 Figure 3.4 Comparison of the experimental and computed forces for Test 1 at the same input displacement 21 Figure 3.5 Displacement record for Test 2 22 Figure 3.6 The hysteretic model B N W for Test 2 23 Figure 3.7 The experimental hysteresis for Test 2 23 Figure 3.8 Comparison of experimental and computed restoring forces for Test 2 at the same input displacement 24 Figure 3.9 Displacement record for Test 3a 25 Figure 3.10 The hysteretic model B N W for Test 3a 26 Figure 3.11 The experimental hysteresis HYST for Test 3a 26 x Figure 3.12 Comparison of experimental and computed restoring forces for Test 3a at the same input displacement 27 Figure 3.13 Displacement record for Test 3b 28 Figure 3.14 The hysteretic model B N W for Test 3b 29 Figure 3.15 The experimental hysteresis HYST for Test 3b 29 Figure 3.16 Comparison of experimental and computed restoring forces for Test 3b at the same input displacement 30 Figure 3.17 Displacement record for Test 3c 31 Figure 3.18 The hysteretic model B N W for Test 3c 32 Figure 3.19 The experimental hysteresis HYST for Test 3c 32 Figure 3.20 Comparison of experimental and computed restoring forces Test 3c at the same input displacement 33 Figure 4.1 Nonlinear, SDOF system under cyclic loading 35 Figure 4.2 The hysteresis for a 20.0kg mass, PGA=1.0g, type of duration= Short 48 Figure 4.3 The hysteresis for a 20.0kg mass, PGA=1.0g, type of duration= Intermediate 49 Figure 4.4 The hysteresis for a 20.0kg mass, PGA=1.0g, type of duration= Long 50 Figure 4.5 The hysteresis for a 35.0kg mass, PGA=1.0g, type of duration= Short 51 Figure 4.6 The hysteresis for a 35.0kg mass, PGA=1.0g, type of duration= Intermediate 52 Figure 4.7 The hysteresis for a 35.0kg mass, PGA=1.0g, type of duration= Long 53 Figure 4.8 The hysteresis for a 50.0kg mass, PGA=1.0g, type of duration= Short 54 Figure 4.9 The hysteresis for a 50.0kg mass, PGA=1.0g, type of duration= Intermediate 55 xi Figure 4.10 The hysteresis for a 50.0kg mass, PGA=1.0g, type of duration= Long 56 Figure A l . 1 Linear interpolation to determine x 61 xn Acknowledgement I would like to thank my supervisor, Dr. Ricardo O. Foschi, for his guidance and support, which has been responsible for so much of my graduate education. Many thanks are due to my friends, Felix Yao and Jianzen Zhang, for their assistance with instrumentation and software. xii i Chapter 1 In t roduct ion 1.1 Prel iminary Remarks The effect of seismic loading on structures has become a main concern due to the number of collapses or incidents of substantial damage during recent earthquakes. Timber structures generally have satisfactory performance during seismic events; nevertheless, they may also show substantial damage. Little theoretical understanding of the actual wood structural behaviour and few field or experimental data has hindered the investigations of the wood structure performance under cyclic loading. Fasteners are one of the most important elements in a timber structure, and the inelastic behavior of a fastener under seismic loading is a determining factor of the response of the whole structure. Hence, the dynamic behaviour and modeling of the fasteners is a main issue for study and the focus of the present thesis. 1.2 Objective A n d Scope The objective of this research is to compare the response of a fastener predicted by two different hysteretic models: HYST (Foschi, 2000) and Bouc-Wen-Baber-Noori Model (BNW) (Foliente et al, 1995). HYST is a sophisticated finite-element model whereas BNW, which involves 13 parameters, is one of the best available mathematical models. The B N W parameters are calibrated by a particular cyclic displacement input, and the calibration results are used to compute the response of the fasteners for other histories or seismic excitations. HYST is used to compute the fastener response for the same inputs, and finally, the results from the two models are analyzed and compared. 1 1.3 Thesis Organization Section 1.4 in Chapter 1 will describe the model HYST and B N W in details. Chapter 2 is divided into two major sections. The first part wil l discuss the method for estimating the B N W parameters. The second area of discussion focuses on a model fitting to an earthquake record. A n evaluation of the accuracy of B N W is discussed in Chapter 3 and 4. Chapter 3 will evaluate the accuracy of the restoring forces with different ground displacement records, and the model will be tested and analyzed with different ground acceleration records, structural masses, and intensities in Chapter 4. Finally, a summary, conclusions, and recommendations for further research are given in Chapter 5. 1.4 Model Descriptions Exact solutions for the nonlinear, cyclic response of a fastener are not available, particularly because the behaviour shows complicated hysteretic properties, with complex force-deformation patterns such as non-conservative energy dissipation, pinching, and stiffness or strength degradation, as shown in Figure 1.1. The lack of closed form solutions for hysteretic random oscillators requires the use of approximate numerical methods, such as the finite-element model HYST or the empirical mathematical model BNW. 2 The Hysteresis Of A Fastener Under Cyclic Loading D e f o r m a t i o n ( m m ) Fig. 1.1 Load-displacement relationship of a fastener under cyclic events 1.4.1 Finite-element Model (HYST) HYST is a finite element-based mechanics model, developed by Foschi (2000). Figures 1.2 and 1.3 show, respectively, the fastener to be studied and the finite-element representation with the degrees of freedom u(x) and w(x). u(x) is axial deformation of the fastener whereas w(x) is the corresponding lateral deformation. A beam finite-element formulation is used for the member, using higher order interpolating shape functions (5 t h degree polynomial for w(x) and 3" degree polynomial for u(x)) in order to reduce the number of required elements. The advantage of this model is that, starting from basic material properties, it automatically adapts to any input displacement history. 