Equivalent rocking systems: Fundamental rocking parameters

Transcription

1 Equivalent rockin systems: Fundamental rockin parameters M.J. DeJon University of Cambride, United Kindom E.G. Dimitrakopoulos The Hon Kon University of Science and Technoloy SUMMARY Early analytical investiations into the rockin behavior of structures assumed a simple model involvin a sinle riid rockin block on a riid half-space. This paper investiates the use of rockin response spectra, analoous to linear elastic response spectra but instead derived from this simple rockin model, to predict the rockin response of more complicated structures. Fundamental rockin parameters are clarified, and examples of rockin spectra for trionometric round acceleration impulses are reviewed. Two specific rockin structures are addressed: the stone masonry arch and the stone masonry spire. Effective rockin parameters are derived for each of these structures, and the equivalence of the rockin response is investiated. Results indicate the utility of the fundamental rockin parameters for conductin a first order prediction of rockin response, althouh limitations are also discussed. While the focus is on masonry structures, the methodoloy applies to a wider class of structures which rock durin earthquakes. Keywords: rockin, analytical dynamics, masonry, response spectra, near-field round motion 1. INTRDUCTIN The response of rockin structures to round motion has lon captured the fascination of researchers. The majority of this research has focused on the sinle rockin block, often inspired by the seminal work of Housner (1963), who developed a simple analytical model to describe rockin behavior. Subsequent research has involved experimental (e.. Aslam et al. 1978), analytical (e.. Yim et al. 198, Zhan and Makris 21), and computational techniques (e.. Winkler et al. 1995), includin advanced mechanical models which predict complicated bouncin, slidin and rockin behavior (e.. Auusti and Sinopoli 1992, Lipscombe and Pellerino 1993). While these studies effectively define rockin behavior, connectin rockin block theory with the response of real structures in earthquakes remains a sinificant challene. For elastic structures, a sinle deree of freedom (SDF) linear elastic oscillator is typically used as the fundamental structure when determinin the dynamic response. Earthquake enineers typically determine relevant mode shapes, and then use response spectra (derived usin the SDF elastic oscillator) to directly determine the seismic response. Use of the response spectra only requires the enineer to determine the natural frequency and dampin of the structure. In an analoous fashion, the aim of this paper is to investiate the use of rockin spectra, derived from the SDF rockin block, to describe the response of more complex rockin systems. In particular, the paper will focus on two masonry structures which tend to rock durin earthquakes: the masonry spire and the masonry arch. Bein comprised of numerous blocks, each with multiple derees of freedom, a major challene is to determine how to simplify these complex structures as SDF systems. In addition, SDF simplifications are only valid for limited round motions, and can only predict lobal mechanisms and collapse. Thus, the focus will be on short duration pulse-type round motions

2 which may be dominant in near-source earthquakes, and which are detrimental to rockin stability (Zhan and Makris 21). More complex slidin, bouncin and rockin behavior between individual blocks, which may occur due to lon-duration earthquake shakin, would have to be considered by other means. In the followin sections, the simple rockin block model proposed by Housner (1963) will be reviewed, alon with a recent study (Dimitrakopoulos and DeJon 212a) which proposes closed-form solutions for rockin response spectra for sinusoidal impulse round motions. Subsequently, the analytical formulation for the masonry spire and the masonry arch will be presented, alon with the parameters which overn rockin motion in each case. Finally, equivalent rockin structures will be discussed, alon with the limitations of the approach. 2. SIMPLE RCKING STRUCTURES Accordin to the formulation of Housner (1963), rockin motion is described by rotation about alternate bottom corners of the block in Fi. 1 (left). The equations of motion which describe this rockin about points and can be written as: I MRsin Mx Rcos I MRsin Mx Rcos (2.1) where x is the horizontal round acceleration, I is the mass moment of inertia of the block about point, R and are defined in Fi. 1 (left), and is the rockin anle with the positive rotation indicated. Note that rockin motion of the block does not commence until the round acceleration exceeds a minimum manitude, defined as: x tan (2.2) Rearranin Eqn. 2.1 and makin use of the sn() function yields: p (2.3) 2 u sin sn cos sn c.. x R z x x R 1 Fiure 1. Geometry of two simple riid rockin structures.

