Earthquake Duration and Damping Effects on Input Energy
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Earthquake Duration and Damping Effects on Input Energy G. Ghodrati Amiri1, G. Abdollahzadeh Darzi2 and M. Khanzadi3 1Associate Professor, Email:[email protected] 2Ph.D. Candidate 3Assistant Professor Center of Excellence for Fundamental Studies in Structural Engineering, College of Civil Engineering, Iran University of Science & Technology, Tehran, Iran Abstract: According to performance-based seismic design method by using energy concept, in this paper it is tried to investigate the duration and damping effects on elastic input energy due to strong ground motions. Based on reliable Iranian earthquake records in four types of soils, structures were analyzed and equivalent velocity spectra were computed by using input energy. These spectra were normalized with respect to PGA and were drawn for different durations, damping ratios and soil types and then effects of these parameters were investigated on these spectra. Finally it was concluded that in average for different soil types when the duration of ground motions increases, the input energy to structure increases too. Also it was observed that input energy to structures in soft soils is larger than that for stiff soils and with increasing the stiffness of the earthquake record soil type, the input energy decreases. But damping effect on input energy is not very considerable and input energy to structure with damping ratio about 5% has the minimum value. Keywords: PGA; duration of strong ground motion; damping ratio; input energy spectrum. 1. Introduction Researches considered that among various types of energy the input energy, EI which is Design of structures with performance-based a very stable parameter of structural design especially based on energy is one of response, is suitable for energy-based design. Downloaded from ijce.iust.ac.ir at 22:42 IRST on Saturday October 2nd 2021 the methods noticed by researches in the Input energy is a measure of energy that recent years [1-8]. Unlike present methods earthquake inputs to structures during the which are based on peak accelerations that ground motion. Based on definition, two ignore duration and hysteretic behavior types of input energy exist [9], relative input B effects on structural design, design based on energy, EI and absolute input energy, EI . The B energy not only considers the effects of these difference between EI and EI is the effect of parameters, but also it can describe treatment the rigid body translation of structure. If EI B of structures during an earthquake, better and EI are evaluated at the end of ground than present methods. motion duration, it can be considered that they differ in the very short and very long In this method formed on the basis of energy period ranges only. concepts and energy balance equation, energy criterion expresses that the structure In a structure if its supply energy was larger collapses when the amount of capacity than its imposed energy, all the energy energy lower than the demand energy to exerted to the structure during ground motion dissipate through inelastic deformations is is dissipated through damping and hysteretic exerted. cyclic behavior of structure at the end of 14 International Journal of Civil Engineerng. Vol. 5, No. 1, March 2007 ground motion. This inelastic cyclic behavior [12], Akiyama [13] and Nakashima et al. [14] causes partial damages in structures and with observed that damping does not have a accumulation of these partial damages, significant influence on the earthquake input structures collapse. Therefore the input energy. energy demand can be assumed as a reliable Results of the study by Bruneau and Wang tool to predict the seismic hazard and seismic [15] indicated that damping ratios smaller design. than 5% have a minor influence on the input energy. Thus evaluation of influence of structural Rahnama and Manuel [16] computed the parameters, characteristics of earthquake input energy for ductility ratios of 2 and 6 records and soil conditions on input energy with 5% damping for six sets of 19 are important. accelerations each, actual records and simulated records with the same duration 5, In this paper, the effects of ground motion 10, 15 and 20 s. They considered that when duration, structural damping and soil type on duration increases, the input energy input energy were studied, by use of 110 increases. Iranian earthquake records. Because these Khashaee et al. [17] computed the relative records are collected from different input energy for 10 accelerations with short earthquakes with various PGA's, in order that duration (shorter than 8 s) and 10 with long the effects of earthquake PGA in result of