SEISMIC RESPONSE OF BUILDINGS WITH NON-UNIFORM STIFFNESS MODELED AS CANTILEVERED SHEAR BEAMS

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1 10NCEE Tenth U.S. National Conference on Earthquake Engineering Frontiers of Earthquake Engineering July 1-5, 014 Anchorage, Alaska SEISMIC RESPONSE OF BUILDINGS WITH NON-UNIFORM STIFFNESS MODELED AS CANTILEVERED SHEAR BEAMS Andres Alonso-Rodriguez 1 and Eduardo Miranda ABSTRACT In this study, the dynamic characteristics of cantilever shear beams with a uniform mass and parabolic reduction of lateral stiffness along its length are examined. Closed-form expressions to mode shapes and frequencies of vibration are developed in terms of first and second order Legendre functions. It is shown that proposed expressions provide dynamic characteristics that agree with those computed with detailed finite element models, therefore, showing that they are the true solutions to the problem at hand. The effect of reductions in the lateral stiffness on seismic response is then evaluated by considering the first five modes of vibration. It is found that the effects of reduction of lateral stiffness are relatively small when the lateral stiffness in the top is smaller than about seventy percent of the lateral stiffness at the bottom (fixed end), but become significant for larger reductions in lateral stiffness. Effects are more important for the derivative of the mode shapes, which is critical to the estimation of interstory drift demands in buildings when modeled as shear beams. The use of the novel closed form equations to the dynamic properties of the non-uniform shear beam is illustrated when computing the seismic response of the SAC 0 steel moment resisting frame building, when subjected to the Chi-Chi earthquake 076 record and Cathedral College, Christchurch earthquake record. 1 Business intelligence Analyst, Seguros Mundial, Bogotá, Colombia andalon@gmail.com Professor, Dept. of Civil and Environmental Engineering, Stanford University, CA emiranda@stanford.edu Alonso-Rodriguez A., Miranda E. Seismic response of buildings with non-uniform stiffness modeled as cantilevered shear beams. Proceedings of the 10 th National Conference in Earthquake Engineering, Earthquake Engineering Research Institute, Anchorage, AK, 014.

2 Seismic Response of buildings with-non uniform stiffness modeled as cantilevered shear beams Andres Alonso-Rodríguez 1 and Eduardo Miranda ABSTRACT In this study, the dynamic characteristics of cantilever shear beams with a uniform mass and parabolic reduction of lateral stiffness along its length are examined. Closed-form expressions to mode shapes and frequencies of vibration are developed in terms of first and second order Legendre functions. It is shown that proposed expressions provide dynamic characteristics that agree with those computed with detailed finite element models, therefore, showing that they are the true solutions to the problem at hand. The effect of reductions in the lateral stiffness on seismic response is then evaluated by considering the first five modes of vibration. It is found that the effects of reduction of lateral stiffness are relatively small when the lateral stiffness in the top is smaller than about seventy percent of the lateral stiffness at the bottom (fixed end), but become significant for larger reductions in lateral stiffness. Effects are more important for the derivative of the mode shapes, which is critical to the estimation of interstory drift demands in buildings when modeled as shear beams. The use of the novel closed form equations to the dynamic properties of the nonuniform shear beam is illustrated when computing the seismic response of the SAC 0 steel moment resisting frame building, when subjected to Chi-Chi earthquake 076 record and Cathedral College, Christchurch earthquake record. Introduction Shear beams have been used for many years to study the dynamic response of soil deposits and buildings that are subjected to lateral loading. For the case in which the soil deposit has uniform properties along its depth or for buildings with uniform lateral stiffness closed form solutions to periods and mode shapes correspond to those of the axial vibration of uniform beams, as the underlying differential equation is the same and are readily available in most textbooks (Clough and Penzien [1]; Weaver, Timoshenko and Young []). However, for the case of non-uniform shear beams only a few closed-form solutions exist corresponding to certain variations of mass and shear stiffness along the length of the beam, and unfortunately most of them are not suitable for the analyses of buildings whose lateral deformations are similar to those of shear beams because the variation of lateral stiffness for which closed-form solution have been developed differs significantly from the reduction of lateral stiffness in buildings. 1 Business Intelligence Analyst, Seguros Mundial. Bogotá, Colombia andalon@gmail.com Professor, Dept. of Civil and Environmental Engineering, Stanford University, CA emiranda@stanford.edu Alonso-Rodriguez A, Miranda E. Seismic response of buildings with non-uniform stiffness modeled as cantilevered shear beams. Proceedings of the 10 th National Conference in Earthquake Engineering, Earthquake Engineering Research Institute, Anchorage, AK, 014.