3 In this research, HYST results are assumed equivalent to experimental results. The parameters in this model involve material properties, such as elastic modulus and yield stress of the fastener, and the compression (embedment) characteristics of the medium. Using the principle of virtual work, the deformation of a fastener has to obey: \aSe+ \p(\w\)-r-rSw = FSw J x \w\ (1.1) where, w stress in the fastener corresponding strain lateral deformation p(w) = reaction of the medium (force/unit length) , shown in Figure 1.2 F = lateral load applied to point Q F(t) p(w) M p(w) Fig. 1.2 Fastener subjected to lateral loads; degrees of freedom u(x) and w(x) 4 u(k) w(x) y Fig. 1.3 Finite element between nodes i and j Introducing the shape functions for w(x) and u(x), Foschi (2000) describes the final corresponding, matrix formulation. The embedment loads p(w) are assumed to follow a 5 relationship as in Figure 1.4. The force p(w) starts from zero and increases with initial stiffness K . As w increases, the rate of change or stiffness decreases, until the corresponding deformation reaches D m a x . The load at D m a x is defined as the maximum value, p m a x . Qi is the asymptotic tangent slope of the curve at Figure 1.4, and Qo is the y-intercept of the tangent. After reaching Dmax, the load begins to drop, and the forcing function p(w) has to pass through a point M shown in Figure 1.4. The coordinates of M are parameters of the model, given by the constants Qj and Q 3 . When the load is removed, the force and deformation decrease along a line PDo parallel to the initial stiffness. The deformation will not return to zero after the load has been removed, and this indicates that a gap between the fastener and the medium, D 0 , has been developed. Since the fastener moves either to the left or to the right during seismic excitations, the model uses a relationship like Figure 1.4 for each of the sides. 1.4.2 Empirical Mathematical Model (BNW) There are several mathematical models for representation of hysteretic behaviour. These normally involve parameters that have to be obtained by calibration to test data. Although the computational time to obtain the response is very short, calibrating the parameters may be lengthy. If the experimental displacements do not provide sufficient information, such as pinching or stiffness degradation of the fastener, the parameters controlling these characteristics may not be properly calibrated. The best and more versatile model of this kind is known as BNW. B N W involves 13 parameters and the relationship between force F and displacement A is given as a first order 6 differential equation, shown in Equation (1.2). The hysteretic response is obtained by integration of this equation, given a history A(t). dF l-v(Bsgn(A)\F\n~lF + y\F\") — = h(F){ ^ B V ; | 1 ^J -Li} (1.2) dA n where, sgn() = signum function i.e. sgn(a) gives -1,0 or 1 depending on whether a is respectively negative, zero, or positive. h(F) = 1.0 - £ exp[-(Fsgn(A) - qFuf I (1.3) and £ 0) = £ I 0 [1.0 - exp(-/>*)] (1.4) C,\ - the severity of pinching or magnitude of initial drop in dF/dA Z2(e) = (¥o+8¥s)(A. + ^ ) (1-5) <^ 2 = controls the spreading of the pinching spread £"10 = measure of total slip q = a fraction of F u as the pinching level p = controls the rate of initial drop in slope y/0 = controls to the amount of pinching 8 = constant specified for the desired rate of pinching evolution X = controls the rate of change of CiasC\ changes The ultimate value of F, F u , can be expressed as: F =[ f Furthermore, the hysteretic function s is written as: 7 e = ]fhdA = (1 - a)k j>JA (1.7) where, k = initial stiffness a = the ratio of final tangent stiffness to initial stiffness The strength- and stiffness-degradation parameters v and n are written as follows: v(e) = 1.0 + Sve (1.8) rj(e) = 1.0+ Sne (1.9) where, 8V = constant for strength degradation 8 n = constant for stiffness degradation 1.4.2.1 Parameters Constraints A l l 13 parameters except y must be positive, and a, 8 ,^ 8V, ^io, y, <!> m& $ v cannot be greater than one. The parameter n should be greater than or equal to one. In addition, the hysteretic shape parameters (3 and y should be chosen such that + y > 0 and y - B < 0 . The parameters in the B N W must be fitted so that the output from the integration of Equation (1.2), for a given history A(t), matches an experimental result for the same history. The calibration procedure is shown in Chapter 2. 8 Chapter 2 Parameter Estimation And BNW Model Fitting To An Experimental Record 2.1 Introduction The estimation of B N W parameters requires a force history and the corresponding displacement history (Yao, 1985). With these given inputs, the parameters can be obtained by optimization to produce a model matching the system output as close as possible, at least matching the chosen calibration data input. 2.2 Parameter Estimation The unknown parameters in B N W are estimated by minimizing the discrepancy between the computed forces and the experimental forces. This is done in the context of the Least Squares method, and the objective function is given as follows: n D ~ ^ theory, (2.1) where, Ftheory. = theoretical restoring force computed by B N W at point i experimental restoring force, in this thesis, the output from HYST for the same calibration input point i D = total discrepancy n = no. of data points 9 2.3 OPT Routine The parameters are optimized by a function minimization computer program called OPT. This optimization does not depend on function gradients but uses a search scheme between given variable bounds. It is necessary to enter the following information: • Initial model parameters, as components of a vector X . • The upper and lower bound ( X U and XB) of each unknown parameter. • The radius of the search zone, X, around the initial, random variable vector, X . • The maximum