Lesson 4-5: Dynamics of Vibration & Attenuation (CE 410) - PDF

BICOL UNIVERSITY COLLEGE OF ENGINEERING LEGAZPI CITY CE 410 EARTHQUAKE ENGINEERING LESSON 4 – 5: DYNAMICS OF VIBRATION, ATTE...

BICOL UNIVERSITY COLLEGE OF ENGINEERING LEGAZPI CITY CE 410 EARTHQUAKE ENGINEERING LESSON 4 – 5: DYNAMICS OF VIBRATION, ATTENUATION & TIME HISTORY PART 2 “This presentation is not for distribution outside this subject and to be used solely for this course subject – CE 413.” PREPARED BY: ANNA G. BILARO, MSCE Faculty ANNA G. BILARO, MSCE FACULTY CE 413 – EARTHQUAKE ENGINEERING Slide No. 1  OBJECTIVES: After the discussion in this chapter, the student will be able to: 1. Learn the seismic design approaches. 2. Learn the configurations and properties of free vibration response of undamped systems. 3. Derive the equation of motion and its alternative formulation. 4. Determine the weight of the structure as SDOF. 5. Determine the stiffness of the structure. 6. Determine the natural angular frequency of a structure. 7. Determine the natural period of a structure. ANNA G. BILARO, MSCE FACULTY CE 413 – EARTHQUAKE ENGINEERING Slide No. 2  Seismic Design Approaches: Static Analysis Formulated by estimating an equivalent lateral seismic loads as a fraction of the total system weight. The fraction is determined from a notional earthquake, which is based on the risk posed by the seismicity of the site and is captured in a single graph known as the response spectrum. Dynamic Analysis The procedure for this analysis models the load effect of earthquakes more accurately by using either a set of specific earthquake ground motion time- history records (response time-history analysis procedure) or the effect of same notional earthquake as the equivalent static analysis (the design response spectrum). ANNA G. BILARO, MSCE FACULTY CE 413 – EARTHQUAKE ENGINEERING Slide No. 3  In the response time history analysis procedure, a linear or nonlinear mathematical model of the system is used to determine the system response (displacement and accelerations, which are used to determine the internal loading in each member) at each increment of time for a suite of ground motion acceleration time histories. With the full history of the response, the absolute maximum internal loads can be determined by combining the results of the set of ground motion acceleration time histories; these loads can be used to design each member. The modal response spectrum analysis procedure uses established methods of structural dynamics to determine system vibration mode shapes and their associated natural periods, which along with the response spectrum are used to establish the maximum structural response. ANNA G. BILARO, MSCE FACULTY CE 413 – EARTHQUAKE ENGINEERING Slide No. 4  UNDAMPED SINGLE SINGLE-DEGREE-OF-FREEDOM SYSTEM The concept of DOF is central to the understanding of vibration theory. A DOF in the context of vibration theory is an independent displacement or rotation of a point on a structure, which means that even the simplest case could potentially have an infinite number of DOFs. Determination of DOFs depends on the idealization of the structural system. The ends of a member are often points of interests. ANNA G. BILARO, MSCE FACULTY CE 413 – EARTHQUAKE ENGINEERING Slide No. 5  SDOF: Frame with rigid connections between columns and beam. Support – 0 DOFs; connections – 3 DOFs. Assumptions: Fig. 1.0 Idealized single DOF system for a portal frame a. beams much stiffer than columns b. columns move in parallel c. axial deformations of the columns and beam are relatively small. No damping in the system. Fig. 2.0 a) and b) Idealized structures (FBD) ; c) vibration due to initial displacement. ANNA G. BILARO, MSCE FACULTY CE 413 – EARTHQUAKE ENGINEERING Slide No. 6  FREE VIBRATION RESPONSE OF UNDAMPED SYSTEMS Free vibration is caused by initial conditions (displacement and velocity) and not by applied force. To characterize the free vibration motion of the portal frame idealized as an SDOF system, we formulate a mathematical model in terms of the displacement as a function of time by establishing an equation of motion for the system. Theories: a. Application of equilibrium equation to FBD using D’Alembert’s principle b. Newton’s 2nd Law, F = ma. ANNA G. BILARO, MSCE FACULTY CE 413 – EARTHQUAKE ENGINEERING Slide No. 7  FREE VIBRATION RESPONSE OF UNDAMPED SYSTEMS