Start with the equation of motion for a linear multi-degree of freedom system with base ground excitation:
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1 SE 80 Earthquake Enneern November 3, 00 STEP-BY-STEP PROCEDURE FOR SETTING UP A SPREADSHEET FOR USING NEWMARK S METHOD AND MODAL ANALYSIS TO SOLVE FOR THE RESPONSE OF A MULTI-DEGREE OF FREEDOM (MDOF) SYSTEM Start wth the equaton of moton for a lnear mult-deree of freedom system wth base round exctaton: mu& + cu& + ku = m& u& Usn Modal Analyss, we can rewrte the ornal coupled matrx equaton of moton as a set of un-coupled equatons. L & q + ζω q& + ω q & = u, =,,, NDOF M wth ntal condtons of d (t = 0) = d and v o (t = 0) = v o Note that total acceleraton or absolute acceleraton wll be q& & = && q + & u We can solve each one separately (as a SDOF system), and compute hstores of q and ther tme dervatves. To compute the system response, plu the q vector back nto u = Φq and et the u vector (and the same for the tme dervatves to et velocty and acceleraton). The beauty here s that there s no matrx operatons nvolved, snce the matrx equaton of moton has become a set of un-coupled equaton, each ncludn only one eneralzed coordnate q n. In the spreadsheet, we wll solve each mode n a separate worksheet. Step - Defne System Propertes and Intal Condtons for Frst Mode (A) Ben by settn up the cells for the Mass, Stffness, and Dampn of the SDOF System (F. ). These values are known. abs (B) Set up the cells for the modal partcpaton factor M These values must be determned n advance usn Modal Analyss. L and mode shape φ (F. ). (C) Calculate the Natural Frequency of the SDOF system usn the equaton Mchael Fraser
2 ω = K M (Equaton ) Note: If the system dampn s ven n terms of the Modal Dampn Rato ( ζ ) then the Dampn ( c ) can be calculated usn the equaton: C = ζ ω M (Equaton ) (D) Set up the cells for the Newmark Coeffcents α & β (F. ), whch wll allow for performn a) the Averae Acceleraton Method, use b) the Lnear Acceleraton Method, use α = and α = and β =. 4 β =. 6 (E) Set up cells (F. ) for the ntal dsplacement and velocty (d o and v o respectvely) Mchael Fraser
3 Equaton Equaton Fure : Spreadsheet After Completn Step Mchael Fraser 3
4 Step Set Up Columns for Solvn The Equaton of Moton Usn Newmark s Method Base Exctaton Appled Force Dvded By Mass Relatve Acceleraton Relatve Velocty Relatve Dsplacement Fure : Spreadsheet After Completn Step Place a cell (F. ) for the tme ncrement ( t). Place columns (F. ) for the tme, base exctaton, appled force dvded by mass, relatve acceleraton, relatve velocty, and relatve dsplacement. Mchael Fraser 4
5 Step 3 Enter the Tme t & Appled Force f(t) nto the Spreadsheet t + = t t (Equaton 3) (F. 3) + For the earthquake problem (acceleraton appled to base of the structure), the appled force dvded by the mass s calculated usn: f (t) M L && = u M (Equaton 4) (F. 3) d where, & u& s the appled base acceleraton at step. (Typcally ths s the base exctaton tme hstory) Check the unts of the nput moton fle. They must be compatble wth the unts of the mass, stffness, and dampn! Equaton 3 Equaton 4 Fure 3: Spreadsheet After Completn Step 3 Mchael Fraser 5
6 Step 4 Compute Intal Values of the Relatve Acceleraton, Relatve Velocty, Relatve Dsplacement, and Absolute Acceleraton (A) The Intal Relatve Dsplacement and Relatve Velocty are known from the ntal condtons (F. 4). q(t = 0) = (Equaton 5) d o q &(t = 0) = (Equaton 6) v o (B) The Intal Relatve Acceleraton (F. 4) s calculated usn L M & q (t = 0) = && u ζωv o ω d o (Equaton 7) Equaton 6 Equaton 7 Equaton 5 Fure 4: Spreadsheet After Completn Step 4 Mchael Fraser 6
7 (A) Step 5 Compute Incremental Values of the Relatve Acceleraton, Relatve Velocty, Relatve Dsplacement, and Absolute Acceleraton At Each Tme Step (F. 5) L t && u C q q K t ( β) q tq q M && + & && + & + + q & + = (Equaton 8) m * ( α) + && q tα q& = + q & && + q t + (Equaton 9) t q = & q ( β) + && q t β + q& + + t + q (Equaton 0) Where, the effectve mass, m* = M + C t α + K t β Equaton 9 Equaton 0 Equaton 8 Fure 5: Spreadsheet wth values for the Relatve Acceleraton, Relatve Velocty, and Relatve Dsplacement at Tme Step (B) Then, hhlht columns I, J, & K and rows 4 throuh to the last tme step (n ths example 4003) and Fll Down (Ctrl+D). See Fures 6 and 7. Mchael Fraser 7
8 Fure 6: Hhlhted Cells Mchael Fraser 8
9 Fure 7: Spreadsheet After Flln Down Columns I throuh K Mchael Fraser 9
10 Step 6 Create Addtonal Worksheet for Second Mode Make a copy of the st Mode worksheet by rht clckn on the st Mode tab and selectn Move or Copy (F. 8) Fure 8: Creatn a Copy of st Mode Worksheet Then check the box for Create a copy and clck on OK button (F. 9) Mchael Fraser 0
11 Fure 9: Creatn a Copy of st Mode Worksheet Rename ths worksheet by rht clckn on the st Mode () tab and selectn Rename. Rename ths worksheet nd Mode (F. 0) Enter the approprate values for M, K, C, L, φ, d o, and v o (F. 0). M Mchael Fraser
12 Fure 0: Worksheet for Second Mode Step 7 Repeat Step 6 for Addtonal Modes Step 8 Determne the Response at Each of the Floors Determne the Response of the frst floor usn the equatons: u = Φq u & = Φq& u & = Φq& For example for a DOF structure, the frst floor response s u u & & u u φ q + φq φ q& + φq& φ && q + φ & q = (Equaton ) = (Equaton ) = (Equaton 3) and the second floor response s φ q + φ q = (Equaton 4) Mchael Fraser
13 u & & u = & & (Equaton 5) = (Equaton 6) φ q + φ q φ && q + φ & q T The frst floor absolute acceleraton s & u = && u + & (Equaton 7) u T The second floor absolute acceleraton s u & = && u + & (Equaton 8) u Mchael Fraser 3
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