duration (longer than 18 s) of strong ground each analysis are vanished, and also their motions. They observed that as the duration responses can be added together, the amount of strong ground motions increases, the input of input energy was divided by its PGA and energy also increases. They computed the the spectrum curve of input energy was relative input energy for structures with drawn. This procedure is also performed in damping ratios 0, 2, 5, 10, 20 and 40% for the previous researches on this topic such as three accelerations with short duration and Benavent–Climent's work [2]. three with long duration of strong ground motion. They observed that for damping Downloaded from ijce.iust.ac.ir at 22:42 IRST on Saturday October 2nd 2021 ratios smaller than 5%, damping has little 2. Literature Review influence on the input energy while for damping ratios greater than 5%, damping has Previously many of researches investigated a significant influence on the input energy the duration of ground motion, damping of spectra, particularly for very long natural structure and soil condition influences on periods as the damping increases, the input input energy and obtained different energy increases. conclusions. Zahrah and Hall [11] computed the input 3. Theoretical Background energy for eight earthquake records and they considered that ductility, damping and past- When a viscous damped single mass elastic to-pre yield stiffness ratios have small effects oscillatory system, SDOF, vibrates subject to on the input and hysteretic energies for a a unidirectional horizontal ground motion, its structure with bilinear behavior. equilibrium equation can be expressed as follows [9]. && & Based on many analyses, McKevitt et al. Myt Cy Ky 0 (1) International Journal of Civil Engineerng. Vol. 5, No. 1, March 2007 15 t0 t0 1 2 2 1 2 Where M, C, K and y are mass, viscous & & && & Myt ∫ Cy dt Ky M ∫ yt xg dt damping coefficient, spring constant and 2 0 2 0 c c relative displacement of the mass, Ek Ed Es EI (6) respectively. yt = y+xg is absolute. .. (or total) displacement of the mass. y and yt are first Where: derivation of y and the second derivation of y 1 t0 . t Ec My& 2 , E Cy& 2 dt with respect to time respectively. x and x k t d ∫ g g 2 0 are earthquake ground displacement and ground acceleration, respectively. t0 1 2 c && & Es Ky , EI M ∫ yt xg dt && &&&& 2 0 By letting yt y xg Equation (1) can be B re-written as follows: Here Ek, Ed and Es are absolute kinetic energy, the energy dissipated by damping && & && My Cy Ky Mxg (2) mechanism and the energy absorbed by spring, respectively, and EB is defined as the .. I Where, y is the second derivation of y with absolute input energy. This definition is respect to time. physically meaningful in where the term && Myt represents the inertia force applied to the Therefore the structural system in a moving structure. This force, from Equation (1), is base system can be treated conveniently as equal to restoring force plus damping force, an equivalent system with a fixed base which is the same as the total force applied to B subjected to an effective horizontal dynamic the structure foundation. Therefore E I − . force of magnitude M xg Depending upon represents the work done by the total base whether Equation (1) or (2) is used to derive shear at the foundation through the the energy equation, different definitions of foundation displacement. input energy may result. Method 2 – Derivation of relative energy equation Downloaded from ijce.iust.ac.ir at 22:42 IRST on Saturday October 2nd 2021 Method 1- Derivation of absolute energy If both sides. of Equation (2) are multiplied equation by dy(=y dt) and then integrated over the If both sidesH of Equation (1) are multiplied entire duration of an earthquake,t, Equation by dy(= ydt) and then integrated over the (2) reduces to the following energy balance entire duration of an earthquake, t0, Equation equation: (1) reduces to the following energy balance t0 t0 t0 t0 && & & & & && & equation: ∫ My(ydt) ∫Cy(ydt) ∫ K y(ydt) ∫ Mxg (ydt) 0 0 0 0 t0 t0 t && & & & & ∫ Myt (ydt) ∫∫Cy(ydt) Ky(ydt) 0 (3) (7) 0 00 t0 t0 t0 t0 H H − H &&& & 2 & && & Replacing y by y t xg in the first term of ∫ M yydt ∫Cy dt ∫ K yydt ∫ Mxg ydt (8) Equation (3), then 0 0 0 0 ttt 000 t0 t0 M&y& (y& x& )dt Cy& 2 dt Kyy&dt 0 (4) 1 & 2 & 2 1 2 && & (9) ∫∫∫t t g My ∫Cy dt Ky M ∫ xg ydt 000 2 0 2 0 tttt0000 2 && & & & && & ∫∫∫∫My t yt dt Cy dt Kyydt Myt xg dt 0000 (5) Ek Ed Es EI (10) 16 International Journal of Civil Engineerng. Vol. 5, No. 1, March 2007 − Where: y= xg and yt = y+xg = 0, 1 t0 E My& 2 , E Cy& 2 dt . k d ∫ therefore EB =0 and E = 0.5 M x2 2 0 I I g t0 i.e. the difference between the input energies 1 2 && & Es Ky , EI M ∫ xg ydt and for a structure.