number of random samples used within the search zone. • The total number of repetitions. • The experimental data: deformation and the actual restoring force history First, the system randomly chooses values for the thirteen parameters in X , as the initial vector. Those numbers have to be within the bounds of the corresponding parameters. Then, the system uses the values of X to calculate the total response by integrating the B N W equation and determines the total discrepancy using the equation (2.1). After that, the system will randomly generate a new set of parameters, which are called X N EW , within a zone around the initial XOLD-*NEW =XOLD+R*A* (XL!, - XB;) (2.2) where, XUj = upper bound for i t h parameter XBj = lower bound for i t h parameter R = random number between 0.0 and 1.0 XOLD = set of parameters X corresponding to the initial, anchor point The system will calculate a discrepancy value for each X N E w - If the new value is smaller than the previous one, X N E W will become XOLD, and the search zone will shift to a new anchor X . The 10 process is repeated until all discrepancies D within a zone are greater than the value at the anchor. The entire procedure is repeated to account for different random values of the initial vector X . 2.3.1 Runge-Kutta Method The evaluation of the model response, Fth , requires the integration of the B N W equation. This is done with the Runge-Kutta method. Runge-Kutta (RK) method is one of the best approximation methods to solve the ordinary differential equation y' = f(y). The method achieves the accuracy of a Taylor series approach by predicting a function value at one point in terms of the function value and its derivatives at another point, which is expressed as: J V M ^ . + A E W J (2.3) where, ^ • = v „ + / £ > , T ; . for,- = W (2.4) a ,^ bj = the coefficients s = number of points in the scheme h = step size If ay =0 for i < j, the internal points Yni,..., Y n s can be computed one after the other from (2.4) and the method is known as explicit; otherwise, the method is known as implicit, such as the fourth-order R K method. The advantage of the implicit method is that the global truncation errors are smaller than the explicit, regardless of the value of the step size. Hence the fourth-order R K method was used. This is expressed as: 11 yM =y, +2k2 +2k3+k4)h (2.5) 0 where, *, = / ( W / ) (2-6a) K = / ( ^ + | * . ^ + | M ) (2-6b) K = /(*,• + (2.6c) kA= /{x.+Ky^k.h) (2.6d) h = step size 2.4 Model Fitting To A n Experimental Record The data used for calibration corresponded to a 63.5mm long, 3mm diameter fastener (Figure 2.1) with a cyclic displacement history A(t) at point P as shown in Figure 2.2. HYST was run for this case using the following properties: E = 200.0 kN/mm 2 a = 0.250 kN/mm 2 Qo = 0.250 kN/mm Q i =0.00150 kN/mm 2 Q 2 = 0.50 Q 3 = 1.50 K = 0.350 kN/mm 2 with the resulting hysteresis loop shown in Figure 2.3. 12 p A(t) — • i < 3.0mm 1 63.5mm r Fig. 2.1 The calibration specimen 13 D i s p l a c e m e n t T i m e His to ry Fig. 2.2 Calibration displacement history T h e Hys te re t i c M o d e l C a l c u l a t e d By HYST —^\—L D i s p l a c e m e n t (mm) Fig. 2.3 Calibration displacement history 14 Using A=0.05, Table 2.1 shows the optimization results and the lower and upper bounds used in OPT. Given: ^=0.05 The maximum number of random samples used per zone in the search =100 The total number of repetitions =100 P a r a m e t e r S y m b o l U n i t s L o w e r B o u n d U p p e r B o u n d O p t i m u m P a r a m e t e r n o . X B X U X 1 a - 0.0001 1 0.0001 2 5n - 0.0001 1 0.10169 3 P - 1 2.9535 4 5u - 0.01 1 0.43689 5 Y - -0.1 1 0.83545 6 q - 0.001 1 0.4554 7 d o 0.001 1 0.98985 8 p - 0.1 1.5 1.4046 9 M> - 0.001 1 0.11173 10 8\\i - 0.001 1 0.001 11 X - 0.001 1 0.38677 12 n - 3 8 3.2094 13 sk kN/mm 0.1 3 1.4814 Discrepancy 2.91840 Table 2.1 The calibration result The predicted B N W hysteresis for the calibration history of Figure 2.2 is shown in Figure 2.4, using the optimum parameters from Table 2.1. 15 Hysteretic Model Calculated By BNW —^\—L Displacement (mm) Fig. 2.4 The hysteretic model B N W (Calibration data) Correlation Of The Restoring Forces Restoring Forces Calculated by BNW (kN) Fig. 2.5 Comparison of the experimental and computed restoring forces with the same input displacement 16 Figures 2.3 and 2.4 show that the calculated B N W parameters allow for a close match of the input data, which is also shown in the comparison of the restoring forces at the same displacement level in Figure 2.5. Can these parameters be used with confidence to estimate the response for other input histories? This issue is studied in the following Chapters. 17 Chapter 3 Evaluation O f The Calibrated B N W Model Accuracy For Dif ferent Cyclic Records 3.1 In t roduct ion The purpose of this Chapter is to use the B N W parameters calibrated in Chapter 2 to determine the hysteresis for different cyclic records and to evaluate the accuracy of the model against the corresponding output from HYST. The optimum parameters are listed in Table 3.1. P a r a m e t e r S y m b o l U n i t s O p t i m u m P a r a m e t e r n o . X 1 a - 0.0001 2 Sri - 0.10169 3 P - 2 .9535 4 8u - 0.43689 5 Y - 0 .83545 6 q - 0.4554 7 C10 _ 0 .98985 8 p - 1.4046 9 w - 0.11173 10 8v|/ - 0.001 11 X - 0.38677 12 n - 3.2094 13 k k N / m m 1.4814 Table 3.1 The optimum parameters Three types of displacement records are chosen for the comparison: 1. Displacement peaks increase with time 2. Displacement peaks decrease with time 3. Earthquake-type displacements, considered as cyclic inputs 18 3.2 Results 3.2.1 Test 1: Cyclic History With Increasing Peaks The cyclic history shown in Figure 3.1 was used for the first test. This history shows a sequence of increasing peaks. The B N W output is shown in Figure 3.2, and the calculated output from HYST is shown in Figure 3.3. A comparison of forces at the same displacements is shown in Figure 3.4. Di spi