3 where p is the rockin frequency parameter, which is of critical importance. For the block in Fi. 1, p 3 4R and equals the pendulum frequency of the block when hun about its corner. For slender blocks, Eqn. 2.3 can be rewritten in linear form usin small anle approximations: x (2.4) 2 p sn In this case, aain assumin small anles, the block commences rockin when. Alternatively, the linearized equation of motion (Eqn. 2.4) can be obtained by usin the followin equation for small amplitude vibrations about the point of unstable equilibrium ( = ): 2 2 T V Q 2 2 (2.5) where T and V are the kinetic and potential enery of the system, respectively, and Q is the eneralized forcin function. When the block returns to its initial position ( = ), impact occurs. Still accordin to the formulation of Housner (1963), the enery dissipated at impact can be estimated by conservation of anular momentum about the impactin corner. The enery dissipated is represented in the form of a coefficient of restitution,, which defines the relative rotational velocities before and after impact. For the block in Fi. 1: after before sin 2 (2.6) Clearly Eqn. 2.6 is a simplification of the actual behavior, which may not always be appropriate. Instead, the coefficient of restitution can be directly estimated, as is typically done in non-smooth dynamics formulations (e.. Moreau 1988). Equations 2.3 (or 2.4) and 2.6 completely define the rockin motion. Thus, the only structural parameters necessary to define the response of the rockin block are p,, and. Note that p has units of frequency, and in some way parallels the natural frequency for linear elastic oscillators. Similarly, parallels the dampin in the linear elastic system. However, linear elastic oscillators do not have a point of unstable equilibrium, so is an additional required parameter. As mentioned in the introduction, there is a vast amount of literature on the sinle rockin block. For example, the authors of this paper derived closed-form solutions for the non-dimensional overturnin envelopes and maximum rockin response for rockin blocks subjected to trionometric impulses (Dimitrakopoulos & DeJon 212a). A representative overturnin envelope plot is presented in Fi. 2 (left). This plot defines the sinusoidal impulse characteristics (amplitude a and frequency ) which will cause overturnin of any rockin block (with the iven coefficient of restitution) usin only four equations (labelled Eqn. (i)-(iv) in Fi. 2). Approximate closed-form solutions were also derived for the maximum rockin response under sinusoidal impulses, and are of the followin form: D where D f a p 2 * * max 1 1 1,,, (2.7) * where D f a,,, p. Eqn. 2.7 is plotted in Fi. 2 (riht) for several coefficients of restitution, and compared with the exact response determined numerically. The curve for =.9 is essentially a

4 slice throuh the overturnin plot in Fi. 2 (left) at the location indicated. Further details can be found in Dimitrakopoulos & DeJon (212a), and similar semi-analytical solutions have also been derived for rockin systems with additional viscous dampin (Dimitrakopoulos & DeJon 212b), althouh these must be solved numerically. In order to exploit the results in Fi. 2, and other existin literature reardin the sinle rockin block, it is useful to define other rockin system usin equivalent parameters. For example, perhaps the simplest alternate rockin structure is the rockin point mass supported by riid struts shown in Fi.1 (riht). For this structure, the equations of motion (Eqn. 2.3 or 2.4) are unchaned. However, the frequency parameter is instead p R1, the natural frequency of a simple pendulum, and the 2 coefficient of restitution is instead 1 2sin. The critical anle of rotation is still. Usin these new parameters, the results in Fi. 2 are directly applicable. In fact, these results are enerally applicable for any rockin system which can be defined by the same rockin parameters. Thus, Fi. 2 (riht) is effectively a rockin spectra, analoous to the standard linear elastic response spectra. 15 α 1 a 5 2π/ω η =.9 Eq.(i) Eq.(iii) Eq.(ii) Eq.(iv) p max a.2.1 a 4 to.98 ( step.2) from.82 numerical Eq.(2.7) p Fiure 2. Non-dimensional rockin response due to a sinusoidal round acceleration impulse: overturnin envelopes defined by four curves labelled Eqn. (i)-(iv) (left), and rockin spectra for a rane of values (riht). 3. ALTERNATIVE (CMPLEX) RCKING STRUCTURES The formulation for rockin structures presented above can be used to predict the seismic response of a variety of more complex structures. This section focuses on masonry structures, but the formulation could be applied to a wider array of rockin system The Masonry Spire Stone masonry spires are often extremely slender, and thus vulnerable to overturnin durin earthquakes. Even in the UK, which has only moderate seismic risk, severe spire damae or overturnin is not uncommon. The analytical modelin presented here is part of a larer study on masonry spires (DeJon 212, DeJon & Vibert 212). While the stone masonry spire is comprised of multiple blocks, assume first that the spire can be modeled as a conical shell which rocks about its base (Fi. 3a). If applied for an infinite duration, the fraction of horizontal to vertical round acceleration required to overturn the hollow conical shell is:

5 hc 3rb tan (3.1) H where the eometry is defined in Fi. 3a. Note that this value is 1.5 times reater than for the rockin block. Also, as for the rockin block, hc = when applyin small anle assumptions. The rockin parameter p is defined by: p 2 H (3.2) where = r b / H. Additionally, the coefficient of restitution becomes: cos (3.3) Eqns. 2.3 and 2.4 still hold, and can be used to define the rockin response of the conical shell (nelectin rotation about the vertical axis). However, in reality, masonry structures crack diaonally when loaded laterally (chsendorf et al. 24), makin it necessary to refine the model. Assumin that a diaonal crack forms at the base of a solid spire tip and that a reflectin rockin mechanism results (Fi. 3b), new rockin parameters can be defined. The equations for these parameters become more extensive (not shown), but the resultin parameter values are presented in Fi. 4 for varyin slenderness values ( r b / H ). Note that the effect of thickness is neliible, so Fi. 4 displays the relevant parameters for the entire rane of typical spire eometries. The exact spire eometry (i.e. the diaonal crack anle) must still be determined, but this is addressed in DeJon (212) and compared with computational and experimental results in DeJon & Vibert (212). Reardless, the fundamental equations from section 2 are applicable (for the assumed analytical model), as are the overturnin and response spectra in Fi. 2. (a) (b) max h tip h o < < H/3 H c.m. y x < c.m. R h c R r b Fiure 3. Assumed eometry and rockin mechanisms for two masonry spire models: (a) hollow conical shell, and (b) conical spire with diaonal crackin below a solid spire tip.

6 h c / h o p 2 h o / Fiure 4. Rockin parameters for masonry spires of varyin r b / H which crack diaonally below a solid tip (see Fi. 3b) The Masonry Arch The masonry arch also rocks due to round motion. Althouh the exact dynamic response involves a complicated interaction between multiple blocks, ppenheim (1992) proposed a SDF analytical model to capture the lobal rockin response (see Fi. 5). This model was extended and used effectively to predict experimental rockin arch collapse due to earthquake loadin by DeJon et al. (29). ppenheim (1992) assumed that the masonry arch could be modeled as a SDF four-bar mechanism, allowin the rockin response to be described by the followin equation of motion: 2 M r L r F P x (3.4) where x is the horizontal round acceleration, r is the centre-line arch radius, is the rotation anle in Fi. 5 (left), and the coefficients M(), L(), F(), and P() are nonlinear in In addition, is defined as the rotation anle with respect to the initial eometry ( ). A similar equation describes rockin in the neative direction (Fi. 5, riht). Finally, a coefficient of restitution is required at impact. This can aain either be derived from conservation of momentum equations, as outlined in De Lorenzis et al. (28), or directly specified as an independent parameter. The equation of motion for a four-bar mechanism (e.. Eqn. 3.4) is hihly nonlinear. However, typical arch eometries can sustain relatively small rotations before reachin the point of unstable equilibrium, i.e. the critical rotation anle. Thus, the use of linearized equations for small rockin amplitudes may describe arch rockin with reasonable accuracy. If so, use of linearized equations would allow definition of rockin parameters which make use of rockin block results. Fiure 5. Assumed reflectin SDF rockin arch mechanism.

