SE 8 Fnal Project
Story Shear Frame u m Gven: u m L L m L L EI ω ω Solve for m
Story Bendng Beam u u m L m L Gven: m L L EI ω ω Solve for m 3
3 Story Shear Frame u 3 m 3 Gven: L 3 m m L L L 3 EI ω ω ω 3 u u m m L L Solve for m 3 4
3 Story Bendng u m Beam L Gven: m m L =L =L 3 =L EI ω ω ω 3 u u 3 m L m 3 Solve for m 3 L 3 5
Part : Determnng the unknown mass of your structure Step : Assemble the mass and stffness matrces for your structure For Example: Fnd m, gven m, L = L = h, EI c, & EI b m EI b u u 6 u 5 L m EI c EI b u u 4 u 3 L EI c h 6
m Step : Mass and Stffness Matrces (contnued) m Example (contnued): m = 48. E. I c h 3 4. E. I c 4. E. I c h 3 4. E. I c 6. E. I c 6. E. I c 6E.. I c h 6. E. I c 6E.. I c h 6. E. I c k = h 3 6E.. I c h 6E.. I c h h 3 6. E. I c h 6. E. I c h 6. E. I c h 6. E. I c h 8E.. I c h h E.. I b h. E.. I c h 4E.. I b h. 8E.. I c h h E.. I b h. E.. I c h 4E.. I b h. 4E.. I c h h E.. I c h E.. I b h. 4E.. I b h. 4E.. I c h h E.. I c h E.. I b h. 4E.. I b h. 7
Determnng the unknown mass of your structure Step : Perform Statc Condensaton (f necessary) k k tt t [ k] = t k k kˆ tt = k tt k T t k k t Example (contnued): k tt = 48. E. I c h 3 4. E. I c h 3 4. E. I c h 3 4. E. I c h 3 k t = 6E.. I c h 6E.. I c h 6E.. I c h 6E.. I c h 6E.. I c h 6E.. I c h k = 8E.. I c h E.. I b h. E.. I c h 4E.. I b h. 8E.. I c h E.. I b h. E.. I c h 4E.. I b h. 4E.. I c h E.. I c h E.. I b h. 4E.. I b h. 4E.. I c h E.. I c h E.. I b h. 4E.. I b h. 8
Step : Statc Condensaton (contnued) Example (contnued): kˆ tt = k tt - k t T k - k t = 4. E. 3. Ic 63. Ic. Ib 8. Ib Ic. 8. Ic 36. Ic. Ib 9Ib.. h 3 4. E.. Ic 7. Ic. Ib 9Ib. Ic. 8. Ic 36. Ic. Ib 9Ib.. h 3 4. E.. Ic 7. Ic. Ib 9Ib. Ic. 8. Ic 36. Ic. Ib 9Ib.. h 3 4. E. 4Ic. 8. Ic. Ib 9Ib. Ic. 8. Ic 36. Ic. Ib 9Ib.. h 3 Where kˆ tt s the condensed stffness matrx 9
Determnng the unknown mass of your structure Step 3: Solve the Egen-value problem to determne the determnant of [ k] ω [ m] Example (contnued): [ k] ω [ m] = 4. E. 3. Ic 63. Ic. Ib 8. Ib Ic. ω. m 8. Ic 36. Ic. Ib 9Ib.. h 3 4. E.. Ic 7. Ic. Ib 9Ib. Ic. 8. Ic 36. Ic. Ib 9Ib.. h 3 4. E.. Ic 7. Ic. Ib 9Ib. Ic. 8. Ic 36. Ic. Ib 9Ib.. h 3 4. E. 4Ic. 8. Ic. Ib 9Ib. Ic. ω. m 8. Ic 36. Ic. Ib 9Ib.. h 3
Example (contnued): Step 3: Egen-value Problem (contnued) [ k] ω [ m] = =
Determnng the unknown mass of your structure Step 4: Plug the gven values for m, L, L, EI, & ω nto [ k] ω [ m] =, and solve for the unknown mass. Example (contnued): Gven: m =. kg L =.348 m L =.348 m EI =.8894 n-m ω =.566 rad/s ω = 86.3rad/s
Step 4: Solve for unknown m (contnued) Example (contnued): Plug m, L, L, EI, & ω nto [ k] ω [ m] = Solve for m m = 4. E. Ic. 4. E. Ic 3 6. Ic. E. Ib 4Ic.. ω. m. h 3 8. Ic. ω. m. h 3. Ib 6. Ic. E. Ib 9. ω. m. h 3. Ib h 3. ω 768. E. Ic 3 8. Ic. ω. m. h 3 5. Ic. E. Ib 43. Ic. E. Ib 36. Ic. ω. m. h 3.. Ib 9. ω. m. h 3. Ib m =.949 kg 3