acement T i me Hi st ory Time (s) Fig. 3.1 Displacement record for Test 1 19 .Hysta-d ic Model Calculated By.BNW W ^ Displacement (mm) Fig. 3.2 The hysteretic model B N W for Test 1 The Hysteretic Model Calculated By HYST i — i Displacement (mm) Fig. 3.3 The experimental hysteresis HYST for Test 1 20 "8 3 • o > LL. - 1 -i_ o tl QJ C o r r e l ^ i o n Of T h e R e s t o r i n g F o r c e s 45-degree line R e s t o r i n g F o r c e s C a l c u l ^ e d By B N W ( k H ) Fig. 3.4 Comparison of the experimental and computed forces for Test 1 at the same input displacement 21 3.2.2 Test 2: Cyclic History With Decreasing Peaks Figure 3.5 shows a cyclic history of decreasing peaks. The B N W output is shown in Figure 3.6, and the calculated output from HYST is shown in Figure 3.7. A comparison offerees at the same displacements is shown in Figure 3.8. Displacement Ti me History Ti me (s) Fig. 3.5 Displacement record For Test 2 22 Hysteretic Model Calculated By BNW Displacement (mm) Fig. 3.6 The hysteretic model B N W for Test 2 The Hysteretic Model Calculated By HYST -4 Displacement (mm) Fig. 3.7 The experimental hysteresis HYST for Test 2 23 Restoring Forces CsJculled By BNW (kN J Fig. 3.8 Comparison of experimental and computed restoring forces for Test 2 at the same input displacement 24 3.2.3 Test 3: Earthquake Displacements 3.2.3.1 Test 3a: Displacements W i t h A Strong Mot ion Segment O f Average Durat ion Figure 3.5 shows a displacement history with a strong motion segment of average duration. The results obtained by B N W and HYST are shown in Figures 3.10 and 3.11 respectively, and Figure 3.12 shows the comparison of forces computed by the two models. D i s p l a c e m e n t T i m e H i s t o r y T i m e ( s ) Fig. 3.9 Displacement record for Test 3a 25 The Hysteretic Model Calculated By BNW -4 Displacement (mm) Fig. 3.10 The hysteretic model B N W for Test 3a The Hysteretic Model Calculated By HYST •Dispiabement (mm) Fig. 3.11 The experimental hysteresis HYST for Test 3a 26 5* CD "8 ts "5 o TO z O CO U . X c "C o m m Correlation Of Restoring Forces •46-degree line Restoring Forces Calculated By BNW (kN) Fig. 3.12 Comparison of experimental and computed restoring forces for Test 3a at the same input displacement 27 3.2.3.2 Test 3b: Displacements Correspond To A Shock Of Short Duration The displacement history is shown in Figure 3.13. A big shock happened initially, and the displacement died down fairly quickly after the shock. Figures 3.14 and 3.15 are the corresponding outputs computed by either B N W or HYST, and Figure 3.16 shows the comparison of the results. 28 The Hysteretic Model Calculated By BMW 4h - ^ H Di spl acement (mm) Fig. 3.14 The hysteretic model B N W for Test 3b T h e Hyst e re t i c M o d e l C a l c u l a t e d By HYST 4 ^ Displ acement (mm) Fig. 3.15 The experimental hysteresis HYST for Test 3b 29 Correl ation Of Restoring Forces Restoring Forces Calculated By BNW (kN) Fig. 3.16 Comparison of experimental and computed restoring forces for Test 3b at the same input displacement 30 3.2.3.3 Test 3c: Displacements With Strong Motion Of Long Duration Figure 3.17 shows the displacement history of a long duration earthquake. Figure 3.18 shows the hysteresis loop computed by BNW, and Figure 3.19 is the output from experimental model HYST. A comparison of the computed forces calculated by two different models is shown in Figure 3.20. Displacement Time History Ti me (s) Fig. 3.17 Displacement record for Test 3c 31 T h e H y s t e r e t i c M o d e l C a l c u l a t e d By B N W tr-, D i s p l a c e m e n t ( m m ) Fig. 3.18 The hysteretic model B N W for Test 3c T h e H y s t e r e t i c M o d e l C a l c u l a t e d By HYST * 4 Di sp i a c e m e n t ( m m ) Fig. 3.19 The experimental hysteresis HYST for Test 3c 32 , Correl atio n, Of Th e Resto ring Forces Restoring Forces Calculated By BNW (kN) Fig. 3.20 Comparison of experimental and computed restoring forces for Test 3c at the same input displacement 33 3.3 Discussion Fairly good agreement is observed between B N W and HYST; however, B N W discrepancies increase after the application of a peak of high intensity, taking the fastener into the nonlinear range. Thus, the predicted behaviour after a shock of short duration is not as good after the first peak of the shock. The next Chapter will extend the explorations and to evaluate the influences to the response of a single-degree-of-freedom (SDOF) oscillator under seismic excitations. 34 Chapter 4 Comparison For SDOF Seismic Response 4.1 Introduction Figure 4.1 shows the fastener, with a mass m at point P, subjected to a seismic excitation. If the displacement of the medium is Ag(t), and A(t) is the relative displacement of m with respect to the medium, then: mA + cA + F(A) = - m A (4.1) Ag(t) A(t) mass=m P i Fig. 4.1 Nonlinear, SDOF system under cyclic loading This Chapter presents the response of this SDOF oscillator (Figure 4.1) using either the calibrated B N W model or the HYST model to calculate the hysteretic nonlinear restoring shear force F(A). It is also assumed that there is viscous damping with a constant c. 35 4.2 Procedure A total of 30 acceleration records A (f), all with a peak ground acceleration (PGA) of l.Og and shown in Appendix n, were used for the comparison. They were divided into 3 categories of duration for the strong motion phase: Short, Intermediate, and Long. The power spectral density proposed by Kanai-Tajimi with Clough-Penzien filter was selected to generate these records. A modulation function was applied to introduce non-stationarity and different strong motion durations, from 1 to 30 seconds. Each record was amplified to have PGA of 0.5g, 0.75g, or l.Og in combination with masses of 20.0kg, 35.0kg, or 50.0kg. A l l other parameters and mechanical properties were identical to those used in Chapter 1 and 2. The B N W and the HYST models were used in a dynamic time step solution of Equation (4.1) (Newmark's method with a constant average acceleration as in Appendix I). The outputs were as follows: • Relative maximum displacement A with respect to the medium, in absolute value, • Relative permanent displacement with respect to the medium (i.e. residual displacement), in absolute value. and these are shown, for the different combinations, in Tables 4.1 through 4.6. 