Equation of Motion: Where: m = is the mass of the system k = is the lateral stiffness Derivation of the free vibration response. Fig. 3.0 FBD of the portal frame ANNA G. BILARO, MSCE FACULTY CE 413 – EARTHQUAKE ENGINEERING Slide No. 8  FREE VIBRATION RESPONSE OF UNDAMPED SYSTEMS Shown below is the graph of solution for free vibration response of undamped systems. This motion is described as harmonic (therefore periodic) since it is a function of sine and cosine of the same frequency, n. The time required to complete a full cycle (2π) is known as the natural period of vibration, Tn (s) and is determined as: Fig. 4.0 Free vibration of an undamped SDOF ANNA G. BILARO, MSCE FACULTY CE 413 – EARTHQUAKE ENGINEERING Slide No. 9  FREE VIBRATION RESPONSE OF UNDAMPED SYSTEMS The natural cyclic frequency, fn (hertz or cycles/ s), is defined as the reciprocal of the natural period of vibration, Tn (s), and is proportional to the natural circular frequency, n (rad/s), that is: These three quantities are related and essentially represent the same physical quantity expressed in different units. ANNA G. BILARO, MSCE FACULTY CE 413 – EARTHQUAKE ENGINEERING Slide No. 10  FREE VIBRATION RESPONSE OF UNDAMPED SYSTEMS Alternative Formulation: Additional parameters was introduced, the phase angle, , and the amplitude, u0 and relation of new variable C to constants A and B. Fig. 5.0 Relationship between phase angle and amplitude. ANNA G. BILARO, MSCE FACULTY CE 413 – EARTHQUAKE ENGINEERING Slide No. 11  STRUCTURAL WEIGHT The circular frequency is proportional to the mass, which in turn directly proportional to the weight. For seismic analysis of structures, the weight is defined as the total effective weight, W (ASCE, Min Design Loads for Buildings and other Structures): ANNA G. BILARO, MSCE FACULTY CE 413 – EARTHQUAKE ENGINEERING Slide No. 12  STRUCTURAL WEIGHT Fig. 6.0 Building structure and lumped mass simplified model. ANNA G. BILARO, MSCE FACULTY CE 413 – EARTHQUAKE ENGINEERING Slide No. 13  STRUCTURAL WEIGHT Fig. 7.0 Story weight, Wx for calculating lateral forces. ANNA G. BILARO, MSCE FACULTY CE 413 – EARTHQUAKE ENGINEERING Slide No. 14  STRUCTURAL STIFFNESS To completely characterize the equation of motion for a linear elastic SDOF system, we need the lateral stiffness, k. This variable, along with the mass, is needed to determine the circular frequency., which once obtained can be used to determine the natural period of vibration of the system. The stiffness is defined as the ratio of force (or moment) to displacement (or rotation). 𝑃 𝑘= Δ ANNA G. BILARO, MSCE FACULTY CE 413 – EARTHQUAKE ENGINEERING Slide No. 15  STRUCTURAL STIFFNESS Fig. 8.0 Equivalent Stiffness Constants, k ANNA G. BILARO, MSCE FACULTY CE 413 – EARTHQUAKE ENGINEERING Slide No. 16 REFERENCES: Elnashai, Amr S. and Di Sarno, Luigi. 2008. Fundamentals of Earthquake Engineering. John Wiley & Sons Ltd, United Kingdom. Estrada, Hector and See, Luke S. 2017. Introduction to Earthquake Engineering. CRC Press, Taylor and Francis Group, Florida. Chen, W.F. and Lui, E.M. 2006. Earthquake Engineering for Structural Design. CRC Press, Taylor and Francis Group, Florida. Chopra, Anil K. 2012. Dynamics of Structures, Theory and Applications to Earthquake Engineering., 4th Edition. Prentice Hall, New Jersey. Clough, R and Penzien, J. 2003. Dynamics of Structures, 3rd Ed. Berkeley, USA. Gioncu, Victor and Mazzolani, Federico M. 2011. Earthquake Engineering for Structural Design. Spon Press, New York, USA and Taylor & Francis Group e-Library. Jimenez, Guillermo Alfonso Lopez. 2016. Static and Dynamic Behavior of Pile Supported Structures in Soft Soils. Universite Grenoble Alpes. Kramer, Steven L. 1996. Geotechnical Earthquake Engineering. Prentice Hall, Upper Saddle River, New Jersey. Lectures of Dr. Latha, G. Madhavi from Department of Civil Engineering, Indian Institute of Science. CE 259 Lectures. 2018. Institute of Civil Engineering, UP Diliman, Quezon City. Villamil, C.V., Perez, J.S. and Cayabyab, F.F. 2016. Earthquake Intensity Assessment and Isoseismal Map of the 2013 MW 7.2 Bohol Earthquake. 6th Asia Conference on Earthquake Engineering (6ACEE), Cebu City, Philippines. www.phivolcs.dost.gov.ph ANNA G. BILARO, MSCE FACULTY CE 413 – EARTHQUAKE ENGINEERING Slide No. 17  THANK YOU FOR LISTENING! ANNA G. BILARO, MSCE FACULTY CE 413 – EARTHQUAKE ENGINEERING Slide No. 18