7 Usin Eqn. 2.5, the equation of motion for the rockin arch, linearized about the point of unstable equilibrium ( = ), becomes: I G B x (3.5) eq eq eq where is the rockin rotation defined above and the coefficients I eq, G eq, and B eq are constants derived directly from the arch eometry. Eqn. 3.5 can be rearraned to yield: x 2 p sn ascale (3.6) where: G B eq eq p ; a scale (3.7) I G eq eq Eqn. 3.6 is similar to the linearized equation of motion of the rockin block, with the exception of the a scale term. This term results from a primary difference in the rockin systems. And, althouh Eqn. 3.7 provides viable equations to predict the response, an alternate method of definin a scale is proposed usin the followin dynamic boundary conditions of the system: ; x ; x (3.8) where is aain the horizontal acceleration necessary to initiate motion. Meetin the boundary conditions in Eqn. 3.8 results in a revised definition of a scale : a scale (3.9) Note that, usin Eqn. 3.9, a scale = 1 for the linearized rockin block, and therefore does not appear in Eqn Thus, an additional parameter is necessary to define rockin systems in which. Thus, three new fundamental rockin parameters have been defined for the rockin arch, and are plotted in Fi. 6. Note that the values for p assume r = 1 m, but the plots for and a scale are enerally applicable as they do not depend on scale. A coefficient of restitution is also needed (not shown), but this is unchaned from previous studies. p [rad/s] [derees] [rad] [derees] a scale [derees] Fiure 6. Arch rockin parameters for a rane of arch eometries: arch inclusion anle, = 15º 18º, arch thickness to radius ratio, t a / r =.12.2.

8 (a) B (b).5 (c) A 1..5 x t p r = 1 m = 157.5º t a / r =.15 2t p time [s] C D [rad] [rad] time [s] (d) time [s] nonlinear linear.44 - nonlin.44 - linear.4 - nonlin.4 - linear.36 - nonlin.36 - linear Fiure 7. Comparison of linear and nonlinear arch rockin formulation: (a) arch eometry and mechanism, (b) free rockin response, (c) applied impulse, and (d) impulse response for t p =.36,.4,.44. A comparison between the nonlinear arch rockin response (Eqn. 3.4) and the linearized rockin response (Eqns. 3.6 and 3.9) is shown for the arch investiated by ppenheim (1992) in Fi. 7. The free rockin response of the arch released from a iven rockin anle is shown in Fi. 7b. The free rockin response is independent of a scale, and thus the accuracy of the frequency parameter p (Eqn. 3.7) can be evaluated independently. The results from the nonlinear and linear formulations compare reasonably well, with the rockin frequency predicted by the linear formulation bein slihtly hiher. The linear and nonlinear predictions are compared in Fi. 7d for the impulse shown in Fi. 7c. The rockin rotations aain compare reasonably well. Note that the prediction is worse if the block nears the point of instability, as the response become more nonlinear in that reion. While a more complete evaluation of the accuracy of the linear formulation is onoin, these initial results indicate that the linear formulation is more than adequate when considerin the uncertainly of earthquake loadin manitude and the simplifications made to derive the SDF analytical model Equivalence of the Rockin Response The use of non-dimensional parameters allows a sinle solution to the rockin equations of motion to describe the rockin response of completely different structures to different round impulses. For example, Fi. 8 shows the response of three rockin structures (Fi. 8a) to sinusoidal impulses of different manitudes and frequencies (Fi. 8b). In Fi. 8c, the dimensional responses are quite different, but the non-dimensional responses collapse to a sinle curve due to the equivalent dimensionless terms shown. Thus, it is aain evident how the rockin response spectra, derived from the response of the sinle rockin block, can be used to predict the response of more complex structures.

9 (a) A B C D (b) x.5 block spire arch H =.4 m B =.1 m =.245 rad p = 5.97 rad/s h o = 6 m h c = 3 m r b / h =.1 =.2 rad p = 1.94 rad/s r = 5 m = 16º t / r =.15 =.78 rad p = 2.5 rad/s t [s] (c), [rad].2.1 block spire arch.8.4, ascalea,.7 p t [s] pt Fiure 8. Comparison of rockin response: (a) three different structure with noted eometries, (b) applied impulse for each structure, (c) dimensional response, and (d) identical non-dimensional response. 5. CNCLUSINS Rockin spectra, derived from the rockin response of the sinle block (as in Fi. 2), have been shown to be applicable to both the masonry spire and the masonry arch. In the process, four parameters have been shown to be necessary to predict the linearized rockin response: p,,, and. Thus, a rockin equivalent to the linear elastic response spectra, for structures which can be described as SDF mechanisms, has been exemplified. Clearly there are limitations to the approach described above. It must be emphasized that the intent is to quickly determine the approximate manitude of lobal rockin response, not to predict precise displacements. Even the rockin response of a sinle block is quite sensitive to sliht chanes in round motion for lon duration rockin (e.. Yim et al. 198), particularly if the structure is small relative to the manitude of shakin. However, for near-source records, where the manitude of stron shakin is short (i.e. similar to an impulse), and for relatively larer structures which are less sensitive to hih-frequency acceleration spikes, prediction capabilities are more robust. Thus, while predictin local displacement and damae to individual stones in a masonry structure is impossible, a reasonable prediction of lobal rockin is feasible, and has proved effective when compared to experimental and computational modelin results (DeJon et al. 28, DeJon and Vibert 212). The formulation above is also limited to structures for which elastic effects can be assumed minimal, as is often the case for masonry structures which are relatively stocky. However, a similar approach can be applied for more flexible rockin structures (e.. bride piers), for which elastic effects may not be neliible. Such structures are explored by Acikoz & DeJon (212a,b).