36 4.3 Results 4.3.1 Relative Maximum Displacement With Respect To The Medium (a) Short PGA Acceleration 0.5g 0.75g 1.0g Record # HYST BNW HYST BNW HYST BNW 1 0.078 0.083 0.119 0.124 0.162 0.167 2 0.088 0.095 0.135 0.143 0.136 0.151 3 0.063 0.082 0.097 0.123 0.134 0.164 4 0.091 0.098 0.132 0.145 0.174 0.195 5 0.061 0.073 0.093 0.11.0 0.126 0.148 6 0.062 0.074 0.095 0.111 0.128 0.148 7 0.084 0.092 0.126 0.133 0.168 0.184 8 0.068 0.079 0.107 0.118 0.154 0.157 9 0.058 0.065 0.087 0.097 0.118 0.129 10 0.088 0.103 0.140 0.154 0.200 0.206 (b) Intermediate PGA Acceleration 0.5g 0.75g 1.0g Record # HYST BNW HYST BNW HYST BNW 1 0.078 0.078 0.119 0.118 0.151 0.157 2 0.078 0.086 0.118 0.129 0.160 0.172 3 0.069 0.082 0.103 0.124 0.144 0.165 4 0.070 0.079 0.106 0.118 0.143 0.158 5 0.061 0.070 0.092 0.104 0.124 0.140 6 0.072 0.079 0.109 0.118 0.150 0.158 7 0.060 0.068 0.092 0.102 0.126 0.136 8 0.069 0.081 0.102 0.121 0.135 0.161 9 0.060 0.070 0.090 0.105 0.122 0.140 10 0.077 0.080 0.118 0.120 0.145 0.161 (c) Long PGA Earthquake 0.5g 0.75g 1.0g Record # HYST BNW HYST BNW HYST BNW 1 0.072 0.085 0.110 0.128 0.149 0.173 2 0.069 0.079 0.104 0.119 0.148 0.160 3 0.059 0.067 0.088 0.100 0.118 0.133 4 0.076 0.093 0.117 0.140 0.149 0.166 5 0.075 0.078 0.120 0.117 0.165 0.156 6 0.074 0.077 0.110 0.115 0.146 0.153 7 0.082 0.094 0.139 0.141 0.182 0.189 8 0.079 0.079 0.121 0.118 0.161 0.157 9 0.072 0.087 0.109 0.124 0.164 0.174 10 0.069 0.072 0.103 0.108 0.144 0.144 Table 4.1 Maximum Displacement (in mm) for a 20.0kg mass 37 (a) Short P G A Acceleration 0.5g 0.75g 1.0g Record # H Y S T BNW H Y S T BNW H Y S T BNW 1 0.126 0.151 0.201 0.229 0.326 0.315 2 0.192 0.212 0.327 0.322 0.499 0.440 3 0.145 0.153 0.235 0.229 0.366 0.306 4 0.178 0.188 0.277 0.278 0.418 0.356 5 0.120 0.147 0.187 0.222 0.283 0.304 6 0.122 0.142 0.191 0.212 0.305 0.288 7 0.131 0.152 0.211 0.229 0.337 0.310 8 0.122 0.143 0.195 0.214 0.354 0.280 9 0.104 0.118 0.163 0.178 0.228 0.244 10 0.132 0.148 0.226 0.220 0.312 0.290 (b) Intermediate P G A Acceleration 0.5g 0.75g 1.0g Record # H Y S T BNW H Y S T BNW H Y S T BNW 1 0.133 0.136 0.207 0.204 0.362 0.277 2 0.138 0.161 0.213 0.245 0.305 0.338 3 0.135 0.151 0.213 0.228 0.350 0.316 4 0.128 0.140 0.203 0.212 0.339 0.292 5 0.114 0.125 0.173 0.190 0.248 0.260 6 0.127 0.124 0.190 0.183 0.275 0.238 7 0.124 0.143 0.193 0.216 0.280 0.291 8 0.143 0.147 0.241 0.221 0.385 0.303 9 0.117 0.132 0.185 0.198 0.273 0.266 10 0.153 0.162 0.259 0.250 0.349 0.367 (c) Long P G A Acceleration 0.5g 0.75g 1.0g Record # H Y S T BNW H Y S T BNW H Y S T BNW 1 0.132 0.141 0.209 0.218 0.334 0.320 2 0.122 0.127 0.196 0.194 0.302 0.276 3 0.109 0.129 0.174 0.193 0.277 0.258 4 0.157 0.179 0.265 0.268 0.493 0.443 5 0.145 0.159 0.244 0.244 0.358 0.362 6 0.135 0.152 0.211 0.227 0.330 0.306 7 0.155 0.164 0.307 0.246 0.580 0.381 8 0.129 0.141 0.219 0.213 0.314 0.300 9 0.164 0.179 0.284 0.269 0.381 0.377 10 0.135 0.121 0.213 0.182 0.350 0.243 Table 4.2 Maximum Displacement (in mm) for a 35.0kg mass 38 (a) Short P G A Acceleration 0.5g 0.75g 1.0g Record # H Y S T BNW H Y S T BNW H Y S T BNW 1 0.207 0.218 0.362 0.342 0.738 0.625 2 0.328 0.329 0.673 0.512 1.103 0.763 3 0.209 0.208 0.376 0.295 0.634 0.396 4 0.294 0.269 0.599 0.447 0.852 2.207 5 0.181 0.184 0.349 0.285 0.684 0.434 6 0.162 0.181 0.286 0.282 0.523 0.424 7 0.221 0.238 0.447 0.379 0.932 0.538 8 0.248 0.234 0.437 0.363 0.780 0.492 9 0.160 0.173 0.274 0.257 0.572 0.363 10 0.322 0.301 0.669 0.564 1.332 6.111 (b) Intermediate P G A Acceleration 0.5g 0.75g 1.0g Record # H Y S T BNW H Y S T BNW H Y S T BNW 1 0.246 0.229 0.526 0.358 0.580 0.531 2 0.225 0.231 0.392 0.375 0.700 0.727 3 0.224 0.224 0.325 0.345 0.468 0.485 4 0.193 0.200 0.418 0.310 0.979 0.433 5 0.169 0.167 0.311 0.262 0.549 0.417 6 0.192 0.191 0.272 0.288 0.431 0.388 7 0.154 0.174 0.278 0.258 0.541 0.371 8 0.207 0.215 0.344 0.331 0.539 0.289 9 0.184 0.201 0.353 0.306 1.059 0.428 10 0.266 0.259 0.494 0.418 1.051 0.913 (c) Long P G A Acceleration 0.5g 0.75g 1.0g Record # H Y S T BNW H Y S T BNW H Y S T BNW 1 0.197 0.205 0.495 0.418 0.907 18.408 2 0.172 0.187 0.305 0.291 0.535 12.146 3 0.168 0.181 0.284 0.289 0.630 0.528 4 0.239 0.258 0.507 0.532 0.822 20.968 5 0.231 0.218 0.472 0.337 0.760 9.212 6 0.173 0.188 0.291 0.295 0.568 0.364 7 0.310 0.291 0.515 1.062 1.030 25.682 8 0.200 0.209 0.200 0.312 0.612 35.093 9 0.276 0.287 0.527 1.979 0.843 35.825 10 0.224 0.198 0.325 0.299 0.468 0.567 Table 4.3 Maximum displacement (in mm) for a 50.0kg mass 39 4.3.2 Residual Displacement (a) Short P G A Accel eratic 0.5g 0.75g 1.0g Record # H Y S T B N W HYST B N W H Y S T B N W 1 0.002 0.000 0.004 0.000 0.004 0.003 2 0.000 0.000 0.001 0.000 0.000 0.000 3 0.001 0.000 0.003 0.000 0.005 0.001 4 0.001 0.001 0.002 0.002 0.004 0.004 5 0.001 0.001 0.003 0.001 0.005 0.001 6 0.001 0.001 0.003 0.001 0.005 0.002 7 0.001 0.000 0.002 0.001 0.002 0.001 8 0.001 0.001 0.001 0.001 0.006 0.000 9 0.001 0.000 0.003 0.001 0.005 0.002 10 0.002 0.001 0.001 0.002 0.005 0.004 (b) Intermediate P G A Acceleratic 0.5g 0.75g 1.0g Record # HYST B N W H Y S T B N W H Y S T B N W 1 0.000 0.001 0.003 0.001 0.003 0.001 2 0.003 0.002 0.008 0.003 0.011 0.004 3 0.003 0.000 0.004 0.000 0.007 0.001 4 0.000 0.000 0.001 0.000 0.001 0.000 5 0.002 0.001 0.003 0.002 0.004 0.002 6 0.001 0.001 0.003 0.002 0.003 0.003 7 0.001 0.000 0.001 0.000 0.002 0.000 8 0.000 0.000 0.001 0.000 0.001 0.000 9 0.000 0.000 0.000 0.000 0.000 0.000 10 0.002 0.002 0.003 0.002 0.000 0.001 ( c ) Long P G A Acceleratic 0.5g 0.75g 1-Og Record # HYST B N W H Y S T B N W HYST B N W 1 0.005 0.004 0.009 0.006 0.014 0.008 2 0.003 0.001 0.005 0.002 0.004 