# Similitude Distortion Compensation for a Small Scale Model of a Knee Braced Steel Frame

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3 Fig. 2. block connector Fig. 3. Brace and knee element connectors Fig. 4. A brace and knee element 483

4 yielding is the knee elements which constitute the main source of damping. Therefore the knee element geometric characteristics were kept under the similarity laws and an attempt to duplicate the overall dynamic properties of the structure is made by using an overall simulation which for a given length scales coefficient S l requires: K M p p K = Sl M m m Because of the uniform distribution of the stiffness and masses in the structure, K is taken as the equivalent lateral storey initial stiffness of the knee braced frame and M is the floor mass. Subscripts m and p relate to the model and the prototype respectively. The above equation is used to adjust the model masses to satisfy the scale factor. Analytical Correlation between the Model and the Prototype Fig. 5. Overall view of the small scale model Modified Similitude Requirements The primary objectives of the model testing are to investigate the dynamic behavior of knee braced steel frames and to get a better understanding of the performance of their energy dissipation mechanisms without being interested in duplicating the exact response of a particular real structure, provided that the response of the model is within the range of similar real buildings. Thus an analytical study is carried out in order to quantify the discrepancies between the scaled model and the prototype structure due to the distortion in the similarity requirements. The main members (columns and beams) of the knee braced frame are designed to remain elastic at all times for these tests. The only part that is subject to The discrepancies in response between the small scale model and the prototype structure are quantified through a comparative numerical study of the modal characteristics and the responses of the two structures to El-Centro N-S component earthquake in the elastic and plastic range. The scaled model and the prototype frequencies are within 10% (Table 1) and the mode shapes are very close (Fig. 6). The inter-storey drift is an important response parameter in this testing context of the small scale KBF model and should be kept as close as possible to the prototype value. The scaled maximum lateral displacement of the model are plotted together with those of the prototype in Fig. 7. The overall displacements are in good agreement. The maximum error which is less than 10% occurred first at the level of columns dimensions transition and the error sign changes in the upper storey. The key parameter of testing the KBF small scale model is to assess the energy dissipation capacity of the knee elements and the local ductility to measure the damage that the knee elements may endure under a seismic ground motion. The relevant response parameters are the maximum ductility and the hysteresis loops. As can be seen in Fig. 8 the distribution of the maximum ductility of the knee elements at each storey in the model are higher than those of the prototype but the trend is similar. However, the accumulated plastic rotations of the knee elements are closer except for the 4th transition storey (Fig. 9). It can be stated that the small scale model can portrayed the height distribution of the plastic deformations of the knee elements. 484

5 Fig. 6. Model and prototype mode shapes Fig. 7. Maximum lateral displacement Table 1. Model and prototype natural periods (prototype time) Periods (s) Mode Model (T m ) Model (S l *T m ) Prototype Representative hysteresis loops of knee elements at a lower storey for both the model and the prototype are plotted on Fig. 10. They are slightly different in the way that intermediate moment-rotation hysteresis loops developed in the prototype are wider. In terms of energies, the plot of the time histories of the input energy, the structural damping energy and the dissipated hysteresis energy show that the scaled model energies are matching those of the prototype (Fig. 11). This is very important because the objective of this small scale modeling is to investigate the capability of the knee bracing system in dissipating the imparted energy. 485

6 The time histories of the top floor displacement in Fig. 12 and 13. Show that the prototype and the model oscillations are on phase, but the major peaks of the model are larger in the elastic response. The amplitudes of the vibrations however are comparable in the nonlinear response. Fig. 8. Maximum storey knee element ductility Fig. 9. Storey accumulated plastic rotation of knee element 486

7 (a) (b) Fig. 10. Moment-rotation hysteresis loops of the knee elements (a) Hysteresis loop of the model (b) Hysteresis loop of the prototype Fig. 11. Energy time histories Fig. 12. Linear response of the top floor displacement time histories 487

8 Fig. 13. Nonlinear response of the top floor displacement time histories Conclusion The small scale dynamic model has proved to be a powerful tool in investigating and understanding the structural behavior in many complex situations where analytical techniques are inadequate or to validate the existing analytical procedure by developing mathematical models that correlate well with the experimental results. However true replica models are practically impossible to build and test because of the severe restrictions imposed on the model. Alternate scaling laws have been shown to simulate adequately the behavior of the structure and some particular distortions of the similitude requirements can be accounted for. The artificial mass simulation used in this study was successful in correcting the unsatisfied similitude conditions using an overall mechanical simulation. As demonstrated, the dynamic characteristics as well as the response parameters of primary interest such as lateral deflection, knee bracing ductility and the energy dissipation capacity of the small scale model are within the range of true scale building. Funding Information Research described in this study was financed internally by the University of Bristol (U.K). Author s Contributions All the authors have participated equally in the research work. Ethics This work has not been published elsewhere so there are no ethical issues known to authors that may arise after the publication of this manuscript. References Bourahla, N., Knee bracing system for earthquake resisting steel frames. PhD Thesis, University of Bristol, UK. Dumanoglu, A.A. and R.T. Severn, Dynamic behaviour of CLASP-type buildings. Earthquake Eng. Structural Dynam., 13: DOI: /eqe Li, C.S., S.S.E. Lam, M.Z. Zhang and Y.L. Wong, Shaking table test of a 1:20 scale high-rise building with a transfer plate system. J. Structural Eng., 132: DOI: /(ASCE) (2006)132:11(1732) Sabnis, G.M., H.G. Harris, R.N. White and M.S. Mirza, Structural Modeling and Experimental Techniques. 1st Edn., Prentice Hall, Englewood Cliffs, NJ, pp: 585. Williams, M.S., A. Blakeborough, D. Clement and N. Bourahla, Seismic behaviour of knee braced frames. Structures Build. ICE, 152: DOI: /stbu Zhou, X. and G. Li, Shaking table model test of a steel-concrete composite high-rise building. J. Earthquake Eng., 14: DOI: /

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