10 AKCNWLEDGEMENT Financial support for aspects of this research was provided by the Enineerin and Physical Sciences Research Council of the United Kindom under rant reference number EP/H32657/1. REFERENCES Acikoz, S. and DeJon, M.J. (212a). The interaction of elasticity and rockin in flexible structures allowed to uplift. Earthquake Enineerin and Structural Dynamics (published on-line, doi: 1.12/eqe.2181). Acikoz, S. and DeJon, M.J. (212b). Characterizin the vulnerability of flexible rockin structures to stron round motions. 15 th World Conference on Earthquake Enineerin, Lisbon, Portual. Aslam, M., Godden, W.G. and Scalise, D.T. (1978). Rockin and overturnin of riid bodies to earthquake motions, LBL-7539, Lawrence Berkeley Laboratory, University of California, Berkeley. Auusti, G. and Sinopoli, A. (1992). Modelin the dynamics of lare block structures, Meccanica 27:3, Bray, J. D. and Rodriuez-Marek A. (24). Characterization of forward-directivity round motions in the nearfault reion. Soil Dynamics and Earthquake Enineerin 24:11, Dimitrakopoulos, E.G. and DeJon, M.J. (212a). Revisitin the rockin block: Closed form solutions and similarity laws. Proceedins of the Royal Society A (published on-line, doi: 1.198/rspa ). Dimitrakopoulos, E.G. and DeJon, M.J. (212b). verturnin of retrofitted rockin structures under pulse-type excitations. ASCE Journal of Enineerin Mechanics (published on-line, doi: 1.161/(ASCE)EM ). DeJon, M.J. (212). Seismic response of stone masonry spires: Analytical Modelin, Enineerin Structures (published on-line, doi: 1.116/j.enstruct ). DeJon, M.J., De Lorenzis, L., Adams, S. and chsendorf, J. (28). Rockin stability of masonry arches in seismic reions. Earthquake Spectra 24:4, DeJon M.J. and Vibert C. (212). Seismic response of stone masonry spires: Computational and experimental modelin, Enineerin Structures (published on-line, doi: 1.116/j.enstruct ). De Lorenzis, L., DeJon, M.J. and chsendorf, J. (27). Failure of masonry arches under impulse base motion. Earthquake Enineerin & Structural Dynamics 36, Housner, G.W. (1963). The behavior of inverted pendulum structures durin earthquakes. Bulletin of the Seismoloical Society of America 53:2, Lipscombe, P.R. and Pellerino, S. (1993). Free rockin of prismatic blocks, ASCE Journal of Enineerin Mechanics 119:7, Moreau, J.J. (1988). Unilateral contact and dry friction in finite freedom dynamics. In J.J. Moreau, P.D. Panaiotopoulos (Eds.), Nonsmooth Mechanics and Applications, CISM Courses and Lectures No. 32, pp Wien, New York: Spriner-Verla. chsendorf, J.A., Hernando, J.I., and Huerta S. (24). Collapse of masonry buttresses. ASCE Journal of Architectural Enineerin 1:3, Winkler, T., Meuro, K. and Yamazaki, F. (1996). Response of riid body assemblies to dynamic excitation, Earthquake Enineerin and Structural Dynamics 24:1, Yim, C. S., Chopra, A. K. and Penzien, J. (198). Rockin response of riid blocks to earthquakes. Earthquake Enineerin & Structural Dynamics 8:6, Zhan, J. and Makris, N. (21). Rockin response of free-standin blocks under cycloidal pulses. Journal of Enineerin Mechanics 127:5,