0.004 3 0.002 0.001 0.005 0.002 0.010 0.002 4 0.000 0.001 0.002 0.000 0.077 0.069 5 0.005 0.001 0.004 0.001 0.008 0.001 6 0.001 0.000 0.003 0.001 0.004 0.001 7 0.001 0.001 0.009 0.001 0.013 0.000 8 0.000 0.000 0.001 0.000 0.004 0.000 9 0.002 0.001 0.057 0.052 0.001 0.002 10 0.003 0.001 0.004 0.001 0.007 0.002 Table 4.4 Residual displacement (in mm) for a 20.0kg mass 40 (a) Short P G A Acceleratic 0.5g 0.75g 1.0g Record # H Y S T BNW H Y S T BNW H Y S T BNW 1 0.002 0.000 0.018 0.000 0.046 0.002 2 0.003 0.001 0.017 0.012 0.052 0.002 3 0.001 0.000 0.037 0.002 0.079 0.010 4 0.003 0.001 0.013 0.005 0.058 0.027 5 0.003 0.001 0.002 0.000 0.009 0.006 6 0.004 0.003 0.018 0.009 0.066 0.029 7 0.000 0.007 0.021 0.004 0.085 0.019 8 0.004 0.000 0.004 0.002 0.002 0.009 9 0.004 0.002 0.002 0.005 0.011 0.014 10 0.009 0.000 0.011 0.002 0.029 0.004 P G A Acceleratic 0.5g 0.75g 1.0g Record # H Y S T BNW H Y S T BNW H Y S T BNW 1 0.003 0.001 0.016 0.003 0.075 0.011 2 0.008 0.001 0.008 0.005 0.009 0.018 3 0.000 0.002 0.006 0.005 0.040 0.010 4 0.001 0.000 0.008 0.002 0.028 0.012 5 0.005 0.001 0.008 0.000 0.014 0.006 6 0.001 0.002 0.001 0.002 0.006 0.000 7 0.007 0.000 0.000 0.002 0.003 0.009 8 0.000 0.001 0.003 0.002 0.063 0.008 9 0.000 0.000 0.001 0.001 0.006 0.003 10 0.006 0.002 0.012 0.001 0.004 0.031 (c) Long P G A Acceleratic 0.5g 0.75g LOg Record # H Y S T BNW H Y S T BNW H Y S T BNW 1 0.011 0.005 0.025 0.004 0.061 0.007 2 0.008 0.002 0.012 0.001 0.039 0.013 3 0.005 0.002 0.011 0.001 0.006 0.016 4 0.001 0.003 0.014 0.016 0.025 0.086 5 0.003 0.000 0.012 0.009 0.007 0.067 6 0.005 0.000 0.014 0.001 0.063 0.010 7 0.006 0.002 0.011 0.009 0.056 0.093 8 0.003 0.000 0.014 0.006 0.008 0.024 9 0.008 0.002 0.024 0.001 0.035 0.023 10 0.000 0.000 0.006 0.001 0.040 0.007 Table 4.5 Residual displacement (in mm) for a 35.0kg mass 41 (a) Short P G A Acceleratic 0.5g 0.75g 1.0g Record # H Y S T BNW H Y S T BNW H Y S T BNW 1 0.007 0.001 0.038 0.026 0.059 0.273 2 0.075 0.024 0.207 0.129 0.356 0.415 3 0.013 0.003 0.003 0.000 0.060 0.022 4 0.051 0.005 0.044 0.076 0.095 1.459 5 0.004 0.000 0.044 0.006 0.127 0.037 6 0.008 0.003 0.044 0.005 0.049 0.077 7 0.026 0.005 0.155 0.042 0.437 0.107 8 0.005 0.001 0.007 0.012 0.030 0.042 9 0.002 0.002 0.026 0.005 0.145 0.047 10 0.014 0.026 0.074 0.051 0.381 0.562 P G A Acceleratic 0.5g 0.75g 1.0g Record # H Y S T BNW H Y S T BNW H Y S T BNW 1 0.032 0.003 0.105 0.020 0.021 0.014 2 0.001 0.002 0.026 0.059 0.056 0.417 3 0.020 0.003 0.003 0.004 0.006 0.069 4 0.006 0.000 0.021 0.010 0.005 0.021 5 0.006 0.002 0.057 0.008 0.099 0.076 6 0.006 0.002 0.010 0.002 0.026 0.032 7 0.007 0.002 0.016 0.000 0.011 0.029 8 0.007 0.000 0.020 0.006 0.054 0.019 9 0.001 0.001 0.011 0.008 0.003 0.021 10 0.035 0.009 0.067 0.062 0.231 0.406 (c) Long P G A Acceleratic 0.5g 0.75g 1.0g Record # H Y S T BNW H Y S T BNW H Y S T BNW 1 0.025 0.010 0.022 0.128 0.019 6.674 2 0.011 0.001 0.042 0.006 0.047 0.984 3 0.011 0.000 0.016 0.024 0.059 0.288 4 0.004 0.006 0.023 0.102 0.067 5.478 5 0.011 0.000 0.025 0.020 0.006 2.084 6 0.012 0.002 0.021 0.015 0.067 0.024 7 0.019 0.016 0.002 0.606 0.143 12.870 8 0.008 0.001 0.008 0.003 0.069 13.072 9 0.009 0.001 0.009 0.429 0.041 0.664 10 0.020 0.005 0.003 0.036 0.006 0.286 Table 4.6 Residual displacement (in mm) for a 50.0kg mass 42 4.4 Comparison of the BNW Results And the HYST Results 4.4.1 Maximum Displacement In order to further compare the results, two indicators of agreement were obtained • A linear correlation coefficient p between B N W and HYST results. The root mean square (rms) of the differences as rms = i=1 Tables 4.7 and 4.8 show the linear correlations and the corresponding rms under different strong motion phases, whereas Tables 4.9 and 4.10 show the correlations and rms under different masses and intensities with all the motion phases. Types of Duration Mass (kg) PGA (g) 0.5 0.75 1.0 Short 20 0.944 0.952 0.942 35 0.975 0.962 0.919 50 0.979 0.974 0.768 Intermediate 20 0.833 0.802 0.884 35 0.806 0.781 0.526 50 0.960 0.760 0.478 Long 20 0.739 0.768 0.806 35 0.879 0.822 0.749 50 0.941 0.603 0.558 Table 4.7 Correlation p under different types of motion phase Types of Duration Mass (kg) PGA (g) 0.5 0.75 1.0 Short 20 0.011 0.014 0.017 35 0.019 0.019 0.043 50 0.015 0.090 1.588 Intermediate 20 0.009 0.013 0.016 35 0.013 0.017 0.045 50 0.011 0.072 0.288 Long 20 0.010 0.012 0.013 35 0.015 0.024 0.075 50 0.016 0.495 19.847 Table 4.8 rms under different types of motion phase 43 Mass P G A (g) (kg) 0.5 0.75 1.0 20 0.872 0.877 0.890 35 0.910 0.870 0.787 50 0.965 0.444 0.184 Table 4.9 Correlation p with all different types of motion ph Mass P G A (g) (kg) 0.5 0.75 1.0 20 0.010 0.013 0.015 35 0.016 0.020 0.056 50 0.014 0.293 11.496 Table 4.10 rms with all different types of motion phases 44 4.4.2 Residual Displacement Tables 4.11 and 4.12 show the correlations and rms of residual displacements under different motion phase. Similarly, Table 4.13 and 4.14 show the comparison of two models under different masses and intensities with all types of motion phases. Types of Duration Mass P G A (g) (kg) 0.5 0.75 1.0 Short 20 0.422 0.000 0.175 35 0.000 0.078 0.409 50 0.563 0.792 0.083 Intermediate 20 0.681 0.769 0.784 35 0.031 0.212 0.000 50 0.821 0.394 0.705 Long 20 0.676 0.994 0.985 35 0.638 0.000 0.000 50 0.610 0.000 0.603 e 4.11 Correlation p under different types of motion phase Types of Duration Mass P G A (g) (kg) 0.5 0.75 1.0 Short 20 0.001 0.002 0.003 35 0.004 0.014 0.040 50 0.024 0.049 0.455 Intermediate 20 0.001 0.002 0.003 35 0.004 0.006 0.030 50 0.014 0.034 0.129 Long 20 0.002 0.004 0.006 35 0.004 0.012 0.041 50 0.010 0.237 6.406 Table 4.12 rms under different types of motion phase 45 Mass P G A (g) (kg) 0.5 0.75 1.0 20 0.691 0.983 0.974 35 0.117 0.154 0.002 50 0.592 0.061 0.023 Table 4.13 Correlation p with all different types of motion ph; Mass P G A (g) (kg) 0.5 0.75 1.0 20 0.001 0.003 0.004 35 0.004 0.011 0.038 50 0.017 0.141 3.708 Table 4.14 rms with all