Determnng the unknown mass of your structure Step 5: Perform a self-check on the value of the mass you just found Plug the newly determned mass along wth the remanng ω nto [ k] ω [ m] and verfy that the determnant s stll equal to zero Example (contnued): m = 4. E. Ic. h 3. ω. 4. E. Ic 3 6. Ic. E. Ib 4Ic.. ω. m. h 3 8. Ic. ω. m. h 3. Ib 6. Ic. E. Ib 9. ω. m. h 3. Ib 768. E. Ic 3 8. Ic. ω. m. h 3 5. Ic. E. Ib 43. Ic. E. Ib 36. Ic. ω. m. h 3. Ib 9. ω. m. h 3. Ib m =.949 kg (same as wth ω OK!) 4
Part : Modal Analyss Step : Determne the Mode Shapes φ n ( k - m) u = ω Equaton Contnung wth the Egen-value problem soluton (agan, Matlab does ths, or by hand for a -dof system), for each ω we get an assocated φ n n mode shape. To do ths (for each dentfed ω n ), go ahead and substtute ths ωnfor ω n Eq. above. Upon ths substtuton, you can solve for the correspondng vector u, the components of whch defnes the mode shape. φ n 5
For the Prevous example: ω =.566 rad/s Step : Mode Shapes (contnued) 4. E. 3. Ic 63. Ic. Ib 8. Ib Ic. ω. m 8. Ic 36. Ic. Ib 9Ib.. h 3 4. E.. Ic 7. Ic. Ib 9Ib. Ic. 8. Ic 36. Ic. Ib 9Ib.. h 3 4. E.. Ic 7. Ic. Ib 9Ib. Ic. 8. Ic 36. Ic. Ib 9Ib.. h 3 4. E. 4Ic. 8. Ic. Ib 9Ib. Ic. ω. m 8. Ic 36. Ic. Ib 9Ib.. h 3 =.397. 3 58.566 58.566 4.65.397. 3 58.566 58.566 4.65 φ. = φ Let φ =.; therefore, φ =.4 6
Example (contnued): ω = 86. rad/s Step : Mode Shapes (contnued) 4. E. 3. Ic 63. Ic. Ib 8. Ib Ic. ω. m 8. Ic 36. Ic. Ib 9Ib.. h 3 4. E.. Ic 7. Ic. Ib 9Ib. Ic. 8. Ic 36. Ic. Ib 9Ib.. h 3 4. E.. Ic 7. Ic. Ib 9Ib. Ic. 8. Ic 36. Ic. Ib 9Ib.. h 3 4. E. 4Ic. 8. Ic. Ib 9Ib. Ic. ω. m 8. Ic 36. Ic. Ib 9Ib.. h 3 = 5.933 58.566 58.566 6.69. 3 5.933 58.566 58.566 6.69. 3 φ. = φ Let φ =.; therefore, φ = -.876 7
Example (contnued): Step : Mode Shapes (contnued) Frst Mode Second Mode φ φ φ φ 8
Part : Modal Analyss Step : Determne the Modal Partcpaton Factors NDOF M = m j φ j j= NDOF L = m j φ j j= φ φ φ. =.. φ NDOF L M 9
Example (contnued): Step : Modal Partcpaton Factors (contnued) NDOF M = m j φ j j= M M = mφ + m = mφ + m φ φ = (.)() + (.949)(.4) = 5.67535 kg = (.)() + (.949)(-.876) =.78 kg NDOF L = m j φ j j= L φ + = m m φ L φ + = m m φ = (.)() + (.949)(.4) =.4795 kg = (.)() + (.949)(-.876) =.687 kg
Example (contnued): Step : Modal Partcpaton Factors (contnued) L M =.4795 kg 5.67535 kg =.437 L M =.687 kg.78 kg =.563
Part : Modal Analyss Step 3: Determne K K = ω M K = ω M = (.566 ) (5.67535) = 896.65 K = ωm = (86.) (.78) = 533.435
Step 4: Add Dampng Part : Modal Analyss Now, you can add any modal dampng you wsh (whch s another bg plus, snce you control the dampng n each mode ndvdually). If you choose ζ =. or.5, the equatons become: & q + ξ ω q& + ω q = L M & u g, =,, NDOF 3
Step 5: Solve for q (t) Part : Modal Analyss Solve for q (t) n the above uncoupled equatons (usng a SDOF-type program), and the fnal soluton s obtaned from: u = Φq u & = Φq& u & = Φq& t u & = u&& + & u& g 4
