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American University of Beirut Faculty of Engineering & Architecture Department of Civil and Environmental Engineering CIVE – 616-EARTHQUAKE ENGINEERING “HOMEWORK (2)” Prepared by: Ali El-Hajj ID # 201820917 27th Feb. 2018 Problem 1: Free Vibration Response 1) The free vibration response - displacment time series - of a single-degree-offreedom (SDOF) system is given by the following equation: u(t) = exp (-Ʒ ωn t) [u (0) cos (ωDt) + ú (0) + Ʒ 𝜔𝑛 𝑢 (0) 𝜔𝐷 sin (ωDt)] The MATLAB function called “FREE_VIBRATION.m” is defined to take some of the input parameters of the above equation such as u (0), ú (0), Ʒ, and Tn, then calculates the output time vector (t) at discrete time steps dt =0.02 secs, and the displacment vector u(t) corresponding to each time step. u (0): initial displacment of the system ú (0): initial velocity of the system Ʒ: damping ratio Tn: natural period of the system 2) The script presented in MATLAB file “FREE_VIBRTION_RUN.m” is used to run the function defined in part (1), that computes and plots the free vibration response for systems having the initial conditions u (0) =0, ú (0) =20cm/sec, and Tn=0.5 sec for different damping ratios. The results for damping ratios (Ʒ) of 0, 0.02, and 0.05 are presented in Figure 3 Figure 1, Figure 2, and respectively. 3) The script presented in MATLAB file “FREE_VIBRTION_RUN.m” is used to run the function defined in part (1) that computes and plots the free vibration response for systems having the initial conditions u (0) =0, ú (0) =20cm/sec, and Tn=2 sec for different damping ratios. The results for damping ratios (Ʒ) of 0, 0.02, and 0.05 are presented in Figure 6 respectively. 2 Figure 4, Figure 5, and Figure 1 Figure 2 3 Figure 3 Figure 4 4 Figure 5 Figure 6 5 By comparing figures 1, 2 and 3 for constant period of 0.5s, as the damping ratio increases, the rate at which the motion decay decreases. The same applies for the comparison of figures 4, 5, and 6 for constant period of 2sec. By comparing figures of different periods, as the period increase, the max displacment response increases. 6 Problem 2: Numerical Methods 1) When acceleration time series is given in the form of a defined function, the velocity time series would be obtained by integrating the acceleration function over the time of the series if it had an exact mathematical solution. The deformation time series would also be obtained in a similar procedure by integrating the velocity time series function. Since it is difficult, or even not possible, to define the ground acceleration time series due to earthquake as definite mathematical function, thus approximate numerical integration methods must be used for deriving the velocity and displacment time series, given the ground acceleration. The MATLAB function “GROUND_Acc_Vel_Disp.m” is defined to read the input quantities of earthquake ground acceleration over specified time step dtg for a given earthquake from a text file. The ground acceleration time series must be saved in the text file in the name of “filename.txt” so that the mentioned function reads the input from the text file and uses “cumsum” function for numerical integration to compute the velocity time series, and then the displacement time series for a given earthquake ground acceleration record. The script presented in MATLAB file “GROUND_Acc_Vel_Disp _PLOTS_PEAKS.m” is used to run the above function and plot the ground acceleration, velocity and displacement time series of the Napa 2014 Earthquake. The results are presented in Figure 7, Figure 8, and Figure 9 respectively. 7 Figure 7 Figure 8 8 Figure 9 2) The MATLAB function called “PG_VALUES.m” is defined to take the input of ground acceleration, velocity and displacement time series, and compute the absolute maximum value of each that is the peakground-acceleration (PGA), peak-ground-velocity (PGV), peakground-displacment (PGD). The script presented in MATLAB file “GROUND_Acc_Vel_Disp _PLOTS_PEAKS.m” also gives the peak ground motion parameters for the Napa 2014 Earthquake that are: PGA = 418.0450 cm/sec2 PGV = 81.9091 cm/sec PGD = 33.9694 cm 9 3) The equation of motion for a system subjected to ground motion excitation is given by: 𝐶 𝐾 m ű(t) + 𝑚 ú(t) + 𝑚 u(t) = -m űg(t) (Equation 1) ű(t) + 2 Ʒ ωn ú(t) + ωn2 u(t) = -űg(t) (Equation 2) The solution for this equation can be obtained using the central difference method which is based on a finite difference approximation of the time derivatives of displacement (i.e., velocity and acceleration). Taking constant time steps ∆t, the central difference expressions for velocity and acceleration at time i are [1]: 𝑢(𝑖+1) −𝑢(𝑖−1) úi = űi = (Equation 3) 2∆𝑡 𝑢(𝑖+1) −2 𝑢(𝑖) + 𝑢(𝑖−1) (Equation 4) (∆𝑡)2 REF _Ref507427060 \h \* MERGEFORMAT úi = 𝑢(𝑖+1) −𝑢(𝑖−1) 2∆𝑡 𝑢(𝑖+1)−2 𝑢(𝑖)+ 𝑢(𝑖−1)(∆𝑡)2 (Equation 4) in (Equation 2): 𝑢(𝑖+1) −2 𝑢(𝑖) + 