3 Although non-uniform beams can be discretized and analyzed using numerical methods (e.g., using the finite element method), closed-form solutions allow solutions to the eigenvalue problem expressed as a function of a small number of variables and provide unique insight to the problem. Other advantages of closed-form solutions include: (i) they provide a continuous function of mode shapes thus allowing the possibility of finding analytical solutions to the derivative of the mode shapes for finding exact shear deformations at any point along the length; (ii) provide a benchmark for checking discretized models; (iii) are useful in preliminary design and are particularly useful in parametric studies and for other cases in which, because of the number of different models or analyses to be performed, it is not feasible or practical to use finely discretized models. Therefore, there is a need for closed-form solutions for stiffness variations that resemble those found in buildings. In particular, it is expected that buildings follow a parabolic stiffness variation along height, consequently the aim of this paper is presenting closed-form solution to the dynamic characteristics of non-uniform cantilever shear beams with uniform mass and with a parabolic reduction in lateral stiffness, showing how they can be helpful for studying the seismic behavior of buildings during earthquakes. Approximate Distribution of Lateral Stiffness in Buildings In buildings subjected to earthquake lateral loading, defined according to the equivalent lateral force method (ELF) shear demands decrease with increasing height. For example in the case of buildings designed with the equivalent lateral force method (ELF) the International Building Code IBC, [3] and the Eurocode 8, [4] seismic lateral load at floor level i is computed with: F i = V b N i=1 w i h i k w i h i k (1) Where, w i is the weight of floor i, h i is the height of floor level i measured from its base, N is the number of floors in the building and V b is the base shear. In the Eurocode coefficient k is set equal to one, imposing a linear varying lateral load, Trend also observed in the national building code of Canada [5]. Furthermore, for buildings with no setbacks, the mass is approximately constant along the height of the building. Eq. 1 will indeed define a linear varying lateral load or a trend close to it, which, by simple integration will give rise to a parabolic trend in shear demands: 1 [ 1 ] z V ( x) = Vb ( k + 1) Hdz = Vb x () H x Where x is the length ordinate normalized by its height (x/h). Please note that the normalized story shear at the top of the building decreases with the number of stories but it does not vanish. Consequently, a minimum resistance must be provided for the last level. According to Eq. the following is observed

4 V V b = N + 1 Top (3) Following Eq. 3, and considering how the capacity of the lateral load is curtailed to accommodate the diminished demand of story shear with height, it is expected that stiffness on the building decrease following this pattern: S sh = 1 (1 δ ) (4) ( x) x Where δ is the ratio of the lateral stiffness at the top of the building normalized by the lateral stiffness at the base of the building, A tentative value for it can be found by comparing the results of Eq. 3, 4 when evaluated at x=1, finding: δ = (5) N + 1 Albeit equivalent lateral load methods are not as widely used as before, buildings designed in accordance with response spectrum modal analysis (RSMA) still observe design shear demands following patters that resemble the stiffness variation proposed in Eq. 4. [6] [7]; making the proposed trend a reasonable one for studying behavior of buildings when subjected to earthquake actions. Dynamic Behavior of a Shear Beam with parabolic stiffness variation The differential equation of a shear beam with a stiffness distribution according to Eq. 4 and a uniform distribution of mass is [7]: ( ) x " # 1 1 δ $ % d φ(x) x 1 δ dx [ ] dφ(x) dx +ω τ φ(x) = 0 (6) x is the length coordinate normalized by building height H (). and τ is: ρ0h τ = (7) GA 0 Where GA 0 is the shear stiffness at the base of the beam and ρ 0 is the mass per height. It can be shown [7] that the general solution of Eq. 6 is given by φ(x) = P v ( 1 δ x) + P (0) v Q v (0) Q v ( 1 δ x) (8) Where P v (x) and Q v (x) are the first and second type Legendre functions of zero order and degree ν, where the latter is related to the circular vibration frequency, ω by the following expression:

5 ω τ = v ( v + 1)(1 δ ) (9) The parameter can be found by solving the modal characteristic equation. [ 1 δ Q ( 1 δ ) Q ( 1 δ )] Q (0) [ 1 δ P ( 1 δ ) P ( 1 )] 0 Pv ( 0) v v v v v δ = (10) A plot of the characteristic equation for δ = 5 and δ= 5 is shown in Fig. 1. It can be seen that the characteristic equation is an oscillating function whose roots only depend on ν MCE δ=5# δ=5# ωτ# Figure 1. Modal Characteristic Equation in terms of normalized vibration frequencies for lateral stiffness ratios δ= 5 and 5. Once the mode shapes have been defined, the modal participation factor for each mode i, Γ i can be computed in the following manner [1]: 0 Γi = 1 φ i ( x) dx 0 1 φ ( x) dx i (11) Validation of the solution: In order to verify if the proposed solution is correct, mode shapes and frequencies defined by the proposed closed form solution were compared to those computed with the finite element method (FEM) using a model in which non-uniform shear beams were discretized into 1000 elements. Solutions were assessed for a wide range of values of δ and a comparison of mode shapes and its derivative for the first five modes of vibration were done. Fig. shows an example of such comparisons for the first three modes of vibration for a non-uniform shear beam where the lateral stiffness at the top is 5% of the lateral stiffness at the base. It can be seen that the finite element solutions match the proposed solution indicating the correctness of the proposed closed-form solution.

6 This Study FEM This Study FEM This Study FEM This Study FEM Γφ' Γφ' Figure. FEM verification. Mode shapes (first and third) shown on the left. Mode shapes derivatives on right. δ= 5 Effect of reduction of stiffness on dynamic characteristics As a closed form solution has been proposed it is possible to carry a systematic study of the effects of reductions in lateral stiffness focusing in cases which the lateral stiffness at the top of the building, in particular, smaller than 5%. That as indicated by Eq. 5 becomes important for buildings higher than seven stories. Fig. 3 Shows the variation of modal periods of non-uniform beams normalized by modal periods of uniform beams (δ=). It is shown that period ratios tend to decrease indicating that in non-uniform beams periods of vibration become closer to each other with respect to ratios in uniform models. Furthermore, an approximate linear reduction is observed for reduction in δ up to 5% but for smaller values of δ a sharper effect on period ratios is clearly noticeable. Another interesting observation regarding period ratios is that the reductions are almost the same for all modes. For the case of zero stiffness at the top, the difference between modal periods of the uniform and the non-uniform beams for the fifth mode are just 6% lower than for the second mode. For a value of δ = 5% the difference among both is just %. Τr(δ)/Τr(δ=1)" δ" Figure 3. Effect of reduction of lateral stiffness on modal periods. It should be noted that even though the effects of stiffness reduction in periods of vibrations are relatively small, they could have a larger effect on seismic response, particularly in the case of lightly damped buildings and for buildings subjected to ground motions with narrow band spectra where small changes in frequency can lead to a large change in seismic response. Fig. 4 shows the effects of reductions in lateral stiffness on products of mode shapes and modal participation factors. It should be noted, that unlike mode shapes and modal participation nd 3rd 4th 5th