Step 5: Solve for q (t) (contnued) We wll solve for q (t) usng a modfed verson of the spreadsheet for solvng for the response of a SDOF system usng Newmark s Method 5
Part 3: Spreadsheet for Modal Analyss Step-By-Step Procedure For Settng Up A Spreadsheet For Usng Newmark s Method and Modal Analyss To Solve For The Response Of A Mult-Degree Of Freedom (MDOF) System Start wth the equaton of moton for a lnear mult-degree of freedom system wth base ground exctaton: mu& + cu& + ku = m& u& g 6
Usng Modal Analyss, we can rewrte the orgnal coupled matrx equaton of moton as a set of un-coupled equatons. & q + ζωq& + ω wth ntal condtons of q = L M, =,,, NDOF and Note that total acceleraton or absolute acceleraton wll be & u g d = abs v (t = ) = (t = ) d o v o & q = && q + & u g 7
We can solve each one separately (as a SDOF system), and compute hstores of q and ther tme dervatves. To compute the system response, plug the q vector back nto u = Φq and get the u vector (and the same for the tme dervatves to get 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 ncludng only one generalzed coordnate. q n In the spreadsheet, we wll solve each mode n a separate worksheet. 8
Step - Defne System Propertes and Intal Condtons for Frst Mode (A)Begn by settng up the cells for the Mass, Stffness, and Dampng of the SDOF System (Fg. ). These values are known. (B) Set up the cells for the modal partcpaton factor and mode shape φ (Fg. ). These values must be determned n advance usng Modal Analyss. (C) Calculate the Natural Frequency of the SDOF system usng the equaton L M ω = K M (Equaton ) 9
Step (contnued) Note: If the system dampng s gven n terms of the Modal Dampng Rato ( ζ ) then the Dampng ( C ) can be calculated usng the equaton: C = ζ ω M (Equaton ) (D) Set up the cells for the Newmark Coeffcents α & β (Fg. ), whch wll allow for performng a) the Average Acceleraton Method, use and α = β =. 6 b) the Lnear Acceleraton Method, use α = and β =. 4 (E) Set up cells (Fg. ) for the ntal dsplacement and velocty (d o and v o respectvely) 3
Step (contnued) Equaton Equaton Fgure : Spreadsheet After Completng Step 3
Step Set Up Columns for Solvng The Equaton of Moton Usng Newmark s Method Place a cell (Fg. ) for the tme ncrement ( t). Place columns (Fg. ) for the tme, base exctaton, appled force dvded by mass, relatve acceleraton, relatve velocty, and relatve dsplacement. 3
Step (contnued) Base Exctaton Appled Force Dvded By Mass Relatve Acceleraton Relatve Velocty Relatve Dsplacement Fgure : Spreadsheet After Completng Step 33