different types of motion phases 46 4.4.3 Hysteresis Loops Figures 4.2 through 4.10 (a, b) show the calculated hysteresis loops using either HYST B N W model, for different correlations. Figure 4.2 shows nearly elastic response; however, the nonlinearity increases (Figure 4.8 to 4.10) when the mass reaches 50.0kg. 47 HYST Sh Deformation (mm) (a) HYST B N W j>.2 -a. < s Deformation (mm) (b) B N W Fig. 4.2 The hysteresis for a 20.0kg mass, PGA= LOg, type of duration= Short 48 HYST 0,25, . o.as ' Deforma t ion (mm) (a) HYST BNW &rSS--Deforma t ion (mm) (b) B N W Fig. 4.3 The hysteresis for a 20.0kg mass, PGA=1.0g, type of duration= Intermediate 49 HYST Z ' tu o o LL 0,35 , 0 2 rt 15 •Q.15 0.05 0.1 o.as Deformation (mm) (a) HYST B N W eas-a a ^ ST+S--itVt-aos--0.1 -0.05 0.05 0.15.. -0,15 -3,35 beformatjon (mm) (b) B N W Fig. 4.4 The hysteresis for a 20.0kg mass, PGA=1.0g, type of duration= Long 50 -o . o LL H Y S T • O S - , rt y\ 0 3 0 3 rt . j 0.2 0.3 6*5 -Deformation (mm) (a) HYST BNW ' O.S, | 0.0 cu o IL -O.i . -o.a -0.2 0.1 0.2. 0.4 • & ^ _ L Deformation (mm) (b) B N W Fig. 4.5 The hysteresis for a 35.0kg mass, PGA=1.0g, type of duration= Short 51 HYST 0) 0:1 0.2 0.3 • Deformation (mm) (a) HYST BNW fr*-rt A U . J rW— 0) -0.4 U De fo rma t ion (mm) (b) B N W Fig. 4.6 The hysteresis for a 35.0kg mass, PGA=1.0g, type of duration= Intermediate 52 LL 0.2 A^E-Deformation (mm) (a) HYST BNW 0 . * 1 z O - 0 . 0 - 0 . 2 -0.1 -er4-0.1 0 .2 0 . 3 Deformation (mm) (b) B N W Fig. 4.7 The hysteresis for a 35.0kg mass, PGA= l.Og, type of duration= Long 53 HYST Def o r m i i o n (mm) (b) B N W Fig. 4.8 The hysteresis for a 50.0kg mass, PGA=1.0g, type of duration^ Short 54 HYST 15 0 6 M1 Deformation (mm) (a) HYST BNW D e f o r m a t ion (m m) (b) B N W 4.9 The hysteresis for a 50.0kg mass, PGA=1.0g, type of duration= Intermediate 55 HYST O.S S r e - i Deformation (mm) (a) HYST BNW Deformation (mm) (b) B N W Fig. 4.10 The hysteresis for a 50.0kg, PGA=1.0g, type of duration= Long 56 4.5 Discussion The B N W model gives a fairly accurate prediction of the maximum displacements when the mass and P G A are small. However, the discrepancies increase when the nonlinearity of the response increases, particularly for longer duration earthquakes. In general, residuals are not predicted well by BNW, given that the behaviour is not predicted well after the maximum peak demand has occurred. 57 Chapter 5 Concluding Remarks 5.1 Research Contr ibut ions The contributions of the thesis are summarized as follows: • The objective of this research was to compare the response of a fastener under cyclic loading as predicted by two hysteretic models: HYST and BNW. • The thesis describes the fitting of an experimental hysteresis loop to calibrate the 13 B N W parameters and studies the accuracy of the fitted B N W model for other cyclic histories or seismic excitations. 5.2 Conclusions • B N W gives fairly accurate results for relative maximum displacement with respect to the medium, when the restoring forces and the structural mass are small and the structural response remains elastic. • B N W cannot accurately predict the residual deformation. 5.3 Recommendation For Future Investigations The thesis represents a step in understanding the applicability of the B N W model, and the aspects covered are rather limited. It looked at the use of two models in assessing maximum structural response during an earthquake. The differences found in estimating the response will directly translate into the differences in estimating the reliability of the structure. This should then be extended to determine differences in reliability for different performance levels, in order to fully assess the significance of the variances in hysteretic model perdictions in comparison with the experimental test results. 58 References Baber, T., Noori, M . N . (1985), "Random Vibration Of Degrading, Pinching Systems," J. Engrg Mech., ASCE, 111(8), P. 1010-1026. Chopra, A . K . , (1999), Dynamics Of Structures Theory And Applications To Earthquake Engineering, Prentice Hall, New Jersey. Chapra, S .C, Canale, R.P. (1998), Numerical Methods For Engineers, McGraw-Hill, New York. Dobson, S., Noori, M . , Hou, Z., Dimentberg, M . (1998), " Direct Implementation Of Stochastic Linearization For SDOF Systems With General Hysteresis," Structural Engineering And Mechancis, vol. 6(5), P. 473-484. Foliente, G.C. (1995), "Hysteresis Modeling Of Wood Joints And Structural Systems," Journal Of Structural Engineering, Jun 1995, P. 1013-1021. Foschi, R.O. (2000), "Modeling The Hysteretic Response Of Mechanical Connections For Wood Structures," University of British Columbia, BC. Hairer, E., Lubich, C , Roche, M . (1980), The Numerical Solution of Differential-Algebraic Systems by Runge-Kutta Methods, Springer-Verlag, New York. Heine, C P . (2001), "Simulated Response of Degrading Hysteretic Joints With Slack Behavior," PhD Thesis, Virginia Polytechnic Institute And State University, Blacksburg, V A . Lence, B. , (2002), CIVL 555 Analysis Of Civil Engineering System Class Notes, University Of British Columbia, Vancouver, BC. Rahman, S., Grigoriu, M . , (1994), "Local And Models For Nonlinear Dynamic Analysis Of Multi-story Shear Buildings Subject To Earthquake Loading", Computer And Structures, vol. 53 (3), P. 739-754. Weld, L.D. (1916), Theory Of Errors And Least Squares, The MacMillan Company, New York. 59 Appendix I Newmark's Method For Nonlinear Systems A l . l Theory In 1959, N . M . Newmark developed time-stepping methods based on the following equations: mxM + cxM + F(xM ) = -m(ag (A. 1.1) where, m = lumped mass c = damping constant xM = relative velocity with respect to the medium at step i+1 xM = relative acceleration with respect to the medium at step i+1 F (XJ + I ) = inelastic (hysteretic) restoring force a = ground acceleration at step i+1 The time-stepping relative acceleration and velocity with respect to the medium at step i and are related as