7 factors, this product is independent of how mode shapes are normalized. Fig. 4a shows the effect of stiffness reductions on the value of this product at the top of the model for the first five modes of vibration. Fig 4b shows the value of this product at the top of the model in non-uniform models normalized by the product at the same location but in uniform models. It can be seen that for values δ>5 the effects are relatively small but again become significant as it decreases. Furthermore, the effects are more important as the mode increases. For example, for the fundamental mode the product of the modal participation factor and mode shape at the top of the model increases 16% for the model with vanishing stiffness at the top with respect to the uniform case, increasing from 1.7 to 1.5. However for the second mode differences among modal ordinates at top can be twice, increasing from (-) to (-8). For the third mode the modal ordinate for the beam with vanishing stiffness at top is 8, while for the uniform is only 5, that is an increase of 1.7 times st nd 3rd 4th 5th -1.5 (a) δ# (b) Γφ(δ) Γφ(δ =1) 3.0 Figure 4. Effect of stiffness reduction on modal participation factors. The effect of reduction in lateral stiffness on dynamic characteristic of non-uniform beams is further illustrated on fig. 5 where the effect of reduction in lateral stiffness on the product of mode shapes and modal participation factors is shown. It can be seen that effects are larger for higher modes, where at some heights the effects are substantial for small values of δ. Observation of the effect of the reduction of lateral stiffness on the first mode of vibration shows that as the lateral stiffness at the top of the model approaches zero the mode shape tends to become linear st nd 3rd 4th 5th δ" 1 st Mode nd Mode 3 rd Mode 4 th Mode δ=5# δ=0.10# δ=0# δ=0# Figure 5. Effect of stiffness reduction on the product of mode shapes and modal participation factors δ=5# δ=5# δ=0.10# δ=0.10# δ=0# δ=0# δ=0# δ=0# Fig. 6 shows the effect of reduction in lateral stiffness on the derivative of mode shapes multiplied by modal participation factors. As noted by Miranda and Akkar [8] this product is important for the estimation of interstory drift demands in buildings. In general, the effect on the δ=5# δ=0.10# δ=0# δ=0#

8 derivative of the modes is significantly larger than on the mode indicating that the effect of reduction in lateral stiffness is more important for the estimation of interstory drift demands than for the estimation of acceleration demands. Furthermore, the effects become very large for values of δ smaller than. In general reductions are observed near the base of the model and large increments are observed in the upper part. At the top, however, there is no effect as the shear at the top is zero in the free end and therefore the derivative of the mode shape is null at this location. 1 st Mode nd Mode δ=5$ δ=0.10$ δ=0$ δ=0$ 3 rd Mode δ=5$ δ=0.10$ δ=0$ δ=0$ 0.5 Γφ' δ=5$ δ=0.10$ δ=0$ δ=0$ Γφ' Γφ' Figure 6. Effect of stiffness reduction on mode shape derivatives multiplied by their modal participation factors Study case: SAC 0 Building In order to further assess how the proposed model could represent real buildings, the SAC 0 model was studied, as it was properly designed by a well-recognized structural engineering firm, following state of the art code provisions [9]. Its modal properties are widely available, and thus could be used as a benchmark to assess the proposed shear beam model presented in this work. By performing system identification, with the aim of establishing the δ parameter that would describe in best way the modal properties of the SAC 0, it can be shown how the shear beam with the parabolic stiffness clearly performs well at assessing its behavior. A value of 0.34 was found, which is significantly greater than the one expected with Eq. 4, which is 95. Table 1: Comparison of period ratios associated with the SAC 0 building and its shear beam representation Period Ratio SAC 0 Building Shear beam with δ=0.34 T /T T 3 /T 1 1 Indeed, the shear beam with parabolic stiffness is more flexible than the building. This can be explained by the fact that column and beam sections don't change evenly along building height, rather, do it in a discrete manner, observing the same cross sectional properties along several levels. Thus, is quite remarkable to observe differences less than 10% on period ratios.