Step 3 Enter the Tme t & Appled Force f(t) nto the Spreadsheet t + = t + t (Equaton 3) (Fg. 3) For the earthquake problem (acceleraton appled to base of the structure), the appled force dvded by the mass s calculated usng: f(t) M = L M && u g (Equaton 4) (Fg. 3) where, & u& g s the appled base acceleraton at step. (Typcally ths s the base exctaton tme hstory) m q Check the unts of the nput moton fle. They must be compatble wth the unts of the mass, stffness, and dampng! & u& g 34
Step 3 (contnued) Equaton 3 Equaton 4 Fgure 3: Spreadsheet After Completng Step 3 35
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 (Fg. 4). q(t = ) = d o (Equaton 5) q &(t = ) = v o (Equaton 6) (B) The Intal Relatve Acceleraton (Fg. 4) s calculated usng && q(t L = ) = && u g ζωv M o ω d o (Equaton 7) 36
Step 4 (contnued) Equaton 6 Equaton 7 Equaton 5 Fgure 4: Spreadsheet After Completng Step 4 37
Step 5 Compute Incremental Values of the Relatve Acceleraton, Relatve Velocty, Relatve Dsplacement, and Absolute Acceleraton At Each Tme Step (Fg. 5) (A) && q + = L M && u g + C t && q + q& K m * t ( β) && q + tq& + q (Equaton 8) + ( α) + && q tα q& q & && + = q t + t q = && q & + + + ( β) + && q t β + q t q (Equaton 9) (Equaton ) Where, the effectve mass, m* = M + C t α + K t β 38
Step 5 (contnued) Equaton 9 Equaton Equaton 8 Fgure 5: Spreadsheet wth values for the Relatve Acceleraton, Relatve Velocty, and Relatve Dsplacement at Tme Step 39
Step 5 (contnued) (B) Then, hghlght columns I, J, & K and rows 4 through to the last tme step (n ths example 43) and Fll Down (Ctrl+D). See Fgures 6 and 7. Fgure 6: Hghlghted Cells 4
Step 5 (contnued) Fgure 7: Spreadsheet After Fllng Down Columns I through K 4
Step 6 Create Addtonal Worksheet for Second Mode Make a copy of the st Mode worksheet by rght clckng on the st Mode tab and selectng Move or Copy (Fg. 8) Fgure 8: Creatng a Copy of st Mode Worksheet 4
Step 6 (contnued) Then check the box for Create a copy and clck on OK button (Fg. 9) Fgure 9: Creatng a Copy of st Mode Worksheet 43
Step 6 (contnued) Rename ths worksheet by rght clckng on the st Mode () tab and selectng Rename. Rename ths worksheet nd Mode (Fg. ) L Enter the approprate values for M, K, C,, φ, d o, and v o (Fg. ). M 44
Step 6 (contnued) Fgure : Worksheet for Second Mode 45
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 usng the equatons: u = Φq u & = Φq& u & = Φq& 46
Step 8 (contnued) For example for a DOF structure, the frst floor response s (Fg. ) u u & & u = φ q + φq & φ = φ q + q && φ = φ q + q & & (Equaton ) (Equaton ) (Equaton 3) 47
Step 8 (contnued) and the second floor response s (Fg. ) u u & & u = φ q + φ q & = φ q + φ q && = φ q + φ q & & (Equaton 4) (Equaton 5) (Equaton 6) The frst floor absolute acceleraton s & u && + & (Equaton 7) T = u u g The second floor absolute acceleraton s & u && + & T = u u g (Equaton 8) 48
Step 8 (contnued) Equaton 7 Equaton Equaton Equaton 3 Fgure : Frst Floor Response 49
Step 8 (contnued) Equaton 8 Equaton 6 Equaton 5 Equaton 4 Fgure : Second Floor Response 5