follows: xM = x i + ^ h L ^ (A. 1.2a) 4 = TTTTT (XM ~ xi ~ XA<) ~ xi (A. 1.2b) (Ar) Substitute equations (A. 1.2) into (A. 1.1): r Am 2 . r Am 2 . Am [ 7 + — c]xi+. - [ + — c]x, - ( + c)x, + (Ar) 2 Ar 1 + 1 (Ar) Ar A^ ' ' + F(xM ) - mXi = -m(ag)M (A. 1.3) This is a nonlinear equation because F(XJ+I) is a nonlinear restoring force term. 60 Rearrange the equation (A. 1.3) to define a quantity of \\i that must be zero at the solution: . Am 2 n r Am 2 n .Am T + C\XM - [ T + c i x i ~ ( " ( A r ) At 1 + 1 ( A r ) At ' At + F(XM) - mxM + mias )M ( A l -4) An iteration scheme (Newton) can be set up to find the zero of xM=x* ^ (A1.5) A * * dxM in which, d¥ _ Am +2_c+dF(xM) ( A i 6 ) dxM (At) At dxM Determine the root x, shown in Figure A l . l , by linear interpolation A1.2 Determination Of The Root Of Xj+i Equation ( A l .5) is used to get x 2 from x, which is a guess, and then x for \|/(x)=0 is obtained by successive interpolation. 1. Assume x 0 and F 0 are equal to x i and Fi respectively 2. Determine and ^^/jx using equation (A 1.4) and (A 1.5) 3. Calculate x 2 from equation (A 1.6) and determine \\>2 61 4. Determine v|/ from equation (A1.4) and x from the following equation: 5. If y - y , > 0, \\f]=\\i andxi=x If y/ • y/x < 0, \4/2=vi/ and x2=x 6. Repeat the iterations until the absolute value of x-Xj or \\i is less than 1 .OE-08, then Xi+i= x 62 A p p e n d i x II Inpu t A c c e l e r a t i o n R e c o r d s , N o r m a l i z e d t o peak =1.0g A.2.1 Dura t i on Of S t r o n g S e i s m i c E x c i t a t i o n : S h o r t Acceleration Record (Short#1) T i m e (sec) Acceleration Record (Short #2) T ime (sec) Acceleration Record (Short #3) _15 20 25 30 35 40 T ime (sec) 63 Acceleration Record (Short #4) 3 1 c o 0.5 (0 k_ 0 © O -0.5 O < -1 I ,j I . . 15 ? 0 ? 5 30 35 40 'Mi Time (sec) Acceleration Record (Short #5) Time (sec) Time (sec) J.0 Acceleration Record (Short #6) 25 30 35 40 64 Acceleration Record (Short #7) Time (sec) Acceleration Record (Short #8) 3 1 j c o 0.5 --^5 <Q i_ 0 -0 -0.5 <> 0) u -1 -< Time (sec) Acceleration Record (Short #9) 3 c o (0 © u o < 1 0.5 0 -0.5 -1 0 I 5 10 15 90 Time (sec) 65 66 A.2.2 Duration Of Strong Seismic Excitation: Intermediate Acceleration Record (Intermediate #1) Time (sec) Acceleration Record (lntermediate#2) 0.5 2 0 ft -0.5 u o < -1 lb-Time (Sec) Acceleration Record (lntermediate#3) CT 1 IT 0.5 1 0 jj> -0.5 $ O -1 o < -1.5 , , I. . I . iMTlHVM I T0 45 L-20 25 30 35 40 Time (sec) 67 Acceleration Record (intermediate#4) < -1 Time (sec) Accelerat ion Record (lntermediate#5) Time (sec) Acceleration Record (lntermediate#6) Time 9sec) 68 Acceleration Record (Intermediate #7) Time (sec) Acceleration Record (lntermediate#8) 3 1 § 0.5 2 0 8 -0.5 u < 1 I \ 0 1^ 1 Time (sec) Acceleration Record (lntermediate#9) 69 70 A.2.3 Duration Of Strong Seismic Excitation: Long Acceleration Record (Long #1) 1 I 0 0.5 ro © u u < -0.5 •1.5 Time (sec) Acceleration Record (Long #2) Time (sec) Acceleration Record (Long #3) 40 _50 Time (sec) 71 Acceleration Record (Long #4) Time (sec) 3 1 c o 0.5 re 0 V <D U -0.5 o < -1 ..Ln i i. II IJI.II.I NI.I Ll ,I I y l L J i u L i L . . u » . , "11 I ' l l 1 " ML Time (sec) Acceleration Record (Long #5) _50 Acceleration Record (Long #6) 30 40 Time (sec) 72 Acceleration Record (Long #7) "45 Time (sec) Accelerat ion Record (Long #8) c 1 J 0 0.5 -ro 0 u> 0 -\ 0 -0.5 0 0 < -1 --45 Time (sec) Acceleration Record (long #9) § 1 2 0 o 8 -0.5 * -1 ll I . I. 1 • I 30 as_ ,40 Time (sec) 7 3 74
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A comparison of two hysteretic models in predicting the response of a connector under cyclic loading Lo, Yvonne P. Y. 2002
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Title | A comparison of two hysteretic models in predicting the response of a connector under cyclic loading |
Creator |
Lo, Yvonne P. Y. |
Date Issued | 2002 |
Description | The effect of seismic loading on structures has become a main concern due to the number of collapses or incidents of substantial damage during recent earthquakes. The inelastic behavior of a fastener under seismic loading is a determinant factor of the response of the whole structure; hence, the dynamic behaviour of the fasteners is a main issue for study. The objective of this research is to compare the response of a fastener predicted by a finite-element model (HYST) and the empirical Bouc-Wen-Baber-Noori model (BNW). HYST is a mechanics-based model that uses material properties of the fastener and the characteristics of its surrounding medium to predict the response. BNW is a mathematical model involving 13 parameters. The hysteretic results calculated by this model are obtained by integration of a first order differential equation, given a displacement history A(t). The BNW parameters are calibrated using the response to a particular cyclic displacement history, and then the responses are evaluated for other different histories or seismic excitations. Results are compared to the corresponding HYST outputs. The results show that BNW gives comparable values for relative maximum displacements with respect to the medium, and restoring forces when the response of the structure remains elastic. However, the discrepancies increase with nonlinearities, and the model cannot accurately predict, for example, the residual deformations. |
Extent | 2475163 bytes |
Genre |
Thesis/Dissertation |
Type |
Text |
FileFormat | application/pdf |
Language | eng |
Date Available | 2009-09-22 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
IsShownAt | 10.14288/1.0063650 |
URI | http://hdl.handle.net/2429/13031 |
Degree |
Master of Applied Science - MASc |
Program |
Civil Engineering |
Affiliation |
Applied Science, Faculty of Civil Engineering, Department of |
Degree Grantor | University of British Columbia |
GraduationDate | 2002-11 |
Campus |
UBCV |
Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
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