9 The ability of the model to reproduce modal response of the SAC 0 model really exceeded expectations. Mode shapes widely reported [9] are reproduced with agreements closer than 5% while the uniform model would show differences larger than 5. Furthermore, the proposed model also represents well ground motion response of buildings satisfactorily. For this purpose, a FEM model was elaborated in SAP 000 and then was analyzed by considering two ground motions: Cathedral college caused by 010 Christchurch Earthquake and record TCU 076 observed during 1999 Chi-Chi earthquake. Figure 7. Peak acceleration and drift response of the Sac 0 building, compared with response of their associated non uniform shear beam model; after the cathedral college record, from Christchurch 011 earthquake. Figure 8. Peak acceleration and drift response of the Sac 0 building, compared with response of their associated non uniform shear beam model; after TCU 076 record, after Chi-Chi 1999 earthquake. Clearly, general trends and overall behavior are satisfactorily estimated by the non-uniform shear beam model, peak drifts are somewhat overestimated, as derivatives are calculated, instead of point estimates. Trends on acceleration are also well represented, albeit some overestimation at the last floor for the cathedral college record were observed. On the contrary, trends defined by the uniform model clearly depart from results obtained for the finite element model representation, on both drift and peak acceleration. This way, it is evident that the shear beam with a parabolic stiffness does a better job than the uniform one, at estimate peak drift and acceleration demands of buildings designed for earthquake actions.

10 Conclusions The dynamic characteristics of cantilever shear beams with a uniform mass along the height but with parabolic reduction of lateral stiffness have been studied. It has been shown that this type of variation of stiffness along the height resembles that of buildings designed to resist seismic forces where design lateral forces are equal or similar to inverted triangular distributions and therefore the variation of shear demands is approximately parabolic. A closed-form solution to mode shapes and frequencies of vibration has been developed in terms of first and second order Legendre functions. The solution was verified by comparing the results to those computed with the finite element method using discretized models with 1,000 elements. The effect of reducing the lateral stiffness was studied in the first five modes of vibration. Evaluation of reduction of lateral stiffness was performed on periods of vibration, period ratios, mode shapes, modal participation factors and derivatives of mode shapes. In general, it was found that the effects of reduction of lateral stiffness are relatively small when the lateral stiffness in the top of the beam is larger than 30 percent of the lateral stiffness at the fixed end, but become significant when the reduction in lateral stiffness is larger. Changes on modal ordinates at the top can be 0% higher for the first mode, while increases of 100% and 175% can be plausible for the second and third modes. The latter becomes important for buildings with seven or more stories. Effects are particularly important for the derivative of the mode shapes, which is critical to the estimation of interstory drift demands in buildings when modeled as shear beams, for which differences of two or three times relative to those of the uniform case could be expected for beams with just 5% of stiffness at top, which is plausible on a 0-level building. The closed-form solutions to the non-uniform shear beam model were used to estimate the seismic response of the SAC 0-story steel moment resisting frame building obtaining excellent results. Peak modal response up the third mode is estimated, observing differences less than 5%, compared with an underestimation of 5% associated with the uniform shear beam, showing how a notorious advance could be obtained by considering the solution proposed in this study. Acknowledgments This project was carried during the doctorate studies of the first author, at ROSE SCHOOL (The Earthquake Engineering and Engineering Seismology programme of the graduate school in understanding and managing extremes), part of the IUSS (Istituto di Studi Superiori di Pavia) of Pavia, Italy. Financial support for this work was provided by the Italian Republic, through a Dottorato di Ricerca scholarship awarded to the first author. References 1. Clough R, Penzien J. Dynamics of structures. McGraw-Hill: Singapore, Weaver W, Timoshenko S, Young D. Vibration problems in engineering. John Wiley & sons: Hoboken NJ, 1990.

11 3. International Code Council ICC. 01 International Building Code. Delmar Cengage Learning, Florence: KY, USA, European Committee for Standardization, CEN. Eurocode 8: Design of structures for earthquake resistance Part 1: General Rules, Seismic actions and rules for buildings. CEN: Brussels, Belgium, Chopra A. Dynamics of structures. Prentice Hall: Upper Saddle River NJ, USA, Newmark N. Rosenblueth E. Fundamentals of earthquake engineering. Prentice Hall: Upper Saddle River NJ, Alonso A. Assessment of elastic acceleration and drift demands in buildings through generalized continuous models.ph.d. Thesis. Istituto Universitario di studi superiori di Pavia, Pavia, Italy, Miranda E, Akkar S. Generalized interstory drift demand spectrum. Journal of structural engineering, ASCE 006; 13 (6): Ohtori Y, Chirstenson R, Spencer B, Dyke S. Benchmark control problems for seismically excited nonlinear buildings. Journal of engineering mechanics, ASCE 004; 130 (4) :