𝑢(𝑖−1) (∆𝑡)2 𝑢(𝑖+1) −𝑢(𝑖−1) + 2 Ʒ ωn [ 2∆𝑡 ] + ωn2 u(i) = - űg(i) Rearranging terms in (Equation 5) gives: Ҏ 𝒖(𝒊+𝟏) = Ҟ 1 Ʒ 𝜔𝑛 Ҟ= [ (∆𝑡)2 + Ҏ = -ű𝑔(𝑖) – a 𝑢(𝑖−1) – b 𝑢(𝑖) a = [ (∆𝑡)2 - b= [𝜔𝑛 2 − 1 ∆𝑡 Ʒ 𝜔𝑛 ∆𝑡 ] ] 2 ] (∆𝑡)2 Initial conditions for time step i=0 ű0 = -ű𝑔(0) – 2 Ʒ 𝜔𝑛 ú0 - 𝜔𝑛 2 𝑢(0) 10 (Equation 5) 𝑢−1 = 𝑢0 - ∆t(ú0 ) + (∆𝑡)2 2 (ű0 ) 11 The MATLAB function called “Deformation.m” is defined to read the input of an earthquake ground acceleration time series from a text file as described before, in addition to the input of specified structure’s natural period (Tn), specified damping ratio (Ʒ) or (Z), as defined in the MATLAB library, initial velocity of the system (V0), and initial displacement of the system (U0). This function outputs and plots the deformation time series of the system having the specified parameters, initial conditions and subjected to the given ground motion. The script presented in MATLAB file “Deformation_RUN” is used to compute and plot the deformation time series of the system. The results of systems, having same Ʒ=0.05, ú0 = 0, and 𝑈0 = 0, but with varying natural periods (Tn= 0.28, 0.57, 1.2,2.5) and subjected to same earthquake ground motion recorded during Napa 2014 Earthquake , are plotted in Figure 10, Figure 11, Figure 12, and Figure 13 respectively. The results of the maximum deformation of each system with different values of (Tn) are also presented in Table 1. Table 1 Tn (sec) 0.28 0.57 1.2 2.5 D (cm) 1.1667 6.1274 47.2712 48.05 12 Figure 10 Figure 11 13 Figure 12 Figure 13 14 4) The MATLAB function called “Response_SPECTRA.m” is defined to take the input of a vector of damping ratios (Ʒ) and a vector of natural periods (Tn) for systems of interest or over specified range, in addition to the initial conditions of these systems, u (0), and ú (0). This function calls the previously defined function “Deformation.m” to compute the deformation response for each of the specified periods and damping ratios. It outputs then a vector of deformation of systems as function of periods, which can be used to pot the deformation response spectrum(D). Furthermore, it computes the vectors for plotting velocity response and acceleration response by multiplying each deformation response of each system by 𝜔𝑛 and 𝜔𝑛 2 respectively. 5) The script presented in MATLAB file “RESPONSE_SPECTRA _RUN.m” is used to compute and plot deformation and pseudoacceleration response spectra corresponding to ground motion of Napa 2014 Earthquake. The initial displacment and velocities are u (0) =0, and ú (0) =0. The spectra are plotted for values of natural periods between 0 and 10 secs, with 0.1sec increment, and for damping ratios of 0, 0.02, and 0.05 and are presented in Ошибка! Источник ссылки не найден. and Ошибка! Источник ссылки не найден.. 15 Figure 14 Figure 15 16 6) Earthquake name: Imperial Earthquake date: 6/6/1938 Earthquake magnitude: 5 Station name: El Centro Array #9 Station Rupture distance: 34.98 Km 𝑉𝑠30: 213.44 (m/sec2) Record Sequence number: 4 Horizontal-1 Acc. Filename is RSN4_IMPVALL.BG_B-ELC000.txt Figure 16 17 Figure 17 Figure 18 18 Figure 19 Figure 20 19 Problem 3: Earthquake response 1) (adapted from Chopra, p.9) 20 (a) If the cross section of the beam is much larger than that of the columns, the stiffness of the SDOF system may be approximated by (Equation 1.3.2) K= 254 ∗104 ] 12 [64∗109 ] 24∗[2∗105 ]∗[ = 24,414.063 N/mm = 24,414,063 N/m 𝑚 5000 𝐾 24,414,063 Tn = 2𝜋√ = 2𝜋√ = 0.09 sec Ʒ = 0.05 Using the response spectra as plotted in problem 2.5 and presented in Figure 21 and Figure 22 at time step = 0.05 sec and with ground acceleration scaled to 0.5g: A = 580 cm/sec2 = 5.8 m/sec2 D = 0.00125 m Base shear fs0 = mA = 29 KN Design moment Mb0 = m*A*h = h*fs0 = 116 KN.m 21 Figure 21 Figure 22 22 (b) If the cross section of the beam is smaller than that of the columns, the stiffness of the SDOF system may be approximated by (Equation 1.3.3) K= 254 ∗104 ] 12 [64∗109 ] 6∗[2∗105 ]∗[ = 6,103.516 N/mm = 6,103,516 N/m 𝑚 5000 𝐾 6,103,516 Tn = 2𝜋√ = 2𝜋√ = 0.18 sec Ʒ = 0.05 Using the response spectra as plotted in problem 2.5 and presented in Figure 21 and Figure 22 at time step = 0.05 sec and with ground acceleration scaled to 0.5g: A = 8.8 m/sec2 D = 0.0065 m Base shear fs0 = mA = 44 KN Design moment Mb0 = m*A*h = h*fs0 = 176 KN.m 2) When the rigidity of the beam increase, the stiffness of the frame increases, thus deformation response is reduced, and design forces are increased. On the other hand, as the rigidity of the beam decrease, the stiffness of the frame decrease, thus deformation response is increased, and design forces are reduced. 23 References Dynamics of Structures: Theory and Applications to Earthquak Engineering, by A.K. Chopra, Fourth Edition, Pearson Prentice Hall, 2014 https://ngawest2.berkeley.edu/ Sigmon, Kermit, and Timothy A. Davis. MATLAB primer. CRC Press, 2004. 24